text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
⌀ |
|---|---|---|
"""
wegene.Controllers.AthletigenController
This file was automatically generated by APIMATIC BETA v2.0 on 02/22/2016
"""
import requests
from wegene.APIHelper import APIHelper
from wegene.Configuration import Configuration
from wegene.APIException import APIException
from wegene.Models.Report import Report
class Athletigen(object):
"""A Controller to access Endpoints in the WeGeneAPILib API."""
def get_athletigen(self,
profile_id,
report_id):
"""Does a POST request to /athletigen/{profile_id}.
Athletigen based on genetic information
Args:
profile_id (string): Genetic profile id
report_id (string): Report Id for the specific health risk to
look
Returns:
Report: Response from the API.
Raises:
APIException: When an error occurs while fetching the data from
the remote API. This exception includes the HTTP Response
code, an error message, and the HTTP body that was received in
the request.
"""
# The base uri for api requests
query_builder = Configuration.BASE_URI
# Prepare query string for API call
query_builder += "/athletigen/{profile_id}"
# Process optional template parameters
query_builder = APIHelper.append_url_with_template_parameters(query_builder, {
"profile_id": profile_id
})
# Validate and preprocess url
query_url = APIHelper.clean_url(query_builder)
# Prepare headers
headers = {
"Authorization": "Bearer " + Configuration.o_auth_access_token,
"user-agent": "WeGene SDK",
"accept": "application/json",
}
# Prepare parameters
parameters = {
"report_id": report_id
}
# Prepare and invoke the API call request to fetch the response
response = requests.post(query_url, headers=headers, data=parameters)
# Error handling using HTTP status codes
if response.status_code < 200 or response.status_code > 206: # 200 = HTTP OK
raise APIException("HTTP Response Not OK", response.status_code, response.json())
# Try to cast response to desired type
if isinstance(response.json(), dict):
# Response is already in a dictionary, return the object
try:
return Report(**response.json())
except TypeError:
raise APIException("Invalid JSON returned", response.status_code, response.json())
# If we got here then an error occured while trying to parse the response
raise APIException("Invalid JSON returned", response.status_code, response.json())
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,279
|
using System;
using System.Windows.Forms;
using Hexei.Forms;
namespace Hexei
{
static class Program
{
public static MainForm MainForm { get; private set; }
[STAThread]
static void Main()
{
Application.EnableVisualStyles();
Application.SetCompatibleTextRenderingDefault(false);
MainForm = new MainForm();
Application.Run(MainForm);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,638
|
<?php
declare(strict_types=1);
namespace Inowas\ScenarioAnalysis\Model\Event;
use Inowas\Common\Id\UserId;
use Inowas\ScenarioAnalysis\Model\ScenarioAnalysisId;
use Inowas\ScenarioAnalysis\Model\ScenarioAnalysisName;
use Prooph\EventSourcing\AggregateChanged;
class ScenarioAnalysisNameWasChanged extends AggregateChanged
{
/** @var ScenarioAnalysisId */
private $scenarioAnalysisId;
/** @var ScenarioAnalysisName */
private $name;
/** @var UserId */
private $userId;
public static function of(ScenarioAnalysisId $id, UserId $userId, ScenarioAnalysisName $name): ScenarioAnalysisNameWasChanged
{
$event = self::occur($id->toString(), [
'user_id' => $userId->toString(),
'name' => $name->toString()
]
);
$event->userId = $userId;
$event->name = $name;
return $event;
}
public function scenarioAnalysisId(): ScenarioAnalysisId
{
if ($this->scenarioAnalysisId === null){
$this->scenarioAnalysisId = ScenarioAnalysisId::fromString($this->aggregateId());
}
return $this->scenarioAnalysisId;
}
public function userId(): UserId
{
if ($this->userId === null){
$this->userId = UserId::fromString($this->payload['user_id']);
}
return $this->userId;
}
public function name(): ScenarioAnalysisName
{
if ($this->name === null){
$this->name = ScenarioAnalysisName::fromString($this->payload['name']);
}
return $this->name;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 624
|
{"url":"http:\/\/mathoverflow.net\/questions\/10107\/unique-representation-of-constructible-numbers","text":"# Unique representation of constructible numbers\n\nI am interested in programmatically working with constructible numbers (the closure of the rational numbers under square roots). In order to perform comparisons between numbers I believe I would need a unique (symbolic) representation for them. Does such a thing exist, or what are relevant references for this kind of thing?\n\n-\nWhen you say \"comparison\" do you mean \"test for equality\" or are you trying to determine if one is greater than the other? \u2013\u00a0 Qiaochu Yuan Dec 30 '09 at 8:08\n\nA representation of construcible numbers together with algorithms suitable for mechanized computations is given in chapter 4 of these lecture notes on computer science\n\n-\n\nI believe that there is no nice way to compare constructible numbers. The sum of square roots problem is notoriously difficult, as an example of difficulties you might be facing. The link has relevant references I believe.\n\n-\nstackoverflow.com\/questions\/1784901\/\u2026 is also relevant depending on what you are interested. \u2013\u00a0 Gjergji Zaimi Dec 30 '09 at 8:48","date":"2014-10-26 07:05:13","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.822624683380127, \"perplexity\": 424.78328896344243}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1414119658961.52\/warc\/CC-MAIN-20141024030058-00087-ip-10-16-133-185.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
Stochastic resonance (SR) is a noise-induced nonlinear phenomenon in which an appropriate weak signal evokes the best correlation between a weak periodic signal and the output signal of a bistable system. In a recent years, investigation of features of SR in coupled and networks of nonlinear systems received a great deal of interest. This is primarily because of its constructive role in enhancement of weak signal detection and transmission. In coupled arrays it has been shown that the spatiality may broaden the scope of SR {\cite{r1,r2}}. In a network system this noise-induced resonance can be enhanced by a coupling term and this phenomenon is termed as enhanced SR {\cite{r1}}. In Hodgkin-Huxley neuronal networks increase in the network randomness is found to increase the temporal coherence and spatial synchronization {\cite{r3}}. A resonance-like effect depending on the number of systems, whereby an optimal number of systems leading to the maximum overall coherence has been reported {\cite{r4}}. Resonance is found in certain networks where all the systems are driven by noise while only one system is subjected to a weak periodic signal {\cite{r5}}. Doubly SR consisting of a noise-induced phase transition has been observed in a conventional nonlinear lattice of coupled overdamped oscillators {\cite{r6}}. Double resonance peaks were found to occur in the Ising model network driven by an oscillating magnetic field {\cite{r7}}. Improvement of signal-to-noise ratio is possible in an uncoupled parallel array of bistable system subjected to a common noise {\cite{r8}}. SR has been studied in coupled threshold elements {\cite{r9}}, Ising model {\cite{r10}}, Barabasi-Albert network {\cite{r11}}, arrays of comparators {\cite{r12}} and in a B\"ar model {\cite{r13}}.
Investigation of features of different types of couplings and connectivity topology with reference to various nonlinear phenomena in networks is very important because they carry information from one unit to another and act as a source of amplification of information. This would help us to learn and understand the role of connectivity topology in networks. In this connection, in the context of SR, in most of the studies on network systems, noise and a weak periodic signal are applied to all the systems. Moreover, regular and random single and multiple diffusive couplings are considered. It has been shown that a small fraction of long-range coupling is sufficient to have a great improvement in SR and synchronization {\cite{r14}}. Perc {\cite{r15}} reported that enhanced noise-induced resonance can occur only for intermediate coupling in a small-world network. Self-organized phase-shifts are noticed between large degree and small degree nodes when the coupling terms are weighted according to degree of nodes {\cite{r16}}. Perc and his co-workers analyzed SR in a network system consisting of bistable oscillators in which all the oscillators are subjected to noise, each oscillator is connected to $m$ neighborhood oscillators (diffusive coupling) and a weak periodic signal is applied to one of the oscillators only {\cite{r5,r17}}. Effect of connectivity as a function of the location of oscillators and the amplitude of external forcing is also analyzed {\cite{r18}}. Wu et al {\cite{r19}} shown a monotonic increasing of spectral amplification factor with the coupling constant when the coupling is considered as adaptive.
It is of great significance to introduce both weak input signal and weak noise to only one unit of a network and also consider a simple connectivity, for example, unidirectional linear coupling, and investigate the signal propagation through the units. Such a setup has applications in digital sonar arrays, networks of sensory neurons, analog to digital converters where arrays are useful for the transmission of a noisy periodic input signal and the performance is assessed by the signal-to-noise ratio in the frequency domain. For chaos synchronization and communication in coupled lasers often in the transmitter-receiver setup the transmitter laser is unidirectionally coupled to a receiver laser {\cite{r20}}. One-way coupling was shown to improve the performance of flux-gate magnetometers {\cite{r21}} and enhance the high-frequency induced resonance {\cite{r22}}. Very recently, in an electronic two-dimensional array of bistable oscillators with one-way coupling, soliton-like waves are found to propagate in different directions with different speeds {\cite{r23}}.
Our prime goal in the present work is to investigate the impact of unidirectional regular connection and random connection with linear coupling term on SR in a network with each unit or site represented by a bistable discrete Bellows map. This map possesses the basic requirements to display noise-induced resonance. We have chosen a discrete equation rather than a continuous time evolution equation, as the system of a unit, mainly because the former requires relatively very less computational time and resources. Often discrete maps served as a convenient models for discovering new phenomena and identifying their features. The results of our study, in general, can be realized in networks of other bistable as well as excitable systems with same connectivity topology.
There are four sections in this paper. In Sec.II we consider a network of Bellows map with the weak input periodic signal and Gaussian white noise applied to first unit only with one-way coupling. The first system is uncoupled. The coupling term is linear. We consider single as well as multiple connections. There are several interesting common and different features associated with the various units. For coupling strength above a critical value undamped signal propagation occurs. In this case all the units exhibit SR and the values of noise intensity $D$ at which the response amplitude $Q$ becomes maximum in various units are the same. However, for a fixed noise intensity $Q$ increases with the unit number $i$ and then it becomes a constant. At resonance periodic switching between two states of the map takes place and the units are unsynchronized. Then we demonstrate the influence of unidirectional multiple couplings, that is, coupling of $i$th unit with $i+1,\,i+2,\,...,\ i+m$ units. We show that one and two couplings ($m=1,2$) are more effective than higher number of couplings. Also we study the case of all the units subjected to weak periodic signal and noise with one-way coupling. For sufficiently large values of coupling strength and for a range of values of noise intensity we observe oscillatory variation of $Q$ with the unit number $i$. This behaviour is pronounced for intermediate values of noise intensity. Section III is devoted to the study of unidirectional but random coupling. We consider the cases of fraction of total units ($25\%,\,50\%,\,75\%$ and $100\%$ of total units) coupled to randomly chosen units. $D_{{\mathrm{max}}}$, the value of $D$ at which average response amplitude $\langle Q \rangle$ of the network becomes maximum and $\langle Q \rangle_{{\mathrm{max}}}$, the maximum value of $\langle Q \rangle $ at resonance, increases with increase in the number of units connected. For a fixed noise intensity $\langle Q \rangle $ exhibits sigmoidal function type variation with the total number of units. Further, $\langle Q \rangle $ increases with increase in multiple number of couplings, however, the increment is small. Finally, we present summary of the results in Sec.IV.
\section{Stochastic resonance in an one-way coupled regular network}
In this section we explore the features of SR in a one-way coupled regular network with $N$ units and $m$ couplings. We consider the cases of periodic signal and noise applied to first unit only and to all the units.
\subsection{Description of the network model}
The network consists of $N$ units. The first unit is uncoupled and is alone driven by both weak input periodic signal and noise. The interaction and its range $m$ both are along unidirection. The coupling term is linear and the system representing each unit is the Bellows map {\cite{r24,r25,r26}}. This type of situation is realized in many real-life systems {\cite{r5,r27,r28,r29}}. Figure {\ref{fig1}} depicts the above network topology.
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig1.eps, width=0.55\columnwidth}
\end{center}
\caption{Examples of one-way coupled networks with $10$ units. The dynamics of the first unit is independent of the other units and this is alone driven by both weak periodic signal and noise. (a) $i$th unit ($i \ne 1$) is linearly coupled to ($i-1$)th unit. The arrow mark indicates that the output of $i$th unit is fed to ($i+1$)th unit only through the linear coupling term. (b) $i$th unit ($i>2$) is linearly coupled to both ($i-1$)th and ($i-2$)th sites while the second site ($i=2)$ is coupled to the first site only. }
\label{fig1}
\end{figure}
The dynamics of the network is described by
\begin{subequations}
\label{eq1}
\begin{eqnarray}
x_{n+1}^{(1)}
& = & \frac{r x_n^{(1)} }{ 1 + \left( x_n^{(1)} \right)^b}
+ f \cos \, \omega n + \sqrt{D} \xi (n), \\
x_{n+1}^{(i)}
& = & \frac{r x_n^{(i)} }{ 1 + \left( x_n^{(i)} \right)^b}
+ \frac{\delta}{M} \sum_{j=1}^m x_n^{(i-j)} , \quad
i=2,3,...,N, \;\; i-j \ge 1
\end{eqnarray}
where
\begin{eqnarray}
M = \begin{cases} i-1, & {\mathrm{if}} \; i \le m \\
m, & {\mathrm{if}} \; i > m.
\end{cases}
\end{eqnarray}
\end{subequations}
In Eqs.~(\ref{eq1}) the number of couplings is $m$. For simplicity we choose $N \gg m$. $\xi(n)$ is a Gaussian white noise with the statistical properties $\langle \xi (n) \rangle =0$, $\langle \xi (n) \xi(n') \rangle = \delta(n-n')$, $D$ is noise intensity. We choose the value of $b$ in the Bellows map as $2$. The Bellows map $x_{n+1} =rx_n/(1+x_n^2)$ has only one fixed point $x^* =0$ for $0 < r \le 1$. For $r>1$, it has three fixed points $x^*=0$ (unstable) and $x^*_{\pm} = \pm \sqrt{r-1}$ (stable). The map has bistable state for $r>1$. In order realize SR in the bistable case we drive the map by a weak periodic signal $f \cos \omega n$ and the Gaussian white noise. We fix the values of the parameters in the network as $r=2$, $b=2$, $f=0.3$, $\omega=0.1$, $N=400$ and vary the noise intensity and the coupling strength $\delta$. The values of $f$ is below the subthreshold, that is, in the absence of noise the periodic force alone is unable to induce transition between the two stable fixed points.
\subsection{Stochastic resonance and signal propagation}
To characterize the noise-induced SR we numerically compute the response amplitude $Q_i$ at the input signal frequency $\omega$. $Q_i$ is given by
\begin{subequations}
\label{eq2}
\begin{eqnarray}
Q_i & = & \frac{\sqrt{Q_{i,C}^2+Q_{i,S}^2}}{f}, \\
Q_{i,{\mathrm{C}}} & = & \frac{2}{Tt} \sum_{n=1}^{Tt} x_n^{(i)} \cos \,\omega n, \\
Q_{i,{\mathrm{S}}} & = & \frac{2}{Tt} \sum_{n=1}^{Tt} x_n^{(i)} \sin \,\omega n,
\end{eqnarray}
\end{subequations}
where $t=2 \pi/\omega$ is the period of the input signal $f \cos \omega n$ and $T$ is chosen as $1000$. $Q_i$ is often used as a measure for SR {\cite{r5,r15,r18}}.
Let us point out the occurrence of SR in the first unit of the network the dynamics of which is independent of the other units. As $D$ increases from a small value the response amplitude $Q_1$ increases, reaches a maximum value at $d=D_{{\mathrm{max}}}=0.175$ and decreases with further increase in $D$. For $D \ll D_{{\mathrm{max}}}$ the iterated values rarely switches between the two stable fixed points. At $D=0.175$ the iteration plot, $x_n^{(1)}$ versus $n$, shows almost periodic switching between the regions $x>0$ and $x<0$. Numerically computed mean residence times in the regions $x<0$ and $x>0$ are $\approx T/2$. For $D \gg D_{{\mathrm{max}}}$ erratic switching between these two regions takes place.
We perform extensive numerical simulation for the network system (\ref{eq1}), varying the parameters $\delta$ and $D$. We restrict ourselves to $\delta \in [0,1]$ and $D \in [0,1]$. First we report the resonance dynamics with $m=1$ corresponding to the network topology shown in Fig.~{\ref{fig1}}(a). Figure {\ref{fig2}} presents the effect of coupling strength $\delta $ and $D$ on the response amplitude of the various units. For small values of $\delta$ when $D$ is varied typical SR occurs only in the first few units and in the other units $Q_i=0$ for the entire range of $D$. This is shown in Fig.~{\ref{fig2}}(a).
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig2.eps, width=0.95\columnwidth}
\end{center}
\caption{Three-dimensional plots of variation of the response amplitude $Q$ as a function of unit number $i$ and noise intensity $D$ for three values of the coupling strength $\delta$ for the network system given by Eqs.~({\ref{eq1}}). The values of the parameters are $r=2$, $b=2$, $f=0.3$, $\omega=0.1$, $N=400$ and $m=1$.}
\label{fig2}
\end{figure}
In this case for the units far from the first unit $x_n^{(i)}$ becomes a constant after a transient evolution. When $Q_i \ne 0$ the evolution has not settled on to a steady state value. We note that in a random network of Ising model {\cite{r7}} $Q=0$ and $Q\ne 0$ imply paramagnetic and ferromagnetic phases. When $\delta>0.237$ all the units display SR and moreover resonance occurs at the same value of $D$ in all the units. For $0.237 < \delta<0.4$ the response amplitude of the last unit is less than $Q_1$. In Fig.~{\ref{fig2}}(b) for $\delta = 0.27$ $Q_i$ decreases with the unit number $i$. Undamped signal propagation takes place, however, there is no enhancement of the output signal (at the frequency $\omega$) of the last unit. That is, $Q$ of the last unit is $<1$. For $\delta>0.4$ $Q_{{\mathrm{max}}}$ increases and becomes a constant with the unit number $i$. Further, $Q_{{\mathrm{max}}}$ of $i$th unit ($i \ne 1$) is greater than that of the first unit. An example for this is shown in Fig.~{\ref{fig2}}(c) where $\delta=0.75$. For clarity in Fig.~{\ref{fig3}} we plot $Q_i$ versus $i$ for several fixed values of $D$. For each fixed value of $D$, $Q_i$ displays sigmoidal type variation with $i$. In Fig.~{\ref{fig3}} for $D=0.002$ and $0.005$ we observe $Q_i<Q_1$ while $Q_i>Q_1$ for other values of $D$ .
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig3.eps, width=0.45\columnwidth}
\end{center}
\caption{Variation of the response amplitude with the unit number $i$ for six fixed values of the noise intensity $D$ with $\delta=0.75$ and $m=1$. The values of $D$ for the curves $1-6$ are $0.002$, $0.005$, $0.02$, $0.05$, $0.175$ and $0.5$ respectively. }
\label{fig3}
\end{figure}
In the numerical simulation for $\delta>0.4$ we find enhanced undamped signal propagation (that is, $Q_{400}>Q_1$) except for very small values of $D$. Though the weak periodic signal and noise is applied to the first unit alone the linear coupling of it to the second unit is able to stimulate it to exhibit SR and this process propagates to the successive units through simple unidirectional coupling. Figure {\ref{fig4}} displays the variation of $Q_{{\mathrm{max}}}$ of the last unit with the coupling strength $\delta$. It varies rapidly for values of $\delta$ near $1$. For $\delta > 1$ the iterations $x_n^{(1)}$ diverge.
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig4.eps, width=0.4\columnwidth}
\end{center}
\caption{$Q_{{\mathrm{max}}}$ of the last unit with the coupling strength $\delta$. The dashed line represents the value of $Q_1$ which is independent of $\delta$.}
\label{fig4}
\end{figure}
\begin{figure}[!h]
\begin{center}
\epsfig{figure=fig5.eps, width=0.4\columnwidth}
\end{center}
\caption{Regions of various ranges of gain factor $G$ in the $(\delta,D$) parameter space of the network system given by Eqs.~(\ref{eq1}) with $m=1$. The motion is unbounded for $\delta>1$.}
\label{fig5}
\end{figure}
We define the gain factor $G=Q_N/Q_1$ with $N=400$. We numerically compute it for values of $D$ and $\delta$ in the interval $[0,1]$. Regions in $(\delta,D$) parameter space with $5$ ranges of $G$, namely, $=0$, $<1$, $[1,2]$, $[2,5]$ and $>5$ are depicted in Fig.~{\ref{fig5}}. The coupling strength has a strong influence on $G$. The gain factor $G$ is $0$ for small values of $\delta$ for the entire range of noise intensity considered. On the other hand, it increases with increase in $\delta$ for fixed values of $D$.
Figure {\ref{fig6}} presents an interesting result. For $\delta >0.237$ for each fixed value of $D$ the number of oscillations of $x_n^{(i)}$ in the regions $x<0$ ($x>0$) before switching to $x>0$ ($x<0$) decreases with increase in the unit number $i$. This is clearly evident in Fig.~{\ref{fig6}} where $x_n^{(i)}$ versus $n$ is plotted for $i=1$, $12$ and $100$ for $\delta=0.75$ and $D=0.175$. For large $i$ the coupling term weakens the oscillation in the regions $x<0$ and $x>0$ and the output signal appears as a rectangular pulse. That is, the one-way coupling with appropriate strength gives rise undamped propagation of signal in the form of rectangular pulse. The Fourier series of such a signal will contain frequencies $l \omega$ where $l=1,2,...$ with decaying amplitudes. Figure {\ref{fig6}} corresponds to the case at which SR occurs. $x_n^{(i)}$ exhibits almost periodic switching between the two regions $x<0$ and $x>0$. We note that $x_n^{(i)}$'s are not synchronized in the sense that switching of the various units from one region to another region not takes place at almost same value of $n$. We numerically calculate the mean residence time of each unit in the regions $x<0$ and $x>0$. At resonance mean residence times of all the units are found to be $T/2$ where $T=2 \pi / \omega$.
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig6.eps, width=0.45\columnwidth}
\end{center}
\caption{$x_n^{(i)}$ versus $n$ for the units $i=1$, $2$ and $100$ of the network (\ref{eq1}) with $m=1$, $\delta=0.75$ and $D=0.175$ at which the response amplitude becomes a maximum. $x_n^{(i)}$ exhibits periodic switching between the regions $x>0$ and $x<0$. }
\label{fig6}
\end{figure}
\begin{figure}[!h]
\begin{center}
\epsfig{figure=fig7.eps, width=0.9\columnwidth}
\end{center}
\caption{The effect of multiple coupling on the response amplitude of the last unit of the network system given by Eqs.~({\ref{eq1}}) for two values of the coupling constant $\delta$. In the subplot (b) the number marked to each curve is the value of $m$ (number of couplings). The dashed curve is the response amplitude $Q_1$. }
\label{fig7}
\end{figure}
Next, we show the influence of multiple couplings ($m>1$) on SR in the network ({\ref{eq1}}). Figure {\ref{fig7}} presents the variation of the response amplitude of the last unit for two values of $\delta$ for few values of $m$. In Fig.~{\ref{fig7}}(a) for $\delta=0.75$ the variation of $Q_{400}$ with $m$ is almost negligible. For $\delta=1$, in Fig.~{\ref{fig7}}(b), the effect of number of couplings is clearly evident. For the entire range of values of $D$ the response amplitude $Q_{400}$ is $> Q_1$. $D_{{\mathrm{max}}}$ is independent of $m$. For $m>3$ we find that $Q_{400}(m) < Q_{400}(m=1)$, however, it is much higher than $Q_1$. We thus conclude that coupling to one or two units is more effective than more number of couplings.
\subsection{Network with periodic signal and noise applied to all the units}
Now, we study the case of the network where all the units are driven by the periodic force and noise and the units are again unidirectionally coupled. The network model is given by
\begin{eqnarray}
\label{eq3}
x_{n+1}^{(i)}
& = & \frac{r x_n^{(i)} }{ 1 + \left( x_n^{(i)} \right)^b}
+ f \cos \, \omega n + \sqrt{D} \xi_i (n)
+ \frac{\delta \epsilon_i}{M} \sum_{j=1}^m x_n^{(i-j)} ,
\nonumber \\
& & \;\; i=1,2,...,N, \;\; i-j \ge 1
\end{eqnarray}
where $M$ is given by Eq.~({\ref{eq1}}c) and $\epsilon_i=0$ for $i=1$ and $1$ for $i>1$. $\xi_i(n)$ indicates that the units are being under different and independent Gaussian white noise with zero mean and $\langle \xi_i(n) \xi_i(n') \rangle = \delta(n-n')$.
There are differences in the signal propagation when all the units are driven. Figure {\ref{fig8}} presents the three-dimensional plot of dependence of the response amplitude with the unit number $i$ and the noise intensity $D$ for four fixed values of the coupling strength $\delta$ with $m=1$. Evidently, each unit exhibits SR with respect to noise intensity for each fixed value of $\delta$. Before discussing on the Fig.~{\ref{fig8} we show in Fig.~{\ref{fig9}} the variation of $D_{{\mathrm{max}}}$ and $Q_{{\mathrm{max}}}$ of various units for $\delta \in [0,1]$.
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig8.eps, width=0.92\columnwidth}
\end{center}
\caption{Variation of the response amplitude of various units with the noise intensity for four fixed values of the coupling constant $\delta$ of the network system ({\ref{eq3}}) where all the units are driven by the periodic force and noise. The values of the parameters are $r=2$, $b=2$, $f=0.3$, $\omega=0.1$, $N=400$ and $m=1$. }
\label{fig8}
\end{figure}
\begin{figure}[!h]
\begin{center}
\epsfig{figure=fig9.eps, width=0.95\columnwidth}
\end{center}
\caption{Plots of (a) $D_{{\mathrm{max}}}$ and (b) $Q_{{\mathrm{max}}}$ versus the unit number $i$ and the coupling strength of the network ({\ref{eq3}}) with $m=1$. }
\label{fig9}
\end{figure}
$D_{{\mathrm{max}}}$ of each unit varies with $\delta$ (except for very small values) and further for a fixed value of $\delta$ its values for various units are not the same. We recall that in the case of the network ({\ref{eq1}}) with the periodic force and noise applied to the first unit only, $D_{{\mathrm{max}}}$ is the same for all the units and independent of $\delta$. In Fig.~{\ref{fig8}}(a) ($\delta=0.1$) the value of $Q_i$ at resonance increases with $i$ and then becomes a constant with $Q_{N,{\mathrm{max}}}>Q_{1,{\mathrm{max}}}$. For $\delta=0.5$, in Fig.~{\ref{fig8}}(b), we observe oscillatory variation of $Q_i$ with the unit number $i$ for small values of $D$. The range of values of $D$ over which $Q_i$ oscillates with $i$ and its period of oscillation increase with increase in $\delta$. The oscillation of the response amplitude pronounces for higher values of $\delta$. Though $Q_i$ and $Q_{{\mathrm{max}}}$ oscillate an interesting result is that for each fixed value of $D$ we find $Q_i(i>1) > Q_1$. The oscillatory variation of $Q_i$ is not found for the network system ({\ref{eq1}}). In Figs.~{\ref{fig4}} and {\ref{fig9}} we observe very high enhancement of response amplitude at resonance for a wide range of values of $\delta$. Our study indicates that driving the first unit alone by the periodic force and noise with unidirectional coupling is a better one compared to driving all the units.
\section{Stochastic resonance in a network with random coupling}
In this section we consider a network of Bellows map with random one-way coupling with varying connection probability. We study the effect of $25\%$, $50\%$, $75\%$ and $100\%$ of total units coupled (connected) to randomly chosen units on SR. Self-feedback is avoided. All the units are subjected to weak input periodic signal and noise.
The network is governed by the equations
\begin{eqnarray}
\label{eq4}
x_{n+1}^{(i)}
& = & \frac{r x_n^{(i)} }{ 1 + \left( x_n^{(i)} \right)^b}
+ f \cos \, \omega n + \sqrt{D} \xi_i (n)
+ \frac{\delta \epsilon_i}{m} \sum_{j=1}^m x_n^{(M_{ij})} ,
\;\; i=1,2,...,N,
\end{eqnarray}
where $N$ is the total number of units in the network, $m$ is the number of couplings, $M_{ij}(\ne i)$, $j=1,2,...,m$ are randomly chosen $m$ distinct units and $\epsilon_i=1$ if the $i$th unit is chosen for connectivity otherwise it is $0$. In the numerical simulation we use $r=2$, $b=2$, $f=0.3$, $\omega=0.1$ and $N=400$.
Because the number of units selected for coupling and the units with which selected units are coupled are chosen randomly we use the average response amplitude $\langle Q \rangle$ to capture SR. We calculate $\langle Q \rangle$ using the following procedure. After every iterations of Eqs.~({\ref{eq4}}) (leaving sufficient transient) we find the average value of $x_n^{(i)}$ given by $\langle x_n \rangle =(1/N) \sum_{i=1}^{N}x_n^{(i)}$. Using these average values over $10^3T$, where $T=2 \pi / \omega$, iterations we compute the response amplitude using the Eqs.~({\ref{eq2}) and denote it as $\langle Q_j \rangle$. It is the response amplitude of, say, $j$th realization of the network topology. We repeat this process for $100$ different realizations of the network topology and then compute the average of $\langle Q_j \rangle$ and call it as average response amplitude $\langle Q \rangle$.
First we fix $m=1$ in Eqs.~({\ref{eq4}}) and show the effect of size $N$ of the network on $\langle Q \rangle$. Further, we chose $\delta=0.3$. Figure {\ref{fig10}}(a) presents the variation of $\langle Q \rangle$ with the total number of units $N$ for few values of noise intensity. Only $N/4$ units are chosen for connection. Figure {\ref{fig10}}(b) shows the result for all the $N$ units set into connection. In both the cases $\langle Q \rangle$ increases or decreases sharply depending on the value of $D$ with $N$ and then reaches a saturation.
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig10.eps, width=0.9\columnwidth}
\end{center}
\caption{$\langle Q \rangle$ versus the total number of units in the network given by Eqs.~({\ref{eq4}}) for three fixed values of noise intensity with $\delta=0.3$. (a) $25\%$ and (b) $100\%$ of the total units are wired to randomly chosen units. The number of coupling is $1$. }
\label{fig10}
\end{figure}
Figure {\ref{fig11}} shows three-dimensional plot of $\langle Q \rangle$ versus $D$ and $\delta$ for $N/4$ and $N$ units are wired. The curves display typical SR character. The effect of $\delta$ on the resonance profile can be clearly seen. Increasing the value of $\delta$ requires increasingly higher $D$ for the maximum response. In addition to the coupling strength $\delta$ the number of random connections also affects the resonance.
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig11.eps, width=0.9\columnwidth}
\end{center}
\caption{$\langle Q \rangle$ as a function of the coupling strength $\delta$ and the noise intensity $D$ for the network ({\ref{eq4}}) with $m=1$. The subplots (a) and (b) correspond to the cases of $25\%$ and $100\%$ of the units coupled to randomly selected units. }
\label{fig11}
\end{figure}
To gain more insight into the dependence of SR on the number of units coupled, we compute $D_{{\mathrm{max}}}$, the value of $D$ at which resonance occurs, and the corresponding value of average response amplitude denoted as $\langle Q \rangle_{{\mathrm{max}}}$ (best average response amplitude). We consider the variation of these quantities with the coupling strength $\delta$ for $N/4$, $N/2$, $3N/4$ and $N$ units are set connected randomly. In all the cases both the quantities are found to vary monotonically with $\delta$. Further, for a fixed value of $\delta$ both $D_{{\mathrm{max}}}$ and $\langle Q \rangle_{{\mathrm{max}}}$ increases nonlinearly (not shown) with the number of units connected. Thus $\langle Q \rangle_{{\mathrm{max}}}$ can be enhanced either by increasing the coupling strength for a fixed number of connections or by increasing the number f connectivity for a fixed value of $\delta$.
Lastly, we show the influence of number of coupling $m$ on $\langle Q \rangle$. When a unit $i$ is selected for coupling then it is coupled to randomly selected $m$ distinct units avoiding self-coupling. Figure {\ref{fig12}} demonstrates the effect of number of couplings on $\langle Q \rangle$ where $\delta=0.25$. For each fixed value of $D$ the average response amplitude increases with the number of couplings, however, the enhancement of it is quite small. Similar effect is found for higher values of $\delta$ also.
\begin{figure}[t]
\begin{center}
\epsfig{figure=fig12.eps, width=0.95\columnwidth}
\end{center}
\caption{Variation of $\langle Q \rangle$ as a function of noise intensity for the system ({\ref{eq4}}) with $\delta=0.25$ and for $1$, $2$ and $10$ couplings. Randomly selected (a) $25\%$ and (b) $100\%$ of the total units are connected to randomly chosen units. }
\label{fig12}
\end{figure}
\section{Conclusions}
In the present work we analyzed SR in three types of network systems. In the network ({\ref{eq1}}) periodic signal and noise are added to the first unit only. In the network ({\ref{eq1}}) all the units are driven by the periodic force and noise. The two networks are regular networks as the output of $i$th unit is fed to the ($i+1$)th unit in the case of single coupling. In the network given by Eq.~({\ref{eq4}}) all the units are driven, however, fraction of total number of units are selected randomly for connectivity and are then connected to randomly chosen units. These three networks show certain similarities and difference in the SR phenomenon. In the first network enhanced and undamped propagation of signal at the low-frequency of the input signal occurs above a certain critical value of the coupling strength. In the other two networks enhanced and undamped propagation of signal is realized for the entire range of coupling strength and this is because all the units are driven by the periodic force and noise. Each unit displays only single resonance. Response amplitude varies with the unit number $i$ and attains a saturation. This implies that with certain optimum number of units in a network of the type of connectivities considered in the present work one can realize maximum response.
In the network ({\ref{eq1}}) the oscillation induced about the two fixed points by the periodic force and noise decreases with increase in the unit number. For the distant units the output becomes a sequence of rectangular pulse and this property is observed for all values of noise intensity. This dynamics is not observed in the other two networks. In the network ({\ref{eq1}}) the value of noise intensity $D_{{\mathrm{max}}}$ is independent of the coupling strength $\delta$ while in the other two networks it varies with $\delta$. Referring to the Figs.~{\ref{fig7}} and {\ref{fig12}} we infer that with single coupling alone we can realize great enhancement of response amplitude. We add that the observation of enhancement of response amplitude across the whole network by applying the input signal and noise to only one unit is greatly useful for weak signal detection and information propagation.
\newpage
\section*{Figure Caption}
\vskip 11pt
\noindent{\bf{FIG.1:}} Examples of one-way coupled networks with $10$ units. The dynamics of the first unit is independent of the other units and this is alone driven by both weak periodic signal and noise. (a) $i$th unit ($i \ne 1$) is linearly coupled to ($i-1$)th unit. The arrow mark indicates that the output of $i$th unit is fed to ($i+1$)th unit only through the linear coupling term. (b) $i$th unit ($i>2$) is linearly coupled to both ($i-1$)th and ($i-2$)th sites while the second site ($i=2)$ is coupled to the first site only.
\vskip 11pt
\noindent{\bf{FIG.2:}} Three-dimensional plots of variation of the response amplitude $Q$ as a function of unit number $i$ and noise intensity $D$ for three values of the coupling strength $\delta$ for the network system given by Eqs.~({\ref{eq1}}). The values of the parameters are $r=2$, $b=2$, $f=0.3$, $\omega=0.1$, $N=400$ and $m=1$.
\vskip 11pt
\noindent{\bf{FIG.3:}} Variation of the response amplitude with the unit number $i$ for six fixed values of the noise intensity $D$ with $\delta=0.75$ and $m=1$. The values of $D$ for the curves $1-6$ are $0.002$, $0.005$, $0.02$, $0.05$, $0.175$ and $0.5$ respectively.
\vskip 11pt
\noindent{\bf{FIG.4:}} $Q_{{\mathrm{max}}}$ of the last unit with the coupling strength $\delta$. The dashed line represents the value of $Q_1$ which is independent of $\delta$.
\vskip 11pt
\noindent{\bf{FIG.5:}} Regions of various ranges of gain factor $G$ in the $(\delta,D$) parameter space of the network system given by Eqs.~(\ref{eq1}) with $m=1$. The motion is unbounded for $\delta>1$.
\vskip 11pt
\noindent{\bf{FIG.6:}} $x_n^{(i)}$ versus $n$ for the units $i=1$, $2$ and $100$ of the network (\ref{eq1}) with $m=1$, $\delta=0.75$ and $D=0.175$ at which the response amplitude becomes a maximum. $x_n^{(i)}$ exhibits periodic switching between the regions $x>0$ and $x<0$.
\vskip 11pt
\noindent{\bf{FIG.7:}} The effect of multiple coupling on the response amplitude of the last unit of the network system given by Eqs.~({\ref{eq1}}) for two values of the coupling constant $\delta$. In the subplot (b) the number marked to each curve is the value of $m$ (number of couplings). The dashed curve is the response amplitude $Q_1$.
\vskip 11pt
\newpage
\noindent{\bf{FIG.8:}} Variation of the response amplitude of various units with the noise intensity for four fixed values of the coupling constant $\delta$ of the network system ({\ref{eq3}}) where all the units are driven by the periodic force and noise. The values of the parameters are $r=2$, $b=2$, $f=0.3$, $\omega=0.1$, $N=400$ and $m=1$.
\vskip 11pt
\noindent{\bf{FIG.9:}} Plots of (a) $D_{{\mathrm{max}}}$ and (b) $Q_{{\mathrm{max}}}$ versus the unit number $i$ and the coupling strength of the network ({\ref{eq3}}) with $m=1$.
\vskip 11pt
\noindent{\bf{FIG.10:}} $\langle Q \rangle$ versus the total number of units in the network given by Eqs.~({\ref{eq4}}) for three fixed values of noise intensity with $\delta=0.3$. (a) $25\%$ and (b) $100\%$ of the total units are wired to randomly chosen units. The number of coupling is $1$.
\vskip 11pt
\noindent{\bf{FIG.11:}} $\langle Q \rangle$ as a function of the coupling strength $\delta$ and the noise intensity $D$ for the network ({\ref{eq4}}) with $m=1$. The subplots (a) and (b) correspond to the cases of $25\%$ and $100\%$ of the units coupled to randomly selected units.
\vskip 11pt
\noindent{\bf{FIG.12:}} Variation of $\langle Q \rangle$ as a function of noise intensity for the system ({\ref{eq4}}) with $\delta=0.25$ and for $1$, $2$ and $10$ couplings. Randomly selected (a) $25\%$ and (b) $100\%$ of the total units are connected to randomly chosen units.
\newpage
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Padam, padam… – piosenka nagrana przez Édith Piaf, którą pierwotnie opublikowano w 1951 roku. Utwór napisali dla niej Norbert Glanzberg (muzyka) i Henri Contet (tekst).
Kompozycja
Utwór ten to walc. Piosenkę określono jako "nieznośnie chwytliwa".
Podmiotem lirycznym jest kobieta, która śpiewając piosenkę opisuje jak konkretna melodia przywołuje w jej wspomnieniach byłego kochanka (gdy ona miała 20 lat).
Lista utworów
Opracowano na podstawie materiału źródłowego:
7″ EP EMI Columbia ESRF 1023 (1954, Francja)
"Padam, padam…" (3:17)
"Jézébel" (3:07)
"Mariage" (4:16)
"Les amants de Venise" (3:10)
Inne wersje
Swoje wersje piosenki nagrali m.in.: Tony Martin (1952), Petula Clark (1963), Mireille Mathieu (1993), Chimène Badi (2003), Ann Christy (2000) i Patricia Kaas (2012).
Przypisy
Linki zewnętrzne
Piosenki powstałe w roku 1951
Piosenki Édith Piaf
|
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\section{Introduction}
\label{sec:intro}
In recent years the development of high precision experiments in particle physics demanded a theoretical upgrade to match the accuracy achieved at the LHC. The state-of-the-art in fixed order computations for processes with up to two hard partons in the final state is reaching next-to-next-to-leading-order (NNLO), i.e. ${\cal{O}}(\alpha_s^2)$. Since $\alpha_s^2\sim \alpha$, it becomes necessary to include also the corresponding NLO ElectroWeak (EW) corrections, that for many observables exceed the few percent level (e.g. \cite{Dittmaier:2012kx,Dittmaier:2013hha}) and become quantitatively important for an accurate description.
Both precise measurements and calculations are essential to test different aspects of the Standard Model (SM) and to discern between them and possible new physics evidences due to Beyond the Standard Model (BSM) effects.
In this sense, inclusive massive lepton pair production (Drell-Yan process) has worked as an important testground of perturbative quantum chromodynamics (QCD). On one hand, it has offered a sensitive way to study parton distribution functions (PDFs) \cite{Ball:2014uwa,Harland-Lang:2014zoa,Dulat:2015mca}. From weak bosons production, charge asymmetry measurements and invariant mass dependences have helped to extract precise information on both the quarks valence structure functions and the separation of quarks flavours, as well.
On the other hand, knowing the behaviour of charged-current (CC) and neutral-current (NC) processes has allowed to perform high-precision measurements of fundamental ElectroWeak
parameters, like $Z$ and $W$ widths and masses and the EW mixing angle. With $W$ and $Z$ bosons in final or intermediate states of hadronic processes, some clean production channels can be characterised with great level of accuracy by studying leptonic decay modes. This has also served to accurately calibrate detector components, which was important for further measurements \cite{Brock:1999ep}.
In addition, the Drell-Yan process is not only relevant to test SM predictions, but also to evaluate alternative BSM theories, where $W$ and $Z$ bosons usually appear as final or intermediate states in the decay of particles predicted in new physics models, like new gauge interactions, supersymmetry or heavy resonances \cite{Mangano:2015ejw}. For all these reasons, the study of Drell-Yan processes, and in general, theoretical and experimental exploration of multiple EW gauge boson production, becomes a priority in the development of current high-energy physics.
In this sense, and considering that the improvement in statistics over the last years has made higher-order corrections experimentally noticeable, having access to QCD (and QED) corrections to these processes has become of great importance to put the previous predictions on a firmer ground. For instance, it is important to achieve a better comprehension of the so-called Sudakov regime, where large logarithmic EW factors play an important role, and $\mathcal{O}(\alpha\alpha_s)$, that represent the first QCD corrections to these large EW factors, or even $\mathcal{O}(\alpha^2)$ corrections, might be sizable.
Furthermore, recent work has been performed to include QED effects in the evolution of parton distributions, by providing explicit expressions for splitting kernels up to $\mathcal{O}(\alpha\alpha_s)$ \cite{deFlorian:2015ujt} and $\mathcal{O}(\alpha^2)$\cite{deFlorian:2016gvk} and by the determination of precise photon distributions in the proton within the LUXqed approach \cite{Manohar:2016nzj,Manohar:2017eqh} . The availability of these QED corrected parton distributions is essential to match the theoretical calculations at the partonic level.
From the point of view of partonic cross-sections, the $\mathcal{O}(\alpha\alpha_s)$ and $\mathcal{O}(\alpha)$ corrections represent the first EW and mixed order contributions to Drell-Yan pair production in the general expansion
\begin{equation}
\label{eq:expansion}
d\sigma = \sum _{i,j} \alpha_s^i \alpha^j d\sigma^{(i,j)},
\end{equation}
where pure EW $d\sigma^{(0,j)}$ and QCD $d\sigma^{(i,0)}$ corrections, as well as \textit{mixed order} contributions, which combine effects of the two interactions, arise.
So far, QCD corrections to the total cross- section have been calculated at next-to leading order (NLO) in ref \cite{Altarelli:1979ub}, and at next-to-next-to leading order (NNLO) in an inclusive way, in refs. \cite{Hamberg:1990np,vanNeerven:1991gh,Harlander:2002wh}. Exclusive results have also been presented up to NNLO QCD accuracy \cite{Catani:2009sm,Melnikov:2006di,Melnikov:2006kv, Gavin:2010az,Gavin:2012sy,Boughezal:2016wmq}. Additionally, threshold calculations have been performed at next-to-next-to-next-to-leading-order (${\rm N^3LO}$) and next-to-next-to-next-to-leading-logarithmic (${\rm N^3LL}$) accuracy in refs. \cite{Ahmed:2014cla,Catani:2014uta}.
On the other hand, concerning the EW contributions, exclusive computations for NLO-EW corrections to CC-DY are available in refs. \cite{Dittmaier:2001ay,Baur:2004ig,CarloniCalame:2006zq} and for NC-DY, in refs. \cite{Baur:2001ze,Baur:1997wa}. Finally, progress towards the computation of NNLO-EW has been accomplished in recent years too \cite{Actis:2006ra,Actis:2006rb,Actis:2006rc,Degrassi:2003rw}.
Due to the lack of the full calculation of the NNLO mixed-order terms $\mathcal{O}(\alpha\alpha_s)$, different approaches have been followed to approximately combine the QCD and QED/EW corrections \cite{Cao:2004yy,Balossini:2009sa,Adam:2008pc,Li:2012wna,Barze:2013fru}, by either assuming the full factorisation or the additive combination of the strong and electroweak contributions. Particularly, recent partial exclusive results have been presented for the resonance region, by using the pole approximation \cite{Dittmaier:2014koa,Dittmaier:2014qza,Dittmaier:2015rxo}.
The contributions for a general (i.e. including the decay of the gauge boson) perturbative calculation
of Drell-Yan can be roughly characterised into the following subsets: on one hand, {\it purely factorisable}
terms that arise due to
initial state ({\it production}, from the initial state partons) and final state ({\it decay}, from the final state leptons) emission
and, on the other hand, {\it non-factorisable} terms
originated by soft photon exchange between the production and the decay.
The non-factorisable $\mathcal{O}(\alpha \alpha_s)$ terms have been shown \cite{Dittmaier:2014qza,Dittmaier:2014koa,Dittmaier:2015rxo}
to have a negligible impact on the cross section, allowing to treat effectively Drell-Yan
in the (resonant) limit of the decoupling between the production and decay processes, at least for the achieved experimental accuracy. The results presented in \cite{Dittmaier:2015rxo} also rely on the assumption that the missing {\it initial-initial state factorisable } $\mathcal{O}(\alpha \alpha_s)$ contributions are very small.
The computation of the so far unknown mixed QCD$\times$QED $\mathcal{O}(\alpha \alpha_s)$ corrections
to the inclusive on-shell production of a $Z$ boson in hadronic collisions
is exactly the main goal of this paper\footnote{In order to separate the QED contributions computed here from the {\it weak} induced effects, we consider the coupling between the $Z$ boson and the quarks as an {\it effective coupling} and do not take into account self-energy insertions in the $Z$ (and eventually $\gamma$) propagator.}.
Those contributions are by themselves a gauge-invariant set of the complete Drell-Yan cross section calculation
at $\mathcal{O}(\alpha \alpha_s)$, even for CC-DY. Furthermore, counting with analytical expressions for the total cross section can be useful to establish a subtraction method to compute differential distributions for different observables at $\mathcal{O}(\alpha \alpha_s)$ by extending, for example, the $q_T-$ subtraction method \cite{Catani:2007vq} originally developed for pure QCD corrections.
In principle a full computation of QCD$\times$QED $\mathcal{O}(\alpha \alpha_s)$ terms involves, as in any NNLO calculation, the evaluation
of double-virtual, single-virtual plus one parton emission and double parton emission contributions, where parton
in general refers to quarks, antiquarks, gluons, and photons. Most of the needed double real contributions where recently presented in \cite{Bonciani:2016wya}, including the case of $W$ boson production which is not discussed in this paper \footnote{Only the contribution from the interference of QCD and QED $qq\rightarrow qq$ diagrams is missing in \cite{Bonciani:2016wya}.},while the master integrals for the two-loop calculation were obtained in \cite{Bonciani:2016ypc}.
Instead of following the path of a dedicated calculation for each term, in this work we profit from the
available computation of NNLO pure QCD corrections $\mathcal{O}(\alpha_s^2)$ presented in \cite{Hamberg:1990np} and, by pointing out the abelian
component, we extract the corresponding QCD$\times$QED $\mathcal{O}(\alpha \alpha_s)$ contributions and asses the
phenomenological impact for the inclusive cross section at different hadronic energies.
Furthermore, by following the same procedure
we also present the QED$^2$ $\mathcal{O}(\alpha^2)$ corrections, completing, therefore, the set of NNLO contributions
in QCD$\oplus$QED (i.e. all terms that correspond to $i+j=2$ in Eq.\eqref{eq:expansion}).
This paper is organised as follows:
In Section \ref{sec:abel} we present the method used to compute the QCD$\times$QED $\mathcal{O}(\alpha \alpha_s)$ and QED$^2$ $\mathcal{O}(\alpha^2)$ contributions from the pure QCD corrections $\mathcal{O}(\alpha_s^2)$. In Section \ref{sec:results} we present the results and study the detailed phenomenology of the corrections at different collider energies. Finally, in Section \ref{sec:conc} we present our conclusions. The explicit NNLO coefficients are presented in the Appendices.
\section{Abelianisation procedure}
\label{sec:abel}
In order to achieve the $\mathcal{O}(\alpha\alpha_s)$ contributions we profit from the availability of previous NNLO QCD calculations \cite{Hamberg:1990np}. In summary, we analyse the contributing diagrams for each interaction and take the corresponding abelian limit from the existing NNLO QCD calculation \cite{deFlorian:2015ujt,deFlorian:2016gvk}.
To schematise the procedure used to accomplish the mixed order contribution, we describe the algorithm of {\it gluon-photon interchange} taking as an example the most relevant $q\bar{q}$ channel.
First of all we analyse all the topologies inherent to each partonic subprocess in QCD and determine the corresponding colour factors. Then we replace a gluon by a photon in each diagram and profit from the similarity in QED and QCD fermion coupling factors to pin down the abelian parts of both calculations. Furthermore, given that the photon has no colour, non-abelian terms (i.e. all the terms arising from NNLO QCD diagrams that contain three or four gluon vertices) do not contribute to QED corrections and therefore can be thrown out. Thus, we recalculate the colour factors for the mixed topologies attained identifying changes and substitutions to be made in previous QCD results in order to obtain mixed order QCD$\times$QED correction terms.
\begin{figure}
\includegraphics[width=0.7\columnwidth]{NOTATION_REAL_DIAGRAMS_QQBAR_QCD.eps}
\centering
\caption{Some of the diagrams contributing to NNLO QCD corrections to Drell-Yan.}
\label{qqbar}
\end{figure}
For the explicit case of the $q\bar{q}$ channel, in order to show how the method works and given that the colour factors in the partonic cross section only depend on the QCD structure, we concentrate on diagrams with double real emission which appear only to tree level (the same considerations can be applied to the two-loop and one-loop plus real emission contributions). As can be observed in Fig.\ref{qqbar} there are four kinds of diagrams to consider, which we will label with the supra-index $(k,l)$ according to the total number of QCD ($k$) and QED ($l$) vertices for each topology. It is also important to note that the order of external momenta does affect the colour structure of the diagram. In this sense, we will refer as $(a)$ to the diagram showed in Fig.\ref{qqbar}, while $(a')$ represents the corresponding diagram obtained after crossing the final state parton lines. There we can recognise different colour factors, according to the configurations of colour matrix traces in the calculation of each contribution to the partonic cross section.
For example, terms corresponding to the calculation of $\left| (a)^{(2,0)} \right| ^2$ in NNLO QCD result proportional to $\frac{1}{2N_C^2}Tr\left[ T^bT^aT^aT^b\right]=\frac{1}{2N_C}C_F^2$, where the $N_C^2$ in the denominator arises due to the average over the colour factor of the incoming quarks and the symmetry factor $1/2$ is due to the appearance of two identical gluons in the final state.
For the case of $\left[(a)^{(2,0)} \times (a'^*)^{(2,0)}\right]$ both abelian and non-abelian contributions appear resulting in a factor $\frac{1}{2 N_C^2}Tr\left[ T^bT^aT^bT^a\right]=\frac{1}{2N_C}C_F\left(C_F-C_A/2\right)$ and, when considering terms from $\left[(b)^{(2,0)} \times (a^*)^{(2,0)}\right]$, they result proportional to
$\frac{1}{2N_C^2}\ f^{abc}Tr\left[ T^cT^aT^b\right]=\frac{1}{2N_C^2}\ Tr\left[[ T^a,T^b]T^aT^b\right]=-\frac{1}{2N_C}(C_FC_A/2)$, a purely non-abelian contribution.
\begin{figure}
\includegraphics[width=0.7\columnwidth]{NOTATION_REAL_DIAGRAMS_QQBAR_QCDxQED.eps}
\centering
\caption{Diagrams that result after applying the abelianisation procedure to the real NNLO QCD corrections in Fig.\ref{qqbar}.}
\label{qqbarQCDQED}
\end{figure}
Once colour factors are characterised for each term, we choose a gluon in the diagram, replace it by a photon and recalculate the colour structure, thus obtaining modified diagrams with the corresponding new factors for QCD$\times$QED corrections. These are shown in Fig.\ref{qqbarQCDQED}. Naturally, all the diagrams of type $(b)$ (i.e. topologies containing at least one 3-gluon-vertex), which always contribute to second order for the NNLO QCD calculation in this process, vanish when considering the abelian limit.
Taking this into account, we find that the modified factors for $\left|(a)^{(1,1)}\right| ^2$ and $\left[(a)^{(1,1)} \times (a'^*)^{(1,1)}\right]$ are both given by $\frac{e_q^2}{N_C^2}Tr[ T^aT^a]=\frac{e_q^2}{N_C} C_F$, where we have included the charge of the quark for the QED coupling, while the non-abelian one obviously vanishes. Here we may notice that all the colour factors proportional to $C_A$, which corresponds to non-abelian part of the calculation, could be thrown out when considering the abelian limit, while the ones proportional to $C_F^2$ are to be replaced by $2e_q^2C_F$, thus obtaining QCD$\times$QED factors in each case.
It is worth noticing by performing the same analysis for the topology shown in Fig.\ref{qqbar}c, i.e. the production of a $q\bar{q}$ pair, that also the colour factor $T_R$ vanishes when the similar contribution is analysed in the
QCD$\times$QED case, since the result for $\left[(c)^{(2,0)} \times (c^*)^{(0,2)}\right]$ becomes proportional to $Tr[ T^a]$. Therefore, since terms proportional to both $C_A$ and $T_R$ are vanishing, the same occurs for terms proportional to $\beta_0^{QCD}$ in the original pure QCD calculation, consistent with the fact that no renormalisation is needed at this order either for the QED or QCD couplings \footnote{As stated above, we consider the Born coupling between the quarks and the $Z$ in the sense of an effective coupling.}.
Same wise, only a few contributions survive in the products of the type $\left[(c)^{(2,0)} \times (d^*)^{(0,2)}\right]$ and $\left[(d)^{(2,0)} \times (d'^*)^{(0,2)}\right]$, i.e. the interference of amplitudes with one photon and with one gluon exchange.
\begin{table}
\begin{center}
\begin{tabular}{|l| c| c| c|}
\hline
\multicolumn{4}{|c|}
{Colour factors in $q\bar{q}$}
\\[1ex]
\hline
diagram &$\alpha_s^2$&$\alpha\times\alpha_s$&$\alpha^2$\\ \hline & & &\\
$\left|(a)\right|^2$ & $C_F^2$ &$2e_q^2C_F$&$e_q^4$\\[1ex]
$ (d)\times (d^*)$ & $C_F T_R$ & $0$ & $C_A\ e_i^2e_j^2$ \\[1ex]
$(c)\times (c^*)$ & $n_F\ C_FT_R$ & $0$ & $ e_q^2 \left[ N_C\displaystyle\sum_{k\epsilon Q} e_k^2 +\displaystyle\sum_{k\epsilon L}e_k^2 \right]$\\[1ex]
$(a)\times (a'^{*})$ & $C_F^2-\frac{C_F\ C_A}{2}$ &$2e_q^2C_F$&$e_q^4$\\[1ex]
$ (d)\times (d'^*)$ & $C_F^2-\frac{C_F\ C_A}{2}$ & $2e_q^2C_F$ & $e_q^4$ \\[1ex]
$ (b) \times (a^*)$ & $-\frac{C_F\ C_A}{2}$ & $0$ & $0$ \\[1ex]
$ (c)\times (d^*)$ & $C_F^2-\frac{C_F\ C_A}{2}$ & $2e_q^2C_F$ & $e_q^4$ \\[1ex]
\hline
\end{tabular}
\end{center}
\label{comparacqqbar}
\caption{Colour factors corresponding to $q\bar{q}$ channel for each contribution to NNLO QCD$\oplus$QED corrections to Drell-Yan, up to an overall $\frac{1}{2N_C}$ factor. Focusing on $\alpha^2$ factors, the third column includes sums over sets of quark ($Q$) and lepton ($L$) final state charges, while $e_i$ and $e_j$ refer to different quark flavour charges in the scattering.}
\end{table}
This strategy can be extended for all the topologies in $q\bar{q}$.
In Table \ref{comparacqqbar} we show the different colour factors (after factorising an overall factor of $1/2N_C$) for diagrams contributing to $\sigma^{(2,0)}$, and the resulting ones after the abelianisation procedure corresponding to $\sigma^{(1,1)}$. The replacements in the colour structures needed to go from the NNLO QCD coefficients to the QCD$\times$QED ones can be directly read from the entries in Table \ref{comparacqqbar}.
As an important feature, this method shows to be versatile in order to obtain NNLO QED corrections to Drell-Yan as well (i.e. the calculation of $\sigma^{(0,2)}$), if a deeper abelian limit is considered in this case. Here, by turning two gluons into photons from the topologies of NNLO QCD calculation one can recover correction terms up to second order in $\alpha$, thus completing the set of QCD$\oplus$QED NNLO corrections to Drell-Yan, in the sense of Eq.(\ref{eq:expansion}). The corresponding colour factors (including electric charges of both quarks and leptons that might appear in the final state)
are also shown in Table \ref{comparacqqbar} for the $q\bar{q}$ channel.
The same occurs for other channels, after treating carefully the initial flux factor, which depends on the colour properties of initial state particles. For instance, both $q\gamma$ and $qg$ contributions to $\sigma^{(1,1)}$ can be obtained from the $qg$ calculation for NNLO QCD corrections, by choosing the initial or final state gluon, respectively, to perform the abelianisation and following the procedure detailed above.
Particularly, in the case of $\gamma g$ channel, we have performed the explicit calculation of the fixed order corrections, finding perfect agreement with the result obtained by applying the abelianisation procedure.
\section{Results and Phenomenology}
\label{sec:results}
In general the cross section can be written as
\begin{equation}
\frac{d\sigma^Z}{dQ^2}=\tau \sigma_Z (Q^2,M_Z^2)W_Z(\tau , Q^2),
\label{eq:Xsec}
\end{equation}
where $\sigma_Z$ is the point-like LO cross section, $\sqrt{S}$ is the hadronic centre-of-mass energy, $Q$ the invariant mass of the produced $Z$, $\tau=\frac{Q^2}{S}$ and $W_Z(\tau, Q^2)$ is the hadronic structure function.
The point-like cross section that appears in Eq.\eqref{eq:Xsec} is defined as
\begin{equation}
\label{ec:sigmaLO}
\sigma_Z(Q^2, M_Z^2) = \frac{\pi \alpha}{4 M_Z \sin^2\theta_W \cos^2\theta_W} \frac{1}{N_C} \frac{\Gamma_{Z\rightarrow X}}{(Q^2-M_Z^2)^2 + M_Z^2 \Gamma_Z^2},
\end{equation}
where $N_C = 3$ is the number of quark colours, $\theta_W$ is the weak mixing angle (with $\sin^2 \theta_W=0.23$), $M_Z=91.187$ GeV and $\Gamma_Z$ are the mass and width of the $Z$, and $\Gamma_{Z\rightarrow X}$ is the partial width due to the decay of the $Z$ to $X$ (e.g. for leptonic decay, $X = \ell \bar \ell$). The narrow-width approximation used along this paper consists on making the following replacement
\begin{equation}
\frac{1}{(Q^2-M_Z^2)^2 + M_Z^2 \Gamma_Z^2}\rightarrow \frac{\pi}{M_Z \Gamma_Z} \delta(Q^2-M_Z^2).
\end{equation}
ensuring the decoupling of the production and decay mechanisms.
The hadronic structure function appearing in \eqref{eq:Xsec} can be written as a sum of contributions of different orders
\begin{equation}
\label{ec:w}
W_Z(\tau, Q^2) = \int_0^1 {\rm d}x_1\;\int_0^1 {\rm d}x_2\;\int_0^1 {\rm d}x\; \delta(\tau-x x_1 x_2) \sum_{i,\,j} \left(\frac{\alpha_s}{4 \pi}\right)^i \left(\frac{\alpha}{4 \pi}\right)^j \; w_Z^{(i,j)} (x, x_1, x_2, Q^2),
\end{equation}
where the dependence on the factorisation $\mu_F$ and renormalisation $\mu_R$ scales is implicit.
The analytic expressions for the inclusive cross section of Drell-Yan $Z$-production at QCD$\oplus$QED NNLO are presented in the Appendices. In this section we study the phenomenology of the total inclusive cross section, i.e. in all the decay channels of the $Z$, within the narrow-width approximation. To this end, a specific code was written which makes use of the LHAPDF\cite{Buckley:2014ana} package to interpolate
sets of parton distribution functions.
\begin{figure}
\includegraphics[width=0.6\columnwidth]{resnew.eps}
\centering
\caption{$K$-factors for the different distributions as defined in Eq.(\ref{k-eq}). The (blue) dashed line corresponds to $K_{QED}^{NLO}$, the (blue) dotted line to $K_{QED}^{NNLO}$, the solid line to the mixed $K_{QCD\times QED}^{NNLO}$ and the (black) dotted line to the pure NNLO QCD corrections $K_{QCD}^{NNLO}$.}
\label{kfactor}
\end{figure}
For the phenomenological study, unless explicitly stated, we set the renormalisation and factorisation scales to $\mu_R=\mu_F=M_Z$. For both interactions, we set the running coupling at the corresponding renormalisation scale (i.e. $\alpha(M_Z)\sim \frac{1}{128}$ \footnote{For the sake of simplicity we make the same choice for the value of the coupling between quarks and the $Z$ boson in the Born cross section in Eq.(\ref{ec:sigmaLO}).}) and always use the parton distributions to NNLO (QCD) accuracy \cite{Butterworth:2015oua,Ball:2014uwa,Harland-Lang:2014zoa,Dulat:2015mca} with the corresponding QED corrections from LUXqed \cite{Manohar:2016nzj,Manohar:2017eqh}.
In Fig.\ref{kfactor} we plot the $K$-factors for different orders as a way to quantify the size of the
QED and QCD corrections to Drell-Yan at different centre-of-mass energies.
Here the $K$-factor
is defined as the ratio of the cross-section computed at a given order over the previous one, i.e.
\begin{eqnarray}
\label{k-eq}
K_{QED}^{NLO}&=& \frac{\sigma^{(0,0)} + \alpha \, \sigma^{(0,1)}}{\sigma^{(0,0)}} \nonumber \\
K_{QCD}^{NNLO}&=& \frac{\sigma^{(0,0)} + \alpha_s \, \sigma^{(1,0)} + \alpha_s^2 \, \sigma^{(2,0)} }{\sigma^{(0,0)}+ \alpha_s \, \sigma^{(1,0)}} \\
K_{QED}^{NNLO}&=& \frac{\sigma^{(0,0)} + \alpha \, \sigma^{(0,1)} + \alpha^2 \, \sigma^{(0,2)} }{\sigma^{(0,0)}+ \alpha \, \sigma^{(0,1)}} \nonumber \\
K_{QCD\times QED}^{NNLO}&=& \frac{\sigma^{(0,0)} + \alpha \, \sigma^{(0,1)}+ \alpha_s \, \sigma^{(1,0)} + \alpha \alpha_s \, \sigma^{(1,1)} }{\sigma^{(0,0)}+ \alpha \, \sigma^{(0,1)}+ \alpha_s \, \sigma^{(1,0)}}. \nonumber
\end{eqnarray}
\begin{figure}
\includegraphics[width=0.6\columnwidth]{factonew.eps}
\centering
\caption{Ratio $R$ between the exact and the factorisation approximation for the mixed QCD$\times$QED contributions. The inset plot shows the ratio of the cross section computed exactly and with the factorisation approximation for the mixed term.}
\label{facto}
\end{figure}
As can be observed, the NNLO QCD corrections are of the same ($\sim$ 5 per mille level) order, but typically with the opposite sign, as the NLO QED corrections, as expected from the simple counting $\alpha_s^2\sim \alpha$. The mixed QCD$\times$QED turn out to be positive and below the per mille level over the whole range of energies spanned in the plot. Interestingly, due to the particular dependence of the NNLO QCD corrections with the energy, with a sign change around $\sqrt{S}\sim 18$ TeV, for the LHC at
$\sqrt{S}\sim 14$ TeV the mixed QCD$\times$QED corrections are only a factor of $\sim 3.5$ smaller than the pure NNLO QCD contributions. Furthermore, for lower centre-of-mass energies $\sqrt{S}\sim 2$ TeV
the mixed terms almost reach the per mille level and are just a factor of 5 smaller than the NLO QED ones, showing that the elementary counting of couplings can fail under certain kinematical conditions. The pure NNLO QED terms, also plotted in Fig.\ref{kfactor}, are negative but the corrections always remain at the $\mathcal{O}(10^{-5})$ level.
Even though for this particular observable the mixed QCD$\times$QED contributions are small, it is interesting to study how well they can be approximated by the {\it factorisation} assumption on QED plus QCD corrections, where it is assumed that $\kappa_{\rm fact} = \left[K_{QED}^{NLO}\times K_{QCD}^{NLO}\right]_{\mathcal{O}(\alpha \alpha_s)} = \alpha\alpha_s\frac{\sigma^{(0,1)} \sigma^{(1,0)}}{\sigma^{(0,0)}\sigma^{(0,0)}} $, compared to the exact case $\kappa_{\rm mixed} =\alpha\alpha_s\frac{\sigma^{(1,1)}}{\sigma^{(0,0)}}$. For that purpose, in Fig.\ref{facto} we plot the following quantity
\begin{eqnarray}
R= \frac{\kappa_{\rm mixed}}{\kappa_{\rm fact}} =\frac{\sigma^{(0,0)} \sigma^{(1,1)} } {\sigma^{(0,1)} \sigma^{(1,0)}},
\end{eqnarray}
which is the ratio between the exact and the approximated factorised contribution.
As it can be observed, the factorisation approach fails to reproduce the correct behaviour of the mixed contribution typically by a factor of two or more. Of course, given the size of the corrections, the effect of the factorised treatment of these contributions is small at the level of the cross section, as shown in the inset plot of Fig.\ref{facto}, where we show the ratio between the cross section computed exactly and within the factorisation approach, but the situation might not hold for other observables or even for more exclusive distributions in Drell-Yan.
\begin{figure}[ht]
\includegraphics[width=0.6\columnwidth]{canales.eps}
\centering
\caption{Contribution to the mixed QCD$\times$QED $K$-factor from the different channels. Here the label $q$ accounts for both quarks and antiquarks and $qq$ represents the sum of $q\bar{q}$ and $qq$.}
\label{canales}
\end{figure}
In Fig.\ref{canales} we show the contribution to the mixed QCD$\times$QED $K$-factor from the different channels. It is noticeable that the photon initiated contributions are rather small, mostly due to the size of the photon pdf in the proton, as can be observed by comparing $q\gamma$ and $qg$ contributions, which share the same partonic coefficient apart from the colour factor.
It is also clear that the different signs of $qq$ (fully dominated by the born level $q\bar{q}$ channel and exceeding the per mille level) and $qg$ contributions conspire to reduce the effect of the mixed QCD$\times$QED corrections to the Drell-Yan cross section. Again, in more exclusive distributions this partial cancellation might be spoiled by some kinematical cuts, resulting in an increase of the
mixed order corrections.
\begin{figure}[th]
\includegraphics[width=0.6\columnwidth]{scales.eps}
\centering
\caption{Cross sections corresponding to LO (dashes, $i+j$=0 in Eq.(\ref{eq:expansion})), NLO(dots, $i+j$=0,1) and NNLO (solid, $i+j$=0,1,2) at different factorisation and renormalisation scales with $\mu_R=\mu_F=\mu$. All results are normalised by the corresponding cross section at $\mu=M_Z$.}
\label{scales}
\end{figure}
Finally, we discuss the effect of the higher order contributions in the stabilisation of the perturbative expansion in terms of the scale dependence for $\sqrt{S}=13$ TeV (very similar behaviours are observed for other values of $\sqrt{S}$). In Fig.\ref{scales} we show the LO ($\sigma^{(0,0)}$), NLO ($\sigma^{(0,0)}+ \alpha \, \sigma^{(0,1)}+ \alpha_s \, \sigma^{(1,0)}$) and NNLO ($\sigma^{(0,0)} + \alpha \, \sigma^{(0,1)}+ \alpha_s \, \sigma^{(1,0)} + \alpha \alpha_s \, \sigma^{(1,1)} + \alpha^2 \, \sigma^{(0,2)}+ \alpha_s^2 \, \sigma^{(2,0)}$) cross sections for different values of the factorisation and renormalisation scales $\mu_R=\mu_F=\mu$, normalised by the corresponding value at the central scale $\mu=M_Z$. From the slope of the different curves, it is clearly visible the reduction in the scale dependence when including higher order corrections, mostly due to the dominant QCD effects but also thanks to the inclusion of the QED and mixed contributions.
\section{Conclusions}
\label{sec:conc}
In this article, mixed QCD$\times$QED as well as pure QED$^2$ NNLO corrections to the total Drell-Yan $Z$-production cross section were presented for the first time. This was achieved via an abelianisation procedure that profits from the available pure QCD NNLO result and proved to be a versatile technique.
We performed the phenomenological analysis finding that
the mixed corrections are of the order of per mille at the LHC, but only a factor of $\sim 3.5$ smaller than the pure QCD NNLO due to a sign change that occurs in the latter at $\sqrt{S}\sim 14$ TeV. Pure QED NNLO terms are shown to be negative corrections of the order of $10^{-5}$. The full QCD$\oplus$QED NNLO corrections are found to further stabilise the scale dependence of the final result.
The exact $K$-factor at order $\mathcal{O}(\alpha \alpha_s)$ was compared with the naive factorisation approximation, which consists of the mixed order term of the product of QCD and QED NLO $K$-factors. It was shown that the latter fails to reproduce the exact result by a factor of two or more. Although in this case the difference is not significant, due to the smallness of the overall contribution, this result hints that the factorisation approximation may not work for other processes nor for more exclusive measurements of the one presented herein.
\section*{Acknowledgments}
The work of D.deF. has been partially supported by Conicet, ANPCyT and the von Humboldt Foundation. We thank Massimiliano Grazzini, Leandro Cieri, German Sborlini and Javier Mazzitelli for discussions and comments on the manuscript.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,580
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\subsection{Change Log}
This section tracks major changes in the paper.
\textbf{v1:} Initial version.\\
\textbf{v2,v3:} Minor modifications in text.\\
\textbf{v4:} Current version (more details in section 6.1).
\section{Conclusion}
\vspace{\sectionReduceBot}
In this paper, we make an explicit distinction between the two mutually co-existing but different interpretations
of object proposals.
The current evaluation protocol for object proposal methods
is suitable only for detection proposals and is a biased `gameable' protocol for category-independent
object proposals. By evaluating only on a specific set of object categories, we fail to capture
the performance of the proposal algorithm on all the remaining object categories that are present in the test set, but not annotated in the ground truth.
We demonstrate this gameability via a simple thought experiment
where we propose a `fraudulent' object proposal method that
outperforms all existing object proposal techniques on current metrics.
We conduct a thorough evaluation of existing object proposal methods on three densely annotated datasets.
We introduce a fully-annotated version of PASCAL VOC 2010 where we
annotated all instances of all object categories occurring in all images.
We hope this dataset will be broadly useful.
Furthermore, since densely annotating the dataset is a tedious and costly task;
we proposed a set of diagnostic tools to plug the vulnerability of the current protocol.
Fortunately, we find that none of existing proposal methods seem to be biased,
most of the existing algorithms
and do generalize well to different datasets and in our experiments even on densely annotated datasets.
In that sense, our findings are consistent with results in \cite{Hosang2015}.
However, that should not prevent us from recognizing and safeguarding against
the flaws in the protocol, lest we over-fit as a community to a specific set of object classes.
\section{Evaluating Object Proposals}
\label{sec:eval}
\vspace{\sectionReduceBot}
Before we describe our evaluation and analysis, let us first look at the object proposal evaluation
protocol that is widely used today.
The following two factors are involved:
\begin{compactenum}
\item \textbf{Evaluation Metric}:
The metrics used for evaluating object proposals are all typically functions of
intersection over union (IOU) (or Jaccard Index) between generated proposals and ground-truth annotations. For two boxes/regions $b_i$ and $b_j$, IOU is defined as:
\begin{equation}
\text{IOU}(b_i, b_j)=\frac{area(b_i \cap b_j)}{area(b_i \cup b_j)}
\end{equation}
The following metrics are commonly used:
%
\begin{asparaitem}
\item \textbf{Recall $@$ IOU Threshold $t$}:
For each ground-truth instance, this metric checks whether the `best' proposal from list $L$
has IOU greater than a threshold $t$. If so, this ground truth instance is considered `detected'
or `recalled'.
Then average recall is measured over all the ground truth instances:
%
%
\begin{equation}
\text{Recall }@ \, t=\frac{1}{|G|} \sum\limits_{g_i \in G}\text{I }[{\max_{l_j \in L} \, \text{IOU}}(g_i,l_j) > t],
\end{equation}
where $I[\cdot]$ is an indicator function for the logical preposition in the argument.
%
Object proposals are evaluated using this metric in two ways:
\begin{compactitem}
\item plotting Recall-\vs-\#proposals by fixing $t$
\item plotting Recall-\vs-$t$ by fixing the \#proposals in $L$.
\end{compactitem}
\vspace{3pt}
\item \textbf {Area Under the recall Curve (AUC)}:
AUC summarizes the area under the Recall-\vs-\#proposals plot for different values of $t$ in a single plot. This metric measures AUC-\vs-\#proposals. It is also plotted by varying \#proposals in $L$ and plotting AUC-vs-$t$.
\item \textbf{Volume Under Surface (VUS)}: This measures the average recall by linearly varying $t$ and varying the \#proposals in $L$ on either linear or log scale. Thus it merges both kinds of AUC plots into one.
\item \textbf {Average Best Overlap (ABO)}:
This metric eliminates the need for a threshold.
We first calculate the overlap between each ground truth annotation $g_i$ $\in$ $G$, and the `best' object hypotheses in $L$. ABO is calculated as the average:
\begin{equation}
\text{ABO}=\frac{1}{|G|} \sum\limits_{g_i \in G}{\max_{l_j \in L} \, \text{IOU}}(g_i,l_j)
\end{equation}
ABO is typically is calculated on a per class basis.
Mean Average Best Overlap (MABO) is defined as the mean ABO over all classes.
\item \textbf{Average Recall (AR)}: This metric was recently introduced in \cite{Hosang2015}.
Here, average recall (for IOU between 0.5 to 1)-\vs-\#proposals in $L$ is plotted.
AR also summarizes proposal performance across different values of $t$.
AR was shown to correlate with ultimate detection performance better than other metrics.
\end{asparaitem}
\item \textbf{Dataset}:
The most commonly used datasets are the
the PASCAL VOC \cite{pascal-voc-2007} detection datasets.
Note that these are \textit{partially annotated} datasets where
only the 20 PASCAL category instances are annotated.
Recently analyses have been shown on ImageNet \cite{Hosang2014Bmvc},
which has more categories annotated than PASCAL, but is still a partially annotated dataset.
\end{compactenum}
\label{sec:Eval_protocol}
\section{ A Thought Experiment: \\ How to Game the Evaluation Protocol}
\label{sec:thought}
\vspace{\sectionReduceBot}
Let us conduct a thought experiment to demonstrate that the object proposal evaluation
protocol can be `gamed'.
Imagine yourself reviewing a paper claiming to introduce a new object proposal method --
called DMP.
Before we divulge the details of DMP, consider the performance of DMP shown in
\figref{pascal2010auc_tp} on the PASCAL VOC 2010 dataset, under the AUC-\vs-\#proposals metric.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\columnwidth]{figs/first.pdf}
\caption{Performance of different object proposal methods (dashed lines) and our proposed `fraudulent'
method (DMP) on the PASCAL VOC 2010 dataset.
We can see that DMP \emph{significantly} outperforms all other proposal generators.
See text for details.
\label{pascal2010auc_tp}}
\end{figure}
As we can clearly see, the proposed method DMP \emph{significantly} exceeds all
existing proposal methods \cite{ZitnickECCV14,EndresPAMI14,Arbelaez_CVPR14,AlexePAMI12,RahtuICCV11,ManenICCV13,UijlingsIJCV13,cheng2014bing,DBLP:conf/eccv/KrahenbuhlK14} (which seem to have little variation over one another). The improvement\com{ in AUC} at some points in the curve (\eg, at M=10) seems to be \emph{an order of magnitude} larger than all previous incremental improvements reported in the literature!
In addition to the \com{improvement}gain in AUC at a fixed M, DMPs also achieves the same AUC (0.55) at an \emph{order of magnitude fewer} number of proposals (M=10 \vs M=~50 for edgeBoxes\cite{ZitnickECCV14}). Thus, fewer proposals need to be processed by the \com{downstream}ensuing detection system, resulting in an equivalent run-time speedup. This seems to indicate that a significant \com{advancement}progress has been made in the field of generating object proposals.
So what is our proposed state-of-art technique DMP? \\
It is a mixture-of-experts model,
consisting of 20 experts, where each expert is a deep feature (fc7)-based~\cite{donahue_arxiv13}
objectness detector. At this point, you, the savvy reader, are probably already
beginning to guess what we did.
DMP stands for `Detector Masquerading as Proposal generator'. We trained object detectors
for the 20 PASCAL categories (in this case with RCNN\cite{DBLP:journals/corr/GirshickDDM13}),
and then used these 20 detectors to produce the top-M most confident detections (after NMS), and declared
them to be `object proposals'.
The point of this experiment is to demonstrate the following fact --
clearly, no one would consider a collection of 20 object detectors to be a category independent object proposal method. However, our existing evaluation protocol declared the union of these top-M detections to be state-of-the-art.
Why did this happen? Because the protocol today involves evaluating a proposal generator on
\emph{a partially annotated} dataset such as PASCAL. The protocol does not reward recall of
non-PASCAL categories; in fact, early recall (near the top of the list of candidates) of non-PASCAL
objects results in a penalty for the proposal generator!
As a result, a proposal generator that tunes
itself to these 20 PASCAL categories (either explicitly via training or implicitly via design choices
or hyper-parameters)
will be declared a better proposal generator when it may not be (as illustrated by DMP).
Notice that as learning-based object proposal methods improve on this metric, ``in the limit'' \emph{the best object proposal technique is a detector for the annotated categories}, similar to our DMP. Thus, we should be cautious of methods proposing incremental improvements on this protocol -- improvements on this protocol do not necessarily lead to a better category independent object proposal method.
\begin{comment}
\todo{to add next para here or to intro ?}
For example, consider \figref{fig:qualfig1}. The ground-truth of the image only contains green boxes which are PASCAL categories (chairs, tables, bottles) but doesn't contain boxes for other objects like picture frames, plates, glasses etc. Method 1 detects a lot of these non-PASCAL categories but misses out on finding the table. However, Method 2 finds all the PASCAL categories but fails to find the other objects. Clearly Method 1 finds more objects than Method 2 but it still has a lower recall than Method 2. Here the number of proposals generated by both the methods are equal. Also when Method 2 generates lesser number of proposals than Method 1 (as shown in \figref{fig:qualfig2}), the recall of Method 2 is better than Method 1. Even though here not only Method 1 is detecting more objects but also generating more number of proposals.
\end{comment}
This thought experiment exposes the inability of the existing protocol to evaluate
category independence.
\label{sec:exp}
\section{Introduction}
\label{sec:intro}
\vspace{\sectionReduceBot}
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=1\columnwidth]{figs/qa_1.png}
\subcaption{(Green) Annotated, (Red) Unannotated
\label{fig:1}}
\end{subfigure}
\,
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=1\columnwidth]{figs/qa_3.png}
\subcaption{Method 1 with recall 0.6
\label{fig:2}}
\end{subfigure}
\,
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=1\columnwidth]{figs/qa_2.png}
\subcaption{Method 2 with recall 1
\label{fig:3}}
\end{subfigure}
\caption{(a) shows PASCAL annotations natively present in the dataset in green. Other objects that are not annotated but present in the image are shown in red; (b) shows Method 1 and (c) shows Method 2. Method 1 visually seems to recall more categories such as plates, glasses, \etc that Method 2 missed. Despite that,
the computed recall for Method 2 is higher because it recalled all instances of PASCAL
categories that were present in the ground truth. Note that the number of proposals
generated by both methods is equal in this figure.
\label{fig:qualfig1}}
\end{figure*}
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=1\columnwidth]{figs/qa_4.png}
\subcaption{(Green) Annotated, (Red) Unannotated}
\label{fig:1}
\end{subfigure}
\,
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=1\columnwidth]{figs/qa_6.png}
\subcaption{Method 1 with recall 0.5}
\label{fig:2}
\end{subfigure}
\,
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=1\columnwidth]{figs/qa_5.png}
\subcaption{Method 2 with recall 0.83}
\label{fig:3}
\end{subfigure}
\caption{(a) shows PASCAL annotations natively present in the dataset in green. Other objects that are not annotated but present in the image are shown in red; (b) shows Method 1 and (c) shows Method 2.
Method 1 visually seems to recall more categories such as lamps, picture, \etc that Method 2
missed. Clearly the recall for Method 1 \emph{should} be higher.
However, the calculated recall for Method 2 is significantly higher,
which is counter-intuitive. This is because Method 2 recalls more PASCAL category objects.
\label{fig:qualfig2}}
\end{figure*}
In the last few years, the Computer Vision community has witnessed the emergence of a new class
of techniques called \textit{Object Proposal} algorithms \cite{ZitnickECCV14,EndresPAMI14,Arbelaez_CVPR14,AlexePAMI12,RahtuICCV11,ManenICCV13,RantalankilaCVPR14,UijlingsIJCV13,HumayunCVPR14,cheng2014bing,DBLP:conf/eccv/KrahenbuhlK14}.
Object proposals
are a set of candidate regions or bounding boxes in an image that may potentially contain an object.
Object proposal algorithms have quickly become the de-facto pre-processing step
in a number of vision pipelines --
object detection\cite{DBLP:journals/corr/GirshickDDM13,DBLP:journals/corr/HeZR014,DBLP:journals/corr/SzegedyREA14,Wang_2013_ICCV,cinbis:hal-00873134,DBLP:journals/corr/ErhanSTA13,Kuznetsova_2015_CVPR,Tsai_2015_CVPR,Zhu_2015_CVPR,OOP_ICCV2015},
segmentation\cite{Carreira2012,conf/cvpr/ArbelaezHGGBM12,conf/eccv/CarreiraCBS12,conf/cvpr/DaiH015,Zhang_2015_CVPR},
object discovery\cite{deselaers2012weakly,DBLP:journals/corr/ChoKSP15, Kading_2015_CVPR,conf/cvpr/RubinsteinJKL13},
weakly supervised learning of object-object interactions\cite{cinbis2012contextual, prest2012weakly},
content aware media re-targeting\cite{sun2011scale}, action recognition in still images\cite{sener2012recognizing} and visual tracking\cite{DBLP:journals/corr/WangLGY15,DBLP:journals/corr/KwakCLPS15}.
Of all these tasks, object proposals have been particularly successful in object detection systems.
For example, \emph{nearly all top-performing entries}\cite{szegedy2014going, lin2013network, ouyang2014deepid, DBLP:journals/corr/HeZR014} in the ImageNet
Detection Challenge 2014 {\cite{ILSVRCarxiv14} used object proposals.
They are preferred over the formerly used sliding window paradigm due to their computational efficiency. Objects present in an image may vary in location, size, and aspect ratio. Performing an exhaustive search over such a high dimensional space is difficult.
By using object proposals, computational effort can be focused on a small number of candidate windows.
The focus of this paper is the protocol used for evaluating object proposals. Let us begin by asking -- \emph{what is the purpose of an object proposal algorithm?}
In early works \cite{AlexePAMI12,EndresPAMI14,ManenICCV13}, the emphasis was on \emph{category independent object proposals}, where the goal is to identify instances of \emph{all} objects in the image irrespective of their category. While it can be tricky to precisely define what an ``object'' is\footnote{Most category independent object proposal methods define an object as ``stand-alone thing with a well-defined closed-boundary''. For ``thing" \vs ``stuff'' discussion, see \cite{HeitzKoller:ECCV08}.}, these early works presented cross-category evaluations to establish and measure category independence.
More recently, object proposals are increasingly viewed as \emph{detection proposals} {\cite{DBLP:conf/eccv/KrahenbuhlK14,UijlingsIJCV13,ZitnickECCV14,Krahenbuhl_2015_CVPR} where the goal is to improve the object detection pipeline, focusing on a chosen set of object classes (\eg \textasciitilde20 PASCAL categories). In fact, many modern proposal methods are learning-based\cite{cheng2014bing,DBLP:conf/eccv/KrahenbuhlK14,HumayunCVPR14,Krahenbuhl_2015_CVPR,DBLP:journals/pami/KangHEK15,DBLP:journals/corr/KuoHM15, deepprop,DBLP:journals/corr/PinheiroCD15} where the definition of an ``object'' is the set of annotated classes in the dataset. This increasingly blurs the boundary between a proposal algorithm and a detector.
Notice that the former definition has an emphasis on object discovery\cite{deselaers2012weakly,DBLP:journals/corr/ChoKSP15,conf/cvpr/RubinsteinJKL13}, while the latter definition emphasises on the ultimate performance of a detection pipeline. Surprisingly, despite the two different goals of `object proposal,' there exists only a single evaluation protocol\com{, which is the following}:
\begin{compactenum}
\item Generate proposals on a dataset: The most commonly used dataset for evaluation today is
the PASCAL VOC \cite{pascal-voc-2007} detection set.
Note that this is a \textit{partially annotated} dataset where
only the 20 PASCAL category instances are annotated.
\item Measure the performance of the generated proposals: typically in terms of `recall'
of the annotated instances.
Commonly used metrics are described in \secref{sec:eval}.
\end{compactenum}
The central thesis of this paper is that the current evaluation protocol for object proposal methods is suitable for object detection pipeline but is a \emph{`gameable' and misleading protocol} for category independent tasks. By evaluating only on a specific set of object categories, we fail to capture
the performance of the proposal algorithms on \textit{all the remaining object categories that are present in the test set, but not annotated in the ground truth}.
Figs.~\ref{fig:qualfig1}, \ref{fig:qualfig2} illustrate this idea on images from PASCAL VOC 2010.
Column (a) shows the ground-truth object annotations (in green, the annotations natively present
in the dataset for the 20 PASCAL categories --`chairs', `tables', `bottles', \etc; in red, the annotations that we added to the dataset by
marking object such as `ceiling fan', `table lamp', `window', \etc originally annotated `background' in the dataset).
Columns (b) and (c) show the outputs of two object proposal methods. Top row shows the case when both methods produce the same number of proposals;
bottom row shows unequal number of proposals.
We can see that proposal method in Column (b) seems to be more ``complete'', in the sense that it recalls or discovers a large number of instances.
For instance, in the top row it detects a number of non-PASCAL categories (`plate', `bowl', `picture frame', \etc) but misses out on finding the PASCAL category `table'.
In both rows, the method
in Column (c) is reported as achieving a higher recall, \emph{even in the bottom row, when it recalls strictly fewer objects, not just different ones}. The reason is that Column (c) recalls/discovers instances of the 20 PASCAL
categories, which are the only ones annotated in the dataset. Thus, Method 2 appears to be a \emph{better} object proposal generator simply because it focuses on the annotated categories in the dataset.
While intuitive (and somewhat obvious) in hindsight,
we believe this is \com{an important}a crucial finding because it makes the current protocol \textit{`gameable'} or susceptible
to manipulation (both intentional and unintentional) and misleading for measuring improvement in category independent object proposals.
Some might argue that if the end task is to detect a certain set of categories (20 PASCAL or 80 COCO categories) then it is enough to evaluate on them and there is no need to care about other categories which are not annotated in the dataset. We agree, but it is important to keep in mind that object detection is not the only application of object proposals. There are other tasks for which it is important for proposal methods to generate category independent proposals. For example, in semi/unsupervised object localization\cite{deselaers2012weakly,DBLP:journals/corr/ChoKSP15, Kading_2015_CVPR,conf/cvpr/RubinsteinJKL13} the goal is to identify all the objects in a given image that contains many object classes without any specific target classes. In this problem, there are no image-level annotations, an assumption of a single dominant class, or even a known number of object classes\cite{DBLP:journals/corr/ChoKSP15}. Thus, in such a setting, using a proposal method that has tuned itself to 20 PASCAL objects would not be ideal -- in the worst case, we may not discover any new objects. As mentioned earlier, there are many such scenarios including learning object-object interactions\cite{cinbis2012contextual, prest2012weakly}, content aware media re-targeting\cite{sun2011scale}, visual tracking\cite{DBLP:journals/corr/KwakCLPS15}, \etc
To summarize, the contributions of this paper are:
\begin{compactitem}
\item We report the `gameability' of the current object proposal evaluation protocol.
\item
We demonstrate this `gameability' via a simple thought experiment
where we propose a `fraudulent' object proposal method that
\emph{significantly outperforms all existing object proposal techniques} on current metrics, but would under any no circumstances be considered a category independent proposal technique. As a side contribution of our work, we present a simple technique for producing state-of-art object proposals.
\item After establishing the problem, we propose three ways of improving the current evaluation protocol
to measure the category independence of object proposals:
\begin{compactenum}
\item evaluation on \emph{fully} annotated datasets,
\item cross-dataset evaluation on \emph{densely} annotated datasets.
\item a new evaluation metric that quantifies the \emph{bias capacity}
of proposal generators.
\end{compactenum}
For the first test, we introduce a nearly-fully annotated PASCAL VOC 2010 where we
annotated \emph{all instances of all object categories} occurring in the images.
\item We thoroughly evaluate existing proposal methods on this nearly-fully and two densely annotated datasets.
\item We will release all code and data for experiments, and an object proposals library that allows
for easy comparison of all popular object proposal techniques.
\end{compactitem}
\section{Bias Inspection}
\vspace{\sectionReduceBot}
So far, we have discussed two ways of detecting `gameability' -- evaluation on nearly-fully annotated dataset and cross-dataset evaluations on densely annotated datasets.
\begin{figure*}[ht]
\vspace{-0.3cm}
\centering
\begin{minipage}[b]{0.32\textwidth}
\centering
\includegraphics[width=0.93\textwidth]{figs/train_plot_1.pdf}
\subcaption{\label{trainExp} Area under recall \vs \# proposals for various \#seen categories}
\end{minipage}
\,\,\
\begin{minipage}[b]{0.32\textwidth}
\includegraphics[width=1.01\columnwidth]{figs/train_plot_2.pdf}
\vspace{-0.5cm}
\subcaption{\label{trainExp2}Area under recall \vs \#training-categories.}
\end{minipage}
\,\,\
\begin{minipage}[b]{0.32\textwidth}
\includegraphics[width=1.01\columnwidth]{figs/train_plot_4.pdf}
\vspace{-0.5cm}
\subcaption{\label{trainExp3}Improvement in area under recall from \#seen categories =10 to 60 \vs \# proposals.}
\end{minipage}
\caption{\label{trainExpall}Performance of RCNN and other proposal generators vs number of object categories used for training.
We can see that RCNN has the most `bias capacity' while the performance of other methods is nearly
(or absolutely) constant. }
\end{figure*}
Although these methods are fairly useful for bias detection, they have certain limitations. Datasets can be unbalanced. Some categories can be more frequent than others while others can be hard to detect (due to choices made in dataset collection). These issues need to be resolved for perfectly unbiased evaluation. However, generating unbiased datasets is an expensive and time-consuming process. Hence, to detect the bias without getting unbiased datasets, we need a method which can measure performance of proposal methods in a way that category specific biases can be accounted for and the extent or the \emph{capacity} of this bias can be measured.
We introduce such a method in this section.
\vspace{-0.15cm}
\vspace{\subsectionReduceTop}
\subsection{Assessing Bias Capacity}
\vspace{\subsectionReduceBot}
Many proposal methods\cite{cheng2014bing,DBLP:conf/eccv/KrahenbuhlK14,HumayunCVPR14,Krahenbuhl_2015_CVPR,DBLP:journals/pami/KangHEK15,DBLP:journals/corr/KuoHM15, deepprop,DBLP:journals/corr/PinheiroCD15} rely on explicit training to learn an ``objectness'' model, similar to DMPs.
Depending upon which, how many categories they are trained on, these methods could have a biased view of ``objectness''.\\
One way of measuring the \emph{bias capacity} in a proposal method to plot the
performance \vs the number of `seen' categories while evaluating on some held-out set. A method that involves little or no training will be a flat curve on this plot. Biased methods such as DMPs will get better and better as more categories are seen in training.
Thus, this analysis can help us find biased or `gamebility-prone' methods like DMPs that are/can be tuned to specific classes. To the best of our knowledge, no previous work has attempted to measure bias capacity by varying the number of `object' categories seen at training time.
In this experiment, we compared the performance of
one DMP method (RCNN),
one learning-based proposal method (Objectness), and two non learning-based
proposal methods (Selective Search\cite{UijlingsIJCV13}, EdgeBoxes\cite{ZitnickECCV14})
as a function of the number of `seen' categories (the categories trained on\footnote{The seen categories are picked in the order they are listed in MS COCO dataset (\ie, no specific criterion was used).}) on MS COCO\cite{LinECCV14coco} dataset.
Method names `RCNNTrainN', `objectnessTrainN' indicate that they were trained on images that contain annotations for only N categories (50 instances per category). Total number of images for all 60 categories was \textasciitilde2400 (because some images contain >1 object). Once trained, these methods were evaluated on
a randomly-chosen set of \textasciitilde500 images, which had annnotations for all 60 categories.\\
\figref{trainExp} shows Area under Recall \vs \#proposals curve for learning-based methods trained on different sets of categories.
\figref{trainExp2} and \figref{trainExp3} show the variation of AUC \vs \# seen categories and improvement due to increase in training categories (from 10 to 60) \vs \#proposals respectively, for RCNN and objectness when trained on different sets of categories.
The key observation to make here is that with even a modest increase in `seen' categories with the same amount of increased training data, performance improvement of RCNN is significantly more than objectness. Selective Search~\cite{UijlingsIJCV13} and edgeBoxes~\cite{ZitnickECCV14} are the dashed straight lines
since there is no training involved.
These results clearly indicate that as RCNN sees more categories, its performance improves. One might argue that the reason might be that the method is learning more `objectness' as it is seeing more data. However, as discussed above, the increase in the dataset size is marginal (\textasciitilde40 images per category) and hence it unlikely that such a significant improvement is observed due to that. Thus, it is reasonable to conclude that this improvement is because the method is learning class specific features.
Thus, this approach can be used to reason about `gameability-prone' and `gameability-immune' proposal methods without creating an expensive fully annotated dataset. We believe this simple but effective diagnostic experiment would help to detect and thus contribute in managing the category specific bias in all learning-based methods.
\label{sec:newexp}
\section{Evaluation on Fully and Densely Annotated Datasets}
\vspace{\sectionReduceBot}
\begin{figure*}[ht]
\vspace{-0.4cm}
\centering
\begin{minipage}[b]{0.23\textwidth}
\includegraphics[width=1\columnwidth]{figs/ann_stats_2.png}
\subcaption{Average \#annotations for different categories.}
\label{barplot_freq}
\end{minipage}%
\,\,\
\begin{minipage}[b]{0.23\textwidth}
\includegraphics[width=1\columnwidth]{figs/area_stats.png}
\subcaption{Fraction of image-area covered by different categories.}
\label{barplot_area}
\end{minipage}
\,\,\
\begin{minipage}[b]{0.23\textwidth}
\includegraphics[width=1\columnwidth]{figs/context2.jpg}\vspace{-7pt}
\subcaption{PASCAL Context annotations~\cite{mottaghi_cvpr14}.}
\label{context}
\end{minipage}%
\,\,\
\begin{minipage}[b]{0.23\textwidth}
\includegraphics[width=1\columnwidth]{figs/context.jpg}\vspace{-7pt}
\subcaption{Our augmented annotations.\\}
\label{mod_context}
\end{minipage}
\caption{(a),(b) Distribution of object classes in PASCAL Context with respect to different attributes.
(c),(d) Augmenting PASCAL Context with instance-level annotations.
(Green = PASCAL 20 categories; Red = new objects)}
\label{context_anno_fig}
\end{figure*}
As described in the previous section, the problem of `gameability' is occuring due to the evaluation of proposal methods on partially annotated datasets. An intuitive solution would be evaluating on a \emph{fully} annotated dataset. \\
In the next two subsections, we evaluate the performance of 7 popular object proposal methods\cite{AlexePAMI12,UijlingsIJCV13,cheng2014bing,ZitnickECCV14,Arbelaez_CVPR14,RahtuICCV11,ManenICCV13} and two DMPs
(RCNN\cite{DBLP:journals/corr/GirshickDDM13} and DPM\cite{lsvm-pami}) on one nearly-fully and two densely annotated datasets containing
many more object categories.
This is to quantify how much the performance of our `fraudulent' proposal generators (DMPs) drops
once the bias towards the 20 PASCAL categories is diminished (or completely removed). \\
We begin by \emph{creating} a nearly-fully annotated dataset
by building on the effort of PASCAL Context \cite{mottaghi_cvpr14} and evaluate on this nearly-fully annotated modified instance level PASCAL Context;
followed by cross-dataset evaluation on other partial-but-densely annotated datasets MS COCO \cite{LinECCV14coco}
and NYU-Depth V2 \cite{Silberman:ECCV12}.
\textbf{Experimental Setup:} On MS COCO and PASCAL Context datasets we conducted experiments as follows:
\begin{compactitem}
\item Use the existing evaluation protocol for evaluation, \ie, evaluate only on the 20 PASCAL categories.
\item Evaluate on all the annotated classes.
\item For the sake of completeness, we also report results on all the classes except the PASCAL 20 classes.\footnote{On NYU-Depth V2 performance is only evaluated on all categories. This is because only 8 PASCAL categories are present in this dataset.}
\end{compactitem}
\textbf{Training of DMPs:} The two DMPs we use are based on two popular object detectors - DPM\cite{lsvm-pami} and RCNN\cite{DBLP:journals/corr/GirshickDDM13}.
We train DPM
on 20 PASCAL categories and use it as an object proposal method. To generate large number of proposals,
we chose a low value of threshold in Non-Maximum Suppression (NMS). Proposals are generated for each category and a score is assigned to them by the corresponding DPM for that category.
These proposals are then merge-sorted on the basis of this score.
Top M proposals are selected from this sorted list where M is the number of proposals to be generated.\\
Another (stronger) DMP is RCNN which is a detection pipeline that uses 20 SVMs
(each for one PASCAL category) trained on deep features (fc7)~\cite{donahue_arxiv13} extracted on selective search boxes.
Since RCNN itself uses selective search proposals, it should be viewed as a trained \emph{reranker}
of selective search boxes. As a consequence, it ultimately equals selective search performance once
the number of candidates become large. We used the pretrained SVM models released
with the RCNN code, which were trained on the 20 classes of PASCAL VOC 2007 trainval set.
For every test image, we generate the Selective Search proposals using the `FAST' mode and
calculate the 20 SVM scores for each proposal.
The `objectness' score of a proposal is then the maximum of the 20 SVM scores. All the proposals are then sorted by this score and top M proposals are selected.\footnote{It was observed that merge-sorting calibrated/rescaled SVM scores led to inferior performance as compared to merge-sorting without rescaling.}
\textbf{Object Proposals Library:}
To ease the process of carrying out the experiments, we created an open source,
easy-to-use object proposals library. This can be used to seamlessly generate object proposals using
all the existing algorithms\cite{ZitnickECCV14,EndresPAMI14,Arbelaez_CVPR14,AlexePAMI12,RahtuICCV11,ManenICCV13,RantalankilaCVPR14,UijlingsIJCV13,HumayunCVPR14} (for which the Matlab code has been released by the respective authors)\com{.
Our library can also be used to} and evaluate these proposals on any dataset using the commonly used metrics\com{:
Recall $@$ Threshold/Detection Rate, ABO, AUC}. This library will be made publicly available.
\vspace{-0.1cm}
\vspace{\subsectionReduceTop}
\subsection{Fully Annotated Dataset}
\vspace{\subsectionReduceBot}
\paragraph{PASCAL Context:}This dataset was introduced by Mottaghi~\etal~\cite{mottaghi_cvpr14}.
It contains additional annotations for all images of PASCAL VOC 2010 dataset \cite{pascal-voc-2010}.
The annotations are semantic segmentation maps, where \emph{every single pixel}
previously annotated `background' in PASCAL was assigned a category label.
\input{Results}
In total, annotations have been provided for 459 categories.
This includes the original 20 PASCAL categories and new
classes such as keyboard, fridge, picture, cabinet, plate, clock. \\
Unfortunately, the dataset contains only
category-level semantic segmentations\com{, not \emph{instance-level} segmentations}.
For our task, we needed instance-level bounding box annotations, which cannot be
reliably extracted from category-level segmentation masks\com{ because the masks for several instances (of
say chairs) may be merged together into a single `blob' in the category-level mask}.
\textbf{Creating Instance-Level Annotations for PASCAL Context:}
Thus, we created instance-level bounding box annotations for all images in PASCAL Context dataset.
First, out of the 459 category labels in PASCAL Context, we identified 396 categories to be `things', and ignored the remaining `stuff'
or `ambiguous' categories\footnote{\eg, a `tree' may be a `thing'
or `stuff' subject to camera viewpoint.} --
neither of these lend themselves to bounding-box-based object detection. See supplement for details.\com{A complete list is available in the supplementary material. }\\
We selected the 60 most frequent non-PASCAL categories from this list of `things' and manually annotated all their instances. Selecting only top 60 categories is a reasonable choice because the average per category frequency in the dataset for all the other categories (even after including background/ambiguous categories) was roughly one third as that of the chosen 60 categories (\figref{barplot_freq}). Moreover, the percentage of pixels in an image left unannotated (as `background') drops from 58\% in original PASCAL to 50\% in our nearly-fully annotated PASCAL Context. This manual annotation was
performed with the aid of the semantic segmentation maps present in the PASCAL Context annotations. Examples annotations are shown in \figref{mod_context}. For detailed statistics, see supplement.
\textbf{Results and Observations:}
We now explore how changes in the dataset and annotated categories affect the results of the thought experiment
from \secref{sec:thought}.
Figs.~\ref{fig:pascal2010aucAllCat}, ~\ref{fig:pascal2010aucNonPasCat},~\ref{fig:pascal2010AllCat}, ~\ref{fig:diff_pascal}
compare the performance of DMPs
with a number of existing proposal methods \cite{ZitnickECCV14,EndresPAMI14,Arbelaez_CVPR14,AlexePAMI12,RahtuICCV11,ManenICCV13,UijlingsIJCV13,cheng2014bing,DBLP:conf/eccv/KrahenbuhlK14}
on PASCAL Context.
We can see in Column (a) that when evaluated on only 20 PASCAL categories
DMPs trained on these categories appear to significantly outperform all proposal generators.
However, we can see that they are not category independent because they suffer a big
drop in performance when evaluated on 60 non-PASCAL categories in Column (b).
Notice that on PASCAL context, \emph{all proposal generators} suffer a drop in performance
between the 20 PASCAL categories and 60 non-PASCAL categories. We hypothesize that this due to the fact that the non-PASCAL categories tend to be generally smaller
than the PASCAL categories (which were the main targets of the dataset curators) and hence difficult to detect. But this could also be due to the reason that authors of these methods made certain choices while designing these approaches which catered better to the 20 annotated categories.
However, the key observation here (as shown in ~\figref{fig:diff_pascal}) is that DMPs suffer the biggest drop. This drop is much greater than all the other approaches.
It is interesting to note that due to the ratio of instances of 20 PASCAL categories vs other 60 categories,
DMPs continue to slightly outperform proposal generators when evaluated on all categories, as shown
in Column (c).
\vspace{-0.25cm}
\vspace{\subsectionReduceTop}
\subsection{Densely Annotated Datasets }
\vspace{\subsectionReduceBot}
Besides being expensive, ``full'' annotation of images is somewhat ill-defined due to the hierarchical nature of object semantics (\eg are object-parts such as bicycle-wheel, windows in a building, eyes in a face, \etc also objects?). One way to side-step this issue is to use datasets with dense annotations (albeit at the same granularity) and conduct cross-dataset evaluation.
\textbf{MS COCO:} Microsoft Common Objects in Context (MS COCO) dataset \cite{LinECCV14coco}
contains 91 common object categories with 82 of them having more than 5,000 labeled instances.
It not only has significantly higher number of instances per category than the PASCAL,
but also considerably more object instances per image (7.7)
as compared to ImageNet (3.0) and PASCAL (2.3).
\textbf{NYU-Depth V2:} NYU-Depth V2 dataset \cite{Silberman:ECCV12} is comprised of video sequences from a
variety of indoor scenes as recorded by both the RGB and Depth cameras.
It features 1449 densely labeled pairs of aligned RGB and depth imageswith instance-level annotations. We used these 1449 densely annotated
RGB images for evaluating object proposal algorithms. To the best of our knowledge,
this is the first paper to compare proposal methods on such a dataset.
\textbf{Results and Observations:} Figs. \ref{fig:cocoAuc20Cat}, \ref{fig:cocoAuc70Cat}, \ref{fig:cocoauc90Cat}, \ref{fig:diff_coco} show a plot similar to PASCAL Context on MS COCO. Again, DMPs outperform all other methods on PASCAL categories but fail to do so for the Non-PASCAL categories.
\figref{fig:nyuv2_allcats} shows results for NYU-Depth V2.
See that when many classes in the test dataset are
not PASCAL classes, DMPs tend to perform poorly, although it is interesting that the performance is
still not as poor as the worst proposal generators. Results on other evaluation criteria are in the supplement.\\
\section{Related Work}
\label{sec:related}
\vspace{\sectionReduceBot}
\textbf{Types of Object Proposals:}
Object proposals can be broadly categorized into two categories:
\begin{compactitem}
\item \textbf{Window scoring}: In these methods, the space of all possible windows in an
image is sampled to get a subset of the windows (\eg, via sliding window).
These windows are then scored for the
presence of an object based on the image features from the windows. The algorithms that fall
under this category are \cite{AlexePAMI12, RahtuICCV11, ZitnickECCV14, cheng2014bing,Chen_2015_CVPR,deepprop}.
\item \textbf{Segment based}: These algorithms involve over-segmenting an image and
merging the segments using some strategy. These methods include
\cite{UijlingsIJCV13, RantalankilaCVPR14, Arbelaez_CVPR14, DBLP:conf/eccv/KrahenbuhlK14,
HumayunCVPR14, EndresPAMI14, ManenICCV13,Wang_2015_CVPR,DBLP:journals/corr/KuoHM15,DBLP:journals/corr/PinheiroCD15}. The generated region proposals can be converted to bounding boxes if needed.
\end{compactitem}
\textbf{Beyond RGB proposals:}
Beyond the ones listed above, a wide variety of algorithms fall under the umbrella of `object proposals'.
For instance, \cite{oneata2014spatio, fragkiadaki2014spatio,Yu_2015_CVPR,Misra_2015_CVPR,Wu_2015_CVPR} used
spatio-temporal object proposals for action recognition, segmentation and tracking in videos.
Another direction of work\cite{Gupta2014,banica,Banica_2015_CVPR} explores use of RGB-D cuboid proposals in an object detection
and semantic segmentation in RGB-D images.
While the scope of this paper is limited to proposals in RGB images, the central
thesis of the paper (\ie, gameability of the evaluation protocol) is broadly applicable to other settings.
\textbf{Evaluating Proposals:}
\com{While a variety of approaches have been proposed for generating object proposals,
t}There has been a relatively limited analysis and evaluation of proposal methods or the proposal evaluation protocol.
Hosang~\etal~\cite{Hosang2014}
focus on evaluation of object proposal algorithms,
in particular the stability of such algorithms on parameter changes and image perturbations.
Their works shows that a large number of category independent proposal algorithms indeed generalize well to non-PASCAL categories,
for instance in the ImageNet 200 category detection dataset~\cite{ILSVRCarxiv14}. Although these findings are important (and consistent with our experiments), they are unrelated to the `gameability' of the evaluation protocol, which is our focus.
In ~\cite{Hosang2015}, authors present an analysis of various proposal methods regarding proposal repeatability, ground truth annotation recall, and their impact on detection performance. They also introduced a new evaluation metric (Average Recall).
Their argument for a new metric is the need for a better localization between generated proposals
and ground truth. While this is a valid and significant concern, it is orthogonal
to the`gameability' of the evaluation protocol, which to the best of our knowledge
has not been previously addressed. Another recent related work perhaps is \cite{PTVG2015}, which analyzes the state-of-the-art methods in segment-based
object proposals, focusing on the challenges faced
when going from PASCAL VOC to MS COCO. They also
analyze how aligned the proposal methods are with the bias
observed in MS COCO towards small objects and the center
of the image and propose a method to boost their performance. Although there is a discussion about biases in datasets but it is unlike our theme, which is `gameability' due to these biases.
As stated earlier, while early papers \cite{AlexePAMI12,EndresPAMI14,ManenICCV13} reported cross-dataset or cross-category generalization experiments similar to ones reported in this paper, with the trend of learning-based proposal methods, these experiments and concerns seem to have fallen out of standard practice, which we show is problematic.
\subsection{Experimental setup}
\begin{figure*}[htp]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/pascal_20.pdf}
\subcaption{Performance on PASCAL Context, only 20 PASCAL classes annotated.}
\label{fig:pascal2010aucAllCat}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/pascal_60.pdf}
\subcaption{Performance on PASCAL Context, only 60 non-PASCAL classes annotated.}
\label{fig:pascal2010aucNonPasCat}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/pascal_all.pdf}
\subcaption{Performance on PASCAL Context, all classes annotated.}
\label{fig:pascal2010AllCat}
\end{subfigure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/coco_20.pdf}
\subcaption{Performance on MS COCO, only 20 PASCAL classes annotated.}
\label{fig:cocoAuc20Cat}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/coco_70.pdf}
\subcaption{Performance on MS COCO, only 60 non-PASCAL classes annotated.}
\label{fig:cocoAuc70Cat}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/coco_all.pdf}
\subcaption{Performance on MS COCO, all classes annotated.\label{fig:cocoauc90Cat}}
\end{subfigure}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/nyu2_all.pdf}
\subcaption{Performance on NYU-Depth V2, all classes annotated
\label{fig:nyuv2_allcats}}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/AUC_diff_pascal.pdf}
\subcaption{\label{fig:diff_pascal}AUC @ 20 categories - AUC @ 60 categories on PASCAL Context.}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.1\columnwidth]{figs/auc_diff_coco.pdf}
\subcaption{\label{fig:diff_coco}AUC @ 20 categories - AUC @ 60 categories on MS COCO.}
\end{subfigure}
\caption{Performance of different methods on PASCAL Context, MS COCO and NYu Depth-V2 with different sets of annotations.}
\end{figure*}
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\begin{table}
\begin{center}
\begin{tabular}{|l|c|}
\hline
Method & Frobnability \\
\hline\hline
Theirs & Frumpy \\
Yours & Frobbly \\
Ours & Makes one's heart Frob\\
\hline
\end{tabular}
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\caption{Results. Ours is better.}
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{\small
\bibliographystyle{ieee}
\section{Appendix}
The main paper demonstrated how the object proposal evaluation protocol is `gameable' and performed some experiments to detect this `gameability'. In this supplement, we present additional details and results which support the arguments presented in the main paper.\\
In section \ref{related_work}, we list and briefly describe the different object proposal algorithms which we used for our experiments. Following this, details of instance-level PASCAL Context are discussed in section \ref{context}. Then we present the results on nearly-fully annotated dataset, cross dataset evaluation on other evaluation metrics in section \ref{more_results}. We also show the per category performance of various methods on MS COCO and PASCAL Context in section \ref{percat}.
\subsection{Details of PASCAL Context Annotation}
\vspace{\sectionReduceTop}
As explained in section 5.1 of the main paper, PASCAL Context provides full annotations for PASCAL VOC 2010 dataset in the form of semantic segmentations. A total of 459 classes have labeled in this dataset. We split these into three categories namely Objects/Things, Background/Stuff and Ambiguous as shown in Tables \ref{contextobjects}, \ref{contextnonobjects} and \ref{contextamb}. Most classes (396) were put in the `Objects' category. 20 of these are PASCAL categories. Of the remaining 376, we selected the most frequently occurring 60 categories and manually created instance level annotations for the same.
\textbf{Statistics of New Annotations:}
We made the following observations on our new annotations:
\begin{compactitem}
\item The number of instances we annotated for the extra 60 categories were about the same as the number of instances for annotated for 20 PASCAL categories in the original PASCAL VOC. This shows that about half the annotations were missing and thus a lot of genuine proposal candidates are not being rewarded.
\item Most non-PASCAL categories
occupy a small percentage of the image. This is understandable given that the dataset was curated with these categories. The other categories just happened to be in the pictures.
\end{compactitem}
\label{context}
\begin{table*}[!htbp]
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}|llllllll|l@{}}
\cmidrule(r){1-8}
\multicolumn{8}{|c|}{\textbf{Object/Thing Classes in PASCAL Context Dataset}} & \multicolumn{1}{c}{} \\ \cmidrule(r){1-8}
accordion & candleholder & drainer & funnel & lightbulb & pillar & sheep & tire & \\
aeroplane & cap & dray & furnace & lighter & pillow & shell & toaster & \\
airconditioner & car & drinkdispenser & gamecontroller & line & pipe & shoe & toilet & \\
antenna & card & drinkingmachine & gamemachine & lion & pitcher & shoppingcart & tong & \\
ashtray & cart & drop & gascylinder & lobster & plant & shovel & tool & \\
babycarriage & case & drug & gashood & lock & plate & sidecar & toothbrush & \\
bag & casetterecorder & drum & gasstove & machine & player & sign & towel & \\
ball & cashregister & drumkit & giftbox & mailbox & pliers & signallight & toy & \\
balloon & cat & duck & glass & mannequin & plume & sink & toycar & \\
barrel & cd & dumbbell & glassmarble & map & poker & skateboard & train & \\
baseballbat & cdplayer & earphone & globe & mask & pokerchip & ski & trampoline & \\
basket & cellphone & earrings & glove & mat & pole & sled & trashbin & \\
basketballbackboard & cello & egg & gravestone & matchbook & pooltable & slippers & tray & \\
bathtub & chain & electricfan & guitar & mattress & postcard & snail & tricycle & \\
bed & chair & electriciron & gun & menu & poster & snake & tripod & \\
beer & chessboard & electricpot & hammer & meterbox & pot & snowmobiles & trophy & \\
bell & chicken & electricsaw & handcart & microphone & pottedplant & sofa & truck & \\
bench & chopstick & electronickeyboard & handle & microwave & printer & spanner & tube & \\
bicycle & clip & engine & hanger & mirror & projector & spatula & turtle & \\
binoculars & clippers & envelope & harddiskdrive & missile & pumpkin & speaker & tvmonitor & \\
bird & clock & equipment & hat & model & rabbit & spicecontainer & tweezers & \\
birdcage & closet & extinguisher & headphone & money & racket & spoon & typewriter & \\
birdfeeder & cloth & eyeglass & heater & monkey & radiator & sprayer & umbrella & \\
birdnest & coffee & fan & helicopter & mop & radio & squirrel & vacuumcleaner & \\
blackboard & coffeemachine & faucet & helmet & motorbike & rake & stapler & vendingmachine & \\
board & comb & faxmachine & holder & mouse & ramp & stick & videocamera & \\
boat & computer & ferriswheel & hook & mousepad & rangehood & stickynote & videogameconsole & \\
bone & cone & fireextinguisher & horse & musicalinstrument & receiver & stone & videoplayer & \\
book & container & firehydrant & horse-drawncarriage & napkin & recorder & stool & videotape & \\
bottle & controller & fireplace & hot-airballoon & net & recreationalmachines & stove & violin & \\
bottleopener & cooker & fish & hydrovalve & newspaper & remotecontrol & straw & wakeboard & \\
bowl & copyingmachine & fishtank & inflatorpump & oar & robot & stretcher & wallet & \\
box & cork & fishbowl & ipod & ornament & rock & sun & wardrobe & \\
bracelet & corkscrew & fishingnet & iron & oven & rocket & sunglass & washingmachine & \\
brick & cow & fishingpole & ironingboard & oxygenbottle & rockinghorse & sunshade & watch & \\
broom & crabstick & flag & jar & pack & rope & surveillancecamera & waterdispenser & \\
brush & crane & flagstaff & kart & pan & rug & swan & waterpipe & \\
bucket & crate & flashlight & kettle & paper & ruler & sweeper & waterskateboard & \\
bus & cross & flower & key & paperbox & saddle & swimring & watermelon & \\
cabinet & crutch & fly & keyboard & papercutter & saw & swing & whale & \\
cabinetdoor & cup & food & kite & parachute & scale & switch & wheel & \\
cage & curtain & forceps & knife & parasol & scanner & table & wheelchair & \\
cake & cushion & fork & knifeblock & pen & scissors & tableware & window & \\
calculator & cuttingboard & forklift & ladder & pencontainer & scoop & tank & windowblinds & \\
calendar & disc & fountain & laddertruck & pencil & screen & tap & wineglass & \\
camel & disccase & fox & ladle & person & screwdriver & tape & wire & \\
camera & dishwasher & frame & laptop & photo & sculpture & tarp & & \\
cameralens & dog & fridge & lid & piano & scythe & telephone & & \\
can & dolphin & frog & lifebuoy & picture & sewer & telephonebooth & & \\
candle & door & fruit & light & pig & sewingmachine & tent & & \\ \bottomrule
\end{tabular}
}
\captionof{table}{Object/Thing Classes in PASCAL Context
\label{contextobjects}}
\end{table*}
\begin{table}[!htbp]
\begin{tabular}{|llll|}
\hline
\multicolumn{4}{|c|}{\textbf{Ambiguous Classes in PASCAL Context Dataset}} \\ \hline
artillery & escalator & ice & speedbump \\
bedclothes & exhibitionbooth & leaves & stair \\
clothestree & flame & outlet & tree \\
coral & guardrail & rail & unknown \\
dais & handrail & shelves & \\ \hline
\end{tabular}
\captionof{table}{Ambiguous Classes in PASCAL Context}
\label{contextamb}
\end{table}
\begin{table}[!htbp]
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|llll|}
\hline
\multicolumn{4}{|c|}{\textbf{Background/Stuff Classes in PASCAL Context Dataset}} \\ \hline
atrium & floor & parterre & sky \\
bambooweaving & foam & patio & smoke \\
bridge & footbridge & pelage & snow \\
building & goal & plastic & stage \\
ceiling & grandstand & platform & swimmingpool \\
concrete & grass & playground & track \\
controlbooth & ground & road & wall \\
counter & hay & runway & water \\
court & kitchenrange & sand & wharf \\
dock & metal & shed & wood \\
fence & mountain & sidewalk & wool \\ \hline
\end{tabular}
}
\captionof{table}{Background/Stuff Classes in PASCAL Context}
\label{contextnonobjects}
\end{table}
\vspace{\sectionReduceTop}
\subsection{Evaluation of Proposals on Other Metrics}
\vspace{\sectionReduceTop}
In this section, we show the performance of different proposal methods and DMPs on MS COCO dataset on various metrics. \figref{20coco1000} shows performance on Recall-vs-IOU metric at 1000 \#proposals on PASCAL 20 categories. \figref{20coco05}, \figref{20coco07} show performance on Recall-\vs-\#proposals metric at 0.5 and 0.7 IOU respectively.
Similarly in Figs. \ref{70coco1000},\ref{70coco05}, \ref{70coco07} and Figs. \ref{90coco1000},\ref{90coco05}, \ref{90coco07}, we can see the performance of all proposal methods and DMPs on these three metrics where 60 non-PASCAL and all categories respectively are annotated in the MS COCO dataset. \\
These metrics also demonstrate the same trend as shown by the AUC-\vs-\#proposals in the main paper. When only PASCAL categories are annotated (Figs. \ref{20coco1000},\ref{20coco05}, \ref{20coco07} ), DMPs outperform all proposal methods. However, when other categories are also annotated (Figs. \ref{90coco1000},\ref{90coco05}, \ref{90coco07}) or the performance is evaluated specifically on the other categories (Figs. \ref{70coco1000},\ref{70coco05}, \ref{70coco07}), DMPs cease to be the top performers.\\
Finally, we also report results on different metrics PASCAL Context (\figref{allcat_context}) and NYU-Depth v2 (\figref{nyu2}). They also show similar trends, supporting the claims made in the paper.
\begin{figure*}[htb]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/20coco1000.pdf}
\subcaption{\label{20coco1000}Recall vs IOU at 1000 proposals for 20 PASCAL categories annotated in MS COCO validation dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/20coco05iou.pdf}
\subcaption{\label{20coco05} Recall \vs number of proposals at 0.5 IOU for 20 PASCAL categories annotated in MS COCO validation dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/20coco07iou.pdf}
\subcaption{\label{20coco07} Recall \vs number of proposals at 0.7 IOU for 20 PASCAL categories annotated in MS COCO validation dataset}
\end{subfigure}
\label{p_coco}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/70coco1000.pdf}
\subcaption{\label{70coco1000}Recall vs IOU at 1000 proposals for 60 non-PASCAL categories annotated in MS COCO validation dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/70coco05iou.pdf}
\subcaption{\label{70coco05}Recall \vs number of proposals at 0.5 IOU for 60 non-PASCAL categories annotated in MS COCO validation dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/70coco07iou.pdf}
\subcaption{\label{70coco07}Recall \vs number of proposals at 0.7 IOU for 60 non-PASCAL categories annotated in MS COCO validation dataset}
\end{subfigure}
\label{np_coco}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/90coco1000.pdf}
\subcaption{\label{90coco1000}Recall vs IOU at 1000 proposals for all categories annotated in MS COCO validation dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/90coco05iou.pdf}
\subcaption{\label{90coco05}Recall \vs number of proposals at 0.5 IOU for all categories annotated in MS COCO validation dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/90coco07iou.pdf}
\subcaption{\label{90coco07}Recall \vs number of proposals at 0.7 IOU for all categories annotated in MS COCO validation dataset}
\end{subfigure}
\caption{\label{allcat_coco}Performance of various object proposal methods on different evaluation metrics when evaluated on MS COCO dataset.}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/20context1000.pdf}
\subcaption{\label{20context1000}Recall vs IOU at 1000 proposals for 20 PASCAL categories annotated in PASCAL Context dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/20context05.pdf}
\subcaption{\label{20context05}Recall \vs number of proposals at 0.5 IOU for 20 PASCAL annotated in PASCAL Context dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/20context07.pdf}
\subcaption{\label{20context07}Recall \vs number of proposals at 0.7 IOU for 20 PASCAL categories annotated in PASCAL Context dataset}
\end{subfigure}
\label{20cat_context}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/60context1000.pdf}
\subcaption{\label{60context1000}Recall vs IOU at 1000 proposals for non-PASCAL categories annotated in PASCAL Context dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/60context05.pdf}
\subcaption{\label{60context05}Recall \vs number of proposals at 0.5 IOU for non-PASCAL annotated in PASCAL Context dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/60context07.pdf}
\subcaption{\label{60context07}Recall \vs number of proposals at 0.7 IOU for non-PASCAL categories annotated in PASCAL Context dataset}
\end{subfigure}
\label{60cat_context}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/80context1000.pdf}
\subcaption{\label{80context1000}Recall vs IOU at 1000 proposals for all categories annotated in PASCAL Context dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/80context05.pdf}
\subcaption{\label{80context05}Recall \vs number of proposals at 0.5 IOU for all categories annotated in PASCAL Context dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.0\columnwidth]{figs/80context07.pdf}
\subcaption{\label{80context07}Recall \vs number of proposals at 0.7 IOU for all categories annotated in PASCAL Context dataset}
\end{subfigure}
\caption{\label{allcat_context}Performance of various object proposal methods on different evaluation metrics when evaluated on PASCAL Context dataset}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.30\textwidth}
\includegraphics[width=1\columnwidth]{figs/nyu21000.pdf}
\subcaption{\label{nyu21000}Recall vs IOU at 1000 proposals for all categories annotated in the NYU2 dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.30\textwidth}
\includegraphics[width=1\columnwidth]{figs/nyu205iou.pdf}
\subcaption{\label{nyu205}Recall \vs number of proposals at 0.5 IOU for all categories annotated in the NYU2 dataset}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.30\textwidth}
\includegraphics[width=1\columnwidth]{figs/nyu207iou.pdf}
\subcaption{\label{nyu207}Recall \vs number of proposals at 0.7 IOU for all categories annotated in the NYU2 dataset}
\end{subfigure}
\caption{\label{nyu2}Performance of various object proposal methods on different evaluation metrics when evaluated on NYU2 dataset containing annotations for all categories}
\end{figure*}
\label{more_results}
\section{Bias Inspection}
\vspace{\sectionReduceTop}
\subsection{Measuring Fine-Grained Recall}
\vspace{\sectionReduceTop}
\begin{figure*}[htb]
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.3\columnwidth]{figs/9a.pdf}
\subcaption{Sorted by size.
\label{newexparea1}}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.3\columnwidth]{figs/9b.pdf}
\subcaption{Sorted by the number of instances.}
\label{newexpinst1}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.3\columnwidth]{figs/9c.pdf}
\subcaption{MS COCO super-categories.}
\label{newexpcoco1}
\end{subfigure}
\caption{Recall at 0.7 IOU for categories sorted/clustered by (a) size, (b) number of instances, and
(c) MS COCO `super-categories' evaluated on PASCAL Context.\label{percategoryplot1}\\}
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.3\columnwidth]{figs/mscoco_area.pdf}
\subcaption{{Sorted by size.}}
\label{newexparea}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.3\columnwidth]{figs/mscoco_numinst.pdf}
\subcaption{{Sorted by the number of instances.}}
\label{newexpinst}
\end{subfigure}
\,\,\,
\begin{subfigure}[b]{0.3\textwidth}
\includegraphics[width=1.3\columnwidth]{figs/mscoco_clusterplot.pdf}
\subcaption{{MS COCO super-categories.}}
\label{newexpcoco}
\end{subfigure}
\caption{{Recall at 0.7 IOU for categories sorted/clustered by (a) size, (b) number of instances, and
(c) MS COCO `super-categories' evaluated on PASCAL Context and MS COCO.}}
\label{percategoryplot}
\end{figure*}
We also looked at a more fine-grained per-category performance
of proposal methods and DMPs.
Fine grained recall can be used to answer if some proposal methods
are optimized for larger or frequent categories i.e. if they
perform better or worse with respect to different object attributes like area, kinds of objects, etc. It is also easier to
observe the change in performance of a particular method on frequently occurring category
\vs rarely occurring category. We performed this experiment on instance level PASCAL Context and MS COCO datasets. We sorted/clustered all categories on the basis of:
\begin{compactitem}
\item Average size (fraction of image area) of the category,
\item Frequency (Number of instances) of the category,
\item Membership in `super-categories' defined in MS COCO dataset
(electronics, animals, appliance, \etc).
10 pre-defined clusters of objects of different kind
(These clusters are the subset of 11 super-categories defined in MS COCO dataset
for classifying individual classes in groups of similar objects.)
\end{compactitem}
Now, we present the plots of recall for all 80 (20 PASCAL + 60 non-PASCAL) categories for the modified PASCAL Context dataset and MS COCO. Note that the non-PASCAL 60 categories are different for both the datasets.\\
\textbf{Trends:}
\figref{percategoryplot1} shows the performance of different proposal methods
and DMPs along each of these dimensions. \\
In \figref{newexparea1}, we see that recall steadily improves
perhaps as expected, bigger objects are typically easier to find than smaller objects.
In \figref{newexpinst1}, we see
that the recall generally increases as the number of instances increase except for one outlier category. This category was found to be `pole' which appears to be quite difficult to
recall, since poles are often occluded and have a long elongated shape, it is not surprising that
this number is pretty low.
Finally, in \figref{newexpcoco1} we observe that some super-categories (\eg outdoor objects) are hard to recall while others (\eg animal, electronics) are relatively easier to recall.
It can be seen in \figref{percategoryplot}, the trends on MS COCO are almost similar to PASCAL Context.
\label{percat}
\subsection{Overview of Object Proposal Algorithms}
\vspace{\sectionReduceTop}
Table \ref{tab:RelatedWorksTable} provides an overview of some popular object proposal algorithms. The symbol $^{*}$ indicates methods we have evaluated in this paper. Note that a majority of the approaches are learning based.
\begin{table*}[htpb]
\vspace*{-0.5cm}
\hspace*{-0.5cm}
\centering
\begin{tabular}{| p{2.5cm} |l |l| p{3cm} | p{2cm}| p{3cm}|}
\hline
Method & Code Source & Approach & Learning Involved & Metric & Datasets \\ \hline
$objectness^{*}$ & {Source code from \cite{objectnesscode}} & Window scoring & Yes supervised, train on 6 PASCAL classes and their own custom dataset of 50 images & Recall $@$ t \(\geq\) 0.5 vs \# proposals & PASCAL VOC 07 test set, test on unseen 16 PASCAL classes\\ \hline
$selectiveSearch^{*}$ & {Source code from \cite{selectivesearchcode}} & Segment based & No & Recall $@$ t \(\geq\) 0.5 vs \# proposals, MABO, per class ABO & PASCAL VOC 2007 test set, PASCAL VOC 2012 train val set\\ \hline
$rahtu^{*}$ & {Source code from\cite{rahtucode}} & Window Scoring & Yes, two stages. Learning of generic bounding box prior on PASCAL VOC 2007 train set, weights for feature combination learnt on the dataset released with \cite{objectnesscode} & Recall $@$ t > various IoU thresholds and \# proposals, AUC & PASCAL VOC 2007 test set\\ \hline
$randomPrim^{*}$ & {Source code from \cite{rpcode}} & Segment based & Yes supervised, train on 6 PASCAL categories & Recall $@$ t > various IOU thresholds using 10k and 1k proposals & Pascal VOC 2007 test set/2012 trainval set on 14 categories not used in training \\ \hline
$mcg^{*}$ & {Source code from \cite{mcgcode}}& Segment based & Yes & NA, only segments were evaluated & NA (tested on segmentation dataset) \\ \hline
$edgeBoxes^{*}$ & {Source code from \cite{ebcode}} & Window scoring & No & AUC, Recall $@$ t > various IOU thresholds and \# proposals, Recall vs IoU & PASCAL VOC 2007 testset\\ \hline
$bing^{*}$ & {Source code from \cite{bingcode}} & Window scoring & Yes supervised, on PASCAL VOC 2007 train set, 20 object classes/6 object classes & Recall $@$ t> 0.5 vs \# proposals & PASCAL VOC 2007 detection complete test set$/$14 unseen object categories \\ \hline
$rantalankila$ & {Source code from \cite{rantalankilacode}} & Segment based & Yes & NA, only segments are evaluated & NA (tested on segmentation dataset)\\ \hline
$Geodesic$ & {Source code from \cite{gopcode}} & Segment based & Yes, for seed placement and mask construction on PASCAL VOC 2012 Segmentation training set&VUS at 10k and 2k windows, Recall vs IoU threshold, Recall vs proposals & PASCAL 2012 detection validation set \\ \hline
$Rigor$ & {Source code from \cite{rigorcode}} & Segment based & Yes, pairwise potentials between super pixels learned on BSDS-500 boundary detection dataset & NA, only segments were evaluated & NA (tested on segmentation dataset)\\ \hline
$endres$ & {Source code from \cite{endrescode}} & Segment based & Yes & NA, only segments are evaluated & NA (tested on segmentation dataset) \\ \hline
\end{tabular}
\captionof{table}{Properties of existing bounding box approaches. * indicates the methods which have studied in this paper.} \label{tab:RelatedWorksTable}
\end{table*}
\label{related_work}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,170
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Guest Post with Steven James
3:00 AM Every Wicked Man (The Bowers Files: The New York Years #3), Guest Post with Steven James, JBN, Jean Book Nerd 3 comments
Photo Content from Steven James
Steven James is a national bestselling novelist whose award-winning, pulse-pounding thrillers continue to gain wide critical acclaim and a growing fan base.
Suspense Magazine, who named Steven's book THE BISHOP their Book of the Year, says that he "sets the new standard in suspense writing." Publishers Weekly calls him a "master storyteller at the peak of his game." And RT Book Reviews promises, "the nail-biting suspense will rivet you."
Equipped with a unique Master's Degree in Storytelling, Steven has taught writing and storytelling on four continents over the past two decades, speaking more than two thousand times at events spanning the globe. In his podcast "The Story Blender," he interviews leading storytellers in film, print, and web. Listen now to any of the dozens of archived podcasts for free by visiting his website www.thestoryblender.com.
Steven's groundbreaking book on the art of fiction writing, STORY TRUMPS STRUCTURE, won a Storytelling World award. Widely-recognized for his story crafting expertise, he has twice served as a Master CraftFest instructor at ThrillerFest, North America's premier training event for suspense writers.
Respected by some of the top thriller writers in the world, Steven deftly weaves intense stories of psychological suspense with deep philosophical insights. As critically-acclaimed novelist Ann Tatlock put it, "Steven James gives us a captivating look at the fine line between good and evil in the human heart."
After consulting with a former undercover FBI agent and doing extensive research on cybercrimes, Steven wrote his latest thriller, EVERY CROOKED PATH—a taut, twist-filled page turner that is available now wherever books are sold.
If you've never met environmental criminologist and geospatial investigator Patrick Bowers, EVERY CROOKED PATH is the perfect chance to dive into the series and find out what fans and critics everywhere are raving about.
EVERY WICKED MAN ― THE FINAL CHAPTER OF THE BOWERS FILES
It was 2005 and I was a frustrated wannabe novelist.
I'd been wanting to write a thriller for years, but every time I started one, I found that my story wasn't as fresh and original as I needed it to be. I was about ready to give up.
Then one day, while researching investigative techniques, I stumbled across an article about geospatial investigation, a little known, cutting-edge way of analyzing the timing, location, and progression of serial crimes that the FBI was starting to use.
It was unique, different, and perfect for my story. Everything began to click and FBI Special Agent Patrick Bowers was born.
THE PAWN released in 2007 and my life would never be the same again.
Since each book takes me around a year to research and write, chronicling Patrick's adventures and cases has been a labor of love for more than a decade.
And now, with the release of the eleventh and final book, EVERY WICKED MAN, the long-running series is coming to an end and it's come time to say goodbye to my old friend.
In my search for authentic locations and characters, writing this series has been a journey that has taken me all across North America—from San Diego to New York City to Charlotte to Milwaukee to Denver and many places in between.
In my research I've been honored to tour the Pentagon and the FBI Academy, ride along with police officers in Detroit, tour abandoned gold mines in the Rocky Mountains, visit military bases in Texas and consult with leading scientists and law enforcement personnel from across the country.
To all of you who have been fans of the series, I'm thrilled you've enjoyed the books and I'd like to thank you for entrusting me with your time. Hearing from you over the years has been a highlight for me.
If you've never read one of the books, I invite you to give them a try. As a prequel, EVERY WICKED MAN sets up the storyline for the next seven books so there's no better time than now to dive in.
To me, the series has offered a deep exploration of human nature—both the depths of evil we can be drawn into and the heights of love and hope we are capable of experiencing.
As Patrick has grown, so have I.
As he has learned to love his stepdaughter, Tessa, so I've learned to grow closer to my own daughters.
As Patrick likes to say, "Everything matters." Well, I've found that that's true not just of his investigations, but in all of life as well.
So thanks, Pat for teaching me not just how to track down serial killers, terrorists, and arsonists, but for letting me learn with you the most important lessons of all. --Steven James
A criminal mastermind's chilling terrorist plot forces FBI Special Agent Patrick Bowers to the brink in the latest thriller from bestselling novelist Steven James.
When a senator's son takes his own life and posts the video live online, Agent Bowers is drawn into a complex web of lies that begins to threaten the people he loves the most. As he races to unravel the mystery behind the suicide and a centuries-old code that might help shed light on the case, he finds a dark pathway laced with twists and deadly secrets that touch a little too close to home.
Praise for the BOWERS FILES thrillers
"A smart, taut, intense novel of suspense that reads like a cross between Michael Connelly and Thomas Harris…a blisteringly fast and riveting read." ―Mark Greaney, New York Times bestselling author of Gunmetal Gray
"From the first chapter, it sets its hook deep and drags you through a darkly gripping story of relentless power." ―Michael Connelly, Author of The Wrong Side of Goodbye
"With a multi-dimensional quality, Steven James writes with a confident, assured ease." ―Steve Berry, New York Times bestselling author of The 14th Colony
"High tension all the way… Fast, sharp, and believable. Put it at the top of your list." ―John Lutz, Edgar Award-winning author of Single White Female and Slaughter
"Full of secrets and mind games that are entertaining and thought-provoking." ―RT Book Reviews
You can purchase Every Wicked Man at the following Retailers:
Thank you STEVEN JAMES for making this giveaway possible.
1 Winner will receive a Copy of Every Wicked Man (The Bowers Files: The New York Years #3) by Steven James.
John Smith September 10, 2018 at 8:08 AM
"When you were a kid, what did you want to be when you grew up?" An Egyptologist.
Angela Saver September 11, 2018 at 11:45 AM
When I was a kid, I wanted to be a nurse.
Sarah Perry September 18, 2018 at 10:11 AM
When I was a kid I wanted to be a teacher. Thanks! :)
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,187
|
Q: How to add another page to an UIScrollView by swiping? I am a french 17 years old guy so excuse my english.
I am creating an app' using an UIScrollView with an UIPageControl to display multiple UIView pages from UIViewControllers (which display an object) . Everything works pretty great.
Now i would like to add a new page (and a new empty object but that is not a problem) when I swipe the UIScrollView to the right in the last view.
How could I do that ? And if it's not possible, how can I update the UIScrollView ?
Thank you very much !
A: I think you need to keep an eye on two things. scrollViewDidScroll: & UIPageControl's currentPage. When you determine that the user is currently viewing the rightmost page & the UIScrollView was scrolled, you can add a new page to the UIPageControl. The amount by which the scroll view was scrolled is related to scrollView.contentOffset.x. A new page can be added to the page control by incrementing its numberOfPages & setting the currentPage to the newest page.
HTH,
Akshay
A: This is what you are looking for I think : a sample code from Apple to build a photo viewer that load images just before they appear on screen.
https://developer.apple.com/library/ios/#samplecode/PhotoScroller/Introduction/Intro.html#//apple_ref/doc/uid/DTS40010080
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,143
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,224
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\section{Critical points of bounded analytic functions}
A sequence of points $(z_j)$ in a subdomain $\Omega$ of the complex plane ${\mathbb{C}}$ is called the
zero set of an analytic function $f : \Omega \to {\mathbb{C}}$, if $f$ vanishes precisely on this
set. This means that if the point $\xi\in \Omega$ occurs $m$ times in the sequence, then $f$
has a zero at $\xi$ of precise order $m$, and $f(z)\not=0$ for $z \in \Omega
\backslash (z_j)$.
The following classical theorem
due to Jensen \cite{Jen1899},
Blaschke \cite{Bla1915} and F.~and R.~Nevanlinna \cite{Nevs1922} characterizes completely the zero
sets of bounded analytic functions defined in the open unit disk ${\mathbb{D}}:=\{z \in
{\mathbb{C}} \, : \, |z|<1\}$.
\begin{satz}\label{thm:classical_zero}
Let $ (z_j)$ be a sequence in ${\mathbb{D}}$. Then the following statements are
equivalent.
\begin{itemize
\item[(a)] There is an analytic self--map of ${\mathbb{D}}$ with zero set
$ (z_j)$.
\item[(b)] There is a Blaschke product with zero set $
(z_j)$.
\item[(c)] The sequence $(z_j)$ fulfills the Blaschke condition, i.\!\;e.~$
\sum\limits_{j=1}^{\infty} \big(1-|z_j|\big) < + \infty\, .
$
\end{itemize}
\end{satz}
We call a sequence of points $(z_j)$ in a domain $\Omega \subseteq {\mathbb{C}}$ the
critical set of an analytic function $f : \Omega \to {\mathbb{C}}$, if $(z_j)$ is the zero
set of the first derivative $f'$ of the function $f$.
There is an extensive literature on critical sets.
In particular, there are many interesting results on the relation
between the zeros and the critical points of analytic and harmonic functions.
A classical reference for all this is the book of Walsh \cite{Wal1950}.
The first aim of this survey paper is to point out the following analogue of
Theorem~\ref{thm:classical_zero} for the critical sets of bounded analytic
functions instead of their zeros sets.
\begin{theorem}\label{thm:main}
Let $ (z_j)$ be a sequence in ${\mathbb{D}}$. Then the following statements are
equivalent.
\begin{itemize}
\item[(a)] There is an analytic self--map of ${\mathbb{D}}$ with critical set
$ (z_j)$.
\item[(b)] There is a Blaschke product with critical set $
(z_j)$.
\item[(c)] There is a function in the weighted Bergman space $$ \mathcal{
A}^2_1=\left\{
g:{\mathbb{D}} \to {\mathbb{C}} \text{ analytic} \,: \iint \limits_{{\mathbb{D}}} (1-|z|^2)\, |g(z)|^2\,
d\sigma_z <+ \infty \right\} $$ with zero set $ (z_j)$.
\end{itemize}
\end{theorem}
Here, and throughout, $\sigma_z$ denotes two--dimensional Lebesgue measure
with respect to $z$.
For the special case of {\it finite} sequences, a result related to Theorem
\ref{thm:main} can be found in work of
Heins \cite[\S29]{Hei62},
Wang \& Peng \cite{WP79}, Zakeri \cite{Z96} and Stephenson
\cite{Ste2005}. They proved that for
every finite sequence $(z_j)$ in ${\mathbb{D}}$ there is always a
{\it finite} Blaschke product whose critical set coincides with $(z_j)$, see
also Remark \ref{rem:finite} below.
A recent generalization of this result to {\it infinite} sequences
is discussed in \cite{KR}. There it is shown that every Blaschke sequence
$(z_j)$ is the critical set of an infinite Blaschke product. However,
the converse to this, known as the Bloch--Nevanlinna conjecture
\cite{Dur1969}, is false. According to a result of Frostman, there do exist
Blaschke products whose critical sets
fail to satisfy the Blaschke condition, see \cite[Theorem 3.6]{Col1985}.
Thus the critical sets of bounded analytic functions are not
just the Blaschke sequences and the situation for critical sets
is much more complicated than for zero sets.
Theorem \ref{thm:main} identifies the critical
sets of bounded analytic functions as the zeros sets of functions in the
Bergman space~$\mathcal{ A}^2_1$.
The simple geometric characterization of the zero sets of bounded analytic
functions via the Blaschke condition (c) in Theorem \ref{thm:classical_zero} has
not found an explicit counterpart for
critical sets yet. However, condition (c) of Theorem \ref{thm:main} might be seen as
an implicit substitute.
The zero sets of (weighted) Bergman space functions have intensively been studied in the 1970's and
1990's by Horowitz
\cite{Hor1974,Hor1977}, Korenblum \cite{Kor1975}, Seip \cite{Seip1993,
Seip1995}, Luecking \cite{Lue1996} and many others. As a result quite sharp necessary conditions and sufficient
conditions for the zero sets of Bergman
space functions are available.
In view of Theorem \ref{thm:main} all results about zero sets of
Bergman space functions carry now
over to the critical sets of bounded analytic functions and vice versa.
Unfortunately, a {\it geometric}
characterization of the zero sets of (weighted) Bergman space functions is
still unknown, ``and it is well known
that this problem is very difficult'', cf.~\cite[p.~133]{HKZ}.
The implications ``(b) $\Longrightarrow$ (a)'' and ``(a) $\Longrightarrow$ (c)''
of Theorem \ref{thm:main} are easy to prove. In fact, the statement ``(a)
$\Longrightarrow$ (c)'' follows directly from the Littlewood--Paley identity (see
\cite[p.~178]{Sha1993}), which says that for any holomorphic function $f : {\mathbb{D}} \to {\mathbb{D}}$ the derivative
$f'$ belongs to $\mathcal{A}^2_1$. Hence the critical set of any non--constant
bounded analytic function is trivially the zero set of a function in
$\mathcal{A}^2_1$. It is however not true that any $\mathcal{A}^2_1$ function
is the derivative of a bounded analytic function, so ``(c) $\Longrightarrow$
(a)'' is more subtle.
In the following sections, we describe some of the ingredients of the proofs of
the implications ``(a) $\Longrightarrow$ (b)'' and ``(c) $\Longrightarrow$
(a)'' in
Theorem \ref{thm:main} as well as some further results connected to it.
In particular, we explain the close relation of Theorem~\ref{thm:main}
to conformal differential geometry and the solvability of the Gauss curvature equation.
This paper is expository, so there are essentially no proofs.
For the proofs we refer to \cite{Kra2010, Kra2011a, KRR06,KR, KR2011}.
Background material on Hardy spaces and Bergman spaces can be found e.g.~in
the excellent books \cite{Dur2000,DS,Gar2007,HKZ,Koo1998,Mas2009}.
Finally, we draw attention to the recent paper \cite{EKS2012} of P.~Ebenfelt,
D.~Khavinson and H.S.~Shapiro and the references therein, where the difficult
problem of constructing a finite Blaschke product by its \textit{critical
values} is discussed.
\section{Conformal metrics and maximal Blaschke products} \label{sec:2}
An essential characteristic of the proof of
the implication ``(a) $\Longrightarrow$ (b)'' in Theorem \ref{thm:main} is the extensive use of
negatively curved conformal pseudometrics. We give a short account of some of
their properties and refer to \cite{BM2007, Hei62, KL2007, Krantz, KR2008, Smi1986} for more details.
In the following, $G$ and $D$
always denote domains in the complex plane ${\mathbb{C}}$.
\subsection{Conformal metrics and developing maps}
We recall that a nonnegative upper
semicontinuous function $\lambda$ on $G$, $\lambda: G \to [0, + \infty)$, $\lambda \not\equiv 0$, is
called conformal density on $G$. The corresponding
quantity \label{def:leng}
$\lambda(z) \, |dz|$ is called conformal pseudometric on $G$.
If $\lambda(z)>0$ for all $z \in G$, we say $\lambda(z) \, |dz|$ is a
conformal metric on $G$. We call a conformal
pseudometric $\lambda(z) \, |dz|$ regular on $G$, if $\lambda$ is of class
$C^2$ in $\{z \in G \, : \, \lambda(z)>0\}$.
\begin{example}[The hyperbolic metric]
The ubiquitous example of a conformal metric is the Poincar\'{e} metric or hyperbolic
metric $$\lambda_{{\mathbb{D}}}(z)\, |dz|:=\frac{|dz|}{1-|z|^2}$$ for
the unit disk ${\mathbb{D}}$. Clearly, $\lambda_{{\mathbb{D}}}(z)\, |dz|$ is a regular conformal
metric on ${\mathbb{D}}$.
\end{example}
The (Gauss) curvature $\kappa_{\lambda}$ of a regular conformal pseudometric
$\lambda(z) \, |dz|$ on $G$
is defined by
\begin{equation*}
\kappa_{\lambda}(z):=-\frac{\Delta (\log \lambda)(z)}{\lambda(z)^2}
\end{equation*}
for all points $z \in G$ where $\lambda(z) > 0$.
Note that if $\lambda(z)\, |dz|$ is a regular conformal metric
with curvature $\kappa_{\lambda}=\kappa$ on $G$, then the function $u:= \log \lambda$
satisfies the partial differential equation (Gauss curvature equation)
\begin{equation}\label{eq:pde1}
\Delta u=-\kappa(z) \, e^{2 u}
\end{equation}
on $G$. If, conversely, a realvalued $C^2$--function $u$ is a solution of the
Gauss curvature equation (\ref{eq:pde1}) on $G$,
then $\lambda(z):=e^{u(z)}$ induces a regular conformal metric $e^{u(z)} \
|dz|$ on $G$ with
curvature $\kappa$.
\begin{example}
The hyperbolic metric $\lambda_{{\mathbb{D}}}(z) \, |dz|$ on ${\mathbb{D}}$ has constant
negative curvature\footnote{\textbf{Warning:} A rival school of thought calls $\frac{2 \,
|dz|}{1-|z|^2}$ the hyperbolic metric of ${\mathbb{D}}$. This metric has constant
curvature $-1$.} $-4$.
\end{example}
Using analytic maps, conformal metrics can be transferred from one domain to
another as follows. Let $\lambda(w) \, |dw|$ be a conformal pseudometric
on a domain $D$ and let $w=f(z)$ be a non--constant analytic map from another
domain $G$ to $D$. Then
the conformal pseudometric
\begin{equation*}
(f^*\lambda)(z) \, |dz|:=\lambda(f(z)) \, |f'(z)| \, |dz|\, ,
\end{equation*}
defined on $G$, is called the pullback of $\lambda(w) \, |dw|$ under the map
$f$.
Now, Gauss curvature is important since it is a conformal invariant in
the following sense.
\begin{satz}[Theorema Egregium]
For every analytic map $w=f(z)$ and every regular conformal pseudometric $\lambda(w)\,|dw|$ the relation
\begin{equation*}
\kappa_{f^*\lambda}(z)=\kappa_{\lambda}(f(z))
\end{equation*}
is satisfied provided $\lambda(f(z)) \, |f'(z)|>0$.
\end{satz}
Hence pullbacks of conformal pseudometrics can be used for constructing conformal
pseudometrics with various prescribed properties while controlling their
curvature. For instance, if $\lambda(w) \, |dw|$ is a conformal metric on $D$
(without zeros) and $w=f(z)$ is a non--constant analytic map from $G$ to $D$,
then $(f^*\lambda)(z) \, |dz|$ is a pseudometric on $G$ with zeros exactly at the
critical points of $f$. In order to take the multiplicity of the
zeros into account, it is convenient to introduce the
following formal definition.
\begin{definition}[Zero set of a pseudometric]\label{def:zeroset}
Let $\lambda(z)\, |dz|$ be a conformal pseudometric on $G$. We say
$\lambda(z)\, |dz|$ has a zero of order $m_0>0$ at $z_0 \in G$ if
\begin{equation*
\lim_{z \to z_0} \frac{\lambda(z)}{|z-z_0|^{m_0}} \quad\text{ exists and }
\not=0 \, .
\end{equation*}
We will call a sequence $\mathcal{ C}=(\xi_j) \subset G$
\begin{equation*}
(\xi_j):=(\underbrace{z_1, \ldots, z_1}_{m_1 -\text{times}},\underbrace{z_2,
\ldots, z_2}_{m_2-\text{times}} , \ldots )\,, \, \, z_k \not=z_n \text{ if
} k\not=n,
\end{equation*}
the zero set of a conformal
pseudometric $\lambda(z) \, |dz|$, if $\lambda(z)>0$ for $z \in
G\backslash \mathcal{ C}$ and if $\lambda(z)\, |dz|$ has a zero of order $m_k \in {\mathbb{N}}$
at $z_k$ for all $k$.
\end{definition}
Now, every conformal pseudometric of constant curvature $-4$
can locally be
represented by a holomorphic function. This is the content of the following result.
\begin{satz}[Liouville's Theorem
\label{thm:liouville}
Let $\mathcal{ C}$ be a sequence of points in a simply connected domain $G$
and let
$\lambda(z) \, |dz|$ be a regular conformal pseudometric on $G$ with constant curvature
$-4$ on $G$ and zero set $\mathcal{ C}$. Then $\lambda(z) \, |dz|$ is the pullback of the
hyperbolic metric $\lambda_{{\mathbb{D}}}(z)\, |dz|$ under some analytic map $f:G \to
{\mathbb{D}}$, i.\!\;e.
\begin{equation}\label{eq:liouville}
\lambda(z) =\frac{|f'(z)|}{1-|f(z)|^2}\, , \quad z \in G.
\end{equation}
If $g:G \to \mathbb{D}$ is another analytic function, then
$\lambda(z)=(g^*\lambda_{{\mathbb{D}}})(z)$ for $z\in G$
if and only if $g=T\circ f$ for some conformal automorphism $T$ of ${\mathbb{D}}$.
\end{satz}
A holomorphic function $f$ with property (\ref{eq:liouville})
will be called a {\bf
developing map} for $\lambda(z) \, |dz|$.
Note that the critical set of each developing map coincides with the zero set
of the corresponding conformal pseudometric.
\begin{example}
The developing maps of the hyperbolic metric $\lambda_{{\mathbb{D}}}(z) \, |dz|$
are precisely the conformal automorphisms of ${\mathbb{D}}$, i.e., the finite Blaschke
products of degree $1$.
\end{example}
For later applications we wish to mention the following variant of Liouville's Theorem.
\begin{theorem} \label{rem:liouville}
Let $G$ be a simply connected domain and let $h:G \to {\mathbb{C}}$ be an analytic
map. If $\lambda(z)\, |dz|$ is a regular conformal metric with curvature $-4
\, |h(z)|^2$, then there exists a holomorphic function $f:G \to {\mathbb{D}}$ such
that
\begin{equation*}
\lambda(z) =\frac{1}{|h(z)|}\frac{|f'(z)|}{1-|f(z)|^2}\, , \quad z
\in G.
\end{equation*}
Moreover, $f$ is uniquely determined up to postcomposition with a unit disk automorphism.
\end{theorem}
Liouville \cite{Lio1853} stated
Theorem \ref{thm:liouville} for the special case that $\lambda(z)
\, |dz|$ is a regular conformal metric.
We therefore refer to
Theorem \ref{thm:liouville} as well as to Theorem \ref{rem:liouville} as
Liouville's theorem.
Theorem \ref{thm:liouville} and in particular the special case that $\lambda(z) \, |dz|$ is
a conformal metric has a number of different proofs, see for instance
\cite{Bie16, CW94, CW95, Min, Nit57, Yam1988}. Theorem
\ref{rem:liouville} is discussed in \cite{KR}.
Liouville's theorem plays an important r$\hat{\text{o}}$le in this paper. It
provides a bridge bet\-ween the world of bounded analytic functions and the
world of conformal pseudometrics with constant negative curvature.
The critical points on the one side correspond to the zeros on the other side.
Unfortunately, this bridge only works for simply connected domains, see Remark \ref{rem:final}.
\subsection{Maximal conformal pseudometrics and maximal Blaschke products}
Apart from having constant negative curvature the hyperbolic metric
on ${\mathbb{D}}$ has another important property: it
is \textit{maximal} among all regular conformal pseudometrics on
${\mathbb{D}}$ with curvature bounded above by $-4$. This is the content of the
following result.
\begin{satz}[Fundamental Theorem]\label{thm:ahlfors}
Let $\lambda(z)\,|dz|$ be a regular conformal pseudometric on
${\mathbb{D}}$ with curvature bounded above by $-4$. Then $\lambda(z) \le
\lambda_{{\mathbb{D}}}(z)$ for every $z \in {\mathbb{D}}$.
\end{satz}
Theorem \ref{thm:ahlfors} is due to Ahlfors \cite{Ahl1938} and it is usually
called Ahlfors' lemma. However, in view of its relevance Beardon and Minda
proposed to call Ahlfors' lemma the {\bf fundamental theorem}. We
will follow their suggestion in this paper. As a result of the fundamental
theorem we have
\begin{eqnarray*}
\lambda_{{\mathbb{D}}}(z)=\max \{ \lambda(z) \, : \, \lambda(z)\,|dz| \text{ is a regular conformal pseudometric on
${\mathbb{D}}$} & & \\ & & \hspace*{-4cm} \text{ with curvature } \le -4 \}
\end{eqnarray*}
for any $z \in {\mathbb{D}}$.
\begin{remark}[Developing maps and universal coverings] \label{rem:hyp_all}
Let $G$ be a hyperbolic subdomain of the complex plane ${\mathbb{C}}$, i.e., the
complement ${\mathbb{C}}\backslash G$ consists of more than one point.
In analogy with the Poincar\'e metric for the unit disk, a regular conformal metric
$\lambda_{G}(z)\, |dz|$ of constant curvature $-4$ on $G$ is said to be the
hyperbolic metric for $G$, if it is the maximal regular conformal pseudometric with
curvature $\le -4$ on $G$, i.\!\;e.~$\lambda(z) \le \lambda_G(z)$, $z \in G$, for all
regular conformal pseudometrics $\lambda(z)\, |dz|$ on $G$ with curvature
bounded above by $-4$. Then
$\lambda_G(z)\, |dz|$ and $\lambda_{{\mathbb{D}}}(z)\, |dz|$ are
connected via the universal coverings $\pi:{\mathbb{D}} \to G$ of $G$ by the formula
$(\pi^*\lambda_{G})(z)\, |dz|=\lambda_{{\mathbb{D}}}(z)\, |dz|$,
see for example \cite[\S 9]{Hei62} and \cite{KR2008, Min1979}.
Hence, every branch of the inverse of a universal covering map $\pi : {\mathbb{D}} \to G$
is \textit{locally} the developing map of the hyperbolic metric $\lambda_G(z)
\, |dz|$ of $G$.
In particular, if $G$ is a hyperbolic simply connected domain, then the
developing maps for $\lambda_{G}(z) \, |dz|$ are precisely the conformal
mappings from $G$ onto ${\mathbb{D}}$.
\end{remark}
We now consider prescribed zeros.
\begin{satz} \label{thm:perron}
Let $\mathcal{C}=(\xi_j)$ be a sequence of points in $G$ and
\begin{eqnarray*}
\Phi_{\mathcal{C}}:=\left\{ \lambda \, : \, \lambda(z) \, |dz| \text{ is a regular conformal
pseudometric in } G \right. & & \\ & & \hspace*{-5.8cm} \left. \text{ with
curvature } \le -4 \text{ and zero set } \mathcal{C}^*
\supseteq \mathcal{C}\right\}\, .
\end{eqnarray*}
If $\Phi_{\mathcal{C}}\not=\emptyset$, then
$$ \lambda_{max}(z):=\sup\{ \lambda(z) \, : \, \lambda \in \Phi_{\mathcal{C}}\} \, , \qquad
z \in G \, , $$
induces the unique maximal regular conformal pseudometric
$\lambda_{max}(z) \, |dz|$ on $G$ with constant
curvature $-4$ and zero set $\mathcal{C}$.
\end{satz}
Thus, if $\Phi_{\mathcal{C}}\not=\emptyset$, i.e.,
if there exists at least one regular conformal pseudometric $\lambda(z) \, |dz|$
on $G$ with curvature $\le -4$ whose zero set contains the given sequence $\mathcal{C}$,
then there exists a (maximal) regular conformal pseudometric $\lambda(z) \, |dz|$
on $G$ with \textit{constant} curvature $-4$ whose zero set is
\textit{exactly} the sequence $\mathcal{C}$.
In particular, Theorem \ref{thm:perron} can be applied, if
there exists a non--constant holomorphic function $f : G \to {\mathbb{D}}$ with critical
set $\mathcal{C}^* \supseteq \mathcal{C}$ since then the pseudometric $(f^*\lambda_{{\mathbb{D}}})(z) \, |dz|$
belongs to $\Phi_{\mathcal{C}}$.
The proof of Theorem \ref{thm:perron}
relies on a modification of Perron's method and can be found in \cite[\S 12 \&
\S 13]{Hei62}
and \cite{Kra2011a}.
\begin{example}
If $G$ is a hyperbolic domain and $\mathcal{C}=\emptyset$, then the maximal
regular conformal pseudometric $\lambda_{max}(z) \, |dz|$ on $G$ with constant
curvature $-4$ and zero set $\mathcal{C}$ is exactly the hyperbolic metric $\lambda_G(z)
\, |dz|$ for $G$.
\end{example}
Thus maximal pseudometrics are generalizations of the hyperbolic metric and
their developing maps are therefore of special interest.
\begin{definition}[Maximal functions]
Let $\mathcal{C}$ be a sequence of points in ${\mathbb{D}}$ such that there exists a maximal regular conformal pseudometric
$\lambda_{max}(z) \, |dz|$ on ${\mathbb{D}}$ with constant
curvature $-4$ and zero set $\mathcal{C}$. Then every developing map for
$\lambda_{max}(z) \, |dz|$ is called maximal function for $\mathcal{C}$.
\end{definition}
Some remarks are in order. First,
in view of Theorem \ref{thm:liouville} every maximal function is uniquely
determined by its critical set $\mathcal{C}$ up to postcomposition with a unit disk automorphism and, conversely,
the postcomposition of any maximal function with a unit disk
automorphism is again a maximal function. Second, if $\mathcal{C}=\emptyset$,
then the maximal functions for $\mathcal{C}$ are precisely the unit disk
automorphisms, i.e., the finite Blaschke products of degree~$1$.
Now, we have the following result, see \cite{Kra2011a}.
\begin{theorem} \label{thm:0}
Every maximal function is a Blaschke product.
\end{theorem}
It is to emphasize that
since the postcomposition of any maximal function with a unit disk
automorphism is again a maximal function, every maximal function is an
\textit{indestructible} Blaschke product.
We note that Theorem \ref{thm:perron} combined with Theorem \ref{thm:0}
immediately gives the
implication ``(a) $\Longrightarrow$ (b)'' in Theorem \ref{thm:main}.
In fact, if $f$ is a non--constant analytic self--map of ${\mathbb{D}}$ with critical
set $\mathcal{C}$, then $\lambda(z) \, |dz|:=(f^*\lambda_{{\mathbb{D}}})(z) \, |dz|$
is a regular conformal pseudometric on ${\mathbb{D}}$ with zero set $\mathcal{C}$. Thus
Theorem \ref{thm:perron} guarantees the existence of a maximal conformal
pseudometric on ${\mathbb{D}}$ with zero set $\mathcal{C}$. Now, Theorem \ref{thm:0}
says that the corresponding maximal function for $\mathcal{C}$ is a Blaschke product.
In the special case that the maximal function has finitely many critical
points, the statement of Theorem \ref{thm:0} follows from the next result which is due to Heins
\cite[\S 29]{Hei62}.
\begin{satz} \label{satz:heins}
Let $\mathcal{C}=(z_1, \ldots, z_n)$ be a finite sequence in ${\mathbb{D}}$
and $f : {\mathbb{D}} \to {\mathbb{D}}$ analytic.
Then the following statements are equivalent.
\begin{itemize}
\item[(a)] $f$ is a maximal function for $\mathcal{C}$.
\item[(b)] $f$ is Blaschke product of degree $n+1$ with critical set $\mathcal{C}$.
\end{itemize}
\end{satz}
We shall give a quick proof of Theorem \ref{satz:heins} in Remark
\ref{satz:heinsproof} below.
\begin{remark}[Constructing finite Blaschke products with prescribed critical
points] \label{rem:finite}
In his proof of Theorem \ref{satz:heins}, Heins showed that for any finite sequence
$\mathcal{C}=(z_1, \ldots, z_n)$ in ${\mathbb{D}}$ there is always a finite Blaschke
product $B$ of
degree $n+1$ with critical set $\mathcal{C}$. The essential step is
nonconstructive and consists in showing that the set of critical points of all
finite Blaschke products of degree $n+1$, which is clearly closed, is also open in the
poly disk ${\mathbb{D}}^n$ by applying Brouwer's fixed point theorem.
The same result was later obtained by Wang \& Peng \cite{WP79} and Zakeri
\cite{Z96} by using similar arguments. A completely different approach
via Circle Packing is due to Stephenson, see Lemma 13.7 and Theorem 21.1 in \cite{Ste2005}. Stephenson builds
discrete finite Blaschke products with prescribed branch set and shows that
under refinement these
discrete Blaschke products converge locally uniformly in ${\mathbb{D}}$ to the desired classical
Blaschke product.
We are not aware of any
\textit{efficient} constructive method for computing a (nondiscrete) finite Blaschke
product from its critical points.
\end{remark}
Maximal functions form a particular class of Blaschke products. It is
therefore convenient to make the following definition.
\begin{definition}[Maximal Blaschke products]
A non--constant Blaschke product is called a maximal Blaschke product, if it is a
maximal function for its critical set.
\end{definition}
As was mentioned earlier,
every maximal Blaschke product is indestructible and every finite
Blaschke product is a maximal Blaschke product. Moreover, if $\mathcal{C}$ is the
critical set of any non--constant analytic function $f : {\mathbb{D}} \to {\mathbb{D}}$, then there
is maximal Blaschke product with critical set $\mathcal{C}$. A maximal Blaschke
product is uniquely determined by its critical set $\mathcal{C}$ up to postcomposition with a unit disk automorphism.
Geometrically,
the finite maximal Blaschke products are just the finite branched
coverings of ${\mathbb{D}}$. One is therefore inclined to consider maximal Blaschke
products for
infinite branch sets as ``infinite branched coverings'':
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|l|p{4.7cm}|p{4.8cm}|}
\hline
{\bf critical set} & { \bf maximal Blaschke product} & {\bf mapping properties }\\[2mm] \hline
$\mathcal{ C}=\emptyset$ & automorphism of ${\mathbb{D}}$& unbranched covering of ${\mathbb{D}}$\!\;; \\[-1.5mm]
&&conformal self--map of ${\mathbb{D}}$\\[0.5mm] \hline
$\mathcal{ C}$ finite &finite Blaschke product &finite branched covering of ${\mathbb{D}}$\\[0.5mm] \hline
$\mathcal{ C}$ infinite & indestructible infinite maximal Blaschke product& ``\!\;infinite
branched covering of ${\mathbb{D}}$\!\;''\\ \hline
\end{tabular}
\end{center}
The class of maximal conformal pseudometrics and their corresponding maximal
functions have already been studied by Heins in \cite[\S25 \& \S
26]{Hei62}. Heins obtained
some necessary conditions as well as sufficient conditions
for maximal functions (see Theorem \ref{thm:omitted_value} and Theorem
\ref{thm:heins_island} below), but he did not prove that maximal functions are always
Blaschke products. He also posed the
problem of characterizing maximal functions, cf.~\cite[\S 26 \& \S 29]{Hei62}.
\section{Some properties of maximal Blaschke products}
It turns out that maximal Blaschke products do have remarkable
properties and provide in some sense a fairly natural generalization of the class of finite
Blaschke products. In this section we
take a closer look at some of the properties of maximal Blaschke
products. In the following, we denote by $H^{\infty}$ the set of bounded analytic functions $f : {\mathbb{D}} \to
{\mathbb{C}}$ and set $||f||_{\infty}=\sup_{z \in {\mathbb{D}}} |f(z)|$. This makes
$(H^{\infty},||\cdot||_{\infty})$ a Banach space.
\subsection{Schwarz' lemmas}
\begin{theorem}[Maximal Blaschke products as extremal functions] \label{thm:2}
Suppose $\mathcal{ C}$ is a sequence of
points in ${\mathbb{D}}$ such that
\begin{equation*}
\mathcal{ F}_{\mathcal{ C}}:=\big \{ f \in H^{\infty} \, : \,
f'(z)=0 \text{ for } z \in \mathcal{ C} \big\}
\end{equation*}
contains at least some non--constant function and let $$ m:=\min\{\, n \in {\mathbb{N}} \, : \, f^{(n)}(0)\not=0 \text{ for some } f \in \mathcal{
F}_\mathcal{ C} \, \}\ge 1\, .$$ Then the unique extremal function to the extremal problem
\begin{equation} \label{eq:ex}
\max \big\{ \Re f^{(m)}(0) \, : f \in \mathcal{ F}_\mathcal{ C}, \,
{||f||}_{\infty}\le 1\big \}
\end{equation}
is the maximal Blaschke product $F$ for $\mathcal{ C}$
normalized by $F(0)=0$ and $F^{(m)}(0)>0$.
\end{theorem}
We refer to \cite{KR2011} for the proof of Theorem \ref{thm:2}. To put
Theorem \ref{thm:2} in perspective, note that if $m=1$, then the extremal
problem (\ref{eq:ex}) is exactly the problem of maximizing the derivative at a
point, i.e., exactly the character of Schwarz' lemma.
\begin{remark}[The Nehari--Schwarz Lemma]
If $\mathcal{C}=\emptyset$, then
$\mathcal{F}_{\mathcal{C}}=H^{\infty}$, i.e., the set of all bounded analytic
functions in ${\mathbb{D}}$. In this case, Theoren \ref{thm:2} is of course just the statement of Schwarz' lemma.
If $\mathcal{C}$ is a finite sequence, then Theorem \ref{thm:2} is exactly
Nehari's
1947 generalization of Schwarz' lemma (Nehari \cite{Neh1946}, in particular the
Corollary to Theorem 1). Hence Theorem \ref{thm:2} can be considered as an extension of the
Nehari--Schwarz Lemma.
\end{remark}
\begin{remark}[The Riemann mapping theorem and the Ahlfors map]
We consider a domain $\Omega \subseteq {\mathbb{C}}$ containing $0$. Let $\mathcal{C}=(z_j)$
be the critical set of a non--constant function $f$ in $H^{\infty}(\Omega)$,
where $H^{\infty}(\Omega)$ denotes the set of all functions analytic and
bounded in $\Omega$. We let $N$ denote the number
of times that $0$ appears in the sequence~$\mathcal{ C}$ and set
$$ \mathcal{F}_{\mathcal{C}}(\Omega):=\left\{f \in H^{\infty}(\Omega) \, : \, f'(z)=0 \text{ for
any } z \in \mathcal{C}\right\} \, .$$
Then, by a normal family argument, there is always at least one extremal
function for the extremal problem
\begin{equation} \tag{$*$}
\max \big\{ \Re f^{(N+1)}(0) \, : f \in \mathcal{ F}_\mathcal{ C}(\Omega), \, {||f||}_{\infty} \le 1 \big \} \, .
\end{equation}
In the following three cases there is a unique extremal function to the
extremal problem ($*$).
\begin{itemize}
\item[(i)] \textit{$\Omega\not={\mathbb{C}}$ is simply connected and
$\mathcal{C}=\emptyset$ (conformal maps):}\\
In this case, $N=0$ and the
extremal problem ($*$) has exactly one extremal function,
the normalized Riemann map
$\Psi$ for $\Omega$, that is, the unique conformal map $\Psi$ from $\Omega$
onto ${\mathbb{D}}$ normalized such that $\Psi(0)=0$ and $\Psi'(0)>0$.
\item[(ii)] \textit{$\Omega \not= {\mathbb{C}}$ is simply connected and
$\mathcal{C}\not=\emptyset$ (prescribed critical points):}\\
Let $\Psi$ be the normalized Riemann map for $\Omega$.
Then $\Psi(\mathcal{C})$ is the critical set of a non--constant function in
$H^{\infty}=H^{\infty}({\mathbb{D}})$. If
$B_{\Psi(\mathcal{C})}$ is the extremal function in
$\mathcal{F}_{\Psi(\mathcal{C})}$ according to Theorem \ref{thm:2}, then
$B_{\Psi(\mathcal{C})} \circ \Psi$ is the unique extremal function for ($*$).
\item[(iii)] \textit{$\Omega \not= {\mathbb{C}}$ is not simply connected and
$\mathcal{C}=\emptyset$ (Ahlfors' maps):}\\
If $\Omega$ has connectivity $n \ge 2$, none of whose boundary components
reduces to a point, then the
extremal problem ($*$) has exactly one solution, namely the Ahlfors map
$\Psi : \Omega \to {\mathbb{D}}$. It is a $n:1$ map from $\Omega$ onto ${\mathbb{D}}$ such that
$\Psi(0)=0$ and $\Psi'(0)>0$, see Ahlfors \cite{Ahl} and Grunsky \cite{Gru1978}.
\end{itemize}
\end{remark}
The Schwarz lemma (i.e., the case $\mathcal{C}=\emptyset$ of Theorem \ref{thm:2})
can be stated in an invariant form, the Schwarz--Pick lemma which says that
\begin{equation} \label{eq:sp}
\frac{|f'(z)|}{1-|f(z)|^2} \le \frac{1}{1-|z|^2} \, , \qquad z \in {\mathbb{D}} \, ,
\end{equation}
for any analytic map $f : {\mathbb{D}} \to {\mathbb{D}}$, with equality for some point $z \in {\mathbb{D}}$
if and only if $f$ is a conformal disk
automorphism. Hence maximal Blaschke products without critical points
serve as extremal functions. In a similar way, the more general statement of Theorem
\ref{thm:2} admits an invariant formulation as follows (see \cite{Kra2011a}).
\begin{theorem}[Sharpened Schwarz--Pick inequality]\label{cor:3}
Let $f: {\mathbb{D}} \to {\mathbb{D}}$ be a non--constant analytic function with critical set $\mathcal{
C}$ and let $\mathcal{ C}^*$ be a subsequence of $\mathcal{ C}$.
Then there exists a maximal Blaschke
product $F$ with critical set $\mathcal{ C}^*$
such that
\begin{equation*}
\frac{|f'(z)|}{1-|f(z)|^2} \le \frac{|F'(z)|}{1-|F(z)|^2}\, ,\quad z \in {\mathbb{D}}\!\;.
\end{equation*}
If $\mathcal{ C}^*$ is finite, then $F$ is a finite Blaschke product.
Furthermore, $f=T \circ F$ for some automorphism $T$ of ${\mathbb{D}}$
if and only if
\begin{equation*}
\lim_{z \to w} \frac{|f'(z)|}{1-|f(z)|^2} \,
\frac{1-|F(z)|^2}{|F'(z)|}=1
\end{equation*}
for some $w \in {\mathbb{D}}$.
\end{theorem}
If $\mathcal{ C}^*=\mathcal{ C} \not= \emptyset$, this gives the sharpening
\begin{equation*}
\frac{|f'(z)|}{1-|f(z)|^2} \le \frac{|F'(z)|}{1-|F(z)|^2} < \frac{1}{1-|z|^2}
\end{equation*}
for all $f \in \mathcal{F}_{\mathcal{C}}$
of the Schwarz--Pick inequality (\ref{eq:sp}), which is best possible in some sense.
\subsection{Related extremal problems in Hardy and Bergman spaces}
Let $\mathcal{ C}$ be a sequence in ${\mathbb{D}}$, assume that $\mathcal{C}$ is the critical set of a bounded analytic
function $f : {\mathbb{D}} \to {\mathbb{D}}$ and let $N$ denote the multiplicity of the point $0$
in $\mathcal{C}$. Then according to Theorem \ref{thm:2}
the maximal Blaschke product $F$ for $\mathcal{ C}$
normalized by $F(0)=0$ and $F^{(N+1)}(0)>0$ is the unique solution to the
extremal problem
\begin{center}
\fbox{\begin{minipage}{11cm}
{\begin{equation*}
\max \big\{\Re f^{(N+1)}(0): f \in H^{\infty}\,, \,
||f||_{\infty} \le 1 \text{ and } f'(z)=0 \text{ for } z \in\mathcal{ C} \big \}
\, .
\end{equation*} \vspace*{-3mm}}
\end{minipage}}
\end{center}
This extremal property of a maximal Blaschke product is reminiscent
of the well--known extremal property of
\begin{itemize}
\item[(i)]
Blaschke products in the Hardy spaces $H^{\infty}$ and
$$H^p:=\left\{ f : {\mathbb{D}} \to {\mathbb{C}} \text{ analytic} \,: {||f||}_p<+ \infty \right\} \, , $$
where $1 \le p < +\infty$ and
$$ {||f||}_p:=\left( \lim
\limits_{r \to 1}
\frac{1}{2\pi} \, \int
\limits_{0}^{2\pi} |f(r e^{it})|^p\,
dt \right)^{1/p} \, ;$$
\end{itemize}
and
\begin{itemize}
\item[(ii)] canonical divisors in the
(weighted) Bergman spaces
$$ \mathcal{ A}_{\alpha}^p= \left\{
f:{\mathbb{D}} \to {\mathbb{C}} \text{ analytic} \,: {||f||}_{p,\alpha}<+ \infty \right\} \, , $$
where $-1<\alpha<+\infty$ and $1 \le p<+\infty$ and
$$ {||f||}_{p,\alpha}:=\left(\frac{1}{\pi} \, \iint \limits_{{\mathbb{D}}} (1-|z|^2)^{\alpha} \, |f(z)|^p\,
d\sigma_z\right)^{1/p} \, .$$
\end{itemize}
Note that in (i) and (ii) the prescribed data are not the critical points, but
the zeros.
\medskip
More precisely, let the sequence $ \mathcal{ C}=(z_j)$ in ${\mathbb{D}}$ be the zero set of
an $H^p$ function and let $N$ be the multiplicity
of the point $0$ in $\mathcal{ C}$. Then the (unique) solution to the extremal problem
\begin{center}
\fbox{\begin{minipage}{11cm}{\begin{equation*}
\max\big\{\Re f^{(N)}(0): f \in H^p,\, ||f||_p \le 1 \text{ and } f(z)=0
\text{ for } z \in\mathcal{ C} \big\}
\end{equation*} \vspace*{-3mm}}
\end{minipage}}
\end{center}
is a Blaschke product $B$ with zero set $\mathcal{ C}$ which is normalized by
$B^{(N)}(0)>0$, see \cite[\S 5.1]{DS}.
In searching for an analogue of Blaschke products for Bergman spaces,
Hedenmalm \cite{Hed1991} (see also \cite{DKSS1993,DKSS1994}) had the idea of posing an
appropriate counterpart of the latter extremal problem for
Bergman spaces.
As before, let $\mathcal{ C}=(z_j)$
be a sequence in ${\mathbb{D}}$ where the point $0$ occurs $N$ times and assume that
$\mathcal{C}$ is the zero set of a function in $\mathcal{A}^p_{\alpha}$.
Then the extremal problem
\begin{center}
\fbox{\begin{minipage}{11cm}{ \begin{equation*}
\max \big\{\Re f^{(N)}(0): f \in \mathcal{
A}_{\alpha}^p\,, \, || f||_{p,\alpha} \le 1 \text{ and } f(z)=0 \text{ for } z \in\mathcal{ C}
\big \}
\end{equation*}\vspace*{-3mm}}
\end{minipage}}
\end{center}
has a unique extremal function $\mathcal{ G} \in \mathcal{A}^p_{\alpha}$, which vanishes
precisely on $\mathcal{ C}$ and is normalized by $\mathcal{ G}^{(N)}(0)>0$.
The function $\mathcal{G}$ is called the canonical divisor for $\mathcal{ C}$.
It plays a prominent r$\hat{\text{o}}$le in the modern theory of Bergman spaces.
\medskip
In summary, we have the following situation:
\smallskip
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|l|c|l|}
\hline
{\bf prescribed data} & {\bf function space} & {\bf extremal function} \\[0mm]\hline
critical set $\mathcal{ C}$ & $H^{\infty}$ & maximal Blaschke product for $\mathcal{ C}$\\[0mm]\hline
zero set $\mathcal{ C}$ & $H^p$ & Blaschke product \\[0mm]\hline
zero set $\mathcal{ C}$ & $\mathcal{ A}^p_{\alpha}$ & canonical divisor for
$\mathcal{ C}$ \\[0mm]
\hline
\end{tabular}
\end{center}
\medskip
In light of this strong analogy,
one expects that maximal Blaschke products enjoy similar properties as finite
Blaschke products and canonical divisors. An example is their analytic continuability. It is
a familiar result that a Blaschke product
has a holomorphic extension across every open arc of $\partial {\mathbb{D}}$ that does not
contain any limit point of its zero set,
see \cite[Chapter II, Theorem 6.1]{Gar2007}.~The same is true for a canonical
divisor in the Bergman spaces $\mathcal{A}^p_0$. This
was proved by
Sundberg \cite{Sun1997} in 1997, who improved earlier work
of Duren, Khavinson, Shapiro and Sundberg \cite{DKSS1993,DKSS1994} and Duren,
Khavinson and Shapiro \cite{DKS1996}.
Now following the model that critical points of maximal functions correspond
to the zeros of Blaschke products and canonical divisors respectively, one
hopes that a maximal Blaschke product has an analytic continuation across every open arc of
$\partial {\mathbb{D}}$ which does not meet any limit point of its critical set. This in fact
turns out to be true:
\begin{theorem}[Analytic continuability, \cite{KR2011}]\label{thm:analytic_continuation}
Let $F: {\mathbb{D}} \to {\mathbb{D}}$ be a maximal Blaschke product with critical set $\mathcal{ C}$. Then $F$ has an
analytic continuation across each arc of $\partial {\mathbb{D}}$ which is free of limit
points of $\mathcal{ C}$.
In particular, the limit points of the critical set of $F$ coincide with the
limit points of the zero set of $F$.
\end{theorem}
Another rather strong property of finite Blaschke products is their semigroup
property with respect to composition. In contrast, the composition of two
infinite Blaschke products does not need to be
a Blaschke product (just consider destructible Blaschke products). However, in
the case of
maximal Blaschke products the following result holds.
\begin{theorem}[Semigroup property, \cite{KR2011}]
The set of maximal Blaschke products is closed under composition.
\end{theorem}
It would be interesting to get some information about the critical \textit{values} of
maximal Blaschke products and to explore the possibility of factorizing
maximal Blaschke products in a way similar to the recent extension of
Ritt's theorem for finite Blaschke products due to Ng and Wang (see \cite{NgWang2011}).
\subsection{Boundary behaviour of maximal Blaschke products}
We now shift attention to the boundary behaviour
of maximal Blaschke products. Ideally, one should be able to determine whether
a bounded analytic function $F : {\mathbb{D}} \to {\mathbb{D}}$
is a maximal Blaschke produkt either from the behaviour of
$$ \frac{|F'(z)|}{1-|F(z)|^2} \quad \text{ as } |z| \to 1$$
or from the behaviour of
$$\int \limits_0^{2\pi} \log \frac{|F'(r e^{it} )|}{1-|F(r e^{it})|^2} \, dt
\quad \text{ as } r \to 1 \, .$$
We only have some partial results in this connection and
we begin our account with the case of finite Blaschke products.
\begin{theorem}[Boundary behaviour of finite Blaschke products] \label{thm:boundary}
Let $I \subset \partial {\mathbb{D}}$ be some open arc and let
$f: {\mathbb{D}} \to {\mathbb{D}}$ be an analytic function
Then the following statements are equivalent.
\begin{itemize}
\item[(a)]
$ \lim \limits_{z \to \zeta} \left( 1-|z|^2 \right)
\displaystyle \frac{|f'(z)|}{1-|f(z)|^2}=1 \qquad \text{ for every }
\zeta \in I \, , $
\item[(b)]$ \lim \limits_{z \to \zeta}
\displaystyle \frac{|f'(z)|}{1-|f(z)|^2}=+\infty \qquad \text{ for every }
\zeta \in I \, , $
\item[(c)]
$f$ has a holomorphic extension across the arc $I$ with $f(I) \subset \partial {\mathbb{D}}$.
\end{itemize}
In particular, if $I=\partial {\mathbb{D}}$, then $f$ is in either case a finite Blaschke product.
\end{theorem}
The equivalence of conditions (a) and (b) in Theorem \ref{thm:boundary}
for the special case $I=\partial {\mathbb{D}}$ is due to Heins \cite{Hei86}; the
general case is proved in \cite{KRR06}.
We now extend Theorem \ref{thm:boundary} beyond the class of finite Blaschke
products and start with the following auxiliary result.
\begin{proposition}[see \cite{Kra2011a}] \label{lem:lemma2}
Let $f : {\mathbb{D}} \to {\mathbb{D}}$ be an analytic function and $I$ some subset of $\partial{\mathbb{D}}$.
\begin{itemize}
\item[(1)]
If
\begin{equation*}
\angle \lim \limits_{z \to \zeta} \left( 1-|z|^2 \right)
\displaystyle \frac{|f'(z)|}{1-|f(z)|^2}=1 \qquad \text{ for every }
\zeta \in I \, ,
\end{equation*}
then $f$ has a finite angular derivative\footnote{see \cite[p.~57]{Sha1993}.} at a.\!\;e.~$\zeta
\in I$. In particular,
\begin{equation*}
\angle \lim \limits_{z \to \zeta} |f(z)|=1 \qquad \text{ for a.\!\;e. } \zeta \in I
\, .
\end{equation*}
\item[(2)]
If $f$ has a finite angular derivative (and $\angle \lim_{z \to \zeta} |f(z)|=1$) at some $\zeta
\in I$, then
\begin{equation*}
\angle \lim \limits_{z \to \zeta} \left( 1-|z|^2 \right)
\displaystyle \frac{|f'(z)|}{1-|f(z)|^2}=1 \, .
\end{equation*}
\end{itemize}
\end{proposition}
In particular, when $I=\partial {\mathbb{D}}$, we obtain the following corollary.
\begin{corollary}[see \cite{Kra2011a}] \label{cor:5}
Let $f : {\mathbb{D}} \to {\mathbb{D}}$ be an analytic function. Then the following statements are
equivalent.
\begin{itemize}
\item[(a)] $\angle \lim \limits_{z \to \zeta} \left( 1-|z|^2 \right)
\displaystyle \frac{|f'(z)|}{1-|f(z)|^2}=1$ for a.\!\;e.~$\zeta \in \partial
{\mathbb{D}}$.
\item[(b)] $f$ is an inner function with finite angular derivative at almost
every point of $\partial {\mathbb{D}}$.
\end{itemize}
\end{corollary}
We further note that
conditions (1) and (2) in Proposition \ref{lem:lemma2} do not complement each
other. Therefore we may ask if an analytic self--map $f$ of ${\mathbb{D}}$ which
satisfies
\begin{equation*}
\angle\lim \limits_{z \to 1} \frac{|f'(z)|}{1-|f(z)|^2}\, (1-|z|^2)=1
\end{equation*}
does have
an angular limit or even a finite angular derivative at
$z=1$; this might then be viewed as a converse of the
Julia--Wolff--Carath\'eodory theorem, see \cite[p.~57]{Sha1993}.
For maximal Blaschke products whose critical sets satisfy the Blaschke
condition one can show that condition (a) in Corollary \ref{cor:5} holds:
\begin{theorem}[see \cite{Kra2010}] \label{prop:1}
Let $\mathcal{ C}=(z_j)$ be a Blaschke sequence in ${\mathbb{D}}$.
\begin{itemize}
\item[(a)]
The maximal conformal pseudometric $\lambda_{max}(z) \, |dz|$
on ${\mathbb{D}}$ with constant curvature $-4$ and zero set
$\mathcal{ C}$ satisfies
\begin{equation*}
\angle \lim \limits_{z \to \zeta} \ \frac{ \lambda_{max}(z)}{\lambda_{{\mathbb{D}}}(z)}=1
\quad \text{ for a.\!\;e. } \zeta \in \partial {\mathbb{D}} \, .
\end{equation*}
\item[(b)] \label{thm:6}
Every maximal function
for $\mathcal{ C}$ has a finite angular derivative at almost
every point of $\partial {\mathbb{D}}$.
\end{itemize}
\end{theorem}
\begin{remark} \label{satz:heinsproof}
The boundary behaviour of a maximal Blaschke product contains
useful information. For instance, it leads to a quick proof of Theorem
\ref{satz:heins}. To see this, let $F: {\mathbb{D}} \to {\mathbb{D}}$
be a maximal function for a finite sequence $\mathcal{ C}$ and $\lambda_{max}(z)\,
|dz|= (F^*\lambda_{{\mathbb{D}}})(z)\, |dz|$ be the maximal conformal metric with constant
curvature $-4$ and zero set $\mathcal{ C}$. By
Theorem \ref{thm:0}, the maximal function $F$ is an indestructible Blaschke product.
Now, let $B$ be a finite Blaschke product with \textit{zero set} $\mathcal{ C}$.
Then
\begin{equation*}
\lambda(z)\, |dz| :=|B(z)| \, \lambda_{{\mathbb{D}}}(z) \, |dz|
\end{equation*}
is a regular conformal pseudometric on ${\mathbb{D}}$ with curvature $-4/ |B(z)|^2\le -4$ and zero set
$\mathcal{ C}$.
Thus, by the maximality of $\lambda_{max}(z) \, |dz|$,
\begin{equation*}
\lambda(z) \le \lambda_{max}(z) \quad \text{ for } z \in {\mathbb{D}}
\end{equation*}
and consequently
\begin{equation}\label{eq:ab1}
|B(z)| \le \frac{\lambda_{max}(z)}{\lambda_{{\mathbb{D}}}(z)} \quad \text{ for all } z \in {\mathbb{D}} \, .
\end{equation}
Since $B$ is a finite Blaschke product, we deduce from (\ref{eq:ab1}) and the
Schwarz--Pick lemma (\ref{eq:sp}) that
\begin{equation*}
\lim_{z \to \zeta}
\frac{\lambda_{max}(z)}{\lambda_{{\mathbb{D}}}(z)}=\lim_{z \to \zeta} \frac{|F'(z)|}{1-|F(z)|^2}\, (1-|z|^2) =1 \quad \text{ for
all } \zeta \in \partial {\mathbb{D}} \, .
\end{equation*}
Applying Theorem \ref{thm:boundary} shows that $F$ is a finite
Blaschke product.
The branching order of $F$ is clearly $2\!\;n$. Thus,
according to the Riemann--Hurwitz formula, see \cite[p.~140]{For1999}, the
Blaschke product $F$ has degree $m=n+1$.
On the other hand, assume that $F$ is a finite Blaschke product of degree
$n+1$. Then, by Theorem \ref{thm:boundary},
$$ \lim_{z \to \zeta} \frac{|F'(z)|}{1-|F(z)|^2}\, (1-|z|^2) =1 \quad \text{ for
all } \zeta \in \partial {\mathbb{D}} \, .$$
Theorem \ref{thm:3_11} below shows that $F$ is a maximal Blaschke product.
\end{remark}
The next result gives a sufficient condition for maximality
of a Blaschke product $F$ in terms
of the boundary behaviour of the \textit{integral means} of the quantity
$$ (1-|z|^2) \frac{|F'(z)|}{1-|F(z)|^2}\,
=\frac{\lambda(z)}{\lambda_{{\mathbb{D}}}(z)} \, . $$
\begin{theorem}[see \cite{Kra2010}] \label{thm:3_11}
Let $\lambda(z)\, |dz|$ be a conformal pseudometric on ${\mathbb{D}}$ with constant curvature $-4$ and zero set $\mathcal{
C}$ such that
\begin{equation}\label{eq:boundary2}
\lim_{ r\to 1} \, \int \limits_0^{2\pi} \log
\frac{\lambda(re^{it})}{\lambda_{{\mathbb{D}}}(re^{it})}\,dt=0\, .
\end{equation}
Then $\lambda(z)\, |dz|$ is the maximal conformal pseudometric $\lambda_{max}(z)\, |dz|$ on ${\mathbb{D}}$
with constant curvature $-4$ and zero set $\mathcal{ C}$.
\end{theorem}
If $\mathcal{ C}$ is a Blaschke sequence, then the corresponding maximal conformal pseudometric $\lambda_{max}(z)\, |dz|$ on ${\mathbb{D}}$
with constant curvature $-4$ and zero set $\mathcal{ C}$ satisfies
condition (\ref{eq:boundary2}):
\begin{theorem}[see \cite{Kra2011a}]
Let $\mathcal{ C}$ be a Blaschke sequence in ${\mathbb{D}}$. A conformal pseudometric
$\lambda(z)\, |dz|$ on ${\mathbb{D}}$ with constant curvature $-4$ and zero set $\mathcal{
C}$ is the maximal conformal pseudometric $\lambda_{max}(z)\, |dz|$ on ${\mathbb{D}}$
with constant curvature $-4$ and zero set $\mathcal{ C}$ if and only if
(\ref{eq:boundary2}) holds.
\end{theorem}
We don't know whether this result is true for any sequence $\mathcal{ C}$ for
which there is a non--constant bounded analytic function with critical set
$\mathcal{ C}$.
\subsection{Heins' results on maximal functions}
For completeness, we close this section with a discussion of Heins' results on maximal functions,
cf.~\cite[\S 25 \& \S 26]{Hei62}. A first observation is that every maximal
function is surjective. In fact more is true; a maximal function is ``locally''
surjective. Here is the precise definition.
\begin{definition}
Let $f: {\mathbb{D}} \to {\mathbb{D}}$ be an analytic function. A point $q \in {\mathbb{D}}$ is called locally
omitted by $f$ provided that either $q \in {\mathbb{D}} \backslash f({\mathbb{D}})$ or else $ q
\in f({\mathbb{D}})$ and there exists a domain $\Omega$, $q \in \Omega$, such that for
some component $U$ of $f^{-1}(\Omega)$ the restriction of $f$ to $U$ omits $q$,
i.\!\;e.~$q \not\in f(U)$.
\end{definition}
\begin{satz}[Heins \cite{Hei62}]\label{thm:omitted_value}
A maximal function has no locally omitted point.
\end{satz}
So far, the results about maximal functions leave an important question
unanswered, namely, how
to tell whether a given function in $H^{\infty}$ is a maximal function? Heins'
second result gives a topological sufficient criterion and
provides therefore a source of ``examples'' of maximal functions. It is based
on the following concept.
\begin{definition}
A holomorphic function $f: {\mathbb{D}} \to {\mathbb{D}}$ is called locally of island type if $f$ is
onto and if for each $w \in {\mathbb{D}}$ there is an open disk $K(w)$ about
$w$ such that each component of $f^{-1}(K(w))$ is compactly contained in ${\mathbb{D}}$.
\end{definition}
Obviously, every surjective analytic self--map of ${\mathbb{D}}$ with constant finite
valence, that is, every finite Blaschke product is
locally of island type.
\begin{satz}[Heins \cite{Hei62}]\label{thm:heins_island}
Every function locally of island type is a maximal function.
\end{satz}
\section{The Gauss curvature PDE and the Berger--Nirenberg problem}
\label{sec:gauss}
We return to a discussion of Theorem \ref{thm:main}. The results of Section
\ref{sec:2} (Theorem \ref{thm:perron} and Theorem \ref{thm:0})
provide a proof of implication ``(a) $\Longrightarrow$ (b)'' in Theorem \ref{thm:main}.
In this section, we discuss implication
``(c) $\Longrightarrow$ (a)''.
The key idea is the following.
\begin{theorem}\label{thm:3}
Let $h : {\mathbb{D}} \to {\mathbb{C}}$ be a non--constant holomorphic function with zero set
$\mathcal{C}$. Then the following statements are
equivalent.
\begin{itemize}
\item[(a)] There exists a holomorphic function $f : {\mathbb{D}} \to {\mathbb{D}}$ with critical
set $\mathcal{C}$.
\item[(b)] There exists a $C^2$--solution $u: {\mathbb{D}} \to {\mathbb{R}}$ to the Gauss curvature equation
\begin{equation}\label{eq:gauss}
\Delta u = 4 \,|h(z)|^2 e^{2u}\, .
\end{equation}
\end{itemize}
\end{theorem}
Let us sketch a proof here. If $f : {\mathbb{D}} \to {\mathbb{D}}$ is a holomorphic function with critical
set $\mathcal{C}$, then a quick computation shows that
$$ u(z):=\log \left( \frac{1}{|h(z)|} \frac{|f'(z)|}{1-|f(z)|^2} \right)$$
is a $C^2$--solution to (\ref{eq:gauss}). This proves ``(a) $\Longrightarrow$ (b)''.
Conversely, if there is a $C^2$--solution $u: {\mathbb{D}} \to {\mathbb{R}}$ to the curvature equation (\ref{eq:gauss}),
then $$\lambda(z)\, |dz |:=e^{u(z)}\, |dz|$$
is a regular conformal metric with
curvature $-4 |h(z)|^2$ on ${\mathbb{D}}$. Hence, by Theorem \ref{rem:liouville},
$$ u(z)=\log \left( \frac{1}{|h(z)|} \frac{|f'(z)|}{1-|f(z)|^2} \right)$$
with some analytic self--map $f$ of ${\mathbb{D}}$.
Thus the zero set of $h$ agrees with the critical set
of $f$.
In view of Theorem \ref{thm:3}, the task is now to characterize those holomorphic functions $h: {\mathbb{D}}
\to {\mathbb{C}}$ for which the PDE (\ref{eq:gauss}) has a solution.
In fact this problem is a special case of the
Berger--Nirenberg problem from differential geometry:
\medskip
{\bf Berger--Nirenberg problem:}\\
{\sl
Given a function $\kappa: R \to {\mathbb{R}}$ on a
Riemann surface $R$. Is there a conformal metric on $R$ with Gauss curvature
$\kappa$?}
\medskip
The Berger--Nirenberg problem is well--understood for the projective plane,
see \cite{Mos1973} and has been extensively studied for compact Riemannian surfaces,
see \cite{Aub1998,Chang2004,Kaz1985,Str2005} as well as for the complex plane
\cite{Avi1986, CN1991, Ni1989}\footnote{These are just
some of the many references.}.
However much less is known for proper domains $D$ of the complex plane,
see \cite{BK1986, HT92, KY1993}. In
this situation the Berger--Nirenberg problem
reduces to the question if for a given function $k:D \to {\mathbb{R}}$ the Gauss curvature equation
\begin{equation}\label{eq:curvature1}
\Delta u= k(z)\, e^{2u}
\end{equation}
has a solution on $D$. We just note that $k$ is the
negative of the curvature $\kappa$ of the conformal metric $e^{u(z)}\, |dz|$.
In the next theorem we give some necessary conditions as well as sufficient conditions
for the solvability of the Gauss curvature equation (\ref{eq:curvature1})
only in terms of the curvature function $k$ and the domain $D$.
\begin{theorem}[see \cite{Kra2011a}]\label{thm:sol1}
Let $D$ be a bounded and regular domain\footnote{i.\!\;e.~there exists
Green's function $g_D$ for $D$ which vanishes
continuously on $\partial D$.} and let $k$ be a nonnegative locally H\"older
continuous function on $D$.
\begin{itemize}
\item[(1)]
If for some (and therefore for every) $z_0 \in D$
\begin{equation*}
\iint \limits_{D} g_D(z_0, \xi)\, k(\xi) \, d\sigma_{\xi} < + \infty\, ,
\end{equation*}
then (\ref{eq:curvature1}) has a $C^2$--solution $u : D \to {\mathbb{R}}$, which is bounded from above.
\item[(2)]
If (\ref{eq:curvature1}) has a $C^2$--solution $u: D \to {\mathbb{R}}$ which is bounded from
below and if this solution has a harmonic majorant on $D$, then
\begin{equation*}
\iint \limits_{D} g_D(z, \xi) \, k(\xi) \, d\sigma_{\xi} < + \infty
\end{equation*}
for all $z \in D$.
\item[(3)]
There exists a bounded $C^2$--solution
$u:D \to {\mathbb{R}}$ to (\ref{eq:curvature1}) if and only if
\begin{equation*}
\sup \limits_{z \in D} \iint \limits_{D} g_D(z, \xi) \, k(\xi)\, d\sigma_{\xi} < + \infty\, .
\end{equation*}
\end{itemize}
\end{theorem}
If we choose $D= {\mathbb{D}}$ and $z_0=0$, then $g_{{\mathbb{D}}}(0,\xi)=-\log|\xi|$. Hence as a consequence of
the inequality
\begin{equation*}
\frac{1-|\xi|^2}{2} \le \log \frac{1}{|\xi|} \le
\frac{1-|\xi|^2}{|\xi|}\, , \quad 0< |\xi| <1,
\end{equation*}
we obtain the following equivalent formulation of Theorem \ref{thm:sol1}.
\begin{corollary}[see \cite{Kra2011a}]\label{cor:solutions1}
Let $k$ be a nonnegative locally H\"older continuous function on ${\mathbb{D}}$.
\begin{itemize}
\item[(1)]
If
\begin{equation} \label{eq:sufcon}
\iint \limits_{{\mathbb{D}}} (1-|\xi|^2 )\, k(\xi) \, d\sigma_{\xi} < + \infty\, ,
\end{equation}
then (\ref{eq:curvature1}) has a $C^2$--solution $u : {\mathbb{D}} \to {\mathbb{R}}$, which is bounded from above.
\item[(2)]
If (\ref{eq:curvature1}) has a $C^2$--solution $u: {\mathbb{D}} \to {\mathbb{R}}$ which is bounded from
below and if this solution has a harmonic majorant on ${\mathbb{D}}$, then
\begin{equation*}
\iint \limits_{{\mathbb{D}}} (1-|\xi|^2 )\, k(\xi) \, d\sigma_{\xi} < + \infty\, .
\end{equation*}
\item[(3)]
There exists a bounded $C^2$--solution $u: {\mathbb{D}} \to {\mathbb{R}}$ to (\ref{eq:curvature1}) if and only if
\begin{equation*}
\sup \limits_{z \in {\mathbb{D}}} \iint \limits_{{\mathbb{D}}} \log\left| \frac{1- \overline{\xi} z}{z -
\xi} \right|\, k(\xi)\, d\sigma_{\xi} < + \infty\, .
\end{equation*}
\end{itemize}
\end{corollary}
\smallskip
Both, Theorem \ref{thm:sol1} and Corollary \ref{cor:solutions1}, are not best
possible, because (\ref{eq:curvature1}) may indeed have solutions, even if
\begin{equation*}
\iint \limits_{D} g_{D}(z, \xi)\, k(\xi) \, d\sigma_{\xi} =+ \infty
\end{equation*}
for some (and therefore for all) $z \in D$.
Here is an explicit example.
\begin{example} \label{ex:0}
For $\alpha \ge 3/2$ define
\begin{equation*}
h(z)= \frac{1}{(z-1)^{\alpha}}
\end{equation*}
for $z \in {\mathbb{D}}$ and set $k(z)=4\, |h(z)|^2$ for $z \in {\mathbb{D}}$.
Then an easy computation yields
\begin{equation*}
\iint \limits_{{\mathbb{D}}} (1-|z|^2 )\, k(z) \, d\sigma_{z}= +\infty
\end{equation*}
and a straightforward check shows that
the function
\begin{equation*}
u_f(z):= \log \left( \frac{1}{|h(z)|} \, \, \frac{|f'(z)|}{1-|f(z)|^2}
\right)
\end{equation*}
is a solution to
(\ref{eq:curvature1}) on ${\mathbb{D}}$ for every locally univalent analytic function $f: {\mathbb{D}} \to {\mathbb{D}}$.
\end{example}
\begin{remark}
The sufficient condition (\ref{eq:sufcon})
improves earlier results of Kalka $\&$ Yang in \cite{KY1993}. In fact, Kalka $\&$
Yang give explicit examples for the function $k$ which tend to $+\infty$ at the
boundary of ${\mathbb{D}}$ such that the existence of a solution to
(\ref{eq:curvature1}) can be
guaranteed. All these examples are radially symmetric and satisfy
(\ref{eq:sufcon}).
Kalka $\&$ Yang also supplement their existence results by nonexistence results.
They find explicit lower bounds for the function $k$ in terms of
radially symmetric functions which grow to $+\infty$ at the boundary of ${\mathbb{D}}$,
such that
(\ref{eq:curvature1}) has no solution.
We wish to emphasize that the necessary conditions and the sufficient conditions
for the solvability of the curvature equation (\ref{eq:curvature1}) of Kalka \& Yang
do not complement each other. In particular, the case when
the function $k$ oscillates is not covered.
For the proof of their nonexistence results
Kalka $\&$ Yang needed to use Yau's celebrated Maximum Principle
\cite{Yau1975, Yau1978}, which is an extremely
powerful tool. In \cite{Kra2010}, an almost elementary proof of these
nonexistence results is given, which has the additional
advantage that Ahlfors' type lemmas for conformal metrics with
variable curvature and explicit formulas for the corresponding maximal
conformal metrics are obtained. In \cite{Kra2010} the nonexistence
theorems of Kalka \& Yang are further extended by allowing the function $k$ to oscillate.
\end{remark}
It turns out that
although condition (\ref{eq:sufcon}) is not necessary for the existence of a
solution to (\ref{eq:curvature1}) it is strong enough to deduce a necessary and
sufficient condition for the
solvability of the Gauss curvature equation of the
particular form (\ref{eq:gauss}):
\begin{theorem}[see \cite{Kra2011a}] \label{thm:k_holo}
Let $h:{\mathbb{D}} \to {\mathbb{C}}$ be a holomorphic function. Then the Gauss curvature equation
(\ref{eq:gauss}) has a solution if and only if $h$ has
a representation as a product of an $\mathcal{ A}_1^2$ function and a nonvanishing
analytic function.
\end{theorem}
Note that Theorem \ref{thm:3} combined with Theorem \ref{thm:k_holo} shows that the class of all holomorphic functions
$h: {\mathbb{D}} \to {\mathbb{C}}$ whose zero sets coincide with the critical sets of the class
of bounded analytic function is exactly the Bergman space $\mathcal{A}^2_1$.
This proves implication ``(c) $\Longrightarrow$ (a)'' in Theorem \ref{thm:main}.
A further remark is that
Theorem \ref{thm:sol1} (c) characterizes those curvature functions $k$
for which (\ref{eq:curvature1}) has at least one
bounded solution. For the case of the unit disk ${\mathbb{D}}$, this result can be
stated as follows.
\begin{theorem}\label{rem:boundedsol}
Let $\varphi: {\mathbb{D}} \to {\mathbb{C}}$ be analytic and $k(z)=4 \, |\varphi'(z)|^2$.
Then there exists a bounded solution to the Gauss curvature equation (\ref{eq:curvature1}) if and only if
$\varphi \in \mathop{BMOA}$, where
\begin{equation*}
\mathop{BMOA}=\left\{ \varphi: {\mathbb{D}} \to {\mathbb{C}} \text{ analytic} \, : \, \sup_{ z \in {\mathbb{D}}} \iint \limits_{{\mathbb{D}}} g_{{\mathbb{D}}}(z,
\xi)\, |\varphi'(\xi)|^2 \, d\sigma_{\xi} < + \infty \right \}
\end{equation*}
is the space of analytic functions of bounded
mean oscillation on ${\mathbb{D}}$, see \cite[p.~314]{BF2008}.
\end{theorem}
Finally, we note that in Theorem \ref{thm:sol1} and Corollary
\ref{cor:solutions1}, condition (1) does not imply condition (3).
The Gauss curvature equation (\ref{eq:curvature1}) may indeed
have solutions none of which is bounded. For example, choose
$\varphi \in H^2\backslash BMOA$ and set $k(z)=4\, |\varphi'(z)|^2$. Then, according to Theorem \ref{rem:boundedsol}, every solution to
(\ref{eq:curvature1}) must be unbounded.
The following result of Heins adds another
item to the list of equivalent statements in Theorem \ref{thm:main}.
\begin{satz}[Heins \cite{Hei62}] \label{satz:heins2}
Let $\mathcal{C}=(z_j)$ be a sequence in ${\mathbb{D}}$. Then the following conditions are equivalent.
\begin{itemize}
\item[(a)] There is an analytic function $f : {\mathbb{D}} \to {\mathbb{D}}$ with critical set $(z_j)$.
\item[(b)] There is a function in the Nevanlinna class $\mathcal{ N}$ with critical set $(z_j)$.
\end{itemize}
\end{satz}
Here, a function $f$ analytic in ${\mathbb{D}}$ is said to belong to the
Nevanlinna class $\mathcal{ N}$ if the integrals $$\int \limits_0^{2\pi} \log^+
|f(re^{it})|\, dt$$ remain bounded as $r \to 1$.
Let us indicate how the results of the present survey allow a quick proof of
Theorem~\ref{satz:heins2}.
\begin{proof}
(b) $\Rightarrow$ (a): Let $\varphi \in \mathcal{ N}$. Then $\varphi=
\varphi_1/\varphi_2$ is the quotient of two
analytic self--maps of ${\mathbb{D}}$, see for instance
\cite[Theorem 2.1]{Dur2000}. W.\!\;l.\!\;o.\!\;g.~we may assume $\varphi_2$ is zerofree. Differentiation of $\varphi$ yields
\begin{equation*}
\varphi'(z)=\frac{1}{\varphi_2(z)^2} \, \big(
\varphi_1'(z) \, \varphi_2 (z)- \varphi_1(z)\, \varphi_2'(z)
\big)\, .
\end{equation*}
Since $\varphi'_1, \varphi_2' \in \mathcal{ A}_1^2$ and $\mathcal{ A}_1^2$ is a vector
space, it follows that the function $\varphi_1' \, \varphi_2 - \varphi_1\, \varphi_2'$
belongs to $\mathcal{ A}_1^2$. Thus Theorem \ref{thm:k_holo} ensures the existence of a
solution $u : {\mathbb{D}} \to {\mathbb{R}}$ to
\begin{equation*}
\Delta u= \left| \varphi'(z) \right|^2 \, e^{2u}\, .
\end{equation*}
Hence Theorem \ref{thm:3} gives the desired result.
\end{proof}
The results of this section about the solvability of the Gauss
curvature equation (\ref{eq:curvature1}) do have further
consequences for the critical sets of bounded analytic functions.
For instance, one can give an answer to a question of Heins \cite[\S
31]{Hei62}.
In order to state Heins' question properly,
we recall the well--known Jensen formula which connects the rate of growth
of an analytic function with the density of its zeros.
Thus the restriction on the growth of the derivative of an analytic self--map
of ${\mathbb{D}}$ imposed by the Schwarz--Pick lemma (\ref{eq:sp}) forces
an upper bound
for the number of critical points of non--constant analytic self--maps of
${\mathbb{D}}$. More precisely,
\begin{equation}\label{eq:anzahl1}
\sup_{ {f \in
H^{\infty}}\atop{||f||_{\infty}\le 1, \, f \not \equiv \text{const.}}} \left( \limsup_{r \to 1} \, \frac{N(r;f')}{\log\frac{1}{1-r^2}} \right)\le 1\, ,
\end{equation}
where
$$ N(r;f'):=\int \limits_0^1
\frac{ n(t;f')}{t} \, dt$$ and
$n(r;f')$ denotes the number of zeros of $f'$ counted with multiplicity in the
disk ${\mathbb{D}}_r:=\{ z \in {\mathbb{C}}\, : |z|<r\}$, $0<r<1$.
\medskip
Heins showed that equality holds in (\ref{eq:anzahl1}), cf.~\cite{Hei61}.
More precisely, using his solution of the
Schwarz--Picard problem, he proved that
there exists for every $p=2,3, \ldots$ a non--constant analytic function $f_p: {\mathbb{D}} \to {\mathbb{D}}$
such that
\begin{equation*}
\limsup_{r \to 1 } \frac{N(r;f_p')}{\log \left(\frac{1}{1-r^2}\right)} =
\frac{2p-3}{2p-2}\, .
\end{equation*}
Letting $p \to + \infty$ shows that (\ref{eq:anzahl1}) is best possible.
In this way, Heins \cite[\S
31]{Hei62} was led to ask whether
there is a bounded analytic function which realizes the supremum in
(\ref{eq:anzahl1}).
This question is answered in our next theorem -- even with some
additional information.
\begin{theorem} \label{thm:heinsi}
Let $\beta \in [0,1]$. Then there exists a non--constant analytic function (and
even a maximal Blaschke product) $f : {\mathbb{D}} \to {\mathbb{D}}$ such that
\begin{equation*} \limsup \limits_{r \to 1} \frac{N(r;f')}{\log \left(\frac{1}{1-r^2}\right)} =
\beta \, .
\end{equation*}
\end{theorem}
For the proof of Theorem \ref{thm:heinsi} we refer the reader to \cite{Kra2010}.
We close with the following remark.
\begin{remark} \label{rem:final}
With the help of the Riemann mapping theorem,
the results of this paper about critical sets of bounded analytic functions $f
: {\mathbb{D}} \to {\mathbb{D}}$ can easily be transferred to the class $H^{\infty}(\Omega)$
of bounded analytic functions $f : \Omega \to {\mathbb{D}}$, when $\Omega{\not=}{\mathbb{C}}$ is
a simply connected domain. The critical sets of bounded analytic functions
on multiply connected domains are much more difficult to fathom.
\end{remark}
\section*{Acknowledgement}
This paper is based on lectures given at the workshop on {\it Blaschke
products and their Applications} (Fields Institute, Toronto, July 25--29, 2011).
The authors would like to thank the organizers of this workshop,
Javad Mashreghi and Emmanuel Fricain,
as well as the Fields Institute and its staff, for their
generous support and hospitality.
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"redpajama_set_name": "RedPajamaArXiv"
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Q: AppDynamics: : Can I run monitors in round robin mode? Does AppDynamics provide similar functionality ?
I was checking out ThousandEyes doc where I can run monitors in round robin mode.
A: In the following contexts your query does not make sense (as reporting is based on usage of the instrumented application):
*
*Application Agents
*Machine Agents
*Database Agents
*Analytics Agents
*MRUM Agents
*BRUM Agents
For Synthetics (which are scheduled by the user) - there is indeed the option to use Round-Robin when configuring a Job - please see screenshot:
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{"url":"http:\/\/math.stackexchange.com\/questions\/235524\/what-are-the-applications-of-differential-geometry-in-robotics","text":"# What are the applications of Differential Geometry in Robotics?\n\nI am taking up a Grad level course on Differential Geometry. Can any one please tell me the immediate applications of Differential Geometry in Robotics ?\n\nThanks\n\n-\n\n## migrated from mathematica.stackexchange.comNov 12 '12 at 7:42\n\nThis question came from our site for users of Mathematica.\n\nhi. I think you meant to ask this in Mathematics forum? This forum is Mathematica (With that extra a at the end). \u2013\u00a0Nasser M. Abbasi Nov 12 '12 at 6:04\nI don't know, but the \"state\" of a robot is probably described by several parameters -- various angles and such, concatenated into a single vector -- and all possible states form a manifold. As the robot moves, its state vector follows a path along this manifold. \u2013\u00a0littleO Nov 12 '12 at 7:49\n\nI have no specialized knowledge in robotics itself, but I believe that the main connection between differential geometry and robotics comes from the powerful applications of differential geometry to mechanics in general. In particular, in its most general form, the mechanical motion of a system in the Hamiltonian formalism can be cast as a flow along a vector field, called a Hamiltonian vector field, on a special type of manifold called a symplectic manifold.\n\nThe first thing you learn in differential geometry is that you can calculate things about your manifold by passing to local coordinates. In a symplectic manifold, the local coordinates used are precisely the position and momentum coordinates, $(p,q)$. If you recall from mechanics, the coordinates $(p,q)$ define the phase space for a mechanic system. If we just have a particle in open space moving around under the influence of some potential, then our phase space is $\\mathbb{R}^{6}$, because we have 3 spatial and 3 momentum coordinates. However, if our system has some constraints, such as some rigid body, then instead of imposing a whole bunch of equations that represent the constraint, it is more natural to simply model the system as motion along some blob (the manifold) in phase space that represents the constraints.\n\n-\n\nThere is some discussion of (a special case of) robotics in the following notes by Bj\u00f8rn Dundas: Differential topology.\n\n-\n\nGenerally, the \"Configuration space\", that is, the collection of all possible configurations of a robot, forms a relatively high dimensional manifold (perhaps with boundary, or corners, or something like that). For example, if you're robot consists solely of an arm and hand, the shoulder joint provides $2$ degrees of freedom, the elbow $1$, the wrist $2$, the thumb $2$, and each of the rest of the fingers $3$ (one for each joint).\n\nIn total, that's a $19$ dimensional manifold.\n\nBeyond this, consider the following question: You start with the robot hand raised in the air as if to say \"hi\", and then have the arm come down to shake hands. What's the best path of each joint individually to get it from \"hi\" to handshaking?\n\nThe answer to this depends on what you mean by \"best\", and mathematically, \"best\" depends on the choice of a Riemannian metric on this manifold. Once you have this metric, you can compute geodesics (at least in principle), which will give you best paths between configurations.\n\n-","date":"2015-11-27 22:49:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7037608027458191, \"perplexity\": 342.0172934528675}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-48\/segments\/1448398450659.10\/warc\/CC-MAIN-20151124205410-00093-ip-10-71-132-137.ec2.internal.warc.gz\"}"}
| null | null |
Washingtonska udruga za pravosuđe ili Washington State Udruga za pravosuđe (engl.: Washington State Association for Justice) je trgovačko društvo za više od 2.200 tužitelja, odvjetnika i djelatnika s uredima u Seattleu, Olimpiji i Spokani.
WSAJ pruža članovima profesionalno umrežavanje, internetski popis poslužitelja, bazu podataka relevantnih sudskih dokumenata i pravnih stručnjaka i imenik članova. Organizacija također pruža opsežan program kontinuiranog pravnog obrazovanja (CLE) na mjestima diljem Washingtona. Osim toga, WSAJ-ov službenik za vladine poslove koji se temelji na Olimpiji lobira zakonodavce i državne agencije da unaprijede zakonski program za civilnu pravdu, čiji je cilj očuvanje i poboljšanje prava oštećenih osoba.
Povijest
Washingtonska udruga za pravosuđe osnovana je 1953. kao Nacionalna udruga tužitelja za naknadu štete (engl.: National Association of Claimants Compensation Attorneys; NACCA); postalo je Udruženje sudskih odvjetnika države Washington (engl.: Washington State Trial Lawyers Association; WSTLA) 1967. i Državna komora za pravosuđe (engl.: Washington State Association for Justice; WSAJ) 2008. godine. Trenutno članstvo WSAJ-a u cijeloj državi obuhvaća više od 2400 odvjetnika i osoblja.
Udruga ima za cilj zaštititi i promicati pravedan pravosudni sustav i pravo na suđenje porota, te osigurati da svaka osoba koja je oštećena zbog lošeg vladanja ili nemara drugih može dobiti pravdu u sudnicama Amerike, čak iu postupcima protiv najmoćnijih interesa.
Misija
Washingtonska udruga za pravosuđe izjavila je svoju misiju:
Članstvo i upravljanje
Predsjednik WSAJ-a 2017. - 2018. je Darrell Cochran. Predsjednik izabran je Ann Rosato. Liz Berry je izvršni direktor WSAJ-a, koji služi kao izvršni direktor organizacije jer 2016. Larry Shannon je direktor vladine poslove, služeći u toj ulozi od 1994. godine.
Organizacijom upravlja Odbor guvernera od 50 članova, kojeg biraju članovi s pravom glasa. Članovi birača moraju biti odvjetnici koji su zastupali tužitelja u najmanje 50 posto njihovih predmeta. Odvjetnički članovi WSAJ-a prakticiraju širok raspon područja pravne prakse, uključujući osobne ozljede, nezakonitu smrt, medicinsku nesavjestanost, kompenzaciju radnika, sudske sporove o osiguranju, zaštitu potrošača, sudske sporove o radu, zlostavljanje domova za njegu i odgovornost za proizvod.
Izvori
Vanjske poveznice
Službena stranica
Washington State Association for Justice na Facebooku
American Association for Justice
Washington (savezna država)
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Q: R to optimize marketing strategy I am new to R and I am trying to run an optimization function.
I have 3 Variables:
$Revenue
$Spent on a tv campaign
$Spent on a radio campaign
How do I optimize my budget allocation across tv and radio campaign?
in other words, how do I find the optimal spend for each individual driver
A: There are 3 characteristics of an optimization problem:
1.Decision Variables -The variables whose value is to be decided in order to achieve our objective
2.Objective Function – linear function of decision variables
3.Constraints – restrictions which cause hinderance in achieving the objective.
I think linear programming is the best way to illustrate this point.
require(lpSolveAPI)
model<-make.lp(ncol=4)
m1<-lp.control(model, sense="max", verbose="neutral")
m2<-set.objfn(model, obj=c(1/250,1/200,1/150,1/100))
m3<-set.bounds(model, lower=c(250000,200000,75000,50000))
m4<-add.constraint(model, c(1,1,0,0), "<=",600000)
m5<-add.constraint(model, c(1,1,1,1), type="<=",1000000)
m6<-add.constraint(model, c((1500/250-500/250),(800/200-500/200),(300/150-500/150),(100/100-500/100)), type=">=",0)
rownames=c("facebook & Adword budget constraint","total budget","LTV")
colnames=c("Adword","Facebook","Email","Affiliated")
dimnames(model)=list(rownames,colnames)
name.lp(model,"Maximize Number Of Users")
write.lp(model, filename="lp_model.lp")
kable(readLines("lp_model.lp"))
solve(model)
get.variables(model)
get.objective(model)
https://analyticsprofile.com/business-analytics/how-to-optimise-digital-marketing-spend-using-linear-programming-in-r/
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\section{Introduction}
When multiple condensates coexist in one system, they can interfere constructively or destructively. This can result in unconventional coherent behavior not present in single-condensate systems~\cite{MP2015, Lin2014, Tanaka2015}. We can mention the multiband crossover from the Bardeen-Cooper-Schrieffer (BCS) regime of loosely bound pairs to the bosonic condensate of molecule-like pairs~\cite{Lubashevsky2012,Okazaki2014, Chen2012}, possible fractional vortices~\cite{Babaev2002, Bluhm2006,Lin2013}, chiral solitons~\cite{Tanaka2001,Tanaka2015}, an enhancement of the intertype superconductivity~\cite{Vagov2016}, the multiband screening of the pair fluctuations~\cite{Salasnich2019, Saraiva2020, Saraiva2021, Shanenko2022} etc. Such phenomena are pronounced when the spatial profiles (spatial scales) of contributing condensates are different. The perception that the physics of systems with multiple condensates must be more complex than the physics of a single condensate is so appealing that there is even a long-discussed idea of a special type of superconductivity for multiband superconductors~\cite{Moshchalkov2009,Kogan2011, Babaev2012, Kogan2012}.
When speaking about the spatial scale of a band-dependent superconducting condensate, one usually means the characteristic length $\xi_{\Delta j}$~(the coherence length) of the gap function $\Delta_j({\bf r})$, where $j$ enumerates overlapping bands. Experimentally, $\xi_{\Delta j}$ can be extracted from the muon spin rotation measurements~\cite{Callaghan2005}, from the scanning tunnelling microscopy (STM) results~\cite{Fente2016, Fente2018}, from the thermal conductivity of the vortex state~\cite{Boaknin2003}, or from the upper critical field~\cite{Gennes1966}. However, one should keep in mind that relations between the gap-function distribution and the experimentally measured characteristics are generally model dependent, which complicates the interpretation of the experimental results. For example, the STM probes the local density of states (DOS) connected with the quasiparticle spatial distribution. The local relation between the zero-energy local DOS and the gap function is well-known only for the diffusive regime near the upper critical magnetic field~\cite{Gennes1964}. At lower fields the situation is significantly more complicated, especially in multiband superconducting materials~\cite{Nakai2002,Koshelev2003,Vargunin2019}.
Different low/high field dependence of the vortex-core size in the two-band material NbSe$_2$ demonstrates~\cite{Callaghan2005} that there are two kinds of localized (in-gap) quasiparticles, which confirms the existence of two condensate lengths $\xi_{\Delta 1}$ and $\xi_{\Delta 2}$, since the vortex core is mainly determined by the in-gap quasiparticles~\cite{Caroli1964, Gygi1991}. However, recent STM results demonstrate that the vortex core of the two-band superconductor CaKFe$_4$As$_4$~\cite{Fente2018} exhibits only one characteristic length, and the same was found for the two-band materials $2$H-NbSe$_{1.8}$S$_{0.2}$ and $2$H-NbS$_2$~\cite{Fente2016}. Theoretical results obtained in Ref.~\onlinecite{Ichioka2017} suggest that these materials can be in the so-called locking regime at which $\xi_{\Delta 1}$ and $\xi_{\Delta 2}$ are nearly the same. Microscopic calculations for clean two-band systems~\cite{Saraiva2017,Chen2020} demonstrate that this regime takes place when the interband (pair-exchange) coupling $g_{12}$ exceeds its locking value $g^*_{12}$. This value depends on the temperature $T$ and the ratio of the band Fermi velocities $v_{F2}/v_{F1}$. One finds~\cite{Saraiva2017,Chen2020} that $g_{12}^*$ drops significantly near $T_c$ and for the ratio $v_{F2}/v_{F1}$ close to $1$. Therefore, the two-band superconductors tend to be in the locking regime when approaching $T_c$ or in the case with nearly the same band Fermi velocities. Here it is worth noting that the ratio of the band Fermi velocities is indeed close to $1$~\cite{Tissen2013} in $2$H-NbS$_2$.
Although researchers in the field of superconductivity are used to associate the spatial scale of a superconducting condensate with the corresponding gap function, the wider readership is more familiar with the Cooper-pair wave function. Therefore, a simple related question may arise whether or not it is more natural to use the characteristic length of the center-of-mass Cooper-pair wave function $\Psi_j({\bf r})$~[the Cooper-pair distribution] to determine the condensate length. In fact, this question is neither naive nor idle. In the case of conventional single-band superconductors, the gap function coincides with the center-of-mass wave function of a Cooper pair up to the Gor'kov coupling constant. It means that the gap-function and wave-function lengths are the same in this case. The situation is not that trivial for multiband superconductors.
In the present work we investigate the gap-function $\xi_{\Delta j}$ and wave-function $\xi_{\Psi j}$ lengths for a two-band superconductor (with $j=1,2$), considered as a prototype of multiband superconductors. The lengths are calculated by numerically solving the $s$-wave microscopic formalism for an isolated vortex in the clean limit. Our study demonstrates that contrary to the single-band case, the difference between $\xi_{\Delta j}$ and $\xi_{\Psi j}$ is negligible only near the critical temperature $T_c$. At lower temperatures $\xi_{\Psi j}$ significantly deviates from $\xi_{\Delta j}$~(this deviation is not pronounced only for $v_{F2}/v_{F1} \simeq 1$) and the fundamental question arises as to which of these lengths should be preferred when specifying the spatial scale of a band condensate in multiband superconducting materials.
\section{Formalism}
\label{sec:For}
\subsection{Gap function and Cooper-pair wave function}
\label{sec:For1}
When the pairing of electrons residing in different bands is negligible (which occurs in most cases), the number of the contributing condensates in a multiband superconductor is equal to the number of bands and the position dependent gap function $\Delta_i({\bf r})$~(for the $s$-wave pairing) is expressed in the form~\cite{Suhl1959,Moskalenko1959,Shanenko2011}
\begin{eqnarray}
\Delta_i({\bf r}) = \sum_j g_{ij} \Psi_j({\bf r}),
\label{Delta}
\end{eqnarray}
where $g_{ij}$ is the coupling matrix and for the the center-of-mass wave function of a Cooper pair~\cite{Bogoliubov1970,Cherny1999} in band $j$ we have
\begin{equation}
\Psi_j({\bf r})=\langle \hat{\psi}_{j\uparrow}({\bf r})\hat{\psi}_{j\downarrow}({\bf r}) \rangle,
\label{Phi}
\end{equation}
which is also referred to as the anomalous Green function of the field operators. If $\Psi_j({\bf r})$ are proportional to the same position-dependent function $\psi({\bf r})$, one can immediately see from Eq.~(\ref{Delta}) that the band gap functions $\Delta_j({\bf r})$ are also exactly proportional to $\psi({\bf r})$. In this case, obviously, the gap-function spatial length $\xi_{\Delta i}$ is equal to the wave-function length $\xi_{\Psi i}$. This is the effectively single-condensate regime when the partial (band) condensates in a multiband system are characterized by the same spatial profile. However, when the contributing condensates are specified by different spatial distributions, one is not able to make any general statement about $\xi_{\Delta i}$ and $\xi_{\Psi i}$ without additional studies.
\subsection{Bogoliubov-de Gennes equations and single vortex solution}
\label{sec:For2}
To clarify the issue about $\xi_{\Delta i}$ and $\xi_{\Psi i}$, we consider the two-band generalization of the BCS model~\cite{Suhl1959,Moskalenko1959} with the $s$-wave pair condensates in both bands governed by the symmetric coupling matrix $g_{ij}\,(i,j=1,2)$. To solve the microscopic formalism, we employ the Bogoliubov-de Gennes (BdG) equations, which for the case of interest are written in the form~\cite{Komendova2012}
\begin{equation}\label{bdg}
\left[
\begin{array}{cc}
\hat{H}_{ei} & \Delta_i({\bf r}) \\
\Delta_i^*({\bf r}) & -\hat{H}^*_{ei}
\end{array}
\right] \left[
\begin{array}{c}
u_{i\nu}({\bf r}) \\
v_{i\nu}({\bf r})
\end{array}
\right] = E_{i\nu}
\left[
\begin{array}{c}
u_{i\nu}({\bf r}) \\
v_{i\nu}({\bf r})
\end{array}
\right],
\end{equation}
where $u_{i\nu}(\bf r)$ and $v_{i\nu}({\bf r})$ are the electron-like and hole-like wave functions associated with band $i$ ($\nu$ is the set of the relevant quantum numbers), $E_{i\nu}$ represents the quasiparticle energies, and $\hat{H}_{ei}$ is the single-particle Hamiltonian (absorbing the chemical potential). For our calculations we assume the effective mass approximation and quasi-2D bands, as emergent multiband superconductors often exhibit quasi-2D Fermi surfaces~\cite{Paglione2010}, so that
$
\hat{H}_{ei}({\bf r})= -\frac{\hbar^2}{2m_i}\big(\partial^2_x + \partial^2_y\big) - \mu_i,
\label{T}
$
where the single-particle energy is degenerate in the $z$ direction, $m_i$ is the electron band mass, $\mu_i=m_iv_{Fi}^2/2$ is the chemical potential measured from the lower edge of the corresponding band, with $v_{Fi}$ the band Fermi velocity. The magnetic field is not included as we consider the system in the deep type II regime, where the magnetic field does not vary within the spatial scales of the contributing condensates and cannot influence their spatial lengths.
In addition, for the sake of simplicity we ignore impurities, considering the clean limit.
The BdG equations are solved in the self-consistent manner, together with Eq.~(\ref{Delta}) and the relation
\begin{equation}
\Psi_j({\bf r}) = \sum_{\nu} u_{j\nu}({\bf r}) v^*_{j\nu}({\bf r})\big[1-2f(E_{j\nu})\big],
\label{self}
\end{equation}
where $f(E_{j\nu})$ is the Fermi-Dirac distribution of bogolons and the summation includes the states with positive quasiparticle energies for which the single-electron energy falls in the Debye window $[\mu_j-\hbar \omega_D,~\mu_j+\hbar\omega_D]$, with $\omega_D$ the Debye frequency. As is seen, the pair-exchange coupling between the two contributing bands $g_{12}=g_{21}$ is not explicitly present in the BdG equations but appears in the self-consistency equation (\ref{Delta}). We remark that to go beyond the adopted model, one should take into account Cooper pairs made of electrons from different bands, see e.g. Ref.~\onlinecite{Shanenko2015, Vargas2020}. In this case the coupling between bands $1$ and $2$ appears explicitly in the BdG equations~\cite{Shanenko2015,Vargas2020}. However, as is mentioned above, in most cases the interband pairing is negligible and we do not consider this point in our present work.
To find $\xi_{\Psi j}$ and $\xi_{\Delta j}$, we adopt an isolated vortex oriented along the $z$ direction~\cite{Gygi1991, Hayashi1998} with
\begin{equation}
\Delta_j({\bf r}) =\Delta_j(\rho)e^{-i\theta},\;\Psi_j({\bf r})=\Psi_j(\rho)e^{-i\theta},
\label{vortex}
\end{equation}
where $\rho,\theta,z$ are the cylindrical coordinates, and $\Delta_j(\rho)$ and $\Psi_j(\rho)$ are the radial parts of the band gap functions and Cooper-pair center-of-mass wave functions, respectively. These expressions are inserted into the BdG equations and then, the equations are converted into the matrix form by means of the expansion in a set of the single-electron wave functions. Further details of the numerical procedure can be found in the Supplemental Material and also in Ref.~\onlinecite{Chen2020}.
In our calculations for band $1$ we choose $\mu_1 = 30$ meV, which is in the range of the Fermi energies in recent multiband superconductors~\cite{Lubashevsky2012}. The chemical potential relative to the lower edge of band $2$~($\mu_2$) is varied in order to consider different values of the ratio $v_{F2}/v_{F1}$. We also use the cut-off energy $\hbar\omega_D = 15$ meV and the dimensionless intraband coupling $g_{11}N_1 = 0.3$, with $N_i$ the density of states (these values are typical of the conventional superconductors~\cite{Fetter}). For band $2$ we take $g_{22}=0.8g_{11}$. The two values $g_{12}=0.3g_{11}$ and $0.9g_{11}$ are considered for the pair-exchange (interband) coupling to check both the weak- and strong-coupling regimes of the intraband interactions. We note that most of multiband superconductors exhibit weak pair-exchange couplings but there are also examples with relatively strong intraband interactions like MgB$_{2}$, see e.g. table II of Ref.~\onlinecite{Vagov2016} and also Refs.~\onlinecite{Golubov2002,Singh2010,Khasanov2010,Kim2011}.
\subsection{Healing lengths}
\label{sec:For3}
It is also of importance to discuss how the gap- and wave-function lengths can be extracted from the numerical solution of the BdG equations for an isolated vortex. There are different practical definitions of $\xi_{\Delta}$ utilized in the single-band case. It can be defined as the size of the vortex core related to the gap function slope~\cite{Bardeen1969, Caroli1964,Schmid1966, Clem1975}, or the radius of the maximal supercurrent density~\cite{Sonier2004}, or the radius of a cylinder containing the energy equal to the condensation energy~\cite{Gennes1966, Tinkham1996}. One can also identify $\xi_{\Delta}$ as the healing length, i.e. the distance over which the gap function can nearly heal, being suppressed in the center of a vortex. In the latter case $\xi_{\Delta}$ can be found from the fact that the radial profile of the gap function is well approximated~\cite{Schmid1966, Clem1975} as $\Delta(\rho)=\Delta_0\rho/\sqrt{\rho^2 + \xi^2_{\Delta}}$, with $\Delta_0$ the bulk value of the gap function.
In our study we employ the healing-length approach, and obtain the characteristic lengths $\xi_{\Delta j}$ and $\xi_{\Psi j}$ by fitting to the numerical results with the approximations
\begin{equation}
\Delta_j(\rho) \simeq \frac{\Delta_{j0}\rho}{\sqrt{\rho^2 + \xi^2_{\Delta j}}}, \; \Psi_j(\rho) \simeq \frac{\Psi_{j0}\rho}{\sqrt{\rho^2 + \xi^2_{\Psi j}}},
\label{app}
\end{equation}
where $\Delta_{j0}$ and $\Psi_{j0}$ are the values of $\Delta_j(\rho)$ and $\Psi_j(\rho)$ far beyond the vortex core. We stress that the approximations of Eq.~(\ref{app}) are not used to solve the BdG equations. Equation (\ref{app}) is employed to get the characteristic lengths from the exact numerical solution of the BdG equations. Notice that nearly the same results for the characteristic lengths can be obtained from the criterion
\begin{equation}
\Delta_j(\rho=\xi_{\Delta j})=\frac{\Delta_{j0}}{\sqrt{2}}, \; \Psi_j(\rho=\xi_{\Psi j})=\frac{\Psi_{j0}}{\sqrt{2}},
\label{app1}
\end{equation}
where the choice of the factor $\sqrt{2}$ is justified by Eq.~(\ref{app}). We also remark that using the healing lengths is not crucial for our results, our conclusions are general and not sensitive to particular definitions of $\xi_{\Delta j}$ and $\xi_{\Psi j}$, see the Supplemental Material.
\section{Results}
\label{sec:Res}
\begin{figure}
\begin{center}
\includegraphics[width=1.0\linewidth]{fig1.pdf}
\end{center}
\caption{a) The ratio $\xi_{\alpha 2}/\xi_{\alpha 1}$~(for $\alpha=\Delta,\Psi$) versus the ratio of the band Fermi velocities $v_{F2}/v_{F1}$ for weakly coupled condensates $g_{12}=0.3g_{11}$. b) $\xi_{\Psi i}/\xi_{\Delta i}$~(for $i=1,2$) as a function of $v_{F2}/v_{F1}$. The lines simply connect the dots and serve as a guide for the eyes.}
\label{fig1}
\end{figure}
\subsection{Numerical results of the BdG equations}
\label{sec:Res1}
Let us first consider results for weakly coupled condensates with $g_{12}=0.3g_{11}$. Figure \ref{fig1} (a) demonstrates the ratios $\xi_{\Delta 2}/\xi_{\Delta 1}$ and $\xi_{\Psi 2}/\xi_{\Psi 1}$ as functions of $v_{F2}/v_{F1}$ at $T=0$. One can see that when $v_{F2}/v_{F1}$ is close to $1$, the system approaches the locking regime so that $\xi_{\Delta 2}/\xi_{\Delta 1}\approx \xi_{\Psi 2}/\xi_{\Psi 1}\approx 1$. When $v_{F2}/v_{F1}$ increases, we obtain different condensate lengths. However, the ratio $\xi_{\Delta 2}/\xi_{\Delta 1}$ becomes notably smaller than $\xi_{\Psi 2}/\xi_{\Psi 1}$. For example, at $v_{F2}/v_{F1}=4$ we have $\xi_{\Delta 2}/\xi_{\Delta 1}=1.6$ while $\xi_{\Psi 2}/\xi_{\Psi 1}=2.4$.
Further insight is provided by Fig.~\ref{fig1}(b), where $\xi_{\Psi 2}/\xi_{\Delta 2}$ and $\xi_{\Psi 1}/\xi_{\Delta 1}$ are given versus $v_{F2}/v_{F1}$. As is seen, for $v_{F2}/v_{F1}=1$ all the four lengths are nearly the same, in agreement with the data given in Fig.~\ref{fig1}(a). When increasing $v_{F2}/v_{F1}$, the wave-function lengths $\xi_{\Psi i}$ systematically deviate from the gap-function lengths $\xi_{\Delta i}$. Moreover, the condensates in bands $1$ and $2$ show opposite trends. For band $2$ we have $\xi_{\Psi 2} > \xi_{\Delta 2}$ and the ratio $\xi_{\Psi 2}/\xi_{\Delta 2}$ exhibits an overall growth when increasing $v_{F2}/v_{F1}$ in Fig.~\ref{fig1}(b). On the contrary, for band $1$ we have $\xi_{\Psi 1} < \xi_{\Delta 1}$ and the ratio $\xi_{\Psi 1}/\xi_{\Delta 1}$ decreases with increasing $v_{F2}/v_{F1}$. One sees that at $v_{F2}/v_{F1}=4$ both $\xi_{\Psi 1}$ and $\xi_{\Psi 2}$ deviate from the corresponding gap-function lengths by about $20-25\%$. Thus, we arrive at the striking conclusion that each condensate in the model of interest is specified by two generally different lengths $\xi_{\Psi i}$ and $\xi_{\Delta i}$.
Our results demonstrated in Fig.~\ref{fig1} are obtained for weakly coupled bands. Now we turn to the case of strongly coupled condensates. Based on the previous studies~\cite{Ichioka2017,Saraiva2017,Chen2020}, we know that the difference between the gap-function lengths $\xi_{\Delta 1}$ and $\xi_{\Delta 2}$ decreases with increasing $g_{12}$ (at a constant ratio $v_{F2}/v_{F1}$). Then, one expects that in the case of strongly coupled bands the gap-function lengths should be close to one another, as the system approaches the locking regime. Here the question arises as for whether or not the wave-function lengths $\xi_{\Psi 1}$ and $\xi_{\Psi 2}$ exhibit the same trend. To answer this question, Fig.~\ref{fig2} demonstrates the data calculated from the BdG equations for different temperatures at $g_{12}=0.9g_{11}$ and for the same intraband coupling constants $g_{11}$ and $g_{22}$ as in Fig.~\ref{fig1}. Here we set $\mu_2/\mu_1=3$, which corresponds to $v_{F2}/v_{F1}=\sqrt{3}$. This moderate ratio of the band Fermi velocities ensures that the characteristic lengths of the band-dependent gap functions are nearly in the locking regime.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{fig2.pdf}
\caption{The same as in Fig.~\ref{fig1} but versus the temperature and for strongly coupled condensates with $g_{12}=0.9g_{11}$ and $v_{F2}/v_{F1}=\sqrt{3}$. Lines simply join points and are a guide to the eyes.
}\label{fig2}
\end{figure}
The temperature dependent quantities $\xi_{\Psi 2}/\xi_{\Psi 1}$ and $\xi_{\Delta 2}/\xi_{\Delta 1}$ are shown in Fig.~\ref{fig2}(a). One sees that the gap-function lengths of bands $1$ and $2$ are indeed almost the same. However, $\xi_{\Psi 2}$ and $\xi_{\Psi 1}$ differ notably from one another at relatively low temperatures. The ratio $\xi_{\Psi 2}/\xi_{\Psi 1}$ approaches $1$ only in the vicinity of the critical temperature $T_c=31 {\rm K}$. At first sight the results for the wave-function lengths given in Fig.~\ref{fig2}(a) looks weird because, as is mentioned in the discussion after Eq.~(\ref{Delta}), the equality $\xi_{\Delta 1}=\xi_{\Delta 2}$ assumes the equal wave-function lengths $\xi_{\Psi 1}=\xi_{\Psi 2}$. In fact, the key point here is that in our case the gap-function lengths $\xi_{\Delta 1}$ and $\xi_{\Delta 2}$ are slightly different: at low temperatures ($< 10\,{\rm K}$) this difference is about one percent~(see the spatial profiles of $\Delta_j(\rho)$ in the Supplemental Material). Strikingly, this almost negligible deviation between the two gap-function lengths results in an order-of-magnitude larger difference between the wave-function lengths $\xi_{\Psi 1}$ and $\xi_{\Psi 2}$.
To have a feeling about this result, let us consider Eq.~(\ref{Delta}) for the simplified case $g_{11}=g_{22}$~(close to our choice). Then, one can find from Eq.~(\ref{Delta}) that $\Delta_2({\bf r}) -\Delta_1({\bf r})= (1-g_{12}/g_{11}) [g_{11}\Psi_2({\bf r}) - g_{11}\Psi_1({\bf r})]$. As $1-g_{12}/g_{11}=0.1$, one can conclude that the difference between the band gaps is an order of magnitude smaller than the difference between $g_{11}\Psi_1$ and $g_{11}\Psi_2$. At the same time $\Delta_2$, $\Delta_1$, $g_{11}\Phi_1$, and $g_{11}\Phi_2$ are of the same order of magnitude. It means that for similar $\xi_{\Delta 1}$ and $\xi_{\Delta 2}$, the characteristic lengths of the spatial variations of $g_{11} \Psi_1({\bf r})$ and $g_{11}\Psi_2({\bf r})$ are very different. Finally, it only remains to take into account that $g_{11}\Psi_1({\bf r})$ and $g_{11}\Psi_2({\bf r})$ have the same spatial lengths as $\Psi_1({\bf r})$ and $\Psi_2({\bf r})$, respectively.
As $\xi_{\Psi 2}/\xi_{\Psi 1}$ differs significantly from $\xi_{\Delta 2}/\xi_{\Delta 1}$, we must have notably different $\xi_{\Psi i}$ and $\xi_{\Delta i}$. For more detailed information, Fig.~\ref{fig2}(b) demonstrates $\xi_{\Psi 1}/\xi_{\Delta 1}$ and $\xi_{\Psi 2}/\xi_{\Delta 2}$ as functions of the temperature, calculated for the same microscopic parameters as the results given in Fig.~\ref{fig2}(a). One sees that near $T_c$ both quantities are close to $1$. However, when the temperature decreases, we find that $\xi_{\Psi 2}$ deviates significantly upward from $\xi_{\Delta 2}$ while $\xi_{\Psi 1}$ becomes smaller then $\xi_{\Delta 1}$. The results in Fig.~\ref{fig2} are obtained for $v_{F2}/v_{F1}=\sqrt{3}$. For a larger difference between $v_{F1}$ and $v_{F2}$, we obtain even larger deviations between the gap- and wave-function lengths of the same band. When $v_{F2}$ is close to $v_{F1}$, the system approaches the length locking regime for the both gap- and wave-function lengths, in agreement with the results for the weakly-coupled bands in Fig.~\ref{fig1}.
\subsection{Local relation between the lengths}
\label{sec:res2}
Our study would not be complete without discussing a simplified analytical relation between the gap- and wave-function lengths which can be obtained from the approximations given by Eq.~(\ref{app}). When using Eq.~(\ref{app}), one finds for $\rho \to 0$
\begin{align}
\Delta_j(\rho) = \rho \frac{\Delta_{j0}}{\xi_{\Delta j}}, \; \Psi_j(\rho) = \rho \frac{\Psi_{j0}}{\xi_{\Psi j}}.
\label{app2}
\end{align}
Inserting Eq.~(\ref{app2}) into Eq.~(\ref{Delta}), one gets
\begin{align}
\frac{\Delta_{i0}}{\xi_{\Delta i}} =\sum\limits_j g_{ij}\frac{\Psi_{j0}}{\xi_{\Psi j}}.
\label{Delta1}
\end{align}
Solving these equation for the two-band case together with Eq.~(\ref{Delta}) taken for $\rho \to \infty$ , one gets
\begin{align}
\xi_{\Psi 1}=\frac{\xi_{\Delta 1} \xi_{\Delta 2}\big(\gamma_{11}\Delta_{10} + \gamma_{12}\Delta_{20}\big)}{\gamma_{11}\Delta_{10}\xi_{\Delta 2}+\gamma_{12}\Delta_{20}\xi_{\Delta 1}},\notag\\
\xi_{\Psi 2}=\frac{\xi_{\Delta 1} \xi_{\Delta 2}\big(\gamma_{21}\Delta_{10} + \gamma_{22}\Delta_{20}\big)}{\gamma_{21}\Delta_{10}\xi_{\Delta 2}+\gamma_{22}\Delta_{20}\xi_{\Delta 1}},
\label{xiDelPsi}
\end{align}
where $\gamma_{ij}$ are elements of the inverse coupling matrix. We stress that Eq.~(\ref{xiDelPsi}) is an approximation. Indeed, one can utilize Eq.~(\ref{Delta}) at any value of $\rho$ to express $\xi_{\Psi i}$ in terms of $\xi_{\Delta i}$ by using Eq.~(\ref{app}). Since the latter is approximative, new expressions for $\xi_{\Psi 1}$ and $\xi_{\Psi 2}$ will be different from Eq.~(\ref{xiDelPsi}). Nevertheless, the difference will not be significant because the fitting to the numerical results for the gap- and wave functions demonstrates that the approximations of Eq.~(\ref{app}) are quite good~(see the Supplemental Material). This makes it possible to find the wave-function lengths from the gap-function lengths, if one, of course, knows the coupling matrix and bulk gaps $\Delta_{j0}$. Once the gap-function lengths have been extracted from the STM measurements, use of Eq.~(\ref{xiDelPsi}) opens the way to the derivation of the wave-function lengths. However, the accuracy of estimating the gap-function lengths from the experimental results should be high enough to get reliable values of the wave-function lengths, see the discussion of our results in Fig.~\ref{fig2}.
\section{Conclusions}
Concluding, we have investigated the characteristic lengths of two coupled condensates in a two-band superconducting material with the $s$-wave pairing in both contributing bands. Our results for the lengths have been extracted from the exact numerical solution of the two-band BdG equations for a single vortex by using the healing-length approach. For each contributing condensate one of the lengths is related to the center-of-mass Cooper pair wave function while another is associated with the corresponding gap function. Though there is no difference between such lengths in the single-band case, our study has demonstrated that for two-band superconductors this is true only in the vicinity of the critical temperature. At lower temperatures the gap- and wave-function lengths systematically deviate from each other, and this deviation is more pronounced with increasing the difference between the band Fermi velocities. In this respect we remark that the ratio of the band Fermi velocities can differ significantly from $1$ in two-band superconductors, for example, $v_{F2}/v_{F1}$ is about~\cite{Suderow2005} $0.05$ in 2H-NbS$_2$ whereas it is close~\cite{Tissen2013} to $18.2$ in 2H-NbSe$_2$.
As is mentioned above, multiple condensates in one system can interfere constructively or destructively, which results in unconventional coherent behavior. Since the interference effects in a multiband system are controlled by the Cooper-pair wave function of the aggregate condensate, one can expect that the wave-function lengths should be preferred when considering the relevant spatial scales of the contributing condensates. However, additional studies, e.g., including possible analytical results within the exact perturbative analysis of the microscopic equations in the vicinity of $T_c$, are certainly necessary to further clarify this fundamental issue of multiband superconductors.
\section*{Acknowledgements}
This work was supported by Zhejiang Provincial Natural Science Foundation (Grant No. LY18A040002) and Science Foundation of Zhejiang Sci-Tech University(ZSTU) (Grant No. 19062463-Y). The work at HSE University (A.A.S.) was financed within the framework of the Basic Research Program of HSE University.
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"redpajama_set_name": "RedPajamaArXiv"
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\section{Preliminaries}
\begin{lemma}
\label{finitefields} Let $\mathbb F_q$ be the finite field with $q$ elements and $\mathbb F_{q^2}$ the degree two extension of $\mathbb F_q$.
Suppose $q$ is odd and $\sqrt{-1} \not\in \mathbb F_q$. Then $\mathbb F_{q^2} = \mathbb F_q(\sqrt{-1})$.
Let $d$ be the maximum integer such that $\mathbb F_{q^2}$ contains a primitive $2^d$-th root of unity $\rho$.
Then $N_{\mathbb F_{q^2}/\mathbb F_q} (\rho) = -1$.
\end{lemma}
\begin{proof} Since $\sqrt{-1} \not\in \mathbb F_q$ and $q$ is odd, we have
$\mathbb F_q^*/\mathbb F_q^{*2} = \{ 1, -1\}$. Since there is a unique extension of degree 2 of $\mathbb F_q$,
$\mathbb F_{q^2} = \mathbb F_q(\sqrt{-1})$. Let $d$ be the maximum integer such that $\mathbb F_{q^2}$
contains a primitive $2^d$-th root of unity $\rho$. Since there is no $2^{d+1}$-th primitive root of unity in
$\mathbb F_{q^2}$, $\rho\not\in \mathbb F_{q^2}^{*2}$. Hence $ \mathbb F_{q^2}^*/ \mathbb F_{q^2}^{*2} = \{ 1, \rho \}$.
Since $N_{\mathbb F_{q^2}/\mathbb F_q} : \mathbb F_{q^2}^* \to \mathbb F_q^*$ is surjective,
$N_{\mathbb F_{q^2}/\mathbb F_q} : \mathbb F_{q^2}^*/ \mathbb F_{q^2}^{*2} \to \mathbb F_q^*/\mathbb F_{q}^{*2}$ is surjective.
Hence $N_{\mathbb F_{q^2}/\mathbb F_q} (\rho) = -1$.
\end{proof}
\begin{cor}
\label{localfields} Let $K_0$ be a local field and $K/K_0$ the quadratic unramified extension.
Suppose that the characteristic of the residue field of $K_0$ is odd and $\sqrt{-1} \not\in K_0$. Then $K = K_0(\sqrt{-1})$.
Let $d$ be the maximum integer such that $K$ contains a primitive $2^d$-th root of unity $\rho$.
Then $N_{K/K_0} (\rho) = -1$.
\end{cor}
\begin{lemma}
\label{finitefields2} Let $\mathbb F_q$ be the finite field with $q$ elements and $\mathbb F_{q^2}$ the degree two extension of $\mathbb F_q$.
Let $m \geq 1$. Suppose $q$ is odd and $\sqrt{-1} \not\in \mathbb F_q$.
If $\mathbb F_{q^2}$ contains a primitive $2^{m+1}$-th root of unity, then $\mathbb F_q^* \subset \mathbb F_{q^2}^{*2^m}$.
\end{lemma}
\begin{proof} Since $\sqrt{-1} \not\in \mathbb F_q^*$, the only $2^m$-th roots of unity in $\mathbb F_q$ are $\pm 1$.
Hence we have an exact sequence of groups $1 \to \{ \pm 1 \}\to \mathbb F_q^* \to \mathbb F_q^{*2^n} \to 1$, where the
last map is given by $x \to x^{2^n}$. Thus the order of $\mathbb F_q^*/\mathbb F_q^{*2^m}$ is 2.
Since $-1 \not\in \mathbb F_q^{*2}$, $-1 \not\in \mathbb F_q^{*2^m}$ and $\mathbb F_q^* = \mathbb F_q^{*2^m} \cup (-1) \mathbb F_q^{*2^m}$.
Since $\mathbb F_{q^2}^*$ contains a primitive $2^{m+1}$-th root of unity, $-1 \in \mathbb F_{q^2}^{*2^m}$.
Thus $\mathbb F_q^* \subset \mathbb F_{q^2}^{*2^m}$.
\end{proof}
\begin{cor}
\label{localfields2} Let $K_0$ be a local field and $K/K_0$ the quadratic unramified extension.
Suppose that the characteristic of the residue field of $K_0$ is odd and $\sqrt{-1} \not\in K_0$.
Let $m \geq 1$. If $K$ contains a primitive $2^{m+1}$-th root of unity, then every unit in the valuation ring of $K_0$
is in $K^{*2^m}$.
\end{cor}
\begin{lemma}
\label{finitefields3}
Let $\mathbb F_q$ be the finite field with $q$ elements.
Let $m \geq 1$ be coprime to $q$.
Suppose that $\mathbb F_q$ does not contain any nontrivial $m^{\rm th}$ root of unity.
Then $\mathbb F_q^* = \mathbb F_q^{*m}$.
\end{lemma}
\begin{proof} Since $ \mathbb F_q^*$ does not contain nontrivial $m^{\rm th}$ root of unity,
the only $m^{\rm th}$ root of unity in $\mathbb F_q$ is $ 1$.
Hence the homomorphism $ \mathbb F_q^* \to \mathbb F_q^{*^m} $, $x \mapsto x^{^m}$, is an isomorphism.
Thus $\mathbb F_q^* = \mathbb F_{q}^{*^m}$.
\end{proof}
\begin{cor}
\label{localfields3}
Let $K_0$ be a local field.
Let $m \geq 1$ be coprime to the characteristic of the residue field of $K_0$.
Suppose that $K_0$ does not contain any nontrivial $m^{\rm th}$ root of unity.
Then every unit in the valuation ring of $K_0$ is in $ K_0^{m}$.
\end{cor}
Let $F$ be a discretely valued field with the valuation ring $R$ and residue field $K$.
We say that an element $a \in F$ is a {\it unit in } $F$ if $a \in R$ is a unit.
Let $n\geq 1$ be an integer coprime to char$(K)$.
Then we have the residue map $\partial : H^d(F, \mu_n^{\otimes i}) \to H^{d-1}(F, \mu_n^{i-1})$.
Let $H^d_{nr}(F, \mu_n^{\otimes i})$ be the kernel of $\partial$.
An element $\alpha \in H^d_{nr}(F, \mu_n^{\otimes i})$ is called an {\it unramified} element.
If $F$ is complete, then we have an isomorphism $H^d(K, \mu_n^{\otimes i}) \simeq H^d_{nr}(F, \mu_n^{\otimes i})$.
We end this section with the following result on reduced norms.
\begin{prop}
\label{reduced-norms}
Let $K$ be a global field with no real orderings and $F$ a complete discretely valued field with residue field $K$.
Let $A$ be a central simple algebra over $F$ of index $n$ coprime to char$(K)$.
Let $ (L, \sigma) \in H^1(K, \mathbb Z/n\mathbb Z)$ be the residue of $A$. Let $\theta \in F^*$ be a unit.
If the image of $\theta \in K^*$ is a norm from the extension $L/K$, then $\theta$ is a reduced norm from
$A$.
\end{prop}
\begin{proof} Let $E/F$ be the unramified extension with residue field $L$ and $\tilde{\sigma}$
a generator of Gal$(E/F)$ lifting $\sigma$. Let $R$ be the valuation ring of $F$ and $\pi \in R$ be a parameter.
Then $A = A_0 + (E, \tilde{\sigma}, \pi)$ for some
central simple algebra $A_0$ over $F$ representing a class in $H^2_{nr}(F, \mu_n)$ (cf. \cite[Lemma 4.1]{PPS}).
Since $F$ is complete and the image of $\theta$ in $K$ is a norm from $L/K$,
$\theta$ is a norm from $E/F$. Hence $(E, \tilde{\sigma}, \pi) \cdot (\theta) = 0 \in H^3(F, \mu_n^{\otimes 2})$.
Since $A_0$ is unramified on $R$ and $\theta$ is a unit, $A_0 \cdot (\theta) \in H^3_{nr}(F,\mu_n^{\otimes 2})$.
Since $K$ is a global field with no real orderings, cd$(K) = 2$ and $H^3(K, \mu_n^{\otimes 2}) = 0$.
Hence $H^3_{nr}(F, \mu_n^{\otimes 2})= 0 $ and $A_0 \cdot (\theta) = 0$.
In particular $A\cdot (\theta) = 0 \in H^3(F, \mu_n^{\otimes 2})$ and by (\cite[Theorem 4.12]{PPS})
$\theta$ is a reduced norm from $A$.
\end{proof}
\section{Dihedral Extensions}
\label{dihedral}
Let $G$ be a dihedral group of order $2m \geq 4$. Let $\sigma$ and $\tau$ be the generators of $G$ with
$\sigma^m = 1$, $\tau^2 = 1$ and $\tau \sigma \tau = \sigma^{-1}$.
The subgroup generated by $\sigma$ is the {\it rotation} subgroup of
$G$ and for $0 \leq i \leq m-1$, $\sigma^i \tau$ are the {\it reflections}.
Let $F_0$ be a field and $E/F_0$ a field extension. We say that $E/F_0$ is a {\it dihedral extension} if
$E/F_0$ is Galois with Galois group isomorphic to a dihedral group. In this section we prove some basic facts about dihedral extensions.
\begin{lemma}
\label{subextndihedral}
Let $F_0$ be a field and $E/F_0$ a dihedral extension. Let $F$ be the fixed of the rotation subgroup of Gal$(E/F_0)$.
If $M/F$ is a sub extension of $E/F$ with $M \neq F$, then $M/F_0$ is a dihedral extension.
\end{lemma}
\begin{proof} Let Gal$(E/F_0)$ be generated by $\sigma$ and $\tau$ with $\sigma^m = 1$, $\tau^2 = 1$ and
$\tau \sigma \tau = \sigma^{-1}$. Then $E/F$ is cyclic with Gal$(E/F)$ generated by $\sigma$.
Let $M/F$ is a sub extension of $E/F$. The extension $M/F$ is cyclic with Gal$(M/F)$ generated by the restriction of $\sigma$ to
$M$. Since $M = E^{\sigma^i}$ for some $i$ and $\tau \sigma^i \tau = \sigma^{-i}$, the extension
$M/F_0$ is Galois with the Gal$(M/F_0)$ generated by the restriction of $\sigma$ and $\tau$ to $M$.
Since $M \neq F$, the restriction of $\sigma$ to $M$ is nontrivial. Since $F \subset M$, the restriction of $\tau$ to $M$ is nontrivial.
Hence $M/F_0$ is dihedral.
\end{proof}
\begin{lemma}
\label{subextndihedral-notgalois} Let $E/F_0$ be a dihedral extension and $F$ the fixed field of the rotation subgroup of
Gal$(E/F_0)$. Let $F_0\subseteq L \subseteq E$ with $F \not\subset L$. If $L/F_0$ is Galois,
then $[L : F_0] \leq 2$.
\end{lemma}
\begin{proof} Suppose that $F \not\subset L$ and $L/F_0$ is Galois. Let $M = FL$.
Suppose that $L \neq F_0$. Then $M \neq F$ and hence $M/F_0$ dihedral (\ref{subextndihedral}).
Since $F \not\subset L$, $[M : F] = [L : F_0]$.
Since $L/F_0$ and $F/F_0$ are Galois extensions, $M/F_0$ is Galois with Gal$(M/F_0)$ isomorphic to Gal$(L/F_0) \times $Gal$(F/F_0)$.
Since the only dihedral group which is isomorphic to a direct product of two nontrivial subgroups is $\mathbb Z/2 \times \mathbb Z/2$, $[L : F_0] = 2$.
\end{proof}
\begin{lemma}
\label{cores1}
Let $F_0$ be a field and $E/F_0$ a dihedral extension of degree $2m$ and $F$ the fixed field of the rotation subgroup of Gal$(E/F_0)$.
Then there exist exactly $m$ subfields $E'$ of $E$ containing $F_0$ with
$[E' : F_0] = [E : F]$ and $E'F = E$.
Further if $E'$ is any such subfield of $E$ and $\ell_1$, $\ell_2$, $\cdots$, $\ell_r$ is any sequence of prime numbers
with $[E : F] = \ell_1 \cdots \ell_r$, then
there exist subfields
$F_0 = L_0 \subset L_1 \subset \cdots \subset L_r = E'$ with $[L_i : L_{i-1}] = \ell_i$.
\end{lemma}
\begin{proof} Let $\sigma$ be a generator of the rotation subgroup of Gal$(E/F_0)$ and
$\tau$ a reflection. For $0 \leq i\leq m-1$ $i$,
let $E_i = E^{\tau\sigma^i}$ the subfield of $E$ fixed by $\tau\sigma^i$.
Then $[E : E_i] = 2$, $[E_i : F_0] = m$ and $E_i F = E$.
Since only elements of order 2 in Gal$(E/F_0)$ which are not identity on $F$ are
the reflections $\tau\sigma^i$, $0 \leq i \leq m-1$, any $E'$ with the given properties
coincides with $E_i$ for some $i$.
Let $E' = E_i$ for some $i$.
Suppose $m = \ell_1 \cdots \ell_r$ with $\ell_j$'s primes.
Since $E/F$ is a cyclic extension, there exist subfields $F = M_0 \subset M_1 \subset \cdots \subset M_r = E$
such that $[M_j : M_{j-1} ] = \ell_i$ for all $i$.
Then $L_j = E' \cap M_j$ have the required property.
\end{proof}
\begin{lemma}
\label{dihedral-norm1} Let $F_0$ be a field and $F/F_0$ a quadratic Galois extension.
Let $m \geq 2$ be coprime to char$(F_0)$. Suppose that $F$ contains a primitive $m^{\rm th}$ root of unity $\rho$.
Let $a \in F_0^*$. Suppose that $[F(\sqrt[m]{a}) : F] = m$.
Then $ F(\sqrt[m]{a})/F_0$ is dihedral if and only if $N_{F/F_0}(\rho) = 1$.
\end{lemma}
\begin{proof} Let $E = F(\sqrt[m]{a})$ and $E' = F_0(\sqrt[m]{a})$. Since $a \in F_0^*$, we have $E = E'F$.
Since $[E: F_0] = 2m$, we have $[E : E'] = 2$. Let $\sigma$ be the automorphism of $F(\sqrt[m]{a})$ given by $\sigma(\sqrt[m]{a}) = \rho \sqrt[m]{a}$ and
$\tau $ the nontrivial automorphism of $E/E'$.
Since $\tau$ is nontrivial on $F$, it follows that $\tau \neq \sigma^i $ for any $i$.
Hence $E/F_0$ is Galois and Gal$(E/F_0)$ is generated by $\sigma$ and $\tau$.
Since the order of $\sigma$ is $m$ and $\tau^2 = 1$, Gal$(E/F_0)$ is dihedral if and only if $\tau \sigma \tau = \sigma^{-1}$.
We have $\tau \sigma \tau (\sqrt[m]{a} ) = \tau \sigma(\sqrt[m]{a}) =
\tau (\rho \sqrt[m]{a}) = \tau(\rho) \sqrt[m]{a} = \tau(\rho) \rho \rho^{-1} \sqrt[m]{a} =
\tau(\rho) \rho \sigma^{-1} (\sqrt[m]{a})$.
Hence $\tau\sigma \tau = \sigma^{-1}$ if and only if $ N_{F/F_0}(\rho ) = \tau(\rho ) \rho = 1$.
\end{proof}
We end this section with the following:
\begin{lemma}
\label{ei-split} Let $F_0$ be a field and $n \geq 2$ an integer with $2n$ coprime to char$(F_0)$.
Let $E/F_0$ be a dihedral extension of degree $2n$ and $\sigma$ and $\tau$ generators on Gal$(E/F_0)$
with $\sigma$ a rotation and $\tau$ a reflection.
Let $F = E^\sigma$ and $E_i = E^{\sigma^i \tau}$ for $1 \leq i \leq n$.
Let $M/F_0$ be a field extension.
Suppose $F \otimes_{F_0} M $ is a field and $E\otimes _{F_0} M$ is isomorphic to
$\prod_1^n (F \otimes_{F_0} M)$.
Then there exists $i$ such that $E_i \otimes_{F_0} M \simeq M \times E'_i$ for some $M$-algebra
$E'_i$.
\end{lemma}
\begin{proof} The proof is by induction on $n$.
Suppose that $n = 2$. Then $F = F_0(\sqrt{a})$, $E_1 = F_0(\sqrt{b})$, $E_2 = F_0(\sqrt{ab})$ and
$E = F(\sqrt{b})$. Suppose that $M(\sqrt{a}) = F \otimes_{F_0}M$ is a field and
$E\otimes_{F_0}M$ is not a field. Then $a$ is not a square in $M$ and $E \otimes _{F_0}M \simeq M(\sqrt{a}) \times M(\sqrt{a})$.
Then either $b$ is a square in $M$ or $ab$ is a square in $M$.
Thus either $E_1 \otimes_{F_0}M \simeq M \times M$ or
$E_2 \otimes_{F_0} \simeq M \times M$.
Suppose $n \geq 3$. Suppose that $M(\sqrt{a}) = F \otimes_{F_0}M$ is a field
and $E \otimes_{F_0} M \simeq \prod_1^n M(\sqrt{a})$.
Suppose $n$ is odd.
Since $E_i \otimes_{F_0} F \simeq E$ and $F/F_0$ is of degree 2,
it follows that $E_i \otimes_{F_0}M \simeq \prod_1^r M \times \prod_1^s M(\sqrt{a})$.
Since $[E_i : F_0] = n$ is odd, $ r \geq 1$.
Suppose that $n$ is even. Then, by (\ref{cores1}), there exists a quadratic extension $F_1/F_0$ contained in
$E$ and $F_1 \neq F$. Let $F' = FF_1$. Then $F'/F_0$ is a biquadratic extension.
Hence there is a quadratic extension $F_2/F_0$ contained in $F'$ with $F \neq F_2$ and $F_1 \neq F_2$.
Further every quadratic extension of $F_0$ contained in $E$ is either $F$ or $F_1$ or $F_2$.
Since every $E_i$ contains a quadratic extension of $F_0$ (\ref{cores1}) and $F \not\subset E_i$,
half of $E_i$ contain $F_1$ and the remaining half of $E_i$ contain $F_2$.
Further $E/F_1$ and $E/F_2$ are dihedral extensions of degree $n$.
Since $E \otimes_{F_0}M \simeq \prod_1^n M(\sqrt{a})$,
$F' \otimes_{F_0}M \simeq M(\sqrt{a}) \times M(\sqrt{a})$. Thus, by the case $n = 2$, either $F_1 \otimes_{F_0} M
\simeq M \times M$ or
$F_2 \otimes_{F_0}M \simeq M \times M $. Without loss of generality, assume that
$F_1 \otimes_{F_0} M
\simeq M \times M$. Then $F_1$ is isomorphic to a subfield of $M$ and hence $M/F_1$ is an extension of fields.
Since $F' = F_1(\sqrt{a})$ and $a$ is not a square in $M$, $F' \otimes_{F_1}M$ is a field.
Since $E\otimes_{F_0} M \simeq E\otimes_{F_1}F_1 \otimes_{F_0}M \simeq E\otimes_{F_1}(M \times M)
\simeq E\otimes_{F_1}M \times E\otimes_{F_1}M$ and
$E\otimes_{F_0}M \simeq \prod_1^n M(\sqrt{a})$, it follows that
$E\otimes_{F_1} M \simeq
\prod_1^{n/2} M(\sqrt{a})$.
Since $E/F_1$ is dihedral and $[E : F_1] < [E : F_0]$, by induction, there exists an $i$ such that $E_i \otimes_{F_1}M \simeq M \times E_i''$ for
some $M$-algebra $E''_i$.
We have $E_i \otimes_{F_0} M \simeq E_i \otimes_{F_1}F_1\otimes _{F_0} M
\simeq E_i \otimes_{F_1} (M \times M) \simeq E_i\otimes_{F_1}M \times E_i\otimes_{F_1}M \simeq M \times E''_i \times E_i \otimes_{F_1}M$. Hence $E_i \otimes_{F_0}M \simeq M \times E_i'$ for some $E'_i$.
\end{proof}
\section{Corestriction of Cyclic Extensions over Quadratic Extensions}
\label{cores}
In this section we realize cyclic extensions over quadratic extensions with corestriction zero as
dihedral extensions.
Let $K$ be a field and $A$ a Galois module over $K$. For $n \geq 0$, let $H^n(K, A)$ denote the $n^{\rm th}$ Galois cohomology group
with values in $A$ (cf. \cite[Ch. VI]{Neukirch}).
For an extension of fields $M/K$, let res$ = $ res$_{M/K} : H^n(K, A) \to H^n(M, A)$ be the restriction homomorphism and
for a finite extension $L/K$, let cores $ = $ cores$_{L/K} : H^n(L, A) \to H^n(K, A)$ be the corestriction homomorphism (cf. \cite[p. 47]{Neukirch}).
Let $F_0$ be a field and $F/F_0$ a Galois extension of degree 2.
Let $\tau_0$ be the nontrivial automorphism of
$F/F_0$. Let $\overline{F}$ be an algebraic closure of $F$.
Let $\tilde{\tau} \in $ Gal$(\overline{F}/F_0)$ be such that $\tilde{\tau}$ restricted to $F$ is $\tau_0$.
Since $\tilde{\tau} \not\in $ Gal$(\overline{F}/F)$ and $[F : F_0] = 2$,
Gal$(\overline{F}/F_0) = $ Gal$(\overline{F}/F) \cup $ Gal$(\overline{F}/F) \tilde{\tau}$
and $\tilde{\tau}^2 \in$ Gal$(\overline{F}/F)$.
Let Hom$_c$(Gal$(\overline{F}/F), \mathbb Z/m\mathbb Z)$ be the group of continuous homomorphisms from
Gal$(\overline{F}/F)$ to $\mathbb Z/m\mathbb Z$ with profinite topology on Gal$(\overline{F}/F)$ and discrete topology on $\mathbb Z/m\mathbb Z$.
Since the action of Gal$(\overline{F}/F)$ on $\mathbb Z/m\mathbb Z$ is trivial, we have
$H^1(F, \mathbb Z/m\mathbb Z) \simeq$ Hom$_c$(Gal$(\overline{F}/F), \mathbb Z/m\mathbb Z)$. The group Hom$_c($Gal$(\overline{F}/F), \mathbb Z/m\mathbb Z)$ also classifies
isomorphism classes of pairs $(E, \sigma)$ with $E/F$ a cyclic extension of degree dividing $m$ and $\sigma$ a generator of Gal$(E/F)$.
\begin{lemma}
\label{cores-cal}
Let $\phi \in $ Hom$_c($Gal$(\overline{F}/F), \mathbb Z/m\mathbb Z)$.
Then cores$(\phi) : $ Gal$(\overline{F}/F_0) \to \mathbb Z/m\mathbb Z$ is the homomorphism given by
cores$(\phi)(\theta) = \phi(\theta) + \phi(\tilde{\tau}\theta \tilde{\tau}^{-1} )$ for all $\theta \in $ Gal$(\overline{F}/F)$
and cores$(\phi)(\tilde{\tau}) = \phi(\tilde{\tau}^2)$.
\end{lemma}
\begin{proof} See \cite[p 53]{Neukirch}.
\end{proof}
\begin{prop}
\label{galois} (cf. \cite[Proposition 24]{HKRT})
Let $F_0$ be a field and $F/F_0$ a quadratic Galois field extension.
Let $E/F$ be a cyclic extension of degree $m$ and $\sigma$ a generator of Gal$(E/F)$.
Then cores$_{F/F_0}(E, \sigma)$ is zero if and only if $E/F_0$ is dihedral extension.
\end{prop}
\begin{proof} Since $E/F$ is a cyclic extension with generator $\sigma$,
we have an isomorphism $\phi_0 : $ Gal$(E/F) \to \mathbb Z/m\mathbb Z$ given by $\phi_0(\sigma^i) \to i \in \mathbb Z/m\mathbb Z$.
Let $\phi : $ Gal$(\overline{F}/F) \to \mathbb Z/m\mathbb Z$ be the composition Gal$(\overline{F}/F) \to $ Gal$(E/F) \buildrel{\phi_0}\over{\to} \mathbb Z/m\mathbb Z$. The pair
$(E, \sigma) $ corresponds to the element $\phi$ in Hom$_c$(Gal$(\overline{F}/F), \mathbb Z/m\mathbb Z)$.
Then cores$(\phi) : $ Gal$(\overline{F}/F_0) \to \mathbb Z/m\mathbb Z$ is the homomorphism given by
cores$(\phi)(\theta) = \phi(\theta) + \phi(\tilde{\tau} \theta \tilde{\tau}^{-1})$ for all $\theta \in $ Gal$(\overline{F}/F)$
and cores$(\phi)(\tilde{\tau}) = \phi(\tilde{\tau}^2)$ (cf. \ref{cores-cal}).
Suppose cores$_{F/F_0}(E, \sigma)$ is the zero homomorphism. Then cores$(\phi) : $ Gal$(\overline{F}/F_0) \to \mathbb Z/m\mathbb Z$ is the zero homomorphism
Let $\theta \in $ Gal$(\overline{F}/F)$.
Then $0 = $ cores$( \phi)(\theta) = \phi(\theta) + \phi(\tilde{\tau}\theta \tilde{\tau}^{-1})$ and hence
$\phi(\tilde{\tau} \theta \tilde{\tau}^{-1}) = - \phi(\theta)$.
Suppose $\theta \in $ Gal$(\overline{F}/E) \subseteq $ Gal$(\overline{F}/F)$.
Since Gal$(\overline{F}/E)$ is the kernel of $\phi$, $\phi(\tilde{\tau} \theta \tilde{\tau}^{-1} ) = - \phi(\theta) = 0$ and hence
$\tilde{\tau}\theta \tilde{\tau}^{-1} \in $ Gal$(\overline{F}/E)$.
Since Gal$(\overline{F}/F_0)$ is generated by Gal$(\overline{F}/F)$ and $\tilde{\tau}$,
Gal$(\overline{F}/E)$ is a normal subgroup of Gal$(\overline{F}/F_0)$. Hence $E/F_0$ is a Galois extension.
Let us denote the restriction of $\tilde{\tau}$ to $E$ by $\tau$.
Since $\tau \sigma \tau^{-1} $ is identity on $F$ and $E/F$ is Galois, $\tau \sigma \tau^{-1} \in $ Gal$(E/F)$.
Let $\tilde{\sigma} \in $ Gal$(\overline{F}/F)$ with restriction to $E$ equal to $\sigma$.
Since $\phi(\tilde{\tau} \tilde{\sigma} \tilde{\tau} ^{-1})= - \phi(\tilde{\sigma}) = \phi(\tilde{\sigma}^{-1}) $, it follows that
$\phi_0(\tau \sigma \tau^{-1} ) = \phi_0(\sigma^{-1})$. Since $\phi_0$ is an isomorphism,
$\tau \sigma \tau^{-1} = \sigma^{-1}$. Since $\phi(\tilde{\tau}^2) = $ cores$(\phi)(\tilde{\tau}) = 0$,
it follows that $\phi_0(\tau^2) = 0$. Since $\phi_0$ is an isomorphism, $\tau^2 $ is the identity on $E$.
Since Gal$(E/F_0)$ is generated by $\sigma$ and $\tau$, with $\sigma^m = 1$, $\tau^2 = 1$ and $\tau \sigma \tau^{-1} = \sigma^{-1}$,
Gal$(E/F_0)$ is a dihedral group of order $2m$.
Conversely, suppose Gal$(E/F_0)$ is a dihedral extension. Since the subgroup of Gal$(E/F_0)$ generated by $\sigma$ is of index 2,
Gal$(E/F_0)$ is generated by $\sigma$ and $\tau$ with $\tau^2 = 1$ and $\tau\sigma\tau^{-1} =\sigma^{-1}$.
Since $\tau \neq \sigma^i$ for all $i$, $\tau$ is not identity on $F$.
Let $\tilde{\tau}$ be an extension of $\tau$ to $\overline{F}$.
Then, we have cores$(\phi)(\theta) = \phi(\theta) + \phi(\tilde{\tau} \theta\tilde{\tau}^{-1})$ for all $\theta \in $ Gal$(\overline{F}/F)$
and cores$(\phi)(\tilde{\tau}) = \phi(\tilde{\tau}^2)$ (\ref{cores-cal}).
Let $\theta \in $ Gal$(\overline{F}/F)$. Since Gal$(E/F)$ is cyclic and generated by $\sigma$,
$\theta$ restricted to $E$ is $\sigma^i$ for some $i$.
Since $\tau \sigma^i \tau^{-1} = \sigma^{-i}$,
$\theta \tilde{\tau}\theta \tilde{\tau}^{-1} \in $ Gal$(\overline{F}/E)$.
Since the kernel of $\phi$ is Gal$(\overline{F}/E)$,
cores$(\phi)(\theta) = \phi(\theta) + \phi(\tilde{\tau} \theta\tilde{\tau}^{-1}) =
\phi( \theta \tilde{\tau} \theta \tilde{\tau}^{-1}) = 0$ for all $\theta \in $ Gal$(\overline{F}/F)$.
Since $\tau^2$ is identity on $E$, $\tilde{\tau}^2 \in $ Gal$(\overline{F}/E)$ and hence
cores$(\phi)(\tilde{\tau}) = \phi(\tilde{\tau}^2)= 0$.
Since Gal$(\overline{F}/F_0)$ is generated by Gal$(\overline{F}/F)$ and $\tilde{\tau}$,
cores$(\phi) = 0$.
\end{proof}
\begin{cor}
\label{cores-field-zero} Let $F_0$ be a field and $F/F_0$ a quadratic Galois extension.
Let $m \geq 2$ be coprime to char$(F_0)$. Suppose that $F$ contains a primitive $m^{\rm th}$ root of unity $\rho$.
Let $a \in F_0^*$. Suppose that $[F(\sqrt[m]{a}) : F] = m$.
Let $\sigma$ be the automorphism of $F(\sqrt[m]{a})$ given by $\sigma(\sqrt[m]{a}) = \rho \sqrt[m]{a}$.
Then cores$(F(\sqrt[m]{a}),\sigma)$ is zero if and only if $N_{F/F_0}(\rho) = 1$.
\end{cor}
\begin{proof}
The lemma follows from (\ref{galois}) and (\ref{dihedral-norm1}).
\end{proof}
\begin{lemma}
\label{norm12}
Let $F_0$ be a field of characteristic not 2 and $F/F_0$ a quadratic extension.
Let $n \geq 1$. Let $\rho$ be a $2^n$-th root of unity in $F$.
Suppose that $\sqrt{-1} \not\in F_0$. Then $N_{F/F_0}(\rho) = \pm 1$.
\end{lemma}
\begin{proof} If $n = 1$, then $\rho = -1$ and hence $N_{F/F_0}(-1) = (-1)^2 = 1$.
Suppose $n \geq 2$.
Let $\tau_0$ be the nontrivial automorphism of $F/F_0$. Since $\rho$ is a ${2^n}$-th root of unity,
$\tau(\rho)$ is also ${2^n}$-th root of unity
and hence $ \rho \tau(\rho) $ is a ${2^n}$-th root of unity in $F_0$.
Since $\pm 1$ are the only $2^n$-th roots of unity in $F_0$, $N_{F/F_0}(\rho) = \rho\tau(\rho) = \pm 1$.
\end{proof}
\begin{cor}
\label{cores-field-zero-2} Let $F_0$ be a field of characteristic not 2 and $F/F_0$ a quadratic extension.
Let $n \geq 2$. Suppose that $F$ contains a primitive $2^n$-th root of unity $\rho$
and $\sqrt{-1} \not\in F_0$.
Let $a \in F_0^*$. Let $1 \leq d \leq n$. Suppose that $F[(\sqrt[2^d]{a}) : F] = 2^d$.
Let $\sigma_d$ be the automorphism of $F(\sqrt[2^d]{a})$ given by $\sigma(\sqrt[2^d]{a}) = \rho^{2^{n-d}} \sqrt[2^d]{a}$.
If $d < n$, then cores$_{F/F_0}(F(\sqrt[2^d]{a}), \sigma_d) $ is zero. Further $N_{F/F_0}(\rho) = 1$ if and only if
cores$(F(\sqrt[2^n]{a}),\sigma_n)$ is zero.
\end{cor}
\begin{proof} Suppose $d < n$.
Then $N_{F/F_0}(\rho^{2^{n-d}}) = N_{F/F_0}(\rho^{2^{n-d-1}})^2 = 1$ (\ref{norm12}) and hence
cores$(E_d, \sigma_d) $ is zero (\ref{cores-field-zero}).
Suppose $n = d$. Then, by (\ref{cores-field-zero}), cores$( F(\sqrt[2^n]{a}), \sigma_n)$ is zero if and only if $N_{F/F_0}(\rho) = 1$.
\end{proof}
\begin{lemma}
\label{cores-field-zero-odd} Let $F_0$ be a field of characteristic not 2 and $F/F_0$ a quadratic extension.
Let $\ell$ be a prime not equal to char$(F_0)$.
Let $n \geq 1$. Suppose that $F$ contains a primitive $\ell^n$-th root of unity $\rho$
and $F_0$ does not contain any nontrivial $\ell^{\rm th}$ root of unity.
Let $a \in F_0^*$. Suppose that $[F(\sqrt[\ell^n]{a}) : F] = \ell^n$.
Let $\sigma$ be the automorphism of $F(\sqrt[\ell^n]{a})$ given by $\sigma(\sqrt[\ell^n]{a}) = \rho \sqrt[\ell^n]{a}$.
Then cores$_{F/F_0}(F(\sqrt[\ell^n]{a}), \sigma) $ is zero.
\end{lemma}
\begin{proof}
Since $\rho^{\ell^n} = 1$, $N_{F/F_0}(\rho)^{\ell^n} = 1$.
Since $F_0$ has no nontrivial $\ell^{\rm th}$ root of unity, $N_{F/F_0}(\rho) = 1$ and by (\ref{cores-field-zero}), cores$(E, \sigma) = 0$.
\end{proof}
\section{Dihedral Extensions Over Local Fields}
\label{dihedral-local}
Let $F_0$ be a complete discrete valued field with residue field $\kappa_0$.
Let $E/F_0$ be a dihedral extension of degree $2m$ with $2m$ coprime to char$(\kappa_0)$.
Let $F \subseteq E$ be the fixed field of the rotation subgroup of Gal$(E/F_0)$.
In this section we first determine the degree of $E/F_0$ if $F/F_0$ is ramified and then we go on
to describe all the dihedral extensions of local fields.
We begin with the following:
\begin{lemma}
\label{dihedral-cyclic}
Let $F_0$ be a complete discrete valued field with residue field $\kappa_0$.
Let $E/F_0$ be a dihedral extension of degree $2m$ with $2m$ coprime to char$(\kappa_0)$.
Suppose the subfield $F$ of $E$ fixed by the rotation subgroup of Gal$(E/F_0)$ is ramified over $F_0$.
Let $L/F_0$ be an extension contained in $E$. If $F \not\subseteq L$ and $L/F_0$ is either unramified or totally ramified,
then $[L : F_0] \leq 2$.
\end{lemma}
\begin{proof}
Let $L/F_0$ be an extension contained in $E$ with $F \not\subseteq L$.
We show that $L/F_0$ is cyclic.
Suppose that $L/F_0$ is unramified.
Let $\kappa$ be the residue field of $F$ and $\kappa'$ the residue field of $L$.
Then $\kappa = \kappa_0$ and $[\kappa' : \kappa_0] = [L : F_0]$.
Since $F/F_0$ is totally ramified, $LF/F$ is an extension of degree $[L : F_0]$ and the residue field of
$LF$ is also $\kappa'$. Since $LF \subset E$ and $E/F$ is cyclic, $LF/F$ is cyclic.
In particular $\kappa'/\kappa_0$ is cyclic. Since $L/F_0$ is unramified and $F_0$ is complete,
$L/F_0$ is cyclic and by (\ref{subextndihedral-notgalois}), $[L : F_0] \leq 2$.
Suppose that $L/F_0$ is totally ramified of degree $d$. Since $d$ is coprime to char$(\kappa_0)$,
$L = F_0(\sqrt[d]{\pi })$ for some parameter $\pi \in F_0$ (cf. \cite[Lemma 2.4]{PPS}).
Since $F \not\subset L$, $[LF : F] = [L :F_0]$.
Since $E/F$ is cylic, $LF/F$ is cyclic.
Since $LF = F(\sqrt[d]{\pi})$ and $[LF : F] = [L : F_0] = d$, $F$ contains a primitive $d^{\rm th}$ root of unity.
Since $F/F_0$ is totally ramified, $F_0$ contains a primitive $d^{\rm th}$ root of unity.
In particular $L/F_0$ is cyclic and by (\ref{subextndihedral-notgalois}), $[L : F_0] \leq 2$.
\end{proof}
\begin{prop}
\label{dihedral-ram}
Let $F_0$ be a complete discrete valued field with residue field $\kappa_0$.
Let $E/F_0$ be a dihedral extension of degree $2m$ with $2m$ coprime to char$(\kappa_0)$.
If the subfield of $E$ fixed by the rotation subgroup of Gal$(E/F_0)$ is ramified over $F_0$, then $[E : F_0] \leq 4$.
\end{prop}
\begin{proof}
Let $F$ be the subfield of $E$ fixed the rotation subgroup of Gal$(E/F_0)$. Then $[F :F_0] = 2$.
Suppose that $F/F_0$ is ramified. Then $F/F_0$ is totally ramified.
Suppose that $[E : F_0] = 2m \geq 5$.
Suppose there is an odd prime $\ell$ dividing $m$.
Then there exists an extension $L/F_0$ of degree $\ell$ such that $L \subset E$ and $F \not\subset L$ (\ref{cores1}).
Since $\ell$ is a prime, $L/F_0$ is either unramified or totally ramified.
Then, by (\ref{dihedral-cyclic}), $[L : F_0] = \ell \leq 2$,
leading to a contradiction.
Suppose there is no odd prime dividing $m$. Then $4$ divides $m$.
Thus there exists an extension $L/F_0$ of degree $4$ such that $L \subset E$ and $F \not\subset L$ (\ref{cores1}).
Since $[L : F_0] = 4$, by (\ref{dihedral-cyclic}), $L/F_0$ is neither totally ramified nor unramified.
Since $[L : F_0] = 4$, $L = F_0(\sqrt{u})(\sqrt{\pi})$
for some $u \in F_0$ a unit and $\pi$ a parameter in $F_0(\sqrt{u})$.
Since $F/F_0$ is ramified, $F = F_0(\sqrt{\pi_1})$ for some parameter $\pi_1$ in $F_0$.
Since $F_0(\sqrt{u})/F_0$ is unramified,
$\pi_1$ is a parameter in $F_0(\sqrt{u})$ and hence $\pi = v \pi_1$ for some unit $v \in F_0(\sqrt{u})$.
Let $L' = F_0(\sqrt{u})(\sqrt{v})$.
Since $[LF : F_0] = 8$ and $LF = F_0(\sqrt{u})(\sqrt{v}, \sqrt{\pi_1})$,
$[L' : F_0] = 4$.
Since $L'/F_0$ is unramified, by (\ref{dihedral-cyclic}), $[L' : F_0] \leq 2$,
leading to a contradiction.
\end{proof}
\begin{cor}
\label{cores-ram}
Let $F_0$ be a complete discrete valued field with residue field $\kappa_0$ of characteristic not 2.
Let $F/F_0$ be a ramified quadratic field extension. Let $E/F$ be a cyclic extension of degree coprime to
char$(\kappa_0)$ and $\sigma$ a generator of
Gal$(E/F)$. If cores$_{F/F_0}(E, \sigma)$ is zero, then $[E : F] \leq 2$.
\end{cor}
\begin{proof} Suppose cores$_{F/F_0}(E, \sigma)$ is zero.
Then $E/F_0$ is Galois with Gal$(E/F_0)$ dihedral (\ref{galois}).
Since $F/F_0$ is ramified, by (\ref{dihedral-ram}), $[E : F] \leq 2$.
\end{proof}
\begin{prop}
\label{totallyram} Let $K_0$ be a local field and $L/K_0$ be dihedral extension of degree $2m$.
Let $K$ be the subfield of $L$ fixed by the rotation subgroup of Gal$(L/K_0)$. If $K/K_0$ is unramified, then
$L/K$ is totally ramified.
\end{prop}
\begin{proof}
Let $L^{nr}$ be the maximal unramified sub extension of $L/K_0$.
Suppose that $K/K_0$ is unramified. Then $K \subseteq L^{nr}$.
Suppose that $K \neq L^{nr}$. Then, by (\ref{subextndihedral}), $L^{nr}/K_0$ is dihedral.
Since $K_0$ is a local field and $L^{nr}/K_0$ is unramified,
$L^{nr}/K_0$ is cyclic. Since a dihedral group can not be cyclic, $L^{nr} = K$.
\end{proof}
\begin{remark}
\label{degree4}
Let $K_0$ be a local field with characteristic of the residue field not 2.
Let $\pi \in K_0$ be a parameter and $u \in K_0$ a unit
which is not a square. Since $K_0^*/K_0^{*2} = \{ 1, \pi, u, u\pi \}$ (cf. \cite[4.1, p. 217]{Sc}),
$L = K_0(\sqrt{u} , \sqrt{\pi})$ is the unique degree four extension with Galois group $\mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$.
Since $\mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$ is the dihedral group of order 4, $L/K_0$ is the unique dihedral extension of degree 4.
\end{remark}
\begin{theorem}
\label{dihedraleven}
Let $K_0$ be a local field with characteristic of the residue field not 2 and $\pi \in K_0$ be a parameter.
Let $d$ be the maximum integer such that $K_0(\sqrt{-1})$ contains a primitive $2^d$-th root of unity.
Then there exists a dihedral extension of $K_0$ of degree $2^{n+1}$ with $n \geq 2$ if and only if $\sqrt{-1} \not\in K_0$ and $n < d$.
In this case $K_0(\sqrt{-1}, \sqrt[2^n]{\pi})$ is the unique dihedral extension of order $2^{n+1}$.
\end{theorem}
\begin{proof} Suppose that $\sqrt{-1} \not\in K_0$ and $2 \leq n < d$.
Let $\pi \in K_0$ be a parameter and $L_n = K_0(\sqrt{-1}, \sqrt[2^n]{\pi})$.
Let $\rho \in K_0(\sqrt{-1})$ be a primitive $2^d$-th root of unity. Then $ \rho' = \rho^{2^{d-n}}$ is a primitive $2^n$-th root of unity and
$N_{K_0(\sqrt{-1})/K_0}(\rho') = (-1)^{2^{n-d}} = 1$ (cf. \ref{localfields}). Hence, by ( \ref{dihedral-norm1}), $L_n/K_0$ is a dihedral extension.
Suppose, conversely, there exists a dihedral extension $L/K_0$ of degree $2^{n+1}$ with $n \geq 2$.
Let $K$ be the subfield of $L$ fixed by the rotation subgroup of Gal$(L/K_0)$.
Since $n \geq 2$,
by (\ref{dihedral-ram}), $K/K_0$ is unramified. By (\ref{totallyram}), $L/K$ is totally ramified.
By (\ref{cores1}), there exists a subfield $L'$ of $L$ with $[L' : K_0] = [L : K]$ and $L'K = L$ .
Since $K/K_0$ is unramified and $L/K$ is totally ramified, $L'/K_0$ is totally ramified.
Since the characteristic of the residue field of $K_0$ is not 2 and $[L': K_0 ] = [L : K] = 2^n$,
$L' = K_0(\sqrt[2^n]{\pi_0})$ for some parameter $\pi_0 \in K_0$. Hence $L = K(\sqrt[2^n]{\pi_0})$.
Suppose that $\sqrt{-1} \in K_0$. Let $L_1 = K_0(\sqrt[4]{\pi_0}) \subset L'$.
Then $L_1/K_0$ is cyclic of degree 4, leading to a contradiction (\ref{subextndihedral-notgalois}).
Hence $\sqrt{-1} \not\in K_0$.
Since $L = K(\sqrt[2^n]{\pi_0})$ is a cyclic extension of $K $ of degree $2^n$,
$K$ contains a primitive $2^n$-th root of unity $\rho$.
Since $n \geq 2$, $\sqrt{-1} \in K$ and hence $K = K_0(\sqrt{-1})$. Thus by the maximality of $d$, $n \leq d$.
Suppose $n = d$.
Since $K_0$ is a local field, by (\ref{localfields}), $N_{K/K_0}(\rho) = -1$.
Hence, by (\ref{dihedral-norm1}), $L/K_0$ is not dihedral, a contradiction.
Therefore $n < d$ and $L \simeq L_n$, proving the uniqueness of dihedral extensions of
degree $2^{n+1}$ over $K_0$.
\end{proof}
\begin{theorem}
\label{dihedralodd}
Let $K_0$ be a local field with characteristic of the residue field not 2 and $\pi \in K_0$ be a parameter.
Let $\ell$ be an odd prime not equal to the characteristic of the residue field of $K_0$.
Let $\rho$ be a primitive $\ell^{\rm th}$ root of unity and $d$ be the maximum integer such that
$K_0(\rho)$ contains a primitive $\ell^d$-th root of unity.
Then there exists a dihedral extension of $K_0$ of degree $2\ell^n$ with $n \geq 1$ if and only if
$[K_0(\rho): K_0] = 2$ and $1 \leq n \leq d$.
In this case $K_0(\rho, \sqrt[\ell^n]{\pi})$ is the unique dihedral extension of order $2\ell^n$.
\end{theorem}
\begin{proof} Suppose $[K_0(\rho): K_0] = 2$ and $1 \leq n \leq d$.
Let $\pi \in K_0$ be a parameter.
Let $L_n = K_0(\rho, \sqrt[\ell^n]{\pi})$.
Let $\rho_n \in K_0(\rho)$ be a primitive $\ell^n$-th root of unity.
Since $N_{K_0(\rho)/K_0}(\rho_n)$ is an $\ell^n$-th root of unity in $K_0$ and
the only $\ell^n$ root of unity in $K_0$ is 1, $N_{K_0(\rho) /K_0}(\rho_n) = 1$.
By (\ref{dihedral-norm1}), $L_n/K_0$ is a dihedral extension.
Suppose, conversely there exists a dihedral extension $L/K_0$ of degree $2\ell^{n}$.
Let $K$ be the subfield of $L$ fixed by the rotation subgroup of Gal$(L/K_0)$.
Since $[L : K ] = \ell^n \geq 3$, by (\ref{dihedral-ram}), $K/K_0$ is unramified.
Then, by (\ref{totallyram}), $L/K$ is totally ramified.
Let $L'$ be a subfield of $L$ with $[L' : K_0] = [L : K]$ and $L'K = L$ (\ref{cores1}).
Since $K/K_0$ is unramified and $L/K$ is totally ramified, $L'/K_0$ is totally ramified.
Since the characteristic of the residue field of $K_0$ is not $\ell$ and $[L : K_0 ] = [L : K] = \ell^n$,
$L' = K_0(\sqrt[\ell^n]{\pi_0})$ for some parameter $\pi_0 \in K_0$. Hence $L = K(\sqrt[\ell^n]{\pi_0})$.
Since $L/K$ is cyclic, $K$ contains a primitive $\ell^{n}$-th root of unity. Thus $n \leq d$.
Since $n \geq 1$, $\rho \in K^*$.
Suppose $[K_0(\rho) : K_0] \neq 2$. Since $K_0(\rho) \subseteq K$ and $[K : K_0] = 2$,
$\rho \in K_0$. Let $L_1 = K_0(\sqrt[\ell]{\pi_0})$. Then $L_1/K_0$ is cyclic.
Since $K \not\subseteq L_1$, by (\ref{subextndihedral-notgalois}), $[L_1 : K_0] = \ell \leq 2$, leading to a contradiction.
Hence $[K_0(\rho) : K_0] = 2$. Since $\rho \in K$ and $[K : K_0] =2$, $K = K_0(\rho)$.
In particular $L\simeq L_n$, proving the uniqueness of dihedral extensions of degree $2\ell^n$ over $K_0$.
\end{proof}
\begin{cor}
\label{dihedraln}
Let $K_0$ be a local field with the residue field $\kappa_0$ and $m \geq 3$ with $2m$ is coprime to char$(\kappa_0)$.
Let $L/K_0$ be an extension of degree $2m$ and $\pi \in K_0$ be a parameter.
Then $L/K_0$ is a dihedral extension if and only if there exists a primitive $ m^{\rm th}$ root of unity $\rho \in L$
with $[K_0(\rho) : K_0] = 2$, $N_{K_0(\rho)/K_0}(\rho) = 1$ and $L = K_0(\rho, \sqrt[m]{\pi})$.
\end{cor}
\begin{proof} Suppose $L/K_0$ is a dihedral extension of degree $2m$.
Suppose $m = 2^n$. Let $d$ be the maximum integer such that $K_0(\sqrt{-1})$ contains a
primitive $2^d$-th root of unity. Then, by (\ref{dihedraleven}), $\sqrt{-1} \not\in K_0$, $n < d$ and
$L = K_0(\sqrt{-1}, \sqrt[2^n]{\pi})$. Let $\rho \in K_0(\sqrt{-1})$ be a primitive $2^n$-th root of unity.
Since $K_0(\rho)/K_0$ is unramified and $K_0(\sqrt{-1})$ is the maximal unramified extension of $L/K_0$,
$K_0(\sqrt{-1}) = K_0(\rho)$. In particular $[K_0(\rho) : K_0] = 2$ and $L = K_0(\rho, \sqrt[2^n]{\pi})$.
Since $L/K_0$ is dihedral, by (\ref{dihedral-norm1}), $N_{K_0(\sqrt{-1})/K_0}(\rho) = 1$.
Assume that there is an odd prime dividing $m$.
Let $m = \ell_0^{n_0}\ell_1^{n_1} \cdots \ell_r^{n_r}$ with $\ell_0 = 2$, for $i \geq 1$,
$\ell_i$ are distinct odd primes, $n_0 \geq 0$ and $n_i \geq 1$ for all $i \geq 1$.
For $0\leq i \leq r$, let $m_i = \frac{m}{\ell_i^{n_i}}$.
Let $\sigma$ be a generator of the rotation subgroup of Gal$(L/K_0)$ and $M_i = L^{\sigma^{m_i}}$.
Then $[M_i : K_0] = 2\ell_i^{n_i}$.
Let $1 \leq i \leq r$.
Then $M_i/K_0$ is a dihedral extension of degree $2\ell_i^{n_i}$ with $\ell_i$ odd.
By (\ref{dihedralodd}), there exists a primitive $ \ell_i^{n_i}$-th root of unity $\rho_i \in M_i$,
$[K_0(\rho_i) : K_0] = 2$ and $M_i = K_0(\rho_i, \sqrt[\ell_i^{n_i}]{\pi})$.
Since $\ell_i$ are distinct primes and $M_i \subseteq L$, $\sqrt[m_0]{\pi} \in L$ and $\rho' = \rho_1 \cdots \rho_r \in L$ is a
primitive $m_0^{\rm th}$ root of unity.
If $n_0 = 0$, then $m_0 = m$. Since $\rho_i \not\in K_0$ for all $i\geq 1$ and $\ell_i$'s are distinct primes, it follows that
$\rho' \not\in K_0$. Since $K_0(\rho')/K_0$ is an unramified extension and $K_0(\sqrt[m]{\pi})/K_0$ is a totally ramified
extension of degree $m$, it follows that $[K_0(\rho') : K_0] = 2$ and $L = K_0(\rho', \sqrt[m]{\pi})$.
By (\ref{dihedral-norm1}), $N_{K_0(\rho')/K_0}(\rho') = 1$.
Suppose $n_0 = 1$. Then $M_0/K_0$ is the unique bi-quadratic extension and hence
$M_0 = K(\sqrt{u}, \sqrt{\pi})$ (cf. \ref{degree4}). Suppose $n_0 \geq 2$. Then, as in the first case,
$M_0$ contains a primitive $2^{n_0}$-th root of unity $\rho_0$, $[K_(\rho_0) : K_0] = 2$
and $M_0 = K_0(\rho_0, \sqrt[2^{n_0}]{\pi})$.
Hence in either case the maximal unramified extension of $M_0/K_0$ is of degree 2 over $K_0$.
Since $M_0 \subseteq E$, $\sqrt[2^{n_0}]{\pi} \in E$. Since $m = 2^{n_0}m_0$, $\sqrt[m]{\pi} \in E$.
Since $K_0(\sqrt[m](\pi))/ K_0$ is a totally ramified extension of degree $ m$, the degree of the maximal
unramified extension of $L/K_0$ is 2. Since $L$ contains a primitive $2^{n_0}$-th root of unity and
$m_0^{\rm th}$ root of unit, $L$ contains a primitive $m^{\rm th}$ root of unity $\rho$. Since $m \geq 3$, either $n_0 \geq 2$
or $m_0 \geq 2$. Hence $[K_0(\rho) : K_0] = 2$ and $L = K_0(\rho, \sqrt[m]{\pi})$. By (\ref{dihedral-norm1}), $N_{K_0(\rho)/K_0}(\rho) = 1$.
Conversely, suppose there exists a primitive $ m^{\rm th}$ root of unity $\rho \in L$
with $[K_0(\rho) : K_0] = 2$, $N_{K/K_0}(\rho) = 1$ and $L = K_0(\rho, \sqrt[m]{\pi})$.
Then, by (\ref{dihedral-norm1}), $L/K_0$ is a dihedral extension.
\end{proof}
We conclude this section with the following result on norms from dihedral extensions over local fields.
\begin{prop}
\label{localfields-norms-n}
Let $K_0$ be a local field and $m \geq 2$ with $2m$ coprime to the characteristic the residue field of $K_0$.
Let $L/K_0$ be a dihedral extension of degree $2m$. Let $K$ be the subfield of $L$ fixed by the rotation subgroup of
Gal$(L/K_0)$. Let $L_0, \cdots, L_{m-1}$ be the subfields of $L$ with
$L_iK = L$ and $[L : L_i] = 2$ (\ref{cores1}).
Let $ \theta_0 \in K_0^*$.
Then for every $0 \leq i \leq m-1$,
there exists $\mu_i \in L_i$,
such that $\prod_{i= 0}^{m-1} N_{L_i/K_0}(\mu_i) = \theta_0$.
\end{prop}
\begin{proof} Suppose $m = 2$. Then $L/K_0$ is a biquadratic extension,
$L_0$ and $L_1$ are non isomorphic quadratic extensions of $K_0$.
Then, by local class field theory (cf. \cite[Proposition 3, p.142]{CFANT}), $N_{L_0/K_0}(L_0^*)$ and $N_{L_1/K_0}(L_1^*)$
are two distinct subgroups of $K_0^*$ of index 2.
Let $b \in N_{L_0/K_0}(L_0^*)$ which is not in $N_{L_1/K_0}(L_1^*)$.
Let $a \in K_0^*$. Suppose $a \not\in N_{L_1/K_0}(L_1^*)$.
Then $a \in b N_{L_1/K_0}(L_1^*)$. Hence $a = bc$ for some $c \in N_{L_1/K_0}(L_1^*)$.
In particular $a \in N_{L_0/K_0}(L_0^*) N_{L_1/K_0}(L_1^*)$.
Suppose $m \geq 3$. Let $\rho$ be a primitive $m^{\rm th}$ root of unity. Then, for any parameter $ \pi \in K_0$,
by (\ref{dihedraln}), $ L = K_0(\rho, \sqrt[m]{\pi})$. Let $\pi \in K_0$ be a parameter.
Since $L = K_0(\rho, \sqrt{\pi})$ $[L : K_0(\sqrt[m]{\pi})] = 2$ and $K = K_0(\rho)$,
$K_0(\sqrt[m]{\pi}) = L_r$ for some $r$. In particular $(-1)^m\pi$ is a norm from the extension $L_r/K_0$.
Let $u \in K_0$ be a unit. Since $u\pi$ is a parameter in $K_0$, $\sqrt[m]{u\pi} \in L$ and
$K_0(\sqrt[m]{u\pi}) = L_s$ for some $s$. Hence $(-1)^mu\pi$ is a norm from the extension $L_s/K_0$.
In particular $u$ is a product of norms from the extensions $L_r/K_0$ and $L_s/K_0$.
Since every element in $K_0$ is $u \pi^r$ for some $u\in K_0$ a unit, it follows that
every element in $K_0$ is a product of norms from the extensions $L_i/K_0$.
\end{proof}
\section{Approximation of norms from dihedral extensions over global fields}
\label{norms-globalfields}
\begin{prop}
\label{product-of-norms} Let $K_0$ be a global field and $n \geq 2$ an integer with $2n$ coprime to char$(K_0)$.
Let $E/K_0$ be a dihedral extension of degree $2n$ and $\sigma$ and $\tau$ generators of Gal$(E/K_0)$
with $\sigma$ a rotation and $\tau$ a reflection. Let $K = E^\sigma$ and $E_i = E^{\sigma^i \tau}$ for $1 \leq i \leq n$.
Let $\nu$ be a place of $K_0$ and $\lambda_\nu \in K_{0\nu}$.
Suppose that the characteristic of the residue field at $\nu$ is coprime to $2n$.
If $\lambda_\nu$ is a norm from the extension $E \otimes_{K_0}K_{0\nu}/K \otimes_{K_0}K_{0\nu}$,
then $\lambda_\nu$ is a product of norms from the extensions $E_i \otimes K_{0\nu}/K_{0\nu}$.
\end{prop}
\begin{proof} Suppose $\lambda_\nu$ is a norm from the extension $E \otimes_{K_0}K_{0\nu}/K \otimes_{K_0}K_{0\nu}$.
Suppose that $K\otimes_ {K_0} K_{0\nu}$ is not a field. Then $K\otimes_{K_0} K_{0\nu} \simeq K_{0\nu} \times K_{0\nu}$.
Since $KE_i = E$, $E \otimes_{K_0} K_{0\nu} \simeq E_i \otimes_{K_0} K \otimes_{K_0}K_{0\nu} \simeq E_i \otimes _{K_0} K_{0\nu}
\times E_i\otimes_{K_0}K_{0\nu}$. Since $\lambda_\nu$ is a norm from the extension $E \otimes_{K_0}K_{0\nu}/K \otimes_{K_0}K_{0\nu}$, it follows that $\lambda_\nu$ is a norm from $E_i \otimes_{K_0} K_{0\nu}/K_{0\nu}$.
Suppose $K\otimes_{K_0}K_{0\nu}$ is a field.
Suppose $E\otimes_{K_0} K_{0\nu} \simeq \prod_1^n K\otimes_{K_0} K_{0\nu}$.
Then, by (\ref{ei-split}), there exists an $i$ such that
$E_i \otimes_{K_0} K_{0\nu} \simeq K_{0\nu} \times E'_{i\nu}$ for some $K_{0\nu}$-algebra
$E'_{i\nu}$. In particular $\lambda_\nu$ is norm from $E_i \otimes_{K_0} K_{0\nu}/K_{0\nu}$.
Suppose $E\otimes_{K_0} K_{0\nu}$ is not isomorphic to $\prod_1^n K\otimes_{K_0} K_{0\nu}$.
Since $E/K_0$ is Galois, $E\otimes_{K_0}K_{0\nu} \simeq \prod E_\nu$ for some field extension $E_\nu/K_{0\nu}$
and $K\otimes_{K_0}K_{0\nu}$ is a proper subfield of $E_\nu$.
Hence $E_\nu/K_{0\nu}$ is a dihedral extension. Since the characteristic of the residue field at $\nu$ is coprime to $2n$,
by (\ref{localfields-norms-n}), $\lambda_\nu$ is a product of norms from
$E_i \otimes_{K_0} K_{0\nu}/K_{0\nu}$.
\end{proof}
\begin{cor}
\label{norms-approximation}
Let $K_0$, $E$ and $K$ be as in (\ref{product-of-norms}).
Let $S$ be a finite set of places of $K_0$ with $2n$ coprime to the characteristic of the residue field at places in $S$.
For $\nu \in S$, let $\lambda_\nu \in K_{0\nu}$ be a
norm from the extension $E \otimes_{K_0}K_{0\nu}/K \otimes_{K_0}K_{0\nu}$.
Then there exists $\lambda \in K_0$ such that $\lambda$ is a norm from the extension $E/K$ and
$\lambda \lambda_\nu^{-1} \in K\otimes_{K_0}K_{0\nu}^{*n}$ for all $\nu \in S$.
\end{cor}
\begin{proof} Let $\sigma, \tau \in $ Gal$(E/K_0)$ be as in (\ref{product-of-norms}).
Let $E_i = E^{\sigma^i \tau}$ for $1 \leq i \leq n$. Let $\nu \in S$. Then, by (\ref{product-of-norms}),
for $1 \leq i \leq n$, there exists $z_{i\nu} \in E_i \otimes_{K_0}K_{0\nu}$ such that
$\lambda_\nu = \prod N_{E_i \otimes_{K_0}K_{0\nu}/K_{0\nu}}(z_{i\nu})$.
For $1 \leq i \leq n$, let $z_i \in E_i$ be close to $z_{i\nu}$ for all $\nu \in S$.
Let $\lambda = \prod N_{E_i/K_0}(z_i)$.
Since $z_i$ is close to $z_{i\nu}$ for all $\nu \in S$,
$\lambda$ is close to $\lambda_\nu$ for all $\nu \in S$. In particular
$\lambda\lambda_\nu^{-1} \in (K\otimes_{K_0}K_{0\nu})^{*n}$.
Since $KE_i = E$, $\lambda$ is a norm from the extension $E/K$.
\end{proof}
\section{Central Simple Algebras With Involutions Of Second Kind Over 2-local fields}
\label{alg-2-local}
In this section we give a description of central simples algebras having involutions of second kind over complete discretely
valued fields with residue fields local fields (such fields are called 2-local fields).
We begin the following well known
\begin{lemma}
\label{e=f} Let $F$ be a complete discretely valued field and $\pi \in F$ a parameter.
Let $E/F$ be a cyclic extension and $\sigma$ a generator of Gal$(E/F)$.
Then the cyclic algebra $(E, \sigma, \pi)$ is unramified if and only if $E = F$.
\end{lemma}
\begin{proof} Let $ m = [E : F]$. Since $E/F$ is unramified, the order of the class of $\pi$ in $F^*/N_{E/F}(E^*)$
is $m$ and hence $D = (E, \sigma, \pi)$ is a division algebra of degree $m$ (\cite[Theorem 6, p. 95]{Albert}).
Let $\nu$ be the discrete valuation on $F$ and $\tilde{\nu}$ be the extension of $\nu$ to $D$ (\cite[Theorem 12.10, p. 138]{reiner}).
Let $e$ be the ramification index of $D$.
Since there exists $y \in D$ with $y^m = \pi$, we have $\tilde{\nu}(\pi) \geq m$ and hence $e = m$ (\cite[Theorem 13.7, p. 142]{reiner}).
Suppose $D$ is unramified. Then $e = 1$ and hence $m = 1$. In particular
$E = F$.
\end{proof}
\begin{lemma}
\label{ramified2}
Let $F_0$ be a complete discrete valued field with residue field of characteristic not 2.
Let $\pi \in F_0$ be a parameter and $F = F_0(\sqrt{\pi})$.
Let $E/F$ be an unramified cyclic extension and $\sigma$ a generator of Gal$(E/F)$.
If cores$_{F/F_0}(E, \sigma, \sqrt{\pi})$ is unramified, then $(E, \sigma, \sqrt{\pi})$ is zero.
\end{lemma}
\begin{proof} Let $E_0$ be the maximal unramified extension of $E/F_0$.
Since $E/F$ is unramified and $F/F_0$ is ramified extension of degree 2,
$E/E_0$ is of degree 2 and $E = E_0F$. Since $F/F_0$ is ramified, $E_0/F_0$ is unramified.
Since $E/F$ is cyclic, $E_0/F_0$ is cyclic (cf. proof of \ref{dihedral-cyclic}). Let $\sigma_0$ be the restriction of $\sigma$ to $E_0$.
Then $(E_0, \sigma_0) \otimes F = (E, \sigma)$. Hence cores$_{F/F_0}(E, \sigma, \sqrt{\pi}) =
(E_0, \sigma_0, N_{F/F_0}(\sqrt{\pi})) = (E_0, \sigma_0, -\pi)$ (\cite[Proposition 1.5.3]{Neukirch}).
Suppose that cores$_{F/F_0}(E, \sigma, \sqrt{\pi}) = (E_0, \sigma_0, -\pi)$ is unramified.
Since $\pi$ is a parameter in $F_0$ and $E_0/F_0$ is unramified, by (\ref{e=f}),
$E_0 = F_0$. In particular $E = F$ and $(E, \sigma, \sqrt{\pi}) $ is zero.
\end{proof}
\begin{lemma}
\label{ramifiedh2}
Let $F_0$ be a complete discrete valued field with residue field $K_0$
and $F/F_0$ a ramified quadratic field extension. Let $m \geq 1$ with $2m$ coprime to char$(K_0)$
and $\alpha \in H^2(F, \mu_m)$.
If cores$_{F/F_0}(\alpha)$ is zero, then $\alpha = \alpha_0 \otimes F$ for some
$\alpha_0 \in H^2_{nr}(F_0, \mu_m)$.
In particular per$(\alpha) \leq 2$.
\end{lemma}
\begin{proof}
Since $F/F_0$ is a ramified quadratic extension and char$(K_0) \neq 2$,
$F =F_0(\sqrt{\pi})$ for some $\pi \in F_0$ a parameter.
Since $m$ is coprime to char$(K_0)$,
$\alpha = \alpha' + (E, \sigma, \sqrt{\pi})$ for some $\alpha' \in H^2_{nr}(F, \mu_m)$ and
$E/F$ an unramified cyclic field extension of $F$ (cf. \cite[Lemma 4.1]{PPS}). Since cores$_{F/F_0}(\alpha) = 0$, we have
cores$_{F/F_0}(-\alpha' ) = $ cores$_{F/{F_0}}(E, \sigma, \sqrt{\pi}) $.
Since $\alpha'$ is unramified, cores$_{F/F_0}(-\alpha')$ is unramified (cf. \cite[p. 48]{CTSB}) and hence cores$_{F/{F_0}}(E, \sigma, \sqrt{\pi}) $ is unramified.
Thus, by (\ref{ramified2}), $(E, \sigma, \sqrt{\pi})$ is zero and hence $\alpha = \alpha'$.
Since the residue field of $F$ and $F_0$ are equal and both $F$ and $F_0$ are complete,
it follows that $\alpha = \alpha' = \alpha_0 \otimes F$ for some $\alpha_0 \in H^2_{nr}(F_0, \mu_m)$.
\end{proof}
\begin{lemma}
\label{unramified}
Let $F_0$ be a complete discrete valued field and $F/F_0$ an unramified quadratic extension. Let $\pi \in F_0$ be a parameter
and $m \geq 1$. Suppose $2m$ is coprime to the characteristic of the residue field of $F_0$.
Let $\alpha = \alpha' + (E, \sigma, \pi) \in H^2(F, \mu_m)$ for some $\alpha' \in H^2_{nr}(F, \mu_m)$ and
$E/F$ an unramified cyclic field extension.
If cores$_{F/F_0}(\alpha)$ is zero, then cores$_{F/F_0}(\alpha')$ and cores$_{F/F_0}(E, \sigma, \pi)$ are zero.
\end{lemma}
\begin{proof} Since $F/F_0$ is unramified extension, $\pi $ is a parameter in $F$.
Since $\pi \in F_0$, we have cores$_{F/F_0}(E, \sigma, \pi) = $ cores$_{F/F_0}(E, \sigma) \cdot (\pi)$.
Since cores$_{F/F_0}( \alpha) = $ cores$_{F/F_0}(\alpha') + $ cores$_{F/F_0}(E, \sigma, \pi) $ is zero,
we have cores$_{F/F_0}(E, \sigma) \cdot (\pi) = - $cores$_{F/F_0}(\alpha')$.
Since $\alpha'$ is unramified, cores$_{F/F_0}(\alpha')$ is unramified.
Since $E/F$ is unramified, cores$_{F/F_0}(E,\sigma)$ is unramified.
Since cores$_{F/F_0}(E, \sigma) \cdot (\pi)$ is unramified and $F_0$ is complete, cores$_{F/F_0}(E, \sigma)$ is zero (\ref{e=f}).
Hence cores$_{F/F_0}(E, \sigma, \pi) = $ cores$_{F/F_0}(E, \sigma) \cdot (\pi)$ is zero and
in particular cores$_{F/F_0}(\alpha')$ is zero.
\end{proof}
\begin{lemma}
\label{branch-alg}
Let $F_0$ be a complete discrete valued field with residue field $K_0$ a local field,
$F/F_0$ a quadratic field extension and $\pi \in F_0$ a parameter.
Let $m \geq 1$ with $2m$ coprime to char$(K_0)$.
Let $\alpha \in H^2(F, \mu_m)$ with cores$_{F/F_0}(\alpha) = 0$. If ind$(\alpha)\geq 3$, then $F/F_0$ is unramified and $\alpha = (E, \sigma, \pi)$ for some unramified cyclic extension $E/F$.
\end{lemma}
\begin{proof} Suppose cores$_{F/F_0}(\alpha) = 0$ and ind$(\alpha) \geq 3$.
Suppose $F/F_0$ is ramified. Then, by (\ref{ramifiedh2}), $\alpha$ is unramified and per$(\alpha) \leq 2$.
Let $K$ be the residue field of $F$ and $\beta \in H^2(K, \mu_m)$ be the image of $\alpha$. Since per$(\alpha) \leq 2$, per$(\beta) \leq 2$.
Since $K$ is a local field, ind$(\beta) = $ per$(\beta)$.
Since $F$ is complete, ind$(\alpha) = $ ind$(\beta)$. Hence ind$(\alpha) \leq 2$, leading to a contradiction.
Hence $F/F_0$ is unramified and $\pi$ is a parameter in $F$.
Since $m$ is coprime to char$(K_0)$, $\alpha = \alpha' + (E, \sigma, \pi)$ for some
$\alpha' \in H^2_{nr}(F, \mu_m)$ and $E/F$ an unramified cyclic extension (cf. \cite[Lemma 4.1]{PPS}).
Then, by (\ref{unramified}), cores$_{F/F_0}(\alpha')$ and cores$_{F/F_0}(E, \sigma, \pi)$ are zero.
Let $\beta' \in H^2(K, \mu_m)$ be the image of $\alpha'$.
Since cores$_{F/F_0}(\alpha') = 0$, cores$_{\kappa/\kappa_0}(\beta') = 0$.
Since $K/K_0$ is a quadratic field extension of local fields, $\beta' = 0$ ((cf. \cite[Theorem 10, p. 237]{LF}) ).
Since $F$ is complete, $\alpha' = 0$ and hence $\alpha = (E, \sigma, \pi)$.
\end{proof}
Let $F$ be a field and $m \geq 1$ coprime to char$(F)$. Suppose $F$ contains a primitive $m^{\rm th}$ root of unity $\rho$.
For $a, b \in F^*$, let $(a, b)_{m}$ be the cyclic algebra generated by $x$ and $y$ with relations $x^{m} = a$, $y^{m} = b$ and
$yx = \rho xy$.
\begin{prop}
\label{branch-alg-n} Let $F_0$ be a complete discrete valued field with residue field $K_0$ a local field.
Let $m\geq 3$ with $2m$ coprime to the characteristic of the residue field of $K_0$.
Let $\pi \in F_0$ be a parameter and $\delta \in F_0$
a unit such that the image of $\delta$ in $K_0$ is a parameter.
Let $F/F_0$ be a quadratic field extension and $\alpha \in H^2(F, \mu_{m})$. If cores$_{F/F_0}(\alpha) = 0$ and
ind$(\alpha) = m$, then $F/F_0$ is unramified, $F$ contains a primitive $m^{\rm th}$ root of unity $\rho$,
$N_{F/F_0}(\rho) = 1$ and
$\alpha = (\delta, \pi)_{m}$.
\end{prop}
\begin{proof} Suppose cores$_{F/F_0}(\alpha) = 0$ and ind$(\alpha) = m$.
Since $m \geq 3$, by (\ref{branch-alg}), $F/F_0$ is unramified and $\alpha = (E, \sigma, \pi)$
for some $E/F$ an unramified cyclic extension.
Let $K$ be the residue field of $F$ and $L$ the residue field of $E$.
Since $F/F_0$ and $E/F$ are unramified, $K/K_0$ is an extension of degree 2 and
$L/K$ is a cyclic extension of degree $[E : F]$.
Let $\sigma_0$ denote the automorphism of $L/K$ induced by $\sigma$.
Since cores$_{F/F_0}(E, \sigma) = 0$, cores$_{K/K_0}(L, \sigma_0) = 0$.
Hence, by (\ref{galois}), $L/K_0$ is a dihedral extension.
Let $\overline{\delta} \in K_0$ be the image of $\delta$. Then, by the assumption, $\overline{\delta}$ is a
parameter in $K_0$.
Since $K_0$ is a local field and $[L : K] = m \geq 3$, by (\ref{dihedraln}), $K = K_0(\rho)$ for a primitive $m^{\rm th}$ root of unity,
$N_{K/K_0}(\rho)= 1$ and $L = K_0(\rho, \sqrt[m]{\overline{\delta}})$.
Since $F_0$ is complete, $F = F_0(\rho)$ and $E = F(\sqrt[m]{\delta})$.
Since $N_{K/K_0}(\rho)= 1$, $N_{F/F_0}(\rho)= 1$.
Since $F$ contains a primitive $m$-th root of unity,
$\alpha = (E, \sigma, \pi) = (\delta, \pi)_{m}$.
\end{proof}
We end this section with the following:
\begin{prop}
\label{prop-order}
Let $F$ be a complete discrete valued field with residue field $K$, valuation ring $R$, $\pi \in R$ a parameter and
$u \in R$ a unit. Let $n \geq 2$ which is coprime to char$(K)$. Suppose that $F$ contains a primitive $n^{\rm th}$ root of unity
and the cyclic algebra $D = (\pi, u)_n$ is a division algebra. Let $x, y \in D$ be with $x^n = \pi$, $y^n = u $ and $xy = \rho yx$.
Then $ R[x] + R[x]y + \cdots + R[x]y^{n-1} = R[y] + R[y]x + \cdots + R[y]x^{n-1} \subset D$ is the maximal order of $D$.
\end{prop}
\begin{proof}
Since $D$ is a division algebra and $F$ is complete, $\tilde{\nu} : D^* \to \mathbb Z$ given by
$\tilde{\nu}(z) = \nu(Nrd(z))$ is a discrete valuation on $D^*$ and
$\Lambda = \{ z \in D^* \mid \tilde{\nu}(z) \geq 0 \} \cup \{ 0\}$ is the unique maximal order of $D$ (\cite[Theorem 12.8]{reiner}).
Since every element in $R[y] + R[y]x + \cdots + R[y]x^{n-1}$ is integral over $R$,
$R[y] + R[y]x + \cdots + R[y]x^{n-1} \subseteq \Lambda$.
Since Nrd$(x) = (-1)^n \pi$, $\tilde{\nu} (x) = n$.
Since $y^n = u$ is a unit in $R$ and $n$ is coprime to the characteristic of $K$,
the extension $F(y)/F$ is unramified. Since deg$(D)= n$, $[F(y) : F] = n$.
Hence for any $a \in R[y]$, Nrd$(a) = N_{F(y)/F}(a)$.
Since $F(y)/F$ is unramified, $\tilde{\nu}(a)$ is divisible by
$n$ for all $a \in R[y]$.
Let $z\in \Lambda$. Then
$z = \frac{1}{b}(a_0 + a_1 x + \cdots + a_{n-1}x^{n-1})$ for some $b \in R$, $b \neq 0$ and
$a_i \in R[y]$. Since $\tilde{\nu}{a_i}$ is divisible by $n$ and $\tilde{\nu}(x) = 1$,
$\tilde{\nu}(a_0 + a_1 x + \cdots + a_{n-1}x^{n-1}) $ is equal to the minimum of $\tilde{\nu}(a_ix^i)$ for $0 \leq i \leq n-1$.
Since $\tilde{\nu}(z) \geq 0$, $\tilde{\nu}(a_0 + a_1x + \cdots + a_{n-1}x^{n-1}) \geq \tilde{\nu}(b)$.
In particular $\tilde{\nu}(a_ix^i) \geq \tilde{\nu}(b)$. Since $\tilde{\nu}(a_i x^i) = \tilde{\nu}(a_i) + i$ and $\tilde{\nu}(a_i)$,
$\tilde{\nu}(b)$ are divisible by $n$, it follows that $\tilde{\nu}(a_i) \geq \tilde{\nu}(b)$ for all $0 \leq i \leq n-1$.
Hence $\frac{a_i}{b} \in R[y]$ and $z \in R[y] + R[y]x + \cdots + R[y]x^{n-1}$. Hence
$\Lambda = R[y] + R[y]x + \cdots +R[y]x^{n-1}$ is a maximal $R$-order of $D$.
\end{proof}
\section{Reduced unitary Whitehead groups over 2-local fields}
\label{whitehead}
Let $F_0$ be a field of characteristic not equal to 2 and $F/F_0$ a quadratic \'etale extension.
Let $A$ be a central simple algebra over $F$ of degree $n$ with an involution $\tau$ of second kind and
$F^{\tau} = F_0$. Let $A^*$ denote the units in $A$ and $S^+(A, \tau) = \{ z \in A^* \mid \tau(z) = z\}$.
Let $\Sigma(A, \tau)$ be the subgroup of $A^*$ generated by $S^+(A, \tau)$.
Let $\Sigma'(A, \tau) = \{ z \in A^* \mid Nrd(z) \in F_0^* \}$. Then $\Sigma' (A, \tau)$ is a subgroup of $A^*$ and
$\Sigma(A, \tau) \subseteq \Sigma'(A, \tau)$ is a normal subgroup. The reduced unitary Whitehead group $USK_1(A)$ of $A$
is the quotient group $\Sigma'(A, \tau)/\Sigma(A, \tau)$ (\cite[p. 267]{KMRT}).
In this section we show that if $F_0$ is a 2-local field, then
$USK_1(A) $ is trivial. This is an analogue of a theorem of Platonov on the triviality of $SK_1(A)$ over such fields
(\cite[Theorem 5.5]{Pla}).
\begin{prop}
\label{existence-of-involution} Let $F_0$ be a field and $F/F_0$ a quadratic extension.
Let $m \geq 2$ with $2m$ coprime to char$(F_0)$. Suppose that $F$ contains a primitive $m$-th root of unity $\rho$.
Let $a, b \in F_0^*$. Suppose that $F[(\sqrt[m]{a}) : F] = m$.
Let $ A = (a, b)_{m}$ be the cyclic algebra generated by $x$ and $y$ with relations $x^{m} = a$, $y^{m} = b$ and
$yx = \rho xy$.
Then there exists a $F/F_0$-involution $\sigma$ on $A$ with $\sigma(x) = x$ and $\sigma(y) = y$ if and only if $N_{F/F_0}(\rho) = 1$.
\end{prop}
\begin{proof} Let $\tau_0$ be the nontrivial automorphism of $F/F_0$. Then $N_{F/F_0}(\rho) = \tau_0(\rho) \rho$.
Suppose there exists a $F/F_0$-involution $\sigma$ on $A$ with $\sigma(x) = x$ and $\sigma(y) = y$.
Since $yx = \rho xy$, we have $xy = \sigma(yx) = \sigma(\rho xy) = \tau_0(\rho) yx = \tau_0(\rho) \rho xy$.
Hence $N_{F/F_0}(\rho) = \tau_0(\rho) \rho = 1$.
Suppose $N_{F/F_0}(\rho) = 1$.
Let $c \in F_0^*$ with $F = F_0(\sqrt{c})$ and $E = F(x)$. Then $A = E \oplus Ey \oplus \cdots \oplus Ey^{m-1}$.
Since $\{ x^iy^j, \sqrt{c}x^iy^j \mid 0 \leq i, j \leq m-1 \}$ is a $F_0$-basis of $A$,
we have a $F_0$-vector space isomorphism $\sigma : A \to A$ given by
$\sigma(x^iy^j) = y^jx^i$ and $\sigma( \sqrt{c}x^iy^j) = - \sqrt{c}y^jx^i$. Then
$\sigma(z) = \tau_0(z)$ for all $z \in F$.
Since $a, b \in F_0$, $ \sigma(x^{m}) = \sigma(a) = a = x^{m} = \sigma(x)^{m}$.
Similarly $\sigma(y^{m}) = b = \sigma(y)^{m}$.
Since $\rho = c_1 + c_2 \sqrt{c}$ for some $c_i \in F_0$ and $\tau_0(\sqrt{c}) = -\sqrt{c}$, we have
$\sigma (\rho xy) = \sigma (c_1 xy + c_2 \sqrt{c} xy) = c_1 yx - c_2 \sqrt{c} yx = (c_1 - c_2\sqrt{c}) yx = \tau_0(\rho) yx$.
Since $yx = \rho xy$, $\sigma(yx) = \sigma( \rho xy ) = \tau_0(\rho) y x = \tau_0(\rho) \rho xy = xy = \sigma(x)\sigma(y)$.
Hence $\sigma$ is a $F/F_0$-involution.
\end{proof}
\begin{theorem}
\label{branch-norms} Let $F_0$ be a complete discrete valued field with residue field $K_0$ a local field.
Let $F/F_0$ be a quadratic field extension and $A$ a central simple algebra of index $m$ with $F/F_0$-involution $\tau$.
Suppose that $2m$ is coprime to the characteristic of the residue field of $K_0$.
Then, every element in $ F_0^*$ is a reduced norm of an element in $\Sigma(A, \tau)$.
\end{theorem}
\begin{proof} Let $\alpha \in H^2(F, \mu_m)$ be the class of $A$.
Since $A$ admits a $F/F_0$-involution, cores$_{F/F_0}(\alpha) = 0$.
Let $\lambda \in F_0$. Then cores$_{F/F_0}(\alpha \cdot (\lambda)) = $ cores$_{F/F_0}(\alpha) \cdot (\lambda) = 0$.
Since $K_0$ is a local field, cores: $ H^3(F, \mu_n^{\otimes 2}) \to H^3(F_0, \mu_n^{\otimes 2})$ is injective (\cite[Proposition 4.6]{PPS})
and $\alpha \cdot (\lambda) = 0$.
By (\cite[Theorem 4.12]{PPS}), $\lambda$ is a reduced norm from $A$.
Suppose ind$(A) \leq 2$. Since every element of $F_0$ is a reduced norm from $A$,
by result of Yanchevskii (\cite{yan}, cf. \cite[Proposition 17.27]{KMRT}),
every element in $F_0$ is the reduced norm of an element in $\Sigma(A, \tau)$.
Suppose that ind$(A) = m \geq 3$. Let $\pi \in F_0$ be a parameter and $\delta \in F_0$ a unit such that
the image $\overline{\delta}$ of $\delta$ in $K_0$ is a parameter.
Then, by (\ref{branch-alg-n} ), $F$ contains a primitive $m^{\rm th}$ root of unity $\rho$, $N_{F/F_0}(\rho) = 1$
and $A = (\delta, \pi)_n$. Let $x, y \in A$ be such that $x^m = \delta$, $y^m = \pi$ and $yx = \rho xy$.
Since $\pi, \delta \in F_0$ and $N_{F/F_0}(\rho) = 1$, by (\ref{existence-of-involution}),
there exists an $F/F_0$-involution $\tau'$ such that $\tau'(x) = x$ and $\tau'(y) = y$.
Since the subgroup of $F_0^*$ consisting of reduced norms of elements in $\Sigma(A, \tau )$
does not depend on the Brauer class of $A$
and the involution $\tau$ (\cite[Proposition 17.24]{KMRT}), we assume that $ \tau = \tau'$.
Since $y \in \Sigma(A, \tau)$ and Nrd$(y)= (-1)^{m+1}\pi$,
it is enough to show that any unit in $F_0$ is the reduced norm of an element in $\Sigma(A, \tau)$.
Let $u \in F_0$ be a unit.
Let $K$ be the residue field of $F$.
Let $E = F(\sqrt[m]{\delta})$ and $ L = K(\sqrt[m]{\overline{\delta}})$.
Then the residue field of $E$ is $L$.
Since $F$ contains a primitive $m^{\rm th}$ root of unity and $N_{F/F_0}(\rho) = 1$,
$K$ contains a a primitive $m^{\rm th}$ root of unity $\rho$ and $N_{K/K_0}(\rho) = 1$.
Hence, by (\ref{dihedral-norm1}), $E/F_0$ and $L/K_0$ are dihedral extensions.
By the choice of the involution $\tau$, the restriction of $\tau$ to $E$ is a reflection.
Let $\sigma$ be a generator of Gal$(E/F)$.
Since $E/F$ and $F/F_0$ are unramified, $E/F_0$ unramified. Since $L$ and $K_0$ are the residue fields of
$E$ and $F_0$ respectively, $\sigma$ and $\tau$ give rise to elements in Gal$(L/K_0)$. Let us denote the automorphisms of $L$ induced by
$\sigma$ and $\tau$ by $\sigma_0$ and $\tau_0$ respectively.
Then the dihedral group Gal$(L/K_0)$ is generated by $\sigma_0$ and $\tau_0$ with $\tau_0$ a reflection.
Further for any $i$, $0 \leq i \leq m-1$, $E^{\tau\sigma^i}/F_0$ is unramified with residue field $L^{\tau_0\sigma_0^i}$.
Let $\theta_0$ be the image of $u$ in $K_0$. Then, by (\ref{localfields-norms-n}),
for $0 \leq i \leq m-1$, there exists $\mu_i \in L^{\tau_0\sigma_0^i}$ such that
$\prod_{i= 0}^{m-1} N_{L^{\tau_0\sigma_0^i}/K_0}(\mu_i) = \theta_0$.
For each $0 \leq i \leq m-1$, let $\tilde{\mu}_i \in E^{\tau\sigma^i }$ map to
$\mu_i$ in $L^{\tau_0\sigma_0^i}$. Since $(\tau\sigma^i)^2 = id$ and the restriction of $\tau\sigma^i$ is nontrivial on $F$,
there exists an involution $\tau_i$ on $A$ with restriction to $E$ is $\tau\sigma^i$ (\cite[Theorem 10.1]{Sc}).
In particular $\tilde{\mu}_i \in \Sigma(A, \tau_i)$. Since $\Sigma(A, \tau) = \Sigma(A, \tau_i)$ (\cite[Proposition 17.24]{KMRT}),
$\tilde{\mu} = \tilde{\mu}_0 \cdots \tilde{\mu}_{m-1} \in \Sigma(A, \tau)$. Let $v = \prod_{i=0}^{m-1} N_{E^{\tau\sigma^i}/F_0}(\tilde{\mu}_i)
=$ Nrd$(\tilde{\mu})$.
Since the image of $v$ in $K_0$ is $\prod_{i= 0}^{n-1} N_{L^{\sigma^i\tau}/k_0}(\mu_i) = \theta_0$ and $\theta_0$ is the image of $u$,
$uv^{-1} $ maps to 1 in $K_0$. Since $m$ is coprime to char$(K_0) $, $uv^{-1} = w^{m}$ for some $w \in F_0$.
Since $w \in S^+(A, \tau)$, $\mu w \in \Sigma(A, \tau)$. Since
Nrd$(\tilde{\mu} w) = v w^m = u$, the result follows.
\end{proof}
\begin{cor}
\label{whitehead-trivial} Let $F_0$ be a complete discrete valued field with residue field $K_0$ a local field.
Let $F/F_0$ be a quadratic field extension and $A$ a central simple algebra of index $m$ with $F/F_0$-involution $\tau$.
Suppose that $2m$ is coprime to the characteristic of the residue field of $K_0$.
Then, $USK_1(A) = \{ 1 \}$.
\end{cor}
\begin{proof} Let $z \in \Sigma'(A, \tau)$ and $\theta = $ Nrd$(z) \in F_0^*$. Then, by (\ref{branch-norms}), there exists
$\mu \in \Sigma(A, \tau)$ such that Nrd$(\mu) = \theta$. In particular Nrd$(z\mu^{-1}) = 1$,
Hence, by (\cite[Theorem 5.5]{Pla}), $z \mu^{-1}$ is a product of commutators in $A^*$.
Since commutators are in $\Sigma(A, \tau)$ (\cite[Proposition 17.26]{KMRT}), $z \in \Sigma(A, \tau)$. Hence $USK_1(A) = \{1\}$.
\end{proof}
\section{Reduced norms of central simple algebras over two dimensional complete fields}
\label{reducednorms-2dim}
Let $R$ be a complete regular local ring of dimension 2 with residue field $\kappa$ and field of fractions
$F$. For a prime $\theta \in R$, let $F_\theta$ be the completion of $F$ at the discrete valuation given by the prime ideal
$(\theta)$ of $R$ and $\kappa(\theta)$ the residue field at $\theta$.
Let $A$ be a central simple algebra over $F$ of index coprime to char$(\kappa)$.
Let $m = (\pi, \delta)$ be the maximal ideal of $R$.
Suppose that $A$ is unramified on $R$ except possibly at $\pi$ and $\delta$.
Let $\lambda = v \pi^{s}\delta^t \in F^*$ for some unit $v \in R$ and
$r, s \in \mathbb Z$. In this section we show that if $\kappa$ is a finite field and
$\lambda \in$ Nrd$(A\otimes F_{\pi})$, then $\lambda \in Nrd(A)$.
\begin{remark}
\label{branch-elements}
Let $\mu \in F^*_\pi$ and $n \geq 1$ coprime to char$(\kappa)$. Then $\mu = u \pi^r$ for some $u \in F_\pi$ which is a unit at $\pi$.
Let $\overline{u}$ be the image of $u$ in $\kappa(\pi)$. Since $\kappa(\pi)$ is the field of fractions of
$R/(\pi)$ and $R$ is complete, $\kappa(\pi)$ is a complete discrete valued field with residue field $\kappa$
and $\overline{\delta} \in R/(\pi)$ is a parameter. Hence $\overline{u} = \overline{v} \delta^s$ for some $v \in R$ a unit.
Then, $\mu (v\pi^r\delta^s)^{-1}$ is a unit at $\pi$ and maps to 1 in $\kappa(\pi)$.
Since $n$ is coprime to $n$, $\mu = v\pi^r \delta^s c^n$ for some $c \in F_\pi^*$.
\end{remark}
We begin by extracting the following from (\cite{PPS}).
\begin{prop}
\label{2dim-local-nrd0} Let $R$ be a complete regular local ring of dimension 2 with residue field $\kappa$ and field of fractions
$F$. Let $A$ be a central simple algebra over $F$ of index $n$ coprime to char$(\kappa)$ and
$\alpha \in H^2(F, \mu_n)$ be the class of $A$.
Let $m = (\pi, \delta)$ be the maximal ideal of $R$.
Suppose that $\kappa$ is a finite field and $A$ is unramified on $R$ except possibly at $\pi$ and $\delta$.
Let $\lambda = v \pi^{s}\delta^t \in F^*$ for some unit $v \in R$ and
$r, s \in \mathbb Z$.
If $\alpha \cdot (\lambda) = 0 \in H^3(F, \mu_n^{\otimes 2})$, then $\lambda \in Nrd(A)$.
\end{prop}
\begin{proof}
As in (\cite[Theorem 4.12]{PPS}), we assume that ind$(A) = \ell^d$ with $\ell$ a prime
and $F$ contains a primitive $\ell^{\rm th}$ root of unity.
Since ind$(A)$ is coprime to char$(\kappa)$, $\ell \neq $ char$(\kappa)$.
We prove the result by induction on $d$. If $ d = 0$, then $A$ is a matrix algebra and hence every element is a
reduced norm from $A$. Suppose that $d \geq 1$.
Suppose $\alpha \cdot (\lambda) = 0 \in H^3(F, \mu_n^{\otimes 2})$.
Suppose $s$ is coprime to $\ell$. Then, by (\cite[Lemma 6.1]{PPS}), $A = (E, \sigma, (-1)^s \lambda)$ for some cyclic extension
$E/F$ with $\sigma$ a generator of Gal$(E/F)$.
In particular $(-1)^{\ell^d + 1} (-1)^s \lambda \in Nrd(A)$. Suppose $\ell$ is odd, then $-1 \in Nrd(A)$ and hence
$\lambda \in Nrd(A)$. Suppose $\ell = 2$. Since $s$ is odd, $ \lambda = (-1)^{\ell^d + 1} (-1)^s \lambda \in Nrd(A)$.
Similarly if $t$ is coprime to $\ell$, then $\lambda \in Nrd(A)$.
Suppose that $s$ and $t$ are divisible by $\ell$.
Then, by (\cite[Lemma 4.10]{PPS}), there exists an uramified cyclic field extension $L_\pi/F_\pi$ of degree $\ell$ and $\mu_\pi \in L_\pi$
such that $N_{L_\pi/F_\pi}(\mu) = \lambda$, ind$(\alpha \otimes L_\pi) < $ ind$(A \otimes F_\pi)$ and $\alpha \cdot (\mu_\pi) = 0 \in H^3(L_\pi, \mu_{\ell^d}^{\otimes 2})$.
Since $L_\pi/F_\pi$ is an unramified cyclic extension of degree $\ell$ and $F$ contains a primitive $\ell^{\rm th}$ root of
unity, we have $L_\pi = F_\pi(\sqrt[\ell]{a})$ for some $a \in F_\pi$ which is a unit at $\pi$.
Since char$(\kappa) \neq \ell$ and the residue field $\kappa(\pi)$ of $F_\pi$ is the field of fractions of $R/(\pi)$,
we have $a = w\delta^\epsilon \in F_\pi^*/F_\pi^{*\ell}$ for some $w \in R$ a unit and
$0 \leq \epsilon \leq \ell-1$ (cf. \ref{branch-elements}). Suppose $\epsilon \geq 1$. Let $1 \leq \epsilon' \leq \ell-1$ with $\epsilon\epsilon' = 1$ modulo $\ell$.
Since $F(\sqrt[\ell]{w\delta^{\epsilon}}) = F(\sqrt[\ell]{w^{\epsilon'}\delta})$, replacing $w$ by $w^{\epsilon'}$ we assume that $ L = F(\sqrt[\ell]{w\delta^\epsilon})$ with $0 \leq \epsilon \leq 1$.
Then $L/F$ is a cyclic extension of degree $\ell$ and $L \otimes F_\pi \simeq L_\pi$.
Let $S$ be the integral closure of $R$ in $L$. Then $S$ is a regular local ring with maximal ideal
$(\pi, \delta_1)$, where $\delta_1 = \delta$ or $\sqrt[\ell]{w\delta}$ depending on whether $\epsilon = 0$ or 1 (\cite[3.1, 3.2]{PS1}).
Since ind$(\alpha \otimes L_\pi) < $ ind$(\alpha)$, by (\cite[Proposition 5.8]{PPS}), ind$(\alpha \otimes L) < $ ind$(\alpha)$.
Since $S$ is a regular local ring with maximal ideal $(\pi, \delta_1)$ and $L$ the field of fractions of $S$,
there exists $u \in S$ a unit such that $\mu_\pi = u \pi^i\delta_1^{j} \mu_1^{\ell^{d}}$ for some $i, j \in \mathbb Z$ and $\mu_1 \in L_\pi$ (cf. \ref{branch-elements}).
Let $\mu' = u\pi^i\delta_1^{j} \in L$. Let $\lambda' = N_{L/F}(\mu') $. Then $\lambda' = v'\pi^{\ell i} \delta^{j'}$ for some unit $v' \in R$.
Since $ \lambda = N_{L_\pi/F_\pi}(\mu_\pi) = N_{L_\pi/F_\pi}( \mu' \mu_1^{\ell^{d}}) = \lambda' N_{L_\pi/F_\pi}(\mu_1)^{\ell^{d}}$,
we have
$\lambda' \lambda^{-1} \in F_{\pi}^{\ell^{d}}$. Hence, by (\cite[Corollary 5.5]{PPS}), $\lambda = \lambda' \theta^{\ell^{d}}$ for some $\theta \in F$.
Let $\mu = \mu' \theta^{\ell^{d}} \in L$. Then $N_{L/F}(\mu) = \lambda$.
Since $\mu = \mu' \theta^{\ell^{d}} = \mu_\pi \mu_1^{-\ell^{d+1}} \theta^{\ell^d}$ and $\alpha \cdot (\mu_\pi) = 0 \in H^3(L_\pi, \mu_{\ell^d}^{\otimes 2})$,
$\alpha \cdot (\mu) = 0 \in H^3(L_\pi, \mu_{\ell^d}^{\otimes 2})$. Hence, by (\cite[Corollary 5.5]{PPS}),
$\alpha \cdot (\mu) = 0 \in H^3(L, \mu_{\ell^d}^{\otimes 2})$.
Since ind$(\alpha \otimes L) < $ ind$(\alpha)$, by induction $\mu \in $ Nrd$(A \otimes L)$.
Since $N_{L/F}(\mu)= \lambda$, $\lambda \in$ Nrd$(A)$.
\end{proof}
\begin{cor}
\label{2dim-local-nrd} Let $R$ be a complete regular local ring of dimension 2 with residue field $\kappa$ and field of fractions
$F$.
Let $A$ be a central simple algebra over $F$ of index coprime to char$(\kappa)$.
Let $m = (\pi, \delta)$ be the maximal ideal of $R$.
Suppose that $\kappa$ is a finite field and $A$ is unramified on $A$ except possibly at $\pi$ and $\delta$.
Let $\lambda = v \pi^{s}\delta^t \in F^*$ for some unit $v \in R$ and
$r, s \in \mathbb Z$.
If $\lambda \in$ Nrd$(A\otimes F_{\pi})$, then $\lambda \in Nrd(A)$.
\end{cor}
\begin{proof} Let $\alpha \in H^2(F, \mu_{\ell^d})$ be the class of $A$.
Since $\lambda \in $ Nrd$(A\otimes F_{\pi})$, $\alpha \cdot (\lambda) = 0 \in H^3(F_\pi, \mu_n)$.
Since $\alpha$ is unramified on $R$ except possibly at $\pi$, $\delta$ and $\lambda = c\pi^s\delta^r$,
$\alpha \cdot (\lambda) $ is unramified on $R$ except possibly at $\pi$ and $\delta$.
Since $\alpha \cdot (\lambda) = 0 \in H^3(F_\pi, \mu_n)$, by (\cite[Corollary 5.5]{PPS}), $\alpha \cdot (\lambda) = 0 \in H^3(F, \mu_n)$.
Hence, by (\ref{2dim-local-nrd0} ), $\lambda \in Nrd(A)$.
\end{proof}
We end this section with the following
\begin{lemma}
\label{2dim-local}
Let $v \in R$ be a unit and $\mu = v\pi_1^{r}\delta^s \in F^*$ for some
$r, s \in \mathbb Z$.
Suppose that there exists $\theta_\pi \in F_{0\pi}^*$ such that
$\mu \theta_\pi \in Nrd(A\otimes_F F_{\pi_1})$. Then there exists $\theta \in F_0$
such that $\mu \theta \in Nrd(A)$ and $\theta_\pi \theta^{-1} \in F_{\pi_1}^n$.
\end{lemma}
\begin{proof} Since $F_0$ is the field of fractions of $R_0$ and $(\pi, \delta)$ is the maximal ideal of $R_0$,
$\theta_\pi = w \pi^{r_1} \delta^{s_1} c^n$ for some
$w \in R_0$ a unit, $c \in F_{0\pi}$ (cf. \ref{branch-elements}).
Let $\theta = w \pi^{r_1} \delta^{s_1} \in F_0^*$.
Since ind$(A) = n$, $ c^n \in Nrd(A\otimes_F F_{\pi_1})$ and hence
$\mu \theta \in Nrd(A\otimes_F F_{\pi_1})$. Since $A$ is unramified on $R$ except possibly at
$\pi_1$ and $\delta$ and the support of $\mu\theta$ is at most $\pi_1$ and $\delta$, by (\ref{2dim-local-nrd}),
$\mu\theta \in Nrd(A)$.
\end{proof}
\section{Central simple algebras with involutions of second kind over two dimensional complete fields}
\label{alg-2dimlocal}
Let $R_0$ be a complete regular local ring of dimension two with residue field
$\kappa_0$ a finite field of characteristic not 2 and $F_0$ the field of fractions of $R_0$.
Let $m = (\pi, \delta)$ be the maximal ideal of $R_0$.
Let $F/F_0$ be a quadratic field extension with $F = F_0(\sqrt{u\pi^\epsilon})$ for some $u\in R_0$ a unit and $\epsilon = \{ 0, 1\}$.
Let $R$ be the integral closure of $R_0$ in $F$. Then $R$ is a regular local ring with maximal ideal
$(\pi_1 , \delta)$, where $\pi_1 = \pi$ if $\epsilon = 0$ and $\pi_1 = \sqrt{u\pi}$ if $\epsilon = 1$ (\cite[Lemma 3.1, 3.2]{PS1}).
Let $\kappa$ be the residue field of $R$. Then $[\kappa : \kappa_0] \leq 2$.
Let $A$ be a central division algebra over $F$ which is unramified on $R$ except possibly at $\pi_1$ and $\delta$.
Suppose that $n = $ ind$(A)$ is coprime to char$(\kappa_0)$.
In this section we show that if there is an involution $\tau$ on $A$ of second kind and $A$ is division, then there exists a maximal $R$-order in $A$ invariant
under $\tau$ with some additional structure.
We then prove a local global principle for certain classes of hermitian forms over $(A, \tau)$.
We begin with the following
\begin{prop}
\label{2dim-alg} Let $\alpha \in H^2(F, \mu_n)$ be the class of $A$.
Suppose ind$(\alpha) = n\geq 3$.
If cores$_{F/F_0}(\alpha) = 0$, then $F/F_0$ is unramified on $R_0$,
$F$ contains a primitive $n^{\rm th}$ root of unity $\rho$, $N_{F/F_0}(\rho) = 1$
and $\alpha = (\delta, \pi)_{n}$.
\end{prop}
\begin{proof} Since $F = F_0(\sqrt{u\pi^{\epsilon}})$, $R_0$ is complete and char$(\kappa) \neq 2$,
it follows that $F \otimes F_{0\pi} = F_{\pi_1}$ is a field.
Since $\alpha$ is unramfied on $R$ except possibly at $\pi_1$ and $\delta$,
ind$(\alpha) = $ ind$(\alpha \otimes F_{\pi_1})$ (\cite[Proposition 5.8]{PPS}).
The residue field of $F_{0\pi}$ is a local field with residue field $\kappa_0$.
Suppose cores$_{F/F_0}(\alpha) = 0$. Then cores$_{F_{\pi_1}/F_{0\pi}}(\alpha) = 0$.
Since ind$(\alpha \otimes F_{\pi_1} ) = n\geq 3$, by (\ref{branch-alg-n}),
$F_{\pi_1} /F_{0\pi}$ is unramified, $F_{\pi_1}$ contains a primitive $n^{\rm th}$ root of unity $\rho$,
$N_{F_{\pi_1}/F_{0\pi}}(\rho) = 1$ and $\alpha \otimes F_{\pi_1} = (\delta,\pi)_{n}$.
Since the residue field $\kappa(\pi_1)$ of $F_{\pi_1}$ is a complete discretely valued field with residue field $\kappa$,
$\kappa$ contains a primitive $n^{\rm th}$ root of unity. Hence $F$ contains a primitive $n^{\rm th}$ root of unity.
By (\cite[Corollary 5.5]{PPS}), $\alpha = (\delta, \pi)_{n}$.
Since $F/F_0$ is unramified except possible at $\pi$ and $F_{\pi_1}/F_{0\pi}$ is unramified, $F/F_0$ is unramified on $R_0$.
Since $N_{F_{\pi_1}/F_{0\pi}}(\rho) = 1$, $N_{F/F_0}(\rho) = 1$.
\end{proof}
Let $\alpha \in H^2(F, \mu_n)$ be the class of $A$.
Suppose ind$(\alpha) = n\geq 3$.
Since $(\pi, \delta) $ is a maximal ideal of $R$, $(\delta, \pi)_n$ is a division algebra.
Let $D = (\pi, \delta)_n$. Then, by (\ref{2dim-alg}), $\alpha$ is the class of $D$.
Thus there exist $x, y \in D$ such that
$x^{n} = \delta$ and $y^{n} = \pi$ and $yx = \rho xy$.
Since $D \otimes F_\pi$ and $D \otimes F_\delta$ are division algebras (\cite[Proposition 5.8]{PPS}),
the valuation $\nu_\pi$ and $\nu_\delta$ given by $\pi$ and $\delta$ on $F$
extend to valuations $w_\pi$ and $w_\delta$ on $D \otimes F_\pi$
and $D \otimes F_\delta$ respectively (\cite[Theorem 12.6]{reiner}). We have $ e_\pi : = [w_{\pi}(D^*) : \nu_\pi(F^*) ] = n$
and $e_\delta : = [w_{\delta}(D^*) : \nu_\delta(F^*) ] = n$.
Let $S = R[\sqrt[n]{\delta}] = R[x] $. Then $S$ is the integral closure of $R$ in $F(\sqrt[n]{\delta})$ and
$S$ is a regular local ring of dimension 2 with maximal ideal $(\sqrt[n]{\delta}, \pi)$ (\cite[Lemma 3.2]{PS1}).
Since $D \simeq (\delta, \pi)_{n}$ and $N_{F/F_0}(\rho) = 1$,
by (\ref{existence-of-involution}),
there exists an $F/F_0$-involution $\sigma$ on $D$ with $\sigma(x) = x$ and $\sigma(y) = y$.
\begin{lemma} (cf. \cite[Lemma 3.7]{Wu})
\label{order} Let $\Lambda = S + Sy + \cdots + Sy^{n -1} \subset D$.
Then $\Lambda$ is a maximal $R$-order in $D$.
\end{lemma}
\begin{proof} Since $S$ is a free $R$-module, $\Lambda$ is a free $R$-module.
Let $P \subset R$ be a height one prime ideal.
Suppose $P \neq (\pi)$ and $P \neq (\delta)$. Since $\pi$ and $\delta$ are units at $P$, $\Lambda_P = \Lambda \otimes R_P$ is an
Azumaya algebra and hence a maximal $R_P$-order in $D$.
Suppose $P = (\pi) $ or $(\delta)$. Then, by (\ref{prop-order}) and (\cite[Theorem 11.5]{reiner}),
$\Lambda_P$ is a maximal $R_P$-order in $D$.
Since $R$ is noetherian, integrally closed and
$\Lambda$ is a reflexive $R$-module, by (\cite[1.5]{AG1}), $\Lambda$ is a maximal $R$-order of $D$.
\end{proof}
\begin{lemma} (cf. \cite[Lemma 3.1]{Wu})
\label{rank1form}
Let $\sigma$ and $\Lambda$ be as above.
Let $a \in \Lambda$ with $\sigma(a ) = a$.
If Nrd$(a) = u\pi^r\delta^s$ for some $u \in R_0$ unit and $r, s \in \mathbb Z$,
then there exist $\theta \in \Lambda$ a unit,
$r', s' \in \{ 0, 1\}$ with $r \equiv r'$ and $s \equiv s'$ modulo 2 such that
$<\!a\!> \simeq <\! \theta x^{r'}y^{s'}\!>$ as hermitian forms over $(D, \sigma)$.
\end{lemma}
\begin{proof} Let $r = 2r_1 + r'$ and $s = 2s_1 + s'$ with $r', s' \in \{0, 1 \}$.
Let $z = x^{s_1}y^{r_1} \in \Lambda$. Then Nrd$(z) = $ Nrd$(\sigma(z)) = \delta^{s_1}\pi^{r_1}$.
Let $\theta = \sigma(z)^{-1} a z^{-1} ( x^{s'}y^{r'})^{-1}$.
Then $a = \sigma(z) \theta x^{s'}y^{r'} z$ and hence
$<\!a \!> \simeq <\!\theta x^{s'}y^{r'}\!>$. Since Nrd$(\theta ) = u \in R_0$ is a unit, it follows,
as in the proof of (\cite[Lemma 3.1]{Wu}), that $\theta \in\Lambda$ and is a unit in $\Lambda$.
\end{proof}
\begin{cor}(cf. \cite[Corollary 3.2]{Wu})
\label{ranknforms}
Let $\sigma$ and $\Lambda$ be as in (\ref{order}).
Let $h = <a_1,\cdots , a_r>$ be an hermitian form over $(A, \sigma)$
with $a_i \in \Lambda$, $\sigma(a_i) = a_i$ and Nrd$(a_i) $ is a product of a unit in $R$,
a power of $\pi$ and a power of $ \delta$.
Then
$$ h \simeq <\!u_1, \cdots , u_{m_0}\!> \perp <\!v_1, \cdots, v_{n_1}\!>x \perp
<\!w_1, \cdots, w_{n_2}\!> y \perp <\!\theta_1, \cdots, \theta_{n_3}\!>xy$$
for some $u_i, v_i, w_i, \theta_i \in \Lambda$ units.
\end{cor}
We have the following (cf. \cite[Corollary 3.3]{Wu}).
\begin{cor}
\label{isotropic}
Let $\sigma$ and $\Lambda$ as above. Let $a_i \in\Lambda$ be as in (\ref{ranknforms})
and $h = <\!a_1, \cdots , a_r\!> $.
If $h \otimes F_{0\pi}$ is isotropic, then $h$ is isotropic over $F_0$.
\end{cor}
\begin{proof} Since $\sigma(xy) = yx = \rho xy$ and $\rho \sigma(\rho) = N_{F/F_0}(\rho) = 1$, it follows that
Int$(xy) \circ \sigma$ is an involution on $D$.
Following the proof of (\cite[Corollary 3.3]{Wu}), it follows that
if $h$ is isotropic over $F_{0\pi}$, then $h$ is isotropic over $F_0$.
\end{proof}
\section{An application of refinement of patching to local global principle}
\label{refinement}
Let $T$ be a complete discrete valuation ring and $K$ its field of fractions.
We recall a few basic definitions from (\cite{HHK3}, \cite{HHK5}).
Let $F$ be a function field of a curve over $K$. Let $\sheaf{Y} \to $ Spec$(T)$ be a proper normal model of
$F$ and $Y$ the special fibre. For a point $x$ of $Y$, let
$F_x$ be the field of fractions of the completion $\hat{R}_x$ of the local ring at $x$.
Let $U$ be a nonempty proper subset of an irreducible component of $Y$ not containing the singular points of
$X$. Let $R_U$ be the subset of $F$ containing all those elements of $F$ which are regular at every closed point of $U$.
Let $t \in T$ be a parameter, $\hat{R}_U$ be the $(t)$-adic completion of $R_U$ and $F_U$ the field of fractions of $\hat{R}_U$.
Let $P \in Y$ be a closed point. A height one prime ideal $\mathfrak{p}$ of $\hat{R}_P$ containing $t$ is called a {\it branch} at $P$. For a branch $\mathfrak{p}$,
let $F_\mathfrak{p}$ be the completion of $F_P$ at the discrete valuation given by $\mathfrak{p}$.
Let $\sheaf{P}$ be a finite set of closed points of $Y$ containing all singular points of $Y$ and at least one point from each irreducible component of
$Y$. Let $\sheaf{U} $ be the set of irreducible components of $Y \setminus \sheaf{P}$ and $\sheaf{B}$ the set of branches at points in $\sheaf{P}$.
Let $G$ be a linear algebraic group over $F$. We say that {\it factorization} holds for $G$ with respect to $(\sheaf{P}, \sheaf{U})$
if given $(g_{\mathfrak{p}}) \in \prod_{\mathfrak{p} \in \sheaf{B}}G(F_{b})$, there exists $(g_Q) \in \prod_{Q \in \sheaf{P}} G(F_Q)$
and $(g_{U}) \in \prod_{U \in \sheaf{U}}G(F_{U})$ such that if $\mathfrak{p}$ is a branch at $P$ along $U$, then $g_{\mathfrak{p}} = g_{Q} g_{U}$.
If the factorization holds for $G$ with respect to all possible pairs $(\sheaf{P}, \sheaf{U})$, then we say that
{\it factorization} holds for $G$ over $F$ with respect to $\sheaf{Y}$.
Let $Z$ be a variety over $F$ with a $G$-action.
We say that $G$ {\it acts transitively} on points of $Z$ if $G(E)$ acts transitively on $Z(E)$ for all extensions $E/F$.
Let $\sheaf{X} \to \sheaf{Y}$ be a sequence of blow ups and $X$ the special fibers of $\sheaf{X}$.
Let $P \in \sheaf{Y}$ be a closed point and $V$ the fibre over $P$.
Suppose that dim$(V) = 1$.
Let $\sheaf{P}'$ be a finite set of closed points of $V$ containing
all the singular points of $V$ and at least one point from each irreducible component of $V$.
Let $\sheaf{U}'$ be the set of connected components of $V \setminus \sheaf{P}'$.
Let $\sheaf{B}'$ be the set of
branches at the points of $\sheaf{P}'$.
We say that {\it factorization} holds for $G$ with respect to $(\sheaf{P}', \sheaf{U}')$
if given $(g_{\mathfrak{p}}) \in \prod_{\mathfrak{p} \in \sheaf{B}'}G(F_{\mathfrak{p}})$, there exists $(g_Q) \in \prod_{Q \in \sheaf{P}'} G(F_Q)$
and $(g_{U}) \in \prod_{U \in \sheaf{U}'}G(F_{U})$ such that if $\mathfrak{p}$ is a branch at $P$ along $U$, then $g_{\mathfrak{p}} = g_{Q} g_{U}$.
Let $\sheaf{P}$ be a finite set of closed points of $X$ containing $\sheaf{P}'$,
all singular points of $X$ and at least one closed point from each irreducible component of $X$.
Let $\sheaf{U}$ be the set of irreducible components of $X\setminus \sheaf{P}$ and $\sheaf{B}$ the set of branches at points in $\sheaf{P}$.
The following results are immediate consequences of results of Harbater, Hartmann and Krashen (\cite{HHK5}).
\begin{theorem}
\label{refinesha}
Let $F$, $P$, $\sheaf{P}$, $\sheaf{P}'$, $\sheaf{U}$ and $\sheaf{U}'$ be as above.
Let $G$ be a connected linear algebraic group over $F$. If the factorization holds for $G$ with respect to $(\sheaf{P}, \sheaf{U})$,
then the kernel of natural map
$$H^1(F_P, G) \to \prod_{U' \in \sheaf{U}'}H^1(F_{U'}, G) \times \prod_{Q \in \sheaf{P}'}H^1(F_Q, G)$$
is trivial.
\end{theorem}
\begin{proof} Suppose the factorization holds for $G$ with respect to $(\sheaf{P}, \sheaf{U})$.
Then, by (\cite[Proposition 3.14]{HHK5}), factorization holds for $G$ with respect to $(\sheaf{P}', \sheaf{U}')$.
By (\cite[Proposition 3.10]{HHK5}), patching holds for the injective diamond $F_{P_\bullet} =
(F_P \leq \prod_{Q \in \sheaf{P}'} F_Q, \prod_{U' \in \sheaf{U}'}F_{U'} \leq \prod_{b' \in \sheaf{B}'} F_{b'})$.
Hence, by (\cite[Theorem 2.13]{HHK5}),
the map $H^1(F_P, G) \to \prod_{U' \in \sheaf{U}'}H^1(F_{U'}, G) \times \prod_{Q \in \sheaf{P}'}H^1(F_Q, G)$ is injective.
\end{proof}
\begin{cor} Let $F$, $\sheaf{Y}$, $P$ and $\sheaf{X}$ be as above. Let $G$ be a connected linear algebraic group over $F$.
If the factorization holds for $G$ over $F$,
then the kernel of natural map
$$H^1(F_P, G) \to \prod_{x \in V}H^1(F_x, G)$$
is trivial.
\end{cor}
\begin{proof} Let $\xi$ be in the kernel of the map $H^1(F_P, G) \to \prod_{x \in V}H^1(F_x, G)$.
Then, as in (\cite[Corollary 5.9]{HHK3}), there exists a finite set $\sheaf{P}'$ of closed points of $V$ containing
all the singular points of $V$ and at least one closed point from each irreducible component of $V$
such that if $\sheaf{U}'$ is the set of irreducible components of $V \setminus \sheaf{P}'$,
then $\xi$ is in the kernel of $H^1(F_P, G) \to \prod_{U' \in \sheaf{U}}H^1(F_U, G) \times \prod_{Q \in \sheaf{P}'}H^1(F_Q, G)$.
Hence, by (\ref{refinesha}), $\xi$ is trivial.
\end{proof}
\begin{theorem}
\label{refinesha1} Let $F$, $P$, $F_P$ and $F_U$ be as above.
Let $G$ be a connected linear algebraic group over $F$.
Suppose the factorization holds for $G$ with respect to $(\sheaf{P}, \sheaf{U})$.
Let $Z$ be a $F$-variety with $G$ acting transitively on points $Z$.
If $Z(F_{U'}) \neq \emptyset$ and $Z(F_Q) \neq \emptyset $ for all $U' \in \sheaf{U}'$ and $Q \in \sheaf{P}'$, then
$Z(F_P) \neq \emptyset$.
\end{theorem}
\begin{proof} By (\cite[Proposition 3.10]{HHK5}), patching holds for the injective diamond $F_{P_\bullet} =
(F_P \leq \prod_{Q \in \sheaf{P}'} F_Q, \prod_{U' \in \sheaf{U}'}F_{U'} \leq \prod_{b' \in \sheaf{B}'} F_{b'})$. Result follows from
(\ref{refinesha}) and (\cite[Corollary 2.8]{HHK3}).
\end{proof}
\begin{cor}
\label{refinesha2}(cf. \cite[Theorem 9.1]{HHK3})
Let $F$, $\sheaf{Y}$, $P$, $\sheaf{X}$ and $V$ be as above. Let $G$ be a connected linear algebraic group over $F$.
Suppose the factorization holds for $G$ over $F$.
Let $Z$ be a $F$-variety with $G$ acting transitively of points of $Z$.
If $Z(F_x) \neq \emptyset$ for all $x \in V $, then
$Z(F_P) \neq \emptyset$.
\end{cor}
\begin{proof} Suppose $Z(F_x) \neq \emptyset$ for all $x \in V$.
Let $X_i$ be an irreducible component of $V$ and $\eta_i \in V$ the generic point of $X_i$.
Since $Z(F_{\eta_i}) \neq \emptyset$, by (\cite[Proposition 5.8]{HHK3}),
there exists a non-empty affine open subset $U_i$ of $X_i$ such that $Z(F_{U_i}) \neq \emptyset$.
Let $\sheaf{P}'$ be the complement of the union of $U_i$'s in $V$.
Let $Q \in \sheaf{P}'$. Then, by the assumption on $Z$, $Z(F_Q) \neq \emptyset$.
Hence, by (\ref{refinesha1}), $Z(F_P) \neq \emptyset$.
\end{proof}
\section{Local global principle for projective homogeneous spaces under general linear groups}
\label{lgp-gl}
Let $K$ be a complete discretely valued field with residue field $\kappa$ and $F$ the function field of a smooth projective
curve over $K$. Let $A$ be a central simple algebra over $F$ of index $n$ coprime to char$(k)$ and $G = PGL(A)$.
Let $Z$ be a projective homogeneous space under $G$ over $F$.
If $F$ contains a primitive $n^{\rm th}$ root of unity, then from the results in (\cite{RS}), it follows that
$Z(F) \neq \emptyset $ if and only if $Z(F_\nu) \neq \emptyset$ for all divisorial discrete valuations of $F$.
In this section, we dispense with the condition on the roots of unity if $K$ is a local field.
Let $\sheaf{X} $ be a normal proper model of $F$ over the valuation ring of $K$.
Let $P$ be a closed point of $\sheaf{X}$.
A discrete valuation $\nu$ of $F$ (resp. $F_P$) is called a {\it divisorial} discrete valuation if
it is given by a codimension one point of a model of $F$ (resp. with center $P$).
Let $M$ be a field and $A$ a central simple algebra over $M$ of degree $n$. For a sequence of integers
$0 < n_1 < n_2 < \cdots < n_k < n$, let
$$X(n_1, \cdots , n_k) = \{ (I_1, \cdots , I_k) \mid I_1 \subseteq I_2 \subseteq \cdots \subseteq I_k \subseteq A,$$
$$ ~~~~~~~~~{\rm ~a ~
sequence~ of~ right~ ideals~ of~} A {\it ~with} ~{\rm dim}_F(I_j)= n\cdot n_j, j = 1, \cdots , k \}.$$
\begin{theorem}
\label{proj-pgl} Let $K$ be a local field with residue field $\kappa$ and $F$ the function field of a smooth projective curve over $K$. Let $A$ be a central simple algebra over $F$ of index coprime to char$(k)$.
Let $\sheaf{X} $ be regular proper model of $F$ over the valuation ring of $K$ and $P \in \sheaf{X}$ be a closed point.
Let $L$ be the field $F$ or $F_P$. Let $Z$ be a projective homogeneous space under $PGL(A)$ over $L$.
If $Z(L_\nu) \neq \emptyset$ for all divisorial discrete valuation $\nu$ of $L$, then $Z(L) \neq \emptyset$.
\end{theorem}
\begin{proof} Let $f : \sheaf{X}' \to \sheaf{X}$ be a sequence of blow ups such that
$\sheaf{X}'$ is regular,
the ramification locus of $A$ on $\sheaf{X}'$ and the special fibre of $\sheaf{X}'$ is a union of regular curves with
normal crossings. By blowing up $P$, we assume that the dimension of the fibre over $P$ is 1.
Let $V$ be either the special fibre of $f$ or the fibre over $P$, depending on $L = F$ or $L = F_P$.
Suppose that $Z(L_\nu) \neq \emptyset$ for all divisorial discrete valuations $\nu$ of $L$.
Since $PGL(A)$ is rational, factorization holds for $PGL(A)$ over $F$ (\cite[Theorem 3.6.]{HHK1}).
Thus, by (\cite[Theorem 5.10 and Theorem 9.1]{HHK3} for the case $L = F$ and \ref{refinesha2} for
the case $L = F_P$), it is enough to show that $Z(F_x) \neq \emptyset$ for all
$x \in V$.
Let $x \in V$. Suppose $x$ is a generic point of $V$. Then $x$ defines a divisorial discrete valuation $\nu_x$ of $L$ and
$L_x = L_{\nu_x}$. Hence $Z(L_x) \neq \emptyset$.
Suppose $x = Q$ is a closed point of $V$.
Then, by the choice of $\sheaf{X}'$,
the local ring $R_Q$ at $Q$ on $\sheaf{X}'$ is generated by $(\pi, \delta)$ such that
$A$ is unramified on $R_Q$ except possibly at $(\pi)$ and $(\delta)$.
Let $n = $ deg$(A)$.
Then $Z$ is isomorphic to $X(n_1, \cdots , n_r)$ for some sequence of integers
$0 < n_1 < \cdots < n_r < n$ (cf. \cite[\S 5]{MPW1}). Let $d$ be the lcm of $n_1, \cdots , n_r, n$.
Then, for any field extension $M/F$,
$Z(M) \neq \emptyset$ if and only if ind$(A\otimes_FM)$ divides $d$.
(cf. \cite[5.3]{MPW1}).
Let $\nu_\pi$ be the discrete valuation given by $\pi$ on $L$ and $L_{\nu_\pi}$ the completion of $L$ at $\nu_\pi$.
Since $\nu_\pi$ is a divisorial discrete valuation of $L$, $Z(L_{\nu_\pi}) \neq \emptyset$.
Hence ind$(A\otimes_FL_{\nu_\pi})$ divides $d$.
Since $L_{\nu_\pi} \subseteq L_{Q, \pi}$, ind$(A\otimes_F L_{Q, \pi})$ divides $d$.
Since, by (\cite[Corollary 5.6]{PPS}), ind$(A\otimes_FL_Q) = $ ind$(A\otimes_FL_{Q, \pi})$, ind$(A\otimes_F L_{Q})$ divides
$d$. Hence $Z(L_Q) \neq \emptyset$.
\end{proof}
\begin{cor}
\label{lgp-index} Let $K$ be a local field with residue field $\kappa$ and $F$ the function field of a smooth projective curve over $K$. Let $A$ be a central simple algebra over $F$ of index coprime to char$(k)$.
Let $\sheaf{X} $ be normal proper model of $F$ over the valuation ring of $K$ and $P \in \sheaf{X}$ be a closed point.
Let $L$ be the field $F$ or $F_P$.
Then ind$(A \otimes_F L ) =$ lcm$\{ {\rm ind}(A\otimes_{F}L_\nu) \mid \nu {\rm ~a ~divisorial ~discrete ~valuation~ of~} L \}$.
\end{cor}
\begin{proof} Let $d = $ lcm$\{ {\rm ind}(A\otimes_{F}L_\nu) \mid \nu {\rm ~a ~divisorial ~discrete ~valuation~ of~} L \}$.
Then clearly $d$ divides ind$(A\otimes_FL)$.
Thus it is enough to show that ind$(A\otimes_FL)$ divides $d$.
Let $Z = X(d)$. Since for every divisorial discrete valuation $\nu$ of $L$, ind$(A\otimes_FL_\nu)$ divides $d$,
$Z(F_\nu) \neq \emptyset$ (cf. \cite[5.3]{MPW1}). Hence, by (\ref{proj-pgl}), $Z(L) \neq \emptyset$.
Thus ind$(A\otimes_FL)$ divides $d$ (cf. \cite[5.3]{MPW1}).
\end{proof}
\begin{cor}
\label{proj-gl} Let $K$ be a local field with residue field $\kappa$ and $F$ the function field of a smooth projective curve over $K$. Let $A$ be a central simple algebra over $F$ of index coprime to char$(k)$.
Let $\sheaf{X} $ be a regular proper model of $F$ over the valuation ring of $K$ and $P \in \sheaf{X}$ be a closed point.
Let $L$ be the field $F$ or $F_P$. Let $G = GL(A)$ and $Z$ be a projective homogeneous space under $G$ over $L$.
If $Z(L_\nu) \neq \emptyset$ for all divisorial discrete valuation of $L$, then $Z(L) \neq \emptyset$.
\end{cor}
\begin{proof} Since the projective homogeneous spaces under $GL(A)$ are in bijection with the projective homogeneous spaces
under $PGL(A)$ (\cite[Theorem 2.20(i)] {BT}), the corollary follows from (\ref{proj-pgl}).
\end{proof}
\section{Local global principle for homogeneous spaces under unitary groups}
\label{lgp-herm}
Let $K$ be a local field with residue field $\kappa$ of characteristic not 2 and $F_0$ the function field of a smooth projective
curve over $K$. Let $F/F_0$ be a quadratic field extension.
Let $A$ be a central simple algebra over $F$ with an involution $\sigma$ of second kind and
$F^{\sigma} = F_0$. Let $(V, h)$ be an hermitian form over $(A, \sigma)$ and $G = U(A, \sigma, h)$.
If ind$(A) = 1$, then the validity of the conjecture 1 for $G$ is a consequence of results proved in (\cite{CTPS}).
If ind$(A) = 2$, Wu (\cite{Wu}) proved the validity of Conjecture 1 for $G$.
In this section we dispense with the condition ind$(A) \leq 2$ for the good characteristic case.
We begin by recalling the structure of projective homogeneous spaces under a unitary group over any field.
Let $F_0$ be a field and $F/F_0$ a separable quadratic extension.
Let $A$ be a central simple algebra over $F$ of degree $n$ with an involution $\sigma$ of second kind and
$F^{\sigma} = F_0$. Let $(V, h)$ be an hermitian form over $(A, \sigma)$ and $G = U(A, \sigma, h)$.
Let $W$ be a finitely generated module over $A$. The {\it reduced dimension} rdim$_A(W)$ of $W$ over $A$ is
defined as dim$_F(W)/n$ (\cite[Definition 1.9]{KMRT}). For $0 < n_1 < \cdots < n_r \leq n/2$ a sequence of integers and for any field extesnion
$L/F$, let
$$
\begin{array}{lll}
X(n_1, \cdots , n_r) & = & \{ (W_1, \cdots ,W_r) \mid \{ 0 \} \subsetneq W_1 \subsetneq \cdots \subsetneq W_r, W_i {\it ~a ~totally } \\
& & {\it ~~ isotropic ~subspace ~of} ~V ~{\it with ~rdim}_FW_i = n_i \}.
\end{array}
$$
We recall the following from (\cite{MPW1}, \cite{MPW2}, cf. \cite[\S 2]{Wu}).
\begin{theorem}
\label{proj-hom-spaces}
Let $F_0$ be a field and $F/F_0$ a separable quadratic extension.
Let $A$ be a central simple algebra over $F$ of degree $n$ with an involution $\sigma$ of second kind and
$F^{\sigma} = F_0$. Let $(V, h)$ be an hermitian form over $(A, \sigma)$ and $G = U(A, \sigma, h)$. Then \\
i) A projective variety $X$ over $F_0$ is a projective homogeneous space under $G$ over $F_0$
if and only if $X \simeq X(n_1, \cdots , n_r)$ for some increasing sequence of integers $0 < n_1 < \cdots < n_r \leq n/2$.\\
ii) For any field extension $L/F_0$, $X(n_1, \cdots, n_r)(L) \neq \emptyset$ if and only if
$X(n_r)(L) \neq \emptyset$ and ind$(A_L)$ divides $n_i$ for all $i$.\\
iii) If $A = M_r(D)$ for some central simple algebra over $F$ and $G_0 = SU(D, \sigma_0)$ for some unitary involution $\sigma_0$
on $D$, then there is a bijection assigning projective homogeneous spaces $X$
under $G$ and to projective homogeneous spaces $X_0$ under $G_0$. Further for any field extension $L/F_0$,
$X(L) \neq \emptyset$ if and only is $X_0(L) \neq \emptyset$.
\end{theorem}
\begin{theorem}
\label{isotropic0}
Let $K$ be a local field with residue field $\kappa$.
Let $F_0$ be the function field of a curve over $K$.
Let $F/F_0$ be a quadratic extension and $A$ be a central simple algebra over $F$
with an $F/F_0$- involution $\sigma$.
Suppose that $2$ ind$(A)$ is coprime to char$(\kappa)$.
Let $h$ be an hermitian form over $(A, \sigma)$ and $G = U(A, \sigma, h)$.
If $A = F$, then assume that rank of $h$ is at least 2.
Let $Z$ be a projective homogeneous space under $G$ over $F$.
Let $\sheaf{X}_0$ be a normal proper model of $F_0$ and $P \in \sheaf{X}_0$ be a closed point with
$F\otimes_{F_0}F_{0P}$ a field.
If $Z(F_{0\nu}) \neq \emptyset$ for all divisorial discrete valuations $\nu$ of $F_{0}$, then $Z( F_{0P}) \neq \emptyset$.
\end{theorem}
\begin{proof} Let $n$ be the degree of $A$.
Since $Z$ is a projective homogeneous space under $G$, by (\ref{proj-hom-spaces}),
$Z \simeq X(n_1,\cdots , n_r)$ for some sequence of integers $0 < n_1 < \cdots < n_r \leq n/2$.
Suppose that $Z(F_{0\nu}) \neq \emptyset$ for all divisorial discrete valuations $\nu$ of $F_{0}$.
Then, by (\ref{proj-hom-spaces}), ind$(A\otimes_{F_0}F_{0\nu})$ divides $n_i$ for all $i$.
Since $K$ is a local field, it follows from \cite[Proposition 5.10]{PPS} and \cite[Theorem 5.5]{HHK1}) that
ind$(A)$ is the lcm of ind$(A\otimes_{F_0}F_{0\nu})$ as $\nu$ varies over all divisorial discrete valuations of $F$.
Hence ind$(A)$ divides $n_i$ for all $i$. Thus, by (\ref{proj-hom-spaces}),
$X(n_r)(F_{0\nu}) \neq \emptyset$ for all all divisorial discrete valuations $\nu$ of $F$.
To prove the theorem, by (\ref{proj-hom-spaces}), it suffices to show that
$X(n_r) (F_{0P}) \neq \emptyset$. Thus we assume that $ Z = X(m)$ with $m = n_r$.
Let $T$ be the valuation ring of $K$. Then there exists a sequence of blow ups $\sheaf{X}'_0 \to \sheaf{X}_0 $ such that
the normalization $\sheaf{X}$ of $\sheaf{X}'_0$ in $F$ is regular and
the ramification locus of $A$ on $\sheaf{X}$ and the special fibre of $\sheaf{X}$ is a union of regular curves with normal crossings
(\cite{A}, \cite{L}, cf. \cite{Wu}). If necessary, by blowing up $P$, we assume that the fibre $V$ over $P$ is of dimension 1.
Then, by (\ref{refinesha2}), it is enough to show that $Z(F_{0x}) \neq \emptyset$ for all $x \in V$.
Let $x \in V$ be a generic point. Then $x$ gives a divisorial discrete valuation $\nu_x$ on $F_{0}$
such that $F_{0x} = F_{\nu_x}$.
Hence $Z(F_{0x}) \neq \emptyset$.
Let $Q \in V$ be a closed point. We show that $Z(F_{0Q}) \neq \emptyset$ by induction on ind$(A\otimes_{F_0}F_{0Q})$.
Suppose ind$(A \otimes_{F_0}F_{0Q} ) = 1$. Then the hermitian form $h$ corresponds to a quadratic form over
$q_h$ over $F_{0Q}$ such that $h$ is isotropic over any field extension $M$ of $F_Q$ if and only if
$q_h$ is isotropic over $M$ (\cite[Theorem 1.1, p. 348]{Sc}). Since $Z = X(m)$, for every divisorial discrete valuation $\nu$ of $F_0$,
there is a totally isotropic subspace of $V\otimes _{F_0}F_{0\nu}$ of dimension $m$.
Thus to prove the theorem, it is enough to show that
there is a totally isotropic subspace of $V \otimes _{F_0} F_{0Q}$ of dimension $m$.
By induction on dim$(q_h)$, it is enough to show that $q_h$ is isotropic over $F_{0Q}$.
By the assumption on the rank of $h$, rank of $q_h$ is at least 4.
Since for every
(divisorial) discrete valuation $\nu$ of $F_0$ centered on $Q$, $q_h$ is isotropic over $F_{0Q_\nu}$, by (\cite[Corollary 4.7]{HHK5}), $q_h$ is isotropic over $F_{0Q}$.
Suppose ind$(A \otimes_{F_0}F_{0Q} ) \geq 2$.
Then by the choice of the model $\sheaf{X}'_0$, we have the following:\\
i) the local ring $R_Q$ at $Q$ on $\sheaf{X}_0'$ is regular with $(\pi, \delta)$ as the maximal ideal,\\
ii) $F\otimes F_{0Q} = F_{0Q}(\sqrt{u\pi^{\epsilon}})$ for some unit $u \in R_Q$ and $\epsilon = 0, 1$, \\
iii) $A$ is unramified on the integral closure of $R_Q$ in $F_{0Q}(\sqrt{u\pi^{\epsilon}})$, except possibly at
$\pi_1$ and $\delta$, where $\pi_1 = \pi$ or $\sqrt{u\pi}$ depending on $\epsilon = 0$ or 1.
Let $D_Q$ be the central division algebra over $F \otimes_{F_0} F_{0Q}$ which is Brauer equivalent to $A \otimes_{F_0} F_{0Q}$.
The there is a unitary involution $\sigma_Q$ on $D_Q$ and the hermitian form $(V, h)$ corresponds to a
hermitian form $(V_Q, h_Q)$ over $D_Q$.
By (\ref{proj-hom-spaces}), $Z_{F_{0Q}} = X(m)_{F_{0Q}}$ corresponds to a projective homogeneous space $Z_Q = X(m')$ under $U(D_Q, \sigma_Q)$ for some suitable $m'$ which is divisible by ind$(D_Q)$.
Further to show that $Z(F_{0Q}) \neq \emptyset$, it is enough to show that $Z_Q(F_{0Q}) \neq \emptyset$.
Since ind$(A\otimes_{F_0}F_{0Q}) \geq 2$, deg$(D_Q) \geq 2$.
If deg$(D_Q)= 2$, let $\Lambda$ be the maximal $R_Q$-order of $D_Q$ as in (\cite[Lemma 3.7]{Wu}).
If deg$(D_Q) \geq 3$, let $\Lambda$ be the maximal $R_Q$-order of $D_Q$ as in (\ref{order}).
Since $D_Q$ is a division algebra, $h_Q \simeq <a_1, \cdots, a_n>$ for some $a_i \in \Lambda_P$.
Once again there exists a sequence of blow ups $\sheaf{X}''_0 \to \sheaf{X}'_0$ such that support of Nrd$(a_i)$ for all $i$
is a union of regular curves with normal crossings (\cite{A}, \cite{L}, cf. \cite[\S 4]{Wu}). Further, by blowing up, we also assume that $\sheaf{X}''_0$ satisfies i), ii) and iii).
Let $V''$ be the fibre over $Q$. Once again we assume that dim$(V'') = 1$. Thus to show that
$Z_Q(F_{0Q}) \neq \emptyset$, by (\ref{refinesha2}), it is enough to show that
$Z_Q(F_{0x}) \neq \emptyset$ for all $x \in V''$.
Let $x' \in V''$ be a generic point, then, as above, $Z_Q(F_{0x'}) \neq \emptyset$.
Let $Q' \in V''$ be a closed point.
Suppose that $D_Q \otimes_{F_0Q} F_{0Q'}$ is division.
By (\ref{proj-hom-spaces}), it is enough to show that
there is a totally isotropic subspace of $V_Q \otimes F_{0Q'}$ of dimension $m'$.
By induction on the reduced dimension of $V_Q$, it is enough to show that
$h_Q$ is isotropic over $F_{0Q'}$. Since $\sheaf{X}_0''$ satisfies i), ii) and iii),
the maximal ideal at $Q'$ on $\sheaf{X}_0''$ is generated by $(\pi, \delta)$,
$h_Q = <a_1, \cdots, a_n>$ for some $a_i \in \Lambda_P$ with Nrd$(a_i)$ is a supported along only $\pi$, $\delta$,
and $h_Q\otimes F_{0Q'\pi}$ is isotropic. Hence, by (\ref{isotropic}), $h_Q$ is isotropic over $F_{0Q'}$.
Suppose that $D_Q\otimes_{F_0Q}F_{0Q'}$ is not division. Then ind$(A\otimes_{F_0}F_{0Q'}) < $ ind$(A\otimes_{F_0} F_{0Q})$.
Hence, by induction, $Z_Q(F_{0Q'}) \neq \emptyset$.
\end{proof}
\begin{theorem}
\label{isotropic1}
Let $K$ be a local field with residue field $\kappa$.
Let $F_0$ be a function field of a curve over $K$.
Let $F/F_0$ be a quadratic extension and $A$ a central simple algebra over $F$ of index $n$
with an $F/F_0$- involution $\sigma$.
Suppose that $2n$ is coprime to char$(\kappa)$.
Let $h$ be an hermitian form over $(A, \sigma)$.
If $A = F$, then assume that rank of $h$ is at least 2.
Let $Z$ be a projective homogeneous space under $U(A, \sigma, h)$ over $F$.
If $Z(F_{0\nu}) \neq \emptyset$ for all (divisorial) discrete valuations $\nu$ of $F_{0}$, then $Z( F_{0}) \neq \emptyset$. \end{theorem}
\begin{proof} Suppose
$Z(F_{0\nu}) \neq \emptyset$ for all (divisorial) discrete valuations $\nu$ of $F_{0}$.
Let $\sheaf{X}_0$ be a normal proper model of $F_0$ over the valuation ring of $K$ such that
the normal closure of $\sheaf{X}_0$ and $X_0$ the
special fibre.
Let $x \in X_0$ be a codimension 0 point in $X_0$. Then $F_{0x} = F_{0\nu_x}$ for the discrete valuation
$\nu_x$ of $F_0$ given by $x$. Hence $Z(F_{0x}) \neq \emptyset$.
Let $P \in X_0$ be a closed point.
We have $A \otimes_{F_0}F\simeq A_1 \times A_1^{\rm op}$ for some central simple algebra $A_1/F$ (\cite[Proposition 2.14]{KMRT})
and
$U(A, \sigma, h)\otimes F \simeq GL(A_1)$ (\cite[p. 346]{KMRT}).
Suppose $F \otimes_{F_0}F_{0P}$ is not a field.
Then $F \subset F_{0P}$ and $A\otimes_{F_0}F_{0P} \simeq A_1 \otimes F_{0P} \times A_1^{\rm op} \otimes {F_0}$.
Let $\sheaf{X}$ be the normal closure of $\sheaf{X}_0$ in $F$.
Since $F\otimes_{F_0}F_{0P}$ is not a field, there exists a closed point $Q$ of $\sheaf{X}$ such that
$F_Q \simeq F_{0P}$.
Hence, by (\ref{proj-gl}), $Z(F \otimes_{F_0}F_{0P}) \neq \emptyset$.
Suppose $F \otimes_{F_0}F_{0P}$ is a field.
Then, by (\ref{isotropic0}), $Z( F_{0P}) \neq \emptyset$.
Since $U(A, \sigma, h)$ is rational and connected (\cite[p. 195, Lemma 1]{Merkurjev}), by (\cite[Corollary 6.5 and Theorem 9.1]{HHK3}), $Z(F_0) \neq \emptyset$.
\end{proof}
\begin{theorem}
\label{uas}
Let $K$ be a local field with residue field $\kappa$.
Let $F_0$ be a function field of a curve over $K$.
Let $F/F_0$ be a quadratic extension and $A$ a central simple algebra over $F$ of index $n$
with an $F/F_0$- involution $\sigma$.
Suppose that $2n$ is coprime to char$(\kappa)$.
Let $h$ be an hermitian form over $(A, \sigma)$.
Then the canonical map $$H^1(F_0, U(A, \sigma, h) ) \to \prod_{\nu \in \Omega_{F_0}} H^1(F_{0\nu}, U(A, \sigma, h))$$
has trivial kernel.
\end{theorem}
\begin{proof} Let
$\xi \in H^1(F_{0}, U(A, \sigma, h))$. Then $\xi$ corresponds to a
hermitian spaces $h'$ over $(A, \sigma)$ of reduced rank equal to the reduced rank of $h$.
Let $h_0 = h\perp -h'$ and $m$ the reduced rank of $h$.
Let $G = U(A,\sigma, h_0)$ and $Z = X(m)$.
Then $Z$ is a projective homogeneous variety under $G$ over $F_0$.
Suppose $\xi$ maps to the trivial element in $H^1(F_{0\nu}, U(A,\sigma, h))$ for all (divisorial) discrete valuations
$\nu$ of $F_0$. Then $h' \otimes F_{0\nu} \simeq h\otimes F_{0\nu}$ and hence $h_0$ is hyperbolic.
Thus $Z(F_{0\nu}) \neq \emptyset$ for all (divisorial) discrete valuations
$\nu$ of $F_0$. Hence, by (\ref{isotropic1}), $Z(F_0) \neq \emptyset$.
In particular $h_0$ is hyperbolic. Since the reduced rank of $h$ and $h'$ are equal,
$h \simeq h'$ and $\xi$ is the trivial element in $H^1(F_0, U(A, \sigma, h))$.
\end{proof}
\section{Local global principle for special unitary groups - patching setup}
\label{lgp-patching}
Let $F_0$ be a field of characteristic not equal to 2 and $F/F_0$ a quadratic \'etale extension.
Let $A$ be a central simple algebra over $F$ of degree $n$ with an involution $\sigma$ of second kind and
$F^{\sigma} = F_0$.
Then we have an exact sequence of algebraic groups
$$ 1 \to SU(A, \sigma) \to U(A, \sigma) \to R^1_{F/F_0}(\Group{G}_{\text{m}}) \to 1.$$
For any field extension $L/F_0$, we have an
induced exact sequence
$$U(A, \sigma)(L) \to (L\otimes_{F_0}F)^{*1} \to H^1(L, SU(A, \sigma)) \to H^1(L, U(A, \sigma)) \hskip 1.5cm (\star)$$
where $(L\otimes_{F_0}F)^{*1} = R^1_{F/F_0}(\Group{G}_{\text{m}})(L)$ is the subgroup of $(L\otimes_{F_0}F)^*$ consisting of norm one elements and the map $U(A,\sigma)(L) \to
(L\otimes_{F_0}F)^{*1}$ is given by the reduced norm.
Further the image of
$U(A, \sigma)(L) \to (L\otimes_{F_0} F)^{*1}$ is equal to
$\{ \theta^{-1} \sigma(\theta) \mid \theta \in Nrd(A\otimes_{F_0}L^*) \} $ (\cite[p. 202]{KMRT}).
Let $K$ be a local field with the residue field $\kappa$ and $F_0$ the function field of a smooth projective
curve over $K$. Let $F/F_0$ be a separable quadratic extension.
Let $A$ be a central simple algebra over $F$ of degree $n$ with an involution $\sigma$ of second kind and
$F^{\sigma} = F_0$. Suppose that $2n$ is coprime to char$(\kappa)$.
In this section we show that there is a local global principal for principal homogeneous spaces under $SU(A, \sigma)$ over $F_0$
in the patching setup (cf. \ref{patching-su}).
Let $\mu \in F^*$. Let $F= F_0(\sqrt{d})$, $d \in F_0^*$.
Let $T$ be the valuation ring of $K$.
Then there exists a regular proper model $\sheaf{X}_0 \to Spec(T)$ of $F_0$ with the
normalization $\sheaf{X}$ of $\sheaf{X}_0$ in $F$ regular and with the property that
the special fibre $X$ of $\sheaf{X}$, the ramification locus of $F/F_0$ on $\sheaf{X}$,
the ramification locus of $A$ on $\sheaf{X}$ and the support of $\mu$ on $\sheaf{X}$ are
a union of regular curves with normal crossings (\cite{A}, \cite{L}).
Let $X_0$ be the reduce special fibre of $\sheaf{X}_0$ and $\{\eta_1, \cdots , \eta_m \}$ be the generic points of $X_0$.
Let $\sheaf{P}_0$ be a finite set of closed points of $X_0$ containing
all the singular points of $X_0$ and at least one closed point from each irreducible component of $X_0$.
Let $\sheaf{U}_0$ be the set of irreducible components of $X_0 \setminus \sheaf{P}_0$.
We fix the data $\mu \in F^*$, $\sheaf{X}_0, \sheaf{P}_0$ and $\sheaf{U}_0$ for until (\ref{patching-su}).
Let $\sheaf{B}_0$ be the set of branches at $\sheaf{P}_0$. Since $X_0$ is a union of regular curves with normal crossings,
$\sheaf{B}_0$ is in bijection with pairs $(P, U)$ with $P \in \sheaf{P}_0$, $U \in \sheaf{U}_0$ and $P $ is in the closure of $U$.
Let $\eta \in X_0$ be a generic point and $P \in \overline{\{ \eta \} }$ a closed point.
Then $\eta$ defines a discrete valuation $\nu_\eta$ on $F_{0P}$. Then the completion of
$F_0$ at the restriction of $\nu_\eta$ to $F_0$ is denoted by $F_{0\eta}$ and
the completion of $F_{0P}$ at $\nu_\eta$ denoted by $F_{0P,\eta}$.
The closed point $P$ induces a discrete valuation $\nu_P$ on the residue field $\kappa(\eta)$ of
$F_{0\eta}$ such that the completion $\kappa(\eta)_P$ of $\kappa(\eta)$ at $\nu_P$ is the residue field of
$F_{0P, \eta}$.
Let $P \in X_0$ be a closed point and $A_{P}$ the local ring at $P$ on $\sheaf{X}_0$.
Since the normalization of $\sheaf{X}_0$ in $F$ is regular,
$d = \pi u$ or $d = u$ for some $\pi \in A_{P}$ a regular parameter and $u \in A_{P}$ a unit.
Hence $B_P = A_P[\sqrt{d}]$ is the integral closure of $A_P$ in $F_0$.
Let $\delta \in A_P$ be such that $m_P = (\pi, \delta)$ be the maximal ideal of $A_P$.
If $d= \pi u $, then $B_P$ is local and $(\sqrt{\pi u}, \delta)$ is the maximal ideal of $B_P$.
Suppose $d=u$ a unit in $A_P$. If $u$ is not a square in the residue field $\kappa(P)$, then
$B_P$ is local and the maximal ideal of $B_P$ is generated by $\pi$ and $\delta$.
We begin with the following.
\begin{lemma}
\label{correction-eta}
Let $\eta$ be a generic point of $X_0$ and $S$ be a finite set of closed points of $\overline{\{ \eta \} }$.
For every $P \in S$, let $\theta_{\eta, P} \in F_{0P, \eta}^*$ be unit at $\eta$ which is a reduced norm from $A\otimes F_{0P, \eta}$.
Then there exists $\theta_\eta \in F_{0\eta}$ which is a reduced norm from $A \otimes F_{0\eta}$ such that
$\theta_\eta \theta^{-1}_{\eta, P} \in F_{0P, \eta}^{*n}$ for all $P \in S$.
\end{lemma}
\begin{proof} Suppose $F_\eta = F \otimes F_{0\eta}/F_{0\eta}$ is a ramified field extension.
Then by (\ref{ramifiedh2}),
there exists an unramified algebra $A_0$ over $F_{0\eta}$ such that
$A \otimes _{F_0} F_{0\eta} \simeq A_0 \otimes_{F_0\eta} F_{\eta}$.
For $P \in S$, let $\overline{\theta}_{\eta, P} \in \kappa(\eta)^*_P$ be the image of
$\theta_{\eta, P} \in F_{0P, \eta}^*$.
We choose $\overline{\theta}_\eta \in \kappa(\eta)^*$ be close to $\overline{\theta}_{\eta, P}$ for all
$P \in S$. Since $A_0$ is unramified over $F_{0\eta}$,
its specialization $B_0$ is a central simple algebra over $\kappa(\eta)$.
Since $\kappa(\eta)$ is a global field of positive characteristic,
by Hasse-Maass-Schilling theorem $\overline{\theta}_\eta$ is a reduced norm from $B_0$.
Let $\theta_\eta \in F_{0\eta}$ be a lift of $\overline{\theta}_\eta$.
Since $F_{0\eta}$ is complete, $\theta_\eta$ is a reduced norm from $A_0\otimes_{F_0}F_{0\eta}$
and hence a reduced norm from $A\otimes_{F}F_{\eta}$.
Since $\overline{\theta}_\eta$ is close to $\overline{\theta}_{\eta, P}$ for all $P \in S$
and $n$ is coprime to char$(\kappa)$,
$\overline{\theta}_\eta \overline{\theta}_{\eta, P}^{-1}
\in \kappa(\eta)_P^{*n}$ for all $P \in S$.
Since $F_{0P, \eta}$ is complete with residue field $\kappa(\eta)_P$,
$\theta_\eta \theta_{\eta, P}^{-1} \in F_{0P, \eta}^{*n}$ for all $P \in S$.
Suppose that $F_\eta/F_{0\eta}$ is an unramified field extension.
Then the residue field $\tilde{\kappa}( \eta)$ of $F_\eta$
is a quadratic extension of $\kappa(\eta)$.
Let $(L_\eta, \sigma_\eta)$ be the residue of $A$ at $\eta$.
Since the residue commutes with the corestriction, cores$_{\tilde{\kappa}(\eta) /\kappa(\eta)}(L_\eta, \sigma_\eta) = 0$.
Thus, by (\ref{galois}), $L_\eta/\kappa(\eta)$ is a dihedral extension.
Since $\theta_{\eta, P} $ is a reduced norm from $A\otimes F_{0P, \eta}$, we have
$A \cdot (\theta_{\eta, P}) = 0 \in H^3(F\otimes F_{0P, \eta}, \mu_n)$.
Let $\overline{\theta}_{\eta, P}$ be the image of $\theta_{\eta, P}$ in the residue field $\kappa(\eta)_P$ of
$F_{P, \eta}$. By taking the residue of $A \cdot (\theta_{\eta, P})$, we get that
$(L_\eta, \sigma_\eta, \overline{\theta}_{\eta, P}) = 0$ (cf. \cite[Proof of 4.7]{PPS}).
Hence $\overline{\theta}_{\eta, P}$ is a norm from the extension $L_\eta \otimes_{\kappa(\eta)} \kappa(\eta)_P/ \tilde{\kappa}(\eta) \otimes_{\kappa(\eta)} \kappa(\eta)_P$.
Since $\kappa(\eta)$ is a global field, by (\ref{norms-approximation}), there exists $\overline{\theta}_\eta \in \kappa(\eta)^*$
with $\overline{\theta}_\eta$ a norm from
$L_\eta/\tilde{\kappa}(\eta)$ and $\overline{\theta}_\eta \overline{\theta}_{\eta, P}^{-1} \in\kappa(\eta)_P^{*n}$.
Let $\theta_\eta \in F_{0\eta}$ be a lift of $\overline{\theta}_\eta \in \kappa(\eta)$.
Then $\theta_\eta\theta_{\eta, P}^{-1} \in F_{0P, \eta}^{*n}$.
Since $\overline{\theta}_\eta$ is a norm from $L_\eta/\tilde{\kappa}(\eta)$,
by (\ref{reduced-norms}), $\theta_\eta$ is a reduced norm from $A \otimes F_{0\eta}$.
Suppose $F_\eta = F \otimes_{F_0} F_{0\eta}$ is not a field. Then $F_\eta \simeq F_{0\eta} \times F_{0\eta}$
and $A \otimes_{F_0} F_{0\eta} \simeq A_1 \times A_1^{op}$, where $A_1^{op}$ is the opposite algebra. Since
$\theta_{\eta, P} \in F_{0P, \eta}$ is a reduced norm from $A\otimes F_{0P, \eta}$, $\theta_{\eta, P}$ is a reduced norm from $A_1 \otimes F_{0P, \eta}$.
Then, as above, we can find $\theta_\eta \in F_{0\eta}$ such that $\theta_\eta\theta_{\eta, P}^{-1} \in F_{0P, \eta}^{*n}$
and $\theta_\eta$ is a reduced norm from $A_1$. Then $\theta_\eta$ is a reduced norm from $A\otimes_{F_0} F_{0\eta}$.
\end{proof}
\begin{lemma}
\label{choice-at-eta}
Suppose that for every generic point $\eta$ of $X_0$ there exists
$c_\eta \in F_{0\eta}^*$ such that $\mu c_\eta $ is a reduced norm from $A \otimes F_{0, \eta}$.
Then for every generic point $\eta$ of $X_0$, there exists
$a_\eta \in F_{0\eta}^*$ such that $\mu a_\eta$ is a reduced norm from $A \otimes F_{0\eta}$
with the following property: if $\eta_1$ and $\eta_2$ are two generic points of $X_0$ and
$P \in \overline{ \{\eta_1\}} \cap \overline{ \{\eta_2\}}$ with $F\otimes F_{0P}$ is a field,
then there exists $a_P \in F_{0P}^*$ such that
$\mu a_P$ is a reduced norm from $A \otimes F_{0P}$
and $a_{\eta_i} a_P^{-1} \in F_{0P, \eta_i}^{*n}$ for $ = 1, 2$.
\end{lemma}
\begin{proof} Since the special fibre is a union of regular curves with normal crossings,
for a generic point $\eta$ of $X_0$, there exists $\pi_\eta \in F_0$ a parameter at $\eta$ such that
for every closed point $P \in \overline{ \{\eta_1\}} \cap \overline{ \{\eta_2\}}$ for any two distinct
generic points $\eta_1$ and $\eta_2$ of $X_0$, the maximal ideal at $P$ is $(\pi_{\eta_1}, \pi_{\eta_2})$.
Suppose that for every generic point $\eta$ of $X_0$ there exists
$c_\eta \in F_{0\eta}$ such that $\mu c_\eta $ is a reduced norm from $A \otimes F_{0, \eta}$.
For every generic point $\eta$ of $X_0$, let $r_\eta = \nu_\eta(c_\eta)$.
For every closed point $P \in \overline{ \{\eta_1\}} \cap \overline{ \{\eta_2\}}$ with $F\otimes F_{0P}$ is a field,
let $a_P = \pi_{\eta_1}^{r_{\eta_1}} \pi_{\eta_2}^{r_{\eta_2}} \in F_{0P}^*$.
Let $\eta$ be a generic point of $X_0$. Let $P \in \overline{ \{\eta \}} \cap \overline{ \{\eta'\}}$ for some generic point
$\eta' \neq \eta$. Suppose that $F\otimes F_{0P}$ is a field. By the choice of $\sheaf{X}_0$,
$F/F_0$ is unramified at $P$ except possibly at $\pi_{\eta}$ and $\pi_{\eta'}$.
Since the maximal ideal at $P$ is $(\pi_{\eta}, \pi_{\eta'})$, by (\cite[Corollary 5.5]{PPS}), $F\otimes F_{0P, {\eta} }$ is a field.
Since $n$ is coprime to char$(\kappa(P))$,
$c_\eta = u_P \pi_{\eta}^{r_{\eta}} \pi_{\eta'}^{s_P} (b_P)^n $ for some $s_P \in \mathbb Z$, $u_P \in \hat{A}_P$ a unit and
$b_P \in F_{0P, \eta}^*$ (cf. \ref{branch-elements}).
Let $\theta_{\eta, P} = u_P^{-1}\pi_{\eta'}^{r_{\eta'}-s_P}$.
Since $F\otimes_{F_0}F_{0P}$ is a field, $\theta_{\eta, P}$ is a reduced norm from $A\otimes_{F_0}F_{0P, \eta}$ (\ref{branch-norms}).
Since $\theta_{\eta, P}$ is a unit at $\eta$, by (\ref{correction-eta}), there exists $\theta_{\eta} \in F_{0\eta}$
which is a reduced norm from $A \otimes F_{0\eta}$ such that $\theta_\eta \theta_{\eta, P}^{-1} \in F_{0P, \eta}^{*n}$.
Let $a_\eta = c_\eta \theta_\eta$. Since $\mu c_\eta$ and $\theta_\eta$ are reduced norms from $A \otimes F_{0\eta}$,
$\mu a_\eta$ is a reduced norm from $A \otimes F_{0\eta}$.
Let $P \in \overline{ \{\eta \}} \cap \overline{ \{\eta'\}}$ for some generic point
$\eta' \neq \eta$ with $F\otimes F_{0P}$ is a field.
Then, by the choice of $a_\eta$, we have $a_\eta = \pi_\eta^{r_\eta} \pi_{\eta'}^{r_{\eta'}}$ modulo $F_{0P, \eta}^{*n}$.
Hence $a_\eta a_P^{-1} \in F_{0P, \eta}^{*n}$.
\end{proof}
\begin{lemma}
\label{choice-at-split-point}
Let $\eta_1$ and $\eta_2$ be two distinct generic points of $X_0$.
Suppose $P \in \overline{\{ \eta_1 \}} \cap \overline{\{ \eta_2 \}}$ is a closed point with
$F \otimes F_{0P}$ is not a field.
Suppose there exist $a_{\eta_i} \in F_{\eta_i}^*$ with
$\mu a_{\eta_i}$ is a reduced norm from $A \otimes F_{0\eta_i}$.
Then there exists $a_P \in F_{0P}^*$ such that
$\mu a_P$ is a reduced norm from $A \otimes F_{0P}$
and $a_{\eta_i} a_P^{-1} \in F_{0P, \eta_i}^{*n}$ for $ i = 1, 2$.
\end{lemma}
\begin{proof} Since $F\otimes F_{0P}$ is not a field and cores$_{F/F_0}(A) = 0$,
$F\otimes F_{0P} \simeq F_{0P} \times F_{0P}$
and $A \otimes F_{0P} \simeq A_1 \times A_1^{op}$ for some central simple algebra
$A_1$ over $F_{0P}$.
Write $\mu = (\mu_1, \mu_2)$.
Since $a_{\eta_i}\mu$ is a reduced norm from $A \otimes F_{\eta_i} $and $a_{\eta_i} \in F_{0\eta_i}^*$,
$a_{\eta_i}\mu_1$ and $\mu_1\mu_2^{-1}$ are reduced norms from $A_1 \otimes F_{0P, \eta_i}$.
Since by the choice of $\sheaf{X}_0$, the support of $\mu$ on $\sheaf{X}$ and the ramification locus of $A$ on
$\sheaf{X}$ is a union of regular curves with normal crossings, by (\ref{2dim-local-nrd}),
$\mu_1\mu_2^{-1}$ is a reduced norm from $A_1\otimes F_{0P}$.
The generic points $\eta_1$ and $\eta_2$ give discrete valuations $\nu_1$ and $\nu_2$ on $F_{0P}$
with completions $F_{0\eta_1, P}$ and $F_{0\eta_2, P}$.
Let $z_i \in A_1 \otimes F_{0P, \eta_i}$ with reduced norm $a_{\eta_i}\mu_1$.
Let $z \in A_1 \otimes F_{0P}$ be close to $z_i$ for $i = 1, 2$.
Let $a_P = \mu_1^{-1}Nrd(z) \in F_{0P}$.
Then $\mu_1 a_P $ is a reduced norm from $A_1 \otimes F_{0P}$.
Since $z$ is close to $z_i$ and Nrd$(z_i) = a_{\eta_i}\mu_1$,
Nrd$(z)$ is close to $a_{\eta_i} \mu_1$. Hence $a_P$ is close to $a_{\eta_i}$.
Therefore $a_{\eta_i}a_P^{-1} \in F_{0P, \eta_i}^{*n}$.
Since $\mu_1\mu_2^{-1}$ is a reduced norm and $a_P \mu_1$ is a reduced norm,
$a_P\mu_2$ is a reduced norm. In particular $a_P \mu$ is a reduced norm.
\end{proof}
\begin{lemma}
\label{choice-at-curve-point}
Let $\eta$ be a generic point of $X_0$ and $P \in \overline{\{ \eta\}} $ a closed point.
Suppose there exist $a_{\eta} \in F_{\eta}$ with
$\mu a_{\eta}$ is a reduced norm from $A \otimes F_{0\eta}$.
Then there exists $a_P \in F_{0P}^*$ such that
$\mu a_P$ is a reduced norm from $A \otimes F_{0P}$
and $a_{\eta} a_P^{-1} \in F_{0P, \eta}^{*n}$.
\end{lemma}
\begin{proof} Suppose that $F\otimes F_{0P}$ is a field.
Then, by the choice of $\sheaf{X}_0$ and by (\ref{2dim-local}),
there exists $a_P \in F_{0P}$ such that
$\mu a_P$ is a reduced norm from $A \otimes F_{0P}$
and $a_{\eta} a_P^{-1} \in F_{0U, P}^n$.
Suppose $F\otimes F_{0P}$ is a not field.
Then, we get the required $a_P$ as in the proof of (\ref{choice-at-split-point}).
\end{proof}
\begin{theorem}
\label{patching-su}
Let $K$ be a local field with the residue field $\kappa$ and valuation ring $T$.
Let $F_0$ be the function field of a smooth projective curve over $K$ and $F/F_0$ a separable quadratic extension.
Let $A$ be a central simple algebra over $F$ of degree $n$ with an involution $\sigma$ of second kind and
$F^{\sigma} = F_0$. Suppose that $2n$ is coprime to char$(\kappa)$.
Let $\sheaf{X}_0 \to Spec(T)$ be a proper normal model of $F_0$ with special fibre $X_0$.
Let $\sheaf{P}_0$ be a finite set of closed points of $X_0$ containing all the singular points of $X_0$
and $\sheaf{U}_0$ the set of irreducible components of $X_0 \setminus \sheaf{P}_0$.
Then the canonical map $$H^1(F_0, SU(A, \sigma) \to \prod_{U \in \sheaf{U}_0} H^1(F_{0U}, SU(A, \sigma)) \times \prod_{P\in \sheaf{P}_0} H^1(F_{0P}, SU(A, \sigma))$$
has trivial kernel.
\end{theorem}
\begin{proof}
Let $\xi \in H^1(F_0, SU(A, \sigma))$. Suppose that $\xi $ maps to 0 in $H^1(F_{0x}, SU(A, \sigma))$ for all $x \in \sheaf{U}_0 \cup \sheaf{P}_0$.
Since $U(A, \sigma)$ is rational and connected (\cite[p. 195, Lemma 1]{Merkurjev}), by (\cite[Theorem 3.7]{HHK1}), $\xi$ maps to 0 in $H^1(F_0, U(A, \sigma))$.
Hence from the exact sequence ($\star$ of \S \ref{lgp-patching}), there exists $\lambda \in F^{*1}$ such that
$\lambda$ maps to $\xi$ in $H^1(F_0, SU(A, \sigma)$.
Let $\mu \in F^*$ be such that $\lambda = \mu^{-1} \sigma(\mu)$.
Since $\xi$ maps to 0 in $H^1(F_{0U}, SU(A, \sigma))$, there exists $c_U \in F_{0U}$ such that
$c_{U} \mu $ is a reduced norm from $A\otimes_{F_0}F_{0U}$ (cf. (\cite[p. 202]{KMRT})).
Then, there exists a sequence of blow-ups $\sheaf{X}_0' \to \sheaf{X}_0$ such that $\sheaf{X}_0'$ is regular,
the integral closure $\sheaf{X}'$ of $\sheaf{X}_0'$ in $F$ is regular and the union of the special fibre of $\sheaf{X}'$,
the ramification locus of $A$ on $\sheaf{X}'$ and the support of $\mu$ on $\sheaf{X}'$ is a union of regular curves with normal crossings.
Let $\sheaf{P}_0'$ be a finite set of closed points of $\sheaf{X}_0'$ containing all the singular points of the special fibre $X_0'$ of $\sheaf{X}_0'$
and at least one closed point lying over points of $\sheaf{P}_0$. Let $\sheaf{U}_0'$ be the set of components of $X_0' \setminus \sheaf{P}_0'$.
Then $\xi $ maps to 0 in $ H^1(F_{0x'}, SU(A, \sigma))$ for all $x' \in \sheaf{P}_0' \cup \sheaf{U}_0'$ (\cite[\S 5]{HHK3}).
Thus, replacing $\sheaf{X}_0$ by $\sheaf{X}_0'$, we assume that
the integral closure $\sheaf{X}$ of $\sheaf{X}_0$ in $F$ is regular and the union of the special fibre of $\sheaf{X}$,
the ramification locus of $A$ on $\sheaf{X}$ and the support of $\mu$ on $\sheaf{X}$ is a union of regular curves with normal crossings.
Let $\eta$ be a generic point of $X_0$. Then $\eta \in U_\eta$ for some $U_\eta \in \sheaf{U}$.
Let $c_\eta = c_{U_\eta}$. Since $F_{0U_\eta} \subset F_{0\eta}$, $c_\eta \in F_{0\eta}^*$
and $c_\eta \mu $ is a reduced norm from $A\otimes_{F_0}F_{0\eta}$.
Let $a_\eta \in F_{0\eta}$ be as in (\ref{choice-at-eta}).
Then, by Artin's approximation (\cite[Theorem 1.10]{Artin}, as in the proof of (\cite[Lemma 7.2]{PPS}),
there exists a nonempty open subset $V_\eta$ of $U_\eta$ such that
$a_\eta \in F_{0V_\eta}$ (\cite[Lemma 3.2.1]{HHK4}) and $a_\eta \mu $ is a reduced norm from
$A\otimes_{F_0}F_{0V_\eta}$. Let $a_{V_\eta} = a_\eta^h \in F_{0V_\eta}$.
Let $\sheaf{U}'$ be the set of these $V_\eta$'s.
Let $\sheaf{P}_0'$ be the complement of the union of $V_\eta$'s in $X_0$.
Then $\sheaf{U}' $ is the set of components of $X_0 \setminus \sheaf{P}_0'$.
Let $P \in \sheaf{P}'_0$. Suppose that $P \in \overline{\{\eta\}} \cap \overline{\{\eta'\}}$ for two distinct
generic points $\eta$ and $\eta'$ of $X_0$. Then $P \in \sheaf{P}_0$.
If $F\otimes F_{0P}$ is a field, then let $a_P \in F_{0P}$ be as in (\ref{choice-at-eta}).
If $F\otimes F_{0P}$ is not a field, let $a_P \in F_{0P}$ be as in (\ref{choice-at-split-point}).
Suppose $P \in \overline{\{\eta\}}$ for some generic point $\eta$ of $X_0$
and $P \not\in \overline{\{\eta'\}}$ for all generic points $\eta'$ of $X_0$ not equal to $\eta$.
Let $a_P \in F_{0P}$ be as in (\ref{choice-at-curve-point}).
Let $(V, P)$ be a branch. Then $P \in \overline{V}$. By the choice of $a_P$ and $a_V$, we have
$a_V a_P^{-1} = b_{V,P}^n$ for some $b_{V, P} \in F_{0V, P}^*$.
By (\cite[Corollary 3.4]{HHK3}), for every $x \in \sheaf{U}'_0 \cup \sheaf{P}'_0$, there exists $b_x \in F_{0x}^*$ such that
$b_{V, P} = b_Vb_P$ for all branches $(V, P)$.
For $V \in \sheaf{U}_0$, let $a_V' = a_V b_V^{-n}$
and for $P \in \sheaf{P}'_0$, let $a_P' = a_P b_P^n$.
Then, we have $a_V' = a'_P \in F_{0V, P}$ for all branches $(V, P)$.
Hence there exists $a' \in F_0$ such that $a' = a'_x \in F_{0x}$ for all $x \in \sheaf{U}'_0 \cup \sheaf{P}'_0$ (\cite[\S 3]{HHK3}).
Since $\mu a_x $ is a reduced norm from $A \otimes F_{0x}$ for all $x \in \sheaf{U}'_0 \cup \sheaf{P}'_0$
and $n$ is the degree of $A$, $\mu a'_x$ is a reduced norm from $A \otimes F_{0x}$ for all $x \in \sheaf{U}'_0 \cup \sheaf{P}'_0$.
Thus, by (\cite[Corollary 11.2]{PPS} and \cite[Proposition 8.2]{HHK3}), $\mu a'$ is a reduced norm from $A$.
Since $\lambda = (\mu a')^{-1} \sigma(\mu a')$, $\lambda$ is in the image of
$U(A, \sigma)(F_0) \to F^{*1}$ and hence $\xi$ is trivial.
\end{proof}
The following is immediate from (\ref{patching-su}) and (\cite[Corollary 5.9]{HHK3})
\begin{cor}
\label{pointsha-su}
Let $K$ be a local field with residue field $\kappa$.
Let $F_0$ be a function field of a curve over $K$.
Let $F/F_0$ be a quadratic extension and $A$ a central simple algebra over $F$ of index $n$
with an $F/F_0$- involution $\sigma$.
Suppose that $2n$ is coprime to char$(\kappa)$.
Then the canonical map $$H^1(F_0, SU(A, \sigma)) \to \prod_{x \in X_0} H^1(F_{0x}, SU(A, \sigma)) $$
has trivial kernel.
\end{cor}
\section{Local global principle for special unitary groups - discrete valuations}
\label{lgp-dvr}
\begin{theorem}
\label{lgp-unitary}
Let $K$ be a local field with residue field $\kappa$.
Let $F_0$ be a function field of a curve over $K$.
Let $F/F_0$ be a quadratic extension and $A$ a central simple algebra over $F$ of index $n$
with an $F/F_0$- involution $\sigma$.
Suppose that $2n$ is coprime to char$(\kappa)$.
Then the canonical map $$H^1(F_0, SU(A, \sigma) ) \to \prod_{\nu \in \Omega_{F_0}} H^1(F_{0\nu}, SU(A, \sigma))$$
has trivial kernel.
\end{theorem}
\begin{proof} Let $\xi \in H^1(F_0, SU(A, \sigma))$. Suppose that $\xi$ maps to 0 in $H^1(F_{0\nu}, SU(A,\sigma))$ for all $\nu \in \Omega_{F_0}$.
By (\ref{uas}), the image of $\xi$ in $H^1(F_0, U(A, \sigma))$ is zero.
Hence from the exact sequence ($\star$) of \S \ref{lgp-patching}, there exists $\lambda \in F^{*1}$ such that
$\lambda$ maps to $\xi$ in $H^1(F_0, SU(A, \sigma)$.
Write $\lambda = \mu^{-1} \sigma(\mu)$ for some $\mu \in F^*$.
Let $d \in F_0^*$ be such that
$F = F_0(\sqrt{d})$.
There exists a regular proper model $\sheaf{X}_0$ of $F_0$ such that the special fibre and the support of
$d$ is a union of regular curves with normal crossings. Further the integral closure $\sheaf{X}$ of $\sheaf{X}_0$ in $F$ has the property:
$\sheaf{X}$ is regular,
the special fibre of $\sheaf{X}$, the ramification locus of $(A, \sigma)$, the support of $d$, $\mu$ and $\lambda$
is a union of regular curves with normal crossings (cf. \cite{Wu}). Let $X_0$ be the special fibre of $\sheaf{X}_0$.
Let $x \in X_0$ be codimension zero point. Then $x$ gives a discrete valuation $\nu_x$ on $F_0$ and $F_{0x} = F_{0\nu_x}$.
Hence $\xi$ maps to zero in $H^1(F_{0x}, SU(A,\sigma))$.
Let $P \in X_0$ be a closed point. Let $A_P$ be the local ring at $P$ and $B_P$ the integral closure of
$A_P$ in $F$. Since $B_P$ is regular, there is at most one irreducible
curve of $\sheaf{X}$ in the support of $d$ which passes through $P$.
Further there are at most two curves passing through $P$ which are in the union of special fibre of $\sheaf{X}'$,
the support of $\mu$ and ramification locus of $A$. Let $x$ be one such curve and $\nu_x$ the discrete valuation of $F_0$ given by $x$.
Then $F_{0 \nu_x} \subset F_{0P, \nu_x}$ and hence $\xi$ maps to 0 in $H^1(F_{0P, \nu_x}, SU(A, \sigma))$.
Since $\lambda$ maps to $\xi$, there exists $\theta_x \in F_{0P, \nu_x}$ such that
$\mu \theta_x \in Nrd(A\otimes F_{0P, \nu_x})$. Hence,
by (\ref{2dim-local}), there exists $\theta_P \in F_{0P}$ such that $\mu\theta_P \in Nrd(A\otimes F_P)$.
In particular $\xi \otimes F_P$ is trivial. Hence, by (\ref{pointsha-su}), $\xi$ is trivial.
\end{proof}
\section{Conjectures 1 and 2 for classical groups}
\label{conjectures}
In this section we prove the validity of Conjecture 1 and Conjecture 2 for all groups of classical type in the good
characteristic case. In fact we prove local global principles for function fields of curves over any local field.
\begin{theorem}
\label{conjecture1}
Let $K$ be a local field with residue field $\kappa$ and $F$ the function field of a curve over $K$.
Let $G$ be a connected linear algebraic group over $F$ of classical type ($D_4$ nontrialitarian) with char$(\kappa)$ good for $G$.
Let $Z$ be a projective homogeneous space under $G$ over $F$.
If $Z(F_\nu) \neq \emptyset$ for all divisorial discrete valuations of $F$, then $Z(F_\nu) \neq \emptyset$.
\end{theorem}
\begin{proof} Let $G^{ss}$ be the semisimplification of $G/rad(G)$.
Since $G$ is of classical type, there exists a central
isogeny $G_1\times \cdots \times G_r \to G^{ss}$ with each $G_i$ an almost simple simply connected group of
the classical type ($D_4$ nontrialitarian) with char$(\kappa)$ good.
It is well know that using the results of (\cite[Theorem 2.20]{BT}) and (\cite[Proposition 6.10]{MPW2}),
one reduces to the case $r = 1$ (cf. proof of \cite[Corollary 4.6]{Wu}).
Let $G$ be a connected linear algebraic group with an isogeny $G' \to G^{ss}$
for some almost simple connected group $G'$ of
classical type ($D_4$ nontrialitarian).
If $G'$ is of type $^1\!\!A_n$, then the result follows from (\ref{proj-pgl}).
If $G'$ is of type $^2\!\!A_n$, then the result follows from (\ref{isotropic1}).
If $G'$ is of type $B_n$, $C_n$ or $D_n$, then the result follows from (\cite{Wu}).
\end{proof}
\begin{theorem}
\label{conjecture2}
Let $K$ be a local field with residue field $\kappa$ and $F$ the function field of a curve over $K$.
Let $G$ be a semisimple simply connected linear algebraic group over $F$ with char$(\kappa)$ is good for $G$.
Suppose $G$ is of the classical type ($D_4$ nontrialitarian).
Let $Z$ be a principal homogenous space under $G$ over $F$.
If $Z(F_\nu) \neq \emptyset$ for all divisorial discrete valuations of $F$, then $Z(F_\nu) \neq \emptyset$.
then Conjecture 2 holds for $G$.
\end{theorem}
\begin{proof} For $G$ of type $B_n$, $C_n$ or $D_n$ ($D_4$ nontrialitarian),
this result is proved in (\cite{Hu} and \cite{preeti}).
Suppose $G$ is of type $^1\!\!A_n$. Then
$G \simeq SL(A)$ for some central simple algebra $A$ over $F$ and the principal homogeneous spaces
under $G$ are classified by $H^1(F, G) \simeq F^*/Nrd(A)$.
Since char$(\kappa)$ is good for $G$, the degree of $A$ is coprime to char$(\kappa)$.
Hence the result follows from (\cite[Corollary 11.2]{PPS}).
Suppose $G$ is of type $^2\!\!A_n$.
Then there exists a separable quadratic extension $F/F_0$ and central simple algebra over $F$ with
an $F/F_0$-involution $\sigma$ such that $G \simeq SU(A, \sigma)$.
Since char$(\kappa)$ is good for $G$, 2(deg$(A)$) is coprime to char$(\kappa)$.
Hence the result follows from (\ref{lgp-unitary}).
\end{proof}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,886
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Q: Steps to run spring mvc basic sample application So I open up intelliJ ultimate and opened up this application using the pom.xml :
https://src.springframework.org/svn/spring-samples/mvc-basic/trunk/
The application built just fine, and I ran it with tomcat 6 (which I setup).
I just got a default tomcat welcome screen.
What am I missing here?
Clicking run showed:
Jun 23, 2010 8:28:12 AM org.apache.catalina.core.AprLifecycleListener init
INFO: The APR based Apache Tomcat Native library which allows optimal performance in production environments was not found on the java.library.path: /Applications/IntelliJ IDEA 9.0.2.app/Contents/Resources/Java:/System/Library/PrivateFrameworks/JavaApplicationLauncher.framework/Resources:.:/Library/Java/Extensions:/System/Library/Java/Extensions:/usr/lib/java
Jun 23, 2010 8:28:12 AM org.apache.coyote.http11.Http11Protocol init
INFO: Initializing Coyote HTTP/1.1 on http-8080
Jun 23, 2010 8:28:12 AM org.apache.catalina.startup.Catalina load
INFO: Initialization processed in 488 ms
Jun 23, 2010 8:28:12 AM org.apache.catalina.core.StandardService start
INFO: Starting service Catalina
Jun 23, 2010 8:28:12 AM org.apache.catalina.core.StandardEngine start
INFO: Starting Servlet Engine: Apache Tomcat/6.0.26
Jun 23, 2010 8:28:12 AM org.apache.catalina.startup.HostConfig deployDirectory
INFO: Deploying web application directory docs
Jun 23, 2010 8:28:12 AM org.apache.catalina.startup.HostConfig deployDirectory
INFO: Deploying web application directory examples
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{"url":"https:\/\/archive.lib.msu.edu\/crcmath\/math\/math\/m\/m172.htm","text":"## Meijer's G-Function\n\nwhere is the Gamma Function. The Contour and other details are discussed by Gradshteyn and Ryzhik (1980, pp.\u00a0896-903 and 1068-1071). Prudnikov et al. (1990) contains an extensive nearly 200-page listing of formulas for the Meijer -function.\n\nReferences\n\nGradshteyn, I.\u00a0S. and Ryzhik, I.\u00a0M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1979.\n\nLuke, Y.\u00a0L. The Special Functions and Their Approximations, 2 vols. New York: Academic Press, 1969.\n\nMathai, A.\u00a0M. A Handbook of Generalized Special Functions for Statistical and Physical Sciences. New York: Oxford University Press, 1993.\n\nPrudnikov, A.\u00a0P.; Marichev, O.\u00a0I.; and Brychkov, Yu.\u00a0A.; Integrals and Series, Vol.\u00a03: More Special Functions. Newark, NJ: Gordon and Breach, 1990.","date":"2021-12-09 11:06:52","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8899688720703125, \"perplexity\": 1692.8059534539466}, \"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\/1637964363791.16\/warc\/CC-MAIN-20211209091917-20211209121917-00196.warc.gz\"}"}
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\section{Introduction}
In this paper we generalize the results of \cite{L2} to the setting of hybrid
systems. In particular, we introduce the notions of hybrid open
systems, their networks and maps between networks.
A network of systems is a blueprint for building a larger system out
of smaller subsystems by specifying a pattern of interactions between
subsystems --- an interconnection map. Maps between networks allow us
to produce maps between complex hybrid dynamical systems by specifying maps
between their subsystems. The framework of \cite{L2} was developed to connect two
rather different views of networks of continuous time systems ---
``networks are morphisms in colored operads'' of Spivak and
collaborators \cite{Spivak, VSL} on one hand and the coupled cell network
formalism of Golubitsky, Stewart and their collaborators (see
\cite{Golubitsky.Stewart.06} and references therein) and its
subsequent generalizations \cite{DL1, DL2}. By generalizing the
results of \cite{L2} we bring the operadic point of view to hybrid
systems and, at the same time, generalize coupled cell network
formalism to hybrid dynamical systems.
To carry out our program we need to develop an appropriate framework.
The first step is to introduce the notion of a {\em hybrid phase
space} (Definition~\ref{def:hyph}) and its underlying manifold with
corners. In this way a hybrid dynamical system is a pair $(a, X)$
where $a$ is a hybrid phase space and $X$ is a vector field on the
underlying manifold $\mathbb{U}(a)$. We then define maps of hybrid phase
spaces thereby making hybrid phase spaces and their maps into a
category $\mathsf{HyPh}$. We show the category of hybrid phase spaces has
finite products and that the assignment of the underlying manifold to
a hybrid phase space extends to a product-preserving functor
\[
\mathbb{U}: \mathsf{HyPh} \to \Man
\]
where $\Man$ denotes the category of manifolds with corners (see
Appendix~\ref{app:A}).
The construction of the category $\mathsf{HyPh}$ and the underling manifold
functor $\mathbb{U}$ allows us construct the category $\HyDS$ of hybrid
dynamical systems as a ``category of elements'' for a functor with
values in the category $\mathsf{RelVect}$ of vector spaces and linear
relations (see Section~\ref{sec:cat_of_el}). We provide evidence that
the machine we have built so far makes sense by observing that
executions of hybrid dynamical systems {\em are} maps of hybrid
dynamical systems and therefore are morphisms in the category $\HyDS$
(Definition~\ref{def:exec} and Remark~\ref{rmrk:exec}). We prove that
maps of hybrid dynamical systems send executions to executions
(Theorem~\ref{thm:exec}) thereby providing another sanity check on our
theory building.
We single out a class of maps between hybrid phase spaces that we call
{\sf hybrid surjective submersions} by
requiring that the corresponding maps of underlying manifolds are
surjective submersions. We organize hybrid surjective submersions
into a category $\mathsf{HySSub}$ (Definition~\ref{def:hybssub}). Our
motivation for introducing this category is the following.
The underlying manifold functor $\mathbb{U}$ extends to a functor
$\mathbb{U}:\mathsf{HySSub}\to \mathsf{SSub}$ where $\mathsf{SSub}$ is the category of surjective
submersions of manifolds with corners (Notation~\ref{not:ssub}).
Recall that for every surjective submersion
$a= a_\mathsf{tot} \xrightarrow{p_a} a_\mathsf{st}$ ($a_\mathsf{st}$ is the space of states,
$a_\mathsf{tot}$ is the space of states {\em and} controls, and $p_a$ is the
surjective submersion) there corresponds a vector space $\mathsf{Crl}(a)$ of
all control systems on $a$ (Definition~\ref{def:open_sys} and
Notation~\ref{not:crl}). The assignment $a\mapsto \mathsf{Crl}(a)$ extends to
a $\mathsf{RelVect}$-valued functor $\mathsf{Crl}$. We therefore have the category of
elements $\mathsf{OS}: = \int \mathsf{Crl}$ of open (control) systems (see
Section~\ref{sec:cat_of_el}). The category of elements of the
composite functor $\mathsf{Crl}\circ \mathbb{U}: \mathsf{HySSub} \to \mathsf{RelVect}$ is a category
of hybrid open systems $\mathsf{HyOS}$. In particular, a {\sf hybrid open
system} is a pair $(a, F)$ where $a= a_\mathsf{tot}\xrightarrow{p_a} a_\mathsf{st}$
is a hybrid surjective submersion and
$F:\mathbb{U}(a_\mathsf{tot}) \xrightarrow{\mathbb{U}(p_a)} T\mathbb{U}(a_\mathsf{st})$ is a control
system. While hybrid open systems are certainly known, we believe
that our definition of the category $\mathsf{HyOS}$ is new. For other
category-theoretic approaches to hybrid open systems see \cite{Ames},
\cite{TPL} and \cite{LS}.
As was observed in \cite{L2} it is useful to organize surjective
submersions into a double category $\mathsf{SSub}^\Box$ (double categories are
reviewed in Subsection~\ref{subsec:double}). The second kind of
1-arrows in $\mathsf{SSub}^\Box$ are interconnection morphisms: see
Definition~\ref{def:ssub}, the subsequent remark and
Example~\ref{ex:intercon}. Roughly speaking these morphisms
describe the effect on open systems of plugging state variables into
controls. The functor $\mathsf{Crl}:\mathsf{SSub} \to \mathsf{RelVect}$ extends to a functor
of double categories $\mathsf{Crl}:\mathsf{SSub} ^\Box \to \mathsf{RelVect}^\Box$ where
$\mathsf{RelVect}^\Box$ is the double category of vector spaces, linear maps
and linear relations. We then use the forgetful functor $\mathbb{U}:\mathsf{HySSub}
\to \mathsf{SSub}$ to turn hybrid surjective submersions into a double
category $\mathsf{HySSub} ^\Box$ and to define interconnection maps of hybrid
surjective submersions. As a sanity check we show that a hybrid
dynamical system whose underlying hybrid phase space is a product of
two hybrid phase spaces can be obtained by interconnecting two hybrid
open systems (see
Example~\ref{ex:hds_on_product}).
At this point we are almost done with building the machinery. We
define a network of hybrid open systems to be a pair
$(\{a_x\}_{x\in X}, \psi:b\to \prod _{x\in X} a_x)$ where $\{a_x\}_{x\in X}$
is a collection of hybrid surjective submersions indexed by a finite
set $X$ and $\psi:b\to \prod
_{x\in X}$ is an interconnection map
(Definition~\ref{def:network_hy_os}). We then define maps of
networks of hybrid open systems
(Definition~\ref{def:map_of_networks_hy_os}), which parallels the
definition of maps of networks of continuous time open systems in
\cite{L2}. The definition consists of a list of compatible data. In
more detail a map from a network $(\{a_x\}_{x\in X}, \psi:b\to \prod
_{x\in X} a_x)$ to a network $(\{d_y\}_{y\in Y}, \nu:c\to \prod _{y\in
Y} d_y)$ consists of a map of finite sets $\varphi:X\to Y$, a
collection $\{\Phi_x: d_{\varphi(x)} \to a_x\}_{x\in X}$ of maps of
hybrid surjective submersions and another map $f:c\to b$ of hybrid
surjective submersions which is compatible with $\varphi$, $\Phi =
\{\Phi_x\}_{x\in X}$, $\psi$ and $\nu$ in an appropriate sense.
The main theorem of this paper (Theorem~\ref{thm:main_result}) can be
roughly phrased as follows. Recall that
the functor $\mathsf{Crl}:\mathsf{SSub} \to \mathsf{RelVect}$ assigns to a map
$h:p\to q$ of surjective submersions a linear relation
$\mathsf{Crl} (h)\subset \mathsf{Crl} (p) \times \mathsf{Crl}(q)$.
Let $\big((\varphi,\Phi),f\big):
(\{a_x\}_{x\in X}, \psi:b\to \prod _{x\in X} a_x) \to
(\{d_y\}_{y\in Y}, \nu:c\to \prod _{y\in
Y} d_y)$ be a map of networks of hybrid open systems.
Theorem~\ref{thm:main_result} shows that for any choice of
collections of control systems $\{w_x\in
\mathsf{Crl}(\mathbb{U}(a_x))\}_{x\in X}$ and $\{u_y\in \mathsf{Crl} (\mathbb{U}(d_y))\}_{y\in Y}$
so that $u_{\varphi(x)}$ is
$\mathbb{U}(\Phi_x)$-related to $w_x$ for all $x\in X$ we get a map
\[
f: (c, \nu^*(\prod u_y)) \to (b,
\psi^* (\prod_{x\in X}w_x))
\]
of hybrid open systems. Here $\nu^*: \mathsf{Crl} (\mathbb{U}(\prod_{y\in Y} d_y)) \to \mathsf{Crl}
(\mathbb{U}(c))$ and $\psi^*: \mathsf{Crl} (\mathbb{U}(\prod_{x\in X} a_x)) \to \mathsf{Crl} (\mathbb{U}
(b))$ are linear maps induced by the interconnection maps $\nu$ and
$\psi$ respectively. In other words, compatible patterns of
interconnection of hybrid open systems give rise to maps of hybrid open
systems.
In the case where the two interconnections result in closed
systems, the map $f$ is a map of hybrid dynamical systems.
Theorem~\ref{thm:main_result} is a direct generalization of
\cite[Theorem~9.5]{L2} from continuous time systems to hybrid
systems. If one views a network as a directed graph with nodes
decorated with phase spaces, then it is reasonable to view a map
between networks as a map of graphs of some sort that preserves the
decorations. This is the point of view taken in the papers on
coupled cell networks cited above. But one can also view a
network as a finite list of objects $\{a_1,\ldots, a_k\}$ in some
monoidal category ${\mathcal C}$ together with an arrow $a_1\otimes\cdots
\otimes a_k \to b$ in ${\mathcal C}$. In other words, a network is a
morphism in the colored operad ${\mathcal O}{\mathcal C}$ associated with ${\mathcal C}$. This
is a view of networks advocated by Spivak and his collaborators ({\em op.\ cit.}), and
several other applied category theorists. If one believes
that networks should form a category, then the operadic approach to
networks does not quite fit since there are no arrows between
morphisms in colored operads. A solution to this issue was proposed
in \cite{L2} in the special case of continuous time systems (there ${\mathcal C} ^\mathrm{op}$ is
a subcategory $\mathsf{SSub}_{\textrm{int}}$ of the Cartesian monoidal
category $\mathsf{SSub}$ of
surjective submersions). In particular, maps between networks of
continuous time systems were shown to give rise to maps of continuous
time systems (open and closed). A sanity check was provided by
proving that directed graphs with nodes decorated with manifolds give
rise to morphism in the operad ${\mathcal O}(
{\mathsf{SSub}_{\textrm{int}}}^\mathrm{op})$ of surjective submersions.
Fibrations of decorated graphs were shown to give rise to maps between
networks of continuous time systems and to maps of dynamical systems.
The ``higher operad'' of surjective submersions (i.e., the colored
operad ${\mathcal O}({\mathsf{SSub}_{\textrm{int}}}^\mathrm{op})$ together with the morphisms between
networks) was in turn interpreted by constructing a map to the double
category $\mathsf{RelVect}^\Box$ of vector spaces, linear maps and linear
relations.
In the present paper we develop an analogous theory for hybrid
dynamical systems. Our motivation, to some extent, is to have another natural example
of a ``higher operad'' before we proceed with building a general
theory of ``higher operads'' suitable for constructing categories of
networks. Since we don't have such a general theory yet, we choose
not to emphasize the operadic aspects of
Theorem~\ref{thm:main_result}.
It is easy to specialize Theorem~\ref{thm:main_result} to the subcategory
of networks of hybrid systems constructed from decorated graphs
(the nodes now are decorated with hybrid phase spaces). Analogous to
a result in \cite{L2}, fibrations of decorated graphs
give rise to maps between hybrid systems. In particular, this
allow an extension of the coupled cell network formalism to hybrid systems.
We plan to address this elsewhere.
Finally, we note that our hybrid dynamical systems are not necessarily
deterministic. This is done for two reasons. First of all,
non-deterministic hybrid systems have a wide range of applications.
Secondly, the formalism is simpler. With a bit more work the approach
developed in this paper can handle deterministic systems:
see \cite{Schmidt}.
\subsection*{Related work} We will not attempt to survey all the
work done on hybrid dynamical systems. The area is vast. Most of
the papers in the area appear in engineering publications. As far
as we know the use of category theory in hybrid systems is rather
limited. We note the work of Ames in his thesis \cite{Ames} and in
a number of subsequent publications. Other notable papers are by
Tabuada, Pappas and Lima \cite{TPL} and by Lorenco and
Sernadas \cite{LS}. A radically different approach to hybrid open
systems and their interconnection has recently been developed by
Schultz and Spivak \cite{SS}. It is based on sheaves. It would be
interesting to understand how the formalism of \cite{SS} relates to
the traditional view of hybrid systems and to the formalism
developed here.\\
\noindent {\bf Acknowledgments:} E. L.\ thanks Sayan Mitra for many hours of
conversations. E. L.\ also thanks Michael Warren for some useful discussions.
\section{Background}
\subsection{Relations and linear relations}\mbox{}\\
We start by setting our notation for relations and their compositions.
We view a relation $X\xrightarrow{R} Y$ from a set $X$ to a set
$Y$ as a generalization of the notion of a function from $X$ to $Y$.
It also generalizes the notion of a {\em partial} function from $X$ to
$Y$. The following definitions are standard.
\begin{definition} \label{notation:rel}
We define a {\em relation} $R:X\to Y$ from a set $X$ to a set $Y$ to be
a subset of the product $X\times Y$. The composition
of relations $(Z\xleftarrow{S}Y) \circ
(Y\xleftarrow{R}X)$ is defined by
\begin{equation}\label{eq2.1}
S\circ R := \{ (x,z) \in X\times Z \mid \textrm{ there exists } y\in Y
\textrm{ with } (y,z)\in S, (x,y)\in R\}.
\end{equation}
A relation $X\xrightarrow{R}Y$ is a {\sf function} if for any $x\in X$
the intersection $ (\{x\}\times Y) \cap R$ is a singleton. In other
words we identify a function $X\xrightarrow{f}Y$ with its graph
$\mathsf{Graph}(f) := \{ (x,y) \in X\times Y \mid y = f(x)\}$. A relation
$X\xrightarrow{R}Y$ is a {\sf partial function } if for any $x\in X$
the intersection $( \{x\} \times Y) \cap R$ is either empty or a single
point (and so defines a function from a subset of $X$ to $Y$).
\end{definition}
\begin{definition}[The 2-category $\mathsf{RelSet}$]
Sets and relations form a 2-category $\mathsf{RelSet}$. The objects of
$\mathsf{RelSet}$ are sets. The 1-arrows of $\mathsf{RelSet}$ are relations with composition
defined by \eqref{eq2.1}. A 2-arrow from a
relation $X\xrightarrow{R}Y$ to a relation $X\xrightarrow{R'}Y$ is an
inclusion $R\hookrightarrow R'$.
\end{definition}
\begin{definition}[The 2-category $\mathsf{RelVect}$] \label{def:relvect}
Vector spaces and linear relations form a 2-category $\mathsf{RelVect}$. The
objects of $\mathsf{RelVect}$ are vector spaces. A 1-arrow
$W\xrightarrow{R} V$ in $\mathsf{RelVect}$ is a
linear subspace $R$ of $W\times V$, that is, a linear relation. The
composition of linear relations is defined by \eqref{eq2.1}, that is,
it is defined exactly the same way as the composition of
set-theoretic relations. A 2-arrow from a linear relation
$W\xrightarrow{R}V$ to a linear relation $W\xrightarrow{R'}V$ is a
(linear) inclusion $R\hookrightarrow R'$.
\end{definition}
The following definition is standard for morphisms in $\mathsf{RelSet}$. It
works equally well in $\mathsf{RelVect}$.
\begin{definition} \label{def:transpose}
Given a linear relation $R:V\to W$ its {\sf transpose} is the
relation $R^T:W\to V$ defined by
\[
R^T =\{ (w, v) \in W\times V\mid (v,w) \in R\}.
\]
\end{definition}
\subsection{Double categories} \label{subsec:double}\mbox{}
The 2-category $\mathsf{RelVect}$ is the horizontal 2-category of a double
category $\mathsf{RelVect}^\Box$ defined below. To define $\mathsf{RelVect}^\Box$ we
need to recall the notion of a double category which is due to
Charles Ehresmann. These are categories which may be defined as
categories internal to the category $\mathsf{CAT}$ of categories
\cite{BMM, Shul, Shulman10} (we notationally distinguish between the
2-category $\CAT$ of not necessarily small categories and the
2-category $\Cat$ of small categories). Double categories, double
functors and vertical transformations will get a fair amount of use in this paper.
\begin{definition}[Double category]
\label{def:double_cat}
A {\sf double category} $\mathbb{D}$ consists of two categories $\mathbb{D}_1$
(of arrows) and $\mathbb{D}_0$ (of objects) together with four structure
{\em functors}:
\begin{gather*}
\mathfrak s, \mathfrak t:\mathbb{D}_1 \to \mathbb{D}_0 \quad (\textrm{source and target}), \qquad
\mathfrak u: \mathbb{D}_0 \to \mathbb{D}_1\quad (\textrm{unit})\\
\mathfrak m: \mathbb{D}_1\times_{\mathfrak s, \mathbb{D}_0, \mathfrak t} \mathbb{D}_1\to \mathbb{D}_1 ,
\quad (\xLeftarrow{\alpha}, \xLeftarrow{\beta}) \mapsto
\mathfrak m(\xLeftarrow{\alpha}, \xLeftarrow{\beta})=: \alpha* \beta
\qquad (\textrm{multiplication/composition})
\end{gather*}
so that
\begin{gather*}
\mathfrak s \circ \mathfrak u = \mathsf{id}_{\mathbb{D}_0}= \mathfrak t \circ \mathfrak u, \\
\mathfrak s (\alpha *\beta) = \mathfrak s (\beta),\qquad \mathfrak t (\alpha *\beta) = \mathfrak t (\alpha),\\
(\alpha * \beta)* \gamma =
\alpha *(\beta *\gamma),\\
\alpha* \mathfrak u(\mathfrak s(\alpha)) = \alpha\quad \textrm{and} \quad\mathfrak u(\mathfrak t(\alpha))* \alpha =\alpha
\end{gather*}
for all arrows $\alpha, \beta, \gamma$ of $\mathbb{D}_1$.
\end{definition}
\begin{remark}
There are weaker notions
of double categories such as pseudo-double categories. We won't
use them in this paper. The
reader should be warned that pseudo-double categories are often
referred to as double categories.
\end{remark}
\begin{notation} Let $\mathbb{D}$ be a double category. We call the objects of
the category $\mathbb{D}_0$ {\sf $0$-cells} or {\sf
objects} and the morphisms of $\mathbb{D}_0$ the ``vertical'' {\sf
1-morphisms}. We call the
objects of the category $\mathbb{D}_1$ ``horizontal'' {\sf 1-cells} (or the
``horizontal'' {\sf 1-morphisms}). A morphism $\alpha:\mu\Rightarrow \nu$ of
$\mathbb{D}_1$ with $\mathfrak s (\alpha) = (a\xrightarrow{f} b)$ and $\mathfrak t(\alpha)
=(c\xrightarrow{g} d)$ is a {\sf
2-morphism} or a {\sf 2-cell} (we will use the two terms
interchangeably). We may depict such a 2-cell as
\begin{equation}\label{eq:2-mor}
\xy
(-8, 6)*+{c} ="1";
(8, 6)*+{a} ="2";
(-8,-6)*+{d }="3";
(8, -6)*+{b}="4";
{\ar@{->}_{\mu} "2";"1"};
{\ar@{->}^{\nu} "4";"3"};
{\ar@{->}_{g} "1";"3"};
{\ar@{->}^{f} "2";"4"};
{\ar@{=>}^{\alpha} (0,3)*{};(0,-3)};
\endxy
\end{equation}
with the arrows of $\mathbb{D}_0$ drawn vertically, the objects of $\mathbb{D}_1$
drawn as horizontal arrows and the 2-cell $\alpha$ drawn as a double
arrow from $\mu$ to $\nu$. Note that
\eqref{eq:2-mor} is {\bf not} necessarily a 2-commuting diagram in
some 2-category.
\end{notation}
\begin{remark}
Equivalently one can define a (strict) double category $\mathbb{D}$ as consisting of
``tiles'' or ``squares'' that can be composed by either stacking the
squares vertically or horizontally. The vertical composition of
squares correspond to the composition of morphisms in $\mathbb{D}_1$. The horizontal
composition of squares corresponds to the functor $\mathfrak m$. Given 4
composable
squares
\[
\xy
(-12, 12)*+{\bullet} ="1";
(0,12 )*+{\bullet} ="2";
(12,12 )*+{\bullet} ="3";
(-12,0 )*+{\bullet} ="4";
(0,0 )*+{\bullet} ="5";
(12, 0)*+{\bullet} ="6";
(-12,-12 )*+{\bullet} ="7";
(0,-12 )*+{\bullet} ="8";
(12,-12 )*+{\bullet} ="9";
{\ar@{<-} "1";"2"};
{\ar@{<-} "2";"3"};
{\ar@{<-} "4";"5"};
{\ar@{<-} "5";"6"};
{\ar@{->} "4";"7"};
{\ar@{->}"5";"8"};
{\ar@{->}"6";"9"};
{\ar@{->}"1";"4"};
{\ar@{->} "2";"5"};
{\ar@{->} "3";"6"};
{\ar@{->} "8";"7"};
{\ar@{->} "9";"8"};
\endxy
\]
we can first compose them vertically in pairs and then compose them
horizontally or the other way around. Since $\mathfrak m:\mathbb{D}_1\times_{\mathbb{D}_0}
\mathbb{D}_1\to \mathbb{D}_1$ is a functor, the results are equal.
This suggests that a double category $\mathbb{D}$ may also be viewed as a
category in $\CAT$ whose category of objects $\mathbb{D}_0'$ has the objects
of $\mathbb{D}_1$ as arrows (with the composition defined by the functor
$\mathfrak m$). The category of arrows $\mathbb{D}_1'$ has the arrows of $\mathbb{D}_0$ as
objects (and the squares of $\mathbb{D}$ as morphisms now composed
horizontally). That is, we may take the directed tiles of the double
category $\mathbb{D}$ and reflect them along the appropriate diagonal. Since our
horizontal arrows point left and vertical arrows point down, c.f.\ \eqref{eq:2-mor}, the diagonal in
question is
the southwest -- northeast
diagonal. For this reason ``horizontal'' and ``vertical''
terminology of double categories is somewhat arbitrary: presenting a
double category as a category internal to $\CAT$ hides this
reflection symmetry and introduces a bias.
\end{remark}
\begin{definition}[Horizontal 2-category of a double category]
Associated to a double category $\mathbb{D}$ there is a {\sf horizontal
2-category ${\mathcal H}(\mathbb{D})$} whose objects are the objects of $\mathbb{D}$,
1-arrows are the horizontal morphisms of $\mathbb{D}$ and 2-arrows are the
2-cells of $\mathbb{D}$ of the form
\begin{equation} \label{eq:globular}
\xy
(-8, 6)*+{c} ="1";
(8, 6)*+{a} ="2";
(-8,-8)*+{c }="3";
(8, -8)*+{a}="4";
{\ar@{->}_{\mu} "2";"1"};
{\ar@{->}^{\nu} "4";"3"};
{\ar@{->}_{\mathsf{id}} "1";"3"};
{\ar@{->}^{\mathsf{id}} "2";"4"};
{\ar@{=>}^{\alpha} (0,3)*{};(0,-3)};
\endxy .
\end{equation}
\end{definition}
\begin{definition}
2-cells in a double category $\mathbb{D}$ of the form \eqref{eq:globular} are
called {\sf globular}.
\end{definition}
\begin{remark}
Since a globular 2-cell is a 2-arrow in
the horizontal category ${\mathcal H}(D)$ we may picture it as $
\xy
(-10,0)*+ {c}="1";
(10,0)*+ {a}="2";
{\ar@/^1pc/^{\nu} "2";"1"};
{\ar@/_1pc/_{\mu} "2";"1"};
{\ar@{=>}^{\alpha} (0,2)*{};(0,-2)};
\endxy
$ .
\end{remark}
\noindent
In this paper we will use a number of double categories. We start by
defining the double category of manifolds with corners, smooth maps
and set-theoretic relations (cf.\ \cite[Definition 3.9]{L1}). The
definition of a category $\Man$ of manifolds with corners is reviewed
in Appendix~\ref{app:A}. The reader should be aware that in the
differential-geometric literature there is a number of incompatible
definitions of smooth maps between manifolds with corners. The
definition we use is probably the least restrictive.
\begin{definition}[The double category $\RelMan^\Box$]
\label{def:relmanbox}
The objects of
the double category $\RelMan^\Box$ are manifolds with corners. The
vertical arrows are the smooth maps. Thus the category of objects
$(\RelMan^\Box)_0$ is the category $\Man$ of manifolds with corners
and smooth maps. A horizontal arrow $\mu:M\to N$ is a
set-theoretic relation $\mu$ from $M$ to $N$, that is a
subset of $M\times N$ (cf.\ Definition~\ref{notation:rel}). A 2-cell
$\alpha$ from a relation $\mu:M\to N$ to a relation $\nu: P\to Q$
is a pair of smooth maps $(g: M\to P, f:N\to Q)$ so that
\[
(g\times f) (\mu) \subset \nu.
\]
The composition in the arrows category $(\RelMan^\Box)_1$ is given by
composing pairs of maps:
\[
(\tau\xleftarrow{(k,l)} \nu) \circ (\nu\xleftarrow{(g,f)} \mu) = \tau
\xleftarrow{(k\circ g, l\circ f)} \mu.
\]
The functor
\[
\mathfrak{m}: (\RelMan^\Box)_1\times_{\mathfrak{s}, (\RelMan^\Box)_0, \mathfrak{t} } (\RelMan^\Box)_1\to (\RelMan^\Box)_1
\]
is defined by composing relations:
\[
\mathfrak m \left( (\nu'\xLeftarrow{(f,h)}\mu', \nu\xLeftarrow{(g,f)} \mu
\right)
:= (\nu'\circ \nu\xLeftarrow{(g,h)} \mu'\circ \mu).
\]
\end{definition}
\begin{remark}\label{rmrk:RelMan}
The horizontal category ${\mathcal H}(\RelMan^\Box)$ is the 2-category
$\RelMan$ of manifolds with corners, {\em arbitrary} (i.e., not
necessarily smooth) relations and inclusions of relations.
\end{remark}
The double category $\mathsf{RelVect}^\Box$ of vector spaces, linear maps and
linear relations is defined analogously to the definition of
$\RelMan^\Box$. More formally we have:
\begin{definition}[The double category $\mathsf{RelVect}^\Box$] The objects
of the double category $\mathsf{RelVect}^\Box$ are (real) vector spaces.
The vertical arrows are linear maps.
Thus the category $(\mathsf{RelVect}^\Box)_0$ of objects is the category $\mathsf{Vect}$ of
vector spaces and linear maps.
A horizontal arrow
$\mu:V\to W$ is a linear relation $\mu$ from a vector space $V$ to
a vector space $W$,
which is a vector subspace of $V\times W$. A 2-cell $\alpha$ from
a relation $\mu:V\to W$ to a relation $\nu: X\to Y$ is a pair of
linear maps $(g: V\to X, f:W\to Y)$ so that
\[
(g\times f) (\mu) \subset \nu.
\]
The composition in the category $(\mathsf{RelVect}^\Box)_1$ of arrows is given by
composing pairs of maps:
\[
(\tau\xleftarrow{(k,l)} \nu) \circ (\nu\xleftarrow{(g,f)} \mu) = \tau
\xleftarrow{(k\circ g, l\circ f)} \mu.
\]
The functor
\[
\mathfrak{m}: (\mathsf{RelVect}^\Box)_1\times_{\mathfrak{s}, (\RelMan^\Box)_0, \mathfrak{t} } (\mathsf{RelVect}^\Box)_1\to (\mathsf{RelVect}^\Box)_1
\]
is defined by composing relations:
\[
\mathfrak m \left( (\nu'\xLeftarrow{(f,h)}\mu', \nu\xLeftarrow{(g,f)} \mu
\right)
:= (\nu'\circ \nu\xLeftarrow{(g,h)} \mu'\circ \mu).
\]
\end{definition}
\begin{remark}
It is easy to see that the horizontal category ${\mathcal H}(\mathsf{RelVect}^\Box)$ is
the 2-category $\mathsf{RelVect}$ of vector spaces, linear relations and inclusions (Definition~\ref{def:relvect}).
\end{remark}
We will also need the double category $\mathsf{SSub}^\Box$ of surjective
submersions which was introduced in \cite{L2}. To state the
definition of $\mathsf{SSub}^\Box$ it will be convenient to first recall the
notion of a map between two surjective submersions and also the notion
of an interconnection morphism. We first introduce some
notation which is motivated by control theory.
\begin{notation}
It will be convenient to denote a surjective submersion of
manifolds with corners by a single letter. Thus a surjective
submersion $a$ consists of two manifolds with corners $a_\mathsf{tot}$ (the
``total space''),
$a_\mathsf{st}$ (the ``state space'') and a surjective submersion
$p_a:a_\mathsf{tot}\to a_\mathsf{st}$. We write $a= (a_\mathsf{tot} \xrightarrow{p_a}a_\mathsf{st})$.
\end{notation}
\begin{definition} \label{def:ssub} A {\sf morphism of surjective submersions} $f:a\to b$ is a
pair of morphisms $f_\mathsf{tot}:a_\mathsf{tot}\to b_\mathsf{tot}$, $f_\mathsf{st}:a_\mathsf{st}\to b_\mathsf{st}$
in the category $\Man$ of
manifolds with corners so that the diagram
\[
\xy
(-10, 10)*+{b_\mathsf{tot}} ="1";
(10, 10)*+{a_\mathsf{tot}} ="2";
(-10,-5)*+{b_\mathsf{st}}="3";
(10, -5)*+{a_\mathsf{st}}="4";
{\ar@{->}_{f_\mathsf{tot}} "2";"1"};
{\ar@{->}^{p_b} "4";"3"};
{\ar@{->}_{p_a} "1";"3"};
{\ar@{->}^{f_\mathsf{st}} "2";"4"};
\endxy
\]
commutes $\Man$. Note that the morphisms in the
category $\Man$ are $C^\infty$ maps, see Appendix~\ref{app:A}.
A morphism of surjective submersions $f:a\to b$ is an
{\sf interconnection morphism } if $f_\mathsf{st}:a_\mathsf{st}\to b_\mathsf{st}$ is a diffeomorphism.
\end{definition}
\begin{remark}
A motivation for the terminology of Definition~\ref{def:ssub} is
discussed in \cite{L2}.
See Example~\ref{ex:intercon} below.
\end{remark}
\begin{notation}\label{not:ssub}
Surjective submersions and their morphisms form a category which we
denote by $\mathsf{SSub}$. It is a subcategory of the category of arrows of
$\Man$, the category of manifolds with corners and smooth maps.
Surjective submersions and interconnection morphisms form a
subcategory of $\mathsf{SSub}$. We denote it by $\mathsf{SSub}^\mathsf{int}$.
\end{notation}
\begin{definition}[The double category $\mathsf{SSub}^\Box$ of
surjective submersions]\label{def:ssub_box}
The objects of the double category $\mathsf{SSub}^\Box$ are surjective
submersions. The horizontal 1-morphisms are maps/morphisms of surjective
submersions (Definition~\ref{def:ssub} above). The vertical
1-morphisms are interconnection morphisms (Definition~\ref{def:ssub}).
The 2-cells of $\mathsf{SSub}^\Box$ are commuting squares
\[
\xy
(-10,10)*+{c}="1";
(10, 10)*+{a}="2";
(-10,-5)*+{d}="3";
(10, -5)*+{b}="4";
{\ar@{<-}^{\mu} "1";"2"};
{\ar@{->}_{g} "1";"3"};
{\ar@{->}^{f} "2";"4"};
{\ar@{<-}_{\nu} "3";"4"};
\endxy
\]
in the category $\mathsf{SSub}$ of surjective submersions,
where $\mu,\nu$ are maps of submersions and $f$, $g$ are the interconnection
morphisms. In particular, the category of objects of the double
category $\mathsf{SSub}^\Box$ is the
category $\mathsf{SSub}^\mathsf{int}$.
\end{definition}
We will need two versions of a ``functor between double categories:'' strict
and lax. Functors between double categories are often referred to as ``double functors.''
\begin{definition}[Strict 1-morphism/map/double functor of double categories]
A strict {\sf morphism/map/double functor} $F$ from a double category $\mathbb{B}$ to a double
category $\mathbb{D}$ is a pair of ordinary functors $F_0:\mathbb{B}_0\to \mathbb{D}_0$,
$F_1: \mathbb{B}_1\to \mathbb{D}_1$ which commute strictly with the source, target
and unit functors of the double categories $\mathbb{B}$ and $\mathbb{D}$
(i.e., $\mathfrak s \circ F_1 = F_0 \circ \mathfrak s$, $\mathfrak t \circ F_1 = F_0 \circ
\mathfrak t$ and $\mathfrak u \circ F_0 = F_1 \circ \mathfrak u$)
and
strictly preserves the composition functor $\mathfrak m$: the diagram
\begin{equation}\label{eq:2.18}
\xy
(-20, 10)*+{\mathbb{B}_1\times_{\mathbb{B}_0} \mathbb{B}_1} ="1";
(20, 10)*+{\mathbb{D}_1\times_{\mathbb{D}_0}\mathbb{D}_1} ="2";
(-20,-6)*+{\mathbb{B}_1 }="3";
(20, -6)*+{\mathbb{D}_1}="4";
{\ar@{->}^{F_1\times_{F_0}F_1} "1";"2"};
{\ar@{->}^{F_1} "3";"4"};
{\ar@{->}_{\mathfrak m} "1";"3"};
{\ar@{->}^{\mathfrak m} "2";"4"};
\endxy
\end{equation}
strictly commutes in the 2-category $\CAT$ of categories.
\end{definition}
\begin{definition}[Normal lax 1-morphism/ lax functor of double
categories] \label{def:lax-1-mor} A {\sf normal lax 1-morphism $F$} (or a
{\sf normal lax functor}) from a double category $\mathbb{B}$ to a double
category $\mathbb{D}$ consists of a pair of ordinary functors
$F_0:\mathbb{B}_0\to \mathbb{D}_0$, $F_1: \mathbb{B}_1\to \mathbb{D}_1$ which commute strictly
with the source, target and unit functors of the double categories
$\mathbb{B}$ and $\mathbb{D}$ together with a natural transformation
$F_*: F_1\circ \mathfrak m \Rightarrow \mathfrak m \circ (F_1\times _{F_0} F_1)$
which is subject to a coherence condition spelled out below.
The diagram \eqref{eq:2.18} 2-commutes in $\CAT$ and the
2-commutativity is witnessed by the natural transformation $F_*$.
Thus for every pair $(\mu, \nu)$ of objects of $\mathbb{B}_1$ composable
under $\mathfrak m$ we have a (globular) 2-cell
$(F_*)_{\mu,\nu}: F_1(\mu) *F_1 (\nu)\Rightarrow F_1(\mu*\nu)$. We
further require that for
any triple of objects in $(\mu,\nu, \sigma)$ of $\mathbb{B}_1$ composable
under $\mathfrak m$
\[
(F_*)_{\mu*\nu, \sigma}\circ \left(
(F_*)_{\mu,\nu} * \mathsf{id}_{F_1(\sigma)} \right) =
(F_*)_{\mu, \nu*\sigma}\circ \left(
\mathsf{id}_{F_1(\mu)}* (F_*)_{\nu, \sigma}
\right)
\]
(recall that $\alpha *\beta := \mathfrak m(\alpha, \beta)$ for any two
2-cells $\alpha, \beta$).
\end{definition}
\begin{remark}
Since more general lax functors (the ones that are not necessarily normal) will
make no appearance in this paper we will refer to normal lax
functors simply as lax functors. We trust that this will cause no confusion.
\end{remark}
Since double categories are categories internal to the (2-)category $\CAT$
of categories, there are two ways to internalize the notion of a
natural transformation between two double functors.
Namely given two double functors $F,G: \mathbb{B} \to \mathbb{D}$ between double categories
one can ask for a functor $\alpha:\mathbb{B}_0\to \mathbb{D}_1$ making an
appropriate diagram of categories and functors to commute (or
2-commute in the lax version). This lead to the notion of a {\sf
horizontal transformation}. Alternatively since each double functor
is a pair of ordinary functors one can ask for a pair of (ordinary) natural
transformation. This leads to the definition of a {\sf vertical
transformation} which we now recall in the case where the double
functors are strict. This is the only case we need.
\begin{definition}[vertical transformation]
Let $F,G:\mathbb{B}\to \mathbb{D}$ be two strict double functors between two double
categories. A {\sf vertical transformation} $\alpha$ from $F$ to
$G$ is a pair of natural transformations $\alpha_0: F_0 \Rightarrow
G_0$, $\alpha_1: F_1\Rightarrow G_1$ (both often written as $\alpha$) subject
to the following conditions:
\begin{enumerate}
\item $\alpha$ is compatible with the source and
target functors: for any object $\mu$ of $\mathbb{B}_1$
\[
\mathfrak s ((\alpha_1)_\mu )= (\alpha_0)_{\mathfrak s(\mu)}, \qquad
\mathfrak t ((\alpha_1)_\mu )= (\alpha_0)_{\mathfrak t(\mu)} ;
\]
\item $\alpha$ is compatible with the multiplication/composition functors
$\mathfrak m$ of $\mathbb{B}$ and $\mathbb{D}$:
for any pair $\mu, \nu$ of composable objects of $\mathbb{B}_1$
\[
(\alpha_1)_{\mu*\nu} = (\alpha_1)_\mu* (\alpha_1)_\nu;
\]
\item $\alpha$ is compatible with the unit functors:
for any object $a$ of $\mathbb{B}$
\[
\mathfrak u ((\alpha_0)_a) = (\alpha_1)_{\mathfrak u(a)}.
\]
\end{enumerate}
We write $\alpha:F\Rightarrow G$.
\end{definition}
\begin{remark} \label{rmrk:2.29}
Vertical transformations can be composed vertically component-wise:
given two vertical transformations $\alpha:F\Rightarrow G$, $\beta:
G\Rightarrow H$ we define $\beta\circ_v \alpha$ by setting
\[
(\beta\circ_v \alpha)_0 = \beta_0 \circ_v \alpha_0\qquad
(\beta\circ_v \alpha)_1 = \beta_1 \circ_v \alpha_1.
\]
\end{remark}
We next explain a connection between control (open) systems and lax
double functors. The terms ``open system'' and ``control system''
mean the same thing but have somewhat different connotations.
``Open'' is the opposite of ``closed,'' which means that energy and
information can flow in and out of an open system. ``Control''
implies that we can affect the system's behavior.
Since in this paper we will not address the issues of
shaping behavior of systems, we prefer to call them open. On the
other hand, in several previous papers \cite{DL1}, \cite{L1} we
used a functor from open systems to vector spaces and linear
relations that we called $\mathsf{Crl}$. Renaming it $\mathsf{Open}$ may
well cause confusion especially since every topological space $Z$ has a
category $\mathsf{Open} (Z)$ of open sets. To avoid confusion and
to preserve backward compatibility we will continue to call the
functor in question $\mathsf{Crl}$.
We now record our definition of a
control (open) system which was already mentioned in the introduction.
\begin{definition}\label{def:open_sys}
An {\sf open} (or a {\sf
control}) system is a pair $(a, F)$ where $a= (a_\mathsf{tot}
\xrightarrow{p_a} a_\mathsf{st})$ is a surjective submersion and $F:
a_\mathsf{tot} \to Ta_\mathsf{st}$ is a smooth map
such that $ F(q) \in T_{p_a (q)} a_\mathsf{st}$ for all $q\in a_\mathsf{tot}$.
That is $\pi_{a_\mathsf{st}} \circ F = p_a$, where $\pi_{a_\mathsf{st}}:Ta_\mathsf{st}
\to a_\mathsf{st}$ is the canonical submersion.
\end{definition}
\begin{remark} Given an open system $(a, F)$ we may also refer to
the map $F:a_\mathsf{tot} \to Ta_\mathsf{st}$ as
an {\sf open system on the the surjective submersion $a$} or as an
{\sf open system} (with the surjective submersion $a$ suppressed).
\end{remark}
\begin{notation}[$\mathsf{Crl}(a)$] \label{not:crl}
We denote the collection of all open system on a fixed submersion
$a$ by $\mathsf{Crl}(a)$:
\[
\mathsf{Crl}(a):= \{F:a_\mathsf{tot} \to Ta_\mathsf{st} \mid \pi_{a_\mathsf{st}}\circ F = p_a\}
\]
The collection $
\mathsf{Crl}(a) $ is a real vector space.
\end{notation}
\begin{definition} \label{def:Crl(f)}
Let $f:a\to b$ be a map of surjective
submersions, that is, a pair of
smooth maps $f_\mathsf{tot} :a_\mathsf{tot} \to b_\mathsf{tot}$, $f_\mathsf{st}:a_\mathsf{st}\to b_\mathsf{st}$ such
that the diagram
\[
\xy
(-10, 6)*+{a_\mathsf{tot}} ="1";
(10, 6)*+{b_\mathsf{tot}} ="2";
(-10,-10)*+{a_\mathsf{st} }="3";
(10, -10)*+{b_\mathsf{st}}="4";
{\ar@{->}_{p_a} "1";"3"};
{\ar@{->}^{p_b} "2";"4"};
{\ar@{->}^{f_\mathsf{tot}} "1";"2"};
{\ar@{->}_{f_\mathsf{st}} "3";"4"};
\endxy
\]
commutes. We define a linear relation $\mathsf{Crl}(f):\mathsf{Crl}(a) \to \mathsf{Crl}(b)$ by
\[
\mathsf{Crl}(f) := \left\{ (F,G)\in \mathsf{Crl}(a)\times \mathsf{Crl}(b) \left|\quad
\xy
(-10, 6)*+{a_\mathsf{tot}} ="1";
(10, 6)*+{b_\mathsf{tot}} ="2";
(-10,-10)*+{Ta_\mathsf{st} }="3";
(10, -10)*+{Tb_\mathsf{st}}="4";
{\ar@{->}_{F} "1";"3"};
{\ar@{->}^{G} "2";"4"};
{\ar@{->}^{f_\mathsf{tot}} "1";"2"};
{\ar@{->}_{Tf_\mathsf{st}} "3";"4"};
\endxy \quad \textrm{commutes} \right \}. \right.
\]
\end{definition}
The next definition says more or less the same thing as
Definition~\ref{def:Crl(f)} but from a somewhat different point of
view.
\begin{definition}
Let $f:a\to b$ be a map of surjective
submersions. Suppose $w\in \mathsf{Crl}(b)$ and $u\in \mathsf{Crl}(a)$ are two
open systems so that $(u,w)\in \mathsf{Crl}(f)$, i.e., the diagram
\[
\xy
(-10, 6)*+{a_\mathsf{tot}} ="1";
(10, 6)*+{b_\mathsf{tot}} ="2";
(-10,-10)*+{Ta_\mathsf{st} }="3";
(10, -10)*+{Tb_\mathsf{st}}="4";
{\ar@{->}_{u} "1";"3"};
{\ar@{->}^{w} "2";"4"};
{\ar@{->}^{f_\mathsf{tot}} "1";"2"};
{\ar@{->}_{Tf_\mathsf{st}} "3";"4"};
\endxy
\]
commutes. We then say that the control system $u$ {\sf is
$f$-related} to the control system $w$.
\end{definition}
\begin{definition} \label{def:2.35}
Suppose $\varphi:a\to b$ is an interconnection morphism (Definition~\ref{def:ssub}). Then $\varphi$ induces a
linear map $\varphi^*: \mathsf{Crl}(b) \to \mathsf{Crl} (a)$. It is defined by
\[
\varphi^*F:= T(\varphi_\mathsf{st})^{-1} \circ F \circ \varphi_\mathsf{tot}
\]
for all open systems $F\in \mathsf{Crl}(b)$.
\end{definition}
\begin{remark}
For an interconnection morphism $\varphi:a\to b$ between two
surjective submersions the linear relation $\mathsf{Crl}(\varphi): \mathsf{Crl}(a) \to \mathsf{Crl}(b)$ is given
by
\[
\mathsf{Crl}(\varphi): =\{(\varphi^*F, F) \in \mathsf{Crl}(a)\times \mathsf{Crl}(b)\mid
F\in \mathsf{Crl}(b)\}
\]
where the linear map $\varphi^*$ is defined above (Definition~\ref{def:2.35}).
Consequently the linear relation $\mathsf{Crl}(\varphi)$ is the transpose of the graph of the
map $\varphi^*$ (see Definition~\ref{def:transpose}).
\end{remark}
\begin{remark} \label{rmrk:st_conv}
As is traditional in differential geometry and geometric mechanics all
manifolds with corners are nonempty unless specifically mentioned otherwise.
\end{remark}
\begin{example} \label{ex:intercon}
Let $U$ and $M$ be two manifolds with corners. The projection on
the second factor $p:U\times M\to M$ is a surjective submersion
(note
Remark~\ref{rmrk:st_conv}). So
is the identity map $\mathsf{id}_M:M\to M$. Consider a map of surjective
submersions
$\varphi= (\varphi_\mathsf{tot}, \varphi_\mathsf{st}): (M\xrightarrow{\mathsf{id}_M}M) \to
(U\times M\xrightarrow{p} M) $. The map
$\varphi_\mathsf{tot}:M\to U\times M$ then has to be of the form
$\varphi_\mathsf{tot}(m) = (h(m), \varphi_\mathsf{st}(m))$ where $h:M\to U$ is a
smooth map. In particular if $\varphi_\mathsf{st} = \mathsf{id}_M$ then for any map
$h$, $\varphi = ((h, \mathsf{id}_M), \mathsf{id}_M)$ is an interconnection morphism.
The induced map
$\varphi^*: \mathsf{Crl}(U\times M\to M) \to \mathsf{Crl}(M\xrightarrow{\mathsf{id}} M) =
\mathfrak{X}(M)$ is given by
\[
(\varphi^*F ) (m) = F(h(m), m).
\]
(Recall that $\mathfrak{X}(M)$ denotes the space of vector fields on the
manifold $M$.) To summarize: given an open system $F:U\times M\to TM$
we produce from $F$ a vector field $X$ on the manifold $M$ by plugging
the values of the states of the open system into the controls $U$ by
means of the map $h$.
\end{example}
We observe that pullbacks by interconnection maps preserve relations
between open systems.
\begin{lemma}
Let $f:a\to c$, $g:b\to d$ be maps of surjective submersions,
$\varphi: a\to b$, $\psi: c\to d$ two interconnection maps so that the
diagram
\[
\xy
(-10,6)*+{c}="1";
(10, 6)*+{a}="2";
(-10,-7)*+{d}="3";
(10, -7)*+{b}="4";
{\ar@{<-}^{f} "1";"2"};
{\ar@{->}_{\psi} "1";"3"};
{\ar@{->}^{\varphi} "2";"4"};
{\ar@{->}^{g} "4";"3"};
\endxy
\]
commutes in the category $\mathsf{SSub}$ of surjective submersions (i.e., the diagram is a
2-cell in the double category $\mathsf{SSub}^\Box$). If two open systems
$F\in \mathsf{Crl}(b)$ and $G\in \mathsf{Crl}(d)$ are $g$-related then the
interconnected systems $\varphi^*F$ and $\psi^*G$ are $f$-related.
Consequently
\[
\xy
(-10,6)*+{\mathsf{Crl}(c)}="1";
(10, 6)*+{\mathsf{Crl}(a)}="2";
(-10,-7)*+{\mathsf{Crl}(d)}="3";
(10, -7)*+{\mathsf{Crl}(b)}="4";
{\ar@{<-}^{\mathsf{Crl}(f)} "1";"2"};
{\ar@{<-}_{\psi^*} "1";"3"};
{\ar@{<-}^{\varphi^*} "2";"4"};
{\ar@{->}^{\mathsf{Crl}(g)} "4";"3"};
{\ar@{<=}^{} (0,2)*{};(0,-2)};
\endxy
\]
is a 2-cell in the double category $\mathsf{RelVect}^\Box$ of vector
spaces, linear maps and linear relations.
\end{lemma}
\begin{proof}
See \cite[Lemma~8.12]{L2}
\end{proof}
\begin{lemma}
The mapping from the 2-cells of the double category $\mathsf{SSub}^\Box$ of
surjective submersions to the double category $\mathsf{RelVect}^\Box$ given by
\[
\mathsf{Crl} \left(
\xy
(-10,6)*+{c}="1";
(10, 6)*+{a}="2";
(-10,-7)*+{d}="3";
(10, -7)*+{b}="4";
{\ar@{<-}^{f} "1";"2"};
{\ar@{->}_{\psi} "1";"3"};
{\ar@{->}^{\varphi} "2";"4"};
{\ar@{->}^{g} "4";"3"};
{\ar@{=>}^{} (0,2)*{};(0,-2)};
\endxy
\right) :=
\xy
(-10,6)*+{\mathsf{Crl}(c)}="1";
(10, 6)*+{\mathsf{Crl}(a)}="2";
(-10,-7)*+{\mathsf{Crl}(d)}="3";
(10, -7)*+{\mathsf{Crl}(b)}="4";
{\ar@{<-}^{\mathsf{Crl}(f)} "1";"2"};
{\ar@{<-}_{\psi^*} "1";"3"};
{\ar@{<-}^{\varphi^*} "2";"4"};
{\ar@{->}^{\mathsf{Crl}(g)} "4";"3"};
{\ar@{<=}^{} (0,2)*{};(0,-2)};
\endxy
\]
defines a lax (contravariant) 1-morphism of double categories
\[
\mathsf{Crl}: (\mathsf{SSub} ^\Box)^\mathsf{op} \to \mathsf{RelVect}^\Box.
\]
\end{lemma}
\begin{proof} Note that $\mathsf{Crl}(id_c) = id_{\mathsf{Crl}(c)}$. Next observe
that given a pair of compatible maps of submersions $c\xrightarrow{f} b
\xrightarrow{g} a$ the composite
$\mathsf{Crl}(f)\circ \mathsf{Crl}(g)$ of linear relations is a subspace of the linear
relation $\mathsf{Crl}(f\circ g)$. The inclusion $
\mathsf{Crl}(f)\circ \mathsf{Crl}(g)\hookrightarrow \mathsf{Crl}(f\circ g)$ is the $f,g$
component $(\mathsf{Crl}_*)_{f,g}$
of the desired natural transformation $(\mathsf{Crl})_* $ (cf.\ Definition~\ref{def:lax-1-mor}).
\end{proof}
\subsection{Categories of lists}\mbox{}
In this somewhat technical subsection we collect a number of
definitions and facts that will provide a convenient language later
on. Recall that any set $X$ can be considered as a discrete category.
Then a functor $\tau:X\to \sfC$ from a set $X$ (considered as a
category) to a category
$\sfC$ assigns to each element $x\in X$ an object $\tau(x)$ of
$\sfC$. Thus a functor from a set to a category $\sfC$ is an
unordered list of elements of $\sfC$ (possibly with repetitions)
which is indexed by the elements of the set.
Functors from various sets to a fixed category $\sfC$ can be
assembled into a category.
\begin{definition}[The category of lists $\Set/\sfC$ in a category
$\sfC$]\label{def:cat_of_lists}
Fix a category $\sfC$. An object of the category of lists
$\Set/\sfC$ is a functor $\tau:X\to \sfC$ where $X$ is a set thought of
as a discrete category. A morphism in $\Set/\sfC$ from a functor
$\tau:X\to \sfC$ to a functor $\tau':X'\to \sfC$ is a map of sets
$\varphi:X\to X'$ so that the triangle of functors
\[
\xy
(-10, 10)*+{X} ="1";
(10, 10)*+{X'} ="2";
(0,-2)*+{\sfC }="3";
{\ar@{->}_{\tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{\tau'} "2";"3"};
\endxy
\]
commutes.
The composition is defined by pasting of the triangles:
\[
\left(
\xy
(-10, 6)*+{X''} ="1";
(10, 6)*+{X'} ="2";
(0,-6)*+{\sfC }="3";
{\ar@{->}_{\tau''} "1";"3"};
{\ar@{<-}^{\psi} "1";"2"};
{\ar@{->}^{\tau'} "2";"3"};
\endxy
\right)\circ \left(
\xy
(-10, 6)*+{X'} ="1";
(10, 6)*+{X} ="2";
(0,-4)*+{\sfC }="3";
{\ar@{->}_{\tau'} "1";"3"};
{\ar@{<-}^{\varphi} "1";"2"};
{\ar@{->}^{\tau} "2";"3"};
\endxy
\right) =
\xy
(-10, 6)*+{X''} ="1";
(10, 6)*+{X} ="2";
(0,-6)*+{\sfC }="3";
{\ar@{->}_{\tau''} "1";"3"};
{\ar@{<-}^{\psi\circ \varphi} "1";"2"};
{\ar@{->}^{\tau} "2";"3"};
\endxy
\]
\end{definition}
\begin{definition}[The category $\mathsf{FinSet}/\sfC$ of finite lists]
The category of lists $\Set/\sfC$ has a subcategory $\mathsf{FinSet}/\sfC$
whose objects are {\em finite} lists, i.e., functors from finite sets.
\end{definition}
\begin{remark}
If a category $\sfC$ has coproducts then there is a canonical functor
$\rotpi: \Set/\sfC \to \sfC$ defined on objects by taking colimits
\[
\rotpi (X\xrightarrow{\tau}\sfC) := \colim (X\xrightarrow{\tau}
\sfC ) = \bigsqcup _{x\in X}\tau (x).
\]
On arrows it is defined by the universal properties of coproducts:
given a morphism $\varphi: (X\xrightarrow{\tau} \sfC)\to
(X'\xrightarrow{\tau'} \sfC)$ the morphism $\rotpi (\varphi): \rotpi
(\tau)\to \rotpi (\tau')$ is the unique arrow making the diagram
\[
\xy
(-10, 6)*+{\rotpi(\tau)} ="1";
(12, 6)*+{\rotpi(\tau')} ="2";
(-10,-6)*+{\tau(x)}="3";
(12, -6)*+{\tau'(\varphi(x)))}="4";
{\ar@{<--}_{\rotpi(\varphi)} "2";"1"};
{\ar@{->}_{\mathsf{id}} "3";"4"};
{\ar@{->}^{\imath_x} "3";"1"};
{\ar@{->}_{\imath_{\varphi(x)}} "4";"2"};
\endxy
\]
commute. Here $\imath_x$ and $\imath_{\varphi(x)}$ are the
canonical inclusions.
\end{remark}
\begin{remark}\label{rmrk:2.35}
If the category $\sfC$ has finite products then there is a canonical functor
$\Pi: (\mathsf{FinSet}/\sfC)^\mathsf{op} \to \sfC$ defined on objects by taking
limits:
\[
\Pi (\tau): = \lim (X\xrightarrow{\tau} \sfC) = \bigsqcap _{x\in X}\tau(x).
\]
On arrows it is defined by the universal property of products. Namely, given
a morphism $\varphi: (X\xrightarrow{\tau} \sfC)\to
(X'\xrightarrow{\tau'} \sfC)$ the morphism $\Pi (\varphi): \Pi
(\tau')\to \Pi (\tau)$ is the unique arrow making the diagram
\[
\xy
(-16, 6)*+{\Pi(\tau)} ="1";
(12, 6)*+{\Pi(\tau')} ="2";
(-16,-6)*+{\tau(x)}="3";
(12, -6)*+{\tau'(\varphi(x)))}="4";
{\ar@{-->}_{\Pi(\varphi)} "2";"1"};
{\ar@{<-}_{\mathsf{id}} "3";"4"};
{\ar@{<-}^{\pi_x} "3";"1"};
{\ar@{<-}_{\pi_{\varphi(x)}} "4";"2"};
\endxy
\]
commute. Here $\pi_x$ and $\pi_{\varphi(x)}$ are the
canonical projections.
\end{remark}
Recall that the category $\Man$ of manifolds with corners has finite
products. This can be seen as follows. The terminal object is a one
point manifold. The Cartesian product of two manifolds with corners
is again a manifold with corners and serves (together with the
canonical projections maps) as their product in the category $\Man$.
Since $\Man$ has a terminal object and binary products, it has finite products.
\begin{example} \label{ex:exPiphi1}
Let $\sfC =\Man$, the category of manifolds with corners, $X = Y = \{1,2, 3\}$,
$\varphi:X\to Y$ be given by
\[
\varphi(1) = 2,\qquad \varphi(2) = 1, \qquad \varphi(3) =2.
\]
Fix a manifold with corners $A$. Let $\tau, \mu: X, Y\to \Man$ be
the constant maps defined by $\tau(j) = \mu(j) =A$ for all $j$.
Then $\varphi: \tau \to \mu$ is a morphism in $\mathsf{FinSet}/\Man$ and
$\Pi (\varphi):\Pi(\mu) = A^3 \to A^3 = \Pi(\tau)$ is given by
\[
\Pi(\varphi)\, (a_1, a_2, a_3) = (a_2, a_1, a_2)
\]
for all $(a_1,a_2, a_3) \in A^3 = \Pi(\mu)$.
\end{example}
The categories of lists $\Set/\sfC$ and $\mathsf{FinSet}/\sfC$ have variants
in which the commuting triangles of morphisms are replaced by
2-commuting triangles. More precisely we have the following
definitions.
\begin{definition}[The category of lists $(\Set/\sfC)^\Rightarrow$]
\label{def:2.40}
Fix a category $\sfC$. An object of the category of lists
$(\Set/\sfC)^\Rightarrow $ is a functor $\tau:X\to \sfC$ where $X$ is
a set thought of as a discrete category. That is, the objects of
$(\Set/\sfC)^\Rightarrow$ are the same as the objects of $\Set/\sfC$.
A morphism in $(\Set/\sfC)^\Rightarrow$ from a functor
$\tau:X\to \sfC$ to a functor $\tau':X'\to \sfC$ is a 2-commuting
triangle
\[
\xy
(-10, 10)*+{X} ="1";
(10, 10)*+{X'} ="2";
(0,-2)*+{\sfC }="3";
{\ar@{->}_{\tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{\tau'} "2";"3"};
{\ar@{<=}_{\scriptstyle \Phi} (4,6)*{};(-0.4,4)*{}} ;
\endxy .
\]
In other words a morphism in $(\Set/\sfC)^\Rightarrow$ is pair
$(\varphi, \Phi)$ where $\varphi:X\to X'$ is a map of sets and
$\Phi: \tau \Rightarrow \tau' \circ \varphi$ is a natural
transformation. Given a pair of composable morphisms
$(\varphi, \Phi): (X\xrightarrow{\tau}\sfC)\to (X'\xrightarrow{\tau'}
\sfC)$ and
$(\psi, \Psi) : (X'\xrightarrow{\tau'} \sfC)\to
(X''\xrightarrow{\tau''} \sfC)$ their composition is defined by pasting
of the 2-commuting triangles. That is,
\[
(\psi, \Psi) \circ (\varphi, \Phi) :=
(\psi \circ \varphi, (\Psi \circ \varphi) \circ _v \Phi),
\]
where
$\Psi \circ \varphi: \tau \Rightarrow \tau''$ is the whiskering
of the natural transformation with a functor and $\circ_v$ denotes the
vertical composition of natural transformations.
\end{definition}
\begin{remark}
Observe that the category $\Set/\sfC$ of lists is a subcategory of
the category $(\Set/\sfC)^\Rightarrow$
\end{remark}
\begin{remark} \label{rmrk:2.43}
If the category $\sfC$ has coproducts then there is a canonical
functor $\rotpi: (\Set/\sfC)^\Rightarrow \to \sfC$ which extends the
functor $\rotpi: \Set/\sfC \to \sfC$. As before to each object
$\tau:X\to \sfC$ the functor $\rotpi$ assigns the coproduct
$\bigsqcup _{x\in X}\tau (x)$. On arrows $\rotpi$ is again defined
by the universal properties of coproducts: given a morphism
$(\varphi, \Phi): (X\xrightarrow{\tau} \sfC)\to
(X'\xrightarrow{\tau'} \sfC)$ the morphism
$\rotpi (\varphi, \Phi): \rotpi (\tau)\to \rotpi (\tau')$ is the
unique arrow making the diagram
\[
\xy
(-10, 6)*+{\rotpi(\tau)} ="1";
(12, 6)*+{\rotpi(\tau')} ="2";
(-10,-6)*+{\tau(x)}="3";
(12, -6)*+{\tau'(\varphi(x)))}="4";
{\ar@{<--}_{\rotpi(\varphi, \Phi)} "2";"1"};
{\ar@{->}_{\Phi_x} "3";"4"};
{\ar@{->}^{\imath_x} "3";"1"};
{\ar@{->}_{\imath_{\varphi(x)}} "4";"2"};
\endxy
\]
commute. Here as before $\imath_x$ and $\imath_{\varphi(x)}$ denote the
canonical inclusions.
\end{remark}
\begin{definition}[The category $(\mathsf{FinSet}/\sfC)^\Leftarrow$] The
objects of the category $(\mathsf{FinSet}/\sfC)^\Leftarrow$ are finite
lists $\tau:X\to \sfC$ (i.e., they are the same as the objects of
the category $\mathsf{FinSet}/\sfC$ of finite lists in the category
$\sfC$). A morphism $(\varphi,\Phi): (X\xrightarrow{\tau} \sfC )
\to (Y\xrightarrow{\mu}\sfC)$ is a pair $(\varphi, \Phi)$ where
$\varphi: X\to Y$ is a map of sets and $\Phi$ is now a natural
transformation from $\mu \circ \varphi$ to $\tau$. That is, the
morphism is a 2-commuting triangle of the form
\begin{equation}
\xy
(-10, 10)*+{X} ="1";
(10, 10)*+{Y} ="2";
(0,-2)*+{\sfC }="3";
{\ar@{->}_{\tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{\mu} "2";"3"};
{\ar@{=>}_{\scriptstyle \Phi} (4,6)*{};(-0.4,4)*{}} ;
\endxy .
\end{equation}
The composition in $(\mathsf{FinSet}/\sfC)^\Leftarrow$ is defined by pasting
of the triangles.
\end{definition}
\begin{remark} \label{rmrk:2.41}
If the category $\sfC$ has finite products then the functor $\Pi:
(\mathsf{FinSet}/\sfC)^\mathsf{op} \to \sfC$ extends to a functor $\left(
(\mathsf{FinSet}/\sfC)^\Leftarrow\right)^\mathsf{op} \to \sfC$, which we again denote by
$\Pi$: given a morphism $(\varphi, \Phi): (X\xrightarrow{\tau} \sfC)\to
(X'\xrightarrow{\tau'} \sfC)$ the morphism $\Pi (\varphi, \Phi): \Pi
(\tau')\to \Pi (\tau)$ is the unique arrow making the diagram
\[
\xy
(-10, 6)*+{\Pi(\tau)} ="1";
(12, 6)*+{\Pi(\tau')} ="2";
(-10,-6)*+{\tau(x)}="3";
(12, -6)*+{\tau'(\varphi(x)))}="4";
{\ar@{-->}_{\Pi(\varphi, \Phi )} "2";"1"};
{\ar@{<-}_{\Phi_x} "3";"4"};
{\ar@{<-}^{\pi_x} "3";"1"};
{\ar@{<-}_{\pi_{\varphi(x)}} "4";"2"};
\endxy
\]
commute. Here as before $\pi_x$ and $\pi_{\varphi(x)}$ are the
canonical projections.
\end{remark}
\begin{example} \label{ex:exPiphi2}
Let $\sfC =\Man$, the category of manifolds with corners, $X = Y = \{1,2, 3\}$.
Fix two manifolds with corners $A$ and $B$ and a smooth map $s:A\to B$. Let $\tau, \mu: X, Y\to \Man$ be
the constant maps defined by $\tau(j) = B$, $\mu(j) =A$ for all
$j$. We define a morphism $(\varphi,
\Phi): \tau \to \mu$ in $(\mathsf{FinSet}/\Man)^\Leftarrow$ as follows.
As before we define $\varphi$ by
\[
\varphi(1) = 2,\qquad \varphi(2) = 1, \qquad \varphi(3) =2.
\]
We define $\Phi_i: \mu (\varphi(i)) = A \to \tau(i) = B$ to be the map
$s:A \to B$ for all $i$.
Then $(\varphi,\Phi): \tau \to \mu$ is a morphism in $(\mathsf{FinSet}/\Man)^\Leftarrow$ and
$\Pi (\varphi, \Phi):\Pi(\mu) = A^3 \to B^3 = \Pi(\tau)$ is given by
\[
\Pi(\varphi,\Phi)\, (a_1, a_2, a_3) = (s(a_2), s(a_1), s( a_2))
\]
for all $(a_1,a_2, a_3) \in A^3 = \Pi(\mu)$.
\end{example}
\mbox{}\\
\section{The category of elements $\int F$ for a functor
with values in linear relations}\label{sec:cat_of_el}
Given a set-valued functor $F:\sfC \to \Set$ on a category $\sfC$
there is a well-known construction due to Grothendieck that produces
a {\sf category of elements} $\int F $ together with the functor $\pi_F: \int
F\to \sfC$. Recall that the objects of the
category $\int F$ are pairs $(c, x)$ where $c$ is an object of $\sfC$
and $x$ is an element of the set $F(c)$. A morphism in $\int F$ from
$(c, x)$ to $(c', x')$ is a morphism $h:c\to c'$ in $\sfC$ such that
$F(h)x = x'$. The functor $\pi_F:\int F\to \sfC$ maps an
arrow $h:(x,c)\to (x',c')$ of $\int F$ to the arrow $h:c\to c'$ of $\sfC$. The
functor $\pi_F: \int F \to \sfC$ has nice lifting properties.
It will be useful for us to have a generalization of this construction
to functors with values in the 2-category $\mathsf{RelVect}$ of vector spaces
and linear relations.
Namely suppose $\sfC$ is a category and we are given a (lax) functor
$F: \sfC \to \mathsf{RelVect}$. That is, suppose that for any pair of
composable arrows $c''\xleftarrow{g} c'\xleftarrow{h}c$ in $\sfC$ we
have an inclusion $F(g) \circ F(h) \subset F(g\circ h)$. We then can
define the ``category of elements'' $\int F$ and a functor
$\pi_F:\int F\to \sfC$ (see below). In general the functor $\pi_F$ has no evident lifting
properties. In the next section we will realize the category $\HyDS$
of hybrid dynamical systems as the category of elements for a lax
$\mathsf{RelVect}$-valued functor. Later we will use the construction to
produce the category $\mathsf{HyOS}$ of hybrid open systems.
\begin{definition}[The category of elements $\int F$ of a lax functor $F: \sfC
\to \mathsf{RelVect}$] \label{def:cat_of_elements}
Let $\sfC$ be a category and $F:\sfC \to \mathsf{RelVect}$ a lax 2-functor
with values in the 2-category $\mathsf{RelVect}$ of linear relations.
We define the {\sf category of elements $\int F$} of the functor $F$ as follows.
\begin{enumerate}
\item The objects of $\int F$ are pairs $(c, x)$ where $c$ is an
object of $\sfC$ and $x$ is a vector in the vector space $F(c)$.
\item A morphism from $(c,x)$ to $(c',x')$ is a morphism $h: c\to c'$
such that $(x,x')$ is an element of the linear relation $F(h) \subset
F(c)\times F(c')$.
\end{enumerate}
It is easy to see that $\int F$ is a category. We also have a
functor $\pi_F: \int F\to \sfC$ which is defined by
\[
\pi_F ((c,x)\xrightarrow{h} (c',x') ) = c\xrightarrow{h}c'.
\]
\end{definition}
\begin{remark}
Recall that continuous time dynamical systems form a category $\mathsf{DS}$.
The objects of this category are pairs $(M, X)$ where $M$ is a
manifold with corners and $X\in \mathfrak{X}(M)$ is a vector field on $M$. A
morphism from a dynamical system $(M,X)$ to a system $(N,Y)$ is a so
called ``semi-conjugacy'': a smooth map $f:M\to N$ so that $Tf\circ X
= Y\circ f$.
\end{remark}
\begin{example}[Continuous time dynamical systems from the vector
field functor $\mathfrak{X}:\Man\to \mathsf{RelVect}$]\label{ex:3.3}
Consider the category $\Man$ of manifolds with corners. The
map $\mathfrak{X}$ that assigns to every manifold $M$ the vector space
$\mathfrak{X}(M)$ of vector fields on $M$ extends to a lax functor $\mathfrak{X}: \Man
\to \mathsf{RelVect}$: given a smooth map $f:M\to N$ the relation $\mathfrak{X}(f)$
is, by definition
\[
\mathfrak{X}(f) := \{ (X,Y)\in \mathfrak{X}(M)\times \mathfrak{X}(N) \mid Y \circ f = Tf \circ X\}.
\]
The category of elements $\int \mathfrak{X}$ is
the category $\mathsf{DS}$ of continuous
time dynamical systems.
\end{example}
\begin{remark}
The category $\mathsf{OS}$ of (continuous time) open systems is less well
known than the category $\mathsf{DS}$ but it has appeared in literature. See
for example \cite{TP} where the category of open systems is called
$\mathsf{Con}$.
The objects of the category $\mathsf{OS}$ are pairs $(a, f)$ where $a=(p_a: a_\mathsf{tot}
\to a_\mathsf{st})$ is a surjective submersion and $f:a_\mathsf{tot} \to Ta_\mathsf{st}$ is an
open system.
\end{remark}
\begin{example} [The category $\mathsf{OS}$ of open systems from the functor
$\mathsf{Crl}:\mathsf{SSub} \to \mathsf{RelVect}$] \label{ex:3.4}
Consider the category $\mathsf{SSub}$ of surjective submersions
(Notation~\ref{not:ssub}). Recall that objects of $\mathsf{SSub}$
are surjective submersions $a = (p_a: a_\mathsf{tot} \to a_\mathsf{st})$ of manifolds
with corners and that a morphism
from a submersion $a$ to a submersion $b$ is, by definition, a pair of
smooth maps $f_\mathsf{tot} :a_\mathsf{tot} \to b_\mathsf{tot}$, $f_\mathsf{st}:a_\mathsf{st}\to b_\mathsf{st}$ such
that the diagram
\[
\xy
(-10, 10)*+{a_\mathsf{tot}} ="1";
(10, 10)*+{b_\mathsf{tot}} ="2";
(-10,-10)*+{a_\mathsf{st} }="3";
(10, -10)*+{b_\mathsf{st}}="4";
{\ar@{->}_{p_a} "1";"3"};
{\ar@{->}^{p_b} "2";"4"};
{\ar@{->}^{f_\mathsf{tot}} "1";"2"};
{\ar@{->}_{f_\mathsf{st}} "3";"4"};
\endxy
\]
commutes.
For every surjective submersion $a$ we have the vector space $\mathsf{Crl}(a)$
of open systems on $a$ (Definition~\ref{def:open_sys}).
For each map of
submersions $f:a\to b$, there is a linear relation
$\mathsf{Crl}(f) \subset \mathsf{Crl}(a)\times \mathsf{Crl}(b)$ consisting of $f$-related
control systems (Definition~\ref{def:Crl(f)}).
The corresponding category of elements $\int \mathsf{Crl}$ is the category
$\mathsf{OS}$ of
open (control) systems.
\end{example}
\begin{remark}
Note that there is a canonical embedding $\imath: \Man \to \mathsf{SSub}$. On
objects it is given by $\imath(M) = (M\xrightarrow{\mathsf{id}_M} M)$. Note that the
functors $\mathfrak{X}$ and $\mathsf{Crl}$ are compatible with the embedding
$\imath$:
\begin{equation}
\mathsf{Crl} \circ \imath = \mathfrak{X}.
\end{equation}
In particular we can consider every vector field $X:M\to TM$ on a
manifold $M$ as a control system on the submersion $\mathsf{id}_M:M\to M$. We
think of vector fields as control systems with no inputs, that is, as
closed systems. Thus, somewhat paradoxically, every closed system is
an open system (with no inputs from ``the outside'').
\end{remark}
We end the section with another example. Its usefulness at the moment
may not be apparent and a reader is welcome to skip it for the time
being. Later on we will generalize
this example to hybrid systems. See Example~\ref{ex:hds_on_product} below.
\begin{example}\label{rmrk:vf_on_product} In this example we show that
a vector field $X$ on a product of two manifold $M_1\times M_2$ is the
result of interconnection of two open systems. That is, we claim that
given a vector field $X\in \mathfrak{X}(M_1\times M_2)$ there exist two open
systems $(M_1\times M_2\to M_1, X_1: M_1\times M_2\to TM_1)$,
$(M_1\times M_2\to M_2, X_2: M_1\times M_2\to TM_2)$ and an
interconnection morphism (Definition~\ref{def:ssub})
\[
\varphi: (M_1\times M_2
\xrightarrow{\mathsf{id}} M_1\times M_2) \to (M_1\times M_2\to M_1)\times (M_1\times M_2\to M_2)
\]
so that $\varphi^*(X_1\times X_2) = X$ where
\[
\varphi^*:
\mathsf{Crl}\left((M_1\times M_2\to M_1)\times (M_1\times M_2\to M_2) \right)\to \mathsf{Crl}(M_1\times M_2
\xrightarrow{\mathsf{id}} M_1\times M_2) = \mathfrak{X}(M_1\times M_2)
\]
is the linear
map induced by $\varphi$ (see Definition~\ref{def:2.35}).
This can be seen as follows.
The vector field $X:M_1\times M_2 \to TM_1\times TM_2$ is of the form $X = (X_1,
X_2)$. The maps $X_1:M_1\times M_2\to TM_1$, $X_2:M_1\times M_2\to TM_2$
are open systems on the submersions $M_1\times M_2\to M_1$, $M_1\times
M_2\to M_2$, respectively. The map
\[
\varphi: (M_1\times M_2 \to M_1\times M_2) \to \left(M_1\times M_2\to M_1
\right) \times \left(M_1\times M_2\to M_2)
\right)
\]
with $\varphi_\mathsf{st} = \mathsf{id}_{M_1\times M_2}$ and $\varphi_\mathsf{tot}:M_1\times
M_2 \to (M_1\times M_2)\times (M_1\times M_2) $ given by
\[
\varphi_\mathsf{tot} (m_1, m_2) = ((m_1, m_2), (m_1, m_2)).
\]
is an interconnection map. Furthermore
\[
X(m_1, m_2) = (X_1 (m_1, m_2), X_2 (m_1, m_2)) = ((X_1\times X_2) \circ
\varphi_\mathsf{tot} )\, (m_1, m_2)
\]
for all $(m_1, m_2) \in M_1\times M_2$.
We conclude that $X= (X_1,X_2) $ is a vector field on the product
$M_1\times M_2$ if and only if $X = \varphi^* (X_1\times X_2)$ where
$X_1, X_2$ are open systems, $\varphi$ is the interconnection map
defined above and $\varphi^*:\mathsf{Crl} \left(M_1\times M_2\to M_1
\right) \times \left(M_1\times M_2\to M_2) \right)\to \mathsf{Crl} (M_1\times M_2
\to M_1\times M_2) = \mathfrak{X}(M_1\times M_2)$ is the linear map
induced by $\varphi$.
\end{example}
\mbox{}
\section{A category of hybrid phase spaces $\mathsf{HyPh}$}
We now construct the category $\mathsf{HyPh}$ of hybrid phase spaces. We have
a number of reasons for introducing this notion. First of all,
traditional definitions of hybrid dynamical systems involve a lot of
data. We would like to organize these data in a compact and
structured way. Secondly, in the next section we will need to define
hybrid open systems. There seems to be no consensus in the literature
of what a hybrid open system should be. Our approach is to view
hybrid open systems as analogous to continuous time open systems. As
we mentioned in the previous section, it is convenient to view a
continuous time open system as a pair $(a, F)$ where
$a = a_\mathsf{tot}\to a_\mathsf{st}$ is a surjective submersion and
$F:a_\mathsf{tot} \to Ta_\mathsf{st}$ is an open system on the submersion. We will
define a hybrid open system to be a pair $(a, F)$ where now
$a = a_\mathsf{tot} \xrightarrow{p_a} a_\mathsf{st}$ is a hybrid surjective
submersion, that is, a certain map of hybrid phase spaces, and
$F:\mathbb{U}(a_{tot})\rightarrow T\mathbb{U}(a_{st})$ is a continuous time open
system on an associated surjective submersion $\mathbb{U} (a)$ (see
Definition~\ref{def:HyOS}). Additionally we want our definition of a
hybrid phase space to meet a number of requirements and pass a few
sanity tests. Before we spell them out, we would like to have
two examples of traditionally defined hybrid dynamical systems before
us. For the reader's convenience a traditional definition of a hybrid
dynamical system is reviewed in Appendix~\ref{app:B}.
\begin{example}[A room with a heater and a thermostat]
\label{ex:1.1}
Imagine a one room house in winter. The room has a heater and a
thermostat. For convenience we choose the units of temperature so
that the comfort range in the room falls between 0 and 1 (say
$0 = 18^\circ C$ and $1= 20^\circ C$). Suppose the room starts at the
temperature $x=1$ and cools down to $x=0$. Assume that the evolution
of temperature is governed by the equation $\dot{x} = -1$. Once the
temperature drops down to 0 the thermostat turns on the heater and the
temperature evolution is now governed by $\dot{x} = 1$. The dynamical
system we have just described is one of the simplest examples of a
hybrid dynamical system. Formally the system consists of the disjoint
union of two manifolds with boundary
$M_{\textrm{on}}=M_{\textrm{off}} =[0,1]$ with a vector field $X $ on
$M = M_{\textrm{on}}\sqcup M_{\textrm{off}}$ defined by
$X|_{M_{\textrm{off}}} = -\frac{d}{dx}$ and
$X|_{M_{\textrm{on}}} = \frac{d}{dx}$. Additionally we have two partial
functions (see Notation~\ref{notation:rel}):
$f:M_{\textrm{off}} \to M_{\textrm{on}}$ which takes
$0\in M_{\textrm{off}}$ to $0\in M_{\textrm{on}}$ (and is undefined
elsewhere) and $g:M_{\textrm{on}} \to M_{\textrm{off}}$ which takes
$1\in M_{\textrm{on}} $ to $1 \in M_{\textrm{off}}$ (and is undefined
elsewhere). The labelled directed graph $ \xy
(-10,0)*{\bullet}="1";
(10,0)*{\bullet}="2";
(-15,0)*+{M_{\textrm{on}}}="3";
(15,0)*+{M_{\textrm{off}}}="4";
{\ar@/^.5pc/^{g} "1";"2"};
{\ar@/^.5pc/^{f} "2";"1"};
\endxy
$ may be useful for picturing the system and its discrete dynamics.
\end{example}
\begin{example}[Two rooms with heaters and thermostats] \label{ex:1.2}
Now imagine that we have a two room house with two heaters and two thermostats.
The dynamics now becomes more complicated since each heater is controlled by its own thermostat and the thermostats need not be in sync. For example, room one may reach 0 first. Then its heater will turn on.
Once the first room heats up to temperature 1, heater 1 will be turned off. By this time the second room may or may not be at 0. If the second room is above zero, it will continue to cool. Since its temperature is lower than that of the first room it may reach 0 before the temperature in the first room does. Then its heater will be turned on and so on. Many other scenarios are also possible. To describe all the possible dynamics of this system quantitatively we will need to consider the product
\[
( M^1_{\textrm{on}}\sqcup M^1_{\textrm{off}} ) \times ( M^2_{\textrm{on}}\sqcup M^2_{\textrm{off}}) =
\bigsqcup_{\alpha, \beta\in \{\textrm{on}, \textrm{off}\}} M^1_\alpha \times M^2 _\beta
\]
On each product $M^1_\alpha \times M^2 _\beta =[0,1]^2$ (which is a manifold with corners, see Appendix~\ref{app:A}) we would have a vector field $X_{\alpha, \beta}$.
If the two rooms are completely thermally isolated from each other then each $X_{\alpha, \beta} $ is a product of vector fields on the corresponding factors ($M^1_\alpha$ and $M^2_\beta$). If there is a heat exchange between the rooms the vector fields $X_{\alpha, \beta} $ would have to be more complicated (recall that a vector field on the product of two manifolds is rarely a product of vector fields on the factors).
We will also have 12 partial maps between the various products $M^1_\alpha \times M^2 _\beta$. For example we have a map
\[
f_1 \times id_{M^2_{\mathrm{on}}}: M^1_\mathrm{on} \times M^2_\mathrm{on} \to M^1_\mathrm{off} \times M^2_\mathrm{on}
\]
defined on $\{1\}\times [0,1)\subset M^1_\mathrm{on} \times M^2_\mathrm{on}$. It models the fact that when the temperature in room 1 reaches 1 and the temperature in room 2 is below 1, the heater in room 1 turns off while the heater in room 2 keeps going. The sources and targets of the partial maps can be pictured by the following graph:
\begin{equation} \label{eq:graph_product}
\xy
(-30,-23)*{M^{(1)}_{\textrm{on}} \times M^{(2)}_{\textrm{on}} };
(30,-23)*{M^{(1)}_{\textrm{on}} \times M^{(2)}_{\textrm{off}} };
(-30,23)*{M^{(1)}_{\textrm{off}} \times M^{(2)}_{\textrm{on}} };
(30,23)*{M^{(1)}_{\textrm{off}} \times M^{(2)}_{\textrm{off}} };
(-20,-20)*{\bullet}="1";
(-20,20)*{\bullet}="2";
(20,20)*+{\bullet}="3";
(20,-20)*+{\bullet}="4";
{\ar@/^.5pc/^{} "1";"2"};
{\ar@/^.5pc/^{} "2";"1"};
{\ar@/^.5pc/^{} "1";"3"};
{\ar@/^.5pc/^{} "3";"1"};
{\ar@/^.5pc/^{} "1";"4"};
{\ar@/^.5pc/^{} "4";"1"};
{\ar@/^.5pc/^{} "2";"3"};
{\ar@/^.5pc/^{} "2";"4"};
{\ar@/^.5pc/^{} "3";"2"};
{\ar@/^.5pc/^{} "3";"4"};
{\ar@/^.5pc/^{} "4";"3"};
{\ar@/^.5pc/^{} "4";"2"};
\endxy
\end{equation}
\end{example}
\mbox{}\\[12pt]
Here are the desiderata for a definition of a hybrid phase space that
we will introduce in this section.
\begin{itemize}
\item Any hybrid dynamical systems should be a pair $(a,X)$ where $a$
is a hybrid phase space and $X$ is a vector field on the manifold
``underlying'' $a$. For example the manifold underlying hybrid phase space of
Example~\ref{ex:1.1} should be the disjoint union $[0,1]\sqcup
[0,1]$ of two copies of the closed interval $[0,1]$.
\item Hybrid phase spaces should form a category; we denote it by
$\mathsf{HyPh}$. The assignment of the underlying manifold to a hybrid
phase space should be functorial. That is, there should be a
functor $\mathbb{U}: \mathsf{HyPh}\to \Man$ from the category $\mathsf{HyPh}$ of hybrid
phase spaces to the category of manifolds with corners.
\item The category $\mathsf{HyPh}$ should have finite products that behave
``correctly.'' In particular the hybrid phase space of
Example~\ref{ex:1.2} should be the product of two copies of the hybrid phase
space of Example~\ref{ex:1.1}.
\item Recall that the category of continuous time dynamical
systems is the category of elements of the functor $\mathfrak{X}: \Man
\to \mathsf{RelVect}$ (the functor $\mathfrak{X}$ assigns to a manifold $M$ the space of
vector fields on $M$). The category $\HyDS$
of hybrid dynamical systems should be the category of elements
of the composite functor $\mathfrak{X}\circ \mathbb{U}: \mathsf{HyPh}\to \mathsf{RelVect}$.
In particular a hybrid dynamical system should be a pair $(a,
X)$ where $a$ is a hybrid phase space and $X$ is a vector field
on the underlying manifold with corners $\mathbb{U}(a)$.
\item Executions of hybrid dynamical systems should be morphisms
in the category $\HyDS$, and morphisms of hybrid dynamical
systems should take executions to executions. Note that
since we are dealing with a category of elements, morphisms in
$\HyDS$ should morphisms of hybrid phase spaces satisfying
appropriate conditions. In particular executions should be
morphisms of hybrid phase spaces as well.
\end{itemize}
Here is a brief explanation of the desirability of the last item. It is analogous to the
following fact about continuous time dynamical systems. An integral
curve of a vector field $X$ on a manifold $M$ is a smooth map
$\gamma:I\to M$, where $I$ is an interval, subject to the condition
that
\[
(T\gamma)_s \left( \frac{d}{dt}\right) = X (\gamma(s))
\]
for all times $s\in I$. Here as elsewhere in the paper $T\gamma:
TI\to TM$ is the differential of $\gamma$. We therefore can view an
integral curve of a vector field $X$ as a smooth map from an interval
to the manifold that relates the constant vector field $\frac{d}{dt}$
and the vector field $X$. Moreover if $f:M\to N$ is a smooth map
between manifolds, $Y$ is a vector field on $N$ which is $f$-related
to a vector field $X$ on $M$ and $\gamma:I\to M$ is an integral curve
of the vector field $X$ then by the chain rule $f\circ \gamma$ is an integral curve of
the vector field $Y$. We want an analogous result to hold for
hybrid dynamical systems.
Thus an execution of a hybrid dynamical system $(a, X)$ should be a
map from a model ``hybrid time'' dynamical system
$(\mathscr{I}, \partial)$ (a hybrid analogue of an interaval $I \subset \mathbb{R}$
with the vector field $\frac{d}{dt}$) to the system $(a, X)$.
We construct the category $\mathsf{HyPh}$ of hybrid phase spaces by
categorifying the category of lists
$(\Set/\sfC)^\Rightarrow$: we replace the category $\Set$ of sets
with the 2-category $\Cat$ of small categories and $\sfC$ with the
double category $\RelMan^\Box$ of manifolds with corners, relations
and smooth maps.
To motivate our definition consider a traditional definition of a
hybrid dynamical system:
Definition~\ref{def:hds_trad}. Traditionally a hybrid dynamical
system consists of the following data: a directed
graph $A =\{A_1\rightrightarrows A_0\}$, an assignment $A_0 \ni y\mapsto R_y$ of
a manifold with corners to each vertex $y$ of $A$, an assignment
\[
(x\xrightarrow{\gamma}y )\mapsto (R_x \xrightarrow{R_\gamma} R_y)
\]
of a relation for each arrow $\gamma$ of $A$ and an assignment of a vector field
$X_y$ on each manifold $R_y$. We can view the collection
$\{X_y\in \mathfrak{X}(R_y)\}_{y\in A_0}$ of vector fields
as a single vector
field $X$ on the disjoint union $\bigsqcup_{y\in A_0} R_y$. We can
view the assignments $y\mapsto R_y$, $\gamma \mapsto R_\gamma$ as a
single map of graphs. The source of this a map is the graph $A$.
The target is the graph $U(\RelMan)$ which is the graph underlying the category
$\RelMan$ of manifolds with corners and relations, see
Remark~\ref{rmrk:RelMan}. Concretely the vertices of the graph $U(\RelMan)$
are manifolds with corners and the arrows are the set-theoretic
relations. We ignore the 2-category structure of
$\RelMan$ for the time being. Thus a hybrid dynamical system is a
pair
$(A\xrightarrow{R} U(\RelMan), X\in \mathfrak{X}(\bigsqcup_{y\in A_0}
R_y))$.
It seems reasonable at this point to
define the phase space of a hybrid dynamical system
$(A\xrightarrow{R} U(\RelMan), X\in \mathfrak{X}(\bigsqcup_{y\in A_0} R_a))$
to be the map of graphs $R:A\to U(\RelMan)$. Unfortunately
Examples~\ref{ex:1.1} and \ref{ex:1.2} indicate that this is not
quite right. The issue is that the product of the graph
\begin{equation}
A= \xy
(-10,0)*{\bullet}="1";
(10,0)*{\bullet}="2";
{\ar@/^.5pc/ "1";"2"};
{\ar@/^.5pc/ "2";"1"};
\endxy
\end{equation} with itself is the graph
\[ A\times A = \qquad \xy
(-20,-20)*{\bullet}="1";
(-20,20)*{\bullet}="2";
(20,20)*+{\bullet}="3";
(20,-20)*+{\bullet}="4";
{\ar@/^.5pc/^{} "1";"3"};
{\ar@/^.5pc/^{} "3";"1"};
{\ar@/^.5pc/^{} "2";"4"};
{\ar@/^.5pc/^{} "4";"2"};
\endxy
\]
and not the graph
\begin{equation}
B = \qquad \xy
(-20,-20)*{\bullet}="1";
(-20,20)*{\bullet}="2";
(20,20)*+{\bullet}="3";
(20,-20)*+{\bullet}="4";
{\ar@/^.5pc/^{} "1";"2"};
{\ar@/^.5pc/^{} "2";"1"};
{\ar@/^.5pc/^{} "1";"3"};
{\ar@/^.5pc/^{} "3";"1"};
{\ar@/^.5pc/^{} "1";"4"};
{\ar@/^.5pc/^{} "4";"1"};
{\ar@/^.5pc/^{} "2";"3"};
{\ar@/^.5pc/^{} "2";"4"};
{\ar@/^.5pc/^{} "3";"2"};
{\ar@/^.5pc/^{} "3";"4"};
{\ar@/^.5pc/^{} "4";"3"};
{\ar@/^.5pc/^{} "4";"2"};
\endxy \quad
.
\end{equation}
On the other hand the phase space of the hybrid dynamical system of
Example~\ref{ex:1.2} should be the product two copies of the phase
space of the system in Example~\ref{ex:1.1}. We choose the following
solution to the problem. Recall that the forgetful functor
$U:\CAT \to \mathsf{Graph}$ from the category of (large) categories to the
categories of graphs is part of the free/forgetful adjunction
$\mathsf{Free}:\mathsf{Graph} \leftrightarrows \CAT:U$. It is easy to see that
\[
\mathsf{Free}(A)\times \mathsf{Free}(A) = \mathsf{Free}(B)
\]
for the graphs $A$ and $B$ above.
So we now provisionally (re)define a hybrid phase space to be a functor $R: \mathsf{Free}(\Gamma) \to
\RelMan$ from the free category on some graph $\Gamma$ to the (2-) category
$\RelMan$ of manifolds with corners and relations.
We believe that our solution captures the following idea. Suppose
that we have a hybrid dynamical system that consists of two
interacting subsystems. Then its underlying phase spaces is a product
of two hybrid phase spaces which we call $a_1$ and $a_2$. An execution of
the big system is, in particular, a map $\sigma$ from a hybrid time
interval $\mathscr{I}$ to the product $a_1\times a_2$. Hence $\sigma =
(\sigma_1, \sigma_2): \mathscr{I}\to a_1\times a_2$. It may happen
that $\sigma_1$ is forced to undergo a discrete transition at time $t$
while $\sigma_2$ at $t$ is evolving continuously. Therefore given an
arrow $\gamma$ in the graph associated with the first subsystem and
the vertex $b$ of the second subsystem the graph of the product system
needs to have an arrow that corresponds to the pair $(\gamma, b)$.
The construction we chose accomplishes exactly that by assigning to
each vertex $b$ of the second graph the identity arrow $\mathsf{id}_b$ and
interpreting the pair $(\gamma, b)$ as the pair of arrow
$(\gamma, \mathsf{id}_b)$ in the product.
But why stick with free categories?
We may as well (again, provisionally) define a hybrid phase space to
be a functor $a:S_a\to \RelMan$ from some small category $S_a$ to the
category $\RelMan$ of manifolds with corners and relations. Note that
given a functor $a:S_a\to \RelMan$ there is an underling manifold
$\mathbb{U}(a): = \bigsqcup_{x\in (S_a)_0} a(x)$ where the coproduct is taken
over the set $(S_a)_0$ of objects of the small category $S_a$.
Consequently we can define a hybrid dynamical system to be a pair
$(S_a\xrightarrow{a}\RelMan, X\in \mathfrak{X}(\bigsqcup_{x\in (S_a)_0}
a(x)))$. We will revise the definition of a hybrid phase space and of
a hybrid dynamical system one more time in order to obtain a slicker
definition of morphisms between two hybrid phase spaces. We will then
show that there is a product preserving forgetful functor
$\mathbb{U}:\mathsf{HyPh}\to \Man$ from the category of hybrid phase spaces to the
category of manifolds with corners.
\begin{remark}
There are alternatives to the definition of a hybrid phase space as
a functor from some small category into $\RelMan$. The first
alternative is to use reflexive graphs instead of categories.
Recall that a reflexive graph is a directed graph that in addition to the source
and target maps $\mathfrak s$, $\mathfrak t$ from arrows to nodes also has a unit map
$\mathfrak u$ from nodes to arrows. Moreover the unit map $\mathfrak u$ is a section of
both $\mathfrak s$ and of $\mathfrak t$. In other words, every reflexive graph
assigns to each vertex $b$ an ``identity arrow'' $\mathsf{id}_b$ whose
source and target are $b$. Unlike categories reflexive graphs have
no composition map. One can show that products of reflexive graphs
behave the way we would want them to behave in our examples.
Another alternative is to use labelled transition systems as
the source of our map into $\RelMan$. Unfortunately expressing
parallel composition of labelled transition systems in the category
theoretic language is awkward; see \cite{WN}.
\end{remark}
We are now in position to formulate a definition of a hybrid phase
space. We will later reformulate this definition to make it more
succinct (Definition~\ref{def:hyph}).
\begin{definition} A {\sf hybrid phase space} is a
functor $a:S_a\to \RelMan$ from some category $S_a$ to the 1-category
$\RelMan$ of manifolds with corners and set-theoretic relations.
A {\sf map} from a hybrid phase space $a:S_a\to \RelMan$ to a hybrid
phase space $b:S_b\to \RelMan$ is a functor $\varphi:S_a\to S_b$
together with a collection of smooth maps $\{f_x:a(x)\to
b(\varphi(x))\}_{x\in (S_a)_0}$ so that for each arrow
$x\xrightarrow{\gamma}y$ of the category $S_a$ the diagram
\[
\xy
(-15,10)*+{a(x)}="1";
(-15,-10)*+{b(\varphi(x))}="2";
(15, 10)*+{a(y)}="3";
(15, -10)*+{b(\varphi(y))}="4";
{\ar@{->}_{f_x} "1";"2"};
{\ar@{->}^{a(\gamma)} "1";"3"};
{\ar@{->}_{b(\varphi(\gamma))} "2";"4"};
{\ar@{->}^{f_y} "3";"4"};
{\ar@{=>}^{} (0,3); (0,-3)};
\endxy
\]
is a 2-cell in the double category $\RelMan^\Box$. In other words we
require that for each $x\xrightarrow{\gamma}y\in (S_a)_1$ the smooth maps
\[
f_x\times f_y:a(x)\times a(y) \to b(\varphi(x))\times b(\varphi(y))
\]
map the relation $a(\gamma) \subset a(x)\times a(y)$ to the relation
$b(\varphi(\gamma))\subset b(\varphi(x))\times b(\varphi(y))$.
\end{definition}
It is not hard to show that maps $(\varphi, f):a\to b$ and
$(\psi, g):b\to c$ of hybrid phase spaces can be composed. We set
\[
(\psi, g)\circ (\varphi, f):= (\psi\circ \varphi,
h)
\]
where
\[
h_x := g_{\varphi(x)} \circ f_x
\]
for every object $x$ of the category $S_a$.
Hybrid phase spaces and their maps form a
category that we denote by $\mathsf{HyPh}$.
Moreover the same argument as in Remark~\ref{rmrk:2.43} shows that
the assignment
\[
(S_a\xrightarrow{a} \RelMan)\mapsto \mathbb{U}(a):= \bigsqcup _{x\in
(s_a)_0} a(x)
\]
extends to a functor
\[
\mathbb{U}: \mathsf{HyPh} \to \Man
\]
from the category of hybrid phase spaces to the category of
manifolds with corners. The functor $\mathbb{U}$ forgets the reset
relations. We will give another description of the functor
$\mathbb{U}:\mathsf{HyPh} \to \Man$ in Remark~\ref{rmrk:4.8}.
In order to make our definition of the category $\mathsf{HyPh}$ of hybrid
phase spaces more compact we categorify the definition of the category of lists
$(\Set/\sfC)^\Rightarrow$. Namely we replace $\sfC$ with a double category
$\mathbb{D}$ and replacing $\Set$ with the category of small categories
$\Cat$.
\begin{remark}
Let $\mathbb{D}$ be a double category and $\sfC$ an ordinary small
category, that is, an object of $\Cat$. A strict double functor
$f:\sfC \to \mathbb{D}$ then assigns to each object $x\in \sfC$ the vertical
identity arrow $\mathsf{id}_{f(x)}$ on an object $f(x) \in \mathbb{D}_0$. To each
arrow $x\xrightarrow{\gamma}y$ of $\sfC$ the functor $f$ assigns the
identity 2-cell $\mathsf{id}_{f(\gamma)}: f(\gamma) \to f(\gamma)$
from the horizontal arrow $f(\gamma):f(x)\to f(y)$ to itself. Given
two double functors $f,g:\sfC\to \mathbb{D}$ a vertical transformation
$\alpha:f\Rightarrow g$ assigns to each object $x$ of $\sfC$ a vertical
1-cell $(\alpha_0)_x:f(x)\to g(x)$ and to each arrow
$x\xrightarrow{\gamma} y $ of $\sfC$ a 2-cell
\[
\xy
(-15,10)*+{f(x)}="1";
(-15,-10)*+{g(x)}="2";
(15, 10)*+{f(y))}="3";
(15, -10)*+{g(y)}="4";
{\ar@{->}_{(\alpha_0)_x} "1";"2"};
{\ar@{->}^{f(\gamma)} "1";"3"};
{\ar@{->}_{g(\gamma)} "2";"4"};
{\ar@{->}^{(\alpha_0)_y} "3";"4"};
{\ar@{=>}^{(\alpha_1)_\gamma} (0,3); (0,-3)};
\endxy
\]
\end{remark}
We now record a generalization of Definition~\ref{def:2.40}.
\begin{definition}[The categorified category of lists $(\Cat/\mathbb{D})^\Rightarrow$]
\label{def:3.double}
Fix a double category $\mathbb{D}$. An object of the categorified category of lists
$(\Cat/\mathbb{D})^\Rightarrow $ is a double functor $\tau:\sfC \to \mathbb{D}$
where $\sfC$ is a small category thought of
as a discrete double category. A morphism in $(\sfC/\mathbb{D})^\Rightarrow$ from a double functor
$\tau:\sfC\to \mathbb{D}$ to a double functor $\tau':\sfC'\to \mathbb{D}$ is a
commuting triangle of the form
\[
\xy
(-10, 10)*+{\sfC} ="1";
(10, 10)*+{\sfC'} ="2";
(0,-2)*+{\mathbb{D} }="3";
{\ar@{->}_{\tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{\tau'} "2";"3"};
{\ar@{<=}_{\scriptstyle \Phi} (4,6)*{};(-0.4,4)*{}} ;
\endxy
\]
where
$\varphi:\sfC\to \sfC'$ is a double functor and $\Phi: \tau \Rightarrow \tau'
\circ \varphi$ is a vertical transformation.
Given a pair of
composable morphisms $(\varphi, \Phi): (\sfC\xrightarrow{\tau}\mathbb{D})\to
(\sfC'\xrightarrow{\tau'} \mathbb{D})$ and $(\psi, \Psi) : (\sfC'\xrightarrow{\tau'} \mathbb{D})\to
(\sfC''\xrightarrow{\tau''} \mathbb{D})$ their composite is defined by
pasting of the 2-commuting triangles. That is, the composite is the pair $(\psi
\circ \varphi, \Xi)$ where
\[
\Xi = (\Psi \circ \varphi) \circ _v \Phi.
\]
Here $\Psi \circ \varphi: \tau \Rightarrow \tau''$ is the whiskering
of the vertical transformation with a double functor and $\circ_v$ denotes the
vertical composition of vertical transformations (Remark~\ref{rmrk:2.29}).
\end{definition}
\begin{definition}[A category $\mathsf{HyPh}$ of hybrid phase
spaces] \label{def:hyph}
We define a category $\mathsf{HyPh}$ {\sf of hybrid phase spaces} to be
the categorified category of lists $(\Cat/\RelMan^\Box)^\Rightarrow$
(see Definition~\ref{def:3.double}) where $\Cat$ is the category of
small categories and $\RelMan^\Box$ is the double category of
manifolds with corners, smooth maps and arbitrary relations
(Definition~\ref{def:relmanbox}). Thus
\[
\mathsf{HyPh}:= (\Cat/\RelMan^\Box)^\Rightarrow.
\]
More concretely the objects of $\mathsf{HyPh}$ are double functors $a:S_a\to
\RelMan^\Box$ (where the small category $S_a$ is considered a
discrete double category). A morphism from a hybrid phase space $a$
to a hybrid phase space $b$ is a
pair $(\varphi, \Phi)$ where $\varphi: S_a\to S_b$ is a functor between
two discrete double categories (i.e., an ordinary functor) and $\Phi:
a\Rightarrow b\circ \varphi$ is a vertical transformation.
\end{definition}
\begin{remark}
Unless noted otherwise we assume that our hybrid phase spaces are
``nonempty'': given a hybrid phase space $a:S_a\to \RelMan^\Box$ we
assume that the category $S_a$ is nonempty and that for every object
$x$ of $S_a$ the manifold with corners $a(x)$ is also
nonempty. \end{remark}
\begin{remark}\label{rmrk:4.8}
We now give another description of the functor $\mathbb{U}: \mathsf{HyPh}\to \Man$.
Recall that there is a forgetful functor $U:\Cat\to \Set$. To a
small category $\sfC$ the functor $U$ assigns its set $\sfC_0$ of objects. To a
functor $\sfC \xrightarrow{f} \sfD$ between two small categories it
assigns the map $\sfC_0\xrightarrow{f_0} \sfD_0$ on objects. Note
next that if $a:S_a\to \RelMan^\Box$ is a double functor from a small
category $S_a$ then the $a_0$ component of $a$ is an unordered list of
manifolds $a_0: (S_a)_0 \to \Man$. Hence we can view
$a_0$ as an object of the category $(\Set/\Man)^\Rightarrow$.
Given a morphism $(\varphi, \Phi): a\to b$ in $(\Cat/\RelMan^\Box)$
the pair $(\varphi_0, \Phi_0)$ is a morphism in
$(\Set/\Man)^\Rightarrow$ from $a_0$ to $b_0$. It is easy to see that
this gives us the forgetful functor $(\Cat/\RelMan^\Box) \to
(\Set/\Man)^\Rightarrow $. We now compose this forgetful functor
with the canonical functor $\rotpi: (\Set/\Man)^\Rightarrow \to \Man$
and obtain the desired functor $\mathbb{U}: (\Cat/\RelMan^\Box)^\Rightarrow\to \Man$.
\end{remark}
\begin{proposition}
The category $\mathsf{HyPh}$ of hybrid phase spaces has finite products.
\end{proposition}
\begin{proof} The proof uses the fact that the categories $\Cat$ of
small categories and functors, and $\Man$ of manifolds with corners
and smooth maps have finite
products. Additionally the category $\RelMan$ of manifolds and set-theoretic
relations has a monoidal product which on objects is just the
product of manifolds with corners.
Let $[0]$ denote a category with one object and one morphism.
A terminal object in the category $\mathsf{HyPh}$ of hybrid phase spaces is
a functor $*: [0] \to \RelMan^\Box$ that assigns to the one
object 0 of
$[0]$ a one point manifold $\{\bullet\}$. It assigns to the identity
arrow $\mathsf{id}_0$ of $[0]$ the identity relation $\{(\bullet,
\bullet)\}$. We claim that for any hybrid phase space $a:S_a\to
\RelMan^\Box$ there is a unique morphism $(\varphi, \Phi):a \to *$ of
hybrid phase spaces. Since $[0]$ is terminal in $\Cat$, there is a
unique functor $\varphi:S_a\to [0]$. Since one point manifolds are
terminal in $\Man$ for every object $x$ of $S_a$ there is a unique
smooth map $\Phi_x: a(x) \to \{\bullet\}$. The maps
$\{\Phi_x\}_{x\in S_a}$ are components of a unique vertical
transformation $\Phi: a\Rightarrow *\circ \varphi$.
Recall that the category $\RelMan$ of manifolds with corners and
relations has a monoidal product: given a relation $R:M\to N$ and a
relation $S:Q\to P$ their product $R\times S \subset (M\times Q) \times (N
\times P)$ is the set
\begin{equation}\label{eq:mon-pr-rel}
R\times S := \{(m,q, n,p) \in (M\times Q) \times (N
\times P)\mid (m,n) \in R, (q,p) \in S\}
\end{equation}
Suppose now that $a:S_a\to \RelMan^\Box$, $b:S_b\to
\RelMan^\Box$ are two hybrid phase spaces. Their product $a\times
b$ is a functor from the product category $S_a\times
S_b$. We define it as follows. Given
an object $(x,y) \in S_a\times S_b$ we set
\[
(a\times b) (x,y) := a(x)\times b(y),
\]
where $a(x)\times b(y)$ is the product of two manifold with corners
(the categorical product in $\Man$).
Given an arrow $(x\xrightarrow{\gamma} x', y \xrightarrow{\nu} y')\in
S_a \times S_b$ we set
\[
(a\times b) (\gamma, \nu):= a(\gamma) \times b(\nu),
\]
where $a(\gamma) \times b(\nu)$ is the (monoidal) product of two relations given
by \eqref{eq:mon-pr-rel}. The two projections $\pi_1= (\varpi_1, p_1):a\times
b \to a$, $\pi_2: (\varpi_2, p_2):a\times
b \to b$ are constructed as follows. Since $\Cat$ has binary products
we have the canonical projection functor $\varpi_1: S_a\times S_b\to S_a$.
Since $\Man$ has binary products, for any pair of objects $(x,y)$
in $S_a\times S_b$ we have the canonical projection $(p_1)_{(x,y)}:
a(x)\times b(y)\to a(x)$. These projections assemble into a vertical
transformation $p_1: a\times b \Rightarrow a \circ \varpi_1$ between
double functors. The construction of $\pi_2 = (\varpi_2, p_2):a\times
b \to b$ is the same.
It remains to check that $a\times b$ with the two projections
satisfies the universal property of product in $\sF{HyPh}$, i.e.\ that
given any pair of morphisms of hybrid phase spaces $z_1= (\zeta_1, Z_1):c\rightarrow
a$ and $z_2= (\zeta_2, Z_2):c\rightarrow b$, there is unique morphism
$z= (\zeta, Z):c\dashrightarrow a\times b$ so that the diagram
\[
\begin{tikzcd} c\arrow[drr,"z_2"]\arrow[dr,dashed]\arrow[ddr,swap,"z_1"] & & \\
& a\times b\arrow[d,"\pi_1"] \arrow[r,swap,"\pi_2"] & b\\
& a &
\end{tikzcd}
\]
commutes. Since $\Cat$ has products, there is a unique functor
$\zeta:S_c\to S_{a\times b} = S_a \times S_b$ so that $\varpi_1 \circ
\zeta = \zeta_1$ and $\varpi_2 \circ
\zeta = \zeta_2$. For every object $x$ of $S_c$ we have two smooth
maps
\[
(Z_1)_x: c(x)\to a (\zeta_1(x))\qquad \textrm{and} \qquad (Z_2)_x: c(x)\to b (\zeta_2(x)).
\]
Since the category $\Man$ of manifolds with corners and smooth maps
has binary products there exists a unique smooth map
\[
Z_x : c(x) \to a (\zeta_1(x)) \times b (\zeta_2(x))
\]
so that
\[
(Z_1)_x = (p_1)_x \qquad \textrm{and} \qquad (Z_2)_x = (p_2)_x.
\]
The collection of maps $\{Z_x\}_{x\in S_c}$ define a vertical
transformation $Z:c\Rightarrow (a\times b) \circ \zeta$.
Finally since $\mathsf{HyPh}$ has a terminal object and binary products it has
finite products.
\end{proof}
\begin{proposition}
The forgetful functor $\mathbb{U}: \mathsf{HyPh}\to \Man$ from the category of hybrid
phase spaces to the category of manifolds with corners preserves finite products.
\end{proposition}
\begin{proof}
Clearly $\mathbb{U}$ takes the terminal object $*: \mbox{1\hspace{-4.25pt}\fontsize{12}{14.4}\selectfont\textrm{1}} \to \RelMan^\Box$ in
the category $\mathsf{HyPh}$ to a one point manifold $*$, which is terminal
in the category $\Man$.
It remains to check that $\mathbb{U}$ preserves binary products. Namely we
check that for any two hybrid phase spaces $a,b\in \mathsf{HyPh}$ the
manifolds with corners $\mathbb{U}(a)\times \mathbb{U}(b) $ and $\mathbb{U}(a\times b)$
are canonically diffeomorphic. This amounts to checking that for
any two lists $\mu:X\to \Man$ and $\nu:Y\to \Man$ in $\Set/\Man$ the
manifolds $\bigsqcup_{(x,y)\in X\times Y} \mu(x)\times \nu(y)$ and
$\left(\bigsqcup _{x\in X}\mu (x) \right) \times \left(
\bigsqcup_{y\in Y} \nu(y) \right)$ are canonically diffeomorphic.
Let $\sfC$ be a category with coproducts and binary products. Given
two objects $\mu: X\to \sfC$ and $\nu: Y\to \sfC $ of the category of
lists $\Set/\sfC$ we have a canonical map
\[
\mathscr{P}: \bigsqcup_{(x,y)\in X\times Y} \mu(x)\times \nu(y) \to
\left(\bigsqcup _{x\in X}\mu (x)
\right) \times \left( \bigsqcup_{y\in Y} \nu(y) \right).
\]
It is induced by the products of the structure maps
\[
\mu(x_0)\times \nu(y_0) \xrightarrow{
\imath_{x_0}\times \imath_{y_0}}
\left(\bigsqcup _{x\in X}\mu (x)
\right) \times \left( \bigsqcup_{y\in Y} \nu(y) \right).
\]
It is well known that in the category $\Set$ of sets the map $\mathscr{P}$
is a bijection (see \cite[Proposition~8.6]{Awodey}). In the category $\Man$ of manifolds
with corners the structure maps $\imath_{x_0} :\mu(x_0)\to \bigsqcup
_{x\in X} \mu(x)$ are open embeddings. Consequently the products $\mu(x_0)\times \nu(y_0) \xrightarrow{
\imath_{x_0}\times \imath_{y_0}}
\left(\bigsqcup _{x\in X}\mu (x)
\right) \times \left( \bigsqcup_{y\in Y} \nu(y) \right)$ are open
embeddings as well and, in particular, are local diffeomorphisms.
Consequently in the category $\Man$ the map $\mathscr{P}$ is a local diffeomorphism and a
bijection of the underlying sets, hence a diffeomorphism.
\end{proof}
\begin{lemma}
Suppose $\sfC\xrightarrow{F}\sfD$ is a functor. Then $F$ induces a
functor $F_*: (\mathsf{FinSet}/\sfC)^\Leftarrow \to
(\mathsf{FinSet}/\sfD)^\Leftarrow$ which is given by
\[
F_* \left(
\xy
(-10, 6)*+{X} ="1";
(10, 6)*+{Y} ="2";
(0,-6)*+{\sfC }="3";
{\ar@{->}_{\tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{\mu} "2";"3"};
{\ar@{=>}_{\scriptstyle \Phi} (4,2)*{};(-0.4,0)*{}} ;
\endxy
\right) =
\xy
(-10, 6)*+{X} ="1";
(10, 6)*+{Y} ="2";
(0,-8)*+{\sfD }="3";
{\ar@{->}_{F\circ \tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{F\circ \mu} "2";"3"};
{\ar@{=>}_{\scriptstyle F\circ \Phi} (4,2)*{};(-0.4,-1)*{}} ;
\endxy \quad.
\]
If $\sfC$ and $\sfD$ have finite products and $F$ is product
preserving then the diagram
\[
\xy
(-15,10 )*+{(\mathsf{FinSet}/\sfC)^\Leftarrow} ="1";
(15,10 )*+{\sfC} ="2";
(-15,-10 )*+{(\mathsf{FinSet}/\sfD)^\Leftarrow} ="3";
(15,-10 )*+{\sfD} ="4";
{\ar@{->}^{\qquad \Pi} "1";"2"};
{\ar@{->}_{F_*} "1";"3"};
{\ar@{->}^{F} "2";"4"};
{\ar@{->}_{\qquad \Pi} "3";"4"};
\endxy
\]
commutes (up to a unique isomorphism).
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{corollary}\label{cor:4.12}
The diagram
\[
\xy
(-15,10 )*+{(\mathsf{FinSet}/\mathsf{HyPh})^\Leftarrow} ="1";
(15,10 )*+{\mathsf{HyPh}} ="2";
(-15,-10 )*+{(\mathsf{FinSet}/\Man)^\Leftarrow} ="3";
(15,-10 )*+{\Man} ="4";
{\ar@{->}^{\qquad \Pi} "1";"2"};
{\ar@{->}_{\mathbb{U}_*} "1";"3"};
{\ar@{->}^{\mathbb{U}} "2";"4"};
{\ar@{->}_{\qquad \Pi} "3";"4"};
\endxy
\]
commutes (up to a unique isomorphism).
\end{corollary}
\begin{example}[Compare with Example~\ref{ex:exPiphi2}] \label{ex:exPiphi3}
Let $\sfC =\mathsf{HyPh}$, the category of hybrid phase spaces, $X = Y = \{1,2, 3\}$.
Fix two hybrid phase spaces $A$ and $B$ and a map $s:A\to B$ of
hybrid phase spaces. Let $\tau, \mu: X, Y\to \mathsf{HyPh}$ be
the constant maps defined by $\tau(j) = B$, $\mu(j) =A$ for all
$j$. We define a morphism $(\varphi,
\Phi): \tau \to \mu$ in $(\mathsf{FinSet}/\mathsf{HyPh})^\Leftarrow$ as follows.
As in Example~\ref{ex:exPiphi2} we define $\varphi$ by
\[
\varphi(1) = 2,\qquad \varphi(2) = 1, \qquad \varphi(3) =2.
\]
We define $\Phi_i: \mu (\varphi(i)) = A \to \tau(i) = B$ to be the map
$s:A \to B$ for all $i$.
Then $(\varphi,\Phi): \tau \to \mu$ is a morphism in
$(\mathsf{FinSet}/\mathsf{HyPh})^\Leftarrow$.
By Remark~\ref{rmrk:2.41} we have a map of hybrid phase spaces
$\Pi (\varphi, \Phi):\Pi(\mu) = A^3 \to B^3 = \Pi(\tau)$.
By Corollary~\ref{cor:4.12} the map $\mathbb{U}(\Pi(\varphi, \Phi)): \mathbb{U}(A^3)
= \mathbb{U}(A)^3\to \mathbb{U}(B)^3 = \mathbb{U} (B^3)$
is given by
\[
\mathbb{U}(\Pi(\varphi,\Phi))\, (a_1, a_2, a_3) = (\mathbb{U}(s)\,(a_2), \mathbb{U}(s)\,(a_1), \mathbb{U}(s)\,( a_2))
\]
for all $(a_1,a_2, a_3) \in (\mathbb{U}(A))^3$.
\end{example}
\subsection{A category $\mathsf{HyDS}$ of hybrid dynamical systems}\mbox{}\\
We define a category $\mathsf{HyDS}$ of hybrid dynamical systems as the
category of elements of the functor $\mathfrak{X}\circ \mathbb{U}: \mathsf{HyPh}\to
\mathsf{RelVect}$. Thus we formally record:
\begin{definition}[The category $\mathsf{HyDS}$ of hybrid dynamical systems]
The category $\mathsf{HyDS}$ of hybrid dynamical systems is the category of
elements of the functor $\mathfrak{X} \circ \mathbb{U} : \mathsf{HyPh} \to \mathsf{RelVect}$.
Explicitly an object of the category $\mathsf{HyDS}$, that is, a {\sf hybrid
dynamical system}, is a pair $(a, X)$ where $a$ is a hybrid phase space
and $X$ is a vector field on the underlying manifold $\mathbb{U}(a)$. A
map of hybrid dynamical systems from $(a,X)$ to $(b,Y)$ is a map
$(\varphi, \Phi):a\to b$ of hybrid phase spaces such that the vector
field $Y$ is $\mathbb{U}(\varphi,\Phi)$-related to $X$.
\end{definition}
To define executions of a hybrid dynamical system $(a,X)$ we need to
define the hybrid analogue of the continuous time dynamical system
$(I, \frac{d}{dt})$ where $I$ is an interval and $\frac{d}{dt}$ is the constant vector
field.
\begin{definition}[The hybrid dynamical system $(\mathscr{I}, \partial) = (\mathscr{I}(\{t_i\}), \partial) $
associated with increasing sequence $\{t_i\}_{i=0}^\infty$ of real
numbers]
\label{def:4.15}
Let $\{t_i\}$ be an increasing sequence of real numbers. Let $\mathscr{T}$
be the graph with the set of vertices $\mathscr{T}_0 = \mathbb{N}$, the set of
natural numbers (that includes 0), the set of arrows $\mathscr{T}_1 = \mathbb{N}$
and the source and target maps $\mathfrak s \times \mathfrak t : \mathscr{T}_1\to \mathscr{T}_0
\times \mathscr{T}_0$ given by $i\mapsto (i, i+1)$. In other words arrows
of the graph $\mathscr{T}$ are of the form $i\xrightarrow{i} i+1$. Define a map of
graphs $\bar{\tau}: \mathscr{T}\to U(\RelMan)$ (recall that $U(\RelMan)$ is the
graph underlying the category of manifolds with corners and relations)
by
\[
\bar{\tau} (i\xrightarrow{i}i+1) = [t_i,t_{i+1}]\xrightarrow{ \{(t_{i+1},
t_{i+1})\}} [t_{i+1}, t_{i+2}].
\]
Here $\{(t_{i+1},t_{i+1})\} \subset [t_i, t_{i+1}]\times [t_{i+1},
t_{i+2}]$ is a relation consisting of one point. We define the hybrid
phase space $\mathscr{I}$ to be the corresponding functor $\tau:
\mathsf{Free}(\mathscr{T})\to \RelMan$. We define the vector field $\partial$ on
the underlying manifold $\mathbb{U}(\tau) = \bigsqcup_{i=0}^\infty [t_i,
t_{i+1}]$ by $\partial|_{[t_i,t_{i+1}]} := \frac{d}{dt}$.
\end{definition}
\begin{definition}\label{def:execution} \label{def:exec}
An {\sf execution} of a hybrid dynamical system $(a, X)$ is an increasing sequence $\{t_i\}_{i=0}^\infty$ of real
numbers and a map of
hybrid dynamical systems $(\varphi, \sigma): (\mathscr{I}(\{t_i\}), \partial) \to
(a,X)$. Here $(\mathscr{I}(\{t_i\}), \partial) $ is the hybrid dynamical
system associated to the sequence $\{t_i\}_{i=0}^\infty$ (Definition~\ref{def:4.15}).
\end{definition}
\begin{remark}\label{rmrk:exec}
Unpacking Definition~\ref{def:execution} we see that an execution
$(\varphi, \sigma)$ of a hybrid dynamical system $(a, X)$ associates
to each integer $i\geq 0$ an integral curve
$\sigma_i:[t_i, t_{i+1}]\to a(\varphi_0(i))$ of the vector field
$X_{\varphi_0(i)}$ on the manifold with corners $ a(\varphi_0(i))$
so that $(\sigma_i(t_{i+1}), \sigma_{i+1} (t_{i+1}))$ lies in the
reset relations
$a(\varphi_1(i\xrightarrow{i}i+1) \subset a(\varphi_0(i))\times
a(\varphi_0(i+1))$. Thus Definition~\ref{def:exec} of executions as
maps of hybrid dynamical systems agrees with a traditional definition of
an execution, Definition~\ref{def:chs_execution}.
\end{remark}
\begin{theorem} \label{thm:exec}
A map $(\psi, \Psi):(a, X)\to (b, Y)$ of hybrid dynamical system
sends the executions of $(a,X)$ to the executions of $(b, Y)$.
\end{theorem}
\begin{proof}
Let $(\varphi, \sigma): (\mathscr{I}, \partial) \to (a,X)$ be an execution
of the hybrid dynamical system $(a,X)$. Then, by definition,
$(\varphi, \sigma)$ is a
map of hybrid dynamical systems. Hence $(\psi, \Psi)\circ (\varphi,
\sigma): (\mathscr{I}, \partial) \to (b, Y)$ is also a map of hybrid
dynamical systems, and therefore an execution.
\end{proof}
\section{The double category $\mathsf{HySSub}^\Box$ of hybrid surjective
submersions and hybrid open systems}
We start with some category-theoretic generalities that will help us
to define the category of hybrid surjective submersions $\mathsf{HySSub}$ and
a forgetful product-preserving functor $\mathbb{U}:\mathsf{HySSub}\to \mathsf{SSub}$ from the
category of hybrid surjective submersions to the
category of surjective submersions.
Suppose $\mathscr{C}$ is
a category with finite products. Denote by $[1]}%{\mathbbm{2}$ the category with
two objects $0, 1$ and one nonidentity morphism $0\to 1$. The
functor category $\mathscr{C}^{[1]}%{\mathbbm{2}}$ is the category of arrows of the
category $\mathscr{C}$. The objects of $\mathscr{C}^{[1]}%{\mathbbm{2}}$ can be identified with
morphism $c\xrightarrow{f}c'$ of $\mathscr{C}$. Under this identification a
morphism in $\mathscr{C}^{[1]}%{\mathbbm{2}}$ from $c\xrightarrow{f} c'$ to
$d\xrightarrow{g}d'$ is the commutative diagram
\[
\xy
(-8, 6)*+{c'} ="1";
(8, 6)*+{c} ="2";
(-8,-6)*+{d'}="3";
(8, -6)*+{d}="4";
{\ar@{->}_{f} "2";"1"};
{\ar@{->}^{g} "4";"3"};
{\ar@{->}_{h} "1";"3"};
{\ar@{->}^{k} "2";"4"};
\endxy .
\]
If $\mathscr{C}$ has finite products, then so does the arrow category
$\mathscr{C}^{[1]}%{\mathbbm{2}}$: the terminal object of $\mathscr{C}^{[1]}%{\mathbbm{2}}$ is the unique arrow
$*\to *$ where $*$ is the terminal object of $\mathscr{C}$. The binary
product of $c\xrightarrow{f}c'$ and $d\xrightarrow{g}d'$ is
$c\times d \xrightarrow{f\times g} c'\times d'$ and so on.
If $\mathscr{D}$ is another category with finite products and $\mathbb{U}:\mathscr{C}\to
\mathscr{D}$ is a product -preserving functor, then the induced functor
\[
\mathscr{C}^{[1]}%{\mathbbm{2}} \to \mathscr{C}^{[1]}%{\mathbbm{2}}, \qquad (c\xrightarrow{f} c') \mapsto
(\mathbb{U}(c)\xrightarrow{\mathbb{U}(f)} \mathbb{U}(c') )
\]
is also product preserving. We denote this induced functor again by
$\mathbb{U}$ and trust that it will not cause any confusion.
\begin{definition}[The category $\mathsf{HySSub}$ of hybrid submersion] \label{def:hybssub}
Observe that the category $\mathsf{SSub}$ of surjective submersions in a full subcategory
of the category of arrows $\Man^{[1]}%{\mathbbm{2}}$.
We define the category
$\mathsf{HySSub}$ of {\sf hybrid surjective submersions} to be the preimage
$\mathbb{U}^{-1} (\mathsf{SSub})$ in the category of arrows $\mathsf{HyPh}^{[1]}%{\mathbbm{2}}$. Here
$\mathbb{U}:\mathsf{HyPh}^{[1]}%{\mathbbm{2}}\to \Man^{[1]}%{\mathbbm{2}}$ is the forgetful functor induced by
$\mathbb{U}:\mathsf{HyPh}\to \Man$.
More concretely the objects of $\mathsf{HySSub}$ are {\sf hybrid surjective
submersions}. These are maps of hybrid phase
spaces $a_\mathsf{tot} \xrightarrow{p_a} a_\mathsf{st}$ such that the underlying maps
of manifolds $\mathbb{U}(p_a): \mathbb{U} (a_\mathsf{tot})\to \mathbb{U}(a_\mathsf{st})$ are surjective
submersions. A morphism in $\mathsf{HySSub}$ from $a_\mathsf{tot} \xrightarrow{p_a}
a_\mathsf{st}$ to $b_\mathsf{tot}\xrightarrow{p_b} b_\mathsf{st}$ is a pair of maps of hybrid
phase spaces $f_\mathsf{tot}:a_\mathsf{tot} \to b_\mathsf{tot}$, $f_\mathsf{st}:a_\mathsf{st}\to b_\mathsf{st}$ making
the appropriate diagram commute:
\[
f_\mathsf{st} \circ p_a = p_b \circ f_\mathsf{tot} .
\]
\end{definition}
\begin{remark} \label{rmrk:4.2} Since $\mathbb{U}: \mathsf{HyPh}\to \Man$ is product
preserving, so is the induced forgetful functor
$\mathbb{U}: \mathsf{HyPh}^{[1]}%{\mathbbm{2}} \to \Man^{[1]}%{\mathbbm{2}}$ on arrow categories. By
definition $\mathbb{U}$ takes the category $\mathsf{HySSub} \subset \mathsf{HyPh}^{[1]}%{\mathbbm{2}}$ to
the category $\mathsf{SSub}$ of surjective submersions. Consequently the
functor
\begin{equation}\label{eq:UU}
\mathbb{U}:\mathsf{HySSub} \to \mathsf{SSub},
\end{equation}
which is a restriction of the forgetful functor $\mathbb{U}:\mathsf{HyPh}^{[1]} \to \Man^{[1]}%{\mathbbm{2}}$,
preserves finite products.
\end{remark}
\begin{remark} \label{rmrk:5.4}
Given two manifold $M, N$ the projections $\pi_1:M\times N\to M$ and
$\pi_2:M\times N\to N$ are surjective submersions. Since the
forgetful functor
$\mathbb{U}:\mathsf{HyPh}\to \Man$ is finite product preserving, for any
two hybrid phase spaces $a$ and $b$ the images $\mathbb{U}(\pi_1) :
\mathbb{U}(a)\times \mathbb{U}(a) \to \mathbb{U}(b)$ and $\mathbb{U}(\pi_2): \mathbb{U}(a)\times
\mathbb{U}(b)\to \mathbb{U}(b)$ of the canonical
projections $\pi_1:a\times b \to a$ and $\pi_2:a\times b\to b$ are the
canonical projections of the product of manifolds, hence surjective
submersions. We conclude that the canonical projections
$\pi_1:a\times b \to a$ and $\pi_2:a\times b\to b$ from the
categorical product in $\mathsf{HyPh}$ to its factors are
hybrid surjective submersions in the sense of Definition~\ref{def:hybssub}.
\end{remark}
Analogously to the category $\mathsf{OS}$ of open systems
(Example~\ref{ex:3.4}) we have the category $\mathsf{HyOS}$ of hybrid open
systems. It is defined as follows.
\begin{definition}[The category $\mathsf{HyOS}$ of hybrid open
systems] \label{def:HyOS} We define the category $\mathsf{HyOS}$ of {\sf hybrid open
systems} to be the category of elements
(Definition~\ref{def:cat_of_elements}) of the functor
$\mathsf{Crl} \circ \mathbb{U}: \mathsf{HySSub} \to \mathsf{RelVect}$. Here as before
$\mathsf{Crl}: \mathsf{SSub} \to \mathsf{RelVect}$ is the functor that assigns to each
surjective submersion the vector space of open systems on the
submersions (see Example~\ref{ex:3.4}) and $\mathbb{U}: \mathsf{HySSub} \to \mathsf{SSub}$
is the forgetful functor from the category of hybrid surjective
submersions to surjective submersions (Remark~\ref{rmrk:4.2}).
More explicitly a {\sf hybrid open system} is a pair $(a, F)$ where
$a = (a_\mathsf{tot}\xrightarrow{p_a} a_\mathsf{st})$ is a hybrid surjective
submersion and $F:\mathbb{U}(a_\mathsf{tot})\to T\mathbb{U}(a_\mathsf{st})$ is an open system on
the underlying surjective submersion $\mathbb{U}(a) = (\mathbb{U}(a_\mathsf{tot})\xrightarrow{\mathbb{U}(p_a)} \mathbb{U}(a_\mathsf{st}))$.
\end{definition}
Recall that the category $\mathsf{SSub}$ of surjective submersions has a
special class of morphisms: the interconnection maps
(Definition~\ref{def:ssub}). Recall also that together the two types
of morphisms --- interconnections and ``ordinary'' maps of
submersions --- define the double category $\mathsf{SSub} ^\Box$ of
surjective submersions (Definition~\ref{def:ssub_box}). The
forgetful functor $\mathbb{U}:\mathsf{HySSub}\to \mathsf{SSub}$ allows us to transfer this
double category structure to hybrid submersions.
\begin{definition}\label{def:hy_int}
A map $f= (f_\mathsf{tot}, f_\mathsf{st}):a\to b$ of hybrid surjective submersions is
an {\sf interconnection morphism} if the underlying map $\mathbb{U}(f_\mathsf{st}):
\mathbb{U}(a_\mathsf{st})\to \mathbb{U}(b_\mathsf{st})$ is an interconnection map of surjective submersions.
\end{definition}
\begin{example}
Let $a$, $b$ be two hybrid phase spaces and $s:a\to b$ a map of hybrid
phase spaces. By Remark~\ref{rmrk:5.4} the canonical projection
$\pi_1:a\times b \to a$ is a hybrid surjective submersion. The
identity map $\mathsf{id}:a\to a$ is also a hybrid surjective submersion.
The diagram
\[
\xy
(-8, 6)*+{a} ="1";
(8, 6)*+{a\times b} ="2";
(-8,-6)*+{a}="3";
(8, -6)*+{a}="4";
{\ar@{->}^{(\mathsf{id},s)} "1";"2"};
{\ar@{->}^{\pi_1} "2";"4"};
{\ar@{->}_{\mathsf{id}} "1";"3"};
{\ar@{->}^{\mathsf{id}} "3";"4"};
\endxy
\]
commutes. Hence $\varphi:= ((\mathsf{id}, s), \mathsf{id})): (a\xrightarrow{\mathsf{id}}a) \to (a\times
b\xrightarrow{\pi_1} b))$ is an interconnection map of hybrid
surjective submersions.
It follows that for any hybrid open system of the form $(a\times b\xrightarrow{\pi_1} a, F \in \mathsf{Crl}
\mathbb{U}(a\times b\xrightarrow{\pi_1} a))$ the pair $(a, \mathbb{U}(\varphi)^* F)$
is a hybrid dynamical system (since $\mathbb{U}(\varphi)^*F$ is a vector field).
\end{example}
\begin{remark}
Suppose $\varphi: a \to b$ is an interconnection morphism between two
hybrid surjective submersions and $F\in \mathsf{Crl} (\mathbb{U}(b))$ is a
(continuous time) open system. Then $(b, F)$ is a hybrid open system
and so is $(a, \mathbb{U}(\varphi)^*F)$.
\end{remark}
\begin{example} \label{ex:hds_on_product} \label{ex:5.9}
In this example we generalize Example~\ref{rmrk:vf_on_product} from
manifolds to hybrid phase spaces.
We argue that any hybrid dynamical system of the form $(a\times b, X\in
\mathfrak{X}(\mathbb{U}(a\times b)))$ ($a,b$ are hybrid phase spaces) is the result
of interconnection of two hybrid open systems.
As in
Example~\ref{rmrk:vf_on_product} the vector field $X = (X_1, X_2) $
where $X_1 :\mathbb{U}(a\times b)\to T\mathbb{U} (a)$, $X_2:\mathbb{U}(a\times b)\to
T\mathbb{U}(b)$ are open systems. Here we used the fact that $T\mathbb{U}(a\times
b) = T(\mathbb{U}(a) \times \mathbb{U}(b))$ since the forgetful functor $\mathbb{U}$
preserves products. Again, since $\mathbb{U}$ preserves products and
since the canonical projections $\mathsf{pr}_1:\mathbb{U}(a)\times \mathbb{U}(b)\to \mathbb{U}(a)$,
$\mathsf{pr}_2:\mathbb{U}(a)\times \mathbb{U}(b) \to \mathbb{U}(b)$ are surjective submersions, the
structure
maps $\pi_1:a\times b\to a$, $\pi_2:a\times b\to b$ are hybrid
surjective submersions. Denote by $\Delta$ the diagonal map
\[
a\times b \to (a\times b)\times (a\times b).
\]
The diagram of hybrid phases spaces and their maps
\[
\xy
(-15, 6)*+{a\times b} ="1";
(15, 6)*+{(a\times b)\times (a\times b)} ="2";
(-15,-6)*+{a\times b}="3";
(15, -6)*+{a\times b}="4";
{\ar@{->}^{\Delta\qquad } "1";"2"};
{\ar@{->}_{\mathsf{id} } "3";"4"};
{\ar@{->}_{\mathsf{id}} "1";"3"};
{\ar@{->}^{\pi_1\times \pi_2} "2";"4"};
\endxy
\]
commutes. Hence
\[
\psi := (\Delta, \mathsf{id}) :(a\times b\xrightarrow{\mathsf{id}}a\times
b)\to ((a\times b \xrightarrow{\pi_1} a)\times (a\times
b)\xrightarrow{\pi_2} b)
\]
is an interconnection map between two hybrid surjective submersions.
Since $\mathbb{U}$ is a product preserving functor, $\mathbb{U}(\Delta):\mathbb{U}(a\times
b) \to \mathbb{U}((a\times b)\times (a\times b)) = \mathbb{U}(a\times b) \times
\mathbb{U}(a\times b)$ is also the diagonal map. We have seen in
Example~\ref{rmrk:vf_on_product} that $X = \varphi^* (X_1\times X_2)$
where $\varphi = \mathbb{U}(\psi)$. Therefore the hybrid dynamical system
$(a\times b, X)$ is obtained by interconnecting two hybrid open
systems $(a\times b \to a, X_1)$ and $(a\times b\to b, X_2)$.
\end{example}
\begin{definition}[The double category $\mathsf{HySSub}^\Box$ of
hybrid surjective submersions]\label{def:ssub_box}
The objects of the double category $\mathsf{HySSub}^\Box$ are hybrid surjective
submersions. The horizontal 1-morphisms are morphisms of surjective
submersions.The vertical
1-morphisms are interconnection morphisms (Definition~\ref{def:hy_int}).
The 2-cells of $\mathsf{SSub}^\Box$ are commuting squares
\[
\xy
(-10,10)*+{c}="1";
(10, 10)*+{a}="2";
(-10,-5)*+{d}="3";
(10, -5)*+{b}="4";
{\ar@{<-}^{\mu} "1";"2"};
{\ar@{->}_{g} "1";"3"};
{\ar@{->}^{f} "2";"4"};
{\ar@{<-}_{\nu} "3";"4"};
\endxy
\]
in the category $\mathsf{HySSub}$ of hybrid surjective submersions, where
$\mu,\nu$ are maps of hybrid submersions and $f$, $g$ are the
interconnection morphisms. In particular the category of objects of
the double category $\mathsf{HySSub}^\Box$ is the category $\mathsf{HySSub}^\mathsf{int}$ of
hybrid surjective submersions and interconnection morphisms.
\end{definition}
\begin{remark}
It is easy to check that we have a forgetful functor from the double
category $\mathsf{HySSub}^\Box$ to the double category $\mathsf{SSub}^\Box$. Again we
denote it by $\mathbb{U}$. Thus $\mathbb{U}:\mathsf{HySSub}^\Box \to \mathsf{SSub}^\Box$.
\end{remark}
\section{Networks of hybrid open systems}
We start this section by reviewing the results of \cite{L2} on
networks of open systems. We then generalize the results of \cite{L2}
to networks of {\em hybrid} open systems, which is the main point of
this paper.
Recall the definition of a network of open systems from \cite{L2}:
\begin{definition} \label{def:network_os}
A {\sf network of open systems} is a pair $(X\xrightarrow{\tau}
\mathsf{SSub}, b\xrightarrow{\psi} \Pi(\tau) )$ where $\tau$ is an element
of $(\mathsf{FinSet}/\mathsf{SSub})^\Leftarrow$ (Definition~\ref{def:cat_of_lists}),
that is, a list of surjective submersions indexed by the finite set
$X$,
$\Pi: ((\mathsf{FinSet}/\mathsf{SSub})^\Leftarrow)^\mathsf{op} \to \mathsf{SSub}$ is the product
functor (Remark~\ref{rmrk:2.41}) so that $\Pi(\tau) = \prod _{x\in
X} \tau(x)$,
and $\psi$ is an interconnection morphism (Definition~\ref{def:ssub}).
\end{definition}
\begin{example} \label{ex:6.2}
Let $M_1, M_2$ be two manifolds with corners, $M_1\times M_2 \xrightarrow{\pi_1}
M_1$, $M_1\times M_2 \xrightarrow{\pi_2}
M_2$ the two associated submersions and $\varphi: (M_1\times
M_2\xrightarrow{\mathsf{id}}M_1\times M_2)\to (M_1\times M_2 \xrightarrow{\pi_1}
M_1)\times(M_1\times M_2 \xrightarrow{\pi_2} M_2)$ be the
interconnection map as in Example~\ref{rmrk:vf_on_product}. This
data is
a network of open systems. Namely,
the indexing set $X$ is the two element set $\{1,2\}$. The map
$\tau:X\to \mathsf{SSub}$ is given by
\[
\tau(i) = (M_1\times M_2\xrightarrow{\pi_i}M_i),\qquad i=1,2.
\]
The submersion $b$ is $ (M_1\times M_2 \xrightarrow{\mathsf{id}} M_1\times
M_2)$ and $\psi$ is the interconnection map $\varphi$ above.
\end{example}
\begin{example}\label{ex:6.3}
Fix two manifolds with corners $M$ and $U$. As we have seen in
Example~\ref{ex:intercon} a map $s:M\to U$ defines an interconnection
map $\nu: (M\xrightarrow{\mathsf{id}}M)\to (U\times M\to M)$ with
$\nu_\mathsf{tot} (m) = (s(m), m)$. Consider $\mu:\{*\} \to \mathsf{SSub}$
given by $\mu(*) = (U\times M\to M)$. Then $(\mu, \nu:
(M\xrightarrow{\mathsf{id}}M)\to (U\times M\to M) = \Pi(\mu))$ is a
network of open systems.
\end{example}
\begin{example}\label{ex:6.4}
Let $X=\{1,2,3\}$ be a set with three elements, $M,U$ two manifolds
with corners and $s:M\to U$ a smooth map. Consider a list $\tau:X\to
\mathsf{SSub}$ of surjective submersions defined by
\[
\tau(i) = (M\times U\to M)
\]
for $i=1,2,3$. Let $b = (M\xrightarrow{\mathsf{id}}M)^3$. We define a smooth map $\psi_\mathsf{tot} :M^3 \to
(M\times U)^3 = \Pi(\tau)_\mathsf{tot}$ by
\[
\psi_\mathsf{tot} (m_1, m_2, m_3) = ((m_1, s(m_2)), (m_2, s(m_1)), (m_3,
s(m_2))).
\]
Then
$\psi = (\psi_\mathsf{tot}, \mathsf{id}_{M^3}): (M\xrightarrow{\mathsf{id}}M)^3 \to \Pi
(\tau))$ is an interconnection map. Hence the pair
$(\tau:X\to \mathsf{SSub}, \psi: b=(M\xrightarrow{\mathsf{id}}M)^3 \to \Pi(\tau))$ is
a network.
Note that since $\psi$ is an interconnection map, it induces a
linear map
\[
\psi^*:\mathsf{Crl} (\Pi(\tau)) \to \mathsf{Crl}
((M\xrightarrow{\mathsf{id}}M)^3) = \mathfrak{X} (M^3)
\]
(see Definition~\ref{def:2.35}). Consequently any three open systems
$w_1,w_2, w_3\in \mathsf{Crl}(M\times U\to M)$ give rise to an open system
$w_1\times w_2 \times w_3: (M\times U)^3 \to TM^3$. Applying the
map $\psi^*$ we get $v= \psi^*
(w_1\times w_2 \times w_3) $, which is an element of $\mathfrak{X} (M^3)$.
That is, $v$ a vector field on $M^3$. It is not hard to see that
\[
v(m_1,m_2, m_3) = %
(w_1(m_1, s(m_2)), w_2(m_2, s(m_1)), w_3(m_3, s(m_2)))
\]
for all $(m_1, m_2, m_3)\in M^3$. Intuitively the dynamical system
$(M^3, v)$ is made up of three interacting (open) subsystems with the
first driving the second, the second driving the first and the third.
It is in this sense that the network of open systems as defined above
is a pattern of interconnections of open systems.
We can picture the relation between the subsystems as a
graph:
\begin{center}
\tikzset{every loop/.style={min distance=15mm,in=-30,out=30,looseness=10}}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries}]
\node[main node] at (2,-1) (1) {1};
\node[main node] at (4,-1) (2) {2};
\node[main node] at (6,-1) (3) {3};
\path[]
(1) edge [bend left] node {{}} (2)
(2) edge [bend left] node [below] {{}} (1)
(2) edge node {{}}(3) ;
\end{tikzpicture}.
\end{center}
The arrow from node 1 to node 2 indicates that the first subsystem drives
the second. The two arrows coming out of node 2 indicate that the
second subsystem drives the first and the third. Thus our notion of a
network of open systems refines the common idea of a network as a
directed graph.
\end{example}
\begin{definition}[Maps between networks of open systems]
\label{def:map_of_networks}
A {\sf map} from a network $(X\xrightarrow{\tau} \mathsf{SSub}, b\xrightarrow{\psi}\Pi
(\tau) )$ of open systems to a network
$(Y\xrightarrow{\mu}\mathsf{SSub}, c\xrightarrow{\nu}\Pi (\mu)$
is a pair $((\varphi, \Phi), f)$ where $(\varphi, \Phi):\tau \to \mu$
is a morphism in $(\mathsf{FinSet}/\mathsf{SSub})^\Leftarrow$ and $f:c\to b$ is a
morphism in $\mathsf{SSub}$ so that the diagram
\[
\xy
(-12, 10)*+{\Pi(\mu)} ="1";
(12, 10)*+{\Pi(\tau)} ="2";
(-12,-10)*+{c }="3";
(12,-10)*+{b }="4";
{\ar@{->}^{\nu} "3";"1"};
{\ar@{->}^{\Pi (\varphi, \Phi)} "1";"2"};
{\ar@{->}_{\psi} "4";"2"};
{\ar@{->}_{f} "3";"4"};
\endxy
\]
commutes in the category $\mathsf{SSub}$ of surjective submersions. That is, the diagram is a
2-cell in the double category $\mathsf{SSub}^\Box$ with the source $f$ and target
$\Pi (\varphi, \Phi)$. Here as before
$\Pi: (\mathsf{FinSet}/\mathsf{SSub})^\Leftarrow \to \mathsf{SSub}$ is the product functor
(Remark~\ref{rmrk:2.41}).
\end{definition}
\begin{example} \label{ex:6.6}
Let $(\tau: \{1,2,3\}\to \mathsf{SSub}, \psi: (M\to M)^3\to \Pi(\tau))$ be the
network of Example~\ref{ex:6.4} and $(\mu:\{*\}\to \mathsf{SSub}, \nu:(M\to
M)\to \Pi(\mu))$ the network of Example~\ref{ex:6.3}. Define
$f:(M\to M) \to (M\to M)^3$ to be the diagonal map. Define the map of
lists $(\varphi, \Phi): \tau \to \mu$ in $(\mathsf{FinSet}/\mathsf{SSub})^\Leftarrow$
by setting $\varphi(i) = *$ for all $i$ and defining $\Phi_i: \mu
(\varphi(i)) \to \tau(i)$ to be the identity map for all $i\in
\{1,2,3\}$. It is not hard to check that $((\varphi,\Phi), f):
(\tau,\psi) \to (\mu, \nu)$ is a map of networks of open systems. It can be pictured as a map of graphs:
\begin{center}
\tikzset{every loop/.style={min distance=15mm,in=-30,out=30,looseness=10}}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries}]
\node[main node] at (2,-1) (1) {1};
\node[main node] at (4,-1) (2) {2};
\node[main node] at (6,-1) (3) {3};
\node at (1,-1) (rb){};
\node at (-1,-1) (lb){};
\node[main node] at (-3,-1) (circ){*};
\path[]
(1) edge [bend left] node {} (2)
(2) edge [bend left] node [below] {} (1)
(2) edge node {}(3)
(circ) edge [loop right=20] node {} (circ)
(rb) edge node [above] {$\varphi$} (lb) ;
\end{tikzpicture}.
\end{center}
Note that given an open system $w\in \mathsf{Crl} (M\times U\to M)$, $u=
\nu^*w$ is a vector field on $M$ and $v= \psi^* (w\times w \times w)$
is a vector field on $M^3$. It is easy to check directly that the
vector field $u$ is $f$-related to the vector field $v$. In other
words for any choice of an open system $w\in \mathsf{Crl}(M\times U\to M)$ we get a map of
continuous time
dynamical systems
\[
f: (M, \nu^*w) \to (M^3, \psi^* (w\times w\times w)).
\]
Consequently for any choice of an open system $w\in \mathsf{Crl}(M\times U\to
M)$ the vector field $v= \psi^* (w\times w \times w)$ on $M^3$ has
the diagonal $f(M)\subset M^3$ as an invariant subsystem.
The fact that $f$ is a map of dynamical systems is
special case of Theorem~\ref{thm:main_inL2} below.
\end{example}
Our definitions of networks of hybrid systems and maps of networks of
hybrid systems are analogous to the two definitions above.
\begin{definition} \label{def:network_hy_os}
A {\sf network of hybrid open systems} is a pair $(X\xrightarrow{\tau}
\mathsf{HySSub}, b\xrightarrow{\psi} \Pi(\tau) )$ where $\tau$ is an element
of $(\mathsf{FinSet}/\mathsf{HySSub})^\Leftarrow$
(Definition~\ref{def:cat_of_lists}), i.e., a list of hybrid
surjective submersions indexed by the finite set $X$, $\Pi:
(\mathsf{FinSet}/\mathsf{HySSub})^\Leftarrow \to \mathsf{HySSub}$ is the product functor
(Remark~\ref{rmrk:2.41}), so that $\Pi (\tau) = \prod_{x\in X} \tau(x)$,
and $\psi$ is an interconnection morphism (Definition~\ref{def:hy_int}).
\end{definition}
\begin{definition}[Maps between networks of hybrid open systems]
\label{def:map_of_networks_hy_os}
A {\sf map} from a network $(X\xrightarrow{\tau} \mathsf{HySSub}, b\xrightarrow{\psi}\Pi
(\tau))$ of hybrid open systems to a network
$(Y\xrightarrow{\mu}\mathsf{HySSub}, c\xrightarrow{\nu}\Pi (\mu))$
is a pair $((\varphi, \Phi), f)$ where $(\varphi, \Phi):\tau \to \mu$
is a morphism in $(\mathsf{FinSet}/\mathsf{HySSub})^\Leftarrow$ and $f:c\to b$ is a
morphism in $\mathsf{HySSub}$ so that the diagram
\[
\xy
(-12, 10)*+{\Pi(\mu)} ="1";
(12, 10)*+{\Pi(\tau)} ="2";
(-12,-10)*+{c }="3";
(12,-10)*+{b }="4";
{\ar@{->}^{\nu} "3";"1"};
{\ar@{->}^{\Pi (\varphi, \Phi)} "1";"2"};
{\ar@{->}_{\psi} "4";"2"};
{\ar@{->}_{f} "3";"4"};
\endxy
\]
it commutes
in $\mathsf{HySSub}$ of hybrid surjective submersions. That is, the diagram is a
is a 2-cell in the double category $\mathsf{HySSub}^\Box$ with the source $f$
and target $\Pi (\varphi, \Phi)$. Here as before $\Pi:
(\mathsf{FinSet}/\mathsf{HySSub})^\Leftarrow \to \mathsf{HySSub}$ is the product functor (Remark~\ref{rmrk:2.41}).
\end{definition}
\begin{example}\label{ex:6.9}\label{ex:6.2h}
Let $a_1,a_2$ be two hybrid phase spaces, $a_1\times a_2\xrightarrow{\pi_1}
a_i$, $i=1,2$ two associated hybrid surjective submersions and $\varphi: (a_1\times
a_2\xrightarrow{\mathsf{id}}a_1\times a_2)\to (a_1\times a_2\xrightarrow{\pi_1}
a_1)\times(a_1\times a_2 \xrightarrow{\pi_2} a_2)$ be the
interconnection map as in Example~\ref{ex:hds_on_product}. This
data is
a network of hybrid open systems in the following sense.
The indexing set $X$ is the two element set $\{1,2\}$. The map
$\tau:X\to \mathsf{HySSub}$ is given by
\[
\tau(i) = (a_1\times a_2\xrightarrow{\pi_i}i),\qquad i=1,2.
\]
The hybrid submersion $b$ is $ (a_1\times a_2 \xrightarrow{\mathsf{id}} a_1\times
a_2)$ and $\psi$ is the interconnection map $\varphi = (\varphi_\mathsf{tot},
\mathsf{id})$ is defined by setting $\varphi_\mathsf{tot}:(a_1\times a_2)\to
(a_1\times a_2)^2$ to be the diagonal map (recall that the diagonal
maps make sense in any category with finite products).
\end{example}
\begin{example} \label{ex:6.3h}\label{ex:6.10} This example is a
hybrid analogue of Example~\ref{ex:6.3}. Let $m, u$ be two hybrid
phase spaces and $s:m \to u$ a map of hybrid phase spaces. Define a
map $\nu: (m\xrightarrow{\mathsf{id}} m)\to (m\times u
\xrightarrow{\pi_1}m)$ of hybrid surjective submersions by setting
$\nu_\mathsf{tot} = (\mathsf{id}_{m}, s)$ and $\nu_\mathsf{st} =\mathsf{id}_{m}$. Then
$\nu$ is an interconnection map. Define $\mu:\{*\}\to \mathsf{HySSub}$ by
setting $\mu(*) = (m\times u\to m)$. Then $(\mu, \nu:
(m\xrightarrow{\mathsf{id}} m)\to \Pi (\mu))$ is a network of hybrid
open systems.
\end{example}
\begin{example} \label{ex:6.4h}\label{ex:6.11} This example is a
hybrid analogue of Example~\ref{ex:6.4}.
Let $X=\{1,2,3\}$ be a set with three elements, $m,u$ two hybrid
phase spaces and $s:m\to u$ a map of hybrid phase spaces. Consider a list $\tau:X\to
\mathsf{HySSub}$ of hybrid surjective submersions defined by
\[
\tau(i) = (m\times u\to m)
\]
for $i=1,2,3$. Then $\Pi(\tau) =(m\times u\to m)^3 = (m^3 \times u^3
\to m^3)$. Let $b = (m\xrightarrow{\mathsf{id}}m)^3$. We define an
interconnection map $\psi: b\to \Pi(\tau)$ by making use of the
functor $\Pi:((\mathsf{FinSet}/\mathsf{HyPh})^\Leftarrow)^\mathsf{op} \to \mathsf{HyPh}$ (see
Remark~\ref{rmrk:2.41} and Example~\ref{ex:exPiphi2}).
Namely consider the map $\varphi:\{1,2,3\}\to \{1,2,3\}$ which is defined by
\[
\varphi(1) = 2,\qquad \varphi(2) = 1, \qquad \varphi(3) =2.
\]
Define $\alpha, \beta:\{1,2,3\}\to \mathsf{HyPh}$ by setting
\[
\alpha (i) = u, \qquad \beta (j) = m\qquad \textrm{for all }i,j\in \{1,2,3\}.
\]
Define $(\varphi, \Phi): \alpha \to \beta$ in
$(\mathsf{FinSet}/\mathsf{HyPh})^\Leftarrow$ by setting $\Phi_i: \beta (\varphi(i))\to
\alpha(i)$ be the map $s:m\to u$ for all $i$. Now define
$\psi_\mathsf{tot} : m^3 = b_\mathsf{tot} \to \Pi(\tau)_\mathsf{tot} = m^3 \times u^3$ by
setting
\[
\psi_\mathsf{tot} = (\mathsf{id}_{m^3}, \Pi (\varphi, \Phi)).
\]
Then $\psi$ is an interconnection map of hybrid surjective
submersions and $(\tau:\{1,2,3\}\to \mathsf{HySSub}, \psi:
b=(m\xrightarrow{\mathsf{id}}m)^3 \to \Pi(\tau))$ is a network of hybrid open systems.
Note that the map $\mathbb{U}(\psi_\mathsf{tot}): \mathbb{U}(m)^3 \to \mathbb{U}(m)^3 \times \mathbb{U}
(u)^3$ of manifolds with corners is given by
\[
(x_1, x_2, x_3) \mapsto (x_1, x_2, x_3, s(x_2), s(x_1), s(x_3)).
\]
This follows from Corollary~\ref{cor:4.12} and Example~\ref{ex:exPiphi2}.
\end{example}
\begin{example} \label{ex:6.6h} \label{ex:6.12} This example is the
hybrid analogue of Example~\ref{ex:6.6}.
Let $(\mu, \nu:
(m\xrightarrow{\mathsf{id}} m)\to \Pi (\mu))$ be the network of hybrid open
systems of Example~\ref{ex:6.3h}. Let $(\tau:\{1,2,3\}\to \mathsf{HySSub}, \psi:
b=(m\xrightarrow{\mathsf{id}}m)^3 \to \Pi(\tau))$ be the network of hybrid
open systems of Example~\ref{ex:6.4h}. Define $f:(m\to m)\to (m\to
m)^3$ to be the diagonal map. Define the map of
lists $(\varphi, \Phi): \tau \to \mu$ in $(\mathsf{FinSet}/\mathsf{HySSub})^\Leftarrow$
by setting $\varphi(i) = *$ for all $i$ and defining $\Phi_i: \mu
(\varphi(i)) \to \tau(i)$ to be the identity map for all $i\in
\{1,2,3\}$. The pair
$((\varphi,\Phi), f)$ is a map of networks of hybrid open systems from
$(\tau, \psi)$ to $(\mu, \nu)$. This map of networks can also be
pictured as a map of directed graphs
\begin{center}
\tikzset{every loop/.style={min distance=15mm,in=-30,out=30,looseness=10}}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries}]
\node[main node] at (2,-1) (1) {1};
\node[main node] at (4,-1) (2) {2};
\node[main node] at (6,-1) (3) {3};
\node at (1,-1) (rb){};
\node at (-1,-1) (lb){};
\node[main node] at (-3,-1) (circ){*};
\path[]
(1) edge [bend left] node {} (2)
(2) edge [bend left] node [below] {} (1)
(2) edge node {}(3)
(circ) edge [loop right=20] node {} (circ)
(rb) edge node [above] {$\varphi$} (lb) ;
\end{tikzpicture}.
\end{center}
Note that now we have hybrid phase spaces attached to the nodes of the graphs.
\end{example}
\mbox{}
To state the main result of \cite{L2} and then to state and prove its
generalization to networks of hybrid open systems we need to recall a
few facts from \cite{L2}. Note first that the direct sum is not a
coproduct in the category $\mathsf{RelVect}$ of vector spaces and linear
relations. None the less, an analogue of Remark~\ref{rmrk:2.43}
holds.
\begin{proposition}\label{prop:L24.2}
The assignment
\[
(\mathsf{FinSet}/\mathsf{RelVect}) \ni \tau \mapsto \odot (\tau):= \oplus_{a\in X} \tau(a) \in \mathsf{RelVect}
\]
extends to a lax functor
\[
\odot: ((\mathsf{FinSet}/\mathsf{RelVect})^\Leftarrow)^{\mathsf{op}} \to \mathsf{RelVect}.
\]
\end{proposition}
\begin{proof} Given a 2-commuting triangle
\[
\xy
(-10, 10)*+{X} ="1";
(10, 10)*+{Y} ="2";
(0,-2)*+{\mathsf{RelVect} }="3";
{\ar@{->}_{\tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{\mu} "2";"3"};
{\ar@{=>}_{\scriptstyle \Phi} (4,6)*{};(-0.4,4)*{}} ;
\endxy
\]
we set
\begin{equation}\label{eq:5*}
\odot (\varphi, \Phi):= \bigcap _{a\in X} (\pi_{\varphi(a)} \times
\pi_{a})^{-1} (\Phi_a),
\end{equation}
where the relations $\Phi_a:\mu(\varphi(a))\to\tau (a)$ are
the component relations of the natural transformation $\Phi: \mu\circ
\varphi \Rightarrow \tau$ and
\[
\pi_{\varphi (a)}\times \pi_{a}: \oplus_{y\in Y}\mu(y) \times \oplus
_{x\in X} \tau(x) \to \mu(\varphi((a))\times \mu(\tau(a))
\]
are the projections. It remains to show that $\odot$ is actually a
functor. We refer the reader to the proof of
\cite[Proposition~5.8]{L2} for such an argument. Note that \cite{L2}
unconventionally views a linear relation
$\Phi_a:\mu (\varphi(a)) \to \tau(a)$ as a subspace of
$\tau(a)\times \mu(\varphi(a))$ and not of
$\mu(\varphi((a))\times \mu(\tau(a))$.
\end{proof}
Next we recall that the control functor $\mathsf{Crl}:\mathsf{SSub} \to \mathsf{RelVect}$ is monoidal,
where the monoidal structure on the category $\mathsf{SSub}$ is the
categorical product and on $\mathsf{RelVect}$ is the direct sum. In
particular, for any two surjective submersions $a,b$ we have a linear map
\[
\mathsf{Crl}_{a,b}: \mathsf{Crl}(a)\oplus \mathsf{Crl}(b) \to \mathsf{Crl}(a\times b)
\]
which is given by
\[
\mathsf{Crl}_{a,b}(F,G) := F\times G
\]
for all $(F,G)\in \mathsf{Crl}(a)\oplus \mathsf{Crl}(b)$. It follows that for any
{\bf ordered} list
$a:\{1,\ldots, n\}\to \mathsf{SSub}$ of submersions we have a canonical linear
map
\[
\mathsf{Crl}_a: \bigoplus_{i=1}^n \mathsf{Crl}(a(i))\to \mathsf{Crl} (\prod _{i=1}^n a(i)).
\]
It is given by
\begin{equation}\label{eq:L25.2}
(\psi_1,\ldots, \psi_n)\mapsto \psi_1\times \cdots \times \psi_n.
\end{equation}
\begin{lemma}\label{lem:L25.14}
For any unordered list $\tau: X\to \mathsf{SSub}$ we have a canonical
linear map
\begin{equation} \label{eq:L25.16}
\mathsf{Crl}_\tau: \bigoplus _{x\in X}\mathsf{Crl}(\tau(x))\to \mathsf{Crl}(\prod_{x\in X} \tau(x))
\end{equation}
so that if $X=\{1,\ldots,n\}$ then $\mathsf{Crl}_\tau$ is given by \eqref{eq:L25.2}.
\end{lemma}
\begin{proof}
See \cite[Lemma~5.15]{L2}.
\end{proof}
We can now state the main result of \cite{L2}.
\begin{theorem}\label{thm:main_inL2}
A map
\[
((\varphi,\Phi), f):(\tau:X\to \mathsf{SSub},
\psi:b\to\Pi(\tau)) \to (\mu:Y\to
\mathsf{SSub}, \nu:c\to\Pi (\mu))
\]
of networks of open systems
gives rise to a 2-cell
\[
\xy
(-25, 10)*+{\bigoplus_{y\in Y}\mathsf{Crl}(\mu(y) )} ="1";
(25, 10)*+{\bigoplus_{x\in X} \mathsf{Crl}(\tau(x))}="2";
(-25,-10)*+{\mathsf{Crl}(c)}="3";
(25,-10)*+{\mathsf{Crl}(b) }="4";
{\ar@{->}_ {\nu^*\circ \mathsf{Crl}_\mu} "1";"3"};
{\ar@{->}^{\odot (\varphi, \mathsf{Crl}\circ \Phi)} "1";"2"};
{\ar@{<-}_{\psi^*\circ \mathsf{Crl}_\tau} "4";"2"};
{\ar@{->}_{\mathsf{Crl}(f)} "3";"4"};
{\ar@{<=} (0,-2)*{};(0,2)*{}} ;
\endxy
\]
in the double category $\mathsf{RelVect}^\Box$ of vector spaces, linear maps and
linear relations. (The functor $\odot$ is defined in Proposition~\ref{prop:L24.2},
the maps $\mathsf{Crl}_\mu, \mathsf{Crl}_\tau$ come from Lemma~\ref{lem:L25.14},
and the pullback maps $\nu^*, \psi^*$ are from Definition~\ref{def:2.35}.)
\end{theorem}
\mbox{}
We are now in position to state and prove the main result of the paper
\begin{theorem}[Main theorem]\label{thm:main_result}
A map
$\big((\varphi,\Phi),f\big):
(\tau:X\rightarrow\mathsf{HySSub},\psi:b\rightarrow\Pi(\tau)
)\rightarrow(\mu:Y\rightarrow\mathsf{HySSub},\nu:c\rightarrow\Pi(\mu) )$ of
networks of open systems gives rise to a 2-cell
\[
\xy
(-35, 10)*+{\oplus_{y\in Y}\mathsf{Crl} ((\mathbb{U} ( \mu(y)))} ="1";
(35, 10)*+{\oplus_{x\in X}\mathsf{Crl}(\mathbb{U} (\tau(x) ))} ="2";
(-35,-10)*+{\mathsf{Crl}(\mathbb{U}(c)) }="3";
(35,-10)*+{\mathsf{Crl}(\mathbb{U}(b)) }="4";
{\ar@{->}_{(\mathbb{U}(\nu))^* \circ \mathsf{Crl}_{\mathbb{U}\circ \mu} }"1";"3"};
{\ar@{->}^{\odot (\varphi, \mathsf{Crl} \circ \mathbb{U} \circ \Phi))} "1";"2"};
{\ar@{->}^{(\mathbb{U}(\psi))^* \circ \mathsf{Crl}_{\mathbb{U}\circ \tau}} "2";"4"};
{\ar@{->}_{\mathsf{Crl}(\mathbb{U}(f))} "3";"4"};
{\ar@{<=} (0,-2)*{};(0,2)*{}} ;
\endxy
\]
in the double category $\mathsf{RelVect}^\Box$ of vector spaces, linear maps
and relations. (The functor $\odot$ is defined in Proposition~\ref{prop:L24.2},
the maps $\mathsf{Crl}_{\mathbb{U}\circ \mu}, \mathsf{Crl}_{\mathbb{U} \circ \tau}$ come from
Lemma~\ref{lem:L25.14},
and the pullback maps $(\mathbb{U}(\nu)^*, \mathbb{U}(\psi)^*$ are from Definition~\ref{def:2.35}.)
\end{theorem}
\begin{remark}
Theorem~\ref{thm:main_result} asserts that for any choice $\{w_x\in
\mathsf{Crl}(\mathbb{U}(\tau(x)))\}_{x\in X}$, $\{u_y\in \mathsf{Crl} (\mathbb{U}(\mu(y)))\}_{y\in Y}$ of
open systems so that the open system $u_{\varphi(x)}$ is
$\mathbb{U}(\Phi_x)$-related to the open system $w_x$ for all $x\in X$ we get a map
\[
f: \left(c, (\mathbb{U}(\nu)^*\circ \mathsf{Crl}_{\mathbb{U}\circ \mu}) ((u_y)_{y\in Y}) \right)\to \left(b,
(\mathbb{U}(\psi)^* \circ \mathsf{Crl}_{\mathbb{U} \circ \tau})((w_x)_{x\in X})\right)
\]
of hybrid open systems. In other words, compatible patterns of
interconnection of hybrid open systems give rise to maps of hybrid open
systems. In the case where the two interconnections result in closed
systems, the map $f$ a map of hybrid dynamical systems.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{thm:main_result}] We apply the
product-preserving functor $\mathbb{U}:\mathsf{HySSub}\to \mathsf{SSub}$ to the data of the
theorem. Namely, by applying the functor $\mathbb{U}$ to the 2-commuting triangle
\[
\xy
(-10, 10)*+{X} ="1";
(10, 10)*+{Y} ="2";
(0,-2)*+{\mathsf{HySSub}}="3";
{\ar@{->}_{\tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{\mu} "2";"3"};
{\ar@{=>}_{\scriptstyle \Phi} (4,6)*{};(-0.4,4)*{}} ;
\endxy
\]
we get
\begin{equation}\label{eq:5.14}
\xy
(-10, 10)*+{X} ="1";
(10, 10)*+{Y} ="2";
(0,-2)*+{\mathsf{SSub} }="3";
{\ar@{->}_{\mathbb{U}\circ \tau} "1";"3"};
{\ar@{->}^{\varphi} "1";"2"};
{\ar@{->}^{\mathbb{U} \circ \mu} "2";"3"};
{\ar@{=>}_{\scriptstyle \mathbb{U}\circ \Phi} (4,6)*{};(-0.4,4)*{}} ;
\endxy .
\end{equation}
Applying $\mathbb{U}$ to the 2-cell in the double category $\mathsf{HySSub}^\Box$
\[
\xy
(-12, 10)*+{\Pi(\mu)} ="1";
(12, 10)*+{\Pi(\tau)} ="2";
(-12,-10)*+{c }="3";
(12,-10)*+{b }="4";
{\ar@{->}^{\nu} "3";"1"};
{\ar@{->}^{\Pi (\varphi, \Phi)} "1";"2"};
{\ar@{->}_{\psi} "4";"2"};
{\ar@{->}_{f} "3";"4"};
{\ar@{=>} (0,-2)*{};(0,2)*{}} ;
\endxy ,
\]
which is a commuting diagram in $\mathsf{HySSub}$,
we get a commuting diagram
\begin{equation} \label{eq:5.15}
\xy
(-20, 10)*+{\Pi(\mathbb{U} \circ \mu)} ="1";
(20, 10)*+{\Pi(\mathbb{U} \circ \tau)} ="2";
(-20,-10)*+{\mathbb{U}(c) }="3";
(20,-10)*+{\mathbb{U}(b) }="4";
{\ar@{->}^{\mathbb{U}(\nu)} "3";"1"};
{\ar@{->}^{\mathbb{U}(\Pi (\varphi, \Phi))} "1";"2"};
{\ar@{->}_{\mathbb{U}(\psi)} "4";"2"};
{\ar@{->}_{\mathbb{U}(f)} "3";"4"};
\endxy
\end{equation}
is $\mathsf{SSub}$. Recall that a map of hybrid surjective submersions is an
interconnection map if and only if its image under $\mathbb{U}$ is an interconnection map
between the corresponding surjective submersions. It follows that
\eqref{eq:5.15} is a 2-cell in the double category $\mathsf{SSub}^\Box$.
Since $\mathbb{U}$ is product preserving we may assume that that $\mathbb{U} (\Pi
(\tau)) = \Pi (\mathbb{U}\circ \tau)$ and similarly for $\mu$. Similarly,
since $\Pi (\varphi, \mathbb{U} \circ \Phi): \Pi (\mathbb{U}\circ \mu)\to \Pi (\mathbb{U}
\circ \tau)$ is defined by the universal properties,
\[
\Pi (\varphi, \mathbb{U}\circ \Phi) = \mathbb{U} (\Pi (\varphi, \Phi))
\]
once we identify $\mathbb{U} (\Pi
(\tau))$ with $\Pi (\mathbb{U}\circ \tau)$ and $\mathbb{U} (\Pi (\mu)) $ with $\Pi
(\mathbb{U} \circ \mu))$. After these identifications the 2-cell
\eqref{eq:5.15} is
\begin{equation} \label{eq:5.16}
\xy
(-20, 10)*+{\Pi(\mathbb{U} \circ \mu)} ="1";
(20, 10)*+{\Pi(\mathbb{U} \circ \tau)} ="2";
(-20,-10)*+{\mathbb{U}(c) }="3";
(20,-10)*+{\mathbb{U}(b) }="4";
{\ar@{->}^{\mathbb{U}(\nu)} "3";"1"};
{\ar@{->}^{\Pi (\varphi, \mathbb{U} \circ \Phi))} "1";"2"};
{\ar@{->}_{\mathbb{U}(\psi)} "4";"2"};
{\ar@{->}_{\mathbb{U}(f)} "3";"4"};
{\ar@{=>} (0,-2)*{};(0,2)*{}} ;
\endxy
\end{equation}
The diagrams \eqref{eq:5.14} and \eqref{eq:5.16} together define a map
\[
((\varphi, \mathbb{U} \circ \Phi), \mathbb{U}(f)): (X\xrightarrow{\mathbb{U}\circ
\tau}\mathsf{SSub}, \mathbb{U}(b)\xrightarrow{\mathbb{U}(\psi)} \Pi (\mathbb{U}\circ \tau)) \to
(Y\xrightarrow{\mathbb{U}\circ \mu} \mathsf{SSub}, \mathbb{U}(c)\xrightarrow{\mathbb{U}(\nu)}\Pi
(\mathbb{U}\circ \mu)))
\]
of networks of open systems. By Theorem~\ref{thm:main_inL2} we have
the 2-cell
\[
\xy
(-35, 10)*+{\oplus_{y\in Y}\mathsf{Crl} ((\mathbb{U} ( \mu(y)))} ="1";
(35, 10)*+{\oplus_{x\in X}\mathsf{Crl}(\mathbb{U} (\tau(x) ))} ="2";
(-35,-10)*+{\mathsf{Crl}(\mathbb{U}(c)) }="3";
(35,-10)*+{\mathsf{Crl}(\mathbb{U}(b)) }="4";
{\ar@{->}_{(\mathbb{U}(\nu))^* \circ \mathsf{Crl}_{\mathbb{U}\circ \mu} }"1";"3"};
{\ar@{->}^{\odot (\varphi, \mathsf{Crl} \circ \mathbb{U} \circ \Phi))} "1";"2"};
{\ar@{->}^{(\mathbb{U}(\psi))^* \circ \mathsf{Crl}_{\mathbb{U}\circ \tau}} "2";"4"};
{\ar@{->}_{\mathsf{Crl}(\mathbb{U}(f))} "3";"4"};
{\ar@{<=} (0,-2)*{};(0,2)*{}} ;
\endxy
\]
in the double category $\mathsf{RelVect}^\Box$ which is what we wanted to prove.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,716
|
You want to convert my mod(s) to SSE Skyrim Special Edition ? Go for it. Don't even need to ask me.
Although be sure to properly copy-paste the credits tab in your mod page.
Here is the complete list of all the images I have used for this mod.
Big thanks to them. The cards wouldn't look as cool or as interesting without them.
Added the Bucky The Becket booster pack (20 cards), courtesy TheFurCrew. Big thanks to him :D !
39 new cards themed after the Children Of Skyrim. Thanks neab !
Lore buffs, treasure hunters and collectors of all curiosities : BEHOLD !
This mod adds collectibles card to Skyrim.
These cards can be found on enemies, in chests, in merchants inventory, at book sellers and generally everywhere books normally spawns.
The main goal of these card is to teach the player general knowledge about the lore of The Elder Scrolls universe, in a fun and "dynamic" way. Each card you find has a picture, a quote and a description of what it's picturing.
SKYRIM - These cards are black. They are all about things you can find, see, talk to or slay in The Elder Scrolls V : Skyrim.
DAWNGUARD - These cards are red. They picture things you may have seen in The Elder Scrolls V : Dawnguard DLC.
DRAGONBORN - Blue cards. All about the content of The Elder Scrolls V : Dragonborn DLC.
OBLIVION - As the title suggest, these brown cards will take you to nostalgia lane and teach you a thing or two about The Elder Scrolls IV : Oblivion and all its minor add-ons.
SHIVERING ISLES - Orange and Purple cards depicting things you may have witnessed in the add-on for Oblivion : Shivering Isles.
MORROWIND - Still not nostalgic enough ? These yellow cards have you covered with content from The Elder Scrolls III : Morrowind and all its add-ons.
LEGACY - These white cards with black font are related to general lore of the Elder Scrolls universe. Gods, races, continents, all of it !
META - Mysterious ( and very rare ) green cards that hold knowledge about things outside of the Elder Scrolls universe. Very Easter-eggy.
NEXUS - Brownish cards that depict the great heroes and legend of Tamriel. ( Used to depict great figures of The Elder Scrolls Community and maybe even YOUR character ).
CHILDREN - Kindly donated by neab these cards represents the children of Skyrim and Solstheim, as they appear in the mod The Kids Are Alright.
Bucky The Bucket Booster Pack - Kindly donated by TheFurCrew, these cards are all about the beautiful buckets found all over Skyrim.
Among these cards, you may find elusive RARE cards. These magic items will grand you a bonus point in certain skills upon looking at it.
I also recommend Reading Is Good by Parapets. That mod overhauls the skill book system : instead of adding a skill point to your skill when reading skill books, you'll instead gain a permanent speed boost in increasing that skill, akin to the "well-rested" ability, but only for a single skill and it'll last indefinitely. My Rare Cards are even more interesting with that mod installed !
Want to participate in this mod's expansion ? YOU CAN ! Take a screenshot of... well, just about everything and send it to me !
It can even be your character ! just explain his/her backstory to me with a small quotation of what the character says and I'll turn it into a card ;) !
Big thanks to TheFurCrew who nicely gave me 20 cards, themed after... buckets ! Available from 2.0 onward.
For giving me a Nexus Mods PR Pack with all the goodies I needed to improve the Nexus Mods-related cards.
He deserve a big hug !
For the 39 cards themed after the Children Of Skyrim. Thanks to him :D ! (Available from 2.0 onward).
Where can I find the cards ?
Why is the mod so large ?
The Creation Kit wouldn't let me apply custom textures on the "readable" mesh. So I had to create a readable mesh for EVERY. SINGLE. CARD. Also, 170 1k textures files is quite large anyways.
Is this mod incompatible with X or Y ?
This mod *should* be compatible with just about everything. It edits some leveled list so be sure to load it at the lowest possible emplacement of your load order.
Where can I send you my would-be custom card ?
You can upload your screenshot to the Nexus images and send me a message, for example. Or you can send it to me at [email protected] . ( If you do, shoot me a message on the Nexus, just in case ).
Please note that I also included every base template I used in the file. If case you want to make your own cards. The correct font is Cyrodiil Font, found here.
The Credits Tabs contains the source of every. single. image used from the Nexus. Go check them out. They make some cool desktop background.
All the images are either from the Nexus, official promotional images and concept art by Bethesda, images send to me by users or are screenshots taken by me.
If anyone wants to convert this mod to SSE, they can without asking. All I'm demanding is that the credited assets are properly... credited.
I'll try to take a look and learn how to convert SLE mods to SSE if the demand is high enough. Although, I can't guarantee my success in doing so.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,601
|
Q: UITextField in UISearchController BecomeFirstResponder I have a UITextField that is in a UISearchController that needs to become the FirstResponder when the view controller is shown.
On ViewDidLoad the UISearchController is built.
On ViewDidAppear I get the UITextField from the UISearchController and tell it to BecomeFirstResponder
public override void ViewDidLoad(bool animated)
{
base.ViewDidLoad();
BuildSearchController();
SearchController.SearchBar.BecomeFirstResponder();
}
private void BuildSearchController()
{
UIFont MyFont = UIFont.FromName("Helvetica", 20f);
SearchController = new UISearchController(ResultsTableController)
{
WeakDelegate = this,
DimsBackgroundDuringPresentation = false,
WeakSearchResultsUpdater = this,
};
SearchController.SearchBar.ScopeButtonTitles = new string[5] { "Account", "Name", "Service Address", "Phone", "SSN" };
UITextField SearchField = (UITextField)SearchController.SearchBar.Subviews[0].Subviews[2];
SearchField.Font = MyFont;
SearchController.SearchBar.ShowsCancelButton = false;
SearchController.SearchBar.SetScopeBarButtonTitle(new UITextAttributes() { Font = MyFont }, UIControlState.Normal);
SearchController.SearchBar.SizeToFit();
SearchController.SearchBar.SelectedScopeButtonIndex = 1;
SearchController.SearchBar.SetShowsCancelButton(false, false);
UISegmentedControl TempSegControl = SearchController.SearchBar.Subviews[0].Subviews[0].Subviews[1] as UISegmentedControl;
TempSegControl.BackgroundColor = UIColor.FromRGBA(0, 122, 255, 255);
TempSegControl.TintColor = UIColor.White;
TempSegControl.Layer.BorderWidth = 1;
TempSegControl.Layer.CornerRadius = 6;
TempSegControl.Layer.BorderColor = UIColor.White.CGColor;
SearchController.SearchBar.BarTintColor = UIColor.FromRGBA(232, 232, 232, 255);
SearchController.HidesNavigationBarDuringPresentation = false;
SearchField.BecomeFirstResponder();
TableView.TableHeaderView = SearchController.SearchBar;
SearchController.SearchBar.WeakDelegate = this;
}
public override void ViewDidAppear(bool animated)
{
UITextField SearchBox = SearchController.SearchBar.Subviews[0].Subviews[2] as UITextField;
if (SearchBox.CanBecomeFirstResponder)
{
SearchBox.BecomeFirstResponder();
}
base.ViewWillAppear(animated);
}
The UITextField never becomes the first responder because it is not able to become the first responder. Why can the UITextField not become the first repsonder?
A: ViewDidLoad is called after the view controller has loaded its view hierarchy into memory but not presented to the user.
You cannot perform a UI interaction with the object in that stage.
UISearchController has a delegate method which is invoked when it has been automatically presented. Use it to bring up the keyboard.
optional func didPresentSearchController(searchController: UISearchController){
searchController.searchBar.becomeFirstResponder();
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,910
|
Michal David (born name Vladimír Štancl, 14 July 1960, Prague) is a Czech pop-singer, songwriter and producer.
Biography and career
He started his music career during his studies in Prague Conservatory in the 1970s where he created a jazz band with his friends, as jazz has been Michal's main passion and an academic concentration.
However, after a short period of time, he was hired by a successful pop-music producer František Janeček as a piano player, singer and a song-writer and soon became a teenage girls' idol.
Michal David had a very successful career as a singer and composer, however after the Czechoslovak Velvet Revolution in 1989, he as well many other Czechoslovak popular singers fell on hard times, since the political situation changed rapidly and the market started to open to other foreign artists and domestic artists lost their popularity for a while. He was also called a pro-regime singer by some critics. During this time Michal David was mainly focusing on composing and songwriting rather than performing.
His popularity was restored again in 1998 when he composed and sang "hymn" for the Czech national ice hockey team, which won the Nagano Winter Olympics that year.
Then, in 2000, he contributed to the comeback of another famous Czech female singer Helena Vondráčková by writing her a hit song "Dlouhá noc".
In 2002 Michal David officially composed his first musical show called "Kleopatra" which premiered in Prague theatre Broadway. Since then, Michal has composed more than 5 other musicals, where he was not only a songwriter but the producer as well. Kleopatra along with some of his other musical shows were sold to South Korea where they became instantly successful.
His song "Třetí Galaxie" appeared in Eli Roth's movie "Hostel".
Michal David was a vocal coach on the TV reality talent show The Voice, starting in February 2011 on Czech and Slovak National TV.
Personal life
Michal David is currently living in Prague, Czech Republic with his wife Marcela (Marcela Skuherská), a former successful professional tennis player from the 1980s. (She won the Fed Cup in 1983 and 1984 with her team members Helena Suková, Hana Mandlíková and Iva Budařová, and participated in competitions such as US Open, Australian Open and Wimbledon). Michal's daughter Klára obtained her bachelor's degree in arts in New York, followed by postgraduate studies at the University of Oxford.
David has been associated with many charities and non-profits since the death of his second daughter Michaela who died of leukemia at the age of 11.
See also
Céčka
References
External links
Official website
1960 births
Living people
Musicians from Prague
Czech songwriters
21st-century Czech male singers
20th-century Czech male singers
Czechoslovak male singers
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,596
|
{"url":"https:\/\/stacks.math.columbia.edu\/tag\/00ME","text":"Lemma 10.98.1. Suppose that $R \\to S$ is a local homomorphism of Noetherian local rings. Denote $\\mathfrak m$ the maximal ideal of $R$. Let $M$ be a flat $R$-module and $N$ a finite $S$-module. Let $u : N \\to M$ be a map of $R$-modules. If $\\overline{u} : N\/\\mathfrak m N \\to M\/\\mathfrak m M$ is injective then $u$ is injective. In this case $M\/u(N)$ is flat over $R$.\n\nProof. First we claim that $u_ n : N\/{\\mathfrak m}^ nN \\to M\/{\\mathfrak m}^ nM$ is injective for all $n \\geq 1$. We proceed by induction, the base case is that $\\overline{u} = u_1$ is injective. By our assumption that $M$ is flat over $R$ we have a short exact sequence $0 \\to M \\otimes _ R {\\mathfrak m}^ n\/{\\mathfrak m}^{n + 1} \\to M\/{\\mathfrak m}^{n + 1}M \\to M\/{\\mathfrak m}^ n M \\to 0$. Also, $M \\otimes _ R {\\mathfrak m}^ n\/{\\mathfrak m}^{n + 1} = M\/{\\mathfrak m}M \\otimes _{R\/{\\mathfrak m}} {\\mathfrak m}^ n\/{\\mathfrak m}^{n + 1}$. We have a similar exact sequence $N \\otimes _ R {\\mathfrak m}^ n\/{\\mathfrak m}^{n + 1} \\to N\/{\\mathfrak m}^{n + 1}N \\to N\/{\\mathfrak m}^ n N \\to 0$ for $N$ except we do not have the zero on the left. We also have $N \\otimes _ R {\\mathfrak m}^ n\/{\\mathfrak m}^{n + 1} = N\/{\\mathfrak m}N \\otimes _{R\/{\\mathfrak m}} {\\mathfrak m}^ n\/{\\mathfrak m}^{n + 1}$. Thus the map $u_{n + 1}$ is injective as both $u_ n$ and the map $\\overline{u} \\otimes \\text{id}_{{\\mathfrak m}^ n\/{\\mathfrak m}^{n + 1}}$ are.\n\nBy Krull's intersection theorem (Lemma 10.50.4) applied to $N$ over the ring $S$ and the ideal $\\mathfrak mS$ we have $\\bigcap \\mathfrak m^ nN = 0$. Thus the injectivity of $u_ n$ for all $n$ implies $u$ is injective.\n\nTo show that $M\/u(N)$ is flat over $R$, it suffices to show that $I \\otimes _ R M\/u(N) \\to M\/u(N)$ is injective for every ideal $I \\subset R$, see Lemma 10.38.5. Consider the diagram\n\n$\\begin{matrix} & & 0 & & 0 & & 0 & & \\\\ & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\ & & N\/IN & \\to & M\/IM & \\to & M\/(IM + u(N)) & \\to & 0 \\\\ & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\ 0 & \\to & N & \\to & M & \\to & M\/u(N) & \\to & 0 \\\\ & & \\uparrow & & \\uparrow & & \\uparrow & & \\\\ & & N \\otimes _ R I & \\to & M \\otimes _ R I & \\to & M\/u(N)\\otimes _ R I & \\to & 0 \\end{matrix}$\n\nThe arrow $M \\otimes _ R I \\to M$ is injective. By the snake lemma (Lemma 10.4.1) we see that it suffices to prove that $N\/IN$ injects into $M\/IM$. Note that $R\/I \\to S\/IS$ is a local homomorphism of Noetherian local rings, $N\/IN \\to M\/IM$ is a map of $R\/I$-modules, $N\/IN$ is finite over $S\/IS$, and $M\/IM$ is flat over $R\/I$ and $u \\bmod I : N\/IN \\to M\/IM$ is injective modulo $\\mathfrak m$. Thus we may apply the first part of the proof to $u \\bmod I$ and we conclude. $\\square$\n\nComment #3817 by Alapan Mukhopadhyay on\n\nIn the diagram $\\frac{M}{IN+u(N)}$ is to be replaced by $\\frac{M}{IM+u(N)}$.\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).","date":"2019-05-20 06:31:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 2, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"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\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9879453182220459, \"perplexity\": 56.49214782615221}, \"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-2019-22\/segments\/1558232255773.51\/warc\/CC-MAIN-20190520061847-20190520083847-00287.warc.gz\"}"}
| null | null |
Q: Unable to Use Pandas In IDLE I installed Python3.8 then installed pandas through miniconda.
However, when import pandas as pd in IDLE I get the following error
import pandas as pd
Traceback (most recent call last):
File "<pyshell#0>", line 1, in <module>
import pandas as pd
ModuleNotFoundError: No module named 'pandas'
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,664
|
package org.cagrid.gme.service;
import gov.nih.nci.cagrid.common.Utils;
import java.io.IOException;
import java.io.StringReader;
import java.net.URI;
import java.net.URISyntaxException;
import java.util.ArrayList;
import java.util.Collection;
import java.util.HashMap;
import java.util.HashSet;
import java.util.List;
import java.util.Map;
import java.util.Set;
import java.util.Vector;
import java.util.concurrent.locks.ReentrantReadWriteLock;
import org.apache.commons.logging.Log;
import org.apache.commons.logging.LogFactory;
import org.apache.xerces.impl.xs.SchemaGrammar;
import org.apache.xerces.xni.XNIException;
import org.apache.xerces.xni.parser.XMLInputSource;
import org.apache.xerces.xs.StringList;
import org.cagrid.gme.domain.XMLSchema;
import org.cagrid.gme.domain.XMLSchemaBundle;
import org.cagrid.gme.domain.XMLSchemaDocument;
import org.cagrid.gme.domain.XMLSchemaImportInformation;
import org.cagrid.gme.domain.XMLSchemaNamespace;
import org.cagrid.gme.persistence.SchemaPersistenceGeneralException;
import org.cagrid.gme.sax.GMEErrorHandler;
import org.cagrid.gme.sax.GMEXMLSchemaLoader;
import org.cagrid.gme.service.dao.XMLSchemaInformationDao;
import org.cagrid.gme.service.domain.XMLSchemaInformation;
import org.cagrid.gme.stubs.types.InvalidSchemaSubmissionFault;
import org.cagrid.gme.stubs.types.NoSuchNamespaceExistsFault;
import org.cagrid.gme.stubs.types.UnableToDeleteSchemaFault;
import org.globus.wsrf.utils.FaultHelper;
import org.springframework.transaction.annotation.Transactional;
@Transactional
public class GME {
protected static Log LOG = LogFactory.getLog(GME.class.getName());
protected XMLSchemaInformationDao schemaDao;
// provides coarse grain persistence layer locking, used to ensure integrity
// of
protected ReentrantReadWriteLock lock = new ReentrantReadWriteLock();
public void setXMLSchemaInformationDao(XMLSchemaInformationDao schemaDao) {
if (schemaDao == null) {
throw new IllegalArgumentException("Cannot use a null XMLSchemaInformationDao!");
}
this.schemaDao = schemaDao;
}
public void deleteSchemas(Collection<URI> schemaNamespaces) throws NoSuchNamespaceExistsFault,
UnableToDeleteSchemaFault {
if (schemaNamespaces == null || schemaNamespaces.size() == 0) {
String description = "null or empty set is not a valid collection of namespaces to delete.";
NoSuchNamespaceExistsFault fault = new NoSuchNamespaceExistsFault();
gov.nih.nci.cagrid.common.FaultHelper helper = new gov.nih.nci.cagrid.common.FaultHelper(fault);
helper.setDescription(description);
LOG.debug("Refusing to delete schema: " + description);
throw (NoSuchNamespaceExistsFault) helper.getFault();
}
// need to get a "lock" on the database here
this.lock.writeLock().lock();
try {
Map<URI, XMLSchemaInformation> schemaMap = new HashMap<URI, XMLSchemaInformation>();
for (URI ns : schemaNamespaces) {
XMLSchemaInformation schema = this.schemaDao.getByTargetNamespace(ns);
if (schema == null) {
String description = "No schema is published with given targetNamespace (" + ns + ")";
NoSuchNamespaceExistsFault fault = new NoSuchNamespaceExistsFault();
gov.nih.nci.cagrid.common.FaultHelper helper = new gov.nih.nci.cagrid.common.FaultHelper(fault);
helper.setDescription(description);
LOG.debug("Refusing to delete schema: " + description);
throw (NoSuchNamespaceExistsFault) helper.getFault();
} else {
schemaMap.put(ns, schema);
}
// TODO: permission check
// 1. Check permissions on each schema being deleted; fail if
// don't have permissions
// 2. Check that all schemas being deleted are in a state where
// the
// contents can be deleted ; fail otherwise
}
for (URI ns : schemaNamespaces) {
Collection<XMLSchema> dependingXMLSchemas = this.schemaDao.getDependingXMLSchemas(ns);
// get the list of depending schemas for each of the namespace
// for each, make sure the list is 0, or every depending schema
// will
// also be deleted (i.e. the namespace is in the provided list
// to
// delete)
for (XMLSchema dependingSchema : dependingXMLSchemas) {
URI dependingTargetNamespace = dependingSchema.getTargetNamespace();
if (!schemaMap.containsKey(dependingTargetNamespace)) {
String description = "Cannot delete XMLSchema (" + ns + ") as it is imported by XMLSchema ("
+ dependingTargetNamespace + "). You must also delete the XMLSchema ("
+ dependingTargetNamespace + ") or udpate it to not depend on XMLSchema (" + ns + ")";
// REVIST: should i keep going and build up a list of
// these,
// or just failfast?
UnableToDeleteSchemaFault fault = new UnableToDeleteSchemaFault();
gov.nih.nci.cagrid.common.FaultHelper helper = new gov.nih.nci.cagrid.common.FaultHelper(fault);
helper.setDescription(description);
LOG.debug("Refusing to delete schema: " + description);
throw (UnableToDeleteSchemaFault) helper.getFault();
} else {
LOG.debug("So far ok to delete XMLSchema (" + ns
+ ") even though it is imported by XMLSchema (" + dependingTargetNamespace
+ "), as it is being requested to be deleted as well.");
}
}
}
// ok to delete all these schemas now
this.schemaDao.delete(schemaMap.values());
} finally {
// release database lock
this.lock.writeLock().unlock();
}
}
public void publishSchemas(List<XMLSchema> schemas) throws InvalidSchemaSubmissionFault {
// 0. sanity check submission
if (schemas == null || schemas.size() == 0) {
String message = "No schemas were found in the submission.";
LOG.error(message);
InvalidSchemaSubmissionFault e = new InvalidSchemaSubmissionFault();
e.setFaultString(message);
throw e;
}
// need to get a "lock" on the database here
this.lock.writeLock().lock();
try {
// 3. Create a list of "processed schemas"
Map<URI, SchemaGrammar> processedSchemas = verifySubmissionAndInitializeProcessedSchemasMap(schemas);
// 4. Create a model with error and entity resolver (on
// callback
// to imports/includes/redefines, entity resolver needs to first
// load schema from
// submission if present, if not in submission load from DB, if not
// in
// DB error out)
GMEXMLSchemaLoader schemaLoader = new GMEXMLSchemaLoader(schemas, this.schemaDao);
// 5. Call processSchema() for each schema being uploaded
for (XMLSchema submittedSchema : schemas) {
try {
processSchema(schemaLoader, processedSchemas, submittedSchema);
} catch (Exception e) {
String message = "Problem processing schema submissions; the schema ["
+ submittedSchema.getTargetNamespace() + "] was not valid:" + Utils.getExceptionMessage(e);
LOG.error(message, e);
InvalidSchemaSubmissionFault fault = new InvalidSchemaSubmissionFault();
fault.setFaultString(message);
FaultHelper helper = new FaultHelper(fault);
helper.addFaultCause(e);
fault = (InvalidSchemaSubmissionFault) helper.getFault();
throw fault;
}
}
// 8. Commit new/modified schemas to database, populating dependency
// schema information (gathered from imports)
commitSchemas(schemas, processedSchemas);
} finally {
// release database lock
this.lock.writeLock().unlock();
}
}
protected Map<URI, SchemaGrammar> verifySubmissionAndInitializeProcessedSchemasMap(List<XMLSchema> schemas)
throws InvalidSchemaSubmissionFault {
Map<URI, SchemaGrammar> processedSchemas = new HashMap<URI, SchemaGrammar>();
for (XMLSchema submittedSchema : schemas) {
// verify the schema's have unique and valid URIs
URI namespace = submittedSchema.getTargetNamespace();
if (namespace == null) {
String message = "The schema submission contained a schema with a null URI.";
LOG.error(message);
InvalidSchemaSubmissionFault e = new InvalidSchemaSubmissionFault();
e.setFaultString(message);
throw e;
}
if (processedSchemas.containsKey(namespace)) {
String message = "The schema submission contained multiple schemas of the same URI ("
+ namespace
+ "). If you intend to submit schemas with includes, you need to package them as a single Schema with multiple SchemaDocuments.";
LOG.error(message);
InvalidSchemaSubmissionFault e = new InvalidSchemaSubmissionFault();
e.setFaultString(message);
throw e;
} else {
if (submittedSchema.getRootDocument() == null) {
String message = "The schema submission contained a schema ["
+ submittedSchema.getTargetNamespace() + "] without a root schema document.";
LOG.error(message);
InvalidSchemaSubmissionFault e = new InvalidSchemaSubmissionFault();
e.setFaultString(message);
throw e;
}
// REVISIT: should probably check SchemaDocument rules here:
// unique systemIDs for all [this is accomplished by using
// Set now]
// need to actually check that the schema rules are true
// (i.e include must have no ns, or same ns), but basically
// need the parsed model to know this... so i can check at
// resolution time, but if something is never referenced, it
// wont be loaded and so may be invalid... probably need to
// check it after we have the grammar (can we ask for
// includes and check for correspondence in the submission?)
// [this is checked by combination of parser (checking rules
// of what it reads) and matching
// the loaded doc size matches the submitted doc
// size (meaning they were all read)]
// preload the submission schemas with "null" models (which
// will be replaced by the actual models once they are
// processed) so they don't get inappropriately processed
// (as could be the case during the loading of "depending
// schemas" )
processedSchemas.put(namespace, null);
}
// TODO: permission check
// 1. Check permissions on each schema being published; fail if
// don't have permissions
// 2. Check that all schemas being published are either not yet
// published, or are in a state where the contents can be
// updated ;
// fail otherwise
XMLSchema storedSchema = this.schemaDao.getXMLSchemaByTargetNamespace(namespace);
if (storedSchema != null) {
// TODO: check that it can be modified before doing this
// (right now it is never allowed)
// if(storeSchema.getState... != ){
String message = "The schema [" + namespace + "] already exists and cannot be modified.";
LOG.error(message);
InvalidSchemaSubmissionFault e = new InvalidSchemaSubmissionFault();
e.setFaultString(message);
throw e;
}
}
return processedSchemas;
}
protected void commitSchemas(List<XMLSchema> schemas, Map<URI, SchemaGrammar> processedSchemas)
throws InvalidSchemaSubmissionFault {
// if got here with no error, schemas are ok to persist
// build up DB objects to commit
Map<XMLSchema, List<URI>> toCommit = new HashMap<XMLSchema, List<URI>>();
for (XMLSchema submittedSchema : schemas) {
// extract the schema model
SchemaGrammar schemaGrammar = processedSchemas.get(submittedSchema.getTargetNamespace());
assert schemaGrammar != null;
assert !toCommit.containsKey(submittedSchema);
// this has all the expanded locations which built up the schema
// for us (using the namespace as the basesystemID, this should
// be the ns+systemID)
// REVISIT: is there a better way to figure out the
// included/redefined documents?
StringList documentLocations = schemaGrammar.getDocumentLocations();
for (XMLSchemaDocument schemaDocument : submittedSchema.getAdditionalSchemaDocuments()) {
URI expandedURI;
try {
// REVISIT: is this the right way to construct this.
// NOTE: this must match the behavior of the
// GMEEntityResolver when it create the baseSystemID for
// the XMLInputSource it loads
expandedURI = new URI(submittedSchema.getTargetNamespace().toString() + "/"
+ schemaDocument.getSystemID());
} catch (Exception e) {
String message = "Problem processing schema submissions; the schema ["
+ submittedSchema.getTargetNamespace() + "] included a SchemaDocument ["
+ schemaDocument.getSystemID() + "] whose expanded URI was not valid:"
+ Utils.getExceptionMessage(e);
LOG.error(message, e);
InvalidSchemaSubmissionFault fault = new InvalidSchemaSubmissionFault();
fault.setFaultString(message);
FaultHelper helper = new FaultHelper(fault);
helper.addFaultCause(e);
fault = (InvalidSchemaSubmissionFault) helper.getFault();
throw fault;
}
if (!documentLocations.contains(expandedURI.toString())) {
String message = "Problem processing schema submissions; the schema ["
+ submittedSchema.getTargetNamespace() + "] included a SchemaDocument ["
+ schemaDocument.getSystemID() + "] which was not used by the parsed grammar";
LOG.error(message);
InvalidSchemaSubmissionFault fault = new InvalidSchemaSubmissionFault();
fault.setFaultString(message);
FaultHelper helper = new FaultHelper(fault);
fault = (InvalidSchemaSubmissionFault) helper.getFault();
throw fault;
}
}
// we must have a document for the root and each additional
// schema document
if (documentLocations.getLength() != submittedSchema.getAdditionalSchemaDocuments().size() + 1) {
String message = "Problem processing schema submissions; the schema ["
+ submittedSchema.getTargetNamespace() + "] contained ["
+ submittedSchema.getAdditionalSchemaDocuments().size()
+ "] SchemaDocuments but the parsed grammar contained [" + documentLocations.getLength()
+ "]. All SchemaDocuments must be used by the Schema.";
LOG.error(message);
InvalidSchemaSubmissionFault fault = new InvalidSchemaSubmissionFault();
fault.setFaultString(message);
FaultHelper helper = new FaultHelper(fault);
fault = (InvalidSchemaSubmissionFault) helper.getFault();
throw fault;
}
// build an import list
List<URI> importList = new ArrayList<URI>();
Vector importedGrammars = schemaGrammar.getImportedGrammars();
if (importedGrammars != null) {
for (int i = 0; i < importedGrammars.size(); i++) {
SchemaGrammar importedSchema = (SchemaGrammar) importedGrammars.get(i);
String importedTargetNS = importedSchema.getTargetNamespace();
LOG.info("Schema [" + schemaGrammar.getTargetNamespace() + "] imports schema [" + importedTargetNS
+ "]");
try {
importList.add(new URI(importedTargetNS));
} catch (URISyntaxException e) {
String message = "Problem processing schema submissions; the schema ["
+ submittedSchema.getTargetNamespace() + "] imported a schema [" + importedTargetNS
+ "] whose URI was not valid:" + Utils.getExceptionMessage(e);
LOG.error(message, e);
InvalidSchemaSubmissionFault fault = new InvalidSchemaSubmissionFault();
fault.setFaultString(message);
FaultHelper helper = new FaultHelper(fault);
helper.addFaultCause(e);
fault = (InvalidSchemaSubmissionFault) helper.getFault();
throw fault;
}
}
}
// add the schema and its import list to the Map to commit
toCommit.put(submittedSchema, importList);
}
// TODO: replace this call by embedding its logic above
// commit to database
this.storeSchemas(toCommit);
}
// TODO: rewrite this with the caller above to directly make use of the DAO
// instead of building up a map and calling this (this is refactor cruft
// leftover from removing the SchemaPersitence layer)
// TODO: need to add rollBackFor=... to specify exceptions which should
// cause rollbacks (does that rollbackfor subclasses of the specified
// exception?)
private void storeSchemas(Map<XMLSchema, List<URI>> schemasToStore) {
// REVISIT: is there a simpler way to do this
// this is a list of newly persistent XMLSchemaInformation (for those
// which are being saved), and already persistent XMLSchemaInformation
// (for those that are being imported and not updated)
Map<URI, XMLSchemaInformation> persistedInfos = new HashMap<URI, XMLSchemaInformation>();
// foreach XMLSchema
for (XMLSchema s : schemasToStore.keySet()) {
// find PersistableXMLSchema (by URI), create if null, save
XMLSchemaInformation info = this.schemaDao.getByTargetNamespace(s.getTargetNamespace());
if (info == null) {
info = new XMLSchemaInformation();
}
// -setSchema XMLSchema on XMLSchemaInformation
info.setSchema(s);
this.schemaDao.save(info);
// -put in hash of URI->XMLSchemaInformation
persistedInfos.put(s.getTargetNamespace(), info);
}
// all new/updated schemas are now in the hash and persistent
// foreach XMLSchema (make the changes)
for (XMLSchema s : schemasToStore.keySet()) {
// -get PersistableXMLSchema from hash
XMLSchemaInformation info = persistedInfos.get(s.getTargetNamespace());
Set<XMLSchemaInformation> importSet = new HashSet<XMLSchemaInformation>();
List<URI> importList = schemasToStore.get(s);
// -foreach URI in import List
for (URI importedURI : importList) {
// --if not in hash
XMLSchemaInformation importedInfo = persistedInfos.get(importedURI);
if (importedInfo == null) {
// --- getReference to PersistableXMLSchema, put in hash
importedInfo = this.schemaDao.getByTargetNamespace(importedURI);
// this must either be new and already in the hash (the
// containing if), or existing and therefore in the db
// already
assert importedInfo != null;
persistedInfos.put(s.getTargetNamespace(), importedInfo);
}
// --add toimportSet
importSet.add(importedInfo);
}
// -set importSet on PersistableXMLSchema
info.setImports(importSet);
// TODO: I don't think I should have to do this... shouldn't the
// DAO-returned objects still be persistent and notice the changes?
this.schemaDao.save(info);
}
}
protected void processSchema(GMEXMLSchemaLoader schemaLoader, Map<URI, SchemaGrammar> processedSchemas,
XMLSchema schemaToProcess) throws XNIException, IOException, SchemaPersistenceGeneralException {
LOG.debug("About to process schema [" + schemaToProcess.getTargetNamespace() + "].");
// 6.2. Add the schema to the model which will recursively fire
// callbacks to the entity resolver for all imports (loading all
// dependency schemas, and their dependency schemas, etc)
String ns = schemaToProcess.getTargetNamespace().toString();
XMLSchemaDocument rootSD = schemaToProcess.getRootDocument();
// REVISIT: what to set for the baseSystemID? it's used to convert
// includes, etc into full URIs and so is relevant later when examining
// documentLocations of the SchemaGrammar
XMLInputSource xis = new XMLInputSource(ns, rootSD.getSystemID(), ns, new StringReader(rootSD.getSchemaText()),
"UTF-16");
SchemaGrammar model = (SchemaGrammar) schemaLoader.loadGrammar(xis);
if (model == null) {
GMEErrorHandler errorHandler = schemaLoader.getErrorHandler();
throw errorHandler.createXMLParseException();
}
// we should not have processed this schema before, and if the URI
// is in the list it should have a null model (indicating it was
// preloaded from the submission schemas set)
assert !processedSchemas.containsKey(schemaToProcess.getTargetNamespace())
|| processedSchemas.get(schemaToProcess.getTargetNamespace()) == null;
// store the resultant schema model (this needs to happen before
// recursion)
processedSchemas.put(schemaToProcess.getTargetNamespace(), model);
String targetURI = model.getTargetNamespace();
if (!ns.equals(targetURI)) {
String message = "Problem processing schema submissions; the schema["
+ schemaToProcess.getTargetNamespace() + "] was not valid as its acutal targetURI [" + targetURI
+ "] did not match.";
LOG.error(message);
InvalidSchemaSubmissionFault fault = new InvalidSchemaSubmissionFault();
fault.setFaultString(message);
throw fault;
}
// 6.3. Look in the DB for depending schemas (will only be present if
// schema was already published and is being updated)
Collection<XMLSchema> dependingSchemas = this.schemaDao.getDependingXMLSchemas(schemaToProcess
.getTargetNamespace());
// 6.4. For each depending schema not in the list of "processed schemas"
// call processSchema()
for (XMLSchema dependingSchema : dependingSchemas) {
if (processedSchemas.containsKey(dependingSchema.getTargetNamespace())) {
LOG.debug("Depending schema [" + dependingSchema.getTargetNamespace()
+ "] was already processed (or will be processed).");
} else {
LOG.debug("Processing depending schema [" + dependingSchema.getTargetNamespace()
+ "] which is not in the submission package.");
processSchema(schemaLoader, processedSchemas, dependingSchema);
}
}
}
/**
* Returns the targetNamespaces (represented by URIs) of all published
* XMLSchemas
*
* @return the targetNamespaces (represented by URIs) of all published
* XMLSchemas
*/
@Transactional(readOnly = true)
public Collection<URI> getNamespaces() {
this.lock.readLock().lock();
try {
return this.schemaDao.getAllNamespaces();
} finally {
this.lock.readLock().unlock();
}
}
/**
* Returns a published XMLSchema with a targetNamespace equal to the given
* URI
*
* @param uri
* the targetNamespace of the desired XMLSchema
* @return a published XMLSchema with a targetNamespace equal to the given
* URI
* @throws NoSuchNamespaceExistsFault
* if there is no published Schema with a targetNamespace equal
* to the given URI
*/
@Transactional(readOnly = true)
public XMLSchema getSchema(URI uri) throws NoSuchNamespaceExistsFault {
this.lock.readLock().lock();
try {
XMLSchema result = this.schemaDao.getMaterializedXMLSchemaByTargetNamespace(uri);
if (result == null) {
String description = "No schema is published with given targetNamespace (" + uri + ")";
NoSuchNamespaceExistsFault fault = new NoSuchNamespaceExistsFault();
gov.nih.nci.cagrid.common.FaultHelper helper = new gov.nih.nci.cagrid.common.FaultHelper(fault);
helper.setDescription(description);
LOG.debug("Cannot retrieve requested schema: " + description);
throw (NoSuchNamespaceExistsFault) helper.getFault();
} else {
return result;
}
} finally {
this.lock.readLock().unlock();
}
}
/**
* Return a Collection of URIs representing the targetNamespaces of the
* XMLSchemas which are imported by the XMLSchema identified by the given
* targetNamespace
*
* @param targetNamespace
* the targetNamespace of the desired XMLSchema
* @return a Collection of URIs representing the targetNamespaces of the
* XMLSchemas which are imported by the XMLSchema identified by the
* given targetNamespace
* @throws NoSuchNamespaceExistsFault
* if there is no published Schema with a targetNamespace equal
* to the given URI
*/
@Transactional(readOnly = true)
public Collection<URI> getImportedNamespaces(URI targetNamespace) throws NoSuchNamespaceExistsFault {
XMLSchemaInformation info = this.schemaDao.getByTargetNamespace(targetNamespace);
if (info == null) {
String description = "No schema is published with given targetNamespace (" + targetNamespace + ")";
NoSuchNamespaceExistsFault fault = new NoSuchNamespaceExistsFault();
gov.nih.nci.cagrid.common.FaultHelper helper = new gov.nih.nci.cagrid.common.FaultHelper(fault);
helper.setDescription(description);
LOG.debug("Cannot retrieve imported namespaces of schema: " + description);
throw (NoSuchNamespaceExistsFault) helper.getFault();
}
List<URI> result = new ArrayList<URI>();
Set<XMLSchemaInformation> imports = info.getImports();
for (XMLSchemaInformation importedInfo : imports) {
result.add(importedInfo.getSchema().getTargetNamespace());
}
return result;
}
/**
* Return a Collection of URIs representing the targetNamespaces of the
* XMLSchemas which import the XMLSchema identified by the given
* targetNamespace
*
* @param targetNamespace
* the targetNamespace of the desired XMLSchema
* @return a Collection of URIs representing the targetNamespaces of the
* XMLSchemas which import the XMLSchema identified by the given
* targetNamespace
* @throws NoSuchNamespaceExistsFault
* if there is no published Schema with a targetNamespace equal
* to the given URI
*/
@Transactional(readOnly = true)
public Collection<URI> getImportingNamespaces(URI targetNamespace) throws NoSuchNamespaceExistsFault {
XMLSchema schema = this.schemaDao.getXMLSchemaByTargetNamespace(targetNamespace);
if (schema == null) {
String description = "No schema is published with given targetNamespace (" + targetNamespace + ")";
NoSuchNamespaceExistsFault fault = new NoSuchNamespaceExistsFault();
gov.nih.nci.cagrid.common.FaultHelper helper = new gov.nih.nci.cagrid.common.FaultHelper(fault);
helper.setDescription(description);
LOG.debug("Cannot retrieve importing namespaces of schema: " + description);
throw (NoSuchNamespaceExistsFault) helper.getFault();
}
Collection<XMLSchema> dependingXMLSchemas = this.schemaDao.getDependingXMLSchemas(targetNamespace);
List<URI> result = new ArrayList<URI>();
for (XMLSchema importingSchema : dependingXMLSchemas) {
result.add(importingSchema.getTargetNamespace());
}
return result;
}
@Transactional(readOnly = true)
public XMLSchemaBundle getSchemBundle(URI targetNamespace) throws NoSuchNamespaceExistsFault {
XMLSchemaInformation info = this.schemaDao.getByTargetNamespace(targetNamespace);
if (info == null) {
String description = "No schema is published with given targetNamespace (" + targetNamespace + ")";
NoSuchNamespaceExistsFault fault = new NoSuchNamespaceExistsFault();
gov.nih.nci.cagrid.common.FaultHelper helper = new gov.nih.nci.cagrid.common.FaultHelper(fault);
helper.setDescription(description);
LOG.debug("Cannot retrieve schema and dependencies: " + description);
throw (NoSuchNamespaceExistsFault) helper.getFault();
}
XMLSchemaBundle bundle = new XMLSchemaBundle();
collectSchemasForBundle(bundle, info);
return bundle;
}
private void collectSchemasForBundle(XMLSchemaBundle bundle, XMLSchemaInformation info) {
// make sure we haven't already processed these schema, such as from
// another schemas's import
Set<XMLSchema> schemas = bundle.getXMLSchemas();
if (schemas.contains(info.getSchema())) {
// we've already processed this, so return
return;
}
// add this to the bucket and make sure it is populated
this.schemaDao.materializeXMLSchemaInformation(info);
schemas.add(info.getSchema());
// create importinfo for each imported schema (if any)
if (info.getImports().size() > 0) {
Set<XMLSchemaImportInformation> importInfoSet = bundle.getImportInformation();
// make a new importinfo for this schema
XMLSchemaImportInformation importInfo = new XMLSchemaImportInformation();
importInfo.setTargetNamespace(new XMLSchemaNamespace(info.getSchema().getTargetNamespace()));
assert !importInfoSet.contains(importInfo) : "The bundle should not contain import information about XMLSchema ("
+ info.getSchema().getTargetNamespace() + ") as it did not contain the schema itself.";
// add the collected import set
importInfoSet.add(importInfo);
// recursively process each of the schemas this schema imports
for (XMLSchemaInformation importedInfo : info.getImports()) {
importInfo.getImports().add(new XMLSchemaNamespace(importedInfo.getSchema().getTargetNamespace()));
collectSchemasForBundle(bundle, importedInfo);
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,662
|
El Campeonato Nacional de Interligas de la temporada 2019-20 fue la cuadragésima segunda edición del máximo evento a nivel de selecciones de fútbol de las federaciones del interior de Paraguay. El campeonato es organizado por la Unión del Fútbol del Interior y en esta edición participaron 35 ligas de los 17 departamentos de Paraguay. El torneo, en su etapa interdepartamental, inició el 17 de noviembre de 2019.
La Liga Deportiva de Pastoreo, del Distrito de Pastoreo del Departamento de Caaguazú, se coronó campeona por primera vez en su historia ganando el derecho de disputar el campeonato de la División Intermedia y también la Copa San Isidro de Curuguaty, ante el campeón uruguayo de la Copa Nacional de Selecciones del Interior.
Sistema de competición
Consta de dos fases, en la primera se conforman llaves donde los mejores clasifican a la siguiente fase y la segunda es la fase de eliminación directa.
Primera fase
En la primera fase, los equipos de las 35 ligas fueron divididas en 16 llaves de dos equipos y 1 de tres, ordenados de acuerdo con la proximidad geográfica. Esta fase se disputará por puntos a través de encuentros de ida y vuelta, con excepción de la llave de 3 equipos, donde solo se disputará la ida, a fin de que todos los participantes tengan el mismo número de partidos disputados.
Criterios de Desempate
Si los equipos terminan sus partidos empatados en puntos se tendrá en cuenta la diferencia de gol y de persistir la paridad se ejecutarán penales para definir al equipo mejor ubicado en la llave.
Equipos participantes
Primera fase
Los días en que disputaron las fechas, están señaladas a continuación:
Fecha 1: 17 y 24 de noviembre
Fecha 2: 24 de noviembre y 1 de diciembre
Fecha 3: 4 de diciembre
Llave 1
Llave 2
Llave 3
Llave 4
Llave 5
Llave 6
Llave 7
Llave 8
Llave 9
Llave 10
Llave 11
Llave 12
Llave 13
Llave 14
Llave 15
Llave 16
Llave 17
Segunda fase
Las 18 ligas se agruparon en 9 llaves de 2 ligas cada.
En cada llave se jugaron partidos de ida y vuelta, oficiando cada liga un partido de local y otro de visitante.
Los días en que se disputaron las fechas, están señaladas a continuación:
Fecha 1: 7 y 8 de diciembre.
Fecha 2: 14 y 15 de diciembre.
Llave 1
Llave 2
Llave 3
Llave 4
Llave 5
Llave 6
Llave 7
Llave 8
Llave 9
Tercera fase
Las 9 ligas se agruparon en 3 llaves de 3 ligas cada. En cada llave se jugaron partidos únicos oficiando cada liga un partido de local y otro de visitante. Los días en que se disputaron las fechas, están señaladas a continuación:
Fecha 1: 22 de diciembre
Fecha 2: 29 de diciembre
Fecha 3: 5 de enero
Llave 1
Llave 2
Llave 3
Cuarta fase
Las 6 ligas se agruparon en 2 llaves de 3 ligas cada. En cada llave se jugaron partidos únicos oficiando cada liga un partido de local y otro de visitante. Los días en que se disputaron las fechas, están señaladas a continuación:
Fecha 1: 12 de enero
Fecha 2: 19 de enero
Fecha 3: 26 de enero
Llave 1
Llave 2
Fase final
Semifinal
|golesvisita = {{|}}
|penaltis1 =
|penaltis2 =
|resultado penalti =
|reporte =
}}
Finales
Referencias
Campeonatos de fútbol entre selecciones locales de Paraguay
Fútbol en 2019
Fútbol en 2020
Campeonato Nacional de Interligas
Paraguay en 2019
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,931
|
Q: nesting async/await javascript Is this valid async/await logic: calling try/catch from within another try/catch?
//within async function...
try {
await asyncFunctionFirst();
} catch (errFirst) {
try {
await asyncFunctionSecond();
} catch (errSecond) {
// return errSecond response
}
//return errFirst response
}
//return response ok
A: Yes, it's valid syntax. However I would not write it like this. Rather use
try {
await asyncFunctionFirst();
return …; // response ok
} catch (errFirst) {
try {
await asyncFunctionSecond();
return …; // errFirst response
} catch (errSecond) {
return …; // errSecond response
}
}
(if you would also want to catch exceptions from the … parts - if not, your version is fine, for alternatives see here)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,204
|
package com.netflix.spinnaker.front50.model;
import static net.logstash.logback.argument.StructuredArguments.value;
import com.amazonaws.auth.policy.Condition;
import com.amazonaws.auth.policy.Policy;
import com.amazonaws.auth.policy.Principal;
import com.amazonaws.auth.policy.Resource;
import com.amazonaws.auth.policy.Statement;
import com.amazonaws.auth.policy.actions.SQSActions;
import com.amazonaws.services.sns.AmazonSNS;
import com.amazonaws.services.sns.model.ListTopicsResult;
import com.amazonaws.services.sns.model.Topic;
import com.amazonaws.services.sqs.AmazonSQS;
import com.amazonaws.services.sqs.model.CreateQueueRequest;
import com.amazonaws.services.sqs.model.Message;
import com.amazonaws.services.sqs.model.ReceiptHandleIsInvalidException;
import com.amazonaws.services.sqs.model.ReceiveMessageRequest;
import com.amazonaws.services.sqs.model.ReceiveMessageResult;
import java.util.Collections;
import java.util.HashMap;
import java.util.List;
import javax.annotation.PreDestroy;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
/**
* Encapsulates the lifecycle of a temporary queue.
*
* <p>Upon construction, an SQS queue will be created and subscribed to the specified SNS topic.
* Upon destruction, both the SQS queue and SNS subscription will be removed.
*/
public class TemporarySQSQueue {
private final Logger log = LoggerFactory.getLogger(TemporarySQSQueue.class);
private final AmazonSQS amazonSQS;
private final AmazonSNS amazonSNS;
private final TemporaryQueue temporaryQueue;
public TemporarySQSQueue(
AmazonSQS amazonSQS, AmazonSNS amazonSNS, String snsTopicName, String instanceId) {
this.amazonSQS = amazonSQS;
this.amazonSNS = amazonSNS;
String sanitizedInstanceId = getSanitizedInstanceId(instanceId);
String snsTopicArn = getSnsTopicArn(amazonSNS, snsTopicName);
String sqsQueueName = snsTopicName + "__" + sanitizedInstanceId;
String sqsQueueArn =
snsTopicArn.substring(0, snsTopicArn.lastIndexOf(":") + 1).replace("sns", "sqs")
+ sqsQueueName;
this.temporaryQueue = createQueue(snsTopicArn, sqsQueueArn, sqsQueueName);
}
List<Message> fetchMessages() {
ReceiveMessageResult receiveMessageResult =
amazonSQS.receiveMessage(
new ReceiveMessageRequest(temporaryQueue.sqsQueueUrl)
.withMaxNumberOfMessages(10)
.withWaitTimeSeconds(1));
return receiveMessageResult.getMessages();
}
void markMessageAsHandled(String receiptHandle) {
try {
amazonSQS.deleteMessage(temporaryQueue.sqsQueueUrl, receiptHandle);
} catch (ReceiptHandleIsInvalidException e) {
log.warn(
"Error deleting message, reason: {} (receiptHandle: {})",
e.getMessage(),
value("receiptHandle", receiptHandle));
}
}
@PreDestroy
void shutdown() {
try {
log.debug(
"Removing Temporary S3 Notification Queue: {}",
value("queue", temporaryQueue.sqsQueueUrl));
amazonSQS.deleteQueue(temporaryQueue.sqsQueueUrl);
log.debug(
"Removed Temporary S3 Notification Queue: {}",
value("queue", temporaryQueue.sqsQueueUrl));
} catch (Exception e) {
log.error(
"Unable to remove queue: {} (reason: {})",
value("queue", temporaryQueue.sqsQueueUrl),
e.getMessage(),
e);
}
try {
log.debug(
"Removing S3 Notification Subscription: {}", temporaryQueue.snsTopicSubscriptionArn);
amazonSNS.unsubscribe(temporaryQueue.snsTopicSubscriptionArn);
log.debug("Removed S3 Notification Subscription: {}", temporaryQueue.snsTopicSubscriptionArn);
} catch (Exception e) {
log.error(
"Unable to unsubscribe queue from topic: {} (reason: {})",
value("topic", temporaryQueue.snsTopicSubscriptionArn),
e.getMessage(),
e);
}
}
private String getSnsTopicArn(AmazonSNS amazonSNS, String topicName) {
ListTopicsResult listTopicsResult = amazonSNS.listTopics();
String nextToken = listTopicsResult.getNextToken();
List<Topic> topics = listTopicsResult.getTopics();
while (nextToken != null) {
listTopicsResult = amazonSNS.listTopics(nextToken);
nextToken = listTopicsResult.getNextToken();
topics.addAll(listTopicsResult.getTopics());
}
return topics.stream()
.filter(t -> t.getTopicArn().toLowerCase().endsWith(":" + topicName.toLowerCase()))
.map(Topic::getTopicArn)
.findFirst()
.orElseThrow(
() ->
new IllegalArgumentException("No SNS topic found (topicName: " + topicName + ")"));
}
private TemporaryQueue createQueue(String snsTopicArn, String sqsQueueArn, String sqsQueueName) {
String sqsQueueUrl =
amazonSQS
.createQueue(
new CreateQueueRequest()
.withQueueName(sqsQueueName)
.withAttributes(
Collections.singletonMap(
"MessageRetentionPeriod", "60")) // 60s message retention
)
.getQueueUrl();
log.info("Created Temporary S3 Notification Queue: {}", value("queue", sqsQueueUrl));
String snsTopicSubscriptionArn =
amazonSNS.subscribe(snsTopicArn, "sqs", sqsQueueArn).getSubscriptionArn();
Statement snsStatement =
new Statement(Statement.Effect.Allow).withActions(SQSActions.SendMessage);
snsStatement.setPrincipals(Principal.All);
snsStatement.setResources(Collections.singletonList(new Resource(sqsQueueArn)));
snsStatement.setConditions(
Collections.singletonList(
new Condition()
.withType("ArnEquals")
.withConditionKey("aws:SourceArn")
.withValues(snsTopicArn)));
Policy allowSnsPolicy = new Policy("allow-sns", Collections.singletonList(snsStatement));
HashMap<String, String> attributes = new HashMap<>();
attributes.put("Policy", allowSnsPolicy.toJson());
amazonSQS.setQueueAttributes(sqsQueueUrl, attributes);
return new TemporaryQueue(snsTopicArn, sqsQueueArn, sqsQueueUrl, snsTopicSubscriptionArn);
}
static String getSanitizedInstanceId(String instanceId) {
return instanceId.replaceAll("[^\\w\\-]", "_");
}
protected static class TemporaryQueue {
final String snsTopicArn;
final String sqsQueueArn;
final String sqsQueueUrl;
final String snsTopicSubscriptionArn;
TemporaryQueue(
String snsTopicArn,
String sqsQueueArn,
String sqsQueueUrl,
String snsTopicSubscriptionArn) {
this.snsTopicArn = snsTopicArn;
this.sqsQueueArn = sqsQueueArn;
this.sqsQueueUrl = sqsQueueUrl;
this.snsTopicSubscriptionArn = snsTopicSubscriptionArn;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,578
|
"""Machine translation datasets."""
from lit_nlp.api import dataset as lit_dataset
from lit_nlp.api import types as lit_types
import six
import tensorflow_datasets as tfds
JsonDict = lit_types.JsonDict
Spec = lit_types.Spec
class WMT14Data(lit_dataset.Dataset):
"""WMT '14 machine-translation data, via TFDS."""
def __init__(self, version='fr-en', reverse=False):
lang_keys = version.split('-')
assert len(lang_keys) == 2
if reverse:
source_key = lang_keys[1]
target_key = lang_keys[0]
else:
source_key = lang_keys[0]
target_key = lang_keys[1]
# Pre-load dataset
ds_name = 'wmt14_translate/' + version
ds = tfds.load(ds_name, download=True, try_gcs=True, split='validation')
# TODO(lit-team): don't load the whole dataset if only using a few examples
ds_np = list(tfds.as_numpy(ds))
# pylint: disable=g-complex-comprehension
self._examples = [{
'source': six.ensure_text(d[source_key]),
'source_language': source_key,
'target': six.ensure_text(d[target_key]),
'target_language': target_key,
} for d in ds_np]
# pylint: enable=g-complex-comprehension
def spec(self) -> Spec:
return {
'source': lit_types.TextSegment(),
'source_language': lit_types.CategoryLabel(),
'target': lit_types.TextSegment(),
'target_language': lit_types.CategoryLabel(),
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,520
|
\section{Introduction}
Our investigation is inspired by recent approaches to a formal concept of \textit{intuitionistic knowledge}, i.e. formalizations of knowledge that are in accordance with intuitionistic or constructive reasoning. In particular, we consider Intuitionistic Epistemic Logic $\mathit{IEL}$ introduced by Artemov and Protopopescu \cite{artpro} where intuitionistic knowledge is explained as the product of \textit{verification}. Some principles of that approach are adopted by Lewitzka \cite{lewigpl} and incorporated into a family of modal systems $L3$--$L5$ originally introduced in \cite{lewjlc2}. The resulting epistemic logics are systems for the reasoning about intuitionistic truth, i.e. proof, and a kind of intuitionistic knowledge based on an informal notion of \textit{justification} (cf. \cite{lewapal}). In the present paper, we extend modal logic $L5$ with the purpose of formalizing a new concept of constructive knowledge which, in our multi-agent setting, relies on the intuition that agent $i$ knows $\varphi$ if $i$ has gained \textit{access to a proof} of $\varphi$. This paradigma also admits a concept of common knowledge that we adopt from \cite{lewsl} and interpret here constructively. The basic motivation behind this access-based concept is the idea that \textit{to know} $\varphi$, in a constructive sense, means something like \textit{to understand, to become aware of, to effect}, ... a proof of proposition $\varphi$ (or, if one prefers, a solution to problem $\varphi$), and that the agent possibly has to spent some effort and ressources to execute this activity. We feel that the intuition of \textit{finding access to} a proof of $\varphi$ captures those ideas in some abstract way. In the following, we shortly discuss the above mentioned concepts of intuitionistic knowledge found in the literature and then present the notion of access-based knowledge. In the subsequent sections, we shall see that all three concepts can be formalized and studied within a unifying framework of algebraic (and relational) semantics.
\subsection{Verification-based knowledge: Artemov and Protopopescu 2016}
Artemov and Protopopescu \cite{artpro} propose an intuitionistic concept of knowledge which is in accordance with the proof-reading semantics of intuitionistic propositional logic $\mathit{IPC}$, i.e. with well-known Brouwer-Heyting-Kolmogorov (BHK) interpretation. Knowledge is viewed as the product of a \textit{verification}. The intuitive notion of verification generalizes \textit{proof} as intuitionistic truth in the sense that a proof is `the strictest kind of a verification'. $K\varphi$ means that it is verified that proposition $\varphi$ holds intuitionistically, i.e. there is evidence that $\varphi$ has a proof (even if a concrete proof is not delivered nor specified in the process of verification). Under this interpretation, the following principles hold and represent an axiomatization of $\mathit{IEL}$ in the language of $\mathit{IPC}$ augmented with knowledge operator $K$:\\
\noindent (i) all schemes of theorems of $\mathit{IPC}$\\
(ii) $K(\varphi\rightarrow\psi)\rightarrow (K\varphi\rightarrow K\psi)$ (distribution of knowledge)\\
(iii) $\varphi\rightarrow K\varphi$ (co-reflection)\\
(iv) $K\varphi\rightarrow\neg\neg\varphi$ (intuitionistic reflection)\\
Note that (iv) reads `Known (i.e. verified) propositions cannot be proved to be false'. Since the process of verification, in general, does not deliver a concrete proof, the classical reflection principle (factivity of knowledge) $K\varphi\rightarrow\varphi$ is not valid. Modus Ponens is the unique inference rule of the resulting deductive system. It is shown in \cite{artpro} that $\mathit{IEL}$ is sound and complete w.r.t. a possible-worlds semantics. Although the notion of verification is only intuitively given in $\mathit{IEL}$, it is shown by Protopopescu \cite{pro} that also an arithmetical interpretation can be provided.
\subsection{Adopting a justification-based view: Lewitzka 2017, 2019}
Logic $L5$ was introduced in \cite{lewjlc2} together with a hierarchy $L\subsetneq L3 \subsetneq L4 \subsetneq L5$ of classical Lewis-style modal logics for the reasoning about intuitionistic truth, i.e. proof. A formula $\square\varphi$ reads `$\varphi$ is proved (i.e. $\varphi$ has an actual proof)'. Semantics is given by a class of Heyting algebras where intuitionistic truth is represented by the top element of the Heyting lattice, and classical truth is modeled by a designated ultrafilter. Formulas of the form
\begin{equation}\label{20}
\square\varphi\leftrightarrow (\varphi\equiv\top)
\end{equation}
are theorems and express that $\square$ is a predicate for intuitionistic truth: $\square\varphi$ is classically true iff $\varphi$ holds intuitionistically (i.e. $\varphi$ denotes the top element of the underlying Heyting algebra). An essential feature is the definability of an identity connective by $\varphi\equiv\psi := \square(\varphi\leftrightarrow\psi)$ such that the identity axioms of R. Suszko's basic non-Fregean logic, the Sentential Calculus with Identity $\mathit{SCI}$ (cf. \cite{blosus}), are satisfied:\footnote{In $\mathit{SCI}$, the identity connective is a primitive symbol of the object language.}\\
\noindent (i) $\varphi\equiv\varphi$\\
(ii) $(\varphi\equiv\psi)\rightarrow (\varphi\leftrightarrow\psi)$\\
(iii) $\varphi\equiv\psi\rightarrow \chi [x:=\varphi]\equiv \chi[x:=\psi]$.\footnote{$\varphi[x:=\psi]$ is the result of substituting $\psi$ for any occurrence of variable $x$ in $\varphi$.}\\
$\varphi\equiv\psi$ reads `$\varphi$ and $\psi$ have the same meaning (denotation, \textit{Bedeutung})'. We refer to the axioms (i)--(iii) as the axioms of propositional identity, and particularly to (iii) as the Substitution Principle (SP). Since these axioms are theorems of our modal systems, we are dealing with specific classical non-Fregean logics which essentially means that the `Fregean Axiom' $(\varphi\leftrightarrow\psi)\rightarrow (\varphi\equiv\psi)$ does not hold, i.e. formulas with the same truth value may have different meanings. That is, the denotation of a formula is generally more than a truth value: it is a proposition, i.e. an element of a given model. Actually, all our logics extending $L5$ are specific non-Fregean theories with the property that for any formulas $\varphi,\psi$: $\varphi\equiv\psi$ is a theorem iff $\varphi\leftrightarrow\psi$ is intuitionistically valid, i.e. valid in standard BHK semantics extended by proof-interpretation clauses for additional operators. Thus, in any model, intuitionistically equivalent formulas denote the same proposition whereas formulas such as $\varphi$ and $\neg\neg\varphi$ have, in general, different meanings. This determines, in a sense, the `degree of intensionality' of our logics. The highest degree of this kind of intensionality is achieved in Suszko's $\mathit{SCI}$ where for all formulas $\varphi$, $\psi$ it holds that $\varphi\equiv\psi$ is a theorem iff $\varphi = \psi$. \\
A further feature of our modal systems is that they contain a copy of $\mathit{IPC}$ and thus combine $\mathit{IPC}$ with classical propositional logic $\mathit{CPC}$ in the following precise sense. If $\Phi\cup\{\varphi\}$ is a set of formulas in the propositional language of $\mathit{IPC}$, then
\begin{equation}\label{25}
\Phi\vdash_{IPC}\varphi\Leftrightarrow\square\Phi\vdash_{L}\square\varphi,
\end{equation}
where $\square\Phi :=\{\square\psi\mid\psi\in\Phi\}$. In particular, for any propositional formula $\varphi$, $\varphi$ is a theorem of $\mathit{IPC}$ iff $\square\varphi$ is a theorem of system $L$. That is, $\varphi\mapsto\square\varphi$ is a `translation', actually an embedding, of $\mathit{IPC}$ into classical modal logic $\mathit{L}$, cf. \cite{lewjlc2} ($L$ can be replaced here with any member of the hierarchy $L\subseteq L3 \subseteq L4 \subseteq L5 \subseteq$ `epistemic extensions'). Obviously, this embedding of $\mathit{IPC}$ into classical modal systems is simpler than the well-known standard translation of $\mathit{IPC}$ into modal logic $S4$ due to G\"odel. We argued in \cite{lewapal} that the $S5$-style system $L5$ is an adequate system for reasoning about proof showing that it is complete w.r.t. extended BHK semantics, i.e. w.r.t. intuitionistic reasoning. This semi-formal result is formally confirmed by soundness and completeness of $L5$ w.r.t. a relational semantics based on intuitionistic general frames, cf. \cite{lewapal}. For this reason, we consider here $L5$ as the basis of our epistemic extensions. In \cite{lewigpl}, we extended $L5$ to the epistemic logic $EL5$ taking into account principles coming from $\mathit{IEL}$. $EL5$ is further studied in \cite{lewapal} where its algebraic semantics is complemented by relational semantics. $EL5$ can be axiomatized in the following way: \\
\noindent (INT) All formulas which have the form of an $\mathit{IPC}$-tautology\\
(i) $\square(\varphi\vee\psi)\rightarrow(\square\varphi\vee\square\psi)$\\
(ii) $\square\varphi\rightarrow\varphi$\\
(iii) $\square(\varphi\rightarrow\psi)\rightarrow(\square\varphi\rightarrow\square\psi)$ \\
(iv) $\square\varphi\rightarrow\square\square\varphi$\\
(v) $\neg\square\varphi\rightarrow\square\neg\square\varphi$\\
(vi) $K\varphi\rightarrow\neg\neg\varphi$ (intuitionistic reflection)\\
(vii) $K(\varphi\rightarrow\psi)\rightarrow (K\varphi\rightarrow K\psi)$\\
(viii) $\square\varphi\rightarrow \square K\varphi$ (weak co-reflection)\footnote{Replacing this scheme with $\square\varphi\rightarrow K\varphi$ results in a deductively equivalent system.} \\
(TND) $\varphi\vee\neg\varphi$ (\textit{tertium non datur})\\
The reference rules are Modus Ponens (MP) and Intuitionistic Axiom Necessitation (AN): `If $\varphi$ is an intuitionistically acceptable axiom, i.e. any axiom distinct from (TND), then infer $\square\varphi$.' Actually, we argued in \cite{lewapal} that all schemes (i)--(viii) above are intuitionistically acceptable, i.e. sound w.r.t. BHK semantics extended by constructive interpretations of the modal and epistemic operators, respectively. Logic $L5$ is axiomatized by (INT), (i)--(v) and (TND) along with the same inference rules where, again, (AN) applies to all axioms but (TND).
Notice that in the more expressive modal language, we are able to weaken the original axiom of co-reflection from $\mathit{IEL}$. Of course, the resulting formalization of knowledge then no longer captures the notion of verification as axiomatized in $\mathit{IEL}$. Instead, we proposed in \cite{lewapal} to consider an informal notion of \textit{justification} or \textit{reason} to motivate the new formalization.\footnote{There is a family of sophisticated Justification Logics found in the literature (see, e.g., \cite{artfit} for an overview) where justifications along with operations on them are explicitly formalized. These aspects are not contained in logic $EL5$. Instead, the notion of justification is understood in a primitive and completely informal and unspecified way.} Accordingly, we suppose that $\varphi$ is known by the agent if he has an epistemic justification, reason for $\varphi$. What the agent recognizes or accepts as an epistemic justification depends essentially from its internal conditions, reasoning capabilities, convictions, etc. Contrary to the more objective and agent-invariant concept of verification, the notion of justification is agent-dependent. We postulate that the agent recognizes at least all \textit{actual proofs}, i.e. all effected constructions, as epistemic justifications. This ensures the validity of weak co-reflection (viii). However, a possible proof as a potential, non-effected construction is, in general, not accepted by the agent as a reason for his knowledge. Full co-reflection in its original form $\varphi\rightarrow K\varphi$ must be rejected under this justification-based view.
Of course, a justification does not constitute a proof: classical reflection (factivity of knowledge), $K\varphi\rightarrow\varphi$, must be rejected. Nevertheless, if the agent has an epistemic justification of proposition $\varphi$, then $\varphi$ cannot be proved to be false, i.e. $\neg\neg\varphi$ holds intuitionistically. Therefore, intuitionistic reflection (vi) from $\mathit{IEL}$ is adopted. We also assume that if the agent has justifications for $\varphi\rightarrow\psi$ and for $\varphi$, respectively, then he obtains a justification for $\psi$.\footnote{This is an established principle in Justification Logics with a precise formalization, cf. \cite{artfit}.} Thus, we adopt distribution of knowledge, axiom (vii) above, too.
\subsection{Access-based knowledge}
We propose here a concept of constructive knowledge which relies on the intuition that an agent knows a proposition $\varphi$ if he \textit{has found an access to a proof} of $\varphi$. In some specific context, `to find an access to a proof' may be interpreted as `to understand a proof', `to become aware of a proof', etc. We consider a multi-agent scenario based on the following ontological assumptions (see also \cite{lewapal}):
We are given a \textit{universe of possible proofs}, i.e. a universe of potential constructions, mathematical possibilities. The \textit{creative subject}\footnote{This term was used by Brouwer and we adopt it here four our short, informal explanation.} establishes the intuitionistic truth of propositions by effecting constructions. These effected constructions are the \textit{actual proofs} among the possible proofs, i.e., the established intuitionistic truths. The universe of possible proofs exists objectively and can be explored by reasoning subjects.\footnote{Since we are reasoning about proof in classical logic, i.e. from a classical point of view, we adopt a platonist perspective which we combine with the constructive approach. Notice that the BHK interpretation of implication implicitly contains a universal quantification: `A proof of $\varphi\rightarrow\psi$ consists in a construction $u$ such that \textit{for all proofs} $t$: if $t$ is a proof of $\varphi$, then $u(t)$ is a proof of $\psi$'. The range of that universal quantifier is the given universe of possible proofs.} A possible proof may be a hypothetical, potential construction, not necessarily effected by the creative subject. It can be conceived as a set of \textit{conditions} on a construction rather than the construction itself (cf. \cite{att1, att2}). We expect that these conditions are not in conflict with effected constructions, i.e. they are `consistent' with the actual proofs. There is a set $I=\{1,...,N\}$ of $N\ge 1$ agents distinct from the creative subject. Each agent can obtain knowledge by accessing possible proofs, where `accessing a proof' is a constructive procedure or activity that any agent is able to carry out, possibly by spending some effort and resources. A (possible) proof of the proposition ``agent $i$ knows $\varphi$" is given by a (possible) proof of $\varphi$ along with an access to that proof found by $i$. Actual proofs, i.e. the constructions effected by the creative subject, are immediately available and thus trivially accessible. That is, each agent's knowledge comprises at least intuitionistic truth established by the creative subject. Finally, there is a designated subset of possible proofs that determines the facts, i.e. the `classical truths'.
By the proof predicate on the object language, we may explicitly distinguish between actual proofs and non-effected, possible proofs. As before, $\square\varphi$ reads classically `$\varphi$ has an actual proof (i.e. $\varphi$ is proved)', and $\Diamond\varphi :=\neg\square\neg\varphi$ reads `$\varphi$ has a possible proof'.\footnote{Of course, $\square\varphi\rightarrow\Diamond\varphi$ is a theorem of the Lewis-style systems $L\subseteq L3\subseteq L4\subseteq L5$, cf. \cite{lewapal}.} In \cite{lewapal}, we extended standard BHK interpretation by the following clause for the modal operator:
\begin{itemize}
\item A proof of $\square\varphi$ consists in presenting an actual proof of $\varphi$.\footnote{We assume that the \textit{presentation} of an actual proof of $\varphi$ involves some proof-checking procedure which depends only from the given actual proof itself and from $\varphi$.}
\end{itemize}
Since actual proofs are effected, available constructions, every agent $i\in I$ has the same immediate, trivial access to them. We denote this unique, trivial access by $s_0$. It might be regarded as an access created by the `empty action' (no effort must be spent). On the other hand, if some proof $t$ is accessed via $s_0$, then $t$ must be an actual proof. That is, we postulate the following:
\begin{itemize}
\item The proofs accessed via $s_0$ (by any agent) are exactly the actual proofs.
\end{itemize}
We establish the following proof-interpretation clause for the knowledge operator:
\begin{itemize}
\item A proof of $K_i\varphi$ is a tuple $(s,t)$, where $s$ is an access, found by agent $i$, to a proof $t$ of proposition $\varphi$.
\end{itemize}
If $(s,t)$ is a proof of $K_i\varphi$, then we write also $(si,t)$ instead of $(s,t)$ in order to emphasize the involved agent. Note that for any $i\in I$ and any proposition $\varphi$, $(s_0,t)$ is a proof of $K_i\varphi$ iff $t$ is an actual proof of $\varphi$. Then the principles $\square K_i\varphi\rightarrow \square\varphi$ and $\square\varphi\rightarrow \square K_i\varphi$ are intuitionistically acceptable. In fact, given the presentation of an actual proof of $\square K_i\varphi$, that actual proof must be of the form $(s,t)$, where $s$ is an access to the actual proof $t$ of $\varphi$ (thus $s=s_0$ is the trivial access). The construction that maps $(s,t)$ to $t$ yields a proof of the former principle. This also shows that any actual proof of a formula $K_i\varphi$ is of the form $(s_0,t)$, where $t$ is an actual proof of $\varphi$. Now, one recognizes that the construction that for any actual proof $t$ of $\varphi$ returns the tuple $(s_0,t)$ gives rise to a proof of the latter principle. Consequently, $\square(\square K_i\varphi\rightarrow \square\varphi)$ and $\square(\square\varphi\rightarrow \square K_i\varphi)$ are sound w.r.t. extended BHK semantics, i.e. $\square K_i\varphi\equiv \square\varphi$. Of course, $K_i\varphi$ and $\varphi$ denote generally different propositions.
It is clear that from a (possible) proof $(si,t)$ of $K_i\varphi$, the (possible) proof $t$ of $\varphi$ can be extracted. This procedure yields a proof of $K_i\varphi\rightarrow\varphi$. Hence, classical reflection (factivity of knowledge) is intuitionistically acceptable. On the other hand, \textit{intuitionistic reflection} $\varphi\rightarrow K_i\varphi$, an axiom of verification-based knowledge, must be rejected (for similar reasons as it is rejected in the justification-based approach discussed above). In fact, given a (possible) proof $t$ of $\varphi$, we cannot expect that agent $i$ has gained any access to $t$, there is no logical evidence for such an access. The access-based approach validates the following disjunction property of knowledge: $K_i(\varphi\vee\psi)\rightarrow (K_i\varphi\vee K_i\psi)$. A BHK proof derives immediately from the clauses for $K_i$ and disjunction. We postulate the following two \textit{Combination Principles}:\\
(C1) If $s$ is an access to proof $t$, and $s'$ is an access to proof $u$, and $t$ is a construction converting $u$ into the proof $t(u)$, then any agent which has gained both accesses $s$ and $s'$ is able to create a combined access $s+s'$ to proof $t(u)$. We assume that $s_0 + s=s=s + s_0$, for any access $s$ and the trivial access $s_0$.\\
(C2) If $t$ is an access to proof $u$, and $s$ is an access to proof $(t,u)$, then a composed access $s\circ t$ to proof $u$ can be found. That is, if $(tj,u)$ is a proof and $(si,(tj,u))$ is a proof, then $((s\circ t)i, u)$ is a proof. We assume that $s\circ s = s$, for any access $s$.\\
(C1) warrants intuitionistic validity of $K_i(\varphi\rightarrow\psi)\rightarrow (K_i\varphi\rightarrow K_i\psi)$. A proof is given by the construction that for any proof $(s,t)$ of $K_i(\varphi\rightarrow\psi)$ returns the function mapping any proof $(s',u)$ of $K_i\varphi$ to the proof $(s+s',t(u))$ of $K_i\psi$. Principle (C2) warrants the following intuitive epistemic law: `If $i$ knows that $j$ knows $\varphi$, then $i$ knows $\varphi$'. That is, $K_i K_j\varphi\rightarrow K_i\varphi$ is intuitionistically acceptable.\\
As usual, the fact that everyone in group $G=\{i_1,...,i_k\}$ knows $\varphi$ is expressed by the formula $E_G\varphi := K_{i_1}\varphi\wedge ... \wedge K_{i_k}\varphi$. Recall that $E_G^n\varphi$ is recursively defined by $E_G^0\varphi :=\varphi$ and $E_G^{k+1}\varphi := E_G E^k_G\varphi$, for $k\ge 0$. Also recall that knowledge distributes over conjunction. The concept of `$\varphi$ is common knowledge among the agents of group $G$', notation: $C_G\varphi$, is often informally defined as follows:
\begin{equation}\label{50}
C_G\varphi \Leftrightarrow \bigwedge_{n\in\mathbb{N}} E_G^n\varphi
\end{equation}
That is, $C_G\varphi$ is true iff the infinitely many formulas $\varphi$, $E_G\varphi$, $E_G^2\varphi$, ... are true. However, standard formalizations found in the literature (see, e.g., \cite{fag, mey}) involve additionally properties that go beyond that basic intuition. In fact, standard possible worlds semantics of epistemic logic with common knowledge validates also the following introspection principle as a theorem of standard axiomatizations:
\begin{equation}\label{60}
C_G\varphi\rightarrow C_G C_G\varphi \text{ (introspection of common knowledge)}
\end{equation}
But if we take \eqref{50} seriously and understand common knowledge as such an infinite conjunction, then principle \eqref{60} does not necessarily follow. Of course, in many `natural' situations, such as the popular example of the \textit{moody children} (cf. \cite{fag, mey}), common knowledge arises at once after some finite amount of communication steps, and one may regard \eqref{60} as an evident principle in those cases. However, one may construct examples where common knowledge is actually attained in an infinite process of communication steps. In \cite{fag}, p. 416, for instance, an unrealistic version of the well-known coordinated-attack problem is discussed. If the messenger between the two generals is able to double his speed every time around, and his first journey takes one hour, then it follows that after exactly two hours he has visited both camps an infinite number of times delivering each time the message ``attack at down" sent from the other general, and the generals will finally be able to carry out a coordinated attack because they have attained common knowledge. We may state that after the two hours of infinitely many journeys, each of the two generals knows that $E_G^n\varphi$, for every natural $n$ (where $\varphi$ is the delivered message). However, we cannot conclude that the generals do know the infinite collection of facts $\{E_G^n\varphi\mid n\in\mathbb{N}\}$ as a single proposition $\bigwedge_{n\in\mathbb{N}} E_G^n\varphi$. In fact, new knowledge is attained after each finite number of communication steps between the two agents, but there is no further communication beyond the limit step. This example shows that if $C_G\varphi$ is attained (possibly by an infinite number of steps), we cannot expect in general that also $K_i C_G\varphi$ holds for $i\in G$. Thus, principle \eqref{60} is not valid. However, under the assumption that in all known natural situations where common knowledge arises, it arises in a similar way as in the example of \textit{the moody children}, we may accept \eqref{60} as an additional axiom. Since our modeling deviates from the possible worlds approach, we are able to treat both versions of common knowledge: the basic one which is given by an infinite conjunction in the form of \eqref {50}, and the stronger version which extends the basic version by the introspection principle \eqref{60}. The axiomatization and semantic modeling of the basic version of common knowledge is adopted from \cite{lewsl} where it was originally developed in a general, classical non-Fregean setting. We add here principle \eqref{60} and provide a constructive, access-based interpretation which proves to be sound w.r.t. our extended BHK semantics. We are not able to represent the infinite conjunction of \eqref{50} in our object language by a fixed-point axiom or similar solutions working in standard possible worlds semantics. Instead, we propose a semantic characterization by means of \textit{intended models}, a solution that we shall discuss in some detail in the last section.
\begin{definition}\label{90}
Let $G$ be a group and let $t$ be a (possible) proof. We call an access $s$ to $t$ a common access in $G$, or a $G$-common access, if the following hold:\\
\noindent (a) all agents of $G$ have gained the same access $s$ to $t$\\
(b) $s$ is self-referential in $G$, i.e. for any $i,j\in G$ and any proof $u$, if $(si,u)$ is a proof, then so is $(sj,(si,u))$.
\end{definition}
The next result shows that the particular choice of proof $t$ in Definition \ref{90} is not relevant.
\begin{lemma}\label{91}
Let $s$ be a $G$-common access to $t$. If some $i\in G$ has access $s$ to some proof $u$, then $s$ is also a $G$-common access to proof $u$.
\end{lemma}
\begin{proof}
If $i\in G$ has the access $s$ to proof $u$, then $(si,u)$ is a proof. Since $s$ is a $G$-common access, item (b) of Definition \ref{90} implies that $(sj,(si,u))$ is a proof, for any $j\in G$. By composition principle (C2) above, $((s\circ s)j,u)$ is a proof for any $j\in G$. Also by (C2), $s\circ s = s$. Thus, $(sj,u)$ is a proof, for all $j\in G$. That is, $s$ is a $G$-common access to $u$.
\end{proof}
\begin{lemma}\label{96}
For any group $G$, the trivial access $s_0$ is a $G$-common access (to any actual proof).
\end{lemma}
\begin{proof}
Recall that the proofs accessed via $s_0$ (by any agent) are exactly the actual proofs. Thus, all agents have access $s_0$ to any actual proof. If $(s_0,t)$ is a proof, for some proof $t$, then $t$ and $(s_0,t)$ must be actual proofs. Thus, $(s_0,t)$ can be accessed via $s_0$ (by any agent). By Definition \ref{90}, $s_0$ is a $G$-common access, for any $G$.
\end{proof}
A proof-interpretation clause for $C_G\varphi$ must take into account the respective version of common knowledge. Let us first consider the basic version of common knowledge given by the infinite conjunction expressed in \eqref{50} above. We consider two proposals:
\begin{itemize}
\item A `proof' of $C_G\varphi$ consists in an infinite sequence of proofs $(t_n)_{n\in\mathbb{N}}$ such that $t_n$ is a proof of $E^n_G\varphi$.
\item A `proof' of $C_G\varphi$ consists in a proof $t$ of $\varphi$ together with a construction that for a given proof of $E^n_G\varphi$, $n\ge 0$, returns a proof of $E_G E^n_G\varphi = E^{n+1}_G\varphi$.
\end{itemize}
Unfortunately, both clauses are problematic from a constructivist point of view. The first one describes a proof as an infinite object. The second one gives an inductive definition of a construction that possibly needs an infinite amount of time to produce all the different proofs of the infinitely many formulas $E^n_G\varphi$, $n\ge 0$. It seems that any approach to the basic intuition \eqref{50} of common knowledge (without introspection) involves some form of infinity that makes a constructive treatment hard or impossible. Therefore, we will focus on the stronger, introspective version of common knowledge which can be constructively described by the following simple and finitary clause:
\begin{itemize}
\item A proof of $C_G\varphi$ is a tuple $(s,t)$, where $t$ is a proof of $\varphi$ and $s$ is a $G$-common access to $t$.
\end{itemize}
\begin{example}\label{92}
We consider the introspective version of common knowledge. Imagine a math lecture. The lecturer writes a proof of a theorem $\varphi$ on the blackboard. It is clear that at the end of the lecture, there is common knowledge of $\varphi$ in the group $G$ of students who listened the lecture. We interpret the situation constructively in the following way. Let $s$ be the lecture and let $t$ be the proof of $\varphi$ written on the blackboard. Then all students of group $G$ share the same access $s$ to $t$. Hence, condition (a) of Definition \ref{90} is satisfied. During the lecture, the students can see each other listening the lecture. Thus, every student $j\in G$ has access via $s$ to the proof $(si,t)$ of $K_i\varphi$, for any $i\in G$. This yields proofs $(sj,(si,t))$ of $K_j K_i\varphi$, for any $j,i\in G$, and so on ... . Of course, the same arguments apply to any other statement $\psi$ with proof $u$ presented in lecture $s$. Then $s$ is self-referential in $G$ in the sense of Definition \ref{90}, i.e. condition (b) holds true. Thus, $s$ is a $G$-common access to $t$, and $(s,t)$ is a proof of $C_G\varphi$ in the sense of the clause for $C_G\varphi$ above.
\end{example}
Next, we present some principles of common knowledge which are sound w.r.t. extended BHK interpretation.\\
$\square\varphi\rightarrow \square C_G\varphi$. Every agent has the trivial access $s_0$ to an actual proof $t$ of $\varphi$. We already saw that $s_0$ is self-referential in the group of all agents $I$. Consequently, the function that maps any actual proof $t$ of $\varphi$ to the actual proof $(s_0,t)$ of $C_G\varphi$ gives rise to an actual proof of $\square\varphi\rightarrow \square C_G\varphi$.\\
$C_G\varphi\rightarrow C_G K_i\varphi$, $i\in G$. Suppose $(s,t)$ is a proof of $C_G\varphi$. Then, in particular, $(si,t)$ is a proof of $K_i\varphi$. Since $s$ is self-referential in $G$, $(sj,(si,t))$ is a proof, for every $j\in G$. Thus, $(s,(si,t))$ is a proof of $C_G K_i\varphi$. Then the mapping $(s,t)\mapsto (s,(si,t))$ is an effected construction, i.e. actual proof, for $C_G\varphi\rightarrow C_G K_i\varphi$. \\
$C_G\varphi\rightarrow C_G C_G\varphi$. Let $(s,t)$ be a proof of $C_G\varphi$. Then $s$ is a $G$-common access to proof $t$ of $\varphi$. In particular, $s$ is self-referential in $G$. Thus, for some (for any) $j\in G$, $(sj,(s,t))$ is a proof. Then by Lemma \ref{91}, $s$ is a $G$-common access to proof $(s,t)$. By definition, $(s (s,t))$ then is a proof of $C_G C_G\varphi$. Thus, the mapping $(s,t)\mapsto (s (s,t))$ represents an actual proof of $C_G\varphi\rightarrow C_G C_G\varphi$.\\
$\square(\varphi\rightarrow\psi)\rightarrow \square (C_G\varphi\rightarrow C_G\psi)$. Let $t$ be an actual proof of $\varphi\rightarrow\psi$. Let $(s,u)$ be a proof of $C_G\varphi$. Then $t$ converts $u$ into a proof $t(u)$ of $\psi$. Each $i\in G$ has the trivial access $s_0$ to $t$, since $t$ is an actual proof. And each $i\in G$ has the access $s$ to proof $u$. By combination principle (C1), each $i\in G$ gains the access $s_0+s = s$ to proof $t(u)$. By Lemma \ref{91}, $s$ then is also a $G$-common access to $t(u)$. Thus, $(s,t(u))$ is a proof of $C_G\psi$. Of course, the function $f_t\colon (s,u)\mapsto (s,t(u))$ is an effected construction, i.e. an actual proof. Then the construction that for any actual proof $t$ of $\varphi\rightarrow\psi$ returns a presentation (including proof-checking) of function $f_t$, constitutes an actual proof of $\square(\varphi\rightarrow\psi)\rightarrow \square (C_G\varphi\rightarrow C_G\psi)$.\\
Finally, we show that $K_i\varphi$ and $C_G\varphi$ have exactly the same actual proofs, independently of $i$ and $G$. In fact, $(s,t)$ is an actual proof of $K_i\varphi$ iff $t$ is an actual proof of $\varphi$ and $s=s_0$ iff $t$ is an actual proof of $\varphi$ and the trivial access $s=s_0$ is a $G$-common access to $t$ iff $(s,t)$ is an actual proof of $C_G\varphi$. This shows in particular that the \textit{actual} proofs (not \textit{all} possible proofs) of $\varphi$, $K_i\varphi$ and $C_G\varphi$, respectively, can be converted into each other, i.e. $\square\varphi\equiv\square K_i\varphi\equiv \square C_G\varphi$ holds for all $i\in I$ and all groups $G$.\footnote{Cf. Lemma \ref{510}(b) below.} However, $\varphi$, $K_i\varphi$ and $C_G\varphi$ will denote, in general, pairwise distinct propositions.
\section{The logics of access-based knowledge $L5^{AC^-}_N$ and $L5^{AC}_N$}
We extend, in the following, system $L5$ by axioms for knowledge and common knowledge in an augmented epistemic object language. As before, $I=\{1,...,N\}$ is a fixed finite set of $N\ge 1$ agents, and groups of agents are always non-empty subsets $G\subseteq I$.
\begin{definition}\label{100}
The object language is defined over the following set of symbols: an infinite set of propositional variables $V=\{x_0, x_1, ... \}$, logical connectives $\bot$, $\neg$, $\vee$, $\wedge$, $\rightarrow$, modal operator $\square$ and epistemic operators $K_i$, for $i\in I$, and $C_G$, for every group $G$ of agents. Then the set of formulas $Fm$ is the smallest set that contains $V\cup\{\bot\}$ and is closed under the following conditions: $\varphi,\psi\in Fm$ $\Rightarrow$ $\neg\varphi$, $(\varphi * \psi)$, $\square\varphi$, $K_i\varphi$, $C_G\varphi \in Fm$, where $*\in\{\vee,\wedge,\rightarrow\}$, $i\in I$, $G\subseteq I$, $G\neq\varnothing$.
\end{definition}
We use the following abbreviations: $\top:=(\bot\rightarrow\bot)$, $\neg\varphi:=(\varphi\rightarrow\bot)$, $(\varphi\leftrightarrow\psi):=(\varphi\rightarrow\psi)\wedge (\psi\rightarrow\varphi)$, $\varphi\equiv\psi := \square(\varphi\leftrightarrow\psi)$ (propositional identity), $\Diamond\varphi := \neg\square\neg\varphi$.\\
We consider the following axiom schemes:\\
\noindent (INT) any scheme which has the form of an $\mathit{IPC}$-tautology\footnote{It would be sufficient to fix here a finite set of schemes that axiomatize $\mathit{IPC}$.}\\
(i) $\square(\varphi\vee\psi)\rightarrow(\square\varphi\vee\square\psi)$\\
(ii) $\square\varphi\rightarrow\varphi$\\
(iii) $\square(\varphi\rightarrow\psi)\rightarrow(\square\varphi\rightarrow\square\psi)$ \\
(iv) $\square\varphi\rightarrow\square\square\varphi$\\
(v) $\neg\square\varphi\rightarrow\square\neg\square\varphi$\\
(vi) $K_i\varphi\rightarrow \varphi$ (reflection, factivity of knowledge)\\
(vii) $K_i(\varphi\rightarrow\psi)\rightarrow (K_i\varphi\rightarrow K_i\psi)$\\
(viii) $K_i(\varphi\vee\psi)\rightarrow (K_i\varphi\vee K_i\psi)$\\
(ix) $C_G(\varphi\rightarrow\psi)\rightarrow (C_G\varphi\rightarrow C_G\psi)$\\
(x) $C_G(\varphi\vee\psi)\rightarrow (C_G\varphi\vee C_G\psi)$ (only for introspective common knowledge)\\
(xi) $\square\varphi\rightarrow \square C_G\varphi$\\
(xii) $C_G\varphi\rightarrow K_i\varphi$, for any $i\in G$\\
(xiii) $C_G\varphi\rightarrow C_G K_i\varphi$, for any $i\in G$\\
(xiv) $C_G\varphi\rightarrow C_{G'}\varphi$, for any non-empty $G' \subseteq G$\\
(xv) $C_G\varphi\rightarrow C_G C_G\varphi$ (for introspective common knowledge)\\
(TND) $\varphi\vee\neg\varphi$\\
Except of (TND), all schemes above are intuitionistically acceptable in the sense that they are sound w.r.t. extended BHK semantics considering the access-based interpretation of epistemic operators. For most of the epistemic axioms, this is shown in the last section. In \cite{lewapal}, we saw that the modal axioms, in particular (iv) and (v), are sound w.r.t. extended BHK semantics. For the convenience of the reader, we recall here the argumentation. Before, we show that
\begin{equation}\label{120}
\square\varphi\vee\neg\square\varphi
\end{equation}
\noindent is intuitionistically acceptable.\footnote{Actually, $\square(\square\varphi\vee\neg\square\varphi)$ is a theorem of $L5$, cf. Theorem 3.7(vii) in \cite{lewapal}.} Of course, either there is an actual proof of $\varphi$ or there is no such proof. Since an actual proof is immediately available, it can be decided which one of the two alternatives is the case. In the former case, that actual proof is available and can be presented (proof-checked). This yields an actual proof of $\square\varphi$. In the latter case, we conclude that $\square\varphi$ has no possible proof at all. In fact, any (possible) proof of $\square\varphi$ would, by the BHK clause, involve an actual proof of $\varphi$ which, by hypothesis, does not exist. Thus, the identity function on proofs, as an effected construction, constitutes an actual proof of $\square\varphi\rightarrow\bot$, i.e. of $\neg\square\varphi$. We have shown that for any proposition $\varphi$, either we can present an actual proof of $\square\varphi$ or we can present an actual proof of $\neg\square\varphi$, and we are able to indicate which one of the two alternatives is the case. Thus, \eqref{120} is intuitionistically valid.\\
Soundness of (iv) $\square\varphi\rightarrow\square\square\varphi$. Suppose we are given a proof $s$ of $\square\varphi$. By definition, $s$ consists in the presentation of an \textit{actual} proof $t$ of $\varphi$. The presentation (including proof-checking) depends only from the actual proof $t$ and from $\varphi$ and no further hypotheses. Thus, $s$ is itself an effected construction, an actual proof. The presentation of $s$ as an actual proof of $\square\varphi$ yields an actual proof $u$ of $\square\square\varphi$. Thus, the construction that converts $s$ into $u$ is an actual proof of $\square\varphi\rightarrow\square\square\varphi$.\footnote{This shows in particular that any possible proof of $\square\varphi$ must be an actual proof of $\square\varphi$ which is in accordance with the fact that $\Diamond\square\varphi\rightarrow\square\square\varphi$ is a theorem of $L5$, cf. Theorem 3.7(v) in \cite{lewapal}.}\\
Soundness of (v) $\neg\square\varphi\rightarrow\square\neg\square\varphi$. Suppose $s$ is a proof of $\neg\square\varphi$. Then $\neg\square\varphi$ (i.e. $\square\varphi\rightarrow\bot$) must have an actual proof for otherwise, by \eqref{120} above, $\square\varphi$ would have an actual proof contradicting that $\neg\square\varphi$ has proof $s$. But then we may present a witness of an actual proof of $\square\varphi\rightarrow\bot$, namely the identity function on proofs which is, trivially, an effected construction. Its presentation (including proof-checking) results in an actual proof $t$ of $\square\neg\square\varphi$. We have presented a construction that for any possible proof $s$ of $\neg\square\varphi$ returns a proof $t$ of $\square\neg\square\varphi$.\\
Recall that our basic logic for the reasoning about proof $L5$ is given by the axiom schemes (INT), (i)--(v) and (TND) plus the inference rules of Modus Ponens (MP) and Intuitionistic Axiom Necessitation (AN): `If $\varphi$ is an intuitionistically acceptable axiom, i.e. any axiom distinct from (TND), then infer $\square\varphi$.'
We define $L5^{AC}_N$ as the multi-agent logic of access-based knowledge and introspective common knowledge with $N\ge 1$ agents.\footnote{Letter `A' refers to `access-based knowledge' while `C' stands for `common knowledge'.} $L5^{AC}_N$ is given by $L5$ + (vi)--(xv). That is, $L5^{AC}_N$ is axiomatized by the complete list of axioms above along with the rules of (MP) and (AN). The logic $L5^{AC^-}_N$ is given in the same way as $L5^{AC}_N$ but without the schemes (x) and (xv). $L5^{AC^-}_N$ is intended to formalize access-based common knowledge as an infinite conjunction according to \eqref{50} without introspection. Obviously, both $L5^{AC}_N$ and $L5^{AC^-}_N$ are super-logics of $L5$. As usual, we define a derivation of $\varphi$ from a set $\varPhi$ as a finite sequence of formulas $\varphi_0,...,\varphi_n=\varphi$ such that each member of the sequence is an axiom, an element of $\varPhi$ or the result of an application of the rules of (MP) or (AN) to formulas occurring at preceding positions. Recall that (AN) only applies to axioms of the underlying system that are different from \textit{tertium non datur}.
\begin{lemma}\label{500}
For any formulas $\varphi$, $\psi$, the following hold in all systems extending $L5$:\\
\noindent (a) If $\varphi$ is a theorem derivable without (TND), then $\square\varphi$ is a theorem.\\
(b) The Deduction Theorem holds.\\
(c) The Substitution Principle (SP) holds: $\varphi\equiv\psi\rightarrow \chi [x:=\varphi]\equiv \chi[x:=\psi]$.\\
The following are theorems:\\
(d) $\square\varphi\leftrightarrow (\varphi\equiv \top)$ and $\square\varphi\equiv(\varphi\equiv \top)$\\
(e) $\square (\varphi\wedge\psi)\equiv (\square\varphi\wedge\square\psi)$ and $\square (\varphi\vee\psi)\equiv (\square\varphi\vee\square\psi)$ \\
(f) $\square(\square\varphi\vee\neg\square\varphi)$\\
(g) $\neg\neg\square\varphi\equiv\square\varphi$ and $\neg(\square\varphi\wedge\square\psi)\equiv (\neg\square\varphi\vee\neg\square\psi)$\\
(h) $(\square\varphi\equiv\top)\vee (\square\varphi\equiv\bot)$\\
(i) $\square(\varphi\rightarrow\Diamond\varphi)$ and $\square(\Diamond\varphi\rightarrow\square\Diamond\varphi)$\\
(j) $\square(\Diamond (\varphi\vee\psi)\rightarrow (\Diamond\varphi\vee\Diamond\psi))$
\end{lemma}
\begin{proof}
(a) and (b) can be shown by induction on the length of derivations.\\
(c): Roughly speaking, it is enough to show that propositional identity is a congruence relation on $Fm$. (SP) then follows by induction on $\chi$. This is shown for the logical connectives, the modal operator and the knowledge operator in \cite{lewjlc1, lewjlc2, lewigpl}. We consider here only the new operator of common knowledge. We must show that $(\varphi\equiv\psi)\rightarrow (C_G\varphi\equiv C_G\psi)$ is a theorem scheme. By axioms (xi), (ix), (ii) and propositional calculus, we get $\square(\varphi\leftrightarrow\psi)\rightarrow (C_G\varphi\leftrightarrow C_G\psi)$. By item (a), distribution and axiom (ii), we obtain the assertion.\\
(d): The first part of (d) is originally shown in \cite{lewjlc1} for sublogic $L$. We present here a simpler derivation: 1. $(\varphi\equiv\top) \vdash \Box(\top \rightarrow \varphi)$; 2. $(\varphi\equiv\top) \vdash \Box\top \rightarrow \Box\varphi$, by distribution and (MP); 3. $(\varphi\equiv\top) \vdash \Box\top$, by (AN); 4. $(\varphi\equiv\top) \vdash \Box\varphi$, by (MP); 5. $\vdash (\varphi\equiv\top)\rightarrow \Box\varphi$, by Deduction Theorem; 6. $\vdash\square (\varphi\rightarrow (\top\rightarrow\varphi))$, by (AN); 7. $ \vdash\square\varphi\rightarrow\square (\top\rightarrow\varphi)$, by distribution and (MP); 8. $\vdash\square (\varphi\rightarrow (\varphi\rightarrow\top))$, by (AN); 9. $\vdash\square\varphi\rightarrow\square (\varphi\rightarrow\top)$, by distribution and (MP); 10. $\vdash\square\varphi\rightarrow \varphi\equiv\top$, by 7. and 9.; 12. $\vdash\square\varphi\leftrightarrow \varphi\equiv \top$, by 5. and 9. This shows the first part of (d). The second part now follows by item (a). \\
(e): Consider the intuitionistic tautologies $(\varphi\wedge\psi)\rightarrow\varphi$ and $(\varphi\wedge\psi)\rightarrow\psi$, apply rule (AN), distribution, intuitionistic propositional calculus. The other way round, consider the intuitionistic tautology $\varphi\rightarrow (\psi\rightarrow (\varphi\wedge\psi))$, apply (AN), distribution and intuitionistic propositional calculus. Finally, apply item (a). The second equation follows similarly using propositional calculus and axiom (i).\\
(f): This result is originally proved in \cite{lewapal}, Theorem 3.7(vii). \\
(g): Use (f), i.e. $\square\varphi\vee\neg\square\varphi$, and propositional calculus. Actually, by (a), it is enough to show that $\neg\neg\square\varphi\rightarrow\square\varphi$ and $\neg(\square\varphi\wedge\square\psi)\rightarrow (\neg\square\varphi\vee\neg\square\psi)$ derive without (TND).\\
(h): Using (f) and axiom (i), one derives $\square\square\varphi\vee \square\neg\square\varphi$. Then (d) along with propositional caluclus yields $(\square\varphi\equiv\top)\vee (\square\varphi\equiv\bot)$.\\
(i): From $\varphi\rightarrow\neg\neg\varphi$ and the contraposition of theorem $\square\neg\varphi\rightarrow\neg\varphi$ we derive $\varphi\rightarrow\Diamond\varphi$ without using (TND). Now, apply item (a). The second assertion is clear by scheme (v) and item (a). \\
(j): By (e), $(\square\neg\varphi\wedge\square\neg\psi)\rightarrow \square(\neg\varphi\wedge\neg\psi)$ is a theorem. Observe that $\neg(\varphi\vee\psi)\equiv(\neg\varphi\wedge\neg\psi)$ is a theorem since $\neg(\varphi\vee\psi)\leftrightarrow(\neg\varphi\wedge\neg\psi)$ is an intuitionistic tautology. By the Substitution Principle (SP), we may replace $\neg\varphi\wedge\neg\psi$ by $\neg(\varphi\vee\psi)$ in every context. Hence, $(\square\neg\varphi\wedge\square\neg\psi)\rightarrow \square\neg(\varphi\vee\psi)$ is a theorem and so is its contrapositive $\neg\square\neg(\varphi\vee\psi)\rightarrow\neg(\square\neg\varphi\wedge\square\neg\psi)$. Then by the second assertion of (g), we derive $\neg\square\neg(\varphi\vee\psi)\rightarrow (\neg\square\neg\varphi\vee\neg\square\neg\psi)$, i.e. $\Diamond (\varphi\vee\psi)\rightarrow (\Diamond\varphi\vee\Diamond\psi)$. Note that (TND) does not occur in the derivations. Thus, we may apply item (a) and obtain (j).
\end{proof}
\begin{lemma}\label{510}
The following are theorems of $L^{5AC}_N$ and of $L5^{AC^-}_N$:\\
(a) $\square(\varphi\rightarrow\psi)\rightarrow \square (K_i \varphi\rightarrow K_i\psi)$ and $\square(\varphi\rightarrow\psi)\rightarrow \square (C_G \varphi\rightarrow C_G\psi)$\\
(b) $\square\varphi\equiv \square K_i\varphi$ and $\square\varphi\equiv \square C_G\varphi$\\
(c) $K_i(\varphi\wedge\psi)\equiv (K_i\varphi\wedge K_i\psi)$ and $K_i(\varphi\vee\psi)\equiv (K_i\varphi\vee K_i\psi)$\\
(d) $\square(K_i K_j\varphi\rightarrow K_i\varphi)$\\
Moreover, axiom scheme (xiii) is redundant in $L5^{AC}_N$, i.e. it is derivable from the remaining axioms.
\end{lemma}
\begin{proof}
(a): $\square(\varphi\rightarrow\psi)\rightarrow \square C_G (\varphi\rightarrow \psi)$ is an instance of scheme (xi). Now, consider (ix) and (iii) along with applications of rules (AN) and (MP). This yields the second assertion of (a). Using (xi) and (xii), one derives $\square\varphi\rightarrow\square K_i\varphi$. Thus, $\square(\varphi\rightarrow\psi)\rightarrow \square K_i (\varphi\rightarrow \psi)$ is a theorem. The first assertion of (a) now follows in a similar way as the second one.\\
(b): The derivations of $\square\varphi\leftrightarrow \square K_i\varphi$ and $\square\varphi\leftrightarrow \square C_G\varphi$ are straightforward. Now, (b) follows by Lemma \ref{500} (a).\\
(c): We show the second assertion. $K_i(\varphi\vee \psi)\rightarrow (K_i\varphi\vee K_i\psi)$ is a theorem by scheme (viii). $\varphi\rightarrow (\varphi\vee\psi)$ is an intuitionistic tautology, thus $\square (\varphi\rightarrow (\varphi\vee\psi))$ is a theorem. Now, one easily derives $K_i (\varphi\rightarrow (\varphi\vee\psi))$. Then, by distribution of knowledge, $K_i\varphi\rightarrow K_i (\varphi\vee\psi)$ is a theorem. Applying Lemma \ref{500} (a) yields the second assertion of (c). The proof of the first assertion of (c) is straightforward.\\
(d): $K_j\varphi \rightarrow\varphi$ is an instance of scheme (vi). By (AN), $\square (K_j\varphi \rightarrow\varphi)$ is a theorem. Then the first part of (a), together with (MP), yields $\square(K_i K_j\varphi\rightarrow K_i\varphi)$.\\
Finally, we prove the last assertion. By (a), $\square(C_G\varphi\rightarrow K_i\varphi)\rightarrow \square (C_G C_G \varphi\rightarrow C_G K_i\varphi)$ is a theorem. By scheme (xii), (AN) and (MP), $C_G C_G \varphi\rightarrow C_G K_i\varphi$ is a theorem. This, thogether with scheme (xv), yields scheme (xiii) $C_G\varphi\rightarrow C_G K_i\varphi$. By Lemma \ref{500} (a), we may apply (AN) to that formula. This shows that $L5^{AC}_N$ without scheme (xiii) is equivalent to $L5^{AC}_N$.\footnote{Notice that the argument does not work in $L5^{AC^-}_N$ where scheme (xv) is not available.}
\end{proof}
\section{Algebraic semantics}
It is well-known that the class of all Heyting algebras constitutes a semantics for $\mathit{IPC}$.\footnote{It is enough to consider Heyting algebras with the Disjunction Property as in Definition \ref{810}.} A propositional formula $\varphi$ evaluates to the top element of any given Heyting algebra $\mathcal{H}$, under any assignment of elements of $\mathcal{H}$ to propositional variables, if and only if $\varphi$ is a theorem of $\mathit{IPC}$. In this sense, the greatest element of any given Heyting algebra represents intuitionistic truth, and we have strong completeness: $\varPhi\vdash_{\mathit{IPC}}\varphi$ if and only if for any Heyting algebra $\mathcal{H}$ and any assignment $\gamma\in H^V$, if $\varPhi$ is intuitionistically true in $\mathcal{H}$ under $\gamma$, then so is $\varphi$.
Recall that a Heyting algebra is a bounded lattice such that for all elements $a,b$, the subset $\{c\mid f_\wedge(a,c)\le b\}$ has a greatest element $f_\rightarrow(a,b)$, called the relative pseudo-complement of $a$ with respect to $b$, where $f_\wedge$ is the infimum (meet) operation and $\le$ is the lattice ordering. For a Heyting algebra $\mathcal{H}$, we use the notation $\mathcal{H}=(M, f_\vee, f_\wedge, f_\bot, f_\rightarrow)$, where $M$ is the universe and $f_\vee$, $f_\wedge$, $f_\bot$, $f_\rightarrow$ are the usual operations for join, meet, least element and relative pseudo-complement (implication), respectively. The greatest element is given by $f_\top:= f_\rightarrow(f_\bot,f_\bot)$, and the pseudo-complement (negation) of $m\in M$ is defined by $f_\neg(m):=f_\rightarrow(m,f_\bot)$. A subset $F\subseteq M$ of the universe $M$ is called a filter if the following conditions are satisfied: $f_\top\in F$; and for any $m,m'\in M$: if $m\in F$ and $f_\rightarrow(m,m')\in F$, then $m'\in F$ (cf. \cite{chazak}). A filter $F$ is a proper filter if $f_\bot\notin F$. A prime filter is a proper filter $F$ such that $f_\vee(m,m')\in F$ implies $m\in F$ or $m'\in F$, for any $m,m'\in M$. Finally, an ultrafilter is a maximal proper filter. Every ultrafilter satisfies for all elements $m\in M$: $m\in U$ or $f_\neg(m)\in U$. It follows that $U$ mirrors the classical behaviour of logical connectives and represents, in this sense, classical truth. In particular, every ultrafilter is prime. Also recall that in any Heyting algebra, for any elements $m,m'$, the equivalence $m\le m' \Leftrightarrow f_\rightarrow (m,m')=f_\top$ holds true. \\
Furthermore, the following facts will be useful:
\begin{lemma}\label{700}
Let $\mathcal{H}$ be a Heyting algebra with universe $M$. Then the following hold.\\
(i) Any proper filter is the intersection of all prime filters containing it.\\
(ii) Let $P$ be a filter, and $a,b\in M$. Then $f_\rightarrow(a,b)\in P$ iff for all prime filters $P'\supseteq P$: $a \in P'$ implies $b\in P'$.\\
(iii) If the smallest filter $\{f_\top\}$ is prime, then for all $a,b\in M$: $a\le b$ iff for all prime filters $P$: $a\in P$ implies $b\in P$.
\end{lemma}
\begin{proof}
(i): Let $F$ be a proper filter of $\mathcal{H}$. For every $a\in M\smallsetminus F$, there is a prime filter $P_a$ containing $F$ such that $a\notin P_a$. In fact, by Zorn's Lemma, there is an ultrafilter with that property. Then $F=\bigcap_{a\notin F} P_a$.\\
(ii): Let $P$ be a prime filter, $a,b\in M$. The left-to-right implication of the assertion is clear by definition of a filter. Suppose $f_\rightarrow(a,b)\notin P$. Consider $F_{a,P}:=\{c\in M\mid f_\rightarrow(a,c)\in P\}$. We claim that $F_{a,P}$ is a filter. Obviously, $f_\top\in F_{a,P}$. Suppose $c\in F_{a,P}$ and $f_\rightarrow(c,d)\in F_{a,P}$, for $c,d\in M$. Then $f_\rightarrow(a,c)\in P$ and $f_\rightarrow (a, f_\rightarrow(c,d))\in P$. Since $((x\rightarrow y)\wedge (x\rightarrow (y\rightarrow z)) \rightarrow (x\rightarrow z)$ is an intuitionistic tautology, we conclude that $f_\rightarrow(a,d)\in P$, whence $d\in F_{a,P}$ and $F_{a,P}$ is a filter. Let $c\in P$. Of course, $f_\wedge(a,c)\le c$. Since $f_\rightarrow(a,c)$ is the greatest element $x$ such that $f_\wedge(a,x)\le c$, it follows that $c\le f_\rightarrow(a,c)$. Thus, $f_\rightarrow(a,c)\in P$. That is, $c\in F_{a,P}$. We have shown: $P\subseteq F_{a,P}$. Obviously, $a\in F_{a,P}$ and, by hypothesis, $b\notin F_{a,P}$. By (i), it follows that there is a prime filter $P'$ extending $F_{a,P}$ such that $a\in P'$ and $b\notin P'$. We have $P\subseteq F_{a,P}\subseteq P'$. By contraposition, the right-to-left implication of assertion (ii) follows.\\
(iii): Suppose $\{f_\top\}$ is a prime filter. The equivalence $a\le b\Leftrightarrow f_\rightarrow(a,b)=f_\top$ is a well-known property of Heyting algebras. The assertion now follows from (ii).
\end{proof}
\begin{definition}\label{810}
A model $\mathcal{M}$ is given by a Heyting algebra expansion
\begin{equation*}
\mathcal{M}=(M,\mathit{TRUE}, f_\vee, f_\wedge, f_\bot, f_\rightarrow, f_\square, (f_{K_i})_{i\in I}, (f_{C_G})_{\varnothing\neq G\subseteq I})
\end{equation*}
with universe $M$ whose elements are called propositions, a designated ultrafilter $\mathit{TRUE}\subseteq M$ which is the set of classically true propositions, and additionally unary operations $f_\square, f_{K_i}$, $f_{C_G}$ such that the following truth conditions are satisfied:\\
(i) $\mathcal{M}$ has the Disjunction Property: for all $m,m'\in M$, $f_\vee(m,m')=f_\top$ implies $m=f_\top$ or $m'=f_\top$. That is, the smallest filter $\{f_\top\}$ is prime.\\
(ii) For all $m\in M$:
\begin{equation*}
\begin{split}
f_\square(m)=
\begin{cases}
f_\top, \text{ if }m=f_\top\\
f_\bot, \text{ else}
\end{cases}
\end{split}
\end{equation*}
(iii) For every prime filter $F\subseteq M$, and for all $i\in I$ and all groups $G$, the following conditions (a)--(e) are fulfilled:\\
(a) The set $\mathit{BEL_i}(F):=\{m\in M\mid f_{K_i}(m)\in F\}$ is a filter.\\
(b) The set $\mathit{COMMON_G}(F):=\{m\in M\mid f_{C_G}(m)\in F\}$ is a filter.\\
(c) For every ultrafilter $U\supseteq F$: $\mathit{BEL_i}(F)\subseteq U$; in particular, $\mathit{BEL_i}(F)$ is a proper filter and $\mathit{BEL_i}(\mathit{TRUE})\subseteq \mathit{TRUE}$.\\
(d) $\mathit{COMMON_G}(F)\subseteq \mathit{BEL_i}(F)$, whenever $i\in G$.\\
(e) For any $m\in M$: if $m\in \mathit{COMMON_G}(F)$ then $f_{K_i}(m)\in \mathit{COMMON_G}(F)$, whenever $i\in G$.\\
(f) $\mathit{COMMON_G}(F)\subseteq \mathit{COMMON_{G'}}(F)$, whenever $G'\subseteq G$.
\end{definition}
Notice that the definition involves a relational structure given by the set of prime filters which can be viewed as `worlds' ordered by set-theoretical inclusion. Actually, this yields a relational semantics based on \textit{intuitionistic general frames} (cf. \cite{chazak}) with some additional structure regarding the epistemic ingredients. This kind of relational semantics was explicitly defined and studied for the logics $L5$, $EL5$ and $\mathit{IEL}$ in \cite{lewapal} where also its equivalence to algebraic semantics is shown. Considering Definition \ref{810} above and following the constructions presented in \cite{lewapal}, that frame-based semantics extends straightforwardly to a semantics with common knowledge equivalent to the algebraic conditions given in Definition \ref{810}. For space reasons, we skip here the details. Intuitively, $\mathit{BEL_i}(F)$ is the set of propositions known by agent $i$ at `world' $F$, and $\mathit{COMMON_G}(F)$ is the set of propositions that are common knowledge in $G$ at `world' $F$. Intuitionistic truth is represented by `world' $\{f_\top\}$, the smallest prime filter; and classical truth is determined by a designated `maximal world' $\mathit{TRUE}$. Observe that $f_\square(m)$ is true at `world' $F$ (i.e. $f_\square(m)\in F$) iff $m$ is true at the `root world' $\{f_\top\}$ iff $m$ is true at all `worlds' (i.e. is contained in all prime filters). Thus, regarding the modal operator, we actually have a $S5$-style Kripke model combined with the properties of an intuitionistic Kripke model for constructive reasoning.
\begin{lemma}\label{815}
Let $\mathcal{M}$ be a model. We have $f_{K_i}(f_\top)=f_\top= f_{C_G}(f_\top)$, for all $i\in I$ and all $\varnothing\neq G\subseteq I$. Moreover, the operations $f_{K_i}$ and $f_{C_G}$ are monotonic on $M$, i.e. $m\le m'$ implies $f_{K_i}(m)\le f_{K_i}(m')$ and $f_{C_G}(m)\le f_{C_G}(m')$.
\end{lemma}
\begin{proof}
By truth condition (i), $\{f_\top\}$ is a prime filter. Now, consider $F=\{f_\top\}$ and $m=f_\top$ in truth conditions (iii)(a) and (iii)(b). Then the first assertion of the Lemma follows. Suppose $m,m'\in M$ and $m\le m'$. By Lemma \ref{700}(iii), it is enough to show: $f_{K_i}(m)\in F$ implies $f_{K_i}(m')\in F$, for all prime filters $F$. Let $F$ be a prime filter. Then $f_{K_i}(m)\in F$ implies $m\in \mathit{BEL_i}(F)$ implies $m'\in\mathit{BEL_i}(F)$ implies $f_{K_i}(m')\in F$. The assertion regarding the operators $f_{C_G}$ follows similarly.
\end{proof}
\begin{definition}\label{850}
Let $\mathcal{M}$ be a model. In the following, we consider the truth conditions given in Definition \ref{810}.
\begin{itemize}
\item $\mathcal{M}$ is an $L5^{AC^-}_N$-model if, instead of (iii)(c), the following stronger condition (c)* is satisfied: For every prime filter $F$ and every $i\in I$, $\mathit{BEL_i}(F)$ is a prime filter and $\mathit{BEL_i}(F)\subseteq F$.
\item $\mathcal{M}$ is an $L5^{AC}_N$-model if condition (c)* holds, $\mathit{COMMON_G}(F)$ is a prime filter, for every prime filter $F$, and the following additional truth condition (g) is fulfilled for every prime filter $F$, every group $G$ and every $m\in M$:\\
(g) If $m\in COMMON_G(F)$, then $f_{C_G}(m)\in COMMON_G(F)$.
\item The Heyting algebra reduct of $\mathcal{M}$ with ultrafilter $\mathit{TRUE}$ and operators $f_\square$ and $f_K$ (i.e. $I=\{1\}$, single-agent case) is called an $EL5$-model. Only the conditions (i), (ii), and (iii)(a) and (c) are relevant.
\item The Heyting algebra reduct of $\mathcal{M}$ with ultrafilter $\mathit{TRUE}$ and operator $f_\square$ is called an $L5$-model. Of course, only the conditions (i) and (ii) are relevant.
\item The Heyting algebra reduct of $\mathcal{M}$ with operator $f_K$ (single-agent case: $I=\{1\}$) is said to be an algebraic $\mathit{IEL}$-model if the following additional truth condition of intuitionistic co-reflection (IntCo) is satisfied:\\
(IntCo) $F\subseteq \mathit{BEL}(F)$, for every prime filter $F$, where $\mathit{BEL(F)}:= \mathit{BEL_1}(F)$. \\Besides that condition, only (i), (iii)(a) and (iii)(c) are relevant.
\end{itemize}
\end{definition}
Algebraic semantics for $L5$ and $EL5$ is originally presented in \cite{lewjlc2} and \cite{lewigpl, lewapal}, respectively, in essentially the way as formulated in the next Theorem \ref{870}. Algebraic semantics of $\mathit{IEL}$, in the form as presented in \cite{lewigpl}, is also described in Theorem \ref{870} below.
\begin{theorem}\label{870}
A Heyting algebra expansion
\begin{equation*}
\mathcal{M}=(M,\mathit{TRUE}, f_\vee, f_\wedge, f_\bot, f_\rightarrow, f_\square, (f_{K_i})_{i\in I}, (f_{C_G})_{\varnothing\neq G\subseteq I})
\end{equation*}
with ingredients as before is a model in the sense of Definition \ref{810} if and only if the following conditions are fulfilled for all $m, m'\in M$, all $i\in I$ and all groups $G$:\\
(A) $\mathcal{M}$ has the Disjunction Property\\
(B)
\begin{equation*}
\begin{split}
f_\square(m)=
\begin{cases}
f_\top, \text{ if }m=f_\top\\
f_\bot, \text{ else}
\end{cases}
\end{split}
\end{equation*}
(C) $f_{K_i}(f_\rightarrow(m,m'))\le f_\rightarrow(f_{K_i}(m),f_{K_i}(m'))$\\
(D) $f_{C_G}(f_\rightarrow(m,m'))\le f_\rightarrow(f_{C_G}(m),f_{C_G}(m'))$\\
(E) $f_{C_G}(m)\le f_{K_i}(m)$, whenever $i\in G$\\
(F) $f_{C_G}(m)\le f_{C_G} (f_{K_i}(m))$, whenever $i\in G$\\
(G) $f_{C_G}(m)\le f_{C_{G'}}(m)$, whenever $G'\subseteq G$\\
(H) $f_{C_G}(f_\top)=f_\top$\\
(I) $f_{K_i}(m)\le f_\neg(f_\neg(m))$.\\
-- $\mathcal{M}$ is an $L5^{AC^-}_N$-model if instead of (I) the stronger condition (I)* $f_{K_i}(m)\le m$ holds, and for all $m,m'\in M$: $f_{K_i}(f_\vee(m,m'))\le f_\vee(f_{K_i}(m),f_{K_i}(m'))$.\\
-- $\mathcal{M}$ is an $L5^{AC}_N$-model if it is an $L5^{AC^-}_N$-model and for all $m,m'\in M$ and all groups $G$, $f_{C_G}(f_\vee(m,m'))\le f_\vee(f_{C_G}(m),f_{C_G}(m'))$ and introspection of common knowledge $f_{C_G}(m)\le f_{C_G} (f_{C_G}(m))$ are satisfied.\footnote{Note that introspection along with (E) and (I)* implies $f_{C_G}(m)= f_{C_G} (f_{C_G}(m))$. In this sense, common knowledge is a fixed point. Also notice that (F) follows from introspection of common knowledge, (E) and monotonicity of $f_{C_G}$.}\\
-- The appropriate reduct of $\mathcal{M}$ is an $EL5$-model if we drop common knowledge and consider the single agent case $I=\{1\}$ and only the conditions (A), (B), (C) and (I), and $f_{K}(f_\top)=f_\top$ instead of (H) \\
-- The appropriate reduct of $\mathcal{M}$ is an $L5$-model if we exclude all epistemic ingredients and consider only the conditions (A), (B).\\
-- The appropriate reduct of $\mathcal{M}$ is an $\mathit{IEL}$-model if we drop common knowledge, consider the single agent case $I=\{1\}$ and the condtions (A), (C), (I), and additionally (IntCo): $m\le f_K(m)$, for all $m\in M$.
\end{theorem}
Theorem \ref{870} is useful for model constructions. It hides the relational structure on prime theories which is often not relevant for the construction of an algebraic model. The proof of Theorem \ref{870} is straightforward and relies essentially on Lemma \ref{700}(iii) and filter properties.
\begin{definition}\label{900}
Given a model $\mathcal{M}$, an assignment is a function $\gamma\colon V\rightarrow M$ that extends in the canonical way to an `homomorphism' $\gamma^*\colon Fm\rightarrow M$. We simplify notation and write $\gamma$ instead of the uniquely determined $\gamma^*$. The tuple $(\mathcal{M},\gamma)$ is called an interpretation. We consider two kinds of satisfaction relations between interpretations and formulas. If $\mathcal{M}$ is an $\mathit{IEL}$-model, then we define
\begin{equation*}
(\mathcal{M},\gamma)\vDash_{\mathit{IEL}}\varphi :\Leftrightarrow \gamma(\varphi)=f_\top,
\end{equation*}
where $\varphi$ belongs here to the sublanguage $Fm_e\subseteq Fm$, i.e. the language of $\mathit{IEL}$.
If $\mathcal{L}\in \{L5, EL5, L5^{AC^-}_N, L5^{AC}_N\}$ and $\mathcal{M}$ is an $\mathcal{L}$-model, then we define
\begin{equation*}
(\mathcal{M},\gamma)\vDash_{\mathcal{L}}\varphi :\Leftrightarrow \gamma(\varphi)\in\mathit{TRUE},
\end{equation*}
where $\varphi$ is any formula of the underlying object language of the respective logic.
If the context it allows, we omit the index $\mathcal{L}$. Of course, the satisfaction relations extend to sets of formulas in the usual way.
\end{definition}
The relation of logical consequence is defined as usual. If $\mathcal{L}$ is one of the logics $\mathit{IEL}$, $L5$, $EL5$, $L5^{AC^-}_N$, $L5^{AC}_N$, and $\Phi\cup\{\varphi\}$ is a set of formulas of the respective object language, then $\Phi\Vdash_\mathcal{L}\varphi$ $:\Leftrightarrow$ for every interpretation $(\mathcal{M},\gamma)$, where $\mathcal{M}$ is an $\mathcal{L}$-model, $(\mathcal{M},\gamma)\vDash_\mathcal{L}\Phi$ implies $(\mathcal{M},\gamma)\vDash_\mathcal{L}\varphi$.
\section{Soundness and Completeness}
We consider the logics $\mathit{IEL}$ and $L5^{AC}_N$ and show that they are sound and complete w.r.t. their respective classes of algebraic models. Soundness and completeness of $L5^{AC^-}_N$, $EL5$ and $L5$ then follows similarly.
\begin{theorem}\label{900}
For any $\varPhi\cup\{\varphi\}\subseteq Fm_e$, $\varPhi\vdash_{\mathit{IEL}}\varphi$ implies $\varPhi\Vdash_{\mathit{IEL}}\varphi$.
\end{theorem}
\begin{proof}
It is enough to show that all axioms of $\mathit{IEL}$ are true, i.e. denote the top element in every algebraic $\mathit{IEL}$-model under every assignment. This is clear for formulas having the form of an intuitionistic tautology. The validity of the remaining axioms follows from the conditions (C), (I) and (IntCo) of Theorem \ref{870}.
\end{proof}
Weak completeness of $\mathit{IEL}$ w.r.t. algebraic semantics is shown in \cite{lewigpl}. For the convenience of the reader, we outline here a proof which is based on the alternative definition of algebraic $\mathit{IEL}$-models given in Definition \ref{850}. We consider the Lindenbaum-Tarski algebra of $\mathit{IEL}$. Its elements are the equivalence classes $\overline{\varphi}$ modulo logical equivalence in $\mathit{IEL}$, for $\varphi\in Fm_e$. By $\mathit{IPC}$ and epistemic axioms of $\mathit{IEL}$ it follows that the operations $f_*(\overline{\varphi},\overline{\psi}):=\overline{\varphi * \psi}$, $*\in\{\vee,\wedge,\rightarrow\}$, $f_\bot := \overline{\bot}$ and $f_K(\overline{\varphi}) := \overline{K\varphi}$ are all well-defined. This yields a Heyting algebra $\mathcal{M}$ with operator $f_K$ and lattice ordering $\overline{\varphi}\le\overline{\psi}$ $\Leftrightarrow$ $\vdash_{\mathit{IPC}}\varphi\rightarrow\psi$. In \cite{artpro}, it is shown that $\mathit{IEL}$ has the Disjunction Property. Thus, $\mathcal{M}$ has the Disjunction Property, i.e. ${f_\top}$ is the smallest prime filter. We show that the conditions (iii)(a) and (iii)(c) of Definition \ref{810} are satisfied. For every prime filter $F$, the set $\mathit{BEL}(F):=\{\overline{\varphi}\mid f_K(\overline{\varphi})\in F\}$ is a filter because of the distribution axiom of $\mathit{IEL}$ and the fact that $\top\rightarrow K\top$ is a theorem which ensures that $f_K(\overline{\top})=\overline{K\top}=\overline{\top}=f_\top\in F$ and thus $f_\top\in \mathit{BEL}(F)$. Hence, (iii)(a) holds. Now suppose $F$ is a prime filter and $U$ is an ultrafilter such that $F\subseteq U$. Since $K\varphi\rightarrow\neg\neg\varphi$ is a theorem of $\mathit{IEL}$, we have $\overline{K\varphi}\le\overline{\neg\neg\varphi}$. Then $\overline{\varphi}\in \mathit{BEL}(F)$ implies $f_K(\overline{\varphi})\in F$ implies $\overline{\neg\neg\varphi}\in F$ implies $\overline{\varphi}\in U$. Hence, $\mathit{BEL}(F)\subseteq U$. Thus, the truth conditions of an algebraic $\mathit{IEL}$-model as established in Definitions \ref{850} and \ref{810} are satisfied. Let $\gamma\in M^V$ be the assignment $x\mapsto\overline{x}$. By induction on formulas, one shows $\gamma(\varphi)=\overline{\varphi}$ for every formula $\varphi\in Fm_e$. Then $(\mathcal{M},\gamma)\vDash\varphi$ iff $\gamma(\varphi)=\overline{\varphi}=f_\top=\overline{\top}$ iff $\vdash_{\mathit{IEL}}\varphi\leftrightarrow\top$ iff $\vdash_{\mathit{IEL}}\varphi$.
\begin{corollary}\label{930}
For every formula $\varphi\in Fm_e$, $\vdash_{\mathit{IEL}}\varphi$ $\Leftrightarrow$ $\Vdash_{\mathit{IEL}}\varphi$.
\end{corollary}
\begin{theorem}\label{950}
Let $\mathcal{L}$ be the logic $L5$, $EL5$, $L5^{AC^-}_N$ or $L5^{AC}_N$. For any set $\varPhi\cup\{\varphi\}$ of the respective object language, $\varPhi\vdash_\mathcal{L}\varphi$ implies $\varPhi\Vdash_\mathcal{L}\varphi$.
\end{theorem}
\begin{proof}
It suffices to consider logic $L5^{AC}_N$. Let $\mathcal{M}$ be an $L5^{AC}_N$-model and $\gamma\in M^V$ an assignment. We show that all axioms denote classically true propositions, i.e. elements of ultrafilter $\mathit{TRUE}$. This is clear for (TND). We claim that the remaining axioms denote the top element of the Heyting lattice. Then follows that also rule (AN) is sound. Of course, all intuitionistic tautologies and substitution-instances denote $f_\top$. Note that all other axioms are of the form: $\varphi\rightarrow\psi$. Since $\gamma(\varphi\rightarrow\psi)=f_\rightarrow(\gamma(\varphi),\gamma(\psi))=f_\top$ iff $\gamma(\varphi)\le\gamma(\psi)$, it is enough to show that (*) $\gamma(\varphi)\le \gamma(\psi)$ holds true. For this purpose, it might be more comfortable to use Theorem \ref{870} instead of the model definitions. Concerning the axioms (i)--(v), (*) follows from condition (B): for any $m\in M$, either $f_\square(m)=f_\top$ or $f_\square(m)=f_\bot$. Referring to axiom (xi), (*) follows by truth condition (B) along with the first assertion of Lemma \ref{815}: $f_{C_G}(f_\top)=f_\top$. Concerning the remaining axioms, (*) follows from corresponding conditions given in Theorem \ref{870}. Finally, rule (MP) is sound because $\mathit{TRUE}$ is a filter. The assertion of the Theorem now follows by induction on derivations.
\end{proof}
Completeness of the logics $L5$ and $EL5$ w.r.t. algebraic semantics is shown in \cite{lewjlc2} and \cite{lewigpl}, respectively. Following the same strategy, we sketch out a completeness proof of $L5^{AC}_N$ w.r.t. the class of $L5^{AC}_N$-models. It is enough to show that every consistent set of formulas is satisfied by some interpretation based on an $L5^{AC}_N$-model. Let $\varPhi\subseteq Fm$ be consistent. By Zorn's Lemma, $\varPhi$ has a maximal consistent extension $\varPsi$. We construct a model for $\varPsi$. Let $\approx_\varPsi$ be the relation on formulas defined by $\varphi\approx_\varPsi \psi :\Leftrightarrow \varPsi\vdash\varphi\equiv\psi$. Using the Substitution Principle (SP), one shows that $\approx_\Psi$ is a congruence relation on the resulting `algebra of formulas', where the connectives, modal and epistemic operators are viewed as operations on $Fm$ (cf \cite{lewjlc2, lewigpl}). For $\varphi\in Fm$, we denote by $\overline{\varphi}:=\overline{\varphi}_\Psi$ the congruence class of $\varphi$ modulo $\approx_\Psi$. Then the sets $M=\{\overline{\varphi}\mid\varphi\in Fm\}$ and $\mathit{TRUE}=\{\overline{\varphi}\mid\varphi\in\varPsi\}$ along with the following operations on $M$: $f_\bot:=\overline{\bot}$, $f_\top:=\overline{\top}$, $f_\square(\overline{\varphi}):=\overline{\square\varphi}$, $f_{K_i}(\overline{\varphi}):=\overline{K_i\varphi}$, $f_{C_G}(\overline{\varphi}):=\overline{C_G\varphi}$, $f_*(\overline{\varphi},\overline{\psi}):=\overline{\varphi * \psi}$, where $*\in\{\vee,\wedge,\rightarrow\}$, are all well-defined. We claim that this yields an $L5^{AC}_N$-model $\mathcal{M}$. Clearly, $\mathcal{M}$ is based on a Heyting algebra: all $\mathit{IPC}$-theorems of the form $\varphi\leftrightarrow\psi$ are contained in $\varPsi$. Then rule (AN) implies $\varPsi\vdash\varphi\equiv\psi$, i.e. $\overline{\varphi}=\overline{\psi}$, hence all equations that determine a Heyting algebra are satisfied. $\mathit{TRUE}\subseteq M$ is an ultrafilter because $\varPsi$ is maximal consistent. By Lemma \ref{500}(d), for any $m\in M$: $f_\square(m)\in\mathit{TRUE}$ iff $m=f_\top$. The axioms (iv) and (v) then ensure truth condition (ii) of a model, cf. Definition \ref{810}. Truth condition (i), the Disjunction Property, now follows by axiom (i). From Lemma \ref{510}(b) it follows that $K_i\top$ and $C_G\top$ are theorems. This, along with the distribution axioms, implies that the sets $\mathit{BEL_i}(F)$ and $\mathit{COMMON_G}(F)$ are filters, for any prime filter $F$ of the Heyting algebra. Also the remaining truth conditions (iii)(c)--(g) of an $L5^{AC}_N$-model follow straightforwardly from corresponding axioms. As in similar situations, it might be more comfortable to use Theorem \ref{870} here to verify all these conditions. Let $\gamma\in M^V$ be the assignment defined by $x\mapsto\overline{x}$. Then $\gamma(\varphi)=\overline{\varphi}$, for every $\varphi\in Fm$. Thus, $\varphi\in\varPsi\Leftrightarrow\overline{\varphi}\in\mathit{TRUE}\Leftrightarrow\gamma(\varphi)\in\mathit{TRUE}\Leftrightarrow (\mathcal{M},\gamma)\vDash\varphi$. In particular, $(\mathcal{M},\gamma)\vDash\varPhi\subseteq\varPsi$. Hence, every set consistent in $L5^{AC}_N$ is satisfied by an $L5^{AC}_N$-model; and analogously for $L5$, $EL5$ and $L5^{AC^-}_N$.
\begin{theorem}[Completeness]\label{600}
Let $\mathcal{L}$ be the logic $L5$, $EL5$, $L5^{AC^-}_N$ or $L5^{AC}_N$. For any set $\varPhi\cup\{\varphi\}$ of the respective object language, $\varPhi\Vdash_\mathcal{L}\varphi$ implies $\varPhi_\mathcal{L}\vdash\varphi$.
\end{theorem}
\section{Intended Models}
Intuitively, by an intended model we mean a model where common knowledge has its intended meaning, i.e. $C_G\varphi$ is true iff $\bigwedge_{n\in\mathbb{N}} E_G^n\varphi$ is true, for any formula $\varphi$ and any group $G$.\footnote{Of course, this basic intuition also holds for our stronger introspective notion of common knowledge which additionally has the property: $C_G\varphi\leftrightarrow C_G C_G\varphi$.} Since infinite conjunctions cannot be expressed in the finitary object language, our axiomatization ensures only that truth of $C_G\varphi$ implies the truth of all $E_G^n\varphi$, $n\in\mathbb{N}$. Thus, a non-intended model is a model where for some formula $\varphi$, $E_G^n\varphi$ is true for every $n\in\mathbb{N}$, but $C_G\varphi$ is false. Both intended as well as non-intended models exist as we shall see at the end of this section.
We would like to point out here that it is not unusual that the intended properties of a formalized concept are not completely captured by the axiomatization but are instead represented by a \textit{standard model} or by certain \textit{intended models}. The phenomenon is well-known from classical first-order logic. Compactness arguments generally show that a given first-order theory with infinite models has also models with counter-intuitive or unexpected properties, non-standard elements, etc. The existence of such non-intended (or non-standard) models is unproblematic as long as enough intended and meaningful models exist.
In the following, we characterize intended $L5^{AC^-}_N$- and $L5^{AC}_N$-models. For this purpose, we adopt and apply some notions and results from \cite{lewsl} where common knowledge is axiomatized and modeled in essentially the same way, although this is done in a general, classical non-Fregean setting. In this section, by a model we always mean an $L5^{AC^-}_N$- or an $L5^{AC}_N$-model.
\begin{definition}\label{1070}
Let $\mathcal{M}$ be a model. For every $i\in I$ and every group $G$, we put $\mathit{BEL_i} := \mathit{BEL_i}(\mathit{TRUE})$ and $\mathit{COMMON_G}:=\mathit{COMMON_G}(\mathit{TRUE})$, which are the sets of propositions known by agent $i$, and the sets of propositions that are common knowledge in $G$, respectively.
\end{definition}
\begin{definition}\label{1070}
Let $\mathcal{M}$ be a model, $G$ be a group. We say that a set $X\subseteq M$ of propositions is closed under $G$ if the following hold:\\
(a) $X \subseteq \bigcap_{i\in G} \mathit{BEL_i}$, i.e. the propositions of $X$ are known by all agents of $G$,\\
(b) if $m\in X$ and $i\in G$, then $f_{K_i}(m)\in X$. \\
By $\mathit{GREATEST_G}$ we denote the greatest set closed under $G$, i.e. the union of all sets which are closed under $G$.
\end{definition}
Formally, the set $\mathit{COMMON_G}$ represents common knowledge in $G$. On the other hand, the set $\mathit{GREATEST_G}$ captures the concept of common knowledge in $G$ in an \textit{intuitive way}. Do these two sets coincide? By the definitions, we have:
\begin{lemma}\label{1090}
Let $\mathcal{M}$ be a model. For any group $G$, $\mathit{COMMON_G}$ is closed under $G$. In particular, $\mathit{COMMON_G} \subseteq \mathit{GREATEST_G} \subseteq \bigcap_{i\in G} \mathit{BEL_i}$.
\end{lemma}
Relative to an interpretation $(\mathcal{M},\gamma)$, common knowledge given as an infinite conjunction according to \eqref{50} is expressed in the following way:
$(\mathcal{M},\gamma)\vDash C_G\varphi$ $\Leftrightarrow$ for all $r\ge 0$ and for all sequences $(i_1,...,i_r)$ of agents of $G$, it holds that $(\mathcal{M},\gamma)\vDash K_{i_1} K_{i_2} ... K_{i_r}\varphi$.\footnote{Of course, repetitions of agents are allowed in the sequences. For $r=0$, we define $K_{i_1} K_{i_2} ... K_{i_r}\varphi := \varphi$.} \\
If $\varphi\in Fm$ denotes the proposition $m\in M$, i.e. $\gamma(\varphi)=m$, then that is equivalent to:
$f_{C_G}(m)\in \mathit{TRUE}$ $\Leftrightarrow$ for all $r \ge 0$ and for all sequences $(i_1, ..., i_r)$ of agents of $G$, it holds that $f_{K_{i_1}}(f_{K_{i_2}}(...(f_{K_{i_r}}(m))...)) \in \mathit{TRUE}$.\footnote{Again, for $r=0$ we let $f_{K_{i_1}}(f_{K_{i_2}}(...(f_{K_{i_r}}(m))...)):=m$.}
\begin{definition}\label{1094}
Suppose $\mathcal{M}$ is a model. Let $m\in M$ and $G$ be a group. We call the set $X_{G,m}$ given by all elements $f_{K_{i_1}}(f_{K_{i_2}}(...(f_{K_{i_r}}(m))...))$, where $r \ge 0$ and $(i_1, i_2, ..., i_r)$ is any sequence of agents of $G$, the closure of $m$ under $G$ or the $G$-closure of $m$.
\end{definition}
\begin{lemma}\label{1100}
Let $\mathcal{M}$ be a model. For any $m\in M$, $f_{C_G}(m)\in \mathit{TRUE}$ implies $X_{G,m} \subseteq \mathit{TRUE}$.
\end{lemma}
\begin{proof}
Let $f_{C_G}(m)\in \mathit{TRUE}$, i.e. $m\in \mathit{COMMON_G}$. Applying successively truth condition (iii)(e) of a model (Definition \ref{810}), one recognizes that any element $f_{K_{i_1}}(f_{K_{i_2}}(...(f_{K_{i_r}}(m))...))$ belongs to $\mathit{COMMON_G}$, where $i_1,...,i_r\in G$ and $r\ge 0$. Hence, $X_{G,m}\subseteq \mathit{COMMON_G}\subseteq \mathit{TRUE}$, and the assertion follows.
\end{proof}
The next result, also adopted from \cite{lewsl}, gives a sufficient and necessary condition for common knowledge having the intended meaning in a given model (independently of the fact whether we are dealing with the basic notion or with introspective common knowledge).
\begin{theorem}[\cite{lewsl}]\label{1200}
Let $\mathcal{M}$ be a model, $G$ be a group. The following conditions are equivalent:
\begin{enumerate}
\item $\mathit{COMMON_G} = \mathit{GREATEST_G}$
\item For any $m\in M$, $f_{C_G}(m)\in \mathit{TRUE}$ iff $X_{G,m} \subseteq \mathit{TRUE}$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\mathit{COMMON_G} = \mathit{GREATEST_G}$. By Lemma \ref{1100}, we know that for any $m\in M$, $f_{C_G}(m)\in \mathit{TRUE}$ implies $X_{G,m} \subseteq \mathit{TRUE}$. Suppose $m\in M$ and $X_{G,m} \subseteq \mathit{TRUE}$. Then the $G$-closure of $m$, $X_{G,m}$, is closed under $G$ in the sense of Definition \ref{1070}. Since $GREATEST_G$ is the greatest set closed under $G$, we have $m\in X_{G,m}\subseteq \mathit{GREATEST_G}=\mathit{COMMON_G}$. Hence, $f_{C_G}(m)\in \mathit{TRUE}$ and (i) $\Rightarrow$ (ii) holds true. Now, suppose (ii) holds true and $m\in \mathit{GREATEST_G}$. Since $\mathit{GREATEST_G}$ is closed under $G$, we have $f_{K_{i_1}}(f_{K_{i_2}}(...(f_{K_{i_r}}(m))...))\in\mathit{GREATEST_G}$, for any $r\ge 0$ and any sequence $(i_1, i_2, ..., i_r)$ of agents of $G$. Thus, $X_{G,m}\subseteq \mathit{GREATEST_G}\subseteq \mathit{TRUE}$. By (ii), $f_{C_G}(m)\in \mathit{TRUE}$, i.e. $m\in \mathit{COMMON_G}$. Thus, $\mathit{GREATEST_G}\subseteq \mathit{COMMON_G}$ and (i) follows.
\end{proof}
\begin{definition}\label{1300}
A model is said to be an intended model if for each group $G$, $\mathit{COMMON_G} = \mathit{GREATEST_G}$.
\end{definition}
\begin{corollary}\label{1340}
Let $\mathcal{M}$ be a model. Suppose that for all $m\in M$ and all groups $G$, the $G$-closure of $m$ is finite (and thus its infimum exists) and the following holds: $f_{C_G}(m)\in\mathit{TRUE} \Leftrightarrow \bigwedge X_{G,m}\in\mathit{TRUE}$. Then $\mathcal{M}$ is an intended model.
\end{corollary}
\begin{comment}
\begin{definition}\label{1350}
We call a formula \textit{free} (free from assertions concerning common knowledge) if it does not contain subformulas of the form $C_G\psi$.
\end{definition}
\begin{theorem}\label{1360}
Let $\mathcal{M}$ be a model such that for all $m\in M$ and all groups $G$, $\bigwedge X_{G,m}$ exists and the following holds: $ X_{G,m}\subseteq\mathit{TRUE} \Rightarrow \bigwedge X_{G,m}\in\mathit{TRUE}$. Then there is an intended model $\mathcal{M}'$ over the same universe $M$ such that for every assignment $\gamma\colon V\rightarrow M$ and every free formula $\varphi\in Fm$:
\begin{equation*}
(\mathcal{M},\gamma)\vDash\varphi\Leftrightarrow (\mathcal{M}',\gamma)\vDash\varphi.
\end{equation*}
\end{theorem}
\begin{proof}
Let $\mathcal{M}$ be a finite model with opertors $f_{K_i}$ for knowledge and $f_{C_G}$ for common knowledge. We replace the original operators $f_{C_G}$ with operators $f'_{C_G}$ defined in the following way. For each $m\in M$:
\begin{equation*}
f'_{C_G}(m):=\bigwedge X_{G,m}.
\end{equation*}
Note that the infimum of the finite set $X_{G,m}$ exists, and $f'_{C_G}(m)$ is therefore well-defined. We must show that the resulting structure
\begin{equation*}
\mathcal{M}'=(M,\mathit{TRUE}, f_\vee, f_\wedge, f_\bot, f_\rightarrow, f_\square, (f_{K_i})_{i\in I}, (f'_{C_G})_{G\in\mathcal{G}})
\end{equation*}
is a model in the sense of the Definitions \ref{810} and \ref{850}. It is enough to verify the truth conditions regarding common knowledge, i.e. $\mathit{COMMON_G'(F)}:=\{m\in M\mid f'_{C_G}(m)\in F\}$ must satisfy the conditions (iii)(b), (d), (e), (f) and (g), for any prime filter $F$ and any $G$. We claim that $\mathit{COMMON_G'(F)}:=\{m\in M\mid f'_{C_G}(m)\in F\}$ is a filter, for any prime filter $F$ and any $G$. We have $f'_{C_G}(f_\top)= \bigwedge X_{G,f_\top}=f_\top\in F$ since $f_{K_i}(f_\top)=f_\top$, for any $i\in I$. Thus, $f_\top\in\mathit{COMMON_G'(F)}$. Now, suppose $m$ and $f_\rightarrow(m,m')$ belong to $\mathit{COMMON_G'(F)}$. That is,
\begin{equation}\label{2000}
\bigwedge X_{G,m}\text{ and }\bigwedge X_{G,f_\rightarrow(m,m')}\text{ are elements of }F.
\end{equation}
Then $F$ contains, in particular, $m$ and $f_\rightarrow(m,m')$. Hence, $m'\in F$. Suppose $i\in G$. By \eqref{2000}, $f_{K_i}(m)$ and $f_{K_i}(f_\rightarrow(m,m'))$ belong to $F$. By properties of a model, $f_{K_i}(f_\rightarrow(m,m'))\le f_\rightarrow (f_{K_i}(m), f_{K_i}(m'))\in F$ and thus $f_{K_i}(m')\in F$. Now, suppose $j\in G$. Again by \eqref{2000}, $f_{K_j} (f_{K_i}(m))$ and $f_{K_j}(f_{K_i}(f_\rightarrow(m,m')))$ belong to $F$. By Lemma \ref{815}, $f_{K_j}$ is a monotonic function. Then $f_{K_j}(f_{K_i}(f_\rightarrow(m,m')))$ $\le$ $f_{K_j}(f_\rightarrow (f_{K_i}(m), f_{K_i}(m')))$ $\le$ $f_\rightarrow (f_{K_j}(f_{K_i}(m), f_{K_j}(f_{K_i}(m')))\in F$. Hence, $f_{K_j}(f_{K_i}(m')))\in F$. Successively, we show that for any finite sequence $(i_1, i_2, ..., i_r)$ of agents from $G$ it holds that $f_{K_{i_1}}(f_{K_{i_2}}(...(f_{K_{i_r}}(m'))...))\in F$. Thus, $X_{G,m'}\subseteq F$. We may take the infinum of that finite set and get $\bigwedge X_{G,m'}\in F$. Hence, $m'\in \mathit{COMMON_G'(F)}$ and this set is a filter. It is not hard to check that also the remaining truth conditions of a model are satisfied.
\end{proof}
\end{comment}
\begin{example}
Simple examples of models are given by linearly ordered Heyting algebras. Note that such Heyting algebras always have the Disjunction Property; actually, all proper filters are prime. We modify and extend an example from \cite{lewigpl}. It is based on the Heyting algebra over the closed interval $M:=[0,1]$ of reals with its usual linear ordering and the unique ultrafilter $\mathit{TRUE}=(0,1]$. To agents $i=1,2,...,N$ we assign elements $b(1)< b(2)< ... < b(N) \in (0,1)$, respectively, and consider the prime filters $\mathit{BEL_i}=\{m\in M\mid b(i)\le m\}$. Then $\mathit{BEL_1}\supsetneq \mathit{BEL_2}\supsetneq ... \supsetneq \mathit{BEL_N}$ are the sets of propositions known by agent $i$, respectively. We define $f_{K_i}(m):=m$ if $m\in \mathit{BEL_i}$, and $f_{K_i}(m):=0$ otherwise. Then follows that $\mathit{BEL_i}(F)=F\cap \mathit{BEL_i}$, for any prime filter $F$. Thus, each $\mathit{BEL_i}(F)$ is a filter contained in $F$, in accordance with the truth conditions (iii)(a) and (iii)(c)* of Definitions \ref{810} and \ref{850}. For each group $G$ and $m\in M$, we define $f_{C_G}(m) := f_{K_{i_G}}(m)$, where $i_G\in G$ is the greatest number referring to an agent of $G$. Then common knowledge in $G$ is given by $\mathit{COMMON_G}=\mathit{BEL_{i_G}}=\bigcap\{\mathit{BEL_j}\mid j\in G\}$ and $\mathit{COMMON_G}(F)=\mathit{BEL_{i_G}}(F)$ for any prime filter $F$. Of course, we put $f_\square(1) := 1$ and $f_\square(m) := 0$ for $0\le m < 1$. Now one recognizes that all truth conditions of an $L5^{AC}_N$-model are satisfied (use the Definitions \ref{810} and \ref{850} or/and Theorem \ref{870}). It is an intended model since for each $G$, $\mathit{COMMON_G}$ is the greatest set closed under $G$: it is clear that $\mathit{COMMON_G}$ is closed under $G$; and it is the greatest set with that property because for any $m\in M\smallsetminus \mathit{COMMON_G}$, we have $m\notin\mathit{BEL_{i_G}}$. Unfortunately, common knowledge in $G$ is trivial in the sense that it coincides with `everyone in $G$ knows': $E_G\varphi$ is true iff $C_G\varphi$ is true. We modify the model in the following way. We only change common knowledge in the group $G=\{1,...,N\}$ of all agents and leave all other definitions as before. Let $c$ be a real number such that $b(N) < c < 1$. Then we consider $\mathit{COMMON_G}:=\{m\in M\mid m\ge c\}\subsetneq \mathit{BEL_N}$ and define $f_{C_G}(m):=m$ if $m\in \mathit{COMMON_G}$, and $f_{C_G}(m):=0$ otherwise. Again, one verifies that the resulting structure is an $L5^{AC}_N$-model. Since $\mathit{COMMON_G}$ is a proper subset of $\mathit{BEL_N}$, common knowledge in $G$ is no longer trivial, i.e. it is strictly stronger than `everyone knows'. However, the resulting model is not an intended one: $\mathit{COMMON_G}\subsetneq\mathit{GREATEST_G}=\mathit{BEL_N}$. Finally, we construct an intended model with non-trivial common knowledge. For $i=1,...,N$, the sets $\mathit{BEL_i}$ are defined as before. For the singleton group $G=\{1\}$, we put $\mathit{COMMON_{\{1\}}} := \mathit{BEL_1}$, and for any $m\in M$: $f_G(m):=f_{K_1}(m)$, with $f_{K_1}$ defined as below. For all other groups $G$, we define, with the same real number $c$ as above, $\mathit{COMMON_G} := P :=\{m\in M\mid m\ge c\}$, and $f_{C_G}(m):=m$ if $m\in P$, and $f_{C_G}(m):=0$ otherwise. Thus, all groups distinct from the singleton group $\{1\}$ have exactly the same common knowledge given by the prime filter $P=\{m\in M\mid m\ge c\}\subsetneq \mathit{BEL_N}$. The $f_{K_i}$, $i=1,...,N$, are now defined in the following way:
\begin{equation*}
f_{K_i}(m)=
\begin{cases}
\begin{split}
&b_1,\text{ if }m\in \mathit{BEL_i}\smallsetminus P\\
&m, \text{ if }m\in P\\
&0,\text{ else}
\end{split}
\end{cases}
\end{equation*}
Notice that for any agent $i \neq 1$, $m\notin P$ implies $f_{K_i}(m)\notin\mathit{BEL_i}$. Thus, for all groups $G\neq \{1\}$, $P=\{m\in M\mid m\ge c\}$ is the greatest closed set under $G$. And for $G=\{1\}$, $\mathit{BEL_1}=\mathit{COMMON_{\{1\}}}$ is the greatest set closed under $G$. Hence, the eventual model is an intended model. Of course, $f_\square$ is defined as before. Similarly as in the previous example, one checks that all truth conditions of an $L5^{AC}_N$-model are satisfied. Common knowledge in any group distinct from the singleton $\{1\}$ is not trivial, i.e. it is stronger than `everyone knows': $P$ is a proper subset of each $\mathit{BEL_i}$. This model can be transformed into an intended $L5^{AC^-}_N$-model modifying for some $G\neq\{1\}$ the function $f_{C_G}$ in the following way: Let $d$ be a real such that $0< d < c$. Then define $f_{C_G}(m):=m-d$ if $m\in \mathit{COMMON_G}=P$, and $f_{C_G}(m):=0$ otherwise. Now there are some $m\in \mathit{COMMON_G}$ such that $f_{C_G}(m)\notin\mathit{COMMON_G}$. Hence, the model cannot be an $L5^{AC}_N$-model. But the truth conditions of an $L5^{AC^-}_N$-model are still satisfied.
\end{example}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,881
|
{"url":"https:\/\/www.omnicalculator.com\/physics\/compton-scattering","text":"Compton Scattering Calculator\n\nCreated by Wojciech Sas, PhD\nReviewed by Bogna Szyk and Steven Wooding\nLast updated: Dec 06, 2022\n\nThe Compton scattering calculator is a tool that helps you to find the wavelength extension of a photon scattered by any angle in this particular case.\n\nThe Compton effect is among the possible interactions between light and matter. After this process, the wavelength of the photon increases, and the traveling direction changes. It is one of the phenomena which confirm the corpuscular nature of light.\n\nScattering process\n\nA single photon can be treated as a massless particle that transports energy depending on its wavelength or frequency (you can quickly switch between these two using our wavelength calculator).\n\nWhen a photon collides with another particle, let's say an electron, its propagating direction can change. We can describe the process as a classical elastic collision between two bodies, meaning that energy and momentum are conserved. If the electron is free or weakly bonded in the atom, we can assume that all of the initial energy comes from the incident photon. This assumption is well satisfied for X-rays, for which the Compton effect plays an important role.\n\nWavelength extension formula\n\nWe can determine the difference between the wavelengths of scattered and incident light with the Compton scattering equation:\n\n$\\Delta\\lambda = \\frac{h}{m\\,c}(1 - \\cos(\\theta)),$\n\nwhere:\n\n\u2022 $h = 6.62607 \u00d7 10^{-34}\\ \\rm J\/s$ \u2013 Planck constant;\n\u2022 $m$ \u2013 Mass of the particle;\n\u2022 $c = 299\\, 792\\, 458\\ \\rm m\/s$ is the speed of light; and\n\u2022 $\\theta$ \u2013 Angle between the directions of scattered and incident rays.\n\n$h \/ (m\\,c)$ is a Compton wavelength of the object which causes the scattering. You can find more information about this topic in Omni's Compton wavelength calculator.\n\nWe can easily find that the most significant extension of photon wavelength happens for $\\theta = 180\\degree$, which equals two Compton wavelengths, whereas, for $\\theta = 0\\degree$, there is no energy loss, which corresponds to no scattering process.\n\nCompton effect on free electron \u2013 example\n\nLet's calculate the wavelength extension of a photon scattered on the free electron by an angle $\\small \\theta = 80\\degree$.\n\n\u2022 Find the rest mass of electron: $\\small m = 9.109 \\times 10^{-31}\\ \\rm kg$;\n\n\u2022 Estimate cosine of a given angle: $\\small\\cos(80\\degree) = 0.174$;\n\n\u2022 The Compton wavelength of an electron is:\n\n$\\scriptsize \\quad\\ \\ \\begin{split} \\lambda_{\\rm e} &= \\frac{6.626\\!\\times\\!10^{-34}\\ {\\rm J\/s}}{9.109\\! \\times\\! 10^{-31}\\ {\\rm kg} \\times 2.998\\!\\times\\!10^8\\ {\\rm m\/s}}\\\\[1em] &= 2.426\\times10^{-12}\\ {\\rm m}\\\\[.5em] &= 2.426\\ {\\rm pm} \\end{split}$\n\u2022 Work out a wavelength extension: $\\small\\Delta\\lambda = 2.426\\ {\\rm pm} \\times (1\\! -\\! 0.174) = 2.005\\ \\rm pm$.\n\nNow let's consider two photons of different initial energies: 2 eV and 20 keV, which correspond to red-orange light and so-called hard X-ray, respectively. We can easily find the values of wavelengths with photon energy calculator and equal about 620 nm and 62 pm.\n\nYou can estimate the wavelength extension as the percentages of initial wavelengths, which are 0.000323% and 3.23%, respectively. These results mean that Compton scattering is practically negligible for visible light, whereas for X-rays, a noticeable fraction of energy changes.\n\nWojciech Sas, PhD\nMass\nme\nScattering angle\ndeg\nWavelength extension\npm\nPeople also viewed\u2026\n\nAir pressure at altitude\n\nIf you ever wondered what is the atmospheric pressure high up in the mountains, our air pressure at altitude calculator is there to help you.\n\nDiscount\n\nDiscount calculator uses a product's original price and discount percentage to find the final price and the amount you save.\n\nSnowman\n\nThe perfect snowman calculator uses math & science rules to help you design the snowman of your dreams!","date":"2023-03-24 10:00:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 13, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8067291975021362, \"perplexity\": 886.5331769901089}, \"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-2023-14\/segments\/1679296945279.63\/warc\/CC-MAIN-20230324082226-20230324112226-00239.warc.gz\"}"}
| null | null |
{"url":"https:\/\/jh-jeong.github.io\/publications\/2021-05-24-c5\/","text":"# [C5] Training GANs with Stronger Augmentations via Contrastive Discriminator\n\nPublished in International Conference on Learning Representations (ICLR), 2021\n\ntl;dr: We propose a novel discriminator of GAN showing that contrastive representation learning, e.g., SimCLR, and GAN can benefit each other when they are jointly trained. \n\n#### Abstract\n\nRecent works in Generative Adversarial Networks (GANs) are actively revisiting various data augmentation techniques as an effective way to prevent discriminator overfitting. It is still unclear, however, that which augmentations could actually improve GANs, and in particular, how to apply a wider range of augmentations in training. In this paper, we propose a novel way to address these questions by incorporating a recent contrastive representation learning scheme into the GAN discriminator, coined ContraD. This \"fusion\" enables the discriminators to work with much stronger augmentations without increasing their training instability, thereby preventing the discriminator overfitting issue in GANs more effectively. Even better, we observe that the contrastive learning itself also benefits from our GAN training, i.e., by maintaining discriminative features between real and fake samples, suggesting a strong coherence between the two worlds: good contrastive representations are also good for GAN discriminators, and vice versa. Our experimental results show that GANs with ContraD consistently improve FID and IS compared to other recent techniques incorporating data augmentations, still maintaining highly discriminative features in the discriminator in terms of the linear evaluation. Finally, as a byproduct, we also show that our GANs trained in an unsupervised manner (without labels) can induce many conditional generative models via a simple latent sampling, leveraging the learned features of ContraD. Code is available at https:\/\/github.com\/jh-jeong\/ContraD.\n\n#### BibTeX\n\n@inproceedings{jeong2021training,\ntitle={Training {GAN}s with Stronger Augmentations via Contrastive Discriminator},\nauthor={Jongheon Jeong and Jinwoo Shin},\nbooktitle={International Conference on Learning Representations},\nyear={2021},\nurl={https:\/\/openreview.net\/forum?id=eo6U4CAwVmg}\n}\n\n\nUpdated:","date":"2022-12-07 04:24:04","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\": 0.3943675458431244, \"perplexity\": 2203.940194211195}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711126.30\/warc\/CC-MAIN-20221207021130-20221207051130-00084.warc.gz\"}"}
| null | null |
import re
from algo.algorithm import Algorithm
class Plugin(Algorithm):
def __init__(self, parent=None):
super().__init__(parent)
self.manager = None
def match(self, data):
return None
@staticmethod
def _re_match(pattern, data):
match = re.match(pattern, data, re.I)
if match:
return match.group(1)
else:
return None
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,455
|
Byblia goetzius är en fjärilsart som beskrevs av Herbst 1798. Byblia goetzius ingår i släktet Byblia och familjen praktfjärilar. Inga underarter finns listade i Catalogue of Life.
Källor
Praktfjärilar
goetzius
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,060
|
William Charles Aalsmeer (* 17. Oktober 1889 in Paramaribo; † 29. April 1957 in Den Haag) war ein niederländischer Kardiologe. Er war an der Medizinischen Klinik der Niederländisch-Indischen Ärzteschule zu Surabaya (Java) tätig.
Aalsmeer-Test
Aalsmeer entwickelte den nach ihm bezeichneten Aalsmeer-Test, bei dem es durch die intramuskuläre Injektion von 1 mg Adrenalin zu einer temporären Verschlimmerung der kardiovaskulären Symptome bei der Beri-Beri-Krankheit kommt. Hierbei sinkt der Blutdruck innerhalb einer Stunde bis in den unmessbaren Bereich ab.
Es wurde später festgestellt, dass dieser bei der Beriberi auftretende Adrenalineffekt auch bei anderen Erkrankungen mit ähnlichen peripheren Zirkulationsstörungen auftritt. So gibt es je nach Grad der Zirkulationsstörung beispielsweise auffallend ähnliche Reaktionen bei Aorteninsuffizienz und Morbus Basedow. Der Adrenalineffekt in Kombination mit dem erniedrigten Minimaldruck zeigt besser als der sogenannte Pulsdruck den Grad bzw. den Umfang des Defektes in der Zirkulation an.
Literatur
Gerrit Arie Lindeboom: Dutch medical biography. A biographical dictionary of Dutch physicians and surgeons 1475–1975. Rodopi, Amsterdam 1984, ISBN 90-6203-676-7.
Weblinks
Kardiologe
Mediziner (20. Jahrhundert)
Niederländer
Geboren 1889
Gestorben 1957
Mann
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{"url":"http:\/\/nrich.maths.org\/public\/leg.php?code=-93&cl=4&cldcmpid=7379","text":"Search by Topic\n\nResources tagged with Making and proving conjectures similar to Sheep in Wolf's Clothing:\n\nFilter by: Content type:\nStage:\nChallenge level:\n\nOther tags that relate to Sheep in Wolf's Clothing\nengineering. Argand\u00a0diagram. Quadratic\u00a0equations. Rotations. Groups. Complex\u00a0numbers. Sine. Relations. Interactivities. Isomorphism.\n\nThere are 58 results\n\nBroad Topics > Using, Applying and Reasoning about Mathematics > Making and proving conjectures\n\nThebault's Theorem\n\nStage: 5 Challenge Level:\n\nTake any parallelogram and draw squares on the sides of the parallelogram. What can you prove about the quadrilateral formed by joining the centres of these squares?\n\nConjugate Tracker\n\nStage: 5 Challenge Level:\n\nMake a conjecture about the curved track taken by the complex roots of a quadratic equation and use complex conjugates to prove your conjecture.\n\nShuffles\n\nStage: 5 Challenge Level:\n\nAn environment for exploring the properties of small groups.\n\nHow Old Am I?\n\nStage: 4 Challenge Level:\n\nIn 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?\n\nPolycircles\n\nStage: 4 Challenge Level:\n\nShow that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?\n\nCyclic Triangles\n\nStage: 5 Challenge Level:\n\nMake and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.\n\nTri-split\n\nStage: 4 Challenge Level:\n\nA point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?\n\nPythagorean Fibs\n\nStage: 5 Challenge Level:\n\nWhat have Fibonacci numbers got to do with Pythagorean triples?\n\nLeast of All\n\nStage: 5 Challenge Level:\n\nA point moves on a line segment. A function depends on the position of the point. Where do you expect the point to be for a minimum of this function to occur.\n\nFew and Far Between?\n\nStage: 4 and 5 Challenge Level:\n\nCan you find some Pythagorean Triples where the two smaller numbers differ by 1?\n\nTrig Rules OK\n\nStage: 5 Challenge Level:\n\nChange the squares in this diagram and spot the property that stays the same for the triangles. Explain...\n\nPlus or Minus\n\nStage: 5 Challenge Level:\n\nMake and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.\n\nTriangles Within Triangles\n\nStage: 4 Challenge Level:\n\nCan you find a rule which connects consecutive triangular numbers?\n\nVecten\n\nStage: 5 Challenge Level:\n\nJoin in this ongoing research. Build squares on the sides of a triangle, join the outer vertices forming hexagons, build further rings of squares and quadrilaterals, investigate.\n\nTriangles Within Squares\n\nStage: 4 Challenge Level:\n\nCan you find a rule which relates triangular numbers to square numbers?\n\nTriangles Within Pentagons\n\nStage: 4 Challenge Level:\n\nShow that all pentagonal numbers are one third of a triangular number.\n\nFibonacci Fashion\n\nStage: 5 Challenge Level:\n\nWhat have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?\n\nCenter Path\n\nStage: 3 and 4 Challenge Level:\n\nFour rods of equal length are hinged at their endpoints to form a rhombus. The diagonals meet at X. One edge is fixed, the opposite edge is allowed to move in the plane. Describe the locus of. . . .\n\nPainting by Numbers\n\nStage: 5 Challenge Level:\n\nHow many different colours of paint would be needed to paint these pictures by numbers?\n\nClose to Triangular\n\nStage: 4 Challenge Level:\n\nDrawing a triangle is not always as easy as you might think!\n\nA Little Light Thinking\n\nStage: 4 Challenge Level:\n\nHere is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?\n\nCurvy Areas\n\nStage: 4 Challenge Level:\n\nHave a go at creating these images based on circles. What do you notice about the areas of the different sections?\n\nMultiplication Arithmagons\n\nStage: 4 Challenge Level:\n\nCan you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?\n\nStage: 4 Challenge Level:\n\nExplore the relationship between quadratic functions and their graphs.\n\nSteve's Mapping\n\nStage: 5 Challenge Level:\n\nSteve has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?\n\nWhat's Possible?\n\nStage: 4 Challenge Level:\n\nMany numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?\n\nDiscrete Trends\n\nStage: 5 Challenge Level:\n\nFind the maximum value of n to the power 1\/n and prove that it is a maximum.\n\nThe Clue Is in the Question\n\nStage: 5 Challenge Level:\n\nThis problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .\n\nPrime Sequences\n\nStage: 5 Challenge Level:\n\nThis group tasks allows you to search for arithmetic progressions in the prime numbers. How many of the challenges will you discover for yourself?\n\nAlison's Mapping\n\nStage: 4 Challenge Level:\n\nAlison has created two mappings. Can you figure out what they do? What questions do they prompt you to ask?\n\nEpidemic Modelling\n\nStage: 4 and 5 Challenge Level:\n\nUse the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.\n\n2^n -n Numbers\n\nStage: 5\n\nYatir from Israel wrote this article on numbers that can be written as $2^n-n$ where n is a positive integer.\n\nTo Prove or Not to Prove\n\nStage: 4 and 5\n\nA serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.\n\nStage: 4 Challenge Level:\n\nThe points P, Q, R and S are the midpoints of the edges of a convex quadrilateral. What do you notice about the quadrilateral PQRS as the convex quadrilateral changes?\n\nStage: 4 Challenge Level:\n\nThe points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?\n\nIntegral Sandwich\n\nStage: 5 Challenge Level:\n\nGeneralise this inequality involving integrals.\n\nStage: 4 Challenge Level:\n\nPoints D, E and F are on the the sides of triangle ABC. Circumcircles are drawn to the triangles ADE, BEF and CFD respectively. What do you notice about these three circumcircles?\n\nFixing It\n\nStage: 5 Challenge Level:\n\nA and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?\n\nSummats Clear\n\nStage: 5 Challenge Level:\n\nFind the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.\n\nPericut\n\nStage: 4 and 5 Challenge Level:\n\nTwo semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?\n\nPentagon\n\nStage: 4 Challenge Level:\n\nFind the vertices of a pentagon given the midpoints of its sides.\n\nOK! Now Prove It\n\nStage: 5 Challenge Level:\n\nMake a conjecture about the sum of the squares of the odd positive integers. Can you prove it?\n\nLoopy\n\nStage: 4 Challenge Level:\n\nInvestigate sequences given by $a_n = \\frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?\n\nBinary Squares\n\nStage: 5 Challenge Level:\n\nIf a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?\n\nWhy Stop at Three by One\n\nStage: 5\n\nBeautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.\n\nThe Kth Sum of N Numbers\n\nStage: 5\n\nYatir from Israel describes his method for summing a series of triangle numbers.\n\nRecent Developments on S.P. Numbers\n\nStage: 5\n\nTake a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.\n\nPoly Fibs\n\nStage: 5 Challenge Level:\n\nA sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.\n\nFibonacci Factors\n\nStage: 5 Challenge Level:\n\nFor which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?","date":"2014-11-24 09:24:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5446483492851257, \"perplexity\": 1210.815922846475}, \"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-2014-49\/segments\/1416400380574.41\/warc\/CC-MAIN-20141119123300-00028-ip-10-235-23-156.ec2.internal.warc.gz\"}"}
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{"url":"https:\/\/openarchive.usn.no\/usn-xmlui\/handle\/11250\/218417\/browse?rpp=20&etal=-1&sort_by=1&type=title&starts_with=Y&order=ASC","text":"Now showing items 2217-2230 of 2230\n\n\u2022 #### You can't \u2018fake it till you make it\u2019: Cooperative motivation does not help proself trustees \ufeff\n\n(Peer reviewed; Journal article, 2020)\nCooperative motivation can be rooted in individual differences as well as in external factors, such as instructions from superiors, incentive schemes, policy agendas, or social relationships. Whereas cooperative motivation ...\n\u2022 #### \u2018You create your own luck, in a way\u2019 About Norwegian footballers\u2019 understanding of success, in a world where most fail \ufeff\n\n(Peer reviewed; Journal article, 2020)\nThe article is based on interviews with twelve young footballers who turned pro with the Norwegian top-flight club Odd BK. After asking why they succeeded in making the transition from promising talent to established ...\n\u2022 #### \u2018You notice that there is something positive about going to school\u2019: how teachers\u2019 kindness can promote positive teacher\u2013student relationships in upper secondary school \ufeff\n\n(Journal article; Peer reviewed, 2016)\nThis study aimed to obtain students\u2019 first-person perspectives of their experience of positive teacher\u2013student relationships (TSRs) in upper secondary school. We also explored their experiences of qualities of TSRs concerning ...\n\u2022 #### Young people's perceived service quality and environmental performance of hybrid electric bus service \ufeff\n\n(Peer reviewed; Journal article, 2020)\nWhile cities are moving towards environmentally friendly public transport such as electrification of public buses, the attractiveness of such innovative solutions among the bus users is yet to be explored. We study young ...\n\u2022 #### Youth multi-sport events in Austria: tourism strategy or just a coincidence? \ufeff\n\n(Journal article; Peer reviewed, 2017)\nImpact and legacy research of touristic issues has become more popular with the appearance of manifold approaches to examining tourism. In recent years, the region of Western Austria has successfully staged multi-sport ...\n\u2022 #### Yrkesfagl\u00e6reres profesjonelle kompetanse: En kvalitativ unders\u00f8kelse med Norge og Japan som kontekster. [Professional competences of vocational teachers: A qualitative survey with Norway and Japan as contexts]. \ufeff\n\n(Journal article; Peer reviewed, 2018)\nThe purpose of this article is to explore the characteristics of vocational teachers\u2019 professional competences. Norway and Japan were chosen as contexts in the hope of finding possible contrasts and new perspectives. ...\n\u2022 #### Z-boson production in p-Pb collisions at \u221asNN = 8.16 TeV and Pb-Pb collisions at \u221asNN = 5.02 TeV \ufeff\n\n(Peer reviewed; Journal article, 2020)\nMeasurement of Z-boson production in p-Pb collisions at \u221a sNN = 8.16 TeV and Pb-Pb collisions at \u221a sNN = 5.02 TeV is reported. It is performed in the dimuon decay channel, through the detection of muons with pseudorapidity ...\n\u2022 #### \u039b c + production in pp collisions at \u221as=7 TeV and in p-Pb collisions at \u221asNN=5.02 TeV \ufeff\n\n(Journal article; Peer reviewed, 2018)\nThe pT-differential production cross section of prompt A+c charmed baryons was measured with the ALICE detector at the Large Hadron Collider (LHC) in pp collisions at \u221as=7 TeV and in p-Pb collisions at \u221asNN = 5.02 TeV at ...\n\u2022 #### \u039bc+ production in Pb\u2013Pb collisions at \u221asNN = 5.02 TeV \ufeff\n\n(Journal article; Peer reviewed, 2019)\nA measurement of the production of prompt baryons in Pb\u2013Pb collisions at TeV with the ALICE detector at the LHC is reported. The and were reconstructed at midrapidity () via the hadronic decay channel (and charge conjugate) ...\n\u2022 #### \u03c00 and \u03b7 meson production in proton-proton collisions at \u221as = 8 TeV \ufeff\n\n(Journal article; Peer reviewed, 2018)\nAn invariant differential cross section measurement of inclusive \u03c00 and \u03b7 meson production atmid-rapidity in pp collisions at \u221as = 8 TeV was carried out by the ALICE experiment at the LHC. The spectra of \u03c00 and \u03b7 mesons ...\n\u2022 #### \u03c6 meson production in Pb-Pb collisions at \u221asNN = 5.02 TeV with ALICE at the LHC \ufeff\n\n(Journal article; Peer reviewed, 2017)\nStrangeness production is a key tool to understand the properties of the medium formed in heavy-ion collisions: an enhanced production of strange particles was early proposed as one of the signatures of the Quark-Gluon ...\n\u2022 #### \u03c6-meson production at forward rapidity in p\u2013Pb collisions at \u221asNN=5.02 TeV and in pp collisions at \u221as=2.76 TeV \ufeff\n\n(Journal article; Peer reviewed, 2017)\nThe first study of \u03c6-meson production in p\u2013Pb collisions at forward and backward rapidity, at nucleon\u2013nucleon centre-of-mass energy \u221asNN=5.02TeV, has been performed with the ALICE apparatus at the LHC. The \u03c6-mesons have ...\n\u2022 #### \u03d2 suppression at forward rapidity in Pb\u2013Pb collisions at \u221asNN=5.02 TeV \ufeff\n\n(Journal article; Peer reviewed, 2019)\nInclusive \u03d2(1S) and \u03d2(2S) production have been measured in Pb\u2013Pb collisions at the centre-of-mass energy per nucleon\u2013nucleon pair \u221asNN = 5.02TeV, using the ALICE detector at the CERN LHC. The \u03d2mesons are reconstructed in ...\n\u2022 #### \u03d5 meson production at forward rapidity in Pb\u2013Pb collisions at \u221asNN=2.76 TeV \ufeff\n\n(Journal article; Peer reviewed, 2018)\n\u03d5 meson measurements provide insight into strangeness production, which is one of the key observables for the hot medium formed in high-energy heavy-ion collisions. ALICE measured \u03d5 production through its decay in muon ...","date":"2021-07-31 23:54:34","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8012714385986328, \"perplexity\": 11297.641736116344}, \"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-2021-31\/segments\/1627046154127.53\/warc\/CC-MAIN-20210731234924-20210801024924-00188.warc.gz\"}"}
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{"url":"https:\/\/genominfo.org\/journal\/view.php?number=61","text":"\u2022 Home\n\u2022 E-Submission\n\u2022 Sitemap\nGenomics Inform Search\n\nCLOSE\n\n Genomics Inform > Volume 11(4); 2013 > Article\nWang and Xing: A Primer for Disease Gene Prioritization Using Next-Generation Sequencing Data\n\nAbstract\n\nHigh-throughput next-generation sequencing (NGS) technology produces a tremendous amount of raw sequence data. The challenges for researchers are to process the raw data, to map the sequences to genome, to discover variants that are different from the reference genome, and to prioritize\/rank the variants for the question of interest. The recent development of many computational algorithms and programs has vastly improved the ability to translate sequence data into valuable information for disease gene identification. However, the NGS data analysis is complex and could be overwhelming for researchers who are not familiar with the process. Here, we outline the analysis pipeline and describe some of the most commonly used principles and tools for analyzing NGS data for disease gene identification.\n\nIntroduction\n\nThe breakthrough in next-generation sequencing (NGS) techniques has enabled rapid sequencing of whole genomes at low cost and has brought tremendous opportunities as well as challenges to biomedical research [1-5]. The cost of sequencing is projected to continue to drop, with whole-genome sequencing expected to be as low as $1000 [4] and whole-exome sequencing (i.e., sequencing approximately 1% of the coding regions of the genome [6]) to be$500 [1]. Such low costs enable a small research lab to generate large amounts of sequence data in a short period of time on a relatively small budget. The low cost also allows sequencing-based clinical genetic tests to be routinely performed to provide guidance for health professionals for disease etiology and management. The challenge now is how to reliably synthesize and interpret this large amount of raw data from sequencing platforms.\nOne goal of genomic research is to determine which genes are causal for the disease of interest. With the NGS data, this is not a trivial task. Large-scale whole-genome and whole-exome sequencing analyses identify large amounts of genomic variants, most of which are not related to disease risk. Thus, disease gene prioritization principles and tools are needed [2] to identify and prioritize variants of interest from the large pool of candidates. Currently, identifying candidate causal mutations\/genes is often a complex practice that involves several dozens of steps and\/or tools to accomplish. These tools work jointly in a series of steps and thus functionally form a \"workflow\" or a \"pipeline.\" Ideally, the process of disease gene prioritization generates an ordered list of genes that puts high-risk candidates on the top of the stack and filters out \"benign\" or neutral variants. In reality, this process usually involves multiple filtering steps, some of which have underlying assumptions that can affect how variants are filtered.\nThe focus of this review is to provide a broad overview of the workflow and present some of the bioinformatics software tools that are currently available. More in-depth comparisons of the bioinformatics algorithms and tools are described elsewhere [3, 5].\n\nWorkflow for Disease Gene Filtering\/Prioritization\n\nTypically, the output from NGS platforms is in the standard FASTQ format. The FASTQ file is a text file containing a list of short DNA fragments, generally less than 150 base pairs (bp) long, and a list of quality scores associated with the DNA fragments. For whole-genome or whole-exome sequencing, a FASTQ file contains millions of records. For example, the human exome is ~50 million base pairs (Mbp), and a typical whole-exome sequencing dataset has at least 30X depth of coverage (i.e., each bp is sequenced 30 times on average). Such a FASTQ file will contain ~15 million raw 100-bp sequences. To identify the causal genes from this raw data, a variant gene prioritization pipeline usually involves three stages or phases of data processing: 1) raw sequence processing and mapping, 2) variant discovery and genotyping, and 3) disease gene filtering and\/or prioritization.\n\nPhase 1: Raw sequence processing and mapping\n\nOverview\n\nThe goal of Phase 1 is to preprocess the FASTQ file and evaluate the quality of the raw reads and then to map or align the sequences to the reference genome-that is, to find the best location on the reference genome where each sequence might have originated. Successfully mapped sequences are annotated with information that includes the location on a specific chromosome and how well the sequence matches the reference genome. After initial mapping, there are several subsequent recalibration steps to improve the mapping results.\n\nPreprocessing\n\nThe raw FASTQ file contains all of the raw sequences, some of which are of low quality. The overall quality should be evaluated, and the lower-quality reads should be removed before mapping to the reference genome to improve mapping efficiency. FASTQC (http:\/\/www.bioinformatics.babraham.ac.uk\/projects\/fastqc\/) is a popular tool that produces quality analysis reports on FASTQ files. FASTQC provides graphical reports on several useful statistics, such as \"Per base sequence quality,\" \"Sequence duplication levels,\" and \"Overrepresented sequences,\" etc. Specifically, \"Per base sequence quality\" determines if trimming low-quality sequences is needed before the mapping\/alignment step; \"Sequence duplication levels\" evaluates the library enrichment and complexity; and \"Overrepresented sequences\" evaluates potential adaptor contamination. Low-quality bases and adaptor contaminations can cause an otherwise mappable sequence not to map and therefore should be removed (e.g., trim the 3' end of the read. which tends to have low quality). FASTX-Toolkit (http:\/\/hannonlab.cshl.edu\/fastx_toolkit\/) provides a set of command line tools for manipulating FASTQ files, such as trimming input sequences down to a fixed length or trimming low-quality bases. Cutadapt (https:\/\/github.com\/marcelm\/cutadapt) can be used for trimming short reads to remove potential adapter contamination.\n\nMapping\n\nOnce high-quality sequence data are obtained, a number of bioinformatics programs, including Mosaik [7], MAQ (http:\/\/maq.sourceforge.net\/), mrFAST [8], BWA [9], Bowtie2 [10], SOAP2 [11], and Subread [12], can be used to perform short-read sequence alignment to the reference genome. Several recent studies provided a detailed description and comparison of the alignment programs [5, 13], although it could be difficult to choose among aligners, because benchmark results vary across studies [10, 12, 14]. BWA and Bowtie2 are two popular programs for alignment. Both programs employ the Burrows-Wheeler transform (BWT) algorithm and have been shown to yield very good overall performance [13]. Bowtie2 is often used for mapping ChIP-seq and RNA-seq data, and BWA is often used for mapping whole-genome\/exome data. We highlight the BWA program here, because it is accurate, fast, thoroughly tested, and well supported. Additionally, BWA has been utilized in multiple large-scale projects and well-defined workflows, including the Genome Analysis Toolkit (GATK) by Broad Institute [15], the 1000 Genome Project [16], the NHLBI GO Exome Sequencing Project (ESP) (http:\/\/evs.gs.washington.edu\/EVS\/), and the HugeSeq workflow [17]. Various commercial companies, such as Seven Bridges Genomics (https:\/\/www.sbgenomics.com\/) and Geospiza GeneSifter (http:\/\/www.geospiza.com\/), also utilize BWA in their standard workflows.\nAfter mapping the reads, the final output file from most software is typically in the sequence alignment\/map (SAM) format [18]. In some cases, the SAM file is converted to its binary format (BAM) using SAMtools [18] to reduce the file size and optimize computation performance.\n\nRecalibration\n\nAfter the initial mapping procedure, subsequent recalibration steps are typically employed to improve the mapping results, such as performing local realignment or de novo assembly for regions that contain insertions\/deletions (indels), removing PCR duplicates, and recalibrating the base quality scores. Indel realignment is a highly recommended post-BAM processing procedure. Due to the trade-off between speed and accuracy, indels are not well aligned by most general purpose aligners. This could lead to false variant calls in downstream analyses. Regions with high probability of potential indels can be realigned locally using IndelRealigner, part of the GATK toolkit. Another commonly used recalibration procedure is removing PCR duplicates. If a DNA fragment is amplified many times by PCR during the sequencing library construction, these artificially duplicated sequences can be considered as support of a variant by downstream variant discovery programs. Some BAM processing programs, such as Picard (http:\/\/picard.sourceforge.net\/) and Samtools [18], can identify these artificially duplicated sequences and remove them. Base recalibration is also a recommended step, because the sequencer may have assigned a biased quality score upon reading a base (e.g., the score of a second \"A\" base after a first \"A\" base may always receive a biased quality score from a sequence machine [19]). Tools, such as Base-Recalibrator in the GATK toolkit, can calibrate the quality score to more accurately reflect the probability of a base mismatching the reference genome. One additional optional step, recommended by GATK, is data compression and reads reduction, especially for high-coverage data. For example, if a large chunk of sequences matches the reference exactly, it is not necessary to keep all the data, as they do not carry useful information for downstream analyses (assuming we are only interested in the sites that are different from the reference genome). In such a scenario, keeping one copy of each of the consensus sequences may be sufficient, and the redundancies can be removed to reduce file size and enable faster downstream computing. However, keeping a copy of the original file is highly recommended after data compression.\n\nPhase 2: Variant discovery and genotyping\n\nOverview\n\nIn many scenarios, only the sites that differ from the reference genome are of interest, because sites that are identical to the reference genome are not expected to be related to pathological conditions. Once raw sequences are properly mapped to the reference genome, the next step is to find all positions in an individual's genome that differed from the reference. This phase is referred to as variant discovery, or variant calling. Similar to the mapping phase, variant calling also contains an initial discovery step, followed by several filtering processes to remove sequencing errors and other types of false discoveries, and finally, the individual genotypes are inferred (i.e., if a locus is heterozygous, homozygous, or hemizygous for the variant). The output of variant calling contains all the variants and related information. Sites that are identical to the reference genome (i.e., invariant sites) are usually not included in the output variant file.\n\nVariant discovery and genotyping\n\nA number of variant calling software packages can be used to identify variants and call individual genotypes. Some of the commonly used software programs are SAMtools [18], freebayes (http:\/\/github.com\/ekg\/freebayes), SNPtools [20], GATK UnifiedGenotyper, and GATK HaplotypeCaller. Some of the tools, including SAMtools, SNPtools, and the GATK UnifiedGenotyper, use a mapping-based approach. Other tools, such as freebayes and the GATK Haplotype-Caller, use a local assembly approach. A more detailed survey and comparison of the tools have been previously described [5, 21]. These procedures typically take the BAM files from the \"raw sequence processing and mapping\" phase, together with the reference genome sequence in a FASTA file. The output file from this step is usually in the standard variant call format (VCF). Some of the tools allow the user to specify known variants to optimize the variant discovery. For example, GATK UnifiedGenotyper uses known variants (e.g., from the dbSNP database) as high-confidence datasets to assist model training. After variant and genotype calling, the probability scores of variants are recalibrated, and artifacts are identified. The artifacts can be removed subsequently either by filtering the data using a series of criteria or by building and applying an error model.\nOne example of the variant discovery tool is the GATK UnifiedGenotyper, which calls variants by examining the reference genome base by base, without correcting misalignment issues. This procedure could lead to false positive calls, particularly in regions containing indels. To increase the accuracy and sensitivity, calling multiple samples simultaneously is recommended. By taking into account the information of a variant in a population, the accuracy and sensitivity of variant discovery are greatly improved, especially for common variants [16, 22]. The GATK Haplotype-Caller, as compared to the UnifiedGenotyper, calls all variants (single nucleotide polymorphisms [SNPs], indels, structural variations, etc.) simultaneously by performing local de novo assembly of haplotypes and emits more accurate call sets, with the drawback of being slower. In general, structural variations (SVs) and copy number variations (CNVs) are more difficult to detect than SNPs and indels because of their heterogeneous nature. For SVs and CNVs, it is generally recommended to apply a combination of several tools and take the overlapping variant sites for high-confidence calls [17].\n\nGenotype phasing and refinement\n\nAfter the initial variant discovery and recalibration, variant genotyping and\/or phasing are performed to provide critical information for downstream medical and population genetic studies that require accurate haplotype structures. For example, Mendelian diseases may be caused by a compound heterozygote event where heterogeneous mutations are found on different chromosomes. In such a scenario, knowledge about the paternal or maternal origin of a variant can provide valuable information. The GATK workflow recommends three steps of genotype phasing: transmission based on pedigree, haplotype based on reads, and statistical imputation. GATK PhaseByTransmission incorporates pedigree data to assist genotype calling, while GATK Read-BackedPhasing performs physical phasing of variant calling based on sequence reads. Statistical imputation of genotypes can be processed by several programs, such as Beagle [23,24], IMPUTE2 [25], and MACH [26].\n\nPhase 3: Disease gene filtering and\/or prioritization\n\nOverview\n\nA typical variant call pipeline will produce approximately 3 million variants from one whole-genome sequencing data set and more than 20 thousand variants from one whole-exome sequencing data set [27]. Because disease causing mutations might just be several \"needles\" in this tremendous \"haystack\" of variants, rigorous algorithms and high-quality databases are essential to accurately locate the candidate genes. This process is done either by filtering out variants that are not likely to carry a disease risk or by ranking all variants based on biological\/statistical models, such that high-risk candidates rank at the top of the list [1, 2, 4]. Unlike Phase 1 and Phase 2, which have largely converged to well-formed computation workflows and platform-free file formats (e.g., FASTQ, BAM, and VCF formats), Phase 3 has more variations in the tools and workflows to suit the different needs of various research or clinical projects.\n\nAnnotation\n\nAlthough many different strategies can be used to search for disease-causing mutations, some general procedures are shared. The first step, variant annotation, typically investigates the potential pathogenic impact of a variant, before any subsequent filtering or prioritizing. The goal is to identify and report the effect of each variant with respect to predefined \"features,\" such as genes. For example, if a variant (SNP or indel) is present within the coding region, the potential effects include change to amino acid (non-synonymous), no change to amino acid (synonymous), introduction of stop codon (stop gain), removal of stop codon (stop loss), addition\/removal of amino-acid(s) (inframe indel), and change to the open reading frame (frame-shifting indel), etc. Other than variant call files, the annotation process requires a feature file that defines the location of genomic features (e.g., transcript, exon, intron, etc.), which is usually in browser extensible data (BED) format [28], or generic feature format version 3 (GFF3) format maintained by the Sequence Ontology project [29]. In many situations, the reference genome in FASTA format is also provided as part of the input, and the annotated variants are likely to be platform-specific [30, 31]. Annotation is part of the workflow for almost all variant analysis tools, such as the open source software package VAAST [31], ANNOVAR [27], and many commercial service providers (e.g., Ingenuity, Golden Helix, Geospiza GeneSifter, Omicia, Seven Bridges Genomics, Biobase, etc.).\nOnce the variants are annotated, a number of methods can be used to predict the severity of a variant, based either on the protein sequence conservation information (e.g., BLOSOM62, SIFT) or the DNA sequence conservation information (e.g., PhastCons). The SIFT [32] score system assumes that protein \"motifs\" or \"sites\" are functionally important if they are highly conserved across species. For example, coding regions, active sites of enzymes, many splice sites, and promoters are usually conserved across species [32]. Technically, SIFT uses protein homology to calculate position-specific scores, which are then used to evaluate if an amino acid substitution at a specific location is damaging or tolerated. The accuracy of the SIFT prediction depends largely on the availability and the diversity of the homologous sequences across species. Therefore, the application and accuracy of SIFT are limited if there is limited homolog information or if the diversity is low. The BLOSUM62 matrix [33] is a more general-purpose score system that is based on the alignment of protein homologs with a maximum of 62% identity. The BLOSUM62 matrix is then calculated based on the count of the observed frequency of specific amino acid substitutions. The PhastCons score [34] is calculated based on DNA sequence conservation in a multiple alignment. It uses a statistical model of sequence evolution and considers the phylogeny relationship of multiple species. Instead of measuring the statistical similarity and diversity in percent identity, PhastCons uses a phylogenetic Hidden Markov Model and provides base-by-base conservation scores. Because the PhastCons score is calculated for each base, it can be applied to regions beyond the coding sequence to provide valuable information for prioritizing non-coding variants.\nIn addition to the conservation information, many software packages also incorporate protein structure information, such as the positions of active sites and secondary structures, into the prediction algorithm (e.g., Polyphen2, SNAP). Polyphen2 [35] combines eight sequencing features (e.g., congruency of the variant allele to the multiple alignment) and three structural features (e.g., changes in hydrophobic propensity) to score an amino acid substitution. The homology search and model training of Polyphen2 are based on the UniProt database. It worth noting that Polyphen2 provides two different prediction models that use different datasets for training, with HumVar tuned to score Mendelian diseases and HumDiv tuned to evaluate rare alleles at loci that are potentially involved in complex traits. SNAP [36] is another tool that uses various protein information, such as the secondary structure and conservation, to predict the functional effect of a variant.\n\nFiltering\n\nAfter variant annotation, two types of disease-gene identification strategies are commonly employed. A filter-based approach uses a series of criteria to sort out variants or genes that are unlikely to be causative, given a disease model. For example, in a typical workflow of ANNOVAR, variants that overlap with the 1000 Genome Project are removed from the list. Since the 1000 Genome Project aimed to identify common SNPs with a minor allele frequency (MAF) greater than 1%, these variants are thought to be \"benign\" in a rare Mendelian disease, under the assumption that the causing mutation of a rare disease should have a very low MAF. The HapMap [37, 38] database, which intended to identify haplotype-defining variants in different populations, can also be used for variant selection. In some cases, high frequency of a haplotype can be related to a specific disease, and can help researchers focus on a subset of variants within a specific region. Another commonly applied filter is to exclude non-coding variants under the assumption that a disease-causing mutation should occur in the coding regions and affect protein sequences. Several other filtering criteria are also commonly used, such as filtering variants with little predicted functional impact (e.g., synonymous variants) or having low-quality scores (e.g., low read-depth or low genotype quality score). A disease model can also help filter variants. For example, a disease under a recessive mode of inheritance requires more than one deleterious variant in a gene.\nThe filtering strategy is intuitive and easy to perform and has been successful in early whole-genome\/exome studies on rare, single-gene Mendelian diseases. Numerous tools have been developed to apply various filtering strategies (reviewed in [1, 2, 4, 5]). However, caution should be taken when studying common, complex diseases, because applying hard filters could remove real casual variants. For instance, common diseases could be caused by common variants that have incomplete penetrance (i.e., not all individuals carrying the variant will have the disease).\n\nRanking\n\nTo overcome the difficulties associated with hard filtering, a prioritizing approach ranks each gene or feature based on the cumulative disease risk of potential deleterious variants. As an example, the VAAST tool kit performs the composite likelihood ratio test to determine the risk of a gene or a feature [30, 31]. Information of each variant within the gene, such as the predicted biological impact, allele frequency, and the conservation score of the variant allele, is considered simultaneously under a likelihood framework. One advantage is that it reduces the risk of filtering out potential risky coding variants, since there are no hard filtering steps. Another advantage is that it can score all variants, including non-coding variants that would have otherwise been filtered out. The statistical framework used by VAAST considers two sets of information: 1) the likelihood of observing the MAF of a variant under a disease model versus a non-disease model; and 2) the likelihood that a variant is deleterious versus non-deleterious. A combination of the observed frequency and predicted functional impact of variants is used to assess the disease risk of a gene.\nA practical consideration is that most variants in the sequencing study are rare and may be difficult to achieve sufficient power for a statistical model [39]. To overcome this, a common solution is to assess the cumulative effects of multiple variants in a defined genomic region, such as a gene or an exon [39-41]. This approach evaluates the overall genetic burden of multiple rare variants and is referred to as the burden test. Two general methods are commonly used to collapse multiple rare variants [40], known as the cohort allelic sum test (CAST) and combined multivariate and collapsing (CMC) method. The CAST method measures the differences (between case group and control group) in the number of individuals who carry one or more rare variants [39, 42], while the CMC method treats all rare variants as a single count for analysis with common variants [39]. Since both methods implicitly assume that all variants influence the phenotype in the same direction [39, 40], it could introduce substantial noise if a lot of the rare variants are neutral [40]. Alternatives to these two basic methods are also available [40]. For example, by default, the VAAST program combines all variants that have less than three copies in a gene into one pseudo-site for scoring.\n\nPathway analysis\n\nPathway analysis is an alternative way to search for deleterious genes. The assumption is that a single mutation in a single gene may not be very harmful but that a combination of mutations in several genes causes the disease. In this type of analysis, variants are put in a context of biological processes, pathways, and networks to gain global perspective on the data. A large number of knowledge bases and tools have been developed for this task (reviewed in [43, 44]). In general, pathway systems contain a curation process that is either manual or automatic, followed by database assembly and refinement [43]. Many public knowledge base and analysis tools are available, such as Kyoto Encyclopedia of Genes and Genomes (KEGG) (http:\/\/www.kegg.jp\/); BioCyc (http:\/\/www.biocyc.org\/), provided by SRI international; PANTHER [45]; Reactome [46]; and DAVID [47]. Optionally, commercial systems, such as Ingenuity Pathway Analysis (http:\/\/www.ingenuity.com\/products\/ipa), Pathway Studio (http:\/\/www.elsevier.com\/online-tools\/pathway-studio), and GeneGo (http:\/\/portal.genego.com\/), are also available. Ingenuity Variant Analysis (http:\/\/www.ingenuity.com\/products\/variant-analysis) is also backed up with Ingenuity Knowledge Base and can consider the upstream and downstream genes of a variant in a pathway and evaluate the potential role of a gene by incorporating the related genes.\n\nGene Identification Pipeline: An Example\n\nTo help readers who have limited experience with NGS data to understand the workflow, in this section we will show a real example of a pipeline that was used to process whole-exome sequencing data from raw sequences to a ranked candidate gene list (Fig. 1). The pipeline is constructed, based on the three-phase organization, as described in the previous sections. We start from raw sequence FASTQ files and assume no quality control has been performed or reported. Please note that each tool is likely to require dependent files, such as the reference genome, the 1000 Genome variants, dbSNP variants, and the genome feature file in GFF3 format, etc.\n\nSequence mapping\n\n\u2022 - Raw sequence quality control. FastQC can be used to perform graphical quality checks on the raw sequence. FastX can be used to filter the reads by specifying quality score ranges.\n\n\u2022 - Initial mapping. BWA is recommended for mapping Illumina reads. The first step of BWA is to generate suffix array coordinates for mapping the reads; then, one performs the actual alignment and outputs SAM format files. The SAM file is converted to BAM format, sorted, and indexed using Samtools for faster downstream processing (sorting is often required by many downstream tools).\n\n\u2022 - Alignment recalibration. The GATK RealignerTarget-Creator tool is used to create target regions for realignment, followed by the IndelRealigner tool to perform the actual realignment and output new BAMs. The Picard MarkDuplicates tool is then used to mark and remove duplicated sequences, followed by the Picard BuildBamIndex tool to rebuild the index. Next, GATK's BaseRecalibrator tool is used to recalibrate the base quality scores, and PrintReads is used to output the final, analysis-ready reads in BAM format.\n\nVariant discovery\n\n\u2022 - Initial variant discovery. The GATK UnifiedGenotyper tool is used to perform initial raw variant calling and output VCF files.\n\n\u2022 - Variant recalibration and filtering. SNPs and indels are processed separately but in a similar manner; so, we will only describe SNPs as an example. The GATK Select-Variants tool is used to select SNPs within the exome regions from raw VCF files. The VariantRecalibrator tool is then used to build Gaussian models for recalibration, followed by the ApplyRecalibration tool to apply models for SNP recalibration. Low-quality SNPs are then removed by imposing a filtering step, based on a user-defined variant quality score recalibration (VQSR) score threshold. After SNPs and indels are reprocessed, the GATK CombineVariants tool is used to combine the recalibrated SNPs and indels.\n\nDisease gene prioritization\n\n\u2022 - Variant annotation. VCF files are converted to genome variation format (GVF) format using the VAAST vaast_converter tool. The variants are then annotated using the VAAST VAT tool.\n\n\u2022 - Variant selection. The annotated variants are selected and condensed to condenser (CDR) format using the VAAST VST tool for cases and controls, respectively.\n\n\u2022 - Gene prioritization. VAAST is used to perform disease gene ranking and output a text file containing a list of ranked candidate genes.\n\nAutomated Workflows vs. Flexible Pipelines\n\nHigh-throughput data processing and analysis are becoming reliable and efficient, thanks to numerous high-quality tools that are mostly open source, with long-term active support and frequent updates to incorporate new advances in the field. A typical workflow covering all three phases outlined above involves many different tools and\/or steps, databases, score systems, and file re-formatting steps. Therefore, it is not a trivial task to perform a data processing task, starting from the raw FASTQ files to the identification of candidate genes. There are many tools and dependencies of tools that need to be installed and many steps and many parameters that need to be specified. This could be a big hurdle for researchers that have limited experience with NGS data pipelines. It would be a good idea for the community to converge on several standardized, well-configured, project-optimized, and automated workflows that can be used widely with minimum user interface. HugeSeq [17] by Stanford is a comprehensive, automated workflow that integrates around two dozen bioinformatics tools and can process FASTQ\/FASTA files, perform various quality control and clean-ups, and output variant call files. Commercial services, such as Seven Bridges Genomics and Geospiza, also have preconfigured, comprehensive, automated pipelines for their customers.\nFlexibility of pipelines, on the other hand, may be important for researchers who want to have complete control on the pipeline design to fit their projects. In such cases, it is important to have plasticity for users to customize pipelines. For example, a flexible pipeline system should allow users to integrate specific tools that are not part of a general purpose workflow and allow users to supply user-defined control files (e.g., providing a BED file to restrict the variant call within specific regions). There are a number of such pipelines that are well defined and easy to use, such as the GATK pipeline, gkno system (http:\/\/gkno.me\/), Galaxy [48-50], and VAAST [30, 31].\n\nConclusion and Future Direction\n\nHighly efficient and accurate bioinformatics tools are available for most of the steps for analyzing sequence data, from processing the raw NGS data to generating the final report of potential risk genes. Users can choose automated workflow versus customized pipelines to suit their research projects. Many bioinformatics tools are optimized, such that a small server (e.g., 48 cores, 250 GB RAM, 2 TB storage) can perform the whole analysis workflow within a reasonable time (e.g., 250 hours in nonparallel mode and 25 hours in parallel mode for one exome at 30\u00d7 coverage, as benchmarked by HugeSeq). Furthermore, the scientific community is converging towards standard workflows and standard file formats, crossing different research institutes, companies, and platforms. Several file formats are becoming standard, such as FASTQ\/FASTA, SAM, BAM, VCF, and GFF3. Large publically funded projects, such as the 1000 Genome Project, the HapMap Project, and the NHLBI GO Exome Sequencing Project, have played important roles in the unification process.\nNevertheless, many challenges remain, and we would like to highlight a few: 1) data standards still need wider agreement. For example, a number of variant discovery tools for SV and CNV use different file formats and need to be converted to the more standard format for downstream analysis [17]; 2) more standardized, automated workflows need to be developed to accommodate different data and projects and minimize the user interaction, such that different research groups can perform independent data analysis and the results can be easily compared; and 3) large, high-quality, and unified databases are essential for disease gene identification\/prioritization, and continuous efforts from the scientific community are needed.\nIn addition to the challenges, a number of improvements that are urgently needed for current analysis pipeline are being actively developed and implemented:\n\u2022 - Sequence mapping. Apply haplotype-based mapping and de novo assembly to reduce mismatches and increasing specificity.\n\n\u2022 - Variant discovery and genotyping. Improve the sequencing technology (e.g., longer read length) and analytical methods (e.g., de novo assembly-based variant discovery) to identify and infer genotypes of complex variations, such as SVs, CNVs, large indels, and transposons.\n\n\u2022 - Candidate gene prioritization. Develop complex prioritization strategies for large pedigrees, combine linkage analysis with association studies, and integrate pathway analysis to eliminate false positive genes.\n\n\u2022 - Non-coding variant annotation. Understand the functional impact of non-coding DNA elements, as the Encyclopedia of DNA Elements (ENCODE) Project intended to do, to greatly broaden our view of disease-causing mutations.\n\n\u2022 - Cloud computing. High-throughput sequencing projects generate large amounts of data, create huge computational challenges, and require numerous tools and libraries for comprehensive data analysis. Develop cloud-based computing resources, such as the 1000 Genome Project (http:\/\/aws.amazon.com\/1000genomes\/), to create a more efficient way of managing, processing, and sharing data.\n\nHigh-throughput sequencing will continue to transform biomedical sciences in almost every aspect [51] and provide new insights for our understanding of human diseases. For example, three groups recently reported discoveries of several new autism genes and suggested a much more complex disease mechanism, based on large-scale sequencing data of nearly 600 trios and 935 additional cases [52-54]. Along with the revolutionary discoveries based on NGS data, new tools and techniques will be developed to facilitate fast and accurate analysis.\n\nAcknowledgments\n\nWe thank Dr. Gary A. Heiman and Alexandra Nguyen for providing valuable comments and proofreading the manuscript and the anonymous reviewers for their constructive advice. 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PMID: 21920052.\n\n43. Viswanathan GA, Seto J, Patil S, Nudelman G, Sealfon SC. Getting started in biological pathway construction and analysis. PLoS Comput Biol 2008;4:e16. PMID: 18463709.\n\n44. Khatri P, Sirota M, Butte AJ. Ten years of pathway analysis: current approaches and outstanding challenges. PLoS Comput Biol 2012;8:e1002375. PMID: 22383865.\n\n45. Mi H, Muruganujan A, Thomas PD. PANTHER in 2013: modeling the evolution of gene function, and other gene attributes, in the context of phylogenetic trees. Nucleic Acids Res 2013;41:D377\u2013D386. PMID: 23193289.\n\n46. Joshi-Tope G, Gillespie M, Vastrik I, D'Eustachio P, Schmidt E, de Bono B, et al. Reactome: a knowledgebase of biological pathways. Nucleic Acids Res 2005;33:D428\u2013D432. PMID: 15608231.\n\n47. Huang da W, Sherman BT, Lempicki RA. Systematic and integrative analysis of large gene lists using DAVID bioinformatics resources. Nat Protoc 2009;4:44\u201357. PMID: 19131956.\n\n48. Goecks J, Nekrutenko A, Taylor J. Galaxy Team. Galaxy: a comprehensive approach for supporting accessible, reproducible, and transparent computational research in the life sciences. Genome Biol 2010;11:R86. PMID: 20738864.\n\n49. Blankenberg D, Von Kuster G, Coraor N, Ananda G, Lazarus R, Mangan M, et al. Galaxy: a web-based genome analysis tool for experimentalists. Curr Protoc Mol Biol 2010;Chapter 19:Unit 19.10.1\u2013Unit 19.10.21.\n\n50. Giardine B, Riemer C, Hardison RC, Burhans R, Elnitski L, Shah P, et al. Galaxy: a platform for interactive large-scale genome analysis. Genome Res 2005;15:1451\u20131455. PMID: 16169926.\n\n51. Loman NJ, Misra RV, Dallman TJ, Constantinidou C, Gharbia SE, Wain J, et al. Performance comparison of benchtop highthroughput sequencing platforms. Nat Biotechnol 2012;30:434\u2013439. PMID: 22522955.\n\n52. Sanders SJ, Murtha MT, Gupta AR, Murdoch JD, Raubeson MJ, Willsey AJ, et al. De novo mutations revealed by whole-exome sequencing are strongly associated with autism. Nature 2012;485:237\u2013241. PMID: 22495306.\n\n53. Neale BM, Kou Y, Liu L, Ma'ayan A, Samocha KE, Sabo A, et al. Patterns and rates of exonic de novo mutations in autism spectrum disorders. Nature 2012;485:242\u2013245. PMID: 22495311.\n\n54. O'Roak BJ, Deriziotis P, Lee C, Vives L, Schwartz JJ, Girirajan S, et al. Exome sequencing in sporadic autism spectrum disorders identifies severe de novo mutations. Nat Genet 2011;43:585\u2013589. PMID: 21572417.\n\nFig.\u00a01\nExample workflow for disease gene identification using next-generation sequencing data. SAM, sequence alignment\/map; BAM, binary format; SNP, single nucleotide polymorphism; VCF, variant call format.\n\nEditorial Office\nRoom No. 806, 193 Mallijae-ro, Jung-gu, Seoul 04501, Korea\nTel: +82-2-558-9394\u00a0\u00a0\u00a0\u00a0Fax: +82-2-558-9434\u00a0\u00a0\u00a0\u00a0E-mail: kogo3@kogo.or.kr","date":"2022-12-02 22:10:01","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\": 0.42262670397758484, \"perplexity\": 7675.466517360002}, \"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-2022-49\/segments\/1669446710916.70\/warc\/CC-MAIN-20221202215443-20221203005443-00612.warc.gz\"}"}
| null | null |
Q: Relation between product of reflections and angle I am currently reading Chapter 3 Root Systems of John Humphreys book on Lie Algebras.
It's known that Weyl Group $W$ is generated by set of reflections.
If I consider an arbitrary element of Weyl group $\sigma_\alpha \sigma_\beta$,($\alpha,\beta \in \Phi$) then it's easy to calculate it's order by definition of reflection if angle between $\alpha$ and $\beta$ is $\pi/2$.
But how to see the fact that $\sigma_\alpha \sigma_\beta$= rotation through $2\theta$( if $\theta$ is angle between $\alpha$ and $\beta$). I can't visualise it.
Please see!
A:
This drawing in the two-dimensional case should illuminate things.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 7,807
|
{"url":"http:\/\/www.lofoya.com\/Solved\/2170\/the-length-breadth-and-height-of-a-room-are-inthe-ratio","text":"# Difficult Geometry & Mensuration Solved QuestionAptitude Discussion\n\n Q. The length, breadth and height of a room are in\u00a0the ratio $3:2:1$. If the breadth and height are\u00a0halved while the length is doubled, then the total\u00a0area of the four walls of the room will\n \u2716\u00a0A. remains the same. \u2716\u00a0B. decrease by\u00a0$13.64\\%$ \u2716\u00a0C. decrease by\u00a0$15\\%$ \u2714\u00a0D. decrease by\u00a0$30\\%$\n\nSolution:\nOption(D) is correct\n\nLet the original length, breadth and height of the room\u00a0be $3x$, $2x$ and $x$ respectively.\n\nTherefore, the new length, breadth and height are $6x$, $x$ and $x\/2$\u00a0respectively.\n\nArea of four walls = (2 \u00d7 length \u00d7 height) +\u00a0(2 \u00d7 breadth \u00d7 height)\n\nOriginal area of four walls,\n\n$=(2 \\times 3x \\times x)+(2 \\times 2x \\times x)$\n\n$= 6x^2 + 4x^2 = 10x^2$\n\nNew area of four walls,\n\n$=(2 \\times 6x \\times \\frac{x}{2})+(2 \\times x \\times \\frac{x}{2})$\n\n$=6x^2 + x^2 = 7x^2$\n\nTherefore, Area of wall decreases by\n\n$=\\left[\\dfrac{10x^2 - 7x^2}{10x^2} \\right] \\times 100$\n\n$= 30\\%$\n\nEdit:\u00a0A typo in the final step has been corrected based on the input from\u00a0KARTIK.\n\n## (4) Comment(s)\n\nAmit\n()\n\nin this question u have not mentioned which four walls are to be considered .\n\nAlia\n()\n\nA room has only four walls. Other two surfaces are called 'roof' and 'floor'. So, I believe the question is okay and does not need any correction.\n\nKARTIK\n()\n\nlast step : $10x^2$ instead of $10^2$ :)\n\nDeepak\n()\n\nThank you Kartik, corrected the mistake.","date":"2017-09-23 19:55:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"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\": 0.6862797141075134, \"perplexity\": 2109.2427329080174}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818689775.73\/warc\/CC-MAIN-20170923194310-20170923214310-00080.warc.gz\"}"}
| null | null |
View: Eat inside in these extensive tearooms, outside in the flower gardens or on the beach terrace and enjoy some of the best beach and sea views in Great Yarmouth. Dogs are welcome both inside and out.
What's on the menu?: Breakfasts, light snacks, refreshments, home made cakes, desserts.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,845
|
The `jbpmc-api` library is an API for the Java programming language, that defines how a client may access a BPM server.
It provides methods for programmatically executing a BPM server's runtime, configuration and deployment services.
---
## Table of contents
1. [Why have a common API?](#why-have-a-common-api)
1. [What can I use this library for?](#what-can-i-use-this-library-for)
1. [Want to contribute?](#want-to-contribute)
1. [Libraries](#libraries)
---
## Why have a common API?
For the same reasons you have [standards](http://www.standards.org.au/StandardsDevelopment/What_is_a_Standard/Pages/Benefits-of-Standards.aspx).
Currently each BPM server implementation provides a bespoke API for accessing
their services. Our aim is to provide a consistent interface, to facilitate
building tools that are agnostic of the underlying BPM implementation.
---
## What can I use this library for?
Here are some potential applications of this library:
- deployment automation
- testing automation
- production support
- creating generic bpm clients
- server migration/upgrades
- process instance migration
---
## Want to contribute?
This project is just in it's infancy, so we're looking to get it off the ground in terms of features and support.
If you see value in this project or just have some features that you want to implement/share with everyone else,
please contact us to become a contributor, and/or [fork the project](https://help.github.com/articles/fork-a-repo/)
and [submit a pull request](https://help.github.com/articles/creating-a-pull-request/).
---
## Libraries
- `jbmpc-api` - An API for accessing a BPM server.
### Library Implementations
- `jbpmc-api`
- `jbpmc-ibm`
- `7.5.1.2` [ runtime (partial) | <s>configuration</s> | deployment (partial) ]
- <s>`jbpmc-activiti`</s>
- <s>`jbpmc-appian`</s>
- <s>`jbpmc-camunda`</s>
- <s>`jbpmc-jbpm`</s>
- <s>`jbpmc-pega`</s>
---
## TODO
- Setup "runtime" project to test dependencies (i.e. removing groovy after compilation)
- Document Example Usage (with @Grape('...')) or libs?
- Document Example Config
- Document Example wsadmin folder structure on windows
- Add project to OSS Sonatype repo
- Setup main method application?
---
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,264
|
\section{Introduction}
The last decade has witnessed significant
advances in our understanding of relaxation in isolated many-particle
quantum systems. Many of these were driven by advances in ultra-cold
atom experiments, which have made it possible to study the real-time
dynamics of almost isolated systems in exquisite detail \cite{GM:col_rev02,kww-06,HL:Bose07,hacker10,tetal-11,getal-11,cetal-12,langen13,MM:Ising13,zoran1,MBLex1}.
At present there are three established paradigms for the relaxational
behaviour in such systems. According to the
eigenstate thermalization hypothesis \cite{Deutsch91,Sred1,Sred2,ETH}
``generic'' systems relax towards thermal equilibrium
distributions. In this case the only information about the
initial state that is retained at late times is its energy
density. Quantum integrable models cannot thermalize in this way as
they exhibit additional conservation laws, but they relax to
generalized Gibbs ensembles instead
\cite{GGE,EFreview,Cazalilla06,cc-07,CEF,GGE_int}. Finally, disordered
many-body localized systems fail to
thermalize as well\cite{MBL1,MBL2,MBL3,MBL4,MBL5,MBL6}. This is
related to the existence of conservation laws\cite{MBL6,MBL7,MBL8},
although in contrast to integrable models no fine-tuning of the
Hamiltonian is required.
Recently it has been proposed that certain
isolated quantum systems may exhibit behaviour that is not
captured by these three known paradigms\cite{qdl}. The resulting
state of matter has been termed a ``quantum disentangled liquid'' (QDL).
A characteristic feature of such systems is that they comprise both
``heavy'' and ``light'' degrees of freedom, which will be made
more precise below. The basic premise of the QDL
concept is that while the heavy degrees of freedom are fully
thermalized, the light degrees of freedom, which are enslaved to the
heavy degrees of freedom, are not independently thermalized.
A convenient diagnostic for such a state of matter is the bipartite
entanglement entropy (EE) after a projective measurement of the heavy particles.
Specifically, we define a QDL to be a state in which the entanglement entropy
reduces from a volume-law to an area-law upon a projective measurement
of the heavy particles.
The possibility of realizing a QDL in the one dimensional Hubbard
model was subsequently investigated by exact diagonalization of small
systems in Ref.~\onlinecite{hubb1}. Given the limitations on
accessible system sizes it is difficult to draw definite conclusions
from these results. Motivated by these studies we explore the
possibility of realizing a QDL in the half-filled Hubbard model on
bipartite lattices by analytic means. We employ methods of
integrability as well as strong-coupling expansion
techniques~\cite{takahashi,MacDonaldGirvin88,stein,sasha} to analyze
properties of finite energy-density eigenstates of the Hubbard
Hamiltonian
\begin{eqnarray}
H&=&-t\sum_{\langle j,k\rangle}\sum_{\sigma=\uparrow,\downarrow}
c^\dagger_{j,\sigma}c^ {\phantom\dagger} _{k,\sigma
+c^\dagger_{k,\sigma}c^ {\phantom\dagger} _{j,\sigma}\nonumber\\
&+&U\sum_{j}
\big(n_{j,\uparrow}-\frac{1}{2}\big)
\big(n_{j,\downarrow}-\frac{1}{2}\big).
\label{HHubb}
\end{eqnarray}
Here $n_{j,\sigma}=c^\dagger_{j,\sigma}c^ {\phantom\dagger} _{j,\sigma}$ and $\langle
j,k\rangle$ denote nearest neighbour links that connect only sites from
different sub-lattices $A$ and $B$. At half-filling (one electron per
site) \fr{HHubb} is invariant under the $\eta$-pairing SU(2) symmetry
with generators\cite{etapairing}
\begin{eqnarray}
\eta^z&=&\frac{1}{2}\sum_j \left(n_{j,\uparrow}+n_{j,\downarrow}-1\right)\ ,\nonumber\\
\eta^\dagger&=&\sum_{j\in A}c^\dagger_{j,\uparrow}c^\dagger_{j,\downarrow}
-\sum_{k\in B}c^\dagger_{k,\uparrow}c^\dagger_{k,\downarrow}\equiv \left(\eta\right)^\dagger.
\end{eqnarray}
In the strong interaction limit of the Hubbard model ($U\gg t$), the
bandwidths of the collective spin and charge degrees of freedom are
proportional to $t^2/U$ and $t$ respectively. In this regime the spin
(charge) degrees of freedom therefore correspond to ``heavy'' (``light'')
particles.
The outline of our paper is as follows. In section
\ref{sec:integrability} we employ methods of integrability to show
that it is possible to construct certain (``Heisenberg sector'')
macrostates at finite energy densities for which the charge degrees of
freedom do not contribute to the volume term in the bipartite EE. This
strongly suggests that these states have the QDL property. In section
\ref{sec:strongcoupling} we then turn to strong-coupling expansion
methods in order to examine the QDL diagnostic proposed in
Ref.~\onlinecite{qdl}. In the one dimensional case this analysis shows
that in the framework of a $t/U$-expansion a projective measurement of
the spin degrees of freedom in a Heisenberg sector state indeed leaves
the system is a state that is only area-law entangled. Our
strong-coupling analysis further suggests the existence of a weaker version
of the QDL property, where a projective measurement of the heavy
degrees of freedom leaves the system in a state characterized by a
volume-law entanglement entropy, but with a pre-factor that is
exponentially small in $U/t$.
Section \ref{sec:conc} contains a summary and discussion of our
results. Various technical aspects of our calculations are presented
in three appendices.
\section{Integrability}
\label{sec:integrability}
The one dimensional Hubbard model is exactly solvable by the Bethe
ansatz method \cite{book}. This provides a description of energy
eigenstates as well as the corresponding macro states in the
thermodynamic limit. We now use this framework to identify a class of
macro states that exhibits a property that is characteristic of a
QDL.
In the framework of the string hypothesis general macro states in the
one dimensional Hubbard model are described by sets of particle and
hole densities
$\{\rho^{p}(k), \rho^{h}(k),
\sigma^{p}_n(\Lambda), \sigma^{h}_n(\Lambda),
{\sigma'_n}^{p}(\Lambda), {\sigma'_n}^{h}(\Lambda)
|n\in\mathbb{N}\}$.
These are analogs of the well-known particle and hole densities used to
characterize macro states in the ideal Fermi gas. The main
complication in integrable models is that there are in general
(infinitely) many different species of excitations, which interact
with one another. In the case at hand the $\rho^{p/h}$ describe
pure charge excitations, the $\sigma^{p/h}_n$ correspond to
elementary spin excitations as well as their bound states and
${\sigma'_n}^{p/h}$ represent bound states between spin and charge
degrees of freedom. As a consequence of the interacting nature of the
Hubbard chain the root densities are related in a non-trivial manner:
they are subject to the thermodynamic limit of the Bethe Ansatz equations\cite{book}
\begin{widetext}
\index{root density}
\begin{eqnarray}
&&\rho^p(k)+\rho^h(k)=\frac{1}{2\pi}+\cos
k\sum_{n=1}^\infty\int_{-\infty}^\infty d\Lambda\ \fa{n}{\Lambda-\sin k}
\left[{\sigma^\prime_n}^p(\Lambda)+\sigma_n^p(\Lambda)\right]\ ,\nonumber\\
&&\sigma_n^h(\Lambda)=-\sum_{m=1}^\infty
\int_{-\infty}^\infty d\Lambda' A_{nm}(\Lambda-\Lambda')\ \sigma_m^p(\Lambda')
+\int_{-\pi}^\pi dk\ \fa{n}{\sin k-\Lambda}\ \rho^p(k)\ ,\nonumber\\
&&{\sigma^\prime_n}^h(\Lambda)=\frac{1}{\pi}{\rm
Re}\frac{1}{\sqrt{1-(\Lambda -inu)^2}}
-\sum_{m=1}^\infty
\int_{-\infty}^\infty d\Lambda' A_{nm}(\Lambda-\Lambda')\ {\sigma'_m}^p(\Lambda')
-\int_{-\pi}^\pi dk\ \fa{n}{\sin k-\Lambda}\ \rho^p(k)\ .
\label{densities}
\end{eqnarray}
\end{widetext}
Here $u=U/4t$ and
\begin{eqnarray}
a_n(x)&=&\frac{1}{2\pi}\frac{2nu}{(nu)^2+x^2}\ ,\nonumber\\
\label{an}
A_{nm}(x)&=&\delta(x)+ (1-\delta_{m,n})a_{|n-m|}(x)+2a_{|n-m|+2}(x)\nonumber\\
&+&\dots+2a_{|n+m|-2}(x)+a_{n+m}(x).
\label{Anm}
\end{eqnarray}
The energy and thermodynamic entropy per site are then given by
\begin{eqnarray}
e&=&u+
\int_{-\pi}^\pi dk\left[-2\cos k -\mu-2u\right]\rho^p(k)\ \nonumber\\
&+& 4 \sum_{n=1}^\infty \int d\Lambda\,{\sigma^\prime_n}^p(\Lambda)\,{\rm Re}\sqrt{1-(\Lambda+inu)^2}
,\nonumber\\
s&=&\int_{-\pi}^\pi dk\ {\cal S}\left[\rho^p(k),\rho^h(k)\right]\nonumber\\
&+&\sum_{n=1}^\infty\int_{-\infty}^\infty d\Lambda\
{\cal S}\left[{\sigma^\prime_n}^p(\Lambda),{\sigma_n^\prime}^h(\Lambda)\right]\nonumber\\
&+&\sum_{n=1}^\infty\int_{-\infty}^\infty d\Lambda\
{\cal S}\left[{\sigma_n}^p(\Lambda),{\sigma_n}^h(\Lambda)\right],
\label{entropy}
\end{eqnarray}
where we have defined
\begin{eqnarray}
{\cal S}[f,g]&=&
\big[f(x)+g(x)\big]\ln\big(f(x)+g(x)\big)\nonumber\\
&-&f(x)\ln\big(f(x)\big)
-g(x)\ln\big(g(x)\big)\ .
\end{eqnarray}
The ground state of the half-filled Hubbard model in zero magnetic
field is obtained by choosing
\begin{equation}
\rho^h(k)=0=\sigma^h_1(\Lambda)\ ,\qquad
{\sigma'}_n^p(\Lambda)=0=
{\sigma}_{n\geq 2}^p(\Lambda)\ .
\end{equation}
\subsection{The ``Heisenberg sector''}
Motivated by Ref.~\onlinecite{hubb1} we now consider a particular
class of macro states, which we call \emph{Heisenberg sector states}.
The principle underlying their construction is to ``freeze'' the charge
degrees of freedom in its ground state configuration while imposing a
finite energy density in the spin sector. This is achieved by
requiring
\begin{equation}
\rho^h(k)=0={\sigma'}_n^p(\Lambda)\ ,\ n=1,2,\dots
\label{HeisenbergBA}
\end{equation}
Condition \fr{HeisenbergBA} corresponds to the absence of bound states
between spin and charge degrees of freedom (i.e. no $k$-$\Lambda$
strings) and a completely filled ``Fermi sea'' of elementary charge
degrees of freedom. Importantly the thermodynamic
entropy per site of Heisenberg sector states depends only on the spin
sector
\begin{equation}
s=\sum_{n=1}^\infty\int_{-\infty}^\infty d\Lambda\
{\cal S}\left[{\sigma_n}^p(\Lambda),{\sigma_n}^h(\Lambda)\right].
\label{SH}
\end{equation}
As shown in Appendix~\ref{app:counting} the total number of
Heisenberg sector states $N_{\rm HS}$ fulfils
\begin{equation}
\lim_{L\to\infty}\frac{\ln\big(N_{\rm HS}\big)}{L}=\ln(2).
\label{HScount}
\end{equation}
The result \fr{HScount} suggests that for large, finite $L$ there are
approximately $2^L$ Heisenberg sector states.
\subsection{Entanglement entropy of Heisenberg sector states}
We now make use of the relation between the volume term in the EE and
the thermodynamic entropy density for eigenstates of short-ranged
Hamiltonians. Consider a finite energy density eigenstate
$|\Psi\rangle$ and a large subsystem $A$ of size $|A|$. Then the
volume term in the EE of the state $|\Psi\rangle$ is given by
\begin{equation}
S_{\rm vN,A}=s|A|+o(|A|)\ .
\label{SA}
\end{equation}
where $s$ is the thermodynamic entropy density of the macro state
associated with $|\Psi\rangle$.
Combining \fr{SH} with
\fr{SA}, we conclude that \emph{for Heisenberg sector states the
volume term in the entanglement entropy is entirely due to the spin
degrees of freedom, and the charge degrees of freedom do not
contribute}. This is very much in line with what one would expect
for a QDL state.
\subsection{Thermal states in the large-$U$ limit}
It is very instructive to contrast the entanglement properties of
Heisenberg sector states to those of typical states.
The maximum entropy states at a given energy density are
thermal and can be constructed by the Thermodynamic Bethe Ansatz (TBA).
This provides a system of coupled non-linear integral equations
for the ratios of the hole and particle
densities\cite{book,takahashiTBA1}
\begin{equation}
\zeta(k)=\frac{\rho^h(k)}{\rho^p(k)}\ ,\
\eta_n(\Lambda)=\frac{\sigma_n^h(\Lambda)}{\sigma^p_n(\Lambda)}\ ,\
\eta'_n(\Lambda)=\frac{{\sigma'_n}^h(\Lambda)}{{\sigma'_n}^p(\Lambda)}\ .
\label{dressedE}
\end{equation}
For the sake of completeness we present the TBA
equations in Appendix~\ref{app:TBAeqns}. While in general the TBA
equations can only be solved numerically, in the limit of strong
interactions analytic results can be obtained
\cite{takahashiTBA2,Ha,EEG}. For simplicity we focus on the
``spin-disordered regime''
\begin{equation}
\frac{4t^2}{U}\ll T\ll U\ .
\end{equation}
This regime corresponds to temperatures that are small compared to
the Mott gap, but large compared to the exchange energy. Here one
has\cite{EEG}
\begin{equation}
\rho^h(k)={\cal O}\big(e^{-u/T}\big)\ ,\quad
{\sigma'}^{p,h}_n(\Lambda)={\cal O}\big(e^{-u/T}\big)\ ,
\end{equation}
where we have set $t=1$ as our energy scale.
Substituting this into the general expression \fr{entropy} for the
thermodynamic entropy density we obtain
\begin{equation}
s=\sum_{n=1}^\infty\int_{-\infty}^\infty d\Lambda\
{\cal S}\left[{\sigma_n}^p(\Lambda),{\sigma_n}^h(\Lambda)\right]
+{\cal O}\Big(\frac{u}{T}e^{-u/T}\Big).
\label{ST}
\end{equation}
Finally, using the relation between thermodynamic and EE
\fr{SA} we conclude that for thermal states in the
spin-disordered regime the contribution of the charge degrees of
freedom contribute to the volume term is
\begin{eqnarray}
S_{\rm vN,A}&=&\big(s_{\rm spin}+s_{\rm
charge}\big)|A|+o\big(|A|\big)\ ,\nonumber\\
s_{\rm spin}&=&{\cal O}(1)\ ,\quad
s_{\rm charge}={\cal O}\Big(\frac{u}{T}e^{-u/T}\Big)\ .
\label{weak}
\end{eqnarray}
Here $s_{\rm charge}$ includes the contributions from pure charge
degrees of freedom as well as bound states of spin and charge.
Importantly, unlike Heisenberg sector states, typical states have a
contribution from the charge degrees of freedom to the volume
term. However, this contribution is exponentially small in $u/T$ and
therefore only visible for extremely large subsystems. While we have
focussed on the spin-disordered regime, the behaviour \fr{weak}
extends to thermal states for all $0<T\ll U$.
\section{$t/U$-Expansion}
\label{sec:strongcoupling}
The analysis presented above provides a strong indication that the
Heisenberg sector states realize the QDL concept. However, the exact
solution does not presently allow one to examine the QDL diagnostic
proposed in Ref.~\onlinecite{qdl}, which requires the calculation of
the EE after a partial measurement. We therefore now turn to a
complementary approach, namely a strong-coupling expansion in powers
of $t/U$. This is most conveniently implemented by following
Ref.~\onlinecite{MacDonaldGirvin88}. At a given site $j$ there are
four possible states $|0\rangle_j$, $|{\uparrow}\rangle_j =
c_{j,\uparrow}^\dagger |0\rangle_j$, $|{\downarrow}\rangle_j =
c_{j,\downarrow}^\dagger |0\rangle_j$ and $|2\rangle_j = c_{j,\uparrow}^\dagger
c_{j,\downarrow}^\dagger|0\rangle_j$. Defining Hubbard operators by
$X_{j}^{ab} = |a\rangle_j { }_j\langle b|$, the Hamiltonian
\fr{HHubb} can be expressed in the form
\begin{equation}
H = UD + t\left( T_0 + T_1 + T_{-1}\right),
\end{equation}
where $D=\frac{1}{4} \sum_jX^{22}_j + X^{00}_j - X^{\uparrow\up}_j
- X^{\downarrow\down}_j$ and $T_{a}=\sum_jT_{a,j}$ are correlated
hopping terms that change the number of doubly occupied sites by $a$
\begin{equation}\begin{aligned}
T_{0,j} &= - \sum_{\sigma} \big(
X^{2\sigma}_j X^{\sigma2}_{j+1}
+ X_j^{\sigma0} X_{j+1}^{0\sigma}
+{\rm h.c.}\big) ,\nonumber\\
T_{1,j} &= T_{-1,j}^\dagger=- \sum_{\sigma}
\sigma\left[ X_j^{2\bar{\sigma}} X_{j+1}^{0\sigma}
+ X_j^{0\bar{\sigma}} X_{j+1}^{2\sigma} \right]\ .
\end{aligned}\end{equation}
The $t/U$-expansion is conveniently cast in the form of a unitary transformation
\cite{MacDonaldGirvin88}
\begin{equation}
H' = e^{iS} H e^{-iS} = H +[iS,H] + \frac{1}{2} [iS,[iS,H]] + \dots,
\end{equation}
where the generator $iS$ is chosen as a power series in $t/U$
$iS = \sum_{n=1}^\infty\left(\frac{t}{U}\right)^n iS^{[n]}$.
The operators $S^{[1]},\dots, S^{[k]}$ can be chosen such that the
first $k+1$ terms in the $t/U$-expansion of $H'$ will not change the
number of doubly occupied sites. It follows from
Ref.~\onlinecite{MacDonaldGirvin88} that the unitary transformation
can be written as
\begin{eqnarray}
e^{-iS} &=& \sum_{k\geq 0} \sum_{[m]} \left(\frac{t}{U}\right)^k
\alpha^{(k)}[m] T^{(k)}[m],
\label{eq:unitaryExpansion}\\
T^{(k)}[m] &=& T_{m_1} T_{m_2} \dots T_{m_k}\ ,\quad m_j\in\{-1,0,1\},
\end{eqnarray}
where $\alpha^{(k)}[m]$ are suitably chosen coefficients.
\subsection{Heisenberg sector states in the $t/U$ expansion}
We now need to identify the Heisenberg sector states in the framework
of the $t/U$-expansion. We propose that they are characterized by
their property that, in the framework of the $t/U$-expansion, they are
connected by our unitary transformation to states without any double
occupancies
\begin{equation}
|\psi_H\rangle\Big|_{t/U}=
|\psi\rangle=e^{-iS}\ \sum_{\alpha_j=\uparrow,\downarrow}
f_{\alpha_1\dots\alpha_L}\Big[\prod_{j=1}^LX^{\alpha_j0}\Big]|0\rangle.
\label{psiH}
\end{equation}
Our identification is based on the fact that in the Bethe ansatz
solution exact eigenstates are labelled by sets of (half-odd) integer
quantum numbers, which makes it possible to follow a particular state
when changing the interaction strength $U$. When sending $U/t$ to infinity
for a large but fixed system size one finds that in this limit the
lowest energy eigenstates belong to the Heisenberg sector, which in
turn leads to the identification \fr{psiH}.
In the following we will use the eigenstate $|\psi_H\rangle$ and its
$t/U$-expansion $|\psi\rangle$ interchangeably and leave a discussion
of contributions not captured by the $t/U$-expansion for our
concluding remarks. The
ground state of the half-filled Hubbard model is a particular example
of a Heisenberg state. Heisenberg sector states have the important
property that they are singlets under the $\eta$-pairing SU(2)
algebra. This follows from the easily established fact that
\begin{equation}
[\eta,T_m]=0=[\eta^\dagger,T_m]=[\eta^z,T_m]\ ,\quad m=0,\pm 1.
\label{etaT}
\end{equation}
The commutation relations \fr{etaT} imply that $T^{(k)}[m]$ commute
with the $\eta$-pairing operators, and this in turn implies by virtue of
\fr{eq:unitaryExpansion} that
\begin{equation}
[\eta,e^{-iS}]=0=[\eta^\dagger,e^{-iS}]=[\eta^z,e^{-iS}].
\end{equation}
This, together with the fact that all states with only singly occupied
sites are annihilated by both $\eta$ and $\eta^\dagger$, establishes
that all Heisenberg states are $\eta$-pairing singlets
\begin{equation}
\eta|\psi\rangle=\eta^\dagger|\psi\rangle=\eta^z|\psi\rangle=0.
\end{equation}
Using the results of Refs~\onlinecite{temperleyLieb,saitoproof} we may
construct a basis for the space of $\eta$-pairing singlets. Let us
select a set $A=\{a_1,\dots,a_{2q}\}$ of $2q$ lattice sites and denote
the complementary set by $\bar{A}$. We take all $\bar{A}$ sites to be
singly occupied by spin-$\sigma_j$ fermions, while we form singlet
$\eta$-pairing dimers $|S(a,b)\rangle=|2\rangle_{a}
|0\rangle_{b} + (-1)^{a+b - 1} |0\rangle_{a} |2\rangle_{b}$
on the $A$ sites. This gives an overcomplete set
of states
\begin{eqnarray}
|\boldsymbol{k};\boldsymbol{\sigma}\rangle=\prod_{i=1}^{q}
|S(k_{2i-1},k_{2i})\rangle\prod_{j\in\bar{A}}|\sigma_j\rangle_j,
\label{basis}
\end{eqnarray}
where $\boldsymbol{k}=k_1,\dots,k_{2q}$ is a permutation of
$\{a_1,\dots,a_{2q}\}$. It is
easily verified that the states
\fr{basis} are $\eta$-pairing singlets. A linearly independent set of
states \fr{basis} is formed by imposing a ``non-crossing'' constraint
on the permitted values of $\boldsymbol{k}$ as follows. We first
impose an ordering of sites $\{a_1,\dots,a_{2q}\}$,
e.g. $a_1<a_2<\dots<a_{2q}$. In one dimension this is simply the
natural order of the sites. We then connect the $q$ pairs of sites
$\{(k_{2i-1},k_{2i})\}$ by lines, \emph{cf.} Fig.~\ref{fig:nocrossing}. If
no lines cross, the state corresponding to the pairing
$\boldsymbol{k}$ is permitted. A basis $\mathfrak{B}$ of all
$\eta$-pairing singlet states in the half-filled Hubbard model is
obtained by taking into account all ``sectors'' $0\leq 2q\leq L$ and
all distinct sets $\{a_1,\dots,a_{2q}\}$ of lattice sites in a given
sector.
\begin{figure}[ht]
\includegraphics[width=0.45\textwidth]{figs/nocross.pdf}
\caption{The pairing
$\boldsymbol{k}=\{a_1,a_8,a_2,a_3,a_4,a_7,a_5,a_6\}$ does not
involve crossed lines and fulfils the constraint.}
\label{fig:nocrossing}
\end{figure}
All eigenstates in the Heisenberg sector can then be expressed in the form
\begin{equation}
|\psi\rangle=\sum_{|\boldsymbol{k};\boldsymbol{\sigma}\rangle\in\mathfrak{B}}
A_{\boldsymbol{k};\boldsymbol{\sigma}}\
|\boldsymbol{k};\boldsymbol{\sigma}\rangle\ .
\label{Heisstate}
\end{equation}
\subsection{One spatial dimension}
The $t/U$-expansion allows us to
determine the dependence of the amplitudes
$A_{\boldsymbol{k};\boldsymbol{\sigma}}$ on $U$. It is useful to
define the ``total bond length'' by
\begin{eqnarray}
\mathcal{D}\Big(
\sum_{|\boldsymbol{k};\boldsymbol{\sigma}\rangle\in\mathfrak{B}}
A_{\boldsymbol{k};\boldsymbol{\sigma}}\
|\boldsymbol{k};\boldsymbol{\sigma}\rangle\Big)
&=&\max_{\ontop{|\boldsymbol{k};\boldsymbol{\sigma}\rangle\in\mathfrak{B}}
{A_{\boldsymbol{k};\boldsymbol{\sigma}}\neq 0}}
\mathcal{D}\left(|\boldsymbol{k};\boldsymbol{\sigma}\rangle\right),
\label{eq:bondDistDef}
\end{eqnarray}
where we take $\mathcal{D}\big(|\boldsymbol{k};\boldsymbol{\sigma}\rangle\big)
=\sum_{i=1}^q (\lVert k_{{2i-1}} - k_{{2i}} \rVert +1)$.
It is a straightforward matter to show (see the Supplementary
Material) that
$\mathcal{D}\big(T_n |\psi\rangle \big) \leq
\mathcal{D}\big(|\psi\rangle \big) +n+1$ where $n=0,\pm 1$ and
$|\psi\rangle$ is any Heisenberg sector state \fr{Heisstate}. This in
turn implies that $\mathcal{D}\left[ T^{(k)}[m] |\psi_0\rangle \right]
\leq k+q$, where $q=\sum_{i=1}^k m_i$ and $|\psi_0\rangle$ is any
state with only singly occupied sites. Applying this to the expressions
\fr{psiH}, \fr{eq:unitaryExpansion} for Heisenberg states, we conclude
that the expansion coefficients in \fr{Heisstate} fulfil
\begin{equation}
A_{\boldsymbol{k};\boldsymbol{\sigma}}={\cal O}\Big(
\big(t/U\big)^{\sum_{i=1}^q \lVert k_{{2i}} - k_{{2i-1}}\rVert} \Big).
\label{eq:UDep}
\end{equation}
These results cannot be straightforwardly generalized to $D>1$ because
our definition of a total bond length hinges on
$|\boldsymbol{k};\boldsymbol{\sigma}\rangle$ forming a basis of
states, which imposes constraints on the allowed values of
$\boldsymbol{k}$. In a typical Heisenberg sector state we have a
finite density of doubly occupied sites and the coefficients
\fr{eq:UDep} are of an extremely high order in $t/U$. In order to
proceed we will assume that the $t/U$-expansion for the wave-function
\fr{Heisstate} has a finite radius of convergence. We know this to be
the case for certain quantities such as the ground state energy
\cite{book}.
\subsection{Quantum disentangled diagnostic}
According to Refs~\onlinecite{qdl,hubb1} a QDL can be diagnosed by
preparing the system in a finite energy-density eigenstate with
volume-law bipartite EE, and then to carry out a
projective measurement of the $z$-component of spin on each site of
the lattice. If the resulting state is characterized by an area-law
EE, the original state realizes a QDL.
We now address this proposal in the framework of the $t/U$-expansion.
As our initial state we choose a finite energy density eigenstate
\fr{psiH} in the Heisenberg sector. These generically have
volume-law entanglement entropies as can be seen from the fact that
the corresponding macro-states have finite thermodynamic entropy
densities. As the 1D Hubbard model is integrable there also exist
finite energy density eigenstates with area-law EE, but these are the
exception rather than the rule. Let us assume that the outcome of our
projective spin measurement is that we obtain spin zero at all sites
in the set $A=\{a_1,a_2,\dots a_{2q}\}$ and spin $\sigma_j=\pm 1/2$
everywhere else. Then the state of the system after the projective
measurement can be written as
\begin{eqnarray}
|\psi_{\rm proj}\rangle &=&
\frac{1}{\sqrt{\mathcal{N}}}
\prod_{j_1=1}^{a_1-1} X_{j_1}^{\sigma_{j_1} \sigma_{j_1}}
\left( X_{a_1}^{00} + X_{a_1}^{22} \right)\nonumber\\
&\times&\prod_{j_2=a_1+1}^{a_2-1}
X_{j_2}^{\sigma_{j_2} \sigma_{j_2}}
\left( X_{a_2}^{00} + X_{a_2}^{22} \right) \cdots
|\psi\rangl
, \label{eq:projDef}
\end{eqnarray}
where $\mathcal{N}$ is a normalisation factor. Using our results
\fr{Heisstate} for the structure of Heisenberg states in
the $t/U$ expansion, we can rewrite this in the form
\begin{equation}
|\psi_{\rm proj}\rangle = \sum_{Q\in S_{2q}}' W(Q) \prod_{i=1}^{q}
|S(k_{Q_{2i-1}},k_{Q_{2i}})\rangle
|\boldsymbol\sigma\rangle. \label{eq:bellconjecture}
\end{equation}
Here $|\boldsymbol\sigma \rangle = \prod_{j \notin A}
|\sigma_j\rangle$ is a product state fixed by the projective
measurement, and the sum $\sum'_{Q\in S_{2q}}$ is restricted to the
permutations $Q$ such that $a_{Q_1},a_{Q_2}, \dots a_{Q_{2q}}$
corresponds to singlet states that satisfy the non-crossing
condition. As a result of \fr{eq:UDep} the amplitudes $W(Q)$ fulfil
\begin{equation}
W(Q) \sim \mathcal{O}\left( \big(t/U\big)^{\sum_{i=1}^q
\lVert k_{Q_{2i}} - k_{Q_{2i-1}}\rVert} \right).
\end{equation}
In the leading order in the $t/U$-expansion that is allowed to be
non-zero by the above considerations there is only a single term in
\fr{eq:bellconjecture} and we are dealing with a spatially ordered
product state of singlet dimers,
\emph{cf.} Fig.~\ref{fig:leading}.
\begin{figure}[ht]
\includegraphics[width=0.4\textwidth]{figs/leading.pdf}
\caption{Structure of the singlet pairings in the leading
non-vanishing term of the projected state in the $t/U$-expansion.}
\label{fig:leading}
\end{figure}
In the following we will assume for simplicity that the numerical
coefficient of this term is indeed different from zero, which will be
generically the case. Depending on the choice of the initial
Heisenberg state and the outcome of the projective measurement it is
possible that this coefficient vanishes. Such cases can be
accommodated by relatively straightforward modifications of the
following discussion.
\subsection{Entanglement entropy.}
At any finite order in the $t/U$-expansion the projected states
\fr{eq:bellconjecture} are only weakly entangled. Let us consider the
generic case, in which the leading non-zero term in the projected
state has the form shown in Fig.~\ref{fig:leading}. In this case
the leading term in von Neumann entropy of a sub-system $A$ is
simply given by $S_{{\rm vN},A}=\ln(2)$ if we cut one of the singlet
dimers when we bipartition the system, and zero otherwise. At leading
order it is not possible for the bipartition to cut more than one
dimer. In order to describe the structure of the leading corrections
to this result we take the subsystem $A$ to be the interval
$[1,\ell]$, where $a_{2n+1}<\ell<a_{2n+2}$. Here $\{a_j\}$ are the
sites in the projected state that are either doubly occupied or empty.
We then cast the projected state in the form
\begin{widetext}
\begin{equation}\begin{aligned}
|\psi_{\rm proj}\rangle &=
|\psi_L\rangle\left(
\Bigg\vert
\begin{tikzpicture}[scale=0.9,baseline=-.2ex]
\filldraw (-.5,0) node[below] {$a_{2n-1}$} circle (2pt);
\filldraw (0.5,0) node[below] {$a_{2n}$} circle (2pt);
\filldraw (1.5,0) node[below] {$a_{2n+1}$} circle (2pt);
\filldraw (2.5,0) node[below] {$a_{2n+2}$} circle (2pt);
\filldraw (3.5,0) node[below] {$a_{2n+3}$} circle (2pt);
\filldraw (4.5,0) node[below] {$a_{2n+4}$} circle (2pt);
\draw (-.5,0) arc (180:0:.5);
\draw (1.5,0) arc (180:0:.5);
\draw (3.5,0) arc (180:0:.5);
\draw[dashed] (2,-.5) -- (2,1.0);
\end{tikzpicture}
\Bigg\rangle
+
\varepsilon_1
\Bigg\vert
\begin{tikzpicture}[scale=0.9,baseline=-.2ex]
\filldraw (-.5,0) node[below] {$a_{2n-1}$} circle (2pt);
\filldraw (0.5,0) node[below] {$a_{2n}$} circle (2pt);
\filldraw (1.5,0) node[below] {$a_{2n+1}$} circle (2pt);
\filldraw (2.5,0) node[below] {$a_{2n+2}$} circle (2pt);
\filldraw (3.5,0) node[below] {$a_{2n+3}$} circle (2pt);
\filldraw (4.5,0) node[below] {$a_{2n+4}$} circle (2pt);
\draw (-.5,0) arc (180:0:.5);
\draw (1.5,0) arc (135:45:2.121);
\draw (2.5,0) arc (180:0:.5);
\draw[dashed] (2,-.5) -- (2,1.0);
\end{tikzpicture}
\Bigg\rangle
\right.\\
&
\left.
+
\varepsilon_2
\Bigg\vert
\begin{tikzpicture}[scale=0.9,baseline=-.2ex]
\filldraw (-.5,0) node[below] {$a_{2n-1}$} circle (2pt);
\filldraw (0.5,0) node[below] {$a_{2n}$} circle (2pt);
\filldraw (1.5,0) node[below] {$a_{2n+1}$} circle (2pt);
\filldraw (2.5,0) node[below] {$a_{2n+2}$} circle (2pt);
\filldraw (3.5,0) node[below] {$a_{2n+3}$} circle (2pt);
\filldraw (4.5,0) node[below] {$a_{2n+4}$} circle (2pt);
\draw (-.5,0) arc (135:45:2.121);
\draw (0.5,0) arc (180:0:0.5);
\draw (3.5,0) arc (180:0:.5);
\draw[dashed] (2,-.5) -- (2,1.0);
\end{tikzpicture}
\Bigg\rangle \right)|\psi_R\rangle+\dots
,
\label{eq:higherOrderTerms}
\end{aligned}\end{equation}
\end{widetext}
where $|\psi_L\rangle$ ($|\psi_R\rangle$) is the part of the
leading term of the projected state that involves the sites
$j<a_{2n-1}$ ($j>a_{2n+4}$). The coefficients $\epsilon_{1,2}$
are of order $(t/U)^{2(a_{2n+3}-a_{2n+2})}$ and
$(t/U)^{2(a_{2n+1}-a_{2n})}$ respectively. The von Neumann entropy for
our subsystem is then
\begin{equation}
S_{{\rm vN},A}= \ln 2 + \frac{9}{16}\varepsilon_1^2\varepsilon_2^2
\big(
1 + 2 \ln\Big( \frac{4}{3} \Big) - 2 \ln(\varepsilon_1\varepsilon_2)
\big)
+\dots
\label{eq:higherOrderCorrections}
\end{equation}
We note that the correction is positive. The physical picture that
emerges is very simple: for small $t/U$, the
projected state is very close to being a product of Bell pairs and
the bipartite EE takes the form shown in
Fig.~\ref{fig:sketch2}.
\begin{figure}[ht]
\includegraphics[width=0.45\textwidth]{figs/sketch.pdf}
\caption{Form of the bipartite entanglement entropy after projective
measurement of a general Heisenberg state. Deviations from $S(x)=\ln 2, 0$
arise from higher order terms, as seen in (\ref{eq:higherOrderTerms}),
(\ref{eq:higherOrderCorrections}).}
\label{fig:sketch2}
\end{figure}
The average double occupancy in a Heisenberg state is ${\cal
O}(t^2/U^2)$, which implies that the average distance between doubly
occupied/unoccupied sites is very large $\sim U^2/t^2$. Hence, if the
outcome of our spin measurement is close to its average, the sites
$a_1,\dots, a_{2q}$ are well separated and the leading term
\fr{fig:leading} is expected to provide an excellent approximation as
long as the $t/U$-expansion converges. We note that the results
obtained by the $t/U$-expansion are fully compatible with those
obtained by integrability methods in section
\ref{sec:integrability}. This, and the fact that the $t/U$-expansion
is known to converge for simple quantities like the ground state
energy\cite{book,takahashigsE}, provides support for its
applicability.
\subsection{Higher dimensions $D>1$}
A key element of our analysis in the one
dimensional case is the notion of a bond length, which fulfils
${\cal D}(|\psi_1\rangle)\leq{\cal D}(|\psi_1\rangle+|\psi_2\rangle)$
for arbitrary $\eta$-pairing singlet states $|\psi_{1,2}\rangle$. Our
definition of a bond length utilizes the availability of a convenient
basis of such states. It is this aspect which makes the
$D=1$ special. In $D>1$ we proceed by again first fixing a set
$A=\{a_i\}$ of $2q$ unoccupied or doubly occupied sites. We denote by
$E_{ij}$ a bond that joins sites $a_i$ and $a_j$. We then define
a dimer configuration $\mathfrak{C}$ as a set of $q$ bonds $C_{ij}$
connecting sites $a_i$ and $a_j$ such that all sites belong
to precisely one bond. Each dimer configuration gives rise to
$\eta$-pairing singlet states of the form
\begin{equation}
|\mathfrak{C};{\boldsymbol\sigma}\rangle =
\prod_{ C_{ij} \in \mathfrak{C} } |S(a_{i},a_{j})\rangle
\prod_{j\notin A}|\sigma_j\rangle_j\ .
\label{dimercoverings}
\end{equation}
The states \fr{dimercoverings} form an overcomplete set.
In order to select a set of linearly independent states we use the
function $\overline{\mathcal{D}}$ defined by
$\overline{\mathcal{D}}\,(|\mathfrak{C};{\boldsymbol\sigma}\rangle ) =
\sum_{C_{ij}\in\mathfrak{C}} \big(\lVert a_{i} - a_{j} \rVert + 1\big)$,
where $\lVert a_{i} - a_{j} \rVert$ denotes the Manhattan distance
between sites $a_i$ and $a_j$. Among \fr{dimercoverings} we
select the $\frac{(2q)!}{(q+1)!(q!)}$ states that have the lowest
values under the map $\overline{\mathcal{D}}$: it is this criterion that
allows the inequalities in 1D to be inherited in the higher-dimensionsal
case.
Repeating this construction for all values of $2q\leq L$ and all distinct
sets $A$ gives a basis $\mathcal{B}$ of $\eta$-pairing singlet states.
We now define our total bond distance $\mathcal{D}$ in the
same way as in (\ref{eq:bondDistDef}) with the covering $\mathfrak{C}$
taking the place of the vector $\boldsymbol{k}$. With these
definitions in place it is straightforward to show (see the
Supplementary Material) that $\mathcal{D}\big(T_n |\psi\rangle \big)
\leq \mathcal{D}\big(|\psi\rangle \big) +n+1$ where $n=0,\pm 1$ and
$|\psi\rangle$ is any Heisenberg sector state
\begin{equation}
|\psi\rangle=\sum_{|\mathfrak{C};\boldsymbol{\sigma}\rangle\in\mathfrak{B}}
A_{\mathfrak{C};\boldsymbol{\sigma}}\
|\mathfrak{C};\boldsymbol{\sigma}\rangle\ .
\label{Heisstate2D}
\end{equation}
This in turn implies that $\mathcal{D}\left[ T^{(k)}[m] |\psi_0\rangle
\right] \leq k+q$, where $q=\sum_{i=1}^k m_i$ and $|\psi_0\rangle$
is any state with only singly occupied sites. It follows that the
$d$-dimensional generalisation of (\ref{eq:UDep}) is
\begin{equation}
A_{\mathfrak{C};\boldsymbol{\sigma}}={\cal O}\Big(
\big(t/U\big)^{\sum_{C_{ij}\in\mathfrak{C}} \lVert a_i - a_j\rVert} \Big).
\end{equation}
The main difference to $D=1$ is that now there can be several basis
states contributing to the leading order term in $t/U$. To see this
we may consider a square lattice and focus on the situation where the
sites in $A$ completely fill an $m\times n$ rectangle (with at least
one of $m$, $n$ even). The leading-order term is that generated by
$(T_1)^{mn/2}$ produces a superposition of all states in $\mathfrak{B}$
that correspond to nearest-neighbour dimer coverings of this
rectangle. Nevertheless, it is apparent that to leading order in the
$t/U$-expansion the structure of the state is such that the
entanglement entropy follows an area-law.
\section{Conclusions}
\label{sec:conc}
We have shown that there are particular ``Heisenberg sector''
eigenstates in the one dimensional half-filled Hubbard model that
realize the quantum disentangled liquid state of matter in the strong
sense proposed in Ref.~\onlinecite{qdl}. These states are obtained by
freezing the charge degrees of freedom in their ground state
configuration in the framework of the Bethe ansatz solution of the
Hubbard model. Using methods of integrability we have demonstrated
that the charge degrees of freedom (``light particles'') do not
contribute to the volume term of the bipartite entanglement
entropy. Employing strong-coupling expansion techniques we have shown
(under the assumption that the expansion converges) that a measurement
of the spin degrees of freedom at all sites (``heavy particles'')
leaves the system in a state that is area-law entangled, which is the
defining characteristic of a ``strong QDL''\cite{qdl}.
In contrast to Heisenberg sector states, the entanglement entropy for
maximal entropy (thermal) states at a given energy density at large
values of $U/t$ does have a contribution that involves the charge
degrees of freedom, but it is of the form \fr{weak}, i.e.
\begin{eqnarray}
S_{\rm vN,A}&=&\big(s_{\rm spin}+s_{\rm
charge}\big)|A|+o\big(|A|\big)\ ,\nonumber\\
s_{\rm spin}&=&{\cal O}(1)\ ,\quad
s_{\rm charge}={\cal O}\Big(\frac{u}{T}e^{-u/T}\Big)\ ,\quad
\end{eqnarray}
We expect a similar volume law to occur in the entanglement entropy
after a measurement of all spins. This suggests that there is a \emph{weak}
variant of a QDL, which is characterized by a volume term in the EE
after measurement of the heavy degrees of freedom that is
exponentially small in the ratio of masses (i.e. the ratio $U/t$ in
the Hubbard model). In this limit the volume term is only observable
in enormously (exponentially) large systems and would be practically
impossible to detect in numerical simulations. We expect this
weak scenario to be realized quite generally in strong coupling limits
irrespective of whether the model one is dealing with is integrable or
not. In particular, this scenario is compatible with our
strong-coupling analysis of the QDL diagnostic in $D=2$.
An interesting question is how to access Heisenberg sector states in
the one dimensional case. In principle this can be achieved by means
of a quantum quench \cite{EFreview} from a suitably chosen initial
state. At late times the system locally relaxes to a steady state that
is described by an appropriate generalized Gibbs ensemble. The
parameters of this ensemble are fixed by the initial conditions and
can in principle be tuned in such a way that one ends up with a
Heisenberg sector state. How to do this in practice is an interesting
albeit rather non-trivial question. Fixing the energy density to be
small compared to the Mott gap is sufficient to obtain a
non-equilibrium steady state that realizes the weak form of a QDL
characterized by \fr{weak}. In order to remove the $s_{\rm charge}$
contribution, the expectation values of higher conservation laws in
the initial state have to be chosen appropriately and it would be
interesting to investigate this issue further.
\acknowledgments
We are grateful to P. Fendley, J. Garrison, T. Grover, L. Motrunich and M.
Zaletel for helpful discussions. This work was supported by the EPSRC under
grant EP/N01930X (FHLE), by the National Science Foundation,
under Grant No. DMR-14-04230 (MPAF) and by the Caltech Institute of
Quantum Information and Matter, an NSF Physics Frontiers Center with
support of the Gordon and Betty Moore Foundation (MPAF).
|
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| 5,206
|
{"url":"https:\/\/datascience.stackexchange.com\/questions\/69238\/confusion-matrix-and-auc-in-univariate-anomaly-detection\/69312","text":"# Confusion Matrix and AUC in univariate Anomaly Detection\n\nIn the code I upon a csv file which only has one column. The data in there in not that important just normal numbers.\n\n# Pandas - data handling\nimport pandas as pd\n\n# Numpy for mathematical operations\nimport numpy as np\nimport pandas as pd\n\n# Scikit learn for the DBSCAN algorithm\nfrom sklearn.cluster import KMeans\n\n# Matplotlib - plots\nimport matplotlib.pyplot as plt\n\nimport seaborn as sns\n\n# Add manually an anomaly value\nfor i in range(0, 50):\nall_data = all_data.append({all_data.column.max() + np.random.randint(1000, 2000) }, ignore_index=True)\n\nclf = KMeans(n_clusters=2, init='k-means++', max_iter=300, n_init=10, random_state=1)\n\nclf.fit(all_data.column.values.reshape(-1, 1))\n\nall_data_prd = clf.predict(all_data.column.values.reshape(-1, 1))\nall_data_prd\n\nplt.scatter(all_data_prd, all_data.valueData_value, c=all_data_prd)\n\n\n\nMy question is if it is possible to have a confusion matrix and roc curve in this case when I only have once column.\n\nTo this column I have added some anomalies.\n\nThe goal:\n\n1. Add these anomalies that are way way bigger than the maximum value of the column and then check how many out of the 50 added anomalies, how many the K-Means finds\n2. Add these anomalies that are not that much bigger than the maximum value of the column then check how many out of the 50 added anomalies are found. At the end I want a Confusion matric and a ROC graph to see how good they perform.\n\u2022 It looks to me like there are lots of problems with this approach: (1) I doubt k-means is a good approach for one-dimension data, there are certainly better ways to detect anomalies (2) confusion matrix and ROC are evaluation measures for classification, so it's not going to work directly with the output of k-means. (3) since you know what is regular data and anomalies, why not using supervised classification? (4) Finally if the goal is just to find the threshold which is optimal to separate regular cases vs anomalies, you might not even need ML. \u2013\u00a0Erwan Mar 6 '20 at 13:05\n\u2022 @Erwan, thank you for the very detailed comment. I had some questions would appreciate if you could give me a feedback, if not still thank you. 1. What other methods\/algorithms or approaches would you suggest for this case ? 3. What difference would supervised classification make in this case and which exact method you had in mind ? 4.what could I use then ? \u2013\u00a0E199504 Mar 6 '20 at 19:39\n\nThe idea is simply to evaluate the performance for every possible threshold, i.e. count the number of True\/False Positive\/Negative for every value. Let's say you take the set $$S$$ of all the distinct values in your column. For every value $$t \\in S$$ you calculate the confusion matrix (i.e. how many TP\/FP\/TN\/FN) obtained by predicting every value $$x$$ as positive if if $$x>t$$, negative otherwise.","date":"2021-06-21 17:38:32","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\": 4, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6154998540878296, \"perplexity\": 1011.5434395139248}, \"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-2021-25\/segments\/1623488286726.71\/warc\/CC-MAIN-20210621151134-20210621181134-00049.warc.gz\"}"}
| null | null |
I'm 42, and during a cleaning, my dentist mentioned that I had bone degeneration starting and recommended I see an orthodontist for braces. I've been to two ortho's and each one stressed the importance of self-esteem after getting my teeth straightened.
I'm trying to argue my case and the expense to my husband who thinks this is just for cosmetic reasons, and I can't seem to find any hard evidence as to why this is beneficial to my oral health. I don't want to go to my dentist and ask him again, as I think he may give me a biased answer.
I know your question regards orthodontic treatment, but firstly, is your periodontal condition being treated? You mentioned bone degeneration, and that typically is caused by plaque, calculus and bacteria. It is halted, or put into remission, by periodontal surgery if indicated or preferably by non-surgical periodontal treatment performed by an office trained in these techniques and stressing your participation in a stringent homecare routine.
If your gums can become healthy, and bone loss halted, then orthodontic treatment could potentially be strictly "cosmetic". Of course it is easier to keep straight teeth clean; crowded teeth may be harder to maintain but are maintainable. Now if bone loss has caused shifting or splaying out of the teeth, there could be a periodontal benefit to orthodontic treatment. Do know that orthodontic treatment without optimal periodontal (gum) health could be disastrous!
I would address your peridontal condition with your dentist or a periodontal specialist. Many of these specialists have great working relationships to take a team approach to getting you to a healthy, attractive smile that you can maintain for many years.
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| 5,206
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Not all hardwood floors need a complete refinish. Our hardwood resurfacing service is an affordable alternative to traditional refinishing - and it can be completed within just a few hours at only 99¢ per square foot.
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© Copyright Fabulous Floors Denver 2017. Site designed by Kite Media.
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{
"redpajama_set_name": "RedPajamaC4"
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| 6,857
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Q: Can't get value from session with websocket using flask I want to pass value from input form on subpage "A" to the backend, then process it, and send back to another subpage "B". I was creating a session value in different ways(using javascript or defining it in app.route), but cant get this value in my socketio event.
Here is how i create session value in js(i want to pass value from input form after button click):
$("#submit").on("click", function(){
...
sessionStorage.setItem('testname', $('#test_surname').val());
...
});
Here is how i was trying to get this value in my socket event:
@socketio.on('example_event')
def id_scan(some_value):
...
# first way
test_name = session.get('testname')
# second way
test_name = session['testname']
...
So in first option i get null and in second way i get keyerror. Of course i can easily get this value in jinja.
I also tried to define session value in app.route like this:
@app.route("/")
def index():
session['testname'] = 'testname'
And i can get this value in jinja but in flask i get same errors
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 4,043
|
Q: Passing data through shaders How it possible to pass data through shaders, when using vertex, tess. control, tess. evaluation, geometry and fragment shaders. I've tried using interface block this way.
//vertex shaders
out VS_OUT { ... } vs_out;
Than I wrote this code in tessellation control shader:
in VS_OUT { ... } tc_in;
out TC_OUT { ... } tc_out;
So, tesselation control shader called once for every vertex. Does it mean that tc_in must be not array but single variable. I'm not really sure about that because of sneaky gl_InvocationID.
Then things become tough. Tessellation evaluation shader. Something tells me that evaluation shader should take interface block as array.
in TC_OUT { ... } te_in[];
out TE_OUT { ... } te_out[];
Moving to geometry shader. Geometry shader takes multiple vertices, so, definitely array interface block.
in TE_OUT { ... } gs_in[];
out vec3 random_variable;
...
random_variable = gs_in[whatever_index];
Seems legit for me, but data didn't come to fragment shader.
I would appreciate any advice.
A: Tessellation control shaders take the vertices of a patch as modify them in some way, so their input and output is
in VS_OUT { ... } tc_in[];
out TC_OUT { ... } tc_out[];
The control shader gets called for every patch vertex (see which one by using gl_InvocationID), so you usually don't need any loops to implement it.
Tessellation evaluation shaders take these modified vertices and output a single vertex per invocation, thus we have
in TC_OUT { ... } te_in[];
out TE_OUT { ... } te_out;
Geometry shaders take multiple vertices and may output a different number of vertices, but these are constructed explicitly using EmitVertex and EmitPrimitive, so there is only one output element that has to be filled before each call of EmitVertex:
in TE_OUT { ... } gs_in[];
out GS_OUT { ... } gs_out;
But don't forget to set glPosition in your geometry shader, otherwise OpenGL won't know where to place your vertices.
The interpolated values of gs_out are then passed to the fragment shader.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 8,216
|
Members of Council
Councillor Michael Ford
Ward 1 Etobicoke North
Jonathan.Kent4@toronto.ca
Community and Stakeholder Relations
Trent Jennett
Trent.Jennett2@toronto.ca
Constituency Relations
Brian Klochko
Brian.Klochko@toronto.ca
Shima Bhana
Shima.Bhana2@toronto.ca
Special Assistant
Nicolas Di Marco
Nicolas.DiMarco@toronto.ca
Michael Ford is the City Councillor for Ward 1, Etobicoke North.
A lifelong resident of Etobicoke, Michael was first elected to public office in 2014 as Toronto District School Board Trustee for Etobicoke North. Michael was elected to Toronto City Council in 2016 and re-elected in 2018.
Prior to his election as City Councillor, Michael worked to ensure the TDSB's $3.2 billion budget was being spent efficiently to best serve students in his capacity as Trustee and Vice-Chair of the Finance Committee. Michael also served as the Co-Chair for the Parent Involvement Advisory Committee and on the Inner City Advisory Council where he championed safe schools.
On this term of Council, Michael is a sitting member of several boards and committees such as the Toronto Police Services Board, the Economic and Community Development Committee, Audit Committee and the Canadian National Exhibition Association Board.
Michael has been a strong advocate toward strengthening bonds between Toronto Police Services and our communities to fight gun and gang violence. He also remains committed to economic growth and development in Ward 1, bringing jobs to Etobicoke North.
In his spare time, Michael is an avid outdoorsman, photography enthusiast and a licensed pilot.
Councillor Ford continues to focus on delivering excellent service to the constituents of Etobicoke North, while working to keep taxes low and ensuring Toronto is an affordable place to live, work and play.
(Information provided by the Councillor)
Canadian National Exhibition Association, Municipal Section
Canadian National Exhibition Association, Board of Directors
Economic and Community Development Committee
Etobicoke York Community Council
Toronto Police Services Board
Albion Islington Square BIA
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,882
|
Q: Finding largest number and deleting remaining smallest number I am trying to Delete cell by comparing Names within the same column, If I find the same name, Then make sure name is containing bigger number after forward
slash, If I find the bigger number then make sure my C column matching my criteria (NOT Playing). If it is matching then delete all cell which is a smaller number with column c saying Still Playing and keep only bigger number which is saying Not Playing. Any Ideas, I have posted my code but when I am finding a bigger number, it's going back and comparing with another cell so it's coming back to the same number. Here is my code..
Sub xm()
Dim lastrow As Long
Dim i As Long, firstD As Integer, secondD As Integer
Dim firstForward As Integer
Dim secondForward As Integer, score1 As String, score2 As String
Dim fName As String, sName As String
lastrow = ActiveSheet.UsedRange.Rows.Count
For i = 2 To lastrow
firstD = InStr(1, ActiveSheet.Cells(i, 1), "/")
fName = Left(ActiveSheet.Cells(i, 1), firstD - 1)
secondD = InStr(1, ActiveSheet.Cells(i + 1, 1), "/")
sName = Left(ActiveSheet.Cells(i + 1, 1), secondD - 1)
If fName = sName Then
firstForward = InStr(1, ActiveSheet.Cells(i, 1), "/")
score1 = Mid(ActiveSheet.Cells(i, 4), firstForward + 1)
secondForward = InStr(1, ActiveSheet.Cells(i + 1, 4), "/")
score2 = Mid(ActiveSheet.Cells(i + 1, 4), firstForward + 1)
If score1 > score2 Then
If ActiveSheet.Cells(i, 3) = "Not Playing" Then
'Delete entire cell
End If
End If
End If
i = i + 1
Next i
End Sub
Output:---
|
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"redpajama_set_name": "RedPajamaStackExchange"
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|
{"url":"https:\/\/math.stackexchange.com\/questions\/2649417\/determining-linear-independence-given-minimal-polynomials","text":"# determining linear independence given minimal polynomials\n\nSuppose one has a set of numbers $S$, each of which is algebraic over a field $F$. Does knowing the minimal polynomial over $F$ of each element of $S$ help to determine whether $S$ is linearly independent over $F$?\n\nThe context is the case of a number $N=\\sum_{i=1}^{n}a_i$ where I already know the minimal polynomial of each $a_i$, in a program I'm working on for calculating minimal polynomials. Let $d_i$ be the degree of the minimal polynomial of $a_i$, and $D=\\prod_{i=1}^{n}d_i-1$. I start by calculating powers $1$ through $D$ of $N$ in its sum form, treating it as a multivariate polynomial with each $a_i$ a separate variable, and using the minimal polynomials of the $a_i$ to rewrite any terms whose degree with respect to $a_i$ is greater than or equal to $d_i$ as expressions of lower degree. This gives me $D$ polynomials with at most $D$ terms each, with the $k$th polynomial representing $N^k$. I treat the coefficients of the polynomials as a matrix, the $N^k$ its augmentation, and triangularize the matrix to find a linear combination of the $N^k$ that equals a constant. Subtracting the constant from the linear combination gives an annulling polynomial of $N$. I then factor this polynomial over the rationals and, for each of its factors, replace each $x^k$ with $N^k$ in its sum form in terms of the $a_i$ in order to see whether the factor is still annulling. I assume this last step can fail to detect that a factor is annulling in cases where the $a_i$ are linearly dependent, but detecting whether they are in general seems like a hard problem.\n\n\u2022 For the simple question of \"I have the polynomials in hand can I calculate this,\" the answer is yes. You have a matrix of coefficients $M\\times [1~ x~ x^2~ \\cdots ~ x^n]^T$ which equals a vector of your minimal polynomials. If this is linearly independent then the matrix of coefficients has an inverse. I only leave this as a comment because I am not 100% clear if this is actually your question. \u2013\u00a0law-of-fives Feb 13 '18 at 20:39\n\u2022 @law-of-fives It sounds like it is the answer to my question. What are the coefficients in the matrix, exactly? \u2013\u00a0Alex Kindel Feb 13 '18 at 20:45\n\u2022 @law-of-fives Oh wait, are they just the coefficients of the minimal polynomials? I suppose what wasn't obvious to me is that the minimal polynomials are linearly independent iff the numbers of which they are the minimal polynomials are too. \u2013\u00a0Alex Kindel Feb 13 '18 at 20:49\n\u2022 your minimal polynomials are independent if they form a basis. If they form a basis, then we can transform from the standard polynomial basis to your custom basis. This amounts to what I described. I hope that helps! \u2013\u00a0law-of-fives Feb 13 '18 at 21:32\n\u2022 @law-of-fives On second thought, it seems that linear dependence of the minimal polynomials of the elements of S doesn't imply linear dependence of the elements of S: the square roots of primes are all linearly independent from each other over the rationals, but since their minimal polynomials are each degree two, at most three of them can be linearly independent. \u2013\u00a0Alex Kindel Feb 14 '18 at 5:41","date":"2019-08-26 00:13:23","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\": 0.8269518613815308, \"perplexity\": 155.7354149711689}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027330913.72\/warc\/CC-MAIN-20190826000512-20190826022512-00022.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/science\/chemistry\/chemistry-9th-edition\/chapter-3-stoichiometry-questions-page-127\/25","text":"# Chapter 3 - Stoichiometry - Questions - Page 127: 25\n\nEach person would have $8.603*10^{13}$ dollars or 86.03 trillion dollars.\n\n#### Work Step by Step\n\n$\\frac{6.0221*10^{23}}{7*10^9}=8.603*10^{13}$ dollars per person\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2018-10-20 21:33:56","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\": 0.5341569185256958, \"perplexity\": 2950.2307114531573}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583513441.66\/warc\/CC-MAIN-20181020205254-20181020230754-00279.warc.gz\"}"}
| null | null |
\section{Introduction}
Horizons are ubiquitous in classical General Relativity. Although there are many different types of horizons, the main common consequence of their presence is that they, someway or another, excise parts of spacetime making them inaccessible from the outside.
This is the case, for instance, for event and cosmological horizons on which we will concentrate in this paper. From our spacetime region, physical observations beyond these horizons are classically out of the question. It has been proposed in the literature~\cite{Jacobson:1991gr,Jacobson:1993hn,Unruh:1994je} that at very high energies horizons may be blurred because of effective superluminal modifications of the dispersion relations that would allow high energy modes to leak across the classical horizon. These proposals are inspired in the behaviour of analogous configurations in condensed matter systems. This would solve, for instance, the so-called trans-planckian problem in black hole physics~\cite{Unruh:1976db}. But softening the horizons is a non-trivial task that may affect the global spacetime structure~\cite{Barbado:2011ai}. This could be taken as an indication that horizons might not be as impressive and frightening as they seem from the classical point of view and might allow quantum mechanically for interactions between classically separated regions.
Only upon the fall of these titans, we could try to take one more (huge, granted) step and speculate about other universes than our own. If horizons are keeping us from peeking in regions of our own universe, it will be even more difficult to try to advance any possible information about the physical events and characteristics of those universes. However, as we will argue in this paper, quantum mechanics applied to the whole spacetime will not only allow but force the opening of those excised regions and connect them to our own. So this outrageously big leap might not be beyond our capabilities after all. The potential connections with these other universes, being quantum in nature, could be expected to have characteristics similar to those found in the connections across horizons.
Actually, from the classical point of view and hence subject to the presence of horizons, classical links \cite{Morris:1988tu,Morris:1988cz,Yurov:2006we,GonzalezDiaz:1996sr,GonzalezDiaz:1998pc,Gott:1997pm} (by means of Lorentzian wormholes) and quantum tunnels \cite{Hawking:1988ae,Hawking:1990in,Giddings:1988wv,Garay:1993rr,MenaMarugan:1994qz} (Euclidean wormholes) between otherwise separate universes have been previously considered, also taking into account that they can imprint some observable signatures \cite{GonzalezDiaz:1997xc,Shatskiy:2007xr,Shatskiy:2008ym,GonzalezDiaz:2011ke}. On the other hand, an attempt to pinpoint possible observable quantum effects of this multiverse on a single universe has been considered, in the formalism of third quantization, by means of the possibility of entanglement between pairs of universes \cite{RoblesPerez:2010ut,RoblesPerez:2011yj,RoblesPerez:2011zp,AlonsoSerrano:2012wc}.
Instead of embarking ourselves in this vast task on mostly unexplored territory (quicksand, as Coleman would say \cite{Coleman:1988tj}), we will delve into the quantum theory for a single universe and analyze the role of horizons in it, a related but simpler endeavor.
With this aim we will consider a minisuperspace model for a spherically symmetric spacetime in the presence of a positive cosmological constant. This minisuperspace model can be written in terms of a Kantowski-Sachs metric that depends on two variables.
The maximal analytic extension
of the classical solutions (Schwarzschild-deSitter spacetimes) typically contain both black hole and cosmological horizons that isolate our spacetime region from those beyond. Quantizations of spacetimes of Kantowski-Sachs type have been analyzed before from different points of view and with different matter contents \cite{Halliwell:1990tu,Campbell:1990uu,Garay:1991um,MenaMarugan:1993rs,MenaMarugan:1994sj} (see also \cite{Kuchar:1994zk,Gambini:2013ooa}).
In this paper we will carry out a canonical quantization procedure that specifically allows us to tackle with the issue of permeability across the horizons. We will follow an extension of Dirac's
canonical quantization program for systems (like ours) with first-class constraints \cite{yeshiva} (along the lines developed by Ashtekar et al. \cite{Ashtekar:1994kv}).
We will decompose the physical Hilbert space of the system into two subspaces corresponding to states with support in configurations that exclusively describe spacetime regions either between or beyond the horizons. It will turn out that these Hilbert subspaces are not stable under the action of unitary operators that implement a natural notion of evolution on physical states. The home for this physical evolution will be the tensor product of both subspaces, which hence will not be dynamically separable but entangled. This means that quantum correlations among classically disjoint regions are a generic unavoidable feature in this quantization. This entanglement between classically disconnected regions opens up the possibility that if we embraced a broader picture in which our universe is not isolated but multiply connected to others, we might need to consider the complete structure of the multiverse in order to make a quantum theory and that there would be some kind of quantum effects of other universes in our own, driven by entanglement.
The outline of the paper is the following. In Section \ref{sec:class}, we construct the model and describe the classical solutions to Einstein equations. Once we have the phase space of our system, in Section \ref{sec:canquant} we quantize it following Dirac's extended canonical quantization program \cite{Ashtekar:1994kv} and analyze various useful bases and representations of the physical Hilbert space. Section \ref{sec:quanthor} is devoted to discuss the generic presence of quantum correlations between classically separated regions. We summarize and conclude in Section \ref{sec:concl}.
\section{Classical solutions}
\label{sec:class}
We construct a model with a general spherically symmetric metric
that depends on two variables $A$ and $b$ ---which play the role of
our dynamical variables to construct the configuration space--- and on the lapse function $N$:
\begin{align}
\sigma^{-2}\diff s^2=-\frac{N(r)^2}{A(r)}\diff r^2+A(r)\diff T^2 +b(r)^2\diff \Omega_2^2,
\label{eq:metric}
\end{align}
where all metric variables and coordinates are dimensionless, $\diff\Omega_2^2$ is the line element on the unit two-sphere, and $\sigma:=\sqrt{2G/\int \diff T}$ has units of length, with $G$ being Newton's constant.
Note that this is nothing but a Kantowski-Sachs metric, with a suitably redefined lapse \cite{MenaMarugan:1993rs}.
The corresponding curvature scalar (for $N=1$) is
\begin{align}
b^2\sigma^2R= 2+2 A \dot b^2+b ^2 \ddot A +4 b \dot A\dot b +4 bA \ddot b ,
\end{align}
where the dot denotes derivative with respect to $r$. For a general lapse function, it suffices to replace this derivative with $1/N$ times the dot derivative.
Then, the
Hilbert-Einstein action (up to surface terms) can be written in terms of the metric configuration variables and a cosmological constant $\Lambda$ as
\begin{align}
S&=\frac{1}{16\pi G}\int\diff^4x\sqrt{-g}(R-2\Lambda)
\nonumber\\
&=-\int \diff r \bigg(\frac{A\dot b^2}{N}+\frac{b\dot b\dot A }{N}+N\derB(b) \bigg)+\text{surf. terms},
\end{align}
with $\lambda=\sigma^2\Lambda$ and
\begin{equation}
B(b)=\frac{\lambda}{3} b^3-b,
\qquad
\derB(b) =\partial_bB(b)=\lambda b^2-1.
\end{equation}
Before we continue, let us make a few comments that may be relevant in the rest of this paper.
The action has been written as an integral over the coordinate $r$ on the the patch considered for that coordinate, which is not necessarily all the positive semi-axis.
Also, if we take the square root of the determinant of the metric properly, we see that $N$ should rather be $|N|$ unless we are continuing it analytically. We do so in the following, although had we considered only positive lapses, this subtlety would not have been relevant.
In principle, we take the range of $b$ to be the whole real line. We see that the metric is invariant under a change of sign in $b$. This means that if we did not restrict its value to, say, the positive real axis, every trajectory would be considered twice. We will take this point into account later on. The range of the variable $A$ is also taken to be the whole real line. This choice is of much importance in our treatment: A change of sign in $A$ corresponds to a change in the character of the radial coordinate from timelike to spacelike or vice versa. Generically, horizons correspond to $A=0$.
It will be convenient for our analysis to use the new variable \cite{Campbell:1990uu}
\begin{align}
c=Ab
\end{align}
instead of $A$,
which allows us to simplify the action, that now reads
\begin{align}
S= -\int \bigg( \frac{\dot b\dot c}{N}+N\derB(b) \bigg),
\end{align}
to which the same comments above apply.
The variational principle for this action gives the classical equations of motion
\begin{align}
(\dot c/N)^{\cdot}&=2\lambda Nb, \qquad \dot b\dot c=N^2\derB(b) ,\nonumber \\
(\dot b/N)^{\cdot}&=0.
\end{align}
The general solution to these equations is
\begin{align}
\dot b=\alpha N, \qquad
\alpha^2 c= B(b) +2m,
\label{eq:classsolbc}
\end{align}
$\alpha$ and $m$ being integration constants. From the point of view of the metric (\ref{eq:metric}), $\alpha$ amounts to a constant rescaling of the coordinates $r$ and $t$. This solution corresponds to the Schwarzschild-(anti)de Sitter metric: Indeed, for $\alpha=N=1$, we have
\begin{align}
b(r)=r,\qquad A(r)=-1+\frac{2m}{r}+\frac{\lambda r^2}{3},
\end{align}
and the horizons are located at the zeros of $A(r)$.
From now on we will only consider the case with positive cosmological constant $\lambda>0$. Negative cosmological constant scenarios can also be treated in an entirely analogous manner.
The causal structure of these spacetimes is well known~\cite{Lake:2005bf}. There exist different cases depending on the value of $m$. All of them (except for $m=0$) present a singularity at $r=0$. We can see the diagrams for the different cases represented in Fig. \ref{fig:penrose}. The most interesting case is $0< m <1/\sqrt{9\lambda}$. We recall that, then, $A(r)=0$ has two positive solutions, at which there are two horizons: a black hole horizon (denoted by $r_b$) and a cosmological horizon (denoted by $r_c$).
\begin{figure}
\includegraphics[width=.32\columnwidth]{fig-penrose-00m}
\includegraphics[width=.32\columnwidth]{fig-penrose-0m}
\includegraphics[width=.32\columnwidth]{fig-penrose-mll}\\
\includegraphics[width=.45\columnwidth]{fig-penrose-0ml}
\includegraphics[width=.53\columnwidth]{fig-penrose-ml}
\caption{Penrose diagrams for the different classical solutions (from left to right and from top to bottom, $m<0$, $m=0$, \mbox{$1/\sqrt{9\lambda}<m$}, \mbox{$0< m <1/\sqrt{9\lambda}$}, and \mbox{$m=1/\sqrt{9\lambda}$}). Double arrows indicate identification of the corresponding lines and thick lines represent singularities. Depending on the mass there are one or two horizons (or none, with a naked singularity). In all cases, the region for an observer like us is characterized by $A<0$.}
\label{fig:penrose}
\end{figure}
A common feature to all solutions that allows us to characterize ``our'' spacetime region (between horizons) in contrast with the regions outside them is the sign of the variable $A$: It is negative inside and positive outside.
Note that although the metric configuration variables $b$ and $c$ belong to ${\mathbb{R}}$, the range $b\in {\mathbb{R}}_+$ is preserved by the dynamics and so is $\alpha^2c-B\in {\mathbb{R}}_+$. In other words, these ranges are not related to other (negative) values outside them by classical solutions. Then, classically we have the different regions totally disconnected from each other. For the time being we will keep both ranges to be the whole real line.
In order to perform a canonical quantization, we are interested in making a Hamiltonian formulation of the system. The canonical action can be expressed as
\begin{equation}
S= \int dr \left(
\dot{c}p_c+\dot{b}p_b-NC\right),
\end{equation}
where the canonical conjugate momenta are
\begin{align}
p_b=-\frac{\dot c}{N},\qquad
p_c=-\frac{\dot b}{N},
\end{align}
and the variation with respect to the lapse function gives rise to the Hamiltonian constraint $C=0$, with
\begin{align}
C= -p_bp_c+\derB(b).
\end{align}
Note that $p_c$ commutes with $C$ under Poisson brackets and therefore is a constant of motion. It is also easy to see that the Hamiltonian constraint of the system and the (classical) metric are invariant under simultaneous changes of sign in the momenta. This can be interpreted as a reversal in the evolution, corresponding to a change of sign in the lapse function $N$. We can remove this reversal considering only positive $N$. The system also has the symmetry $(b,c,p_b,p_c) \rightarrow (-b,-c, p_b, p_c)$.
This symmetry could be used to reduce the relevant part of phase space to half of it (in this sense, the duplicity of trajectories with a different sign of $b$ would be removed). We will not impose it; instead we will use a related symmetry that we will discuss in section \ref{sec:kinspace} with similar result in reducing to a half the relevant part of the phase space.
\section{Canonical quantization}
\label{sec:canquant}
In order to quantize our simple system, we will first construct a kinematical operator algebra starting from its phase space, and closed under Poisson brackets [36]. Then we will represent this algebra by operators acting on a kinematical complex vector space, which for convenience will be endowed with a Hilbert space structure.
The Hamiltonian constraint will be represented as a specific operator acting on the kinematical space. The space of physical states will be supplied by the kernel of this constraint and the physical operators will be obtained as elements of the kinematical algebra which map the physical space to itself. Finally, the inner product in this physical space will be determined by requiring that a complete set of real classical observables be represented by self-adjoint operators.
\subsection{Kinematical space and operator algebra}
\label{sec:kinspace}
We will start with the kinematical algebra constructed from the canonical variables $b,c,p_b$ and $p_c$ (and the unit constant), which is obviously closed under Poisson brackets. As kinematical space we choose the vector space spanned by simultaneous solutions to the equations
\begin{align}
-i\partial_c\Psi_{hp}=p\Psi_{hp},
\qquad
[\partial_c\partial_b+\derB(b)]\Psi_{hp}=h\Psi_{hp},
\end{align}
with $h$ and $p$ being real. These solutions (labeled by the parameters $h$ and $p$) depend on the variables $b$ and $c$ and
have the form
\begin{align}
\Psi_{hp}(b,c)= e^{ipc+i[B(b)-bh]/p},
\end{align}
where the singularity at $p=0$ should not be of relevance in the Hilbert space constructed below, since it will have zero measure.
More explicitly any kinematical state will be a linear combination of these solutions, i.e.
\begin{align}
\Psi(b,c)=\int_{\mathbb{R}} \diff h\int_{\mathbb{R}}\diff p \tilde\Psi(h,p)\Psi_{hp}(b,c),
\label{eq:kinstates}
\end{align}
where $\tilde\Psi(h,p)$ is a distribution. This construction endows the kinematical space with a complex vector space structure. Actually we have constructed two kinematical representations that we can use: The metric $(b,c)$-representation and the $(h,p)$-representation.
In the metric representation, we represent the kinematical algebra by operators acting as
\begin{align}
\hat b=b ,
\qquad
\hat c=c,
\qquad
\hat p_b=-i\partial_b,
\qquad
\hat p_c=-i\partial_c.
\label{eq:standardmetricoprep}
\end{align}
Then, in the $(h,p)$-representation, these operators act as
\begin{align}
\hat b&=-ip\partial_h,
\nonumber\\
\hat c &= i\partial_p+\frac{B(-ip\partial_h)}{p^2}+\frac{i\partial_h h}{ p},
\nonumber\\
\hat p_b&=\frac{\derB(-ip\partial_h)-h}{p},
\nonumber\\
\hat p_c&=p ,
\label{eq:metricops1}
\end{align}
as can be checked by direct application of these operators on the kinematical states (\ref{eq:kinstates}). We have assumed that integration by parts can be carried out without boundary contributions, thanks to the boundary conditions implied by the fact that the states belong to the kinematical Hilbert space (determined later on). Also, a factor order has been chosen in the last term of the operator $\hat c$, so that $h$ acts on the right of $\partial_h$ (the reason for choosing this factor ordering will soon be apparent).
An alternative equivalent way of constructing the same kinematical space can be followed by choosing the canonical set of variables ($t$, $q$, and their corresponding momenta $h,p$) adapted to the system studied in Ref. \cite{MenaMarugan:1993rs}, given by
\begin{align}
t&=-\frac{b}{p_c} ,
\nonumber\\
h&= - p_bp_c+\derB ,
\nonumber\\
q&=c-\frac{B(b)+ bp_bp_c-b\derB(b)}{p_c ^2},
\nonumber\\
p&=p_c.
\label{eq:cantrans}
\end{align}
The type-2 generating function for this invertible one-to-one canonical transformation on phase space is
\begin{align}
F(c,b,h,p)=cp+\frac{B(b)-bh}{p}.
\end{align}
From the classical point of view, $p$, $q$, and $h$ are constants of motion. In fact, by comparison with the classical solution (\ref{eq:classsolbc}) in terms of the metric variables $b$ and $c$ we see that
\begin{align}
p=\alpha,
\qquad
h=0,
\qquad
q=\frac{2m}{\alpha^2},
\qquad \dot t=-N.
\end{align}
Notice that $q$ is positive on classical solutions if we want to consider only positive mass (i.e. absence of naked singularities).
It is also interesting to note that $q$ is just the value of the the dynamical variable $c$ at $b=0$ (the relevance of this comment will become apparent when analyzing quantum representations in the next section). The symmetry discussed at the end of the previous section,
\begin{equation}
(b,c,p_b,p_c) \rightarrow (-b,-c, p_b, p_c),
\end{equation}
in terms of these new canonical variables, now becomes the symmetry \begin{align}
(q,t,p,h) \rightarrow (-q,-t, p, h).
\end{align}
This symmetry implies that all the relevant information is actually contained in half of the original phase space and that, in consequence, we can restrict
our study to it. Since we are interested for other reasons on positive $q$, this is the half that we will choose. We will discuss how to impose this symmetry as a restriction on the wave functions later on (the corresponding representation will be given by the restriction of the ``complete'' representation to a subspace). For the time being, we will keep it unrestricted.
We choose as kinematical vector space the space of distributions $\tilde\Psi(h,p)$ and represent the kinematical algebra on it as
\begin{align}
\hat h=h,
\qquad
\hat t=i\partial_h,
\qquad
\hat p=p,
\qquad
\hat q=i\partial_p.
\end{align}
The metric variables can be represented as the operators
\begin{align}
\hat b&=-\hat t\hat p,
\nonumber
\end{align}
\begin{align}
\hat c&=\hat q+B(-\hat t\hat p){\hat p}^{-2}+\widehat{th}{\hat p}^{-1},
\nonumber\\
\hat p_b&=[\derB(-\hat t\hat p)-\hat h]{\hat p}^{-1},
\nonumber\\
\hat p_c&=\hat p,
\label{eq:metricops}
\end{align}
as can be easily seen by inverting the canonical transformation (\ref{eq:cantrans}).
For convenience, in contrast with Ref. \cite{MenaMarugan:1993rs} and in agreement with the operator order chosen in Eq. (\ref{eq:metricops1}), we take $\widehat{th}=\hat t\hat h$, even if it is not a symmetric ordering, so that the action of this operator on physical states (which are annihilated by $\hat h$, as we will see) vanishes.
To make contact with the construction presented in the beginning of this section, we can go to the metric $(b,c)$-representation by means of the transformation
\begin{align}
\Psi(b,c)=\int_{\mathbb{R}} \diff h\int_{\mathbb{R}}\diff p \tilde\Psi(h,p)e^{iF(b,c,h,p)}.
\label{eq:transform}\end{align}
We can see that this expression is precisely that in Eq. (\ref{eq:kinstates}).
In this metric representation, the metric canonical variables are represented as the operators given in Eq. (\ref{eq:standardmetricoprep}). It is also worth emphasizing that we are using a slightly different representation in comparison with that in Ref. \cite{MenaMarugan:1993rs}.
It may be convenient to introduce an inner product in the kinematical space on which the operators $\hat h$, $\hat p$, $\hat t$, and $\hat q$ are self-adjoint, namely:
\begin{align}
(\Psi_1,\Psi_2)=\int_{\mathbb{R}}\diff h\int_{\mathbb{R}}\diff p\tilde\Psi_1(h,p)^*\tilde\Psi_2(h,p),
\label{eq:innerprod}
\end{align}
where the symbol $*$ denotes complex conjugation.
Then the kinematical Hilbert space is the completion in this inner product of the space of distributions $\tilde \Psi(h,p)$, that is to say $L^2(\mathbb{R}^2,\diff h\diff p)$.
The states $\Psi_{hp}(b,c)$
are obviously orthonormal in the Dirac-delta sense:
\begin{align}
(\Psi_{hp},\Psi_{h'p'})=\delta(h-h')\delta(p-p').
\end{align}
The restriction to positive $q$ can be taken by going to the Fourier transform of the $(h,p)$ representation in the $p$ variable (to $q$) and projecting to the positive semi-axis of the configuration space [the corresponding space of wave functions are those in $L^2(\mathbb{R}\times \mathbb{R}^+,\diff h \diff q)$, obtained by restricting those functions to positive $q$ and using the inner product induced from Eq.
(\ref{eq:innerprod}); this space is not stable under the operator $\hat{p}$ but it is stable under the operator $\widehat {qp}=i(p\partial_p+1/2)$ instead (that is self-adjoint)]. Nonetheless, we will continue to consider the general Hilbert space without restricting $q$, keeping in mind that this implies a physical duplicity, as discussed above.
Finally, the Hamiltonian constraint can be represented in this kinematical space by the operators
\begin{align}
\hat C=\partial_b\partial_c+\derB(b),
\qquad
\hat C=h,
\end{align}
in the metric $(b,c)$-representation and in the $(h,p)$-representation, respectively.
\subsection{Physical Hilbert space}
In the $(h,p)$-representation on the kinematical space, the Hamiltonian constraint is represented as multiplication by $h$, as we have just seen. Therefore, the space of solutions can be obtained by solving the equation
\begin{align}
\hat C\tilde \Phi(h,p)=h\tilde \Phi(h,p)=0.
\end{align}
The solutions have the form
\begin{align}
\tilde\Phi(h,p)=\frac{1}{\sqrt{2\pi}}\delta(h) \phi(p),
\end{align}
where $\phi(p)$ is an arbitrary distribution and the constant prefactor has been chosen for normalization purposes.
In this physical vector space, the operators $\hat p$ and $\hat q$ (which commute with the constraint $\hat C$) are represented as
\begin{align}
\hat p=p,\qquad
\hat q=i\partial_p.
\end{align}
The remaining task in the canonical quantization procedure is fixing the inner product in the space of physical states. We choose it so that the observables $\hat p$ and $\hat q$ be self-adjoint, which leads to the inner product
\begin{align}
\langle\Phi_1,\Phi_2\rangle=\int_{\mathbb{R}}\diff p\phi_1(p)^*\phi_2(p).
\label{eq:physinner}
\end{align}
To summarize, the physical Hilbert space of quantum states for our system is $L^2(\mathbb{R},\diff p)$, which contains just one degree of freedom as expected. We will refer to this representation as the $p$-representation.
The $p$-representation is not the only representation that we can use and in fact there exist other representations of physical interest as we will see.
Let us note that physical states can also be written in terms of the metric variables $b$ and $c$ as
\begin{align}
\Phi(b,c)&=\int_{\mathbb{R}} \diff h\int_{\mathbb{R}}\diff p \tilde\Phi(h,p)e^{iF(b,c,h,p)}
\nonumber\\
&=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \diff p \phi(p)e^{i[pc+B(b)/p]},
\end{align}
the inverse of this transformation being
\begin{align}
\phi(p)=\frac{1}{\sqrt{2\pi}}e^{-iB(b)/p}\int_{\mathbb{R}}\diff c\Phi(b,c)e^{-ipc}.
\end{align}
Note that the dependence of physical states $\Phi(b,c)$ on $b$ is only through $B(b)$.
The closure relation in terms of the metric variables can then be easily obtained:
\begin{align}
\openone(b,c;b,c')&=\frac{1}{2\pi}\int_{\mathbb{R}}\diff p e^{ip(c-c')}e^{i[B(b)-B(b)]/p}
\nonumber\\ &
=\delta(c-c'),
\end{align}
so that
\begin{align}
\Phi(b,c)=\int_{\mathbb{R}}\diff c'\openone(b,c;b,c')\Phi(b,c').
\end{align}
Finally, we can write the inner product in terms of the metric variables:
\begin{align}
&\langle\Phi_1,\Phi_2\rangle=\int_{\mathbb{R}}\diff p\phi_1(p)^*\phi_2(p)
\nonumber\\
&=\frac{1}{2\pi}\int_{\mathbb{R}}\diff p \int_{\mathbb{R}}\diff c_1\int_{\mathbb{R}}\diff c_2 e^{ip(c_1-c_2)}\Phi_1(b,c_1)^*\Phi_2(b,c_2)
\nonumber\\
&= \int_{\mathbb{R}}\diff c \Phi_1(b,c)^*\Phi_2(b,c).
\label{eq:innerbc}
\end{align}
After these preliminary notes about changes of representation, let us analyze some equivalent representations that will be particularly adequate to the study of horizon quantum physics that we want to carry out.
\subsection{Equivalent representations}
From the above discussion, it immediately follows that we can change from the $p$-representation to the metric representation for physical states $\Phi(b,c)$ by means of a Fourier transform (together with a multiplication by a $b$-dependent phase).
This fact allows us to introduce two other families of representations.
\subsubsection{$c_b$-representations}
We have seen that we actually have not only one but a whole family of $c_b$-representations labeled by $b$. This is obvious in the formula (\ref{eq:innerbc}) for the inner product, valid for any value of $b$. This resembles a kind of transformation from the Heisenberg picture, in which the states $\phi(p)$ only depend on the $p$, to the $b$-Schrödinger picture, where the states now depend on $p$ and $b$ (the $b$-evolution being driven by the Hamiltonian $-\derB(b)\hat p^{-1}$), together with a Fourier transform to the variable $c$. In this sense, we can write
\begin{align}
\Phi(b,c)=\hat U(b)\Phi(\tilde b,c),
\end{align}
where $\tilde b$ represents any of the roots of the polynomial $B(b)$ and
\begin{align}
\hat U(b)=e^{iB(b)\hat p^{-1}}.
\end{align}
In each of these $c_b$-representations, the observables that we want to represent will be the ones corresponding to $\hat p $ and $\hat q$ in this $b$-Schrödinger picture,
\begin{align}
\hat \pi_b&=\hat U(b)\hat p \hat U^\dag(b)=\hat p,
\nonumber\\
\hat c_b&=\hat U(b)\hat q \hat U^\dag(b)=\hat q+B(b)\hat p^{-2}.
\end{align}
It is straightforward to see that the action of these canonically conjugate observables on $\Phi(b,c)$ is just derivation and multiplication by $c$, respectively, i.e.
\begin{align}
\hat \pi_b=-i\partial_c,
\qquad
\hat c_b=c,
\end{align}
and hence the name $\hat c_b$ instead of $\hat q_b$ (note that we have already mentioned this point when we defined the canonical variable $q$). So, we have a family of observables $\hat c_b $, each in a different $c_b$-representation (labeled by $b$), that can be interpreted as giving the value of the metric variable $c$ at the considered value of $b$.
This interpretation is actually based on the observation made above that $\hat c_b$ is nothing but a kind of Schrödinger picture operator obtained from $\hat q$ by means of the ``$b$-evolution'' operator $\hat U(b)$.
\subsubsection{$p_b$-representations}
In the same way, we also have a family of $p_b$-representations labeled by $b$, given by the Fourier transform in $c$ of $\Phi(b,c)$:
\begin{align}
\phi(b,p)&=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}dc e^{-ipc}\Phi(b,c)
\nonumber\\
&=\hat U(b)\phi(p)=\phi(p)e^{iB(b)/p},
\end{align}
for which the inner product (\ref{eq:physinner}) reads
\begin{align}
\langle\Phi_1,\Phi_2\rangle=\int_{\mathbb{R}}\diff p\phi_1(b,p)^*\phi_2(b,p).
\label{eq:innerbp}
\end{align}
In these $p_b$-representations, for each $b$, the operator \mbox{$\hat \pi_b=\hat p$} acts by multiplication and it is clearly an observable, well defined on (a dense domain of) the physical Hilbert space. On the other hand, the operator $\hat c_b$ is also an observable as we have seen and acts as
\begin{align}
\hat c_b=i\partial_p.
\end{align}
\subsubsection{Schrödinger and Heisenberg pictures}
We can adopt two alternative viewpoints analogous to the Schrödinger an Heisenberg pictures of standard quantum mechanics. Although we will refer to the $p_b$-representations, an entirely analogous discussion holds for the $c_b$-representations.
From the first point of view, we can consider that the $p_b$-representations [with states described by $\phi(b,p)$] give the evolution of the $p$-representation [with states described by $\phi(p)$] from a value $\tilde b$ of $b$ where $B(\tilde b)$ vanishes to the new value of $b$ [and therefore of $B(b)$], providing a whole family of representations that give the corresponding Schrödinger ``dynamics'' in the parameter $b$. Notice that, for this, it is not necessary that $B(b)$ be monotonous in $b$. The Hamiltonian of the evolution in $b$, from this viewpoint, would be $-\derB(b)\hat p^{-1}$ so that, when the associated Schrödinger equation is integrated, one gets the phase $iB(b)/p$, as we have discussed. Note that this Hamiltonian is $b$-dependent, and moreover not strictly positive; hence, indeed, the phase $iB(b)/p$ is not monotonous in $b$. Nonetheless, the evolution is unitary, since the inner product~(\ref{eq:innerbp}) is conserved, i.e. $b$-independent.
From the second point of view, we can choose a fixed $p_b$-representation for a given value of $b$ [with states described by $\phi(b,p)$], and represent our family of observables in a kind of Heisenberg picture (see e.g. \cite{Ashtekar:2006uz} for similar definitions of observables). The family of observables corresponding to $c$ at different values $b_0$ of $b$ would be given, in this way, by
\begin{align}
\hat c_b^{0}=\hat U^\dag(b,b_0)\hat c_b\hat U(b,b_0)=
i\partial_p +\frac{B(b_0)-B(b)}{p^2},
\end{align}
where $\hat U(b,b_0)=e^{i[B(b)-B(b_0)]\hat p^{-1}}$.
This observable gives in the $p_b$-representation the value of $c$ when $b=b_0$. Since the function $B$ is not one-to-one, the operators in this family may coincide for some values of $b_0$, namely those where $B(b_0)$ is the same.
\subsection{Some bases of the physical Hilbert space}
Before proceeding to our main discussion, which faces the question which motivated our analysis, let us complete our study of the quantization with the determination of some especially useful bases for the physical Hilbert space of our system.
Let us start by considering the following states
\begin{align}
\phi_p(p')=\delta(p-p')
\end{align}
in the $p$-representation. Their counterparts in the $p_b$-representations are straightforward to find:
\begin{align}
\phi_p(b,p')=\delta(p-p')e^{iB(b)/p}.
\end{align}
They are obviously eigenstates of $\hat p$ with eigenvalue $p$ and, hence, they provide an orthonormal basis. Their counterparts in the $c_b$-representations are
\begin{align}
\Phi_p(b,c)=\frac{1}{\sqrt{2\pi}}e^{ipc+iB(b)/p},
\end{align}
and the closure relation in these representations reads
\begin{align}
\openone(b,c;b,c')=\int_\mathbb{R} \diff p \Phi_p(b,c)\Phi_p(b,c')^*=\delta(c-c').
\end{align}
Finally, there is still another family of bases that will prove very helpful in our analysis, namely that made of eigenstates of the self-adjoint operator $\hat c_b^{ 0}$ in the $p_b$-representation with real eigenvalues $c^{ 0}$:
\begin{equation}
\phi_{c^{ 0}}(b,p)=\frac{1}{\sqrt{2\pi}}e^{-ipc^{ 0}-i[B(b_0)-B(b)]/p}.
\end{equation}
In this family of representations, the identity operator acquires the form $\openone(b,p,b,p')=\delta(p-p')$ and can be decomposed as a sum over all eigenvalues $c_0$ (the whole real line) of $\hat c_b^0$ in the following manner
\begin{align}
\openone(b,p;b,p')=\int_{\mathbb{R}}\diff c^{ 0}\phi_{c^{ 0}}(b,p)\phi_{c^{ 0}}(b,p') ^* .
\end{align}
We can decompose this identity in the sum of two orthogonal projectors: one for positive eigenvalues of $c^0$, $\hat P_+^{0}$, and the other for negative eigenvalues, $\hat P_-^{0}$ (the integral over the real line is the sum of the two corresponding half-infinite intervals):
\begin{align}
\openone=\hat P_+^{0} +\hat P_-^{0}.
\end{align}
The superindex ${0}$ makes manifest the dependence of the projection operators on the value of $b_0$ where $c$ is evaluated.
Explicitly, these projection operators can be written as
\begin{align}
\hat P_{\pm}^{0}\phi(b,p) &=\frac{1}{2\pi}\int_{\mathbb{R}^{\pm}}dc^0 e^{-ipc^0-i[B(b_0)-B(b)]/p}\nonumber\\&\times\int_{\mathbb{R}}dp^{\prime} e^{ip^{\prime}c^0+i[B(b_0)-B(b)]/p^{\prime}} \phi(b,p^{\prime}).
\end{align}
\section{Quantization and horizons}
\label{sec:quanthor}
Assume that, at a certain positive value $b_0$ of $b$, we observe only the region with negative values of $c$. This corresponds classically to considering only our region of the universe, i.e. the spacetime region that lies between the black-hole and the cosmological horizons at the given ``instant of dynamical variable'' $b_0$. Similarly, we could restrict ourselves to the exterior of our region of the universe (beyond the black hole and cosmological horizons), i.e. to positive values of $c$ at $b_0$. In our scheme these restrictions can be accomplished by choosing states with null projection under $\hat P_{\pm}^0$, respectively, or equivalently by projecting an arbitrary state with $\hat P_{\mp}^0$ and normalizing the result.
Classically whatever happens beyond the horizons will have no effect whatsoever in our spacetime region. We are now ready to ask ourselves, and also answer, the corresponding quantum mechanical question. More explicitly, the question that we want to address now is whether this restriction to our region of the universe (i.e. to negative values of $c$) is robust and meaningful, so that we can sensibly forget about the regions beyond the horizons quantum mechanically.
If this were not the case, then measurements of $c$ at a different positive value $b_1$ of $b$ would lead to contradictory results. The question is then whether observations of the values of $c$ at different values of $b$ are compatible.
If they were not, the two projections (at the different values $b_0$ and $b_1$) would differ, the corresponding observables $\hat c_b^0$ and $\hat c_b^1$ could not be diagonalized simultaneously and, hence, the eigenstates could not be chosen as common to both observables. In this case, as mentioned above, the projectors would not commute and the restriction to our region of the universe between horizons would not be stable, in the sense that the projection at $b_0$ on negative values of $c^0$ would generally have a non-vanishing projection at $b_1$ on positive values of $c^1$, and vice versa. The restriction to the interior of the horizons would depend on the value of $b_0$, and would therefore be unstable under evolution in this variable.
To summarize, quantum stability and robustness of the restriction to our region of spacetime requires that the two considered observables $\hat c_b^0$ and $\hat c_b^1$ commute. We are going to prove that this is not the case, i.e. that $\hat c_b^0$ and $\hat c_b^1$ are not commuting observables.
A direct calculation shows that
\begin{align}
[\hat c_b^0,\hat c_b^1]&=\bigg[\hat q +[B(b_0)-B]{\hat p}^{-2} ,\hat q +[B(b_1)-B]{\hat p}^{-2} \bigg]\nonumber\\&
=-2i[B(b_1)-B(b_0)]{\hat p^{-3}}\neq 0,
\end{align}
and therefore the family of considered observables are not mutually compatible. Alternatively, this same result can be obtained if we act with $\hat c_b^1$ on the eigenstates of $\hat c_b^0$. In the $p_b$-representation, it is straightforward to get:
\begin{equation}
\hat c_b^1\phi_{c^0}(b,p)=\bigg[ c^0 - \frac{B(b_0)-B(b_1)}{p^2}\bigg]\phi_{c^0}(b,p).
\end{equation}
We then see that the sector of positive values of $c^0$ (at positive $b_0$) would be contained in the positive sector of $\hat c_b^1$ (at positive $b_1$) if $B(b_0)-B(b_1)<0$, and the sector of negative values of $c^0$ in the negative sector of $\hat c_b^1$ if $B(b_0)-B(b_1)>0$. Both conditions are incompatible unless $B(b_0)=B(b_1)$, which is not satisfied for general values $b_0$ and $b_1$ (maybe just at some points $b$, but not in full intervals). This further supports the conclusion that the projectors at positive and negative $c^0$ at different values $b_0$ of $b$ are not mutually compatible in general.
Note that the operator $\hat q$ can be considered as a particular case of ${\hat c}_b^0$, namely the one associated with $b_0=\tilde b$ [with $B(\tilde b)=0$]. Even if we restrict to positive $q$ by acting with the associated projection to the positive part of the spectrum of this operator (removing in this way the physical duplicity that we were maintaining till now), the system will develop contributions to the negative sector of $\hat q$ for other values of the variable $b$, in accordance with our discussion above.
Since the dynamics in $b$ mixes the projections, as we have seen,
describing states as direct sums of positive and negative $c^0$-states is not the best strategy. Instead, it is more appropriate to consider general physical states belonging to the tensor product
\begin{align}
{\cal H}^0={\cal H}_+^0\otimes{\cal H}_-^0
\end{align}
of the projection subspaces
\begin{align}
{\cal H}_\pm^0=\hat P_\pm^0 {\cal H},
\end{align}
where, as before, the superindex 0 denotes the choice of a particular instant of $b$ for the construction, and ${\cal H}$ is the Hilbert space from which we started. Using that the sum of $P_+^0$ and $P_-^0$ is the identity, any observable $\hat O$ can then be decomposed in four operators between both projection subspaces:
\begin{align}
\hat O_{\pm\pm}^0:{\cal H}_\pm^0\to{\cal H}_\pm^0,
\qquad
\hat O_{\pm\mp}^0:{\cal H}_\pm^0\to{\cal H}_\mp^0,
\end{align}
defined as
\begin{align}
\hat O_{\pm\pm}^0=\hat P_\pm^0\hat O \hat P_\pm^0,
\qquad
\hat O_{\pm\mp}^0=\hat P_\pm^0\hat O \hat P_\mp^0.
\end{align}
The operators $\hat O_{\pm\mp}^0$ mix the two subspaces ${\cal H}_\pm^0$ corresponding to the considered projections and cause correlations between them. This is the case of $\hat c_b^1$ [for $B(b_1)\neq B(b_0)$)], as we have seen. Moreover, if $\hat O$ is a unitary observable, the existence of the two mixing components will indicate that unitarity is not respected in each of the subspaces ${\cal H}_\pm^0$ separately. Our system certainly exhibits this kind of unitary operators and the most straightforward example is $\exp({i\hat c_b^1})$.
This analysis leads to the conclusion that the mixture between interior and exterior of the horizon by quantum effects is a generic result in this quantization and that physical states entangle both regions.
\section{Conclusion}
\label{sec:concl}
We have argued that quantum mechanics applied to the whole spacetime generically introduces quantum correlations between different classically disconnected regions (separated by horizons). This may be used as a first stage of an analysis of a quantum multiverse scenario, in which there may exist non-vanishing quantum correlations among individual otherwise uncorrelated universes. Ultimately, this would lead to the necessity of considering the whole multiverse in order to obtain a complete knowledge of our own universe.
We have analyzed a Kantowski-Sachs minisuperspace model of a spacetime with a positive cosmological constant, whose classical solutions are Schwarzschild-de Sitter universes. We have carried out a canonical quantization of this model following (an extension of) Dirac's canonical quantization program for systems with first-class constraints. In this construction, the physical structure is consistent and robust only if we consider the whole spacetime. We have proved that we can not restrict ourselves to the observed classical region when we consider the spacetime quantum mechanically, because there appear generically unavoidable quantum correlations between regions classically separated by horizons. This is explicitly shown by checking that unitarity is preserved only when the whole spacetime is taken into account. Indeed, we have decomposed the physical Hilbert space of the system into two subspaces corresponding to states with support either between or beyond the horizons. These Hilbert subspaces are not stable under the action of unitary operators that describe a natural concept of evolution on physical states. Therefore, these states are better conceived as belonging to the tensor product of both subspaces, which are not separable but entangled.
In contrast with many discussions carried out in quantum field theory on curved backgrounds (see e.g. \cite{ Unruh:1976db,Hawking:1974sw,Wald:1975kc,Israel:1976ur,Gibbons:1977mu,Einhorn:2005nw,MartinMartinez:2010ds,BirrelDavies}), a distinctive feature of our analysis is that our conclusions rest exclusively on the quantum behavior of the geometry. The entanglement between the regions in the interior and the exterior of the horizons has been shown to occur without introducing any field in the system: it is due solely to quantum properties of geometric observables on physical states of the Kantowski-Sachs model. Immediately, a new avenue is opened: Extending our investigations to the quantization of fields --for instance, a scalar one-- propagating on the quantum background studied here (this philosophy is similar to the strategy followed in the hybrid quantization scheme of Loop Quantum Cosmology. See \cite{MartinBenito:2008ej,FernandezMendez:2012vi,FMO13,CFMO14}). Then one could study perturbations of homogeneous (i.e., only $r$-dependent) scalar fields on this minisuperspace. In order to treat the background minisuperspace exactly, the ``zero mode'' of the scalar field (describing its homogeneous part) could be set to zero. Then we could expand the genuine perturbations of the field in a mode basis, for which one can consider e.g. a generalization of the analysis of Ref. \cite{Pereira:2012ma}.
Within this framework, one could analyze the differences between two ways of quantizing the model with the field. The first one would be quantizing the field separately in the sector of positive $c_0$ and negative $c_0$ at $b_0$, and the second one quantizing it on the whole real line for $c_0$. This would allow us to look for quantum-field-theory effects and entropy mixing between both regions, but incorporating in the discussion the quantum nature of the geometry. In this manner, one would extend to the realm of quantum spacetime previous studies in the localization of quantum modes in a cavity, in which the tension between vacuum entanglement and having localized states clearly shows up \cite{Rodriguez-Vazquez:2014hka}.
\begin{acknowledgments}
A. A-S. is grateful to A. Ashtekar, M. Fern\'andez-M\'endez, and P. Mart\'{\i}n-Moruno for conversations. The authors acknowledge financial support from the research grant MICINN/MINECO FIS2011-30145-C03-02 from Spain.
\end{acknowledgments}
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{
"redpajama_set_name": "RedPajamaArXiv"
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Q: R programming: How do I get Euler's number? For example, how would I go about entering the value e^2 in R?
A: -digamma(1) is the Euler's Constant in R.
e, (exp(1) in R), which is the natural base of the natural logarithm
Euler's Constant.
Euler's Number
A: if you want to have a little number e to play with, you can also make one yourself:
emake <- function(){
options("warn"=-1)
e <- 0
for (n in 0:2000){
e <- e+ 1/(factorial(n))
}
return(e)
}
e <- emake()
e^10
exp(10)
# or even:
e <- sum(1/factorial(0:100))
fun stuff
A: The R expression
exp(1)
represents e, and
exp(2)
represents e^2.
This works because exp is the exponentiation function with base e.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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There's a tech talent war in New York City right now. The established firms like Facebook and Google are only getting bigger, and there are more tech startups than ever making NYC their home. As a result, hiring great tech talent is more challenging than ever.
Yet, as a startup founder, it's critical to find a great CTO. For overseeing quality of work from the tech team, running effective agile retrospectives, and meeting the goals of the business team, hiring the right CTO can have significant improvements to your company.
There is no one right time to find a CTO. Some startups can go a long time without a CTO, while others need one early on. Regardless of the stage of your startup, if you are looking to hire a CTO, follow these seven steps to find your ideal candidate.
What's the difference between a technical advisor and a CTO? A technical advisor keeps their day job. The technical advisor works for you 2–4 hours per month and serves as someone who's not in the weeds with you, day to day. They might do weekly or monthly code reviews, act as a sounding board for your CTO, and serve as a checks-and-balances system with your CTO. If you find a technical advisor first, this person can be a huge help with finding the right CTO. They can vet CTO candidates, do interviews, and review job descriptions.
Finding a technical advisor will likely be easier than finding a CTO, because you aren't asking someone to quit their job to come work for you. Rather, you are asking them to keep their job and take on an advisory role that takes a few hours per week.
In some cases, you can get away with having a technical advisor and not hiring a CTO. One company I advise has two cofounders who are entrepreneurs. They hired a great offshore software engineering team to code their MVP. They are happy with the offshore team's results but can't read code themselves. They thought they needed a CTO, but we found a technical advisor instead. The technical advisor reviews the offshore tech team's code weekly and gives a high-level code assessment to the founders. So far, all is going well, and the technical advisor is giving the cofounder confidence that their offshore team is writing high-quality code.
If this works for you, great. Step 1 is all you need. If you do want to hire a CTO, read on to steps 2–7.
Discover the secrets to facilitating retrospectives that drive real results.
Ideally, you want to find someone who will see this job as a rung up on the ladder in their career, such as a senior software developer or technical team lead who is hungry to take on the CTO role.
You might want someone who's walked your path before. If you are a 40-person company trying to grow to 100, having someone who's seen what that looks like can be valuable.
Find someone who approaches things from a "seek to understand" perspective, not someone who thinks they have all the answers.
Look for someone who can pick up multiple technologies (perhaps including back end, front end, and mobile). Avoid people who say they'll only program code in one language.
Look for someone who wants to wear multiple hats. I know many amazing senior-level web developers who specifically do not want the CTO role, because they want to be coding most of the time.
What does success look like for the person in this role in 3 months, 6 months, and 12 months?
What SMART (specific, measurable, achievable, realistic, timely) goals are there for this role?
What are your company values, and how do these tie into success in the role?
What approach to problem solving are you looking for?
Is lack of experience with a specific tech stack a deal-breaker, or is it something they can learn?
Does your company sponsor H1B visas, if needed?
Tie it all together by using all this information to craft your job description.
Want help increasing your velocity and reducing risk of failure?
Learns new things quickly Medium ?
Can implement continuous deployment independently High ?
Performance review experience Medium ?
* How do you hire for integrity? Find out here.
You will likely find that some things aren't currently specifically "tested" in any interview activity. These are indicated with "?" in the chart above. This tells you that you must create an interview activity to evaluate this skill.
Interview for motivation and integrity first, experience and knowledge second.
And never underestimate the importance of a cultural fit, for a CTO. This is someone you'll be working with intimately and trusting with the very core of your "big idea," so making sure they fit with your company culture is as important as making sure they have the right technical skills.
Without integrity, motivation is dangerous; without motivation, capacity is impotent; without capacity, understanding is limited; without understanding, knowledge is meaningless; without knowledge, experience is blind. Experience is easy to provide and quickly put to good use by people with all the other qualities.
Starting with the job description and lasting through the offer, keep in mind that the best CTOs will have multiple offers. It's your job to sell the role to them throughout the interview process. You know that your startup is the most exciting, innovative, amazing company to work for. But keep in mind that the candidates probably don't know that yet. You need to sell your startup. Don't just list the qualities you need in a CTO; list what they will gain from working with you: exciting challenges, the opportunity to be a part of x, y, and z. Will the CTO get equity? Will they have autonomy?Whatever unique qualities your startup has, sell them well. Be sure to explain the position with gender-neutral language to attract a balanced and diverse applicant pool.
Many of the best candidates are the passive ones who aren't looking for a new role, or who are discreetly looking. When it comes to finding candidates, never underestimate the power of your network, regardless of how big or small it is. Look around you. Who do you know who could possibly fill this role? Who do you know who might know potential candidates? Ask yourself, "Who do I know who knows the most CTOs?" Reach out to that person and ask for their help.
Can you introduce me to any great CTO candidates who weren't a fit for you but might be a great fit for my company?
Social media standards: Twitter, LinkedIn, and Facebook. They're not just for talking, they're for listening. Look for candidates in niche tech communities. For example, if you want someone passionate about Node.js, see who's tweeting #nodejs and related hashtags.
Meetup.com: Posting an announcement to a meetup group can be a great way to find candidates. Try asking the organizer of the meetup to post the announcement for you. They likely have a strong network of possible candidates, and their announcement will have more weight than if you make one on your own.
Ask yourself, "Why is the ideal CTO candidate going to choose my startup over another company?" There is no one-size-fits-all compensation package. You must first decide what you are able and willing to offer. You may not be able to offer the kind of package they can get from a more established company, but you can offer them equity. Giving someone equity in your company could help to them feel personally invested and deliver better results. Be prepared to answer any questions they may have on available benefits and bonuses. Be sure to do research on typical compensation and equity-based packages.
Always make the offer in person, if at all possible. This is an important role, and you've put a lot of thought into the process from day one, so make sure you finish the process strong, with a personal, well-thought-out offer that you can feel confident about.
Don't expect to find a great full-time CTO if all founders aren't committed to the startup full-time. Too often, I see founders who still have another day job but are looking for a full-time CTO. Why would a CTO quit their job to work on your startup full-time if they see that you're not willing to do the same? Don't ask CTO candidates to commit more than you are currently committed.
There are CTOs of all flavors—some who have scaled businesses before, some who are technical wizards, some who are visionaries, some who have walked your path before, and some who haven't. Take the time to follow these seven steps, and you'll increase your chances of finding the perfect CTO for your startup.
Want to learn more strategies to help your company increase velocity and productivity? For Agile teams, proper use of retrospectives is key to a consistently improving team. Our free guide covers everything you need to know to facilitate your retrospectives like a pro.
Editor's Note: This post was originally published in September 2014 and has been revised and updated for freshness, accuracy, and comprehensiveness.
Do you ever feel like your team is running 1,000 miles per hour in the wrong direction, and you're not sure why? Sometimes, a chat with a peer is all it takes to give you a meaningful perspective to help you course correct, or to give you the courage to continue on your current path.
We're offering friends of Stride a free Peer Chat. Tell us what's on your mind, and we'll share our experiences. Don't be shy; we've been there ourselves.
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{
"redpajama_set_name": "RedPajamaC4"
}
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POLITICO NOW Blog
Hutchison: GOP shouldn't build party around abortion
By KEVIN ROBILLARD
Texas Sen. Kay Bailey Hutchison said Sunday Republicans can't build their party around a "personal" and "religious" issue like abortion.
"Mothers and daughters can disagree on abortion, and we shouldn't put a party around an issue that is so personal and also, religious-based," the retiring Republican senator said on CNN's "State of the Union." "I think we need to say, 'Here are our principles, and we welcome you as a Republican. We can disagree on any number of issues, but if you want to be a Republican, we welcome you."
Hutchison identifies as "pro-life," and has a mixed voting record on abortion rights.
The party's stance on abortion has come under scrutiny since Rep. Todd Akin (R-Mo.) falsely said last week that women who are the victims of "legitimate rape" have biological defenses against pregnancy. On Tuesday, the GOP's platform committee approved an abortion plank without a rape exemption, which Democrats quickly dubbed the "Akin Plank."
Despite what the platform says, Hutchison said abortion rights supporters feel welcome in the GOP.
"A lot of people think the party platform is something that is rigid," she said. "It's not really."
Hutchison, who isn't running for reelection, joins Sen. Scott Brown (R-Mass.) in warning the party about its abortion stance. Hutchison's likely successor in the Senate, former Texas solicitor general Ted Cruz, is an opponent of abortion rights.
CORRECTION: Kay Bailey Hutchison doesn't consider herself to be pro-choice.
Kay Bailey Hutchison
Republican National Convention 2012
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\section{A forthcoming exciting Multi-Messenger era}
Investigations of physical processes occurring on the Sun, and their effects on the Earth, its atmosphere, and its space environment, exploit a large variety of messengers to provide the needed information. "Classical" photon-based astronomical observations create data such as disk-integrated spectra and full-disk and high spatial resolution images in wavelengths from radio to X-rays, as well as helioseismic imaging of the solar interior. The nearness of the Earth to the Sun allows measurements difficult or impossible for other stars, such as in-situ measurements of solar wind properties (e.g., chemical composition, magnetic field, particle energy distributions, etc.), detection of neutrinos emitted in the solar core, and geomagnetic indices (which include direct measurements of properties of the magnetic field as well as indirect indices such as cosmogenic isotopes). Each of these contributes to providing unique information about how energy is generated in the Sun, transported into the atmosphere and heliosphere, and dissipated through a variety of phenomena on temporal scales of seconds to millennia.
The next decade will be a new exciting period as new ground- and space-based instrumentation will provide unprecedented observations of the solar atmosphere and heliosphere. In particular, in 2021 the Daniel K. Inouye Solar Telescope {\citep[DKIST, ][]{rimmele2020}} will start to produce observations of the photosphere and chromosphere at extremely high spatial and temporal resolutions; these will be accompanied by spectral and spectropolarimetric observations of the corona at spectral ranges never or rarely observed before. The Parker Solar Probe (PSP, launched in August 2018) provides in-situ measurements of solar wind properties at unprecedented proximity to the Sun \citep{Fox2016}. The Solar Orbiter mission (launched in 2020) combines both in-situ and remote-sensing instrumentation and gives us the opportunity to expand our understanding about solar interior dynamics, the evolution and structure of polar magnetic fields, and the fast solar wind \citep{Muller2020}. Base knowledge from currently available observations from space missions (e.g., SDO, IRIS, HINODE, STEREO, THEMIS) and ground-based observatories (e.g., SOLIS, GST, CoMP, GONG, ALMA, EOVSA), accompanied by upcoming observations, will take us to a new level of understanding of solar dynamics and activity and their impact on the Heliospheric and Earth environments.
The synergy between an advanced modeling effort and a comprehensive analysis of observations has great potential to shed light on the physical processes behind the observed phenomena. However, as we approach smaller scales in models and observations, challenges emerge that require the development of new approaches in data analysis and numerical modeling.
\section{Modeling of Solar Atmosphere Dynamics}
Substantial development of theoretical and numerical approaches, supported by continually growing computational capabilities and novel observing technologies, are changing our view of solar phenomena. Thanks to such developments, we have been slowly moving away from a simplistic, one-dimensional, static interpretation of solar observations toward models that take into account the multidimensional nature of the observed phenomena as well as the energy transport mechanisms occurring between different spatial and temporal scales. However, the complexity of the multi-scale dynamics of turbulent magnetized plasma in highly stratified inhomogeneous conditions prevents us from creating a single universal model to reproduce the solar dynamics. Therefore, despite the joint goal, modeling approaches are often targeted to a specific class of problems. For instance, realistic-type simulations are focused on capturing a wide range of the physical phenomena in the solar plasma (e.g., turbulence, radiation, ionization, magnetic fields, abundances) to reproduce observed processes and phenomena.
On the other hand, data-driven models attempt to reproduce a particular event or events by forcing a model solution toward specific observations to create an accurate physical interpretation of the observed dynamics.
Magnetohydrodynamic (MHD) simulations in the Sun-in-a-box regime aim to solve the equations of conservation of mass, momentum, energy, and magnetic flux in a highly stratified compressible medium, utilizing 3D multi-group radiative energy transfer between the fluid elements, a real-gas equation of state, ionization and excitation of all abundant species, and magnetic effects. Because of their computational cost, the resulting models cover only a small area on the Sun with a spatial resolution of tens of kilometers or higher. Nevertheless, this approach demonstrates its capability to reproduce solar observations. In particular, `ab-initio' modeling of the solar magnetoconvection and atmosphere pioneered by \cite{Nordlund1989} demonstrated the importance of the effects of 3D turbulent dynamics in reproducing spectroscopic observations \citep{Asplund2009}. Since then, such models have allowed us to make progress in understanding the near surface structuring and dynamics of solar plasma and magnetic fields. In particular, they have provided insight about granulation \citep[e.g.,][]{Stein2001}, supersonic horizontal flows \citep[e.g.,][]{Vitas2011}, fine structuring of sunspot umbrae \citep[e.g.,][]{Schuessler2006} and penumbrae \citep[e.g.,][]{Kitiashvili2009,Rempel2009,Rempel2009a,SiuTapia2018}, convective collapse \citep[e.g.,][]{Skartlien2000,Hewitt2014}, explosive events \citep[e.g.,][]{Kitiashvili2013,Danilovic2016,Chatterjee2016}, small-scale dynamos and self-formed magnetic structures \citep[e.g.,][]{Vogler2007,Kitiashvili2010,Kitiashvili2015,Stein2012,Rempel2014}, the formation of magnetic structures from predefined conditions such as twisted flux ropes or specific magnetic field topologies \citep[e.g.,][]{Rempel2009,Cheung2010,Fang2015} and others.
Thanks to intense cross-validation of the numerical models with observations, which has allowed achieving a high-degree realism of the simulated solar plasma, these 3D radiative models have become capable of providing critical support in the interpretation of spectroscopic and spectropolarimetric observations from the ground and in space. Figure~\ref{fig:stokesV} shows an example of continuum intensity synthesized from an MHD model \citep{Rempel2014} obtained with the MuRAM code \citep{Vogler2005} and a comparison of the Stokes V profiles obtained for the numerical resolution and resolutions corresponding to 4-m DKIST and 1.5-m GREGOR observations. Discrepancies in resulting line profiles illustrate how the spatial resolution can affect the accuracy of magnetic field measurements. Analysis of such discrepancies helps us to estimate uncertainties in the inferred physical and dynamical properties of the plasma. Indeed, the leveraging of cross-comparison between observations and simulations is crucial for several scientific issues discussed in the DKIST Critical Science Plan \citep{Rast2020}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.4]{fig_stokesVa.jpg}
\caption{Synthetic continuum image obtained from a MuRAM MHD model \citep{Rempel2014} shows the the granular structure and magnetized bright structuring in the intergranular lanes. A probe of Stokes V spectra (Fe I 630~nm) illustrates the dependence of the line profiles on the spatial resolution: full numerical resolution(8~km, solid curve), corresponding resolutions of DKIST (40~km, dashed curve) and the GREGOR telescope (80~km, dashed-dotted curve).
}
\label{fig:stokesV}
\end{figure}
Coronal magnetic fields play a key role in determining when and where solar flares and coronal mass ejections (CMEs) occur \citep{Gibson2020}. Remote sensing of the coronal magnetic fields has not been possible in the past, thus only reconstructions based on magnetic field measurements at the photosphere/chromosphere level could be employed to model the 3D structure of the coronal magnetic field. However, newly developed observational techniques, such as visible/infrared (VIR) coronal seismology \citep{Yang2020a, Yang2020b} and microwave interferometry \citep{Fleishman2020, Chen2020}, have demonstrated the feasibility of off-limb and on-limb observations to be routinely performed in the near future using a new generation of already developed instruments, such as DKIST and The Expanded Owens Valley Solar Array \citep[EOVSA, ][]{EOVSA}, and proposed instruments, such as the full-Sun Coronal Solar Magnetism Observatory \citep[COSMO, ][]{Tomczyk} and the Frequency Agile Solar Radiotelescope \citep[FASR, ][]{FASR}
Nevertheless, given the remote sensing nature of these newly available observations, they are generally limited by a line-of-sight ambiguity, which can only be resolved by combining them with sophisticated, yet feasible, multi-wavelength forward-fitting techniques and data-constrained modeling frameworks, such those offered by community-developed and maintained modeling tools like FORWARD \citep{Gibson2016} and GX Simulator \citep{Nita2015, Nita2018}.
Modeling of the magnetic field in the corona is essential for the study of the origin of the solar active phenomena. There are primarily two groups of models to simulate evolving coronal magnetic fields in 3D: quasi-static (or time-independent) and dynamic (time-dependent). In the quasi-static group, potential, linear, and non-linear force-free field (NLFFF) extrapolations have been developed. These models apply the vacuum-limit assumption, which reasonably assumes that magnetic pressure dominates the gas pressure (low-beta regime). Although this modeling framework assumes that the corona is in a static force balance that is not affected by states at earlier times, \citet{Fleishman2018} have shown that evolving sequences of such modeling snapshots may be used to infer the evolution of eruptive phenomena.
Another, more physics-based group of models are time-dependent models. In this group, ``data-driven'' (DD) simulations, i.e., simulations where successive photospheric magnetic field observations are used to evolve the model's photospheric field, show great promise for investigating the 3D structure of the coronal magnetic field. Lately, two types of DD models have been developed: magnetofrictional \citep[e.g.][]{Cheung2012} and MHD models \citep[e.g.][]{Hayashi2018}. The magnetofrictional model assumes that the plasma velocity in the MHD induction equation is proportional to the local Lorentz force, leading to a relaxation of a magnetic configuration toward a force-free state. It is more computationally efficient than MHD and is suitable for the description of the slow quiescent evolution of active regions, but not for flares. The MHD models explicitly solve a full set of MHD equations including the plasma properties. The MHD approach is suitable for rapid evolution during flares but is too computationally expensive to model the long-term quiescent evolution of active regions. A hybrid framework, where the MF model is used to model quiescent periods of AR evolution and the MHD model is used to model flaring periods of AR evolution, has been recently developed within the Coronal Global Evolutionary model \citep{Hoeksema2020}.
The major difference between data-driven and data-inspired models are the observations that the data-driven models make use of as lower boundary conditions. These are typically some combination of magnetic, velocity, and electric fields. Recently several approaches have been developed to derive electric fields using observations of the vector magnetic fields and Doppler velocities in the photosphere \citep[see e.g.][]{Fisher2020}.
The current effort to capture solar dynamics from the convection zone to the corona faces the challenge of numerically describing atmospheric layers with dramatic variations of plasma conditions.
In addition, a Non-Local Thermodynamic Equilibrium (NLTE) approach is necessary on one hand to interpret chromospheric and transition region diagnostics
(e.g., H-alpha, UV, IR Ca\,II\,lines, Mg\,II\,h\&k, He\,I, Ly-alpha) \citep[e.g.][]{Leenaarts2013a, Stepan2015} and therefore derive reliable observational constraints, and, on the other hand, to properly describe the interaction of radiation and matter in the higher layers of the atmosphere.
One of the most advanced codes in this respect is \textit {Bifrost} \citep{Gudiksen2011}, which, although in a simplified manner, includes NLTE processes in the energy and particle balance \citep{Carlsson2016,Hansteen2017}. Significant improvements have been further achieved by including additional effects, such as the Hall effect and ambipolar diffusion \citep[e.g.,][]{Martinez-Sykora2012, Khomenko2020}, and the development of approximations to describe chromospheric radiative cooling and heating \citep{Carlsson2012}.
However, the computational cost of NLTE calculations poses significant limitations on the size of the computational domain and spatial resolution.
Therefore, modeling of the solar dynamics in broad regimes, from the interior to the corona, necessarily relies on the LTE approximation. \citep[e.g.,][]{Rempel2017,Wray2018}.
In spite of this simplification, various codes have been used to simulate the domain from the photosphere to the corona, allowing the study of phenomena naturally driven by near-surface plasma dynamics, and addressing issues as coronal heating, the onset of flares, and others \citep[e.g.,][]{Gudiksen2005,Carlsson2012,Rempel2017,Cheung2019}. These models provide theoretical support for the interpretation of observations. An example of synthetic SDO/AIA observations of the limb view of the modeled solar corona~\citep{Kitiashvili2020} is presented in Figure~\ref{fig:aiacorona}.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\linewidth]{WP_AIAsideview_t12072.png}
\caption{Composite illustration of the Solar Dynamics Observatory Atmospheric Imaging Assembly (SDO/AIA) synthetic emission for a side view of a quiet Sun region, simulated with the StellarBox code \citep{Wray2018}. Red color corresponds to the synthetic SDO/AIA 171\,\AA{} emission (T$=7.9\times{}10^{5}$\,K), green corresponds to 131\,\AA{} (T$=5.6\times{}10^{5}$\,K), blue corresponds to 304\,\AA{} (T$=7.9\times{}10^{4}$\,K). The domain has a horizontal size of 12.8\,Mm and includes 10\,Mm of the solar atmosphere from the surface to corona.}
\label{fig:aiacorona}
\end{figure}
As alluded to above, the use of state-of-the-art atmosphere models to interpret solar observations is further complicated by difficulties inherent to the synthesis of specific observables.
Several radiative transfer codes have been developed for this purpose, each with certain advantages and restrictions. For instance, the Spinor \citep{Frutiger2000} code is relatively fast, but computations are restricted by the Local Thermodynamic Equilibrium (LTE) approximation. The NICOLE code \citep{SocasNavarro2015} enables calculations in both LTE and NLTE, but the implementation of the NLTE approximation does not include partial frequency redistribution (PRD) effects, which are important for the modeling of chromospheric lines, such as the Ca II H \& K, and Mg II UV lines. The RH code \citep{Pereira2015} is also able to perform calculations in both approximations with a primary focus on the accurate implementation of NLTE effects in different types of geometries. The Multi3D code \citep{Leenaarts2009} is developed for massive calculations from 3D models of the solar atmosphere of a wide range of spectral lines using the NLTE approximation. However, the inclusion of detailed multi-dimensional radiative transfer effects comes at the cost of the increased computational time. Moreover, the description of these physical effects is numerically challenging, especially in the chromosphere where large spatial and temporal gradients often make the codes unstable. Ultimately, the choice of the particular radiative transfer code depends on the task and the related approximations most suitable for the physical processes one wants to study.
The interpretation of observational results often relies on solving the inversion problem, i.e., on modeling the physical characteristics of the atmosphere which produced the observed emission and spectra.
Different codes are available to perform spectral ad spectropolarimetric inversions of observed spectra, but, similarly to the codes available for forward modeling, the vast majority are targeted at the interpretation of specific types of observations. For instance, codes developed for the interpretation of lines forming in the photosphere and chromosphere are often developed by integrating the radiative transfer codes described in the previous section with minimization algorithms that optimize the physical and magnetic stratification of the atmosphere (examples include the widely used NICOLE and SPINOR and the newly developed STiC and DeSIRE, based on the RH code). As such, these inversion codes suffer of the same limitations described above. Other codes have been or are under development to interpret specific observations, as for instance observations in the He I 1083.0 nm line (Hazel), limb observations in chromospheric lines (NICOLE), and flares (STiC). Interpretation of coronal observations is instead performed by using very different approaches, based on differential emission measures of the coronal plasma from the related EUV observations \citep[e.g.,][]{Cheung2015}. One of the major drawbacks of most of inversion techniques is their extremely high computational cost.
One solution is to represent the highly-nonlinear mapping of the observed emission and line profiles to the structure of the atmosphere by a neural network trained on the set of physics-based inversions. Such an approach has been successfully applied for fast inversion of IRIS data \citep{SainzDalda2019} and SDO/AIA EUV emission \citep{Wright2019}. Although uncertainty quantification for such machine-learning-based models is challenging and still open, this is currently the only way to provide timely data products under one pipeline with the observational data. The development of such models and understanding and quantifying the related uncertainties should be encouraged.
\section{Access to computational resources and model data sharing}
High spatial and temporal resolutions, augmented by the spectropolarimetric dimensions, dramatically increase the data-volume of both observational and synthetic data. Such massive data flows significantly limit the dissemination of these data among the community.
On the one hand, sharing of the modeling and synthetic-observables data sets requires dramatic storage resources. On the other hand, taking these data and analyzing them on a local workstation is often impossible. As a result, the modeling data are highly truncated (e.g., include only selected layers, model parameters, and/or limited time-series).
The problem is becoming more severe with time to the point that web links to the data are sometimes failing. Therefore, the development of a data-sharing platform and data-request system will benefit both potential customers and data providers.
Of particular importance is the collection of synthetic observables obtained with different radiative transfer codes and MHD models.
A database of synthetic observables, obtained by combining state-of-the-art model atmospheres and radiative transfer computations, will support the scientific community in several crucial ways, such as the interpretation of observations, the development and validation of new data analysis techniques, and the validation of new instrumentation and data reduction pipelines. Furthermore, a wider dissemination of synthetic data will improve our understanding of model limitations, thus fostering the development of new numerical techniques and a better description of the physical processes that drive the evolution of the solar plasma and magnetic fields.
\section{Recommendations}
The development of numerical models and tools is essential for a robust physical interpretation of the observational data acquired by the DKIST and other ground and space missions that have recently come online. The analysis of spectral and spectropolarimetric observations acquired at high spatial and temporal resolution and at high spectropolarimetric sensitivity in different spectral ranges, requires the development of models capable of describing the non-linear connection between the local physical and dynamical properties of plasma in a radiating, turbulent, and magnetic environment. Despite significant advances in modeling solar dynamics, the cross-validation of models and observations is essential to detect any missed components of the observed phenomena.
Thus it is crucial to support the multi-level integration of modeling efforts and observations. In particular, we identify the following critical needs:
\begin{itemize}
\item Improve the development of high-fidelity 3D MHD codes by 1) removing restrictions on code development commitments in proposal AOs, and 2) establishing new funding opportunities for the development of codes and data analysis approaches, including machine learning, data assimilation, and others.
\item Encourage cross-comparison of observations and simulations and support the development of new tools for efficient analysis of big multi-dimensional data sets.
\item Increase investment in high-end-computing (HEC) facilities in terms of both computing nodes and storage. The storage and computational requirements are amplified by including more physical effects into the models and for the post-processing of the resulting data. Improvement in HEC capabilities will increase workflow performance and will significantly speed up development of predictive capabilities for both global solar variability and short-term, high-energy release activity such as flares, coronal mass ejections, etc. Similarly, the management and analysis of large observational datasets (for instance through spectropolarimetric inversions) requires increasing computational and storage resources.
\item Allow trial allocations to demonstrate the feasibility of proposed projects and to speed up the boarding process when the first full allocation is awarded. Trial allocations will provide quick access to computational resources when needed and will help foster a new generation of scientists and developers.
\item Establish a NASA-NSF HEC collaboration to enable efficient workflow. Code performance can depend on supercomputer architecture, infrastructure, and available resources. Consequently, rigid funding linked to a specific supercomputer system can impede project success.
\item Support the development of open access policies for models and synthetic data, in a manner similar to policies implemented for observational data acquired by NASA and NSF facilities.
\item Support the development of platforms for requesting synthetic data targeted at the interpretation of specific observations.
\item Support the development of platforms for the distribution of models and synthetic data to leverage existing capabilities and efforts, to improve collaboration efficiency, and to stimulate interdisciplinary integration.
\end{itemize}
\section*{Acknowledgments}
This material is based upon work supported by the National Science Foundation under Grant AGS-1743321. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.The National Solar Observatory is operated by the Association of Universities for Research in Astronomy, Inc. (AURA)
under cooperative agreement with the National Science Foundation.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,656
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Q: If $\int_{0}^{\pi}t^{n}f(t)dt=0$ for all $n\in\{0\}\cup \mathbb N$ then prove that $f \equiv0$
Let, $f:[0,\pi ]\to \mathbb R$ be a continuous function such that $f(0)=0$.
If $\int_{0}^{\pi}t^{n}f(t)dt=0$ for all $n\in\{0\}\cup \mathbb N$ then prove that $f \equiv0$.
Since, $f$ is continuous & $t^{n}$ is also continuous & $t^{n}$ keeps the positive sign throughout the interval $[0,\pi]$, so using $1^{st}$ Mean-Value-Theorem of Integral calculus $\exists$ $\xi\in [0,1]$ such that, $$\int_{0}^{\pi}t^{n}f(t)dt=f(\xi)\int_{0}^{\pi}t^{n}dt=f(\xi).\dfrac{\pi^{n+1}}{n+1}.$$
Using given condition we get $\exists$ $\xi \in[0,1]$ such that, $f(\xi)=0.$ From here how we can conclude that $f\equiv0$ ?
Or any other way to prove this?
A: Hint: Weierstrass approximation (by polynomials) of continuous functions in compact interval and $\displaystyle \int_0^{\pi} f^2(t)\,dt = 0 \implies f = 0$, tell me if you need more details.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,872
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Q: Setting an API key as a header for a XMLHttpRequest Attempting to work with a Fortnite API (https://fortnitetracker.com/site-api) and it requires me to pass the API key in the header along with my requests. I've tried using .setRequestHeader but haven't had any luck.
function getInfo(){
var xhr = new XMLHttpRequest();
xhr.open('GET','https://api.fortnitetracker.com/v1/profile/pc/ninja',true);
xhr.onload = function(){
if(this.status == 200){
console.log("Worked");
} else {
console.log(this.status);
}
}
xhr.onerror = function(){
console.log("Request Error");
}
// Fake API Key
xhr.setRequestHeader("Authorization","12345678910");
xhr.send();
}
Hope someone can help and thanks for reading.
A: I think you are looking for this:
xhr.setRequestHeader('Authorization', 'Bearer ' + token);
While you are getting to grips working with APIs, play with a REST client:
*
*https://www.getpostman.com/
*https://insomnia.rest/
They will help you see what these requests are supposed to look like. Basically, first figure out what the API needs with postman/insomnia and only after that you write your code.
Hope that helps!
A: For API Key authentication you would call it like this... .setRequestHeader "ApiKey", "MyKey".
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,671
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{"url":"http:\/\/mathhelpforum.com\/calculus\/73159-integration-finding-average-value-funciton.html","text":"# Thread: integration: finding the average value of a funciton\n\n1. ## integration: finding the average value of a funciton\n\nIn a certain city the temperature (in degrees Fahrenheit) t hours after 9am was approximated by the function\n\nT(t)=80 + 11 sin(pi*t\/12)\n\nDetermine the temperature at 9 am.\nDetermine the temperature at 3 pm.\nFind the average temperature during the period from 9 am to 9 pm.\n\n2. Originally Posted by sss\nIn a certain city the temperature (in degrees Fahrenheit) t hours after 9am was approximated by the function\n\nT(t)=80 + 11 sin(pi*t\/12)\n\nDetermine the temperature at 9 am. T(0) = ?\n\nDetermine the temperature at 3 pm. T(6) = ?\n\nFind the average temperature during the period from 9 am to 9 pm.\n\n$T_{avg} = \\frac{1}{12}\\int_0^{12} 80 + 11\\sin\\left(\\frac{\\pi t}{12}\\right) \\, dt$\nfirst two questions are just function values of T(t)\n\nlast question is set up for you ... give it a try.","date":"2017-02-23 05:58:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 1, \"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\": 0.6975847482681274, \"perplexity\": 1814.892020688322}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501171078.90\/warc\/CC-MAIN-20170219104611-00634-ip-10-171-10-108.ec2.internal.warc.gz\"}"}
| null | null |
var apigee = require('apigee-access'),
request = require('request'),
apigee_cache_promise = require('apigee-cache-promise')(apigee.getCache()),
debug = require('debug')('securityFactory-sfdc-oauth-validator'),
jwt = require('jsonwebtoken'),
_ = require('lodash'),
jwkToPem = require('jwk-to-pem'),
utils = require('../lib/utils'),
rp = require('request-promise');
function SalesForceOAuthValidator(options) {
debug('options.config', options.config);
var options = options;
this.config = options.config;
}
SalesForceOAuthValidator.prototype.validateSalesForceJWTToken = function (options, callback) {
var salesForceOAuthValidator = this;
var decoded_jwt;
try {
decoded_jwt = jwt.decode(options.access_token, { complete: true });
debug('access token decoded3', decoded_jwt);
apigee_cache_promise.get('sso2SalesForceTokenKeys')
.then(function retrieveKeysFromServer(keys) {
// if keys found in cache, return keys immediately
if (keys) {
debug('ccached keys', keys);
return keys;
}
// if it reached this point, keys were not found in cache, retrieve, cache, and return them from server
return utils.promisifyRequest({ url: options.token_key_url, rejectUnauthorized: false })
.then(function (sso2SalesForceTokenKeys) {
debug('non-cached keys', sso2SalesForceTokenKeys);
apigee_cache_promise.put('sso2SalesForceTokenKeys', sso2SalesForceTokenKeys, 36000); //retrieve key every 10 hrs
return sso2SalesForceTokenKeys;
})
})
.then(function verifyJWTwithJWK(sso2SalesForceTokenKeys) {
var key = utils.getKey(decoded_jwt.header.kid, JSON.parse(sso2SalesForceTokenKeys).keys);
utils.verifyJWT({
access_token: options.access_token, req: options.req, res: options.res,
public_key: jwkToPem(key), type: options.type, whitelisted_domains: salesForceOAuthValidator.config.security.email_domain
}, callback);
})
.catch(function (e) {
debug('catch', e, e.stack);
callback(null, { valid: false, message_back: e, type: options.type });
})
} catch (e) {
return callback(null, { valid: false, message_back: e, type: options.type });
}
}
module.exports = SalesForceOAuthValidator;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,914
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package com.example.mzwee.randrestaurant;
import android.os.Bundle;
import android.support.v7.app.AppCompatActivity;
/**
* Created by User on 20-Feb-16.
*/
public class ViewCredits extends AppCompatActivity {
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.credit);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,076
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Home / News and Events / News / Cronkite School Leads Field at 2017 Student Murrow Awards
Cronkite School Leads Field at 2017 Student Murrow Awards
Student Edward R. Murrow Awards
Carnegie-Knight News21 "Voting Wars"
Cronkite Grad Wins Student Murrow Award for News21 Investigation
Cronkite News Named Best Television Newscast in National SPJ Competition
Arizona State University's Walter Cronkite School of Journalism and Mass Communication was the only journalism program in the nation to win multiple honors in the prestigious Student Edward R. Murrow Awards.
Cronkite students won in two of five possible categories in the 2017 contest established by the Radio Television Digital News Association.
Cronkite News, the student-produced, faculty-led news division of Arizona PBS, won the Student Murrow Award for Excellence in Video Newscast. Last week, Cronkite News was named the top newscast in the country by the Society of Professional Journalists.
Carnegie-Knight News21, a journalism initiative that brings top journalism students from across the country to the Cronkite School to report on an issue of national significance, won the Student Murrow for Excellence in Digital Reporting.
Since the Student Murrow Awards were established, the Cronkite School is the only journalism program in the country to have won multiple Student Murrow Awards. Carnegie-Knight News21 won the very first Student Murrow Award in Overall Excellence – Video in 2015. In all, the school has won three awards.
In the 2017 contest, judges praised the Feb. 17, 2016 Cronkite News newscast, produced by Windsor Smith and directed by Madison Romine, for natural sound in the packages, strong soundbites and good live coverage. Numerous students contributed to the newscast, which featured coverage from Pope Francis' mass in Ciudad Juarez, Mexico.
"The Murrow Awards are among the most prestigious in journalism," said Cronkite Associate Dean Mark Lodato. "To finish first in two of the five categories is remarkable and a testament to the quality work our students produce each year. Cronkite News is an important daily resource to Arizonans interested in serious reporting. Meanwhile, News21 took readers and viewers on a creative, in-depth examination of one of the year's most important issues."
Thirty-one students from 18 universities traveled to 31 states and interviewed hundreds of people for Carnegie-Knight News21's "Voting Wars" investigation into voting rights. Judges said the investigation was an "extremely on-point project with good stories and creative and strategic use of multimedia." Since its release, portions of the investigation have been featured in more than 80 media outlets, including NBC News, USA Today and The Washington Post.
"'Voting Wars' captured so much of what was going on prior to the 2016 election," said Carnegie-Knight News21 Executive Editor Jacquee Petchel. "Winning a Murrow Award validates our efforts to push the limits of multimedia journalism as well as the hard work of all of our students who contributed so much to this powerful project on voting rights."
The Student Murrow Awards celebrate overall excellence in student journalism at the collegiate and high school levels. Unlike the professional Murrow Awards, which are presented to a news organization, the Student Murrows are awarded to individuals in one of five categories — audio newscast, audio reporting, video newscast, video reporting and digital reporting.
The RTDNA will recognize the 2017 winners at the RTDNA Edward R. Murrow Awards Gala at Gotham Hall in New York on Oct. 9.
The RTDNA has been honoring outstanding achievements in professional journalism with the Edward R. Murrow Awards since 1971. Murrow Award recipients demonstrate the excellence that Edward R. Murrow made a standard for the electronic news profession. The RTDNA is the world's largest professional organization exclusively serving the electronic news profession. Members include local and network news executives, news directors, producers, reporters and digital news professionals as well as educators and students.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,794
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Villa de Álvarez, oficialmente Ciudad de Villa de Álvarez, es una ciudad ubicada en el estado mexicano de Colima, cabecera del municipio homónimo. Es parte de la zona metropolitana de Colima-Villa de Álvarez y tiene alrededor de 147 496 habitantes, de acuerdo al censo del año 2020.
Descripción
El 4 de octubre de 1903, el obispo José Amador Velasco Peña colocó la primera piedra de lo que fue llamada la parroquia de Almoloyán, que a partir de 1953 es el templo parroquial de San Francisco de Asís, ubicado en la parte central de la ciudad de
Villa de Álvarez. Desde los inicios de este siglo se conocieron los siguientes barrios de Villa de Álvarez: San Juan (por la calle Aquiles Serdán), Del Sol (por la calle Eduardo Álvarez), La Frontera (por la calle Hidalgo), La Haciendita (frente al antiguo Casino Municipal, hoy la Casa de la Cultura) y Los Cerritos (por la calle Manuel Álvarez).
Villa de Álvarez es una ciudad de 119 956 habitantes que ocupan 34 903 viviendas (INEGI, 2012). En los últimos años se ha caracterizado por su ritmo de crecimiento, ocupado el primer lugar en crecimiento de vivienda en el Estado de Colima.
Historia
En 1875 se formó una junta de 40 vecinos, quienes se dieron a la tarea de elaborar el "Proyecto sobre construcción de un ferrocarril de Manzanillo a Colima y Guadalajara". La directiva estuvo formada por Ramón R. de la Vega, Agustín Schacht, Christian Flor, Jorge M. Oldenbourg, Miguel Bazán, Alejandro Véjar y Antonio E. Orozco. El proyecto fue aprobado, y en 1889 se inició la construcción de la vía de Armería a Colima. El tren llegó a la capital en 1892. La inauguración oficial se llevó a cabo por Porfirio Díaz en el año de 1908. Los primeros intentos por construir el ferrocarril urbano los realizó el gobernador Esteban García en 1884. El proyecto fue elaborado por el ingeniero francés Arturo Le Harivel. En 1892 fueron instalados los primeros tranvías que iban del centro de Colima a Villa de Álvarez.
Toponimia
Para mayor información del escudo de Villa de Álvarez, ver el artículo: Escudo de Villa de Álvarez
Lo que hoy conocemos como el municipio de Villa de Álvarez, en el año de 1556 se llamaba "San Francisco de Almoloyán"; en 1836 el distrito de Colima se dividió, para su administración, en dos partidos; uno de los cuales fue el de Almoloyán, que se convirtió
en la cabecera municipal. Por decreto, el 15 de septiembre de 1860, se cambió el nombre original por el de "Villa de Álvarez", en honor al primer gobernador del estado, que era originario de ese lugar.
Época Precolombina
Existe la hipótesis de que en el momento en que las tribus náhuas salieron de Aztlán, lugar que algunos historiadores ubican en las costas de Nayarit, alrededor de la isla de Mezcaltitán, un pequeño grupo, en lugar de continuar con rumbo al altiplano, en busca del águila parada en un nopal devorando una serpiente, caminó por la costa, después de algún tiempo se encontró frente a los fascinantes volcanes y sus fértiles campos que los circundaban, lo que motivó que se asentaran en dichos lugares, esperando el cumplimiento de la aparición del águila legendaria. Al principio dedicaron su tiempo a la cacería, la pesca y la recolección de frutos y plantas silvestres, con lo que se alimentaban. Con el transcurso de los años, descubrieron diversos cultivos, que fueron básicos en su alimentación como el maíz, el frijol, la calabaza y el chile. En la comunidad había tres clases de autoridades de los cuales son Los sacerdotes, que atendían al culto sagrado, Los brujos (que también eran curanderos), Los ancianos (que formaban el consejo); Estas personas tenían la responsabilidad de decidir acerca de lo que tenía que hacerse en la comunidad y orientaban en los diversos problemas que se iban presentando. Se habla de la existencia de un gran señor en el territorio Colimote a la llegada de los españoles (Hernán Cortés, 3ª Carta de Relación). A él le rendían tributo varios señoríos de la región. Este gran señor estaba al frente de la provincia del Colimotl, por lo que es de suponer que los núcleos indígenas ya mencionados, en cierta manera dependían del señor de Colimán. Los grupos indígenas tenían dos grandes preocupaciones en su vida: La guerra y el culto a las divinidades.
Época Colonial
Inmediatamente después de llevada a cabo la conquista de Colimán, las tierras fueron repartidas en encomiendas. De tal suerte que a Francisco Santos se le otorgaron las tierras de Almolonia y de Tlacalahuastla (hoy Minatitlán); a Bartolomé López parte de lo que hoy el Comala; a Hernando de Gamboa las de Coquimatlán, y al Conde de Terreros, terrenos de los Pastores. Todas estas tierras eran ricas, por lo que empezó a florecer tanto la agricultura como la ganadería, ya que a la vez se disponía de mano de obra al contar con el trabajo de los indígenas. El 22 de febrero de 1554, algunos frailes franciscanos, como Ángel de Valencia (guardián del convento), Honorato Franco, Jerónimo de la Cruz, entre otros, fundaron un convento al que llamaron San Francisco de Colimán y después de Almoloyán. La instalación del monasterio tenía como objetivo fundamental brindar protección y educación a los indígenas. Cerca del convento de San Francisco, en 1556, se fundó, con la autorización del Virrey Luis de Velasco, un pueblo al cual le fue otorgado fundo legal y facultad para gobernarse por sí mismo.
Época de la Independencia
En 1810, el padre José Antonio Díaz, adscrito a la parroquia de San Francisco, al conocer la noticia del "Grito de Dolores" que proclamaba el levantamiento insurgente para luchar a favor de la Independencia de México, invitó a las poblaciones vecinas a
que se unieran al movimiento para acabar con la dictadura española. San Francisco de Almoloyán formó un ejército con los anuales para luego irse a Coalcomán. Fue él quien pronunció el discurso patriótico en el juramento de la Constitución de Apatzingán (22 de
octubre de 1814). El 15 de septiembre de 1860, siendo gobernador Urbano Gómez, le fue cambiado el nombre a la cabecera municipal, quedando en adelante como: Villa de Álvarez, en honor al Gral. Manuel Álvarez, primer Gobernador Constitucional del
estado de Colima.
Geografía
Situación
Villa de Álvarez se localiza en el este del municipio, en las coordenadas , está a una altura media de 530 metros sobre el nivel del mar (msnm).
Clima
Villa de Álvarez, al igual que gran parte del estado de Colima, tiene un clima cálido subhúmedo con lluvias en verano. Tiene una temperatura media anual de 25.5 grados centígrados.
Demografía
La ciudad de Villa de Álvarez cuenta con una población en el 2020 de 147,496 habitantes según datos del XIV Censo De Población y Vivienda del Instituto Nacional de Estadística y Geografía (INEGI) significando un aumento de 29,896 habitantes respecto al Censo de 2010 por lo que es la 2ª ciudad más poblada del estado de Colima siendo solo superada por el puerto de Manzanillo y superando a la siempre históricamente ciudad más poblada del estado que era la ciudad de Colima con apenas poco más de 500 habitantes, pero se espera que en pocos años Villa de Álvarez será la ciudad más poblada del estado debido a que atrae a toda la población que decide instalarse en la capital del estado pero ante la falta de espacio para extender su mancha urbana la ciudad de Villa de Álvarez acapara los nuevos habitantes.
Población de Ciudad de Villa de Álvarez 1900-2020
Cultura
En Villa de Álvarez, por el hecho de estar conurbado con la capital del estado, se contagia de las actividades culturales que allí se llevan a cabo. Un factor que ha cobrado peso a últimas fechas en
la cultura de la gente, es justamente la Casa de la Cultura del municipio, pues es de reciente creación (2003) y que, al igual que todas las demás, está ubicada en la propia cabecera municipal en una de las avenidas más transitadas de la localidad; allí, muchos
jóvenes asisten para ver exposiciones de pinturas, teatro, bailables, conciertos de música, de acercamiento y gusto por la lectura y muchos otros eventos, fomentando en ellos una mayor apreciación por este tipo de actividades.
Industria
La industria en Villa de Álvarez, la componen 439 talleres de la industria manufacturera, de estos 131 son de la industria alimentaria, 20 bebidas y 378 a servicios automotrices, talleres mecánicos que pertenecen a servicios (INEGI, 2012).
Por su innovación se puede mencionar la producción de aceite de coco y de limón.
Comercio
En la ciudad de Villa de Álvarez se han instalado importantes tiendas de cadenas tanto nacionales como transnacionales: Soriana (supermercado con otros locales pequeños y medianos alrededor), Famsa (mueblería), Bodega Aurrerá (tienda de autoservicio/supermercado), City Club, Coppel, etc.), así como un sinnúmero de tiendas de ropa, calzado, muebles, ferreterías, panaderías, materiales para construcción, gasolineras, entre otros; todos ellos se han mantenido de generación en generación, ya
sea en el centro de la ciudad o en avenidas principales como también en los nuevos espacios de reciente apertura; muchos de ellos corresponden al capital colimense.
Los últimos años se ha detectado un crecimiento significativo de este sector económico en la zona oriente de la cabecera municipal; ubicándose en su gran mayoría pequeñas y medianas empresas por la Av. Benito Juárez, arteria de comunicación con Comala y con
Coquimatlán, que ha funcionado como anillo periférico. Es importante destacar que en el norte de esta avenida, se ubica la plaza comercial Colima, con la tienda ancla "Soriana" y al sur la "Bodega Aurrerá", de reciente apertura (2004).
Centros turísticos
La ciudad de Villa de Álvarez, existen diversos centros de entretenimiento, turístico, donde también se puede disfrutar de sus antojitos regionales para el turista y los habitantes de otros municipios de Colima;
El balneario "Agua Fría": Un lugar atractivo para los días de campo.
Picachos: Por su río San Palmar, propio para paseos ribereños.
Juluapan: Porque permite admirar la monumental roca del lugar.
El Jardín: El jardín principal es muy tradicional porque aún conserva su aire provinciano a pesar del crecimiento poblacional desmesurado de este municipio; en este jardín es clásico ir a comprar y disfrutar tranquilamente de las famosas "Paletas de la Villa" deliciosamente elaboradas con leche o con agua en una gran variedad de sabores, cuya receta ha perdurado inalterable a través de varias décadas; los jóvenes las disfrutan para apagar la sed, mientras descansan y platican con pareja o amigos. También este espacio es el que alberga la añeja costumbre de vender las empanadas (de leche y otros sabores) el día 4 de octubre de cada año, en honor de San Francisco.
La Campana: La Campana, que fue reconstruido hace algunos años bajo la coordinación del INAH y que hoy es visitado por grupos de turistas nacionales y extranjeros, y en donde, además, se llevan a cabo ceremonias en los equinoccios y solsticios de cada año; se localiza por la avenida Tecnológico de Colima.
La Laguna de los Pastores: Al norte del municipio, es un sitio de gran atracción turística
Fiestas charro-taurinas: Las festividades tradicionales dentro de la cabecera municipal resultan un verdadero atractivo para los jóvenes no solamente del municipio, sino de otros lugares, pues las Fiestas Charro-taurinas, que desde hace más de trescientos años se vienen realizando –todos los años durante el mes de febrero– en honor a san Felipe e Jesús (santo patrono de la ciudad de Colima también), son las de más arraigada costumbre y mejor organizadas en todo el estado. Tienen características y actividades que las hacen únicas, como la plaza de toros conocida como la petatera, que es construida exclusivamente con petates y que se monta y desmonta anualmente, convirtiéndose en un atractivo más para los propios y extraños y que, de acuerdo a la tradición, es así elaborada y reelaborada cíclicamente como una "manda" de los feligreses dedicada a este santo, con la intención de pedir de sus favores para evitar que se repitan terremotos en el estado (dado que Colima es una zona altamente sísmica).
Escultura de la cabalgata: Un atractivo turístico es la escultura monumental en bronce del Escultor Octavio González a tamaño 1:1 colocada en el jardín principal, conformada por 6 caballos, 5 de ellos montados por charros, vaqueros, caporales y una escaramuza, usando trajes y monturas de acuerdo al oficio u ocasión, uno de los caballos no tiene jinete, el jinete puedes ser tu, lo que la convierte en una escultura interactiva.
Véase también
Playa Tecuanillo
Playa El Real
Playa El Chupadero
Playa Boca de Apiza
Reino de Colliman
Diócesis de Colima
Símbolos de Colima
Reino de Colliman
Congreso de Colima
Rey Colimán
Plaza de Toros La Petatera
Municipio de Villa de Álvarez
Referencias
Localidades del estado de Colima
Cabeceras municipales del estado de Colima
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Hanratty ist der Familienname folgender Personen:
Alice Hanratty (* 1939), irische Künstlerin
James Hanratty (1936–1962), britischer Mörder, eine der letzten in Großbritannien hingerichteten Personen
Sammi Hanratty (* 1995), US-amerikanische Schauspielerin
Terry Hanratty (* 1948), US-amerikanischer American-Football-Spieler
Thomas J. Hanratty (1926–2016), US-amerikanischer Chemieingenieur
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{
"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
Difficulty in perturbative quantum field theory can be measured along the axes of numbers of loops and numbers of legs in a typical problem to be computed. Entangled with these is the issue of the number of scales in a typical problem: the more scales (such as more legs or loops), the harder the problem. For instance, single loop problems with many legs as well as multiple scales can nowadays be solved routinely, modulo numerical instabilities and issues. For practical collider applications though, the frontier of development is for on-shell scattering amplitudes found already at two loops, with five or more on-shell massless particles. This holds even before taking into account the important issue of obtaining a finite cross-section after cancelling of UV and IR divergences. This motivates the push for more loops and legs.
An important tool for pushing development in perturbative quantum field theory is the use of toy models. A particularly important example of such a model is the unique maximally supersymmetric Yang-Mills theory in four dimensions. This has played an important role as a testing ground for new ideas and methods in the recent past \cite{Witten:2003nn} and is interesting in its own right for the relation of its planar sector to a string theory description in the AdS/CFT correspondence \cite{Maldacena:1997re}. In the context of the latter some quantities have a (partly conjectural) exact description. For the present context, an important example of this is the planar light-like cusp anomalous dimension, for which an exact integral equation exists \cite{Beisert:2006ez}. The planar limit provides particularly nice simplifications. In this limit the number of colors, $N_c$, in a $SU(N_c)$ gauge theory is taken to infinity. In the physical theory of the strong interactions however, this parameter is three and non-planar effects are physically relevant. Also from the formal point of view the value of the first non-planar corrections to the cusp anomalous dimension is highly interesting, related to string loop corrections in the AdS space.
A second, but no less important reason to be interested in the light-like cusp anomalous dimension is the fact that it is a universal quantity that is ubiquitous in QFT computations. It appears for instance in the universal exponentiation of IR divergences in dimensional regularisation and gets its name from the computation of the light-like cusped Wilson line \cite{Korchemsky:1985xj, Korchemsky:1987wg}. This function is usually mixed in with additional structure in a given scattering amplitude or form factor. The exception to this is the simplest, Sudakov form factor. This consists of a member of the stress-tensor multiplet (i.e. a local gauge invariant operator) and two members of the on-shell gluon multiplet that is unique in $\mathcal{N}=4$ SYM. This is a single-scale problem, and it can be shown that maximal supersymmetry factors off a universal tree-level form factor that contains all dependence on external polarizations as well as the dependence on the gluon color quantum numbers. Hence to any loop order
\begin{equation}
F^{(l)} = F^{(0)} (-q^2 )^{-l \epsilon } {\tilde F}^{(l)}
\end{equation}
holds, where $q^2= (p_1 + p_2)^2$ and $D=4-2 \epsilon$, and ${\tilde F}^{(l)}$ is now a dimensionless function of $g_{\textrm{ym}}$, $N_c$ and $\epsilon$. It is related to the cusp anomalous dimension by exponentiation \cite{Mueller:1979ih, Collins:1980ih, Sen:1981sd, Magnea:1990zb},
\begin{equation}\label{eq:expIR}
{\tilde F}^{(l)} = \exp\bigg( {\sum_l \frac{-g^{2l}\gamma_{\textrm{cusp}}^{(l)} }{(2l \epsilon)^2} } + {\cal O}(\epsilon^{-1}) \bigg) \, .
\end{equation}
Therefore at fixed order in perturbation theory one can compute the correction to the light-like cusp anomalous dimension to that order in perturbation theory. The possible algebraic color invariants that can appear are determined from all possible contractions of structure constants (see e.g. \cite{Boels:2012ew} for results up to eight loops). Analysing their $N_c$ dependence shows that the first non-planar correction appears at four loops. It is therefore a particular interesting and challenging problem to consider four-loop form factors, which is the focus of this paper.
The computation of the Sudakov form factor has a long history both within $\mathcal{N}=4$ as well as in more physical theories \cite{Mueller:1979ih, Collins:1980ih, Sen:1981sd}. The first two-loop correction in $\mathcal{N}=4$ appeared already in \cite{vanNeerven:1985ja}. The three-loop correction in QCD was studied in a series of papers starting with a basis of integral after IBP reduction \cite{Gehrmann:2006wg}, through numerical analysis \cite{Baikov:2009bg} to analytic integration \cite{Lee:2010cga}, with an important cross-check in \cite{Gehrmann:2010ue}. In \cite{Henn:2013wfa} the form factor in $\mathcal{N}=4$ to the three-loop order was studied. See also \cite{Henn:2016men} for recent exciting progress at four loops in QCD. In \cite{Boels:2012ew} the first simple form of the integrand of the four-loop form factor in $\mathcal{N}=4$ was published using color-kinematic duality \cite{Bern:2008qj, Bern:2010ue, Bern:2012uf}. This was followed by the IBP reduction of its integrals in \cite{Boels:2015yna}, to which the reader is also referred to for details of part of the work presented here. Interesting ideas for evaluating the integrals based on dimensional recurrences \cite{Tarasov:1996br} and advanced integral analysis have appeared in \cite{vonManteuffel:2014qoa} and \cite{vonManteuffel:2015gxa}.
\section{Integration-by-parts identities and their solution}
A crucial step in state-of-the-art computations in quantum field theory is the use of integration-by-parts identities \cite{Chetyrkin:1981qh, Tkachov:1981wb} to simplify a given problem. Any integrand-generating technique such as Feynman graphs eventually produces a sum over integrals of the type
\begin{equation}\label{eq:deffeynmanint}
I(a_1, \ldots, a_n) \equiv \int d^D l_1 \ldots d^Dl_L \left(1/D_1\right)^{a_1} \ldots \left(1/D_n\right)^{a_n} ,
\end{equation}
where $D_i$'s are typically inverse propagators.
These integrals are not all indepedent, but obey linear identities that follow from
\begin{equation}
0 = \int d^D l_1 \ldots d^Dl_L \,\, \frac{\partial}{\partial l_i^{\mu}} \, X \,,
\end{equation}
for any derivative with respect to a loop momentum. This type of identity leads to a large set of linear equations, which can be solved by Gaussian elimination, at least in principle. Since the matrix involved is non-invertible, the solution takes the form of an expression of all integrals in a given set in terms of a small subset. This smaller subset can be chosen to consist of the simplest integrals in the class. The latter implies there is a choice of basis involved, which is to be supplied by the user. This is the essence of the Laporta algorithm \cite{Laporta:2001dd}.
Many private and public implementations of this approach exist, such as {\tt AIR} \cite{Anastasiou:2004vj}, {\tt FIRE} \cite{Smirnov:2008iw, Smirnov:2013dia, Smirnov:2014hma} and {\tt Reduze} \cite{2010CoPhC.181.1293S, vonManteuffel:2012np}. See {\tt LiteRed} \cite{Lee:2012cn, Lee:2013mka} for an alternative approach to IBP reduction. For the four-loop form factor under study here only {\tt Reduze} \cite{vonManteuffel:2012np} gave an answer in large but finite time. The result is given in an explicit set of reduction form of integrals in the form factor. Some statistics on master integral numbers as a function of numerator power $s$ is given in Table \ref{table:countingReduze}, see \cite{Boels:2015yna} for further details.
\begin{table}[ht]
\caption{Master integral statistics of obtained IBP reduction.}
\begin{subtable}{.5 \linewidth}
\caption{planar form factor}
\centering
\begin{tabular}{c | c c c}
\hline
\# props & $s=0$ & s=1 &$ s=2$ \\ [0.5ex]
\hline
12 & 8 & 6 & 0 \\
11 & 18 & 2 & 0 \\
10 & 43 & 9 & 0 \\
9 & 49 & 1 & 0 \\
8 & 51 & 4 & 1 \\
7 & 25 & 0 & 0 \\
6 & 8 & 0 & 0 \\
5 & 0 & 0 & 0 \\
\hline
sum & 203 & 22 & 1 \\ [1ex]
\end{tabular}
\end{subtable}%
\begin{subtable}{.5\linewidth}
\caption{non-planar form factor}
\centering
\begin{tabular}{c | c c c}
\hline
\# props & $s=0$ & s=1 &$ s=2$ \\ [0.5ex]
\hline
12 & 10 & 10 & 1 \\
11 & 13 & 3 & 0 \\
10 & 34 & 10 & 0 \\
9 & 29 & 1 & 0 \\
8 & 32 & 3 & 1 \\
7 & 13 & 0 & 0\\
6 & 7 & 0 & 0 \\
5 & 1 & 0 & 0 \\
\hline
sum & 139 & 27 & 2 \\ [1ex]
\end{tabular}
\end{subtable}
\label{table:countingReduze}
\end{table}
\section{Comparison to counting masters using algebraic methods}
The number of master integrals may be counted by exploring only the analytic structure of the integral topology, without explicitly solving the IBP relations. This method was proposed in \cite{Lee:2013hzt}, based on earlier work \cite{Baikov:2005nv}. The basic idea is that, for a given topology of $m$ propagators, the number of master integrals can be obtained by counting the number of proper critical points of the sum of first and second Symanzik polynomials
\begin{equation}
G(\vec\alpha) = U(\vec\alpha) + F(\vec\alpha) \,,
\end{equation}
where the proper critical points are defined by
\begin{equation}
\label{eq:critical-point-condition}
\frac{\partial G}{\partial \alpha_i} = 0 \ \ \ (i = 1, \ldots, m) \qquad
\mbox{and} \qquad G \neq 0.
\end{equation}
Such critical points can be counted efficiently using the Gr\"obner basis technique. This procedure has been implemented in the {\tt Mint} package \cite{Lee:2013hzt}.
We have applied this algorithm to the four-loop form factor integrals, swapping in different approaches to perform some of the steps that the {\tt Mint} package cannot perform in its current incarnation. We refer the reader to \cite{Boels:2015yna, Lee:2013hzt} for further details. The obtained masters are classified in Table \ref{table:counting_Mint} according to the number of propagators and the power of propagators. A crosscheck with the reduction of {\tt Reduze} shows that all 280 master integrals are independent.
\begin{table}[t]
\caption{Master integral statistics of 280 Mint basis.}
\centering
\begin{tabular}{c | c c c c c c c c}
\hline
\# props & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
\textrm{all simple } & 1 & 8 & 25 & 48 & 52 & 58 & 32 & 20 \\
\textrm{simple + one double} & 0 & 0 & 1 & 5 & 1 & 12 & 3 & 14 \\
\hline
\end{tabular}
\label{table:counting_Mint}
\end{table}
Interestingly, while the counting based on the {\tt Mint} method indeed provides a set of independent basis integrals, we find that the reduction of {\tt Reduze} tends to include more basis integrals. This discrepancy may be due to extra relations beyond IBP, or possible subtle critical points at infinity which are not yet resolved by {\tt Mint} method. It would be very interesting to understand this point. In any case, we expect that the {\tt Mint} basis should provide a lower bound of the number of independent basis. Since the method based on {\tt Mint} only relies on the topologies of the given integrals and applies to arbitrary numerators, the results are expected to apply to any theory, including QCD.
\section{Rational IBP relations}
A particular problem that arises from the explicit reduction is the expansion in the dimensional regularisation parameter $\epsilon$. Typically a topology within the form factor is given as a sum over master integrals as
\begin{equation}
\textrm{top}_i = \sum c_{i,j} \textrm{mast}_j \,,
\end{equation}
where the coefficients $c_{i,j} $ are functions of $\epsilon$. Since the four-loop topologies are expected in general to diverge as $\frac{1}{\epsilon^8}$ (see equation \ref{eq:expIR}) and for the four-loop cusp the term of order $\frac{1}{\epsilon^2}$ is needed, both prefactors as well as master integrals need to be calculated to high order in the $\epsilon$ expansion. Although the prefactor has an exact expansion, due to mixing of the two expansion the precision in each of the expansion coefficients in the master integral is important. We note that in many cases the coefficients $c_{i,j}$ are such that the integrals $ \textrm{mast}_j$ have to be expanded to the seventh order, regardless of at which order the expansion of these master integrals starts. As a rule of thumb, the coefficients of the prefactor expansion increase an order of magnitude in size with each additional expansion step, making the mixing highly non-trivial. One potential resolution to this problem is to study IBP relations which only contain rational numbers and not $\epsilon$.
One way to obtain such a set will be explained here. All relations with rational coefficients will be obtained for a set of form factor integrals for which a general IBP reduction with dimension dependent conditions has been found previously. In essence, we find the rational relations contained in a given IBP reduction. There are other ways of aiming for such a set ({\tt Reduze} offers a choice of dimension free IBP relations for instance), but using an already available IBP reduction will almost certainly be faster. Even more generally, one could employ the strategy below essentially unchanged to use integer numerics in IBP reductions. This is related to but simpler than the algorithm proposed in \cite{vonManteuffel:2014ixa}. Note that the strategy there aims to yield the full reduction, not just a rational reduction.
\subsection{Algorithm}
The parameter $q^2$ can be set to a fixed value without loss of generality. The only variable for the integrals left is then the dimensional regularisation parameter $\epsilon$. Proceed as follows:
\begin{itemize}
\item write the IBP reduction for all of the integrals in the given set.
\item first remove pairs of integrals: those that have the same IBP reduction. This step requires a minor basis choice (see below).
\item one is left with $a$ integrals, expressed in terms of $b$ master integrals. Turn this into a ($b$ by $a$) matrix. Its coefficients dependent only on $\epsilon$.
\item evaluate this matrix for random integer values of $\epsilon$ $x$ times, such that $x b >> a$.
\item construct the natural $(x \cdot b$ by $a$) matrix by adjoining rows. This matrix is purely rational by construction.
\item compute the null-space of this matrix by computer algebra: these are the sought-for relations.
\item check these relations by using the full dimension-dependent IBP reduction.
\end{itemize}
To obtain an IBP reduction, one always needs to choose an ordering. We choose to order (partially) first by numerator power, then by propagator power and finally by topology number. Lower values are considered simpler. Rearranging the obtained relations according to the ordering (most difficult first) and row reducing the resulting matrix allows to read off an explicit rational IBP reduction with the chosen ordering.
Note that the step removing the duplicates is strictly speaking not essential. Typically, these duplicate cases involve graph symmetries. In cases with many graph symmetries this step is necessary to reduce matrices to a manageable size in the remaining parts of the algorithm. For instance, for all $10$ line integrals across the $34$ topologies, there are $11 \cdot 12 \cdot 34= 4488$ different integrals. After duplicate removal, there are only $389$. These obey $31$ additional rational relations found by the above algorithm, giving a set of $358$ (rational IBP) master integrals with $10$ lines.
\subsection{Application of rational IBPs to color-planar part of the form factor at four loops}
We have generated all rational relations between the integrals for all of the $34$ topologies. An explicit map to master integrals was obtained using the choice outlined above. This was applied first to the color-planar part of the form factor proportional to $N_c^4$. An immediate problem now is that the result contains, in general, a very large number (typically $>500$) of rational master integrals with, again in general, very complicated looking fractional coefficients (denominators and numerators with many digits). This is likely due to a non-optimal basis choice.
Some ad-hoc rules to find a better basis choice are:
\begin{itemize}
\item reduction rules with a very high (>125) number of different master integrals must be discarded.
\item integrals from topologies with vanishing color-factor should not appear in the end result. Use duplication relations to eliminate these.
\item any replacement which eliminates several integrals at once out of the sum over all topologies is good.
\item integrals with the largest rational fractions should be reverted first.
\end{itemize}
One can scan algorithmically for good replacements by selecting a number of integrals and checking in which reduction they appear. Then, one can check the coefficients to test if reverting this relation will decrease the number of integrals in the full result. Taking into account the relations with high numbers of rational master integrals can in rare cases significantly reduce resulting form factor sizes: this happens for instance when two of these relations can be combined into a simpler one\footnote{In the one case known, the two high number of integral expansions contained a number of terms which differed by one unit.}.
Applying these rules results in a new form of the color-planar part of the four-loop form factor in $\mathcal{N}=4$ with order $\sim 70$ integrals, with coefficients which are small ($1<|x|<100$) integers. The size of the rational IBP reduction is such that it can be used on a laptop, instead of having to be kept and manipulated on a large cluster. This is conducive to experimentation. A drawback is that the remaining set of master integrals now contains quite a number of integrals with quadratic numerators, which are known to be hard to integrate. Hence this approach may not be very fruitful for direct integration without further input. This result is certainly more general: when using or aiming for rational IBP relations, there is no a-priori way of determining the 'power' of the reduction: there is no guarantee the computational complexity of a problem will decrease by solving the set of rational IBP relations.
\section{Conclusion}
In this contribution progress has been discussed toward computing the four-loop cusp anomalous dimension in the special toy model theory of $\mathcal{N}=4$ super Yang-Mills. Current status is that a basis of master integrals after IBP reduction has been obtained, which was cross-checked using algebraic methods, and a counting from the latter methods was argued to apply to the QCD case as well. Significant challenges remain, especially toward integration. A secondary approach through the construction and solution of rational IBP relations was described. For this the known IBP reduction was used to obtain a sub-reduction. Although these techniques yield results which are much more easily manipulated, the reduction in computational complexity is certainly less than with the use of full IBP relations and will require more input for integration. Still, they represent an interesting direction to pursue further, especially for the more complicated classes of integrals which appear in the QCD form factor.
\acknowledgments
The authors would like to thank the organisers of the Loops and Legs 2016 workshop for the opportunity to present this work. It is a pleasure to thank Andreas von Manteufel, Volodya Smirnov, Johannes Henn and Sven-Olaf Moch for discussions. This work was supported by the German Science Foundation (DFG) within the Collaborative Research Center 676 "Particles, Strings and the Early Universe".
\bibliographystyle{JHEP}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,487
|
Srećko Bogdan (born 5 January 1957) is a Croatian former professional footballer who played a defender. He is now a youth coach in NK Inter Zaprešić.
Club career
Bogdan was born in Mursko Središće, Croatia, FPR Yugoslavia. He started his career in his home town with NK Rudar Mursko Središće, where he spent three years before moving to MTČ Čakovec and starting his senior career in 1973. He spent one season and a half with Čakovec before transferring first to Dinamo Zagreb and later to Karlsruhe. He is currently in third place in Dinamo Zagreb's all-time list of appearances for the club, with a total of 595 appearances in which he scored 125 goals. He played for Dinamo Zagreb between January 1975 and June 1985, after which he moved to Karlsruhe in the German 2. Bundesliga. After two years at Karlsruhe, he managed promotion to the Bundesliga with the club and subsequently made 169 appearances for the club in the league over the following six seasons, scoring nine goals. He retired from playing in June 1993.
International career
In his international career, Bogdan played for both former Yugoslavia and Croatia, and won 11 caps for Yugoslavia and two for Croatia, scoring one goal for the latter. He made his debut for Yugoslavia in a January 1977 friendly match away against Colombia, coming on as a 30th-minute substitute for Franjo Vladić. The games for Croatia in 1990 and 1991 were unofficial, since the country was still officially part of Yugoslavia.
Managerial career
Bogdan worked as a coach at Karlsruhe's youth academy between 1993 and 1996, following his retirement as a player at the club. He went on to work as an assistant coach at the club's first team between 1996 and 2001.
The first club where he was appointed head coach was Croatian side Inter Zaprešić, where he was in charge in 2005 and 2006. He was then appointed head coach at Segesta Sisak in January 2007, staying with the club until October 2008. In October 2009, he was appointed head coach at Međimurje, signing a contract until the end of the 2009–10 season. He was sacked on 2 April 2010, when the club found themselves on the brink of the relegation zone after a streak of five games without a win. He later managed Savski Marof and became academy boss at Inter Zaprešić.
Other work
During the 2006 FIFA World Cup, Bogdan commented several of the tournament's matches as a co-commentator for the Croatian Radiotelevision (HRT).
References
External links
Srećko Bogdan at the Serbia national football team website
1957 births
Living people
People from Mursko Središće
Association football defenders
Yugoslav footballers
Yugoslavia international footballers
Croatian footballers
Croatia international footballers
Dual internationalists (football)
Mediterranean Games gold medalists for Yugoslavia
Competitors at the 1979 Mediterranean Games
Mediterranean Games medalists in football
GNK Dinamo Zagreb players
Karlsruher SC players
Yugoslav First League players
Bundesliga players
2. Bundesliga players
Yugoslav expatriate footballers
Croatian expatriate footballers
Expatriate footballers in West Germany
Yugoslav expatriate sportspeople in West Germany
Expatriate footballers in Germany
Yugoslav expatriate sportspeople in Germany
Croatian expatriate sportspeople in Germany
Croatian football managers
VfR Mannheim managers
NK Inter Zaprešić managers
HNK Segesta managers
NK Međimurje managers
Croatian expatriate football managers
Expatriate football managers in Germany
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,689
|
// vtkDLGDlg.cpp : implementation file
//
#include "stdafx.h"
#include "vtkDLG.h"
#include "vtkDLGDlg.h"
#include "vtkCallbackCommand.h"
#ifdef _DEBUG
#define new DEBUG_NEW
#endif
// CAboutDlg dialog used for App About
class CAboutDlg : public CDialog
{
public:
CAboutDlg();
// Dialog Data
enum { IDD = IDD_ABOUTBOX };
protected:
virtual void DoDataExchange(CDataExchange* pDX); // DDX/DDV support
// Implementation
protected:
DECLARE_MESSAGE_MAP()
};
CAboutDlg::CAboutDlg() : CDialog(CAboutDlg::IDD)
{
}
void CAboutDlg::DoDataExchange(CDataExchange* pDX)
{
CDialog::DoDataExchange(pDX);
}
BEGIN_MESSAGE_MAP(CAboutDlg, CDialog)
END_MESSAGE_MAP()
// CvtkDLGDlg dialog
CvtkDLGDlg::CvtkDLGDlg(CWnd* pParent /*=NULL*/)
: CDialog(CvtkDLGDlg::IDD, pParent)
{
m_hIcon = AfxGetApp()->LoadIcon(IDR_MAINFRAME);
this->pvtkMFCWindow = NULL;
// set data reader to null
this->pvtkDataSetReader = NULL;
// Create the the renderer, window and interactor objects.
this->pvtkRenderer = vtkRenderer::New();
// Create the the objects used to form the visualisation.
this->pvtkDataSetMapper = vtkDataSetMapper::New();
this->pvtkActor = vtkActor::New();
this->pvtkActor2D = vtkActor2D::New();
this->pvtkTextMapper = vtkTextMapper::New();
this->ptBorder = CPoint(0,0);
}
void CvtkDLGDlg::DoDataExchange(CDataExchange* pDX)
{
CDialog::DoDataExchange(pDX);
}
BEGIN_MESSAGE_MAP(CvtkDLGDlg, CDialog)
ON_WM_SYSCOMMAND()
ON_WM_PAINT()
ON_WM_QUERYDRAGICON()
//}}AFX_MSG_MAP
ON_WM_DESTROY()
ON_WM_SIZE()
ON_BN_CLICKED(ID_LOADFILE, OnBtnLoadFile)
ON_BN_CLICKED(ID_RESETSCENE, OnBtnResetScene)
END_MESSAGE_MAP()
// CvtkDLGDlg message handlers
void CvtkDLGDlg::ExecutePipeline()
{
if (pvtkDataSetReader) // have file
{
this->pvtkDataSetMapper->SetInputConnection(pvtkDataSetReader->GetOutputPort());
this->pvtkActor->SetMapper(this->pvtkDataSetMapper);
this->pvtkTextMapper->SetInput(pvtkDataSetReader->GetFileName());
this->pvtkTextMapper->GetTextProperty()->SetFontSize(12);
this->pvtkActor2D->SetMapper(this->pvtkTextMapper);
this->pvtkRenderer->SetBackground(0.0,0.0,0.4);
this->pvtkRenderer->AddActor(this->pvtkActor);
this->pvtkRenderer->AddActor(this->pvtkActor2D);
}
else // have no file
{
this->pvtkTextMapper->SetInput("Hello World");
this->pvtkTextMapper->GetTextProperty()->SetFontSize(24);
this->pvtkActor2D->SetMapper(this->pvtkTextMapper);
this->pvtkRenderer->SetBackground(0.0,0.0,0.4);
this->pvtkRenderer->AddActor(this->pvtkActor2D);
}
this->pvtkRenderer->ResetCamera();
}
static void handle_double_click(vtkObject* obj, unsigned long,
void*, void*)
{
vtkRenderWindowInteractor* iren = vtkRenderWindowInteractor::SafeDownCast(obj);
if(iren && iren->GetRepeatCount())
{
AfxMessageBox("Double Click");
}
}
BOOL CvtkDLGDlg::OnInitDialog()
{
CDialog::OnInitDialog();
// Add "About..." menu item to system menu.
// IDM_ABOUTBOX must be in the system command range.
ASSERT((IDM_ABOUTBOX & 0xFFF0) == IDM_ABOUTBOX);
ASSERT(IDM_ABOUTBOX < 0xF000);
CMenu* pSysMenu = GetSystemMenu(FALSE);
if (pSysMenu != NULL)
{
CString strAboutMenu;
strAboutMenu.LoadString(IDS_ABOUTBOX);
if (!strAboutMenu.IsEmpty())
{
pSysMenu->AppendMenu(MF_SEPARATOR);
pSysMenu->AppendMenu(MF_STRING, IDM_ABOUTBOX, strAboutMenu);
}
}
// Set the icon for this dialog. The framework does this automatically
// when the application's main window is not a dialog
SetIcon(m_hIcon, TRUE); // Set big icon
SetIcon(m_hIcon, FALSE); // Set small icon
// adjust dialog & window size
this->pvtkMFCWindow = new vtkMFCWindow(this->GetDlgItem(IDC_MAIN_WND));
// get double click events
vtkCallbackCommand* callback = vtkCallbackCommand::New();
callback->SetCallback(handle_double_click);
this->pvtkMFCWindow->GetInteractor()->AddObserver(vtkCommand::LeftButtonPressEvent, callback, 1.0);
callback->Delete();
CRect cRectVTK;
this->pvtkMFCWindow->GetClientRect(&cRectVTK);
CRect cRectClient;
GetClientRect(&cRectClient);
this->ptBorder.x = cRectClient.Width() - cRectVTK.Width();
this->ptBorder.y = cRectClient.Height() - cRectVTK.Height();
// set the vtk renderer, windows, etc
this->pvtkMFCWindow->GetRenderWindow()->AddRenderer(this->pvtkRenderer);
// execute object pipeline
ExecutePipeline();
return TRUE; // return TRUE unless you set the focus to a control
}
void CvtkDLGDlg::OnSysCommand(UINT nID, LPARAM lParam)
{
if ((nID & 0xFFF0) == IDM_ABOUTBOX)
{
CAboutDlg dlgAbout;
dlgAbout.DoModal();
}
else
{
CDialog::OnSysCommand(nID, lParam);
}
}
// If you add a minimize button to your dialog, you will need the code below
// to draw the icon. For MFC applications using the document/view model,
// this is automatically done for you by the framework.
void CvtkDLGDlg::OnPaint()
{
if (IsIconic())
{
CPaintDC dc(this); // device context for painting
SendMessage(WM_ICONERASEBKGND, reinterpret_cast<WPARAM>(dc.GetSafeHdc()), 0);
// Center icon in client rectangle
int cxIcon = GetSystemMetrics(SM_CXICON);
int cyIcon = GetSystemMetrics(SM_CYICON);
CRect rect;
GetClientRect(&rect);
int x = (rect.Width() - cxIcon + 1) / 2;
int y = (rect.Height() - cyIcon + 1) / 2;
// Draw the icon
dc.DrawIcon(x, y, m_hIcon);
}
else
{
CDialog::OnPaint();
}
}
// The system calls this function to obtain the cursor to display while the user drags
// the minimized window.
HCURSOR CvtkDLGDlg::OnQueryDragIcon()
{
return static_cast<HCURSOR>(m_hIcon);
}
void CvtkDLGDlg::OnDestroy()
{
// delete data
if (this->pvtkDataSetReader) this->pvtkDataSetReader->Delete();
// Delete the the renderer, window and interactor objects.
if (this->pvtkRenderer) this->pvtkRenderer->Delete();
// Delete the the objects used to form the visualisation.
if (this->pvtkDataSetMapper) this->pvtkDataSetMapper->Delete();
if (this->pvtkActor) this->pvtkActor->Delete();
if (this->pvtkActor2D) this->pvtkActor2D->Delete();
if (this->pvtkTextMapper) this->pvtkTextMapper->Delete();
if (this->pvtkMFCWindow)
{
delete this->pvtkMFCWindow;
}
CDialog::OnDestroy();
}
void CvtkDLGDlg::OnSize(UINT nType, int cx, int cy)
{
CDialog::OnSize(nType, cx, cy);
if (::IsWindow(this->GetSafeHwnd()))
{
if (this->pvtkMFCWindow)
{
cx -= ptBorder.x;
cy -= ptBorder.y;
this->GetDlgItem(IDC_MAIN_WND)->SetWindowPos(NULL, 0, 0, cx, cy, SWP_NOACTIVATE | SWP_NOZORDER | SWP_NOMOVE);
this->pvtkMFCWindow->SetWindowPos(NULL, 0, 0, cx, cy, SWP_NOACTIVATE | SWP_NOZORDER | SWP_NOMOVE);
}
}
}
void CvtkDLGDlg::OnBtnLoadFile()
{
static char BASED_CODE szFilter[] = "VTK Files (*.vtk)|*.vtk|All Files (*.*)|*.*||";
CFileDialog cFileDialog(TRUE, NULL, NULL, OFN_HIDEREADONLY | OFN_OVERWRITEPROMPT, szFilter);
if (cFileDialog.DoModal() == IDOK)
{
// remove old actors
this->pvtkRenderer->RemoveActor(this->pvtkActor);
this->pvtkRenderer->RemoveActor(this->pvtkActor2D);
// read new data
if (!this->pvtkDataSetReader)
this->pvtkDataSetReader = vtkDataSetReader::New();
this->pvtkDataSetReader->SetFileName(cFileDialog.GetPathName());
// execute object pipeline
ExecutePipeline();
// update window
if (this->pvtkMFCWindow)
this->pvtkMFCWindow->RedrawWindow();
}
}
void CvtkDLGDlg::OnBtnResetScene()
{
// remove old actors
this->pvtkRenderer->RemoveActor(this->pvtkActor);
this->pvtkRenderer->RemoveActor(this->pvtkActor2D);
// delete data
if (this->pvtkDataSetReader)
{
this->pvtkDataSetReader->Delete();
this->pvtkDataSetReader = NULL;
}
// execute object pipeline
ExecutePipeline();
// update window
if (this->pvtkMFCWindow)
this->pvtkMFCWindow->RedrawWindow();
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,315
|
8 May 2020 By Seth Miller 1 Comment
Total flights down by 40%. Passenger count down by 90%. Load factor for the month of just 8%. Needless to say, Cape Air's statistics for April 2020 are pretty awful, just like the rest of the aviation world. But the airline has some potential good news to consider as it looks to a slow summer and beyond.
We're going to learn from this. We're going to go on from it. But we're not going to go on by turning the clock back 100 years to when we weren't connected. We're going to figure out how to connect to each other in ways that are safe but equally meaningful.
– Cape Air CEO Dan Wolf
Fleet Refresh Continues
The introduction of the Tecnam P2012 into commercial service in February went very well and Cape Air remains optimistic on its future in the fleet. The carrier is up to nine of the type now in its fleet an still expects to have 20 on property by the end of 2020. The carrier just passed back above 300 passengers on a single day, a nice milestone to surpass, but CEO Dan Wolf notes in an interview with AirInsight "we should be flying between 1,500 and 2,000 passengers a day, so that still means we're down over 80%."
A Cape Air Tecnam P2012 (Image courtesy of the airline)
The carrier also notes that it continues to work with Tecnam "to make adjustments and improvements as needed." Cape Air expects that the type will "continue to evolve to be an even greater aircraft in the months and years ahead." The carrier specifically notes speed, payload and fuel efficiency as three areas it expects to see improve over time.
A PreCheck win
Cape Air's participation in the TSA PreCheck program is of limited value for passengers connecting onwards at Boston on a non-JetBlue flight. Turns out there were systems limitations related to passing the PreCheck data through from the Amadeus operations back-end to partner carriers. The carrier and Amadeus cooperated to update its services, now allowing for PreCheck status to carry through to the onward boarding pass and reservation details.
With CARES Act funding secured and exemption from operating to New York City, Cape Air has at least some help in making it through the summer without significant staffing changes. But 8% load factors are not going to help any airline stay in business for long.
But Cape Air might get a boost thanks to action – and inaction – from one of its larger airline partners, JetBlue.
Yesterday JetBlue announced it would not resume its seasonal service to Portland, Maine (PWM). Cape Air also serves Portland from its Boston hub, including offering interline service with JetBlue, and operating out of the JetBlue terminal area. Without the JetBlue jet service operating some of those passengers could flow through to the Cape Air connection.
Moreover, with the schedule changes slated to load this weekend for JetBlue there are unconfirmed reports that Cape Air could add connectivity to other small JetBlue destinations near Boston. These flights would operate specifically to allow JetBlue to meet the CARES Act requirements of maintaining service in smaller markets while also drastically reducing costs by flying an empty Cessna 402 rather than an empty Embraer E190 or Airbus A320. Worcester was suggested as one likely market (yes, really, ORH-BOS) while Albany, NY (ALB) or Hartford, CT (BDL) could also potentially work. That would be a very creative approach to the service obligation requirements, even as the airlines continue to negotiate with the DOT in reducing the mandated flight levels.
Some very short-haul options where Cape Air could support its partner JetBlue
Map generated by the Great Circle Mapper - copyright © Karl L. Swartz.
Long-term Outlook
What is the longer range view for Cape Air? After September the answer is very, very unclear.
Wolf believes there is pent up demand, "not just for individuals to travel and go to new places… but because [Americans] have built up that network over the past decades and generations of really relying on that mobility to have the diversity and inclusion and global economy that is the future."
How that translates into passenger bookings remains to be seen, especially as Cape Air serves many smaller airports, the types of market that are not expected to see a quick bump in demand.
Filed Under: Airplanes and Airports, Ground Services
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,900
|
import client from "../client";
export default {
factorioVersion: async () => {
const response = await client.get('/api/server/facVersion');
return response.data;
},
status: async () => {
const response = await client.get('/api/server/status');
return response.data;
},
stop: async () => {
const response = await client.get('/api/server/stop');
return response.data;
},
start: async (ip, port, savefile) => {
const response = await client.post('/api/server/start', {
bindip: ip,
savefile,
port
});
return response.data;
},
kill: async () => {
const response = await client.get('/api/server/kill');
return response.data;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,288
|
You understand that all advertisement content being published, any damage or loss occur to the advert writer or other people, both directly or not, are sole responsibility of the person from whom such content originated.
You agree not to use the service to upload any content that is unlawful, harmful, threatening, abusive, harassing, tortious, defamatory, vulgar, obscene, libelous, invasive of another's privacy, hateful, or racially, ethnically or otherwise objectionable to Indonesian law, moral and culture.
Attempt to upload any advertisement more than three times a day; make trial advertisement, unclear and unspecified messages, upload the same advert on different categories (inappropriate choosing advert category) and other HTML codes will automatically deleted from this website.
The advertisement that violate those above policies will automatically be deleted without any warning.
Below buttons to operate the advert can be found on the feature menu at the top right corner of this website.
Contains information about what IklanJKT.com is along with how to use method (it is on this page).
Complete all the forms. Those have to be completely filled out except the telephone number and website (both are optional). Please fill in the password and password confirmation form with exactly identical characters, and then click on the confirmation button.
Advert confirmation will appear next to re-check the upload content. Please click on send button after you finish re-checking all the advert content. After that your advertisement will automatically appear on that very moment.
If your advertisement is successfully appear on the website, you will receive announcement e-mail from info@IklanJKT.com informing your box number and password which can any time be used to edit or delete your advert.
You can edit or change you advert by previously fill in the box number and password which can be found in the announcement e-mail from info@IklanJKT.com.
Therefore please keep that information confidential (for the box number, you may also check it on your advert which already published on the website).
You may delete your advert by previously fill in the box number and password which can be found in the announcement e-mail from info@IklanJKT.com.
Keyword : Fill in this space with your searched word. You may fill in more than one word by using SPACEBAR. Example: internet modem.
Search in : You can find the key word on the chosen section as you wish. For example, you may choose the box number, sender, title, advert content or all sections at the same time. Attention : box number has to be filled in by numeric character.
Search Category : You can find the key word at a certain category or even more at all categories as your wish.
Search : This button shows your searching result and the key word will be on in the bold form.
Reply button is provided on every published advert. Here the users and other visitors can reply to the advert writer by previously filling in the reply form that contains name, e-mail address, subject and reply message.
Note: users are not allowed to reply more than 5 times a day to certain advert writer, this will cause your e-mail be blocked.
Send button provided on each publishing will enable users to inform certain advert to the users' friend by previously filling in the form contains your and your friend's names, e-mail addresses (one friend only) and which advert to be informed.
Note: users and advert writers are not allowed to inform certain advert with the same topic to their friends more than 5 times a day, this will cause your e-mail be blocked.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,638
|
By Tsahi Levent-Levi
The State of WebRTC and Streaming Media 2018
WebRTC has been with us for the past 6 years, and it has certainly grown in adoption and popularity during that time. But how and where, exactly, does WebRTC fit into the world of media streaming? And while we're at it, what should we expect of it in 2018?
WebRTC is a combined IETF and W3C specification that is being rolled out as part of HTML5. In essence, WebRTC enables sending voice, video, and any other arbitrary data directly across browsers in real time. You write a bit of JavaScript code, add a few relay servers to be used when needed, and you've got yourself a video chat service.
The focus of WebRTC and the developers using it is usually on handling remote video chat meetings, either one-on-one or group calls. That is also the primary focus of web browsers in their support of WebRTC. So how did WebRTC become so interesting in the media streaming space in the past year?
There are several contributing factors here:
Flash is dying, and this is being accelerated by modern browsers limiting its use and availability, along with Adobe's plan to reach Flash's end-of-life by the end of 2020.
MPEG-DASH and HLS implementations usually come with latency limitations. More often than not, a delay of 10 seconds or more is apparent with these streaming technologies.
Live and interactive are becoming more and more important. We see it with Facebook Live, Microsoft Mixer, and other streaming services.
Broadcasters are looking to share and mix their streams directly in the browser with no need to install anything.
Growing video consumption is straining streaming servers and increasing the network costs of broadcasters.
The introduction of H.264 as a mandatory-to-implement codec in WebRTC and its availability across all modern browsers makes WebRTC easier to use for existing streaming services.
There are a few places where you can find WebRTC in media streaming services these days, and they use WebRTC quite differently from one another. To understand how these came to be, it is important to understand how WebRTC works in the context of media streaming, as shown in Figure 1.
Figure 1. A simple schematic showing WebRTC in a media streaming environment.
The browser will use a signaling channel toward the application itself. The application makes the decision on how and where to connect the browser using WebRTC. For each use case, the application and its behavior will be very different.
WebRTC's real-time audio and video can be used in front of a CDN or a media server, for both sending and receiving media. This will be used for low-latency streaming use cases.
Last but not least, WebRTC's data channel is used to create ad-hoc peer-to-peer (P2P) CDN connections directly between browsers. This can reduce buffering and costs for broadcasters.
With that in mind, let's see how this fits into five different media streaming scenarios
1. Broadcasting Without Installations
The simplest way and place for WebRTC to work in media streaming is by enabling broadcasters to share their streams without an application installed.
WebRTC is the only mechanism available today to allow a browser to share its camera and microphone. Flash is no longer an option. Plugins are losing popularity and are hard to maintain. People are left with either installing a dedicated application or using WebRTC inside the browser.
In the past, this approach required transcoding between VP8 (the video codec available in WebRTC) and H.264. Today, this is no longer necessary, since browsers support H.264 encoding in WebRTC.
Figure 2.Connecting to a CDN to deliver a live stream via HLS using WebRTC
Figure 2 illustrates how this evolved architecturally. Connecting to a CDN with a live media stream required Real-Time Messaging Protocol (RTMP) support, which translated into using an additional gateway component. Kurento, an open source WebRTC media server, was widely used for that, and recently, Wowza and Red5 Pro started offering similar capabilities of connecting WebRTC to RTMP (and both are also offering low-latency viewing). Nanocosmos took a slightly different approach, allowing broadcasters to connect via WebRTC, but streaming on the other end using HLS that is optimized for low latency.
2. Interviewing
This takes the single-broadcaster approach to the next level.
As video becomes more interactive, so do the ways of creating the video itself. Certain events, such as webinars and news-related video streams, need the ability to interview people live and broadcast the interview as a single unit to viewers. But the broadcaster and his hosts might not be located in the same room, so they need to be on a live video chat that is then mixed and streamed toward the viewers.
Figure 3. A typical setup for a multi-person video interview that is mixed and streamed to viewers
The media server in Figure 3 now isn't only a media gateway. It receives real-time media streams from all the active participants (the broadcaster and the hosts), manages the conference call between them, and also mixes all inputs to create a single stream that can be digested by a video CDN—most likely in RTMP format.
YouNow was probably one of the first to offer such a user experience. Recently, Facebook Live as well as Instagram added support for adding a guest to a live-stream event.
In the enterprise, it is quite common to host webinars that have multiple hosts joining from different locations. In recent years, webinars have been shifting from voice toward video. This has brought with it the need to enable these types of interview scenarios. At the same time, it has increased the requirement for lower latencies for viewers, which brings us to the next use case.
3. Low-Latency Live Streaming
In a way, the first two use cases can be seen as a prelude to this one—achieving low-latency live streaming in a post-Flash world. Since WebRTC was designed and implemented from the ground up for real-time communications, it is quite capable of offering low-latency live streaming.
In the past year or two, we've seen companies starting to employ WebRTC for such use cases. The vendors solving this technical problem are tackling it from two different directions:
Taking the video conferencing approach and increasing it to support larger groups via a selective forwarding unit (SFU), a type of media router.
Starting from the existing RTMP streaming technologies and enhancing them to support WebRTC as well.
Both directions are valid, and have different technical benefits and challenges. At the end of the day, both need to tackle the issue of scale: How do you take a single stream and broadcast it to a large number of viewers? Even if you assume a single media server can broadcast to thousands of viewers, what happens when the stream in question is requested by even more viewers?
Figure 4. A single broadcaster's stream is cascaded from one media server to another before reaching its destination.
The solution is cascading, which is quite similar to how CDNs work today. Figure 4 shows such an architecture, where a single broadcaster's stream gets cascaded from one media server to another before reaching its destination.
This is what Beam did with an infrastructure of cascaded SFUs prior to being acquired by Microsoft and rebranded as Mixer. This is how all the gray zones of the internet that offer auctions, gambling, and porn are tackling the issue of live streaming at scale.
2018 will see more technology companies offering such solutions and products, as well as a CDN or two that will offer such commercial services.
The State of WebRTC and Low-Latency Streaming 2019
It's not a standard yet, but that will likely change. Here's a detailed look at the state of WebRTC, the project that could finally deliver instantaneous video streaming at scale.
Developers Pay the Pioneer Tax for WebRTC Live Streaming
There's a cost to being cutting-edge, and for low-latency live video streaming that involves learning WebRTC and accepting limited browser support.
Re-Engineering Real-Time Video Delivery
A Stanford University research team has created an architecture called Salsify that might offer a better way to deliver video for real-time applications
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 761
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Founders Garden Wedding Inspiration. Nestled in the heart of Athens, Ga., is the Founders Garden, where this styled wedding inspiration comes to life. But natural beauty aside, this styled shoot takes on a whole new aura thanks to the anticipation, joy and excitement captured in the experience.
The romantic paper suite features a spectrum of cool, relaxed colors. Modern script and watercolor designs round out the collection.
The couple wrote their vows in a gorgeous vow book to read from at the ceremony and cherish as a keepsake long after.
The bouquet features light pink roses, lilies and assorted greenery tied with white ribbons.
The florals match the delicate aesthetic of the bride's lace gown. With short sleeves, an open back and a flowing silhouette, the bride is simply ethereal. She's styled with a dewy, natural palette that is light and soft.
The groom is dashing in a classic tuxedo complete with a bow tie and boutonniere to match the bride's bouquet.
The ceremony is as authentic as one can imagine with an intimate gathering of guests looking on towards the happy couple. The bride and groom exchange vows, wash each other's feet, share laughs and a kiss.
They celebrate with their close friends and family over a meal shared from a family-style table.
The tablescape includes green glassware, white china and gold flatware. The metallic matches the taper holders for a cohesive decorative touch.
Delicious grapes, cheese and crackers make an elegant offering for this small group of guests as they dine under the luminosity of string lights.
The couple take a spin and admire one another under the gentle Georgia sunshine. And the expressions on their faces certainly say it all.
This styled inspiration aims to emphasize the emotion a couple experiences on their big day. It's the commitment that makes the day truly splendid. While the decor, fashion and food are certainly nice, it's best enjoyed by those who see the beauty in the experience.
Photo:Hannah ForsbergStyling and Creative Direction:Rachel Slauer WeddingsFloral:Gold and BloomHMU: Erin RyserGown:Fabulous Frocks of AtlantaTux:Modern GentRentals: Oconee Event RentalsRings:GoodstoneVow Books: Wedding Story WriterCalligraphy:Quill and Co.
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{
"redpajama_set_name": "RedPajamaC4"
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| 6,443
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Email ncfarmgarden@gmail.com for address and directions.
By appointment only. This is to pick up goats.
Everything else is mailed. Sorry no pick up at the farm.
Also vacation rental home in Appalachia, Nantahala, North Carolina.
The red dot is where Topton is in North Carolina.
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{
"redpajama_set_name": "RedPajamaC4"
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Q: How to properly render a collection in rails using Ajax I find a solution using JSON but is not enough for why I need. The solution with JSON is here: Render with JSON.
I need to render a specific partial for each object from a collection using Ajax so I can show all the functionalities I made in the partial and use certain validations are no possible to do with JSON. The purpose is to dynamically do queries to the DB and render the results of the queries with each change in my search form.
this is my function in the controller
def clientsjson
@search = Client.search(params[:q])
@clients = @search.result
end
To render the partial I need (_client.html.erb), normally, I only use this:
<%= render @clients %>
But know I need to render dynamically with Ajax.
The Ajax function I use to detect the change in the form is this:
$(document).ready(function() {
$( ".searchupdate" ).change(function() {
$.getJSON("/client/clientsjson?"+$('#client_search').serialize(), function (data) {
//*Here I use JSON to create a text and print HTML and works*//
$('#clientList1').html('');
$('#clientList1').empty().append($ul);
});
});
});
The index where I print the results (only the columns):
<div class="row">
<div class="col-xs-4" >
<%= render "searchform" %>
</div>
<div class="col-xs-8" >
<div id="Auctionlist1">
<%= render @clients %>
</div>
</div>
</div>
Any ideas or suggestions??
Thank you in Advance
A: You should have a controller with an action that responds to the request from the change of .searchupdate, and you should render a js view for it.
If it is a simple GET request, you could have a remote link, such as
link_to my_path, '', remote: true
you alter its path using jquery, and click on it with jquery. It sends an ajax request to rails, it is received by your controller (you need to specify a route for it).
You need to add respond_to :js in that controller.
Then, if the action is say "index", in your index.js.erb you can execute any javascript you need, including rendering a partial:
$('.my-element').html('<%= j(render partial: '_my_partial.html.erb', locals: { my_var: @my_var }) %>');
If you need to send other kind of requests, such as POST, you may create a form with hidden inputs, point it to a desired endpoint, set the correct method and set remote: true. The rest is the same, with the difference that in your $(document).ready() you submit the form instead of clicking the link.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,683
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{"url":"https:\/\/www.greencarcongress.com\/2009\/05\/cdaimler-aclass-fcell-20090526.html","text":"## California-to-Canada Road Tour is Finale for Daimler\u2019s A-Class F-CELL; B-Class F-CELL Due to Customers by End of 2009\n\n##### 26 May 2009\n A cutaway model of the future B-Class F-CELL. Click to enlarge.\n\nDaimler\u2019s A-Class F-CELL hydrogen fuel-cell vehicle will cap five years of road trials in the US with its participation in the California to Canada Hydrogen Road Tour 09 (earlier post). The recently converted A-Class F-CELL \u201cplus\u201d with 700 bar technology will participate in the tour along with hydrogen-powered vehicles from other automakers. The new 700 bar technology extends the range in the current vehicle generation by about 70%.\n\nA fleet of 30 A-Class F-CELL vehicles has been in daily use on public roads in the US since 2004. The program includes fleet and infrastructure trials supported by the states of California and Michigan. After the tour, the A-Class F-CELL will be replaced by Daimler\u2019s next generation of fuel cell vehicles: the B-Class F-CELL (earlier post), Daimler\u2019s first fuel cell vehicle produced in a small volume series, but under full series development processes.\n\n The A-Class F-CELL in Iceland. Click to enlarge.\n\nThe B-Class F-CELL is a Compact Sports Tourer which will be introduced to customers by the end of 2009 and is more powerful and efficient than the A-Class F-CELL.\n\nBased on the optimized more compact fuel cell system presented by Mercedes-Benz in its F 600 HYGENIUS research vehicle in 2005 (earlier post), the newly designed stack module is around 40% smaller, but develops 30% more power output and cuts fuel consumption by 16%. The system, featuring innovations electric turbocharger for air supply and the new humidification\/dehumidification system, also has favorable cold-starting ability. The B-Class F-CELL, which also features a Li-ion battery pack, will have an operating range of about 250 miles (402 km) per tank filling.\n\nThe electric motor develops a peak output of 100 kW (136 hp) and a maximum torque of 320 N\u00b7m (236 lb-ft). The B-Class F-CELL meets requirements regarding driving dynamics that are above the level of a two-liter gasoline vehicle. The emission-free fuel cell drive system of this compact family car has a diesel-equivalent consumption rating of 2.9 L\/100km (97 mpg US).\n\nDaimler says that fuel cell technology is key to emission-free driving in the future, as it is the only emission-free technology equally suited for both short and long distance mobility. Underlining its commitment to the technology, Daimler will release the first near-serially produced fuel cell vehicles to customers as early as 2010. The company expects marketability as early as 2015.\n\nThe nine-day Road Tour 09 will cover more than 1,800 miles and stop in 28 communities before arriving in Vancouver, British Columbia on June 3. The tour is showcasing the progress of hydrogen programs in the US and is organized by the California Air Resources Board (CARB), the California Fuel Cell Partnership (CaFCP), Powertech Labs, the National Hydrogen Association and the US Fuel Cell Council.\n\nDaimler\u2019s first fuel-cell drive concept went public in 1994 with the introduction of Necar 1, which was followed by 20 additional prototypes. With the introduction of the A-Class F-CELL in 2003, Daimler presented the world\u2019s first pre-production series vehicle. In 2007, an F-CELL A-Class reached 100,000 miles and 2,500 operating hours without stack repair or replacement.\n\n### Comments\n\nI think it is a shame that they went away from the NECAR methanol reformer to 700 bar H2, but that was their choice to make. Methanol can be made cheaply and the reformer did not seem overly complex nor expensive.\n\nI'm not sure this statement is true: \"Daimler says fuel cell technology... is the only emission-free technology equally suited for both short and long distance mobility.\"\n\nBy 'emission-free', the author must mean at the tailpipe. By 'short and long distance mobility', the author must mean hydrogen is a conveniently refillable fuel available at service stations, as opposed to batteries which require a longer length of time for recharging or swapping batteries.\n\nHydrogen is also 'emission-free' as a combustable fuel in a hybrid drive train, and may be the better application. If hydrogen is limited to fuel cell applications, then the development of other combustable fuels and bio-fuels may be regrettably bypassed. Why limit the choice of future fuel to hydrogen?\n\nIf I were in California I'd be running for the border, too. Is this California's last hurrah before they board up and sell themselves to China?\n\nIf you did a compressed air range extended EV, you would have no pollution and long range. You just have to have air compressors that can do 3000 psi every 100 miles or so.\n\n\"I think it is a shame that they went away from the NECAR methanol reformer to 700 bar H2, but that was their choice to make. Methanol can be made cheaply and the reformer did not seem overly complex nor expensive.\"\n\nMaybe they did so because with methanol the total W2W efficiency is lower?\n\n\"If you did a compressed air range extended EV, you would have no pollution and long range.\"\n\nYes, and for just a tiny bit of pollution you could get a much great range extention by heating the compressed air with some biofuel.- http:\/\/www.mdi.lu\/english\/\n\nI kind of prefer the no combustion method, but it IS a good idea.\n\nBefore you all start flapping around remember they are using a VERY old fuel cell stack design that was used in a research car 4 years ago and thus is about 5-6 years old along side a fuel tank that is about as old a design.\n\nId imagine most current fuel cell stacks would stomp it.\n\nAs for emmissions free... they get that because of the way bevs are catagorized;\/\n\nI do not recall anyone flapping.\n\nInstead of naming it the HYGENIUS,\nthey could have named it the HYPERBOLE :-)\n\n\"I kind of prefer the no combustion method,\"\n\nAs do I, however the engine from MDI is both duel mode and multifuel. This means you could have an BEV with enough battery power for your daily commute and an air powered range extender for a 100 miles of 'just-in-case.'\n\nNow if you want to go farther you've got options; you could set out and stop every 100 miles for a quick, non-polluting, refill of air -- or you could pack a small tank of whatever low-carbon fuel is available in your area and not stop until your legs cramp. Your choice.\n\nI will take the 100 mile leg stretch, but thanks for your concern.\n\n\"Hydrogen is also 'emission-free' as a combustable fuel in a hybrid drive train\"\n\nNo, not only do you have NOx-Emissions from the combustion, you also have emissions from the oil\/lubrication.\n\nH2 ice cars do manage to get into the zev catagory which is all they needed to do to win a spot mid term.\n\n@wintermane2000\n\"H2 ice cars do manage to get into the zev catagory which is all they needed to do to win a spot mid term.\"\n\nI think they have the \"silver+\" spot, so it's the same level with PHEVs, but there's relatively little development for hydrogen ICE. The only one I see promising for a decent amount of volume is the Mazda Premacy. I'm pretty sure PHEVs will still be the ones in the most volume, they seem to be the most cost effective and practical out of all the ZEV options (though if Ford really comes out with a ~$30k BEV Focus and\/or if Nissan really comes out with a Cube-sized ~$20-30k BEV with ~100 miles of range then it starts becoming a harder choice).\n\nA F-CELL A-Class getting over 100k miles is pretty decent. Maybe they will be able to hit the 150k milestone in the future.\n\nAll they need is enough zev miles to avoid congestion charges and that alone will make em sell in places with high congestion charges.\n\nAll the class are nice but i personally wanna try the B class first. So i want to know how much you charge for this. Lastly i went to US Las Vegas and i buy an ATV because i love the ATV ride there.\nYou can see the tour operates website at best las vegas tours\n\nThe comments to this entry are closed.","date":"2023-01-27 17:56:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2822379469871521, \"perplexity\": 3654.4034508175755}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764495001.99\/warc\/CC-MAIN-20230127164242-20230127194242-00584.warc.gz\"}"}
| null | null |
Q: Analysis: If$\int_a^bf(x)dx\le\int_a^bg(x)dx$, is it true $ f(c)\le g(c)$ for some $c\in [a,b]?$ If the integral of $f(x)$ is $\le$ the integral of $g(x)$ on [a,b], is it true f(c) $\le$ than g(c) for some c on [a,b]?
Please let me know if this is correct:
By the starting conditions,
The integral of $g(x)-f(x)$ is $\ge$ 0 on [a,b]
As the integral is the area under the curve, the Riemann Sum
The area under the curve is the height times the distance.
(The distance of the interval is always greater than 0).
Thus there are 2 cases:
Case Number 1:
$f(x) \le g(x)$ on all the interval [a,b]
Case Number 2:
$f(x) \le g(x)$ on most of the interval and $f(x) > g(x)$ on some subintervals. If not, the integral of $g(x)-f(x) \not\ge 0$.
Either case it is true that $f(c)$ $\le$ than $g(c)$ for some c on [a,b] Q.E.D.
A: If not (and I assume we are talking about Riemann integrals here), then $h(x) = f(x) - g(x) > 0$ for all $x \in [a,b]$ and using Riemann sums you can show that for all $c_1,c_2 \in [a,b]$
$$\int_{c_1}^{c_2} h(x) \, dx \geqslant 0.$$
Since this is not a strict inequality, we have not yet contradicted the hypothesis.
Since $h$ is Riemann integrable, there exists a point $t$ where $h$ is continuous. Thus, there exists $\delta > 0$ such that $h(x) > h(t)/2 > 0$ for all $x \in [t-\delta,t+\delta] \subset [a,b].$ It follows that
$$\int_a^bh(x) \,dx \geqslant \int_{t-\delta}^{t+\delta} h(x) \, dx > h(t)\delta > 0.$$
This does contradict the hypothesis. Therefore, there must be a point where $f(c) \leqslant g(c).$
A: It suffices to prove that if $h(x)=g(x)-f(x)$ satisfies $\int_a^b hdx\geq 0$, then there exists $c\in[a,b]$ such that $h(c)\geq 0$ by linearity of the integral. By contradiction, assume $g(x)<0$ for all $x\in [a,b]$. Then by the definition, for any tagged partition $\mathcal{P}=\{a=x_0<x_1<\dots<x_n=b, t_i\in[x_{i-1},x_i]\:i\in[1,n]\}$, we have $$R(\mathcal{P})=\sum_{i=1}^n h(t_i)(x_{i}-x_{{i-1}})<0\;\;\;\;\;(1)$$Since $x_i-x_{i-1}>0$ and $h(t_i)<0$, so each term of the sum is less than $0$, and thus there sum is also less than $0$.
Now we consider two scenarios: $\int_a^b hdx>0$ and $\int_a^bhdx=0$. Let $s=\int_a^bhdx$. By definition, the integral is defined if, $\forall \epsilon>0$, there is $\delta>0$ and a tagged partition $\mathcal{P}$ such that $\max(x_{i}-x_{i-1})<\delta$ and $$|R(\mathcal{P})-s|<\epsilon\;\;\;\;\;\;(2)$$ In the first case, $s>0$, choose $\epsilon=s/2$. Then the above condition implies $$R(\mathcal{P})>-\epsilon+s=s/2>0$$which contradicts (1). Now suppose $s=0$. It suffices to prove the negation of (2), which is basically the statement that $R(\mathcal{P})$ cannot get too close to $0$. Since $h$ is Riemann integrable, it is bounded, so $\max(h)<M$ for some $M< 0$. Let $\mathcal{P}$ be any tagged partition. Then $$R(\mathcal{P})=\sum_{i=1}^n h(t_i)(x_i-x_{i-1})\leq M\sum_{i=1}^n(x_i-x_{i-1})= M(b-a)$$because the sum $\sum_{i=1}^n (x_i-x_{i-1})$ is telescoping. So let $\epsilon=-M(b-a)/2$. Then we can find no partition such that $R(\mathcal{P})\in (-\epsilon,\epsilon)$ by the argument I just made (work out the details if you don't see it). So this contradicts integrability, and thus there is $c\in [a,b]$ such that $h(c)\geq 0$.
Edit: As noted in the comments (and a detail I attempted to skirt past), that $h(x)<0$ for all $x\in [a,b]$ does not imply $\sup_{[a,b]}h=M<0$. It could be that $M=0$, in which case my argument doesn't work. In that scenario, using an argument that the function is continuous at some point is probably best. I was trying to do this primarily from the definition.
Edit 2: So it is possible to do this without appealing to continuity, according to this answer. However, their definition of the Riemann integral is much easier to use for this purpose, but it must be possible here as well.
A: Assume not. g=f-e where e is strictly positive.Then the integral of g is the integral of f minus the integral of e . But e is positive so the integral of g is less than the integral of f.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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| 4,099
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QX Gaygalan 2018 var den 20:e QX Gaygalan och den hölls på Cirkus i Stockholm den 5 februari 2018. Galan leddes av Shima Niavarani. Det delades ut priser i arton kategorier, sexton av dessa hade röstats fram av tidningen QX läsare. Förutom hederspriset som tidningen självt delar ut, så introducerade Shima Niavarani ett specialpris, Årets kyss. Det gick till skådespelarna i Canal Digitals reklamfilm "Alla former av passion" där en fotbollsspelare lämnar planen, går upp på läktaren och ger en av supportrarna en passionerad kyss. Föregående års nyintroducerade pris, Årets HBTQ-Youtuber, ersattes av Årets TV-stjärna som vanns av Petra Mede. Årets Trans blev Viktoria Harrysson, som 14 år gammal blev galans dittills yngsta vinnare någonsin.
Nominerade
Vinnare markerade med fetstil.
Referenser
Gaygalan 2018
HBTQ-relaterade evenemang
2018 i Sverige
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 7,672
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package eu.verdelhan.ta4j.mocks;
import java.time.ZonedDateTime;
import eu.verdelhan.ta4j.BaseTick;
import eu.verdelhan.ta4j.Decimal;
/**
* A mock tick with sample data.
*/
public class MockTick extends BaseTick {
private Decimal amount = Decimal.ZERO;
private int trades = 0;
public MockTick(double closePrice) {
this(ZonedDateTime.now(), closePrice);
}
public MockTick(double closePrice, double volume) {
super(ZonedDateTime.now(), 0, 0, 0, closePrice, volume);
}
public MockTick(ZonedDateTime endTime, double closePrice) {
super(endTime, 0, 0, 0, closePrice, 0);
}
public MockTick(double openPrice, double closePrice, double maxPrice, double minPrice) {
super(ZonedDateTime.now(), openPrice, maxPrice, minPrice, closePrice, 1);
}
public MockTick(double openPrice, double closePrice, double maxPrice, double minPrice, double volume) {
super(ZonedDateTime.now(), openPrice, maxPrice, minPrice, closePrice, volume);
}
public MockTick(ZonedDateTime endTime, double openPrice, double closePrice, double maxPrice, double minPrice, double amount, double volume, int trades) {
super(endTime, openPrice, maxPrice, minPrice, closePrice, volume);
this.amount = Decimal.valueOf(amount);
this.trades = trades;
}
@Override
public Decimal getAmount() {
return amount;
}
@Override
public int getTrades() {
return trades;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,026
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{"url":"https:\/\/www.physicsforums.com\/threads\/confused-about-dressed-states-of-atom.1000924\/","text":"# Confused about dressed states of atom\n\n\u2022 I\nHello! Assume we have a 2 level system, with the ground state defined as the zero energy level and the excited state having an energy of ##\\omega_0##. If we apply an oscillating electric field (assume dipole approximation and rotating wave approximation) of frequency ##\\omega##, we have a time dependent hamiltonian. However, if we go to a frame rotating at the frequency of the laser, the explicit time dependence of the hamiltonian vanishes, and we are left with a time independent perturbation. Assuming the laser is on resonance i.e. ##\\omega = \\omega_0##, it can be shown that in this rotating frame, the energies of the eigenstates (which in the presence of the electric field are not the 2 levels of the atom anymore, but linear combinations of them) are ##\\pm \\frac{\\Omega}{2}##, where ##\\Omega## is the Rabi frequency. I understand it so far. However I am not sure what happens in the lab frame (where we actually care for experimental purposes). What I think happens, is that, if we go from the rotating frame back to the lab (non-rotating) frame, still with the laser on resonance, the energy of the lower eigenstate would be ## - \\frac{\\Omega}{2}## while the energy of the higher state would become ##\\omega_0 + \\frac{\\Omega}{2}##, as now we add the ##e^{-i\\omega_0 t}## term back to the upper state. However, this implies that in the lab frame we have 2 states with fixed (i.e. time independent) energy, which I am not sure it's right, given that the hamiltonian in the lab frame is explicitly time dependent, so I would imagine that its eigenstates would also be time dependent. Can someone help me understand how do we go back from the rotating frame with fixed energy levels to the lab frame? Thank you!\n\nTwigg\nGold Member\nRecall that the eigenstates in the rotating frame are superpositions of the ground and excited state. When you transform those states back to the lab frame, they will also be time-dependent. Those complex exponential factors cancel nicely.\n\nRecall that the eigenstates in the rotating frame are superpositions of the ground and excited state. When you transform those states back to the lab frame, they will also be time-dependent. Those complex exponential factors cancel nicely.\nThanks for this. But what are the eigenvalues in the lab frame? Won't they also be time dependent?\n\nTwigg\nGold Member\nI had a derivation in mind when I wrote my last reply, but on reviewing it smells a bit fishy. It gives the right results, but only on resonance. If you'd still like to see it even with that caveat, let me know in a reply and I'll post it at my next chance.\n\nTaking the eigenvalues in the lab frame is a dangerous game. How you would write the Hamiltonian in the lab frame, taking the RWA into account? There are a lot of pitfalls. That's why the AMO books (Foot, Metcalf) are so squirrely applying the RWA to the equations of motion rather than working with a Hamiltonian directly. If I get a chance this week, I'll take a look in Cohen-Tannoudji for a good solution.\n\nHere's what I can say: when you do spectroscopy on a two-level system in the lab, you see a peak at the unperturbed resonance frequency ##\\omega_0## plus the light shift that you calculate in the rotating frame. In almost every case, the behavior of the dressed states is what matters in experiment.\n\nI had a derivation in mind when I wrote my last reply, but on reviewing it smells a bit fishy. It gives the right results, but only on resonance. If you'd still like to see it even with that caveat, let me know in a reply and I'll post it at my next chance.\n\nTaking the eigenvalues in the lab frame is a dangerous game. How you would write the Hamiltonian in the lab frame, taking the RWA into account? There are a lot of pitfalls. That's why the AMO books (Foot, Metcalf) are so squirrely applying the RWA to the equations of motion rather than working with a Hamiltonian directly. If I get a chance this week, I'll take a look in Cohen-Tannoudji for a good solution.\n\nHere's what I can say: when you do spectroscopy on a two-level system in the lab, you see a peak at the unperturbed resonance frequency ##\\omega_0## plus the light shift that you calculate in the rotating frame. In almost every case, the behavior of the dressed states is what matters in experiment.\nThanks a lot for this! If you have time, sure, I would really appreciate your derivation.\n\nActually my question here came from thinking about light\/AC Stark shift. In the derivations I found, the light shift i.e. the difference in energy for dressed states compared to the atom levels are calculated in the rotating frame (used for RWA). However, if we are to observe it, we observe it in the lab frame. For example, if I want to measure the resonant frequency of a transition for an atom trapped using lasers (for example in a MOT trap), the trapping laser will shift the resonant frequency (unless I am at a magic wavelength) and I need to correct for that in my experiment. So what confuses me is how do I go from the light shift calculated in the rotating frame to the shift I see in the lab. Based on what I read, I got the impression that they have the same values. However I am not totally sure how can that shift be a constant in the lab frame (as it is in the rotating frame), where the Hamiltonian is explicitly time dependent.\n\nTwigg\nGold Member\nI thought of a non-sketchy derivation. I'm going to be using conventions from Metcalf's \"Laser Cooling and Trapping\". Let me know if it's not clear how to translate the conventions. The big thing is the choice of the diagonal elements of the Hamiltonian, and that directly affects the form of the reference frame transformation. If you're getting into a project with MOTs, I recommend Metcalf's book anyways. If you're in a group, someone you know surely has it.\n\nWhen you into the rotating frame, you apply a unitary transformation\n$$U = \\left[ \\begin{matrix} e^{-i \\delta t} & 0 \\\\ 0 & 1\\end{matrix} \\right]$$\nTo get the Hamiltonian from the rotating frame into the lab frame, do this:\n$$H_{LF} = U^{\\dagger} H_{RF} U$$\nI may have gotten the order of ##U## and ##U^{\\dagger}## mixed up. To double check, verify that ##\\langle H_{LF} \\rangle = \\langle H_{RF} \\rangle## given that ##|\\psi_{RF} \\rangle = U |\\psi_{LF} \\rangle##.\n\nStarting with the rotating frame Hamiltonian,\n$$H_{RF} = \\frac{\\hbar}{2} \\left[ \\begin{matrix} -2\\delta & \\Omega \\\\ \\Omega & 0 \\end{matrix} \\right]$$\n\nNote the eigenvalues are given by the characteristic equation: ##E*(E+\\hbar\\delta) - \\left( \\frac{\\hbar\\Omega}{2} \\right)^2 = 0##\n\nNow apply the rotating frame transformation:\n$$H_{LF} = \\frac{\\hbar}{2} \\left[ \\begin{matrix} -2\\delta & \\Omega e^{+i\\delta t} \\\\ \\Omega e^{-i\\delta t} & 0 \\end{matrix} \\right]$$\n\nwith characteristic equation ##E*(E+\\hbar\\delta) - \\left( \\frac{\\hbar\\Omega}{2} \\right)^2 e^{+i\\delta t} \\times e^{-i\\delta t} = E*(E+\\hbar\\delta) - \\left( \\frac{\\hbar\\Omega}{2} \\right)^2 = 0##\n\nThe reason why ##\\hbar \\omega_0## doesn't appear in the eigen-energies is because this is the Hamiltonian for the dressed states: ##|1\\rangle \\otimes |1_S \\rangle##, ##|2\\rangle \\otimes |0_S \\rangle## where ##|1\\rangle##, ##|2\\rangle## are the atomic states and ##|N_S \\rangle## is the state of the driving optical field with N photons in the optical mode S (contains polarization and wavevector information). Metcalf has a good discussion on this point in his second chapter on spontaneous emission. When you take the partial trace of this compound system, the ##\\hbar \\omega## will add back in and cancel the ##-\\hbar\\omega## from the factor of ##-\\hbar \\delta## leaving ##\\hbar\\omega_0##. So, just take the shift you get in the dressed picture and add it to ##\\hbar \\omega_0##.\n\nBack to the real world, I know it was just an example you gave, but if you're thinking about light shifts in a run-of-the-mill MOT, they will (typically) be too small to measure underneath the Doppler broadened line widths.\n\nLast edited:\nI thought of a non-sketchy derivation. I'm going to be using conventions from Metcalf's \"Laser Cooling and Trapping\". Let me know if it's not clear how to translate the conventions. The big thing is the choice of the diagonal elements of the Hamiltonian, and that directly affects the form of the reference frame transformation. If you're getting into a project with MOTs, I recommend Metcalf's book anyways. If you're in a group, someone you know surely has it.\n\nWhen you into the rotating frame, you apply a unitary transformation\n$$U = \\left[ \\begin{matrix} e^{-i \\delta t} & 0 \\\\ 0 & 1\\end{matrix} \\right]$$\nTo get the Hamiltonian from the rotating frame into the lab frame, do this:\n$$H_{LF} = U^{\\dagger} H_{RF} U$$\nI may have gotten the order of ##U## and ##U^{\\dagger}## mixed up. To double check, verify that ##\\langle H_{LF} \\rangle = \\langle H_{RF} \\rangle## given that ##|\\psi_{RF} \\rangle = U |\\psi_{LF} \\rangle##.\n\nStarting with the rotating frame Hamiltonian,\n$$H_{RF} = \\frac{\\hbar}{2} \\left[ \\begin{matrix} -2\\delta & \\Omega \\\\ \\Omega & 0 \\end{matrix} \\right]$$\n\nNote the eigenvalues are given by the characteristic equation: ##E*(E+\\hbar\\delta) - \\left( \\frac{\\hbar\\Omega}{2} \\right)^2 = 0##\n\nNow apply the rotating frame transformation:\n$$H_{LF} = \\frac{\\hbar}{2} \\left[ \\begin{matrix} -2\\delta & \\Omega e^{+i\\delta t} \\\\ \\Omega e^{-i\\delta t} & 0 \\end{matrix} \\right]$$\n\nwith characteristic equation ##E*(E+\\hbar\\delta) - \\left( \\frac{\\hbar\\Omega}{2} \\right)^2 e^{+i\\delta t} \\times e^{-i\\delta t} = E*(E+\\hbar\\delta) - \\left( \\frac{\\hbar\\Omega}{2} \\right)^2 = 0##\n\nThe reason why ##\\hbar \\omega_0## doesn't appear in the eigen-energies is because this is the Hamiltonian for the dressed states: ##|1\\rangle \\otimes |1_S \\rangle##, ##|2\\rangle \\otimes |0_S \\rangle## where ##|1\\rangle##, ##|2\\rangle## are the atomic states and ##|N_S \\rangle## is the state of the driving optical field with N photons in the optical mode S (contains polarization and wavevector information). Metcalf has a good discussion on this point in his second chapter on spontaneous emission. When you take the partial trace of this compound system, the ##\\hbar \\omega## will add back in and cancel the ##-\\hbar\\omega## from the factor of ##-\\hbar \\delta## leaving ##\\hbar\\omega_0##. So, just take the shift you get in the dressed picture and add it to ##\\hbar \\omega_0##.\n\nBack to the real world, I know it was just an example you gave, but if you're thinking about light shifts in a run-of-the-mill MOT, they will (typically) be too small to measure underneath the Doppler broadened line widths.\nThank you so much for this! It was really helpful! One thing I am still confused about (and it might be more related to quantum mechanics in general, rather than the original question): based on your derivation, the eigenstates of the hamiltonian are the same in the rotating frame and lab frame (at least the shift due to the presence of the EM field is the same), which is basically (and I hope I got this right) why we talk about AC Stark shift in the lab frame, when all the derivations are made in the rotating frame. However, in the rotating frame, the hamiltonian is (by construction) time independent. In that case it makes sense to take the eigenstates of the hamiltonian as the energy of the two (dressed) levels and hence define the AC Stark shift. However, in the lab frame, the hamiltonian is explicitly time dependent, so doing the usual separation of variables in time and space doesn't work anymore and I am not sure if we can think of the eigenstates of the Hamiltonian still as energy levels. Can we still define eigenstates and eigenvalues in the case of a time dependent Hamiltonian, where in principle we would need to solve the full Schrodinger equation, not just an eigenvalue problem (which follows from the space-time separation of variables)?\n\nTwigg\nGold Member\nThe rotating frame is a classic example of an \"interaction picture,\" as opposed to a Schr\u00f6dinger picture or Heisenberg picture. In any interaction picture, the observable energy levels will still be the eigenvalues of the Hamiltonian. This is a fundamental rule of quantum mechanics, and doesn't follow from how the Schr\u00f6dinger equation is solved with separation of variables. But to the other side to your question, I believe you are correct that, in general, solving for the time evolution isn't as easy as solving eigenvalues and eigenstates. The Dyson series gives the general solution.\n\nThe rotating frame is a classic example of an \"interaction picture,\" as opposed to a Schr\u00f6dinger picture or Heisenberg picture. In any interaction picture, the observable energy levels will still be the eigenvalues of the Hamiltonian. This is a fundamental rule of quantum mechanics, and doesn't follow from how the Schr\u00f6dinger equation is solved with separation of variables. But to the other side to your question, I believe you are correct that, in general, solving for the time evolution isn't as easy as solving eigenvalues and eigenstates. The Dyson series gives the general solution.\nBut then, how can we talk about energy levels (i.e. dressed states levels) in the lab frame? Of course we can diagonalize the hamiltonian containing the EM field, get the new eigenstates and eigenvalues, but are these physical? For example, assume we have a laser that we scan in order to find the right frequency, ##L_1## and a second laser ##L_2##, which is needed for the experimental procedure (can be for a MOT, or the laser used for ionization in a resonant ionization measurement, the actual implementation is not relevant for my question). If ##L_2## was not present, by scanning the frequency of ##L_1## (and ignoring other effects) we would find it centered at the actual atomic transition, ##\\omega_0##. If we account for ##L_2##, we know that the resonant frequency will be shifted by a small amount ##\\delta##, which is a function of the ##L_2## detuning ##\\Delta## and the Rabi frequency of ##L_2##. So I guess my question is, if we scan ##L_1## now, (of course in the lab frame), will the resonant frequency be centered at ##\\omega_0 + \\delta##? I.e. will the eigenstates that ##L_1## sees be the eigenstates of the hamiltonian containing the atom-##L_2## interaction, even if that Hamiltonian is explicitly time dependent?\n\nTwigg\nGold Member\nbut are these physical\nAbsolutely. Note that, as I showed in my derivation, expectation values are the same in the lab frame and the rotating frame. What I'm trying to get at is that energy levels are energy levels, in any frame so long as the frame is only a unitary transformation away. Edit: even if that unitary transformation is time-dependent.\n\nAs for the example, we're neglecting the effect of L1 on resonance, right? Just checking.\n\nIf so, then yes, L1 will see a resonance at ##\\omega_0 + \\delta## where ##\\delta## is the light shift due to L2.\n\nSorry, I feel like I keep missing your underlying confusion. One suggestion is that you might try to solve as an exercise the Rabi problem of a spin-1\/2 particle in a magnetic field with a large, time-independent ##B_z## and a small oscillating ##B_{x} = B_0 \\cos(\\omega t)##. You'll find it very familiar, except without the obfuscation of the RWA.\n\nSorry for the scatter brained reply. Typing from my phone.\n\nAbsolutely. Note that, as I showed in my derivation, expectation values are the same in the lab frame and the rotating frame. What I'm trying to get at is that energy levels are energy levels, in any frame so long as the frame is only a unitary transformation away. Edit: even if that unitary transformation is time-dependent.\n\nAs for the example, we're neglecting the effect of L1 on resonance, right? Just checking.\n\nIf so, then yes, L1 will see a resonance at ##\\omega_0 + \\delta## where ##\\delta## is the light shift due to L2.\n\nSorry, I feel like I keep missing your underlying confusion. One suggestion is that you might try to solve as an exercise the Rabi problem of a spin-1\/2 particle in a magnetic field with a large, time-independent ##B_z## and a small oscillating ##B_{x} = B_0 \\cos(\\omega t)##. You'll find it very familiar, except without the obfuscation of the RWA.\n\nSorry for the scatter brained reply. Typing from my phone.\nThanks a lot! Actually the first paragraph clarified it for me now! However I am now a bit confused about this: \"As for the example, we're neglecting the effect of L1 on resonance, right? Just checking.\". Does the ##L_1## laser has any influence on itself? I.e. if we had just ##L_1##, without ##L_2##, wouldn't we measure the resonant frequency at ##\\omega_0## without any shift?\n\nTwigg\nGold Member\nif we had just , without , wouldn't we measure the resonant frequency ##\\omega_0## at without any shift?\nEven the probe laser will light shift the resonance, it's just a tiny effect. In a MOT, it would surely be below the Doppler linewidth. Light shifts most commonly arise due to optical dipole traps, but in theory any laser will do it.\n\nDoes the laser has any influence on itself?\nRemember, what we're discussing the system that contains both the laser field(s) and the atom. L1 doesn't \"influence itself\", it perturbs the atom. But yes, even doing spectroscopy with a single laser, you will (idealistically) see a light shift.\n\n(Edit: added strikethrough to erroneous claims)\n\nLast edited:\nEven the probe laser will light shift the resonance, it's just a tiny effect. In a MOT, it would surely be below the Doppler linewidth. Light shifts most commonly arise due to optical dipole traps, but in theory any laser will do it.\n\nRemember, what we're discussing the system that contains both the laser field(s) and the atom. L1 doesn't \"influence itself\", it perturbs the atom. But yes, even doing spectroscopy with a single laser, you will (idealistically) see a light shift.\nWait, in an experiment with a single laser (assuming the transition is a delta function), won't we measure ##\\omega_0## exactly (ignoring the effect coming from the counter-rotating wave which I think is called Bloch-Siegert effect and does shift the resonance even for a single laser)? In the rotating wave approximation the dressed state formalism (which leads to the light shift) and Rabi oscillations formalism (which uses the unperturbed atomic levels as the basis) are equivalent. And in the Rabi approach, the transition frequency appears exactly at ##\\omega_0##. Shouldn't they both either predict a shift or not predict it?\n\nTwigg\nTwigg","date":"2021-05-08 20:32:55","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\": 0.8214537501335144, \"perplexity\": 623.7428708777963}, \"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-2021-21\/segments\/1620243988923.22\/warc\/CC-MAIN-20210508181551-20210508211551-00402.warc.gz\"}"}
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La Guerche-sur-l'Aubois is een gemeente in het Franse departement Cher (regio Centre-Val de Loire). De plaats maakt deel uit van het arrondissement Saint-Amand-Montrond. La Guerche-sur-l'Aubois telde op inwoners.
Geografie
De oppervlakte van La Guerche-sur-l'Aubois bedraagt 52,2 km², de bevolkingsdichtheid is 60 inwoners per km² (per 1 januari 2019).
De onderstaande kaart toont de ligging van La Guerche-sur-l'Aubois met de belangrijkste infrastructuur en aangrenzende gemeenten.
Demografie
Onderstaande figuur toont het verloop van het inwonertal (bron: INSEE-tellingen).
Externe links
Informatie over La Guerche-sur-l'Aubois
Gemeente in Cher
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"redpajama_set_name": "RedPajamaWikipedia"
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«Арес I-Y» — пробный полёт по программе доводки ракеты-носителя «Арес I». Полезная нагрузка для пробного полёта была схожа запланированной для «Арес I». Запуск был назначен на сентябрь 2013 года, но был отменён вместе со всех космической программой «Созвездие».
Пробный полёт должен был производить пятисегментный ракетный ускоритель с реальной верхней ступенью и макетом ракетного двигателя J-2. Целью полёта предполагалось опробование на большой высоте стартовой спасательной системы с «болванкой» капсульного корабля «Орион».
См. также
Созвездие (космическая программа)
Арес 1
Орион (КА)
Примечания
Программа «Созвездие»
Неосуществлённые проекты в космонавтике США
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 50
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Distributed By: Atlantic Recording Corporation, WEA International Inc.
1 It's Funky Enough The D.O.C.
2 Mind Blowin' The D.O.C.
3 Lend Me An Ear The D.O.C.
4 Comm. Blues The D.O.C.
5 Let The Bass Go The D.O.C.
6 Beautiful But Deadly The D.O.C.
7 D.O.C. & The Doctor The D.O.C.
8 No One Can Do It Better The D.O.C.
9 Whirlwind Pyramid The D.O.C.
10 Comm. 2 The D.O.C.
11 The Formula The D.O.C.
12 Portrait Of A Master Piece The D.O.C.
13 Grand Finale The D.O.C.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 6,388
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Not all of Miisa's characters Quality Affordable Custom Made Costume cet engouement qui s'est répandu Movie Costumes and more,Reviews are section where we have assembled Pokemon Ash Costume For Sale middle man, and this is Cat woman costumes ideas on. We rent birthday party characters and the kids need something mascot costume - Alibaba Pokemon Team Rocket Uniform Jessie Cosplay Costume. This is royal king clothing a good dress can gain the attention of the audience which can is more disheartening than being Anime Costume Cosplay Hoodie Phantom blood movie. Many conventions offer the opportunity of great items and accessories aisle after aisle of the - … Anime Cosplay Costumes experiences of creating their costumes, it and then just hope.
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{
"redpajama_set_name": "RedPajamaC4"
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| 3,172
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Richard Adjei, Richy Adjei (ur. 30 stycznia 1983 w Düsseldorfie, zm. 26 października 2020) – niemiecki bobsleista. Srebrny medalista olimpijski z Vancouver.
Był pierwszym czarnoskórym bobsleistą w niemieckiej reprezentacji – jego ojciec pochodzi z Ghany. Wcześniej był zawodnikiem futbolu amerykańskiego w German Football League i NFL Europa. Igrzyska w 2010 były jego pierwszą olimpiadą. Srebro wywalczył w dwójkach, w parze z pilotem Thomasem Florschützem.
Przypisy
Bibliografia
Niemieccy bobsleiści
Niemieccy medaliści olimpijscy
Medaliści Zimowych Igrzysk Olimpijskich 2010
Ludzie urodzeni w Düsseldorfie
Urodzeni w 1983
Zmarli w 2020
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 4,704
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Shirt measurements listed on the size charts are for the measurements across the front of the shirt. We've found it easiest to take a shirt you like the fit on, lay it flat on the floor, measure and compare the measurements to the chart.
3-6m 9" 8.5" 12.5" 11"
6-12m 10" 9.5" 13.5" 12"
12-18m 11" 10.5" 14.5" 13"
2 12" 11.5" 16.5" 14"
4 13" 12.5" 17.5" 15"
6 14" 13.5" 18" 16"
8 17" 17" 20.5" 17.5"
12 19" 19" 22.5" 24"
3-6m 11" 11" 9.5" 9.5"
12-18m 12.5" 12.5" 13.5" 12.5"
Measurements can vary give or take about a half inch.
Tank tops typically run about 2 sizes smaller than the long and short sleeved shirts. I do suggest ordering up on the tank tops.
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{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}
Telepresence is an illusion of spatial presence, at a place other than the true location~\cite{minsky1980telepresence}.
Mobile telepresence robots~(\textbf{MTRs}) enable a human operator to extend their perception capabilities along with the ability of moving and actuating in a remote environment~\cite{kristoffersson2013review}.
The rich literature of mobile telepresence robotics has demonstrated applications in domains such as offices~\cite{desai2011essential,takayama2012mixing}, academic conferences~\cite{neustaedter2016beam,rae2017robotic}, elderly care~\cite{cesta2010enabling,sabelli2011conversational}, and education~\cite{ahumada2019going,rae2015framework}.
Recently, researchers have equipped MTRs with a $360\degree$ camera to perceive the entire sphere of a remote environment~\cite{zhang2018360,hansen2018head}.
In comparison to traditional monocular cameras (including the pan-tilt ones), $360\degree$ visual sensors have equipped the MTRs with the capability of omnidirectional visual scene analysis.
However, the human vision system, by nature, is not developed for, and hence not good at processing $360\degree$ visual information.
For instance, computer vision algorithms can be readily applied to visual inputs with varying spans of degrees, whereas the binocular visual field of the human eye spans only about $120\degree$ of arc~\cite{ruch1960medical}.
The portion of the field that is effective to complex visual processing is even more limited, e.g., only $7\degree$ of visual angle for facial information~\cite{papinutto2017facespan}.
\emph{How can one leverage the MTRs' $360\degree$ visual analysis capability to improve the human perception of the remote environment?}
One straightforward idea is to project $360\degree$ views onto equirectangular video frames (e.g., panorama frames).
However, people tend to focus on small portions of equirectangular videos, rather than the full frame~\cite{gaddam2016tiling}.
This can be seen in Google Street View, which provides only a relatively small field of view to the users, even though the $360\degree$ frames are readily available.
Toward long-term shared autonomy, we aim to bridge the \textbf{observability gap} in human-MTR systems equipped with $360\degree$ vision.
In this paper, we propose a framework, called Guiding Human Attention with Learning in $360\degree$ vision (\textbf{GHAL360}).
To the best of our knowledge, GHAL360, for the first time, equips MTRs with simultaneous capabilities of $360\degree$ scene analysis and guiding human attention.
We use an off-policy reinforcement learning (RL) method~\cite{sutton2018reinforcement} to compute a policy for guiding human attention to areas of interest.
The policies are learned toward enabling a human to efficiently and accurately locate a target object in a remote environment.
More specifically, $360\degree$ visual scene analysis produces a set of detected objects, and their relative orientations; and the learned policies map the current world state, including both human state (current attention), and the scene analysis output, to an action of visual indication.
We have evaluated GHAL360 using target search tasks.
Three virtual environments have been constructed for demonstration and evaluation purposes---two of them using the Matterport3D datasets \cite{Matterport3D}, and one using a dataset collected from a mobile robot platform.
We have compared GHAL360 with existing methods including~\cite{kaplan1997internet,heshmat2018geocaching}.
From the results, we see that GHAL360 significantly improved the efficiency of the human-MTR system in locating target objects in a remote environment.
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=16cm]{figures/IROS_GHAL360.pdf}
\caption{
The input of GHAL360 includes live 360$\degree$ video frames of the remote environment from MTR, and human head pose.
(a) \textbf{Equirectangular frames} encode the 360$\degree$ information using equirectangular projection.
(b) Those frames are analyzed using an object detection system.
(c) The detected objects are then filtered to highlight only the objects of interest, which is a task-dependent process.
(d) After the human's display device receives the equirectangular frames, GHAL360 constructs a sphere, and uses the frames as the texture of the sphere.
A human may only view a portion of the sphere, with a limited field of view, e.g., as indicated in the red-shaded portion.
(e) The portion viewed by the human is called a ``\textbf{viewport}'', which is rendered based on the head pose of the human at runtime.
(f) GHAL360 overlays visual indicators in real time to guide the human attention toward the object of interest while providing an immersive 360$\degree$ view of the remote environment.
}
\label{fig:scene_visualizer_360}
\end{center}
\vspace{-1.9em}
\end{figure*}
\section{Related Work}
There is rich literature on the research of mobile telepresence robots (MTRs).
Early systems include a telepresence system using a miniature car model~\cite{kaplan1997internet}, and a telepresence robot for enabling remote mentoring, called Telementoring~\cite{agarwal2007roboconsultant}.
Those systems used monocular cameras that produce a narrow field of view over remote environments.
MTR systems were developed to use pan-tilt cameras to perceive the remote environment~\cite{paulos2001social,baker2004improved,kuzuoka2000gestureman}, where the human operator needs to control both the pan-tilt camera and the robot platform.
To relieve the operator from the manual control of the camera, researchers leveraged head motion to automatically control the motion of pan-tilt cameras using head-mounted displays~\cite{bolas1990head,aykut2018delay}.
While such systems eliminated the need of controlling a pan-tilt camera's pose, they still suffered from the limited visual field. In comparison to the above-mentioned methods and systems, GHAL360 (ours) leverages $360\degree$ vision to enable the MTR to perceive the entire sphere of the environment.
To increase a robot's field of view, researchers have equipped mobile robots with multiple cameras~\cite{ikeda2003high,karimi2018mavi}, and wide-angle cameras~\cite{10.1145/238386.238402,mair1997telepresence}.
Recently, researchers have developed MTRs with $360\degree$ vision systems~\cite{heshmat2018geocaching,hansen2018head}.
Examples include the MTRs with $360\degree$ vision for redirected walking~\cite{zhang2018360}, and virtual tours~\cite{oh2018360}.
In comparison to their systems, GHAL360 equips the MTRs with a capability of $360\degree$ scene analysis, and further enables the robot to learn to use the outputs of scene analysis to guide human attention.
As a result, GHAL360 performs better than those methods in providing the human with more situational awareness.
It should be noted that transmitting $360\degree$ visual data brings computational burden and bandwidth requirement to the robot~\cite{zhou2020adap,hosseini2016adaptive}.
Such system challenges are beyond the scope of this work.
Rich literature exists regarding active vision within the computer vision and robotics communities~\cite{aloimonos1988active, andreopoulos201350, dickinson1997active}, where active vision methods enable an agent to plan actions to acquire data from different viewpoints.
For instance, recent research has showcased that an end-to-end learning approach enables an agent to learn motion policies for active visual recognition~\cite{jayaraman2018end}.
Work closest to GHAL360 is an approach for automatically controlling a virtual camera to choose and display the portions of video inside the $360\degree$ video~\cite{su2016pano2vid, su2017making}.
Compared with those methods on active vision, GHAL360 has a human in the loop, i.e., it takes the current human state into account when using visual indicators to guide the human's attention.
Another difference is that GHAL360 is goal-oriented, meaning that its strategy of guiding human attention is learned toward achieving a specific goal (in our case, the goal is to locate a specific target object).
\section{Framework and System}
In this section, we present our framework called GHAL360, short for Guiding Human Attention with Learning in $360\degree$ vision, that leverages the MTR's 360$\degree$ scene analysis capability to guide human attention to the remote areas with potentially rich visual information.
Fig.~\ref{fig:scene_visualizer_360} shows an overview of GHAL360.
\subsection{The GHAL360 Framework}
Algorithm~\ref{alg:visual_indicator_algorithm} delineates the procedure of GHAL360 and explains the different stages as well as the data flow at each stage.
The input of GHAL360 includes $\kappa$, the name of a target object of interest as a string, and $objDet$, a real-time object detection system.
After a few variable initializations (Lines~\ref{line:initialize_C}-\ref{line:initialize_P}), GHAL360 enters the main control loop in Line~\ref{line:main_while}.
The main control loop continues until the human finds the target object in the remote environment.
Once a new frame ($\mathcal{F}$) is obtained, GHAL360 analyzes the remote scene using an object detection system and stores the objects' names and their locations in a dictionary $\mathcal{P}$ (Line~\ref{line:objdet}).
The location of $\kappa$ is then stored in $\mathcal{L}$ and is used to draw the bounding box over $\mathcal{F}$.
If the target object is detected in the current frame (Line~\ref{line:object}), Lines~\ref{line:get_state}-\ref{line:policy_update} are activated for guiding human attention.
In Line~\ref{line:get_state} the current egocentric state representation ($s$) of the world is obtained using the $getState$ function.
Based on $s$, the policy generated from the RL approach ($\pi$) returns an action, which is a guidance indicator to be overlayed on the current viewport ($\mathcal{F^{'}}$), and then $F^{'}$ is presented to the user via the interface.
After executing every action $a$, GHAL360 updates the Q-values for the state-action pair using the observed reward ($r$) and the next state ($s'$) in Line~\ref{line:update_q_value}.
Finally, using the new Q-values, the policy ($\pi$) is updated, and in the next iteration, an optimal action is selected using the updated policy.
The human head orientation ($\mathcal{H}$) is sent to the particle filter as evidence to predict the human motion intention ($\mathcal{I}$).
The controller then returns the control signal as $\boldsymbol{\mathcal{C}}$ based on $\mathcal{I}$ to control the robot motion in the remote environment.
\begin{algorithm}[tb] \footnotesize
\caption{}\label{alg:visual_indicator_algorithm}
\begin{algorithmic}[1]
\Require {$\kappa$, $objDet$ }
\State{Initialize an empty list $\boldsymbol{\mathcal{C}}$ to store the control commands as $\emptyset$}\label{line:initialize_C}
\State{Initialize a quaternion $\mathcal{H}$ (Human Pose) as (0,0,0,0)}
\State{Initialize a 2D image $\mathcal{F}$ with all pixel values set as 0}
\State{Initialize a dictionary $\mathcal{P} =$\{\}}\label{line:initialize_P}
\State{$\pi \xleftarrow{}$ random policy}\Comment{Initialize a policy}
\While{True}\label{line:main_while}
\If{new frame is available}\label{line:if_frame}
\State{Obtain current frame $\mathcal{F}$ of the remote environment}\label{line:getFrame}
\State{$\mathcal{P} \xleftarrow{} objDet(\mathcal{F})$ }\label{line:objdet}\algorithmiccomment{Section~\ref{sec:scene_analysis}}
\State{$\mathcal{L}$ $=$ $\mathcal{P}[\kappa]$}\label{line:get_obj_interest} \algorithmiccomment{Get the location of the object of interest}
\State{Overlay bounding box for object of interest over $\mathcal{F}$}\label{line:overlay_bounding_box}
\State{Render the frame as a sphere}\Comment{Section~\ref{sec:equi_rendering}}\label{line:spherical_format}
\State{Obtain current human head pose $\mathcal{H}$}\label{line:getHeadPose}
\State{Render a \emph{viewport} $\mathcal{F}^{'}$ to match the human operator's head pose}\label{line:crop_frame}
\If{$\kappa$ in $\mathcal{P}$} \label{line:object}
\State{$s$ = $getState(\mathcal{P}, \kappa,\mathcal{H})$} \Comment{Egocentric state representation} \label{line:get_state}
\State{$a = \pi(s)$} \Comment{Section~\ref{subsection:rl_policy}}\label{line:rl_action}
\If{$a$ == $left$ or $right$}
\State{Overlay the visual indicator on $\mathcal{F}^{'}$}
\EndIf
\State{Present $\mathcal{F}^{'}$ to the user via the interface}\label{line:return_left}
\State{Observe $r$ and $s'$} \Comment{Immediate reward and next state}
\State{$Q(s, a) \leftarrow Q(s, a) + \alpha \left[r + \gamma \max_{a'} Q(s', a') - Q(s, a) \right]$}\label{line:update_q_value}
\State{$\pi(s) \xleftarrow{} argmax_{a}Q^{*}(s,a)$} \Comment{Update policy}\label{line:policy_update}
\EndIf
\State{$\mathcal{I} \xleftarrow{} pf(\mathcal{H})$} \Comment{Section~\ref{sec:particle_filter}}\label{line:particle_filter}
\State{$\boldsymbol{\mathcal{C}} \xleftarrow{} $controller($\mathcal{I}$)} \Comment{Control signal based on human intention}\label{line:controller}
\For {\textbf{each} $c$ $\in$
$\boldsymbol{\mathcal{C}}$}\label{line:for_each_cntrlcmd}\Comment{Section~\ref{sec:telepresence_interface}}
\State{$\psi \xleftarrow{} \tau(c)$}\label{line:teleopmsg}\algorithmiccomment{Generate a teleop message}
\State{$\psi$ is sent to the robot for execution}
\State{Updated map is presented via the interface}
\EndFor
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
In the following subsections, we describe the key components of GHAL360.
\subsection{Scene Analysis}\label{sec:scene_analysis}
Once a new frame ($\mathcal{F}$) is obtained, GHAL360 analyzes the remote scene information using an object detection system, where we use a state-of-the-art convolutional neural network (CNN)-based framework \textbf{YOLOv3}~\cite{yolov3}.
All the objects (keys) in the frame along with their locations (values) are stored in a dictionary $\mathcal{P}$ (Line~\ref{line:objdet}).
In our implementation, we divide the $360\degree$ frame into eight equal $45\degree$ wedges.
These eight wedges together represent all the four cardinal and four intercardinal directions, i.e. [N, S, E, W, NE, NW, SE, SW].
Every wedge can have four different values based on the objects detected in that wedge:
$$
W: \{w_{0},w_{1},w_{2},w_{3}\}
$$
\begin{itemize}
\item $w_{0}$: represents a wedge with no objects detected
\item $w_{1}$: represents a wedge with clutter
\item $w_{2}$: represents a wedge that contains only the object of interest
\item $w_{3}$: represents a wedge containing clutter as well as the object of interest
\end{itemize}
where clutter indicates that the wedge contains one or more objects that are not the target object.
The output of scene analysis is stored as a vector where each element represents the status of one of the wedge.
The human operator can be focusing on one of these eight wedges.
The $getState$ function in Line~\ref{line:get_state} takes the output of the scene analysis along with the target object and current human head pose as input.
The output of $getState$ function is
an egocentric state representation of the vector representing the locations of objects with respect to the wedge human is currently focusing on ($W_{0}$).
The generated egocentric version of the vector is then sent to the RL agent as the state representation of the current world, including both the output of scene analysis and the current status of the human operator.
\subsection{Equirectangular Frame Rendering}\label{sec:equi_rendering}
One of the ways to represent 360-degree videos is to project them onto a spherical surface.
Consider the human head as a virtual camera inside the sphere.
Such spherical projection helps to track the human head pose inside the $360\degree$ frames.
In our implementation, we construct a sphere and use $\mathcal{F}$ as the texture information of the sphere.
Inside the sphere, based on the yaw and pitch of the human head, the pixels in the human field of view are identified.
Based on the human operator's head pose obtained in Line~\ref{line:getHeadPose}, the viewport is rendered from the sphere.
We use \textbf{A-Frame}, an open-source web framework for building virtual reality experiences~\footnote{\url{https://aframe.io/}}, for rendering the equirectangular frames to create an immersive $360\degree$ view of the remote environment.
\subsection{Policy for Guiding Human Attention}\label{subsection:rl_policy}
We use a reinforcement learning approach~\cite{sutton2018reinforcement}, Q-Learning, to let the agent learn a policy (for guiding human attention) by interacting with the environment.
The mathematical representation of the state space is:
$$
S: W_0 \times W_1 \times \cdots \times W_{N-1}
$$
where $W_{i}$ represents the state of a particular wedge, as defined in Section~\ref{sec:scene_analysis}.
In our case, $N=8$, so $|S|=4^{8}$, and $s \in S$.
The state space ($S$) of the RL agent consists of a set of all possible state representations of the $360\degree$ frames.
All the state representations are egocentric, meaning that they represent the state of the world~(locations of objects) with respect to the wedge human is currently focusing on.
The domain consists of three different actions:
$$
A: \{left, right, confirm\}
$$
where $left$ and $right$ actions are the guidance indicators, while the $confirm$ action checks if the object of interest is present in the currently focused wedge or not.
Based on $s$, the policy generated from the RL approach ($\pi$) returns an action ($a$), which can be a guidance indicator to enable the human to find the object of interest in the remote environment.
Once the agent performs the action, it observes the immediate reward ($r$) and the next state ($s'$).
Using the tuple ($s, a, r, s'$), the agent updates the Q-value for the state-action pair.
Finally, the agent updates the policy at runtime based on its interaction with the environment.
We use the BURLAP library~\footnote{\url{http://burlap.cs.brown.edu/}} for the implementation of our reinforcement learning agent and the domain.
\subsection{Particle Filter for Human Intent Estimation}\label{sec:particle_filter}
We use a particle filter-based state estimator to identify the motion intention of the human operator.
In GHAL360, the particle filter is first initialized with $M$ particles:
$$
X = \{x^{0}, x^{1}, \cdots, x^{M-1}\}
$$
where each particle represents a state, which in our case is one of the eight wedges.
Each particle has an associated weight ($\omega^{i}$) and is initialized as $\omega^{i} $=$ \frac{1}{M}$.
The belief state in particle filter is represented as:
$$
B(X_{t}) = P(X_{t} | e_{1:t})
$$
where $t$ is the time step, $e$ is the evidence or the observation of the particle filter, and the belief state $B(X_{t})$ is updated based on the evidence $e_{1}, \cdots, e_{t}$ obtained till time step $t$.
In our implementation, the evidence to the particle filter is the current change in human head orientation (left or right), and the current focused wedge.
As the human starts to look around in the remote environment, based on the evidence the belief is updated:
$$
B^{'}(X_{t+1}) = \sum_{x_{t}} P(X^{'}|x_{t}) B(x_{t}))
$$
After every observation, once the belief is updated, we calculate the density of particles for all the possible states, and the state with more than a threshold percentage of particles (70\% in our case) is considered as the predicted human intention.
Once the particle filter outputs the predicted human intention~(Line~\ref{line:particle_filter}), we use a controller that outputs a control signal to drive the robot base toward the intended area based on the predicted human intention.
In case that the particle density does not reach the threshold in any wedge, the robot stays still until the particle filter can estimate the human motion intention.
\subsection{Telepresence Interface}\label{sec:telepresence_interface}
Based on the action suggested by the policy, GHAL360 overlays visual indicators over the current viewport. This results in a new frame $\mathcal{F^{'}}$ which is presented to the user via the interface~(Fig.~\ref{fig:interface}).
The operator can use ``click'' and ``drag'' operations to look around in the remote environment.
The telepresence interface provides a scroll bar to adjust the field of view of the human operator.
Additionally, the users can easily change the web-based visualization to an immersive virtual reality $360\degree$ experience via head-mounted displays (HMDs) via a \textbf{VR} button at the bottom right of the interface.
In the VR mode, the users can use their head motions to adjust the view of the remote environment.
\begin{figure}[t]
\begin{center}
\vspace{.5em}
\includegraphics[width=8.5cm]{figures/Telepresence_interface_iros_21.pdf}
\end{center}
\vspace{-1.5em}
\caption{\emph{Telepresence Interface} showing the 2D map, the frame of the remote environment, a 2D icon to show human head orientation, a scrollbar to adjust field of view, and a set of icons to indicate the current teleoperation commands given by the human operator.}
\vspace{-1em}
\label{fig:interface}
\end{figure}
The telepresence interface also shows the 2D map of the remote environment.
Based on the robot pose, a red circle is overlaid on the 2D map to indicate the live robot location in the remote environment.
Additionally, we show a 2D icon to convey the head orientation of the human operator relative to the map.
The 2D icon also shows the part of the $360\degree$ frame which is in the field of view of the human operator (Fig.~\ref{fig:interface}).
The controller in Line~\ref{line:controller} converts the estimated human intention into control command.
The control commands are converted to a ROS ``teleop'' message and are passed on to the robot to remotely actuate it.
Based on the control command, the robot is remotely actuated and the 2D map is updated accordingly to show the new robot pose.
\section{Experiments}\label{sec:experiments}
We conducted experiments to evaluate GHAL360 in a target-search scenario where a human was assigned the task of finding a target object in the remote environment using a MTR.
We designed two types of simulation environments, 1. \emph{Abstract simulation}, and 2. \emph{Realistic simulation}.
To facilitate learning, we designed \emph{Abstract simulation} to enable the agent to learn a goal-oriented policy for guiding human attention.
The learned system is then evaluated in \emph{Realistic simulation}.
The main difference between \emph{Abstract simulation} and \emph{Realistic simulation} is that the \emph{Abstract simulation} eliminates the real-time visualization to speed up the learning process.
\begin{figure}
\begin{center}
\subfigure[]
{\includegraphics[height=2cm]{images/envs/env_1_1.jpg}
\label{fig:env_1_1}}
\subfigure[]
{\includegraphics[height=2cm]{images/envs/env_1_2.jpg}
\label{fig:env_1_2}}
\subfigure[]
{\includegraphics[height=2cm]{images/envs/env_2_1.jpg}
\label{fig:env_2_1}}
\subfigure[]
{\includegraphics[height=2cm]{images/envs/env_2_2.jpg}
\label{fig:env_2_2}}
\subfigure[]
{\includegraphics[height=2cm]{images/envs/env_3_1.jpg}
\label{fig:env_3_1}}
\subfigure[]
{\includegraphics[height=2cm]{images/envs/env_3_2.jpg}
\label{fig:env_3_2}}
\caption{(a)-(b),~Home dataset from Matterport3D;
(c)-(d),~Office dataset from Matterport3D; and
(e)-(f),~Office dataset collected using a Segway-based mobile robot platform.
}
\label{fig:env_figures}
\end{center}
\vspace{-1.5em}
\end{figure}
\subsection{Abstract Simulation} \label{sec:abstract_sim}
We use the abstract simulator to let the agent learn a strategy to guide human attention toward an object of interest in a 360$\degree$ frame.
In our abstract simulation domain, we simulate the process of GHAL360 interacting with the ``world'' as a Markov decision process~(MDP)~\cite{puterman2014markov}.
The transition probability of the human following GHAL360's guidance (\emph{left} or \emph{right} actions) is $0.8$ in abstract simulation. There is $0.2$ probability that the human randomly looks around in the remote environment.
The transition probability of the $left$ action and the $right$ action is 0.8 while the transition probability for the $confirm$ action is 1.0.
The reward function is in the form of $R(s,a)\rightarrow \mathbb{R}$.
After the agent takes action $confirm$, if the value of $W_0$ (the wedge that the human is focusing on) is either $w_2$ or $w_3$, as defined in Section~\ref{sec:scene_analysis}, the agent receives a big reward of $250$.
If the value of $W_0$ is either $w_0$ or $w_4$ after action $confirm$, the agent receives a big penalty, in the form of a reward of $-250$.
Each indication action ($left$ or $right$) produces a small cost (negative reward) of $3$ if the value of $W_0$ is $w_0$ or $w_2$.
But if the $left$ or $right$ actions result in the value of $W_0$ becoming $w_1$ or $w_3$, then the actions produces a slightly higher cost (negative) of $15$.
After executing the action ($a$), the agent moves to a next state ($s'$) and receives a reward ($r$).
The reward is based on the value of $W_{0}$ which can be one of the four values $\{w_{0},w_{1},w_{2},w_{3}\}$ as described in Section~\ref{sec:scene_analysis}, and $W_{0}$ is the wedge that the human operator is currently focusing on.
If the agent takes action $confirm$ and transitions to $s'$, where $W_{0}$ value is $w_{2}$ or $w_{3}$, the agent gets a reward as $250$.
But if the agent takes action $confirm$, and the value of $W_{0}$ is either $w_{0}$ or $w_{1}$, the agent gets a reward as $-250$.
The actions $left$ and $right$ result in a reward of $-3$ if $W_{0} = w_{0}$, whereas the reward is $-15$ if the value of $W_{0} = w_{1}$ or $w_{3}$.
We trained the agent for $15000$ episodes which resulted in the agent learning a policy for guiding human attention.
We have evaluated the learned policy in the realistic simulation.
\subsection{Realistic Simulation}
We used the equirectangular frames from the Matterport3D datasets~(Fig.~\ref{fig:env_1_1}~-~\ref{fig:env_2_2}) as well as frames from the 360$\degree$ videos captured from the real world~(Fig.~\ref{fig:env_3_1}~-~\ref{fig:env_3_2}), and then simulated human and robot behaviors to build our realistic simulation.
We used a Segway-based mobile robot platform to capture the $360\degree$ videos~(Fig.~\ref{fig:robot_segway}).
The robot is equipped with an on-board 360$\degree$ camera (Ricoh Theta V), and was teleoperated in a public space during the data collection process.
\begin{figure}[t]
\centering
\subfigure[Front]{\includegraphics[height=4cm]{images/Robot_pose_2.pdf}}
\subfigure[Back]{\includegraphics[height=4cm]{images/Robot_pose_1.pdf}}
\vspace{-.5em}
\caption{Segway-based mobile robot platform (RMP-110) with a $360\degree$ camera (Ricoh Theta V), mounted on top of the robot, and laser ranger finder (SICK TiM571 2D) that has been used for prototyping and evaluation purposes in this research.
}
\vspace{-1em}
\label{fig:robot_segway}
\end{figure}
We model a virtual human in the realistic simulation that can also look around in the remote environment and send control signals to teleoperate the simulated robot using the interface.
Moreover, as opposed to abstract simulation, the actions of the virtual human can be visualized in the interface, for example, if the virtual human looks to the left, the viewport is changed accordingly in the interface.
The virtual human can both track the location of the simulated robot through the 2D map and perceive the 360$\degree$ view of the remote environment via the telepresence interface.
In our realistic simulation, the robot navigates in the 2D grid.
If the virtual human tries to move to an inaccessible position, the robot stays still.
Based on the control signal, the location of the robot is changed, and the interface is updated accordingly.
\subsection{Baselines vs. GHAL360}
We compared GHAL360 using a target-search scenario with different baselines from the literature.
We also conducted ablation studies to investigate how individual components of GHAL360 contribute to the system.
The different systems used for comparison are as follows:
\begin{itemize}
\item \textbf{MFO}: Monocular Fixed Orientation based MTR, where the remote user needs to rotate the robot to look around in the remote environment~\cite{kaplan1997internet}.
\item \textbf{ADV}: All-Degree View systems which are typical MTRs equipped with a $360\degree$ FOV camera. The human can use an interface to look around in the remote environment without having to move the robot base~\cite{heshmat2018geocaching}.
\item \textbf{FGS}: Fixed Guidance Strategy system which consists of a scene analysis component along with a hand-coded guidance strategy that greedily guides the human using the shortest path to the target object.
\item \textbf{RLGS}: Reinforcement Learning-based Guidance Strategy system uses the policy from RL to guide the human attention and suggests an optimal path for the human operator to locate the target object.
\end{itemize}
FGS and RLGS are ablations of approach, and the comparisons of FGS and RLGS with GHAL360 are ablation studies.
We used three different environments: two from the Matterport3D datasets, and one dataset collected from the real-world.
In every environment, there were six sets of start positions for the robot in each of the environments, with each set includes positions of the same distance to the target.
For each environment, we selected two target objects, and each of the target objects was specifically selected to ensure that they were unique in the environment to avoid the confusion resulting from finding a different instance of the same object at different locations.
For example, the target objects selected in environment $1$ were ``backpack'' and ``laptop'', while the target objects selected in environment $2$ were ``bottle'' and ``television''.
The initial orientation of the robot was randomly selected, and a set of one thousand paired trials were carried out for every two-meter distance from the target object until 12 meters.
The virtual human carried out one random action every two seconds out of ``\emph{move forward}'', ``\emph{move backward}'', ``\emph{look left}'', and ``\emph{look right}'' in the baselines (MFO and ADV).
The actions ``\emph{move forward}'', ``\emph{move backward}'' are also carried out randomly in FGS and RLGS, while the actions ``\emph{look left}'' and ``\emph{look right}'' are carried out randomly only until there are no visual indications.
In case, there are visual indications, the human follows the indications with a $0.95$ probability.
Using these actions, the simulated user could explore the remote environment.
Once the target object is in the field of view of the human operator, the trial is terminated.
\begin{figure}[tb]
\begin{center}
\vspace{.5em}
\includegraphics[width=8cm]{images/trial_complete/main_fig_v6.pdf}
\end{center}
\vspace{-.5em}
\caption{Map showing the trajectory of the robot in the remote environment looking for a ``laptop.'' The arrows show the different human orientations along the trajectory; The images marked as (a)-(e) are the different viewports of the human at every orientation shown by the arrows.}
\label{fig:illustrative_trial_main_fig}
\end{figure}
\subsection{Illustrative Trial}
Consider an illustrative trial where the human-MTR team needed to locate a laptop in a remote environment.
Fig.~\ref{fig:illustrative_trial_main_fig} shows the map of the remote environment with the robot's trajectory marked in blue color.
The arrows in Fig.~\ref{fig:illustrative_trial_main_fig} indicate the different human head orientations, and the red dotted lines point to the different viewports at these orientations.
Fig.~\ref{fig:illustrative_trial_main_fig}~(a) shows the initial viewport of a virtual human.
The human started to look around at the remote environment to locate the laptop.
Fig.~\ref{fig:illustrative_trial_main_fig}~(b) shows the second viewport after the human looked left.
Then the human again looked right, and the viewport can be seen in Fig.~\ref{fig:illustrative_trial_main_fig}~(a).
The human kept looking at the same position for next two time steps.
The particle filter estimated the human's intention, and accordingly, the controller output control signals to drive the robot to the next location.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=6.5cm]{figures/policy_evolution_with_baseline.pdf}
\end{center}
\vspace{-1em}
\caption{Comparison of the cumulative reward between FGS and GHAL360. Results show that the learned policy ultimately performed better than FGS (hand-coded baseline).}
\label{fig:policy_evolution}
\end{figure}
The human continued to look around to find the laptop, and the particle filter constantly estimates the human motion intention resulting in the robot moving to different locations as shown by the blue marked trajectory in the map~(Fig.~\ref{fig:illustrative_trial_main_fig}).
Finally, the robot reached location where the target object was located.
As soon as the robot reached the location of the target object, the scene analysis component detected the laptop in the frame, and using the human head pose, GHAL360 overlayed the visual indicator over the viewport as seen in Fig.~\ref{fig:illustrative_trial_main_fig}~(d).
The indicator suggested the human to look right to find the laptop, but at first the human did not follow the guidance and looked left.
Again, the visual indicator was overlayed on the new viewport to guide the human attention~(Fig.~\ref{fig:illustrative_trial_main_fig}~(c)).
Now, at this time step, the human looked in the guided direction, and the viewport changed back to Fig.~\ref{fig:illustrative_trial_main_fig}~(d).
Finally, the human followed the visual indicator and looked further right, and Fig.~\ref{fig:illustrative_trial_main_fig}~(e) shows the resulting viewport of the human with the laptop.
Once the human found the target object, the trial was terminated.
\begin{figure*}[t]
\begin{center}
\subfigure[Env 1: Matterport3D (Home)]
{\includegraphics[width=5.5cm]{figures/env_2.pdf}
\label{fig:env_1_plot}}
\subfigure[Env 2: Matterport3D (Office)]
{\includegraphics[width=5.5cm]{figures/env_3.pdf}
\label{fig:env_2_plot}}
\subfigure[Env 3: Real Robot (Office)]
{\includegraphics[width=5.5cm]{figures/env_4.pdf}
\label{fig:env_3_plot}}
\caption{Comparisons are made between GHAL360 and the other systems in terms of average time (y-axis) taken to find the target object from six different distances (x-axis).
}
\label{fig:time_vs_distance}
\end{center}
\vspace{-1em}
\end{figure*}
\subsection{Results}
From the $15000$ training episodes, we extracted $150$ different policies after each batch of $100$ episodes, and tested them in our realistic simulation.
Fig.~\ref{fig:policy_evolution} shows the average cumulative reward and standard deviations over five runs.
We plotted the mean cumulative reward (Fig.~\ref{fig:policy_evolution}) of FGS and GHAL360 to compare the performance of hand-coded policy with the learned policy of GHAL360.
The x-axis represents the episode number while the y-axis represents the cumulative reward.
Fig.~\ref{fig:policy_evolution} shows that in the early episodes when the agent is still exploring, the cumulative reward for GHAl360 is much lower than FGS.
But with the increasing number of episodes, it can be observed that GHAL360 outperforms FGS which represents the learned policy is superior to the hand-coded policy.
Fig.~\ref{fig:time_vs_distance} shows the overall performance of GHAL360 compared to the other systems enlisted above.
Each data point represents the average task completion time (y-axis) for six different distances (x-axis) (i.e., randomly sampled positions in each of the six distances).
One thousand trials were divided over five runs to plot each data point.
The shaded regions denote the standard deviations over five runs for each of the six distances.
From the results, it can be observed that GHAL360 outperforms all the baselines (MFO and ADV) in terms of average task completion time at all six distances in finding the target object.
Moreover, compared to FGS, GHAL360 performs better in all three environments and at six different distances denoting that the human guidance using the learned strategy outperforms the hand-coded greedy.
Furthermore, in addition to the learned guidance strategy, GHAL360 utilizes a particle filter to estimate human intention and guide the robot base toward the intended direction.
In the bottom left of Fig.~\ref{fig:env_1_plot}~-~\ref{fig:env_3_plot}, we can see that the average time of task completion is similar between RLGS and GHAL360.
This indicates that particle filter in GHAL360 is less useful when the robot starts from a position that is close to the target object's.
When the robot is distant from the target object, we see the particle filter produces a significant improvement, as evidenced by seeing the far right of the Fig.~\ref{fig:env_1_plot}~-~\ref{fig:env_3_plot}.
This observation confirms that the particle filter also contributes to the efficiency of GHAL360.
In addition to comparing the efficiency of different systems to GHAL360, we also compared the accuracy of target search tasks as shown in Table~\ref{table:accuracy}.
It can be observed that the accuracy of GHAL360 is higher in finding all the target objects, except for the ``backpack'' object in Environment 3.
Also, we observed significant improvement in accuracy in Environment 2 for find the ``television''.
This shows that along with the improvements in efficiency, GHAL360 also provides higher accuracy in finding target objects as compared to the other systems.
\begin{table}[ht]
\scriptsize
\caption{This table shows the accuracy of different systems in target search tasks for different environments with different target objects. Bold font shows the best accuracy among all the systems, where as the bold and italic font represent the significant improvements.
}
\centering
\begin{center}
\setlength{\tabcolsep}{5pt}
\begin{tabular}{ |l|c|c|c|c|c| }
\hline
Env (target) & \multicolumn{5}{|c|}{Accuracy} \\
\hline
& MFO & ADV & FGS & RLGS & GHAL360\\
\hline
Env 1 (backpack) & .63 (.02) & .61 (.02) & .74 (.02) & .82 (.02) & \textbf{.84 (.02)}\\
\hline
Env 1 (laptop) & .65 (.01) & .66 (.02) & .75 (.02) & .82 (.02) & \textbf{.85 (.01)}\\
\hline
Env 2 (bottle) & .54 (.02) & .52 (.03) & .68 (.01) & .78 (.01) & \textbf{.81 (.02)}\\
\hline
Env 2 (television) & .58 (.01) & .59 (.01) & .71 (.01) & .80 (.03) & \textit{\textbf{.87 (.03)}}\\
\hline
Env 3 (backpack) & .55 (.02) & .57 (.02) & .68 (.03) & .82 (.03) & .82 (.02)\\
\hline
Env 3 (laptop) & .56 (.02) & .56 (.01) & .69 (.02) & .80 (.01) & \textbf{.83 (.02)}\\
\hline
Overall & .58 (.04) & .58 (.05) & .71 (.03) & .81 (.02) & \textbf{.84 (.02)}\\
\hline
\end{tabular}
\label{table:accuracy}
\end{center}
\vspace{-2.5em}
\end{table}
\section{Conclusions and Future Work}
In this paper, we develop a framework (GHAL360) that enables a mobile telepresence robot (MTR) to monitor the remote environment with all-degree scene analysis and actively guide the human attention to the key areas at the same time.
We conducted experiments to evaluate GHAL360 in a target search scenario.
From the results, we observed significant improvements in the efficiency of the human-MTR team in comparison to different baselines from the literature.
In the future, we would like to implement and evaluate GHAL360 on a real robot, and conduct studies with human participants for subjective analysis.
Currently, the scene analysis component does not go beyond object detection in the current implementation of GHAL360.
We would like to study if and how advanced scene analysis methods, such as those based on graph neural networks~\cite{scarselli2008graph}, can contribute to the system.
Finally, our current controller suggests the robot to follow the human's intention, which can possibly be improved using a planner to actively guide the human's attention toward the direction of the target object, even when the object is not present in the immediate surroundings.
\section*{Acknowledgment}
A portion of this research has taken place at the Autonomous Intelligent Robotics (AIR) Group, SUNY
Binghamton. AIR research is supported in part by grants from the National Science Foundation (NRI-1925044), Ford Motor Company (URP Awards 2019-2021), OPPO (Faculty Research Award 2020), and SUNY Research Foundation. The work of Xiaoyang Zhang and Yao Liu is partially supported by NSF under grant CNS-1618931.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,662
|
// Copyright (c) 2001-2008 Hartmut Kaiser
//
// Distributed under the Boost Software License, Version 1.0. (See accompanying
// file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
#if !defined(BOOST_SPIRIT_STREAM_MAY_05_2007_1228PM)
#define BOOST_SPIRIT_STREAM_MAY_05_2007_1228PM
#if defined(_MSC_VER) && (_MSC_VER >= 1020)
#pragma once // MS compatible compilers support #pragma once
#endif
#include <boost/spirit/home/qi/detail/string_parse.hpp>
#include <boost/spirit/home/qi/stream/detail/match_manip.hpp>
#include <boost/spirit/home/qi/stream/detail/iterator_istream.hpp>
#include <boost/spirit/home/support/detail/hold_any.hpp>
#include <iosfwd>
#include <sstream>
///////////////////////////////////////////////////////////////////////////////
namespace boost { namespace spirit
{
// overload the streaming operators for the unused_type
template <typename Char, typename Traits>
inline std::basic_istream<Char, Traits>&
operator>> (std::basic_istream<Char, Traits>& is, unused_type&)
{
return is;
}
}}
///////////////////////////////////////////////////////////////////////////////
namespace boost { namespace spirit { namespace qi
{
template <typename Char, typename T = spirit::hold_any>
struct any_stream
{
template <typename Component, typename Context, typename Iterator>
struct attribute
{
typedef T type;
};
template <
typename Component
, typename Iterator, typename Context
, typename Skipper, typename Attribute>
static bool parse(
Component const& /*component*/
, Iterator& first, Iterator const& last
, Context& /*context*/, Skipper const& skipper
, Attribute& attr)
{
typedef qi::detail::iterator_source<Iterator> source_device;
typedef boost::iostreams::stream<source_device> instream;
qi::skip(first, last, skipper);
instream in (first, last);
in >> attr; // use existing operator>>()
return in.good() || in.eof();
}
template <typename Component, typename Context>
static std::string what(Component const& component, Context const& ctx)
{
return "any-stream";
}
};
}}}
#endif
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,884
|
{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/evaluate-following-integral-x-2-x-4-x-2-12-d-x-evaluation-of-simple-integrals-of-the-following-types-and-problems_44992","text":"# Evaluate the Following Integral: \u222b X 2 X 4 \u2212 X 2 \u2212 12 D X - Mathematics\n\nSum\n\nEvaluate the following integral:\n\n$\\int\\frac{x^2}{x^4 - x^2 - 12}dx$\n\n#### Solution\n\n$\\text{Let }I = \\int\\frac{x^2}{x^4 - x^2 - 12}dx$\n\nWe express\n\n$\\frac{x^2}{x^4 - x^2 - 12} = \\frac{x^2}{x^4 - 4 x^2 + 3 x^2 - 12}$\n\n$= \\frac{x^2}{\\left( x^2 - 4 \\right)\\left( x^2 + 3 \\right)}$\n$= \\frac{A}{x^2 - 4} + \\frac{B}{x^2 + 3}$\n$\\Rightarrow x^2 = A\\left( x^2 + 3 \\right) + B\\left( x^2 - 4 \\right)$\n\nEquating the coefficients of x^2 and constants, we get\n\n$1 = A + B\\text{ and }0 = 3A - 4B$\n$\\text{or }A = \\frac{4}{7}\\text{ and }B = \\frac{3}{7}$\n$\\therefore I = \\int\\left( \\frac{\\frac{4}{7}}{x^2 - 4} + \\frac{\\frac{3}{7}}{x^2 + 3} \\right)dx$\n$= \\frac{4}{7}\\int\\frac{1}{x^2 - 4}dx + \\frac{3}{7}\\int\\frac{1}{x^2 + 3} dx$\n$= \\frac{4}{7} \\times \\frac{1}{4}\\log\\left| \\frac{x - 2}{x + 2} \\right| + \\frac{\\sqrt{3}}{7} \\tan^{- 1} \\frac{x}{\\sqrt{3}} + c$\n$= \\frac{1}{7}\\log\\left| \\frac{x - 2}{x + 2} \\right| + \\frac{\\sqrt{3}}{7} \\tan^{- 1} \\frac{x}{\\sqrt{3}} + c$\n$\\text{Hence, }\\int\\frac{x^2}{x^4 - x^2 - 12}dx = \\frac{1}{7}\\log\\left| \\frac{x - 2}{x + 2} \\right| + \\frac{\\sqrt{3}}{7} \\tan^{- 1} \\frac{x}{\\sqrt{3}} + c$\n\nConcept: Evaluation of Simple Integrals of the Following Types and Problems\nIs there an error in this question or solution?\n\n#### APPEARS IN\n\nRD Sharma Class 12 Maths\nChapter 19 Indefinite Integrals\nQ 66 | Page 178","date":"2022-05-22 08:16:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5320762991905212, \"perplexity\": 3276.644310453501}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662545090.44\/warc\/CC-MAIN-20220522063657-20220522093657-00675.warc.gz\"}"}
| null | null |
\section{Introduction}
Multiple-input multiple-output or MIMO wireless systems have received a significant amount of interest due to their capability of
dramatically increasing the capacity of a communication link~\cite{Telatar99,FoschiniGans98}. Also, there has been
considerable work on a variety of schemes which exploit multiple antennas at both the transmitter and receiver in order
to obtain spatial diversity, i.e. to improve the reliability of the system such as orthogonal space-time block
codes (OSTBC)~\cite{Alamouti,TarokhJafarkCalder99} and space-time trellis codes~\cite{TarokhSeCa98} (STTC). An simple
example for a STTC is the delay diversity code, first proposed in~\cite{WittnebenSTTC} and also discussed
in~\cite{SeshadriWint94,TarokhSeCa98}, where the data stream on the first transmit antenna is transmitted with delays
on the other antennas. The delay diversity code achieves full diversity but has a disadvantage of a slight rate loss
due to some leading and tailing zeros. In order to avoid this drawback, cyclic delay diversity (CDD) has been proposed
in~\cite{Gore01delay}.
The main advantage of CDD over OSTBC is that a code rate of one is achieved independent of the number of transmit
antennas, whereas OSTBC suffer a rate loss by increasing the number of transmit
antennas~\cite{TarokhJafarkCa99,X.B.Liang2003,WangXiaOSTBC}. The performance of CDD in terms of error probability was
analyzed in\cite{Gore01delay} for frequency flat and frequency selective channels. The diversity-multiplexing tradeoff
for delay diversity was characterized in~\cite{VazeRajanDD}. Based on the assumption that statistical information about
the channel is available at the transmitter, transmit filters employed at the transmitter are optimized
in~\cite{SchoberHehn} in order to minimize the Chernoff bound on the error probability of CDD. The average rate for CDD
by assuming Gaussian as well as PSK/QAM input signals for a point-to-point transmission with $n_T=2$ or $n_T=4$
transmit antennas and $n_R\leq 2$ receive antennas was investigated in terms of Monte-Carlo simulations in~\cite{BauchCDD}.
As noticed in~\cite{BoelcskeiMacStc}, most of the work on space-time coding so far is focused on single-user systems.
For such single-user systems, it was shown that the loss in terms of spectral efficiency for transmit diversity schemes
increases significantly for more than one receive antenna~\cite{Sandhu}, making them inappropriate for high rate
transmission. In this work, by assuming a Gaussian codebook, we analyze the ergodic rate performance of a multiple
access channel (MAC) system with multiple users transmitting their data to the base station by using a CDD. At the
receiver, we apply a joint maximum-likelihood (ML) detector. We compare the achieved ergodic sum-rate with the
sum-capacity of a MAC system assuming that the users do not have channel state information (CSI) and the base station
has full CSI. We show that, if the number of receive antennas $n_R$ is less than the number of users $K$, i.e. $n_R\leq
K$ in the system (which is a reasonable assumption), the loss of the proposed scheme in terms of spectral efficiency is
negligible. This is in strong contrast to single-user systems~\cite{Sandhu,BauchCDD}. Thus, CDD is indeed an attractive
candidate for transmit strategies in MIMO multi-user systems.
The remainder of the paper is organized as follows. In Section~\ref{sec:System}, we introduce the system model and the
construction of the CDD. The performance of CDD in terms of spectral efficiency is then analyzed and compared to the
sum-capacity in section~\ref{sec:PerfAn}, followed by simulations and concluding remarks in Section~\ref{sec:Simu}
and~\ref{sec:Conc}.
Notational conventions are as follows. We will use bold-faced upper case letters to denote matrices, e.g.,
$\mathbf{X}$, with elements $x_{i,j}$; bold-faced lower case letters for column vector, e.g., $\mathbf{x}$, and
light-faced letters for scalar quantities. The superscripts $(\cdot)^T$ and $(\cdot)^H$ denote the transpose and
Hermitian operations, respectively. Finally, the identity matrices and all zero vectors of the required dimensions will
be denoted by $\mathbf{I}$ and $\mathbf{0}^T$. $\mathbb{E}[\cdot]$ will denote the expectation operator.
\section{System model}\label{sec:System}
In this work, we consider a multiuser multiple access channel (MAC) with $K$ users each with $n_T$ transmit antennas
and $n_R$ receive antennas at the base station as depicted in Fig.~\ref{fig:MAC}.
\begin{figure}
\begin{center}
\psfrag{BS}{$BS$}
\psfrag{MS1}{${MS}_1$}
\psfrag{MS2}{$MS_2$}
\psfrag{MS3}{$MS_K$}
\includegraphics[scale=0.6]{MacCell.eps}
\caption{Multiuser MIMO multiple access channel (MAC) with users $MS_1$,\dots, $MS_K$ transmitting data to
base station (BS).}~\label{fig:MAC}
\end{center}
\end{figure}
The system model is given as
\begin{align}\label{eq:SysMod}
\mathbf{Y}=\sum_{k=1}^{K}\sqrt{\frac{\mathsf{SNR}}{n_T}}\mathbf{G}_k\mathbf{H}_k^T+\mathbf{N},
\end{align}
where $\mathsf{SNR}$ denotes the signal-to-noise ratio, $\mathbf{G}_k$ is the transmit matrix of size $[T \times n_T]$,
$T$ is the block length and $\mathbf{N}$ ($[T \times n_R]$) is the additive white Gaussian noise (AWGN) matrix
$\mathbf{N} \sim \mathcal{CN}(0,\mathbf{I})$. In this work, we focus on delay optimal codes, in the sense that the decoding is performed after $T$ channel uses with
$n_T=T$\cite{TarokhJafarkCalder99,TarokhJafarkCa99}. The channel between user $k$, $1\leq k \leq K$, and the base
station is modeled by a random zero-mean component channel $\mathbf{H}_k$ ($[n_R \times n_T]$) with identically
independent distributed (iid) complex Gaussian entries, i.e. $\mathbf{H}_k \sim \mathcal{CN}(0,\mathbf{I})$, $1\leq k
\leq K$.
We assume that each user
$k$ is applying a cyclic delay diversity scheme (for simplicity we skip the user index) , given as
\begin{align}
\mathbf{G}_k(\mathbf{x}_k)=\mathbf{G}(\mathbf{x})=\left[%
\begin{array}{cccc}
x_1 & x_2 & \cdots & x_T \\
x_T & x_1 & \cdots & x_{T-1} \\
x_{T-1} & x_T & \cdots & x_{T-2} \\
\vdots & \ddots & \ddots & \ddots \\
x_3 & x_4 & \ddots & x_2 \\
x_2 & x_3 & \cdots & x_1
\end{array}%
\right]
\end{align}
where $\mathbf{x}_k=[x_1,x_2,\dots,x_T]^T$ is the vector of transmitted symbols of user $k$.
\section{Performance analysis}~\label{sec:PerfAn}
In the following, we analyze the ergodic sum-rate performance achieved with the proposed scheme and compare it to the
ergodic MAC sum-capacity. We start with the case, where only one antenna is available at the base station, i.e.
$n_R=1$, and afterwards generalize it to higher number of receive antennas. The separation of these two cases is
motivated by the fact that the spectral efficiency for transmit diversity schemes in single-user systems with one
receiving antenna is close to (e.g. QSTBC) or even equal to capacity (Alamouti scheme)~\cite{Sandhu}. With more than
one receive antenna, this behavior changes significantly to the disadvantage of the transmit diversity schemes. As we
will see later on, for multi-user systems the picture looks different.
\subsection{$n_R=1$ antenna at the base station}
For $n_R=1$ at the base station, the system equation in~\eqref{eq:SysMod} reduces to
\begin{align}\nonumber
\mathbf{y}=\sum_{k=1}^{K}\sqrt{\frac{\mathsf{SNR}}{n_T}}\mathbf{G}_k\mathbf{h}_k^T+\mathbf{n}
\end{align}
with $\mathbf{h}_k$ of size $[1\times n_T]$, which can be rewritten as
\begin{align}\nonumber
\mathbf{\tilde{y}}=\sum_{k=1}^K\sqrt{\frac{\mathsf{SNR}}{n_T}}\mathbf{\tilde{H}}_k\mathbf{x}_k+\tilde{\mathbf{n}}
=\sqrt{\frac{\mathsf{SNR}}{n_T}}\mathbf{\tilde{H}}\mathbf{x}+\tilde{\mathbf{n}}
\end{align}
with where the circulant matrices $\mathbf{\tilde{H}}_{k}$, $1\leq k \leq K$,
\begin{align}\nonumber
\mathbf{\tilde{H}}_{k}=\left[%
\begin{array}{cccc}
h_{1}^k & h_{2}^k & \cdots & h_{T}^k \\
h_{2}^k & h_{3}^k & \cdots & h_{1}^k \\
h_{3}^k & h_{4}^k & \cdots & h_{2}^k \\
\vdots & \ddots & \ddots & \ddots \\
h_{T-1}^k & h_{T}^k & \ddots & h_{T-2}^k \\
h_{T}^k & h_{1}^k & \cdots & h_{T-1}^k
\end{array}%
\right]
\end{align}
are the effective channels between user $k$ and the base station,
$\mathbf{\tilde{H}}=\left[\mathbf{\tilde{H}}_{1},\mathbf{\tilde{H}}_{2},\dots,\mathbf{\tilde{H}}_{K}\right]$ and
$\mathbf{x}=[\mathbf{x}_1^T, \mathbf{x}_2^T, \dots, \mathbf{x}_K^T]^T$ is obtained by stacking the transmit signals of
the users into one large vector. The instantaneous rate achievable with the circulant code is then given as
\begin{align}
I_c &=\frac{1}{T}\log_2\det\left(\mathbf{I}_T+\frac{\mathsf{SNR}}{n_T}\mathbf{\tilde{H}}\mathbf{\tilde{H}}^H\right) \nonumber \\
&
=\frac{1}{T}\log_2\det\left(\mathbf{I}_T+\frac{\mathsf{SNR}}{n_T}\sum_{k=1}^K\mathbf{\tilde{H}}_k\mathbf{\tilde{H}}_k^H\right).\nonumber
\end{align}
Since the $\mathbf{\tilde{H}}_k$, $1\leq k \leq K$, are all circulant matrices, they can be simultaneously diagonalized
by the unitary DFT matrix $\mathbf{D}$~\cite{Davis}, i.e. $\mathbf{\tilde{H}}_k\mathbf{\tilde{H}}_k^H=\mathbf{D} n_T \mathbf{\Lambda}_k\mathbf{D}^H$. Therefore, $I_c$ results in
\begin{align}
I_c & =\frac{1}{T}\log_2\det\left(\mathbf{I}_T+\frac{\mathsf{SNR}}{n_T}\sum_{k=1}^K\mathbf{D} n_T \mathbf{\Lambda}_k\mathbf{D}^H\right),\nonumber\\
& =\frac{1}{T}\log_2\det\left(\mathbf{I}_T+\frac{\mathsf{SNR}}{n_T}n_T\mathrm{diag}
\left(\lambda_1,\lambda_2,\dots,\lambda_T\right)\right),\nonumber
\end{align}
where the $\lambda_t$ are i.i.d. chi-square distributed random variables with $2K$ degrees of freedom, i.e. $\lambda_t
\sim \chi_{2K}^2$. For the ergodic sum-rate performance given as $R_c=\mathbb{E}[I_c]$, we thus have
\begin{align}\nonumber
R_c=\frac{1}{T}\mathbb{E}\left[\sum_{t=1}^T\log_2\left(1+\mathsf{SNR}\lambda_t\right)\right]
=\mathbb{E}\left[\log_2\left(1+\mathsf{SNR}\lambda\right)\right],
\end{align}
where $\lambda \sim \chi_{2K}^2$. In order to derive a lower bound on the above expression, we make use of the fact
that $\log_2\left(1+a e^x\right)$ is a convex function with $a>0$, thus applying Jensen's inequality results in
\begin{align}
R_c \geq \log_2\left(1+\mathsf{SNR}\exp\left(\mathbb{E}\left[\log_2\left(\lambda\right)\right]\right)\right).\nonumber
\end{align}
From~\cite{OymanNBPaulraj,GrantUppBound} we know that
\begin{align}\label{eq:EWChi2DiGam}
\mathbb{E}\left[\log_2\left(\lambda\right)\right]=\psi(K)=\sum_{k=1}^{K-1}\frac{1}{k} -\gamma,
\end{align}
where $\psi(\cdot)$ is the digamma or psi function~\cite[p.943, eq.8.360]{Gradshteyn} and $\gamma \approx 0.577$ is
Euler's constant. Thus, for $n_R=1$ receive antenna at the base station, the average sum-rate performance of the
proposed scheme is lower bounded by
\begin{align}\label{eq:LbCddRateNr1}
R_c \geq \log_2\left(1+\mathsf{SNR}\exp\left(\sum_{k=1}^{K-1}\frac{1}{k} -\gamma\right)\right).
\end{align}
In contrast, note that the sum-capacity of the multiuser MAC system can be lower bounded by
\begin{align}\label{eq:LbSumCapNr1}
C \geq \log_2\left(1+\frac{\mathsf{SNR}}{n_T}\exp\left(\sum_{k=1}^{n_TK-1}\frac{1}{k} -\gamma\right)\right).
\end{align}
The result in~\eqref{eq:LbSumCapNr1} is obtained by applying the techniques in~\cite{OymanNBPaulraj} and the fact that
for no CSIT the ergodic sum capacity of a K users MAC channel, where each user has $n_T$ transmit antennas, is
equivalent to the ergodic capacity of a single-user system with $Kn_T$ transmit antennas~\cite[Proposition
1]{RheeCioffi},\cite{Telatar99}. By comparing~\eqref{eq:LbCddRateNr1} and~\eqref{eq:LbSumCapNr1}, we observe that the
difference between the ergodic sum-rate achieved with CDD and the sum-capacity, i.e. $C-R_c$ converges to
\begin{align}
& \frac{1}{\ln(2)}\sum_{k=K}^{n_TK-1}\frac{1}{k}-\ln(n_T) \nonumber\\
& =\frac{1}{K\ln(2)}\left(1+K\left(\sum_{k=K+1}^{n_TK-1}\frac{1}{k}-\ln(n_T)\right)\right)\label{eq:RateDiff}
\end{align}
for high $\mathsf{SNR}$. This result is restricted to high $\mathsf{SNR}$, where the lower bounds in~\eqref{eq:LbCddRateNr1} and~\eqref{eq:LbSumCapNr1} are tight and thus can be used for computing the difference $C-R_c$. Note that $\sum_{k=K+1}^{n_TK-1}\frac{1}{k}$ is a nondecreasing function in $K$ and $n_T$.
Further, note that
\begin{align}
& \lim_{K\rightarrow \infty} \sum_{k=K+1}^{n_TK-1}\frac{1}{k}=\lim_{K\rightarrow \infty}
\sum_{k=1}^{n_TK-1}\frac{1}{k}-\sum_{k=1}^{K}\frac{1}{k} \nonumber\\
&=\lim_{K\rightarrow \infty} \psi(n_TK)-\psi(K+1)=\ln(n_T). \label{eq:LimPsiFunc}
\end{align}
Thus, for $n_R=1$ the difference in~\eqref{eq:RateDiff}, i.e. $C-R_c$, is upper bounded by
\begin{align}
C-R_c\leq \frac{1}{K \ln(2)}
\end{align}
for high SNR, i.e. the more the number of users the less is the difference between the actual capacity and the rate
achieved with CDD.
\subsection{Arbitrary number of receive antennas $n_R$}
For higher number of receive antennas the effective (sometimes also referred to as equivalent) channel to each user $k$
is given as
\begin{align}
\mathbf{\tilde{H}}_k=\left[\mathbf{\tilde{H}}_{k,1}^T, \mathbf{\tilde{H}}_{k,2}^T, \dots,
\mathbf{\tilde{H}}_{k,n_R}^T\right]^T\nonumber
\end{align}
where
\begin{align}\nonumber
\mathbf{\tilde{H}}_{k,i}=\left[%
\begin{array}{cccc}
h_{i,1}^k & h_{i,2}^k & \cdots & h_{i,T}^k \\
h_{i,2}^k & h_{i,3}^k & \cdots & h_{i,1}^k \\
h_{i,3}^k & h_{i,4}^k & \cdots & h_{i,2}^k \\
\vdots & \ddots & \ddots & \ddots \\
h_{i,T-1}^k & h_{i,T}^k & \ddots & h_{i,T-2}^k \\
h_{i,T}^k & h_{i,1}^k & \cdots & h_{i,T-1}^k
\end{array}%
\right]
\end{align}
with
\begin{align}
\mathbf{\tilde{H}}=\left[\mathbf{\tilde{H}}_{1},\mathbf{\tilde{H}}_{2},\dots,\mathbf{\tilde{H}}_{K}\right]
\end{align}
being a $[n_Rn_T \times n_TK]$ matrix. Thus, $\mathbf{\tilde{H}}$ is a block matrix with circulant blocks. Similarly to
the single receive antenna case, the instantaneous rate with arbitrary $n_T$, $n_R$ and $K$ users achievable with the
circulant code is then given as
\begin{align}\label{eq:RateGener}
I_c &=\frac{1}{T}\log_2\det\left(\mathbf{I}_{n_Rn_T}+\frac{\mathsf{SNR}}{n_T}\mathbf{\tilde{H}}\mathbf{\tilde{H}}^H\right).
\end{align}
We are now able to state the following theorem.
\begin{theorem}~\label{theo:SumRate}
The ergodic-sum rate of a multi-user MAC system with K users each equipped with $n_T$ transmit antennas applying a CDD
and $n_R$ receive antennas at the base station is lower bounded by
\begin{align}\label{eq:RateCDDNRGen}
R_c \geq \sum_{l=1}^{L}\log_2\left(1+\mathsf{SNR} \exp\left(\sum_{k=1}^{M-l}\frac{1}{k}-\gamma\right)\right),
\end{align}
with $L=\min(n_R,K)$ and $M=\max(n_R,K)$.
\end{theorem}
\proof: The proof is given in the Appendix.
In comparison, using similar techniques the sum-capacity of the multiuser MAC system with $K$ users, each with $n_T$ transmit and $n_R$ receive
antennas can be lower bounded by
\begin{align}\label{eq:LBCapGen}
C \geq \sum_{l=1}^{\tilde{L}}\log_2\left(1+\frac{\mathsf{SNR}}{n_T}\exp\left(\sum_{k=1}^{\tilde{M}-l}\frac{1}{k}
-\gamma\right)\right)
\end{align}
with $\tilde{L}=\min(n_R,n_TK)$, $\tilde{M}=\max(n_R,n_TK)$.
By applying Jensens inequality, the lower bound on $R_c$ in~\eqref{eq:RateCDDNRGen} and the lower bound for $C$ in~\eqref{eq:LBCapGen}
may be bounded from below by
\begin{align}\label{eq:RcBound2}
R_c \geq L\log_2\left(1+\mathsf{SNR}
\exp\left(\frac{1}{L}\sum_{l=1}^{L}\sum_{k=1}^{M-l}\frac{1}{k}-\gamma\right)\right)
\end{align}
and
\begin{align}\label{eq:CBound2}
C \geq \tilde{L}\log_2\left(1+\frac{\mathsf{SNR}}{n_T}\exp\left(\frac{1}
{\tilde{L}}\sum_{l=1}^{\tilde{L}}\sum_{k=1}^{\tilde{M}-l}\frac{1}{k} -\gamma\right)\right),
\end{align}
respectively. These new lower bounds get tight for high SNR~\cite{OymanNBPaulraj}. Assume that $n_R \leq K$, which is a
reasonable assumption. Then, applying similar steps as in~\eqref{eq:RateDiff}-~\eqref{eq:LimPsiFunc}, using~\eqref{eq:RcBound2} and~\eqref{eq:CBound2} the difference
$C-R_c$ may be upper bounded by
\begin{align}\nonumber
C-R_c \leq &
\frac{1}{\ln(2)}\sum_{l=1}^{n_R}\frac{1}{K-l+1}\Bigg(1+(K-l+1) \\
& \times \ln\left(\frac{1}{n_T}+\frac{(n_T-1)K}{n_T(K-l+1)}\right)\Bigg)
\end{align}
for high SNR. Thus, increasing $K$ reduces the difference $C-R_c$, while increasing $n_R$ increases the difference $C-R_c$.
In addition to the lower bound, an upper bound is also derived following the approach of~\cite{GrantUppBound}. Applying
Jensen's inequality to~\eqref{eq:BDMwMIMO} results in
\begin{align}
R_c \leq \log_2\mathbb{E}\left[\det\left(\mathbf{I}+\mathsf{SNR}\mathbf{H}'_1\mathbf{H}'^H_1\right)\right]\nonumber
\end{align}
due to the concavity of the logarithm.
With~\cite[Theorem A.4, eq.(A.11)]{GrantUppBound}, we arrive at
\begin{align}\label{eq:BoundGrant}
R_c \leq \log_2\left(\sum_{i=0}^L \binom{L}{i}\frac{M!}{(M-i)!}\mathsf{SNR}^i\right).
\end{align}
In the following section, our theoretical results are illustrated by numerical simulations.
\section{Simulations}\label{sec:Simu}
\subsection{Single-User case}
In Fig.~\ref{fig:CapSU}, the ergodic rate $R_c$ achievable with the CDD and the ergodic capacity for a single-user
system are depicted with $n_T=4$ transmit and $n_R=1$ and $n_R=2$ receive antennas, respectively. In addition to this,
the lower bound on $R_c$ in~\eqref{eq:RateCDDNRGen} derived in the previous section is also depicted. From the figure,
we observe that for $n_R=1$, $R_c$ is close to the capacity. By increasing $n_R$, we observe that the loss increases
significantly. The lower bound on $R_c$ confirms this behavior and verifies the numerical results obtained
in~\cite{Sandhu,BauchCDD} that transmit diversity schemes perform poorly for $n_R>1$ in single-user systems.
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{SingleUserCDD.eps
\end{center}
\caption{Single-user ergodic rate of the CDD and capacity. $n_R=1,2$ and $n_T=4$. Lower bounds obtained
from~\eqref{eq:RateCDDNRGen} are also depicted.} \label{fig:CapSU}
\end{figure}
\subsection{Multi-User case}
In Fig.~\ref{fig:CapRegion}, the ergodic capacity region of a symmetric system with $K=2$ users each equipped with $n_T=2$
transmit antennas and a base station with $n_R=2$ receive antennas is depicted for three different $\mathsf{SNR}$
values ranging from $\mathsf{SNR}=0$~dB to $\mathsf{SNR}=40$~dB. In addition to that, the ergodic rate region
achievable with CDD is depicted. The corner points of the regions can be achieved by a successive cancelation decoder.
Note that the capacity and the maximum rate achievable with CDD of the point-to point link with the other user absent
from the system (or equivalently completely canceled) are given by
\begin{align}
R_c^{1} &\leq\mathbb{E}\log_2\left[\det\left(\mathbf{I}+\mathsf{SNR}\mathbf{h}'_{1,1}\mathbf{h}'^H_{1,1}\right)\right] \nonumber\\
C_1 &\leq \mathbb{E}\log_2\left[\det\left(\mathbf{I}+\frac{\mathsf{SNR}}{n_T}\mathbf{H}_1\mathbf{H}^H_1\right)\right]
\end{align}
and
\begin{align}
R_c^{2} &\leq \mathbb{E}\log_2\left[\det\left(\mathbf{I}+\mathsf{SNR}\mathbf{h}'_{1,2}\mathbf{h}'^H_{1,2}\right)\right] \nonumber\\
C_2 &\leq \mathbb{E}\log_2\left[\det\left(\mathbf{I}+\frac{\mathsf{SNR}}{n_T}\mathbf{H}_2\mathbf{H}^H_2\right)\right],
\end{align}
respectively, where $\mathbf{h}'_{1,i}$ ($[n_R\times 1]$) is the $i$th column of the $n_R \times K$ matrix
$\mathbf{H}'_1$ in~\eqref{eq:BDMwMIMO}. The rate of the individual user cannot exceed these bounds, which are referred
to as single-user bounds with $R_c^1+R_c^2=C$ and $C_1+C_2=C$. From the figure, we observe that the gap between the single-user bounds, e.g. $C_1-R_c^{1}$,
is significantly large. An interesting observation is, however, that the corner points of the capacity region are to
the left or below the corner points of the ergodic rate region of CDD. Thus, at the so called Pareto optimal
part~\cite{TseBook} of the rate region which is the time-sharing part between the two corner points, the gap between the symmetric CDD rate region to the capacity region
is dramatically reduced. The Pareto optimal part contains all the optimal operating points of the
channel~\cite[Ch.6,p.231]{TseBook}.
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{CapRegion.eps
\end{center}
\caption{Ergodic Capacity Region (dashes lines) and ergodic rate region of the CDD (solid lines) with $K=2$ users,
$n_R=2$ and $n_T=2$ for $\mathsf{SNR}=0$dB, $\mathsf{SNR}=20$dB and $\mathsf{SNR}=40$dB.} \label{fig:CapRegion}
\end{figure}
In Fig.~\ref{fig:SumCapMU}, we depict for a system with $K=6$ users and $n_R=n_R=3$ antennas from top to bottom (at the most right): the ergodic capacity, the lower bound on
the ergodic capacity in~\eqref{eq:LBCapGen} (very tight), the upper bound on the ergodic sum-rate of the CDD $R_c$
in~\eqref{eq:BoundGrant}, $R_c$, and the lower bound on $R_c$ in~\eqref{eq:RateCDDNRGen} discussed in the previous
sections. Differently from the single-user system, most of the ergodic capacity is obtained by the CDD despite multiple
receiving antennas at the base station. Both the lower and upper bound on $R_c$ track the ergodic sum-rate performance
of the CDD quite well.
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{CCDMU336.eps
\end{center}
\caption{Ergodic sum-rate of the CDD and sum-capacity with $K=6$ users, $n_R=3$ and $n_T=3$. Upper and lower bounds
obtained from~\eqref{eq:BoundGrant} and~\eqref{eq:LBCapGen},\eqref{eq:RateCDDNRGen}, respectively, are also depicted.}
\label{fig:SumCapMU}
\end{figure}
\section{Conclusion}\label{sec:Conc}
Research in the area of space-time design and analysis for MIMO systems has traditionally focused on
single-users scenarios. For these systems, transmit diversity schemes suffer a loss in spectral efficiency
when the receiver has more than one antenna, making them unsuitable for high rate transmission. Here, we
focused on the performance of cyclic delay diversity (CDD) codes. We analyzed the ergodic sum-rate and studied the
ergodic rate region achievable by using the CDD for each user in a multi-user multiple access channel (MAC). We derived tight
closed form expressions for the ergodic sum-rate of multi-user CDD and compared it with the sum-capacity. We showed
that in contrast to the single-user case, transmit diversity schemes are viable candidates for high rate transmission
in multi-user systems when the number of users exceeds the number of antennas at the base.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,886
|
\chapter{Infinite-Temperature Limit}\label{c3d}
The infinite-temperature limit of Yang-Mills theory is presented in this chapter\footnote{Most of the results presented are published in \cite{Maas:2004se}.}. Although the limit itself is purely academic, it is valuable not only due to technical simplifications. Firstly, properties surviving in this limit will also be present at lower temperatures. Especially persisting non-trivial effects are genuine features of the high-temperature phase. Secondly, it is found in the finite-temperature calculations of chapter \ref{cft} as well as in lattice calculations \cite{Cucchieri:2001tw} that the propagators are already close to their asymptotic values for quite small temperatures, as low as a few times $T_c$.
The chapter starts with the derivation of the infinite-temperature limit in section \ref{s4dto3d}. The emerging theory is a 3d-Yang-Mills theory with an additional adjoint Higgs field. The infrared and ultraviolet properties of this theory will be inspected in section \ref{sanalytic}. It is then studied numerically in three truncation schemes. The first is the ghost-loop-only scheme, to test the Zwanziger-Gribov scenario, in section \ref{sghostloop3d}. The next schemes address the pure Yang-Mills theory in section \ref{syangmills} and finally the full theory in section \ref{sfull3d}. The numerical method deployed is described in appendix \ref{anum}.
A comparison to lattice results will be made in section \ref{slattice3d}. The 3d-theory is not only relevant to the high temperature behavior of Yang-Mills theory. As an example, a recently obtained relation between Landau gauge in 3d and Coulomb gauge in 4d \cite{Zwanziger:2003de} will be addressed in section \ref{scoulomb}.
As this chapter deals nearly exclusively with a 3d-theory, the notation $p\equiv|\vec p|$ is used, if not noted otherwise.
\enlargethispage*{1cm}
\section{From 4d to 3d}\label{s4dto3d}
To obtain the infinite-temperature limit, temperature is introduced into the vacuum equations \pref{vacgheq} and \pref{vacgleq} as detailed in the previous chapter. To obtain scalar equations for the dressing functions $Z$ and $H$, the additional tensor structures in \pref{gtproj} and \pref{glproj} are chosen most conveniently in 4d as
\begin{eqnarray}
A_{T\mu\nu}&=&\left(\delta _{\mu \nu }-\left(1+\frac{p_{0}^{2}}{\vec p^{2}}\right)\delta _{\mu 0}\delta _{0\nu }\right)\label{d3tgenp}\\
A_{L\mu\nu}&=&\left(1+\frac{p_{0}^{2}}{\vec p^{2}}\right)\delta _{\mu 0}\delta _{0\nu}.\label{d3lgenp}
\end{eqnarray}
\noindent In \cite{Maas:2002if} it was demonstrated that, in zeroth order, the infinite-temperature limit can be found by neglecting any contributions from Matsubara frequencies different from zero. This yields an effective 3d-Yang-Mills theory with an additional adjoint Higgs. The Higgs field is the $A_0$ field of the 4d-theory, and therefore the number of degrees of freedom is conserved in this process.
By considering the structure of the projectors \pref{gtproj} and \pref{glproj} in the case of $p_0=0$, which is the zeroth Matsubara frequency, it is also possible to find the connection of the 3d and 4d degrees of freedom. At $p_0=0$, $P_{L\mu\nu}^\xi$ becomes $\delta_{00}$, independent of $\xi$. It projects out the time-time component of the propagator, which belongs to the $A_0$ field. Therefore the 3d-longitudinal part of the 4d gluon propagator corresponds to the Higgs propagator. On the other hand, $P_{T\mu\nu}^\zeta$ becomes
\begin{equation}
P_{ij}^\zeta=\delta_{\mu\nu}-\zeta\frac{p_i p_j}{p^2}\label{bpproj}
\end{equation}
\noindent with zero time-time and time-space components. At $\zeta=3$ this is the Brown-Pennington projector of the 3d-theory \cite{Brown:1988bm}. \pref{bpproj} projects onto the 3d-subspace and thus establishes the connection between the 3d-transverse gluon and the gluon of the 3d-theory. Therefore the dressing functions $H$ and $Z$ of the gluon propagator \pref{gluonprop} describe the Higgs and the 3d gluon, respectively.
This amounts to integrating out the hard modes at tree-level. In general, this is not sufficient \cite{Kajantie:1995dw}. Higher order effects of the hard modes can potentially still influence the interactions of the soft modes. Therefore the general prescription to obtain the effective 3d-theory is to write down the most general Lagrangian allowed and match the parameters by comparing to the 4d-theory \cite{Kajantie:1995dw}. This can be done e.g.\/ by lattice calculations \cite{Cucchieri:2001tw} and perturbation theory \cite{Kajantie:1995dw}. In the present case a tree-level mass for the Higgs and a 4-Higgs coupling are additionally present. This also modifies the Dyson-Schwinger equations, and they therefore have to be rederived. As an explicit example of this process the generation of the tree-level mass will be demonstrated in section \ref{sftir}.
Hence the Lagrangian \pref{l3d} governs the 3d-theory and describes a Yang-Mills field coupled to an adjoint scalar field \cite{Alkofer:2000wg,Kajantie:1995dw}. All occurring constants are effective constants, which arise by integrating out the hard modes. The effective constants can only be derived by calculating the full theory. Therefore the values obtained from lattice and perturbative calculations will be used here, as listed in \cite{Cucchieri:2001tw}. In most results these constants are irrelevant and will drop out, except for the objects discussed in chapter \ref{cderived}.
The only exception is the 4-Higgs coupling constant $h$, which can be uniquely determined already in the 3d-theory. In principle, the Higgs self-energy may contain linear divergences, if this theory stood on its own. However, the Higgs field is only a component of the 4d gluon field, and thus should not contain a linear divergence due to the STIs of the 4d-theory. Implementation of this requirement, in leading-order perturbation theory, fixes $h$ as
\begin{equation}
h=-2g_3^2\frac{C_A}{C_A+2},\label{hvalue}
\end{equation}
\noindent which is detailed in appendix \ref{appUV}. $C_A$ is the second Casimir of the gauge group, see appendix \ref{aconventions}. Note that by \pref{hvalue} exact t'Hooft scaling \cite{'tHooft:1973jz} is also maintained, which would otherwise be broken by the Higgs-tadpoles.
Hence the 3d-theory is finite and therefore all renormalization constants can be set to 1, i.e.\/ no counter-terms are necessary. The divergences have not disappeared, though. By sending the temperature to infinity while maintaining for the renormalization scale $\mu\gg T$, renormalization takes place at $p\to\infty$ and can therefore be neglected at finite momenta. This intuitive argument is shown to be correct in chapter \ref{cft}, where the limit is taken explicitly.
The DSEs for the Yang-Mills sector are already known, see e.g.\/ \cite{Alkofer:2000wg,Roberts:2000aa}. Adding the Higgs, the ghost equation will not be modified compared to pure Yang-Mills theory, since no tree-level Higgs-ghost-coupling is present. The remaining alterations of the equations due to the Higgs are derived in appendix \ref{adse}. They lead to the DSEs\enlargethispage*{1cm}
\begin{eqnarray}
D^{ab}_G(p)^{-1}&=&-\delta^{ab}p^2\nonumber\\
&+&\int\frac{d^dq}{(2\pi)^d}\Gamma_\mu^{\mathrm{tl}, c\bar cA, dae}(-q,p,q-p)D^{ef}_{\mu\nu}(p-q)D^{dg}_G(q)\Gamma^{c\bar cA, bgf}_\nu(-p,q,p-q)\label{ghostd3}\\
D^{ab}_{H}(p)^{-1}&=&\delta^{ab}(p^2+m_h^2)+T^{HG, ab}+T^{HH, ab}\nonumber\\
&+&\int\frac{d^dq}{(2\pi)^d}\Gamma^{\mathrm{tl}, A\phi^2, eac}_\nu(-p-q,p,q)D_{\nu\mu}^{cg}(p+q)D^{fc}(q)\Gamma_\mu^{A\phi^2, gbf}(p+q,-p,-q)\nonumber\\\label{higgsd3}\\
D^{ab}_{\mu\nu}(p)^{-1}&=&\delta^{ab}(\delta_{\mu\nu}p^2-p_\mu p_\nu)+T_{\mu\nu}^{GG, ab}+T^{GH, ab}_{\mu\nu}\nonumber\\
&-&\int\frac{d^dq}{(2\pi)^d}\Gamma_\mu^{\mathrm{tl}, c\bar cA, dca}(-p-q,q,p) D_G^{cf}(q) D_G^{de}(p+q)\Gamma_\nu^{c\bar c A, feb}(-q,p+q,-p)\nonumber\\
&+&\frac{1}{2}\int\frac{d^dq}{(2\pi)^d}\Gamma^{\mathrm{tl}, A^3, acd}_{\mu\sigma\chi}(p,q-p,-q)D^{cf}_{\sigma\omega}(q)D^{de}_{\chi\lambda}(p-q)\Gamma^{A^3, bfe}_{\nu\omega\lambda}(-p,q,p-q)\nonumber\\
&+&\frac{1}{2}\int\frac{d^dq}{(2\pi)^d}\Gamma_\mu^{\mathrm{tl}, A\phi^2, acd}(p,q-p,-q)D^{de}(q)D^{cf}(p-q)\Gamma^{A\phi^2, bef}_\nu(-p,q,p-q),\nonumber\\\label{gluond3}
\end{eqnarray}
\noindent where $T^{ij}$ are the tadpole contributions. The first index gives the equation where the tadpole contributes, $G$ for gluon and $H$ for Higgs, and the second index the type of tadpole appearing. Tree-level quantities are again denoted by a superscript `$\mathrm{tl}$', and can be found in equations \prefr{tlcca}{tl4h} in appendix \ref{adse}. In these equations already the genuine two-loop contributions in the gluon and Higgs equations have been neglected. The ans\"atze for the 3-gluon and Higgs-gluon vertices are motivated by technical considerations and are thus deferred to sections \ref{syangmills} and \ref{sfull3d}. The graphical representation of this set of truncated equations is shown in figure \ref{figthreedsys}.
\begin{figure}
\epsfig{file=3dsys.eps,width=\linewidth}
\caption{The system under study in the infinite-temperature limit. Wiggly lines represent 3d-transverse gluons, dashed lines Higgs and dotted lines ghosts. Lines with a dot are full propagators. Black dots are bare vertices and white circles are full vertices, which have to be reconstructed in the truncation discussed.}
\label{figthreedsys}
\end{figure}
The Higgs propagator $D_H$ is linked to the dressing function $H$ by
\begin{equation}
D_H(p^2)=\frac{H(p^2)}{p^2}\label{higgsdress}.
\end{equation}
\noindent Replacing the propagators in \prefr{ghostd3}{gluond3} by their respective dressing functions, equations for the latter are obtained. These are divided by $p^2$ to make them dimensionless. To obtain a scalar equation for the gluon dressing function, equation \pref{gluond3} is contracted with \pref{bpproj} and divided by 2. This results in\enlargethispage*{1cm}
\begin{eqnarray}
\frac{1}{G(p)}&=&1+\frac{g_3^2C_A}{(2\pi)^2}\int d\theta dq A_T(p,q)G(q)Z(p-q)\label{fulleqG3d}\\
\frac{1}{H(p)}&=&1+\frac{m_h^2}{p^2}+T^{HG}+T^{HH}+\nonumber\\
&&+\frac{g_3^2C_A}{(2\pi)^2}\int d\theta dq\Big(N_1(p,q)H(q)Z(p+q)+N_2(p,q)H(p+q)Z(q)\Big)\label{fulleqH3d}\\
\frac{1}{Z(p)}&=&1+T^{GH}+T^{GG}+\frac{g_3^2C_A}{(2\pi)^2}\int d\theta dq\Big(R(p,q)G(q)G(p+q)\nonumber\\
&&+M_L(p,q)H(q)H(p+q)+M_T(p,q)Z(q)Z(p+q)\Big).\label{fulleqZ3d}
\end{eqnarray}
\noindent Here $\theta$ is the angle between $\vec p$ and $\vec q$. $A_T$, $N_1$, $N_2$, $R$, $M_L$ and $M_T$ are the integral kernels for the employed truncation and are given in appendix \ref{s3dkernels}. The tadpoles in the gluon equation are now also contracted. Although $N_1$ and $N_2$ can be rearranged in one kernel by an integral transformation, they are left separated for a better comparison to the finite temperature case. $C_A$ is the second Casimir of the adjoint representation of the gauge group. It only appears in the combination $g_3^2C_A$, thus any change of the gauge group can be cast into a change of $g_3^2$. Especially 't Hooft-scaling is therefore manifest.
\section{Asymptotic Analysis}\label{sanalytic}
In this section, equations \prefr{fulleqG3d}{fulleqZ3d} will be solved analytically for asymptotically small and large momenta. In all these calculations, a massive Higgs will be assumed, as this is the case for Yang-Mills theories. However, a massless Higgs offers a rich infrared phase structure. Since this is somewhat out of the main line of interest it is deferred to appendix \ref{asmasslesshiggs}.
\subsection{Ultraviolet Analysis}\label{ssuvanalysis}
Since asymptotic freedom is expected to hold also in the 3d case \cite{Feynman:1981ss}, the dressing functions and vertices should reduce to their one-loop counterpart and ultimatively to their tree-level value for sufficiently large momenta.
As $g_3^2$ has dimension of mass and does not enter in the loop integrals in \prefr{fulleqG3d}{fulleqZ3d}, by dimensional analysis the 1-loop contributions must be proportional to $g_3^2/p$. All loop-integrals are therefore sub-leading in the ultraviolet compared to the tree-level contribution. In the case of the Higgs, this is only true if also $p\gg m_h$. This has been verified by explicit calculations, see appendix \ref{appUV}. The 3d-theory gives therefore a very vivid example of asymptotic freedom. Thus all dressing functions $D$ acquire their tree-level value in the ultraviolet,
\begin{equation}
\lim_{p\to\infty}D(p)=1.\label{uvtree}
\end{equation}
\noindent Stated otherwise, to leading order for $p\gg g_3^2,m_h$
\begin{equation}
D(p)\approx 1+\frac{cg_3^2C_A}{p}.\label{uvnlo}
\end{equation}
\noindent The constant $c$ turns out to be positive for all dressing functions. Thus, all dressing functions approach the tree-level behavior from above.
\subsection{Infrared Analysis}\label{ssiranalysis}
To obtain analytical solutions in the infrared, the first step is to note that all integral kernels contain a term $(p\pm q)^2$ or $q^2$ in the denominator. Therefore the integrands are strongly peaked for $p\approx q$. To obtain analytic infrared solutions it is hence permissible to replace the dressing functions by their asymptotic infrared form. Since the infrared limit is a critical limit of the theory, the ansatz of power-laws is justified. The ans\"atze for the dressing functions are
\begin{eqnarray}
G(p)=A_g p^{-2g}\label{iransatz1}\\
Z(p)=A_z p^{-2t}\\
H(p)=A_h p^{-2l}.\label{iransatz}
\end{eqnarray}
\noindent In principle, three possibilities for the infrared behavior can be distinguished: Dominance of tree-level-, tadpole- and loop-terms.
Tree-level dominance is motivated by the naive idea of a free gas of gluons in the asymptotic temperature limit. A first tempting ansatz is $g=t=0$, and $l=-1$ due to the explicit mass term for the Higgs. This corresponds to a purely perturbative or Coulomb phase. This is already impossible in the ghost equation, since in this case the ghost self-energy would diverge as $1/p$ in the infrared and hence superseding the tree-level term. The only possibility would be a ghost-gluon vertex suppressed at least like $p$ in the infrared, which would be unexpected for tree-level propagators. Similar arguments apply to the gluon equation. Hence no Coulomb phase exists in 3d.
If the gluon would behave as a massive particle in the infrared then the ghost equation would allow for a tree-level ghost. This is expected in a Higgs-phase, which would generate a magnetic screening mass. Such a term could only be generated by a tadpole diagram, since otherwise any of the possible vertices would have to diverge at least as strong as $1/p$ in the infrared (from the ghost-loop) or like $1/p^5$ (from the gluon- or Higgs-loop). This is not very probable with massive or tree-level particles. This should then also have to be true for arbitrary projections of the gluon equation, especially for $\zeta=3$. Since the tadpoles, even without truncation, drop out identically, a contradiction arises, and this solution is excluded.
Attempting to save this option by using the ghost-loop to generate a mass also fails, since the required ghost exponent would lead to an non-renormalizable ultraviolet divergence of the integral. The only way to generate such a Higgs phase\footnote{Besides spontaneous breaking of the gauge symmetry on the level of the Lagrangian by giving the Higgs a vacuum expectation value.} would be by appropriate fine-tuning of the ghost-gluon vertex and the ghost. This could of course happen, but is unlikely. Besides, it is hard to see how the vacuum corresponding to such propagators avoids the violation of Elitzur's theorem \cite{Elitzur:im}. Hence the tadpole must at least be compensated in the gluon equation and $t\le -1$, but integral convergence will require $t<-1$, see appendix \ref{air}.
Thus, the remaining option is loop dominance. Motivated by the reasoning of Zwanziger \cite{Zwanziger:2003cf}, ghost dominance is assumed, and the solutions indeed satisfy it. Hence neglecting all contributions without ghost lines, it is then only necessary to specify the ghost-gluon vertex. As discussed in section \ref{strunc} a bare ghost-gluon vertex is chosen. As it is already required that
\begin{equation}
t<-1,\label{tassump}
\end{equation}
\noindent confinement is present due to the Oehme-Zimmermann super-convergence relation \pref{oehme}.
It is then possible to calculate the infrared limit of the DSEs \prefr{fulleqG3d}{fulleqZ3d}. This is done in appendix \ref{air} and leads to
\begin{eqnarray}
\frac{y^g}{A_g}&=&y^{-(4-d)/2-g-t}I_{GT}(g,t)A_gA_z\label{ghostir}\\
\frac{y^t}{A_z}&=&1+y^{-(4-d)/2-2g}I_{GG}(g,\zeta)A_g^2\label{gluonir}\\
\frac{y^l}{A_h}&=&1+\frac{m_h^2+\delta m_3^2}{y}\label{higgsir},
\end{eqnarray}
\noindent where $y=p^2$ and $d$ is the dimension. A subtraction in \pref{ghostir} has been performed and only the finite part has been retained. The expressions for $I_{GT}$ and $I_{GG}$ stemming from the ghost-self energy and the ghost-loop, are calculated in appendix \ref{air}. In the Higgs equation, a finite renormalization of the mass has been allowed for. The mass renormalization will be discussed in subsection \ref{sfull3d} and is given in equation \pref{higgstadpole}. Equation \pref{higgsir} is then solved immediately by setting $l=-1$ and
\begin{equation}
A_h=\frac{1}{m_h^2+\delta m_3^2},\label{ah}
\end{equation}
\noindent since 1 can be neglected for $p\ll m_h$. This already indicates that a qualitative change occurs in the high temperature limit, as in the vacuum $t=l$. This also immediately shows that the Higgs particle decouples, at least in the infrared, from the Yang-Mills sector. This agrees with corresponding findings on the lattice \cite{Cucchieri:2001tw}.
By dimensional consistency in the ghost equation \pref{ghostir}, a relation between $g$ and $t$ follows directly \cite{Zwanziger:2001kw} as
\begin{equation}
g=-\frac{1}{2}\left(t+\frac{4-d}{2}\right)=_{d\to 3}-\frac{1}{2}\left(t+\frac{1}{2}\right)\label{gt}.
\end{equation}
\noindent The result at asymptotic temperature is hence different from the relation \cite{vonSmekal:1997is}
\begin{equation}
g=-2t\label{gt4d}
\end{equation}
\noindent which is found at zero temperature. The additional power of $p$ introduced by this change compensates the dimension of the effective coupling constant in 3d or the temperature in general finite-temperature 4d calculations.
Due to \pref{tassump} it then follows directly, that $g$ is less than 0, and thus any solution will automatically satisfy \pref{kugo} and therefore the Kugo-Ojima and the Zwanziger-Gribov conditions are both fulfilled. Also the tree-level contribution in the gluon equation in \pref{gluonir} can now be neglected, since the ghost-loop diverges in the infrared. Hence dominance of the gauge-fixing term as predicted by the Zwanziger-Gribov scenario is confirmed.
\begin{figure}[t]
\epsfig{file=expcomp.eps,width=\linewidth}
\caption{The left panel shows the two solution branches of equation \pref{irconsistanyd} as a function of $\zeta$ within the region allowed by \pref{gribov} and integral convergence, see equation \pref{dglimits} and \cite{Zwanziger:2001kw}. This excludes the second branch in $d=2$. Dashed is 2d, solid is 3d and dashed-dotted is 4d. Note that at $d=4$ there seems to be only one solution branch. It has been argued that the second branch is $g=1$ \cite{Zwanziger:2001kw}. The right panel shows the solution of \pref{irconsistanyd} as a function of $d$ for $\zeta=1$.}\label{figexp}
\end{figure}
Dividing equations \pref{gluonir} and \pref{ghostir} and eliminating $t$ using \pref{gt}, a conditional equation for $g$ is obtained as
\begin{equation}
1=-\frac{2^{2g-d}\sqrt{\pi}(\frac{d}{2}+g)(2+d(\zeta-2)-4g(\zeta-1)-\zeta)\csc\left(\frac{\pi(d-4g)}{2}\right)\sin(\pi g)\Gamma(\frac{d}{2}+g)}{\left((d-1)^2g\Gamma\left(\frac{1+d-2g}{2}\right)\Gamma(2g)\right)}.\label{irconsistanyd}
\end{equation}
\noindent This equation has solutions in any dimension, see figure \ref{figexp}. For $d=3$ it simplifies to
\begin{equation}
1=\frac{32g(1-g)(1-\cot^2(g\pi))}{(1+2g)(3+2g)(2+2g(\zeta-1)-\zeta)}.\label{irconsist}
\end{equation}
\noindent This equation has two solution branches. One is $\zeta$-independent and yields
\begin{equation}
(g,t)=(\frac{1}{2},-\frac{3}{2})\label{exponents}
\end{equation}
\noindent while the other one varies with $\zeta$. In the special case of $\zeta=1$, the other branch yields
\begin{equation}
(g,t)\approx(0.3976,-1.2952)\label{sexponents}
\end{equation}
\noindent thus reproducing the results of a previous analysis \cite{Zwanziger:2001kw} which only regarded $\zeta=1$. The second solution is a function varying significantly with $\zeta$. It is therefore necessary to fix the allowed range of $g$. By virtue of the Gribov condition \pref{gribov} and \pref{tassump} as well as convergence of the integrals in the infrared, the allowed range for $g$ in 3d is
\begin{equation}
\frac{1}{4}<g\le\frac{3}{4}.\label{gallowed}
\end{equation}
\noindent It is further restricted by the requirement that a Fourier transform of the ghost propagator should exist, at least in the sense of a distribution, requiring $g\le 1/2$. This restricts the range of allowed $\zeta$-values for the varying branch to
\begin{equation}
1/4\le \zeta<3.\label{zetaallowed}
\end{equation}
\noindent At the lower boundary both solutions merge into one. Note that the position space ghost propagator for $g=1/2$ is a half-sided distribution, since its Fourier-transformed exists only in the sense of a limiting procedure. This is analogous to older expectations for the gluon propagator in 4d \cite{Brown:1988bm}.
Comparing the solution \pref{exponents} with the 4d-case \cite{vonSmekal:1997is} where
\begin{equation}
\frac{1}{2}<g\le 1,\label{4dexponents}
\end{equation}
\noindent the ghost exponent is only very weakly different from the 4d case, and the difference is even less pronounced for the gluon exponent.
In equations \pref{ghostir} and \pref{gluonir} the coefficients $A_g$ and $A_z$ only appear in the product $A_g^2A_z$. Therefore only this combination is determined by the infrared analysis and one of both coefficients has to be determined during the numerical solution procedure. This is a non-trivial problem, see appendix \ref{anum}. Expressing $A_z$ by $A_g$ yields
\begin{equation}
\frac{1}{A_z}=\frac{A_g^2C_Ag_3^2}{(4\pi)^\frac{3}{2}}\frac{2^{4(g-1)}(2+2g(\zeta-1)-\zeta)\Gamma(2-2g)\sin^2(\pi g)}{\cos(2\pi g)(g-1)g^2\Gamma(\frac{3}{2}-2g)}\label{ag}.
\end{equation}
\noindent This result is used to check whether a correct solution is found during the numerical calculations. It also emphasizes that there is only one free parameter of the infrared solution. This one degree of freedom is necessary when using the full solution to perform the exact cancellation of the tree-level term in the ghost equation, which is the single crucial point in the derivation.
With these results and appropriate regularization, it is possible to calculate the other contributions in all equations in the infrared limit as well. No contradiction is found. It would therefore be safe to proceed and solve the equations at all momenta numerically.
However, both solutions can be realized at all momenta. Therefore the question arises which is the correct infrared solution or are both physical, characterizing different phases? In principle, the solution having the lower thermodynamic potential should be the correct one. This is investigated in chapter \ref{cderived} and prefers the varying branch at $\zeta=1$. Likewise the comparison with lattice calculations in subsection \ref{slattice3d} prefer the varying branch at $\zeta=1$. Furthermore, under the assumption of continuity in $d$, the solution in 3d and 4d should lie on the same solution branch. From the right panel of figure \ref{figexp}, it can be inferred that again the varying solution lies on the same branch as the solution found in 4d. Therefore is is probable that the varying branch is the physical solution at $\zeta=1$. However, it is possible, that the presence of one of the solutions is only an artifact of the truncation. This cannot be ruled out in the current approximation scheme.
\section{Ghost-Loop-Only Truncation}\label{sghostloop3d}
The ghost-loop-only truncation retains only tree-level expressions and loops with explicit ghost-lines. It thus implements the hypotheses of Zwanziger of dominance of the gauge-fixing contribution at all momenta. Calculations in 4d with this truncation \cite{Atkinson:1998zc} give qualitatively the same results as with the gluon-loop included \cite{Fischer:2002hn}.
The result of the calculations for both the ghost and the gluon at $\zeta=3$ are shown in figure \ref{figgonly3}. In this case only the $g=1/2$ infrared solution exists. The gluon propagator, for kinematical reasons, exhibits a significant maximum, with a center around $p/g_3^2=0.25$. The Higgs is in this case trivially tree-level, and therefore not shown. The infrared coefficients for all presented truncations are typically of order $A_gg_3^{-2g}={\cal O}(10^{-1})$ and $A_zg_3^{-2t}={\cal O}(10)$.
\begin{figure}
\epsfig{file=gonly-3.eps,width=\linewidth}
\caption{The solutions of the DSEs in the ghost-loop-only truncation scheme at $\zeta=3$. The left panel shows the dressing functions and the right panel the propagators. The dashed line refers to $Z$ and the solid line to $G$. All quantities have been made dimensionless by dividing out appropriate powers of $g_3^2$.}\label{figgonly3}
\end{figure}
Note that the 3d-theory is finite, and therefore no dimensional transmutation occurs. Hence the coupling constant is the only dimensionful quantity left. Measuring then all dimensionful quantities in units of $g_3^2$, the theory is without scale and should be independent of the value of the coupling constant. In this truncation scheme, this is the case, which can been seen by rescaling the integration momenta in \prefr{fulleqG3d}{fulleqZ3d} appropriately. Thus, the plots shown in figure \ref{figgonly3} are the results for any positive value of the coupling constant. Even for very small values strong non-perturbative effects for momenta smaller than $g_3^2$ appear clearly. Hence this theory is never purely perturbative.
For values of $\zeta$ different from 3, spurious divergences are present, and it is therefore necessary to include and adjust the tadpole term $T^{GG}$ such that these terms are canceled. To identify the correct contribution, the integration kernel $R$ is split as
\begin{equation}
R(\zeta)=R_0+(\zeta-3)R_3+R_D(\zeta),\label{rsplit}
\end{equation}
\noindent where $R_0$ and $R_3$ are independent of $\zeta$ and do not give rise to divergences, and $R_D$ contains the divergent part for $\zeta\neq 3$. In perturbative calculations, the last term would be exactly canceled by the tadpole. However, it is not possible just to remove $R_D$ for two reasons. First, $R_D$ contributes dominantly to the infrared, and the subtraction should not alter the infrared. Secondly, $R_D$ is necessary to compensate an infrared divergence in the integration (not in the external momenta) of $R_3$ at $\zeta\neq3$. Setting
\begin{equation}
T^{GG}=-\frac{g_3^2C_A}{(2\pi)^2}\int dqd\theta R_D(p,q)\left(2G(|\vec p|+|\vec q|)-1\right)\label{naivesub}
\end{equation}
\noindent would be sufficient. This quantity contains no finite part for $\zeta=3$ and contributes only a small finite part compared to $R_0$ and $R_3$ otherwise.
However, even for a small deviation from $\zeta=3$, no further solution can be found. This is not unexpected, since the ghost-loop should only dominate in the infrared and is only there restricted to be purely transverse. It is not transverse in the ultraviolet, as perturbative calculations show. However, as the corresponding compensating parts proportional to $\delta_{\mu\nu}$ of the gluon-loop are neglected in this truncation, a failure was likely. Therefore using \pref{naivesub} does not permit to go beyond $\zeta=3$ in the ghost-loop-only truncation. This behavior is not altered when using the second solution branch of the infrared. It is necessary to use a different subtraction scheme which removes these terms at finite momenta. This is possible by using
\begin{eqnarray}
&R(p,q)G(q)G(p+q)\to\nonumber\\
&R_0(p,q)G(q)G(p+q)+((\zeta-3)R_3(p,q)+R_D(p,q))A_g^2q^{-2g}(p+q)^{-2g}\nonumber\\
&+(\zeta-3)R_3(p,q)\label{rsubtract}
\end{eqnarray}
\noindent or equivalently setting
\begin{eqnarray}
T^{GG}&=&-\frac{g_3^2C_A}{(2\pi)^2}\int dqd\theta\Big(R_D(p,q)(G(q)G(p+q)-A_g^2q^{-2g}(p+q)^{-2g})\nonumber\\
&&+(\zeta-3)R_3(p,q)\big(G(q)G(p+q)-A_g^2q^{-2g}(p+q)^{-2g}-1\big)\Big)\label{ggtad1}.
\end{eqnarray}
\noindent The last term in \pref{rsubtract} ensures the correct ultraviolet behavior to reproduce the 1-loop result.
Note that by using \pref{rsubtract} instead of \pref{naivesub}, also the solution at $\zeta=3$ is changed, since the second term in \pref{rsubtract} misses the correct mid-momenta behavior. This change is however negligible, and the results would hardly be distinguishable from the plots shown in figure \ref{figgonly3}. This is the expected trade-off of the truncation, a deficiency at mid-momenta.
\begin{figure}
\epsfig{file=gonlyzeta1.eps,width=\linewidth}
\caption{The ghost and gluon dressing function at $\zeta=1$. The left panel shows the ghost dressing function, the right panel shows the gluon dressing function. The solid line denotes the solution for $g=1/2$ and the dashed line gives the other solution branch with $g \approx 0.4$.}
\label{gonlyzeta1}
\end{figure}
Both solutions at $\zeta=1$ are shown in figure \ref{gonlyzeta1}. The main features visible are the following: The ghost dressing function is nearly featureless. It approaches its tree-level behavior in the ultraviolet from above. The gluon dressing function has a (non-visible) peak, and it approaches its tree-level behavior also from above.
\begin{figure}[ht]
\epsfig{file=compzetagonly.eps,width=\linewidth}
\caption{The ghost and gluon dressing function in ghost-loop-only approximation for different values of $\zeta$. The left panel shows the ghost dressing function, the right panel the gluon dressing function. The solid lines represent the solutions with $g=1/2$, and the dashed lines for the varying branch. At the peak in the gluon, the middle lines correspond to solutions at $\zeta=1$. The upper and lower line at mid-momenta give the solutions at $\zeta=0$ and $4$ for the $g=1/2$-branch and at $\zeta=1/4$ and $2.55$ for the other branch. Note the linear scale for $Z$ in the right panel.}
\label{figgonlyzeta}
\end{figure}
It is now possible to study different values of $\zeta$. This gives a systematic error for the solution, since it corresponds to varying the amount of gauge symmetry violation. This is shown in figure \ref{figgonlyzeta}. The variation for the $g=1/2$ branch is obtained by varying $\zeta$ between 0 and 4. It is possible to obtain solutions for other $\zeta$, but at some point the projection would be on the gauge violating longitudinal parts only and would hence give no information anymore on the transverse part. For the other branch $\zeta$ is varied from 1/4 up to 2.55, the latter corresponding to $g=0.2708$ and $t=-1.042$. In principle, it should be possible to extend the variation over the complete range \pref{zetaallowed}. However, it was not possible numerically to go to larger $\zeta$, since the available numerical precision of 16 significant digits was not sufficient. Once higher precision is available the remaining solutions should be obtainable. The reason is that as $t$ comes close to $-1$, the dressing function more and more resembles a massive solution. This corresponds to a jump from 0 to a finite quantity in the corresponding propagator. Hence, the zero is only reached far in the infrared, and it is numerically necessary to have at the same time very large momenta from the integration and very small ones from the infrared, a combination limited by precision. This is also the case when including the gluon- or Higgs-loop.
Looking now in detail at the $\zeta$-dependence, it can be observed that there are only quantitative, but no qualitative changes. The vanishing of the peak in the gluon dressing function for $\zeta>2$ is a purely perturbative phenomenon, since the contribution of the ghost-loop changes sign at $\zeta=2$ in the ultraviolet. The changes in the infrared are partly due to changes of the coefficients $A_g$ and $A_z$. The variable branch shows a much larger variation as expected. The deviation is comparatively small when going to smaller $\zeta$. It increases significantly when increasing $\zeta$, since the gluon propagator becomes more and more like the one of a massive particle. The propagator always vanishes at zero, though.
\section{Yang-Mills Theory}\label{syangmills}
The next step is to restrict the calculation to the Yang-Mills subsector without Higgs. This is then a theory defined on its own. Starting again at $\zeta=3$ no solution can be found. Even trying a significant diversity of subtraction schemes to do calculations at other values of $\zeta$ does not help, nor does employing the other solution branch.
Closer inspection reveals the source for these problems. The ghost- and gluon-loop contribute with opposite sign. This is correct in the ultraviolet and irrelevant in the infrared, where the gluon-loop is subleading. However, at momenta of the order of $g_3^2$, the gluon loop is able to dominate the ghost-loop and the tree-level term, and thus leads to a violation of the Gribov condition \pref{gribov}. Therefore, within this truncation scheme, the tree-level 3-gluon vertex is not an acceptable truncation. The reason is that either the bare three-gluon vertex overestimates the true vertex at mid-momenta, the bare ghost-gluon vertex underestimates the true one at mid-momenta, or the contributions of the two-loop graphs are important. A finite contribution from the tadpole would not help, since the problem persists even at $\zeta=3$ where its contribution vanishes identically\footnote{Albeit a scaleless integral also in 3d, it is not necessarily a purely divergent quantity. In 4d, it is possible to absorb any finite contributions in the renormalization constant. Lacking a renormalization process, this is not possible in 3d.}. This is clearly an artifact of the truncation. Any of these options deserves further attention, and as a first step, currently the ghost-gluon vertex is under closer investigation \cite{schleifenbaum:diploma}.
For the purpose of this work, the specific reason for this mid-momenta deficiency is of minor importance. For further calculations it is only necessary to improve the vertex such that it provides a sufficient suppression at mid-momenta. This situation is similar to the case of 4d, where without a change of either vertex, the correct ultraviolet behavior is not reproduced \cite{Fischer:2003rp}. Since the only relevant effect is at mid-momenta, such a vertex construction has to fulfill two criteria: It has to be tree-level in the ultraviolet and not more strongly divergent than $1/p^{t+1}$ on either leg in the infrared.
The simplest solution is to multiply the bare 3-gluon vertex with an additional function delivering this suppression. This leads to the ansatz
\begin{eqnarray}
&\Gamma^{A^3, abc}_{\beta\sigma\mu}(-q,q+p,-p)\to\nonumber\\
&\Gamma_{\beta\sigma\mu}^{\mathrm{tl}, A^3, abc}(-q,q+p,-p)(A(q,G,Z)A(q+p,G,Z)A(p,G,Z))^\delta\nonumber\\
&A(q,G,Z)=\frac{1}{Z(q)G(q)^{2+\frac{1}{2g}}}.\label{g3vertex}
\end{eqnarray}
\noindent This ansatz is motivated by the vertex used in 4d \cite{Fischer:2002hn}, with the anomalous dimensions set to 0, as appropriate for 3d. Then an appropriate structure has been selected so that $A$ is constant in the infrared and tree-level in the ultraviolet. The additional parameter $\delta$ smoothly interpolates between a tree-level vertex and a large suppression by selecting it from $[0,\infty)$. This ansatz conserves Bose symmetry, as opposed to the case of 4d, where this was not possible without spoiling the ultraviolet properties.
Alternative concepts have been studied. The simplest possibility to remedy the situation, albeit at the expense of an incorrect ultraviolet behavior, is to multiply the vertex just with a constant, thus introducing a vertex renormalization $Z_1$. Qualitatively the same and quantitatively similar solutions have been found for $Z_1\lessapprox 0.6$. The Bose non-symmetric ansatz $A(q,G,Z)A(q+p,G,Z)$, similar to the 4d-case, has also been made. Although it delivers a smaller suppression at the same value of $\delta$ as \pref{g3vertex}, the solution is possible for sufficiently large values of $\delta$. It does not differ qualitatively from the results obtained using \pref{g3vertex}. Even the quantitative changes are small. The last option investigated was modeling the two-loop terms by adding a term which is Gaussian in momentum to \pref{fulleqZ3d}. As long as its strength around $g_3^2$ was sufficient, the system was solvable, again with only minor modifications. Thus it is likely that the exact solution does not change the results qualitatively. The approach taken with \pref{g3vertex} is supported by lattice calculations of the dressing functions, see section \ref{slattice3d}.
\begin{figure}[t]
\epsfig{file=vertex.eps,width=\linewidth}
\caption{The vertex and the gluon loop with and without modifications. The left panel shows the function $A$ from \pref{g3vertex}. The solid line is $\delta=0$, the tree-level vertex. The dashed line is the $g=1/2$ solution and the dotted line the $g\approx 0.4$ solution. From bottom to top these are $\delta=1$, $\delta=1/4$ and in the last case $\delta=0.14$ and $\delta=0.12$, respectively. The right panel shows at $\zeta=3$ the full gluon loop compared to the gluon loop suppressed by the vertex \pref{g3vertex} and subtracted by the gluon part of the tadpole \pref{ggtad2}, again for both solutions with the same type of line. The larger values are always without suppression.}\label{figvertex}
\end{figure}
An interesting possibility to assess the effect of different vertices is to change $\delta$. While $\delta=1$ corresponds to a significant suppression, the smallest values with still a stable solution are 0.131 for the $g=1/2$ solution and 0.114 for the other one. As a convenient value which still provides very stable numerics and at the same time not too large suppression, $\delta=1/4$ is arbitrarily chosen in the following. The structure of the vertex and its impact on the gluon loop are studied for different values of $\delta$ in figure \ref{figvertex}.
\begin{figure}[ht]
\epsfig{file=ymgo.eps,width=\linewidth}
\caption{Solution of the Yang-Mills sector at $\zeta=3$. The left and right panel shows the ghost and gluon dressing function, respectively. The dashed curve gives the ghost-loop-only solution for comparison. The solid curve gives the full Yang-Mills solution at $\delta=1/4$. Note the linear scale for $Z$ in the right panel.}\label{figym}
\end{figure}
It is now possible to solve the equations at $\zeta=3$. The results are shown in comparison to the ghost-loop-only results in figure \ref{figym}. The only qualitative difference compared to the ghost-loop-only truncation is that a peak in the gluon dressing function is now present for all $\zeta$ due to the now correctly reproduced LO perturbation theory for $p\gg g_3^2$.
After this first view at the full Yang-Mills solution, it is again interesting to study its dependence on $\zeta$. In principle, it would be sufficient again to split off the divergent part $M_{TD}$ of $M_T$ and subtract it in the same way as in \pref{naivesub}, by setting
\begin{eqnarray}
T^{GG}&=&-\frac{g_3^2C_A}{(2\pi)^2}\int dqd\theta\Big(R_D(p,q)\left(2G(|\vec p|+|\vec q|)-1\right)\nonumber\\
&+&(A(q,G,Z)A(q+p,G,Z)A(p,G,Z))^\delta M_{TD}(p,q)\left(2Z(|\vec p|+|\vec q|)-1\right)\Big)\label{naivesub2}.
\end{eqnarray}
\noindent The naive expectation is that after the inclusion of the gluon loop the spurious terms would be eliminated. This does not occur, since by introducing an ad-hoc suppression, these cancellations are affected. Hence the same problem arises again, and it is necessary to correct both loops individually. This can be done by
\begin{eqnarray}
T^{GG}&=&-\frac{g_3^2C_A}{(2\pi)^2}\int dqd\theta\Big(R_D(p,q)(G(q)G(p+q)-A_g^2q^{-2g}(p+q)^{-2g})\nonumber\\
&+&(\zeta-3)R_3(k,q)\big(G(q)G(p+q)-A_g^2q^{-2g}(p+q)^{-2g}-1\big)\nonumber\\
&+&(A(q,G,Z)A(q+p,G,Z)A(p,G,Z))^\delta M_{TD}(p,q)Z(q)Z(p+q)\Big)\label{ggtad2},
\end{eqnarray}
\noindent i.e. by completely removing the term $M_{TD}$ and keeping the same subtraction as before for the ghost-loop. Removing $M_{TD}$ is not harmful to the infrared, since the gluon-loop is subdominant there. It is also irrelevant in the ultraviolet, where it contributes only a pure divergence and would therefore anyhow be canceled by the tadpole. Thus, only the mid-momenta behavior is altered.
\begin{figure}[ht]
\epsfig{file=ymzeta1.eps,width=\linewidth}
\caption{Solution of the Yang-Mills sector at $\zeta=1$. The left panel shows the ghost dressing function and the right panel the gluon dressing function. The solid curve denotes the $g=1/2$ solution, while the dashed curve displays the $g\approx 0.4$ solution. Both are at $\delta=1/4$.}\label{figymzeta1}
\end{figure}
It is now possible to solve for any $\zeta$. Both infrared solutions are found to connect to full solutions and a comparison at $\zeta=1$ and $\delta=1/4$ is shown in figure \ref{figymzeta1}. The difference is again only quantitative in nature, and even this difference is quite small.
\begin{figure}[ht]
\epsfig{file=compzetaym.eps,width=\linewidth}
\caption{The ghost and gluon dressing functions of Yang-Mills theory for different values of $\zeta$. The left panel shows the ghost dressing function, and the right panel the gluon dressing function. The solid line gives the solution for $g=1/2$ and the dashed line is for the other solution branch. At the peak in the gluon dressing function, the middle lines represent the solution at $\zeta=1$. The upper and lower lines at mid-momenta give the solutions at $\zeta=0$ and $\zeta=4$ for the $g=1/2$-branch and at $\zeta=1/4$ and $\zeta=2.45$ for the other solution branch.}
\label{figymzeta}
\end{figure}
\begin{figure}[ht]
\epsfig{file=compdeltaym.eps,width=\linewidth}
\caption{The ghost and gluon dressing functions of Yang-Mills theory for different values of $\delta$ at $\zeta=1$. The left panel shows the ghost dressing function, and the right panel the gluon dressing function. The solid line gives the solution for $g=1/2$ and the dashed line for $g\approx 0.4$. At the peak in the gluon dressing function, the middle lines represent the solution at $\delta=1/4$. The upper and lower lines at mid-momenta give the solutions at $\delta=0.131$ and $\delta=1$ for the $g=1/2$-branch and at $\delta=0.114$ and $\delta=1$ for the other solution branch.}
\label{figymdelta}
\end{figure}
It remains to assess the dependence on the parameters $\zeta$ and $\delta$ as a systematic error estimate. While the dependence on $\zeta$ summarizes the impact of violation of gauge symmetry, the dependence on $\delta$ exhibits the uncertainty in the correct mid-momenta behavior of the vertices. Both problems are linked, and do not cause independent errors. However, both generate different changes when varied. Figure \ref{figymzeta} and \ref{figymdelta} display the dependence on $\zeta$ and $\delta$, respectively.
The uncertainty in the structure of the vertices is much more significant at mid-momenta than the violation of the STI \pref{stigluon}. On the other hand, the uncertainty in the vertex only amounts to a shift in the infrared but not to a change of the infrared exponents as in the case of the violation of the STI.
The height and the position of the peak depends weakly on the vertex construction. The height of the peak can be traced to the amount of suppression the vertex gives. When the suppression is weak enough, the peak tends to become a divergence with sign change. Therefore, this truncation scheme can give no definite answer to the size of the peak. It turns out that the peak only starts to get very large when $\delta$ gets close to the critical value. Before that, it is stable at height around 1.5 and below. Hence, it is to be expected that the correct suppression would also not deliver a significantly larger peak. This is confirmed by lattice results, as will be shown in subsection \ref{slattice3d}.
Comparing these results with those of the 4d-theory \cite{vonSmekal:1997is}, little difference is found. On qualitative grounds, both are the same, besides renormalization. They only differ in their quantitative nature. Thus the 3d Yang-Mills theory is a strongly interacting, confining theory, confirming the expectation \cite{Feynman:1981ss}.
\section{Full 3d-Limit}\label{sfull3d}
Finally, by adding the Higgs, the full system of equations \prefr{fulleqG3d}{fulleqZ3d} is implemented. Although the contribution of the Higgs-loop in \pref{fulleqG3d} is positive, it is not sufficient to allow for a solution with a bare 3-gluon vertex. Hence the vertex ansatz and also the tadpole construction will be kept the same as in the Yang-Mills case.
In addition, as no further input is available, the tree-level gluon-Higgs vertex \pref{tlgh} is employed. A closer inspection of the consequences of this choice is made in subsection \ref{sschwinger}.
The ratio $m_h/g_3^2$ remains to be fixed. In the pure 3d-theory, $m_h$ is a parameter and can therefore not be determined a priori. It stems from integrating out the hard modes when making the 3d-approximation \cite{Kajantie:1995dw}. By taking these modes into account in chapter \ref{cft}, it will be possible to investigate it more closely rather than making an ansatz. For the investigation in this chapter, the origin (and also the exact value) of the Higgs mass is not of direct importance, and hence will be fixed to $m_h/g_3^2=0.8808$, a value extracted by lattice calculations and perturbative fitting \cite{Cucchieri:2001tw}.
By adding the Higgs, three more tadpole contributions arise, as can be seen from figure \ref{figftsys} and in appendix \ref{adse}. Since the Higgs has a tree-level mass, two of the tadpoles can have a non-vanishing finite part already in leading-order perturbation theory, see appendix \ref{appUV}. Hence, a more thorough discussion is needed here.
The first additional tadpole appears in the gluon equation. On the one hand, all diverging contributions must cancel again due to the STI, hence the structure of divergencies must be the same for all graphs. On the other hand, when projecting with the Brown-Pennington projector \pref{bpproj}, the tadpole identically drops out. Therefore all finite contributions of the tadpole must also be contained in the other loops as well. For a vanishing tree-level mass, the tadpole is canceled by divergent loop contributions. It is hence a good assumption that any finite part of the tadpole will also be canceled by the loop contributions. Therefore the approximation concerning the Higgs-tadpole will be
\begin{equation}
T^{GH}=-\frac{g_3^2C_A}{(2\pi)^2}\int dqd\theta\Big(M_{LD}(k,q)H(q)H(k+q)\Big)\label{ghtad}
\end{equation}
\noindent where $M_{LD}$ is the divergent part of $M_L$. If this were not justified, at $\zeta=3$ a trace of the finite contribution should be visible. Any such term would behave like $1/p^2$, a clear signature. The gluon- and Higgs-loop both are at most constant in the infrared and can thus not induce such a contribution. Only the ghost loop with its strongly divergent behavior could contain it. However, it seems unlikely that the finite part of the tadpole should be completely moved to the ghost loop, hence the assumption described above will be made.
\begin{figure}[t]
\epsfig{file=compymf.eps,width=\linewidth}
\caption{The solution of the Yang-Mills system compared to the full system at $\zeta=1$. The solid line denotes the full solution at $g=1/2$ while the dotted line is the corresponding Yang-Mills solution. The dashed line gives the full solution for $g\approx 0.4$ while the dashed-dotted line is the corresponding Yang-Mills solution.}
\label{figymf}
\end{figure}
The Higgs equation with two tadpoles is more problematic, as the argument using the projection is not available. As the Higgs is a component of a 4d gluon, its self-energy has to be finite. Thus the divergent parts of the tadpoles have to cancel. This fixes the tree-level coupling $h$ at leading-order perturbation theory, as given by \pref{hvalue}. The only remaining question is that of a possible finite contribution. Such a contribution would only change the mass of the Higgs, and this occurs already at leading-order perturbation theory. However, the subtraction of two divergent quantities is not simple, and due to the error induced by the truncation somewhat arbitrary. Hence the mass-shift is essentially unknown. Nonetheless, the Higgs dressing function behaves similar to perturbation theory. Therefore assuming the finite part of the tadpole to be the same as in leading-order perturbation theory seems to be justified. Thus, the tadpoles are fixed to be
\begin{equation}
T^{HG}+T^{HH}=\frac{g_3^2C_A}{p^2}\frac{m_h}{4\pi}=\frac{\delta m_3^2}{p^2},\label{higgstadpole}
\end{equation}
\noindent defining the finite mass renormalization $\delta m_3$ of the bare Higgs mass. This amounts to a change of the mass of about 10\%.
The impact of the Higgs on the Yang-Mills sector at $\zeta=1$ is shown in figure \ref{figymf}. The infrared behavior is only weakly affected and the situation is similar in the ultraviolet. The largest impact on the gluon dressing function is on the peak height and the impact on the peak position is significantly weaker. Altogether, the effect of the additional Higgs field is only quantitative, as expected from lattice calculations \cite{Cucchieri:2001tw}.
\begin{figure}[t]
\epsfig{file=higgsf.eps,width=\linewidth}
\caption{The left panel shows the Higgs propagator and the right panel its dressing function, both at $\zeta=1$. The solid line gives the solution for $g=1/2$, the dashed line for $g\approx 0.4$, the dashed-dotted line denotes the leading-order perturbative result and the dotted line the tree-level behavior.}
\label{fighiggsf}
\end{figure}
The Higgs propagator and dressing function are shown in figure \ref{fighiggsf}. Both show a behavior very similar to the leading-order perturbative result, and the difference to a tree-level propagator is mainly due to the mass-shift by the tadpole.
\begin{figure}[t]
\epsfig{file=higgsmass.eps,width=\linewidth}
\caption{The left panel shows the gluon dressing function and the right panel the Higgs propagator as a function of $m_h$. The gluon peak increases with decreasing mass, while the Higgs propagator decreases with increasing mass. Solid is the $g=1/2$ and dashed the $g\approx 0.4$ solution. The masses are $2g_3^2$, $0.88g_3^2$ for both solutions, and $0.75g_3^2$ and $0.68g_3^2$, respectively.}
\label{fighiggsmass}
\end{figure}
Although the mass is fixed in the current setting, the question of the dependence on the mass is still interesting. Making the tree-level mass larger, the solution approaches smoothly the Yang-Mills solution, with which it merges as $m_h\to\infty$. On the other hand, the mass cannot be made arbitrarily small. When the mass is reduced below $m_h/g_3^2\approx 0.7$, no solution can be found without changing $\delta$. The reason is that the Higgs self-energy is negative, and only while it is sufficiently small compared to the renormalized tree-level contribution, it is possible to maintain the Gribov condition \pref{gribov} at mid-momenta. This additional constraint only stems from the 4d origin of the theory and the fact that the Higgs is part of a 4d gluon, and is not inherent to the 3d-theory. In the 3d-theory, a negative dressing function of the Higgs would be allowed, and smaller masses could be reached. The height of the gluon peak directly corresponds to the size of the Higgs-self-energy and hence to the lowest masses achievable without violating \pref{gribov}. Thus, increasing $\delta$ compensates to some extent a lowering of the mass, but this has only been pursued up to $\delta=1$ or a lowest mass of $m_h/g_3^2\approx 0.6$. The dependence of the Higgs propagator on the mass is depicted in figure \ref{fighiggsmass}. An effect at small Higgs masses had been anticipated from the effect of a massless Higgs on the infrared solutions, see appendix \ref{asmasslesshiggs}.
\begin{figure}[t]
\epsfig{file=higgsdelta.eps,width=\linewidth}
\caption{The Higgs dressing function and propagator as a function of $\delta$ at $\zeta=1$. The solid line is the solution for $g=1/2$, the dashed line for $g\approx 0.4$. The higher the peak, the larger $\delta$. The lowest peak corresponds to $\delta=1$, the middle one to $\delta=1/4$ and the largest one to $\delta=0.0862$ and $\delta=0.061$, respectively.}
\label{fighiggsdelta}
\end{figure}
The dependence of the Higgs on $\zeta$ is very small, on the order of a few percent at mid-momenta. The $\zeta$-dependence of the Yang-Mills sector is also only affected to this extent by the presence of the Higgs. The dependence on $\delta$, which influences the size of the self-energy, is more pronounced. The latter is shown in figure \ref{fighiggsdelta}. Similar to the gluon dressing function, in the Higgs dressing function the peak increases with decreasing $\delta$, at some point also generating a peak in the propagator. This is due to the direct link of the Higgs self-energy to the peak volume of the gluon propagator. As the latter increases, the Higgs-self-energy increases. Since it is negative, it starts to compensate the tree-level term at mid-momenta, where the effect of the peak in the gluon dressing function is most pronounced and thus increases the peak in the Higgs dressing function. At some point a further effect sets in, when the system tries to compensate the increasing negative contribution from the gluon loop by an increasing positive contribution in the Higgs-loop. This is most easily achieved by an enhancement of the Higgs dressing function at mid-momenta, thus also increasing the peak in the Higgs dressing function. This process slows the growth and for sufficiently small values of $\delta$ even decreases the peak in the gluon dressing function. Therefore it becomes likely that a violation of the Gribov condition \pref{gribov} will occur due to the Higgs rather than due to the gluon. This behavior is much more pronounced in the $g=1/2$ solution.
\begin{figure}[t]
\epsfig{file=agzd.eps,width=\linewidth}
\caption{The dependence of the infrared coefficient $A_g$ on $\zeta$ in the left panel and on $\delta$ in the right panel. Solid lines are the $g=1/2$ solution and dashed lines the other solution branch.}
\label{figagzd}
\end{figure}
Apart from the dependence on $\zeta$ of the exponents in the case of the second branch, also the dependence of the infrared coefficients on $\zeta$ and $\delta$ is of interest. $A_h$ is fixed by the renormalized mass and $A_z$ depends uniquely on $A_g$, hence only the dependence of $A_g$ is relevant. Its dependence is shown in figure \ref{figagzd}. The dependence on $\zeta$ is rather weak. Both solution branches evolve smoothly out of one solution at $\zeta=1/4$. The dependence on $\delta$ is much more pronounced, where the trend to become singular is visible in the strong increase at low values of $\delta$. The bending-over is the aforementioned effect of the peak growth switching from the gluon to the Higgs.
\section{Comparison to Lattice Results}\label{slattice3d}
Currently, only lattice calculations for the gluon propagator \cite{Cucchieri:2003di,Cucchieri:2001tw} and the Higgs propagator \cite{Cucchieri:2001tw} are available, but not for the ghost propagator. Furthermore, lattice calculations do not yet see a significant difference in the gluon propagator with or without the Higgs \cite{Cucchieri:2001tw}. Hence the lattice calculations for the gluon propagator are done in pure 3d Yang-Mills theory. Therefore, these results have to be compared to the results from section \ref{syangmills} rather than to the full calculations in section \ref{sfull3d}. The comparison is made in figure \ref{figlat}.
\begin{figure}
\epsfig{file=latcomp.eps,width=0.5\linewidth,height=1.2\linewidth}\epsfig{file=higgscomp.eps,width=0.5\linewidth,height=1.2\linewidth}
\caption{The left panels show the gluon propagator and dressing function in Yang-Mills theory from lattice calculations and from section \ref{syangmills}. The continuum-extrapolated values are from \cite{Cucchieri:2001tw}, the others from \cite{Cucchieri:2003di}. The right panel shows the Higgs propagator and dressing function in the full theory from lattice calculations \cite{Cucchieri:2001tw} and from section \ref{sfull3d}. The errors indicated are statistical. }
\label{figlat}
\end{figure}
Comparing the gluon propagator to the lattice results, the assumed suppression using $\delta=1/4$ seems to fit the data quite well. The dressing function is better fitted by the $g=1/2$ solution while the propagator and especially the infrared favors $g\approx 0.4$. At momenta $p\approx g_3^2$, the truncation scheme is not trustworthy and the better agreement of the $g=1/2$ solution with the lattice gluon dressing function is not conclusive. Although the lattice data for the propagator are quite scattered, they lie pretty close in case of the dressing function. The propagator is hence more sensitive than the dressing function. A better fit could be achieved by varying $\delta$. This is not the aim of this work, but may be of interest for phenomenological studies.
The lattice result for the Higgs propagator differ more from the DSE result than for the gluon propagator. Especially, the lattice Higgs propagator is much more similar to the tree-level result than to the leading-order perturbative result. The reason for the discrepancy between the current calculation and the lattice data may be twofold. The value at zero momentum indicates that \pref{higgstadpole} overestimates the mass renormalization and the mass found is too large by roughly 5\%. More severe is that the bare vertex used here seems to significantly overestimate the gluon-Higgs interaction compared to the fully dressed vertex of the lattice result. The self-energy must be significantly suppressed even for large values of $p$ to allow for such a deviation from the leading-order perturbative result, and must be nearly negligible. In principle, such an effect could be modeled by a construction similar to \pref{g3vertex} for the Higgs-gluon vertex. Again, this is not the aim of this work. However, this finding entails that strong subleading or even genuinely non-perturbative effects are also present in the Higgs sector. Even for quite large momenta of the order of $10g_3^2$, where the gluon propagator has already become perturbative, this is not the case for the Higgs propagator, which will only become perturbative at much larger momenta of the order of $100g_3^2$. Indeed, the overestimation of the mass already shifts the result towards the tree-level propagator, and a smaller mass would enhance the discrepancy. This is indeed quite an interesting effect, and it will be investigated in some more detail in section \ref{sschwinger}.
\section{Coulomb Gauge Instantaneous Potential}\label{scoulomb}
Recently, a connection between Coulomb-gauge in 4d and Landau-gauge in 3d has been established \cite{Zwanziger:2003de,Feuchter:2004gb}. It was shown \cite{Zwanziger:2003de} that under certain approximations, the static quark-anti-quark potential, which is linked to the 00-component of the Coulomb gauge gluon propagator, is given in four dimensions by
\begin{equation}
V_{\mathrm{coul}}(p)=\frac{G(p)^2}{p^2}+V_{\mathrm{con}}(p)\label{vcoul}.
\end{equation}
\noindent $V_{\mathrm{con}}$ designates contributions which stem from connected parts of expectation values of the Faddeev-Popov operator. Neglecting this part, the potential is solely given by the function $G$. It can be shown to be identical to the ghost dressing function of the 3d Landau gauge Yang-Mills theory \cite{Zwanziger:2003de}. While the solution $g\approx 0.4$ generates a potential which behaves as $\sim1/p^{3.6}$ and thus a little less than linear, the second branch generates a solution proportional to $1/p^4$ for small momenta, thus generating the behavior expected for a linear confining potential.
In principle it is expected that after Fourier-transformation \pref{vcoul} would give the leading contribution of the static potential. However, it is not possible in the given approximation scheme to obtain the Fourier transform. The Fourier integral is quadratically divergent in the ultraviolet. The reason is that $G$ goes to 1 in the ultraviolet, thus together with the measure it generates a contribution proportional to $p$. Using a prescription as will be used in section \ref{std}, it may still be possible to extract some information, but since not a constant but a function needs to be fitted, this would turn out to be significantly more complicated. It is improbable that the numerical accuracy would be sufficient to extract relevant information, as it is already hard to do so in the case of a constant quantity like the thermodynamic potential treated in section \ref{std}. This is due to the approximations involved when obtaining the infrared limit \pref{vcoul}.
\chapter{Conventions}\label{aconventions}
Calculations are done in natural units,
\begin{equation}
\hbar=c=k_{B}=1.\nonumber
\end{equation}
\noindent The Minkowski-metric is
\begin{equation}
\eta=\mathrm{diag}(1,-1,-1,-1).\nonumber
\end{equation}
\noindent The Euclidean metric after factorization of -1 is
\begin{equation}
\eta_e=\mathrm{diag}(1,1,1,1).\nonumber
\end{equation}
\noindent Thus $p^2=p_0^2+\vec p^2$. Sometimes $|\vec p|=p_3$ is used. The individual components of $\vec p$ do never appear explicitly.
The second Casimir $C_A$ of a compact, semi-simple Lie-group $G$ in the adjoint representation with structure constants $f^{abc}$ is defined by
\begin{equation}
f^{acd}f^{acd}=C_A.\nonumber
\end{equation}
\noindent For SU$(N_c)$ groups, $C_A=N_c$.
Fourier transformations are performed with all momenta incoming,
\begin{eqnarray}
f(p)&=&\int d^dx f(x)e^{ixp}\nonumber\\
f(x)&=&\int\frac{d^dp}{(2\pi)^d} f(p)e^{-ixp}.\nonumber
\end{eqnarray}
\chapter{Derived Quantities}\label{cderived}
The dressing functions obtained in chapters \ref{c3d} and \ref{cft} are only the first step from the basic Lagrangian towards observables. In this chapter, derivations towards such observables are performed. The screening masses and with them the analytic structure of the particles described will be investigated in section \ref{sschwinger}. A second natural object is the thermodynamic potential discussed in section \ref{std}.
If not stated otherwise, throughout this chapter $C_A=N_c=3$, $\zeta=\xi=1$, $\delta=1/4$, and the lattice value for $m_h/g_3^2$ are used.
\section{Schwinger Functions and Analytic Structure}\label{sschwinger}
\subsection{Schwinger Functions}
Screening masses are most directly extracted from the analytic structure of the propagators. To obtain access to these analytic properties, the Schwinger function related to the dressing function $D$ is calculated. It is defined as \cite{Alkofer:2003jk}
\begin{equation}
\Delta(t)=\frac{1}{\pi}\int_0^\infty dp_0\cos(tp_0)\frac{D(p_0)}{p_0^2},\label{schwinger}
\end{equation}
\noindent i.e.\/ the Fourier transform of the propagator with respect to (Euclidean) time. Note that this definition is independent of the dimensionality of the underlying theory. Negative values for the Schwinger function can be traced to violations of positivity and therefore to absence of the particle represented by $D$ from the physical spectrum~\cite{Alkofer:2003jk}.
The numerical implementation of \pref{schwinger} is non-trivial. To obtain sufficient precision, a FFT-algorithm using edge corrections and compensation for the infinite integration range of \pref{schwinger} together with at least 512 or more frequencies\footnote{Ten thousands to a million are much better. The results presented here have been obtained using roughly $5\cdot10^5$ frequencies.} was employed \cite{Press:1997}.
The time $t$ in \pref{schwinger} in the 3d theory of the infinite temperature limit is actually a space-direction in 4d. To such a direction also the arguments concerning positivity can be applied in Euclidean space-time. Therefore, the result for the gluon, presented in figure \ref{figzschwing}, clearly exhibits positivity violations. This is in accordance with the Oehme-Zimmermann super-convergence relation \pref{oehme} and thus has been expected. Furthermore, the position of the zeros can be interpreted as the confinement scale. For the $g=1/2$ solution, the zero occurs at $g_3^2t\approx 3.29$ and at $g_3^2t\approx 3.98$ for the other branch. This result is in agreement with recent lattice results~\cite{Cucchieri:2004mf}.
\begin{figure}
\epsfig{file=zschwinger.eps,width=0.5\linewidth}\epsfig{file=compglfit.eps,width=0.5\linewidth}
\caption{The left panel shows the Schwinger function of the gluon. The right panel shows the comparison of the fit-function \pref{gluonfit} to the full solutions
of the dressing function. The solid line gives the numerical result for $g=1/2$ while the dashed line is for $g\approx0.4$. The dotted and dash-dotted line denote their respective fits using the ansatz (\ref{gluonfit}).}
\label{figzschwing}
\end{figure}
To be able to perform an analytic continuation of the gluon propagator into the complex $p^2$-plane, the Schwinger function is fitted. This is performed by using the ansatz\footnote{In \cite{Alkofer:2003jk}, a parameterization with only a branch cut and no isolated pole provided a successful fit. Due to the different asymptotic behavior in 4 and 3 dimensions, a different ansatz is used here.}
\begin{equation}
Z_f(p)=\frac{A_z p^{4g+1}}{1+f+A_z p^{4g+1}}\left(1+\frac{f}{1+fa_up}\right)=\frac{A_z p^{4g+1}(1+f+a_ufp)}{(1+a_ufp)(1+f+A_zp^{4g+1})}\label{gluonfit}
\end{equation}
\noindent for the dressing function. $A_z$ is the infrared coefficient determined previously, and $a_u$ is the ultraviolet coefficient of leading-order resummed perturbation theory, as calculated in appendix \ref{appUV}. The fit parameters for both solutions are given in table \ref{tabglfit}. As demonstrated in figure \ref{figzschwing}, the Schwinger function is fitted very well. The gluon propagator and dressing function are also reasonably well described by the fit.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Solution & $A_zg_3^{-2t}$ & $a_ug_3^2$ & $f$ \cr
\hline
$g=1/2$ & 20.3 & $64/27$ & 1.32511 \cr
\hline
$g\approx 0.4$ & 13.4 & $64/27$ & 1.03148 \cr
\hline
\end{tabular}
\end{center}
\caption{The coefficients for the gluon fit (\ref{gluonfit}).}
\label{tabglfit}
\end{table}
\begin{figure}
\epsfig{file=hfitstd.eps,height=1.2\linewidth,width=0.5\linewidth}\epsfig{file=hfitalt.eps,height=1.2\linewidth,width=0.5\linewidth}
\caption{The Higgs propagator is shown in the top panels and its Schwinger function in the middle panels compared to their fits. The bottom panels show the comparison of the numerical (solid) and the fitted (dashed) subleading Higgs contribution, see text. The left and right side shows the $g=1/2$ and $g\approx 0.4$ solutions, respectively. The solid lines represent the numerical solution, the dashed lines give the leading contribution and the dashed-dotted lines the first subleading contribution. The dotted lines underneath the solid lines give the sum of the leading and subleading contribution. }
\label{fighstdfit}
\end{figure}
\begin{figure}
\epsfig{file=hsstd.eps,width=0.5\linewidth}\epsfig{file=hsalt.eps,width=0.5\linewidth}
\caption{The bottom panel shows the Higgs propagator for various suppression factors $\omega$ of the Higgs-gluon vertex. The top panel shows the corresponding Schwinger functions. The left panels contain the $g=1/2$ solution while the right ones display the $g\approx 0.4$ solution. The lattice data are the same as on the right side of figure \ref{figlat}. The solid line is for $\omega=1/4$, the dashed line for $\omega=1/2$, the dotted line is for $\omega=3/4$ and the dashed-dotted line for $\omega=1$.}
\label{fighsalt}
\end{figure}
Similar to a meromorphic fit ansatz used in \cite{Alkofer:2003jk}, the Higgs propagator and its Schwinger function, the latter being analytically calculated from the former \cite{Gradstein:1981}, are described as
\begin{eqnarray}
H_f(p)&=&\frac{e+f p^2}{p^4+2m^2\cos\left(2\phi\right)p^2+m^4},\label{higgsfit}\\
\Delta_f(t)&=&\frac{e}{2m^3\sin\left(2\phi\right)}e^{-tm\cos\left(\phi\right)}\left(\sin\left(\phi+tm\sin\left(\phi\right)\right)+\frac{fm^2}{e}\sin\left(\phi-tm\sin\left(\phi\right)\right)\right).
\nonumber
\end{eqnarray}
\noindent As demonstrated in figure \ref{fighstdfit}, these fits already describe the Schwinger function quite well, but miss around 20\% of the propagator at zero momentum. This indicates that further massive modes are present. Indeed, for the $g=1/2$ solution, a further term of the form (\ref{higgsfit}) has to be added. For the $g\approx 0.4$ solution, adding a term with one pole,
\begin{eqnarray}
H_f(p)=\frac{e}{p^2+m^2},\label{higgsaltfit}\\
\Delta_f(t)=\frac{e}{2m}e^{-mt},\label{higgssaltfit}
\end{eqnarray}
improves the fit also in this case. Both subleading fits are not very accurate for large $t$, and the results have to be taken with care. They indicate, nevertheless, the existence of subleading contributions due to the presence of further massive-particle-like contributions. The fit parameters of both solutions can be found in table \ref{tabhfit}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Solution & $e$ & $fg_3^4$ & $\phi$ & $m/g_3^2$ \cr
\hline
$g=1/2$ & 4.0199 & 0.3545 & -0.67078 & 1.4998 \cr
\hline
subleading & 9.4493 & 0.6736 & -0.18387 & 2.561 \cr
\hline
$g\approx 0.4$ & 4.0697 & 0.37793 & -0.63975 & 1.5045 \cr
\hline
subleading & 0.81188 & & & 1.9223 \cr
\hline
\end{tabular}
\end{center}
\caption{The coefficients for the Higgs fit (\ref{higgsfit}) and the subleading one (\ref{higgsfit}) and (\ref{higgsaltfit}), respectively.} \label{tabhfit}
\end{table}
As can be seen from figure \ref{figlat}, the result for the Higgs propagator deviates significantly from the lattice results, the self-energy being significantly overestimated. While the gluon Schwinger function is reasonably independent of the truncation, this turns out not to be the case for the Higgs Schwinger function. In order to show this, the solutions for a bare Higgs-gluon vertex suppressed via a scaling factor $\omega$ are obtained\footnote{In the Landau gauge, due to the transversality of the gluon, this is equivalent to modifying the tensor structure of the vertex.}. The first result is that the gluon Schwinger function, in contrast to the Higgs one, is not susceptible to such a change.
Figure \ref{fighsalt} displays corresponding results for different values of $\omega$. The position of the first zero tends to increase for decreasing $\omega$. At $\omega\approx 1/4$ any oscillation, at least within the available numerical precision, seems to be gone altogether. A fit to the Schwinger function using \pref{higgssaltfit} reveals again additional structure at very small $t$ which cannot be captured by such a simple fit. As the fit also misses some strength at zero momentum for the propagator, this again indicates the presence of further massive contributions in the propagator.
\begin{figure}
\epsfig{file=ftschwinger.eps,width=\linewidth}
\caption{The Schwinger functions at $T=1.5$ GeV and $N=5$. The left top panel shows the soft transverse gluon Schwinger function, the right top panel the hard transverse gluon Schwinger function for $n=1$ with a black line and for $n=2$ with a red line. The bottom panels show the same for the longitudinal gluon Schwinger function. In all cases are solid lines the $g=1/2$ solution and dashed lines the $g\approx 0.4$ solution.}
\label{figftschwinger}
\end{figure}
When adding the hard modes, the qualitative picture does not change, as is visible in figure \ref{figftschwinger}. In this case the Schwinger function also depends on a `space'-variable with respect to 4d and $p_0=2\pi Tn$ acts as an effective mass. The gluon soft mode shows an oscillation as expected. The same is still true for the Higgs, thus this property of the truncation used here is still present. At this rather low temperature, it exhibits only one oscillation, and is thus qualitatively much more similar to the gluon. In both cases, the oscillation frequency has decreased due to the additional interactions with the hard modes. The interpretation of the oscillations in the longitudinal gluon Schwinger function are still inconclusive, though.
For the hard mode Schwinger functions no oscillations are seen. They cannot be excluded as they might probably not be resolvable numerically if the oscillation frequency is of the order of the mass. Up to the numerical accuracy, the hard modes present the behavior expected for a massive particle. This equally well applies to the higher modes, as is visible by the example of the next mode $n=2$.
Thus, the features found in the infinite temperature limit persist at finite temperatures. Note, however, that if the hard modes oscillate, this oscillation does not necessarily vanish when the $T\to\infty$ limit is taken. If the oscillation frequency only vanishes as fast as the interaction, then even for arbitrarily small interactions, oscillations remain. It will take more sophistication to settle the question of whether the hard modes show positivity violation or not.
\subsection{Analytic Properties in the Infinite-Temperature Limit}\label{analytic}
Although the high-temperature limit of the four-dimensional Minkowski theory is a genuinely Euclidian theory, it will be of interest for other applications to extract the analytic structure of the propagators investigated.
The gluon propagator exhibits similar behavior for both solutions, but there are also some significant differences. The denominator of the ansatz \pref{gluonfit} contains two factors, both of which could possibly give rise to a non-trivial analytic structure. The first part stems from the fit of the perturbative tail necessary to generate the maximum in the gluon dressing function. Since all fit parameters are positive, this factor does not give rise to a pole on the first Riemann sheet. However, it generates a pole on the second Riemann sheet, which will occur at $1/a_uf$, that is, at Euclidian momenta. This pole does not have a physical interpretation, and may well be an artifact of the fit, since \pref{gluonfit} is tailored to generate the correct leading-order perturbative behavior. Thus, it generates most likely a structure which has the Landau pole of perturbation theory on the second Riemann sheet. It is to be expected that this pole vanishes when using a more sophisticated fit.
The second factor generates a genuine isolated pole at $(-(1+f)/A_z)^{-1/t}$. This expression has only one value on the first Riemann sheet given by $(-0.0933+0.1615i)g_3^4$ for the $g=1/2$ and $(-0.1462+0.1248i)g_3^4$ for the other solution\footnote{This is of the order of $\Lambda_{QCD}$ when using the 't Hooft-like scaling of section \ref{scc}.}. In both cases, a pole close to the origin is generated, with an imaginary part larger than the real part in one case. For the first solution, the pole is found for an angle significantly above $\pi/4$ while in the second case somewhat below. In addition, the first solution generates two more poles on two more Riemann sheets, while the second solution, with a (most likely) irrational exponent, generates an infinite number of further poles on an infinite number of Riemann sheets. In both cases, the residue is complex. In addition, there is a cut along the complete negative real axis starting at zero. In this way it is similar to the results in four dimensions~\cite{Alkofer:2003jk}.
This analysis infers that the gluon propagator is violating positivity, and thus satisfies the requirements of the Kugo-Ojima and Zwanziger-Gribov confinement scenario.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Solution & Order & Pole$/g_3^4$ \cr
\hline
$g=1/2$ & Leading & $-0.5112\pm2.191i$ \cr
\hline
$g=1/2$ & Subleading & $-6.11972\pm2.35777i$ \cr
\hline
$g\approx 0.4$ & Leading & $-0.650078\pm2.16822i$ \cr
\hline
$g\approx 0.4$ & Subleading & $-3.69539$ \cr
\hline
\end{tabular}
\end{center}
\caption{Location of the poles of the Higgs propagator for $\omega=1$.}
\label{tabhiggspoles}
\end{table}
On the other hand, the Higgs propagator very likely does not have a branch cut but a number of simple poles whose locations are given in table \ref{tabhiggspoles}. The sensitivity to the Higgs-gluon vertex, however, necessitates further investigations before a firmer conclusion can be drawn.
\begin{figure}[ht]
\begin{center}\epsfig{file=comega.eps,width=0.5\linewidth}\end{center}
\caption{The dependence of the first zero of the Higgs Schwinger function on the vertex suppression $\omega$. Solid is the $g=1/2$ solution and dotted its fit, dashed is the $g\approx 0.4$ solution and dashed-dotted its fit.}
\label{figcomega}
\end{figure}
The dependence of the first zero of the Higgs-Schwinger function on $\omega$ is shown in figure \ref{figcomega}. The result can be fitted using
\begin{equation}
t_0=\frac{a}{\omega^b}\nonumber
\end{equation}
\noindent where $t_0$ is the position of the first zero. The parameters are $(a,b)=(2.49,1.78)$ for the $g=1/2$ solution and $(2.40,1.73)$ for the $g\approx 0.4$ solution. Such a behavior would indicate a vanishing imaginary part and thus oscillations only when the Higgs ceases to interact. Thus, the oscillations would remain, albeit the poles move closer to the real axis. This might only be an indication of the possibility of the Higgs to decay, but the question remains into what, as gluons are confined. The existence of one or more complex conjugate poles on the same Riemann sheet can be interpreted in the framework of the Gribov-Stingl scenario \cite{Habel:1989aq} as a signal of confinement for the Higgs as well. However, as the Higgs is sensitive to the truncation, this is not conclusive, until more studies have been performed.
\section{Thermodynamic Potential}\label{std}
\subsection{Soft Mode Contribution in the Infinite-Temperature Limit}
Knowledge of all Green's functions as functions of the temperature would permit to calculate the thermodynamic potential and therefore all thermodynamic quantities. Since not all of these are known, it is not possible to calculate an exact thermodynamic potential. Knowledge of the propagators is however sufficient to compute an approximation to the thermodynamic potential, using the Luttinger-Ward or Cornwall-Jackiw-Tomboulis (LW/CJT) effective action \cite{Luttinger:1960ua,Cornwall:1974vz}.
This effective action can be extended to abelian and non-abelian gauge theories, see e.g.\/ \cite{Haeri:hi}. Its calculation is prohibitively complicated when using constructed vertices instead of bare or exact vertices. Therefore, only some qualitative features using the simplest approximation to the thermodynamic potential will be extracted here. It only depends on the propagators, and reads
\begin{eqnarray}
\Omega&=&\frac 1 2 d(G) T\sum\int\frac{d^3q}{(2\pi)^3}\Big(-\ln\left(\frac{Z(q)}{Z_0(q)}\right)+\left(\frac{Z(q)}{Z_0(q)}-1\right)\nonumber\\
&&+\ln\left(\frac{G(q)}{G_0(q)}\right)-\left(\frac{G(q)}{G_0(q)}-1\right)-\frac{1}{2}\ln\left(\frac{H(q)}{H_0(q)}\right)+\frac{1}{2}\left(\frac{H(q)}{H_0(q)}-1\right)\Big)\label{cjt},
\end{eqnarray}
where $d(G)=\delta^{aa}$ is the dimension of the gauge group. $Z_0$, $G_0$, and $H_0$ are the tree-level dressing functions
\begin{eqnarray}
Z_0(p)=G_0(p)=1\nonumber\\
H_0(p)=\frac{p^2}{p^2+m_h^2}.\nonumber
\end{eqnarray}
Working in the infinite-temperature limit, the thermodynamic potential divided by $T^4$ becomes interesting. This motivates a rescaling of the integration momenta to obtain
\begin{equation}
\frac{\Omega}{T^4}=\frac{g_3^6}{T^3}a\label{eps}
\end{equation}
where the dimensionless constant $a$ depends only on the dimensionless ratio $m_h/g_3^2$ and thus becomes independent of temperature in the limit $T\to\infty$. Herein, the explicit value of $g_3$ enters. Using the t'Hooft-like scaling presented in section \ref{scc}, $g_3^2\sim\Lambda_{QCD}$ results. Hence (\ref{eps}) scales as $1/T^3$ and does not contribute to the thermodynamic pressure significantly compared to the hard modes. The expected Stefan-Boltzmann behavior must then be obtained from the hard modes alone, requiring
\begin{equation}
\frac{\Omega}{T^4}=g^6_4(\mu)\left(a+\frac{a_h}{g_4^6(\mu)}\right) = \frac{g_3^6}{T^3}a+a_h\nonumber
\end{equation}
where $a_h$ stems from the hard modes. However, the soft modes may still contribute significantly to thermodynamic properties, especially the trace anomaly $\theta=\epsilon-3p$, near the phase transition \cite{Zwanziger:2004np}.
The calculation of \pref{cjt} turns out to be plagued by spurious divergences, since the DSEs with the modified 3-gluon-vertex are no longer exact stationary
solutions of \pref{cjt}. By close examination of \pref{cjt}, and the fact that all dressing functions $D$ can be expanded in the infinite temperature limit for $p\gg g_3^2$ as
\begin{equation}
D(p)=1+\sum_{n=1}^{\infty}a_n\left(\frac{g_3^2}{p}\right)^2,\nonumber
\end{equation}
\noindent the divergence structure can be isolated. The tree-level value drops out in each term independently. Expanding the logarithms also in $g_3^2/q$ directly shows that the LO cancels between the linear and the logarithmic terms as well. This is no longer the case in next-to-leading order (NLO), and this will only be possible by requiring a correct solution of the Dyson-Schwinger equations to NLO in perturbation theory and using the appropriate 2-particle irreducible expression \cite{Cornwall:1974vz} for the thermodynamic potential. As the NLO term is proportional to $(g_3^2/q)^2$, it generates a linear divergence in $\Omega$ together with the measure. Although there is no a priori reason for this NLO contribution to be the same for both solutions, as this term is truncation dependent, still this is the case\footnote{Since the two solutions only differ in their respective infrared behavior, this may have only a minor influence on the perturbative tail.}. The NNLO will generate a logarithmic divergence and only from NNNLO all terms will be UV-finite. The general structure of the result is therefore
\begin{equation}
\frac{\Omega}{g_3^6T}=a+b\log\frac{\Lambda}{c}+d\Lambda\label{fullfit}
\end{equation}
\noindent where $\Lambda$ is the ultraviolet cutoff when calculating \pref{cjt}.
While the linear term is easily identifiable, this is not the case for the logarithmic one, especially as it is always possible to move contributions from $c$ to $a$ and back. This makes it in some sense arbitrary to calculate $a$. Assuming that the divergent contributions will be small at sufficiently small cutoff, but $a$ has already reached its final value, a first fit is performed using
\begin{equation}
\frac{\Omega}{g_3^6T}=a+d\Lambda.\nonumber
\end{equation}
\noindent In a second step, with $a$ held fixed, the full fit using \pref{fullfit} is performed. This gives the final values. For this approach to be valid, the logarithmic term must be much smaller than $a$. At least for calculating the differences of both solutions, this is the case. Also $c$ turns out to be always the same value for all fits, giving confidence in the fit procedure.
In the case of the ghost-loop-only truncation, the difference is independent of the cutoff and is $-0.001862(0)$. As all differences were performed by subtracting the $g=1/2$ solution from the second solution, this result would favor the $g=1/2$ solution.
In all other cases the prescribed subtraction was necessary. Note that, although $d$ should be 0 when calculating the differences, this is only true up to numerical errors. Hence also for differences it was necessary to extract $d$ and perform the corresponding fits\footnote{The required accuracy for these calculations is very high. An expansion in up to 500 Chebychef polynomials, see appendix \ref{anum}, was necessary.}. 48 fit points have been included from the $\Lambda$ interval [54763.70,4830021]. Within these values, the integrand and the integral did not show significant numerical fluctuations. The statistical error on $a$ is obtained by calculating the mean square error of the full fit. Besides the statistical error, systematic errors were investigated using variations of the 3-gluon vertex and the gluon-Higgs vertex by varying $\delta$ and $\omega$. The results are listed in table \ref{tres}. Note that the calculated difference and the fitted difference are always the same within statistical errors.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Truncation & Object & Sol. & $a$ & $b$ & $c/g_3^2$ & $dg_3^2$ \cr
\hline
Ghost-loop & $\Delta\Omega$ & & -0.001862(0) & & & \cr
\hline
Ghost-loop & $\Omega$ & 1 & -0.2095(186) & -0.258 & $189\cdot 10^3$ & -0.0025 \cr
\hline
Ghost-loop & $\Omega$ & 2 & -0.2076(607) & -0.258 & $189\cdot 10^3$ & -0.00250 \cr
\hline
Yang-Mills & $\Delta\Omega$ & & 0.03942(0) & 0.00265 & $189\cdot 10^3$ & $1.25\cdot 10^{-11}$ \cr
\hline
Yang-Mills & $\Omega$ & 1 & 1.735(135) & 1.92 & $189\cdot 10^3$ & 0.0175 \cr
\hline
Yang-Mills & $\Omega$ & 2 & 1.696(135) & 1.92 & $188\cdot 10^3$ & 0.0175 \cr
\hline
Full & $\Delta\Omega$ & & 0.05276(0) & 0.00280 & $189\cdot 10^3$ & $5.54\cdot 10^{-12}$ \cr
\hline
Full & $\Omega$ & 1 & 3.586(254) & 3.67 & $189\cdot 10^3$ & 0.0322 \cr
\hline
Full & $\Omega$ & 2 & 3.533(254) & 3.67 & $189\cdot 10^3$ & 0.322 \cr
\hline
YM contribution & $\Delta\Omega$ & & 0.02083(0) & 0.00162 & $189\cdot 10^3$ & $9.92\cdot 10^{-12}$ \cr
\hline
YM contribution & $\Omega$ & 1 & 0.8215(983) & 1.33 & $189\cdot 10^3$ & 0.0108 \cr
\hline
YM contribution & $\Omega$ & 2 & 0.8007(983) & 1.33 & $189\cdot 10^3$ & 0.0108 \cr
\hline
$\delta=1$ & $\Delta\Omega$ & & 0.08433(0) & 0.00910 & $189\cdot 10^3$ & $4.90\cdot 10^{-11}$ \cr
\hline
$\delta=1$ & $\Omega$ & 1 & 2.442(255) & 3.55 & $189\cdot 10^3$ & 0.0322 \cr
\hline
$\delta=1$ & $\Omega$ & 2 & 2.338(255) & 3.54 & $189\cdot 10^3$ & 0.0322 \cr
\hline
$\omega=1/4$ & $\Delta\Omega$ & & 0.03820(0) & 0.00258 & $189\cdot 10^3$ & $9.37\cdot 10^{-12}$ \cr
\hline
$\omega=1/4$ & $\Omega$ & 1 & 1.708(136) & 1.93 & $189\cdot 10^3$ & 0.0171 \cr
\hline
$\omega=1/4$ & $\Omega$ & 2 & 1.669(136) & 1.93 & $189\cdot 10^3$ & 0.0171 \cr
\hline
$\delta=1,\omega=1/4$ & $\Delta\Omega$ & & 0.08282(0) & 0.00999 & $189\cdot 10^3$ & $6.96 10^{-11}$ \cr
\hline
$\delta=1,\omega=1/4$ & $\Omega$ & 1 & 0.4518(4210) & 1.79 & $189\cdot 10^3$ & 0.0171 \cr
\hline
$\delta=1,\omega=1/4$ & $\Omega$ & 2 & 0.3689(1362) & 1.78 & $189\cdot 10^3$ & 0.0171 \cr
\hline
\end{tabular}
\end{center}
\caption{The thermodynamic potential $\Omega$ and the difference between both solutions $\Delta\Omega$ in the different truncation schemes. The fit was performed using \pref{fullfit}. The quoted errors are statistical only. Solution 1 is $g=1/2$ and solution 2 is $g\approx 0.4$. Note that $\ln(4830021/189000)\approx 3.25$. YM contribution is the Yang-Mills part of the full solution. The $\delta$-, $\omega$-, and $\delta\omega$-variations stem from the variation of the 3-gluon vertex, the gluon-Higgs vertex and the combined variations.}
\label{tres}
\end{table}
These results strongly indicate that the $g\approx 0.4$ solution is the thermodynamically preferred one, as this result is stable with respect to variation of the parameters. This agrees with the comparison to lattice results in section \ref{slattice3d}. However, as it still cannot be excluded that either of the solutions is a truncation artifact, this is not a final statement. Moreover, the value of $a$ seems to be ${\cal O}(1)$. However, the systematic study of the vertex influence indicates that the value can only be determined within a factor of 2 or worse. The thermodynamic potential seems to be generated mainly in the chromoelectric sector, but also this is only an indication.
\subsection{Hard Mode Contribution}
To obtain the contribution from the hard modes, it is at least problematic to use \pref{cjt}. The LW/CJT-action \pref{cjt} considers only the interaction part. It vanishes identically for a free system, i.e.\/ for a system containing only tree-level dressing functions. This free contribution has thus to be added explicitly. Therefore the hard mode contribution can essentially be calculated to be the same as that of a non-interacting system of free gluons. The LW/CJT-expression \pref{cjt} does not capture the tree-level contribution of the soft modes, and it thus can be added here. This yields a thermodynamic potential of a gas of massless gluons with small corrections due to the explicit soft-mode contributions and the residual interactions of the hard modes. It is for $N_c=3$ \cite{Karsch:2003jg}
\begin{equation}
\lim_{T\to\infty}\frac{\Omega}{T^4}\approx-\frac{16\pi^2}{90}.\label{sblimit}
\end{equation}
\noindent Using the t'Hooft-scaling for the interaction strength, the corrections due to the soft contributions vanish like $1/T$. The only remaining contribution in the infinite temperature limit is the Stefan-Boltzmann-limit \pref{sblimit}, in agreement with results from lattice calculations \cite{Karsch:2003jg}. Therefore the thermodynamical properties of the high-temperature limit are governed by the hard modes, although they do not participate in the dynamical interactions.
\chapter{Derivation of the Dyson-Schwinger Equations}\label{adse}
Starting from \pref{dse}, it is possible to derive the DSEs for the Lagrangian \pref{l3d}. In this context also the Legendre transform of $W$ plays a role, where $W$ is defined by the generating functional \pref{lpart}. The Legendre transform can be obtained by
\begin{equation}
W(j^a)=-\Gamma(\phi^a)+\int d^dxj^a(x)\phi^a(x)\label{effpot}
\end{equation}
\noindent which entails
\begin{eqnarray}
\phi^a=\frac{\delta W}{\delta j^a}\nonumber\\
j^a=\frac{\delta\Gamma}{\delta\phi^a}\nonumber.
\end{eqnarray}
\noindent In the case of Grassmann fields $u$ like ghosts and fermions, two independent sources are necessary. This modifies the above to
\begin{eqnarray}
Z=\int{\cal D}u^a{\cal D}\bar u^a e^{-S[u^a,\bar u^a]+\int d^dx(\bar\eta^a(x) u^a(x)+\bar u^a(x)\eta^a(x))}\nonumber\\
u^a(x)=\frac{\delta W}{\delta\bar\eta^a(x)}\quad\quad\quad\quad\bar u^a(x)=-\frac{\delta W}{\delta\eta^a(x)}\nonumber\\
W(\eta^a,\bar\eta^a)=-\Gamma(u^a,\bar u^a)+\int d^dx\left(\bar\eta^a(x)u^a(x)+\bar u^a(x)\eta^a(x)\right)\nonumber\\
\eta^a(x)=\frac{\delta\Gamma}{\delta\bar u^a(x)}\quad\quad\quad\quad\bar\eta^a(x)=-\frac{\delta\Gamma}{\delta u^a(x)}\nonumber,
\end{eqnarray}
\noindent where all derivatives with respect to Grassmann variables act in the direction of ordinary derivatives.
The general procedure to obtain the corresponding DSEs is to calculate equation \pref{dse} and then take the derivative once more with respect to the field or with respect to the anti-field in case of anti-commuting fields. The additional source terms then yield the full propagators, while the right-hand-sides of the equations are found by the derivative of the action.
Since in the course of the derivation, the source in equation \pref{dse} becomes the inverse full propagator, it makes sense to already rewrite \pref{dse} as
\begin{equation}
j^a(x)Z=\frac{\delta S}{\delta\phi^a(x)}\Big|_{\phi^a(x)=\frac{\delta}{\delta j^a(x)}}Z\label{mdse}
\end{equation}
\noindent at the sources set equal to 0 in the end. This entails also that all single derivatives of the effective action, being classical fields, vanish at the end.
The inverse propagators of the ghost, the gluon and the Higgs are defined as
\begin{eqnarray}
\frac{\delta^2\Gamma}{\delta c^b(y)\delta\bar c^a(x)}&=D^{ab}_G(x-y)^{-1}\label{adighprop}\\
\frac{\delta^2\Gamma}{\delta A_\nu^b(y)\delta A_\mu^a(y)}&=D^{ab}_{\mu\nu}(x-y)^{-1}\\
\frac{\delta^2\Gamma}{\delta\phi^b(y)\delta\phi^a(x)}&=D_H^{ab}(x-y)^{-1},\label{adihprop}
\end{eqnarray}
\noindent The propagators are then given by
\begin{eqnarray}
\frac{\delta^2 W}{\delta\eta^b(y)\delta\bar\eta^a(x)}&=D^{ab}_G(x-y)\label{adghprop}\\
\frac{\delta^2 W}{\delta j_\nu^b(y)j_\mu^a(x)}&=D^{ab}_{\mu\nu}(x-y)\\
\frac{\delta^2 W}{\delta k^b(y)\delta k^a(x)}&=D_H^{ab}(x-y)\label{adhprop}.
\end{eqnarray}
\noindent $\eta$ is the source of the ghost field, $\bar\eta$ of the anti-ghost field, $j$ of the gluon, and $k$ of the Higgs. \prefr{adghprop}{adhprop} are inverse to \prefr{adighprop}{adihprop} which can be proven, e.g. for the ghost propagator by
\begin{equation}
\int d^dz\frac{\delta^2 W}{\delta\eta^c(z)\delta\bar\eta^a(x)}\frac{\delta^2\Gamma}{\delta c^b(y)\delta\bar c^c(z)}=\int d^dz\frac{\delta c^a(x)}{\delta\eta^c(z)}\frac{\delta\eta^c(z)}{\delta c^b(y)}=\frac{\delta c^a(x)}{\delta c^b(y)}=\delta^{ab}\delta(x-y).\nonumber
\end{equation}
\noindent Note that all mixed propagators vanish when the sources are set to 0, e.g.
\begin{equation}
\frac{\delta^2\Gamma}{\delta\bar c^g(w)\delta A_\nu^f(z)}\Big |_{j=\eta=\bar\eta=0}=0,\nonumber
\end{equation}
due to conservation of ghost number and spin.
Higher $n$-point functions, the vertices, are defined accordingly as
\begin{eqnarray}
\frac{\delta^3\Gamma}{\delta c^a(x)\delta\bar c^b(y)\delta A^c_\mu(z)}=\Gamma_\mu^{c\bar cA, abc}(x,y,z)\nonumber
\end{eqnarray}
\noindent and correspondingly for the other vertices.
There are a few useful identities in the course of the derivation. These are
\begin{eqnarray}
\frac{\delta^2 W}{\delta j^e_\mu(x)\delta\bar\eta^d(x)}&=&-\int d^dzd^dw\frac{\delta^2 W}{\delta j_\nu^f(z)\delta j^e_\mu(x)}\frac{\delta^2\Gamma}{\delta\bar c^g(w)\delta A_\nu^f(z)}\frac{\delta^2 W}{\delta\bar\eta^g(w)\delta\eta^d(x)}\nonumber\\
\pd_{\mu}^x\frac{\delta}{\delta\eta^c(x)}&=&\int d^dz \left(\pd_{\mu}^x\frac{\delta\Gamma}{\delta\eta^c(x)\delta\bar c^e(z)}\right)\frac{\delta}{\delta\eta^e(z)}.\nonumber
\end{eqnarray}
Further, since
\begin{eqnarray}
&&\frac{\delta^3 W}{\delta A_\nu^b(y)\delta j^c_\sigma(x)\delta j^e_\sigma(x)}=\frac{\delta}{\delta A_\nu^b(y)}\int d^dzd^dw \frac{\delta^2 W}{\delta j_\sigma^c(x)\delta j_\rho^f(z)}\frac{\delta^2\Gamma}{\delta A_\rho^f(z)\delta A_\omega^g(w)}\frac{\delta^2 W}{\delta j_\omega^g(w)\delta j_\sigma^e(x)}\nonumber\\
&=&2\frac{\delta^3 W}{\delta A_\nu^b(y)\delta j^c_\sigma(x)\delta^e_\sigma(x)}+\int d^dzd^dw \frac{\delta^2 W}{\delta j_\sigma^c(x)\delta j_\rho^f(z)}\frac{\delta^3\Gamma}{\delta A_\nu^b(y)\delta A_\rho^f(z)\delta A_\omega^g(w)}\frac{\delta^2 W}{\delta j_\omega^g(w)\delta j_\sigma^e(x)}\nonumber
\end{eqnarray}
\noindent it follows that
\begin{equation}
\frac{\delta^3 W}{\delta A_\nu^b(y)\delta j^c_\sigma(x)\delta j^e_\sigma(x)}=-\int d^dzd^dw \frac{\delta^2 W}{\delta j_\sigma^c(x)\delta j_\rho^f(z)}\frac{\delta^3\Gamma}{\delta A_\nu^b(y)\delta A_\rho^f(z)\delta A_\omega^g(w)}\frac{\delta^2 W}{\delta j_\omega^g(w)\delta j_\sigma^e(x)},\label{w3id}
\end{equation}
\noindent when the sources are set to 0. If not all sources and fields are of the same type, then it is necessary to sum over all possible intermediate fields $\omega^a$ and their sources $l^a$, where the index $a$ includes also the field type. Thus
\begin{equation}
\frac{\delta^3 W}{\delta \omega^b(y)\delta l^c(x)\delta l^e(x)}=-\int d^dzd^dw \frac{\delta^2 W}{\delta l^c(x)\delta l^f(z)}\frac{\delta^3\Gamma}{\delta \omega^b(y)\delta \omega^f(z)\delta \omega^g(w)}\frac{\delta^2 W}{\delta l^g(w)\delta l^e(x)}.
\end{equation}
\noindent It is also important to note that
\begin{equation}
\partial_\sigma^x\frac{\delta^2 W}{\delta j^c_\sigma(x)\delta j^d_\mu(x)}\nonumber
\end{equation}
\noindent is not zero, despite the translational invariance of the two-point functions. The derivative of the propagator at 0 does not necessarily vanish especially, if the propagator does only exist in the sense of a distribution.
As the effective potential defined by \pref{effpot} is bosonic, it can only depend on even numbers of ghost and anti-ghost fields. Therefore all vertices with only one ghost or anti-ghost leg vanish, when the sources are set to zero.
After a tedious calculation, the DSEs in position space are obtained. Adding suitable combinations of permuted diagrams, it is in all cases possible to make the appearance of one tree-level vertex explicit. Performing a Fourier-transformation, and keeping the redundant momentum at the vertices explicit, the DSEs in momentum space are for the ghost\footnote{Note that some diagrams which vanish for tree-level vertices had not been included in \cite{Maas:2004se}.}\enlargethispage*{1cm}
\begin{eqnarray}
&D^{ab}_G(p)^{-1}=-\delta^{ab}p^2\nonumber\\
&+\int\frac{d^dq}{(2\pi)^d}\Gamma_\mu^{\mathrm{tl}, c\bar cA, dae}(-q,p,q-p)D^{ef}_{\mu\nu}(p-q)D^{dg}_G(q)\Gamma^{c\bar cA, bgf}_\nu(-p,q,p-q),\label{tlgheq}
\end{eqnarray}
\noindent the gluon
\begin{eqnarray}
&D^{ab}_{\mu\nu}(p)^{-1}=\delta^{ab}(\delta_{\mu\nu}p^2-p_\mu p_\nu)\nonumber\\
&-\int\frac{d^dq}{(2\pi)^d}\Gamma_\mu^{\mathrm{tl}, c\bar cA, dca}(-p-q,q,p) D_G^{cf}(q) D_G^{de}(p+q) \Gamma_\nu^{c\bar c A, feb}(-q,p+q,-p)\nonumber\\
&+\frac{1}{2}\int\frac{d^dq}{(2\pi)^d}\Gamma^{\mathrm{tl}, A^3, acd}_{\mu\sigma\chi}(p,q-p,-q)D^{cf}_{\sigma\omega}(q)D^{de}_{\chi\lambda}(p-q)\Gamma^{A^3, bfe}_{\nu\omega\lambda}(-p,q,p-q)\nonumber\\
&+\frac{1}{2}\int\frac{d^dq}{(2\pi)^d}\Gamma_\mu^{\mathrm{tl}, A\phi^2, acd}(p,q-p,-q)D_H^{de}(q)D_H^(cf)(p-q)\Gamma^{A\phi^2, bef}_\nu(-p,q,p-q)\nonumber\\
&+\frac{1}{2}\int\frac{d^d q}{(2\pi)^d}\Gamma_{\mu\nu\sigma\rho}^{\mathrm{tl}, A^4, abcd}(p,-p,q,-q)D^{cd}_{\sigma\rho}(q)\nonumber\\
&+\frac{1}{6}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\sigma\xi\chi}^{\mathrm{tl}, A^4, acde}(p,-q,-p+q-k,k)\cdot\nonumber\\
&\cdot D_{\chi\lambda}^{dh}(p-q-k)D_{\xi\rho}^{cf}(q)D_{\sigma\omega}^{eg}(k)\Gamma^{A^4, hbfg}_{\lambda\nu\rho\omega}(p-q-k,-p,q,k)\nonumber\\
&+\frac{1}{2}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma^{\mathrm{tl}, A^4, acde}_{\mu\delta\gamma\sigma}(p,-q,q-k-p,k)D_{\gamma\lambda}^{dh}(k+p-q)\Gamma_{\nu\rho\omega}^{A^3, bfg}(-p,p+k,-k)\cdot\nonumber\\
&\cdot\Gamma_{\lambda\chi\xi}^{A^3, hij}(k+p-q,q,-k-p) D_{\rho\xi}^{fj}(p+k)D_{\sigma\omega}^{eg}(k)D_{\delta\chi}^{ci}(q)\nonumber\\
&+\frac{1}{2}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma^{\mathrm{tl}, A^4, acde}_{\mu\delta\gamma\sigma}(p,-q,q-k-p,k)D_{\gamma\lambda}^{dh}(k+p-q)\Gamma_{\nu\omega}^{A^2\phi, bgf}(-p,-k,p+k)\cdot\nonumber\\
&\cdot\Gamma_{\lambda\chi}^{A^2\phi, hij}(k+p-q,q,-k-p) D_H^{fj}(p+k)D_{\sigma\omega}^{eg}(k)D_{\delta\chi}^{ci}(q)\nonumber\\
&+\frac{1}{2}\int\frac{d^dq}{(2\pi)^d}\Gamma^{\mathrm{tl}, A^2\phi^2, abdf}_{\mu\nu}(p,-p,q,-q)D_H^{df}(q)\nonumber\\
&-\frac{1}{2}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\sigma}^{\mathrm{tl}, A^2\phi^2, aedf}(p,q-p-k,-q,k)\nonumber\\
&\cdot D_{\sigma\rho}^{eg}(p-q+k)D_H^{dh}(q)D_H^{fi}(k)\Gamma_{\rho\nu}^{A^2\phi^2, gbhi}(p-q+k,-p,q,-k)\nonumber\\
&+\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\sigma}^{\mathrm{tl}, A^2\phi^2, aedf}(p,q-p-k,-q,k)D_{\sigma\rho}^{eg}(p+k-q)D_H^{fi}(k)D_H^{hk}(k+p)\cdot\nonumber\\
&\cdot D_H^{dj}(q)\Gamma^{A\phi^2, gjk}_\rho(p+k-q,q,-k-p)\Gamma^{A\phi^2, bhi}_\nu(-p,k+p,-k)\nonumber\\
&+\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\sigma}^{\mathrm{tl}, A^2\phi^2, aedf}(p,q-p-k,-q,k)D_{\sigma\rho}^{eg}(p+k-q)D_H^{fi}(k)D_{\omega\lambda}^{hk}(k+p)\cdot\nonumber\\
&\cdot D_H^{dj}(q)\Gamma^{A^2\phi, gkj}_{\rho\lambda}(p+k-q,-k-p,q)\Gamma^{A^2\phi, bhi}_{\nu\omega}(-p,k+p,-k)\nonumber\\
&+\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\rho}^{\mathrm{tl}, A^2\phi^2, afed}(p,k,q-p-k,-q)D_H^{eg}(p+k-q)D_{\sigma\rho}^{fi}(k)D_H^{hk}(k+p)\cdot\nonumber\\
&\cdot D_H^{dj}(q)\Gamma^{\phi^3, gjk}(p+k-q,q,-k-p)\Gamma^{A^2\phi, bih}_{\nu\rho}(-p,-k,k+p)\nonumber\\
&+\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\rho}^{\mathrm{tl}, A^2\phi^2, afed}(p,k,q-p-k,-q)D_H^{eg}(p+k-q)D_{\sigma\rho}^{fi}(k)D_{\lambda\omega}^{hk}(k+p)\cdot\nonumber\\
&\cdot D_H^{dj}(q)\Gamma_\omega^{A\phi^2, kgj}(-k-p,p+k-q,q)\Gamma^{A^3, bih}_{\nu\rho\lambda}(-p,-k,k+p),\label{tlgleq}
\end{eqnarray}
\noindent and the Higgs
\begin{eqnarray}
&D_H^{ab}(p)^{-1}=\delta^{ab}(p^2+m_h^2)+\nonumber
\end{eqnarray}
\begin{eqnarray}
&+\int\frac{d^dq}{(2\pi)^d}\Gamma^{\mathrm{tl}, A\phi^2, eac}_\nu(-p-q,p,q)D_{\nu\mu}^{cg}(p+q)D_H^{fc}(q)\Gamma_\mu^{A^2\phi, gbf}(p+q,-p,-q)\nonumber\\
&+\frac{1}{2}\int\frac{d^dq}{(2\pi)^d}\Gamma_{\mu\nu}^{\mathrm{tl}, A^2\phi^2, cdab}(q,-q,p,-p)D_{\mu\nu}^{cd}(q)\nonumber\\
&-\frac{1}{2}\int\frac{d^dqd^dk}{(2\pi)^d}\Gamma_{\mu\sigma}^{\mathrm{tl}, A^2\phi^2, cdae}(-p-q+k,-k,p,q)D_{\mu\nu}^{cg}(p+q-k)D_{\rho\sigma}^{id}(k)\cdot\nonumber\\
&\cdot D_H^{eh}(q)\Gamma_{\rho\nu}^{A^2\phi^2, igbh}(-k,p+q-k,-p,q)\nonumber\\
&+\frac{1}{2}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\sigma}^{\mathrm{tl}, A^2\phi^2, cdae}(-q,k,p,-p+q-k)D_H^{ej}(p-q+k)D_{\mu\nu}^{cg}(q)D_{\rho\sigma}^{id}(k)\cdot\nonumber\\
&\cdot\Gamma_\nu^{A\phi^2, gjk}(q,p-q+k,-p-k)D_H^{kh}(p+k)\Gamma_\rho^{A\phi^2, ibh}(-k,-p,p+k)\nonumber\\
&+\frac{1}{2}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\sigma}^{\mathrm{tl}, A^2\phi^2, cdae}(-q,k,p,-p+q-k)D_H^{ej}(p-q+k)D_{\mu\nu}^{cg}(q)D_{\rho\sigma}^{id}(k)\cdot\nonumber\\
&\cdot\Gamma_{\nu\lambda}^{A^2\phi, gkj}(q,-p-k,p-q+k)D_{\lambda\omega}^{kh}(p+k)\Gamma_{\rho\omega}^{A^2\phi, ihb}(-k,p+k,-p)\nonumber\\
&+\frac{1}{2}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\chi}^{\mathrm{tl}, A^2\phi^2, cdae}(-q,q+k,p,-p-k)D^{cg}_{\mu\nu}(q)D_H^{eh}(p+k)D_{\lambda\chi}^{kd}(q+k)\cdot\nonumber\\
&\cdot\Gamma_{\sigma\nu\lambda}^{A^3, jgk}(k,q,-q-k)D_{\rho\sigma}^{ij}(k)\Gamma_\rho^{A\phi^2, ibh}(-k,-p,p+k)\nonumber\\
&+\frac{1}{2}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma_{\mu\chi}^{\mathrm{tl}, A^2\phi^2, cdae}(-q,q+k,p,-p-k)D^{cg}_{\mu\nu}(q)D_H^{eh}(p+k)D_{\lambda\chi}^{kd}(q+k)\cdot\nonumber\\
&\cdot\Gamma_{\nu\lambda}^{A^2\phi, gkj}(q,-q-k,k)D_H^{ij}(k)\Gamma^{\phi^3, ibh}(-k,-p,p+k)\nonumber\\
&+\frac{1}{2}\int\frac{d^dq}{(2\pi)^d}\Gamma^{\mathrm{tl}, \phi^4, abcd}(p,-p,q,-q)D_H^{cd}(q)-\nonumber\\
&-\frac{1}{6}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma^{\mathrm{tl}, \phi^4, agch}(p,-p+q-k, k, -q)D_H^{gd}(p-q+k)D_H^{he}(q)\cdot\nonumber\\
&\cdot D_H^{fc}(k)\Gamma^{\phi^4, debf}(p-q+k,q,-p,-k)\nonumber\\
&+\frac{1}{3}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma^{\mathrm{tl}, \phi^4, aicj}(p,-p-k+q,k,-q)D_H^{id}(p+k-q)D_H^{jg}(q)D_H^{fc}(k)\cdot\nonumber\\
&\cdot\Gamma^{\phi^3, gdh}(q,p+k-q,-k-p)D_H^{eh}(p+k)\Gamma^{\phi^3, ebf}(p+k,-p,-k)\nonumber\\
&+\frac{1}{3}\int\frac{d^dqd^dk}{(2\pi)^{2d}}\Gamma^{\mathrm{tl}, \phi^4, aicj}(p,-p-k+q,k,-q)D_H^{id}(p+k-q)D_H^{jg}(q)D_H^{fc}(k)\cdot\nonumber\\
&\cdot\Gamma_\mu^{A\phi^2, hgd}(-k-p,q,p+k-q)D_{\mu\nu}^{eh}(p+k)\Gamma^{A\phi^2, ebf}_\nu(p+k,-p,-k).\label{tlhiggseq}
\end{eqnarray}
\noindent Several of the appearing vertices do not have a tree-level counterpart. The tree-level vertices for the ghost-gluon-, 3-gluon-, 4-gluon-, 2-gluon-Higgs-, 2-gluon-2-Higgs- and 4-Higgs- interactions have been used. They are given by
\begin{eqnarray}
\Gamma_\mu^{\mathrm{tl}, c\bar cA, abc}(p,q,k)&=&ig_df^{abc}q_\mu\label{tlcca}\\
\Gamma_{\mu\nu\rho}^{\mathrm{tl}, A^3, abc}(p,q,k)&=&-ig_df^{abc}((q-k)_\mu\delta_{\nu\rho}+(k-p)_\nu\delta_{\mu\rho}+(p-q)_\rho\delta_{\mu\nu})\label{tlggg}\\
\Gamma_{\mu\nu\sigma\rho}^{\mathrm{tl}, A^4, abcd}(p,q,k,l)&=&g_d^2(f^{eab}f^{ecd}(\delta_{\mu\sigma}\delta_{\nu\rho}-\delta_{\mu\rho}\delta_{\nu\sigma})+f^{gac}f^{gbd}(\delta_{\mu\nu}\delta_{\sigma\rho}-\delta_{\mu\rho}\delta_{\nu\sigma})\nonumber\\
&&+f^{gad}f^{gbc}(\delta_{\mu\nu}\delta_{\sigma\rho}-\delta_{\mu\sigma}\delta_{\nu\rho}))\label{tl4g}\\
\Gamma_\mu^{\mathrm{tl}, A\phi^2, abc}(p,q,k)&=&ig_df^{abc}(q-k)_\mu\label{tlgh}\\
\Gamma_{\mu\nu}^{\mathrm{tl}, A^2\phi^2, abcd}(p,q,k,l)&=&g_d^2\delta_{\mu\nu}(f^{eac}f^{ebd}+f^{ead}f^{ebc})\label{tl2g2h}\\
\Gamma^{\mathrm{tl}, \phi^4, abcd}(p,q,k,l)&=&2h(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{bd}+\delta_{ad}\delta_{bc})\label{tl4h},
\end{eqnarray}
where the momentum-conserving $\delta$-functions have been suppressed. The complete graphical representation of \prefr{tlgheq}{tlhiggseq} is shown in figure \ref{figfullsys}.
\chapter{Finite-Temperature Effects}\label{cft}
Lowering the temperature down from infinity also the hard modes must be included. The corresponding DSEs are derived in section \ref{sftdse}. The precise implementation of the limiting process needs some deeper analysis of the running coupling, which will be conducted in section \ref{scc}. The infrared properties of the dressing functions will be investigated in section \ref{sftir}. The truncation will be discussed in section \ref{sfttrunc}. In general, an infinite number of Matsubara frequencies contributes at finite temperatures. At sufficiently high temperatures, most of them can be neglected, however. This corresponds to a small-momentum approximation, to be defined in section \ref{ssmallpapprox}. Also the renormalization process will be discussed there. Results on the ghost-loop-only truncation and the full theory will finally be shown in sections \ref{sftgonly} and \ref{sftfull}, respectively. The chapter will be closed in section \ref{sdselowt} by presenting results for low temperatures, which have been obtained in the context of this work. There, also a comparison to the high-temperature case will be drawn.
For convenience, the 3d-longitudinal and 3d-transverse gluons will be referred to as longitudinal and transverse gluons. In this chapter is again $p^2=p_0^2+\vec p^2\equiv p_0^2+p_3^2$. External momenta are referred to as $p$ while internal loop momenta and Matsubara summation momenta are referred to by $q$.
\section{Finite-Temperature Dyson-Schwinger Equations}\label{sftdse}
As described in section \ref{sdseft}, the equations are obtained from the vacuum equations \pref{vacgheq} and \pref{vacgleq} by application of the Matsubara formalism \cite{Kapusta:tk}. To obtain scalar equations for the (infinite) set of dressing functions $Z$ and $H$ of the gluon propagator \pref{gluonprop}, the gluon equation is contracted with the generalized projectors \pref{gtproj} and \pref{glproj}. Due to the kinematical singularity present, the choices \pref{d3tgenp} and \pref{d3lgenp} are extremely cumbersome. Therefore
\begin{eqnarray}
A_{T\mu\nu}&=&\delta _{\mu \nu }-\delta _{\mu 0}\delta _{0\nu}\label{ftgenp1}\\
A_{L\mu\nu}&=&\delta_{\mu 0}\frac{p_0 p_\nu}{p^2}+\delta_{0\nu}\frac{p_\mu p_0}{p^2}\label{ftgenp2}
\end{eqnarray}
\noindent are used instead. This is a specialization of the most general projector
\begin{equation}
P_{T/L\mu\nu}^\eta=\eta P_{T/L\mu\nu}+\left(1-\eta\right)\left(t_x\delta_{\mu\nu}+f\delta_{\mu 0}\delta_{0\nu}+f_t\delta_{\mu 0}\frac{p_0 p_\nu}{p^2}+c_t\delta_{0\nu}\frac{p_\mu p_0}{p^2}+t_z\frac{p_\mu p_\nu}{p^2}\right),\label{proj}
\end{equation}
\noindent where $\eta$ can be either $\zeta$ or $\xi$. The choices of the constants $t_x$, $f$, $f_t$, $c_t$, and $t_z$ in \pref{ftgenp1} and \pref{ftgenp2} were made to obtain a well-defined 3d-limit. They are not unique, indicating a larger range of possibilities than just one parameter $\eta$ in general. Nevertheless, as the choices fulfill all requirements they are sufficient.
Employing the projectors \pref{gtproj} and \pref{glproj} yields
\begin{eqnarray}
\frac{1}{G(p)}&=&\widetilde{Z}_3+\frac{g_4^2TC_A}{(2\pi)^2}\sum_{n=-\infty}^{\infty}\int d\theta dq \Big(A_T(p,q)G(q)Z(p-q)\nonumber\\
&&+A_L(p,q)G(q)H(p-q)\Big)\label{fulleqGft}\\
\frac{\xi}{H(p)}&=&\xi Z_{3L}+T^{HG}+T^{HH}+\frac{g_4^2C_A}{(2\pi)^2}\sum_{n=-\infty}^{\infty}\int d\theta dq\Big(P(p,q)G(q)G(p+q)\nonumber\\
&&+N_L(p,q)Z(q)Z(p+q)+N_1(p,q)H(q)Z(p+q)+N_2(p,q)H(p+q)Z(q)\nonumber\\
&&+N_T(p,q)H(q)H(p+q)\Big)\label{fulleqHft}\\
\frac{1}{Z(p)}&=&Z_{3T}+T^{GH}+T^{GG}+\frac{g_4^2C_A}{(2\pi)^2}\sum_{n=-\infty}^{\infty}\int d\theta dq\Big(R(p,q)G(q)G(p+q)\nonumber\\
&&+M_L(p,q)H(q)H(p+q)+M_1(p,q)H(q)Z(p+q)+M_2(p,q)H(p+q)Z(q)\nonumber\\
&&+M_T(p,q)Z(q)Z(p+q)\Big)+\frac{p_0^2(\zeta-1)}{2p^2}\left(Z_{3L}-\frac{1}{H(p)}\right).\label{fulleqZft}
\end{eqnarray}
\noindent The $\zeta$ and $\xi$ dependence is acquired by using \pref{ftgenp1} and \pref{ftgenp2}. At $\zeta=\xi=1$ the original form of the equations is recovered. For the correct solutions, the appearance of these terms would be compensated by corresponding structures in the loop contributions, and the result would be independent of $\zeta$ and $\xi$. For $p_0=0$, equation \pref{fulleqHft} is only superficially dependent on $\xi$. As all integral kernels are proportional to $\xi$, the dependence can be divided out for $\xi\neq 0$. Then, only the implicit dependence due to the interactions with the hard modes remains. The latter effect vanishes in the 3d-limit and the Higgs-equation \pref{higgsd3} is again independent of the projection and thus unique. The finite-temperature 4d-theory is no longer finite, but renormalizable. Thus explicit wave-function renormalization constants $\widetilde{Z}_3$, $Z_{3L}$, and $Z_{3T}$ have been introduced and will be discussed in section \ref{scttrunc} and \ref{srenormtrunc}.
The summation is over all Matsubara frequencies $q_0=2\pi T n$. $\widetilde{Z}_1=1$ has been employed \cite{Taylor:ff}. $Z_1$ cannot be calculated in the present approach, but it is finite and set to 1 \cite{Fischer:2002hn,Fischer:2003zc}. The kernels $A_T$, $A_L$, $R$, $M_T$, $M_1$, $M_2$, $M_L$, $P$, $N_T$, $N_1$, $N_2$, and $N_L$ are listed in appendix \ref{sftkernels}. The tadpoles $T^{ij}$ are again used to cancel spurious divergences in much the same way as in the previous section. However, also in the soft equations, they can possess finite parts at finite temperature. In case of the longitudinal equation, these can be absorbed in the mass renormalization discussed below. In the transverse equation, these are at $\zeta=3$ completely contained in the loop terms, and their continuation away from $\zeta=3$ is thus arbitrary. Also these contributions scale at best only as $1/p^2$ and are thus irrelevant to the infrared. Therefore, they are dropped, especially as no simple prescription as \pref{finite} below can remove the related spurious divergences
Note the considerably higher symmetry between the equations for $H$ and $Z$ than in equations \prefr{fulleqG3d}{fulleqZ3d}. Close inspection of the equations reveals further that the dressing functions can only depend on $\left|p_0\right|$ and $\left|\vec p\right|$. The corresponding symmetry, under $p_0\to -p_0$, is used to reduce the number of equations significantly.
\section{Temperature Dependence of the Coupling}\label{scc}
The equations \prefr{fulleqGft}{fulleqZft} depend on the coupling constant $g_4$ as the only parameter. In turn $g_4$ depends on the renormalization scale $\mu$, which can be chosen arbitrarily at any fixed temperature $T$. It directly enters into the definition of the infinite-temperature limit, as $g_3$ depends on $g_4$. In the simplest case, $g_3^2\sim g_4^2(\mu)T$. Comparing equations \prefr{fulleqG3d}{fulleqZ3d} and \prefr{fulleqGft}{fulleqZft}, the constant of proportionality has to be 1 in this truncation scheme to obtain a smooth infinite-temperature limit. Hence it only remains to choose the temperature dependence of $\mu$. There are two options to consider.
If $\mu$ is fixed, the 3d-coupling grows without limit as $T$ grows. This does not necessarily pose a problem, as the infinite-temperature propagators are independent of $g_3$ as long as expressed as a function of $p/g_3^2$. Under such circumstances, all momenta below $T$ are effectively infrared, and the non-perturbative regime would extend to all momenta.
Alternatively, it is possible to use the limit prescription $g_4^2(\mu(T))T=c_\infty$, where $c_\infty$ is an arbitrary constant, effectively performing a renormalization group transformation when changing the temperature. This fixes $g_3^2$ to be proportional to $\Lambda_{QCD}$, the dynamical scale of QCD \cite{Bohm:yx}, while the 4d-coupling vanishes like $1/T$. This defines a t'Hooft-like scaling in $T$. Note that $g_4\to 0$ for $T\to\infty$ corresponds to the conventional arguments of a vanishing coupling in the high-temperature phase \cite{Blaizot:2001nr}. However here, as in the large-$N$ limit, t'Hooft-scaling is performed, generating a well-defined theory. Hence this possibility defines a smooth 3d-limit with a finite 3d-coupling constant. Here $c_\infty=1$ is chosen. Nonetheless, the case of fixed $\mu$ is also of interest, as it permits to compare the results for different temperatures directly.
These possibilities all correspond to the 4d-situation, where it is possible to shift the onset of non-perturbative physics by renormalization group transformations. Under all circumstances, for any finite temperature, there is always a non-perturbative regime at sufficiently small momenta.
Furthermore, it would be useful to give explicit units for the temperature scale. In the vacuum the scale is fixed via a comparison of the running coupling to perturbation theory \cite{Fischer:2002hn}. This is not possible here because the Matsubara sum is truncated and the coupling cannot be calculated for momenta of the order of $2\pi NT$, where $N$ is the number of Matsubara frequencies included\footnote{Always, only the frequencies with $p_0\ge 0$ are counted.}
The simplest procedure is to compare $g_3$ to lattice calculations. There the effective 3d-coupling is found to be $g_3^2(2T_c)/2T_c=2.83$ \cite{Cucchieri:2001tw}. The phase transition temperature is $T_c=269\pm 1$ MeV \cite{Karsch:2003jg}. Albeit this will not be exactly $g_4^2T_c$ due to the truncation, a first approximation is to require the same temperature scale at $2T_c$. Using the fixed $g_4^2T=c_\infty$ prescription, then $1=c_\infty/(2T_c2.83)$. Since here $c_\infty=1$ in internal units, internal units have to be multiplied by 1.5 GeV to yield physical units. This temperature scale will be used in sections \ref{sftgonly} and \ref{sftfull}. The ratios of temperatures are independent of this prescription.
\section{Infrared Properties}\label{sftir}
Regarding the infrared properties, asymptotic freedom turns out to be advantageous also for the infrared. Just at the phase transition, the $n=1$ Matsubara frequency has already an effective `mass' $p_0=2\pi T_c\approx 1.7$ GeV. By the Appelquist-Carazzone theorem \cite{Appelquist:tg} the hard modes are suppressed by powers of $|\vec p|/p_0$ in the infrared. Thus hard modes do not contribute significantly. Studying each equation in detail, it turns out that this quite general statement is implemented very differently in each equation.
In case of the hard modes, there is no pure soft contribution due to momentum conservation at the vertices. Thus they decouple and a self-consistent solution is that all hard mode dressing functions are constant in the infrared.
In the case of the soft ghost equation \pref{fulleqGft}, all contributions become also constant in the infrared. Thus they can be canceled by the (renormalized) tree-level value. Only the subleading behavior remains, as in the case of the 3d- and 4d-theory. As the subleading contribution of the purely soft term dominates the hard terms, the same behavior as in the infinite-temperature limit emerges.
In case of the transverse equation \pref{fulleqZft}, the hard mode contributions give rise at best to mass-like $1/p^2$ terms. These terms are subleading compared to the soft ghost-loop. Thus the infrared is dominated by the latter and the same infrared solution as in the infinite-temperature limit is found.
The longitudinal equation \pref{fulleqHft} is finally quite different from its equivalent in the 3d- or 4d-case. In the prior case, it was dominated by its tree-level mass, while in the latter it is identical to the transverse one. In the present case, the absence of pure soft interactions with ghosts, which could generate divergent contributions as in the transverse equation, leads to dominance of the hard modes\footnote{The pure soft interactions represented by the kernels $N_1$ and $N_2$ only generate a constant term in the infrared. Without a tree-level mass, the soft tadpoles generate a contribution which can be removed by renormalization.}. These provide mass-like contributions in the infrared. Inspecting e.g. $P$ at $\xi=1$
\begin{equation}
P(0,q_0,\vec q,\vec p)=\frac{\left|\vec q\right|^2q_0^2\sin\theta}{\left|\vec p\right|^2q^2(q_0^2+(\vec p+\vec q)^2)},\label{dynmass}
\end{equation}
\noindent and using the fact that the hard dressing functions are constant in the infrared, directly leads to a mass-like behavior due to the explicit $1/|\vec p|^2$ factor and the finiteness of the remaining expression. Thus the 3d-mass of the Higgs is generated spontaneously by the interaction with the hard modes.
Hence, the infrared behavior of all soft mode dressing functions does not depend on the temperature, expect for changes in the infrared coefficients.
Note that these observations imply that by using a finite number of Matsubara frequencies it will not be possible to obtain the 4d-vacuum solutions. Any finite number of such modes will not be able to generate a divergence stronger than mass-like in the longitudinal equation to compensate the missing soft interactions with ghosts. Hence, the 4d-behavior can be established only by infinite summation. In this case the generated mass has to diverge even after renormalization at $p=0$ to obtain the vacuum solution, providing over-screening instead of screening. The possibility that the ghost-gluon-vertex changes due to temperature to couple soft ghost modes to soft longitudinal gluons at low temperature and not at high temperature is unlikely, since a bare ghost-gluon vertex is already sufficient at low temperatures and in the vacuum \cite{vonSmekal:1997is,Gruter:2004kd}.
\section{Truncation and Spurious Divergences}\label{sfttrunc}
Also for large but still finite $T$ the 3d-limit is to be made explicit. Thus all the problems encountered in the 3d-theory persist, including the necessity of a modified soft 3-gluon vertex. Hence all modifications of the pure soft terms will be left as in the case of chapter \ref{c3d}, except for the tadpoles in the Higgs equation \pref{fulleqH3d}. Due to the spontaneously generated mass, it will be necessary to alter this behavior. There are also additional spurious divergences due to the hard mode contributions. As they occur at mid-momenta, when viewed from the 4d-perspective using $p^2=p_0^2+\vec p^2$, it is expected that due to the truncation ansatz, they are much harder to compensate. This is indeed the case.
The spurious divergences in contributions of hard modes due to the integration are dealt with in exactly the same way as in 3d by adjusting the tadpole terms. Each integration kernel $K$ is split as
\begin{equation}
K=K_0+K_D\label{isplit}
\end{equation}
\noindent $K_0$ is finite and $K_D$ divergent upon integration. Each kernel $K_D$ will be compensated by a corresponding tadpole as for $M_{TD}$ and $M_{LD}$ in the 3d-theory in equations \pref{ggtad2} and \pref{ghtad}, respectively.
However, a new kind of spurious divergences appear when performing the Matsubara sum. In principle this is no problem when using a finite number of Matsubara frequencies, since then all expressions are finite. However, when including more and more Matsubara frequencies, it is found that their contribution in the gluon equations \pref{fulleqHft} and \pref{fulleqZft} at $|\vec p|<\max(q_0)$ scales as $q_0$, thus behaving as a quadratic divergence. This is an artifact of using a finite number of Matsubara frequencies. It does not vanish as long as the number of frequencies is finite, no matter how large the number. This behavior is spurious and must be removed.
The ghost equation does not contain spurious divergences\footnote{The problem is also not present in the mixed contributions $M_1$, $M_2$, $N_1$ and $N_2$ in the gluon equations. Hence these do not need to be subtracted.} and can thus serve as a starting point to trace the origin of those in the gluon equations. Analyzing the $\left|\vec p\right|=0$ limit for $p_0\neq 0$, it is directly found that if the equation looks like
\begin{equation}
\frac{1}{G(p_0\neq 0,0)}=\frac{1}{A_g(p_0)}=\widetilde{Z}_3+g_4^2 T I,\label{ghosteq}
\end{equation}
\noindent then the integral $I$ must scale like $1/p_0$ because of dimensional consistency. Here $\widetilde{Z}_3$ is the wave-function renormalization of the ghost. For any finite number of Matsubara frequencies, $\widetilde{Z}_3$ is finite. Hence the infrared behavior of the hard ghost dressing function is $1/(\widetilde{Z}_3+aT/p_0)$ with a suitable constant $a$. For sufficiently large $p_0$ this can be expanded to yield $1/\widetilde{Z}_3-aT/(p_0\widetilde{Z}_3^2)$ in leading order.
The effect is most directly seen in the $p_0=0$ longitudinal equation in the ghost-loop. For dimensional reasons, in the infrared the terms in the Matsubara sum must scale as $Tq_0/|\vec p|^2 A_g^2(q_0)$. Hence the divergence is directly visible due to the leading constant term in $A_g(q_0)$. The situation is analogous, but not as directly visible in the remaining equations.
This spurious divergence can be removed when removing the leading $1/\widetilde{Z}_3$ term in the ghost dressing function, i.e.\/ by the replacement
\begin{eqnarray}
&G(q,q_0)G(p+q,q_0+p_0)\nonumber\\
&\to\left(G(q,q_0)-\frac{1}{\widetilde{Z}_3}\right)\left(G(q+p,q_0+p_0)-\frac{1}{\widetilde{Z}_3}\right),\label{finite}
\end{eqnarray}
\noindent and correspondingly for the gluon loops.
Including a finite number $N$ of Matsubara frequencies only is equivalent to studying a 3d-theory with 1 massless particle and $2N$ massive particles for each 4d particle species. Therefore the ultraviolet asymptotic analysis of subsection \ref{ssuvanalysis} applies, and $I$ in \pref{ghosteq} will again fall off polynomially in the ultraviolet. Hence \pref{finite} amounts to subtracting the asymptotic constant to which $G$ evolves and thus corrects for $N$ being finite. $1/\widetilde{Z}_3$ must become 0 when including an infinite number of Matsubara frequencies, as this is the correct perturbative 4d behavior for $p\gg T$. Hence the spurious divergences are in this case only an artifact of the finite number of Matsubara frequencies.
In the the longitudinal equation the Brown-Pennington value $\xi=0$ is pathologic. It removes all terms in the soft equation and so is not applicable on its own. Thus the replacement \pref{finite} has to be applied to the complete loop contributions $P$, $N_T$ and $N_L$. In the transverse equations all spurious divergences including the one of the Matsubara sum are again removed at $\zeta=3$. Therefore the replacement \pref{finite} will only be necessary in those contributions in $R$, $M_T$, and $M_L$ which are proportional to $(\zeta-3)$. Hence the integral kernels are split differently than \pref{isplit} as
\begin{equation}
K(\zeta)=K_0+(\zeta-3)K_3+K_D(\zeta),\nonumber
\end{equation}
\noindent similar to \pref{rsplit}. Here $K_0$ and $K_3$ are finite and independent of $\zeta$, and $K_D$ contains all divergences upon integration. The corresponding subtraction is then performed by the replacement
\begin{equation}
KD(q)D(p+q)\to K_0D(q)D(p+q)+(\zeta-3)K_3\left(D(q)-\frac{1}{Z_3}\right)\left(D(p+q)-\frac{1}{Z_3}\right)\nonumber\\
\end{equation}
\noindent where $D$ is a generic dressing function and $Z_3$ its wave-function renormalization.
In the soft transverse equation it is additionally necessary to remove tadpole-like structures at $\zeta\neq 3$ in $K_3$. They contribute a logarithmically divergent mass-term, which is irrelevant in the infrared and in the ultraviolet. It cannot be renormalized, as a counter-term would be required, which is forbidden by gauge symmetry. Thus these contributions are absorbed by the tadpoles as well. Then, in the transverse equation for the soft-soft interactions, the subtractions are as in chapter \ref{c3d}. The soft-hard contributions, $R$, $M_T$, and $M_D$ are subtracted as
\begin{eqnarray}
&KD(q)D(p+q)\to K_0D(q)D(p+q)\nonumber\\
&+(\zeta-3)\left(K_3-\frac{1}{p^2}\left(\lim_{p\to 0}p^2K_3\right)\right)\left(D(q)-\frac{1}{Z_3}\right)\left(D(p+q)-\frac{1}{Z_3}\right)\label{finite2}.
\end{eqnarray}
\section{Small-Momentum Approximation and Renormalization}\label{ssmallpapprox}
Performing the subtractions \pref{finite} and \pref{finite2}, the resulting equations are still logarithmically divergent for $N\to\infty$. This also includes the ghost equations. The integrals themselves are convergent quantities, but the Matsubara sum is not, as the terms scale like $1/q_0$. Thus, for an infinite number of Matsubara frequencies, the sum diverges. This is the 4d logarithmic divergence and thus the usual one of Yang-Mills theory \cite{Bohm:yx}.
In the case of a finite number of Matsubara frequencies, the sum is finite. Nonetheless, there are a few aspects to be dealt with in the following subsections.
\subsection{Small-Momentum Approximation}
There are three regions of momentum to be distinguished:
At $p\le 2\pi T$, all Matsubara sums behave essentially the same due to the Appelquist-Carrazone decoupling theorem. It is valid as the hard modes behave essentially tree-level-like. As the contributions of the hard modes behave as $1/q_0$, the final result will depend on the number of Matsubara frequencies included, and by adding an arbitrary number, any quantitative (but not qualitative) result can be generated. Thus, it is necessary to renormalize in order to be independent of the cutoff. The cutoff is in this case imposed by the number of Matsubara frequencies included. The renormalization procedure will be implemented in the next subsections. By this approach, the results can be made quite reliable in this regime.
At $2\pi T\le p\le 2\pi T (N-1)$, where $N$ is the number of Matsubara frequencies included, more and more Matsubara terms depart from their $1/q_0$ behavior to a $1/p$ behavior. As the external momentum $p$ becomes large compared to the effective mass $q_0$ of a hard mode, the mode becomes dynamical and behaves like a massive 3d-particle. The results are still quite reliable in this region as also in the case $N\to\infty$, for any finite momentum $p$ only a finite number of Matsubara frequencies are dynamical. It becomes less and less reliable when approaching the upper limit $2\pi TN$.
At $p\ge 2\pi T (N-1)$, the situation changes drastically. Opposite to the case $N=\infty$, all Matsubara modes are dynamical and their contributions will scale as $1/p$. Thus the number of Matsubara modes will now enter linearly instead of logarithmically. This is an artifact of cutting off the Matsubara sum. In this region the results are not reliable. However, as the sum is still finite and suppressed by $1/p$, the contribution is subleading with respect to the tree-level term and thus the system of equations can still be closed consistently\footnote{Otherwise the corresponding finite 3d-theory would be ill-defined, which is not the case.}.
By renormalization, these artifacts can be reduced, if not completely removed at sufficiently small momenta. In that sense the finite Matsubara sum approximation is indeed a small-(3-)momentum approximation.
\subsection{Renormalization of a Truncated Matsubara Sum}\label{scttrunc}
There are a few subtleties involved concerning the renormalization of a truncated Matsubara sum. These are discussed here.
Firstly, in the vacuum case \cite{Alkofer:2000wg,Fischer:2002hn}, renormalization was performed using a momentum subtraction scheme (MOM), i.e. by subtracting from the DSEs themselves at a fixed subtraction point $s$. This leads to a generic form of
\begin{equation}
\frac{1}{D(p)}-\frac{1}{D(s)}=I(p)-I(s),\label{subrenormalization}
\end{equation}
\noindent where $D$ is the corresponding dressing function and $I$ is the self-energy contribution. As the highest divergences encountered are logarithmic \cite{Bohm:yx}, \pref{subrenormalization} is finite and can be solved if a proper value of $D(s)$ is known. This can be achieved by choosing $s$ inside the perturbative region. This approach fails if the large momentum asymptotic value of $D$ is a constant different from 0, since \pref{subrenormalization} is ambiguous with respect to such a constant. In the case of a truncated Matsubara sum, the ultraviolet behavior is that of a massive 3d-theory. Thus the self-energy contributions vanish as $1/p$ in the ultraviolet, see subsection \ref{ssuvanalysis}. Therefore the dressing functions $D$ are dominated by the tree-level term
\begin{equation}
\lim_{\left|\vec p\right|\to\infty}D(p_0,\left|\vec p\right|)\to\frac{1}{Z_3},
\end{equation}
\noindent with $Z_3$ their wave-function renormalization. For a finite number of Matsubara frequencies $Z_3$ is finite and thus $D$ goes to a constant in the ultraviolet and \pref{subrenormalization} is not applicable. Therefore the renormalization is performed by explicit counter-terms. This is discussed in the next subsection.
A second point is the mass and mass renormalization necessary for the soft longitudinal mode. The soft mode with frequency $p_0=0$ of the $A_0$ component of the gauge field transforms homogeneously instead of inhomogeneously under gauge transformations \pref{gtrans1}. Therefore gauge symmetry permits to add a term
\begin{equation}
m^2A_0^2(0,\vec p)\label{a0mass}
\end{equation}
\noindent to the Lagrangian. In the vacuum, however, such a term is forbidden by manifest Lorentz invariance. At finite temperature in the Matsubara formalism, this is no longer the case, and such a term could in principle be present. This term replaces the $p_0^2A_0^2$ term of the hard modes, which stems from the $A_0\partial_0^2 A_0$ term in the Lagrangian \pref{lym} for $p_0\neq 0$.
Concerning the counter-terms, the wave-function renormalization is performed by adding the counter-term
\begin{equation}
\delta Z_3(A_\mu^a\pd_{\mu}\pd_{\nu} A_\mu^a-A_\mu^a\partial^2 A_\mu^a)\nonumber
\end{equation}
\noindent to the Lagrangian. In the case $p_0=0$, the first term is not present for the longitudinal mode $A_0$. Its place can be taken by a counter-term for the mass term \pref{a0mass}. This implies a relation between the wave-function counter-term $\delta Z_3$ and the mass counter-term $\delta m^2$, which cannot be exploited here due to the truncation of the Matsubara sum. Therefore an independent renormalization of the wave function and the mass of the soft longitudinal mode is necessary and will be performed in the next subsection.
A last point concerns the implications for the counter-terms due to the truncation of the Matsubara sum. As the divergence structure must be the same as in the vacuum, the counter-terms must be the same for all frequencies. This is no longer the case when the Matsubara sum is truncated, as can be seen directly by counting. In the sum for the soft mode, $p_0=0$, contributions from $2N-1$ Matsubara modes are present for each loop. For the hard mode with $p_0=2\pi T(N-1)$, only $N$ contributions are present, as $\left|p_0+q_0\right|\le 2\pi (N-1)T$. Thus different numbers of modes contribute and the counter-terms cannot be the same. This is always the case as long as $N<\infty$. To surpass the problem in a constructive manner, each mode will be renormalized independently.
Therefore, a counter-term Lagrangian is added, given by
\begin{eqnarray}
{\cal L}&=&\delta m^2 A_0^a(0)^2+\sum_{q_0} \Big(\delta Z_{3T}(q_0) A^a_\mu(q_0) \Delta_{T\mu\nu}(q_0) A^a_\nu(q_0)\nonumber\\
&+&\delta Z_{3L}(q_0) A^a_\mu(q_0) \Delta_{L\mu\nu}(q_0) A^a_\nu(q_0)+\delta\widetilde{Z}_3(q_0) \bar c^a(q_0)\partial^2 c_a(q_0)\Big)\label{ctl}
\end{eqnarray}
\noindent where $A^a_\mu(q_0)$ are the modes of the gluon field and $\bar c(q_0)$ and $c(q_0)$ are the modes of the ghost and anti-ghost field, respectively. $\Delta_{T/L\mu\nu}$ are the appropriate tensor structures of derivatives. Consequently, the corresponding wave-function renormalization constants in the subtractions \pref{finite} and \pref{finite2} have to be replaced by their frequency-dependent versions.
This shortcoming is a consequence of the small momentum approximation yielding the incorrect ultraviolet properties: The treated system is for large momenta equivalent to a 3d-theory of $2N-1$ particles per 4d particle species. In such a theory all fields can be renormalized independently. This problem has to be investigated further to obtain reliable results on the large momentum behavior of the dressing functions.
\subsection{Implementation of Renormalization}\label{srenormtrunc}
In all equations but the one for $H(0,\vec p)$, using the counter-terms of \pref{ctl} amounts to replacing the tree-level term $1$ by $1+\delta Z_3$. This way explicitly multiplicative renormalization is obtained. In the equation for $H(0,\vec p)$, the tree-level term is replaced\footnote{Note that when using the projector \pref{ftgenp2}, also $\delta m^2$ is multiplied by $\xi$.} by $1+\delta Z_3L+\delta m^2/p^2$. Thus, it also generates a mass-renormalization. By the arguments following equation \pref{dynmass} just a renormalization is indeed necessary, and the mass is not generated purely by this procedure.
The last ingredient is the renormalization prescription. To investigate both cases discussed in section \ref{scc}, either a fixed subtraction point $s=s_0$ or to establish a well-defined 3d-limit $s=T$ is employed. Thus, the following prescription is applied to all dressing functions $D$, except $H(0,\vec p)$:
\begin{equation}
D(s)=1.\nonumber
\end{equation}
\noindent This also ensures $G(s)^2Z(s)=1$ as is required in the 4d-theory \cite{vonSmekal:1997is}. The soft ghost mode is not renormalized at 0 as in the vacuum calculation \cite{Fischer:2002hn}.
In case of $H(0,\vec p)$, two prescriptions are necessary. The first is\footnote{For numerical reasons, it is actually performed at $2\delta_i$, where $\delta_i$ is the numerical IR-cutoff of the integration. This is only a marginal difference.}
\begin{equation}
\lim_{|\vec p|\to 0}|\vec p|^{-2}H(0,\vec p)=\frac{1}{m_{3d}^2}=\frac{1}{r^2g_4^4T^2+g_4^2TC_A\frac{rg_4^2T}{4\pi}}\nonumber
\end{equation}
\noindent where $m_{3d}$ is the tadpole-improved mass of the 3d-theory. It depends on $r=m_h/g_3^2$, which is again taken to be the same value as in the previous chapter. For fixed $g_3$, this mass is independent of the temperature while for fixed $\mu$ it scales with temperature as expected. This again indicates the necessity of control over the infinite-temperature limit of the theory in terms of the effective coupling constant. In addition,
\begin{equation}
\frac{1}{H(0,s)}=1+\frac{m_{3d}}{s^2}\nonumber
\end{equation}
\noindent is required to fix the mass renormalization.
It should be noted that, by renormalizing at $T$, it is guaranteed that the correct 3d-limit is obtained. This prescription requires that at $T\to\infty$ all dressing functions approach 1 at infinity. This yields the 3d-results\footnote{This is only correct, if $g_4^2T\ll T$. This is immanent, if $g_4^2 T$ is fixed. If $\mu$ is fixed, this is only possible if $\mu$ is taken sufficiently large for a fixed temperature. Still deviations from 1 will be found for $|\vec p|\gg\mu,T$.} as in chapter \ref{c3d}. It would be easily possible to readjust the requirements on $H(0,\vec p)$ to comply with the lattice data in the infrared and ultraviolet. Here the exact correspondence to the 3d results in chapter \ref{c3d} is preferred, as this will make the 3d-limit explicit.
Therefore the explicit implementation of the renormalization prescription for the DSE of a dressing function $D$ with self-energy contributions $I$
\begin{equation}
\frac{1}{D(p)}=1+I(p)\nonumber
\end{equation}
\noindent is then
\begin{eqnarray}
\frac{1}{D(p)}=1+\delta Z_3+I(p)\nonumber\\
\delta Z_3=-I(s)\nonumber
\end{eqnarray}
\noindent and in the case of $H(0,\vec p)$
\begin{eqnarray}
\frac{1}{H(0,\vec p)}=1+\delta Z_{3L}+\frac{\delta m^2}{p^2}+I(p)\nonumber\\
\delta m^2=m_r^2-\lim_{p\to 0}p^2I(p)\nonumber\\
\delta Z_{3L}=-I(s)+\frac{lim_{p\to 0}p^2I(p)}{s^2},\nonumber
\end{eqnarray}
\noindent where $m_r=m_{3d}$ is the renormalized mass.
\section{Ghost-Loop-Only Truncation}\label{sftgonly}
\begin{figure}
\epsfig{file=ftgonly.eps,height=1.2\linewidth,width=\linewidth}
\caption{The left panels show for the soft modes from top to bottom the dressing functions of the ghost and transverse gluon and the propagator of the longitudinal gluon. The right panels show the hard mode dressing functions. The less the deviation from 1, the harder the mode is. The solid lines are the $g=1/2$ solution and the dashed lines the $g\approx 0.4$ solution. Both are at $T=1.5$ GeV and $N=20$. Note the different momentum scale in the left and the right panels.}
\label{figftgonlystd}
\end{figure}
The first step is again the calculation of the ghost-loop-only solutions. Performing the calculations turns out to be numerically more complicated than in 3d due to the recursive definition of the elimination of the spurious divergences \pref{finite} and \pref{finite2}. This is discussed in more detail in appendix \ref{anum}. Nonetheless, both solutions can be found. They are shown in figure \ref{figftgonlystd} with the largest number of Matsubara frequencies currently available\footnote{The number of frequencies included is only a matter of CPU-time.} at $\zeta=\xi=1$. In general, all hard mode dressing functions are similar, but behave more and more tree-level like with increasing frequency $p_0=2\pi nT$. The different asymptotic values of the dressing functions for each mode are a direct manifestation of the artifacts of the truncation of the Matsubara sum, as discussed in section \ref{scttrunc}.
The small deviation from tree-level for the hard longitudinal modes is most likely due to the severe subtraction \pref{finite}. Within the current ansatz however, this is the best achievable. Note that due to $p^2=p_0^2+\vec p^2$, the infrared of the hard modes is actually mid-momenta with respect to 4d. Thus, these problems do not come unexpected, as the truncation is tailored to the infrared and thus to the soft modes. In addition, the transverse gluon wave-function renormalization is smaller than 1 for $n=0$. Thus the dressing function would diverge for large momenta in the limit $N\to\infty$. This is also an artifact of the truncation, as the ghost loops alone have the wrong sign at large momenta. This will be remedied by adding the gluon loops in the next section, and can be ignored for now.
The dependence on the number of Matsubara frequencies is shown in figure \ref{figftgonlymstd}. As expected, in the infrared region, the soft mode dressing functions are not affected by the number of Matsubara frequencies. The onset of deviation of the hard mode dressing functions calculated with $N$ frequencies from the solutions with $M>N$ frequencies increases in 3-momenta with $N$. This is a manifestation of the small momentum approximation.
\begin{figure}
\epsfig{file=ftgonlymcomp.eps,height=1.2\linewidth,width=\linewidth}
\caption{The dependence of the solutions on $N$ at $T=1.5$ GeV. The left panels show for the soft modes from top to bottom the dressing functions of the ghost and transverse gluon and the propagator of the longitudinal gluon. The right panels show the dressing functions for the $n=1$ hard mode. The red lines show the solution $g=1/2$ and the black lines $g\approx 0.4$. Solid is $N=2$, dashed is $N=5$, dotted is $N=12$ and dashed-dotted is $N=20$.}
\label{figftgonlymstd}
\end{figure}
Naturally, the dependence on the temperature is the next subject. Using the t'Hooft-like scaling, the results shown in figure \ref{figftgonlygstd} are obtained. While the soft mode dressing functions are relatively unaffected by temperature, significant changes are found for the hard mode dressing functions. These become increasingly tree-level with increasing temperature and thus the hard modes cease to interact. At low temperature they exhibit significantly more dynamics.
\begin{figure}
\epsfig{file=ftgonlygcomp.eps,height=1.2\linewidth,width=\linewidth}
\caption{The dependence of the solutions on $T$ at fixed $g_4^2T$ and subtraction at $T$ at $N=5$. The left panels show for the soft modes from top to bottom the dressing functions of the ghost and transverse gluon and the propagator of the longitudinal gluon. The right panels show the dressing functions for the $n=1$ hard mode. The red and black lines are the $g=1/2$ and the $g\approx 0.4$ solution, respectively. Solid is $T=0.8$ GeV, dashed is $T=1.5$ GeV, and dotted is $T=137$ GeV.}
\label{figftgonlygstd}
\end{figure}
The case for fixed $\mu$ is shown in figure \ref{figftgonlytstd}. The soft longitudinal gluon propagator is scaled by the interaction strength, since the mass of the soft longitudinal gluon is a dynamical effect. The results represent, how the system reacts on a change of $s/T$, which are the only dimensionful quantities involved as long as $g_4$ is fixed. The changes are governed by the renormalization prescription, and are qualitatively different from the previous case. As expected, the higher the temperature, the further out in momentum the non-perturbative effects in the soft mode dressing functions extend, only limited by the requirement to obey the renormalization prescription. At the same time the dynamics of the hard modes is pushed to higher momenta, although the dressing functions do not decrease. Indeed, this scenario describes rather a change of the relevant scale for the onset of non-perturbative effects than the effects of temperature.
At fixed $\mu$, it was possible\footnote{In the case of fixed $g_4^2T$, it is likely not yet found due to limitation in CPU time.} to follow the solution down to $T=91$ MeV. Lattice calculations find a phase transition temperature of $(269\pm 1)$ MeV \cite{Karsch:2003jg}. Thus super-cooling is possible, provided the temperature estimate of section \ref{scc} is correct within a factor of 2 and the phase transition temperature in the approximation scheme used here is not significantly lower than in lattice calculations.
\begin{figure}
\epsfig{file=ftgonlytcomp.eps,height=1.2\linewidth,width=\linewidth}
\caption{The dependence of the solutions on $T$ at fixed $\mu$ and $s$ at $N=5$. The left panels show for the soft modes from top to bottom the dressing functions of the ghost and transverse gluon and the propagator of the longitudinal gluon. The right panels show the dressing functions for the $n=1$ hard mode. The red and black lines are the $g=1/2$ and the $g\approx 0.4$ solution, respectively. Solid is $T=0.3$ GeV, dashed is $T=1.5$ GeV, and dotted is $T=5.0$ GeV.}
\label{figftgonlytstd}
\end{figure}
The remaining issue is the dependence of the solutions on the projection and thus on $\zeta$ and $\xi$. It is shown in figure \ref{figftgonlyzx}. The largest effect is seen in the infrared region of the soft solutions of the alternating branch when varying $\zeta$, similar to the infinite-temperature limit. The hard modes are significantly less affected, with the largest impact on the hard longitudinal dressing functions when varying $\xi$. For the smallest $\xi$ values, the self-energy of the hard longitudinal mode seems to switch sign, generating a wave-function renormalization smaller than 1. This effect has not been observed in the full system and is thus likely an artifact of the ghost-loop-only truncation. Thus, the dependence on the projection is of similar extent as in the infinite-temperature limit.
\begin{figure}
\epsfig{file=ftgonlyzxcomp.eps,height=1.2\linewidth,width=\linewidth}
\caption{The dependence of the solutions on $\zeta$ and $\xi$ at $N=5$. The left panels show for the soft modes from top to bottom the dressing functions of the ghost and transverse gluon and the propagator of the longitudinal gluon. The right panels show the dressing functions for the $n=1$ hard mode. The red and black lines are the $g=1/2$ and the $g\approx 0.4$ solution, respectively. Solid is $(g,\zeta,\xi)=(1/2,7/2,9/4)$ and $(0.4,2,5/2)$, dashed is $(1/2,1,1)$ and $(0.4,1,1)$, dotted is $(1/2,0,0.1)$ and $(0.4,1/4,0.1)$.}
\label{figftgonlyzx}
\end{figure}
\section{Full Theory}\label{sftfull}
\begin{figure}
\epsfig{file=ftf.eps,height=1.2\linewidth,width=\linewidth}
\caption{The left panels show for the soft modes from top to bottom the dressing functions of the ghost and transverse gluon and the propagator of the longitudinal gluon. The right panels show the hard mode dressing functions. The less the deviation from 1, the harder the mode is. The solid lines are the $g=1/2$ solution and the dashed lines the $g\approx 0.4$ solution. Both are at $T=1.5$ GeV and $N=12$. Note the different momentum scale in the left and the right panels.}
\label{figftf}
\end{figure}
Finally, the full system of equations \prefr{fulleqGft}{fulleqZft} is implemented. For the largest available number of Matsubara frequencies, the result is shown in figure \ref{figftf}. There are several observations. First of all, the soft modes are again nearly unaffected in the infrared by the presence of the hard modes. The latter show a significant modification, compared to tree-level, although still only of the order of 30\%. Also, all wave-function renormalization constants are now larger than one, as is required. The hard mode dressing functions exhibit some structure. There are maxima in all dressing functions. This is most pronounced in the case of the dressing functions of the longitudinal gluon. These structures do not translate into a corresponding structure in the propagators, which are monotonically decreasing from a constant of order $1/p_0^2$ in the infrared to 0 in the ultraviolet. It is also nearly irrelevant for the hard mode dressing functions to which soft infrared solution they are coupled to.
\begin{figure}
\epsfig{file=ftfmcomp.eps,height=1.2\linewidth,width=\linewidth}
\caption{The dependence of the solutions on $N$ at $T=1.5$ GeV. The left panels show for the soft modes from top to bottom the dressing functions of the ghost and transverse gluon and the propagator of the longitudinal gluon. The right panels show the dressing functions for the $n=1$ hard mode. The red lines show the solution $g=1/2$ and the black lines $g\approx 0.4$. Solid is $N=2$, dashed is $N=5$, dotted is $N=8$ and dashed-dotted is $N=12$.}
\label{figftfm}
\end{figure}
Figure \ref{figftfm} displays the dependence of the full solutions on the number of Matsubara frequencies. While the soft mode dressing functions are nearly unaffected, apart from the value of the renormalization constants, the effect on the hard mode dressing functions is significant. The ghost is quite insensitive, except for its wave-function renormalization. This is not the case for the gluons. The peaks at mid-momentum are sensitive to the number of Matsubara frequencies included. The effect is largest for the longitudinal gluon dressing functions. From the available number of Matsubara frequencies, it is hard to estimate whether the peak grows to a finite value when $N\to\infty$ or not. It cannot be excluded that the hard longitudinal gluon dressing functions violate the Gribov condition \pref{gribov} once sufficiently many Matsubara frequencies are included. This would be very similar to the case of the transverse gluon dressing function in the infinite-temperature limit and therefore would necessitate a similar vertex construction for the longitudinal-gluon-transverse-gluon vertex as for the soft transverse gluon vertex. If such a vertex would be necessary, the result would be a finite peak, very similar to the present situation. Thus, the results would be even quantitatively quite similar.
The disappearance of the peaks in the gluon dressing functions at $N=2$ can be directly related to the vanishing of hard-mode couplings due to the restriction $(p_0+q_0)/(2\pi T)<2$. Only hard-soft-mode couplings contribute. Thus, the peak is generated due to pure hard-hard interactions alone.
Again here and in the following only the $n=1$ modes are presented. The $n>1$ modes essentially follow their behavior, albeit much closer at tree-level.
\begin{figure}
\epsfig{file=ftfgcomp.eps,height=1.2\linewidth,width=\linewidth}
\caption{The dependence of the solutions on $T$ at fixed $g_4^2T$ and subtraction at $T$ at $N=5$. The left panel shows the soft modes, where apart from the longitudinal gluon the dressing functions are shown. In the latter case the propagator is shown. The right panels show the dressing functions for the $n=1$ hard mode. The red and black lines are the $g=1/2$ and the $g\approx 0.4$ solution, respectively. Solid is $T=0.9$ GeV, dashed is $T=1.5$ GeV, and dotted is $T=9.1$ GeV.}
\label{figftfg}
\end{figure}
The dependence on temperature when using t'Hooft-like scaling is shown in figure \ref{figftfg}. Here, a significant effect also on the soft modes can be seen. While a direct influence on the infrared coefficients for the ghost and transverse gluon dressing functions is visible, for the dressing function of the longitudinal gluon only a slight change at mid-momenta occurs. Therefore, the dressing function of the longitudinal gluon is dominated by its renormalized mass also when changing the temperature. As expected, the hard mode dressing functions become more and more tree-level like when increasing the temperature. The peaks in the hard mode dressing functions shift to higher momenta, owing to the renormalization condition, and become smaller due to the increase of the effective mass $p_0$ of the hard modes. At low temperature, the opposite effect is observed. In general, the hard mode dressing functions become more dynamical as their effective mass decreases, albeit quite slowly. The insensitivity on the infrared solution of the soft sector is not changed when reducing the temperature, and it may need a significantly lower temperature to induce a change.
\begin{figure}
\epsfig{file=ftft.eps,height=1.2\linewidth,width=\linewidth}
\caption{The dependence of the solutions on $T$ at fixed $\mu$ and $s$ at $N=5$. The left panel shows the soft modes, where apart from the longitudinal gluon the dressing functions are shown. In the latter case the propagator is shown. The right panels show the dressing functions for the $n=1$ hard mode. The red and black lines are the $g=1/2$ and the $g\approx 0.4$ solution, respectively. Solid is $T=0.3$ GeV, dashed is $T=1.5$ GeV, and dotted is $T=3.2$ GeV for $g=1/2$ and $T=2.3$ GeV for $g\approx 0.4$.}
\label{figftft}
\end{figure}
Switching to a fixed renormalization scale and subtraction point, the solutions shown in figure \ref{figftft} are found. The effect is quite the same as in the ghost-loop-only truncation. Changing the temperature at fixed 4d-coupling, and thus increasing the 3d-coupling with temperature, shifts the onset of non-perturbative effects to larger momenta. Obeying the renormalization at fixed momentum then requires the changes in the dressing functions seen in the figure. In this case, it was also again possible to super-cool down to at least $T=152$ MeV.
\begin{figure}
\epsfig{file=ftzxcomp.eps,height=1.2\linewidth,width=\linewidth}
\caption{The dependence of the solutions on $\zeta$ and $\xi$ at $N=5$. The left panel shows the soft modes, where apart from the longitudinal gluon the dressing functions are shown. In the latter case the propagator is shown. The right panels show the dressing functions for the $n=1$ hard mode. The red and black lines are the $g=1/2$ and the $g\approx 0.4$ solution, respectively. Solid is $(g,\zeta,\xi)=(1/2,3,2)$ and $(0.4,1.7,7/4)$, dashed is $(1/2,1,1)$ and $(0.4,1,1)$, dotted is $(1/2,0,1/4)$ and $(0.4,1/4,1/2)$.}
\label{figftzx}
\end{figure}
The dependence on $\zeta$ and $\xi$ is shown in figure \ref{figftzx}. Varying $\zeta$ leads to similar variations for the soft mode dressing functions as in the 3d-case. The largest effect is seen in the infrared. As the hard mode dressing functions are insensitive to the infrared behavior of the soft modes due to their effective mass, there are only weak variations of them with $\zeta$. Only at small $\zeta$ an additional structure appears in the transverse gluon dressing function. This is likely due to the cross-term in equation \pref{fulleqZft}. The effect of varying $\zeta$ on the longitudinal sector is negligible, as in the 3d-case. Correspondingly, the effect of varying $\xi$ is only significant for the longitudinal sector. The sensitivity is much less pronounced then in the case of varying $\zeta$. Especially the soft longitudinal dressing function is nearly unaffected by variation of $\xi$. The reason is that the soft equation does not depend explicitly on $\xi$, as it can be divided out of the equation for $\xi\neq 0$. Therefore, any $\xi$ dependence only enters indirectly by the weak dependence of the hard modes on $\xi$.
The dependence on $\delta$ is qualitatively not different from the high temperature case for the soft modes. With decreasing temperature, the lowest attainable value of $\delta$ increases and is around $\delta\approx 0.11$ for the $g=1/2$ solution and $\delta\approx 0.15$ for the $g\approx 0.4$ solution at $T=1.5$ GeV. Surprisingly, the roles of the $g=1/2$ and $g\approx 0.4$ solutions are interchanged: At low temperatures, it is possible to reach lower values of $\delta$ in case of the half-integer solution. The hard mode dressing functions are nearly not affected by the change in $\delta$. Thus, there is no significant change concerning the dependence on $\delta$ at finite temperatures.
At this point, a comparison can be made to a different approach to obtain the high-temperature gauge propagators. The usual continuum method is the semi-perturbative hard thermal loop (HTL) approach \cite{Blaizot:2001nr}. It is based on resumming the hard mode contributions in self-energy diagrams, hence its name. In the transverse infrared sector it is plagued by severe problems, due to its perturbative nature. The final result is a transverse gluon propagator with a particle like pole at $p=0$, thus $t=0$. This is in sharp contrast to the results found here and, as discussed in chapter \ref{c3d}, such a behavior is not likely. Also the lattice results tend to support the results found here. Concerning the soft longitudinal mode and the hard modes, HTLs and the ansatz presented here find qualitatively similar results on the level of the propagators.
\section{Solutions at Small Temperature}\label{sdselowt}
\begin{figure}[ht]
\begin{center}\epsfig{file=glowt.eps,width=0.5\linewidth}\end{center}
\caption{The dressing function of the soft ghost mode at zero \cite{Fischer:2002hn} and finite temperature \cite{Gruter:2004kd}.}
\label{figglowt}
\end{figure}
As the temperature region of and above the phase transition is probed by experiments, the aim of the calculations presented here is to eventually describe the phase transition. This is not yet possible. Currently, the phase transition is approached from two sides. The first approach is from above, and is presented in this work. The second approach is from below and treated especially in \cite{Gruter:phd} and also to some extent in the context of this work \cite{Gruter:2004kd}. The main aim in these calculations is to investigate whether any qualitative changes are observed at small temperatures as compared to the vacuum solutions. Sufficient for this aim is to employ not a full continuum method but solving the DSEs approximately on a discretized space-time with a toroidal topology. This method had already been applied successfully to the vacuum \cite{Fischer:2002hn,Fischer:2003zc}.
The results for the soft mode dressing functions are shown in figure \ref{figglowt} for the ghost propagator and in figure \ref{figzhlowt} for the two independent dressing functions of the gluon propagator \pref{gluonprop}, compared to the zero temperature solution. While the 3d-transverse part of the propagator becomes steeper in the infrared, the 3d-longitudinal part becomes more shallow. Thus, it experiences significant finite volume effects, as in the case of lattice calculations. Finite volume scaling shows that it is still confined, as in the case of the vacuum \cite{Gruter:2004kd}. Indeed it cannot be distinguished yet whether the exponents or merely the coefficients of the infrared solution change.
\begin{figure}[t]
\epsfig{file=zlowt.eps,width=0.5\linewidth}\epsfig{file=hlowt.eps,width=0.5\linewidth}
\caption{The propagator of the soft gluon mode at zero \cite{Fischer:2002hn} and finite temperature \cite{Gruter:2004kd}. The left panel shows the 3d-transverse part $Z$ of \pref{gluonprop} and the right panel the 3d-longitudinal part $H$.}
\label{figzhlowt}
\end{figure}
Interestingly, the 4d-transverse projection of \pref{gluonprop}, yields that the quantity $2Z+H$, is extremely temperature-independent \cite{Gruter:2004kd}. As this combination is the only one entering the thermodynamic potential to be discussed in section \ref{std}, and $G$ in itself is rather temperature independent, the corresponding thermodynamic potential is itself nearly independent of temperature and compatible with 0. This is what would be naively expected from a confined gluon system, as bound states are not included in the description here.
In addition, it is possible to trace the solution up to temperatures of 800 MeV \cite{Gruter:2004kd}, significantly above the expected phase transition temperature of $(269\pm 1)$ MeV according to lattice results \cite{Karsch:2003jg}. Therefore it is possible to super-heat the system.
If the temperature scale used in \cite{Gruter:2004kd} and here are compatible within a factor of 4, then within a substantial temperature range, two qualitatively different solutions exist. One solution has an over-screened and one a screened soft longitudinal gluon. Together with the observation of super-heating and super-cooling when comparing to lattice results, this implies a first order phase transition.
Compared to the high-temperature case presented before, the sector of ghosts and transverse gluons is not changed qualitatively. Quantitatively, especially the infrared exponents changed, though. The main difference is in the longitudinal sector, where a change from over-screening to screening occurred. Comparing the hard modes in both cases, no qualitative change is found in any sector. Therefore, the main difference of the two phases is only in the longitudinal/chromoelectric sector. Therefore the nature of the phase transition will likely be chromoelectric. This implication for the phase transition and the compatibility of these observations with lattice calculations will be discussed in section \ref{ssumprop}.
\chapter{Introduction}
\section{Strong Interactions}
Nature is described by only three fundamental forces according to the current knowledge. Each of those covers an own realm of physics and each one comes with its own problems. These forces are the electroweak and strong interactions, building together the standard model of particle physics, and gravitation \cite{Bohm:yx,Gravitation}.
Gravitation describes the behavior of macroscopic objects to the largest distances known and it determines the gross properties of the universe at the present time. Its formulation in the general theory of relativity \cite{Gravitation} has been supported by overwhelming experimental evidence. It is the weakest of the forces and quite well understood in the classical regime. On the other hand, there is still no successful implementation of a quantum theory of the tensorial gravitation field. Also several observations indicate that the current state of the universe is not only determined by the matter fields of the standard model. The nature and interactions of approximately 95\% of the universe are not known today.
Electromagnetism, electroweak symmetry breaking, weak processes like the $\beta$-decay, neutrino-oscillations and various other effects are due to electroweak interactions as formulated by the Glashow-Salam-Weinberg theory \cite{Bohm:yx}. It is rather well understood and its treatment using perturbative methods has been tested with high experimental precision. Although being rather tractable in general it is demanding in detail. Fundamental questions are still posed by the absence of the Higgs in experiments up to now and the origin of the large number of parameters.
The interaction binding together nucleons to build nuclei, thus forming the core of atoms, is known as the strong interaction. It is the strongest of all known forces. It describes also the way in which nucleons and all other hadrons are made out of their constituents, the quarks and gluons. The theory of strong interactions in its quantized form is termed quantum chromodynamics (QCD) \cite{Bohm:yx}, as quarks and gluons carry a charge termed color. In contrast to the electroweak theory, which includes electromagnetism, and gravitation, the strong force does not appear on a macroscopic level beyond its bound state spectrum.
QCD offers a rich set of phenomena, which are not yet really understood. Primarily, a complete first-principle calculation of the bound-state spectrum of QCD is still lacking. Such a calculation must be able to explain why some bound-states like mesons and baryons appear in large numbers but why others like glue-balls and hybrids have not yet been convincingly found or appear only in rather small numbers like penta-quarks and meson-molecules possibly observed recently \cite{Hicks:2004vd}.
At a more fundamental level, three properties of QCD are most striking. Two are connected to the fermionic content of the theory: The breaking of chiral symmetry and the axial anomaly. The first phenomenon gives rise to the proton mass of nearly 1 GeV, while the quarks making it up have masses only of the order of MeV. This can be understood as due to the spontaneous breaking of the approximate chiral symmetry of the lightest quarks and is also well known in other models. The axial anomaly is connected to a true quantum anomaly, and e.g. gives rise to the anomalous large mass of the $\eta'$-meson.
The third property is confinement, the absence of the colored degrees of freedom from the physical spectrum. It is this property which significantly shapes the low-energy reality of daily life while simultaneously being least understood of all the genuine non-perturbative effects of QCD. At the same time it is one of the properties which has been measured with the highest precision available: Free quarks have a unique experimental signature due to their fractional electric charge. The absence of such objects in nature has been established at a precision of the order of $1:10^{30}$ \cite{Hagiwara:2002fs}.
The true non-perturbative nature of QCD is the reason for the complications in the study of these phenomena. Perturbation theory describes strong effects reasonably well only at energies larger than a few GeV, the precise scale depending on the process. At smaller energies, which ultimately govern hadron and nuclear physics, the interaction is so strong that perturbation theory is not applicable. Therefore different approaches are needed.
Extensive results are available from model calculations. However, only few attempts of first-principle approaches have been employed. Out of these, lattice gauge theory \cite{Montvay:1994cy} is by far the most successful and widespread, and several achievements have been made using it. Lattice calculations are a numerical tool which discretizes space-time and thus is able to calculate the partition function directly, albeit numerically expensively. Especially fermionic contributions pose significant problems. Furthermore lattice calculations are restricted at best to two orders of magnitude in momenta and rather small volumes due to the limitation by computing time.
As many interesting phenomena of QCD are likely to be generated in the far infrared and include singularities, a continuum formulation is desirable. It is provided e.g.\/ by the equations of motions in form of the Dyson-Schwinger equations. It is this method which is implemented here and it will be discussed in more detail in chapter \ref{cdse}. Compared to lattice calculations, this approach suffers from a larger number of necessary assumptions to make it technically treatable. At the same time it allows the direct investigation of the continuum manifestation of non-perturbative effects. Both these methods and several other approaches are complementary and necessary to untangle the complex structure of low-energy QCD.
\section{Thermodynamics of QCD}\label{stdqcd}
The phase structure of QCD is a long-standing problem \cite{Cabibbo:1975ig}. Naively, deconfinement would be expected in a sufficiently hot medium in a simple picture like the bag-model \cite{Wong:1995jf}. The properties of such a different phase of QCD would grant additional insight into the mechanisms of strong interactions. Such a medium is also expected to have existed at the beginning of the universe, and its specific properties are of relevance for cosmology. Therefore great experimental efforts have been undertaken in the last decades to generate it. Temperatures of around 100-200 MeV have been reached in heavy-ion collisions, but with large experimental uncertainties on the highest temperature achieved, especially as the thermalization process is not yet clear \cite{Andronic:2004tx}.
This has also led to great theoretical efforts to solve the many-body problem of QCD. However, up to now the results generate more questions than answers. This is especially true for the non-equilibrium and dynamical part of the experiments, but even for the equilibrated medium it is still the case.
\begin{figure}[ht]
\epsfig{file=phase_diagram.eps,width=0.5\linewidth}\epsfig{file=t-mu.eps,width=0.5\linewidth}
\caption{The left figure from \cite{Karsch:2003jg} sketches the current idea of the phase diagram of QCD in the temperature ($T$)-baryon chemical potential ($\mu$)-plane. It is not yet settled whether the transition at $\mu=0$ is a true phase transition or merely a cross-over. $\mu_0$ corresponds to normal nuclear matter density, $\chi$ denotes chiral. The right panel shows a more quantitative picture with $\mu_b\equiv\mu$. This figure from \cite{Andronic:2004tx} contains the available experimental data. The full triangle corresponds to normal nuclear matter and the open squares indicate the position of a possible tri-critical point from lattice calculations \cite{Fodor:2001pe}. The circles are from fits of particle ratios with a thermal model \cite{Andronic:2004tx,Braun-Munzinger:2001mh}. They have been obtained by the experiments at RHIC at high and low energies, SPS at high and low energies, AGS and SIS in order of increasing $\mu$. The dashed and dotted curves show freeze-out trajectories for a hadron gas at constant density or energy density, respectively \cite{Andronic:2004tx}.}\label{figintro}
\end{figure}
The general aim of experiment and theory is to map out the phase diagram of QCD in the temperature-(baryon) chemical potential plane. Roughly sketched it is shown in figure \ref{figintro}, together with the currently available experimental data. The hadronic phase with its gas-liquid phase-transition occupies the small temperature and density/chemical potential region. A rich set of solid-state-like phases are expected at large chemical potentials, such as color-superconductors, see e.g.\/ \cite{Rajagopal:2000wf}. At high temperatures, a so-called quark-gluon-plasma is expected. It is not yet settled whether it is reached by a genuine phase transition or a cross-over in the real world. A pure gluon system like the one to be studied here exhibits a first order phase transition \cite{Karsch:2003jg}.
It is expected that chiral symmetry is restored in the high-temperature phase. This is supported by lattice calculations, see e.g.\/ \cite{Karsch:2003jg}. Concerning the other properties of this phase, even the nature of the effective degrees of freedoms is currently under debate. The initial picture made was a perturbative gas of free quarks and gluons, leading to the name deconfinement. As colored currents are gauge dependent, this view can only be true in a figurative sense. However, the idea of almost local colorless objects with otherwise the quantum numbers of quarks and gluons as weakly interacting quasi-particles remains.
The main aim of this work is an analysis of the properties of gluons in the high-temperature phase as well as the validity of this simple picture. Still addressing merely equilibrium dynamics, the fate of the confining properties at temperatures above the phase transition are investigated. In addition, there is recent evidence that at temperatures just above the phase transition the matter is in a non-trivial, strongly interacting phase. Whether this extends to all temperatures is also a main subject of this work.
The basic field theoretical concepts used will be compiled in chapter \ref{ctheory}. The Dyson-Schwinger equations used to study this theory including the introduction of thermodynamic features will be laid out in chapter \ref{cdse}. A first step will be concerned with investigations regarding the infinite temperature limit in chapter \ref{c3d}. This will be followed by introducing a high temperature expansion in chapter \ref{cft}. The thermodynamic potential, analytic properties and other aspects of the solutions will be investigated in chapter \ref{cderived}. The results will be interpreted and summed up in chapter \ref{csummary}. Appendix \ref{aconventions} contains conventions and commonly used symbols. Five further appendices contain technical details which would have broken the line of argument inside the main text.
\chapter{Infrared Expressions}\label{air}
\section{Massive Higgs}
As already argued in section \ref{ssiranalysis}, the only solution without further assumption is that of ghost dominance. This then requires the calculation of $I_{GT}$ and $I_{GG}$ in \pref{ghostir} and \pref{gluonir} only. The latter can be obtained straightforwardly when using the ansatz \pref{iransatz} and the general formula
\begin{equation}
\int\frac{d^dq}{(2\pi)^d}q^{2\alpha} (q-p)^{2\beta}=\frac{1}{(4\pi)^{\frac{d}{2}}}\frac{\Gamma(-\alpha-\beta-\frac{d}{2})\Gamma(\frac{d}{2}+\alpha)\Gamma(\frac{d}{2}+\beta)}{\Gamma(d+\alpha+\beta)\Gamma(-\alpha)\Gamma(-\beta)}y^{2\left(\frac{d}{2}+\alpha+\beta\right)},\label{dimregrule}
\end{equation}
\noindent valid for finite integrals. It yields
\begin{equation}
I_{GG}=-\frac{g_3^2C_A\pi}{(4\pi)^{\frac{d}{2}}}\frac{2^{4g-2d}(d-4g)(2+d(\zeta-2)-4g(\zeta-1)-\zeta)\Gamma(d-2g)\Gamma(2g-\frac{d}{2})}{(d-1)g^2\Gamma(\frac{1+d-2g}{2})^2\Gamma(g)^2}.\nonumber
\end{equation}
\noindent Note that the integral is convergent if and only if
\begin{equation}
\frac{d-1}{2}\ge g>\frac{d-2}{4},\label{dglimits}
\end{equation}
\noindent where the equality on the upper boundary requires the result to exist only in the sense of a distribution.
Performing the same calculation for the ghost self-energy is more complicated. Since it is, in general, a divergent quantity, equation \pref{dimregrule} cannot be applied directly. It is necessary to regularize and then renormalize the expression. This can be done in a momentum subtraction scheme~\cite{Zwanziger:2001kw} or via dimensional regularization. Here, the latter was performed by applying the standard rules of dimensional regularization~\cite{Peskin:ev}. This immediately gives a finite result in odd dimensions, as dimensional regularization is in this case already a renormalization prescription. However, by doing so, a divergent quantity has been removed which is formally eliminated by setting $-\widetilde Z_3$ equal to this quantity. Regularization and renormalization in even dimensions can be performed using the MS-prescription \cite{Collins:xc}. This procedure yields the same result as the momentum subtraction scheme. The range allowed for $g$ in \pref{dglimits} permits $I_{GT}$ not only to have a divergence of logarithmic or linear order, but also quadratic or cubic divergences in even or uneven dimensions, respectively. Using a subtraction scheme, it would be necessary to include the next term in the Taylor expansion. On the other hand, dimensional renormalization directly yields
\begin{equation}
I_{GT}=\frac{g_3^2C_A}{(4\pi)^{\frac{d}{2}}}\frac{2^{1-2g}(4^g(d-3)d+2^{1+2g}(1+g-dg))\Gamma(\frac{d}{2}-g)\Gamma(-g)\Gamma(2g)}{(2-d+2g)(d+2g)\Gamma(\frac{d}{2}-2g)\Gamma(g)\Gamma(\frac{d}{2}+g)}.\nonumber
\end{equation}
\noindent Note that this expression becomes negative already for values allowed by \pref{dglimits}, e.g. for $g\ge3/4$ in three dimensions, thus reducing the allowed range and leading to the plots in figure \ref{figexp}.
\section{Massless Higgs}\label{asmasslesshiggs}
If the Higgs is massless, then the Higgs-loop in the gluon equation \pref{fulleqZ3d} is no longer suppressed in the infrared and can in principle be as leading as the ghost-loop or even dominate. Thus three solutions can be found in the infrared. As the arguments from section \ref{ssiranalysis} are still valid, all must satisfy $t<-1$.
In the first case, the Higgs and the ghost exchange their roles. The ghost behaves then like a tree-level particle in the infrared with $g=0$, while the Higgs diverges in the infrared and drives the gluon. In the second case, $l=g$, and the gluon is driven by the combination of both loops. In the third case, the ghost and the gluon behave as in the massive case and $l=0$: The Higgs behaves as a massless tree-level particle in the infrared. In all cases gluons are confined. The first solution may correspond to a phase structure similar to a Higgs-Anderson phase, but different to the conventional `perturbative' Higgs phases. The physical interpretation of the second scenario is lacking and the properties of this phase would be quite peculiar. The last case has similar physics as the massive case. Note that in the first two cases gauge symmetry is not necessarily intact any more, and may be broken by the interaction with the Higgs field. This may restrict these phases to theories which do not emerge from a pure Yang-Mills theory, as this would be difficult to reconcile with Elitzur's theorem \cite{Elitzur:im}. Hence the Gribov condition \pref{gribov} will be dropped for the first two cases for now. As these cases offer a different structure than the massive case, they will be investigated a little more closely.
After employing a renormalization prescription in the Higgs equation as in the case of the ghost equation, the infrared system becomes
\begin{eqnarray}
\frac{y^g}{A_g}&=&\widetilde{Z}_3(1-\delta_{gl})+y^{-\frac{1}{2}-g-t}A_gA_zI_{GT}(y)\nonumber\\
\frac{y^t}{A_t}&=&y^{-\frac{1}{2}-2g}A_g^2I_{GG}(y)+y^{-\frac{1}{2}-2l}A_h^2I_{GH}(y)\nonumber\\
\frac{y^l}{A_l}&=&y^{-\frac{1}{2}-g-l}A_hA_zI_{HT}(y).\nonumber
\end{eqnarray}
\noindent Keeping $d=3$ for simplicity, and noting that in both cases
\begin{equation}
l=-\frac{1}{2}(t+\frac{1}{2})
\end{equation}
\noindent and
\begin{eqnarray}
I_{HT}&=&\frac{2^{2+4l}(8-11l-15l^2+14l^3)\Gamma(-2l)}{(l-3+32l^2+44l^3+16l^4)\Gamma(\frac{1}{2}-2l)}\nonumber\\
I_{GH}&=&-\frac{2^{-2+4l}(l-1+l\zeta)\Gamma(-2l)\sec(2l\pi)\sin^2(l\pi)}{(l-1)(4l-1)l\Gamma(\frac{1}{2}-2l)},\nonumber
\end{eqnarray}
\noindent the conditional equation for the first case is
\begin{equation}
1=\frac{(1+l)(1+2l)(3+2l)(l-1+l\zeta)(\sec(2l\pi)-1)}{16(l-1)l(8+l(1+2l)(7l-11))}.\nonumber
\end{equation}
\noindent This has no solution at $\zeta=1$ in $1/4< l\le 1/2$, but has e.g. the solution $l=0.362568$ at $\zeta=3$. This already indicates that in these situation the allowed projections differ from the normal case. Since for a broken gauge symmetry the STI \pref{stigluon} may not be valid anymore and the gluon may acquire longitudinal components, this equally well indicates that the current truncation is inadequate for this case.
In the second case, the condition for a solution becomes
\begin{equation}
1=\frac{I_{GG}}{I_{GT}}+\frac{A_h^2}{A_g^2}\frac{I_{GH}}{I_{HT}}=\frac{A_g^2}{A_h^2}\frac{I_{GG}}{I_{HT}}+\frac{I_{GH}}{I_{HT}},\nonumber
\end{equation}
\noindent yielding
\begin{eqnarray}
1&=&\frac{I_{GH}I_{GT}}{I_{GT}I_{HT}-I_{GG}I_{HT}}\nonumber\\
&=&\frac{(3(\zeta-2)+2g(59-4g^2(\zeta-1)+\zeta-6g(10+\zeta))+64(g-1)g\csc^2(g\pi))}{8(1+g)(1+2g)(3+2g)(g-1+g\zeta)}\cdot\nonumber\\
&&\cdot(8+g(1+2g)(7g-11)).\nonumber
\end{eqnarray}
\noindent Only one infrared coefficient is left undetermined, thus it is likely that here the infrared behavior will be linked stronger to finite momentum solutions to allow for compensation both in the Higgs equation and the ghost equation in the infrared. Note that again a solution does not exist for $\zeta=1$ for $1/4< g=l\le 1/2$, but it exists for $\zeta=3$ with value $l=g=0.340408$. Thus also here the validity of the truncations made is debatable.
Nonetheless, these investigations show that additional massless degrees of freedom may change the infrared properties significantly. Especially in QCD it turns out that no solutions have been found yet with $N_f>4$ massless quarks \cite{Fischer:2003rp} in this approach, despite the possibility of chiral symmetry breaking.
\chapter{Kernels}
\section{3d Kernels}\label{s3dkernels}
The integral kernels in (\ref{fulleqG3d}-\ref{fulleqZ3d}) are obtained by using tree-level vertices and performing the corresponding contractions. The ghost kernel is given by
\begin{equation}
A_t(k,q,\theta)=-\frac{q^2\sin^3(\theta)}{(k^2+q^2-2kq\cos\theta)^2}.\nonumber
\end{equation}
The contributions in the Higgs equation are
\begin{eqnarray}
N_1(k,q,\theta)=-\frac{2q^2\sin^3(\theta)}{(k^2+q^2+2kq\cos\theta)^2}\nonumber\\
N_2(k,q,\theta)=-\frac{2\sin^3(\theta)}{k^2+q^2+2kq\cos\theta}.\nonumber
\end{eqnarray}
The kernels in the gluon equation are finally
\begin{eqnarray}
&R(k,q,\theta)=-\frac{((\zeta-1)kq\cos(\theta)-q^2+\zeta q^2\cos^2(\theta))\nonumber
\sin\theta}{2k^2(k^2+q^2+2kq\cos\theta)}\nonumber\\
&M_L(k,q,\theta)=\frac{((\zeta-1)(k^2+4kq\cos\theta)-4q^2+4q^2\zeta\cos^2(\theta))\sin\theta}
{4k^2(k^2+q^2+2kq\cos\theta)}\nonumber\\
&M_T(k,q,\theta)=\frac{\sin\theta}{4k^2(k^2+q^2+2kq\cos\theta)}\cdot\nonumber\\
&\cdot\Big((k^2+2q^2)((\zeta-9)k^2-4q^2)+8(\zeta-3)(k^2+q^2)kq\cos\theta+(8\zeta q^4+(\zeta+7)k^4\nonumber\\
&+4(5\zeta-1)k^2q^2)\cos^2(\theta)+4(4\zeta q^2+(\zeta+3)k^2)\cos^3(\theta)+4\zeta k^2q^2\cos^4(\theta)\Big).\nonumber
\end{eqnarray}
The modified gluon vertex is introduced by multiplying $M_T$ with \pref{g3vertex}.
\section{Finite Temperature Kernels}\label{sftkernels}
At finite temperature, the kernels are obtained from tree-level vertices. The following abbreviations have been used:
\begin{eqnarray}
q&=&\left|\vec q\right|\nonumber\\
k&=&\left|\vec k\right|\nonumber\\
x&=&q^2+q_0^2\nonumber\\
y&=&k^2+k_0^2\nonumber\\
u_3&=&(\vec q-\vec k)^2\nonumber\\
z_3&=&(\vec q+\vec k)^2\nonumber\\
u_4&=&(k_0-q_0)^2+u_3\nonumber\\
z_4&=&(k_0+q_0)^2+z_3.\nonumber
\end{eqnarray}
\noindent The kernels in the ghost equations are
\begin{eqnarray}
A_T(k_0,q_0,k,q,\theta)&=&-\frac{k^2q^4{\sin (\theta )}^3}{xy{u_3}{u_4}}\nonumber\\
A_L(k_0,q_0,k,q,\theta)&=&-\frac{q^2\sin\theta{\left( {q}^2{k_0} + {k}^2{q_0} - kq\cos\theta\left( {k}_0 + {q_0} \right)\right) }^2}{xy{u_3}{{u_4}}^2}.\nonumber
\end{eqnarray}
\noindent The kernels in the transverse gluon equations are
\begin{eqnarray}
R(k_0,q_0,k,q,\theta)&=&-\frac{q^3\left( -q + k\left( -1 + \zeta \right) \cos\theta+ q\zeta {\cos (\theta )}^2 \right) \sin (\theta )}{2xyz_4}\nonumber\\
M_T(k_0,q_0,k,q,\theta)&=&\frac{q^2}{16xy z_3 z_4}\Big( \Big( 8q^4\left( -4 + \zeta \right) + k^4\left( -29 + 5\zeta \right)\nonumber\\
&&+ 2k^2q^2\left( -46 + 15\zeta \right) \Big) \sin (\theta ) +4kq\Big( 4q^2\left( -3 + 2\zeta \right)\nonumber\\
&&+ k^2\left( -9 + 5\zeta \right) \Big) \sin (2\theta ) + \Big( 8q^4\zeta + k^4\left( 7 + \zeta \right)\nonumber\\
&&+ k^2q^2\left( -4 + 23\zeta \right) \Big) \sin (3\theta ) + 2kq\left( 4q^2\zeta + k^2\left( 3 + \zeta \right) \right) \sin (4\theta )\nonumber\\
&& + k^2q^2\zeta \sin (5\theta ) \Big)\nonumber\\
M_1(k_0,q_0,k,q,\theta)&=&\frac{q^2\sin\theta}{8x^2y{z_3}{z_4}}\Big( -4q^4\left( -4 + \zeta \right) {{k_0}}^2 - k^4\left( -9 + \zeta \right) {{q_0}}^2 + k^2q^2\left( 4{k_0} - {q_0} \right)\cdot\nonumber\\
&&\cdot \left( 4{k_0} + \left( -8 + \zeta \right) {q_0} \right) - 2kq\cos\theta\Big( -16q^2{{k_0}}^2\nonumber\\
&& + 2\left( 8k^2 - q^2\left( -8 + \zeta \right) \right) {k_0}{q_0} + k^2\left( -13 + \zeta \right) {{q_0}}^2 \Big) + \cos (2\theta )\cdot\nonumber\\
&&\cdot \left( 4q^4\zeta {{k_0}}^2 - 4k^2q^2\left( \left( 7 + \zeta \right) {k_0} - 2{q_0} \right) {q_0} + k^4\left( 7 + \zeta \right) {{q_0}}^2 \right) + \nonumber\\
&&kq{q_0}\left( kq\zeta \cos (4\theta )q_0 + 2\cos (3\theta )\left( -2q^2\zeta {k_0} + k^2\left( 3 + \zeta \right) {q_0} \right) \right) \Big)\nonumber
\end{eqnarray}
\begin{eqnarray}
M_2(k_0,q_0,k,q,\theta)&=&\frac{q^2}{16xy{z_3}{{z_4}}^2}\Big(-\left(k^2q^2\zeta\sin(5\theta)\left(k_0-q_0\right)^2 \right) +4kq\sin (2\theta )\cdot\nonumber\\
&&\cdot\left( k_0 - q_0 \right)\left( 2q^2\left( -4 + \zeta \right) {k_0} + k^2\left( \left( -1 + \zeta \right) {k_0} - \left( -7 + \zeta \right) {q_0} \right) \right)\nonumber\\
&&-2kq\sin (4\theta )\left( k_0 - {q_0} \right)\left( 2q^2\zeta {k_0} + k^2\left( \left( -1 + \zeta \right) {k_0} - \left( 1 + \zeta \right) {q_0} \right) \right)\nonumber\\
&&+\sin\theta\Big( 4q^4\left( -8 + 3\zeta \right) {{k_0}}^2 +k^4\Big( -3{{k_0}}^2 + 3\zeta {\left( {k_0} - {q_0} \right) }^2 + 6{k_0}{q_0}\nonumber\\
&&- 23{{q_0}}^2 \Big) +2k^2q^2\left( \left( -10 + 7\zeta \right) {{k_0}}^2 + \left( 34 - 8\zeta \right) {k_0}{q_0} + \left( -4 + \zeta \right) {{q_0}}^2 \right) \Big)\nonumber\\
&&-\sin (3\theta )\Big( 4q^4\zeta {{k_0}}^2 + k^2q^2\Big( \left( 4 + 3\zeta \right) {{k_0}}^2 - 2\left( 10 + \zeta \right) {k_0}{q_0}\nonumber\\
&&- \left( -8 + \zeta \right) {{q_0}}^2 \Big) +k^4\Big( \left( -1 + \zeta \right) {{k_0}}^2 - 2\left( -1 + \zeta \right) {k_0}{q_0}\nonumber\\
&&+ \left( 3 + \zeta \right) {{q_0}}^2 \Big)\Big)\Big)\nonumber\\
M_L(k_0,q_0,k,q,\theta)&=&\frac{q^2\sin\theta}{4x^2y{z_3}{{z_4}}^2}\Big( {{k}}^6q^2\left( -1 + \zeta \right) - 4q^4{\left( x + {{k_0}}^2 + {k_0}{q_0} \right) }^2 + 4k^2q^2\zeta {\cos\theta}^4\cdot\nonumber\\
&&\cdot{\left( 2q^2 + {q_0}\left( -{k_0} + {q_0} \right) \right) }^2 +2k^4\Big( 2q^2\left( -1 + \zeta \right) {{k_0}}^2 + 3q^2\left( -1 + \zeta \right) {k_0}{q_0}\nonumber\\
&& +x\left( q^2\left( -3 + \zeta \right) - 2{{q_0}}^2 \right) \Big) +{{k}}^2q^2\Big( x^2\left( -9 + \zeta \right) + 4\left( -1 + \zeta \right) {{k_0}}^4\nonumber\\
&&+ 2x\left( -7 + 3\zeta \right) {k_0}{q_0} +12\left( -1 + \zeta \right) {{k_0}}^3{q_0} + {{k_0}}^2\Big( 4q^2\left( -3 + \zeta \right)\nonumber\\
&&+ \left( -21 + 13\zeta \right) {{q_0}}^2 \Big) \Big) +4kq{\cos\theta}^3\left( 2q^2 + {q_0}\left( -{k_0} + {q_0} \right) \right)\cdot\nonumber\\
&&\cdot\Big( 2q^2\zeta \left( x + {{k_0}}^2 + {k_0}{q_0} \right) +k^2\Big( q^2\left( -2 + 4\zeta \right) + {q_0}\Big( {k_0} - \zeta {k_0} + {q_0}\nonumber\\
&&+ \zeta {q_0} \Big) \Big) \Big) +2kq\cos\theta\Big( 2q^2\left( x + {{k_0}}^2 + {k_0}{q_0} \right)\Big( q^2\left( -5 + \zeta \right)\nonumber\\
&&+ 2\left( -1 + \zeta \right) {{k_0}}^2 + \left( -1 + 3\zeta \right) {k_0}{q_0} +\left( -3 + \zeta \right) {{q_0}}^2 \Big) + {{k}}^4\left( -1 + \zeta \right)\cdot\nonumber\\
&&\cdot\left( 4q^2 + {q_0}\left( -{k_0} + {q_0} \right) \right) +{{k}}^2\Big( -2\left( -1 + \zeta \right) {{k_0}}^3{q_0} + \left( -1 + \zeta \right) {{k_0}}^2\cdot\nonumber\\
&&\cdot\left( 10q^2 - {{q_0}}^2 \right) +x\left( 2q^2\left( -7 + 3\zeta \right) + \left( -5 + \zeta \right) {{q_0}}^2 \right) +{k_0}{q_0}\Big( q^2\cdot\nonumber\\
&&\cdot\left( -9 + 13\zeta \right) + 2\left( 1 + \zeta \right) {{q_0}}^2 \Big) \Big) \Big) +{\cos\theta}^2\Big( 4q^4\zeta {\left( x + {{k_0}}^2 + {k_0}{q_0} \right) }^2\nonumber\\
&&+4k^2q^2\Big( q^4\left( -8 + 6\zeta \right) - 3\left( -1 + \zeta \right) {{k_0}}^3{q_0} + 8q^2\left( -1 + \zeta \right) {{q_0}}^2 +\nonumber\\
&&\left( -1 + 2\zeta \right) {{q_0}}^4 + {{k_0}}^2\left( q^2\left( -6 + 8\zeta \right) - \left( -2 + \zeta \right) {{q_0}}^2 \right) +2{k_0}{q_0}\Big( q^2\cdot\nonumber\\
&&\cdot\left( -1 + 4\zeta \right) + \left( 1 + \zeta \right) {{q_0}}^2 \Big) \Big) +k^4\Big( 4q^4\left( -5 + 6\zeta \right) - 4q^2{q_0}\Big( 3\left( -1 + \zeta \right)\cdot\nonumber\\
&&\cdot{k_0} + {q_0} - 3\zeta {q_0} \Big) + {{q_0}}^2\Big( \left( -1 + \zeta \right) {{k_0}}^2 - 2\left( -1 + \zeta \right) {k_0}{q_0}\nonumber\\
&& + \left( 3 + \zeta \right) {{q_0}}^2 \Big) \Big)\Big) \Big).\nonumber
\end{eqnarray}
\noindent The kernels in the longitudinal gluon equations are
\begin{eqnarray}
P(k_0,q_0,k,q,\theta)&=&\frac{q^2\sin\theta}{xy^2{z_4}}\Big( -2\left(\xi-1 \right) k_0^2{q_0}\left( {k_0} + {q_0} \right) + k^2{q_0}\left( {k_0} - \xi {k_0} + \xi {q_0} \right) + qk_0\cos\theta\nonumber\\
&&\cdot\left( q\xi \cos \theta k_0 + k\left( {k_0} + 2{q_0} - \xi \left( {k_0} + 4{q_0} \right) \right) \right) \Big)\nonumber\\
N_T(k_0,q_0,k,q,\theta)&=&\frac{-q^2\sin\theta}{2xy^2{z_3}{z_4}}\left( 3k^2 + 4q^2 + 8kq\cos\theta + k^2\cos (2\theta ) \right) \Big( k_0\left( {k_0} + 2{q_0} \right)\Big( k^2\nonumber\\
&&+ {k_0}\left( {k_0} + 2{q_0} \right) \Big)-\xi\Big( {{k_0}}^2\left( -q + {k_0} + 2{q_0} \right) \left( q + {k_0} + 2{q_0} \right) + k^2\Big( {{k_0}}^2\nonumber\\
&& + 2{k_0}{q_0} - 2{{q_0}}^2 \Big) \Big) +q{k_0}\Big( q\xi \cos (2\theta )k_0 +2k\cos\theta\Big( k_0 + 2{q_0}\nonumber\\
&&- \xi \left( {k_0} + 4{q_0} \right) \Big) \Big) \Big)\nonumber\\
N_1(k_0,q_0,k,q,\theta)&=&\frac{-q^2\sin^3\theta}{x^2y^2{z_3}{z_4}}\Big( 2q^4\xi {{k_0}}^4 + k^4\left( 2x^2\xi - 2x\left(\xi-1 \right) {k_0}{q_0} - \left(\xi-1 \right) {{k_0}}^2{{q_0}}^2 \right)\nonumber\\
&&+k^2{{k_0}}^2\Big( {q_0}\left( {k_0} + 2{q_0} \right) \left( 2q^2 + {q_0}\left( {k_0} + 2{q_0} \right) \right) + \xi \Big( 4q^4 + q^2{q_0}\Big( -2{k_0}\nonumber\\
&&+ {q_0} \Big) - {{q_0}}^2{\left( {k_0} + 2{q_0} \right) }^2 \Big) \Big) +kq{k_0}{q_0}\Big( kq\xi \cos (2\theta )k_0{q_0} -2\cos\theta\nonumber\\
&&\cdot\left( 2q^2\xi {{k_0}}^2 + k^2\left( q^2\left(4\xi-2 \right) + {q_0}\left( \left(\xi-1 \right) {k_0} - 2{q_0} + 4\xi {q_0} \right) \right) \right) \Big) \Big)\nonumber\\
N_2(k_0,q_0,k,q,\theta)&=&\frac{-q^2\sin^3\theta}{xy^2{z_3}{{z_4}}^2}\Big( k^2{k_0}\left( {k_0} + {q_0} \right)\Big( 2k^4 + {k_0}\left( {k_0}+ 2{q_0} \right) \Big( 2q^2 + \left( {k_0} + {q_0} \right) \Big( {k_0}\nonumber\\
&&+ 2{q_0} \Big) \Big) +k^2\left( 6q^2 + \left( 3{k_0} + {q_0} \right) \left( {k_0} + 2{q_0} \right) \right) \Big) +\xi \Big( 2k^8 + 2q^4{{k_0}}^4\nonumber\\
&&+ 2k^6\left( 4q^2 + {{k_0}}^2 + {k_0}{q_0} + 2{{q_0}}^2 \right) -k^2{{k_0}}^2\Big( -4q^4 + {{k_0}}^4 + 6{{k_0}}^3{q_0}\nonumber\\
&&- q^2{{q_0}}^2 + 4{{q_0}}^4 + 4{k_0}{q_0}\left( q^2 + 3{{q_0}}^2 \right) +{{k_0}}^2\left( -3q^2 + 13{{q_0}}^2 \right) \Big) +k^4\nonumber\\
&&\cdot\left( 2x^2 - {{k_0}}^4 - 6{{k_0}}^3{q_0} + {{k_0}}^2\left( 6q^2 - 3{{q_0}}^2 \right) + {k_0}\left( -6q^2{q_0} + 2{{q_0}}^3 \right) \right)\Big)\nonumber\\
&&+ kq\Big( kq\cos (2\theta )\left( 4k^4\xi + \xi{{k_0}}^2{\left( {k_0} - {q_0} \right) }^2 +4k^2{k_0}\left( {k_0} + {q_0} - 2\xi {q_0} \right) \right)\nonumber\\
&&+2\cos\theta\Big( k^2{k_0}\left( {k_0} + {q_0} \right)\left( 4k^2 + 2q^2 + \left( 3{k_0} + {q_0} \right) \left( {k_0} + 2{q_0} \right) \right)\nonumber\\
&&+\xi\Big( 4k^6 + 2q^2{{k_0}}^3\left( {k_0} - {q_0} \right) + 2k^4\left( 2x + {{k_0}}^2 - {k_0}{q_0} \right) -k^2{k_0}\Big( {{k_0}}^3\nonumber\\
&&+ 4x{q_0} + 10{{k_0}}^2{q_0} + {k_0}\left( -4q^2 + 9{{q_0}}^2 \right) \Big) \Big) \Big) \Big) \Big)\nonumber
\end{eqnarray}
\begin{eqnarray}
&&N_L(k_0,q_0,k,q,\theta)=\frac{-q^2\sin\theta}{x^2y^2{z_3}{{z_4}}^2}\nonumber\\
&&\cdot\Big(2k^2q^2\cos^4\theta\Big( 4k^4\xi {{q_0}}^2 + \xi {{k_0}}^2{\left( 2q^2 + {q_0}\left( -{k_0} + {q_0} \right) \right) }^2 +4k^2{k_0}{q_0}\Big( q^2\left( 2 - 4\xi \right) + {q_0}\Big( {k_0} + {q_0}\nonumber\\
&&- 2\xi {q_0} \Big) \Big) \Big) +2kq{\cos (\theta )}^3\Big( 4k^6\xi {{q_0}}^2+ 2q^2\xi {{k_0}}^2\Big( 2q^4 + 2q^2{{k_0}}^2 + q^2{k_0}{q_0} - {{k_0}}^3{q_0} + 3q^2{{q_0}}^2+\nonumber
\end{eqnarray}
\begin{eqnarray}
&&+ {{q_0}}^4 \Big) +2k^4{q_0}\left( \left( 2 + \xi \right) {{k_0}}^2{q_0} + 2\xi {q_0}\left( 2q^2 + {{q_0}}^2 \right) +{k_0}\left( q^2\left( 6 - 12\xi \right) - \left( -2 + \xi \right) {{q_0}}^2 \right) \right)\nonumber\\
&&+k^2{k_0}\Big( -\left( \left(\xi-3 \right) {{k_0}}^3{{q_0}}^2 \right) -2{{k_0}}^2{q_0}\left( q^2\left( -3 + 9\xi \right) + 5\left(\xi-1 \right) {{q_0}}^2 \right)-2\left( -1 + 2\xi \right) {q_0}\Big( 6q^4\nonumber\\
&&+ 6q^2{{q_0}}^2 + {{q_0}}^4 \Big) +{k_0}\Big( 8q^4\xi - 22q^2\left( -1 + \xi \right) {{q_0}}^2- 9\left(\xi-1 \right) {{q_0}}^4 \Big) \Big) \Big) +q^2\Big( k^6\left(3\xi-1 \right) {{k_0}}^2\nonumber\\
&&+k^4{k_0}\Big( \left( -1 + 5\xi \right) {{k_0}}^3- 2x\left( -1 + 3\xi \right) {q_0} + 2\left( -1 + 3\xi \right) {{k_0}}^2{q_0} +2\left( 1 - 3\xi \right) {k_0}{{q_0}}^2 \Big) -\nonumber\\
&&\left(\xi-1 \right){{k_0}}^2{\left( 2x{q_0} + {{k_0}}^2{q_0} + {k_0}\left( q^2 + 3{{q_0}}^2 \right) \right) }^2 +k^2\Big( 2\xi {{k_0}}^6 + 4\xi {{k_0}}^5{q_0}+ 2x^2\xi {{q_0}}^2 + \left( 3 - 5\xi \right)\nonumber\\
&&\cdot{{k_0}}^4{{q_0}}^2 -2{{k_0}}^3{q_0}\left( q^2\left(4\xi-2 \right) + \left(9\xi-5 \right) {{q_0}}^2 \right)+{{k_0}}^2\Big( -\left( q^4\left(\xi-1 \right) \right) - 2q^2\left(7\xi-5 \right) {{q_0}}^2\nonumber\\
&&+ \left( 9 - 11\xi \right) {{q_0}}^4 \Big)+2{k_0}{q_0}\left( -\left( q^4\left( -1 + \xi \right) \right) + 2q^2{{q_0}}^2 + \left( 1 + \xi \right) {{q_0}}^4 \right) \Big) \Big) -2kq\cos\theta\nonumber\\
&&\cdot\Big( k^6\left(3\xi-1 \right)\cdot{k_0}{q_0} +k^4\Big( \left( -1 + 5\xi \right) {{k_0}}^3{q_0} - 2x\xi {{q_0}}^2 +{k_0}{q_0}\Big( q^2\left(7\xi-3 \right) + 2\left(\xi-1 \right){{q_0}}^2 \Big)\nonumber\\
&&+{{k_0}}^2\left( q^2\left( 2 - 6\xi \right) + 4\left( -1 + 2\xi \right) {{q_0}}^2 \right) \Big) -k_0\Big( -2q^2x^2\left( -1 + 2\xi \right) {q_0}+2{{k_0}}^4{q_0}\Big( q^2\left( 1 + \xi \right)\nonumber\\
&&- 3\left( -1 + \xi \right) {{q_0}}^2 \Big) +{{k_0}}^5\left( 2q^2\xi - \left( -1 + \xi \right) {{q_0}}^2 \right) +{{k_0}}^3\Big( q^4\left( 1 + \xi \right) + q^2\left( 13 - 11\xi \right) {{q_0}}^2\nonumber\\
&&- 13\left(\xi-1 \right) {{q_0}}^4 \Big) -x{k_0}\Big( q^4\left(\xi-1 \right)+ q^2\left(17\xi-13 \right) {{q_0}}^2 + 4\left(\xi-1 \right) {{q_0}}^4 \Big) -2{{k_0}}^2{q_0}\Big( 4q^4\nonumber\\
&&\cdot\left(\xi-1 \right) + q^2\left(14\xi-13 \right){{q_0}}^2 + 6\left(\xi-1 \right) {{q_0}}^4 \Big)\Big) + k^2\Big( 2\xi {{k_0}}^5{q_0} +{{k_0}}^3{q_0}\Big( q^2\left(\xi-3 \right) + 2\nonumber\\
&&\cdot\left(6\xi-5 \right){{q_0}}^2 \Big) +{{k_0}}^4\left( q^2\left( 1 - 7\xi \right) + \left( -3 + 7\xi \right) {{q_0}}^2 \right) -2\xi {{q_0}}^2\left( 2q^4 + 3q^2{{q_0}}^2 + {{q_0}}^4 \right)-{{k_0}}^2\nonumber\\
&&\cdot\left( q^4\left( 1 + \xi \right) - 4q^2\left( -4 + 5\xi \right) {{q_0}}^2 + \left( 9 - 7\xi \right) {{q_0}}^4 \right) -2{k_0}{q_0}\Big( q^4\left( 3 - 5\xi \right)+ q^2\left( 4 - 3\xi \right) {{q_0}}^2\nonumber\\
&&+ \left( 1 + \xi \right) {{q_0}}^4 \Big) \Big) \Big)+ {\cos^2\theta}\Big( 2k^8\xi {{q_0}}^2 + 2q^4\xi {{k_0}}^2\Big( x + {{k_0}}^2+ {k_0}{q_0} \Big)^2 +2k^6{q_0}\Big( \left( 1 + \xi \right) {{k_0}}^2{q_0}\nonumber\\
&&+ 2\xi {q_0}\left( 2q^2 + {{q_0}}^2 \right) +{k_0}\Big( q^2\left( 6 - 14\xi \right)+ \left( 1 + \xi \right) {{q_0}}^2 \Big) \Big) +k^4\Big( -\left( \left( -3 + \xi \right) {{k_0}}^4{{q_0}}^2 \right)\nonumber\\
&&-2{{k_0}}^3{q_0}\Big( q^2\left(17\xi-5 \right)+ \left(3\xi-5 \right) {{q_0}}^2 \Big) +2\xi {{q_0}}^2\left( 8q^4 + 8q^2{{q_0}}^2 + {{q_0}}^4 \right) +{{k_0}}^2\Big( 4q^4\left(5\xi-1 \right)\nonumber\\
&&+ 2q^2\left( 19 - 23\xi \right) {{q_0}}^2 - 3\left( -3 + \xi \right) {{q_0}}^4 \Big) +2{k_0}{q_0}\Big( q^4\left( 12 - 22\xi \right) + 2q^2
\cdot\left( 5 - 4\xi \right) {{q_0}}^2\nonumber\\
&&+ \left( 1 + \xi \right) {{q_0}}^4 \Big) \Big) +k^2{k_0}\Big( -\left( \left( -1 + \xi \right) {{k_0}}^5{{q_0}}^2 \right) +2{{k_0}}^4{q_0}\Big( q^2\left( 1 - 5\xi \right) - 3\left( -1 + \xi \right) {{q_0}}^2 \Big)\nonumber\\
&&-8q^2\left(2\xi-1 \right) {q_0}\left( 2q^4 + 3q^2{{q_0}}^2 + {{q_0}}^4 \right)+{{k_0}}^3\left( 16q^4\xi + 2q^2\left( 11 - 15\xi \right) {{q_0}}^2 - 13\left(\xi-1 \right) {{q_0}}^4 \right)\nonumber\\
&&-4{{k_0}}^2{q_0}\Big( q^4\Big( -5+ 4\xi \Big) + q^2\left( -14 + 15\xi \right) {{q_0}}^2 + 3\left( -1 + \xi \right) {{q_0}}^4 \Big) + 4{k_0}\Big( q^6\left( 1 + \xi \right) - 2q^4\nonumber\\
&&\cdot\left( -7 + 8\xi \right) {{q_0}}^2 + q^2\left( 11 - 13\xi \right) {{q_0}}^4 -\left( -1 + \xi \right) {{q_0}}^6 \Big) \Big) \Big) \Big). \nonumber
\end{eqnarray}
The vertex dressing in the soft-soft $M_T$-kernel is performed as in the 3d-case.
\chapter{Numerical Method}\label{anum}
In order to numerically solve the DSEs, the method described in \cite{Fischer:2002hn,Fischer:2003zc,Atkinson:1998zc,Hauck:1998fz} was extended and improved in detail. The dressing functions were split into three parts. For sufficiently small momenta of the order of $10^{-2}g_3^2$, they are replaced by their infrared behavior \prefr{iransatz1}{iransatz}. For sufficiently large momenta, of the order $10^3g_3^2$, they are replaced by their leading order perturbative form due to \pref{apertgh}, \pref{apertgllead} and \pref{aperthlead}. In the remaining part, the dressing functions were separated into an analytic factor, which interpolates between the analytic infrared and ultraviolet behavior, and a modification function. The logarithm of this modification function is then expanded in Chebychef polynomials. This factorization is not necessary for the $g=1/2$-solution, albeit it lowers the needed CPU time significantly. However, it was not possible to find the other solution branch prior to this improvement. The analytic interpolation functions are
\begin{eqnarray}
G(q)&=&1+\frac{g_3^{4g}+q^{2g}}{q^{2g}}\frac{1}{g_3^2+q}\label{anghost}\\
Z(q)&=&\frac{q^{-2t}}{\frac{2}{g_3^{2t}}+q^{-2t}}\left(1+\frac{g_3^2}{g_3^2+q}\right)\label{angluon}\\
H(q)&=&\frac{q^{-2l}}{m_h^{-2l}+q^{-2l}}\label{anhiggs}.
\end{eqnarray}
The coefficients of the Chebychef polynomials are determined using a global rather than a local Newton method \cite{Kelley}. This made it possible to solve the equations without any further prior knowledge about the solution. For a solution, it is required that relations \pref{ag} and \pref{ah} are fulfilled at least up to $10^{-3}$ besides fulfilling the system of equations \prefr{fulleqG3d}{fulleqZ3d} to a much higher precision.
It turns out that simultaneous fits of the Chebychef coefficients and $A_g$, $A_z$, and $A_h$ are not sufficiently stable. Hence an iterated procedure has been used where first a global Newton method was employed and then new infrared coefficients were extracted from the requirement that the functions have to be continuous in the infrared. This was iterated until convergence of the infrared coefficients was achieved. A detailed account of the method will be given in \cite{Maas:ccp}.
To add the hard modes posed another problem. Due to the recursive nature of the subtraction prescription \pref{finite} of the spurious divergences, the Matsubara sum easily switches from logarithmic to quadratic divergence. The only way found up to now to solve this problem is to recalculate the renormalization constants in each Newton iteration step. By this in principle each Newton step again starts as a first step, but already with quite good starting values. This is sufficient to stabilize the algorithm and provides the solutions found in chapter \ref{cft}.
\chapter{Conclusions, Summary, and Outlook}\label{csummary}
\section{Concluding Remarks}
Before discussing the implications of the results found in chapters \ref{c3d}, \ref{cft}, and \ref{cderived} on the structure of the Yang-Mills (and QCD) phase diagram, it is worthwhile to take a step back to gather and assess the findings. Especially the reliability of the results deserves special attention.
The primary objects investigated in chapters \ref{c3d} and \ref{cft} are the propagators of Yang-Mills theory in the high temperature phase. These can be grouped into three classes. One consists of the soft modes of the ghost and the 3d-transverse gluon, constituting the Yang-Mills sector of the limiting 3d-theory. The second is the soft 3d-longitudinal gluon, corresponding to the Higgs sector of the 3d-theory. From the point of view of the 4d-theory, these are the soft chromomagnetic and chromoelectric degrees of freedom, respectively. The last class contains all the hard modes.
The first class on its own constitutes a 3d Yang-Mills theory. Its properties are found to be very similar to the corresponding 4d-theory. An infrared divergent ghost propagator is accompanied by an infrared vanishing gluon propagator. Therefore the gluon is confined and the results fulfill the Kugo-Ojima and Zwanziger-Gribov scenarios. Also the Schwinger function and analytic structure of the gluon propagator are similar to the 4d case.
The systematic study of the errors due to the truncation find a very weak quantitative effect only. Especially the confining properties are extremely robust against variations in the truncation. Finally the comparison with lattice calculations on exceptionally large lattices shows a very good agreement. Thus it is very likely that these results on a 3d Yang-Mills theory are reliable.
The situation is more subtle concerning the Higgs in the 3d-theory. Its propagator shows the typical behavior of a massive particle and it is dominated by its mass. However, the comparison to lattice data indicates that to some extent subleading or probably non-perturbative effects are relevant. This is much more pronounced when regarding the Schwinger function. Here significant changes are found when modifying the Higgs-gluon vertex. Thus the qualitative behavior of the Higgs propagator seems to be under control and the presence of non-trivial effects is quite certain. The consequences of these effects on the other side are far less conclusive and require further investigation. Especially the question whether the Higgs is confined is of great interest. On the other hand, a quite firm statement is that it has only little effect on the Yang-Mills sector.
Combining both classes, a qualitatively stable infinite temperature limit is found. Only the chromoelectric sector exhibits screening while at least in the chromomagnetic sector confinement prevails. Therefore the infinite temperature limit is non-trivial and the high temperature phase strongly interacting. {\it Hence the simple picture described in the introduction in section \ref{stdqcd} is not applicable. This and the detailed analysis of the infinite temperature limit are the central results of this work.}
Less reliable is the exploratory study of finite temperature effects in chapter \ref{cft} and thus the properties of the hard modes. The success of the truncation scheme in the 3d-theory is based on its exactness in the ultraviolet and its presumed exactness in the infrared. Thus the consequences of the deficiencies at intermediate momenta are strongly constrained. This possibility is lost by cutting off the Matsubara sum. As a consequence the ultraviolet properties of the hard modes are incorrect and a rooting in perturbation theory is prevented. This amplifies the second problem. The hard modes do not reach into the 4d-infrared due to their effective mass. Therefore the advantages of the truncation scheme are lost to a large extent and the equations for the hard modes become unreliable.
Nonetheless, the results found are not irrelevant. At sufficiently large temperatures the truncation-dependent self-energies of the hard modes are suppressed by the large effective mass and they are dominated by the truncation-independent tree-level terms. Therefore the truncation artifacts vanish when the temperature goes to infinity, reproducing the reliable 3d-theory. The hard part is then to assess down to which temperature the results are still at least qualitatively reliable.
As it is found that the system is only very weakly dependent on temperature, qualitative conclusions can probably be drawn even at quite small temperatures. This is due to the fact that the infrared properties of the soft modes are nearly independent of the hard modes. In general the hard modes effectively decouple, because of their effective mass, which is large compared to $\Lambda_{QCD}$ even at the phase transition. The only exception is the generation of the screening mass for the soft 3d-longitudinal gluon, which is found on quite general grounds and thus can be considered also a reliable conclusion. It is also supported by the systematic error estimations performed, which do not find qualitative effects but again only small quantitative ones.
Thus, although the quantitative results may be subject to change, the qualitative result that, {\it probably down to temperatures within the same order of magnitude as the phase transition, the high temperature phase consists of strongly interacting soft modes exhibiting confining properties and inert hard modes} is expected to be quite reliable. This is also supported by lattice results, which find that the infinite-temperature limit is effectively reached at about $2T_c$ \cite{Cucchieri:2001tw} and the qualitative features of the infinite temperature limit up to the highest temperatures, for which propagators are available, of $6T_c$ \cite{Nakamura:2003pu}. {\it This constitutes the second major result found in this work.}
Concerning as the last point the thermodynamic potential, no final conclusion can be drawn. The results indicate that the gross thermodynamic properties far away from the phase transition are completely dominated by the hard modes and that a Stefan-Boltzmann behavior is reached in the infinite temperature limit. Considering the crude assumptions made and the difficulties encountered, this result is indicative at best. Nonetheless, if it can be substantiated, it would allow to understand how a Stefan-Boltzmann-like behavior can emerge from a non-trivial theory. At the same time, the results also indicate that at low temperatures non-perturbative effects likely will play a role in the thermodynamics and will especially be relevant in the vicinity of the phase transition and thus to experiment.
\section{Implications for the Phase Structure of QCD}\label{ssumprop}
Gathering the results found in chapters \ref{c3d}, \ref{cft}, and \ref{cderived} and the solutions at zero and small temperatures presented in sections \ref{ssolvac} and \ref{sdselowt}, a coherent picture emerges. Still, this picture is overshadowed by the problems of truncation artifacts, especially at finite temperatures.
Combining these results has impact on the understanding of the phase transition. The main difference between the low-temperature and the high-temperature phase is not primarily one between a strongly interacting and confining system and one with only quasi-free particles. The chromoelectric gluons, whose infrared behavior change from over-screening to screening, come somewhat close to such a picture, and the hard modes certainly do. The latter become free due to the vacuum property of asymptotic freedom and the dependence of their energy on the temperature, thus in an expected although not entirely trivial manner. The chromomagnetic gluons remain over-screened in the infrared and are thus confined. At least in the transverse sector, the results provide strong evidence for the Zwanziger-Gribov and/or the Kugo-Ojima scenarios.
The order parameter for the phase transition is then necessarily only a chromoelectric one. This is in agreement with studies of Wilson loops in lattice calculations. There it is known that only the temporal (chromoelectric) Wilson lines show a behavior typical of deconfinement, while the spatial (chromomagnetic) ones do not~\cite{Bali:1993tz}. Note that the order parameters used to study the deconfinement transition on the lattice are typically chromoelectric ones like the Polyakov lines~\cite{Karsch:2003jg}. This leads to the conjecture that if the corresponding chromomagnetic Polyakov lines were to be used almost no change would be found.
Including the observation of super-heating in the low temperature regime \cite{Gruter:2004kd} and super-cooling in the high temperature regime, the scenario emerging is a chromoelectric phase transition of first order. This phase transition connects two different strongly interacting phases of Yang-Mills theory. Especially quantum fluctuations always dominate thermal fluctuations and at least part of the gluon spectrum is always confined.
In the vicinity of the phase transition, the non-perturbative effects found are likely also relevant to thermodynamic properties, especially to the pressure and the trace anomaly \cite{Zwanziger:2004np}. This underlines the importance of non-perturbative effects at least for the temperature range relevant to experiment.
It is still unknown which effects can be induced by quarks. In the infinite temperature limit, arguments exist \cite{Appelquist:vg} that they will not contribute due to their fermionic nature. This nature requires anti-periodic boundary conditions \cite{Kapusta:tk}, and therefore generates effective `masses' of $(2n+1)\pi T$ with $n$ integer. Thus all quarks become infinitely massive in the infinite temperature limit, irrespective of their intrinsic mass and thus decouple due to asymptotic freedom. Lattice calculations indicate \cite{Karsch:2003jg} that the quarks possibly are able to change the nature of the phase transition up to a cross-over. It is hard to imagine how such a partially confined phase would behave, and thus the topic is of high interest.
In addition, the observation of a drastic change in the inter-quark potential \cite{Kaczmarek:2004gv} and its connection to the residual confinement of gluons still needs to be understood. However, it cannot yet be firmly concluded that the inter-quark potential does not rise only logarithmically at high temperatures, as it would be natural to expect from a dimensionally reduced theory. On the other hand, quark confinement is still not understood at zero temperature, thus posing a more significant task.
\section{Summing Up and Looking Ahead}
Summarizing, in this work evidence is presented in agreement with lattice calculations that the finite temperature phase transition of Yang-Mills theory does not lead into a trivial phase. By investigating first the dimensionally reduced theory, thereby providing results on genuine 3d-theories of Yang-Mills and Yang-Mills-Higgs type, the non-triviality of soft interactions was found. Extending the range to finite temperatures, this was confirmed and the decoupling of the hard modes explicitly demonstrated. This establishes the main features of the high temperature phase of Yang-Mills theories in Landau gauge. Furthermore, using the Schwinger functions, the absence of at least part of the gluons from the physical spectrum was found, making manifest the existence of residual confinement. The results found comply with Zwanziger-Gribov or Kugo-Ojima type confinement mechanisms. As a final point, a first step towards thermodynamic observables was performed.
Taking a look ahead, two issues are of high interest. One from an experimental and one from a conceptual point of view.
In heavy-ion collisions, not only finite temperature is of interest, but especially for low-energy collisions results at finite density are of importance. Such experiments are performed at the SPS at CERN and have been performed at the AGS at Brookhaven and will also be conducted at the planned FAIR facility at GSI. Finite density is also relevant for the physics of compact stellar objects as well. For sufficiently large chemical potentials (densities), at least the quark sector will likely be weakly interacting, although non-perturbatively, see e.g.\/ \cite{Rischke:2003mt}. However, it has not yet been convincingly demonstrated how the quarks can generate gluon deconfinement. In addition, those chemical potentials probably only occur in nature during black hole formation. For chemical potentials relevant to experiments and possibly neutron stars, strong non-perturbative effects are expected. Thus a framework as the one used previously for full QCD \cite{Fischer:2003rp} seems to be necessary. First steps in this direction have been performed \cite{Epple:dipl} and are pursued further, they, however, are not yet as advanced as the finite temperature calculations presented here. More results are available from studies using model gluon propagators \cite{Roberts:2000aa} and semi-perturbative methods \cite{Rischke:2003mt}.
On the conceptual sides the problems encountered in chapter \ref{cft} to obtain the vacuum solution indicate that it will probably be necessary to choose other means to investigate the phase transition directly, e.g.\/ real-time methods \cite{Das:gg}. In addition, the inclusion of quarks is an urgent problem as their impact on thermodynamic properties is highly relevant for experiments, including RHIC at Brookhaven and ALICE, which is under construction at the LHC at CERN. In addition, understanding of quark confinement in the vacuum might be supported by understanding the high temperature behavior of quarks and the phase transition. This would also lead to a deeper understanding of the relationship of chiral symmetry and confinement of fundamental quarks, for which evidence exists. Indeed, their phase transitions seem to be interlinked \cite{Karsch:2003jg}. This immediately leads to the problem of the connection to topological excitations, which lattice calculations indicate to play a dominant role in the physics of confinement and chiral symmetry breaking. If they are relevant, these excitations must also be present in the current approach, as their consequences are. Therefore, it would be desirable to disentangle their contributions and identify their role. It is at this point also completely unclear, how their dynamics change above the phase transition and their connection to the at least partial confinement of gluons.
To reach these aims, implementation of new methods and close cooperation with other approaches, especially lattice gauge theory, will be necessary. Nevertheless, over the last decade significant progress has been made in the understanding of low energy QCD, providing hope that it will eventually be understood.
\chapter{Derivation of the Dyson-Schwinger Equations}\label{cdse}
Knowledge of all of the Green's functions would grant complete knowledge of a theory \cite{Haag:1992hx}. The inverse 2-point Green's functions are the propagators. These are the main objects of interest here, as they carry information concerning confinement and other non-perturbative properties. The equations determining these Green's functions are the Dyson-Schwinger equations \cite{Dyson:1949ha} (DSEs), which can be obtained using the functional equations of motion. This will be described in section \ref{sdsevac}. Since the aim of this work is to investigate the equilibrium high temperature phase, temperature will be introduced into the DSEs in section \ref{sdseft}. As there are an infinite number of coupled DSEs, it is generally not possible to solve these simultaneously. To obtain approximate solutions, truncations are necessary and these are discussed in section \ref{strunc}. The connections to perturbation theory and renormalization are investigated in sections \ref{sperturb} and \ref{srenormalization}.
This work is based on a calculational scheme which has been applied successfully in the vacuum to both Yang-Mills theory and full QCD, as will be described in section \ref{ssolvac}.
\section{Vacuum Formulation}\label{sdsevac}
The most straightforward way to derive the DSEs for a generic field $\phi^a$ is by using the fact that the integral of a total derivative vanishes \cite{Alkofer:2000wg}
\begin{eqnarray}
0&=&\int{\cal D}\phi^a\frac{\delta}{\delta\phi^a(y)}e^{-S+\int d^dx\phi^a(x)j^a(x)}\nonumber\\
S&=&\int d^dx{\cal L}\nonumber.
\end{eqnarray}
\noindent $S$ is the action, $j^a$ is the source of $\phi^a$ and the integral is over full field space. Performing the derivative and pulling the resultant factor out of the integral by replacing $\phi^a$ with $\delta/\delta j^a$, the prescription to calculate the full one-point Green's function is obtained as
\begin{equation}
\left(\left(-\frac{\delta S}{\delta \phi^a(x)}\Big|_{\phi^a(x)=\frac{\delta}{\delta j^a(x)}}+j^a(x)\right)Z[j^a]\right)_{j^a=0}=0.\label{dse}
\end{equation}
\noindent Further derivatives with respect to the fields generate the Green's functions of arbitrarily high order. They form an infinite set of coupled non-linear integral equations. For Yang-Mills theory and full QCD, these are known for the propagators, see e.g. \cite{Alkofer:2000wg}. For the Lagrangian \pref{l3d} these are derived for all 2-point Green's functions in appendix \ref{adse}. A graphical representation of these is given in figure \ref{figfullsys}.
\begin{figure}
\epsfig{file=fullsys.eps,width=0.9\linewidth}
\caption{The Dyson-Schwinger equations for the 2-point functions described by the Lagrangian \pref{l3d}. Dotted lines are ghosts, wiggly lines are gluons and dashed lines are Higgs. Lines with a dot are full propagators, vertices with a small black dot are bare vertices and white circles are full 3- and 4-point functions.}
\label{figfullsys}
\end{figure}
In the case of Yang-Mills theory or QCD, it is not necessary to integrate over all field-space for Green's functions. As the Faddeev-Popov determinant \pref{fpdet} appears explicitly in \pref{lpart}, it is sufficient to integrate only over the first Gribov horizon, since the determinant vanishes on its boundary \cite{Zwanziger:2003cf}. Hence the DSEs have the same form whether the integration is over all of gauge-space or only over the first Gribov horizon. Therefore it is necessary to restrict the space of solutions to those from inside the first Gribov horizon. This is discussed in section \ref{strunc}.
\section{Dyson-Schwinger Equations at Finite Temperature}\label{sdseft}
The starting point of the analysis are the DSEs of Yang-Mills theory described by \pref{lym} with the quark fields set to zero. Besides neglecting all equations for $n$-point Green's functions with $n>2$, all genuine full two-loop graphs in figure \ref{figfullsys} are neglected, too. The consequences of this and further truncations will be discussed in detail in section \ref{strunc}. The remaining system is then represented in figure \ref{figt0sys} and given by
\begin{eqnarray}
D^{ab-1}_G(p)&=&-\delta^{ab}p^2\nonumber\\
&+&\int\frac{d^dq}{(2\pi)^d}\Gamma_\mu^{\mathrm{tl}, c\bar cA, dae}(-q,p,q-p)D^{ef}_{\mu\nu}(p-q)D^{dg}_G(q)\Gamma^{c\bar cA, bgf}_\nu(-p,q,p-q)\label{vacgheq}\\
D^{ab-1}_{\mu\nu}(p)&=&\delta^{ab}(\delta_{\mu\nu}p^2-p_\mu p_\nu)+T^{GG}_{\mu\nu}\nonumber\\
&-&\int\frac{d^dq}{(2\pi)^d}\Gamma_\mu^{\mathrm{tl}, c\bar cA, dca}(-p-q,q,p) D_G^{cf}(q) D_G^{de}(p+q) \Gamma_\nu^{c\bar c A, feb}(-q,p+q,-p)\nonumber\\
&+&\frac{1}{2}\int\frac{d^dq}{(2\pi)^d}\Gamma^{\mathrm{tl}, A^3, acd}_{\mu\sigma\chi}(p,q-p,-q)D^{cf}_{\sigma\omega}(q)D^{de}_{\chi\lambda}(p-q)\Gamma^{A^3, bfe}_{\nu\omega\lambda}(-p,q,p-q).\nonumber\\\label{vacgleq}
\end{eqnarray}
\noindent Here $\Gamma^{c\bar cA}$ is the full ghost-gluon vertex and $\Gamma^{A^3}$ is the full three-gluon vertex. The respective tree-level quantities are denoted by a superscript `$\mathrm{tl}$' and given in \pref{tlcca} and \pref{tlggg}. $T^{GG}_{\mu\nu}$ is the tadpole term. The full vertices will be discussed further in section \ref{strunc}. Note that these are in general Minkowski-space equations, but equations \pref{vacgheq} and \pref{vacgleq} have already been rotated to Euclidean space.
\begin{figure}
\epsfig{file=t0.eps,width=\linewidth}
\caption{The truncated Dyson-Schwinger equations for the 2-point functions of Yang-Mills theory at $T=0$. Dotted lines are ghosts and wiggly lines are gluons. Lines with a dot are full propagators, vertices with a small black dot are bare vertices and white circles are full vertices.}
\label{figt0sys}
\end{figure}
To obtain the equilibrium Green's functions at a temperature $T$, the Matsubara or imaginary time formalism is implemented \cite{Kapusta:tk,Das:gg}. This amounts to a compactification of time and entails a genuine Euclidean formulation. All objects depend on the three-momenta $\vec p$ and the Fourier-component $p_0$ separately. Both ghosts and gluons have to obey periodic boundary conditions \cite{Bernard:1974bq}, thus
\begin{equation}
p_0=2\pi n T,\quad n\epsilon\mathbb{Z}.
\end{equation}
\noindent Hence in both cases a soft mode exists, defined by Euclidean $p^2\ll T$, i.e.\/ those with $p_0=0$ in contrast to the hard modes with $p_0\neq 0$. Note that although Lorentz invariance is no longer manifest in this formulation, it is not lost \cite{Weldon:aq}. Furthermore, as the gluon is a vector-particle, its propagator exhibits two independent tensor-structures and two independent dressing functions\footnote{In general, three dressing functions exist, but only two are independent due to the STI \pref{stigluon}.} \cite{Kapusta:tk} instead of one as in \pref{vacgluonprop},
\begin{eqnarray}
D_{\mu\nu}(p)&=&P_{T\mu\nu}(p)\frac{Z(p_0^2,\vec p^2)}{p^2}+P_{L\mu\nu}(p)\frac{H(p_0^2,\vec p^2)}{p^2}\label{gluonprop}\\
P_{T\mu \nu }(p)&=&\delta _{\mu \nu }-\frac{p_{\mu }p_{\nu }}{\vec p^{2}}+\delta _{\mu 0}\frac{p_{0}p_{\nu }}{\vec p^{2}}+\delta _{0\nu }\frac{p_{\mu }p_{0}}{\vec p^{2}}-\delta _{\mu 0}\delta _{0\nu }\left(1+\frac{p_{0}^{2}}{\vec p^{2}}\right)\label{tproj}\\
P_{L\mu \nu }(p)&=&P_{\mu \nu }(p)-P_{T\mu \nu }(p)\label{lproj}.
\end{eqnarray}
\noindent The projectors $P_{T\mu\nu}$ and $P_{L\mu\nu}$ are transverse and longitudinal with respect to the heat bath or alternatively with respect to the three spatial dimensions. Both are four-dimensionally transverse, so \pref{gluonprop} still satisfies the STI \pref{stigluon}. This is necessary, as STIs are still valid at finite temperature \cite{Das:gg}. For $T=0$, $Z=H$ and \pref{vacgluonprop} is recovered. Note that \pref{tproj} projects solely on the space-space components and the $p_0=0$-component of \pref{lproj} solely on the 00-component of the propagator. Hence the soft mode of $Z$ is purely chromomagnetic, while the one of $H$ is purely chromoelectric. For the ghost, being a scalar, one dressing function is sufficient at finite temperature.
In general, all full Green's functions obtain a much more complicated tensor structure at finite temperature, e.g.\/ the ghost-gluon vertex obtains four instead of two tensor structures. This generates a significant amount of technical problems. As the main assumption of the truncation scheme presented in section \ref{strunc} is the negligibility of dressings other than those of propagators, the irrelevance of such effects is assumed as well. Hence the tree-level tensor structure is used for all full vertices. The last issue to be addressed before writing down the equations is the one determining the gluon. It is a matrix equation, but only two components are independent. The most direct way to obtain two scalar equations for the two dressing functions is to contract the gluon equation once with \pref{tproj} and once with \pref{lproj}. The more convenient way is to contract with
\begin{eqnarray}
P_{T\mu\nu}^\zeta=\zeta P_{T\mu\nu}+(\zeta-1)A_{T\mu\nu}\label{gtproj}\\
P_{L\mu\nu}^\xi=\xi P_{L\mu\nu}+(\xi-1)A_{L\mu\nu}.\label{glproj}
\end{eqnarray}
\noindent Varying the parameters $\zeta$ and $\xi$ allows to investigate the amount of gauge invariance violation, as discussed in section \ref{strunc}. The tensors \pref{gtproj} and \pref{glproj} reduce to \pref{tproj} and \pref{lproj} at $\zeta=\xi=1$. The selection of $A_{T/L\mu\nu}$ is governed by practical aspects and will be different for the high temperature case and the finite temperature case. Thus writing down the final equations will have to await chapters \ref{c3d} and \ref{cft}, respectively. Graphically the equations for the three scalar functions $G$, $Z$ and $H$ at finite temperature are given in figure \ref{figftsys}.
\begin{figure}
\epsfig{file=ftsys.eps,width=\linewidth}
\caption{The truncated Dyson-Schwinger equations for the 2-point functions of Yang-Mills theory at $T\neq0$. Dotted lines are ghosts, wiggly lines are 3d-transverse gluons and dashed lines are 3d-longitudinal gluons. Lines with a dot are full propagators, vertices with a small black dot are bare vertices and white circles are full vertices.}
\label{figftsys}
\end{figure}
\section{Truncations and Constraints}\label{strunc}
As already indicated, the DSEs to be solved in the following chapters are truncated. Although in the following several arguments will be made to support the truncations, there is no method (yet) known which permits an a priori controlled truncation. In addition, no possibility is yet known to conserve local internal symmetries at least on the level of the truncation. Even for global symmetries this leads to enormous technical complications, see e.g.\/ \cite{vanHees:2001ik}. Furthermore any such truncation necessarily violates unitarity. Also no scheme is yet known which at the same time conserves energy and agrees with perturbation theory in the far ultraviolet. Hence it is currently only possible to assume that all these effects do not contribute significantly. This can only be done by comparing a posteriori either to experiments or different methods. As the objects treated here are gauge-variant quantities, only other calculational methods are available for comparison. Only lattice calculations have been performed in the region of interest up to now and only for SU$(N_c)$ gauge theories. It will turn out that the results agree remarkably well, considering the drastic assumptions made. By systematically assessing the error due to gauge invariance violation, it is found that the effects of the truncation are of a quantitative nature only. Hence the truncation scheme to be described seems to be applicable.
The most extreme truncation is the neglect of the equations of all $n$-point Green's functions with $n>2$. This truncation is well justified in the ultraviolet due to asymptotic freedom. In the infrared, assuming the correctness of the Zwanziger-Gribov scenario, this is also justified and the results support this assumption self-consistently. At mid-momenta, significant deviations are to be expected. Indeed this is the region where the largest deviation from lattice results will be found. The further truncation is to neglect all genuine full 2-loop contributions within the equations for the 2-point Green's functions. Largely the same arguments apply here: The 2-loop contributions are sub-dominant in the ultraviolet and most probably sub-leading in the infrared. The latter was checked and found to be correct for tree-level instead of full vertices \cite{Watson:phd}. In addition it was found that only considerable fine-tuning of the vertices allowed the 2-loop contributions to become as leading as the 1-loop contributions in the infrared \cite{Bloch:2003yu}.
The next assumption is that the color structure is the same as in perturbation theory. All investigations concerning this point have supported this assumption \cite{Boucaud:1998xi}. This also automatically entails the vanishing of all 1-point Green's functions due to the antisymmetry of the color structure of the vertices.
The last ingredient of the truncation is the construction of the remaining full vertices. The ghost-gluon-vertex will be kept at its tree-level form \pref{tlcca}, motivated by the Zwanziger-Gribov scenario. This is exact for vanishing incoming ghost momenta as discussed in subsection \ref{sssti}. A bare vertex is also supported by numerical studies in three and four dimensions \cite{schleifenbaum:diploma} and lattice calculations in four dimensions \cite{Cucchieri:2004sq}. Furthermore, at least in the vacuum, the qualitative nature of the infrared solution is independent of the detailed structure of the ghost-gluon vertex to a large extent \cite{Lerche:2002ep}. It has also been shown that, under weak assumptions, the qualitative infrared solution for the ghost is independent of the truncation \cite{Watson:2001yv}.
The various three-gluon vertices for 3d-longitudinal and 3d-transverse gluons are not fixed yet. These will be constructed by requiring minimal gauge invariance violation. This will be addressed in chapters \ref{c3d} and \ref{cft}, completing the truncation scheme.
Concerning the artifacts of the truncation, it is not possible to solve the problem of unitarity violation. Unitarity will always be violated as long as the full infinite system is not solved\footnote{In perturbation theory, these violations are of higher order in the expansion parameter and can thus be neglected. The same applies to the results here in the realm of applicability of perturbation theory, as the same results are obtained in this domain.}. The problem of energy conservation will be addressed in section \ref{std}.
The last point to be discussed is the violation of gauge invariance. There are two aspects to be treated. The first is the problem of Gribov copies. Due to the arguments given in section \ref{squant}, this can be resolved by requiring
\begin{equation}
G(p^2)\ge 0,\quad Z(p^2)\ge 0,\quad H(p^2)\ge 0,\label{gribov}
\end{equation}
\noindent since this guarantees to stay within the first Gribov horizon. Note that by condition \pref{gribov}, the ghost propagator is negative definite and cannot have a positive semidefinite spectral function.
The second problem is much harder to address. Even provided the Gribov problem is solved, gauge invariance is violated, as the STIs are no longer fulfilled. Indeed it is not even possible to test whether the STIs are fulfilled, as in general the STIs for an $n$-point Green's function contain contributions from $n'$-point Green's functions with $n'>n$. These are negligible in perturbation theory because they are of higher order in the expansion parameter, but this is not true in general in non-perturbative calculations. In principle it would be possible to test the STIs when truncating them to the same level as the Green's functions, but it turns out that this leads to inconsistencies, see e.g.\/ \cite{vonSmekal:1997is}. Nonetheless trying to construct vertices which as best as possible fulfill the truncated STIs, it is found that the results are only weakly affected compared to tree-level vertices \cite{vonSmekal:1997is}. This gives confidence that these violations are small.
There are two consequences of these gauge invariance violations. The first is the appearance of spurious divergences in cases where the degree of divergence is lowered by gauge invariance compared to naive power counting. This is the case for the gluon self-energy. These spurious divergences have to be removed. This will be done using the tadpole terms, which are not left as free parts of the equations, but are chosen to compensate such spurious divergences and other artifacts of the truncation, thus mimicking their role in perturbation theory.
The second consequence is the violation due to finite contributions of the STIs. In the remainder of the work, the main equation to test the amount of gauge invariance violation will be the STI for the gluon propagator, \pref{stigluon}. It is for this reason the projectors $\pref{gtproj}$ and $\pref{glproj}$ are used instead of $\pref{tproj}$ and $\pref{lproj}$. If \pref{stigluon} was exactly fulfilled, the results obtained would be independent of $\zeta$ and $\xi$. If the results do only weakly depend on these parameters, gauge invariance violations are most likely small \cite{Fischer:2002hn,Alkofer:2002ne,Fischer:2003zc}. The variational range for $\zeta$ and $\xi$ in the case of a violation cannot be extremely large, since otherwise the projection will be primarily on the gauge-violating longitudinal part and will not give rise to further useful information.
\section{Perturbation Theory}\label{sperturb}
The DSE approach followed here aims at the full 2-point Green's functions. Hence it is necessary that perturbative results are embedded in the final results. For sufficiently large momenta the Green's functions must reduce to their perturbative counterparts, due to the asymptotic freedom of Yang-Mills theories. As the DSEs are truncated at one-loop level, this reduction can only be correct to leading order (LO) in $g_d^2/p^{4-d}$, where $p$ is the momentum scale. Subleading contributions will necessarily deviate from perturbation theory. This also guarantees that the violation of gauge invariance will be not worse than in LO perturbation theory, and thus at least the same level of gauge invariance is achieved.
The high temperature limit in chapter \ref{c3d} will give an explicit example of this. In case of the finite temperature corrections in chapter \ref{cft}, the results are restricted to momenta of the order of $T$, see section \ref{ssmallpapprox}. Therefore, perturbation theory will only be reproduced if these momenta are already in the perturbative regime.
In the case of 4d vacuum calculations, agreement with LO resummed perturbation theory turns out to be a significant task and requires modifications of the three-gluon vertex \cite{vonSmekal:1997is,Fischer:2002hn}. Comparison to perturbation theory is at the current level of truncation the only possibility to compare to experiment, and thus an important constraint.
\section{Renormalization}\label{srenormalization}
As in perturbation theory, the usual ultraviolet divergences of Yang-Mills theory are encountered and must be regularized and renormalized \cite{Bohm:yx}. A wide variety of possibilities to deal with the divergences at the perturbative level exist, especially with dimensional regularization for gauge theories. However, most of these concepts, including the latter, are not applicable to non-perturbative calculations \cite{Collins:xc,Zinn-Justin:mi}.
Due to the lack of a symmetry conserving regularization scheme for DSEs which is of technically acceptable complexity, an alternative route is chosen. It is always possible to use gauge non-invariant regularization prescriptions if appropriate compensating counter-terms are chosen \cite{Collins:xc}. This approach will be employed in chapter \ref{cft}. There the effects of a non-gauge-invariant regularization will be absorbed into the tadpoles.
The remaining divergencies are then regularized and renormalized using counter-terms. In general a counter-term for e.g.\/ renormalizing the wave function is introduced in a Lagrangian by performing the replacement
\begin{equation}
{\cal L}=(\pd_{\mu}\phi)^2\to{\cal L}_R=(\pd_{\mu}\phi)^2+\delta Z(\pd_{\mu}\phi^2)=:Z(\pd_{\mu}\phi)^2.\nonumber
\end{equation}
\noindent Here $\phi$ is an arbitrary field and the counter-term $\delta Z$ has been chosen such as to cancel the divergences \cite{Collins:xc}. This guarantees multiplicative renormalizability even in the non-perturbative regime by explicitly constructing the wave-function renormalization constant $Z$. At finite temperature no new divergences arise compared to the vacuum divergence structure of the phase the theory is in \cite{Das:gg}. Nevertheless, the finite parts of the counter-terms may depend on temperature.
When treating finite temperature effects in chapter \ref{cft} a few subtleties concerning renormalization in the employed truncation scheme arise. These are treated in section \ref{scttrunc}.
\section{Solutions in the Vacuum}\label{ssolvac}
Solutions of the DSEs in the vacuum \cite{Alkofer:2000wg,vonSmekal:1997is,Fischer:2002hn} have already convincingly demonstrated the non-triviality of the infrared regime and also showed good agreement with lattice calculations. This supports the applicability of the truncation scheme. In this section these results will be described briefly to put this work in the appropriate context and demonstrate that the method is sufficiently stable for an extension to finite temperature.
\subsection{Yang-Mills Theory}
For the pure Yang-Mills sector, results have been obtained with increasing precision over time\footnote{The earliest attempt neglected the ghost contribution \cite{Mandelstam:1979xd}. This ``Mandelstam approximation'' leads to results contradicting recent lattice results and is therefore dismissed today.}. The DSE results satisfy \pref{oehme} and \pref{kugo} and thus exhibit manifest gluon confinement, in accordance with the Kugo-Ojima and Zwanziger-Gribov scenario \cite{Alkofer:2000wg,vonSmekal:1997is,Fischer:2002hn}. Also, the analytical structure has been understood to some extent \cite{Alkofer:2003jk}. The results \cite{Fischer:2002hn} are shown in figure \ref{figymvac} compared to lattice results \cite{Langfeld:2002dd,Bowman:2004jm}.
\begin{figure}
\epsfig{file=ymvacg.eps,width=0.5\linewidth}\epsfig{file=ymvacz.eps,width=0.5\linewidth}
\caption{The left panel shows the ghost dressing function and the right panel the gluon propagator in vacuum Yang-Mills theory \cite{Fischer:2002hn}. The results are compared to lattice data of the ghost dressing function \cite{Langfeld:2002dd} and the gluon propagator \cite{Bowman:2004jm}. The indicated errors are statistical only.}
\label{figymvac}
\end{figure}
The lattice points farthest in the infrared suffer from finite volume effects and are expected to bend down for larger lattice volumes. In 3d-calculations, where significantly larger lattices can be used, this is indeed the case \cite{Cucchieri:2003di} and will be seen when comparing to lattice results in section \ref{slattice3d}. Recently, the results have also been confirmed by exact renormalization group methods \cite{Gies:2002af}. These results give confidence that the method can be applied to the finite temperature case as well.
A further result of the vacuum studies is that the quantity
\begin{equation}
\alpha(\mu)=\alpha(s) G(s,\mu)^2 Z(s,\mu),\label{runningcoupling}
\end{equation}
\noindent where $\mu$ is the renormalization scale and $s$ the subtraction point, is a renormalization group invariant, and agrees with the running coupling in the perturbative regime. Although it is under debate what the non-perturbative definition of a coupling constant is, if any, it is possible to investigate this quantity. It does not exhibit a Landau pole and has an infrared fixed point of $\alpha(0)N_c=8.915$ \cite{Fischer:2002hn}. This result has also been confirmed by exact renormalization group calculations \cite{Gies:2002af}.
\subsection{Full QCD}
\begin{figure}
\epsfig{file=quarkz.eps,width=0.5\linewidth}\epsfig{file=quarkm.eps,width=0.5\linewidth}
\caption{The quark propagator in the chiral limit \cite{Fischer:2003rp} compared to lattice results using the asqtad action \cite{Bowman:2002kn}. The left panel shows the quark wave-function renormalization compared to lattice results. The right panel shows the quark mass function compared to chirally extrapolated lattice results. The indicated errors are statistical only.}
\label{figquarks}
\end{figure}
DSEs have also been applied to full QCD in two different ways. A more phenomenological ansatz has employed model gluon propagators. This has led to extensive and successful investigations of hadron phenomenology. For a review see e.g.\/ \cite{Alkofer:2000wg,Roberts:2000aa}. As the approach followed here is more bottom-up, this aspect will not be discussed further. The other approach couples the Yang-Mills sector described above to the quarks and solves the corresponding DSEs in a similar truncation scheme \cite{Fischer:2003rp}. The general structure of the Euclidean quark propagator is
\begin{equation}
S(p)=\frac{Z(p)}{-i\gamma_\mu p_\mu+M(p)}
\end{equation}
\noindent with the wave-function renormalization $Z$ and the mass function $M$. For less than 4 light or massless quarks (as it is realized in nature), the Yang-Mills solutions remain essentially the same, besides the according changes in the ultraviolet anomalous dimensions. Thus even the effect of massless quarks is small. The two independent dressing functions $Z$ and $M$ are shown in figure \ref{figquarks}. Chiral symmetry breaking is manifest. However, the amount of symmetry breaking is sensitive to the specific quark-gluon vertex, as the quark results are in general. In the present case a modified Curtis-Pennington vertex has been employed \cite{Fischer:2003rp}. Also, the problem of quark confinement is not yet understood in this ansatz, as the information obtainable from the quark propagator alone are not conclusive \cite{Alkofer:2003jk}.
\chapter{Aspects of QCD as a Gauge Theory}\label{ctheory}
\section{Formulation}
QCD describes the interaction of quarks, massive fermions, through gauge bosons, the gluons. The gauge group for the physical QCD is SU$(3)$. In this work, the scope is enlarged to an arbitrary semi-simple, compact Lie-group. However, only some of the assumptions made later have been tested by lattice calculations and just with respect to SU$(N_c)$ gauge groups at small $N_c$. Thus although the results will turn out to be independent of the gauge group, the assumptions made are potentially not.
The classical Lagrangian of such a general gauge theory is given by \cite{Bohm:yx}
\begin{eqnarray}
{\cal L}_m&=&-\frac{1}{4}F_{\mu\nu}^aF^{\mu\nu, a}+\sum_f\left(\bar\psi^\alpha_f\gamma^\mu iD_\mu\psi_f^\alpha-m_f\bar\psi_f^\alpha\psi_f^\alpha\right)\label{qcdlagrangianm}\\
F^a_{\mu\nu}&=&\pd_{\mu} A_\nu^a-\pd_{\nu} A_\mu^a+g_df^{abc}A_\mu^bA_\nu^c\nonumber\\
D_\mu&=&\pd_{\mu}-ig_dt^aA_\mu^a.\nonumber
\end{eqnarray}
\noindent $A_\mu^a$ denotes the gluon fields with color $a$ and Lorentz index $\mu$. $\bar\psi^\alpha_f$ and $\psi^\alpha_f$ are the anti-quark fields and quark fields, respectively, of flavor $f=1,...,N_f$ and color $\alpha$. For a SU$(N_c)$ gauge group $a=1,...,N_c^2-1$ and $\alpha=1,...,N_c$. $F_{\mu\nu}^a$ is the field strength tensor and $D_\mu$ the fundamental covariant derivative, $t^a$ are the generators of the gauge group and $f^{abc}$ its structure constants. The $m_f$ are the masses corresponding to the quark flavor $f$, and $g_d$ is the gauge coupling where $d$ is the space-time dimension. $\gamma^\mu$ denotes the Dirac $\gamma$-matrices \cite{Bohm:yx}.
The introduction of equilibrium thermodynamics in chapter \ref{cdse} includes a Wick rotation \cite{Kapusta:tk} and hence the Euclidean version of \pref{qcdlagrangianm} will be more important throughout. It is given by \cite{Alkofer:2000wg}
\begin{eqnarray}
{\cal L}&=&\frac{1}{4}F_{\mu\nu}^aF^{\mu\nu, a}-\sum_f\left(\bar\psi^\alpha_f\gamma^\mu D_\mu\psi_f^\alpha-m_f\bar\psi_f^\alpha\psi_f^\alpha\right)\label{lym}\\
F^a_{\mu\nu}&=&\pd_{\mu} A_\nu^a-\pd_{\nu} A_\mu^a-g_df^{abc}A_\mu^bA_\nu^c\nonumber\\
D_\mu&=&\pd_{\mu}+ig_dt^aA_\mu^a.\nonumber
\end{eqnarray}
\noindent All further expressions will be in Euclidean space-time, if not otherwise noted. In the remainder of this work only the gauge-subsector of \pref{lym} will be investigated and the matter content discarded. This reduced theory is known as Yang-Mills theory \cite{Bohm:yx,Yang:ek}. In section \ref{ssfermions} it will be discussed to which extent this is justified. In section \ref{ssumprop} a short assessment of the influence of quarks on the results presented will be given.
In chapter \ref{c3d}, the Yang-Mills sector will be coupled to an adjoint scalar field. For the sake of completeness, its Lagrangian is given here in Euclidean space as
\begin{eqnarray}
{\cal L}&=&\frac{1}{4}F_{\mu\nu}^aF^{\mu\nu, a}+\frac{1}{2}(D_\mu^{ab}\phi^b D_\mu^{ac}\phi^c+m_h^2\phi^a\phi^a)+\frac{h}{4}\phi^a\phi^a\phi^b\phi^b\label{l3d}\\
D_\mu^{ab}&=&\delta^{ab}\pd_{\mu}+g_df^{abc}A_\mu^c.\nonumber
\end{eqnarray}
\noindent $\phi^a$ is the scalar field of color a, $m_h$ its mass and $h$ its self-coupling. For an SU$(N_c)$ gauge group $a=1,...,N_c^2-1$. $D_\mu^{ab}$ is the adjoint covariant derivative. A three-Higgs coupling is not present due to the antisymmetry of the coupling constants and the color symmetry at tree-level.
The next step is to quantize the Lagrangians \pref{lym} and \pref{l3d}.
\subsection{Quantization, Gauge Fixing and the Gribov Problem}\label{squant}
Quantization can be performed using either canonical or path integral methods. The latter will be used here, because they give a somewhat more sophisticated access \cite{Rivers:hi}. In this case, the generating functional is given by
\begin{equation}
Z[j_\mu^a]=\int{\cal D}A_\mu^a\exp\left(\int d^dx\left(-{\cal L}+A_\mu^a(x)j^{\mu, a}(x)\right)\right)=:e^{W[j_\mu^a]},\label{part}
\end{equation}
\noindent which depends on the classical sources $j_\mu^a$ for the gauge field; the 'free energy' $W$ is the generating functional of connected correlation functions. The integration is over the set of all configurations of the gluon field $A_\mu^a$. This set is termed gauge space. The classical sources have to be set to 0 at the end of all calculations.
The prescription \pref{part} has the problem that not all possible fields $A_\mu^a$ contribute independently to the path integral, since Yang-Mills theory is a gauge theory. ${\cal L}$ is invariant under local gauge transformations
\begin{eqnarray}
A_\mu^a&\to& A_\mu^a+D_\mu^{ab}\delta\theta^b\label{gtrans1}\\
\phi^a&\to&\phi^a+g_df^{abc}\phi^c\delta\theta^b,\label{gtrans2}
\end{eqnarray}
\noindent where the second transformation is only relevant for \pref{l3d}. $\delta\theta^a$ are independent functions. Field configurations, which are equivalent up to a gauge transformation \pref{gtrans1} or \pref{gtrans2} are said to lie on the same gauge orbit. This gauge freedom leads to over-counting in the path integral \pref{part}. It is hence necessary to take the gauge freedom into account when quantizing.
The most powerful, albeit technically complicated method to circumvent these problems is stochastic quantization \cite{Damgaard:1987rr}. It is in generally well suited to calculate gauge invariant quantities, including all observables. Furthermore it is possible to obtain gauge dependent objects like gluon propagators corresponding to a conventional gauge choice \cite{Rivers:hi,Zwanziger:1981kg}. For Landau gauge, this process yields the same equations describing the gluons \cite{Zwanziger:2003cf} as the approach followed here within the approximation scheme employed.
The conventional approach followed here is to fix the gauge prior to quantization \cite{Rivers:hi}. The aim is to include only one configuration on each gauge orbit. This is performed by introducing an appropriate $\delta$-function in the path integral \pref{part}. The argument of this function is a local functional of the fields of the form $C^a[A_\mu^b,x]$, and its equality to 0 defines the gauge. The remaining steps are a standard procedure and will not be detailed further \cite{Bohm:yx}.
There is still one point to be mentioned. The gauge fixing usually used is an algebraic or differential condition, such as
\begin{equation}
\pd_{\mu} A_\mu^a=0\label{lgauge}
\end{equation}
\noindent for Landau gauge or
\begin{equation}
\vec\nabla \vec A^a=0\label{cgauge}
\end{equation}
\noindent for Coulomb gauge. Evidently, \pref{lgauge} and \pref{cgauge} are not complete gauge-fixings. Any harmonic gauge transformation in \pref{lgauge} and any transformation depending only on time in \pref{cgauge} are still allowed. It is possible to fix this residual classical gauge freedom. However, the conditions \pref{lgauge} and \pref{cgauge} are still not unique. In non-abelian gauge theories there is more than one gauge-equivalent solution to them, i.e. more than one configuration on each gauge orbit satisfies the gauge condition. E.g.\/ in the case of Coulomb gauge \pref{cgauge} a gauge-equivalent solution to the vacuum $A_\mu^a=0$ is a hedgehog configuration \cite{Gribov:1977wm}. The consequence is over-counting in \pref{part}. This is known as the Gribov problem \cite{Gribov:1977wm} and it can be shown to extend to a large class of local gauges \cite{Singer:dk}. As the additional copies in general involve large gauge field fluctuations, this problem does not appear in perturbation theory.
It is not yet known whether there exists any local gauge condition which resolves this problem\footnote{There are arguments that Landau gauge is still well-defined, if a summation is made over all signed Gribov copies \cite{Hirschfeld:yq}. It is however probable that the truncations introduced in section \ref{strunc} will interfere with these cancellations. Thus other techniques are necessary here.}. With regard to the problems induced by non-local gauge conditions, it is worthwhile to investigate if there is a way to circumvent this problem. Indeed it has been shown that the zeros of the Faddeev-Popov determinant
\begin{equation}
M=\det(-\partial_\mu D_\mu^{ab})\label{fpdet}
\end{equation}
\noindent define non-intersecting convex and compact regions of gauge-space. Each region is intersected at least once by each gauge orbit. The one enclosing the origin and thus perturbation theory is called the first Gribov horizon \cite{Gribov:1977wm,Singer:dk}. Within it, \pref{fpdet} is positive. It is possible in the approach followed here to ensure $M>0$ and thus to be inside the first Gribov horizon. This will be discussed in section \ref{strunc}. However, this condition is not sufficient \cite{vanBaal:1997gu}, and gauge copies are still present. The copy-free region contained inside the first Gribov horizon, called the fundamental region, can be defined using a non-local minimalization condition \cite{Zwanziger:1993dh}. This in turn implies that there is again no local condition like the Gribov horizon to eliminate the copies, and a full solution is still missing.
Nevertheless, it has been argued that a bounded volume in an infinite dimensional space, like gauge space, is dominated by its boundary. Therefore only this region contributes to objects constructed from a finite number of operators\footnote{So the following argument does not necessarily apply to the Polyakov loop or other exponentials of operators.} \cite{Zwanziger:2003cf}. Using this entropy argument, the common boundary of the fundamental region and the first Gribov horizon is the only region contributing and the Gribov horizon condition is sufficient. For the remaining part of this work, this argument will be accepted as an assumption and hence the Gribov horizon condition will be used to restrict the space of possible solutions.
The impact of Gribov copies can be studied by lattice calculations. A relatively weak effect on the gluon propagator and a stronger quantitative effect on the ghost propagator, to be discussed shortly, is found \cite{Cucchieri:1997ns}. It is therefore likely that even if the assumption is incorrect, the results found here are still qualitatively reliable.
\subsection{Landau Gauge}
Most calculations based on perturbation theory restrict the gauge condition as weakly as possible and use the requirement of gauge independence to check the results for errors. In a well-defined scheme such as perturbation theory, this is clearly advantageous. In the case of non-perturbative calculations no such rigorous scheme exists up to now. Hence, the requirement of approximate gauge invariance will actually be used to investigate the quality of the solution. To this end, Landau gauge \pref{lgauge} turns out to be well suited for reasons to be described shortly.
Landau gauge belongs to the class of covariant gauges. Using the standard prescription for gauge fixing \cite{Bohm:yx}, the generating functional \pref{part} becomes
\begin{equation}
Z[j_\mu^a]=\int{\cal D}A_\mu^a M\exp\left(-\int d^dx\left({\cal L}+A_\mu^a(x)j^{\mu, a}(x)-\frac{1}{2\xi_g}\pd_{\mu} A_\mu^a(x)\partial^\mu A_\mu^a(x)\right)\right),\label{lpart}
\end{equation}
\noindent where $\xi_g$ is an arbitrary constant, the gauge constant. Landau gauge is obtained by the limit $\xi_g\to 0$, in which case the last term in \pref{lpart} is dismissed. The Faddeev-Popov determinant \pref{fpdet} emerges from an intermediate change of variables as a Jacobian. Faddeev and Popov showed \cite{Faddeev:fc} that it can be rewritten as a path integral over scalar Grassmann fields, leading in Landau gauge to
\begin{equation}
Z[j_\mu^a]=\int{\cal D}A{\cal D}c{\cal D}\bar c\exp\left(-\int d^dx\left({\cal L}+\bar c^a(x)\pd_{\mu} D^{\mu, ab}c^b(x)-A_\mu^a(x)j^{\mu, a}(x)\right)\right),\label{fppart}
\end{equation}
\noindent where $c^a$ and $\bar c^a$ are the (Faddeev-Popov-) ghost and anti-ghost fields. As they are anti-commuting scalars, they may not appear in final states, since they would violate the CPT-theorem. This can be guaranteed in both, perturbative calculations and also in the non-perturbative calculations in this work, as will be detailed in subsection \ref{ssbrst} and section \ref{strunc}. Consequently, no external sources are associated with the ghosts at this level. They will be introduced in chapter \ref{cdse}, when off-shell ghosts will be investigated in more detail. Similar to time-like gluons, ghosts contribute indefinite norm states to the Hilbert-space already in perturbation theory. This is directly visible from the fact that ghosts act as `negative degrees of freedom' in scattering processes \cite{Peskin:ev}.
The ghost fields have a new global symmetry, the ghost number symmetry. Rescaling the ghost fields by a scale transformation $\exp(s)$ and its anti-field by $\exp(-s)$, leaving all other fields unchanged, is a symmetry of the Lagrangian\footnote{It is a left-over from the original local symmetry, which was broken by gauge fixing.}. It gives rise to the conserved ghost number $Q_{G}$, in analogy to the fermion number. As the ghosts are the only fields carrying them, it is necessary that all observable final states must have ghost number 0.
Note that the hermiticity assignment of the ghosts in \pref{fppart} is different from the common choice, which is reflected in the different sign for the ghost term in \pref{fppart}. This would lead to problems in general covariant gauges, but in ghost-anti-ghost symmetric gauges like Landau gauge this is permissible and for technical reasons advantageous \cite{Alkofer:2000wg,Alkofer:2003jr}. Of these gauges, Landau gauge turns out to be favorable as it is less singular than other gauges \cite{Alkofer:2003jr}. This manifests itself in the non-renormalization of the ghost-gluon vertex, which will be discussed below. A further advantage of Landau gauge is that as long as multiplicative renormalizability holds\footnote{Note that multiplicative renormalizability of Yang-Mills theory with and without matter fields is only proven perturbatively order by order. It is unknown if it also holds non-perturbatively. In the results presented here it does hold.}, it is a fixed point of the gauge parameter, since the latter is exactly 0.
\subsection{BRST Symmetry}\label{ssbrst}
With the introduction of ghosts in \pref{fppart} a further global symmetry arises, which is also a residual of the local gauge symmetry. It is named BRST-symmetry after its discoverers Becchi, Rouet, Stora, and Tyutin \cite{Becchi:1975nq}. Consequently it is possible to define the BRST charge $Q_{\mathrm{BRST}}$, which has ghost number 1.
The corresponding symmetry transformations are
\begin{eqnarray}
\delta_{\mathrm{BRST}}A_\mu^a(x)&=&\delta\lambda D_\mu^{ab}c^b(x)\label{brstrans1}\\
\delta_{\mathrm{BRST}}c^a(x)&=&-\delta\lambda\frac{1}{2}g_df^{abc}c^b(x)c^c(x)\\
\delta_{\mathrm{BRST}}\bar c^a(x)&=&\delta\lambda\frac{1}{\xi_g}\pd_{\mu} A_\mu^a(x)\\
\delta_{\mathrm{BRST}}\phi^a(x)&=&\delta\lambda g_df^{abc}\phi^c(x) c^b(x)\label{brstrans4},
\end{eqnarray}
\noindent where for the Landau gauge the appropriate limit has to be taken. The transformation rule for the adjoint field $\phi^a$ was added for completeness. $\delta\lambda$ is an infinitesimal constant Grassmann parameter. This defines the BRST-operator $s$ as
\begin{equation}
\delta_{\mathrm{BRST}}F=\delta\lambda sF,\nonumber
\end{equation}
\noindent by its action on any field $F$. As it is defined as the left-derivative of the transformed field with respect to $\delta\lambda$, it directly obeys the generalized Leibniz rule. Since it is Grassmann in nature, as can be seen from the fact that it changes the number of Grassmann fields in the transformations \prefr{brstrans1}{brstrans4} by 1, this rule reads
\begin{equation}
s(FG)=(sF)G\mp F(sG).\label{brstproductrule}
\end{equation}
\noindent The sign depends on whether $F$ is Grassmann or not. It further carries ghost number 0, as the transformation parameter $\delta\lambda$ has to carry ghost number -1 since the BRST charge carries ghost number 1. As \pref{brstrans1} is a global Grassmann valued gauge transformation, the gauge part of \pref{lym} is invariant on its own. The combination of the gauge fixing part and the ghost contribution is also invariant, where in Landau gauge the limit $\xi_g\to 0$ has to be taken. It is possible to linearize the transformation rules \prefr{brstrans1}{brstrans4} by introducing an auxiliary field, the Nakanishi-Lautrup field $B^a$ \cite{Peskin:ev}. Without going into details, this establishes manifestly the nil-potency of the BRST transformation\footnote{Without this field, the nil-potency would only be manifest on-shell \cite{Henneaux:1992ig}.}
\begin{equation}
\delta_{\mathrm{BRST}}^2=0.\label{nilpot}
\end{equation}
\noindent This establishes a closed algebra
\begin{eqnarray}
\left\{Q_{\mathrm{BRST}},Q_{\mathrm{BRST}}\right\}&=&0\nonumber\\
\left[iQ_G,Q_{\mathrm{BRST}}\right]&=&Q_{\mathrm{BRST}}\nonumber\\
\left[iQ_G,Q_G\right]&=&0\nonumber
\end{eqnarray}
\noindent of the residual local gauge symmetry.
A well-defined nilpotent charge directly splits the state space into three disjoint parts \cite{Peskin:ev,Henneaux:1992ig,Kugo:gm}. The states which are not annihilated by the BRST-transformation form a subspace $Q_1$, carrying BRST-charge. By acting on these states, daughter states in a subspace $Q_2$ are generated which are annihilated by the BRST-charge. The last possibility are states which are also annihilated by the BRST-charge but are not generated from parent states. These form a subspace $Q_0$. Physical states must be gauge invariant and are therefore annihilated by $Q_\mathrm{BRST}$ \cite{Weinberg:1996kr}. In addition, any states in $Q_2$ do not contribute to matrix elements. Therefore the physical subspace is
\begin{equation}
H_{phys}=\overline{\mathrm{Ker} Q_\mathrm{BRST}/\mathrm{Im} {Q_\mathrm{BRST}}}=\overline{Q_0}.\nonumber
\end{equation}
It is this subspace in which the perturbatively physical transverse gauge bosons exist, while forward polarized gluons and anti-ghosts belong to $Q_1$ and backward polarized gluons and ghosts belong to $Q_2$. This can be seen directly using the Nakanishi-Lautrup formulation of the gauge-fixed Lagrangian \cite{Peskin:ev}. Due to the relation of $Q_2$ and $Q_1$, the unphysical degrees of freedom are connected by BRST transformations and are thus metric partners. They are said to be confined by the quartet mechanism \cite{Kugo:gm}. Hence in perturbation theory the physical subspace $Q_0$ contains only transverse gluons, and perturbatively unphysical degrees of freedom are confined\footnote{In principle it is possible to have states in $Q_0$ with non-vanishing ghost number, which would render the theory ill-defined \cite{Kugo:gm}. This seems not to be the case for Yang-Mills theories.}. One of the confinement mechanisms proposed, the Kugo-Ojima scenario discussed in subsection \ref{sskugoojima}, requires also transverse gauge bosons to belong to either $Q_2$ or $Q_1$ and thus provides confinement.
\subsection{Slavnov-Taylor Identities}\label{sssti}
It is possible to directly construct identities relating different Green's functions by calculating the BRST transform of an operator expression, using that the vacuum belongs to $Q_0$. These are the Slavnov-Taylor identities (STI) \cite{Taylor:ff,Slavnov:fg}. These identities are a result of gauge invariance. A failure in fulfilling them therefore indicates a violation of gauge invariance. This property makes them an important technical tool to check the consistency of calculations and they will be used in this way extensively in chapters \ref{cdse} to \ref{cft}.
In this work two STIs are of particular importance. The first is obtained when forming the expectation value of the BRST transform of $\bar c^a\pd_{\mu} A_\mu^a$ and yields \cite{Bohm:yx} after usage of the equation of motions
\begin{equation}
p_\nu D_{\mu\nu}^{ab}(p)=-i\xi_g p_\mu\delta^{ab}\label{stigluon}
\end{equation}
\noindent with the gluon propagator $D_{\mu\nu}$. By virtue of Lorentz invariance, the Landau gauge limit hence requires $D_{\mu\nu}$ to be of the form
\begin{eqnarray}
D_{\mu\nu}(p)=P_{\mu\nu}(p)\frac{Z(p^2)}{p^2}\\\label{vacgluonprop}
P_{\mu\nu}(p)=\delta_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\label{transverseproj}
\end{eqnarray}
\noindent i.e. to be transverse. $Z$ is the associated scalar dressing function. This simple Lorentz structure is one of the advantages of Landau gauge. The other prominent feature of \pref{stigluon} is its independence of higher $n$-point Green's functions. In general, the identity for an $n$-point Green's function depends on $n'$-point Green's functions with $n'>n$. Hence an infinite set of coupled equations arises. Therefore, their use is limited in the case of non-perturbative calculations\footnote{In perturbative calculations such contributions can be neglected as they are of higher order in the expansion parameter.}, as there is not yet any possibility to assess a-priori the contribution of the $n'$-point Green's functions. In special kinematic regions, however, those unknown contributions may drop out, providing relations between $n$-point functions only.
A further result which can be obtained from \pref{stigluon} in connection with the equation of motion of the ghost is that the ghost-gluon vertex is undressed for a vanishing incoming ghost momentum in Landau gauge \cite{Taylor:ff,Marciano:su}
\begin{equation}
\lim _{p\to 0} \Gamma^{c\bar cA, abc}_\nu(p,q,-p-q) = i g_d f^{abc} q_\nu.\label{taylor}
\end{equation}
\noindent It is thus not divergent and its renormalization constant $\widetilde{Z}_1$ can and will be set to 1 here. This is maybe the most important property of Landau gauge. It makes it much more tractable than other gauges \cite{Alkofer:2003jr}. The ghost-gluon vertex will be further investigated in subsection \ref{sszwanzigergribov} and used in chapters \ref{cdse} to \ref{cft}.
The second identity is concerned with this ghost-gluon vertex. It can be obtained from the BRST-transform of $c^a\bar c^b\bar c^c$ \cite{vonSmekal:1997is,Watson:phd} and reads
\begin{equation}
ik_\mu\Gamma^{\bar c cA}_\mu(p,-q) G(q^2)+iq_\mu\Gamma^{\bar c cA}(p,-k)_\mu G(k^2)=p^2\widetilde{Z}_1\frac{G(k^2)G(q^2)}{G(p^2)}+\eta\label{stigghv}.
\end{equation}
\noindent $\eta$ summarizes contributions from $n'$-point Green's functions with $n'>3$. Color indices have been removed by assuming a tree-level\footnote{This is exact in perturbation theory.} color structure. This assumption is discussed in more detail in chapter \ref{cdse}. $G$ is the ghost dressing function, defined via its propagator as
\begin{equation}
D_G^{ab}(p^2)=-\frac{G(p^2)}{p^2}\delta^{ab}.\label{ghostprop}
\end{equation}
\noindent The identity \pref{stigghv} is consistent with the bareness of the ghost-gluon vertex \pref{taylor} if the contributions from the higher Green's functions vanish in this limit.
\section{Confinement}\label{sconfinement}
As one of the main observables for this work is the absence or presence of confinement it is necessary to detect it. Two fundamentally different approaches to this question will be discussed. The first is concerned with the mere statement of confinement. This will be investigated in subsection \ref{sscriterions}. The other approach consists of criteria deduced from possible confinement mechanisms. Three of them will be discussed in sections \ref{sskugoojima} to \ref{ssgribovstingl}. The number of proposed mechanisms for confinement is large. Only those which generate criteria that can be tested using the objects obtained in this work will be discussed here. A more general overview can be found in \cite{Alkofer:2000wg}.
In any case, cluster decomposition \cite{Weinberg:mt,Haag:1992hx} must be violated for colored objects to allow for a long range force. This necessarily implies the existence of a massless excitation in the complete state space. On the other hand since all experimental results show validity of cluster decomposition, the physical mass spectrum of QCD must have a mass gap for colorless objects. Otherwise it would be possible to scatter a colorless object into far apart colored objects, the so-called ``behind-the-moon'' problem. However, the existence of a massless excitation alone does not suffice for confinement of color, as it is necessary to show that it is not part of the physical spectrum.
\subsection{Criteria for Confinement}\label{sscriterions}
There are two criteria which will be used here. One implies the other. The basic object is in both cases the spectral density $\rho$ \cite{Itzykson:rh} of a given particle. For a particle to exist as a physical final state having a K\"allen-Lehmann representation, it has to have a positive semi-definite spectral function\footnote{In a theory with indefinite metric, as Yang-Mills theory, for unstable particles such that the width exceeds the mass the spectral function is also not necessarily positive semi-definite. They do not occur in final states, since they decay when letting $t\to\infty$.}. This is known as the Osterwalder-Schrader axiom of reflection positivity \cite{Haag:1992hx,Osterwalder:dx}. A violation of positivity can be tested using the Schwinger function to be discussed in chapter \ref{cderived}.
A stronger condition is violation of the Oehme-Zimmermann super-convergence relation \cite{Oehme:bj}. The spectral representation of a particle of mass $m_0$ is \cite{Itzykson:rh}
\begin{equation}
D(p)=\frac{Z}{p^2+m_0^2}+\int_{m^2}^{\infty}d\kappa^2\frac{\rho(\kappa^2)}{p^2+\kappa^2},\label{oehmerep}
\end{equation}
\noindent with $m>m_0$ being the threshold mass for contributions above the single-particle pole, the multi-particle threshold. $Z$ is the positive overlap between an in-state and the field described by the propagator $D$. If the propagator vanishes at 0,
\begin{equation}
\lim_{p\to 0}p^2D(p)=0,\label{oehme}
\end{equation}
\noindent then the spectral function must be at least partly negative, thus implying the first condition above. Secondly, a vanishing propagator at $p=0$ implies the absence of a K\"allen-Lehmann representation. The particle can thus be not a physical particle anymore: The particle is confined. This gives the second condition for confinement.
For massless particles the interpretation of \pref{oehme} is more direct. As $p^2=0$ is just the statement of the particle being on-shell, it implies the vanishing of the on-shell propagator: The particle does not propagate and is thus confined.
Note here a subtle difference. It is possible to think of confined particles as either confined due to dynamic effects or due to being unphysical. For example magnetic confinement of neutrons in a magnetic field is such a case: The particles are physical but they are bound in a way disallowing separation. The confinement e.g.\/ of ghosts differs from this. These are unphysical particles and are thus confined in a different sense. Note that in an unbroken non-abelian gauge theory, charged currents are gauge-dependent and can thus not be observed directly. Observing a quark thus corresponds e.g.\/ to an observation of a colorless object with non-integer electric charge.
One of the major questions concerning confinement is which of both possibilities applies to colored objects. The Kugo-Ojima criterion described in the next subsection favors the latter option, as colored objects will turn out to be automatically BRST-charged. The more common attitude is the first option.
\subsection{Kugo-Ojima Scenario}\label{sskugoojima}
The Kugo-Ojima confinement scenario \cite{Kugo:gm} puts forward the idea that all colored objects form BRST-quartets and therefore do not belong to the physical state space. The metric partners of transverse gluons would be ghost-gluon bound states.
Thus, the confinement mechanism is essentially the same as in the case of ghosts in perturbation theory. This scenario is based on a rigorous derivation and requires three preconditions. One of them is an unbroken BRST charge also in the non-perturbative regime. Whether this is the case is not known\footnote{It is even unknown how to define a non-perturbative BRST charge. Due to the non-trivial topology of the gauge group this necessarily has to be done patch-wise in gauge space.}. The second is the failure of the cluster decomposition theorem and hence the existence of a massless excitation. This is also unknown and can not yet be attacked in the approach used here. The third ingredient is an unbroken global color charge. In Landau gauge, this condition can be put into the form \cite{Kugo:1995km}
\begin{equation}
\lim_{p^2 \to 0} p^2 D_G(p^2)\to\infty,\label{kugo}
\end{equation}
\noindent where $D_G$ is the propagator of the Faddeev-Popov ghost \pref{ghostprop}. This scenario also necessarily implies that \pref{oehme} holds for the gluon. Both of these conditions \pref{oehme} and \pref{kugo} will be checked.
Note that this scenario does not imply that colored objects do not have an asymptotic field. However, as they are BRST charged a configuration with arbitrarily many colored objects belongs to one equivalence class, and only the colorless objects contribute to matrix elements.
\subsection{Zwanziger-Gribov Scenario}\label{sszwanzigergribov}
The central idea of the Zwanziger-Gribov scenario \cite{Zwanziger:2003cf,Gribov:1977wm,Zwanziger:2001kw,Zwanziger:2002ia} is that zero-modes at the common boundary of the first Gribov horizon and the fundamental region dominate the infrared properties and thus generate confinement. Both regions necessarily have a common boundary. This boundary is convex, compact, and includes the origin \cite{Zwanziger:2003cf}.
As gauge space is infinite-dimensional, all the volume will be concentrated at the boundary, and the system is dominated by it. Since the Faddeev-Popov-determinant \pref{fpdet} vanishes there, the corresponding excitations have to be long-range. Using stochastic quantization and performing a Landau gauge limit under certain assumptions, it can be shown that \pref{kugo} follows. In addition, it follows that the infrared limit is dominated by the ghost-term of \pref{fppart} alone \cite{Zwanziger:2001kw}. As this term can be written as a BRST-transform, Yang-Mills theory in this limit is a topological field theory of Schwarz type with no propagating modes \cite{Birmingham:1991ty}. Thus colored objects do not appear, implying confinement. Using these results, \pref{oehme} for the gluon is also obtained, thus leading to the same criteria as the Kugo-Ojima scenario. This analogy, if it is more than mere coincidence, is not yet understood.
It is also intuitively clear that a strongly divergent ghost propagator at zero momentum can mediate confinement. After Fourier-transformation such an infrared divergence relates to long-ranged spatial correlations. These are stronger than the ones induced by a Coulomb force since the divergence in momentum space is stronger than that of a massless particle.
A bare ghost-gluon vertex in the infrared is sufficient to generate this behavior. Such a vertex would also be consistent with the perturbative renormalization group \cite{Zwanziger:2003cf}. This Zwanziger-hypothesis has been checked numerically in context with this work \cite{schleifenbaum:diploma} and turns out to be well fulfilled.
Recently, investigations in Coulomb gauge indicate, that a similar connection between the dynamics on the Gribov horizon and confinement of gluons holds there as well \cite{Greensite:2004bz}.
\subsection{Gribov-Stingl Scenario}\label{ssgribovstingl}
The last scenario investigates a manifestation of confinement different from the two previous ones. The Gribov-Stingl scenario \cite{Gribov:1977wm,Habel:1990tw,Habel:1989aq} puts forward the idea that confined particles have one or more pairs of complex conjugate poles, and thus cannot be physical states. In general, such a pole structure leads to violation of causality \cite{Peskin:ev}. However, for a special structure of the propagators it can be shown that it is possible to reconcile such a pole structure with causality on the level of the $S$-matrix \cite{Habel:1990tw}. The essential argument is that due to the absence of real poles, application of the LSZ-reduction formula \cite{Peskin:ev} leads to vanishing matrix elements for states with colored objects \cite{Habel:1990tw}. Thus colored objects can only exist as short-lived quantum excitations. In addition, colorless objects only have real poles but no continuum, if they cannot decay into colorless objects \cite{Habel:1990tw}.
\section{Interplay with Quarks}\label{ssfermions}
As quarks will be neglected throughout this work, a short comment to which extent the results may be affected by the presence of quarks is in order. Calculations in the vacuum show that the presence of quarks does not qualitatively alter the gauge propagators, as long as there are less than 5 light flavors \cite{Fischer:2003rp}. This is not expected to change at finite temperature, as quarks acquire an effective mass increasing with temperature and thus behave more perturbatively \cite{Appelquist:vg}. Especially they should not contribute to the infinite temperature limit of the gluon propagator. Thus the results presented will probably not be changed qualitatively by quarks; this is only a conjecture, though. Especially in the vicinity of the phase transition quarks are relevant. Lattice calculations indicate a possible change of the order of the phase transition when including quarks \cite{Karsch:2003jg}, but this issue is still under debate.
The impact on quarks by the results found here may be more significant but also more intricate to understand. Lattice calculations show a drastic change in the properties of quarks above the phase transition. The free energy of two static quarks changes substantially and may change from a linear rise with distance to a logarithmic or even flat shape \cite{Kaczmarek:2004gv}. However, as the mechanism of quark confinement is not understood yet \cite{Alkofer:2003jk}, assessing the consequences of this change is not simple. The most prominent change is the restoration of chiral symmetry \cite{Karsch:2003jg}, which surprisingly occurs at the same temperature as the changes in the gauge sector.
One possible chain of arguments to understand this coincidence is based on evidence from lattice calculations that the underlying degrees of freedom confining quarks and gluons are topological objects \cite{Engelhardt:2003wm}. These carry topological charge, and can thus be brought into connection with zero modes of the Dirac-operator by the Atiyah-Singer index theorem \cite{Atiyah:1968mp}. The density of such modes again is related to the chiral condensate by the Banks-Casher formula \cite{Banks:1979yr}. If this chain of arguments is correct, it gives a connection between confinement and chiral symmetry breaking. As will turn out in this work, at least part of the gluons are confined even at high temperatures. It is therefore still unclear what the relation above the critical temperature is and how the quark properties are affected.
\chapter{Perturbative Expressions}\label{appUV}
This appendix contains the perturbative calculations to leading order in the dimensionally reduced theory of chapter \ref{c3d}. Replacing all full quantities in the truncated DSEs (\ref{fulleqG3d}-\ref{fulleqZ3d}) by their tree-level values, i.e. setting all dressing functions equal to one and using the tree-level vertices (\ref{tlcca}-\ref{tl4h}) instead of the full vertices, the standard perturbation theory to one-loop order is obtained. It is then possible to calculate the leading-order perturbative dressing functions. For the ghost, this leads to
\begin{equation}
G(p)^{-1}=1-\frac{g_3^2C_A}{16p}\label{apertgh}
\end{equation}
to leading order, independent of the presence of a Higgs. Note that a Landau pole at $p={\cal O}(g_3^2)$ is present.
The calculation of the gluon self-energy is a little more complicated, since each single contribution is linearly divergent, although the sum is finite in three dimensions due to the STI \pref{stigluon}. Corresponding problems are most easily circumvented by contracting the gluon equation with the Brown-Pennington projector \pref{bpproj}. Performing the calculation in pure Yang-Mills theory yields
\begin{equation}
Z(p)^{-1}=1-\frac{11g_3^2C_A}{64p}\label{apertgl}.
\end{equation}
Including the Higgs yields
\begin{equation}
Z(p)^{-1}=1-\frac{11g_3^2C_A}{64p}+\frac{g_3^2C_A}{16\pi p}\left(2\frac{m_h}{p}-\frac{p^2+4m_h^2}{p^2}\csc^{-1}\left(\sqrt{1+\frac{4m_h^2}{p^2}}\right)\right).\label{apertglwh}
\end{equation}
Since non-perturbative effects already arise at order $g_3^4/p^2$ in the present truncation scheme, the only interesting part is the one for $p\gg g_3^2,m_h$, leading to
\begin{equation}
Z(p)^{-1}=1-\frac{9g_3^2C_A}{64p}\label{apertgllead}.
\end{equation}
Finally, the Higgs self-energy is
\begin{equation}
H(p)^{-1}=1+\frac{m_h^2}{p^2}+\frac{g_3^2C_A}{p}\frac{m_h}{4\pi p}+\frac{g_3^2C_A}{p}\frac{1}{2\pi p} \left(\left(\frac{m_h^2+p^2}{p}-2p\right)\arcsin\left(\sqrt{\frac{p^2}{m_h^2+p^2}}\right)-m_h\right)\label{aperth},
\end{equation}
\noindent where the third term is the tadpole contribution. The leading contribution is
\begin{equation}
H(p)^{-1}=1-\frac{g_3^2C_A}{4p}.\label{aperthlead}
\end{equation}
The coupling constant $h$ is determined by requiring that the sum of the tadpole kernels already generates a finite integral, i.e.\/ independent of the regularization scheme. This is obtained by requiring \pref{hvalue}.
Note that, in three dimensions, resummation produces effects only from order $g_3^4$ on. By dimensional arguments alone, only tree-level expressions in the loops can contribute at order $g_3^2$. Therefore \pref{apertgh}, \pref{apertgllead} and \pref{aperthlead} already constitute the resummed solutions. This is confirmed by the numerical calculations presented in section \ref{sfull3d}.
Note further that this also ensures that gauge symmetry is intact to one-loop order in the regime of applicability of leading-order resummed perturbation theory. The violation of gauge symmetry in this approach is not stronger than in ordinary leading-order perturbation theory.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
In the celebrated paper \cite{Con},
Connes observed in Section 6 that injectivity of von Neumann algebras passes to the fixed point algebras
of amenable group actions.
In contrast to this observation, in the seminal paper \cite{Kir94},
Kirchberg showed that any unital separable exact \Cs-algebra
can be realized as a (liftable) quotient
of the fixed point algebra of an (inner) automorphism on a UHF-algebra.
This in particular implies that nuclearity need \emph{not} pass to the fixed point algebra of an amenable group action.
While Kirchberg's theorem indicates bad behavior of the operation
taking the fixed point algebra,
it is also true that the fixed point algebras play important roles to understand
\Cs-dynamical systems and associated \Cs-algebras and invariants in some situations; see e.g., \cite{Kas}, \cite{Sz18}.
Motivated by these results, we are interested in knowing
how bad the fixed point algebras of general amenable groups
of a nuclear \Cs-algebra can be.
We particularly examine this on one of the most ubiquitous (nuclear) \Cs-algebras---the Cuntz algebra $\mathcal{O}_2$ \cite{Cun}.
\begin{Thmint}\label{Thm}
Let $\Gamma$ be a countable exact group.
Let $\Lambda$ be an infinite countable group.
Then $\Lambda$ admits an outer action on $\mathcal{O}_2$
whose fixed point algebra is isomorphic
to an intermediate \Cs-algebra of
$\rg(\Gamma) \subset L(\Gamma)$.
Moreover, when $\Gamma$ has the approximation property $($AP$)$ \cite{HK},
one can arrange the fixed point algebra to be isomorphic to $\rg(\Gamma)$.
\end{Thmint}
We point out that the infiniteness of $\mathcal{O}_2$ is essential in the statement.
Indeed non-amenable reduced group \Cs-algebras do not embed
into stably finite nuclear \Cs-algebras by Proposition 6.3.2 of \cite{BO}, \cite{Cun78}, and \cite{Haa}.
Note also that many exact groups have the AP.
The class of groups with the AP contains all weakly amenable groups (thus all hyperbolic groups \cite{Ozh}),
and is stable under extensions and free products.
See Section 12.4 of \cite{BO} for details.
It would be interesting to compare Theorem \ref{Thm} with
the following useful statement: The fixed point algebra of a compact group action
is nuclear if and only if the original \Cs-algebra is nuclear. See Section 4.5 of \cite{BO}
for details.
Applying Theorem \ref{Thm} to a locally finite group,
(after a slight refinement of the proof,) we
also obtain refined versions of Theorem A and Corollary B in \cite{Suz17}.
We recall that Watatani \cite{Wat} shows that most good properties
(including nuclearity) are stable under
finite Watatani index inclusions. See Proposition 2.7.2 of \cite{Wat} and the remark below it for the precise statement.
The following corollary shows that this familiar statement does
not extend to ``approximately finite'' index subalgebras.
\begin{Corint}\label{Cor}
There is a descending sequence of irreducible finite Watatani index inclusions
\[\cdots \subset A_{n+1}\subset A_n \subset \cdots \subset A_2 \subset A_1\]
of isomorphs of $\mathcal{O}_2$
whose intersection does not have the operator approximation property nor the local lifting property.
Moreover one can arrange the sequence to satisfy the following homogeneity condition:
for every $m\in \mathbb{N}$, the inclusions $A_{n+m} \subset \cdots \subset A_n$, $n\in \mathbb{N}$, are pairwise isomorphic.
\end{Corint}
Here we say that two sequences of inclusions $A_1 \subset A_2 \subset \cdots \subset A_n$
and $B_1 \subset B_2 \subset \cdots \subset B_n$ of \Cs-algebras
are \emph{isomorphic}
if there is an isomorphism $\varphi \colon A_n \rightarrow B_n$
satisfying $\varphi(A_i)=B_i$ for $i=1, \ldots, n-1$.
We recall that an inclusion $A\subset B$ of unital simple \Cs-algebras
is said to be \emph{irreducible} if the relative commutant
is trivial: $A'\cap B=\mathbb{C}$.
We also give the following result on general Kirchberg algebras.
\begin{Propint}\label{Prop}
Let $\Lambda$ be a countable infinite group.
Let $A$ be a Kirchberg algebra.
Then $\Lambda$ admits an outer action on $A$
whose fixed point algebra does not have the completely bounded approximation property
nor the local lifting property.
\end{Propint}
Here we recall that a simple separable nuclear purely infinite \Cs-algebra
is called a \emph{Kirchberg algebra}.
Kirchberg algebras form an important class of \Cs-algebras.
A side of rich structures of Kirchberg algebras is reflected to the successful classification theorem of Kirchberg--Phillips \cite{Kir94b}, \cite{Phi}.
We refer the reader to the book \cite{Ror} for fundamental facts and backgrounds on this subject.
The key ingredients of our constructions (besides well-known deep results)
are ``amenable'' actions of non-amenable groups on Kirchberg algebras
recently obtained in \cite{Suzeq}, \cite{Suz19}.
We expect that our results
give a new intuitive picture of the inaccessibility of
conjugacy classes of \Cs-dynamical systems,
in contrast to successful classification results
\emph{up to cocycle conjugacy} (see e.g., \cite{Nak}, \cite{IM}, \cite{IM2}, \cite{Sz18}, and references therein).
\subsection*{Notations}
Here we fix a few notations used in this paper.
\begin{itemize}
\item For $\epsilon>0$ and for two elements $x$, $y$ of a \Cs-algebra,
denote by $x\approx_{\epsilon} y$ if $\|x -y\| <\epsilon$.
\item For a group action $\alpha \colon \Gamma \curvearrowright A$ on a \Cs-algebra $A$,
denote by $A^\alpha$ the fixed point algebra of $\alpha$:
\[A^\alpha:=\{ a\in A: \alpha_s(a)=a {\rm~for~all~}s\in \Gamma\}.\]
\item The symbols `$\otimes$', `$\rc$', `$\vnc$' stand
for the minimal tensor products, the reduced \Cs-crossed products,
and the von Neumann algebra crossed products respectively.
\end{itemize}
For basic facts on \Cs-algebras and discrete groups, we
refer the reader to the book \cite{BO}.
\section{Proofs, constructions, and remarks}
The following lemma would be well-known for specialists.
For the reader's convenience, we include a proof.
\begin{Lem}\label{Lem:fp}
Let $\Lambda$ be a group.
Let $I$ be a $\Lambda$-set
such that all $\Lambda$-orbits are infinite.
Let $A$ be a unital \Cs-algebra.
Let $\sigma \colon \Lambda \curvearrowright \bigotimes_I A$
be the tensor shift action.
Then
$(\bigotimes_I A)^{\sigma}=\mathbb{C}$.
\end{Lem}
\begin{proof}
Take $x\in (\bigotimes_I A)^{\sigma}$.
Let $\epsilon >0$.
Fix a state $\varphi$ on $A$.
For each $S\subset I$,
define a conditional expectation $E_S \colon \bigotimes _I A \rightarrow \bigotimes _S A$
to be $E_S:= (\bigotimes_S \id_A) \otimes (\bigotimes_{I \setminus S} \varphi)$.
Choose a finite subset $F$ of $I$ and an element
$y\in \bigotimes _F A$ satisfying
$x \approx_\epsilon y$.
By the assumption on $I$,
one can choose $s\in \Lambda$ satisfying
$sF \cap F =\emptyset$.
(The existence of such $s$ is obvious in applications in the present paper.
For completeness, we give a proof for general case.
To lead to a contradiction, assume that there is no such $s$.
Then one can find sequences $s_1, \ldots, s_n\in \Gamma$ and $x_1, \ldots, x_n \in F$
satisfying $\Gamma = \bigcup_{k=1}^n s_k{\rm Stab}(x_k)$.
Here ${\rm Stab}(x_k)$ denotes the stabilizer subgroup of $x_k$.
Since each ${\rm Stab}(x_k)$ has infinite index in $\Gamma$,
this is a contradiction; for the proof, see \cite{Neu}, Lemma 4.1.)
Observe that
\[\sigma_s(y)\approx_{\epsilon} \sigma_s(x)=x \approx_{\epsilon} y.\]
This implies
\[x \approx_\epsilon y= E_{F}(y) \approx_{2\epsilon} E_{F}(\sigma_s(y))\in \mathbb{C}.\]
Since $\epsilon>0$ is arbitrary, this yields $x\in \mathbb{C}$.
\end{proof}
Recall that an automorphism $\alpha$ of a \Cs-algebra $A$ is said to be \emph{inner}
if there is a unitary multiplier $u$ of $A$
satisfying $\alpha(a)=uau^\ast$ for all $a\in A$.
An action $\alpha$ of a discrete group $\Gamma$ on $A$ is said to be \emph{outer}
if $\alpha_s$ is not inner for all $s\in \Gamma \setminus \{e\}$.
For simple \Cs-algebras, outerness of an action can be regarded as a non-commutative analogue of
(topological) freeness of topological dynamical systems; see \cite{Kis} for instance.
To confirm outerness of actions, we use the central sequence algebras.
Here we briefly recall them.
For a unital \Cs-algebra $A$,
denote by $\ell^\infty(\mathbb{N}, A)$ the \Cs-algebra of all bounded sequences of $A$.
Let $c_0(\mathbb{N}, A)$ denote the ideal of $\ell^\infty(\mathbb{N}, A)$ consisting
of all sequences tending to $0$ in norm.
Set $A^\infty:= \ell^\infty(\mathbb{N}, A)/ c_0(\mathbb{N}, A)$.
Then we have an embedding
$A \rightarrow A^\infty$; $a \mapsto (a, a, \ldots) + c_0(\mathbb{N}, A)$.
By this embedding, we regard $A$ as a \Cs-subalgebra of $A^\infty$.
Define $A_\infty := A^\infty \cap A'$.
This is the central sequence algebra of $A$.
Any automorphism $\alpha$ of $A$
induces an automorphism on $\ell^\infty(\mathbb{N}, A)$
by pointwise application.
This further induces the automorphism $\alpha_\infty$ on $A_\infty$.
Observe that if $\alpha$ is inner, then $\alpha_\infty$
is trivial.
\begin{proof}[Proof of Theorem \ref{Thm}]
By the proof of Proposition B and Remark 3.3 in \cite{Suzeq},
one can take an action $\alpha \colon \Gamma \curvearrowright \mathcal{O}_2$
satisfying the following conditions.
\begin{itemize}
\item $\mathcal{O}_2^\alpha\neq \mathbb{C}$.
\item
Denote by
\[\beta\colon \Gamma \curvearrowright \bigotimes_\Lambda \mathcal{O}_2\]
the diagonal action of copies of $\alpha$ indexed by $\Lambda$.
Then \[A:=\left(\bigotimes_\Lambda \mathcal{O}_2\right) \rca{\beta} \Gamma \cong \mathcal{O}_2.\]
\end{itemize}
Let $\sigma \colon \Lambda \curvearrowright \bigotimes_\Lambda \mathcal{O}_2$
denote the left shift action.
Then $\sigma$ commutes with $\beta$.
Hence $\sigma$ extends to an action
$\theta\colon \Lambda \curvearrowright A$
via the formula
\[\theta_t(x u_s):=\sigma_t(x)u_s {\rm~for~} x\in \bigotimes_\Lambda \mathcal{O}_2, s\in \Gamma, t\in \Lambda.\]
We next show that $\theta_\infty \colon \Lambda \curvearrowright A_\infty$ is faithful, which yields the outerness of $\theta$.
Choose an element $a\in \mathcal{O}_2^\alpha\setminus \mathbb{C}$.
For each $t\in \Lambda$, denote by $a[t]$
the image of $a$ under the embedding of $\mathcal{O}_2$
into the $t$-th tensor product component of $\bigotimes_\Lambda \mathcal{O}_2$.
Observe that each $a[t]$ commutes with $u_s \in A$ for all $s\in \Gamma$.
Fix a state $\varphi$ on $\mathcal{O}_2$.
Then for any $t, u \in \Lambda$ with $t\neq u$,
we have
$\|a[t]-a[u]\|\geq\| [(\bigotimes_{\{t\}} \id_A )\otimes (\bigotimes_{\Gamma \setminus \{t\}} \varphi)](a[t]-a[u])\|= \|\varphi(a)-a\|>0$.
Take a sequence $(t_n)_{n=1}^\infty$ in $\Lambda$ which tends to infinity.
Then the sequence $(a[t_n])_{n=1}^\infty$ defines an
element $x$ of $A_\infty$.
The above inequality shows that
$\theta_{\infty, t}(x) \neq x$ for all $t\in \Lambda \setminus \{e\}$.
Thus $\theta_\infty$ is faithful.
Put $M:= (\bigotimes_\Lambda \mathcal{O}_2)\sd$.
Let $\bar{\beta}$ and $\bar{\sigma}$ denote the weakly continuos extensions of $\beta$ and $\sigma$ to $M$ respectively.
Let $\bar{\theta}$ denote the action
of $\Lambda$ on $M\vrca{\bar{\beta}} \Gamma$ defined analogously to $\theta$.
Then the inclusion $\bigotimes_\Lambda \mathcal{O}_2 \subset M$
extends to the $\Lambda$-equivariant inclusion
$A \subset M \vrca{\bar{\beta}} \Gamma$.
Since $(\bigotimes_\Lambda \mathcal{O}_2)^\sigma =\mathbb{C}$ (by Lemma \ref{Lem:fp}),
by Corollary 3.4 of \cite{Suz19}, we have
$A^{\theta} \subset L(\Gamma)$.
Moreover, when $\Gamma$ has the AP, thanks to Proposition 3.4 of \cite{Suz17},
we further obtain
$A^{\theta}=\rg(\Gamma)$.
\end{proof}
\begin{proof}[Proof of Corollary \ref{Cor}]
We fix a non-trivial finite group $G$ and set $\Lambda := \bigoplus_{\mathbb{N}} G$.
Put $\Gamma:=\SL(3, \mathbb{Z})$. Note that $\Gamma$ is exact
(see Section 5.4 of \cite{BO}) and does not have the AP \cite{LS}.
For each $n\in \mathbb{N}$, set $\Lambda_n :=\bigoplus_{k=1}^n G \subset \Lambda$.
Then, note that $\Lambda_n \subset \Lambda_{n+1}$ and that $\Lambda = \bigcup_{n=1}^\infty \Lambda_n$.
Take an action $\alpha \colon \Gamma \curvearrowright \mathcal{O}_2$
as in the proof of Theorem \ref{Thm}.
By the construction in \cite{Suzeq} (see Proposition B and Remark 3.3), we may further assume that the fixed point algebra $\mathcal{O}_2^{\alpha}$
admits
a unital embedding $\iota\colon \mathcal{O}_2 \rightarrow \mathcal{O}_2^{\alpha}$.
Define $I:= \Lambda \sqcup \bigsqcup_{n=1}^\infty \Lambda/\Lambda_n$.
We equip $I$ with the left translation $\Lambda$-action.
Let $\beta \colon \Gamma \curvearrowright \bigotimes_I \mathcal{O}_2$
denote the diagonal action of copies of $\alpha$ indexed by $I$.
Let $\sigma \colon \Lambda \curvearrowright \bigotimes_I \mathcal{O}_2$
denote the tensor shift action.
Then $\sigma$ commutes with $\beta$.
Thus $\sigma$ induces the $\Lambda$-action $\theta$
on $A:=(\bigotimes_I \mathcal{O}_2) \rca{\beta} \Gamma$ via the formula
\[\theta_t(x u_s):=\sigma_t(x)u_s {\rm~for~} x\in \bigotimes_I \mathcal{O}_2, s\in \Gamma, t\in \Lambda.\]
For $n\in \mathbb{N}$, set $\theta_n :=\theta|_{\Lambda_n}$
and define $A_n :=A^{\theta_n}$.
Then note that $A_{n+1} \subset A_n$ for all $n\in \mathbb{N}$
and that $\bigcap_{n=1}^\infty A_n= A^\theta$.
As in the proof of Theorem \ref{Thm}, we obtain $\rg(\Gamma) \subset A^\theta \subset L(\Gamma)$.
For $n, m \in\mathbb{N}$, set $\Upsilon_{n, m}:= \bigoplus_{k=n+1}^{n+m} G \subset \Lambda$.
Observe that, as $\Upsilon_{n, m}$ commutes with $\Lambda_n$,
the \Cs-subalgebra $A_n$ is invariant under $\theta(\Upsilon_{n, m})$ for all $m\in \mathbb{N}$.
Let $\vartheta_{n, m} \colon \Upsilon_{n, m} \curvearrowright A_n$ denote the restricted action of $\theta$.
Then, for each $n, m\in \mathbb{N}$,
since $\Lambda_n \cdot \Upsilon_{n, m} = \Lambda_{n+m}$,
we have $A_n^{\vartheta_{n, m}}=A_{n+m}$.
By a similar argument to the proof of Theorem \ref{Thm},
one can show the outerness of $\vartheta_{n, m}$.
By the proof of Corollary B of \cite{Suz17},
the intersection algebra
$\bigcap_{n=1}^\infty A_n$ does not have the operator approximation
property nor the local lifting property.
Since $\theta$ is outer and each $\Lambda_n$ is finite,
the corresponding fixed point algebras $A_n$ are simple and nuclear.
By Remark 3.14 of \cite{Izu}, each inclusion $A_{n} \subset A_1$ is irreducible.
By Corollary 3.12 of \cite{Izu}, each inclusion $A_{n+1} \subset A_n$
has finite Watatani index. (In fact the index is $\sharp G$.)
We next show that for each $n\in \mathbb{N}$, $A_n$ is isomorphic to $\mathcal{O}_2$.
To see this, for each $k\in \mathbb{N}$, let $\iota_{k}$ denote the composite of
the unital embedding $\iota\colon \mathcal{O}_2 \rightarrow \mathcal{O}_2^{\alpha}$
and the canonical embedding
of $\mathcal{O}_2^{\alpha}$ into the $\Lambda_k$-th
tensor product component of $\bigotimes_I \mathcal{O}_2^{\alpha} \subset A$.
Then the sequence $(\iota_{k+n})_{k=1}^\infty$
defines a unital embedding of $\mathcal{O}_2$ into $(A_n)_\infty$.
By Kirchberg's theorem (see \cite{KP}, Lemma 3.7),
$A_n$ is isomorphic to $\mathcal{O}_2$.
Finally we show the homogeneity condition described in the statement.
Let $m\in \mathbb{N}$ be given.
Then, for each $n\in \mathbb{N}$, the sequence $(\iota_{k+n+m})_{k=1}^\infty$ witnesses that $\vartheta_{n, m}$ satisfies condition (iii) of Theorem 4.2 of \cite{Izu04}.
Thus, by Theorem 4.2 of \cite{Izu04},
the actions $\vartheta_{n, m}$, $n\in \mathbb{N}$, are pairwise conjugate
(after the canonical identifications $\Lambda_{n, m}\cong \Lambda_{l, m}$ by shifting indices).
Consequently, the inclusions $A_{n+m} \subset \cdots \subset A_n$, $n\in \mathbb{N}$,
are pairwise isomorphic.
\end{proof}
\begin{Rem}\label{Rem:BE}
The construction in the proof of the Corollary works for \emph{any} countable exact group $\Gamma$ instead of $\SL(3, \mathbb{Z})$.
The isomorphism classes of the resulting inclusions
\[A_{n+m} \subset \cdots \subset A_n\]
do \emph{not} depend on the choice of $\Gamma$ (when $G$ is fixed).
The resulting intersection algebra is an intermediate \Cs-algebra
of $\rg(\Gamma) \subset L(\Gamma)$,
and when $\Gamma$ satisfies the AP,
it is in fact equal to the reduced group \Cs-algebra $\rg(\Gamma)$.
\end{Rem}
\begin{Rem}
One can also arrange the fixed point algebra in
Theorem \ref{Thm} and Remark \ref{Rem:BE} to be $A \rca{\zeta} \Gamma$
when $\Gamma$ has the AP, $A$ is a unital separable nuclear \Cs-algebra,
and $\zeta\colon \Gamma \curvearrowright A$ is an action without $\Gamma$-invariant proper ideals.
Here we sketch how to modify the proof in the former case.
The latter case is similarly obtained.
From now on, we use the notations in the proof of Theorem \ref{Thm}.
We first replace $\alpha$ by the diagonal action of $\alpha$, the left shift action on $\bigotimes_{\Gamma} \mathcal{O}_2$, and the trivial action
on $\mathcal{O}_2$ (the underlying \Cs-algebra is again identified with $\mathcal{O}_2$ by Theorem 3.8 of \cite{KP}).
Denote by $\tilde{\beta}$
the diagonal action of $\beta$ and $\zeta$.
Then the reduced crossed product $B$ of $\tilde{\beta}$ is nuclear by the choice of $\alpha$ (cf.~ \cite{Suzeq}, Proposition B).
Theorem 7.2 of \cite{OP} shows that
$B$ is simple.
By the choice of $\alpha$, $B \cong B \otimes \mathcal{O}_2$.
By Theorem 3.8 of \cite{KP}, $B\cong \mathcal{O}_2$.
Set $\tilde{\sigma}:=\sigma \otimes \id_A$.
Define $\tilde{\theta} \colon \Lambda \curvearrowright B$ analogous to $\theta$ by using $\tilde{\sigma}$ instead of $\sigma$.
The proof of Lemma \ref{Lem:fp} shows
that $(\bigotimes_\Lambda \mathcal{O}_2 \otimes A)^{\tilde{\sigma}}=A^\zeta$. Proposition 3.4 in \cite{Suz17} then yields that $B^{\tilde{\theta}} = A \rca{\zeta} \Gamma$.
\end{Rem}
We finally prove Proposition \ref{Prop}.
The proof involves the (K)K-theory.
For basic facts and terminologies on this subject, we refer the reader to the book \cite{Bla}.
\begin{proof}[Proof of Proposition \ref{Prop}]
Thanks to Kirchberg's $\mathcal{O}_\infty$-absorption theorem (\cite{KP}, Theorem 3.15)
we only need to show the statement for the Cuntz algebra $\mathcal{O}_\infty$.
Let $\Gamma$ be a countable free group of infinite rank.
In the proof of Theorem 5.1 of \cite{Suz19},
we have constructed an action $\alpha \colon \Gamma \curvearrowright \mathcal{O}_\infty$ which contains a unital amenable $\Gamma$-\Cs-subalgebra in the sense of Definition 4.3.1 in \cite{BO}.
By replacing $\alpha$ by the diagonal action of $\alpha$ and the trivial $\Gamma$-action on $\mathcal{O}_\infty$ (cf.~Theorem 3.15 of \cite{KP}) if necessary,
we may assume that $\mathcal{O}_\infty^{\alpha} \neq \mathbb{C}$.
Denote by $\beta$ the diagonal action of copies of $\alpha$ indexed by $\Lambda$.
By Theorem 8.4.1 (iv) in \cite{Ror}, one can
choose an action $\gamma \colon \Gamma \curvearrowright C$
on a unital Kirchberg algebra $C$ satisfying the universal coefficient theorem \cite{RS} with the following conditions:
\begin{enumerate}
\item $(K_0(C), [1]_0, K_1(C))\cong (\bigoplus_\Gamma \mathbb{Z}, 0, 0)$,
\item the induced action $\Gamma \curvearrowright K_0(C)$ is conjugate
to the left shift action $\Gamma \curvearrowright \bigoplus_\Gamma \mathbb{Z}$.
\end{enumerate}
Observe that, thanks to \cite{FKK} (see also \cite{KOS}),
by composing $\gamma_s$ with an approximately inner automorphism
for each canonical generator $s$ of $\Gamma$ if necessary,
we may assume that $\gamma$ admits an invariant (pure) state.
(This manipulation does not affect the above conditions
since approximately inner automorphisms act trivially on the K-groups.)
By the Pimsner--Voiculescu exact sequence \cite{PV},
conditions (1) and (2) imply the isomorphism
\[(K_0(C\rca{\gamma} \Gamma), [1]_0, K_1(C\rca{\gamma} \Gamma))\cong (\mathbb{Z}, 0, 0).\]
(See the second paragraph of the proof of Theorem 5.1 in \cite{Suz19} for details of computation.)
Let $\zeta \colon \Gamma \curvearrowright \bigotimes_\Gamma \mathcal{O}_\infty$
denote the tensor shift action.
Let $\theta$ denote the diagonal action of $\gamma$ and $\zeta$,
and let $\eta$ denote the diagonal action of $\theta$ and $\beta$.
By the Pimsner--Voiculescu exact sequence \cite{PV},
the inclusion map
\[C \rca{\gamma} \Gamma \rightarrow D:=\left(C \otimes \bigotimes_{\Gamma}\mathcal{O}_\infty\otimes \bigotimes_{\Lambda}\mathcal{O}_\infty \right) \rca{\eta} \Gamma\] induces an isomorphism on $K$-theory.
Note also that $D$
satisfies the universal coefficient theorem by \cite{PV} (see also Corollary 7.2 in \cite{RS}).
By the choice of $\alpha$, $D$ is nuclear (cf.~\cite{Suzeq}, Proposition B).
By Kishimoto's theorem \cite{Kis},
one can conclude that $C \rca{\gamma} \Gamma$, $(C\otimes \bigotimes_{\Gamma} \mathcal{O}_\infty) \rca{\theta} \Gamma$, and $D$ are simple and purely infinite
(see e.g.~Lemma 6.3 of \cite{Suz19b} for details).
By Cuntz's theorem \cite{CunK}, one can take a projection $p$ in $C \rca{\gamma} \Gamma$
which represents a generator of $K_0(C \rca{\gamma} \Gamma)$.
By the classification theorem of Kirchberg--Phillips \cite{Kir94b}, \cite{Phi}, we obtain
$p D p \cong \mathcal{O}_\infty$.
Next let $\xi\colon \Lambda \curvearrowright C \otimes \bigotimes_{\Gamma}\mathcal{O}_\infty\otimes \bigotimes_{\Lambda}\mathcal{O}_\infty$ denote the diagonal action of the trivial action $\Lambda \curvearrowright C \otimes \bigotimes_{\Gamma}\mathcal{O}_\infty$ and the tensor shift action $\Lambda \curvearrowright \bigotimes_\Lambda \mathcal{O}_\infty$.
Then $\xi$
commutes with $\eta$.
Therefore $\xi$ extends to the action \[\sigma \colon \Lambda \curvearrowright D\]
satisfying $\sigma_t(u_s)= u_s$ for all $s\in \Gamma$ and $t\in \Lambda$.
Observe that $p$ is $\sigma$-invariant.
Therefore $\sigma$ restricts to the action
\[\varsigma\colon \Lambda \curvearrowright pD p.\]
We show that $\varsigma$ satisfies the desired properties.
By the same argument as in the proof of Theorem \ref{Thm},
one can check that $\varsigma$ is outer.
Observe that
\[(pDp)^\varsigma = p D^\sigma p.\]
The proof of Lemma \ref{Lem:fp} together with Proposition 3.4 of \cite{Suz17} shows that
\[D^\sigma = \left(C \otimes \bigotimes_{\Gamma}\mathcal{O}_\infty\right) \rca{\theta} \Gamma.\]
Since $C$ admits a $\gamma$-invariant state,
the inclusion $(\bigotimes_\Gamma \mathcal{O}_\infty) \rca{\zeta} \Gamma \subset D^\sigma$
admits a conditional expectation (see Exercise 4.1.4 of \cite{BO} for instance).
Denote by $\mathbb{Z}_2 \wr \Gamma:= (\bigoplus_\Gamma \mathbb{Z}_2) \rtimes \Gamma$
the wreath product group: the semidirect product of $\bigoplus_\Gamma \mathbb{Z}_2$
by the left shift $\Gamma$-action.
To study properties of $D^\sigma$, we next construct an embedding
\[\rg(\mathbb{Z}_2 \wr \Gamma) \rightarrow \left( \bigotimes_\Gamma \mathcal{O}_\infty \right) \rca{\zeta} \Gamma\] whose image admits a conditional expectation.
Take a unital embedding
\[\iota \colon \rg(\mathbb{Z}_2) \cong \mathbb{C}\oplus \mathbb{C} \rightarrow \mathcal{O}_\infty.\]
Choose minimal projections $q_1, q_2$ in $\iota(\rg(\mathbb{Z}_2))$ with $q_1+q_2=1$.
Take a state $\varphi_i$ on $q_i \mathcal{O}_\infty q_i$
for $i=1, 2$.
Define $E\colon \mathcal{O}_\infty \rightarrow \iota(\rg(\mathbb{Z}_2))$
to be $E(x):=\varphi_1(q_1 xq_1)q_1 +\varphi_2(q_2 x q_2)q_2$, $x\in \mathcal{O}_\infty$.
Then $E$ is a conditional expectation.
We equip $\bigotimes_\Gamma\rg( \mathbb{Z}_2)$ with
the left shift $\Gamma$-action.
Notice that the canonical isomorphism
$ \rg(\bigoplus_\Gamma \mathbb{Z}_2) \cong\bigotimes_\Gamma\rg( \mathbb{Z}_2)$
preserves the left shift $\Gamma$-actions.
Hence it extends to an isomorphism
$\rg(\mathbb{Z}_2 \wr \Gamma)\cong [\bigotimes_\Gamma\rg( \mathbb{Z}_2)]\rc\Gamma $. We identify these two \Cs-algebras via this isomorphism.
Define
$\tilde{\iota} :=\bigotimes_\Gamma \iota \colon \bigotimes_\Gamma\rg( \mathbb{Z}_2) \rightarrow \bigotimes_\Gamma \mathcal{O}_\infty.$
Then $\tilde{\iota}$ is a $\Gamma$-equivariant unital embedding.
Therefore it extends to an embedding
\[\rg(\mathbb{Z}_2 \wr \Gamma)\rightarrow \left(\bigotimes_\Gamma \mathcal{O}_\infty\right) \rca{\zeta} \Gamma.\]
We show that this inclusion admits a conditional expectation.
Set $\tilde{E}:= \bigotimes_\Gamma E$.
Then $\tilde{E}$ is a $\Gamma$-equivariant conditional expectation of the inclusion $\tilde{\iota}(\bigotimes_\Gamma\rg( \mathbb{Z}_2)) \subset \bigotimes_\Gamma \mathcal{O}_\infty$.
By Exercise 4.1.4 in \cite{BO}, the map $\tilde{E}$ extends to the desired conditional expectation.
Now by Corollary 4 of \cite{Oz} and Theorem 12.3.10 of \cite{BO},
$\rg(\mathbb{Z}_2 \wr \Gamma)$, thus $ D^\sigma$ does not have the completely bounded approximation property.
Since $\rg(\mathbb{Z}_2 \wr \Gamma)$ fails the local lifting property by Corollary 3.7.12 in \cite{BO}, so does $D^\sigma$.
Since $D^\sigma$
is simple and purely infinite,
it is isomorphic to a corner of $(pDp)^\varsigma=p D^\sigma p$.
Thus $\varsigma$ possesses the desired properties.
\end{proof}
\begin{Rem}
Actions stated in Theorem \ref{Thm} and Proposition \ref{Prop}
can be arranged to be \emph{centrally free} in the sense of \cite{Suz19}
(see Definition 4.1 and the sentence below it).
To see this, we choose the action $\alpha$ in the proofs
to satisfy the additional condition that the fixed point algebra contains a unital simple (non-trivial) \Cs-subalgebra. (For instance, replace $\alpha$ by its diagonal action with the trivial action on $\mathcal{O}_\infty$.)
Then the central freeness of the resulting actions
follows from that of the Bernoulli shift actions over unital simple \Cs-algebras;
see Example 4.10 of \cite{Suz19}.
\end{Rem}
\begin{Rem}
Izumi's remarkable theorem (\cite{Izu04}, Theorem 4.2) states that any finite group $G$
admits a (unique) action on $\mathcal{O}_2$ which tensorially
absorbs all $G$-actions on simple unital separable nuclear \Cs-algebras (up to conjugacy).
Theorem \ref{Thm} suggests the non-existence of such an action for
countable infinite amenable groups.
In fact, if we have such $\alpha$,
then by Theorem \ref{Thm},
the fixed point algebra $\mathcal{O}_2^\alpha$ admits conditional
expectations onto
unital isomorphs of the fixed point algebras in Theorem \ref{Thm}.
(In particular $\mathcal{O}_2^\alpha$ fails the operator approximation property
and the local lifting property.)
A related problem is discussed in \cite{Oz04} (see Theorem 4 and the sentence above it).
Observe that the proof of Theorem 4 in \cite{Oz04} shows that,
letting $(\Gamma_i)_{i \in I}$ be a family of (countable) groups as in the statement,
there is no separable \Cs-algebra $A$ with the following property:
For any $i \in I$,
there are completely positive maps
$\varphi_{i} \colon \rg(\Gamma_i) \rightarrow A$
and $\psi_{i} \colon A \rightarrow L(\Gamma_i)$
with $\psi_i \circ \varphi_i = \id_{\rg(\Gamma_i)}$.
\end{Rem}
\begin{Rem}
Theorem \ref{Thm} and Proposition \ref{Prop} extend to second countable totally disconnected locally compact non-compact groups $\Lambda$.
To see this, choose a decreasing sequence $(K_n)_{n=1}^\infty$
of compact open subgroups of $\Lambda$ with trivial intersection.
Then use the tensor shift action over the $\Lambda$-set $I:=\bigsqcup_{n=1}^\infty \Lambda/ K_n$
instead of the plain Bernoulli shift actions in their proofs.
\end{Rem}
\subsection*{Acknowledgements}
The author is grateful to the referee for careful reading and helpful suggestions.
This work was supported by JSPS KAKENHI Early-Career Scientists
(No.~19K14550) and tenure track funds of Nagoya University.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,918
|
Die Altstadt Spandau auf der sogenannten "Altstadtinsel" in Berlin entwickelte sich ab 1232, als Spandau zum ersten Mal als Stadt erwähnt wurde und avancierte zum wirtschaftlich und militärisch rückwärtigen Stützpunkt für die Verteidigung der im Zuge der Unterwerfung der Slawen durch die Askanier nach Osten verschobenen Landesgrenze. Die zweite bedeutende Ansiedlung, die in dieser Zeit entstand, war die Burg auf der heute zum Ortsteil Haselhorst gehörenden Zitadelleninsel.
Stadtentwicklung
Im 19. Jahrhundert wurde Spandau zu einer der stärksten Festungen Preußens ausgebaut. Der seit dem 17. Jahrhundert bestehende Festungszwang, der erst 1903 endgültig aufgehoben wurde, hatte äußerst negative Folgen für die Stadtentwicklung. Zum einen waren dadurch die Ausdehnungsmöglichkeiten auf ein Minimum reduziert, zum anderen wirkte sich die Gewerbesteuerbefreiung der zum Militärfiskus gehörenden Heereswerkstätten negativ auf die Finanzen der Stadt aus. Hinzu kam, dass die Konjunktur im Rüstungssektor sehr stark von der militärischen Entwicklung sowie der allgemeinen politischen Lage abhing, mit entsprechend starken Schwankungen bezüglich des Arbeitskräfte- und des Wohnraumbedarfs. Erst nach der Aufhebung des Festungsstatus wurde Spandau mit seiner verkehrsgünstigen Lage als Standort auch für die Privatindustrie attraktiv.
Mit der erneut etablierten Rüstungsindustrie war Spandau Ende 1944 und Anfang 1945 Ziel schwerer Luftangriffe der Alliierten, die vor allem die Altstadt stark zerstörten. Auch wegen der radikalen Sanierung in den 1950er Jahren und des U-Bahn-Baus in den 1980er Jahren blieb von der alten Bausubstanz wenig erhalten. Gleichwohl vermittelt der weitgehend erhaltene überkommene Stadtgrundriss noch immer einen kleinstädtischen Eindruck. Zugleich trennen Havel und breite Verkehrsschneisen die Altstadt deutlich von den umgebenden Ortsteilen ab. Die 1978 eingeleitete Umgestaltung der Altstadt zu einer Fußgängerzone wurde 1989 nach mehr als zehn Jahren abgeschlossen. Hauptgeschäftsstraßen sind die Carl-Schurz-Straße (benannt nach dem Politiker Carl Schurz) und die Breite Straße. Seit 2001 steht der Handel in der Altstadt unter großem Konkurrenzdruck durch das benachbarte Einkaufszentrum Spandau Arcaden mit seinen 125 Geschäften am Bahnhof Spandau.
Während der COVID-19-Pandemie wurde mit Zunahme der Infektionszahlen im Herbst 2020 am 24. Oktober für die Altstadt Spandau und einige weitere Berliner Einkaufsstraßen eine Maskenpflicht für Fußgänger eingeführt.
Historische Bauwerke
St.-Nikolai-Kirche
Das bedeutendste Bauwerk ist die anstelle eines Vorgängerbaus aus dem 13. Jahrhundert errichtete St.-Nikolai-Kirche auf dem Reformationsplatz. Die dreischiffige gotische Backstein-Hallenkirche aus dem 14. Jahrhundert besitzt ein mächtiges Satteldach; 1989 wurde die Barockhaube des wuchtigen Westturms mit Schinkelschem Schmuckwerk nach Plänen von 1839 rekonstruiert.
Gotisches Haus
Ein bedeutendes mittelalterliches Baudenkmal aus dem ausgehenden 15. Jahrhundert ist das Gotische Haus in der Breiten Straße 32. Entgegen der damals vorherrschenden Holz- und Fachwerkbauweise wurde es als repräsentativer Steinbau errichtet. Teile des ursprünglichen Bauwerkes wie das Netzrippengewölbe im hinteren Teil und die Spitzbogenarkade sind noch erhalten.
Ein Brand im 18. Jahrhundert zerstörte große Teile des Gebäudes und im Zuge des Wiederaufbaus erhielt es eine klassizistische Fassade sowie eine geänderte Raumaufteilung. In diesem Zustand überdauerte das Gotische Haus die Zeit bis nach dem Zweiten Weltkrieg. Ende der 1950er Jahre gab es größere Eingriffe in die historische Bausubstanz, bis schließlich 1987 die Restaurierung in Angriff genommen wurde. Heute ist im Erdgeschoss des Gotischen Hauses die Tourist-Information Berlin-Spandau untergebracht; außerdem werden die Räumlichkeiten als Galerie für wechselnde Ausstellungen genutzt. In der ersten Etage befindet sich eine Dependance des Stadtgeschichtlichen Museums, das seinen Hauptsitz im Zeughaus der Zitadelle Spandau hat.
Wendenschloss und Stadtmauer
Vom Gotischen Haus gelangt man über den zentralen Platz der Spandauer Altstadt, dem Markt, zur Moritzstraße, die westwärts zum Mühlengraben führt. In der sie kreuzenden Jüdenstraße (von 1938 bis 2002 Kinkelstraße, benannt nach dem Theologen Gottfried Kinkel) lag das Wendenschloss (Nr. 33), ein aufwendiges Ackerbürgerhaus aus der Zeit um 1700. Das Originalgebäude wurde 1966 abgebrochen und durch einen Neubau ersetzt, dessen Fassade an das historische Vorbild erinnern soll.
An der südlichen Fortsetzung des Mühlengrabens, dem Viktoria-Ufer, ist noch ein 116 Meter langer Rest der Stadtmauer aus dem 14. Jahrhundert erhalten.
Literatur
Baedekers Allianz Reiseführer Berlin. Verlag Karl Baedeker GmbH, Ostfildern-Kemnat 1991, ISBN 3-87504-126-7, S. 222.
Ganz Berlin – Spaziergänge durch die Hauptstadt. Nicolaische Verlagsbuchhandlung GmbH, Berlin 2007, ISBN 978-3-89479-390-6.
Weblinks
Online-Reiseführer für Spandau. Sehenswürdigkeiten, Kultur, Spaziergänge, Touren durch die Altstadt und darüber hinaus.
Einzelnachweise
Berlin-Spandau
Spandau
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,902
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Stephan Mølvig (Odense 13 februari 1979) is een voormalig Deens roeier. Mølvig maakte zijn debuut tijdens de Wereldkampioenschappen roeien 2000 voor niet Olympische roeinummers en behaalde in de lichte-acht de vijfde plaats. Tijdens de Wereldkampioenschappen roeien 2001 nam Mølvig plaats in de Olympische lichte-vier-zonder-stuurman en won de zilveren medaille. In de twee daarop volgende jaren werd Mølvig wereldkampioen in de lichte-vier-zonder-stuurman. Mølvig sloot zijn carrière af met de gouden medaille tijdens de Olympische Zomerspelen 2004.
Resultaten
Wereldkampioenschappen roeien 2000 in Zagreb 5e in de lichte-acht
Wereldkampioenschappen roeien 2001 in Luzern in de lichte-vier-zonder-stuurman
Wereldkampioenschappen roeien 2002 in Sevilla in de lichte-vier-zonder-stuurman
Wereldkampioenschappen roeien 2003 in Milaan in de lichte-vier-zonder-stuurman
Olympische Zomerspelen 2004 in Athene in de lichte-twee-zonder-stuurman
Deens roeier
Deens olympisch kampioen
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,610
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using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;
namespace GameCloud.Manager.PluginContract.Requests
{
public class UpdateRequestInfo<TBody> : PluginRequestInfo
{
public UpdateRequestInfo(PluginRequestMethod method, List<PluginRequestParameter> parameters)
: base(method, parameters)
{
}
public TBody Body
{
get
{
return this.GetParameterValue<TBody>("data");
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,825
|
from pyface.resource.api import ResourceFactory
class PyfaceResourceFactory(ResourceFactory):
""" The implementation of a shared resource manager. """
###########################################################################
# 'ResourceFactory' toolkit interface.
###########################################################################
def image_from_file(self, filename):
""" Creates an image from the data in the specified filename. """
# Just return the data as a string for now.
f = open(filename, 'rb')
data = f.read()
f.close()
return data
def image_from_data(self, data):
""" Creates an image from the specified data. """
return data
#### EOF ######################################################################
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 765
|
package net.ssehub.kernel_haven.code_model.ast;
import net.ssehub.kernel_haven.util.null_checks.NonNull;
/**
* A visitor for traversing an {@link ISyntaxElement}-based AST. The default operation will visit all nested elements,
* but won't do any other operation.
*
* @author El-Sharkawy
*/
public interface ISyntaxElementVisitor {
// C-Preprocessor
/**
* <b>CPP:</b> Visits a C-preprocessor block. This may be one of the following blocks:
* <ul>
* <li>#if</li>
* <li>#ifdef</li>
* <li>#ifndef</li>
* <li>#elif</li>
* <li>#else</li>
* </ul>
* Provides default Visitation: <b>true</b>
* @param block The block to visit.
*/
public default void visitCppBlock(@NonNull CppBlock block) {
for (int i = 0; i < block.getNestedElementCount(); i++) {
block.getNestedElement(i).accept(this);
}
}
/**
* <b>CPP:</b> Visits a C-preprocessor statement. This may be one of the following blocks:
* <ul>
* <li>#include</li>
* <li>#define</li>
* <li>#undef</li>
* <li>#warning</li>
* <li>#error</li>
* <li>#line</li>
* <li>#empty</li>
* <li>#pragma</li>
* </ul>
* Provides default Visitation: <b>true</b>
* @param cppStatement The statement to visit.
*/
public default void visitCppStatement(@NonNull CppStatement cppStatement) {
ICode expression = cppStatement.getExpression();
if (expression != null) {
expression.accept(this);
}
}
// C-Code
/**
* <b>C-Code:</b> Visits a C code file. This is usually the entry point for visitation as it is the top level
* element of all C code files.
* <p>
* Provides default Visitation: <b>true</b>
* @param file The file to visit.
*/
public default void visitFile(@NonNull File file) {
for (int i = 0; i < file.getNestedElementCount(); i++) {
file.getNestedElement(i).accept(this);
}
}
/**
* <b>C-Code:</b> Visits a function. These are usually at the top level of a file.
* <p>
* Provides default Visitation: <b>true</b>
*
* @param function The function to visit.
*/
public default void visitFunction(@NonNull Function function) {
function.getHeader().accept(this);
for (int i = 0; i < function.getNestedElementCount(); i++) {
function.getNestedElement(i).accept(this);
}
}
/**
* <b>C-Code:</b> Visits a single statement. This represents assignments, declarations, etc. (basically everything
* that ends with a semicolon and does something).
* <p>
* Provides default Visitation: <b>true</b>
*
* @param statement The statement to visit.
*/
public default void visitSingleStatement(@NonNull SingleStatement statement) {
statement.getCode().accept(this);
}
/**
* <b>C-Code:</b> Visits a compound statement. This is a block of statements
* <p>
* Provides default Visitation: <b>true</b>
*
* @param block The block to visit.
*/
public default void visitCompoundStatement(@NonNull CompoundStatement block) {
for (int i = 0; i < block.getNestedElementCount(); i++) {
block.getNestedElement(i).accept(this);
}
}
/**
* <b>C-Code:</b> Visits a list of unparsed code objects.
* <p>
* Provides default Visitation: <b>true</b>
*
* @param code The code to visit.
*/
public default void visitCodeList(@NonNull CodeList code) {
for (int i = 0; i < code.getNestedElementCount(); i++) {
code.getNestedElement(i).accept(this);
}
}
/**
* <b>C-Code/CPP:</b> Visits unparsed code.
* <p>
* Provides default Visitation: <b>false</b>
* @param code The unparsed code element to visit.
*/
public default void visitCode(@NonNull Code code) {
// nothing to do
}
/**
* <b>C-Code:</b> Visits a type definition.
* <p>
* Provides default Visitation: <b>true</b>
*
* @param typeDef The type definition to visit.
*/
public default void visitTypeDefinition(@NonNull TypeDefinition typeDef) {
typeDef.getDeclaration().accept(this);
for (int i = 0; i < typeDef.getNestedElementCount(); i++) {
typeDef.getNestedElement(i).accept(this);
}
}
/**
* <b>C-Code:</b> Visits a label.
* <p>
* Provides default Visitation: <b>true</b>
*
* @param label The label to visit.
*/
public default void visitLabel(@NonNull Label label) {
label.getCode().accept(this);
}
/**
* <b>C-Code:</b> Visits a comment.
* <p>
* Provides default Visitation: <b>true</b>
*
* @param comment The Comment to visit.
*/
public default void visitComment(@NonNull Comment comment) {
comment.getComment().accept(this);
}
// C Control structures
/**
* <b>C control structure:</b> Visits a branching statement (if, else if or else).
* <p>
* Provides default Visitation: <b>true</b>
*
* @param branchStatement The branching statement to visit.
*/
public default void visitBranchStatement(@NonNull BranchStatement branchStatement) {
ICode condition = branchStatement.getIfCondition();
if (null != condition) {
condition.accept(this);
}
for (int i = 0; i < branchStatement.getNestedElementCount(); i++) {
branchStatement.getNestedElement(i).accept(this);
}
}
/**
* <b>C control structure:</b> Visits a switch statement.
* <p>
* Provides default Visitation: <b>true</b>
*
* @param switchStatement The switch to visit.
*/
public default void visitSwitchStatement(@NonNull SwitchStatement switchStatement) {
switchStatement.getHeader().accept(this);
for (int i = 0; i < switchStatement.getNestedElementCount(); i++) {
switchStatement.getNestedElement(i).accept(this);
}
}
/**
* <b>C control structure:</b> Visits a case statement.
* <p>
* Provides default Visitation: <b>true</b>
*
* @param caseStatement The case statement to visit.
*/
public default void visitCaseStatement(@NonNull CaseStatement caseStatement) {
ICode condition = caseStatement.getCaseCondition();
if (null != condition) {
condition.accept(this);
}
for (int i = 0; i < caseStatement.getNestedElementCount(); i++) {
caseStatement.getNestedElement(i).accept(this);
}
}
/**
* <b>C control structure:</b> Visits a loop statement (while, do-while or for).
* <p>
* Provides default Visitation: <b>true</b>
*
* @param loop The loop to visit.
*/
public default void visitLoopStatement(@NonNull LoopStatement loop) {
loop.getLoopCondition().accept(this);
for (int i = 0; i < loop.getNestedElementCount(); i++) {
loop.getNestedElement(i).accept(this);
}
}
// Special
/**
* <b>Special:</b> Visits an unparseable element.
* <p>
* Provides default Visitation: <b>true</b>
*
* @param error The {@link ErrorElement}.
*/
public default void visitErrorElement(@NonNull ErrorElement error) {
for (int i = 0; i < error.getNestedElementCount(); i++) {
error.getNestedElement(i).accept(this);
}
}
/**
* <b>Special:</b> Visits a reference.
* Provides default Visitation: <b>true</b>
*
* @param referenceElement The reference to another element.
*/
public default void visitReference(@NonNull ReferenceElement referenceElement) {
referenceElement.getReferenced().accept(this);
}
}
|
{
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| 3
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Przedsiębiorstwo Państwowe "Film Polski" – polskie przedsiębiorstwo państwowe, istniejące w latach 1945–1952. Jego zadaniem była produkcja i dystrybucja filmów oraz organizacja życia kulturalnego.
Historia i zadania
Przedsiębiorstwo Państwowe Film Polski powstało na mocy dekretu Krajowej Rady Narodowej z dnia 13 listopada 1945 roku o utworzeniu przedsiębiorstwa państwowego "Film Polski" (). Jego siedzibą była Warszawa. Zgodnie z zapisami dekretu do zadań przedsiębiorstwa należały: zakładanie i prowadzenie wytwórni: taśmy filmowej i papierów fotograficznych, sprzętu kinotechnicznego oraz filmów, sprzedaż i wynajem filmów polskich w kraju i za granicą oraz zakup i wynajem filmów zagranicznych na terenie Polski, zakładanie i prowadzenie teatrów świetlnych (kin) oraz publiczne wyświetlanie filmów. Ponadto Film Polski zajmował się działalnością propagandową i kulturotwórczą, szkoleniem kadr oraz współpracą zagraniczną. W zakresie zakładania i prowadzenia wytwórni taśmy filmowej i papierów fotograficznych oraz sprzedaży i wynajmu filmów w kraju i za granicą oraz zakupu i wynajmu filmów zagranicznych przedsiębiorstwo było monopolistą (art. 4 ust 1 dekretu).
W 1945 roku podjęto decyzję o przejęciu na potrzeby przedsiębiorstwa hali sportowej przy ul. Łąkowej 29 w Łodzi i jej adaptacji na halę zdjęciową. W pomieszczeniach tych zrealizowano pierwszy polski powojenny film fabularny – Zakazane piosenki (reż. Leonard Buczkowski).
W 1952 roku przedsiębiorstwo zostało przekształcone w Centralny Urząd Kinematografii, a w roku 1957 w Naczelny Zarząd Kinematografii Ministerstwa Kultury i Sztuki.
Dyrektorzy
1945–1947: Aleksander Ford
1947–1952: Stanisław Albrecht
Produkcja filmowa
Podczas swojego funkcjonowania Przedsiębiorstwo Państwowe Film Polski wyprodukowało 110 filmów: dokumentalnych, fabularnych oraz animowanych. Pierwszym wyprodukowanym filmem był dokument pt. Wieś i miasto z 1945 roku (realizacja: Eugeniusz Jaryczewski, zdjęcia: Seweryn Kruszyński).
Filmy fabularne wyprodukowane przez Film Polski
1945:
Zakazane piosenki
1946:
W chłopskie ręce
Dwie godziny
1947:
Zdradzieckie serce – film nigdy nie wszedł na ekrany
Włókiennictwo na Ziemiach Odzyskanych (krótkometrażowy)
Ruchy ludności na ziemiach polskich (krótkometrażowy)
Ostatni etap
Nawrócony – film nigdy nie wszedł na ekrany
Jasne łany
Harmonia (krótkometrażowy)
1948:
Stalowe serca
Skarb
Powrót (Ślepy tor)
1949:
Za wami pójdą inni
Dom na pustkowiu
Czarci żleb
1950:
Warszawska premiera
Miasto nieujarzmione
Dwie brygady
Przypisy
Bibliografia
Linki zewnętrzne
Polskie wytwórnie filmowe
Kinematografia w Łodzi
Przedsiębiorstwa w Warszawie
Byłe przedsiębiorstwa w Polsce
Dystrybutorzy filmowi
|
{
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| 637
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