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Although diabetic, leading Azerbaijani journalist on hunger strike in prison
OCCRP
Reporters Without Borders (RSF) is extremely concerned about Afgan Mukhtarli, an imprisoned Azerbaijani journalist who has begun a hunger strike despite being diabetic. RSF holds the Azerbaijani authorities responsible for his fate and calls on the international community to press for his immediate release.
This well-known investigative journalist is in great danger because, although suffering from both diabetes and hypertension, he began a hunger strike on 22 September in protest against the arbitrary treatment that both he and his lawyer are receiving from the prison authorities.
"He is aware that this hunger strike is putting his life in danger," his wife, Leyla Mustafayeva told RSF. "He has taken this decision despite everything because the judicial procedures offer him no hope." The only response from the prison authorities to his hunger strike has been to place him in solitary confinement.
Mukhtarli was sentenced to six years in prison on an absurd charge after being abducted and brought back by force from self-imposed exile in neighbouring Georgia in May 2017. As if that were not enough, the prison authorities have done everything possible to cut him off from the world, limiting visits and appeals to the utmost and putting pressure on his fellow inmates.
Mukhtarli finally decided he could stand no more after his lawyer, Nemat Kerimli, was searched on entering Mukhtarli's cell on 20 September and again on leaving, when prison officials demanded to see the notes he had taken during the meeting. Kerimli says that when he refused, the guards slammed him against a wall, grabbed his bag and locked him up for three quarters of an hour while they read his notes.
Kerimli had already been prevented from visiting Mukhtarli at the end of August. Mukhtarli suggests that this latest harassment could be a reprisal for the visits he received during the summer from two senior European officials. He is demanding an end to the impunity enjoyed by the prison staff, starting with the prison's deputy director, Emin Aliyev.
"We are extremely concerned about Afgan Mukhtarli and we hold the Azerbaijani authorities responsible for his fate," said Johann Bihr, the head of RSF's Eastern Europe and Central Asia desk.
"From abduction in Georgia to harassment in prison, the injustice to which he is being subjected is intolerable. We call on the Azerbaijani government to release him as a matter of urgency, or else we will do everything possible to have international sanctions placed on everyone involved in persecuting him."
Mukhtarli was kidnapped on 29 May 2017 in the Georgian capital, Tbilisi, where he had been living for three years, often writing about high-level government corruption in Azerbaijan. Against all evidence, the Azerbaijani authorities claimed that he was arrested with 10,000 euros in his pockets after entering Azerbaijan illegally. Georgia's investigation into his abduction has not so far yielded any conclusive results.
Azerbaijan is ranked 166th out of 180 countries in RSF's 2019 World Press Freedom Index. The leading critical media outlets have been silenced or have had to relocate abroad, the main independent websites are blocked and at least six journalists are currently in prison.
Legal framework and justice systemViolence against journalistsAdvocacy
RSF and partners urge the authorities to investigate the harassment campaign against Emilia Șercan
Four months after the journalist Emilia Șercan filed complaints for the publication of her private pictures and the alleged leak of key elements of the case, RSF and partners are calling on the prosecution to commit to pursue the investigations.
Legal framework and justice systemDigital space and democracyNews
Media bill poses major threat to press freedom in northern Cyprus
The administration has approved a bill that threatens journalists with arbitrary prosecution. RSF urges the local parliament to reject it and instead prefer European methods for promoting reliable information over Turkey's repressive ones.
Disinformation and propagandaInternational laws and governanceDigital space and democracyModels and good practicesNews
RSF's proposals for ambitious, innovative European Media Freedom Act
RSF calls on the European Commission, which is drafting a European Media Freedom Act, to be ambitious in its efforts to safeguard and ensure respect for this fundamental freedom at a time when it faces increased threats and unprecedented challenges. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,523 |
Die Internationale Gesellschaft für Regenbogenfische (IRG) ist ein Verein, dessen Mitglieder sich aquaristisch und wissenschaftlich mit den australisch-ozeanischen Vertretern der Süßwasserfische, hauptsächlich den Regenbogenfischen, beschäftigt. Besonderes Ziel der IRG ist zur Arterhaltung und artgerechten Haltung und Aufzucht dieser Fische beizutragen. Der Verein mit Sitz in Düsseldorf wurde am 7. Juni 1986 auf Initiative von Harro Hieronimus gegründet. Die IRG ist Mitglied im Bundesverband für fachgerechten Natur- und Artenschutz (BNA).
Die etwa 500 Mitglieder kommen überwiegend aus Europa. Innerhalb Deutschlands haben sich sechs Regionalgruppen organisiert, daneben gibt es Ländergruppen in Belgien, den Niederlanden, Norwegen, Österreich, der Schweiz und der Tschechischen Republik.
Zeitschrift
Der Verein gibt seit 1986 viermal jährlich die Vereinszeitschrift Regenbogenfisch (zurzeit Ausgaben in deutscher und niederländischer Sprache) mit Fachbeiträgen sowie gelegentlich thematische Sonderhefte heraus.
Aktivitäten
Die Mitglieder treffen sich jährlich im Juni abwechselnd in Deutschland und einem anderen europäischen Staat zu einem mehrtägigen Jahrestreffen. Es dient dem intensiven Austausch zwischen Hobby und Wissenschaft durch Fachvorträge und eine internationale Fischbörse.
Neben der Publikation von Fachbüchern und Artikeln in aquaristischen Fachzeitschriften dokumentieren Mitglieder der IRG auch die Lebensräume der Fische in der Natur, entdecken neue Arten und beteiligen sich an deren wissenschaftlicher Beschreibung sowie an Importen. Sie arbeiten dabei mit anderen Organisationen, wie der Australia New Guinea Fishes Association (ANGFA) und Ichthyologen zusammen. Beispielsweise werden Gewebeproben von in Aquarien gehaltenen Fischen zur Verfügung gestellt für genetische Studien zur Diversität der Regenbogenfische.
Mitglieder sind beteiligt, den in der Natur ausgestorbenen Eachamsee-Regenbogenfisch (Melanotaenia eachamensis) zu erhalten. Dieser Endemit aus dem australischen Lake Eacham im nördlichen Queensland wurde erst 1978 entdeckt, 1982 wissenschaftlich beschrieben und war bereits 1989 durch für den Angelsport in den See ausgesetzte andere Fischarten ausgerottet. Der Lebensraum für Melanotaenia eachamensis ist nicht mehr zurückzugewinnen. Weltweit wird die Art phaenotypisch von Institutionen und Liebhabern erhalten.
Weblinks
Offizielle Website
Einzelnachweise
Aquaristik
Verein (Düsseldorf)
Gegründet 1986 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,786 |
P.F. Chang's in Lakewood affected by security breach
by: Chuck Hickey
Posted: Aug 4, 2014 / 08:40 AM MDT / Updated: Aug 4, 2014 / 09:34 PM MDT
The P.F. Chang's restaurant in Belmar shopping area in Lakewood was one of 33 in the country that were hit by a security breach, the company announced Monday, Aug. 4, 2014.
SCOTTSDALE, Ariz. -- Restaurant chain P.F. Chang's China Bistro said Monday it has identified 33 locations, including its restaurant in Lakewood, that were involved in a security breach involving credit and debit card data.
In a statement by P.F. Chang's CEO Rick Federico, the company was contacted by the Secret Service on June 10 that the chain might have been compromised involving credit and debit card data, reportedly stolen from 33 restaurants around the country. The security breach occurred from Oct. 19 to June 11.
Federico said the company initiated an internal investigation, which discovered credit and debit card numbers, names and expiration dates of some customers' cards that were stolen.
The investigation determined the locations, and specific date time frames the card-processing system was compromised for each. However, specific card-holders have not been determined.
The Lakewood restaurant is in the Belmar shopping area at 7210 W. Alameda Ave. and the breach there occurred from April 10 to June 11.
Customers can learn more about identity protection services being provided for potentially affected guests at pfchangs.com/security. Customers can also call 1-877-412-7152.
Boebert says she won't attend inauguration, says Trump is 'one of the greatest' US presidents | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,596 |
Q: AngularJS Form custom validation. Cannot read property '$validators' of undefined I want to create a custom form validation, using AngularJS. That form should have input and select elements. The form should be valid, when either imputs are empty or both filled with some values. Here is the view:
<form name="recipientsForm" novalidate>
<md-input-container>
<label>Name</label>
<input name="name" type="text" ng-model="relationship.name" value="" empty-or-both-filled="relationship.relationshipType">
<div ng-messages="recipientsForm.name.$error">
<div ng-message="emptyOrBothFilled">Enter name.</div>
</div>
</md-input-container>
<md-input-container>
<md-select name="type" placeholder="Select your relation... " ng-model="relationship.relationshipType" empty-or-both-filled="relationship.name">
<md-option ng-repeat="type in relationshipTypes" value="{{type.relationshipType}}">
{{type.name}}
</md-option>
</md-select>
<div ng-messages="recipientsForm.type.$error">
<div ng-message="emptyOrBothFilled">Pick relationship.</div>
</div>
</md-input-container>
</form>
And here is the directive:
(function () {
'use strict';
angular
.module('app')
.directive('emptyOrBothFilled', [emptyOrBothFilled]);
function emptyOrBothFilled() {
return {
restrict: 'A',
required: 'ngModel',
scope: {
targetNgModel: '=emptyOrBothFilled'
},
link: function($scope, element, attrs, ngModel) {
ngModel.$validators.emptyOrBothFilled = function(val) {
var isValueEmpty = !val;
var isTargetEmpty = !$scope.targetNgModel;
return (isTargetEmpty && isValueEmpty) || (!isTargetEmpty && !isValueEmpty);
}
$scope.$watch('targetNgModel', function() {
ngModel.$validate();
})
}
}
}
})();
Prompt, please, why do I get this error:
TypeError: Cannot read property '$validators' of undefined
at link (http://localhost:3000/app/shared/directives/EmptyOrBothFilled.js:17:24)
A: It should be
require: 'ngModel',
not
required: 'ngModel',
in the directive specification.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 47 |
\section{Introduction}
Topological states of matter have recently attracted immense scientific interest
which was in particular boosted by the theoretical
prediction\cite{BernevigZhang, BernevigHughesZhang, FuKaneMele, MooreBalents,
Roy} and subsequent experimental discovery\cite{KonigMolenkampetc, Hsieh} of
two-dimensional (2D) and three-dimensional (3D) time reversal invariant
topological insulators.
In their bulk, topological
insulators\cite{HasanKaneReview,QiZhang,SchnyderLudwig08,Kitaev09} (TI) are
electronic band insulators characterized by a topological invariant which
accounts for the non-trivial structure of the Bloch states. In contrast, the
interface between two topologically different phases (e.g. TI - vacuum) hosts
gapless, extended boundary states. Their appearance is topologically protected
via the bulk-boundary correspondence.\cite{Gurarie} In retrospect we understand
that the quantum Hall effect (QHE)\cite{vonKlitzing} at given quantized
transverse conductance was the first example of a topological insulator: The
Landau levels provide the bulk band gap which is accompanied by the topological
TKNN\cite{TKNN82} number and the protected chiral edge states.
In contrast to the QHE, the newly discovered 2D and
3D topological insulators require the absence of magnetic
field and rely on strong spin-orbit interaction. Further, their topological
invariant takes only values in $\mathbb Z_2$ (contrary to the TKNN integer). The
2D TI phase (also known as quantum spin Hall state) was experimentally
identified by the characteristic quantized value $2 e^2/h$ of the
two-point conductance in HgTe quantum wells.\cite{KonigMolenkampetc} The discriminating feature of all
3D TI is the massless Dirac states on the 2D boundary which were first
spectroscopically detected in BiSb\cite{Hsieh} alloys and subsequently in many
other materials.\cite{HasanKaneReview} To present date, various experimental
groups confirmed predominant surface state transport (for a review see Ref.
\onlinecite{Culcer}), in particular elucidating ambipolar field-effect\cite{Steinberg,CheckelskyHor2011,KimFuhrer2012,HongCui2012, KimQuiziFuhrer2012} and the typical QHE-steps of Dirac
electrons,\cite{ChengChen2010,HanaguriSasagawa2010,AnalytisFisher2010,SacepeMorpurgo2011,BruehneHgTe} Aharonov-Bohm oscillations\cite{PengCui2010,XiuWang2011,DufouleurGiraud2013} as well as weak antilocalization (WAL) corrections in the magnetoconductivity data.\cite{SteinbergPRB,ZhangLi2011,Chen} Moreover, several transport experiments reveal the importance of electron-electron interactions in 3D TI materials.\cite{Chen,Wang,Liu}
Inspired by recent experimental advances, we present here a detailed analysis of
interference and interaction corrections to conductivity in the most conventional setup for
transport experiments: the slab geometry, in which the 3D TI films are rather
thin (down to $\sim 10$ nm) although still thick enough to support well
separated surface states. As we will explain in more detail, the long-range
Coulomb interaction between the two major surfaces plays an important role. We
derive the quantum corrections to conductivity in the diffusive regime by taking
into consideration the WAL effect as well as
corrections of Altshuler-Aronov type \cite{AltshulerAronov}
induced by inter- and intrasurface interaction. We consider the
general situation of different surfaces subjected to different random potentials, mismatch in carrier densities
and unequal dielectric environment.
The present paper constitutes a natural extension of the previous
work\cite{OGM2010} by three of the authors in which a single 3D TI surface was
analyzed. It was found that the interplay of topological protection and
interaction- and interference-induced conductivity corrections drives the system
into a novel critical state with longitudinal conductance of the order of
$e^2/h$. As we show below, the intersurface interaction in a thin
TI slab makes the overall picture much more complex and crucially affects the
ultimate infrared behavior.
In another recent paper, \cite{BurmistrovGornyiTikhonov} two of us were involved in the
theoretical investigation of inter- and intrawell interaction effects in double quantum
well heterostructures studied experimentally in Ref. \onlinecite{Minkov}. Let us point out key
differences between the present paper and that work. First,
in Ref.~\onlinecite{BurmistrovGornyiTikhonov} only
equal carrier densities were considered. Second, disorder was assumed to be the
same in both quantum wells (and thus completely correlated). This affects the
soft-mode content of the low-energy theory. Third, quantum
wells host electrons with spin degeneracy which can be lifted by a
magnetic field. As a consequence, i) electrons in double quantum well fall into a symmetry class different
from that of 3D TI and ii) more interaction channels have to be included.
These {subtleties} affect in a crucial way the renormalization group (RG) flow: according to the analysis
of Ref.~\onlinecite{BurmistrovGornyiTikhonov}, the interwell interaction becomes irrelevant at
low energies, which is opposite to what we find in the two-surface TI model in
the present paper. Finally, the TI problem shows topology-related effects that
{were} absent in the double quantum well structure.
As in the two preceding works, we here use the interacting, diffusive non-linear
sigma model (NL$\sigma$M) approach to capture the diffusive low-energy physics.
Quantum corrections to the longitudinal conductivity $\sigma$ are obtained by
renormalization of this effective action in the one loop approximation (i.e.
perturbatively in $1/\sigma$ but exactly in interaction amplitudes). The
interacting NL$\sigma$M was originally developed by
Finkel'stein in the eighties \cite{Finkelstein1983,Finkelstein1984} (for review articles see Ref.s \onlinecite{Finkelstein1990,BelitzKirkpatrick,Finkelstein2010}). {In addition to perturbative RG treatment (which can also be performed diagrammatically \cite{CastellaniDiCastro}) it also allows one to incorporate topological effects} and was thus a
fundamental tool for understanding the interplay of disorder and interactions in
a variety of physical problems, including the superconducting
transition in dirty films, \cite{Finkelstein1987,FinkelsteinSITreview} the integer
QHE, \cite{PruiskenBaranov,PruiskenBurmi} and the metal-insulator transition in
Si MOSFETs. \cite{PunnooseFinkelstein}
Analyzing the RG equations for the thin 3D TI film, we find that (in contrast to the previous work on the double quantum well structure) the intersurface interaction is relevant in the
RG sense. The system flows towards a metallic fixed point at which even two
originally different surfaces are characterized by the same conductivities. As we discuss in detail below, the hallmark of the intersurface
interaction in 3D TI transport experiments is a characteristic non-monotonic
temperature dependence of the conductivity. In contrast to the case
of decoupled surfaces, due to the intersurface interaction, quantum corrections to the conductivity depend on the carrier densities.
The paper is structured as follows. In Sec.~\ref{sec:Formalism}
we expose in detail the theoretical implications of a typical experimental slab
geometry setup, demonstrate the relevance of intersurface interaction and
introduce the microscopic fermionic Hamiltonian. Subsequently (Sec.
\ref{sec:towardsNLSM}), we use the non-Abelian bosonization technique to map the
fermionic theory on the ($\mathbf U\left (1\right )$-) gauged, interacting
NL$\sigma$M with $\mathbb Z_2$ topological term. Here we also discuss the Fermi
liquid treatment of generally strong electron-electron interactions. Next, we
renormalize the NL$\sigma$M in Sec.~\ref{sec:1loopRG}. {Sections \ref{sec:towardsNLSM} and \ref{sec:1loopRG} contain both pedagogical explanations and important details for experts. Readers purely interested in the results can jump to} Sec. \ref{sec:AnalysisofRG}, where the RG flow and
the implied phase diagram are analyzed.
Detailed predictions for typical experiments can be found in
Sec.~\ref{sec:experiment}. We close the paper by
summarizing our results and discussing prospects for
future work in Sec.~\ref{sec:conclusions}.
\section{Topological insulator slabs: Experimental setup and theoretical model}
\label{sec:Formalism}
\subsection{Setup}
\label{sec:setup}
In this work we analyze the effect of interaction on transport properties of
strong 3D topological insulator thin films in the diffusive regime. While we
mainly focus on the theoretically most interesting case of purely surface
transport,
we also show that our theory can easily be extended to a case when only a
part of the sample is in the topological phase, i.e. one has a
conduction through a topologically protected surface spatially separated from a
thick (bulk) conducting region.
A typical experimental setup is shown in Fig.~\ref{fig:setup}.
Our analysis is
valid in the regime where the penetration depth of
surface states $a$ is small with respect to the film thickness $d$. We
therefore neglect intersurface tunneling (which would destroy the topological
protection).
Further, we assume the disorder correlation length (depicted by the range of
the impurity potentials) to be small $\xi \ll d$. We treat a generic case
when the vicinity to the coat or, respectively, to the substrate may induce
a different degree of disorder on the top and bottom surfaces. We thus consider the corresponding mean free
paths $l_1$ and $l_2$ as two independent parameters.
Moreover, we also allow the chemical potentials $\mu_1$ and $\mu_2$ on the two
surfaces to be different. {(By convention we set $\mu_s = 0$ at the Dirac point. Here and below $s = 1,2$ denotes the surface index.)}
The chemical potentials may be experimentally controlled by means of
electrostatic gates. As has been stated above, we mostly focus on the situation
where both
$\mu_1$ and $\mu_2$ lie well within the bulk gap $\Delta_{\rm bulk}$. The
extension
of our results to the experimentally important regime when only one of
chemical potentials is located within the bulk gap, $\vert \mu_1 \vert \ll
\Delta_{\rm bulk} \lesssim \vert \mu_2 \vert$, can be found in section
\ref{sec:surfacebulk}.
If the electrostatic gates are present and too close\footnote{Closer than the
typical length scale $L_E$ of the system, see Eq. \eqref{eq:Diffregime2}.} to
the sample, Coulomb interaction is externally screened and the electron-electron
interaction is purely short range. However, such an experimental scenario is
a rare
exception from the rule. Therefore, in the main text we assume sufficiently
distant gates and concentrate on the limit of long-range Coulomb interaction. In
addition we derive general RG equations (Appendix \ref{sec:RGderiv})
which allow us to explore the crossover from the long-range case
to the short-range one, see Appendix \ref{sec:shortrange}. Qualitatively,
the RG flow for a sufficiently strong short-range interaction in the
case of externally screened surfaces turns out to be similar to the flow in
the absence of external screening.
Since we assume that the thickness $d$ of the sample is much smaller than its
other linear dimensions, we neglect contributions of four side faces of the
slab (whose area is proportional to $d$).
\begin{figure}
\includegraphics[scale=.44]{Setupwithgates}
\caption{Scheme of a typical experimental setup. The hierarchy of length scales
is explained in the main text.}
\label{fig:setup}
\end{figure}
The goal of the present analysis is to study conduction properties of thin 3D TI films in the
diffusive regime, i.e., at energy scales $E$ far below the elastic
scattering rates $1/\tau_{s}$ of both surfaces,
\begin{equation}
E \ll \min_{s = 1,2} \hbar/ {\tau_{s}}\, . \label{eq:Diffregime1}
\end{equation}
In turn the elastic scattering rates are assumed to be small compared to the chemical potentials
\begin{equation}
\hbar/\tau_{s} \ll \vert \mu_s \vert. \label{eq:weakdisorder}
\end{equation}
In experiment $E$ is set
by the AC frequency ($E = \hbar \omega$) or by temperature ($E = k_B T$),
whichever of the two is larger. Equation \eqref{eq:Diffregime1} is equivalent to
the hierarchy of length scales
\begin{equation}
l \ll L_E \label{eq:Diffregime2}\,,
\end{equation}
where we have introduced the maximal
mean free path $l = \max_{s=1,2} l_s$ and the length scale $L_E = \min_{s=1,2}
(\hbar D_s / E)^{1/2}$, with $D_s$ being
the diffusion coefficients for the two surfaces.
\subsection{Interaction}
\label{sec:IAimportant}
Can Coulomb interaction between the top and bottom surface states play an important role in
the experiment? To answer this question, we compare the sample thickness with
all natural length scales of the system: the screening length $l_{\rm scr}$, the
(maximal) mean free path $l$ and the experimentally tunable scale $L_E$.
The Coulomb interaction is (throughout the paper underlined symbols denote
$2\times 2$ matrices in the surface space)
\begin{equation}
\underline U_0 \left ( \v r \right )= \frac{e^2}{\epsilon}
\left (\begin{array}{cc}
\frac{1}{r} & \frac{1}{\sqrt{ r^2 + d^2}} \\
\frac{1}{\sqrt{r^2 + d^2}} & \frac{1}{r} \label{eq:Coulombnoelstat}
\end{array} \right ).
\end{equation}
The two dimensional vector $\v r$ connects the two dimensional positions of the
particles, $r=|{\bf r}|$, $e$ is the charge of the electrons, and $\epsilon$
denotes the effective dielectric constant.
Fourier transformation and RPA-screening leads to \cite{ZhengMacDonald,
KamenevOreg, Flensberg, BurmistrovGornyiTikhonov} ($U \equiv U\left ( \v q
\right ) \equiv 2\pi e^2/\epsilon q$)
\begin{equation}
\underline U_{\rm scr} \left ( \v q \right ) = \frac{\underline U}{1-
\left (\Pi_1 + \Pi_2 \right )U + U^2 \Pi_1 \Pi_2 \left (1 - e^{-2dq}\right )}
\label{eq:URPA}
\end{equation}
with
\begin{equation}
\underline U = U \left (\begin{array}{cc}
1-\Pi_2 U \left (1 - e^{-2dq}\right ) & e^{-dq} \\
e^{-dq} & 1-\Pi_1 U \left (1 - e^{-2dq}\right )
\end{array} \right ) . \notag
\end{equation}
Here $\Pi_s$ is the polarization operator of the surface states.
In the present section we will concentrate on the statically screened
interaction potential. In this limit the polarization operator is determined by
the thermodynamic density of states: $ \Pi_{s}\left (\omega = 0, \v q\right ) =
-\nu_s$.
In the diffusive regime defined by the condition \eqref{eq:Diffregime2}, the
wavevector $q$ satisfies the inequality $1/L_E \ll q \ll 1/l$.
Therefore, in a sample of thickness $d \gg L_E$ we always have $d q \gg 1$ and
the two surfaces decouple,
\begin{equation}
\underline U_{\rm scr} \stackrel{d \gg L_E}{=} 2\pi \frac{e^2}{\epsilon} \left
(\begin{array}{cc}
\frac{1}{q + \kappa_1} & 0 \\
0 & \frac{1}{q + \kappa_2}
\end{array} \right ), \label{eq:decoupledsurf}
\end{equation}
where $\kappa_s = 2\pi e^2\nu_s/\epsilon$ is the inverse Thomas-Fermi
screening length for a single surface $s$. A universal form of the
Altshuler-Aronov correction to conductivity induced by the
Coulomb interaction \cite{AltshulerAronov, OGM2010} arises in the unitary limit
when one can neglect $q$ as compared with $\kappa_s$ in
Eq.\eqref{eq:decoupledsurf}. The unitary limit is achieved if $\kappa_s^{-1} \ll
l$ (the meaning of this condition {as well as the complementary case} are
discussed in section \ref{sec:scattchannels}).
In the opposite limit of a small interlayer distance, $d \ll l$, we can
approximate $e^{-dq} \approx 1$ in the whole diffusive regime. This implies
\begin{eqnarray}
\underline U_{\rm scr} \stackrel{d \ll l}{=} & \displaystyle \frac{2\pi
e^2}{\epsilon}\frac{1}{q +\kappa_1 + \kappa_2 + 2d\kappa_1 \kappa_2 \left
(1-qd\right )}\notag \\
& \times \left
(\begin{array}{cc}
1+2 \kappa_2 d & 1 \\
1 & 1+2 \kappa_1 d
\end{array} \right ). \label{eq:coupledsurf}
\end{eqnarray}
At the first glance, it looks as if also negative interaction potential was
possible. However, this is not the case as shall be explained in what follows.
Depending on the hierarchy of the lengthscales $\kappa_1^{-1}, \kappa_2^{-1}$
and $d$ the following scenarios are conceivable:
First, consider $\kappa_s d \ll 1$ for both $s = 1$ and $s =2$. In this case,
the $q$ dependence of the interaction potential implies the definition of the
coupled layer screening length $l_{\rm scr}$:
\begin{equation}
\left (\underline U_{\rm scr}\right )_{ss'}\left (\v q\right ) \sim \frac{1}{q +
\kappa_1 + \kappa_2}\ \ \Rightarrow\ \ l_{\rm scr} = \frac{1}{\kappa_1 +
\kappa_2}.
\label{eq:screeninglength}
\end{equation}
If in addition the condition $l_{\rm scr} \ll l$ is fulfilled,
the Coulomb interaction potential \eqref{eq:coupledsurf} becomes
``overscreened'' ($q$-independent) for all diffusive momenta $q \ll
l^{-1}$.
Second, assume that $\kappa_s d \gg 1$ for at least one surface. Then the
$q$-dependence of $\underline U_{\rm scr}$ is always negligible and thus the
notion of coupled layer screening length is meaningless. It is worthwhile to
remark that, as expected, the potential
\eqref{eq:coupledsurf} reduces to the decoupled form
\eqref{eq:decoupledsurf} in the limit when $\kappa_s^{-1} \ll d$ for both
surfaces (which also implies that $\kappa_s^{-1} \ll l$).
In this paper we derive the conductivity
corrections in the unitarity limit of $q$-independent interaction, see Eqs. \eqref{eq:RGeqs}. As expected, in the
limit of decoupled surfaces, $\kappa_s^{-1} \ll d$, they reproduce the previous
result \cite{OGM2010}, while whenever $d \ll \kappa_1^{-1}$ or $d \ll
\kappa_2^{-1}$ novel conductivity corrections induced by
intersurface electron-electron interaction emerge.
Finally, in the intermediate regime $l \ll d \ll L_E$ the scale-dependent
conductivity can be obtained by the following two-step RG analysis. First, one
integrates the single-surface RG equations starting from the shortest scale $l$
up
to the intersurface distance $d$. After this, one uses the running
coupling constants at scale $d$ as starting values for the coupled-surface RG flow
and integrates these RG equations up to the scale $l_E$.
Different regimes discussed above are shown schematically in Fig.
\ref{fig:regimediag} in the parameter plane $d$ -- $\kappa^{-1}$. For
simplicity, we assume there the two surfaces have comparable screening lengths:
$\kappa_1^{-1} \sim \kappa_2^{-1}$.
\begin{figure}
\includegraphics[scale=.55]{Regimediagram2}
\caption{Sketch of the regimes discussed in the main text for
the case of comparable screening lengths, $\kappa^{-1}_1 \sim \kappa^{-1}_2$
(denoted by $\kappa^{-1}$). The regimes \textbf{I} and \textbf{II} correspond to
effectively decoupled surfaces (studied in Ref. \onlinecite{OGM2010}), { while in regimes \textbf{III} and \textbf{IV} intersurface interaction is important.} The conductivity corrections in \textbf{I} {and \textbf{III}} are due to
``overscreened'' Coulomb interaction. In contrast, in \textbf{II} {and \textbf{IV}} this type of
corrections sets in only in the low-energy regime where the running length scale
(i.e. the typical scale $L_E$) exceeds the screening length.}
\label{fig:regimediag}
\end{figure}
In the end of the paper, Sec.~\ref{sec:experiment}, we analyze in
detail the regions and limits of applicability of our theory with respect to
representative experimental setups. In particular, we show that the hierarchy
of scales $d \ll l \ll L_E$ is realistic.
In order to illustrate the importance of intersurface
interaction (i.e. the relevance of the inequality $d \lesssim \kappa_s^{-1}$)
under realistic conditions, we show in Fig.~\ref{fig:screeninglength} a
dependence of the screening length on the Fermi momentum.
The density of states for the linear (Dirac) spectrum is $\nu (\mu_s) =
k^{(s)}_F / 2\pi \hbar v_F$, where $k^{(s)}_F$ is the Fermi wave vector of
the $s$-th surface state and $v_F$ the Fermi velocity. Therefore
\begin{equation}
\kappa_{s}^{-1} = \frac{1}{\alpha} \frac{1}{k^{(s)}_F } .
\label{eq:dcondition}
\end{equation}
We introduced the dimensionless parameter $\alpha = {e^2}/{\epsilon \hbar
v_F}$ which is the effective coupling constant of the Coulomb interaction and
is equal to ${c}/{\epsilon v_F}$ times the fine structure constant of
quantum electrodynamics.
Clearly, $\alpha$ plays the same role as the
dimensionless density parameter $r_s$ in conventional theories of electrons in
parabolic bands. We will assume that the interaction is not too
strong, $\alpha\lesssim 1$; otherwise the system may become unstable, see
a discussion at the end of Sec.~\ref{sec:Microham}.
The dashed red curve in Fig.~\ref{fig:screeninglength} represents the lower
bound (corresponding to $\alpha = 1$) of $\kappa_s^{-1}$ as a function of
$k^{(s)}_F$. The actual value of
$\kappa_s^{-1}$ for an exemplary case of Bi$_2$Se$_3$ (experimental parameters
can be found in Table \ref{tab:expvaluesBise} below) is depicted by the blue
solid curve. We see that the screening length can by far
exceed the thickness of the topological insulator slab. Indeed, the Bi$_2$Se$_3$
experiments \cite{Wang, Liu, Chen, Steinberg} are performed on probes of
thickness $d \simeq 1 - 100$ nm. For this material, our assumption of separate
gapless surface states (no tunneling) is both numerically\cite{Linder} and
experimentally\cite{Ando} shown to be valid down to $d \simeq 10$ nm (blue
horizontal dashed line). Thus, relevant experimental values of $d$ in the
experiments of interest range from $ d\simeq 10$ nm up to $ d\simeq 100$ nm.
On the other hand,
surface electrons have a maximal Fermi wavevector of
$k_F \sim 0.1/\text{\AA}$ associated with $\mu = \Delta_{\text{bulk}} = 0.3$ eV,
see blue vertical dashed line. For the lowest concentration, increase of the
screening length is limited by disorder. In this way, we estimate the range
of $\kappa_s^{-1}$ as 20-200 nm, so that the condition $\kappa_s^{-1} > d$ can
be easily fulfilled. This is particularly the case for relatively thin films
($d \simeq 10$ nm) and in the vicinity of surface Dirac point.
\begin{figure}
\includegraphics[scale=.45]{screeninglengthsingle}
\caption{Plot of the single surface screening length $\kappa_s^{-1}$. The red curve (large dashes) is the lower
bound (corresponding to $\alpha = 1$) of the screening length. The solid, blue curve is the screening length for Bi$_2$Se$_3$ film with experimental parameters given in Table \ref{tab:expvaluesBise} {in Sec.\ref{sec:2materials}}. For the latter, the required minimal thickness
and maximal Fermi momentum are also depicted (dotted blue lines). The
disorder-induced regularization of the divergence at small Fermi momentum is
schematically represented by the black dot-dashed curve. }
\label{fig:screeninglength}
\end{figure}
The above analysis proves the relevance of the intersurface electron-electron interaction.
In fact, in course of this analysis we have made several simplifying
assumptions that require certain refinements; we list them for the reader's
benefit. First, in general, the
coating material ($\epsilon_1$), the topological insulator ($\epsilon_2$), and the substrate ($\epsilon_3$) are all dielectrica with different dielectric constants $\epsilon_1\neq\epsilon_2\neq\epsilon_3$. In order to
determine the exact Coulomb interaction, one has to solve the electrostatic
problem of a point charge in such a sandwich structure of dielectrica,\cite{Profumo2010,Katsnelson2011,CoulombDragCarrega} see
Appendix \ref{sec:Elstat}.
Second, the long-range Coulomb interaction is accompanied by short-range
contributions, which, in particular, induce corrections to the
polarization operator which affect the screening length. More precise
calculations taking Fermi liquid corrections into account can be found in
Section \ref{sec:cleanFLMaintext} and Appendix \ref{sec:cleanFL}.
Finally, we neglected the dependence of the Fermi velocity $v_F$ on the
chemical potential $\mu_s$, see
Sec.~\ref{sec:Microham}. However, all these refinements do not modify our
conclusion of the importance of interaction between the surface states. We now
proceed with presentation of the field-theoretical formalism that will allow us
to explore the problem.
\subsection{Microscopic Hamiltonian}
\label{sec:Microham}
\begin{figure}
\includegraphics[scale=.15]{2cones12}
\caption{Pictographic representation of the microscopic model: Diffusively
propagating surface states at different chemical potentials which interact with
each other by means of long-range Coulomb interaction.
}
\label{fig:Microscopicaction}
\end{figure}
The model under consideration {is schematically depicted in Fig. \ref{fig:Microscopicaction}. It is described in path integral technique
\begin{equation}
\mathcal Z = \int \mathcal D \left [\bar \psi, \psi \right ] \; e^{-S\left [\bar \psi, \psi \right ]}
\end{equation}
by} the following microscopic Matsubara action:
\begin{equation}
S\left [\bar \psi, \psi \right ] = \int_{\tau, \v x} \bar \psi \left
(\partial_\tau + \text H_0 + \text H_{\rm dis} \right ) \psi + S_{\rm int}.
\label{eq:Highenergyaction}
\end{equation}
The notation $\int_{\tau, \v x} = \int d^2x \int_0^\beta d \tau$ will be used throughout the article, where, as usual, $\beta = 1/T$ is the inverse temperature. If not specified otherwise, we set
Boltzmann's constant, Planck's constant, and the speed of light $k_B =
\hbar = c = 1$ in the remainder.
The fermionic fields $ \bar \psi \left (\v x, \tau\right ) = \left ( \bar
\psi^\uparrow_1, \bar \psi^\downarrow_1, \bar \psi^\uparrow_2, \bar
\psi^\downarrow_2\right )$
and $ \psi \left (\v x, \tau\right ) = \left ( \psi^\uparrow_1,
\psi^\downarrow_1, \psi^\uparrow_2, \psi^\downarrow_2 \right )^T$
describe the spinful ($\uparrow,\downarrow$) excitations living
on surfaces $s=1$ and $s=2$. The one particle Hamiltonian which characterizes
the surface $s$ is
\begin{equation}
\left (\text H_0 + \text H_{\rm dis}\right )_s = \left( V_s(\v x ) -\mu_s \right
)
\otimes \mathbf I_{\sigma} +
i(-)^{s} v^{(s)}_F
\nabla \wedge \vec \sigma
, \label{eq:OneParticleHam}
\end{equation}
where $\mathbf I_\sigma$ is the unit matrix in spin space and
we define $\v a \wedge \v b = a_x b_y - a_y b_x$.
The disorder potentials $V_s\left ( \v x \right )$ for two surfaces are
assumed to be white-noise distributed and uncorrelated:
\begin{equation}
\left \langle V_s(\v x ) V_{s'}(\v x' ) \right \rangle = \frac{\delta \left
(\v x - \v x '\right ) \delta_{ss'}}{\pi \nu_s \tau_s}.
\end{equation}
The disorder strengths $1/\pi \nu_s \tau_s$ may be different for two
surfaces.
It is worth emphasizing the following physical implications of this
Hamiltonian.
\begin{itemize}
\item First, the model (and its analysis below) corresponds to the general case
in which the chemical potentials $\mu_1$, $\mu_2$ and hence the carrier
densities of the two surfaces may differ.
\item Second, since the disorder potentials are different for two
surfaces, no inter-surface diffuson and cooperon modes will arise.
Note that the considered model of fully uncorrelated disorder correctly describes the low-energy physics of the majority of experimental setups, even in the presence of moderate inter-surface correlations of disorder. Indeed, any mismatch in chemical potentials and/or disorder configurations leads to an energy gap in the inter-surface soft modes. Two physical regimes are conceivable:
\begin{itemize}
\item[(i)] almost identical surfaces in almost fully correlated random potentials, $\vert \mu_1 -\mu_2 \vert \ll 1/\tau_s$ and $ \left \langle \left [ V_1(\v x ) - V_{2}(\v x' ) \right ]^2 \right \rangle \ll \sum_{s=1,2} \left \langle V_s(\v x ) V_{s}(\v x' ) \right \rangle$;
\item[(ii)] all other parameter regimes, when at least one of the conditions in (i) is not fulfilled.
\end{itemize}
Our model is designed for the case (ii), where the gap is
comparable to the elastic scattering rate and inter-
surface soft modes do not enter the diffusive theory
at all. It also applies to the case (i) in the ultimate large-scale limit (i.e. at energy scales below the gap). In this case there will be, however, an additional, intermediate regime in the temperature dependence (or AC frequency dependence) which is not considered in our work.
\item Third, $\vec \sigma$ in Eq.\eqref{eq:OneParticleHam} in general does not describe the physical spin. For example,
in Bi$_2$Se$_3$ structures the effective spin $\sigma$ is determined by a linear
combination of real spin and the parity (band) degrees of freedom. The mixing
angle depends on how the crystal is cut. \cite{ZhangKaneMele} In this case also
the Fermi velocity becomes anisotropic.
\item Fourth, because of interaction effects, the true dispersion relation is not
linear but contains logarithmic corrections (or more generally is subjected to ``ballistic'' RG \cite{GonzalezGuineaVozmedianoGraphene01,FosterAleiner08,SheehySchmalian07}) which leads to dependence of the Fermi
velocity on the chemical potential. This is reflected in the notation
$v_F^{(s)} \equiv v_F (\mu_s)$.
\item Similarly, also the strength of the disorder may be substantially different for
both surfaces, so that the (quantum) mean free times $\tau_s$ are considered
as two independent input parameters. This is primarily because the vicinity to
the substrate or, respectively, to the coating material makes the impurity
concentration on both surfaces a priori different.
In addition, $\tau_s$ acquire renormalization corrections, leading to a
logarithmic dependence on $\mu_s$. \cite{AleinerEfetovGraphene,OGM2006Graphene,SOGM09,FosterAleiner08}
\item The (pseudo-)spin
texture on the top and bottom surfaces is opposite (denoted by the factor $(-)^{s}$).
\item Finally, in some materials (in particular, in Bi$_2$Te$_3$), the Dirac cone is
strongly warped. We neglect the warping as it does not affect the main result of
this paper, namely the (universal) RG equations.
Recently,\cite{diffusionandwarping} it has been shown that warping only
influences the dephasing length (i.e., the lengthscale at which the RG flow is
stopped).
\end{itemize}
The interaction is mediated by the Coulomb potential, see
Eq.~\eqref{eq:Coulombnoelstat} and Appendix \ref{sec:Elstat}. With the
definition $\rho_s\left (\tau, \v x\right ) = \bar \psi_s\left (\tau, \v x
\right ) \psi_s\left (\tau, \v x \right )$ the corresponding contribution to the
action is given by
\begin{equation}
S_{\rm int} = \frac{1}{2} \sum_{ss'}\int_{\tau, \v x, \v x'} \rho_s\left (\tau,
\v
x\right ) U_{0,ss'}\left (\vert \v x - \v x'\right \vert)\rho_{s'}\left (\tau,
\v x'\right ). \label{eq:S_Coulsimple}
\end{equation}
For equal surfaces ($v_F^{(1)}=v_F^{(2)}$), a simple rescaling of equations
\eqref{eq:Highenergyaction} and \eqref{eq:S_Coulsimple} shows that the effective
coupling to the Coulomb interaction is $\alpha$. It can, in general, become
of the order of unity. Since the perturbation theory is insufficient in such a
case, we adopt the more general, yet phenomenological, Fermi liquid theory
to access the behavior for energies down to the elastic scattering rates
$\tau^{-1}_{1,2}$, see Sections \ref{sec:scattchannels},
\ref{sec:cleanFLMaintext} and Appendix \ref{sec:cleanFL}). This (clean)
Fermi liquid theory will then be a starting point for the interacting
diffusive problem at energies below the elastic scattering rate.
If the interaction becomes too strong, it might in principle drive the system
into a phase with spontaneously broken symmetry.\cite{SitteRoschFritz2013} Examples are the Stoner
instability \cite{Peres} as well as more exotic phenomena such as
topological exciton condensation, \cite{TEC} which is specific to 3D TI thin
films.
Throughout our analysis, we assume that the system is not in a vicinity of
such an instability. To our knowledge, this assumption is consistent with all
transport experiments on 3D TI slabs addressed in this work.
\section{Sigma-model description}
\label{sec:towardsNLSM}
We are interested in the low-energy (low-temperature, long-length-scale) physics
of the 3D TI problem defined by Eqs.~\eqref{eq:Diffregime1} and
\eqref{eq:weakdisorder}. This physics is controlled by coupled diffuson and
cooperon modes. In this Section we derive the effective field
theory -- diffusive non-linear $\sigma$ model -- that describes
the system in this regime.
\subsection{Symmetries of the action}
\label{sec:symmetries}
The structure of the effective low-energy theory, the diffusive NL$\sigma$M,
is controlled by symmetries of the microscopic action. The information
about other microscopic details enters the theory only via the values of the
coupling constants. We thus begin by analyzing symmetries of the problem.
First, our system obeys the time reversal symmetry $H = \sigma_y H^T \sigma_y$.
Second, we assume no intersurface tunneling, i.e., the particle
number is conserved in each surface separately. This
implies invariance of the action with respect to $\mathbf{U}(1) \times
\mathbf{U}(1)$ transformations (global in space and time).
The presence of Coulomb interaction promotes the $\mathbf{U}(1)$ symmetry in the
total-density channel, $\rho_1+\rho_2$, to transformations which are local in time but global in
space. In other words, rotations of fermionic fields, $\bar \psi_s\left (\tau, \v x
\right ) \to \bar \psi_s\left (\tau, \v x
\right ) \exp{[-i \chi_s(\tau)]}$, $\psi_s\left (\tau, \v x \right )\to\exp{[i \chi_s(\tau)]}\psi_s\left (\tau, \v x \right )$, with equal phases $\chi_1\left (\tau\right
)=\chi_2\left (\tau\right )$ leave the action
\eqref{eq:Highenergyaction} invariant. This is a special case of ``$\mathcal
F$-invariance'' \cite{MishandlingI} and has important consequences for the
present problem. The $\mathcal F$-invariance (it is intimately linked to gauge invariance) generally states that in each
channel with long-range interaction, time-dependent but spatially constant
$\mathbf{U}(1)$ rotations are symmetries of the action. In our problem, as it
follows from the $q\to 0$ limit of the Coulomb interaction:
\begin{equation}
\underline{U}\left (\v q\right ) \stackrel{q\rightarrow 0}{\propto} \frac{1}{q}
\left (\begin{array}{cc}
1 & 1 \\
1 & 1
\end{array} \right ), \label{eq:q0limitt}
\end{equation}
only the interaction between the total densities is long-ranged.
The structure of Eq. \eqref{eq:q0limitt} remains true also in the case of asymmetric dielectric
environment, see Appendix \ref{sec:appendixCoulomb}.
To make the time-reversal symmetry explicit, we define
particle-hole bispinors by
combining $\psi$ and $\bar{\psi}$ fields. \cite{EfetovLarkinKhmel, BelitzKirkpatrick} In the momentum space the
bispinors read
\begin{equation}
\Phi_{n}\left (\v k\right ) = \frac{1}{\sqrt 2} \left (\begin{array}{c}
\bar \psi_{n}\left (- \v k\right ) ^T \\
i \sigma_y \psi_{n}\left (\v k\right )
\end{array} \right )
\end{equation}
and
\begin{equation}
\bar \Phi_{n}\left (\v k\right ) = \left [C \Phi_{n}\left (-\v k\right )\right
]^T \text{ with } C = i \sigma_y \tau_x ,
\end{equation}
where $n$ is the index associated to the fermionic Matsubara frequency $i
\epsilon _n$, and $\tau$ matrices act in the particle-hole space.
This allows us to rewrite the one-particle Hamiltonian as
\begin{equation}
S^{\rm free} = -\sum_n \int_{\v k} \bar \Phi_{n} \left (\v k\right ) \left (i
\epsilon_n - H^T\left (- \v k \right ) \right ) \Phi_{n} \left (\v k\right ).
\end{equation}
It is convenient to perform a rotation of bispinors
\begin{equation}
\eta = \sqrt {\tau_x} \Phi,
\end{equation}
where $\sqrt {\tau_x} = e^{-i\pi/4} (\mathbf I_\tau + i\tau_x)/\sqrt{2}$.
The free action then takes the form
\begin{eqnarray}
S^{\rm free} &=& -\sum_s \int_{\v x} \eta^T_s \Big \lbrace \left [i \hat
\epsilon - V_s + \mu_s \right ] \left ( - i \sigma_y \right ) \\
&+& \left. (-)^{s+1} v^{(s)}_F\left (\partial_x - i
\partial_y \sigma_z \right ) \right\} \eta_s \label{eq:Sofeta} .
\end{eqnarray}
The Matsubara frequency summation is incorporated into the scalar product
$\eta^T \left (\dots \right ) \eta$. In these notations, $\hat \epsilon$ is
a diagonal matrix in the Matsubara space consisting of entries $\epsilon_n$.
In order to perform the average over disorder, we replicate the theory $N_R$
times. Furthermore, in order to implement the $\mathbf
U\left (1\right )$-gauge invariance in the framework of the NL$\sigma$M,
we apply a double cutoff truncation procedure with $N_M \ll N^\prime_M$
for the Matsubara frequencies.\cite{MishandlingI}
Here $N'_M$ and $N_M$ are the numbers of retained Matsubara harmonics for fast
(electrons of the original theory) and slow (diffusons and cooperons
of the NL$\sigma$M) degrees of freedom, respectively.
As a consequence, $\eta$ becomes a $\left (2_s \times 2_\sigma \times 2_\tau
\times 2 N'_M
\times N_R\right )$-dimensional Grassmannian vector field. Except for the
frequency term, the free action \eqref{eq:Sofeta} is manifestly invariant
under global orthogonal rotations of the kind
\begin{equation}
\eta_s \rightarrow \left (O_s \otimes \mathbf I_\sigma \right ) \eta_s \text{
with } O_s \in \mathbf{O}\left (2_\tau \times 2 N'_M \times N_R\right ).
\end{equation}
Since the surfaces are fully decoupled in the absence of interactions,
the rotations $O_1$ and $O_2$ of the fields corresponding to the top and bottom surfaces are
completely independent.
\subsection{Quasiclassical conductivity}
\label{sec:semiclassical}
To obtain the quasiclassical conductivity, we first find the fermionic
self-energy within the self-consistent Born approximation (SCBA):
\begin{equation}
\Sigma^s_n = \frac{-2i\sigma_y}{\pi \nu_s \tau_s} \left \langle \eta_{\v x,s}
\eta_{\v x,s}^T\right\rangle_{\text{SCBA}}.
\label{eq:SCBA}
\end{equation}
Here $\left \langle \dots \right \rangle_{\text{SCBA}}$ denotes the
self-consistent treatment, i.e. a shift $\mu_s \rightarrow \mu_s + \Sigma^s_n$
in the fermionic propagator. Equation \eqref{eq:SCBA} yields for the imaginary
part of the self-energy ${\rm Im}(\Sigma^s_n) = (i / 2\tau_s) \text{sgn}(n)$.
The quasiclassical Drude DC conductance of the non-interacting problem in the
absence of a magnetic field is
\begin{equation}
\sigma^D_s = 2\pi \nu_s D_s \frac{e^2}{h} ,\label{eq:Drude}
\end{equation}
with $D_s = (v^{(s)}_F)^2 \tau_s$. Note that the transport time is twice
the quantum mean free time $\tau_s$. In the diagrammatic language, this is
a consequence of vertex corrections.
\subsection{Fermionic currents and bosonization rules}
To derive the NL$\sigma$M, we use the method of non-Abelian bosonization.
\cite{Witten, NersesyanTsvelikWenger94, NersesyanTsvelikWenger95, ASZ,
AltlandGraphene} An advantage of this approach is that non-trivial
topological properties of the Dirac fermions are translated
into the field theory in a particularly transparent way.
In the first step, the kinetic term (Sec. \ref{sec:NONABBOS}) is bosonized.
Subsequently, we bosonize also the terms induced by the chemical potential, disorder
and frequency (Sec. \ref{sec:freeNLSM}). Since only interaction couples the two
surfaces, we omit the surface index $s$ in Sec. \ref{sec:NONABBOS} and Sec.
\ref{sec:freeNLSM}. This index is restored
later in Sec.~\ref{sec:interactingNLSM} where the interaction is included.
Local left ($\eta_{\uparrow} \rightarrow O_{L} \eta_{\uparrow}$) and
right ($\eta_{\downarrow} \rightarrow O_{R} \eta_{\downarrow}$) rotations define the
left and right currents. The bosonization rules for these currents as well as
for the mass term are
\begin{subequations}
\begin{align}
j_{+} = v_F \eta_{\uparrow} \eta_{\uparrow} ^T &\leftrightarrow
\frac{1}{8\pi}\left (O \partial_+ O^T\right ), \label{eq:leftcurrent} \\
j_{-} = v_F \eta_{\downarrow} \eta_{\downarrow} ^T &\leftrightarrow
\frac{1}{8\pi}\left (O^T \partial_- O\right ), \label{eq:rightcurrent} \\
\eta_{\uparrow} \eta_{\downarrow}^T &\leftrightarrow i \lambda O,
\label{eq:thirdbosonrule}
\end{align}
\label{eq:Bosonrules}
\end{subequations}
where $\partial_\pm = \partial_x \pm i \partial_y$. The energy scale $\lambda$
is of the order of the ultraviolet (UV) cutoff and is introduced here for
dimensional
reasons; see Sec.~\ref{sec:freqboson} and \ref{sec:Densityresponse} for a
discussion of its physical meaning. Note that in general, the UV cutoff is different for the top and bottom surfaces,
$\lambda_1 \neq \lambda_2$. Further, $O$ is an orthogonal $\left (2_\tau \times 2
N'_M
\times N_R\right )\times \left (2_\tau \times 2 N'_M \times N_R\right )$
{matrix field}. Below we will need the following constant matrices in this space
\begin{eqnarray}
\Lambda^{\tau_1 \tau_2; \alpha \beta}_{nm} &=& \text{sgn}\left (n \right )
\delta^{\tau_1 \tau_2} \delta^{\alpha \beta} \delta_{nm} , \notag \\
\hat \eta^{\tau_1 \tau_2; \alpha \beta}_{nm} &=& n \delta^{\tau_1 \tau_2}
\delta^{\alpha \beta} \delta_{nm} , \label{eq:matrixdefs} \\
\left (I^{\alpha_0}_{n_0} \right )^{\tau_1 \tau_2; \alpha \beta}_{nm} &=&
\delta^{\tau_1 \tau_2} \delta^{\alpha_0 \alpha} \delta^{\alpha_0 \beta}
\delta_{n-m,n_0}. \notag
\end{eqnarray}
Here and throughout the paper we use a convention that
$\alpha, \beta \in \left \lbrace 0, N_R \right \rbrace$ denote replicas and
$m, n \in \left \lbrace - N'_M, \dots , N'_M - 1 \right \rbrace$ Matsubara
indices.
The double cutoff regularization scheme \cite{MishandlingI} prescribes that
matrices $O$ have non-trivial matrix elements $O_{nm}$ only for low-energy
excitations $n,m \in \left \lbrace - N_M, \dots , N_M - 1 \right \rbrace$ and
stay equal to the origin $O_0$ of the $\sigma$ model manifold outside this
low-energy region. As explained below, $O_0 = \Lambda$.
\subsection{Bosonization of the kinetic part}
\label{sec:NONABBOS}
The kinetic part of \eqref{eq:Sofeta} is nothing but the Euclidean counterpart
of the model considered {in} Ref. \onlinecite{Witten}. Upon
non-Abelian bosonization it
yields the Wess-Zumino-Novikov-Witten (WZNW) action
\begin{equation}
S_{\rm WZNW} = \int_{\v x} \frac{1}{16\pi} \text{tr} \nabla O \nabla O^{-1} +
\frac{i}{24
\pi} \Gamma_{WZ} ,
\label{eq:S-WZNW}
\end{equation}
where $\Gamma_{WZ}$ is the Wess-Zumino (WZ) term
\begin{equation}
\Gamma_{WZ} = \int_{\v x, w} \epsilon_{\mu \nu \rho} \text{tr} \left [\left ( \tilde O^{-1}
\partial_\mu \tilde O\right ) \left (\tilde O^{-1} \partial_\nu \tilde O \right
)\left (\tilde O^{-1} \partial_\rho \tilde O\right )\right ],
\label{eq:WZW}
\end{equation}
where $\epsilon_{\mu\nu\rho}$ denotes the Levi-Civita symbol.
The definition of the WZ term involves an auxiliary coordinate $w \in \left [0,
1 \right ]$ and smooth fields $\tilde O \left (\v x , w \right )$ satisfying
$\tilde
O \left ( \v x, w = 0 \right ) = {\rm const} $ and $\tilde O \left (\v x , w
=1\right
) = O \left (\v x \right )$.
As a result the compactified two-dimensional coordinate space $\mathbb{R}^2 \cup
\left \lbrace \infty \right \rbrace \simeq \mathbb{S}^2$ is promoted to the
solid 3-ball $\mathbb{B}^3$ (i.e., the ``filled'' sphere).
\subsection{Free NL$\sigma$M of class AII}
\label{sec:freeNLSM}
\subsubsection{Disorder, frequency, and the chemical potential}
\label{sec:freqboson}
The action (\ref{eq:S-WZNW}) is the bosonized counterpart of the second
(proportional to velocity) term of the microscopic action \eqref{eq:Sofeta}.
Let us now consider the first term in Eq.~\eqref{eq:Sofeta} which carries
information about the chemical potentials, frequency and random potential.
Bosonization of the terms with frequency and the chemical potential in the microscopic action \eqref{eq:Sofeta} yields
\begin{align}
\delta S
=
2 \int_{\v x} \text{tr} \left [\left (i\hat \epsilon +\mu\right )\eta_{\uparrow}\eta_{\downarrow}^T\right ]
\leftrightarrow
- 2\lambda \int_{\v x} \text{tr} \left (\hat \epsilon - i\mu\right)O .
\label{eq:disordermassterms-frmu}
\end{align}
Upon disorder averaging and bosonization, the term with random potential
provides the following contribution to the field theory:
\begin{align}
\delta S_\textrm{dis} &=
- \frac{1}{\pi \nu \tau} \int_{\v x}\left (\text{tr}
\eta_\uparrow \eta_\downarrow^T\right )^2
+ \frac{1}{\pi \nu \tau} \int_{\v x}\text{tr} (\eta_{\uparrow}
\eta_{\downarrow}^T)^2 \notag\\
&\leftrightarrow
\frac{\lambda^2}{\pi\nu\tau}\int_{\v x} \left (\text{tr} O\right )^2
\notag \\&
+\frac{\lambda^2}{2\pi \nu \tau} \int_{\v x} \text{tr} \left (O^T-O\right )^T\left
(O^T-O\right ).
\label{eq:disordermassterms}
\end{align}
As we see, disorder induces mass terms for
$O$-matrices. Both mass terms in Eq. \eqref{eq:disordermassterms} are strictly non-negative.
Therefore, they are minimized by arbitrary traceless symmetric orthogonal matrix. It is convenient to choose the specific saddle-point solution as
\begin{equation}
O = \Lambda .
\label{eq:Lamda}
\end{equation}
This saddle-point solution coincides with the SCBA. Indeed, Eq. \eqref{eq:SCBA} can be written as
\begin{align}
\frac{i}{2\tau} \Lambda {\otimes \textbf 1}_\sigma&= \frac{2}{\pi \nu \tau} \left \langle \left (\begin{array}{cc}
-\eta_\downarrow \eta_\uparrow^T & -\eta_\downarrow \eta_\downarrow^T \\
\eta_\uparrow \eta_\uparrow^T & \eta_\uparrow \eta_\downarrow^T
\end{array} \right )\right\rangle_{\text{SCBA}} \notag \\
& \leftrightarrow \frac{2}{\pi \nu \tau} \left \langle \left (\begin{array}{cc}
i\lambda O^T & \frac{-1}{8\pi v_F} O^T \partial_- O \\
\frac{1}{8\pi v_F} O \partial_+ O^T & i\lambda O
\end{array} \right )\right \rangle .
\end{align}
It is solved by the saddle-point solution \eqref{eq:Lamda} provided the auxiliary UV energy scale $\lambda$ introduced in
Eq.~(\ref{eq:Bosonrules}) is related to the density of states (i.e., to the chemical
potential),
\begin{equation}
\lambda = \frac{\pi \nu}{4} = \frac{\vert \mu \vert}{8 v_F^2}. \label{eq:lambdamu}
\end{equation}
We will rederive this relation from a different viewpoint below, see
Sec.~\ref{sec:Densityresponse}.
Equation (\ref{eq:Lamda}) is not the only solution of the saddle point
equation. It is easy to see that rotations
\begin{equation}
O\rightarrow O^T_{\text{soft}} O O_{\text{soft}},\; O_{\text{soft}}\in \mathbf G=
\mathbf{O}\left (2_\tau \times 2N_M\times N_R\right )
\end{equation}
leave the mass term unaffected. On the other hand, the saddle-point $O=\Lambda$
is invariant under rotations from a smaller group, $O_{\text{soft}}\in \mathbf{K}=\mathbf
O\left (2_\tau \times N_M\times N_R\right )\times \mathbf O\left ( 2_\tau \times
N_M\times N_R\right )$. This can be understood as a breakdown of symmetry
$\mathbf G\to \mathbf K$.
We thus obtain a non-trivial manifold of saddle-points
annihilating the mass term. Allowing for a slow variation of $O_{\text{soft}}$
and restricting other terms in the action to this manifold, we will obtain the
NL$\sigma$M action.
\subsubsection{Free NL$\sigma$M with $\mathbb Z_2$ topological term}
\label{sec:Z2term}
As we have just discussed, we keep only the soft modes
\begin{equation}
Q = O^T_{\text{soft}} \Lambda O_{\text{soft}} \text{ with }
O_{\text{soft}} \in
\mathbf{G} . \label{eq:defQ}
\end{equation}
The subscript $_{\text{soft}}$ will be omitted in the remainder. The NL$\sigma$M
manifold ${\cal M}=\mathbf G/ \mathbf K$.
We also rename the coupling constants according to the conventional notation of
diffusive NL$\sigma$Ms and restore the surface index $s$,
\begin{equation}
S^{\rm free} = \sum_s \int_{\v x} \frac{\sigma_s}{16} \text{tr} \left (\nabla
Q_s\right )^2 - 2 \pi T z_s \text{tr} \left [\hat \eta Q_s\right ]+ i S^{(\theta)}_s.
\label{eq:freeNLSM}
\end{equation}
As will become clear from linear response theory (Sec. \ref{sec:Kubo}),
$\sigma_s$ measures the DC conductivity of surface $s$ (in units $e^2/h$). Its
bare value is the Drude conductance depending on the chemical potential $\mu_s$, as can be directly verified, see Appendix
\ref{sec:appendixBosonization}. The coupling constants $z_s$ determine the
renormalization of the specific heat.
The non-trivial second homotopy group of the NL$\sigma$M manifold $\pi_2 ({\cal
M})= \mathbb Z_2$ allows for topological excitations (instantons), similarly
to the QHE theory. A crucial difference is that in the QHE case the second
homotopy group is $\mathbb Z$, so that any integer topological charge (number
of instantons) is allowed. Contrary to this, in the present case
any configuration of an even number of instantons can be continuously deformed
to the trivial, constant vacuum configuration. Therefore, the theta term
$S^{(\theta)}_s$ appearing in \eqref{eq:freeNLSM} only distinguishes between an
even ($S^{(\theta)}_s = 0 \; \text{mod} \; 2\pi$) and odd ($S^{(\theta)}_s = \pi
\;\text{mod}\; 2\pi$) number of instantons.
Such a $\mathbb Z_2$ theta term $S^{(\theta)}$ does not appear in the case of
usual metals with strong spin-orbit coupling; it results from the
Dirac-fermion nature of carriers and is a hallmark of topologically protected
metals (in our case, the surface of a topological insulator).
The topological term flips the sign of the
instanton effects (as compared to the case of a
usual metal with spin-orbit interaction) from localizing to
delocalizing.
Thus, the theta term translates the protection against Anderson
localization into the NL$\sigma$M approach.
We are now going to show that $S^{(\theta)}_s$ is nothing but
the WZ term (obtained from non-Abelian bosonization) restricted to the smaller
symmetry group:
\begin{equation}
S^{(\theta)}_s = \frac{1}{24 \pi} \left . \Gamma_{WZ,s} \right \vert_{\tilde
O_s\left (\v x, w = 1\right ) = Q_s\left (\v x\right ) = Q_s^T\left (\v x\right
)}.
\label{eq:thetaterm}
\end{equation}
Note that, since the second homotopy group of the NL$\sigma$M manifold is
non-trivial, the definition of the WZ term requires that away from $w =
1$ the extended fields can take values in the big orthogonal group $\mathbf G$.
To show that Eq.~(\ref{eq:thetaterm}) is indeed the $\mathbb
Z_2$ theta-term, we proceed in the same way as was recently done for symmetry
class CII.\cite{KOPM2012} First of all, it is straightforward to check that
$S^{(\theta)}_s$ is invariant under small variations of the sigma-model field,
$Q_s \rightarrow Q'_s = Q_s + \delta Q_s$ ($Q_s'^2 = \mathbf{1} = Q_s^2$). Thus,
$S^{(\theta)}_s$ only depends on the topology of the field configuration.
This immediately implies that it is zero in the topologically trivial sector.
In order to proof that $S^{(\theta)}_s$ also returns the correct value
$S^{(\theta)}_s = \pi \; (\text{mod} \; 2\pi)$ in the topologically non-trivial
sector, it is sufficient to insert a single instanton into $S^{(\theta)}_s$.
Instantons are field configurations that per definition can not be continuously
deformed into the vacuum configuration. Introducing the third dimension and
allowing the field to take values in the entire orthogonal group we can
continuously shrink the instanton in the $w=1$ sphere to the constant at $w =
0$. A necessary condition for this untwisting to happen is that
for some subinterval of $\left (0,1\right )$ the field leaves the NL$\sigma$M
manifold for the larger orthogonal group.
A direct calculation shows that the group volume covered while untwisting
indeed yields the value $iS^{(\theta)}_s = i\pi$, see Appendix
\ref{sec:Instanton}).
There have been alternative derivations of the $\mathbb Z_2$ term before
\cite{RyuZ2, OGM2007}. Viewing this theta term as a symmetry-broken WZ-term,
Eq.~(\ref{eq:thetaterm}), yields a
local expression for it and implies the following advantages. First, this form
is very useful for understanding the crossover between 3D topological insulators
of class DIII and AII. Second and more importantly, an analysis of response of
the system to an external electric field requires coupling of the diffusive
matter fields to $\mathbf{U}(1)$ gauge potentials. In particular, one should
gauge the topological term, which can be done in a standard way by using a
local expression for it. We will show in Section \ref{sec:EdotB} that such a
procedure yields the correct linear response theory for the anomalous quantum
Hall effect of Dirac fermions.
In addition to a non-trivial second homotopy group $\pi_2$, the sigma model
manifold of the class AII possesses also a non-trivial first homotopy group,
$\pi_1({\cal M}) = \mathbb Z_2$. For this reason, the RG flow in 2D systems
of class AII (as well as in other classes with a non-trivial $\pi_1$ group,
namely AIII, BDI, CII, and DIII) is affected by vortices, as was shown in
Ref.~\onlinecite{KOPM2012}. In the case of AII (and DIII) class these are $\mathbb
Z_2$ vortices,\cite{KOPM2012} i.e., a vortex is identical to an anti-vortex.
In a recent work \cite{FuKaneVortices} it was argued that such vortices are
crucial for establishing localization in the class AII. {Conversely, the robustness of a non-localized state on the
surface of a {weak} topological insulator and of the critical state separating 2D trivial and topological insulator were explained by vanishing of the
corresponding fugacity.}
On the surface of a strong 3D TI, the effect of vortices is erased by the
$\mathbb Z_2$ topological term, in the same way as argued
previously\cite{KOPM2012} for the case of the symmetry class CII.
Specifically, due to the $\mathbb Z_2$ theta term, the vortices acquire an
internal degree of freedom which, upon averaging, annihilates the
contribution of vortices to renormalization. For this reason, the vortices
{need not} be taken into account in the present context.
\subsection{Interacting NL$\sigma$M}
\label{sec:interactingNLSM}
In the previous subsection we have derived the diffusive non-linear sigma model
for non-interacting particles. The next step is to include the electron-electron
interactions.
\subsubsection{Interacting Fermi gas}
We concentrate first on the case of a weak Coulomb interaction ($\alpha \ll
1$).
At length scales larger than the screening length the interaction is effectively
pointlike:
\begin{equation}
S_{\rm int} = \frac{T}{2} \sum_{m,\alpha;ss'} \int_{\v x} \text{tr} {\left (I_m^\alpha
\psi_s \bar\psi_s\right )} U^q_{ss'}{\left (I_{-m}^\alpha \psi_{s'}
\bar\psi_{s'}\right )}
\end{equation}
where $ U^q_{ss'}$ is the ``overscreened'' Coulomb interaction matrix i.e., the $q
\rightarrow 0$ limit of Eq. \eqref{eq:coupledsurf} (for its generalization in case of an
asymmetric dielectric environment, see Appendix \ref{sec:Elstat}). We use the
bosonization rule
\begin{align}
\text{tr} I_m^\alpha \psi_s \bar \psi_s & = \text{tr} I_m^\alpha \left (1-\tau_y\right )\eta_{s,\uparrow} \eta_{s,\downarrow}^T -\text{tr} I_m^\alpha\left (1-\tau_y\right ) \eta_{s,\downarrow} \eta_{s,\uparrow}^T \notag \\
&\leftrightarrow i \lambda \left [ \text{tr} I_m^\alpha\left (1-\tau_y\right ) \left
(O_s + O_s^T\right )\right ].
\end{align}
When disorder is introduced, the matrices $O$ become restricted to the
sigma-model manifold ${\cal M}$, and we obtain
\begin{equation}
S_{\rm int} {=} - \lambda^2 8T \sum_{m,\alpha;ss'} \int_{\v x} {\text{tr}\left
[J_{-m}^\alpha Q_s\right ]} U^q_{ss'} {\text{tr}\left [J_{m}^\alpha Q_{s'}\right ]}.
\label{eq:bosonizedIA}
\end{equation}
Here we have defined $J_n^\alpha = I_n^\alpha \frac{1+\tau_y}{2}$.
As has been already emphasized, we want to treat the general case of strong
interactions up to $\alpha \sim 1$. Therefore, in the
following (and in more detail in Appendix \ref{sec:cleanFL}), we present the
Fermi liquid (FL) treatment of strongly interacting surface states of a thin 3D TI film.
\subsubsection{Effective spinless theory}
One of the most striking peculiarities of the surface states of 3D topological
insulators is their Rashba-like kinetic term. As a consequence, spin and
momentum are locked in a manner visualized in Fig.~\ref{fig:cone}. Such
states are called helical; one associates helicity eigenvalues $+1$ ($-1$)
with states with positive (respectively, negative) kinetic energy.
As has been stated above, we will be interested in the low energy regime $E \ll
\vert \mu_{1,2} \vert$. Hence, at each of the surfaces only one type of helical states
represents dynamical low energy degrees of freedom, while the other one is suppressed by a
mass $\approx 2 \vert \mu_{1,2} \vert$. Therefore, we project onto the appropriate helicity
eigenstate of each surface using the following projection operator
\begin{equation}
\mathcal{P}_{s} = \vert \mu_s, \v p \rangle \langle \mu_s, \v p \vert \text{
with } \vert \mu_s, \v p \rangle = \frac{1 }{\sqrt{2}} \left (\begin{array}{c}
1 \\
i \text{sgn}\mu_s \; e^{i \phi( \v p )}
\end{array} \right ),
\end{equation}
where we have defined the polar angle $\phi$ of the momentum, $p_x \equiv \vert
\v p \vert \cos \phi$ and $p_y \equiv \vert \v p \vert \sin \phi$.
The clean single-particle action becomes effectively spinless:
\begin{equation}
S^{(s)}_0 = -\sum_s \int_{\v p} \bar{\zeta}_{s} \left ( \v p \right ) \left
[i \hat \epsilon + \text{sgn}\left(\mu_s\right) \left( \vert \mu_s \vert - v_F^s
\vert \v p \vert \right ) \right ] \zeta_{s} \left ( \v p \right ),
\label{eq:spinlessSkin}
\end{equation}
where $\zeta_{s}$, $\bar \zeta_{s}$ are
the fields associated with the helicity eigenstates, $\zeta_{s} = \langle
\mu_s, \v p \vert \psi_{s}$ and $\bar \zeta_{s} = \bar \psi_{s,\sigma}\vert
\mu_s, \v p \rangle$.
\begin{figure}
\includegraphics[scale=.25]{cone}
\caption{Schematic representation of the Dirac cone and the strong Rashba spin
orbit coupling.
If the chemical potential (black plane) is large compared to the typical energy
scale $E$ (e.g., temperature), only one kind of helical states can take part in
the dynamics.
}
\label{fig:cone}
\end{figure}
\subsubsection{Scattering channels}
\label{sec:scattchannels}
In the presence of a Fermi surface, the electron-electron interaction at low
energies decouples into separate scattering channels defined by small
energy-momentum transfer and by the tensor structure in the surface space:
\begin{equation}
S_{\rm int} = - \frac{T}{2} \int_{P_1, P_2, K} \sum_{\alpha} \left
[\mathcal{O}^{IA}_{0+1} + \mathcal{O}^{IA}_2 + \mathcal{O}^{IA}_c\right ]
\label{eq:realIA}
\end{equation}
with
\begin{eqnarray}
\mathcal{O}^{IA}_{0+1} &=& \sum_{s_1s_2}\left [\bar \zeta^\alpha_{s_1} \left
(P_1\right ) \zeta^\alpha_{s_1} \left (P_1 + K\right )\right ] \notag \\ &&
\times \Gamma^{0+1,q}_{s_1,s_2;\hat p_1, \hat p_2} \left [\bar \zeta^\alpha_{s_2}
\left (P_2\right ) \zeta^\alpha_{s_2} \left (P_2 - K\right )\right ],
\end{eqnarray}
\begin{eqnarray}
\mathcal{O}^{IA}_2 &=& \sum_{s_1s_2}\left [\bar \zeta^\alpha_{s_1} \left
(P_2\right ) \zeta^\alpha_{s_1} \left (P_1 + K\right )\right ] \notag \\ &&
\times \Gamma^{2,q}_{s_1,s_2;\hat p_1, \hat p_2} \left [\bar \zeta^\alpha_{s_2}
\left (P_1\right ) \zeta^\alpha_{s_2} \left (P_2 - K\right )\right ],
\end{eqnarray}
and
\begin{eqnarray}
\mathcal{O}^{IA}_c &=& \sum_{s_1s_2} \left [\bar \zeta^\alpha_{s_1} \left
(P_2\right ) \zeta^\alpha_{s_1} \left (-P_1 + K\right )\right ] \notag \\ &&
\times \Gamma^{c,q}_{s_1,s_2;\hat p_1, \hat p_2} \left [\bar \zeta^\alpha_{s_2}
\left (-P_2+K\right ) \zeta^\alpha_{s_2} \left (P_1\right )\right ].
\end{eqnarray}
Here the
capital letters denote 2+1 momenta. The smallness of $K = \left (\omega_m, \v
q\right )$ means that the following conditions hold $\left (\omega_m, \vert \v q \vert \right) \ll \left (
\vert \mu_{s} \vert,p^{(s)}_{F}\right )$ for both $s = 1,2$. We emphasize that all
``Dirac factors'' of 3D surface electrons are included in the angular dependence
of the scattering amplitudes (subscripts $\Gamma_{\hat p_1,\hat p_2}$).
We refer to the three scattering channels as small angle scattering channel
($\Gamma^{0+1}$), large angle scattering channel ($\Gamma^2$), and the Cooper channel
($\Gamma^c$). The quantities entering Eq.~\eqref{eq:realIA} are the static
limit of the corresponding scattering amplitude, $\Gamma\left (\omega_m = 0, \v q
\right )$. They already include static screening and do not acquire any
tree-level corrections due to disorder. \cite{Finkelstein1990, Finkelstein2010}
Exemplary diagrams are given in figures \ref{fig:Gamma0} -- \ref{fig:Gammac}.
There, the small angle scattering amplitude is subdivided into its one Coulomb
line reducible part ($\Gamma^0$) and irreducible part ($\Gamma^1$) such that
\begin{equation}
\Gamma^{0+1}=\Gamma^{0}+\Gamma^{1}.
\end{equation}
The irreducible part $\Gamma^1$
also includes the short range interaction induced by the finite thickness of the 3D TI film (see Appendix \ref{sec:Elstat} and \ref{sec:BareGamma}).
For the short-range interaction amplitudes ($\Gamma^1$, $\Gamma^2$, $\Gamma^c$),
the static limit coincides with the ``q-limit'' $\Gamma^q = \lim_{q\rightarrow 0
}\Gamma\left (\omega_m = 0 , \v q \right )$, see also Appendix
\ref{sec:cleanFL}. It should be kept in mind that for the one-Coulomb-line-reducible part $\Gamma^{0}$ (it is long-ranged) the ``q-limit'' $\Gamma^{0,q}$ is only a valid
approximation if the mean free path $l$ exceeds the screening length. This applies to most realistic situations. (In the opposite
case
$\Gamma^0$ is parametrically small. {On top of this, the $q$-dependence of the Coulomb potential implies a strong scale dependence of both conductivity corrections and the interaction amplitude until the running scale reaches} the screening length at which $\Gamma^{0} \approx \Gamma^{0,q}$ is again
justified.)
We conclude this section with a side remark concerning the topological exciton
condensation. \cite{TEC} In order to find the conventional pole structure of the
FL Green's functions for the case $\text{sgn}(\mu_s) =-1$ one needs to transpose the
bilinear form in action \eqref{eq:spinlessSkin} and swap the notation $\zeta_s
\left (\epsilon_n\right )\leftrightarrow \bar \zeta_s \left (-\epsilon_n\right
)$. If $\text{sgn}(\mu_1 \mu_2) =-1$, this interchange of notations obviously happens in only
one surface. In this case, the large-angle scattering amplitude $\Gamma^{2}_{12}$ and the
Cooper-channel amplitude $\Gamma^c_{12}$ are interchanged. Even though this procedure illustrates the analogy between exciton condensation (divergence in $\Gamma^{2}_{12}$) and Cooper instability (divergence in $\Gamma^c_{12}$), in the
following we
choose to keep our original notation of $\zeta_s$ and $\bar \zeta_s$ also in the case of $\mu_s <0$.
\begin{figure}
\includegraphics[scale=.4]{Gamma0}
\caption{An example of contribution to a one Coulomb-line reducible small angle scattering
amplitude. Independently of $\text{sgn}\left (\mu_s\right )$, ingoing arrows
denote fields $\zeta_s$, outgoing arrows $\bar \zeta_s$.}
\label{fig:Gamma0}
\end{figure}
\begin{figure}
\includegraphics[scale=.4]{Gamma1}
\caption{An example of contribution to a one-Coulomb-line irreducible small-angle scattering
amplitude.}
\label{fig:Gamma1}
\end{figure}
\begin{figure}
\includegraphics[scale=.4]{Gamma2}
\caption{An example of contribution to a large-angle scattering amplitude.}
\label{fig:Gamma2}
\end{figure}
\begin{figure}
\includegraphics[scale=.4]{Gammac}
\caption{An example of contribution to a scattering amplitude in the Cooper channel.}
\label{fig:Gammac}
\end{figure}
\subsubsection{Clean Fermi liquid theory}
\label{sec:cleanFLMaintext}
A systematic treatment of the scattering amplitudes involves the field-theory of
the FL\cite{Nozieres_Luttinger_1962, AbrikosovGorkovDzyaloshinski,
LandauLifshitz9} (see Appendix \ref{sec:cleanFL}.) It is valid down to energy
scales $\sim\tau^{-1}_{1,2}$ and therefore constitutes the starting point for
the effective diffusive theory at lower energies, $T \ll \tau^{-1}_{1,2}$.
In contrast to the Green's function of the free theory, in the FL the
exact electronic propagator contains both a singular and a regular part. The
singular part (``quasiparticle pole'') includes a renormalized dispersion
relation and its residue is no more equal to unity but rather is $a_s \in \left
(0,1\right )$. As usual in the context of disordered FLs,\cite{Finkelstein1990} we absorb the quasiparticle residue by rescaling
the fermionic fields and redefining the scattering amplitude.
The conservation of the particle number separately in each of the two
surfaces leads to the following Ward identities:
\begin{equation}
\Pi^\omega_{s_1,s_2} \equiv \lim_{\omega_m \rightarrow 0} \Pi_{s_1,s_2} \left
(\omega_m,\v q = 0 \right ) = 0
\label{eq:Piomega}
\end{equation}
and
\begin{equation}
\Pi^q_{s_1,s_2} \equiv \lim_{\vert \v q \vert \rightarrow 0} \Pi_{s_1,s_2}
\left (\omega_m = 0,\v q \right ) = -\frac{\partial N_{s_1}}{\partial
\mu_{s_2}}.
\label{eq:Piq}
\end{equation}
Since these identities reflect the gauge invariance, they can not be altered
during the RG procedure. Thus, the static polarization operator is always given
by the compressibility $\partial N_{s_1}/\partial \mu_{s_2}$.
The FL theory in a restricted sense contains only
short range interactions $\Gamma^1$, $\Gamma^2$ and $\Gamma^c$. For electrons in
metals, one has also to include the long-range Coulomb interaction. Following
Ref. \onlinecite{Nozieres_Luttinger_1962}, the associated scattering amplitude
$\Gamma^0$ is obtained by means of static RPA-screening of Coulomb interaction
with the help of the FL renormalized polarization operator and
triangular vertices (see Fig.~\ref{fig:Gamma0def}). In Appendix
\ref{sec:cleanFL} we explicitly perform the formal FL treatment. This determines the interaction amplitudes at ballistic scales. They will serve as bare coupling constants of the diffusive NL$\sigma$M (see Sec. \ref{sec:barevalues}).
We now turn our attention to the disordered FL. This will allow us to find out which of the interaction channels give
rise to soft modes within our problem.
\begin{figure}
\includegraphics[scale=.5]{Gamma0screened}
\caption{A diagram contributing to $\Gamma^0$.}
\label{fig:Gamma0def}
\end{figure}
\subsubsection{Diffusive Fermi liquid theory}
The full amplitudes $\Gamma^{0+1}\left (K\right )$, $\Gamma^2\left (K\right )$
and $\Gamma^c\left (K\right )$ contain, among others, diagrams describing
multiple particle-hole (in the Cooper channel, particle-particle) scattering
(see Appendix \ref{sec:cleanFL}). The very idea of dirty FL lies in
replacing the dynamic part of these particle-hole (particle-particle) sections
by their diffusive counterpart.\cite{Finkelstein1990, Finkelstein2010} In
particular, only the zeroth angular harmonic of the scattering amplitudes
survives in the diffusive limit.
The scattering amplitude $\Gamma^2_{12}$ (as well as $\Gamma^c_{12}$) contains
only particle-hole (respectively, particle-particle) sections consisting of
modes from
opposite surfaces of the topological insulator. Since we assume the disorder to
be uncorrelated between the surfaces, these modes will not become diffusive
and are hence not of interest for the present investigation. We therefore do not
consider $\Gamma^2_{12}$ and $\Gamma^c_{12}$ any longer. As one can see from figures \ref{fig:Gamma0} - \ref{fig:Gamma2}, the large angle scattering amplitudes $\Gamma^2_{11}$ and
$\Gamma^2_{22}$ cannot be distinguished from the small angle scattering amplitudes $\Gamma^{0+1}_{11}$ and
$\Gamma^{0+1}_{22}$, respectively. We incorporate the effect of $\Gamma^2_{11}$ and
$\Gamma^2_{22}$ into the ``singlet channel'', which has the following matrix structure in the surface space
\begin{equation}
\underline \Gamma^{\rho} = \left (\begin{array}{cc}
\Gamma^{0+1-2}_{11} & \Gamma^{0+1}_{12} \\
\Gamma^{0+1}_{12} & \Gamma^{0+1-2}_{22}
\end{array} \right ). \label{eq:Gammasinglet}
\end{equation}
Here we used
\begin{equation}
\Gamma^{0+1-2}=\Gamma^{0+1}-\Gamma^{2}.
\end{equation}
The intrasurface Cooper channel interaction {$\Gamma^c_{ss}$} will be also neglected. Its bare value is repulsive for the Coulomb
interaction, so that the Cooper renormalization on ballistic scales $1/\tau \ll
E \ll \vert \mu \vert$ renders it small on the UV scale of the diffusive theory (i.e., at
the mean free path). Within the diffusive RG of a single 3D TI surface it
quickly becomes of the order of $1/\sqrt{\sigma}$ and thus negligible (see Ref. \onlinecite{Finkelstein1990} and
supplementary material of Ref. \onlinecite{OGM2010}).
{Consequently we drop the Cooper channel amplitude and do not consider the superconductive instability in this work.\footnote{The superconducting instability in a disordered system of Dirac fermions on a 3D TI surface (in the absence of density-density interactions) was recently addressed in Ref. \onlinecite{NandkishoreSondhi2013}.} For the opposite case of attraction in the Cooper channel Coulomb interaction suppresses the transition temperature $T_c$ .\cite{Finkelstein1987} The difference between Coulomb and short-range repulsive interaction was addressed in Ref. \onlinecite{BurmistrovGornyiMirlin12}.}
\subsubsection{Bosonization of Fermi Liquid}
The non-Abelian bosonization relies on the Dirac nature of the 2D
electrons
and on the associated non-Abelian anomaly. On the other hand, for $\alpha \sim
1$ the spectrum of the system gets strongly renormalized by interaction.
An appropriate description in such a situation is the FL
theory which is restricted to fermionic excitations close to the Fermi level.
So, one can ask whether the result of non-Abelian bosonization remains
applicable for $\alpha \sim 1$. The answer is yes, for the follwing reasons.
All terms of the bosonized theory except for the $\mathbb Z_2$ theta term are
determined by fermionic excitations close to the Fermi energy. Therefore, they
equally hold for the FL if the coupling constants are appropriately
redefined in terms of the corresponding FL parameters.
On the other hand, the $\mathbb Z_2$ theta term is a consequence of
the chiral anomaly and thus the only term determined by energies far from $\mu$.
However, it is well known that anomalies in quantum field theories are
insensitive to interactions. Hence, the $\mathbb Z_2$ term in the diffusive
NL$\sigma$M persists even for $\alpha \sim 1$. This follows also from
the key property of the FL state: its spectrum is
adiabatically connected to the free spectrum. This implies that
topological implications remain unchanged.
To summarize, the only difference between the NL$\sigma$M for the
weakly interacting
Fermi gas ($\alpha \ll 1$) and the FL ($\alpha \sim 1$) is
the replacement of the interaction strength by the appropriate FL
constant,
\begin{equation}
\underline U^q \rightarrow - \underline \Gamma^\rho \notag
\end{equation}
in Eq.~\eqref{eq:bosonizedIA}.
\subsubsection{Bare value of scattering amplitudes}
\label{sec:barevalues}
According to the formal FL treatment (Appendix
\ref{sec:appendixCoulomb}), the singlet-channel interaction amplitude is given
by
\begin{equation}
\underline \nu \underline \Gamma^{\rho} \underline \nu = - \underline \nu -
\frac{\det \underline \Pi^q}{\Pi^q_{11} +\Pi^q_{22}+2\Pi^q_{12}} \left
(\begin{array}{cc}
1 & -1 \\
-1 & 1
\end{array} \right ), \label{eq:Coulombcond}
\end{equation}
where $(\underline \nu)_{ss'} = \nu_s \delta_{ss'}$ and
\begin{equation}
\underline \Pi^q = - \underline \nu - \underline \nu \left (\begin{array}{cc}
\Gamma^{1-2}_{11} & \Gamma^{1}_{12} \\
\Gamma^{1}_{12} & \Gamma^{1-2}_{22}
\end{array} \right ) \underline \nu. \label{eq:barePolop}
\end{equation}
Here $\Gamma^{1-2}=\Gamma^{1}-\Gamma^{2}$.
The remarkably simple matrix structure of $\underline \nu + \underline \nu
\underline \Gamma^{\rho} \underline \nu$ is actually due to the presence of the
long-range Coulomb interaction. This fact will be explained by means
of $\mathcal
F$-invariance in section \ref{sec:Finv}. It has very important consequences for
the RG flow in the diffusive regime, see Sec.~\ref{sec:Densityresponse}.
\subsubsection{Action of NL$\sigma$M}
We are now in a position to present the full action of the diffusive interacting
NL$\sigma$M for the problem under consideration:
\begin{equation}
\label{eq:TOTALNLSM}
S = \sum_s \left [S^{({\rm kin})}_s + i S^{(\theta)}_s\right ] +
S^{(\eta\,+\,{\rm int})}.
\end{equation}
It contains the kinetic term
\begin{equation}
S^{({\rm kin})}_s = \frac{\sigma_s}{16} \int_{\v x} \text{tr} \left (\nabla Q_s\right
)^2
\end{equation}
and the ${\mathbb Z}_2$ theta term
\begin{equation}
S^{(\theta)}_s = \frac{1}{24 \pi} \left . \Gamma_s \right \vert_{\tilde
O_s\left (\v x, w = 1\right ) = Q_s\left (\v x\right ) = Q_s^T\left (\v x\right
)}
\label{eq:Stheta}
\end{equation}
for each of the surfaces, as well as the frequency and interaction terms,
\begin{eqnarray}
S^{(\eta\,+\,{\rm int})} &=& -\pi T \left [\sum_s 2z_s \text{tr} \hat \eta Q_s \right . \notag
\\
& - & \left . \sum_{ss';n,\alpha} \text{tr} \left [J_n^\alpha Q_s\right ]
\Gamma_{ss'}\text{tr} \left [J_{-n}^\alpha Q_{s'}\right ]\right ] .
\label{eq:IANLSM}
\end{eqnarray}
Here we have introduced the notation
\begin{equation}
\Gamma_{ss'} = \frac{8}{\pi} \lambda_s \Gamma^\rho_{ss'}\lambda_{s'}.
\label{eq:defGammaNLSM}
\end{equation}
\subsection{Inclusion of scalar and vector potentials into the NL$\sigma$M}
In this subsection, we investigate consequences of the gauge invariance for the
interacting NL$\sigma$M.
\subsubsection{Electromagnetic gauge invariance}
\label{sec:elmaggauge}
We include the scalar potential $\Phi_s$ and the vector potential
$A_{\mu,s}$ for surface $s$ in the microscopic action
\eqref{eq:Highenergyaction}
by means of covariant derivatives. This makes the action
gauge-invariant, i.e., unchanged under
local $\mathbf{U}(1)$-rotations of the fermionic fields $\psi$ and $\bar \psi$
accompanied by the corresponding gauge transformation of the potentials.
Note that locality implies independent rotations on the top and bottom surfaces of the TI film.
The rotations of $\psi$-fields imply the following rotation of bispinors:
\begin{equation}
\eta_s\left (\v x\right ) \rightarrow W_s \eta_s \left (\v x\right ),
\label{eq:etarotation}
\end{equation}
where
\begin{equation}
W_s = \left [e^{-i \hat \chi_s^T} \frac{1 +\tau_y}{2} + e^{i \hat \chi_s}
\frac{1 -\tau_y}{2} \right ]
\label{eq:deofofW}
\end{equation}
and we use the following convention for hatted matrices: $\hat a \equiv
\sum_{n,\alpha} a_n^\alpha I_n^\alpha$.
Let us recall that the $\eta_s$ fields are considered as
vectors in the Matsubara space. Upon introducing replica indices in the theory,
the $\mathbf{U}(1)$ rotation angles and correspondingly the gauge potentials
get replicated as well.
\subsubsection{$\mathcal{F}$-algebra and $\mathcal{F}$-invariance}
\label{sec:Finv}
As a direct consequence of \eqref{eq:etarotation}, $Q$-matrices transform under
a gauge transformation $\chi_s$ in the following way:
\begin{equation}
Q_s \rightarrow W_s Q_s W_s^T. \label{eq:Wrot}
\end{equation}
Under such rotations, in the limit $N_M^\prime, N_M \to \infty$,
$N_M/N_M^\prime \to 0$, the frequency term acquires the correction
\cite{MishandlingI}
\begin{equation}
\delta_\chi \text{tr} \hat \eta Q_s = 2 \sum_{n, \alpha} \left [in \chi_{s,n}^\alpha
\text{tr} J_{-n}^\alpha Q_s - n^2 \chi_{s,n}^\alpha \chi_{s,-n}^\alpha\right ],
\label{eq:gaugetraforQ}
\end{equation}
while the factors entering the interaction term vary as follows:
\begin{equation}
\delta_\chi \text{tr} J_n^\alpha Q_s = -i2 n \chi_{s,n}^\alpha.
\end{equation}
As explained in Sec. \ref{sec:symmetries}, the presence of the Coulomb
interaction
implies invariance of the fermionic action \eqref{eq:Highenergyaction} under
a simultaneous rotation in both surfaces by the same spatially constant
(``global'') but
time-dependent $\mathbf{U}(1)$-phase even \textit{without} inclusion of gauge
potentials (``$\mathcal F$-invariance''). This symmetry has to be
preserved on NL$\sigma$M level, implying that
\begin{equation}
\left (\underline{z} + \underline{\Gamma}\right )\left (\begin{array}{c}
1 \\
1
\end{array} \right ) = 0 .
\label{eq:Finv0}
\end{equation}
Here $(\underline z)_{ss'} = z_s \delta_{ss'}$. Since the intersurface
interaction is symmetric, $\Gamma_{12}=\Gamma_{21}$, Eq. \eqref{eq:Finv0}
yields
\begin{equation}
\underline{z} + \underline{\Gamma} = \text{const.} \times \left
(\begin{array}{cc}
1 & -1 \\
-1 & 1
\end{array} \right ) \label{eq:Finv}.
\end{equation}
This relation is consistent with Eq. \eqref{eq:Coulombcond}. However, contrary
to Eq. \eqref{eq:Coulombcond}, the relation \eqref{eq:Finv} is {manifestly}
imposed by the symmetry (``$\mathcal{F}$-invariance'') of the action
\eqref{eq:TOTALNLSM}. It should therefore remain intact under RG flow.
\subsubsection{Gauging the NL$\sigma$M and linear-response theory}
\label{sec:Kubo}
Generally, the requirement of gauge invariance prescribes the correct
coupling to the scalar and vector potentials in the action of the NL$\sigma$M,
Eq. \eqref{eq:TOTALNLSM}.
In particular, in the kinetic term one has to replace $\partial_\mu Q_s
\rightarrow D_{\mu,s} Q_s$ with the long derivative $D_\mu$ of the form
\begin{equation}
D_{\mu,s} Q_s \equiv \partial_\mu Q_s + \sum_{n,\alpha} i A^\alpha_{\mu,s, -n}
\left [J_n^\alpha - \left (J_n^\alpha\right )^T,Q_s\right ].
\end{equation}
For simplicity, the electron charge is absorbed into the vector potential here and in the following subsection.
As the theory is non-local in the imaginary time, the inclusion of the scalar
potential is non-linear. The corresponding term that should be added to the
NL$\sigma$M
\eqref{eq:TOTALNLSM} reads
\begin{eqnarray}
S^{\Phi} &=& -2 \sum_{n\alpha,ss'} {\Phi_{n,s}^\alpha} {\left (\underline
z+\underline \Gamma\right )_{ss'}} \text{tr} J_{n}^{\alpha}Q_s \notag \\ &+&
\frac{1}{\pi T} \sum_{n\alpha,ss'} {\Phi_{n,s}^\alpha} {\left (\underline
z+\underline \Gamma\right )_{ss'}} \Phi_{-n, s}^\alpha .
\end{eqnarray}
The inclusion of the scalar and vector potentials allow us to express the
density-density correlation function and the conductivity in terms of the
matrix fields $Q_s$ by means of the linear-response theory.
In particular, a double differentiation of the partition function with respect
to the scalar
potential yields the density-density response,
\begin{eqnarray}
\Pi^{\rm RPA}_{ss'}\left (\omega_n, \v q\right ) &=& -\frac{2}{\pi} \left
(z+\Gamma\right )_{ss'} \notag \\
&& + 4 T\sum_{s_1,s_2} \left (z+\Gamma\right )_{ss_1} \left \langle \text{tr}
J_{n}^{\alpha}Q_{s_1}\left (\v q\right )\right . \times \notag \\
&& \times \left . \text{tr} J_{-n}^{\alpha}Q_{s_2}\left (- \v q\right )\right \rangle
\left (z+\Gamma\right )_{s_2s'} .
\label{eq:Kubodensity}
\end{eqnarray}
Here $\left \langle ... \right \rangle$ denotes average with respect to the action
\eqref{eq:TOTALNLSM}. The superscript $^{\rm RPA}$ emphasizes that the quantity appearing in the total density-density response includes RPA resummation. It is thus one-Coulomb-line-reducible and only its irreducible part
corresponds to the polarization operator.
In the same spirit, we obtain the expression for the conductivity (in units
of $e^2/h$) at a finite, positive frequency $\omega_n$:
\begin{equation}
\sigma'_{ss'}\left (\omega_n\right ) = B^{(s)}_1 \delta_{ss'} + B^{(ss')}_2.
\label{eq:Kubocurrent}
\end{equation}
Here we introduced two correlators:
\begin{equation}
B^{(s)}_1 = \frac{\sigma_s }{8n} \left \langle \text{tr} \left [J_n^\alpha - \left
(J_n^\alpha\right )^T,Q_s\right ]\left [J_{- n}^\alpha - \left
(J_{-n}^\alpha\right )^T,Q_s\right ]\right \rangle \label{eq:defB1}
\end{equation}
and
\begin{eqnarray}
\hspace*{-0.5cm} B^{(ss')}_2 &=& \frac{\sigma_s\sigma_{s'}}{128 n} \int_{\v x -
\v x'} \sum_{\mu = x,y} \notag \\
&& \left \langle \text{tr}\left \lbrace \left [J_n^\alpha - \left (J_n^\alpha\right
)^T,Q_s\right ] \partial_\mu Q_s \right \rbrace_{\v x} \times \right . \notag
\\
&& \times \left . \text{tr}\left \lbrace \left [J_{-n}^\alpha - \left
(J_{-n}^\alpha\right )^T,Q_{s'}\right ] \partial_\mu Q_{s'} \right \rbrace_{\v
x'} \right \rangle. \label{eq:defB2}
\end{eqnarray}
Substituting the saddle-point value $Q_s = \Lambda$, we obtain the classical value
$\sigma'_{ss'} \left (\omega_n\right ) = \sigma_{s}\delta_{ss'}$. Hence the
dimensionless coupling constant of the NL$\sigma$M has been identified with the
physical conductivity in units of $e^2/h$.
\subsubsection{Gauging the theta term and anomalous quantum Hall effect}
\label{sec:EdotB}
The local expression of the $\mathbb Z_2$ theta term, i.e, the WZW-term, Eq.
\eqref{eq:Stheta}, also allows of inclusion of gauge
potentials. \cite{PolyakovWiegman, DiVecchia, Faddeev,
Gerasimov, Nekrasov, Smilga} However, the
situation is more subtle here. Specifically, it turns out that the contribution
of non-singular gauge potentials to the topological term $S^{(\theta)}$
vanishes. We explicitly show this in Appendix \ref{sec:appendixBosonization}.
The situation changes when the time-reversal symmetry is broken (at least,
in some spatial domain at the surface) by a random or/and unform magnetic
field.
Subjected to a strong magnetic field, 3D TI surface states display the
characteristic quantum Hall effect of Dirac electrons \cite{BruehneHgTe,NomuraQHE}
with quantized transverse conductance
\begin{equation}
\sigma_{xy} = g \left (n \pm \frac{1}{2}\right )\frac{e^2}{h}, \; n \in \mathbb
Z,
\label{eq:anomalous_QHE}
\end{equation}
where $g$ is the degeneracy of Dirac electrons, e.g., $g = 2$ for two 3D TI
surfaces. {It is intimately linked to the topological magnetoelectric effect.\cite{QiZhangTME,QiZhangMonopole,EssinMooreVanderbilt09,PesinMacDonald12}}
Theoretically, the anomalous quantum Hall effect was explained and discussed in a previous
work by three of the authors. \cite{OGMQHE} We will explain in the following how to understand it in the
framework of the
linear response theory within the NL$\sigma$M. As it turns out, the crucial
point is that gauge potentials drop from $S^{(\theta)}$.
We first briefly recall the NL$\sigma$M field theory describing the
ordinary integer
QHE (i.e., for electrons with quadratic dispersion). It contains Pruisken's
theta
term, \cite{Pruisken} which assumes the following form upon inclusion of the vector potential: \cite{MishandlingI}
\begin{subequations}
\begin{align}
S^{\text{QHE}}=&\frac{ \vartheta}{16\pi} \int_{\v x} \epsilon_{\mu\nu} \text{tr} Q_U
\partial_\mu Q_U \partial_\nu Q_U
\label{eq:Pruiskenterm}\\
&+ \frac{ i \vartheta}{4\pi} \int_{\v x} \epsilon_{\mu\nu} \text{tr} \partial_\mu \hat
A_\nu Q_U \label{eq:FQ} \\
&+ \frac{\vartheta}{4\pi} \int_{\v x}
\epsilon_{\mu\nu}
\sum_{n,\alpha} n A_{\mu,n}^\alpha A_{\nu,-n}^\alpha.
\label{eq:ChernI}
\end{align}\label{eq:QHE}
\end{subequations}
Here $Q_U = U^{-1}\Lambda U$ with $U \in \mathbf{U}\left (2 N_M \times
N_R\right )$, $\epsilon_{\mu\nu} = -\epsilon_{\nu\mu}$ is the 2D antisymmetric symbol ($\epsilon_{xy} \stackrel{\text{def.}}{=} 1$), and $\vartheta$ is the theta angle of the Pruisken's NL$\sigma$M. We
emphasize, that the last two terms (Eqs.~\eqref{eq:FQ} and \eqref{eq:ChernI})
determine the effective electromagnetic response and thus prescribe the relation
between the physical observable $\sigma_{xy}$ (in units of $e^2/h$) and the theta angle
$\vartheta$. In particular, $\vartheta/2\pi$ is identified as the bare value of the Hall conductance.\cite{PruiskenBurmistrovCPN-1}
Let us now turn to a single Dirac surface state. As has been discussed above,
all gauge potentials drop from $S^{(\theta)}$. Let us first add a random magnetic
field (keeping zero average magnetic field) to the gauged NL$\sigma$M.
This implies a breakdown of the symmetry:
\begin{equation}
\mathcal{M} \rightarrow \frac{\mathbf{U}\left (2
N_M N_R\right )}{\mathbf{U}\left ( N_M N_R\right )\times\mathbf{U}\left (N_M
N_R\right )}.
\end{equation}
The $\mathbb Z_2$ theta term becomes the Pruisken's theta
term \cite{BocquetSerbanZirnbauer} (recall $\theta = \pi\, \text{mod}\, 2\pi$)
\begin{equation}
S^{(\theta)}_{U}=\frac{\theta}{16\pi} \int_{\v x} \epsilon_{\mu\nu} \text{tr} Q_U
\partial_\mu Q_U \partial_\nu Q_U. \label{eq:Pruiskentermcriticality}\\
\end{equation}
We emphasize that {together with the gauged kinetic term} $S^{(\theta)}_{U}$ is the complete gauged theory, no extra terms of
the type
\eqref{eq:FQ} and \eqref{eq:ChernI} appear. Being topological, the Pruisken's theta
term is invariant under smooth $\mathbf U\left (1\right )$ rotations. Recall
that exactly the terms \eqref{eq:FQ} and \eqref{eq:ChernI} provided a link
between $\vartheta$ and $\sigma_{xy}$ in the conventional (non-Dirac) QHE
setting. Their absence in Eq. \eqref{eq:Pruiskentermcriticality} is thus
physically very natural: without a net magnetic field the Hall conductivity is
zero.
We consider now the case when the average magnetic field is non-zero. The
action of the NL$\sigma$M describing a Dirac fermion is then given by a sum of
Eqs.~\eqref{eq:QHE} and \eqref{eq:Pruiskentermcriticality}.
The renormalization of the action of the NL$\sigma$M is governed by the full theta angle $\vartheta + \theta$. On the other hand, only $\vartheta$ is related with the bare value of $\sigma_{xy}$. Then standard arguments for the quantization of the Hall conductivity \cite{PruiskenBurmi} leads to the result
(\ref{eq:anomalous_QHE}) for the anomalous QHE.
\section{One-loop RG}
\label{sec:1loopRG}
In the preceding section we have derived the diffusive {NL$\sigma$M},
Eqs. \eqref{eq:TOTALNLSM}. We will now investigate its behavior under
renormalization. This will allow us, in particular, to deduce the scale
dependence of the conductivity. The most important steps of the calculation are
presented in the main text; further details can be found in Appendix
\ref{sec:RGderiv}.
We calculate the renormalization of the NL$\sigma$M parameters within the
linear-response formalism (rather than the background-field method). This is
favorable since it implies a more direct physical interpretation of the
NL$\sigma$M coupling constants. Furthermore, this way one can in principle
treat simultaneously different infrared regulators, such as temperature or
frequency. However,
for the sake of clarity of presentation we restrict ourselves to a purely
field-theoretical regularization scheme and add a mass term to the action
\begin{equation}
S_{L} = -\sum_{s=1,2}\frac{\sigma_s L^{-2}}{8} \int_{\v x} \text{tr} \Lambda Q_s.
\label{eq:Massterm}
\end{equation}
The connection between the running length scale $L$ and the physical regulators
temperature or frequency was analyzed in Ref. [\onlinecite{MishandlingII}].
Roughly speaking, in the presence of a single infrared scale $E$, e.g. when calculating DC conductance at finite
temperature and assuming an infinite sample, one can replace $L$ by
$L_E$ in the results.
We will calculate all UV-divergent contributions in the dimensional regularization scheme.
This allows us to preserve the local
$\text{O}\left (2_\tau \times N_M \times N_R\right ) \times \text{O}\left
(2_\tau \times N_M \times N_R\right )$-symmetry of the $Q$-matrix
\eqref{eq:defQ} and to ensure the renormalizability of the theory.
\subsection{Diffusive propagators}
\label{sec:Propagators}
We employ the exponential parametrization of the matrix fields $Q_s = \Lambda
\exp W_s$. The antisymmetric fields
$$W_s = \left
(\begin{array}{cc}
0 & q_s \\
-q_s^T & 0
\end{array} \right )
$$
anticommute with $\Lambda$.
Further, we define a set of real matrices in the particle-hole space: $\tilde
\tau_\mu \equiv 2^{-1/2} \left (\mathbf{1}, \tau_x, i \tau_y, \tau_z \right
)$. This allows us to introduce the fields $q^{(\mu)} \equiv \text{tr}^{\tau} q \tilde
\tau_\mu^T$, where $\text{tr}^\tau$ is the trace in the particle-hole space only.
With these definitions at hand, we expand the action, Eqs.~\eqref{eq:TOTALNLSM}
and \eqref{eq:Massterm}, to quadratic order in $q^{(\mu)}$ and obtain the
NL$\sigma$M propagators that describe the diffusive motion in the particle-hole
(diffusons) and particle-particle (cooperons) channels.
The fields $q^{(1)}$ and $q^{(3)}$ describe cooperons. Their propagator is
unaffected by interaction (since we have discarded the interaction in the
Cooper channel),
\begin{eqnarray}
\left \langle \left [q_s^{(\mu)}(\v p)\right ]_{m_1m_2}^{\alpha_1\alpha_2} \left
[q_{s'}^{(\nu)}(-\v p)\right ]_{n_1n_2}^{\beta_1\beta_2} \right \rangle = \frac{4}{\sigma_s} D_s
\left (\omega_{n_{12}},\v p\right ) \delta_{ss'} \notag \\
\hfill\times \delta_{\mu\nu} \delta_{n_1m_1}\delta_{n_2m_2}\delta_{\alpha_1\beta_1}
\delta_{\alpha_2\beta_2}\bigl (\delta_{\mu 1} + \delta_{\mu 3} \bigr ) , \hspace{0.3cm}{}\,
\end{eqnarray}
where
\begin{equation}
\bigl [ D_s\left (\omega_{n_{12}},\v p\right ) \bigr ]^{-1}= \v
p^2 + L^{-2} + \frac{4z_s}{\sigma_s} \omega_{n_{12}}.
\label{eq:usualpropagator}
\end{equation}
The Matsubara indices $n_1$, $m_1$ are non-negative, while the indices
$n_2$, $m_2$ are negative; we have also defined $n_{12} \equiv n_1-n_2 > 0$ and
$m_{12} \equiv m_1-m_2 > 0$.
Next, we consider the diffusons $q^{(0)}$ and $q^{(2)}$. Their Green's function,
written as a matrix in surface space, is
\begin{eqnarray}
&\left \langle \left [q^{(\mu)}_s(\v p)\right ]_{m_1m_2}^{\alpha_1\alpha_2}
\left [q^{(\nu)}_{s'}(-\v p)\right ]_{n_1n_2}^{\beta_1\beta_2} \right \rangle
= \displaystyle \frac{4}{\sigma_s} D_s
\left (\omega_{n_{12}},\v p\right )
\delta_{\mu\nu} \notag \\
& \times\delta_{n_{12},m_{12}}\delta_{\alpha_1\beta_1}
\delta_{\alpha_2\beta_2}\bigl (\delta_{\mu 0} + \delta_{\mu 2} \bigr ) \notag \\
&\times
\left [\delta_{n_1m_1} \delta_{ss'}
- \displaystyle\frac{8\pi T}{\sigma_{s'}} \delta^{\alpha_1\alpha_2} \left (\underline{\Gamma D^c}\left (\omega_{n_{12}},\v p\right )
\right )_{ss'}\right ].
\label{eq:totaldiffusonpropagator}
\end{eqnarray}
Here we have introduced
\begin{equation}
\bigl [D^c\left (\omega_{n_{12}},\v p\right )\bigr ]^{-1}_{ss'} = D_s^{-1}\left (\omega_{n_{12}},\v p\right )\delta_{ss'} + \frac{4 \omega_{n_{12}}}{\sigma_s}
\Gamma_{ss'} .
\label{eq:AApropagator}
\end{equation}
\subsection{RG invariants}
\label{sec:Densityresponse}
The bare action contains, aside from the mass $L^{-1}$, seven running coupling
constants: $\sigma_1$, $\sigma_2$, $z_1$, $z_2$, $\Gamma_{11}$, $\Gamma_{22}$
and $\Gamma_{12}$. We are now going to show that three linear combinations of
them are conserved under RG. To this end we evaluate the density-density
response \eqref{eq:Kubodensity} at the tree level:
\begin{equation}
\underline{\Pi}^{\rm RPA}\left (\omega, \v p\right ) = -\frac{2}{\pi} \left
[\underline{z} + \underline{\Gamma}\right ]\bigl (1-4\omega \underline{\sigma^{-1} D^c}\left (\omega, \v p\right )\left
[\underline{z} + \underline{\Gamma}\right ]\bigr )
\label{eq:Pired}
\end{equation}
where $(\underline{\sigma})_{ss'}=\sigma_s\delta_{ss'}$. There is no need for infrared
regularization here and we therefore omit the mass term \eqref{eq:Massterm}.
On the other hand, the density-density response function can be obtained from
the fermionic formulation of the theory, see Appendix \ref{sec:Piredappendix}:
\begin{equation}
\underline{\Pi}^{\rm RPA} = \left
[\underline{\Pi}^q - \underline{\nu\Gamma}^0 \underline \nu\right ]\left (1+
\omega \underline \Delta^{\Gamma}\left (\omega, \v p\right )\left [\underline{\Pi}^q -
\underline{\nu\Gamma}^0 \underline \nu\right ]\right ),
\label{eq:fermionicPired}
\end{equation}
where
\begin{equation}
\Delta^{\Gamma} \left (\omega, \v p\right ) = \left
[\underline{\nu D} \v p ^2 + \omega\left ( \underline {\nu} + \underline{\nu}
\underline{\Gamma}^{\rho,q} \underline{\nu}\right ) \right ]^{-1}
\end{equation}
The equality of {Eqs.} \eqref{eq:Pired} and \eqref{eq:fermionicPired}
{relates two functions of momentum and frequency. In the static limit, we find}
the following {constraint} connecting the NL$\sigma$M coupling constants with
physical FL parameters:
\begin{eqnarray}
\frac{2}{\pi}\left (\underline{z} + \underline{\Gamma}\right ) = -
\underline{\Pi^q} + \underline{\nu}\underline{\Gamma^0}\underline{\nu}.
\label{eq:constraint1}
\end{eqnarray}
{Next, from comparison of momentum dependence in Eqs. \eqref{eq:Pired} and \eqref{eq:fermionicPired},} we find {the Einstein relation:} $\sigma_s = 2\pi\nu_s D_s$. Accordingly, $\sigma$ measures the
conductance in units of $e^2/h$, consistently with what has been found
in Secs.~\ref{sec:semiclassical} and \ref{sec:Kubo}.
In view of gauge invariance (Sec.
\ref{sec:cleanFLMaintext}), the static polarization operator
entering Eq.~(\ref{eq:constraint1}) is nothing but the
compressibility
$$\Pi^{ss',q} = - \frac{\partial N_s}{\partial \mu_{s'}}.$$
Its value is not renormalized because it can be expressed as a derivative of a
physical observable with respect to the chemical potentials. On ballistic
scales the chemical potential enters
logarithmically divergent corrections only as the UV cutoff of the
integrals. In the diffusive regime, the UV cutoff is provided by the scattering
rates $\tau^{-1}_s \ll \vert \mu_s \vert$. Therefore, diffusive contributions to the
derivative with respect to the chemical potential
vanish.\cite{Finkelstein1990} Since $\underline \nu \underline \Gamma^0
\underline \nu$ only depends on $\underline \Pi^q$ (see Appendix
\ref{sec:appendixCoulomb}) it is not renormalized as well.
Therefore, the right-hand side of
\eqref{eq:constraint1} is not renormalized and hence neither is its
left-hand-side, i.e., $\underline z + \underline \Gamma$.
This matrix constraint yields three RG invariants: $z_1 + \Gamma_{11}$,
$z_2 + \Gamma_{22}$, and $\Gamma_{12}$. Thus, only four out of seven
NL$\sigma$M parameters are independent running coupling constants. We
emphasize that, in contrast to Eq. \eqref{eq:Finv}, this reasoning is valid also
in the absence of long-range interaction.
Finally, let us evaluate {Eq.} \eqref{eq:constraint1} on the bare level. Expressing the static polarization operator as $\underline \Pi^q = - \underline \nu -
\underline{\nu}\underline{\Gamma^{1-2}}\underline{\nu}$ {and using} the definition of $z_s$ in Sec. \ref{sec:freqboson}
one can {find the following relations for} the bare values
\begin{equation}
\frac{4\lambda_s}{\pi} \equiv \frac{2}{\pi} z_s = \nu_s .
\label{eq:relrel0}
\end{equation}
Equivalently, the same relationship between $\lambda_s$ and $\nu_s$
can be obtained by comparing the bare definition of $\underline \Gamma$ [Eq.
\eqref{eq:defGammaNLSM}] with the right hand side of \eqref{eq:constraint1}.
{The relation \eqref{eq:relrel0}} has been foreseen earlier on the basis of SCBA, see Eq.
\eqref{eq:lambdamu}. In conclusion, the SCBA and the density response
independently show that the UV cutoff scale for the bosonization is
automatically set by the chemical potential (which is also very natural from
the physical point of view).
\subsection{Renormalization of conductivities}
\subsubsection{{Correlator $B_1$}}
We will first analyze the correlator $B^{(s)}_1$, Eq. \eqref{eq:defB1}. The
one-loop correction is determined by the expansion to second order in
$q^{(\mu)}$. The tensor structure in particle-hole space implies that the
diffuson contribution ($\mu = 0,2$) vanishes. The classical
value together with the cooperon contribution ($\mu = 1,3$) is
\begin{equation}
B^{(s)}_1 = \sigma_s +2 \int_{\v p }D_s(\omega_n,\v p) .
\label{eq:usualHLN1}
\end{equation}
We evaluate this term in the announced regularization scheme:
\begin{eqnarray}
B^{(s)}_1 &=& \sigma_s + 2 \text{I}^{(2+\epsilon)}_1\\
&=& \sigma_s + \frac{1}{2\pi} \left [-\frac{2}{\epsilon} + 2\ln {L}/{l}
+ \text{const.}\right ] .
\label{eq:usualHLN}
\end{eqnarray}
For dimensional reasons we have introduced the reference length scale $l$,
which for the present diffusive problem is set by the mean free path $l =
\max_{s=1,2}l_s$. We have further evaluated the following standard
dimensionless integral
\begin{eqnarray}
\text I^{(D)}_1 &\equiv & l^{D-2}\int \frac{d^D p}{\left (2 \pi\right )^D}
\frac{1}{\v p^2 + L^{-2}} \notag \\
&=& \frac{ \left (\frac{l^2}{L^2}\right )^{\frac{D}{2}-1}}{\left (4\pi\right
)^{\frac{D}{2}}} \Gamma \left (1- \frac{D}{2}\right ) \notag \\
&\stackrel{D=2+\epsilon}{=}& \frac{1}{4\pi}\left [-\frac{2}{\epsilon} + 2\ln
{L}/{l} + \ln 4\pi - \boldsymbol \gamma + \mathcal{O}\left (\epsilon\right )\right ],
\notag
\end{eqnarray}
where $\boldsymbol \gamma \approx 0,577 $ is the Euler-Mascheroni constant.
The logarithmic term in Eq.~\eqref{eq:usualHLN} is nothing but the
well-known weak-antilocalization effect.\cite{HLN}
\subsubsection{{Correlator $B_2$}}
Next we turn our attention to $B^{(ss')}_2$, Eq.~\eqref{eq:defB2}. Because of
the presence of gradients {it} does not contribute neither at classical nor at
tree level. Furthermore, due to the absence of the Cooper channel and the
uncorrelated disorder on the {top and bottom} surfaces, there are no quantum corrections to
the transconductance $\sigma_{12}$.
The correlator $B^{(ss')}_2$ can be recast into the form (see Appendix \ref{sec:RGderiv})
\begin{eqnarray}
B_2^{(ss')} &=& \frac{16\delta_{ss'}}{n\sigma_s}\int_{\v p} \v p^2 \sum_{\omega_m>0} \omega_m \notag \\
&&\times \Bigl [ \left ( \underline{D\Gamma D^c}\right )_{ss} \left (\omega_{m},
{\v p}\right )D_s \left (\omega_{m+n}, {\v p}\right ) \notag \\
&& - \left ( \underline{D\Gamma D^c}\right )_{ss} \left (\omega_{m+n}, {\v
p}\right )D_s \left (\omega_{m+2n}, {\v p}\right ) \Bigr ] .
\label{eq:AAcorr1}
\end{eqnarray}
For its evaluation it is instructive to separate
contributions stemming from intrasurface interaction $\Gamma_{ss}$ and
intersurface interaction $\Gamma_{12}$. This leads to
\begin{eqnarray}
B_2^{ss'} &=& -4 \delta_{ss'} \left (\underbrace{1 -
\frac{1+\gamma_{ss}}{\gamma_{ss}} \ln \left ( 1 + \gamma_{ss}\right
)}_{\text{single surface}} \right .\notag \\
& +& \left .\underbrace{\left (1+\gamma_{ss}\right )\left (\frac{\ln \left ( 1 +
\gamma_{ss}\right )}{\gamma_{ss}} - \frac{\ln \left ( 1 + \tilde
\gamma_{ss}\right )}{\tilde \gamma_{ss}} \right )}_{\text{intersurface
interaction}} \right ) I_2^{(2+\epsilon)} \notag \\
&=& -\frac{\delta_{ss'}}{\pi}\left (1 - \frac{1+\gamma_{ss}}{\tilde \gamma_{ss}} \ln \left
( 1 + \tilde \gamma_{ss}\right )\right )\times \notag \\
&& \times \left [- \frac{2}{\epsilon} + 2 \ln L/l + \text{const}\right ] .
\label{eq:AAcorr}
\end{eqnarray}
We have introduced $\gamma_{ss} = \Gamma_{ss}/z_s$, $\tilde{\gamma}_{11}
= \gamma_{11} + (\sigma_1/\sigma_2) (1+\gamma_{11} )$ and
$\tilde{\gamma}_{22}= \gamma_{22} + (\sigma_2 / \sigma_1)
(1+\gamma_{22} )$. Note that in the limit of $z_2 + \Gamma_{22} = 0$
[which corresponds to $\Gamma_{12} =0$ in view of \eqref{eq:Finv}] we recover
the well-known conductivity corrections to $\sigma_{11}$ for a single surface
(see also Sec. \ref{sec:surfacebulk}).
Further, in Eq.~(\ref{eq:AAcorr}) we have evaluated the second standard
diverging integral
\begin{eqnarray}
\text I^{(D)}_2 &\equiv & l^{D-2}\int \frac{d^D p}{\left (2 \pi\right )^D}
\frac{\v p^2}{\left (\v p^2 + L^{-2}\right )^2} \notag \\
&=& \frac{ \left (\frac{l^2}{L^2}\right )^{\frac{D}{2}-1}}{\left (4\pi\right
)^{\frac{D}{2}}} \frac{D}{2}\Gamma \left (1- \frac{D}{2}\right ) \notag \\
&\stackrel{D=2+\epsilon}{=}& \frac{1}{4\pi}\left [-\frac{2}{\epsilon} + 2\ln
{L}/{l} + \ln 4\pi -1 - \boldsymbol \gamma + \mathcal{O}\left (\epsilon\right )\right ].
\notag
\end{eqnarray}
\subsection{Renormalization of the interaction amplitudes}
The renormalization of the interaction amplitudes, or equivalently, of Finkelstein
parameters $z_s$, is intimately linked to the renormalization of the specific
heat. \cite{CastellaniDiCastro86} This is because the scale (e.g., temperature)
dependence of
the total thermodynamic potential $\Omega$ is governed by the scale dependence of $z_s$. In the
present case of coupled surfaces we can only extract the correction to the
sum $z_1 + z_2$ from the (one-loop) correction to the {total} {thermodynamic potential}: \cite{MishandlingII}
\begin{equation}
z'_1 +z'_2 = \frac{1}{2\pi \text{tr} \eta \Lambda} \frac{\partial}{\partial T}
\frac{\Omega}{T} .
\label{eq:z12def}
\end{equation}
At the classical level Eq. \eqref{eq:z12def} yields the relation $z'_1 +z'_2 = z_1+z_2$.
Evaluating the quantum corrections in Eq. \eqref{eq:z12def}, we find
\begin{equation}
\left (z'_1 +z'_2\right ) = \left (z_1 +z_2\right ) + 2 \sum_{s=1,2} \Gamma_{ss}
\int_{\v p} D_s\left (0, \v p\right ) .
\end{equation}
As the correction is a sum of contributions from the two
opposite surfaces, it is natural to assume that the parameters $z_s$ are
renormalized $separately$ (and without intersurface interaction effects):
\begin{eqnarray}
z_s' &=&z_s + 2 \Gamma_{ss}
\int_{\v p} D_s\left (0, \v p\right ) \notag \\
&=&z_s + 2 \frac{\Gamma_{ss}}{\sigma_s} \text{I}^{(2+\epsilon)}_1
\notag \\
&=& z_s + \frac{1}{2\pi} \frac{\Gamma_{ss}}{\sigma_s} \left [-\frac{2}{\epsilon}
+ 2 \ln L/l + \text{const}\right ]. \label{eq:Renormz}
\end{eqnarray}
We have directly proven this assumption of separate $z_s$ renormalization by
the background field method. \footnote{Instead of considering the
renormalization of $z_s$ one can equivalently consider the renormalization of
$\Gamma_{ss}$. It is governed by the interaction term $S^{int}$ in Eq.
\eqref{eq:IANLSM}. Within the background field method two types of contributions
can arise. First, there is $\left \langle S^{int}\right
\rangle_{\text{fast}}$. This term does not involve a frequency integration.
Because disorder is uncorrelated between the surfaces, $\Gamma_{11}$ and
$\Gamma_{22}$ are renormalized separately. This is described by Eq.
\eqref{eq:Renormz}. All possible further contributions at this order would arise from $\left \langle \left
(S^{int}\right ) ^2\right \rangle_{\text{fast}}$. This term generates so-called
ring diagrams.\cite{Finkelstein1990} We have explicitly checked
that the ring diagrams vanish in one-loop approximation.}
\subsection{The one-loop RG equations}
Applying the minimal subtraction scheme to Eqs.~\eqref{eq:usualHLN}, \eqref{eq:AAcorr} and \eqref{eq:Renormz},
{we derive the one-loop perturbative RG equations}:
\begin{subequations}
\begin{align}
\T \B \frac{d\sigma_1}{dy} & = - \frac{2}{\pi} F\left
(\gamma_{11},\frac{\sigma_1}{\sigma_2}\right ),\\
\T \B \frac{d\sigma_2}{dy} & = - \frac{2}{\pi} F\left
(\gamma_{22},\frac{\sigma_2}{\sigma_1}\right ), \\
\T \B \frac{d\gamma_{11}}{dy} &= - \frac{\gamma_{11} \left (1+\gamma_{11}\right
)}{\pi \sigma_1}, \label{eq:RGgamma11} \\
\T \B \frac{d\gamma_{22}}{dy} &= - \frac{\gamma_{22} \left (1+\gamma_{22}\right
)}{\pi \sigma_2},
\label{eq:RGgamma22}
\end{align}
\label{eq:RGeqs}
\end{subequations}
where {$y = \ln L/l$, $\gamma_{ss} = \Gamma_{ss}/z_s$, $l =
\max_{s=1,2}l_s$ and
\begin{equation}
F\left (\gamma,x\right ) = \frac{1}{2} - \frac{1+\gamma}{x \left
[1+\gamma\left (1+ \frac{1}{x}\right )\right ]} \text{ln}\left [\left (1+x\right
)\left (1 + \gamma\right )\right ].
\end{equation}
We recall that
$\Gamma_{12}$, $z_1 + \Gamma_{11}$ and $z_2 + \Gamma_{22}$ are not
renormalized. {We mention that the mass $L^{-1}$ acquires a quantum correction
\cite{MishandlingII} but it does not affect the one-loop renormalization of the
other parameters $\sigma_s$, $z_s$ and $\Gamma_{ss'}$.}
For an alternative presentation of the RG equations \eqref{eq:RGeqs} we introduce the total
conductivity $\sigma = \sigma_1 + \sigma_2$ and the ratio of the conductivities
of the two surfaces $t = \sigma_1/\sigma_2$. In terms of these parameters the RG
equations take the following form:
\begin{widetext}
\begin{subequations}
\begin{align}
\T \B \frac{d\sigma}{dy} & = - \frac{2}{\pi} \left \lbrace 1 - \frac{1}{t}
\frac{1+\gamma_{11}}{1+\gamma_{11}\left (1+ \frac{1}{t}\right )} \text{ln}\left
[\left (1+{t}\right )\left (1 + \gamma_{11}\right )\right ] - {t}
\frac{1+\gamma_{22}}{1+\gamma_{22}\left (1+ {t}\right )} \text{ln}\left [\left
(1+\frac{1}{t}\right )\left (1 + \gamma_{22}\right )\right ] \right \rbrace,
\\
\T \B \frac{dt}{dy} & = - \frac{2}{\pi} \frac{1+t}{\sigma}\left \lbrace
\frac{1-t}{2} - \frac{1}{t} \frac{1+\gamma_{11}}{1+\gamma_{11}\left (1+
\frac{1}{t}\right )} \text{ln}\left [\left (1+{t}\right )\left (1 +
\gamma_{11}\right )\right ] + {t^2} \frac{1+\gamma_{22}}{1+\gamma_{22}\left (1+
{t}\right )} \text{ln}\left [\left (1+\frac{1}{t}\right )\left (1 + \gamma_{22}\right
)\right ] \right \rbrace ,\\
\T \B \frac{d\gamma_{11}}{dy} &= - \left (1+\frac{1}{t}\right )\frac{\gamma_{11}
\left (1+\gamma_{11}\right )}{\pi \sigma}, \label{eq:RGeqs:gamma11}\\
\T \B \frac{d\gamma_{22}}{dy} &= - \left (1+t\right )\frac{\gamma_{22} \left
(1+\gamma_{22}\right )}{\pi \sigma} . \label{eq:RGeqs:gamma22}
\end{align}
\label{eq:RGeqssigmatot}
\end{subequations}
\end{widetext}
\section{Analysis of the RG equations}
\label{sec:AnalysisofRG}
It is worthwhile to remind the reader that the RG equations \eqref{eq:RGeqs}
describe the quantum corrections to conductivity due to the interplay of two
distinct effects. First, they contain weak-antilocalization corrections (WAL)
$\delta \sigma^{WAL}_s = (1/\pi) \ln L/l$ due to quantum
interference in a disordered system with the strong spin-orbit coupling.
Second, these are interaction-induced contributions of Altshuler-Aronov (AA)
type, including effects of both, long-range and short-range interactions. The
result \eqref{eq:RGeqs} was obtained perturbatively to leading order in
$1/\sigma_s \ll 1$ but it is exact in the singlet interaction
amplitudes. While these equations describe the experimentally most relevant
case of Coulomb interaction, in Appendix \ref{sec:shortrange} we also
present the RG equations for the case of short-range interaction.
Equations \eqref{eq:RGeqs} which determine the flow of the coupling constants
$\sigma_1, \sigma_2, \gamma_{11}$ and $\gamma_{22}$ imply a rich phase diagram
in the four-dimensional parameter space. Before discussing the general
four-dimensional RG flow we highlight the simpler case of two equal
surfaces.
\subsection{Two equal surfaces}
\label{sec:twoequalsurfaces}
Equal surfaces are defined by $\sigma_1 = \sigma_2 = \sigma/2$,
$\gamma_{11} = \gamma_{22} = \gamma$ and, because of Eq. \eqref{eq:Finv},
$\gamma_{12} = -1 - \gamma$. It can be checked that the plane of identical surfaces
is an attractive fixed plane of the four dimensional RG-flow (see Appendix
\ref{sec:stableequalsurfaces}). The RG equations for the
two coupling constants $\sigma$ and $\gamma$ are
\begin{subequations}
\begin{align}
\T \B \frac{d\sigma}{dy} &= - \frac{2}{\pi} \left [ 1 - \frac{2+2\gamma}{1+2\gamma} \text{ln} \left (2 + 2 \gamma\right )\right ] , \label{eq:equallayers1} \\
\T \B \frac{d\gamma}{dy} &= - \frac{2\gamma \left (1+\gamma\right )}{\pi \sigma}. \label{eq:equallayers2}
\end{align}
\label{eq:equallayers}
\end{subequations}
Experimentally, the case of equal surfaces is realized if both surfaces are
characterized by the same mean free path and the same carrier density and,
furthermore, if the dielectric environment of the probe is symmetric
($\epsilon_1 = \epsilon_3$).
\subsubsection{Flow Diagram within the fixed plane}
\begin{figure}
\includegraphics[scale=.4]{equallayerslargewithtext.pdf}
\caption{RG flow for equal surfaces in the parameter space $\sigma$ (total
conductivity) and $\gamma$ (intra-surface interaction strength). Here and in all
following RG diagrams, arrows indicate the flow towards the infrared.}
\label{fig:RGflowequallayers}
\end{figure}
The RG flow within the $\sigma$--$\gamma$ plane is depicted in
Fig.~\ref{fig:RGflowequallayers}. The green vertical fixed line at $\gamma = -1$
corresponds to the case of two decoupled surfaces (recall $\gamma_{12} = -1
- \gamma$), and reproduces the result of Ref. \onlinecite{OGM2010} for a single
surface of 3D TI. In this limit the total correction to the conductivity is
negative and obeys the universal law
\begin{equation}
\delta \sigma_{\gamma = -1} = 2 \times \frac{2}{\pi} \left
(\underbrace{1/2}_{\rm WAL} - \underbrace{1}_{\rm AA}\right ) \ln L/l=
-\frac{2}{\pi}
\ln L/l.
\end{equation}
The line of decoupled surfaces is repulsive, as can be seen from Eq.
\eqref{eq:equallayers2}. Flowing towards the infrared, the conductivity first
decreases before turning up again while the system approaches the second fixed
line at $\gamma = 0$. Note that on this line $\gamma_{12} = -1$: the intrasurface
interaction has died out, but the intersurface interaction is maximal. Here the
conductivity correction is positive indicating the flow into a metallic state:
\begin{equation}
\delta\sigma_{\gamma=0} = 2 \times \frac{2}{\pi} \left
(\underbrace{1/2}_{\rm WAL} -
\underbrace{\left [1 - \ln 2 \right ]}_{\rm interaction}\right )\ln L /l .
\label{eq:equallayersgamma0}
\end{equation}
The flow on this fixed line is towards the perfect-metal point
$$
\left (1/\sigma^*, t^*,
\gamma_{11}^*,\gamma_{22}^* \right )= \left (0,1,0,0\right ),
$$
As discussed below, see Sec. \ref{sec:attractiveplane},
this is the only attractive fixed point even in the case of the general four
dimensional RG flow.
On the $\gamma = 0$ fixed line the intersurface interaction reduces the strength
of the WAL effect but it is not strong enough to reverse the
behavior.
The region $\gamma >0$ corresponds to attractive interaction in the singlet channel and is
shown on the flow diagram for the sake of completeness.
\subsubsection{Typical bare values and crossover scale}
Typically, before renormalization the intersurface interaction $\gamma_{12}$ is
weaker than or equal to the intrasurface interaction $\gamma$. This implies that
its bare value $\gamma_0$ takes values in the range between $\gamma_0 = -1$
(decoupled surfaces, i.e. $\gamma_{12,0} = 0$) and $\gamma_0=-1/2 =
\gamma_{12,0}$. For small $\alpha$ we can approximate $\gamma_0$ by its RPA
value:
\begin{equation}
\gamma_0 = - \frac{1}{2} - \frac{1}{2} \frac{\kappa d}{1 + \kappa d}.
\label{eq:gammaRPA}
\end{equation}
Here $d$ is the system thickness and $\kappa = 2\pi\frac{e^2 }{\epsilon_2} \nu$
the inverse single surface screening length obtained for the general symmetric
situation: $\epsilon_1 = \epsilon_3 \neq \epsilon_2$, see Appendix
\ref{sec:Elstat}.
Note that at $\kappa d = 0$ the conductivity corrections due to WAL and AA
exactly compensate each other:
$$
\delta \sigma_{\gamma=-1/2} = \frac{2}{\pi} \left (2 \times
\underbrace{1/2}_{\rm WAL} - \underbrace{1}_{\rm AA}\right ) \ln L/l = 0,
$$
as can also be seen in Fig. \ref{fig:RGflowequallayers}.
Typically $ \kappa d> 0$ or, as already explained on general grounds,
$-1 < \gamma_0 < - 1/2$. Then the most drastic consequence of intersurface
interaction is the non-monotonic temperature (or length) dependence: the
conductivity first decreases with lowering $T$ but eventually the sign of
$d\sigma/dT$ changes and the system is ultimately driven into the
metallic phase. It is natural to ask for the temperature scale, which is
associated with this sign change. The scale $y_*$ at which
the conductivity reaches its minimum can be extracted from
Eqs.~\eqref{eq:equallayers} and is expressed by the integral
\begin{equation}
y_* = - \frac{\pi \sigma_0}{2} \int_{\gamma_0}^{\gamma_*}
\frac{d\gamma'}{\gamma'}\frac{1+\gamma_0}{(1+\gamma')^2} \left
[\frac{\gamma'}{\gamma_0}\right ]^{1-2\ln 2} e^{2\left [f\left (\gamma'\right )
- f\left (\gamma_0\right )\right ]},
\label{eq:ystar}
\end{equation}
where $f\left (x\right ) = \text{Li}_2\left (-x\right ) - \text{Li}_2\left
(-\left (1+2x\right )\right )$, Li$_2$ is the dilogarithm, and $\gamma_*
= -1/2$.
Numerical integration of \eqref{eq:ystar} yields the crossover length scale or
temperature $y_* = \ln L_*/l = 1/2 \ln T_0/T_*$. Its dependence on the bare
values $\sigma_0$ and $\gamma_0$ is plotted in Fig. \ref{fig:PlotT}. Using Eq.
\eqref{eq:gammaRPA} one can also investigate the dependence of $y_*$ on $\kappa d$
instead of $\gamma_0$ (see inset in Fig.\ref{fig:PlotT}).
\begin{figure}
\includegraphics[scale=.55]{PlotTbetainset2.pdf}
\caption{Temperature scale associated with the minimum of $\sigma$ as a
function of the bare values $\sigma_0$ and $\gamma_0$. Inset: the same quantity as a
function of $\sigma_0$ and $\kappa d$.}
\label{fig:PlotT}
\end{figure}
\subsubsection{Role of topology: Dirac electrons vs. electrons with quadratic dispersion in the presence of
spin-orbit interaction}
The perturbative RG equations \eqref{eq:RGeqs} and \eqref{eq:equallayers} are
valid for $\sigma \gg 1$. Instanton effects are suppressed by $\exp(-2
\pi\sigma)$ in this region and we therefore neglected them.
{As has been discussed in Sec. \ref{sec:Z2term}, in the diffusive NL$\sigma$M of
Dirac electrons, the $\mathbb Z_2$ theta term reflects the topological
protection from Anderson localization. This term is absent in the case of
non-topological symplectic metals (NTSM) such as electrons with quadratic
dispersion subjected to strong spin-orbit coupling.\footnote{{The $\mathbb Z_2$
theta term is also absent for the critical state separating 2D trivial and
topological insulator. Such a state can be realized, in particular, on a
surface of a weak 3D topological insulator. Despite the absence of theta term
it is protected from Anderson localization due to topological reasons
\cite{RingelKrausStern12, OGM2012,
FuKaneVortices} (see Sec. \ref{sec:Z2term}) and hence do not fall into our
definition of non-topological symplectic metals.}} The presence (respectively,
absence) of the topological term results in the opposite signs of the instanton
contribution in the two cases.}
However, as instantons are suppressed, our perturbative result is
equally applicable to the surfaces of a 3D TI and, for example, to a
double-quantum-well structure in a material with strong spin-orbit coupling.
Here we discuss non-perturbative differences between the two problems.
Let us start from the case of decoupled surfaces (green line, i.e. $\gamma =
-1$, in Fig.~\ref{fig:Comparison}). This limiting case has been analyzed
before \cite{OGM2010}. For {NTSM} localizing AA corrections overcome the
WAL effect and the system always flows towards localization (Fig.
\ref{fig:Comparison}, left). In contrast, for TI the topological protection implies
$d\sigma/dy > 0$ for small $\sigma$ and hence an attractive fixed point at
$\sigma \sim 1$ (Fig. \ref{fig:Comparison}, right).
As has been explained, the $\gamma=-1$ line is unstable with respect to the
intersurface interaction and the system eventually flows towards the
antilocalizing red line at $\gamma=0$. Let us now analyze this fixed line. The
fact that conductivity corrections \eqref{eq:equallayersgamma0} are
positive stems back to the (non-interacting) WAL effect. Its contribution $2
\times (1/\pi) \ln L/l$ is independent of $\sigma$ only for $\sigma
\gg 1$. For {NTSM} it decreases with decreasing $\sigma$
and eventually becomes negative at the metal-insulator transition (MIT) point
$\sigma_{\rm MIT} \approx 2 \times 1.42\, e^2/h$. \cite{OhtsukiSlevinKramer2004, MarkosSchweitzer, EversMirlinRMP}
(As explained above, Sec. \ref{sec:Z2term}, in a recent
investigation\cite{FuKaneVortices} the crucial role of $\mathbb Z_2$ vortices
for this MIT was pointed out.)
Qualitatively, the picture of the MIT survives the presence of interactions,
which even enhance the tendency to localization. Therefore, for the double layer
system of {NTSM} we expect the antilocalizing
RG flow on the
$\gamma=0$ line to turn localizing below some $\sigma_{\rm MIT}\sim 1$. This MIT
point is indicated by a dot in the left panel of Fig.~\ref{fig:Comparison}.
In contrast, for the surfaces of a topological insulator the system is
topologically protected from Anderson localization, \cite{OGM2007}
i.e., the beta function $d\sigma/dy$ bends up when $\sigma
\rightarrow 0$. There is a numerical evidence \cite{bardarson, Nomura} that
in a non-interacting case this happens without any intermediate fixed points.
Again, the arguments are qualitatively unchanged by the presence of (pure
intersurface) interaction and this scenario is expected to hold also on the red
$\gamma=0$ line of the thin 3D TI film, see Fig.~\ref{fig:Comparison}, right.
(Strictly speaking, one cannot rule out a possibility that in the presence of
interaction there emerge intermediate fixed points but we assume the simplest
possible flow diagram consistent with large- and small-conductivity behavior.)
The interpolation between the limiting cases of decoupled surfaces and
maximally interacting surfaces produces the two phase diagrams shown in Fig.
\ref{fig:Comparison}.
For a {double layer of NTSM}, there is a separatrix connecting the weak-coupling,
decoupled layers fixed point $\left (\gamma, 1/\sigma\right ) = \left
(-1,0\right )$ with the critical MIT point $\left (\gamma, 1/\sigma\right ) \sim
\left (0,1\right )$ that we introduced above. (Strictly speaking, we cannot
exclude the possibility that this fixed point might lie slightly off the
$\gamma =0$ line.) Below the separatrix the conductivity renormalizes down to
$\sigma = 0$, i.e. the system is in the Anderson-localized phase. In
contrast, above the separatrix the characteristic non-monotonic conductivity
behavior leads to the metallic state. As the horizontal position in the phase
diagram is controlled by the parameter $\kappa d$, we predict a
quantum phase transition between metal and insulator as a function of the
interlayer distance. On the other hand, in the case of the coupled top and bottom surfaces of a thin 3D TI film the flow is always towards the metallic phase. The critical point of decoupled
surfaces at $\gamma=-1$ with $\sigma \sim 1$ is unstable in the direction of $\gamma$.
It is worth recalling that in this paper we neglected the tunneling between the opposite surfaces of the 3D TI. If such a tunneling is included, it introduces a corresponding exponentially small temperature scale below which the two surfaces behave as a single-layer NTSM. This would imply a crossover to localizing behavior at such low temperatures.
\begin{figure}
\begin{minipage}[b]{0.5\linewidth
\includegraphics[scale=0.2]{usualspinfulmetal.pdf}
\end{minipage
\begin{minipage}[b]{0.5\linewidth
\includegraphics[scale=0.2]{topologicalmetal.pdf}
\end{minipage}
\caption{Comparison between expected RG-flow for a double layer system of NTSM (left) and the coupled surfaces of a thin 3D TI film (right).}
\label{fig:Comparison}
\end{figure}
\subsection{General RG flow}
We now turn our attention to the complete analysis of RG equations
\eqref{eq:RGeqs} which, in general, describe the case of different
carrier density, disorder and interaction strength on the top and bottom surfaces of a 3D TI film.
The renormalization of interaction parameters $\gamma_{11}$ and $\gamma_{22}$,
Eqs.~\eqref{eq:RGeqs:gamma11} and \eqref{eq:RGeqs:gamma22}, determines four
fixed planes of the RG flow:
\begin{itemize}
\item \underline{$\gamma_{11} = -1 = \gamma_{22}$.} Repulsive fixed plane of two
decoupled surfaces with only intrasurface Coulomb interaction.
This problem has been studied in Ref.~\onlinecite{OGM2010}.
\item \underline{$\gamma_{11} = 0$, $\gamma_{22} = -1$ or vice versa}. Fixed
plane describing a 3D TI film with strongly different surface population. This case
in analyzed in Sec. \ref{sec:surfacebulk} below.
\item \underline{$\gamma_{11}=0=\gamma_{22}$}. Attractive fixed plane.
Intrasurface interaction has died out and only intersurface interaction
survived. This case is analyzed in Sec. \ref{sec:attractiveplane} below.
\end{itemize}
Concerning the repulsive fixed planes, one should keep in mind that the
renormalization of interaction amplitudes is suppressed by the small factor
$1/\sigma$. Therefore even if the conditions on $\gamma_{11}$ and $\gamma_{22}$
are only approximately fulfilled the behavior in the fixed plane dictates the
RG flow in a large temperature/frequency window. RPA-estimates of the bare values of interaction amplitudes can be found in Appendix \ref{sec:BareGamma}.
We also remind the reader that the RG equations describing the model with
finite-range interaction (and thus the whole crossover between the
problem with Coulomb interaction and the non-interacting system)
is discussed in Appendix \ref{sec:shortrange}.
\subsubsection{Strongly different surface population}
\label{sec:surfacebulk}
We investigate here the fixed plane of Eqs.~\eqref{eq:RGeqs} in which
$\gamma_{11} = 0$ and $\gamma_{22} = -1$. (Clearly, the reversed situation
$\gamma_{11}=-1$ and $\gamma_{22}=0$ is completely analogous.) Both fixed planes
are ``saddle-planes'' of the RG flow, i.e., they are attractive in one of the
$\gamma$-directions and repulsive in the other.
Before analyzing this fixed plane, it is worth explaining why
this limit is of significant interest for gate-controlled transport experiments,
in particular, those on Bi$_2$Se$_3$. As for this material the Fermi energy
is normally located in the bulk conduction band, an electrostatic gate is
conventionally used to tune the chemical potential into the bulk gap and hence
to bring the system into a topologically non-trivial regime. A situation as
depicted in Fig. \ref{fig:setupwithbulk} is then believed
to arise in a certain range of gate voltages: \cite{SteinbergPRB} one of the two
surfaces (here surface 1) is separated by a depletion region from
a relatively thick bulk-surface layer.
Recently, \cite{OGM2012, Glazman, Krueckl} disorder-induced interference
corrections for 3D TI bulk electrons have been investigated
theoretically: \footnote{It is worthwhile to repeat that in the diffusive regime
of typical experiments on thin films the 3D TI bulk electrons are subjected to 2D diffusive motion.} While at small length scales additional symmetries of the
Hamiltonian provide non-trivial localization behavior, at sufficiently large
scales the usual WAL effect sets in. The strong coupling between electron states
in the conducting part of the bulk and at surface 2 does not alter this
universal low-energy property. In
conclusion, at sufficiently large length scales the symplectic class
NL$\sigma$M, Eq.~\eqref{eq:TOTALNLSM}, is the adequate description of such a
system (under the assumption of negligible tunneling between
surface 1 and the conducting part of the bulk).
Since the bulk-surface layer has a much higher carrier density than the carrier density on the spatially separated surface 1 we can expect that $\kappa_2\gg\kappa_1$. Provided $\kappa_1 d\ll 1$ the
electron-electron interaction on the spatially separated surface 1 is effectively
screened out such that $|\gamma_{11}|\approx(\kappa_1/\kappa_2)(1+2\kappa_2 d)\ll 1$ (see Eq. \eqref{eq:coupledsurf}). Conversely, the
effect of screening by electrons on the surface 1 is negligible
for Coulomb interaction of the bulk states: $1+\gamma_{22}\approx \kappa_1/\kappa_2\ll 1$.
\begin{figure}
\includegraphics[scale=.4]{Setupwithgatesandbulk.pdf}
\caption{Typical scenario for gate-controlled transport experiments: A
topologically protected surface separated from a thick bulk-surface layer.}
\label{fig:setupwithbulk}
\end{figure}
Substituting $\gamma_{11}=0$ and $\gamma_{22}=-1$ into Eqs. \eqref{eq:RGeqssigmatot}, we find that the RG equations in this fixed plane are as follows
\begin{subequations}
\begin{align}
\T \B \frac{d\sigma}{dy} & = - \frac{2}{\pi} \left \lbrace 1 - \frac{1}{t}
\ln (1+{t}) \right \rbrace, \label{eq:RGeqsunequal:sigma}\\
\T \B \frac{dt}{dy} & = - \frac{2}{\pi} \frac{1+t}{\sigma}\left \lbrace
\frac{1-t}{2} - \frac{1}{t} \ln (1+{t})\right \rbrace .
\label{eq:RGeqsunequal:t}
\end{align}
\label{eq:RGeqsunequal}
\end{subequations}
They can equivalently be written in terms of conductivities $\sigma_1$ and
$\sigma_2$:
\begin{subequations}
\begin{align}
\T \B \frac{d\sigma_1}{dy} & = - \frac{2}{\pi} \left \lbrace \frac{1}{2} -
\frac{\sigma_2}{\sigma_1} \text{ln}\left [1+\frac{\sigma_1}{\sigma_2}\right ] \right
\rbrace, \label{eq:RGeqsunequalsigma12:sigma1} \\
\T \B \frac{d\sigma_2}{dy} & = - \frac{1}{\pi}.
\label{eq:RGeqsunequalsigma12:sigma2}
\end{align}
\label{eq:RGeqsunequalsigma12}
\end{subequations}
We emphasize that the limit $\gamma_{11} = 0$ and $\gamma_{22} = -1$ is very peculiar. Indeed, due to the relation
\eqref{eq:Finv}, this limit implies that the condition $z_1/z_2=0$ holds. Equations \eqref{eq:RGeqsunequal} and \eqref{eq:RGeqsunequalsigma12}
are written under assumption that the ratio $t = \sigma_1/\sigma_2$ is finite in spite of the fact that $z_1/z_2=0$. In the experiment it corresponds to the case in which $\kappa_1/\kappa_2\ll 1$ but the ratio $D_1/D_2 \gg 1$ where $D_s=\sigma_s/4z_s$ is the diffusion coefficient.
Equations \eqref{eq:RGeqsunequalsigma12} become decoupled for $\sigma_1/\sigma_2 =
0$. Then, as expected, $\delta \sigma_1 =
\frac{1}{\pi}\ln L /l$ (WAL, no interaction on the surface 1) and $\delta \sigma_2 =
-\frac{1}{\pi} \ln L /l$ (WAL and AA due to Coulomb interaction on the surface 2).
However, the line $t=0$ is unstable. As one can see from Eq. \eqref{eq:RGeqsunequal:t}, due to the very same quantum
corrections the initially small parameter $t= \sigma_1 / \sigma_2$
increases under RG. The ultimate limit of the perturbative RG flow is $\sigma
\rightarrow 0$ and $t \rightarrow \infty$, see Fig.~\ref{fig:unnequallimit}.
The scale dependence of $\sigma_1$ is non-monotonous; the position of the
corresponding maximum is determined by zeros of the right-hand-side of
Eq.~\eqref{eq:RGeqsunequalsigma12:sigma1} shown by a green line in the right
panel of Fig.~\ref{fig:unnequallimit}.
As has been already emphasized, the perturbative RG equations are
sufficient only
in the regime of large $\sigma_{s}$. We now discuss the topological effects at
small values of conductivities. In the limit $\gamma_{11} = 0$, $\gamma_{22} = -1$ the
renormalization of $\sigma_2$ is {\it exactly} independent of the surface 1.
Indeed, {in the conductivity corrections,} the two surfaces influence each other only via mutual
RPA screening. In the NL$\sigma$M description the interaction amplitudes in the
full action \eqref{eq:TOTALNLSM} and hence in the propagators \eqref{eq:AApropagator} (diffusons and cooperons) fully account for this effect.
Since the layer 2 includes a
single TI surface, we know that
$\sigma_2$ is topologically protected and flows towards $\sigma_2^*$ of the
order of the quantum of conductance (``interaction-induced
criticality'' \cite{OGM2010}). Before this happens, the flow of $\sigma_1$
becomes reversed from antilocalizing to localizing, see
Eq.~\eqref{eq:RGeqsunequalsigma12:sigma1}.
However, since the surface 1 is
also topologically protected, its states can not be strongly localized and
$\sigma \rightarrow \sigma_1^*>0$.\footnote{A priori $\sigma_1^*$
and $\sigma_2^*$ are different although we cannot exclude a possibility that
they might be equal.} Thus, both surfaces are at the quantum critical points with
conductivities of order $e^2/h$. The conclusion concerning the surface 1 is
particularly remarkable: even though $\gamma_{11}=0$, there is
``intersurface-interaction-induced criticality'' on the surface 1.
\begin{figure}
\begin{minipage}[b]{0.45\linewidth}
\includegraphics[scale=0.42]{unequallimitalpha_1.pdf}
\end{minipage}
\begin{minipage}[b]{0.45\linewidth}
\includegraphics[scale=0.49]{unequallimitsigma1sigma2alpha_1.pdf}
\end{minipage}
\caption{Perturbative RG flow in the fixed plane $\gamma_{11} = 0$, $\gamma_{22}
= -1$. In the experimentally motivated scenario (Fig. \protect \ref{fig:setupwithbulk}),
the flow starts at $ t= \sigma_{1}/\sigma_{2} \ll 1$. The green line in the
right panel is a line of zeros of the right-hand-side of
Eq.~\eqref{eq:RGeqsunequalsigma12:sigma1}; it determines the maximum
in the RG flow of $\sigma_1$. }
\label{fig:unnequallimit}
\end{figure}
\subsubsection{Attractive fixed plane}
\label{sec:attractiveplane}
According to Eqs.~\eqref{eq:RGeqs:gamma11} and \eqref{eq:RGeqs:gamma22}, any
$\gamma_{ss} \notin \left \lbrace0,-1\right \rbrace$ is renormalized to zero.
The $\gamma_{11}=\gamma_{22}=0$ is thus an attractive fixed plane of the
general RG flow. The flow within this plane has the form
determined by the following RG equations
\begin{subequations}
\begin{align}
\T \B \frac{d\sigma}{dy} & = - \frac{2}{\pi} \left \lbrace 1 - \frac{1}{t}
\text{ln}\left [1+{t}\right ] - {t} \text{ln}\left [1+\frac{1}{t}\right ] \right \rbrace,
\\
\T \B \frac{dt}{dy} & = - \frac{2}{\pi} \frac{1+t}{\sigma}\left \lbrace
\frac{1-t}{2} - \frac{1}{t} \text{ln}\left [1+{t}\right ] + {t^2} \text{ln}\left
[1+\frac{1}{t}\right ] \right \rbrace ,
\label{eq:RGeqs:attractiveplane:t}
\end{align}
\label{eq:RGeqs:attractiveplane}
\end{subequations}
or, equivalently,
\begin{subequations}
\begin{align}
\T \B \frac{d\sigma_1}{dy} & = - \frac{2}{\pi} \left \lbrace \frac{1}{2} -
\frac{\sigma_2 }{\sigma_1} \text{ln}\left [1+\frac{\sigma_1}{\sigma_2}\right ] \right
\rbrace ,\\
\T \B \frac{d\sigma_2}{dy} & = - \frac{2}{\pi} \left \lbrace \frac{1}{2} -
\frac{\sigma_1}{\sigma_2 } \text{ln}\left [1+\frac{\sigma_2}{\sigma_1}\right ]\right
\rbrace .
\end{align}
\label{eq:RGeqs:attractiveplanesigma12}
\end{subequations}
\begin{figure}
\begin{minipage}[b]{0.5\linewidth
\includegraphics[scale=0.44]{differentlayers_WAL.pdf}
\vspace*{.2cm}
\end{minipage
\begin{minipage}[b]{0.5\linewidth
\includegraphics[scale=0.46]{differentlayers_WAL_sigma12plane.pdf}
\end{minipage}
\caption{The RG flow in the attractive fixed plane $\gamma_{11} = 0 =
\gamma_{22}$. The zero of Eq. \eqref{eq:RGeqs:attractiveplane:t} is displayed by
the red line.}
\label{fig:attractiveplane}
\end{figure}
Even though the single-surface conductivities $\sigma_s$ display
non-monotonic behavior within this plane, eventually all quantum corrections are
antilocalizing, see Fig.~\ref{fig:attractiveplane}. The ratio of conductivities
flows to the symmetric situation $t = \frac{\sigma_1}{\sigma_2} =1$, as has been
discussed in Sec.~\ref{sec:twoequalsurfaces}. We reiterate that at the
corresponding fixed line the WAL
effect is competing with a contribution of the opposite sign due to
intersurface interaction. While the WAL wins, the antilocalizing flow is slower
than for free electrons, see Eq. \eqref{eq:equallayersgamma0}.
\subsubsection{General RG flow}
After having analyzed the RG flow in various fixed planes,
we briefly discuss the general {RG} flow. According to
Eqs.~\eqref{eq:RGgamma11} and \eqref{eq:RGgamma22}, there is a single
attractive fixed point of the overall RG flow -- the metallic fixed point with
zero intrasurface interaction, $\sigma_1 = \sigma_2 \to \infty$ and $\gamma_{11}
= \gamma_{22} = 0$. On the other hand, for the values of $\gamma_{ss}$ close to $-1$ the corresponding conductivity $\sigma_s$ is first subjected to localizing quantum
corrections and will thus show a non-monotonic behavior towards
antilocalization. There also exists a range of initial parameters for the RG flow for which
the conductivity at one surface demonstrates monotonous antilocalizing behavior, while the conductivity in the other surface flows in the described non-monotonous manner.
\section{Discussion and experimental predictions}
\label{sec:experiment}
In the preceding Section we have performed a general analysis of the
renormalization {group} flow determined by the RG equations \eqref{eq:RGeqs}. The
purpose of the present Section is to apply these results to specific
experimentally relevant materials.
\subsection{Parameters}
As explained in Sec.~\ref{sec:setup}, the RG equations \eqref{eq:RGeqs} apply
in the case of the following hierarchy of length scales:
\begin{subequations}
\begin{align}
l &\ll L_E, \label{eq:diffusiveregimecond}\\
d &\ll l. \label{eq:intersurfaceimportant}
\end{align}
In order to deal with $q$-independent interaction amplitudes, an additional
requirement occurs in the case $\kappa_s d \ll 1$ for both $s =1$ and $s=2$:
\begin{equation}
l_{\rm scr} \ll L_E. \label{eq:AAexists}
\end{equation}
In view of condition \eqref{eq:diffusiveregimecond}, the constraint
\eqref{eq:AAexists} is fulfilled in the entire diffusive regime if $l_{\rm scr}
\ll l$.
Further, we have assumed that the intersurface tunneling is negligible; the
corresponding condition reads
\begin{align}
a &\ll d. \label{eq:notunneling}
\end{align}
\end{subequations}
In this Section, we will concentrate on the case when the RG scale is set by
temperature, $L_E = l_T$.
We recall the definition of the length scales entering the above conditions: $l
= \max_{s=1,2} l_s$ is the larger mean free path,
$l_T = \min_{s=1,2} \sqrt{D_s/kT}$ the smaller thermal length, $d$ the sample
thickness, $a$ the penetration depth, $\kappa_s$ the inverse Thomas-Fermi screening
length for the surface $s$ and $l_{\rm scr}$ the total screening length for the
3D TI film.
The situation in which only one of the two surfaces is in the diffusive regime,
while the other one is in the ballistic regime (i.e. $T\tau_1 \ll 1$ and
$T\tau_2 \gg 2$ or vice versa) is also a conceivable and interesting
scenario. However, we do not address it in the present paper.
The effect of intersurface interaction becomes prominent if the
sample thickness does not exceed too much at least one of the single surface
screening lengths $\kappa_s^{-1}$. As discussed above (Sec.
\ref{sec:AnalysisofRG}), this condition implies that the bare values of
interaction $\gamma_{11}$ and $\gamma_{22}$ are not too close to $-1$.
It is useful to present expressions for the length scales appearing in the
conditions \eqref{eq:diffusiveregimecond}-\eqref{eq:notunneling} in terms of
standard parameters characterizing samples in an experiment.
For simplicity, we assume $v_F^{(1)}=v_F^{(2)}$ and $\tau_1 = \tau_2 $ in these
formulas.
The densities of states (DOS) and inverse screening lengths for the top and
bottom surfaces are
\begin{equation}
\nu_s = \sqrt{\frac{n_s}{\pi v_F^2}},\qquad \kappa_s \equiv \frac{2\pi e^2}{\epsilon_2} \nu_s = 2\pi
\alpha\sqrt{\frac{n_s}{\pi}} \label{eq:kappas1}
\end{equation}
where $n_s$ are the corresponding electron densities.
If the electron densities for each surface separately are not known, the total
density $n_{\text{tot}} = n_1 + n_2$ can be used to estimate the {DOS} and the
screening lengths:
\begin{equation}
\nu_1^2 + \nu_2^2 = \frac{n_{\text{tot}}}{\pi v_F^2},\qquad \kappa_1^2 +
\kappa_2^2 = \left (2\pi\alpha\right )^2 \frac{n_{\text{tot}}}{\pi}.
\label{eq:kappas2}
\end{equation}
The mean free path can be expressed as
\begin{equation}
l = v_F \tau_{tr} = \frac{\sigma}{\pi v_F \left (\nu_1 + \nu_2\right )}.
\label{eq:MFpath}
\end{equation}
The thermal length in the diffusive regime is given by
\begin{equation}
l_T = \sqrt{\frac{D}{kT}} = \sqrt{\frac{v_F l}{2 k T}}
= \sqrt{\frac{\sigma}{kT 2\pi \left ( \nu_1 + \nu_2\right ) }}
\end{equation}
Hence, the condition \eqref{eq:diffusiveregimecond} is fulfilled for
temperatures
\begin{equation}
kT \ll kT_{\text{Diff}},
\label{eq:Tmax}
\end{equation}
where
\begin{equation}
kT_{\text{Diff}}
= \frac{v_F}{2l}
= \frac{1}{\sigma [e^2/h ]} \left (
\frac{v_F^2}{2}\right )2\pi (\nu_1 + \nu_2)
\end{equation}
is the temperature scale at which the diffusion sets in.
In order to obtain $l_{\rm scr}$ entering Eq.~(\ref{eq:AAexists}), we have to
consider the full (inter- and
intrasurface) Coulomb interaction, see Appendix \ref{sec:Elstat}.
As explained in Sec. ~\ref{sec:IAimportant} it is only a meaningful quantity
provided $\kappa_s d \ll 1$. Taking into account the influence of the
surrounding dielectrics, we find
\begin{equation}
l_{\rm scr} = \frac{\epsilon_1 + \epsilon_3}{2\epsilon_2} \frac{1}{\kappa_1 + \kappa_2}.
\label{eq:screeninglengthexp}
\end{equation}
When deriving Eq.~\eqref{eq:screeninglengthexp}, we assumed for simplicity that
$\epsilon_2 \lesssim \epsilon_1 + \epsilon_3$. Regarding the experimental setups discussed in Sec. \ref{sec:2materials}, this condition is well fulfilled for
Bi$_2$Se$_3$ but only marginally for HgTe. Thus in the latter case
Eq.~\eqref{eq:screeninglengthexp} should be considered as a rough estimate.
Finally, to check the validity of the condition \eqref{eq:notunneling}, one
needs to know the value of the penetration depth $a$. The latter can be
estimated from the condition
\begin{equation}
\frac{v_{F,\perp} p_\perp}{\Delta_{\text{bulk}}} \sim 1 ,
\end{equation}
where $p_\perp \sim 1/a$ denotes typical momenta perpendicular to the surface.
Provided $v_{F,\perp} \sim v_{F}$, it yields
\begin{equation}
a \sim
\frac{v_{F}}{\Delta_{\text{bulk}}}.
\label{eq:a_estimate}
\end{equation}
We are now going to consider two exemplary materials for 3D TIs: Bi$_2$Se$_3$
and strained HgTe. We shall estimate numerically all the
relevant parameters and present characteristic plots for temperature
dependence of conductivities.
\subsection{Exemplary 3D TI materials}
\label{sec:2materials}
\subsubsection{Bi$_2$Se$_3$}
\begin{table}
\begin{tabular}{|l|c|}
\hline Fermi velocity & $v_F \sim 5 \times 10^{5} $m/s \\
\hline Bulk gap & $\Delta_{\text{bulk}} \sim 0.3$ eV \\
\hline Sample thickness & $d \sim$ 10 nm \\
\hline Dielectric properties & \begin{tabular}{r l} Coat: & $\epsilon_1 \sim1$ \\
3D TI (Bi$_2$Se$_3$): & $\epsilon_2 \sim 100$ \\
Substrate (SrTiO$_3$): & $\epsilon_3 \sim10^3 - 10^4$ \end{tabular} \\
\hline Carrier density & $n_{\text{tot}} \sim 3 \times 10^{12}$ cm$^{-2}$ \\
\hline Mobility & $\mu_{el}\sim100 \dots 1000$ cm$^{2}/$V$\cdot$s \\
\hline Sheet resistance & $1/\sigma \sim 0.097 \, h/e^2$ at $T\sim 50$ mK \\
\hline \hline Effective coupling &$\alpha \sim 4 \times 10^{-2}$\\
\hline Chemical potential & $\mu_1^2 + \mu_2^2 = \left (0.2\, \text{eV}\right )^2$ \\
\hline Penetration depth &$a \sim 1$ nm \\
\hline Mean free path &$l \sim 24 \dots 34$ nm\\
\hline Diff. temperature & $T_\text{Diff} \sim 80 \dots 57$ K\\
\hline Screening length & $\kappa_1^2 + \kappa_2^2 \sim \left (37 \text{ nm} \right )^{-2}$\\
\hline Scr. length (total) & $l_{\rm scr} \sim 132 \dots 186 \text{ nm}, \; \text{for} \; \epsilon_3 = 10^3$\\
\hline \hline Bare interaction (RPA) & \begin{tabular}{r l} top surface: & $\gamma_{11} \sim -0.6 \dots -1 $ \\
bottom surface: & $ \gamma_{22} \sim - 0.6 \dots 0$ \end{tabular}\\
\hline
\end{tabular}
\caption{Experimental values of sample parameters at the point
of the minimal carrier density and associated length scales for transport
experiments on Bi$_2$Se$_3$ films of Refs. [\onlinecite{Chen,YQLi}]. The dots ``$\dots$'' separate values for the
symmetric ($n_1 = n_2$) and totally asymmetric ($n_1 = n_{\text{tot}}, n_2 = 0$)
cases. The bare interaction amplitudes are estimated in the random phase
approximation (RPA).}
\label{tab:expvaluesBise}
\end{table}
Bi$_2$Se$_3$ is currently the most conventional material for experimental
realization of the 3D TI phase. Typical experimental data (extracted from the
point of the minimal conductance in Refs. \onlinecite{Chen,YQLi}) is summarized
in
the upper part of Table \ref{tab:expvaluesBise}. Using Eqs.~\eqref{eq:kappas1}
-- \eqref{eq:a_estimate} we can estimate the hierarchy of length scales (lower
part of the same Table). One can see that all of the requirements of validity
for our theory are fulfilled for length scales above $l_{\rm
scr}$ [temperatures below $T_{\max} = 2.6 \dots 1.9 $K, see condition
\eqref{eq:AAexists}.] \footnote{The only assumption that can not be
directly verified on the basis of the quoted experimental data is the absence of
complete intersurface correlations of disorder. We remind the reader that in
the case of $\mu_1 = \mu_2$ a completely correlated disorder implies extra soft
modes \cite{BurmistrovGornyiTikhonov}. We see however no reason for such perfect
correlations between impurities at opposite surfaces of a {3D TI film}. }
From the experimental data, the ratio of carrier densities is not known.
Therefore, we show in Fig. \ref{fig:BiSe} the expected temperature dependence of
total conductivity for various values of this ratio. Clearly, the behavior
strongly differs from the case of decoupled surfaces (dashed line). First,
the slope of $d\sigma/d\ln T$ is considerably smaller. Second, one observes a
clear curvature of the dependence $\sigma(\ln T)$ which is a manifestation of
the non-monotonicity. (For the parameters used in the plot the minimum of
$\sigma$ occurs at still lower temperatures.) This curvature is especially
pronounced for strongly different surfaces.
\begin{figure}
\includegraphics[scale=0.7]{BiSeSigma_of_T.pdf}
\caption{Theoretical prediction for the temperature dependence of the total
conductivity in thin Bi$_2$Se$_3$ films.}
\label{fig:BiSe}
\end{figure}
It should be mentioned that the substrate used in Ref. \onlinecite{Chen} has a
strongly temperature-dependent dielectric function $\epsilon_3$ since SrTiO$_3$
approaches a ferroelectric transition at low temperatures. This could
result in a temperature dependence of effective gate voltage and consequently of
carrier density. The resulting classical temperature dependence of
conductivity (and interaction constants) would mask the quantum effects
described in our analysis. However, in the presence of the gating field, the
temperature dependence of $\epsilon_3$ saturates at low temperatures. This
motivates the presentation in Fig. \ref{fig:BiSe} where we assumed independent of temperature
$\epsilon_3 = 1000$.
\subsubsection{Strained HgTe}
\begin{table}
\begin{tabular}{|l|c|}
\hline Fermi velocity & $v_F \sim 5 \times 10^{5} $m/s \\
\hline Bulk gap & $\Delta_{\text{bulk}} \sim 0.022$ eV \\
\hline Sample thickness & $d \sim$ 70 nm \\
\hline Dielectric properties & \begin{tabular}{r l} Coat:& $\epsilon_1 \sim1$ \\
3D TI (HgTe): & $\epsilon_2 \sim 21$ \\
Substrate (CdTe): & $\epsilon_3 \sim 10 $ \end{tabular}\\
\hline Carrier density & \begin{tabular}{r l} top surface: & $n\sim 4.8 \times
10^{11}$ cm$^{-2}$ \\
bottom surface: & $n\sim 3.7 \times 10^{11}$ cm$^{-2}$ \end{tabular}\\
\hline Mobility & $\mu_{el}\sim 34 000$ cm$^2$/V$\cdot$s \\
\hline Sheet resistance & $1/\sigma \sim 0.04\, h/e^2$ at $T=50$ mK \\
\hline \hline Effective coupling & $\alpha \sim 0.21$\\
\hline Chemical potential & \begin{tabular}{r l} top surface: & $\mu_1 \sim 0.08$ eV \\
bottom surface: & $\mu_2\sim 0.07$ eV \end{tabular}\\
\hline Penetration depth & $a \sim 15$ nm\\
\hline Mean free path & $l \sim 108$ nm \\
\hline Diff. temperature & $T_\text{Diff} \sim 18$ K\\
\hline Screening length & \begin{tabular}{r l} top surface: & $\kappa_1^{-1} \sim 19.53 $ nm \\
bottom surface: & $ \kappa_2^{-1} \sim 22.24$ nm \end{tabular}\\
\hline \hline Bare interaction (RPA) & \begin{tabular}{r l} top surface: &
$\gamma_{11} \sim -0.893 $ \\
bottom surface: & $ \gamma_{22} \sim - 0.878$ \end{tabular}\\
\hline
\end{tabular}
\caption{Typical experimental values for transport experiments on HgTe films of Refs. [\onlinecite{BaarsSorger1972,
BruehneHgTe}].}
\label{tab:expvaluesHgTe}
\end{table}
Another very promising 3D TI material is strained HgTe. The presence
of Dirac-like surface states was experimentally confirmed by the odd
series of QHE plateaus, as well as by ARPES \cite{BruehneHgTe}. While the
transport experiment indicates dominant surface conduction, the extracted
carrier density appears to be too large for a pure surface theory with linear
spectrum, yielding the value of the chemical potential $\mu$ larger than the gap
{$\Delta_{\rm bulk}$}, see Table \ref{tab:expvaluesHgTe}. (The role of the bulk
conduction band as well as the parabolic bending of the dispersion was also
discussed within an independent magneto-optical study by the same
experimental group. \cite{Hancock}) Thus, it remains to be clarified under what experimental
conditions the strained HgTe sample is in the true TI regime (i.e., the
bulk contribution to transport is negligible). Notwithstanding this point and
motivated by the excellent surface transport data, we apply our theory to the
HgTe experiment, see Fig. \ref{fig:HgTe}. In spite of the considerable thickness
of the probe, the effect of intersurface interaction is clearly visible: the
slope of $d\sigma/d\ln T$ is considerably smaller than it is expected for decoupled
surfaces.
\begin{figure}
\includegraphics[scale=0.8]{HgTeSigma_of_T.pdf}
\caption{Theoretical prediction for the temperature dependence of the total
conductivity in thin films of strained HgTe.}
\label{fig:HgTe}
\end{figure}
\subsection{Hallmarks of surface transport and interactions}
We briefly summarize now our most salient predictions for experimental
signatures of surface transport in 3D TI with an intersurface interactions.
\begin{itemize}
\item As already exploited in 3D TI experiments, \cite{Chen} the magnetoconductance formula \cite{HLN} for the total conductivity is
\begin{equation}
\delta \sigma \left (B\right ) = - \frac{e^2}{2\pi h} \sum_{s=1,2}\left [ \psi \left
(\frac{1}{2}+\frac{B^{(s)}_{\phi}}{B}\right ) - \ln \left (\frac{B^{(s)}_{\phi}}{B}\right
)\right ] ,
\end{equation}
where the characteristic magnetic field $B^{(s)}_{\phi}=\hbar/(4eD_s^{(s)}\tau^{(s)}_\phi)$ is determined by the diffusion coefficient $D_s^{(s)}$ and the phase breaking time $\tau^{(s)}_\phi$ for the surface $s$. The function $\psi$ denotes the digamma function here.
\item The characteristic effect of intersurface
interaction is the non-monotonous temperature dependence of conductivity (see
Fig. \ref{fig:exemplarynonmonotonicity}, top).
It may happen that in experimentally accessible temperature window
this effect manifests itself only
as a deviation of the conductance slope
\begin{equation}
\delta \sigma\left (T\right ) = {\; \frac{e^2}{h}} c \ln T
\end{equation} from the value $c=1/\pi$ characteristic
for two decoupled surfaces accompanied by some bending of the curve $\sigma(\ln
T)$, see Figs.~\ref{fig:BiSe} and \ref{fig:HgTe}.
The ultimate low-$T$ behavior of the coupled system is always antilocalizing
and following the universal law
\begin{equation}
\delta \sigma \left (T\right ) ={\; \frac{e^2}{\pi h}}
\left (1-2 \ln 2 \right ) \ln T .
\end{equation} However, depending on the
parameters, this asymptotics may become valid at very low temperatures
only.
\item The strength of intersurface interaction is governed by the parameters
$\kappa_1 d$ and $\kappa_2 d$, where $\kappa$ is the screening length.
Therefore, in contrast to usual, single surface conductivity corrections, the
predicted effect strongly depends on the carrier density (see Fig.
\ref{fig:exemplarynonmonotonicity}, bottom).
\end{itemize}
It is also possible to access the intersurface induced quantum corrections in
the frequency dependence of the AC conductivity (by the simple replacement $T
\rightarrow \omega$ in $\delta \sigma \left (T\right )$ if $\omega \gg T$).
\begin{figure}
\begin{minipage}{1\linewidth}
\includegraphics[scale=0.8]{variousnonmonotonicities.pdf}
\end{minipage}
\begin{minipage}{1\linewidth}
\includegraphics[scale=.85]{deltasigmaofn.pdf}
\end{minipage}
\caption{{\it Top:} Conductivity corrections for low carrier
concentration. The total electron concentration in units of $10^{12} \text{cm}^{-2}$ is equal to $0.55, 0.48, 0.41, 0.34, 0.27$
from bottom to top.
The characteristic non-monotonous behavior is clearly seen; deviations
from the behavior of decoupled surfaces are very strong. {\it Bottom:}
Carrier-density dependence of conductivity corrections. The non-trivial
dependence is entirely due to the intersurface interaction: in the case of the
decoupled surfaces, the conductivity correction would be constant as a function
of density, $\sigma\left (0.02 K\right ) - \sigma\left (50 K\right ) \approx -
2.49 e^2/h$.
We used the values of the parameters $d$, $v_F$ and $\alpha$ as in Table
\ref{tab:expvaluesBise}
for Bi$_2$Se$_3$. Further, we assumed the case of equal surfaces ($n_{tot} = 2 n$) and $T_{\text{Diff}} =
1/2\tau = 50 K$. }
\label{fig:exemplarynonmonotonicity}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
In this paper, we have investigated interference and interaction
effects in the surface state conductivity of 3D topological
insulator slabs. We have taken into account the electron-electron
interaction within the top and bottom surfaces of a slab and between
them. These two surfaces were in general assumed to be characterized by
different carrier densities and scattering rates, and by
asymmetric dielectric environment.
Our field-theoretical analysis was based on the interacting non-linear sigma
model approach describing the system at length scales above the mean
free path. We demonstrated how this effective theory can be obtained from the non-Abelian
bosonization. In particular, we have shown that upon inclusion of potential
disorder the Wess-Zumino term generates a local expression
for the $\mathbb Z_2$ theta term. The appearance of this topological term is the
hallmark of the Dirac surface states; it is absent in conventional 2D metals of
the same symmetry class. We have further analyzed the $\mathbf{U}(1)$-gauged
sigma model that describes a coupling to the external electromagnetic field.
This has allowed us to connect the physical linear-response characteristics of
the problem and the sigma-model {coupling constants}. We have also analyzed the effect of
breaking of time-reversal symmetry, namely, the anomalous quantum Hall effect
of Dirac electrons.
It is worth emphasizing that our theory treats the general situation of
potentially strong
interactions and thus went beyond perturbation theory. We have thus developed
the Fermi liquid theory of the strongly correlated double layer system in the
ballistic and diffusive regime.
We renormalized the interacting NL$\sigma$M of the two surfaces in the one-loop
approximation and obtained the RG equations, Eq.~\eqref{eq:RGeqs}. This way we
have determined the temperature (or else, frequency, or length scale) dependence
of the conductivities of both surfaces.
The RG is controlled by a large conductivity, $k_F l \gg 1$.
Our calculations are exact in the singlet interaction amplitudes, while
contributions due to a repulsive Cooper interaction are parametrically small and
can be neglected.
Inspecting the RG equations, we showed that intersurface
interaction is relevant in the RG sense and the limiting case of decoupled
surfaces is therefore unstable. The rich flow diagram has been analyzed in
detail. For fully decoupled surfaces the system flows into an
intermediate-coupling fixed point (``interaction-induced criticality''). This
point is, however, unstable with respect to the intersurface coupling.
The flow is then towards a single attractive fixed point
which is ``supermetallic'', $\sigma \to \infty$, and at which even
originally different surfaces have the same transport properties, $\sigma_1 =
\sigma_2$, see Figs. \ref{fig:RGflowequallayers} and
\ref{fig:attractiveplane}. Further, this fixed point is characterized by
vanishing intrasurface and finite intersurface interaction.
Typically, this fixed point is reached via a characteristic
non-monotonous temperature dependence of conductivity.
Our perturbative results are equally applicable to weak topological
insulator\cite{RingelKrausStern12,KobayashiOhtsukiImura13} thin films and to non-topological double
layer
systems with spin-orbit interaction. {For the latter type of structures,} we
have also discussed the difference
compared to the strong TI films which is in non-perturbative
topological effects, see a comparison of the flow diagrams in
Fig.~\ref{fig:Comparison}.
While in the TI case these effects lead to a topological
protection of the surface states from Anderson localization, a conventional
(non-topological) double layer system undergoes a metal-insulator
transition which is tuned by the ratio of interlayer distance and screening
length.
Finally, we have estimated parameters and presented explicit predictions
for the temperature dependence of the conductivity for typical
experimental setups based on Bi$_2$Se$_3$ and strained HgTe materials.
Before closing, we discuss perspectives for further research. First,
experimental studies of temperature dependence of conductivity of 3D
topological insulators for different positions of chemical potentials would be
highly useful. A comparison of such experimental data with our theoretical
predictions would allow one to judge whether the system is in the truly
topological phase. Second, more work is needed on effects of local breaking of time-reversal
symmetry in TI slabs.
{Third, it is known that Coulomb interaction in electronically decoupled
double-layer systems induces a finite but typically small
transconductance $\sigma_{12}$.\cite{SolomonDrag, GramilaDrag, CoulombDragCarrega, MichaelDrag}
However, the side walls of 3D TI films connect the two major surfaces, which
might be a serious obstacle for performing Coulomb drag experiments.
Fourth, {in view of recent experimental progress,\cite{WrayTopSC}} it would be
interesting to perform an RG analysis for a superconducting
counterpart of the system that we have explored, namely, for surface states of a
3D topological superconductor with spin-orbit interaction (class DIII).\cite{FosterYuzbashyan2012}
\section{Acknowledgements}
We thank J. Smet, Y.Q. Li, A. Finkel'stein, M. Sch\"utt and U. Briskot for
useful discussions. The work was supported by BMBF, DAAD, DFG SPP 1666, RAS
programs ``Quantum mesoscopic and disordered systems'', ``Quantum Physics of Condensed Matter'', ``Fundamentals of nanotechnology and nano materials'', RFBR grants No.
11-02-12126 and No. 12-02-00579, Russian President grant No. MK-4337.2013.02,
and by the Russian Ministry of Education and Science under contract No. 8678.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,689 |
/*
* Minio Cloud Storage, (C) 2014-2016 Minio, Inc.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package cmd
import (
"net/rpc"
router "github.com/gorilla/mux"
)
// Set up an RPC endpoint that receives browser related calls. The
// original motivation is for propagating credentials change
// throughout Minio cluster, initiated from a Minio browser session.
const (
browserPeerPath = "/browser/setauth"
)
// The Type exporting methods exposed for RPC calls.
type browserPeerAPIHandlers struct {
AuthRPCServer
}
// Register RPC router
func registerBrowserPeerRPCRouter(mux *router.Router) error {
bpHandlers := &browserPeerAPIHandlers{}
bpRPCServer := rpc.NewServer()
err := bpRPCServer.RegisterName("BrowserPeer", bpHandlers)
if err != nil {
return traceError(err)
}
bpRouter := mux.NewRoute().PathPrefix(minioReservedBucketPath).Subrouter()
bpRouter.Path(browserPeerPath).Handler(bpRPCServer)
return nil
}
| {
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The 1996 Rexona Cup – Singles was a tennis event played on outdoor clay courts at the Am Rothenbaum in Hamburg in Germany that was part of Tier II of the 1996 WTA Tour. The 1996 Rexona Cup tournament was held from April 29 through May 5, 1996.
Gigi Fernández and Martina Hingis were the defending champions but lost in the final 4–6, 7–6, 6–4 against Arantxa Sánchez Vicario and Brenda Schultz-McCarthy.
Seeds
Champion seeds are indicated in bold text while text in italics indicates the round in which those seeds were eliminated. The top two seeded teams received byes into the quarterfinals.
Arantxa Sánchez Vicario / Brenda Schultz-McCarthy (champions)
Gigi Fernández / Martina Hingis (final)
Conchita Martínez / Patricia Tarabini (semifinals)
Kristie Boogert / Miriam Oremans (semifinals)
Draw
External links
ITF tournament edition details
1996 WTA Tour
WTA German Open | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,013 |
Although the main importance of this dialogue is to tell Death Watch that they are neither with the Jedi, nor Count Dooku, I actually found the final line to be very fascinating. At the beginning of the last episode in which we saw Darth Maul, when his brother got excited about a bunch of credits: "Look Brother – a fortune!" and Maul responded sharply: "True fortune will be the demise of Kenobi. Credit chips are meaningless without a plan to survive. The Jedi are hunting us. We will be forced to make a stand." Now, it seems Maul either has changed from solely seeking revenge on Obi-Wan Kenobi and seeking fortune and power or he merely stated that to tell Pre Viszla. I'm guessing he was actually speaking earnestly and has now turned his focus to a much broader – and bigger – scope. Wow!
Following this dialogue, Pre Viszla turns away, satisfied, and commands the medical droid: "Repair this one's legs and do what you can for the other", ensuring that they will be properly healed.
In so doing, Savage Oppress gets a new arm, while Darth Maul's legs, which had been some weird-looking mechanical contraption, are now normal-looking (at the very least, they can fit into pants). However, they are also shorter than what he had before, so he's shorter. Not only that, but he's shorter than his brother now. This makes him less intimidating (think about the beginning of "Revival" (the previous episode in which we see Maul), where Maul uses his mechanical "foot" to step on the face of Oppress). | {
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/applied-mathematics\/elementary-technical-mathematics\/chapter-12-section-12-5-circles-exercise-page-410\/20","text":"## Elementary Technical Mathematics\n\n$A=239~in^{2}$\nThe area of the side of the tire is the area of the outer circle minus the area of the inner circle. We use the area of a circle: $A=\\frac{\\pi*d^{2}}{4}$ With the two different diameters and subtract the difference. Thus, we have: $A=\\frac{\\pi*dout^{2}}{4}-\\frac{\\pi*din^{2}}{4}=\\frac{\\pi*(23in)^{2}}{4}-\\frac{\\pi*(15in)^{2}}{4}=238.76in^{2}\\approx239~in^2$","date":"2021-05-08 11:11:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7575613260269165, \"perplexity\": 148.8414365797571}, \"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\/1620243988858.72\/warc\/CC-MAIN-20210508091446-20210508121446-00448.warc.gz\"}"} | null | null |
CREATE TABLE `py_users` (
`nick` TEXT NOT NULL UNIQUE,
`ident` TEXT NOT NULL,
`host` TEXT NOT NULL,
`passphrase` TEXT NOT NULL,
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{"url":"https:\/\/math.meta.stackexchange.com\/questions\/4012\/problem-with-cases-enviroment","text":"# Problem with cases enviroment\n\nI am not able to pinpoint where the problem is - it seems that [] are the culprit there. Anyway only one of the following almost identical codes is rendered. (At the moment.)\n\n$$\\Gamma(x)=\\begin{cases} [0,1]\\setminus\\mathbb Q; &x<0 \\\\ [0,1]; & x\\ge 0 \\end{cases}$$\n\n\n$$\\Gamma(x)=\\begin{cases} [0,1]\\setminus\\mathbb Q; &x<0 \\\\ [0,1]; & x\\ge 0 \\end{cases}$$\n\n$$\\Gamma(x)=\\begin{cases} (0,1)\\setminus\\mathbb Q; &x<0 \\\\ (0,1); & x\\ge 0 \\end{cases}$$\n\n\n$$\\Gamma(x)=\\begin{cases} (0,1)\\setminus\\mathbb Q; &x<0 \\\\ (0,1); & x\\ge 0 \\end{cases}$$\n\n$$\\Gamma(x)=\\begin{cases} [0,1]\\setminus\\mathbb Q; &x<0 \\\\ \\left[0,1\\right]; & x\\ge 0 \\end{cases}$$\n\n\n$$\\Gamma(x)=\\begin{cases} [0,1]\\setminus\\mathbb Q; &x<0 \\\\ \\left[0,1\\right]; & x\\ge 0 \\end{cases}$$\n\nI've noticed this in an old post of mine. Although I cannot be absolutely sure, I believe it was rendered ok, when I posted it (quite some time ago). There is some discussion in the comments bellow my answer, which means that I've returned to that post a few times, so I would have probably noticed the problem.\n\nA screenshot - just in case this is browser-dependent or this behavior will change.\n\n\u2022 It also happens on $\\LaTeX$ sometimes; I run into this sort of problem all the time when I have equations involving commutators. It thinks you are giving it an optional parameter. \u2013\u00a0Arturo Magidin Apr 20 '12 at 14:21\n\nThe problem is that MathJax is incorrectly thinking that your \\\\ is followed by an optional parameter because the next line starts with an open bracket (note that \\\\[dimen] is used to add space between lines). Normally, optional parameters can have space before the [, and new-lines count as space, but for \\\\[dimen] it should not be allowed. This is a bug that will be fixed in the next release of MathJax, but for now you can use\n$$\\Gamma(x)=\\begin{cases} [0,1]\\setminus\\mathbb Q; &x<0 \\\\ {[0,1]}; & x\\ge 0 \\end{cases}$$\n`\n$$\\Gamma(x)=\\begin{cases} [0,1]\\setminus\\mathbb Q; &x<0 \\\\ {[0,1]}; & x\\ge 0 \\end{cases}$$","date":"2021-05-13 09:18:23","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\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.859036922454834, \"perplexity\": 667.0334111851138}, \"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\/1620243990584.33\/warc\/CC-MAIN-20210513080742-20210513110742-00174.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.skepticalcommunity.com\/viewtopic.php?f=3&t=141&start=140","text":"## The POPE against..........\n\nHot topics in delusion and rationalization.\nthaiboxerken\nPosts: 135\nJoined: Sun Jun 13, 2004 9:51 pm\nLocation: Oregon, and it rocks!\nCarlos to Latin wa baka desu, ne?\nCarlos is on my ignore list.\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nPyrrho wrote: Yes, they educate. It is an educational foundation.\nAre they educating people to be skeptic the way his leader is?\nPyrrho wrote: Yes, James Randi, leader of the JREF, is a skeptic.\nIf James Randi is a skeptic , are the belief and faith in GOD compatible with skepticism?\nPyrrho wrote: Yes, this board is related to skepticism.\nOf course it is . It is a \"community\" formed at the JREF forum.\nPyrrho wrote: A complicated question. There are users here who also post at the JREF Forum. There are users here who are probably members of the JREF. This site exists because the JREF Forum will be eliminating parts of itself that it does not want. Other than that, there is no relation between this forum and the JREF. The JREF does not own this forum, nor does it administer this forum. The JREF itself has nothing to do with this forum.\nBut it is stilll a \"community\" formed around James Randi and the JREF.\nYou forgot to tell the people that also donate money to the JREF , buy merchandise related to the JREF and go to the annual reunion TAM.\n\nPyrrho wrote: Some of them are, some of them are not.\nMost of the members until now are related to the JREF forum.\nWhich mmebers do you think are not related to the JREF forum until now?\nLess than five? More than five?\nPyrrho wrote: I do not know. Why don't you post a poll and ask them?\nIf you don't know if the majority of members are supporters of the ideas of James Randi \/JREF , then I am missing something.\nAre you a supporter?\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nPyrrho wrote:\nMr. Randi\/JREF in the official page of the JREF said this\n\nhttp:\/\/www.randi.org\/jr\/050903.html\nCan a person be both a skeptic and a person of faith?\n\nThe answer is, Mr. Randi and I agree, a resounding YES.\n\nIs Faith something paranormal and \/or supernatural?\nYou have not defined \"faith\", but I assume you mean \"faith in God\". Faith is the combination of human emotion and intellect. It is neither paranormal nor supernatural. It is human.\nJREF\/Randi and Hal were the ones who defined faith. Under that concept was my question directed.\n\nPyrrho wrote: It is a personal decision. I would not call it skepticism; however, if a person has faith yet continues to question his faith, he is practicing skepticism.\nFaith in God is faith in GOD, under that circunstances that faith don't need to question the existence of God. Is the principle of all religions.\nPyrrho wrote: A person may have developed \"faith\" as a result of skeptical questioning of his beliefs, thoughts, and outside events. Again, if one continues to question one's faith, one may still be practicing skepticism. This can be considered natural.\nMaybe your response is not related to my question. Or my question was not clear enough.\nSince Faith in God is natural , is God paranormal or supernatural?\nPyrrho wrote: Some people never question their faith. This is called \"true belief\". Others question their faith, whether that faith is in God or in science.\nWe are talking about people with Faith in God. A \"true belief\" according to some \"skeptick\" ideas.\nPyrrho wrote:\nAre not they promoting, given their own example, to be a person of faith and also an skeptic?\nThat's pretty much what Hal Bidlack says in his article.\nThen you agree they(JREF) are promoting to be a person a faith and also an skeptic.\nPyrrho wrote: The consequences are that people such as you will complain about it. Also, that the JREF might not be seen as an organization that promotes atheism. which it isn't.\nAre you telling us that James Randi is not an atheist and that maybe this is a kind of strategy that will reach donations from true believers?\n\nThanks,\nCarlos\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nCarlos wrote:\nPyrrho wrote: Yes, they educate. It is an educational foundation.\nAre they educating people to be skeptic the way his leader is?\nI don't know.\nPyrrho wrote: Yes, James Randi, leader of the JREF, is a skeptic.\nIf James Randi is a skeptic , are the belief and faith in GOD compatible with skepticism?\nThe second part of your question is not conditional upon the first. Whether or not James Randi is a skeptic, belief and faith in God are incompatible with skepticism.\nPyrrho wrote: Yes, this board is related to skepticism.\nOf course it is . It is a \"community\" formed at the JREF forum.\nThis forum is the bastard child of a misbegotten notion to excise content that the JREF does not like from the JREF Forum. This forum has no official ties to the JREF. If by \"community\" you mean a group of people who post on the forum, some are skeptics and some are not skeptics. It is not a \"skeptics only\" forum; the JREF Forum is not a \"skeptics only\" forum either.\nPyrrho wrote: A complicated question. There are users here who also post at the JREF Forum. There are users here who are probably members of the JREF. This site exists because the JREF Forum will be eliminating parts of itself that it does not want. Other than that, there is no relation between this forum and the JREF. The JREF does not own this forum, nor does it administer this forum. The JREF itself has nothing to do with this forum.\nBut it is stilll a \"community\" formed around James Randi and the JREF.\nYou forgot to tell the people that also donate money to the JREF , buy merchandise related to the JREF and go to the annual reunion TAM.\nThe JREF Forum is the \"community formed around James Randi and the JREF\". This \"community\" exists because it is no longer wanted by the JREF. What's this \"you forgot\" business? That wasn't part of your question.\nPyrrho wrote: Some of them are, some of them are not.\nMost of the members until now are related to the JREF forum.\nWhich mmebers do you think are not related to the JREF forum until now?\nLess than five? More than five?\nI don't know. I won't guess. People from the JREF Forum are here because on July 8 they will not be able to post content at the JREF Forum that does not fit the undefined criteria of the \"JREF Mission\". The content we post has been defined as being unrelated to the JREF Mission, so how can this forum be related to the JREF? The JREF does not want to be related to this forum.\nPyrrho wrote: I do not know. Why don't you post a poll and ask them?\nIf you don't know if the majority of members are supporters of the ideas of James Randi \/JREF , then I am missing something.\nAre you a supporter?\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nI believe in asking the right people the right questions. If you want to know what people here think, post a poll. Why make me a spokesman for other people? Why should I know if they support the JREF or not? Ask them yourself if you truly want the answer.\nPyrrho wrote:\nMr. Randi\/JREF in the official page of the JREF said this\n\nhttp:\/\/www.randi.org\/jr\/050903.html\nCan a person be both a skeptic and a person of faith?\n\nThe answer is, Mr. Randi and I agree, a resounding YES.\n\nIs Faith something paranormal and \/or supernatural?\nYou have not defined \"faith\", but I assume you mean \"faith in God\". Faith is the combination of human emotion and intellect. It is neither paranormal nor supernatural. It is human.\nJREF\/Randi and Hal were the ones who defined faith. Under that concept was my question directed.\nI answered your question. Faith in god is not paranormal or supernatural. It is uniquely human behavior.\nPyrrho wrote: It is a personal decision. I would not call it skepticism; however, if a person has faith yet continues to question his faith, he is practicing skepticism.\nFaith in God is faith in GOD, under that circunstances that faith don't need to question the existence of God. Is the principle of all religions.\nFaith is not permanent. People lose faith all the time, every day. It is just as subject to question as any other human behavior.\nPyrrho wrote: A person may have developed \"faith\" as a result of skeptical questioning of his beliefs, thoughts, and outside events. Again, if one continues to question one's faith, one may still be practicing skepticism. This can be considered natural.\nMaybe your response is not related to my question. Or my question was not clear enough.\nSince Faith in God is natural , is God paranormal or supernatural?\nThat was not the question you asked. I did answer the question you did ask. As to this question, \"Is God paranormal or supernatural?\" I can only say that I do not know, because I do not know if God exists. If for the sake of discussion we assume that God does exist, we see that God is paradoxical -- in order to exist, God must be natural...and in order to exist, God must be supernatural. We could spin around the paradox indefinitely and never come to a conclusion.\n\nFor my own part, I think God is a figment of human imagination, and is thus merely natural, and does not objectively exist outside of the human imagination.\nPyrrho wrote: Some people never question their faith. This is called \"true belief\". Others question their faith, whether that faith is in God or in science.\nWe are talking about people with Faith in God. A \"true belief\" according to some \"skeptick\" ideas.\nSome people never question their Faith in God. This is called \"true belief.\" Others question their Faith in God, even if they have not discarded their Faith in God. Questioning your faith is skepticism.\nPyrrho wrote:\nAre not they promoting, given their own example, to be a person of faith and also an skeptic?\nThat's pretty much what Hal Bidlack says in his article.\nThen you agree they(JREF) are promoting to be a person a faith and also an skeptic.\nI am agreeing that that is pretty much what Hal Bidlack says in his article. It is an article containing his opinions; it is no more a promotion of faith in God than Randi's opinions about religion are a promotion of atheism. They are merely opinions. Promoting something requires more activity than writing opinions. For example, selling books about atheism could be considered promotion of atheism; selling books about deism could be considered promotion of deism. The JREF does not do either of those things, or other similar promotion activities.\nPyrrho wrote: The consequences are that people such as you will complain about it. Also, that the JREF might not be seen as an organization that promotes atheism. which it isn't.\nAre you telling us that James Randi is not an atheist and that maybe this is a kind of strategy that will reach donations from true believers?\n\nThanks,\nCarlos\nI do not know if James Randi is an atheist. The JREF is more than James Randi. Its stated purposes do not contain language that promotes atheism. Its activities do not include actions that promote atheism. Its purpose is to promote critical thinking about the paranormal, the normal, and other things, not just about religion or God.\n\nIf James Randi wanted donations from true believers, he'd probably start a church. You're speculating, Carlos. Be careful -- someone might think you were a skeptic.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote: Yes, they educate. It is an educational foundation.\nAre they educating people to be skeptic the way his leader is?\nI don't know.\nYou don't know if the leader' ideas are the examples of any organization?\nYou don't know if a skeptic like James Randi , leader of the JREF don't pretend to form and educate people under an skeptical point of view?\n\nPyrrho wrote: The second part of your question is not conditional upon the first. Whether or not James Randi is a skeptic, belief and faith in God are incompatible with skepticism.\nThen, their own quote is incompatible with skepticism.\nPyrrho wrote: This forum is the bastard child of a misbegotten notion to excise content that the JREF does not like from the JREF Forum. This forum has no official ties to the JREF. If by \"community\" you mean a group of people who post on the forum, some are skeptics and some are not skeptics. It is not a \"skeptics only\" forum; the JREF Forum is not a \"skeptics only\" forum either.\nYes , sometimes I see this forum as a bastard child of the JREF forum.\nSometimes I see it like an aborted child .\nAnyway is a community formed at the JREF forum.\nPyrrho wrote: The JREF Forum is the \"community formed around James Randi and the JREF\". This \"community\" exists because it is no longer wanted by the JREF. What's this \"you forgot\" business? That wasn't part of your question.\n\nSince you wrote this :\"There are users here who also post at the JREF Forum. There are users here who are probably members of the JREF. This site exists because the JREF Forum will be eliminating parts of itself that it does not want.\"\nThen I just added this :You forgot to tell that are also members that donate money to the JREF , buy merchandise related to the JREF and go to the annual reunion JREF TAM.\nPyrrho wrote: I don't know. I won't guess. People from the JREF Forum are here because on July 8 they will not be able to post content at the JREF Forum that does not fit the undefined criteria of the \"JREF Mission\". The content we post has been defined as being unrelated to the JREF Mission, so how can this forum be related to the JREF? The JREF does not want to be related to this forum.\nIt is related in the way the same posters now here were part of the JREF forum.\nPyrrho wrote:\nIf you don't know if the majority of members are supporters of the ideas of James Randi \/JREF , then I am missing something.\nAre you a supporter?\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nI believe in asking the right people the right questions. If you want to know what people here think, post a poll. Why make me a spokesman for other people? Why should I know if they support the JREF or not? Ask them yourself if you truly want the answer.\nYou didn't anwer my questions.\nPyrrho wrote: Faith is not permanent. People lose faith all the time, every day. It is just as subject to question as any other human behavior.\nIs skepticism other human behavior?\nIf it is so , can you lose it all the time?\nPyrrho wrote:\nMaybe your response is not related to my question. Or my question was not clear enough.\nSince Faith in God is natural , is God paranormal or supernatural?\nThat was not the question you asked. I did answer the question you did ask. As to this question, \"Is God paranormal or supernatural?\" I can only say that I do not know, because I do not know if God exists. If for the sake of discussion we assume that God does exist, we see that God is paradoxical -- in order to exist, God must be natural...and in order to exist, God must be supernatural. We could spin around the paradox indefinitely and never come to a conclusion.\nI already told you that maybe my question was not clear enough.\nBut now your answer telling us you don't know what God is (paranormal or supernatural without evidence ), under an skeptic point of view , can be interpretated that you are asumming that God is real and natural.\n\nFor the real skeptics , God is something that has no proof.\nFor the ones who have Faith in God , God is something that vcan not be questioned.\nPyrrho wrote: For my own part, I think God is a figment of human imagination, and is thus merely natural, and does not objectively exist outside of the human imagination.\n\nHere is something that can be interpretated as a double speech from a confessed skeptic like you. Read the above.\nPyrrho wrote: I am agreeing that that is pretty much what Hal Bidlack says in his article. It is an article containing his opinions; it is no more a promotion of faith in God than Randi's opinions about religion are a promotion of atheism. They are merely opinions. Promoting something requires more activity than writing opinions. For example, selling books about atheism could be considered promotion of atheism; selling books about deism could be considered promotion of deism. The JREF does not do either of those things, or other similar promotion activities..\nWhen an article on the official page of the JREF , states what their main leaders think , then it is a promotion.\nPyrrho wrote:\nAre you telling us that James Randi is not an atheist and that maybe this is a kind of strategy that will reach donations from true believers?\nI do not know if James Randi is an atheist. The JREF is more than James Randi. Its stated purposes do not contain language that promotes atheism. Its activities do not include actions that promote atheism. Its purpose is to promote critical thinking about the paranormal, the normal, and other things, not just about religion or God.\n\nIf James Randi wanted donations from true believers, he'd probably start a church. You're speculating, Carlos. Be careful -- someone might think you were a skeptic.\nThe JREF is formed around the main figure of James Randi. He is the leader and the main representant of the JREF.\nHe is a confessed athiest skeptic telling you that a person can be both a skeptic and a person of faith......in God.\n\nJames Randi\/JREF doesn't need a church to collect donations or to sell merchandise or souvenirs. It is called the JREF. The strategies are different , the results are the same.\n\nThanks,\nCarlos\nlatinijral\nPosts: 488\nJoined: Tue Jun 08, 2004 3:27 am\nCarlos wrote:\nMRC_Hans wrote:I will make an exception and answer a Carlos post:\n\nCarlos, there is no contradiction between faith and scepticism, quite the contrary.\n\nTo say \"I have faith in this\" means \"I do not know, but I choose to accept this\"; a true skeptical stance.\n\nIn contrast, believers say \"I know this is so\". That is not a skeptical stance because without proof, you cannot know.\n\nComprendes?\n\nHans\nHi MRC Hans :\nI know that when things are getting diffcult to the JREFers you need to come in their rescue. Besides you are another JREFer.\nI understand that your ignore buttom just make you be more curious.\n\nThe point here is that FAITH in something that you pray to obtain something.\n\nThat something that you pray to , is ( under the skeptic perspective ), unknown , something without any evidence or proof of existence , something not scientifically explained , something that has to be questioned.\n\nYes , we are talking about that kind of Faith.\nhttp:\/\/www.randi.org\/jr\/050903.html\n\nThen that FAITH is a true belief.\n\nSo please tell me . under what circunstances you accept that a true believer is also an skeptic?\nCan a person be both a skeptic and a person of faith?\n\nThe answer is, Mr. Randi and I agree, a resounding YES.\nWe have already Thaiboxerken point of view , he is in disagree with the JREF \"skeptical\" announce.\nI understand his bad\/insulting style , since I was the one who was questioning him and made him think.\n\nWhat it counts is how SKEPTICISM can easily be involved in a double speech.\n\nAre you saying that you completely agree with the \"skeptical\" position of the JREF regarding that kind of faith?\n\nWhat about the persons that thinks that is just a marketing strategy in order to collect money from true believers(aka woo woos)?\nIs that an honest position?\n\nTheres is a difference about being sceptic and being skeptic.\n\nI hope you \"comprende\".\n\nThanks,\nCarlos\nTo Carlos and Pyrro: ..in the copy of Kramers letter I see in the left column a board of advisors:\n\nASTROLOGY\nGeofre Dean (Australia)\nASTRONOMY\nJack Horkheitmer (Florida)\nASTROPHYISICS\nJavier Amentler (Spain)\nCHEMISTRY\nRolf Manne (Norway)\nEDUCATION\nRicahrd Dawkins (England)\nELECTRONICS\nGiles-Maurice (Belgium)\nFORENsics\nAlexander Jason (Califormia)\nORGANIC CHEMISTRY\nLuigi Garlasshelli (Itlay)\nPHYSICS\nRobert Park (Maryland)\nSTATISTICS\nChip Denman (Maryland)\nUFO AUTHORITY\nRovert Sheaffer (San Jose California\n\nFOREINGN REPRESENTATIVES\nMassimo Pedorro (Italy)[\/quote]\nI love you all !!!\nPure skeptic\nDoctor X\nPosts: 73562\nJoined: Fri Jun 04, 2004 8:09 pm\nTitle: Collective Messiah\nTaiKickboxerKen-San:\n\nHont\u00f5 desu!\n\n--J.D.\nMob of the Mean: Free beanie, cattle-prod and Charley Fan Club!\n\"Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out.\" \u2013 Don\nDocX: FTW. \u2013 sparks\n\"Doctor X wins again.\" \u2013 Pyrrho\n\"Never sorry to make a racist Fucktard cry.\" \u2013 His Humble MagNIfIcence\n\"It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone.\" \u2013 clarsct\n\"I'd leave it up to Doctor X who has been a benevolent tyrant so far.\" \u2013 Grammatron\n\"Indeed you are a river to your people.\nShit. That's going to end up in your sig.\" \u2013 Pyrrho\n\"Try a twelve step program and accept Doctor X as your High Power.\" \u2013 asthmatic camel\n\"just like Doc X said.\" \u2013 gnome\n\nWS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X6!!!!!!\nMRC_Hans\nPosts: 519\nJoined: Fri Jun 04, 2004 8:11 pm\nLocation: Denmark\nCarlos wrote:\n\nHi MRC Hans :\nI know that when things are getting diffcult to the JREFers you need to come in their rescue.\n\nThank you, but you overestimate my importance. However, I don't see the JREFers in any kind of trouble. Certainly not from you.\n\nBesides you are another JREFer.\n\nDepends on your definition of a JREFer.\n\nI understand that your ignore buttom just make you be more curious.\n\nIt is for my convinience. I'm not a slave of it, however.\n\nThe point here is that FAITH in something that you pray to obtain something.\n\nThat is not my definiton of faith.\n\nThat something that you pray to , is ( under the skeptic perspective ), unknown , something without any evidence or proof of existence , something not scientifically explained , something that has to be questioned.\n\nFaith is when you choose to accept something even if you have not seen evidence. This can be because you simply want to, or because you assume that evidence does exist, or no doubt other reasons. Praying has nothing to do with it.\n\nSo please tell me . under what circunstances you accept that a true believer is also an skeptic?\n\nHaving faith is not the same as being a \"true believer\". But, of course it is possible to be a true beleiver in some things and a skeptic about others.\n\nFor instance, even ardent true believers are usually hard-core skeptics when it comes to buying a used car .\n\nWhat about the persons that thinks that is just a marketing strategy in order to collect money from true believers(aka woo woos)?\nIs that an honest position?\n\nEhh? Who is collecting money from woowoos? What on earth are you talking about??\n\nTheres is a difference about being sceptic and being skeptic.\n\nNot according to my dictionary.\n\nI hope you \"comprende\".\n\n?????\n\nThanks,\nCarlos\nFrankly, I think you are getting weirder and weirder.\n\nHans\n[i]Fly pretty, anyone can fly safe...[\/i]\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nMRC_Hans wrote:\nThank you, but you overestimate my importance. However, I don't see the JREFers in any kind of trouble. Certainly not from you.\nSince you claimed I were in your ignore list. is this your second post to me after you put me in ignore?\nMRC_Hans wrote: Depends on your definition of a JREFer.\nA JREF er is a JREF er. You tell us what kind do you think you are.\nMRC_Hans wrote: It is for my convinience. I'm not a slave of it, however.\nI know you read all my posts . And I know you always like to claim I am in your ignore list.\n\nMRC_Hans wrote: Faith is when you choose to accept something even if you have not seen evidence. This can be because you simply want to, or because you assume that evidence does exist, or no doubt other reasons. Praying has nothing to do with it.\nPeople that has faith ..........prays to the \"unknown\".\nAnd you know what kind of faith we are talking about.\nMRC_Hans wrote: For instance, even ardent true believers are usually hard-core skeptics when it comes to buying a used car\nDon't mix skepticism with scepticism.\nMRC_Hans wrote: Ehh? Who is collecting money from woowoos? What on earth are you talking about??\nUnder a skeptic \"definition\" , what is a person who prays to something that has not a evidence? It is \"woo woo\" the way you prefer to call them or not?\n\nThanks,\nCarlos\nSkeeve\nPosts: 13375\nJoined: Wed Jun 09, 2004 7:35 am\nCarlos wrote:Under a skeptic \"definition\" , what is a person who prays to something that has not a evidence? It is \"woo woo\" the way you prefer to call them or not?\n\nThanks,\nCarlos\nYes, he's mean. Really, really mean.\n\nCarlos, why are you mean?\n\nWhy do you play pretend skeptic?\n\nWhy don't you just take your lens flare and go home? Everybody, even me, now, can tell what you're on about.\nThen Skank Of America could start in...\nMRC_Hans\nPosts: 519\nJoined: Fri Jun 04, 2004 8:11 pm\nLocation: Denmark\nCarlos wrote:\nMRC_Hans wrote:\nThank you, but you overestimate my importance. However, I don't see the JREFers in any kind of trouble. Certainly not from you.\nSince you claimed I were in your ignore list. is this your second post to me after you put me in ignore?\nOnce I engage in a discussion, I find it most polite to conduct it to some kind of conclusion, no matter whom the other part is.\nMRC_Hans wrote: Depends on your definition of a JREFer.\nA JREF er is a JREF er. You tell us what kind do you think you are.\nMRC_Hans wrote: It is for my convinience. I'm not a slave of it, however.\nI know you read all my posts . And I know you always like to claim I am in your ignore list.\nMRC_Hans wrote: Faith is when you choose to accept something even if you have not seen evidence. This can be because you simply want to, or because you assume that evidence does exist, or no doubt other reasons. Praying has nothing to do with it.\nPeople that has faith ..........prays to the \"unknown\".\nAnd you know what kind of faith we are talking about.\nI dont see what praying has to do with it. And I'm not going to play guessing games with you.\nMRC_Hans wrote: For instance, even ardent true believers are usually hard-core skeptics when it comes to buying a used car\nDon't mix skepticism with scepticism.\nIn my dictionary, there is no difference between the two words.\nMRC_Hans wrote: Ehh? Who is collecting money from woowoos? What on earth are you talking about??\nUnder a skeptic \"definition\" , what is a person who prays to something that has not a evidence? It is \"woo woo\" the way you prefer to call them or not?\nYou don't make sense.\n\nThanks,\nCarlos[\/quote]\n\nHans\n[i]Fly pretty, anyone can fly safe...[\/i]\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nCarlos wrote:\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote: Yes, they educate. It is an educational foundation.\nAre they educating people to be skeptic the way his leader is?\nI don't know.\nYou don't know if the leader' ideas are the examples of any organization?\nYou don't know if a skeptic like James Randi , leader of the JREF don't pretend to form and educate people under an skeptical point of view?\nI don't know if the JREF is educating people to be the kind of skeptic James Randi is, because I have not attended any of their seminars, I have not attended any of the \"Amaz!ng Meetings\", I have not purchased any of their books, and so on. I do not know.\n\nYou appear to be asking rhetorical questions. If you have statements to make, make them. Asking me rhetorical questions won't work.\nPyrrho wrote: The second part of your question is not conditional upon the first. Whether or not James Randi is a skeptic, belief and faith in God are incompatible with skepticism.\nThen, their own quote is incompatible with skepticism.\nWhich quote is that? Who is \"they\"?\nPyrrho wrote: This forum is the bastard child of a misbegotten notion to excise content that the JREF does not like from the JREF Forum. This forum has no official ties to the JREF. If by \"community\" you mean a group of people who post on the forum, some are skeptics and some are not skeptics. It is not a \"skeptics only\" forum; the JREF Forum is not a \"skeptics only\" forum either.\nYes , sometimes I see this forum as a bastard child of the JREF forum.\nSometimes I see it like an aborted child .\nAnyway is a community formed at the JREF forum.\nMore or less. Come July 8, there will be no \"community\" at the JREF Forum.\nPyrrho wrote: The JREF Forum is the \"community formed around James Randi and the JREF\". This \"community\" exists because it is no longer wanted by the JREF. What's this \"you forgot\" business? That wasn't part of your question.\n\nSince you wrote this :\"There are users here who also post at the JREF Forum. There are users here who are probably members of the JREF. This site exists because the JREF Forum will be eliminating parts of itself that it does not want.\"\nThen I just added this :You forgot to tell that are also members that donate money to the JREF , buy merchandise related to the JREF and go to the annual reunion JREF TAM.\nDo not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nPyrrho wrote: I don't know. I won't guess. People from the JREF Forum are here because on July 8 they will not be able to post content at the JREF Forum that does not fit the undefined criteria of the \"JREF Mission\". The content we post has been defined as being unrelated to the JREF Mission, so how can this forum be related to the JREF? The JREF does not want to be related to this forum.\nIt is related in the way the same posters now here were part of the JREF forum.\nThat's your opinion. Please remember that the JREF does not want to be connected to the material posted here. They specifically said that it was detrimental to their organization, and must be removed from their Forum.\nPyrrho wrote:\nIf you don't know if the majority of members are supporters of the ideas of James Randi \/JREF , then I am missing something.\nAre you a supporter?\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nI believe in asking the right people the right questions. If you want to know what people here think, post a poll. Why make me a spokesman for other people? Why should I know if they support the JREF or not? Ask them yourself if you truly want the answer.\nYou didn't anwer my questions.\nAre you a supporter?\nNo.\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nNo.\nPyrrho wrote: Faith is not permanent. People lose faith all the time, every day. It is just as subject to question as any other human behavior.\nIs skepticism other human behavior?\nIf it is so , can you lose it all the time?\nYes, skepticism is human behavior. Yes, people can stop being skeptical.\nPyrrho wrote:\nMaybe your response is not related to my question. Or my question was not clear enough.\nSince Faith in God is natural , is God paranormal or supernatural?\nThat was not the question you asked. I did answer the question you did ask. As to this question, \"Is God paranormal or supernatural?\" I can only say that I do not know, because I do not know if God exists. If for the sake of discussion we assume that God does exist, we see that God is paradoxical -- in order to exist, God must be natural...and in order to exist, God must be supernatural. We could spin around the paradox indefinitely and never come to a conclusion.\nI already told you that maybe my question was not clear enough.\nBut now your answer telling us you don't know what God is (paranormal or supernatural without evidence ), under an skeptic point of view , can be interpretated that you are asumming that God is real and natural.\n\nFor the real skeptics , God is something that has no proof.\nFor the ones who have Faith in God , God is something that vcan not be questioned.\nI am giving you an honest answer to your question. I am a skeptic; I do not know if God exists, or if God is natural, or if God is supernatural. I do not have enough information in order to make a decision one way or the other. If I were to declare that \"God does not exist,\" that is a decision based on insufficient evidence. If I were to declare that \"God exists,\" that, too, would be a decision based on insufficient evidence. Your personal interpretation of my answer is only your opinion. Once again, I ask you not to put words in my mouth. You do not speak for me.\nPyrrho wrote: For my own part, I think God is a figment of human imagination, and is thus merely natural, and does not objectively exist outside of the human imagination.\n\nHere is something that can be interpretated as a double speech from a confessed skeptic like you. Read the above.\nNo interpretation is necessary. I have given my opinion. Please stop trying to wring a meaning you prefer out of my words.\nPyrrho wrote: I am agreeing that that is pretty much what Hal Bidlack says in his article. It is an article containing his opinions; it is no more a promotion of faith in God than Randi's opinions about religion are a promotion of atheism. They are merely opinions. Promoting something requires more activity than writing opinions. For example, selling books about atheism could be considered promotion of atheism; selling books about deism could be considered promotion of deism. The JREF does not do either of those things, or other similar promotion activities..\nWhen an article on the official page of the JREF , states what their main leaders think , then it is a promotion.\nAt the time it was written, Hal was not a \"leader\" of the JREF. He was only a friend of Randi who ran the JREF Forum for Randi, and who helped with the JREF meeting. Now, of course, he is a director of the JREF, so his words carry more weight. I maintain that a \"promotion\" has to be more than the expression of an opinion. Weighed against the many words of Randi's many commentaries, Hal's opinion isn't much.\nPyrrho wrote:\nAre you telling us that James Randi is not an atheist and that maybe this is a kind of strategy that will reach donations from true believers?\nI do not know if James Randi is an atheist. The JREF is more than James Randi. Its stated purposes do not contain language that promotes atheism. Its activities do not include actions that promote atheism. Its purpose is to promote critical thinking about the paranormal, the normal, and other things, not just about religion or God.\n\nIf James Randi wanted donations from true believers, he'd probably start a church. You're speculating, Carlos. Be careful -- someone might think you were a skeptic.\nThe JREF is formed around the main figure of James Randi. He is the leader and the main representant of the JREF.\nHe is a confessed athiest skeptic telling you that a person can be both a skeptic and a person of faith......in God.\n\nJames Randi\/JREF doesn't need a church to collect donations or to sell merchandise or souvenirs. It is called the JREF. The strategies are different , the results are the same.\n\nThanks,\nCarlos\n\"Confessed\"...as if being an atheist is a sin or a crime...\"confessed\"...as if being a skeptic is a sin or a crime...\n\nThere's nothing wrong with the JREF seeking donations. Hal Bidlack's article does not ask for money from people who believe in God.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nlatinijral wrote:\nCarlos wrote:\nMRC_Hans wrote:I will make an exception and answer a Carlos post:\n\nCarlos, there is no contradiction between faith and scepticism, quite the contrary.\n\nTo say \"I have faith in this\" means \"I do not know, but I choose to accept this\"; a true skeptical stance.\n\nIn contrast, believers say \"I know this is so\". That is not a skeptical stance because without proof, you cannot know.\n\nComprendes?\n\nHans\nHi MRC Hans :\nI know that when things are getting diffcult to the JREFers you need to come in their rescue. Besides you are another JREFer.\nI understand that your ignore buttom just make you be more curious.\n\nThe point here is that FAITH in something that you pray to obtain something.\n\nThat something that you pray to , is ( under the skeptic perspective ), unknown , something without any evidence or proof of existence , something not scientifically explained , something that has to be questioned.\n\nYes , we are talking about that kind of Faith.\nhttp:\/\/www.randi.org\/jr\/050903.html\n\nThen that FAITH is a true belief.\n\nSo please tell me . under what circunstances you accept that a true believer is also an skeptic?\nCan a person be both a skeptic and a person of faith?\n\nThe answer is, Mr. Randi and I agree, a resounding YES.\nWe have already Thaiboxerken point of view , he is in disagree with the JREF \"skeptical\" announce.\nI understand his bad\/insulting style , since I was the one who was questioning him and made him think.\n\nWhat it counts is how SKEPTICISM can easily be involved in a double speech.\n\nAre you saying that you completely agree with the \"skeptical\" position of the JREF regarding that kind of faith?\n\nWhat about the persons that thinks that is just a marketing strategy in order to collect money from true believers(aka woo woos)?\nIs that an honest position?\n\nTheres is a difference about being sceptic and being skeptic.\n\nI hope you \"comprende\".\n\nThanks,\nCarlos\nTo Carlos and Pyrro: ..in the copy of Kramers letter I see in the left column a board of advisors:\n\nASTROLOGY\nGeofre Dean (Australia)\nASTRONOMY\nJack Horkheitmer (Florida)\nASTROPHYISICS\nJavier Amentler (Spain)\nCHEMISTRY\nRolf Manne (Norway)\nEDUCATION\nRicahrd Dawkins (England)\nELECTRONICS\nGiles-Maurice (Belgium)\nFORENsics\nAlexander Jason (Califormia)\nORGANIC CHEMISTRY\nLuigi Garlasshelli (Itlay)\nPHYSICS\nRobert Park (Maryland)\nSTATISTICS\nChip Denman (Maryland)\nUFO AUTHORITY\nRovert Sheaffer (San Jose California\n\nFOREINGN REPRESENTATIVES\nMassimo Pedorro (Italy)\n[\/quote]\nRobert Sheaffer is a debunker of UFOs.\n\nhttp:\/\/www.debunker.com\/ufo.html\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nSkeeve\nPosts: 13375\nJoined: Wed Jun 09, 2004 7:35 am\nPyrrho wrote: Do not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nI'm sorry, Mr. Pyrrho, but I've come to the conclusion that he is a very mean person, and that your request will be honored only in the breach.\n\nI am sorry to say that I haven't the patience or the tenacity to actually attempt a discussion with this fellow Carlos, based simply on his behavior in this thread, and on his and that latin.... person's interjections and backslapping that appears here and there whenever they think they have a chance of an argument.\n\nPerhaps you know better than I do, but I fear you won't be treated civilly, sir.\nThen Skank Of America could start in...\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nSkeeve wrote:\nPyrrho wrote: Do not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nI'm sorry, Mr. Pyrrho, but I've come to the conclusion that he is a very mean person, and that your request will be honored only in the breach.\n\nI am sorry to say that I haven't the patience or the tenacity to actually attempt a discussion with this fellow Carlos, based simply on his behavior in this thread, and on his and that latin.... person's interjections and backslapping that appears here and there whenever they think they have a chance of an argument.\n\nPerhaps you know better than I do, but I fear you won't be treated civilly, sir.\n[Clouseau]\nYes...yes...I know that, yes...the old circular argument ploy. What a pity that my detective's instincts told me this long ago, yes...\n[\/Clouseau]\n\nYeah, well, I've dealt with them before. Look at it this way, it keeps them off the streets.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote: Yes, they educate. It is an educational foundation.\nAre they educating people to be skeptic the way his leader is?\nI don't know.\nYou don't know if the leader' ideas are the examples of any organization?\nYou don't know if a skeptic like James Randi , leader of the JREF don't pretend to form and educate people under an skeptical point of view?\nI don't know if the JREF is educating people to be the kind of skeptic James Randi is, because I have not attended any of their seminars, I have not attended any of the \"Amaz!ng Meetings\", I have not purchased any of their books, and so on. I do not know.\nI accept your answer that you don't know if the skeptic James Randi don't pretend to form and educate people under an skeptical point of view.\n\nI accept the answer and the excuses gave by an ex-administrator of the JREF forum.\nPyrrho wrote:\nPyrrho wrote: The second part of your question is not conditional upon the first. Whether or not James Randi is a skeptic, belief and faith in God are incompatible with skepticism.\nThen, their own quote is incompatible with skepticism.\nWhich quote is that? Who is \"they\"?\nThe quote is this one : \"Can a person be both a skeptic and a person of faith?\n\nThe answer is, Mr. Randi and I agree, a resounding YES.\n\nhttp:\/\/www.randi.org\/jr\/050903.html\n\nAnd they are : James Randi ( leader of the JREF ) and Hal Bidlack ( actual director of the JREF )\n\nPyrrho wrote:\nPyrrho wrote: This forum is the bastard child of a misbegotten notion to excise content that the JREF does not like from the JREF Forum. This forum has no official ties to the JREF. If by \"community\" you mean a group of people who post on the forum, some are skeptics and some are not skeptics. It is not a \"skeptics only\" forum; the JREF Forum is not a \"skeptics only\" forum either.\nYes , sometimes I see this forum as a bastard child of the JREF forum.\nSometimes I see it like an aborted child .\nAnyway is a community formed at the JREF forum.\nMore or less. Come July 8, there will be no \"community\" at the JREF Forum.\nPyrrho wrote: The JREF Forum is the \"community formed around James Randi and the JREF\". This \"community\" exists because it is no longer wanted by the JREF. What's this \"you forgot\" business? That wasn't part of your question.\n\nSince you wrote this :\"There are users here who also post at the JREF Forum. There are users here who are probably members of the JREF. This site exists because the JREF Forum will be eliminating parts of itself that it does not want.\"\nThen I just added this :You forgot to tell that are also members that donate money to the JREF , buy merchandise related to the JREF and go to the annual reunion JREF TAM.\nDo not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nPlease tell me what words don't belong to you and what words do you think I am atributing to you.\nIf you show me I made a mistake , I will apologize to you.\nPyrrho wrote:\nPyrrho wrote: I don't know. I won't guess. People from the JREF Forum are here because on July 8 they will not be able to post content at the JREF Forum that does not fit the undefined criteria of the \"JREF Mission\". The content we post has been defined as being unrelated to the JREF Mission, so how can this forum be related to the JREF? The JREF does not want to be related to this forum.\nIt is related in the way the same posters now here were part of the JREF forum.\nThat's your opinion. Please remember that the JREF does not want to be connected to the material posted here. They specifically said that it was detrimental to their organization, and must be removed from their Forum.\nI understand the JREF technicism.\nI was refering about the other connection.\nFine , thanks.\nPyrrho wrote:\nAre you a supporter?\nNo.\nWhere you a voluntary administrator of the JREF forum?\nCan that action be interpretaded as a support the JREF recieved or that you made?\n\nPyrrho wrote:\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nNo.\nIf you don't believe in polls , why you asked me do do a poll?\nYou already knew I think polls is not an skeptical way to know a truth , as I assume you do.\n\nPyrrho wrote:\nPyrrho wrote: Faith is not permanent. People lose faith all the time, every day. It is just as subject to question as any other human behavior.\nIs skepticism other human behavior?\nIf it is so , can you lose it all the time?\nYes, skepticism is human behavior. Yes, people can stop being skeptical.\nBased in your quote there is corelation between faith an skepticism.\nThen : Skepticism is not permanent. People lose skepticism , every day. It is just as subject to question as any other human behavior.\n\nCan we conclude that skepticism is not a Philosophy?\n\nPyrrho wrote:\nPyrrho wrote:\nMaybe your response is not related to my question. Or my question was not clear enough.\nSince Faith in God is natural , is God paranormal or supernatural?\nThat was not the question you asked. I did answer the question you did ask. As to this question, \"Is God paranormal or supernatural?\" I can only say that I do not know, because I do not know if God exists. If for the sake of discussion we assume that God does exist, we see that God is paradoxical -- in order to exist, God must be natural...and in order to exist, God must be supernatural. We could spin around the paradox indefinitely and never come to a conclusion.\nI already told you that maybe my question was not clear enough.\nBut now your answer telling us you don't know what God is (paranormal or supernatural without evidence ), under an skeptic point of view , can be interpretated that you are asumming that God is real and natural.\n\nFor the real skeptics , God is something that has no proof.\nFor the ones who have Faith in God , God is something that vcan not be questioned.\nI am giving you an honest answer to your question. I am a skeptic; I do not know if God exists, or if God is natural, or if God is supernatural. I do not have enough information in order to make a decision one way or the other. If I were to declare that \"God does not exist,\" that is a decision based on insufficient evidence. If I were to declare that \"God exists,\" that, too, would be a decision based on insufficient evidence. Your personal interpretation of my answer is only your opinion. Once again, I ask you not to put words in my mouth. You do not speak for me.\nI am not .\nYou were the one who wrote that is a paradox , and you knew we were talking about an skeptical definition of God. It was your opinion , not mine.\n\nPyrrho wrote:\nPyrrho wrote: For my own part, I think God is a figment of human imagination, and is thus merely natural, and does not objectively exist outside of the human imagination.\n\nHere is something that can be interpretated as a double speech from a confessed skeptic like you. Read the above.\nNo interpretation is necessary. I have given my opinion. Please stop trying to wring a meaning you prefer out of my words.\nI wrote that it can be interpretated. Remember that interpretation is just another human behavior.\n\nPyrrho wrote:\nPyrrho wrote: I am agreeing that that is pretty much what Hal Bidlack says in his article. It is an article containing his opinions; it is no more a promotion of faith in God than Randi's opinions about religion are a promotion of atheism. They are merely opinions. Promoting something requires more activity than writing opinions. For example, selling books about atheism could be considered promotion of atheism; selling books about deism could be considered promotion of deism. The JREF does not do either of those things, or other similar promotion activities..\nWhen an article on the official page of the JREF , states what their main leaders think , then it is a promotion.\nAt the time it was written, Hal was not a \"leader\" of the JREF. He was only a friend of Randi who ran the JREF Forum for Randi, and who helped with the JREF meeting. Now, of course, he is a director of the JREF, so his words carry more weight. I maintain that a \"promotion\" has to be more than the expression of an opinion. Weighed against the many words of Randi's many commentaries, Hal's opinion isn't much.\nI mantain that is a promotion gave by the leader and the actual director of the JREF on the official web page of the JREF.\n\nThe commentaries of the leaders of any organization are reflects what the organization is , when those commentaries are related to goals of the organization.\nPyrrho wrote:\nPyrrho wrote:\nAre you telling us that James Randi is not an atheist and that maybe this is a kind of strategy that will reach donations from true believers?\nI do not know if James Randi is an atheist. The JREF is more than James Randi. Its stated purposes do not contain language that promotes atheism. Its activities do not include actions that promote atheism. Its purpose is to promote critical thinking about the paranormal, the normal, and other things, not just about religion or God.\n\nIf James Randi wanted donations from true believers, he'd probably start a church. You're speculating, Carlos. Be careful -- someone might think you were a skeptic.\nThe JREF is formed around the main figure of James Randi. He is the leader and the main representant of the JREF.\nHe is a confessed athiest skeptic telling you that a person can be both a skeptic and a person of faith......in God.\n\nJames Randi\/JREF doesn't need a church to collect donations or to sell merchandise or souvenirs. It is called the JREF. The strategies are different , the results are the same.\n\nThanks,\nCarlos\n\"Confessed\"...as if being an atheist is a sin or a crime...\"confessed\"...as if being a skeptic is a sin or a crime...\n\nThere's nothing wrong with the JREF seeking donations. Hal Bidlack's article does not ask for money from people who believe in God.\nThere is nothing wrong that James Randi is a self confessed atheist.\nAlso there is nothing wrong for seeking donations , it doesn't matter if the ones who ask for it are an official church , a cult , a foundation or whatever.\n\nThanks,\nCarlos\nlatinijral\nPosts: 488\nJoined: Tue Jun 08, 2004 3:27 am\nSkeeve wrote:\nI'm sorry, Mr. Pyrrho, but I've come to the conclusion that he is a very mean person, and that your request will be honored only in the breach.\n\nI am sorry to say that I haven't the patience or the tenacity to actually attempt a discussion with this fellow Carlos, based simply on his behavior in this thread, and on his and that latin.... person's interjections and backslapping that appears here and there whenever they think they have a chance of an argument.\n\nPerhaps you know better than I do, but I fear you won't be treated civilly, sir.\nWhat is your conclution about the Popes commentary...or you are only giving \"palmaditas en el poto\" to Pyrro?\nI love you all !!!\nPure skeptic\nSkeeve\nPosts: 13375\nJoined: Wed Jun 09, 2004 7:35 am\nlatinijral wrote:\nSkeeve wrote:\nI'm sorry, Mr. Pyrrho, but I've come to the conclusion that he is a very mean person, and that your request will be honored only in the breach.\n\nI am sorry to say that I haven't the patience or the tenacity to actually attempt a discussion with this fellow Carlos, based simply on his behavior in this thread, and on his and that latin.... person's interjections and backslapping that appears here and there whenever they think they have a chance of an argument.\n\nPerhaps you know better than I do, but I fear you won't be treated civilly, sir.\nWhat is your conclution about the Popes commentary...or you are only giving \"palmaditas en el poto\" to Pyrro?\nAlexander Pope? Pope John XXIII? Who and what are you talking about? As I have discarded the need to have an old man in a robe, one who has sworn off both woman and life as it were, to tell me how to live my maritial life, I don't think that what any Pope, meaning head of the Roman Universal Church, has much meaning at all.\n\nIf that is what you mean, he is not my Pope, does not lead me, and I have nothing much to say about his humanly fallable ponderings. I am, sir, not particularly interested in what he has to say, as long as he does not try to force me to live by his 2000 years antiquated rules for the survival of the group.\nThen Skank Of America could start in...\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nPyrrho wrote: Robert Sheaffer is a debunker of UFOs.\n\nhttp:\/\/www.debunker.com\/ufo.html\nPyrrho:\n\nThanks,\nCarlos\nhgc\nPosts: 166\nJoined: Mon Jun 14, 2004 4:47 pm\nSkeeve wrote:\nPyrrho wrote: Do not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nI'm sorry, Mr. Pyrrho, but I've come to the conclusion that he is a very mean person, and that your request will be honored only in the breach.\n\nI am sorry to say that I haven't the patience or the tenacity to actually attempt a discussion with this fellow Carlos, based simply on his behavior in this thread, and on his and that latin.... person's interjections and backslapping that appears here and there whenever they think they have a chance of an argument.\n\nPerhaps you know better than I do, but I fear you won't be treated civilly, sir.\nYou are correct in your assessment of these characters. It is the likes of Carlos and his pet monkey latininjuredanus that turned me into the crusty coot that I am.\n\nAs you know, I am well past civility myself.\nSkeeve\nPosts: 13375\nJoined: Wed Jun 09, 2004 7:35 am\nhgc wrote:\nSkeeve wrote:\nPyrrho wrote: Do not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nI'm sorry, Mr. Pyrrho, but I've come to the conclusion that he is a very mean person, and that your request will be honored only in the breach.\n\nI am sorry to say that I haven't the patience or the tenacity to actually attempt a discussion with this fellow Carlos, based simply on his behavior in this thread, and on his and that latin.... person's interjections and backslapping that appears here and there whenever they think they have a chance of an argument.\n\nPerhaps you know better than I do, but I fear you won't be treated civilly, sir.\nYou are correct in your assessment of these characters. It is the likes of Carlos and his pet monkey latininjuredanus that turned me into the crusty coot that I am.\n\nAs you know, I am well past civility myself.\nWell, as you may be able to observe in one or another of these threads, it is possible for a person to be a crusty old coot when dealing with floating detritus like Carlos, and be a polite, civil person when dealing with someone who can carry on a dialog.\n\nCarlos wrote another lie\nHe is such a phony,\nCaught some lens flare on his film,\nAnd turned it to baloney!\n\nCarlos, you're a little jerk\nCarlos, you're a phony,\nCarlos we all know your kind,\nThen Skank Of America could start in...\nSkeptoid\nPosts: 1296\nJoined: Fri Jun 04, 2004 5:28 am\nLocation: Wisconsin\nActually, Skeeve, most of us came to the conclusion that the object in question was a bird, as the flapping of wings was quite evident in the video.\nSkeeve\nPosts: 13375\nJoined: Wed Jun 09, 2004 7:35 am\nSkeptoid wrote:Actually, Skeeve, most of us came to the conclusion that the object in question was a bird, as the flapping of wings was quite evident in the video.\nOh, sorry, Carlos, I guess, then, you were just trying to give us all the bird!\nThen Skank Of America could start in...\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nCarlos wrote:\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote: Yes, they educate. It is an educational foundation.\nAre they educating people to be skeptic the way his leader is?\nI don't know.\nYou don't know if the leader' ideas are the examples of any organization?\nYou don't know if a skeptic like James Randi , leader of the JREF don't pretend to form and educate people under an skeptical point of view?\nI don't know if the JREF is educating people to be the kind of skeptic James Randi is, because I have not attended any of their seminars, I have not attended any of the \"Amaz!ng Meetings\", I have not purchased any of their books, and so on. I do not know.\nI accept your answer that you don't know if the skeptic James Randi don't pretend to form and educate people under an skeptical point of view.\n\nI accept the answer and the excuses gave by an ex-administrator of the JREF forum.\nVery well.\nPyrrho wrote:\nPyrrho wrote: The second part of your question is not conditional upon the first. Whether or not James Randi is a skeptic, belief and faith in God are incompatible with skepticism.\nThen, their own quote is incompatible with skepticism.\nWhich quote is that? Who is \"they\"?\nThe quote is this one : \"Can a person be both a skeptic and a person of faith?\n\nThe answer is, Mr. Randi and I agree, a resounding YES.\n\nhttp:\/\/www.randi.org\/jr\/050903.html\n\nAnd they are : James Randi ( leader of the JREF ) and Hal Bidlack ( actual director of the JREF )\nDo they mean \"faith\" as in practicing a religion, or \"faith\" as in holding a belief without question? It's not clear to me what they mean. If by \"faith\" they mean a belief without question, it's incompatible with skepticism. If by \"faith\" they mean practicing a religion, that's not necessarily incompatible.\nPyrrho wrote:\nPyrrho wrote: This forum is the bastard child of a misbegotten notion to excise content that the JREF does not like from the JREF Forum. This forum has no official ties to the JREF. If by \"community\" you mean a group of people who post on the forum, some are skeptics and some are not skeptics. It is not a \"skeptics only\" forum; the JREF Forum is not a \"skeptics only\" forum either.\nYes , sometimes I see this forum as a bastard child of the JREF forum.\nSometimes I see it like an aborted child .\nAnyway is a community formed at the JREF forum.\nMore or less. Come July 8, there will be no \"community\" at the JREF Forum.\nPyrrho wrote: The JREF Forum is the \"community formed around James Randi and the JREF\". This \"community\" exists because it is no longer wanted by the JREF. What's this \"you forgot\" business? That wasn't part of your question.\n\nSince you wrote this :\"There are users here who also post at the JREF Forum. There are users here who are probably members of the JREF. This site exists because the JREF Forum will be eliminating parts of itself that it does not want.\"\nThen I just added this :You forgot to tell that are also members that donate money to the JREF , buy merchandise related to the JREF and go to the annual reunion JREF TAM.\nDo not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nPlease tell me what words don't belong to you and what words do you think I am atributing to you.\nIf you show me I made a mistake , I will apologize to you.\nThe words \"You forgot to tell, etc.\" imply that the statement is something I would have made, but forgot. It's your statement; make it yourself. Don't say that I forgot to say it.\nPyrrho wrote:\nPyrrho wrote: I don't know. I won't guess. People from the JREF Forum are here because on July 8 they will not be able to post content at the JREF Forum that does not fit the undefined criteria of the \"JREF Mission\". The content we post has been defined as being unrelated to the JREF Mission, so how can this forum be related to the JREF? The JREF does not want to be related to this forum.\nIt is related in the way the same posters now here were part of the JREF forum.\nThat's your opinion. Please remember that the JREF does not want to be connected to the material posted here. They specifically said that it was detrimental to their organization, and must be removed from their Forum.\nI understand the JREF technicism.\nI was refering about the other connection.\nFine , thanks.\nPyrrho wrote:\nAre you a supporter?\nNo.\nWhere you a voluntary administrator of the JREF forum?\nCan that action be interpretaded as a support the JREF recieved or that you made?\nI was a voluntary administrator of the JREF Forum. That action was support given to the JREF by me. I no longer serve in that capacity. My support has ended.\nPyrrho wrote:\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nNo.\nIf you don't believe in polls , why you asked me do do a poll?\nYou already knew I think polls is not an skeptical way to know a truth , as I assume you do.\nSo don't post a poll. Just ask the other people what they think.\nPyrrho wrote:\nPyrrho wrote: Faith is not permanent. People lose faith all the time, every day. It is just as subject to question as any other human behavior.\nIs skepticism other human behavior?\nIf it is so , can you lose it all the time?\nYes, skepticism is human behavior. Yes, people can stop being skeptical.\nBased in your quote there is corelation between faith an skepticism.\nThen : Skepticism is not permanent. People lose skepticism , every day. It is just as subject to question as any other human behavior.\n\nCan we conclude that skepticism is not a Philosophy?\nSkepticism is a philosophy; it is a very old philosophy.\nPyrrho wrote:\nPyrrho wrote:\nMaybe your response is not related to my question. Or my question was not clear enough.\nSince Faith in God is natural , is God paranormal or supernatural?\nThat was not the question you asked. I did answer the question you did ask. As to this question, \"Is God paranormal or supernatural?\" I can only say that I do not know, because I do not know if God exists. If for the sake of discussion we assume that God does exist, we see that God is paradoxical -- in order to exist, God must be natural...and in order to exist, God must be supernatural. We could spin around the paradox indefinitely and never come to a conclusion.\nI already told you that maybe my question was not clear enough.\nBut now your answer telling us you don't know what God is (paranormal or supernatural without evidence ), under an skeptic point of view , can be interpretated that you are asumming that God is real and natural.\n\nFor the real skeptics , God is something that has no proof.\nFor the ones who have Faith in God , God is something that vcan not be questioned.\nI am giving you an honest answer to your question. I am a skeptic; I do not know if God exists, or if God is natural, or if God is supernatural. I do not have enough information in order to make a decision one way or the other. If I were to declare that \"God does not exist,\" that is a decision based on insufficient evidence. If I were to declare that \"God exists,\" that, too, would be a decision based on insufficient evidence. Your personal interpretation of my answer is only your opinion. Once again, I ask you not to put words in my mouth. You do not speak for me.\nI am not .\nYou were the one who wrote that is a paradox , and you knew we were talking about an skeptical definition of God. It was your opinion , not mine.\n\nPyrrho wrote:\nPyrrho wrote: For my own part, I think God is a figment of human imagination, and is thus merely natural, and does not objectively exist outside of the human imagination.\n\nHere is something that can be interpretated as a double speech from a confessed skeptic like you. Read the above.\nNo interpretation is necessary. I have given my opinion. Please stop trying to wring a meaning you prefer out of my words.\nI wrote that it can be interpretated. Remember that interpretation is just another human behavior.\n\nPyrrho wrote:\nPyrrho wrote: I am agreeing that that is pretty much what Hal Bidlack says in his article. It is an article containing his opinions; it is no more a promotion of faith in God than Randi's opinions about religion are a promotion of atheism. They are merely opinions. Promoting something requires more activity than writing opinions. For example, selling books about atheism could be considered promotion of atheism; selling books about deism could be considered promotion of deism. The JREF does not do either of those things, or other similar promotion activities..\nWhen an article on the official page of the JREF , states what their main leaders think , then it is a promotion.\nAt the time it was written, Hal was not a \"leader\" of the JREF. He was only a friend of Randi who ran the JREF Forum for Randi, and who helped with the JREF meeting. Now, of course, he is a director of the JREF, so his words carry more weight. I maintain that a \"promotion\" has to be more than the expression of an opinion. Weighed against the many words of Randi's many commentaries, Hal's opinion isn't much.\nI mantain that is a promotion gave by the leader and the actual director of the JREF on the official web page of the JREF.\n\nThe commentaries of the leaders of any organization are reflects what the organization is , when those commentaries are related to goals of the organization.\nPyrrho wrote:\nPyrrho wrote:\nAre you telling us that James Randi is not an atheist and that maybe this is a kind of strategy that will reach donations from true believers?\nI do not know if James Randi is an atheist. The JREF is more than James Randi. Its stated purposes do not contain language that promotes atheism. Its activities do not include actions that promote atheism. Its purpose is to promote critical thinking about the paranormal, the normal, and other things, not just about religion or God.\n\nIf James Randi wanted donations from true believers, he'd probably start a church. You're speculating, Carlos. Be careful -- someone might think you were a skeptic.\nThe JREF is formed around the main figure of James Randi. He is the leader and the main representant of the JREF.\nHe is a confessed athiest skeptic telling you that a person can be both a skeptic and a person of faith......in God.\n\nJames Randi\/JREF doesn't need a church to collect donations or to sell merchandise or souvenirs. It is called the JREF. The strategies are different , the results are the same.\n\nThanks,\nCarlos\n\"Confessed\"...as if being an atheist is a sin or a crime...\"confessed\"...as if being a skeptic is a sin or a crime...\n\nThere's nothing wrong with the JREF seeking donations. Hal Bidlack's article does not ask for money from people who believe in God.\nThere is nothing wrong that James Randi is a self confessed atheist.\nAlso there is nothing wrong for seeking donations , it doesn't matter if the ones who ask for it are an official church , a cult , a foundation or whatever.\n\nThanks,\nCarlos\nIt's wrong if the donations are sought under false pretenses, such as the frauds who pretend to work miracles on sick people.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nSundog\nPosts: 2578\nJoined: Mon Jun 07, 2004 4:27 pm\nDang, Pyrrho, when you finally show up you don't fiddle around, do you?\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nCarlos wrote:\nPyrrho wrote: Robert Sheaffer is a debunker of UFOs.\n\nhttp:\/\/www.debunker.com\/ufo.html\nPyrrho:\n\nThanks,\nCarlos\nI don't know. Why don't you ask him personally?\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nSundog wrote:Dang, Pyrrho, when you finally show up you don't fiddle around, do you?\nI'm tone deaf. Can't play a note.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nDoctor X\nPosts: 73562\nJoined: Fri Jun 04, 2004 8:09 pm\nTitle: Collective Messiah\nBecomes a drummer.\n\nWe do not need notes. . . .\n\n--J.D.\nMob of the Mean: Free beanie, cattle-prod and Charley Fan Club!\n\"Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out.\" \u2013 Don\nDocX: FTW. \u2013 sparks\n\"Doctor X wins again.\" \u2013 Pyrrho\n\"Never sorry to make a racist Fucktard cry.\" \u2013 His Humble MagNIfIcence\n\"It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone.\" \u2013 clarsct\n\"I'd leave it up to Doctor X who has been a benevolent tyrant so far.\" \u2013 Grammatron\n\"Indeed you are a river to your people.\nShit. That's going to end up in your sig.\" \u2013 Pyrrho\n\"Try a twelve step program and accept Doctor X as your High Power.\" \u2013 asthmatic camel\n\"just like Doc X said.\" \u2013 gnome\n\nWS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X6!!!!!!\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nPyrrho wrote:\nCarlos wrote:\nI accept your answer that you don't know if the skeptic James Randi don't pretend to form and educate people under an skeptical point of view.\n\nI accept the answer and the excuses gave by an ex-administrator of the JREF forum.\nVery well.\nYes, it is.\nPyrrho wrote:\nThe quote is this one : \"Can a person be both a skeptic and a person of faith?\n\nThe answer is, Mr. Randi and I agree, a resounding YES.\n\nhttp:\/\/www.randi.org\/jr\/050903.html\n\nAnd they are : James Randi ( leader of the JREF ) and Hal Bidlack ( actual director of the JREF )\nDo they mean \"faith\" as in practicing a religion, or \"faith\" as in holding a belief without question? It's not clear to me what they mean. If by \"faith\" they mean a belief without question, it's incompatible with skepticism. If by \"faith\" they mean practicing a religion, that's not necessarily incompatible.\nAsk them. Or try to understand what they mean.\nPyrrho wrote:\nPyrrho wrote:\n\nDo not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nPlease tell me what words don't belong to you and what words do you think I am atributing to you.\nIf you show me I made a mistake , I will apologize to you.\nThe words \"You forgot to tell, etc.\" imply that the statement is something I would have made, but forgot. It's your statement; make it yourself. Don't say that I forgot to say it.\nThen I was not putting words in your mouth.\nYou are correct , it is more polite to say that I was just helping you all to remember that are also members that donate money to the JREF , buy merchandise related to the JREF and go to the annual reunion JREF TAM.\nWe were talking about all kind of relationship of the members of this board and the JREF.\nPyrrho wrote:\nCarlos wrote:\n\nAre you a supporter?\nNo.\nI was a voluntary administrator of the JREF Forum. That action was support given to the JREF by me. I no longer serve in that capacity. My support has ended..\nThen you were a supporter of the JREF.\nI mean :what were your reasons to don't continue supporting them?\nI think you were doing a great honest job administrating that forum.\n\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote:\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nNo.\nSo don't post a poll. Just ask the other people what they think.\nUhh.I asked you since you seem to knew.\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote:\nYes, skepticism is human behavior. Yes, people can stop being skeptical.\nBased in your quote there is corelation between faith an skepticism.\nThen : Skepticism is not permanent. People lose skepticism , every day. It is just as subject to question as any other human behavior.\n\nCan we conclude that skepticism is not a Philosophy?\nSkepticism is a philosophy; it is a very old philosophy.\nI though you said skepticism is just another human behavior.\nAs well as Faith.\nWhat is the philosophy of Faith?\nWhat is the philosophy of Skepticism?\nI am not refering about the old Greeck school of Scepticism .\n\nPyrrho wrote:\n\nThere is nothing wrong that James Randi is a self confessed atheist.\nAlso there is nothing wrong for seeking donations , it doesn't matter if the ones who ask for it are an official church , a cult , a foundation or whatever.\nIt's wrong if the donations are sought under false pretenses, such as the frauds who pretend to work miracles on sick people.\nIf they are frauds then the law must caught them and send them to prison.\nBut the reality is other.\n\nMaybe because it is just a matter of faith and skepticism involved .\nJust human behaviors as you wrote it.\n\nThanks,\nCarlos\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote: Robert Sheaffer is a debunker of UFOs.\n\nhttp:\/\/www.debunker.com\/ufo.html\nPyrrho:\n\nThanks,\nCarlos\nI don't know. Why don't you ask him personally?\nYes , I know you don't know.\nI know , since you posted a link and told us who this JREF advisor is , that there is nothing related to that case in the link you posted.\n\nI know that James Randi never wrote about that topic also.\nI know that web is still full of interpretations about that case.\n\nThanks,\nCarlos\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nCarlos wrote:\nPyrrho wrote:\nCarlos wrote:\nI accept your answer that you don't know if the skeptic James Randi don't pretend to form and educate people under an skeptical point of view.\n\nI accept the answer and the excuses gave by an ex-administrator of the JREF forum.\nVery well.\nYes, it is.\nIt is it.\nPyrrho wrote:\nThe quote is this one : \"Can a person be both a skeptic and a person of faith?\n\nThe answer is, Mr. Randi and I agree, a resounding YES.\n\nhttp:\/\/www.randi.org\/jr\/050903.html\n\nAnd they are : James Randi ( leader of the JREF ) and Hal Bidlack ( actual director of the JREF )\nDo they mean \"faith\" as in practicing a religion, or \"faith\" as in holding a belief without question? It's not clear to me what they mean. If by \"faith\" they mean a belief without question, it's incompatible with skepticism. If by \"faith\" they mean practicing a religion, that's not necessarily incompatible.\nAsk them. Or try to understand what they mean.\nPyrrho wrote:\nPyrrho wrote:\n\nDo not put words in my mouth, Carlos. Say what you have to say, but don't pretend that it is I who must say it. Be honest.\nPlease tell me what words don't belong to you and what words do you think I am atributing to you.\nIf you show me I made a mistake , I will apologize to you.\nThe words \"You forgot to tell, etc.\" imply that the statement is something I would have made, but forgot. It's your statement; make it yourself. Don't say that I forgot to say it.\nThen I was not putting words in your mouth.\nYou are correct , it is more polite to say that I was just helping you all to remember that are also members that donate money to the JREF , buy merchandise related to the JREF and go to the annual reunion JREF TAM.\nWe were talking about all kind of relationship of the members of this board and the JREF.\nPyrrho wrote:\nCarlos wrote:\n\nAre you a supporter?\nNo.\nI was a voluntary administrator of the JREF Forum. That action was support given to the JREF by me. I no longer serve in that capacity. My support has ended..\nThen you were a supporter of the JREF.\nI mean :what were your reasons to don't continue supporting them?\nI think you were doing a great honest job administrating that forum.\nI am not in a \"bad mood\" toward the JREF. I just don't support it with my time and effort anymore. I still support skepticism.\n\nI had to quit being the adminstrator because I did not have time to do the job properly. I expressed that concern to Hal before May 4.\n\nI don't participate at the JREF anymore because of the way the change was done at the JREF Forum. I was angry about it, now I am not. I am only saddened that in one single minute all the weeks of effort toward making it a \"friendlier\" and more reasonable forum was wasted.\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote:\nDo you believe in the results of polls ?, I mean this kind of \"skeptics\" polls.\nNo.\nSo don't post a poll. Just ask the other people what they think.\nUhh.I asked you since you seem to knew.\nPyrrho wrote:\nCarlos wrote:\nPyrrho wrote:\nYes, skepticism is human behavior. Yes, people can stop being skeptical.\nBased in your quote there is corelation between faith an skepticism.\nThen : Skepticism is not permanent. People lose skepticism , every day. It is just as subject to question as any other human behavior.\n\nCan we conclude that skepticism is not a Philosophy?\nSkepticism is a philosophy; it is a very old philosophy.\nI though you said skepticism is just another human behavior.\nAs well as Faith.\nWhat is the philosophy of Faith?\nWhat is the philosophy of Skepticism?\nI am not refering about the old Greeck school of Scepticism .\nIt can be a \"behavior\" and a \"philosophy\". The two are not mutually exclusive.\nPyrrho wrote:\n\nThere is nothing wrong that James Randi is a self confessed atheist.\nAlso there is nothing wrong for seeking donations , it doesn't matter if the ones who ask for it are an official church , a cult , a foundation or whatever.\nIt's wrong if the donations are sought under false pretenses, such as the frauds who pretend to work miracles on sick people.\nIf they are frauds then the law must caught them and send them to prison.\nBut the reality is other.\n\nMaybe because it is just a matter of faith and skepticism involved .\nJust human behaviors as you wrote it.\n\nThanks,\nCarlos\nNobody knows the whole truth.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nPyrrho wrote: I am not in a \"bad mood\" toward the JREF. I just don't support it with my time and effort anymore. I still support skepticism.\n\nI had to quit being the adminstrator because I did not have time to do the job properly. I expressed that concern to Hal before May 4.\n\nI don't participate at the JREF anymore because of the way the change was done at the JREF Forum. I was angry about it, now I am not. I am only saddened that in one single minute all the weeks of effort toward making it a \"friendlier\" and more reasonable forum was wasted.\nDo you think it was not a reasonable and friendler JREF forum before you were not administrating it? I mean in Hal's administration.\nPyrrho wrote: It can be a \"behavior\" and a \"philosophy\". The two are not mutually exclusive.\nThose were not my questions.\nWhat is the philosophy of Faith?\nWhat is the philosophy of Skepticism?\nI am not refering about the old Greeck school of Scepticism .\n\nPyrrho wrote:\n\nIf they are frauds then the law must caught them and send them to prison.\nBut the reality is other.\n\nMaybe because it is just a matter of faith and skepticism involved .\nJust human behaviors as you wrote it.\nNobody knows the whole truth.\nYou said it. It is just your skeptic opinion.\n\nThanks,\nCarlos\nSkeeve\nPosts: 13375\nJoined: Wed Jun 09, 2004 7:35 am\nCarlos wrote:You said it. It is just your skeptic opinion.\n\nThanks,\nCarlos\nCarlos, what is your evidences on the 10 commandments? Why don't you reply to Doctor X in the thread about the 10 commandments?\n\nIt's very enlightening, and it's much more certain that Johnathan Livingston Twin Towers...\n\nI am sorry, I'd thought you were selling lense flare when it turns out you were trying to sell seagull. They're hardly in short supply, you know.\nThen Skank Of America could start in...\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nCarlos wrote:\nPyrrho wrote: I am not in a \"bad mood\" toward the JREF. I just don't support it with my time and effort anymore. I still support skepticism.\n\nI had to quit being the adminstrator because I did not have time to do the job properly. I expressed that concern to Hal before May 4.\n\nI don't participate at the JREF anymore because of the way the change was done at the JREF Forum. I was angry about it, now I am not. I am only saddened that in one single minute all the weeks of effort toward making it a \"friendlier\" and more reasonable forum was wasted.\nDo you think it was not a reasonable and friendler JREF forum before you were not administrating it? I mean in Hal's administration.\nI've said enough about that on other forums. It doesn't matter what it was. What is is now is what matters.\nPyrrho wrote: It can be a \"behavior\" and a \"philosophy\". The two are not mutually exclusive.\nThose were not my questions.\nWhat is the philosophy of Faith?\nBelief without question.\nWhat is the philosophy of Skepticism?\nI am not refering about the old Greeck school of Scepticism .\nContemporary skepticism is defined here:\n\nhttp:\/\/www.utm.edu\/research\/iep\/s\/skepcont.htm\nPyrrho wrote:\n\nIf they are frauds then the law must caught them and send them to prison.\nBut the reality is other.\n\nMaybe because it is just a matter of faith and skepticism involved .\nJust human behaviors as you wrote it.\nNobody knows the whole truth.\nYou said it. It is just your skeptic opinion.\n\nThanks,\nCarlos\nYes. It is only my opinion.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nSkeeve wrote:\nCarlos wrote:You said it. It is just your skeptic opinion.\n\nThanks,\nCarlos\nCarlos, what is your evidences on the 10 commandments? Why don't you reply to Doctor X in the thread about the 10 commandments?\n\nIt's very enlightening, and it's much more certain that Johnathan Livingston Twin Towers...\n\nI am sorry, I'd thought you were selling lense flare when it turns out you were trying to sell seagull. They're hardly in short supply, you know.\nNo, it was probably a common starling. Probably a deep roller.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nCarlos\nPosts: 669\nJoined: Fri Jun 18, 2004 10:40 pm\nPyrrho wrote:\nCarlos wrote:\n\nDo you think it was not a reasonable and friendler JREF forum before you were not administrating it? I mean in Hal's administration.\nI've said enough about that on other forums. It doesn't matter what it was. What is is now is what matters.\nI asked you because you wrote here about your efforts to make that forum \"friendlier\" and more reasonable. So my question was related to that quote in order to know how it was before you entered.\nIn my opinion , in your short period administarting the JREF forum, you made it more reasonable and you were honest .\n\nPyrrho wrote:\nCarlos wrote:\n\nWhat is the philosophy of Faith?\nBelief without question.\nThat is the definition of faith.\nPyrrho wrote:\nCarlos wrote:\nWhat is the philosophy of Skepticism?\nI am not refering about the old Greeck school of Scepticism .\nContemporary skepticism is defined here:\n\nhttp:\/\/www.utm.edu\/research\/iep\/s\/skepcont.htm\nJust another definition.\n\nPyrrho wrote:\nCarlos wrote:\nYou said it. It is just your skeptic opinion.\nYes. It is only my opinion.\nThen is not a fact that nobody knows the whole truth.\nOr maybe it is.\n\nThanks,\nCarlos\nlatinijral\nPosts: 488\nJoined: Tue Jun 08, 2004 3:27 am\nCarlos wrote:\nPyrrho wrote: Robert Sheaffer is a debunker of UFOs.\n\nhttp:\/\/www.debunker.com\/ufo.html\nPyrrho:\n\nThanks,\nCarlos\nMaybe Kramer and Co. forgot to send it?\nI love you all !!!\nPure skeptic\nDoctor X\nPosts: 73562\nJoined: Fri Jun 04, 2004 8:09 pm\nTitle: Collective Messiah\nMaybe he forgot to reply to Kramer:\n\nEvidences\n\n--J.D.\nMob of the Mean: Free beanie, cattle-prod and Charley Fan Club!\n\"Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out.\" \u2013 Don\nDocX: FTW. \u2013 sparks\n\"Doctor X wins again.\" \u2013 Pyrrho\n\"Never sorry to make a racist Fucktard cry.\" \u2013 His Humble MagNIfIcence\n\"It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone.\" \u2013 clarsct\n\"I'd leave it up to Doctor X who has been a benevolent tyrant so far.\" \u2013 Grammatron\n\"Indeed you are a river to your people.\nShit. That's going to end up in your sig.\" \u2013 Pyrrho\n\"Try a twelve step program and accept Doctor X as your High Power.\" \u2013 asthmatic camel\n\"just like Doc X said.\" \u2013 gnome\n\nWS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X6!!!!!!\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nAh, I see. They forgot Carlos even existed.\n\nBut, they should allow Carlos to post on their forum. If they're going to tell their side of the story on that forum, Carlos should be allowed to tell his side of the story.\nThe flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus.\nDoctor X\nPosts: 73562\nJoined: Fri Jun 04, 2004 8:09 pm\nTitle: Collective Messiah\nPerhaps when they received no evidence his claim existed they moved on to more likely claims.\n\n--J.D.\nMob of the Mean: Free beanie, cattle-prod and Charley Fan Club!\n\"Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out.\" \u2013 Don\nDocX: FTW. \u2013 sparks\n\"Doctor X wins again.\" \u2013 Pyrrho\n\"Never sorry to make a racist Fucktard cry.\" \u2013 His Humble MagNIfIcence\n\"It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone.\" \u2013 clarsct\n\"I'd leave it up to Doctor X who has been a benevolent tyrant so far.\" \u2013 Grammatron\n\"Indeed you are a river to your people.\nShit. That's going to end up in your sig.\" \u2013 Pyrrho\n\"Try a twelve step program and accept Doctor X as your High Power.\" \u2013 asthmatic camel\n\"just like Doc X said.\" \u2013 gnome\n\nWS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X6!!!!!!\nPyrrho\nPosts: 30924\nJoined: Sat Jun 05, 2004 2:17 am\nTitle: Man in Black\nLocation: Division 6\nDoctor X wrote:Perhaps when they received no evidence his claim existed they moved on to more likely claims.\n\n--J.D.\nHis application existed, at least. He never advanced to the \"claimant\" level.\n\nBah. I don't want to rehash this stuff. Better for Carlos if he talks directly to the persons who were involved, which weren't us.\nThe flash of light you saw in the sky was not a UFO. 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Scoil Naomh Pádraig, Curragh Camp.
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BERNARD BLAZES TO VICTORY AT MILLROSE GAMES
ASTONISHING!!!……..On Saturday night in New York City, Scoil Naomh Pádraig's Bernard Ibirogba wrote his name into the history books of the world famous Millrose Games. Bernard, who was competing against the fastest 8 year olds on the east coast of America, stormed to a phenomenal win in the closest run race of the day. His victory in The Fastest Kid on the Bock 55m race makes Bernard the first Irish winner at the famed track and field competition since Mark Carroll won the Wannamaker Mile 17 years ago. Our little sprint sensation joins a famous list of past Irish winners at the meet including Eamon Coghlan, Ronnie Delaney and Marcus O'Sullivan.
Bernard's weekend didn't start as planned as his flight to New York was diverted to Washington D.C. before eventually arriving into the Big Apple at 1.00am (6:00am Irish time). Bernard didn't allow the door to door journey of almost 24 hours derail his ambitions of winning, he just smiled and embraced every moment of the trip. After a good night's sleep, Bernard set off to the Amory Track on 168th street for his final training session. His final workout included some light drills and some fine tuning of his starting technique. A couple of hours later he was centre stage at the official press conference before finishing the day out at the official Millrose Games Banquet where he dined with Olympic 1500m Champion, Matt Centrowitz. Some might consider this an overwhelming experience for an 8 year old but this wasn't the case for Bernard, he loved every second of it and his infectious smile lit up New York City.
When race day arrived on Saturday, Bernard entered the sold out 5000 capacity stadium at 1:00pm. He checked in for his race, picked up his bib and then sat down to soak up the atmosphere. Like an eagle in the sky, he studiously observed the professional athletes going through their pre-race rituals. An hour before his race, he set off for the warm up track to rub shoulders with the pros and at this point, Bernard and the many Olympic champions were now on a first name basis with each other. Bernard completed his pre-race warm and drills just as he would have done in the Curragh.
At 3:11pm, Bernard laced up his racing spikes and stepped onto the line to compete against America's fittest and fastest. The starting gun fired and Bernard shot out from his crouching start but he slightly trailed the leader. As with previous races, Bernard hit full stride at the halfway mark and he began to close in on first place. He gained precious ground with every step and stormed over the line in a sensational 8.41 seconds, the very same time as his closest competitor, Phil Kamara. Who won? Nobody knew…..The entire crowd anxiously awaited the result of the photo finish but after a few nervous moments the winner was announced…..BERNARD IBIROGBA - MILLROSE CHAMPION!!!
This incredible moment was witnessed by Bernard's proud parents Kudi and Abiodun who made the trip to New York. Congratulations must go their way for always supporting and encouraging Bernard. On behalf of Bernard, his family and Scoil Naomh Pádraig, we would all like to express our immense gratitude to Dermot McDermott and Caoimhe Ní Mhurcú who have created this wonderful opportunity through the Believe and Achieve initiative. Without any sponsorship, Dermot and Caoimhe have funded the entire trip out of their own pockets and they left no stone unturned to create an amazing weekend for an amazing boy. Scoil Naomh Pádraig would also like to congratulate Emily Kelly of Donegal who ran an excellent race to finish in 3rd place in the girls' Fastest Kid on the Block race. Emily and Bernard were great ambassadors for Ireland.
Bernard would also like to thank his school teammates and classmates who have encouraged him throughout the year and in particular, Ife Ojo, who has trained very closely with Bernard over the past month. In addition, Bernard would like to thank all the staff at Scoil Naomh Pádraig, including teachers Mr Healy, Miss Collins, Miss Sheehan and Mr Connolly who have all regularly joined the running team on their runs over the Curragh Plains. Finally, the school would like to thank the Defence Forces for use of their track, Alan Ward (Emerald Facility Services), Jim Aughney (Dublin Marathon) and Jill Bradbury (Bradburys Irish School Meals) who have all made generous contributions to the school's cross country programme over the past three years.
The final word goes to our hardworking hero, Bernard Ibirogba. We are all so proud of you and you deserve every success. WELL DONE CHAMP!!!
Bernard has gone viral - see press coverage below.
Race Video: https://www.youtube.com/watch?v=t2yXOXVE3Uc
Believe and Achieve Initiative: https://twitter.com/believe9achieve?lang=en
Irish Independent:http://www.independent.ie/sport/other-sports/athletics/speedsters-match-americas-fastest-kids-at-millrose-games-35445447.html
Irish Runner Magazine:http://www.irishrunner.ie/young-irish-light-up-millrose-games/
Balls.ie: https://www.balls.ie/athletics/bernard-ibirogba-358921
The42.ie: http://www.the42.ie/bernard-ibirogba-new-york-millrose-games-3237213-Feb2017/
Irish Examiner: http://www.irishexaminer.com/breakingnews/sport/watch-8-year-old-irish-boy-win-fastest-kid-on-the-block-title-in-new-york-776971.html
Kildare Nationalist http://www.kildare-nationalist.ie/2017/02/10/bernards-fastest-feet-take-him-all-the-way-to-new-york/
Ife Ojo leads school to 1-2-3 clean sweep at the South Dublin Cross Country League
The school cross country club continues to grow from strength to strength and this was clear to see by all this past Thursday. A small squad of 23 boys set off for Clondalkin Park with hopes of scoring their first points in their respective races. On the back of months of hard training in all weather conditions, the boys were eager to test themselves against some of the larger schools of South Dublin. Although small in size, this is a squad high in quality and this quality shone through on the day as all expectations were surpassed.
First up were the 3rd class boys who were split into 3 races of 40 boys each. Ross Fagan (2nd class), making his debut for the school team, lead the charge with a courageous run and he was rewarded with an excellent 5th place finish. As Ross finished inside the top 10, he put the first points on the board for Scoil Naomh Pádraig. Christian and Adam Behan were soon to follow and both finished in the top 20.
Next up was Race 2 and this was undoubtedly the race of the day. Within 5 seconds of the gun sounding, Scoil Naomh Pádraig vests swarmed to the front of the field with Ife Ojo storming into the lead. Chasing hot on his heels was Dylan O'Reilly and remarkably Eoin Molloy (2nd class) sat in 3rd place. As the race approached the closing stages, Ife and Dylan O'Reilly ran well clear of the chasers and the only question now was who would take gold. Both boys were neck and neck coming around the final bend but a final surge by Ife was enough to take the win. With Dylan taking a comfortable 2nd place, Eoin Molloy showed great stamina to make it a 1-2-3 finish. This was an incredible achievement and the icing was put on the cake when Cian Browne showed great pacing ability to work his way through the main pack to finish in 5th place. The result of this race alone meant that the team banked 36 points (12+10+8+6), a new record score for the school. Other notable performers in the 3rd class category were Seán Byrne, Ryan Sage, Cillian Keane, Dylan Comerford, Ben O'Connor, Scott Murphy, Daryl Byrne, Scott Doyle and Bradley Cox.
As the day progressed, the harriers from the Curragh continued to excel and earned further points through the great efforts of Teighan Ryan Klimuk and captain Mikey Ijoma. The ever improving Kian McGowan ran another solid race to earn a top 20 finish. Rhys Hayden, Cormac Doyle and Tyon Drazi closed the day with fine runs. A final score of 49 points smashed our previous best of 30 points; well done to all on our best performance yet. Onwards and upwards!
Mini Moores come to Scoil Naomh Pádraig
Moorefield GAA kindly sent 3 of their top coaches to Scoil Naomh Pádraig to give us a glimpse of what would be in store for any of the boys who decide to join the Mini Moores. Led by past pupil and Senior Footballer James Lonergan, ably assisted by fellow coaches Colin and Anthony, all 4 classes took part in fun skills sessions. Anyone looking to find out more about the Mini Moores, contact Philip (087-2023580) or Alan (0876505013) or come to their First Touch Open and Registration Day (for children born in 2010/11/12) in Moorefield GAA on Saturday 25th February where you can meet the coaches and your team mates. Training for other year groups also begins on Sat 25th.
Bernard set to take on America's finest in New York City's 110th Millrose Games
At 3:11pm (8:11pm Irish time) on Saturday 11th February, Scoil Naomh Pádraig's Bernard Ibirogba will lace up his racing spikes and the toe the line against America's fastest 8 year old boys at the world famous Millrose Games. Having become Irish National Champion last December, Bernard will pull on the Irish vest for the very first time and compete in the Fastest Kid on the Block 55 metre sprint. With 12 Olympic Gold Medallists competing at the same meet, this will undoubtedly be a very proud moment for Bernard, his family and friends. The entire school community of Scoil Naomh Pádraig are so proud of Bernard and we all wish him the best of luck in New York City. As often said during our training sessions; RUN LIKE THE CLAPPERS BERNARD!!!
Info on Bernard's race is below
Live webcast available:
http://www.usatf.tv/gprofile.php?mgroup_id=45365&do=videos&video_id=192217
To watch the webcast, a login must be created. Subscribe at http://plus.runnerspace.com/ at a monthly cost of $12.99. The subscription must be cancelled within a month to avoid any further charges. A subscriber can only view from one device so it may be a fun idea to team up and have a Millrose party with a neighbour! If subscribing to the live webcast, Bernard's race will be on the first TV slot as shown below (12:00PM to 4:00PM). Bernard's race is scheduled to start at 3:11pm New York Time (8:11pm Irish Time). The professional races will begin shortly afterwards at 4:00pm (9:00pm Irish Time).
If you are unable to watch the race live, we will endeavour to publish the race on the school website as soon as possible.
Facebook updates available on Believe and Achieve:
https://www.facebook.com/believe9achieve/
Millrose Games Website: http://www.nyrrmillrosegames.org/
Race fields (to be updated shortly): http://www.nyrrmillrosegames.org/fields-2017/
Race schedule: http://www.nyrrmillrosegames.org/meet-schedule/
Results available on Armory Track website: http://www.armorytrack.com/
Bradburys Bakery and Restaurant supporting Bernard Ibirogba in NYC
With less than a week to go to the world's longest running and most prestigious indoor track and field competition, Scoil Naomh Pádraig's Bernard Ibirogba has been fine tuning his training in preparation for the biggest moment of his prosperous running career. Before Bernard jets off to the Big Apple, Scoil Naomh Pádraig, Bernard and his family would like to express their gratitude to Jill Bradbury of Bradburys Bakery, Newbridge for kindly making a generous contribution towards Bernard's travel and accommodation costs. Last week, Jill kindly took time out of her day to come into the school and visit Bernard and wish him best of luck in New York. Further information on Bradburys extensive range of products can be found on their website: http://www.bradburys.ie/
Further details of Bernard's race including a live stream link will be posted during the week.
The latest news and events from Scoil Naomh Pádraig in the Curragh Camp
Greenschools
Mr Healy | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,902 |
Q: on setState render a new component I have an analog clock
using
https://github.com/zackargyle/react-analog-clock
I have a textbox that fetches city name and timezone on autocomplete,
*
*Default clock2 loaded with offset +8
*On select of city in textbox
*Fetch City Name
*Fetch Timezone
*Re-render the clock2
How can I code that?
public render(): React.ReactElement<Props> {
return (
<div className={styles.time}>
<AnalogClock theme={Themes.light} width={120} gmtOffset=${this.state.gmtoffset} /><br/>
{this.state.wxLoc}
</div>
);
I attempted
this.showClock2(this.state.gmtoffset);
////////////////////////
private showClock2(offset="+8"){
let show=`<AnalogClock theme={Themes.light} width={120} gmtOffset=${offset} /><br/>
{this.state.wxLoc}`;
return show;
}
//other codes
public render(): React.ReactElement<Props> {
return (
<div className={styles.time}>
{this.showClock2()}
</div>
);
it shows Literal Code, Is it possible to Render the <AnalogClock/> ?
A: private showClock2(offset="+8") {
ReactDOM.render(<AnalogClock theme={Themes.light} width={120} gmtOffset={offset} />, document.getElementById('root'))
}
Add in the public render and it works
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,633 |
Stock #: 900826 Exterior Color: Eclipse Interior Color: Black Onyx Body Type: 4dr Car Transmission: Automatic Drivetrain: RWD Engine: Gas/Electric I4 2.0L/ Title Condition: Clear City 20 mpg Hwy 21 mpg Actual mileage will vary with options, driving conditions and driving habits. Vehicle Description Clean CARFAX. Recent Arrival! Odometer is 11458 miles below market average!Reviews:* If you desire a rare car with stunning looks, a luxurious interior, and eco-conscious powertrain and sensibilities, the Fisker Karma fits the bill. Think of it as the green car for the jet set. Source: * Plug-in-hybrid fuel economy; smooth and ample electric power; sharp handling; head-turning style; earth-friendly cabin materials. Source: EdmundsAlthough every reasonable effort has been made to ensure the accuracy of the information contained on this... site, absolute accuracy cannot be guaranteed. This site, and all information and materials appearing on it, are presented to the user as is without warranty of any kind, either expressed or implied. All vehicles are subject to prior sale. Price does not include applicable tax, title, license, processing and/or documentation fees, and destination charges. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,407 |
package org.zstack.header.vm;
import org.zstack.header.cluster.ClusterInventory;
import org.zstack.header.host.HostInventory;
import org.zstack.header.host.HostState;
import org.zstack.header.host.HostStatus;
import org.zstack.header.message.APIReply;
import org.zstack.header.rest.RestResponse;
import java.sql.Timestamp;
import java.util.List;
import static org.codehaus.groovy.runtime.InvokerHelper.asList;
/**
* Created by xing5 on 2016/5/14.
*/
@RestResponse(fieldsTo = {"hosts=hostInventories", "clusters=clusterInventories"})
public class APIGetVmStartingCandidateClustersHostsReply extends APIReply {
private List<HostInventory> hostInventories;
private List<ClusterInventory> clusterInventories;
public List<HostInventory> getHostInventories() {
return hostInventories;
}
public void setHostInventories(List<HostInventory> hostInventories) {
this.hostInventories = hostInventories;
}
public List<ClusterInventory> getClusterInventories() {
return clusterInventories;
}
public void setClusterInventories(List<ClusterInventory> clusterInventories) {
this.clusterInventories = clusterInventories;
}
public static APIGetVmStartingCandidateClustersHostsReply __example__() {
APIGetVmStartingCandidateClustersHostsReply reply = new APIGetVmStartingCandidateClustersHostsReply();
String clusterUuid = uuid();
HostInventory hi = new HostInventory ();
hi.setAvailableCpuCapacity(2L);
hi.setAvailableMemoryCapacity(4L);
hi.setClusterUuid(clusterUuid);
hi.setManagementIp("192.168.0.1");
hi.setName("example");
hi.setState(HostState.Enabled.toString());
hi.setStatus(HostStatus.Connected.toString());
hi.setClusterUuid(uuid());
hi.setZoneUuid(uuid());
hi.setUuid(uuid());
hi.setTotalCpuCapacity(4L);
hi.setTotalMemoryCapacity(4L);
hi.setHypervisorType("KVM");
hi.setDescription("example");
hi.setCreateDate(new Timestamp(System.currentTimeMillis()));
hi.setLastOpDate(new Timestamp(System.currentTimeMillis()));
reply.setHostInventories(asList(hi));
ClusterInventory cl = new ClusterInventory();
cl.setName("cluster1");
cl.setUuid(clusterUuid);
cl.setZoneUuid(uuid());
cl.setCreateDate(new Timestamp(System.currentTimeMillis()));
cl.setLastOpDate(new Timestamp(System.currentTimeMillis()));
cl.setHypervisorType("KVM");
reply.setClusterInventories(asList(cl));
return reply;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,822 |
Cedar of Lebanon - What do you know?
Removing blutak from rough side of burl?
Anyone going to Tambourine Turnfest???
Who has a Record Lathe?
Washer behind a Nova chuck?
Opinions please oh mighty gurus!
Wee chuck for my new Record??
Woodturners Guild - Lalor Park. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,199 |
{"url":"https:\/\/www.satyenkale.com\/pubs\/near-optimal-algorithms-for-online-matrix-prediction\/","text":"Elad Hazan, Satyen Kale and Shai Shalev-Shwartz\nIn SIAM Journal on Computing (SICOMP), 2017. Proceedings of 25th Conference on Learning Theory (COLT), 2012\n\nIn several online prediction problems of recent interest the comparison class is composed of matrices with bounded entries. For example, in the online max-cut problem, the comparison class is matrices which represent cuts of a given graph and in online gambling the comparison class is matrices which represent permutations over $$n$$ teams. Another important example is online collaborative filtering in which a widely used comparison class is the set of matrices with a small trace norm. In this paper we isolate a property of matrices, which we call $$(\\beta,\\tau)$$-decomposability, and derive an efficient online learning algorithm, that enjoys a regret bound of $$\\tilde{O}(\\sqrt{\\beta \\tau T})$$ for all problems in which the comparison class is composed of $$(\\beta,\\tau)$$-decomposable matrices. By analyzing the decomposability of cut matrices, triangular matrices, and low trace-norm matrices, we derive near optimal regret bounds for online max-cut, online gambling, and online collaborative filtering. In particular, this resolves (in the affirmative) an open problem posed by Abernethy (2010) and Kleinberg et al (2010). Finally, we derive lower bounds for the three problems and show that our upper bounds are optimal up to logarithmic factors. In particular, our lower bound for the online collaborative filtering problem resolves another open problem posed by Shamir and Srebro (2011).","date":"2022-10-01 23:38:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7936402559280396, \"perplexity\": 574.5789637339147}, \"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-2022-40\/segments\/1664030336978.73\/warc\/CC-MAIN-20221001230322-20221002020322-00578.warc.gz\"}"} | null | null |
"""Retriever script for Zanne et al. Global wood density database.
"""
import os
import sys
import xlrd
from retriever.lib.templates import Script
from retriever.lib.models import Table
from retriever.lib.excel import Excel
class main(Script):
def __init__(self, **kwargs):
Script.__init__(self, **kwargs)
self.name = "Zanne et al. Global wood density database."
self.shortname = "GWDD"
self.urls = {"GWDD": "http://datadryad.org/bitstream/handle/10255/dryad.235/GlobalWoodDensityDatabase.xls?sequence=1"}
self.tags = ["Taxon > Plants", "Spatial Scale > Global",
"Data Type > Observational"]
self.ref = "http://datadryad.org/resource/doi:10.5061/dryad.234"
self.description = "A collection and collation of data on the major wood functional traits, including the largest wood density database to date (8412 taxa), mechanical strength measures and anatomical features, as well as clade-specific features such as secondary chemistry."
self.citation = "Chave J, Coomes DA, Jansen S, Lewis SL, Swenson NG, Zanne AE (2009) Towards a worldwide wood economics spectrum. Ecology Letters 12(4): 351-366. http://dx.doi.org/10.1111/j.1461-0248.2009.01285.x"
self.addendum = """ *Correspondence for updates to the database: G.Lopez-Gonzalez@leeds.ac.uk
For descriptions of the database, see Chave et al. 2009. Towards a worldwide wood economics spectrum. Ecology Letters. Identifier: http://hdl.handle.net/10255/dryad.234
Below we list the rules of use for the Global wood density database.
These are developed based on the rules of use for the Glopnet dataset (www.nature.com/nature/journal/v428/n6985/full/nature02403.html) and Cedar Creek LTER and Related Data (http://www.lter.umn.edu/cgi-bin/register).
If you would like to use the Global wood density database, we request that you:
1. Notify the main address of correspondence (Gaby Lopez-Gonzalo) if you plan to use the database in a publication.
2. Provide recognition of the efforts of this group in the assembly of the data by using the citation for the database above.
3. Recognize that these data were assembled by the group for various analyses and research questions. If any of these uses overlap with your interests, you recognize that group has precedence in addressing these questions."""
def download(self, engine=None, debug=False):
Script.download(self, engine, debug)
self.engine.download_file(self.urls["GWDD"], "GlobalWoodDensityDatabase.xls")
filename = os.path.basename("GlobalWoodDensityDatabase.xls")
book = xlrd.open_workbook(self.engine.format_filename(filename))
sh = book.sheet_by_index(1)
rows = sh.nrows
#Creating data table
lines = []
for i in range(1, rows):
row = sh.row(i)
if not all(Excel.empty_cell(cell) for cell in row):
this_line = {}
def format_value(s):
s = Excel.cell_value(s)
return str(s).title().replace("\\", "/").replace('"', '')
for num, label in enumerate(["Number", "Family", "Binomial", "Wood_Density",
"Region", "Reference_Number"]):
this_line[label] = format_value(row[num])
lines.append(this_line)
table = Table("data", delimiter="\t")
table.columns=[("Number" , ("pk-int",) ),
("Family" , ("char",) ),
("Binomial" , ("char",) ),
("Wood_Density" , ("double",) ),
("Region" , ("char",) ),
("Reference_Number" , ("int",) )]
table.pk = 'Number'
table.contains_pk = True
gwdd = []
for line in lines:
gwdd_data = [line["Number"],
line["Family"],
line["Binomial"],
line["Wood_Density"],
line["Region"],
line["Reference_Number"]]
gwdd.append(gwdd_data)
data = ['\t'.join(gwdd_line) for gwdd_line in gwdd]
self.engine.table = table
self.engine.create_table()
self.engine.add_to_table(data)
#Creating reference table
lines = []
sh = book.sheet_by_index(2)
rows = sh.nrows
for i in range(1, rows):
row = sh.row(i)
if not all(Excel.empty_cell(cell) for cell in row):
this_line = {}
def format_value(s):
s = Excel.cell_value(s)
return str(s).title().replace("\\", "/").replace('"', '')
for num, label in enumerate(["Reference_Number", "Reference"]):
this_line[label] = format_value(row[num])
lines.append(this_line)
table = Table("reference", delimiter="\t")
table.columns=[("Reference_Number" , ("pk-int",) ),
("Reference" , ("char",) )]
table.pk = 'Reference_Number'
table.contains_pk = True
gwdd = []
for line in lines:
gwdd_ref = [line["Reference_Number"],
line["Reference"]]
gwdd.append(gwdd_ref)
data = ['\t'.join(gwdd_line) for gwdd_line in gwdd]
self.engine.table = table
self.engine.create_table()
self.engine.add_to_table(data)
return self.engine
SCRIPT = main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,883 |
Home Rugby Ronan O'Gara and Joe Rokocoko Question Level Of Leadership In The All Blacks Team
Ronan O'Gara and Joe Rokocoko Question Level Of Leadership In The All Blacks Team
Sean McMahon July 7, 2017
It's no secret that the All Blacks have lost some huge players in recent years – Dan Carter, Richie McCaw, Kevin Mealamu, Ma'a Nonu, Conrad Smith – that's a massive amount of caps and experience to lose from a team, even one with has massive depths of talent such as New Zealand.
One of the main talking points this week ahead of the final Test at Eden Park is the lack of experience in the All Blacks side which Steve Hansen has named. Both Jordie Barrett and Ngani Laumape are both starting their first game for the All Blacks. Even Anton Lienert-Brown only has 12 caps to his name.
Nevertheless, the All Blacks are a formidable outfit regardless and they really should have beaten the Lions last weekend, even when they were reduced to 14 men for the majority of the game.
That loss, according to Ronan O'Gara writing in his column for the Irish Examiner, could be down to a lack of leadership in the All Blacks side, he writes:
Racing 92 returned to pre-season training this week and I had an interesting conversation with Joe Rokocoko about leadership. He noted that in the pressure moments of the second Test, the Lions were dotted with leaders — Wyn-Jones, the entire back-row, Davies, Farrell, Sexton, and Murray. Then he named the coterie of leaders that New Zealand rugby has lost in recent times — from Carter to McCaw, from Nonu to Mealamu.
After the loss of Sonny Bill Williams as well, there was a sense, he felt, that the All Blacks ran on autopilot for a while, but when the Lions cranked it up late on, were there enough leaders in black?
It's an interesting opinion and one which may hold some truth – there will be a huge Lions support at the All Blacks' fortress of Eden Park tomorrow, which is something the players would not be used too. They will be under a level of pressure which they have not experienced too many times before.
How they react to that pressure will likely contribute to the outcome of this game and the series.
All Blacks ronan o'gara steve hansen 2017-07-07
Sean McMahon
About Sean McMahon
Sean is Head of Pundit Arena Rugby. Contact him on Twitter here:
@Sean_McMahon89 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,518 |
\section{Introduction}
Markov processes have been widely used in many different applications for modeling random phenomena. In some problems more general stochastic processses (e.g., reciprocal processes or CM processes) are required. CM and reciprocal processes have been used in many different applications, including stochastic mechanics, intent inference, image processing, trajectory modeling, and acausal systems \cite{Levy_1}--\cite{Krener1}. Dynamic models of CM, reciprocal, and Markov processes play a very important role in the application of these processes. This paper elaborates on the relationship between these models.
Gaussian CM processes were introduced in \cite{Mehr}. Reciprocal processes were introduced in \cite{Bernstein} related to a problem posed by E. Schrodinger \cite{Schrodinger_1}--\cite{Schrodinger_2}. Later, reciprocal processes were studied more in \cite{Slepian}--\cite{Moura}. Dynamic models and characterizations of NG $CM_L$ and $CM_F$ sequences were obtained in \cite{CM_Part_I_Conf}. A Gaussian sequence is reciprocal if and only if (iff) it is both $CM_L$ and $CM_F$ \cite{CM_Part_II_A_Conf}. A dynamic model with locally correlated dynamic noise governing the NG reciprocal sequence was presented in \cite{Levy_Dynamic}. Dynamic models with white dynamic noise governing the NG reciprocal sequence were developed in \cite{CM_Part_II_A_Conf}.
Consider stochastic sequences defined over $[0,N]=\lbrace 0,1,\ldots,N \rbrace$. For convenience, let the index be time. A sequence is Markov iff conditioned on the state at any time $k$, the subsequences before and after $k$ are independent. A sequence is reciprocal iff conditioned on the states at any two times $k_1$ and $k_2$, the subsequences inside and outside the interval $[k_1,k_2]$ are independent. In other words, ``inside" and ``outside" are independent given the boundaries. A sequence is $CM_L$ ($CM_F$) iff conditioned on the state at time $N$ ($0$), the sequence is Markov over $[0,N-1]$ ($[1,N]$). The subscripts ``$L$" (``$F$") is used because the conditioning is at the \textit{last} (\textit{first}) time of the interval.
The Markov sequence is a special class of reciprocal sequences. The reciprocal sequence is a special class of $CM_L$/$CM_F$ sequences. Thus, evolution of a Markov sequence can be governed by a Markov/reciprocal/$CM_L$/$CM_F$ model. Similarly, evolution of a reciprocal sequence can be governed by a reciprocal/$CM_L$/$CM_F$ model. Therefore, a CM sequence can have more than one model. These models are \textit{equivalent} in the sense that they govern the same sequence (i.e., the sequences governed by the models have the same distribution). Also, sometimes only a forward (backward) model is available when the corresponding backward (forward) one is desired or required. These forward and backward models are also equivalent since they govern the same sequence. In some cases, the above definition of equivalent models is not sufficient because it is only about distributions, not each sample path. It is only equivalent for the set, not element-wise equivalent. The two-filter smoothing approach is an example, where to verify the conditions required for derivation, one needs relationship between sample paths of dynamic noises and boundary values of forward and backward Markov models for the same sample path of the sequence \cite{Wax_Kailath}--\cite{Alan_Wilskey}. Given a model and a sample path of its dynamic noise and boundary values\footnote{For a forward (backward) Markov model, boundary values mean the initial (final) values.} corresponding to an arbitrary sample path of the sequence, it is desirable to obtain an equivalent model and a sample path of its dynamic noise and boundary values leading to the same sample path of the sequence. In other words, it is desirable to find relationship between the dynamic noises and boundary values (of two equivalent models) leading to the same sample path of the governed sequence. Therefore, such models with an explicit relationship between their sample paths are said to be \textit{explicitly sample-equivalent}. It is important to find relationships between the equivalent models because one model can be more easily applicable than the other in some applications. For example, the reciprocal model of \cite{Levy_Dynamic} is driven by colored noise and not necessarily easy to apply for trajectory modeling \cite{DD_Conf}--\cite{DW_Conf}. But the equivalent recirpocal $CM_L$ model of \cite{CM_Part_II_A_Conf} is driven by white noise and its application is straightforward.
Determination of a NG backward Markov model based on its forward model has been the topic of several papers \cite{MB_1}--\cite{Verghese}. Equivalence of the backward Markov model in \cite{MB_1}--\cite{MB_4} was derived based on equality of the second moments calculated by forward and backward models, which does not deal with specific sample paths. \cite{Verghese} presented an explicitly equivalent backward Markov model only for forward models with nonsingular state transition matrices. In the case of a singular state transition matrix, the derivation of \cite{Verghese} does not provide explicit equivalence (i.e., the relationship between sample paths of the dynamic noises and the boundary values) of forward and backward models. In such a case, \cite{Verghese} only provides parameters of the backward model. Given a NG Markov model, an approach was presented in \cite{Levy_Dynamic} for the determination of an explicitly equivalent reciprocal model with locally correlated dynamic noise governing the same Markov sequence.
The main contributions of this paper are as follows. It discusses relationships between dynamic models governing NG $CM_L$, $CM_F$, reciprocal, and Markov sequences. A unified approach is presented, such that within these classes given a model, any explicitly equivalent model can be obtained. As a special case, a backward Markov model explicitly equivalent to a forward Markov model can be obtained. Unlike \cite{Verghese}, this approach works for both singular and nonsingular state transition matrices. The explicitly equivalent reciprocal model obtained in \cite{Levy_Dynamic} can be derived by our approach.
Section \ref{Definitions} reviews definitions and models of $CM_L$, $CM_F$, reciprocal, and Markov sequences. Also, definition of explicitly sample-equivalent models is presented. Section \ref{General_Approach} presents an approach to determining explicitly equivalent models. In Section \ref{Examples}, explicitly sample-equivalent forward and backward Markov models, and explicitly sample-equivalent $CM_L$ and reciprocal models are obtained. Section \ref{Summary} contains a summary and conclusions.
\section{Definitions and Preliminaries}\label{Definitions}
Throughout the paper we consider sequences defined over $[0,N]$. The following conventions are used:
\begin{align*}
[i,j]& \triangleq \lbrace i,i+1,\ldots ,j-1,j \rbrace, i,j \in [0,N], i<j\\
[x_k]_{i}^{j} & \triangleq \lbrace x_k, k \in [i,j] \rbrace\\
[x_k] & \triangleq [x_k]_{0}^{N}\\
x & \triangleq [x_0, \ldots , x_N]'
\end{align*}
Also, $C$ is the covariance matrix of the whole sequence $[x_k]$. The symbol ``$\setminus$" is used for set subtraction.
\begin{definition}
Two dynamic models are \textit{equivalent} if they govern the same sequence (i.e., their sequences have the same distribution).
\end{definition}
\begin{definition}\label{Explicit_Equivalent}
Two dynamic models are \textit{explicitly sample-equivalent} if, given a sample path of the dynamic noise and the boundary values of one model for an arbitrary sample path of the governed sequence, a sample path of the dynamic noise and the boundary values of the other model leading to the same sample path of the governed sequence is given explicitly.
\end{definition}
In other words, if the relation between sample paths of dynamic noises and boundary values of two equivalent models is provided (so that the two models generate the same sample path), they become explicitly equivalent.
Forward and backward Markov, reciprocal, $CM_L$, and $CM_F$ models of \cite{Levy_Dynamic}, \cite{CM_Part_I_Conf}, \cite{CM_Part_II_A_Conf} are reviewed first. Let $[x_k]$ be a zero-mean NG sequence.
\subsection{Markov Model}
$[x_k]$ is Markov iff
\begin{align}
x_k&=M_{k,k-1}x_{k-1}+e^M_{k}, k \in [1,N] \label{Markov_Dynamic_Forward}\\
x_0&=e^M_0\label{M_BC}
\end{align}
where $[e^M_k]$ is a zero-mean white NG sequence with covariances $M_k$.
We have
\begin{align*}
\mathcal{M}x&=e^M\\
e^M &= [(e^M_0)' , \ldots , (e^M_N)']'
\end{align*}
where $\mathcal{M}$ is the nonsingular matrix
\begin{align}\label{M}
\left[ \begin{array}{cccccc}
I & 0 & 0 & \cdots & 0 & 0\\
-M_{1,0} & I & 0 & \cdots & 0 & 0\\
0 & -M_{2,1} & I & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & -M_{N-1,N-2} & I & 0\\
0 & 0 & 0 & \cdots & -M_{N,N-1} & I
\end{array}\right]
\end{align}
A NG sequence with covariance matrix $C$ is Markov iff $C^{-1}$ is tri-diagonal given by $\eqref{CML}$ below with $D_0=\cdots=D_{N-2}=0$.
\subsection{Backward Markov Model}
$[x_k]$ is Markov iff
\begin{align}
x_{k}&=M^B_{k,k+1}x_{k+1}+e^{MB}_{k}, k \in [0,N-1] \label{Markov_Dynamic_Backward}\\
x_N&=e^{MB}_N\label{MB_BC}
\end{align}
where $[e^{MB}_k]$ is a zero-mean white NG sequence with covariances $M^B_k$.
We have
\begin{align*}
\mathcal{M}^Bx&=e^{MB}\\
e^{MB} &=[(e^{MB}_0)' , \ldots , (e^{MB}_{N})']'
\end{align*}
where $\mathcal{M}^B$ is the nonsingular matrix
\begin{align}\label{MB}
\left[ \begin{array}{cccccc}
I & -M^{B}_{0,1} & 0 & \cdots & 0 & 0\\
0 & I & -M^B_{1,2} & 0 & \cdots & 0\\
0 & 0 & I & -M^B_{2,3} & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & 0 & I & -M^B_{N-1,N}\\
0 & 0 & 0 & \cdots & 0 & I
\end{array}\right]
\end{align}
\subsection{Reciprocal Model}
$[x_k]$ is reciprocal iff
\begin{align}
R^0_kx_k-R^-_{k}x_{k-1}-R^+_{k}x_{k+1}=e^R_k, k \in [1,N-1] \label{Reciprocal_Dynamic}
\end{align}
where $[e^R_k]_0^{N}$ is a zero-mean NG sequence with $E[e^R_k(e^R_{k})']=R^0_k$, $E[e^R_k(e^R_{k+1})']=-R^+_k$, $E[e^R_k(e^R_j)']=0, |k-j|>1$, $R^+_k=(R^-_{k+1})'$, and the boundary conditions
\begin{align}
R^0_0x_0-R^-_0x_N-R^+_0x_1&=e^R_0\label{Reciprocal_Cyclic_BC1}\\
R^0_Nx_N-R^-_Nx_{N-1}-R^+_Nx_0&=e^R_N\label{Reciprocal_Cyclic_BC2}
\end{align}
where $E[e^R_N(e^R_0)']=-R^+_N$ and $R^-_0=(R^+_N)'$, and the parameters of model $\eqref{Reciprocal_Dynamic}$ and boundary conditions $\eqref{Reciprocal_Cyclic_BC1}$--$\eqref{Reciprocal_Cyclic_BC2}$ lead to a nonsingular sequence.
We have
\begin{align*}
\mathcal{R}x&=e^R\\
e^R &= [(e^R_0)' , \ldots , (e^R_N)']'
\end{align*}
where $\mathcal{R}$ is the nonsingular matrix
\begin{align}\label{R}
\left[ \begin{array}{cccccc}
R^0_0 & -R^+_0 & 0 & \cdots & 0 & -R^-_0\\
-R^-_{1} & R^0_1 & -R^+_1 & 0 & \cdots & 0\\
0 & -R^-_{2} & R^0_2 & -R^-_2 & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & -R^-_{N-1} & R^0_{N-1} & -R^+_{N-1}\\
-R^+_N & 0 & 0 & \cdots & -R^-_{N} & R^0_N
\end{array}\right]
\end{align}
A NG sequence with covariance matrix $C$ is reciprocal iff its $C^{-1}$ is cyclic tri-diagonal given by $\eqref{CML}$ with $D_1=\cdots=D_{N-2}=0$.
The reciprocal model $\eqref{Reciprocal_Dynamic}$ (with its boundary conditions) is well-posed if its parameters lead to a nonsingular covariance matrix for the whole sequence $[x_k]$.
We call model $\eqref{Reciprocal_Dynamic}$ the reciprocal model, to distinguish it from reciprocal $CM_L$/$CM_F$ models, defined below.
\subsection{Forward $CM_c$ Models}
$[x_k]$ is $CM_c$ iff
\begin{align}
x_k&=G_{k,k-1}x_{k-1}+G_{k,c}x_c+e_k, k \in [1,N] \setminus \lbrace c \rbrace
\label{CML_Dynamic_Forward}\\
x_c&=e_c, \quad x_0=G_{0,c}x_c+e_0 \, \, (\text{for} \, \, c=N) \label{CML_Forward_BC2}
\end{align}
where $[e_k]$ is a zero-mean white NG sequence with covariances $G_k$.
For $c=0$,
\begin{align}
\mathcal{G}^Fx&=e^F\label{Lx=e}\\
e^F& = [e_0' , \ldots , e_N']'\nonumber
\end{align}
where $\mathcal{G}^F$ is the nonsingular matrix
\begin{align}\label{F}
\left[ \begin{array}{cccccc}
I & 0 & 0 & \cdots & 0 & 0\\
-2G_{1,0} & I & 0 & \cdots & 0 & 0\\
-G_{2,0} & -G_{2,1} & I & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
-G_{N-1,0} & 0 & \cdots & -G_{N-1,N-2} & I & 0\\
-G_{N,0} & 0 & 0 & \cdots & -G_{N,N-1} & I
\end{array}\right]
\end{align}
For $c=N$,
\begin{align}
\mathcal{G}^Lx&=e^L\label{Lx=e}\\
e^L& =[e_0' , \ldots , e_N']'\nonumber
\end{align}
where $\mathcal{G}^L$ is the nonsingular matrix
\begin{align}\label{L_2}
\left[ \begin{array}{cccccc}
I & 0 & 0 & \cdots & 0 & -G_{0,N}\\
-G_{1,0} & I & 0 & \cdots & 0 & -G_{1,N}\\
0 & -G_{2,0} & I & 0 & \cdots & -G_{2,N}\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & \cdots & -G_{N-1,N-2} & I & -G_{N-1,N}\\
0 & 0 & 0 & \cdots & 0 & I
\end{array}\right]
\end{align}
A NG sequence with covariance matrix $C$ is $CM_L$ ($CM_F$) iff its $C^{-1}$ is $CM_L$ ($CM_F$), defined as follows.
\begin{definition}
A symmetric positive definite matrix is $CM_L$ if it has form $\eqref{CML}$ and $CM_F$ if it has form $\eqref{CMF}$:
\begin{align}
\left[
\begin{array}{ccccccc}
A_0 & B_0 & 0 & \cdots & 0 & 0 & D_0\\
B_0' & A_1 & B_1 & 0 & \cdots & 0 & D_1\\
0 & B_1' & A_2 & B_2 & \cdots & 0 & D_2\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\
0 & \cdots & 0 & B_{N-3}' & A_{N-2} & B_{N-2} & D_{N-2}\\
0 & \cdots & 0 & 0 & B_{N-2}' & A_{N-1} & B_{N-1}\\
D_0' & D_1' & D_2' & \cdots & D_{N-2}' & B_{N-1}' & A_N
\end{array}\right]\label{CML}\\
\left[
\begin{array}{ccccccc}
A_0 & B_0 & D_2 & \cdots & D_{N-2} & D_{N-1} & D_{N}\\
B_0' & A_1 & B_1 & 0 & \cdots & 0 & 0\\
D_2' & B_1' & A_2 & B_2 & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\
D_{N-2}' & \cdots & 0 & B_{N-3}' & A_{N-2} & B_{N-2} & 0\\
D_{N-1}' & \cdots & 0 & 0 & B_{N-2}' & A_{N-1} & B_{N-1}\\
D_{N}' & 0 & 0 & \cdots & 0 & B_{N-1}' & A_N
\end{array}\right]\label{CMF}
\end{align}
\end{definition}
Here $A_k$, $B_k$, and $D_k$ are matrices in general.
\begin{remark}
$[x_k]$ is reciprocal iff it obeys $\eqref{CML_Dynamic_Forward}$--$\eqref{CML_Forward_BC2}$ and
\begin{align}
G_k^{-1}G_{k,c}=G_{k+1,k}'G_{k+1}^{-1}G_{k+1,c}
\label{CML_Condition_Reciprocal}
\end{align}
$\forall k \in [1,N-2]$ for $c=N$, and $\forall k \in [2,N-1]$ for $c=0$. Moreover, $[x_k]$ is Markov iff we also have, for $c=N$,
\begin{align}
G_0^{-1}G_{0,N}=G_{1,0}'G_1^{-1}G_{1,N}
\end{align}
and for $c=0$,
\begin{align}
G_{N,0}=0
\end{align}
\end{remark}
\subsection{Backward $CM_c$ Models}
$[x_k]$ is $CM_c$ iff
\begin{align}
x_k&=G^B_{k,k+1}x_{k+1}+G^B_{k,c}x_c+e^{B}_k, k \in [0,N-1] \setminus \lbrace c\rbrace
\label{CML_Dynamic_Backward}\\
x_c&=e^{B}_c, \quad x_{N}=G^{B}_{N,c}x_c+e^{B}_{c} \, \, (\text{for} \, \, c=0) \label{CML_Backward_BC2}
\end{align}
where $[e^{B}_k]$ is a zero-mean white NG sequence with covariances $G^B_k$.
\begin{remark}
$[x_k]$ is reciprocal iff it obeys $\eqref{CML_Dynamic_Backward}$--$\eqref{CML_Backward_BC2}$ and
\begin{align}
(G^B_{k+1})^{-1}G^B_{k+1,c}=(G^B_{k,k+1})'(G^B_{k})^{-1}G^B_{k,c}
\label{CMF_Condition_Reciprocal_B}
\end{align}
$\forall k \in [1,N-2]$ for $c=0$, and $\forall k \in [0,N-3]$ for $c=N$. Moreover, $[x_k]$ is Markov iff we also have, for $c=0$,
\begin{align}
G^B_N)^{-1}G^{B}_{N,0}=(G^B_{N-1,N})'(G^B_{N-1})^{-1}G^B_{N-1,0}
\end{align}
and for $c=N$,
\begin{align}
G^{B}_{0,N}=0
\end{align}
\end{remark}
A forward/backward $CM_L$/$CM_F$ model of a reciprocal (Markov) sequence is called a reciprocal (Markov) forward/backward $CM_L$/$CM_F$ model.
\section{Determination of Explicitly Equivalent Dynamic Models: A Unified Approach}\label{General_Approach}
Let $[x_k]$ be a CM sequence governed by a model presented in Section \ref{Definitions}. We have
\begin{align}
T x&=\xi, \quad \xi =[\xi_{0}' , \ldots , \xi_{N}']' \label{Model}
\end{align}
where the vector $\xi$ includes the dynamic noise and the boundary values and $P=\text{Cov}(\xi)$. Matrix $T$ is determined by parameters of the model. Note that $T$ and $P$ for a model have a specific form. So, parameters of equivalent models (in terms of each other) can be determined easily (see the first step of Proposition \ref{Equivalent_Construction_Lemma} below). $T$ is nonsingular for the forward and the backward $CM_L$, $CM_F$, and Markov models. Also, since the sequence is assumed to be nonsingular, $T$ is nonsingular for the reciprocal model, too \cite{Levy_Dynamic}.
By Definition \ref{Explicit_Equivalent}, explicit equivalence is mutual, i.e., if model 2 is explicitly equivalent to model 1, then model 1 is also explicitly equivalent to model 2. We have the following proposition for the construction of explicitly equivalent models.
\begin{proposition}\label{Equivalent_Construction_Lemma}
Let a forward, backward $CM_L$, $CM_F$, reciprocal, or Markov model be given by
\begin{align}
T_1x=\xi \label{Model_1}
\end{align}
where $\xi =[\xi_{0}' , \ldots , \xi_{N}']'$ includes the dynamic noise and the boundary values, and the covariance of $\xi$ is $P_1$. Any explicitly equivalent forward or backward $CM_L$/$CM_F$/reciprocal/Markov model, denoted by
\begin{align}
T_2x=\zeta \label{Model_2}
\end{align}
with $\zeta=[\zeta_{0}' , \ldots , \zeta_{N}']'$ containing the corresponding dynamic noise and boundary values with covariance $P_2$, is constructed in two steps:
(1) The parameters of the equivalent model are determined based on\footnote{Due to the special structures of $T_1$, $P_1$, $T_2$, and $P_2$, parameters of model 2 can be easily obtained in terms of parameters of model 1 using $\eqref{Step_1}$. Then, having the parameters of model 2, $T_2$ and $P_2$ are known.}
\begin{align}
T_2'P_2^{-1}T_2=
T_1'P_1^{-1}T_1\label{Step_1}
\end{align}
Therefore, $T_2$ and $P_2$ can be obtained given their forms.
(2) The relationship between the dynamic noises and the boundary values of the two models are obtained through
\begin{align}
T_2'(P_2)^{-1}\zeta = T_1'(P_1)^{-1}\xi \label{Step_2}
\end{align}
\end{proposition}
\begin{proof}
Step (1): The inverse of the covariance matrix ($C^{-1}$) of the sequence governed by model $\eqref{Model_1}$ is calculated as $E[(T_1x)(T_1x)']=E[\xi \xi ']$, which leads to
\begin{align}\label{Inverse_C_1}
C^{-1}=T_1'(P_1)^{-1}T_1
\end{align}
The inverse of the covariance matrix of the sequence governed by $\eqref{Model_2}$ can be calculated as
\begin{align}\label{Inverse_C_2}
C^{-1}=T_2'(P_2)^{-1}T_2
\end{align}
In order for the two models to be explicitly equivalent, their governed sequences must have the same covariance matrix; thus we have $\eqref{Step_1}$ (i.e., the two models are equivalent). Due to the special structures of $T_1$, $P_1$, $T_2$, and $P_2$, parameters of model 2 can be easily obtained in terms of parameters of model 1 using $\eqref{Step_1}$ ($\eqref{MfbP_1}$--$\eqref{MfbP_5}$ below show such calculations for equivalent Markov models). Then, $P_2$ and $T_2$ are known. Note that parameters of model 2 calculated by $\eqref{Step_1}$ (in terms of those of model 1) are unique. It can be easily verified based on $\eqref{Step_1}$ for all models. This uniqueness can be also concluded from the definition of conditional expectation. Now, given $P_2$ and $T_2$, the relation between $\xi$ and $\zeta$ should be determined so that the two models are \textit{explicitly} equivalent.
Step (2): Let $P _2$ and $T _2$ be given. We show how $\eqref{Step_2}$ leads to an explicitly equivalent model. First, we show that $\zeta$ obtained by $\eqref{Step_2}$ has the desired dynamic noise and boundary values, i.e., its covariance is $P_2$. By $\eqref{Step_2}$, we have
\begin{align*}
T_2'&(P_2)^{-1}\text{Cov}(\zeta)(P_2)^{-1}T_2\\ &\quad \quad \quad =T_1'(P_1)^{-1}\text{Cov}(\xi )(P_1)^{-1}T_1
\end{align*}
Then, substituting $\text{Cov}(\xi)=P_1$, we obtain
\begin{align*}
\text{Cov}(\zeta)&=P_2(T_2')^{-1}T_1'(P_1)^{-1}P_1
(P_1)^{-1}T_1(T_2)^{-1}P_2\\
&=P_2(T_2')^{-1}T_1'
(P_1)^{-1}T_1(T_2)^{-1}P_2
\end{align*}
Since $T_1'P_1^{-1}T_1=
T_2'P_2^{-1}T_2$, we have $\text{Cov}(\zeta)=P_2(T_2')^{-1}T_2'(P_2)^{-1}T_2(T_2)^{-1}P_2=P_2$,
which means that $\zeta$ has the desired dynamic noise and boundary values.
Second, we show that assuming $\eqref{Step_2}$ holds, two models $\eqref{Model_1}$ and $\eqref{Model_2}$ generate the same sample path of the sequence. Substituting $\eqref{Model_1}$ into $\eqref{Step_2}$, we obtain $T_1'(P_1)^{-1}T_1 x = T_2'(P_2)^{-1}\zeta$. Then, $P_2(T_2')^{-1}(T_1'(P_1)^{-1}T_1) x = \zeta \label{1}$. Pre-multiplying both sides by $(T _2)^{-1}$, we have
\begin{align}
(T_2'(P_2)^{-1}T_2)^{-1}(T_1'(P_1)^{-1}T_1) x = (T_2)^{-1}\zeta \label{2}
\end{align}
Since $T_1'P_1^{-1}T_1=
T_2'P_2^{-1}T_2$, we get $T_2 x = \zeta$. Therefore, $\eqref{Model_2}$ and $\eqref{Model_1}$ are explicitly equivalent if $\eqref{Step_2}$ holds.
\end{proof}
The first step of Proposition \ref{Equivalent_Construction_Lemma}, $\eqref{Step_1}$, is to determine an equivalent model, and the second step, $\eqref{Step_2}$, makes the model explicitly equivalent.
By Proposition \ref{Equivalent_Construction_Lemma}, given a model, one can construct an explicitly equivalent model. Assume that two equivalent models generate the same sample path of the governed sequence. What is the relation between sample paths of their dynamic noises and boundary values? The next proposition answers this question.
\begin{proposition}\label{Equivalent_Relation_Proposition}
Let two equivalent forward, backward $CM_L$, $CM_F$, reciprocal, or Markov models be given as
\begin{align}
T_1x=\xi \label{Dynamic_1}\\
T_2x=\zeta \label{Dynamic_2}
\end{align}
where $\xi=[\xi_{0}' , \ldots ,\xi_{N}']'$ and $\zeta=[\zeta_{0}' , \ldots ,\zeta_{N}']'$ contain their dynamic noises and boundary values, with covariances $P_1$ and $P_2$, respectively, and the nonsingular matrices $T_1$ and $T_2$ are determined by the parameters of the corresponding models. If the two models generate the same sample path of the governed sequence, the relationship between sample paths of their dynamic noises and boundary values is as $\eqref{Step_2}$.
\end{proposition}
\begin{remark}
By $\eqref{Step_1}$, $\eqref{Step_2}$ is equivalent to
\begin{align}\label{Condition_Equivalent_2}
T_1^{-1}\xi = T_2^{-1}\zeta
\end{align}
\end{remark}
Although $\eqref{Condition_Equivalent_2}$ looks simpler, for the construction of explicitly equivalent models, $\eqref{Step_2}$ is preferred. This is explained as follows. It can be seen that the matrices $P_i$, $i=1,2$, in $\eqref{Step_2}$ corresponding to the forward or backward $CM_L$, $CM_F$, and Markov models are block-diagonal, and their inverses are easily calculated. Also, for the reciprocal model, no calculation is needed because we have $P=T$ \cite{Levy_Dynamic}. However, calculation of the inverse of $T_i$, $i=1,2$, in $\eqref{Condition_Equivalent_2}$ is not straightforward in general.
Note that Proposition \ref{Equivalent_Construction_Lemma} and Proposition \ref{Equivalent_Relation_Proposition} are not restricted to (forward/backward) $CM_L$, $CM_F$, reciprocal, and Markov models. The results work for other models satisfying the required conditions.
\section{Examples of Explicitly Sample-Equivalent Models}\label{Examples}
For illustration, two examples of explicitly sample-equivalent models obtained using the approach of Proposition \ref{Equivalent_Construction_Lemma} are presented.
\subsection{Forward and Backward Markov Models}
Given a forward Markov model $\eqref{Markov_Dynamic_Forward}$--$\eqref{M_BC}$ for $[x_k]$, by $\eqref{Step_1}$ parameters of a backward Markov model $\eqref{Markov_Dynamic_Backward}$--$\eqref{MB_BC}$ for $[x_k]$ are obtained in terms of those of the forward one as follows. For $k=2, 3, \ldots, N$,
\begin{align}
(M_0^B)^{-1}=&M_0^{-1}+M_{1,0}'M_1^{-1}M_{1,0}\label{MfbP_1}\\
M_{0,1}^B=&M_0^BM_{1,0}'M_1^{-1}\label{MfbP_2}\\
(M_{k-1}^B)^{-1}=&M_{k-1}^{-1}+M_{k,k-1}'M_{k}^{-1}M_{k,k-1}- \nonumber \\
&(M_{k-2,k-1}^B)'(M^B_{k-2})^{-1}M_{k-2,k-1}^B \label{MfbP_3}
\end{align}
\begin{align}
M^B_{k-1,k}=&M^B_{k-1}M_{k,k-1}'M_{k}^{-1} \label{MfbP_4}\\
(M^B_N)^{-1}=&M_{N}^{-1}-(M^B_{N-1,N})'(M^B_{N-1})^{-1}M^B_{N-1,N}\label{MfbP_5}
\end{align}
Then, by $\eqref{Step_2}$, the relationship between dynamic noises and boundary values of the two models is:
\begin{align}
(M^B_0)^{-1}e^{MB}_0=&M_0^{-1}e^M_0-M_{1,0}'M_1^{-1}e^M_1\label{Mfb_1}\\
(M^B_{k})^{-1}e^{MB}_{k}=&(M^B_{k-1,k})'(M^B_{k-1})^{-1}e^{MB}_{k-1} + M_{k}^{-1}e^M_{k} \nonumber\\
-&M_{k+1,k}'M_{k+1}^{-1}e^M_{k+1}, k \in [1,N-1] \label{Mfb_2}\\
(M^B_N)^{-1}e^{MB}_N=&(M^B_{N-1,N})'(M^B_{N-1})^{-1}e^{MB}_{N-1}+
M_{N}^{-1}e^M_{N}\label{Mfb_3}
\end{align}
By the above equations, given a backward model, one can also obtain the explicitly equivalent forward model. $\eqref{MfbP_1}$--$\eqref{MfbP_5}$ and $\eqref{Mfb_1}$--$\eqref{Mfb_3}$ give explicitly equivalent forward and backward models no matter if the state transition matrix is singular or nonsingular. Based on $\eqref{Mfb_1}$--$\eqref{Mfb_3}$, we can verify the required condition in the derivation of the two-filter smoother \cite{Wax_Kailath}--\cite{Alan_Wilskey}, for singular and nonsingular state transition matrices.
\subsection{Reciprocal $CM_L$ and Reciprocal Models}
Let a $CM_L$ model governing a reciprocal sequence $[x_k]$ be given. Taking the first step of Proposition \ref{Equivalent_Construction_Lemma}, parameters of the reciprocal model governing $[x_k]$ are obtained from parameters of the $CM_L$ model as follows.
\begin{align}
R^0_0&=G_0^{-1}+G_{1,0}'G_1^{-1}G_{1,0}\label{CML_1}\\
R^0_k&=G_k^{-1}+G_{k+1,k}'G_{k+1}^{-1}G_{k+1,k}\label{CML2}, k \in [1,N-2] \\
R^0_{N-1}&=G_{N-1}^{-1}\label{CML3}\\
R^0_{N}&=G_N^{-1}+\sum _{k=1}^{N-1} G_{k,N}'G_k^{-1}G_{k,N} + G_{0,N}'G_0^{-1}G_{0,N}\label{CML_2}\\
R^+_k&=G_{k+1,k}'G_{k+1}^{-1}, k \in [0,N-2] \label{CML5}\\
R^+_{N-1}&=G_{N-1}^{-1}G_{N-1,N}\label{CML6}\\
R^-_0&=G_0^{-1}G_{0,N}-G_{1,0}'G_1^{-1}G_{1,N}\label{CML_3}
\end{align}
Then, taking the second step of Proposition \ref{Equivalent_Construction_Lemma}, the relationship between dynamic noises and boundary values of the two models is as follows. For $\eqref{CML_Dynamic_Forward}$--$\eqref{CML_Forward_BC2}$, we have
\begin{align}
e^R_0=&G_0^{-1}e_0-G_{1,0}'G_1^{-1}e_1\label{LR2_1}\\
e^R_k=&G_k^{-1}e_k-G_{k+1,k}'G_{k+1}^{-1}e_{k+1}, k \in [1,N-2]\label{LR_2}\\
e^R_{N-1}=&G_{N-1}^{-1}e_{N-1}\label{LR_3}\\
e^R_N=&-\sum _{k=1}^{N-1}G_{k,N}'G_k^{-1}e_k+G_N^{-1}e_N-G_{0,N}'G_0^{-1}e_0\label{LR2_4}
\end{align}
By the above equations, given a reciprocal model, one can obtain the explicitly equivalent reciprocal $CM_L$ model. This is important because the reciprocal $CM_L$ model can be more easily applicable than the reciprocal model.
\section{Summary and Conclusions}\label{Summary}
Relationships between dynamic models governing different classes of Gaussian conditionally Markov (CM) sequences (including Markov, reciprocal, and so-called $CM_L$ and $CM_F$ sequences) have been studied. One CM sequence can obey different models. Given one, it is desirable to obtain other models for the same sequence. Two models are called \textit{equivalent} if they govern the same random sequence (i.e., their sequences have the same distribution). In some problems it is not sufficient to have only equivalent models---we need to know the relationship between dynamic noises and boundary values for the equivalent models to have the same sample path of the governed sequence. Two equivalent models are \textit{explicitly sample-equivalent} if such a relationship is given.
A unified approach has been presented, such that given a forward, backward $CM_L$, $CM_F$, reciprocal, or Markov model, any explicitly equivalent such model can be obtained. This approach does not require any assumptions (e.g., nonsingularity) about the matrix coefficients of the models. So, unlike \cite{Verghese} (which is restricted to nonsingular state transition matrices), the proposed approach can be used to obtain a backward Markov model explicitly equivalent to a forward Markov model with either a singular or nonsingular state transition matrix.
\subsubsection*{Acknowledgments}
Research was supported by NASA Phase03-06 through grant NNX13AD29A.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,254 |
Q: Regular Expression such that Should contain atleast one uppercase , one lower case , one numeric and one special character - either $,!,@ or # only Right now I am able to write ^.*(?=.{6,})(?=.*\d)(?=.*[a-z])(?=.*[A-Z])(?=.*[$!@#]).*$ but the problem is, it accepts all other special character also I just want it to accept $!@#
A: You can use:
^(?=.*\d)(?=.*[a-z])(?=.*[A-Z])(?=.*[$!@#])[$!@#a-zA-Z\d]{6,}$
RegEx Demo
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,372 |
Q: Problem with deployment of Spring Boot Web app: No errors but app still fails I am trying to run a Spring Boot app on Heroku. I have followed their tutorials and tried to deploy my app but for some unknown reason it keeps failing.
2019-11-08T12:23:50.484165+00:00 app[web.1]:
2019-11-08T12:23:50.484215+00:00 app[web.1]: . ____ _ __ _ _
2019-11-08T12:23:50.484401+00:00 app[web.1]: /\\ / ___'_ __ _ _(_)_ __ __ _ \ \ \ \
2019-11-08T12:23:50.484501+00:00 app[web.1]: ( ( )\___ | '_ | '_| | '_ \/ _` | \ \ \ \
2019-11-08T12:23:50.484593+00:00 app[web.1]: \\/ ___)| |_)| | | | | || (_| | ) ) ) )
2019-11-08T12:23:50.484680+00:00 app[web.1]: ' |____| .__|_| |_|_| |_\__, | / / / /
2019-11-08T12:23:50.484770+00:00 app[web.1]: =========|_|==============|___/=/_/_/_/
2019-11-08T12:23:50.486599+00:00 app[web.1]: :: Spring Boot :: (v2.2.1.RELEASE)
2019-11-08T12:23:50.486673+00:00 app[web.1]:
2019-11-08T12:23:50.816454+00:00 app[web.1]: 2019-11-08 12:23:50.809 INFO 4 --- [ main] P.p.ProgresstrackerappApplication : Starting ProgresstrackerappApplication v0.0.1-SNAPSHOT on 8f560338-e240-4b5d-afdf-be7d44992eb6 with PID 4 (/app/target/progresstrackerapp-0.0.1-SNAPSHOT.jar started by u28233 in /app)
2019-11-08T12:23:50.819527+00:00 app[web.1]: 2019-11-08 12:23:50.819 INFO 4 --- [ main] P.p.ProgresstrackerappApplication : No active profile set, falling back to default profiles: default
2019-11-08T12:23:53.772610+00:00 app[web.1]: 2019-11-08 12:23:53.772 INFO 4 --- [ main] o.s.b.w.embedded.tomcat.TomcatWebServer : Tomcat initialized with port(s): 8080 (http)
2019-11-08T12:23:53.791185+00:00 app[web.1]: 2019-11-08 12:23:53.790 INFO 4 --- [ main] o.apache.catalina.core.StandardService : Starting service [Tomcat]
2019-11-08T12:23:53.791563+00:00 app[web.1]: 2019-11-08 12:23:53.791 INFO 4 --- [ main] org.apache.catalina.core.StandardEngine : Starting Servlet engine: [Apache Tomcat/9.0.27]
2019-11-08T12:23:53.897317+00:00 app[web.1]: 2019-11-08 12:23:53.897 INFO 4 --- [ main] o.a.c.c.C.[Tomcat].[localhost].[/] : Initializing Spring embedded WebApplicationContext
2019-11-08T12:23:53.897452+00:00 app[web.1]: 2019-11-08 12:23:53.897 INFO 4 --- [ main] o.s.web.context.ContextLoader : Root WebApplicationContext: initialization completed in 2928 ms
2019-11-08T12:23:54.261713+00:00 app[web.1]: 2019-11-08 12:23:54.261 INFO 4 --- [ main] o.s.s.concurrent.ThreadPoolTaskExecutor : Initializing ExecutorService 'applicationTaskExecutor'
2019-11-08T12:23:54.413274+00:00 app[web.1]: 2019-11-08 12:23:54.412 INFO 4 --- [ main] o.s.b.a.w.s.WelcomePageHandlerMapping : Adding welcome page template: index
2019-11-08T12:23:54.709609+00:00 app[web.1]: 2019-11-08 12:23:54.709 INFO 4 --- [ main] o.s.b.w.embedded.tomcat.TomcatWebServer : Tomcat started on port(s): 8080 (http) with context path ''
2019-11-08T12:23:54.715140+00:00 app[web.1]: 2019-11-08 12:23:54.714 INFO 4 --- [ main] P.p.ProgresstrackerappApplication : Started ProgresstrackerappApplication in 5.801 seconds (JVM running for 6.938)
2019-11-08T12:24:00.589583+00:00 heroku[router]: at=error code=H20 desc="App boot timeout" method=GET path="/" host=progresstrackerapp.herokuapp.com request_id=68c7754f-2c39-4cf8-b207-426af344f40c fwd="xxx.xxx.xxx.xxx" dyno= connect= service= status=503 bytes= protocol=https
I edited out my IP from the text. Any idea what might cause the timeout?
EDIT 1:
Some things i did: I added the Procfile by myself in the root directory of the app since without it my app would crash
web java -jar target/progresstrackerapp-0.0.1-SNAPSHOT.jar
Also for the commands I did :
git add .
git commit -m "s1"
git push heroku master
Build of the app was fine and no problem appeared there
heroku ps:scale:1 // this resulted in the problem
EDIT 2:
I found the problem.It seems that I needed to specify an environment variable PATH. Heroku needs this variable to be able to deploy on it since it is the developer's job to include it. That is what fixed it for me.
A: Make sure you're binding to the correct port by passing -Dserver.port=$PORT to your Procfile command
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,485 |
Rituale Satanum è il primo album in studio del gruppo black metal finlandese Behexen, pubblicato nel 2000.
Tracce
Formazione
Hoath Torog - voce
Gargantum - chitarre
Lunatic - basso
Horns - batteria
Collegamenti esterni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,981 |
Иоаннис Теофилопулос (; , Лангадиа, Аркадия — , Лангадиа, Аркадия) — греческий моряк, участник Освободительной войны Греции 1821—1829 годов, чьё имя стоит в историографии рядом с именами капитанов брандеров Димитриоса Папаниколиса и Константина Канариса.
Биография
Необычным в биографии этого моряка является тот факт, что он родился и вырос в далёкой от морских традиций горной глубинке полуострова Пелопоннес, в селе Лангадиа, епархия Гортиния, ном Аркадия.
Нет информации когда и как он стал моряком, но к началу Греческой революции 1821 года, Теофилопулос был среди экипажей острова Псара и имел кличку Цакалос ( — шакал. Здесь имеет значение твёрдолобый, упрямый) и Каравояннос (греч.Καραβόγιαννος — свободный перевод: «Яннис не сходящий на берег»). Теофилопулос был рулевым первого брандера войны, который под командованием Д. Папаниколиса сжёг 27 мая 1821 года в бухте Эрессос, остров Лесбос турецкий фрегат, что положило начало эпопее греческих брандеров
. Мужество пелопоннесца Теофилопулоса было отмечено заявлением парламента острова Псара, от 6 июня 1821 года.
Теофилопулос принял участие во всех операциях псариотов вдоль малоазийского побережья до входа в Дарданеллы, у островов Тенедос и Самос, у Кассандра (полуостров).
В феврале 1822 года Теофилопулос принял участие в морском сражении без победителей, у города Патры, где флотом псариотов командовал Николис Апостолис.
В марте — мае того же года Теофилопулос принял участие в двух походах к острову Хиос.
Канарис говорил что Теофилопулос служил безвозмездно и уходил от раздоров. После резни на Хиосе (см. Хиосская резня), Канарис, задавшись целью отомстить туркам, призвал Теофилопулоса рулевым на свой брандер. 6 июня 1822 года брандер Канариса, с рулевым Теофилопулосом, сжёг турецкий флагман на рейде Хиоса. Роль Теофилопулоса в этом деле имела решающаее значение. Последними с брандера спрыгнули Теофилопулос и Канарис.
В августе 1822 года Теофилопулос вернулся на Пелопоннес и продолжил своё участие в войне на суше, воюя под командованием военачальников К. Делияннис и Теодороса Колокотрониса.
После освобождения Теофилопулос был назначен начальником милиции в городе Триполи (1830).
В 1865 году, принимая участие в комитете переписи живых участников Освободительной войны, Теофилопулос подписывает сертификаты как майор.
Умер Теофилопулос в глубокой бедности, в 1885 году, в своём родном селе.
Память
На настенной росписи в зале Венизелос, Элефтериос греческого парламента, группа немецких художников в 1836 году, основываясь на работе немецкого скульптора Ludwig Michael Schwanthaler, изобразила Теофилопулоса знаменосцем рядом с Канарисом.
В стихотворении Теодороса Фалез-Колокотрониса Теофилопулос описывается следующим образом:
Был горным львом он
дельфином в море
при имени его тряслись
и турки и в Алжире.
В 1976 году ВМФ Греции назвал его именем одно из двух, построенных в Перама, Пирей, судов обслуживания маяков -" Каравояннос " (А-479) (греч. ΠΦΑ ΚΑΡΑΒΟΓΙΑΝΝΟΣ (Α-479)).
В селе Лангадиа установлен бюст Теофилопулоса, работы греческого скульптора Ясона Пападимитриу
Примечания
Источники
Δημ. Φωτιάδη, Κανάρης, εκδ. Δωρικός
Ο Μωραΐτης πυρπολητής του 1821, Αγγέλα Κυριακοπούλου, Ε-Ιστορικά, Ελευθεροτυπίας (23.3.2000)
Участники Греческой революции. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,840 |
A MUSICAL ABOUT STAR WARS Returns For Two Weeks Off-Broadway
Returning to the production are the shows original stars and creators; Taylor Crousore, Scott Richard Foster and Emily McNamara.
by A.A. Cristi
BroadwayWorld.com Dec. 2, 2021
Off-Broadway's hilariously irreverent musical comedy; A Musical About Star Wars, which was forced to close in 2020, is excited to announce they are returning for a strictly-limited two week engagement at The American Theatre of Actors. With performances beginning Friday, Dec 17th the musical comedy celebrates the blockbuster film franchise, nerd culture, cosplay, and comic-con in a show-within-a-show that is a little meta and hella rad.
A long time ago, in a galaxy far far away...a Staten Island Blockbuster Video to be exact...Two Star Wars fanatics, Scott and Taylor, wrote the most epic Star Wars musical ever. On their endless quest to perform it at Comic Con - a restraining order filed by actor Warwick Davis stands in their way - they have opted to perform it on the glamorous off-Broadway stages instead! However, dark forces are looming over their production, and her name is Emily. "Do Scott and Taylor prevail? Does the show make it to Comic Con? The answers are less important than the fun time you'll have getting there! May the FUN be with you!" - Theasy.com
The show opened on May The Fourth in 2019 at Theatre Row and within weeks transferred to the larger (and aptly named) St. Luke's Theatre. The show closed prematurely in March of 2020 due to a small global pandemic that you might have heard about.
Returning to the production are the shows original stars and creators; Taylor Crousore, Scott Richard Foster and Emily McNamara. Created by Tom D'Angora and written by D'Angora, Crousore, and Foster, the musical features an original score and lyrics by Billy Recce. The production is directed by Tom and Michael D'Angora, with choreography by Alex Ringler, production design by Brendan McCann, costumes by McCann and William Bailey, lighting design by Erik Petersen, music direction by Ed Goldscheider, and stage managed by Brent Jones. The original cast album is available through Broadway Records.
A Musical About Star Wars plays at The American Theatre of Actors located at 314 W 54th St between 8-9th Ave. The show has a varying schedule. For more information and to purchase tickets visit www.AMusicalAboutStarWars.com.
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Danny Wolohan to Assume the Role of Samuel Byck in ASSASSINS
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Off-Broadway SHOWS
Sign-Up for Off-Broadway News | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,045 |
Description: Coloriage Barbie Jeux De Coloriage De Barbie Inspirant Coloriage Voiture Beau Jeux De from the above resolutions which is part of the Coloriage. Download this image for free in HD resolution the choice "download button" below. If you do not find the exact resolution you are looking for, then go for a native or higher resolution.
49 Photos of "Coloriage Barbie Jeux De Coloriage De Barbie Inspirant Coloriage Voiture Beau Jeux De"
Related Posts of "Coloriage Barbie Jeux De Coloriage De Barbie Inspirant Coloriage Voiture Beau Jeux De" | {
"redpajama_set_name": "RedPajamaC4"
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Our Supreme Court Brief
Dear Patriot,
On February 13, 2012, the Project Liberty Amicus brief was filed on behalf of 295 individual citizens asserting their Constitutional rights seeking a declaration from the United States Supreme Court that Obamacare is unconstitutional and should be declared void in its entirety.
Thank you for your steadfast support, your courage and your love of liberty.
Here is a copy of your brief which was filed on your behalf by The Justice foundation and a wonderful team of cooperating attorneys. I want to particularly thank Steven Fitschen of the National Legal Foundation, and Kathleen Cassidy Goodman for their assistance on the brief. Special thanks to Professor Steven Presser of Northwestern University and also other lawyers who contributed significantly to the writing and research were Eric Welter and John Lindsay Bower. They have and deserve our gratitude. Eric, Lindsay, and Kathleen are Distinguished Fellows of The Justice Foundation and have worked on other projects as well.
The case will now be argued before the Supreme Court for three days, for a total of six hours of oral argument, in March 2012. This amount of oral argument is unprecedented in modern times, although the Supreme Court used to spend a whole day on important cases and sometimes more.
Our brief agreed with the State Attorney General's and the National Federation of Independent Business that the Act was unconstitutional on Commerce Clause grounds. However, we did not spend much time on this, since numerous other parties had briefed that issue in great detail. Such a brief on our part would have been merely cumulative, a "me, too" brief. The unique aspect of our brief, and which is a contribution that either we alone or few were able to make, but which was appreciated by other parties, is that we added the additional claim that if the Act was not struck down on Commerce Clause grounds in its entirety, then the religious liberty of individuals who oppose abortion on religious and moral grounds would be violated.
The timing of President Barak Obama and Kathleen Sibelius' directive forcing Catholics and other persons of faith to provide insurance that covers abortion pills, and sterilization, against their conscience, greatly benefits our brief. I believe that this vast overreaching by the federal government will encourage the conservative members of the Court to strike down the entire healthcare law. If the Act is overturned, then the regulations fall, since they were only authorized to implement Obamacare.
The Appendix has your names. We did not use last names because so many people indicated they did not want to use their full names. In addition, we used some people's quotes from their contracts in the early pages of the brief to show the interest of Amici. We could not show everyone's comments because of page limitation.
Please click here to view our complete press release.
It has been an honor to serve you. The only thing that can be done now is prayer for the Supreme Court to do the right thing. Pray for our nation that the God who gave us liberty will have mercy on us.
Allan E. Parker | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,578 |
package io.requery.android.sqlite;
import android.database.Cursor;
import java.io.ByteArrayInputStream;
import java.io.InputStream;
import java.io.Reader;
import java.io.StringReader;
import java.math.BigDecimal;
import java.net.MalformedURLException;
import java.net.URL;
import java.sql.Array;
import java.sql.Blob;
import java.sql.Clob;
import java.sql.Date;
import java.sql.NClob;
import java.sql.Ref;
import java.sql.ResultSet;
import java.sql.ResultSetMetaData;
import java.sql.RowId;
import java.sql.SQLDataException;
import java.sql.SQLException;
import java.sql.SQLFeatureNotSupportedException;
import java.sql.SQLWarning;
import java.sql.SQLXML;
import java.sql.Statement;
import java.sql.Time;
import java.sql.Timestamp;
import java.sql.Types;
import java.util.Calendar;
import java.util.Map;
/**
* {@link ResultSet} implementation using Android's {@link Cursor} interface.
*
* @author Nikhil Purushe
*/
public class CursorResultSet extends NonUpdateableResultSet implements ResultSetMetaData {
private final Statement statement;
private final Cursor cursor;
private final boolean closeCursor;
private int lastColumnIndex;
public CursorResultSet(Statement statement, Cursor cursor, boolean closeCursor) {
if(cursor == null) {
throw new IllegalArgumentException("null cursor");
}
this.statement = statement;
this.cursor = cursor;
this.closeCursor = closeCursor;
cursor.moveToPosition(-1);
}
@Override
public boolean absolute(int row) throws SQLException {
return cursor.moveToPosition(row - 1);
}
@Override
public void afterLast() throws SQLException {
cursor.moveToLast();
cursor.moveToNext();
}
@Override
public void beforeFirst() throws SQLException {
cursor.moveToPosition(-1);
}
@Override
public void cancelRowUpdates() throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public void clearWarnings() throws SQLException {
}
@Override
public void close() throws SQLException {
if (closeCursor) {
cursor.close();
}
}
@Override
public void deleteRow() throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public int findColumn(String columnName) throws SQLException {
int index = cursor.getColumnIndex(columnName);
if (index == -1) {
throw new SQLDataException("no column " + columnName);
}
return index + 1;
}
@Override
public boolean first() throws SQLException {
return cursor.moveToFirst();
}
@Override
public Array getArray(int columnIndex) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public Array getArray(String colName) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public InputStream getAsciiStream(int columnIndex) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public InputStream getAsciiStream(String columnName) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public BigDecimal getBigDecimal(int columnIndex) throws SQLException {
String value = getString(columnIndex);
return value == null ? null : new BigDecimal(value);
}
@Override
public BigDecimal getBigDecimal(int columnIndex, int scale) throws SQLException {
String value = getString(columnIndex);
BigDecimal result = value == null ? null : new BigDecimal(value);
if(result != null) {
result = result.setScale(scale, BigDecimal.ROUND_DOWN);
}
return result;
}
@Override
public BigDecimal getBigDecimal(String columnName) throws SQLException {
return getBigDecimal(findColumn(columnName));
}
@Override
public BigDecimal getBigDecimal(String columnName, int scale) throws SQLException {
return getBigDecimal(findColumn(columnName));
}
@Override
public InputStream getBinaryStream(int columnIndex) throws SQLException {
return new ByteArrayInputStream(getBytes(columnIndex));
}
@Override
public InputStream getBinaryStream(String columnName) throws SQLException {
return getBinaryStream(findColumn(columnName));
}
@Override
public Blob getBlob(int columnIndex) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public Blob getBlob(String columnName) throws SQLException {
return getBlob(findColumn(columnName));
}
@Override
public boolean getBoolean(int columnIndex) throws SQLException {
return getInt(columnIndex) > 0;
}
@Override
public boolean getBoolean(String columnName) throws SQLException {
return getBoolean(findColumn(columnName));
}
@Override
public byte getByte(int columnIndex) throws SQLException {
return (byte) getShort(columnIndex);
}
@Override
public byte getByte(String columnName) throws SQLException {
return getByte(findColumn(columnName));
}
@Override
public byte[] getBytes(int columnIndex) throws SQLException {
lastColumnIndex = columnIndex;
return cursor.getBlob(columnIndex - 1);
}
@Override
public byte[] getBytes(String columnName) throws SQLException {
return getBytes(findColumn(columnName));
}
@Override
public Reader getCharacterStream(int columnIndex) throws SQLException {
return new StringReader(getString(columnIndex));
}
@Override
public Reader getCharacterStream(String columnName) throws SQLException {
return new StringReader(getString(columnName));
}
@Override
public Clob getClob(int columnIndex) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public Clob getClob(String colName) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public int getConcurrency() throws SQLException {
return CONCUR_READ_ONLY;
}
@Override
public String getCursorName() throws SQLException {
return null;
}
@Override
public Date getDate(int columnIndex) throws SQLException {
lastColumnIndex = columnIndex;
if(cursor.isNull(columnIndex - 1)) {
return null;
}
return new Date(cursor.getLong(columnIndex - 1));
}
@Override
public Date getDate(int columnIndex, Calendar cal) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public Date getDate(String columnName) throws SQLException {
return getDate(findColumn(columnName));
}
@Override
public Date getDate(String columnName, Calendar cal) throws SQLException {
throw new SQLFeatureNotSupportedException();
}
@Override
public double getDouble(int columnIndex) throws SQLException {
lastColumnIndex = columnIndex;
return cursor.getDouble(columnIndex - 1);
}
@Override
public double getDouble(String columnName) throws SQLException {
return getDouble(findColumn(columnName));
}
@Override
public int getFetchDirection() throws SQLException {
return FETCH_FORWARD;
}
@Override
public int getFetchSize() throws SQLException {
return 0;
}
@Override
public float getFloat(int columnIndex) throws SQLException {
lastColumnIndex = columnIndex;
return cursor.getFloat(columnIndex - 1);
}
@Override
public float getFloat(String columnName) throws SQLException {
return getFloat(findColumn(columnName));
}
@Override
public int getInt(int columnIndex) throws SQLException {
lastColumnIndex = columnIndex;
return cursor.getInt(columnIndex - 1);
}
@Override
public int getInt(String columnName) throws SQLException {
return getInt(findColumn(columnName));
}
@Override
public long getLong(int columnIndex) throws SQLException {
lastColumnIndex = columnIndex;
return cursor.getLong(columnIndex - 1);
}
@Override
public long getLong(String columnName) throws SQLException {
return getLong(findColumn(columnName));
}
@Override
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{"url":"https:\/\/puzzling.stackexchange.com\/questions\/59550\/the-three-rules","text":"# The three rules\n\nThe black rule says: The solution is easy, but not \"easy\".\nThe red rule says: The solution is simple, but not \"simple\".\nThe golden rule says: The solution is straightforward, but not \"straightforward\".\n\nWhat is the solution?\n\nAll three of those words, Easy, Simple and Straightfoward, translate to \"einfach\" in German (The German flag being described by the colored rules), which I presume is the answer. Also, not sure if intentional, but a rule can be a type of line (see http:\/\/printwiki.org\/Rule_Line or https:\/\/en.wikipedia.org\/wiki\/Ruled_paper), potentially meaning the \"rules\" are lines on the flag.\n\n\u2022 Very good. That is indeed the correct answer. \u2013\u00a0infinitezero Jan 19 '18 at 21:51\n\nThe German flag is composed of the colors black, red and gold.\n\nIf that is the solution, it's indeed straightforward, easy and simple, but I think there should be more to it.\n\n\u2022 This is a nice hint. \u2013\u00a0infinitezero Jan 19 '18 at 22:08\n\nThe answer's already posted, but given the form of the question, I have to wonder if an alternate form of the solution is\n\n\"the solution\".\n\n\u2022 I don't see why. Can you please elaborate? \u2013\u00a0infinitezero Jan 20 '18 at 10:42","date":"2019-09-22 11:09:20","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\": 0.9106526970863342, \"perplexity\": 1028.7036031642433}, \"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-2019-39\/segments\/1568514575484.57\/warc\/CC-MAIN-20190922094320-20190922120320-00067.warc.gz\"}"} | null | null |
\chapter{Manuscript IV: Recurrent Activity from Active Asteroid (248370) 2005~QN173: A Main-belt Comet}
\chaptermark{Recurrent Activity from Active Asteroid (248370) 2005~QN173}
\label{chap:2005QN173}
\acresetall
Colin Orion Chandler\footnote{\label{QN:nau}Department of Astronomy and Planetary Science, Northern Arizona University, PO Box 6010, Flagstaff, AZ 86011, USA}, Chadwick A. Trujillo$^\mathrm{\ref{QN:nau}}$, Henry H. Hsieh\footnote{Planetary Science Institute, 1700 East Fort Lowell Rd., Suite 106, Tucson, AZ 85719, USA, Institute of Astronomy and Astrophysics, Academia Sinica, P.O.\ Box 23-141, Taipei 10617, Taiwan}
\textit{This is the Accepted Manuscript version of an article accepted for publication in Astrophysical Journal Letters. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at }\url{https://iopscience.iop.org/article/10.3847/2041-8213/ac365b}\textit{.}
\doublespacing
\section{Abstract}
\label{QN:Abstract}
We present archival observations of main-belt asteroid (248370)~2005~QN$_{173}$ (also designated 433P) that demonstrate this recently discovered active asteroid (a body with a dynamically asteroidal orbit displaying a tail or coma) has had at least one additional apparition of activity near perihelion during a prior orbit. We discovered evidence of this second activity epoch in an image captured 2016 July 22 with the DECam on the 4~m Blanco telescope at the Cerro Tololo Inter-American Observatory in Chile. As of this writing, (248370)~2005~QN$_{173}$ is just the 8th active asteroid demonstrated to undergo recurrent activity near perihelion. Our analyses demonstrate (248370)~2005~QN$_{173}$ is likely a member of the active asteroid subset known as main-belt comets, a group of objects that orbit in the main asteroid belt that exhibit activity that is specifically driven by sublimation. We implement an activity detection technique, \textit{wedge photometry}, that has the potential to detect tails in images of solar system objects and quantify their agreement with computed antisolar and antimotion vectors normally associated with observed tail directions. We present a catalog and an image gallery of archival observations. The object will soon become unobservable as it passes behind the Sun as seen from Earth, and when it again becomes visible (late 2022) it will be farther than 3~au from the Sun. Our findings suggest (248370)~2005~QN$_{173}$ is most active interior to 2.7~au (0.3~au from perihelion), so we encourage the community to observe and study this special object before 2021 December.
\section{Introduction}
\label{QN:introduction}
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth]{qnFiles/finalPlot_r5to60_dT1_2_3_4_5_6_7_8_9_10_im.png}
\caption{The $126''\times126''$ thumbnail image (left) shows (248370)~2005~QN$_{173}$ (green dashed arrow) at center with a tail (white arrows) oriented towards 5 o'clock. This 89~s $z$-band exposure captured with the DECam is the only image in which we could unambiguously identify activity. We conducted wedge photometry (right) that shows the tail orientation is $251.3^\circ\pm1.4^\circ$ (blue star), in close agreement with the $251.6^\circ$ antisolar angle (yellow $\odot$) and the $251.7^\circ$ antimotion vector (red $v$) as computed by JPL Horizons. The plot shows counts radially outward from the the object center at (0,0).}
\label{QN:fig:wedgephot}
\end{figure*}
Active asteroids are objects that are dynamically asteroidal but that display comet-like activity such as a tail or coma \citep{hsiehActiveAsteroidsMystery2006}. Activity may be caused by mechanisms unrelated to volatiles (e.g., impact, rotational disruption) or by sublimation as is typically the case with comets. Sublimation driven active objects provide key insights into the present-day volatile distribution in our solar system, as well as clues about the origins of those volatiles and how they arrived on Earth \citep{hsiehPopulationCometsMain2006}. These objects have been persistently difficult to study because of the small numbers detected to date: fewer than 30 active asteroids, of which roughly half are thought to exhibit sublimation driven activity; see \citealt{chandlerSAFARISearchingAsteroids2018} for a summary.
When the aforementioned sublimation driven activity is connected with a main-belt asteroid, the object is classified as a main-belt comet (MBC). MBCs are often characterized by activity near perihelion and the absence of activity elsewhere in the orbit \citep{hsiehMainbeltCometsPanSTARRS12015,agarwalBinaryMainbeltComet2017,hsieh2016ReactivationsMainbelt2018}, suggesting that the primary activity mechanism is sublimation of volatiles such as water ice \citep{snodgrassMainBeltComets2017}.
By contrast, stochastic events like impacts may result in comet-like activity but, in such cases, the appearance of activity is expected to cease once the material dissipates. Roughly 60\% of known active asteroids have been observed to display activity during only a single apparition \citep{chandlerSAFARISearchingAsteroids2018}.
Asteroid (7968), now comet 133P/Elst-Pizarro, was the first active main-belt asteroid to be discovered. While it was unclear at the time whether the activity was sublimation driven \citep{boehnhardtComet1996N21996,boehnhardtImpactInducedActivityAsteroidComet1998} or due to a one-time event \citep{tothImpactgeneratedActivityPeriod2000}, subsequent apparitions showing activity indicated sublimation was the cause \citep{hsiehStrangeCase133P2004,hsiehReturnActivityMainbelt2010}. This example illustrates the importance of detecting additional activity epochs
Asteroid (248370)~2005~QN$_{173}$ is a 3.2$\pm$0.4~km diameter \citep{hsiehPhysicalCharacterizationMainbelt2021} outer main-belt asteroid ($a$=3.075~au, $e$=0.226, $i$=$0.067^\circ$) that has a 5.37~yr orbit that ranges from a perihelion distance of $q$=2.374~au to an aphelion distance of $Q$=3.761~au. The object first drew particular attention when it was reported as active on 2021 July 9 \citep{fitzsimmons2483702005QN1732021}. Subsequently, Zwicky Transient Facility (ZTF) data were used to help constrain the activity onset to between 2020 July 10 and 2021 June 11 \citep{kelley2483702005QN2021}.
We set out to locate archival astronomical images of (248370)~2005~QN$_{173}$ in order to characterize prior activity. We made use of solar system object thumbnails (small image cutouts like Figure \ref{QN:fig:wedgephot}) derived from publicly available archival data. We previously demonstrated how our data sources, such as the Dark Energy Camera (DECam), are well suited to discovering and characterizing active objects \citep{chandlerSAFARISearchingAsteroids2018,chandlerSixYearsSustained2019,chandlerCometaryActivityDiscovered2020a}.
Here we report activity of (248370)~2005~QN$_{173}$ on 2016 July 22 \citep{chandler2483702005QN2021a}, an apparition prior to the 2021 outburst. We describe the process by which the activity was identified and examine the implications of this discovery.
\section{Second Activity Epoch}
\label{QN:sec:secondActivityEpoch}
In order to find an additional activity epoch for (248370)~2005~QN$_{173}$, we searched, assessed, and analyzed publicly available archival image data, building upon the methods of \cite{chandlerSAFARISearchingAsteroids2018,chandlerSixYearsSustained2019,chandlerCometaryActivityDiscovered2020a}.
\subsection{Data Acquisition}
\label{QN:subsec:datamining}
To locate archival images of (248370)~2005~QN$_{173}$, we queried our own database of publicly available observation metadata \citep[see][]{chandlerSAFARISearchingAsteroids2018}. This database, which updates daily, includes observing details such as sky coordinates, exposure time, and filter selection. Additionally, we searched Palomar Transient Factory (PTF) and ZTF data through 2021 August 31 through online search tools (listed in Appendix \ref{QN:sec:equipQuickRef}) as well as a ZTF Alert Stream search and retrieval tool we created for this purpose. All instruments and data sources we made use of are listed in Appendix \ref{QN:sec:equipQuickRef}, and we note that some data were found or retrieved via more than one pathway.
\clearpage
\pagestyle{empty}
\atxy{\dimexpr1in}{.5\paperheight}{\rotatebox[origin=center]{270}{\thepage}}
\begin{sidewaystable}[h]
\centering
\footnotesize
\caption{(248370)~2005~QN$_{173}$ Observations}
\begin{tabular}{cclcrcccrrccl}
Image\footnote{Label in image gallery figures.} & Obs. Date\footnote{UT observing date in year-month-day format.} & Source & N\footnote{Number of images.} & Exp. [s]\footnote{Exposure time for each image.} & Filter & $V$\footnote{Apparent $V$-band magnitude (Horizons).} & r [au]\footnote{Heliocentric distance.} & STO [$^\circ$]\footnote{Sun--target--observer angle.} & $\nu$ [$^\circ$]\footnote{True anomaly.} & \%$_{Q\rightarrow q}$\footnote{Percentage to perihelion $q$ from aphelion $Q$.} & Act?\footnote{Activity observed.} & Archive\\
\hline
a & 2004-07-08 & MegaPrime & 3 & 180 & \textit{i } & 20.7 & 2.74 & 17.6 & 287.5 & 73\% & N & CADC,* \\
b & 2005-06-08 & SuprimeCam & 3 & 60 & \textit{W-J-VR} & 21.0 & 2.42 & 23.8 & 18.6 & 96\% & N & CADC,SMOKA \\
c & 2010-06-14 & Pan-STARRS1 & 2 & 30 & \textit{z } & 20.2 & 2.42 & 21.1 & 339.2 & 96\% & N & CADC \\
d & 2010-08-02 & Pan-STARRS1 & 1 & 45 & \textit{i } & 19.0 & 2.39 & 3.9 & 353.5 & 99\% & N & CADC \\
e & 2010-08-05 & Pan-STARRS1 & 1 & 40 & \textit{r } & 18.9 & 2.39 & 2.5 & 355.4 & 99\% & N & CADC \\
f & 2010-08-06 & Pan-STARRS1 & 1 & 43 & \textit{g } & 18.8 & 2.39 & 2.0 & 354.7 & 99\% & N & CADC \\
g & 2010-08-28 & PTF & 2 & 60 & \textit{r } & 19.2 & 2.39 & 8.4 & 1.2 & 99\% & N & IRSA/PTF \\
h & 2010-08-31 & Pan-STARRS1 & 2 & 45 & \textit{i } & 19.3 & 2.39 & 9.7 & 2.1 & 99\% & N & CADC \\
i & 2010-09-01 & PTF & 2 & 60 & \textit{r } & 19.3 & 2.39 & 10.1 & 2.3 & 99\% & N & IRSA/PTF \\
j & 2010-09-06 & Pan-STARRS1 & 2 & 43 & \textit{g } & 19.5 & 2.39 & 12.2 & 3.8 & 99\% & N & CADC \\
k & 2010-09-12 & Pan-STARRS1 & 2,2 & 43,40 & \textit{g,r } & 19.6 & 2.40 & 14.5 & 5.6 & 98\% & N & CADC \\
l & 2010-09-15 & PTF & 2 & 60 & \textit{r } & 19.7 & 2.39 & 15.5 & 6.5 & 99\% & N & IRSA/PTF \\
m & 2010-10-30 & Pan-STARRS1 & 2 & 30 & \textit{z } & 20.6 & 2.41 & 23.8 & 19.6 & 97\% & N & CADC \\
n & 2011-07-14 & Pan-STARRS1 & 1 & 40 & \textit{r } & 21.8 & 2.86 & 15.8 & 84.2 & 65\% & N & CADC \\
o & 2011-11-24 & Pan-STARRS1 & 2,2 & 43,40 & \textit{g,r } & 20.6 & 3.15 & 4.0 & 109.3 & 44\% & N & CADC \\
p & 2011-11-30 & Pan-STARRS1 & 2 & 45 & \textit{i } & 20.4 & 3.16 & 1.8 & 110.3 & 43\% & N & CADC \\
q & 2011-12-01 & Pan-STARRS1 & 2 & 43 & \textit{g } & 20.4 & 3.16 & 1.4 & 110.5 & 43\% & N & CADC \\
r & 2014-03-01 & OmegaCam & 5 & 360 & \textit{r } & 21.4 & 3.53 & 7.0 & 218.2 & 17\% & N & CADC,ESO \\
s & 2016-07-22 & DECam & 1 & 89 & \textit{z } & 21.2 & 2.59 & 22.7 & 56.5 & 84\% & Y & CADC,* \\
t & 2019-07-03 & DECam & 9 & 40 & \textit{VR } & 22.6 & 3.55 & 14.6 & 216.9 & 15\% & N & CADC,* \\
u & 2020-02-04 & DECam & 1 & 38 & \textit{r } & 21.6 & 3.16 & 18.1 & 248.9 & 43\% & N & CADC,* \\
v & 2020-02-10 & DECam & 1 & 199 & \textit{z } & 21.8 & 3.15 & 18.2 & 249.9 & 44\% & N & CADC,* \\
w & 2020-04-25 & ZTF & 1,1 & 30,30 & \textit{g,r } & 20.0 & 2.99 & 4.4 & 263.3 & 55\% & N & IRSA/ZTF \\
x & 2020-05-18 & ZTF & 1,1 & 30,30 & \textit{g,r } & 19.9 & 2.93 & 4.7 & 267.7 & 60\% & N & IRSA/ZTF \\
x & 2020-05-27 & ZTF & 1,1 & 30,30 & \textit{g,r } & 20.1 & 2.91 & 8.4 & 269.5 & 61\% & N & IRSA/ZTF \\
x & 2020-06-11 & ZTF & 1,3 & 30,30 & \textit{g,r } & 20.3 & 2.88 & 13.3 & 272.5 & 63\% & N & IRSA/ZTF \\
x & 2020-06-14 & ZTF & 1,1 & 30,30 & \textit{g,r } & 20.4 & 2.87 & 14.1 & 273.1 & 64\% & N & IRSA/ZTF \\
x & 2020-06-17 & ZTF & 2,2 & 30,30 & \textit{g,r } & 20.4 & 2.87 & 14.9 & 273.8 & 64\% & N & IRSA/ZTF \\
x & 2020-06-20 & ZTF & 2,2 & 30,30 & \textit{g,r } & 20.5 & 2.86 & 15.7 & 274.4 & 65\% & N & IRSA/ZTF \\
x & 2020-06-23 & ZTF & 2,1 & 30,30 & \textit{g,r } & 20.5 & 2.85 & 16.5 & 275.0 & 65\% & N & IRSA/ZTF \\
x & 2020-06-26 & ZTF & 4,1 & 30,30 & \textit{g,r } & 20.6 & 2.85 & 17.2 & 275.6 & 65\% & N & IRSA/ZTF\\
\hline
\end{tabular}
%
\footnotesize
\raggedright
Note \textit{W-J-VR} is a single wide-band filter. See Appendix \ref{QN:sec:equipQuickRef} for image source and archive details.
* indicates the data were obtained from our local repository.
Table entries with multiple comma-separated values contain groups of exposures taken with different filters.
\label{QN:tab:observations}
\end{sidewaystable}
\clearpage
We identified candidate images where (248370)~2005~QN$_{173}$ was expected to be within the field of view (FOV) based on observation times, pointing centers, and FOV sizes and orientations, downloaded associated data, and extracted image cutouts. We organized data by instrument and observation date and summarize observation details in Table \ref{QN:tab:observations}.
\begin{figure*}
\centering
\begin{tabular}{ccccc}
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\begin{overpic}[width=0.17\linewidth]{qnFiles/2005_QN173_2010-10-30_04.51.47.177000_rings.v3.skycell.1044.013.wrp.z.55499_20223_chip1_30arcsec_NuElnoArrows_overlaid.png_2stacked_mean_185pix_overlaid.png} \put (5,7) {\huge\color{green} \textbf{m}}\end{overpic} &
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\begin{overpic}[width=0.17\linewidth]{qnFiles/2005_QN173_2011-11-30_07.30.59.895000_rings.v3.skycell.1781.056.wrp.i.55895_31279_chip1_126arcsec_NuElnoArrows_overlaid.png_4stacked_mean_185pix_overlaid.png}\put (5,7) {\huge\color{green} \textbf{p}}\end{overpic} &
\begin{overpic}[width=0.17\linewidth]{qnFiles/2005_QN173_2011-12-01_11.39.11.507000_rings.v3.skycell.1781.057.wrp.g.55896_48515_chip1_30arcsec_NuElnoArrows_overlaid.png_2stacked_mean_185pix_overlaid.png}\put (5,7) {\huge\color{green} \textbf{q}}\end{overpic} &
\begin{overpic}[width=0.17\linewidth]{qnFiles/2005_QN173_2014-03-01_06.13.46.527000_OMEGA.2014-03-01T06.13.46.527_chip25-ESO_CCD_85_30arcsec_NuEl_19.4dRA_-7.7dDec_5stacked_sum_70pix_overlaid.png}\put (5,7) {\huge\color{green} \textbf{r}}\end{overpic} &
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\end{tabular}
\caption{Archival images of (248370)~2005~QN$_{173}$ with the best activity detection potential (i.e., sufficient depth and observing conditions) for (248370)~2005~QN$_{173}$. For all images, north is up, east is left, the FOV is $30''\times30''$, the antisolar (yellow -$\odot$) and antimotion (red -$v$) directions are shown with the origin at image center. See Appendix \ref{QN:sec:equipQuickRef} for instrument and archive details. Panel (s) is from the only thumbnail in which we could identify activity unambiguously (Figure \ref{QN:fig:wedgephot}). (a) 2004 July 8 MegaPrime 3$\times$180~s $i$ band. (b) 2005 June 8 SuprimeCam 3$\times$60~s \textit{W-J-VR} band. (c) 2010 June 14 PS1 2$\times$60~s $z$ band. (d) 2010 August 2 PS1 1$\times$45~s $i$ band. (e) 2010 August 5 PS1 1$\times$40~s $r$ band. (f) 2010 August 6 PS1 1$\times$43~s $g$ band. (g) 2010 August 28 PTF 2$\times$60~s $r$ band. (h) 2010 August 31 PS1 $\times$45~s $i$ band. (i) 2010 September 1 PTF 2$\times$60~s $r$ band. (j) 2010 September 6 PS1 2$\times$43~s $g$ band. (k) 2010 September 12 PS1 2$\times$40~s $r$ band + 2$\times$43~s $g$ band. (l) 2010 September 15 PTF 2$\times$60~s $r$ band. (m) 2010 October 30 PS1 2$\times$30~s $z$ band. (n) 2011 July 14 PS1 1$\times$40~s $r$ band. (o) 2011 November 24 PS1 2$\times$40~s $r$ band + 2$\times$43~s $g$ band. (p) 2011 November 30 PS1 2$\times$45~s $i$ band. (q) 2011 December 1 PS1 2$\times$43~s $g$ band. (r) 2014 January 31 OmegaCAM 5$\times$360~s $r$ band. (s) 2016 July 22 1$\times$89~s \textit{z} band. (t) 2019 July 3 DECam 9$\times$40~s \textit{VR} band. (u) 2020 February 4 DECam 1$\times$38~s \textit{r} band. (v) 2020 February 10 DECam 1$\times$199~s $z$ band. (w) 2020 April 25 ZTF 1$\times$30~s \textit{g} band + 1$\times$30~s \textit{r} band. (x) 2020 May 18, 27 + 2020 June 11, 14, 17, 20, 23, \& 26 ZTF 9$\times$30~s \textit{r} band + 12$\times$30~s \textit{g} band.}
\label{QN:fig:gallery}
\end{figure*}
We extracted eighty-one $480\times480$ pixel thumbnail images (such as the image shown in Figure \ref{QN:fig:wedgephot}) in which we could confidently identify (248370)~2005~QN$_{173}$ (Table \ref{QN:tab:observations}). We coadded images from the same instrument when observations were close enough in time for computed tail orientation to be in close agreement such that coaddition could enhance activity, if present. The thumbnails with the ``best activity detection potential'' -- meaning the images were judged to have observing conditions (e.g., seeing) and depth (i.e., magnitude limit) amenable to activity detection -- are shown in Figure \ref{QN:fig:gallery}.
To allow for uniform spatial comparisons and to magnify the region of interest around (248370)~2005~QN$_{173}$, all thumbnail images in Figure \ref{QN:fig:gallery}
are displayed with $30''\times30''$ fields of view.
\subsection{Image Assessment}
\label{QN:subsec:imageassessment}
We vetted each thumbnail by visually confirming (248370)~2005~QN$_{173}$ was visible. In cases where the object could not be readily identified, we employed our pipeline to produce comparison thumbnails derived from DECam data that showed the same region of sky, instrument, broadband filter, and exposure time, but from epochs when the object was not in the FOV. We made use of Gaia DR2 \citep{gaiacollaborationGaiaDataRelease2018} and Sloan Digital Sky Survey Release 9 (SDSS~DR-9) catalogs \citep{ahnNinthDataRelease2012} to visually validate World Coordinate System of images within the SAOImageDS9 Vizier \citep{ochsenbeinVizieRDatabaseAstronomical2000} catalog query system.
We next identified vetted thumbnails that were suitable for coaddition by clustering images based on instrument and date. For compatible image sets that included multiple broadband filters, we carried out coaddition among matching filters as well as combining all images. Finally, we visually examined the results and flagged images with potential activity.
We found a single image with clear evidence of activity (Figure \ref{QN:fig:wedgephot}) in an 89~s $z$-band exposure captured 2016 July 22 by Dustin Lang and Alistair Walker as part of the DECam Legacy Survey \citep[DECaLS;][]{deyOverviewDESILegacy2019}. This discovery makes (248370)~2005~QN$_{173}$ the ninth recurrently active main-belt asteroid to be identified to date. The other objects, 238P/Read, 259P/Garradd, 288P, 311P/PANSTARRS, 313P/Gibbs, 324P/La Sagra, (6478)~Gault, and (7968)~Elst-Pizarro, have all demonstrated recurrent activity near perihelion, with the exception of (6478)~Gault \citep{chandlerSixYearsSustained2019}.
We measured the tail length to be about $2\farcmin14$ ($2.4\times10^5$~km) in this image but a longer tail may well have been revealed with a longer exposure (see \citet{hsiehPhysicalCharacterizationMainbelt2021} for 2021 apparition tail measurement). Applying our wedge photometry technique (Section \ref{QN:subsec:wedgephotometry}), we produced a diagnostic plot (Figure \ref{QN:fig:wedgephot}) and measured a position angle on the sky of $251.3^\circ\pm1.4^\circ$ for the tail, in close agreement with the Horizons computed $251.6^\circ$ antisolar and $251.7^\circ$ antimotion vectors.
\subsection{Wedge Photometry Tail Tool}
\label{QN:subsec:wedgephotometry}
We crafted a new algorithm to (a) identify potentially active objects by detecting likely tails, and (b) quantify alignment between an observed tail and predicted antisolar and antimotion vectors, which are commonly associated with tail direction. Currently, the tool is designed to analyze single tails $<15^\circ$ in angular extent, though we plan to address multiple tails and comae in the future. The technique, which we refer to as \textit{wedge photometry}, sums all pixel values in a variable-width wedge bound between an inner and outer radius and identifies wedges containing excess flux relative to other wedges, if present. A similar approach was used in \cite{2011Icar..215..534S} but we have made improvements in angular resolution and algorithmic efficiency. Excess flux within a particular wedge may indicate the presence of a tail, and testing tail alignment with antisolar and antimotion vectors provides additional weight that a detected tail is real. Here we focused on quantifying tail orientation and position angle agreement.
To optimize the process, we convert Cartesian pixel coordinates ($x$,$y$) to polar coordinates ($r$,$\theta$) where the central thumbnail pixel is defined as (0,0). The resulting three-dimensional array has columns $r$, $\theta$, and $c$ (counts).
For a series of wedge sizes $\theta$ ($1^\circ$--$10^\circ$ in $1^\circ$ increments) we summed pixel values in annular segments spanning an angle $\pm\theta/2$ along a radial component $r$ from an inner bound, $r_0=5$ pixels, to a maximum of $r_\mathrm{max}=50$ pixels ($13''$ for our DECam data), as given by
\begin{equation}
c(\theta, \Delta\theta) = \sum_{\theta=-\Delta\theta/2}^{\theta=+\Delta\theta/2} \sum_{r=r_0}^{r=r_\mathrm{max}} c(r, \theta).
\end{equation}
\noindent We further optimize the procedure by selecting for the target a starting radius $r_0$ outside the FWHM, and choosing a maximum radius $_\mathrm{max}$ that allows for a wedge length long enough to ensure that all bins have sufficient counts to avoid necessitating resampling of any individual pixels. Thus, pixels are assigned to wedges based solely on their precise pixel center coordinate, and any fractional flux from a pixel that spans a wedge boundary is automatically assigned to the wedge containing the pixel center coordinate. We then compute the mean and standard deviation of the resulting counts for each $\theta$ to compare with the predicted antisolar and antimotion vectors.
We produce a polar plot (Figure \ref{QN:fig:wedgephot}) to aid assessing relative radial flux distribution. Most position angles have a $1\sigma$ of $\sim$200 counts. The tail is clearly identified by our algorithm at $> 7 \sigma$ for several $\Delta \theta$ wedge sizes.
\section{Main-belt Comet Classification}
\label{QN:sec:mainBeltComet}
Once we had identified a previous activity epoch we set out to determine if (248370)~2005~QN$_{173}$ could be an MBC.
\subsection{Prerequisites}
\label{QN:subsec:mbc}
For (248370)~2005~QN$_{173}$ to qualify as an MBC it must (1) be an active asteroid, (2) orbit within the Main Asteroid Belt, and (3) exhibit sublimation driven activity.
(1) To qualify as an active asteroid, a body must typically meet three criteria (see \citealt{jewittActiveAsteroids2012} for discussion): (i) A coma or tail must have been observed visually (as is the case in this work) or, potentially, through alternate means such as spectroscopy \citep[e.g.,][]{kuppersLocalizedSourcesWater2014,busarevNewCandidatesActive2018} or detecting magnetic field enhancements \citep[e.g.,][]{russellInterplanetaryMagneticField1984}. (ii) The semi-major axis $a$ must not be exterior to that of Jupiter ($a_\mathrm{J}\approx$ 5.2~au) as is the case for comets and active Centaurs \citep{jewittActiveCentaurs2009}; (248370)~2005~QN$_{173}$ has $a=$3.1~au. And (iii) the Tisserand parameter with respect to Jupiter, $T_\mathrm{J}$, must be greater than 3; this is because objects with $T_\mathrm{J}<3$ are canonically considered cometary and $T_\mathrm{J}>3$ asteroids \citep{vaghiOriginJupiterFamily1973,vaghiOrbitalEvolutionComets1973}.
$T_\mathrm{J}$ describes how an orbit is related to Jupiter by
\begin{equation}
T_\mathrm{J} = \frac{a_\mathrm{J}}{a} + 2 \sqrt{\frac{a\left(1-e^2\right)}{a_\mathrm{J}}}\cos\left(i\right).
\end{equation}
\noindent where $e$ is the eccentricity and $i$ is the orbital inclination. $T_\mathrm{J}$ for (248370)~2005~QN$_{173}$ is 3.192 and thus it qualifies as asteroidal. (248370)~2005~QN$_{173}$ properties are provided in Appendix \ref{QN:sec:ObjectData}, and are established with this criterion.
(2) (248370)~2005~QN$_{173}$ orbits between 2.4~au and 3.76~au and thus does not cross the orbits of either Mars or Jupiter. With a semi-major axis of 3.1~au, (248370)~2005~QN$_{173}$ is an outer main-belt asteroid orbiting between the Kirkwood gaps corresponding to the 7:8 and 2:1 mean motion resonances with Jupiter.
(3) Recurrent activity near perihelion is diagnostic of volatile sublimation as the most likely mechanism responsible for the observed activity \citep[e.g.,][]{hsiehOpticalDynamicalCharacterization2012}. However, other underlying causes of recurrent activity are known, so this point warrants further investigation.
\subsection{Activity Mechanism}
\label{QN:subsec:mechanism}
We demonstrated in Section \ref{QN:sec:secondActivityEpoch} that (248370)~2005~QN$_{173}$ has been active during at least two epochs. This helps rule out activity mechanisms such as \textit{impact events} (e.g., (596)~Scheila; \citealt{bodewitsCollisionalExcavationAsteroid2011,ishiguroObservationalEvidenceImpact2011,moreno596ScheilaOutburst2011}) that are only expected to produce one-time outbursts but which can expel dust and produce comet-like activity. Aside from \textit{temperature-correlated volatile sublimation} (which we examine further in Section~\ref{QN:subsec:tempestimation}) other mechanisms for producing recurrent activity have been proposed.
\textit{Rotational destabilization} causes dust to be flung from a body in a potentially multiepisodic manner, as may be the case for (6478)~Gault \citep{chandlerSixYearsSustained2019,kleynaSporadicActivity64782019}. Taxonomic classification can help diagnose rotational destabilization, as with $S$-type (6478)~Gault, because activity from desiccated asteroid classes is unlikely to be sublimation driven. As discussed in Section \ref{QN:subsec:tempestimation}, the taxonomic class of (248370)~2005~QN$_{173}$ is not yet known but it is likely a C-type asteroid. An accurate rotation period for (248370)~2005~QN$_{173}$ is currently unavailable, and as such, we can neither confirm nor rule out destabilization as a contributing factor to the observed activity at this time.
\textit{Rubbing binaries} is a hypothetical scenario whereby two merging asteroids repeatedly collide and eject material. Proposed as a possible mechanism for the activity of 311P/PANSTARRS \citep{hainautContinuedActivity20132014}, the rubbing binary scenario has yet to be confirmed for that object \citep{jewittNucleusActiveAsteroid2018} or any other. As of this writing, there is no evidence that (248370)~2005~QN$_{173}$ is a binary asteroid, and activity spans two epochs separated by 5 yr, so we would expect merging processes to have either finished or that the binary orbit would have stabilized (see \citealt{jewittNucleusActiveAsteroid2018} for additional discussion concerning dissipation timescales). Therefore we find it unlikely that rubbing causes the activity associated with (248370)~2005~QN$_{173}$.
Geminid meteor stream parent (3200)~Phaethon undergoes extreme temperature changes ($\sim$600~K) and peaks at 800~K to 1100~K, well above the 573~K serpentine-phyllosilicate decomposition threshold \citep{ohtsukaSolarRadiationHeatingEffects2009}. These temperatures likely induce \textit{thermal fracture} \citep{licandroNatureCometasteroidTransition2007,kasugaObservations1999YC2008} leading to mass shedding \citep{liRecurrentPerihelionActivity2013,huiResurrection3200Phaethon2017}.
Two mechanisms, thermal fracture and temperature-correlated volatile sublimation warrant further inquiry into the thermophysical properties of (248370)~2005~QN$_{173}$.
\subsection{Temperature Estimation}
\label{QN:subsec:tempestimation}
Estimating temperatures experienced by (248370)~2005~QN$_{173}$ aids us in understanding direct thermal effects (e.g., thermal fracture) as well as assessing long-term volatile survival, especially water. For these reasons, we computed temperatures for an airless body over the course of an orbit similar to that of (248370)~2005~QN$_{173}$.
Following \citet{hsiehMainbeltCometsPanSTARRS12015}, the energy balance equation for a gray body on which water ice sublimation is occurring is
\begin{equation}
{F_{\odot}\over r_h^2}(1-A) = \chi\left[{\varepsilon\sigma T_{eq}^4 + L f_D\dot m_{w}(T_{eq})}\right]
\label{QN:equation:sublim1}
\end{equation}
where $r_h$ is the object's heliocentric distance, $T_\mathrm{eq}$ is the equilibrium surface temperature, $F_{\odot}=1360$~W~m$^{-2}$ is the solar constant, $r_h$ is in au, $A=0.05$ is the assumed Bond albedo of the body, $\chi$ accounts for the distribution of solar heating over the object's surface, $\sigma$ is the Stefan--Boltzmann constant, and $\varepsilon=0.9$ is the assumed effective infrared emissivity, and $L=2.83$~MJ~kg$^{-1}$ is the latent heat of sublimation of water ice (which we approximate here as being independent of temperature), $f_D$ represents the reduction in sublimation efficiency caused by mantling, where $f_D=1$ in the absence of a mantle, and $\dot m_w$ is the water mass-loss rate due to sublimation of surface ice.
In this equation, $\chi=1$ corresponds to a flat slab facing the Sun, known as the subsolar approximation, and produces the maximum expected temperature for an object,
while $\chi=4$ applies to objects with fast rotation or low thermal inertia, known as the isothermal approximation, and produces the minimum expected temperature for an object.
Next, the sublimation rate of ice into a vacuum can be computed using
\begin{equation}
\dot m_{w} = P_v(T) \sqrt{\mu\over2\pi k T}
\label{QN:equation:sublim2}
\end{equation}
where $\mu=2.991\cdot 10^{-26}$~kg is the mass of one water molecule, and $k$ is the Boltzmann constant, and the equivalent ice recession rate, $\dot \ell_{i}$, corresponding to $\dot m_{w}$ is given by
$\dot \ell_{i} = \dot m_{w}/ \rho$,
where $\rho$ is the bulk density of the object.
Finally, the Clausius--Clapeyron relation,
\begin{equation}
P_v(T) = 611 \times \exp\left[{{\Delta H_\mathrm{subl}\over R_g}\left({{1\over 273.16} - {1\over T}}\right)}\right]
\label{QN:equation:sublim3}
\end{equation}
gives the vapor pressure of water, $P_v(T)$, in Pa, where $\Delta H_\mathrm{subl}=51.06$~MJ~kmol$^{-1}$ is the heat of sublimation for ice to vapor and $R_g=8314$~J~kmol$^{-1}$~K$^{-1}$ is the ideal gas constant.
Solving these three equations iteratively, one can calculate the equilibrium temperature of a gray body at a given heliocentric distance on which water ice sublimation is occurring.
In Figure~\ref{QN:fig:ActivityTimeline}, we plot the object's expected equilibrium temperature over several orbit cycles, as computed by solving the system of equations above. We plot temperatures computed using both $\chi=1$ and $\chi=4$ to show the full range of possible temperatures.
\begin{figure*}[ht]
\centering
\begin{tabular}{c}
\hspace{-4mm}\includegraphics[width=0.85\linewidth]{qnFiles/Activity_dOnly_noYears.pdf}\\
\hspace{9mm}\includegraphics[width=0.92\linewidth]{qnFiles/248370_2005_QN173_at807_noDates.pdf}\\
\hspace{-4mm}\includegraphics[width=0.845\linewidth]{qnFiles/Activity_yearMarkersOnly.pdf}\\
\hspace{-1mm}\includegraphics[width=0.845\linewidth]{qnFiles/newTforChis.pdf}
\end{tabular}
\caption{(248370)~2005~QN$_{173}$ heliocentric distance (top plot), observability timeline (middle plot) and temperature (bottom plot), beginning the year of our first archival data (2004) through 2022.
Top: triangles represent positive (filled red) and negative (unfilled blue) activity detections. Markers indicate when the object was inbound (downward pointing triangles) or outbound (upward pointing triangles). Table \ref{QN:tab:observations} lists observation details.
Middle: apparent $V$-band magnitude (solid green line) and our ``observability'' metric (yellow dashed line) that represents hours during a given UT observing date the object is above $> 15^\circ$ elevation while the Sun is below the horizon. Peaks in apparent magnitude coinciding with observability occur during opposition events, and observability troughs indicate solar conjunctions when (248370)~2005~QN$_{173}$ was only above the horizon during daylight. Perihelion (orange $q$) and aphelion (blue $Q$) events are indicated.
Bottom: temperature $T$~(K) by date for different $\chi$ values, where $\chi=1$ (top line) represents a ``flat slab'' and $\chi=4$ (bottom line) an isothermal body.
}
\label{QN:fig:ActivityTimeline}
\end{figure*}
Figure~\ref{QN:fig:ActivityTimeline} (top) shows two parameters key to observing (248370)~2005~QN$_{173}$ between 2015 and 2023: apparent $V$-band magnitude and ``observability,'' which we define as the number of hours an object remains above $15^\circ$ elevation during nighttime for a given UT observing date. This plot informs us of potential observational biases or geometric effects that may bias activity detection, such as preferential activity discovery during opposition events, as was the case with (6478)~Gault \citep{chandlerSixYearsSustained2019}.
Figure~\ref{QN:fig:ActivityTimeline} (bottom) illustrates (248370)~2005~QN$_{173}$'s heliocentric distance and temperature (as computed using Equations \ref{QN:equation:sublim1}-\ref{QN:equation:sublim3}) over time, plus dates of observed activity and images where no visible activity was conclusively identified. Throughout its entire orbit the surface of (248370)~2005~QN$_{173}$ is consistently warmer than 145~K, the temperature above which water ice is not expected to survive over timescales on the order of the age of the solar system \citep{schorghoferLifetimeIceMain2008,snodgrassMainBeltComets2017}.
However, it is possible for water ice to remain preserved over long (Gyr) timescales on small asteroids at depths as shallow as a few centimeters to 30~cm below the surface \citep{schorghoferLifetimeIceMain2008,prialnikCanIceSurvive2009}, where present-day activity may be triggered by meter-scale impactors that excavate subsurface ice. We note that water ice has been detected on the surface of main-belt asteroid (24) Themis \citep{campinsWaterIceOrganics2010,rivkinDetectionIceOrganics2010}, but the mechanism by which that water ice is able to persist on its surface -- likely requiring continual replenishment from subsurface volatile reservoirs -- is not well understood, and furthermore may not have the same effectiveness on kilometer-scale objects like (248370)~2005~QN$_{173}$ as it does on the 200~km diameter (24) Themis.
The surface temperature of (248370)~2005~QN$_{173}$ varies at most between 145~K and 190~K over its orbit (Figure \ref{QN:fig:ActivityTimeline}), far less than the 600~K temperature swings peaking at 800--1000~K described in Section \ref{QN:subsec:mechanism}. We consider it is unlikely that thermal fracture is the primary cause of (248370)~2005~QN$_{173}$'s activity.
\subsection{Nondetection of Activity}
\label{QN:subsec:nullresults}
The two known epochs of activity for (248370)~2005~QN$_{173}$ both occurred when the object was interior to a heliocentric distance of 2.7~au. However, (248370)~2005~QN$_{173}$ was observed in 2005 and 2010 when the object was interior to 2.7~au but no activity was detected. We believe the circumstances of these epochs preclude a definitive assessment of activity.
The 2010 Pan-STARRS1 data suffer from image artifacts that are significant enough to obscure activity that may have been present. The 2005 SuprimeCam observations should have been well suited to detecting activity as the 8~m Subaru telescope has a large aperture, exposure times (60~s) were sufficiently long, the \textit{W-J-VR} filter covered a broad wavelength range, and the object was well placed in the sky in terms of airmass/elevation during the observations. However, extinction varied significantly over the observing sequence as the summit log for that night\footnote{\url{https://smoka.nao.ac.jp/calendar/slog/2005/slog_20050608.txt}} indicated that conditions were windy with cirrus clouds and the differential image motion monitor measured significant seeing variation (roughly $0\farcsec8$ to $>3''$)\footnote{\url{https://smoka.nao.ac.jp/calendar/subaruseeing/20050608.gif}}, potentially contributing to a considerable reduction in our ability to detect activity. All sources in the field could be matched to SDSS~DR-9 stars but the faintest stars we were able match to the SDSS~DR-9 catalog were $r\approx21.3$, very similar to the JPL Horizons computed $V$=21.0 for (248370)~2005~QN$_{173}$. The Subaru Exposure Time Calculator estimates an equivalent $r$-band exposure would deliver a signal-to-noise ratio of 66 for a source of equivalent magnitude, but we estimate the images are at best $\sim$0.5 mag deeper than necessary to detect the object and thus we find it unlikely that activity would be detectable unless the object was undergoing a significant outburst at the time.
Although we cannot definitively rule out the presence of activity in 2005 or 2010 from these observations, another possibility is that a triggering event (e.g., impact, rotational destabilization) that started (248370)~2005~QN$_{173}$'s current activity occurred between 2005 June and 2016 July. This would explain the object's apparent inactivity and activity on each of those dates, respectively.
\subsection{Main-belt Comet Membership}
\label{QN:subsec:mbcmembership}
Given the above reasoning, we find it most likely that the activity associated with (248370)~2005~QN$_{173}$ is sublimation driven, in which case the object is an MBC. However, we emphasize that in order to rule out rotational destabilization as the root cause of the observed activity, additional work is needed. Moreover, confirmation of a third activity epoch would lend further evidence favoring sublimation as the primary agency of activity.
\section{Summary and Future Work}
\label{QN:sec:summary}
We harvested eighty-one images of (248370)~2005~QN$_{173}$ (also designated 433P) spanning thirty-one observing dates. We found clear evidence of a previous activity epoch on 2016 July 22. We provide a catalog of archival observations along with an image gallery. Making use of wedge photometry -- a novel tail detection and characterization tool we introduce in this Letter -- we measure tail orientation to be in close agreement with the antisolar and antimotion vectors computed by Horizons. We showed that (248370)~2005~QN$_{173}$ is a likely member of the MBCs, a group of active asteroids orbiting within the Main Asteroid Belt that are active due to volatile sublimation.
The current observing window for this object ends around 2021 December, and when it returns in late 2022 it will be over 3~au from the Sun and less likely to show activity. We did not find any images showing (248370)~2005~QN$_{173}$ active at beyond 3~au, so we call on observers to make use of the present activity apparition while it is still possible to do so. Continued monitoring to study the evolution of the tail's brightness, including surface brightness measurements, can lead to better characterization of ejected dust grain sizes and total mass loss during this apparition. Once activity subsides, time-series observations to measure a rotation period will be especially useful for diagnosing rotational breakup. Preliminary color measurements suggest (248370)~2005~QN$_{173}$ is a C-type asteroid \citep{hsiehPhysicalCharacterizationMainbelt2021}, but a robust taxonomic classification would help further solidify our assessment of the underlying activity mechanism.
\section{Acknowledgements}
\label{QN:sec:acknowledgements}
The authors thank the anonymous referee whose comments greatly improved the quality of this Letter.
We thank Dr.\ Mark Jesus Mendoza Magbanua (University of California San Francisco) for his frequent and timely feedback on the project.
The authors express their gratitude to
Prof. Mike Gowanlock (NAU),
Dr. Annika Gustafsson (NAU, Lowell Observatory, Southwest Research Institute),
Jay Kueny (Steward Observatory),
and the Trilling Research Group (NAU), all of whom provided invaluable insights which substantially enhanced this work. The unparalleled support provided by Monsoon cluster administrator Christopher Coffey (NAU) and the High Performance Computing Support team facilitated the scientific process.
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under grant No.\ (2018258765). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. C.O.C., H.H.H. and C.A.T. also acknowledge support from the NASA Solar System Observations program (grant 80NSSC19K0869).
Computational analyses were run on Northern Arizona University's Monsoon computing cluster, funded by Arizona's Technology and Research Initiative Fund. This work was made possible in part through the State of Arizona Technology and Research Initiative Program.
World Coordinate System (WCS) corrections facilitated by the \textit{Astrometry.net} software suite \citep{langAstrometryNetBlind2010}.
This research has made use of data and/or services provided by the International Astronomical Union's Minor Planet Center.
This research has made use of NASA's Astrophysics Data System.
This research has made use of The Institut de M\'ecanique C\'eleste et de Calcul des \'Eph\'em\'erides (IMCCE) SkyBoT Virtual Observatory tool \citep{berthierSkyBoTNewVO2006}.
This work made use of the {FTOOLS} software package hosted by the NASA Goddard Flight Center High Energy Astrophysics Science Archive Research Center.
This research has made use of SAOImageDS9, developed by Smithsonian Astrophysical Observatory \citep{joyeNewFeaturesSAOImage2006}.
This work made use of the Lowell Observatory Asteroid Orbit Database \textit{astorbDB} \citep{bowellPublicDomainAsteroid1994,moskovitzAstorbDatabaseLowell2021}.
This work made use of the \textit{astropy} software package \citep{robitailleAstropyCommunityPython2013}.
This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A\&M University, Financiadora de Estudos e Projetos, Funda\c{c}\~{a}o Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Funda\c{c}\~ao Carlos Chagas Filho de Amparo \`{a} Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico and the Minist\'{e}rio da Ci\^{e}ncia, Tecnologia e Inova\c{c}\~{a}o, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones En\'{e}rgeticas, Medioambientales y Tecnol\'{o}gicas–Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgen\"ossische Technische Hochschule (ETH) Z\"urich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci\`{e}ncies de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit\"{a}t M\"{u}nchen and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A\&M University.
Based on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory (NOAO Prop. ID 2016A-0190, PI: Dey), which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.
The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID \#2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID \#2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID \#2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program ``The Emergence of Cosmological Structures'' Grant \# XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant \# 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant \# 11433005). The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO.
Based in part on data collected at Subaru Telescope and obtained from the SMOKA, which is operated by the Astronomy Data Center, National Astronomical Observatory of Japan \citep{2002ASPC..281..298B}.
\section{Appendix}
\subsection{Equipment and Archives}
\label{QN:sec:equipQuickRef}
\begin{sidewaystable}
\caption{Equipment and Archives}
\centering
\footnotesize
\begin{tabular}{llclcccccc}
Instrument & Telescope & Pixel Scale & Location & NOIR & ESO & IRSA & SMOKA & SSOIS & STScI \\
& & [$''$/pix] & & & & &\\
\hline
\hline
DECam & 4 m Blanco & 0.263 & Cerro Tololo, Chile & S,R & & & & S & \\
OmegaCAM & 2.6 m VLT Survey & 0.214 & Cerro Paranal, Chile & & R & & & S & \\
GigaPixel1 & 1.8 m Pan-STARRS1 & 0.258 & Haleakalā, Hawaii & & & & & S & R \\
MegaPrime & 3.6 m CFHT & 0.185 & Mauna Kea, Hawaii & & & & & S,R & \\
PTF/CFHT 12K& 48" Samuel Oschin & 1.010 & Mt. Palomar, California & & & S,R & & & \\
SuprimeCam & 8.2 m Subaru & 0.200 & Mauna Kea, Hawaii & & & & R & S & \\
ZTF Camera & 48" Samuel Oschin & 1.012 & Mt. Palomar, California & & & S,R & & & \\
\end{tabular}
\raggedright
\\
\footnotesize{
R indicates repository for data retrieval. S indicates search capability.\\
NOIR: NSF National Optical Infrared Labs AstroArchive (\url{https://astroarchive.noirlab.edu}).\\
ESO: European Space Organization Archive (\url{https://archive.eso.org}).\\
IRSA: NASA/CalTech Infrared Science Archive (\url{https://irsa.ipac.caltech.edu}).\\
SMOKA: NAOJ Subaru-Mitaka-Okayama-Kiso Archive Science Archive (\url{https://smoka.nao.ac.jp}).\\
SSOIS: CADC Solar System Object Image Search (\url{https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/ssois/}).\\
STScI: Space Telescope Science Institute (\url{https://www.stsci.edu/}).}
\label{QN:tab:equipAndArchives}
\end{sidewaystable}
Table \ref{QN:tab:equipAndArchives} lists the instruments and telescopes used in this work, along with their respective pixel scales, locations, and data archives.
\subsection{(248370) 2005 QN173 Data}
\label{QN:sec:ObjectData}
We provide current information regarding (248370)~2005~QN$_{173}$ below (Table \ref{QN:table:qnproperties}).
\begin{sidewaystable}[ht]
\centering
\caption{(248370)~2005~QN$_{173}$ Properties}
\begin{tabular}{lll}
Parameter & Value & Source\\
\hline\hline
Discovery Date & 2005 August 29 & Minor Planet Center\\
Discovery Observers & Near-Earth Asteroid Tracking (NEAT) & Minor Planet Center\\
Discovery Observatory & Palomar & Minor Planet Center\\
Activity Discovery Date & 2021 July 7 & CBET 4995\\
Activity Discoverer(s) & A. Fitzsimmons / ATLAS & CBET 4995 \citep{fitzsimmons2483702005QN1732021}\\
Orbit Type & Outer Main-belt & IMCCE, AstOrb\\
Taxonomic Class & C-type (unconfirmed) & \citet{hsiehPhysicalCharacterizationMainbelt2021}\\
Diameter & $D=3.6$~km; 3.4$\pm$0.4~km & Horizons, {\citet{harrisAsteroidsThermalInfrared2002}} ; \citet{hsiehPhysicalCharacterizationMainbelt2021}\\
Absolute $V$-band Magnitude & $H=16.02$ & Horizons\\
Geometric Albedo & 0.054 & Horizons, \citet{mainzerNEOWISEDiametersAlbedos2019}\\
Rotation Period & unknown & \\
Orbital Period & $P=5.37$ yr & Horizons \\
Semimajor Axis & $a=3.075$ au & Horizons\\
Eccentricity & $e=0.226$ & Horizons\\
Inclination & $i=0.067^\circ$ & Horizons\\
Longitude of Ascending Node & $\Omega=174.28$ & Minor Planet Center\\
Mean Anomaly & $M=8.79^\circ$ & Minor Planet Center\\
Argument of Perihelion & $\omega=146.09^\circ$ & Horizons\\
Perihelion Distance & $q=2.374$ au & Horizons\\
Aphelion Distance & $Q=3.761$ au & Horizons\\
Tisserand Parameter w.r.t. Jupiter & $T_J=3.192$ & AstOrb\\
\end{tabular}
\label{QN:table:qnproperties}
\end{sidewaystable}
\chapter{Manuscript III: Cometary Activity Discovered on a Distant Centaur: A Nonaqueous Sublimation Mechanism}
\chaptermark{Nonaqueous Cometary Activity Discovered on a Distant Centaur}
\label{chap:2014OG392}
\acresetall
Colin Orion Chandler\footnote{\label{og:nau}Department of Physics \& Astronomy, Northern Arizona University, PO Box 6010, Flagstaff, AZ 86011, USA}, Jay K. Kueny$^\mathrm{\ref{og:nau},}$\footnote{Lowell Observatory, 1400 W Mars Hill Rd, Flagstaff, AZ 86001, USA}, Chadwick A. Trujillo$^\mathrm{\ref{og:nau}}$, David E. Trilling$^\mathrm{\ref{og:nau}}$, William J. Oldroyd$^\mathrm{\ref{og:nau}}$
\textit{This is the Accepted Manuscript version of an article accepted for publication in Astrophysical Journal Letters. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at }\url{https://iopscience.iop.org/article/10.3847/2041-8213/ab7dc6}\textit{.}
\doublespacing
\section{Abstract}
Centaurs are minor planets thought to have originated in the outer solar system region known as the Kuiper Belt. Active Centaurs enigmatically display comet-like features (e.g., tails, comae) even though they orbit in the gas giant region where it is too cold for water to readily sublimate. Only 18 active Centaurs have been identified since 1927 and, consequently, the underlying activity mechanism(s) have remained largely unknown up to this point. Here we report the discovery of activity emanating from Centaur 2014~OG$_{392}${}, based on archival images we uncovered plus our own new observational evidence acquired with the Dark Energy Camera (Cerro Tololo Inter-American Observatory Blanco 4~m telescope), the Inamori-Magellan Areal Camera \& Spectrograph (Las Campanas Observatory 6.5~m Walter Baade Telescope), and the Large Monolithic Imager (Lowell Observatory 4.3~m Discovery Channel Telescope). We detect a coma as far as 400,000 km from 2014~OG$_{392}${}, and our novel analysis of sublimation processes and dynamical lifetime suggest carbon dioxide and/or ammonia are the most likely candidates for causing activity on this and other active Centaurs. We find 2014~OG$_{392}${} is optically red, but CO$_2$ and NH$_3$ are spectrally neutral in this wavelength regime so the reddening agent is as yet unidentified.
\textbf{Keywords:} Centaurs (215), Comae (271), Comet tails (274), Astrochemistry (75)
\section{Introduction}
\label{og:sec:introduction}
Prior to the mid-20th century, comets were thought to be the only astronomical objects with tails or comae. Unsurprisingly, then, the first two active Centaur discoveries---29P/Schwassman--Wachmann~1 \citep{schwassmannNEWCOMET1927} and 39P/Oterma \citep{otermaNEWCOMETOTERMA1942}---were initially classified as comets.
\begin{figure}[H]
\centering
\includegraphics[width=0.5\columnwidth]{ogFiles/Thumb_DEEP_stackA}
\caption{2014~OG$_{392}${} (dashed arrow) displays a coma (short arrows) during our 2019 August 30 observations. Stack of $4\times250$~s DECam exposures.}
\label{og:fig:stacked}
\end{figure}
\noindent In 1949 the discovery of the first active asteroid, (4015)~Wilson--Harrington (also designated 107P), blurred the dividing line between asteroid and comet \citep{cunninghamPeriodicCometWilsonHarrington1950}. In 1977 (2060)~Chiron was discovered \citep{kowalSlowMovingObjectKowal1977}, the first member of the population now known as Centaurs. (2060)~Chiron was later found to be active, making it the first object to be identified as a Centaur prior to activity discovery \citep{meechAtmosphere2060Chiron1990}.
We adopt the Centaur classification system \citep{jewittActiveCentaurs2009} that defines Centaurs as objects (1) with perihelia and semi-major axes between the orbits of Jupiter ($\sim$5 au) and Neptune ($\sim$30~au) and (2) not in 1:1 mean-motion resonance with a giant planet (as is the case for the Trojans). We distinguish between Centaurs and Jupiter-Family Comets (following \citealt{levisonLongTermDynamicalBehavior1994}) via the Tisserand parameter with respect to Jupiter, given by
\begin{equation}
T_\mathrm{J} = \frac{a_\mathrm{J}}{a} + 2\sqrt{\left(1-e^2\right)\frac{a}{a_\mathrm{J}}}\cos(i),
\label{og:eq:TJ}
\end{equation}
\noindent with eccentricity $e$, inclination $i$, and the semi-major axes of the body and Jupiter $a$ and $a_\mathrm{J}$, respectively. Centaurs have $T_\mathrm{J}\geqslant3$ whereas Jupiter-Family Comets range between $2<T_\mathrm{J}<3$.
\begin{table*}
\centering
\footnotesize
\caption{Active Centaurs}
\label{og:tab:activecentaurs}
\hspace{40mm}Orbital Elements\hspace{25mm}Activity Discovery\\
\begin{tabular}{l|rrrr|crccr}
Object Name or Designation & $P$ & $a$ & $q$ & $Q$ & $r$ & $\%_{T\rightarrow q}$ & $M_V$ & Date & Ref.\\
& (yr) & (au) & (au) & (au) & (au) & & & (UT) & \\
\hline
Chiron~(95P) & 50.5 & \ 6.0 & 8.5 & 18.9 & 11.8 & 68 & 17.0 & 1989 Apr 10 & 1\\
Echeclus~(174P) & 35.3 & 10.8 & 5.9 & 15.6 & 13.1 & 25 & 21.1 & 2005 Dec 04 & 2\\
29P/Schwassmann--Wachmann 1 & 14.8 & \ 6.0 & 5.5$^\dagger$ & \ 6.6 & \ 6.0 & 53 & 15.3 & 1927 Nov 15 & 3\\
39P/Oterma & 19.5 & \ 7.2 & 3.4$^\dagger$ & \ 9.0 & \ 3.5 & 99 & 15.1 & 1942 Feb 12 & 4\\
165P/LINEAR & 76.4 & 18.0 & 6.8 & 29.3 & \ 6.9 & 100 & 19.4 & 2000 Jan 09 & 5\\
166P/NEAT & 51.9 & 13.9 & 8.6 & 19.2 & \ 8.6 & 100 & 19.6 & 2001 Oct 15 & 6\\
167P/CINEOS & 64.8 & 16.1 & 11.8 & 20.5 & 12.2 & 96 & 20.7 & 2004 Jun 07 & 7\\
P/2005~S${2}$ (Skiff) & \ 22.5 & \ 8.0 & \ 6.4 & \ 9.5 & \ 6.5 & 98 & 19.7 & 2005 Sep 16 & 8\\
P/2005~T$_{3}$ (Read) & \ 20.6 & \ 7.5 & \ 6.2 & \ 8.8 & \ 6.2 & 100 & 20.7 & 2005 Aug 07 & 9\\
P/2011~C${2}$ (Gibbs) & \ 20.0 & \ 7.4 & \ 5.4 & \ 9.3 & \ 5.5 & 97 & 20.3 &2011 Feb 12 & 10\\
C/2011~P${2}$ (PanSTARRS) & \ 30.6 & \ 9.8 & \ 6.2 & 13.4 & \ 6.3 & 98 & 20.3 & 2011 Aug 03 & 11\\
P/2011~S${1}$ (Gibbs) & \ 25.4 & \ 8.6 & \ 6.9 & 10.4 & \ 7.5 & 82 & 21.0 &2011 Sep 18 & 12\\
C/2013~C${2}$ (Tenagra) & \ 64.4 & 16.1 & \ 9.1 & 23.0 & \ 9.8 & 96 & 19.1 &2013 Feb 14 & 13\\
C/2013~P${4}$ (PanSTARRS) & \ 56.8 & 14.8 & \ 6.0 & 23.6 & \ 6.3 & 98 & 19.5 &2013 Aug 15 & 14\\
P/2015~M${2}$ (PanSTARRS) & 19.3 & \ 7.2 & \ 5.9& \ 8.5 & \ 5.9 & 100 & 19.5 & 2015 Jun 28 & 15\\
C/2015~T${5}$ (Sheppard--Tholen)& 147.9 & 28.0 & \ 9.3 & 46.6 & \ 9.4 & 100 & 22.3 &2015 Oct 13 & 16\\
C/2016~Q${4}$ (Kowalski) & \ 69.0 & 16.8 & \ 7.1 & 26.5 & \ 7.5 & 98 & 20.1 & 2016 Aug 30 & 17\\
2003~QD$_{112}$ & \ 82.8 & 19.0 & \ 7.9 & 30.1 & 12.7 & 57 & 21.7 & 2004 Oct 10 & 18\\
2014~OG$_{392}${} & \ 42.5 & 12.2 & \ 10.0 & 14.4 & 10.6 & 86 &21.1 & 2017 Jul 18 & 19\\
\end{tabular}
\raggedright
\footnotesize
\vspace{1mm}
\textbf{Notes.} $P$: orbital period; $a$: semi major axis; $q$: perihelion--distance; $Q$: aphelion distance; $r$: heliocentric distance; $\%_{T\rightarrow q}$: fractional perihelion-aphelion distance (Equation \ref{og:eq:percentperi}); $M_V$: apparent $V$-band magnitude. $Q$ computed via $Q=a(1+e)$ when otherwise unavailable. Asteroid parameters provided by the Minor Planet Center. Heliocentric distance and apparent magnitude courtesy of JPL Horizons \citep{giorginiJPLOnLineSolar1996}.
$^a$ Original value(s) from activity discovery epoch adopted where available; otherwise, values adopted from more recent epoch(s). Reference points to a source that discusses activity of the object.
\textbf{References.} 1:\cite{meechAtmosphere2060Chiron1990}, 2:\cite{choi605582000EC2006}; 3:\cite{schwassmannNEWCOMET1927}, 4:\cite{otermaNEWCOMETOTERMA1942}, 5:\cite{greenComets165P20002005}, 6:\cite{pravdoComet2001T42001}, 7:\cite{romanishinCOMET2004PY422005}, 8:\cite{gajdosComet2005S22005}, 9:\cite{readComet2005T32005}, 10:\cite{gibbsComet2011C22011}, 11:\cite{wainscoatComet2011P22011a}, 12:\cite{gibbsComet2011S12011}, 13:\cite{holvorcemComet2013C22013}, 14:\cite{wainscoatComet2013P42013}, 15:\cite{bacciComet2015M22015}, 16:\cite{tholenComet2015T52015}, 17:\cite{kowalskiCOMET2016Q42016}, 18:\cite{jewittActiveCentaurs2009}, 19:this work
\end{table*}
Centaurs are thought to have migrated inward from the Kuiper Belt (see review; \citealt{morbidelliCometsTheirReservoirs2008}), a region that spans 30~au (Neptune's orbital distance) to 50~au. Neptune Trojans may also serve as a Centaur reservoir \citep{hornerNeptuneTrojansNew2010}. Centaurs all orbit exterior to the 3~au water ice line so they cannot readily undergo sublimation. Surprisingly, though, 18 Centaurs ($\sim$ 4\% of known Centaurs) have been found to display prominent comet-like features such as comae (e.g., Fig. \ref{og:fig:stacked}) or tails; these are the active Centaurs. Table \ref{og:tab:activecentaurs} lists the known active Centaurs along with key physical parameters and discovery circumstances.
Our understanding of active Centaurs has been limited because of their faint apparent magnitudes (the mean apparent magnitude $m_V$ at discovery is $\sim$20; Table \ref{og:tab:activecentaurs}), since it is necessary to probe several magnitudes fainter in order to reliably detect activity via telescopic imaging. Spectroscopy has been used with some success to identify cometary activity originating from asteroids \citep{busarevNewCandidatesActive2018} but this method requires even brighter targets than detection by imaging. Discovering activity on Centaurs is observationally challenging because they are faint, telescope time-intensive, and because they are rare. Active centaurs are discovered, on average, within $\sim$10\% of their perihelion distance (Table \ref{og:tab:activecentaurs}) where they are significantly brighter and, importantly, warmer.
Another significant obstacle to understanding active Centaurs stems from the extreme cold found at their orbital distances. Water and methanol ices have been detected on the surfaces of $\sim$10 Centaurs, but only one of these, (2060)~Chiron, has also been visibly active (see review, \citealt{peixinhoCentaursComets402020}). At surface temperatures less than 150~K and pressures below $\sim10^{-12}$ bar many thermodynamical properties (e.g., enthalpy of sublimation) of volatile ices are not well known from laboratory experiments \citep{fraySublimationIcesAstrophysical2009a}. Moreover, ices may exist in two or more different structural forms; energy from the H$_2$O crystalline--amorphous state transition may even play a role in generating activity \citep{jewittActiveCentaurs2009}.
\section{Mining Archival Data}
\label{og:sec:archivaldata}
In order to overcome the observational challenges discussed in Section \ref{og:sec:introduction} we began by searching archival images captured with the 0.5~gigapixel \acf{DECam} on the Blanco 4~m telescope at the Cerro Tololo Inter-American Observatory in Chile. Archival data from this facility allow the detection of faint activity because of the relatively large aperture and because a large number of objects serendipitously imaged by the instrument can be searched.
We identified Centaurs in our own proprietary database cataloging the NSF's National Optical-Infrared Astronomy Research Laboratory (NSF's OIR Lab, formerly NOAO) public \ac{DECam} archive following the methodology outlined in \cite{chandlerSAFARISearchingAsteroids2018}. Our general approach was to correlate image celestial coordinate and temporal data with object ephemeris services such as NASA JPL Horizons \citep{giorginiJPLOnLineSolar1996} and IMCCE SkyBot (\citet{berthierSkyBoTNewVO2006}; see also the acknowledgements).
We (1) extracted event information from the entire \ac{DECam} public archive database, (2) submitted objects to SkyBot or matched against ephemerides produced via the Minor Planet Center and/or Horizons, and then (3) carried out a database query to identify potential images containing Centaurs.
After (4) downloading the data, we (5) checked each chip for the presence of the Centaur to ensure the object was visible and free of imaging complications (e.g., gaps between chips, scattered light from bright stars, cosmic rays). Finally, we (6) adhered to the routine outlined in \cite{chandlerSAFARISearchingAsteroids2018} where, following image file retrieval of 2014~OG$_{392}${} from the archive, we extracted Flexible Image Transport System (FITS) and Portable Network Graphics (PNG) thumbnails (480$\times$480 pixel images). We subjected these thumbnails to image processing techniques in order to assist by-eye analysis.
While examining each Centaur PNG thumbnail image by eye we flagged any with apparent activity for later analysis. FITS thumbnail images corresponding to those flagged were subjected to additional image processing techniques in an effort to enhance image quality, especially comae contrast.
To ascertain potential heliocentric distance effects we made use of a simple metric \citep{chandlerSAFARISearchingAsteroids2018}, $\%_{T\rightarrow q}$, which describes how close to perihelion ($q$) an object's distance ($d$) is relative to its aphelion distance ($Q$):
\begin{equation}
\%_{T\rightarrow q} = \left(\frac{Q - d}{Q-q}\right)\cdot 100\mathrm{\%}.
\label{og:eq:percentperi}
\end{equation}
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth]{ogFiles/Plot_2014OG392observability}
\footnotesize\caption{2014~OG$_{392}${} activity timeline beginning 2012 September (\ac{DECam} first light) to present. Red stars show when we found visible activity. The orbital period is $\sim$42 yr so neither perihelion (2021 December 3) nor aphelion are visible on this plot. The solid green line (left vertical axis) shows the geocentric apparent $V$-band magnitude of 2014~OG$_{392}${}. Dashed lines (right vertical axis) indicate the number of nighttime hours with elevation $> 15^\circ$ for the southern hemisphere \ac{DECam} (blue; site code: 807) and for the northern hemisphere DCT (orange; site code: G37). The overlaid histogram (vertical blue bars and right axis) shows the number of thumbnail images captured during one calendar month. Note that in all instances when observability was high and many thumbnails were present, activity was observed.
}
\label{og:fig:ActivityTimeline}
\end{figure*}
From \ac{DECam} archival data we extracted $\sim20$ thumbnail images of 2014~OG$_{392}${}; Figure \ref{og:fig:ActivityTimeline} shows the number of thumbnails obtained along with the predicted apparent $V$-band magnitude and observability of 2014~OG$_{392}${}. In images from 2017, July and August, we spotted what appeared to be activity emanating from 2014~OG$_{392}${} (see gallery; Figure \ref{og:fig:archivalimages}); at that time the object was 10.60~au from the Sun.
\section{Follow-up Observing}
\label{og:sec:observations}
To confirm the presence of activity we used the same \ac{DECam} instrument and made additional observations on UT 2019 August 30. Fig.\ 1 shows 2014~OG$_{392}${} with a telltale coma revealed by a combined 1000~s exposure. Figure \ref{og:fig:newobserations} contains a gallery showing the four constituent 250 s \ac{DECam} exposures, plus two images where isophotal contours were overplotted to help identify coma extent for each of the first two exposures (Figure \ref{og:fig:isophotalcontours}).
We made use of three observatories for follow-up observations of 2014~OG$_{392}${}: (1) NSF's OIR Labs \ac{DECam} with $VR$ filter on the Blanco 4~m telescope at the Cerro Tololo Inter-American Observatory in Chile (2) WB4800-7800 filtered imaging with the Magellan 6.5~m Walter Baade Telescope equipped with the Inamori-Magellan Areal Camera \& Spectrograph (IMACS) at the Las Campanas Observatory on Cerro Manqui, Chile, and (3) $g$, $r$, and $i$ filter images taken with the \ac{LMI} at the Lowell Observatory 4.3~m Discovery Channel Telescope (DCT) in Arizona, USA. Galleries showing our Magellan images and DCT images are shown in Figure \ref{og:fig:magellanobservations} \ref{og:fig:dctobservations}
respectively. A log of observations is provided in Appendix \ref{og:sec:observationdetails}. Astrometric calibration was performed using the \textit{astrometry.net} \citep{langAstrometryNetBlind2010} and/or \textit{PhotometryPipeline} \citep{mommertPHOTOMETRYPIPELINEAutomatedPipeline2017} software packages.
\section{Simulating Dynamical Lifetime}
\label{og:sec:dynamicallifetimesimulation}
Determining the total mass loss possible for different volatiles requires knowledge of the dynamical lifetime of 2014~OG$_{392}${} in the Centaur region (where both perihelion distance and semi-major axis are between 5 and 30~au). To this end we made use of the REBOUND $N$-body integrator to model the orbits of 2014~OG$_{392}${} and giant planets Jupiter, Saturn, Uranus, and Neptune \citep{reinHybridSymplecticIntegrators2019}. We also carried out 25 simulations of 2014~OG$_{392}${}, each with an orbital clone derived from the orbital uncertainties published by the Minor Planet Center. From these dynamical integrations, we found that the lifetime of 2014~OG$_{392}${} spans the range of 13,000--1.8 million years, roughly in agreement with prior work \citep{liuInvestigationOriginCentaurs2019}.
\section{Sublimation Modeling}
\label{og:sec:sublimationmodeling}
In order to better assess potential processes responsible for 2014~OG$_{392}${} activity, we computed equilibrium temperatures and modeled mass-loss rates for seven astrophysically relevant ices: ammonia (NH$_3$), carbon dioxide (CO$_2$), carbon monoxide (CO), methane (CH$_4$), methanol (CH$_3$OH), nitrogen (N$_2$), and water (H$_2$O).
Object distance is the primary factor in determining potential ice sublimation effects. We began with a simple sublimation model \citep{hsiehMainbeltCometsPanSTARRS12015} well suited to gaining broad insight into the observed activity from 2014~OG$_{392}${}; we expanded the procedure to apply more generally to other volatile ices. As we do not know the composition of 2014~OG$_{392}${} we cannot make use of a more comprehensive model which includes effects of, for example, porosity, tortuosity, or crystal structure \citep{schorghoferLifetimeIceMain2008a}. Moreover, 2014~OG$_{392}${} is undoubtedly not composed of a single ice, and mixtures of ices can exhibit behavior uncharacteristic for any lone constituent \citep{grundySolarGardeningSeasonal2000}. For the limiting case of an inert gray body orbiting at a distance $R$ from the Sun (measured in au)
\begin{equation}
\frac{F_\odot}{R^2}(1-A)=\chi \epsilon \sigma T_\mathrm{eq}^4
\end{equation}
\noindent where the fiducial solar flux $F_\odot$ is 1360 W m$^{-2}$, $A$ is the Bond albedo (we choose 0.1 as representative for Centaurs; \citealt{peixinhoCentaursComets402020}), $\epsilon$ is the infrared emissivity of the ice (set here as 0.9), $T_\mathrm{eq}$ is the equilibrium temperature of the body, and $\sigma$ is the Stefan--Boltzmann constant ($5.670\times10^{-8}$W m$^{-2}$ K$^{-4}$). Here $\chi$ is a factor that describes the rotational and axial tilt effects on how much flux is received from the Sun: $\chi=1$ indicates the maximum heating scenario where the body is a ``slab'' facing the Sun at all times; $\chi=\pi$ describes a body that rotates quickly with no axial tilt with respect to the Sun; and $\chi=4$, which we adopt here, is used for a fast-rotating (on the order of a few hours) isothermal body in thermodynamic equilibrium.
Here ``fast-rotating'' means that the rotation period of the object is short compared to the thermal wave propagation time \citep{schorghoferLifetimeIceMain2008a,hsiehMainbeltCometsPanSTARRS12015}.
We next consider an energy balance that incorporates sublimation in addition to blackbody radiation \citep{hsiehMainbeltCometsPanSTARRS12015}:
\begin{equation}
\frac{F_\odot}{R^2} (1-A) = \chi\left[\epsilon\sigma T^4 + L f_\mathrm{D}\dot{m}_\mathrm{S}(T)\right]
\label{og:eq:sublimationfull}
\end{equation}
\noindent where $f_\mathrm{D}$ is the ``diffusion barrier factor'' that describes how much emission is blocked by overlaying material (e.g., regolith), and $L$ the latent heat of sublimation. The mass-loss rate $\dot{m}_\mathrm{S}(T)$ is given by
\begin{equation}
\dot{m} = P_\mathrm{v}(T)\sqrt{\frac{\mu}{2\pi k T}}
\end{equation}
\noindent with $\mu$ the SI mass of one molecule, and $k$ the Boltzmann constant of $1.38069\times10^{-23}$ J K$^{-1}$. The vapor pressure (in Pa) of the substance can be related to temperature by the Clausius--Clapeyron relationship
\begin{equation}
P_\mathrm{v}(T) = e_\mathrm{S} \exp \left[\frac{\Delta H_\mathrm{subl}}{R_\mathrm{g}}\left(\frac{1}{T_\mathrm{triple}}-\frac{1}{T}\right)\right]
\end{equation}
\noindent in which $e_\mathrm{S}$ is the saturation vapor pressure (in Pa) of the substance at the triple-point temperature $T_\mathrm{triple}$, $\Delta H_\mathrm{subl}$ is the heat of sublimation of the substance (in kJ mol$^{-1}$), and $R_\mathrm{g}$ is the ideal gas constant ($\rm 8.341~J/mol\cdot K$).
Solving Equation \ref{og:eq:sublimationfull} for heliocentric distance $R$ (in au) yields
\begin{equation}
R(T) = \sqrt{\frac{F_\odot (1-A)}{\chi\left[\epsilon\sigma T^4 + L f_\mathrm{D}\dot{m}_\mathrm{S}(T)\right]}}.
\end{equation}
Energy of sublimation values \citep{lunaNewExperimentalSublimation2014b} and triple-point temperatures and pressures \citep{fraySublimationIcesAstrophysical2009a} were incorporated as needed. To validate our model we computed the mass-loss rate for (2060)~Chiron assuming $\chi=4$, an albedo of $0.057$, a diameter of 206 km, and an orbit ranging from 8.47 au at perihelion to 18.87~au at aphelion. Our (2060)~Chiron model validation results were in rough agreement with the 0.5--20 kg s$^{-1}$ mass-loss rate reported by \citet{womackCODistantlyActive2017}
\begin{figure*}
\centering
\includegraphics[width=0.85\linewidth]{ogFiles/Plot_MassLossByDA10kai4}
\footnotesize \caption{ Mass-loss rates for seven different astrophysically relevant ices on an isothermal ($\chi=4$) body; water (H$_2$O) and methanol (CH$_3$OH) ices have been detected on Centaurs. Orbital distances of Jupiter, Saturn, Uranus, and Neptune are indicated about the top axis. The current 10.11~au heliocentric distance of 2014~OG$_{392}${} is indicated by a vertical black bar, bracketed by perihelion (9.97 au) and aphelion (14.40~au) distances (leftmost and rightmost dashed vertical lines, respectively). Over the course of one orbit (between the vertical dashed lines), water and methanol never appreciably sublimate and carbon monoxide (CO), methane (CH$_4$), and molecular nitrogen (N$_2$) sublimate at high and relatively constant rates; we rule out all of these molecules as potential causes of activity. (The shallow slopes of CO, CH$_4$, and N$_2$ extend beyond 50~au [not shown], which informs us the mass loss would have begun long before 2014~OG$_{392}${} became a Centaur.) However, over the course of one orbit the sublimation rates for CO$_2$ and NH$_3$ vary substantially, presumably producing significant variation in visible activity. Order-of-magnitude estimates of mass-loss-rate upper limits for the dynamical lifetime of 2014~OG$_{392}${} are shown as horizontal dotted lines. Only CO$_2$ and NH$_3$ have sublimation rates near these limits.}
\label{og:fig:masslossrates}
\end{figure*}
We use our computed dynamical lifetime to circumstantially constrain the molecule(s) responsible for the sublimation of 2014~OG$_{392}${}. Fig. \ref{og:fig:masslossrates} shows, over the orbit of 2014~OG$_{392}${}, the mass-loss rates for the different ices determined via modeling and validated through laboratory measurements. If 2014~OG$_{392}${} has an albedo of 10\%, similar to that measured for other Centaurs (see review; \citealt{peixinhoCentaursComets402020}), then the body is about 20~km in diameter (see Section \ref{og:sec:absmaganddiameter}). Assuming a spherical body of low density in the range of 1--3~g cm$^{-1}$ suggests a reasonable body mass of $4.2--12.6\times 10^{15}$~kg and a surface area of $3.1 \times 10^{8} \mbox{ m}^2$. Thus, the 13,000--1.8 Myr dynamical lifetime of 2014~OG$_{392}${} suggests a maximum orbit-averaged mass-loss rate in the range of $7.1\times10^{-7}$ to $\rm 3.3\times 10^{-5}\ kg/m^{2}/s$ (horizontal dashed lines in Figure \ref{og:fig:masslossrates}) before the body would be entirely lost due to sublimation.
\section{Colors}
\label{og:sec:colors}
The archival data and our confirmation observations did not contain enough information to determine colors, so we obtained six 300~s exposures of 2014~OG$_{392}${} in a $g$-$r$-$i$ filter sequence at the DCT (Section \ref{og:sec:observations}). We made use of the \textit{PhotometryPipeline} software package \citep{mommertPHOTOMETRYPIPELINEAutomatedPipeline2017} to automate astrometry using SCAMP \citep{bertinAutomaticAstrometricPhotometric2006} which made use of the Vizier catalog service \citep{ochsenbeinVizieRDatabaseAstronomical2000} Gaia Data Release 2 catalog \citep{collaborationGaiaDataRelease2018}, and photometric image calibration using solar stars from the Sloan Digital Sky Survey Data Release 9 (SDSS-DR9) catalog \citep{ahnNinthDataRelease2012}. We carried out manual aperture photometry using the Aperture Photometry Tool \citep{laherAperturePhotometryTool2012a}.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{ogFiles/DataAndModelSBRRP.pdf}
\caption{\footnotesize Surface brightness radial profiles of 2014~OG$_{392}${} and a nearby SDSS-DR9 catalog solar-type star (J004840.66-022335.6) are plotted along with a model fit for each object. After subtracting the background flux from the two profiles we normalized the standard star profile to the peak of the 2014~OG$_{392}${} profile. The coma flux tapers from 125 counts to background (0 counts) at $\rho\simeq60$ pixels, or $4.3\times10^5$~km. We estimate there are $\sim5.8\times10^{17}$ particles in the coma assuming a grain radius of 1~mm; for a density of 1~g cm$^{-3}$ the total mass is $2.4\times10^{15}$~g. Data from our 300~s $g$-band exposure taken on UT 2019 December 30 2:29 using the LMI on the Lowell Observatory 4.3~m DCT.}
\label{og:fig:sbrp}
\end{figure}
Prior to analysis we examined all thumbnail images showing activity emanating from 2014~OG$_{392}${} to ensure no significant background sources were blended with the nucleus. To help us identify unseen contaminators we measured and modeled surface brightness radial profiles of 2014~OG$_{392}${} (Figure \ref{og:fig:sbrp}) and a nearby solar-type star, using the Aperture Photometry Tool. The radial profile itself (i.e., not the model) was used to identify flux contribution by unseen background sources; we rejected images in which the nucleus or nearby coma was significantly contaminated. We note that we identified at least one background source within the coma in all of our images, although for color measurement we were able to use an aperture small enough (5 pixel radius) to exclude all resolvable background objects.
We measured 2014~OG$_{392}${} apparent magnitudes to be $g=21.99\pm0.018$, $r=21.19\pm0.016$, and $i=20.81\pm0.018$. We compared our colors of $g-r=0.80\pm0.024$ and $r-i=0.39\pm0.024$ to SDSS reported solar colors of $g-r=0.44\pm0.02$ and $r-i=0.11\pm0.02$\footnote{\url{http://www.sdss.org/dr12/algorithms/ugrizvegasun}}. Centaur colors are often reported in Johnson $B$ -- $R$ colors (see, e.g., \citealt{teglerTWOCOLORPOPULATIONS2016}), so we computed the $B$ -- $R$ color for 2014~OG$_{392}${} via \cite{jesterSloanDigitalSky2005a} transformations. We found $B$ -- $R$ = \ogColor{}, which is about one magnitude redder than the Sun, and red according to the classification system of \cite{teglerTWOCOLORPOPULATIONS2016} (see discussion in Section \ref{og:sec:discussion}).
\section{Absolute Magnitude and Diameter Estimation}
\label{og:sec:absmaganddiameter}
To gauge the overall spatial extent of the coma we examined the radial surface brightness profiles of 2014~OG$_{392}${} and nearby solar-type star J004840.66-022335.6 (see Section \ref{og:sec:colors}). We fit the profiles to the model
\begin{equation}
S(r) = A + Br + Cr^2 + Dr^3 + Er^4 + F e^{-\frac{r^2}{2\sigma^2}}
\end{equation}
\noindent as described in \cite{gwynSSOSMovingObjectImage2012a}.
After subtracting the sky flux from each profile and each model we scaled the star to the peak flux of the 2014~OG$_{392}${} radial profile. Figure \ref{og:fig:sbrp} shows the radial profiles and their corresponding models plotted; we estimate the coma returns to sky background flux levels at $\sim$60 pixels from the aperture center, thus the coma extent is $\sim4.3\times10^5$ km. The FWHM of 2014~OG$_{392}${} was $13.62\pm0.37$ pixels (3\farcsec2$\pm$0\farcsec09), whereas the star FWHM was $6.05\pm0.05$ pixels ($1.45\pm0\farcsec012$).
As reported in Section \ref{og:sec:sublimationmodeling}, the coma is likely present throughout the orbit of 2014~OG$_{392}${}. As a result, prior absolute ($H$) magnitude estimates would have included the excess flux caused by the coma, as evinced in Figure \ref{og:fig:sbrp}. To estimate the absolute nuclear magnitude of 2014~OG$_{392}${} we compared the ratio of the total (nucleus + coma) flux (blue line and circles, Figure \ref{og:fig:sbrp}) to the scaled stellar flux (orange line and triangles, Figure \ref{og:fig:sbrp}). We estimate the coma accounts for 0.75 and 1.1 magnitudes of the observed $r$-band and $g$-band fluxes, respectively, implying the nucleus apparent magnitudes are $m_r=21.9$ and $m_g=23.1$.
The absolute magnitude of an asteroid, $H$, is commonly used to estimate the size of small bodies in the solar system . $H$ is defined as equal to the apparent $V$-band magnitude of an object observed at a heliocentric distance $R=1$ au, a geocentric distance $\Delta=1$ au, and a phase angle $\alpha=0^\circ$. Here we employ the International Astronomical Union defined \citep{swingsTransactionsInternationalAstronomical1986} $H$ -- $G$ magnitude system approximated from \cite{bowellApplicationPhotometricModels1989}:
\begin{equation}
V = 5 \log \left(R \Delta\right) + H - 2.5 \log \left[\left(1-G\right)\Phi_1 + G \Phi_2\right]
\label{og:eq:HG}
\end{equation}
\noindent where the phase function $\Phi$ is given by
\begin{equation}
\Phi_i = \exp\left[-A_i\tan\left(\alpha/2\right)^{B_i}\right]; i=1,2
\end{equation}
\noindent with constants $A_1=3.33$, $A_2=1.87$, $B_1=0.63$, and $B_2=1.22$.
We make use of the relationships put forth by \cite{jesterSloanDigitalSky2005a} to derive Johnson $V=22.4$ from our $g$ and $r$ nuclear magnitudes. The JPL Horizons ephemerides service \citep{giorginiJPLOnLineSolar1996} provided $G=0.150$ (the standard assumed slope for dark surfaces), $r=10.10$~au, $\Delta=10.01$~au, and $\alpha=5\fdegree 58$ for UT 2019 December 30. Via Equation~\ref{og:eq:HG} we find $H=11.3$, $0.5$ magnitudes fainter than reported by the Minor Planet Center and JPL Horizons.
\cite{harrisRevisionRadiometricAlbedos1997a} provide a convenient method to approximate object diameter $D$,
\begin{equation}
D = \frac{1329}{\sqrt{G}}\times 10^{-H/5},
\end{equation}
\noindent which, for 2014~OG$_{392}${}, gives $D\approx20$~km.
\section{Coma Dust Analysis}
\label{og:sec:dustproductionmodeling}
To facilitate comparing our 2014~OG$_{392}${} dust-related metrics with other works we adopt the instrument and aperture-independent cometary dust production parameter described by \cite{ahearnCometBowell1980b1984a}. The metric, $Af\rho$ (units of cm), combines the mean albedo $A$ of ejecta grains within an aperture of radius $\rho$ (in cm), scaled by the filling factor $f$ (unitless), which describes how much of the aperture area ($\pi\rho^2$) is filled by $N$ grains of cross section area $\sigma$ (in cm$^2$),
\begin{equation}
f = \frac{N(\rho)\sigma}{\pi\rho^2}.
\label{og:eq:f}
\end{equation}
We measured $Af\rho$ (following the method outlined by \citealt{shiResearchActivityMain2019b}) via
\begin{equation}
A f \rho = 4 R^2 \Delta^2 10^{0.4(m_{\odot,F} - m_{\mathrm{OG},F})} \rho^{-1}
\end{equation}
\noindent where $R$ is the 2014~OG$_{392}${} heliocentric distance in au, $\Delta$ is the geocentric distance of 2014~OG$_{392}${} in cm, and, for filter $F$, $m_{\odot,F}$ and $m_{\mathrm{OG},F}$ are the magnitudes of the Sun and 2014~OG$_{392}${}, respectively. For $m_{\odot,F}$ we made use of solar apparent Vega magnitudes\footnote{\url{http://mips.as.arizona.edu/~cnaw/sun.html}} (see \citet{willmerAbsoluteMagnitudeSun2018} for details) in Table \ref{og:tab:AppMagFilter}:
\begin{table}
\centering
\caption{Solar Apparent Magnitude by Filter}
\begin{tabular}{cc}
Filter & $m_{\odot,F}$\\
\hline
SDSS-$g$ & -26.34\\
SDSS-$r$ & -27.04\\
SDSS-$i$ & -27.38\\
\end{tabular}
\label{og:tab:AppMagFilter}
\end{table}
To estimate the number of particles $N$ within our measured $Af\rho$ we can substitute Equation \ref{og:eq:f} into the equality $Af\rho=Af\rho$
\begin{equation}
Af=A\frac{N(\rho)\sigma}{\pi\rho^2}\rho
\end{equation}
\noindent and solve for $N(\rho)$,
\begin{equation}
N(\rho) = Af\rho \frac{\pi\rho}{A\sigma}.
\label{og:eq:Nrho}
\end{equation}
To quantify the total number of particles in the coma $N_\mathrm{tot}$ we can scale the aperture of Equation \ref{og:eq:Nrho} to the 60 pixel aperture containing the entire coma, $\rho_\mathrm{max}$,
\begin{equation}
N(\rho_\mathrm{max}) = Af\rho \frac{\pi\rho_\mathrm{max}^2}{A \sigma \rho}.
\label{og:eq:Nmax}
\end{equation}
\noindent Recall the quantity $Af\rho$, here, is a measured value, so the quantities $A\rho$ do not cancel in Equation \ref{og:eq:Nmax}.
Four of our observations, Images 15-18 (details in Appendix \ref{og:sec:observationdetails}) were suitable for directly measuring $Af\rho$. We found $Af\rho=487\pm 12$~cm with an aperture of $4.3\times10^5$~km. With the albedo adopted for our sublimation modeling ($A=0.1$) and a 1~mm radius grain, the coma around 2014~OG$_{392}${} is composed of roughly $5.8\times10^{17}$ particles. Assuming a grain density of 1~g~cm$^{-3}$ the total coma mass is $\sim2.4\times10^{15}$~g.
\section{Discussion}
\label{og:sec:discussion}
The activity we observed spans more than two years, which rules out impact-driven activity. We determined that the two ices previously detected on Centaurs, water and methanol, would not appreciably sublimate at any point in 2014~OG$_{392}${}'s orbit and so should still be present in solid form on the surface (Figure \ref{og:fig:masslossrates}). Moreover, CO, N$_2$ and CH$_4$ are highly volatile and sublimate at temperatures low enough that their supply is likely depleted, though reservoirs could still be trapped below the surface. We reiterate our model encompasses single-species ices subjected to the thermodynamic conditions outlined in Section \ref{og:sec:sublimationmodeling}; heterogeneous ice environments may alter sublimation chemistry (see, e.g., \citealt{grundySolarGardeningSeasonal2000}), as can single-species state transitions (e.g., energy released during crystallization of amorphous water ice; see, e.g., \citealt{jewittActiveCentaurs2009}).
We find that the molecule(s) most likely to drive the observed activity is either $\rm CO_2$ and/or possibly $\rm NH_3$. Neither would have sublimated appreciably at Kuiper Belt distances prior to 2014~OG$_{392}${} becoming a Centaur. Interestingly, both of these substances sublimate at rates that vary by over two orders of magnitude over the course of a 2014~OG$_{392}${} orbit, peaking at perihelion. As a result we predict 2014~OG$_{392}${} will become less active post-perihelion. This further implies that all other active Centaurs should follow this trend, with peak sublimation near perihelion and a significant drop in outgassing for most of their orbits.
We determined 2014~OG$_{392}${} is at present roughly one magnitude redder than the Sun at visible wavelengths. However, we were only able to obtain two images in each filter, so uncertainty could be improved upon with additional observations. Our color measurements inexorably included the coma; future observations during a quiescent period (should one exist) would allow for color measurements of the bare nucleus. We did, however, attempt to better estimate the $H$ magnitude by subtracting the coma measured in the radial surface brightness profiles. We found 2014~OG$_{392}${} has $H\approx11.3$, 0.5 magnitudes fainter than previously reported. The $H$ magnitude implies a radius of about 20~km when assuming a slope parameter $G=0.15$ as is typical for a dark surface.
In our images of 2014~OG$_{392}${} background sources were typically present in the coma and/or blended with the nucleus, but from four images we were able to directly measure dust properties. Assuming a 10\% albedo and a grain radius of 1~mm we estimate the coma contains roughly $5.8\times10^{17}$ particles. If the grain density is 1~g cm$^{-3}3$, the total mass is $\sim2.4\times10^{15}$~g, or $\sim0.01$\% the total mass of 2014~OG$_{392}${}. If the coma mass is indeed of this scale, 2014~OG$_{392}${} must be eroding very quickly, undergoing new activity, or the ejecta is accumulating faster than it is escaping. Our measured $Af\rho$ of $487\pm12$~cm is comparable to other Centaurs active at the same orbital distance as 2014~OG$_{392}${}: C/2011~P2~(PANSTARRS) with $Af\rho=161\pm 4$~cm at $\sim 9$~au \citep{epifaniNucleusActiveCentaur2017}, and for 166P~(NEAT) $Af\rho=288\pm 19$~cm at $\sim 12$~au \citep{shiCCDPhotometryActive2015}.
Centaurs are sometimes classified as either gray or red depending on whether the object has a $B$-$R$ color closer to $\sim$1.2 or $\sim$1.7, respectively (see \citealt{teglerColorsCentaurs2008a} and \citealt{peixinhoCentaursComets402020} reviews for in-depth discussions). We find our derived $B$-$R$ color of \ogColor{} consistent with the red classification. Notably both molecules we find viable for sublimation are spectrally neutral in visible wavelengths so the reddening agent is as yet unidentified.
2014~OG$_{392}${} will remain observable through 2020 February and will again be observable beginning around 2020 August. We anticipate imaging and spectroscopy will yield further insight into the nature of these rare objects. We wish to emphasize further lab work is needed to characterize sublimation processes of volatiles under low pressure and temperature regimes.
\section{Acknowledgments}
We thank Dr.\ Mark Jesus Mendoza Magbanua (University of California San Francisco) for his frequent and timely feedback on the project. J.K.K. acknowledges support from Northern Arizona University through a startup award administered by TD Robinson. Mark Loeffler (NAU) and Patrick Tribbett (NAU) helped us interpret lab results. Stephen Tegler (NAU) provided extensive insight into the nuances of Centaur colors. Kazuo Kinoshita provided cometary elements that saved us considerable time and energy. The authors express their gratitude to Mike Gowanlock (NAU), Cristina Thomas (NAU), and the Trilling Research Group (NAU), all of whom provided insights that substantially enhanced this work. Thank you to Stephen Kane (University of California Riverside), Dawn Gelino (NASA Exoplanet Science Institute at California Institute of Technology), and Jonathan Fortney (University of California Santa Cruz) for encouraging the authors to pursue this work. The unparalleled support provided by Monsoon cluster administrator Christopher Coffey (NAU) and his High Performance Computing Support team greatly facilitated the work presented here.
\ This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under grant No.\ 2018258765. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
\ Computational analyses were carried out on Northern Arizona University's Monsoon computing cluster, funded by Arizona's Technology and Research Initiative Fund.
\ This work was made possible in part through the State of Arizona Technology and Research Initiative Program.
\ This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. We also employed their solar system Object Search \citep{gwynSSOSMovingObjectImage2012a}.
\ This research has made use of data and/or services provided by the International Astronomical Union's Minor Planet Center.
\ This research has made use of NASA's Astrophysics Data System.
\ This research has made use of the The Institut de M\'ecanique C\'eleste et de Calcul des \'Eph\'em\'erides (IMCCE) SkyBoT Virtual Observatory tool \citep{berthierSkyBoTNewVO2006}.
\ Simulations in this Letter made use of the REBOUND code, which is freely available at http://github.com/hannorein/rebound.
\ This work made use of the {FTOOLS} software package hosted by the NASA Goddard Flight Center High Energy Astrophysics Science Archive Research Center.
\ This research has made use of SAO Image DS9, developed by Smithsonian Astrophysical Observatory \citep{joyeNewFeaturesSAOImage2003}. This work made use of the Lowell Observatory Asteroid Orbit Database \textit{astorbDB} \citep{moskovitzModernizingLowellObservatory2019a}.
\ This work made use of the \textit{astropy} software package \citep{robitailleAstropyCommunityPython2013}.
\ This project used data obtained with the \acf{DECam}, which was constructed by the \acf{DES} collaboration. Funding for the \ac{DES} Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A\&M University, Financiadora de Estudos e Projetos, Funda\c{c}\~{a}o Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Funda\c{c}\~ao Carlos Chagas Filho de Amparo \`{a} Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico and the Minist\'{e}rio da Ci\^{e}ncia, Tecnologia e Inova\c{c}\~{a}o, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energ\'{e}ticas, Medioambientales y Tecnol\'{o}gicas–Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgen\"ossische Technische Hochschule (ETH) Z\"urich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci\`{e}ncies de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit\"{a}t M\"{u}nchen and the associated Excellence Cluster Universe, the University of Michigan, NSF's National Optical-Infrared Astronomy Research Laboratory, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A\&M University.
\ This work is based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory (Prop. IDs 2019A-0337, PI: Trilling; 2014B-0404, PI: Schlegel), which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.
These results made use of the Discovery Channel Telescope at Lowell Observatory. Lowell is a private, non-profit institution dedicated to astrophysical research and public appreciation of astronomy and operates the DCT in partnership with Boston University, the University of Maryland, the University of Toledo, Northern Arizona University and Yale University. The Large Monolithic Imager was built by Lowell Observatory using funds provided by the National Science Foundation (AST-1005313). This Letter includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile.
\section{Appendix}
\subsection{Activity Observation details}
\label{og:sec:observationdetails}
\begin{table}
\centering
\caption{Activity Observations}
\begin{tabular}{cccrc}
\# & Instrument & Date/Time & Exp. & Filter\\
& & (UT) & [s]\ & \\
\hline
1 & DECam$^1$ & 2017-07-18 09:27 & 137 & $z$ \\
2 & DECam$^1$ & 2017-07-18 10:20 & 250 & $z$ \\
3 & DECam$^1$ & 2017-07-22 05:37 & 79 & $g$ \\
4 & DECam$^1$ & 2017-07-25 06:25 & 60 & $r$ \\
5 & DECam$^1$ & 2017-07-25 06:32 & 52 & $r$ \\
6 & DECam$^1$ & 2017-08-20 04:48 & 67 & $r$ \\
7 & DECam$^2$ & 2019-08-30 09:54 & 250 & \textit{VR}\\
8 & DECam$^2$ & 2019-08-30 09:58 & 250 & \textit{VR}\\
9 & DECam$^2$ & 2019-08-30 10:03 & 250 & \textit{VR}\\
10 & DECam$^2$ & 2019-08-30 10:08 & 250 & \textit{VR}\\
11 & IMACS & 2019-12-27 00:54 & 300 & WB4800-7800\\
12 & IMACS & 2019-12-27 01:01 & 300 & WB4800-7800\\
13 & IMACS & 2019-12-27 01:36 & 600 & WB4800-7800\\
14 & LMI & 2019-12-30 02:08 & 300 & $g$\\
15 & LMI & 2019-12-30 02:17 & 300 & $r$\\
16 & LMI & 2019-12-30 02:23 & 300 & $i$\\
17 & LMI & 2019-12-30 02:29 & 300 & $g$\\
18 & LMI & 2019-12-30 02:35 & 300 & $r$\\
19 & LMI & 2019-12-30 02:41 & 300 & $i$\\
\end{tabular}
$^1$Program 2014B-0404 (PI: Schlegel)\\
$^2$Program 2019A-0337 (PI: Trilling)\\
\label{og:tab:observations}
\end{table}
Table \ref{og:tab:observations} provides a listing of the observations used in this work.
\subsection{Thumbnail Gallery}
\label{og:sec:thumbnailgallery}
\begin{figure}
\begin{tabular}{ccc}
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_HARVEST_2017071809271zmin2pIovX} &
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_HARVEST_2017071810202zmin3pIovX} &
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_HARVEST_2017072205374gmin3pIovX}\\
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_HARVEST_2017072506251rmin3pIovX} &
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_HARVEST_2017072506324rmin3pIovX} &
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_HARVEST_2017082004482rmin3pIovX} \\
\end{tabular}
\caption{\textit{DECam Archival Images}. Top left: UT 2017-Jul-18 09:27 -- 137~s $z$-band. Top center: UT 2017-Jul-18 10:20 -- 250~s $z$-band. Top right: UT 2017-Jul-22 05:37 -- 79~s $g$-band. Bottom left: UT 2017-Jul-25 06:25 -- 60~s $r$-band. Bottom center: UT 2017-Jul-25 06:32 -- 52~s $r$-band. Bottom right: UT 2017-Aug-20 04:48 -- 67~s $r$-band. All Images: The coma (green arrows) was exceptionally faint in all of these DECam archival images of 2014~OG$_{392}${} (indicated by dashed red arrows) but nevertheless they prompted us to obtain follow-up observations.}
\label{og:fig:archivalimages}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.4\linewidth]{ogFiles/Thumb_DEEP_954_alt.png} & \includegraphics[width=0.4\linewidth]{ogFiles/Thumb_DEEP_958_alt.png}\\
\includegraphics[width=0.4\linewidth]{ogFiles/Thumb_DEEP_1003_alt.png} & \includegraphics[width=0.4\linewidth]{ogFiles/Thumb_DEEP_1008_alt.png}
\end{tabular}
\caption{\textit{New DECam Observations Gallery}. Top left: UT 9:54. Top right: UT 9:58. Bottom left: UT 10:03; Bottom right: UT 10:08. All images: (1) dashed red arrow points to 2014~OG$_{392}${}, (2) green arrows highlight the comae if visible, (3) observing date was UT 2019 August 30, (4) filter was \textit{VR}, (5) exposure time was 250~s. The apparent decrease in coma prominence was the result of increasing background noise as images were taken into twilight.}
\label{og:fig:newobserations}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{ogFiles/Thumb_DEEP_954_contours} & \includegraphics[width=0.45\linewidth]{ogFiles/Thumb_DEEP_958_contours}\\
\end{tabular}
\caption{\textit{Isophotal Contours.} Isophotal contours indicate the extent and irregularity of the coma emanating from 2014~OG$_{392}${} (dashed arrows), especially when contrasted with background objects (yellow arrows) presenting relatively symmetric radial profiles. These two 250 s \textit{VR}--band exposures were taken at 9:54 (left) and 9:58 (right) during our 2019 August 30 follow-up campaign.}
\label{og:fig:isophotalcontours}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_Magellan_pift0020c2_minmax_log_invert_arrows.png} & \includegraphics[width=0.30\linewidth]{ogFiles/Thumb_Magellan_pift0021c2_minmax_log_invert_arrows.png} & \includegraphics[width=0.30\linewidth]{ogFiles/Thumb_Magellan_pift0024c2_minmax_log_invert_arrows.png} \\
\end{tabular}
\textit{New Magellan Observations Gallery}. \caption{2014~OG$_{392}${} imaged December 27, 2019 via the Magellan 6.5m Baade Telescope using the WB4800-7800 filter on the Inamori-Magellan Areal Camera \& Spectrograph (IMACS) at Las Campanas Observatory on Cerro Manqui, Chile. The three images reveal an apparent coma (green arrows) emerging from the object (red dashed arrow) and were taken at 300~s (left, center) exposures and one 600~s exposure (right).}
\label{og:fig:magellanobservations}
\end{figure}
\begin{figure}
\begin{tabular}{ccc}
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_DCT_2014OG392lmi0052Healed.png} & \includegraphics[width=0.30\linewidth]{ogFiles/Thumb_DCT_2014OG392lmi0053Healed.png} & \includegraphics[width=0.30\linewidth]{ogFiles/Thumb_DCT_2014OG392lmi0054Healed.png} \\
\includegraphics[width=0.30\linewidth]{ogFiles/Thumb_DCT_2014OG392lmi0055Healed.png} & \includegraphics[width=0.30\linewidth]{ogFiles/Thumb_DCT_2014OG392lmi0056Healed.png} & \includegraphics[width=0.30\linewidth]{ogFiles/Thumb_DCT_2014OG392lmi0057Healed.png} \\
\end{tabular}
\caption{\textit{New DCT Observations Gallery.} 2014~OG$_{392}${} imaged December 30, 2019, via the Lowell Observatory 4.3~m Discovery Channel Telescope (Arizona, USA) using the \acf{LMI}. Green arrows trace out a diffuse coma and a dashed red arrow points to the nucleus in each of the six images. Each exposure in the two $g$-$r$-$i$ sequences (top and bottom rows) was 300~s long.}
\label{og:fig:dctobservations}
\end{figure}
Figure \ref{og:fig:archivalimages} shows six of the archival images in which we originally spotted what appeared to be activity emanating from 2014~OG$_{392}${}. We obtained confirmation first through DECam observations (Figure \ref{og:fig:newobserations}); the coma is more readily apparent in the isophotal contours shown in Figure \ref{og:fig:isophotalcontours}. Figure \ref{og:fig:magellanobservations} shows two additional we took at Magellan provided additional confirmation. Figure \ref{og:fig:dctobservations} shows six images of 2014~OG$_{392}${} we captured a the DCT which enabled us to perform color measurement and radial surface brightness profiling.
\chapter{Manuscript V: Migratory Outbursting Quasi-Hilda Object 282P/(323137) 2003 BM80}
\chaptermark{Migratory Outbursting Quasi-Hilda Object 282P/(323137) 2003 BM80}
\label{chap:282P}
\acresetall
Colin Orion Chandler\footnote{\label{282P:nau}Department of Astronomy and Planetary Science, Northern Arizona University, PO Box 6010, Flagstaff, AZ 86011, USA}, William J. Oldroyd$^\mathrm{\ref{282P:nau}}$, Chadwick A. Trujillo$^\mathrm{\ref{282P:nau}}$
\textit{This is a preliminary version of an article submitted for publication in Astrophysical Journal Letters. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it.}
\doublespacing
\newcommand{282P}{282P}
\newcommand{\objnameBMFull}{282P/(323137)~2003~BM$_{80}$}
\newcommand\blfootnote[1]{%
\begingroup
\renewcommand\thefootnote{}\footnote{#1}%
\addtocounter{footnote}{-1}%
\endgroup
}
\newcounter{obsnotelabel}
\newcommand{\obsnote}[1]{\refstepcounter{obsnotelabel}\label{#1}}
\setcounter{obsnotelabel}{0}
\section{Abstract}
\label{282P:Abstract}
We report object \objnameBMFull{} is undergoing a sustained activity outburst, lasting over 15 months thus far. These findings stem in part from our \acs{NASA} Partner Citizen Science project \textit{Active Asteroids} (\url{http://activeasteroids.net}), which we introduce here.
We acquired new observations of 282P{} via our observing campaign (\acf{VATT}, \acf{LDT}, and the Gemini South telescope), confirming 282P{} was active on UT 2022 June 7, some 15 months after 2021 March images showed activity in the 2021--2022 epoch.
We classify 282P{} as a member of the \acp{QHO}, a group of dynamically unstable objects found in an orbital region similar to, but distinct in their dynamical characteristics to, the Hilda asteroids (objects in 3:2 resonance with Jupiter). Our dynamical simulations show 282P{} has undergone at least five close encounters with Jupiter and one with Saturn over the last 180 years. 282P{} was most likely a Centaur or \ac{JFC} 250 years ago. In 350 years, following some 15 strong Jovian interactions, 282P{} will most likely migrate to become a \ac{JFC} or, less likely, an \acl{OMBA} orbit. These migrations highlight a dynamical pathway connecting Centaurs and \acp{JFC} with Quasi-Hildas and, potentially, active asteroids. Synthesizing these results with our thermodynamical modeling and new activity observations, we find volatile sublimation is the primary activity mechanism. Observations of a quiescent 282P{}, which we anticipate will be possible in 2023, will help confirm our hypothesis by measuring a rotation period and ascertaining spectral type.
\blfootnote{Based on observations obtained at the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\'{i}a e Innovaci\'{o}n (Argentina), Minist\'{e}rio da Ci\^{e}ncia, Tecnologia, Inova\c{c}\~{o}es e Comunica\c{c}\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).}
\blfootnote{Magellan telescope time was granted by \acs{NSF}'s \acs{NOIRLab}, through the \ac{TSIP}. \ac{TSIP} was funded by \ac{NSF}.}
\section{Introduction}
\label{282P:introduction}
Volatiles are vital to life as we know it and are critically important to future space exploration, yet basic knowledge about where volatiles (e.g., H$_2$O, CO, CH$_4$) are located within our own solar system is still incomplete. Moreover, the origin of solar system volatiles, including terrestrial water, remains inconclusive. Investigating sublimation-driven active solar system bodies can help answer these questions \citep{hsiehPopulationCometsMain2006}.
We define volatile reservoirs as a dynamical class of minor planet that harbors volatile species, such as water ice. Comets have long been known to contain volatiles, but other important reservoirs are coming to light, such as the active asteroids -- objects on orbits normally associated with asteroids, such as those found in the main-belt, that surprisingly display cometary features such as tails and/or comae \citep{jewittActiveAsteroids2015a}. Fewer than 30 active asteroids have been discovered \citep{chandlerSAFARISearchingAsteroids2018} since the first, (4015)~Wilson-Harrington, was discovered in 1949 \citep{cunninghamPeriodicCometWilsonHarrington1950} and, as a result, they remain poorly understood. One scientifically important subset of active asteroids consists of members that display recurrent activity attributed to sublimation: the \acp{MBC} \citep{hsiehMainbeltCometsPanSTARRS12015}. An important diagnostic of indicator sublimating volatiles, like water ice, is recurrent activity near perihelion \citep{hsiehOpticalDynamicalCharacterization2012,snodgrassMainBeltComets2017}, a feature common to the \acp{MBC} \citep{hsiehMainbeltCometsPanSTARRS12015,agarwalBinaryMainbeltComet2017,hsieh2016ReactivationsMainbelt2018}. Fewer than 10 recurrently active \acp{MBC} have been discovered (though others exhibit activity attributed to sublimation), and as a result we know very little about this population.
Another potential volatile reservoir, active Centaurs, came to light after comet 29P/Schwassmann-Wachmann 1 \citep{schwassmannNEWCOMET1927} was identified as a Centaur following the 1977 discovery of (2060)~Chiron \citep{kowalSlowMovingObjectKowal1977}. Centaurs, found between the orbits of Jupiter and Neptune, are cold objects thought to primarily originate in the Kuiper Belt prior to migrating to their current orbits (see review, \citealt{jewittActiveCentaurs2009}). The dynamical properties of these objects are discussed in Section \ref{282P:sec:dynamicalClassification}. Fewer than 20 active Centaurs have been discovered to date, thus they, like the active asteroids, are both rare and poorly understood.
In order to enable the study of active objects in populations not typically associated with activity (e.g., \acp{NEO}, main-belt asteroids), we created a Citizen Science project designed to identify roughly 100 active objects via volunteer identification of activity in images of a known minor planets. The Citizen Science paradigm involves concurrently crowdsourcing tasks yet too complex for computers to perform, while also carrying out an outreach program that engages the public in a scientific endeavor. Launched in Fall 2021, our \ac{NSF} funded, \acs{NASA} partner program \textit{Active Asteroids}\footnote{\url{http://activeasteroids.net}} immediately began yielding results.
\begin{figure*}[ht]
\centering
\begin{tabular}{cccc}
\begin{overpic}[width=0.24\linewidth]{282Pfiles/093_0323137H_20210314-08071_4179806_8.8y_qC_Ob_090s_i-3_p2_ovX.png}\put (5,7) {\huge\color{green} \textbf{a}}\put (45,8) {\large\color{green} \textbf{2021-03-14}}\end{overpic} &
\begin{overpic}[width=0.24\linewidth]{282Pfiles/282P20210331UT2208sec12x120stellar_Jaeger_crop2.png}\put (5,7) {\huge\color{green} \textbf{b}}\put (45,8) {\large\color{green} \textbf{2021-03-31}}\end{overpic} &
\begin{overpic}[width=0.24\linewidth]{282Pfiles/282P_20210404_2250_5B_5min_1600_R_Fichtl_crop2.png}\put (5,7) {\huge\color{green} \textbf{c}}\put (45,8) {\large\color{green} \textbf{2021-04-04}}\end{overpic} &
\begin{overpic}[width=0.24\linewidth]{282Pfiles/20220607utGMOSS_MADS_blue_arrows.png}\put (5,7) {\huge\color{green} \textbf{d}}\put (45,8) {\large\color{green} \textbf{2022-06-07}}\end{overpic}\\
\begin{overpic}[width=0.24\linewidth]{282Pfiles/2003_BM80_2012-03-28_09.15.27.160000_1534624p_chip23-ccd22_126arcsec_NuEl.png}\put (5,7) {\huge\color{green} \textbf{e}}\put (45,8) {\large\color{green} \textbf{2012-03-28}}\end{overpic} &
\begin{overpic}[width=0.24\linewidth]{282Pfiles/2003_BM80_2013-05-05_06.13.23.267946_c4d_130505_061323_opi_r_v1_chip42-N11_126arcsec_NuEl.png}\put (5,7) {\huge\color{green} \textbf{f}}\put (45,8) {\large\color{green} \textbf{2013-05-05}}\end{overpic} &
\begin{overpic}[width=0.24\linewidth]{282Pfiles/2003_BM80_2013-06-13_10.35.07.260000_1631654p_chip23-ccd22_126arcsec_NuEl.png}\put (5,7) {\huge\color{green} \textbf{g}}\put (45,8) {\large\color{green} \textbf{2013-06-13}}\end{overpic} &
\begin{overpic}[width=0.24\linewidth]{282Pfiles/2003_BM80_2021-03-17_06.13.35.614647_c4d_210317_061335_opi_i_v1_chip60-N29_126arcsec_NuEl.png}\put (5,7) {\huge\color{green} \textbf{h}}\put (45,8) {\large\color{green} \textbf{2021-03-17}}\end{overpic}
\end{tabular}
\caption{
Top row: four images, spanning 15 months, showing \objnameBMFull{} activity during the recent 2021--2022 activity epoch.
\textbf{(a)} Epoch II thumbnail image of 282P{} was classified as ``active'' by 14 of 15 volunteers of our Citizen Science project \textit{Active Asteroids}, a NASA Partner program. This 90~s $i$ band image was taken with the Dark Energy Camera on UT 2021 March 14, Prop. ID 2019A-0305 (\acs{PI} Drlica-Wagner).
\textbf{(b)} Epoch II, 12$\times$300~s co-added exposures imaged by Michael Jäger with a QHY600 camera on a 14'' Newtonian telescope in Weißenkirchen, Austria. Image reproduced with permission of Michael Jäger.
\textbf{(c)} Epoch II 5$\times$300~s co-added images captured by Roland Fichtl using a CDS cooled Canon 5D Mark III camera on a 16'' Newtonian telescope in Engelhardsberg, Germany. Image reproduced with permission of Roland Fichtl.
\textbf{(d)} For this most recent Epoch II image we co-added six 120~s $g'$ band images of 282P{} (green dashed arrow) we acquired on UT 7 June 2022 with the \ac{GMOS} imager on the 8.1~m Gemini South telescope (Prop. ID GS-2022A-DD-103, \acs{PI} Chandler); a tail is clearly visible (orange arrows).
Bottom row: Archival images of 282P{} that show clear evidence of activity. For each 126'$\times$126' thumbnail image, north is up and east is left. With the center of each image as the origin, the antisolar (yellow -$\odot$) and antimotion (red -$v$) directions (often correlated with tail appearance) are indicated. 282P{} is indicated by the green dashed arrow, and visible activity is marked by the white arrows.
\textbf{(e)} Epoch I image from UT 2012 March 28 MegaPrime 120~s $r$ band, Prop. ID 12AH16 (\acs{PI} Wainscoat).
\textbf{(f)} Epoch I image from UT 2013 May 5 DECam 150~s $r$ band, Prop. ID 2013A-0327 (\acs{PI} Rest).
\textbf{(g)} Epoch I image from UT 2013 June 13 MegaPrime 120~s $r$ band, Prop. ID 13AH09 (\acs{PI} Wainscoat).
\textbf{(h)} Epoch II image from UT 2021 March 17 DECam 90~s $i$ band, Prop. ID 2019A-0305 (\acs{PI} Drlica-Wagner).
}
\label{282P:fig:282P}
\end{figure*}
\objnameBMFull{}, hereafter 282P{}, was originally discovered as 2003~BM$_{80}$ on UT 2003 Jan 31 by Brian Skiff of the \ac{LONEOS} survey, and independently as 2003~FV$_{112}$ by \ac{LINEAR} on UT 2003 Apr 18. 282P{} was identified to be active during its 2012--2013 epoch (centered on its perihelion passage) in 2013 \citep{bolinComet2003BM2013}, at which time 282P{} was given the additional identifier 282P. Here, we introduce an additional activity epoch, spanning 2021--2022.
In this work we (1) present our \ac{NASA} Partner Citizen Science project \textit{Active Asteroids}, (2) describe how volunteers identified activity that led to our investigation into 282P{}, (3) present (a) archival images and (b) new observations of 282P{} that show it has undergone periods of activity during at least two epochs (2012--2013 and 2021--2022) spanning consecutive perihelion passages, (4) classify 282P{} as a \ac{QHO}, (5) explore the migratory nature of this object through dynamical modeling, including identification of a dynamical pathway between \acp{QHO} and active asteroids, and (6) determine volatile sublimation as the most probable activity mechanism.
\section{Citizen Science}
\label{282P:subsec:citsci}
We prepared thumbnail images (e.g., Figure \ref{282P:fig:282P}a) for examination by volunteers of our NASA Partner Citizen Science project \textit{Active Asteroids}, hosted on the Zooniverse\footnote{\url{https://www.zooniverse.org}} online Citizen Science platform. First we extract thumbnail images from publicly available \ac{DECam} archival images using a pipeline, \ac{HARVEST}, first described in \cite{chandlerSAFARISearchingAsteroids2018} and expanded upon in \cite{chandlerSixYearsSustained2019,chandlerCometaryActivityDiscovered2020a,chandler2483702005QN2021}. We optimize the Citizen Science process by excluding thumbnail images based on specific criteria, for example when (a) the image depth is insufficient for detecting activity, (b) no source was detected in the thumbnail center, and (c) too many sources were in the thumbnail to allow for reliable target identification.
Our workflow is simple: we show volunteers an image of a known minor planet and ask whether or not they see evidence of activity (like a tail or coma) coming from the object at the center of the image, as marked by a reticle (Figure \ref{282P:fig:282P}a). Each thumbnail is examined by at least 15 volunteers to minimize volunteer bias. To help train volunteers and validate that the project is working as intended, we created a training set of thumbnail images that we positively identified as showing activity, consisting of comets and other active objects, such as active asteroids. Training images are injected at random, though the interval of injection decays over time so that experienced volunteers only see a training image 5\% of the time.
We take the ratio of ``positive for activity'' classifications to the total number of classifications the object received, as a score to estimate the likelihood of the object being active. Members of the science team visually examines all images with a likelihood score of $\ge$80\% and flag candidates that warrant archival image investigation and telescope follow-up (Section \ref{282P:sec:observations}). We also learn of activity candidates through Zooniverse forums where users interact with each other, moderators, and our science team. Volunteers can share images they find interesting which has, in turn, led us directly to discoveries.
As of this writing, over 6,600 volunteers have participated in \textit{Active Asteroids}. They have conducted over 2.8$\times10^6$ classifications, completing assessment of over 171,000 thumbnail images. One image of 282P{} from UT 2021 March 14 (Figure \ref{282P:fig:282P}a) received a score of 93\% after 14 of 15 volunteers classified the thumbnail as showing activity. A second image from UT 2021 March 17 (Figure \ref{282P:fig:282P}h) was classified as active by 15 of 15 volunteers, providing additional strong evidence of activity from 2021 March.
\section{Observations}
\label{282P:sec:observations}
\subsection{Archival Data}
\label{282P:susbec:archivalData}
\begin{figure*}[ht]
\centering
\begin{tabular}{c}
\hspace{5mm}\includegraphics[width=0.80\linewidth]{282Pfiles/observability_Activity_dOnly.png}\\
\hspace{9mm}\includegraphics[width=0.92\linewidth]{282Pfiles/observability_323137_2003_BM80_at807_G37.pdf}\\
\hspace{-5mm}\includegraphics[width=0.85\linewidth]{282Pfiles/observability_TvsDate.png}
\end{tabular}
\caption{282P{} heliocentric distance (top), observability (middle) and temperature (bottom), from 2012 through 2024.
\textbf{Top:} activity detections (triangles) are marked as positive (filled red) and negative (unfilled blue) detections and as either
inbound ($\blacktriangledown$) or outbound ($\blacktriangle$).
Observations are cataloged in Table \ref{282P:tab:observationsTable}.
\textbf{Middle:} Our observability metric for \acf*{CTIO}, site code 807 (yellow dashed line) and the \acf*{LDT}, site code G37 (blue dashed line), depicting the number of hours 282P{} was observable ($>15^\circ$ between sunset and sunrise) during a given \ac{UT} observing date. Opposition events and conjunctions result in concurrent maxima and minima, respectively. Also indicated are perihelion (orange $q$) and aphelion (blue $Q$) passages.
\textbf{Bottom:} Modeled temperature by date for the thermophysical extremes: a ``flat slab'' ($\chi=1$, top line), and an isothermal body ($\chi=4$, bottom line).
}
\label{282P:fig:ActivityTimeline}
\end{figure*}
For each candidate active object stemming from \textit{Active Asteroids} we conduct an archival data investigation, following the procedure described in \cite{chandler2483702005QN2021}. For this task, we query public astronomical image archives and identify images which may show 282P{} in the \ac{FOV}. We download the data, extract thumbnail images centered on 282P{}, and visually examine all images to search for evidence of activity.
After visually inspecting $>400$ thumbnail images we found 57 images (listed in Table \ref{282P:tab:observationsTable}) in which we could confidently identify 282P{} in the frame. The remaining images either did not probe faintly enough, did not actually capture 282P{} (e.g., 282P{} was not on a detector), or suffered from image artifacts that made the image unsuitable for activity detection. The 57 images span 22 observing dates; nine dates had at least one image we ascertained showed probable activity, five from the 2012--2013 epoch and four dates from the 2021--2022 apparition. Section \ref{282P:sec:observations} provides a complete listing of observations used in this work.
Figure \ref{282P:fig:ActivityTimeline} shows three plots with shared $x$-axes (years).
Apparent magnitude and observability (the number of hours an object is above the horizon and the Sun is below the horizon) together provide insight into potential observational biases. For example, observations for detecting activity are ideal when 282P{} is brightest, near perihelion, and observable for many hours in an observing night. When contrasting hemispheres, this plot makes it clear that some periods (e.g., 2016 -- 2020) are more favorable for observations in the northern hemisphere, whereas other observation windows (e.g., 2013 -- 2015, 2022) are better suited to southern hemisphere facilities.
\subsection{Follow-up Telescope Observations}
\label{282P:subsec:telescopeobservations}
\paragraph{Magellan} During twilight on UT 2022 March 7 we observed 282P{} with the \ac{IMACS} instrument \citep{dresslerIMACSInamoriMagellanAreal2011} on the Magellan 6.5~m Baade telescope located atop Las Campanas Observatory (Chile). We successfully identified 282P{} in the images, however 282P{} was in front of a dense part of the Milky Way
preventing us from unambiguously identifying activity. We used these observations to inform our Gemini \ac{SNR} calculations.
\paragraph{VATT} On UT 2022 April 6 we observed 282P{} with the 1.8~m \ac{VATT} at the \ac{MGIO} in Arizona (Proposal ID S165, \ac{PI} Chandler). 282P{} was in an especially dense part of the galaxy so we conducted test observations to assess the viability of activity detection under these conditions. We concluded object detection would be challenging and activity detection essentially impossible in such a dense field.
\paragraph{LDT} On UT 2022 May 21 we observed 282P{} with the \ac{LDT} in Arizona (PI: Chandler). Finding charts indicated 282P{} was in a less dense field compared to our \ac{VATT} observations, however we were hardly able to resolve 282P{} or identify any activity because the field was still too crowded.
\paragraph{Gemini South} On UT 2022 June 7 we observed 282P{} with the \ac{GMOS} South instrument \citep{hookGeminiNorthMultiObjectSpectrograph2004,gimenoOnskyCommissioningHamamatsu2016} on the 8.1~m Gemini South telescope located atop Cerro Pachón in Chile (Proposal ID GS-2022A-DD-103, \acs{PI} Chandler). We timed this observation to take place during a $\sim$10 day window when 282P{} was passing in front of a less dense region of the Milky Way. We acquired eighteen images, six each in $g'$, $r'$, and $i'$. Activity was clearly visible in the reduced data in all filters, with activity appearing strongest in $g'$ (Figure \ref{282P:fig:282P}d). Our observations confirmed 282P{} was still active, 15 months after the 2021 archival data, evidence supporting sublimation as the most likely cause for activity (Section \ref{282P:sec:mechanism}).
\section{Dynamical Modeling}
\label{282P:subsec:dynamicalmodeling}
We analyzed 282P{} orbital characteristics in order to (1) determine its dynamical class (Section \ref{282P:sec:dynamicalClassification}), and (2) inform our activity mechanism assessment (Section \ref{282P:sec:mechanism}).
We simulated a cloud of 500 282P{} orbital clones, randomly drawn from Gaussian distributions centered on the current fitted parameters of 282P{}, with widths corresponding to uncertainties of those fits (Table \ref{282P:tab:ObjectData} lists parameters and associated uncertainties), as reported by \acs{JPL} Horizons \citep{giorginiJPLOnLineSolar1996}.
We modeled the gravitational influence of the Sun and the planets (except Mercury) on each orbital clone using the \ac{IAS15} N-body integrator \citep{reinIAS15FastAdaptive2015}, typically accurate to machine precision, with the \texttt{REBOUND} Python package \citep{reinREBOUNDOpensourceMultipurpose2012,reinHybridSymplecticIntegrators2019}. We ran simulations 1,000 years forward and backward through time. Longer integrations were unnecessary because dynamical chaos ensues prior to $\sim$200 years ago and after $\sim$350 years into the future. Beyond these times the orbit of 282P{} is not deterministic due to observational uncertainties.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.36\linewidth]{282Pfiles/dynamics_orb_323137_10e3years.png} & \includegraphics[width=0.48\linewidth]{282Pfiles/dynamics_a_323137_10e3years.png}\\
(a) & (b)\\
\\
\includegraphics[width=0.48\linewidth]{282Pfiles/dynamics_e_323137_10e3years.png} & \includegraphics[width=0.48\linewidth]{282Pfiles/dynamics_i_323137_10e3years.png}\\
(c) & (d)\\
\end{tabular}
\caption{
Results from dynamical integration of 282P{} orbital clones. For all plots, time $t=0$ corresponds to UT 2022 January 21. Jovian and Saturnian close encounters prevent accurate orbital parameter determination outside $-180\lesssim t\lesssim300$ yrs, given known orbital uncertainties.
\textbf{(a)} Orbital diagram for 282P{} and nearby planets.
\textbf{(b)} Semi-major axis $a$ evolution.
\textbf{(c)} Eccentricity $e$ evolution.
\textbf{(d)} Inclination $i$ evolution.
}
\label{282P:fig:orbitevolution1}
\end{figure*}
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.48\linewidth]{282Pfiles/dynamics_dJ_323137_10e3years.png} & \includegraphics[width=0.48\linewidth]{282Pfiles/dynamics_dS_323137_10e3years.png}\\
(a) & (b)\\
\\
\includegraphics[width=0.48\linewidth]{282Pfiles/dynamics_r_323137_10e3years.png} & \includegraphics[width=0.48\linewidth]{282Pfiles/dynamics_TJ_323137_10e3years.png}\\
(c) & (d)\\
\end{tabular}
\caption{
Additional results from dynamical integration of 282P{} orbital clones. For each plot, time $t=0$ is UT 2022 January 21. Close encounters with Jupiter and Saturn are so significant that orbital elements cannot be accurately determined before/after $-180\lesssim t\lesssim300$ yrs, given orbital uncertainties.
\textbf{(a)} Distance between Jupiter and 282P{} as a function of time. Indicated Hill radii provide references for the degree of orbit alteration imparted by a close encounter. For reference, the semi-major axes of two Jovian moons are shown: Callisto, the outermost Galilean satellite, and Sinope \citep{nicholsonDiscoveryNinthSatellite1914}, a likely captured \citep{gravPhotometricSurveyIrregular2003a} distant irregular and retrograde Jovian moon.
\textbf{(b)} Distance between Saturn and 282P{} as a function of time. The semi-major axis of the irregular Saturnian moon Phoebe, believed to be captured through close encounter \citep{johnsonSaturnMoonPhoebe2005,jewittIrregularSatellitesPlanets2007}, is given for reference.
\textbf{(c)} Heliocentric distance $r$ evolution.
\textbf{(d)} Tisserand parameter with respect to Jupiter $T_\mathrm{J}$ (Equation \ref{282P:eq:TJ}), where the horizontal orange line representing $T_\mathrm{J}=3$ indicates the widely-adopted boundary between comet-like and asteroid-like orbits.
}
\label{282P:fig:orbitevolution2}
\end{figure*}
Results from the dynamical evolution of the 282P{} orbital clones are shown in Figure \ref{282P:fig:orbitevolution1} and Figure \ref{282P:fig:orbitevolution2}. For all plots, time $t=0$ corresponds to \ac{JD} 2459600.5 (UT 2022 Jan 21) and time ranges from $t=-250$ to $t=+350$ (1772--2372 AD). Horizontal lines at distances of one, three, and five Hill radii (Equation \ref{282P:eq:rH}) from Jupiter and Saturn are shown in Figure \ref{282P:fig:orbitevolution2} panels a and b. The Hill Radius \citep{hillResearchesLunarTheory1878} $r_H$ is a metric of orbital stability and indicates the region where a secondary body (e.g., a planet) has dominant gravitational influence over a tertiary body (e.g., a moon), with both related to a primary body, such as the Sun. At pericenter, the Hill radius of the secondary body can be approximated as
\begin{equation}
r_\mathrm{H} \approx a(1-e)(m/3M)^{1/3},
\label{282P:eq:rH}
\end{equation}
\noindent where $a$, $e$, and $m$ are the semi-major axis, eccentricity and mass of the secondary (Jupiter or Saturn in our case), respectively, and $M$ is the mass of the primary (here, the Sun). Close passages of a small body within a few Hill radii of a planet are generally considered to be significant perturbations and may drastically alter the orbit of the small body (see \citealt{hamiltonOrbitalStabilityZones1992} Section 2.1.2 for discussion).
From $\sim$180 years ago until $\sim$300 years in the future, the orbit of 282P{} is well-constrained in our simulations. Figure \ref{282P:fig:orbitevolution2}a illustrates that 282P{} has roughly 10 close encounters (within $\sim$2 au) with Jupiter, and one with Saturn, over the range $-250<t<350$ yr. These encounters have a strong effect on the semi-major axis $a$ of 282P{} (Figure \ref{282P:fig:orbitevolution1}b), and, as illustrated by Figure \ref{282P:fig:orbitevolution2}d, a noticeable influence on its Tisserand parameter with respect to Jupiter $T_\mathrm{J}$,
\begin{equation}
T_\mathrm{J} = \frac{a_\mathrm{J}}{a} + 2\cos(i)\sqrt{\frac{a}{a_\mathrm{J}}\left(1-e^2\right)},
\label{282P:eq:TJ}
\end{equation}
\noindent where $a_\mathrm{J}$ is the semi-major axis of Jupiter and $a$, $e$ and $i$ are the semi-major axis, eccentricity and inclination of the body, respectively. $T_\mathrm{J}$ essentially describes an object's close approach speed to Jupiter or, in effect, the degree of dynamical perturbation an object will experience as a consequence of Jovian influence. $T_\mathrm{J}$ is often described as invariant \citep{kresakJacobianIntegralClassificational1972} or conserved, meaning that changes in orbital parameters still result in the same $T_\mathrm{J}$, although, in practice, its value does change slightly as a result of close encounters (see Figure \ref{282P:fig:orbitevolution2}d).
Due to the small Jupiter-centric distances of 282P{} during these encounters, compounded by its orbital uncertainties, the past orbit of 282P{}, prior to $t\approx-180$ yrs, is not deterministic. Dynamical chaos is plainly evident in all panels as orbital clones take a multitude of paths within the parameter space, resulting in a broad range of possible orbital outcomes due only to slight variations in initial 282P{} orbital parameters.
A consequential encounter with Saturn occurred around 1838 ($t\approx-184$~yr; Figure \ref{282P:fig:orbitevolution2}b), followed by another interaction with Jupiter in 1846 ($t=-176$ yr; Figure \ref{282P:fig:orbitevolution2}a). After these encounters 282P{} was a \ac{JFC} (100\% of orbital clones) with a semi-major axis between Jupiter's and Saturn's semi-major axes (Figure \ref{282P:fig:orbitevolution1}b), and crossing the orbits of both planets (Figure \ref{282P:fig:orbitevolution2}c). These highly perturbative passages placed 282P{} on the path that would lead to its current Quasi-Hilda orbit.
In 1940 ($t=-82$~yr), 282P{} had a very close encounter with Jupiter, at a distance of 0.3~au -- interior to one Hill radius. As seen in Figure \ref{282P:fig:orbitevolution1}a, this encounter dramatically altered 282P{}'s orbit, shifting 282P{} from an orbit primarily exterior to Jupiter to an orbit largely interior to Jupiter (Figure \ref{282P:fig:orbitevolution1}b). This same interaction also caused 282P{}'s orbit to migrate from Jupiter- and Saturn-crossing to only a Jupiter-crossing orbit (Figure \ref{282P:fig:orbitevolution2}c). This step in the orbital evolution of 282P{} also changed its $T_\mathrm{J}$ (Figure \ref{282P:fig:orbitevolution2}d) to be close to the traditional $T_\mathrm{J}=3$ comet--asteroid dynamical boundary. At this point in time, 282P{} remained a \ac{JFC} (100\% of orbital clones) despite its dramatic change in orbit.
Around $t\approx200$ yr, 282P{} crosses the $T_\mathrm{J}=3$ boundary dividing the \ac{JFC}s and the asteroids on the order of 10 times. Although no major changes in the orbit 282P{} occur during this time, because of the stringency of this boundary, relatively minor perturbations result in oscillation between dynamical classes.
After a major encounter with Jupiter around 2330 AD ($t\approx308$ yrs), dynamical chaos again becomes dominant and remains so for the rest of the simulation. Following this encounter, the orbit of 282P{} does not converge around any single solution. Slight diffusion following the previous several Jupiter passages are also visible in Figure \ref{282P:fig:orbitevolution1}b-d and Figure \ref{282P:fig:orbitevolution2}a-d, and these also add uncertainty concerning encounters around 2301 to 2306 ($t\approx280$ to $285$ yrs). Although we are unable to precisely determine past and future orbits of 282P{} outside of $-180\lesssim t\lesssim300$ because of dynamical chaos, we are able to examine the fraction of orbital clones that finish the simulation (forwards and backwards) on orbits associated with different orbital classes.
\section{Dynamical Classifications: Past, Present and Future}
\label{282P:sec:dynamicalClassification}
Minor planets are often classified dynamically, based on orbital characteristics such as semi-major axis. 282P{} was labeled a \ac{JFC} by \cite{hsiehMainbeltCometsPanSTARRS12015}, in agreement with a widely adopted system that classifies objects dynamically based on their Tisserand parameter with respect to Jupiter, $T_\mathrm{J}$ (Equation \ref{282P:eq:TJ}).
Via Equation \ref{282P:eq:TJ}, Jupiter's $T_\mathrm{J}$ is 2.998 given $a_\mathrm{J}=5.20$, $e_\mathrm{J}=0.049$, and $i_\mathrm{J}=0.013$. Notably, objects with $T_\mathrm{J}>3$ cannot cross the Jovian orbit, thus their orbits are entirely interior or exterior to Jupiter's orbit \citep{levisonCometTaxonomy1996}.
Objects with $T_\mathrm{J}<3$ are considered cometary \citep{levisonCometTaxonomy1996}, while those with $T_\mathrm{J}>3$ are not \citep{vaghiOriginJupiterFamily1973,vaghiOrbitalEvolutionComets1973}, a classification approach first suggested by \cite{carusiHighOrderLibrationsHalleyType1987,carusiCometTaxonomy1996}. \acp{JFC} have $2<T_\mathrm{J}<3$ (see e.g., \citealt{jewittActiveCentaurs2009}),
and Damocloids and have $T_\mathrm{J}<2$ \citep{jewittFirstLookDamocloids2005}.
We note, however, that the traditional $T_\mathrm{J}$ asteroid -- \ac{JFC} -- Damocloid continuum does not include (or exclude) \acp{QHO}.
As discussed in Section \ref{282P:introduction}, we adopt the \cite{jewittActiveCentaurs2009} definition of Centaur, which stipulates that a Centaur has an orbit entirely exterior to Jupiter, with both $q$ and $a$ interior to Neptune, and the body is not in 1:1 resonance with a planet. 282P{} has a semi-major axis $a=4.240$~au, well interior to Jupiter's $a_\mathrm{J}=5.2$~au. This disqualifies 282P{} as presently on a Centaurian orbit.
Active objects other than comets orbiting interior to Jupiter are primarily the active asteroids, defined as (1) $T_\mathrm{J}>3$, (2) displaying comet-like activity, and (3) orbiting outside of mean-motion resonance with any of the planets. This last stipulation rules out the Jupiter Trojans (1:1 resonance) and the Hildas (3:2 resonance with Jupiter), even though both classes have members above and below the $T_\mathrm{J}=3.0$ asteroid--comet transition line. We compute $T_\mathrm{J}=2.99136891\pm(3.73\times10^{-8})$ for 282P{} (see Table \ref{282P:tab:ObjectData} for a list of orbital parameters). These values do not exceed the traditional $T_\mathrm{J}=3$ cutoff; thus 282P{} cannot be considered an active asteroid in its current orbit. \acp{MBC} are an active asteroid subset defined as orbiting entirely within the main asteroid belt \citep{hsiehMainbeltCometsPanSTARRS12015}. Figure \ref{282P:fig:orbitevolution2}c shows that 282P{}'s heliocentric distance does not stay within the boundaries of the Asteroid Belt (i.e., between the orbits of Mars and Jupiter), and so 282P{} does not qualify as a \ac{MBC}.
\begin{figure*}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.32\linewidth]{282Pfiles/corotating_with_jupiter_Elst-Pizarro.pdf} &
\includegraphics[width=0.32\linewidth]{282Pfiles/corotating_with_jupiter_67P.pdf} &
\includegraphics[width=0.32\linewidth]{282Pfiles/corotating_with_jupiter_Chiron.pdf}\\
(a) & (b) & (c)\\
\\
\includegraphics[width=0.32\linewidth]{282Pfiles/corotating_with_jupiter_Hilda.pdf} &
\includegraphics[width=0.32\linewidth]{282Pfiles/corotating_with_jupiter_246P.pdf} &
\includegraphics[width=0.32\linewidth]{282Pfiles/corotating_with_jupiter_282P.pdf} \\
(d) & (e) & (f)\\
\end{tabular}
\caption{The orbital motion of minor planets (blue lines) as seen in the reference frame corotating with Jupiter (orange lines at right edge of plots).
(a) \ac{MBC} (7968)~Elst-Pizarro (133P).
(b) \ac{JFC} 67P/Churyumov-Gerasimenko (previously visited by the \ac{ESA} Rosetta Spacecraft).
(c) Centaur (2060)~Chiron (95P).
(d) (153)~Hilda, the namesake of the Hilda dynamical class, in the 3:2 interior mean-motion resonance with Jupiter.
(e) Quasi-Hilda 246P/\acs{NEAT}, also designated 2010~V$_2$ and 2004~F$_3$.
(f) Our object of study, 282P{}, in its Quasi-Hilda orbit.
}
\label{282P:fig:corotatingFrame}
\end{figure*}
Blurring the lines between \ac{JFC} and Hilda is the Quasi-Hilda regime. A Quasi-Hilda, also referred to as a \ac{QHO}, \ac{QHA} \citep{jewittOutburstingQuasiHildaAsteroid2020}, or \ac{QHC}, is a minor planet on an orbit similar to a Hilda \citep{tothQuasiHildaSubgroupEcliptic2006,gil-huttonCometCandidatesQuasiHilda2016}. Hildas are defined by their 3:2 interior mean-motion resonance with Jupiter, however Quasi-Hildas are not in this resonance, though they do orbit near it. Quasi-Hildas likely migrated from the \ac{JFC} region (see discussion, \citealt{jewittOutburstingQuasiHildaAsteroid2020}). We favor the term \ac{QHO} or \ac{QHA} over \ac{QHC}, given that fewer than 15 Quasi-Hildas have been found to be active, while the remainder of the $>270$ identified Quasi-Hildas \citep{gil-huttonCometCandidatesQuasiHilda2016} have not been confirmed to be active. Notable objects from the Quasi-Hilda class are 39P/Oterma \citep{otermaNEWCOMETOTERMA1942}, an object that was a Quasi-Hilda prior to 1963, when a very close (0.095~au) encounter with Jupiter redirected the object into a Centuarian orbit. Another notable Quasi-Hilda was D/Shoemaker-Levy~9, which famously broke apart and impacted Jupiter in 1994 (e.g., \citealt{weaverHubbleSpaceTelescope1995}).
Quasi-Hildas have orbital parameters similar to that of the Hildas, approximately $3.7 \lesssim a \lesssim 4.2$~au, $e\le0.3$, and $i\le20^\circ$. In rough agreement, 282P{} has $a=4.24$~au, $e=0.188$, and $i=5.8^\circ$ (Table \ref{282P:tab:ObjectData}). Hildas are also known for their trilobal orbits as viewed in the Jupiter corotating frame (caused by their residence in the 3:2 interior mean motion resonance with Jupiter), especially the namesake asteroid (153)~Hilda (Figure \ref{282P:fig:corotatingFrame}d). Because (153)~Hilda is in a stable 3:2 resonant orbit with Jupiter, its orbit remains roughly constant, with a small amount of libration over time. By contrast, Quasi-Hildas like 246P/\acs{NEAT} (Figure \ref{282P:fig:corotatingFrame}e) are near the same resonance and show signs of this characteristic trilobal pattern, however their orbits drift considerably on timescales of hundreds of years. 282P{} (Figure \ref{282P:fig:corotatingFrame}f) also displays a typical Quasi-Hilda orbit as viewed in the Jupiter corotating reference frame.
In the past, prior to 250~yr ago, 52\% (260) of the 500 orbital clones were \acp{JFC}, 48\% (239) were Cenaturs, 5\% (26) were already \acp{QHO}, and one (0.2\%) was an \ac{OMBA}. The most probable scenario prior to 250 years ago was that was either a \ac{JFC} or Centaur, both classes that trace their origins to the Kuiper Belt (see reviews, \citealt{morbidelliKuiperBeltFormation2020} and \citealt{jewittActiveCentaurs2009}, respectively).
In the future, after 350 years time, 81\% (403) of clones become \acp{JFC}, 18\% (90) remain \acp{QHO}, 14\% (69) become \acp{OMBA}, and 5.6\% (28) return to Centaurian orbits. Clearly the most likely scenario is that 282P{} will become a \ac{JFC}, however there are still significant possibilities that 282P{} remains a \ac{QHO} or becomes an active \ac{OMBA}.
\section{Thermodynamical Modeling}
\label{282P:sec:thermo}
In order to understand the approximate temperature ranges that 282P{} experiences over the course of its present orbit in order to (1) understand what role, if any, thermal fracture may play in the activity we observe, and (2) evaluate the likelihood of ices surviving on the surface, albeit with limited effect because of the narrow window ($\sim$500 years) of dynamically well-determined orbital parameters available (Section \ref{282P:subsec:dynamicalmodeling}).
Following the procedure of \cite{chandler2483702005QN2021} (originally adapted from \citealt{hsiehMainbeltCometsPanSTARRS12015}), we compute the surface equilibrium temperature $T_\mathrm{eq}$ for 282P{} as a gray airless body. $\chi$ describes the distribution of heat over the surface of a body, with $\chi=1$ the isothermal approximation (i.e., a fast-rotating body) and $\chi=4$ the ``slab'' case, where an object has one side that always faces the Sun; these cases result in the minimum and maximum expected temperatures for the body, respectively.
Solving Equations 3 -- 5 of \cite{chandler2483702005QN2021} (energy balance for a gray airless body, sublimation mass loss rate, and the Clausius--Clapeyron relationship) for the body's heliocentric distance $r_\mathrm{h}$ (in au) as a function of equilibrium temperature $T_\mathrm{eq}$ and $\chi$, where
$A=0.05$ is the assumed typical bond albedo,
$\Delta H_\mathrm{subl}=51.06\ \mathrm{MJ}/\mathrm{kmol}$ is the ice-to-gas heat of sublimation,
$\epsilon = 0.9$ is the assumed typical effective infrared emissivity,
$f_\mathrm{D}=1$ accounts for sublimation efficiency dampening due to mantling (unity in the absence of mantle),
$F_\odot$ is the solar constant 1360~$\mathrm{W}/\mathrm{m}^2$,
$L=2.83\ \mathrm{MJ}/\mathrm{kg}$ is the latent heat of H$_2$O ice (approximated here as temperature independent),
$R_\mathrm{G}=8314\ \mathrm{J}\ \mathrm{kmol}^{-1}\ \mathrm{K}^{-1}$ is the ideal gas constant,
and
$\sigma=5.67037\times10^{-8}\ \mathrm{W}\ \mathrm{m}^{-2}\ \mathrm{K}^{-4}$ is the Stefan--Boltzmann constant:
\begin{equation}
r_\mathrm{h}(T_\mathrm{eq},\chi) = \frac{F_\odot\left(1-A\right)\chi^{-1}}{\epsilon\sigma T_\mathrm{eq}^4 + L f_\mathrm{D} \cdot 611\ e^{\frac{\Delta H_\mathrm{subl}}{R_\mathrm{G}}\left(\frac{1}{273.16\mathrm{K}} - \frac{1}{T_\mathrm{eq}}\right)}}
\label{282P:eq:teq}
\end{equation}
We translate Equation \ref{282P:eq:teq} to a function of equilibrium temperature $T_\mathrm{eq}$ by computing $r_\mathrm{h}$ for an array of values (100~K to 300~K in this case), then fit a model to these data with a \texttt{SciPy} \citep{virtanenSciPyFundamentalAlgorithms2020} (Python package) univariate spline. Using this model we compute 282P{} temperatures for 282P{} heliocentric distances from perihelion and aphelion
with this function to arrive at temperatures for 282P{} over the course of its orbit.
Figure \ref{282P:fig:ActivityTimeline} (bottom panel) shows the temperature evolution for the maximum and minimum solar heating distribution scenarios ($\chi=1$ and $\chi=4$, respectively) for 282P{} from 2012 through 2024. Temperatures range between roughly 175~K and 220~K for $\chi=1$, or 130~K and 160~K for $\chi=4$, with a $\sim45$~K maximum temperature variation in any one orbit. 282P{} spends some ($\chi=4$) or all ($\chi=1$) of its time with surface temperatures above 145~K. Water ice is not expected to survive above this temperature on Gyr timescales \citep{schorghoferLifetimeIceMain2008,snodgrassMainBeltComets2017}, however we showed in Section \ref{282P:subsec:dynamicalmodeling} that, prior to $\sim80$ years ago, 282P{} had a semi-major axis of $a>6$~au, a region much colder than 145~K. Even if 282P{} had spent most of its life with temperatures at the high end of our computed temperatures ($>220$~K), water ice can survive on Gyr timescales at shallow (a few cm) depths \citep{schorghoferLifetimeIceMain2008,prialnikCanIceSurvive2009}. Some bodies, such as (24)~Themis, have been found to have surface ices \citep{campinsWaterIceOrganics2010,rivkinDetectionIceOrganics2010} that suggest that an unknown mechanism may replenish surface ice with subsurface volatiles. In this case the ice lifetimes could be greatly extended.
\section{Activity Mechanism}
\label{282P:sec:mechanism}
Infrequent stochastic events, such as impacts (e.g., (596)~Scheila, \citealt{bodewitsCollisionalExcavationAsteroid2011,ishiguroObservationalEvidenceImpact2011,moreno596ScheilaOutburst2011}), are highly unlikely to be the activity mechanism given the multi-epoch nature of the activity we identified in this work. Moreover, it is unlikely that activity ceased during the 15 month interval between the UT 2021 March 14 archival activity and our UT 7 June 2022 Gemini South activity observations (Section \ref{282P:sec:observations}), when 282P{} was at a heliocentric distance $r_\mathrm{H}=3.548$~au and $r_\mathrm{H}$=3.556~au, respectively, and 282P{} was only closer to the Sun in the interim. Similarly, our archival data shows activity lasted $\sim15$ months during the 2012 -- 2013 apparition.
Recurrent activity is most commonly caused by volatile sublimation (e.g., 133P, \citealt{boehnhardtComet1996N21996,hsiehStrangeCase133P2004}) or rotational instability (e.g., (6478)~Gault, \citealt{kleynaSporadicActivity64782019,chandlerSixYearsSustained2019}). Rotational instability is impossible to rule out entirely for 282P{} because its rotation period is unknown. However, (1) no activity attributed to rotational stability for any object has been observed to be continuous for as long as the 15 month episodes we report, and (2) rotational instability is not correlated with perihelion passage. It is worth noting that there are not yet many known objects with activity attributed to rotational disruption, so it is still difficult to draw firm conclusions about the behavior of those objects. In any case it would be useful to measure a rotation period for 282P{} to help assess potential influence of rotational instability in the observed activity of 282P{}. The taxonomic class of 282P{} is unknown, but should 282P{} be classified as a member of a desiccated spectral class (e.g., S-type), then sublimation would not likely be the underlying activity mechanism. Color measurements or spectroscopy when 282P{} is quiescent would help determine its spectral class.
A caveat, however, is that many of our archival images were taken when 282P{} was significantly fainter than the images showing activity
(Figure \ref{282P:fig:ActivityTimeline}),
thereby making activity detection more difficult than if 282P{} was brighter. Consequently, archival images showing 282P{} were predominitely taken near its perihelion passage. The farthest evidently quiescent image of 282P{} was taken when it was at $\sim$4~au (Figure \ref{282P:fig:ActivityTimeline}). Thus we cannot state with total certainty that 282P{} was inactive elsewhere in its orbit.
Thermal fracture can cause repeated activity outbursts. For example, (3200)~Phaethon undergoes 600~K temperature swings, peaking at 800~K -- 1100~K, exceeding the serpentine-phyllosilicate decomposition threshold of 574~K \citep{ohtsukaSolarRadiationHeatingEffects2009}, and potentially causing thermal fracture \citep{licandroNatureCometasteroidTransition2007,kasugaObservations1999YC2008} including mass loss \citep{liRecurrentPerihelionActivity2013,huiResurrection3200Phaethon2017}. Temperatures on 282P{} reach at most $\sim220$~K (Figure \ref{282P:fig:ActivityTimeline}), with $\sim45$~K the maximum variation. Considering the relatively low temperatures and mild temperature changes we (1) consider it unlikely that 282P{} activity is due to thermal fracture, and (2) reaffirm that thermal fracture is generally considered a nonviable mechanism for any objects other than \acp{NEO}.
Overall, we find volatile sublimation on 282P{} the most likely activity mechanism, because (1) it is unlikely that an object originating from the Kuiper Belt such as 282P{} would be desiccate
, (2) archival and new activity observations are from when 282P{} was near perihelion (Figure \ref{282P:fig:ActivityTimeline}), a characteristic diagnostic of sublimation-driven activity \citep[e.g.,][]{hsiehOpticalDynamicalCharacterization2012}, and (3) 15 months of continuous activity has not been reported for any other activity mechanism (e.g., rotational instability, impact events) to date, let alone two such epochs.
\section{Summary and Future Work}
\label{282P:sec:summary}
This study was prompted by Citizen Scientists from the NASA Partner program \textit{Active Asteroids} classifying two images of 282P{} from 2021 March as showing activity. Two additional images by astronomers Roland Fichtl and Michael Jäger brought the total number of images (from UT 2021 March 31 and UT 2021 April 4) to four. We conducted follow-up observations with the Gemini South 8.1~m telescope on UT 2022 June 7 and found 282P{} still active, indicating it has been active for $>15$ months during the current 2021 -- 2022 activity epoch. Our archival investigation revealed the only other known apparition, from 2012--2013, also spanned $\sim15$ months. Together, our new and archival data demonstrate 282P{} has been active during two consecutive perihelion passages, consistent with sublimation-driven activity.
We conducted extensive dynamical modeling and found 282P{} has experienced a series of $\sim5$ strong interactions with Jupiter and Saturn in the past, and that 282P{} will again have close encounters with Jupiter in the near future. These interactions are so strong that dynamical chaos dominates our simulations prior to 180 years ago and beyond 350 years in the future, but we are still able to statistically quantify a probable orbital class for 282P{} prior to $-180$ yr (52\% \acp{JFC}, 48\% Centaur) and after $+350$ yr (81\% \acp{JFC}, 18\% \ac{QHO}, 14\% \ac{OMBA}). We classify present-day 282P{} as a \acf{QHO}.
We carried out thermodynamical modeling that showed 282P{} undergoes temperatures ranging at most between 135~K and 220~K, too mild for thermal fracture but warm enough that surface water ice would not normally survive on timescales of the solar system lifetime. However, 282P{} arrived at its present orbit recently; prior to 1941 282P{} was primarily exterior to Jupiter's orbit and, consequently, sufficiently cold for water ice to survive on its surface. Given that both activity apparitions (Epoch I: 2012 -- 2013 and Epoch II: 2021 -- 2022) each lasted over 15 months, and both outbursts spanned perihelia passage, we determine the activity mechanism to most likely be volatile sublimation.
Coma likely accounts for the majority of the reflected light we observe emanating from 282P{}, so it is infeasible to determine the color of the nucleus and, consequently, 282P{}'s spectral class (e.g., C-type, S-type). Measuring its rotational period would also help assess what (if any) role rotational instability plays in the observed activity. Specifically, a rotation period faster than the spin-barrier limit of two hours would indicate breakup.
Most images of 282P{} were taken when it was near perihelion passage (3.441~au), though there were observations from Epoch I that showed 282P{} clearly, without activity, when it was beyond $\sim$4~au. 282P{} is currently outbound and will again be beyond 4~au in mid-2023 and, thus, likely inactive; determining if/when 282P{} returns to a quiescent state would help bolster the case for sublimation-driven activity because activity occurring preferentially near perihelion, and a lack of activity elsewhere, is characteristic of sublimation-driven activity.
282P{} is currently observable, especially from the southern hemisphere, however the object is passing in front of dense regions of the Milky Way until the end of 2022 November (see Lowell \texttt{AstFinder}\footnote{\url{https://asteroid.lowell.edu/astfinder/}} finding charts). 282P{} will be in a less dense region of the Milky Way and be observable, in a similar fashion to our Gemini South observations (Section \ref{282P:sec:observations}) on UT 2022 September 26 for $\sim$12 days, carefully timed for sky regions with fewer stars. As Earth's orbit progresses around the Sun, 282P{} becomes observable for less time each night through 2022 November, until UT 2022 December 26, when it becomes observable only during twilight. Observations during this window would help constrain the timeframe for periods of quiescence.
\section{Acknowledgements}
\label{282P:sec:acknowledgements}
We thank Dr.\ Mark Jesus Mendoza Magbanua of \ac{UCSF} for his frequent and timely feedback on the project. Many thanks for the helpful input from Henry Hsieh of the \ac{PSI} and David Jewitt of \ac{UCLA}.
The authors express their gratitude to Prof. Mike Gowanlock (\acs{NAU}), Jay Kueny of \ac{UA} and Lowell Observatory, and the Trilling Research Group (\acs{NAU}), all of whom provided invaluable insights which substantially enhanced this work. The unparalleled support provided by Monsoon cluster administrator Christopher Coffey (\acs{NAU}) and the High Performance Computing Support team facilitated the scientific process.
We thank Gemini Observatory Director Jennifer Lotz for granting our \ac{DDT} request for observations, German Gimeno for providing science support, and Pablo Prado for observing. Proposal ID GS-2022A-DD-103, \acs{PI} Chandler.
The VATT referenced herein refers to the Vatican Observatory's Alice P. Lennon Telescope and Thomas J. Bannan Astrophysics Facility. We are grateful to the Vatican Observatory for the generous time allocations (Proposal ID S165, \acs{PI} Chandler). We especially thank Vatican Observatory Director Br. Guy Consolmagno, S.J. for his guidance, Vice Director for Tucson Vatican Observatory Research Group Rev.~Pavel Gabor, S.J. for his frequent assistance, Astronomer and Telescope Scientist Rev. Richard P. Boyle, S.J. for patiently training us to use the \ac{VATT} and for including us in minor planet discovery observations, Chris Johnson (\ac{VATT} Facilities Management and Maintenance) for many consultations that enabled us to resume observations, Michael Franz (\acs{VATT} Instrumentation) and Summer Franks (\ac{VATT} Software Engineer) for on-site troubleshooting assistance, and Gary Gray (\ac{VATT} Facilities Management and Maintenance) for everything from telescope balance to building water support, without whom we would have been lost.
This material is based upon work supported by the \acs{NSF} \ac{GRFP} under grant No.\ 2018258765. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the \acl{NSF}. The authors acknowledge support from the \acs{NASA} Solar System Observations program (grant 80NSSC19K0869, PI Hsieh) and grant 80NSSC18K1006 (PI: Trujillo).
Computational analyses were run on Northern Arizona University's Monsoon computing cluster, funded by Arizona's \ac{TRIF}. This work was made possible in part through the State of Arizona Technology and Research Initiative Program.
\acf{WCS} corrections facilitated by the \textit{Astrometry.net} software suite \citep{langAstrometryNetBlind2010}.
This research has made use of data and/or services provided by the \ac{IAU}'s \ac{MPC}.
This research has made use of \acs{NASA}'s Astrophysics Data System.
This research has made use of The \acf{IMCCE} SkyBoT Virtual Observatory tool \citep{berthierSkyBoTNewVO2006}.
This work made use of the \texttt{FTOOLS} software package hosted by the \acs{NASA} Goddard Flight Center High Energy Astrophysics Science Archive Research Center.
\ac{SAO} \ac{DS9}: This research has made use of \texttt{\acs{SAO}Image\acs{DS9}}, developed by \acl{SAO} \citep{joyeNewFeaturesSAOImage2006}. \acf{WCS} validation was facilitated with Vizier catalog queries \citep{ochsenbeinVizieRDatabaseAstronomical2000} of the Gaia \ac{DR} 2 \citep{gaiacollaborationGaiaDataRelease2018} and the \acf{SDSS DR-9} \citep{ahnNinthDataRelease2012} catalogs.
This work made use of AstOrb, the Lowell Observatory Asteroid Orbit Database \textit{astorbDB} \citep{bowellPublicDomainAsteroid1994,moskovitzAstorbDatabaseLowell2021}.
This work made use of the \texttt{astropy} software package \citep{robitailleAstropyCommunityPython2013}.
Based on observations at \ac{CTIO}, \acs{NSF}'s \acs{NOIRLab} (\acs{NOIRLab} Prop. ID 2019A-0305; \acs{PI}: A. Drlica-Wagner, \acs{NOIRLab} Prop. ID 2013A-0327; \acs{PI}: A. Rest), which is managed by the \acf{AURA} under a cooperative agreement with the \acl{NSF}.
This project used data obtained with the \acf{DECam}, which was constructed by the \acf{DES} collaboration. Funding for the \acs{DES} Projects has been provided by the US Department of Energy, the US \acl{NSF}, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute for Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A\&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Enérgeticas, Medioambientales y Tecnológicas–Madrid, the University of Chicago, University College London, the \acs{DES}-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, \acs{NSF}'s \acs{NOIRLab}, the University of Nottingham, the Ohio State University, the OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, \ac{SLAC} National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A\&M University.
These results made use of the \acf{LDT} at Lowell Observatory. Lowell is a private, non-profit institution dedicated to astrophysical research and public appreciation of astronomy and operates the \acs{LDT} in partnership with Boston University, the University of Maryland, the University of Toledo, \acf{NAU} and Yale University. The \acf{LMI} was built by Lowell Observatory using funds provided by the \acf{NSF} (AST-1005313).
\ac{VST} OMEGACam \citep{arnaboldiVSTVLTSurvey1998,kuijkenOmegaCAM16k16k2002,kuijkenOmegaCAMESONewest2011} data were originally acquired as part of the \ac{KIDS} \citep{dejongFirstSecondData2015}.
The \acs{Pan-STARRS}1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the \acs{Pan-STARRS} Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the \ac{LCOGT} Network Incorporated, the National Central University of Taiwan, the \acl{STScI}, the \acl{NASA} under Grant No. NNX08AR22G issued through the Planetary Science Division of the \acs{NASA} Science Mission Directorate, the \acf{NSF} Grant No. AST-1238877, the University of Maryland, \ac{ELTE}, the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.
Based on observations obtained with MegaPrime/MegaCam, a joint project of \ac{CFHT} and \ac{CEA}/\ac{DAPNIA}, at the \ac{CFHT} which is operated by the \acf{NRC} of Canada, the Institut National des Science de l'Univers of the \acf{CNRS} of France, and the University of Hawaii. The observations at the \acf{CFHT} were performed with care and respect from the summit of Maunakea which is a significant cultural and historic site.
Magellan observations made use of the \ac{IMACS} instrument \citep{dresslerIMACSInamoriMagellanAreal2011}.
This research has made use of the \acs{NASA}/\ac{IPAC} \ac{IRSA}, which is funded by the \acl{NASA} and operated by the California Institute of Technology.
\subsection{Facilities}
Astro Data Archive,
Blanco (DECam),
CFHT (MegaCam),
Gaia,
Gemini-South (GMOS-S),
IRSA,
LDT (LMI),
Magellan: Baade (TSIP),
PO:1.2m (PTF, ZTF),
PS1,
Sloan,
VATT (VATT4K),
VST (OmegaCAM)
\subsection{Software}
{\tt astropy} \citep{robitailleAstropyCommunityPython2013},
{\tt astrometry.net} \citep{langAstrometryNetBlind2010},
{\tt FTOOLS}\footnote{\url{https://heasarc.gsfc.nasa.gov/ftools/}},
{\tt IAS15} integrator \citep{reinIAS15FastAdaptive2015},
{\tt JPL Horizons} \citep{giorginiJPLOnLineSolar1996},
{\tt Matplotlib} \citep{hunterMatplotlib2DGraphics2007},
{\tt NumPy} \citep{harrisArrayProgrammingNumPy2020},
{\tt pandas} \citep{mckinneyDataStructuresStatistical2010,rebackPandasdevPandasPandas2022},
{\tt REBOUND} \citep{reinREBOUNDOpensourceMultipurpose2012,reinHybridSymplecticIntegrators2019},
{\tt SAOImageDS9} \citep{joyeNewFeaturesSAOImage2006},
{\tt SciPy} \citep{virtanenSciPyFundamentalAlgorithms2020},
{\tt Siril}\footnote{\url{https://siril.org}},
{\tt SkyBot} \citep{berthierSkyBoTNewVO2006},
{\tt termcolor}\footnote{\url{https://pypi.org/project/termcolor}},
{\tt tqdm} \citep{costa-luisTqdmFastExtensible2022},
{\tt Vizier} \citep{ochsenbeinVizieRDatabaseAstronomical2000}
\clearpage
\thispagestyle{empty}
\section{Appendix}
\label{282P:sec:appendix}
\clearpage
\footnotesize
\atxy{\dimexpr1in}{.5\paperheight}{\rotatebox[origin=center]{270}{\thepage}}
\singlespacing
\begin{sidewaystable}
\caption{Table of Observations}
\label{282P:tab:observationsTable}
\centering
\begin{tabular}{cccccrccccrrc}
Fig.$^\mathrm{a}$ & Act.$^\mathrm{b}$ & Obs. Date$^\mathrm{c}$ & Source & $N^\mathrm{d}$ & Exp. [s]$^\mathrm{e}$ & Filter(s) & V$^\mathrm{f}$ & $r$ [au]$^\mathrm{g}$ & STO [$^\circ$]$^\mathrm{h}$ & $\nu$ [$^\circ$]$^\mathrm{i}$ & \%$_{Q\rightarrow q}^\mathrm{j}$ & Note$^\mathrm{k}$\\
\hline\hline
& & 04 Feb 2011 & PS1 & 2 & 40 & $r$ & 19.7 & 4.26 & 3.3 & 258.1 & 84\% & \ref{20120224} \\
& & 16 Feb 2012 & PS1 & 1 & 45 & $i$ & 19.3 & 3.69 & 8.9 & 306.4 & 95\% & \ref{20120224} \\
& & 24 Feb 2012 & PS1 & 1,1 &43, 40 & $g$, $r$ & 19.2 & 3.68 & 6.9 & 307.6 & 95\% & \obsnote{20120224}\ref{20120224} \\
& & 26 Feb 2012 & PS1 & 2 & 40 & $r$ & 19.2 & 3.67 & 6.3 & 307.9 & 96\% & \ref{20120224}\\
\ref{282P:fig:282P}e & Y & 28 Mar 2012 & MegaPrime & 2 & 120 & $r$ & 18.9 & 3.64 & 3.2 & 312.6 & 96\% & \obsnote{20120328}\ref{20120328}\\
& Y & 05 Jul 2012 & OmegaCAM & 4 & 240 & $i$ & 20.1 & 3.54 & 16.1 & 328.0 & 98\% & \obsnote{20120705}\ref{20120705} \\
& & 14 Apr 2013 & PS1 & 2 & 45 & $i$ & 19.3 & 3.47 & 12.8 & 14.9 & 99\% & \ref{20120224}\\
& & 22 Apr 2013 & PS1 & 2 & 30 & $z$ & 19.2 & 3.47 & 11.1 & 16.3 & 99\% & \ref{20120224}\\
\ref{282P:fig:282P}f & Y & 05 May 2013 & \acs{DECam} & 2 & 150 & $r$ & 19.5 & 3.48 & 12.9 & 318.4 & 97\% & \obsnote{20130505}\ref{20130505} \\
& Y & 15 May 2013 & PS1 & 1 & 43 & $g$ & 18.8 & 3.48 & 5.2 & 20.1 & 99\% & \ref{20120224}\\
\ref{282P:fig:282P}g & Y & 13 Jun 2013 & MegaPrime & 10 & 120 & $r$ & 18.8 & 3.50 & 4.9 & 24.8 & 99\% & \obsnote{20130613}\ref{20130613} \\
& & 03 Aug 2013 & PS1 & 2 & 80,60 & $y$, $z$ & 19.6 & 3.54 & 15.3 & 33.0 & 98\% & \ref{20120224}\\
& & 11 Jun 2014 & PS1 & 2 & 45 & $i$ & 20.0 & 3.95 & 12.5 & 78.1 & 90\% & \ref{20120224}\\
& & 14 Aug 2014 & PS1 & 3 & 45 & $i$ & 19.4 & 4.05 & 3.3 & 86.1 & 88\% & \ref{20120224}\\
& & 15 Aug 2014 & PS1 & 4 & 45 & $i$ & 19.4 & 4.04 & 3.5 & 86.2 & 88\% & \ref{20120224}\\
& & 04 Jan 2021 & \acs{ZTF} & 1 & 30 & $r$ & 20.0 & 3.63 & 15.7 & 312.5 & 96\% & \obsnote{20210104}\ref{20210104}\\
& & 07 Jan 2021 & \acs{ZTF} & 1 & 30 & $g$ & 20.0 & 3.63 & 15.7 & 312.9 & 96\% & \ref{20210104}\\
& & 09 Jan 2021 & \acs{ZTF} & 1 & 30 & $r$ & 20.0 & 3.62 & 15.7 & 313.2 & 96\% & \ref{20210104}\\
\ref{282P:fig:282P}a & Y & 14 Mar 2021 & \acs{DECam} & 1 & 90 & $i$ & 18.9 & 3.55 & 6.1 & 323.1 & 98\% & \obsnote{20210314}\ref{20210314} \\
\ref{282P:fig:282P}h & Y & 17 Mar 2021 & \acs{DECam} & 1 & 90 & $i$ & 18.9 & 3.55 & 5.2 & 323.6 & 98\% & \ref{20210314} \\
\ref{282P:fig:282P}b & Y & 31 Mar 2021 & QHY600 & 1 & 2160 & UV/IR & 18.5 & 3.54 & 0.7 & 325.9 & 98\% & \obsnote{20210331}\ref{20210331}\\
\ref{282P:fig:282P}c & Y & 04 Apr 2021 & CDS-5D & 1 & 1500 & (none) & 18.5 & 3.54 & 0.5 & 326.4 & 98\% & \obsnote{20210404}\ref{20210404} \\
& & 07 Mar 2022 & \acs{IMACS} & 5 & 10 & WB4800-7800&20.0 & 3.48 & 16.3 & 22.3 & 99\% & \obsnote{20220307}\ref{20220307}\\
& & 21 May 2022 & \acs{LDT} & 3 & 90 & VR, $i$ & 19.1 & 3.54 & 8.3 & 34.4 & 98\% & \obsnote{20220521}\ref{20220521}\\
\ref{282P:fig:282P}d & Y & 07 Jun 2022 & \ac{GMOS}-S & 6,6,6 & 120 & $g$, $r$, $i$&18.8&3.56 & 3.8 & 37.2 & 98\% & \obsnote{20220607}\ref{20220607}\\
\end{tabular}
\footnotesize
\noindent
\raggedright\\
$^\mathrm{a}$Figure showing the image. $^\mathrm{b}$Activity identified in image(s). $^\mathrm{c}$UT date of observation. $^\mathrm{d}$Number of images. $^\mathrm{e}$Exposure time. $^\mathrm{f}$Apparent $V$-band magnitude (Horizons). $^\mathrm{g}$Heliocentric distance. $^\mathrm{h}$Sun--target--observer angle.
$^\mathrm{i}$True anomaly.
$^\mathrm{j}$Percentage to perihelion $q$ from aphelion $Q$, defined by $\%_{T\rightarrow q} = \left(\frac{Q - r}{Q-q}\right)\cdot 100\mathrm{\%}$.
$^\mathrm{k}$Note number. \\
Notes:
\ref{20120224}: PS1 is the \acf{Pan-STARRS} One.
\ref{20120328}: Prop. ID 12AH16, \acs{PI} Wainscoat.
\ref{20120705}: Prop. ID 177.A-3016(D), \acs{PI} Kuijken.
\ref{20130505}: \acf{DECam}; Prop. ID 2013A-0327, \acs{PI} Rest.
\ref{20130613}: Prop. ID 13AH09, \acs{PI} Wainscoat.
\ref{20210104}: \acf{ZTF}; Prop. ID 1467501130115, \acs{PI} Kulkarni; data acquired through \acs{ZTF} Alert Stream service \citep{pattersonZwickyTransientFacility2019}.
\ref{20210314}: Prop. ID 2019A-0305, \acs{PI} Drlica-Wagner.
\ref{20210331}: Michael Jäger (Weißenkirchen, Austria), QHY600 \ac{CCD} on a 14'' Newtonian, .
\ref{20210404}: Roland Fichtl (Engelhardsberg, Germany), Central DS brand modified cooled Canon 5D Mark III on a 0.4~m f/2.5 Newtonian; \url{http://www.dieholzhaeusler.de/Astro/comets/0282P.htm}.
\ref{20220307}: \acf{IMACS}; PI Trujillo.
\ref{20220521}: \acf{IMACS}; PI Trujillo.
\ref{20220607}: \acf{GMOS}; Prop. ID GS-2022A-DD-103, \acs{PI} Chandler.
\end{sidewaystable}
\normalsize
\doublespacing
\clearpage
\thispagestyle{empty}
\atxy{\dimexpr1in}{.5\paperheight}{\rotatebox[origin=center]{270}{\thepage}}
\begin{sidewaystable}
\caption{Equipment and Archives}
\label{282P:tab:equipQuickRef}
\centering
\footnotesize
\begin{tabular}{llclccccc}
Instrument & Telescope & Pixel Scale & Location & \texttt{AstroArchive} & \acs{ESO} & \acs{SSOIS} & \acs{STScI} & \acs{IRSA}\\
& & ['/pix] & & & & &\\
\hline
\hline
\acs{DECam} & 4.0~m Blanco & 0.263 & Cerro Tololo, Chile & S,R & & S & \\
\acs{GMOS}-S & 8.1~m Gemini South & 0.080 & Cerro Pachón, Chile & &\\
\acs{IMACS} & 6.5~m Baade & 0.110 & Las Campanas, Chile & & & \\
OmegaCAM & 2.6~m \acs{VLT} Survey & 0.214 & Cerro Paranal, Chile & & R & S & \\
GigaPixel1 & 1.8 m \acs{Pan-STARRS}1 & 0.258 & Haleakalā, Hawaii & & & S & R \\
\acs{LMI} & 4.3~m \acs{LDT} & 0.120 & Happy Jack, Arizona & & & \\
MegaPrime & 3.6~m \acs{CFHT} & 0.185 & Mauna Kea, Hawaii & & & S,R & \\
\acs{PTF}/\acs{CFHT}12K & 48" Samuel Oschin & 1.010 & Mt. Palomar, California & & & & & S,R \\
\acs{ZTF} Camera & 48" Samuel Oschin & 1.012 & Mt. Palomar, California & & & & & S,R \\
\acs{VATT}4K \acs{CCD} & 1.8~m \acs{VATT} & 0.188 & Mt. Graham, Arizona & & & &\\
\end{tabular}
\raggedright
\footnotesize{\\
R indicates repository for data retrieval. S indicates search capability.\\
\texttt{AstroArchive}: \ac{NSF} \ac{NOIRLab} \texttt{AstroArchive} (\url{https://astroarchive.noirlab.edu}).\\
\ac{ESO}: \acl{ESO} (\url{https://archive.eso.org}).\\
\ac{IRSA}: \acs{NASA}/CalTech \ac{IRSA} (\url{https://irsa.ipac.caltech.edu}).\\
\acs{PTF}: The \ac{PTF}.
\acs{SSOIS}: The \ac{SSOIS} (\citealt{gwynSSOSMovingObjectImage2012}, \url{https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/ssois/}).\\
\ac{STScI}: \url{https://www.stsci.edu/}.
}
\label{282P:tab:equipAndArchives}
\end{sidewaystable}
\doublespacing
\clearpage
\atxy{\dimexpr1in}{.5\paperheight}{\rotatebox[origin=center]{270}{\thepage}}
\begin{sidewaystable}
\caption{282P/(323137) 2003 BM80 Data}
\label{282P:tab:ObjectData}
\singlespacing
\centering
\footnotesize
\begin{tabular}{lll}
Parameter & Value & Source\\
\hline\hline
Designations & (323137), 2003~BM$_{80}$, 2003~FV$_{112}$, 282P & \acs{JPL} \acs{SBDB}, \acs{MPC}\\
Discovery Date & 2003 January 31 & \acs{JPL} \acs{SBDB}, \acs{MPC}\\
Discovery Observer(s) & \ac{LONEOS} & \acs{JPL} \acs{SBDB}, \acs{MPC}\\
Discovery Observatory & Lowell Observatory & \acs{JPL} \acs{SBDB}, \acs{MPC}\\
Discovery Site & Anderson Mesa Station, Arizona & \acs{JPL} \acs{SBDB}, \acs{MPC}\\
Discovery Site Code & 688 & \acs{MPC} \\
Activity Discovery Date & 2013 June 12 & \acs{CBET} 3559 \citep{bolinComet2003BM2013}\\
Activity Discoverer(s) & Bryce Bolin, Larry Denneau, Peter Veres & \acs{CBET} 3559 \citep{bolinComet2003BM2013}\\
Orbit Type & \acf{QHO} & this work\\
Diameter & $D=$3.4$\pm$0.4~km & {\citet{harrisAsteroidsThermalInfrared2002}}\\
Absolute $V$-band Magnitude & $H=13.63$ & \acs{MPC} (MPO648742)\\
Geometric Albedo & Unknown & \\
Assumed Geometric Albedo & 4\% & \cite{snodgrassSizeDistributionJupiter2011}\\
Rotation Period & Unknown & \\
Orbital Period & $P=8.732\pm(2.174\times10^{-7})$~yr & \acs{JPL} \acs{SBDB} \\
Semi-major Axis & $a=4.240\pm(7.039\times10^{-8})$~au & \acs{JPL} \acs{SBDB}\\
Perihelion Distance & $q=3.441\pm(3.468\times10^{-7})$~au & \acs{JPL} \acs{SBDB}\\
Aphelion Distance & $Q=5.039\pm(8.366\times10^{-8})$~au & \acs{JPL} \acs{SBDB}\\
Eccentricity & $e=0.188\pm(7.790\times10^{-8})$ & \acs{JPL} \acs{SBDB}\\
Inclination & $i=5.812^\circ\pm(1.166^\circ\times10^{-5})$ & \acs{JPL} \acs{SBDB}\\
Argument of Perihelion & $\omega=217.626^\circ\pm(7.816^\circ\times10^{-5})$ & \acs{JPL} \acs{SBDB}\\
Longitude of Ascending Node & $\Omega=9.297^\circ\pm(5.974^\circ\times10^{-5})$ & \acs{JPL} \acs{SBDB}\\
Mean Anomaly & $M=9.979^\circ\pm(3.815^\circ\times10^{-5})$ & \acs{JPL} \acs{SBDB}\\
Tisserand Parameter w.r.t. Jupiter & $T_\mathrm{J}=2.99136891\pm\left(3.73\times10^{-8}\right)$ & this work\\
Orbital Solution Date & 2021 October 8 & \acs{JPL} \acs{SBDB}\\
\end{tabular}
\raggedright\footnotesize
Notes:
\ac{CBET} \footnote{\url{http://www.cbat.eps.harvard.edu}}.
\ac{JPL} \ac{SBDB} is the \acs{NASA} \acs{JPL} \acl{SBDB}\footnote{\url{https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html}}.
\acs{MPC} is the \acl{MPC}\footnote{\url{https://minorplanetcenter.net}}.
\end{sidewaystable}
\doublespacing
\chapter*{Acknowledgements}
\textit{Note: Manuscript-specific acknowledgements are found at the end of each corresponding chapter.}
\section{In Memoriam}
\textit{In memoriam} Jean E. Buethe (Flagstaff, Arizona), Nadine G. Barlow (Northern Arizona University), Adam P. Showman (University of Arizona) and Kazuo Kinoshita (Japan), all of whom encouraged and influenced my work.
\section{Indigenous Land Acknowledgement} Virtually all of the work described in this dissertation took place on lands important to indigenous cultures, but which has overwhelmingly been stolen from them. Here is an incomplete list of communities impacted in places where I have done work for this dissertation. \textbf{You can help by supporting organizations such as the Southwest Indian Relief Council (}\url{http://www.nativepartnership.org}) https://www.yakama.com (\url{https://engage.collegefund.org}\textbf{).}
The \ac{NAU} Mountain Campus is in Flagstaff, Arizona at the base of the San Francisco Peaks. This area is home to many peoples, including the Havasupai Tribe\footnote{\url{https://theofficialhavasupaitribe.com}}, Hopi Tribe\footnote{\url{https://www.hopi-nsn.gov}}, Hualapai Tribe\footnote{\url{https://hualapai-nsn.gov}}, Kaibab Band of Paiutes\footnote{\url{https://kaibabpaiute-nsn.gov}}, the Navajo Nation\footnote{\url{https://www.navajo-nsn.gov}}, the San Juan Southern Paiute Tribe\footnote{\url{https://www.sanjuanpaiute-nsn.gov}}.
I have traveled through areas of the Zuni Tribe\footnote{\url{https://www.ashiwi.org}} and the Yavapai-Prescott Indian Tribe\footnote{\url{https://www.ypit.com}}.
I have observed at \ac{MGIO}, a mountain stolen from the San Carlos Apache\footnote{\url{http://www.sancarlosapache.com/home.htm}} and sacred to others, including the White Mountains Apache Tribe\footnote{\url{http://www.wmat.us/}}.
I have traveled through and/or attended conferences and workshops on land in Arizona taken from the Tohono O'Odham Nation\footnote{\url{http://www.tonation-nsn.gov/}}, Gila River Indian Community\footnote{\url{https://www.gilariver.org/}} and undoubtedly many others. Agencies that have provided me with funding also benefit directly and indirectly from ceded lands, such as those of the Yakama Nation\footnote{\url{https://www.yakama.com}}.
\section{Individuals and Groups}
I thank my partner, Dr.\ Mark Jesus Mendoza Magbanua (University of California San Francisco), for his frequent feedback, critical insights, and constant encouragement. I thank my parents, Arthur and Jeanie Chandler, for their continued support that enabled my academic endeavors, and for having the foresight to give me ``Orion'' as a middle name. I also appreciate Corey and Atsuko, Charles and Van, and Julie and Praxis for their enthusiasm all along the way. Many thanks to my best friend, Robert Reich, for his continuous encouragement and compassion. Thanks also to Jerome, Rushmore and Ziggy Diaz for being great friends.
Thanks to my former business partner and great friend Bill Bowker (Fog City Mac, LLC) for allowing me to move on and pursue my path in astronomy.
Many thanks to all of my committee members for guiding me through this entire process. A special thanks to Chad, David, and Ty, all of whom provided crucial guidance and support during my time at \ac{NAU}. A special thank you to Prof. Cristina Thomas of \ac{NAU} for including me in \ac{DART} work that helped me to become a better photometrist.
Many thanks to Dr. Annika L. Gustafsson of \ac{NAU}, Lowell Observatory, and \ac{SwRI}, for countless hours together in classes, teaching, sharing world-class telescopes, attending conferences, and collaborating on research. Much appreciation goes to Aaron Weintraub of \ac{NAU} for being the best of friends and a supportive colleague from start to finish of this work. Thanks to both for encouraging me to go ``all in'' on my \ac{NSF} \ac{GRFP}. I am eternally grateful for James D. Windsor of \ac{NAU} for great times, helping me balance my life again through cycling and, along with his wife Cheyann, for literally saving my life.
Many, many thanks to Will Oldroyd for collaborating on papers, proposals, observing, hosting workshops, attending conferences, cooking, praying, and catching mice. A very special thanks to Jay Kueny (Steward Observatory, University of Arizona) for his commitment to the Citizen Science project and all that you taught me along the way. Thanks also for stepping in to help observe at the last minute on more than one occasion and at more than one observatory. Thanks also to Will Burris of \ac{SDSU} for your help with \textit{Active Asteroids} classification analysis.
A special thank you to \textit{Active Asteroids} forum Moderator Elisabeth Baeten (Belgium) who has greatly enhanced the success of our Citizen Science. Thank you also Cliff Johnson (Zooniverse) and Marc Kuchner (NASA), both of whom provided invaluable insights into Citizen Science and encouraged this project to move forward.
Thank you to the \ac{LSST} community for welcoming me, especially Ranpall Gill (Rubin Observatory), Henry Hsieh (Planetary Science Institute), Meg Schwamb (Queen's University Belfast), Agata Rożek (University of Edinburgh), and Mario Jurić and Andrew Connolly of the \ac{DiRAC} Institute and University of Washington).
Thank you Nathan Smith and Lori Pigue for independently inspiring and fueling my interest in meteorites and minerals. Thank you Schuyler Borges (\ac{NAU}) for helping me to become a better member of the community and inspiring me to continue advocating for diversity, equity and inclusivity. Thanks Tony and Catherine (\acs{NAU}) for being great colleagues and for the fun times on the slopes and in the pool. Thank you Trevor Cotter (NAU, McGill University) for all the adventures and, especially, for introducing me to cross country skiing. Thanks to fellow \acs{NAU} students Erin Aadland, Haylee Archer, Dan Avner, Lauren Biddle, Helen Eifert, Anna Engle, Oriel Humes, Joel Johnson, Ari Koeppel, David Kelly, Sarah Lamm, Audrey Martin, Lauren McGraw, Robyn Meier, Alissa Roegge, Raaman Nair, Kathryn Neugent, Garrett Thompson, and Mike Zeilnhofer for being supportive colleagues and classmates. A special thanks to Christian Joey Tai Udovicic for listening to my qualifying examination talk many times.
Thank you to my friends from the \ac{SFSU} \ac{PAC} for your encouragement, especially Daniel Steckhahn of \ac{UCB}, Ryan Rickards-Vaught of \ac{UCSD}, Sarah Deveny of {TTU}, and Michelle Howard. Many thanks to \ac{SFSU} faculty for their inspiration, especially Joseph Baranco, Adrienne Cool, Jeff Greensite, Ron Marzke, Weining Man, Chris McCarthy, and Barbara Neuhauser. Caroline Alcantra also helped me with all things administrative at \ac{SFSU} and I am eternally grateful. Thank you Alan Fisk (\ac{SFSU}) and the \ac{DPRC} for your guidance and helping me succeed.
Thank you Phil Massey (Lowell Observatory) for teaching me to become a better astronomer. Thank you Will Grundy, Stephen Levine, Jenn Hanley, Michael West, Diedre Hunter, Jeff Hall, Nick Moskovitz, Audrey Thirouin, Joe Llama, Lisa Prato, Gerard van Belle, Amanda Bosch, and Dave Schleicher for welcoming me at Lowell and vastly improving my experience in astronomy.
Thank you to Guy Consolmagno, Rich Boyle, Paul Gabor, Gary Gray and Chris Johnson for all of your support at the Vatican Observatory. Many thanks to Jenny Power of \ac{MGIO} for all the support with \ac{LBT} observation planning and execution. Thanks to everyone at the \ac{LBT} for accommodating us on-site on multiple occasions, especially Rick Hansen and David Huerta. A special thanks to all of the telescope operators at the \acf{LDT} for all their help through the years. Thanks also to Matt Holman (Harvard) and Jerome Berthier (\ac{IMCCE}) for their support with the \ac{MPC} and SkyBot, respectively.
Thank you to \ac{NAU} Pres. Rita Cheng for supporting the Astronomy and Planetary Science Ph.D. program at \ac{NAU}, the \ac{NAU}--\ac{LDT} partnership, and the Presidential Fellowship Program, all of which were essential to this work. Thank you Maribeth Watwood for overseeing this program and many others.
Thank you Ed Anderson (\ac{NAU}), Lisa Chien (\ac{NAU}), John Kistler (\ac{NAU}), David Koerner (\ac{NAU}), and Mark Salvatore (\ac{NAU}) for helping me to become a better teacher. Thanks Anna Engle (\ac{NAU}) for collaborating with teaching during the pandemic. Thanks to my many undergraduate teaching assistants, especially Anna Ross-Browning (University of Iowa) and Lisa Matrecito (\ac{NAU}), for directly enabling me to be a better teacher. Thanks to all of my astronomy and physics students at \ac{NAU} who all enriched my experience at \ac{NAU}. Thanks to department administrators Elizabeth Massey, Judene McLane, and Alix Ford for all their help throughout my time at \ac{NAU}. Thanks also to Lara Schmit (Merriam Powell Institute, \ac{NAU}) for being a great neighbor during the pandemic and for all your advice in navigating \ac{NAU} systems.
The unparalleled support given by Monsoon high performance computing cluster administrator Christopher Coffey of \ac{NAU}, as well as the High Performance Computing Support team, truly facilitate the scientific process throughout all of my work.
Thank you to Barry Lutz (\ac{NAU}) for funding and supporting the \ac{BLT} which was part of my education as well as my teaching endeavors. Thank you Stephen Tegler (\ac{NAU}) for your insights into Centaur colors and observational techniques, and Mark Loeffler (\ac{NAU}) for encouraging me to question sources to their very beginnings and the value of astrochemistry. Thank you Christopher Edwards (\ac{NAU}) for helping me better understand spectroscopy and advocating for me and my colleagues. Thanks also Chris Mann (\ac{NAU}) for helping me better understand the optics behind the telescopes I rely upon. Thanks to Devon Burr and Josh Emery for all your encouragement and advocacy at \ac{NAU}.
The Trilling Research Group at \ac{NAU} has always been an invaluable resource for insights which substantially enhanced this work. A special thanks to members Annika Gustaffson (\ac{SwRI}), Andy J López Oquendo (\ac{NAU}), Ryder Sluss (\ac{NAU}), Joey Chatelain of \ac{LCO}, Samuel Navarro-Meza (University of Mexico, \ac{NAU}), Daniel Kramer (\ac{NAU}), Andrew McNeil (\ac{NAU}), Maggie McAdam of \ac{SOFIA}, Connor Auge (University of Hawaii), and Michael Mommert (\ac{NAU}).
The HabLab research group (Ty Robinson) at \ac{NAU} and \ac{UA} is full of science and support. A special thanks to members Arnaud Salvadore, Amber Young, Megan Gialluca (\ac{UW}), Malik Bossett, Chris Wolfe, Patrick Tribbett, and Shih-Yun Tang.
Thank you Stephen Kane (University of California Riverside) for encouraging me to pursue my academic dreams and laying the foundation for successful research.
Thank you Erik Mamajek (NASA \ac{JPL})for invaluable advice that helps keep my focus on science goals.
Thank you Michael Way (NASA Goddard), Brandon Cruickshank (\ac{NAU}), Greg McGuffey, Kathy \& John Eastwood, all of whom inspire balance between outdoor activity and other pursuits.
Thank you to Hannah Nuñez and Dr. Matt Wise of \ac{NAU} for helping to keep me healthy during my time at \ac{NAU}. Thanks also Fr. Matt Lowry and the \ac{NAU} Catholic Jacks for providing a welcoming spiritual environment on campus.
\section{Citizen Scientists}
\label{ack:sec:citSci}
I wish to thank the following individuals for their efforts on the \textit{Active Asteroids} project. These individuals represent our most active classifiers, classified thumbnail images included in this work, or both. A special thanks goes to our top classifier,
Michele T. Mazzucato, FRAS (Sesto Fiorentino, Italy).
Thank you A. J. Raab (Seattle, USA),
Alice Juzumas (São Paulo, Brazil),
Angelina A. Reese (Sequim, USA),
Arttu Sainio (Järvenpää, Finland),
Bill Shaw (Fort William, Scotland),
\texttt{@Boeuz} (Penzberg, Germany),
Brenna Hamilton (DePere, USA),
Brian K Bernal (Greeley, USA),
Carl Groat (Okeechobee, USA),
Clara Garza (West Covina, USA),
C. J. A. Dukes (Oxford, United Kingdom),
Dr. David Collinson (Mentone, Australia),
Edmund Frank Perozzi (Glen Allen, USA),
\texttt{@EEZuidema} (Driezum, Netherlands),
Dr. Elisabeth Chaghafi (Tübingen, Germany),
\texttt{@graham\_d} (Hemel Hempstead, England),
Ivan A. Terentev (Petrozavodsk, Russia),
Juli Fowler (Albuquerque, USA),
Leah Mulholland (Peoria, USA)
M. M. Habram-Blanke (Heidelberg, Germany),
Marvin W. Huddleston (Mesquite, USA),
Michael Jason Pearson (Hattiesburg, USA),
Milton K. D. Bosch, MD (Napa, USA),
\texttt{@mitch} (Chilliwack, Canada),
Patricia MacMillan (Fredericksburg, USA),
Petyerák Jánosné (Fót, Hungary),
R. Banfield (Bad Tölz, Germany),
Scott Virtes (Escondido, USA),
Sergey Y. Tumanov (Glazov, Russia),
Stikhina Olga Sergeevna (Tyumen, Russia),
Thorsten Eschweiler (Übach-Palenberg, Germany),
Tiffany Shaw-Diaz (Dayton, USA),
Timothy Scott (Baddeck, Canada),
and
Virgilio Gonano (Udine, Italy).
\section{Funding}
This material is based upon work supported by the \acl{NSF} \acl{GRFP} under grant No.\ (2018258765). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. C.O.C., H.H.H. and C.A.T. also acknowledge support from the NASA Solar System Observations program (grant 80NSSC19K0869).
This work was supported in part by NSF awards 1461200, 1852478, and 1950901 (\acs{NAU} REU program in astronomy and planetary science).
\section{General Acknowledgements}
Computational analyses were run on Northern Arizona University's Monsoon computing cluster, funded by Arizona's Technology and Research Initiative Fund. This work was made possible in part through the State of Arizona Technology and Research Initiative Program. World Coordinate System (WCS) corrections facilitated by the \textit{Astrometry.net} software suite \citep{langAstrometryNetBlind2010}.
This research has made use of data and/or services provided by the International Astronomical Union's Minor Planet Center.
This research has made use of NASA's Astrophysics Data System.
This research has made use of The Institut de M\'ecanique C\'eleste et de Calcul des \'Eph\'em\'erides (IMCCE) SkyBoT Virtual Observatory tool \citep{berthierSkyBoTNewVO2006}.
This work made use of the {FTOOLS} software package hosted by the NASA Goddard Flight Center High Energy Astrophysics Science Archive Research Center.
This research has made use of SAOImageDS9, developed by Smithsonian Astrophysical Observatory \citep{joyeNewFeaturesSAOImage2006}.
This work made use of the Lowell Observatory Asteroid Orbit Database \textit{astorbDB} \citep{bowellPublicDomainAsteroid1994,moskovitzAstorbDatabaseLowell2021}.
This work made use of the \textit{astropy} software package \citep{robitailleAstropyCommunityPython2013}.
This project used data obtained with the \acf{DECam}, which was constructed by the \acf{DES} collaboration. Funding for the \acs{DES} Projects has been provided by the US Department of Energy, the US \acl{NSF}, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute for Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A\&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Enérgeticas, Medioambientales y Tecnológicas–Madrid, the University of Chicago, University College London, the \acs{DES}-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ciències de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, \acs{NSF}'s \acs{NOIRLab}, the University of Nottingham, the Ohio State University, the OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, \acs{SLAC} National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A\&M University.
Based on observations at \acl{CTIO}, \acs{NSF}'s \acs{NOIRLab}
(
NOAO Prop. ID 2012B-0001, PI: Frieman;
NOAO Prop. ID 2013A-0327, PI: Rest;
NOAO Prop. ID 2013B-0453, PI: Sheppard;
NOAO Prop. ID 2013B-0536, PI: Allen;
NOAO Prop. ID 2014A-0303, PI: Sheppard;
NOAO Prop. ID 2014A-0479; PI: Sheppard
NOAO Prop. ID 2014B-0303, PI: Sheppard;
NOAO Prop. ID 2014B-0404, PI: Schlegel;
NOAO Prop. ID 2015A-0351, PI: Sheppard;
NOAO Prop. ID 2015A-0620, PI: Bonaca;
NOAO Prop. ID 2015B-0265, PI: Sheppard;
NOAO Prop. ID 2016A-0190, PI: Dey;
NOAO Prop. ID 2016A-0401, PI: Sheppard;
NOAO Prop. ID 2016B-0288, PI: Sheppard;
NOAO Prop. ID 2017A-0367, PI: Sheppard;
NOAO Prop. ID 2017B-0307, PI: Sheppard;
NOAO Prop. ID 2019A-0272, PI: Zenteno;
NOAO Prop. ID 2019A-0305, PI: Drlica-Wagner;
NOAO Prop. ID 2019A-0337, PI: Trilling;
NOAO Prop. ID 2019B-0323, PI: Zenteno
),
which is managed by the \acf{AURA} under a cooperative agreement with the \acl{NSF}.
This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.
The Legacy Surveys consist of three individual and complementary projects: the Dark Energy Camera Legacy Survey (DECaLS; Proposal ID \#2014B-0404; PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID \#2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Survey (MzLS; Prop. ID \#2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF's NOIRLab; the Bok telescope, Steward Observatory, University of Arizona; and the Mayall telescope, Kitt Peak National Observatory, NOIRLab. The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du'ag (Kitt Peak), a mountain with particular significance to the Tohono O'odham Nation. BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Priority Research Program ``The Emergence of Cosmological Structures'' Grant \# XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance. The BASS is also supported by the External Cooperation Program of Chinese Academy of Sciences (Grant \# 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant \# 11433005). The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by the National Aeronautics and Space Administration. The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; and by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to NOAO.
These results made use of the Lowell Discovery Telescope (LDT) at Lowell Observatory. Lowell is a private, non-profit institution dedicated to astrophysical research and public appreciation of astronomy and operates the LDT in partnership with Boston University, the University of Maryland, the University of Toledo, Northern Arizona University and Yale University. The Large Monolithic Imager was built by Lowell Observatory using funds provided by the National Science Foundation (AST-1005313). NIHTS was funded by NASA award \#NNX09AB54G through its Planetary Astronomy and Planetary Major Equipment programs.
Based in part on data collected at Subaru Telescope and obtained from the SMOKA, which is operated by the Astronomy Data Center, National Astronomical Observatory of Japan \citep{2002ASPC..281..298B}.
Plots were primarily created using the Python tool \texttt{Matplotlib} \citep{hunterMatplotlib2DGraphics2007}.
Python mathematical and array operations were frequently executed using the Python \texttt{NumPy} tool \citep{harrisArrayProgrammingNumPy2020}. We made use of Pandas dataframes \citep{mckinneyDataStructuresStatistical2010,rebackPandasdevPandasPandas2022}. We made use of the \texttt{Rebound} dynamical simulator \citep{reinREBOUNDOpensourceMultipurpose2012,reinHybridSymplecticIntegrators2019} with the \texttt{Mercury} \citep{chambersHybridSymplecticIntegrator1999} and \texttt{IAS15} \citep{reinIAS15FastAdaptive2015} integrators. We made use of the \texttt{SciPy} Python suite \citep{virtanenSciPyFundamentalAlgorithms2020}.
We used \texttt{Theli3}\footnote{\url{https://github.com/schirmermischa/THELI}} \citep{schirmerTHELIConvenientReduction2013} for some of our data reduction.
\chapter*{Copyright}
\section{Published Works}
The following copyright statements apply to each respective published manuscript.
\vspace{3mm}
\noindent\textbf{Chapter \ref{chap:SAFARI} -- Manuscript I}
\\
\noindent \textit{Searching Asteroids for Activity Revealing Indicators (SAFARI)} \citep{chandlerSAFARISearchingAsteroids2018}:
\begin{quote}
\textit{This is the Accepted Manuscript version of an article accepted for publication in Publications of the Astronomical Society of the Pacific. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at} \url{https://iopscience.iop.org/article/10.1088/1538-3873/aad03d}\textit{.}
\end{quote}
\noindent\textbf{Chapter \ref{chap:Gault} -- Manuscript II}
\\
\noindent \textit{Six Years of Sustained Activity from Active Asteroid (6478)~Gault} \citep{chandlerSixYearsSustained2019}
\begin{quote}
\textit{This is the Accepted Manuscript version of an article accepted for publication in Astrophysical Journal Letters. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at} \url{https://iopscience.iop.org/article/10.3847/2041-8213/ab1aaa}\textit{.}
\end{quote}
\noindent\textbf{Chapter \ref{chap:2014OG392} -- Manuscript III}
\\
\noindent \textit{Cometary Activity Discovered on a Distant Centaur: A Nonaqueous Sublimation Mechanism} \citep{chandlerCometaryActivityDiscovered2020a}
\begin{quote}
\textit{This is the Accepted Manuscript version of an article accepted for publication in Astrophysical Journal Letters. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at }\url{https://iopscience.iop.org/article/10.3847/2041-8213/ab7dc6}\textit{.}
\end{quote}
\noindent\textbf{Chapter \ref{chap:2005QN173} -- Manuscript IV}
\\
\textit{Recurrent Activity from Active Asteroid (248370) 2005~QN173: A Main-belt Comet} \citep{chandlerRecurrentActivityActive2021a}
\begin{quote}
\textit{This is the Accepted Manuscript version of an article accepted for publication in Astrophysical Journal Letters. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at }\url{https://iopscience.iop.org/article/10.3847/2041-8213/ac365b}\textit{.}
\end{quote}
\section{Unpublished Works}
The remaining chapters will be submitted to the American Astronomical Society journals for publication as soon as possible:
\\
\noindent\textbf{Chapter \ref{chap:282P} -- Manuscript V}
\\
\textit{Active Asteroid Origin Insights from Transition Object (323137) 2003 BM80}
\chapter*{Dedication}
This dissertation is dedicated to my partner, Mark Jesus Mendoza Magbanua, and to my parents, Arthur and Jeanie Chandler; this work would not have been possible without their encouragement and support.
\chapter{Manuscript II: Six Years of Sustained Activity from Active Asteroid (6478)~Gault}
\chaptermark{Six Years Sustained Activity from Active Asteroid (6478)~Gault}
\label{chap:Gault}
\acresetall
Colin Orion Chandler\footnote{\label{GaultNAU}Department of Physics \& Astronomy, Northern Arizona University, PO Box 6010, Flagstaff, AZ 86011, USA}, Jay Kueny$^\mathrm{\ref{GaultNAU}}$, Annika Gustafsson$^\mathrm{\ref{GaultNAU}}$, Chadwick A. Trujillo$^\mathrm{\ref{GaultNAU}}$, Tyler D. Robinson$^\mathrm{\ref{GaultNAU}}$, David E. Trilling$^\mathrm{\ref{GaultNAU}}$
\textit{This is the Accepted Manuscript version of an article accepted for publication in Astrophysical Journal Letters. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at} \url{https://iopscience.iop.org/article/10.3847/2041-8213/ab1aaa}\textit{.}
\doublespacing
\section{Abstract}
\label{Gault:Abstract}
We present archival observations demonstrating that main belt asteroid (6478)~Gault has an extensive history of comet-like activity. Outbursts have taken place during multiple epochs since 2013 and at distances extending as far as ~2.68 au, nearly aphelion. (6478)~Gault is a member of the predominately S-type (i.e., volatile-poor) Phocaea family; no other main belt object of this type has ever shown more than a single activity outburst. Furthermore, our data suggest this is the longest duration of activity caused by a body spinning near the rotational breakup barrier. If activity is indeed unrelated to volatiles, as appears to be the case, (6478) Gault represents a new class of object, perpetually active due to rotational spin-up.
\textit{Keywords:} minor planets, asteroids: individual ((6478) Gault) --- comets: individual ((6478) Gault)
\section{Introduction}
\label{Gault:introduction}
Active asteroids like (6478)~Gault (Figure~\ref{Gault:fig:gault2013a}, this work) are dynamically asteroidal objects but they uncharacteristically manifest cometary features such as tails or comae \citep{hsiehActiveAsteroidsMystery2006}. With only $\sim$20 known to date (see Table 1 of \citealt{chandlerSAFARISearchingAsteroids2018}), active asteroids remain poorly understood, yet they promise insight into solar system volatile disposition and, concomitantly, the origin of water on Earth \citep{hsiehPopulationCometsMain2006}.
Active asteroids are often defined as objects with (1) comae, (2) semimajor axes interior to Jupiter, and (3) Tisserand parameters with respect to Jupiter $T_\mathrm{J}>3$; $T_\mathrm{J}$ describes an object's orbital relationship to Jupiter by
\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{gaultFiles/Gault130928T022337ooigd2.png}
\caption{(6478)~Gault (dashed green circle) displays a prominent tail (indicated by white arrows) during this 2013 September 28 apparition when (6478)~Gault was halfway between perihelion and aphelion. This 90 s $g$-band exposure reached $\sim$7 mag fainter than (6478)~Gault. The anti-Solar direction ($-\odot$; yellow) and negative heliocentric velocity vector ($-\vec{v}$; red) are shown.}
\label{Gault:fig:gault2013a}
\end{figure}
\begin{equation}
T_\mathrm{J} = \frac{a_\mathrm{J}}{a} + 2 \sqrt{\frac{a\left(1-e^2\right)}{a_\mathrm{J}}}\cos\left(i\right)
\end{equation}
\noindent where $a_\mathrm{J}$ is the semimajor axis of Jupiter, $e$ is the eccentricity, and $i$ is the inclination; see \cite{jewittActiveAsteroids2012} for a thorough treatment. Main belt comets are a subset of active asteroids dynamically constrained to the main asteroid belt and thought to have volatile-driven activity (see e.g., \citealt{snodgrassMainBeltComets2017} for an in-depth discussion). It is worth pointing out that some objects have had multiple classifications, for instance (3552) Don Quixote has an asteroid designation due to its low activity and has been called a near-Earth asteroid \citep{mommertDiscoveryCometaryActivity2014} but has a $T_\mathrm{J}$ of 2.3 which indicates it is more properly a Jupiter family comet.
Discovering these objects has proven observationally challenging. The first active asteroid, (4015)~Wilson-Harrington, was discovered in 1949 \citep{harrisCometNotesComet1950}. In the mid-1980s a connection between bow-shock magnetic field disturbances detected by the \textit{Pioneer} spacecraft suggested (2201)~Oljato was leaving behind a distant comet-like gas trail \citep{kerrCouldAsteroidBe1985}, even if not detected at the object itself \citep{russellInterplanetaryMagneticField1984}. Despite many efforts (see e.g., \citealt{chamberlin4015WilsonHarrington22011996}) it was not until the 1996 discovery of activity in (7968)~Elst-Pizarro that another active asteroid was visually identified \citep{elstComet1996N21996}. Though initially impact appeared a possible cause (e.g., \citealt{tothImpactgeneratedActivityPeriod2000}), when activity recurred \citep{hsiehReturnActivityMainbelt2010} it was more indicative of volatile sublimation.
A significant complication hindering our understanding of active asteroids arises when assessing underlying activity mechanisms: causes are neither few nor mutually exclusive (see \citealt{jewittActiveAsteroids2012} for a comprehensive overview). Responsible primary processes include volatile sublimation (e.g., 133P/Elst-Pizarro, \citealt{hsiehReturnActivityMainbelt2010}), impact events (e.g., (596) Scheila, \citealt{bodewitsCollisionalExcavationAsteroid2011,morenoDustEnvironmentMainBelt2011}), rotational breakup (e.g., 311P/PanSTARRS, \citealt{jewittExtraordinaryMultitailedMainbelt2013,morenoIntermittentDustMass2014}), thermal fracture (e.g., (3200) Phaethon, discussed below), and cryovolcanism (e.g., (1) Ceres, \citealt{kuppersLocalizedSourcesWater2014,witzeBrightSpotsCeres2015}). Physical interaction, or ``rubbing binary,'' has been proposed as a primary mechanism in the case of 311P (\citealt{hainautContinuedActivity20132014}, cf. \citealt{jewittNucleusActiveAsteroid2018}). Secondary mechanisms such as electrostatic gardening, physical properties like chemical makeup, and geometric effects (e.g., the opposition effect) may influence our ability to reliably detect and quantify outbursts.
One crucial diagnostic indicator of the underlying activity mechanism is whether or not activity recurs. If activity is observed on only one occasion (i.e., a single apparition), then the object may have experienced a recent impact event. Expelled material and/or exposed volatiles sublimating may both cause comae or tails to appear. Activity would then cease once ejecta dissipated or the volatile supply is exhausted, reburied, or refrozen.
Recurrent activity is typically associated with volatile sublimation. For example, Geminid Meteor Shower parent body (3200) Phaethon is thought to undergo thermal fracture during the rapid temperature changes accompanying its perihelion passages \citep{liRecurrentPerihelionActivity2013} where it experiences temperatures $>800$ K \citep{ohtsukaSolarRadiationHeatingEffects2009}. Fracture events may directly expel material in addition to exposing volatiles for sublimation.
Thermally induced activity is thought to increase with decreasing heliocentric distance; that is, the closer a body is to the Sun, the more likely an outburst is to occur.
Active asteroids are more likely to exhibit activity during perihelion passage (see Table 1 of \citealt{chandlerSAFARISearchingAsteroids2018}). Notable exceptions where activity was discovered at distances far from perihelion include 311P/PanSTARRS \citep{jewittExtraordinaryMultitailedMainbelt2013} and (493) Griseldis \citep{tholenEvidenceImpactEvent2015}. Activity in ``traditional'' comets has been reported at distances that are substantially farther than the main asteroid belt, for instance Comet C/2010 U3 (Boattini) at 27~au \citep{hui2010U3Boattini2019}. Of the $\sim$20 active asteroids known to date, 16 are carbonaceous (i.e., C-type) but only four are believed to be composed of silicate-rich non-primitive material (i.e., \citealt{demeoExtensionBusAsteroid2009} S-type taxonomy): (2201) Oljato (Apollo-orbit), 233P/La Sagra (Encke-orbit), 311P/PanSTARRS (inner main belt), and 354P/LINEAR (outer main belt).
(6478)~Gault activity was first reported in 2019 January \citep{smith6478Gault2019}. Ensuing follow-up observations \citep{maury6478Gault2019} confirmed activity with subsequent reports \citep{leeEarlyDetection64782019,yeContinuedActivity64782019} providing evidence of ongoing activity. \cite{yeMultipleOutburstsAsteroid2019} reported that multiple outbursts actually began in 2018 December. Analysis of dust emanating from (6478)~Gault via Monte Carlo tail brightness simulations indicate the current apparition, comprised of two outbursts, could have begun as early as 2018 November 5 \citep{morenoDustPropertiesDoubletailed2019}. Simultaneous to our own work, \cite{jewittEpisodicallyActiveAsteroid2019} reported three tails with independent onsets, the earliest being 2018 October 28.
We set out to determine if any data in our local repository of National Optical Astronomy Observatory (NOAO) Dark Energry Camera (DECam) images showed signs of activity. The $\sim$500 megapixel DECam instrument on the Blanco 4 m telescope situated on Cerro Tololo, Chile, probes faintly ($\sim$24 mag) and, as we demonstrated in \cite{chandlerSAFARISearchingAsteroids2018}, it is well-suited to detect active asteroids. We produced novel tools taking into account (1) orbital properties of (6478)~Gault (summarized in Appendix \ref{Gault:subsec:GaultData}) and (2) observational properties (e.g., apparent magnitude, filter selection, exposure time) to find ideally suited candidate images.
\section{Methods}
\label{Gault:sec:methods}
We searched our own in-house database of archival astronomical data (e.g., observation date, coordinates) in order to locate images that are likely to show (6478)~Gault. Our database includes the entire NOAO DECam public archive data tables along with corresponding data from myriad sources (e.g., NASA JPL Horizons \citealt{giorginiJPLOnLineSolar1996}; see also the Acknowledgements).
\section{Locating Candidate Images}
\label{Gault:subsec:candidates}
We began our search for (6478)~Gault by making use of a fast grid query in R.A. and decl. space. We then passed these results through a more accurate circular filter prescribed for the DECam image sensor arrangement. Lastly, we computed image sensor chip boundaries precisely to ensure that the object fell on a sensor rather than, for example, gaps between camera chips. This progressively more precise query approach cut down image search time by orders of magnitude.
\section{Observability Assessment}
\label{Gault:subsec:observability}
We created a reverse exposure time calculator to estimate how faintly (i.e., the magnitude limit) candidate images probed. After applying color coefficient corrections (see \citealt{willmerAbsoluteMagnitudeSun2018} for procedure details) we transformed the color-corrected magnitudes to the absolute bolometric system used by the DECam exposure time calculator\footnote{\url{http://www.ctio.noao.edu/noao/node/5826}}. These steps enabled us to compute differences between apparent magnitude and the specific magnitude limit of the DECam exposure so that we could produce a list of images where (6478)~Gault could be detected.
\section{Thumbnail Extraction}
\label{Gault:subsec:acquisition}
We downloaded the image files containing (6478)~Gault from the NOAO archive and, following the procedures of \citet{chandlerSAFARISearchingAsteroids2018}, we extracted flexible image transport system (FITS) thumbnails of (6478)~Gault. We then performed image processing to enhance contrast before finally producing portable network graphics (PNG) image files for inspection.
\section{Image Analysis}
\label{Gault:subsec:imageanalysis}
We visually inspected our (6478)~Gault thumbnail images to check for signs of activity. PNG thumbnails with activity indicators were examined in greater detail via the corresponding FITS thumbnail image.
To assess the influence of heliocentric distance on activity level we employed a simple metric (see \cite{chandlerSAFARISearchingAsteroids2018} ``$\mathrm{\%}_\mathrm{peri}$'' for motivation) describing how far from perihelion $q$ the target $T$ was located (at distance $d$) relative to its aphelion distance $Q$ by
\begin{equation}
\%_{T\rightarrow q} = \left(\frac{Q - d}{Q-q}\right)\cdot 100\mathrm{\%}.
\label{Gault:eq:percentperi}
\end{equation}
\vspace{2mm}
\section{Results}
\label{Gault:sec:results}
\begin{table*}
\caption{(6478)~Gault Archival DECam Observations Examined}
\footnotesize
\centering
\begin{tabular}{ccrlrcccccrr}
Activity & UT Date & Time & Processing & $t_\mathrm{exp}$ & Filter & $m_\mathrm{lim}$ & $m_V$ & $\Delta m$ & $r$ & $\%_{T\rightarrow q}$ & $\angle_\mathrm{STO}$ \\
& & & & (s) & & & & & (au) & & ($^\circ$)\\
\hline\hline
& 2013 Sep 22 & 3:03 & R, I, Re & 45 & Y & 21.0 & 17.2 & -4 & 2.27 & 54 & 3.41 \\
$*$ & 2013 Sep 28 & 2:23 & R, I, I, Re & 90 & g & 24.2 & 17.0 & -7 & 2.28 & 52 & 0.45 \\
$*$ & 2013 Oct 13 & 3:06 & I, I, I, Re & 90 & i & 23.3 & 17.1 & -6 & 2.32 & 49 & 8.21 \\
& 2013 Oct 13 & 3:08 & I, I, Re & 90 & z & 22.7 & 17.1 & -6 & 2.32 & 49 & 8.21 \\
$*$ & 2016 Jun 09 & 4:45 & R, I, Re & 96 & r & 23.9 & 16.8 & -7 & 1.86 & 100 & 22.38 \\
$*$ & 2016 Jun 10 & 4:40 & R, I, Re & 107 & g & 24.2 & 16.8 & -7 & 1.86 & 100 & 22.48 \\
& 2017 Oct 23 & 8:57 & R, I, Re & 80 & z & 22.7 & 18.8 & -4 & 2.66 & 10 & 14.79 \\
& 2017 Nov 11 & 7:13 & R, I, Re & 80 & z & 22.7 & 18.5 & -5 & 2.68 & 8 & 10.92 \\
$*$ & 2017 Nov 12 & 5:14 & R, I, Re & 111 & g & 24.3 & 18.5 & -6 & 2.68 & 8 & 10.81 \\
\hline
\end{tabular}
\raggedright
\footnotesize
\vspace{1mm}
\textbf{Note.} Process types: Raw (R), InstCal (I), Resampled (Re); $r$: Sun-target distance; $\%_{T\rightarrow q}$: target distance toward perihelion from aphelion (Equation \ref{Gault:eq:percentperi}); $t_\mathrm{exp}$: exposure time; $m_\mathrm{lim}$: estimated exposure magnitude limit; $m_V$: (6478)~Gault apparent $V$-band magnitude; $\Delta m$: $m - m_\mathrm{lim}$; $\angle_\mathrm{STO}$: Sun-Target-Observer (phase) angle.
Thumbnails are included in Appendix \ref{Gault:subsec:ThumbnailGallery}.
\label{Gault:tab:observations}
\end{table*}
We successfully extracted thumbnails from 9 archival observations of (6478)~Gault; see Table \ref{Gault:tab:observations} for details. Most data were available in raw and calibrated form (``InstCal'' and ``Resampled'' are described by \citealt{darkenergysurveycollaborationDarkEnergySurvey2016}) allowing us to extract $\sim$30 thumbnail images in total. Figure~\ref{Gault:fig:gault2013a} shows (6478)~Gault in 2013 with a pronounced tail in the 6 o'clock direction. Figure~\ref{Gault:fig:thumb2016} \textbf{(a)} and Figure~\ref{Gault:fig:thumb2016} \textbf{(b)} show (6478)~Gault in 2016 and 2017, respectively. Additional images may be found in Appendix \ref{Gault:subsec:ThumbnailGallery}.
\begin{figure*}[ht]
\centering
\includegraphics[width=1.0\linewidth]{gaultFiles/GaultOrbit}
\caption{(6478)~Gault activity timeline beginning with DECam operation commencement (2012 September) to present. Red stars show when we found visible activity; the blue pentagon represents the current apparition where prominent activity has been seen. Above the top axis are marked perihelion ($q$) and aphelion ($Q$) events. The solid green line indicates the apparent $V$-band magnitude of (6478)~Gault as viewed from Earth. The dashed yellow line shows our ``observability'' metric, defined as the number of hours per UT observing date meeting both of the following conditions possible for DECam: (1) elevation $> 15^\circ$, and (2) the refracted solar upper-limb elevation was $< 0^\circ$ (i.e., nighttime). Peaks in apparent magnitude coinciding with peaks in observability indicate opposition events; conversely, secondary magnitude peaks aligned with observability troughs highlight solar conjunctions, i.e., when (6478)~Gault was ``behind'' the Sun as viewed from Earth. All activity has been observed near opposition events. Also, activity was seen at every epoch in our data. The histogram (vertical blue bars) indicate the number of thumbnails that we extracted for a given observing month.}
\label{Gault:fig:ActivityTimeline}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{gaultFiles/GaultActivity}
\caption{Positive detections of (6478)~Gault activity with DECam as a function of heliocentric distance $r$ (au) and surface temperature $T$ (K). Our activity observations are indicated by red stars, whereas the current apparition is represented by the blue pentagon. Distance and temperature of (6478)~Gault perihelion $q$ (orange dashed line) and aphelion $Q$ (blue dashed-dotted line) events are shown. During the course of one full orbit, (6478)~Gault is exposed to temperatures greater than 165~K. As a result, (6478)~Gault is consistently subjected to temperatures that are too high for water ice to form at the 5 au ice formation distance \citep{snodgrassMainBeltComets2017}.}
\label{Gault:fig:tempsanddistance}
\end{figure}
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{gaultFiles/Gault160610T044331ooigv1.png} & \includegraphics[width=0.45\linewidth]{gaultFiles/Gault171112T051403ooigv1c.png}\\
\textbf{(a)} & \textbf{(b)}\\
\end{tabular}
\caption{\textbf{(a)} A tail (indicated by white arrows) at $\sim$8 o'clock is seen in this 107-second $g$-band exposure imaged June 10, 2016. The yellow arrow indicates anti-Solar ($-\odot$) direction and red the negative heliocentric velocity vector ($-\vec{v}$). \textbf{(b)} (6478)~Gault top seen on November 12, 2017. The 111 second exposure in the $g$-band delivered a flux limit 6 magnitudes fainter than (6478)~Gault, revealing a faint tail ($\sim$2 o'clock, indicated by white arrows) and coma. The yellow arrow indicates anti-Solar ($-\odot$) direction and the red arrow negative heliocentric velocity ($-\vec{v}$). Dashed green circles outline (6478)~Gault and white arrows have been placed perpendicular to any observed activity.}
\label{Gault:fig:thumb2016}
\end{figure*}
Figure~\ref{Gault:fig:ActivityTimeline} summarizes our observed activity. We found activity at least once in every set of observations and no correlation with distance. We plotted apparent $V$-band magnitude (solid green line) and found that all periods of activity were observed near opposition events. We further define ``observability'' (dashed yellow line) as when the object was (1) above 15$^\circ$ elevation, and (2) visible outside of daylight hours. This allowed us to assess potential observational biases specific to the southern hemisphere where our data were collected. As demonstrated by the coinciding of apparent magnitude maxima with spikes in apparent magnitude, the primary observability factor was solar elongation.
Figure~\ref{Gault:fig:tempsanddistance} shows how (6478)~Gault varies in both temperature $T$ and distance $r$ through time. Indicated are our activity observations (red stars) and the current apparition (blue pentagon). Temperature varies between $\sim$165 K at aphelion $Q$ (blue dash-dotted line) and $\sim$200K at perihelion $q$ (orange dotted line).
We define persistent activity as activity detectable across contiguous sets of observations spanning at least two epochs, even if activity is not visible in every image (due to, for example, exposure time and/or filter selection). We also expect to see activity at all positions throughout the orbit where (6478)~Gault may be detectable by DECam, given appropriate observing parameters (e.g., exposure time, filter selection).
\section{Discussion}
\label{Gault:sec:discussion}
Most active asteroids, like comets, are composed of low-albedo (i.e.,~dark), primitive material allowing for sublimation or release of volatiles to occur when the body is heated during close passages with the Sun \citep{hsieh2016ReactivationsMainbelt2018}.
Of the $\sim$20 known active asteroids, four belong to the S-type asteroid taxonomy defining non-primitive silicate-rich material \citep{demeoExtensionBusAsteroid2009}. For these four objects, the causes of activity, when identifiable, are thought to be rotational breakup or impact. Rotational breakup and impact events are consistent with single apparitions or short-lived activity. Furthermore, \cite{hsiehAsteroidFamilyAssociations2018} found that processed material bodies, such as S-types, are more likely to become active due to disruption, while primitive material bodies, such as C-types, can become active via multiple mechanisms due to their volatile abundances.
(6478)~Gault has been identified as a core member of the Phocaea Family \citep{knezevicProperElementCatalogs2003}. The Phocaea family is dominated by 75\% S-types, followed by 15\% C-types, and 10\% a mix of other asteroid taxonomies \citep{carvanoSpectroscopicSurveyHungaria2001}. While this work was in review, \cite{smith6478Gault2019, jewittEpisodicallyActiveAsteroid2019} reported color measurements suggesting that (6478)~Gault is closer in taxonomic class to a C-type body, rather than an S-type. However, gases were not detected in their spectra, suggesting that sublimation may not be the underlying cause.
Sustained activity near perihelion normally can point to sublimation driven activity, but we observe activity nearly at aphelion during opposition. We do see variability in activity intensity, but we observe activity in (6478)~Gault in at least one image in each set of observations in our DECam data set, suggesting that the target is perpetually active. As a result, we conclude there is no correlation between distance and activity for (6478)~Gault.
Because we observe persistent activity, impact-driven disruption seems unlikely as we would expect the timescale for the activity to be relatively short. The most probable cause for activity has been presented as disruption due to rotational breakup of (6478)~Gault \citep{morenoDustPropertiesDoubletailed2019, yeMultipleOutburstsAsteroid2019, yeContinuedActivity64782019} due to the Yarkovsky--O'Keefe--Radzievskii--Paddack (YORP) effect spin-up \citep{kleynaSporadicActivity64782019}; see \cite{mcneillBrightnessVariationDistributions2016}, \cite{lowryDirectDetectionAsteroidal2007}, and \cite{bottkeYarkovskyYorpEffects2006} for detailed explanations of YORP forces.
Rotational breakup holds for an S-type composition where we would anticipate landslides or surface material redistribution caused by rapid rotation near the 2.2 hr spin rate barrier, which is consistent with the measured $\sim$2 hr light curve period reported by \cite{kleynaSporadicActivity64782019}. We predict that (6478)~Gault will continue to show signs of activity as it has for the last 6 years in a relatively steady state.
We do not expect catastrophic disruption of (6478) Gault (cf.\ \citet{morenoDustPropertiesDoubletailed2019}).
The activity observed in (6478)~Gault over multiple epochs and throughout its orbit make (6478)~Gault the first known sustained-activity active asteroid in the main asteroid belt. As a likely S-type asteroid, this is also the first time that we have observed a sustained active body at the rotational barrier for such an extended duration. If activity is in fact not volatile-related, then Gault is a new class of object, perpetually active due to spin-up. We encourage continued monitoring of both the lightcurve and activity level of (6478) Gault, as well as photometric color observations or spectra to further explore its composition.
\section{Acknowledgements}
\label{Gault:subsec:GaultAcknowledgements}
The authors thank the anonymous referee whose comments greatly improved the quality of this Letter.
We wish to thank Nick Moskovitz (Lowell Observatory), who strongly encouraged us to pursue this report. We thank Dr.\ Mark Jesus Mendoza Magbanua (University of California San Francisco) for his frequent and timely feedback on the project. Many thanks to David Jewitt and Man-To Hui (Univeristy of California Los Angeles) for their helpful comments. The authors express their gratitude to Jonathan Fortney (University of California Santa Cruz), Mike Gowanlock (NAU), Cristina Thomas (NAU), and the Trilling Research Group (NAU), all of whom provided invaluable insights which substantially enhanced this work. The unparalleled support provided by Monsoon cluster administrator Christopher Coffey (NAU) and his High Performance Computing Support team facilitated the scientific process wherever possible.
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No.\ (2018258765). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Computational analyses were run on Northern Arizona University's Monsoon computing cluster, funded by Arizona's Technology and Research Initiative Fund. This work was made possible in part through the State of Arizona Technology and Research Initiative Program. ``GNU's Not Unix!'' (GNU) Astro \textit{astfits} \citep{akhlaghiNoisebasedDetectionSegmentation2015} provided command-line FITS file header access.
This research has made use of data and/or services provided by the International Astronomical Union's Minor Planet Center. This research has made use of NASA's Astrophysics Data System. This research has made use of the The Institut de M\'ecanique C\'eleste et de Calcul des \'Eph\'em\'erides (IMCCE) SkyBoT Virtual Observatory tool \citep{berthierSkyBoTNewVO2006}. This work made use of the {FTOOLS} software package hosted by the NASA Goddard Flight Center High Energy Astrophysics Science Archive Research Center. This research has made use of SAO ImageDS9, developed by Smithsonian Astrophysical Observatory \citep{joyeNewFeaturesSAOImage2003}. This work made use of the Lowell Observatory Asteroid Orbit Database \textit{astorbDB} \citep{bowellPublicDomainAsteroid1994,moskovitzAstorbLowellObservatory2018}. This work made use of the \textit{astropy} software package \citep{robitailleAstropyCommunityPython2013}.
This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A\&M University, Financiadora de Estudos e Projetos, Funda\c{c}\~{a}o Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Funda\c{c}\~ao Carlos Chagas Filho de Amparo \`{a} Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico and the Minist\'{e}rio da Ci\^{e}ncia, Tecnologia e Inova\c{c}\~{a}o, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones En\'{e}rgeticas, Medioambientales y Tecnol\'{o}gicas–Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgen\"ossische Technische Hochschule (ETH) Z\"urich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci\`{e}ncies de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit\"{a}t M\"{u}nchen and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A\&M University.
Based on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory (NOAO Prop. IDs 2012B-0001, PI: Frieman; 2014B-0404, PI: Schlegel), which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.
\section{Appendix}
\subsection{Gault Data}
\label{Gault:subsec:GaultData}
For reference we provide essential information regarding (6478) below.
\begin{center}
\textbf{Properties of (6478)~Gault}
\begin{tabular}{lll}
Parameter & Value & Source\\
\hline\hline
Discovery Date & 1988 May 12 & \citet{schmadelDictionaryMinorPlanet2003}\\
Discovery Observers & C. S. \& E. M. Shoemaker & \citet{schmadelDictionaryMinorPlanet2003}\\
Discovery Observatory & Palomar & \citet{schmadelDictionaryMinorPlanet2003}\\
Activity Discovery Date & & \citet{smith6478Gault2019}\\
Alternate Designations & 1988 JC1, 1995 KC1 & NASA JPL Horizons\\
Orbit Type & Inner Main Belt & NASA JPL Horizons\\
Family & Phocaea (Core Member) & {\citet{knezevicProperElementCatalogs2003}}\\
Taxonomic Class & S & via Phocaea association\\
Diameter & $D=4.5$ km & {\citet{harrisAsteroidsThermalInfrared2002}}\\
Absolute $V$-band Magnitude & $H=14.4$ & NASA JPL Horizons\\
Slope Parameter & $G=0.15$ & NASA JPL Horizons\\
Orbital Period & $P=3.5$ yr & NASA JPL Horizons \\
Semimajor Axis & $a=2.305$ au & NASA JPL Horizons\\
Eccentricity & $e=0.1936$ & NASA JPL Horizons\\
Inclination & $i=22.8113^\circ$ & NASA JPL Horizons\\
Longitude of Ascending Node & $\Omega=183.558$ & Minor Planet Center\\
Mean Anomaly & $M=289.349^\circ$ & Minor Planet Center\\
Argument of Perihelion & $\omega=83.2676^\circ$ & NASA JPL Horizons\\
Perihelion Distance & $q=1.86$ au & NASA JPL Horizons\\
Aphelion Distance & $Q=2.75$ au & NASA JPL Horizons\\
Tisserand Parameter w.r.t. Jupiter & $T_J=3.461$ & astorbDB\\
\end{tabular}
\end{center}
\subsection{Thumbnail Gallery}
\label{Gault:subsec:ThumbnailGallery}
For all thumbnails, red arrows indicate the negative motion vector $-\vec{v}$ of (6478)~Gault; yellow arrows point away from the Sun ($-\odot$). When possible, (6478)~Gault has been circled with a dashed green line and white arrows placed perpendicular to any observed activity. Areas outsize of chip boundaries appear black in color.
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{gaultFiles/Gault130922T030341ooiYv2.png} & \includegraphics[width=0.45\linewidth]{gaultFiles/Gault131013T030641ooiiv1.png}\\
\end{tabular}
\footnotesize
Left panel: 2013 September 22 3:03 (UT); 45 s $Y$-band. Right panel: 2013 October 13 03:06 (UT); 90 s $i$-band.
\end{center}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{gaultFiles/Gault131013T030842ooizd1.png} &
\includegraphics[width=0.45\linewidth]{gaultFiles/Gault160609T044810ooirv1.png} \\
\end{tabular}
\footnotesize
Left panel: 2013 October 13 03:08 (UT); 90 s $z$-band. Right panel: 2016 June 09 04:45 (UT); 96 s $r$-band.
\end{center}
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{gaultFiles/Gault171023T085749ooizv1.png} &
\includegraphics[width=0.45\linewidth]{gaultFiles/Gault171111T071346ooizv1.png} \\
\end{tabular}
\begin{center}
\footnotesize
Left panel: 2017 October 23 08:57 (UT); 80 s $z$-band. Right panel: 2017 November 11 07:13 (UT); 80 s $z$-band.
\end{center}
\chapter{Comprehensive Discussion of Methods and Materials}
\label{chap:methods}
\acresetall
\doublespacing
Each individual manuscript included in this dissertation includes descriptions of the methods utilized for the work therein. Here I provide updates, additional depth, and unified descriptions.
\section{HARVEST Pipeline}
\label{methods:sec:methods:pipeline}
The pipeline that ultimately produces thumbnails for examination by Citizen Scientists, myself, and my science team, is called \ac{HARVEST}. The pipeline was first described in the \ac{SAFARI} proof-of-concept (\citealt{chandlerSAFARISearchingAsteroids2018}, Chapter \ref{chap:SAFARI}). \ac{HARVEST} has evolved substantially since then into the pipeline described here.
\subsection{Pipeline Overview}
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth]{methods/HARVEST_SAFARI_flow.pdf}
\caption{The \acf{HARVEST} daily pipeline. The primary eight steps are sequential. Tasks within each step start from the top of the enclosing rectangle and finish at the bottom. Green shaded regions indicate processes that may be executed in parallel. Each step and the constituent tasks are described in text.}
\label{methods:fig:harvestFlow}
\end{figure*}
As shown in Figure \ref{methods:fig:harvestFlow}, the \ac{HARVEST} pipeline runs in a series of steps, from Step 1 to Step 8 (described in detail below). In Step 1 we query public astronomical image archives for metadata that describes observations (e.g., \acs{UT} observing date, sky coordinates, exposure time). Concurrently we query external services, such as the \ac{MPC}, for new and updated information about minor planets, such as designations, orbital parameters (e.g., semi-major axis, eccentricity). In Step 2 we estimate the depth (faintness reached) for each image, exclude duplicate datafiles that stemmed from the same observation, assign dynamical classes to objects, and compute parameters such as Tisserand's parameter with respect to Jupiter. In Step 3 we query \texttt{SkyBot} to determine which minor planets may be in each field discovered in Step 1; here we also estimate the detection viability for each object identified by SkyBot. In Step 4 we begin downloading data from the astronomical archives, then determine the sky coordinates for all \ac{CCD} chips. In Step 5 we extract thumbnail images of the desired minor planets from the archival data. During Step 6 we examine the thumbnail images, excluding those that we deem unlikely to yield results via our Citizen Science project. In Step 7 we run automated reporting code that describe, for example, how many thumbnail images per square degree on sky we have in \ac{HARVEST}. Finally, in Step 8, we conduct maintenance tasks such as purging datafiles that have completed all stages of the pipeline.
The \ac{HARVEST} pipeline is run daily and is a blend of serial and parallel processes, visually depicted in Figure \ref{methods:fig:harvestFlow}. All steps are carried out on Monsoon, the \ac{NAU} High Performance Computing cluster. All parallel tasks and most serial tasks are executed using the Slurm scheduler \citep{yooSLURMSimpleLinux2003}
\ac{HARVEST} is primarily written in Python (v3.x) with parent shell scripts executing code via Slurm scripts. Binary components are entirely supplied by the Anaconda\footnote{\url{https://www.anaconda.com/}} environment and thus require no manual compilation. Most operations involve a \texttt{My\acs{SQL}}\footnote{\url{https://www.mysql.com/}} relational database of my own design, which I describe in the next section.
\subsubsection{Overarching Principles}
Three critical overarching principles apply throughout the entire \ac{HARVEST} pipeline:
\paragraph{A. Task Counting} An optimization that applies to virtually every element of the pipeline is task counting. This determines how much work is necessary in order to (a) evaluate whether or not Slurm is necessary and, if so, (b) assess how many Slurm jobs are needed for each task. Task counting is also used in reporting to ensure the daily pipeline is not falling behind in any area (e.g., catalog queries). The number of jobs launched depends on factors such as the amount of memory needed and the intensity of disk access required. We arrived at these values through trial and error during the development of the \ac{HARVEST} pipeline.
\paragraph{B. Exclusion} Nearly every database table and most tasks involve either excluding data we determine is unfit for use, or ignoring previously excluded data. Exclusion is discussed in in Sections \ref{methods:subsubsec:imageArchives}, \ref{methods:subsubsec:sourceAnalysis}, and \ref{methods:subsubsec:datafileRoutines}.
\paragraph{C. Self-awareness of Elapsed Computing Time} To account for the finite time allotted for each process, all tasks executed through the \ac{HARVEST} pipeline keep track of elapsed time and periodically compute how much time is remaining. This is primarily accomplished through environment variables, but a secondary layer of signaling via the operating system and/or Slurm allows tasks to be systematically warned they are almost out of time. This awareness enables (a) tasks to safely wrap up whatever they are doing (e.g., write data to disk), (b) the pipeline to understand the underlying tasks were not finished, and (c) Slurm scheduler optimization, especially back-fill (making use of leftover compute time on the compute cluster).
\paragraph{D. Self-limiting Service Calls} Tasks that call external services (e.g., \ac{JPL}, \ac{MPC}) may unduly tax host servers. Services typically determine the origin of queries by one or two methods: the originating \ac{IP} address and/or (if supported) an identifier (e.g., email address) embedded in an \ac{API} call. Some external services may ``blacklist'' (block) queries, usually from a specific \ac{IP} address. To address this, we limit the number of queries per minute based on rates we ascertained via documentation and communicate with external service providers as needed.
\paragraph{E. Logging} Extensive logging is crucial for diagnosing problems and, importantly, easily assessing what (if any) tasks need to be repeated. Thus we output information such as how far a task has progressed and what item or objects are being processes.
\paragraph{F. Notifications} Mobile device notifications allow me to monitor the pipeline and attend to problems promptly. For example, a program will monitor jobs submitted for execution and send one of the science team that the jobs are complete.
\paragraph{G. Dynamically Interactive Code} Interactive elements are eliminated from the automated pipeline, but interactive elements play a crucial role in parts of the system that are executed outside of the pipeline, such as choosing a replacement for a bad datafile. We designed and implemented a system that enables tasks to ``know'' whether or not they are being executed in a supervised mode amenable to interaction. Otherwise, tasks running on compute nodes would ask for user input which is impossible to acquire.
\paragraph{H. Permanence} A completely dynamic system can be flexible and change information, for example updating a minor planet's designation. This is possible internally in \ac{HARVEST}, but some properties must remain fixed once set because of Citizen Science involvement. For example, a thumbnail name cannot change once assigned, because the thumbnail name is the identifying feature that allows linkage between the Citizen Science project and \ac{HARVEST}.
\paragraph{I. Minor Planet Identity Management} Managing names, identifications, and designations has proven to be one of the most difficult aspects of the pipeline. Requesting a specific solar system object occurs during the vast majority of pipeline tasks. Challenges include handling identity when an object breaks apart (e.g., P/2016 J1-A and P/2016 J1-B, \citealt{huiSplitActiveAsteroid2017}) or, more commonly, deciding how to handle objects that become consolidated (linked, e.g., P/2022 C$_4$ = P/2010 LK$_{36}$ = P/2016 MD = P/2016 PM$_1$ WISE-PANSTARRS, \citealt{fitzsimmonsCBAT5137COMET2022}) or retracted (e.g., 2011~UH$_{413}$ and 2013~QQ$_{95}$, \citealt{deenMPEC2020N22RETRACTION2020}).
\subsection{HARVEST Step 1: Catalog Queries}
The first step of my pipeline is to query catalog services that provide details about telescopic observations and information about solar system objects.
\subsubsection{Image Archives}
\label{methods:subsubsec:imageArchives}
We query astronomical image archives for metadata, especially \ac{UT} observation date/time, sky coordinates, broadband filter, and the release date when the data becomes publicly available. At this stage we perform several initial checks and exclude images that do not meet specific criteria:
\begin{enumerate}
\item \textbf{Airmass:} Pointings must be below 3 airmasses.
\item \textbf{Moon Separation:} Observations with Moon separations $<4^\circ$ are excluded; we compute the Moon separation ourselves, with the distance computed between the Moon's center and the center of the telescope pointing provided in the image archive metadata.
\item \textbf{Invalid Coordinates:} We exclude observations with illegal coordinates (e.g., \ac{RA} above 360$^\circ$, e.g., \ac{DECam} archive filename c4d\_201123\_040730\_ooi\_i\_v1.fits.fz).
\item \textbf{Processing Type:} We exclude raw data, stacked (co-added) images, and non-image data (e.g., data quality masks). We exclude RAW data because (a) activity is harder to detect, and (b) the embedded \ac{WCS} are insufficient for our thumbnail purposes (i.e., the object may not appear near the center of a thumbnail image). Stacked data are excluded because they often produce images that eliminate the object or introduce image artifacts likely to complicate activity detection.
\item \textbf{Broadband Filter:} We keep broadband filters (e.g., Sloan) and exclude others (e.g., H$\alpha$) that are unlikely to show faint activity. The retained filters are U, B, V, R, I, u, g, r, i, z, Y, J, H, K, M, VR, V+R, g+r+i, g+i, and variants therein (e.g., i variants include DECam-i, SDSS-i, and MegaPrime-i).
\end{enumerate}
We note that it is not uncommon in big data applications, such as with the \ac{HARVEST} processing astronomical image archive metadata, to encounter rare but problematic peculiarities such as sky coordinates outside of the defined range (e.g., RA$>360^\circ$) or representing an impossible elevation for a telescope to reach. Hence we implement safeguards to screen for these data which would otherwise cause problems for software not designed to handle these situations.
\paragraph{NSF's NOIRLab AstroArchive} The \ac{NSF} \ac{NOIRLab} AstroArchive\footnote{\url{https://astroarchive.noirlab.edu}} is the primary source for data submitted for volunteer examination, \ac{DECam} image data from the Blanco 4~m telescope at the \ac{CTIO} in Chile. We also incorporate the \ac{KPNO} 4~m telescope archival data that AstroArchive hosts. Note: DECam images and metadata were hosted at the \ac{NOAO} prior to their consolidation into \ac{NOIRLab}.
\paragraph{CADC} The \ac{CADC}\footnote{\url{http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca}} hosts archival data and provides search tools for many astronomical instruments. At present we query the \ac{CADC} for \ac{CFHT} MegaPrime and \ac{WIRCam} metadata.
\subsubsection{Minor Planet Data}
\label{methods:subsec:objData}
These tasks query external services for minor planet data including orbital parameters, designations, and rotation periods.
\paragraph{Minor Planet Center} The \ac{MPC} provides tabular data describing objects (asteroids and comets) and their orbits. The \ac{MPC} also serves as our first line of inquiry for object numbers, names, and provisional designations.
\paragraph{Kinoshita Comet Pages} The late Kazuo Kinoshita was an amateur astronomer in Japan who computed comet orbits and maintained a website\footnote{\url{https://jcometobs.web.fc2.com}} of comets. These pages included some orbital elements (e.g., perihelion distance) and name permutations we were unable to locate anywhere else. Kazuo Kinoshita passed away in 2021 July.
\paragraph{Ondrêjov Discoveries} This website, hosted at \ac{ASU} \footnote{\url{http://www.asu.cas.cz/~asteroid/news/numbered.htm}}, contains a list of discoveries made at Ond\^{r}ejov Observatory (site code 557). These names may not be present in \ac{MPC} or \ac{JPL} Horizons data, either because the object has a slightly different spelling or the object has a name completely unknown to the \ac{MPC} and Horizons. Importantly, the \ac{IMCCE} Quaero Service \footnote{\url{https://ssp.imcce.fr/webservices/ssodnet/api/quaero/}}, which serves the names returned by \ac{IMCCE} SkyBot (described below in Section \ref{methods:subsubsec:skyBot}), may return names that are found at this website but not necessarily at the \ac{MPC} or \ac{JPL} Horizons (Section \ref{methods:objectSpecificData}).
\paragraph{Astorb} The Lowell Observatory \ac{AstOrb} \footnote{\url{https://asteroid.lowell.edu/main/astorb/}} \citep{bowellPublicDomainAsteroid1994,moskovitzAstorbDatabaseLowell2021} contains object designations, orbital classifications, and related orbital elements (e.g., semi-major axis). This dataset is primarily used to provide orbital elements we were unable to attain elsewhere, and as an additional source for minor planet designations.
\subsection{HARVEST Step 2: New Data Handling}
\label{methods:subsec:newDataHandling}
\subsubsection{Datafile-specific Operations}
\paragraph{Magnitude Estimates} For computationally expedient and consistent estimation of depth we first compute a rough estimated magnitude depth for a given combination of exposure time, telescope/instrument, and broadband filter. The method we use is telescope-specific because whenever possible we use the observatory-supplied \ac{ETC}, sometimes referred to as an \ac{ITC}. In cases where we are unable to use an observatory-supplied \ac{ETC} we make use of the \ac{DECam} \ac{ETC}\footnote{\url{https://noirlab.edu/science/documents/scidoc0494}}, replacing the default mirror effective surface as needed. We compute an estimated magnitude reachable with a \ac{SNR} of 10. For filters that are not already $V$ band we offset the filter-specific depth to a generic $V$-band depth by applying an offset from known solar apparent Vega magnitudes\footnote{\url{http://mips.as.arizona.edu/~cnaw/sun.html}} \citep{willmerAbsoluteMagnitudeSun2018}. The purpose of this step is to match the $V$-band magnitudes supplied by ephemeris services (e.g., \ac{JPL} Horizons) for later use in determining the viability of activity detection (Delta Magnitude, Section \ref{methods:subsec:fieldAssesment}). To be clear, we only need a very rough estimate of depth for the purpose of gauging activity detection viability for each exposure.
\paragraph{Version Selection} involves choosing a single datafile for thumbnail extraction (Section \ref{methods:subsec:thumbnailExtraction}). All others are excluded as they are essentially duplicates. This process is necessary for two reasons: (1) some images have multiple versions of a specific processing application, or (2) multiple viable image processing techniques are available for a single observation.
\subsubsection{Object-Specific Data Handling}
\label{methods:objectSpecificData}
\paragraph{Object Class} A solar system object can qualify as a member of more than one dynamical class. For example, all Apollo asteroids are types of \acp{NEO} and some (but not all) Mars-crossing asteroids are \acp{NEO}. At present we assign a primary class to each solar system object. Classes in \ac{HARVEST} are Comet, Amor, Apollo, Aten, Mars-crosser, \ac{IMB}, \ac{MMB}, \ac{OMB}, Cybele, Hungaria, \ac{JFC}, Hilda, Trojan, Centaur, Damocloid, \ac{TNO}/\ac{KBO}, or Interstellar Object. The \ac{MPC} and \ac{JPL} \ac{SBDB} do not contain all of these dynamical classes (e.g., Damocloid) so we take additional steps to reclassify objects. For example, we perform a check to reclassify objects labeled as Centaurs with a $T_\mathrm{J}<2$ as Damocloids, and $2<T_\mathrm{J}<3$ as \ac{JFC}s. These distinctions help us to collect thumbnail images from our library and organize them by dynamical class for efficient examination. The default position is to assume the class provided with the \ac{IMCCE} SkyBot results. We generally do not reclassify an object once a thumbnail image is produced because the object class is contained (as a code) in the thumbnail name for convenience, and the thumbnail names cannot be changed once submitted to \textit{Active Asteroids} for Citizen Scientist classification.
\paragraph{NASA JPL Object Data} In order to optimize calculations and reduce query calls to \ac{JPL} we maintain our own internal database of NASA \ac{JPL} provided solar system object-specific parameters, such as rotation period (used for observation planning) and semi-major axis $a$. Our queries to \ac{JPL} come in two forms: the \ac{JPL} Horizons ephemeris service\footnote{\url{https://ssd.jpl.nasa.gov/horizons/}} and the \ac{JPL} Small Body Database Lookup service\footnote{\url{https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html}}, both of which return object-specific information.
\paragraph{Solar System Object Parameters} During this phase we attempt to collect a unified set of object-specific dynamical properties, namely semi-major axis $a$, inclination $i$, eccentricity $e$, perihelion distance $q$, and aphelion distance $Q$. At this stage we compute the Tisserand parameter \citep{tisserandMecaniqueCeleste1896} with respect to Jupiter, $T_\mathrm{J}$, commonly used as a metric to quantify the gravitational interaction between Jupiter and a small body given their respective orbits \citep{kresakJacobianIntegralClassificational1972}. The parameter is given by
\begin{equation}
T_\mathrm{J} = \frac{a_\mathrm{J}}{a} + 2\sqrt{\left(1-e^2\right)\frac{a}{a_\mathrm{J}}}\cos(i),
\label{eq:TJ}
\end{equation}
\noindent where $e$ represents osculting eccentricity, $i$ orbital inclination, $a$ the semi-major axis of the body, and $a_\mathrm{J}$ is Jupiter's semi-major axis.
\subsection{HARVEST Step 3: Assessing Fields}
\label{methods:subsec:fieldAssesment}
These tasks are specific to a Field record, defined as a unique combination of an observing date/time and sky coordinates. Multiple field records may exist for a single observation as sky coordinates are adjusted during the astrometry phase of the pipelines used with, for example, \ac{DECam} data.
\subsubsection{SkyBot}
\label{methods:subsubsec:skyBot}
\paragraph{Query} The \ac{IMCCE} SkyBot service\footnote{\url{https://ssp.imcce.fr/webservices/skybot/}} \citep{berthierSkyBoTNewVO2006} allows a user to determine which solar system objects (if any) may be present in an astronomical image given (1) the \ac{UT} date/time, (2) \ac{RA} and \ac{Dec} sky coordinates, and (3) an angular shape that specifies the size of the area to search. The shape chosen ideally minimizes area not covered by an imaging detector. Given the roughly circular shape of the \ac{DECam} chip arrangement, we primary use the radius (1.1$^\circ$) of the 2.2$^\circ$ \ac{FOV} to conduct a \textit{cone search}. We also use rectangular queries for instruments more suitable to this configuration, for example \ac{CFHT} MegaCam. Queries are carried out using the online SkyBot \ac{API}.
\paragraph{Resubmitting Fields}
For each query submitted, SkyBot returns a unique ticket ID. Along with the date of the query we keep track of the age of each SkyBot Ticket record. We limit the number of SkyBot queries per day, however we try not to allow Ticket dates to exceed 90 days of age. We institute these policies to accommodate new object discovery as well as improvements to object orbital elements. We wrote a program that will flag specific Ticket records for resubmitting to SkyBot based on orbital parameters, however the 90 day age limit has proven far more computationally efficient and fewer overall queries to the SkyBot servers.
\subsubsection{Delta Magnitudes}
\label{methods:subsubsec:deltaMagLim}
During the SkyBot step we compute a simple metric that describes how much brighter (or fainter) an object will appear as compared to the depth of the exposure we computed during Magnitude Estimates (Section \ref{methods:subsec:newDataHandling}. The metric is
\begin{equation}
\Delta_\mathrm{mag} = V_\mathrm{JPL} - V_\mathrm{ITC},
\end{equation}
\noindent where $V_\mathrm{JPL}$ is the Horizons provided apparent $V$-band magnitude computed for the object at the time of exposure, and $V_\mathrm{ITC}$ is the $V$-band magnitude depth we computed for this exposure, as described in Section \ref{methods:subsec:newDataHandling}. $\Delta_\mathrm{mag}<0$ indicates the object will likely appear bright enough in the image to be detected, whereas $\Delta_\mathrm{mag}>0$ would not. We set a maximum $\Delta_\mathrm{mag}$ of $-1$ for all detections. Note: the apparent magnitude is computed by Horizons, however the object may appear brighter or fainter, for example, in cases of activity outburst or prior mass loss, respectively. At the time of this writing, roughly 57\% ($\sim$21 million) of all potential minor planet detections in \ac{HARVEST} were excluded because of the $\Delta_\mathrm{mag}$ threshold.
\subsubsection{Trail Length}
\label{methods:subsec:deltaPixels}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{methods/trails/0541518C_20140922-00053_0889277_0.0y_qA_OE_620s_T-7_pI_ovX_100pix.png} & \includegraphics[width=0.45\linewidth]{methods/trails/0617278N_20140327-06345_0709151_1.3y_qj_Od_600s_T-6_pI_ovX_280pix.png} \\
\end{tabular}
\caption{Examples of trails resulting from high rates of apparent motion on the sky. Images credit: this work.
\textbf{Left:} Comet C/2014~QU$_2$ (PANSTARRS) as observed by \ac{DECam} on \ac{UT} 2014 September 22 (Prop. ID 2014B-0303, PI Sheppard). Even with a trail length of about 100 pixels, activity is still clearly visible in the upward direction.
\textbf{Right:} Apollo class \ac{NEO} 2014~EH$_{45}$ observed by \ac{DECam} on \ac{UT} 2014 March 27 (Prop. ID 2014A-0479, PI Sheppard) with an approximately 280 pixel length trail. The variable brightness seen along the length of the trail is diagnostic of fast rotation.
}
\label{methods:fig:trails}
\end{figure}
\textit{Delta pixels} is a metric we compute that describes the number of pixels an object appears to travel across the FOV (Figure \ref{methods:fig:trails}), given the pixel scale of the instrument and the object's apparent rate of motion on the sky. We use this value to screen, for example, images for use in Citizen Science training and the Field Guide, described in Section \ref{methods:sec:citsci}.
This value may be used in the future to exclude thumbnails with long (e.g., $>15$ pixel) trails that may be confusing to volunteers and lead to activity detection false positives. Alternatively, in response to \textit{Active Asteroid} volunteer comments, we may use this value to help search for fast-rotating asteroids known as fast-rotators (Figure \ref{methods:fig:trails}). These trails may appear to have variable width, or even appear dashed, depending on the physical geometry of the object and its rotation speed.
\subsection{HARVEST Step 4: Thumbnail Preparations}
\label{methods:subsec:thumbPrep}
The thumbnail preparation tasks in this section are conducted in parallel because we do not wait for datafiles to download. Instead, we work with the datafiles already on disk, and return to incomplete or missing datafiles during a subsequent pipeline execution. This avoids long delays induced if we wait for large quantities of data to finish downloading from archives.
\subsubsection{Data Download}
During this phase we perform several tasks: (1) check datafiles flagged as ``on disk'' to ensure they are actually present and, if not, flag the datafile as ``to download.'' We generate bash scripts for daemons (persistent tasks) dedicated to downloading data. Datafiles are typically flagged for download for three reasons: (1) new data was added to an astronomical archive that does not have a proprietary period, (2) datafiles known by \ac{HARVEST} with proprietary periods that have since ended, and (3) a new solar system object was found in a datafile we previously had finished processing and purged from disk.
\subsubsection{File Inspection}
\label{methods:subsec:datafileIntrospection}
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{methods/mosaic/c4d_180318_063808_ooi_g_v2_mosaic_smaller.png}
\caption{This mosaic shows the \ac{CCD} chip arrangement of the \acf{DECam} instrument. Image data is not recorded anywhere outside of the \ac{CCD} chips (rectangles). Areas between chips are commonly referred to as ``chip gaps.''}
\label{methods:fig:mosaic}
\end{figure}
\paragraph{Chip Corners} Here we examine in parallel downloaded image data and populate a database table with the sky coordinates (RA, Dec) of each corner of each camera chip. This step provides an order-of-magnitude speed increase (as compared with manually reading the bounds from an image file) when determining which solar system objects do not fall on a chip, either because they are on a chip gap, or on a chip that is not functional at the time of observation (as is the case for some \ac{DECam} images). Moreover, with the corner coordinates recorded, we do not need to download a datafile previously processed and purged to check chip corners because the coordinates are already cached in our database. Chips gaps are the spaces between the \ac{CCD} chips of cameras that use multiple detectors; these gaps do not record any image data.
\paragraph{Header Archiving} As part of Chip Corners we also write a compressed text file to disk containing all (primary and chip-specific) header information for each archival image file. This allow us to make use of data held in headers even after the parent datafile has been purged.
\subsection{HARVEST Step 5: Thumbnail Extraction}
\label{methods:subsec:thumbnailExtraction}
This stage consists of two steps executed in parallel. We process up to 50 datafiles simultaneously, which stays below our estimated \ac{HPC} cluster file system throughput limit.
\subsubsection{FITS Thumbnails}
We extract \ac{FITS} format ``cutouts'' (extracted region of image data) of each unexcluded SkyBot result (Section \ref{methods:subsubsec:skyBot}). We first execute a \ac{SQL} query of our internal database table that stores records of chip corners (Section \ref{methods:subsec:datafileIntrospection}) to determine which chip should contain the image of the object. If the object does not fall within the bounds of any chip, the thumbnail and associated SkyBot records are excluded, so (1) we do not attempt to extract the thumbnail image again, and (2) our database queries remain optimized, and (3) statistical reports reflect current state of \ac{HARVEST} (e.g., number of thumbnail images processed). If the object is located within the bounds of a camera chip then we perform an image cutout that produces a $480\times480$ pixel thumbnail image in \ac{FITS} format. We handle boundary conditions (e.g., the object falls near a chip edge) by filling empty pixels coordinates with \texttt{NULL} values.
Crucial for future analyses, our Thumbnail Extraction Tool adjusts the \ac{WCS} to match the cutout image prior to writing the \ac{FITS} file to disk. \ac{WCS} in \ac{FITS} headers provide a reference that effectively translates pixel coordinates to sky coordinates. For convenience and record completeness we also copy global header values (e.g., observation date and time) and the headers associated with the corresponding chip to the resulting \ac{FITS} format thumbnail image file.
\subsubsection{PNG Thumbnails}
We convert the \ac{FITS} format thumbnail to a \ac{PNG} image file. In order to maximize the likelihood of activity detection, we carry out an iterative rejection technique to determine the best contrast range for the image (see Chapter \ref{safari:procedure} for details).
\subsection{HARVEST Step 6: Post Thumbnails}
Source Analysis, which is carried out in parallel, must conclude prior to the Counting steps, which may all be performed in parallel.
\begin{figure*}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.32\linewidth]{methods/exclusion/empty.png} & \includegraphics[width=0.32\linewidth]{methods/exclusion/crowded.png} & \includegraphics[width=0.32\linewidth]{methods/exclusion/blended_0687902U_20190429-10134_1456120_15.y_qB_Od_30.s_z-2_pI_ovX.png} \\
(a) & (b) & (c)
\end{tabular}
\caption{The three causes for excluding a thumbnail during the Source Analysis section of the \acf{HARVEST} pipeline.
(a) Empty center: there is no object at the center of the frame to inspect.
(b) Crowded field: there are too many sources in the center 270$\times$270 pixels, making identification of the target difficult.
(c) Blended sources: multiple overlapping sources are seen here, making it difficult to identify which source (if any) is the target.
}
\label{methods:fig:exclusionBySourceAnalysis}
\end{figure*}
\subsubsection{Source Analysis}
\label{methods:subsubsec:sourceAnalysis}
We make use of \texttt{SExtractor} \citep{bertinSExtractorSourceExtractor2010} to determine the pixel coordinates of each point source within each thumbnail image. At this stage we exclude thumbnail images for cases which would be confusing to volunteers or unlikely to lead to an activity detection. The three exclusion causes are given with numbers current as of 10 July 2022, when \ac{HARVEST} had a total of 22,004,739 unexcluded thumbnail images, spanning all instruments. An example from each scenario is provided in Figure \ref{methods:fig:exclusionBySourceAnalysis}.
(1) No source was detected within the center $20\times20$ pixels of the thumbnail image (Figure \ref{methods:fig:exclusionBySourceAnalysis}a). This resulted in 16\% of thumbnails being excluded, 4,248,133 in total. \texttt{SExtractor} detects even faint sources, so if it cannot identify an object at the center of the frame we do not expect volunteers to be able to either. This may be due to poor observing conditions or cases where the object did not appear directly in the center of the frame because, for example, there was a high positional uncertainty.
(2) Too many ($>150$) sources were in the center $270\times270$ pixels of the thumbnail image (Figure \ref{methods:fig:exclusionBySourceAnalysis}b). The 952,289 thumbnails excluded for this reason represents another 4\% reduction in the total number of viable thumbnails. This situation makes identifying which object is the target confusing for volunteers.
(3) When there are too many ($>5$) overlapping (blended) sources at the center of the frame (Figure \ref{methods:fig:exclusionBySourceAnalysis}c) it can be very difficult to determine which source is the object of interest. This is the least common cause for thumbnail exclusion, with just 84,697 (0.4\%) thumbnails excluded. We exclude these images because it can be exceptionally difficult to identify the object of interest in the image.
\subsubsection{Source Tallying}
\label{methods:subsubsec:counting}
\paragraph{Objects per Field} At this point all exclusion routines have finished executing. We tally how many unexcluded solar system objects are present in each field. This step optimizes our analysis of the distribution of on-sky of solar system objects (see Chapter \ref{chap:discussion}).
\paragraph{SkyBot Source Density} We tally unexcluded SkyBot results associated with each SkyBot Ticket record. Tickets include a unique identification for the query performed, and this metric is used for later internal reporting. Tracking tickets is also crucial for troubleshooting with, for example, the \ac{IMCCE}.
\paragraph{Observation Source Tally Consolidation} We consolidate the above tallies into the parent Observation records in our database to optimize analysis tasks. For example, this optimization reduces the compute time for plotting the sky distribution of minor planets in our database (Figure \ref{discussion:fig:asteroidsOnSky}) by orders of magnitude.
\subsection{HARVEST Step 7: Reporting}
\paragraph{SkyBot Reports} Here we determine the age of SkyBot tickets and create an overview of the status of all fields. This provides a status of SkyBot queries within \ac{HARVEST} and can be diagnostic of problems (e.g., if many tickets are much older than 90 days). These reports are periodically reviewed by our science team
\paragraph{Objects per Field Plot} This plot (Figure \ref{discussion:fig:asteroidsOnSky}) shows how many solar system objects are present for each telescope pointing in our database. These data derive from the tallies executed in Step 6 (Section \ref{methods:subsubsec:counting}). The Objects per Field Plot has many uses, including revealing gaps in sky coverage which, in turn, result in observational biases. For example, this plot shows how far north and to what degree our sky coverage extends from our predominately southern hemisphere originating observations.
\subsection{HARVEST Step 8: Maintenance}
\subsubsection{Datafile Routines}
\label{methods:subsubsec:datafileRoutines}
\paragraph{Datafile Checks} We inspect downloaded datafiles to ensure datafile integrity; this additional check is a secondary stage, prompted by flags set during \ac{HARVEST} Step 4: Thumbnail Preparations (Section \ref{methods:subsec:thumbPrep}). Here we check \ac{FITS} standards for each image data and header extension of the file, as well as a basic file size check. Datafiles that we determine are corrupt we attempt to download a second time. Should the second datafile also be corrupt, we flag the datafile as ``bad'' in our database and attempt to find a replacement, either a different version or another acceptable image processing type.
\paragraph{Datafile Exclusion by Property} This task excludes datafile records based on specific properties. At present the only task is excluding datafiles with exposure times $<1$~s, including \texttt{NULL}, 0~s, and negative exposure time values.
\paragraph{Purge Datafiles} As we do not have the storage space to keep a copy of all data we download from astronomical data archives we must delete files once we have finished carrying out all steps of the \ac{HARVEST} pipeline.
\subsection{Auxiliary Procedure: Comparison Images}
\label{methods:subsec:comparisonImages}
\begin{figure*}[h]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{methods/comparison/100_0648610H_20151219-02525_1672035_7.3y_qC_OH_090s_z-2_pI_ovX.png} & \includegraphics[width=0.45\linewidth]{methods/comparison/0648610H_20151219-02525_1672035_7.3y_qC_OH_090s_z-2_pI_ovX_POIMid53_2016-01-01_01.40.14_c4d_160101_014146_ooi_r_v1_chip8-S21_240pix_NuElnoArrows.png} \\
\end{tabular}
\caption{Comparison between two identical areas on sky, imaged with similar circumstances (e.g., instrument, exposure time, broadband filter), but taken at times far apart enough that the solar system object would not appear in the comparison image. Images credit: this work.
\textbf{Left: } This thumbnail image of \acf{JFC} 2015~TC$_1$ was taken on \ac{UT} 2015 December 19 (Prop. ID 2012B-0001, \acs{PI} Frieman) and submitted for examination by \textit{Active Asteroids} (\url{http://activeasteroids.net}) volunteers; all volunteers classified this image as showing activity, and several users also shared this image in the Talk forums as a potentially active object.
\textbf{Right: } This image of the same field was captured \ac{UT} 2016 January 1 (Prop. ID 2012B-0001, \acs{PI} Frieman) probes somewhat fainter than the original image on the left. The comparison illustrates that there is no background galaxy or other phenomenon present at the exact coordinate where 2015~TC$_1$ was seen.
}
\label{methods:fig:comparison}
\end{figure*}
A common situation arises where a thumbnail shows what appears to be activity, but the activity is ambiguous. For example, Citizen Scientists highlighted a thumbnail image (Figure \ref{methods:fig:comparison}) as having activity, through discussion forums as well as classification. To ascertain whether or not suspected activity is actually a background source (e.g., galaxy), a comparable image (e.g., same area on sky, similar depth) could be useful, so long as enough time has elapsed such that the solar system object is no longer in the field; this could be from minutes to days later, depending on the object's apparent rate of motion.
I created a tool, which we employ on a case-by-case basis, to address this scenario by finding \textit{comparison images} to accompany the image with potential activity. My tool employs a systematic approach to find the closest match as possible, prioritizing (in approximate order) (1) enough time has elapsed to ensure the object is not in the \ac{FOV} based on the object's apparent rate of motion, (2) instrument, (3) delta magnitude limit (Section \ref{methods:subsec:fieldAssesment}), (4) broadband filter, and (5) processing type. (Processing types vary by instrument. For \ac{DECam} we make use of InstCal and Resampled images. The latter incorporates more processing but is not available for all images in the archive.) Crucially, the tool relies on our internal database of chip corner coordinates (Section \ref{methods:subsec:datafileIntrospection}). Once potential comparison image sources have been identified (I set the default to be five comparison images) then the datafiles are flagged for download and download scripts generated. Once downloads have finished we then run a separate tool that produces the thumbnail cutout images.
\section{Citizen Science Project}
\label{methods:sec:citsci}
We chose the Citizen Science platform \textit{Zooniverse}\footnote{\url{https://www.zooniverse.org}} for our project, named \textit{Active Asteroids}\footnote{\url{http://activeasteroids.net}}. Zooniverse has a proven track record of success and, importantly, they provided customization and support that facilitates any hosted project's success.
The overall process, from inception to launch, is as follows:
\begin{enumerate}
\item Prepare project on Zooniverse (see sections below)
\item Conduct an initial ``Beta Release'' to test project viability
\item Formally launch the project (i.e. open to the public)
\item In a cyclic fashion we continue to
\subitem a. Interact with volunteers via online forums and Zooniverse messages, the internal Zooniverse inter-user communication system similar to email
\subitem b. Download and analyze results
\subitem c. Prepare and upload the next set of images
\subitem d. Notify volunteers that additional work is available via Twitter and an email newsletter sent by Zooniverse
\subitem e. Conduct scientific investigation (Section \ref{sec:methods:followup}) into the resulting candidates as they become available
\end{enumerate}
\subsection{Project Components}
\label{methods:subsec:projectComponents}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{methods/citsci/landing/landingMinimal.png}
\caption{Landing page for the \textit{Active Asteroids} Citizen Science project hosted at Zooniverse. The logo we designed is at the top-left. The horizontal blue text box displays updates we provide to volunteers. The green box indicates the project status; as indicated, the project was complete, so uploading another Subject Set (collection of images) would be the next step in our workflow.}
\label{methods:fig:citsci:landing}
\end{figure}
Zooniverse provides a standard framework for each project. All users start at the project landing page, shown in Figure \ref{methods:fig:citsci:landing}. From there Citizen Scientists can begin classifying (working) immediately or navigate to one of the other pages, described below.
\subsection{Workflow}
\label{methods:subsec:workflow}
\begin{figure}
\centering
\includegraphics[width=0.75\linewidth]{methods/citsci/Zooniverse_Active_Asteroids_workflow_dark_close_crop.png}
\caption{The \textit{Active Asteroids} project workflow is simply to ask whether or not volunteers can see activity emanating from the object at the center of the screen, indicated by the green reticle. Citizen Scientists can also tag thumbnails for discussion in Talk forums or collect them in their own albums.}
\label{methods:fig:workflow}
\end{figure}
The \textit{Workflow} describes the task we are asking the Citizen Scientists to perform. The \textit{Active Asteroids} workflow is concise: we ask volunteers whether or not they see activity emanating from the object at the center of a thumbnail image, such as the one shown in Figure \ref{methods:fig:workflow}.
\subsubsection{Tutorial}
\begin{figure}
\centering
\begin{tabular}{cccc}
\includegraphics[width=0.22\linewidth]{methods/citsci/tutorial/1_welcome.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/tutorial/2_hotOrNot.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/tutorial/3_tails.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/tutorial/4_comae.png}\\
(1) & (2) & (3) & (4) \\
\\
\includegraphics[width=0.22\linewidth]{methods/citsci/tutorial/5_lookAlikes.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/tutorial/6_uncertain.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/tutorial/7_training.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/tutorial/8_noObject.png}\\
(5) & (6) & (7) & (8)\\
\end{tabular}
\caption{The \textit{Active Asteroids} Tutorial. (1) ``Welcome'' provides a project overview. (2) ``Hot or Not'' describes the workflow. (3) ``Tails'' describes tails with an example. (4) ``Comae'' introduces volunteers to the less familiar coma morphology. (5) ``Activity Look-a-Likes'' describes common false positive scenarios. (6) ``Feeling uncertain?'' explains what to do if a classification is ambiguous. (7) ``One last thing...'' lets users know about the injected training images. (8) This panel, requested by volunteers, describes a common situation where the object cannot be conclusively identified.}
\label{methods:fig:tutorial}
\end{figure}
Figure \ref{methods:fig:tutorial} shows the eight panels of the project tutorial which is shown to volunteers the first time they participate in the project. The tutorial is also available at all times in a panel of the classification workflow window.
\subsubsection{Field Guide}
\begin{figure}
\centering
\begin{tabular}{cccc}
\includegraphics[width=0.22\linewidth]{methods/citsci/fieldGuide/trailsArtificial.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/fieldGuide/crowdedFields.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/fieldGuide/donutes.png} & \includegraphics[width=0.22\linewidth]{methods/citsci/fieldGuide/partialImages.png} \\
(1) & (2) & (3) & (4)\\
\end{tabular}
\caption{Four examples from the Field Guide of the \textit{Active Asteroids} project. (1) ``Trails (artificial),'' includes an example satellite trail and a a example of a telescope tracking problem that results in all sources in a field becoming trailed. (2) ``Crowded Fields'' shows an example of a situation we try to automatically screen out from the project, but which may occasionally make it through our filter. (3) ``Donuts'' discusses this optical issue and includes a donut with activity indicators. (4) ``Partial Images (Edges and Corners)'' explains what happens when an object falls near one or two chip edges and includes an example.}
\label{methods:fig:fieldGuide}
\end{figure}
The Field Guide is available at all times during the project classification workflow, and can be left open at the side of the window. The guide provides example images and discussion concerning myriad scenarios volunteers may encounter. As of this writing, the topics covered are
\begin{itemize}
\item \textbf{Asteroids (object of interest)} explains the object should be at center, as a point source or trailed.
\item \textbf{Trails (natural)} discusses trailing and fast-rotators.
\item \textbf{Tails (object of interest)} describes possible tail morphology, including multiple tails.
\item \textbf{Comae} explains how comae are diffuse (as compared to a tail) and provides an example.
\item \textbf{Missing Object} acknowledges this situation arises and describes how these can make it past our vetting process.
\item \textbf{Crowded Fields} shows examples of fields with many sources, with and without an active object.
\item \textbf{Blurry images} states that these low quality images should probably be skipped.
\item \textbf{Galaxies} shows several galaxy examples, include some juxtaposed with active objects.
\item \textbf{Cosmic Rays} defines the phenomenon and gives examples of thumbnails with cosmic rays.
\item \textbf{Trails (artificial)} briefly discusses satellite trails and telescope tracking issues.
\item \textbf{Saturation and Scattered Light} explains how scattered light could be misinterpreted as activity.
\item \textbf{Background Object of Note (and size!)} gives examples of interesting clusters and dust clouds that are large in size given the angular size of the thumbnail.
\item \textbf{Donuts} are displayed, including one showing activity, and we discuss the donut phenomenon.
\item \textbf{Dead Columns} gives examples of columnar artifacts intersecting with the object at the center of the thumbnail.
\item \textbf{Partial Images (Edges and Corners)} supplies examples of chip edge and chip corner cases that may be confusing.
\item \textbf{Object width (object of interest)} discusses why point sources appear to have different widths.
\end{itemize}
\subsubsection{About}
\paragraph{Research} This page provides further discussion of the science justification, a description of the thumbnail production pipeline, explains some of the common challenges for the project, justifies why Citizen Science is needed, and provides a proof-of-concept statement along with links to papers that have resulted from the project and preparations.
\paragraph{The Team} Provides pictures and brief biographical sketches describing team members. At the time of this writing, the \textit{Core Research Team} is made up of (1) this author (Colin Orion Chandler) of \ac{NAU}, (2) Co-founder Chad Trujillo of \ac{NAU}, (3) Co-founder Jay Kueny of Lowell Observatory and \ac{UA}, Project Scientist Will Oldroyd of \ac{NAU}, and Project Scientists Will Burris of \ac{SDSU}. \textit{Contributors} is comprised of Annika Gustaffson of \ac{SwRI}. The \textit{Science Advisory Board} includes Henry Hsieh of the \ac{PSI}, Mark Jesus Mendoza Magbanua of \ac{UCSF}, Michael Gowanlock of \ac{NAU}, David Trilling of \ac{NAU}, and Ty Robinson of \ac{UA} and \ac{NAU}. Our \textbf{Moderator} is Elisabeth Baeten (Belgium).
\paragraph{Results} The Results page currently lists the \ac{SAFARI} proof-of-concept (Chapter \ref{chap:SAFARI}, \citealt{chandlerSAFARISearchingAsteroids2018}), the (6478)~Gault recurrent activity finding (Chapter \ref{chap:Gault}, \citealt{chandlerSixYearsSustained2019}), and our 2014~OG$_{392}$ active Centaur discovery (Chapter \ref{chap:2014OG392}, \citealt{chandlerCometaryActivityDiscovered2020a}). Additional discoveries will be posted on the Results page as they are published and/or announced.
\paragraph{FAQ} The \ac{FAQ} page answers some of the many questions we encountered during project preparations and after project launch.
\subsection{Talk}
\label{methods:subsec:talk}
A facility common to all Zooniverse projects are online forums, known as \textit{Talk}. Here volunteers can, for example, discuss thumbnail images they find interesting, or ask questions. Talk also provides a space where volunteers can build relationships with each other and the scientists. Crucial to the success of Talk are forum Moderators. These individuals help mediate interactions between individuals and answer questions. Our lead forum Moderator is Elisabeth Baeten (see Acknowledgements).
Surprisingly, volunteers posting on Talk became one of two primary paths which facilitate identifying candidate active objects, the other path being our analysis of classification data. Examples include 2017~QN$_{84}$ (Figure \ref{discussion:fig:2017QN84}) and 2015 TC$_1$ (Figure \ref{methods:fig:comparison}). Typically an image that is brought to our attention via Talk are also classified as ``active'' by the majority of volunteers who classified the image.
\subsection{Subject Sets}
\begin{table}
\centering
\caption{Composition of First Citizen Science Subject Set}
\label{methods:tab:CitSciMakeup}
\begin{tabular}{lrr}
\multicolumn{1}{c}{Kind} & \multicolumn{1}{l}{Number} & \multicolumn{1}{l}{Percentage} \\
\hline\hline
Damocloid & 343 & 3.2\% \\
Centaur & 433 & 4.1\% \\
\ac{JFC} & 600 & 5.6\% \\
\ac{KBO}/\ac{TNO} & 564 & 5.3\% \\
Hungaria & 750 & 7.0\% \\
Main-belt Inner & 1500 & 14.1\% \\
Main-belt Middle & 1500 & 14.1\% \\
Main-belt Outer & 1500 & 14.1\% \\
Main-belt Cybele & 750 & 7.0\% \\
\ac{NEO} & 600 & 5.6\% \\
Mars-crosser & 500 & 4.7\% \\
Trojan & 1000 & 9.4\% \\
Hilda & 600 & 5.6\% \\
\hline
\textbf{Total} & \textbf{10640} & \textbf{100.0\%}
\end{tabular}
\end{table}
A Subject Set is a Zooniverse element that contains both data (images in our case) and associated metadata for use in the Citizen Science project. By project launch we had uploaded one \textit{Training} subject set (Section \ref{methods:subset:trainingSet}) and one \textit{Test} subject set (comprised of images needing classification) containing roughly 10,000 thumbnail images. Since launch we have submitted an additional 7 Test subject sets. Zooniverse limits the number of subjects a project can contain using a quota system. Quotas are automatically increased with, for example, successful classification of subjects.
\subsubsection{Thumbnail Selection}
\label{methods:subsubsec:thumbnailSelection}
I wrote a Thumbnail Selection Tool to associate thumbnail records in our database with a numbered subject set. The tool allows selection by object class (e.g., outer main-belt asteroid) and delta magnitude (see Section \ref{methods:subsec:fieldAssesment}). There are options to limit the number of thumbnail images per object class, by unique solar system object, or both. These options allow us to provide a variety of objects for Citizen Scientists to look at. Otherwise the thumbnail images would favor bright objects over fainter (and thus more rare) farther objects such as Centaurs.
\paragraph{Subject Set Composition}
In general we limit the number of images per object to one, with the exception of object classes of which we have few observations (e.g., Centaur) which would quickly diminish to zero thumbnails in a batch because of their relative paucity. For the majority of object classes the limit of one image per object stems from my own experience examining thumbnails that indicates the likelihood of a single thumbnail showing activity, and another thumbnail from approximately the same observing \ac{UT} date does not, is very low. Moreover, we limit bias by showing each unique solar system object once, especially main-belt asteroids, prior to showing the same object to volunteers a second time. The ideal scenario would involve all images being examined by volunteers, however this may take years at the current rate of examination (12,000 to 120,000 classifications per day) and the continued \ac{DECam} thumbnail output of $\sim$5,000 thumbnails per day.
I have kept the ratio of object classes roughly the same for each batch, though the exact numbers necessarily varies as certain object classes (e.g., Centaurs) are limited in number because they are, for example, distant and faint. Table \ref{methods:tab:CitSciMakeup} shows the composition of the first test subject set provided for the 2021 August 31 project launch.
\paragraph{Delta Magnitude Limit}
We (the science team) considered prioritizing thumbnails showing objects computed to appear especially bright in the image as computed by our Delta Magnitude metric (Section \ref{methods:subsubsec:deltaMagLim}). However, this would lead to an observational bias favoring objects that appear brighter because of any combination of larger size, closer distance, higher albedo, or favorable phase angle. Additionally, we hoped to avoid volunteer fatigue that could result if we favored the most promising images early in the project, then later only provided images with a lower probability of activity detection. Concern that there may not be enough volunteer interest if the activity occurrence rate was too low was evidently unwarranted, in part because (a) we provide a training set that injects known active images, and (b) there is a significant fraction of images that indeed appear to show activity, be it real or perceived.
\paragraph{Subject Set Size}
A factor that has a significant impact on the overall flow and timeline of the project is the quantity of subjects contained within a given subject set. Because volunteers are shown subjects at random, it takes exponentially longer (in calendar time) for each subject (image) to be \textit{retired} -- the state when an image has been examined by the preset number of volunteers (15) -- for a subject size of 100,000 thumbnails than 10,000 thumbnails. However, even though a smaller subject set will be completed in less time, there is added overhead in preparing subject sets, uploading batches of images, sending out volunteer calls to action, and analyzing the results. At the time of this writing we prefer subject sets containing between 17,000 and 25,000 images.
\subsubsection{Subject Set Preparation}
\label{methods:subsec:subjSetPrep}
I created a \ac{HARVEST} Subject Set Preparation tool to (1) locate images that were flagged as part of a given Citizen Science batch (subject set), (2) copy the thumbnail images to a directory dedicated to permanently storing images uploaded to \textit{Active Asteroids}, and (3) add a green reticle, as shown in Figure \ref{methods:fig:workflow}.
This tool also creates the required manifest that must accompany the images when they are uploaded to Zooniverse. The manifest holds information we later use to identify the thumbnail image in our database, linking it to a specific thumbnail record that represents a unique combination of a SkyBot result and datafile from an astronomical image archive.
\subsubsection{Classification}
\label{methods:citsci:classification}
Once a subject set has been uploaded to Zooniverse the subject set can be assigned to our active Workflow (Section \ref{methods:subsec:workflow}). At this point, the project status will change to reflect the overall completion status. Following Zooniverse advice, we leave all subject sets marked as active; thus when we add additional subjects, the overall completion percentage does not start over at 0\%. Thus far we have not added an additional subject set until all objects have been fully classified (``retired''), so are not amending any subject set that is in the process of being examined.
The general workflow to notify Citizen Scientists that new work is ready is to (1) adjust the announcement banner on the landing page (Figure \ref{methods:fig:citsci:landing}), (2) tweet the event from the project Twitter account (\texttt{@ActiveAsteroids}), and (3) send out an email ``newsletter'' to all past volunteers of the project via the Zooniverse platform. In practice, we have yet to send an email as of this writing. The first two actions have thus far sufficed to accomplish our goals promptly. One other modality, (4) press releases, coincided with project launch\footnote{\url{http://activeasteroids.net}} and seemed effective at drawing new participants to \textit{Active Asteroids}. We note that effectiveness of publicity is impossible to measure as these engagements occurred prior to, during, and following project launch. We intend to issue press releases in the future with new publications and/or announcements in conjunction with our host institutions and NASA.
\subsection{Training Set}
\label{methods:subset:trainingSet}
\begin{table}
\centering
\caption{Expert Activity Scoring Index}
\begin{tabular}{cl}
Score & Description \\
\hline\hline
0 & missing from center or unidentifiable\\
1 & visible as a point-source\\
2 & vaguely fuzzy\\
3 & fuzzy, but likely not activity-related\\
4 & suggestive of activity but inconclusive\\
5 & likely active, but some ambiguity remains\\
6 & likely activity, not very ambiguous\\
7 & definitely active, medium-strength indicators\\%; anyone in active body study would conclude the object is active\\
8 & definitely active, strong signs of activity\\% that any solar system scientists would recognize as activity\\
9 & activity so obvious that a single image suffices\\
\end{tabular}
\label{methods:tab:activityScore}
\end{table}
I manually examined over 10,000 images of comets and other active bodies output by the \ac{HARVEST} pipeline. For each image we ranked apparent activity on an integer scale from 0 -- 9, following my own rough classification scheme (Table \ref{methods:tab:activityScore}). We imported these scores into the \ac{HARVEST} database, then prepared a subject set consisting only objects with scores of five or higher (meaning these images definitely showed activity; Section \ref{intro:subsec:visibleActivity}). These became the Training Subject Set for the \textit{Active Asteroids} project.
The primary purpose of the training set is to help teach Citizen Scientists how to recognize activity. As volunteers classify thumbnails in the primary workflow, training images are shown at an interval that is inversely proportional to how many images a volunteer has classified, as determined by Zooniverse. The decaying rate is necessary for (and was requested by) volunteers that become highly proficient and no longer need (or wish) to see training images. The probability a user is shown a training image, $P_\mathrm{T}$, is given by
\begin{equation}
P_\mathrm{T}(N) =
\left\{
\begin{array}{rr}
50\% & 1 \le N \le\; 10\;\\
20\% & 10 < N \le\; 50\\
10\% & 50 < N \le 100\\
5\% & 100 < N < \infty\: \:
\end{array}
\right\}
\end{equation}
\noindent where $N$ is the number of classifications a user has carried out in \textit{Active Asteroids}. The training set also serves to validate that the project is working as intended. It was clear during our project's Beta Review that volunteers were adept at identifying activity.
\section{Follow-up Campaign}
\label{sec:methods:followup}
Once we have identified activity candidates we conduct follow-up study, first in the form of an archival investigation. Then, if warranted, we may pursue observations at telescope facilities to further study the object.
\subsection{Archival Investigation}
\label{methods:subsec:archivalInvestigation}
This process is documented extensively in Chapter \ref{QN:sec:secondActivityEpoch} \citep{chandlerRecurrentActivityActive2021a}. Here, I provide a brief synopsis of the overall process and discuss additional methods and sources we use that were not part of the aforementioned work.
\subsubsection{Step 1: Supplemental Searches}
I carry out searches independent of the \ac{HARVEST} pipeline to find additional archival data for a given solar system object, through (1) three automated supplemental pipelines we wrote and (2) manual query of two sources. This auxiliary pipeline is especially useful for the study of specific objects, where we consider it worth spending extra time to find the maximum amount of data available for an object of interest. However there are many drawbacks when compared with the \ac{HARVEST} pipeline. For example, the overwhelming majority (roughly 90\%, depending on circumstances such as dynamical class) of data are unusable, either not probing faint enough to see the object, or the object is not on a camera chip. Moreover, accessing myriad archives with data from myriad instruments requires us to manually intervene in the pipeline to, for example, address errors generated by \ac{FITS} files that do not conform to the \ac{FITS} standard, or to conduct astrometry in order to embed a valid \ac{WCS} that we need to extract a thumbnail image of the target object. All of these added steps also require additional administrative overhead, from managing downloading data to keeping a separate directory structure organized with results.
\paragraph{CADC SSOIS} This pipeline queries the \ac{CADC} \ac{SSOIS}\footnote{\url{https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/ssois/}}. Results include some overlap with \ac{HARVEST} (i.e., \ac{DECam}, MegaPrime, \ac{KPNO} instruments), which are useful for validating \ac{HARVEST} results. Additional instrument archives searched include \ac{SOAR}, \ac{ESO} instruments (e.g., \ac{VST} OMEGACam, \ac{VISTA} {VIRCam}), \ac{NEAT} \ac{GEODSS}, \ac{SDSS}, Subaru SuprimeCam and \ac{HSC}, and \ac{WISE}.
\paragraph{IRSA} This pipeline queries the NASA/CalTech \ac{IRSA}\footnote{\url{https://irsa.ipac.caltech.edu}} archive, which searches both \ac{ZTF} and \ac{PTF} data for moving objects via their \ac{MOST}.
\paragraph{ZTF Alert Stream} This pipeline downloads all \ac{ZTF} alert stream \citep{pattersonZwickyTransientFacility2018} data then prunes out all data unrelated to solar system objects. Alert packets have already been matched to known solar system objects, making these data simple to search. The alerts include image data as well as metadata such as apparent magnitude.
\paragraph{Manual Queries} We manually query the \ac{KOA}\footnote{\url{https://koa.ipac.caltech.edu/cgi-bin/KOA/nph-KOAlogin}} via their \ac{MOST}. We also initiate queries via the \ac{CATCH} tool\footnote{\url{https://catch.astro.umd.edu/}} which provides quick search and image delivery from multiple archives, including \ac{NEAT} \citep{pravdoNearEarthAsteroidTracking1999} and SkyMapper \citep{kellerSkyMapperTelescopeSouthern2007}.
\subsubsection{Step 2: Data Acquisition}
During the aforementioned pipeline processes we automatically generate download scripts that are added to the same queue that executes downloads for the \ac{HARVEST} pipeline. However, several sources require additional steps to download archival image data.
\paragraph{ESO} \ac{ESO} results, including \ac{VST} OMEGACam and \ac{VIRCam}, can be downloaded in an automated fashion
We generate a separate set of bash scripts we designed to handle downloading, sorting, and preprocessing of \ac{ESO} data.
\paragraph{SMOKA} The \ac{SMOKA}\footnote{\url{https://smoka.nao.ac.jp}} archive \citep{babaDevelopmentSubaruMitakaOkayamaKisoArchive2002} that serves SuprimeCam and \ac{HSC} data requires an account to request data and carry out downloads. The system is queue-based, with email notifications upon data retrieval readiness. We optimize the process by constructing a custom URL that requests all specific datafiles we need at once. After the request is processed, \ac{SMOKA} sends an email that includes a one-line bash script for downloading the data. Once the data are downloaded, we have separate scripts that prepare the data for analysis. We do not reduce (e.g., flatten) the data, though this is a step we are considering for future work.
\paragraph{CASU} Data hosted at the \ac{CASU} Astronomical Data Centre\footnote{\url{http://casu.ast.cam.ac.uk/casuadc/}}, especially \ac{INT} \ac{WFC}, require data be requested from their queue-based service. Once the data are available an email notice is sent. From this point we produce shell scripts to download the data,
conduct astrometry, and prepare the data for additional analysis.
\subsubsection{Step 3: Astrometry}
My thumbnail extraction code requires embedded \ac{WCS} of sufficient quality to allow for the target object to appear at or near the center of the thumbnail images we extract. Many archive/instrument combinations (e.g., \ac{PS1}, \ac{ZTF}) provide data with excellent astrometry via embedded \ac{WCS}, however some archives either provide no \ac{WCS} at all, or \ac{WCS} with insufficient or unreliable precision. We perform astrometry as needed via Astrometry.net \citep{langAstrometryNetBlind2010} on the \ac{NAU} Monsoon computing cluster. Astrometry.net makes use of source catalogs, including the Gaia Data Release 2 catalog \citep{gaiacollaborationGaiaDataRelease2018}, and \ac{SDSS} \citep{ahnNinthDataRelease2012}.
\subsubsection{Step 4: Thumbnails}
This tool extracts \ac{FITS} and \ac{PNG} thumbnails with a uniform \ac{FOV} that we specify, the default being 126''$\times$126''. We maintain a database of instrument and telescope parameters so that we can extract thumbnail images (via \ac{HARVEST}-derived code) with North pointing up and East pointing left, standard astronomical orientation. We query \ac{JPL} Horizons and plot symbols indicating the anti-Solar and anti-motion vectors commonly associated with tail direction, such as those shown in Figure \ref{intro:fig:activity}. An additional optional step co-adds \ac{FITS} format thumbnails to help enhance signal for activity searches.
\subsection{New Telescopic Observations}
\label{methods:subsec:telescopicObservations}
\begin{table}
\caption{Telescopes Utilized}
\footnotesize
\begin{tabular}{lllllcc}
Instrument & Telescope & Diameter [m] & Observatory & Location & Country & Site Code \\
\hline
AltaU-47 & \acs{BLT} & 0.5 & \acs{ARO} & Flagstaff, Arizona & USA & 687 \\
\ac{DECam} & Blanco & 4.0 & \acs{CTIO} & Cerro Tololo & Chile & 807 \\
\acs{GMOS}-S & Gemini South & 8.1 & Gemini & Cerro Pachon & Chile & I11 \\
\acs{IMACS} & Baade & 6.5 & Magellan & Las Campanas & Chile & 304 \\
\acs{LBCB}, \acs{LBCR} & \acs{LBT} & 8.5$\times$2 & \acs{MGIO} & Mt. Graham, Arizona & USA & G83 \\
\acs{LMI}, \acs{NIHTS} & \acs{LDT} & 4.3 & Lowell Observatory & Happy Jack, Arizona & USA & G37 \\
VATT4K & \acs{VATT} & 1.8 & \acs{MGIO} & Mt. Graham, Arizona & USA & 290
\end{tabular}
\raggedright
\footnotesize
Definitions: \acf{BLT}, \acf{ARO}, \acf{DECam}, \acf{CTIO}, \acf{GMOS}, \acf{IMACS}, \acf{LBCB}, \acf{LBCR}, \acf{LBT}, \acf{MGIO}, \acf{LMI}, \acf{NIHTS}, \acf{LDT}, \acf{VATT}.
\label{methods:tab:facilities}
\end{table}
Over the course of the work contained in this dissertation our team made use of telescopes for activity searches and follow-up study. Table \ref{methods:tab:facilities} lists the instruments and associated facilities used for this work.
\chapter{Overall Discussion}
\label{chap:discussion}
\acresetall
In order to help fill key knowledge gaps about solar system volatiles, such as where volatiles are found throughout the solar system, we set out to find objects like active asteroids and active Centaurs, bodies that display cometary properties such as tails and comae even though they are not classified as comets. In furtherance of this goal we designed and launched a Citizen Science project, \textit{Active Asteroids}, that carries out an outreach program of public engagement while concurrently identifying new members of known and unknown active minor planet classes. Before and after launch, \textit{Active Asteroids} has led to discoveries ranging from identifying new active asteroids to uncovering new epochs of activity associated with known active objects. Some of these discoveries are in this dissertation, while others will be the subject of future investigations by our team.
Prior to constructing the Citizen Science project, we carried out a proof-of-concept (Chapter \ref{chap:SAFARI}) that demonstrated the suitability of \ac{DECam} data for identifying activity emanating from minor planets. First we created a pipeline, \ac{HARVEST}, that extracts small thumbnail images centered on a known solar system object. Without \textit{a priori} knowledge of which object was shown in each thumbnail image, we visually examined all 15,600{} thumbnails we had extracted from 35,640{} \ac{FITS} files. Of the 11,703{} unique minor planets visible in the thumbnails, three were previously identified as active asteroids. However, (1)~Ceres and (779)~Nina have not been observed to be visibly active from Earth. The third, (62412)~2000~SY$_{178}$ \citep{sheppardDiscoveryCharacteristicsRapidly2015}, we successfully identified as active in a thumbnail image (Figure \ref{safari:fig:62412}). This one in 11,703{} occurrence rate agreed with other works that estimated activity takes place in approximately one in ten thousand asteroids \citep{jewittActiveAsteroids2015a,hsiehMainbeltCometsPanSTARRS12015}.
\begin{figure*}
\centering
\includegraphics[width=1.0\linewidth]{discussion/AsteroidsOnSkyHist_18432889tot_fldLim100000000_360x120bins_gnuplot2DECam.png}
\caption{The distribution in each square degree on sky of the roughly 18 million automatically vetted thumbnail images \ac{HARVEST} produced.}
\label{discussion:fig:asteroidsOnSky}
\end{figure*}
My \ac{NSF} \ac{GRFP} proposal was selected for funding and we expanded the \ac{HARVEST} pipeline to work with all publicly available \ac{DECam} archival image data. We implemented a vetting scheme to exclude images that, for example, did not probe faintly enough to detect activity. Figure \ref{discussion:fig:asteroidsOnSky} shows the distribution on sky of the $\sim$18 million automatically vetted thumbnail images in the \ac{HARVEST} database.
In 2019 January asteroid (6478)~Gault was observed to be active \citep{smith6478Gault2019}. We made use of \ac{HARVEST} to find additional images of Gault in \ac{DECam} archival data. We discovered Gault had been active during two prior orbits: 2013 and 2016. We introduced a new metric, observability, to help identify potential biases by describing the number of hours an object is observable at a specific location for a given UT observing date. Notably, Gault's activity appeared unrelated to heliocentric distance (Chapter \ref{Gault:subsec:imageanalysis}), indicating the activity was probably not caused by volatile sublimation.
Gault is a core member of the Phocaea family \citep{knezevicProperElementCatalogs2003}, a group of predominantly (75\%) S-type asteroids \citep{carvanoSpectroscopicSurveyHungaria2001}. S-type asteroids are composed primarily of non-primitive, desiccated silica \citep{demeoExtensionBusAsteroid2009}, material highly unlikely to harbor volatile substances. The majority of active asteroids found to date (see Table \ref{safari:Table:TheAAs}) have been composed of primitive materials (e.g., C-type, which are carbonaceous and volatile-rich), amenable to sublimation-driven activity associated with perihelion passage \citep{hsieh2016ReactivationsMainbelt2018}. Very few (around four, or $\sim$20\%) known active asteroids are S-type, and their activity is thought to be caused by impact, such as (596)~Scheila \citep{bodewitsCollisionalExcavationAsteroid2011,jewittHubbleSpaceTelescope2011}, or rotational instability, as with 311P/PanSTARRS, \citep{jewittExtraordinaryMultitailedMainbelt2013}, rather than sublimation \citep{hsiehAsteroidFamilyAssociations2018}.
Given the combination of (1) Gault's probable desiccated S-type spectral class, (2) rotational breakup as the likely activity mechanism \citep{morenoDustPropertiesDoubletailed2019, yeMultipleOutburstsAsteroid2019,yeContinuedActivity64782019,kleynaSporadicActivity64782019}, and (3) our discovery of recurrent activity, we proposed that Gault is a new type of active asteroid, one that is persistently active because of spin-up induced rotational instability. We also identified Gault as the first active asteroid with recurrent and sustained activity throughout its orbit. We anticipated that Gault would remain active for some time, and although it did not catastrophically disintegrate (cf.\ \citet{morenoDustPropertiesDoubletailed2019}), Gault became inactive in 2020 \citep{2021MNRAS.505..245D} and has remained so through at least 7 January 2022.
During ongoing Citizen Science project preparations we found indicators suggestive of activity in thumbnails of Centaur 2014~OG$_{392}$ (Figure \ref{og:fig:archivalimages}). We carried out an observing campaign that confirmed the presence of activity based on a result stemming from the \ac{HARVEST} pipeline. We uncovered archival images of 2014~OG$_{392}$ that indicated it had been active for more than two years, effectively ruling out stochastic events (e.g., impact).
We introduced a new technique that synthesizes sublimation modeling with dynamical modeling to help estimate which molecules are most likely responsible for activity. My simple sublimation model is not intended to account for the complex behaviors of ice mixtures \citep{grundySolarGardeningSeasonal2000} or amorphous--crystalline ice transitions \citep{jewittActiveCentaurs2009}, rather it treats bodies as airless and uniformly covered in a single species of ice. We found carbon dioxide and/or ammonia most likely to be sublimating, while other surface species would have sublimated completely over its 13 kyr to 1.8 Myr dynamical lifetime (carbon monoxide, molecular nitrogen, methane), or were unable to appreciably sublimate at all (water, methanol).
Colors we measured from our observations revealed that 2014~OG$_{392}$ was about one magnitude redder than the Sun in the visible spectrum and, with a $B$-$R$ color of \ogColor{}, the object is considered a red Centaur \citep{peixinhoCentaursComets402020}. However, both species we identified as likely to be responsible for sublimation (carbon dioxide, ammonia) are spectrally neutral in visible wavelengths, as many ices are, indicating the involvement of an as-yet unidentified reddening agent. The \acp{TNO} also have an unknown reddening agent \citep{boehnhardtVisibleNearIRObservations2001}. We computed a new absolute magnitude of $H\approx11.3$, fainter than previously reported by about 0.5 magnitudes. This $H$ value indicates a diameter of $\sim$20~km, assuming a slope $G=0.15$ as is typical for a dark surface \citep{bowellApplicationPhotometricModels1989}. We measured a coma mass of roughly $\sim2.4\times10^{15}$~g, assuming a grain density of 1~g/cm$^3$, 1~mm grain radii, and an albedo of $A=0.1$.
In 2021 July, prior to launching the \textit{Active Asteroids} Citizen Science project, activity was discovered coming from (248370) 2005~QN$_{173}$ \citep{fitzsimmons2483702005QN1732021}. We conducted an archival investigation (described in Chapters \ref{methods:subsec:archivalInvestigation} and \ref{QN:sec:secondActivityEpoch}) and located 81 images (spanning 31 observations) in which the object was positively identifiable. From these we found a single image (Figure \ref{QN:fig:wedgephot}) from UT 2016 July 22 that clearly showed the object had been active during this orbit. Our analysis indicated that (248370)~2005~QN$_{173}$ is most likely a member of the \acp{MBC} because of (1) the recurrent nature of its activity, (2) its orbit with the Main Asteroid Belt, and (3) its likely C-type spectral class \citep{hsiehPhysicalCharacterizationMainbelt2021}.
Along with my archival investigation we introduced a tail angle measurement tool, wedge photometry, that may also be used for activity detection. We showed the tail orientation of (248370)~2005~QN$_{173}$ was in close agreement with the anti-Solar and anti-Motion vectors (coincident at the time) as computed by \ac{JPL} Horizons.
We launched the Citizen Science project \textit{Active Asteroids}\footnote{\url{http://activeasteroids.net}}, a NASA Partner\footnote{\url{https://science.nasa.gov/citizenscience}}, on 31 August 2022. The initial subject set of $\sim$18,000 thumbnail images was completely classified by volunteers within days of launch. Although we did produce statistical evidence that volunteers could identify activity based on the training se
, discoveries of activity from objects not yet known to be active came to light within 24 hours of launch.
One example, 2015~TC$_1$ (Figure \ref{methods:fig:comparison}), is a \ac{JFC}. We had coincidentally observed this object previously with the twin 8.5~m \ac{LBT} on 2021 April 4 (Prop. ID AZ-2021A-506, PI Oldroyd). Observations were carried out in poor weather conditions and the object was not near perihelion, and we did not find any conclusive evidence of activity. The next perihelion passage for 2015~TC$_1$ takes place in 2022 December and, per the \ac{JPL} \ac{SBDB}, the object will have an especially close (0.1~au) approach to Jupiter in November 2145.
\begin{figure*}[h]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{discussion/2017_QN84/0888084H_20171223-07553_1283382_7.3y_qb_OC_600s_r-5_pI_ovX_arrows.png} & \includegraphics[width=0.45\linewidth]{discussion/2017_QN84/0888084H_20171223-07553_1283382_7.3y_qb_OC_600s_r-5_pI_ovX_POIMid29_2017-12-24_07.35.46_c4d_171224_073046_ooi_r_v1_chip2-S30_240pix_NuElnoArrows.png} \\
\end{tabular}
\caption{
\textbf{Left:} This image of \ac{JFC} 2017~QN$_{84}$ (green dashed arrow), originally taken 2017 December 23 (Prop. ID 2017B-0307, PI Sheppard), was flagged as active by 14 of 15 \textit{Active Asteroids} (\url{http://activeasteroids.net}) volunteers.
\textbf{Right:} this comparison image, from an image captured on 2017 December 24 (Prop. ID 2017B-0307, PI Sheppard), shows that there is no comparable object or extended feature (e.g., a galaxy) that could be mistaken for the activity visible in the image on the left (orange arrows).}
\label{discussion:fig:2017QN84}
\end{figure*}
Another example of a Citizen Science--informed discovery was activity emanating from 2017~QN$_{84}$ (Figure \ref{discussion:fig:2017QN84}). 14 of 15 volunteers classified a thumbnail of this \ac{JFC} as showing activity. Unfortunately, this was the only image available of the object, which makes identifying activity more challenging. However, as seen in Figure \ref{discussion:fig:2017QN84}, a comparison image (technique described in Chapter \ref{methods:subsec:comparisonImages}) indicated that a chance diffuse background object was not responsible for the observed activity. The object had just completed its 2017 perihelion passage, and it was relatively bright at $V\approx20$.
We are currently investigating other potentially active minor planets. Here we mention both anticipated and unexpected impediments to confirming the presence of activity are worth mentioning here. As described above, (1) not all images have unambiguous activity, and (2) there may only be a single image with activity indicators. Moreover, (3) some objects may not be observable (i.e., below the horizon at night
. (4) Objects may be very faint, and thus difficult or impossible to observe even with the largest telescopes; this is especially problematic for detecting comae and tails which may appear several magnitudes fainter than the nucleus. (5) An object may now be far from perihelion and, consequently, less likely to be active; depending on the object's orbit and time elapsed from the thumbnail image, it may take years or decades for an object to again approach perihelion. (6) Objects with poorly constrained orbits may be lost by the time we identify activity and set out to observe the object. (7) A candidate may be transiting a rich area on sky (e.g., the Milky Way), potentially for months, effectively preventing activity detection and hampering study of the object. (8) Objects with strong activity indicators may still not meet the ``new comet'' \ac{MPC} requirements, as described in Chapter \ref{intro:subsec:visibleActivity}.
\begin{figure}
\centering
\includegraphics[width=4in]{discussion/deviationsFromColin.pdf}
\caption{Agreement between the percentage of Citizen Scientists that identified a thumbnail as showing activity, and my own measure of how active an object appears in an image. Zero represents complete agreement, negative numbers indicates a high fraction of volunteers identified a thumbnail as active as compared to my score, and positive numbers indicate that my activity score was higher than the fraction of volunteers who classified the image as showing activity.}
\label{discussion:fig:colinVsCitSci}
\end{figure}
We plan to derive many useful data products from the \textit{Active Asteroids} project. For example, a training dataset stems from the images we classified for the project training set (Chapter \ref{methods:subset:trainingSet}). This dataset will have two dimensions: (1) my own scores (described in Chapter \ref{methods:subset:trainingSet}), and (2) high numbers ($>1000$) of volunteer classifications (because training data are classified many times). Another data product will contain classification statistics accompanying all images submitted for examination. These types of products will be of use for for surveys such as the \ac{LSST} \citep{verac.rubinobservatorylsstsolarsystemsciencecollaborationScientificImpactVera2021}. For example, an outstanding need is for tools capable of automatically detecting activity, especially for petabyte-scale datasets \citep{kelleyCommunityChallengesEra2021}. \Ac{ML}--assisted activity detection tools can be trained using products resulting from our own classifications as well as those of Citizen Science volunteers \citep{breivikDataSoftwareScience2022}, though it is worth mentioning that the quality of the data are important to the effectiveness of \ac{ML}-based algorithms \citep{kuminskiCombiningHumanMachine2014}. Figure \ref{discussion:fig:colinVsCitSci} shows relative agreement between the two systems (expert inspection, Section \ref{methods:subset:trainingSet}) and the fraction of volunteers that classified a training thumbnail as active.
\begin{figure}
\centering
\includegraphics[width=0.45\linewidth]{discussion/binary/0600085K_20190222-08414_1583206_266y_qb_Oa_070s_z-1_pI_ovX_crop_arrow.png}
\caption{
The two components (green dashed and solid orange arrows) of \acf{TNO} 2012~KU$_{50}$ are fully resolved in this 600~s \textit{VR} filter \ac{DECam} image from UT 2014 March 27 (Prop. ID 2014A-0479, PI Sheppard).
}
\label{discussion:fig:additionalAvenues}
\end{figure}
An avenue of inquiry stemming from \textit{Active Asteroids} preparations unexpectedly supplementing of our original science goals is the discovery of companions to minor planets, including \acp{TNO} and Jupiter Trojans. \ac{TNO} binaries are of special interest because they hold potential clues to the formation of the trans-Neptunian region and the solar system as a whole. At present we are preparing a paper about one such discovery \citep{chandler5540992012KU502021} made during project preparations, that of \ac{TNO} 2012~KU$_{50}$ (Figure \ref{discussion:fig:additionalAvenues}). Companions are readily distinguishable in \ac{DECam} data, especially for widely separated objects in good seeing conditions. Binaries are especially prominent in image sequences of a given object, thus one potential avenue to facilitate companion discovery would be a Citizen Science workflow that presents animated \ac{GIF} files to volunteers and asks if they see a co-moving companion to the object at the center of the images.
\section{Project Status and Assessment}
\label{discussion:sec:projStatus}
\subsection{Volatile Distribution}
\label{discussion:subsec:volatiles}
Here we consider the present-day volatile distribution and delivery of water to Earth. This is by no means meant to be an exhaustive review or in-depth investigation; rather, this section serves as a rough guide. A word of caution: as stated in Chapter \ref{chap:intro}, the purpose of this dissertation and project is to discover more active objects to enable meaningful study of active objects as populations; at present, the numbers remain low (see Section \ref{discussion:subsubsec:activeAsteroidCompleteness}). As an example, the 1 in 10,000 occurrence rate of active objects in the Asteroid Belt has been extrapolated from studies as small as $\sim$10,000 minor planets \citep{chandlerSAFARISearchingAsteroids2018}, less than 1\% of the Asteroid Belt population.
\subsubsection{Terrestrial Water Origins}
\label{discussion:subsubsec:volatilesOnEarth}
We start by considering the total mass of water on Earth today, $M_{\mathrm{W},\oplus}$. We can roughly estimate this quantity via
\begin{equation}
M_{\mathrm{W},\oplus} \approx M_{\mathrm{W},\mathrm{accretion}} + M_{\mathrm{W},\mathrm{delivered}} - M_{\mathrm{W},\mathrm{lost}},
\end{equation}
\noindent where $M_{\mathrm{W},\mathrm{accretion}}$ is the mass of water on Earth that formed \textit{in situ} during accretion, $M_{\mathrm{W},\mathrm{lost}}$ is the mass of water lost from Earth (due to, for example, a large-scale impact event such as the one that formed the Moon), and the delivered water mass $M_{\mathrm{W},\mathrm{delivered}}$, described by
\begin{equation}
M_{\mathrm{W},\mathrm{delivered}} \approx \sum_{s=1}^{s=n} N_s \cdot \frac{4}{3}\pi \left(\bar{r}_s\right)^3 \cdot \bar{\rho}_s \cdot \bar{f}_{s,\mathrm{W}} \cdot f_{s,\mathrm{impact}},
\end{equation}
\noindent where, for a given source class $s$ (e.g., comets, asteroid
), $N_s$ is the number of objects, $\bar{r}_s$ is the average radius, $\bar{\rho}_s$ is the average density, $\bar{f}_{s,\mathrm{W}}$ is the average fraction of material that is water, and $f_{s,\mathrm{impact}}$ the fraction of those objects that impacted Earth through present-day. A more robust treatment would take into account the size, mass, and compositional distributions through time for each of these elements, but these are very poorly known and they are outside the scope of this dissertation.
Our efforts focus primarily on active body populations as volatile sources ($N_s$), so for the purpose of a broad examination of an active body class let us only consider the modern Asteroid Belt. Modern Earth holds around $5\times10^{-4} M_\oplus$ of water (e.g., \citealt{morbidelliSourceRegionsTime2000} and references therein), where $1M_\oplus\approx 6 \times10^{24}$~kg. The most massive active asteroid, (1) Ceres, is also the most massive body in the Asteroid Belt, with a mass of $1.6\times10^{-4} M_\oplus$. If active asteroids are composed of 50\% water (probably a very generous estimate), then the mass of water on Ceres would be $M_\mathrm{W,Ceres}\approx 0.8\times10^{-4} M_\oplus$. The mass of water on Earth could be measured as $M_{\mathrm{W},\oplus}\approx6M_\mathrm{W,Ceres}$. Considering the hypothesized population of $\sim$100 active asteroids ($\sim$1/10,000 of the $>1$~km population), for them to contain enough water to account for all of modern Earth's water they would each need to be roughly 6\% the mass of Ceres. These bodies would be comparable in size to (10)~Hygeia ($\sim$300~km diameter), the fourth largest asteroid in the solar system. To date the vast majority of recurrently active sublimation-driven active asteroids have diameters $< 10$~km (e.g., 133P/Elst-Pizarro at 3.8~km; \citealt{hsiehAlbedosMainBeltComets2009}), with the notable exception of (1) Ceres. This calculation, while rudimentary, nevertheless leaves little doubt that the active asteroids alone cannot at present hold enough water to supply all of the terrestrial water found on Earth today. However, active asteroids may have played a more significant role in the past given its much larger (150 to 250 times) population in the primordial solar system \citep{bottkeFossilizedSizeDistribution2005} .
\subsection{Citizen Science Project}
\label{discussion:subsec:citSciStatus}
The NASA Partner project \textit{Active Asteroids} was launched on the Citizen Science platform Zooniverse on 2022 August 31. Here we discuss the overall status of the project, make some rough estimates about what the project can deliver based on performance thus far, and consider potential enhancements to the project.
\subsubsection{Thumbnail Classification Rate and Completeness}
\label{discussion:subsubsec:classificationRate}
As of this writing, classifications have taken place at approximate rates varying between 12,000 classifications/day and 120,000 classifications/day. The exact cause of the variation is as-yet undetermined, but factors may include media coverage, social media activity (e.g., tweets from \@ActiveAsteroids), time of year (e.g., summer break), and timing of electronic newsletters to project volunteers. Our goal is to increase participation over time, and we hope the upcoming publications stemming from the project, including peer-reviewed journal articles and press releases, will help. Meanwhile we assess completeness making use of the aforementioned rates.
\paragraph{Time to Complete Existing Thumbnails} At present \ac{HARVEST} has produced about 18 million algorithmically vetted thumbnail images. Given each thumbnail is examined by 15 volunteers, the total number of classifications needed to examine the entire dataset is $15\times1.8\times10^7=2.7\times10^8$ classifications. At the maximum rate (120,000 classifications/day) this works out to 2,250 days, or about six years.
\paragraph{Staying Current} \ac{DECam} started operations in 2012 September, and data are released regularly through the \ac{NOIRLab} AstroArchive. The average number of vetted thumbnails produced per day is $\sim$5000, averaged over all dates, not just dates the telescope acquired data. It is worth noting that, in addition to normal weather and engineering telescope time losses, \ac{COVID-19} restrictions led to telescope shutdown for an extended period in 2020, thus reducing the overall average data output. At the maximum classification rate (120,000 classifications/day), it would require 15 hours of volunteer classifications per day stay current.
\paragraph{Time to Completion and Staying Current} Considering \textbf{Staying Current} would require 15 hours/day at the maximum classification rate, only nine hours (0.375 days) remain daily for classifying the existing data pool, or 120,000 classifications/day $\times$ 0.375 = 45,000 classifications/day. Each image is classified 15 times, so the rate in images/day is 3000. To classify. all 18 million thumbnails at this rate would require $>$16 years. Afterwards, the project would be able to stay current with \ac{DECam} output at 75,000 classifications/day, lower than the maximum classification rate seen to date.
\paragraph{Triage Options} Parameters used to select thumbnails for Citizen Science classification (Section \ref{methods:subsubsec:thumbnailSelection}) begin with constraining ranges of $\Delta_\mathrm{mag}$, the number of magnitudes fainter an exposure probes than the apparent magnitude of the minor planet (recall $\Delta_\mathrm{mag} = V_\mathrm{JPL} - V_\mathrm{ITC}$, with $V_\mathrm{JPL}$ the Horizons-provided apparent $V$-band magnitude for the object and $V_\mathrm{ITC}$ the $V$-band magnitude depth we computed for the exposure; Section \ref{methods:subsubsec:deltaMagLim}). The number of existing and new thumbnails needing classification can be reduced significantly by altering this parameter alone. For example, requiring $\Delta_\mathrm{mag}<-2$ (instead of the default $\Delta_\mathrm{mag}<-1$) reduces the number of thumbnails to be classified from 18 million to 16 million. Requiring $\Delta_\mathrm{mag}<-3$ lowers the number of thumbnails for classification to 9 million, a 50\% reduction in the classification workload as compared to the original 18 million thumbnails. This action would exacerbate the existing bias towards observing bright objects which, in turn, favors images of objects close to perihelion and disfavors distant objects such as Centaurs. For example, assuming the typical magnitude limit of \ac{DECam} is $\sim V=23$ (Chapter \ref{chap:SAFARI}), requiring $\Delta_\mathrm{mag}<-3$ results in images showing objects brighter than $V=20$.
\begin{table}
\centering
\caption{Classification Rates and Potential Optimizations}
\label{discussion:tab:ratesAndCompleteness}
\begin{tabular}{c}
\textbf{This Work}\\
\begin{tabular}{ccrrrllrr}
Description & $N_\mathrm{current}$ & $t_\mathrm{current,min}$ & $t_\mathrm{current,ave}$ & $N_\mathrm{daily}$ & $t_\mathrm{daily,min}$ & $t_\mathrm{daily,ave}$ & $t_\mathrm{all,min}$ & $t_\mathrm{all,ave}$ \\
& & [days] & [days] & & [days] & [days] & [days] & [days]\\
\hline
Vetted & 1.57E+7 & 1960 & 23500 & 4300 & 0.54 & 6.51$^b$ & 4280 & $\infty$\\
$\Delta_\mathrm{mag}<-2$ & 1.11E+7 & 1380 & 16600 & 3073 & 0.384 & 4.61$^b$ & 2250 & $\infty$\\
$\Delta_\mathrm{mag}<-3$ & 6.10E+6 & 763 & 9150 & 1693 & 0.212 & 2.54$^b$ & 967 & $\infty$ \\
$\%_{q\rightarrow Q} \ge$ 80\% & 2.16E+6 & 270 & 3240 & 600 & 0.075 & 0.9 & 292 & 32500 \\
Wedge Phot.$^a$ & 2.16E+5 & 27 & 324 & 60 & 7.50E-03 & 0.09 & 27 & 356 \\
Mach. Learn.$^a$ & 2.16E+3 & 0.27 & 3 & 0.60 & 7.50E-05 & 0.0009 & 0.27 & 3.25
\end{tabular}\\
\\
\textbf{LSST}\\
\begin{tabular}{ccrr}
Description & $N_\mathrm{daily}$ & $t_\mathrm{daily,min}$ & $t_\mathrm{daily,ave}$\\
& & [days] & [days]\\
\hline
\acs{LSST} baseline & 1.36E+7 & 1700$^b$ & 20400$^b$ \\
\acs{HARVEST} Vetting & 4.08E+6 & 510$^b$ & 6120$^b$ \\
$\Delta_\mathrm{mag}<-2$ & 2.89E+6 & 361$^b$ & 4330$^b$ \\
$\Delta_\mathrm{mag}<-3$ & 1.59E+6 & 199$^b$ & 2386$^b$ \\
$\%_{q\rightarrow Q} \ge$ 80\% & 5.65E+5 & 71$^b$ & 846 $^b$ \\
Wedge Phot.$^a$ & 5.65E+5 & 7$^b$ & 85$^b$ \\
Mach. Learn.$^a$ & 5.65E+3 & 0.1 & 0.8
\end{tabular}
\end{tabular}
\\
\raggedright $N_\mathrm{current}$ indicates the number of \ac{DECam} images in the \ac{HARVEST}-produced database. Daily indicates the average number of thumbnail images produced on a daily basis. Minimum (min) and average (ave) values estimated based on peak (120,000) and mean (10,000) classifications per day, respectively. Vetted: the number of \ac{DECam} thumbnail images after vetting by the \acf{HARVEST} pipeline (Section \ref{methods:sec:methods:pipeline}). $\Delta_\mathrm{mag}$ is how faint an exposure probes versus the apparent magnitude of the object (Section \ref{methods:subsubsec:deltaMagLim}); relative brightness of the object in contrast with the overall exposure depth increases in the negative direction. $^a$Not easily implemented. $^b$Will never catch up. Hypothetical application of the Wedge Photometry is a technique that searches for tails (Section \ref{QN:fig:wedgephot}). Hypothetical application of \acf{ML}-based activity detection technique. The baseline number of \acf{LSST} thumbnails per night, estimated from an average of 8,000 minor planets per field, with 1,700 fields per night, depending on final cadence selection.
\end{table}
Table \ref{discussion:tab:ratesAndCompleteness} shows the impact of applying different optimization parameters to the \ac{HARVEST}-produced thumbnail images as well as rough estimates for implications for the \ac{LSST}. The number of vetted images in the \ac{HARVEST} dataset is roughly 16 million as of these calculations. Applying a $\Delta_\mathrm{mag}<-3$ constraint reduces the number of images by a factor of two, and likewise if we only allow thumbnail images showing objects at or above 80\% of the distance from aphelion to perihelion $\%_{q\rightarrow Q}$ (Section \ref{og:eq:percentperi}). To illustrate potential benefits of additional screening tools, a hypothetical factor of ten improvement is shown for Wedge Photometry (Section \ref{QN:fig:wedgephot}) application, and a factor of 100$\times$ improvement for an as yet undeveloped \ac{ML}-based activity detection tool. Even at our average classification rate of 10,000 classifications/day, the entire \ac{DECam} dataset would be processed and current in $\sim$3.25 days.
A hypothetical scenario for \ac{LSST} is also shown in Table \ref{discussion:tab:ratesAndCompleteness}. Although the final cadence has yet to be confirmed, we compute a baseline number of thumbnails from $\sim$1,700 fields imaged per night with an average of $\sim$8,000 minor planets per field \citep{ivezicLSSTScienceDrivers2019} as $1400\ \mathrm{fields/night}\ \cdot 8000\ \mathrm{objects/field}\ = 1.36\times10^7$ thumbnail images per night. \ac{HARVEST} has an approximate 70\% reduction in thumbnail images following our vetting routines, which still leaves over 45 million minor planets imaged per night. The only scenario in which the Citizen Scientists can keep up with the \ac{LSST} is if the classifications are kept at the rate of 120,000 classifications/night and all of the hypothetical vetting tools provide a combined 1,000$\times$ reduction in the number of thumbnail images needing examination over this work. It is important to note that classification should commence immediately because it will be highly impractical to ``catch up'' after \ac{LSST} has started acquiring data. It is also worth remembering that all times are dependent on classification rates, thus improvements would result from increased participation in the Citizen Science project.
\paragraph{Human Lifetimes} Classification requires time contributed by human volunteers, so we take care to optimize the time spent on our project (e.g., excluding thumbnails with an object we compute to be too faint to identify activity; Section \ref{methods:subsubsec:deltaMagLim}). One measure of how much time is required for a given task is \textit{human lifetimes}, which is 73.4 years as of 2019 \citep{worldhealthorganizationWHOMethodsData2019}, or $\sim4\times10^7$~s. Classifications take, on average, 4.7~s per thumbnail. Assuming very generously that each person is awake and classifying for 2/3 of their day, and classifies for 70 years of their life, then
\begin{equation}
\frac{70\ \mathrm{yr}}{\mathrm{life}} \cdot\frac{5\ \mathrm{classifications}}{\mathrm{s}} \cdot\frac{365 \mathrm{d}}{1~\mathrm{yr}} \cdot{\frac{2}{3}}\cdot \frac{24\ \mathrm{hr}}{\mathrm{d}} \cdot\frac{60\ \mathrm{min}}{\mathrm{hr}}\cdot\frac{60\ \mathrm{s}}{\mathrm{min}} \approx \frac{10^8\ \mathrm{classifications}}{\mathrm{life}},
\end{equation}
\noindent or roughly 100 million classifications per lifetime. Via Table \ref{discussion:tab:ratesAndCompleteness}, \textit{Active Asteroids} \ac{DECam} data could be completed in $\sim0.1$ lifetimes (7 yr). However, \ac{LSST} in the same \ac{HARVEST} vetted state would require, on average, $\sim0.03$ lifetimes per day to classify all of the thumbnail images. Given the nominal survey duration for \ac{LSST} is 10 years (3,650 days), over 120 human lifetimes would be required to classify all of the \ac{HARVEST} vetted thumbnail images. Even with the $\Delta_\mathrm{mag}<-3$ and $\%_{q\rightarrow Q} \ge$ 80\% filters applied, the \ac{LSST} data would still require 16 lifetimes (at 565,000 thumbnails/day for 10 years) without the as-yet unimplemented Wedge Photometry and/or \ac{ML} filters.
\subsubsection{Active Asteroid Detection Rate and Completeness}
\label{discussion:subsubsec:activeAsteroidCompleteness}
As discussed in Chapter \ref{chap:intro}, the occurrence rate of activity among asteroids (due to any cause) is estimated to be very roughly around 1 in 10,000. Considering only the Asteroid Belt, there are roughly 1 million asteroids $>1$~km in diameter (e.g., \citealt{tedescoInfraredSpaceObservatory2002}). This suggests there should be roughly 100 active asteroids in the Asteroid Belt alone. As of this writing about 20 have been observed with visual evidence of activity, so there should be about 80 undiscovered active asteroids in the Asteroid Belt.
Thus far \textit{Active Asteroids} volunteers have discovered one previously unknown active asteroid (paper in preparation) out of the $\sim$78,000 Asteroid Belt objects examined. While this seems to be a rate lower than 1 in 10,000, we have (so far) provided images of objects without selecting for proximity to perihelion passage, so it is likely one or more active objects have been examined but during their presumably quiescent period near aphelion. If we continue to provide images unbiased with respect to orbital position, then volunteers would need to examine 6.2 million thumbnails before identifying 80 active asteroids. Given current classification rates (Section \ref{discussion:subsubsec:classificationRate}) we intend to start prioritizing objects near perihelion passage, so we expect the detection frequency to increase.
\subsubsection{Project Plans}
\label{discussion:projectPlans}
A large number of discoveries stem from the \textit{Active Asteroids} project and \ac{HARVEST} pipeline, such as four published peer-reviewed publications thus far, one article currently in review, and three additional manuscripts are in preparation. These results make it clear the project is successful and should continue. Moreover, the public engagement aspect of the project should not be overlooked, with over 6,000 volunteers participating to date. As discussed in Section \ref{discussion:subsubsec:classificationRate}, the project could use additional participation to achieve completeness sooner, but adjusting thumbnail selection parameters can help in the interim. \ac{HARVEST} has already been upgraded to incorporate other instrument archives, such as the \ac{CFHT} MegaPrime. It is evident that additional selection criteria are necessary to further filter the thumbnail images selected for \textit{Active Asteroids} in order to keep the quantity viable with current classification rates.
One possible way to enhance thumbnail selection for Citizen Scientist examination would be to introduce additional automated detection techniques that identify possible activity and, in combination, result in an activity likelihood score. This will require combining existing detection techniques, such as \ac{PSF} analysis (Section \ref{intro:sec:activityDiscovery}), Wedge Photometry (Section \ref{QN:subsec:wedgephotometry}), and \ac{ML}-based image recognition informed by classification data from \textit{Active Asteroids}. Without this additional activity likelihood vetting, projects that produce massive amounts of data, such as \ac{LSST}, will not be able to benefit fully from Citizen Science projects like \textit{Active Asteroids}.
It is worth noting that the project did take several years to develop, and that involvement by this author and others were critical to project success. The vast majority of work on the \ac{HARVEST} pipeline, the Citizen Science project, the follow-up archival and telescope observations, are analyses performed by one individual (this author), with assistance provided by at least one additional member of the science team (Section \ref{methods:sec:citsci}) at any given time. For ongoing operations, project moderators (Section \ref{methods:subsec:talk}) play a key role in interfacing with volunteers through online forums. Citizen Science project development required additional efforts and involved one additional person (Jay Kueny), and development of an analysis pipeline to optimize classification data evaluation was carried out by another individual (Will Burris). The high volume of \textit{Active Asteroids} discoveries warrants additional full-time commitment by at least one more person, bringing the number of full-time dedicated individuals needed to run the project efficiently to at least two. More would be necessary if, for example, development of new methods are needed.
\subsubsection{Next Steps}
\label{discussion:nextSteps}
The multi-pronged approach to finding and characterizing active solar system bodies presented in this work has proven effective and continues to yield results. Discoveries stemming from the \textit{Active Asteroids} project arise each time we upload a new subject set, with seven batches completed as of this writing. At an activity occurrence rate of roughly one in 10,000 asteroids, there should be around 100 active asteroids in the Asteroid Belt. We hope to double the number of known active asteroids within the next two years by finding an additional 30 active asteroids. This will allow for a more comprehensive study of active bodies, thus enabling the small body community to draw meaningful conclusions about the solar system volatile distribution, the potential volume of volatiles held within each reservoir, and the implications for the origin of terrestrial water. Anyone who can see images on an internet-connected device can participate in \textit{Active Asteroids} by visiting \url{http://activeasteroids.net}.
\chapter{Introduction}
\label{chap:intro}
\acresetall
\section{Definitions}
\label{intro:sec:definitions}
It is important to define some terms that will be used throughout this work. ``Comet'' as a class of object is very loosely defined, with many individuals adopting essentially personal definitions of the term. For example, some people will consider anything seen with a tail as being a comet. Other people will only require that an object have an orbit typical of a comet class, such as \ac{JFC}, and the object will be called a comet even if activity has never been seen. It is not uncommon to mix visual and dynamical elements to define a type of comet, for example the Manx comets \citep{meech2013P2Pan2014}, defined as on a comet-like orbit but displaying little to no tail. For this work I define comets as objects that (1) exhibit activity and (2) are on orbits typically associated with dynamical classes with the word ``comet'' in them, including comet, long-period comet, short-period comet, \ac{JFC}, and \ac{QHC}.
As my adopted definition of comet is not based purely on orbital properties, I use the terms ``class'' or ``group'' to refer to collections of minor planets with common traits, most commonly orbital properties, but not necessarily limited to these attributes. I refer to volatile reservoirs as object classes with two or more objects known to harbor volatiles. These are a reservoir in the sense that comets and asteroids are two reservoirs thought to have supplied water to Earth in the past. Classes of so-called ``transition objects'' (defined in this work as objects only temporarily belonging to a particular class of objects) -- for example Centaurs (defined in Section \ref{intro:sec:activeCentaurs}) -- can still be viable reservoirs because these regions are being continually replenished with new objects (Centaurs in this example).
\section{Background}
\label{intro:sec:background}
Volatiles, such as water, are essential for life as we know it, yet fundamental knowledge about these materials, such as their present-day location within our solar system, and even what they are, is incomplete. This information is crucial for future space exploration and to help answer outstanding fundamental questions about the solar system, including Earth. For example, where did Earth's water originate? It is generally accepted that some water was already present when Earth formed, and that some additional quantity was delivered later, but not even a rough ratio is known. Comets were thought to be the only objects that delivered water post-formation, but the consensus today is that comets alone cannot account for the volume of water we find on Earth (see review, \citealt{alexanderOriginInnerSolar2017}). One explanation is that comets represent just one of multiple ``volatile reservoirs,'' classes of solar system bodies that harbor volatiles on or below their surfaces. Today, asteroids are considered likely contributors to the volatile budget on Earth.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\columnwidth]{introduction/2005QN173/2005_QN173_2021-12-08_02.01.44.513000_m20211208UT_image0029.new_chip0_126arcsec_NuEl.png} &
\includegraphics[width=0.45\columnwidth]{introduction/coma/C2009_F4_2014-01-13_03.31.51.895688_c4d_140113_033418_ooi_z_ls9_chip31-N1_126arcsec_NuEl.png}
\end{tabular}
\caption{Activity in the form of tails or a coma. Images credit: this work.
\textbf{Left:} 2005~QN$_{173}$ imaged on \ac{UT} 2021 December 8 at the \acf{VATT} at the \acf{MGIO} in Arizona (Prop. ID S165, PI Chandler). This image reveals that the object has two tails: one pointing away from the Sun (yellow $-\odot$) and one pointing opposite of the object's apparent direction of motion (red $-v$). These are the two directions most commonly associated with tails. \textbf{Right:} Comet C/2007 F4 (McNaught) displays a prominent coma in this image captured on \acs{UT} 2014 January 13 (Prop. ID 2012B-0001, PI Frieman).}
\label{intro:fig:activeObjects}
\end{figure*}
Comets are known for their remarkable displays of \textit{cometary activity} (Figure \ref{intro:fig:activeObjects}), like a tail or a shroud of material known as a coma (plural: comae). This activity is typically associated with volatile sublimation, the direct phase transition of a material from solid to gas (e.g., when dry ice turns into carbon dioxide gas) at conditions typical on Earth. However, mechanisms other than sublimation can result in mass loss that takes the form of tails or comae. Here we use the term ``activity'' to describe any situation where a body loses material to space.
Surprisingly, comets are not the only objects known to display comet-like activity. (See review by \cite{jewittAsteroidCometContinuum2022} for a comprehensive discussion on the increasingly blurred lines between comets and asteroids.) As a result, other groups of objects, such as active asteroids and active Centaurs (discussed below), may represent viable volatile reservoirs in their own right. However, in sum fewer than 50 members of these active object groups have been discovered since the first active asteroid was identified in 1949 \citep{harrisCometNotesComet1950} and, as a result, it is virtually impossible to draw robust conclusions about the amount and type of volatiles held by these groups. In order to enable the study of potential volatile reservoirs as populations we set out to create a platform that facilitates discovering many additional active bodies, with a long-term goal to increase the numbers of known active minor planets by a factor of two or more. Here I describe the platform we created, as well as several discoveries we made along the way, including five that resulted in peer-reviewed publications \citep{chandlerSAFARISearchingAsteroids2018,chandlerSixYearsSustained2019,chandlerCometaryActivityDiscovered2020a,chandler2483702005QN2021}. We also include a link\footnote{\url{http://activeasteroids.net}} to the online component of the platform, thereby enabling \textit{you} to participate in this exciting scientific endeavor.
\section{Layout of the Solar System}
\label{intro:sec:layoutOfSolarSystem}
\begin{figure}
\centering
\includegraphics[width=0.85\linewidth]{introduction/solarSystemCaltech/PIA05569_FULL_38pct.png}
\caption{The solar system seen from four different scales. The inner solar system (top-left) is shown with the orbit of Jupiter (red circle) as the outermost orbit. Zooming out (top-right) provides a view of the outer solar system, which includes the orbit of Pluto and the Kuiper Belt. Zooming out again (bottom-right) the orbit of (90377)~Sedna can be seen, which gives context to the view of the inner Oort Cloud (bottom-left), the location from which most comets are thought to originate. Image credit: NASA/CalTech.}
\label{intro:fig:solarSystem}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}{clrrS[table-format=3.3]}
Symbol & Name & $a$ & $e$ & Mass\\
& & [au] & & [M$_\oplus$]\\
\hline\hline
\mercury & Mercury & 0.5 & 0.21 & 0.06\\
\venus & Venus & 0.7 & 0.01 & 0.81\\
$\oplus$ & Earth & 1.0 & 0.02 & 1.00\\
\mars & Mars & 1.5 & 0.09 & 0.11\\
\jupiter & Jupiter & 5.2 & 0.05 & 318\\
\saturn & Saturn & 9.5 & 0.05 & 95\\
\uranus & Uranus & 19.2 & 0.05 & 15\\
\neptune & Neptune & 30.1 & 0.01 & 17\\
\end{tabular}
\caption{Orbital elements for the major planets of the solar system: the semi-major axis $a$, orbital eccentricity $e$, and the mass of the body in Earth masses $M_\oplus$.}
\label{intro:tab:planets}
\end{table}
Kepler's First Law states that the planets orbit in ellipses (ovals). The average distance from the Sun to an object over one complete orbit is equal to the semi-major axis ($a$), the distance between the center of an ellipse and the farthest point from the center. Astronomers typically measure distances of planets and small solar system bodies in terms of \textit{astronomical units} (au), defined as the average distance between the centers of Earth and the Sun.
Eccentricity $e$ describes how elongated an orbit is, with $e=0$ being a perfect circle, and $e=1$ a parabola and thus not a closed loop. The planets typically have low eccentricity (Table \ref{intro:tab:planets}), with a median of $e=$0.05, while comets typically have high eccentricity, on average $e=$0.9.
Other objects also orbit the Sun, and these are collectively referred to as minor planets (or small solar system bodies), or dwarf planets (e.g., Sedna). As of 1 July 2022 there are roughly 1.2 million known minor planets, of which fewer than 4,000 are comets. There are two circumstellar ``belts'' containing large numbers of minor planets. The Asteroid Belt (Figure \ref{intro:fig:solarSystem}) is found between the orbits of Mars and Jupiter, roughly between 2~au and 4~au. The Asteroid Belt has a mass of less than one thousandth of Earth's mass (e.g., \citealt{krasinskyHiddenMassAsteroid2002}). The Kuiper Belt (sometimes referred to as the Edgeworth-Kuiper Belt or the Trans-Neptunian Belt) extends from the orbit of Neptune (30~au) to around 50~au. Notably, the Kuiper belt is some 20 times wider in radial extent that the Asteroid Belt, and at one tenth the mass of Earth the Kuiper Belt is 200 times more massive than the Asteroid Belt \citep{gladmanStructureKuiperBelt2001,pitjevaMassesMainAsteroid2018,diruscioAnalysisCassiniRadio2020}.
To date four active minor planet classes (other than comet classes) have members known to display comet-like activity. These are (1) comets, (2) main-belt asteroids, (3) Centaurs (icy bodies orbiting between 5~au and 30~au), (3) \acfp{NEO}, also known as \acfp{NEA},
(4) \acp{QHO}, a type of asteroid with orbits similar to the Hildas (which are in 3:2 orbital resonance with Jupiter), and (4) one interstellar object, designated 2I/Borisov \citep{borisovMPEC2019R106COMET2019}.
Comets are thought to originate from two sources: (1) the Kuiper Belt, and (2) the Oort cloud, a spherical cloud of objects orbiting between 2,000~au to 200,000~au from the Sun \citep{oortStructureCloudComets1950}. Comets are classified by two different means. The first is by recognizing their activity, an approach dating back thousands of years (see catalog by \citealt{kronkCometographyCatalogComets1999}), a technique still valid today. Notably, this definition is not based on orbital characteristics at all, and thus the class ``comet'' is not necessarily dynamically derived. Comets are also identified based on properties of their orbits through essentially two systems of dynamical classification.
(1) The period-based comet classifications are: (i) Hyperbolic comets, which may be interstellar in origin. These have enough momentum to leave the solar system, and so they do not have a period. (ii) Long-period comets have orbits longer than 200 years. (iii) Halley-type comets, named after the famed Halley's Comet, have periods ranging between 20 and 200 years. (4) Short-period comets have periods less than 20 years. These are also sometimes referred to as \acfp{JFC}.
(2) The other system for classifying comets makes use of an orbital metric that describes a body's close approach speed to Jupiter, and can be considered a descriptor of how strongly an orbit is influenced by Jupiter. The metric is known as Tisserand's Parameter with respect to Jupiter ($_\mathrm{J}$), and is described in detail in Chapter \ref{safari:sec:introduction}. Orbits constrained by $T_\mathrm{J}$ include \acp{JFC} (the $T_\mathrm{J}$ definition), having orbits that are strongly influenced by Jupiter.
\section{Active Asteroids}
\label{intro:sec:activeAsteroids}
For a more in-depth discussion of Active Asteroids, see Chapter \ref{safari:sec:introduction}.
\begin{figure}[h]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{introduction/4015_Wilson-Harrington/eso9212b_crop_smallest_arrows.png} & \includegraphics[width=0.45\linewidth]{introduction/29P_2011-02-09_12.05.17.973000_PTF_201102095037_i_p_scie_t120517_u012812468_f02_p100222_c08_chip0_126arcsec_NuEl.png}\\
\end{tabular}
\caption{\textbf{Left:} This image, taken on 1949 November 19, shows the first asteroid activity discovered, emanating from (4015)~Wilson Harrington (green dashed arrow) in the form of a tail (red arrows). Image Credit: ESO and Palomar Observatory (\url{https://www.eso.org/public/images/eso9212b/}). \textbf{Right:} The first active Centaur discovered (retroactively, following the 1977 discovery of the first identified Centaur, (2060) Chiron \citealt{kowal1977UB1977}), 29P/Schwassmann-Wachmann~1 \citep{schwassmannNEWCOMET1927}, shown here in a 60~s $R$-band image acquired with the \ac{PTF} Mosaic imager on \ac{UT} 2011 February 9 at \ac{PTF} (Prop. ID, PI Kulkarni). 29P was first identified as active in 1927 \citep{schwassmannNEWCOMET1927}, but the object was not considered a Centaur until after the discovery of Centaur (2060)~Chiron in 1977. Image Credit: this work.}
\label{intro:fig:famousAAs}
\end{figure}
The first asteroid observed with cometary features was \ac{NEO} (4015)~Wilson-Harrington \citep{cunninghamPeriodicCometWilsonHarrington1950}. Astronomers identified a clear tail in images taken in 1949 (Figure \ref{intro:fig:famousAAs}). However, when astronomers were able to observe the object again, no activity was seen. Since then no conclusive evidence of further activity has been detected, despite considerable efforts \citep{degewij1979VAPhysical1980,chamberlin4015WilsonHarrington22011996,licandroSpitzerObservationsAsteroidcomet2009,ishiguroSearchCometActivity2011,urakawaPhotometricObservations107P2011}.
In 1996, nearly five decades later, the modern era of active asteroids was ushered in with the discovery of an active object orbiting within the asteroid belt, 133P/Elst-Pizarro \citep{elstComet1996N21996}. This object has been observed to be repeatedly active, especially near perihelion \citep{hsiehReturnActivityMainbelt2010} -- the point in an object's orbit when it is closest to the sun. Repeated periodic activity when an object is near to the Sun, the warmest period of an object's orbit, is strong evidence that the activity is sublimation-driven, and this object became the first to be designated a \ac{MBC}. The \acp{MBC} are a subset of active asteroids that orbit within the Asteroid Belt and exhibit sublimation-driven activity \citep{hsiehPopulationCometsMain2006}. As of this writing, sublimation is inferred -- activity has been too weak to confirm the presence of volatiles spectroscopically, despite efforts to do so (e.g., \citealt{hsiehObservationalDynamicalCharacterization2012}). Fewer than ten \acp{MBC} have been found, although there are additional candidates suspected of being \ac{MBC} members.
\section{Active Centaurs}
\label{intro:sec:activeCentaurs}
Centaurs are cold bodies that originate from the Kuiper Belt (see review \citealt{morbidelliCometsTheirReservoirs2008}). Confusingly, there are multiple discrepant definitions of Centaur. In this work we adopt the definition of \cite{jewittActiveCentaurs2009} that defines Centaurs as (1) objects with semi-major axes and perihelion distances that both fall between the orbits of Jupiter (5.2~au) and Neptune (30~au), and (2) the object must not be in a resonant orbit with a giant planet. A \textit{resonant orbit} is defined as any situation when two bodies orbiting a parent body (the Sun in this case) share a similar orbit (1:1 ratio) or have mean orbital periods that are in integer ratios of each other (e.g., 3:2). For example, Jupiter Trojans (Figure \ref{intro:fig:solarSystem}) are co-orbital with Jupiter, leading and following Jupiter in its orbit by 60$^\circ$, so these are not classified as Centaurs.
Centaurs, being significantly more distant than main-belt asteroids and \acp{NEO}, are much fainter and, consequently, harder to detect. Unlike active asteroids, active Centaurs were first identified belatedly from objects previously classified as comets. The first known active Centaur is often cited as being 29P/Schwassmann-Wachmann~1 (Figure \ref{intro:fig:famousAAs}), discovered in 1927 \citep{schwassmannNEWCOMET1927} and, at the time, considered a comet. Notably, Centaur 2020~MK$_4$, an object which has an orbit very similar to that of 29P, was recently found to be active \citep{delafuentemarcosActiveCentaur20202021}. In 1977 the prototype Centaur (2060)~Chiron was discovered \citep{kowalSlowMovingObjectKowal1977}, an object that was itself later found to be active \citep{meechAtmosphere2060Chiron1990}.
\section{Dynamical Evolution}
\label{intro:sec:transitionObjects}
Orbits of all bodies in the solar system change continuously because of the influence of gravity imparted by other objects. Minor planets may experience gravitational perturbations that result in their orbit classification changing entirely, for example from Centaur to \ac{JFC}. We define bodies that are in the process of migrating from one dynamical class to another as \textit{transition objects}.
My favorite transition object, 39P/Oterma, was discovered by Liisi Oterma at Turku (Finland) in 1943 \citep{otermaNEWCOMETOTERMA1942}. At the time of discovery, 39P/Oterma had a perihelion distance of 3.4~au and a semi-major axis of 4.0~au, placing it interior to the Centaur region. At the time, 39P/Oterma was either a \ac{QHC} or \ac{JFC}. However, on \ac{UT} 1963 April 12, 39P/Oterma had a close encounter (0.095~au) with Jupiter that dramatically altered its orbit. Ever since, 39P/Oterma has had a perihelion distance and semi-major axis exterior to 5.4~au, placing this object firmly within the Centaurian orbital regime.
A recent example is P/2019 LD2 (ATLAS), an object in an orbit similar to a Jupiter Trojan. Jupiter Trojans lead and trail Jupiter by 60$^\circ$ in orbit, however P/2019 LD2 is not presently in either of those locations. P/2019 LD2 (ATLAS) was most likely a Centaur before it arrived in its current orbit, and will return to a Centaurian orbit in 2028, followed by a \ac{JFC} orbit in 2063 \citep{hsiehTransientJupiterTrojanlike2021}.
In Chapter \ref{chap:282P} we present a study of 282P/(323137) 2003 BM$_{80}$, and object we classified as a \ac{QHO}. Prior to 180 years ago 282P was likely a Centaur or possibly a \ac{JFC}. After numerous close encounters with Jupiter, 282P migrated inward and was captured in a Quasi-Hilda orbit, which is an orbit with properties similar to the Hilda group that is in 3:2 resonance with Jupiter. Over the next 300 years or so 282P will undergo more encounters with Jupiter before it probably migrates to a \ac{JFC} orbit.
\section{Activity Detection Techniques}
\label{intro:sec:activityDiscovery}
Prior to the invention of the telescope, cometary activity was discovered with the naked eye. Documented discoveries date back thousands of years, with written records of lost comets beginning with comet D/-674, and comets still familiar today starting with 1P/Halley, first recorded in -239 B.C. \citep{kronkCometographyCatalogComets1999}. Here I discuss different modalities of activity detection, limiting the discussion to active asteroids and active Centaurs. Two important notes to bear in mind: (1) not all techniques have yielded new active body discoveries, and (2) some techniques have yet to validate activity claims through empirical visible activity identification (i.e., see a tail or coma), though some (but not all) disclaim that the objects highlighted should be considered candidates. These considerations are addressed as appropriate below.
\subsection{Visually Observed Activity}
\label{intro:subsec:visibleActivity}
Visual identification of a tail and/or coma remains the gold standard of activity detection. The \ac{MPC} adds additional requirements for comet discovery, namely that the activity must be visible in at least two images taken during one observing night, and that two sets of images, preferably from adjacent observing dates, be submitted.
There can be a great deal of uncertainty when searching for activity indicators like coma(e) and/or tail(s). For example, background galaxies and image artifacts can masquerade as activity, especially when looking at just one image instead of a sequence where the solar system object moves against the sky. To account for ambiguity I created an informal system to describe the level of apparent activity in an image, ranging from 0 (unable to locate the object at all) to 9 (any individual shown a single image would have no doubt whatsoever that they are looking at cometary activity), described in Section \ref{methods:subset:trainingSet}.
Myriad techniques have been used to enhance images to bring out additional detail in the tails. In the same way modern image or photo editing tools can minimize shadows or enhance contrast, images of activity can be enhanced to bring out more detail. Another common technique is to add multiple images together (co-addition), sometimes referred to as stacking, thereby strengthening the overall image signal.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.39\linewidth]{introduction/enceladus_nasaJplSTSCI/42_PIA08386_cropped.png} & \includegraphics[width=0.6\linewidth]{introduction/bennu/imagesasteroid20191205PIA23554-16_cocAdjusted.jpeg} \\
&
\end{tabular}
\caption{Activity discovered by fly-by and orbiter spacecraft missions may not be detectable from Earth.
(a) Geysers of water reaching high above the surface of Enceladus were imaged by the Cassini spacecraft. Image Credit: NASA/JPL/Space Science Institute.
(b) Gravel-sized particles (left) being ejected from the surface of (101955)~Bennu were captured by the cameras aboard the OSIRIS-REx spacecraft. Image Credit: NASA/Goddard/University of Arizona/Lockheed Martin
}
\label{intro:fig:activity}
\end{figure}
Activity detection is strongly influenced by measurement sensitivity. I normally think of activity detection as roughly falling into two categories: remote sensing and \textit{in situ}. This dissertation focuses primarily on the study of activity detected from Earth, however several objects have been found to be active once visited by spacecraft. For example, (101955)~Bennu had not been suspected of being active, but upon arrival of the \ac{OSIRIS-REx} mission spacecraft in 2018, unexpected activity was documented by the cameras aboard the spacecraft (Figure \ref{intro:fig:activity}). Thus Bennu is definitely active, but that activity has never been observed from Earth. Similarly, some planetary moons have been found to be active by spacecraft, for example Enceladus \citep{spencerCassiniEncountersEnceladus2006}, as shown in Figure \ref{intro:fig:activity}. Pluto and Ceres represent special cases where an an atmosphere (or exosphere was detected remotely first, then by spacecraft later. In the case of Pluto, an atmosphere was detected remotely in 1989 \citep{elliotPlutoAtmosphere1989}, then later studied up close by the New Horizons spacecraft flyby in 2015 \citep{sternPlutoSystemInitial2015}. In the case of Ceres, the first asteroid discovered \citep{piazziRisultatiOsservazioniNuova1801}, water vapor was discovered by \cite{kuppersLocalizedSourcesWater2014} with Herschel telescope observations. Following the arrival of the Dawn spacecraft at Ceres, activity in the form of water vapor was detected \citep{nathuesSublimationBrightSpots2015,thangjamHazeOccatorCrater2016}, although a later study by \cite{schroderResolvedSpectrophotometricProperties2017} did not find any evidence of sublimation.
\subsection{Brightness}
\label{intro:subsec:brightness}
Approaches connected with measuring the brightness of an object are often used in conjunction with other evidence to support a claim of activity, to allow for additional analyses, or both. There are three primary techniques involving brightness that have been used to find potentially active asteroids, two of which have yielded proven results.
\paragraph{Discrepant Brightness} This method involves looking for unexpected brightening of objects which could be caused by activity reflecting additional light (see \citealt{cikotaPhotometricSearchActive2014} for an example of a broad application). This is achievable by measuring how much light the asteroid reflects, and then comparing this result with an expected value. Note, however, that the source for expected values must provide precision photometry for a source with a well-measured phase function, ideally whilst inactive; sources such as \ac{JPL} Horizons, while highly convenient, are widely considered only accurate to within a couple of magnitudes. Objects with activity should reflect more light, and thus should appear brighter than expected.
\paragraph{Point Spread Function Analysis} The shape and size of points in an image is called a \ac{PSF}. By comparing the \ac{PSF} of an object in an image to a comparably bright star in the same image can reveal if the width of a source is wider than expected, indicating that the source is extended (i.e., elongated). Finding asteroids that have unexpected broadening can be used for detecting activity (e.g., \citealt{hsiehMainbeltCometsPanSTARRS12015}). The measurement of excess breadth is often used as direct evidence of activity, and measurements of the excess flux can reveal important information about the activity, such as the amount of material in a coma (this can also be considered image analysis, with the source of data being photons). See Figure \ref{og:fig:sbrp} for an example.
\subsection{Spectroscopic Indicators}
Spectroscopy is the technique that measures markers in refracted light, as with a prism. The general idea is to identify features in spectra that reveal an object is active, whether the material be composed of volatiles or dust. However, to date spectroscopy has yet to identify an active asteroid that has a tail or coma visible from Earth.
This technique has been used successfully before to identify an asteroid as active. A notable example is the case of (1)~Ceres (e.g., \citealt{kuppersLocalizedSourcesWater2014}), albeit confirmation \citep{nathuesSublimationBrightSpots2015,thangjamHazeOccatorCrater2016} from the visiting Dawn mission spacecraft
is disputed \citep{schroderResolvedSpectrophotometricProperties2017}. Moreover, spectroscopic study of minor planets has successfully identified surface ices in the past, including on bodies known to be active (e.g., (2060)~Chiron; see review, \citealt{peixinhoCentaursComets402020}). However, sublimation from \acp{MBC} has never been spectroscopically confirmed; see review by \cite{snodgrassXshooterSearchOutgassing2017}.
Recently a group has applied a spectroscopic technique to attempt to identify active asteroids, most recently (24)~Themis and (449)~Hamburga \citep{busarevSimultaneousSublimationActivity2021}. Alas, to date none have been observed to display a visibly identifiable tail or coma, despite archival and observational efforts by astronomers, including archival and observational efforts by our team. In 2018 (162173)~Ryugu, one of the objects identified as active by this technique \citep{busarevNewCandidatesActive2018}, was visited by the \ac{JAXA} spacecraft Hayabusa2 \citep{watanabeHayabusa2MissionOverview2017} for a sample return mission. No activity was reported, though there was evidence that the predominately dehydrated Ryugu \citep{sugitaGeomorphologyColorThermal2019} had spun rapidly in the past \citep{watanabeHayabusa2ArrivesCarbonaceous2019} which could have resulted in mass shedding.
\subsection{Non-gravitational Acceleration}
All objects in the solar system experience acceleration due to the force of gravity. Unexplained acceleration can be caused by the gravity of an unknown body, such as the perturbations that resulted in the discovery of Neptune (see account by \citealt{standage2000neptune}) or, more recently, the hypothesized Planet 9 \citep{trujilloSednalikeBodyPerihelion2014,batyginPlanetNineHypothesis2019}. However, not all unexplained acceleration is caused by gravity.
Sources of non-gravitational acceleration include the Yarkovsky effect (first measured on (6489) Golevka, \citealt{chesleyDirectDetectionYarkovsky2003}), a force resulting from imparted solar radiation received by a body being reemitted later as thermal radiation.
These types of forces are negligible over short timescales, yet some objects have demonstrably experienced changes in their orbits that could not be readily explained. For example, interstellar object 1I/\hbox to.666\okinalen{\hss`\hss}Oumuamua{} evidently experienced non-gravitational acceleration \citep{micheliNongravitationalAccelerationTrajectory2018}, possibly attributable to unseen activity \citep{seligmanAnomalousAcceleration1I2019}.
Activity can provide one potential source of non-gravitational acceleration, for example jets of gas. Thus, in principle, it should be possible to search for activity by scrutinizing the orbits of small solar system bodies and looking for unexplained changes. A recent example tries to link two bodies, 2019~PR$_2$ and 2019~QR$_6$, to cometary activity \citep{fatkaRecentFormationLikely2022}, though no visible activity has been conclusively observed as of this writing.
\subsection{Meteor Showers}
This technique has yet to identify a new active asteroid, but active asteroid (3200)~Phaethon has been identified as the apparent parent of the Geminid meteor showers \citep{whippleIAUC388119831983}. In addition to the Geminid Meteor Stream, Phaethon shares an orbit with 2005~UD \citep{ohtsukaApolloAsteroid20052006} and 1999~YC \citep{ohtsukaApolloAsteroid19992008}, which suggests all co-orbital elements may have originated from the breakup of a single parent body. The case of Phaethon implies that it may be possible to connect other meteor streams back to parent bodies (e.g., \citealt{dumitruAssociationMeteorShowers2017}) that may themselves be active. See also reviews by \cite{babadzhanovExtinctCometsAsteroidmeteoroid2015,yeMeteorShowersActive2018}.
\subsection{Magnetic Anomalies}
This approach has only been used once, to claim activity coming from (2201)~Oljato or an outgassing debris trail in its orbit \citep{russellInterplanetaryMagneticField1984,kerrCouldAsteroidBe1985}. A series of interplanetary magnetic field enhancements were measured by the Pioneer spacecraft that was orbiting Venus. These events were correlated with the passage of Oljato during 7 of the 11 magnetic anomalies, with the likelihood the anomalies were coincidental given as 4 in $10^4$. However, despite some further evidence supporting Oljato activity \citep{cochranSpectroscopyAsteroidsUnusual1986,mcfaddenEnigmaticObject22011993,chamberlin4015WilsonHarrington22011996}, all activity associated with Oljato to date has been inferred, rather than directly observed in the form of a coma or tail. The event may have been transient as Oljato's orbit is considered chaotic \citep{milaniDynamicsPlanetcrossingAsteroids1989}.
\section{Activity Mechanisms}
\label{intro:sec:mechanisms}
Volatile sublimation is not the only cause of activity we observe. Moreover, the myriad mechanisms potentially responsible for observed activity are not mutually exclusive, and one activity mechanism may trigger another or occur simultaneously. Some events are stochastic (one-off), while other mechanisms are recurrent by nature. Consequently, recurrent activity is an important diagnostic indicator when ascertaining an underlying activity mechanism.
Below is a listing of mechanisms that may result in the kind of activity we observe associated with active asteroids and active Centaurs. See also the reviews of \cite{jewittActiveAsteroids2015a} and \cite{jewittActiveCentaurs2009}, as well as Table \ref{safari:Table:TheAAs}.
\subsection{Volatile Sublimation}
In the same way that dry ice goes directly from a solid to a gas on Earth's surface, ices can sublimate in space to great effect, releasing volatiles and ejecting dust and rocky material. The primary activity mechanism of comets is volatile sublimation, and these bodies have been studied at length from Earth and \textit{in situ} with spacecraft visits (e.g., Rosetta mission to 67P/Churyumov–Gerasimenko, \citealt{glassmeierRosettaMissionFlying2007,sierksNucleusStructureActivity2015}). In addition to water ice, carbon dioxide, carbon monoxide, ammonia, methane, nitrogen and other molecules have been detected on asteroids and Centaurs (see Chapter \ref{og:sec:introduction}).
To be clear, volatiles need not be on the surface in order to sublimate, however remote detection of ices on inactive bodies requires ice to be present on the surface. These bodies may have reservoirs just under their surfaces or buried far below. Any group of bodies that harbors ice, no matter where that material is located on or within a body, represents a volatile reservoir. However, some bodies may not have any ices at all -- especially silica-rich asteroids known as S-type asteroids. Silica, on Earth commonly associated with desiccant and sand, is typically dehydrated and thus not expected to contain volatiles.
Sublimation requires some form of energy change to take place, with energy imparted directly by the Sun being the most ubiquitous. The closer a body gets to the Sun, the more energy it receives and, as a result, activity becomes more likely if ice is present. Many recurrently active objects, especially comets, are observed to be preferentially active as they get closer to the Sun. Energy may also come from other sources, such as tidal heating (e.g., Europa, \citealt{greenbergTectonicProcessesEuropa1998}) or ice phase transitions (see review, \citealt{jewittActiveCentaurs2009}).
Different substances sublimate at different temperatures. Water ice, for example, will not appreciably sublimate at the orbital distances where Centaurs are found, but it can readily sublimate on bodies found in the asteroid belt. As a consequence the lifetime of ices varies by orbital distance so that, for example, it could be expected that water ice could survive on a body orbiting at 5~au but carbon monoxide and methane would have been depleted long ago \citep{schorghoferLifetimeIceMain2008,snodgrassMainBeltComets2017}. As described in Chapter \ref{og:sec:sublimationmodeling}, this knowledge can be leveraged to help identify which material(s) are most likely responsible for observed sublimation-driven activity.
\subsection{Rotational Instability}
All minor planets in our solar system rotate to some extent. Bodies that rotate rapidly can actually break apart or lose loose surface material to space. Small solar system bodies are susceptible to being ``spun up'' over time by the Sun by a process known as the \ac{YORP} effect (see e.g., \citealt{bottkeYarkovskyYorpEffects2006,lowryDirectDetectionAsteroidal2007} for details about \ac{YORP} forces). Consequently, it is possible that rotational instability can lead to recurrent activity that is unrelated to sublimation \citep{jewittEpisodicEjectionActive2015,chandlerSixYearsSustained2019}. Even if activity is recurrent, the onsets of activity would be uncorrelated with perihelion distance.
Rotational instability can lead to sublimation involvement if, for example, a breakup or landslide exposes previously buried volatiles that subsequently sublimate. Activity may cease again if the volatiles become smothered by settling material that was previously ejected.
\subsection{Thermal Fracture}
Heating solid materials can cause fracture. On Earth we see this happen, for example, when pouring boiling water on a frozen car windshield. Fracture is caused when stress induced by temperature change (e.g., expansion of heated material) overcomes the tensile strength of adjoining material \citep{jewittActiveAsteroids2015a}. Depending on the orbit of an object, this could take place repeatedly, especially when the object is close to the Sun. This action alone may eject material into space and result in the appearance of cometary activity. Moreover, these fractures may exposure sequestered volatiles that subsequently sublimate.
In the case of (3200)~Phaethon, an active asteroid that is thought to be responsible for the Geminid meteor stream \citep{whippleIAUC388119831983}, the body's temperatures reaches roughly 1100~K (1520$^\circ$~F) at its 0.14~U perihelion \citep{ohtsukaSolarRadiationHeatingEffects2009}, temperatures that can causes thermal fracture \citep{licandroNatureCometasteroidTransition2007,kasugaObservations1999YC2008} and in turn may result in mass loss \citep{liRecurrentPerihelionActivity2013,huiResurrection3200Phaethon2017}. The \ac{JAXA} mission \ac{DESTINY+}, scheduled to launch in 2024, is designed to provide more insights into Phaethon and its activity \citep{ozakiMissionDesignDESTINY2022}.
\subsection{Impact}
\label{intro:subsubsec:impact}
I divide impact events into two categories: significant events involving one or more large impactors (meter to kilometer scale), and micrometeorite impacts that involve multiple impacts by very small impactors typically of order 1~cm and smaller.
Two important cases of impact are worth mentioning here. (1) (596)~Scheila is widely considered the seminal example of an impact-driven asteroid activity event \citep{bodewitsCollisionalExcavationAsteroid2011,ishiguroObservationalEvidenceImpact2011,moreno596ScheilaOutburst2011}. (2) The first active asteroid discovered, (4015)~Wilson-Harrington (Figure \ref{intro:fig:famousAAs}) is thought to have undergone a significant impact event because the object was never observed to be active again, despite searches spanning over seventy years (e.g., \citealt{chamberlin4015WilsonHarrington22011996}).
Recently the \ac{OSIRIS-REx} spacecraft arrived at asteroid (101955)~Bennu for a sample return mission. Cameras aboard the spacecraft recorded what appeared to be gravel and other particulate leaving and returning to the surface (Figure \ref{intro:fig:activity}). Micrometeorite impacts have been suggested as a potential cause \citep{laurettaEpisodesParticleEjection2019,bottkeMeteoroidImpactsSource2020,hergenrotherIntroductionSpecialIssue2020}, though thermal fracture and other mechanisms are still being investigated.
\textit{Gardening} describes a process by which micrometeorite impacts overturn the outermost layer of a body. First described on the Moon (e.g., \citealt{chapmanLunarCrateringErosion1970} and references therein), gardening theory can be used to place limits on the amount of mass shed by myriad processes, including electrostatic lofting (see below), micrometeorite impacts (impact gardening) and photons (e.g., solar gardening; \citealt{grundySolarGardeningSeasonal2000}). Although impact gardening has been correlated with absolute ages only on the Moon \citep{gaultMixingLunarRegolith1974}, impact gardening serves an important function of bringing ices closer to a body's surface \citep{schorghoferPredictionsDepthtoiceAsteroids2016}, thus increasing the availability of material for sublimation.
\subsection{Cryovolcanism}
Just as volcanoes can eject molten material, cryovolcanoes can eject liquid and/or gaseous material from a cold body. Saturn's moon Enceladus is also thought to undergo cryovolcanism (Figure \ref{intro:fig:activity}) resulting from tidal heating \citep{nimmoShearHeatingOrigin2007}, first observed as plumes by the Cassini spacecraft in 2005 (e.g., \citealt{spencerCassiniEncountersEnceladus2006}). Asteroid (1)~Ceres is thought to undergo cryovolcanic activity (e.g., \citealt{soriVanishingCryovolcanoesCeres2017,nathuesRecentCryovolcanicActivity2020}), most likely driven by radioactive heating \citep{mccordCeresItsOrigin2011}, but evidence of cryovolcanic activity was only confirmed after the arrival of the Dawn spacecraft. To date it is unclear whether or not any activity on active asteroids or active Centaurs is primarily due to cryovolcanism, though cryovolcanism has been reported as responsible for the activity of 29P/Schwassmann-Wachmann~1 \citep{milesDiscreteSourcesCryovolcanism2016}.
\subsection{Radiation Pressure Sweeping}
Solar wind exerts a force that can, in principle, sweep particles off of the surface of an airless body, especially one with a small gravitational field \citep{jewittActiveAsteroids2015a}. This effect may play an important role as a secondary action, carrying away material ejected via other means that would have otherwise settled back on the surface. This is thought to play an important role for (3200)~Phaethon given its close (0.14) perihelion passage where radiation pressure is significant \citep{jewittActivityGeminidParent2010}. To date this effect has not been directly measured at an active asteroid, though the \ac{DESTINY+} mission may help us better understand the mechanisms at play on Phaethon.
\subsection{Electrostatic Lofting}
This mechanism was first observed by Apollo astronauts in the 1960 as a ``lunar horizon glow'' \citep{rennilsonSurveyorObservationsLunar1974,wangDustChargingTransport2016}. The electrostatic forces behind this mechanism may be powerful enough to eject material from the surface of small airless bodies such as asteroids. Should the material be lofted without sufficient energy for escape, a second activity mechanism (e.g., radiation pressure sweeping) could help carry the material away. Electrostatic lofting is a weak phenomenon, so it is unclear if this mechanism could result in activity detectable at distances farther than spacecraft orbit. However, \cite{sonnettLimitsSizeOrbit2011} suggest very low-level activity on a broad scale ($\sim$5\% of main-belt asteroids, based on a study of $\sim1000$ asteroids from \citealt{masieroThousandAsteroidLight2009}) which may involve electrostatic lofting \citep{jewittActiveAsteroids2015a}.
\subsection{Phyllosilicate Dehydration}
Phyllosilicates are a class of minerals that are characterized by layering, such as mica or smectite clays. Hydrated phyllosilicates have volatiles like water trapped between layers. Laboratory studies of meteorites rich in hydrated phyllosilicates reveal that these volatiles can be released with significant energy when heated sufficiently (e.g., \citealt{gibsonInorganicGasRelease1974}). On large scales this modality may be the underlying cause of thermal fracture, but on small scales this release of energy has the potential to eject material from the surface of an airless body. This mechanism may be at play on asteroid (101955)~Bennu \citep{laurettaEpisodesParticleEjection2019} because the surface has an abundance of hydrated phyllosilicates \citep{hamiltonEvidenceWidespreadHydrated2019}.
\subsection{Binary Rubbing}
The two bodies of a binary asteroid may eventually spiral in and become a contact binary. The physical interaction between the rubbing binaries could cause material to be shed from the surfaces, resulting in a coma or tail. However, this mechanism has yet to be conclusively identified as the cause behind the activity of any known active asteroid, though it has been proposed as one possible explanation for the activity of active asteroid P/2013~P5 \citep{hainautContinuedActivity20132014}.
\section{Citizen Science Project}
\label{intro:sec:citSciProject}
Note: For completeness, this section provides a cursory introduction only. The project is described at length in Chapter \ref{methods:sec:citsci}.
Citizen Science is a paradigm that aims to accomplish scientific goals while simultaneously engaging the public by seeking assistance from volunteers to accomplish tasks that are too numerous for individuals or small groups to complete, and which are also too complex for computers to handle. As described in Chapter \ref{methods:subsec:workflow}, our root method is to ask volunteers whether or not they see a tail or coma in images of known minor planets (e.g., asteroids, Centaurs) extracted from publicly available \ac{DECam} data. Once images are examined we can conduct follow-up investigation and study, as described in Chapter \ref{methods:subsec:archivalInvestigation}.
Zooniverse\footnote{\url{https://zooniverse.org}} is an online Citizen Science platform, known for its highly successful inaugural project \textit{Galaxy Zoo} \citep{lintottGalaxyZooMorphologies2008} that launched in 2007. We selected Zooniverse to host our project because of their proven ability to host and support Citizen Science projects. Our Citizen Science project \textit{Active Asteroids}, a \ac{NASA} Partner, launched on 31 August 2021. The project immediately began yielding results, and volunteers exhausted our original pool of data in just a few days. Since launch, over six thousand volunteers have completed roughly 2.5 million classifications of some two hundred thousand images.
\section{Manuscript Introduction}
\label{sec:intro:manuscripts}
We published numerous discoveries during preparations for the projec
. Here I chronologically introduce the manuscripts included in this dissertation, provide a brief synopsis of each, and describe how they relate to the overall dissertation theme of detecting and characterizing active minor planets via astroinformatics and/or Citizen Science. Key points are indicated by \textbf{bold} typeface.
As discussed at the start of this chapter, I identified a need to identify additional active minor planets in order to facilitate their study. From the start we considered launching a Citizen Science project to assist with this tasks, however it seemed logical to first carry out a proof-of-concept to ensure we could, in fact, supply images of known solar system objects to Citizen Scientists who would check for activity like tails and comae. \acf{SAFARI} was the title for our proof-of-concept \citep{chandlerSAFARISearchingAsteroids2018}, provided in Chapter \ref{chap:SAFARI}. We began by creating a software \textbf{pipeline} that produces thumbnail images (small cutouts from a larger image) from \ac{DECam} data, each displaying a known minor planet at the center. This pipeline became the foundation of our \ac{HARVEST} pipeline, discussed at length in Chapter \ref{methods:sec:methods:pipeline}.
From 594 \ac{DECam} images we extracted a total of 35,640 thumbnail images that contained 11,703 unique solar system objects. We examined all of these thumbnails visually (by eye) and identified activity emanating from what turned out to be one already known active asteroid: (62412) 2000~SY$_{178}$. Identifying an active object in our data served as our \textbf{proof-of-concept}, demonstrating that \ac{DECam}, with its wide ($2.2^\circ\times2.2^\circ$) field of view and large 4~m aperture that probes very faintly, is well-suited for finding active bodies.
We estimated an \textbf{activity occurrence rate} of 1 in 11,000 objects is active, in rough agreement with past studies that found a rate of roughly 1 in 10,000 \citep{jewittActiveAsteroids2015a,hsiehMainbeltCometsPanSTARRS12015}.
As part of our included background review we constructed a \textbf{comprehensive table} listing all active asteroids, along with details such as orbital distance, number of activity epochs, and diagnosed or suspected activity mechanism(s).
With the proof-of-concept a success we set out to improve upon the \ac{HARVEST} pipeline, for example to query the \ac{DECam} public archive to provide us with a plentitude of data, and to query the archive daily in order to keep our library of minor planet thumbnails up-to-date. Another capability we added to \ac{HARVEST} was the ability to quickly provide us with thumbnail images from our repository of a single solar system object. An opportunity arose for us to make use of this feature after a telegram announced asteroid (6478)~Gault was active \citep{smith6478Gault2019}. For the work we would publish, \textit{Six Years of Sustained Activity from Active Asteroid (6478)~Gault} \citep{chandlerSixYearsSustained2019}, provided in Chapter \ref{chap:Gault}), we produced thumbnails using this \ac{HARVEST} feature and found evidence that Gault had been active during at least two prior orbits, thus Gault experienced \textbf{recurrent activity}. We also introduced \textit{observability}, a metric describing how many hours per night an object is observable from a given location (Figure \ref{Gault:fig:ActivityTimeline}). This metric highlights observational biases (e.g., observer location) that can influence analyses such as activity mechanism diagnosis, as well as allowing us to assess how many images of a given object we have in our archive.
The onset of sublimation-driven activity typically occurs preferentially near perihelion, but we showed that Gault's activity was not correlation with heliocentric distance.
We carried out simple \textbf{thermal modeling} to estimate the temperatures experienced by Gault over the course of its orbit, and found it consistently too warm for water ice to have survived. In sum, we found activity unlikely to be sublimation-driven because (a) Gault is from a desiccated asteroid family (Phocaea), and (b) we found activity was unrelated to heliocentric distance. We proposed Gault may represent a \textbf{new type of active object}: recurrently active due to rotational spin-up.
Following the aforementioned expansion of the \ac{HARVEST} pipeline to work with all publicly available \ac{DECam} data we set out to examine large collections of thumbnails, on the order of 10,000 or more. The purpose of this exercise was to identify potential problems that would manifest in the thumbnails, such as handling chip gaps (Section \ref{methods:sec:methods:pipeline}) and enhancing contrast (Section \ref{methods:subsec:thumbnailExtraction}). While examining a thumbnail collection composed of Centaurs we noticed visible evidence of activity in images of Centaur 2014~OG$_{392}$, an object not yet known to be active. Our \textbf{discovery of an active Centaur} would become the foundation for our work which culminated in the publication of \textit{Cometary Activity Discovered on a Distant Centaur: A Nonaqueous Sublimation Mechanism} \citep{chandlerCometaryActivityDiscovered2020a}, Chapter \ref{chap:2014OG392} of this dissertation.
Our discovery represented a milestone for our project as our first active object discovery, but first we needed to confirm the activity was real and not an image artifact. To this end, we first acquired follow-up observations with the \ac{DECam} instrument on the Blanco 4~m telescope at \ac{CTIO} in Chile, the same instrument from which our archival images originated. Co-adding the four 250~s exposures revealed evidence of a coma. We next employed the \ac{IMACS} instrument on the 6.5~m Walter Baade Telescope atop the Las Campanas Observatory in Chile, and these images provided strong evidence of activity. Finally we made use of the \ac{LMI} on the 4.3~m Lowell Observatory \ac{DCT}, now called the \ac{LDT}. Here we were able to measure colors of 2014~OG$_{392}$ that revealed it is optically red.
With the images we acquired we were able to detect a coma composed of $\sim2.4\times10^{12}$~kg extending to at least a distance of 400,000~km from 2014~OG$_{392}$.
We introduced \textbf{a novel approach for estimating which molecular species are most likely responsible for observed activity}. We found carbon dioxide ice and/or ammonia ice the most likely to be sublimating, yet these two materials are optically grey. Thus an as yet unknown reddening agent must must be present to account for the red color of 2014~OG$_{392}$. Upon publication of our work, the \ac{MPC} gave this object the new designation of C/2014~OG392~(\acs{PANSTARRS}). (Note: although provisional names like 2014~OG$_{392}$ are retained as part of comet designation (except for short-period comets), the numeric portion at the end of the designation is no longer represented with subscript text.)
Just prior to launching our Citizen Science project, a telegram announced the discovery of a new active asteroid, (248370) 2005~QN$_{173}$ \citep{fitzsimmons2483702005QN1732021}. We carried out an archival investigation into (248370) 2005~QN$_{173}$ that ultimately led to our publication \textit{Recurrent Activity from Active Asteroid (248370) 2005~QN173: A Main-belt Comet} \citep{chandlerRecurrentActivityActive2021a}, dissertation Chapter \ref{chap:2005QN173}. We found a single \ac{DECam} image from 2016 July 22 (Figure \ref{QN:fig:wedgephot}) that unambiguously showed a tail emanating from (248370) 2005~QN173. Here \textbf{we introduced Wedge Photometry}, a tool that measures tail angle for (a) comparison with anti-Solar and anti-motion angles computed by \ac{JPL} Horizons (Figure \ref{intro:fig:activeObjects}), and (b) activity detection techniques. Our Wedge Photometry tool found the tail orientation to be $251.3\pm1.3^\circ$, in close agreement with the 251.6$^\circ$ and 251.7$^\circ$ orientations computed by the \ac{JPL} Horizons service.
The archival image showing activity we uncovered provided proof that the object had been active during at least one prior orbit. Thus \textbf{(248370) is recurrently active}, having undergone activity during at least two separate orbits. Moreover, we found (248370) was preferentially active near perihelion, only the 8$^\mathrm{th}$ such main-belt asteroid known to exhibit this behavior. Recurrent activity near perihelion is a strong diagnostic indicator of volatile sublimation as the underlying activity mechanism. This combination of recurrent sublimation-driven activity of a main-belt asteroid is evidence that \textbf{(248370) is most likely a member of the \acp{MBC}}. After we announced our discovery via electronic telegram \citep{chandler2483702005QN2021}, (248370) 2005~QN$_{173}$ was assigned an additional designation: comet 433P.
At this point in time (31 August 2022) we successfully launched our \ac{NSF} funded \ac{NASA} Partner Citizen Science project \textit{Active Asteroids} (Section \ref{methods:sec:citsci}) and we started to identify candidate active objects and make discoveries. For example, \textbf{volunteers overwhelming classified two \ac{DECam} images of 282P/(323137) 2003~BM$_{80}$ from 2021 as active}. Additionally, Citizen Scientists classified an image of 282P from 2013, then the only published activity epoch, as active. As described in \textit{Migratory Outbursting Quasi-Hilda Object 282P/(323137) 2003 BM80} (Chapter \ref{chap:282P}, paper submitted to Astrophysical Journal Letters), we carried out a multifaceted study of 282P, also designated 2003~FV$_{112}$, consisting of an archival investigation, telescope follow-up observations, dynamical modeling, and thermodynamical modeling. Notably, this work represents the first peer-reviewed publication to stem from our \textit{Active Asteroids} Citizen Science project.
Our archival investigation yielded additional images of the object that showed evidence of activity. The last images of 282P activity we identified were over a year prior to the Citizen Science project discovery, so we set out to conduct an observational campaign with the goals of (1) determining whether or not 282P was still active and, if so, (2) evaluating ongoing activity for changes in morphology (e.g., shape, extent). Unfortunately, 282P was transiting the Milky Way, which meant there were too many stars to effectively identify activity indicators. We were awarded a Gemini \ac{DDT} observation at Gemini South that we timed to take place during an 11 day window when 282P was passing in front of a less dense region of the Galaxy. The program was successfully executed, yielding 18 images of 282P, and we saw an unmistakable tail (Figure \ref{282P:fig:282P}). These images, in combination with our archival evidence of activity starting in March 2021, indicated that \textbf{282P had been active for at least 15 months}. Furthermore, we found 282P activity preferentially occurs near perihelion passage, typical of sublimation-driven activity, although additional study is needed to confirm this hypothesis.
Our orbital simulations (dynamical modeling) revealed that 282P only recently ($<200$ yr ago) arrived at its present orbit. \textbf{We determined that 282P is a member of the rare ($<$15) active Quasi-Hilda class} (e.g., \citealt{tothQuasiHildaSubgroupEcliptic2006,gil-huttonCometCandidatesQuasiHilda2016}), also referred to as \acfp{QHO}, \acfp{QHC} or \acfp{QHA}. While Hilda asteroids orbit with a 3:2 resonance with Jupiter, \acp{QHC} are only loosely bound to the same region, and not necessarily in 3:2 resonance with Jupiter, as discussed at length in Chapter \ref{chap:282P}. Active Quasi-Hildas like 282P are rare with fewer than 15 active Quasi-Hildas have been found to date. 282P experienced strong interactions with Jupiter and Saturn that dramatically altered its orbit. Of special note, our simulations revealed that 282P used to orbit primarily beyond Jupiter, but now it orbits predominitely interior to Jupiter's orbit. Moreover, 282P will undergo a Jovian interactions in roughly 300 years that will again substantially change its orbit. Giant planet perturbations are so strong, in fact, that dynamical chaos prior to 200 years ago and 300 years in the future prevent exact determination of 282P's origin and future orbital regime. However, when evaluating the possible outcomes of the forward and backward simulations, we identified \acp{JFC} and Centaurs as potential origins of 282P, and that 282P will likely become a \ac{JFC} in the future, thought there is also a chance it may become an active asteroid. Thus \textbf{282P reveals a potential pathway that informs us about the origins of some active asteroids.}
Although outside the scope of this dissertation, we mention here that we are actively working on additional discoveries stemming from the \textit{Active Asteroids} project. These include newfound active asteroids, active Centaurs, \acp{QHC}, \acp{JFC}, and companions to objects such as \acp{TNO}. Anyone interested can partake in this exciting scientific journey by participating in \textit{Active Asteroids}\footnote{\url{http://activeasteroids.net}}.
\chapter*{Preface}
The included manuscript chapters have been written to appear in peer-reviewed scientific journals. Redundancy in text is a result of reformatting to conform to University formatting requirements. References have been consolidated into a unified bibliography that appears at the end of this dissertation.
The following chapters have already been published:
Chapter \ref{chap:SAFARI} -- Manuscript I: ``Searching Asteroids for Activity Revealing Indicators (SAFARI)'' \citep{chandlerSAFARISearchingAsteroids2018}
Chapter \ref{chap:Gault} -- Manuscript II: ``Six Years of Sustained Activity from Active Asteroid (6478)~Gault'' \citep{chandlerSixYearsSustained2019}
Chapter \ref{chap:2014OG392} -- Manuscript III: ``Cometary Activity Discovered on a Distant Centaur: A Nonaqueous Sublimation Mechanism'' \citep{chandlerCometaryActivityDiscovered2020a}
Chapter \ref{chap:2005QN173} -- Manuscript IV: ``Recurrent Activity from Active Asteroid (248370) 2005~QN173: A Main-belt Comet'' \citep{chandlerRecurrentActivityActive2021a}
The remaining chapters have been submitted to the American Astronomical Society journals for publication as soon as possible:
Chapter \ref{chap:282P} -- Manuscript V: ``Migratory Outbursting Quasi-Hilda Object 282P/(323137) 2003 BM80''
\chapter{Searching Asteroids for Activity Revealing Indicators (SAFARI)}
\label{chap:SAFARI}
\acresetall
Colin Orion Chandler\footnote{\label{safari:nau}Department of Physics \& Astronomy, Northern Arizona University, PO Box 6010, Flagstaff, AZ 86011, USA}, Anthony M. Curtis\footnote{Department of Physics, University of South Florida ISA 2019, Tampa, FL 33620, USA}, Michael Mommert$^\mathrm{\ref{safari:nau}}$, Scott S. Sheppard\footnote{Department of Terrestrial Magnetism, Carnegie Institution for Science, 5241 Broad Branch Road. NW, Washington, DC 20015, USA}, Chadwick A. Trujillo$^\mathrm{\ref{safari:nau}}$
\textit{This is the Accepted Manuscript version of an article accepted for publication in Publications of the Astronomical Society of the Pacific. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at} \url{https://iopscience.iop.org/article/10.1088/1538-3873/aad03d}\textit{.}
\doublespacing
\section*{Abstract}
\label{safari:Abstract}
Active asteroids behave dynamically like asteroids but display comet-like comae. These objects are poorly understood, with only about 30 identified to date. We have conducted one of the deepest systematic searches for asteroid activity by making use of deep images from the Dark Energy Camera (DECam) ideally suited to the task. We looked for activity indicators amongst 11,703{} unique asteroids extracted from 35,640{} images. We detected three previously-identified active asteroids ((62412), (1) Ceres and (779) Nina), though only (62412) showed signs of activity. Our activity occurrence rate of 1 in 11,703{} is consistent with the prevailing 1 in 10,000 activity occurrence rate estimate. Our proof of concept demonstrates 1) our novel informatics approach can locate active asteroids and 2) DECam data are well-suited to the search for active asteroids.
\textit{Keywords:} minor planets, asteroids: general -- methods: analytical -- techniques -- image processing
\section{Introduction}
\label{safari:sec:introduction}
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{safariFiles/P2013R3}
\caption{Active asteroid P2013/R3 was imaged in October 2013 while undergoing a breakup (into components A-D) likely caused by rotational instability. The antisolar and negative heliocentric velocity vector arrows are labeled $\odot$ and $-V$, respectively. Reprinted Figure 2 of \cite{jewittAnatomyAsteroidBreakup2017}.}
\label{safari:fig:ExampleAsteroid}
\end{figure}
\begin{table*}
\footnotesize
\caption{The Active Asteroids (1 of 2)}
\label{safari:Table:TheAAs}
\begin{tabular}{lccrrclcrrclc}
Asteroid Name & $a$ & $e$ & $i$ & Orbit & $T_\mathrm{J}$ &$P$ & $q$ & $R$ & $f$ & $\mathrm{\%}_\mathrm{peri}$ & Act. & Cause\\
& (au) & & ($^\circ$) & & & (years) & (au) & (au) & ($^\circ$) & (\%) & (N) & \\
\rlabel{Ceres} (1) Ceres & 2.77 & 0.08 & 10.6 & MB & 3.310 & 4.6 & 2.60 & 2.72 & 279.3 & 62 & 3+ & \sublimate{}\ , $\bigwedge$ \\
\rlabel{Adeona} (145) Adeona & 2.67 & 0.14 & 12.6 & MB & 3.331 & 2.28 & 2.29 & 2.69 & 258.8 & 47 & 2 & \sublimate \\
\rlabel{Constantia} (315) Constantia & 2.24 & 0.17 & 2.4 & MB & 3.614 & 3.36 & 1.86 & 1.94 & 315.9 & 92 & 0$^\dagger$ & (?) \\
\rlabel{Griseldis} (493) Griseldis & 3.12 & 0.18 & 15.2 & OMB & 3.140 & 5.5 & 2.57 & 3.34 & 122.4 & 31 & 1 & $\leftrightsquigarrow$ \\
\rlabel{Scheila} (596) Scheila & 2.93 & 0.16 & 14.7 & OMB & 3.209 & 5.01 & 2.45 & 3.11 & 239.2 & 90 & 1 & $\leftrightsquigarrow$ \\
\rlabel{Interamnia} (704) Interamnia & 3.06 & 0.15 & 17.3 & MB & 3.148 & 5.35 & 2.59 & 2.62 & 19.6 & 97 & 1 & \sublimate \\
\rlabel{Nina} (779) Nina & 2.66 & 0.23 & 14.6 & MB & 3.302 & 4.35 & 2.06 & 2.15 & 36.9 & 93 & 1 & \sublimate \\
\rlabel{Ingrid} (1026) Ingrid & 2.25 & 0.18 & 5.4 & MB & 3.597 & 3.38 & 1.85 & 2.23 & 97.5 & 16 & 0$^\dagger$ & (?) \\
\rlabel{Beira} (1474) Beira & 2.73 & 0.49 & 26.7 & Mars & 3.033 & 4.52 & 1.39 & 1.57 & 310.9 & 93 & 1 & \sublimate \\
\rlabel{Oljato} (2201) Oljato & 2.17 & 0.71 & 2.5 & Apollo & 3.298 & 3.21 & 0.62 & 0.88 & 73.1 & 92 & 1 & (?) \\
\rlabel{Phaethon} (3200) Phaethon & 1.27 & 0.89 & 22.2 & Apollo & 4.510 & 1.43 & 0.14 & 0.14 & 5.1 & 87 & 3 & $\bigodot$ \\
\rlabel{DonQuixote} (3552) Don Quixote & 4.26 & 0.71 & 31.1 & Amor & 2.315 & 8.78 & 1.24 & 1.23 & 343.6 & 100 & 2 & \sublimate,(?) \\
\rlabel{Aduatiques} (3646) Aduatiques & 2.75 & 0.11 & 0.6 & MB & 3.336 & 4.57 & 2.46 & 2.56 & 309.0 & 90 & 0$^\dagger$ & (?) \\
\rlabel{WilHar} (4015) Wil.-Har. & 2.63 & 0.63 & 2.8 & Apollo & 3.082 & 4.26 & 0.97 & 1.17 & 51.0 & 95 & 2$^\ddagger$ & \sublimate{}\ , (?) \\
\rlabel{24684} (24684) 1990 EU4 & 2.32 & 0.08 & 3.9 & MB & 3.572 & 3.53 & 2.13 & 2.28 & 277.9 & 77 & 0$^\dagger$ & (?) \\
\rlabel{35101} (35101) 1991 PL16 & 2.60 & 0.18 & 12.3 & MB & 3.365 & 4.17 & 2.12 & 2.86 & 227.0 & 21 & 0$^\dagger$ & (?) \\
\rlabel{62412} (62412) & 3.15 & 0.08 & 4.7 & OMB & 3.197 & 5.6 & 2.90 & 3.06 & 74.5 & 68 & 1 & $\circlearrowleft$ \\
\rlabel{Ryugu} (162173) Ryugu & 1.19 & 0.19 & 5.9 & Apollo & 5.308 & 1.3 & 0.96 & 1.08 & 288.4 & 8 & 1 & \sublimate \\
\rlabel{GO98} (457175) & 3.96 & 0.28 & 15.6 & OMB & 2.926 & 7.89 & 2.85 & 3.28 & 66.0 & 81 & 1 & (?) \\
\rlabel{ElstPizarro} 133P/Elst--Pizarro & 3.16 & 0.16 & 1.4 & OMB & 3.184 & 5.63 & 2.66 & 2.65 & 21.7 & 100 & 4 & \sublimate \\
\rlabel{176P} 176P/LINEAR & 3.20 & 0.19 & 0.2 & OMB & 3.166 & 5.71 & 2.58 & 2.59 & 10.1 & 1 & 1 & (?) \\
\rlabel{233P} 233P/La Sagra & 3.04 & 0.41 & 11.3 & Encke & 3.081 & 5.28 & 1.78 & 2.01 & 309.1 & 91 & 1 & (?) \\
\rlabel{238P} 238P/Read & 3.16 & 0.25 & 1.3 & OMB & 3.154 & 5.64 & 2.37 & 2.42 & 26.5 & 97 & 3 & \sublimate \\
\rlabel{259P} 259P/Garradd & 2.73 & 0.34 & 15.9 & MMB & 3.217 & 4.51 & 1.81 & 1.85 & 27.6 & 99 & 2 & \sublimate \\
\rlabel{288P} 288P (300163) & 3.05 & 0.20 & 3.2 & OMB & 3.204 & 5.32 & 2.44 & 2.45 & 12.2 & 99 & 2 & \sublimate \\
\rlabel{311P} 311P/PS & 2.19 & 0.12 & 5.0 & IMB & 3.661 & 3.24 & 1.94 & 2.15 & 272.8 & 58 & 2 & $\circlearrowleft$\ , \large{\textbf{:}} \\
\rlabel{313P} 313P/Gibbs & 3.16 & 0.24 & 11.0 & Encke & 3.132 & 5.62 & 2.42 & 2.40 & 8.0 & 100 & 2 & \sublimate \\
\rlabel{324P} 324P/La Sagra & 3.10 & 0.15 & 21.4 & OMB & 3.100 & 5.45 & 2.62 & 2.64 & 20.0 & 98 & 2 & \sublimate \\
\rlabel{331P} 331P/Gibbs & 3.00 & 0.04 & 9.7 & OMB & 3.229 & 5.21 & 2.88 & 3.10 & 140.4 & 11 & 2 & $\leftrightsquigarrow$, $\circlearrowleft$ \\
\rlabel{348P} 348P/PS & 3.17 & 0.30 & 17.6 & OMB & 3.062 & 5.63 & 2.18 & 2.51 & 60.8 & 83 & 1 & (?) \\
\rlabel{354P} 354P/LINEAR & 2.29 & 0.12 & 5.3 & OMB & 3.583 & 3.47 & 2.00 & 2.01 & 12.2 & 99 & 1 & $\circlearrowleft$,$\circledast$ \\
\rlabel{358P} 358P & 3.15 & 0.24 & 11.1 & Encke & 3.135 & 5.59 & 2.39 & 2.42 & 7.5 & 99 & 2 & \sublimate{}\ , (?) \\
\rlabel{P2013R3} P/2013 R3 & 3.03 & 0.27 & 0.9 & OMB & 3.184 & 5.28 & 2.20 & 2.22 & 14.0 & 99 & 1 & $\circlearrowleft$, \sublimate{} \\
\rlabel{P2015X6} P/2015 X6 & 2.75 & 0.17 & 4.6 & MMB & 3.318 & 4.57 & 2.28 & 2.64 & 274.5 & 62 & 1 & $\circlearrowleft$ \\
\rlabel{P2016G1} P/2016 G1 & 2.58 & 0.21 & 11.0 & MMB & 3.367 & 4.15 & 2.04 & 2.52 & 264.7 & 56 & 1 & $\leftrightsquigarrow$ \\
\rlabel{P2016J1} P/2016 J1 & 3.17 & 0.23 & 14.3 & OMB & 3.113 & 5.65 & 2.45 & 2.46 & 345.9 & 99 & 1 & $\circlearrowleft$, \sublimate{} \\
\end{tabular}\\
\\
Orbital parameters retrieved from the Minor Planet Center and \ac{JPL} Horizons. \\
$a$: semimajor axis; %
$e$: eccentricity; %
$i$: inclination; %
Orbit: Inner, Mid, Outer, \& Main Belt (IMB, MMB, OMB, MB); %
$T_\mathrm{J}$:Tisserand parameter with respect to Jupiter; %
$P$:Orbital Period; %
$q$:Perihelion distance; %
$R$: Heliocentric discovery distance.
$f$: True anomaly. %
$\mathrm{\%}_\mathrm{peri}$:Percentage towards perihelion. %
Act.: Number of times object reported active.
$^\dagger$Authors declare object a candidate (activity not yet confirmed). %
$^\ddagger$\cite{ferrin2009ApparitionMethuselah2012} argue (4015) was also active in 1992, 1996, 2008, and 2009-2010. %
\sublimate{}\hspace{0.5mm}Sublimation; %
$\circlearrowleft$\hspace{0.25mm}Rotational Breakup; %
$\leftrightsquigarrow$\hspace{0.25mm}Impact; %
$\bigwedge$\hspace{0.25mm}Cryovolcanism; %
\protect{\large{\textbf{:}}}\hspace{0.25mm}Binary; %
$\bigodot$\hspace{0.25mm}Thermal Fracture; %
$\circledast$\hspace{0.25mm}Dust Model; %
(?)\hspace{0.25mm}Unknown %
\end{table*}
\clearpage
\setcounter{table}{0}
\begin{table*}
\footnotesize
\centering
\caption{The Active Asteroids (2 of 2) \label{safari:Table:TheAAs2}}
\begin{tabular}{llclcccc}
Asteroid Name & Family & $1^\mathrm{st}$Act & Facility & Method & Last & Visit & Refs\\
& & (years) & & & (years) & &\\
(1) Ceres & None & 2014 & Herschel & Spec. & 2017 & Yes & [\ref{Ceres}]\\
(145) Adeona & Adeona & 2017 & Terksol & Spec. & 2016 & No & [\ref{Adeona}]\\
(315) Constantia & Flora & 2013 & MPCAT & Phot. & 2013 & No & [\ref{Constantia}]\\
(493) Griseldis & Eunomia & 2015 & Subaru & Visual & 2015 & No & [\ref{Griseldis}]\\
(596) Scheila & None & 2010 & CSS & Visual & 2010 & No & [\ref{Scheila}]\\
(704) Interamnia & None & 2017 & Terksol & Spec. & 2012 & No & [\ref{Interamnia}]\\
(779) Nina & $\cdots$ & 2017 & Terksol & Spec. & 2012 & No & [\ref{Nina}]\\
(1026) Ingrid & Flora & 2013 & MPCAT & Phot. & 2013 & No & [\ref{Ingrid}]\\
(1474) Beira & $\cdots$ & 2017 & Terksol & Spec. & 2012 & No & [\ref{Beira}]\\
(2201) Oljato & $\cdots$ & 1984 & Pioneer & Mag. & 1984 & No & [\ref{Oljato}]\\
(3200) Phaethon & Pallas & 2009 & STEREO & Visual & 2017 & Yes & [\ref{Phaethon}]\\
(3552) Don Quixote & $\cdots$ & 2009 & Spitzer & Visual & 2018 & No & [\ref{DonQuixote}]\\
(3646) Aduatiques & $\cdots$ & 2013 & MPCAT & Phot. & 2013 & No & [\ref{Aduatiques}]\\
(4015) Wil.-Har. & $\cdots$ & 1949 & Palomar & Visual & 1979$^\ddagger$ & No & [\ref{WilHar}]\\
(24684) 1990 EU4 & $\cdots$ & 2013 & MPCAT & Phot. & 2013 & No & [\ref{24684}]\\
(35101) 1991 PL16 & Eunomia & 2013 & MPCAT & Phot. & 2013 & No & [\ref{35101}]\\
(62412) & Hygiea & 2015 & DECam & Visual & 2014 & No & [\ref{62412}]\\
(162173) Ryugu & Clarissa & 2017 & MMT & Spec. & 2017 & Yes & [\ref{Ryugu}]\\
(457175) & Hilda & 2017 & CSS & Visual & 2017 & No & [\ref{GO98}]\\
133P/Elst--Pizarro & Themis & 1996 & ESO & Visual & 2014 & No & [\ref{ElstPizarro}]\\
176P/LINEAR & Themis & 2009 & HTP & Visual & 2011 & No & [\ref{176P}]\\
233P/La Sagra & $\cdots$ & 2009 & LSSS & Visual & 2009 & No & [\ref{233P}]\\
238P/Read & Gorchakov & 2005 & SW & Visual & 2016 & No & [\ref{238P}]\\
259P/Garradd & $\cdots$ & 2008 & SS & Visual & 2017 & No & [\ref{259P}]\\
288P (300163) & Themis & 2011 & PS & Visual & 2017 & No & [\ref{288P}]\\
311P/PS & Behrens & 2013 & PS & Visual & 2014 & No & [\ref{311P}]\\
313P/Gibbs & Lixiaohua & 2014 & CSS & Visual & 2015 & No & [\ref{313P}]\\
324P/La Sagra & Alauda & 2011 & LSSS & Visual & 2015 & No & [\ref{324P}]\\
331P/Gibbs & Gibbs & 2012 & CSS & Visual & 2014 & No & [\ref{331P}]\\
348P/PS & $\cdots$ & 2017 & PS & Visual & 2017 & No & [\ref{348P}]\\
354P/LINEAR & Baptistina & 2010 & LINEAR & Visual & 2017 & No & [\ref{354P}]\\
358P & Lixiaohua & 2012 & PS & Visual & 2017 & No & [\ref{358P}]\\
P/2013 R3 & Mandragora & 2013 & PS & Visual & 2013 & No & [\ref{P2013R3}]\\
P/2015 X6 & Aeolia & 2015 & PS & Visual & 2015 & No & [\ref{P2015X6}]\\
P/2016 G1 & Adeona & 2016 & PS & Visual & 2016 & No & [\ref{P2016G1}]\\
P/2016 J1 & Theobalda & 2016 & PS & Visual & 2016 & No & [\ref{P2016J1}]\\
\end{tabular}
$1^\mathrm{st}$Act: Year activity discovered. %
Facility: Facility originally reporting activity.
Last: As of January 2018 submission. %
Refs: Object-specific references in Appendix \ref{safari:ObjectReferences}. %
CSS:Catalina Sky Survey; %
ESO:European Space Observatory 1-metre Schmidt %
HTP:Hawaii Trails Project; %
LINEAR:LIncoln Near-Earth Asteroid pRogram; %
LSSS:La Sagra Sky Survey; %
MPCAT:Minor Planet Catalog %
PS:Pan-STARRS; %
SS:Siding Spring; %
SW:Spacewatch %
\end{table*}
\clearpage
\begin{table}
\caption{AA Mass-loss Mechanisms}
\centering
\begin{tabular}{lrr}
\hline\hline
Suspected Mechanism & \ N$^*$ & \%\\
\hline
Sublimation & 15 & 44\\
Rotational Breakup & 7 & 21\\
Impact\ /\ Collision & 4 & 12\\
Thermal Fracturing & 1 & 3\\
Cryovolcanism & 1 & 3\\
Binary Interaction & 1 & 3\\
Unknown & 5 & 15\\
\end{tabular}\\
\raggedright
$^*$ Objects with multiple mechanism are counted more than once; objects listed in Table \ref{safari:Table:TheAAs} as candidates were not included in this computation.
\label{safari:Table}
\end{table}
Active asteroids appear to have tails like comets (Figure \ref{safari:fig:ExampleAsteroid}) but follow orbits predominately within the main asteroid belt. Although the first active asteroid (Wilson--Harrington) was discovered in 1949 \citep{cunninghamPeriodicCometWilsonHarrington1950}, 27 of the 31 objects (87\%) were identified as active in the last decade (Table \ref{safari:Table:TheAAs}). Asteroid activity is thought to be caused by several different mechanisms, combinations of which are undoubtedly at work (e.g., an impact event exposing subsurface ice to sublimation). The number of times (i.e., orbits) an object has displayed activity (Table 1: Act.) is especially diagnostic of the mechanism (Table 1: Cause). A singular (non-recurring) event likely originates from an impact event, e.g., (596)~Scheila. Rotational breakup, as in P/2013~R3 of Figure \ref{safari:fig:ExampleAsteroid}, may be a one-time catastrophic event, or a potentially repeating event if, for example, only a small piece breaks free but the parent body remains near the spin breakup limit. Ongoing or recurrent activity has been observed $\sim$15 times, e.g., 133P/Elst--Pizarro, and is suggestive of sublimation or, in the case of (3200)~Phaethon, thermal fracture. These last two mechanism (sublimation and thermal fracture) should be more likely to occur when an object is closer to the Sun, i.e. perihelion (Table 1:$q$). The Sun-object distance (Table 1: $R$) indicates the absolute distance, but it is can be simpler to consider how close (Table 1: \%$_\mathrm{peri}$) to perihelion the object was when activity was first observed (Table 1: 1$^\mathrm{st}$Act), where 100\% represents perihelion ($q$) and 0\% indicates aphelion:
\begin{equation}
\mathrm{\%}_\mathrm{peri} = \left[1-\left(\frac{d_\mathrm{disc}-d_\mathrm{peri}}{d_\mathrm{ap}-d_\mathrm{peri}}\right)\right]\cdot 100\mathrm{\%}
\end{equation}
\noindent where $d_\mathrm{disc}$ is the heliocentric object distance at the activity discovery epoch, $d_\mathrm{peri}$ the perihelion distance, and $d_\mathrm{ap}$ the aphelion distance.
While the term ``\ac{MBC}'' often refers to this sublimation-driven subset of active asteroids, we use the more inclusive ``active asteroid'' term throughout this paper. We aimed to include all objects termed ``active asteroids'' in the literature for completeness, but we only include objects which have provided observable signs of activity. Objects known to host surface water ice but which have yet to shown signs of activity, such as (24) Themis \citep{rivkinDetectionIceOrganics2010, campinsWaterIceOrganics2010}, are outside the scope of this paper.
Orbital characteristics also provide insight into the dynamical evolution and even the composition of an object. Objects with conspicuously similar orbital properties may have originated from a catastrophic disruption event that created a family (Table 1:Family) of asteroids \citep{hirayamaGroupsAsteroidsProbably1918}. More generally, asteroids can be categorized (Table 1:Orb.) as interior to the Main Asteroid Belt, within the Main Asteroid Belt (and further subdivided into inner, mid, and outer main belt as IMB, MMB, and OMB respectively), or exterior to the Main Asteroid Belt (e.g., Kuiper belt). Objects interior to the Main Asteroid Belt, including Near Earth Objects (NEOs), include Earth-crossing (Apollo), Earth-orbit nearing (Amor), and Mars-crossing asteroids. Objects whose orbits are similar to Comet 2P/Encke are said to be Encke-type.
The Tisserand parameter $T_\mathrm{J}$ (Table \ref{safari:Table:TheAAs}$T_\mathrm{J}$) describes the degree to which an object's orbit is influenced by Jupiter:
\begin{equation}
T_\mathrm{J} = \frac{a_\mathrm{J}}{a} + 2\sqrt{\left(1-e^2\right)\frac{a}{a_\mathrm{J}}}\cos(i).
\label{safari:eq:TJ}
\end{equation}
\noindent The orbital elements are given by $a_\mathrm{J}$ the orbital distance of Jupiter (5.2 AU), plus the semi-major axis $a$, eccentricity $e$, $i$ the inclination (Table 1). For the case where $a=a_\mathrm{J}$ you can see $T_\mathrm{J}=3$. Asteroids in the main-belt are typically inside the orbit of Jupiter (i.e. $a<a_\mathrm{J}$) and usually have $T_\mathrm{J}>3$ \citep{jewittActiveAsteroids2014}; however, as Equation \ref{safari:eq:TJ} indicates, it is the combination of all three free parameters ($a$, $e$, $i$) which describes the magnitude of Jovian influence on the object's orbit. One active asteroid definition also constrains membership to objects whose orbits are interior to Jupiter but whose Tisserand parameters are $>$ 3.08 \citep{jewittActiveAsteroids2014}.
Objects not identified in the literature as active asteroids, yet still appear orbitally asteroidal (e.g., Comet 2P/Enke), are not included in this paper, but objects with $T_\mathrm{J}<3$ which are identified in the literature as active asteroids (e.g., (3552) Don Quixote), are included; see e.g., \cite{hsiehPopulationCometsMain2006,tancrediCriterionClassifyAsteroids2014} for further discussion on distinguishing objects within this regime.
We would like to understand active asteroids in part because they may hold clues about solar system formation and the origin of water delivered to the terrestrial planets. The recent discovery of interstellar asteroid \hbox to.666\okinalen{\hss`\hss}Oumuamua{} \citep{bacci2017U12017} intensifies interest in understanding our own indigenous asteroid population in order to better understand and characterize ejectoids we encounter in the future, an estimated decadal occurrence \citep{trillingImplicationsPlanetarySystem2017}. There has also long been an interest mining asteroids for their metals, and water could prove an invaluable resource providing, for example: energy, rocket fuel, breathable oxygen, and sustenance for plant and animal life \citep{olearyMiningApolloAmor1977,dicksonCongressApprovesSolar1978,kargelMetalliferousAsteroidsPotential1994,forganExtrasolarAsteroidMining2011,hasnainCapturingNearEarthAsteroids2012,lewickiPlanetaryResourcesAsteroid2013,andrewsDefiningSuccessfulCommercial2015}.
Our knowledge of active asteroids has been limited due to small sample size: only $\sim$20 active asteroids have been discovered to date \citep{jewittActiveAsteroids2015a}. As such, the statistics presented in Table \ref{safari:Table} are poorly constrained (e.g., the thermal fracturing rate is based upon a single object: (3200) Phaethon). Spacecraft visits have been carried out or planned to a number of the active asteroids (Table 1: Visit), and while we may learn a great deal from these individual objects, spacecraft visits will not substantially increase the number of known active asteroids. While spectroscopy has recently shown potential for discovering activity, the overwhelming majority of activity detections have been made by visual examination (Table 1:Method). One notable exception was the 1984 (2201) Oljato outburst first detected by magnetic field disturbances (2201) Oljato outburst \citep{russellInterplanetaryMagneticField1984}.
\begin{table}
\small
\centering
\caption{Surveys which have discovered AAs.}
\begin{tabular}{lccc}
AA Discovered by & AAs & Limit & Operation\\
\hspace{4mm}(survey name) & (N) & (mag) & (years)\\
\hline
Catalina Sky Survey & 5 & 22$^a$ & 1998$^f$ --\\
La Sagra Survey & 2 & 17$^b$ & 2008$^g$--\\
LINEAR & 1 & 19.6$^c$ & 1997$^h$ --\\
Pan-STARRS & 8 & 22.7$^d$ & 2008$^i$ --\\
Spacewatch & 2 & 21.7$^e$ & 1981$^j$ --\\
Total & 18 & $\cdots$ & 98\\
\hline
\end{tabular}
\raggedright
$^a$\cite{drakeFirstResultsCatalina2009};
$^b$estimated from aperture;
$^c$\cite{sesarExploringVariableSky2011,stokesLincolnNearEarthAsteroid2000};
$^d$\cite{chambersPanSTARRS1Surveys2016};
$^e$\cite{larsenSearchDistantObjects2007};
$^f$\cite{larsonCatalinaSkySurvey1998};
$^g$\cite{stossJ75SagraSky2011};
$^h$\cite{stokesLincolnNearEarthAsteroid2000};
$^i$\cite{jedickePanSTARRSFirstSolar2008};
$^j$\cite{gehrelsFaintCometSearching1981}
\label{safari:Surveys}
\end{table}
\begin{table*}
\centering
\caption{Active Asteroid Hunting Surveys \& Occurrence Rate Estimates}
\footnotesize
\begin{tabular}{llclclrl}
Survey & Source & Zone & Activity & $N^\dag$ & Limit & Objects & Method\\
& & & ($N$ per $10^6$) & (mag) & &\\
\hline
\cite{cikotaPhotometricSearchActive2014} & MPC & MBA & \hspace{6mm}$\cdots$ & 1 & 16.7 & 330K & Photometric Excess\\
\cite{gilbertUpdatedResultsSearch2010} & CFHT & MBA & \hspace{6mm}$40\pm 18$ & 3 & 22.5$^a$ & 25K & By-Eye\\
\cite{hsiehHawaiiTrailsProject2009} & HTP & OMB & \hspace{6mm}$\cdots$ & 1 & 26 & 600 & By-Eye\\
\cite{hsiehMainbeltCometsPanSTARRS12015} & \footnotesize{Pan-STARRS} & OMB & \hspace{6mm}$96$ & 4 & 22.6 & 300K & PSF\\
SAFARI (this work) & DECam & MBA & \hspace{6mm}$80$ & 1 & 24.3 & 11K & By-Eye\\
\cite{sonnettLimitsSizeOrbit2011} & TALCS & MBA & \hspace{2mm}$<2500$ & 0 & 24.3 & 1K & Excess Sky Flux\\
\cite{waszczakMainbeltCometsPalomar2013} & PTF & MBA & \hspace{2mm}$<30$ & 0 & 20.5 & 220K & Extendedness\\
\hline
\end{tabular}
\raggedright
CFTS: Canada-France-Hawaii Telescope; DECam: Dark Energy Camera; HTP: Hawaii Trails Project; MPC: Minor Planet Center; PTF: Palomar Transient Factory; Pan-STARRS: Panoramic-Survey Telescope And Rapid Response System; TALCS: Thousand Asteroid Light Curve Survey; MBA: Main Belt Asteroids; OMB: Outer Main Belt; $^\dag$Includes known AAs;
$^a$\cite{gilbertSearchingMainbeltComets2009}; PSF: Point Spread Function
\label{safari:aahunts}
\label{safari:ActivityRates
\end{table*}
We chose to visually examine (``by-eye'') images of active asteroids because this technique has so far produced the greatest yield. Other methods have been applied (Table \ref{safari:Surveys}) but with varied degrees of success. \cite{cikotaPhotometricSearchActive2014} examined a large number of objects and searched for unexpected deviations in object brightness; this technique positively identified one known active asteroid, but (so far) the other candidates (\ref{safari:Table:TheAAs}) have not been observed to be active. \cite{sonnettLimitsSizeOrbit2011} examined the regions immediately surrounding asteroids, searching for photometric excess (i.e., a photon count above the sky background level). \cite{waszczakMainbeltCometsPalomar2013} formulated a way to quantify ``extendedness'' of Palomar Transient Factory objects, with a 66\% comet detection rate and a 100\% Main Belt Comet detection efficiency. \cite{hsiehMainbeltCometsPanSTARRS12015} compared point spread function (PSF) widths between background stars and other objects and flagged exceptionally large PSF radii for further follow-up. All of the aforementioned techniques rely upon visual inspection for confirmation of activity. Spectroscopic detection of activity has also been carried out (Table \ref{safari:Table:TheAAs2}), but so far only (1) Ceres has been observed to be visually active in follow-up, and, in that case, in situ by the Rosetta spacecraft orbiting it. Hayabusa 2 recently arrived at (162173) Ryugu but as of yet no tail or coma has been observed
Conservative activity occurrence rates of $>$1 in 10,000 are constrained by the magnitude limits of prior surveys \citep{jewittActiveAsteroids2015a}. We reached past the 17-22.7 magnitude limits of previous large-sky surveys (Table \ref{safari:Surveys}) by making use of existing \ac{DECam} data \citep{sheppardNewExtremeTransNeptunian2016} probing a magnitude fainter than other large-sky active asteroid survey. Note that while we are sensitive to more distant populations (e.g., Centaurs, Trans-Neptunian Objects), 99.7\% of our population is from the main asteroid belt.
We set out to determine the viability of \ac{DECam} data for locating active asteroids. We aimed to create a novel, streamlined pipeline for locating known asteroids within our dataset. We planned to examine our new library of asteroid thumbnails to find active asteroids and to test published asteroid activity occurrence rates (Table \ref{safari:ActivityRates}). We applied our technique to 35,640{} \ac{DECam} images ($\sim$5 Tb) to produce 15,600{} thumbnail images comprising 11,703{} unique objects. We examined the asteroid thumbnails by-eye to identify signs of activity. We show our technique can be applied to an orders-of-magnitude larger publicly-available dataset to elevate active asteroids to a regime where they can be studied as a population.
\section{Methods}
\label{safari:methods}
\begin{figure}
\centering
\includegraphics[height=8in]{safariFiles/SAFARIflow.png}
\caption{The SAFARI workflow.}
\label{safari:flow}
\end{figure}
\subsection{Dark Energy Camera} We made use of data taken by the \ac{DECam} instrument on the 4-meter Blanco telescope at the Cerro Tololo Inter-American Observatory in Chile. The instrument has a $\sim$3 square degree field of view, capturing data via a mosaic of 62 \ac{CCD} chips, each $2048\times 4096$ with a pixel scale of 0\farcsec263/pixel \citep{darkenergysurveycollaborationDarkEnergySurvey2016}. Our data consisted of $594\times 2.2$ Gb frames in the VR filter ($500\pm10$ nm to $760\pm10$ nm), each containing $62\times 33$ Mb subsets of data, one per \ac{CCD}. The mean seeing across all images was 1\farcsec.14$\pm$ 0\farcsec13. We made use of software which required each multi-extension \ac{FITS} file be split into its 62 constituent parts, which we refer to as images for the remainder of this paper. Note: some files contained only 61 chips due to an instrument hardware malfunction.
\subsection{High Performance Computing}
We utilized \textit{Monsoon}, the Northern Arizona University (NAU) High Performance Computing (HPC) computing cluster. \textit{Monsoon} uses the \textit{Slurm Workload Manager} \citep{yooSLURMSimpleLinux2003} software suite to manage the 884 Intel Xeon processors to deliver up to 12 teraflops of computing power. The majority of our tasks each utilized 8 cores and 48 Gb of memory. The online supplement contains the complete listing of requirements necessary for each task.
\subsection{photometrypipeline} We utilized the \textit{photometrypipeline} \citep{mommertPHOTOMETRYPIPELINEAutomatedPipeline2017} software package to carry out source extraction via \textit{Source Extractor} \citep{bertinSExtractorSoftwareSource1996,bertinSExtractorSourceExtractor2010}, photometry and astrometry via \textit{SCAMP} \citep{bertinAutomaticAstrometricPhotometric2006,bertinSExtractorSourceExtractor2010}, and asteroid identification via \textit{SkyBot} \citep{berthierSkyBoTNewVO2006} and \textit{Horizons} \citep{giorginiStatusJPLHorizons2015}. We chose the \textit{Anaconda}\footnote{\url{www.anaconda.com}} \textit{Python} programming language distributions (versions 2.7 and 3.5) and the \textit{Python} package \textit{AstroPy} \citep{robitailleAstropyCommunityPython2013}.
\subsection{Procedure}
\label{safari:procedure}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.46\linewidth]{safariFiles/CDpp} & \includegraphics[width=0.46\linewidth]{safariFiles/CDir}\vspace{-2mm}\\
\vspace{1mm}
(\textbf{a}) & (\textbf{b})\\
\includegraphics[width=0.46\linewidth]{safariFiles/gif_2012_YU2_1} & \includegraphics[width=0.46\linewidth]{safariFiles/gif_2012_YU2_2}\vspace{-2mm}\\
(\textbf{c}) & (\textbf{d})\\
\end{tabular}
\caption{Two asteroid thumbnail contrast selection approaches are shown in \textbf{a} and \textbf{b}. \textbf{a}) The \textit{photometrypipeline} thumbnail shows increased dynamic range. \textbf{b}) Iterative Rejection sacrifices some dynamic range (notice especially the edges of the center galaxy and the spiral galaxy to its upper-right) in favor of recovering more objects, many of which are not visible in \textbf{a} that can be easily seen in \textbf{b}. \textbf{c} \& \textbf{d}) Asteroid 2012 YU2 is shown in two frames comprising one animated \ac{GIF} file.}
\label{safari:AnimatedGif}
\label{safari:ContrastComparison}
\label{safari:ExposureTimes}
\end{figure}
\begin{enumerate}[itemsep=2pt,leftmargin=10pt]
\item \textit{Image Reduction}-- We employed standard image reduction techniques where each frame was bias subtracted, then flat-fielded using a combination of twilight flats and a master flat; full details of our imaging techniques can be found in \cite{sheppardNewExtremeTransNeptunian2016}.
\item \textit{Splitting Multi-Extension FITS Files}-- \ac{DECam} produces multi-extension FITS files, where each extension contains data from one \ac{CCD}; because \textit{photometrypipeline} was incompatible with this format, we split each file into 62 separate \ac{FITS} files via the \textit{FTOOLS} \citep{blackburnFTOOLSFITSData1995} software package. We replicated global and extension headers for each output file to preserve metadata required for our image processing.
\item \textit{Coordinate Correction}-- Each DECam image came pre-encoded with right ascension (RA) and declination (Dec) information indicating the coordinates of the telescope pointing center. We shifted the RA \& Dec of each remaining \ac{CCD} to their true coordinate values. The RA \& Dec offsets used for each \ac{CCD} are provided with the online supplement.
\item \textit{World Coordinate System Purging}-- We discovered World Coordinate System (WCS) headers encoded in the \ac{FITS} files were preventing \textit{photometrypipeline} and/or \textit{astrometry.net} from performing astrometry. We were able to resolve the issue by purging all WCS header information as part of our optimization process. The header record names are listed in the online supplement.
\item \textit{WCS Population via astrometry.net}-- We installed the \textit{astrometry.net} \citep{langAstrometryNetBlind2010} v0.72 software suite on \textit{Monsoon}. We processed all 35,640{} \ac{FITS} files to retrieve coordinate information for each image by matching the image to one or more index files (catalogs of stars for specific regions of sky, designed for astrometric solving).
\item \textit{photometrypipeline Image Processing}-- We performed source extraction, photometry, astrometry and image correction via the \textit{photometrypipeline} software suite.
\item \textit{Identifying Known Asteroids} We identified known asteroid in our data by making use of \textit{pp\_distill}, a module of \textit{photometrypipeline}.
\item \textit{FITS Thumbnail Generation}-- We extracted the RA, Dec, and $(x,y)$ pixel coordinates of each object. We then produced $480\times480$ pixel, lossless, \ac{FITS} format asteroid thumbnails, each a small image centered on an asteroid. For cases where the object was too close ($<$240 pixels) to one or more image edges, we found it best to use the \textit{NumPy}\footnote{\url{www.numpy.org}} \textit{Python} routine to ``roll'' the image array; the technique shifts an array as if it were wrapped around a cylinder. For example: array [0, 1, 2, 3] rolled left by 1 would result in array [1, 2, 3, 0].
\item \textit{Create PNG Thumbnails}-- We used an iterative-rejection technique to compute contrast parameters, then produced \ac{PNG} image files via \textit{MatPlotLib}\footnote{\url{www.matplotlib.org}}.
\item \textit{Animated GIF Creation} We combined thumbnails of asteroids observed more than once (Figure \ref{safari:DataStats}c) to create animated \ac{GIF} files (Figure \ref{safari:AnimatedGif}) using the \textit{Python Image Library}\footnote{\url{www.pythonware.com/products/pil/}} software package. There are a number of advantages to this inspection approach, including 1) the opportunity to inspect one asteroid at multiple epochs, 2) activity may not occur at every epoch, and 3) activity may be easier to spot if the inspector has the opportunity to become familiar with an object (e.g., the general shape or streak pattern), even if only briefly.
\item \textit{Examination of Image Products} -- Three authors served as asteroid thumbnail inspectors. Each inspector conducted a procedure consisting of rapid by-eye examination of asteroid thumbnails and animated \acp{GIF}, covering each thumbnail at least once. We flagged thumbnails and animations containing potential active asteroids for a later en masse review.
\end{enumerate}
\section{Results}
\label{safari:results}
\begin{figure*}
\centering
\begin{tabular}{cc}
\begin{tabular}{cc}
\includegraphics[width=0.2\linewidth]{safariFiles/Figure2} & \includegraphics[width=0.2\linewidth]{safariFiles/Figure3}\vspace{-2mm}\\
(\textbf{a}) & (\textbf{b})\vspace{1mm}\\
\includegraphics[width=0.19\linewidth]{safariFiles/Figure4} & \includegraphics[width=0.21\linewidth]{safariFiles/MagsCum1}\vspace{-2mm}\\%was Figure1 for b prior to 12/12/17 COC
(\textbf{c}) & (\textbf{d})\\
\end{tabular} & \hspace{-10mm}
\begin{tabular}{c}
\includegraphics[width=0.5\linewidth]{safariFiles/Figure6.pdf}\vspace{-2mm}\\
(\textbf{e})\\
\end{tabular}
\end{tabular}
\caption{\textbf{a}) Exposure time distribution in our data. \textbf{b}) Histogram of apparent magnitudes for known asteroids we identified in our dataset. \textbf{c}) Observations per object; the 15,600{} asteroid thumbnails contained 11,703{} unique objects, 3029{} of which were observed more than once. \textbf{d}) Cumulative histogram showing the depth of magnitudes (stars and asteroids) found in our dataset. 50\% of our images reached a magnitude of $m_\mathrm{R}=23.7$. Sources with a signal-to-noise ratio of $<$5:1 were not included. \textbf{e}) Asteroids encountered shown in geocentric ecliptic space, where $\lambda$ and $\beta$ are the ecliptic longitude and latitude, respectively. Distinct patches sum to $\sim$1000 $\mathrm{deg}^2$, as described in the text. Milky Way coordinates were retrieved from the \textit{D3-Celestial} (\url{http://ofrohn.github.io}) software suite.}
\label{safari:DataStats}
\end{figure*}
\textit{Pipeline} -- We created a pipeline (Figure \ref{safari:flow}) that takes as its input DECam multi-extension \ac{FITS} files, and returns individual asteroid thumbnails and animated \ac{GIF} files. The initial total compute time requested across all tasks was 13,000 hours (1.5 compute-years), but after optimization (see Optimization section below) only $\sim$500 compute hours were required. See the online supplement for a comprehensive table of resources utilized during this project.
\textit{Image Products} We extracted 15,600{} asteroid thumbnails from 35,640{} DECam images ($\sim$2 Tb total). Most of our data consisted of exposure times $>$300s (Figure \ref{safari:DataStats}a). These longer integration times allowed us to probe deeper (fainter), with asteroids captured down to $25^\mathrm{th}$ magnitude (Figure \ref{safari:DataStats}b). Each of the 11,703{} unique objects identified in our data were observed between 1 and 5 times, with 3029{} objects imaged more than once (Figure \ref{safari:DataStats}c).
To compute our coverage area on sky (depicted in Figure \ref{safari:DataStats}e) we employed a nearest neighbor algorithm to identify the distinct (non-overlapping) regions of our dataset. Two fields were considered overlapping if their center-to-center distance was $<1.8$ degrees, the width of one DECam field. We computed our coverage to be $\sim200$ distinct 3 $\mathrm{deg}^2$ patches comprising $\sim$1000 square degrees.
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{safariFiles/62412HsvMitchell.png}
\caption{Asteroid (62412) shown with the ``hsv'' colormap and Mitchell interpolation. The asteroid is at the center of the frame and the tail can be seen between the dashed lines.}
\label{safari:fig:62412}
\end{figure}
\textit{Active Asteroids} -- We imaged one asteroid previously discovered to be active \citep{sheppardDiscoveryCharacteristicsRapidly2015}: (62412). The object shows activity in our image (Figure \ref{safari:fig:62412}; see the online supplement for additional image color map and interpolation permutations) and we were able to identify activity in two other DECam frames that were not part of this work. \cite{sheppardDiscoveryCharacteristicsRapidly2015} confirmed activity with Magellan Telescope follow-up observations. We also imaged two other objects listed as active: (1) Ceres and (779) Nina but neither showed signs of activity.
\textit{Optimization} -- The final pipeline resulted from a series of iterative optimizations carried out with a subset of our large dataset. These optimizations produced order-of-magnitude reductions in compute time, and improved successful pipeline completion from the initial $\sim$35\% to the final 94\%. The implemented optimizations and their results are broken down below by number (matched to the corresponding procedure number of Section \ref{safari:procedure}). The final optimized \textit{Slurm} parameters used on \textit{Monsoon} can be found in the online supplement
\begin{enumerate}[itemsep=2pt,leftmargin=10pt]
\item \textit{Image Reduction}: No optimization needed.
\item \textit{File-Splitting}: Splitting each multi-extension \ac{FITS} file into 62 separate \ac{FITS} files resulted in a larger number of smaller tasks which were better suited for parallel processing.
\item \textit{Coordinate Correction}: Coordinate corrections proved cumbersome and inefficient, so we added \textit{astrometry.net} to our pipeline.
\item \textit{WCS Purging}: We identified mismatched distortion coefficients as the primary culprit behind roughly 1/3 of our images failing \textit{photometrypipeline} analysis. We purged all World Coordinate System (WCS) headers, allowing us to employ \textit{astrometry.net} which increased our overall throughput and output.
\item \textit{astrometry.net Astrometry}: We cached all ($\sim$32 Gb) astrometry index files (described in Section \ref{safari:procedure} item 5) locally so \textit{astrometry.net} would not be dependent on the speed of the internet connection and file host. We optimized the \textit{astrometry.net} computation by supplying the following parameters we extracted from our \ac{FITS} files. Providing a pixel scale range ($\sim$0\farcsec25/pixel to $\sim$0\farcsec28/pixel) and R.A./decl. values narrowed the range of indices that required searching. We found a 15$''$ search radius further reduced computation time without impacting image recognition efficacy. We disabled \textit{astrometry.net} plotting due to a \textit{Slurm} incompatibility, and computation time decreased further still. We found submitting \textit{astrometry.net} ``solve-field'' tasks directly to \textit{Slurm} was much faster. All but 41 images successfully matched for astrometry on first pass, and we improved \textit{astrometry.net} image recognition speed roughly tenfold.
\item \textit{photometrypipeline}: Proper configuration of prerequisite software and \textit{photometrypipeline} proved crucial; the online supplement contains the necessary parameters we used. We made minor modifications to the \textit{photometrypipeline} code, described in the online supplement. We found out \textit{astropy} was using home directory temporary storage space, a fatal error for systems with enforced quotas; the home storage space was also slower than the scratch space. Proper configuration reduced computation time and increased the pipeline success rate.
\item \textit{Known Asteroid Identification} We added an initial \textit{SkyBot} query to identify the asteroids within each image. We then populated the requisite \texttt{OBJECT} \ac{FITS} header keyword in each of our images, thereby enabling us to call \textit{Horizons} to locate asteroids in our images and provide accurate astrometry. Prepending the \textit{SkyBot} query and populating the \texttt{OBJECT} keyword enabled us to run asteroid identification tasks in parallel, reducing processing time by three orders-of-magnitude.
\item \textit{FITS Thumbnails}: We ``rolled'' images (described in Section \ref{safari:procedure} item 8) so we could create full-sized ($480\times480$ pixel) thumbnails. While thumbnails sometimes looked peculiar when rolled, this method preserved image statistics used to compute the narrow range of contrast achieved in the next section.
\item \textit{PNG Thumbnails}: While \textit{photometrypipeline} does output thumbnails by default, we were unable to see enough detail with the default scaling. Therefore, we employed an iterative rejection technique. Figures \ref{safari:ContrastComparison} a and \ref{safari:ContrastComparison} b compare the two contrast ranges. For each of the 15,600{} asteroid thumbnails, we chose to output different colormap/interpolation combinations: two modes of interpolation (Mitchell--Netravali balanced cubic spline filter and one set unfiltered), each in 11 color schemes (afmhot, binary, bone, gist\_stern, gist\_yarg, gray, hot, hsv, inferno, Purples, and viridis), examples of which are shown in the online supplement.
The optimized dynamic ranges allowed faint trails to become more visible. These colormap/interpolation schemes gave us, as thumbnail inspectors, the ability to choose a comfortable theme for use while searching thumbnails for asteroid activity, thereby increasing our productivity.
\item \textit{Animated GIFs}: We produced animated \acp{GIF} enabling an alternative inspection format.
\item \textit{Examination}: We uncovered common sources of false positives (discussed in Section \ref{safari:FalsePositives}) and incorporated their presence into our visual examination procedures, resulting in a streamlined examination process while simultaneously reduced the number of false-positives.
\end{enumerate}
\section{Discussion}
\label{safari:discussion}
We set out to determine if DECam data would provide a suitable pool from which to search for active asteroids. We crafted a method to extract asteroid thumbnails from DECam data, and the large number of asteroids encountered (11,703{}) along with the exceptional depth our images probed (Figures \ref{safari:DataStats}b and \ref{safari:DataStats}d) indicate our data are well-suited to locating active asteroids.
\subsection{Population Traits}
\begin{table}
\centering
\small
\caption{SAFARI Asteroid Populations}
\begin{tabular}{lclccrr}
\hline\hline
Zone & $R_i$ & $a_{p_i}$ & $a_{p_o}$ & $R_o$ & \multicolumn2c{SAFARI}\\
& \footnotesize{(A:J)} & \footnotesize{(au)} & \footnotesize{(au)} & \footnotesize{(A:J)} & (N) & (\%)\ \\
\hline
Int. & $\cdots$ & 0 & 2.064 & 4:3 & 115 & 1\\
IMB & 4:3 & 2.064 & 2.501 & 3:1 & 3,605 & 26\\
MMB & 3:1 & 2.501 & 2.824 & 5:2 & 5,358 & 39\\
OMB & 5:2 & 2.824 & 3.277 & 2:1 & 4,599 & 33\\
Ext. & 2:1 & 3.277 & $\infty$ & $\cdots$ & 162 & 1\\
Total$^*$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & 13,839 & 100\\
\hline
\end{tabular}
\label{safari:tab:zones}
\raggedright
\hspace{1.7in}\footnotesize{Int., Ext.,: Interior, Exterior to the main belt}\\
\hspace{1.7in}\footnotesize{IMB, MMB, OMB: Inner, Mid, Outer Main Belt}\\
\hspace{1.7in}\footnotesize{$a_{p_i}$,$a_{p_o}$: inner, outer proper semi-major axis}\\
\hspace{1.7in}\footnotesize{A:J Asteroid:Jupiter; $R_i$, $R_o$: inner/outer resonances}\\
\hspace{1.7in}\footnotesize{$^*$Not included: 791 objects with unknown parameters.}
\end{table}
As indicated by Figures \ref{safari:DataStats}a-d, the population imaged during our survey were subject to selection effects caused by the depth ($\bar{m}_\mathrm{R}=23.7$) of our survey (e.g., closer objects would have appeared as long trails which would have been difficult to identify with our pipeline). We classified the objects following the procedure of \cite{hsiehAsteroidFamilyAssociations2018}; we categorized our population as Inner Main Belt (IMB), Mid Main Belt (MMB), and Outer Main Belt (OMB), plus two additional regions: ``Interior'' (to the IMB) and ``Exterior'' (to the OMB). Table \ref{safari:tab:zones} indicates the boundaries, along with their Asteroid:Jupiter (A:J) resonances.
The synthetic proper semi-major axis $a_p$ aims to minimize the influence of transient perturbations \citep{knezevicSyntheticProperElements2000}. We made use the \textit{AstDyn-2}\footnote{\url{http://hamilton.dm.unipi.it/astdys}} online catalog service \citep{knezevicProperElementCatalogs2003} in determining proper orbital parameters for asteroids in our dataset (Table \ref{safari:tab:zones}).
Our target (object) aperture photometry was computed with a fixed diameter of 10 pixels, though photometric calibration was performed with an aperture radius determined by curve-of-growth analysis (see \citet{mommertPHOTOMETRYPIPELINEAutomatedPipeline2017} for details). To determine the surface brightness limit of our catalog we first computed the limit $SB$ of each image
\begin{equation}
SB_\mathrm{lim} = \frac{\sum_{k=1}^{k=N} \left(m_{0_k} - 2.5 \log_{10}\left(n \sigma_{\mathrm{bg}_k}\sqrt{1/A}\right)\right)}{N},
\end{equation}
\noindent where $m_0$ is the photometric zero point (determined by \textit{PhotometryPipeline}), $n$ the order of detection level for background noise standard deviation $\sigma_\mathrm{bg}$, and $A$ is the area of one pixel in square arcseconds (\citealt{hsiehSurfaceBrightnessLimits2018}, personal communication). The DECam camera had a pixel scale of 0\farcsec263/pixel, give a pixel area
\begin{equation}
A = (0\farcsec 263)^2 = 0.069169\ \mathrm{arcseconds}^2.
\end{equation}
For our surface brightness analysis we made use of $N=32,790$ chips for which we had been able to determine a photometric zero point. We computed the $3\sigma$ mean surface brightness limit of our dataset to be ${SB}_\mathrm{lim}=26.44\pm 0.24$ mag/$\mathrm{arcsec}^2$.
\subsection{Occurrence Rates}
We also aimed to validate the published asteroid activity occurrence rates of Table \ref{safari:ActivityRates}. Occurrence rates have been conservatively set at 1 in 10,000 (for all main belt asteroids), with the limiting magnitude of surveys the primary bottleneck. As shown in Figure \ref{safari:DataStats}d, the DECam instrument reaches an average magnitude of 24 \citep{sheppardNewExtremeTransNeptunian2016}, an unprecedented depth for large area active asteroid surveys. While our complete dataset was consistent with the 1:10,000 activity occurrence rate estimate, it is somewhat surprising we did not discover additional asteroidal activity.
\cite{hsiehMainbeltCometsPanSTARRS12015} postulated many active asteroids could be continuously active throughout their orbits (not just at perihelion), but with weaker activity. We expected then to find active asteroids more frequently in our search, given the objects we observed were indeed of a fainter magnitude (Figure \ref{safari:DataStats}b), though our outer main belt occurrence rate ($\sim$1:4000) was slightly higher than that reported by \cite{hsiehMainbeltCometsPanSTARRS12015} which is in line with their prediction. Small number statistics may have contributed to the possible discrepancy, and it is plausible we missed activity indications due to the limitations of visual inspection which were further compounded by an increased prevalence of background sources compared to shallower surveys. The use of a point spread function (PSF) comparison technique (e.g., \citet{hsiehMainbeltCometsPanSTARRS12015} or a photometric search (e.g., \citet{cikotaPhotometricSearchActive2014}) could help us identify candidates, features we plan to investigate in future work.
\subsection{False Positives}
\label{safari:FalsePositives}
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{safariFiles/432345FP2}\\
\caption{Common potential false-positives encountered in an asteroid thumbnail. (\textbf{a}) This thumbnail includes 4 potential false-positive sources: (\textbf{A}) Asteroid (432345). \textbf{B}) \textit{Scattered light} from a bright star trails toward the lower-left corner. (\textbf{C}) An \textit{extended source}, such as this edge-on galaxy, can present itself as coma if close to an asteroid. (\textbf{D}) \textit{Cosmic rays} with variable morphology are common throughout our images; they can look like trails if they align with a star as in this case. (\textbf{E}) \textit{Juxtaposed} objects can masquerade as active asteroids, especially when a bright object is near one or more progressively dimmer objects along the direction of apparent motion.}
\label{safari:FalsePositives1}
\end{figure}
We found false-positive management to be a formidable task, with specific mechanisms responsible for creating false-positives recurring throughout the project. For the rare cases where one of the authors involved in inspecting thumbnails found potential activity in an asteroid thumbnail, we checked other interpolation and color schemes, other thumbnails of the same asteroid, and the animated \ac{GIF} if available. We checked frames showing the same region on the sky, including original \ac{CCD} images, for background sources or image artifacts. What follows is a discussion of the primary culprits in order to convey the challenges faced during by-eye inspection (which is subjective by nature).
\textit{Juxtaposition} -- Figure \ref{safari:FalsePositives1}A marks asteroid (432345); the object is in close proximity to a galaxy, which, if juxtaposed in a confusing manner, could give the appearance of a coma. \ref{safari:FalsePositives1}D shows how a cosmic ray can be juxtaposed with a star. Figure \ref{safari:FalsePositives1}E demonstrates how multiple objects may appear to be an extended source.
\textit{Extended Sources} -- Extended sources, especially galaxies, were present in a myriad of orientations and configurations. They can appear like active asteroids, as in the edge-on galaxy shown in Figure \ref{safari:FalsePositives1}C. For a given brightness, galaxies occupied more sky area in a frame than other types of natural (i.e., non-artifact) objects and were more likely to be juxtaposed with other objects.
\textit{Scattered Light} -- Figure \ref{safari:FalsePositives1}B is scattered light associated with an especially bright star; the flare originates from the star and tapers off the further the ``tail'' gets from the source. While obvious in Figure \ref{safari:FalsePositives1}, the ``tail'' can be more difficult to identify as scattered light if the source is outside of the thumbnail.
\textit{Cosmic Rays} -- Cosmic rays (e.g. Figure \ref{safari:FalsePositives1}D) are common throughout our images, most of which have exposure times of 300 seconds or longer (see Figure \ref{safari:ExposureTimes}a). Figure \ref{safari:FalsePositives1} D demonstrates how cosmic rays may not appear as straight lines, and they may seem to connect two or more objects together.
\textit{Poor Seeing} -- Images with poor and/or rapidly varying seeing conditions suffered from fuzziness (potentially coma-like) and elongation implying a trailed object (e.g., an asteroid).
\subsection{Limitations of By-eye Inspection}
\label{safari:HumanLimitations}
As proof-of-concept for future projects making use of larger datasets, we sought a general understanding of our throughput as thumbnail inspectors. It is worth noting we did not impose time limits upon ourselves. We noted markedly different inspection rates, with the time required to inspect all thumbnails ranging from 2 to 6 hours. Furthermore, our attention spans varied, with inspection sessions lasting roughly between 10 minutes to 3 hours before requiring a break. The false positive handling described above undoubtedly impacted our image examination efficacy to some degree. Given these challenges, it is evident a computational approach to screen for potential active asteroids (through e.g., PSF comparison) would improve our detection rate.
\subsection{Asteroid Selection}
\label{safari:AsteroidSelection}
We examined only known asteroids during this work, but certainly many unknown asteroids are present within our data. Future efforts involving Citizen Scientists could locate these objects and quantify previously unrecognized biases inherent to locating activity among known asteroids. We used observations from a southern observatory, and while there may be little to no effect on observed activity occurrence rates, we acknowledge this selection effect nonetheless.
\subsection{Future Work}
\label{safari:FutureWork}
A broader study of the efficacy of human inspectors should be carried out if employing a larger number of inspectors. Injecting artificial active asteroids into the datasets would enable quantifying detection rates. The enormous datasets (2M+ thumbnails) we plan to generate will necessitate the deployment of a Citizen Science project, an endeavor that would thoroughly flush out these detection rates.
Citizen Science endeavors enable scientists to analyze otherwise prohibitively large datasets, with the added benefit of providing the scientific community with invaluable outreach opportunities proven to engage the public and spark far-reaching interest in science. \textit{Zooniverse}\footnote{\url{www.zooniverse.org}}, designed with the average scientist in mind, facilitates deployment of crowd-sourcing science projects. Volunteers are enlisted to interpret data too complex for machines, but accomplishable by anyone with minimal training. \textit{Zooniverse} has a proven track record, with notable successes such as \textit{Galaxy Zoo} which, within 24 hours of launch, reached 70,000 identifications/hour \citep{coxDefiningMeasuringSuccess2015}. While traditional and social media coverage undoubtedly boosted the performance of \textit{Galaxy Zoo} and other exemplary Citizen Science projects, the platform is designed to facilitate such exposure, especially through social media connectivity.
Our aim is to expand our survey to a second, comparably sized dataset already in-hand. We will first explore strategies to quantify active asteroid candidacy through computational techniques such as PSF comparison. We will then use the combined datasets to design, implement and test a Citizen Science project. We plan to start with a moderate ($\sim$ 10 member) group of thumbnail inspectors consisting of undergraduate and graduate students, whose feedback will inform the documentation and training system which is crucial to the success of a Citizen Science project. We subsequently intend to expand our dataset to the entire DECam public archive, at which point we would open our analysis system to public participation. We hope to incorporate machine learning into our pipeline as a means of reducing the number of thumbnails sent to the Citizen Science project or to help locate candidates missed at any point in the process.
\section{Summary}
\label{safari:summary}
We have developed an approach for finding active asteroids, rare objects visually like comets but dynamically like asteroids. We show DECam data are suitable for active asteroid searches. The approach involved processing 35,640{} \ac{FITS} files and extracting 15,600{} asteroid thumbnails (small images centered on an asteroid) consisting of 11,703{} unique objects. Upon visual examination of all thumbnails, we identified one previously known active asteroid (62412); our discovery rate of 1 in 11,703{} is consistent with the currently accepted active asteroid occurrence rate of 1 in 10,000. We did observe (1) Ceres and (779) Nina, though the former is a special case of \textit{a priori} activity knowledge \citep{ahearnWaterVaporizationCeres1992,kuppersLocalizedSourcesWater2014}, and neither object has ever shown signs of activity visible from Earth; as we did not observe activity in either object, we did not include them in our activity occurrence rate estimate. From our proof-of-concept study, we conclude a significantly larger survey should be carried out to locate active asteroids, finally placing them into a regime where they may be studied as a population.
\section{Acknowledgements}
\label{safari:acknowledgements}
The authors thank the referee, Henry Hsieh (Planetary Science Institute), whose thorough and thoughtful feedback vastly improved the quality of this work. Prof. Ty Robinson of Northern Arizona University (NAU) helped us keep this project a priority and provided fresh perspectives on scientific dilemmas. Dr. Mark Jesus Mendoza Magbanua (University of California San Francisco) whose insights greatly improved the quality of the paper. Annika Gustaffson (NAU) provided frequent input and encouraged us to move this project forward. The enthusiastic support provided by Monsoon supercomputer system administrator Christopher Coffey (NAU) was essential in overcoming countless technical challenges. The Trilling Research Group (NAU) provided insight and feedback about our data visualization techniques. Prof. Mike Gowanlock (NAU) inspired numerous computational techniques which reduced our analysis compute time.
Computational analyses were run on Northern Arizona University's Monsoon computing cluster, funded by Arizona's Technology and Research Initiative Fund. This work was made possible in part through the State of Arizona Technology and Research Initiative Program. Michael Mommert was supported in part by NASA grant NNX15AE90G to David E. Trilling.
This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A\&AS 143, 23 \citep{ochsenbeinVizieRDatabaseAstronomical2000}. This research has made use of data and/or services provided by the International Astronomical Union's Minor Planet Center. This research has made use of NASA's Astrophysics Data System. This research has made use of the The Institut de M\'ecanique C\'eleste et de Calcul des \'Eph\'em\'erides (IMCCE) SkyBoT Virtual Observatory tool \citep{berthierSkyBoTNewVO2006}. This work made use of the {FTOOLS} software package hosted by the NASA Goddard Flight Center High Energy Astrophysics Science Archive Research Center.
This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A\&M University, Financiadora de Estudos e Projetos, Funda\c{c}\~{a}o Carlos Chagas Filho de Amparo, Financiadora de Estudos e Projetos, Funda\c{c}\~ao Carlos Chagas Filho de Amparo \`{a} Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico and the Minist\'{e}rio da Ci\^{e}ncia, Tecnologia e Inova\c{c}\~{a}o, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones En\'{e}rgeticas, Medioambientales y Tecnol\'{o}gicas–Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgen\"ossische Technische Hochschule (ETH) Z\"urich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci\`{e}ncies de l'Espai (IEEC/CSIC), the Institut de Física d'Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit\"{a}t M\"{u}nchen and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, the Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A\&M University.
Based on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory (NOAO Prop. IDs 2015A-0351
2016B-0288, 2017A-0367, 2015B-0265, 2013B-0453, 2014B-0303, 2016A-0401 and 2014A-0479; PI: Scott Sheppard), which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.
\clearpage
\section{Appendix}
\subsection{Object-specific References}
\label{safari:ObjectReferences}
SPK-ID are found at the JPL Horizons Small Bodies Database (\url{https://ssd.jpl.nasa.gov/sbdb.cgi}).
\begin{itemize}
\setlength\itemsep{0.01mm}
\item[\ref{Ceres}.] (1) Ceres, 1943 XB, A899 OF, SPK-ID=2000001; Activity Discovered:\cite{ahearnWaterVaporizationCeres1992,kuppersLocalizedSourcesWater2014}; Mechanism: \cite{kuppersLocalizedSourcesWater2014}; Activity Obs.: 1 (1992) -- \cite{ahearnWaterVaporizationCeres1992}, 2 (2011-2013) -- \cite{kuppersLocalizedSourcesWater2014, nathuesSublimationBrightSpots2015}, 3 (2015-2016) -- \cite{thangjamHazeOccatorCrater2016,nathuesEvolutionOccatorCrater2017,landisConditionsSublimatingWater2017,rothConstraintsWaterVapor2018}; Visit: Dawn \citep{russellDawnArrivesCeres2016}; Absence of Family Association: \cite{rivkinCaseMissingCeres2014,hsiehAsteroidFamilyAssociations2018}; Additional: \cite{tuSublimationdrivenExosphericModel2014, witzeBrightSpotsCeres2015, hayneThermalStabilityIce2015, nathuesSublimationBrightSpots2015, liSurfaceAlbedoSpectral2016, rothConstraintsExosphereCeres2016, prettymanExtensiveWaterIce2017, mckayObservationalConstraintsWater2017, nathuesOxoCraterCeres2017, landisConditionsSublimatingWater2017}
\item[\ref{Adeona}.] (145) Adeona, SPK-ID=2000145; Activity Discovery: \cite{busarevMaterialCompositionAssessment2016}; Mechanism: \cite{busarevMaterialCompositionAssessment2016}; Activity Obs.: 1 (2012) -- \cite{busarevMaterialCompositionAssessment2016}$^\ast$; Visit: Dawn (cancelled)\footnote{\scriptsize{\url{https://www.nasa.gov/feature/new-horizons-receives-mission-extension-to-kuiper-belt-dawn-to-remain-at-ceres}}}; Additional: \cite{busarevNewCandidatesActive2018
\item[\ref{Constantia}.] (315) Constantia, SPK-ID=2000315; Candidacy: \cite{cikotaPhotometricSearchActive2014}; Flora family association: \cite{alfvenAsteroidalJetStreams1969}
\item[\ref{Griseldis}.] (493) Griseldis, 1902 JS, A915 BB, SPK-ID=2000493; Activity Discovery: \cite{tholenEvidenceImpactEvent2015}; Activity Obs.: 1 (2015) -- \cite{tholenEvidenceImpactEvent2015,seargentWeirdCometsAsteroids2017}; Unknown impactor size: \cite{huiNongravitationalAccelerationActive2017}; Absence of Family Association: \cite{hsiehAsteroidFamilyAssociations2018}
\item[\ref{Scheila}.] (596) Scheila, 1906 UA, 1949 WT, SPK-ID=2000596; Activity Discovery: \cite{larson596Scheila2010}; Mechanism: \cite{jewittHubbleSpaceTelescope2011, bodewitsCollisionalExcavationAsteroid2011, yangNearinfraredObservationsCometlike2011a, moreno596ScheilaOutburst2011, ishiguroObservationalEvidenceImpact2011, ishiguroInterpretation596Scheila2011, hsiehOpticalDynamicalCharacterization2012, husarikRelativePhotometryPossible2012, neslusanDustProductivityImpact2016}; Activity Obs.: 1 (2010-2011) -- \cite{jewittHubbleSpaceTelescope2011, bodewitsCollisionalExcavationAsteroid2011, yangNearinfraredObservationsCometlike2011a, ishiguroObservationalEvidenceImpact2011, hsiehOpticalDynamicalCharacterization2012, husarikRelativePhotometryPossible2012,neslusanDustProductivityImpact2016}; Absence of Family Association: \cite{hsiehAsteroidFamilyAssociations2018}
\item[\ref{Interamnia}.] (704) Interamnia, 1910 KU, 1952 MW, SPK-ID=2000704; Activity Discovery, Mechanism: \cite{busarevMaterialCompositionAssessment2016}; Activity Obs.: 1 (2012) -- \cite{busarevMaterialCompositionAssessment2016}$^\ast$; Absence of Family Association: \cite{rivkinCaseMissingCeres2014}; Shape Model: \cite{sato3DShapeModel2014}; Additional: \cite{busarevNewCandidatesActive2018}
\item[\ref{Nina}.] (779) Nina, 1914 UB, A908 YB, A912 TE, SPK-ID=2000779; Activity Discovery, Mechanism: \cite{busarevMaterialCompositionAssessment2016}; Activity Obs.: 1 (2012) -- \cite{busarevMaterialCompositionAssessment2016}$^\ast$, 2 (2016) -- \cite{busarevNewCandidatesActive2018}
\item[\ref{Ingrid}.] (1026) Ingrid, 1923 NY, 1957 UC, 1963 GD, 1981 WL8, 1986 CG2, 1986 ES2, SPK-ID=2001026; Candidacy: \cite{cikotaPhotometricSearchActive2014}; Follow-up Observation (negative): \cite{betzlerPhotometricObservations10262015}; Flora family association: \cite{alfvenAsteroidalJetStreams1969}; Additional: \cite{nakano1026Ingrid1986,busarevNewCandidatesActive2018}
\item[\ref{Beira}.] (1474) Beira, 1935 QY, 1950 DQ, SPK-ID=2001474; Activity Discovery: \cite{busarevMaterialCompositionAssessment2016}; Mechanism: \cite{busarevMaterialCompositionAssessment2016}; Activity Obs.: 1 (2012) -- \cite{busarevMaterialCompositionAssessment2016}$^\ast$; Chaotic Cometary Orbit: \cite{hahnAsteroidsCometaryOrbits1985}; Additional: \cite{busarevNewCandidatesActive2018}
\item[\ref{Oljato}.] (2201) Oljato, 1947 XC, 1979 VU2, 1979 XA, SPK-ID=2002201; Activity Discovery: \cite{russellInterplanetaryMagneticField1984}; Activity Obs.: 1 (1984) -- \cite{russellInterplanetaryMagneticField1984}, Negative (1996) -- (\cite{chamberlin4015WilsonHarrington22011996}; Visit: \cite{perozziBasicTargetingStrategies2001}; Additional: \cite{kerrCouldAsteroidBe1985, mcfaddenEnigmaticObject22011993, connorsUnusualAsteroid22012016}
\item[\ref{Phaethon}.] (3200) Phaethon, 1983 TB, SPK-ID=2003200; Activity Discovery: \cite{battams3200Phaethon2009}; Mechanism: ; Activity Obs.: Negative -- \cite{chamberlin4015WilsonHarrington22011996,hsiehSearchActivity32002005}, 1 (2009) -- \cite{battams3200Phaethon2009,jewittActivityGeminidParent2010} 2 (2012) -- \cite{liRecurrentPerihelionActivity2013,jewittDustTailAsteroid2013}, 3 (2016) -- (\cite{huiResurrection3200Phaethon2017}; Visit: Destiny Plus \citep{iwataStudiesSolarSystem2016}; Pallas Family Association: \cite{todorovicDynamicalConnectionPhaethon2018}; Additional: \cite{jewittActivityGeminidParent2010, ryabovaPossibleEjectionMeteoroids2012, liRecurrentPerihelionActivity2013, jewittDustTailAsteroid2013, ansdellRefinedRotationalPeriod2014, jakubikMeteorComplexAsteroid2015, hanusNearEarthAsteroid32002016, sarliDESTINYTrajectoryDesign2017}
\item[\ref{DonQuixote}.] (3552) Don Quixote, 1983 SA, SPK-ID=2003552; Activity Discovery, Mechanism: \cite{mommertDiscoveryCometaryActivity2014}; Activity Obs.: 1 (2009) -- \cite{mommertDiscoveryCometaryActivity2014}, (2018) -- \cite{mommertCBET4502201803292018}; Chaotic Cometary Orbit (as 1983 SA): \cite{hahnAsteroidsCometaryOrbits1985}
\item[\ref{Aduatiques}.] (3646) Aduatiques, 1985 RK4, 1979 JL, 1981 WZ6, SPK-ID=2003646; Candidacy: \cite{cikotaPhotometricSearchActive2014}; Follow-up (inconclusive): \cite{sosaoyarzabalPhotometricSearchActivity2014}
\item[\ref{WilHar}.] (4015) Wilson--Harrington, 1979 VA, 107P, SPK-ID=2004015; Activity Discovery: \cite{cunninghamPeriodicCometWilsonHarrington1950}; Activity Obs.: 1 (1949) -- \cite{cunninghamPeriodicCometWilsonHarrington1950}, 2 (1979) -- \cite{degewij1979VAPhysical1980}, Negative (1992) -- \cite{bowell40151979VA1992a}, Negative (1996) \cite{chamberlin4015WilsonHarrington22011996}, Negative (2008) -- \cite{licandroSpitzerObservationsAsteroidcomet2009}, Negative (2009-2010) -- \cite{ishiguroObservationalEvidenceImpact2011,urakawaPhotometricObservations107P2011}, 3-6 (1992, 1996, 2008, 2009-2010) \cite{ferrin2009ApparitionMethuselah2012}; Visits: Failed \citep{raymanDeepSpaceExtended2001}, Concept \citep{sollittMissionConcepts40152009}; Chaotic Cometary Orbit (as 1979 VA): \cite{hahnAsteroidsCometaryOrbits1985}; Additional: \cite{harrisCometNotesComet1950, vanbiesbroeckCometNotesComet1951, helin1979VaPossibleCarbonaceous1980, helinDiscoveryUnusualApollo1981, osipRotationState4015WilsonHarrington1995, fernandezAnalysisPOSSImages1997}
\item[\ref{24684}.] (24684), 1990 EU4, 1981 UG28, SPK-ID=2024684; Candidacy: \cite{cikotaPhotometricSearchActive2014}
\item[\ref{35101}.] (35101) 1991 PL16, 1998 FZ37, SPK-ID=2035101; Candidacy: \cite{cikotaPhotometricSearchActive2014}; Eunomia Family Association: \cite{cikotaPhotometricSearchActive2014}
\item[\ref{62412}.] (62412), 2000 SY178, SPK-ID=2062412; Activity Discovery: \cite{sheppardDiscoveryCharacteristicsRapidly2015}; Activity Obs.: 1 (2014) \cite{sheppardDiscoveryCharacteristicsRapidly2015}; Hygiea Family Association: \cite{sheppardDiscoveryCharacteristicsRapidly2015,hsiehAsteroidFamilyAssociations2018}
\item[\ref{Ryugu}.] (162173) Ryugu, SPK-ID=2162173; Activity Discovery, Mechanism, Activity Obs.: 1 (2007) -- \cite{busarevNewCandidatesActive2018}$^\ast$; Visit: Hayabusa 2 \citep{tsudaSystemDesignHayabusa2013a}; Clarissa Family Association \cite{campinsOriginAsteroid1621732013,lecorreGroundbasedCharacterizationHayabusa22018}; Thermal Inertia: \cite{liang-liangInvestigationThermalInertia2014}; Additional: \cite{suzukiInitialInflightCalibration2018, pernaSpectralRotationalProperties2017}
\item[\ref{GO98}.] (457175), 2008 GO98, 362P, SPK-ID=2457175; Activity Discovery: \cite{kimNewObservationalEvidence2017}; Activity Obs.: 1 (2017) \cite{masi4571752008GO2017}; Hilda Family Association: \cite{warnerLightcurveAnalysisHilda2018}; Additional: \cite{sato4571752008GO2017, yoshimoto4571752008GO2017, birtwhistle4571752008GO2017, bacci4571752008GO2017, bell4571752008GO2017, bryssinck4571752008GO2017}
\item[\ref{ElstPizarro}.] 133P/Elst--Pizarro, (7968), 1979 OW7, 1996 N2, SPK-ID=2007968; Activity Discovery: \cite{elstComet1996N21996}; Mechanism: \cite{hsiehStrangeCase133P2004, jewittHubbleSpaceTelescope2014}; Activity Obs.: 1 (1996) \cite{elstComet1996N21996}, 2 (2002) \cite{hsiehStrangeCase133P2004}, Negative (2005) \cite{tothSearchCometlikeActivity2006}, 2 (2007) \cite{hsiehReturnActivityMainbelt2010, bagnuloPolarimetryPhotometryPeculiar2010, rousselotNearinfraredSpectroscopy133P2011}, 3 (2013) \cite{jewittHubbleSpaceTelescope2014}; Visit: Castalia \citep{snodgrassCastaliaMissionMain2017}; Themis Family Association: \cite{boehnhardtImpactInducedActivityAsteroidComet1998}; Additional: \cite{tothImpactgeneratedActivityPeriod2000, ferrinSecularLightCurves2006, prialnikCanIceSurvive2009}
\item[\ref{176P}.] 176P/LINEAR, (118401), P/1999 RE$_{70}$, 2001 AR7, SPK-ID=2118401; Activity Discovery: \cite{hsiehComet1999RE2006,hsiehHawaiiTrailsProject2009}; Mechanism: \cite{hsiehSearchReturnActivity2014}; Activity Obs.: 1 (2005) \cite{hsiehComet1999RE2006}, Negative (2006-2009) \cite{hsiehPhysicalPropertiesMainbelt2011}, Negative (2011) \cite{hsiehSearchReturnActivity2014}; Themis Family Association: \cite{hsiehHawaiiTrailsProject2009,hsiehAsteroidFamilyAssociations2018} Additional: \cite{hsiehAlbedosMainBeltComets2009,licandroTestingCometNature2011,deval-borroUpperLimitWater2012}
\item[\ref{233P}.] 233P (La Sagra), P/2009 W$_{50}$, 2005 JR71, SPK-ID=1003062; Activity Discovery: \cite{mainzerComet2009WJ2010}, Activity Obs.: 1 (2009) \cite{mainzerComet2009WJ2010}; Absence of Family Association: \cite{hsiehAsteroidFamilyAssociations2018}
\item[\ref{238P}.] 238P/Read, P/2005 U1, 2010 N2, SPK-ID=1001676; Activity Discovery: \cite{readComet2005U12005}; Activity Obs.: 1 (2005) \cite{readComet2005U12005}, 2 (2010) \cite{hsiehMainbeltComet238P2011}, 3 (2016) \cite{hsiehReactivationsMainBeltComets2017}; Gorchakov Family Association: \cite{hsiehAsteroidFamilyAssociations2018}; Former Themis Family Association: \cite{haghighipourDynamicalConstraintsOrigin2009}; Additional: \cite{hsiehPhysicalPropertiesMainBelt2009, pittichovaComet2005U12010}
\item[\ref{259P}.] 259P/Garradd, 2008 R1, SPK-ID=1002991; Activity Discovery: \cite{garraddComet2008R12008}; Mechanism: \cite{jewittMainBeltComet20082009}; Activity Obs.: 1 (2008) \cite{garraddComet2008R12008}, 2 (2017) \cite{hsiehComet259PGarradd2017,hsiehReactivationsMainBeltComets2017}; Absence of Family Association: \cite{hsiehAsteroidFamilyAssociations2018}; Additional: \cite{kossackiMainBeltComet2012, maclennanNucleusMainbeltComet2012, kleynaFaintMovingObject2012}
\item[\ref{288P}.] 288P, (300163), 2006 VW139, SPK-ID=2300163; Activity Discovery: \cite{hsiehComet2006VW2011}; Activity Obs.: 1 (2011) \cite{hsiehComet2006VW2011}, 2 (2016-2017) \cite{agarwalBinaryMainbeltComet2017,hsiehReactivationsMainBeltComets2017}; Themis Family Association: \cite{hsiehDiscoveryMainbeltComet2012,hsiehAsteroidFamilyAssociations2018}; Additional: \cite{hsiehDiscoveryMainbeltComet2012, novakovic2006VW139Mainbelt2012, agarwalHubbleKeckTelescope2016}
\item[\ref{311P}.] 311P/Pan-STARRS, P/2013 P5, SPK-ID=1003273; Activity Discovery: \cite{micheliComet2013P52013}; Mechanism: \cite{jewittExtraordinaryMultitailedMainbelt2013, morenoIntermittentDustMass2014, hainautContinuedActivity20132014, jewittEpisodicEjectionActive2015}; Activity Obs.: 1 (2013-2014) \cite{micheliComet2013P52013,jewittEpisodicEjectionActive2015}; Behrens Family Association: \cite{hsiehAsteroidFamilyAssociations2018}
\item[\ref{313P}.] 313P/Gibbs, P/2014 S4, 2003 S10, SPK-ID=1003344; Activity Discovery: \cite{gibbsComet2014S42014}; Mechanism, Activity Obs.: 1 (2003) \cite{nakanoComet2003S102014,skiffComet2014S42014,huiArchivalObservationsActive2015}, 2 (2015) \cite{jewittNucleusMassLoss2015}; Lixiaohua Family Association: \cite{hsiehMainbeltComet20122013,hsiehSublimationDrivenActivityMainBelt2015,hsiehAsteroidFamilyAssociations2018}; Additional: \cite{jewittNewActiveAsteroid2015, hsiehSublimationDrivenActivityMainBelt2015, pozuelosDustEnvironmentMainBelt2015}
\item[\ref{324P}.] 324P/La Sagra, P/2010 R2, 2015 K3, SPK-ID=1003104; Activity Discovery: \cite{nomenComet2010R22010}; Activity Obs.: 1 (2010-2011) \cite{nomenComet2010R22010,hsiehObservationalDynamicalCharacterization2012}, Negative (2013) \cite{hsiehNucleusMainbeltComet2014}, 2 (2015) \cite{hsiehReactivationMainbeltComet2015,jewittHubbleSpaceTelescope2016}; Alauda Family Association: \cite{hsiehAsteroidFamilyAssociations2018}; Additional: \cite{morenoDustEnvironmentMainBelt2011,hsiehObservationalDynamicalCharacterization2012,hsiehNucleusMainbeltComet2014,hsiehReactivationMainbeltComet2015}
\item[\ref{331P}.] 331P/Gibbs, P/2012 F5, SPK-ID=1003182; Activity Discovery: \cite{gibbsComet2012F52012}; Mechanism: \cite{stevensonCharacterizationActiveMain2012, drahusFastRotationTrailing2015}; Activity Obs.: 1 (2012) \cite{gibbsComet2012F52012}, 2 (2015) \cite{drahusFastRotationTrailing2015}; Gibbs family association: \cite{novakovicDiscoveryYoungAsteroid2014}; Additional: \citep{stevensonCharacterizationActiveMain2012,morenoShortdurationEventCause2012}
\item[\ref{348P}.] 348P, P/2017 A2, P/2011 A5 (PANSTARRS), SPK-ID=1003492; Activity Discovery: \cite{wainscoatComet2017A22017}; Activity Obs.: 1 (2017) \cite{wainscoatComet2017A22017}; Absence of Family Association: \cite{hsiehAsteroidFamilyAssociations2018}
\item[\ref{354P}.] 354P/LINEAR, P/2010 A2, 2017 B5, SPK-ID=1003055; Activity Discovery: \cite{birtwhistleComet2010A22010}; Activity Obs.: 1 (2010) \cite{birtwhistleComet2010A22010,jewittComet2010A22010}; Baptistina Family Association: \cite{hsiehAsteroidFamilyAssociations2018}; Additional: \cite{morenoWatericedrivenActivityMainBelt2010,jewittRecentDisruptionMainbelt2010,snodgrassCollision2009Origin2010,jewittPreDiscoveryObservationsDisrupting2011,hainaut2010A2LINEAR2012,kimMultibandOpticalObservation2012,agarwalDynamicsLargeFragments2012,kleyna2010A2LINEAR2013,jewittLargeParticlesActive2013,agarwalDynamicsLargeFragments2013,kimNewObservationalEvidence2017,kimAnisotropicEjectionActive2017}
\item[\ref{358P}.] 358P/PanSTARRS, P/2012 T1, 2017 O3, SPK-ID=1003208; Activity Discovery: \cite{wainscoatComet2012T12012}; Activity Obs.: 1 (2012) \cite{wainscoatComet2012T12012}, 2 (2017) \cite{kimNewObservationalEvidence2017}; Mechanism: \cite{hsiehMainbeltComet20122013}; Lixiaohua Family Association: \citep{hsiehMainbeltComet20122013,hsiehAsteroidFamilyAssociations2018}; Additional: \cite{morenoDustEnvironmentMainBelt2013, orourkeDeterminationUpperLimit2013, snodgrassXshooterSearchOutgassing2017}
\item[\ref{P2013R3}.] P/2013 R3 (Catalina-Pan-STARRS), SPK-ID=1003275 (P/2013 R3-A SPK-ID=1003333, P/2013 R3-B SPK-ID=1003334); Activity Discovery: \cite{bolinComet2013R32013,hillComet2013R32013}; Activity Obs.: 1 (2013-2015) \cite{bolinComet2013R32013,hillComet2013R32013,jewittAnatomyAsteroidBreakup2017}; Mandragora Family Association: \cite{hsiehAsteroidFamilyAssociations2018}; Additional: \cite{jewittDisintegratingAsteroid20132014, hirabayashiConstraintsPhysicalProperties2014}
\item[\ref{P2015X6}.] P/2015 X6 (Pan-STARRS), SPK-ID=1003426; Activity Discovery: \cite{lillyComet2015J22015}; Activity Obs.: 1 (2015) \cite{lillyComet2015J22015,tubbioloComet2015X62015,morenoDustLossActivated2016}; Aeolia Family Association: \cite{hsiehAsteroidFamilyAssociations2018}
\item[\ref{P2016G1}.] P/2016 G1 (Pan-STARRS), SPK-ID=1003460; Activity Discovery: \cite{werykCOMET2016G12016}; Mechanism: \cite{morenoEarlyEvolutionDisrupted2016}; Activity Obs.: 1 (2016) \cite{werykCOMET2016G12016,morenoDisruptedAsteroid20162017}; Adeona Family Association: \cite{hsiehAsteroidFamilyAssociations2018}
\item[\ref{P2016J1}.] P/2016 J1 (Pan-STARRS), P/2016 J1-A (SPK-ID=1003464), P/2016 J1-B (SPK-ID=1003465); Activity Discovery: \cite{wainscoat2016J1Panstarrs2016}; Activity Obs.: 1 (2016) \cite{wainscoat2016J1Panstarrs2016,huiSplitActiveAsteroid2017}; Theobalda Family Association: \cite{hsiehAsteroidFamilyAssociations2018}
\end{itemize}
$^\ast$: Under review.
\chapter{Acronyms}
\label{chap:acronyms}
\begin{acronym}
\input{Acronyms}
\end{acronym}
\singlespacing
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,600 |
All-in-one EHR and practice management solutions using the power of data to improve clinical and financial results.
We focus on optimizing practice profitability so you can focus on patient care.
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For the third year in a row, we are a Software Advice FrontRunners® in Top Rated EHR.
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Helping You focus on patient care. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,304 |
Stenoglene roseus is a moth in the family Eupterotidae. It was described by Druce in 1886. It is found in Burundi, the Democratic Republic of Congo (Katanga, East Kasai, Bas Congo), Ivory Coast, Kenya, Malawi, Mozambique, Nigeria and Uganda.
Description
The wingspan is about 54 mm. Both wings are pale golden ochreous, the forewings with a broadish postmedian, concave, rusty transverse band, broadening as it approaches the inner margin near the tornus. The hindwings are immaculate.
References
Moths described in 1886
Janinae
Moths of Sub-Saharan Africa | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 872 |
Jakob Hegner ist der Name folgender Personen:
* Jakob Hegner (1882–1962), österreichischer Drucker, Verleger und Übersetzer
Jakob Hegner (Musiker) (* ≈1996), deutscher Fusionmusiker
Jakob Meinrad Hegner (1813–1879), Schweizer Politiker | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,808 |
<h1 align="center">:whale:A collection of algorithms that are implemented in PHP:whale: </h1>
<p align="center">
<a href="https://github.com/PuShaoWei/arithmetic-php#简易结构">
<img src="https://img.shields.io/badge/php-done-brightgreen.svg" alt="php">
</a>
<a href="https://github.com/PuShaoWei/arithmetic-php">
<img src="https://img.shields.io/github/issues-pr-raw/arithmetic-php/cdnjs.svg">
</a>
<a href="https://github.com/PuShaoWei/arithmetic-php">
<img src="https://img.shields.io/codacy/grade/e27821fb6289410b8f58338c7e0bc686.svg">
</a>
<a href="https://github.com/PuShaoWei/arithmetic-php">
<img src="https://img.shields.io/travis/rust-lang/rust.svg">
</a>
<a href="https://github.com/PuShaoWei/arithmetic-php">
<img src="https://img.shields.io/github/license/mashape/apistatus.svg">
</a>
</p>
<p align="center"> <a href="./README.md">中文版</a> <p>
## Simple structure,
```
├──Package
│ ├── Sort
│ │ ├── BubbleSort.php
│ │ ├── QuickSort.php
│ │ ├── ShellSort.php
│ │ ├── MergeSort.php
│ │ ├── InsertSort.php
│ │ └── SelectSort.php
│ │
│ ├── Query 查找篇
│ │ ├── BinaryQuery.php
│ │ ├── InseertQuery.php
│ │ ├── FibonacciQuery.php
│ │ ├── BFSQuery.php
│ │ ├── Kmp.php
│ │ ├── DijkstraQuery.php
│ │ └── QulickQuery.php
│ │
│ └── Other 其他
│ ├── MonkeyKing.php
│ ├── DynamicProgramming.php
│ ├── Fibonacci.php
│ ├── StealingApples.php
│ ├── HanoiGames.php
│ ├── BidirectionalQueue.php
│ ├── ColorBricks.php
│ ├── GetCattle.php
│ ├── OnlyNumbers.php
│ ├── Interval.php
│ ├── Maze.php
│ ├── AntsClimb.php
│ ├── Encryption.php
│ ├── ElevatorDispatch.php
│ ├── kmp.php
│ ├── TraversalOfBinary.php
│ ├── PointInTriangle.php
│ └── BigSmallReplace.php
│ └── Knapsack.php
│ └── Solution.php
│ └── RotationSort.php
│ └── Square.php
│ └── Prim.php
│ └── CartesianProduct.php
│ └── Square.php
│ └── Judge.php
│ └── Factorial.php
│
├──LICENSE
└──README.md
```
## What to do?
```
To record their understanding algorithms, data structure, the process of simple comprehensive and detailed as possible, let the learning algorithm using flexible, refueling(ง •̀_•́)ง
```
## logarithmic
log<sub>10</sub>100 It's equivalent to saying, "how many tens do you multiply?" the answer is, of course, two
so log<sub>10</sub>100=2,The logarithmic operation is the inverse of the power operation
| left | right |
| ------------------ | --------------------- |
| 2<sup>3</sup> = 8 | log<sub>2</sub>8 = 3 |
| 2<sup>4</sup> = 16 | log<sub>2</sub>16 = 4 |
| 2<sup>5</sup> = 32 | log<sub>2</sub>32 = 5 |
If you don't, we won't wait for you
## The elapsed time
Take binary search for example, how much time can you save by using it? Simply look for the Numbers and if the list contains 100 Numbers, you need to guess 100 times.
In other words, the number of guesses is the same as the length of the list, which is called linear time, while binary search is different if the list contains 100 elements
It takes up to seven times, and if the list contains four billion digits, it should be guessed 32 times, while the running time of the subsearch is logarithmic time `O(log)`
## Big O notation
The big O notation is a special representation of how fast the algorithm can be. There's a diaosi. In fact, you often have to copy other people's code.
In this case, you know how fast these algorithms are
- The running time of the algorithm increases at different speeds
- For example, the difference between a simple find and a binary search
| element | Easy to find | Binary search |
| ------------- | ------------ | ------------- |
| 100 | 100ms | 7ms |
| 10000 | 10s | 14ms |
| 1 000 000 000 | 11day | 30ms |
- ` O ` said hair is pointed out that how fast algorithms, such as list contains ` n ` element, a simple search need to check each element, so you need to perform ` n ` time operations
Using large ` O ` said ` O (n) to make this operation `, binary search need to perform log<sub>n</sub> using large ` O ` said to`O(log n)`
- Some common big O runtime
- O(log n) ,It's also called log time, and this algorithm includes binary algorithms
- O(n),Also known as linear time, this algorithm includes simple lookups.
- O(n * log n) Quick sort
- O(n<sub>2</sub>),Selection sort
- O(n!) Factorial time
- Here is the point
- The speed of the algorithm is not the time, but the growth of operands
- When we talk about the speed of the algorithm, what we're talking about is how fast will it run as the input increases
- The running time of the algorithm is expressed in large O notation
- O(log n) is faster than O (n), and the more elements that need to be searched, the more the former is faster than the latter
## A simple comparison of recursion and loops:
1. From a procedural point of view, the recursion manifests itself as calling itself, and the loop does not have this form.
2. Recursive proceed from the ultimate goal of the problem, and gradually to a complex problem into a simple problem, and simple question solution and complicated problem, at the same time the presence of the benchmark, can eventually get a problem, is the reverse. And the circulation is from the simple question, step by step forward development, finally get the question, is positive.
3. Any cycle can be represented by recursion, but it is necessary to use the loop to achieve recursion (except for one-way recursion and tail recursion), and the stack structure must be introduced to stack the stack.
4.In general, non-recursive efficiency is higher than recursion. And recursive function calls are expensive and recursive times are limited by stack size.
## Progressive learning
1. Fork 我的项目并提交你的 `idea`
2. Pull Request
3. Merge
## 纠错
If you find something wrong, you can initiate a [issue](https://github.com/PuShaoWei/designPatterns-go/issues)or [pull request](https://github.com/PuShaoWei/designPatterns-go/pulls),I will correct it in time
> 补充:发起pull request的commit message请参考文章[Commit message 和 Change log 编写指南](http://www.ruanyifeng.com/blog/2016/01/commit_message_change_log.html)
## Contributors
Thanks for the issue or pull request of the following friends:
- [hailwood ](https://github.com/hailwood)
- [zhangxuanru](https://github.com/zhangxuanru)
- [ifreesec](https://github.com/ifreesec)
- [openset](https://github.com/openset)
- [Neroxiezi](https://github.com/Neroxiezi)
## License
MIT
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,606 |
То́ро-крихітка (Phyllastrephus debilis) — вид горобцеподібних птахів родини бюльбюлевих (Pycnonotidae). Мешкає в Східній Африці. До 2009 року гірські торо вважалися конспецифічними з торо-крихітками.
Підвиди
Виділяють два підвиди:
P. d. rabai Hartert, E & Van Someren, 1921 — південно-східна Кенія, північно-східна Танзанія;
P. d. debilis (Sclater, WL, 1899) — від південно-східної Танзанії до східного Зімбабве і південного Мозамбіку.
Поширення і екологія
Торо-крихітки живуть у вологих рівнинних тропічних лісах, сухих саванах і чагарникових заростях.
Примітки
Посилання
Lowland tiny greenbul - Species text in The Atlas of Southern African Birds.
Бюльбюлеві
Птахи, описані 1899
Птахи Кенії
Птахи Танзанії
Птахи Зімбабве
Птахи Мозамбіку | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,902 |
\section{Introduction}
Conceptual meanings of the wave equations dependence on the exact solutions have been considered in \cite{b1,b2,b3,b4}. The derived physical quantities such as eigenfunctions and eigenvalues of these equations can be either compared to the experimental or to the results obtained from other methods. The exact solutions of the wave equations can be used as criteria in other numerical and theoretical methods. The evolutions of non relativistic quantum particles are usually described by
using the Schr\"{o}dinger equation while for the relativistic quantum particles, one has to deal with the correct equation of motion, such as Klein-Gordon or Dirac Equations, depending on the particle's spin. These equations have been investigated via different methods. Usually, the mass parameter in the above mentioned wave equations has been considered to be a constant. Recently, an increasing interest has been devoted to solving quantum wave equations with position-dependent
mass. The Schr\"{o}dinger equation with position-dependent mass distribution was initially proposed by Von Roos \cite{b5}. In certain physical systems, the effective mass parameter should be position-dependent to be consistent with the experimental data \cite{b6}. In this context, the Schr\"{o}dinger equation with different phenomenological potentials and appropriate mass distributions has been investigated using various methods \cite{b7,b8,b9,b10}. For some molecular Hamiltonians, the energy spectra and eigenfunctions of the position-dependent mass particles have been derived \cite{b11}. According to Ref.\cite{b11}, particles with a mass are more likely to tunnel than ordinary ones. The use of this effective mass formalism has been considered for the dynamics of the electrons in inhomogeneous crystals for many years \cite{b12,b13}. It has been also applied in many different fields of physics, such as helium clusters\cite{b14}, semiconductors \cite{b15,b16,b17}, quantum dot\cite{b18}, quantum liquids\cite{b19}, atomic nuclei\cite{b20,b21,b22}.
In this work, we are going to consider the three dimensional time independent Schr\"{o}dinger equation within the effective mass formalism. This paper is organized as follows. In section II, after some preliminaries, the separation of variables is carried out for the deformed Schr\"{o}dinger equation with non-central potential in spherical coordinates.
In section III, we introduce a new non-central potential and we investigate the scattering states solutions as well as the phase shifts under this effective potential . Finally, Sec. IV is devoted to the conclusion.
\section{Variable Separation of Hamiltonian Considering Non-Central Potential in Position-Dependent Mass Formalism }
\numberwithin{equation}{section}
Theoretical background of the position-dependent effective mass formalism (PDEMF) was recently has been considered in Refs \cite{b23,b24}. In the PDEMF, for Schr\"{o}dinger equation, the mass operator $m(x)$ and momentum operator $\vec{p}=-i\hbar\vec{\nabla}$ no longer commute. Therefore, several ways exist for generalizing the usual form of the kinetic energy operator $\vec{p}^2/2m_0$, and consequently the Hamiltonian, in order to obtain a Hermitian operator to describe the quantum state of a physical system that is not trivial. In order to avert any specific choices, one can use the general form of the Hamiltonian originally proposed by Von Roos \cite{b25}.
In Ref. \cite{b26} by choosing the position-dependent mass $m(\vec{r})=\frac{m_0}{f(\vec{r})^2}$ where $m_0$ is a constant mass and $f(r)$ represents a deforming function, the authors obtain a new form of Hamiltonian, and in the special case, this Hamiltonian reduces to the very common BenDaniel-Duke form \cite{b27}.
In the spherical coordinates $\vec{r}=\{r=||\vec{r}||,\theta,\varphi\}$ with $f(\vec{r})=f(r)$, follow separation of variables is customary used to obtain the wave function as
$\psi(r,\theta,\varphi)=\frac{1}{r}\frac{U(r)}{f(r)}Y_{(\ell)}^{(\Lambda)}(\theta,\varphi)$, with $Y_{(\ell)}^{(\Lambda)}(\theta,\varphi)=\Theta(\theta)\Phi(\varphi)$, for a potential taking the form \cite{b26},
\begin{equation}
V(r,\theta,\varphi)=V_1(r)+\frac{f(r)^2}{r^2}V_2(\theta)+\frac{f(r)^2}{r^2sin^2(\theta)}V_3(\varphi)
\label{E9}
\end{equation}
where $V_1(r)$, $V_2(\theta)$ and $V_3(\varphi)$ are arbitrary functions depending on specific arguments.\\
Then, the position dependent mass Schr\"{o}dinger equation with the non central potential defined in (\ref{E9}) can be transformed into a separate system in all three coordinates \cite{b26}:
\begin{eqnarray}
\label{E10}
\bigg[\frac{d^2}{dr^2}+\frac{2m_0}{\hbar^2}\left(\frac{E-V_1(r)}{f(r)^2}\right)-\frac{L^2}{r^2}- \bar{F}(r,\lambda,\delta) \bigg]U(r)=0, \ r\in[0,\infty]
\end{eqnarray}
where $\bar{F}(r,\lambda,\delta) =\frac{(2-\delta-\lambda)}{f(r)}\left(\frac{f''(r)}{2}+\frac{f'(r)}{r}\right) \left(\left(\frac{1}{2}-\delta\right)\left(\frac{1}{2}-\lambda\right)-\frac{1}{4}\right) \left(\frac{f'(r)}{f(r)}\right)^2$
\begin{align}
&\left[\frac{d^2}{d\theta^2}+cot(\theta)\frac{d}{d\theta}+L^2-\frac{\Lambda^2}{sin^2(\theta)}-\frac{2m_0}{\hbar^2}V_2(\theta)\right]\Theta(\theta)=0, \theta\in[0,\pi]
\label{E11}\\
&\left[\frac{d^2}{d\varphi^2}-\frac{2m_0}{\hbar^2}V_3(\varphi)+\Lambda^2\right]\Phi(\varphi)=0,\ \varphi\in[0,2\pi]
\label{E12}
\end{align}
where $\Lambda^2$ and $L^2=\ell(\ell+1)$ are real and dimensionless separation constants. The components of the wavefunction are also constrained to satisfy the boundary conditions: $U(0)=U(\infty)=0$ for the bound states, or $U(0)=0$ for the continuous states, $ \Phi(\varphi)=\Phi(\varphi+2\pi)$, while $\Theta(0)$ and $\Theta(\pi)$ are finite.
\section{ Scattering state solutions and Phase Shifts}
In this section, we consider a particle influenced by a new non central potential, dubbed the double ring-shaped polynomial field potential, obtained from Eq.(\ref{E9}) with:
\begin{align}
& V_1(r)=a+b\cdot r+c\cdot r^2, \label{E14} \\
& V_2(\theta)=\left( \frac{B}{sin^2(\theta)}+\frac{A(A-1)}{cos^2(\theta)}\right)
\label{E15}
\\
& V_3(\varphi)=\left( \frac{\alpha^2D(D-1)}{sin^2(\alpha\varphi)}+\frac{\alpha^2C(C-1)}{cos^2(\alpha\varphi)}\right)
\label{E16}
\end{align}
where the parameters are chosen as $ A,C,D > 1; a,b, B\geq0; c=\frac{1}{2}m_0\omega^2$, $\alpha=1,2,3,\cdots$.
When $a=b=0$ and $D=C=1$, the potential reduces to a double-ring-shaped oscillator potential. Also, when $a=b=B=0$ and $A=C = D = 1$, it reduces to a spherical oscillator potential, considered as one of the most important models in classical and quantum physics.
In the subsequent subsection, we are going to study the scattering states of the Schr\"{o}dinger equation with the double ring-shaped polynomial field potential in the spherical coordinates.
\subsection{Exact solutions of the first angular equation}
We start our investigation with the angular $\varphi$ part of the Schr\"{o}dinger equation.
After introducing the shape form of the potential shown in Eq. (\ref{E16}) into Eq.(\ref{E12}) we get,
\begin{equation}
\left[\frac{d^2}{d\varphi^2}+\Lambda^2-\frac{2m_0}{\hbar^2}\left(\frac{\alpha^2D(D-1)}{sin^2(\alpha\varphi)}+\frac{\alpha^2C(C-1)}{cos^2(\alpha\varphi)}\right)\right]\Phi(\varphi)=0
\label{E17}
\end{equation}
By defining a new variable $x=\sin(\alpha\varphi)^2$, this equation transforms into,
\begin{equation}
\frac{d^2\Phi(x)}{dx^2}+\frac{\frac{1}{2}-x}{x(1-x)}\frac{d\Phi(x)}{dx}+\frac{(-\xi_1^2x^2+\xi_2^2x-\xi_3^2)}{x^2(1-x)^2}\Phi(x)=0
\label{E18}
\end{equation}
with
\begin{align}
& \xi_1^2=\frac{\Lambda^2}{4\alpha^2},\label{E19}\\
& \xi_2^2=\frac{m_0}{2\hbar^2}\left(D(D-1)-C(C-1)\right)+\frac{\Lambda^2}{4\alpha^2},\label{E20}\\
& \xi_3^2=\frac{m_0}{2\hbar^2}D(D-1)
\label{E21}
\end{align}
According to the Nikiforov-Uvarov procedure, the resulting energy eigenvalues are,
\begin{eqnarray}
n_{\varphi}^2+\left(2n_{\varphi}+1\right)\left( \left(\frac{1}{16}+\left(\xi_1^2+\xi_3^2-\xi_2^2\right)\right)^{\frac{1}{2}}+\left(\frac{1}{16}+\xi_3^2\right)^{\frac{1}{2}}+\frac{1}{2}\right)+\nonumber \\
2\left(\frac{1}{16}+\left(\xi_1^2+\xi_3^2-\xi_2^2\right)\right)^{\frac{1}{2}}\left(\frac{1}{16}+\xi_3^2\right)^{\frac{1}{2}}+\left(2\xi_3^2-\xi_2^2-\frac{1}{8}\right)=0
\label{E22}
\end{eqnarray}
Substituting $\xi_1$, $\xi_2$ and $\xi_3$ by their expressions given in Eq. (\ref{E19}), Eq. (\ref{E20}) and Eq. (\ref{E21}) respectively, we finally derive the exact formula of $\Lambda$
\begin{equation}
\Lambda=\pm\alpha\left(\frac{\sqrt{1+\frac{8m_0C(C-1)}{\hbar^2}}}{2}+\frac{\sqrt{1+\frac{8m_0D(D-1)}{\hbar^2}}}{2}+2n_{\varphi}+1\right),\ n_{\varphi}=0,1,2,\cdots
\label{E23}
\end{equation}
which exactly reproduce the result reported in \cite{b26}.
The corresponding eigenfunctions of Eq. (\ref{E18}) read as,
\begin{equation}
\Phi(x)=x^{\frac{1}{2}+\left(\frac{1}{16}+\xi_3^2\right)^{\frac{1}{2}}}\left(1-x\right)^{\frac{1}{4}+\left(\frac{1}{16}+\xi_1^2+\xi_3^2-\xi_2^2\right)^{\frac{1}{2}}}
P_{n_{\varphi}}^{\left( \left(\frac{1}{4}+4\xi_3^2\right)^{\frac{1}{2}}, \left(\frac{1}{4}+4\left(\xi_1^2+\xi_2^2-\xi_3^2\right) \right)^{\frac{1}{2}} \right)}\left(1-2x\right)
\label{E24}
\end{equation}
where $P_{n}^{(a,b)}(z)$ is the generalised Jacobi functions.
\subsection{Exact solutions of the second angular equation}
The substitution of the potential (\ref{E14}) into Eq. (\ref{E10}) leads to the following differential equation :
\begin{equation}
\left[\frac{d^2}{d\theta^2}+cot(\theta)\frac{d}{d\theta}+L^2-\frac{\Lambda^2}{sin^2(\theta)}-\frac{2m_0}{\hbar^2}\left\lbrace \frac{B}{sin^2(\theta)}+\frac{A(A-1)}{cos^2(\theta)}\right\rbrace \right]\Theta(\theta)=0
\label{E25}
\end{equation}
To solve this equation, we also introduce the transformation $z=\cos(\theta)^2$, so we obtain,
\begin{equation}
\frac{d^2\Theta(z)}{dz^2}+\frac{1-\frac{3}{2}z}{z(1-z)}\frac{d\Theta(z)}{dz}+\frac{\left(-\chi_1^2z^2+\chi_2^2z-\chi_3^2\right)}{z^2(1-z)^2}\Theta(z)=0,
\label{E26}
\end{equation}
with
\begin{align}
&\chi_1^2=\frac{L^2}{4}=\frac{\ell(\ell+1)}{4}
\label{E27}
\\
&\chi_2^2=\frac{m_0}{2\hbar^2}\left(B-A\left(A-1\right)\right)+\frac{1}{4}\left(L^2+\Lambda^2\right)
\label{E28}
\\
&\chi_3^2=\frac{m_0}{2\hbar^2}B+\frac{\Lambda^2}{4}
\label{E29}
\end{align}
Like Eq. (\ref{E18}), the eigenfunctions of Eq. (\ref{E26}) are the generalised Jacobi functions,
\begin{equation}
\Theta(z)=z^{\chi_3}\left(1-z\right)^{\frac{1}{4}+\left(\frac{1}{16}+\chi_1^2+\chi_3^2-\chi_2^2\right)^{\frac{1}{2}}}P_{n_{\theta}}^{\left( 2\chi_3 , \left(\frac{1}{4}+4\left(\chi_1^2+\chi_2^2-\chi_3^2\right) \right)^{\frac{1}{2}} \right)}\left(1-2z\right)
\label{E30}
\end{equation}
and the corresponding eigenvalues are solutions of the equation,
\begin{eqnarray}
&\frac{1}{2}n_{\theta}+n_{\theta}^2+\left(2n_{\theta}+1\right)\left(\left(\frac{1}{16}+\chi_1^2+\chi_3^2-\chi_2^2\right)^{\frac{1}{2}}+\chi_3+\frac{1}{4}\right)+2\chi_3\left(\frac{1}{16}+\chi_1^2+\chi_3^2-\chi_2^2\right)^{\frac{1}{2}}\nonumber \\&+2\chi_3^2-\chi_2^2=0
\label{E31}
\end{eqnarray}
Substituting $\chi_1$, $\chi_2$ and $\chi_2$ by their expressions shown in Eq. (\ref{E27}), Eq. (\ref{E28}) and Eq. (\ref{E29}) respectively, we finally obtain the full expression of $\ell$:
\begin{equation}
\ell=\frac{1+\sqrt{1+\frac{8m_0A(A-1)}{\hbar^2}}}{2}+\sqrt{\Lambda^2+\frac{2m_0B}{\hbar^2}}+2n_{\theta},\ n_{\theta}=0,1,2,\cdots
\label{E32}
\end{equation}
which again coincides with the formula derived in \cite{b26} using the asymptotic iteration method.
\subsection{Scattering Phase Shifts}
In order to study the scattering state and phase shifts in the problem of position-dependent mass Schr\"{o}dinger equation with the double ring shaped polynomial field potential given by Eq.(\ref{E13}), we must define the deformation function $f(r)$. Thus, in this section, we use a simple linear representation \cite{b26},
\begin{equation}
f(r)=1+f_0 r,\
\label{E33}
\end{equation}
By substituting into Eq. (\ref{E10}) and by using the potential (\ref{E14}), the confluent forms of Heun's differential equation show up:
\begin{equation}
\left[\frac{d^2}{dr^2}+\frac{2m_0}{\hbar^2(1+f_0r)^2}\left(E-a-cr^2-br-\frac{\hbar^2}{2m_0}P\right)-\frac{Q}{r(1+f_0r)}-\frac{L^2}{r^2}\right]U(r)=0
\label{HeunE}
\end{equation}
with
\begin{eqnarray}
P=\left[\left(\frac{1}{2}-\lambda\right)\left(\frac{1}{2}-\delta\right)-\frac{1}{4}\right]f_0,\ Q=\left[2-\delta-\lambda\right]f_0
\end{eqnarray}
This equation is not easy to solve, however, because we are dealing with scattering states, we can safely replace $1+f_0r$ by $f_0r$ in Eq. (\ref{HeunE}) with $f_0 > 0$. Consequently, the above Heun's equation simplifies to the following differential equation,
\begin{equation}
\left[\frac{d^2}{dr^2}+\left(-\frac{\ell'\left(\ell'+1\right)}{r^2}+\bar{K}^2+\frac{2\bar{\Lambda}}{r}\right)\right]U_{n,\ell'}(r)=0,
\label{E34}
\end{equation}
with
\begin{align}
& \ell'\left(\ell'+1\right)=\ell(\ell+1)-\frac{2m_0}{\hbar^2}\left(\frac{E-a}{f_0^2}\right)+\left(\frac{1}{2}-\delta\right)\left(\frac{1}{2}-\lambda\right)-(\lambda+\delta)+\frac{7}{4}
\label{E35}
\\
& \bar{\Lambda}=-\frac{m_0}{\hbar^2}\frac{b}{f_0^2}
\label{E36}
\\
& \bar{K}^2= -\frac{m_0}{\hbar^2}\frac{2c}{f_0^2}
\label{E37}
\end{align}
Notice here that the $\ell'$ parameter plays the role of the orbital angular momentum in problems with spherical central potentials.
Having in mind the boundary conditions of the scattering states, i.e. $U_{n,\ell'}(r=0)=0$, we use the following ansatz for the asymptotic behavior of the wave function at the origin:
\begin{equation}
U_{n,\ell'}(r)=A\cdot (\bar{K}r)^{\ell'+1}e^{i\bar{K}r}\xi_{n,\ell'}(r)
\label{E38}
\end{equation}
Insertion of Eq. (\ref{E38}) into Eq. (\ref{E34}) results in
\begin{equation}
\left[r\frac{d^2}{dr^2}+\left(2\ell'+2i\bar{K}r+2\right)\frac{d}{dr}+\left(2\bar{\Lambda}+2i\bar{K}\left(\ell'+1\right)\right)\right]\xi_{n,\ell'}(r)=0
\label{E39}
\end{equation}
If in addition we introduce a new variable $s =- 2i\bar{K}r$, then Eq. (\ref{E39}) can be alternatively written as
\begin{equation}
\left[s\frac{d^2}{ds^2}+\left(2\ell'+2-s\right)\frac{d}{ds}+\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}}\right)\right]\xi_{n,\ell'}(s)=0,
\label{E40}
\end{equation}
with $s=|s|e^{-i\frac{\pi}{2}}$. Equation (\ref{E40}) is just the confluent Hypergeometric equation. General form of confluent Hypergeometric can be written as
\begin{equation*}
z\frac{{{d^2}w}}{{d{z^2}}} + (b - z)\frac{{dw}}{{dz}} - aw = 0,
\end{equation*}
where $a$ and $b$ are constant. The Solution of above equation can be written with the aid of Kummer's functions as
\begin{equation*}
M\left( {a,b,z} \right) = \sum\limits_{n = 0}^\infty {\left( {\frac{{{a^{\left( n \right)}}{z^n}}}{{{b^{\left( n \right)}}n!}}} \right) = {}_1{F_1}} \left( {a,b,z} \right),
\end{equation*}
comparing \eqref{E40} with the general form of confluent Hypergeometric differential equation results in
, as $r\rightarrow 0$
\begin{equation}
\xi_{n,\ell'}(s)={}_1F_1\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}},2\ell'+2,-2i\bar{K}r\right).
\label{E41}
\end{equation}
Therefore, our analytical expression of the radial wave function for the scattering states is obtained by substituting Eq.\eqref{E41} into Eq.\eqref{E38} :
\begin{equation}
U_{n,\ell'}(r)=A\cdot (\bar{K}r)^{\ell'+1}e^{i\bar{K}r}{}_1F_1\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}},2\ell'+2,-2i\bar{K}r\right).
\label{E42}
\end{equation}
Now, we want to obtain the asymptotic behavior of the wave function for $r\rightarrow 0$, then calculate the normalization constant and the phase shifts. To this end, we use the transformation formulae for confluent Hypergeometric function when $s\rightarrow \infty$:
\begin{equation}
{}_1F_1\left(\alpha,\gamma,s\right)=\frac{\Gamma(\gamma)}{\Gamma(\alpha)}e^ss^{\alpha-\gamma}+\frac{\Gamma(\gamma)}{\Gamma(\gamma-\alpha)}e^{\pm i\pi\alpha}s^{-\alpha}
\label{E43}
\end{equation}
where $"+"$ and $"-"$ correspond to $arg(s)\in]-\pi/2,3\pi/2[$ and $arg(s)\in]-3\pi/2,\pi/2[$ respectively.
By subtituting $s=|s|e^{-i\frac{\pi}{2}}$, Eq. (\ref{E43}) is re-expressed as
\begin{equation}
{}_1F_1\left(\alpha,\gamma,s\right)=\frac{\Gamma(\gamma)}{\Gamma(\alpha)}e^s|s|^{\alpha-\gamma}e^{-i(\alpha-\gamma)\pi/2}+\frac{\Gamma(\gamma)}{\Gamma(\gamma-\alpha)}e^{ i\pi\alpha/2}|s|^{-\alpha}
\label{E44}
\end{equation}
from which we obtain
\begin{eqnarray}
&{}_1F_1\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}},2\ell'+2,-2i\bar{K}r\right)\nonumber = \frac{\Gamma(2\ell'+2)}{\Gamma(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}})}e^{-2i\bar{K}}\left(2\bar{K}r\right)^{-\left(\ell'+1+i\bar{\Lambda}/\bar{K}\right)}
e^{i\left(\ell'+1+i\bar{\Lambda}/\bar{K}\right)\pi/2} \nonumber\\
&+\frac{\Gamma(2\ell'+2)}{\Gamma(\ell'+1+i\frac{\bar{\Lambda}}{\bar{K}})}\left(2\bar{K}r\right)^{-\left(\ell'+1-i\bar{\Lambda}/\bar{K}\right)}
e^{-i\left(\ell'+1-i\bar{\Lambda}/\bar{K}\right)\pi/2}
\label{E45}
\end{eqnarray}
Because
\begin{equation}
\Gamma\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}}\right)=\left| \Gamma\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}}\right)\right|e^{i\delta'}, \quad \delta_{\ell}'=arg\left(\Gamma\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}}\right)\right)
\label{E46}
\end{equation}
and
\begin{equation}
\Gamma\left(\ell'+1+i\frac{\bar{\Lambda}}{\bar{K}}\right)=\left| \Gamma\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}}\right)\right|e^{-i\delta'}
\label{E47}
\end{equation}
where $\delta'$ is a real number. Eq. (\ref{E45}) becomes
\begin{eqnarray}
&{}_1F_1\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}},2\ell'+2,-2i\bar{K}r\right)\nonumber = \frac{\Gamma(2\ell'+2)}{\Gamma(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}})}\left(\frac{e^{-\frac{\bar{\Lambda}\pi}{2\bar{K}}}\cdot e^{-i\bar{K}r} }{\left(2\bar{K}r\right)^{\ell'+1}}\right) \nonumber\\ &\times 2\sin\left(\bar{K}r+\bar{\Lambda}\ln(2\bar{K}r)/\bar{K}+\delta'-\ell'\pi/2-\pi/2\right)
\label{E48}
\end{eqnarray}
By putting Eq. (\ref{E48}) into Eq. (\ref{E42}), we get,
\begin{equation}
U_{n,\ell'}(r\rightarrow \infty)=A\cdot \frac{\Gamma(2\ell'+2)}{\Gamma(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}})}\left(\frac{e^{-\frac{\bar{\Lambda}\pi}{2\bar{K}}} }{\left(2\right)^{\ell'+1}}\right) 2\sin\left(\bar{K}r+\bar{\Lambda}\ln(2\bar{K}r)/\bar{K}+\delta'-\ell'\pi/2\right)
\label{E49}
\end{equation}
On the other hand, using the asymptotic behavior
\begin{equation}
U_{n,\ell}(r\rightarrow \infty)= 2\sin\left(\bar{K}r+\bar{\Lambda}\ln(2\bar{K}r)/\bar{K}+\delta_{\ell}-\ell\pi/2\right)
\label{E50}
\end{equation}
and comparing the arguments of sine term in Eqs. \eqref{E49} and \eqref{E50}, one can then derive the phase shifts,
\begin{equation}
\delta_{\ell}=\frac{\pi(\ell-\ell')}{2}+\delta_{\ell}', \
\label{E51}
\end{equation}
where $\ell'$ is given by Eq. (\ref{E35}), and the normalization constant also can be evaluated by comparison of coefficients of sine term in Eqs. \eqref{E49} and \eqref{E50} as
\begin{equation}
A=\frac{\Gamma\left(\ell'+1-i\frac{\bar{\Lambda}}{\bar{K}}\right)}{\Gamma(2\ell'+2)}2^{(\ell'+1)}e^{\frac{\pi\bar{\Lambda}}{2\bar{K}}}.
\label{E52}
\end{equation}
\section{ Conclusion}
In this paper, we have considered the time-independent Schr\"{o}dinger equation with a position-dependent effective mass in non central potential. Using the potential form proposed in \cite{b26}, we separated the deformed Schr\"{o}dinger equation in all coordinates. The radial solution is then obtained, as well as the exact analytical angular solution. We have also studied the scattering states of the deformed Schr\"{o}dinger equation under a non central effective potential and derived the energy eigenvalues and the normalization constant of the radial wave functions, as well as the scattering phase shifts.
\section*{ Acknowledgment}
The authors, to be great pleasure in thanking the referee for his/her helpful comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,785 |
Q: How to enable multiple logon remote desktop in Windows 7 In Vista you could copy the 2k8 server .dll for remote desktop and it would let you use remote desktop on the computer without logging off the active user.
Does this work for Windows 7? Has anyone tried?
A: To enable concurrent sessions you can use this patch or copy the files from Server 2k8 (the patch does the same thing, 32 and 64-bit DLLs are in the zip file). Then follow these directions:
*
*Once downloaded, extract the files into a directory (for the purposes of
this guide, it will be assumed that
the files have been extracted to the
folder C:\Win7RDP )
*Open Windows Explorer to the above folder
*Right Click on "install.cmd" and select "Run as Administrator"
*Wait for the script to run entirely. At the end, you should see
something similar to below:
source
A: The patch from Mission Guide (mentioned above) doesn't help me. But I've found RDP patcher where you can patch and unpatch it and enable/disable options whenever you want.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,633 |
import unittest
from mock import patch
from xsvm.parser import parse_line, load_into_memory
from xsvm.vm import Processor
import xsvm.instructions
class ProcessorTestCase(unittest.TestCase):
def setUp(self):
self.cpu = Processor()
def test_fetch_valid_instruction(self):
load_into_memory(self.cpu.memory, ["mov r0, #1", "mov r1, #5"])
inst_1 = self.cpu.fetch_instruction()
self.assertEqual(self.cpu.register_bank.get("pc"), 1)
self.assertEqual(self.cpu.instructions_executed, 0)
self.assertEqual(inst_1.mnemonic, "mov")
self.assertEqual(inst_1.operands[0].value, "r0")
self.assertEqual(inst_1.operands[1].value, 1)
inst_2 = self.cpu.fetch_instruction()
self.assertEqual(self.cpu.register_bank.get("pc"), 2)
self.assertEqual(inst_2.mnemonic, "mov")
self.assertEqual(inst_2.operands[0].value, "r1")
self.assertEqual(inst_2.operands[1].value, 5)
def test_fetch_non_instruction(self):
self.assertRaises(RuntimeError, lambda: self.cpu.fetch_instruction())
@patch.object(xsvm.instructions, 'exec_nop')
def test_execute_instruction(self, mock):
nop_instruction = parse_line("nop")
processor = Processor()
processor.execute_instruction(nop_instruction)
self.assertTrue(mock.called)
self.assertEqual(processor.instructions_executed, 1)
@patch.object(xsvm.instructions, 'exec_nop')
def test_halting(self, mock):
nop_instruction = parse_line("nop")
processor = Processor()
processor.halt()
processor.execute_instruction(nop_instruction)
self.assertFalse(mock.called)
self.assertEqual(processor.instructions_executed, 0)
def test_exiting_on_halt(self):
load_into_memory(self.cpu.memory, ["nop", "nop", "swi #0", "nop"])
self.cpu.execute_until_halted()
self.assertEqual(self.cpu.instructions_executed, 3)
self.assertEqual(self.cpu.register_bank.get("pc"), 3)
if __name__ == '__main__':
unittest.main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,920 |
Seven (1974) es un álbum de la banda británica de jazz fusion Soft Machine.
Historia
El séptimo álbum de estudio fue uno de transición para la banda en varios aspectos.
Sólo Mike Ratledge quedaba de la era "clásica" (la anterior a Six), y estaba perdiendo interés en componer
Entró Roy Babbington, el tercer exmiembro de Nucleus en sumarse a Soft Machine
Fue el último álbum con título numerado (es decir, el próximo se habría llamado Eight) y el último con el sello CBS
Además de esto, las composiciones de Karl Jenkins se vuelven más minimalistas, sencillas y repetitivas, lo que es visto como una "involución" para algunos. El grupo fue presionado por la discográfica a presentar nuevo material en una gira en EE.UU., por lo que todos los temas fueron escritos en poco tiempo excepto "Down the road", anterior a las sesiones. Ratledge fue criticado por copiar la canción "Follow Your Heart" de John McLaughlin en su "Days Eye", mientras Jenkins se "copió" a sí mismo en el tema "Penny Hitch", prácticamente igual a "Soft Weed Factor" del disco anterior.
El repertorio de Seven fue tocado pocas veces en vivo. A finales de 1973 comenzaría una nueva etapa para Soft Machine, cuando el baterista John Marshall sugirió contratar a un guitarrista, el primero en más de cinco años.
Canciones
"Nettle Bed" (Karl Jenkins) – 4:47
"Carol Ann" (Jenkins) – 3:48
"Day's Eye" (Mike Ratledge) – 5:05
"Bone Fire" (Ratledge) – 0:32
"Tarabos" (Ratledge) – 4:32
"D.I.S." (John Marshall) – 3:02
"Snodland" (Jenkins) – 1:50
"Penny Hitch" (Jenkins) – 6:40
"Block" (Jenkins) – 4:17
"Down the Road" (Jenkins) – 5:48
"The German Lesson" (Ratledge) – 1:53
"The French Lesson" (Jenkins) – 1:01
Personnel
Roy Babbington - bajo
Mike Ratledge - teclados
Karl Jenkins - oboe, saxofón barítono y soprano, (Fender Rhodes & Hohner) piano eléctrico
John Marshall - batería, percusión
Véase también
Discografía de Soft Machine
Referencias
Enlaces externos
Noisette (en inglés)
Álbumes de Soft Machine
Álbumes de 1973
Álbumes de jazz rock
Álbumes en inglés | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,783 |
import { Readable } from 'stream'
import { Submittable, Connection } from 'pg'
import Cursor from 'pg-cursor'
interface QueryStreamConfig {
batchSize?: number
highWaterMark?: number
rowMode?: 'array'
types?: any
}
class QueryStream extends Readable implements Submittable {
cursor: any
_result: any
handleRowDescription: Function
handleDataRow: Function
handlePortalSuspended: Function
handleCommandComplete: Function
handleReadyForQuery: Function
handleError: Function
handleEmptyQuery: Function
public constructor(text: string, values?: any[], config: QueryStreamConfig = {}) {
const { batchSize, highWaterMark = 100 } = config
super({ objectMode: true, autoDestroy: true, highWaterMark: batchSize || highWaterMark })
this.cursor = new Cursor(text, values, config)
// delegate Submittable callbacks to cursor
this.handleRowDescription = this.cursor.handleRowDescription.bind(this.cursor)
this.handleDataRow = this.cursor.handleDataRow.bind(this.cursor)
this.handlePortalSuspended = this.cursor.handlePortalSuspended.bind(this.cursor)
this.handleCommandComplete = this.cursor.handleCommandComplete.bind(this.cursor)
this.handleReadyForQuery = this.cursor.handleReadyForQuery.bind(this.cursor)
this.handleError = this.cursor.handleError.bind(this.cursor)
this.handleEmptyQuery = this.cursor.handleEmptyQuery.bind(this.cursor)
// pg client sets types via _result property
this._result = this.cursor._result
}
public submit(connection: Connection): void {
this.cursor.submit(connection)
}
public _destroy(_err: Error, cb: Function) {
this.cursor.close((err?: Error) => {
cb(err || _err)
})
}
// https://nodejs.org/api/stream.html#stream_readable_read_size_1
public _read(size: number) {
this.cursor.read(size, (err: Error, rows: any[]) => {
if (err) {
// https://nodejs.org/api/stream.html#stream_errors_while_reading
this.destroy(err)
} else {
for (const row of rows) this.push(row)
if (rows.length < size) this.push(null)
}
})
}
}
export = QueryStream
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,789 |
\section{Introduction}
TV\,Ret was reported by Kinman et al. (\cite{kinm+91}), who were
observing lightcurves of RR\,Lyr stars in the outer halo of the
Large Magellanic Cloud. In one of the objects, which was named R1 in their
list, they noticed an increase in brightness, which they interpreted
as a possible dwarf nova outburst, and thus classified the object as a
cataclysmic variable.
As such, it was added in the 72nd name-list of variable stars as TV\,Ret
(Kazarovets \& Samus \cite{kaza+95}), and entered the catalogue of cataclysmic
variables (Downes et al. \cite{down+01}).
We performed spectroscopic observations in order to confirm this
classification. Surprisingly, the
object turned out to be a narrow emission--line galaxy of nearly 0.1 redshift.
This class of galaxies consists of two different types:
starburst or H\,II-galaxies and the narrow--emission line AGNs, which
are either Seyfert 2 galaxies or LINERs.
In the first case, the emission lines origin in gas that is photo-ionised
by young, hot OB stars present in the star--forming regions.
In the case of AGNs the emission lines are formed in the Narrow Line Region
which is comprised of clouds ionised by radiation from the central engine
of the galaxy.
Several empirical methods have been developed
to distinguish between these two ionisation mechanisms, mainly by
comparing the line--ratios in the spectrum
(see e.g. Baldwin et al. \cite{bald+81};
Veilleux \& Osterbrock, \cite{veil+87};
Dessauges-Zavadsky et al., \cite{dess+00}).
We attempt such a distinction and also derive the physical parameters
of the galaxy such as size, luminosity, metallicity, and gas
temperature and density.
\section{Observations and data reduction}
\begin{table}[b]
\caption{\label{obstab} All observations obtained for this investigation
listed with
their date, the telescope/instrument combination, the used grism and
slit width, and the exposure time.}
\begin{tabular}{c c c c}
\hline
\hline
\noalign{\smallskip}
Date & Telescope/Instrument & Grism/Slit & Exp-time [s] \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
2003-09-26 & 3.6m / EFOSC2 & Gr\#4 / 1.0$^{\prime\prime}$ & 1800 \\
2004-11-19 & 3.6m / EFOSC2 & Gr\#11 / 1.0$^{\prime\prime}$ & $3\times 900$ \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
The observations were performed on 2003-09-26 and on 2004-11-19 using
EFOSC2 (Buzzoni et al. \cite{buzz+84})
at the 3.6 m telescope at La Silla, Chile.
In Table \ref{obstab} the observational
parameters are summarised.
The standard reduction of the data was performed using IRAF.
The bias was subtracted and the data were divided by a flat field,
which was normalised by fitting Chebyshev functions of high order.
The spectra were optimally extracted (Horne \cite{horn86}).
Wavelength calibration yielded a final resolution of 1.1\,nm\ FWHM
for the 2003 data and 1.2\,nm\ FWHM for the 2004 data.
Flux calibration was performed only for the 2003 spectrum, using the
spectrophotometric standard LTT\,7379 which was observed
with an airmass difference of 0.05.
For all further analysis, if not especially indicated,
the MIDAS package and self--written routines were used.
\section{Results}
\begin{figure}
\rotatebox{-90}{\resizebox{!}{8.7cm}{\includegraphics{spec_tvret_1.ps}}}
\caption{\label{spec1} The flux--calibrated spectrum of TV Ret taken
in September 2003.
The upper plot shows the full intensity range;
in the lower plot the intensity range has been decreased to
better display the weak emission lines and the slope of the continuum.}
\end{figure}
\begin{figure}
\rotatebox{-90}{\resizebox{!}{8.7cm}{\includegraphics{spec_tvret_2.ps}}}
\caption{\label{spec2} The spectrum of TV Ret taken in November 2004
with the continuum
normalised to unity. The upper plot shows the full intensity range;
in the lower plot the intensity range has been decreased to
better display the weak emission lines.}
\end{figure}
\begin{figure}
\resizebox{8.7cm}{!}{\includegraphics{acq.ps}}
\caption{\label{finding} The R--band acquisition image shows the
$3 \times 3$\,arcmin
surroundings of TV\,Ret (in the centre) and a zoom on the object;
north is up, east is left.
With the good seeing of 0.6" the object appears clearly extended.
On the right side, the contours of TV Ret (upper plot) and a star
for comparison are plotted.}
\end{figure}
\begin{table*}
\caption{\label{linetab} Observed wavelengths, equivalent widths,
and FWHM of all identified emission
lines in the 2002 and 2003 spectrum of TV\,Ret. For 2003 the line
flux is also given.
Note that the uncertainty of the line flux describes the
uncertainty of the relative flux in the line and does not include the
photometric error. The line flux has been dereddened using the reddening laws
for H\,II regions and for AGNs (see text for details).}
\begin{tabular}{l r r r r r r r r r}
\hline
\hline
\noalign{\smallskip}
& \multicolumn{6}{c}{September 2003} & ~ ~ ~ ~ & \multicolumn{2}{c}{November 2004}\\
Transition &
\multicolumn{1}{c}{$\lambda_{\rm obs}$ [nm]} &
\multicolumn{1}{c}{$-W$ [nm]} &
\multicolumn{1}{c}{$ F [10^{-18}$\,W\,m$^{-2}]$} &
\multicolumn{1}{c}{$F_{\rm SB\,dered}$} &
\multicolumn{1}{c}{$F_{\rm AGN\,dered}$} &
\multicolumn{1}{c}{$ 100\cdot F_{\rm SB\,dered} / F_{\rm H\beta}$} & &
\multicolumn{1}{c}{$\lambda_{\rm obs}$ [nm]} &
\multicolumn{1}{c}{$-W$ [nm]} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
H$_\alpha$ & 719.47 & 46.0 & 3.51(3) & 4.55 & 3.85 & 285(10) & & 719.42 & 46.1 \\
H$_\beta$ & 532.97 & 10.5 & 1.08(2) & 1.60 & 1.24 & 100(4) & & 533.16 & 9.0 \\
H$_\gamma$ & 475.97 & 3.5 & 0.46(2) & 0.72 & 0.54 & 45(3) & & 476.14 & 3.4 \\
H$_\delta$ & 449.76 & 1.8 & 0.26(5) & 0.42 & 0.31 & 26(6) & & 450.01 & 1.4 \\
H$_\epsilon$ & 434.94 & 1.8 & 0.26(2) & 0.42 & 0.31 & 26(3) & & 435.48 & 1.2 \\
H$_8$ & 425.83 & & & & & & & 426.68 \\
{[}S\,II]: 673.1 & 737.81 & 3.1 & 0.24(2) & & & 15(2) & & 737.58 & \\
{[}S\,II]: 671.7 & 736.36 & 3.7 & 0.29(2) & 0.68 & 0.58 & 18(2) & & 736.20 & 6.8 \\
{[}N\,II]: 658.4 & 721.70 & 3.6 & 0.28(3) & 0.36 & 0.31 & 23(3) & & 721.64 & 3.9 \\
{[}O\,I]: $630.0^{(1)}$&691.20&2.3& 0.17(2) & 0.22 & 0.19 & 14(2) & & & 1.6 \\
He\,I: 587.56 & 644.12 & 2.2 & 0.20(2) & 0.27 & 0.22 & 17(2) & & 644.20 & 2.0 \\
{[}O\,III]: 500.7 & 548.95 & 47.2 & 5.23(2) & 7.61 & 5.97 & 477(15) & & 549.04 & 50.4 \\
{[}O\,III]: 495.9 & 543.69 & 16.0 & 1.73(2) & 2.53 & 1.98 & 159(6) & & 543.82 & 17.1 \\
{[}O\,III]: 436.3 & 478.53 & 0.6 & 0.07(1) & 0.11 & 0.08 & 7(1) & & 478.57 & 0.5 \\
{[}Ne\,III]: 386.8& 423.44 & 3.1 & 0.48(3) & 0.78 & 0.57 & 49(4) & & 424.42 & 3.0 \\
{[}O\,II]: 372.7 & & & & & & 169(15) & & 408.93 & 16.2\\
\noalign{\smallskip}
\hline
\end{tabular}\\
(1): blend of several O\,I lines.
\end{table*}
\subsection{Classification}
In Fig.~\ref{spec1} the flux--calibrated spectrum observed in 2003
is plotted. The spectrum is clearly dominated by strong emission lines.
The continuum matches the spectral energy distribution (SED) of a late A or
early F type
star; the corresponding temperature is $T_{\rm eff} = 7500$\,K.
In Fig. \ref{spec2},
the normalised spectrum from 2004 is plotted, which shows the same
emission lines as the 2003 spectrum and an additional line at 408.9\,nm,
which was outside the spectral range of the 2003 data.
Both spectra show clearly that the object is
not a cataclysmic variable, as the typical emission lines for this kind
of object are not present at their rest wavelengths.
Instead, we find several strong emission lines,
which turned out to be redshifted Balmer lines as well as
O\,I, O\,III, N\,II, S\,II, and Ne\,III. The properties of these lines
are given in Table~\ref{linetab}.
We averaged the individual shifts of these
lines to find the redshift of the object as $z = 0.0964(2)$.
The redshift, which indicates an extragalactic object, as well as the
strength of the ionised lines, are best interpreted if we assume the
object to be an emission line galaxy,
which, however, is unresolved in previous images.
In Fig.~\ref{finding}, a $3\times 3$\,arcmin R--image is plotted,
obtained under good seeing conditions of 0.6\,arcsec.
In this image, the
object appears extended and slightly elongated. The size can be estimated to
about 0.7$\times$0.9\,arcsec.
\subsection{Physical properties of the galaxy}
For all further calculations, we use the standard flat cosmology with
$\Omega_M = 0.27$ and $H_0 = 72\,\rm km\,s^{-1}\,Mpc^{-1}$
(Spergel et al. \cite{sper+03}).
From $z = 0.0964(2)$
we derive the angular size distance
$D_{\rm ang} = 358$\,Mpc. Thus, assuming that we see most of it,
the size of the galaxy is
$1.2\times 1.6$\,kpc, which matches the size of a rather small
dwarf galaxy.
The magnitude of TV\,Ret is given as $B=20.5$ (Kinman et al. \cite{kinm+91}).
The absolute B--magnitude can be derived using
\begin{equation}
M_B = B - 5\log{D_{\rm lum}} +5 -K
\end{equation}
with the luminosity distance calculated as $D_{\rm lum}=430$\,Mpc,
and the k--correction
$K = -2.5 \log{\left((1+z)^{-1} L_{\lambda (1+z)^{-1}} / L_{\lambda}\right)}$
(e.g. Oke \& Sandage \cite{oke+68}).
We use the slope
$L_{\rm 401nm}/L_{\rm 440nm} = 1.15$ as derived from the spectrum
plotted in Fig.\ref{spec1},
and derive the absolute magnitude
of the galaxy as $M_B = -17.5$. This value is typical of massive
dwarf galaxies (see e.g. Melisse \& Israel \cite{meli+94}) and thus confirms
the above classification. However, the
surface brightness of
$\sigma_B = 20.0$\,mag/arcsec$^2$
is rather high.
\subsubsection{Starburst or AGN?}
\begin{table}
\caption{\label{loglinerat} The logarithm of the line ratios is given for
the two reddening laws, starburst (SB) and AGN,
as derived from $\rm H\alpha / H\beta$.
The forbidden line fluxes correspond to the following transitions:
O\,III: 500.7\,nm, N\,II: 658.3\,nm, S\,II: 671.6\,+\,673.1\,nm, and
O\,I: 630.0\,nm.}
\begin{tabular}{c c c c c c}
\hline
\hline
\noalign{\smallskip}
& $\rm \log{\frac{[O\,III]}{H\beta}}$ & $\rm \log{\frac{[N\,II]}{H\alpha}}$ & $\rm \log{\frac{[S\,II]}{H\alpha}}$ & $\rm \log{\frac{[O\,I]}{H\alpha}}$ & $\rm \frac{H\alpha}{H\beta}$ \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
SB & 0.68 & -1.10 & -0.83 & -1.32 & 2.85 \\
AGN & 0.68 & -1.09 & -0.82 & -1.31 & 3.10 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{table}
A more detailed classification of the galaxy can be done by analysing the
emission lines.
Apart from the features at 478.5\,nm and 737\,nm, which are clearly blends of
{[}O\,I] and {[}SII\,673.1]/{[}SII\,671.7] respectively, the lines are not resolved
within our resolution of 1.1\,nm,
and therefore are narrow--emission lines.
Thus, TV\,Ret could be either an H\,II galaxy or a narrow--line AGN.
We use the flux--ratio of the different emission lines to find the
mechanism by which the emission lines are produced following the method
described by Veilleux \& Osterbrock (\cite{veil+87}).
To have a comparable data set, we also determined the reddening in the
same way as these authors, using the
H$\alpha$/H$\beta$ flux--ratio and the Whitford reddening curve parameterised
by Miller \& Mathews (\cite{mill+72}). We measure
$F({\rm H\alpha}) / F({\rm H\beta}) = 3.25$. If we assume
a starburst galaxy with a recombination value
$I({\rm H\beta}) / I({\rm H\alpha})=2.85$, the reddening
is $E(B-V) = 0.13$. If we assume an AGN with
$I({\rm H\beta}) / I({\rm H\alpha})=3.1$, we find a rather low value of
$E(B-V) = 0.05$. For both cases, the dereddened flux values are listed
in Table~\ref{linetab}. Although the dereddened flux values are different,
the logarithms of the line ratios used for the classification are similar
(see Table~\ref{loglinerat}). This is expected, as the line ratios were chosen
for being insensitive to the reddening, i.e. are close in wavelength.
We compared the line ratios of the forbidden lines (see Table~\ref{loglinerat})
with the values given by Veilleux \& Osterbrock
(\cite{veil+87}), i.e. their figures 1--6. In all these diagnostic diagrams,
TV\,Ret lies close to the
border between AGNs and H\,II region--like objects, which is mainly due
to the high value of
$\rm {[}O\,III]\lambda 500.7 / H\beta$.
In $\rm {[}O\,III] / H\beta$ versus
$\rm {[}N\,II] / H\alpha$ and $\rm {[}O\,III] / H\beta$
versus $\rm {[}S\,II] / H\alpha$,
it lies on the side of the H\,II region--like objects, while for
$\rm {[}O\,III] / H\beta$ versus $\rm {[}O\,I] / H\alpha$
it lies on the side of the AGNs.
Comparing the line ratios with the diagnostic diagrams of
Dessauges-Zavadsky (\cite{dess+00}) yields similar conclusions.
We can definitely exclude that TV\,Ret is a LINER, but it is on the
border between the Seyfert 2 and the starburst galaxies, although the
latter appears slightly more likely.
\subsubsection{Temperature and metallicity}
\begin{table}
\caption{\label{parameter} Physical parameters of the HII galaxy}
\begin{centering}\begin{tabular}{rrl}
&
&
\tabularnewline
\hline
\hline
\multicolumn{1}{c}{Quantity }&
\multicolumn{1}{c}{Value}&
\multicolumn{1}{c}{Notes}\tabularnewline
\hline
$T({\rm OIII})$&
$13400_{938}^{969}$ K &
\tabularnewline
$n({\rm SII})$&
$239_{177}^{433}$ cm$^{-3}$ &
\tabularnewline
${\rm O}^{++}/{\rm H}^{+}$ &
$(7.02\pm0.19)\cdot 10^{-5}$ &
average $\lambda5007$ and $\lambda4959$ \tabularnewline
${\rm O}^{+}/{\rm H}^{+}$&
$(2.39\pm0.90)\cdot 10^{-5}$ &
$\lambda3727$\tabularnewline
${\rm O/H}$ &
$(9.41\pm0.92)\cdot 10^{-5}$ &
TOT\tabularnewline
$12+\log({\rm O/H})$ &
$7.97\pm0.04$ &
\tabularnewline
${\rm N+/O+}$&
$(9.87\pm6.48)\cdot 10^{-2}$ &
\tabularnewline
$\log({\rm N/O})$ &
$-1.01\pm0.22$&
\tabularnewline
${\rm N/H}$&
$(2.36\pm2.44)\cdot 10^{-6}$&
\tabularnewline
${\rm S}^{+}/{\rm H}^{+}$ &
$(5.69\pm0.28)\cdot 10^{-7}$&
average $\lambda6716$ and $\lambda6731$\tabularnewline
\hline
\end{tabular}\par\end{centering}
\end{table}
For a first indication, we compared the line ratios with the models of
Ferland \& Netzer (\cite{ferl+83}) and Evans \& Dopita (\cite{evan+85}),
which are overplotted on the diagnostic diagrams of Veilleux \& Osterbrock
(\cite{veil+87}), and find a metallicity
of 0.1 times solar and an ionisation temperature of 45000\,K.
A more sophisticated computation of the abundances have been done
following the so-called direct method
(e.g. Osterbrock \cite{oste89}). The {[}OIII] region electron
temperature $T_{{\rm e}}$, electron density $n_{{\rm e}}$, and abundances
were computed using tasks within the NEBULAR package of IRAF.
The temperature was computed using the {[}OIII] lines ratio
$(4959+5007)/4363$,
and the density using the {[}SII] line ratio $6716/6731$. Using these
values for $T_{{\rm e}}$ and $n_{{\rm e}}$, the ionic abundances
were computed with the IONIC task, with central wavelengths,
line ratios and errors taken from Table~\ref{linetab}.
The total oxygen abundance
was computed as ${\rm O/H=O^{++}/H^{+}+O^{+}/H^{+}}$, i.e. neglecting
the usually small contribution from ${\rm O^{+3}}$ (Skillman \& Kennicutt
\cite{skil+93}). In the case of sulfur, the abundance
was computed as ${\rm S/H = ICF \times (S^{+}+S^{++})}$,
with the ionisation correction factor (ICF) computed as
${\rm ICF=[1-(1-O^{+}/O)^{\alpha}]^{-1/\alpha}}$.
This expression for the ICF was first proposed by Stasinska
(\cite{stas78})
with $\alpha=3$, but we used the value $\alpha=2.6$ which reproduces
Garnett's (\cite{garn89}) photo-ionisation models better in the
range ${\rm O^{+}/O>0.2}$. Nitrogen abundances were computed
assuming ${\rm (N/O)=(N^{+}/O^{+})}$, and then using the nitrogen
to oxygen ratio in the equation ${\rm (N/H)=(N/O)}\times{\rm (O/H)}$.
In Table~\ref{parameter}, the derived parameters are listed for TV\,Ret.
We find that the average metallicity is about 0.12 solar, in agreement
with the values derived from the model of Ferland \& Netzer (\cite{ferl+83}).
The gas temperature derived from the {[}OIII] lines is $1.3(1)\cdot 10^4$\,K,
the density derived from the
{[}S\,II] lines is about 240/cm$^3$.
These values again confirm TV\,Ret as an H\,II galaxy.
\subsection{The outburst}
The reason for the classification of TV\,Ret as a cataclysmic variable
was the outburst that the object underwent in February 1977 and that
was observed over several days (Kinman et al. \cite{kinm+91}).
In Fig.~\ref{lc}, the B--magnitudes and heliocentric Julian Date
that they give in Table 5 are plotted. They estimated the
outburst amplitude as $\Delta m = 3.8$\,mag.
Note that there is no tight
observational constraint on the duration of the outburst. The rise happened
between JD\,2443156 and JD\,2443182, and hence lasted 26 days at the most.
However, the decline was not observed, and thus the actual duration of the
outburst is unknown. We can only
say that the object is back in quiescence during our observations 26 years
later.
If this was an outburst in a foreground object, by chance superimposed on the
galaxy,
we can estimate an upper limit for the brightness of this object.
With a $S/N\approx10$ of our spectra, we would be able to see an
object about three times
fainter than the galaxy. Since no trace of such a forground object is
visible in the spectrum, it must be fainter than $\approx$22\,mag.
In that case, the
amplitude of the outburst was at least 5.5\,mag, leaving two
explanations, a dwarf nova or a nova outburst. In the case of a dwarf nova, the
object would be in quiescence now, close to 22\,mag, but would show strong
Balmer emission lines.
No emission is found at the restframe wavelength of these lines, making
this possibility very unlikely. Note that we cannot refute the possibility
of a dwarf--nova super--outburst which would imply that the quiescent object
is even fainter, and the Balmer emission are no longer observable.
In the case of a nova, the outburst amplitude itself would have been larger,
so that the object is not visible in quiescence. However, even faint novae
have an absolute magnitude of $M_B = -7$ (e.g. Della Valle \& Livio,
\cite{dell+95}), yielding a lower limit of its
distance as 450\,kpc and placing it far outside our Galaxy.
\begin{figure}
\rotatebox{-90}{\resizebox{!}{8.7cm}{\includegraphics{lc.ps}}}
\caption{\label{lc} The B--magnitude of TV\,Ret plotted against
the heliocentric Julian Date. The data are taken from Table 5 of
Kinman et al. (\cite{kinm+91}). The smaller plot is a zoom on the
last 40 days, when the outburst happened.
}
\end{figure}
In the following, we assume that the outburst originates in the galaxy
and is not a chance transient from a foreground object. The
luminosity $L_{\rm out}$ of the outburst can then be estimated via
\begin{equation}
\Delta m = -2.5 \log{\frac{L_{\rm gal} + L_{\rm out}}{L_{\rm gal}}}
\end{equation}
and thus results in $L_{\rm out} = 32.1\,L_{\rm gal}$.
The absolute B--magnitude at the maximum is $M_B = -21.3$. While the
explanation of a supernova seems natural, the luminosity of the
outburst is too high. The brightest supernovae of Type Ia
have an absolute B--magnitude around $M_B = -19.8$ (e.g. Germany et
al.~\cite{germ+04}). Even if we allow for a large K--correction of
0.4\,mag (e.g. Hamuy et al. \cite{hamu+93}), the outburst would still
be about 1\,mag too bright for a Type Ia supernova. Furthermore,
supernovae are not known to vary on short time scales. The lightcurve
observed by Kinman et al., instead, varies around 0.8\,mag and on time
scales of hours both during quiescence and during outburst (see
Fig.~\ref{lc}).
There are however unusual supernovae that are
extremely luminous. The brightest one is SN 1999as (Knop et
al. \cite{knop+99}), a type-Ic hypernova that reached an absolute V
magnitude brighter than -21, i.e. similar to the magnitude that was
found for TV\,Ret. However, the probability for such objects is
rather low.
Even more extreme are pair-production supernovae
(Scannapieco et al., \cite{scan+05}), which can reach $M_B = -21$. However,
pair-production supernovae have never been observed and they are
thought to occur in environments with metallicities that are several
orders of magnitude lower than the metallicity of the host galaxy of
TV\,Ret.
The variation on short time scales indicates a compact source, so an AGN
seems more likely. However, they do not generally show such strong variations
in the optical.
While X-ray variability on short timescales is a common phenomenon in AGNs,
optical intra-night variability of up to 10\% is only found in luminous
Quasars with $M_B < -24.5$
(Gupta \& Joshi \cite{gupt+05}; Stalin et al. \cite{stal+05}).
More extreme variabilities are only exhibited in Radio-loud AGN with
ultra--relativistic jets and there is no indication that TV\,Ret is
such an object. Radio and/or x-ray observations of this galaxy
would help to decide
on the presence of an AGN and thus also on the possible nature of the
outburst.
There is a possible third explanation, if we assume that the accuracy of the
measurements of Kinman et al. is not the 0.03\,mag that they claim, but
rather of the order of 0.3\,mag. In that case, the short--term variation
would not be real
and for the outburst magnitude one would have to subtract the average
magnitude values of outburst and quiescence magnitudes instead of the
extreme values.
This would lower the
outburst amplitude by 0.4\,mag, putting it closer to the possible range of
a bright supernova. However, the method that the authors used is
well--known and the quoted errors seem reasonable for it.
Still, the fact that the variation is also
present during the quiescence phase might indicate that it is a scatter
in the measurements rather than a real variation. On the other hand, the
scatter seems to be slightly larger during outburst, where one would actually
expect the higher accuracy.
We know of no photometric monitoring of TV\,Ret after 1977, which would
be an important observation. If the
short--term variation was confirmed by such measurements, this would be
a strong indication for the presence of a compact source, i.e. an AGN.
If no variation was detected, the probability would grow that the
variation in 1977 is just the uncertainty in the measurements.
All in all, the reason for the outburst remains a mystery. Too many possible
explanations exist, and none of them is really convincing. Unless the
original measurements have rather high uncertainties, we can
conclude that the outburst was some rare and unusual event that is not
comparable with normal sources of variability.
\section{Conclusions}
We refute the classification of TV\,Ret as a cataclysmic variable.
The object is a narrow--emission line galaxy, probably of H\,II type, although
we cannot rule out the contribution of an AGN to the ionisation field.
The size of $\approx$1.2\,kpc and the absolute magnituse of $M_B = -17.5$
places the object among the dwarf galaxies.
Assuming that we see most of the galaxy, it is probably a blue compact dwarf
with ongoing star formation.
The metallicity is 0.12 solar, in agreement with an H\,II galaxy.
We have no final conclusion concerning the observed outburst. If associated
with the galaxy, it is about 1\,mag brighter than normal supernovae of type Ia.
One would thus prefer to explain the outburst via a foreground transient.
However,
the two possible explanations -- dwarf nova or nova -- have their drawbacks as
well.
\acknowledgement{
This research has
made use of the Simbad database operated at CDS, Strasbourg, France.
We thank the referee for valuable comments.
}
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[Updated 11/28/18 10:52 a.m. See below.] Shine Medical Technologies, a Janesville, WI-based company that's working to resume domestic production of a vital medical isotope, said Tuesday that Deerfield Management, a New York-based investing firm, has agreed to pour up to $150 million into Shine's coffers in the coming years.
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A nuclear reactor in Ontario, Canada that for decades was used to produce molybdenum-99 was shuttered in March. Several Eastern Hemisphere nations, such as Australia, South Africa, and the Netherlands, also make molybdenum-99; however, importing the isotope is costly because about 1 percent of finished product is lost each hour due to its 66-hour half-life.
Shine said Tuesday it plans to use some of the money from Deerfield to build a 57,000-square-foot manufacturing plant near its headquarters in southeastern Wisconsin. Two years ago, the Nuclear Regulatory Commission gave Shine the green light to build the facility. Shine plans to start construction in early 2019, and begin commercial production of molybdenum-99 there in 2021. Before that can happen, Shine first needs to obtain an operating license from the commission, as Katrina Pitas, Shine's vice president of business development, explained in 2016.
In February, Shine said it had finished building a prototype production facility that it plans to use to test its technology. According to messages and documents Shine executives sent to shareholders earlier this year, Deerfield indicated Shine would only receive the full nine-figure investment once the company constructs a "building shell" for the larger plant, proves that its process for producing molybdenum-99 works, and meets other pre-determined milestones. | {
"redpajama_set_name": "RedPajamaC4"
} | 586 |
\section{Introduction}
We consider a bounded, open and simply connected set
$\Omega\subset\mathbb R^2$ with smooth boundary. We suppose that $\Omega$
models an inhomogeneous superconducting sample submitted to an
applied external magnetic field. The energy of the sample is given
by the so called pinned Ginzburg-Landau functional,
\begin{equation}\label{eq-2D-GLf}
\mathcal E_{\kappa,H,a,B_{0}}(\psi,\mathbf A)= \int_\Omega\left( |(\nabla-i\kappa
H\mathbf A)\psi|^2+\frac{\kappa^2}{2}(a(x,\kappa)-|\psi|^2)^2\right)\,dx
+\kappa^2H^2\int_{\Omega}|\curl\mathbf A-B_0|^2\,dx\,.
\end{equation}
Here $\kappa$ and $H$ are two positive parameters such that $\kappa$ describes the properties of the material, and $H$ measures the variation of the intensity of the applied magnetic field. The modulus $|\psi|^{2}$ of the wave function
(order parameter) $\psi\in H^1(\Omega;\mathbb C)$ measures the density of the superconducting electron Cooper pairs. The magnetic potential
$\mathbf A$ belongs to $H^1_{\Div}(\Omega)$ where
\begin{equation}\label{eq-2D-hs} H^1_{\Div}(\Omega)=\{\mathbf A=(\mathbf A_{1},\mathbf A_{2})\in
H^1(\Omega)^{2}~:~\Div \mathbf A=0~{\rm in}~\Omega \,,\,\mathbf A\cdot\nu=0~{\rm
on}\,
\partial\Omega \,\}\,,
\end{equation}
with $\nu$ being the unit interior normal vector of
$\partial\Omega$.\\
The function $\kappa H\curl\mathbf A$ gives the induced magnetic field.
When $\psi\equiv0$ and $(\psi,\mathbf A)$ is a minimizer or a critical point of the functional, we call this pair normal state. In our case it is easy to see normal minimizers (if any) are necessarily in the form $(0,\mathbf A)$ with $\mathbf A$ in $H^1_{\Div} (\Omega)$ such that $\curl\mathbf A=B_{0}$. This solution is unique
and denoted by $\mathbf F$. A natural question will be to determine under which condition this normal solution is a minimizer.
The function $B_{0}\in C^{\infty}(\overline{\Omega})$ is the intensity of the external magnetic field which is variable in our problem. Let
\begin{equation}\label{gamma}
\Gamma=\{x\in\overline{\Omega}: B_{0}(x)=0\}\,.
\end{equation}
We assume that either $\Gamma$ is empty or that $B_{0}$ satisfies :
\begin{equation}\label{B(x)}
\left\{
\begin{array}{ll}
|B_{0}| + |\nabla B_0 | >0&\mbox{ in } \overline{\Omega}\\
\nabla B_{0}\times\vec{n}\neq 0 &\mbox{ on } \Gamma\cap\partial\Omega\,.
\end{array}
\right.
\end{equation}
The assumption in \eqref{B(x)} implies that for any open set $\omega$ relatively compact in $\Omega$, $\Gamma\cap\omega$ is either empty, or consists of a union of smooth curves.
The energy $\mathcal E_{\kappa,H,a,B_{0}}$ considered here is slightly different from the classical Ginzburg-Landau energy in the sense that there is a varying term denoted by $a(x,\kappa)$ penalizing the variations of the order parameter $\psi$ and called the pinning term. This term arises also naturally in the microscopic derivation of the Ginzburg-Landau theory from BCS theory (see \cite{FHSS})
without any a priori assumption on the sign of $a$.\\
In this paper, we will assume that the pining term $a$ satisfies:
\begin{ass}\label{assumption}
The function $a(x,\kappa)$ is real, defined on $\overline{\Omega}\times[\kappa_{0},+\infty)$, and satisfies for some $\kappa_0 >0$ the following assumptions:
\begin{enumerate}
\item[$(A_{1})$] \begin{equation} \label{a1} \forall \kappa\geq \kappa_0\,, a(\cdot,\kappa)\in C^{1}(\overline{\Omega})\,.\end{equation}
\item[$(A_{2})$]
\begin{equation}\label{a2}
\sup_{x\in\overline{\Omega},\,\kappa\geq\kappa_{0}}| a(x,\kappa)| <+\infty \,.
\end{equation}
\item[$(A_{3})$]
\begin{equation}\label{a3}
\sup_{{x\in\overline{\Omega},\,\kappa \geq \kappa_{0}}} |\nabla_{x}\,a(x,\kappa)|< +\infty \,.
\end{equation}
\item[$(A_{4})$]
There exists a positive constant $C_{1}$, such that,
\begin{equation}\label{a4}
\forall \kappa\geq \kappa_0\,,\qquad \mathcal{L} \left(\partial\{a(x,\kappa)>0\}\right)\leq C_{1}\,\kappa^{\frac{1}{2}}\,,
\end{equation}
where $\mathcal{L}$ is the "length" of $\partial\{a(x,\kappa)>0\}$ in $\Omega$ in a sense that will be explained in \eqref{defA4}.
\end{enumerate}
\end{ass}
Let us introduce for later use,
\begin{equation}\label{def:L}
L(\kappa) = \sup_x |\nabla_{x}\,a(x,\kappa)| \,,
\end{equation}
\begin{equation}\label{def:sup-a}
\overline{a}=\sup_{x\in\overline{\Omega},\,\kappa\geq\kappa_{0}}a(x,\kappa)
\end{equation}
and
\begin{equation}\label{def:inf-a}
\underline{a}=\inf_{x\in\overline{\Omega},\,\kappa\geq\kappa_{0}}a(x,\kappa).
\end{equation}
The assumption in ($A_{3}$) gives a uniform control for any $\kappa$ of the oscillation of $a(.,\kappa)$ which will be made precise later by an assumption on $L(\kappa)$.
Notice that the normal state $(0,\mathbf F)$ is a critical point of the functional in \eqref{eq-2D-GLf}.
It is standard, starting from a minimizing sequence, to prove the existence of minimizers in $ H^1(\Omega;\mathbb C)\times
H^1_{\Div}(\Omega)$ of the functional $\mathcal E_{\kappa,H,a,B_{0}}$. A minimizer $(\psi,\mathbf A)$ of \eqref{eq-2D-GLf} is a weak solution of
the Ginzburg-Landau equations,
\begin{equation}\label{eq-2D-GLeq}
\left\{
\begin{array}{llll}
-(\nabla-i\kappa H\mathbf A)^2\psi=\kappa^2\, (a(x,\kappa)-|\psi|^2)\, \psi&{\rm in}&
\Omega&(a)
\\
-\nabla^{\bot}\curl(\mathbf A-\mathbf F)=\displaystyle\frac1{\kappa
H}\IM(\overline{\psi}\,(\nabla-i\kappa
H\mathbf A)\psi) &{\rm in}&\Omega&(b)\\
\nu\cdot(\nabla-i\kappa H\mathbf A)\psi=0&{\rm
on}&\partial\Omega&(c)\\
\curl\mathbf A=\curl\mathbf F&{\rm on}&\partial\Omega&(d)\,.
\end{array}\right.
\end{equation}
Here, $\curl\mathbf A=\partial_{x_1}\mathbf A_{2}-\partial_{x_2}\mathbf A_{1}$ and
$\nabla^{\bot}\curl\mathbf A=(\partial_{x_2}(\curl\mathbf A),
-\partial_{x_1}(\curl\mathbf A)).$
Let us introduce the magnetic Schr\"odinger operator in an open set $\widetilde{\Omega}$ in $\mathbb R^2$:
\begin{equation}\label{def:P}
P_{A,V}^{\widetilde{\Omega}}=-(\nabla-iA)^{2}+V(x)\,,
\end{equation}
where $A\in H^{1}_{\Div}(\widetilde{\Omega})$ and $V$ is a continuous function bounded from below.\\
The form domain of $P_{A,V}^{\widetilde{\Omega}}$ is
$$
\mathcal{V}(\widetilde{\Omega})=\{u\in L^{2}(\widetilde{\Omega})\,,\quad (\nabla-iA)u\in L^{2}(\widetilde{\Omega})\,,\quad(V+C)^{\frac{1}{2}}u\in L^{2}(\widetilde{\Omega})\}\,,
$$
and its operator domain is given by
$$
D(P_{A,V}^{\widetilde{\Omega}}):=\{u\in\mathcal{V}(\widetilde{\Omega})\,,\quad P_{A,V}^{\widetilde{\Omega}}u\in L^{2}(\widetilde{\Omega}),\quad \nu\cdot(\nabla-i A)u=0~{\rm on}~\partial\widetilde{\Omega}\}\,.
$$
Then, $\eqref{eq-2D-GLeq}_{a,c}$ reads $$P_{A,V}^{\Omega}\,\psi=-\kappa^{2}\,|\psi|^{2}\psi\,,$$ with $A=\kappa H \mathbf A$, $\psi \in D(P_{A,V}^{\Omega})$ and $V=-\kappa^{2}\,a\,$.\\
There are many papers on the Ginzburg-Landau functional with a
pinning term, most of them study the influence of the pinning term
on the location of {\it vortices}, i.e. the zeros of the minimizing
order parameter. For the functional without a magnetic field (i.e.
$B_0=0$ in \eqref{eq-2D-GLf}), the influence of the pinning term is
studied in \cite{LM} and more recently in \cite{DSM} and the
references therein. The pinning term (i.e. the function $a$) in
\cite{LM} is a step function independent of $\kappa$; more
complicated $\kappa$-dependent periodic step functions are
considered in \cite{DSM}. The magnetic version of the functional in
\cite{LM} is studied in \cite{A.K, K-cocv}.
In \cite{ASS}, Aftalion, Sandier and Serfaty considered a {\bf
smooth} and $\kappa$-dependent pinning term $a$ satisfying:
\begin{enumerate}
\item[$(H_{1})$]
$L(\kappa) \ll\kappa H.$
\item[$(H_{2})$] There exist a continuous function $a(x)$, a positive constant $a_{0}$ and, for all $\kappa\geq 0$, there exist two functions $
\sigma(\kappa)=\textit{o}\left(\left(\ln\left|\ln\frac{1}{\kappa}\right|\right)^{-\frac{1}{2}}\right)
$
and $ \beta(x,\kappa)\geq 0$ such that,
$$\displaystyle \min_{B(x,\sigma(\kappa))} \beta(x,\kappa)=0\,, \qquad a(x,\kappa)= a(x)+\beta(x,\kappa)\,,\qquad{\rm and}\qquad 0< a_{0}\leq a(x)\leq 1\,.$$
\end{enumerate}
The study contains the case when $a(x,\kappa)=a(x)$ ($\beta=0$) but
also cases with a $\kappa$- control of the $x$-oscillation of
$\beta(\cdot,\kappa)$ which could increase with $\kappa$. In the
scales of this paper, the results in \cite{ASS} are valid
when the parameter
$H$ is of order $\frac{|\ln\kappa|}{\kappa}$ as $\kappa \longrightarrow +\infty$.
Extending the discussion, the
functional in \eqref{eq-2D-GLf} is close to models of Bose-Einstein
condensates (see e.g. \cite{AfAB, AlBr}).
In this paper, we will analyze how the pinning term appears in
the asymptotics of the energy in the presence of a strong external
variable magnetic field (see Theorem~\ref{thm-2D-main} below). Also,
we discuss the influence of the pinning on the asymptotic expression
of the third critical field $H_{C_3}$ (see Theorems~\ref{thm:HC3}
and \ref{thm:HC3-vr}).
We focus on the regime of large values of $\kappa$, $\kappa\rightarrow+\infty$ and we study the ground state energy defined as follows,
\begin{equation}\label{eq-2D-gs}
{E_0}(\kappa,H,a,B_{0})=\inf\big\{ \mathcal
E_{\kappa,H,a,B_{0}}(\psi,\mathbf A)~:~(\psi,\mathbf A)\in H^1(\Omega;\mathbb C)\times
H^1_{\Div}(\Omega)\big\}\,.
\end{equation}
More precisely, we give an asymptotic estimate which is valid in the simultaneous limit $\kappa\longrightarrow+\infty$ and $H(\kappa)\longrightarrow+\infty$ with the constraint that $\frac{H(\kappa)}{\kappa}$ remains asymptotically of uniform size, that is satisfying
\begin{equation}\label{cond-H}
\lambda_{\min}\leq \frac{H(\kappa)}{\kappa}\leq\lambda_{\max}\qquad(\kappa\geq\kappa_{0})\,,
\end{equation}
where $\lambda_{\min},\,\lambda_{\rm max}$ are positive constants such that $\lambda_{\min}<\lambda_{\max}$.\\
The behavior of ${E_0}(\kappa,H,a,B_{0})$ involves a function $\hat{f}:[0,+\infty)\longrightarrow[0,\frac{1}{2}]$ introduced in \cite[Theorem~2.1]{KA2}. The function $\hat{f}$ is increasing, continuous and $\hat{f}(b)=\frac{1}{2}$, for all $b\geq 1$.
\begin{thm}\label{thm-2D-main}
Suppose that Assumption~\ref{assumption} and \eqref{cond-H} hold,
and
\begin{equation}
L(\kappa)= \mathcal O(\kappa^{\frac{1}{2}})\qquad{\rm as}~\kappa \rightarrow +\infty\,.
\end{equation}
The ground state energy in \eqref{eq-2D-gs} satisfies
\begin{multline}\label{eq-2D-thm}
{E_0}(\kappa,H,a,B_{0})=\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,,\qquad{\rm as}~\kappa\longrightarrow+\infty\,.
\end{multline}
\end{thm}
When $\Omega\cap\{a(x,\kappa)>0\}=\varnothing$, we obtain directly from \eqref{eq-2D-gs}
$$
\mathcal E_{\kappa,H,a,B_{0}}(\psi,\mathbf A)\geq\frac{\kappa^{2}}{2}\int_{\Omega}a(x,\kappa)^{2}\,dx = \mathcal E_{\kappa,H,a,B_{0}}(0,\mathbf F)\,.
$$
Hence the minimizer of $\mathcal E_{\kappa,H,a,B_{0}}$ is the normal state. In physical terms, this case corresponds to the case when we are above the critical temperature.
We will describe later cases when the remainder term in \eqref{eq-2D-thm} is indeed small compared with the leading order term (see Section~\ref{examples}).
The assumptions in Theorem~\ref{thm-2D-main} contain the case when
the function $a$ is constant and equals~$1$, which was proved in
\cite{KA} under Assumption~\eqref{cond-H}.
Along the proof of Theorem~\ref{thm-2D-main}, we obtain an estimate
of the `magnetic energy' as follows:
\begin{corol}\label{corol-2D-main}
Under the assumptions of Theorem~\ref{thm-2D-main}, we have
\begin{equation}
(\kappa H)^2\int_{\Omega}|\curl\mathbf A-B_0|^2\,dx=\textit{o}(\kappa^{2})\,,\qquad{\rm as}~\kappa\longrightarrow+\infty\,.
\end{equation}
\end{corol}
If $\mathcal D$ is a domain in $\Omega$,
we introduce the local energy in $\mathcal D$ of $(\psi,\mathbf A) \in H^1(\Omega;\mathbb C)\times H^1_{\Div}(\Omega)$ by:
\begin{equation}\label{eq-GLe0}
\mathcal E_{0}(\psi,\mathbf A;a,\mathcal{D})=\int_{\mathcal{D}}|(\nabla -i\kappa
H\mathbf A)\psi|^2\,dx+\frac{\kappa^{2}}{2}\int_{\mathcal{D}}(a(x,\kappa)-|\psi|^{2})^2\,dx\,.
\end{equation}
The next theorem gives an estimate of the local energy $\mathcal E_{0}(\psi,\mathbf A;a,\mathcal{D})$.
\begin{theorem}\label{lc-en}
Under the assumptions of Theorem~\ref{thm-2D-main}, if $(\psi,\mathbf A)$
is a minimizer of \eqref{eq-2D-GLf} and $\mathcal{D}$ is regular set
such that $\mathcal{\overline{D}}\subset\Omega$, then
\begin{multline}\label{eq-lc-en}
\mathcal E_{0}(\psi,\mathbf A;a,\mathcal{D})=\kappa^{2}\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,,\qquad{\rm as}~\kappa\longrightarrow+\infty\,.
\end{multline}
\end{theorem}
Theorem~\ref{lc-en} will be useful in the proof of the next theorem which gives the asymptotic behavior of the order parameter $\psi$, when $(\psi,\mathbf A)$ is a global minimizer.
\begin{theorem}\label{est-psi-main}
Under the assumptions of Theorem~\ref{thm-2D-main}, if $(\psi,\mathbf A)$ is a
minimizer of \eqref{eq-2D-GLf} and $\mathcal{D}$ is a regular set such that $\overline{\mathcal{D}}\subset \Omega$, then
\begin{equation}\label{est-psi-D}
\int_{\mathcal{D}}|\psi(x)|^{4}\,dx=-\int_{\mathcal{D} \cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)-1\right\}\,dx+\textit{o}\left(1\right)\,,\quad{\rm as}~\kappa\longrightarrow+\infty\,.
\end{equation}
\end{theorem}
Formula \eqref{est-psi-D} indicates that $\psi$ is asymptotically localized in the region where $a>0$. When $a(x,\kappa)=1$, Theorem~\ref{est-psi-main} was proved in \cite{KA}.
The techniques that we are going to use here are inspired from those of \cite{KA} and \cite{KA2} (where the case $a=1$ was treated). At a technical level, our proof is slightly different than the proofs in \cite{KA,FK2,SS} since we do not use the uniform elliptic estimates. These important estimates are frequently used in the papers about the Ginzburg-Landau functional (see \cite{FH1}) with a constant pinning term. They appeared first in \cite{LP} and were then extended to the full regime in \cite{FH2}.
Compared with other papers studying the pinned functional, one
novelty here is that the pinning term has no definite sign, another
one being the consideration of a variable (and a potentially
vanishing) applied magnetic field.
The rest of this paper is devoted to the study of third critical field, i.e. the field above which the normal state $(0,\mathbf F)$ is the only critical point of the functional in \eqref{eq-2D-GLf}, in the case when the pining term $a$ is independent of $\kappa$ (i.e. $a(x,\kappa)=a(x)$). We define the set:
\begin{equation}\label{def:Ncp}
\mathcal{N}^{\rm cp}(\kappa)=\{H>0: \mathcal E_{\kappa,H,a,B_{0}}~\text{\rm has a non-normal critical point}\}\,.
\end{equation}\label{def:N}
Notice that the above set is bounded (see Theorem~\ref{thm:GP}). We also introduce the two sets:
\begin{equation}
\mathcal{N}(\kappa)=\{H>0:\mathcal E_{\kappa,H,a,B_{0}}~\text{\rm has a non-normal minimizer}\}\,.
\end{equation}
\begin{equation}\label{def:Nloc}
\mathcal{N}^{\rm loc}(\kappa)=\{H>0:\mu_{1}(\kappa,H)<0\}\,.
\end{equation}
Here, $\mu_{1}(\kappa,H)$ is the ground state energy of the semi-bounded quadratic form
\begin{equation}\label{Quad}
\mathcal{Q}_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}(\phi)=\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf F)\phi|^{2}-\kappa^{2}\,a(x,\kappa)|\phi|^{2}\right)\,dx\,,
\end{equation}
i.e.
\begin{equation}\label{def:mu1}
\mu_{1}(\kappa,H)=\inf_{\substack{\phi\in H^{1}(\Omega)\\ \phi\neq 0}}\left(\frac{\mathcal{Q}_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}(\phi)}{\|\phi\|^{2}_{L^{2}(\Omega)}}\right)\,.
\end{equation}
Note that $\mu_{1}(\kappa,H)$ is the lowest eigenvalue of $P_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}$. Here, we refer to \cite{CR,KIH,JPM,XB-KH} for previous contributions.\\
We introduce the following critical fields (cf. e.g.\cite{FH3,LP})\,.
\begin{align}
&\overline{H}_{C_3}^{cp}(\kappa)=\sup\,\mathcal{N}^{cp}(\kappa)\,,\qquad\underline{H}_{C_3}^{cp}(\kappa)=\inf\,(\mathbb R_{+}\setminus\mathcal{N}^{cp}(\kappa))\label{def:HC3-o}\,,\\
&\overline{H}_{C_3}(\kappa)=\sup\,\mathcal{N}(\kappa)\,,\qquad\quad\underline{H}_{C_3}(\kappa)=\inf\,(\mathbb R_{+}\setminus\mathcal{N}(\kappa))\,,\label{def:HC3}\\
&\overline{H}_{C_3}^{loc}(\kappa)=\sup\,\mathcal{N}^{loc}(\kappa)\,,\qquad\underline{H}_{C_3}^{loc}(\kappa)=\inf\,(\mathbb R_{+}\setminus\mathcal{N}^{loc}(\kappa))\label{def:HC3-u}\,.
\end{align}
Below $\underline{H}_{C_3}$, normal states will loose their stability and above $\overline{H}_{C_3}$, the normal state is (up to a gauge transformation) the only critical point of the functional in \eqref{eq-2D-GLf}.\\
Our aim is to determine the asymptotics of all the critical fields as $\kappa\longrightarrow+\infty$. This involves spectral quantities related to three models depending on $\Gamma$ being empty or not. \\
Let us introduce
$$\displaystyle\Theta_{0}=\inf_{\xi\in\mathbb R} \mu(\xi)\,,$$
where $\mu$ is the lowest eigen value of the operator
$$
\mathfrak{h}^{N,\xi}:=-\frac{d^2}{dt^2}+(t+\xi)^2\qquad{\rm in}~L^{2}(\mathbb R_+)\,,
$$
subject to the Neumann boundary condition $u'(0)=0$.
\begin{theorem}\label{thm:HC3}
Suppose that $\Gamma=\{x\in\Omega: B_{0}(x)=0\}=\varnothing$ and that $a\in C^{1}(\overline{\Omega})$ satisfies $\{a>0\}\neq\varnothing$. Then, as $\kappa\longrightarrow+\infty$, all the six critical fields satisfy an asymptotic expansion in the form:
\begin{equation}
H_{C_3}(\kappa)=\max\left(\sup_{x\in\Omega}\frac{a(x)}{|B_{0}(x)|},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}|B_{0}(x)|}\right)\,\kappa+\mathcal{O}(\kappa^{\frac{1}{2}})\,.
\end{equation}
\end{theorem}
We introduce
\begin{equation}\label{lambda0}
\lambda_{0}=\inf_{\tau\in \mathbb R} \lambda(\tau)\,,
\end{equation}
where $\lambda(\tau)$ is the lowest eigenvalue of the selfadjoint realization of the differential operator
\begin{equation}\label{defM}
M(\tau) = -\frac{d^2}{dt^2} +\frac 14 (t^2+2\tau)^2\qquad{\rm in}~L^{2}(\mathbb R)\,.
\end{equation}
We consider, for any $\theta\in(0,\pi)$ the bottom of the spectrum $\lambda(\mathbb R_{+}^{2},\theta)$ of the operator
\begin{equation}\label{def:lambda-theta}
P_{\mathbf A_{\rm app,\theta},0}^{\mathbb R^{2}_{+}}\quad{\rm with}\quad\mathbf A_{\rm app,\theta}=-\left(\frac{x^{2}_{2}}{2}\cos\,\theta,\frac{x^{2}_{1}}{2}\sin\,\theta \right)\,.
\end{equation}
\begin{theorem}\label{thm:HC3-vr}
Suppose that $\Gamma=\{x: B_{0}(x)=0\}\neq\varnothing$, that
\eqref{B(x)} holds and that $a\in C^{1}(\overline{\Omega})$ satisfies $\{a>0\}\neq\varnothing$. As
$\kappa\longrightarrow+\infty$, the six critical fields in
\eqref{def:HC3-o}-\eqref{def:HC3-u} satisfy the asymptotic
expansion:
$$
H_{C_3}(\kappa)=\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+\mathcal{O}\left(\kappa^{\frac{7}{4}}\right)\,.
$$
Here $\theta(x)$ denotes the angle between $\nabla B_{0}(x)$ and the inward normal vector $-\nu(x)$.
\end{theorem}
\subsection*{Organization of the paper} The rest of the paper is split into twelve sections. Section~\ref{2} analyzes the model problem with a constant magnetic field and a constant pinning term. Section~\ref{upperbound} establishes an upper bound on the ground state energy. Section~\ref{section:P.E.} contains useful estimates on minimizers. The estimates in Section~\ref{section:P.E.} are used in Section~\ref{section5} to establish a lower bound of the ground state energy and to finish the proof of Theorem~\ref{thm-2D-main}, Corollary~\ref{corol-2D-main} and Theorem~\ref{lc-en}. In Section~\ref{examples}, we discuss the conclusion in Theorem~\ref{thm-2D-main} by providing various examples of pinning terms obeying Assumption~\ref{assumption}. Section~{7} is devoted to the proof of Theorem~\ref{est-psi-main}. Section~\ref{GP} generalizes a theorem of Giorgi-Phillips concerning the breakdown of superconductivity under a large applied magnetic field. Sections~\ref{Section:4} and \ref{10} are devoted to the proof of Theorem~\ref{thm:HC3}. The proof of Theorem~\ref{thm:HC3-vr} is the purpose of Sections~\ref{Section:Asympt-m1-vanish} and \ref{12}.
\subsection* {Notation.}
{Throughout the paper, we use the following notation:}
\begin{itemize}
\item If $b_{1}(\kappa)$ and $b_{2}(\kappa)$ are two positive functions on $[\kappa_{0},+\infty)$, we write $b_{1}(\kappa)\ll b_{2}(\kappa)$ if $b_{1}(\kappa)/b_{2}(\kappa)\to0$ as $\kappa\to\infty$.
\item If $b_{1}(\kappa)$ and $b_{2}(\kappa)$ are two functions with $b_{2}(\kappa)\not=0$, we write $b_{1}(\kappa)\sim b_{2}(\kappa)$\\
if $b_{1}(\kappa)/b_{2}(\kappa)\to1$ as $\kappa\to\infty$.
\item If $b_{1}(\kappa)$ and $b_{2}(\kappa)$ are two positive functions, we write $b_{1}(\kappa)\approx b_{2}(\kappa)$
if there exist positive constants $c_1$, $c_2$ and $\kappa_0$ such
that $c_1b_{2}(\kappa)\leq b_{1}(\kappa)\leq c_2b_{2}(\kappa)$ for all
$\kappa\geq\kappa_0$.
\item Let $a_{+}(\widetilde{x}_{0},\kappa)=[a(\widetilde{x}_{0},\kappa)]_{+}$ and $a_{-}(\widetilde{x}_{0},\kappa)=[a(\widetilde{x}_{0},\kappa)]_-$ where, for any $x\in\mathbb R$, $[x]_+=\max(x,0)$ and $[x]_{-}=\max(-x,0)$.
\item Given $R>0$ and $x=(x_{1},x_{2})\in\mathbb R^{2}$, $Q_{R}(x)=(-R/2+x_{1},R/2+x_{1})\times(-R/2+x_{2},R/2+x_{2})$ denotes the
square of side length $R$ centered at $ x=(x_1,x_2)$ and we write $Q_{R}=Q_{R}(0)$.
\end{itemize}
\section{A reference problem}\label{2}
The reference problem is obtained by freezing the pinning term and the magnetic field. This approximation will appear to be reasonable in squares avoiding the boundary and the zero set $\Gamma$ of the magnetic field $B_0$.
\subsection{A useful function}\label{uf}
Consider $R>0$, $b>0$, $\zeta\in\{-1,+1\}$ and $\alpha\in\mathbb R\,$. We define the following Ginzburg-Landau energy with constant magnetic field on $H^1(Q_R)$ by
\begin{equation}\label{eq-GL-F}
u\mapsto F^{\zeta,\alpha}_{b,Q_{R}}(u)=\int_{Q_{R}}\left(b|(\nabla-i\zeta\mathbf A_0)u|^2+\frac{1}{2}\left(\alpha-|u|^2\right)^{2}\right)\,dx\,,
\end{equation}
where
\begin{equation}\label{eq-hc2-mpA0}
\mathbf A_{0}(x)=\frac1{2}(-x_2,x_1)\,,\qquad\forall\,x=(x_1,x_2)\in\mathbb R^{2}\,.
\end{equation}
We have two cases according to the sign of $\alpha$\,:~\\
\textbf{Case~1}. $\alpha>0$:~\\
We notice that
\begin{equation}\label{change-F}
F^{\zeta,\alpha}_{b,Q_{R}}(u)=\alpha^{2} F^{\zeta,1}_{\widetilde{b},Q_{R}}(\widetilde{u})\,,
\end{equation}
where
\begin{equation}\label{def-b-u}
\widetilde{b}=\frac{b}{\alpha}\qquad\text{and}\qquad\widetilde{u}=\frac{u}{\sqrt{\alpha}}\,.
\end{equation}
We introduce the two ground state energies
\begin{eqnarray}
e_{N}(b,R,\alpha)=\inf\left\{F^{+1,\alpha}_{b,Q_{R}}(u): u\in H^{1}(Q_{R};\mathbb C)\right\}\label{eN}\\
e_{D}(b,R,\alpha)=\inf\left\{F^{+1,\alpha}_{b,Q_{R}}(u): u\in H^{1}_{0}(Q_{R};\mathbb C)\right\}\label{eD}\,.
\end{eqnarray}
As $F^{+1,\alpha}_{b,Q_{R}}(u)=F^{-1,\alpha}_{b,Q_{R}}(\overline{u})$, it is immediate that,
\begin{equation}\label{F+=F-}
\inf F^{+1,\alpha}_{b,Q_{R}}(u)=\inf F^{-1,\alpha}_{b,Q_{R}}(u)\,.
\end{equation}
Using \eqref{eN} and \eqref{eD}, we get from \eqref{change-F}
\begin{equation}\label{eNa=eN}
e_{N}(b,R,\alpha)=\alpha^{2}\,e_{N}\left(\frac{b}{\alpha},R,1\right)=\alpha^{2}\,e_{N}\left(\frac{b}{\alpha},R\right)\,,
\end{equation}
and
\begin{equation}\label{eDa=eD}
e_{D}(b,R,\alpha)=\alpha^{2}\,e_{D}\left(\frac{b}{\alpha},R,1\right)=\alpha^{2}\,e_{D}\left(\frac{b}{\alpha},R\right)\,.
\end{equation}
As a consequence of \eqref{change-F} and \eqref{def-b-u}, $\widetilde{u}$ is a minimizer of $F^{\zeta,1}_{\widetilde{b},Q_{R}}$ if and only if $u$ is a minimizer of $F^{\zeta,\alpha}_{b,Q_{R}}$. In particular any minimizer of $F^{\zeta,\alpha}_{b,Q_{R}}$ satisfies
\begin{equation}\label{up-u-a}
|u|\leq \sqrt{\alpha}\,.
\end{equation}
Recall from \cite[Theorem~2.1]{FK2} that,
\begin{equation}\label{f(x)}
\displaystyle \hat{f}\left(\mathfrak{b}\right)=\lim_{R\longrightarrow\infty}\frac{e_{D}(\mathfrak{b},R)}{R^{2}}\,.
\end{equation}
The next proposition was proved in \cite[Lemma~2.2, Proposition~2.4]{KA2} in the case $\alpha=1$. It's present form can be deduced immediately from \eqref{eNa=eN}.
\begin{prop}\label{pro-f(b)}
For all $M>0$, there exist universal constants $C_{M}$ and $R_{M}$ such that $\forall R\geq R_{M},\,\forall\,b>0,\,\forall\,\alpha>0$ such that $\displaystyle 0<\frac{b}{\alpha}\leq M$, we have
\begin{equation}\label{eN>eD}
e_{N}(b,R,\alpha)\geq\,e_{D}\left(b,R,\alpha\right)-C_{M}\alpha^{2}R\left(\frac{b}{\alpha}\right)^{\frac{1}{2}}
\end{equation}
\begin{equation}\label{est-f(b)-eD}
\alpha^{2}\hat{f}\left(\frac{b}{\alpha}\right) \leq \frac{e_{D}(b,R,\alpha)}{R^{2}}\leq\alpha^{2}\hat{f}\left(\frac{b}{\alpha}\right)+C_{M}\frac{\alpha^{\frac{3}{2}}\sqrt{b}}{R}.
\end{equation}
\end{prop}
\textbf{Case~2}. $\alpha\leq 0\,$:~\\
When $\alpha\leq 0$, we write $\alpha=-\alpha_{0}$, $\alpha_{0}\geq 0$ and \eqref{eq-GL-F} becomes
\begin{equation}\label{eq-GL-F2}
F^{\zeta,\alpha}_{b,Q_{R}}(u)=\int_{Q_{R}}\left(b|(\nabla-i\zeta\mathbf A_0)u|^2+\frac{1}{2}\left(\alpha_{0}+|u|^2\right)^{2}\right)\,dx\,.
\end{equation}
It is clear that,
$$F^{\zeta,\alpha}_{b,Q_{R}}(u)\geq \frac{1}{2}\alpha_{0}^{2} R^{2}\qquad{\rm and}\qquad F^{\zeta,\alpha}_{b,Q_{R}}(0)=\frac{1}{2}\alpha_{0}^{2} R^{2}\,.$$
As a consequence, we have
$$
\frac{1}{2}\alpha_{0}^{2}R^{2}\leq e_{D}(b,R,\alpha)\leq F^{\zeta,\alpha}_{b,Q_{R}}(0)=\frac{1}{2}\alpha_{0}^{2}R^{2}\,.
$$
When $\alpha=0$, it is easy to show that
$$F^{\zeta,\alpha}_{b,Q_{R}}(u)=0\,.$$
Notice that the only minimizer of $F^{\zeta,\alpha}_{b,Q_{R}}$ is $u=0\,$.
Thus, for any $\alpha\leq 0\,$, we obtain
\begin{equation}\label{F=}
\frac{e_{D}(b,R,\alpha)}{R^{2}}=\frac{1}{2}\alpha^{2}\,.
\end{equation}
\section{Upper bound of the energy}\label{upperbound}
The aim of this section is to give an upper bound of the ground state energy ${E_0}(\kappa,H,a,B_{0})$ introduced in \eqref{eq-2D-gs} under Assumption~\eqref{cond-H}.
For this we cover $\Omega$ by (the closure of) disjoint open squares $(Q_{\ell}(\gamma))_{\gamma}$ whose centers $\gamma$ belong to a square lattice $\Gamma_\ell= \ell \mathbb Z \times \ell \mathbb Z$.
We will get an upper bound by matching together approximate minimizers, in each square $Q_{\ell}(\gamma)$ contained in $\Omega$, obtained by freezing the pinning term and the magnetic field at a suitable point $\tilde \gamma$. The size $\ell$ of the square will be chosen as a function of $\kappa$. We start with estimates in a given square $Q_\ell (x_0)$ and will take later $x_0=\gamma\,$.\\
{\bf About Assumption $(A_4)$.}\\
We first explain what was meant in Assumption $(A_4)$. By $\mathcal L(\partial\{a >0\}) \leq C_1 \kappa^\frac 12$ we mean the existence of $C_2 >0$ and $\kappa_0$ such that:
\begin{equation}\label{defA4}
\forall \kappa \geq \kappa_0\,,\, \forall \ell \leq C_2 \kappa^{-\frac 12}\,,\, {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{a >0\} \cap \Omega \neq \emptyset\} \leq C_1 \kappa^\frac 12 \ell^{-1}\,.
\end{equation}
~\\
Using Assumption~\eqref{def:L}, for any $\widetilde{x}_{0}\in \overline{Q_{\ell}(x_{0})}$ and $\kappa \geq \kappa_0$, we observe that,
\begin{equation}\label{app-a}
|a(x,\kappa)-a(\widetilde{x}_{0},\kappa)|\leq \left(\sup_x |\nabla_{x}\,a(x,\kappa)|\right)\,|x-x_{0}|\leq \frac{\ell}{\sqrt{2}}\,L (\kappa) \,,\qquad\forall x\in Q_{\ell}(x_{0})\,.
\end{equation}
\begin{definition}[$\rho$-admissible]\label{rho-adm}
Let $\rho\in(0,1)$. We say that triple $(\ell,x_0,\widetilde{x}_{0})$ is $\rho$-admissible if $\overline{Q_{\ell}(x_0)}\subset\{|B_{0}|>\rho\}\cap\Omega$ and $\widetilde{x}_{0}\in\overline{Q_{\ell}(x_{0})}$. In this case, we also say that the pair $(\ell,x_{0})$ is $\rho$-admissible and the corresponding square $Q_{\ell}(x_{0})$ is $\rho$ admissible.
\end{definition}
We recall from \cite[Section~3]{KA2} the definition of the test function,
\begin{equation}\label{def-w2}
\widetilde{w}_{\ell,x_0,\widetilde{x}_{0}}(x)=\begin{cases}
e^{i\kappa H\varphi_{x_{0},\widetilde{x}_{0}}}\widetilde{u}_{R}\left(\frac{R}{\ell}(x-x_{0})\right)&{\rm if}~x\in Q_{\ell}(x_0)\subset\{B_{0}>\rho\}\cap\Omega\\
e^{i\kappa H\varphi_{x_{0},\widetilde{x}_{0}}}\overline{\widetilde{u}}_{R}\left(\frac{R}{\ell}(x-x_{0})\right)&{\rm if}~x\in Q_{\ell}(x_0)\subset\{B_{0}<-\rho\}\cap\Omega\,,
\end{cases}
\end{equation}
where $\widetilde{u}_{R}\in H^{1}_{0}(\Omega)$ is a minimizer of $F^{+1,1}_{b,Q_{R}}$ satisfying by \eqref{up-u-a} $|\widetilde{u}_{R}|\leq 1$ and $\varphi_{x_{0},\widetilde{x}_{0}}$ is the function introduced in \cite[Lemma~A.3]{KA} that satisfies
\begin{equation}\label{F-A}
|\mathbf F(x)-B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_{0})-\nabla\varphi_{x_{0},\widetilde{x}_{0}}(x)|\leq C\, \ell^{2},\,\qquad\,\,\forall x \in Q_{\ell}(x_{0})\,.
\end{equation}
Here $B_{0}=\curl\mathbf F$ and $\mathbf A_{0}$ is the magnetic potential introduced in \eqref{eq-hc2-mpA0}.
Let us introduce the function:
\begin{equation}\label{def-w}
w_{\ell,x_0,\widetilde{x}_{0}}(x)=\sqrt{a_{+}(\widetilde{x}_{0},\kappa)}\,\widetilde{w}_{\ell,x_0,\widetilde{x}_{0}}(x)\,, \qquad \forall x\in Q_{\ell}(\widetilde{x}_{0})\,.
\end{equation}
Using the bound $|\widetilde{w}_{\ell,x_0,\widetilde{x}_{0}}|\leq1$, which is immediately deduced from the bound of $|\widetilde{u}_{R}|$, we get from \eqref{def-w},
\begin{equation}\label{up-w}
|w_{\ell,x_0,\widetilde{x}_{0}}|^{2} \leq a_{+}(\widetilde{x}_{0},\kappa)\,.
\end{equation}
\begin{prop}\label{pp-up-Eg}
Under Assumptions~\eqref{B(x)}-\eqref{a3}, there exist positive constants $C$ and $\kappa_{0}$ such that if $\kappa\geq\kappa_{0}$, $\ell\in(0,1)$, $\delta\in(0,1)$, $\rho>0$, $\ell^{2}\kappa H\rho>1$ and $(\ell,x_{0},\widetilde{x}_{0})$ is a $\rho$-admissible triple, then,
\begin{multline}\label{up-Eg-eq}
\frac{1}{|Q_{\ell}(x_0)|}\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a,Q_{\ell}(x_0))\leq (1+\delta)\kappa^{2}\left[a_{+}(\widetilde{x}_{0},\kappa)^{2}\hat{f}\left(\frac{H\,|B_{0}(\widetilde{x}_{0})|}{\kappa\,a_{+}(\widetilde{x}_{0},\kappa)}\right)+\frac{1}{2}a_{-}(\widetilde{x}_{0},\kappa)^{2}\right]\\
+C\left(\frac{1}{\kappa\ell}+\delta^{-1}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{2}\ell^{4}\right)\kappa^{2}\,.
\end{multline}
\end{prop}
\begin{proof}~\\
Let
\begin{equation}\label{def-Rb}
R=\ell\sqrt{\kappa H |B_{0}(\widetilde{x}_{0})|}\qquad{\rm and}\qquad b=\frac{H\,|B_{0}(\widetilde{x}_{0})|}{\kappa}\,.
\end{equation}
First we estimate $\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}(a(x,\kappa)-|w_{\ell,x_0,\widetilde{x}_{0}}|^{2})^{2}\,dx$ from above. Using \eqref{app-a}, we get the existence of a constant $C>0$ such that for any $\delta\in(0,1)$ and any $\kappa\geq \kappa_{0}$,
\begin{align}\label{up-a}
\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(x,\kappa)-|w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\right)^{2}\,dx&\leq (1+\delta)\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(\widetilde{x}_{0},\kappa)-|w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\right)^{2}\,dx\nonumber\\
&\qquad\qquad\qquad+(1+\delta^{-1})\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(\widetilde{x}_{0},\kappa)-a(x,\kappa)\right)^{2}\,dx\nonumber\\
&\leq(1+\delta)\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(\widetilde{x}_{0},\kappa)-|w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\right)^{2}\,dx\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C\delta^{-1}\kappa^{2}\ell^{4}L(\kappa)^{2}\,.
\end{align}
The estimate of $\int_{Q_{\ell}(x_{0})}|(\nabla-i\kappa H \mathbf F)w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\,dx$ from above is the same as in \cite[Proposition~3.1]{KA2}. We have
\begin{align}\label{up-F}
&\int_{Q_{\ell}(x_{0})}|(\nabla-i\kappa H \mathbf F)w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\,dx\nonumber\\
&\qquad\qquad\leq (1+\delta)\int_{Q_{\ell}(x_0)}\left|\left(\nabla-i\kappa H (B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}})\right)w_{\ell,x_0,\widetilde{x}_{0}}\right|^{2}\,dx\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C\delta^{-1}\kappa^{4}\ell^{6}|\, w_{\ell,x_0,\widetilde{x}_{0}}|^{2}\,.
\end{align}
From \eqref{def:sup-a}, by collecting \eqref{up-a}, \eqref{up-F} and \eqref{up-w}, we find that,
\begin{multline}\label{1up-loc-en}
\mathcal E_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a,Q_{\ell}(x_0))\leq(1+\delta)\mathcal E_{0}\big(w_{\ell,x_0,\widetilde{x}_{0}},\,B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0)\big)\\
+C\delta^{-1}(\kappa^{2}\ell^{4}L(\kappa)^{2}+\kappa^{4}\ell^{6}\,a_{+}(\widetilde{x}_{0},\kappa))\,.
\end{multline}
As we did in \cite{KA2}, we use the change of variable $y=\frac{R}{\ell}(x-x_0)$ and obtain
\begin{align*}
&\mathcal E_{0}\big(w_{\ell,x_0,\widetilde{x}_{0}},\,B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0)\big)\\
&=\int_{Q_{R}}\left[a_{+}(\widetilde{x}_{0},\kappa)\left|\left(\frac{R}{\ell}\nabla-i\frac{R}{\ell}\zeta_{\ell}\,\mathbf A_{0}(y)\right)\widetilde{u}_{R}(y)\right|^{2}+\frac{\kappa^{2}}{2}\left(a(\widetilde{x}_{0},\kappa)-a_{+}(\widetilde{x}_{0},\kappa)\left|\widetilde{u}_{R}(y)\right|^2\right)^2\right]\frac{\ell^2}{R^2}dy.
\end{align*}
Here, we denote by $\zeta_{\ell}$ the sign of $B_{0}(x_{0})$.\\
We distinguish between two cases:\\
\textbf{Case~1:} When $a(\widetilde{x}_{0},\kappa)>0$, we get
$$
\mathcal E_{0}\big(w_{\ell,x_0,\widetilde{x}_{0}},B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0)\big)=\frac{a(\widetilde{x}_{0},\kappa)^{2}}{b}F^{\zeta_{\ell},1}_{b/a(\widetilde{x}_{0},\kappa),Q_{R}}(\widetilde{u}_{R})\,.
$$
From \eqref{F+=F-} and \eqref{eNa=eN}, we obtain,
\begin{equation}\label{2up-loc-en}
\mathcal E_{0}\big(w_{\ell,x_0,\widetilde{x}_{0}},B_{0}(\widetilde{x}_{0})\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0)\big)=\frac{1}{b}e_{D}(b,R,a(\widetilde{x}_{0},\kappa))\,.
\end{equation}
As a consequence of the upper bound in \eqref{est-f(b)-eD}, the ground state energy $e_{D}(b,R,a(\widetilde{x}_{0},\kappa))$ in \eqref{2up-loc-en} is bounded for all $b>0$ and $R\geq 1$ by:
\begin{align}\label{F=eD}
e_{D}(b,R,a(\widetilde{x}_{0},\kappa))&\leq a(\widetilde{x}_{0},\kappa)^{2}\,R^{2}\,\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)+C_{M}a(\widetilde{x}_{0},\kappa)^{\frac{3}{2}}\,R\,\sqrt{b}\,.
\end{align}
With the choice of $R$ in \eqref{def-Rb}, we have effectively $R\geq 1$ which follows from the assumption $R\geq \ell\sqrt{\kappa H \rho}>1$.\\
We get from \eqref{2up-loc-en} and \eqref{F=eD} the estimate
\begin{multline}\label{3up-loc-en}
\mathcal E_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\zeta_{\ell}|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_0)+\nabla\varphi_{x_{0},\widetilde{x}_{0}};a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\leq a(\widetilde{x}_{0},\kappa)^{2}\frac{R^{2}}{b}\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)\\
+C_{M}\frac{a(\widetilde{x}_{0},\kappa)^{\frac{3}{2}}\,R}{\sqrt{b}}\,,
\end{multline}
with $(b,R)$ defined in \eqref{def-Rb}.\\
By collecting the estimates in \eqref{1up-loc-en}-\eqref{3up-loc-en} we get,
\begin{multline}\label{up-E0}
\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\leq (1+\delta)\, a(\widetilde{x}_{0},\kappa)^{2}\, \frac{R^{2}}{b}\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)\\
+C_{M}\frac{\overline{a}^{\frac{3}{2}}\,R}{\sqrt{b}}+C\delta^{-1}(\kappa^{2}\ell^{4}L(\kappa)^{2}+\kappa^{4}\ell^{6}\overline{a})\,.
\end{multline}
Here, we have used the fact that $\displaystyle a(\widetilde{x}_{0},\kappa)\leq\sup_{x\in\overline{\Omega},\,\kappa\geq\kappa_{0}}a(x,\kappa)=\overline{a}\,$.\\
\textbf{Case~2:} When $a(\widetilde{x}_{0},\kappa)\leq 0\,$, we have,
$$
\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))=\frac{\kappa^2}{2}\int_{Q_{\ell}(x_0)}a(x,\kappa)^{2}\,dx\,.
$$
From \eqref{app-a}, we get the existence of a constant $C>0$ such that for any $\delta\in(0,1)$,
\begin{equation}\label{up-E0-2}
\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\leq (1+\delta)\, \frac{\kappa^{2}}{2} \, a(\widetilde{x}_{0},\kappa)^{2}\ell^{2}
+C\,\delta^{-1}\,\kappa^{2}\ell^{4}L(\kappa)^{2}\,.
\end{equation}
\textbf{The results of cases 1-2}, we obtain,
\begin{multline}
\mathcal{E}_{0}(w_{\ell,x_0,\widetilde{x}_{0}},\mathbf F;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\leq (1+\delta)\kappa^{2}\left[a_{+}(\widetilde{x}_{0},\kappa)^{2}\hat{f}\left(\frac{H\,|B_{0}(\widetilde{x}_{0})|}{\kappa\,a_{+}(\widetilde{x}_{0},\kappa)}\right)+\frac{1}{2}a_{-}(\widetilde{x}_{0},\kappa)^{2}\right]\ell^{2}\\
+C\left(\frac{\kappa}{\ell}\,\overline{a}^{\frac{3}{2}}+\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{4}\ell^{4}\,\overline{a}\right)\ell^{2}\,,
\end{multline}
which finishes the proof of Proposition~\ref{pp-up-Eg}.
\end{proof}
\begin{app}\label{app:1}~\\
We select $\ell,\,\rho,\,\delta$ and the constraint on $L(\kappa)$ as follows:
\begin{equation}\label{choice-ell-rho}
\ell=\kappa^{-\frac{7}{12}}\,,\qquad\rho=\kappa^{-\frac{17}{24}}\,,\qquad L(\kappa)\leq C\,\kappa^{\frac{1}{2}}\,.
\end{equation}
and
\begin{equation}\label{choice-delta}
\delta=\kappa^{-\frac{1}{12}}
\end{equation}
Under Assumption~\eqref{cond-H}, this choice permits to verify the assumptions in Proposition~\ref{pp-up-Eg} and to obtain error terms of order $\textit{o}(\kappa^{2})$. We have indeed as $\kappa\longrightarrow\infty$
$$\frac{\kappa}{\ell}=\kappa^{\frac{19}{12}}\ll\kappa^{2}\,,$$
$$\delta^{-1}\kappa^{2}\ell^{2}\,L(\kappa)^{2}\leq\kappa^{\frac{23}{12}}\ll\kappa^{2}\,,$$
$$\delta^{-1}\kappa^{4}\ell^{4}=\kappa^{\frac{21}{12}}\ll \kappa^{2}\,,$$
$$\ell^{2}\kappa H \rho=\kappa^{\frac{3}{24}}\gg 1\,.$$
\end{app}
\begin{theorem}\label{up-Eg}
Under Assumptions~\eqref{B(x)}-\eqref{a4}, if \eqref{cond-H} holds and $L(\kappa)\leq C\,\kappa^{\frac{1}{2}}$, then, the ground state energy ${E_0}(\kappa,H,a,B_{0})$ in \eqref{eq-2D-gs} satisfies
\begin{multline}\label{up-Eg-eq1}
{E_0}(\kappa,H,a,B_{0})\leq \kappa^{2}\int_{\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\,\hat{f}\left(\frac{H\,|B_{0}(x)|}{\kappa\,a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx+\textit{o}(\kappa^{2})\,,\quad{\rm as}~\kappa\longrightarrow\infty\,.
\end{multline}
\end{theorem}
\begin{proof}
Let $\ell\in(0,1)$, $\delta$ and $\rho$ be chosen as in \eqref{choice-ell-rho} and \eqref{choice-delta}.
We consider the lattice $\Gamma_{\ell}:=\ell\mathbb Z\times\ell\mathbb Z$ and write, for $\gamma \in\Gamma_{\ell}$, $ Q_{\gamma,\ell}=Q_{\ell}(\gamma)$. In the next decomposition we keep the $\rho$-admissible boxes $Q_\ell(\gamma)$ in $\Omega$ which in addition are either contained in $\{a >0\}$ or in $\{ a \leq 0\}$. Hence we introduce \begin{equation}\label{def-card-squares}
\mathcal I_{\ell, \rho}^{+}=\left\{\gamma; \,\overline{Q_{\gamma, \ell}}\subset \Omega\cap\left\{|B_{0}|>\rho\,; a>0\right\}\right\},\quad \mathcal I_{\ell, \rho}^{-}=\left\{\gamma; \,\overline{Q_{\gamma, \ell}}\subset \Omega\cap\left\{ |B_{0}|>\rho\,; a\leq 0\right\}\right\}\,,
\end{equation}
and
\begin{equation}
N^{+}={\rm card}~ \mathcal I_{\ell, \rho}^{+}\,,\qquad N^{-}={\rm card}~ \mathcal I_{\ell, \rho}^{-}\,.
\end{equation}
Under Assumption \eqref{a4}, we have,
\begin{equation}\label{N}
N^{+}+N^{-}=|\Omega|\ell^{-2}+\mathcal{O}(\kappa^{\frac{1}{2}}\ell^{-1}+\ell^{-1} + \rho\ell^{-2})\,,\qquad{\rm as}~\kappa\rightarrow+\infty\,.
\end{equation}
In \eqref{N}, $\kappa^{\frac{1}{2}}\ell^{-1}$ appears when treating the boundary of the set $\{a(x,\kappa)>0\}$ (using Assumption $(A_4)$ as explained in \eqref{defA4}), $\ell^{-1}$ appears in the treatment of the boundary and $\rho\ell^{-2}$ appears when treating the neighborhood of $\Gamma$.\\
In each $\rho$-admissible $Q_\ell(\gamma)$, we consider some $\widetilde \gamma$ (to be chosen later) such that $(\ell, \gamma,\widetilde{\gamma})$ be a $\rho$-admissible triple. We consider $w_{\ell,\gamma,\widetilde{\gamma}}$ and extend it by $0$ outside of $Q_{\gamma,\ell}$, keeping the same notation for this extension. Then we define
\begin{equation}
s(x)=
\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}\cup\mathcal{I}_{\ell,\rho}^{-}} \,w_{\ell,\gamma,\widetilde{\gamma}}(x)\,.
\end{equation}
We compute the Ginzburg-Landau energy of the test configuration $(s,\mathbf F)$ in $\Omega$. Since $\curl \mathbf F=B_{0}\,$, we get,
\begin{align}\label{sumE1}
\mathcal{E}_{\kappa,H,a,B_{0}}(s,\mathbf F,\Omega)=\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}\cup\mathcal{I}_{\ell,\rho}^{-}}\mathcal{E}_{0}(w_{\ell,\gamma,\widetilde{\gamma}},\mathbf F;a(\widetilde{\gamma},\kappa),Q_{\gamma, \ell})\,.
\end{align}
Notice that for any $\widetilde{\gamma}\in Q_{\gamma,\ell}\,$, $a(\widetilde{\gamma},\kappa)$ satisfies \eqref{app-a} with $x=\gamma$ and $\widetilde{x}_{0}=\widetilde{\gamma}\,$, and $B_{0}(\widetilde{\gamma})$ satisfies \eqref{F-A}. We recall that $\hat{f}$ is a continuous, non-decreasing function (see \cite[Theorem~2.1]{KA2}) and that $B_{0}$ and $ a(\cdot,\kappa)$ are in $C^{1}$. Then, in each box $Q_{\gamma,\ell}$, we select $\widetilde{\gamma}\in \overline{Q_{\gamma,\ell}}$ such that
$$
|a(\widetilde{\gamma},\kappa)|^{2}\,\hat{f}\left(\frac{H\,B_{0}(\widetilde{\gamma})}{\kappa\,a(\widetilde{\gamma},\kappa)}\right)=\inf_{\widehat{\gamma}\in Q_{\gamma,\ell}} |a(\widehat{\gamma},\kappa)|^{2}\,\hat{f}\left(\frac{H\,B_{0}(\widehat{\gamma})}{\kappa\,a(\widehat{\gamma},\kappa)}\right)\quad(\text{if}~ \gamma\in \mathcal I_{\ell, \rho}^{+})
$$
and
$$
|a(\widetilde{\gamma},\kappa)|^{2}=\inf_{\widehat{\gamma}\in Q_{\gamma,\ell}} |a(\widehat{\gamma},\kappa)|^{2}\quad(\text{if}~ \gamma\in \mathcal I_{\ell, \rho}^{-})\,.
$$
Using Proposition~\ref{pp-up-Eg} and noticing that $|Q_{\gamma,\ell}|=\ell^2$, we get the existence of $C>0$ such that, for any $\delta\in(0,1)$
\begin{multline}\label{sumE2}
\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}\cup\mathcal{I}_{\ell,\rho}^{-}}\mathcal{E}_{0}(w_{\ell,\gamma,\widetilde{\gamma}},\mathbf F;a(\widetilde{\gamma},\kappa),Q_{\gamma, \ell})\leq \kappa^{2}(1+\delta)\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}} \inf_{\widehat{\gamma}\in Q_{\gamma,\ell}} [a(\widehat{\gamma},\kappa)]_{+}^{2}\,\hat{f}\left(\frac{H\,B_{0}(\widehat{\gamma})}{\kappa\,a(\widehat{\gamma},\kappa)}\right)\ell^{2}\\
+\kappa^{2}(1+\delta)\sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{-}} \inf_{\widehat{\gamma}\in Q_{\gamma,\ell}} \frac{[a(\widehat{\gamma},\kappa)]_{-}^{2}}{2}\ell^{2}+\, C\, \sum_{\gamma\in\mathcal{I}_{\ell,\rho}^{+}\cup\mathcal{I}_{\ell,\rho}^{-}}\left(\frac{\kappa}{\ell}+\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{4}\ell^{4}\right)\,\ell^{2}\,.
\end{multline}
We recognize the lower Riemann sum of the function $x\longmapsto [a(x,\kappa)]_+^{2}\,\hat{f}\left(\frac{H\,B_{0}(x)}{\kappa\,a(x,\kappa)}\right)$ in $(\cup_{\gamma\in\mathcal{I}^{+}_{\ell,\rho}}Q_{\gamma,\ell})$ and the function $x\longmapsto [a(x,\kappa)]_-^{2}$ in $(\cup_{\gamma\in\mathcal{I}^{-}_{\ell,\rho}} Q_{\gamma,\ell})$ . Notice that $\{\cup_{\gamma\in \mathcal{I}_{\ell, \rho}}Q_{\gamma, \ell}\}\subset\Omega$. Thanks to Application~\ref{app:1}, using \eqref{N} and the non negativity of $\hat f$, we get by collecting \eqref{sumE1}-\eqref{sumE2} that,
\begin{equation}\label{final-est}
\mathcal{E}_{\kappa,H,a,B_{0}}(s,\mathbf F,\Omega)\leq\kappa^{2}\int_{\{a(x,\kappa)>0\}} a(x,\kappa)^{2} \hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+C\,\kappa^{\frac{23}{12}}\,.
\end{equation}
Since $(\psi,\mathbf A)$ is a minimizer of the functional $\mathcal E_{\kappa,H,a,B_{0}}$ in \eqref{eq-2D-GLf}, we get
$$
{E_0}(\kappa,H,a,B_{0})\leq \mathcal{E}_{\kappa,H,a,B_{0}}(s,\mathbf F,\Omega)\,.
$$
This finishes the proof of Theorem~\ref{up-Eg}.
\end{proof}
\section{A priori estimates of minimizers}\label{section:P.E.}
The aim of this section is to give a priori estimates for the solutions of the Ginzburg-Landau equations \eqref{eq-2D-GLeq}. In the case when $a(x,\kappa)=1$ the starting point is an $L^{\infty}$ estimate of $\psi$. This estimate can be easly extended in the general case considered in this paper when $\eqref{eq-2D-GLeq}_{a}$ and $\eqref{eq-2D-GLeq}_{c}$ hold. Let us introduce:
\begin{equation}
\overline{a}(\kappa)= \sup_{x\in\overline{\Omega}}a(x,\kappa)\,.
\end{equation}
\begin{prop}\label{prop-psi<a}
Let $\kappa>0$; if $(\psi,\mathbf A)$ is a critical point (see \eqref{eq-2D-GLeq}), then,
\begin{equation}\label{eq-psi<supa}
|\psi(x)|^{2}\leq \max\left\{\overline{a}(\kappa),0\right\}\,,\qquad\forall x\in\overline{\Omega}\,.
\end{equation}
\end{prop}
\begin{proof}
We distinguish between two cases:\\
\textbf{Case 1:} \textbf {$\displaystyle\overline{a}(\kappa)\leq 0\,$.}\\
Multiplying the equation for $\psi$ in \eqref{eq-2D-GLeq}$_a$ by $\overline{\psi}$ and integrating over $\Omega$, we get
\begin{equation}\label{eq:A-psi}
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\psi|^{2}\,dx=\kappa^{2}\int_{\Omega}(a(x,\kappa)-|\psi|^{2})|\psi|^{2}\,dx\,.
\end{equation}
Since $(a(x,\kappa)-|\psi|^{2})\leq -|\psi|^{2}$, we obtain that $ |\psi|^{2}=0\,$ almost everywhere.\\
\textbf{Case 2:} \textbf{ $\displaystyle\overline{a}(\kappa)>0\,$}.\\
We will show that $\psi\in C^{0}(\overline{\Omega})$. In fact, $(\psi,\mathbf A)$ satisfies \eqref{eq-2D-GLeq}$_{a}$, $\psi\in L^{p}(\Omega)$ for all $2\leq p<+\infty$ and $\mathbf A\in H_{\rm div}^{1}(\Omega)\hookrightarrow L^{p}(\Omega)$. Thus, $\psi \in W^{2,q}(\Omega)$ for all $q<2$. As a consequence of the continuous Sobolev embedding of $W^{j+m,q}(\Omega)$ into $C^{j}(\overline{\Omega})$ for any $q>\frac{2}{m}$, we obtain that $\psi\in C^{0}(\overline{\Omega})$. Define for any $\kappa>0$ the following open set:
\begin{equation}\label{omega+}
\Omega_{+}=\left\{x\in\Omega:\,|\psi(x)|>\sqrt{\overline{a}(\kappa)}\right\}\,,
\end{equation}
and the following functions on $\Omega_{+}$
$$\phi=\frac{\psi}{|\psi|}\,,\qquad \widehat{\psi}=\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}\phi\,.$$
It is clear that
$$\nabla\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}=1_{\Omega_{+}}\nabla\left(|\psi|-\sqrt{\overline{a}(\kappa)}\right)=1_{\Omega_{+}}\nabla|\psi|\,.$$
Notice that $\psi\in H^{1}(\Omega)$, so applying \cite[Proposition~3.1.2]{FH1}, we get the property that\break $\displaystyle\nabla\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}\in L^{2}(\Omega)$, which implies that $\displaystyle\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}\in H^{1}(\Omega)$.\\
We introduce an increasing cut-off function $\chi\in C^{\infty}(\mathbb R)$ such that,
\begin{equation}\label{chi}
\chi(t)=\left\{
\begin{array}{ll}
0 & \text{for }~t\leq\frac{1}{4}\displaystyle\sqrt{\overline{a}(\kappa)}\\
1& \text{for}~t\geq\frac{3}{4}\displaystyle\sqrt{\overline{a}(\kappa)}\,,
\end{array}
\right.
\end{equation}
and define
\begin{equation}\label{phi1}
\widehat{\phi}=\chi(|\psi|)\frac{\psi}{|\psi|}\,.
\end{equation}
Since $\chi(|\psi|)\frac{\psi}{|\psi|}$ is smooth with bounded derivatives and $\psi\in H^{1}(\Omega)$, the chain rule gives that $\widehat{\phi}\in H^{1}(\Omega)\,.$
Furthermore,
\begin{equation}\label{phi2}
(\nabla-i\kappa H \mathbf A)\widehat{\psi}=1_{\Omega_{+}}\widehat{\phi}\,\nabla|\psi|+\displaystyle\left[|\psi|-\sqrt{\overline{a}(\kappa)}\right]_{+}(\nabla-i\kappa H\mathbf A)\widehat{\phi}.
\end{equation}
Using \eqref{chi} and \eqref{phi1}, we get
\begin{equation}\label{phi3}
1_{\Omega_{+}}(\nabla-i\kappa H \mathbf A)\psi=1_{\Omega_{+}}(\nabla-i\kappa H\mathbf A)(|\psi|\widehat{\phi})=1_{\Omega_{+}}\{\widehat{\phi}\,\nabla|\psi|+|\psi|(\nabla-i\kappa H\mathbf A)\widehat{\phi}\}\,.
\end{equation}
We have on $\Omega_{+}$ that $|\phi|=|\widehat{\phi}|=1\,$. Therefore
\begin{align*}
\phi\nabla\overline{\phi}+\overline{ \phi\nabla\overline{\phi}}&=\phi\nabla\overline{\phi}+\overline{\phi}\nabla\phi\\
&=\nabla|\phi|^{2}\\
&=0\,.
\end{align*}
So, $\RE(1_{\Omega_{+}}\phi\nabla\overline{\phi})=0\,$. This implies by using \eqref{phi2} and \eqref{phi3} that
$$
\RE\left\{\overline{(\nabla-i\kappa H\mathbf A)\widehat{\psi}}\cdot(\nabla-i\kappa H\mathbf A)\psi \right\}=
1_{\Omega_{+}}\left(|\nabla|\psi||^{2}+\left(|\psi|-\sqrt{\overline{a}(\kappa)}\right)|\psi||(\nabla-i\kappa H\mathbf A)\widehat{\phi}|^{2}\right).
$$
Multiplying $\eqref{eq-2D-GLeq}_{a}$ by $\overline{\widehat{\psi}}$ and using $\eqref{eq-2D-GLeq}_{c}$, it results from an integration by parts over $\Omega$ that
\begin{align*}
0&=\RE\left\{\int_{\Omega} \overline{(\nabla-i\kappa H\mathbf A)\widehat{\psi}}(\nabla-i\kappa H\mathbf A)\psi+\overline{\widehat{\psi}}(|\psi|^{2}-a)\psi\,dx \right\}\\
&\geq \RE\left\{\int_{\Omega} \overline{(\nabla-i\kappa H\mathbf A)\widehat{\psi}}(\nabla-i\kappa H\mathbf A)\psi+\overline{\widehat{\psi}}\left(|\psi|^{2}-\overline{a}(\kappa)\right)\psi\,dx \right\}\\
&\geq \int_{\Omega_{+}} |\nabla|\psi||^{2}+\left(|\psi|-\overline{a}(\kappa)\right)|\psi||(\nabla-i\kappa H\mathbf A)\widehat{\phi}|^{2}\\
&\qquad\qquad\qquad\qquad+\left(|\psi|+\sqrt{\overline{a}(\kappa)}\right)\left(|\psi|-\sqrt{\overline{a}(\kappa)}\right)^{2}|\psi|\,dx\,.
\end{align*}
Since the integrand is non-negative in $\Omega_{+}$, we easily conclude that $\Omega_{+}$ has measure zero, and consequently, we get that $|\psi|\in L^{\infty}(\Omega)$\,.\\
Since $\Omega_{+}$ has measure zero and $\psi\in C^{0}(\overline{\Omega})$, we get
$$
|\psi(x)|^{2}\leq \overline{a}(\kappa) \,,\qquad\forall x\in\overline{\Omega}\,.
$$
\end{proof}
\begin{corol}
Let $\kappa>0$; If $(\psi,\mathbf A)\in H^1(\Omega;\mathbb C)\times H^1_{\Div}(\Omega)$ is a critical point, we have,
\begin{equation}\label{eq-psi<a}
|\psi(x)|^{2}\leq \max\left\{\overline{a},0\right\}\,,\qquad\forall x\in\overline{\Omega}\,,
\end{equation}
where $ \overline{a}= \sup_{\kappa} \overline{a}(\kappa)$ was introduced in \eqref{def:sup-a}.
\end{corol}
The following estimates play an essential role in controlling the errors resulting from various approximations (see Section~\ref{section5}). These estimates are simpler than the delicate elliptic estimates in \cite{FH2} and \cite{LP}.
\begin{prop}\label{pr-est}
Suppose that \eqref{cond-H} holds. Let $\beta\in(0,1)$. There exist positive constants $\kappa_{0}$ and $C$ such that, if $\kappa\geq\kappa_{0}$ and $(\psi, \mathbf A)$ is a minimizer of \eqref{eq-2D-GLf}, then
\begin{equation}\label{eq-curlAF}
\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq \frac{C}{H}\,.
\end{equation}
\begin{equation}\label{est-A-F}
\|\mathbf A-\mathbf F\|_{H^{2}(\Omega)}\leq \frac{C}{H}\,,
\end{equation}
\begin{equation}\label{est-A-F2}
\|\mathbf A-\mathbf F\|_{C^{0,\beta}(\overline{\Omega})}\leq \frac{C}{H}\,.
\end{equation}
Here we recall that $\mathbf F$ is the magnetic potential defined by
\begin{equation}\label{div-curlF}
\curl \mathbf F = B_0\,,\, \mathbf F \in H^1_{\Div}(\Omega)\,.
\end{equation}
\end{prop}
\begin{proof}
Under Assumption~\eqref{cond-H}, Theorem~\ref{up-Eg} yields
\begin{align}
&\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq\frac{1}{\kappa H}{E_0}(\kappa,H,a,B_{0})^{\frac{1}{2}}\nonumber\\
&\qquad\qquad\leq\frac{1}{\kappa H}\left(\kappa^{2}\,\int_{\{a(x,\kappa)>0\}} \,a(x,\kappa)^{2} \hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}\,a(x,\kappa)^{2}\,dx\right)^{\frac{1}{2}}\,.
\end{align}
Using \eqref{a2} and the bound $\hat{f}(b)\leq\frac{1}{2}$, we get,
\begin{equation}\label{est-curlAF}
\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq\frac{C}{H}\,.
\end{equation}
As in \cite[Proposition~4.1]{KA2}, we prove that
\begin{equation}\label{est-F2}
\|\mathbf A-\mathbf F\|_{H^{2}(\Omega)}\leq\frac{C}{H}\,.
\end{equation}
Now, the estimate in $C^{0,\beta}$-norm is a consequence of the continuous Sobolev embedding of $H^{2}(\Omega)$ in $C^{0,\beta}(\overline{\Omega})$.
\end{proof}
\section{Lower bounds for the global and local energies} \label{section5}
In this section, we suppose that $\mathcal{D}$ is an open set with smooth boundary such that $\overline{\mathcal{D}}\subset\Omega$ (or $\mathcal{D}=\Omega$). We will give a lower bound of the ground state energy ${E_0}(\kappa,H,a,B_{0})$ introduced in \eqref{eq-2D-gs}.
\begin{prop}\label{prop-lb}
Under Assumptions~\eqref{B(x)}-\eqref{a3}, there exist for all $\beta\in(0,1)$ positive
constants $C$ and $\kappa_{0}$ such that if $\kappa\geq\kappa_{0}$, $\ell \in (0,\frac 12)$, $\delta\in(0,1)$, $\rho>0$, $\ell^{2}\kappa H \rho>1$, $(\psi,\mathbf A)$ is a minimizer of \eqref{eq-2D-GLf}, $h\in C^{1}(\overline{\Omega})$, $\|h\|_{\infty}\leq 1$ and $(\ell,x_{0},\widetilde{x}_{0})$ is a $\rho$-admissible triple, then,
\begin{multline}\label{lb}
\frac{1}{|Q_{\ell}(x_{0})|}\mathcal E_0(h\psi,\mathbf A;a,Q_{\ell}(x_0))\geq(1-\delta)\kappa^{2}\left\{ a_{+}(\widetilde{x}_{0},\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(\widetilde{x}_{0})|}{a_{+}(\widetilde{x}_{0},\kappa)}\right)+\frac{1}{2}a_{-}(\widetilde{x}_{0},\kappa)^{2}\right\}\\
-C\kappa^{2}\left(\delta^{-1}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{2}\ell^{4}+\delta^{-1}\ell^{2\beta}+(\kappa\ell)^{-1}+\ell\,L(\kappa)\right)\,,
\end{multline}
where $L(\kappa)$ is introduced in \eqref{def:L}.
\end{prop}
\begin{proof}
We distinguish between two cases according to the sign of $a(\widetilde{x}_{0},\kappa)$.\\
\textbf{We begin with the case when $a(\widetilde{x}_{0},\kappa)\leq 0\,$.} We have,
\begin{align*}
\mathcal E_0(h\psi,\mathbf A;a,Q_{\ell}(x_0))&=\int_{Q_{\ell}(x_0)}|(\nabla -i\kappa
H\mathbf A)h\psi|^2\,dx+\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)}(a(x,\kappa)-|h\psi|^{2})^2\,dx\\
&\geq\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)} a(x,\kappa)^{2}\,dx-\kappa^{2}\int_{Q_{\ell}(x_0)}a(x,\kappa)|h\psi|^{2}\,dx\,.
\end{align*}
Using \eqref{app-a}, \eqref{eq-psi<a} and the assumptions on $h$, the simple decomposition $a(x,\kappa)=a(\widetilde{x}_{0},\kappa)+(a(x,\kappa)-a(\widetilde{x}_{0},\kappa))$ yields for any $\delta\in(0,1)$
\begin{align}\label{up-a2}
\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)} a(x,\kappa)^{2}\,dx&\geq (1-\delta)\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)} a(\widetilde{x}_{0},\kappa)^{2}\,dx\nonumber\\
&\qquad\qquad\qquad+(1-\delta^{-1})\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_0)} (a(x,\kappa)-a(\widetilde{x}_{0},\kappa))^{2}\,dx\nonumber\\
&\geq (1-\delta)\,\frac{\kappa^{2}}{2}\,a(\widetilde{x}_{0},\kappa)^{2}\,|Q_{\ell}(x_0)|-C\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}\,|Q_{\ell}(x_0)|\,,
\end{align}
and
\begin{align}\label{up-apsi}
-\kappa^{2}\int_{Q_{\ell}(x_0)} a(x,\kappa)|h\psi|^{2}\,dx&\geq -\kappa^{2}\int_{Q_{\ell}(x_0)} a(\widetilde{x}_{0},\kappa)|h\psi|^{2}\,dx-C\,\ell\,L(\kappa)\,\kappa^{2}\,|Q_{\ell}(x_0)|\nonumber\\
&\geq-C\,\ell\,L(\kappa)\,\kappa^{2}\,|Q_{\ell}(x_0)|\,.
\end{align}
Collecting \eqref{up-a2} and \eqref{up-apsi}, we get,
\begin{equation}\label{es of A2}
\frac{1}{|Q_{\ell}(x_{0})|}\mathcal E_0(h\psi,\mathbf A;a,Q_{\ell}(x_0))\geq (1-\delta)\,\frac{\kappa^{2}}{2}\,a(\widetilde{x}_{0},\kappa)^{2}-C\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}-C'\,\ell\,L(\kappa)\,\kappa^{2}\,.
\end{equation}
\textbf{Now, we treat the case when $a(\widetilde{x}_{0},\kappa)>0\,$.} Let $\phi_{x_{0}}(x)=(\mathbf A(x_0)-\mathbf F(x_0))\cdot x$, where $\mathbf F$ is the magnetic potential introduced in \eqref{div-curlF}. Using the estimate of $\|\mathbf A-\mathbf F\|_{C^{0,\beta}(\Omega)}$ given in Proposition~\ref{pr-est}, we get for any $\beta\in (0,1)$ the existence of a constant $C$ such that for all $x\in Q_{\ell}(x_0)$,
\begin{equation}\label{alpha}
|\mathbf A(x)-\nabla\phi_{x_0}-\mathbf F(x)|\leq C\,\frac{\ell^{\beta}}{H}\,.
\end{equation}
Let $\widetilde{x}_{0}\in\overline{ Q_{\ell}(x_{0})}$ and $\varphi=\varphi_{x_{0},\widetilde{x}_{0}}+\phi_{x_{0}}$ with $\varphi_{x_{0},\widetilde{x}_{0}}$ satisfying \eqref{F-A}. We define the function in $Q_{\ell}(x_{0})$,
\begin{equation}\label{defu}
u(x)=e^{-i\kappa H\varphi}h\psi(x)\,.
\end{equation}
Similarly to \eqref{up-a}, we have, for any $\delta\in(0,1)$,
\begin{align}\label{lw-a}
\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(x,\kappa)-|h\psi|^{2}\right)^{2}\,dx\geq(1-\delta)\frac{\kappa^{2}}{2}\int_{Q_{\ell}(x_{0})}\left(a(\widetilde{x}_{0},\kappa)-|h\psi|^{2}\right)^{2}\,dx-C\delta^{-1}\kappa^{2}\ell^{4}L(\kappa)^{2}\,.
\end{align}
Using the same techniques as in \cite[Lemma~4.1]{KA}, we get, for any $\beta\in(0,1)$,
\begin{multline}\label{lw-A}
\int_{Q_{\ell}(x_{0})}|(\nabla-i\kappa H \mathbf A)h\psi|^{2}\,dx\geq (1-\delta)\int_{Q_{\ell}(x_{0})}|(\nabla-i\kappa H(\zeta_{\ell}|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_{0})+\nabla\varphi(x)))h\psi|^{2}\,dx\\
-C\delta^{-1}(\kappa H)^{2}\left(\ell^{4}+\frac{\ell^{2\beta}}{H^{2}}\right)\int_{Q_{\ell}(x_0)}|h\psi|^2\,dx\,.
\end{multline}
Thus, by collecting \eqref{lw-a} and \eqref{lw-A}, using \eqref{a3}, \eqref{eq-psi<a} and $\|h\|_{L^{\infty}(\Omega)}\leq 1$, we get
\begin{multline}\label{es of A}
\mathcal E_0(h\psi,\mathbf A;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\geq (1-\delta)\mathcal
E_0(e^{-i\kappa H\varphi}h\psi(x),\zeta_{\ell}|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_{0});a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\\
-C\delta^{-1}\kappa^{2}\ell^{4}L(\kappa)^{2}
-C_{1}\delta^{-1}\kappa^{2}H^{2}\left(\ell^{4}+\frac{\ell^{2\beta}}{H^{2}}\right)\ell^{2}\,.
\end{multline}
Let $R$ and $b$ be as in \eqref{def-Rb}. Let us introduce the function $v_{\ell,x_0,\widetilde{x}_{0}}$ in $Q_{R}$ as follows:
\begin{equation}\label{def-v}
v_{\ell,x_0,\widetilde{x}_{0}}(x)=\begin{cases}
u\left(\frac{\ell}{R} x+x_{0}\right)&{\rm if}~x\in Q_{R}\subset\{B_{0}>\rho\}\cap\Omega\\
\overline{u}\left(\frac{\ell}{R} x+x_{0}\right)&{\rm if}~x\in Q_{R}\subset\{B_{0}<-\rho\}\cap\Omega\,,
\end{cases}
\end{equation}
where $u$ is defined in \eqref{defu}.\\
Similarly to \eqref{2up-loc-en}, we use the change of variable $y=\frac{R}{\ell}(x-x_{0})$ and get
\begin{equation}\label{E0<}
\mathcal E_0(e^{-i\kappa H\varphi}h\psi(x),\zeta_{\ell}\,\kappa H|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_0);a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))=\frac{1}{b} F^{+1,a(\widetilde{x}_{0},\kappa)}_{b,Q_{R}}(v_{\ell,x_{0},\widetilde{x}_{0}})\,,
\end{equation}
where $F^{+1,a(\widetilde{x}_{0},\kappa),}_{b,Q_{R}}$ is introduced in \eqref{eq-GL-F}.\\
Since $v_{\ell,x_{0},\widetilde{x}_{0}}\in H^{1}(Q_{R})$ then, using \eqref{eN>eD} and \eqref{est-f(b)-eD}, we get
\begin{align}\label{F>f}
\frac{1}{b}F^{+1,a(\widetilde{x}_{0},\kappa)}_{b,Q_{R}}(v_{\ell,x_{0},\widetilde{x}_{0}})&\geq \frac{1}{b} e_{N}\left(b,R,a(\widetilde{x}_{0},\kappa)\right)\nonumber\\
&\geq \frac{1}{b}e_{D}\left(b,R,a(\widetilde{x}_{0},\kappa)\right)-C_{M}\,a(\widetilde{x}_{0},\kappa)^{\frac{3}{2}}\frac{R}{\sqrt{b}}\nonumber\\
&\geq a(\widetilde{x}_{0},\kappa)^{2}\frac{R^{2}}{b}\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)-\widehat{C}_{M}\,\frac{R}{\sqrt{b}}\,.
\end{align}
Inserting \eqref{F>f} into \eqref{E0<}, we get
\begin{multline}\label{E0<2}
\mathcal E_0(e^{-i\kappa H\varphi}h\psi(x),\zeta_{\ell}\,\kappa H|B_{0}(\widetilde{x}_{0})|\mathbf A_{0}(x-x_0);a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\geq a(\widetilde{x}_{0},\kappa)^{2}\frac{R^{2}}{b}\hat{f}\left(\frac{b}{a(\widetilde{x}_{0},\kappa)}\right)\\
-\widehat{C}_{M}\frac{R}{\sqrt{b}}\,.
\end{multline}
Having in mind \eqref{def-Rb} and \eqref{E0<2}, we get from \eqref{es of A},
\begin{multline}\label{E0<4}
\frac{1}{|Q_{\ell}(x_{0})|}\mathcal E_0(h\psi,\mathbf A;a(\widetilde{x}_{0},\kappa),Q_{\ell}(x_0))\geq(1-\delta)\kappa^{2}a(\widetilde{x}_{0},\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(\widetilde{x}_{0})|}{a(\widetilde{x}_{0},\kappa)}\right)\\
-C\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}
-C_{1}\delta^{-1}\kappa^{2}H^{2}\left(\ell^{4}+\frac{\ell^{2\beta}}{H^{2}}\right)
-C_{2}\frac{\kappa}{\ell}\,.
\end{multline}
The estimates in \eqref{es of A2} and \eqref{E0<4} achieve the proof of Proposition~\ref{prop-lb}.
\end{proof}
\begin{app}
We keep the same choice of $\ell$, $\rho$, $L(\kappa)$ and $\delta$ as in \eqref{choice-ell-rho}, \eqref{choice-delta} and choose:
\begin{equation}\label{choice-delta-alpha}
\beta=\frac{3}{4}\,.
\end{equation}
This choice and Assumption~\eqref{cond-H} permit to have the assumptions in Proposition~\ref{prop-lb} satisfied and make the error terms in its statement
of order $\textit{o}(\kappa^{2})$.
We have as $\kappa\longrightarrow\infty\,$,
$$\delta^{-1}\kappa^{4}\ell^{4}=\kappa^{\frac{21}{12}}\ll \kappa^{2}\,,$$
$$\delta^{-1}\kappa^{2}\ell^{2\beta}=\kappa^{\frac{29}{24}}\ll \kappa^{2}\,,$$
$$\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}=\kappa^{\frac{23}{12}}\ll\kappa^{2}\,,$$
$$\frac{\kappa}{\ell}=\kappa^{\frac{19}{12}}\ll \kappa^{2}\,,$$
$$
\ell\,L(\kappa)\,\kappa^{2}=\kappa^{\frac{23}{12}}\ll\kappa^{2}\,,
$$
$$\ell^{2}\kappa H\rho=\kappa^{\frac{3}{24}}\gg 1\,.$$
\end{app}
The next theorem presents a lower bound of the local energy in a relatively compact smooth domain $\mathcal{D}$ in $\Omega$. We deduce the lower bound of the global energy by replacing $\mathcal{D}$ by $\Omega$.
\begin{theorem}\label{lw-Eg}~\\
Under Assumptions~\eqref{B(x)}-\eqref{a4}, if \eqref{cond-H} holds, $L(\kappa)\leq C\,\kappa^{\frac{1}{2}}$ with $C>0$, $h\in C^{1}(\overline{\Omega})$, $\|h\|_{\infty}\leq 1$, $(\psi,\mathbf A)$ is a minimizer of \eqref{eq-2D-GLf} and $\mathcal D$ an open set in $\Omega$, then as $\kappa\longrightarrow+\infty$,
\begin{multline}\label{fianl-Eg1}
\mathcal{E} (h\psi,\mathbf A;a,B_{0},\mathcal{D})\geq\mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D}) \geq \kappa^{2}\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,.
\end{multline}
\end{theorem}
\begin{proof}
The proof is similar to the one in Theorem~\ref{up-Eg} and we keep the same notation. Let
$$
\mathcal{D}^{+}_{\ell,\rho}={\rm int}\left(\cup_{\gamma\in \mathcal I_{\ell, \rho}^{+}} \overline{Q_{\gamma,\ell}}\right)\qquad{\rm and}\qquad\mathcal{D}^{-}_{\ell,\rho}={\rm int}\left(\cup_{\gamma\in \mathcal I_{\ell, \rho}^{-}} \overline{Q_{\gamma,\ell}}\right)\,,
$$
where $\gamma\in \mathcal I_{\ell, \rho}^{+}$ and $\gamma\in \mathcal I_{\ell, \rho}^{-}$ are introduced in \eqref{def-card-squares}.\\
Thanks to Proposition~\ref{prop-lb}, we can easily prove the existence of positive constant $C$ such that for any $\delta\in(0,1)$ and $\beta\in(0,1)$,
\begin{multline*}
\mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D})\geq \kappa^{2}(1-\delta)\left\{\int_{\mathcal{D}^{+}_{\ell,\rho}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\right.\\
\left.+\frac{1}{2}\int_{\mathcal{D}^{-}_{\ell,\rho}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}-C\,r(\kappa,\ell,\delta,\rho,L(\kappa),\beta)\,,
\end{multline*}
where
\begin{equation}\label{asympr}
r(\kappa,\ell,\delta,\rho,L(\kappa),\beta)=\kappa^{2}\ell+\kappa^{2}\rho+\frac{\kappa}{\ell}+\delta^{-1}\kappa^{2}\ell^{2}L(\kappa)^{2}+\delta^{-1}\kappa^{4}\ell^{4}+\delta^{-1}\kappa^{2}\ell^{2\beta}+\ell\,L(\kappa)\,\kappa^{2}\,.
\end{equation}
Notice that using the regularity of $\partial\mathcal{D}$, \eqref{B(x)} and \eqref{a4} (see \eqref{defA4}), we get the existence of constants $C_{1}>0$ and $C_{2}>0$ such that,
\begin{equation}\label{eq:D/D+-}
\forall \ell\leq C_{2}\,\kappa^{-\frac{1}{2}}\,,\quad \forall \rho\in(0,1)\,,\qquad |\mathcal{D}\setminus\mathcal{\mathcal{D}^{+}_{\ell,\rho}}|+ |\mathcal{D}\setminus\mathcal{\mathcal{D}^{-}_{\ell,\rho}}|\leq C_{1}(\kappa^{\frac{1}{2}}\,\ell+\rho)\,.
\end{equation}
This implies by using \eqref{a3} and the upper bound $\hat{f}\leq \frac{1}{2}$,
\begin{multline}\label{eq:1st-main}
\int_{\mathcal{D}^{+}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\geq\int_{\mathcal{D}^{+}_{\ell,\rho}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
-\frac{1}{2}\,\overline{a}\,|\mathcal{D}\setminus\mathcal{\mathcal{D}_{\ell,\rho}}|
\end{multline}
and
\begin{equation}\label{eq:2nd-main}
\frac{1}{2}\int_{\mathcal{D}^{-}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\geq\frac{1}{2}\int_{\mathcal{D}_{\ell,\rho}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx-\frac{1}{2}\,\overline{a}\,|\mathcal{D}\setminus\mathcal{\mathcal{D}^{-}_{\ell,\rho}}|\,,
\end{equation}
where $\overline{a}$ is introduced in \eqref{def:sup-a}.\\
Collecting \eqref{eq:1st-main} and \eqref{eq:2nd-main}, using Assumptions \eqref{a2} and \eqref{eq:D/D+-}, we find that,
\begin{multline}\label{fianl-E0}
\mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D})\geq \kappa^{2}(1-\delta)\left\{\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\right.\\
\left.+\frac{1}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}-C\,\hat r(\kappa,\ell,\delta,\rho,L(\kappa),\beta)\,,
\end{multline}
where $\hat r(\kappa,\ell,\delta,\rho,L(\kappa),\beta)$ satisfies \eqref{asympr}.\\
Under Assumption~\eqref{cond-H}, the choice of the parameters $\rho$, $\ell$, $L(\kappa)$ in \eqref{choice-ell-rho}, $\delta$ in \eqref{choice-delta} and $\beta$ in \eqref{choice-delta-alpha}, implies that all error terms are of lower order compared to $\kappa^{2}$.\\
As a consequence of \eqref{cond-H}, the inequality \eqref{fianl-E0} becomes as $\kappa\longrightarrow+\infty$
\begin{multline}\label{fianl-E01}
\mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D})\geq \kappa^{2}\left\{\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{1}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}\\
+\textit{o}(\kappa^{2})\,.
\end{multline}
Moreover, we know that
$$\mathcal{E}(h\psi,\mathbf A;a,B_{0},\mathcal{D})\geq \mathcal{E}_{0} (h\psi,\mathbf A;a,\mathcal{D})\,.$$
This achieves the proof of Theorem~\ref{lw-Eg}.
\end{proof}
As we now show, Theorem~\ref{lw-Eg} permits to achieve the proof of two statements presented in the introduction:
\begin{proof}[\textbf{Proof of Corollary~\ref{corol-2D-main}}]~\\
If $(\psi,\mathbf A)$ is a minimizer of \eqref{eq-2D-GLf}, we have
\begin{equation}\label{eq-glob-en}
{E_0}(\kappa,H)=\mathcal E_0(\psi,\mathbf A;a,\Omega) + (\kappa H)^2
\int_{\Omega} |\curl\big(\mathbf A - \mathbf F\big)|^2\,dx \,,
\end{equation}
where $\mathcal E_{0}(\psi,\mathbf A;a,\Omega)$ is defined in \eqref{eq-GLe0}.\\
Using \eqref{eq-2D-thm} and \eqref{fianl-E01} (with $\mathcal{D}=\Omega$), then under Assumption~\eqref{cond-H} as $\kappa\longrightarrow+\infty$
\begin{equation}\label{eq-2D}
\mathcal E_{0}(\psi,\mathbf A;a,\Omega)=\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,.
\end{equation}
Putting \eqref{eq-2D} and \eqref{eq-2D-thm} into \eqref{eq-glob-en}, we finish the proof of Corollary~\ref{corol-2D-main}.
\end{proof}
~\\
\begin{proof}[\textbf{Proof of Theorem~\ref{lc-en}}.]~\\
Noticing that \eqref{fianl-E01} is valid when $h=1$ and $\mathcal{D}$ replaced by $\mathcal{\overline{D}}^{c}:=\Omega \setminus \overline{\mathcal D}$ for any open domain $\mathcal{D}\subset\Omega$ with smooth boundary, then we get:
\begin{multline}\label{fianl-E01c}
\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{\overline{D}}^{c})\geq \kappa^{2}\left\{\int_{\mathcal{\overline{D}}^{c}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\right.\\
\left.+\frac{1}{2}\int_{\mathcal{\overline{D}}^{c}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}+\textit{o}(\kappa^{2})\,.
\end{multline}
We can decompose $\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{D})$ as follow:
$$
\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{D})=\mathcal{E}_{0} (\psi,\mathbf A;a,\Omega)-\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{\overline{D}}^{c})\,.
$$
Using \eqref{eq-2D} and \eqref{fianl-E01c}, we get
\begin{multline}\label{fianl-E01-l}
\mathcal{E}_{0} (\psi,\mathbf A;a,\mathcal{D})\leq \kappa^{2}\left\{\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{1}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\right\}\\
+\textit{o}(\kappa^{2})\,.
\end{multline}
\end{proof}
\section{study of examples}\label{examples}
In this section, we will describe situations where the remainder term in \eqref{eq-2D-thm} is indeed small as $\kappa \rightarrow +\infty$ compared with the leading order term
\begin{equation}\label{def:leading-term}
E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0}):=\kappa^{2} \left(\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{1}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx\right)\,,
\end{equation}
where,
\begin{equation}\label{def:sigma}
\sigma=\frac{H}{\kappa}\,.
\end{equation}
Note that $0<\lambda_{\min}\leq\sigma\leq\lambda_{\max}$, so that $\sigma$ will be considered as an independent parameter in $[\lambda_{\min}\,,\,\lambda_{\max}]$.\\
We will also explore, case by case how one can verify Assumption $(A_4)$ as formulated precisely in \eqref{defA4}.
\subsection{The case of a $\kappa$-independent pinning}~\\
\begin{prop}
Suppose \eqref{B(x)} and \eqref{cond-H} hold. Let $a(x,\kappa)=a(x)$ where $a(x)\in C^{1}(\overline{\Omega})$ is a function independent of $\kappa$ and satisfies,
\begin{equation}\label{cond-a1}
\left\{
\begin{array}{lll}
\{x\in\Omega:\,a(x)>0\}\neq\varnothing\,,\\
{\rm or}\\
\{x\in\Omega:\,a(x)<0\}\neq\varnothing\,.
\end{array}
\right.
\end{equation}
There exist positive constants $C$ and $\kappa_{0}$ such that,
$$
\forall\kappa\geq\kappa_{0}\,,\qquad E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0})\geq C\,\kappa^{2}\,.
$$
\end{prop}
\begin{proof}
Since $a(x,\kappa)=a(x)$, the energy $E_{\rm g}^{\textbf{L}}$ becomes:
$$
E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0}):=\kappa^{2} \left(\int_{\{a(x)>0\}}a(x)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a(x)}\right)\,dx\\
+\frac{1}{2}\int_{\{a(x)\leq 0\}}a(x)^{2}\,dx\right)\,.
$$
Each term being positive, it is clear that the leading term is positive if $\{x\in\Omega:\,a(x)<0\} \neq\varnothing$.\\
If $ \{x\in\Omega:\,a(x)<0\}=\varnothing$ and $\{x\in\Omega:\,a(x)>0\} \neq\varnothing$, there exist $\rho_{0}>0$, $a_{0}>0$ and a disk $D(x_{0},r_{0})$ such that
$$
D(x_{0},r_{0})\subset\{a(x)>a_{0}\}\cap\{|B_{0}|>\rho_{0}\}\,.
$$
Using the monotonicity of $\hat{f}$ and the bound of $a(x)$ in \eqref{a2}, we may write
\begin{align}\label{est:a-k-ind}
\int_{\{a(x)>0\}}a(x)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x)}\right)\,dx&\geq \int_{D(x_{0},r_{0})} a(x)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a(x)}\right)\,dx\nonumber\\
&\geq \pi\,r_{0}^{2}\,a_{0}^{2}\,\hat{f}\left(\frac{\rho_{0}}{\overline{a}}\sigma\right)\,,
\end{align}
where $\overline{a}$ is introduced in \eqref{def:sup-a}.\\
In particular, when \eqref{cond-H} is satisfied, there exists $\kappa_{0}>0$ such that
\begin{equation}\label{ex1}
\forall\kappa\geq \kappa_{0}\,,\qquad\int_{\{a(x)>0\}}a(x)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x)}\right)\,dx\geq \pi\,r_{0}^{2}\,a_{0}^{2}\,\hat{f}\left(\frac{\rho_{0}}{\overline{a}}\lambda_{\min}\right)\,.
\end{equation}
\end{proof}
\begin{figure}[ht!]
\begin{center}
\includegraphics[scale=1]{1ere.png}
\caption{Schematic representation of $\Omega$ with pinning term independent of $\kappa$ and with variable magnetic field.}\label{example:1}
\end{center}
\end{figure}
\begin{prop}[\textbf{Verification of $(A_{4})$}]
Suppose that the function $a$ satisfies (see Fig.\ref{example:1}),
\begin{equation}\label{cond-a}
\left\{
\begin{array}{ll}
|a| + |\nabla a | >0&\mbox{ in } \overline{\Omega}\,,\\
\nabla a\times\vec{n}\neq 0 &\mbox{on}~ \widetilde{\Gamma}\cap\partial\Omega\,,
\end{array}
\right.
\end{equation}
where $\widetilde{\Gamma}$ defined as follows:
\begin{equation}\label{gamma-tilde}
\widetilde{\Gamma}=\{x\in\overline{\Omega}: a(x)=0\}\,.
\end{equation}
Then Assumption $(A_{4})$ is satisfied.
\end{prop}
\begin{proof}
From \eqref{cond-a}, we observe that,
$$
{\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{a >0\} \neq \emptyset\}={\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \widetilde{\Gamma} \neq \emptyset\}\,.
$$
Let $\epsilon\in(0,1)$, we introduce the domain
$$
D_{\epsilon}=\{x\in\Omega: \dist(x,\widetilde{\Gamma})\leq \epsilon\}\,.
$$
\textbf{Now we give a rough upper bound for the area of $D_\epsilon$.}\\
By assumption $\widetilde\Gamma$ consists of a finite number of connected curves, which are either closed in $\Omega$ or join two points of $\partial\Omega$. Let us consider the first case, we denote by $\widetilde\Gamma^{(1)}$ such a curve. We can parametrize this curve using the standard tubular coordinates $(s,t)$, where $s$ measures the arc-length in $\widetilde\Gamma^{(1)}$ and $t$ measures the distance to $\widetilde\Gamma^{(1)}$ (see \cite[Appendix~F]{FH1} for the detailed construction of these coordinates).
In the neighborhood of $\widetilde{\Gamma}^{(1)}$, we choose one point $\gamma_{0}$ on $\widetilde{\Gamma}^{(1)}$ corresponding to $(0,0)$. Let $N\in\mathbb N$ and $\mathcal{L}$ the length of $\widetilde\Gamma^{(1)}$. We consider for $i=0,...,N$, $s_{i}=\frac{i}{N}\,\mathcal{L}\,\,({\rm modulo}\,\,\mathcal{L}\mathbb Z)$ and $\gamma_{i}=(s_{i},0)$.\\
Notice that, there exists a positive constant $C$ such that,
$$
|\dist(\gamma_{i},\gamma_{i+1})|=(1+\epsilon_{i})|s_{i}-s_{i+1}|\,,\qquad\left(-\frac{C}{N}\leq \epsilon_{i}<0\right)\,.
$$
Thus,
\begin{align*}
\left|\left\{x\in\Omega: \dist\Big(x,\widetilde{\Gamma}^{(1)}\Big)\leq\frac{\mathcal{L}}{N}\right\}\right|&\leq \sum_{i} \left|Q_{\frac{\mathcal{L}}{N}}((s_i,0))\right|\,.
\end{align*}
Coming back to our problem, we select $N=\left[\frac{\mathcal{L}}{\epsilon}\right]$ and we note that $$\frac{\mathcal{L}}{N +1} \leq \epsilon\leq \frac{\mathcal{L}}{N}\,,$$
which implies that,
\begin{align*}
|D_{\epsilon}|&\leq \frac{\mathcal{L}^2}{N}\left(1+\mathcal{O}\left(\frac{1}{N}\right)\right)\\
&\leq \mathcal{L}\,\epsilon\,\left(1+\mathcal{O}\left(\frac{1}{N}\right)\right) = \epsilon \mathcal L (1+ \mathcal O(\epsilon))\,.
\end{align*}
Hence we have shown that,
$$\limsup_{\epsilon\to 0}\frac{|D_{\epsilon}|}{\epsilon}\leq\mathcal{L}\,.$$
In a similar fashion, we prove that
$$\liminf_{\epsilon\to 0}\frac{|D_{\epsilon}|}{\epsilon}\geq\mathcal{L}\,.$$
and, as a consequence, we end up with the following conclusion:
\begin{equation}\label{Area:D}
\lim_{\epsilon\to 0}\frac{|D_{\epsilon}|}{\epsilon}=\mathcal{L}\,.
\end{equation}
Coming back to Assumption $(A_4)$,
we now observe that all the $Q_{\ell}(\gamma)$ touching $\widetilde \Gamma$ are inside $D_{\sqrt{2}\ell}$, hence
we get, by comparison of the area
$$
\ell^2 {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \widetilde{\Gamma} \neq \emptyset\}\leq C\,\ell \,,
$$
and consequently,
there exist positive constants $C_1$, $C_{2}$ and $\kappa_{0}$ such that
$$
\forall \kappa \geq \kappa_0\,,\, \forall \ell \leq C_2 \kappa^{-\frac 12}\,,\, {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{a >0\} \neq \emptyset\}\leq C_{1}\,\ell^{-1}\,,
$$
which is a stronger form of $(A_4)$.
\end{proof}
\subsection{The case with a $\kappa$-dependent oscillation. }
\subsubsection{Preliminaries}
We start with two lemmas which are standard in homogenization theory (see \cite[Section~16-17]{BLP})
\begin{lem}\label{lem:a-}
Let $D\subset\mathbb R^{2}$ be a bounded open set and $\varphi$ be a $\Gamma_{T_{1},T_{2}}$-periodic continuous function in $\mathbb R^{2}$ with $\Gamma_{T_{1},T_{2}}=T_{1}\mathbb Z\times T_{2}\mathbb Z$. There exists a positive constant $M_{0}$ such that if $M\geq M_{0}$, then,
$$
\int_{D}\varphi(M x)\,dx=\frac{|D|}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\varphi(t_{1},t_{2})dt_{1}dt_{2}+\mathcal{O}(M^{-1})\,.
$$
\end{lem}
\begin{lem}\label{lem:a+}
Let $D\subset\mathbb R^{2}$ be a bounded open set
and $\phi:\mathbb R^{2}\times \overline{D} \longrightarrow\mathbb R^{2}$ be a continuous function satisfying:
\begin{equation}\label{1st:cond}
\phi(t+T,x)=\phi(t,x)\,,\qquad\forall T\in T_{1}\mathbb Z\times T_{2}\mathbb Z\,,
\end{equation}
and uniformly Lipschitz, i.e. with the property that there exist constants $C>0$ and $\epsilon_0$, such that,
\begin{equation}\label{2nd:cond}
|\phi(t,x)-\phi(t,\widetilde{x})|\leq C\, |x-\widetilde{x}|\,,\quad \, \forall t \in \mathbb R^2\,, \,\forall x, \widetilde{x}\in\overline{D}, \; {\rm s.t.}\, |x-\widetilde x|<\epsilon_0\,.
\end{equation}
There exists a positive constant $M_{0}$ such that if $M\geq M_{0}$, then,
$$
\int_{D}\phi(M x,x)\,dx=\int_{D}\overline{\phi}(x)\,dx+\mathcal{O}(M^{-1})\,,
$$
where,
\begin{equation}\label{def:phi}
\overline{\phi}(x)=\frac{1}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\phi((t_{1},t_{2}),x)\,dt_{1}dt_{2}\,.
\end{equation}
\end{lem}
\subsubsection{First example:}
\begin{prop}\label{prop:1st-ex}
Suppose that \eqref{B(x)} and \eqref{cond-H} hold. Let $
a(x,\kappa)=\alpha(\kappa^{\frac{1}{2}}\,x)
$
where $\alpha(\cdot)\in C^{1}(\overline{\Omega})$ is a $\Gamma_{T_{1},T_{2}}$-periodic function\footnote{see Fig.~\ref{example:3}}. Then the leading order term $E_{\rm g}^{\textbf{L}}$ defined in \eqref{def:leading-term} satisfies,
$$
E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0})=\kappa^{2} \, \int_{\Omega}\overline{\phi}_+(x)\,dx+\kappa^{2} |\Omega| \, \overline{\phi}_{-} +\textit{o}(\kappa^{2})\,,\quad{\rm as}~\kappa\to+\infty\,.
$$
Here,
$$
\overline{\phi}_{+}(x)=\frac{1}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\alpha_{+}(t_{1},t_{2})^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t_{1},t_{2})}\right)\,dt_{1}dt_{2}\,,
$$
and
$$
\overline{\phi}_{-} = \frac{1}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\alpha_{-}(t_{1},t_{2})^2\,dt_{1}dt_{2}\,.
$$
\end{prop}
\begin{proof}~\\
{\bf We first estimate the second term in \eqref{def:leading-term}.} We apply Lemma~\ref{lem:a-} with $D=\Omega$, $ M=\kappa^\frac{1}{2}$ and $\varphi=\alpha_{-}^2$, we obtain,
$$
\int_{\Omega}a_{-}(x,\kappa)^{2}\,dx=\frac{|\Omega|}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\alpha_{-}(t_{1},t_{2})^{2}\,dt_{1}dt_{2}+\mathcal{O}(\kappa^{-\frac{1}{2}})\,,
$$
and consequently,
$$
\kappa^{2}\int_{\{a(x)\leq 0\}}a(x,\kappa)^{2}\,dx=\kappa^{2}\,\frac{|\Omega|}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}\alpha_{-}(t_{1},t_{2})^2 \,dt_{1}dt_{2}+\mathcal{O}(\kappa^{\frac{3}{2}})\,.
$$
{\bf Now, we estimate the first term in \eqref{def:leading-term}.}
We first prove that $\hat{f}$ is a Lipschitz function in $[\mathfrak{b}_{0},1]$ with $\mathfrak{b}_{0}\in(0,1)$. We consider this restriction because when $\mathfrak{b}\to 0_{+}$ (see \cite[Theorem~2.1]{KA2}), $\hat{f}$ satisfies,
\begin{equation}\label{ashatf}
\hat{f}(\mathfrak{b})=\frac{\mathfrak{b}}{2}\ln\frac{1}{\mathfrak{b}}(1+\textit{o}(1))\,,
\end{equation}
and $\hat{f}$ is not a Lipschitz function at $0$. We recall the definition of $\hat{f}$
$$
\displaystyle \hat{f}\left(\mathfrak{b}\right)=\lim_{R\longrightarrow\infty}\frac{e_{D}(\mathfrak{b},R)}{R^{2}}\qquad(\forall \mathfrak{b}\in[0,1])\,,
$$
where
$$
e_{D}(\mathfrak{b},R)=\inf_{u}F^{+1,+1}_{\mathfrak{b},Q_{R}}(u):=\inf_{u}\int_{Q_{R}}\left(\mathfrak{b}|(\nabla-i\mathbf A_0)u|^2+\frac{1}{2}\left(1-|u|^2\right)^{2}\right)\,dx\,.
$$
From the definition, we can conclude that $\hat f$ is concave and hence locally Lipschitz in $(0,+\infty)$ (see \cite[Theorem~2.35]{MG}). For completion we write below a proof making explicit the Lipschitz constant. For $\mathfrak{b}' >0$, let $u_{\mathfrak{b}',R}\in H^{1}_{0}(Q_{R})$ be a minimizer of $F^{+1,+1}_{\mathfrak{b}',Q_{R}}$. Then for all $\mathfrak{b}\in (0,1)$, we have,
$$
e_{D}(\mathfrak{b},R)\leq F^{+1,+1}_{\mathfrak{b},Q_{R}} (u_{\mathfrak{b}',R})\leq e_{D}(\mathfrak{b}',R)+\|(\nabla-i\mathbf A_0)u_{\mathfrak{b}',R}\|^{2}_{L^{2}(Q_{R})}|\mathfrak{b}-\mathfrak{b}'|\,.
$$
Now, we estimate $\|(\nabla-i\mathbf A_0)u_{\mathfrak{b}',R}\|^{2}_{L^{2}(Q_{R})}$ from above. Coming back to the definition, we get the existence of a positive constant $C$, such that for any $\mathfrak{b}\in[\mathfrak{b}_0,1]$ and for any $\mathfrak{b}'\in[\mathfrak{b}_0,1]$,
\begin{align}\label{eq:est-from-above2}
\|(\nabla-i\mathbf A_0)u_{\mathfrak{b}',R}\|^{2}_{L^{2}(Q_{R})}&\leq \frac{e_{D}(\mathfrak{b}',R)}{\mathfrak{b}'}\nonumber \,.
\end{align}
This implies that,
$$
e_{D}(\mathfrak{b},R)\leq e_{D}(\mathfrak{b}',R)+ \frac{e_{D}(\mathfrak{b}',R)}{\mathfrak{b}'} |\mathfrak{b}-\mathfrak{b}'|\,.
$$
Dividing by $R^2$ and taking the limit as $R\rightarrow +\infty$, we obtain
$$
\hat f (\mathfrak{b}) \leq \hat f (\mathfrak{b}') + \frac{ | \hat f (\mathfrak{b}') |}{\mathfrak{b}'} |\mathfrak{b}-\mathfrak{b}'|\,.
$$
Using the asymptotic behavior of $\hat f$ in \eqref{ashatf} as $\mathfrak{b}'\rightarrow 0_{+}$, we finally obtain the existence of $C$ such that
$$
\hat f (\mathfrak{b}) \leq \hat f (\mathfrak{b}') + C \left(\log \frac {1}{\mathfrak{b}_0}\right) \, |\mathfrak{b}-\mathfrak{b}'|\,,\, \forall \mathfrak{b}, \mathfrak{b}' \mbox{ with } 1 > \mathfrak{b}>\mathfrak{b}_{0} \mbox{ and } 1 > \mathfrak{b}' >\mathfrak{b}_0\,.
$$
Exchanging $\mathfrak{b}$ and $\mathfrak{b}'$, we have proved the
\begin{lemma} $\hat f$ is locally Lipschitz in $(0,+\infty)$. More precisely, there exists $C$ such that for any $\mathfrak{b}_0 >0$,
\begin{equation}
| \hat f (\mathfrak{b}) - \hat f (\mathfrak{b}')| \leq C \,\left(\log \frac {1}{\mathfrak{b}_0}\right) \, |\mathfrak{b}-\mathfrak{b}'|\,,\, \forall \mathfrak{b}, \mathfrak{b}' \mbox{ with } 1>\mathfrak{b}>\mathfrak{b}_0 \mbox{ and } 1 > \mathfrak{b}' >\mathfrak{b}_0\,.
\end{equation}
In addition, we have
\begin{equation}
| \hat f (\mathfrak{b}) - \hat f (\mathfrak{b}')| \leq 2 \, |\mathfrak{b}-\mathfrak{b}'|\,,\, \forall \mathfrak{b}, \mathfrak{b}' \mbox{ with } \mathfrak{b}>\frac 12 \mbox{ and } \mathfrak{b}' > \frac 12\,.
\end{equation}
\end{lemma}
To continue, we consider
$$ \mathbb R^2 \times \Omega_\rho \ni (t,x) \mapsto \phi(t,x)=\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\right)\,,$$
where, $\Omega_{\rho}:=\Omega\cap\{|B_0|>\rho\}$.\\
The periodicity condition in \eqref{1st:cond} is clear. Let us verify the Lipschitz property.
Let
$$
\mathfrak{b}_{0}=\frac{\lambda_{\min}}{\alpha_{0}}\,\rho \,,
$$
where, $\lambda_{\min}$ is introduced in \eqref{cond-H} and $\alpha_{0}=\sup \alpha_{+}(t)$.\\
Let $\epsilon>0$, $\mathcal{I}_{+}=\{t\in\mathbb R: \alpha_{+}(t)\geq \epsilon\}$ and $\mathcal{I}_{-}=\{t\in\mathbb R: \alpha_{+}(t)\leq \epsilon\}$, we distinguish between two cases:\\
\textbf{Case 1:} $( \alpha_{+}(t)\geq \epsilon)$. We observe that for $(x,t) \in \Omega_\rho\times \mathcal{I}_{+}$, we have
$$
\mathfrak{b}_0\leq\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\leq \frac{\sigma\,|B_{0}(x)|}{\epsilon} \,.
$$
Thus, for any $t\in \mathcal{I}_{+}$ and for any $x,x'\in\overline{\Omega}_{\rho}$, we get
\begin{align}\label{eq:lipschitz1}
\left|\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\right)-\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x')|}{\alpha_{+}(t)}\right)\right|&=\alpha_{+}(t)^{2}|\hat{f}\left(\mathfrak{b}\right)-\hat{f}\left(\mathfrak{b}'\right)|\nonumber\\
&\le
C\,\left(\log \frac {1}{\rho}\right)\,\Big||B_{0}(x)|-|B_{0}(x')|\Big|\,.
\end{align}
Therefore, using also the Lipschitz property for $x\mapsto |B_0(x)|$, we get that $\Omega_\rho\ni x \mapsto \phi (t,x)$ is uniformly Lipschitz for $t\in \mathcal{I}_{+}$.\\
\textbf{Case 2:} $( \alpha_{+}(t)\leq \epsilon)$. We observe that for $(x,t) \in \Omega_\rho\times\mathcal{I}_{-}$,
$$
\frac{\sigma\,|B_{0}(x)|}{\alpha_{+}(t)}\geq \frac{\sigma\,|B_{0}(x)|}{\epsilon}\,.
$$
We note that $
\hat{f}(\mathfrak{b})=\frac{1}{2},\,\forall \mathfrak{b}\geq 1$ (see \cite[Theorem~2.1]{FK2}). For this reason we choose
$$
\epsilon=\frac{\lambda_{\min}}{2}\rho \,,
$$
which implies that for $(x,t) \in \Omega_\rho\times \mathcal{I}_{-}$,
$$\frac{\sigma\,|B_{0}(x)|}{\alpha_{+}(t)}\geq 2\qquad{\rm and}\qquad\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\right)=\frac{1}{2}\,.
$$
Thus, for any $t\in\mathcal{I}_{-}$ and for any $x,x'\in\overline{\Omega}_{\rho}$, we get
\begin{align}\label{eq:lipschitz2}
\left|\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{\alpha_{+}(t)}\right)-\alpha_{+}(t)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x')|}{\alpha_{+}(t)}\right)\right|&=\left|\frac{\alpha_{+}(t)^{2}}{2}-\frac{\alpha_{+}(t)^{2}}{2}\right|\nonumber\\
&=0\,.
\end{align}
Hence we get that $\Omega_\rho\ni x \mapsto \phi (t,x)$ is uniformly Lipschitz for $t\in \mathcal{I}_{-}\,$.\\
Now, we apply Lemma~\ref{lem:a+} with $D=\Omega_\rho$ and $M=\kappa^\frac 12$ and we obtain,
\begin{align}\label{eq:1st-ex-2}
\int_{\Omega_{\rho}}a_{+}(x,\kappa)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a_{+}(x,\kappa)}\right)\,dx&=\int_{\Omega_{\rho}}\overline{\phi}(x)\,dx+\mathcal{O}_{\rho}(\kappa^{-\frac{1}{2}})\,,
\end{align}
where $\overline{\phi}$ is introduced in \eqref{def:phi}.\\
Coming back to the integral over $\Omega$, we get, for any $\rho\in(0,\rho_{0})$ and for any $\kappa\geq\kappa_{0}$ with $\rho_{0}$ small enough and $\kappa_{0}$ large enough,
\begin{equation}\label{eq:final}
\int_{\Omega}a_{+}(x,\kappa)^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a_{+}(x,\kappa)}\right)\,dx=\int_{\Omega}\overline{\phi}(x)\,dx+\mathcal{O}(\rho)+\mathcal{O}_{\rho}(\kappa^{-\frac{1}{2}})\,.
\end{equation}
Here, we have used the fact that $\overline{\phi}$ is a bounded function in $\Omega$.
Let us show that the remainder term $s(\kappa)$ in the right hand side in \eqref{eq:final} is $\textit{o}(1)$. The remainder term has the form $s_{1}(\kappa)+s_{2}(\kappa)$ with $s_{1}(\kappa)=\mathcal{O}(\rho)$ and $s_{2}(\kappa)= \mathcal{O}_{\rho}(\kappa^{-\frac{1}{2}})$. Let us show that it is $o(1)$.
Given $\varepsilon > 0$, there exists $\rho_{\varepsilon}> 0$ such that $|s_{1}(\kappa)|\leq \frac{\varepsilon}{2}$, for all $\kappa \geq \kappa_0$. Then, $\rho=\rho_{\varepsilon}$ being chosen, we can find $\kappa_{\varepsilon}\geq \kappa_0$ such that, for any $\kappa\geq\kappa_{\epsilon}$, $|s_{2}(\kappa)|\leq \frac{\varepsilon}{2}$.
\begin{figure}[ht!]
\begin{center}
\includegraphics[scale=1]{3eme2.png}
\caption{Schematic representation of a domain with a $\kappa$-dependent oscillation pinning and with vanishing magnetic field along $\Gamma$.}\label{example:3}
\end{center}
\end{figure}
\end{proof}
\begin{prop}[Verification of $(A_{4})$]
Suppose that the function $\alpha$ defined in Proposition~\ref{prop:1st-ex} satisfies
\begin{equation}\label{cond-alpha}
|\alpha| + |\nabla \alpha | >0\quad\mbox{ in } \mathbb R^2\,.
\end{equation}
Then Assumption $(A_{4})$ is satisfied.
\end{prop}
\begin{proof}
Using \eqref{cond-alpha}, a change of variable $y=\kappa^{\frac{1}{2}}\,x$ and $\gamma'=\kappa^{\frac{1}{2}}\,\gamma$ yields,
\begin{align*}
& {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{x\in\Omega:\,a(x,\kappa)>0\} \neq \emptyset\}\\
&\hspace*{7cm}={\rm card}\,\{ \gamma' \in \Gamma_{\kappa^{\frac{1}{2}}\ell} \cap \kappa^{\frac{1}{2}}\Omega \mbox{ with } Q_{\kappa^{\frac{1}{2}}\ell} (\gamma') \cap \widehat{\Gamma} \neq \emptyset\}\,,
\end{align*}
where,
$$
\widehat{\Gamma}=\{y\in \mathbb R^2 \,|\, \alpha(y)=0\}\,.
$$
Let $\epsilon\in(0,1)$, we introduce the domain
$$
\widehat D_{\epsilon, M}=\{y\in M \,\cdot \,\Omega: \dist(y,\widehat{\Gamma})\leq \epsilon\}\,.
$$
Thanks to \eqref{Area:D} and the periodicity assumption, we get the existence of positive constants $C$, $M_0$ and $\epsilon_0$ such that, for any $\epsilon \in (0,\epsilon_0)$, $M\geq M_0$
$$
|\widehat D_{\epsilon, M}|\leq C\,M\,\epsilon\,.
$$
In the sequel, we choose $M=\kappa^{\frac{1}{2}}$ and $\epsilon=M\,\sqrt{2}\,\ell$. We note that, there exist constants $c > 0$ and $ \kappa_{0}> 0$ such that,
$$
\forall \kappa\geq \kappa_{0}\,,\quad\forall \ell\leq c\,\kappa^{-\frac{1}{2}}\,,\qquad 0< \epsilon\leq \epsilon_0 \,.
$$
We now observe that all the $Q_{\kappa^{\frac{1}{2}}\ell}(\gamma)$ touching $\widehat{\Gamma}$ are inside $\widehat D_{\kappa^{\frac{1}{2}}\,\sqrt{2}\,\ell,\kappa^{\frac{1}{2}}}$, hence
we get, by comparison of the areas
$$
\kappa\,\ell^2 {\rm card}\,\{ \gamma' \in \Gamma_{\kappa^{\frac{1}{2}}\ell} \cap \kappa^{\frac{1}{2}}\Omega \mbox{ with } Q_{\kappa^{\frac{1}{2}}\ell} (\gamma') \cap \widehat{\Gamma}_{\kappa} \neq \emptyset\}\leq C\sqrt{2} \,\kappa\,\ell \,.
$$
There exist positive constants $C_1$ and $C_{2}$, such that,
$$
\forall \kappa \geq \kappa_0\,,\, \forall \ell \leq C_2 \kappa^{-\frac 12}\,,\, {\rm card}\,\{ \gamma \in \Gamma_\ell \cap \Omega \mbox{ with } Q_\ell (\gamma) \cap \partial\{x\in\Omega:\,a(x,\kappa)>0\} \neq \emptyset\}\leq C_{1}\,\ell^{-1}\,.
$$
\end{proof}
\subsubsection{Second example.}
This example was considered by Aftalion, Sandier and Serfaty (see $(H_2)$).
\begin{prop}
Suppose that \eqref{B(x)} and \eqref{cond-H} hold. Let $a(x,\kappa)=a(x)+\beta(x,\kappa)$, where $\beta(x,\kappa)$ is a nonnegative function and $\{ a>0\}\cap\Omega\neq\varnothing$, (see Fig.~\ref{example:2}). There exist positive constants $\tau_1$ and $\kappa_{0}$ such that,
$$
\forall\kappa\geq\kappa_{0}\,,\qquad E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0})\geq \tau_1 \,\kappa^{2}\,.
$$
\end{prop}
\begin{proof}
We can write,
\begin{align}\label{cp-err}
\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx&\geq \kappa^{2}\int_{\{a(x)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\nonumber\\
&\geq \kappa^{2}\int_{\{a(x)>0\}}a(x)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{\overline{a}}\right)\,dx\,.
\end{align}
Here we have used that $\hat{f}$ is increasing, the nonnegativity of $\beta$ to get $a(x,\kappa)\geq a(x)$, Assumption $(A_{2})$ to estimate $\hat{f}$ from below, and $\{a(x)>0\}\subset\{a(x,\kappa)>0\}$.\\
Proceding like in \eqref{est:a-k-ind}, there exist $\tau_{1}>0$ and $\kappa_{0}>0$ such that,
\begin{equation}\label{lb:a-k2}
\forall\kappa\geq \kappa_{0}\,,\qquad\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\geq\tau_{1}\,\kappa^{2}\,.
\end{equation}
\end{proof}
\begin{figure}[ht!]
\begin{center}
\includegraphics[scale=1]{2eme.png}
\caption{Schematic representation of some domain with pinning term dependent of $\kappa$ and with vanishing magnetic field along $\Gamma$.}\label{example:2}
\end{center}
\end{figure}
\subsubsection{Third example:} This example is similar to the previous example, but here we suppose that
$$\beta(x,\kappa)=\alpha(\kappa^{\frac{1}{2}}x)\,,$$
where $\alpha(\cdot)$ is a $\Gamma_{T_{1},T_{2}}$-periodic positive function in $\mathbb R^2$.
\begin{prop}\label{prop:3rd-ex}
Suppose that \eqref{B(x)} and \eqref{cond-H} hold. Let $a(x,\kappa)=a(x)+\alpha(\kappa^{\frac{1}{2}}x)$, where $\alpha(\cdot)$ is a $\Gamma_{T_{1},T_{2}}$-periodic positive bounded function in $\mathbb R^2$, $a(\cdot)\in C^{1}(\overline{\Omega})$ and $\{a<0\}\cap\Omega=\varnothing$. Then the leading order term $E_{\rm g}^{\textbf{L}}$ defined in \eqref{def:leading-term} satisfies,
$$
E_{\rm g}^{\textbf{L}}(\kappa,H,a,B_{0})=\kappa^{2} \, \int_{\Omega}\overline{\phi}(x)\,dx+\textit{o}(\kappa^{2})\,,\quad{\rm as}~\kappa\to+\infty\,.
$$
Here,
$$
\overline{\phi}(x)=\frac{1}{T_{1}T_{2}}\int_{0}^{T_{1}}\int_{0}^{T_{2}}(a(x)+\alpha(t_1,t_2))^{2}\,\hat{f}\left(\sigma\frac{|B_{0}(x)|}{a(x)+\alpha(t_1,t_2)}\right)\,dt_{1}dt_{2}\,.
$$
\end{prop}
The proof of Proposition~\ref{prop:3rd-ex} is similar to that of Proposition~\ref{prop:1st-ex}.
\subsection{Upper bound of the main term.}~\\
It is easy to show that $ E_{\rm g}^{\textbf{L}}$ is less than $C\kappa^{2}$ for some $C>0$. Indeed, using the bound of $a$ in \eqref{a2} and the bound $\hat{f}(b)\leq\frac{1}{2}$, we have,
$$
\kappa^{2}\int_{\{a(x,\kappa)>0\}}a(x,\kappa)^{2}\,\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\leq C\kappa^{2}\,,
$$
and
$$
\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx\leq C\kappa^{2}\,.
$$
\section{Proof of Theorem~\ref{est-psi-main}}\label{7}
The technique that will be used in this proof has been introduced by Helffer-Kachmar in \cite{HK} for the case $a(x,\kappa)=\,1$. The proof is decomposed into three steps:\\
\textbf{Step 1: Case $\mathcal{D}=\Omega\,$.}\\
Let $(\psi,\mathbf A)$ be a solution of \eqref{eq-2D-GLeq}. Thanks to \eqref{eq:A-psi}, we have,
\begin{align*}
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\psi|^{2}\,dx&=\kappa^{2}\int_{\Omega}(a(x,\kappa)-|\psi|^{2})|\psi|^{2}\,dx\\
&=\frac{\kappa^{2}}{2}\int_{\Omega}(a(x,\kappa)^{2}-|\psi (x)|^{4})\,dx-\frac{\kappa^{2}}{2}\int_{\Omega}(a(x,\kappa)-|\psi|^{2})^{2}\,dx\,.
\end{align*}
Having in mind the definition of $\mathcal E_{0}(\psi,\mathbf A;a,\Omega)$, we get,
\begin{equation}
\frac{\kappa^{2}}{2}\int_{\Omega}(a(x,\kappa)^{2}-|\psi (x)|^{4})\,dx=\mathcal E_{0}(\psi,\mathbf A;a,\Omega)\,.
\end{equation}
Using \eqref{eq-2D}, we get that as $\kappa\longrightarrow+\infty$
\begin{multline} \label{previousone}
\frac{\kappa^{2}}{2}\int_{\Omega}(a(x,\kappa)^{2}-|\psi(x)|^{4})\,dx=\kappa^{2}\int_{\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx\\
+\frac{\kappa^{2}}{2}\int_{\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx+\textit{o}\left(\kappa^{2}\right)\,.
\end{multline}
Notice that
$$\int_{\Omega}a(x,\kappa)^{2}\,dx=\int_{\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\int_{\{a(x,\kappa)> 0\}}a(x,\kappa)^{2}\,dx\,.$$
Therefore, dividing \eqref{previousone} by $\kappa^2$, we get
\begin{equation}\label{asy-psi-Omega}
\int_{\Omega}|\psi(x)|^{4}\,dx=-\int_{\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx-1\right\}\,dx+\textit{o}\left(1\right)\,.
\end{equation}
\textbf{Step 2: Upper bound.} \\
Let $\mathcal{D}\subset\Omega$ be a regular domain and, for $\ell \in (0,1)$,
\begin{equation}
\mathcal{D}_{\ell}=\{x\in\mathcal{D}: {\rm dist}(x,\partial\mathcal{D})\geq\ell\}\,.
\end{equation}
We introduce a cut-off function $\chi_{\ell}\in C^{\infty}_{c}(\mathbb R^{2})$ such that
\begin{equation}\label{chi-l}
0\leq \chi_{\ell} \leq 1~{\rm in}~\mathbb R^{2}\,,\quad {\rm supp}\chi_{\ell}\subset \mathcal{D}\,,\quad \chi_{\ell}=1~{\rm in}~\mathcal{D}_{\ell} \quad{\rm and}\quad|\nabla\chi_{\ell}|\leq\frac{C}{\ell}~{\rm in}~\mathbb R^{2}\,,
\end{equation}
where $C$ is a positive constant. We multiply both sides of $\eqref{eq-2D-GLeq}_{a}$ by $\chi_{\ell}^{2}\psi$. It results from an integration by parts that
\begin{align}\label{eq:1}
\int_{\mathcal{D}} \left(|(\nabla-i\kappa H \mathbf A)\chi_{\ell}\psi|^{2}-\kappa^{2}a\,\chi_{\ell}^{2}|\psi|^{2}+\kappa^{2}\chi_{\ell}^{2}|\psi|^{4}\right)\,dx&=\int_{\mathcal{D}}|\nabla\chi_{\ell}|^{2}\, |\psi|^{2}\,dx\nonumber\\
&=\mathcal{O}(\ell^{-1})\,.
\end{align}
Here, we have used the fact that $|\nabla\,\chi_{\ell}|^{2}=\mathcal{O}(\ell^{-2})$, $|\mathcal{D}_{\ell}|=\mathcal{O}(\ell)$ and the bound of $\psi$ in \eqref{eq-psi<a}.\\
We notice that $\chi_{\ell}^{4}\leq\chi_{\ell}^{2}\leq 1$. We add to both sides the term $\frac{\kappa^{2}}{2}\int_{\mathcal{D}}a^{2}\,dx$ to obtain,
$$
\int_{\mathcal{D}} \left(|(\nabla-i\kappa H \mathbf A)\chi_{\ell}\psi|^{2}+\frac{\kappa^{2}}{2}a^{2}-\kappa^{2}a\,|\chi_{\ell}\,\psi|^{2}+\kappa^{2}|\chi_{\ell}\,\psi|^{4}\right)\,dx\leq\,C\,\ell^{-1}+\frac{\kappa^{2}}{2}\int_{\mathcal{D}}a^{2}\,dx\,.
$$
This implies that
$$
\mathcal E_0(\chi_{\ell}\psi,\mathbf A;a,\mathcal{D})\leq
\frac{\kappa^{2}}{2}\int_{\mathcal{D}}(a^{2}-\chi_{\ell}^{4}|\psi|^{4})\,dx+C\,\ell^{-1}\,.
$$
Using \eqref{chi-l}, we get
\begin{align}\label{up-psi42}
\int_{\mathcal{D}}|\psi|^{4}\,dx&= \int_{\mathcal{D}}\chi_{\ell}^{4}|\psi|^{4}\,dx+ \int_{\mathcal{D}}(1-\chi_{\ell}^{4})|\psi|^{4}\,dx\nonumber\\
&\leq \int_{\mathcal{D}}\chi_{\ell}^{4}|\psi|^{4}\,dx+C'\,\ell\,,
\end{align}
and consequently,
\begin{equation}\label{up-psi4}
\mathcal E_0(\chi_{\ell}\psi,\mathbf A;a,\mathcal{D})\leq
\frac{\kappa^{2}}{2}\int_{\mathcal{D}}(a^{2}-|\psi|^{4})\,dx+C(\ell^{-1}+\ell)\,.
\end{equation}
Using \eqref{fianl-E01} with $h=\chi_{\ell}$ and taking the choice of $\ell$ defined in \eqref{choice-ell-rho}, we get, as $\kappa\to+\infty$,
\begin{multline}
\frac{\kappa^{2}}{2}\int_{\mathcal{D}}(a^{2}-|\psi|^{4})\,dx\geq \kappa^{2}\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx+\frac{\kappa^{2}}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}} a(x,\kappa)^{2}\,dx\\
+\textit{o}\left(\kappa^{2}\right)\,.
\end{multline}
Notice that,
$$\int_{\mathcal{D}}a(x,\kappa)^{2}\,dx=\int_{\mathcal{D}\cap\{a(x,\kappa)\leq 0\}}a(x,\kappa)^{2}\,dx+\int_{\mathcal{D}\cap\{a(x,\kappa)> 0\}}a(x,\kappa)^{2}\,dx\,.$$
Therefore,
\begin{multline}
-\frac{\kappa^{2}}{2}\int_{\mathcal{D}}|\psi|^{4}\,dx\geq \kappa^{2}\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)\,dx-\frac{\kappa^{2}}{2}\int_{\mathcal{D}\cap\{a(x,\kappa)> 0\}} a(x,\kappa)^{2}\,dx\\
+\textit{o}\left(\kappa^{2}\right)\,.
\end{multline}
Dividing both sides by $-\frac{\kappa^{2}}{2}$, we obtain, as $\kappa\longrightarrow +\infty\,$,
\begin{equation}\label{up-psi45}
\int_{\mathcal{D}}|\psi|^{4}\,dx\leq-\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)-1\right\}\,dx+\textit{o}\left(1\right)\,.
\end{equation}
\begin{rem}\label{rm-up-psi-ng}
We can replace $\mathcal{D}$ by $\mathcal{\overline{D}}^{c}$ such that the estimate in \eqref{up-psi45} is still true. That is:
\begin{equation}\label{up-psi-ng}
\int_{\mathcal{\overline{D}}^{c}}|\psi|^{4}\,dx\leq-\int_{\mathcal{\overline{D}}^{c}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)-1\right\}\,dx+\textit{o}\left(1\right)\,.
\end{equation}
\end{rem}
\textbf{Step 3: Lower bound.} \\
We can decompose $\int_{\mathcal{D}}|\psi|^{4}\,dx$ as follows:
$$
\int_{\mathcal{D}}|\psi|^{4}\,dx=\int_{\Omega}|\psi|^{4}\,dx-\int_{\mathcal{\overline{D}}^{c}}|\psi|^{4}\,dx
$$
Thanks to Remark~\ref{rm-up-psi-ng}, using the asymptotics in \eqref{asy-psi-Omega} obtained in Step~1 when $\mathcal{D}=\Omega$ and the upper bound in Step~2 , we get
\begin{equation}\label{up-psi-D}
\int_{\mathcal{D}}|\psi|^{4}\,dx\leq-\int_{\mathcal{D}\cap\{a(x,\kappa)>0\}} a(x,\kappa)^{2}\left\{2\hat{f}\left(\frac{H}{\kappa}\frac{|B_{0}(x)|}{a(x,\kappa)}\right)-1\right\}\,dx+\textit{o}\left(1\right)\,.
\end{equation}
\section{Extension of the Giorgi-Phillips Theorem}\label{GP}
In this section we extend a result of Giorgi-Phillips \cite{GP}, in the two cases when the external magnetic field $B_{0}$ is variable (i.e. $\Gamma\neq\varnothing$) and when the external magnetic field $B_{0}$ is constant (i.e. $\Gamma=\varnothing$), with a pinning term.
We recall that the normal solution $(0,\mathbf F)$ is a trivial solution of the Ginzburg-Landau system \eqref{eq-2D-GLeq}. We will show that this solution is a global minimizer, when $\kappa$ and $H$ are sufficiently large. We first establish a priori estimates for a critical point $(\psi, \mathbf A)$ of the G-L-functional.
\subsection{Estimates of $\mathbf A$ and of $\|(\nabla-i\kappa H\mathbf F)\psi\|$.}~\\
We need the following estimates on $\mathbf A$ and on $\|(\nabla-i\kappa H\mathbf F)\psi\|$ which are independent of the assumption of $\Gamma$.
\begin{thm}\label{thm-2D-apriori}
There exist positive constants $C_{1}$, $C_{2}$ and $C_{3}$ such that, if \eqref{a2} hold, $\kappa>0$, $H>0$ and $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq}, then,
\begin{eqnarray}
\|(\nabla-i\kappa H\mathbf A)\psi\|_{L^{2}(\Omega)}\leq C_{1}\,\kappa\, \|\psi\|_{L^{2}(\Omega)}\label{2nd-<}\,,\\
\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq \frac{C_{2}}{H}\, \|\psi\|_{L^{2}(\Omega)}\|\psi\|_{L^{4}(\Omega)}\label{3d-<}\,,\\
\|(\nabla-i\kappa H\mathbf F)\psi\|_{L^{2}(\Omega)}\leq C_{3}\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\label{8d-<}\,.
\end{eqnarray}
\end{thm}
\begin{proof}
\textbf{We first prove \eqref{2nd-<}.} In the case when $\overline{a}\leq 0$ with $\overline{a}$ introduced in \eqref{def:sup-a}, we get using \eqref{eq-psi<a} that $\psi=0$ and hence \eqref{2nd-<} is proved.\\
In the case when $\overline{a}>0$, thanks to \eqref{eq-psi<a}, we have,
\begin{equation}\label{eq:a-psi}
0\leq(\overline{a}-|\psi|^{2})\leq \overline{a}\,.
\end{equation}
We recall that if $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq} then, (see \eqref{eq:A-psi})
$$
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\psi|^{2}\,dx=\kappa^{2}\int_{\Omega}(a(x,\kappa)-|\psi|^{2})|\psi|^{2}\,dx\,.
$$
Using \eqref{a2} and \eqref{eq:a-psi}, we obtain \eqref{2nd-<}.\\
\textbf{Now, we prove \eqref{3d-<}.} We obtain from the equation in \eqref{eq-2D-GLeq}$_{b}$ the following estimate (see \cite[Equation~(11.9b)]{FH1}):
$$
\kappa H\int_{\Omega}|\curl(\mathbf A-\mathbf F)|^{2}\,dx\leq \|(\nabla-i\kappa H\mathbf A)\psi\|_{L^{2}(\Omega)}\, \|(\mathbf A-\mathbf F)\psi\|_{L^{2}(\Omega)}\,.
$$
Using \eqref{2nd-<} and applying H$\rm\ddot{o}$lder's inequality, we get
$$
\kappa H\int_{\Omega}|\curl(\mathbf A-\mathbf F)|^{2}\,dx\leq C\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|\psi\|_{L^{4}(\Omega)}\, \|\mathbf A-\mathbf F\|_{L^{4}(\Omega)}\,.
$$
We get by regularity of the $\curl$-$\Div$ system (see \cite[A.7]{FH1}),
\begin{equation}\label{eq:curl-div}
\|\mathbf A-\mathbf F\|_{H^{1}(\Omega)}\leq C\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\,,
\end{equation}
where $C$ is a positive constant.\\
By the Sobolev embedding theorem, we get,
\begin{align}\label{sb-emb-L4}
\|\mathbf A-\mathbf F\|_{L^{4}(\Omega)}&\leq C_{\rm Sob}\, \|\mathbf A-\mathbf F\|_{H^{1}(\Omega)}\nonumber\\
&\leq \widehat C \, \|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\,.
\end{align}
Consequently,
$$
\kappa H\, \int_{\Omega}|\curl(\mathbf A-\mathbf F)|^{2}\,dx\leq \kappa\, \|\psi\|_{L^{2}(\Omega)}\|\psi\|_{L^{4}(\Omega)}\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\,,
$$
which leads to \eqref{3d-<}.\\
\textbf{Finally, we prove \eqref{8d-<}.}
Using \eqref{3d-<} and \eqref{sb-emb-L4}, H\"older's inequality gives,
\begin{align}\label{5d-<}
\|(\mathbf A-\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}&\leq \|\mathbf A-\mathbf F\|_{L^{4}(\Omega)}^{2}\|\psi\|_{L^{4}(\Omega)}^{2}\nonumber\\
& \leq \frac{C'}{H^{2}}\|\psi\|_{L^{4}(\Omega)}^{4}\|\psi\|_{L^{2}(\Omega)}^{2}\,,
\end{align}
Using \eqref{2nd-<}, \eqref{5d-<} and the bound of $\psi$ above, Young's inequality gives,
\begin{align}
\|(\nabla-i\kappa H\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}&\leq 2\|(\nabla-i\kappa H\mathbf A)\psi\|_{L^{2}(\Omega)}^{2}+2\,(\kappa H)^{2}\|(\mathbf A-\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}\nonumber\\
&\leq 2\,C''\,\kappa^{2}\|\psi\|^{2}_{L^{2}(\Omega)}\,.
\end{align}
\end{proof}
\subsection{The case $\Gamma=\varnothing$.}~\\
For $\xi\in\mathbb R$, we consider the Neumann realization $\mathfrak{h}^{N,\xi}$ in $L^{2}(\mathbb R_+)$ associated with the operator $-\frac{d^2}{dt^2}+(t+\xi)^2$, i.e.
\begin{equation}
\mathfrak{h}^{N,\xi}:=-\frac{d^2}{dt^2}+(t+\xi)^2\,,\qquad \mathcal{D}(\mathfrak{h}^{N,\xi})=\{u\in B^{2}(\mathbb R_{+}): u'(0)=0\}\,,
\end{equation}
where,
$$
B^2(\mathbb R_+)=\{u\in L^{2}(\mathbb R_+): t^p u^{(q)}\in L^{2}(\mathbb R_+), \forall p,q\in\mathbb N~s.t.~p+q\leq 2\}\,.
$$
M. Dauge and B. Helffer \cite{DH} (see also Fournais-Helffer \cite[Proposition~4.2.2]{FH1}) have proved that the lowest eigenvalue $\mu$ of $\mathfrak{h}^{N,\xi}$ admits a minimum $\Theta_{0}$, which is attained at a unique point $\xi_{0}<0$, and satisfies:
\begin{equation}\label{Theta0}
\Theta_{0}=\inf_{\xi} \mu(\xi)=\mu(\xi_0)<1\,.
\end{equation}
Moreover
\begin{equation}
\Theta_{0}=\xi_{0}^2\,.
\end{equation}
We introduce the notation:
\begin{equation}\label{inf:B0}
\inf_{x\in\overline{\Omega}} |B_{0}(x)|=b_{0}\qquad{\rm and}\qquad \inf_{x\in\partial\Omega} |B_{0}(x)|=b'_{0}\,.
\end{equation}
We denote by $\mu^{N}(\mathcal{B}\mathbf F;\Omega)$ the lowest eigenvalue of the $\rm Schr\ddot{o}dinger$ operator $P_{\mathcal{B}\mathbf F,0}^{\Omega}$ (see \eqref{def:P}) with Neumann condition in $L^{2}(\Omega)$:
\begin{equation}\label{muN(kHF)}
\mu^{N}(\mathcal{B}\mathbf F;\Omega)=\inf_{\substack{\psi\in H^{1}(\Omega)\\ \psi\neq 0}}\frac{\langle P_{\mathcal{B}\mathbf F,0}^{\Omega}\,\psi,\psi\rangle}{\|\psi\|^{2}_{L^{2}(\Omega)}}\,.
\end{equation}
In \cite{FH1}, it is proved that
\begin{theorem}\label{thm:FH}
Suppose that $\Omega\subset\mathbb R^2$ is an open bounded set with smooth boundary and $\Gamma=\varnothing$. Then,
\begin{equation}\label{eq:FH}
\lim_{\mathcal{B}\longrightarrow+\infty}\frac{\mu^{N}(\mathcal{B}\mathbf F,\Omega)}{\mathcal{B}}=\min(b_0,\Theta_{0}\,b'_{0})\,.
\end{equation}
\end{theorem}
In the next theorem, we give a simple proof of the result which says that $(0,\mathbf F)$ is the unique minimizer of the functional when $H$ is sufficiently large and when the magnetic field $B_0$ is constant with pinning term.
\begin{theorem}\label{thm:GP2}
Let $\Omega\subset\mathbb R^2$ be a smooth, bounded, simply-connected open set and $\Gamma=\varnothing$. Then, there exist positive constants $C$ and $\kappa_{0}$, such that, if
$$
H\geq C \kappa\,,\qquad\kappa\geq\kappa_{0}\,,
$$
then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.
\end{theorem}
\begin{proof}
We assume that we have a \textbf{non normal} critical point $(\psi, \mathbf A)$ for $\mathcal E_{\kappa,H,a,B_{0}}$. This means that $(\psi, \mathbf A)\in H^{1}(\Omega)\times H^{1}_{{\rm div}}(\Omega)$ is a solution of \eqref{eq-2D-GLeq} and
\begin{equation}\label{psi>0}
\int_{\Omega}|\psi|^{2}\,dx>0\,.
\end{equation}
Therefore, we get from \eqref{eq-psi<a} that,
$$
|\psi(x)|^{2}\leq \overline{a}\,\qquad\forall x\in\overline{\Omega}\,,
$$
where $\overline{a}$ is introduced in \eqref{def:sup-a}.\\
Let
\begin{equation}\label{eq:B=kH}
\mathcal{B}=\kappa H\,.
\end{equation}
Theorem~\ref{thm-2D-apriori} tells us that,
$$
\|(\nabla-i\mathcal{B}\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}\leq C\,\kappa^{2}\,\|\psi\|^{2}_{L^{2}(\Omega)}\,.
$$
Since $\psi$ satisfies \eqref{psi>0}, this implies by assumption that the lowest Neumann eigenvalue\\
$\mu^{N}(\mathcal{B}\mathbf F;\Omega)$ of $P_{\mathcal{B}\mathbf F,0}^{\Omega}$ in $L^{2}(\Omega)$ satisfies,
\begin{equation}\label{muN<}
\mu^{N}(\mathcal{B}\mathbf F;\Omega)\leq C\,\kappa^{2}\,.
\end{equation}
Thanks to Theorem~\ref{thm:FH}, we get the existence of a constant $C>0$, such that, if $H\geq C\,\kappa$, then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.
\end{proof}
\subsection{The case $\Gamma\neq\varnothing$.}~\\
We recall the definition of $\lambda_{0}$ in \eqref{lambda0}, the definition of $\Gamma$ in \eqref{gamma} and for any $\theta\in(0,\pi)$ we recall that $\lambda(\mathbb R_{+}^{2},\theta)$ is the bottom of the spectrum of the operator
$ P_{\mathbf A_{\rm app,\theta},0}^{\mathbb R^{2}_{+}}$, with
$$\mathbf A_{\rm app,\theta}=-\left(\frac{x^{2}_{2}}{2}\cos\,\theta,\frac{x^{2}_{1}}{2}\sin\,\theta \right)\,.$$
Define
\begin{equation}\label{alpha1}
\alpha_{1}(B_{0})=\min\left\{\lambda_{0}^{\frac{3}{2}}\min_{x\in \Gamma\cap\Omega}|\nabla B_{0}(x)|,\min_{x\in \Gamma\cap\partial\Omega}\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|\right\}\,.
\end{equation}
In \cite{XB-KH}, it is proved that
\begin{thm}\label{thm:BK}
Suppose that \eqref{B(x)} holds and $\Gamma\neq\varnothing$. Then
\begin{equation}\label{eq:pan}
\lim_{\mathcal{B}\longrightarrow+\infty}\frac{\mu^{N}(\mathcal{B}\mathbf F,\Omega)}{\mathcal{B}^{\frac{2}{3}}}=\alpha_{1}(B_{0})^{\frac{2}{3}}\,.
\end{equation}
\end{thm}
In the next theorem, we give a simple proof of the result which says that $(0,\mathbf F)$ is the unique minimizer of the functional when $H$ is sufficiently large and when $B_{0}$ is variable. This result was obtained in \cite{GP} for the case with constant magnetic field and with a constant pinning term.
\begin{theorem}\label{thm:GP}
Let $\Omega\subset\mathbb R^2$ be a smooth, bounded, simply-connected open set, the pinning term $a$ satisfying \eqref{a2}, and the magnetic field $B_{0}$ satisfying \eqref{B(x)}. Then, there exist positive constants $C$ and $\kappa_{0}$, such that, if
$$
H\geq C \kappa^{2}\,,\qquad\kappa\geq\kappa_{0}\,.
$$
Then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.
\end{theorem}
\begin{proof}
Similarly to the proof of Theorem~\ref{thm:GP2}, we assume that we have a \textbf{non normal} critical point $(\psi, \mathbf A)$ for $\mathcal E_{\kappa,H,a,B_{0}}$.\\
Therefore, we get from \eqref{8d-<} that,
$$
\mu^{N}(\mathcal{B}\mathbf F;\Omega)\leq C\,\kappa^{2}\quad(\mathcal B=\kappa H)\,.
$$
Thanks to Theorem~\ref{thm:BK}, we get a contradiction, if $ \displaystyle H\geq C \kappa^{2}$ and $C$ is sufficiently large.
\end{proof}
\section{Asymptotics of $\mu_{1}(\kappa,H)$: the case with non vanishing magnetic field}\label{Section:4}
The aim of this section is to give an estimate for the lowest
eigenvalue $\mu_{1}(\kappa,H)$ of the operator $P_{\kappa
H\mathbf F,-\kappa^{2}a}^{\Omega}$ (see \eqref{def:mu1}) in the case when
$\Gamma=\varnothing$ with a $\kappa$-independent pinning (i.e.
$a(x,\kappa)=a(x)$). Recall that the set $\Gamma$ is introduced in
\eqref{gamma}.
\subsection{Lower bound}~\\
Without loss of generality we suppose that $B_{0}>0 \mbox{ in } \overline{ \Omega}$. Our results will be formulated by introducing:
\begin{equation}\label{Lambda11}
\Lambda_{1}(B_{0},a,\sigma)=\min\left\{\inf_{ x\in\Omega}\left\{\sigma\,B_{0}(x)-a(x)\right\},\inf_{ x\in\partial\Omega}\left\{\Theta_{0}\,\sigma\,B_{0}(x)-a(x)\right\}\right\}\,,
\end{equation}
where $\sigma$ is a positive constant.\\
In the case when the pinning term is constant (i.e. $a(x)=a_0$), \eqref{Lambda11} becomes as follows:
$$
\Lambda_{1}(B_{0},a,\sigma)=\sigma\min\left\{\inf_{ x\in\Omega}\left\{B_{0}(x)\right\},\Theta_{0}\,\inf_{ x\in\partial\Omega}\left\{B_{0}(x)\right\}\right\}-a_0\,.
$$
This case was treated by Pan and Kwek \cite{LP1}.\\
Let $\mathcal{Q}_{\mathcal{B}\,\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}$ be the quadratic form of $P_{\mathcal{B}\mathbf F,- \frac{\mathcal{B}}{\sigma}\,a}^{\Omega}$, i.e.
\begin{equation}\label{def:quad}
\mathcal{Q}_{\mathcal{B}\,\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}(\psi)=\int_{\Omega}\left(|(\nabla-i\mathcal{B}\mathbf F)\psi|^{2}-\frac{\mathcal{B}}{\sigma} \,a(x)|\psi|^{2}\right)\,dx\,.
\end{equation}
\begin{prop}\label{prop:mu1-cst}
Let $\Omega\subset\mathbb R^2$ be an open bounded set with smooth boundary, $I$ a closed interval in $(0,+\infty)$ and $\Gamma=\varnothing$. There exist positive constant $C$ and $\mathcal{B}_{0}$ such that if $\sigma\in I$, $\mathcal{B}\geq\mathcal{B}_{0}$, $\psi\in H^{1}(\Omega)\setminus \{0\} $ and $a\in C^{1}(\overline{\Omega})$, then,
\begin{equation}\label{eq:lower-bound-constant}
\frac{\mathcal{Q}_{\mathcal{B}\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}(\psi)}{\|\psi\|_{L^{2}(\Omega)}^{2}}\geq \frac{\mathcal B}{\sigma}\,\,\Lambda_{1}(B_{0},a,\sigma)-C\,\mathcal{B}^{\frac{3}{4}}\,.
\end{equation}
\end{prop}
\begin{proof}
The proof is a consequence of the following inequality that we take
from \cite[Prop.~9.2.1]{FH1},
$$
\forall~\psi\in H^1(\Omega)\,,\quad\int_\Omega|(\nabla-i\mathcal{B}\mathbf F)\psi|^2\,dx\geq \int_\Omega \big(U(x)-\bar CB^{3/4}\big)|\psi|^2\,dx\,,
$$
where \begin{equation}\label{eq:U}
U(x)= \left\{
\begin{array}{ll}
\mathcal B\,B_0(x)&{\rm if~}{\rm dist}(x,\partial\Omega)\geq B^{-3/8}\,,\\
\Theta_0\mathcal B\,B_0(x)&{\rm if~}{\rm dist}(x,\partial\Omega)< B^{-3/8}\,,
\end{array}
\right.
\end{equation}
$\mathcal B\geq \bar{\mathcal B_0}$, $\bar{\mathcal B_0}$ and $\bar
C$ are two constants independent of $\mathcal B$.
Clearly, there exist two constants $C'>0$ and $\mathcal B_0>0$ such
that, for all $\sigma\in I$, we have,
$$
U(x)-\frac{\mathcal B}{\sigma}a(x)\geq\frac{\mathcal B}{\sigma}\Lambda_1(B_0,a,\sigma)-C'B^{3/4}\,.
$$
\end{proof}
Coming back to our initial parameters $\kappa$ and $H$, we obtain:
\begin{theorem}\label{thm:mu1-cst}
Let $\Omega\subset\mathbb R^2$ be an open bounded set with smooth boundary and $\Gamma=\varnothing$. Suppose that \eqref{cond-H} holds and $a\in C^1(\overline{\Omega})$, then,
$$
\mu_{1}(\kappa,H)\geq \kappa^{2}\,\Lambda_{1}\left(B_{0},a,\frac H \kappa\right)+\mathcal{O}(\kappa^{\frac{3}{2}})\,,\qquad{\rm as}\,\kappa\to+\infty\,.
$$
Here, $\Lambda_{1}$ is introduced in \eqref{Lambda11}.
\end{theorem}
\begin{proof}
We apply Proposition~\ref{prop:mu1-cst} with
$$
\mathcal{B}=\kappa H\,,\quad \sigma=\frac{H}{\kappa}\quad{\rm and}\quad I=[\lambda_{\min},\lambda_{\max}]\,.
$$
Let us verify that the conditions of the proposition are satisfied for this choice.\\
It is trivial that $\sigma\in I$. Now, as $\kappa\to+\infty$, we have,
$$
\mathcal{B}=\sigma\,\kappa^{2}\to+\infty\,.
$$
This implies that, as $\kappa\to+\infty$,
$$
\mu_{1}(\kappa,H)\geq \kappa^{2}\,\Lambda_{1}\left(B_{0},a,\frac{H}{\kappa} \right)+\mathcal{O}(\kappa^{\frac{3}{2}})\,.
$$
This finishes the proof of the theorem.
\end{proof}
\subsection{Upper bound}
\begin{prop}[Upper bound in the bulk]\label{prop:up-blk-cst}
Suppose that $\Omega\subset\mathbb R^2$ is an open bounded set with smooth boundary $\partial\Omega$, $\lambda_{\max} >0$ and $\Gamma=\varnothing$. For any $ x_0\in\Omega$, there exist positive constants $C$ and $\mathcal{B}_{0}$ such that, if $\sigma\in (0,\lambda_{\max}]$, $\mathcal{B}\geq\mathcal{B}_{0}$ and $a\in C^{1}(\overline{\Omega})$, then,
\begin{equation}\label{eq:up-blk-cst}
\mu_{\mathcal{B},\sigma}\leq \frac{\mathcal B}{\sigma}\,\left\{\sigma\,B_{0}(x_{0})-a(x_{0})\right\}+C\,\mathcal{B}^{\frac{1}{2}}\,.
\end{equation}
Here,
\begin{equation}\label{def:mu1-cst}
\mu_{\mathcal{B},\sigma}=\inf_{\psi\in H^{1}(\Omega)\setminus \{0\}}\frac{\mathcal{Q}_{\mathcal{B}\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}(\psi)}{\|\psi\|_{L^{2}(\Omega)}^{2}}\,,
\end{equation}
where $\mathcal{Q}_{\mathcal{B}\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}$ is introduced in \eqref{def:quad}.
\end{prop}
\begin{proof}
Thanks to \eqref{def:quad}, we have,
$$
\mathcal{Q}_{\mathcal{B}\mathbf F,-\frac{\mathcal B}{\sigma}\,a}^{\Omega}(u) = \int_{\Omega}|(\nabla-i\mathcal{B}\mathbf F)u(x)|^{2}\,dx - \frac{ \mathcal B}{\sigma}\,\int_{\Omega}a(x)|u(x)|^{2}\,dx \,.
$$
The upper bound of the first term in the right hand side above is based on the construction of Gaussian quasi-mode (see \cite[Subsection~2.4.2]{FH1} for the case with constant pinning) centered at $x_0\in\Omega$,
$$
\varphi_{1}(x)=\,e^{i\,\mathcal{B}\,\phi_{0}}\,\chi\left(\mathcal{B}^{+\frac{1}{2}}(x-x_{0})\right)\,u\left(\sqrt{\mathcal{B}B_{0}(x_0)}\,(x-x_{0})\right)\,.
$$
Here, $\chi$ is a cut-off function equal to $1$ in a neighborhood of $0$ such that ${\rm supp}\,\chi\subset D(0,1)$, the function $\phi_{0}$ satisfies \eqref{F-A} and the function $u$ defined as follows:
$$
u(x)=\frac{\pi^{-\frac{1}{4}}}{\sqrt{2}}e^{-\frac{|x|^2}{2}}\,.
$$
We note that ${\rm supp}\,\varphi_{1}\subset \Omega$ for $\mathcal{B}$ large enough. As in \cite[(2.35)]{FH1}, we get the existence of a positive constant $\mathcal{B}_{0}$ such that, for any $\mathcal{B}\geq\mathcal{B}_{0}$,
\begin{equation}\label{up:FA-cst}
\frac{\int_{\Omega}|(\nabla-i\mathcal{B}\mathbf F)\varphi_{1}(x)|^{2}\,dx}{\int_{\Omega}|\varphi_{1}(x)|^{2}\,dx}\leq \mathcal{B}\,B_{0}(x_0)+\mathcal{O}(\mathcal{B}^{\frac{1}{2}})\,.
\end{equation}
To derive the upper bound for the second term, we use Taylor's formula for the function $a$ near $x_0$,
\begin{equation}\label{eq:app-a-cst}
|a(x)-a(x_0)|\leq C\,\,\mathcal{B}^{-\frac{1}{2}}\,,\qquad\left(x\in D\left(x_0,\mathcal{B}^{-\frac{1}{2}}\right)\right)\,.
\end{equation}
Using \eqref{eq:app-a-cst}, since ${\rm supp}\,\varphi_{1}\subset D\left(x_0,\mathcal{B}^{-\frac{1}{2}}\right)$, we get,
\begin{equation}\label{upp:a1-cst}
-\int_{\Omega}a(x)|\varphi_{1}(x)|^{2}\,dx\leq -a(x_{0})\int_{\Omega}|\varphi_{1}(x)|^{2}\,dx+C\,\mathcal{B}^{-\frac{1}{2}}\,\int_{\Omega}|\varphi_{1}(x)|^{2}\,dx\,,
\end{equation}
and consequently
\begin{equation}\label{upp:a-cst}
-\frac{\mathcal B}{\sigma}\frac{\int_{\Omega}a(x)|\varphi_{1}(x)|^{2}\,dx}{\int_{\Omega}|\varphi_{1}(x)|^{2}\,dx}\leq -\frac{\mathcal B}{\sigma}\,a(x_{0})+C\,\mathcal{B}^{\frac{1}{2}}\,.
\end{equation}
Collecting \eqref{up:FA-cst} and \eqref{upp:a-cst}, we finish the proof of Proposition~\ref{def:mu1-cst}.
\end{proof}
\begin{rem}\label{rem:first term}
When
$$
\inf_{x\in\Omega}\left\{\sigma\,B_{0}(x)-a(x)\right\} < \inf_{ x\in\partial\Omega}\left\{\Theta_{0}\,\sigma\,B_{0}(x)-a(x)\right\}\,,
$$
we notice that, if the infimum of $\sigma\,B_{0}(x)-a(x)$ was attained on $\partial\Omega$, (i.e. there exists $x_0\in\partial\Omega$ such that $\inf_{x\in\Omega}\left\{\sigma\,B_{0}(x)-a(x)\right\} =\sigma\,B_{0}(x_0)-a(x_0)$), we would have,
$$
\sigma\,B_{0}(x_0)-a(x_0) < \Theta_{0}\,\sigma\,B_{0}(x_0)-a(x_0)\,,
$$
which is impossible, since $\Theta_{0}<1$. Hence, we can choose $x_0\in\Omega$, such that,
$$
\sigma\,B_{0}(x_0)-a(x_0)=\inf_{x\in\Omega}\left\{\sigma\,B_{0}(x)-a(x)\right\}\,,
$$
and we apply Proposition~\ref{prop:up-blk-cst} with
$$
\mathcal{B}=\kappa H\quad{\rm and}\quad\sigma=\frac{H}{\kappa}\,.
$$
Thus, we get the existence of a positive constant $\kappa_0$ such
that, if,
\begin{equation}\label{cond:H-cst}
\kappa\geq\kappa_{0}\quad{\rm and}\quad\kappa_{0}\,\kappa^{-1}<H< \lambda_{\max}\,\kappa\,,
\end{equation}
then,
\begin{equation}\label{eq:up-mu1-cst}
\mu_{1}(\kappa,H)\leq \kappa^{2}\,\inf_{x\in\Omega}\left\{\frac{H}{\kappa}\,B_{0}(x)-a(x)\right\}+ \mathcal{O}(\kappa)\,,\qquad{\rm as}\,\kappa\to+\infty\,.
\end{equation}
\end{rem}
\begin{prop}[Upper bound near the boundary]\label{prop:up-bnd-cst}
Suppose that $\Omega\subset\mathbb R^2$ is an open bounded set with a smooth boundary, $\lambda_{\max} >0$ and $\Gamma=\varnothing$. For any $ x_0\in\partial\Omega$ and for any $\sigma\in (0,\lambda_{\max}]$, we have,
\begin{equation}\label{eq:up-bnd-cst}
\mu_{\mathcal{B},\sigma}\leq \frac{\mathcal B}{\sigma}\big(\sigma\,\Theta_{0}\,B_{0}(x_{0})-a(x_0)\big)+ \mathcal{O}(\mathcal{B}^{\frac{1}{2}})\,,\qquad{\rm as}~\mathcal{B}\to+\infty\,.
\end{equation}
Here, $\Theta_{0}$ is introduced in \eqref{Theta0}.
\end{prop}
\begin{proof}
We recall the definition of $\mu_{\mathcal{B},\sigma}$ as follows:
$$
\mu_{\mathcal{B},\sigma}=\inf_{u\in H^{1}(\Omega)\setminus \{0\}} \left(\frac{\int_{\Omega}|(\nabla-i\mathcal{B}\mathbf F)u(x)|^{2}\,dx}{\int_{\Omega}|u(x)|^{2}\,dx}-\frac{\mathcal B}{\sigma}\,\frac{\int_{\Omega}a(x)|u(x)|^{2}\,dx}{\int_{\Omega}|u(x)|^{2}\,dx}\right)\,.
$$
The first term in the right hand side is studied by Helffer-Morame (see \cite[Theorem~9.1]{HM} with $h=\mathcal{B}^{-1}$ and $\mu_{\mathcal{B},\sigma}=\frac{\mu^{(1)}(h)}{h^{2}}$) or Fournais-Helffer (see \cite[Section~9.2.1]{FH1}). They proved for any $x_0\in \partial \Omega$ the existence of $\mathcal B_0$ such that for $\mathcal{B}\geq \mathcal B_0$ one can construct a trial function $\widehat{u}$ such that,
$$
\frac{\int_{\Omega}|(\nabla-i\mathcal{B}\mathbf F)\widehat u(x)|^{2}\,dx}{\int_{\Omega}|\widehat u(x)|^{2}\,dx}\leq \mathcal{B}\,\Theta_{0}\,B_{0}(x_{0})+ \mathcal{O}(\mathcal{B}^{\frac{1}{2}})\,,\qquad{\rm as}~\mathcal{B}\to+\infty \,.
$$
The estimates of the second term in the right hand side are just as in \eqref{upp:a-cst} and this achieves the proof of the proposition.
\end{proof}
\begin{rem}\label{rem:second term}
$\partial \Omega$ being compact, we can choose $ x_0\in\partial\Omega$, such that,
$$
\sigma\,\Theta_{0}\,B_{0}(x_0)-a(x_0)=\inf_{x\in\partial\Omega}\left\{\sigma\,\Theta_{0}\,B_{0}(x)-a(x)\right\}\,,
$$
and we apply Proposition~\ref{prop:up-bnd-cst} with
$$
\mathcal{B}=\kappa H\quad{\rm and}\quad \sigma=\frac{H}{\kappa}\,,
$$
which implies under Assumption~\ref{cond:H-cst} that,
\begin{equation}\label{eq:up-mu1-cst2}
\mu_{1}(\kappa,H)\leq \kappa^{2}\,\inf_{ x\in\partial\Omega}\left\{\frac{H}{\kappa}\,\Theta_{0}\,B_{0}(x)-a(x)\right\}+\mathcal{O}(\kappa)\,,\qquad{\rm as}\,\kappa\to+\infty\,.
\end{equation}
\end{rem}
Remarks~\ref{rem:first term} and ~\ref{rem:second term} lead to the conclusion in:
\begin{theorem}\label{thm:mu1-cst-upp}
Let $\Omega\subset\mathbb R^2$ is an open bounded set with a smooth boundary and $\Gamma=\varnothing$. Suppose that \eqref{cond:H-cst} hold and $a\in C^1(\overline{\Omega})$, we have
$$
\mu_{1}(\kappa,H)\leq \kappa^{2}\,\Lambda_{1}\left(B_{0},a,\frac H \kappa\right)+ \mathcal{O}(\kappa)\,,\qquad{\rm as}\,\kappa\to+\infty\,.
$$
Here, $\Lambda_{1}$ is introduced in \eqref{Lambda11}.
\end{theorem}
Notice that the conclusion in Theorem~\ref{thm:mu1-cst-upp} is valid under the assumption $ \kappa H\geq \mathcal B_0$ with $\mathcal B_0 >0$
sufficiently large. Lemma~\ref{lem-H=kappa} below takes care of the
regime where $\kappa H=\mathcal O(1)$.
\begin{lem}\label{lem-H=kappa}
Let $C_{\max}>0$. Suppose that $\{a>0\}\not=\emptyset$. There exists a constant $\kappa_0>0$ such that, if
$$\kappa\geq \kappa_0\quad{\rm and}\quad 0\leq H\leq C_{\max}\kappa^{-1}\,,$$
then
$$\mu_1(\kappa,H)<0\,.$$
\end{lem}
\begin{rem}\label{rem:h=0}
The conclusion in Lemma~\ref{lem-H=kappa} is valid in both cases
where $\Gamma=\emptyset$ and $\Gamma\not=\emptyset$.
\end{rem}
\begin{proof}[Proof of Lemma~\ref{lem-H=kappa}]~\\
Let $\ell>0$. Choose $x_0\in\Omega$ such that $a(x_0)>0$. We
introduce a cut-off function $\chi_{\ell}\in C_{c}^{\infty}(\mathbb R^{2})$
satisfying:
\begin{equation}\label{def:chil}
0\leq \chi_{\ell} \leq 1~{\rm in}~\mathbb R^{2}\,,\quad {\rm supp}\chi_{\ell}\subset B(x_{0},\ell)\,,\quad \chi_{\ell}=1~{\rm in}~B\left(x_{0},\ell/2\right)\quad{\rm and}\quad |\nabla\chi_{\ell}|\leq \frac{C}{\ell}\,.
\end{equation}
The min-max principle yields,
$$
\mu^{(1)}(\kappa,H)\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}\leq \int_{\Omega}|(\nabla-i\kappa H\mathbf F)\chi_{\ell}|^{2}\,dx-\kappa^{2}\int_{\Omega}a(x)|\chi_{\ell}(x)|^{2}\,dx\,.
$$
Using the assumptions on $\chi_{\ell}$ and the fact that $\mathbf F\in C^{\infty}(\overline{\Omega})$, a trivial estimate is,
\begin{align}\label{up:F-varphi3}
\int_{\Omega}|(\nabla-i\kappa H\mathbf F)\chi_{\ell}|^{2}\,dx&=\int_{B(x_{0},\ell)}|\nabla\chi_{\ell}(x)|^{2}\,dx+\kappa^{2}H^{2}\int_{B(x_{0},\ell)}|\mathbf F\,\chi_{\ell}(x)|^{2}\,dx\nonumber\\
&\leq C\,(1+(\kappa\,H\,\ell)^{2})\,.
\end{align}
We write by Taylor's formula applied to the function $a$ near $x_0$,
\begin{equation}\label{up:a-varphi3}
-\kappa^{2}\int_{\Omega}a(x)|\chi_{\ell}(x)|^{2}\,dx\leq -a(x_0)\,\kappa^{2}\,\ell^{2}+C\,\kappa^{2}\,\ell^{3}\,.
\end{equation}
Collecting \eqref{up:F-varphi3} and \eqref{up:a-varphi3}, we obtain,
$$
\mu^{(1)}(\kappa,H)\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}\leq-a(x_0)\,\kappa^{2}\,\ell^{2}+C(\kappa^{2}\,\ell^{3}+1+(\kappa\,H\,\ell)^{2})\,.
$$
We select $\ell=\kappa^{-\frac{1}{2}}$ and note that $\kappa H<C_{\max}$. We find that,
$$
\mu^{(1)}(\kappa,H)\|\chi_{\ell}\|_{L^{2}(\Omega)}^{2}\leq-a(x_0)\,\kappa+C\Big(\kappa^{\frac{1}{2}}+1+C_{\max}^{2}\kappa^{-1}\Big)\,.
$$
Since $\chi_\ell\not=0$ and $a(x_0)>0$, we deduce that, for $\kappa$
sufficiently large,
$$
\mu^{(1)}(\kappa,H)<0\,.
$$
\end{proof}
\section{Proof of Theorem~\ref{thm:HC3}}\label{10}
\subsection{Analysis of $\underline{H}_{C_3}^{loc}$ and $\overline{H}_{C_3}^{loc}$.}~\\
In this subsection we give a lower bound of the critical field $\underline{H}_{C_3}^{loc}$ (see \eqref{def:HC3-u}) and we give an upper bound of the critical field $\overline{H}_{C_3}^{loc}$ in the case when the magnetic field $B_0$ is constant with a pining term.
\begin{prop}\label{prop:mu1<0}
Suppose that $\{a>0\}\neq\varnothing$ and $\Gamma=\varnothing$. There exist constants $C>0$ and $\kappa_{0}\geq 0$ such that if
\begin{equation}\label{cond:HC3}
\kappa\geq \kappa_{0}\,,\qquad H\leq \kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\,,
\end{equation}
then,
$$
\mu_{1}(\kappa,H)<0\,.
$$
Moreover,
$$
\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\leq \underline{H}_{C_{3}}^{loc}\,.
$$
\end{prop}
\begin{proof}
To apply the previous results, we take
$$\lambda_{max} = \max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right) +1\,.
$$
We have two cases:\\
\textbf{Case 1.} If
$$\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)}>\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\,.$$
then, there exists $x_0\in\Omega$ (the supremum of $\frac{a(x)}{B_{0}(x)}$ can not be attained on the boundary, since $\frac{a(x)}{\Theta_{0}\,B_{0}(x)}>\frac{a(x)}{B_{0}(x)}$), such that,
$$
\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)} = \frac{a(x_0)}{B_{0}(x_0)} \,.
$$
If \eqref{cond:HC3} is satisfied for some $C>0$, then,
$$
\frac{H}{\kappa}\leq \frac{a(x_0)}{B_{0}(x_0)}-C\,\kappa^{-\frac{1}{2}}\,,
$$
that we can write in the form,
$$
\kappa^{2}\left(\frac{H}{\kappa}B_{0}(x_0)-a(x_0)\right)\leq -C\,M \,\kappa^{\frac{3}{2}}\,,
$$
where $M>0$ is a constant independent of $C$.\\
Suppose that $\kappa H\geq \mathcal B_0$ where $\mathcal B_0$ is selected sufficiently large such that we can apply Remark~\ref{rem:first term}. (Thanks to Lemma~\ref{lem-H=kappa}, $\mu_1(\kappa,H)<0$ when $\kappa H<\mathcal B_0$).\\
Remark~\ref{rem:first term} tells us that there exist positive
constants $C_{1}$ and $\kappa_{0}$ such that, for $\kappa\geq
\kappa_{0}$,
\begin{align}
\mu_{1}(\kappa,H)&\leq \kappa^{2}\inf_{x\in\Omega} \left(\frac{H}{\kappa}B_{0}(x)-a(x)\right)+C_{1}\,\kappa\nonumber\\
&\leq \kappa^{2}\left(\frac{H}{\kappa}B_{0}(x_0)-a(x_0)\right)+C_{1}\,\kappa^{\frac{3}{2}}\\
&\leq (C_1-C\,M) \,\kappa^\frac 32 \,.
\end{align}
By choosing $C$ such that $C\,M> C_{1}$, we get,
$$
\mu_{1}(\kappa,H)<0\,.
$$
\textbf{Case 2.} Here, we suppose that
$$\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\geq \sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)}\,.$$
By compactness, there exists $x'_0\in\partial\Omega$, such that,
$$\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\ = \frac{a(x'_0)}{\Theta_{0}\,B_{0}(x'_0)}
$$
If \eqref{cond:HC3} is satisfied for some $C>0$, then,
$$
\kappa^{2}\left(\frac{H}{\kappa}\Theta_{0}\,B_{0}(x'_0)-a(x'_0)\right)\leq -C\,M'\,\kappa^{\frac{3}{2}}\,.
$$
Thanks to Remark~\ref{rem:second term}, we get the existence of positive constants $C_{2}$ and $\kappa_{0}$ such that, for $\kappa\geq \kappa_0$,
\begin{align}
\mu_{1}(\kappa,H)&\leq \kappa^{2}\inf_{x\in\partial\Omega} \left(\frac{H}{\kappa}\,\Theta_{0}\,B_{0}(x)-a(x)\right)+C_{2}\,\kappa\nonumber\\
&\leq \kappa^{2}\left(\frac{H}{\kappa}\,\Theta_{0}\,B_{0}(x'_0)-a(x'_0)\right)+C_{2}\,\kappa^{\frac{3}{2}}\\
&\leq (C_2-C\,M')\,\kappa^\frac 32\,.
\end{align}
By choosing $C$ such that $C\,M'> C_{2}$, we get,
$$
\mu_{1}(\kappa,H)<0\,.
$$
This finishes the proof of the proposition.
\end{proof}
\begin{prop}\label{prop:mu>0}
Suppose that $\{a>0\}\neq\varnothing$, $\lambda_{\max} >0$ and $\Gamma=\varnothing$. There exist constants $C>0$ and $\kappa_{0} > 0$ such that if
\begin{equation}\label{cond:HC3-2}
\kappa\geq \kappa_{0}\,,\qquad \lambda_{\max}\,\kappa\geq H>\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,,
\end{equation}
then,
$$
\mu_{1}(\kappa,H)>0\,.
$$
Moreover,
$$
\overline{H}_{C_{3}}^{loc}\leq\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,.
$$
\end{prop}
\begin{proof}
To apply the previous results, we take
$$\lambda_{min} = \frac 12 \max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right) \,.$$
If \eqref{cond:HC3-2} holds for some $C>0$, then, for any $x\in\Omega$, we have,
\begin{equation}\label{eq:bulk}
\frac{H}{\kappa}B_{0}(x)-a(x)\geq C\,\kappa^{-\frac{1}{2}}\,,
\end{equation}
and, for any $x'\in\partial\Omega$, we have,
\begin{equation}\label{eq:bnd}
\frac{H}{\kappa}\Theta_{0}B_{0}(x')-a(x')\geq C\,\kappa^{-\frac{1}{2}}\,.
\end{equation}
Having in mind the definition of $\Lambda_1$ in \eqref{Lambda11}, the estimates in \eqref{eq:bulk} and in \eqref{eq:bnd} give us that for $\kappa$ large enough,
$$
\Lambda_{1}\left(B_0,a,\frac{H}{\kappa}\right)\geq C\,\kappa^{-\frac{1}{2}}\,.
$$
Thanks to Theorem~\ref{thm:mu1-cst}, we get the existence of positive constants $C'$ and $\kappa_0$ such that, for $\kappa\geq\kappa_0$,
\begin{align}
\mu_{1}(\kappa,H)&\geq \kappa^{2}\,\Lambda_{1}\left(B_{0},a,\frac H \kappa\right)-C'\,\kappa^{\frac{3}{2}}\nonumber\\
&\geq (C-C')\,\kappa^{\frac{3}{2}}\,.
\end{align}
To finish this proof, we choose $C>C'$.
\end{proof}
As a consequence, we have proved Theorem~\ref{thm:HC3} for $\underline{H}_{C_3}^{loc}$ and $\overline{H}_{C_3}^{loc}$
\subsection{Analysis of $\underline{H}_{C_3}^{cp}$ and $\overline{H}_{C_3}^{cp}$.}~\\
In this subsection we give a lower bound of the critical field $\underline{H}_{C_3}^{cp}$ (see \eqref{def:HC3-o}) and we give an upper bound of the critical field $\overline{H}_{C_3}^{cp}$ in the case when the magnetic field $B_0$ is constant with a pining term. We start with a proposition which measures the effect of the localization at the boundary when $H$ is sufficiently large.
\begin{prop}\label{lem:psi-2<4}
Suppose that $\Gamma=\varnothing$ and \eqref{cond:HC3-2} holds. There exists a positive constant $C$, such that if $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq}, then the following estimate holds:
\begin{equation}
\|\psi\|^{2}_{L^{2}(\Omega)}\leq C\,\kappa^{-\frac{3}{8}}\|\psi\|^{2}_{L^{4}(\Omega)}\,.
\end{equation}
\end{prop}
\begin{proof}~\\
The techniques that will be used in this proof are similar with the ones in \cite[Lemma~2.6]{FK2}. If $H$ satisfies \eqref{cond:HC3-2} for some $C>0$, then, for any $x\in\Omega$, we have.
\begin{equation}\label{bulk2}
\kappa\,H\,B_{0}(x)-\kappa^{2}\,a(x)\geq C\,\kappa^{\frac{3}{2}}\,.
\end{equation}
First, we let $\chi\in C^{\infty}(\mathbb R)$ be a standard cut-off function such that
\begin{equation}\label{def:chi-cst}
\chi=1\quad{\rm in}~[1,\infty]\qquad{\rm and }\qquad\chi=0\quad{\rm in}~]-\infty,1/2]\,.
\end{equation}
Next, we define $\lambda=\kappa^{-\frac{3}{4}}$, and $\chi_{\kappa}$ as follows:
\begin{equation}
\chi_{\kappa}(x)=\chi\left(\frac{\dist(x,\partial\Omega)}{\lambda}\right)\,,\qquad \forall x\in\Omega\,.
\end{equation}
Referring to \eqref{eq:1}, we have
\begin{equation}\label{eq:2}
\int_{\Omega} \left(|(\nabla-i\kappa H \mathbf A)\chi_{\kappa}\psi|^{2}-|\nabla\chi_{\kappa}|^{2}|\psi|^{2}\right)\,dx=\kappa^{2}\int_{\Omega}|\chi_{\kappa}|^{2}(a(x)-|\psi|^{2})|\psi|^{2}\,dx\,.
\end{equation}
We estimate $\int_\Omega|(\nabla-i\kappa H\mathbf A)\chi_{\kappa}\psi|^2\,dx$ from below. As in \cite[Proposition~6.2]{HK}, we can prove that,
$$
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\chi_{\kappa}\psi|^2\,dx\geq \kappa\,H\int_\Omega \curl\mathbf F \,|\chi_{\kappa}\psi|^2\,dx-\kappa\,H\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\|\chi_{\kappa}\psi\|_{L^{4}(\Omega)}^{2}\,.
$$
Noticing that $\displaystyle \curl\mathbf F=B_{0}(x)$ and $\displaystyle\|\curl(\mathbf A-\mathbf F)\|_{L^{2}(\Omega)}\leq \frac{c}{H}\|\psi\|_{L^{2}(\Omega)}$, we have,
$$
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\chi_{\kappa}\psi|^2\,dx\geq \kappa\,H\int_\Omega\,B_{0}(x)\,|\chi_{\kappa}\psi|^2\,dx-c\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|\chi_{\kappa}\psi\|_{L^{4}(\Omega)}^{2}\,.
$$
Implementing a Cauchy-Schwarz inequality, we get
\begin{equation}\label{eq:grd-eng}
\int_{\Omega}|(\nabla-i\kappa H\mathbf A)\chi_{\kappa}\psi|^2\,dx\geq \kappa\,H\int_\Omega\,B_{0}(x)\,|\chi_{\kappa}\psi|^2\,dx-c^{2}\,\|\psi\|_{L^{2}(\Omega)}^{2}-\kappa^{2}\|\chi_{\kappa}\psi\|_{L^{4}(\Omega)}^{4}\,.
\end{equation}
Inserting \eqref{eq:grd-eng} into \eqref{eq:2}, we obtain,
$$
\int_\Omega\,\left(\kappa\,H\,B_{0}(x)-\kappa^{2}\,a(x)\right)\,|\chi_{\kappa}\psi|^2\,dx\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla\chi_{\kappa}|^{2}|\psi|^{2}\,dx-\kappa^{2}\int_{\Omega}\left(\chi_{\kappa}^{2}-\chi_{\kappa}^{4}\right)|\psi|^{4}\,dx\,.
$$
As a consequence of \eqref{bulk2}, the inequality above becomes,
$$
C\,\kappa^{\frac{3}{2}}\int|\chi_{\kappa}\psi(x)|^{2}\,dx\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla\chi_{\kappa}|^{2}|\psi|^{2}\,dx-\kappa^{2}\int_{\Omega}\left(\chi_{\kappa}^{2}-\chi_{\kappa}^{4}\right)|\psi|^{4}\,dx\,.
$$
Notice that $-\kappa^{2}\int_{\Omega}\left(\chi_{\kappa}^{2}-\chi_{\kappa}^{4}\right)|\psi|^{4}\,dx\leq 0\,$.\\
Decomposing the integral $\displaystyle\int_{\Omega}|\psi|^{2}\,dx=\int_{\Omega}|\chi_{\kappa}\psi|^{2}\,dx+\int_{\Omega}(1-\chi_{\kappa}^{2})|\psi|^{2}\,dx$, using \eqref{bulk2} and choosing $C$ such that $C\geq 2 c^{2}$, we get,
$$
\frac{1}{2}C\,\kappa^{\frac{3}{2}}\int|\chi_{\kappa}\psi(x)|^{2}\,dx\leq \left(c^{2}+\|\chi'\|_{L^{\infty}(\mathbb R)}^{2}\,\lambda^{-2}\right)\int_{\left\{x\in\Omega:\,\dist(x,\Gamma)\leq\lambda\right\}}|\psi|^{2}\,dx\,.
$$
Recall that $\lambda=\kappa^{-\frac{3}{4}}$, we observe that,
$$
\int|\chi_{\kappa}\psi(x)|^{2}\,dx\leq 4\|\chi'\|_{L^{\infty}(\mathbb R)}^{2} \int_{\left\{x\in\Omega:\,\dist(x,\Gamma)\leq\lambda\right\}}|\psi|^{2}\,dx\,,
$$
and consequently, we get,
$$
\int|\psi(x)|^{2}\,dx\leq\left(4\|\chi'\|_{L^{\infty}(\mathbb R)}^{2}+1\right) \int_{\left\{x\in\Omega:\,\dist(x,\Gamma)\leq\lambda\right\}}|\psi|^{2}\,dx\,.
$$
By choosing $C=\max \left(2\,c^{2},4\|\chi'\|_{L^{\infty}(\mathbb R)}^{2}+1\right)$, we obtain,
$$
\|\psi\|^{2}_{L^{2}(\Omega)}\leq C\,\kappa^{-\frac{3}{8}}\|\psi\|^{2}_{L^{4}(\Omega)}\,.
$$
\end{proof}
\begin{theorem}\label{thm:lb-H}
Supose that $\Gamma=\varnothing$ and $\{a>0\}\neq\varnothing$. There exists $C>0$ and $\kappa_0$ such that, if $H$ satisfies
\begin{equation}\label{cond:HC3-2w}
H>\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,,
\end{equation} then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.\\
Moreover,
$$
\overline{H}_{C_3}^{cp}\leq\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,.
$$
\end{theorem}
\begin{proof}
We first observe that it results from Giorgi-Phillips like Theorem \ref{thm:GP2} that it remains only to prove the theorem under the stronger Assumption \eqref{cond:HC3-2}.
Suppose now that $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq} with $\psi\neq 0$, we observe that,
\begin{equation}\label{l5est}
0<\kappa^{2}\|\psi\|_{L^{4}(\Omega)}^{4}=-\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf A)\psi|^2-\kappa^{2}a(x)|\psi|^{2}\right)\,dx:=\top\,.
\end{equation}
We can write,
\begin{align}\label{main-eq}
-\top&\geq (1-\sqrt{\top}\,\kappa^{-1})\int_{\Omega}|(\nabla-i\kappa H\mathbf F)\psi|^{2}\,dx-\kappa^{2}\,\int_{\Omega}\,a(x)|\psi|^{2}\,dx-\frac{(\kappa H)^{2}}{\sqrt{\top}\kappa^{-1}}\int_{\Omega}|(\mathbf A-\mathbf F)\psi|^{2}\,dx\nonumber\\
&\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-\sqrt{\top}\,\kappa^{-1}\|(\nabla-i\kappa H\mathbf F)\psi\|^{2}_{L^{2}(\Omega)}-\frac{(\kappa H)^{2}}{\sqrt{\top}\kappa^{-1}}\|(\mathbf A-\mathbf F)\psi\|_{L^{2}(\Omega)}^{2}\,.
\end{align}
We reffer to \eqref{8d-<} and \eqref{5d-<}, we have,
\begin{equation}\label{est:top1}
-\top\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-C\,\sqrt{\top}\,\kappa\,\|\psi\|_{L^{2}(\Omega)}^{2}\,.
\end{equation}
Thanks to Proposition~\ref{lem:psi-2<4}, using \eqref{l5est}, we get,
\begin{equation}\label{est:psi1}
\|\psi\|^{2}_{L^{2}(\Omega)}\leq C\,\kappa^{-\frac{11}{8}}\,\sqrt{\top}\,.
\end{equation}
As a consequence of \eqref{est:psi1}, \eqref{est:top1} becomes,
\begin{equation}\label{est:top2}
-\top\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-C'\,\kappa^{-\frac{3}{8}}\,\top\,.
\end{equation}
Having in mind that $\psi\neq 0$ and $\top>0$ (see \eqref{l5est}), we deduce for $\kappa$ sufficiently large $\mu_{1}(\kappa,H)<0$, which is in contradiction with Proposition~\ref{prop:mu>0}. Therefore, we conclude that $\psi=0$, which is what we needed to prove.
\end{proof}
\begin{prop}\label{prop:cp}
Supose that $\Gamma=\varnothing$ and $\{a>0\}\neq\varnothing$. There exists $C>0$ and $\kappa_0$ such that, if $H$ satisfies
\begin{equation}\label{cond:HC3-2w2}
H\leq \kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\,,
\end{equation}
then there exists a solution $(\psi,\mathbf A)$ of \eqref{eq-2D-GLeq} with $\|\psi\|_{L^{2}(\Omega)}\neq 0$.\\
Moreover,
$$
\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\leq \underline{H}_{C_3}^{cp}\,.
$$
\end{prop}
\begin{proof}
We use $(t\psi_{\ast},\mathbf F)$, with $t$ sufficiently small and $\psi_{\ast}$ an eigenfunction associated with $\mu_{1}(\kappa,H)$, as a test configuration for the functional \eqref{eq-2D-GLf}, i.e.
$$
\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf F)\psi_{\ast}|^{2}-\kappa^{2}\,a(x)|\psi_{\ast}|^{2}\right)\,dx=\mu_{1}(\kappa,H)\|\psi_{\ast}\|^{2}_{L^{2}(\Omega)}\,.
$$
Proposition~\ref{prop:mu1<0} tells us that there exists a constant $C$ such that, under Assumption \eqref{cond:HC3-2w2}, $\mu_{1}(\kappa,H)<0\,$.\\
Therefore,
$$
C_{1}(\kappa,H):=\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf F)\psi_{\ast}|^{2}-\kappa^{2}\,a(x)|\psi_{\ast}|^{2}\right)\,dx<0\,.
$$
We can write,
\begin{align*}
\mathcal E_{\kappa,H,a,B_{0}}(t\psi_{\ast},\mathbf F)&=t^{2}\int_{\Omega}\left(|(\nabla-i\kappa H\mathbf F)\psi_{\ast}|^{2}-\kappa^{2}\,a(x)|\psi_{\ast}|^{2}\right)\,dx+t^{4}\,\frac{\kappa^2}{2}\int_{\Omega}|\psi_{\ast}|^{4}\,dx+\frac{\kappa^2}{2}\int_{\Omega}a(x)\,dx\\
&=t^2\left(C_{1}(\kappa,H)+t^{2}\,\frac{\kappa^2}{2}\int_{\Omega}|\psi_{\ast}|^{4}\,dx\right)+ \mathcal E_{\kappa,H,a,B_{0}}(0,\mathbf F)\,.
\end{align*}
We choose $t$ such that,
$$
C_{1}(\kappa,H)+t^{2}\,\frac{\kappa^2}{2}\int_{\Omega}|\psi_{\ast}|^{4}\,dx<0\,.
$$
Thus, we get
$$
\mathcal E_{\kappa,H,a,B_{0}}(t\psi_{\ast},\mathbf F)<\mathcal E_{\kappa,H,a,B_{0}}(0,\mathbf F)\,.
$$
Hence a minimizer, which is a solution of \eqref{eq-2D-GLeq}, will be non-trivial.
\end{proof}
\subsection{End of the proof of Theorem~\ref{thm:HC3}}
First, we will prove the following inclusion,
$$
\mathcal{N}^{\rm loc}(\kappa)\subset \mathcal{N}(\kappa)\,.
$$
We see that if $H\notin \mathcal{N}(\kappa)$, then $(0,\mathbf F)$ is a local minimizer of $\mathcal E_{\kappa,H,a,B_{0}}$. Thus, the Hessian of the functional $\mathcal E_{\kappa,H,a,B_{0}}$ at the normal state $(0,\mathbf F)$ should be positive.\\
For every $(\widetilde{\phi},\widetilde{\mathbf A})\in H^1(\Omega)\times H^1_{\rm div}(\Omega)$ we have,
$$
\mathcal E_{\kappa,H,a,B_{0}}(t\widetilde{\phi},\mathbf F+t\widetilde{\mathbf A})=t^{2}\left[\mathcal{Q}_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}(\widetilde{\phi})+(\kappa H)^{2}\int_{\Omega}|\curl \widetilde{\mathbf A}|^{2}\,dx\right]+\mathcal{O}(t^3)\,.
$$
This implies that the Hessian of the functional $\mathcal E_{\kappa,H,a,B_{0}}$ at the normal state $(0,\mathbf F)$ can be written as follows:
$$
Hess_{(0,\mathbf F)}[\widetilde{\phi},\widetilde{\mathbf A}]=\mathcal{Q}_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}(\widetilde{\phi})+(\kappa H)^{2}\int_{\Omega}|\curl \widetilde{\mathbf A}|^{2}\,dx\,.
$$
Since $Hess_{(0,\mathbf F)}[\widetilde{\phi},\widetilde{\mathbf A}]\geq 0$, we get that $\mu_{1}(\kappa H)\geq 0\,,$ and consequently $H\notin \mathcal{N}^{\rm loc}(\kappa)$.
Hence we obtain the above inclusion.\\
On the other hand, if $(\psi,\mathbf A)$ is a minimizer of the functional in \eqref{eq-2D-GLf} with $\psi\neq 0$, then $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq}, and we have the following inclusion,
$$
\mathcal{N}(\kappa)\subset \mathcal{N}^{\rm cp}(\kappa)\,,
$$
and consequently,
\begin{equation}\label{inclusion-N}
\mathcal{N}^{\rm loc}(\kappa)\subset \mathcal{N}(\kappa)\subset \mathcal{N}^{\rm cp}(\kappa)\,.
\end{equation}
Having in mind the definition of all the critical fields in \eqref{def:HC3-o}, \eqref{def:HC3} and \eqref{def:HC3-u}, we deduce that,
\begin{equation}\label{eq:ov}
\overline{H}_{C_3}^{loc}(\kappa)\leq\overline{H}_{C_3}(\kappa)\leq\overline{H}_{C_3}^{cp}(\kappa)\,,
\end{equation}
Using \eqref{inclusion-N}, we observe that,
$$
\mathbb R^{+}\setminus\mathcal{N}^{\rm cp}(\kappa)\subset\mathbb R^{+}\setminus\mathcal{N}(\kappa)\subset\mathbb R^{+}\setminus\mathcal{N}^{\rm loc}(\kappa)\,.
$$
From the definition of all the critical fields, we conclude that,
\begin{equation}\label{eq:un}
\underline{H}_{C_3}^{loc}(\kappa)\leq\underline{H}_{C_3}(\kappa)\leq\underline{H}_{C_3}^{cp}(\kappa)\,.
\end{equation}
We note that $\underline{H}_{C_{3}}^{loc}\leq\overline{H}_{C_{3}}^{loc}$ and $\underline{H}_{C_{3}}^{cp}\leq\overline{H}_{C_{3}}^{cp}$. Therefore, all the critical fields are contained in the interval $[\underline{H}_{C_{3}}^{loc},\overline{H}_{C_{3}}^{cp}]$.\\
By Proposition~\ref{prop:mu1<0} and Theorem~\ref{thm:lb-H}, we get the existence of positive constants $C$ and $\kappa_0$, such that for $\kappa\geq\kappa_0$,
\begin{multline}
\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)-C\,\kappa^{\frac{1}{2}}\leq \underline{H}_{C_{3}}^{loc}\leq\overline{H}_{C_{3}}^{cp}\\
\leq\kappa\,\max\left(\sup_{x\in\Omega}\frac{a(x)}{B_{0}(x)},\sup_{x\in\partial\Omega}\frac{a(x)}{\Theta_{0}\,B_{0}(x)}\right)+C\,\kappa^{\frac{1}{2}}\,.
\end{multline}
As a consequence, we have proved Theorem~\ref{thm:HC3} for the six critical fields.
\begin{rem}
As in \cite{FH1}, it would be interesting to show that all the critical fields coincide when $\kappa$ is large enough.
\end{rem}
\section{Asymptotics of $\mu_{1}(\kappa,H)$: the case with vanishing magnetic field}\label{Section:Asympt-m1-vanish}
In this section we give an estimate for the lowest eigenvalue
$\mu_{1}(\kappa,H)$ of the operator $P_{\kappa
H\mathbf F,-\kappa^{2}a}^{\Omega}$ (see \eqref{def:mu1}) in the case when
$\Gamma=\varnothing$ with a $\kappa$-independent pinning, i.e.
$a(\kappa,x)=a(x)$. The results in this section are valid under the
assumption $\Gamma\not=\emptyset$, where the set $\Gamma$ is introduced in
\eqref{gamma}. Let
\begin{equation}
\mathcal{B}=\kappa H\qquad{\rm and}\qquad \widehat{\sigma}=\frac{H}{\kappa^2}\,.
\end{equation}
We observe that,
$$
P_{\kappa H\mathbf F,-\kappa^{2}a}^{\Omega}=P_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}\,.
$$
We will give an estimate for the lowest eigenvalue
$\mu_{\mathcal{B},\widehat{\sigma}}$ of
$P_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}$.
After a change of notation, we deduce an estimate for
$\mu_{1}(\kappa,H)$.
\subsection{Lower bound}
In the
absence of a pinning term, that is when $a=1$, Pan and Kwek
\cite{XB-KH} gave the lower bound for the lowest eigenvalue
$\mu(\mathcal{B}\mathbf F)$ of $P_{\mathcal{B}\mathbf F,0}^{\Omega}$ when
$\mathcal{B}\to+\infty$.
In this subsection, we determine a lower bound for $\mu_{1}$ when $\kappa\to+\infty$ and the pinning term is present.\\
We first recall the definition of $\lambda_{0}$ in \eqref{lambda0}, the definition of $\Gamma$ in \eqref{gamma} and for any $\theta\in(0,\pi)$ we recall that $\lambda(\mathbb R_{+}^{2},\theta)$ is the bottom of the spectrum of the operator
$ P_{\mathbf A_{\rm app,\theta},0}^{\mathbb R^{2}_{+}}$, with
$$\mathbf A_{\rm app,\theta}=-\left(\frac{x^{2}_{2}}{2}\cos\,\theta,\frac{x^{2}_{1}}{2}\sin\,\theta \right)\,.$$
We then define for any $\widehat{\sigma}>0$,
\begin{multline}\label{Lambda1}
\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})=\min\left\{\inf_{x\in \Gamma\cap\Omega}\left\{\lambda_{0}\,\Big(\widehat{\sigma}\,|\nabla B_{0}(x)|\Big)^{\frac{2}{3}}-a(x)\right\},\right.\\
\left.\inf_{x\in \Gamma\cap\partial\Omega}\left\{\lambda(\mathbb R^{2}_{+},\theta(x))\,\Big(\widehat{\sigma}\,|\nabla B_{0}(x)|\Big)^{\frac{2}{3}}-a(x)\right\}\right\}\,.
\end{multline}
Here, for $x\in\Gamma\cap\partial\Omega$, $\theta(x)$ denotes the angle between $\nabla B_{0}(x)$ and the inward normal vector $-\nu(x)$.
We start with a proposition that states a lower bound of $\mu_{1}(\kappa,H)$
in the case when $\Gamma\neq\varnothing$.
\begin{prop}\label{prop:mu1-variable}
Let $I$ be a closed interval in $(0,\infty)$. There exist two
positive constants $\mathcal{B}_{0}>0$ and $C>0$ such that if
$\mathcal{B}\geq\mathcal{B}_{0}$, $\widehat\sigma\in I$, $\psi\in
H^{1}(\Omega)\setminus \{0\} $ and $a\in C^{1}(\overline{\Omega})$,
then,
\begin{equation}\label{eq:lower-bound-variable}
\frac{\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}(\psi)}{\|\psi\|^{2}_{L^{2}(\Omega)}}\geq
\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\Big(\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})
-C\mathcal{B}^{-\frac{1}{18}}\Big)\,.
\end{equation}
\end{prop}
\begin{proof}
Let $\ell=B^{-7/29}$. We define the following sets,
\begin{align*}
&D_{1}=\{x\in\Omega: \dist(x,\Gamma)<2\,\ell\}\,,
&D_{2}=\{x\in\Omega: \dist(x,\Gamma)>\ell\}\,.
\end{align*}
Let $h_{j}$ be a partition of unity satisfying
$$
\sum_{j=1}^{2} h_{j}^{2}=1\,,\qquad \sum_{j=1}^{2}|\nabla h_{j}|^{2}\leq C\,\ell^{-2}=C\mathcal B^{14/29}\qquad{\rm and}\qquad \supp h_{j}\subset D_{j}\quad(j\in\{1,2\})\,.
$$
There holds the following decomposition,
\begin{multline}\label{eq:pu}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}} a}^{\Omega}(\psi)
=\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(h_{1}\psi)+\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{2}}(h_{2}\psi)-\sum_{j=1}^{2}\int_{\Omega}|\nabla h_{j}|^{2}|\psi|^{2}\,dx\\
\geq \mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(h_{1}\psi)+\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{2}}(h_{2}\psi)
-C\mathcal B^{14/29}\int_{\Omega}|\psi|^{2}\,dx\,.
\end{multline}
We cover the curve $\Gamma$ by a family of disks
$$D(\omega_{j},\ell)\subset\{x\in\mathbb R^{2}:\dist(x,\Gamma)\leq 2\ell\}\qquad{\rm and}\qquad D_{1}\subset\bigcup_{j} D(\omega_{j},\ell) \qquad\left(\omega_{j}\in \Gamma\right)\,.$$
Consider a partition of unity satisfying
$$
\sum_{j} \chi_{j}^{2}=1\,,\qquad \sum_{j} |\nabla \chi_{j}|^{2}\leq C\,\ell^{-2}\qquad{\rm and}\qquad \supp \chi_{j}\subset D(\omega_{j},\ell)\,.
$$
Moreover, we can add the property that:
$$
{\rm either~supp}\chi_{j}\cap\Gamma\cap\partial\Omega=\varnothing\,,\quad{\rm either}~\omega_{j}\in\Gamma\cap\partial\Omega\,.
$$
We may write,
\begin{equation}\label{eq:main-bulk+bnd}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}} a}^{D_{1}}(h_{1}\psi)=\sum_{int}\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(\chi_{j}h_{1}\psi)+\sum_{bnd}\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(\chi_{j}h_{1}\psi)-\sum_{j}\int_{D_{1}}|\nabla\chi_{j}|^{2}|h_{1}\psi|^{2}\,dx\,,
\end{equation}
where `int' is in reference to the $j$'s such that $\omega_{j}\in\Gamma\cap\Omega$ and `bnd' is in reference to the $j$'s such that $\omega_{j}\in\Gamma\cap\partial\Omega$.\\
For the last term on the right side of \eqref{eq:main-bulk+bnd}, we get using the assumption on $\chi_{j}$:
\begin{equation}\label{eq:est-error}
\int_{D_{1}}|\nabla\chi_{j}|^{2}|h_{1}\psi|^{2}\,dx\leq C\,\ell^{-2}\,\int_{D_{1}}|h_{1}\psi|^{2}\,dx
= C\,\mathcal B^{14/29}\,\int_{D_{1}}|h_{1}\psi|^{2}\,dx\,.
\end{equation}
We have to find a lower bound for
$\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(h_{1}\psi)$
for each $j$ such that $\omega_{j}\in\Gamma\cap\Omega$ and for each
$j$ such that $\omega_{j}\in\Gamma\cap\partial\Omega$. Thanks to
\cite{JPM}, we have,
$$\int_\Omega|(\nabla-i\mathcal B\mathbf F)\chi_jh_1\psi|^2\,dx\geq \mathcal
B^{\frac23}\int_\Omega\Big((\lambda_{0}\,|\nabla B_{0}(\omega_{j})|\Big)^{\frac{2}{3}}-CB^{-1/18}\Big)|\chi_jh_1\psi|^2\,dx\,.
$$
Using Taylor's formula, we can write in every disk $D(w_j,\ell)$,
\begin{equation}\label{eq:aT}
|a(x)-a(w_j)|\leq C\ell=C\mathcal B^{-7/29}\leq C\mathcal B^{-1/18}\,.
\end{equation}
In that way, we get,
\begin{align}\label{eq:lb-bulk-vr}
&\sum_{int}\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(\chi_{j}h_{1}\psi)\nonumber\\
&\quad\geq\sum_{int}\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}} \left(\lambda_{0}\,\Big(\widehat{\sigma}\,|\nabla B_{0}(\omega_{j})|\Big)^{\frac{2}{3}}-a(\omega_{j})-C\mathcal B^{-1/18}
\right) \int|\chi_{j}h_{1}\psi|^{2}\,dx\nonumber\\
&\quad\geq\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\left(
\inf_{x\in \Gamma\cap\Omega}\left\{\lambda_{0}\,\Big(\widehat{\sigma}\,|\nabla B_{0}(x)|\Big)^{\frac{2}{3}}-a(x)\right\}-C\mathcal B^{-1/18}\right)
\sum_{int} \int|\chi_{j}h_{1}\psi|^{2}\,dx\,.
\end{align}
In a similar fashion, the analysis in \cite{JPM} and \eqref{eq:aT}
yields,
\begin{align}\label{eq:lb-bnd-vr}
&\sum_{bnd}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{1}}(\chi_{j}h_{1}\psi)\nonumber\\
&\quad\geq\sum_{bnd}
\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}
\left(\lambda(\mathbb R^{2}_{+},\theta(\omega_{j}))\,\Big(\widehat{\sigma}\,|\nabla B_{0}(\omega_{j})|\Big)^{\frac{2}{3}}-a(\omega_{j})-C\mathcal B^{-1/18}\right)
\int|\chi_{j}h_{1}\psi|^{2}\,dx\nonumber\\
&\quad\geq\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}
\left(\inf_{x\in \Gamma\cap\partial\Omega}\left\{\lambda(\mathbb R^{2}_{+},\theta(x))\,\Big(\widehat{\sigma}\,|\nabla B_{0}(x)|\Big)^{\frac{2}{3}}-a(x)\right\}-C\mathcal B^{-1/18}\right)
\sum_{bnd} \int|\chi_{j}h_{1}\psi|^{2}\,dx\,
\end{align}
We insert \eqref{eq:lb-bulk-vr}, \eqref{eq:lb-bnd-vr}
and \eqref{eq:est-error}
into \eqref{eq:main-bulk+bnd} to obtain,
\begin{equation}\label{eq:main-bulk+bnd-D2}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}} a}^{D_{1}}(h_{1}\psi)\geq \left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\left(\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})
-C\mathcal B^{-1/18}\right)\,\int|h_{1}\psi|^{2}\,dx\,.
\end{equation}
Now, we will bound $\int_\Omega |(\nabla-i\mathcal B\mathbf F)h_{2}\psi|^2\,dx$ from below. Let $\ell_{1}<\ell$, we cover $D_{2}$ by a family of disks
$$D(\omega_{j}',\ell_{1})\subset\{x\in\mathbb R^{2}:\dist(x,\Gamma)\geq \ell_{1}\}\qquad\left(\omega_{j}'\in \overline \Omega\right)\,.$$
Consider a partition of unity satisfying
$$
\sum_{j} \chi_{j}^{2}=1\,,\qquad \sum_{j} |\nabla \chi_{j}|^{2}\leq C\,\ell_{1}^{-2}\qquad{\rm and}\qquad \supp \chi_{j}\subset D(\omega_{j}',\ell_{1})\,.$$
There holds the decomposition formula,
\begin{align}\label{eq:int-bnd}
\int_\Omega |(\nabla-i\mathcal B\mathbf F)h_{2}\psi|^2\,dx&=\sum_{j}\int_\Omega |(\nabla-i\mathcal B\mathbf F)\chi_{j}\,h_{2}\psi|^2\,dx-\sum_{j}\int_{\Omega}|\nabla \chi_{j}|^{2} |h_{2}\psi|^{2}\,dx\nonumber\\
&\geq \sum_{j}\int_\Omega |(\nabla-i\mathcal B\mathbf F)\chi_{j}\,h_{2}\psi|^2\,dx-C \ell_{1}^{-2}\int_{\Omega} |h_{2}\psi|^{2}\,dx\,,
\end{align}
We observe that there exists a gauge function $\varphi_{j}$ satisfying (see \cite[Equation~(A.3)]{KA}),
$$
\left|\mathbf F(x)-(B_{0}(\omega_{j}')\mathbf A_{0}(x-\omega_{j}')+\nabla\varphi_{j})\right|\leq C\,\ell_{1}^{2} \quad{\rm in}~D(\omega_{j}',\ell_{1}')\,.
$$
Using Cauchy-Schwarz inequality, we may write,
\begin{multline*}
\int_\Omega |(\nabla-i\mathcal B\mathbf F)\chi_{j}\,h_{2}\psi|^2\,dx\geq \frac{1}{2}\int_\Omega |(\nabla-i\mathcal{B}\,B_{0}(\omega_{j}')\mathbf A_{0}(x-\omega_{j}'))e^{-i\mathcal{B}\varphi_{j}}\chi_{j}\,h_{2}\psi|^2\,dx\\
-C\,\mathcal{B}^{2}\,\ell_{1}^{4}\int_{\Omega}|\chi_{j}\,h_{2}\psi|^2\,dx\,.
\end{multline*}
We are reduced to the analysis of the Neumann realization of the Schr\"odinger operator with a constant magnetic field equal to $\mathcal{B}\,B_{0}(\omega_{j}')$ in our case.\\
Notice that by the assumption on $h_{2}$, there exist constants $M>0$ and $\mathcal B_0>0$ such that, for all $j$, $|B_{0}(\omega_{j}')|\geq M\,\ell$ in the support of $h_{2}$. Thus,
$$
\forall j,\quad\mathcal{B}|B_{0}(\omega_{j}')|\geq M\,\mathcal{B}\,\ell=M\mathcal B^{22/29}\gg 1\,.
$$
Moreover, the magnetic potentials $\mathbf A_{0}(x)$ and $\mathbf A_{0}(x-\omega_{j}')$ are gauge equivalent since
$$
\mathbf A_{0}(x-\omega_{j}')=\mathbf A_{0}(x)-\mathbf A_{0}(\omega_{j}')=\mathbf A_{0}(x)-\nabla(\mathbf A_{0}(\omega_{j}')\cdot x)\,.
$$
Thanks to Theorem~\ref{thm:FH}, there exists a constant $\mathcal{B}_{0}$ such that, for any $\mathcal{B}\geq \mathcal{B}_{0}$, we write by the min-max principle,
\begin{align}\label{eq:first-term}
\sum_{j}\int_\Omega |(\nabla-i\mathcal B\mathbf F)\chi_{j}\,h_{2}\psi|^2\,dx&\geq \frac{\Theta_0\mathcal{B}\,|B_{0}(\omega_{j}')|}{2}\sum_{int}\int_{\Omega}|\chi_{j}\, h_{2} \psi|^{2}\,dx-C\,\mathcal{B}^{2}\,\ell_{1}^{4}\sum_{int}\int_{\Omega}|\chi_{j}\,h_{2}\psi|^2\,dx\nonumber\\
&\geq \left(\frac{M\Theta_0}2\mathcal{B}\,\ell-C\mathcal{B}^{2}\,\ell_{1}^{4}\right)\sum_{j}\int_{\Omega}|\chi_{j}\,h_{2}\psi|^2\,dx\nonumber\\
&=\left(\frac{M\Theta_0}2\mathcal{B}\,\ell-C\mathcal{B}^{2}\,\ell_{1}^{4}\right)\int_{\Omega}|h_{2}\psi|^2\,dx\,.
\end{align}
Putting \eqref{eq:first-term} into \eqref{eq:int-bnd}, we obtain
\begin{align}\label{eq:D2}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{2}}(h_{2}\psi)&=\int_\Omega |(\nabla-i\mathcal B\mathbf F)h_{2}\psi|^2\,dx-\left(\frac{\mathcal B}{\widehat\sigma}\right)^{2/3}\int_{\Omega} a(x)|h_{2}\psi|^2\,dx\nonumber\\
&\geq \left(\frac{M\Theta_0}2\mathcal{B}\,\ell-C\mathcal{B}^{2}\,\ell_{1}^{4}-C\ell_{1}^{-2}\right)\int_{\Omega}|h_{2}\psi|^2\,dx-\left(\frac{\mathcal B}{\widehat\sigma}\right)^{2/3}\int_{\Omega} a(x)|h_{2}\psi|^2\,dx\,.
\end{align}
We choose $\ell_1=B^{-\rho}$ and $\frac{9}{22}<\rho<\frac{11}{29}$. We observe that,
$$\mathcal{B}^{2}\,\ell_{1}^{4}=\mathcal{B}^{2-4\rho}\ll \mathcal B^{22/29}= \mathcal{B}\,\ell\,,\quad
\ell_{1}^{-2}=B^{2\rho}\ll\mathcal B\,\ell\,,\quad \mathcal{B}^{2/3}\ll \mathcal B^{22/29}=\mathcal B\,\ell\,.$$
In this way, we infer from \eqref{eq:D2}, that there exists a constant $c>0$ such that, for $\mathcal{B}$ sufficiently large,
\begin{equation}\label{eq:D2'}
\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}a}^{D_{2}}(h_{2}\psi)\geq c\mathcal B^{22/9}\int_{\Omega} |h_{2}\psi|^2\,dx
\geq \left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})\int_{\Omega} |h_{2}\psi|^2\,dx\,.
\end{equation}
Collecting \eqref{eq:pu}, \eqref{eq:main-bulk+bnd-D2} and \eqref{eq:D2'}, we finish the proof of Proposition~\ref{prop:mu1-variable}.
\end{proof}
Theorem~\ref{thm:mu1-up-vr} is valid under the assumption that,
\begin{equation}\label{cond:sigma-hat}
\widehat{\lambda}_{\min}\leq \frac{H}{\kappa^2}\leq \widehat{\lambda}_{\max}\,,
\end{equation}
where $0<\widehat{\lambda}_{\min}<\widehat{\lambda}_{\max}<\infty$ are constants independent of $\kappa$ and $H$.
\begin{theorem}\label{thm:mu1-up-vr}
Let $\Omega\subset\mathbb R^2$ is an open bounded set with a smooth boundary and $\Gamma\neq\varnothing$. Suppose that \eqref{cond:sigma-hat} hold and $a\in C^1(\overline{\Omega})$, we have
$$
\mu_{1}(\kappa,H)\geq \kappa^{2}\,\widehat\Lambda_{1}\left(B_{0},a,\frac {H}{\kappa^{2}}\right)+ \mathcal{O}(\kappa^{\frac{11}{6}})\,,\qquad{\rm as}\,\kappa\to+\infty\,.
$$
Here, $\widehat\Lambda_{1}$ is introduced in \eqref{Lambda1}.
\end{theorem}
\begin{proof}
We apply Proposition~\ref{prop:mu1-variable} with
$$
\mathcal{B}=\kappa H\,,\quad \widehat{\sigma}=\frac{H}{\kappa^2}\quad{\rm and}\quad I=[\widehat{\lambda}_{\min},\widehat{\lambda}_{\max}]\,.
$$
Let us verify that the conditions of the proposition are satisfied for this choice.
Thanks to \eqref{cond:sigma-hat}, $\widehat{\sigma}\in I$. Now, as $\kappa\to+\infty$, we have,
$$
\mathcal{B}=\widehat{\sigma}\,\kappa^{3}\to+\infty\,.
$$
This implies that, as $\kappa\to+\infty$,
$$
\mu_{1}(\kappa,H)\geq \kappa^{2}\,\widehat\Lambda_{1}\left(B_{0},a,\frac{H}{\kappa^2}\right)+\mathcal{O}(\kappa^{\frac{11}{6}})\,.
$$
This finish the proof of the theorem.
\end{proof}
\subsection{Upper bound}~\\
The next theorem is a generalization of the results in \cite{XB-KH}
and \cite{JPM} valid when the pinning term $a(\kappa,x)=a(x)$ is
independent of $\kappa$ and non-constant.
We denote by $\mu_{\mathcal B,\widehat{\sigma}}$ the lowest eigenvalue of the operator $P_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}$ i.e.
$$\mu_{\mathcal B,\widehat{\sigma}}=\inf_{\psi\in H^{1}(\Omega)\setminus\{0\}}\frac{\mathcal{Q}_{\mathcal{B}\mathbf F,-\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\,a}^{\Omega}(\psi)}{\|\psi\|^{2}_{L^{2}(\Omega)}}\,.$$
\begin{prop}\label{prop:up-var}
Suppose that $\Gamma\neq\varnothing$ and $\widehat\lambda_{\max}>0$. There exist positive constants $C$ and $B_{0}$ such that, for $\widehat\sigma\in (0,\widehat\lambda_{\max}]$, $a\in C^{1}(\overline{\Omega})$ and $\mathcal{B}\geq\mathcal{B}_0$, we have,
\begin{equation}\label{eq:upper-bound-variable}
\mu_{\mathcal{B},\widehat{\sigma}}\leq
\left(\frac{\mathcal{B}}{\widehat{\sigma}}\right)^{\frac{2}{3}}\Big(\widehat\Lambda_{1}(B_{0},a,\widehat{\sigma})
-C\mathcal{B}^{-\frac{1}{18}}\Big)\,.
\end{equation}
\end{prop}
\begin{proof}
Let $x_0\in\Gamma$. In \cite{XB-KH, JPM}, a quasi-mode $u(\mathcal
B,x_0;x)$ is constructed such that, ${\rm supp}\,u(\mathcal
B,x_0;\cdot)\subset \overline{\Omega}\cap B(0,\mathcal B^{-1/18})$
and,
$$\forall~\mathcal B\geq \mathcal B_0\,,\quad\frac{\displaystyle\int_\Omega|(\nabla-i\mathcal B\mathbf F)u(\mathcal
B,x_0;x)|^2\,dx}{\displaystyle\int_\Omega|u(\mathcal
B,x_0;x)|^2\,dx}\leq \mathcal B^{\frac{2}{3}}\Big(\Lambda(x_0)+C\mathcal
B^{-1/18}\Big)\,,$$ where $\mathcal B_0$ and $C$ are constants independent of the point $x_0$ and the parameter $\mathcal B$, and
$$\Lambda(x_0)=
\left\{
\begin{array}{ll}
\lambda_{0}\,|\nabla B_{0}(x_0)|^{\frac{2}{3}}&{\rm if~}x_0\in\Gamma\cap\Omega\,,\\
\lambda(\mathbb R^{2}_{+},\theta(x_0))\,|\nabla
B_{0}(x_0)|^{\frac{2}{3}}&{\rm if~}x_0\in\Gamma\cap\partial\Omega\,.
\end{array}
\right.
$$
Using the smoothness of the function $a(\cdot)$, we get in the
support of $u(\mathcal B,x_0;\cdot)$,
$$|a(x)-a(x_0)|\leq C\mathcal B^{-1/18}\,.$$
Thus, we deduce that,
$$\frac{\mathcal Q^\Omega_{ \mathcal B\mathbf F,-\left(\frac{\mathcal B}{\widehat\sigma}\right)^{\frac{2}{3}}a }(u(\mathcal B,x_0;\cdot)}{\|u(\mathcal B,x_0;\cdot)\|^2_{L^2(\Omega)}}
\leq \left(\frac{\mathcal
B}{\widehat\sigma}\right)^{\frac{2}{3}}\Big(\widehat\sigma^{\frac{2}{3}}\Lambda(x_0)-a(x_0)+C\mathcal B^{-1/18}\Big)\,.$$
Thanks to the min-max principle, we deduce that,
$$\mu_{\mathcal B,\widehat\sigma}\leq
\left(\frac{\mathcal
B}{\widehat\sigma}\right)^{\frac{2}{3}}\Big(\widehat\sigma^{\frac{2}{3}}\Lambda(x_0)-a(x_0)+C\mathcal B^{-1/18}\Big)\,.$$
Since this is true
for all $x_0\in\Gamma$, we deduce that,
$$\mu_{\mathcal B,\widehat\sigma}\leq
\left(\frac{\mathcal
B}{\widehat\sigma}\right)^{\frac{2}{3}}\Big(\widehat\Lambda_1(B_0,a,\widehat\sigma)+C\mathcal
B^{-1/18}\Big)\,,$$ where $\widehat\Lambda_1(B_0,a,\widehat\sigma)$ is introduced in \eqref{Lambda1}.
\end{proof}
Proposition~\ref{prop:up-var} permits to obtain:
\begin{theorem}\label{thm:mu1-upp-var}
Let $\widehat{\lambda}_{\max}>0$. Suppose that $\Gamma\neq\varnothing$ and $a\in C^1(\overline{\Omega})$. There exist two constants $C_1>0$ and $\kappa_0>0$ such that,
if,
\begin{equation}\label{cond:H-m1<}
\kappa\geq\kappa_0\,,\quad {\rm and}\quad \kappa_0\kappa^{-1}<H<\widehat{\lambda}_{\max}\kappa^2\,
\end{equation}
then
$$
\mu_{1}(\kappa,H)\leq \kappa^{2}\,\widehat\Lambda_{1}\left(B_{0},a,\frac{H}{\kappa^2}\right)+C_1\kappa^{\frac{11}{6}}\,,\qquad{\rm as}\,\kappa\to+\infty\,.
$$
\end{theorem}
\begin{proof}
To apply the results of Proposition~\ref{prop:up-var}, we take $\mathcal{B}=\kappa H$ and $\widehat{\sigma}=\frac{H}{\kappa^2}$. We see for $\kappa$ sufficiently large that $\widehat{\sigma}\in (0,\widehat{\lambda}_{\max})$ and $\mathcal{B}$ large.
\end{proof}
Theorem~\ref{thm:mu1-upp-var} is valid when $\kappa H\geq \kappa_0$ and $\kappa_0$ is sufficiently large.
\section{Proof of Theorem~\ref{thm:HC3-vr}}\label{12}
\subsection{Analysis of $\underline{H}_{C_3}^{loc}$ and $\overline{H}_{C_3}^{loc}$.}~\\
In this subsection we will prove Theorem~\ref{thm:HC3-vr} for $\underline{H}_{C_3}^{loc}$ and $\overline{H}_{C_3}^{loc}$.
We first recall some useful results from \cite{XB-KH} about the relation between the eigenvalues $\lambda_0$ and $ \lambda(\mathbb R_{+}^{2},\theta)$, introduced in \eqref{lambda0} and in \eqref{def:lambda-theta}.
\begin{thm}\label{thm:PK-R2+}~
\begin{enumerate}
\item[(i)] $\lambda(\mathbb R^{2}_{+},0)=\lambda_{0}$\,.
\item[(ii)] If $0<\theta<\pi$, then $\lambda(\mathbb R^{2}_{+},\theta)<\lambda_{0}$.
\end{enumerate}
\end{thm}
The next proposition gives the region where $\mu_{1}(\kappa,H)<0$ that allows us to obtain an information about $\underline{H}_{C_3}^{loc}$ (see \eqref{def:HC3-u}) in the case when the magnetic field $B_0$ is constant with a pining term.
\begin{prop}\label{prop:mu1<0-var}
Suppose that $\{a>0\}\neq\varnothing$ and $\Gamma\neq\varnothing$. There exist constants $C>0$ and $\kappa_{0}\geq 0$ such that if
\begin{equation}\label{cond:HC3-var}
\kappa\geq \kappa_{0}\,,\qquad H\leq \max\left(\sup_{x\in\Gamma\cap\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}-C\,\kappa^{\frac{11}{6}}\,,
\end{equation}
then,
$$
\mu_{1}(\kappa,H)<0\,.
$$
Moreover,
$$
\max\left(\sup_{x\in\Gamma\cap\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}- C\,\kappa^{\frac{11}{6}}\leq \underline{H}_{C_{3}}^{loc}\,.
$$
\end{prop}
\begin{proof}
We have two cases:\\
\textbf{Case 1.} Here, we suppose that,
$$\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}>\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\,.$$
Thanks to the assumption in \eqref{B(x)}, we have, for all $x\in\Gamma\cap\partial\Omega$, $0<\theta(x)<\pi$. Theorem~\ref{thm:PK-R2+} then tells us that,
$$
\forall~x\in\Gamma\cap\partial\Omega\,,\quad \frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}>\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}\,.
$$
Thus, there exists $x_0\in\Omega\cap\Gamma$ such that (the supremum of $\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}$ in $\Gamma\cap\overline{\Omega}$ can not be attained on the boundary),
$$
\sup_{x\in\Gamma\cap\overline\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}= \frac{a(x_0)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x_0)|} \,.
$$
If \eqref{cond:HC3-var} is satisfied for some $C>0$, then,
$$
\frac{H}{\kappa^{2}}\leq\frac{a(x_0)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x_0)|} -C\,\kappa^{-\frac{1}{6}}\,,
$$
that we can write in the form,
\begin{equation}\label{eq:appendix}
\kappa^{2}\left(\lambda_0\left(\frac{H}{\kappa^2}|\nabla B_{0}(x_0)|\right)^{\frac{2}{3}}-a(x_0)\right)\leq -C \,M\,\kappa^{\frac{11}{6}}\,,
\end{equation}
where $M>0$ is a constant independent of $C$.
Suppose that $\kappa H\geq \mathcal B_0$ where $\mathcal B_0$ is selected sufficiently large such that we can apply Theorem~\ref{thm:mu1-upp-var}. (Thanks to Lemma~\ref{lem-H=kappa}, $\mu_1(\kappa,H)<0$ when $\kappa H<\mathcal B_0$).
By Theorem~\ref{thm:mu1-upp-var}, there exist positive constants $C_{1}$ and $\kappa_{0}$ such that, for $\kappa\geq \kappa_{0}$,
\begin{align}
\mu_{1}(\kappa,H)&\leq \kappa^{2}\inf_{x\in\Gamma\cap\overline\Omega} \left(\lambda_0\left(\frac{H}{\kappa^2}|\nabla B_{0}(x)|\right)^{\frac{2}{3}}-a(x)\right)+C_{1}\,\kappa^{\frac{11}{6}}\nonumber\\
&\leq \kappa^{2}\left(\lambda_0\left(\frac{H}{\kappa^2}|\nabla B_{0}(x_0)|\right)^{\frac{2}{3}}-a(x_0)\right)+C_{1}\,\kappa^{\frac{11}{6}}\nonumber\\
&\leq (C_1-C\,M) \,\kappa^\frac {11}{6} \,.
\end{align}
By choosing $C$ such that $C\,M> C_{1}$, we get,
$$
\mu_{1}(\kappa,H)<0\,.
$$
\textbf{Case 2.} Here, we suppose that
$$\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\geq\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|}\,.$$
The assumption in \eqref{cond:HC3-var} and the upper bound in Theorem~\ref{thm:mu1-upp-var} give us, for all $\kappa\geq \kappa_0$, $\kappa H\geq \mathcal B_0$ and $\mathcal B_0$ a sufficiently large constant,
$$
\mu_{1}(\kappa,H)\leq (C_1-C\,\widetilde M)\,\kappa^\frac{11}{6}\,.
$$
where $\widetilde M>0$ is a constant independent of $C$.
By choosing $C$ such that $C\,\widetilde M> C_{1}$, we get,
$$
\mu_{1}(\kappa,H)<0\,.
$$
This finishes the proof of the proposition.
\end{proof}
The next proposition gives us a lower bound of $\overline{H}_{C_3}^{loc}$ (see \eqref{def:HC3-u}). This is obtained by localizing the region where $\mu_{1}(\kappa,H)>0$ holds.
\begin{prop}\label{prop:mu>0-var}
Suppose that $\{a>0\}\neq\varnothing$, $\widehat\lambda_{\max} >0$ and $\Gamma=\varnothing$. There exist constants $C>0$ and $\kappa_{0} > 0$ such that if
\begin{equation}\label{cond:HC3-2-var}
\begin{aligned}
\kappa\geq \kappa_{0}\,,\qquad \widehat\lambda_{\max}\,\kappa&\geq H\\
&>\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,,
\end{aligned}
\end{equation}
then,
$$
\mu_{1}(\kappa,H)>0\,.
$$
Moreover,
$$
\overline{H}_{C_{3}}^{loc}\leq\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,.
$$
\end{prop}
\begin{proof}
Having in mind the definition of $\widehat\Lambda_1$ in \eqref{Lambda1}, under the assumption in \eqref{cond:HC3-2-var}, we get for $\kappa$ large enough,
\begin{equation}\label{eq:appendix2}
\widehat\Lambda_{1}\left(B_0,a,\frac{H}{\kappa^2}\right)\geq C\,M\,\kappa^{-\frac{1}{6}}\,,
\end{equation}
where $M>0$ is a constant independent of the constant $C$.
Thanks to Theorem~\ref{thm:mu1-up-vr}, we get the existence of positive constants $C'$ and $\kappa_0$ such that, for $\kappa\geq\kappa_0$,
$$
\mu_{1}(\kappa,H)\geq (C\,M-C')\,\kappa^{\frac{11}{6}}
$$
To finish the proof, we choose $C$ sufficiently large such that $C\,M>C'$.
\end{proof}
\subsection{Analysis of $\underline{H}_{C_3}^{cp}$ and $\overline{H}_{C_3}^{cp}$.}~\\
Proposition~\ref{prop:est-psi-var} below is an adaptation of an analogous result obtained in \cite{HK} for the functional in \eqref{eq-2D-GLf} with a constant pinning term. Proposition~\ref{prop:est-psi-var} is valid when $\Gamma\not=\emptyset$.
Proposition~\ref{prop:est-psi-var} says that, if $(\psi,\mathbf A)$ is a critical point of the functional in \eqref{eq-2D-GLf} and $H$ is of order $\kappa^2$, then $|\psi|$ is concentrated near the set $\Gamma$.
\begin{prop}\label{prop:est-psi-var}
Let $\varepsilon>0$. There exist two positive constants $C$ and $\kappa_0$ such that, if
\begin{equation}\label{cond:H>kappa}
\kappa\geq\kappa_0\,,\quad H\geq\varepsilon\,\kappa^{2}\,,
\end{equation}
and $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq}, then
\begin{equation}\label{l3est}
\|\psi\|^{2}_{L^2(\Omega)}\leq C\,\kappa^{-\frac{1}{4}}\|\psi\|^{2}_{L^4(\Omega)}\,.
\end{equation}
\end{prop}
\begin{proof}
Let $\lambda=\kappa^{-\frac 12}$ and $\Omega_{\lambda}=\{x\in\Omega:\dist(x,\partial\Omega)>\lambda~\&~\dist(x,\Gamma)>\lambda\}$. We introduce a function $h\in C^{\infty}_{c}(\Omega)$ satisfying
$$
0\leq h \leq 1~{\rm in}~\Omega\,,\quad h=1~{\rm in}~\Omega_{\lambda}\,,\quad {\rm supp}\,h\subset\Omega_{\lambda/2}\,,
$$
and
$$
|\nabla h |\leq \frac{C}{\lambda}\quad{\rm in}~\Omega\,,
$$
where $C$ is a positive constant.\\
Using \eqref{3d-<}, we can prove that (see the detailed proof in
\cite[Eq.~(6.6)]{HK} when $a$ is constant),
$$
\kappa\,H\int_\Omega|B_{0}(x)|\,|h\psi|^2\,dx-c\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|h\psi\|_{L^{4}(\Omega)}^{2}\leq \int_{\Omega}|(\nabla-i\kappa H\mathbf A)h\psi|^2\,dx\,.
$$
Now, the Cauchy-Schwarz inequality yields,
$$
c\,\kappa\,\|\psi\|_{L^{2}(\Omega)}\|h\psi\|_{L^{4}(\Omega)}^{2}\leq c^{2}\|\psi\|_{L^{2}(\Omega)}^{2}+\kappa^2\|h\psi\|_{L^{4}(\Omega)}^{4}\,,
$$
which implies that
\begin{multline*}
\int_\Omega\,\left(\kappa\,H\,|B_{0}(x)|-\kappa^{2}\,a(x)\right)\,|h\psi|^2\,dx\leq\int_{\Omega}|(\nabla-i\kappa H\mathbf A)h\psi|^2\,dx-\kappa^{2}\int_{\Omega}\,a(x)\,|h\psi|^2\,dx\\
+c^{2}\|\psi\|_{L^{2}(\Omega)}^{2}+\kappa^{2}\|h\psi\|_{L^{4}(\Omega)}^{4}\,.
\end{multline*}
We may use a localization formula as the one in \eqref{eq:2} (but with $\chi_\kappa=h$) to write,
\begin{align*}
\int_\Omega\,\left(\kappa\,H\,|B_{0}(x)|-\kappa^{2}\,a(x)\right)\,|h\psi|^2\,dx&\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla h|^{2}|\psi|^{2}\,dx+\kappa^{2}\int_{\Omega}(h^{4}-h^{2})|\psi|^{4}\,dx\\
&\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla h|^{2}|\psi|^{2}\,dx\,.
\end{align*}
Here, we have used the fact that $h^{4}\leq h^{2}$ since $0\leq h\leq 1$.
By assumption \eqref{B(x)}, $|\nabla B_{0}|$ does not vanish on $\Gamma$, hence
\begin{equation}
|B_0(x)|\geq \frac{1}{M}\,\kappa^{-\frac 12}\qquad{\rm in}\quad\{\dist(x,\Gamma)\geq \lambda\}\,,
\end{equation}
for some constant $M>0$.\\
Thus, by using \eqref{def:sup-a} and \eqref{cond:H>kappa}, we get,
$$
\left(\frac{\varepsilon}{M}\,\kappa^{\frac{5}{2}}-\kappa^{2}\,\overline{a}\right)\int_\Omega|h\psi|^2\,dx\leq c^{2}\int_{\Omega}|\psi|^{2}\,dx+\int_{\Omega}|\nabla h|^{2}|\psi|^{2}\,dx\,.
$$
Writing $\displaystyle\int_{\Omega}|\psi|^{2}\,dx=\int_{\Omega}|h\psi|^{2}\,dx+\int_{\Omega}(1-h^{2})|\psi|^{2}\,dx$ and using the assumption on $h$, we have,
$$
\left(\frac{\varepsilon}{M}\,\kappa^{\frac{5}{2}}-\kappa^{2}\,\overline{a}-c^{2}\right)\int_{\Omega}|h\psi(x)|^{2}\,dx\leq (c^{2}+C\,\kappa)\int_{\Omega\setminus\Omega_{\lambda}}|\psi|^{2}\,dx\,.
$$
For $\kappa$ large enough, $\frac{\varepsilon}{M}\,\kappa^{\frac{5}{2}}-\kappa^{2}\,\overline{a}-c^{2}\geq \frac{\varepsilon}{2M}\,\kappa^{\frac{5}{2}}$ and
$$
\int_\Omega| h\psi(x)|^{2}\,dx\leq 2\frac{M}{\varepsilon}C\,\kappa^{-\frac{3}{2}} \int_{\Omega\setminus\Omega_{\lambda}}|\psi|^{2}\,dx\,.
$$
Thanks to the assumption on the support of $h$, we get further,
$$
\int_\Omega|\psi(x)|^{2}\,dx\leq \left(2\frac{M}{\varepsilon}C\,\kappa^{-\frac{3}{2}}+1\right)\int_{\Omega\setminus\Omega_{\lambda}}|\psi|^{2}\,dx\,.
$$
Recall that $\lambda=\kappa^{-\frac{1}{2}}$. The Cauchy Schwarz inequality yields,
$$
\int_{\Omega\setminus\Omega_\lambda}|\psi(x)|^{2}\,dx\leq |\Omega\setminus\Omega_\lambda|^{1/2} \left(\int_{\Omega\setminus\Omega_\lambda} |\psi|^{4}\,dx\right)^{\frac{1}{2}}
\leq C\,\kappa^{-\frac{1}{4}}\left(\int_\Omega |\psi|^{4}\,dx\right)^{\frac{1}{2}}\,.
$$
This finishes the proof of the proposition.
\end{proof}
Now, we can give an upper bound of the critical field $\overline{H}_{C_3}^{cp}$ in the case when $\Gamma\neq\varnothing$ and with a pining term.
\begin{theorem}\label{thm:lb-H-var}
Supose that $\Gamma\neq\varnothing$ and $\{a>0\}\neq\varnothing$. There exists $C>0$ and $\kappa_0$ such that, if $H$ satisfies
\begin{equation}\label{cond:HC3-2w-var}
H>\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,,
\end{equation} then $(0,\mathbf F)$ is the unique solution to \eqref{eq-2D-GLeq}.\\
Moreover,
$$
\overline{H}_{C_3}^{cp}\leq\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,.
$$
\end{theorem}
\begin{proof}
In light of the result in Theorem~\ref{thm:GP}, we may assume the extra condition that $H\leq \lambda_{\max}\kappa^2$ for a sufficiently large constant $\lambda_{\max}$.
We take the constant $C$ in \eqref{cond:HC3-2w-var} as in Proposition~\ref{prop:mu>0-var}. In that way, under the assumption in \eqref{cond:HC3-2w-var}, we have
\begin{equation}\label{eq:prop:mu>0-var''}
\mu_1(\kappa,H)<0\,.
\end{equation}
Suppose now that $(\psi,\mathbf A)$ is a solution of \eqref{eq-2D-GLeq} with $\psi\neq 0$. Similarly, as in the proof of Theorem~\ref{thm:lb-H}, we have,
\begin{equation}\label{est:top1-var}
-\top\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-C\,\sqrt{\top}\,\kappa\,\|\psi\|_{L^{2}(\Omega)}^{2}\,,
\end{equation}
where $\top=\kappa^2\|\psi\|^4_{L^4(\Omega)}$ is introduced in \eqref{l5est}.
To apply the result of Proposition~\ref{prop:est-psi-var}, we take
$$\varepsilon=\frac{1}{2}\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,,
$$
and get,
\begin{equation}\label{est:psi1-var}
\|\psi\|^{2}_{L^{2}(\Omega)}\leq C\,\kappa^{-\frac14}\|\psi\|^2_{L^4(\Omega)}= C\kappa^{-\frac{5}{4}}\,\sqrt{\top}\,.
\end{equation}
Putting \eqref{est:psi1-var} into \eqref{est:top1-var}, we obtain,
$$
-\top\geq \mu_{1}(\kappa,H)\,\|\psi\|_{L^{2}(\Omega)}^{2}-C'\,\kappa^{-\frac{1}{4}}\,\top\,.
$$
We conclude that, for $\kappa\geq\kappa_0$ and $\kappa_0$ a sufficiently large constant, $\mu_{1}(\kappa,H)<0$, which is in contradiction with \eqref{eq:prop:mu>0-var''}. Therefore, we conclude that $\psi=0$.
\end{proof}
Following the argument given in Proposition~\ref{prop:cp}, we get:
\begin{prop}\label{prop:cp-var}
Supose that $\Gamma\neq\varnothing$ and $\{a>0\}\neq\varnothing$. There exists $C>0$ and $\kappa_0$ such that, if $\kappa\geq\kappa_0$ and $H$ satisfies
\begin{equation}\label{cond:HC3-2w2-var}
H\leq \max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}-C\,\kappa^{\frac{11}{6}}\,,
\end{equation}
then there exists a solution $(\psi,\mathbf A)$ of \eqref{eq-2D-GLeq} with $\|\psi\|_{L^{2}(\Omega)}\neq 0$.\\
Moreover,
$$
\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}-C\,\kappa^{\frac{11}{6}}\leq \underline{H}_{C_3}^{cp}\,.
$$
\end{prop}
\subsection*{End of the proof of Theorem~\ref{thm:HC3-vr}}
All the critical fields are contained in the interval $[\underline{H}_{C_{3}}^{loc},\overline{H}_{C_{3}}^{cp}]$ (the proof of this statement is exactly as the one given for \eqref{eq:ov} and \eqref{eq:un}).\\
By Proposition~\ref{prop:mu1<0-var} and Theorem~\ref{thm:lb-H-var}, we get the existence of positive constants $C$ and $\kappa_0$, such that for $\kappa\geq\kappa_0$,
\begin{multline}
\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}-C\,\kappa^{\frac{11}{6}}\leq \underline{H}_{C_{3}}^{loc}\leq\overline{H}_{C_{3}}^{cp}\\
\leq\max\left(\sup_{x\in\Gamma\cap\overline{\Omega}}\frac{a(x)^{\frac{3}{2}}}{\lambda_{0}^{\frac{3}{2}}|\nabla B_{0}(x)|},\sup_{x\in\Gamma\cap\partial\Omega}\frac{a(x)^{\frac{3}{2}}}{\lambda(\mathbb R^{2}_{+},\theta(x))^{\frac{3}{2}}|\nabla B_{0}(x)|}\right)\,\kappa^{2}+C\,\kappa^{\frac{11}{6}}\,.
\end{multline}
As a consequence, we have proved that the asymptotics in Theorem~\ref{thm:HC3-vr} is valid for for the six critical fields in \eqref{def:HC3-o}, \eqref{def:HC3} and \eqref{def:HC3-u}.
\section*{Acknowledgements}
This work is partially supported by a grant from Lebanese University
and a grant of Universit\'e Paris-Sud. I would like to thank my
supervisors \textit{B.Helffer} and \textit{A.Kachmar} for their
support, and \textit{J.P. Miqueu} for the communication of the
preprint \cite{JPM}.
| {
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{"url":"http:\/\/tomsmathjourneys.blogspot.com\/2013\/08\/mathjax.html","text":"## \u00fater\u00fd 27. srpna 2013\n\n### Here is my second take on explaining my multiparent script.\n\nWhat it does? Standard technique in rigging is linking. You link one object A to another object B so that its position and rotation of object A is fixed if we manipulate with object B. Quite often one would like to do converse, so manipulate with A and keep position and orientation of B fixed against A.\u00a0Normally\u00a0you have to destroy the original link(A to B) and link B to the A. Here I propose solution where you don't have to relink those objects. But everything comes with its price, so you loose direct control of those object but to that later.\n\nOk we start by developing such a link constraint between two objects A and B such that you can rotate center of A around center of B and vice versa. The orientation of A and B won't be affected by such rotation, this is necessary because you need some unconstrained controls.\nTo achieve that we have to introduce a helper object H to which will be centers of A and B fixed and rotation of A,B will somehow affect position and orientation of H.\n\nNext we face difficulty that position of A,B at time $$t$$ does depend on rotation history of those objects. Have a look here for an example\u00a0http:\/\/www.youtube.com\/watch?v=lOfaFJq5Wqk\u00a0as you can see rotation of A,B at time $$t$$ cannot fully describe the system at time $$t$$. You have to know the rotation history of A,B to fully determine position of those objects. So we will define exact positions of A,B \u00a0at some time $$t_0$$ and to find out position of A,B at time $$t$$ we will march frame by frame from time the $$t_0$$ to the time $$t$$.\n\nIn opposite to the previous post I won't use quaternions except when I will combine rotations but I will propose different approach with matrix exponential.\n\n\\begin{align}\n& x_A(t) \\quad \\text{position of object A} \\\\\n& x_B(t) \\quad \\text{position of object H} \\\\\n& x_H(t) \\quad \\text{position of object B} \\\\\n& R_A^t \\quad \\text{rotation matrix of object A at time t} \\\\\n& R_B^t \\quad \\text{rotation matrix of object B at time t} \\\\\n& R_H^t \\quad \\text{rotation matrix of object H at time t}\n\\end{align}\n\nAs already said position of centers of A and B are fixed against H, so\n\\begin{align}\nx_A(t) &= x_H(t) + R_H^tp_A \\\\\np_A &= (R_H^{t_0})^{-1}(x_A(t_0) - x_H(t_0))\\\\\n\nx_B(t) &= x_H(t) + R_H^tp_B \\\\\np_B &= (R_H^{t_0})^{-1}(x_B(t_0) - x_H(t_0))\n\n\\end{align}\n\nNow if we rotate A a little bit we want to move with B, but B is fixed against H so it is sufficient to move with H. So just for a moment forget about B and imagine that H is linked to the A. Than if we rotate with A position of H will change as\n\\begin{align}\nx_H( t + \\Delta t ) \u00a0= x_H( t ) + R^{t+\\Delta t}_A (R^t_A)^{-1} (x_H(t)-x_A(t))\n\\end{align}\nand the orientation of H will change as\n\\begin{align}\nR_H^{t + \\Delta t } = \u00a0R^{t+\\Delta t}_A (R^t_A)^{-1} R^t_H\n\\end{align}\nNow if we take account the B's contribution we get equation\n\n\\begin{align}\nx_H( t + \\Delta t ) \u00a0= x_H( t ) + R^{t+\\Delta t}_A (R^t_A)^{-1} (x_H(t)-x_A(t)) \u00a0+ R^{t+\\Delta t}_B (R^t_B)^{-1} (x_H(t)-x_B(t))\n\\end{align}\nBut there is trouble with orientation update \u00a0of H because matrix multiplication is not commutative so we don't know in which order to multiply those matrix. For now we just write something and we will \u00a0explain later what $$\\{ \\cdot , \\cdot \\}$$ means.\n\\begin{align}\nR_H^{t + \\Delta t } = \u00a0\\{R^{t+\\Delta t}_A (R^t_A)^{-1}, R^{t+\\Delta t}_A (R^t_A)^{-1} \\}R^t_H\n\\end{align}\n\nWhat\u00a0\u00a0$$\\{ \\cdot , \\cdot \\}$$ does is that it takes two rotations and produce another new rotation, which somehow captures those two rotations. From this we require from \u00a0$$\\{ \\cdot , \\cdot \\}$$ these identities\n\\begin{align}\n\\{ R_1 , R_2 \\} \u00a0&= \\{ R_2 , R_1 \\} \\\\\n\\{R_1, I \\} &= R_1\n\\end{align}\nwhere $$I$$ is identity(ie no rotation). First says that it does not depend on the order of the matrices and the second say that if we combine some rotation with identity(ie no rotation) we should get the original rotation.\n\nNow to define \u00a0$$\\{ \\cdot , \\cdot \\}$$ \u00a0without quaternions we need to know a little bit about rotations and matrix exponential.\n\nIf you have rotation around axis $$n$$ by angle $$\\omega$$. Than its rotation matrix $$R$$ can be expressed as\n$$R = e^{\\omega [n]_\\times} = \\sum_{k=0}^\\infty \\frac{ \\omega^k}{k!} [n]_\\times^k$$\nwhere $$[n]_\\times$$ is cross-product matrix\u00a0.\nOther way around if you have antisymmetric matrix $$A$$ that $$e^A$$ is rotation matrix.\n\nWe are ready to define \u00a0$$\\{ \\cdot , \\cdot \\}$$. Let's have two rotation matrices $$R_1 = e^{A_1}, R_2 = e^{A_2}$$. Than\n$$\\{ R_1, R_2 \\} = \\{ e^{A_1}, e^{A_2} \\} = e^{A_1+A_2}$$\nObserve that $$A_1+A_2$$ is again antisymmetric matrix so $$e^{A_1+A_2}$$ is rotation matrix. Next $$A_1+A_2 = A_2+A_1$$ therefore \u00a0$$\\{ e^{A_1}, e^{A_2} \\} = \\{ e^{A_2}, e^{A_1} \\}$$. Lastly $$I = e^0$$ so \u00a0$$\\{ e^{A_1}, e^0 \\} = e^{A_1 + 0} = e^{A_1}$$. So \u00a0$$\\{ \\cdot , \\cdot \\}$$\u00a0 satisfy all identities we wanted.\n\n### How to program this then.\n\nWhat are the inputs?\nWe have to specify time $$t_f$$ at which we want to get positions and orientations of A,B,H. Next we have to know positions of A,B,H at time $$t_0$$ and orientation of H at time $$t_0$$. Than we have to know the whole history of rotation matrices of A,B from time $$t_0$$ to time $$t_f$$.\nWhat is the output?\nPosition of A,B,H at time $$t$$ and orientation of H.\n\nSo the code would be something like this:\n\n1. precalculate values \u00a0$$p_A,p_B$$\n\n2. then use update equations, start at time $$t_0$$ and and at time $$t_f$$.\n\\begin{align}\nx_H( t + \\Delta t ) \u00a0&= x_H( t ) \u00a0- R^{t+\\Delta t}_A (R^t_A)^{-1} R_H^t p_A \u00a0- R^{t+\\Delta t}_B (R^t_B)^{-1}R_H^t p_B\\\\\nR_H^{t + \\Delta t } &= \u00a0\\{R^{t+\\Delta t}_A (R^t_A)^{-1}, R^{t+\\Delta t}_A (R^t_A)^{-1} \\}R^t_H\n\\end{align}\n\n3. from valuse $$x_H(t_f), R_H^{t_f}$$ calculate $$x_A(t_f),x_B(t_f)$$","date":"2017-12-15 19:25:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 2, \"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\": 0.9710057377815247, \"perplexity\": 1573.6456883733238}, \"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-2017-51\/segments\/1512948579564.61\/warc\/CC-MAIN-20171215192327-20171215214327-00143.warc.gz\"}"} | null | null |
When a person is incarcerated there are several ways for that individual to post a bond to secure their release. The different types of bonds. Own Recognizance bonds require an individual to sign stating they will return on their appointed court date and is not secured by any other means. a surety bond requires someone of the Judges choosing to sign, most commonly a spouse or parent. But does not require any money at the time of release, however the bond does state a monetary amount and the signer assumes that burden if the court date is missed. An unsecured bond has a monetary amount assigned to it but does not require money at the time of release, this bond is much like a surety bond but can be signed by the person being released. A partially secured bond has a monetary value and a percentage of that must be paid before release. The total amount and the percentage to be paid are both set by the Judge issuing the bond. A property bond has a monetary value and is allowed to be secured through the use of owned property. This can only be done at the courthouse. All bonds shall be paid at the courthouse during the hours of 8-4 Monday thorough Friday. Cash and surety bonds may be paid at the jail when the courthouse is not open and if under $10,000. | {
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San Vicente est l'une des cinq paroisses civiles de la municipalité de Muñoz dans l'État d'Apure au Venezuela. Sa capitale est San Vicente.
Géographie
Démographie
Hormis sa capitale San Vicente, la paroisse civile possède plusieurs localités dont :
Notes et références
Paroisse civile dans l'État d'Apure | {
"redpajama_set_name": "RedPajamaWikipedia"
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What are the top sleep hacks? How to influence circadian rythms and chronobiology? Why shoud you sleep with your socks on? What are the top sleep promoting supplements and herbs? How eating at night influences blood sugar levels? What is the role of fasting in promoting a good night sleep? All this and more in this episode of Biohacker's LIVE Show.
Greg Potter recently handed in his PhD at the University of Leeds where his research focused on sleep, diet, and metabolism. His work has been featured in Reuters, TIME magazine, The Washington Post, Fox News, USA Today, and many other major media outlets. In addition to his PhD, Greg also holds BSc and MSc degrees in exercise physiology from Loughborough University. Currently Greg serves as Content Director at Humanos.me which is an online platform designed to help people improve their health and performance. These are also values Greg lives by as his goal in life is to help as many people as possible to improve their health using non-pharmacological interventions. He is keen on educating the general public about important health topics through various media.
Morris C. J. & al. (2016). Effects of the Internal Circadian System and Circadian Misalignment on Glucose Tolerance in Chronic Shift Workers. J Clin Endocrinol Metab. Mar;101(3): 1066-74.
In the United States, almost 15% of the workforce undertakes shift work. Epidemiological studies indicate that shift work is a risk factor for type 2 diabetes. Shift workers frequently undergo circadian misalignment such as light/dark, wake/sleep, activity/inactivity, and feeding/fasting cycles.
A randomized, crossover study with two 3-day laboratory visits. One protocol included a simulated day shift and the other a simulated night shift.
9 healthy chronic shift workers who had five or more night shifts per month Their diet was equalized consisting 45% carbs, 35% fat, and 20% protein.
Internal circadian time affects glucose tolerance in shift workers. Separately, circadian misalignment reduces glucose tolerance in shift workers, providing a mechanism to help explain the increased diabetes 2 risk in shift workers.
ZERO by the tech entrepreneur Kevin Rose. See also his other app, the meditation app Oak. | {
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{"url":"https:\/\/tex.stackexchange.com\/questions\/458671\/how-to-autoref-an-appendix-in-ieee-template","text":"# How to autoref an appendix in IEEE template\n\nI need to autoref an appendix but I need the autoref to appear as Appendix A not section A. I use an IEEE conference template. It has a specific command for appendices and to add appendices under it, it uses \\section command. Here is my minimal script to illustrate the issue:\n\n\\documentclass[conference]{IEEEtran}\n\\IEEEoverridecommandlockouts\n\\usepackage{amsmath,amssymb,amsfonts}\n\\usepackage{algorithmic}\n\\usepackage{graphicx}\n\\usepackage{textcomp}\n\\usepackage{xcolor}\n\\usepackage{hyperref}\n\n\\begin{document}\n\\title{Title Here}\n\\maketitle\n\\begin{abstract}\nAbstract here.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:intro}\nSome text here. See this (\\autoref{app:cc}). I want it to appear as Appendix but it appears as section.\n\\clearpage\n\\appendices\n\\section{Title(1) here}\\label{app:cc}\nSome text here.\n\n\\end{document}\n\n\nHere is a screenshot:\n\nEDIT:\n\nPlease note that I do not want to override the section command with appendix. I use autoref to sections in other places.\n\nI think the problem is due to IEEEtran using \\appendices rather than the more commonly used \\appendix.\n\nHopefully someone can offer a better solution but in the mean time I have a hacky solution... I've made a new command for referencing appendices, that offers the same output as if \\autoref could detect the \\appendices section properly.\n\nMWE:\n\n\\documentclass[conference]{IEEEtran}\n\\IEEEoverridecommandlockouts\n\\usepackage{amsmath,amssymb,amsfonts}\n\\usepackage{algorithmic}\n\\usepackage{graphicx}\n\\usepackage{textcomp}\n\\usepackage{xcolor}\n\\usepackage{hyperref}\n\n\\newcommand{\\refappendix}[1]{\\hyperref[#1]{Appendix~\\ref*{#1}}}\n\\begin{document}\n\\title{Title Here}\n\\maketitle\n\\begin{abstract}\nAbstract here.\n\\end{abstract}\n\n\\section{Introduction}\\label{sec:intro}\nSome text here. See this (\\refappendix{app:cc}). I want it to appear as Appendix but it appears as section.\n\\clearpage\n\n\\appendices\n\\section{Title(1) here} \\label{app:cc}\nSome text here.\n\n\\end{document}","date":"2019-11-15 10:44:56","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\": 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\": 0.7139756679534912, \"perplexity\": 2800.598612077303}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496668618.8\/warc\/CC-MAIN-20191115093159-20191115121159-00553.warc.gz\"}"} | null | null |
Home » Tesla Roadster Will Have Centerlock Wheels According To Elon Musk
Tesla Roadster Will Have Centerlock Wheels According To Elon Musk
August 19, 2020 by Alex Harrington
Tesla CEO Elon Musk has confirmed on Twitter that the new Tesla Roadster will have centerlock wheels like a small collection of the highest performance cars on the market at the moment.
It's been a long time since we've heard anything about the Roadster after the Cybertruck took center stage not too long ago, and since then we've been yearning to hear more about the upcoming all-electric hypercar. But once it's released, it will be the halo car for Tesla, while looking to outperform any other performance car on the market.
While it was supposed to hit the market this year, obviously COVID has pushed the reveal back, although we're sure there are other behind-the-scenes reasons as to why it's not on the road yet. Despite this, Musk is slowly drip-feeding us information to tide us over.
My favorite is one in tension, other DoF in compression. New Roadster wheels will only have one nut.
— Elon Musk (@elonmusk) August 18, 2020
It will have one nut per wheel, and they will be "huge".
Yes, it will have huge nuts haha
The main advantage of a single centerlock nut over that of the 4 or 5 that you would see on a conventional wheel is speed. It's much faster to remove just one nut instead of 5. It also offers more room for brakes and the like. Formula 1 cars use this kind of technology, as do a small number of high performance supercars on the road today such as the Lamborghini Aventador SV and Porsche 918.
We're very excited to hear more on the Roadster as we close in on its release. Elon boasts that the base spec will be seeing a 0-60mph time of 1.9 seconds while also having a range of 620 miles. Add on the 'SpaceX Package', and the car will come even faster with "cold air thrusters" that will propel the car alongside the electric motors.
We're expecting an update on both the Roadster and the Tesla Semi later this year.
Tags roadster, tesla
Tesla Is Developing A SmartWatch According To FCC Filing
Tesla Powered Mazda RX-7 Could Be The Best Swap Ever Performed | {
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"Heavy Is the Crown" is a song by American rock band Daughtry. It was released on March 18, 2021, as the second single from the band's sixth studio album, Dearly Beloved. It is written by Chris Daughtry, Johnny Cummings, Elvio Fernandes, Scott Stevens and Marti Frederiksen. With a peak of number four, it is Daughtry's highest-charting single on the Mainstream Rock chart.
Charts
Release history
References
2021 singles
2021 songs
Daughtry (band) songs
Songs written by Chris Daughtry
Songs written by Marti Frederiksen
Songs written by Scott Stevens (singer) | {
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Kew Gardens, London
The area started as royal estates and gardens in the 1200s. Throughout the reigns of various kings and queens, land was acquired, more gardens built, buildings rose and fell, and in the 1700s the estates were merged into a large botanical garden. In the 1840s, it was formally declared the royal botanical gardens (which today means while it 'belongs' to the monarch, it's really a treasure of the state).
The gardens are used not only to collect, but to test and verify transplanted species from around the world (to break the monopoly those places had on these plants, like rubber and spices). This proved a very important economic factor as wars for over 100 years were fought over unique plant environments by the English, which is quite costly.
In the past, it fell into disrepair but the last few decades have seen a very concerted effort of conservation and upkeep that has it at an amazing state.
The Arrival - Kew Gardens Station
I took the District Line (you can also take the Overground train line) to Kew Gardens Station, which is a good hike from the gardens, but the closest you can get.
There is a little market area around the station which has a very comforting Victorian style.
Once you get there, the line to get in is crazy, but you can prepurchase your tickets online.
Being geared toward plants, you'll spend most of your time outside and it's a massive amount of land to cover. There are open air trams to drive you around, but I walk to try to lose some of this traveler's fat I get from drinks and sitting on planes all the time.
There are also a few places to eat, but I picked the Orangery, which used to house citrus plants for delicate fruit. Now it's a delightful place to get lunch and there were many options that a picky eater such as myself finds rare in the "fancy" dining establishments usually at nicer visitation spots.
Inside buildings
There are plants that even gulf stream fed England cannot support outside, and they have particular buildings to provide the environment required by the plants in them. The Lily house is one I did not go in, but the Palm House and the Temperate House were both pretty nice to wander though.
Kew Gardens boasts the largest collection of plants in the world, the oldest wrought iron glass house and the largest glass house in the world, a large array of places to eat, have tea, and relax. Visiting is an entire day affair.
And of course there's a gift shop with tons of horticultural items and some other things that have me scratching my head. | {
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Google May Buy Lytro in a Fire Sale to Boost Its VR Plans March 21, 2018 at 11:58 am
Light field pioneer Lytro is reportedly in talks to be acquired by Google for pennies on the dollar. Google will get a lot of talent and a bevy of patents to add to its VR content creation and computational imaging efforts.
Self-Driving Cars Could Use Lasers to See Around Corners March 5, 2018 at 4:21 pm
A team at Stanford University has developed a system that could one day allow your self-driving car to see around corners so it can make earlier, smarter decisions.
Watch Stanford's Freaky New Soft Robot Grow Itself July 21, 2017 at 11:01 am
Not even vaguely humanoid and still managing to land squarely in the middle of the Uncanny Valley, these crazy techno-tentacles from Stanford can grow and propel themselves like living vines. Researchers hope to use them in disaster scenarios.
New Stanford experiment shakes up our model of the brain's spatial orientation system April 6, 2017 at 12:00 pm
Scientists are continuing to close in on the firmware of the human brain — in this case, for its GPS system.
Hacked-together Lego Mindstorms bot kit brings science to a classroom near you March 22, 2017 at 1:00 pm
With an off-the-shelf robotics toy and some inexpensive parts, Stanford researchers have used the LEGO Mindstorms platform as a springboard to create a liquid-handling robotics kit for kids and STEM classes.
Stanford researchers accidentally discover a whole new role for the cerebellum March 20, 2017 at 12:00 pm
Everybody thought that the cerebellum was just for coordinating muscle motion. Everybody was wrong.
Extreme science: the biggest, fastest and hottest breakthroughs of 2016 January 3, 2017 at 4:19 pm
This year we pushed the boundaries of science to new extremes. We saw new levels of supercomputer performance, peeled the plastic off the biggest telescope and solar plant on earth, and processed information via DNA and liquid light.
Phase change memory can operate thousands of times faster than current RAM August 15, 2016 at 1:47 pm
New discoveries about phase change memory show it can switch at picosecond scales — theoretically opening the door to a DRAM replacement thousands of times faster than our current memory technology.
Toyota turns to 'guardian angel' self-driving cars April 12, 2016 at 3:11 pm
Before complete self-driving, Toyota says, comes assisted driving in dangerous situations. A digital guardian angel takes control, avoids the accident, and returns control to the driver.
Stanford invents lithium-ion battery that can't overheat February 2, 2016 at 10:18 am
Lithium-ion batteries are simply the best battery tech we have available today, but they aren't 100% safe. We've all seen what can happen when one of these batteries gets damaged and overheats.
Stanford has created a water-droplet computer June 12, 2015 at 3:00 pm
Stanford wants to herald the age of physical computing — but not to out-perform electronics. | {
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...I start by bringing some milk to a boil (enough to cover the stain), remove it from the stove before it actually boils, soak the stained part of the material in there for approximately 30 minutes, then rinse it with cold water, and finally throw it in the washing machine with the rest of my clothes. Any cycle (yes, even delicate) will do. Works every time for me.
Okay, but how does it work? How does milk make the red go away? No, you're not staining over the red with white. It actually has to with what's in the milk. The stuff in wine that gives it its color, or the phenolic compounds, don't like to hang out in water or other materials as much as they like to party in an organic, fatty phase—like you'd find in milk. So the coloring actually gets absorbed into the fat of the milk itself, leaving your clothing stain-free. And the more fat content the better, so use whole milk if you have it.
The best part about this trick, according to Jasmin, is that you can just keep on drinking when the spill happens. The heat and fat content of the milk should do its thing whenever you get to it. So you don't have to smash the big red "Wine Spill Emergency" button, rip off your clothes, and spend half the party quoting Lady Macbeth in the bathroom.
What's the Best Way to Clean a Red Wine Stain? | {
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In recent years we have been using photopolymerization techniques under ultra violet to serve as a reagent for composite resins.
These painless solutions make it possible to recover the material and aesthetic appearance of a damaged nail.
There are many solutions for dysgractile nails!
For any problem there are solutions both in the modification of the shape and the appearance of the nail (except parasitosis or infection in progress). | {
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A beautiful steel truss bridge, this historic structure has been connecting pedestrians in west end Toronto since 1907. It has the exuberant feeling of Victorian structures of the industrial revolution, and it's exciting that it is simply for walking. There is even a track along one side to push your bike smoothly up the steps.
It passes over the GO train corridor., which is being expanded for the UnionPearson Express, the new train to the airport. The increased capacity of this rail corridor makes apparent the necessity of the electrification campaign that local MPP Jonah Schein is waging to protect air quality. Location.
The West Toronto Railpath which also runs underneath it is a car-free route for cyclists and walkers on their way downtown. It's such a narrow little strip of land, yet so vital and beautiful. | {
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Q: Is it possible to implement a own IASKSettingsReader? I use core-data for saving a dynamic (small) amount of entities. These entities have a property for "Display" and "Push", which i sync with my server for each entity.
Now i want to add InApp-Settings to give the user the possibility, to change these two settings within the core data entity.
Because the behaviour and look should be like the Settings.app, i want to use the InAppSettingsKit-Project for this case.
As i read, this library allows to implement a custom SettingsStore to save the values within core data, but i need to read the entities and settings from core data too. In my opinion, it is not possible to define a own subclassed IASKSettingsReader for use.
The next problem is, that i want to use the plist on top-level to show the main-settings and then my own-store on a sub-level of the settings.
Example:
-> Display Settings (From plist)
--> List of entities with my own reader and store to show toggles
-> Push Settings (From plist)
--> List of entities with my own reader and store to show toggles
-> Version (From plist)
-> About (From plist)
Is it possible to accomplish this goal without writing the whole settings from scratch (Which would be very painful and unflexible)?
Thanks for heading me to the right direction in advance!
------EDIT------
I think a possible solution would be, to save a custom plist in the needed format for InAppSettingsKit on app-start, read them in the sub-menu as source for this childpane, save the settings with a custom SettingsStore into the plist and save the data back to core data in the synchronize method.
What do you think about this approach?
A: The approach you described sounds reasonable. You'll have to tweak the logic to set the path for the plist, though (-locateSettingsFile:). It should also be possible to write a replacement for IASKSettingsReader to dynamically set the field definitions. Alternatively, you could modify the IASKSettingsReader.dataSource property directly (untested, just an idea).
A: I have implemented the approach that i described in the edit of the question.
The whole approach works this way:
*
*Updating the data in core data with content from my server
*Generate two plist files in InAppSettingsKit-like plist-Files within the InAppSettings.bundle
*Implemented a SettingsStoreCoreData SettingsStore. It builds a dictionary with values from core data on init and saves them back to core-data when synchronize is called.
As Ortwin Gentz mentioned, it is possible to write a own IASKSettingsReader. But i think my approach needs less work and i have not to deal with different source types within IASKSettingsKit.
| {
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Suppliers and their Big 3 customers sometimes seem like a grumpy married couple. As partial compensation for their hard-nosed purchasing tactics, the Big 3 are offering more business to suppliers that produce complex component modules.
But everything comes at a price. Suppliers must engineer their own products, and they are expected to cover warranty costs.
Tired of the continual friction, suppliers are seeking more business with transplant manufacturers. But it's a slow, laborious courtship - foreign automakers are loyal to their longtime suppliers.
During a discussion last month with Automotive News, four veteran supplier executives discussed these and other issues. They were interviewed by Editorial Director Peter Brown, Editor David Sedgwick and Staff Reporter Julie Armstrong.
Fran LeVeque, president of front- and rear-end modules at Meridian Automotive Systems Inc. in Dearborn, Mich.
Tim Manganello, CEO of powertrain components supplier BorgWarner Inc. in Auburn Hills, Mich.
Sid Taylor, CEO of steel processor SET Enterprises Inc. in Warren, Mich.
Don Walker, CEO of interiors supplier Intier Automotive in Newmarket, Ontario.
The Big 3 generate less than 60 percent of U.S. vehicle sales. You are primarily Big 3 suppliers. Are you trying to diversify your customer base?
LeVeque: We recognize where the growth is, and we are trying to meet those emerging customers. On the other hand, we've still got to go with the 60 percent. Trucks and sport-utilities will continue to be the high-volume stuff.
Is it difficult to get business with the transplants?
LeVeque: Yes. I think (foreign automakers) are very loyal to the people that brought them to the dance. Incumbency is a very strong sourcing consideration. But we've had some success. It's not easy, and it's not automatic. It takes a lot of time and a lot of investment.
We've done very well with Toyota, Honda and Nissan. Hopefully, we will continue to (do business with) Ford, GM, DaimlerChrysler, BMW, Audi - our traditional customers.
In terms of sharing technology, is Intier willing to sign Ford Motor Co.'s standard contract that has generated such a ruckus? Are you afraid Ford might reveal your trade secrets to competitors?
Walker: I'm not going to get into our internal policies with our customers. But I don't think any of our major customers would purposely share our technology. Does it happen? Yes it does, and I have seen a couple of examples of that personally, which I won't get into.
However, I think that happens at the lower levels of an organization. I think the senior executives don't condone it. Whether it's Ford or anyone else, if they start burning people on technology, word gets out and spreads like wildfire. And then they'll have a big fight on their hands.
careful about what you show them.
Do you have a program to buy components from minority suppliers?
Walker: We track minority purchasing. We've actually performed quite well. We have a program to encourage growth in minority purchasing. We take it seriously. But at the end of the day, nobody can afford to do something that's not cost-competitive.
So we need to continue to work with minority suppliers to make sure they are capable and competitive. It's an important issue for our customers.
But nobody's going to pay more money (just to get) more minority content. You have to make money. And if you don't make money, you can do all these nice things and you won't have a job anymore because the banks and your shareholders will desert you. I think the health of the Big 3 will determine how much support there is for minority initiatives.
Do the Big 3 give you goals for minority purchasing?
Walker: Yeah, they give us specific targets. They set a minimum that they want you to exceed. If you do a good job, they'll recognize you.
Manganello: We have programs, too. We track our minority purchases. We are definitely looking for more. I think we could probably do better at minority purchasing.
We don't sacrifice quality, cost or delivery just for minority purchases. All our suppliers have to meet the same yardstick. But we definitely are interested in minority purchases. Would I like to do more? Yes.
Sid, how effective are minority purchasing programs overall?
Taylor: The minority purchasing programs started off slow, then they mushroomed. They reached a peak, and now they are on the downswing.
Taylor: I'd say it was in the mid- to late 1990s. And I think it's now on the downswing.
Did the downswing coincide with the recession?
Taylor: You have a lot of companies who really aren't committed to minority programs. They don't believe in it; they do it just because they are told. You've got the people who make the purchase orders, and they are doing their own thing.
Are automakers forcing you to cover more warranty costs? And how are you handling that?
Walker: I think warranty costs have been a big focus over the past number of years with our customers.
Part of the reason for going to modules is that you have somebody who is responsible for functionality as well as fit and craftsmanship. So you get a better vehicle. The best programs are win-lose. If you help (the customer), they share (the savings) with you. And if quality gets worse, then we are responsible for it and they are going to ask us to pick up part of the tab. I think it's a reasonable thing to do. We haven't had problems. Our warranty costs are low. So it hasn't been an issue for us.
Do you have warranty insurance? Or are you self-insured?
Walker: Insurance would be extremely expensive. A lot of it comes down to design and upfront control of engineering and manufacturing.
Are automakers continuing to outsource more component engineering to suppliers? And do they want suppliers to produce more modules?
Walker: I believe it's a trend. However, things go in cycles. I believe the car companies need to maintain a core competency from an engineering standpoint. But it depends partly on their manufacturing strategies, so it may change platform to platform and by manufacturing plant.
In the long run, we'll continue to see the car companies take advantage of the expertise the suppliers have.
But if you're going to be a module supplier you'll be asked for price reductions on a large number of parts that you don't make. So you need to be strong in program management. It's a whole different skill set.
What about Meridian? Are automakers continuing to buy more complex modules?
LeVeque: Yeah. We see it in things like door modules where you produce all of the guts of the door together. You can attach it to the door panel, as well. If you have design responsibility upfront, you'll end up with a better product.
Is that your big growth opportunity?
LeVeque: We are seeing more and more vertical integration in the products that we manufacture. So our growth seems to be in related components - all the stuff that goes into the front end.
If we can effectively design a front end while reducing the number of parts, then we can grab some of the adjacent components. So we're seeing added content. And that seems to be our growth.
We've been able to do some vertical integration. Next to our stamping and chrome plating operations, we have big injection machines. We manufacture the plastic components that go onto the steel bumper systems right there in the plant.
On one front end, we manufacture the lights and the fog lamps that go into those systems. So we have continued to grow. Our $50 bumper system is now a $100 bumper system.
Six or seven years ago, there was an article that predicted GM, Ford and the Chrysler group won't make vehicles by 2010 and that Tier 1 suppliers will make vehicles for them. Do you expect that?
Walker: I think there is going to be a market for contract vehicles. But it's going to be either for excess capacity or for low-volume vehicles. It's very capital-intensive. If a car company has a 200,000-vehicle-per-year cotract, I think the automaker will keep on making it. Unless you decide to make vehicles in China, in which case I presume they'll still make them themselves. But one of the fundamental issues that car companies have to deal with, both in Europe and North America, is legacy costs.
No car company can be globally competitive if they have uncompetitive wages, benefits or work rules. The CAW and the UAW understand that, and the unions in Europe understand that, as well. But the unions' leaders are in a difficult position. They want to support the people they represent. However, they also understand that if they ask for too much, they're going to drive their companies out of business. They have to start working smarter. And I see there is a movement that way. All parties need to understand what they have to do to keep the industry healthy and competitive. | {
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Ågestabron är en bro över sjön Magelungen på gränsen mellan Stockholms kommun och Huddinge kommun. Ungefär halva bron ligger i Stockholm och andra halvan i Huddinge. På Stockholmssidan börjar bron i Farsta strand och sträcker sig till Ågesta på Huddingesidan. Ännu på 1970-talet gick Ågestavägen över bron och slutade i norr vid Nynäsvägen. Den delen namnändrades på 1980-talet till Ågesta broväg.
Historik
En stadsplan för nuvarande Ågestabron upprättades 1969 av Stockholms kommun för norra delen och av Huddinge kommun för södra delen. Stadsplanen vann laga kraft i februari 1971. Efter några års byggtid invigdes den nya bron i oktober 1974. Namnet härrör från Ågesta gård. Tidigare fanns där en träbro som byggdes 1924 och bekostades av Lennart Hellstedt, dåvarande ägare till Ågesta. Bron var smal och tillät bara enkelriktad samt trafikljusreglerad motortrafik med maximal 20 km/h. Den äldsta bron var en flottbro för gående som syns på ett vykort från 1918. Den hade en öppningsbar mittdel där gränsen mellan Huddinge socken och Brännkyrka socken gick.
Nuvarande bron är en betongkonstruktion som vilar på sex breda pelare och har en längd av 200 meter. Brospannet är utfört som en lådbalk i efterspänd betong. Den gamla träbron sprängdes med dynamit och ligger kvar på sjöbotten öster om den nuvarande.
På Huddingesidan, direkt intill östra sidan av brofästet ligger Brostugan även kallad Färjsundet, som var ett torp under Ågesta gård och känt sedan 1689. Vid den tiden fanns ingen bro över Magelungen utan en färja med ett färjeställe där torparen samtidigt var färjkarl. I Brostugan bedrevs även krogrörelse. När flottbron kom till (med en öppningsbar del i mitten) fick torparen på Brostugan ta upp bropengar. Brostugan är fortfarande kvar med sitt tak i höjd med brobanan.
Bilder
Se även
Farstanäsbron (Militärbron) för enbart gång- och cykeltrafik över Magelungen.
Källor
Torp och gårdar i Huddinge.
Noter
Externa länkar
Broar i Stockholm
Byggnader i Huddinge kommun
Broar invigda 1974 | {
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Do you see that very first definition there? A person who has defeated all opponents in a competition or series of competitions, so as to hold first place. It's makes sense right? After all if you look at probably the two most popular stick-and/or-ball sports in the United States that is what they use as a Champion. What Super Bowl team does not have the BEST record for their division going into the big game. Same for the World Series- it's a series of games between the two teams with the best records in their respective divisions. It's what NASCAR use to have.
But this year NASCAR messed with the format for the playoffs yet again, changing "The Chase" rules and how you qualify to even be in the chase, how you stay IN the chase, and who has a chance at the big prize at the end of they day. They wrapped it up in a cutesy little marketing package filled with coined terms they hope would be catchy: Chase Grid, Bracket, Eliminator, and special "Chase" paint schemes. The changes that NASCAR made are challenging the very definition of what a champion is because Sunday the NASCAR Sprint Cup Series champion will be decided for 2014 and one of the four drivers who has a chance at the cup has NOT won a race all season. The very idea of that someone who has not won ONE race all season can win a championship is, to use Misty's word, asinine. That is not a championship- that is a lottery.
First I have to come out and say- this is nothing against Ryan Newman. Some say that he was too aggressive with Larson in Phoenix. I didn't see anything there that wasn't racing. But the numbers speak for themselves. Let's look at some of the numbers shall we?
The first four of this "also ran" chart were originally in the chase field as set at the beginning of the chase. *I included Tony Stewart here- because as you all know he is my driver and as much of a Stewart fan as I am- I will be the first to tell you that he's had a craptacular year. He is currently 25th in the standings.
My point is this: two of the drivers who have a chance at the championships have fewer wins, top 5s, top 10s and laps lead than three of the five drivers up there in my "also ran" list for this season (Yes I purposely picked high performers for this list- that was the point). How does this championship format reward wins if four of "also ran" drivers have more wins than two of the drivers actually eligible for the Championship cup this season? You can't even argue with me that this new format rewards consistency over wins because if you measure consistency by adding top 5s, top 10s and laps lead alone- then Tony Stewart- who as I mentioned has had an absolutely terrible year this season- should qualify for the chase before Ryan Newman who is actually eligible to win the championship cup. What does this prove? Only that this new "chase" with it's "playoff style format" is NOT a championship but an esoteric, jump-through-hoops-to-get-in, lottery system.
Beyond going back to a points system like we use to have- I am not sure that it can be. NASCAR thought that fans were tired of dominance and legacy as defined by Jimmie Johnson's championship run so they started changing the playoff format here and there- first by creating the "Chase" and then by creating this new SPRINT CUP Lottery format. That is just ridiculous. What professional sports sanctioning body does that? I am pretty sure how you qualify for the Super Bowl or World Series hasn't really changed.
Misty suggested a wild card in the final race for anyone who let the championship points lead but didn't make the cut. It could be a start to at least making it more like a championship. You have to remember- in NASCAR racing you don't have two teams vying for the prize every weekend- you have 43. This is not a stick-and-ball sport. This is racing.
Maybe go back to a straight points system- but have wins be worth way more points- like 20, 30 or 50, as well as a point a position, with 1 bonus point for leading a lap and 5 for leading the most laps. NASCAR should go back to 2011 and look at Tony Stewart's championship. THAT is the kind of championship they want to see. Two drivers (Tony Stewart and Carl Edwards) battling it out for every point, every position. The swagger, the smack talking drivers. Quotes like "I'd wreck my mother to win a championship." THAT is what fans want. At least that is what THIS fan wants. Regardless of who those drivers are. Watching Homestead that year? I was in a state- Tony ended up with a hole in his grill in the early laps. I was SURE it was over- but they battled back and WON the race and the championship over Carl Edwards. THAT was a great championship year. Fans (and perhaps NASCAR) need to realize- not all races will be close, nor will all championship runs. But this ridiculousness? This is not a championship- this is a lottery and to me- it takes a little bit off the prestige of the cup. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,129 |
\section{Introduction} \label{sec:intro}
The molecular structure of a protein is defined by a vector in $\mathbb{R}^{3N}$ specifying the Cartesian coordinates of the $N$ constituent atoms. Molecular dynamics (MD) simulation is a computational algorithm that propagates the $3N$-dimensional position vectors through time under the action of a force field specifying the interatomic interaction potential \cite{frenkel2001understanding}. At each step of the MD simulation the full $3N$-dimensional configurational state of the system is available. Sophisticated experimental techniques such as X-ray crystallography and cryo-electron microscopy can solve protein structures to near atomic resolution in crystalline or vitrified samples \cite{PRODRG,Chang2011,Roy2008rr}. Proteins, however, are typically not functional under these conditions and these structures cannot capture transitions between metastable states. For example, the \textit{Mycobacterium tuberculosis} protein tyrosine phosphatase PtpB exhibits dynamical transitions between ``closed'' and ``open'' states in which a pair of $\alpha$-helices transiently cover the active site and dynamically protects the active site from oxidative inactivation \cite{flynn2010dynamic}. Single-molecule experimental techniques such as single molecule F\"orster resonance energy transfer (smFRET) can follow protein dynamics by tracking small numbers of experimentally observable intramolecular distances between fluorescent probes conjugated to the target molecule \cite{Chang2011,Roy2008rr,Zerze2014mo}. There are currently no experimental techniques available to follow the dynamical evolution in atomistic detail.
Takens' Delay Embedding Theorem is a result from dynamical systems theory that asserts a sufficiently long and frequently sampled time series of a single system observable can contain sufficient information to reconstruct the full-dimensional state of the system up to an \textit{a priori} unknown diffeomorphism (i.e., smooth and invertible bijective transformation) \cite{Takens,sauer1991embedology,packard1980,Broomhead1986,Cao1998,stark2003delay,complex1,holger,PMID:25733874}. This theoretical result opens the door to reconstructing the molecular coordinates of a protein from single-molecule experimental measurements such as smFRET. We have previously applied Takens' Theorem to synthetic single-molecule time series extracted from molecular dynamics simulations of polymers and proteins to estimate single molecule free energy surfaces (smFES) \cite{Ferg16,Ferg18}. Since the simulations also furnish the full molecular configurations we also estimated the ``true'' smFES from the atomistic molecular simulation trajectory to numerically verify the existence of the \textit{a priori} unknown diffemorphism and place empirical bounds on the degree of perturbation to the true smFES induced by this transformation. In this work, we build upon these foundations using tools from rigid graph theory and artificial neural networks to learn this transformation from the data and also approximate the inverse transformation from the low-dimensional smFES back up to the molecular configuration space. This calibrates a functional approximation mapping a time series in a single system observable to the atomistic molecular configuration, thereby enabling reconstruction of atomistic molecular structures from experimentally-measurable observables. We term this approach Single-molecule TAkens Reconstruction (STAR).
The structure of this paper is as follows. In the next section we describe the methodological details of STAR. We detail the mathematical formulation and numerical solution of each step of the learning problem combining principles and tools from statistical thermodynamics, manifold learning, and rigid graph theory. In Section \ref{sec:results} we present applications of STAR to MD simulations of a \ce{C_{24}H_{50}} polymer chain and the 10-residue artificial mini-protein Chignolin. We train STAR to reconstruct molecular configurations from univariate time series in the head-to-tail distance as a synthetic and idealized smFRET time trace. We demonstrate reconstruction of molecular configurations from novel time series data not present in the training ensemble to a root mean squared deviation (RMSD) accuracy better than 0.2 nm. In Section \ref{sec:concl} we present our conclusions and opportunities for future work.
\section{Methods} \label{sec:meth}
\subsection{Principles of STAR} \label{subsec:STAR}
A cartoon schematic of STAR is presented in Fig.~\ref{fig:pathway}. The objective of this work is to train STAR as an accurate and generalizable model to predict molecular configurations from time series data in experimental observables through the four-step pathway $b \rightarrow d \rightarrow c \rightarrow e \rightarrow f$. Each panel in the figure corresponds to a different representation of the molecular system and the arrows between them correspond to mathematical operations to convert one representation to another. Red arrows correspond to unsupervised learning problems, typically nonlinear dimensionality reduction, blue arrows to supervised learning problems requiring learning of a nonlinear function linking the inputs and outputs, and grey arrows to deterministic operations. In order to learn and approximate each of these functions we require training data for which the molecular configurations (Fig.~\ref{fig:pathway}a) corresponding to each point in the time series in the experimental observable (Fig.~\ref{fig:pathway}b) is known. We obtain the former from atomistic MD simulations and the latter by computing the experimentally-measurable observable corresponding to each frame in the trajectory. In this work, we adopt the head-to-tail ($h2t$) distance as an experimental observable that could, in principle, be measured by smFRET \cite{Roy2008rr,Zerze2014mo}. We emphasize that the MD simulation data is only required to train the model and once the model is trained there is no requirement for any additional molecular calculations. The trained model predicts molecular configurations (Fig.~\ref{fig:pathway}f) from novel time series data (Fig.~\ref{fig:pathway}b) that was not present in the training data via the pathway $b \rightarrow d \rightarrow c \rightarrow e \rightarrow f$. In this work, we generate the novel time series data from additional MD simulations, but, in principle, the STAR model trained and calibrated over MD data could be applied to experimental measurements. We detail the additional sophistications that would be necessary to achieve that goal in the final section of the paper. A separate STAR model must be trained for each molecular system.
\begin{figure*}[t]
\centering
\includegraphics[height=11cm, width=16cm]{Figure1.pdf}
\linespread{1.0}\caption{Cartoon schematic of Single-molecule TAkens Reconstruction (STAR). The trained STAR model predicts molecular configurations of a protein $\hat{\mathbf{r}}(t) \in \mathbb{R}^{3N}$ from univariate time series in an experimentally-measurable observable of the system $v(t) \in \mathbb{R}^1$ through the four-step pathway $b \rightarrow d \rightarrow c \rightarrow e \rightarrow f$. Each panel corresponds to a different representation of the molecular system and arrows between them to mathematical operations. Red arrows represent unsupervised nonlinear manifold learning problems (i.e., nonlinear dimensionality reduction), blue arrows represent supervised learning problems (i.e., function approximation), and grey arrows represent deterministic operations. (a,b) Molecular dynamics simulations sufficiently long to sample the thermally-relevant configurational space provide training trajectories of molecular configurations $\mathbf{r}(t) \in \mathbb{R}^{3N}$ (panel a) and a scalar time series in an experimentally-measurable system observable $v(t) \in \mathbb{R}^1$ (panel b). In this work, we set $v(t)$ to be the head-to-tail ($h2t$) intramolecular distance that is, in principle, measurable by smFRET. (c) Interactions between atoms constrain the molecular trajectory $\mathbf{r}(t) \in \mathbb{R}^{3N}$ to a $k$-dimensional manifold $\mathcal{M} \subset \mathbb{R}^{k} \subset \mathbb{R}^{3N}$ with effective dimensionality $k \ll 3N$. The collective variables $\{\psi_1, \psi_2, \ldots \psi_k\}$ parameterizing the manifold are extracted from the simulation trajectory using diffusion maps. The manifold $\mathcal{M}$ supports the smFES $\beta F(\psi_1, \psi_2, \ldots \psi_k)$. (d) An image of $\mathcal{M}$ and the smFES is obtained from the time series data $v(t) \in \mathbb{R}^1$ by forming a $d$-dimensional delay embedding $\mathbf{y}(t) = [v(t), v(t-\tau), v(t-2\tau), \ldots , v(t-(d-1)\tau)] \in \mathbb{R}^d$ and then applying diffusion maps to learn a parameterization $\{\psi_1^\prime, \psi_2^\prime, \ldots \psi_k^\prime\}$ of a manifold $\mathcal{M}^\prime \subset \mathbb{R}^{k} \subset \mathbb{R}^{d}$. Under technical conditions on $v$, $d$, and $\tau$ discussed in the main text, Takens' Delay Embedding Theorem asserts that the dynamical evolution of $\mathbf{y}(t)$ is $C^1$-equivalent to that of $\mathbf{r}(t)$ and that the manifold $\mathcal{M}^\prime$ is related to $\mathcal{M}$ via diffeomorphic (i.e., smooth, invertible, and bijective) transformation $\Theta : \mathcal{M}^\prime \rightarrow \mathcal{M}$. We learn the transformation $\Theta$ from the training data using a simple artificial neural network. (e,f) The manifold $\mathcal{M} \subset \mathbb{R}^{k}$ contains a low-dimensional projection of $\mathbf{r}(t) \in \mathbb{R}^{3N}$ into $\{\psi_i\}_{i=1}^k$ from which molecular configurations $\hat{\mathbf{r}}(t) \in \mathbb{R}^{3N}$ may be approximately reconstructed. We perform the reconstruction by first learning the pairwise distances between all atoms $\mathbf{d} \in \mathbb{R}^{N(N-1)/2}$ using an artificial neural network (panel e) and then using classical multidimensional scaling to deterministically transform this into the reconstructed atomic coordinates $\hat{\mathbf{r}}(t) \in \mathbb{R}^{3N}$ (panel f). All molecular renderings in this work were constructed using VMD \cite{humphrey1996vmd}.}
\label{fig:pathway}
\end{figure*}
What is the origin of the multi-step pathway in STAR illustrated in Fig.~\ref{fig:pathway}? Why not predict molecular configurations directly from the $h2t$ time series in a single step $b \rightarrow f$? In this work, we exploit the generic low effective dimensionality of molecular systems along with theoretical guarantees offered by Takens' Delay Embedding Theorem to formulate a succession of simpler and typically lower-dimensional learning problems with firm theoretical underpinnings that we solve using appropriate numerical tools. An additional benefit of this perspective is that the approach also explicitly learns the smFES. The trained STAR model therefore furnishes both a prediction of the molecular structure and its location and thermodynamic stability on the smFES. We now detail each step of the STAR approach presented in Fig.~\ref{fig:pathway}.
\subsubsection{Molecular dynamics training data $(\mathbf{r}(t), v(t))$ }
MD simulations of sufficient duration to comprehensively sample the thermally-relevant configurational space of the molecular system are required to provide the initial training data for STAR. Data that do not sample all relevant conformational states and transitions will result in models overfitted to the training data and a poorly calibrated STAR model that is unable to generalize to new data. For the small macro and biomolecular systems considered in this work, simulations of a few hundred ns to a few $\mu$s were sufficient to provide good sampling and generalizability. The simulation trajectory provides a temporally-ordered sequence of snapshots $\mathbf{r}(t) \in \mathbb{R}^{3N}$ (Fig.~\ref{fig:pathway}a). For each snapshot we compute the instantaneous value of an experimentally-accessible observable to define a 1D time series $v(t) \in \mathbb{R}^1$ (Fig.~\ref{fig:pathway}b). In the present work, we set $v(t)$ to be the head-to-tail molecular distance $h2t(t)$ that is, in principle, measurable by smFRET by conjugating the termini of the molecules with fluorescent probes. These time series can therefore be regarded as idealized synthetic smFRET time traces with no measurement noise and arbitrarily high time resolution. These training data are used to train all steps of the STAR pipeline.
\subsubsection{Learning the atomistic manifold $\mathcal{M}$}
Interactions between the constituent atoms of the molecular system generically constrain molecular trajectories $\mathbf{r}(t) \in \mathbb{R}^{3N}$ within the 3N-dimensional Cartesian coordinate space to a so-called intrinsic manifold $\mathcal{M} \in \mathbb{R}^{k} \subset \mathbb{R}^{3N}$ of effective dimensionality $k \ll 3N$ \cite{ferguson2010systematic,garcia1992large, amadei1993essential, hegger2007complex, Zhuravlev2009, das2006low,belkin2003laplacian} (Fig.~\ref{fig:pathway}c). We learn this manifold from the simulation trajectory training data using the diffusion maps (dMaps) nonlinear manifold learning technique \cite{coifman2008diffusion, ferguson2010systematic, das2006low, ferguson2011cpl, coifman2005geometric, coifman2006diffusion, lpbeltrami, nadler2006advances,coifman2005geometric,coifman2006diffusion}. Applying dMaps identifies both the dimensionality $k$ of the latent manifold and the CVs $\{\psi_1, \psi_2, \ldots \psi_k\}$ parameterizing it as nonlinear functions of the atomic coordinates \cite{ferguson2010systematic, ferguson2011cpl, Ferguson_2017, Sidky2020}. The dMap CVs are the leading eigenvectors of a discrete random walk constructed over the $3N$-dimensional snapshots comprising the simulation trajectory. The bandwidth $\epsilon$ of the Gaussian kernel used to construct the random walk can be tuned automatically based on the structure of the data \cite{ferguson2010systematic, coifman2008graph} and the dimensionality of the embedding $k$ defined by a gap in the eigenvalue spectrum \cite{ferguson2010systematic, ferguson2011cpl, Wang2017,coifman2006diffusion,coifman2005geometric}.
Assuming that the distance metric used to measure pairwise similarities between molecular configurations is a good proxy for the kinetic proximity of the configurations over short time scales, these CVs furnish a dynamically meaningful low-dimensional embedding into the slowest relaxing modes of a diffusion process over the data. In particular, Euclidean distances in the embedding correspond to diffusion distances in configurational space that measure the kinetic proximity of any pair of configurations under the action of the random walk \cite{coifman2005geometric,nadler2006advances}. This dynamic interpretability of dMap CVs make them an excellent choice for parameterization of the intrinsic manifold $\mathcal{M}$, but other nonlinear dimensionality reduction or manifold learning techniques such as Isomap \cite{Tenenbaum2000,Saul2006,Li2006,Jwang2011}, locally linear embeddings (LLE) \cite{roweis2000nonlinear,Zhang2006}, local tangent space alignment \cite{Jwang2011} may also be employed.
The only input to the diffusion map algorithm is the definition of a distance metric $k(\mathbf{r}_i,\mathbf{r}_j)$ with which to measure the dissimilarity of pairs of configurations $\left( \mathbf{r}_i, \mathbf{r}_j \right)$ in the trajectory. In order to guarantee the existence of the diffeomorphism, it is critical that this metric be appropriately symmetrized to eliminate any \textit{spatial symmetries} that cannot be distinguished by the choice of experimentally-measurable observable $v(t)$. Our choice of observable $v(t) = h2t(t)$ is invariant under translation, rotation, mirror inversion, and -- for chemically symmetric molecules (e.g., simple polymer chains) -- head-tail inversion of the molecular configuration $\mathbf{r} \in \mathbb{R}^{3N}$. Any representation of the system based on $v(t)$ sacrifices the ability to distinguish changes in the system state under any of these transformations. It is possible to recover some or all of these symmetries under different choices for $v(t)$ or by multiplexed simultaneous measurements $\mathbf{v}(t) = \{v_1(t), v_2(t), \ldots\}$. As is standard practice in measuring molecular similarity, we select $k(\mathbf{r}_i,\mathbf{r}_j)$ to be the rotationally and translationally aligned root mean squared deviation (RMSD), which naturally mods out the rototranslational symmetry using the Kabsh algorithm \cite{kabsch1976solution}, and we eliminate the discrete symmetries by minimizing under mirror inversion,
\begin{align}
k(\mathbf{r}_i,\mathbf{r}_j) &= \underset{\textrm{mirror inversion}}{min} \; RMSD(\mathbf{r}_i,\mathbf{r}_j).
\end{align}
or both mirror and head-tail inversion for chemically symmetric molecules \cite{Ferg18},
\begin{align}
k(\mathbf{r}_i,\mathbf{r}_j) &= \underset{\textrm{head-tail inversion}}{min} \; \underset{\textrm{mirror inversion}}{min} \; RMSD(\mathbf{r}_i,\mathbf{r}_j).
\end{align}
Failure to eliminate all such spatial symmetries induced by the choice of observable $v(t)$ violates Takens' Theorem and can cause the manifolds $\mathcal{M}$ and $\mathcal{M}^\prime$ to no longer be diffeomorphisms related by a well defined and learnable transformation.
The dMap CVs $\{\psi_i\}_{i=1}^k$ define a data-driven parameterization of the $k$-dimensional manifold $\mathcal{M}$. Projection of the full simulation trajectory into these CVs defines an empirical probability distribution $P(\psi_1, \psi_2, \ldots \psi_k)$. An estimate of the smFES mapping out the free energy $F$ over $\mathcal{M}$ is obtained via the statistical mechanical relationship $\beta F(\psi_1, \psi_2, \ldots \psi_k) = -\ln P(\psi_1, \psi_2, \ldots \psi_k) + C$, where $C$ is an arbitrary additive constant and the free energy is de-dimensionalized by the reciprocal temperature $\beta = 1/k_B T$. New molecular configurations $\mathbf{r}_\textrm{new}$ not contained within the data used to construct $\mathcal{M}$ may be projected onto the manifold using the Nystr\"om extension \cite{sonday2009coarse,laing2007coarse,long2019landmark}.
\subsubsection{Learning the Takens' manifold $\mathcal{M}^\prime$}
The previous section detailed a procedure to estimate the intrinsic manifold $\mathcal{M}$ and the smFES it supports by applying manifold learning to a simulation trajectory of the atomistic molecular configuration $\mathbf{r}(t) \in \mathbb{R}^{3N}$. Takens' Delay Embedding Theorem \cite{Takens,sauer1991embedology,packard1980,Broomhead1986,Cao1998,stark2003delay,complex1,holger,PMID:25733874} provides a means to reconstruct an image of the intrinsic manifold $\mathcal{M}^\prime$ (Fig.~\ref{fig:pathway}d) by applying similar operations to delay embeddings of a time series in a single coarse-grained observable $v(t) \in \mathbb{R}^1$ (Fig.~\ref{fig:pathway}b). Takens' Theorem asserts that (i) a $(d \geq (2k+1))$-dimensional delay embedding $\mathbf{y}(t) = [v(t), v(t-\tau), v(t-2\tau), \ldots , v(t-(d-1)\tau)] \in \mathbb{R}^d$ in a generic observable $v(t)$ constructed at a delay time $\tau$ uniquely specifies the instantaneous state of the system, (ii) the dynamical evolution of $\mathbf{y}(t)$ is $C^1$-equivalent (i.e., identical under a smooth and continuous mapping) to that of $\mathbf{r}(t)$, and (iii) the evolution of $\mathbf{y}(t)$ lies on a manifold $\mathcal{M}^\prime$ that is related to $\mathcal{M}$ by a diffeomorphism (i.e., smooth, invertible, and bijective transformation) $\Theta : \mathcal{M}^\prime \rightarrow \mathcal{M}$ with inverse $\Theta^{-1} : \mathcal{M} \rightarrow \mathcal{M}^\prime$ \cite{Ferg18,Takens,sauer1991embedology,packard1980,Broomhead1986,Cao1998,stark2003delay,complex1,holger,Zerze2014mo}. The diffeomorphism is \textit{a priori} unknown but is guaranteed only to stretch and squash the manifold and not tear it or stitch it together in new ways. As such, $\mathcal{M}^\prime$ is a topologically identical image of $\mathcal{M}$ that preserves its continuity and connectivity \cite{Takens,sauer1991embedology, packard1980, Broomhead1986, complex1, holger}.
Takens' Theorem holds for any generic observable of the system $v$ that does not contain any spurious symmetries that are not present in the system itself (i.e., the system is invariant in the observable under particular symmetries), for any delay time $\tau$ that does not introduce temporal aliasing (i.e., is a multiple of a period of the dynamical motion), and for any delay embedding dimensionality $d$ greater than twice the intrinsic dimension of the system $k$ \cite{Takens,sauer1991embedology,holger,letellier1998non,cross2010differential,letellier1996topological,letellier1998non,Ferg16,Ferg18}. In practice, better results are obtained for observables $v$ that are strong functions of all system degrees of freedom and which respond sensitively to the important dynamical motions of the system \cite{Ferg18}, for delay times $\tau$ estimated as the first minimum of the autocorrelation or mutual information of $v(t)$ \cite{Fraser86,MI}, and for delay dimensionalities $d$ estimated using the E$_1$ method of Cao \cite{Cao97,VillaniB,kennel1992determining}. Takens' Theorem still holds when applied to observables $v$ that do contain symmetries not present in the system, but the system may only be reconstructed up to those symmetric operations (\textit{vide supra}), and also to observables of subsystems and under stochastic or deterministic forcing \cite{stark1999delay,stark2003delay}. The latter two generalizations are relevant to the present work because we adopt the head-to-tail molecular distance as our observable $v(t) = h2t(t)$, which is both an observation of the solute subsystem within the full solute-solvent system and is subject to dynamical coupling with solvent and deterministic or stochastic forcing by any attached thermostats, barostats, or other external constraints. Finally, Takens' Theorem may also be applied to multiplexed measurements of several simultaneous observables $\mathbf{v}(t) = \{v_1(t), v_2(t), \ldots\}$ such as multi-channel smFRET measuring multiple intramolecular distances \cite{Roy2008rr,Cao1998}.
The manifold $\mathcal{M}^\prime \subset \mathbb{R}^{k} \subset \mathbb{R}^{d}$ is estimated by applying dMaps to the delay embedding $\mathbf{y}(t) = [v(t), v(t-\tau), v(t-2\tau), \ldots , v(t-(d-1)\tau)]$ parameterized by the $k$ CVs $\{\psi_1^\prime, \psi_2^\prime, \ldots \psi_k^\prime\}$. We apply dMaps under a distance metric $k^\prime(\mathbf{y}_i,\mathbf{y}_j)$ measuring the dissimilarity of pairs of delay vectors that we select to be a simple Euclidean distance metric. This metric requires modification due to the introduction of a spurious symmetry into the system by the temporal ordering of $v(t)$ within the $\mathbf{y}$ vectors induced by the Takens' delay embedding. This spurious symmetry can be understood by considering a hypothetical microstate transition of the system $\mathbf{r}_A(t-\tau) \rightarrow \mathbf{r}_B(t) \rightarrow \mathbf{r}_C(t+\tau)$ and its reverse $\mathbf{r}_C(t-\tau) \rightarrow \mathbf{r}_B(t) \rightarrow \mathbf{r}_A(t+\tau)$, with associated $d$=3-dimensional delay vectors $\mathbf{y}_\mathrm{fwd} = [v_A, v_B, v_C]$ and $\mathbf{y}_\mathrm{bkwd} = [v_C, v_B, v_A]$. Adopting a convention that defines a mapping between system microstates and delay vectors based on the central microstate, both of these delay vectors are associated with microstate $\mathbf{r}_B$ but the delay vectors themselves are not identical. By observing the past and future of $\mathbf{r}_B$ we can distinguish whether it was occupied as part of a forward or reverse transition, and the ``forward'' $\mathbf{r}_B$ and ``backward'' $\mathbf{r}_B$ appear in the Takens' delay embedding as distinct identifiable states. For equilibrium systems obeying detailed balance, the forward and backward transitions between any pair of microstates are equally probable and we should not be able to identify whether occupancy of a particular microstate $\mathbf{r}_B$ resulted from the forward or reverse transition in any such pair. To prevent the occurrence of this symmetry breaking in $\mathcal{M}^\prime$ that remains unbroken in $\mathcal{M}$ we must symmetrize $k^\prime(\mathbf{y}_i,\mathbf{y}_j)$ to eliminate this \textit{temporal symmetry}. This assures that Takens' Theorem is not violated and the manifolds $\mathcal{M}$ and $\mathcal{M}^\prime$ remain diffeomorphic. We have previously demonstrated the importance of eliminating this symmetry in applications of Takens' Theorem to equilibrium molecular systems \cite{Ferg16,Ferg18}. As such, we minimize the distance metric under time reversal, which is equivalent to minimizing under inversion of one of the delay vectors,
\begin{align}
k^\prime(\mathbf{y}_i,\mathbf{y}_j) &= min \left[ \norm{\mathbf{y}_i - \mathbf{y}_j}_2, \norm{\mathbf{y}_i - flip\left(\mathbf{y}_j\right)}_2 \right]
\end{align}
The dMap CVs $\{\psi_i^\prime\}_{i=1}^k$ define a data-driven parameterization of the $k$-dimensional manifold $\mathcal{M}^\prime$ over which we construct the smFES $\beta F^\prime(\psi_1^\prime, \psi_2^\prime, \ldots \psi_k^\prime) = -\ln P^\prime(\psi_1^\prime, \psi_2^\prime, \ldots \psi_k^\prime) + C^\prime$ by compiling the empirical probability distribution $P^\prime(\psi_1^\prime, \psi_2^\prime, \ldots \psi_k^\prime)$ of the delay vectors projected onto the manifold. New delay vectrors $\mathbf{y}_\textrm{new}$ not contained within the data used in the construction of $\mathcal{M}^\prime$ may be projected onto the manifold using the Nystr\"om extension \cite{sonday2009coarse,laing2007coarse,long2019landmark}. We exploit this out-of-sample extension when applying the trained STAR model to new data that was not present during training.
\subsubsection{Learning the diffeomorphism from $\mathcal{M}^\prime$ to $\mathcal{M}$}
Takens' Theorem guarantees the smooth manifolds $\mathcal{M}$ and $\mathcal{M}^\prime$ are related by a diffeomorphism $\Theta : \mathcal{M}^\prime \rightarrow \mathcal{M}$ \cite{Ferg18,Takens,sauer1991embedology,packard1980,Broomhead1986,Cao1998,stark2003delay,complex1,holger,Zerze2014mo}. The theoretical guarantees on the existence of this mapping and its low dimensional nature are a valuable advantage of the multi-step STAR pathway that would be lost by formulating a direct reconstruction of the molecular configurations from the univariate time series. We adopt a convention associating each delay vector $\mathbf{y}(t) = [v(t), v(t-\tau), v(t-2\tau), \ldots, v(t-(d-1)\tau)]$ with the configurational microstate corresponding its central element $\mathbf{r}(t-((d-1)/2)\tau)$. We assert that $d$ be odd in order to make this association unambiguous. This mapping means that the configurations in the leading $t < (d-1)\tau/2$ and trailing $t > (d-1)\tau/2$ periods of the molecular simulation trajectory are not associated with any delay vector and are eliminated from all analyses.
Having defined the associations between the projections of the delay vectors $\mathbf{y}(t)$ on $\mathcal{M}^\prime$ represented as $\{\psi_i^\prime(t)\}_{i=1}^k$ and the projections of the molecular configurations $\mathbf{r}(t)$ on $\mathcal{M}$ represented as $\{\psi_i(t)\}_{i=1}^k$, we define a supervised learning problem between pairs of data points $\left( \{\psi_i^\prime(t)\}_{i=1}^k, \{\psi_i(t)\}_{i=1}^k \right)$ to perform data-driven estimation of the $k$-dimensional to $k$-dimensional diffeomorphism $\Theta : \mathcal{M}^\prime \rightarrow \mathcal{M}$ (Fig.~\ref{fig:pathway}d $\rightarrow$ c). There are many ways to learn and approximate this function, including k-nearest neighbors \cite{Cover1967}, kernel methods \cite{Scholkopf2002}, or local Jacobians \cite{Ferg16,Ferg18,Principlesriemannian}. In this work we employ simple fully-connected feedforward artificial neural networks (ANN) as an easy to train and flexible function approximator \cite{Hassoun1996}. We typically find that networks comprising 4-8 hidden layers each containing $\sim$10$k$ neurons are adequate to furnish high accuracy mappings.
\subsubsection{Learning the reconstruction $\hat{\mathbf{r}}(t)$}
The final step is to learn approximate reconstructions of the molecular configurations $\mathbf{r}(t) \in \mathbb{R}^{3N}$ from their projections $\{\psi_i(t)\}_{i=1}^k \in \mathbb{R}^k$ onto the intrinsic manifold $\mathcal{M}$. If the effective dimensionality of the simulation trajectory is less than or equal to $k$ and the dMap CVs have been properly learned from the simulation trajectory, then the $k$-dimensional subspace spanned by $\{\psi_i(t)\}_{i=1}^k$ is expected to preserve the important configurational variance in $\mathbf{r}(t)$ \cite{ferguson2010systematic,ferguson2011cpl}. As such, the location on the intrinsic manifold $\mathcal{M}$ should contain sufficient information to approximately reconstruct the configurational state of the system $\hat{\mathbf{r}}(t) \in \mathbb{R}^{3N}$ up to any symmetries that have been modded out in its construction (\textit{vide supra}). In this sense, the process $\mathbf{r}(t) \rightarrow \{\psi_i(t)\}_{i=1}^k \rightarrow \hat{\mathbf{r}}(t)$ may be viewed as the concatenation of a low-dimensional encoder furnished by diffusion maps and a decoder to be learned from the data. Again we have the one-to-one mapping of data pairs $\left( \{\psi_i(t)\}_{i=1}^k, \mathbf{r}(t) \right)$ and can formulate and solve this as a supervised learning problem.
We split this learning problem into two steps and instead of learning to predict the Cartesian coordinates of the atoms $\mathbf{r}(t)$ directly from $\{\psi_i(t)\}_{i=1}^k$, we first learn the ordered vector of pairwise distances between all atoms $\mathbf{d}(t) \in \mathbb{R}^{N(N-1)/2}$. Rigid graph theory asserts that specification of all $N \choose 2$ pairwise distances defines the absolute coordinates of the $N$ points up to translation, rotation, and mirror inversion \cite{Euclid,Singer2008}. In practice, calculation of the coordinates from the pairwise distances is easily accomplished using classical multidimensional scaling (cMDS) to form the Gram matrix and compute its eigendecomposition \cite{Dokmanic_2015,CRIPPEN1978449}. In the present application, the translational, rotational, and mirror inversions (and head-tail inversion, where applicable) have all been modded out of the construction of $\mathcal{M}$ and so there is no (additional) information loss in formulating the supervised learning problem as first approximating the functional mapping $\{\psi_i(t)\}_{i=1}^k \rightarrow \mathbf{d}(t)$ (Fig.~\ref{fig:pathway}c $\rightarrow$ e) and then making the deterministic transformation $\mathbf{d}(t) \rightarrow \hat{\mathbf{r}}$ using cMDS (Fig.~\ref{fig:pathway}e $\rightarrow$ f). Unlike the one-step formulation of the reconstruction $\{\psi_i(t)\}_{i=1}^k \rightarrow \hat{\mathbf{r}}$, the two-step formulation eliminates the need to perform any alignment of the molecular configurations with respect to rotation, translation, and mirror inversion since these symmetries are naturally modded out within the pairwise distance matrix. The omission of these alignment operations, either mutually or to some fixed reference structure, carries advantages in avoiding the introduction of noise and approximations into the fitting problem.
We use simple ANNs to learn $\{\psi_i(t)\}_{i=1}^k \rightarrow \mathbf{d}(t)$ from the training data. Typically we find that networks comprising 4-8 hidden layers each containing on the order of up to $\sim$1000$k$ neurons are adequate to furnish high accuracy mappings. The molecular configurations $\hat{\mathbf{r}}(t) \in \mathbb{R}^{3N}$ are reconstructed only up to translational, rotational, and mirror inversion symmetries (and head-tail inversion, for chemically symmetric molecules), and so comparisons of the reconstruction accuracy between $\left( \hat{\mathbf{r}}, \mathbf{r} \right)$ pairs must be performed by mutual alignment under these transformations. The alignment problem corresponds to an orthogonal Procrustes problem \cite{Schnemann1966AGS} that we efficiently solve using the Kabsch algorithm \cite{kabsch1976solution}.
\subsubsection{Deploying the trained STAR model}
The STAR pipeline trained for a particular molecular system may then be used to predict molecular reconstructions $\hat{\mathbf{r}}(t) \in \mathbb{R}^{3N}$ from new time series data $v(t) \in \mathbb{R}^1$ through the four-step pathway $b \rightarrow d \rightarrow c \rightarrow e \rightarrow f$ (Fig.~\ref{fig:pathway}b-f): Takens' delay vectors $\mathbf{y}$ are constructed from the time series $v(t)$, projected onto the Takens' manifold $\mathcal{M}^\prime$ using the Nystr\"om extension \cite{sonday2009coarse,laing2007coarse,long2019landmark}, mapped onto the atomistic manifold $\mathcal{M}$ using the trained ANN, converted into the atomistic pairwise distances matrix $\mathbf{d}$ using another trained ANN, and then finally transformed into the reconstructed molecular configurations $\hat{\mathbf{r}}$ using cMDS. Provided the training data were sufficiently rich to span the thermally-relevant metastable states and transitions in the system and the supervised learning problems are not overfitted, then the trained STAR model should be able to generalize to new time series data from the system that were not included in model training. Importantly, deployment of the trained model does not require any additional molecular simulation data during the deployment phase (Fig.~\ref{fig:pathway}a).
We validate the trained model by extracting time traces of the molecular heat-to-tail distance $v(t) = h2t(t)$ from independent MD simulation trajectories that were not present in the training data and testing the capability of the trained STAR model to reconstruct the molecular configurations $\mathbf{r}(t)$ in the trajectory. A well-trained model should be able to predict molecular configurations from novel testing data with similar accuracies to that in the training data. In applications to real experimental smFRET data, the true molecular configurations would not be available to make this comparison and the model predictions would have to be validated by indirect means. In all cases we seek only to reconstruct the molecular configuration of the solute and do not attempt to predict the coordinates of the water solvent or any counter ions. In principle, this information is -- again subject to the relevant symmetries -- present within the Takens' delay embedding, but is much more challenging to recover due to the permutational fungibility of the water molecules and the relatively weaker influence of solvent coordinates on our choice of solute-centric observable.
\subsection{Molecular Dynamics Simulations} \label{subsec:MD}
\subsubsection{\ce{C_{24}H_{50}}} \label{subsubsec:meth:C24}
MD simulations of \ce{C_{24}H_{50}} were conducted using the Gromacs 4.6 simulation suite \cite{Gromacs} using the TraPPE potential \cite{martin1998transferable} that models each \ce{CH_3} and \ce{CH_2} group as a united atom and the all-atom SPC model of water \cite{water}. Chain topologies were constructed using the PRODRG2 server \cite{PRODRG}. Lennard Jones interactions were smoothly set to zero at a cutoff of 1.4 nm and Lorentz-Berthelot combining rules used to determine dispersion interactions between unlike atoms \cite{allen1989computer}. Electrostatic interactions were treated using particle mesh Ewald \cite{essmann1995smooth} with a real-space cutoff of 1.4 nm and a 0.12 nm reciprocal-space grid spacing that were optimized during runtime. Simulations were conducted in a 5$\times$5$\times$5 nm$^3$ cubic box with periodic boundary conditions that was sufficiently large to prevent self-interactions of the chain through the periodic walls even in its fully-extended configuration. Initial system configurations were generated by placing an initially elongated chain into an empty box and solvating with 4117 water molecules to a density of 1.0 g/cm$^3$. High energy overlaps were eliminated by steepest descent energy minimization to a force threshold of 2000 kJ/mol.nm. Initial particle velocities were sampled from a Maxwell-Boltzmann distribution at 298 K. Simulations were performed in the NPT ensemble at 298 K and 1 bar using a Nos\'e-Hoover thermostat \cite{nose1984unified} and an isotropic Parrinello-Rahman barostat \cite{parrinello1981polymorphic}. Equations of motion were integrated using the leap-frog algorithm \cite{hockney2010computer} with a 2 fs time step. LINCS constraints were used to fix bond lengths to their equilibrium distances for computational efficiency and as required by the TraPPE and SPC potentials \cite{nadler2006advances}. Systems were equilibrated for 1 ns before conducting a 100 ns production run during which system configurations were saved every 0.2 ps. Head-to-tail distances were computed for each frame of the 500,000 frame simulation trajectory to furnish the univariate time series $v(t) = h2t(t)$ in addition to the atomistic simulation trajectory $\mathbf{r}(t)$. The first 20 ns (100,000 frames) of these trajectories provided the $\left( \mathbf{r}(t), v(t) = h2t(t) \right)$ training data used to train the STAR model, and the remaining 80 ns (400,000 frames) reserved for testing. Input files for these simulations are provided in the \blauw{Supplementary Material}.
\subsubsection{Chignolin} \label{subsubsec:meth:Chig}
MD simulations of the 10 residue (166 atom) engineered mini-protein Chignolin (GYDPETGTWG, PDB ID: 1UAO) \cite{Honda2004} were performed by D.E.~Shaw Research using the Desmond simulation suite \cite{4090217} on the Anton supercomputer \cite{Shaw341} and reported in Ref.~\cite{deshaw}. The peptide was modeled using the the CHARMM22$\*$ force field \cite{PMID:21539772} and solvated by $\sim$1900 water molecules \cite{jorgensen1983comparison} in a 4$\times$4$\times$4 nm$^3$ cubic box with periodic boundary conditions. Two Na$^+$ ions were added to maintain charge neutrality. Lennard-Jones interactions were treated with a 0.95 nm cutoff and electrostatic interactions treated by Gaussian Split Ewald \cite{doi:10.1063/1.1839571} employing a 0.95 nm real-space cutoff and a $32 \times 32 \times 32$ cubic grid. Equations of motion were integrated with a 2.5 fs time step. Equlibration runs were conducted in the NPT ensemble at 340 K and maintained by a Nos\'e-Hoover thermostat \cite{nose1984unified,Hoover}. The 106 $\mu$s production run was conducted in the NVT ensemble employing a Nos\'e-Hoover thermostat \cite{nose1984unified,Hoover} and system configurations harvested every 200 ps. The first 5 $\mu$s (25,000 frames) of these trajectories were used to train the STAR model, and the next 15 $\mu$s (75,000 frames) employed for testing.
\section{Results and Discussion} \label{sec:results}
We demonstrate and validate STAR in applications to molecular dynamics simulations of a \ce{C_{24}H_{50}} polyethylene chain and the $\beta$-hairpin engineered mini-protein Chignolin. Simulation data constituting the training and testing data are collected as described in Section \ref{subsec:MD}. The STAR pipeline was trained and deployed as described in Section \ref{subsec:STAR} with details specific to each system provided below. In each case we achieve RMSD reconstruction accuracies on the hold-out test data of better than 0.2 nm.
\subsection{\ce{C_{24}H_{50}}} \label{subsec:res:C24}
\subsubsection{STAR Training}
The training portion of the \ce{C_{24}H_{50}} simulation trajectory comprised 100,000 frames saved at 0.2 ps intervals recording the Cartesian coordinates $\mathbf{r}(t) \in \mathbb{R}^{72}$ of the $N$ = 24 united atoms of the polymer chain (cf.\ Fig.~\ref{fig:pathway}a). In order to run dMaps into local memory, the simulation trajectory was subsampled with a stride of 20 and the excluded frames projected into the manifold \textit{post hoc} using the the Nystr\"om extension \cite{sonday2009coarse,laing2007coarse,long2019landmark}. Application of spatially-symmetrized dMaps to these data employing a kernel bandwidth of $\epsilon$ = $\exp(-3)$ nm exposed a $k$=2-dimensional intrinsic manifold $\mathcal{M} \subset \mathbb{R}^2$ spanned by nonlinear collective variables of the united atom coordinates $\{ \psi_1, \psi_2 \}$ and supporting the smFES $\beta F(\psi_1, \psi_2)$ (cf.\ Fig.~\ref{fig:pathway}c).
A scalar time series in the head-to-tail distance between the two terminal united atoms $v(t) = h2t(t) \in \mathbb{R}^1$ was computed from the MD training trajectory as a synthetic and idealized smFRET time series over the 100,000 frames separated by intervals of 0.2 ps (cf.\ Fig.~\ref{fig:pathway}b). Takens' delay vectors $\mathbf{y}(t) = [h2t(t), h2t(t-\tau), h2t(t-2\tau), \ldots , h2t(t-(d-1)\tau)]$ were constructed at a delay time of $\tau$ = 30 ps (150 time steps) computed as the first minimum in the mutual information of $h2t(t)$ \cite{Fraser86} and delay dimensionality $d$ = 5 computed using the E$_1$ method of Cao \cite{Cao97,VillaniB,kennel1992determining}. This resulted in the construction of 99,400 delay vectors, each associated with a frame in the MD trajectory containing the molecular structure from which the central element in the delay vector was computed. The initial and terminal $(d-1)\tau/2$ = 60 ps (300 frames) of the MD trajectory that were unassociated with any delay vector were dropped from further analyses. Application of temporally-symmetrized dMaps to the delay embedding trajectory, subsampled with a stride of 20 in order to fit into local memory and employing a kernel bandwidth of $\epsilon$ = 1 nm, defined a $k$ = 2-dimensional manifold $\mathcal{M}^\prime \subset \mathbb{R}^2$ spanned by $\{ \psi_1^\prime, \psi_2^\prime \}$. The frames strided over in the application of dMaps were projected into $\mathcal{M}^\prime$ using the Nystr\"om extension and used to estimate the smFES $\beta F^\prime(\psi_1^\prime, \psi_2^\prime)$ (cf.\ Fig.~\ref{fig:pathway}d).
The diffeomorphism $\Theta : \mathcal{M}^\prime \subset \mathbb{R}^2 \rightarrow \mathcal{M} \subset \mathbb{R}^2$ linking the two manifolds was learned using a simple 2-25-25-25-25-2 fully-connected feedforward ANN comprising four hidden layers of 25 neurons (cf.\ Fig.~\ref{fig:pathway}d $\rightarrow$ c). The ANN employed tanh activations and was trained using Adam \cite{kingma2014adam} with a batch size of 500 and a learning rate of 1$\times$10$^{-4}$. Training was terminated after 250 epochs upon plateauing of the validation error.
The function linking the projection of each frame in the MD trajectory into the intrinsic manifold $\{ \psi_1(t), \psi_2(t) \}$ to the $N(N-1)/2$ = 276-dimensional united atom pairwise distance vectors $\mathbf{d}(t) \in \mathbb{R}^{276}$ corresponding to that configuration was learned using a 2-4-187-370-552-276 fully-connected feedforward ANN employing tanh activations and trained using Adam \cite{kingma2014adam} with a batch size of 500 and a learning rate of 1$\times$10$^{-3}$ over 150 epochs (cf.\ Fig.~\ref{fig:pathway}c $\rightarrow$ e). Molecular reconstructions of the united atom coordinates $\hat{\mathbf{r}}(t) \in \mathbb{R}^{72}$ were computed deterministically from each pairwise distance vector using cMDS (cf.\ Fig.~\ref{fig:pathway}e $\rightarrow$ f). The reconstruction accuracy of the trained STAR pipeline (cf.\ Fig.~\ref{fig:pathway}b $\rightarrow$ d $\rightarrow$ c $\rightarrow$ e $\rightarrow$ f) is assessed by computing the RMSD of the predicted $\hat{\mathbf{r}}(t) \in \mathbb{R}^{72}$ and true $\mathbf{r}(t) \in \mathbb{R}^{72}$ configurations under translational, rotational, mirror, and head-tail alignment by the Kabsch algorithm \cite{kabsch1976solution}.
\subsubsection{STAR Deployment}
We illustrate the application of the trained \ce{C_{24}H_{50}} STAR model in Fig.~\ref{fig:C24}. The trained STAR pipeline enables us to associate each value of $h2t(t)$ in the synthetic smFRET time trace with a molecular reconstruction and also a location and stability on the smFES supported by the low-dimensional intrinsic manifold~\cite{ferguson2010systematic,Ferg16}. The RMSD reconstruction accuracy of the trained pipeline applied to the 20 ns training trajectory is RMSD$_\mathrm{train}$ = 8.4 $\times$10$^{-2}$ nm. In an application to the remaining 80 ns testing trajectory that was not part of the training ensemble, the reconstruction accuracy degrades only slightly to RMSD$_\mathrm{test}$ = 8.6 $\times$10$^{-2}$ nm, indicating that the STAR model is well trained and has good generalizability to novel data. We illustrate for five selected points A-E in the $h2t(t)$ time trace (Fig.~\ref{fig:C24}a) their projection onto the smFES $\beta F(\psi_1, \psi_2)$ (Fig.~\ref{fig:C24}b) and their reconstructed $\hat{\mathbf{r}}(t) \in \mathbb{R}^{72}$ and true $\mathbf{r}(t) \in \mathbb{R}^{72}$ molecular structures (Fig.~\ref{fig:C24}c). \blauw{Movie S1} in the \blauw{Supplementary Material} presents an animation of molecular reconstructions and smFES projections for all data in the time series in Fig.~\ref{fig:C24}a.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{Figure2.pdf}
\caption{Application of STAR to \ce{C_{24}H_{50}} polymer chain. (a) Synthetic idealized smFRET time trace of the head-to-tail distance $h2t(t)$ between the terminal united atoms computed over a 100 ns MD trajectory with frames saved every 0.2 ps. The first 20 ns are used for training (orange) and the remaining 80 ns for testing (blue). The molecule undergoes hundreds of folding and unfolding events over the course of the simulation. (b) The intrinsic manifold $\mathcal{M}$ spanned by the dMap CVs $\{ \psi_1(t), \psi_2(t) \}$ and supporting the smFES $\beta F(\psi_1, \psi_2)$. The Gibbs free energy $F$ is dedimensionalized by the inverse temperature $\beta = 1/k_B T$. The arbitrary zero of $F$ is specified to lie at the global free energy minimum. (c) Molecular reconstructions using the trained STAR pipeline of five representative points A-E in the $h2t(t)$ time series spanning the test set. Reconstructions $\hat{\mathbf{r}}$ (red) are superposed on the corresponding true configurations $\mathbf{r}$ (blue) extracted directly from the MD simulation. The head-to-tail distance of the true configuration and the RMSD under translational, rotational, mirror, and head-tail alignment between the true and reconstructed configurations are reported under each image. The STAR prediction of the location of each point on the smFES is illustrated in panel b and the corresponding dimensionless free energy $\beta F$ reported under each image.} \label{fig:C24}
\end{figure*}
Configuration A corresponds to an elongated chain configurations with the preponderance of the backbone dihedrals in the trans state. Configuration C corresponds to a similarly elongated conformation but with a small number of gauche defects that lead to a small degree of curvature in the contour of the chain. The elevated conformational entropy associated with this small degree of curvature allows Configuration C to reside within 0.3 $k_B T$ of the global free energy minimum at $\beta F$ = 0, while Configuration A lies slightly higher at $\beta F$ = 2.7. Configurations B and D correspond to partially collapsed twisted structures $\sim$3 $k_B T$ less stable than the global free energy minimum that lie along the transition pathway between the global minimum at $(\psi_1 \sim 1.2, \psi_2 \sim 2.0)$ containing elongated chains and the weak local minimum at at $(\psi_1 \sim 3.2, \psi_2 \sim -6.0)$ containing hydrophobically collapsed coiled chains. Configuration E is a metastable hydrophobically collapsed coil that lies slightly outside the local minimum at $\beta F$ = 4.8. We note that Configurations B ($h2t$ = 1.8nm) and D ($h2t$ = 2.0 nm) possess similar values of the $h2t$ distance but correspond to very distinct molecular configurations residing at different locations on the smFES that are both accurately reconstructed by STAR. This illustrates the value of using Takens' Theorem to reconstruct molecular configurations that cannot be distinguished from the instantaneous value of the observable alone.
\subsection{Chignolin} \label{subsec:res:Chig}
\subsubsection{STAR Training}
The training portion of the Chignolin trajectory comprised 25,000 frames saved at 200 ps intervals recording the Cartesian coordinates $\mathbf{r}(t) \in \mathbb{R}^{279}$ of the $N$ = 93 heavy atoms of the protein. A $k$ = 2-dimensional intrinsic manifold $\mathcal{M} \subset \mathbb{R}^2$ spanned by $\{ \psi_1, \psi_2 \}$ was constructed by applying dMaps with a kernel bandwidth of $\epsilon$ = $\exp(-3)$ nm to a subsampling of thsi trajectory with a stride of 2. The excluded frames were projected into the manifold using the the Nystr\"om extension. Taken's delay embeddings were constructed from the synthetic smFRET time series recording the distance between the terminal heavy atoms at a delay time $\tau$ = 200 ps (1 time step) \cite{Fraser86} and delay dimensionality $d$ = 11 \cite{Cao97,VillaniB,Cao97,kennel1992determining}. This results in the construction of 24,990 delay vectors. Applying temporally-symmetrized dMaps to the delay embedding trajectory, sub-sampled with a stride of 2 in order to fit into local memory, $\mathbf{y}(t) \in \mathbb{R}^{11}$ with a kernel bandwidth of $\epsilon$ = 1 nm defined a $k$ = 2-dimensional manifold $\mathcal{M}^\prime \subset \mathbb{R}^2$. Again, the excluded frames were projected into the manifold using the the Nystr\"om extension. The diffeomorphism $\Theta$ mapping $\mathcal{M}^\prime$ to $\mathcal{M}$ was approximated by a 2-25-25-25-25-2 ANN trained using Adam \cite{kingma2014adam} with a batch size of 500 and learning rate of 1$\times$10$^{-4}$ over 250 epochs. The function mapping locations on $\mathcal{M}$ to the $N(N-1)/2$ = 4278-dimensional heavy atom pairwise distances vectors $\mathbf{d}(t) \in \mathbb{R}^{4278}$ was learned and approximated by a 2-4-2855-5706-8556-4278 ANN using Adam \cite{kingma2014adam} with a batch size of 500 and learning rate of 1$\times$10$^{-5}$ over 150 epochs. Predictions of heavy atom molecular configurations were computed deterministically from the pairwise distance vectors using cMDS.
\subsubsection{STAR Deployment}
Application of the trained Chignolin STAR model is illustrated in Fig.~\ref{fig:Chig}. The heavy atom reconstruction accuracy over the 5 $\mu$s training trajectory is RMSD$_\mathrm{train}$ = 0.12 nm, that is only slightly diminished to RMSD$_\mathrm{test}$ = 0.14 nm over the 15 $\mu$s test trajectory, again illustrating good generalizability of the trained model. We illustrate for five selected points A-E in the $h2t(t)$ time trace (Fig.~\ref{fig:Chig}a) their projection onto the smFES $\beta F(\psi_1, \psi_2)$ (Fig.~\ref{fig:Chig}b) and their reconstructed $\hat{\mathbf{r}}(t) \in \mathbb{R}^{279}$ and true $\mathbf{r}(t) \in \mathbb{R}^{279}$ molecular structures (Fig.~\ref{fig:Chig}c). \blauw{Movie S2} presents an animation of molecular reconstructions and smFES projections for all data in the time series in Fig.~\ref{fig:Chig}a.
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{Figure3.pdf}
\caption{ Application of STAR to the 10 residue engineered mini-protein Chignolin. (a) Synthetic idealized smFRET time trace of the head-to-tail distance $h2t(t)$ between the terminal heavy atoms computed over a 20 $\mu$s MD trajectory with frames saved every 200 ps. The first 5 $\mu$s are used for training (orange) and the remaining 15 $\mu$s for testing (blue). The molecule undergoes dozens of folding and unfolding events over the course of the trajectory. (b) The smFES $\beta F(\psi_1, \psi_2)$ supported by the intrinsic manifold $\mathcal{M}$ has its arbitrary zero of $F$ specified to lie at the global free energy minimum. (c) Molecular reconstructions using the trained STAR pipeline of five representative points A-E selected from the testing $h2t(t)$ time series. Reconstructions $\hat{\mathbf{r}}$ (red) are superposed on the corresponding true configurations $\mathbf{r}$ (blue) extracted directly from the MD simulation. The head-to-tail distance of the true configuration and the RMSD under translational, rotational, and mirror alignment between the true and reconstructed configurations are reported under each image. The STAR prediction of the location of each point on the smFES is illustrated in panel b and the corresponding dimensionless free energy $\beta F$ reported under each image.} \label{fig:Chig}
\end{figure*}
Configurations A and D correspond to the native hairpin state of the protein in which all native hydrogen bonds are intact and which lie within the deep global free energy minimum. The reconstruction accuracy of the densely sampled native fold is extremely good as indicated by the RMSD $\approx$ 0.05 nm. Configurations B, C, and E represent a sampling of the unfolded ensemble containing a diversity of random coiled states with some or all of the native hydrogen bonds broken. Despite the relatively sparser sampling and larger configurational diversity of the unfolded ensemble, the RMSD reconstruction accuracy is still better than 0.2 nm. This indicates that the training data, despite containing only four folding/unfolding transitions, provides a sufficiently dense and representative sampling of configurational space to enable accurate reconstruction of even transiently visited molecular conformations. It is unsurprising that configurations lying at high free energies (e.g., Configuration E, $\beta F$ = 10.5, RMSD = 0.15 nm) that are very sparsely sampled in the training data have poorer reconstruction accuracies than the densely sampled native configurations (e.g., Configuration A, $\beta F$ = 0.0, RMSD = 0.06 nm). We propose that adaptive sampling techniques to perform targeted sampling of infrequently-visited regions of the manifold may be beneficial in providing more training data to the STAR pipeline and improving the reconstruction accuracy of higher free energy configurations. Configurations B ($h2t$ = 2.1 nm) and C ($h2t$ = 2.4 nm) possess similar values of the $h2t$ distance but constitute two configurationally and thermodynamically distinct members of the Chignolin unfolded ensemble. Again, we observe that STAR accurately reconstructs configurations with similar instantaneous values of the scalar observable but which correspond to structurally different configurations that lie in very different regions of the smFES.
\section{Conclusions} \label{sec:concl}
This work presents the theoretical underpinnings and numerical implementation of an approach Single-molecule TAkens Reconstruction (STAR) to reconstruct molecular configurations from time series in a single experimentally-measurable observable. The basis for the approach rests upon the integration of Takens' Delay Embedding Theorem with tools from manifold learning, statistical thermodynamics, artificial neural networks, and rigid graph theory to extract a representation of the system state from the scalar time series and learn the \textit{a priori} unknown mapping to the molecular configuration from molecular dynamics simulation training data. The trained STAR model can then be applied to novel time series data to predict both the corresponding molecular configurations and their location and stability on the single molecule free energy surface. We have demonstrated and validated the approach in applications to a \ce{C_{24}H_{50}} polyethylene chain and the 10-residue engineered $\beta$-hairpin mini-protein Chignolin. In both cases we demonstrate that trained STAR models can robustly reconstruct the molecular configurations from time series data in the head-to-tail distance with RMSD accuracies better than 0.2 nm.
In this work, we adopt the head-to-tail distance as an experimental observable that can, in principle, be measured by a single molecule experimental technique such as smFRET. The head-to-tail time traces in this work are extracted from MD simulation trajectories in order to test our approach in applications where the ground truth molecular configurations are explicitly available from the simulations. These time traces can therefore be considered to represent synthetic and idealized smFRET data at arbitrarily high temporal resolution and subject to no measurement error or noise. The present work reports a computational proof-of-principle demonstration of STAR in this idealized limit. Applying STAR to real experimental time series must engage a number of concerns surrounding the experimental realities of smFRET measurements including millisecond limits in sampling frequency; shot noise, uncertainties, and unpredictable trajectory lengths due to photobleaching; degraded measurement reliabilities for fluorophores outside of the 2-8 nm range; and conformational perturbations induced by conjugation of the fluorescent probes \cite{Chang2011,Roy2008rr}. Furthermore, the STAR model must be trained and calibrated on MD training data and so the accuracy of the molecular potential functions and degree of sampling of the thermally-relevant configurational space will limit the quality to the trained model. A STAR model calibrated on MD simulation data employing a poor force field or which does not sample all of the experimentally-accessible states and transitions will not perform well when deployed on real experimental time series. In future work, we propose to engage these practical issues empirically by adding noise to the observables extracted from our simulation trajectories, limiting the accessible time resolution, limiting the length of the time series, and exploring the transferability of the trained models between molecular force fields.
We would also like to explore technical innovations to determine optimal experimental observables (e.g., optimal FRET fluorophore placement) for high-accuracy molecular reconstruction \cite{Mittal2018}, the extension of STAR to multiplexed measurements \cite{Cao1998}, and the potential to reconstruct not just the molecular configuration but also the location and orientation of proximate solvent molecules using permutationally invariant representations of the solvent coordinates \cite{pietrucci2020novel,Han_2018}. It would also be of interest to explore the transferability of reconstructions learned under one set of conditions (e.g., temperature, pressure, salt concentration, mutations from wild type) to reconstruct those at another. We would also like to explore the possibility of adaptive sampling wherein the deployed model can identify regions of configurational space where it does not have sufficient training data to make accurate predictions and can conduct additional on-the-fly molecular simulations to supplement its training in these regions. This is anticipated to be particularly important for applications of STAR to large proteins where it is challenging to comprehensively sample the important configurational space. Finally, we also see potential applications of STAR in other applications where it is of interest to reconstruct the state of a dynamical system where it is challenging or impossible to obtain complete information on its state. As such, we envisage potential applications of the approach in fields such as climatology, epidemiology, and ecology.
\section*{Supplementary Material}
See supplementary material for simulation input files for the \ce{C_{24}H_{50}} molecular dynamics simulations, \blauw{Movie S1} showing an an animation of molecular reconstructions and smFES projections for \ce{C_{24}H_{50}}, \blauw{Movie S2} showing an an animation of molecular reconstructions and smFES projections for Chignolin.
\section*{Acknowledgments}
We thank Dr.~G{\"u}l H. Zerze for fruitful discussions. This material is based upon work supported by the National Science Foundation under Grant No.~DMS-1841810. We are grateful to D.E.~Shaw Research for sharing the Chignolin simulation trajectories.
\section*{Data Availability Statement}
Input files for the \ce{C_{24}H_{50}} molecular dynamics simulations are provided in the \blauw{Supplementary Material}. The Chignolin simulation trajectories reported in Ref.~\cite{deshaw} were obtained upon request from D.E.~Shaw Research.
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{"url":"http:\/\/physicshelpforum.com\/electricity-magnetism\/13504-e-2-718281828459-a.html","text":"Physics Help Forum e=2.718281828459\n\n Electricity and Magnetism Electricity and Magnetism Physics Help Forum\n\n Aug 11th 2017, 11:34 AM #1 Junior Member \u00a0 Join Date: Jul 2017 Posts: 27 e=2.718281828459 Euler's number...... the well-known mathematical constant. The number is related to every natural growth process.... But the what is the exact meaning of Euler's number in electromagnetism? How exactly it is used in electromagnetism? And how come it is so necessary? I need very illustrative and INTUITIVE understanding about the use of mathematical constant e in electromagnetism.... I'll be thankful for every reply....\nAug 11th 2017, 01:00 PM \u00a0 #2\nSenior Member\n\nJoin Date: Apr 2015\nLocation: Somerset, England\nPosts: 547\n I'll be thankful for every reply....\nWell you didn't reply to my description of affine geometry I made for you elsewhere !\n\n Aug 11th 2017, 01:28 PM #3 Senior Member \u00a0 Join Date: Aug 2010 Posts: 186 Every exponential can be written in terms of e: $\\displaystyle a^x= e^{ln(a^x)}= e^{x ln(a)}$ so anywhere there is an exponential to any base, we can use \"e\". $\\displaystyle e^x$ happens to have the nice property that its derivative is itself: $\\displaystyle \\frac{de^x}{dx}= e^x$. Last edited by HallsofIvy; Aug 14th 2017 at 09:28 AM.\n Aug 11th 2017, 09:56 PM #4 Senior Member \u00a0 Join Date: Nov 2013 Location: New Zealand Posts: 459 Are you perhaps confusing E, the electric field vector, with e which is euler's number? They are not the same thing. topsquark likes this.\n\n Thread Tools Display Modes Linear Mode","date":"2017-08-23 15:51:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.750258207321167, \"perplexity\": 2774.6198205439036}, \"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-2017-34\/segments\/1502886120573.75\/warc\/CC-MAIN-20170823152006-20170823172006-00497.warc.gz\"}"} | null | null |
\section{Introduction and Main Results}
The search for bridges between many-body physics and quantum information has been very fruitful, and has led to many important discoveries and insights
\cite{amaderAdP,Fazio-Osterloh-VedralRMP}. On the one hand, quantum information concepts have been used to provide further tools to a
many-body physicist, while on the other, realizable many-body physics systems ranging from quantum optics systems to ion traps are
being tried as potential substrates for quantum information processing tasks.
It is rarely possible to treat an interacting many-body system exactly, and hence it is important to obtain approximate methods to
deal with them. The mean field theory (MFT) \cite{MFT-book, Pathria, Mahan}, introduced by P. Weiss in 1907,
is a very useful tool available to many-body physics to examine such systems, for
both classical and quantum interacting many-body systems, with a low computational cost.
\textbf{Main thesis.} We propose that parallel to, but clearly different from, the MF class of theories \cite{MFT-book},
there exists an entanglement mean field class of theories to treat interacting classical many-body systems,
that deals with one-body and two-body physical parameters in its
self-consistency equations.
The mean field class of theories
are
an ultimate form of coarse-graining of the many-body system, in that it reduces the interacting
many-body Hamiltonian to
single body terms, and deals with single-body physical parameters in its self-consistency equstions \cite{MFT-book}.
In contrast, the entanglement mean field class of theories proposes to stop a step before in the coarse-graining process,
and reduces the parent Hamiltonian to a finite number of
two-body terms, and deals with single- as well as two-body terms in its self-consistency equations.
The entanglement mean field class of theories provides us with a tool to go beyond the mean field class, and yet remain in the low-cost bracket. We believe that
the formalism will be useful in
investigation
and control of
many-body systems in several areas including condensed matter, ultra-cold gases, and quantum information.
An interesting
improvement
of the mean field approach
is the cluster mean field theory (CMFT) \cite{CMFT-review}, where one
reduces the many-body system to a fundamental unit (``cluster'') of the many-body lattice, while still retaining footprints
of the many-body parent as an undetermined parameter, although
this undetermined parameter is
still (like in MFT)
a one-site physical quantity (like, magnetization) of
the many-body parent,
and it is determined by the self-consistency relation (CMF equation) equating the parameter to the same one-site quantity obtained from
the cluster. We stress here that the self-consistency relations in \emph{both} MFT and CMFT are
are based on
\emph{one}-site physical quantities, in particular, on magnetization.
To treat critical phenomena in interacting classical spin models inspired by entanglement mean field theory (EMFT), proposed
for quantum systems in Ref. \cite{aaj-bristi-hoch-chhe},
one
reduces the many-body classical system to a two-body one
while retaining imprints of the many-body
parent as an undetermined parameter. In contrast to MFT, in EMFT, the undetermined parameter depends on a two-site physical
quantity (like, two-point correlation) of the many-body parent. This parameter is then determined by the self-consistency
relation (EMFT equation) equating, e.g.
the
two-point correlation of the many-body parent with that of the EMFT-reduced two-body system.
Note that
there is certainly no quantum entanglement \cite{HHHH-RMP} generated by applying the EMFT to classical spin systems.
We stress here that
that the entanglement mean field theory is different from the cluster mean field approach. While the latter uses single-site
physical parameters in its self-consistency equation, the former uses two-site ones.
EMFT is also different from other useful techniques to deal with many-body systems, like the
renormalization group
approaches \cite{egulo-renorm-boi}, with the latter using block decimation techniques on the whole lattice. These differences, both operational and result-wise,
will be further underlined in Sec. \ref{ebar-baRi-jabo-sat-ta-chobbis-baje}.
In this paper, we also present a further improvement of
EMFT to a ``cluster EMFT'' (CEMFT) that reduces the many-body system to a fundamental unit of the
many-body lattice, while retaining impressions of the original many-body system as undetermined parameters. In contrast
to CMFT, in CEMFT, the undetermined parameters depend \emph{both} on one-site and two-site physical quantities (e.g., on magnetization and
two-site correlation) of the many-body parent. These parameters are then determined via \emph{coupled} self-consistency equations (CEMFT equations)
equating e.g. the magnetization of the original many-body system with that of the (CEMFT-reduced) cluster, and
the
two-point correlation of the many-body parent with that of the cluster.
Apart from CMFT, there are several other interesting generalizations of the mean field theory in the literature, including
the Bethe-Peierls-Weiss approximation \cite{BPW},
the Onsager reaction field theory \cite{ORF},
the diagrammatic expansion method \cite{DEM}
the self-consistent correlated field theory \cite{SCCF},
the screened magnetic field theory \cite{SMF},
and
the correlated cluster mean field theory \cite{CCMFT}, to mention a few (see also \cite{MFT-review-kora-boi}).
Improvements of the entanglement mean field theory in these directions are also possible, and
will be pursued later.
Meanwhile, let us note here that all the above exciting
examples, in the MF class of theories,
deal with single-body physical parameters
in the respective self-consistency equations.
In contrast, the EMF class of theories deal with single- as well as two-body physical parameters
in the EMFT class self-consistency equations.
\textbf{MFT vs. EMFT.}
Solving for
magnetization and correlation functions
from
the EMFT and CEMFT equations leads to the prediction of critical phenomena
in the spin models. We apply the EMF and CEMF theories to the nearest-neighbor Ising model in one, two (hexagonal, square, and
triangular), and three (cubic), dimensional lattices. The results are given in Table 1. In all the cases considered, in the different dimensions and
geometries, EMFT gives better predictions over MFT, and CEMFT
is better than
CMFT. (Actually, EMFT is already better than CMFT
in all the cases considered.) In the best case, EMFT is better than MFT by \(68\%\)
and
CEMFT is better than CMFT by 85\%, happening respectively for the hexagonal and square lattice systems. In the worst case,
EMFT is better than MFT by 42\%, and
CEMFT is better than CMFT by 8\%, happening respectively for the triangular and cubic lattice systems.
\section{EMFT for Classical Models}
\label{ebar-baRi-jabo-sat-ta-chobbis-baje}
Before presenting the entanglement mean field theory inspired approach to classical spin models, let us briefly
describe
the mean field theory
for such systems.
Consider the nearest neighbor (classical) Ising model
\begin{equation}
H= -J\sum_{\langle {\vec{i}}{\vec{j}} \rangle}\sigma_{\vec{i}} \sigma_{\vec{j}}
\end{equation}
which represents a system of interacting (classical) spin-1/2 particles (Ising spins) on a \(d\)-dimensional lattice of an arbitrary fixed geometry.
The coupling strength \(J\) is positive, and \(\sigma_{\vec{i}} = \pm 1\) represents the value of the Ising spin
at the site \(\vec{i}\).
\(\langle {\vec{i}} {\vec{j}}\rangle\) indicates that the corresponding sum runs over
nearest neighbor lattice sites only.
The mean field theory consists in assuming that a particular spin, say at \(\vec{i_0}\), is special, and
replacing all other spin operators by their mean values.
Denoting the mean values of the spin operator \(\sigma_{\vec{i}}\) at the site \(\vec{i}\) by \(m\) (average magnetization),
leads to an MFT Hamiltonian \cite{MFT-book}, which we denote as
\( H_{MFT} \).
One then solves the self-consistency equations (mean field equations)
\begin{equation}
m = \sum_{\mathcal{CF}(\mathcal{I})} \sigma \rho^\beta_{MFT},
\end{equation}
for \(m\).
Here \(\rho^\beta_{MFT}\) is the mean field canonical equilibrium state
\(\exp(-\beta H_{MFT})/Z_{MFT}\), \(Z_{MFT} = \sum_{\mathcal{CF}(\mathcal{I})}\exp(-\beta H_{MFT}))\) is the MF partition function,
\(\beta = \frac{1}{k_B T}\), with \(T\)
denoting temperature on the absolute scale, and \(k_B\) the Boltzmann constant.
Here, and in the rest of the paper,
\(\mathcal{CF}(\mathcal{I})\) will denote all Ising configurations of all the spins involved in that particular case.
In the MF equation as well as in the MF partition function, there is just a single spin left, and \(\mathcal{CF}(\mathcal{I})\) denotes the set of the two possibilities
thereof.
Substituting \(m\) in \(H_{MFT}\) and \(\rho^\beta_{MFT}\), one
can
find the single-body physical properties of the system in the mean field limit.
\begin{figure}[h!]
\label{fig-chhobi-prothhom}
\begin{center}
\epsfig{figure=MFTvsEMFTcluster.eps, height=.27\textheight,width=0.45\textwidth}
\caption{(Color online) MF vs. EMF class of theories.
In MFT, a ``magnifying glass'' is put on a single particle of the many-body interacting system, and it leads to a self-consistency
relation involving single-particle parameters.
A different magnifying glass is employed in EMFT, which focusses attention on \emph{two} particles, and leads to a self-consistency relation
involving \emph{two-particle physical parameters}.
Parallely, in CMFT, a fundamental unit (cluster) is chosen from the lattice which then is used to write self-consistency equations, again involving only
single-site parameters. In CEMFT, the same cluster is used, but the self-consistency equations involve both one-site and two-site physical parameters.
}
\end{center}
\end{figure}
The entanglement mean field theory begins by noting that the square of an Ising spin random variable is unity. The two-body interaction Hamiltonian that we
are dealing with, can be thought of an \(N\)-body interaction Hamiltonian (\(N\) being the total number of Ising spins in the system), in each term of which,
all but two random variables are constant (= unity). Let us call it a unit random variable.
Since the square of any Ising random variable is unity, we can replace a unit random variable
on a site that is neighboring the nontrivial interacting spins of an interaction term,
by the square the Ising random variable at that site,
for all the interaction
terms in the Hamiltonian.
Therefore, the term \(\sigma_{k-1,l}\sigma_{k,l}\) in a Hamiltonian on a two-dimensional square lattice can be replaced by
\(\sigma_{k-1,l}\sigma_{k,l}\sigma_{k+1,l}\sigma_{k+1,l}\). The latter can be re-written as \(AB\), with
\(A=\sigma_{k,l}\sigma_{k+1,l}\), and \(B=\sigma_{k-1,l}\sigma_{k+1,l}\). Let us call this Hamiltonian as
\(H_{EMFT}^{inter}\). For a given interaction term, let us call
the site at which the replacement of the unit random variable by the square of the Ising random variable is done
as the dummy site for that interaction term. Given an interaction term, there can be several nearby sites that can act as the dummy site for that term.
So in the case of the term \(\sigma_{k-1,l}\sigma_{k,l}\), \((k+1,l)\) is used as the dummy site.
We then assume
that a certain \emph{pair} of two neighboring spins are ``special''. (See Fig. 1.)
Consider a \(d\)-dimensional lattice with coordination number \(\nu_{co}\).
There will be \(2(\nu_{co}-1)\) terms
in the Hamiltonian \(H_{EMFT}^{inter}\), that will have the special pair, along with
two more Ising spins in two lattice sites (one of which, viz. the dummy site, is different from the special pair sites).
In such a lattice, any one spin in the special pair is connected (via interactions in \(H\)) to \(\nu_{co} -1\) spins. Let this number
(\(\nu_{co} -1\)) be denoted by \(\nu_E\), and be called the EMFT coordination number.
We now
replace
the non-special two-spin interactions (with nearby spins) in all the interaction terms in \(H_{EMFT}^{inter}\) by a
constant multiple of their mean values \(C\).
Physically, the mean value \(C\) represents the nearest neighbor correlation in the corresponding lattice.
Since every interaction in \(H\) connecting to the special pair actually connects to \emph{one spin in the pair}, the non-special two-spin interactions in
each term in \(H_{EMFT}^{inter}\) is
replaced by \(\frac{1}{2}C\).
The EMFT-reduced Hamiltonian, for the nearest neighbor Ising model on a \(d\)-dimensional lattice with EMFT coordination number \(\nu_E\),
will therefore be
\begin{equation}
\label{emft-asol}
{\cal H}_{EMFT} = -\frac{1}{2}J\nu_E C\sigma_{\vec{i}} \sigma_{\vec{j}}
\end{equation}
where we have ignored the terms in the Hamiltonian which will not contribute to the EMFT equations below, and where
we have assumed that the neighboring lattice sites \({\vec{i}}\) and \({\vec{j}}\) are special.
The self-consistency equation (EMFT equation) is
\begin{equation}
C = \sum_{\mathcal{CF}(\mathcal{I})}\sigma_{\vec{i}} \sigma_{\vec{j}} \varrho^\beta_{EMFT},
\end{equation}
for a system at temperature \(T\).
Here \(\rho^\beta_{EMFT}\) is the entanglement mean field canonical equilibrium state
\(\exp(-\beta H_{EMFT})/Z_{EMFT}\), \(Z_{EMFT} = \sum_{\mathcal{CF}(\mathcal{I})}\exp(-\beta H_{EMFT}))\) is the EMF partition function.
The EMFT equation is to be solved
for \(C\)
for obtaining the two-particle physical properties of
\(H\) in the EMFT limit.
In a typical situation, there is a finite temperature \(T_c\), the critical temperature,
that depends on the lattice geometry and dimension, above which
the EMFT equation provides a nontrivial (i.e. nonzero) solution.
The values of the critical temperatures have been obtained for the interacting Ising systems in different dimensions and geometries,
and are given in Table 1. The predictions of the EMF theory are always better than the corresponding ones from MFT in the cases considered,
and almost always more than 50 percent better,
and for the hexagonal (honeycomb) lattice in two dimesnions, EMFT is 68\% better than MFT. See Table 1 for
further details.
In the entanglement mean field theory, the nature of the interactions propagating in the lattice geometry, enters the predictions through the
the existence of an interaction term (\(\sigma_{\vec{i}} \sigma_{\vec{j}}\)) and
a two-site physical parameter (the mean correlation \(C\)) in the EMFT Hamiltonian \(H_{EMFT}\), and their interplay in the self-consistency equation (EMFT equation).
These features are absent in MFT, where the MFT Hamiltonian contains a single Ising random variable and a single-site physical parameter.
Moreover, an EMFT coordination number enters the stage in EMFT, while it is the coordination number in MFT. These differences
lead to the better memory of the EMFT of its many-body parent, and a consequent better performance of the EMFT over MFT.
Note that the EMFT approach is different from the cluster MFT \cite{CMFT-review} as there the fundamental unit changes
with the lattice geometry and dimension, while here
we always work with two special spins regardless of the lattice geometry.
More importantly, the cluster MFT still (i.e., as in MFT) uses single-site properties to
construct the self-consistency equations, while in EMFT, we use two-site properties to do the same.
Additionally, both MFT and CMFT leads to effective Hamiltonians consisting of either a single random varible or a sum over single random variables, while
in EMFT, we deal with effective Hamiltonians that retain interaction terms involving two random variables.
These are some of the operational differences. Result-wise,
a glance at Table 1 reveals that the predictions, for the critical
temperatures of the different models, of EMFT and CMFT, are very different. Indeed, in all the cases considered, EMFT performs better than CMFT, with the
prediction in the case of the three-dimensional cubic lattice being 49\% better.
It is possible to obtain a generalization of the EMFT
approach, towards a ``cluster EMFT'', for a better consideration of the lattice geometry, and we do so in the succeeding section.
The EMF approach is also different from the other techniques to handle many-body models. In particular, it is unlike the
renormalization group approach \cite{egulo-renorm-boi}, where block decimation techniques are used on the whole lattice,
and free energy of the decimated lattice is equated to that of the original. Result-wise, application of the renormalization group to, e.g. the nearest
neighbor Ising model on the square lattice predicts a critical temperature (in units of \(k_B T/J\)) at \(2.55\) \cite{Pathria}.
The EMFT prediction is \(3\). The CEMFT prediction (considered below) is \(2.08\), with the exact value being \(2.27\) \cite{exact-values}
A similar formalism as above works (for both EMFT and cluster EMFT), with suitable modifications, for quantum spins, higher discrete spins, continuous
spins, more complex lattices and
interactions, etc.
Also, both the mean field theory as well as the EMFT has been described for the ferromagnetic cases. The antiferromagnetic
case requires some modifications in the mean field theory, and correspondingly some changes in the EMFT (and CEMFT). These will not be
discussed in this paper.
\section{Cluster EMFT for Classical Models}
\label{teRe-khide-peye-gyachhe}
As has been noted before, the interactions of the many-body parent propagating in the lattice,
are taken care of in the entanglement mean field theory, by the EMFT coordination number, and the interplay of the mean correlation \(C\) and
the interaction term in \(H_{EMFT}\) in the self-consistency equation. We have seen that this gives a better
consideration to the interactions in the parent Hamiltonian than that in MFT.
Towards improving our approximations, we now include the lattice structure along with interactions between spin variables.
We call it the cluster
entanglement mean field theory, and is described as follows.
For definiteness, consider a two-dimensional square lattice. See Fig. 1. A cluster in this case is a fundamental square of four spins.
Let us focus on a particular cluster of four spins.
These four spins interact among themselves
by four interaction terms in \(H\). They are the intra-cluster interactions.
This basic unit, consisting of four spins, also interact with other spins in neighboring
clusters via inter-cluster interaction terms in \(H\). We first consider the intra-cluster terms.
Just like in the case of the entanglement mean field theory, in every intra-cluster interaction term,
we replace a unit random variable at a nearby dummy site by the square of an Ising random variable at that site. The difference is that
the dummy site is now always chosen from the among the sites in the chosen cluster. This is just like in CMFT, where
only the intra-cluster spins are involved in producing the terms of the form \(m\sigma_{\vec{i}}\).
So for the closen square cluster consisting of the sites
formed by rows \(k, k-1\) and columns \(l,l-1\), for the intra-cluster term
\(\sigma_{k-1,l}\sigma_{k,l}\) in the Hamiltonian \(H\),
the site \((k,l-1)\) can act as a dummy site, whereby we obtain the term
\(\sigma_{k-1,l}\sigma_{k,l}\sigma_{k,l-1}\sigma_{k,l-1}\). Similarly as in EMFT, the so-obtained term
can be re-written as \(A_\chi B_\chi\), with
\(A_\chi=\sigma_{k,l}\sigma_{k,l-1}\), and \(B_\chi=\sigma_{k-1,l}\sigma_{k,l-1}\).
We now replace \(B_\chi\) by the unit multiple of the mean value \(C\), and consequently, the contribution of this intra-cluster term to the cluster EMFT Hamiltonian
\(\mathcal{H}_{CEMFT}\) is \(-JC\sigma_{k,l}\sigma_{k,l-1}\).
The entanglement coordination number of EMFT is absent in CEMFT, as the latter itself depends on the lattice geometry, and hence strenthens the
approximation.
The inter-cluster terms are taken care of by replacing them with effective fields at the corresponding spins of the chosen cluster, and
these terms are exactly the
same as in CMFT. The terms in \(H\) that are neither intra- not inter-cluster, do not appear
in the considerations below, and are therefore ignored. Therefore, for a two-dimensional square lattice, with the chosen cluster, the cluster EMFT Hamiltonian is
\begin{equation}
\mathcal{H}_{CEMFT}^{sq} = -JC \sum_{\langle {\vec{i}}{\vec{j}} \rangle_\chi} \sigma_{\vec{i}} \sigma_{\vec{j}}
-2Jm \sum_{\vec{i}_\chi} \sigma_{\vec{i}}
\end{equation}
where the first sum
runs over nearest neighbor sites of the chosen cluster, and the second runs over sites of the same. The factor 2 in the second term comes from
the fact that we are considering a
square lattice, so that every spin in the chosen cluster is connected (via an interaction term in \(H\)) to \emph{two} spins in the neighboring clusters.
Here, \(C\) denotes the nearest neighbor correlation of the lattice under consideration, and \(m\) the corresponding magnetization.
One may similarly find the cluster EMFT Hamiltonian \(H_{CEMFT}\) for other models.
At this point, both \(C\) and \(m\) are undetermined. They are to be solved from the self-consistency relations (CEMFT equations)
equating the correlation \emph{and} the magnetization of the chosen cluster with the corresponding ones of the whole lattice:
\begin{eqnarray}
C= \sum_{\mathcal{CF}(\mathcal{I})}\sigma_{\vec{i}} \sigma_{\vec{j}} \varrho^\beta_{CEMFT}, \nonumber \\
m= \sum_{\mathcal{CF}(\mathcal{I})}\sigma_{\vec{k}} \varrho^\beta_{CEMFT},
\end{eqnarray}
where \(\vec{i}\) and \(\vec{j}\) are any two nearest neighbor sites, and \(\vec{k}\) a particular site, in the chosen cluster.
Here \(\rho^\beta_{CEMFT}\) is the cluster entanglement mean field canonical equilibrium state
\(\exp(-\beta H_{CEMFT})/Z_{CEMFT}\), \(Z_{CEMFT} = \sum_{\mathcal{CF}(\mathcal{I})}\exp(-\beta H_{CEMFT}))\) is the CEMF partition function.
The CEMFT equations form a set of coupled self-consistency relations for \(C\) and \(m\), and their nontrivial solution set exists only after a certain temperature,
which is the critical temperature obtained from the cluster entanglement mean field theory.
The table below gives the predictions for the critical temperatures for the nearest neighbor Ising model in different
lattice geometries and dimensions. In this paper, we have obtained the predictions from entanglement mean field theory and
cluster entanglement mean field theory. The predictions from mean field theory can be obtained, e.g. from Refs. \cite{MFT-book}.
The predictions from cluster mean field theory are obtained in Refs. \cite{CMFT-review} and references therein.
The exact and series results are obtained in Refs. \cite{exact-values} and references therein.
\begin{widetext}
\begin{equation}
\begin{array}{|c||c|c|c||c|c|c||c|}
\hline
\mbox{Lattice} & \mbox{MFT} & \mbox{EMFT} & \mbox{Improvement (\%)} & \mbox{CMFT} & \mbox{CEMFT} & \mbox{Improvement (\%)} & \mbox{Exact/Series} \\
\hline
\mbox{Linear} & 2 & 1 & 50 & 1.28 & 1.05 & 17.97 & 0 \\
\mbox{Hexagonal} & 3 & 2 & 67.57 & 2.335 & 1.21 & 63.53 & 1.52\\
\mbox{Square} & 4 & 3 & 57.8 & 3.5 & 2.08 & 84.55 & 2.27\\
\mbox{Triangular} & 6 & 5 & 42.37 & 5.64 & 3.99 & 82.5 & 3.64\\
\mbox{Cubic} & 6 & 5 & 66.67 & 5.49 & 3.61 & 8.16 & 4.51\\
\hline
\end{array}
\nonumber
\end{equation}
\begin{center}
Table 1:
A comparison of the critical temperatures obtained for the nearest neighbor Ising model in different lattices and geometries.
Except for those in the two columns that mention the improvements, the numbers in the other columns are in units of
\(k_B T/J\). There are two columns with the heading ``Improvement'', of which the left one shows the improvement in the
EMFT prediction over that from MFT, while the right one shows that for CEMFT over CMFT.
\end{center}
\end{widetext}
\section{Conclusion}
We have proposed an entanglement mean field theory for dealing with classical interacting many-body models. Distinct from the mean field approach to
interacting systems, the entanglement mean field one reduces the many-body parent Hamiltonian into a two-body one involving undetermined mean values of
two-site physical parameters of the many-body parent. These undetermined parameters are determined via self-consistency equations between mean values of
the two-body physical quantity of the
reduced Hamiltonian and the many-body parent. We then generalize the concept to a cluster entanglement mean field theory where we work with a
fundamental unit of the lattice. The self-consistency relations in this case are a set of coupled equations of single-site and two-site
physical quantities. Solving these self-consistency equations lead to the predictions of critical temperatures of the models considered, which we then
compare with the previous results. In all the cases considered, in the different geometries and dimensions, the predictions of the entanglement mean
field theory are better than mean field theory (68\% at most, and 42\% at least),
and the same of the cluster entanglement mean field theory are better than cluster mean field theory (85\% at most, and 8\% at least).
| {
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{"url":"https:\/\/www.physicsforums.com\/threads\/whats-this-equation-come-from.689933\/","text":"# What's this equation come from?\n\n1. May 5, 2013\n\n### pattisahusiwa\n\nI have an equation like this,\n\n$\\frac{dZ}{zD\\beta} = \\frac{d}{d\\beta}\\ln Z$,\n\nis it from $\\frac{d}{d\\beta}\\frac{dZ}{Z}$ or from...?\n\nHow we can prove this relation?\n\n2. May 5, 2013\n\n### Office_Shredder\n\nStaff Emeritus\nIs your equation supposed to be\n$$\\frac{1}{Z} \\frac{ dZ}{d\\beta} = \\frac{d}{d\\beta} \\ln(Z)$$\nIf so, this is just the chain rule\n\n3. May 5, 2013\n\n### pattisahusiwa\n\nThank you for quick replay.\n\nYes, your relation is correct too. If this is a chain rule, so can i write them like one in the first thread?\n\n4. May 6, 2013\n\n### pattisahusiwa\n\nHi all, I just want to know that my relation is correct or not?\n\n$$\\frac{1}{Z}\\frac{dZ}{d\\beta} = \\frac{d}{d\\beta}\\int\\frac{dZ}{Z} = \\frac{d}{d\\beta}\\left(\\ln Z\\right)$$","date":"2017-11-25 09:51:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5097372531890869, \"perplexity\": 2487.7681433024363}, \"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-2017-47\/segments\/1510934809746.91\/warc\/CC-MAIN-20171125090503-20171125110503-00306.warc.gz\"}"} | null | null |
Le phare de Minot's Ledge (en {{lang-en|Minot's Ledge Light}}) est un phare actif situé sur , un récif du port de Cohasset dans le Comté de Norfolk (État du Massachusetts).
Il est inscrit au Registre national des lieux historiques depuis le .
Histoire
Le récif de Minots Ledge se situe à environ des villes de Cohasset et de Scituate, au sud-est de Boston.
Le premier phare
En 1843, l'inspecteur de phare I.W.P. Lewis rédigea un rapport sur Minots Ledge, montrant que plus de 40 navires avaient été perdus après avoir heurté le récif de 1832 à 1841, entraînant de lourdes pertes en vies humaines et des dommages matériels. L'incident le plus dramatique fut le naufrage d'un navire St John en avec quatre-vingt-dix-neuf immigrants irlandais, qui se noyèrent tous à la vue de leur nouvelle patrie. Il avait été initialement proposé de construire un phare semblable au phare d'Eddystone, de l'ingénieur John Smeaton, situé au large de la côte sud-ouest de l' Angleterre. Cependant, le capitaine William H. Swift, chargé de la planification de la tour, pensait qu'il était impossible de construire une telle tour sur ce récif en grande partie submergé. Au lieu de cela, il plaida avec succès pour un phare à pile, une structure en forme d'araignée percée dans la roche.
Le premier phare de Minots Ledge a été construit entre 1847 et 1850 et a été mis en service le . Une nuit d', le phare a été frappé par une tempête majeure qui a causé des dégâts dans toute la région de Boston. Le lendemain, seuls quelques pieux courbés ont été trouvés sur le rocher. Deux assistants gardiens sont décédés alors qu'ils gardaient le phare.
Le phare actuel
Jusqu'en 1863, la conception et la construction des phares relevaient du . Il en résulta une rivalité avec le Corps des ingénieurs de l'armée des États-Unis, établi de longue date, qui construisait des fortifications et avait la responsabilité, comme il le fait aujourd'hui, d'améliorer les voies navigables. L'ingénieur en chef du corps des ingénieurs de l'US Army, Joseph Gilbert Totten, a personnellement pris en charge le projet de conception et de construction d'un phare permanent sur le récif.
La conception de Totten était aussi simple qu'efficace. Avec une vaste expérience dans la construction de fortifications, Totten a pleinement apprécié la permanence et la force des constructions en granit. Il a conçu le phare de manière que les fondations du phare soient une solide base en granit pesant des milliers de tonnes. Pour attacher le phare au récif, il avait plusieurs pieux massifs en fer mis en place afin que le phare soit littéralement accroché au récif par son propre poids. Les travaux sur le récif ne pouvaient avoir lieu que dans des conditions météorologiques où il était à marée basse avec une mer calme. La construction a donc pris des années.
Les travaux du phare actuel ont débuté en 1855 et ont été achevé et mis en service le . Avec un coût final de 300.000 $, il s'agissait du phare le plus cher jamais construit aux États-Unis. Le phare est construit de gros et lourds blocs de granit à queue d'aronde, qui ont été coupés et habillés à terre à Quincy et emmenés par bateau sur le chantier. Le phare était équipé d'une lentille de Fresnel de troisième ordre.
Le signal lumineux, un cycle de clignotement de 1, 4 et 3 éclats adopté en 1894, est appelé localement "I LOVE YOU" (1-4-3, un entier naturel étant le nombre de lettres de cette phrase), et il est souvent cité en tant que tel par couples romantiques dans sa gamme.
Informations historiques de l'US Coast Guard
Minots Ledge est l'un des nombreux groupes d'affleurements rocheux au large des côtes de Cohasset et de Scituate et a été le théâtre de nombreux naufrages. Entre 1832 et 1841, il y eut 40 épaves sur ce récif et les récifs voisins. Entre 1817 et 1847, on estima que 40 personnes et 364.000 dollars de biens avaient été perdus dans des épaves de navires aux environs de Minots Ledge, au large de Cohasset, dans le Massachusetts.
En 1843, l'inspecteur I.W.P. Lewis, du service des phares, soulignait le besoin urgent d'un phare sur Minots Ledge. Son jugement a été confirmé par le capitaine William H. Swift, du bureau topographique des États-Unis, qui avait recommandé un phare à piles de fer offrant moins de résistance aux vagues qu'une tour de pierre.
Le récif avait à peine de large et était découvert à marée basse durant 2 à 3 heures par jour. Au printemps 1847, la construction d'une structure légère en fer ajouré de commença sur ce rocher étroit. Les hommes ne pouvaient travailler que par temps très calme, lorsque la marée était à son plus bas. Les travaux ont été effectués à partir d'une goélette qui est restée près du récif avec les ouvriers dormant à bord. Si une tempête menaçait, la goélette revenait dans le port de Cohasset jusqu'à la fin de la tempête.
Neuf trous ont été forés dans la roche, chacun d'une largeur de et d'une profondeur de . Huit ont été placés dans un cercle de de diamètre, avec le neuvième au centre. Des pieux de fer de de diamètre ont ensuite été collés dans chaque trou. Au forage, quatre hommes travaillaient par tranches de 20 minutes à partir d'un support triangulaire reposant sur de lourds espars, qui soutenaient une plate-forme très au-dessus du récif sur lequel étaient installés les équipements de forage.
Tous les matériaux ont été balayés du rocher par deux tempêtes différentes au cours de l'été 1847. Des ouvriers ont été envoyés plusieurs fois dans la mer, mais aucun n'a été noyé. Les travaux devaient être arrêtés pour l'hiver d' et repris au printemps de 1848, mais en septembre de la même année, les neuf trous avaient été forés et les neuf pieux de fer placés. Les pieux extérieurs sont orientés vers le centre sur une circonférence de , à au-dessus de la surface inégale du récif. Celles-ci étaient maintenues horizontalement par des tiges de fer à intervalles de . Les étais prévus pour renforcer la partie inférieure de la tour ont été omis sur la théorie selon laquelle ils diminueraient plutôt que d'accroître la sécurité globale de l'édifice. Cependant, c'est là que ces attelles étaient prévues, et que la structure s'est cassée plus tard.
Un coffrage en fonte, pesant 5 tonnes était fixe au sommet de cet empilage. Les quartiers du gardien ont été érigés au-dessus. Enfin, une salle de lanterne à 16 côtés tout en haut abritait une lentille de Fresnel, avec 15 réflecteurs. La lumière, une balise fixe avec un arc de 210°, a été allumée pour la première fois le .
Le premier gardien, Isaac Dunham, était convaincu que la structure légère n'était pas sûre et a écrit à Washington pour demander son renfort. En l'absence de décision, il démissionne le . Le capitaine John W. Bennett, qui lui succède, se moque ouvertement des craintes de son prédécesseur. Il a embauché de nouveaux assistants, notamment un Anglais nommé Joseph Wilson et un Portugais nommé Joseph Antoine. Deux gardiens restaient au phare en tout temps.
Les renforts de la structure montraient rapidement des signes de tension et devaient constamment être enlevés, emmenés sur le continent, puis renforcés et redressés. Quelques semaines après sa prise de charge, une terrible tempête a changé l'avis de Bennett qui a officiellement signalé la tour en danger. Un comité, chargé d'enquêter, est arrivé alors que la mer était parfaitement calme et est rentré à Boston, décidant que rien ne devrait être fait.
Le , au cours d'une autre tempête terrible, les gardiens décidèrent que la salle de la lanterne était dangereuse et se réfugièrent dans le magasin où ils se mirent à se recroqueviller pendant quatre jours et quatre nuits, ne montant que de temps en temps à la lanterne pour réparer les dégâts causés par la tempête. Les violents mouvements de tangage et de balancement de la tour les ont presque fait tomber des barreaux de l'échelle, quand ils l'empruntaient. Une période relativement calme a suivi au cours de laquelle les attelles ont été resserrées.
Puis le vent d'est a commencé à souffler autour du . Trois jours plus tard, Bennett partit pour le continent. C'était la dernière fois qu'il vit ses deux assistants en vie. Lorsqu'il tenta de revenir le lendemain, une mer trop lourde coulait à Minots Ledge pour permettre sa tentative. La tempête s'intensifia et, le 16, causait des dégâts considérables à terre. À Minots Ledge, les deux assistants ont gardé la cloche de brouillard et les lampes allumées, mais peu avant 16 heures, ils ont lancé une bouteille contenant un message destiné au monde extérieur, au cas où ils ne survivraient pas. La marée haute à minuit a envoyé vague après vague à travers le cadre supérieur de la structure affaiblie. Ce qui s'est réellement passé alors ne sera jamais connu. Probablement vers 23 heures, le support central s'est complètement détaché, laissant la tour de la lanterne de 30 tonnes située au sommet, tenue par l'empilement extérieur. Puis, le , peu avant une heure du matin, le grand phare s'est finalement penché dans la mer. Un par un, les huit pieux de fer se sont cassés jusqu'à ce qu'il ne reste que trois. Les gardiens, réalisant probablement que la fin était proche, ont commencé à frapper furieusement sur la cloche du phare. Cela a été entendu par les habitants. La tour étant penchée, les supports restants cèdent et la grande tour s'enfonce dans l'océan.
Le corps de Joseph Antoine a été retrouvé plus tard à . Joseph Wilson a réussi à atteindre Gull Rock, le prenant probablement pour le continent, ou il est apparemment mort d'épuisement.
Entre 1851 et 1860, Minots Ledge était gardé par un bateau-phare. Les plans pour un nouvel édifice en pierre ont été élaborés entre-temps pour le tableau des phares par le brigadier général Joseph G. Totten. Le même endroit a été choisi et Barton Stone Alexander, des Ingénieurs des États-Unis, a commencé les travaux de construction en .
Le récif a dû être coupé pour recevoir les pierres de fondation et aucun cofferdam ordinaire n'était disponible. En juin, les vieilles fondations de la première tour ont été enlevées. Pendant ce temps, le granit a été coupé et assemblé sur l'île du Gouvernement à Cohasset, où se trouve la maison du gardien de phare. Sept blocs de granit devaient constituer la fondation. Des puits métalliques permanents, d'une hauteur de , ont été installés dans huit des trous dans lesquels l'ancien phare avait été construit, tandis que le trou central était laissé ouvert, afin de former une cavité pour le cercle de base. Plus tard, un puits pour l'eau potable a été construit à partir de cette cavité au milieu de la nouvelle tour.
La structure ancienne a disparu lors d'une violente tempête le , lorsque le voilier New Empire, qui a ensuite atterri à White Head, a percuté la tour provisoire et a démoli l'échafaudage en fer. Donc, au printemps de 1857, le travail a dû être recommencé.
La première pierre a finalement été posée le . Des cofferdams temporaires ont été construits en sacs de sable afin que les blocs de fondation posés à plus de de la surface de la marée la plus basse puissent être collés à la paroi rocheuse du récif. Un cerclage de fer entre les couches a maintenu les pierres de 2 tonnes séparées pendant le durcissement du ciment.
Les crédits totaux de 330.000 dollars ont été entièrement dépensés, à l'exception d'un léger excédent, dans la construction. À la fin de 1859, le trente-deuxième parcours, à au-dessus de la marée basse, avait été atteint et 377 heures de travail effectif de l'équipage avaient été consommées. La dernière pierre a été posée le . L'ensemble de la structure en granit a donc pris cinq ans, mais il lui manque un jour. Le nouveau phare a été terminé à la mi- et la lumière a été mise en service le . Cependant, la lumière n'a pas été allumée régulièrement avant le , lorsque Joshua Wheeler, le nouveau gardien, et deux assistants sont entrés en fonction.
La nouvelle tour de pierre a résisté à tous les orages qui ont suivi. Les ondes les plus fortes ne provoquent qu'une forte vibration. À certaines occasions, les mers ont balayé le sommet de la structure de sans plus de dommages que ceux causés par quelques fenêtres qui fuient ou par une lampe fissurée ou deux.
Le , une nouvelle lanterne clignotante est installée. La lumière a été rendue automatique en 1947. Aujourd'hui, sa lumière de 45.000 candelas, située à 26 mètres au-dessus de l'eau, peut être vue sur 24 kilomètres. Il a été mis en vente en 2009 en vertu de la .
Description
Le phare est une tour cylindrique en granit, avec une galerie et une lanterne de de haut. La tour est non peinte et la lanterne est grise. Il émet, à une hauteur focale de , huit éclat blanc de 0.3 seconde par période de 45 secondes. Sa portée est de 10 milles nautiques (environ 19 km).
Il est aussi équipé d'une corne de brume radiocommandée émettant un blast par période de 10 secondes.
Identifiant : ARLHS : USA-502 ; USCG : 1-0440 - Amirauté : J0360 .
Voir aussi
Notes et références
Lien connexe
Liste des phares du Massachusetts
Liens externes
Massachusetts Lighthouses
Lighthouses of the United States : Northern Massachusetts
Lighthouses of the United States : Southeastern Massachusetts
Massachusetts - ARLHS World List of Lights (WLOL)
Massachusetts - Online list of lights
Phare du XIXe siècle
Scituate (Massachusetts)
Phare inscrit au Registre national des lieux historiques au Massachusetts
Registre national des lieux historiques dans le comté de Norfolk (Massachusetts)
Registre national des lieux historiques en 1987
Historic Civil Engineering Landmark au Massachusetts | {
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# Heißkalte Bekanntschaft
## Annett Bedford
Herrlich glitzernd lag der See vor ihr. Nadja klopfte ihre Hände, die in dicken Fäustlingen steckten, aneinander. Trotz der wärmenden Handschuhe hatte sie klamme Finger. Es war klirrend kalt. Nadjas Atem dampfte in der eisigen Luft. Endlich wieder Schlittschuh laufen!, dachte sie voller Vorfreude, setzte sich auf eine verwitterte Parkbank und zog ihre Schuhe aus. Sofort machte sich die Kälte auch an ihren Zehen bemerkbar und ohne viel Zeit zu verlieren, schlüpfte die junge Frau in ihre Schlittschuhe. Der See trug dickes Eis; es hatte seit Tagen schon Frost gegeben und nun wollte Nadja endlich mal wieder ihre Kreise ziehen. Vorsichtig stakste sie über die zugefrorene Fläche, die am Rand mit leichtem Schnee bedeckt, aber in der Mitte bereits ganz freigeräumt worden war. Bestimmt hatten ein paar Jungs hier schon Hockey gespielt und mit großen Zweigen vorher die Fläche gefegt. Der erste Schritt war wie eine Befreiung. Nadja glitt schwungvoll über das Eis. Sie war in ihrem Element... Mit festem Schritt umrundete Andreas den See. Er war voller Zorn und stapfte wütend durch die winterliche Landschaft. Würde Greta ihm doch einmal richtig zuhören! Gerade hatte er einen überflüssigen und sehr lauten Streit mit seiner Exfreundin hinter sich gebracht. Während Andreas in Gedanken noch einmal die letzte halbe Stunde Revue passieren ließ, sah er aus den Augenwinkeln eine attraktive Frau über den See flitzen. Beseelt schien sie sich auf dem Eis zu bewegen, drehte anmutige Kreise. Ihre schlanke Gestalt und ihre langen dunklen Haare, die unter ihrer Mütze hervorschauten, fesselten Andreas' Blick. Wie eine Elfe kam sie ihm vor und das Zusehen lenkte ihn von seinem Ärger ab. Mit ihren langen Beinen schwebte sie elegant dahin. Aus den Augenwinkeln nahm Nadja eine Gestalt wahr. Sie blickte leicht über ihre Schulter und bemerkte einen großen gut aussehenden Mann, der am Rande des Sees stehen blieb und zu ihr herübersah. Er war dick eingemummelt, doch das, was Nadja unter der Schicht aus Anorak und Schal erahnte, kam ihr äußerst interessant vor. Sie verspürte ein anregendes Prickeln im Körper und gab sich nun bei ihren Schlittschuhkünsten noch mehr Mühe. Albern, dachte sie bei sich, wie ein Teenie. Aber es kam ihr auch spannend vor, zumal ihre letzte intensive Begegnung mit einem Kerl schon einige Monate zurücklag. Gekonnt zog sie ihre Bahnen und gab alles, was sie auf dem Eis zu bieten hatte. Plötzlich knackte es laut. Die Läuferin bekam einen riesigen Schreck. Noch einmal gab es ein krachendes Geräusch und die dicke Schicht unter Nadja schien aufzustöhnen. Bei dem Versuch, abzubremsen, stolperte sie und rutschte auf die Mitte des Sees zu. Wasser kam ihr entgegen. Wasser? Warum?, dachte sie noch und schlitterte durch eine große Pfütze. Unter Nadja brach das Eis... Erschrocken und wie gebannt beobachtete der junge Mann die Szene auf dem See. Noch bevor Nadja es realisiert hatte, erkannte Andreas die Gefahr. Die Frau würde einbrechen! In Sekundenschnelle rief er sich in Erinnerung, was er tun konnte. Sich vorsichtig auf dem Eis bewegen, bis er das Gefühl hatte, es würde ihn nicht mehr tragen, sich hinlegen und dann mit dem ganzen Körper zu der Unglücksstelle rutschen. Ein Stock! Er brauchte einen Stock. Fieberhaft suchte er hinter sich nach einem geeigneten Gegenstand. Als er sich umdrehte, sah er gerade noch, wie Nadja im Eis einbrach. Da! Etwas weiter entfernt entdeckte Andreas einen dicken Ast, griff ihn sich und eilte auf den gefrorenen See, um Nadja zu helfen. Diese war tatsächlich schon klatschnass und wurde von einer unsichtbaren Hand in die Tiefe des Sees gezogen. Nein, das durfte nicht sein! Die Kälte des Wassers nahm ihr den Atem, eine eisige Lähmung überfiel sie. Krampfhaft versuchte sie, sich an dem Rand des Eises festzuhalten, doch das Bedürfnis, einfach loszulassen, wurde immer größer. Ihre Panik wich einer schlafähnlichen Mattigkeit, da hörte sie eine Stimme. "Nimm, los, greif zu! Hierher sehen, hallo!" Nadja wollte gerade ihre Augen schließen, da kam wieder Leben in ihre eiskalten Glieder. Hoffnung! Sie sah einen Stock, der sich immer weiter auf sie zuschob, versuchte, danach zu greifen. Ihre steifgefrorenen Finger glitten ab. Das Eis krachte unter dem Körper ihres Retters. Sie würde auch ihm den Tod bescheren, wenn sie sich nicht mehr anstrengte. Dieser Gedanke ließ sie neue Kraft schöpfen. Mit letztem Willen erreichten ihre Finger den Stock, schlossen sich darum und ruckartig wurde ihr Körper nach vorn gezogen. Eine Hand legte sich fest um ihren Arm. Nadja fühlte, wie ihre Sinne schwanden. Jemand zog sie aufs Eis und schleifte sie vorsichtig hinter sich her. Sie konnte nicht mehr klar denken... Ein warmes Licht, hell und sonnig, drang durch ihre Lider. Vorsichtig öffnete Nadja ihre Augen. Ihr Nacken schmerzte etwas, als sie ihren Kopf drehte. Sie lag in einem Krankenhausbett. Was war passiert? Verdutzt wollte sie sich aufsetzen, da legte sich eine wohltuende Hand auf ihren Arm. "Hallo. Wie geht es dir?" Eine angenehme tiefe Stimme klang an ihr Ohr. Sie wandte sich ihr zu. Der Sprecher war ein gut aussehender Mann Anfang 30, der auf ihrer Bettkante saß und sie sorgenvoll betrachtete. Nadja fielen sofort seine großen braunen Augen mit den unverschämt langen Wimpern auf. Es war der Mann vom See! Dann zitterte sie. Ein inneres Frösteln verursachte Gänsehaut auf ihrem Körper – oder lag das etwa an dem jungen Mann neben ihr? "Ich weiß nicht. Mir ist noch ganz schön kalt. Was genau ist eigentlich passiert?" Nachdem Andreas in kurzen Worten wiedergegeben hatte, wie seine Rettungsaktion zustande gekommen war, wurde Nadja noch etwas blasser. "Dann haben Sie... ähm, hast du mir also das Leben gerettet? Wie kann ich das je wieder gutmachen?" "Indem du mit mir ausgehst, sobald du wieder fit bist. Ich heiße übrigens Andreas..." Zwei Tage später wurde Nadja aus der Klinik entlassen. Eigentlich sollte sie sich noch etwas schonen, aber ihr Verlangen nach einem Rendezvous mit ihrem Retter war so stark, dass sie alle Ermahnungen ihres Arztes in den Wind schlug. Zum Ausgehen fühlte sie sich tatsächlich noch zu wackelig auf den Beinen, aber sie hatte Andreas vorgeschlagen, ihn bei sich zu einem "Retter- Dinner" zu empfangen. Nun war es so weit: Den Tisch hatte sie festlich gedeckt, Kerzen brannten, der Wein war dekantiert. Im Hintergrund lief eine angenehme Musik, die die Kulisse in Nadjas Wohnung klangvoll atmosphärisch unterlegte. Die junge Frau selbst hatte sich ihre Haarpracht locker am Kopf festgesteckt. Kleine freche Locken kringelten sich wie zufällig an ihrem schmalen Hals entlang. Ihre Figur steckte in einem dunkelroten Kleid, dessen Farbe sehr gut mit Nadjas dunkler Haarpracht harmonierte. Aufgeregt und erwartungsvoll sah Nadja auf die Uhr. Jetzt müsste es gleich klingeln. Oft, vielleicht sogar zu oft hatte sie in den letzten Tagen an Andreas gedacht. Lag das an seiner Rettungsaktion? Nein, beruhigte sie sich. Er hätte ihr auch gefallen, wenn sie nicht im Eis eingebrochen wäre. Kurze Zeit später stand der Mann in ihrem kleinen Flur. Spontan fiel Nadja ihm zur Begrüßung um den Hals. Auch er nahm sie in den Arm, was sich, wie Nadja fand, verdammt gut anfühlte. Draußen tobte ein winterliches Unwetter und Andreas musste sich erst mal aus unzähligen Kleiderschichten schälen. Etwas verlegen hielt er Nadja ein kleines verpacktes Schächtelchen hin. Erfreut nahm sie es entgegen. "Lass uns erst auf deine Gesundheit anstoßen, bevor du es öffnest", bat er sie. "Gern. Komm und setz dich." Angemessen bewunderte der Gast den schön gedeckten Tisch und nahm daran Platz. Sie stießen an und Nadja begann neugierig das Geschenk auszupacken. Entzückt entnahm sie der Schatulle einen kleinen Engelsanhänger aus Silber. Andreas wurde plötzlich rot und sagte leise: "Weil ich nicht immer dabei sein kann, wenn du Schlittschuh läufst, dachte ich, du brauchst einen Schutzengel." "Aber Andreas, das ist superlieb, doch eigentlich müsste ich dir was schenken – und nicht umgekehrt." Nadja erhob sich und umrundete den Tisch. Als sie sich vorsichtig über ihn beugte, um sich mit einem kleinen Kuss zu bedanken, legte Andreas auf einmal seine Hände auf ihre Wangen und zog ihr Gesicht zu sich heran. "Ich finde dich toll!", flüsterte er heiser und küsste ihre weichen Lippen. Nadja schluckte. Das ging ja ganz schön flott, fand sie. Doch der zarte Kuss gefiel ihr und erzeugte sofort ein lang vermisstes Kribbeln in ihrem Unterleib. Sanft küsste sie zurück, was gleich etwas fordernder erwidert wurde. Ihre Lippen verschmolzen miteinander und sachte bahnte sich Andreas' Zunge einen Weg in Nadjas Mund. Er zog Nadja auf seinen Schoß. Aus dem anfänglichen Kribbeln in Nadjas Körper wurde langsam ein Feuer, das während der intensiven Züngelei sofort an Kraft gewann. Atemlos löste sie sich nach einigen Minuten. "Sind wir nicht ein bisschen zu schnell? Wir könnten vorher noch was essen..." "Wozu Zeit verlieren und warten?", antwortete Andreas, ebenfalls atemlos. Nadja musste sich eingestehen, dass auch ihr Verlangen nach mehr wuchs. Sie hatte gar keinen Hunger mehr, sondern verspürte pure Lust. Lust auf einen Mann, der ihr zwar das Leben gerettet hatte, den sie aber kaum kannte. Andreas' Mund wanderte über ihren Hals und entlockte seiner Partnerin die ersten leisen Seufzer. In den Lenden pochte es und Andreas spürte, wie sein Penis sich bereits aufrichtete. Während seine Lippen langsam über den Ausschnitt des Kleides glitten und sich einen Weg zu Nadjas bereits abstehenden Brustwarzen suchten, fühlte sie unter ihrem Po, wie sein Bester wuchs. Andreas knabberte an ihrem Busen. Die junge Frau bewegte ihr Hinterteil vorsichtig vor und zurück und rieb so den harten Freund in Andreas' Hose. Das war ja ein ordentlicher Freudenspender! Zwischen Nadjas Schenkeln sammelte sich die Feuchtigkeit. "Wo steht dein Bett?", fragte Andreas leise keuchend. Er stand auf, nahm Nadja auf seine Arme und trug sie nach ihren Anweisungen in ihr Schlafzimmer. Dort angekommen, sank er mit seiner süßen Last auf die Liegewiese. Kundig glitten seine Finger über ihren Körper. Nadja ließ ihre Hände ebenfalls wandern und machte sich nach kurzer Zeit an seiner Gürtelschnalle zu schaffen. Geschickt öffnete sie Knopf und Reißverschluss und bewegte ihre Finger vorsichtig am Rand seiner Boxershorts entlang. Mit gewaltiger Wucht nahm ihre Lust zu. Am liebsten hätte sich Nadja sofort auf ihren Retter gestürzt, aber sie wollte ihn auch nicht verschrecken. Andererseits war Andreas derjenige, der damit angefangen hatte. Nadja wischte ihre Bedenken beiseite. Als sie Andreas' Bauchbehaarung unter ihren Fingerkuppen spürte, gab es für sie kein Halten mehr. Flink rutschte sie auf ihrem Bett etwas weiter runter und beugte sich über den noch versteckten Penis. Sie strich über die große Wölbung unter dem Stoff. Andreas rekelte sich stöhnend unter ihrer Hand. Sie holte das gute Stück hervor – oh, damit würde sie jede Menge Spaß haben! Nadja begann, das Liebeszepter sanft zu massieren und während Andreas lauter stöhnte, verstärkte sie ihren Griff. Dann stülpte sie gekonnt ihren Mund über seine Eichel. Andreas bäumte sich auf. Nadja ließ ihre Zunge über seinen Penis tanzen und be- merkte dabei, wie sie immer feuchter wurde und ihre Schamlippen anschwollen. Plötzlich packte Andreas ihre Hand und zog sie zu sich. "Stopp!", hechelte er. "Noch 30 Sekunden länger und ich kann mich nicht mehr beherschen." "Aber das ist doch schön!", gurrte Nadja zurück. Weiter kam sie nicht, denn ihr Retter verschloss ihr mit einem heißen Kuss den Mund. Die Hand des Mannes strich an Nadjas Silhouette entlang, fasste unter den Rock ihres Kleides und schob seine Finger in ihren winzigen Slip. Dann fand seine Hand ihren Weg in Nadjas Schritt. Vielversprechende Nässe erfühlend, konnte er sich nur mit Mühe zurückhalten. Seine Finger verschafften sich Eintritt zu Nadjas Lustpforte und während die Lady sich unter seiner Verwöhnattacke wand und stöhnte, streifte Andreas sanft ihre Schamlippen, um dann mit seinem Finger in Nadja einzudringen. Rhythmisch bewegte er seine Hand. Sein Daumen massierte dabei ihre Perle und Nadja wurde heiß und kalt. "Ohhh, jaaa, das... ist... guuut!", stöhnte sie. Mit seiner anderen Hand knetete Andreas Nadjas Po. Es dauerte nicht mehr lange, da versteifte sie sich, keuchte laut: "Nicht aufhören! Schneller!", und Andreas dachte, sein Bester müsste vor Wollust bersten. Als Nadja ihrem Orgasmus entgegenschrie, konnte er sich kaum noch bremsen... Nach Luft ringend, schmiegte sich Nadja an ihren Lover. Ihre Hand legte sich zielstrebig in seinen Schritt und griff sich seinen Besten, der ihr hart und stramm entgegenragte. Nadjas warme Finger umschlossen den starken Phallus und verwöhnten ihn durch ein sachtes Auf und Ab. Dabei sorgte sie dafür, dass ihre Finger keine zu große Reibung erzeugten, sondern ihre Zuwendung eher wie intensives Streicheln ausfiel. Andreas war kurz davor, durch die Decke zu gehen. Selten zuvor war er so geil darauf gewesen, in eine Frau einzudringen. Gerade als er dachte, er würde den Verstand verlieren, ließ Nadja von ihm ab. Langsam zog sie ihn aus. Als er nackt vor ihr lag, begann sie, das Areal rund um seinen Penis mit Küssen zu übersäen. Zart streifte ihr Mund über seine Oberschenkel und umkreiste seinen Bauchnabel. Am liebsten hätte Andreas sie gepackt und mit seiner glühenden Lanze beglückt. Obwohl es ihm schwerfiel, übte er sich in Geduld. Nadjas weiche Lippen umfingen erneut seine Eichel und saugten sanft daran. Dann glitt ihr Mund sachte an seinem Stamm hinab. Andreas krallte sich am Bettlaken fest. Nadja war ziemlich gut in dem, was sie gerade tat. Wieder spürte er, wie sich alles in seinem Körper auf dieses eine Gefühl konzentrierte. Ihm trat der Schweiß auf die Stirn. Er musste sich noch beherschen. Doch dann stoppte Nadja in ihren Bewegungen und setzte sich schwungvoll auf das herrliche Zepter. Groß und prall füllte es Nadjas Inneres aus. Andreas stockte der Atem. Ihre Scheide umfing ihn weich, eng und nass. Er betrachtete die wohlproportionierten Kurven seiner Partnerin. Ihr herrlicher Busen wippte auf und ab, als Nadja ihren Körper auf seinem Unterleib hin und her schob. Während sie sich auf ihm bewegte, streichelten ihre Hände über ihre Brust und zwirbelten ihre Nippel, die daraufhin hart abstanden. Nadja wurde schneller. Sie massierte ihren Partner mit ihrer feuchten Höhle, ließ seinen Strammen herausgleiten, um ihn gleich darauf wieder einzufangen. Gemeinsam stöhnte das Paar seine Lust heraus. Die junge Frau verschaffte ihm eine nie gekannte Gier und massierte gleichzeitig ihre Perle an seinem Unterleib. Jetzt waren sie kurz davor, den Gipfel zu erklimmen. Nadja presste ihre Oberschenkel an Andreas' Körper. Eine unglaubliche Hitze ging von beiden Körpern aus, die sich in zunehmendem Tempo aneinanderrieben. Und als draußen ein Sturm einsetzte, kamen Nadja und Andreas gleichzeitig zu einem gewaltigen Höhepunkt...
# Schlüsselworte
## ✳Marie Sonnenfeld
Ihre Kleidung war fast vollständig von Schnee bedeckt und sie fror ein wenig, als sie am Abend nach ihrem langen Spaziergang durch den Schnee wieder vor ihrer Haustür stand. Anna liebte es, in jedem Jahr auf diese Art den Einbruch des Winters zu begrüßen. Sie genoss die Stille, die die Welt in dieser Zeit einzuhüllen schien. Seit einigen Minuten war ihre romantische Stimmung allerdings verflogen, denn sie wühlte verzweifelt in ihren Taschen und suchte nach ihrem Schlüssel. Wo war er nur? Sie war sich sicher, ihn in ihre rechte Jackentasche gesteckt zu haben, als sie vorhin aus dem Haus gegangen war. Dass er sich plötzlich nicht mehr finden ließ, konnte doch gar nicht sein! Wieder kramte sie mit klammen Fingern in ihren Taschen. Nein, nichts. Statt des Schlüsselbundes entdeckte sie inzwischen etwas anderes in ihrer Jackentasche: ein Loch. Die Naht war aufgerissen, ganz unten in der Ecke. Sollte der Schlüssel etwa dort hindurchgefallen sein? Unglücklich machte Anna sich wieder auf den Weg, die Strecke abzusuchen, die sie gerade gegangen war. Wenn er durchgerutscht war, musste er ja noch auf dem Gehweg liegen. Mittlerweile fror sie stärker und sie fürchtete, dass der frisch fallende Schnee ihren Schlüsselbund zudecken könnte. Sie stapfte durch das weiche weiße Element. Es knirschte unter ihren Füßen und die herunterwehenden Schneeflocken benetzten ihre dunklen langen Haare. Wenn ihr nur nicht so kalt wäre... Sie suchte eine weitere Stunde, ohne Ergebnis. Wieder vor der Haustür angekommen, rüttelte sie noch einmal ohne große Hoffnung am Griff. Die Tür blieb zu. Anna atmete tief durch und zog resigniert ihr Handy aus der Tasche. Schweren Herzens tippte sie gerade die ersten Ziffern der Auskunft ein, um sich zu einem Schlüsseldienst durchstellen zu lassen, als sie bemerkte, dass sich jemand neben sie stellte. Von ihrem Telefon sah sie hinunter auf eine Männerhand, die einen Schlüssel in das Schloss der Eingangstür steckte. Erfreut sah sie auf und blickte in das Gesicht ihres Nachbarn Oliver. "Oh, schön, dass du gerade kommst, dann kann ich mit dir hineinhuschen", sagte sie erleichtert zu ihm. Er erwiderte ihr Lächeln und hielt ihr die Tür auf, damit sie vor ihm das Treppenhaus betreten konnte. Sie traten ein paar Mal fest auf, um den gröbsten Schnee von den Schuhen zu lösen. "Kein Problem", sagte er dabei und fügte hinzu: "Was ist passiert? Schlüssel vergessen?" Anna antwortete ihm, dass sie den Schlüssel wohl verloren haben müsse und versuchte dabei, ihre nassen Haare auszuwringen. Dabei klapperte sie vor Kälte mit den Zähnen, worauf sein Blick über ihre durchnässte Kleidung und ihre nassen Haare glitt. "Vielleicht bekomme ich die Tür auf. Wenn du möchtest, versuche ich es gern." Sein Vorschlag klang verlockend und da Anna ohnehin sehr fror und das nüchterne Treppenhaus mit seinem kalten Kunstlicht nicht besonders einladend und gemütlich anmutete, um auf den Schlüsseldienst zu warten, nahm sie sein Angebot gern an. Oliver wohnte noch nicht sehr lange in Hamburg. Er war wegen einer lukrativen Arbeitsstelle hierher gezogen und hatte sich mit seinen Siebensachen vor einigen Monaten häuslich in der ersten Etage dieses hübschen Altbaus eingerichtet. Bereits vom ersten Tag an war ihm dabei seine attraktive Nachbarin, die ihre Wohnung visà- vis hatte, aufgefallen. Er empfand eine große Sympathie für sie und immer, wenn er sie auf dem Flur sah, schlug sein Herz ein wenig schneller. Es machte beinahe Luftsprünge, wenn sie ihn dazu noch auf ihre freundliche Art anlächelte. Ja, und jetzt stand sie durchgefroren, nass und hilfebedürftig vor ihm. Genau das sprach etwas in ihm an und in diesem Moment breitete sich in seinem Bauch ein warmes Gefühl aus. Oliver zog sich die dunkelblaue Fleecemütze vom Kopf und gemeinsam gingen sie die Treppe hinauf. Als sie oben ankamen und vor Annas Tür standen, schaute er noch einmal unauffällig aus dem Augenwinkel zu ihr herüber, während er seine Geldbörse hervorholte. Süß sah sie aus, wie sie sich noch immer in ihre Hände hauchte und sich die Nässe aus ihrem Gesicht wischte, die von ihren Haaren hinuntertropfte. Oliver konnte nur mit Mühe seinen Blick wieder abwenden, zu gern sah er sie an. Es gelang ihm aber doch und er griff in sein Portemonnaie, um seine Scheckkarte hervorzuziehen. Dann dauerte es eine Weile, in der er immer wieder verschiedene Möglichkeiten probierte... Anna sah ihm fasziniert dabei zu. Sie schaute ihn an, wie er dort vor ihrer Tür hockte, und betrachtete ihn eingehender. Warum nur war ihr bis jetzt nicht aufgefallen, wie gut aussehend er eigentlich war? Seine dunklen Haare waren dicht und gut geschnitten, sein Blick konzentriert aus tiefgründigen blauen Augen. Auch fielen ihr seine breiten Schultern heute zum ersten Mal so richtig deutlich auf. Sie konnte es kaum fassen, dass ihr bisher entgangen war, welch attraktiver Nachbar ihr hier gegenüberwohnte. Plötzlich gelang ihm, was er bisher umständlich versucht hatte, und mit einem lauten 'Klack' sprang ihre Tür auf. Anna war beeindruckt. Sie sah ihn noch immer an, staunte und bedankte sich mit einem glücklichen Lächeln bei ihm. Oliver lächelte zurück, zwinkerte ihr freundschaftlich zu. Wenn es mit der Tür zu ihrem Herzen auch so einfach ginge... Oliver hielt ihrem Blick stand. Dabei nahm er wahr, dass es aus ihrer Wohnung vorweihnachtlich nach Orangen, Lebkuchen und Zimt duftete. Es gefiel ihm und versetzte ihn in eine zärtliche Stimmung. Kurz dachte er an seine eigene Junggesellenbude, in der es solche schönen Düfte nicht gab, und hatte in diesem Moment so gar keine Lust, dorthin allein zurückzukehren. Fast magisch fühlte er sich von der Heimeligkeit in Annas' Wohnung und dem tiefen Blick aus ihrem süßen Gesicht angezogen. "Darf ich dich zum Dank auf eine Tasse Adventstee einladen?" Anna fragte ihn mit einem liebevollen Klang in ihrer Stimme. Und ob er wollte! Nur zu gern nahm er ihre Einladung an. Er trank zwar keinen Tee, aber das war ihm in diesem Moment vollkommen gleichgültig. Nur noch nicht von Anna weg, sie noch weiter genießen, einzig darum ging es ihm. Sein Puls ging schneller, als Anna in ihre Wohnung eintrat, den Lichtschalter betätigte und ihn aufforderte, ihr zu folgen. Beide zogen im Flur ihre Winterstiefel aus und Anna bat ihn, doch schon einmal in ihrem Wohnzimmer Platz zu nehmen, während sie sich noch kurz abtrocknen und umziehen wollte. Er machte sich Licht und setzte sich. Vom Sessel aus konnte er sehen, dass Anna bereits im Flur ihr Oberteil und ihre feuchte Jeans auszog und nur in Unterwäsche in das Bad ging. Wenige Sekunden später hörte er ihren Föhn. Oliver schloss seine Augen und rutschte tiefer in den Sessel hinein. Wie hinreißend sie gerade war, als sie sich auszog. Wie bildschön ihr schlanker Körper in ihrer hübschen Wäsche aussah... Wenn er sich vorstellte, dass sie in diesem Augenblick nur in ihren sexy Dessous im Bad stand, fühlte er, wie sehr ihn dieser Gedanke erregte. Er sah sie vor seinem inneren Auge deutlich vor sich, sah ihre prallen Brüste in der Spitze ihres BHs eingebettet und ihren schlanken Po, der sich verführerisch in den weichen Seidenstoff ihres Slips schmiegte. Oliver konnte nicht verhindern, dass sein Penis sich aufzurichten begann und seine Lust auf Sex deutlich zunahm. Wie gern würde er sie berühren, ihren Körper auf diese besonders lustvolle Art wieder aufwärmen. Mit der Innenseite seiner Hand strich er über die Ausbeulung seiner Jeans und stöhnte dabei beinahe lautlos auf. Die Wärme des Föhns tat ihr gut und so genoss Anna es, sich von der warmen Luft trockenpusten zu lassen. Davon abgesehen, freute sie sich auf die Gespräche und die Tasse Tee mit ihrem wundervollen Nachbarn, der im Wohnzimmer auf sie wartete. Ob er ihr gleich auch wieder so bedeutungsvoll in die Augen schauen würde? Was er wohl für sie empfand? Anna war voller Vorfreude und derart tief in Gedanken versunken, dass sie erst bemerkte, dass Oliver ihr Bad betreten hatte, als sie schon seine Hände fühlte, die sich sanft von hinten um ihre Taille legten. Kurz zuckte sie zusammen, dann schaltete sie den Haartrockner aus und drehte sich zu ihm um. Anschauen konnte sie ihn allerdings nicht, da er sein Gesicht bereits in ihrer Halsbeuge vergraben hatte und zärtlich ihre Schulter küsste. "Sorry, nicht erschrecken", murmelte er dabei und fügte hinzu: "Ich konnte nicht widerstehen, du siehst einfach toll aus, Anna!" Sie legte ihren Kopf auf die Seite und berührte so mit ihrer Wange sein Haar. Es fühlte sich wunderbar weich und warm an. Dann drehte sie sich zu ihm um. Oliver nahm ihr Gesicht in seine Hände und streichelte mit seinen Daumen über ihre Wangen. Dabei flüsterte er: "Weißt du, dass ich für dich schwärme, seit ich hier eingezogen bin? Vom ersten Tag an, Anna." Das überraschte sie wirklich, das hatte sie nicht gewusst. Sie schaute ihn verwundert an. Dass sie das nicht bemerkt hatte! Viel Zeit, um darüber nachzudenken, gab er ihr aber nicht mehr, denn in diesem Moment näherte er sich bis auf wenige Zentimeter und stupste mit seiner Nase zärtlich an ihre. Es war nur ein leises Flüstern, welches Oliver vernahm, als Anna ihm sagte, dass sie ihn ebenfalls sehr mochte. Daraufhin verschmolzen sie in einem langen sanften Kuss. Seine Lippen liebkosten dabei ihre und seine Zunge lud zu einem neckischen Spiel mit ihrer ein, worauf sie sich gern einließ. Es war ein Austausch von Wärme und Gefühl, was beide mit sinnlicher Lust durchflutete und sie aufforderte, sich einander zu öffnen und nur allzu bereitwillig aufeinander einzulassen. Sie stöhnten leise, als sie sich zärtlich küssten und streichelten. Als Anna dabei ihre Hüfte bewegte, um ihr Gewicht von einem Bein auf das andere zu verlagern, bemerkte sie seine beachtliche Erektion durch seine Jeans hindurch. Sie fasste sie als ein Kompliment ohne Worte auf. Gleichzeitig törnte es sie sehr an, ihn derart erregt zu erleben. Sie schmusten noch eine ganze Weile im Stehen weiter, bis Oliver leise stöhnend in Annas Ohr flüsterte: "Du bist unglaublich verführerisch, Anna. Und ich habe das Gefühl, dich schon ewig zu kennen. Wenn du wüsstest, wie oft ich mir schon vorgestellt habe, mit dir noch viele Schritte weiterzugehen..." Anna schmiege sich an ihn. "Oh, das ist schön, Oliver, unwahrscheinlich schön!" Ohne auf ihre Worte einzugehen, presste er sie erhitzt und voller Begehren an seinen Körper und bat keuchend: "Bitte schlafe mit mir, Anna, bitte!" Sie sah in seine Augen, sah seinen Wunsch darin brennen und nahm ihn wortlos an die Hand. Annas Lust war inzwischen ebenfalls übermächtig geworden und so war auch für sie der Gedanke, mit Oliver zu verschmelzen, überaus reizvoll. Sie ging mit ihm in ihr Schlafzimmer, wo sie begann, ihn langsam auszuziehen. Sie küsste und streichelte ihn dabei, was Oliver hingebungsvoll erwiderte. Jeder spürte die Hände des anderen weich und forschend auf seiner Haut und immer wieder küssten sie sich innig. Als sie dann nackt unter Annas Bettdecke eng aneinandergeschmiegt schmusten und Oliver fast außer sich vor Lust ihre Brustwarze liebkoste, fand ihre Hand den Weg zu seinem pulsierenden Penis. Heiß und hart reckte er sich ihr entgegen und zärtlich nahm sie ihn in ihre Hand. Im gleichen Moment hörte sie Olivers lustvolles Stöhnen und fühlte seine Hüfte sich ihr entgegenstemmen. Sein Saugen wurde intensiver, worauf ihre Knospe noch härter wurde. Dieses Gefühl strich mit ungeahnter Heftigkeit durch Annas Körper und unwillkürlich stöhnte auch sie auf. Jetzt wagte sich auch Oliver an ihr Intimstes vor, jetzt traute auch er sich, die Pforte zu ihrem Paradies zu öffnen. Nur mit seinen Fingerspitzen streichelte er sich von ihren Schamhaaren abwärts in feuchtere Regionen. Er fühlte die Hitze überdeutlich, die von dort ausging und ihn in eine neue Dimension der Erregung versetzte. Fast hielt er es nicht mehr aus, nur mit Mühe konnte er sich noch zusammennehmen, nicht sofort und ohne zu zögern in Annas feuchte Venus einzudringen. Sie spreizte ihre Schenkel weit für ihn. Oliver schob sich halb auf sie, und zwar so, dass ein Oberschenkel schon zwischen ihren lag und seine Eichel die Innenseite ihres Schenkels streichelte. Olivers Finger fand ihre kleine harte Perle und verspielt kreiste er immer schneller auf ihr. Anna stöhnte und begann sich unter ihm zu winden. Dabei sagte sie ihm, wie groß ihre Lust auf ihn sei und wie sehr sie sich wünschte, ihn endlich in sich zu fühlen. Oliver hob sich auf ihren weichen warmen Körper. Sich auf seinen Ellenbogen abstützend, beugte er sich zu ihrem Gesicht herunter und küsste sie wieder liebevoll, als er gleichzeitig in sie eindrang. Immer weiter, immer tiefer wagte er sich in ihr vor. Es wurde eng um seine Eichel und seinen Schaft und das heiße feuchte Gefühl von Annas enger Scheide nahm beinahe unerträglich erregende Ausmaße an. Oliver stöhnte laut auf und sein Griff um ihren Kopf wurde kurz fester. Sie genoss das Gefühl der Geborgenheit sehr, welches seine Hände, die ihren Kopf hielten, in ihr auslösten. Anna keuchte. Auch für sie war es ein gigantisches Gefühl, von ihm derart wundervoll ausgefüllt zu werden. "Es ist phantastisch, wie genau du hineinpasst, Oliver", raunte sie ihm daher rau entgegen, worauf er nur ein "Oh ja, Anna!" stöhnen konnte. Sie fanden schnell in einen Rhythmus, der sie unaufhaltsam nach vorn trieb und der ihnen Töne entlockte, wie ein Mensch sie wohl nur in Momenten höchster Lust von sich gibt. Sie schliefen voller Gier und Hingabe miteinander und hatten auch hier wieder das Gefühl, schon seit Ewigkeiten eine verschmolzene Einheit aus Vertrauen und Liebe zu sein. Erbarmungslos und von großem Verlangen getrieben, erreichten beide bald ihren Höhepunkt, wobei Oliver versuchte, darauf zu achten, dass Anna vor ihm kam. Es gelang ihm nur mithilfe der letzten Reserven seiner Selbstbeherrschung, die gerade so lange vorhielten, bis Annas kleine harte Klit unter ihm zu zucken und zu beben begann. In dieser Sekunde, in der er es fühlte und Anna beinahe animalisch aufschrie, brachen auch bei ihm alle Dämme. Sein Orgasmus fiel mit der Heftigkeit eines Tornados über ihn herein und er ergoss sich heiß und lang anhaltend. Dabei presste er sich tief in Annas seidige Nässe hinein, die ihn noch immer warm und zuckersüß umfing. Als Oliver sich einige Minuten später wieder aus ihr herauszog und sie dabei glücklich anlachte, strich Anna ihm durchs Haar und fragte ihn, ob er wohl immer noch Lust auf einen Tee habe. Oliver setzte sich neben sie, streichelte ihren Bauch und gestand ihr zaghaft, gar kein besonders großer Tee-Fan zu sein. Dabei grinste er charmant, was Anna auch ein Lächeln entlockte. Sie knuffte ihn dafür fest in die Seite, was der Anfang einer verspielt-verliebten Balgerei auf dem Bett war, die atemlos in den Armen des anderen endete. Ohne ein Wort, aber mit einem tiefen Blick in die Augen, verloren sie sich erneut in einem Kuss, der die Welt anzuhalten schien. Kurze Zeit später hielt jeder einen dampfenden Becher Kaffee in den Händen, von dem sie, in die weichen Kissen zurückgelehnt, vorsichtig schlürften. Gut gelaunt und sich viel aus ihrem Leben erzählend, naschten sie dazu von den Lebkuchen und den Plätzchen, die Oliver schon vorhin mit ihrem würzigen Duft vollkommen eingenommen hatten. Und würde unser frischverliebtes Pärchen nur ein einziges Mal den Blick voneinander abwenden und zum Fenster sehen, würden sie bemerken, dass noch immer die Schneeflocken zu Boden fielen und die Welt in ein weißes Wintermärchen verwandelten.
# Heiße Liebe
## Jenny Prinz
Wie eine Wand aus Hitze schlug mir die Saunaluft entgegen, als ich die Tür öffnete. Ich brauchte einige Sekunden, um mich daran zu gewöhnen, aber dann genoss ich es in vollen Zügen. Zum Glück war ich allein, obwohl alle Zimmer ausgebucht schienen. Vielleicht waren die meisten Touristen schon unterwegs. Ich legte mein Handtuch auf eine der mittleren Holzbänke, setzte mich darauf und spürte förmlich, wie langsam die Entspannung eintrat. Ja, die kommenden zwei Wochen würden nur mir gehören. Ich konnte kaum in Worte fassen, wie ich mich auf zwei Wochen Schnee, Natur und finnische Gastfreundschaft freute... Obwohl – wenn man es genau nahm, gehörte auch eine Portion deutsche Gastfreundschaft dazu, nämlich die meiner Freundin Ulrike. Ulrike hatte sich vor vier Jahren im Sommerurlaub in einen Finnen, Daavid, verliebt und ihn prompt geheiratet. Seitdem lebte sie mit ihrem Mann hier in der Nähe des Pielinen-Sees und betrieb ein kleines Gasthaus mit mehreren Apartements. Und seitdem bestand sie auch darauf, dass ich regelmäßig Gast in ihrem Hause war. Ihr fehlte der Kontakt zu Deutschland und trotzdem sie hier wirklich glücklich war, konnte sie es kaum erwarten, wenn sich eine ihrer Freundinnen aus der Heimat ankündigte. Die Geburt ihres Sohnes hatte dazu geführt, dass sie selbst im Moment überhaupt nicht mehr ans Reisen denken konnte. Entsprechend begeistert war sie, als ich ankündigte, meinen Geburtstag bei ihr in Finnland feiern zu wollen. Ich fand es von jeher deprimierend, mitten im Winter geboren zu sein; Grillfeste und Gartenpartys gab es für mich nie. Stattdessen beschloss ich in diesem Jahr, meinen Geburtstag wenigstens richtig winterlich zu feiern – mit viel Schnee, Skilanglauf und der berühmten finnischen Sauna. Und ich musste zugeben, ich war wahnsinnig gespannt darauf, den kleinen Viljo persönlich kennenzulernen. Ich beneidete Ulrike um ihre Familie. Für mich war der Richtige bisher noch nicht aufgetaucht, dabei wünschte ich mir nichts sehnlicher, als einen lieben Ehemann und vielleicht später auch eigene Kinder zu haben. Meine Freundin hatte mit ihrem Daavid jedenfalls einen richtigen Glücksgriff getan... Langsam liefen die Schweißtropfen an meinem nackten Körper herab. Es war schon eine Weile her, seit ich das letzte Mal in einer Sauna war, und ich wusste nicht, wie lange ich es heute aushalten würde. Aber im Moment empfand ich die Wärme, die meine Muskeln lockerte, noch als überaus angenehm. Während ich so meinen Gedanken nachhing, fragte ich mich auch, welche Überraschung meine Freundin und ihr Mann wohl für mich vorbereitet hatten. Ich dachte an Ulrikes geheimnisvolle E-Mail von letzter Woche, die wohl einzig den Zweck hatte, mich neugierig zu machen: Deine Geburtstagsüberraschung ist organisiert. Du wirst vielleicht Augen machen! Bin gespannt, wie es Dir gefällt. Freue mich jedenfalls wahnsinnig auf Dich. Tervehdyksiä, Ulrike Gedankenversunken und mit geschlossenen Augen bemerkte ich erst, dass ich nicht mehr allein war, als jemand schräg neben mir Platz nahm. Ich spürte die Bewegung mehr, als das ich sie hörte. Ich öffnete die Augen, um mir den neuen Besucher anzusehen, und war im ersten Moment verwirrt. Ich starrte in das schönste Gesicht, dass ich jemals gesehen hatte. "Anteeksi", entschuldigte sich mein Gegenüber. "Ich wollte dich nicht stören." In meinem Bauch breitete sich ein Kribbeln aus. Dieser Mann war einfach umwerfend! Ich war so von seinem Anblick gefesselt, dass ich kaum registrierte, dass er mich auf Deutsch angesprochen hatte. Sein Gesicht war herzförmig, mit vollen Lippen, hinter denen jetzt schneeweiße Zähne zum Vorschein kamen. Er lächelte mich an. Seine Augen waren dunkelbraun, doch ich glaubte, grüne Sprenkel darin zu erkennen. Braune Locken hingen ihm in die Stirn und umrahmten sein hübsches Gesicht. "Du störst nicht", antwortete ich endlich. "Ich war nur so in Gedanken..." Und damit endete unsere Unterhaltung auch schon wieder. Ich wusste nichts zu sagen, betrachtete nur wortlos seinen sehr schlanken Körper. Unter der Haut zeichneten sich die Muskeln deutlich ab. Das Kribbeln in meinem Bauch wanderte ein Stückchen tiefer, ich spürte, wie sich Lust in mir regte. Wie lange war es her, dass ein Mann diese Wirkung auf mich gehabt hatte? Mir war klar, dass es für Finnen völlig selbstverständlich ist, nackt nebeneinander in der Sauna zu sitzen. Und Finne war er, das war offensichtlich. Andererseits war es wohl mehr als unhöflich von mir, ihn so zu betrachten. Ich hatte das Gefühl, dass es mir nicht zustand, ihn derart zu mustern, und doch konnte ich die Augen nicht von ihm abwenden. Der Mann hatte die Augen geschlossen und schien einfach nur die Hitze zu genießen. Ich hatte ja kaum etwas gesagt und auch ihm schien nicht viel an einer Unterhaltung zu liegen. Mein Blick wanderte immer wieder in seinen Schoß. Er hatte einen schönen Penis, groß, aber nicht zu groß. Ich erwischte mich bei der Überlegung, wie er wohl aussehen würde, wenn er aufgerichtet war. Wie es sich anfühlen würde, ihn zu berühren? Das Prickeln in meiner Venus verstärkte sich. Ich stellte mir vor, wie er reagieren würde, wenn ich ihn in die Hand nahm. Wie sich sein Phallus unter meinen Händen mit Blut füllen und hart und fest werden würde. Ich wollte über die seidigweiche Haut streichen, die samtige Eichel mit meinen Fingerspitzen liebkosen. Langsam breitete sich Feuchtigkeit in meiner Mitte aus. Es war verrückt, aber der Gedanke daran, diesen Mann zu berühren und von ihm berührt zu werden, jagte mir einen Schauer über die Haut. Und plötzlich, mitten in meinem Tagtraum, bemerkte ich, dass er die Augen geöffnet hatte und mich ansah. Erwischt! Meine Wangen brannten, und das lag definitiv nicht an der Hitze der Sauna. Ich fühlte mich auf einmal mehr als unwohl, denn sein Lächeln sagte mir, dass er meine Gedanken lesen konnte. Er wusste, dass ich ihn angestarrt hatte und es war wohl überdeutlich, in welche Richtung meine Überlegungen gingen. Abrupt und ohne ein Wort stand ich auf und verließ den Raum. Erst hinterher merkte ich, dass dies die Situation wohl noch alberner erscheinen ließ, als sie ohnehin schon war. Ich hätte mich ohrfeigen können. Als ich mich zurechtgemacht hatte, ging ich hinüber zu Ulrike und Daavid. Inzwischen hatte ich mich wieder gefangen, hoffte aber inständig, diesem anderen Gast nicht so bald wieder über den Weg zu laufen. Stattdessen wollte ich meine Freundin fragen, ob sie Lust auf einen kleinen Schneespaziergang hätte. Ich wollte meinen Urlaub gemütlich beginnen. "Hyvä huomenta", begrüßte Daavid mich. "Guten Morgen", erwiderte ich ebenso und fragte nach Ulrike. "Ulrike musste noch was besorgen; aber hat Joona dich nicht gefunden?" "Wer ist Joona?", fragte ich völlig verdutzt. "Na, mein Bruder... Jetzt sag bloß, Ulrike hat dir nichts erzählt?" Daavid grinste mich breit an, als er mir in seinem holprigen Deutsch erläuterte, was mein Geburtstagsgeschenk sein sollte: eine Schlittenfahrt mit einem Husky-Gespann. Ulrike wusste seit Langem, wie gern ich das einmal mitmachen wollte, hatte mir aber mit keinem Sterbenswörtchen verraten, dass sie eine dreitägige Tour für mich gebucht hatte. Geschweige denn, wen sie als meine Begleitung auserkoren hatte, damit ich mich nicht so allein fühlen würde: Daavids Bruder. Ich kannte ihn noch nicht, nahm aber an, dass er ähnlich nett war wie Ulrikes Ehemann. Vor meinem inneren Auge entstand sofort das Bild eines blonden gemütlichen Teddybärs. "Und da Joona im Moment ohnehin hier bei uns ist, dachten wir, er könnte dir ein wenig die Zeit vertreiben. Schließlich hat Ulrike durch den Kleinen nicht so viel Zeit wie sonst für dich", schloss Daavid. Ich fiel ihm vor Freude um den Hals. Es war toll, wie viel Mühe meine Freunde sich für mich machten und dass ich tatsächlich einmal mit einem Rudel Schlittenhunde unterwegs sein sollte, war für mich etwas ganz Besonderes. Blieb jetzt nur die Frage, wo besagter Joona steckte; dann würde ich eben ihn mitnehmen auf meinen geplanten Spaziergang. Ich wollte ihn schließlich kennenlernen, bevor ich mich auf drei Tage mit ihm im Schnee einrichtete. Lachend machte Daavid sich los. "Da ist er doch." Er deutete zur Tür. Als ich mich umdrehte, wurden meine Knie weich. Lässig lehnte meine Saunabekanntschaft im Türrahmen, dick eingemummelt gegen die schneidende finnische Kälte, eine Wollmütze bis fast zu den Augenbrauen gezogen. "Wollen wir los?" Seine Augen blitzten und ich nickte verlegen. Als ich ihm folgte, wusste ich zuerst gar nicht, wohin. Aber Hauptsache weg von Daavid, der meinen vermasselten ersten Eindruck auf gar keinen Fall mitbekommen sollte. Ein Stückchen vom Haus entfernt blieb Joona stehen und sah mir in die Augen: "Lust auf einen Spaziergang?" Was war das nur, was ich in seinem Blick entdeckte? Ich nickte erneut. Ohne zu fragen griff er nach meiner Hand, die in einem dicken Handschuh steckte. Selbst durch den Stoff hindurch meinte ich, die Hitze seiner Haut zu fühlen. Ich versuchte, mich seinen Schritten anzupassen, denn ich hatte Angst, dass er mich wieder loslassen würde. Dieses Gefühl aus der Sauna war wieder da, fast noch stärker als zuvor. Seine Anwesenheit bescherte mir eine Gänsehaut, ein Kribbeln in meinem Unterleib und doch war es noch mehr. Trotz meiner Befangenheit fühlte ich mich wohl in seiner Nähe. Ich mochte, dass er so schweigsam war und mir gefiel es, wie selbstverständlich er mich berührte. Alles an ihm übte auf mich eine derartige Anziehungskraft aus, dass ich mir mit jedem Schritt nichts sehnlicher wünschte, als von ihm geküsst, gehalten zu werden. Meine Gedanken schweiften schon wieder ab, ich nahm kaum etwas von der herrlichen Winterlandschaft um uns herum war. Mein ganzes Denken war nur auf diesen Mann neben mir gerichtet, der mich zwischendurch immer wieder anlächelte. Und irgendwann hielt ich das Schweigen nicht mehr aus. Ich räusperte mich, bevor ich ansetzte: "Joona?" "Ja?" "Vorhin in der Sauna... das war..." Weiter wusste ich nicht. Plötzlich blieb Joona stehen und wandte sich mir zu. "Ist schon okay", flüsterte er leise mit seiner rauen Stimme. "Du hast mich auch vom ersten Moment an fasziniert." Und dann beugte er sein Gesicht ganz nah zu meinem und küsste mich. Etwas überrumpelt erwiderte ich seinen Kuss. Zärtlich berührten sich unsere kalten Lippen. Joona ließ sich alle Zeit der Welt, als er meine Mundwinkel küsste und dann mit seiner Zungenspitze die Konturen meiner Lippen nachfuhr. Bereitwillig öffnete ich meinen Mund. Unsere Zungen fanden sich und heiße Schauer liefen über meine Haut, während unsere Zungenspitzen den Mund des anderen erforschten und liebevoll miteinander spielten. Ich spürte, wie Joona seine Arme um mich legte. Mein Herz klopfte bis zum Hals, als er mich fest an sich zog. In dem Augenblick wünschte ich, dass wir nicht so dicke Winterjacken tragen würden, sondern ich seinen Körper besser fühlen könnte... Eine ganze Weile standen wir einfach so da, küssten uns und kuschelten uns aneinander. Jeden Zentimeter nackte Haut erforschte Joona mit seinem Mund, liebkoste meine Wangen, meine Ohren, meine Nasenspitze. Die Kälte kroch an mir hinauf, aber für nichts in der Welt hätte ich diese Umarmung gelöst. Mein ganzer Körper brannte vor Verlangen nach diesem Mann, mein Unterleib kribbelte, meine Venus war feucht und mehr als bereit. Ich wollte den Zauber des Augenblicks aber nicht zerstören; zu unwirklich kam mir vor, was hier gerade passierte. Ob ich ihn genauso erregte wie er mich? Als es sachte anfing zu schneien, hob Joona den Kopf und schaute in den Himmel. "Das wird hier langsam zu kalt, Liebes." Liebes? Hatte ich mich verhört? Ein warmes Gefühl durchflutete mich. Liebes klang gut... Liebes klang, als wenn das hier weitergehen würde. Ich hatte das Gefühl, zu ihm zu gehören, so seltsam es für andere auch klingen mochte. Die Nähe zu Joona war so herrlich, dass sie nur noch von einem anderen Empfinden übertroffen wurde: der Sehnsucht, ihn endlich ganz zu fühlen. Mit ihm zu verschmelzen. Meine Lust, die er in mir entfacht hatte, zu stillen. Unsere Blicke trafen sich und wieder einmal schien er mich wortlos zu verstehen. "Pällä!" Er gab mir noch einen kurzen Kuss, legte seinen Arm um meine Schultern und schlug den Rückweg ein. Wir gingen schnell. Meine Füße waren vor Kälte fast taub, doch das war mir egal. Ich wollte nur noch zurück, in mein Zimmer, allein sein mit Joona. Und Joona schien es genauso zu gehen. Ich werde nie das erste Mal mit Joona vergessen. Mit der gleichen Zärtlichkeit, die er schon beim Küssen an den Tag legte, widmete er sich nun meinem ganzen Körper. Schon nach wenigen Augenblicken hatte ich das Gefühl, es nicht mehr erwarten zu können, ihn endlich in mir zu spüren. Doch Joona hatte keine Eile. Jeden Zentimeter meiner Haut verwöhnte er mit Küssen und Streicheleinheiten. Und ich tat es ihm gleich. Sein Körper faszinierte mich über die Maßen und es war unglaublich erregend, zu sehen, wie heftig er auf meinen Mund und meine Finger reagierte. Sein Penis, den ich am Morgen in der Sauna noch heimlich bewundert hatte, sprang mir bereits hart und prall aufgerichtet entgegen, als ich Joona aus Jeans und Slip befreite. Sein heiseres Keuchen an meinem Ohr, als ich meine Hand um seinen Schaft legte und ihn zärtlich massierte, sein Aufstöhnen, als ich mit den Fingerkuppen sanft über die Eichel fuhr – diese lustvollen Laute elektrisierten mich und machten auch mich mit jeder Sekunde heißer. Als Joona das erste Mal meine inzwischen von einer einladenden Nässe überzogenen Lippen mit den Fingern teilte, schrie ich leise auf. Ich klammerte mich an ihn und genoss die sanften Bewegungen, mit denen er meine Venus erforschte und mich langsam, aber sicher zur Ekstase brachte. Doch dann hörte er plötzlich wieder auf. Joonas Zunge zeichnete feuchte Linien auf die sensible Haut meines Busens, sachte saugte er an meinen fest aufgerichteten Brustwarzen. Ich spreizte meine Beine, öffnete mich ihm, so weit es ging. Mein Unterleib drängte sich ihm entgegen, doch Joona ignorierte es; zu sehr genoss er meine Lust. Ich konnte fast keinen klaren Gedanken mehr fassen. Seine Lippen streichelten meinen Bauch, er küsste die weiche Haut unter dem Bauchnabel. Mit angehaltenem Atem wartete ich darauf, dass er endlich, endlich an meiner empfindlichsten Stelle ankam. Doch Joona ließ mich zappeln. Mit fast unerträglicher Langsamkeit reizte er die Innenseiten meines Oberschenkels und küsste zart meine Schamlippen. Ich spürte, dass ich mein Becken instinktiv anhob und er es mit einem leisen Lachen bemerkte. Meine Hände krallten sich in die bunte Tagesdecke unter mir. Ich konnte mich nicht erinnern, jemals so erregt gewesen zu sein. Dieser Mann raubte mir einfach den Verstand. "Joona", flehte ich... und dann fühlte ich seine weiche Zunge an meiner Perle und es war, als würde ich explodieren. Nur wenige Sekunden genügten und ich stöhnte laut auf. Weiße Funken stoben vor meinen Augen, als mich ein heftiger, lang andauernder Orgasmus überrollte. Meine Beine zitterten, doch Joona hielt sie fest. Er sorgte dafür, dass ich seinen Kopf nicht einklemmen konnte und fuhr so lange mit seinen Liebkosungen fort, bis er spürte, dass ich es nicht mehr aushielt. Vollkommen überwältigt lag ich da und hatte trotzdem das Gefühl, noch nicht genug von ihm zu haben. Ich wollte ihn spüren, ganz tief in mir. Und genau das sagte ich ihm in dem Moment. Joona reagierte sofort. Ich glaube, auch er konnte sich kaum noch zurückhalten. Vorsichtig legte er sich auf mich, stützte seine Arme rechts und links neben meinem Kopf ab. Automatisch zog ich die Beine an, um ihm das Eindringen zu erleichtern. Joona drückte seine Eichel an meinen Eingang und begann, sich zärtlich und doch energisch in mich zu schieben. Es war perfekt. Ich fühlte mich völlig ausgefüllt und auf eine einzigartige Weise mit ihm verbunden. Noch nie war es so intensiv für mich gewesen. Ich spannte meine Muskeln an, um ihn so tief wie möglich in mich zu ziehen. Erstaunt keuchte er auf. Als er anfing, sich in mir zu bewegen, passte ich mich seinem Rhythmus an und schon bald verloren wir uns in unseren Bewegungen und in den Emotionen, die sie in uns auslösten. Es würde nicht lange dauern und ich würde ein zweites Mal meinen Gipfel erreichen, das spürte ich schnell. Immer wieder küssten wir uns. Joona flüsterte mir mit seiner unbeschreiblich erotischen rauen Stimme leise Worte zu, die ich nicht verstand, aber ich wusste, dass es Kosewörter waren. Es war mir egal, dass er vor lauter Leidenschaft vergaß, dass ich seine Sprache nicht verstand. Im Gegenteil, es machte ihn nur noch faszinierender, das Erlebnis mit ihm noch einmaliger. Immer wieder drang er tief in mich ein, unsere Körper stießen heftig gegeneinander und wir begannen zu schwitzen. Als es bei mir zum zweiten Mal so weit war, als mich mein Höhepunkt noch einmal schüttelte, stöhnte ich seinen Namen. Und dann war es auch bei ihm so weit: Mit einem letzten ruckartigen Eintauchen in meine heiße nasse Venus hielt er inne und ich spürte das Pulsieren seines Gliedes, als er sich in mir ergoss. Und dann war es vorbei. Joonas Kopf lag in meiner Halsbeuge, seine Finger hatte er mit meinen verschränkt. Schwer lastete sein Gewicht auf meinem Brustkorb, doch es störte mich nicht. Ich wollte ihn so dicht bei mir haben, wie es nur ging. Ganz kurz durchzuckte mich der Gedanke, dass wir uns beide nicht um eine mögliche Schwangerschaft gekümmert hatten. Aber irgendwie war das in diesem Moment gleichgültig. Und es wunderte mich auch nicht, als Joona an meinem Hals murmelte: "Rakastan sinua." "Ich liebe dich auch", erwiderte ich mit der größten Selbstverständlichkeit und begriff, dass es genau das war, was ich schon die ganze Zeit gefühlt hatte. Und ich musste lächeln, als ich daran dachte, dass dies der erste Satz war, den mir Ulrike auf Finnisch beigebracht hatte. Natürlich ohne zu wissen, dass ich ihn jemals in dieser Sprache hören würde. Ulrike und Daavid hatten später am Abend schnell verstanden, was zwischen mir und Joona passiert war. Daavid quittierte dies nur mit einem Lächeln, doch Ulrike zog mich in einem passenden Moment zu sich. Breit grinsend boxte sie mich in die Seite und sagte: "Wusste ich doch, dass es dir gefällt." Es? Verständnislos sah ich sie an. "Na, dein Geburtstagsgeschenk! Ich hab schon lange auf eine passende Gelegenheit gewartet, dir Joona vorzustellen, aber die Idee, ihn zum Schlittenfahren einzuspannen, war einfach grandios. Hätte nicht gedacht, dass es schon auf den ersten Blick funkt, bevor ihr überhaupt unterwegs seid." Sprachlos starrte ich meine Freundin an. "Aber ich wusste, dass ihr zusammenpasst", fügte sie noch selbstzufrieden hinzu und genau den gleichen Satz hörte ich ein Jahr später noch einmal, als sie ihren Schwager und mich das erste Mal in unserer Wohnung in Helsinki besuchte.
# Die Überraschung
## Linda Freese
Schon seit heute Morgen freute Maren sich auf den Abend. Es war der sechste Dezember – Nikolaus. George hatte ihr eine Überraschung versprochen. Maren rätselte, was das wohl für eine Überraschung sein konnte. Natürlich hatte es etwas mit dem Nikolaus zu tun. Vielleicht würde er eine schöne Rute mitbringen und Maren ein wenig züchtigen. War sie in letzter Zeit artig gewesen?, fragte sie sich in Gedanken und kam zu dem Schluss, dass George eigentlich mit ihr zufrieden sein konnte. Sie waren jetzt schon seit fast zwei Jahren ein Paar und George hatte Maren gut erzogen. Sie mochte es, wenn er streng mit ihr war und ihr sagte, was sie zu tun oder zu lassen hatte. Er bestimmte, was sie anzog, wann sie ausging und mit wem, er bestimmte ihr Leben und es war genau das Leben, was Maren sich immer gewünscht hatte. Sie war devot, und das durch und durch. Für sie gab es nur eine Möglichkeit, glücklich zu sein, nämlich die, sich einem Mann zu unterwerfen. Außerdem mochte sie die Schmerzen bei der Züchtigung. Sie liebte es, den Hintern versohlt zu bekommen. Wenn sie die Macht von George spürte, auf dessen Knien sie lag, und er sie gerechterweise maßregelte, dann war das für Maren die reinste Erfüllung. Jetzt stand sie vor dem großen Spiegel im gemeinsamen Schlafzimmer und überlegte, was sie anziehen sollte. George hatte ihr erlaubt, selbst etwas auszusuchen, er meinte nur, sie solle etwas wählen, was ihm gefallen würde. Sie kannte seine Vorlieben und wusste, wonach sie suchen musste. Zielstrebig öffnete sie ihren Kleiderschrank und entschied sich für einen grauen kurzen Faltenrock, schwarze Kniestrümpfe, eine weiße Bluse, eine schwarze Strickjacke und schwarze Lackschuhe. Maren war sich sicher, dass sie George damit richtig antörnen würde. Er mochte es genau so sehr wie sie, wenn sie eine Schuluniform trug. Dazu würde sie sich ihre langen blonden Haare noch zu Zöpfen flechten und dem heutigen Abend stand somit nichts mehr im Wege. Wenn Maren Glück hatte, würde sie vom Nikolaus die Rute bekommen und die Vorfreude darauf ließ sie euphorisch werden. Tänzelnd stand sie nun nackt vor dem verschnörkelten Spiegel und schlüpfte beschwingt in ihre Kleidungsstücke. Unterwäsche zog sie keine an, denn das mochte George überhaupt nicht. Als sie den kurzen Rock über ihre dralle Hüfte gezogen und den Reißverschluss geschlossen hatte, wippte sie mit ihrem Po und schwang den Minirock hin und her. Maren war sehr weiblich und ihre füllige Figur verlieh ihr eine zufriedene glückliche Ausstrahlung. Sie war mit sich und der Welt im Reinen und das strahlte sie auch aus. Voller Zufriedenheit umfasste sie ihre üppigen Brüste und knetete sie so lange, bis ihre dunklen großen Nippel steif wurden. Ein leises Lächeln huschte um ihre Mundwinkel und Maren nahm ihre Bluse und zog sie über. Der Stoff war transparent und durch ihn hindurch konnte Maren ihre erigierten Nippel gut sehen. Nun schlüpfte sie schnell in die Kniestrümpfe, zog sie hoch und tippelte in ihre Lackschuhe, die sie gestern schon auf Hochglanz poliert hatte. Mit geübten Griffen schloss sie das zarte Fersenriemchen und warf sich die schwarze Strickjacke über. Erneut betrachtete sie sich im Spiegel und war äußerst zufrieden mit dem, was sie dort sah. Die Schuluniform verlieh ihr einen jungen verletzlichen Anblick und Maren war voller Esprit und Schwung. Dynamisch schwebte sie fast ins Bad und frisierte ihre lange blonde Mähne. In die zwei Zöpfe, die sie schnell geflochten hatte, band sie schwarze Haarbänder, die bei jeder Bewegung flatterten. Maren war aufgeregt wie ein kleines Kind, aber genau das liebte George an ihr so sehr. Sie hatte sich ein Stück Kindheit und ein bisschen Naivität bewahren können und für George war sie nur die kleine Göre. Wenn er sie so nannte, schwappte eine Woge aus Stolz und Respekt über sie. Sicher dauerte es nicht mehr lange, bis George nach Hause kam, stellte Maren fest, nachdem sie auf die Uhr gesehen hatte. Voller Vorfreude lief sie in der Wohnung herum, als plötzlich das Telefon schrillte. Maren fragte sich, wer da wohl anrief, als sie den Hörer abnahm. Die vertraute Stimme von George war wie Musik in ihren Ohren. "Hallo, Göre", tönte es tief aus dem Hörer und Maren entgegnete: "Hallo, mein Herr", wie sie es gewohnt war. Aber heute war ein besonderer Tag und George meinte zu ihr: "Kleine Göre, heute wirst du mich nur Nikolaus nennen. Hast du verstanden?" "Ja natürlich, Nikolaus, ich habe verstanden", bestätigte sie und ein kalter Schauer lief ihr über den Rücken. "Gut, kleine Göre, dann hör mir jetzt genau zu." George räusperte sich kurz und sprach dann weiter: "Du wirst jetzt deinen Mantel anziehen, den langen, verhülle dich richtig und dann wirst du nach unten kommen. Dort wartet ein Fahrzeug auf dich, ein roter BMW. Ein Fahrer wird dich zu einem Ort bringen, an dem ich auf dich warte. Setze dich nach hinten ins Auto und spreche nicht mit dem Chauffeur. Bis hierhin alles klar?" "Ja Herr, entschuldige, Nikolaus, bis hierhin alles klar", beteuerte Maren und ihre Aufregung wuchs bei dem Gedanken an den unbekannten Ort und das bevorstehende Abenteuer. "Mach dich jetzt auf den Weg und vertraue mir, ich warte auf dich, Göre", wies George sie an. Maren bestätigte noch einmal seine Worte und legte brav den Telefonhörer wieder auf. Sodann schlenderte sie in den Flur, holte aus dem Garderobenschrank ihren langen Mantel und warf ihn sich über. Mit dem Taillengürtel verschloss sie ihn vernünftig und anschließend wickelte sie sich noch einen kurzen Schal um ihren Hals. Ein letzter Blick in den Spiegel und schon war sie aus dem Haus. Schnell sah sie den roten BMW vor ihrem Haus, öffnete die hintere Autotür und stieg elegant in das Fahrzeug ein. Sofort wurde der Motor gestartet und Maren machte es sich bequem. Nachdem sie eine Weile durch die Stadt gefahren waren, hielt der Fahrer vor einem Haus, welches am Ortsrand, weit entfernt von den Siedlungen, stand. Nachdem der Motor wieder ausgeschaltet wurde, wusste Maren instinktiv, dass sie am Ziel angekommen waren. Sie stieg aus und ging auf das Haus zu, als wie von Geisterhand die Haustür geöffnet wurde. Ein Schild wies mit Pfeilen darauf hin, dass Maren eine Treppe hinabsteigen sollte. Mittlerweile war sie so aufgeregt, dass sie zu zittern begann. Artig stieg sie die Stufen hinab und folgte den Hinweisschildern. Unten angekommen, schritt sie durch einen Flur und betrat dann einen Raum. Neugierig sah sie sich um und stellte fest, dass dieses Zimmer ein alter Schulraum sein musste. Es gab eine Tafel, ein Pult und viele Tische für Schüler. Auf dem Pult lag ein Rohrstock, der ihr sehr bekannt vorkam. Maren durchschritt den Raum und legte währenddessen ihren Mantel und den Schal ab und warf beides achtlos über einen Stuhl. Plötzlich knarrte die Eingangstür und Maren fuhr erschrocken zusammen, drehte sich um und erblickte einen 'Nikolaus'. Dort stand übermächtig eine große Person, die in ein Nikolauskostüm gekleidet war, und hielt ein dickes goldenes Buch in der Hand. Als Maren genauer hinsah, erkannte sie den Mann an seinen strahlenden Augen. Vor ihr stand George und grinste mit seinem künstlichen weißen Bart. Gut erzogen begrüßte Maren ihren Herrn: "Guten Abend, Nikolaus." "Hallo, Göre", dröhnte George durch den Raum und ging auf Maren zu. Sein rot-weißes Gewand flatterte hinter ihm her und Maren hoppelte unruhig von einem Bein auf das andere. Als George direkt vor ihr stand, legte er einen Finger unter ihr Kinn und hob es an. So musste sie ihm direkt in die Augen schauen. Allein diese Geste löste in Maren Schauer der Erregung aus und sie bekam eine Gänsehaut. Mit tiefer Stimme sprach der 'Nikolaus' zu ihr: "Wollen doch mal sehen, ob du im letzten Jahr auch artig gewesen bist." Er räusperte sich, holte tief Luft und sagte dann: "Setz dich hier auf den Stuhl, Göre." Dabei deutete er mit seiner Hand auf einen Holzstuhl in der Nähe. Maren setzte sich sofort und legte ihre Hände auf den Tisch vor ihr. Sie war gespannt und versteifte ihre Muskeln. George kam auf sie zu, stellte sich demonstrativ vor sie hin und nahm das Buch, das er in einer Hand gehalten hatte, nun vor seinen Bauch und begann darin zu blättern. Maren sah auf das goldene Buch und erkannte die typische Handschrift ihres Herrn. Im Stillen fragte sie sich, wann er wohl all diese Eintragungen gemacht hatte. Der Nikolaus wählte willkürlich eine Seite in dem großen Buch und las sie, dann blickte er zu Maren und das Spiel begann. "Hier lese ich eine Eintragung von April, da hattest du deine Kniestrümpfe nicht züchtig hochgezogen." "Ja, das ist so richtig, Nikolaus", bestätigte Maren kleinlaut. George blätterte im Buch weiter und fuhr unbeeindruckt fort: "Die nächste Notiz lautet: 'Frechheit deinem Herrn gegenüber' und ist von Juni. Im August hast du die dir übertragenen Aufgaben nicht vollständig erledigt und im September warst du unartig und hast Wäsche getragen, obwohl dein Herr es verboten hatte." Streng und unnachgiebig sah George Maren an und sie senkte sofort züchtig ihren Blick. Das schlechte Gewissen stand ihr ins Gesicht geschrieben und sie schämte sich ihrer Missetaten. Die dröhnende Stimme des Nikolauses hallte durch den Raum: "Was hast du dazu zu sagen, Göre?" Maren nahm all ihren Mut zusammen und plapperte drauflos: "Es stimmt, Nikolaus, ich war unartig. All die Dinge habe ich getan. Bitte bestrafe mich dafür." George ging einen Schritt zurück, blätterte abermals in dem großen Buch und schüttelte missbilligend seinen Kopf, bevor er zu Maren sagte: "Nun, es folgen noch andere Eintragungen mit deinen Vergehen. Du warst ein böses Mädchen im letzten Jahr. Ist das richtig?" "Ja, das ist richtig, Nikolaus", meinte Maren reuevoll, "ich war ein böses Mädchen und habe eine Strafe verdient. Bitte, lieber Nikolaus, bestrafe mich und züchtige mich, damit ich im nächsten Jahr ein braves Mädchen bin." Erneut schüttelte George energisch seinen Kopf, schloss geräuschvoll das goldene Buch und legte es auf den Tisch vor Maren. Durch seinen dicken weißen Bart hindurch tönte er: "Du lässt mir gar keine andere Wahl. Böse Mädchen müssen ordentlich bestraft werden." Dann drehte er sich um und ging nach vorne zum Pult. Dort nahm er den Rohrstock vom Tisch und klatschte damit bedrohlich auf seine geöffnete Handfläche. "Komm her, unartiges Mädchen", raunzte er und blickte streng zu Maren. Sie erhob sich sofort und tippelte zögernd durch das Schulzimmer. Sie hielt ihren Blick demütig gesenkt und spielte mit einer Hand in ihrem Zopf, drehte ihn und das schwarze Haarband flatterte dabei. Als sie vor dem Pult anhielt, sah sie kurz nach oben in Georges Augen und sah darin seine unbändige Macht, die er ausstrahlte. Schnell blickte sie wieder nach unten und erneut lief ein Schauer der Erregung durch ihre Nervenbahnen. Als ihr Herr die aufgekommene Stille mit seiner Stimme durchbrach, zuckte Maren leicht zusammen. Der Nikolaus sagte: "Leg dich mit dem Oberkörper über das Pult." Diese Anweisung ließ keinen Widerstand zu, aber Maren wollte sich auch nicht widersetzen, denn sie hatte die Strafe verdient. Sie beugte sich nach vorne und schlitterte mit ihren Brüsten über den hölzernen Tisch, dabei rutschte ihr kurzer Rock nach oben und ihre pralle Kehrseite leuchtete fast in diesem schummrigen Licht. Maren umfasste mit ihren Händen den Tischrand und hielt sich dort fest. Spannungsgeladen wartete sie auf neue Anweisungen, die prompt folgten. George fragte sie: "Was für eine Strafe wäre wohl angemessen für dein schlechtes Betragen?" "Ich denke", begann Maren, "dass ich den Stock verdient habe, Nikolaus." George streichelte über den bloßen Po von seiner Göre und erwiderte: "So, du denkst, du hast den Stock verdient. Dann soll der Nikolaus dich also mit seiner Rute züchtigen?" Nun vollends darauf bedacht, bald eine gerechte Strafe zu bekommen, denn mittlerweile wuchs ihre Erregung enorm, sagte Maren mit fester Stimme: "Lieber Nikolaus, ich denke, dass ich deine Rute spüren sollte, denn ich war sehr böse." George knallte den glatten Rohrstock abermals in seine Handfläche und Maren zuckte unwillkürlich zusammen. Plötzlich brüllte George fast: "Ich werde dir zeigen, was ich für angebracht halte!" Schon prallte der erste derbe Hieb auf Marens blassen Hintern. Sie schrie lustvoll auf und stöhnte leidenschaftlich. Unter einer brennenden Salve von Stockschlägen, die auf dem fleischigen Hintern seiner Göre landeten, grölte er: "Du warst sehr unartig, Göre." Maren zuckte immer wieder unter den peinigenden Schlägen zusammen und stammelte: "Ja, Nikolaus, ich war böse." Wieder prasselten mehrere glühende Hiebe mit dem geliebten Rohrstock auf ihren nackten Po und hinterließen inzwischen rote Striemen. Die schwarzen Kniestrümpfe von Maren waren heruntergerutscht und sie zappelte auf dem Pult hin und her, wobei ihre Zöpfe wild durcheinanderflogen. Mit einem Male hörten die Hiebe auf und es wurde bedrückend still. Wieder fühlte Maren, wie George liebevoll mit einer Hand über ihren geschundenen Po streichelte und sein Atem hatte sich beschleunigt. Maren fühlte die glühenden Striemen auf ihrem drallen Hintern und spürte dieses erlösende Kribbeln in ihrem Körper, in ihren Nerven. "Steh auf!", lautete der plötzliche Befehl von George. Maren rappelte sich auf und richtete ihre Kleidung, die nun ziemlich durcheinandergeraten war. Sie stand direkt vor dem Nikolaus und stopfte sich ihre Bluse in den kurzen Rock, drehte den Faltenrock ein wenig, damit er wieder richtig saß, und zog ihre Strickjacke gerade. Nach einer Weile schnauzte ihr Herr sie an: "Hast du nicht etwas vergessen? Läuft so ein züchtiges Mädchen herum?" Sein Blick huschte nach unten und er sah auf ihre heruntergerutschten Strümpfe. Da wusste Maren sofort, was er meinte und entschuldigte sich: "Das habe ich nicht bemerkt, Nikolaus, tut mir leid, es wird nicht wieder vorkommen." Schnell bückte sich Maren und zog ihre Strümpfe nach oben, als sie auf einmal den stahlharten Griff in ihrem Nacken fühlte. George hielt sie mit eiserner Hand in der gebückten Position: "Es wird nicht wieder vorkommen? Wie oft hast du das schon gesagt? Was soll ich nur mit dir machen? Dir ist klar, dass ich dich noch mehr züchtigen muss?" Maren seufzte lustvoll und ein angedeutetes Lächeln huschte über ihren schön geschwungenen Mund, als sie erwiderte: "Ja, Nikolaus, ich habe Strafe verdient, ich bin wohl unverbesserlich." "Auch noch frech werden?", grölte George und verstärkte seinen Griff noch einmal. Maren wandt sich unter seiner Pranke, die sie immer noch in demütigender Stellung festhielt. "Manche Mädchen schreien förmlich nach Züchtigung und du bist so eins", stellte er fest und lockerte seinen Griff, bevor er sie losließ. Maren blieb in dieser Position und wartete auf Anweisungen. George schritt durch den Raum und setzte sich auf den Stuhl hinter dem Pult. "Komm her, Göre!", lautete sein Befehl, dem Maren flink gehorchte und auf ihn zukam. George schaute sie an, klatschte mit seiner Hand auf seinen Schoß und deutete damit an, dass Maren sich darauflegen solle. Sie kannte seine Gesten nur zu genau und folgte seinem Befehl ungehend. Sie legte sich bäuchlings auf seine strammen Oberschenkel und schon wieder geriet ihre Schuluniform dabei in Unordnung. Unter ihren Brüsten spürte Maren seine festen Schenkel und ihre dunklen Nippel reagierten darauf, in dem sie sofort steif wurden und sich aufrichteten. Wieder strichen seine groben Hände über ihren entblößten Hintern und rieben über die roten Striemen, die mittlerweile glühten. George begann zu keuchen und meinte zu ihr: "Das ist die richtige Farbe für deinen strammen Hintern, aber ich denke, wir können noch etwas hinzufügen." Kaum hatte er die bedrohlichen Worte ausgesprochen, klatschte seine Hand auf ihre prallen Backen. Er widmete sich intensiv ihrem Po und hielt sie mit der anderen Hand, die er auf ihren Rücken presste, in dieser Position. Maren stöhnte leidenschaftlich und nun spürte sie die heißen Wogen ihrer Lust, die sich in ihrem Unterleib ausbreiteten. George verlangte dann plötzlich: "So, das war zum Aufwärmen, jetzt wirst du mitzählen. Als Strafe für deine Vergehen bekommst du zehn Schläge mit der Hand und danach werde ich noch einmal den Stock einsetzen. Dann sind es 20 Hiebe. Solltest du dich verzählen oder dich weigern, beginne ich von vorne. Ist das klar, Göre?" Überrascht und voller Vorfreude bestätigte Maren ihm, dass sie verstanden habe. Dann begann ihre lustvolle Tortur und bei dieser Erniedrigung bebte Maren unkontrolliert. Sie zählte fleißig mit und versuchte, sich auf die Zahlen zu konzentrieren, was ihr aber immer schwerer fiel. Die Begierde in Maren nahm zu und sie brabbelte nur noch leise die Zahlen vor sich hin. Das gefiel dem Nikolaus nicht und er warnte sie: "Ich kann dich nicht verstehen. Sprich lauter oder ich beginne von vorn." Maren bemühte sich und zählte lauter, während die klatschenden Hiebe weiter ihre Kehrseite zum Sieden brachten. In diesem Moment fühlte sie sich so frei wie sonst nie. Maren brauchte die Züchtigung, um leben zu können. Als sie die zehn Schläge hinter sich hatte, war sie schon ziemlich geschafft und die kurze Pause tat ihr gut. Abermals strich George über ihren wunden Po. Langsam schlug er ihren kurzen Rock noch einmal nach oben, um ihre volle Pracht sehen zu können. Maren zitterte vor Erregung und versuchte, ihre Körperfunktionen mühevoll zu kontrollieren. "Du weißt, was jetzt kommt?", fragte er sie. "Ja, Nikolaus, ich weiß es. Ich bekomme noch Schläge mit dem Stock. Ich bitte dich um meine gerechte Strafe", flehte Maren ihn an. "Die wirst du bekommen, Göre, verlass dich drauf." "Vielen Dank, Nikolaus." George hörte auf, sie zu streicheln, nahm den Rohrstock abermals und legte sofort los. Voller Schreck vergaß Maren fast zu zählen, besann sich aber und beeilte sich "Eins" zu sagen. Ihr draller Po war schon mit roten Striemen übersät. Maren jedoch hielt tapfer durch, denn sie wusste genau, dass sie diese Strafe mehr als verdient hatte. Immer wieder prallte der unnachgiebige Stab auf ihr Fleisch und artig zählte Maren mit. So schön hatte der Nikolaus ihr noch nie den Hintern versohlt und schon jetzt nahm sie sich vor, im kommenden Jahr auch nicht immer ganz so lieb zu sein, damit er auch dann wieder einen Grund hatte, sie zu züchtigen. Viel zu schnell waren die Hiebe mit der Rute vorbei und Maren stöhnte ein letztes Mal lustvoll auf. Noch einmal klatschte Georges Hand auf ihren Hintern und er forderte sie auf: "Steh auf und zieh dich richtig an, Göre." Maren hievte sich hoch und richtete abermals ihre Kleidung, diesmal vergaß sie nicht ihre Kniestrümpfe. Der Nikolaus sah sie wohlwollend an, stand von seinem Stuhl auf und tätschelte liebevoll über ihren Kopf. Vorsichtig nahm er einen ihrer Zöpfe in die Hand, zog ein wenig daran und meinte zu ihr: "Das war dir hoffentlich eine Lehre, Göre. Benimm dich ab sofort besser und sei ein artiges Mädchen." Maren bog ihren Kopf ein wenig zu der Seite, an der George an dem Zopf zog, und erklärte: "Ja, Nikolaus, ich gebe mir Mühe, ein braves Mädchen zu werden." "So, so", brummte George, "wollen mal sehen, ob dir das gelingt." Dabei ließ er ihren Zopf wieder los und schaute sie mit ernstem Blick an. Voller Demut senkte Maren ihr Haupt und hörte inbrünstig seinen Worten zu, als er mit leiser eindrücklicher Stimme zu ihr sagte: "Du wirst dir jetzt deinen Mantel wieder anziehen, mit dem Fahrer nach Hause fahren und dort auf mich warten. Wir haben heute noch was vor. Der Nikolaustag ist noch nicht zu Ende." Maren nickte nur verständig und sah ihm nach, wie er würdevoll das Schulzimmer verließ. Im Stillen dachte sie noch: Nikolaus kann so schön sein!, und freute sich auf zu Hause.
# Der Nikolaus
## Dave Vandenberg
Trübsinnig starrte Ruth aus dem Fenster. Ihr war elend zumute. Dabei war draußen das schönste Winterwetter, das sie sich sonst vorstellen konnte. Dichte weiße Flocken fielen herab. Frau Holle schüttelte ihre Betten aus, erinnerte sie sich ein wenig wehmütig an das schöne Märchen aus ihren Kindertagen. Sie liebte den Winter, diese kalte Jahreszeit. Sie schmolz beim Anblick der weißen Pracht dahin. Besonders der Monat Dezember hatte es ihr angetan. Der Advent versetzte sie normalerweise in fast ausgelassene Stimmung. Sie konnte es schon im Oktober kaum abwarten, endlich die Wohnung zu schmücken, Geschenke einzukaufen, Karten zu schreiben, auf Weihnachtsmärkten herumzuflanieren und eben all das zu tun, was zur Adventszeit dazugehört. Dieses Jahr war alles anders. Ihr Freund hatte sie nach achteinhalb gemeinsamen Jahren verlassen. Das war im Frühjahr gewesen und sie war noch längst nicht darüber hinweg. Er war es auch, der umgehend ausgezogen war, ihr die Wohnung und sogar den meisten Teil dessen überlassen hatte, was sie zusammen angeschafft hatten. Nicht schnell genug hatte ihm die Trennung gehen können, nachdem er diese andere Frau kennengelernt hatte. Tränen stiegen ihr immer noch so oft in die Augen, die sie auch diesmal trotzig wegwischte. Es war wirklich an der Zeit, ihn zu vergessen. Ihre Freundinnen versuchten sie immer wieder zu überreden, mit ihnen auszugehen. "Die Welt ist voll von Männern, die nur darauf warten, abgeschleppt zu werden." Ihre Freundinnen steckten private Enttäuschungen eben besser weg als sie. Heute Abend wollten sie sich alle zum Glühweintrinken treffen. Heute war Nikolaustag. Am sechsten Dezember vor neun Jahren hatte sie ihren Exfreund kennengelernt. Sie hatte plötzlich große Lust, sich zu betrinken... Ruth schaute auf die Uhr. Schon halb sechs. Es schneite und schneite. Um sieben war sie mit den anderen verabredet. Es wurde Zeit, sich ein wenig zurechtzumachen. Sie hatte es mit einem Mal satt, immer noch hinter ihrem Ex herzutrauern. Vor dem großen Schlafzimmerspiegel blieb sie stehen. Sie hatte abgenommen, was ihr gut stand. Sie hatte immer eine hübsche Figur gehabt. Große Brüste, einen flachen Bauch, durchtrainierte Schenkel. Die richtigen Proportionen. Sie gefiel sich eigentlich. Spontan legte sie ihre Hand in ihren Schritt. Sie hatte seit dem Frühjahr keinen Sex mehr gehabt. Plötzlich vermisste sie ihn. Sie bekam Lust, während sie so dastand und sich ansah. Sie würde sich endlich mal wieder etwas Gutes tun müssen... Enger kurzer wollener Rock, ein figurbetonter Pullover, Strümpfe mit Strapshaltern daran und schicke hohe Stiefel. Sie sah gut aus. Etwas mehr Make-up als sonst. Sie lächelte ihrem Spiegelbild zu. Das Klingeln an der Tür irritierte sie. Es konnte keine ihrer Freundinnen sein. Sie waren in der Stadt verabredet. Ruth öffnete neugierig und staunte nicht schlecht. Vor ihr stand ein richtiger Nikolaus. Mit dickem roten Mantel, einer roten Mütze, Handschuhen, einem weißen Vollbart, derben Stiefeln und einem Jutesack über der Schulter. Und das Beste an ihm war: Man sah sofort, dass er wirklich von draußen kam. Er war mit einer feinen weißen Pulverschicht bedeckt und klopfte sich kräftig den Schnee von den Schultern. Sprachlos starrte sie ihn an. Er schien erst gar nicht zu bemerken, dass sie die Tür schon geöffnet hatte. Doch dann blickte er hoch und strahlte fröhlich. Seine Augen funkelten und blitzten vergnügt. Der Job schien ihm Spaß zu machen. "Hoho, wen haben wir denn da Hübsches?" Ruth musste unwillkürlich lachen, angesteckt von so viel Frohsinn. "Bist du die Ruth?" Sie würde sich später fragen, woher er kam und warum er gerade vor ihrer Tür stand. Er rüttelte vielversprechend an dem großen Sack, der schwer über seiner Schulter hing. "Magst du mich für ein Weilchen bei dir aufwärmen lassen?" Natürlich bat sie ihn herein. "Ich glaube, ich habe etwas ganz besonders Gutes für dich dabei." Er klopfte sich, so gut es ging, den Schnee von den Schuhen und folgte ihr in ihre gemütliche Küche. Bevor er irgendetwas sagen konnte, hatte sie schon einen Topf mit Glühwein auf den Herd gestellt. Sie war mit einem Mal rundum guter Laune. Schnell war das weihnachtliche Getränk erhitzt. "Das tut gut!" Sie hatte beiden einen ordentlichen Becher voll gegossen. "Mit der Zeit wird es doch kalt beim draußen Herumlaufen." Sie hatte ihn schon eingehend begutachtet. Sie konnte kaum sein Gesicht erkennen. Es war fast zugewachsen von dem buschigen Vollbart und den dichten Augenbrauen. Und doch war sie sich sicher, dass er attraktiv war. Etwas stämmig gebaut vielleicht, aber sie mochte Männer, die eher korpulent als mager waren. Schöne kräftige Hände hatte er auf jeden Fall. Und strahlend blaue Augen und eine vertrauenerweckend tiefe Stimme. Ruth fühlte sich zu ihm hingezogen. Auf eine aufregende, fast etwas beunruhigende Art, die so gar nichts mit dem Nikolaus selbst zu tun hatte. Er wollte ihr das kleine Geschenk nicht länger vorenthalten. "Allerdings musst du mir zuerst noch dein Gehör schenken." Ruth wartete gespannt. Und dann holte dieser Nikolaus doch tatsächlich zwei Bücher heraus. Ein goldenes und ein schwarzes. Es war alles genauso wie früher, als sie noch an den Nikolaus geglaubt hatte. Fast alles... Er las zuerst aus dem goldenen Buch vor. Sie traute ihren Ohren kaum, was es da alles an guten Dingen über sie zu lesen gab. Und sie hatte so ihre Idee, woher dieser Nikolaus das alles wissen konnte. Und wer ihn geschickt haben könnte. Dann schlug er das schwarze Buch auf und seine Augen verdunkelten sich. Dass sie zu sehr den alten Zeiten nachhinge, zu trübsinnig geworden war, nicht genug an sich denken würde und so ging es weiter und weiter... Ruth hing an seinen Lippen. Alles stimmte und sie konnte kaum abwarten, es zu ändern. Sie fühlte sich wie hypnotisiert von seiner Stimme und unwiderstehlich angezogen von seinem Verhalten. Der Nikolaus sah sie an. "Ich hoffe, du nimmst das alles sehr ernst." Doch beide konnten plötzlich ein Lachen nicht mehr unterdrücken. Er holte aus dem Jutesack ein kleines Päckchen und gab es ihr. Sie sahen sich sehr lange in die Augen. Es kribbelte in ihrem Unterleib. Wohltuend und erregend. Dieser Nikolaus hatte etwas so Männliches an sich, auf das sie viel zu lange schon nicht mehr entsprechend reagiert hatte. Sie zog an dem roten Schleifenband des hübsch eingepackten Päckchens. "Zuerst die Karte, bitte!" Seine Stimme löste mittlerweile eine aufregende Gänsehaut bei ihr aus. Ruth öffnete den kleinen Briefumschlag. "Fang endlich an, dein Leben wieder richtig zu genießen. Unser Geschenk und der Nikolaus werden dir dabei helfen. Alles Liebe, Deine Freundinnen!" Aus dem Geschenkpapier fiel eine Packung Kondome heraus. Ihre Hände zitterten, als der Weihnachtsmann nach ihr griff und sie an sich zog. Einfach so... Er zog ihr Rock und Pullover aus, während er sie küsste, bis sie nur noch in ihrer aufreizenden Unterwäsche vor ihm stand. "Wow!" Er piff anerkennend durch die Zähne. "Mann, siehst du scharf aus!" Mit geübtem Griff befreite er sie auch noch schnell aus ihrem hauchdünnen Slip. Ruth fühlte, wie sich unter seinem Blick die süßeste Feuchtigkeit zwischen ihren Beinen zu sammeln begann. Sie ließ sich von ihm auf ihren Küchentisch heben und spreizte ihre Schenkel etwas, um ihn an ihrer Geilheit teilhaben zu lassen. "Komm her!" Sie fühlte sich endlich mal wieder unwiderstehlich und begehrt. Der Nikolaus öffnete seinen Gürtel, ließ sein dickes rotes Gewand fallen und zog ungeduldig seine wollene Unterhose aus. Das prächtige Glied, das sich darin versteckt gehalten hatte, nahm ihr den Atem. Es war perfekt gebaut. Sie würde es lieben, wenn es tief und ausdauernd in sie eindrang. Er kniete nieder und sie wand sich lüstern unter seinem stechenden Blick. Als er ihre Lust mit seinen Finger aufstöberte, presste sie stöhnend die Hand vor ihren Mund. Er benetzte seine Fingerkuppen mit der Zunge und liebkoste ihre Perle so intensiv, dass sie machtlos gegen die Wollust war, die sich in ihr ausbreitete. Alles, was sich an Frust, Trauer, Wut und Enttäuschung in den letzten Monaten in ihr aufgestaut hatte, fand nun endlich ein angemessenes Ventil. Der Nikolaus verwöhnte ihren lodernden Punkt mit der Zunge, den Lippen. Er biss sogar zärtlich hinein. Und alles war Erotik pur. Ruth krallte sich in seinen Bart, der erstaunlich fest gewachsen war, und zitterte vor Erregung am ganzen Körper. Ihr erster Höhepunkt durchfuhr sie wie ein Blitz. Sie beugte sich weit nach hinten und schob ihm vertrauensvoll ihr Becken entgegen. Nackte Gier loderte in seinem Blick. Er nahm ihre Hände und presste sie fest auf seinen hoch aufgerichteten Liebesspeer. Er war sehr groß. Griffig. Wohlgeformt. Ruth liebte die großen. Aber dieser hier, der ihr auf so ungewöhnliche Weise beschert wurde, der schwebte in anderen Dimensionen. Wäre sie nicht so ausgehungert und gierig danach gewesen, sie hätte gezögert. Er musste die Zweifel in ihrem Blick gesehen haben, denn er beugte sich zu ihr herunter und flüsterte: "Ich werde behutsam mit dir umgehen und du wirst sehen – er wird dir gefallen!" Wie hypnotisiert starrte sie zwischen seine Beine. Bestaunte die üppigen Hoden und griff fast ein wenig ehrfürchtig danach, als sich der Nikolaus mit all seiner männlichen Pracht Stück für Stück in sie hineinschob. Er stimulierte jeden Millimeter ihrer Vagina damit. Sie fühlte sich aus 1.000 winzigen Lustnerven bestehend, die nur darauf gewartet hatten, endlich entzündet zu werden. Ganz vorsichtig versank er tiefer und tiefer in ihr und je weiter er in sie eindrang, umso hemmungsloser nahm sie ihn auf. Sie bebte vor Lust und Gier nach der totalen Befriedigung, die er ihr ohne Zweifel schenken würde. Seine kräftigen Arme packten ihre Beine und legten sie auf seine Schultern, was ihm ermöglichte, noch intensiver mit ihr zu verschmelzen. Sie dankte es ihm mit einem lustvollen Schrei. Und dann stieß er zu. Zuerst ganz langsam und vorsichtig. Dann, als er sicher war, er konnte ihr nur Gutes tun, verstärkte er seine Stöße mit jeder einzelnen Bewegung seines Unterleibes. "Alles okay?", fragte der Nikolaus sie keuchend und fast ein wenig fürsorglich, als sie nach Luft ringen musste, weil ihr die Ekstase den Atem genommen hatte. Sie presste ihre Vagina zusammen, um dieses goldene Geschenk ja nicht wieder davonziehen zu lassen, bevor es ihr nicht alles bescherte, worauf sie zu lange schon verzichtet hatte. Sie sah das begehrliche Leuchten in den funkelnden Augen des Fremden, spürte die Kraft, die aus seinen Lenden strömte, und riss ihn an sich. Konnte nicht genug bekommen von seiner Potenz. Wollte seine Lust bis an seine Grenzen ausschöpfen. Sie war hemmungslos in ihrer Gier nach Befriedigung. Geradezu besessen davon, sich total darin zu verlieren. Als ein heißer Schub der Wollust in ihrem Unterleib explodierte, war Ruth äußerlich und innerlich eine einzig sprudelnde Liebesfreude... Es schneite immer noch in dichten weißen Flocken. Sie konnte gar nicht genug bekommen von der weißen Pracht, die sich auf die Erde legte. Der Nikolaus hatte sich mit einem zärtlichen Kuss von ihr verabschiedet. Er hatte ihr ein unvergessliches Geschenk beschert...
# Winterdienst
## Kristel Kane
Hagen hasste das Mietshaus, in dem er vor zwei Monaten eingezogen war. Mit ihm gab es insgesamt vier Mietparteien im Haus und seine Nachbarn waren allesamt spießig. Auch für Hagen war Sauberkeit wichtig, aber seiner Meinung nach musste das Putzen des Flures und der Außenanlagen nicht militärisch durchexerziert werden. Oft genug war er in den vergangenen Wochen mit seinen Mitbewohnern deswegen aneinandergeraten. Immer wieder wurde er an seine Pflichten erinnert. Es war keineswegs so, dass er nicht putzte, er hielt sich nur nicht immer an den vorgeschriebenen Zeitplan. Hagen sah keine Notwendigkeit darin, immer samstags die Treppe zu putzen, den Keller zu kehren oder die Außenanlage zu pflegen; ein anderer Wochentag tat es in seinen Augen auch. Die Spannungen nahmen allerdings zu, als es Winter wurde und die Schneefälle einsetzten. Jetzt galt es noch, zusätzlich die Gehwege zu räumen. Unnötig, zu erwähnen, dass Hagen auch hierbei ein eigenes Zeitempfinden hatte, um seiner Pflicht nachzukommen. Unsanft wurde Hagen um halb zehn aus dem Bett geklingelt. Schlaftrunken stand er auf. Er rechnete mit dem Paketboten. Also machte sich der junge Mann gar nicht die Mühe, einen Bademantel überzuwerfen. Die eng anliegenden Shorts ließen erahnen, dass der Träger gut bestückt sein musste. Auf jeden Fall trug Hagen stolz seinen Six-Pack zur Schau. Zu seiner Verwunderung fand er nicht den Postboten vor, sondern seine Nachbarin von oben. Wie gewöhnlich wollte sie zu einer ihrer üblichen Standpauken ansetzen. Doch die Tatsache, dass Hagen halbnackt vor ihr stand, ließ sie verstummen. Geniert wandte sie den Blick ab und starrte angestrengt auf den Boden. Er war nicht dumm und wusste, wie sein Body auf Frauen wirkte. Und offensichtlich hatte er denselben Effekt auf Frau Urbaniak, da sie sich eindeutig stark konzentrieren musste, um mit ihrer Schimpftirade fortzufahren. Innerlich musste sie ihm zugestehen, dass er verdammt gut gebaut war; allerdings war es ihre Pflicht, ihn scharf zu rügen, so in einem anständigen Haus die Tür zu öffnen. Hagen blieb gelassen. Je mehr sie mit ihm schimpfte und sich aufregte, umso cooler stellte er sich in Pose. Es gefiel ihm, mit ihr zu spielen. Deshalb entschloss er sich, einen Schritt weiterzugehen. "Frau Urbaniak, Sie haben völlig recht", stimmte er sie milde, obwohl er überhaupt nicht wusste, wovon sie eigentlich gesprochen hatte. "Es wäre mir nur lieb, wenn wir das in meiner Wohnung bei einer Tasse Kaffee besprechen könnten. Mir wird nämlich kalt." Zur Unterstreichung seiner Aussage rieb er schnell über seine Oberarme und spannte dadurch den Bizeps an. Nach höflichem anfänglichem Zögern trat sie doch ein. Hinter ihr stehend, schüttelte er amüsiert den Kopf. Er konnte nicht verstehen, weshalb jemand, der so aussah, derartig verbiestert war. Vermutlich war sie nur eine weitere frustrierte Ehefrau, die außer der gelegentlichen Missionarsstellung nichts geboten bekam. Hagen entschloss sich, mit dem Feuer zu spielen. Galant half er ihr aus dem Mantel. Leicht und wie zufällig drückte er sich gegen sie und vergaß nicht, ihr Parfüm zu loben. Zufällig erriet er die Marke. Frau Urbaniak zeigte sich geschmeichelt und legte ihr feindseliges Benehmen ab. Neugierig blickte sie sich in dem geschmackvoll eingerichteten Wohnzimmer um. Es wirkte modern und elegant. "Bitte setzen Sie sich, Frau Urbaniak. Ich bringe Ihnen gleich eine Tasse Kaffee", sagte er möglichst locker. Nach einer kurzen Pause und mit der versprochenen Tasse Kaffee in der Hand machte er die alles entscheidende Bemerkung. "Finden Sie nicht auch, dass Frau Urbaniak Sie alt macht? Wie wäre es, wenn wir uns mit den Vornamen anreden? Ich heiße Hagen!" Sie errötete. "Sabine", hauchte sie kleinlaut und ertappte sich dabei, wie sie wieder schwärmerisch auf sein Six-Pack starrte. Es fiel ihr zunehmend schwerer, die Unnahbare zu spielen, da sie sich nicht nur von Hagens Äußerem angezogen fühlte. Der Nachbar war das krasse Gegenteil von ihrem Ehemann. René war schmächtig, unsportlich und langweilig. Hagen hingegen besaß einen Körper zum Dahinschmelzen, vollendete Manieren und sah gefährlich gut aus. Er war genau das, wovon eine Frau in den Dreißigern träumte, die in einer leidenschaftslosen Ehe gefangen war. "Schön, dass wir uns endlich mal privat kennenlernen, Sabine!", flötete er, während er ihr die Hand entgegenreichte und ihr einen zarten Kuss auf die Wange hauchte. Für einen kurzen Moment fühlte sie seinen muskelgestärkten Körper gegen den ihrigen gedrückt. Diese unschuldige Berührung reichte aus, um ihre Leidenschaft zu erwecken. Zu gern wollte sie erfahren, wie es sich anfühlte, wenn sich seine Haut gegen ihre rieb. "Entschuldige mich, ich werde mir rasch etwas anziehen. Was für ein Gastgeber bin ich, der sich halbnackt vor einer jungen Dame herumlümmelt?" Die Worte rissen sie aus dem Tagtraum. Sie fühlte sich schuldig, bei ihren intimsten Gedanken ertappt worden zu sein. Die Kombination aus diesen beiden Gefühlen und die sexuelle Vernachlässigung, die sie in ihrer Ehe erfuhr, veranlassten sie, ihm zu folgen. Wie ein scheues Mädchen stand sie im Türrahmen und blickte auf den entblößten Rücken ihres gut gebauten Nachbarn. Hagen hatte sich bereits eine Jeans übergezogen. Der Baumwollstoff umschloss seine knackigen Hinterbacken und schmeichelte dem Po. "Meinetwegen hättest du dich nicht anziehen müssen", sprach sie und versuchte, möglichst viel Sex in ihre Stimme zu legen. Hagen grinste. Eigentlich hatte er nicht erwartet, dass sie zu ihm ins Schlafzimmer kommen würde. Betont langsam drehte er sich um. Seine Gedanken rotierten bei dem Versuch, einen passenden Spruch zu finden. Zu seiner Überraschung war sie ihm bereits einen Schritt voraus. Sabine hatte sich die Bluse aufgeknöpft und sprang ihn an. Mit dieser Attacke hatte Hagen nicht gerechnet. Gemeinsam fielen sie auf sein breites Bett. "Nimm mich, wild und leidenschaftlich. In deinem eigenen Interesse hoffe ich, dass du nicht so eine lahme Ente im Bett bist...!" Um ihr zu demonstrieren, dass sie ihre Entscheidung nicht zu bereuen brauchte, legte er seine Lippen auf die ihren und küsste sie lang und leidenschaftlich. Bald hatte sie alles um sich herum vergessen. Sabine gab sich einfach nur dem Genuss des Momentes hin. Keinen Gedanken verschwendete sie an das Nachher. Zu lange hatte sie ihr sexuelles Verlangen unterdrücken müssen. Sie wollte auch mal etwas riskieren und begehrenswert sein. Der Nachbar schien ihr der rechte Partner dafür zu sein. Und sie wurde nicht enttäuscht! Hagen entpuppte sich als ein ausdauernder und einfallsreicher Liebhaber. Er zog ihr nicht einfach die Hosen herunter und nahm sie in der Missionarsstellung, wie es René zu tun pflegte. Hagen verwöhnte sie und gab ihr das Gefühl, eine Frau zu sein. Seine Hände glitten über ihren Körper und befreiten sie aus ihrer Kleidung. Er zog sie nicht einfach aus, sondern entblätterte sie vielmehr. Die Art und Weise, wie er die Kleidung über ihre Haut zu streifen verstand, vermittelte ihr die Illusion, in edle Stoffe gehüllt zu sein. Dieser Mann verstand es, aus ordinärer Baumwolle ein aufregendes Sexspielzeug entstehen zu lassen. Nackt und sinnestrunken rekelte sie sich in seinen Laken. Hagen schätzte ihre Bereitschaft richtig ein und fesselte sie ohne Gegenwehr mit Handschellen an seinen Bettpfosten. Sabines bisher unterdrückte Leidenschaft sorgte dafür, dass sie sich ihm vollkommen auslieferte. Sie vertraute ihm. Mit ihrem eigenen Schal wurden ihr die Augen verbunden. Es erregte sie, nackt, gefesselt und blind vor ihm zu liegen und nicht zu wissen, was er als Nächstes mit ihr vorhatte. Lüstern blickte Hagen auf ihren Körper. Unter all diesen langweiligen Hausfrauenklamotten hatte sich tatsächlich ein Juwel verborgen. Es war ihm unverständlich, was Sabine in ihrem René gesehen haben musste, als sie ihn heiratete. Rasch schob er diese störenden Gedanken beiseite und konzentrierte sich darauf, seiner Gespielin eine aufregende Lektion in Sachen Sex zu erteilen. Leidenschaftlich küsste er sie, um ihre Lust erneut anzustacheln und zu signalisieren, dass sie sich in gute Hände begeben hatte. Seine Fingerspitzen berührten kaum ihre Haut, als er sie sachte an ihrem Körper entlangführte. Die zarte Berührung beschleunigte ihren Atem. Während sie ihres eigenen Tastsinnes und des Sehens beraubt war, waren ihre verbleibenden Sinne umso aktiver. Deutlich hörte sie das Anreißen eines Streichholzes und das verräterische Knistern des Kerzendochts. Der leichte Schwefelgeruch war unverkennbar. Sabine konnte nur ahnen, was Hagen mit ihr vorhatte. Gespannt wartete sie auf den Moment, in dem das Wachs unweigerlich auf ihre nackte Haut tropfen würde. Die Ungewissheit erregte sie. Heftig erschrak sie und zuckte zusammen, als die Berührung an ihrer Brustwarze erfolgte. Obwohl innerlich darauf vorbereitet, war sie doch von der Aktion überrascht worden. Hagen widmete sich mittlerweile auf die gleiche Weise dem anderen Nippel. Wieder zuckte Sabine zusammen, diesmal allerdings nicht so heftig. Sie konzentrierte sich auf das Gefühl und stellte fest, dass es sich nicht um Wachs handeln konnte. Der Gegenstand war kalt und nass. Hagen hatte einen Eiswürfel benutzt. Das Schmelzwasser lief langsam über ihren Brustkorb den flachen Bauch hinunter und sammelte sich im Bauchnabel. Lustvoll stöhnte Sabine auf, als sie die Lippen ihres Liebhabers auf ihrer Körpermitte spürte. Genussvoll nahm Hagen das Eiswasser mit der Zunge auf. Diese unbekannte Liebkosung versetzte ihren Unterleib in Aufruhr. Sabine verzerrte sich nach seinem harten Penis. Doch stellte Hagen sie auf eine weitere Geduldsprobe. Langsam verschoben sich seine heißen Küsse vom Nabel in Richtung Scheide. Sabine stöhnte vor Erregung, je näher er ihrem dunklen Dreieck kam. Seine Lippen hatte die ihrigen noch nicht berührt, da schrie sie bereits vor Wollust. Willig und gierig öffnete sie ihre Schenkel, um dem Partner den Zugang zu ermöglichen. Ein tiefes und durchdringendes Stöhnen gab Sabine von sich, als die Lippen sich endlich auf ihre intimste Stelle legten und sie zärtlich küssten. Hagen führte seine Hände unter ihren Po, um sie so ein wenig anheben zu können und ihr gleichzeitig die Hinterbacken zu kneten. Die junge Frau schien unter seinen Berührungen zu vergehen. Keine Konventionen hielten sie mehr zurück, Sabine ließ sich gehen und gab sich bedingungslos dem Genuss hin. Hagen verstand es, einer Frau orale Befriedigung zu verschaffen. So weit es ihre Fesslung erlaubte, bäumte sich die Nachbarin auf und ließ sich von den Wogen ihres Orgasmus tragen. Hagen nahm ihr die Augenbinde ab und lächelte sie an, während er ihre Handschellen löste. Sabine übernahm die Initiative und drückte ihren Partner sanft in die Kissen. Zufrieden blickte sie auf seinen steifen Penis, der in bemerkenswerter Größe vor Lust stand. Wie eine wilde Amazone setzte sie sich auf ihn und nahm sein Glied in die Enge ihrer feuchten Höhle auf. Hemmungslos ritt Sabine den attraktiven Nachbarn. Die aufgestaute sexuelle Frustration des Ehelebens erfuhr in dem wilden Auf und Ab eine befriedigende Befreiung. Hagen hielt sich nicht mehr länger zurück und kam laut stöhnend. Befriedigt lagen sie sich in den Armen. Sie küssten und streichelten sich. Verträumt spielte Hagen mit ihrem langen Haar. "Weshalb hattest du mich eigentlich aus dem Bett geklingelt?", erkundigte er sich plötzlich bei ihr. Sabine lachte verlegen. "Eigentlich wollte ich dir die Hölle dafür heißmachen, dass du den Gehweg immer noch nicht freigeschaufelt hast", gestand sie und blickte ihn mit einem Male lüstern an. "Aber darum kann sich René kümmern, wenn er von der Arbeit kommt." Frech-frivol lachte sie auf und gab Hagen einen leidenschaftlichen Kuss. Heftig stimmte er ein. Recht hatte sie; ihm stand im Moment der Sinn wirklich nicht nach Schneeschaufeln.
# Liebe in Norwegen
## Marie Sonnenfeld
Inmitten der fallenden Schneeflocken stand Thomas vor seinem Haus am Briefkasten und holte die Post heraus. Beim Hineingehen blätterte er die Briefe durch. Werbung, Rechnungen und... ja, und ein handgeschriebener Brief. Erst wunderte er sich, sah dann aber, dass er von Ricarda war! Mit fliegenden Fingern riss er ihn auf. Er konnte keine Sekunde länger warten, ihn zu lesen. Er war von seiner jungen und überaus hübschen Frau! Sie hielt sich zur Zeit beruflich in den verschneiten Bergen Norwegens auf und Thomas vermisste sie sehr. Schon fast eine Woche war sie inzwischen weg. Und nun schrieb sie ihm – wie schön! Eigentlich hatte er nicht damit gerechnet, auf diesem Wege von ihr zu hören. Eine SMS ja, ein Anruf auch, aber ein Brief? Thomas zog den Briefbogen aus dem Kuvert und setzte sich an den Küchentisch. Ja, es war eindeutig ihre Handschrift. Thomas hatte Herzklopfen und voller Spannung begann er zu lesen: Mein geliebter Thomas, hier in Norwegen ist alles weiß verschneit und es ist sehr, sehr kalt. Ohne Dich erscheint es mir sogar noch um viele Grade frostiger. Deine Wärme und Deine Liebe fehlen mir. Ich habe das Gefühl, nur zur Hälfte hier zu sein, wenn Du nicht bei mir bist. Wie einsam sind die Abende und Nächte ohne Dich! Du fehlst mir sehr, mein Schatz. Ich vermisse Deinen starken schützenden Körper neben meinem und mir fehlt Deine zärtliche streichelnde Hand auf meiner Haut. Es ist eine brennende Sehnsucht, ein heißes Verlangen, das ich fühle, wenn ich meine Gedanken zu Dir in unser Zuhause wandern lasse. Wie gern würde ich mich wieder mit Dir vereinen, mit Dir eins werden. Ich kann es kaum noch aushalten, Dich nicht zu spüren und nicht Deine Leidenschaft und Dein Begehren zu erleben. Meine Finger, die mich gezielt verwöhnen, wenn die Sehnsucht am größten ist, können Deinen wundervollen mächtigen Körper nicht ersetzen. Voller Erwartung fiebere ich dem Moment entgegen, Dich wieder in mir fühlen zu dürfen, Deinen Körper wieder in unstillbarer Lust auf meinem zu fühlen. Denn auch wenn ich mich an so manchen Abenden in Gedanken an Dich selber streichle, mich allein zu lustvollen Höhepunkten bringe, so ist es doch nicht mit Dir und Deinem maskulinen Temperament zu vergleichen. Ich verzehre mich nach Dir und immer, wenn ich mich allein liebkose, meine süße feuchte Liebesmuschel mit meinen Fingern verwöhne, denke ich einzig an Dich. Bei jedem weichen Hineingleiten, bei jeder sinnlichen Berührung meiner Liebesperle stelle ich mir vor, Du wärst es und würdest bald mit viel Lust in mir explodieren. Oh, ich freue mich so unglaublich auf Dich und Deine Liebe! Auf unseren feurigen Sex, Deine erotische Stimme und Deine wundervollen Augen, mit denen Du mich immer wieder verzauberst. Es dauert nicht mehr lange und dann sind wir wieder vereint, dann werden unsere erhitzten Körper wieder zueinander finden. Ich freue mich so sehr darauf! In Liebe Deine Ricarda Gott, was für ein Brief!, dachte er, nachdem er die Hand mit dem Papier wieder sinken ließ. Thomas war erregt, ihre Worte hatten ihn heiß gemacht und seine Lust geweckt. Deutlich fühlte er, dass seine Jeans im Schritt eng wurde und sein Körper danach verlangte, Entspannung zu bekommen. Wie von selbst fand seine rechte Hand den Weg zu den Knöpfen seiner Jeans. Mit einer gezielten Bewegung zog er sie auseinander und griff in seine Retroshorts hinein. Sofort umschloss seine Faust seinen harten Penis und ohne zu zögern begann er, seine Vorhaut vor- und zurückzureiben. Immer wieder strich er sie über seine sensible Eichel und mit jeder Bewegung wurde sein Verlangen nach einem Orgasmus stärker. Während er sich stöhnend befriedigte, sah er auf Ricardas Worte, las sie immer wieder. Er stellte sich vor, wie sie dort allein in ihrem Hotelbett lag. Ihre Hand zwischen ihren Schenkeln, ihr Rücken durchgedrückt, ihre Finger sich immer wieder in ihre enge seidige Vagina hineindrängend. Er sah sie vor sich, wie sie sich in einem lustvollen Höhepunkt auf dem Laken rekelte und stöhnte. Und wie sie nach ihm verlangte, nach seinem harten Glied, welches voller Begehren auf sie wartete. Fast wie in Großaufnahme erschien ihr Geschlecht feucht und offen vor seinem inneren Auge. Er sah ihre geschwollenen Schamlippen und ihre exponierte Klit deutlich und klar. Thomas stöhnte und rutschte auf dem Stuhl weiter nach unten. Er spürte, dass es nicht mehr lange dauern konnte, bis er in einem phantastischen Höhepunkt kommen würde. Er rieb sich weiter. Zog seine Vorhaut immer schneller und straffer nach hinten, den Schaft dabei fest umschlossen. Ja! Ja, sie will es unbedingt, will unseren Sex, will hart von mir genommen werden! Wie lustvoll sie ist! Es ist geil, so geil! Als diese Gedanken ihn beherrschten, hatte Thomas die Augen geschlossen und den Kopf zurückgelegt. Zielsicher bewegte er sich auf den 'point of no return' zu, jeden Moment würde er ihn erreichen. Er atmete schneller, stöhnte wieder leise. In seiner geöffneten Jeans hatte er seinen Phallus inzwischen aus den Shorts befreit und hielt ihn nun groß und hart umschlossen in seiner Faust. So nah dran, so kurz davor! Thomas stoppte und schaute auf sein Glied, welches prall in seiner Hand stand. Er war bis aufs Äußerste erregt. Noch einmal blickte er in den Brief, las ein letztes Mal die Stelle, in der Ricarda von der sinnlichen Berührung ihrer Liebesperle und dem Hineingleiten schrieb. Dann zog er seine Vorhaut nur noch ein Mal vor und zurück und kam. Von einem lauten Aufstöhnen begleitet, durchbrach Thomas die Barriere und ergoss sich auf seiner Brust. Schwer atmend öffnete er seine Augen und stand auf. Nachdenklich verpackte er sich wieder in seinen Hosen. Lächelnd zog er sich den Pulli über den Kopf und nahm sich vor, ihn nachher in die Wäsche zu geben, als er ihn vorerst auf den Fliesenboden fallen ließ. Thomas setzte sich wieder, schaute aus dem Fenster und beobachtete die tanzenden Schneeflocken. Den Brief hielt er dabei noch immer in seiner linken Hand. Er überlegte nicht lange, schnell stand sein Entschluss fest, zu ihr zu fliegen. Er konnte nicht anders, er musste sie sehen. Sein Gefühl und seine Libido ließen keinen Widerspruch zu. Der Flug nach Oslo war über das Internet schnell gebucht. Bereits am frühen Nachmittag sollte sein Flieger starten. Seine Arbeit, die er als selbstständiger Übersetzer von zu Hause aus erledigte, würde einige Tage auf seinem Schreibtisch liegen bleiben müssen. Eilig warf er ein paar Kleidungsstücke in seine Reisetasche, suchte seine Bad- und Hygiene-Utensilien zusammen und legte sie dazu. Auf dem Weg zum Flughafen dämmerte es bereits und es fielen noch immer zarte Schneeflocken vom Himmel, die aber nicht liegen blieben. Dass es im Winter aber auch immer so früh dunkel werden muss, sinnierte er, als er im Fond des Taxis saß und durch die mit Schneematsch bedeckten Straßen Hamburgs fuhr. Thomas spürte, wie sich die Vorfreude auf Ricarda immer weiter in ihm ausbreitete. Wie sie stetig größer und mächtiger wurde. Und wie auch seine Erregung unaufhaltsam zunahm. Oh ja, er freute sich auf sie und ihren begehrenswerten Körper! Nach der Landung im verschneiten Oslo kaufte er zuerst am Flughafen eine Flasche guten Sekt und suchte sich dann die Buslinie zu ihrem Hotel in den Bergen heraus. Nach einer rund dreistündigen Fahrt durch viel Schnee und eine weiße üppige Landschaft, von der er wegen der Dunkelheit allerdings nicht viel sah, kam er schließlich an. Es war bereits nach 20 Uhr und so kalt, dass sein Atem weiß vor ihm herschwebte. Thomas stand vor dem Hauptportal des Hotels und fühlte diese angenehme Aufregung in seinem Bauch. Ja, gleich würde er sie sehen, würde sie in seine Arme schließen. Im Foyer klopfte er sich den Schnee vom Mantel und stampfte ein paar Mal fest auf, um auch seine Schuhe von den größten Schneeklumpen zu befreien. An der Rezeption bekam er ihre Zimmernummer genannt und wenige Minuten später stand er mit der Flasche Sekt in der Hand vor ihrer Tür. Er klopfte. Gleich darauf öffnete Ricarda und ihr erstauntes Gesicht erschien im Türspalt. Sie schauten sich an. Sahen sich einfach nur stumm in die Augen. Beide vor Verlangen und Sehnsucht innerlich bebend. Ohne ein Wort hielt Ricarda die Tür weit auf. Dieser Einladung folgend, trat Thomas ein. Er stand in Ricardas Hotelzimmer, stellte die Flasche auf dem Tisch ab und hörte die Tür hinter sich wieder sachte ins Schloss fallen. Ja, es war ihr Duft, der den Raum erfüllte und der ihm so wunderbar vertraut war. Er erkannte das Parfüm, welches er ihr zu ihrem ersten Hochzeitstag im letzten Jahr geschenkt hatte. Gott, wie sehr er sie wollte! Wie dringend er sie ganz nah und innig zu spüren verlangte. Sie kam zu ihm und immer noch ohne etwas zu sagen, nahmen sie sich in die Arme. Sie hielten einander fest und genossen ihre Nähe, während draußen dicke Schneeflocken am Fenster vorbeitrieben. Sie standen Arm in Arm da, bis Ricarda die Stille durchbrach, ihn ansah und leise sagte: "Es ist wunderschön, dich zu spüren. Aber wie kommt es, dass du hier bist, Schatz?" "Nach deinem Brief blieb mir nichts anderes übrig, als meinem Gefühl zu folgen. Ich konnte nicht anders, ich musste dich sehen, Süße!" Ricarda lächelte. "Hast du ihn also bekommen? Das ist gut." Thomas nickte stumm und nahm ihr Gesicht in seine Hände, um es innig zu küssen. Seine Zunge strich zart über die Innenseite ihrer Oberlippe und ebenso zart saugte er an ihren Lippen. Ricarda erwiderte seinen Kuss gern, hatte sie ihn doch lange genug entbehrt. Sie tauchten ein in diesen Kuss, der allmählich immer fordernder und leidenschaftlicher wurde. Bald waren sie außer Atem und sahen sich mit einem lustvollen Blick in die Augen an. Ricarda raunte ihm zu, wie heiß er sie machen würde und wie unsagbar gern sie mit ihm schlafen wollte. Ihre Worte gingen Thomas durch und durch. Und auf die Art, wie sie sie sagte, fühlte er als Antwort, dass sein Penis härter und härter wurde. Er legte seine Hände auf ihren Po, der noch in ihrem Business- Kostümrock steckte, und presste sie gegen die harte Ausbeulung seiner Jeans. Dabei küsste er ihren Hals und flüsterte ihr zu, wie riesig auch seine Lust auf sie sei. Seine Hände auf ihrem festen Hinterteil ertasteten den Reißverschluss ihres Rocks. Er zog ihn herunter, der Rock fiel zu Boden und wenig später folgte ihm ihre Kostümjacke. Sie entkleideten sich langsam und von liebevollen Küssen begleitet. Immer darauf bedacht, ihre beinahe unbändige Erregung im Zaum zu halten. Stöhnend und tiefer atmend streichelten sie ihre nackte Haut und gerade, als Ricarda ihren geliebten Mann mit sich zum Bett zog, um sich vollkommen unter ihm fallen lassen zu können, sagte er: "Hey, Süße, was ist mit dem Sekt? Auch Lust darauf?" Ricarda konnte seine Erregung deutlich in seiner Stimme hören. Wollte er es denn nicht auch genauso sehr wie sie? Warum erst noch der Sekt? Sie konnte nicht mehr warten, wollte ihn jetzt sofort. Sein harter Phallus in ihrer Hand, den sie eben noch streichelte, hatte sie noch schärfer gemacht, als sie es ohnehin schon war, und seine Liebkosungen ihrer feuchten und verlangenden Mitte hatten ein Übriges dazu beigetragen, ihre Begierde zu schüren. "Nein, bitte nicht jetzt..." Ihre Stimme zitterte vor Erregung. Es machte Thomas unbeschreiblich an, ihr Verlangen so direkt zu erleben und noch ein klein wenig hinauszuzögern. Den Moment, den sie so sehr herbeisehnte, noch aufzuschieben. "Doch, Liebling, nur einen Schluck auf unser Wiedersehen, hm?" Er presste sie wieder fest an sich und hob sie ein klein wenig hoch, so dass sein pralles Glied ganz unmittelbar vor ihrem seidigen Eingang lag. Seine Eichel drang ein kleines Stück in sie ein, nur die Spitze, nur einen Zentimeter. Ricarda stöhnte auf und Thomas sah angeheizt in ihr Gesicht, in dem ihre Wollust deutlich zu lesen war. Sie spreizte ihre Beine weiter, aber er zog sich wieder zurück. Auch er war am Rande seiner Selbstbeherrschung, aber trotzdem griff er sich die Flasche und zog Ricarda mit sich auf das Bett. Hier saßen sie eng aneinandergeschmiegt, sie feucht, mit einer einladend offenen Mitte, und er mit einer festen großen Erektion. Ihre Blicke gingen tief, als sie sich in die Augen sahen und Thomas die Sektflasche öffnete. Er gab sie ihr und sie nahm einen Schluck, dann reichte sie ihm die Flasche zurück, den Blick die ganze Zeit über nicht abwendend. Auch als Thomas trank, blickten sie sich weiter in die Augen. "Du hast die schönsten Augen der Welt", sagte Thomas leise und strich Ricarda das Haar aus dem Gesicht. Sie lächelte. "Bekomme ich jetzt, was ich mir so sehr wünsche?" Sie formulierte ihre Frage so süß und liebevoll, dass er nicht anders konnte, als zu nicken. Er stellte den Sekt auf dem Nachttisch ab, während Ricarda sich wohlig zurücklegte. Thomas schmiegte sich zwischen ihre geöffneten Schenkel und lustvoll glitt seine warme weiche Zunge über ihre Liebesperle, die sich ihm bereits erwartungsvoll entgegenreckte. Ricarda stöhnte auf. "Ja, es ist sooo gut!" Thomas fuhr damit fort, sie intensiv zu liebkosen. Immer wieder leckte und saugte er. Gleichzeitig streichelte er sanft die weiche Haut an ihrem Damm und kreiste mit der Fingerspitze seines Zeigefingers um ihren geschwollenen Eingang. Ricarda wand sich vor ihm und öffnete sich ihm vollkommen. Thomas konnte es kaum noch ertragen, nicht mit ihr zu schlafen, nicht in sie einzutauchen. Er sah ihr williges Geschlecht genauso vor sich, wie in seiner Phantasie vorhin an ihrem Küchentisch. Er bewegte sein Becken und rieb so seinen vollkommen harten Penis auf dem Laken unter sich. Er musste aufpassen, zu heftig durfte er nicht werden. Obwohl er in diesem Moment nichts lieber getan hätte, als geradewegs auf seinen Orgasmus zuzusteuern. Als er spürte, dass sie bald kommen würde, dass ihr Höhepunkt kurz bevorstand, hatte Thomas das Gefühl, nur noch aus Lust zu bestehen. Er konnte an nichts an- deres mehr denken und auch Ricarda war nur noch von dem einzigen Wusch erfüllt, sich endlich mit ihm zu vereinigen. Thomas wollte ihr die Entscheidung über den Zeitpunkt überlassen und so fragte er, aus ihrem nassen Delta aufsehend: "Willst du kommen oder willst du mich?" "Dich!" Ricarda rief es ihm laut entgegen und er zögerte keine Sekunde länger, sich in höchster Lust auf sie zu schieben und mit einem animierenden Geräusch stöhnend in sie einzudringen. Er schob sich in ihr seidiges Paradies hinein. Ricarda bog ihm ihre Hüfte entgegen. Lange konnte er es nicht zurückhalten. Die Situation war eine so besondere und derart einzigartig, impulsiv und heiß, dass Thomas es nicht mehr gelang, sich mit neutralen Gedanken herunterzukühlen. Er wollte Ricardas Orgasmus und dann selbst auch die Erlösung in einem grandiosen Höhepunkt finden. Noch einmal zog er sich aus ihr zurück und nahm seine Eichel in die Hand. Er rieb sie feucht und heiß über ihre Klit und beschrieb ihr, was er gerade tat. Ricarda stöhnte und wimmerte. Kurz darauf wurde sie von ihrem Orgasmus überrollt. Zufrieden sah Thomas in ihr Gesicht, welches sich entspannte, und betrachtete ihr Kommen als seinen Startschuss. Er hob ihre Beine zu ihren Schultern und legte sich wieder auf sie. Ricarda umfasste ihre Oberschenkel mit ihren Armen und stöhnte laut auf, als er erneut in sie drang. Sie fühlte seinen beachtlichen Phallus in sich gleiten und bewegte sich automatisch in seinem Rhythmus mit. Durch diese Stellung war er sehr tief in ihr, was er bei jeder Bewegung lustvoll fühlte. Nur noch auf seinen Orgasmus zusteuernd und von Wollust und Gier getrieben, schlief Thomas kraftvoll mit ihr. Ihre Körper klatschten gegeneinander und ihr Stöhnen und Keuchen war laut, als er sich bald darauf mit einem animalischen Stöhnen in ihr entlud. Diese Nacht, irgendwo in den Bergen Norwegens, konnte gegensätzlicher nicht sein. In dem Hotelzimmer von Ricarda brannte das Feuer der Liebe, Lust und Leidenschaft und draußen, schon unmittelbar vor ihrem Fenster, tobte der kalte Schneesturm, der klirrenden Frost und Eis aus dem Norden mit sich brachte.
# Winterliches Intermezzo
## Dave Vandenberg
Wir wollten noch eine letzte ausgiebige Loipe fahren, nach der wir eine Pause einlegen würden, um in der kleinen gemütlichen Imbisshütte oben auf dem Berg etwas zu essen. Es war ein sonniger Wintertag mit strahlend blauem Himmel. Zu schade, um auch nur eine einzige Sekunde ungenutzt zu lassen. Marion und ich hatten seit Monaten auf diese zwei Wochen Skiwandern hingearbeitet. Unser Geschäft lief derzeit nicht besonders, obwohl wir gerade in den letzten Wochen nahezu rund um die Uhr gearbeitet hatten. Es war wohl der allgemein schlechten Wirtschaftslage zuzuschreiben. Kein wirklicher Trost. Aber es lag wenigstens nicht an unserem persönlichen Unvermögen, dass so wenig Umsatz zu machen war, versuchten wir uns gegenseitig aufzubauen. Unsere Ehe litt natürlich unter dem geschäftlichen Druck und der ganzen Schufterei. Richtigen Sex hatten wir schon ewig nicht mehr gehabt. Und wenn, dann nur so was Schnelles, für Marion meist Unbefriedigendes. Mehr schafften wir im Moment einfach nicht. Wir hatten gehofft, der Urlaub würde uns helfen, im Bett wieder zueinander zu finden. Nun waren wir schon seit vier Tagen hier und hatten noch nicht miteinander geschlafen. Noch immer zu müde, nicht genug abgeschaltet, der Klimawechsel war daran schuld, redeten wir uns ein. Irgendwie schien sich eine körperliche Barriere zwischen uns aufgebaut zu haben und keiner wusste, wie er sie niederreißen konnte. Sex hatte für uns seine Spontaneität verloren. Die erotische Liebe schien mit einem Mal unglaublich schwerfällig geworden zu sein... Nach dem Essen tranken wir noch einen Kaffee und machten uns bald schon wieder auf zu unserer letzten Loipe des Tages. Wir hatten noch einige Kilometer vor uns und wir waren nicht so schnell auf unseren Skiern, wie wir uns erhofft hatten. Die Loipe führte in spiralförmigen Pfaden weg von dem Berg, durch einen wunderschön verschneiten Fichtenwald. Ich sah Marions schlanke Gestalt vor mir durch den Schnee gleiten. Sie hatte eine gute Figur. Der Ski-Anzug stand ihr hervorragend. Warum verspürte ich nur so wenig Lust darauf, mit ihr zu schlafen? Nach ein paar Kilometern stoppte sie plötzlich so abrupt, dass ich fast auf ihre Skier gefahren wäre. "Was, zum Teufel...? Ich fluchte laut. Sie drehte sich zu mir um, legte den Zeigefinger auf ihren Mund und lauschte. Rechts vor uns bog ein kleiner Pfad ein. Zwei frische Paar Skispuren waren zu sehen. Und dann hörte ich es auch. Ein merkwürdiges Glucksen. Stimmenraunen. Zweideutige Geräusche. Unschlüssig sahen wir uns an. Und dann lächelte meine Frau wissend und bog in den Weg ein und ich fuhr ihr zögernd hinterher. Sie kam mir plötzlich vor wie ein Hund, der eine unwiderstehliche Witterung aufgenommen hat. "He, was...!" Sie war wieder stehen geblieben. Und dann sahen wir es. Auf einer kleinen sonnenbeschienenen Lichtung vergnügten sich zwei Liebende miteinander. Hier mitten im Wald, am helllichten Tag, in winterlicher Kälte. Automatisch duckten wir uns. Marion zog mich hinter einen großen Busch. Vorsichtig bog sie die dick verschneiten Zweige auseinander. Wir starrten nach vorn und konnten es immer noch nicht glauben. Ich schnappte nach Luft. Noch nie hatten wir andere beim Sex beobachtet. Wir hatten uns bis jetzt nicht mal einen Pornofilm angesehen. Das hier war absolutes erotisches Neuland für uns. Der Mann, der außergewöhnlich kräftig schien, hatte sich die Frau auf seine nackte Hüfte gesetzt und schob sie auf seinem Penis gleichmäßig auf und ab. Die oben herum gut bestückte Brünette hatte ihren Ski-Overall über der Brust geöffnet und ließ ihren Partner an ihren spitzen harten Nippeln lutschen. Ihr Kopf lag im Nacken und sie stöhnte hörbar und selig vor sich hin. Die tiefen grunzenden Laute des Mannes unterstrichen seine muskulösen, fast animalischen Bewegungen. Er strahlte eine ungehörige Dominanz aus. Marion zuckte bei jedem seiner Stöße zusammen. Ich bewunderte seine sexuelle Kraft. Wir waren gelähmt vor voyeuristischer Neugierde... Meine Frau blinzelte mich an. Ich hatte meine Hand auf ihr dick verpacktes Hinterteil gelegt, um es zärtlich zu tätscheln. Mein Glied versteifte sich dabei und wurde endlich mal wieder so richtig hart, ohne dass ich nachhelfen musste. Es blitzte plötzlich auf in ihren Augen. Ich kannte dieses Funkeln immer noch, auch wenn ich es schon lange nicht mehr bei ihr gesehen hatte. Wir küssten uns scheu und blickten wieder auf die Lichtung. Das Paar sah aus, als wenn es kurz vorm ekstatischen Finale stehen würde. Marion griff nach meiner Hand. Doch dann hob der Mann die Frau von sich herunter, dreht sie um und beugte sie nach vorn. Sie kniete sich auf ihre dicke Winterjacke und wedelte ziemlich vulgär mit ihrem Hinterteil. Ich konnte es feucht zwischen ihren Pobacken glitzern sehen. Ich schwöre sogar, ich konnte, als sie ihr Hinterteil geschickt ein Stückchen aufwärts bog, einen Moment lang ihre liebeshungrige kleine Erhebung ausmachen. Dieser wirklich obszöne Anblick setzte ein erotisches Strohfeuer in mir frei, das so schnell nicht wieder erlöschen sollte. Als sich der Mann von hinten mit seinem steil aufgerichteten Glied in seine Partnerin hineindrängte, schob ich meine Hand zwischen die Beine meiner Frau und ließ sie dort liegen, wo ich ihre lustbringende Erhebung spüren konnte. Unter meinem sanften Druck wurde sie noch deutlich tastbarer. Ich presste meine Hand darauf, bis sich Marion zu mir umdrehte und ich sah: Das Funkeln in ihren Augen war noch stärker geworden. Es war eine ganz besondere Art von Erregung, die uns jetzt gepackt hatte. Wir schnallten leise unsere Skier ab. Ich nahm Marion bei der Hand und zog sie vorsichtig hinter mir her durch das lichte Gehölz. Bevor sie was sagen konnte, hatte ich sie aufgerichtet und ihren Mund mit einem sehr feuchten Kuss verschlossen, der meine mittlerweile überwältigende Erregung ausdrückte. Ich presste Marion an einen Baumstamm und umklammerte ihre Brüste. Fest und prall fühlten sie sich an. Sogar durch den dicken Stoff ihres Ski-Anzuges hindurch konnte ich ihre aufspringenden Knospen fühlen. Hart und erwartungsvoll begrüßten sie mich. Ich kniff hinein. Nicht sanft, nicht zärtlich, sondern unnachgiebig. Marions Lippen öffneten sich sehnsüchtig. Meine Zunge glitt tief in ihren Mund. Wir hatten uns lange nicht mehr so leidenschaftlich geküsst. Ich zog hektisch den Reißverschluss ihres Ski-Anzuges herunter. Sie genoss das ungestüme Auspacken ihrer Brüste. Dehnte sich unter meinen Berührungen und schnurrte dabei wie eine Katze. Ihr Gesicht hatte endlich mal wieder einen entspannten Ausdruck angenommen. Zufrieden sah sie zu, wie ihre Brüste schließlich ganz entblößt in meinen Händen lagen. Sie hob sie mit ihren Händen für mich an, präsentierte sie wie auf einem Tablett, bereitgestellt zu meiner Verfügung. Die Sonne schien durch die Schnee verhangenen Äste über uns hindurch und rückte ihre gleichmäßig geformten Brüste in ein wunderschönes Licht. Ich begehrte sie mit einer nicht zu beschreibenden sehnsuchtsvollen Gier. Marion fummelte unter meiner Jacke herum, um den Weg zum Verschluss meiner Hose zu suchen. Sie griff nach meiner beachtlichen Erregung und drückte sie unbeherrscht. Fast ein wenig grob. Und gerade das gefiel mir. Obwohl ich mittlerweile ihren Pullover hochgeschoben hatte, fror sie nicht. Im Gegenteil. Die Hitze schoss ihr zwischen die Beine bis in die Brüste hinauf. Ich sog und leckte an ihren Nippeln, die sich hungrig in die klare Winterluft reckten, und konnte meine Fingerspitzen an ihrer Geilheit wärmen. So musste es sein! Wie hatte ich diese Art von Kick vermisst! Endlich schob sie ihre Finger direkt auf mein Glied. Es wuchs noch weiter, während sich ihre Hände immer fester darum schlossen, sich daran rieben. Ich stöhnte begeistert. Ihr Griff war immer noch überwältigend für mich. Unter halb geschlossenen Lidern prüfte meine Frau die Umgebung. Auch ich sah mich kurz vergewissernd um. Es war nichts zu sehen. Beruhigt widmete sie sich nun mit Hingabe meinem großen lüsternen Glied. Sie hockte sich vor mich und nahm es in den Mund. Zärtlich und bestimmend, einfühlsam und herrisch. Ihre Finger kneteten meine Hoden. Ihre Zunge massierte meinen Schaft. Als sie ihre Zähne vorsichtig in meine Eichel grub, hielt ich es nicht mehr länger aus. Ich packte meine Frau, riss sie hoch und drückte sie zurück an den Baumstamm. Sie half mir mit fahrigen Händen, mein Glied zwischen ihre Beine zu manövrieren und es schließlich nach einigen Versuchen in ihre verführerische Feuchtigkeit hineinzuschieben. Es war fast wie unser erstes Mal vor etlichen Jahren. Hastig, ungeduldig, ungestüm und wunderbar. Ich stieß mit unkontrollierten Bewegungen zu. So wie damals, als ich Marion gerade kennengelernt hatte. Meine Hüfte drängte sich vorwärts. Unnachgiebig umklammerte ich ihre Brüste, biss in ihre Warzen, leckte an ihren Höfen und presste meine Knie zwischen ihre, um meinem außer Kontrolle geratenen Glied etwas mehr Bewegungsfreiheit zu gewähren. Die winterweiße Lichtung war angefüllt von lustvollem Seufzen und dem Geruch nach willkürlichem Sex und lang ersehnter Sünde. Immer schneller bewegte ich mich in ihr. Es war nicht nur die Geilheit, es war auch ein bisschen der Nervenkitzel, dass uns jemand entdecken könnte. Meine Frau ließ sich anstecken von meiner hastigen Art, sie zu befriedigen. Ihre Küsse glitten über mein Gesicht hinweg. Sie schüttelte ihren Unterleib vor Gier und dann, mit einem Mal, zog sich ihr Vaginalmuskel heftig zusammen. Ich wusste, sie konnte jeden Moment kommen. Sie griff wieder und wieder nach meinen Hoden, drückte sie und schrie vor Glück, als ich mich tobend in ihr entlud. Marion folgte mir, noch während ich sie mit meinem Liebesnektar überflutete, in einen auch für sie unvergesslichen Höhepunkt. Etwas später schnallten wir unsere Skier wieder an und glitten langsam auf dem kleinen Pfad entlang, vorbei an der Lichtung, auf der uns das fremde Liebespaar so wunderbar sexuell inspiriert hatte. Sie hatten ihren Akt beendet. Wir folgten ihren Spuren zurück auf die eigentliche Loipe. Es hatte ein wenig angefangen zu schneien. Um uns herum wurde uns die schönste Winterimpression beschert, die wir jemals erlebt hatten. Ich drehte mich zu meiner Frau. Sie strahlte mich glücklich an. Ich erwiderte ihr Lächeln. Wir liebten uns. Immer noch. Und wieder. Wir hatten die körperliche Barriere, die sich in der letzten Zeit zwischen uns aufgebaut hatte, niederreißen können. Mit einem außergewöhnlichen Sexerlebnis. Ich nahm mir fest vor, es sollte nicht bei diesem einen erotischen 'Ausrutscher' bleiben...
# Sinnliche Verführung
## Felicia
Du hattest eine Überraschung für mich geplant, ich sollte ein paar Sachen zusammenpacken und dann hast du mir die Augen verbunden. Wir fuhren eine ganze Weile im Auto und ich hatte keine Ahnung, wohin. Irgendwann sagtest du, dass wir endlich angekommen seien und ich durfte die Augenbinde abnehmen. Überall Schnee und ich wusste nicht, wo wir waren. Du lächeltest über das ganze Gesicht und sagtest mir, dass wir das Wochenende über in der Schneehütte deiner Eltern sind. Wir gingen hinein und es war ziemlich kalt. Wir packten unsere Sachen aus und ich war völlig durchgefroren. Du ließest mir ein heißes Bad ein und sagtest, ich solle mich doch darin etwas entspannen. Ich lächelte dich an und gab dir einen liebevollen Kuss. Willst du nicht mit?, fragte ich dich, aber du schütteltest nur lächelnd den Kopf. Geh du nur, ich muss hier noch etwas machen. Ich legte mich in die schöne Wanne und genoss das heiße Wasser. Leicht bekleidet, nur mit einem Hemd, von dem ein paar Knöpfe geschlossen waren, kam ich langsam heraus. Die Hütte war total dunkel und überall standen Kerzen und du hattest Feuer im Kamin angemacht. Vor dem Kamin lag ein schönes weißes Fell und daneben standen Trauben, eine Flasche Sekt und Sahne. Ich lächelte dich an und kam langsam auf dich zu. Leise Musik ertönte aus dem Radio und du strecktest mir deine Hand entgegen. Ich gab dir meine und du zogst mich an dich. Lächelnd trafen sich unsere Blicke, bis deine Lippen sanft die meinen berührten. Liebevoll und sanft küssten wir uns... Wir schmiegten uns aneinander und wogen uns ein wenig zu der Musik, während mein Kopf auf deiner Schulter ruhte und deine Hand zärtlich meinen Rücken hinabfuhr. Du ließest mich langsam auf das Fell nieder und beugtest dich über mich. Wieder trafen sich unsere Lippen und versanken zu einem innigen Kuss. Du gabst mir mein Glas und ich nippte daran, ein Tropfen entglitt meinem Mund, doch du küsstest ihn mir ganz sanft von den Lippen. Sanft streichelte ich dir mit meiner Hand deinen Nacken entlang, hin zu deiner Wange und deinem Mund. Deine Lippen berührten meinen Handrücken und küssten meine Finger. Ich zog dir langsam deinen Pulli über den Kopf und küsste deinen Hals entlang. Du hattest mich nach hinten gelegt und dich über mich gebeugt. Meine Hände streichelten wie ein Windhauch so zart über deinen Rücken. Die Wirbelsäule rauf und runter, immer wieder und ganz zärtlich. Deine Lippen berührten meine und wanderten zu meinem Hals. Mit der Zungenspitze berührtest du meinen Kehlkopf und jede freie Stelle. Deine Hand fuhr unter das Hemd und streichelte zärtlich über meinen Bauch. Ganz langsam hattest du die Knöpfe aufgemacht und auf jede freie Stelle einen kleinen Kuss gegeben. Du erhobst dich wieder und lächeltest mich an. Ich hab dich nur fragend angesehen und da hattest du auch schon eine Traube in der Hand. Du legtest sie in meinen Bauchnabel und hattest die Gegend erst mit deiner Zunge umspielt, um die Traube dann sanft und spielerisch herauszuholen und zu essen. Du kamst nach oben, küsstest mich, und ich konnte den süßen Geschmack der Traube genießen. Du drehtest dich um und hattest plötzlich einen Seidenschal in der Hand und mich fragend angesehen. Ich wusste, was du vorhast, und nickte dir nur leicht entgegen, erhob meinen Kopf und du verbandest mir die Augen. Ich legte mich wieder zurück und spürte dich zwischen meinen Beinen sitzen. Du hattest langsam ein paar Tropfen des Sekts auf meinen Bauch träufeln lassen und sie zärtlich und liebevoll weggeküsst. Immer und immer wieder. In meinen Bauchnabel gabst du auch etwas davon und hast es sanft herausgeleckt. Du standest auf und sagtest, du seiest gleich wieder da. Ich hörte, wie du wegliefst und kurz darauf auch schon wieder da warst. Du knietest dich erneut zwischen meine Beine und ich spürte, wie etwas Kaltes auf meinen Bauch tropfte. Ich bäumte mich kurz auf und dann wurde es noch kälter und nass. Du hattest einen Eiswürfel geholt... Du umspieltest meine Brustwarzen mit dem Eiswürfel, so dass sie sich hart aufrichteten. Umfuhrst meine Lippen und küsstest sie darauf wieder trocken. Als das Eis langsam kleiner wurde, nahmst du es in den Mund und gabst mir einen tiefen und innigen Kuss. Meine Erregung war kaum noch zu halten, ich wollte dich anfassen, berühren und spüren, aber so weit waren wir längst nicht. Du sprühtest von der Sahne auf meine Brustwarzen und lecktest es sanft wieder runter und deine Küsse wanderten zu meinem Bauchnabel. Du sprühtest von der Sahne auf meinen Venushügel und ganz sanft, nur mit der Zungenspitze, machtest du mich wieder sauber. Plötzlich drang ein Finger tief in mich ein und ließ mich leise aufstöhnen. Langsam hattest du angefangen, ihn in mir zu bewegen. Meine Lust wurde dadurch nur immer größer. Deine Lippen und deine Zunge berührten ganz leicht meine Perle, wie ein Windhauch streiften sie darüber. Du spieltest an ihr und sanft pustetest du dagegen. Ich wendete mich hin und her vor Lust und wollte dich endlich auch berühren. Deine Zunge wurde fordernder und dein Finger schneller. Du spürtest, dass ich schon ganz kurz davor war und hörtest plötzlich auf. Deine Küsse wanderten wieder hoch zu meinen Lippen und dein Finger langsam aus mir. Du nahmst mir die Augenbinde ab und lächeltest mich an. Ich war total verdutzt und drehte dich einfach um. Saugte an deinen Brustwarzen und knabberte leicht daran. Meine Hand wanderte zu deiner Hose und öffnete sie. Langsam zog ich sie dir aus und küsste jeden Millimeter deiner Haut, der mir auf dem Weg nach unten begegnete. Ich tröpfelte von dem Sekt auf deinen Bauch und leckte ihn zärtlich wieder ab. Du stöhntest leicht auf, als meine Hand sanft deinen Freund berührte. Ich streichelte zärtlich über ihn, während die andere Hand deine Hose ganz auszog. Ich nahm eine Traube in den Mund, beugte mich über dich und neckte dich damit. Du legtest deine Hand um meinen Nacken, zogst mich hinab, nahmst die Traube mit einem Biss und küsstest mich. Ich widmete mich inzwischen wieder deinem besten Stück. Ich sprühte von der Sahne auf deine Eichel und lutschte sie ganz sanft wieder ab. Du stöhntest leicht auf und meine Hand umfasste sie. Ganz langsam ließ ich sie, von meinen Lippen umschlossen, in meinen Mund gleiten. Meine Zunge umspielte sie und immer wieder saugte ich ganz leicht an ihr. Ich spürte, wie deine Erregung immer größer wurde und ließ mit meinem Mund von dir ab. Ich kam nach oben und küsste liebevoll deine Lippen. Jetzt konntest du es nicht mehr aushalten, zogst mich weiter hoch, so dass ich deinen Freund zwischen meinen Beinen spüren konnte. Ganz langsam drang er in mich ein... Ich begann mich langsam auf dir zu bewegen, während deine Hände zärtlich meine Brüste kneteten. Du presstest mir dein Becken entgegen, so dass ich dich noch tiefer in mir spüren konnte. Ich beugte mich nach hinten und stützte mich auf deinen Beinen ab, während deine Hände sich fest in meinen Po krallten. Immer schneller werdend spürten wir beide, dass wir ganz kurz davor waren. Ich wurde langsam und beugte mich erschöpft nach vorne. Legte meinen Kopf auf deine Schulter und ließ deinen Besten langsam hinaus. Du streicheltest sanft meinen Rücken entlang und lächeltest mir zu. Ich gab dir einen liebevollen Kuss und hauchte dir leise ins Ohr, wie wunderschön es war und dass ich dich lieb hatte. Deine Hand streichelte durch mein Haar, ein Kuss auf meiner Stirn, ich hab dich auch lieb, mein Engel. Wir deckten uns zu und schliefen gemeinsam Arm in Arm ein.
# Spaß im Schnee
## Diane Bertini
Die Abfahrt jagte Sabrina etwas Angst ein. Sie stand auf ihren neuen Skiern noch nicht so sicher wie die anderen jungen Leute, die bereits unter wildem Gekreische und Gejohle den Hang hinabsausten. Gerard, der süßeste Boy, dem die junge Frau je begegnet war, zeigte Mitleid und wartete, bis Sabrina sich ein Herz nahm und die ersten Schwünge Richtung Tal tat. Erst vor wenigen Tagen hatte sie ihren ersten Skikurs erfolgreich abgeschlossen, sich der Anfängergruppe angeschlossen und versuchte nun, Routine zu gewinnen. Sie war am Morgen die Erste auf der Piste und am Abend die Letzte, die sich von Gerard, dem Skilehrer, verabschiedete. Gerard blickte dem etwa 20-jährigen Mädchen begehrlich hinterher. Sie machte eine aufregende Figur in ihrem atemberaubend engen Ski-Anzug, der wie eine zweite Haut saß. Der weiße Anorak stand ihr hervorragend und die braune Hose zeichnete ihre herrlichen Formen nach. Sie hatte schon jetzt eine gute Haltung, bewegte sich rhythmisch und tanzte in recht beachtlichen, eleganten Schwüngen die Piste hinunter. Er holte sie schnell ein, obwohl sie vor ihm immer wieder den leicht pulvrigen Schnee aufwirbelte und ihn so in eine Schneewolke einhüllte. Er beobachtete sie heimlich. Ihr hübsches Gesicht glänzte in der Sonne, ihren Mund umspielte ein zufriedenes Lächeln. Plötzlich allerdings wurde ihr eine Bodenwelle zum Verhängnis, sie verriss einen Ski und stürzte. Sekunden später war Gerard bei ihr. "Hast du dir wehgetan?" Sabrina lächelte. "Glaub nicht", antwortete sie, stützte sich mit den Skistöcken im Schnee ab und versuchte, aufzustehen. Ihr rechter Knöchel schmerzte, trotzdem schüttelte sie den Kopf. "Nein", versicherte sie dem jungen Mann. Sie wollte sich keine Blöße geben und als wehleidige Zicke gelten. Sie schenkte ihm ein zauberhaftes Lächeln und fuhr weiter. Doch Gerard merkte schnell, dass sie sich, obwohl das Gegenteil behauptend, verletzt hatte. Sie stand plötzlich weniger locker auf ihren Skiern, bewältigte die schnellen Schwünge bei Weitem nicht mehr so elegant wie vorher. Da es bis ins Tal jedoch nur noch wenige Meter waren, ließ er die junge Frau erst mal gewähren. Mit einem eleganten Bogen kam er schließlich knapp vor ihr zum Stehen und wollte schon zu einer Ermahnung ansetzen, als er ihre bleichen Wangen bemerkte. Ganz offenbar hatte sie mittlerweile schlimme Schmerzen. "Du solltest für heute Schluss machen", riet Gerard. "Ich bring dich in dein Hotel, okay?" Sabrina nickte nur und hauchte ihm ein leises "Danke" entgegen. Sie öffnete die Skibindung, dann schnell ihre Skistiefel, und war froh, dass das Hotel, in dem sie vor vier Tagen eingecheckt hatte, gleich gegenüber der Talabfahrt lag. So musste sie nur die Straße überqueren. Gerard nahm ihre Ausrüstung entgegen und stellte sie im Skikeller ab. "Ich würde mir dein Bein gern mal ansehen, wenn du nichts dagegen hast", sagte er, während er Sabrina zum Aufzug begleitete. "Vielleicht brauchst du ja einen Arzt." Entsetzt schüttelte Sabrina den Kopf. "Bestimmt nicht. Es piekt nur ein wenig. Wenn ich das Bein nicht belaste, ist es gar nicht so schlimm", beruhigte sie den Skilehrer. Der winkte ab. "Du solltest das Ganze nicht auf die leichte Schulter nehmen, schließlich..." Noch bevor Gerard seinen Satz beenden konnte, öffnete sich der Aufzug. Sabrina ging schräg über den Flur des Hotels und öffnete eine Tür. Höflich hielt sie dem Skilehrer die Tür auf und trat nach ihm in den Raum. Sie zog ihren Anorak aus und ließ sich auf einem breiten Sessel nieder. Dann schlüpfte sie aus ihren bequemen Gesundheitssandalen, die sie im Skikeller gegen die steifen Skistiefel getauscht hatte, zerrte sich den Socken vom rechten Fuß und hielt dem Mann ihr Bein entgegen. "Hmm, der Knöchel ist schon leicht blau. Du solltest einen kühlenden Umschlag anlegen und den Fuß ruhigstellen. Dann ist es morgen sicher viel besser." Sabrina nickte. "Und was empfiehlst du?" "Ich mach dir einen Vorschlag. Ich besorge dir etwas aus der Apotheke, gehe schnell nach Hause, um mich umzuziehen, und helfe dir dann beim Anlegen des Wickels. Vielleicht können wir ja zusammen zu Abend essen?" Sabrina nickte erneut. "Du bist sehr nett, vielen Dank. Ich nehme dein Angebot gerne an. Und solange du unterwegs bist, stelle ich mich unter die Dusche." Gerard erhob sich, drückte dem Mädchen einen Kuss auf die Stirn und verschwand. Sabrina humpelte ins Badezimmer und genoss das warme Wasser, das über ihren Körper rieselte. Der Knöchel schmerzte wirklich ziemlich und sie freute sich, dass Gerard ihr seine Hilfe angeboten hatte. Nach dem Abtrocknen schlug sie sich ein Tuch um die nassen Haare und kramte zwischen ihren mitgebrachten Dessous nach den passenden Stücken. Immerhin hatte Gerard sie vorhin geküsst, wenn auch nur auf die Stirn. Trotzdem war Sabrina bei diesem kumpelhaften Kuss das Herz in die Hose gerutscht. Vielleicht ergibt sich ja heute Abend die Gelegenheit, den hübschen Kerl etwas näher kennenzulernen, überlegte sie. Spontan entschied sie sich für einen verführerischen Balconette-Büstenhalter in Schwarz, der ihre vollen Brüste vorteilhaft zur Geltung brachte, dazu ein ebenso raffiniertes Höschen mit edler Spitze und durchsichtiger Rückansicht. Anschließend schlüpfte sie in einen seidenen, bunt gestreiften Bademantel. Nachdem sie einen Hauch Make-up aufgelegt hatte, betrachtete sie sich zufrieden im Spiegel. Sie hatte die langen schwarzen Haare zu einem lustigen Pferdeschwanz zusammengebunden, der lockige Pony hing ihr kokett in die Stirn. Wenig später klopfte es. Gerard stand vor der Tür, bepackt mit zwei riesigen Tüten. In gespieltem Ernst stöhnte er laut auf, während er die Tüten auf Sabrinas Bett abstellte. "Was macht der Fuß?", erkundigte er sich fürsorglich und zauberte eine braune Apothekerflasche und eine elastische Tapebinde zwischen seinen Besorgungen hervor. "Geht so", antwortete Sabrina. "Was, zum Teufel, hast du da noch alles mitgebracht?", fragte sie grinsend und warf einen Blick auf die noch immer prall gefüllten Papierbeutel. "Ich hielt es für sinnvoll, wenn du den Fuß so wenig wie möglich belastest und wir heute nicht groß ausgehen. Ich dachte mir, wir könnten vielleicht hier eine Kleinigkeit essen", stellte er mit fragendem Unterton fest, während er allerlei Leckereien, zwei Flaschen Rotwein und Süßigkeiten auspackte. Sabrinas Herz tat einen Sprung. Er scheint mich zu mögen, jubelte sie innerlich, sonst hätte er sich nie die Mühe gemacht, uns ein Abendessen zu besorgen. Und der Rotwein spricht eigentlich für sich, sinnierte sie weiter. "Darf ich jetzt deinen Fuß sehen?", riss Gerard das Mädchen aus ihren Überlegungen. Sabrina setzte sich auf die Bettkante und hielt ihm ihr nacktes Bein hin. Vorsichtig streichelte Gerard die bereits angeschwollene Stelle an ihrem Knöchel, tränkte einen weißen Mullstreifen mit Alkohol und legte einen kunstvollen Verband an. "Wo hast du das denn gelernt?", fragte Sabrina ehrlich erstaunt. "Als Skilehrer hat man eben seine Erfahrungen", konterte Gerard. "So was passiert immer wieder. Wir können unsere Schüler ja nicht jedes Mal gleich zum Arzt schicken. Wenn du den Verband schön fleißig alle vier, fünf Stunden auswechselst, bin ich sicher, dass du spätestens übermorgen wieder zum Skifahren kannst." "Aber da verliere ich einen kompletten Tag", beklagte sich Sabrina. Gerard, der zwischen den langen schlanken Beinen Sabrinas auf dem Boden hockte, sah zu ihr hoch. Sein Blick verriet ihr, dass dieser Ausfall nicht das Schlimmste war, was ihr passieren konnte. "Ich hab morgen frei. Wenn du willst, können wir ja gemeinsam etwas unternehmen, das dich auf andere Gedanken bringt", bot er an, während er zärtlich eine Hand auf ihr linkes Knie legte und sie sehnsüchtig anblickte. "An was dachtest du?", fragte Sabrina zurück und tat vollkommen unschuldig, während ihr Herz erneut einen Hüpfer machte. "Na, es sollte natürlich deinen Fuß nicht allzu sehr belasten", gab Gerard zurück, ebenso harmlos, aber mit einem schelmischen Grinsen im Gesicht. Sabrina hatte den Wink verstanden. Sie ging auf sein Spiel ein. "Dann müssen wir uns eine Beschäftigung suchen, die wir im Sitzen oder Liegen ausüben können?" "Genau." Die Hände Gerards wanderten von Sabrinas Knien aus weiter nach oben. Sanft berührte er die weiche Haut an den Innenseiten ihrer Oberschenkel. Sabrina atmete tief ein und schloss erwartungsvoll die Augen. Der attraktive Skilehrer setzte sich neben die junge Frau und nahm sie liebevoll in die Arme. "Ich denke, wir sind uns einig", sagte er schalkhaft und näherte sich ihrem Mund. Ungeduldig erwartete Sabrina seinen Kuss, den sie leidenschaftlich zurückgab. Ihre Zungen spielten miteinander, erforschten gierig den Mund des anderen. Der Mann nestelte an Sabrinas Bademantel herum und löste den Knoten des Gürtels. Er legte die Frau sachte aufs Bett und betrachtete sie eingehend. Sein Glied zuckte heftig, während er seinen Blick über die festen Brüste des Mädchens gleiten ließ. Ihre rosa schimmernden Brustwarzen lugten vorwitzig über den Rand ihres Büstenhalters und er stellte befriedigt fest, dass sie steil aufragten. Gerard neigte seinen Kopf und näherte sich den herrlichen Nippeln. Mit einem leisen Stöhnen küsste er ihre Warzenvorhöfe, fuhr mit seiner Zunge über ihre herrlichen Hügel und saugte schließlich an ihren Brüsten, dass bald auch Sabrinas Atem schneller ging. Wohlig sank sie in die weichen Kissen und fuhr sich mit beiden Händen über den schlanken Leib. Er sah kurz auf und genoss das Schauspiel. "Ja, hilf mir", gab er plötzlich den Ton an. Seine Augen hingen an ihren Fingern, die ihre Brustwarzen liebkosten und weiter hinunterglitten über den wohlgeformten Bauch, hin zu ihrem Dreieck. Sie zog ihre weichen Schamlippen auseinander und gönnte ihm einen Blick auf ihre harte Perle, die heftig pulsierte. Dann verschloss sie das verlockende Dreieck und sah ihm tief in die Augen. Gerards Hände waren nun nicht mehr zu bremsen. Er hatte gesehen, wie feucht Sabrina war, und setzte seinen Mittelfinger als Kundschafter ein, während er sich wieder ihren Brüsten widmete. Er verzehrte sich nach ihr, wollte ihr aber nicht gleich Erfüllung verschaffen, sondern sie erst noch etwas zappeln lassen. Sanft streiften seine Finger über ihren Venushügel und tasteten zart nach ihrer Perle. Vorsichtig strich er darüber, erst mit den Fingern und schließlich – als er nicht länger warten konnte – mit seiner neugierigen Zunge. Die Frau duftete köstlich und wie berauscht kostete er von ihrem süßen Nektar. Schließlich war er so scharf auf sie, dass er beide Hände in ihr nasses Dreieck wühlte, seine Zunge auf ihrem Körper kreisen ließ, sie überall gleichzeitig berührte. Sabrina verfiel in einen Rausch der Sinne. Nie vorher hatte ein Mann sie derart sinnlich, leidenschaftlich und sanft zugleich berührt. Liebestoll streckte sie ihm ihren Unterkörper entgegen und immer tiefer erforschte er ihren Leib mit Zunge, Lippen und Fingern. Dann drehte er sie mit einer schnellen Bewegung auf den Bauch und umfasste mit beiden Händen ihr Hinterteil. Er knetete das weiche Fleisch, vergrub seine Finger in den wohlgeformten Backen und legte sich schließlich leicht auf sie. Das Mädchen reagierte prompt auf sein steil aufgerichtetes erigiertes Glied, das sich an ihr Rückgrat schmiegte. Wie eine Schlange schob sie sich unter dem Mann hin und her und machte ihn schier verrückt mit ihren geschmeidigen Bewegungen. Erneut drehte er sich um, musterte sie aufmerksam und konnte sich ein Lächeln nicht verbeißen. Ihre beinahe fiebrig glänzenden Augen konnten ihre Erregung kaum verbergen, ebenso ihr hübscher Mund. Aufgeregt fuhr sie sich immer wieder über die Lippen und ihre Zungenspitze lockte ihn. Er wusste, auf was sie hinauswollte und kniete sich mit breit gespreizten Beinen über sie. Dann gab er ihr, wonach sie verlangte. Mit einem lauten Stöhnen stülpte Sabrina ihre Lippen über das Geschlecht des Mannes und schloss den Mund. Sie begann unendlich zärtlich die Eichel zu stimulieren und zu lecken. Mit ihren Händen massierte sie sanft und liebevoll seine Hoden. Gerard stöhnte, was sie nur noch mehr antörnte. Immer schneller fuhr sie mit den Lippen über sein festes Glied und hatte den Mann schnell davon überzeugt, dass sie eine hervorragende Geliebte war. Mal schnell, mal quälend langsam, ließ sie dabei ihre Zunge an seinem Penis entlanggleiten, bis er sich ihr mit einem tiefen lustvollen Ächzen entzog. "Ich kann nicht länger warten", bettelte er entschuldigend und sah sie bekümmert an. "Wenn du mich nur noch eine Sekunde länger warten lässt, werde ich verrückt", sagte er, wobei ihm nicht so recht bewusst wurde, dass noch vor wenigen Minuten er derjenige war, der sie zappeln lassen wollte. Sabrina lächelte, griff sich mit beiden Händen in die Kniekehlen und spreizte ihre Beine, so weit sie konnte. Einladend, feucht und verführerisch schimmernd lag ihre Vagina vor ihm. Noch einmal starrte er auf ihr dunkel behaartes Dreieck, stöhnte erneut leise und gierig auf, nahm seinen erigierten Penis in die Hand, rieb ihn kurz und ließ ihn mit einem wohligen Seufzen in Sabrinas heißen Unterleib gleiten. Es war ein wunderbares Gefühl und sein fast übergroßer, mächtiger Penis füllte Sabrina wundervoll aus, ohne ihr wehzutun. Sie spürte seine vorsichtigen Stöße ganz tief in sich, spürte, wie er immer wieder ihren GPunkt berührte, was bisher noch keiner ihrer Liebhaber geschafft hatte. Ein wohliges Schaudern ließ sie erzittern und sie presste sich noch enger an den Mann. Ihre Hände spielten mit seinen Pobacken, näherten sich seinem Anus und tief erregt fragte sich Gerard, ob wohl einer ihrer schlanken Finger in ihn eindringen würde. Er hatte den Gedanken noch nicht zu Ende gedacht, als er spürte, wie sie, sanft und mit ganz leichtem Druck seine geheimsten Wünsche erfüllte. Das Gefühl machte ihn rasend, er bemerkte, dass er, wenn er nicht sofort etwas dagegen unternahm, vorzeitig kommen würde. Mit Hingabe widmete er sich ihren Brüsten, konzentrierte sich jedoch weiter auf das fremdartige, aber unglaublich süße Gefühl, das ihre Finger ihm schenkten. Nur Millimeter erforschten sie sein Innerstes, aber genau das war es, was Gerard noch um einiges geiler werden ließ. Er vergrub seine Fingernägel in ihrem weichen Fleisch, saugte wie ein Ertrinkender an ihren Brüsten und spielte gleichzeitig an ihrer angeschwollenen Perle. Mit seinem rechten Daumen übte er einen ähnlich sanften Druck auf ihr empfindlichstes Körperteil aus wie sie mit ihrem Zeigefinger zwischen seinen Pobacken. Die beiden verstärkten, ohne ein Wort zu wechseln, ihren Rhythmus. Immer schneller versenkte er sein strammes Glied in ihr, immer heftiger wurden seine Stöße. Auch Sabrina reagierte. Sie spreizte ihre Beine ganz breit, ließ den Mann noch weiter in sich hineingleiten und drückte, als sie spürte, wie ein orkanartiger Orgasmus über sie hinwegspülen wollte, ihren Zeigefinger urplötzlich ganz fest in seinen Anus. Gerard reagierte mit einem lauten wollüstigen Stöhnen und schrie seine Lust hinaus. Während ein heftiges Zittern Sabrinas Unterleib erfasste, ergoss sich Gerard in ihr. Minutenlang blieben sie erschöpft nebeneinander liegen und sprachen kein Wort. Sabrina fand als Erste die Sprache wieder. "Ich finde, wir passen prima zusammen", alberte sie grinsend. Der attraktive Skilehrer nickte. "Ja, erschreckend gut", erwiderte er. "Ich bin froh, dass ich morgen einen freien Tag habe..." "Und ich, dass ich mir den Knöchel gezerrt habe", lächelte Sabrina zufrieden, bevor sie ihn leidenschaftlich küsste und versuchte, seine Männlichkeit gleich noch einmal zu aktivieren.
# Der Weihnachtsmann
## Lisa Cohen
Britta verspürte in diesem Jahr wenig Adventsstimmung. Es waren nur noch zwei Wochen bis zum Weihnachtsfest und sie hatte weder ein Geschenk gekauft noch das Haus geschmückt, war zu keinem weihnachtlichen Kaffeetrinken gegangen und hatte noch keinen Glühwein probiert. Sie konnte trotz allen weihnachtlichen Rummels und der festlichen Stimmung, die überall zu spüren war, nicht verdrängen, dass es kriselte in ihrer Ehe. Sie und ihr Mann führten seit einem Jahr eine so genannte Wochenendbeziehung. Er sollte für seine Firma ein Zweigwerk im Süden des Landes aufbauen und war nur noch von Freitagabend bis Sonntagabend bei seiner Familie. Am Anfang hatte es ausgesehen, als wäre es nur eine vorübergehende Sache, aber dann hatten sich Schwierigkeiten eingestellt und er musste bleiben. Für ihn war es anstrengend. Der verantwortungsvolle Job, das ständige Hin- und Herfliegen, die familiären Spannungen, aber es war auch eine Herausforderung, es war spannend und es war lukrativ. Britta hatte nun viel mehr zu tun mit den beiden noch kleinen Kindern und dem großen Haus. Sie fühlte sich oft überfordert. Außerdem fehlte er ihr. Sie hatten immer eine gute Ehe geführt. Das erste halbe Jahr ihrer räumlichen Trennung war noch ganz okay gewesen. Doch mittlerweile waren beide genervt. Es gab immer öfter Streitigkeiten. Besonders dann, wenn Michael mal nicht kommen konnte. Ihre Telefonate wurden seltener und stressiger. Sie hatten jetzt sogar manchmal am Wochenende Streit. Im Bett herrschte natürlich Funkstille. Und mehr als einmal hatte sich Britta schon gefragt, ob ihr Mann seinen heimischen Frust unter der Woche an einer anderen Frau ausließ. "Mami, Mami, nun komm schon. Du hast uns doch den Weihnachtsmann versprochen!" Miriam zog an Brittas Arm und Pit sah sie vorwurfsvoll an. Sie waren in dieses riesige Einkaufszentrum gefahren, weil hier täglich ein Weihnachtsmann die Kinder bescheren sollte. Normalerweise hielt Britta von diesen weihnachtlichen Bombardierungen nichts. Aber der Kinder wegen... Sie hörten seine Stimme schon von Weitem. Tief und wohlklingend, wie es sich für einen Weihnachtsmann gehört. Dann sahen sie ihn auf einem Podest sitzen. Umringt von freudestrahlenden Kindern, die an seinen Lippen hingen. Er war eine imposante Erscheinung. Groß, breitschultrig. Respekt einflößend, aber auch vertrauenerweckend. Britta streckte sich unbewusst ein wenig, als sie auf ihn zugingen. Sie sahen sich einen Moment lang an. Seine Augen funkelten. Er lächelte ein warmes Lächeln. Und in diesem Moment passierte etwas mit Britta, dass sie auch später nicht würde erklären können. Etwas zog sie geradezu magisch an, während sie ihn dabei beobachtete, wie er die Kinder unterhielt. Und immer wieder schenkte er ihr zwischendurch dieses Lächeln. Britta lächelte zurück und lauschte seinen Worten. "Wart ihr denn auch immer brav?" Miriam und Pit starrten ihn verzückt an. Sie waren rundum selig. Zwei Tage später fuhr Britta wieder mit ihren Kindern zum Einkaufszentrum. Und wieder zog der Weihnachtsmann sie in seinen Bann. Sie hatte sogar von ihm geträumt. Es war ein sexueller Traum gewesen. Ein etwas schmutziger sexueller Traum. Sie hatte diese Art von Träumen lange nicht mehr gehabt. Als er sie ein paar Mal sehr direkt anblickte, spürte sie Schmetterlinge in ihrem Bauch. Und dabei wusste sie nicht mal, wie dieser Mann aussah unter seinem Bart und seinem Gewand. Vielleicht war er hässlich und hatte keinen schönen Körper. Vielleicht war er dumm oder arrogant oder... Er zeigte plötzlich auf sie. "Was ist mit eurer hübschen Mutter? War sie denn auch immer brav und gut zu euch Kindern?" Miriam und Pit nickten eifrig. Sie hofften auf mehr Süßigkeiten. "Und was, wenn nicht?", hörte sie sich fragen. Der Weihnachtsmann hypnotisierte sie jetzt fast mit seinem Blick. Durchdringend und ein wenig drohend. Brittas Brustwarzen drückten gegen den Stoff ihrer dicken Winterbluse. Sie fühlte eine Dominanz von diesem Mann ausgehen, der sie sich gerne unvermittelt hingegeben hätte. Eine sexuelle Dominanz, die sie fast greifen konnte. Sie sah seine großen Hände mit den kräftigen Fingern und wünschte sich, sie wäre ihnen ausgeliefert. Michael war immer ein guter Liebhaber gewesen, aber auch immer ein wenig zu lieb, zu zärtlich. Sie hatte ihn im Bett ab und an mal um etwas mehr physische Dominanz ihr gegenüber gebeten und ihn damit meist verunsichert. Dabei hatte sie das untrügliche Gefühl: Im Grunde hätte auch er es gerne mal etwas 'härter' ausprobiert, schämte sich aber zu sehr, das auch zuzugeben. Es blieb ihr also nichts anderes übrig, als seine sanfte Leidenschaft zu akzeptieren. Die Träume von Unterwerfung und devoter Hingabe waren geblieben, auch wenn sie seltener geworden waren. Doch in diesem Moment, als sie in die Augen dieses Weihnachtsmannes sah, fühlte Britta ihre geheimen Wünsche wieder aufbegehren. Als er sie in einem unbeobachteten Moment leise fragte, ob sie wiederkommen würde, nickte sie. Und so kam es, dass Britta fast täglich mit ihren Kindern ins Einkaufszentrum fuhr. Die stummen Blicke zwischen ihr und ihm, dem verhüllten korpulenten Fremden in dem roten Weihnachtsmanngewand, wurden intensiver und begleiteten sie bis in die Nacht hinein, wenn sie allein in ihrem Bett lag und nicht schlafen konnte. Sie wusste nicht, was sie eigentlich wollte. Sie wusste nur, sie wartete auf etwas. Auf etwas, das stärker war als ihre Vernunft und darauf, dass ihre geheimen Wünsche sich vielleicht erfüllen würden. Es war Freitagnachmittag und ihr Mann würde an diesem Wochenende nicht nach Hause kommen. Sie hatte es erstaunlich ruhig aufgefasst und war dabei in Gedanken ganz woanders. Der Weihnachtsmann wartete schon auf sie. Britta konnte es in seinen Augen lesen. Er lächelte sie verschwörerisch an. "Du kommst gerade zur rechten Zeit. Ich mache gleich eine halbe Stunde Pause...", raunte er ihr zu, als sie dicht an ihm vorbeiging und sich etwas abseits des Kindergetümmels setzte. Er roch männlich und sehr dominant. Als er nach seiner Vorstellung zu ihr herüberkam, kribbelte es in ihrem Unterleib wie in einem Ameisennest. Britta fing an zu schwitzen. "Komm so unauffällig wie möglich hinter mir her..." Sie folgte ihm in ein kleines, karg eingerichtetes und unglaublich hässliches Büro. Überall verstreut lag weihnachtliches Zubehör herum. Der Weihnachtsmann verschloss die Tür und packte sie mit einer Bewegung am Arm, die keine Widerrede duldete. "Du bist kein braves Mädchen. Brave Mädchen kommen nicht in mein Büro..." Ihre Brustwarzen richteten sich auf. "Was willst du hier?" Als wenn er das nicht wüsste! Sie zuckte ein wenig hilflos mit den Achseln. Der Weihnachtsmann blickte sie finster an. "Du bist ein böses Mädchen!" Dann band er den breiten Ledergürtel ab, der seinen riesigen Mantel zusammenhielt, und zog gleichzeitig seine wollene lange Unterhose aus. Sein Körper war kein Traum, aber sein Glied war von beachtlicher Größe. Das war fast alles, was sie von ihm wollte. "Leg dich auf den Tisch." Britta schob sich bäuchlings auf den Tisch und wartete zitternd vor Erregung ab. Er fing an, sie leicht durch den Stoff ihres Rockes zu schlagen. Es tat nicht weh. Sie spürte es kaum. Dann streifte er ihren Rock nach oben und schlug sie durch ihren seidenen Slip hindurch. Auf die Strapshalter ihrer Netzstrümpfe, die sie extra für ihn angezogen hatte und die Michael immer zu ordinär fand. Sie spürte jeden einzelnen Schlag und seufzte auf vor Glück. "Du böses Mädchen", murmelte der Weihnachtsmann immer wieder. Sie fing an, vor Gier zu zittern, als er ihren Slip ein Stück herunterzerrte und sie auf die nackten Pobacken schlug. Die Hiebe waren sanft und erregend brutal gleichzeitig. Er keuchte wollüstig und spreizte ungeduldig ihre Schenkel. Als er mit dem Gürtel zwischen ihren Beinen von hinten nach vorne strich und auf ihrer Klit ausgiebig verharrte, wimmerte sie vor Lust. Britta drückte sich gegen das lederne Teil und bewegte ihren Unterleib, um sich stärker daran reiben zu können. Unbändige Erregung durchflutete ihren Körper. Was würde er mit ihr machen? "Dreh dich um!" Sein Ton war barsch und unfreundlich. Sie drehte sich gehorsam auf den Rücken. "Mach deine Beine breit!" Sie winkelte ihre Beine an und presste sie mit den Händen weit auseinander. Er starrte lüstern auf sie herab. "Du bist ordinär und schamlos!" Sie nickte ergeben. "Mach deine Bluse auf." Mit zitternden Fingern knöpfte sie ihre Bluse auf. "Zieh deinen BH aus!" Dann lag sie mit bloßen Brüsten vor ihm und wartete schwer atmend auf seine Behandlung. Der Gürtel glitt mal hart, mal zart über ihre Brüste, umkreiste ihre Nippel und schlug sanft zu. Sie stöhnte vor Lust und bäumte sich seinen Schlägen entgegen. Als der Gürtel in ihren Schritt glitt und zärtlich, aber bestimmt wieder ihre Perle umkreiste, schlug sie wild mit dem Kopf gegen den Tisch. "Wage es nicht, zu kommen,...", brummte der Weihnachtsmann, "bevor ich es dir erlaube!" Britta versuchte, gegen ihre uferlos werdende Lust anzukämpfen, sie ein wenig unter Kontrolle zu behalten, doch es war vergeblich. Sie starrte auf seine Hand, die seinen Penis massierte, ihn noch größer machte, als er schon war, während er sie mit der anderen Hand weiter bestrafte. Britta zitterte nun vor Lust am ganzen Körper. Wenn er bloß nicht aufhören würde!, hoffte sie. "Ich habe dir noch nicht erlaubt, zu kommen! Vergiss das nicht..." Sie nickte mit zusammengebissenen Lippen. Sie würde es erst wagen, zum Höhepunkt zu kommen, wenn er es ihr gestattete. Sie war nicht geübt, ihre Ekstase zu kontrollieren. Sie wollte explodieren, aber noch nicht gleich, denn dann würde die totale Lust auch vorbei sein. Sehnsüchtig stöhnte sie nach Erlösung. Doch er ließ sich Zeit. Hörte auf, sie mit dem Gürtel zu berühren, sondern starrte sie nur an, wie sie sich wand unter ihrer eigenen Lust. Er widmete sich ausgiebig seinem Glied und ließ sie für eine Weile außer Acht, bis sie ihn fast anflehte, er möge ihr mehr Aufmerksamkeit schenken. Da erst wandte er sich gnädigerweise wieder ihrer Erregung zu. Schlug sie in regelmäßigen Abständen. Strich über ihre Brüste, klopfte an ihre Nippel, streichelte sanft ihren Bauch, bevor er sich erneut und jetzt ganz ausführlich ihrer feuchten sensiblen Region zwischen ihren weit gespreizten Schenkeln widmete. "Nicht aufhören... Nicht mehr aufhören!", bettelte sie leise. Britta war auf dem Weg zur totalen Ekstase und sie konnte sie nicht mehr länger hinauszögern. "Versprichst du, im nächsten Jahr brav zu sein?" Der Weihnachtsmann schwang drohend den Gürtel über ihrer Scham. Sie nickte bebend und schloss die Augen, als sich die absolute Lust in ihr ausbreitete. Britta krümmte ihren Körper zusammen, dehnte sich wieder und gab sich ekstatisch seinen letzten Schlägen hin. Als sich ihr Unterleib zusammenzog und die Lust nur so aus ihr herausströmte, fühlte sie, wie sich sein hartes Glied mit ihrer Glut vereinte. Unerbittlich stieß er in sie hinein, bis sie nichts mehr wahrnahm – außer orgiastischer Erlösung... "Mami, Mami, was hast du?" Pit zog energisch an ihrer Hand. Miriam sah sie gelangweilt an. Dort, wo eben noch der Weihnachtsmann gestanden hatte, strömten eilige Kunden über den Platz. Britta starrte ihre Kinder an und wurde rot vor Scham. Sie hatte gerade eine ihrer geheimen erotischen Vorstellungen gehabt. Peinlich berührt griff sie nach ihren Kindern und zog sie mit sich zum Auto. Als ihr Mann kurz vorm Fest endlich nach Hause kam und sie nach ihren Weihnachtswünschen fragte, hatte sie den Mut, ihm ein wenig zu beichten, wovon sie ganz oft träumte. Und wie sehr sie sich das wünschte, wovon sie träumte. Und so kam es, dass am heiligen Abend, als Miriam und Pit zufrieden im Bett lagen, Michael im roten Weihnachtsmanngewand vor seiner Frau stand und sie zumindest ein ganz kleines bisschen dafür bestrafte, dass sie heimlich so ganz und gar unbrave Gedanken gehabt hatte...
# Ein besonderes Trinkgeld
## Miriam Eister
Das Lokal "Zum vollen Krug" machte seinem Namen heute wieder alle Ehre. Der Laden war bis auf den letzten Platz gefüllt mit fröhlich feiernden Menschen. Katja schob sich langsam mit ihrem Tablett voller leerer Gläser an einigen Gästen vorbei. In der Vorweihnachtszeit war es zwar stressig, aber Katja liebte ihren Job. Ein großer Vorteil in dieser Zeit der Firmenweihnachtsfeiern und privaten Gäste war das üppige Trinkgeld. Seit ihrem 16. Lebensjahr verdiente sie sich etwas Taschengeld mit dem Kellnern. Und als sie endlich mit der Schule fertig war, hatte sie sich voll und ganz in diese Arbeit gestürzt. Dieser Job ließ ihr einfach vormittags Zeit für sich und bot ihr abends die Möglichkeit, neue Leute kennenzulernen. Ihre Eltern waren damit zwar nicht einverstanden, dass sie ihr Studium verzögert beginnen wollte, aber Katja war das egal. Beladen mit mehreren Bieren und Mineralwassern steuerte sie den Tisch mit einer Gruppe gut gekleideter und ziemlich ausgelassen feiernder Herren an. Jedoch ein Mann war noch völlig nüchtern. Wahrscheinlich musste er noch fahren. Aus den Augenwinkeln hatte sie gesehen, dass er ziemlich attraktiv war. Groß, sportliche Figur, gepflegter Haarschnitt. Er war der Typ Mann, den sie sicherlich nie ansprechen würde. Sie selbst war nicht hässlich, aber sie fand sich auch nicht sonderlich schön. Meist versteckte sie ihre weiblichen Rundungen unter einer Bluse und einer Hose. Sie stellte die Getränke ab und blickte hoch. Genau in die braunen Augen von IHM! Schnell schaute sie weg. Sie fühlte sich ertappt und Röte kroch ihr über das Dekolleté. "Sie haben heute ganz schön zu schuften, was?" Seine Stimme klang sanft. Sie nickte. "Ja, das erspart mir jedes Fitness-Studio." Katja sammelte die leeren Gläser vom Tisch. Der eine Bierkrug stand ganz am Ende des Tisches. Ihre Möglichkeiten waren entweder den Krug stehen zu lassen oder sich vor diesem attraktiven Mann vorzubeugen, um ranzukommen. Sie atmete tief durch und entschloss sich zu Letzterem. Mit einer gezielten Bewegung schob sie ihren Oberkörper über den Tisch, stützte sich mit einer Hand ab und griff mit der anderen nach dem Glas. Ihr war bewusst, dass der Fremde ihren warmen Körper direkt vor der Nase hatte und auch ihre Unterwäsche blitzen sehen konnte. Sie fühlte seinen Blick förmlich auf dem schwarzen BH ruhen. Mit der gleichen Ruhe richtete sie sich wieder auf und blickte ihn an. "Möchten Sie die Rechnung haben?" Er nickte. Sie wusste, dass er ihr hinterherschaute und es tat ihrem Ego unheimlich gut. Aber wahrscheinlich würde er gleich gehen und vorbei war der schöne Traum. Sie tippte die Rechnung und brachte sie ihm. Es gab wie vorhergesehen großzügiges Trinkgeld. Gerade als sie den Herren einen schönen Abend wünschen und gehen wollte, wurde sie am Handgelenk festgehalten. Sie blickten sich wieder tief in die Augen. "Ich habe Sie beobachtet. Sie machen Ihren Job sehr gut. Ich habe nächstes Wochenende eine private Weihnachtsparty mit zirka 20 Gästen und könnte eine helfende Hand gut gebrauchen. Ich verspreche eine gute Bezahlung. Bei Interesse rufen Sie mich doch einfach an." Damit drückte er ihr einen Zettel in die Hand und wandte sich zum Gehen. Das Haus, vor dem sie stand, war wirklich eindrucksvoll. Gleich am nächsten Tag hatte Katja bei Sören angerufen und ihm ihre Hilfe für seine Feier zugesagt. Sie hatten einen guten Preis ausgemacht und eventuell gab es sogar noch Trinkgeld. Sören stand schon in der Tür. "Komm rein. Ich zeige dir gleich alles und dann muss ich mich aber auch fertigmachen. Du siehst übrigens toll aus!" Bewusst hatte sie sich ein schwarzes kurzes Kleid angezogen. Somit war sie stilvoll, aber dennoch dezent gekleidet. Schnell machte sie sich mit allem vertraut und legte sich einige Dinge zurecht. Von der ersten Etage hörte sie ein Summen und das Rauschen einer Dusche. Katja fand das unglaublich entspannend. Es war ein schöner Abend. Sören hatte einige Geschäftspartner und Freunde zu einer privaten Feier geladen und Katja sollte diese immer mit Getränken versorgen, die leeren Gläser einsammeln. Eigentlich wie in einem Lokal, nur stilvoller. Weit nach Mitternacht gingen die letzten Gäste und ließen die beiden allein zurück. Katja hatte Sören den ganzen Abend heimlich beobachtet. Sein Anzug stand ihm wunderbar und der Duft seines leichten Rasierwassers wehte ihr immer um die Nase. Lässig legte er sein Sakko über die Couch und kam zu ihr. "Vielen Dank. Du warst wunderbar. Jetzt helfe ich dir noch beim Aufräumen, dann ist das Chaos nicht mehr ganz so schlimm." Schnell waren Geschirr, Besteck und Sonstiges im Geschirrspüler verstaut und die Wohnung mit einigen Griffen wieder hergerichtet. Eigentlich war Katjas Job damit erledigt. Doch sie wollte noch nicht gehen. Sören wollte das scheinbar auch nicht, denn er kam mit zwei frisch gefüllten Weingläsern zurück. "Ich hoffe, du bleibst noch etwas." Katja nickte. Sie tranken einen Schluck Wein. Knisternde Spannung breitete sich aus. Im Hintergrund spielte immer noch leise Musik. Plötzlich nahm Sören ihr das Glas ab und küsste sie. Nicht vorsichtig, sondern wohl wissend, dass er ihr gefiel. Erst gab Katja sich diesem Kuss hin, spielte mit seiner Zunge. Doch auf einmal war ihr Verstand glasklar. Sie kannte ihn doch gar nicht! Sie beendete den Kuss und stand auf. "Ich sollte gehen!" Sein Blick war sanft. "Warum?" Katja wurde verlegen. "Nun ja... Ich kenne dich doch eigentlich nicht und..." Sie suchte nach Worten. Er stand auf. Wie ein Raubtier näherte er sich seiner Beute. "Du hast Angst." Keine Frage, sondern eine Feststellung. Sie nickte. "Warum?" Sie holte tief Luft. "Weil ich nicht viel Erfahrung habe. Du bist so attraktiv und hast dieses tolle Haus. Du könntest jede Frau haben!" Statt einer Antwort zog er sie zu sich. "Tanz mit mir!" Wie von alleine passte sie sich seinen Schritten an. Sie fühlte seine Nähe. Seine Körperwärme drang durch sein Hemd und ihr Kleid. Seine Hände lagen auf ihrem Rücken. Er war einfach Mann und ihre Seite als Frau spürte das nur zu deutlich. Und reagierte. Sie wurde weich und lehnte den Kopf an seine Brust. Katja verlor das Zeitgefühl. Warum sollte sie es nicht einfach genießen? Seine Stimme klang rau. "Magst du immer noch gehen?" Katja schaute hoch und schüttelte den Kopf. "Nein. Nicht wirklich." Er lächelte. "Das ist gut. Ich möchte auch nicht, dass du gehst. Ich habe dich beobachtet. Ich würde dich gern kennenlernen und ich gedenke, in dieser Nacht damit anzufangen." Wieder küsste er sie. Diesmal erwiderte sie den Kuss ohne Scheu und Gewissensbisse. Ein bekanntes Geräusch signalisierte ihr, dass er ihren Reißverschluss am Kleid geöffnet hatte. Sie trat einen Schritt zurück und ließ es an den Schultern hinabgleiten. Darunter trug sie halterlose Strümpfe und ein Wäscheset aus dunkelrotem Stoff. Er musste ja nicht wissen, dass sie sich das extra für diesen Abend gekauft hatte. Insgeheim hatte sie gehofft, dass etwas passieren würde. Sören öffnete sein Hemd und ließ es zu Boden fallen. Eine breite Brust, dunkel behaart, kam zum Vorschein. Schnell legte er auch seine Hose achtlos zur Seite. "Komm mit!" Sie gingen die Treppe hinauf ins Schlafzimmer. Dort zog er sie zum Bett. Das Licht der Straßenlaternen war hell genug. Katja öffnete den Verschluss ihres BHs. Er lachte. "So gierig, junge Dame?" Statt einer Antwort schubste sie ihn auf das Bett, kniete sich über ihn und hielt ihm ihre Brust an die Lippen. Eine Einladung, der er unmöglich widerstehen konnte. Seine Hände strichen über ihren Rücken, während seine Lippen ihre Brustwarzen erkundeten. Durch die Schichten seiner und ihrer Unterwäsche spürte sie seine Erregung, aber offensichtlich hatte er vor, sich Zeit zu lassen. Ihre Brust keinen Moment vernachlässigend, wanderten seine Hände jedoch weiter zu den Rundungen ihrer Hinterbacken. Plötzlich rollte er sich herum und sie lag unter ihm, während er sich etwas abstützte. Sie keuchte auf. "Oh mein Gott..." Er bemerkte ihren Blick und lächelte. "Tja, eine kleine Vorliebe von mir. Es kann ganz spannend sein!" Sören hatte einen riesigen Spiegel an der Decke angebracht. Sie sah sich halbnackt auf dem Bett liegend mit einem Mann, der gerade ihren Slip auszog und mit den Lippen tiefer ging. Er spreizte ihre Beine und kostete sie. Sie stellte fest, dass ihr die Aussicht gefiel. Sören leckte ihre Perle und drang gleichzeitig mit einem Finger zärtlich in sie ein. "Du bist schon unheimlich nass." Sie hob die Hüften als Zeichen, dass er weitermachen sollte. Zu dem einen Finger gesellte sich ein zweiter. Im stetigen Rhythmus glitten sie in ihre nasse Enge rein und wieder raus. In ihr übte er leichten Druck nach oben aus. Scheinbar schien er zu wissen, was er tat. Und es wirkte auch. Katja stöhnte heftiger. Zwischen ihren Brüsten sammelten sich einige Schweißtropfen. Sie hob die Hüfte im Takt seiner Fingerbewegungen an. "Bitte..." Sie flehte ihn an. Doch er hatte nicht vor, jetzt schon aufzuhören. Kräftiger leckte er ihre Klitoris und bewegte sich schneller. Er merkte, dass Katja kurz vor einem Höhepunkt stand. Mit einem tiefen kehligen Stöhnen bog sie den Rücken durch und kam. Seine Finger wurden Zeuge der inneren Zuckungen. Sie ließ ihn ihre Lust hören. Schließlich legte er sich wieder neben sie. Sie schmeckte ihren Nektar auf seinen Lippen. Sie griff an seine Shorts und merkte, dass diese zum Bersten gespannt waren und etwas darum bat, erlöst zu werden. "Zieh das aus!" Schnell kam er dem nach. "Nun bist du dran." Diesmal rutschte sie tiefer. Sie nahm seine Erregung in die Hände und rieb erst sanft daran, bewegte die Haut vor und zurück. Sie hatte den Dreh schnell raus, wie fest sie ihn in der Hand halten musste und wie schnell reiben. Schließlich nahm sie seine Männlichkeit in den Mund, kostete den Tropfen an der Eichel. Sören genoss die Aussicht und das Gefühl dieses warmen feuchten Mundes um sein Glied. Ab und zu drehte sie die Hand etwas, so dass immer neuer Reize entstanden. Doch er wollte nicht, dass sie bis zum Schluss ging. Ein kurzes "Stopp!" genügte, und sie hielt lächelnd inne. "Was ist denn los? Haben wir Probleme, uns zu beherrschen?" Er zog sie hoch. Ihr Becken war nun genau über seinem steil nach oben ragenden Glied. Sie ließ sich sinken. Sein Stab verschwand in ihrer Lust. Einen kurzen Moment hielt sie inne, um das Gefühl zu genießen, wie er sie dehnte. Dann begann sie einen aufregenden Ritt. Sie war sehr erregt und hatte kein Bedürfnis nach einer vorsichtigen Nummer. Kräftig ließ sie sich immer wieder auf ihn sinken und bewegte ihre Hüfte. Er stellte seine Füße in die Matratze und kam ihr mit Stößen von seinem Becken entgegen. Sören genoss diese heiße Frau auf sich. Er streichelte ihre Brüste und hielt die Nippel steif. Er spürte, wie sie das Tempo steigerte. Völlig unerwartet hielt er sie in ihrer Bewegung auf und drehte sie auf den Rücken. Ihre Beine legte er sich über die Schultern und drang wieder tief ein. Er drosselte das Tempo und hielt ihre Erregung damit kurz vor dem Höhepunkt. Sie sah ihm in die Augen. Ihre waren dunkel vor Lust. Sören hob die Hüfte so weit an, dass er aus ihr herausglitt, aber nur, um dann wieder ihre ganze Tiefe auszuloten und zu füllen. Beim Herausziehen entstand ein Geräusch, was typisch für schönen scharfen Sex war und was er so liebte. Die feuchte Hingabe einer Frau. Doch er hatte Erbarmen. Schneller stieß er zu. Sein Glied rieb über ihre Klit. Schweiß lief ihr zwischen den Brüsten entlang und ihr Stöhnen war wie Musik in seinen Ohren. Katja gab sich ihm völlig hin. Während sie ihrem Höhepunkt entgegenstöhnte, sich lustvoll unter ihm wand und ihm ihre Fingernägel kräftig über den Rücken zog, spürte auch er, dass er seinen Orgasmus nicht mehr halten konnte. "Oh... ja... ich komme!" Einige Zeit später lagen sie immer noch nackt nebeneinander. "Von wegen, du bist nicht so erfahren...!" Sie lachte. "Das bin ich wirklich nicht. Ich glaube, ich sollte jetzt aber gehen. Es ist schon sehr spät." Sören rollte sich hoch und hielt sie mit seinem Körper auf. "Nein. Ich möchte, dass du bleibst. Wenn du möchtest, darfst du natürlich gehen. Was deine Bezahlung betrifft, einigen wir uns schon. Ich persönlich hatte gehofft, dass ich dein Trinkgeld sein könnte. Was meinst du?" Die Antwort kam prompt. Katja schloss ihn in ihre Arme, küsste ihn tief. Solch ein Trinkgeld war doch mehr wert als alles andere!
# Lieber Weihnachtsmann
## Jenny Prinz
"Jonas Hellmann, guten Tag." Er streckte mir die Hand hin und lächelte mich freundlich an. Ich stand da wie angenagelt und starrte in seine grünen funkelnden Augen. Erst nach Sekunden merkte ich, wie seltsam ich wirken musste und riss mich zusammen. Möglichst professionell ergriff ich seine Hand und schüttelte sie. "Charlotte Seifert", erwiderte ich sachlich und blickte schnell auf die Unterlagen, die vor mir auf dem Schreibtisch lagen. So etwas war mir noch nie passiert. Von der Sekunde an, als der junge Mann in mein kleines Büro gekommen war, fühlte ich mich wie elektrisiert. Ich fühlte mich schlagartig zu ihm hingezogen, auch wenn ich nicht so recht wusste, wieso. Er sah zweifellos gut aus mit seinen strahlenden Augen und den dunklen Locken, die sich auf seinem Kopf kringelten, aber das war es nicht. Vielmehr hatte ich das Gefühl, ihn schon zu kennen. Er war mir irgendwie vertraut und dann noch dieser Blick, mit dem er mich jetzt musterte... Mein ganzer Körper kribbelte und ich wünschte mir im Augenblick nichts sehnlicher, als ihn in aller Ruhe anzusehen und berühren zu können. Stattdessen blickte ich wieder auf und sagte: "Sie sind also der Weihnachtsmann?" Innerlich stöhnte ich auf, als ich merkte, was ich da von mir gegeben hatte. Mein Gegenüber grinste breit; dabei zeigte er makellose schneeweiße Zähne. "Ich meine, Sie sind unsere neue Aushilfe", versuchte ich den Satz zu retten. Er nickte. "Gut, das hat dann ja schnell geklappt. Ich freue mich, dass Sie einspringen konnten, unser üblicher Weihnachtsmann ist leider erkrankt. Und es wäre ja schade, wenn wir die Aktion heute Nachmittag absagen und damit dutzende Kinder enttäuschen würden." Davon, dass es nicht gerade die beste Reklame für unser Kaufhaus war, wenn groß angekündigte Aktionen ausfielen, sagte ich besser nichts. Aber das konnte er sich wohl ohnehin denken. Ich suchte zwischen den Papieren herum, da er einen Personalfragebogen ausfüllen musste. Dabei hielt ich den Blick wieder gesenkt; ich wollte ihm nicht ins Gesicht sehen, da ich Angst hatte, noch mehr aus dem Konzept zu geraten. So kühl wie möglich überreichte ich ihm das Formular, das ich endlich aus einer Mappe gezogen hatte, und bat ihn, mir dies ausgefüllt und unterzeichnet zurückzugeben. "Ihr Kostüm haben Sie mitgebracht?" Wieder nickte er und hob kurz die linke Hand, in der er einen schwarzen Rucksack hielt. Dann fragte er mit einem Lächeln, das mich sofort dahinschmelzen ließ: "Wo darf sich der Weihnachtsmann denn umziehen?" Am liebsten hier, schoss es mir durch den Kopf, doch natürlich sagte ich das nicht. Etwas an seinem Blick gab mir jedoch das Gefühl, dass er vielleicht Gedanken lesen konnte. Er spürte genau, wie verwirrt ich war. Abwartend stand er vor mir, seine leuchtenden Augen auf mich geheftet. "Ich zeige Ihnen die Aufenthaltsräume für das Personal", beantwortete ich seine Frage und stand auf. Mit weichen Knien ging ich an ihm vorbei und führte ihn zu den Umkleideräumen. Ich fühlte mich mit meinen 1,65 Metern unheimlich klein neben diesem großen, gut durchtrainierten jungen Mann. Während ich vor ihm herging, spürte ich seinen Blick im Rücken und ich erwischte mich dabei, meine Hüften beim Gehen etwas mehr zu bewegen als notwendig. Wie lächerlich, schalt ich mich, doch ich konnte nichts dagegen tun. Je länger Jonas Hellmann in meiner Nähe war, desto mehr erregte er mich. Ich spürte deutlich, wie sich meine Brustwarzen aufrichteten und sich Wärme in meinem Schoß ausbreitete. Wie lange war es jetzt her, dass ich mit einem Mann geschlafen hatte? Fünf Monate? Ein halbes Jahr vielleicht? Auf jeden Fall zu lange, stellte ich fest, während sich die Feuchtigkeit in meiner Mitte verteilte und meine Gedanken darum kreisten, wie es sich wohl anfühlen würde, von dem sexy jungen Mann hinter mir geküsst zu werden... Eine Stunde später stand ich im weihnachtlich geschmückten Eingangsbereich des Kaufhauses und sah zu, wie Jonas ein Kind nach dem anderen nach seinen Wünschen befragte. Geduldig warteten die Kleinen neben ihren Eltern, bis sie an die Reihe kamen, obwohl es hier drin stickig heiß war und die meisten wegen der eisigen Kälte, die seit einigen Tagen die ganze Stadt einfror, dicke Anoraks und Mützen trugen. Jonas war aufgrund seiner beachtlichen Größe ein imposanter Weihnachtsmann und ich lächelte über seine tiefe Stimme, mit der er die Kinder ansprach. Sogar mit diesem riesigen Kostüm und dem weißen Rauschebart sah er noch umwerfend aus. Vielleicht lag es aber nur daran, dass ich mir seinen muskulösen Körper, den ich eben noch in Jeans und Shirt gesehen hatte, sehr genau unter dem roten Filzstoff vorstellte. Unruhig verlagerte ich das Gewicht von einem Fuß auf den anderen. Mit verschränkten Armen stand ich da und beobachtete unseren "Neuen". Plötzlich hob Jonas den Blick und sah mich an. Diese tiefgrünen Augen verfingen sich in meinen blauen und mir lief ein Schauer über den Rücken. Er wusste, dass ich hier nicht stand, um ihn zu kontrollieren, sondern dass ich ihn ansehen wollte. Uns beiden war klar, wie erotisch ich ihn fand, und ich hoffte tief in mir drinnen, dass ich auf ihn die gleiche Wirkung hatte. Er sah mich weiterhin an, während er ein kleines Mädchen schwungvoll wieder auf die Füße stellte. Mit einem Zwinkern in meine Richtung hob er dann seine tiefe Stimme und verkündete laut: "So, liebe Kinder, und nun möchte ich euch jemanden vorstellen." Ich stutzte. "Die junge Dame dort drüben heißt Charlotte. Charlotte hat mich dazu eingeladen, euch heute hier zu besuchen. Und deshalb wollen wir uns jetzt bei Charlotte bedanken und sie darf darum einmal auf meinem Schoß sitzen. Erwachsene haben nämlich auch Wünsche", fügte er noch verschwörerisch an die Kinder gewandt hinzu. Ich dachte, ich müsste im Boden versinken. Alle Augen waren auf mich gerichtet und ich wusste, dass ich das jetzt nicht ablehnen konnte. Die Eltern grinsten, während ich mich langsam auf Jonas zubewegte. Einige Kollegen, die ebenfalls einen Blick auf den Weihnachtsmann werfen wollten, schauten verwundert. Als ich vor ihm stand, zog er mich ohne weitere Umschweife auf seinen Schoß und beugte sein Gesicht mit dem künstlichen weißen Bart ganz nah an meines heran. Es fühlte sich gut an, ihm so nah zu sein. Er hatte einen Arm um meine Taille gelegt, um mich auf seinem Knie festzuhalten, und fragte mit seiner "normalen" Stimme leise: "Na, Charlotte, was wünschst du dir vom Weihnachtsmann?" Ich zögerte eine Sekunde und setzte dann alles auf eine Karte. "Einen Mann", antwortete ich flüsternd und ich spürte, wie meine Wangen warm und rot wurden. "So, so", Jonas schien nachzudenken, "einen Mann. Einen bestimmten?", hakte er dann nach. Nun war es ohnehin egal. "Ja", erwiderte ich noch leiser. "Wir haben da seit heute so eine neue Aushilfe, die hätte ich gern." Ich schaute nach unten, ihm jetzt in die Augen zu sehen, hätte ich nicht fertiggebracht. Ich spürte, dass er mich noch ein wenig näher an sich zog und dann tief Luft holte. Mit seiner "Weihnachtsmannstimme" verkündete er dann laut hörbar: "Charlotte hat mir gerade einen ganz besonders schönen Weihnachtswunsch gesagt und ich werde natürlich mein Bestes tun." Sanft schob er mich von seinem Bein herunter und wandte sich an das nächste Mädchen, das ziemlich dicht vor uns stand und an der Reihe war. "Na, komm mal her, Kleine, wie heißt du denn?", fragte er, während ich mich mit zittrigen Beinen an den Leuten vorbeischob, die mich immer noch mit einem breiten Grinsen bedachten. Obwohl mit Sicherheit keiner von ihnen ahnte, WAS ich mir gerade gewünscht hatte... Aufgeregt und unsicher saß ich in meinem Büro. Ich konnte mich auf nichts konzentrieren. Nervös spielte ich mit dem Adventskranz, der ein wenig vorweihnachtliche Stimmung in mein eher tristes Büro bringen sollte. Jonas würde gleich hereinkommen, wenn sein Auftritt vorüber war, um mir noch seine Personalien zu bringen. Das hoffte ich zumindest. Und dann? Ich hatte keine Ahnung, wie er auf meinen bescheuerten Wunsch reagieren würde. Er hatte gesagt, es sei ein "besonders schöner" Wunsch gewesen, aber im Moment kam ich mir eher lächerlich vor. Wie erbärmlich war es denn, wenn eine Frau sich schon einen Mann zu Weihnachten wünschen musste? Andererseits hatte er mit dem Wünschen angefangen und ich hatte das eindeutige Gefühl, dass er einem Flirt nicht abgeneigt war. Und ich fühlte deutlich die Hitze und das Kribbeln in meiner Venus, wenn ich nur an Jonas dachte. Als es an der Tür klopfte, zuckte ich zusammen. Verlegen rief ich: "Herein." Als er vor mir stand, sah er auch nicht mehr so selbstsicher aus wie noch vorhin. Er lächelte mich jedoch an und reichte mir seinen Fragebogen. Ich war aufgestanden, musste allerdings trotzdem zu ihm aufblicken. "Entschuldige wegen vorhin", sagte er zögernd. "Ich wollte dich vor dem Publikum nicht in Verlegenheit bringen." "Kein Problem", erwiderte ich selbstbewusster, als ich mich fühlte. Ich spürte, dass er sich einen Ruck gab. Es freute mich, dass er jetzt mit mir allein doch nicht mehr ganz so lässig war. "Hast du das eben ernst gemeint?", fragte er nun mit heiserer Stimme, während er mir tief in die Augen sah. Ich nickte. Langsam machte er noch einen Schritt auf mich zu und beugte sich zu mir herunter. Und dann spürte ich seine weichen Lippen auf meinen und er küsste mich lange und liebevoll. Unsere Lippen berührten sich erst vorsichtig, dann wurde unser Kuss zärtlicher und intensiver. Sanft schob er seine Zunge zwischen meine Lippen und ich öffnete ihm bereitwillig meinen Mund. Kleine Stromstöße schossen durch meinen Körper und ich musste mich zusammennehmen, um nicht aufzustöhnen. Ich fühlte mich, als hätte ich jahrelang darauf gewartet, wieder berührt und geküsst zu werden. Als er seine Arme um mich schlang, drängte ich mich an ihn und genoss das Gefühl seiner Muskeln, die sich an mich drückten. Ganz fest hielten wir uns umschlungen, während wir alles um uns herum vergaßen und nur noch mit dem jeweils anderen beschäftigt waren. Seine Hände streichelten meinen Po und Rücken, ich hingegen hielt mich in den Locken in seinem Nacken fest. In diesem Moment war alles einfach perfekt; das erste Mal ging ein Weihnachtswunsch wirklich genau so in Erfüllung, wie ich es mir gewünscht hatte. Jonas war genauso erregt wie ich, das war nicht zu übersehen. Sein hart erigierter Penis drückte sich durch die Jeans an meinen Bauch und ich hätte ihn am liebsten sofort in die Hand genommen. Allerdings war mir in meinem Hinterkopf bewusst, dass wir in meinem Büro standen und hier jeden Moment jemand hereinplatzen könnte. So wollte ich es nicht; ich wollte alle Zeit der Welt haben, um es zu genießen. Also löste ich mich nach einer Weile von Jonas, um ihn zu fragen, ob er mit zu mir kommen würde. Am Leuchten seiner Augen war deutlich zu erkennen, dass er nichts lieber wollte. Wir nahmen unsere Jacken, löschten das Licht und verließen das Gebäude durch den Hinterausgang. Zum Glück kam uns niemand entgegen. Draußen hatte es begonnen zu schneien. Wie selbstverständlich nahm Jonas meine Hand und folgte mir, als ich den Weg zu meiner Wohnung einschlug. Mein Herz klopfte mir bis zum Hals und mein Slip war sicherlich schon ebenso feucht wie mein Lustzentrum. Es war eine unglaubliche Geschichte, die ich hier gerade erlebte und doch fühlte es sich absolut perfekt an. Nur noch Minuten trennten mich davon, in Jonas' Armen zu liegen. Ich konnte es nicht mehr erwarten, ihn ganz zu spüren. Und endlich waren wir da. Wie ließen uns kaum Zeit, unsere Kleidung im Flur abzustreifen. Achtlos blieben Schuhe und Jacken liegen, während Jonas begann, meine Bluse aufzuknöpfen. Unsere Lippen fanden sich und unsere Zungen trafen aufeinander. Jonas schmeckte hervorragend. Tief schob er sich in meine Mundhöhle, erforschte jeden Winkel. Ich spürte, wie geschickt er die Häkchen meines BHs öffnete und schon lagen meine festen runden Brüste in seinen kühlen Händen. Sanft zwirbelte er meine Brustwarzen zwischen Daumen und Zeigefinger und ich hielt kurz die Luft an, als die Verbindung zwischen meinen rosigen Knospen und meiner Venus Funken schlug. Ein neuer Schwall salziger Nässe überflutete meine Muschel. Ich küsste immer fordernder, saugte an seiner Unterlippe, während meine Hände nun ebenfalls auf Wanderschaft gingen. Schnell befreite ich ihn von seinen Klamotten, wollte ich doch seine weiche Haut und die festen Muskeln nackt unter meinen Fingern spüren. In meinen Augen stand das Verlangen wahrscheinlich ebenso deutlich wie in seinen. Vorsichtig und mit zitternden Fingern befreite ich zuletzt seinen prachtvollen dicken Penis aus dem engen Slip, den er trug. Jonas stöhnte leise auf, als sich meine Finger um seinen Schaft schlossen und ich mit gleichmäßigen festen Bewegungen begann, ihn zu massieren. "Das ist unglaublich!", seufzte er, als mein Daumen zwischendurch immer wieder über seine Eichel glitt und besonders die empfindliche Stelle an der Unterseite geschickt reizte. Sein Atem ging schnell und ich bemerkte, dass seine Konzentration nachließ, da er völlig von der Erregung eingenommen wurde, die ich ihm mit meiner Hand bescherte. Eine Weile trieb ich dieses Spiel mit ihm, doch dann wollte ich noch einen Schritt weitergehen. Mit einem Lächeln ließ ich mich auf die Knie sinken; Jonas' Blick folgte mir, als ich nun seine Vorhaut straff zurückzog und seine samtige Spitze mit meinen feuchten Lippen umschloss. Er gab einen unartikulierten Ton von sich, als ich ihn in den Mund nahm und fast gleichzeitig begann, ihn mit meiner Zunge zu streicheln. Zärtlich saugte ich an seinem Penis, liebkoste ihn mit Zunge und Lippen. Er starrte von oben auf die Bewegungen meines Kopfes und ich wusste, dass er sich nicht mehr lange würde beherrschen können. Dafür war dieser Reiz zu intensiv. Er keuchte. Ich genoss es, seine Lust immer weiter zu steigern. Meine linke Hand umschloss seine Pobacke und ich krallte mich in den festen Muskel hinein. Er war wirklich ausgezeichnet trainiert, dachte ich bewundernd. Bevor er jedoch zu seinem Höhepunkt kommen konnte, verlangsamte ich meine Bemühungen wieder und hörte schließlich ganz auf. Jonas schien froh über diese Unterbrechung zu sein. Er zog mich aus meiner knienden Position hoch und gab mir erneut einen Kuss. "Das war der Wahnsinn, Charlotte... das ist so gut mit dir", murmelte er in meinem Mund. "Aber jetzt bist du dran!" Und mit diesen Worten hob er mich hoch und trug mich auf seinen Armen ins Wohnzimmer nebenan. Sachte setzte er mich auf dem Sofa ab. Dann kniete er sich zwischen meine gespreizten Beine. Ich wusste, dass meine Vulva feucht glänzte; ich konnte mich nicht erinnern, jemals so erregt gewesen zu sein. Doch da wusste ich noch nicht, was als Nächstes kommen würde. Jonas beugte den Kopf, um nun auch mich oral verwöhnen zu können. Mit den Fingern spreizte er meine Schamlippen, während er mit der Zunge einmal durch mich hindurchleckte. Sein heißer Atem auf meiner Haut ließ mich schaudern und seine weiche Zunge trieb mich schon bei dieser ersten Berührung ein gutes Stück in Richtung meines eigenen Gipfels. "Du schmeckst phantastisch", lobte Jonas mich mit heiserer Stimme und senkte dann endgültig seinen Kopf zwischen meine Oberschenkel. Seine Lippen umschlossen meine vor Lust aufgerichtete Liebesperle und seine Zungenspitze umkreiste sie so liebevoll und zärtlich, dass ich aufstöhnte. Kurz ließ er von ihr ab, um auch meinen Eingang zu verwöhnen, doch schon bald kehrte er zu meinem empfindlichsten Punkt zurück. Gleichmäßig strich seine heiße Zunge über meine Klit. Er achtete darauf, sie nicht zu intensiv zu reizen, sondern schaffte es immer, auf diesem schmalen Grat zu bleiben, der so unendlich geil war, dass ich glaubte, es keine Sekunde länger mehr aushalten zu können. Doch ich hielt es aus. Minutenlang bearbeitete Jonas mich auf diese Weise, rhythmisch spürte ich seine Zungenschläge, die immer dann wieder etwas langsamer wurden, wenn mein Orgasmus in greifbare Nähe rückte. Perfekt balancierte er meine Erregung so aus, dass ich meine Hände irgendwann in seine Haare klammerte und ihn anflehte, mich endlich kommen zu lassen. Die Nässe, die aus mir herauslief, benetzte inzwischen meinen Po, ich konnte spüren, wie sie unter mir auf das Polster lief. Doch Jonas kannte kein Erbarmen. Ich wusste nicht, wo er die Selbstbeherrschung hernahm, sich nicht sofort auf mich zu stürzen und in mich einzudringen. Stattdessen fühlte ich seine Finger, die meine nasse seidige Liebeshöhle dehnten und tief in mich eindrangen. Das war zu viel. Ich stöhnte meine Lust laut heraus, während Jonas anfing, seine beiden Finger im Takt seiner Zunge herauszuziehen und dann wieder tief in mich zu stoßen. Dabei stimulierte er einen Punkt in mir, der mich fast um den Verstand brachte. Es war einfach teuflisch gut. Ich wand mich und keuchte seinen Namen, meine Hände krallten sich ins Sofa unter mir. Wie Feuer raste es durch meinen ganzen Körper und dann, endlich, war es so weit. Ich explodierte in einem gigantischen Feuerwerk, meine Beine zuckten unkontrolliert und meinen Kopf drückte ich mit geschlossenen Augen fest in den Nacken. Ich sah Sterne tanzen und in meinen Ohren rauschte es. Und noch bevor dieser heftige Schauer in mir abgeklungen war, spürte ich, wie Jonas sich aufrichtete und seinen großen schönen Penis mit einem Ruck in mir versenkte. Ich schnappte nach Luft, als ich urplötzlich so vollkommen ausgefüllt wurde. Tief schob Jonas seinen Harten in meine feuchte Liebeshöhle und begann fast augenblicklich mit schnellen ruckartigen Stößen. Er hielt mein Becken und unterstützte so die Intensität, mit der unsere Körper aneinanderklatschten. Es dauerte keine halbe Minute, bis ich das zweite Mal den Gipfel der Lust erklommen hatte. Meine Orgasmen gingen ineinander über und ich schrie leise auf. Die Kontraktionen in meiner Venus sorgten dafür, dass ich mich noch fester um Jonas schloss und er steigerte das Tempo. Lange würde er das so nicht aushalten, das war offensichtlich. "Oh, jaa, jaaaaa...! Gott, Charlotte!", stöhnte er mit tiefer Stimme, als sich sein Körper verkrampfte und er seine Finger in meine Hüften bohrte. Ich sah die Anspannung in seinem Gesicht, obwohl auch ich immer noch ziemlich weggetreten war. Und dann kam auch Jonas, der dabei wieder und wieder meinen Namen stöhnte. Als ich spätabends allein auf diesem Sofa saß, dachte ich über die Ereignisse des Tages nach. Mit einem Grinsen besann ich mich auf die unzähligen Briefe, die ich in meiner Kindheit an den Weihnachtsmann geschrieben hatte. Meistens begannen meine Wunschzettel mit der Erklärung, warum ich unbedingt – und wirklich unbedingt – ein Pony brauchte. Es hatte nie geklappt. Dieses Jahr jedoch würde ein Brief wahrscheinlich so beginnen: Lieber Weihnachtsmann, um mich brauchst Du Dich dieses Jahr nicht zu kümmern. Das schönste Geschenk habe ich nämlich bereits erhalten: Jonas. Du hattest recht, ein Pony brauchte ich nie so dringend. Aber einen Mann, ja, das ist etwas anderes. Vielleicht ist das sogar ein Mann zum Lieben. Verliebt bin ich auf jeden Fall. Morgen sehen wir uns wieder. Mal sehen, wie es weitergeht. Auf Weihnachten freue ich mich aber schon sehr... Deine Charlotte Mein Lächeln wurde breiter, als ich beschloss, dass so ein Brief auf jeden Fall noch einen Nachsatz brauchte. P.S.: Ich hatte den besten Sex meines Lebens. Danke, lieber Weihnachtsmann!
# Schneegestöber
## Lisa Cohen
Es waren noch zwei Tage bis Weihnachten. Und ich war mittendrin im Weihnachtsstress, den ich so verabscheute und der mich in diesem Jahr voll im Griff hatte. Dabei hatte ich alles so schön geplant. Meine Uni hatte ihre Pforten schon vor einer Woche geschlossen. Ich hatte also genug Zeit gehabt, sämtliche Geschenke zu besorgen, alles liebevoll einzupacken und mich auf meinen Besuch bei meiner Familie vorzubereiten. Sogar das Auto hatte ich schon vollgetankt. So perfekt vorbereitet war ich noch nie gewesen. Tja, und dann, als ich noch mal schnell zu einer Freundin fahren wollte, sprang mein alter Golf nicht an. Rührte sich einfach überhaupt nicht. Ich wusste, dass er seine Macken hatte, hatte aber aus finanziellen Gründen die überfällige Reparatur bis aufs nächste Jahr verschieben wollen. Was nun? Ich würde mit der Bahn fahren müssen und wollte lieber nicht darüber nachdenken, wie ich mein gesamtes Gepäck mitnehmen sollte. Doch die Bahn war natürlich komplett ausgebucht. Nichts ging mehr über die Feiertage. Damit hatte ich überhaupt nicht gerechnet. Meine letzte Rettung war die Mitfahrzentrale. Ich hatte das Glück, das ich brauchte. Und konnte nicht mehr wählerisch sein. Der Typ, der in meine Heimat fuhr, wirkte am Telefon recht mürrisch und wortkarg. Aber er versprach, mich zu einem günstigen Mitfahrpreis fast in die Nähe meiner Eltern zu bringen. Das Wetter war launisch an diesem 23. Dezember. Wir fuhren natürlich viel später los, als er zugesagt hatte. Es sollte schneien gegen Abend. Da hätten wir schon am Zielort sein können. Es war fast drei Uhr, als er endlich vor meiner Tür stand. Ich musste meine Klamotten selbst ins Auto packen. Er war kein Gentleman. Redete wenig, was vielleicht auch besser war. Tobias war in meinem Alter und er hätte ein ganz süßer Typ sein können, wenn er nicht so muffig gewesen wäre. Ich versuchte zu schlafen. Leider war der Auspuff von seinem auch nicht mehr ganz taufrischen BMW kaputt. So knatterten wir schweigend über die Autobahn und ich wollte nur schnellstmöglich zu meiner Familie... Es fing an zu schneien. Erst in einzelnen Flocken, dann stärker. Tobias drehte grimmig am Radio, um dem Wetterbericht zu lauschen. Eine Kaltwetterfront würde über uns hinwegziehen. Mit ganz viel Schnee. Meine Stimmung verdüsterte sich ziemlich, als sich die einzelnen hübsch anzusehenden Flocken in dichtem Schneetreiben vervielfältigten. Wir kamen immer langsamer vorwärts. "Hast du eigentlich Winterreifen drauf?" Tobias sah mich fast mitleidig an. Ich brauchte keine Antwort. Dumme Frage. Er war zum Glück ein umsichtiger verantwortungsvoller Fahrer. Er würde mich schon irgendwie sicher nach Hause bringen. Und dann fuhren wir plötzlich in einen Stau. Der Schnee hatte den Verkehr vor uns zum Erliegen gebracht. Wir sahen uns an. Zum ersten Mal so richtig. Er hatte schöne braune Augen. Sein wuscheliges blondes Haar passte gut dazu. Unter seinem dicken Skipullover musste er ganz knackig sein, stellte ich mir vor. Er stieg aus, um die Lage zu checken, und stieg wieder ein – mit langem Gesicht. "Das sieht erst mal nicht gut aus. Vor uns geht nichts mehr." Ich schloss genervt die Augen. Super Einstieg in den Urlaub! Ich hockte mit diesem Typen auf unbestimmte Zeit fest, in seiner alten Kiste, mitten auf der Autobahn. Tobias drehte unablässig am Radio. Wechselte ständig die Sender, bis er endlich die Musik fand, die ihm gefiel. Komischerweise entspannte er sich ein wenig, obwohl wir festsaßen, und wurde sogar etwas redseliger. Eigentlich war er ein cooler sympathischer Typ, wie sich in der nächsten Stunde herausstellte. Er spürte meine wachsende Unruhe und stieg immer mal wieder aus, um die Wetterlage besser einschätzen zu können, aber es war hoffnungslos. Es sah ganz so aus, als würden wir einschneien. Ich sah mich schon Heiligabend auf der Autobahn steckend mit Tobias. Ich hätte mir wahrhaft Besseres vorstellen können. Als wir beide gleichzeitig am Radiosender drehen wollten, berührten sich unsere Hände. Erstaunt sahen wir uns an und konnten es in den Augen des anderen lesen. Es war, als wenn wir einen leichten Stromschlag erhalten hätten, der durch unsere Körper fuhr. Mit einem Mal prickelte es in mir, wenn ich ihn ansah. Und ich wusste, es erging ihm genauso mit mir. Ein paar Minuten herrschte Schweigen. Tobias schaltete den Scheibenwischer mal wieder ein. Es schneite, als wenn es nie wieder aufhören wollte. Er griff nach hinten und holte aus seiner Reisetasche eine Thermoskanne und zwei Becher. Ich konnte es kaum glauben! Er hatte Kaffee mitgenommen und es gab sogar Kekse dazu. Was für ein geiler Typ dieser Tobias doch im Grunde war... Und dann, aus heiterem Himmel und doch nicht mehr so ganz unerwartet, drehte er sich zu mir und sah mir tief in die Augen. "Hast du eigentlich schon mal im Auto...?" Es sollte unser erstes Mal sein. Tobias küsste mich und alles war ganz einfach und unglaublich mit ihm. Seine Zunge spielte mit meiner, als wenn sie das schon immer so getan hätte. Seine Lippen passten hervorragend auf meine und eroberten jeden Zentimeter. Er war ein genialer Küsser, dieser etwas muffige Typ. Ich ließ es willig geschehen, dass er meinen Sitz zurückkurbelte und meine Bluse aufknöpfte, und dankte meiner Eingebung, dass ich sexy Unterwäsche angezogen hatte, obwohl ich nur zu meinen Eltern fuhr. "Du bist wunderschön." Behutsam fingerte er an meinen Brüsten herum, entblößte sie und betrachtete sie fast ehrfürchtig. Dann biss er zärtlich in meine Brustwarzen hinein, leckte genussvoll daran herum, nahm eine Handvoll Brust in seinen Mund und entlockte mir einen ersten sehnsüchtigen Seufzer. Seine Hände schienen erfahren zu sein. Sie streichelten meinen Bauch und alles, was sich unterhalb befindet, während Tobias weiter mit dem Mund meine Nippel verwöhnte, und konzentrierten sich dann zielstrebig auf den verlockenden Inhalt meines Slips. Ich lehnte mich zurück und schloss selig die Augen. Meinetwegen sollten wir einschneien, wenn man sich so die Zeit vertreiben konnte. Als er meine Erregung ganz deutlich spürte, flüsterte er: "Etwas eng hier..." Ich nahm an, er meinte seinen Wagen damit, und schob mein rechtes Bein auf den Türgriff und das linke zwischen Schaltknüppel und Lenkrad. Es wäre das erste Mal, dass ich mich in einem Auto vernaschen lassen würde und ich hatte noch keine Idee, wie das klappen sollte. Der eigentliche Sexakt würde wohl schwierig sein und vielleicht auch gar nicht unbedingt das, was ich wollte. Das Fummeln und Knutschen und gegenseitige Stimulieren war es, was mich anmachte. Tobias konnte sich gar nicht sattsehen an mir. "Ich steh total auf rasiert...", murmelte er und starrte ausgiebig auf meine entblößte Klitoris. Mir wurde heiß. Hitzeschauer rieselten durch meinen Unterleib. Seine Blicke entzündeten meine Lustnerven. Es begann zu brennen, dort, wo seine Augen verweilten. Ich spreizte meine Beine, so gut es eben ging, was nicht gerade weit war, und genoss es, mich einem Mann so schamlos zu präsentieren, den ich gerade mal drei Stunden kannte. Mein prüfender Griff an seinen Reißverschluss versprach das Ersehnte. Ich öffnete seinen Hosenschlitz und massierte die aufsteigende Potenz, die zu allem bereit war. "Geil!!!" Tobias keuchte dankbar. Er hatte eines dieser kompakten stämmigen Glieder – meine bevorzugten Objekte! Und so streckte ich meinen Mund irgendwie nach unten, um es wenigstens mal kurz zu schmecken, was sich in der Enge des Autos als wirklich schwierig herausstellte. Leider musste ich es bald wieder aufgeben, weil ich sonst einen üblen Krampf im Genick bekommen hätte, aber ich hatte Tobias' männlichen Geschmack auf der Zunge und der sorgte für die geilsten Vorstellungen in mir. Geschickt schob ich meine Hand am Schaft des Gliedes hoch und dann wieder herunter, die Eichel gleichzeitig zart mit einem Zeigefinger klopfend, mit den Daumen seine energiegeladenen Hoden drückend. Diese Technik brachte alle Männer meist blitzschnell auf Hochtouren. Tobias war keine Ausnahme. Er stöhnte gleich heftig, spreizte die Beine und drückte sich tief in seinen Sitz, damit er meinen Griff besser genießen konnte. Gleichzeitig öffnete er meine neugierig gewordene Höhle und stimulierte sie, bis sich unanständig viel süße Feuchtigkeit aus weiblicher Lust darin gebildet hatte. Ich keuchte vor Wonne. Nicht nur ich wusste, wie ich ihn anzufassen hatte. Auch Tobias hatte so seine ganz bestimmten Techniken. Mit Daumen und Mittelfinger spreizte er mich aufregend weit, damit sein kräftiger Zeigefinger ungestört meine Perle massieren konnte. Mir entschlüpften ein paar wirklich unanständige Worte, weil er diesen hochsensiblen Punkt nicht nur exakt traf, sondern ihn auch genaustens stimulierte. Ich spürte es sich regen und wachsen unter seiner Zuwendung und sah fasziniert, wie sich mein anspruchsvoller weiblicher Lustpunkt in nur wenigen Augenblicken fast verselbstständigte vor Erregung. Tobias elektrisierte mich vollkommen. Ich dehnte und streckte mich unter diesen fruchtbaren Berührungen, soweit das in dem engen Auto möglich war. Die Scheiben waren beschlagen, unser Atem dampfte. Orgiastische Empfindungen benebelten meine Sinne. Das Glied in meiner Hand wuchs unter meiner Obhut. Und bald schon sendete es die ersten Lusttropfen aus, die mir verrieten, wie es darum stand. "Sollen wir es versuchen?" Und schon drängte er sich irgendwie auf mich. Ich weiß bis heute nicht, wie wir es zustande brachten, uns auf dem engen Beifahrersitz zu vereinen. Alles kam mir verdreht vor, ich hatte das Gefühl, einen Krampf zu bekommen, doch es war mir egal. Als sein Glied an meine Scheide klopfte, zog ich es wie ausgehungert in mich hinein. Wir pressten uns ineinander. Tobias schaffte es sogar, seine Hüften zu bewegen. Es war bestimmt nicht der beste Sex, den ich je gehabt hatte, das ließen die Umstände nicht zu, aber ganz bestimmt der aufregendste. Wir schafften es, für ein paar Minuten ineinander zu bleiben. Sein Glied passte sich wunderbar den Krümmungen meiner Liebeshöhle an. Ich schaffte es auch noch in dieser für mich doch recht schwierigen Position zu verharren, bis Tobias übers ganze Gesicht strahlte und es für einen kurzen Moment noch ein bisschen heftiger in mir drin brodelte. Dann musste ich ihn von mir schieben, bevor ich ihm folgen konnte in die Besteigung des lustvollen Gipfels, und meine Beine nach unten ziehen. Es fing nämlich leider an, fast überall wehzutun, in dieser unbequemen, verzerrten Stellung... Jemand hupte hinter uns, dann noch jemand. Es schien weiterzugehen auf der Autobahn. Tobias schob sich auf seinen Sitz zurück. Sein hocherhobenes Glied guckte noch aus seiner Hose heraus. Ich musste mich abwenden, um ein Lachen zu verkneifen, und kümmerte mich darum, mich irgendwie und schnellstmöglich wieder anzuziehen, was gar nicht so einfach war. Es ging tatsächlich weiter. Erst im Schritttempo, dann schneller. Es schneite nicht mehr. Wir sprachen wenig während der restlichen Fahrt. Jeder hing seinen Gedanken nach. Ab und zu griff Tobias nach meiner Hand und drückte sie fest. Zum Abschied küssten wir uns noch einmal sehr intensiv. Tobias nahm mich eine Woche später wieder mit zurück nach München. In meiner Wohnung holten wir nach, was bei mir im Schneesturm zu kurz gekommen war...
# Am Weihnachtsabend
## Marie Sonnenfeld
Vergnügt packte Nina die letzten Weihnachtsgeschenke ein. Sie freute sich darauf, in diesem Jahr den Heiligabend mit ihrer Familie zu verbringen. Sie war zur Zeit Single und wollte gerade an diesem Abend nicht allein sein. Und da sie ohnehin gern mit ihrer Familie zusammen war, schaute sie dem Fest voller Vorfreude entgegen. Nina hatte sich an diesem Abend besonders schön gemacht. Sie trug ein schwarzes enges Kleid, das ihre schlanke Silhouette wundervoll betonte und sie hatte sich extra zu diesem festlichen Anlass ihre langen hellblonden Haare hochgesteckt. Nachdem sich alle begrüßt und umarmt hatten, holte ihre Mutter auch schon die Gans aus dem Ofen und bat die Familie zu Tisch. Auf der großen Tafel leuchteten stimmungsvoll die Kerzen, als sie dann gemütlich miteinander zu Abend aßen. Sie waren acht Personen, davon drei noch Kinder, die schon vor dem Dessert voller Ungeduld auf ihre Geschenke warteten. Die Erwachsenen aber unterhielten sich wundervoll und hatten sich viel zu erzählen, da Nina ihren Bruder und dessen Frau lange nicht gesehen hatte. Nach dem Essen räumten sie unter ständigem Fragen der Kinder, wann es denn nun die Geschenke geben würde, gemeinsam den Tisch ab. Später saßen alle gemütlich im Wohnzimmer, als es plötzlich kraftvoll gegen die Haustür klopfte. Nina schaute ihre Eltern fragend an. Ihr Vater zwinkerte ihr fast unmerklich zu und sagte im gleichen Augenblick mit gespieltem Erstaunen: "Nanu? Wer kann das denn sein?" Die Kinder wurden stiller und schauten sich verdutzt an. Ninas Vater ging zur Tür und öffnete. Nina hörte eine zweite Männerstimme aus dem Flur. Dann trat er ein: der Weihnachtsmann! Die Kinder waren von einem Moment auf den anderen mucksmäuschenstill. Dort stand er in seinem roten Weihnachtsmann-Kostüm mit den derben schwarzen Stiefeln und hübschen, sanft blickenden, braunen Augen. Mehr war von ihm unter dem vielen weißen Kunsthaar, dem Bart und dem roten Stoff der Mütze nicht zu sehen. Nina war genauso verblüfft wie die Kinder. Das hatte sie auch nicht gewusst. Sie überlegte fieberhaft, wer der Mann unter der Verkleidung wohl sein könnte, aber so sehr sie sich auch bemühte, sie konnte ihn nicht erkennen. Er machte seine Sache gut. Er hatte ein großes schweres Buch dabei, aus dem er zu jedem Kind mit extra tiefer Stimme etwas vorlas. Er ermahnte und lobte und anschließend fragte er nach einem Gedicht, aber da die Kinder aufgeregt und darauf nicht vorbereitet waren, konnte keines von ihnen eines aufsagen. Der Weihnachtsmann lachte tief und Nina konnte sich von seinen fröhlich blitzenden Augen gar nicht losreißen. Sie mochte ihn, er gefiel ihr irgendwie. Nina nahm wahr, dass er zwischendurch immer wieder zu ihr herüberschaute. Und es schien ihr, als würden seine Blicke mit jedem Mal ein wenig länger auf ihr verweilen. Nachdem er aus einem großen Jutesack die Geschenke an die Kinder verteilt hatte und sich nun den Erwachsenen zuwandte, schaute er Nina in die Augen und sagte mit extra tiefer Stimme: "Und du? Warst du auch schön brav?" Dabei zwinkerte er sie lächelnd an. Nina stand vor ihm und war auch ein wenig aufgeregt. Sie lächelte... Auch sie bekam ein Päckchen und der Weihnachtsmann kündigte an, dass auch die anderen Kinder noch warten würden und er nun wieder weitermüsse. Alle verabschiedeten ihn winkend und er stapfte durch den Flur wieder Richtung Haustür. Nina saß da, schaute ihren Vater an und flüsterte: "Wer war das denn?" Ihr Vater zeigte ihr an, doch kurz mit in die Küche zu kommen. Dort erzählte er ihr flüsternd, dass es sich um einen Arbeitskollegen ihres Bruders handeln würde. Aha, daher kannte Nina ihn nicht. Denn an diese tollen Augen hätte sie sich gewiss erinnert! Ihr Vater bat sie außerdem leise, doch schnell noch hinterherzulaufen, um ihm die kleine 'Aufwandsentschädigung' zu überreichen. Er wollte zwar nichts dafür, aber er sollte es dennoch nicht umsonst getan haben. So nahm Nina die Flasche Wein, warf sich schnell ihren Mantel über und lief zu dem Parkplatz um die Ecke, wo er laut ihres Vaters geparkt haben sollte. Noch immer völlig hin und weg dachte Alexander, als er zurück zu seinem Auto ging: Oh wow, das hätte Helge mir aber echt vorher sagen können, dass er eine so attraktive Schwester hat! Er war nach wie vor fasziniert von ihrem Lächeln, ihren Augen und ihrem tollen Körper in diesem atemberaubend engen Kleid. Er fand sie einfach hinreißend. Als Nina am Parkplatz ankam, zog er sich gerade das Weihnachtsmann-Kostüm aus und seine Jacke wieder an. Sie konnte verfolgen, wie er dann damit fortfuhr, das Kostüm in seinem alten Kombi zu verstauen. Jetzt konnte sie seine dunklen dichten Haare statt der rot-weißen Weihnachtsmütze sehen. Nina sprach ihn ein wenig außer Atem an. Sie übergab ihm die Flasche und bedankte sich im Namen ihrer Familie für seinen tollen Auftritt. Alexander konnte es kaum glauben, dass sie hier jetzt wirklich vor ihm stand. Er freute sich unheimlich, sie noch einmal wiederzusehen. Er schlug die Heckklappe seines Autos zu, lachte sie an und erwiderte fröhlich, dass er es wirklich gern getan hätte. Nina wollte sich nun eigentlich wieder verabschieden, konnte sich aber Alexanders Blick nicht entziehen, als er sie mit seinen warmen braunen Augen so intensiv anschaute. Er fand sie unheimlich hübsch. Er war wie in ihren Bann gezogen. Und auch Nina ging es ähnlich. Ihr Herz klopfte schneller, als sie minutenlang einfach nur dastanden und sich in die Augen sahen. Auch sie fühlte sich wie verzaubert von ihm und seiner sympathischen Ausstrahlung. Irgendwann fasste sie sich ein Herz und sagte leise: "Ich glaube, ich sollte so langsam wieder zurückgehen." Alexander schaute sie weiterhin an und sagte ebenso leise: "Schade." Nina wäre so gern noch bei ihm geblieben. Und auch er wünschte sich, dass sie noch nicht wieder gehen würde. Nina fühlte sich in seiner Nähe sehr wohl, das konnte sie jetzt, nach dieser kurzen Zeit, schon spüren. Alexander strich sanft über ihre Wange und sagte zu der jungen Frau: "Darf ich dich um etwas bitten?" Nina war überrascht und sagte vollkommen perplex: "Ja, gern!" Alexander fragte: "Würdest du bitte dein Haar für mich öffnen?" Nina staunte ein wenig, dachte aber dann: Wenn es weiter nichts ist, und schon im nächsten Moment löste sie ihre Haarspange und ihre Haare fielen ihr weich und lang über die Schultern und den Rücken. Alexander konnte sich ein begeistertes "Wow!" nicht verkneifen und Nina musste schmunzeln. Alexander fand sie in ihrem engen Kleid sehr sexy und unglaublich aufregend. Er war erotisiert und konnte nicht anders, als sie zu berühren. Er fragte vorher noch: "Darf ich?" und fasste im nächsten Augenblick auch schon angeheizt in ihre blonde Mähne hinein. Alexander konnte selbst kaum glauben, wie sehr er sich von ihr angezogen fühlte. Er fand sie bildhübsch und einfach atemberaubend. Er verspürte den Wunsch, sie zu küssen, befürchtete aber, dass Nina sich davon bedrängt fühlen würde, da sie sich ja kaum kannten. Trotzdem wagte er es. Er konnte nicht anders, als es jetzt und hier zu tun. Er trat einen Schritt näher an Nina heran und nahm sie unvermittelt in seinen Arm. Nina war erst überrascht, mochte aber das Gefühl, von ihm im Arm gehalten zu werden, sehr gern. Sie genoss es und legte auch ihre Arme um ihn. Sie drückte sich ein wenig fester an ihn. Es tat ihr gut, da sie das Gefühl, von einem kräftigen großen Mann umarmt zu werden, schon sehr vermisst hatte. Alexander spürte ihr Anschmiegen und freute sich darüber. Es löste in ihm ein kribbliges Gefühl aus und er fühlte, wie sein Penis sich langsam verhärtete und steif wurde. Ja, er fand sie heiß, und wie! Alexander schaute ihr in die Augen, sie erwiderte seinen Blick und gleich darauf gaben sie sich einem langen zärtlichen Kuss hin. Ihre Lippen lagen weich aufeinander und sie konnten die Wärme und den Atem des jeweils anderen spüren. Jetzt wurde auch Nina langsam immer erregter. Sie fand ihn auch ungemein attraktiv. Sie mochte seine große dunkle Erscheinung und seine kräftig aussehenden Hände, die jetzt so aufregend auf ihrer Taille unter ihrem Mantel lagen, sehr. Jetzt schob auch sie ihre Hände unter seine Jacke, während sie ihren Kopf zwischen zwei Küssen an seine Schulter legte. Sie mochte ihn gern berühren, sie fühlte sich bei ihm sehr beschützt. Nina war auf dem besten Wege, sich hoffnungslos in ihn zu verlieben. Und genau das genoss sie gerade sehr. Alexander fühlte ihre warmen zarten Hände unter seiner Jacke und das Kribbeln verstärkte sich daraufhin noch weiter. Würde er seine Hüfte gegen ihre drücken, könnte sie seine Erektion spüren, die sich mittlerweile zu ihrer vollen Größe in seiner Jeans aufgebaut hatte. Alexander fühlte mit jeder Faser seines Körpers, dass er große Lust hatte, Lust auf diese Wahnsinnsfrau, deren Vornamen er noch nicht einmal kannte. Und sie seinen auch nicht, fiel ihm ein. Er gab ihr einen sanften Kuss auf ihren Mund und sagte dann leise, während er ihr eine Haarsträhne aus ihrem Gesicht strich: "Vielleicht ist es nicht schlecht, wenn du weißt, wie der Mann heißt, den du gerade so wundervoll küsst?" Dabei grinste er sie an und fügte dann hinzu: "Ich bin Alexander." "Oh, ja, du hast recht", antwortete Nina lachend und stellte sich dann auch mit den Worten "Und ich heiße Nina" vor. Und dann fügte sie noch fragend hinzu: "Und du bist ein Arbeitskollege meines Bruders?" "Ja, das stimmt", sagte Alexander. "Wir arbeiten tatsächlich zusammen in der Tischlerei. Und da ich an diesem Abend allein war und nichts weiter vorhatte, habe ich total gern den Weihnachtsmann für euch gespielt. Und ich fühle gerade in meinem Arm, dass ich dafür schon jetzt hammermäßig belohnt wurde." Dabei lachte er sie freundlich an und gab ihr erneut einen sanften Kuss auf ihre lächelnden Lippen. "Nina", sagte Alexander jetzt wieder etwas ernster, "ich glaube, du wirst von deiner Familie jetzt wirklich langsam vermisst, hm?" Nina stimmte ihm zu, indem sie sagte, dass das wohl tatsächlich so wäre und fragte ihn, was er denn heute noch so vorhätte. Alexander zuckte ratlos mit den Schultern. Er sagte leise: "Ach, nichts weiter." Nina schaute ihn an. Sie fragte: "Und was ist mit deiner Familie?" Er erklärte: "Sie ist nicht sehr groß. Meine Eltern sind vor Jahren nach Asien ausgewandert und Geschwister habe ich nicht. Tja, und solo bin ich zur Zeit auch." Nina schaute ihm in die Augen. Sie nahm ihn wortlos an die Hand und zog ihn mit sich in Richtung ihres Elternhauses. "Hey!", sagte Alexander, während sie gingen. "Das geht doch nicht, ich kann euch doch jetzt, am Weihnachtsabend, nicht stören!" "Aber du störst doch nicht, im Gegenteil!", erwiderte Nina daraufhin. "Meinen Bruder kennst du gut, der Rest der Familie ist auch in Ordnung und außerdem will ich dich auch nicht so einfach wieder gehen lassen", sagte Nina, während sie stehenblieb und ihn noch einmal liebevoll küsste. Und so verbrachten sie einen wundervollen Heiligabend, der gemütlich und fröhlich war. Alexander genoss noch eine große Portion von der Weihnachtsgans und die ganze Zeit über saßen sie dicht nebeneinander, um im Laufe des Abends immer enger zusammenzurücken. Niemandem entgingen die Blicke und die zarten Berührungen, die Nina und Alexander zwischendurch immer wieder austauschten. Alexander war davon fast wie berauscht. Er fühlte sich lustvoll eingefangen von der Sinnlichkeit, die Nina ausstrahlte und von ihrem süßen Duft, der ihm immer wieder in die Nase stieg. Wenn er sich nur vorstellte, mit ihr zu schlafen, mit ihr richtig guten, hemmungslosen Sex zu haben, fühlte er, wie sein Penis sich erneut versteifte. Er hatte so wahnsinnige Lust auf sie, dass er es nicht hätte in Worte fassen können. Und er fühlte noch etwas, etwas besonders Schönes: Er hatte sich in Nina verliebt. Als sich der Abend seinem Ende näherte und sich irgendwann alle herzlich voneinander verabschiedet hatten, schug Alexander vor, doch den geschenkten Rotwein, den er noch im Auto liegen hatte, gemeinsam bei ihm zu trinken, da er gern noch mehr Zeit mit ihr verbringen wollte. Nina wollte diesen schönen Abend auch noch nicht enden lassen und sagte gern zu. So fuhren sie dann in seinem Kombi, der mit einigen Holzteilen und Werkzeug beladen war, zu ihm. Er hatte ein kleines altes Reetdachhäuschen etwas außerhalb der Stadt, das er zusammen mit Ninas Bruder und einigen anderen Kollegen vor Jahren selbst saniert und renoviert hatte. Als Alexander die Haustür aufschloss und Nina hereinbat, spürte sie gleich die gemütliche Atmosphäre dieses kleinen Häuschens. Es duftete nach Orangen und Lebkuchen. Alexander ging als Erstes zum Kaminofen, um Holz nachzulegen. Kurz darauf erfüllte eine wohlige Wärme die Räume. Sie zogen ihre Jacken und Schuhe aus und Nina ging zu ihm und umarmte ihn fest. Sie sagte: "Echt schön hast du es hier!" Worauf er leise in ihr Ohr flüsterte: "Echt schön ist, dass du hier bist!" Er umarmte sie auch fest und drückte ihren Kopf mit einer Hand sanft an seine Schulter. Dabei streichelte er über ihr Haar. Sie schmiegte sich an ihn. Alexander strich mit der anderen Hand auf ihrem engen Kleid an ihrem Körper herunter. Er flüsterte: "Weißt du, dass du mich fast wahnsinnig machst mit deiner tollen Figur und diesem atemberaubenden Kleid?" Nina schaute ihn fragend an. "Wirklich? Oh, das wusste ich echt nicht. Aber es ist toll, das zu hören", sagte sie grinsend und gab ihm einen Kuss auf den Mund. Alexander legte nun beide Hände auf ihren Po und streichelte ihn. Er küsste sie wieder und diesmal fühlte sie seine warme Zunge, die sich vorsichtig ihrer Zungenspitze näherte und liebevoll mit ihr spielte. Als Nina sich an ihn drückte, konnte sie die Verhärtung in seiner Jeans sehr deutlich spüren. Es freute sie, dass er sie scheinbar derart aufregend fand. Ihr ging es ja ebenso, auch sie hatte große Lust auf ihn. Sie strich mit einer Hand über die harte Ausbeulung seiner Hose. Alexander stöhnte leise in ihr Ohr. Das gefiel Nina, davon liefen ihr kleine Schauer der Erregung den Rücken hinunter. Sie machte weiter, streichelte ihn fester. Alexander atmete schneller und legte seine Hand auf ihre, um sie noch stärker auf seinen harten Penis zu drücken. "Oh Gott, Nina, es ist phantastisch!", stöhnte Alexander in ihr Haar hinein. Er hatte seine Hände noch immer auf ihrem Po, begann nun aber, eine Hand weiter nach innen in Richtung ihrer Schenkelinnenseiten zu schieben, sofern es ihm durch den straffen engen Stoff ihres Kleides möglich war. So heizten sie sich immer weiter auf, streichelten und liebkosten sich und zogen sich dabei gegenseitig ihre Kleider aus. Als Nina ihr Kleid abstreifte, war das für Alexander Erotik pur. Er hätte nur beim Zusehen schon kommen können. Er fand sie mehr als heiß und er war erregt, und wie! Alexander holte seine Bettdecke und breitete sie auf dem Holzdielenboden vor dem Kaminofen aus. Dann setzte er sich auf die Decke und zog Nina an einer Hand mit sich herunter. Nina fand es schön bei ihm. Es war gemütlich, behaglich und warm und sie mochte dieses Knistern, das in der Luft lag, diese spannende Leidenschaft, die kaum auszuhalten war. Sie lagen nur in ihrer Unterwäsche auf der Decke und streichelten sich weiterhin sehr zärtlich und liebevoll. Nina streifte ihm seine Boxershorts vom Po und zog sie ihm aus. Dabei gab diese den Blick auf seinen großen harten Penis frei. Sie schaute ihn an, streichelte mit den Fingerspitzen seinen Schaft entlang und zog seine Vorhaut ein paarmal über seine pralle Eichel. Alexander lag auf die Ellenbogen gestützt auf dem Rücken. Er schaute Nina zu und konnte ein Stöhnen nicht unterdrücken. Und als Nina begann, seine Vorhaut vor und zurück zu reiben, ließ er sich auf die Decke zurückfallen und stöhnte ein weiteres Mal auf. Er zog Nina zu sich heran und zwischen zwei leidenschaftlichen Küssen sagte er leise zu ihr: "Oh, Nina, du bist so sexy, einzigartig und einfach große Klasse!" Nina lächelte ihn lieb an, setzte sich auf seinem Bauch auf und öffnete die Verschlusshaken ihres schwarzen BHs, um ihn auf seinen Oberkörper herabfallen zu lassen. Dann nahm sie seine Hände und legte sie jeweils auf ihre Brüste. Nina konnte genau in seinem Blick sehen, wie sehr ihn das in seiner Erregung noch weiter nach vorn kickte. In seinen Augen stand seine Wollust sehr deutlich geschrieben. Es war dieser intensive und dennoch zärtliche Blick, mit dem er sie ansah, und seine großen geöffneten Pupillen, die seine Augen noch dunkler erscheinen ließen. Und auch Nina war ihre Erregung deutlich anzumerken. Alexander konnte ihre Feuchtigkeit deutlich durch ihr Höschen hindurch auf seinem Bauch fühlen und auch ihre Brustwarzen, die er hart und groß unter seinen Händen spürte, verrieten ihre Lust. Er knetete ihre Brüste ein wenig fester. Nina hatte ihre Augen geschlossen und stöhnte leise. Alexander zog sie wieder zu sich herunter, um sie erneut zu küssen. Nina erwiderte seine Küsse feurig und er konnte es vor lauter Lust und Begierde kaum noch aushalten. Er legte seinen Mund an ihr Ohr und sagte leise und rau: "Nina, bitte schlafe mit mir!" Auch Nina wünschte sich in diesem Moment nichts sehnlicher. Auch sie konnte ihre Leidenschaft und ihre Lust kaum noch im Zaum halten und erwiderte leise stöhnend: "Ja, Alex!" Dabei hob sie ihren Po ein wenig nach oben und schob sich weiter nach hinten. Dann ließ sie sich wieder herunter, aber so, dass sie genau über seinem harten Penis innehielt. Sie nahm seine Hand, führte sie an ihre Scheide, damit er den Steg ihres Höschens beiseiteschob. Er tat es gern und was er dann an seinen Fingerspitzen fühlte, war für ihn einfach nur geil. Sie war so nass, dass sie geradezu auslief und ihre Scheide hatte sich geöffnet wie eine reife Frucht. Sie fühlte sich heiß an, heiß und nass. Er konnte sich nicht länger zurückhalten, er musste in sie eindringen. Er drückte sie sanft ein Stück weit herunter, so dass sein Bester sich langsam in ihre enge Vagina hineinschob. Dann legte er seine Hände auf ihre Hüfte und presste sie ganz auf sich. Er stöhnte laut auf. Auch Nina konnte ein Stöhnen nicht zurückhalten, als sie spürte, dass sie mit einem Mal so komplett ausgefüllt wurde. Es war herrlich! Sie schaute in sein Gesicht und lächelte ihn süß an. Alexander lag mit geschlossenen Augen da. Sie bat ihn, sie anzusehen. Er öffnete seine Augen ein kleines Stück, grinste sie schief an, als er ihr Lächeln sah, und schloss seine Augen daraufhin gleich wieder. Er konzentrierte sich darauf, die Kontrolle über seine starke Erregung zu behalten und der Blick in Ninas süßes Gesicht und auf ihre festen Brüste mit den harten Brustwarzen trug nicht unbedingt dazu bei, dass ihm das besser gelang. Nina stützte sich mit ihren Händen auf seinen Schultern ab und begann sich auf ihm, auf seinem harten Phallus, zu bewegen. Immer wieder auf und ab. Manchmal schneller, dann wieder langsamer, aber immer in einem unerbittlich gleichmäßigen Rhythmus. Alexander lag unter ihr, seine Hände in ihrer Taille, seine Muskeln vor Erregung angespannt. Er stöhnte und atmete schneller. Zwischendurch kreiste sie mit ihrem Becken und manchmal hob sie sich so weit hoch, dass er fast herauszurutschen drohte. Aber nur fast, denn sie senkte sich immer wieder rechtzeitig auf ihn herunter, so dass sein harter großer Penis immer wieder tief in ihr versank. So schlief sie eine Weile mit ihm, wobei sich ihre Erregung immer weiter verstärkte. Auch Alexander spürte, dass er sein Kommen so nicht mehr lange zurückhalten konnte und sagte plötzlich atemlos, nachdem er durch eine besonders intensive Reibung durch Ninas Scheide wieder aufstöhnen musste: "Bitte leg du dich nach unten, sonst komme ich jede Sekunde, Nina!" Sie hielt inne, rutschte von seinem harten lustbringenden Penis herunter, was sich für Alexander auch wieder sehr intensiv anfühlte und ihn die Luft anhalten ließ. Sie legte sich neben ihn, ohne ihn loszulassen. Er drehte sich auf sie und streichelte mit dem Daumen ihr Gesicht. "Du bist wunderschön, Nina", sagte er zärtlich. Und während er sie liebevoll küsste, zog er ihr mit der rechten Hand den schwarzen Slip von ihrem Po und streifte ihn ihr ab. Nina half ihm dabei, indem sie ihre Hüfte anhob. Alexander küsste sie erneut, saugte an ihrer Unterlippe und drang wieder in sie ein. Schob sich in ihr williges Geschlecht. Nina stöhnte erneut auf. Sie umfasste seinen stahlharten Penis mit ihren Scheidenmuskeln ganz fest. Gleichzeitig bewegte Alexander sich in ihr. Seine Vorhaut hatte sich ganz nach hinten über seine Eichel gezogen und seine Erektion war so hart und prall, dass er jeden Moment befürchtete zu kommen. Aber er wollte es gern noch weiter genießen, wollte seinen Orgasmus unbedingt noch hinauszögern. Er bewegte sich langsamer, voller Gefühl. Nina lag unter ihm, ihr Gesicht vor Erregung gerötet, ihre Haare zerwühlt. Alexander konnte sie kaum anschauen, ohne dass es ihn wieder näher an seinen Orgasmus heranbrachte. Er war erregt wie schon lange nicht mehr. Er spürte schon die ersten Anzeichen, spürte, dass er nicht mehr weit davon entfernt war, sich in Nina zu ergießen. Sein Gefühl war grandios, er war ihm komplett ausgeliefert. Er konnte an nichts anderes mehr denken. Alexander bewegte sich wieder ein wenig, stieß sanft in Ninas heiße nasse Scheide hinein. Oh, ja, das war gut, so gut! Er tat es noch einmal und immer wieder. Alex schlief jetzt ohne Kompromisse mit ihr. Er wurde heftiger und schneller. Es war so großartig, das Gefühl der totalen Erregung so gnadenlos. Sein Atem ging schneller, er stöhnte leise und hatte beide Hände, auf seine Ellenbogen gestützt, in ihren Haaren vergraben. Nina sah in sein Gesicht, konnte seine Erregung und seine Lust ganz deutlich darin sehen. Ja, er war kurz davor, das sah und spürte sie genau. Es war herrlich! Auch Nina war hochgradig erregt. Auch sie konnte ihren Orgasmus schon in greifbarer Nähe fühlen. Wenn er sich jetzt noch einige Male so bewegen würde, auf genau diese Art mit seinem großen Penis weiter an ihrer Klitoris entlangreiben würde, dann wäre es auch bei ihr so weit. Aber dann hielt Alexander plötzlich inne, sagte, dass er es nicht mehr länger zurückhalten könne und jetzt kommen würde. Dann stöhnte er laut und tief auf und griff noch einmal fester in ihre Haare hinein. Er presste sich in diesem Moment noch stärker auf sie und vergrub sein Gesicht in ihrer Schulter. Er kam und kam. Nina streichelte seinen Rücken, sie genoss diesen Augenblick sehr. Ihn dabei zu sehen, genau in diesem Moment in sein Gesicht zu schauen, und seinen harten, festen, ejakulierenden Penis in sich zu spüren, machte Nina sehr an. Es war einfach überwältigend. Alexander, noch immer außer Atem, öffnete wieder seine Augen und schaute Nina glücklich lächelnd an. Er küsste sie zärtlich und fragte: "Du bist nicht gekommen, hm?" Nina schüttelte den Kopf. "Aber du möchtest doch noch, oder?", fragte er weiter, während er ihr in die Augen schaute. Nina nickte und grinste ihn an. Er grinste auch und sagte: "Du bist ja wohl das Süßeste, das mir jemals untergekommen ist!" Dabei biss er ihr sanft und verspielt in die Schulter und zog seinen Penis vorsichtig aus ihr heraus. Er rutschte ein Stück tiefer, küsste und streichelte ihren Bauch, spielte mit seiner Zunge in ihrem Bauchnabel und küsste sich langsam weiter abwärts. Er strich mit seinen Lippen und seiner Nase über ihre Schamhaare hinweg und schließlich fand seine Zunge Ninas empfindlichsten Punkt. Alexander konnte fühlen, wie hart und geschwollen Ninas Klit noch war. Er fand es schön und begann, ihre Perle mit seiner Zunge zu umspielen und immer wieder mit ihr darüberzustreichen. Nina stöhnte leise und hielt sich an seinem Kopf fest. Sie wand sich unter seiner Zunge und spreizte ihre Schenkel immer weiter. Ab und zu drang Alexander auch ein Stückchen mit seiner Zunge in Ninas nassen Scheideneingang ein, dann wieder saugte er ganz sanft an ihrer Klit. Er fand es wundervoll. Er verwöhnte sie liebevoll, bis sie schließlich ihre Beine ausstreckte, sich kräftiger an ihm festhielt und etwas lauter sagte: "Ja, Alex, ja!" Er leckte weiter über ihre Perle und saugte noch einmal, diesmal etwas kräftiger an ihm. Nina stöhnte auf, streckte sich seinem Mund entgegen und kam. Ihr Orgasmus war intensiv und unbeschreiblich schön. Sie fühlte ihn wie weiche Wellen, die durch ihren Körper liefen, und wie eine Explosion, die in einem watteweichen Fall endete. Alexander wischte sich mit dem Handrücken über den Mund, küsste sich über ihren Bauch und ihre Brust wieder nach oben und legte sich dann ganz nah neben Nina. Sie drehte sich auf die Seite und schmiegte sich in seinen Arm. Sie flüsterten miteinander, sagten sich, wie gern sie einander hatten, wie wundervoll sie es gerade fanden und wie glücklich sie über ihr Kennenlernen an diesem besonderen Abend waren...
# Ausgerutscht!
## Jenny Prinz
Ich fühlte mich so einsam und leer wie noch nie in meinem Leben. Das war die Quittung dafür, dass ich an die Liebe geglaubt und für einen Mann an das andere Ende von Deutschland gezogen war. Unsere Liebe hielt nach dem Umzug noch genau ein halbes Jahr und das Weihnachtsfest, das unser erstes gemeinsames in München werden sollte, war nun mein absoluter Albtraum. Auch wenn ich den Trennungsschmerz inzwischen überwunden hatte und nur noch wütend war, so war ich doch immer noch eine Fremde in dieser großen Stadt, 700 Kilometer entfernt von zu Hause. Ich hatte mir in der Adventszeit einzureden versucht, dass so ein Weihnachten allein auch etwas sehr Besinnliches sein könnte, doch an Heiligabend funktionierte diese Lüge nicht mehr. Ich war einfach nur traurig und mutterseelenallein. Und je länger ich auf meinen Weihnachtsbaum starrte, desto trübsinniger wurde ich. Dagegen musste ich etwas unternehmen. Kurzentschlossen stieg ich in meine gefütterten Winterstiefel, zog eine Daunenjacke an und machte mich auf den Weg. Wohin ich genau wollte, wusste ich nicht, aber es war auch egal, Hauptsache raus aus der geschmückten Wohnung und an die frische Luft. Gerade als ich auf die Straße trat, fing es an zu schneien. Das war wenigstens ein kleiner Vorteil im Süden; wohin man blickte, lag dicker Schnee und ich freute mich wie ein kleines Kind, als ich die leichten Flocken im Schein der Straßenlampen wirbeln sah. Ich liebe Schnee und ich freute mich nun noch mehr auf meinen Spaziergang. Schon ein bisschen besser gelaunt stapfte ich los und versuchte zu vergessen, dass es Heiligabend war und alle anderen gemeinsam mit ihren Lieben in den Wohnzimmern saßen. Nachdem ich eine halbe Stunde gelaufen war, spürte ich die Kälte trotz warmer Kleidung in jeder Faser. Es war unbeschreiblich, wie wenige Menschen auf der Straße waren; ich hatte das Gefühl, die ganze Welt gehörte mir allein. Wahrscheinlich trug das Wetter auch dazu bei, dass sich kaum jemand in diesem kleinen Vorort vor die Tür wagte. Zügig marschierte ich die Straßen entlang, die Hände tief in den Jackentaschen vergraben. Inzwischen konnte ich mich sogar wieder an den hellen Lichtern erfreuen, die in den Fenstern und Eingängen leuchteten. Vielleicht gab es ja irgendwann auch für mich meine wahre große Liebe. Es war nicht nur das Alleinsein, wie ich mir eingestand, es war auch die körperliche Nähe, die mir fehlte. Es war nun schon so lange her, dass ich die Hände und die Lippen eines Mannes gespürt hatte. Ich sehnte mich danach, von einem Mann berührt zu werden, ihn tief in mir zu fühlen. Ich konnte mich nicht erinnern, schon mal so ausgehungert nach Zärtlichkeit gewesen zu sein. Und diese romantische Weihnachtszeit steigerte mein Verlangen noch um ein Hundertfaches. In diese Gedanken versunken bemerkte ich leider zu spät, dass der Fußweg vor mir ziemlich uneben war und außerdem von Eis überkrustet. Ich rutschte aus und bei dem Versuch, mein Gleichgewicht zurückzuerlangen, verdrehte sich mein Knöchel. Ich schrie auf und landete recht unsanft auf meinem Po. Erschrocken blieb ich einige Sekunden sitzen. Der Schmerz in meinem Fußknöchel war heftig. Ob ich wohl aufstehen konnte? Mit zusammengebissenen Zähnen rappelte ich mich mühselig auf. Noch immer schneite es und die Stille, die mir eben noch so wunderschön vorkam, gab mir jetzt ein mulmiges Gefühl. Mit dem Fuß konnte ich auf keinen Fall weiterlaufen und mein Handy lag zu Hause auf dem Wohnzimmertisch. Ich hatte es in meiner Eile, an die frische Luft zu kommen, einfach liegengelassen. Zögernd blickte ich zu dem Haus, vor dem ich gefallen war. Die Fenster waren dunkel und überhaupt sah es wenig einladend aus. Nun ja, wenn einer zu Hause gewesen wäre, hätte er ja ruhig schneeschippen können. Das nächste Haus lag ein Stück entfernt in einem großen Garten. Mit zusammengebissenen Zähnen humpelte ich los. Ich fühlte mich unwohl bei dem Gedanken, jemand anderem gerade heute ins Abendessen zu platzen, aber ein kurzer Telefonanruf würde wohl nicht allzu sehr stören. Ich müsste mir nur kurz ein Taxi rufen... Mein Knöchel brannte wie Feuer, als ich endlich vor der Haustür stand und den Finger auf die Klingel legte. Im Haus ertönte eine Glocke und schon nach wenigen Sekunden öffnete sich die Tür. Und ich blickte in die wärmsten und liebevollsten Augen, die ich jemals gesehen hatte. Ein wenig erstaunt schaute mich der Mann an. Er war vielleicht Anfang vierzig und sah nicht im eigentlichen Sinne gut aus, wie ich sofort feststellte, aber unglaublich sympathisch. Ein kurzer dunkler Vollbart zierte sein markantes Gesicht mit der ein wenig zu groß geratenen Nase. Zur verwaschenen Jeans trug er ein derbes Hemd und eine gemütlich wirkende Strickjacke. Schweigend und lächelnd wartete er, während ich ihn aufmerksam musterte. Erschrocken riss ich mich zusammen und streckte meine Hand aus: "Hallo, ich bin Helena. Und ich habe mir gerade bei meinem Weihnachtsspaziergang den Fuß verknackst. Mein Handy liegt natürlich zu Hause. Kann ich bitte kurz ein Taxi anrufen?" Und ergänzend fügte ich noch schnell hinzu: "Es tut mir total leid, dass ich ausgerechnet heute hier so hereinplatze!" "Gar kein Problem", antwortete er. "Ich bin übrigens Robert. Komm erst mal herein." Dankbar folgte ich der Einladung und humpelte in den Flur. Es war schön, ins Warme zu kommen. "Ich ruf dir ein Taxi, aber das wird bei dem Wetter ein Weilchen dauern. Willst du dich solange ins Wohnzimmer setzen? Möchtest du einen Tee oder so? Du musst ja halb erfroren sein." Seine Fürsorge tat unheimlich gut, doch ich erinnerte mich daran, dass es der 24. Dezember und mein Auftauchen hier wohl denkbar unpassend war. Ich antwortete also, dass ich nicht stören wolle und ich auch gern draußen warten könne. Doch dann sagte Robert etwas, was mich veranlasste, zu bleiben: "Du störst überhaupt nicht. Ich bin allein und finde es, ehrlich gesagt, furchtbar, an so einem Abend einsam vor dem Baum zu sitzen. Dich schickt mir der Himmel – oder der Weihnachtsmann." Bei diesen Worten grinste er, doch ich konnte an seinem Unterton hören, dass er es ernst meinte. Und ich verstand ihn nur zu gut. Er sah mir direkt in die Augen, als er darauf wartete, wie ich auf dieses Angebot reagierte. Und schon wieder verfing ich mich in diesen warmen braunen Augen, die mir durch und durch gingen. Wenn ich nicht aufpasste, würde ich gar nicht mehr gehen wollen, das wurde mir in diesem Augenblick klar. Also senkte ich schnell den Blick, nickte aber. Es klingt wahrscheinlich dämlich, aber als er mir aus der Jacke half, dachte ich, dass der verstauchte Knöchel vielleicht das Beste war, was mir an diesem Tag passierte... Robert und ich verbrachten einen sehr harmonischen und schönen Heiligabend miteinander. Wir unterhielten uns so gut, dass jeder Gedanke an ein Taxi bald vergessen war. Im Wohnzimmer brannte Holz im Kamin, eine kleine, rotgold geschmückte Tanne stand vor einem Bücherregal, das geradezu riesig im Vergleich zum Rest des Raumes wirkte. Ich fühlte mich fast wie zu Hause und mit jeder Sekunde wurde Robert anziehender für mich. Er war intelligent, warmherzig und lustig und ich hätte stundenlang einfach nur dasitzen und ihm zuhören können. Doch auch körperlich wurde er für mich mit jedem Hinsehen attraktiver. Jedes Mal, wenn unsere Blicke sich trafen, steigerte sich mein Verlangen, ihn zu berühren. Ich hätte es vorher nie für möglich gehalten, aber an diesem Abend lernte ich, was mit dem Begriff "Liebe auf den ersten Blick" gemeint war. Schon seine Gegenwart sorgte für das Kribbeln in meinem Körper, das ich lange nicht mehr gespürt hatte. Ich wurde immer unruhiger. Meine Brustwarzen hatten sich aufgerichtet und ich konnte die Hitze in meinem Schoß fühlen. Ich sehnte mich danach, in den Arm genommen zu werden, von ihm geküsst zu werden. Ich konnte meine Unruhe nur schwer überspielen. Robert versorgte meinen Fuß immer wieder mit Eisbeuteln. Mein Knöchel war angeschwollen und Robert hatte darauf bestanden, dass ich das Bein hochlegte und kühlte. Als er sich erneut zu mir auf das Sofa setzte, legte er seine Hand auf meinen Knöchel und strich sanft darüber. Sofort stellten sich alle Härchen an meinem Körper auf. Er ließ mich nicht aus den Augen, als er begann, mein Bein hinaufzustreicheln. Es ging so viel Zärtlichkeit von ihm aus – das hatte es bei Thomas nie gegeben. Und doch spürte ich, dass er angespannt war, da er nicht wusste, wie ich auf seine Berührungen reagieren würde. Liebevoll und aufmunternd lächelte ich ihn an. Ich wollte nicht, dass er aufhörte. Robert schien zu verstehen, denn sein Streicheln erreichte meinen Oberschenkel. In meinem Bauch tanzten 1.000 Schmetterlinge. Ich legte meine Hand auf seine und wir sahen uns in die Augen. Plötzlich stand er auf und kniete sich vor mich. Es war eine rein praktische Entscheidung, da mein Bein zwischen uns störte, doch trotzdem war es auch eine wunderschöne Geste. Langsam näherte er sich meinem Gesicht. Und dann küssten wir uns. Erst berührten sich unsere Lippen nur zaghaft, aber schon bald wurde sein Kuss tiefer und leidenschaftlicher. Ich öffnete mich seiner Zunge, die nun mit meiner spielte. Schon dieser Kuss katapultierte mein Verlangen in ungeahnte Höhen und auch bei Robert spürte ich das Begehren nach mir. Ich fragte mich, wie lange es wohl her war, seit er eine Frau das letzte Mal berührt hatte, und dieser Gedanke machte mich zusätzlich an. Ich schlang meine Arme um seinen Körper und zog ihn zu mir auf das Sofa. Kräftige Muskeln drückten sich an mich und ich spürte die Erektion in seiner Jeans ganz deutlich. Seine Hände schlossen sich um meine Brüste und kneteten sie liebevoll, während seine Daumen ganz gezielt immer wieder über die Brustwarzen glitten. Es zuckte durch meinen ganzen Körper und meine Venus wurde noch feuchter. "Du bist wunderschön, Helena", flüsterte Robert zwischen zwei Küssen. Das hatte mir noch nie ein Mann gesagt. Es ermutigte mich, nun auch ihn fordernder zu berühren und zu streicheln. Ich spürte, dass er sich bereits jetzt zusammenreißen musste, um nicht aufzustöhnen. Er schien ebenso ausgehungert zu sein wie ich. Trotzdem war er so zärtlich, wie ich es noch nie erlebt hatte, als er mich Stück für Stück von meiner Kleidung befreite. Ich selbst stellte mich etwas unbeholfener an, doch endlich hatte auch er sich ausgezogen und ich konnte seine nackte Haut liebkosen. Roberts Körper war unglaublich warm, seine Brusthaare kitzelten an meinem Busen. Wir konnten kaum genug davon bekommen, uns zu streicheln und zu küssen. Seine harte Männlichkeit rieb sich an meinem Venushügel und ich schob meine Hand zwischen uns, um sie mit den Fingern zu umschließen. Sachte massierte ich seinen zwar nicht sonderlich langen, dafür aber angenehm dicken Penis. Es war phantastisch, ihn so zu reizen und dabei die Spannung in seinem Körper zu spüren. Immer wieder begann er, seine Hüften zu bewegen, um so meiner Hand entgegenzukommen, doch dann rief er sich wieder zur Ordnung und hielt inne. Gleichzeitig revanchierte er sich, indem er mit seinen Fingern meine Schamlippen streichelte, sie dann vorsichtig teilte und zärtlich über meine schon vor Erwartung aufgerichtete Klitoris strich. Wie ein Stromschlag durchfuhr mich die erste Berührung meiner empfindsamen Perle und ich zuckte zusammen. Robert zog mich noch fester an sich, hörte jedoch nicht auf, mich zu stimulieren. Die sanft kreisenden Bewegungen zeigten mir, dass er genau wusste, was er da tat. Noch nie hatte ein Mann mich so gekonnt und geduldig verwöhnt. Vor lauter Lust vergaß ich die Bewegungen meiner eigenen Hand und gab mich nur noch seinen geschickten Fingern hin. Ich vergrub meinen Kopf an seinem Hals, während ich immer lauter stöhnte und mein ganzer Unterleib in Flammen stand. Robert spürte, dass ich mich ganz auf die Gefühle in meinem Innersten konzentrierte und begann, mir leise Kosewörter ins Ohr zu flüstern. Er nannte mich 'Liebes' und 'meine Schöne' und beschrieb mir, was es in ihm auslöste, meine Erregung, meine Nässe unter seinen Fingern zu fühlen. Seine Stimme war dunkel und rau und ich glaubte ihm jedes Wort. In rasendem Tempo katapultierte er mich meinem Gipfel entgegen und irgendwann klammerte ich mich an ihm fest und stöhnte seinen Namen laut heraus, während vor meinen Augen die Kerzen des Weihnachtsbaumes verschwammen und ich durch das Universum flog. Erst hinterher spürte ich, dass ich meine Fingernägel in seinen Rücken gebohrt hatte; Robert verzog nicht einmal das Gesicht deshalb. Er hielt mich fest, bis ich mich wieder beruhigt hatte, und als ich das Gesicht zu ihm hob, küsste er mich voller Leidenschaft und Hingabe. Während er danach seine Zunge spielerisch über meine sensible Haut an Hals und Schulter gleiten ließ, schlossen sich meine Hände um seinen Po und zogen seinen Unterleib fest an meinen. Ich hatte noch nicht genug von ihm, meine von Lust überflutete Muschel verlangte nach ihm. Ich wollte seinen herrlichen Phallus in mir spüren, mit ihm verschmelzen, von ihm genommen werden. Robert keuchte auf, als ich seinen Penis zwischen meine Beine drückte und mich unter ihn schob: "Helena, was machst du... ich kann mich gleich nicht mehr beherrschen." Das war genau das, was ich hören wollte. Ich wollte, dass er mich genauso begehrte wie ich ihn. Er sollte sich nicht beherrschen. "Nimm mich", sagte ich leise, während ich ihm tief in die dunklen Augen sah. Und dann war der Damm gebrochen, Roberts Beherrschung wich einer Flut von Leidenschaft. Er griff kurz zwischen uns, um seine Eichel an meinem Eingang zu positionieren, und drang dann aufreizend langsam und doch energisch in mich ein. Ich hatte das Gefühl, dass er mich ganz in Besitz nahm. Mit geschlossenen Augen genoss er jeden Zentimeter, den er weiter in mein nasses Paradies vordrang und er stöhnte: "Du bist so eng... Gott, ist das geil!" Ich fühlte mich herrlich ausgefüllt, als er endlich ganz in mir war, und spannte meine Beckenmuskeln an, um ihn so fest wie möglich zu umschließen. Dies entlockte Robert ein weiteres tiefes Aufstöhnen. Sofort begann er, sich ein Stückchen zurückzuziehen, nur um dann scheinbar noch tiefer in mir zu versinken. Er presste sich so fest an mich, dass meine Klit von seiner Peniswurzel gereizt wurde und ich schon nach wenigen Stößen glaubte, kurz vor dem nächsten Orgasmus zu stehen. So schnell ging es dann allerdings doch nicht. Roberts Bewegungen waren gleichmäßig und fließend, er schaffte es, sich eine ganze Weile zu beherrschen, obwohl offensichtlich war, dass das Gefühl meiner nassen Muschel ihn fast um den Verstand brachte. Die ganze Zeit schwankte ich auf diesem schmalen Grat zwischen totaler Lust und dem quälenden Gefühl, dass noch eine Winzigkeit fehlte bis zum erlösenden Höhepunkt. Auch ich hatte die Augen geschlossen und gab mich nur unserer Erregung hin. Und endlich, als ich ihn schon anflehte, mich kommen zu lassen, steigerte er sein Tempo um die Stöße, die mir zum ersehnten Orgasmus fehlten. Das zweite Mal schien ich zu fliegen und Feuer raste durch meine Adern, konzentrierte sich in meiner Mitte, bevor es langsam verlosch und ich wieder zu Atem kam. In der Sekunde, als mein Höhepunkt abebbte, war es dann bei Robert so weit. Sein Penis schien noch einmal härter zu werden und er stieß hart, fast ruckartig in mich hinein. "Helena, Liebes, oh jaaaaa...!" Mit abwesendem Blick hielt er plötzlich inne, während ich fühlen konnte, wie er sich in mir ergoss. Mich durchlief ein warmes Kribbeln, während ich einen gigantischen Orgasmus auf seinem Gesicht ablesen konnte. Als ich sicher war, dass es vorbei war, zog ich Robert zu mir herunter und nahm ihn fest in den Arm. So wohl hatte ich mich seit Jahren nicht gefühlt: zufrieden, geliebt und geborgen. Mein Knöchel, den ich während unseres Liebesspiels überhaupt nicht mehr wahrgenommen hatte, pochte zwar wieder, aber das war mir egal. Ich würde die schönsten Weihnachtstage meines Lebens haben und ich war sicher, dass es nicht das letzte Weihnachten war, das Robert und ich zusammen verbrachten. Es stimmte schon: Weihnachten war das Fest der Liebe.
# Frohes neues Jahr!
## Kristel Kane
"Ich wünschte, du würdest nicht gehen!" Flehendlich blickte Carmen ihren Mann an und streichelte seine Wange. Sanft nahm er ihre Hand und haucht ihr verliebt einen Kuss auf. "Ich habe auch keine Lust zu gehen. Aber wir können uns das nicht leisten, gerade an einem Tag wie heute!", versuchte er sich zu rechtfertigen. Carmen wusste, dass er recht hatte. Als Frischverheiratete waren sie auf den Sonderschichtenbonus angewiesen. Dennoch schien es nicht fair, dass sie derartige Opfer bringen mussten. Sie waren jung, verliebt und frisch verheiratet. Da kochte das Blut und sie wollten ihr Leben in vollen Zügen genießen. Beide hatten sich so auf die Party heute Nacht gefreut. Schließlich war Silvester, ihr erster Jahreswechsel als Ehepaar. Kurzfristig hatte Jochen das Angebot, Silvester zu arbeiten, angenommen. Die Nachtschicht würde ihnen einen willkommenen finanziellen Zuschuss einbringen. Als Pannenhelfer würde er dafür sorgen, dass niemand in der Silvesternacht auf der Strecke blieb. Insgeheim machte sich Carmen große Sorgen, dass ihm etwas passieren würde, schließlich schneite es heute sehr stark. Wenn man ihn nicht unbedingt brauchte, sollte man den Wagen bei dem Eis und Schnee heute stehen lassen. Es war langsam an der Zeit, dass sich Jochen auf den Weg macht. Schließlich musste er zunächst zum Betriebshof fahren und sein Einsatzfahrzeug in Empfang nehmen. Carmen geleitete ihn zur Tür und küsste ihn auf den Mund. Kaum berührten sich ihre Lippen, bereute er bereits seine Entscheidung, heute zu arbeiten. Die zarten Lippen seiner Frau entfachten Jochens Lust. Aus einem unschuldigen Kuss wurde ein Orkan. Leidenschaftlich nahm er Carmen in die Arme und hielt sie so fest, als wollte er seine junge Braut nie wieder loslassen. Die Zungen vereinigten sich zu einem wilden Tanz. Die Hände streichelten über ihre Körper. Carmen war wie elektrisiert. Zeit und Raum verschwammen. Einzig real war die Lust, die sie auf ihren Partner verspürte. Hastig öffneten ihre Finger die Knöpfe seines Hemdes, um über seine nackte Brust zu fahren. Die starken Muskeln unter ihren Händen zu fühlen, machte sie willenlos. Leidenschaftlich waren Jochens heiße Küsse, die ihren Hals bedeckten. Seine Hände hatten den Saum des Kleides nach oben gerafft und begannen, ihr das Höschen abzustreifen. Carmen half ihm, so gut es ging. Wie ein Püppchen hob er seine zierliche Frau hoch und drückte sie mit verhaltener Kraft gegen die Wand. Lüstern lächelte sie ihn an und öffnete den Bund seiner Hose. Hart und bereit lag der Penis in ihrer Hand. Ohne Schwierigkeiten ließ er sich in ihre liebesbereite Vagina einführen. Beide stöhnten wollüstig auf. Die Position war nicht einfach, doch beide hatten sie so häufig praktiziert, dass sie eine gewisse Routine besaßen. Carmens Rücken presste sich hart gegen die Wand, während das Becken ausreichend Bewegungsfreiheit bekam, Jochens heftige Stöße mitzumachen. Die Liebe der beiden war jung und die Lust aufeinander grenzenlos. Deshalb bemerkten sie in ihrem ekstatischen Treiben nicht, dass die Haustür immer noch sperrangelweit offen stand. Es interessierte sie nicht, dass sie das gesamte Mietshaus an ihrem Liebesspiel teilhaben ließen. Laut waren die Lustschreie, die ihren Orgasmus begleiteten, durch den Hausflur zu hören. Widerwillig lösten sich ihre Lippen voneinander. Die Vernunft und das Pflichtbewusstsein hatten gesiegt. Jochen hatte eine Verantwortung übernommen, die er nicht ohne Weiteres wieder abtreten konnte. Ein letzter sachter Kuss und der junge Mann ging. Traurig schloss Carmen die Tür hinter ihm. Sie konnte nur darauf hoffen, dass er bei dem Schnee und Eis vorsichtig war. Jochen hatte darauf bestanden, dass sie sich einen schönen Abend machte. Sie sollte zur Party gehen und sich amüsieren. Doch es gelang ihr nicht recht, in Stimmung zu kommen. Immer wieder dachte sie an Jochen, der in so einer Nacht an irgendwelchen Autos herumschrauben musste. Jochen konnte es wirklich nicht glauben, zu wie vielen Einsätzen er schon herausgerufen worden war. Es schien, als machten sich die Leute nur in alten klapprigen Wagen auf den Weg zu ihren Partys. Die meisten dieser Autos hatten nur Probleme damit, bei diesen Minustemperaturen anzuspringen. Kurz nach 23 Uhr erreichte ihn ein weiterer Notruf. Diesmal musste er auf die Autobahn. Ein Camper hatte es noch bis zum Rastplatz geschafft und war dann liegen geblieben. Da die Stelle sehr entlegen war, musste Jochen weit fahren, um das Fahrzeug zu erreichen. Kurz vor Mitternacht erreichte er sein Ziel. Der Camper war nicht zu verfehlen, da er das einzige Gefährt auf dem großen verschneiten Parkplatz war. Routinemäßig schaltete Jochen seine orange Warnbeleuchtung ein. Normalerweise stiegen die Leute aus ihren Fahrzeugen, wenn sie diese erkannten. Hier geschah nichts. Jochen hielt an, gab der Zentrale seine Position durch und stieg aus. Immer noch kein Lebenszeichen von dem Fahrer des Fahrzeuges. Die Fahrerkabine war leer. Jochen beschlich ein merkwürdiges Gefühl. Unsicher ging er zu Seitentür, klopfte energisch an und gab sich zu erkennen. Sein Herzschlag beschleunigte sich, als er bemerkte, wie jemand von innen die Tür öffnete. Nervös trat er einen Schritt zurück, immer mit dem unheilvollen Gefühl, dass dies kein normaler Einsatz war. Der eisige Wind riss ihm die Atemwolke von den Lippen. Nicht nur die Kälte ließ ihn frösteln. Ein rascher Blick auf die Uhr verriet ihm, dass der Jahreswechsel kurz bevorstand. In diesem Augenblick wurde ihm bewusst, dass er besser auf das Geld verzichtet hätte und stattdessen bei seiner jungen Frau geblieben wäre. Dieser Notruf konnte nicht echt sein. Die Tür wurde aufgestoßen und schwang weit nach außen. Jochens Atem stockte. Im diffusem Licht der Innenbeleuchtung stand eine junge Frau im hauchdünnen bodenlangen Negligé. Darunter war sie splitternackt. Die harten Nippel krönten die festen runden Brüste, die schmale Taille betonte das runde Becken und das dunkle Dreieck verhieß Fleischesfreuden. Über Jochens Gesicht lief ein Strahlen. "Prost Neujahr!", rief die junge Frau. "Ich dachte, wir sind es uns einfach schuldig, auf das neue Jahr anzustoßen!" Symbolisch streckte sie ihm die Hand entgegen. In diesem Moment war Jochen der glücklichste Mann der Welt, der ungeheuere Einfallsreichtum seiner Frau hatte es doch möglich gemacht, dass sie den Jahreswechsel miteinander verbringen durften. So viel Initiative verlangte nach einer Belohnung und die wollte er ihr zukommen lassen. Rasch stieg er in den wohltemperierten Camper und ließ die kalte Winterwelt draußen. Es wunderte ihn nicht, dass das Liebesnest bereits vorbereitet war. Die Spielwiese war genauso bereit wie er. Carmen hatte sich schon aufs Bett gelegt und lockte mit ihren Reizen. Sie besaß einen atemberaubenden Körper. Hastig entledigte sich Jochen seiner Arbeitskleidung. Nackt legte er sich zu seiner Frau und spürte ihre warme weiche Haut gegen die seine reiben. Der betörende Duft ihres Parfüms erhöhte sein Verlangen nach ihr. Jochen war sich bewusst, dass von ihm eine Kombination von Schweiß und Schmierstoff ausgehen musste. Er wusste aber auch, dass dieser besondere Mix auf Carmen unheimlich anziehend und männlich wirkte. Angetörnt von der verführerischen Anmut seiner Frau, bedeckte er ihren gesamten Körper mit heißen Küssen. Nichts ließ er aus. Er küsste ihre Fersen, die Knie und wanderte die Innenseiten ihrer Schenkel hinauf. Carmen bebte vor Verlangen. Das Becken bewegte sich verheißungsvoll hin und her. In sehnsüchtiger Erwartung der intimen Küsse schwollen die Schamlippen an. Sanft küsste Jochen den Venusmund seiner Frau. Die junge Frau stöhnte auf und rekelte sich wollüstig in den Kissen. Jochens Lippen wanderten weiter über den flachen Bauch und kehrten erneut zu ihrer intimsten Stelle zurück. Zu seiner Verwunderung forderte Carmen ihn auf, sich hinzulegen. Keinesfalls wollte sie sich diesen höchsten aller Genüsse entgehen lassen, allerdings wollte sie diesen mit ihrem Mann teilen. Carmen positionierte sich über Jochen. Zwischen den geöffneten Schenkeln fiel sein Blick auf das Schatzkästchen der Partnerin. Für einige Sekunden schlossen sich seine Augen. Jochen wollte sich ganz dem überwältigenden Gefühl hingeben, die das langsame Eintauchen seines Glieds in Carmens Mundhöhle in ihm auslöste. Ein heißes Prickeln durchströmte ihn bis in die Lenden. Wohlwollend bemerkte Carmen, dass die Größe des Penis immer mehr zunahm. Ihre faszinierende Technik machte es Jochen schwierig, der Partnerin den gewünschten Genuss zu bescheren. Wenn sie so weitermachte, dann würde er sich nicht mehr lange beherrschen können. Als hätte sie seine Gedanken erahnt, ließ sie plötzlich von ihm ab und platzierte sich auf seinem Schoß. Tief blickte sie ihm in die Augen, während sie sich sachte nach unten sacken ließ und dabei den Penis immer tiefer in sich aufnahm. Langsam bewegte sie sich auf und ab. Der alte Camper geriet dadurch in Bewegung und schaukelte auffällig hin und her. Niemand kümmerte sich darum. Die Ekstase, die die beiden durchlebten, machte alles andere um sie herum nebensächlich. Weder Jochen noch Carmen bekamen mit, dass es zu schneien anfing und niemand hörte die explodierenden Raketen. Es kümmerte sie nicht, dass das Feuerwerk das neue Jahr begrüßte. Zeitgleich mit dem Jahreswechsel erlebten sie einen gemeinsamen Orgasmus.
# Piratenbeute
## Miriam Eister
Das Fest war toll. Schon ein paar Monate im Voraus hatte ich mir die Karte für einen originellen Silvesterabend reserviert. Das Motto lautete "Mit Maske ins neue Jahr". Es hieß also sich gut zu verkleiden und mit allerlei Narren, Pfarrern, Clowns und den vielen anderen Leuten kräftig ins neue Jahr zu rutschen. Aber nicht nur das ungewöhnliche Motto hatte mich fasziniert, sondern auch die Location. Gefeiert werden sollte in einer alten Villa, die ein reicher alter Herr mit viel Zeit und Liebe hatte restaurieren lassen. Die Zeitungen waren nach der Fertigstellung voll des Lobes gewesen über die vielen Zimmer, den Dachboden mit Fachwerkcharme und natürlich über den großen Saal. Und in diesem sollte es am heutigen Silvesterabend so richtig rundgehen. Ich freute mich schon sehr auf diese Feier. Im Kostümverleih hatte ich mich einige Tage zuvor für ein barockes Kleid entschieden. Ich wollte immer schon einmal wissen, wie sich so etwas anfühlte. Es passte wie angegossen. Der dunkelgrüne Stoff war angenehm auf der Haut, der Ausschnitt war durch eine Verschnürung sehr schmuckvoll und der weite Rock fiel locker bis auf die Knöchel. Frieren würde ich mit Sicherheit nicht. Auf das Korsett konnte ich verzichten, denn bewegen wollte ich mich ja auch noch. Eine Maske war dank der großen Auswahl beim Verleiher schnell gefunden. Dezent verziert mit einigen Federn und Perlen. Die musste ich mir zwar kaufen, aber sie würde eine Erinnerung an diesen Abend sein. Und keine Schminke würde mein Gesicht verkleben. Zufrieden ging ich heim. Die Feier war in vollem Gange und ich konnte nur immer wieder über die Kostüme der Leute und ihre Kreativität staunen. Die beiden spärlich bekleideten Meernixen schienen dank des Alkohols überhaupt keine Kälte zu spüren. Ich fühlte einen Blick auf mir und drehte mich um. Ein paar Schritte hinter mir stand ein Pirat. Er war mit einer ledernen Hose bekleidet, die Weste saß locker über seinem weiten Hemd, welches leicht offenstand. Es gab den Blick auf einen muskulösen Oberkörper mit ein paar dunklen Brusthaaren frei. Die Augen unter der Maske waren von einem strahlenden Blau, dass mich sofort faszinierte. Verwirrt wandte ich mich wieder von ihm ab. Bestimmt hatte er es auf eine der Meernixen abgesehen, denn mein Kostüm stellte nicht annähernd so viel Haut zur Schau wie das ihre. Als ich noch einmal zurückblickte, war er verschwunden. Die Feier war wirklich toll! Ich hatte mich mit mehreren Leuten angeregt unterhalten, hatte getanzt und ein paar Gläser Sekt zirkulierten in meinem Blut. Immer wieder hatte ich beim Tanzen neugierige Blicke auf mir gespürt. Nur deren Herkunft hatte ich nicht ausmachen können. Insgeheim hoffte ich, dass es der Pirat von vorhin war. Leider gab es die ersten Volltrunkenen schon kurz nach 22 Uhr. Aber für die hatte der Gastgeber einen separaten Ruheraum vorbereiten lassen. Durch Zufall war ich vorhin an diesem Zimmer schon vorbeigekommen. Doch ich wollte noch mehr von dem Haus sehen und begab mich auf Erkundungstour. Viele Zimmer waren verschlossen und ich stieg die Treppe weiter hinauf. In der Hoffnung, dass zumindest der Dachboden offen sein würde, stieß ich vorsichtig gegen die Tür. Sie ging auf. Die kleine Glühbirne brachte etwas Licht und ich war sprachlos. Dieser Raum war riesig! In den Ecken wurden scheinbar einige der alten Möbel zwischengelagert. Der Ausblick aus dem kleinen Fenster war phantastisch und die Holzbalken des Dachbodens rochen angenehm. Vor mir stand ein altes Sofa, abgedeckt mir einem Laken, dahinter ein riesiger Holzschrank. Ich betrachtete den Schrank näher. Was für eine Arbeit, so viele Schnitzereien, so viele Einzelheiten. – Ein Geräusch von der Treppe schreckte mich auf. Was, wenn ich gar nicht hier oben sein dürfte? Schnell löschte ich das Licht und versteckte mich im Schrank. Die Tür zog ich im gleichen Moment leise bis auf einen kleinen Spalt zu, als die Eingangstür aufging. "Komm rein, hier ist keiner." Ich hörte eine Frau kichern. Beide Gestalten blieben vor dem Sofa stehen und mir dröhnte das Blut in den Ohren. Hauptsache, niemand wollte einen Blick in den Schrank werfen! Nein, das wollten sie nicht, denn einige Sekunden später hörte ich tiefe Seufzer und das Rascheln von Kleidung. Neugierig neigte ich meinen Kopf so, dass ich durch den kleinen Spalt schauen konnte. Vor dem Sofa stand eine der Meernixen und zog sich langsam und lustvoll aus. Sie fuhr sich mit den Händen über ihren Körper, berührte ihre Brüste und stellte provozierend ein Bein auf das Sofa. Da saß noch jemand mit dem Rücken zu mir. Ich kam mir seltsam schuldig vor, denn für meine Augen war dieser Strip eigentlich ja nicht gedacht. Aber ich konnte meinen Blick nicht abwenden. Die Nixe rollte ihren Tanga über den Po die langen Beine hinab. Ihre Wangen waren erhitzt und strahlten. Sie wiegte sich zu dem Takt der Musik, die durch das Treppenhaus nach oben drang, und rieb sich mit der Hand zwischen den Beinen. Aber wer war die andere Person? Wer konnte dort so unbeteiligt sitzen und nicht bei der Erlösung dieser Nixe behilflich sein? Einige Sekunden später erhielt ich meine Antwort. Lange schmale Hände fuhren von unten nach oben über den sinnlich dargebotenen Körper. Zarte Lippen nahmen die Brustwarzen in den Mund und knabberten an ihnen. Ich schlug mir die Hand vor den Mund. Das war die andere Meernixe! Die beiden Frauen schienen gut aufeinander eingespielt zu sein und hatten keine Ahnung, dass sie heimlich beobachtet wurden. Unbeabsichtigt zwar, aber dennoch beobachtet. Und was ich sah, gefiel mir. Unterm Rock rieben sich meine Beine unbewusst aneinander. Plötzlich spürte ich die Anwesenheit von einer weiteren Person ganz in meiner Nähe und eine starke Hand schob sich über meinen Mund. Ich hörte ein leises "Pst!", roch einen herben männlichen Duft und erstarrte. Ich war in diesem Schrank nicht allein! Vor lauter Hektik hatte ich das vorhin nicht bemerkt. Ich drehte vorsichtig den Kopf zur Seite und der schwache Lichtschein erhellte diese blauen fesselnden Augen. Der Pirat! Er ließ meinen Mund los, legt mir einen Finger auf die Lippen und schaute an mir vorbei durch den Spalt in der Tür. Er konnte in dieser Situation genauso wenig tun wie ich, aber bis jetzt hatte ich ihm die Sicht auf die beiden Frauen versperrt. Was er durch einen vorsichtigen Schritt hinter mir abänderte. So hatte er einen guten Blick durch den Spalt und mich vor sich fest im Griff. Ich spürte seinen warmen Atem in meinem Nacken. Zwischenzeitlich hatten es sich die beiden Frauen auf dem Sofa bequem gemacht und streichelten sich. Ihre Seufzer zeugten von ihrer Lust und der Anblick war sehr erotisch. Das war zweifellos auch die Meinung des Piraten hinter mir. Seine Hände tasteten die Form meiner Hüften unter dem Kleid ab und legten sich über den Stoff auf meine Scham. Ich legte meine Hände auf seine, um ihn an weiteren Erkundungen zu hindern. Unsere Blicke wurden immer noch von den beiden Meernixen festgehalten. Eine der beiden konnte sich nicht mehr zurückhalten und kam lautstark zu einem Orgasmus. Ihre Finger krallten sich in das Laken, ihre Zunge leckte gierig über die Lippen. Zitternd hob sich ihr Unterkörper der Hand ihrer Gespielin entgegen. Der Pirat seufzte und beide Frauen horchten auf. "Gibt es hier oben Mäuse?" Mit einem Lächeln antwortete die gerade Befriedigte: "Maus hin oder her. Lass uns gehen. Ich möchte mit dir ganz allein und ungestört ins neue Jahr feiern. Komm...!" Schnell ordneten sie ihre wenigen Kleidungsstücke und verließen kichernd den Dachboden. Das Licht ging aus. Ich merkte, wie ich aus dem Schrank geschoben wurde. Warme Hände hielten meine fest. Das Mondlicht reichte aus, um seinen markanten Körper zu erkennen. Der gleiche verführerische Anblick wie vorhin. Sein Blick fesselte mich. "Wir sind allein und ich habe dich schon den ganzen Abend beobachtet. Das Schwingen deiner Hüften, das Scherzen und Lachen. Ich hatte gehofft, dass du den Weg hier hoch finden würdest." Sein Stimme war dunkel und weich. "Scheint so, als hätte ich meine Beute gefunden." Ich wollte mich herumdrehen und gehen, wurde aber daran gehindert. Seine Lippen waren nur einige Zentimeter von meinen entfernt. "Nein, bleib! Ich nehme nur, was mir freiwillig gegeben wird." Ja, ich hatte Lust bekommen. Er näherte sich und ich war diejenige, die handelte. Ich küsste ihn. Meine Füße verloren den Boden und wir bewegten uns in Richtung Sofa. Ich war ziemlich gierig. Ich wollte ihn, und zwar jetzt! Das Spiel der beiden Frauen hatte mich erregt und der Sekt tat sicherlich sein Übriges. Schnell streifte ich ihm die Weste über seine Schultern, fuhr über seine Brust und stieß ihn nach hinten, so dass er sich setzen musste. Seine Finger zogen an der Verschnürung meines Kleides und fanden meine Brüste. Er war genauso gierig wie ich. Seine Hände schoben mein Kleid nach oben und stießen auf meine unbedeckte Scham. Verblüfft schaute er mich an. "Ich mag es, etwas verrucht zu sein. Und schließlich ist dieses Kostüm warm genug." Meine Erklärung ließ ihn auflachen und plötzlich befanden sich meine Pobacken in seinen warmen Händen. Auch ich konnte nicht sehen, was ich tat, aber meine Finger ertasteten seine Männlichkeit. Ich befreite sie aus der engen Hose und liebkoste sie sanft mit den Fingern. "Genug, meine schöne Unbekannte! Ich möchte dich. Spann mich nicht weiter auf die Folter. Lass uns diesen Augenblick genießen. Komm zu mir. Bitte!" Meine Rock bauschte sich über unseren Schößen. Darunter traf Haut auf Haut. Ich schaute ihm tief in die Augen und setzte mich auf ihn. Diese heiße und intime Berührung ließ ihn stöhnen. Ich wagte mich tiefer herab. Seine harte Spitze glitt in mich. Jeden Zentimeter des Eindringens kostete ich aus. Wir hatten uns mit dem Vorspiel kaum Zeit gelassen, waren dafür zu heiß aufeinander. Aber ich war bereit für ihn. Als ich fest auf ihm saß, holte ich mir einen Kuss und fing an, mich zu bewegen. Wer war hier wessen Beute? Seine Hände auf meinen Pobacken drängten mich zu einem kräftigeren Auf und Ab. Ein tiefes Knurren kam aus seiner Brust. Er biss leicht in meine Brustwarzen, ich dafür in sein Ohrläppchen. Er füllte mich völlig aus. Jeden Stoß fühlte ich tief in mir. Seine Arme hielten mich sicher und fest und unsere Blicke trafen sich. Wir waren immer noch durch unsere Masken verhüllt und konnten auch jeden Moment von anderen neugierigen Personen entdeckt werden. Aber ich wollte ihn nicht loslassen. Also spannte ich meine Beckenbodenmuskulatur an und massierte ihn tief in mir. Ich spürte, wie er kurz vor dem Ziel stand. Ich spannte noch ein paarmal fest an, hob meine Hüften leicht, ließ mich nieder und er kam. Ich wurde von seiner Feuchtigkeit überflutet und sein heißer Atem strich mir um den nackten Busen. Ich strich ihm über den Kopf, bis er sich wieder beruhigt hatte. Draußen ertönte Lärm und Jubel. Es war Mitternacht und das Feuerwerk hatte begonnen. Ich löste mich von ihm, ging zum Fenster und beobachtete das Schauspiel. Zwar war ich immer noch erregt, aber es hätte noch eine Weile bis zu meinem Orgasmus gebraucht. "Halt still!" Seine Stimme war nur ein Flüstern. Er stand hinter mir. Ein Arm hielt meine Hüfte fest umschlungen und die andere Hand schob sich zwischen meine Beine. Seine Finger fanden meine Klit. Er strich vorsichtig darüber, umkreiste sie, rieb schneller, fester. Die Augen geschlossen, konnte ich nur noch genießen. Während meines Höhepunkts wurde ich von ihm genauso festgehalten wie er bei seinem von mir. Ich konnte mich ganz meinem inneren Feuerwerk hingeben... Mein Rock fiel mir wieder locker bis auf die Knöchel und die Schleife an meinem Ausschnitt saß perfekt. Im Trubel des Abends hatten mich die anderen nicht sonderlich vermisst. Ich blieb nicht mehr lange auf dem Fest, denn ich wollte dieses schöne Gefühl noch ganz für mich allein auskosten. Weder hatten wir uns ohne Maske gesehen noch hatten wir uns verabschiedet. Als ich eine Woche später das Kleid wieder in den Laden zurückbrachte, hing ein großer Artikel aus der Zeitung im Schaufenster. Das Bild zeigte den Reporter neben einem zufrieden wirkenden Piraten. Meinem Piraten! Ohne Maske. Ich las: "Das war ein rauschendes Fest mit vielen Überraschungen. Ein phantastisches Feuerwerk läutete das neue Jahr ein. Der Hausherr selbst hatte sich als Pirat verkleidet unter die feiernde Menge gemischt. Auch zukünftig wird er den Saal Feierlichkeiten zur Verfügung stellen." Ich war baff! Der Rest des Artikels lobte das Buffet und die Organisation des Ganzen. Zum Schluss wurde nach dem Wunsch des Hausherren für das neue Jahr gefragt. "Ich habe ein Geschenk für eine maskierte Dame im grünen Kleid. Ich wünsche mir, dass mein wunderschöner alter Schrank bei ihr einen neuen Platz findet und hoffe, dass sie den Mut hat, ihn sich abzuholen...!"
# Wildes Silvester
## Priska Apple v
Als Ralf sich von Maria trennte, brach für sie eine Welt zusammen. Sie brauchte Monate, um überhaupt wieder ans Weggehen zu denken. Der erste Versuch, herauszufinden, ob sie einen Abend mit Freunden wieder genießen konnte, sollte der Silvesterabend sein. Ihre beste Freundin Monika hatte Maria zu einer kleinen Neujahrsparty eingeladen. Monika versprach, dass nur sehr wenige ihrer Freunde kommen würden und Maria sagte nach langem Zögern zu. Als der Silvestertag gekommen war, rief Maria Monika an und sagte wieder ab. Monika versuchte alles, sie umzustimmen, doch vergeblich. Maria konnte sich nicht vorstellen, das erste Silvesterfest ohne ihren Freund zu feiern und fröhlich zu sein. Einsam verkroch sie sich in ihrer Wohnung, legte sich mit einer Decke und einem Buch auf die Couch und öffnete eine Flasche Sekt. Als anderthalb Stunden später die zweite Flasche geleert war, kam Maria das Leben nicht mehr ganz so trostlos vor. Plötzlich unternehmungslustig geworden, schwankte sie in ihr Schlafzimmer und zog ein Kleid nach dem anderen aus dem Schrank. Schließlich entschied sie sich für ein besonders kurzes schwarzes Kleid, das sie schon drei Jahre nicht mehr angezogen hatte, zwängte sich in hochhackige Pumps und lief ins Bad. Dort schminkte sie sich, viel zu stark zwar, aber das fiel ihr gar nicht auf. Dann nahm sie ihre Tasche, die Schlüssel und noch eine Flasche Sekt und verließ die Wohnung. Sehr beschwingt stieg sie 15 Minuten später aus dem Taxi und lief die wenigen Stufen zum Haus von Monika hoch. Auf ihr Klingeln wurde die Tür aufgerissen und ihre Freunde begrüßten sie lautstark und begeistert. Maria setzte sich zu den anderen auf den Teppich und bemerkte plötzlich, dass einige ihrer Freunde kaum noch was anhatten. "Was ist denn hier los?", fragte sie. "Wir spielen Strip-Poker, machst du mit?", antwortete Monika. Maria nahm das Glas mit Sekt in Empfang, das jemand ihr reichte, leerte es auf einen Zug und nickte dann zustimmend. Schon wurden die Karten wieder ausgeteilt und das Spiel begann. Die erste Runde verlor Maria haushoch, doch sie empfand das nicht als schlimm, denn sie nahm einfach einen Ohrring heraus und legte ihn zu den anderen Sachen. Fünf Runden später hatte Maria weder Ohrringe noch Schuhe an ihrem Körper und überlegte krampfhaft, was sie nun, da sie erneut verloren hatte, ausziehen könnte. Die Pokerrunde bestand aus fünf Männern und vier Frauen. Die Männer waren fast vollständig bekleidet und hatten bisher nur ihren Schlips oder eine Armbanduhr abgeben müssen. Bei den Frauen sah das schon ganz anders aus. Monika saß bereits in Unterwäsche in der Runde, die beiden anderen hatten keine Oberteile mehr an. Maria hatte keine große Lust, in Unterwäsche dazusitzen, also stand sie auf und wollte ihren Slip ausziehen. Das, so dachte sie, würde ja keiner sehen können unter dem Kleid. Das Dumme war nur, dass sie vergessen hatte, überhaupt einen Slip anzuziehen. Hochrot stand sie da und überlegte verzweifelt, was sie nun ablegen sollte. Erleichtert fiel ihr ein, dass sie noch einen BH anhatte. Sie versuchte, ihn unbemerkt zu öffnen, als Markus sich hinter sie schob und ihr ins Ohr raunte: "Lass mal, ich mach das schon." Zärtlich glitten seine warmen Hände unter ihr Kleid, streichelten ihren Rücken und öffneten langsam den Büstenhalter. Ein Schauer lief über Marias Rücken. Markus setzte sich wieder und grinste sie herausfordernd an. Da löste sich Marias Starre und sie zog den BH unter ihrem Kleid hervor. Schnell drückte ihr noch jemand ein Glas Sekt in die Hand und die nächste Runde begann. Wieder verlor Maria. "Seid nicht böse, aber hier mache ich Schluss", sagte Maria und erhob sich. Empört riefen ihre Freunde durcheinander. Maria schüttelte nur den Kopf und ging in die Küche. Markus folgte ihr. "Willst du mit mir tanzen?", fragte er. Achselzuckend begleitete sie ihn ins Wohnzimmer zurück und legte beide Arme um seine Schultern. Markus presste sie fest an sich und zusammen bewegten sie sich zu den Klängen der leisen Musik. Markus streichelte langsam ihren Rücken hinunter und bemerkte erfreut, wie sich ihre Brustwarzen aufrichteten und gegen seinen Oberkörper drückten. Maria wiegte sich selbstvergessen im Takt der Musik. Markus ließ seine Hand zu ihrem Po hinabgleiten, umschloss ihn sanft und drückte dann fest zu, als er keine Gegenwehr bemerkte. Im Gegenteil, sie presste ihren Unterkörper sofort an ihn und schlang ihr rechtes Bein um seine Hüfte. Beim Tanzen rieb sie ihren Unterleib fest an seiner Hose und er konnte die Erektion nicht mehr verbergen. Maria hob ihren Kopf und ihre Lippen fanden sich zu einem leidenschaftlichen Kuss. Sie ließen sich noch nicht einmal von den Jubelschreien der anderen stören, wenn wieder jemand etwas ausziehen musste. Als die Musik verstummte, blieben Markus und Maria ineinander verschlungen stehen und bewegten sich erst wieder, als ein neues Stück erklang. Markus war so erregt, dass sie seine Beule in der Hose unmöglich ignorieren konnte. Er schob seine Hand unter ihr Kleid und wurde sofort mit ihrer Nässe konfrontiert. "Komm mit", flüsterte er in ihr Ohr und gemeinsam gingen sie ins Schlafzimmer. Dort blieb sie vor dem Bett stehen und er kam langsam auf sie zu. Er nahm ihre Hände, hob sie über ihren Kopf und zog dann das Kleid darüber. Nun stand sie nackt vor ihm und er sah sie begehrlich an. Mit einer Hand ihren Kopf umfassend, küsste er Maria lange und zärtlich. Seine Hand umschloss ihre Brust und spielte mit dem steif aufgerichteten Nippel. Leise stöhnte sie auf. Markus drückte sie hinunter aufs Bett und seine Mund suchte bereits die Brustwarze, die seiner Hand so entgegengekommen war. Er küsste und leckte sie und knabberte mit seinen Zähnen daran, bis die Knospe vollkommen aufgegangen war. Steif aufgerichtet ragten die Nippel ihm entgegen. Sein Mund wollte jedoch noch mehr erkunden. Er glitt zwischen ihre weit gespreizten Schenkel und trank von dem Tau, der Marias Höhle benetzte. Schließlich stand er auf, riss sich die Sachen vom Leib und kam zu Maria aufs Bett. Wieder küssten sie sich lange, während seine Hand zwischen ihren Schenkeln mit der Liebesperle spielte und ihre Hand sich zaghaft um seinen riesigen Schaft schloss. Dann drückte er ihre Beine mit seinen Schenkeln auseinander und drang ohne Vorwarnung in sie ein. Maria stieß einen leisen Schrei aus, als er sie so plötzlich ausfüllte, doch erstaunlich schnell fanden sie ihren Rhythmus. Sie stöhnten beide laut und ihnen war es egal, wer sie hörte. Völlig enthemmt stieß Maria Markus von sich, kniete sich vors Bett und flehte ihn an, sie von hinten zu nehmen. Markus ließ sich das nicht zweimal sagen. Sogleich war er hinter ihr, krallte seine Hand in ihre Haare und nahm sie erbarmungslos. Wilde Schreie entrangen sich Marias Kehle, aber sie nahmen nicht wahr, wie die Tür sich öffnete. Die Freunde wollten sehen, ob etwas passiert war, doch anstatt sich zurückzuziehen, blieben alle in der weit geöffneten Tür stehen und beobachteten interessiert das Geschehen. Der ein oder andere spielte der vor ihm stehenden Frau an den Nippeln oder am Po herum und die Frauen lehnten sich entspannt an sie. Nur der Fünfte ging leer aus, holte aber seinen Penis raus und befriedigte sich mit der Hand. Da die Frauen sowieso kaum noch bekleidet waren, gestalteten sich die nun folgenden Aktionen sehr leicht. Monika, die beim Anblick des sich selbst befriedigenden Freundes feucht zwischen den Beinen wurde, kniete sich schnell vor ihn, nahm seine Hand weg und begann, kräftig an dessen Männlichkeit zu saugen. Der Freund krallte seine Hände in ihre Schultern und bemerkte nicht, wie sich ein anderer Mann hinter Monika kniete und seinen kräftigen Riemen zwischen ihre Schenkel schob. Monika, die die Bewegung mitbekommen hatte, reckte dem Unsichtbaren verlangend ihr Gesäß entgegen und er suchte sich sofort lustvoll seinen Weg in die feuchte Dunkelheit. Auch die anderen hatten sich bereits zu Pärchen zusammengefunden. Einer der Männer stand an der Wand und hatte seine Sexgespielin hochgehoben. Ihre Beine waren um seine Hüften geschlungen und sie bewegten sich in rasantem Tempo. Der andere Mann hatte sich flach auf den Teppich gelegt und die letzte zur Verfügung stehende Frau ritt auf ihm. Der ganze Raum war von einem Stöhnen erfüllt, doch niemand nahm die anderen richtig war. Als Monikas Freund kam und sich langsam erholte, sah er den anderen Mann, der seine Freundin von hinten beglückte. Der Anblick erregte ihn so sehr, dass er sich dem Pärchen an der Wand näherte. Die beiden bewegten sich langsam, da er ja das ganze Gewicht der Frau trug, und so gelang es ihm, seinen mittlerweile wieder steil aufgerichteten Penis in das Hintertürchen der jungen Frau zu schieben. Sie schrie kurz auf, schien es aber zu genießen. Nun verteilte sich ihre Last auf zwei Männer und gemeinsam ließ man sich auf dem Fußboden nieder. Monikas Freund lag an dem Rücken der jungen Frau und stöhnte laut, da sein Teil durch die Enge des Anus stark gerieben wurde. Der andere Mann hatte ein Bein der Frau über seine Hüfte gelegt und fühlte sich in ihrer Vagina noch sehr wohl. Er spielte an ihrer einen Brust und Monikas Freund an der anderen. Maria und Markus waren lautstark zum Orgasmus gekommen. Als sie sich erschöpft aufrichteten, waren sie anfangs schockiert über das wilde Treiben im Zimmer. Sie sahen sich an und mussten plötzlich lachen. Maria stand auf, nahm ihr Kleid und lief ins Wohnzimmer. Auch Markus sammelte seine Sachen ein und zog sich an. Als er ins Wohnzimmer kam, hatte Maria ihre Unterwäsche im Strip-Poker-Haufen gefunden und angezogen. Ohne viele Worte verließen beide gemeinsam die Wohnung. Sie spazierten Hand in Hand zu Marias Haus, blieben immer wieder stehen und küssten sich ausgiebig. Unschlüssig blieben sie dann vor Marias Haus stehen. "Darf ich noch mit rein?", fragte Markus schließlich und Maria nickte. Als sie Kaffee kochte, hörte sie den Wasserhahn im Badezimmer rauschen. Sie ging hinüber und überraschte Markus dabei, wie er Wasser in die Wanne ließ. Er kam sofort zu ihr und küsste sie. Sie schlang ihre Arme um ihn und versank in seinem Kuss. Langsam zog er sie aus und löste sich nur widerwillig von ihren Lippen. Gemeinsam stiegen sie in die Badewanne. Lange lagen sie einfach nur da und lächelten sich an. Dabei spielte sein Zeh mit ihren Nippeln und ihr Fuß massierte seinen Penis. Dann stand sie auf, drehte sich um und kniete in der Badewanne nieder. Sofort war er hinter ihr und beide lachten, denn durch die Enge der Badewanne gelang es ihm nicht gleich, die richtige Position einzunehmen. Doch dann fand er sie und glitt in sie hinein. Nun, beim zweiten Mal, ließen sie sich mehr Zeit. Langsam und sanft bewegte er sich in ihr. Sie stöhnte laut auf und ging prustend unter, als er versuchte, an ihre Brüste zu kommen. Lachend stiegen sie aus der Wanne und rannten nackt und große Schaumspuren auf dem Teppich hinterlassend in Marias Schlafzimmer. Beide fielen aufs Bett und hielten sich erschöpft an den Händen. Schließlich begann Maria sanft Markus' Bauch zu streicheln. Mit geschlossenen Augen genoss er das wohlige Gefühl. Maria beugte sich über Markus und wanderte mit ihrem Mund unzählige Stellen an seinem Körper küssend hinab zu seinem Penis. Zärtlich küsste und leckte sie an ihm, bevor sie ihn ganz in den Mund nahm und daran knabberte. Mit ihrer ganzen Körperlänge legte sie sich auf Markus, ohne dabei seinen Penis aus ihrem Mund gleiten zu lassen. Jetzt hatte Markus ihr Geschlecht unmittelbar über seinem Gesicht. Seine Hände umfassten ihre Pobacken und spreizten dabei ihre Schamlippen. Dann versank seine Zunge zwischen ihren warmen Schenkeln. Langsam und zeitvergessen befriedigten sie sich oral. Sie kamen gemeinsam und ganz still. Maria kuschelte sich in Markus' Arme und zusammen schliefen sie ein.
# Eingeschneit!
## Marie Sonnenfeld
Seit acht Wochen bin ich erst bei der Werbeagentur Bachmann beschäftigt, aber seit dieser Zeit habe ich die wohl erotischsten Träume und die verruchtesten Gedanken meines Lebens. Denn obwohl er mein Chef ist und ich es eigentlich nicht tun sollte, kann ich nicht damit aufhören, fast ausschließlich an Magnus Bachmann zu denken und mir die heißesten Situationen mit ihm vorzustellen. Abends liege ich oft lange wach und male mir dabei in Gedanken aus, dass er mich zärtlich in seine Arme nimmt und gleich darauf leidenschaftlich mit mir schläft. Was ist das nur für eine unglaubliche Kraft, die mich dermaßen stark zu ihm hinzieht? Ich kann es mir nicht erklären, weiß aber, dass es sich ungemein gut anfühlt und ich jede Sekunde genieße, die ich in meiner Phantasie verschwitzt und voller Lust mit ihm verbringe. Bereits auf dem Weg zur Agentur, die ein wenig außerhalb lag, hatte es ziemlich heftig zu schneien begonnen. Ich maß dem aber noch nicht allzu viel Bedeutung bei, da ich zum einen wieder nur an Magnus, zum anderen aber auch an die viele Arbeit dachte, die sich auf meinem Schreibtisch türmte. Der Himmel war grau und ich hatte meine Scheibenwischer angestellt, um durch den dichten Schneefall überhaupt noch die Straße sehen zu können. An dem ausgebauten, ehemaligen Getreidespeicher angekommen, in dem sich seit rund einem Jahr die Werbeagentur befand, stieg ich hastig aus. Dann lief ich schnellen Schrittes durch den frisch gefallenen Schnee zur Eingangstür, die ich schwungvoll aufzog. Auf der Matte trat ich mir den Schnee von den Schuhen und ging die Treppe hinauf in den offenen hellen Büroraum, der nur durch alte Holzbalken in seiner feudalen Größe unterbrochen wurde. Es war warm und duftete schon angenehm nach frischem Kaffee. Ich sah ihn von hinten, als ich den Raum betrat, und sofort fiel mir wieder Magnus' knackiger Hintern auf, der in einer gut sitzenden, schwarzen Jeans steckte. Auf mein "Guten Morgen!" drehte auch er sich zu mir um und schaute mich aus seinen grünen Augen freundlich an, als er meinen Gruß erwiderte. Was für ein unglaublich charismatischer Mann! Wie jeden Tag ging ich auch heute mit klopfendem Herzen zu meinen Schreibtisch und setzte mich. Und auch heute schaute ich neben meiner Arbeit immer wieder zu Magnus herüber – einfach, weil ich nicht schaffte, es nicht zu tun. Als nach mir kein weiterer Kollege mehr im Büro erschien und ich meine Kollegin gegenüber nach dem Grund fragte, teilte sie mir mit, dass einige sich scheuten, bei diesem starken Schneefall hier heraus vor die Stadtgrenze zu fahren und deshalb Überstundenfrei oder einen Tag Urlaub genommen hatten. Dabei fügte sie hinzu, dass sie sich selbst auch bereits Sorgen wegen des Heimwegs machte, da der Schneefall scheinbar gar nicht weniger wurde. Es waren Unmengen von Aufträgen, die ich zu erledigen hatte, und eine Terminsache, die ich gemeinsam mit Magnus bearbeitete und um deren Abgabe wir kämpften, da sie schon am Montag erfolgen sollte und das Projekt sehr umfangreich war. Immer wieder setzten wir uns deshalb zusammen und in jedem dieser Augenblicke atmete ich besonders tief ein, um seinen herben Duft in mich einzusaugen. Seine Nähe machte mich fast wahnsinnig. Ich fühlte seinen Körper ganz unmittelbar und ich liebte es, wenn er mich zwischendurch ansah. Wenn er lachte, dann funkelten seine Augen, und wenn er sich konzentrierte, dann bekam sein Blick so etwas Tiefes, Nachdenkliches. Kurz vor drei am Nachmittag, als ich gerade wieder in unser Projekt vertieft war, stand Magnus plötzlich auf und teilte uns Mitarbeitern seine Besorgnis bezüglich des heftigen Schneefalls mit. Er verkündete, dass jetzt offiziell Büroschluss sei und dass es wohl ratsam wäre, nun nach Hause zu fahren, da er befürchtete, dass es zu fortschreitender Zeit immer schwieriger werden könnte. Hinzu käme, dass es in weniger als einer Stunde dunkel sei und dass der Wetterbericht auch keine Besserung vorhersagen konnte, ganz im Gegenteil. Von nun an herrschte Aufbruchstimmung. Alle packten ihre Sachen zusammen und auch ich überlegte, ob ich mich auf den Heimweg machen sollte. Ich schaute zu Magnus herüber, der sich wieder gesetzt und sogleich erneut in die Arbeit vertieft hatte. Es sah ganz so aus, als würde er noch bleiben, um daran weiterzuarbeiten. Der Blick aus dem Fenster sagte mir, dass es wohl besser wäre, zu fahren, aber mein Pflichtbewusstsein erinnerte mich an den knappen Abgabetermin. Hinzu kam, dass ich zum einen Magnus mit unserem Projekt nicht hängen lassen und ihn zum anderen auch ein wenig damit beeindrucken wollte, dass ich blieb. Nachdem alle gegangen waren, blickte Magnus vielsagend von seinem Bildschirm auf und sagte erfreut zu mir: "Noch da, Patricia? Wie schön!" Wieder dieser Blick, wieder diese funkelnden lachenden Augen! Er ging mir durch und durch und ich wusste, dass mein Bleiben das einzig Richtige und das Beste war, was ich seit Langem getan hatte. Sogleich vertiefte Magnus sich aber wieder in unsere Auftragsarbeit, was ich ihm gleich tat, und so merkten wir nicht, dass es um uns herum zu dämmern begonnen und der Schnee seinen Weg auf die Erde unentwegt fortgesetzt hatte. Zusätzlich war ein kalter Wind aufgekommen, der ebenfalls, von uns nahezu unbemerkt, um das Gebäude pfiff. Ganz nebenbei knipsten wir unsere Schreibtischleuchten an und da ich von meinem Platz aus nicht so gut hinaussehen konnte, stand ich irgendwann später auf, um ans Fenster zu gehen. Ich wollte wissen, was das Wetter machte, und schaute über den schneebedeckten Parkplatz. "Du, Magnus?" "Hmmm?", machte er freundlich, jedoch ohne aufzublicken. "Wir werden nachher ordentlich Mühe haben, unsere Autos freizugraben, sofern wir sie überhaupt wiederfinden", scherzte ich. Jetzt erhob auch er sich und stellte sich zu mir. "Oh ja, allerdings", erwiderte er zur Bestätigung, als auch er aus dem Fenster schaute. Möglichst unauffällig schob ich mich einige Zentimeter näher an ihn heran. Ich weiß bis heute nicht, ob er es registriert hatte oder ob sein Blick aus einem anderen Grunde sehr weich wurde, als er mich anschaute. Wir fühlten gegenseitig unsere Körper, als sie sich in diesem Augenblick berührten. "Wie kommst du mit der Grafik voran, Patricia?", fragte er mich beinahe liebevoll und ich freute mich, ihm mitteilen zu können, dass es gerade sehr gut lief. Zufrieden nickte er und zwinkerte mir zu, als wir uns wieder setzten. Wieder versanken wir in unserer Arbeit, bis ich rund zwei Stunden später auf meine Uhr schaute. "Oh, so spät ist es schon", entfuhr es mir und da ich wegen der verschneiten Straßen nun doch nach Hause wollte, fügte ich hinzu: "Ich glaube, ich werde dann mal so langsam..." Magnus legte den Stift aus der Hand. "Ja, du hast recht. Wir sollten für heute Schluss machen. Ich werde mich morgen wieder daransetzen." In diesem Moment speicherte er seine Datei und klappte sein Powerbook zu. Auch ich fuhr meinen Computer herunter und schaltete ihn aus. Wir nahmen unsere Jacken in die Hand, löschten alle Lichter und machten uns auf den Weg nach unten in die Halle. Dort griff Magnus an die Tür und wollte sie gerade galant nach außen aufschwingen lassen, um mir den Vortritt zu gewähren, als er vergeblich gegen den Griff drückte. Nur wenige Zentimeter ließ sie sich aufschieben, dann aber bremste eine riesige Schneewehe sie ab. "Hey, was ist das denn?", fragte Magnus erstaunt und wir schauten in die Dunkelheit hinaus. Sofort erkannten wir, dass die Schneeverwehungen die Tür vollkommen blockierten. "Zu viel Schnee vor der Tür", bemerkte ich überflüssigerweise, worauf Magnus mich amüsiert anschaute. "Und es gibt keine zweite Tür, oder?", fragte ich sicherheitshalber nach, meinte aber, dass er mir diese Tatsache bei der Haus- und Büroführung an meinem ersten Tag mitgeteilt hatte. Er war bereits wieder auf dem Weg nach oben und rief mir ein "Nein" zu, als er schon auf der halben Treppe war. Ich eilte ihm hinterher und dachte so bei mir, dass ich schon weitaus Schlechteres erlebt hatte, als hier mit meinem überaus attraktiven Chef in unserem warmen Büro eingeschneit zu sein. Hinzu kam, dass wir über einen gefüllten Kühlschrank in der Teeküche und genug Kaffee und Cappuccino verfügen konnten, so dass es uns wohl an nichts mangeln würde. Vorausgesetzt natürlich, er beendete unsere Lage nicht, indem er einfach den Hausmeister oder jemand anderen anrufen würde, der uns dann hier herausholte. Danach sah es allerdings nicht aus, denn als ich oben ankam, lehnte er bereits lässig an einem der Schreibtische und schaute mich bedeutungsvoll an. Seine Jacke hatte er neben sich auf den Boden gleiten lassen, was ich mit meiner auch sogleich tat. Seine Augen lachten und blitzten, als er sagte: "Na, Patricia?" Ich grinste und legte den Kopf auf die Seite. So recht deuten konnte ich seinen Blick und seine herausfordernde Art gerade nicht, aber ich ahnte, dass auch er sich im Moment Unangenehmeres vorstellen konnte als unsere Situation. Das zu erkennen freute mich ungemein und in meinem Unterleib begann es unmissverständlich zu kribbeln. "Komm doch mal her", forderte er mich zu meiner großen Freude auf. Seine Stimme war dabei sanft und er lächelte mir aufmunternd zu. Dass ich diese Aufmunterung gar nicht so dringend brauchte und auch ohne sie nur allzu gern zu ihm kam, konnte er ja nicht wissen. Oder doch? Ich stellte mich relativ dicht vor ihn. Magnus aber legte seine Hände in meine Taille und zog mich noch einige Zentimeter näher zu sich. Dann kam er mit seinem Gesicht ganz nah an meines heran. So nah, dass sich unsere Nasenspitzen berührten. "Patricia, ich mag dich sehr, sehr gern. Weit über das Maß einer Angestellten hinaus. Ich finde, dass du eine tolle Frau bist. Und dass wir zwei hier heute eingeschneit sind, finde ich, ehrlich gesagt, gar nicht so schlimm", flüsterte er ganz leise. Dabei umspielte ein Lächeln seinen Mund. Ich nickte und lächelte auch. Wie schön, dass er genauso dachte wie ich. Magnus nahm mein Gesicht in seine Hände, wobei er mir tief in die Augen sah. "Und ich glaube, dass du so ähnlich auch für mich fühlst. Stimmt das, Patricia?" Mein nickender Kopf in seinen Händen, mein Blick in seine Augen eintauchend. Gott, war das schön! Ich fühlte meinen Herzschlag bis zum Hals und das verräterische Kribbeln in meinen Schoß. Sein Gesicht kam noch näher und unsere Lippen trafen sich weich, als er mich voller Gefühl küsste. Sein Kuss war wunderschön. Unbeschreiblich zärtlich und sanft. Als er seine Lippen wieder von meinen löste, bot ich ihm meine erneut an. Ein Angebot, welches er nicht ablehnen wollte, und so versanken wir in einem weiteren, ebenso hingebungsvollen Kuss. Seine Daumen streichelten die weiche Haut vor meinen Ohren, wenn er seinen Kuss unterbrach, um mir die schönsten Komplimente zuzuflüstern. Als er mir dann aber sagte, wie sexy und überaus begehrenswert er mich fand und dass er schon so manch unanständigen Traum von mir geträumt hatte, sah er mir konstant in die Augen. Sein Blick sprach in diesem Moment von Lust und großem Verlangen, das sah ich deutlich. Meiner auch, wie es schien, denn als hätte mein Blick ihn aufgefordert, legte er seine Hände auf meinen Po und presste mich fest an sich. Als ich seine Erektion dabei fühlte, wurde mir heiß und kalt. Unglaublich, dass mir das hier gerade wirklich passierte! Das war genau das, wovon ich immer geträumt und was ich mir vor dem Einschlafen oft ausgemalt hatte. Das sagte ich ihm, ich wollte, dass er wusste, dass es auch mir so ging. "Oh, Magnus", stöhnte ich leise, "genau das habe ich mir so oft vorgestellt, wenn ich mich vor dem Einschlafen bis zum Orgasmus gestreichelt habe! Dann habe ich immer an dich und deine Erregung gedacht, während meine Berührungen immer intensiver wurden." Magnus' Atem ging schneller und auch er konnte ein leises Stöhnen nicht unterdrücken. "Erzähl weiter, Patricia, bitte!", bat er heiser in mein Ohr flüsternd. Ich fühlte seine Erektion noch größer werden und dadurch verstärkt auch meine Erregung nass und glitschig in mein Höschen sickern. Ja, ich wollte es ihm gern noch näher beschreiben, meine Empfindungen noch bunter ausmalen. "Magnus, meine Klit wurde dabei immer fester und meine Venus immer feuchter. Es war heiß, dabei an dich zu denken. Manchmal habe ich dich geradezu auf mir gefühlt, so nah waren meine Gedanken bei dir. Das waren die Momente, in denen mein Kommen kaum noch aufzuhalten war, Magnus! In denen ich spürte, dass mein Orgasmus mich jeden Moment überrollen könnte, wenn ich meine Perle nur weiterreiben würde." Sein Stöhnen wurde lauter, seine Erektion hatte sich vollständig aufgerichtet. Noch während er mich wieder küsste, hob er mich ein Stückchen nach oben, wobei er meine Mitte über die harte Beule in seiner Jeans rieb. Sein Penis zwischen meinen Schenkeln. Würden wir keine Kleidung tragen, läge seine Eichel ganz unmittelbar vor meiner inzwischen einladend offenen Vagina. "Ich habe es auch schon getan, Patricia, und es war geil!" Er raunte mir diese Worte in mein Ohr und rieb sich dabei aufreizend zwischen meinen Schenkeln. "Dich befriedigt und dabei an mich gedacht?" Ich flüsterte verführerisch. "Hmmmh, ja!", flüsterte er zurück und begann, meine Hose zu öffnen. Ich tat es ihm gleich und schon bald fühlten wir jeweils die Hand des anderen streichelnd an unseren exponierten Geschlechtern. Beide stöhnten wir und bewegten unsere Becken im gleichen Rhythmus. Er dachte dasselbe wie ich. Beide fühlten wir die Lust in uns brennen und beide wussten wir, dass wir uns nicht mehr lange würden beherrschen können. Ich hatte das Gefühl, am Ziel meiner geheimsten und dennoch allerschönsten Träume angelangt zu sein, als Magnus mir stöhnend und vollkommen erotisiert ins Ohr flüsterte, dass er sich nichts sehnlicher wünschte, als mit mir zu schlafen. Durch sein zärtliches Streicheln meiner Liebesperle schon in fremde Welten entrückt, nickte ich fast wie in Trance. Ich war vollkommen von seinem maskulinen und dennoch sanften Liebesspiel eingenommen. Magnus nahm liebevoll meine Hand von seinem pulsierenden Penis, die ihn in Richtung Ziel massierte, und drehte mich vorsichtig um. Ein Bein hob er dabei aus Jeans und Slip heraus, die um meine Knöchel drapiert lagen. Ich verstand und beugte mich voller Erwartung nach vorn auf den Schreibtisch. Ohne zu zögern, legte Magnus seinen großen festen Phallus an meine nasse Muschel. Er drang aber nicht sofort ein, sondern behielt ihn in der Hand und rieb mit seiner Eichel zuvor ein paar Mal über meine nasse Perle hinweg. Ich stöhnte und ließ mein Becken kreisen. Mein Orgasmus schien zum Greifen nah, zu heiß machte mich dieser interessante aufregende Mann. Was er hier mit mir tat, war unbeschreiblich schön. Niemals zuvor wurde ich so liebevoll verwöhnt und dennoch voller ungeduldiger Lust erwartet. Groß, prall und mich vollständig ausfüllend schob er seinen pulsierenden Phallus tief in mich hinein. Langsam und sich immer wieder neu an die seidige Enge um Eichel und Schaft gewöhnend, drang er immer weiter vor. Magnus stöhnte laut und ich spürte deutlich, wie sehr er darum kämpfte, nicht schon jetzt die Kontrolle zu verlieren. "Patricia, du bist so irrsinnig eng, dass ich es kaum noch zurückhalten kann", stöhnte er, als er gerade wieder in der Bewegung innehielt. Ich wusste um die Wirkung meiner engen Vagina auf Männer und ich freute mich, dass sie auch Magnus so ganz besonders verrückt zu machen schien. Sicher lag es aber auch an seinem beachtlich großen Penis, der mich wiederum fast um den Verstand brachte. Tief war er in mir. Seine Hände auf meiner Hüfte, die wollüstig in mein Fleisch griffen, als er gleich darauf begann, sich in mir zu bewegen. Ich stöhnte guttural auf. Fast animalisch drang mein Stöhnen aus mir heraus, als ich seine Bewegung tief in mir spürte. Magnus zog sich weit zurück, nur um sich dann der ganzen Länge nach wieder in meine heiße feste Höhle hineinzuschieben. Auch er stöhnte dabei und war die ganze Zeit über versucht, sich einfach seinem Gefühl hinzugeben. Aber noch hatte er sich im Griff, noch hielt er immer wieder inne, wenn das Gefühl ihn zu überschwemmen drohte. In einem solchen Moment schob Magnus seine rechte Hand nach vorn zu meinem Delta, um mit der Fingerspitze seines Mittelfingers meine Klit zu liebkosen. Als ich seinen Finger wahrnahm, stöhnte ich auf und als ich seine kreisenden Bewegungen genoss, spürte ich schon bald die süßen Zuckungen meines nahenden Höhepunkts. "Magnus, jetzt!", rief ich noch keuchend, als der Orgasmus mich auch schon erfasste und mit sich davontrug. Magnus fühlte mein Beben und Zucken an seiner hochempfindlichen Eichel, die er in diesem Moment an meine geschwollene Klit gelegt hatte. Er stöhnte mit mir zusammen auf und ich fühlte, dass mein Kommen ihn bis an die Grenze seiner Selbstbeherrschung brachte. Noch einmal drang er in mich ein, wartete einen Augenblick und dann bewegte er sich nur noch minimal, um den bevorstehenden Orgasmus noch so lange wie möglich hinauszuzögern. Er streichelte meinen Rücken hinauf und bewegte sein Becken nur noch ganz sachte vor und zurück. Magnus überschüttete mich mit Komplimenten, die er immer wieder durch sein Stöhnen unterbrach, bis er es keine Sekunde länger aushielt und mir keuchend sagte: "Patricia, ich kann nicht mehr, ich schaffe es nicht länger!" Fast zeitgleich mit dem letzten Wort holte er weit aus und zog sich fast vollständig aus mir zurück, nur um gleich darauf ein letztes Mal tief in mich einzutauchen. Sein Atem ging schnell und schwer. Um seinen Genuss zu erhöhen, spannte ich meine Ringmuskulatur an und machte mich so eng ich konnte. Damit hatte Magnus nicht gerechnet und kam ohne jede Verzögerung in genau diesem Augenblick. Er rief seine Lust stöhnend heraus und presste sich dabei tief in meine Muschel hinein. Seine Lust katapultierte sich kraftvoll in mich, was ich als ein berauschendes Feuerwerk der Liebe empfand. Er zog sich noch nicht gleich zurück, sondern blieb noch einen Moment länger mit mir verschmolzen. Er fühlte sich auch jetzt noch unheimlich gut in mir an, was ich ihm liebevoll sagte. Ich hörte ihn hinter mir leise lachen und fühlte seine Hände, die mich wieder sanft streichelten. Irgendwann zog er sich dann aber doch aus mir heraus und wir kleideten uns wieder an, um uns dann gleich dort, wo wir waren, auf unsere Jacken gekuschelt zum Schlafen aneinanderzuschmiegen. Und heute, drei Winter später, sind wir noch immer ein glückliches Paar und denken so manches Mal an unseren ersten, eingeschneiten und dadurch so romantischen Sex zurück.
# Winterlust
## Lisa Cohen
"Ski Heil!" Das hatten mir wohl fast alle Freunde und Bekannten und sogar meine Kollegen gewünscht, bevor ich für eine Woche zum Skifahren wollte. Seit einigen Jahren hatte ich großen Gefallen an diesem Wintersport gefunden und nun endlich mal wieder Gelegenheit, ihn ausgiebig zu genießen. Bekannte von mir hatten einen Gruppenurlaub in den Bergen Tirols geplant. Schon im Sommer des letzten Jahres hatte ich mich dazu angemeldet. Jetzt war Februar und genau die richtige Zeit, dem trüben Wetter zu Hause zu entfliehen und eine Auszeit zu nehmen. Wir würden so um die 30 Leute sein, von denen ich die meisten noch nicht kannte, was die ganze Sache umso spannender machte. Ich brannte darauf, neue Leute und vor allem neue Männer kennenzulernen. Denn ich wollte nicht nur Ski fahren. Ich hoffte, in diesem Urlaub mit etwas Glück auch eine Affäre oder zumindest einen heißen One-Night-Stand zu erleben. Ich war Single und einem erotischen Abenteuer jeglicher Art grundsätzlich sehr aufgeschlossen. Meine Skikünste hielten sich leider in Grenzen, doch es würden für jedes Level Kurse angeboten, hatte man mir versichert. "Es sind bestimmt auch noch ein paar andere Anfänger dabei...", beruhigte man mich. So lange ich nur nicht alleine die Hänge herunterwedeln musste... Die lange Fahrt in das Skigebiet wurde verkürzt durch gute Musik, ausreichend Stimmungsmache und gegenseitigem Beschnuppern. Die meisten Leute schienen wirklich nett und aufgeschlossen zu sein. Und offensichtlich gab es tatsächlich außer mir noch ein paar andere unsichere Skifahrer. Wir sprachen uns gegenseitig gleich mal Mut zu, tranken darauf einen Schnaps und ich fühlte mich wohl. Natürlich hatte ich auch schon mal die männlichen Mitfahrer ein wenig beäugt und mein Auge war an einem Mann hängen geblieben, der ziemlich genau meinem Typ entsprach. Blond, sportlich, helle Augen, ein freundliches Lächeln und ein unglaublich sinnlicher Mund. Wir wechselten ein paar Worte miteinander, dann wandte er sich wieder dem Kartenspiel mit drei anderen Typen zu. Na, ich würde hoffentlich Zeit genug haben, ihm näherzukommen. Der Ort, in den wir fuhren, war einer dieser typischen Wintersportorte. Etwas voll, etwas hektisch, aber wunderschön gelegen und Wetter und Skibedingungen waren vom ersten Tag an ein Traum. Unsere Gruppe hatte eine große Skihütte gemietet, in der wir uns selbst verpflegen würden. Ich teilte mir mit fünf anderen Frauen das Zimmer. Wir passten gut zusammen. Wir würden viel Spaß haben. Am nächsten Morgen stand ich zum ersten Mal seit langer Zeit wieder auf den Brettern. Und es war die absolute Hölle. Ich geriet bis mittags ins Schwitzen und bis nachmittags in fast hysterische Verzweiflung. Und ich hatte nicht das Gefühl, dass ich mich am nächsten Tag entscheidend verbessern würde. Ich hatte es nicht als so schwierig in Erinnerung. Abends fiel ich todmüde und frustriert ins Bett, zu müde, um mich noch um meine Hormone zu kümmern. Der zweite Tag verlief zum Glück etwas vielversprechender. Ich schöpfte neuen Mut. Ich konnte mich nebenbei ein wenig auf die Après- Ski-Annehmlichkeiten konzentrieren. Es gab recht viele attraktive Männer um mich herum und einige flirteten schon bald mit mir. Doch ich hatte meine Wahl bereits getroffen und warf meine Netze aus. Erst sah ich nur zufällig in seine Richtung, dann stellte oder setzte ich mich neben ihn, sprach ihn an und lächelte ein wenig verführerisch, nicht zu viel. Er nickte freundlich, aber unverbindlich zurück und antwortete ebenso auf meine Fragen. Es sah fast ein wenig danach aus, als wenn er kein richtiges Interesse hätte. Es war auch nicht ganz einfach, ihn zu greifen. Tagsüber wedelte er die schwierigsten Berge herab. Er schien ein begnadeter Skifahrer zu sein, hatte ich etwas neidisch feststellen müssen, und abends widmete er sich ausgiebig dem Kartenspiel mit seinen Zimmergenossen. Doch so leicht wollte ich nicht aufgeben. Ich hatte so eine Idee, wie ich ihm doch noch näherkommen konnte. Ich wartete einen günstigen Moment ab, als er etwas geruhsamer als sonst vom Berg herunterkam, und schaffte es irgendwie, ihm direkt vor die Füße zu fahren. Fast wäre er gestürzt, aber natürlich sah er auch dabei immer noch cool aus, während ich wie ein Häufchen Elend vor ihm im Schnee lag und darauf wartete, dass er mich heldenhaft retten würde. Das tat er dann auch. Erst noch ein wenig fluchend, dann aber lächelnd, als er mein zerknirschtes Gesicht sah. Ich war schon immer eine sehr gute Schauspielerin gewesen... "Also Susi! Ich glaube, ich muss dir mal ein wenig unter die Arme greifen..." Sein Lächeln hätte mich leicht wieder von den Beinen hauen können. Aber er hielt mich beruhigend fest und sah mich an, als sähe er mich zum ersten Mal richtig. "Ich mache dir einen Vorschlag. Zu deiner und unserer aller Sicherheit, stehst du von nun an unter meinem persönlichen Schutz und unter meiner Führung. Ich verspreche dir, du lernst das Skifahren, bevor du wieder in den Bus steigst. Hat doch bis jetzt noch jeder geschafft!" Obwohl das nicht sehr ehrenhaft für mich klang, freute ich mich natürlich riesig. Er hatte angebissen... Und er hielt Wort. Er zeigte mir, was es zu zeigen gab, geduldig und ausdauernd. Am letzten Nachmittag des Urlaubes hatte ich dann tatsächlich das Gefühl, ich hätte es zumindest soweit gelernt, dass ich auch dahin fahren konnte, wo ich hinwollte. Dass wir uns bei diesem persönlichen Schutzprogramm näherkamen, war ein wunderbarer Nebeneffekt. Ich begehrte ihn mittlerweile heftig und ich wollte endlich ein Sexabenteuer erleben... Die letzte Abfahrt lag vor uns. Wir fuhren langsam herunter. Strahlender Sonnenschein, blauer Himmel, pulvriger Schnee. Der Tag war überwältigend. Plötzlich schwenkte Chris in eine Seitenloipe. Dort wartete er auf mich. Ich fuhr in einem rasanten Schwung neben ihn und strahlte vor Freude über meinen kunstvollen Stopp. Er lächelte bewundernd. "Du bist wirklich gut geworden. Schade, dass der Urlaub vorbei ist." Er sah mich irgendwie anders an als sonst. "Hast du Lust, ein paar Schritte zu laufen? Ich möchte dir etwas zeigen." Wir schnallten die Skier ab und gingen schweigend durch den wundervollen Winterwald bis zu einer kleinen Lichtung. Dort stand eine offene Skihütte, ein Unterstand für von schlechtem Wetter überraschte Skifahrer. Mein Blut wurde wärmer, mein Herz schlug schneller. Ich wusste natürlich nicht genau, was passieren würde, aber ich war mir sicher, er würde mich zumindest küssen, mit seinen so sinnlichen Lippen, auf die ich schon seit sechs Tagen Lust hatte. Es war eine unglaublich romantische Situation, in der wir uns befanden. Ohne Zögern, ohne etwas Unnötiges zu fragen, zog er mich in die Hütte hinein und an sich heran. Seine sinnlichen Lippen konnten noch sinnlichere Küsse schenken. Ich spürte sie bis in meine Zehenspitzen und sehnte mich danach, in einem schönen warmen Bett zu liegen, denn ich konnte mir nur schwer vorstellen, hier draußen in dieser Kälte mehr mit einem Mann tun zu können, als ihn zu küssen. Ich sollte mich wundern... Chris zog mich hinter eine Art Bretterwand, die zusätzlichen Schutz bei starkem Wind bieten sollte. Er küsste mich wieder entwaffnend sinnlich. Die Kälte um mich herum war vergessen. Als er meine Skijacke ausziehen wollte, half ich ihm dabei. Er entkleidete mich bis auf meinen BH und presste mich gegen die raue Bretterwand. Ganz langsam fuhren seine Finger am Rand meines BHs entlang, taten, als seien sie noch unentschlossen, ob sie mich berühren sollten oder nicht. Immer wenn ich dachte, jetzt sind sie gleich dran an meinen Nippeln, zogen sich die Finger wieder zurück. Und das machte mich richtig an. Er wich spielerisch meinen Küssen aus, schob den BH herunter, doch nur für einen winzigen Moment, um flüchtig an meinen harten Nippeln zu lutschen. Dann rieb er wieder ganz sacht den Stoff meines Büstenhalters an meinen Nippeln. Diese zögernde Verführung, das bewusste Hinhalten machte mich wahnsinnig. Hitze schoss mir in die Brüste, verteilte sich über den Körper abwärts bis zwischen meine Schenkel. Chris' Hände strichen über meinen Leib, öffneten den Reißverschluss meiner Skihose und schafften es irgendwie, bis in meinen Slip vorzudringen. Zitternd ließ ich es zu, dass er mich ganz aus meinem dicken Anzug befreite. Fast nackt stand ich vor ihm. Es war Winter und es war kalt und trotzdem war mir so heiß, als wenn ich in einer Sauna gesessen hätte. Er stimulierte mich mit nur einem Finger genau auf dem Punkt, aus dem die größte Lust fließt. Als Gegenleistung knöpfte ich seinen Anzug auf. Wühlte mich durch die warme Schicht Unterwäsche, um sein bestes Stück zu begrüßen. Es fühlte sich warm und lebendig an und sehr entschlossen, mich zu befriedigen. Chris drehte mich um und bog mich nach vorne. Ich streckte ihm meinen Po entgegen und grätschte meine Beine. Er hob meine Hüfte an, um seinen Liebesstab in der bestmöglichen Position in mich einzuführen. Der Ruck, mit dem er mich in Besitz nahm, beflügelte mich. Heiß füllte sich mein Unterleib und zitterte vor Erregung. Ich presste mich in seinen Schritt und fühlte mich großartig. Es war mein erstes Mal in einer Wetterhütte. Es war unglaublich romantisch und gleichzeitig wahnsinnig geil. Winterliche Stille um uns herum. Nur unser erregtes Stöhnen zerschnitt die kalte Luft. Ich drehte meinen Kopf, um seine Lippen zu spüren, während er mich mit seiner starken Manneskraft befriedigte. Dampf stieg von unseren Körpern auf. Ich gluckste vor Glück. Sein Glied rieb sich ekstatisch an den Wänden meiner Liebeshöhle und peitschte unsere Lust in die Höhe. Es war kein sanfter Akt. Es war ein kurzes wildes Liebesspiel, das wir hier draußen spielten. Seine Stöße waren kurz und hart. Schnell und effektiv. Ich spürte jeden Muskel in mir, jeden Nerv. Mein Körper war bis zum Äußersten angespannt und würde sich nur durch einen angemessenen Höhepunkt wieder entspannen können. Ich schloss die Augen, um meine Sinne nur noch auf das eine zu konzentrieren. Auf den finalen Orgasmus, der sich tief drinnen in meiner Vagina ankündigte, wo es brodelte und kochte wie in einer himmlischen Hölle. Wir standen beide unter Starkstrom, als Chris explodierte. Unsere ungezügelten Liebeslaute waren ganz bestimmt bis tief in den Wald hinein zu hören. Wir lebten unseren sexuellen Hochgenuss verbal aufs Phantasievollste aus und hielten uns eng aneinandergepresst, bis mich die Gänsehaut auf meinem Körper davon überzeugte, mich besser zu lösen aus diesem sinnlichen Akt und mich wieder anzuziehen. "Das war schön!" Ich konnte Chris' Urteil nur bestätigen. Als wir wieder klar denken konnten, sagte er: "Es war übrigens eine schauspielerische Spitzenleistung, wie du mir vor die Füße gefahren bist." Ich schluckte überrascht. Er hatte mich durchschaut. "Aber ich hätte dich auch ohne diesen 'Unfall' noch angesprochen..." Das tat gut zu hören. Wir fuhren mit ausschweifenden Schlenkern sehr befriedigt zurück ins Tal und ließen uns viel Zeit dabei mit unserer Abfahrt. Ich hatte in meinem Urlaub nicht nur endlich Skifahren gelernt. Ich hatte in wunderbarer Natur den wohl schönsten Sexakt meines Lebens erlebt, mit einem Mann, der es wert war, dass ich ihn wiedersehen würde. Vielleicht konnte mehr aus unserer vielversprechenden ersten Begegnung werden...
| {
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} | 4,459 |
Juan Carlos Casasola (Ciudad de México, 4 de mayo de 1960) es un actor, cantante y conductor mexicano, hijo de la primera actriz mexicana Irina Areu. Recientemente protagonizó Cats México 2013-2014 con más de 350 representaciones bajo la producción de Gerardo Quiroz repitiendo el personaje de "Punk Rock Terco" Rum Tum Tugger por el cual obtuvo el reconocimiento "Luminaria de Oro" del Paseo de las Luminarias, como el mejor actor de teatro musical del año.
Actualmente se desempeña como conductor del programa Guerra de chistes y estelariza el musical La Era Del Rock Rock of ages con el personaje del empresario alemán Hertz Klineman el cual lo hace acreedor, por 2.º año consecutivo, al reconocimiento "Luminaria de Oro" del Paseo de las Luminarias, como el mejor actor de teatro musical.
Trayectoria
Juan Carlos Casasola empezó su carrera en 1979 con cuentos infantiles en la compañía "El Mundo de las Maravillas" de Manuel Lozano El zapatero remendón realizando más de 30 cuentos como Blancanieves, Pinocho, El Gato con Botas, La Bella Durmiente, Cenicienta, El Soldadito de Plomo entre otros.
En 1983 participó en el concurso Valores Juveniles Bacardi obteniendo el Quinto Lugar con el grupo "Triángulo" al lado de su hermana Maricarmen Casasola y Jaime Quintero con la canción "Mala Suerte". Productores Carlos Lara y Jesús Monárrez, disquera Musart. Ingresó al CEA Centro de Educación Artística Televisa en el año 1986.
En 1988 realizó "Pedro y el Lobo" producción de Televisa a cargo de Fernando Morett y Cali Domínguez interpretando a "El Lobo" al lado de Eugenio Cobo y Aurora Molina en el Teatro Libanés con más de 300 representaciones.
En 1989 obtiene su primera oportunidad en televisión en la telenovela "Cuando llega el amor" producida por Carla Estrada dándole vida al personaje de "Beto", actuación por la cual es nominado en los premios TVyNovelas como mejor actor antagónico.
Desde ahí hasta la fecha ha participado en varias telenovelas, entre ellas, La pícara soñadora, Dos mujeres, un camino, El vuelo del águila, Tú y yo, Gotita de amor, Locura de amor, Rayito de luz, Carita de ángel, Aventuras en el tiempo, ¡Vivan los niños!, Sin pecado concebido, Entre el amor y el odio, Amarte es mi pecado, Mujer de madera, La fea más bella, Mañana es para siempre, Corona de lágrimas entre otros. Participó en 46 capítulos de Mujer, casos de la vida real.
En el año 1992 se integra al elenco de la exitosa puesta en escena Cats con el personaje de "Punk Rock Terco" Rum Tum Tugger al lado de Manuel Landeta, Susana Zabaleta, María del Sol, Javier Díaz Dueñas
En 1994 forma parte de Vaselina como "Kiko" compartiendo créditos con Alex Ibarra, Irán Castillo, Lolita Cortés, Lorena Rojas, Jean Duverger entre otros.
En 1994 forma parte de "Pinocho, producción de Televisa en el Teatro República al lado de Eduardo Santamarina. Dicha puesta en escena se repitió una vez más en el año 1999.
Participa en el programa Sábado Gigante en 1998–2007 en la sección de humor dentro de la cual surge la idea de crear Guerra de chistes.
Participó en las dos etapas del lamentable show "Sólo para mujeres" en la primera etapa como bailarín y en la segunda etapa como coreógrafo y conductor. Participó también en el show "Piel Caliente" como conductor.
En 2008 realizó su primera producción discográfica llamada "Loredana" producida por Arturo G Álvarez con composiciones y arreglos de Bruno Krajzar, destacado compositor y arreglista croata. Una de las canciones "Chi Sei" (¿Quién eres?) interpretada por Juan Carlos Casasola y Marichelo, fue el tema de una exitosa telenovela italiana. La canción "Loredana" fue la más escuchada en Europa Occidental en 2011. Esta canción fue grabada en español, inglés y croata.
En mayo de ese año es invitado a presentar su disco en la ciudad de Nueva York en el festival del 5 de mayo ante 6 millones de personas con el cual obtiene un reconocimiento por parte de la Asamblea del Estado.
Es co-conductor del programa Guerra de chistes al lado de Juan Carlos Nava "El borrego", Radamés de Jesús y Yered Licona "La Wanders Lover"
Guerra de chistes se transmite por Telehit a más de 80 países y es desde el 2008 hasta la actualidad el programa de mayor rating de dicho canal.
Desde mayo de 2013 y durante más de 350 representaciones, protagoniza Cats México al lado de Manuel Landeta, Filippa Giordano, Olivia Bucio, Lila Deneken, Ana Cirré
Estelariza Rock Of Ages La Era Del Rock al lado de Patricio Borghetti, Ernesto D'Alessio, Mauricio Castillo, Vadhir Derbez, David Cavazos y Laura Cortés.
Filmografía
Telenovelas
Mujeres de negro (2016).... Joel
Corona de lágrimas (2012-2013).... Benjamín Aguilar
Mañana es para siempre (2008-2009).... Graciano
La fea más bella (2006).... Jorge
Barrera de amor (2005).... Pancho
Mujer de madera (2004).... Heriberto
Amarte es mi pecado (2004).... Gonzalo Carrera
¡Vivan los niños! (2002-2003).... Secundino
Entre el amor y el odio (2002).... Catrín
Sin pecado concebido (2001).... Sergio Orozco
Aventuras en el tiempo (2001).... Lic. Chacal
Rayito de luz (2000-2001).... Justino Hernández
Carita de ángel (2000).... Calixto
Locura de amor (2000).... Damián
¡Amigos x siempre! (2000).... Fernando
Soñadoras (1998).... Criminal en prisión
Gotita de amor (1998).... Román Correa
Tú y yo (1996-1997).... Gonzalo
El vuelo del águila (1994-1995).... Francisco Zarco
Dos mujeres, un camino (1993-1994).... Leobardo
Mágica juventud (1992)
La pícara soñadora (1991).... Fausto Medrano
Días sin luna (1990).... Gastón Solís
Cuando llega el amor (1990).... Beto
Series de TV
LOL: Last One Laughing (2022)
El Inframundo (2021)
La voz (2019)
Adictos (2012) 2a Temporada
Adictos (2009) 1a Temporada
Como dice el dicho (2008-Presente).... (8 CAPÍTULOS)
La rosa de Guadalupe (2008-Presente).... (14 CAPÍTULOS)
Guerra de chistes (2008-2019) (Co-Conductor)
Incógnito (2008).... Casasola (Las Tiernas Aventuras de Juan Menchaca)
Ugly Betty (2007).... Lobo
Objetos perdidos (2007).... Manager
Mujer, casos de la vida real (1995 - 2007).... (46 CAPÍTULOS)
La telaraña (1992).... Ismael
Películas
Cometa, Él, su perro y su mundo (2017)
Por tus pugidos nos acacharon (2013)
Un tigre en la cama (2009)
Policía rural (2005)
Perro rabioso III: Tras el rostro (1992).... Samy
Perro rabioso 2 (1991).... Samy
Muerte por partida doble (1991).... Alfonso
Retén (1991)
Infamia (1991)
Malditos amapoleros (1990)
Entre la fe y la muerte (1990)
Premios y nominaciones
Premios TVyNovelas
Reconocimiento "Luminaria de Oro" del Paseo de las Luminarias por segundo año consecutivo como Mejor actor de Teatro Musical del año por "Punk Rock Terco" en CATS MÉXICO 2013/2014 y "Hertz Klineman" en LA ERA DEL ROCK.
Premio Arlequín 2007 por Trayectoria Artística
Premios Telehit: Mejor programa Guerra de chistes
Reconocimiento "Luminaria de Oro" del Paseo de las Luminarias. Mejor actor de Teatro Musical del año por "Punk Rock Terco" en CATS MÉXICO 2013
Galardón Inmortal al programa de Telehit "Guerra de Chistes" por su brillante proyección televisiva que lo hace merecedor a una placa en bronce en el Paseo de las Luminarias.
Premios Telehit 2013: Mejor Programa de Comedia Guerra de chistes
En los 20 años de Telehit recibiendo el premio al mejor programa de comedia Guerra de chistes
De la producción discográfica que realizó en 2008 "Loredana", el sencillo "Esperanza" obtuvo una pre-nominación en los 56th Grammy Awards por Mejor Canción.
Fundación Cultural Plaza Galerías le otorga un reconocimiento por el éxito de "Sólo para Mujeres" !999-2000
Obtiene el premio "Califa de Oro" por el show "Sólo para Mujeres" 1999.
Por ser "Sólo para mujeres" el show más exitoso del año, fueron acreedores a plasmar su imagen en un boleto de la Lotería Nacional el 27 de septiembre de 1999.
Referencias
Enlaces externos
https://web.archive.org/web/20140606214722/http://www.catsmx.com/
Nacidos en Ciudad de México
Actores de televisión de México
Actores de cine de México
Presentadores de televisión de México | {
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} | 3,948 |
Marathon Etna
=============
:date: 2015-11-12 15:20
:category: Résultats
:location: Etna
:eventdate: 2013-11-03
Superbe marathon. 3h35. 800m D+. 10eme sur 116 inscrits. Une de mes 3 plus
belles courses. Seul regret : avoir coincé alors que j'étais 5eme. Repos avant le
dernier objectif, la saintélyon, et la trêve hivernale.
.. image:: http://assets.acr-dijon.org/Julienetna.jpg
Grosse performance de Julien Harson qui confirme une grand année 2013.
| {
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namespace SPALM.SPSF.Library
{
using System;
using System.Collections.Generic;
using EnvDTE;
using SPALM.SPSF.Library.Editors;
public class WebTemplateEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/Templates/Template/Configuration", "", new XmlWebTemplateNodeHandler());
Helpers2.AddExternalItems(dte, list, "/SharePointConfiguration/SiteTemplates/SiteTemplate", "", new XmlNodeHandler("Title", "Id", "DisplayCategory", "Description"));
return list;
}
}
public class ListTemplateEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/ns:Elements/ns:ListTemplate", "http://schemas.microsoft.com/sharepoint/", new XmlInternalListTemplateNodeHandler("DisplayName", "Type", "FeatureId", "Description"));
Helpers2.AddExternalItems(dte, list, "/SharePointConfiguration/ListTemplates/ListTemplate", "", new XmlNodeHandler("DisplayName", "Type", "FeatureId", "Description"));
return list;
}
}
public class ListInstanceEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/ns:Elements/ns:ListTemplate", "http://schemas.microsoft.com/sharepoint/", new XmlInternalListTemplateNodeHandler("DisplayName", "Type", "FeatureId", "Description"));
Helpers2.AddExternalItems(dte, list, "/SharePointConfiguration/ListTemplates/ListTemplate", "", new XmlNodeHandler("DisplayName", "Type", "FeatureId", "Description"));
return list;
}
}
public class BCSModelEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/ns:Model/ns:LobSystems/ns:LobSystem/ns:Entities/ns:Entity", "http://schemas.microsoft.com/windows/2007/BusinessDataCatalog", new BCSNodeHandler(), ".bdcm");
return list;
}
}
public class FeatureEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/ns:Feature", "http://schemas.microsoft.com/sharepoint/", new XmlNodeHandler("Title", "Id", "Scope", "Description"));
Helpers2.AddInternalItems(dte, list, "/ns:feature", "http://schemas.microsoft.com/VisualStudio/2008/SharePointTools/FeatureModel", new XmlNodeHandler("Title", "Id", "Scope", "Description"), ".feature");
Helpers2.AddExternalItems(dte, list, "/SharePointConfiguration/Features/Feature", "", new XmlNodeHandler("Title", "Id", "Scope", "Description"));
return list;
}
}
public class SiteColumnEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/ns:Elements/ns:Field", "http://schemas.microsoft.com/sharepoint/", new XmlNodeHandler("Name", "ID", "", "Description"));
Helpers2.AddExternalItems(dte, list, "/SharePointConfiguration/Fields/Field", "", new XmlNodeHandler("DisplayName", "ID", "Group", "Description"));
return list;
}
}
public class ContentTypeEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/ns:Elements/ns:ContentType", "http://schemas.microsoft.com/sharepoint/", new XmlNodeHandler("Name", "ID", "Group", "Description"));
Helpers2.AddExternalItems(dte, list, "/SharePointConfiguration/ContentTypes/ContentType", "", new XmlNodeHandler("Name", "ID", "Group", "Description"));
return list;
}
}
public class ContentTypeEditorInternal : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, true, list, "/ns:Elements/ns:ContentType", "http://schemas.microsoft.com/sharepoint/", new XmlNodeHandler("Name", "ID", "Group", "Description"));
return list;
}
}
public class ContentTypeGroupEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/ns:Elements/ns:ContentType", "http://schemas.microsoft.com/sharepoint/", new XmlNodeHandler("Group", "Group", "", ""));
Helpers2.AddExternalItems(dte, list, "/SharePointConfiguration/ContentTypes/ContentType", "", new XmlNodeHandler("Group", "Group", "Group", ""));
return list;
}
}
public class CustomActionSitesEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/CustomActionLocations/ActionGroup", "", new XmlNodeHandler("GroupID", "Location", "Location", "Description"));
Helpers2.AddInternalItems(dte, list, "/ns:Elements/ns:CustomActionGroup", "http://schemas.microsoft.com/sharepoint/", new XmlNodeHandler("Id", "Location", "", "Title"));
return list;
}
}
public class DelegateControlEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
Helpers2.AddInternalItems(dte, list, "/ns:Elements/ns:Control", "http://schemas.microsoft.com/sharepoint/", new XmlNodeHandler("Id", "Id", "", ""));
AddRecipeParameters(provider, list, "/RecipeParameters/DelegateControlIds/DelegateControlId", "", new XmlNodeHandler("Value", "Value", "", "Description"));
return list;
}
}
public class CustomActionRightsEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/SPBasePermissions/SPBasePermission", "", new XmlNodeHandler("Name", "Name", "", "Description"));
return list;
}
}
public class CustomActionLocationsEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/CustomActionLocations/ActionGroup", "", new XmlNodeHandler("Location", "Location", "", "Location"));
return list;
}
}
public class CustomActionGroupLocationsEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/CustomActionGroupLocations/ActionGroup", "", new XmlNodeHandler("Location", "Location", "", "Location"));
return list;
}
}
public class CustomActionListToolBarsEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/CustomActionListToolBars/ActionGroup", "", new XmlNodeHandler("Location", "Location", "Location", "Description"));
return list;
}
}
public class CustomActionListEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/CustomActionList/ActionGroup", "", new XmlNodeHandler("GroupID", "Location", "Location", "Description"));
return list;
}
}
public class HideCustomActionLocationsEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/HideCustomActionIDs/ActionGroup", "", new XmlNodeHandler("ID", "Location", "GroupID", "Description"));
return list;
}
}
public class RibbonGroupEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/RibbonGroups/RibbonGroup", "", new XmlNodeHandler("Group", "Group", "Tab", "Group"));
return list;
}
}
public class RibbonControlEditor : TreeViewEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/RibbonControls/RibbonControl", "", new XmlNodeHandler("ID", "ID", "Group", "Tab"));
return list;
}
}
public class RibbonTabEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/RibbonTabs/RibbonTab", "", new XmlNodeHandler("Tab", "Tab", "Tab", "Tab"));
return list;
}
}
public class ProgIdEditor : ListEditor
{
public override List<NameValueItem> GetItems(DTE dte, IServiceProvider provider)
{
List<NameValueItem> list = new List<NameValueItem>();
AddRecipeParameters(provider, list, "/RecipeParameters/ProgIds/ProgId", "", new XmlNodeHandler("Value", "Value", "", "Description"));
return list;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,180 |
LAS High School Player Interview: Tod Phataraprasit, Suffield Academy
Editor's note: LAS often showcases highlight tapes of HS kids who have some talent and we try to promote players in areas where the game is still growing or in its infancy. We cover the big guys and the establishment as well, but we focus on growth and new areas. There are enough people doing the status quo! For this post, we've got an interview with a HS player with interesting roots and a bright future. Let's check it out!
Tod in action for the TLA.
We're talking with international HS talent, Tod Phataraprasit. We caught up with Tod to learn more about him, to hear more about the pressures of playing for his school, and also playing for his country. The Thai senior is an attackman for Suffield Academy, a prep school in Suffield, Connecticut. Although not traditionally known as a big lacrosse feeder school, Suffield has previously won it's league championship, and sent a handful of it's alums to the Division I and Division 3 (For example: Jake Deane who led UMass to the NCAA finals).
Tod Phataraprasit is originally from Thailand, and currently a member of Thailand National lacrosse team. Tod assisted Team Thailand to its initial 10-8 victory over the Hong Kong national team in their debut game last July. Tod started playing lacrosse at Eaglebrook, a small middle school in Deerfield, Massachusetts (some of it's alum includes Peter Striebel (Princeton '07) younger brother of MLL player, and U.S. national team player Matt Striebel). Tod then carried his love for the game of lacrosse to Suffield Academy, following the footpath of fellow alums, and national team teammates such as Payu Nerngchamnong (A), and Peem Chatikawanij (G).
"When I first met Tod, I didn't know much about him. He was a really quiet, and modest guy trying out for the team." said the President of Thailand Lacrosse Association, Payu Nerngchamnong. "We weren't sure how he would fair compare to the other 4 attackmen on the team, since alot of them had significantly more playing experience than Tod. However, he continued to improve, and shined during our practices. As Tod gained his confidence, so did his game on the field. He became a valuable asset to our team during the game against Hong Kong, as his determination and fitness was put to the test during the 80 minutes of lacrosse. Tod is not a flashy type of attackman, but he is consistent, and very disciplined to the task at hand. You won't see him making any mental errors, like throwing a bad pass or giving up a turn over. If he doesn't have the opportunity to go to the goal, he will certainly create one for his teammate. Tod will be a great attackman, and a strong asset to our team comes Denver 2014," Nerngchamnong continued.
Tod in action at Suffield Academy
We also asked David Pillsbury, the head coach of Suffield Varsity Lacrosse team, about Tod. David had this to say about the young attackman: "Each year Tod Phataraprasit has grown as a player, but more so this year. Playing for his national team instilled a great sense of pride and confidence in him." Coach Pillsbury added, "Though we have not started our lacrosse season at Suffield, I was able to see a more confident athlete on the soccer field this fall. I look forward to coaching Tod in the spring and can't wait to see some tricks and skills he picked up from playing at such a high level this summer. "
More action against Hong Kong.
Where do you want to go to college, and why? What do you plan to major in?
Tod Phataraprasit: There are several, but I want to be realistic about my chances, since playing in college had become very competitive in recent years, and not just at the top level. I am looking at schools in the Boston area, but I am not sure where yet. I would like to major in Business Management, and Entrepreneurial Studies or Finance. I've looked at Emmanuel College among others, and they have recently started their lacrosse program. I think it would be a really interesting experience to be a part of helping develop the new program, as I did with Thailand Lacrosse from the very beginning. I'm still leaving my options open, balancing my decision between education and lacrosse.
What steps are you currently taking to get into those colleges?
Tod Phataraprasit: Education is my top priority, so I've been working hard in classes to get admirable grades. I've played, and trained with the national team over the summer. I played soccer in the fall, and during the off season I go to the gym to get ready for the upcoming spring season. I will also be taking part in an indoor lacrosse league during the winter semester so I don't lose my touch. I will be heading back to Thailand during the winter break to continue my training, and practices with the national team.
Have you been getting noticed by coaches?
Tod Phataraprasit: Not really, I never thought I'd have the ability to play in college but my coaches have been very supportive, and told me that I should really consider it. I have been in contact with several coaches so far, and hopefully it all goes well! It's always been difficult for international players to get noticed by college coaches, since we normally have to go home for the summer. And we do not have a lot of chances to go to recruiting camps, or be a part of a summer tournament teams.
How has your HS lax experience been? And what is it like playing at the International level vs. HS level?
Tod Phataraprasit: My experience has been great, I have improved significantly over the years, and gained a lot of knowledge for the game. Lacrosse never stopped growing on me since day one, and I've developed a true passion for the game. Playing at the high school level sure is different from playing at the international level, especially since I haven't even play at the college level yet. Everything is different, the rules, and in all aspect actually.
For example; we normally take a bus to our away game in High school, but at the international level we have to FLY everywhere. Representing your school, and your country are equally as important but very much different. Obviously, playing for my school is an honor, but the feeling I got from having my country's name and the flag on my uniform is just simply amazing. I'm looking forward to play for Thailand in our team's first international tournament, for the ASPAC Championship in New Zealand this upcoming July.
What's your ultimate lacrosse aspiration?
Tod Phataraprasit: I hope to represent my country at the FIL World Game in 2014 in Denver. I definitely want to help Payu spread the sport in Thailand first, and set a good example for the next generation of Thai lacrosse players, so that Thailand can become another lacrosse nation, and make lacrosse a truly global sport… Which I hope will make the rest of the world, and the IOC, finally recognize lacrosse as a sport that should be in the Olympics.
The Thai team huddles up.
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NextGame Week: Johns Hopkins DocumentaryNext | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,719 |
I like the imageries… Time crawls on millipede feet. Love it!!!
time can be slow for those who wait!
Oh, this one is sad, so sad, and heavy. Loved it in your book. The character waits for her unfaithful husband, in a combination of impatience and dread of hearing the same lies/excuses again. One wonders (or at least I do) if someday she will demand the truth from him, or if he will ever honor her feelings enough to tell her the truth. So sad for both of them, though my sympathies lie more with the woman;although both are wounded in this case.
yes, indeed, and sadly so. The reality of sad lives trapped in a relationship where the bond is almost dead and where the self concept and esteem of the woman are the now the principal victims. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,310 |
We wish to introduce ourselves as a caring organization established with a main motive of helping the special children of our loving nation. The special children whom we are gifted to take care are spastic, blind, dumb, Down syndrome, physically handicapped and mentally retarded. They are multi racial aging between 14 to 54 years.
This Centre was inaugurated in May 1997, with 5 children initially. Currently we are taking care of 27 inmates. The Pusat Penjagaan Kanak-Kanak Terencat Akal Kuantan is in a single story rented house, children are permanently staying here. Our objective is to run this home on long term basis to enable these children to be independent when they grow up into adults in the coming future. We organize activities for the inmates OFF and ON to keep them happy and grow. In this way we are trying to help the community and the government to care for the Handicapped and mentally retarded which is also a service to mankind.
we are Tax Exempted Non Profit Organization approved by LHDN - Our Tax Ref: LHDN.01/35/42/51/179-6.5616. | {
"redpajama_set_name": "RedPajamaC4"
} | 794 |
package org.jhotdraw.xml.css;
import java.util.*;
import net.n3.nanoxml.*;
import org.jhotdraw.util.ReversedList;
import org.w3c.dom.Element;
/**
* StyleManager applies styling Rules to an XML DOM.
* This class supports net.n3.nanoxml as well as org.w3c.dom.
*
* @author Werner Randelshofer
* @version $Id: StyleManager.java 718 2010-11-21 17:49:53Z rawcoder $
*/
public class StyleManager {
private java.util.List<CSSRule> rules;
public StyleManager() {
rules = new ArrayList<CSSRule>();
}
public void add(CSSRule rule) {
rules.add(rule);
}
public void applyStylesTo(Element elem) {
for (CSSRule rule : rules) {
if(rule.matches(elem)) {
rule.apply(elem);
}
}
}
public void applyStylesTo(IXMLElement elem) {
for (CSSRule rule : new ReversedList<CSSRule>(rules)) {
if(rule.matches(elem)) {
//System.out.println("StyleManager applying "+rule+" to "+elem);
rule.apply(elem);
}
}
}
public void clear() {
rules.clear();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,248 |
Q: javascript adding multiple variables to mailto body with a for loop This function below is supposed to add some of the variables "mess" into the body of an email message. The "who" and "what" gets read in the email address and subject ok, and the first line of the mess variables "ISBN" gets read into the body as it should...but the rest of the mess variables that should be read into the body does not get entered. Wondering if the for loop is not looping correctly. Just trying to narrow the problem down. Any help would very nice.
function mailOrder()
{
who=document.order.Email.value;
what="order";
var mess = "";
for (var n = 0; n < bookDB.length; n++)
{
bookNum = bookDB[n].quantity;
if (bookNum > 0)
{
mess += "ISBN: " + bookDB[n].number + " ";
mess += "Book Name: " + bookDB[n].title + " ";
mess += "Quantity Ordered: " + bookDB[n].quantity + " ";
mess += "Book Price: " + bookDB[n].price + " ";
mess += " ***************************************************** ";
}
}
mess +="Customer Name: " + document.order.Name.value + " ";
mess +="Email Address: " + document.order.Email.value + " ";
mess +="Address: " + document.order.Address.value + " ";
mess +="City: " + document.order.City.value + " ";
mess +="Country: " + document.order.Country.value + " ";
parent.location.href='mailto:'+who+'?subject='+what+'&body='+mess;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,039 |
\section{Introduction}\label{intro}
The study of steganography is perhaps best motivated by considering an example. Suppose two political protestors Alice and Bob are arrested and put into two widely separated jail cells. The warden allows them to communicate with hand-written letters that he reads before delivering. However, if the warden reads anything in the letters that he finds suspicious (such as a possible escape plan), then he will not deliver the letter. Luckily, Alice and Bob exchanged a secret key before their incarceration. Can Alice and Bob communicate their escape plan to each other without arousing the warden's suspicions? This is where the study of steganography comes into play.
The science of sending information through seemingly innocuous messages has a long history, dating back to at least 440 B.C.\cite{greekstego}. It is worth making clear its differences from cryptography. In cryptography, a secret message (the {\it plaintext}) is encrypted using the shared secret key, and the resulting {\it ciphertext} is then sent to the desired receiver to be decoded. If an eavesdropper (Eve) observes the ciphertext, she cannot decode it without the secret key. However, she will know that there {\it is} a secret message, since Alice is sending apparent gibberish to Bob.
By contrast, if Alice uses a steganographic encoding, she hides the secret message (or {\it stegotext}) into a larger {\it covertext}, which appears to Eve as an innocuous message. The hidden message may or may not be encrypted itself, but the main line of defense is that the eavesdropper is unaware that a message is even being sent.
During WWII, a Japanese spy named Velvalee Dickinson sent classified information to neutral South America. She was a dealer in dolls, and her letters discussed the quantity and type of doll to ship. The covertext was the doll orders, while the concealed stegotext was encoded information about battleship movements \cite{ww2}.
The quantum analogue of cryptography has been widely studied \cite{norbertcrypto}. However, the quantum analogue of steganography is still in a relatively early stage. There have been a number of different proposals for encoding quantum information steganographically, or encoding classical information into quantum states or channels \cite{natoristego,banerjeestego}. In this paper we consider hiding secret messages as error syndromes of a quantum error-correcting code \cite{gea2002hiding}. This approach to quantum steganography has been studied in detail by Shaw and Brun, with explicit encoding and decoding procedures and calculated rates of communication and secret key consumption \cite{shaw2011quantum,shaw2010hiding}. It was shown that such schemes can hide both quantum and classical information, with a quantitative measure of secrecy, even in the presence of a noisy physical channel. When the error rate of the physical channel is lower than the eavesdropper's expectation, it is possible to achieve non-zero asymptotic rates of communication. (If the eavesdropper has exact knowledge of the channel, secret communication may still be possible, but the amount of secret information that can be transmitted in general grows sublinearly with the number of channel uses.)
More recently, a closely related idea has been studied under the name of quantum covert communication \cite{qcovert1,qcovert2,qcovert3,qcovert4,qcovert5}. Many of the ideas in this paper are closely related to steganographic requirements, such as secrecy and recoverability. This is not surprising, since covert quantum communication can be seen as a special case of quantum steganography over noisy quantum channels in the case when the eavesdropper has exact knowledge of the channel, and where Eve assumes the channel is idle (so only noise is being transmitted). Similarly, quantum steganography is a type of covert quantum communication where Eve knows about the covertext communication but not the hidden stegotext, and where Eve may not have perfect knowledge of the channel. The work on covert communication has generally found that, if Eve has exact knowledge of the channel, the amount of secret communication that can be done grows like the square root of the number of channel used.
The goal of this paper is to formalize the assumptions and reasonable conditions of quantum steganography introduced in \cite{shaw2011quantum}, and to give upper bounds on the achievable rates of quantum communication while remaining secure from an eavesdropper's suspicion, for the special case when the true underlying channel is noiseless. Our results include achievability results as well as converse proofs for quantum steganography.
In Section \ref{qstego} we formalize our notion of quantum steganography where secret messages are hidden in the syndromes of an error-correcting code, and outline a specific steganographic encoding where Alice is able to emulate any general quantum channel $\mathcal{N}$ on her encoded secret message and covertext. We work out specific examples for the bit-flip channel and the depolarizing channel, before giving the more general result. In Section \ref{noiseless} we prove upper bounds on the amount of steganographic communication possible, and show that these bounds are asymptotically equal to the rates achieved in the previous section.
The assumption that the physical channel is noiseless greatly simplifies the analysis. However, we believe that the main intuition underlying this approach will apply equally well in the case of a noisy channel. We will end this paper with a discussion of how to extend this work to the case where the physical channel between the two parties is noisy.
\section{Quantum Steganography: Achievability}\label{qstego}
As discussed in the introduction, there have been several approaches to generalizing steganography to the quantum setting. Here we will make explicit the notion of quantum steganography based on syndromes of quantum error-correcting codes. We assume that Eve expects to see quantum information passing through a noisy quantum channel. However, the actual physical channel is assumed to be noiseless. This is obviously an idealized assumption, which greatly simplifies the analysis; we will discuss below how it might be justified at least as an approximation.
Alice wants send a secret message steganographically to Bob. Using her shared secret key, she encodes the stego text into a codeword of a quantum error-correcting code (QECC) with errors applied to it, and sends it to Bob. The codeword encodes an innocent state; the stego text is conveyed in the errors. If Eve were to perform measurements on this codeword, it would be indistinguishable from an innocent encoded covertext that had passed through a given noisy quantum channel to Bob.
Before discussing how to quantify the security of a quantum steganographic protocol, let us make clear what Alice is trying to achieve. Alice wants to encode an innocent covertext state, together with her secret message, into an $N$-qubit codeword in such a way that it cannot be distinguished from the covertext alone encoded into a quantum error-correcting code that has undergone typical errors induced by the quantum channel $\mathcal{N}^{\otimes N}$. The steganographic encoding works by mapping all possible secret messages onto syndromes of the QECC. This encoding is not limited to classical messages: it is possible to encode a quantum state by preparing the codeword in a superposition of different error syndromes.
In analyzing this quantum steganography protocol, we make the following assumptions. Alice is communicating with Bob by a quantum channel that is actually noiseless. But the eavesdropper, Eve, believes that this channel is noisy, perhaps because Alice and Bob have been systematically making the channel appear noisier than it actually is. Because Alice and Bob have been systematically deceiving Eve in this way, we assume that they know (at least fairly closely) what Eve's estimate of the channel is. Before the protocol began, Alice and Bob shared with each other a secret key: an arbitrarily long string of random bits. This key is known only to the two of them. But once the protocol begins, they cannot communicate except through channels that can be monitored by Eve. Alice sends an innocent-looking message to Bob over the channel. This is a covertext state $\rho_c$, encoded into an error-correcting code; it is assumed that the choice of code is known to Eve, and this code should be a plausible choice for the noisy channel that Eve believes exists.
One important caveat for this section: we will be considering the case where the QECC that Alice uses is nondegenerate. That is, each typical error corresponds to a unique error syndrome. This allows Alice to communicate as much steganographic information as possible, and it allows us to ignore the details of which QECC is being used. Methods similar to those in this section should also work for degenerate codes; but in that case, the encoding will be strongly dependent on the properties of the particular code, since the typical errors must first be grouped into equivalent sets, and then the possible messages mapped into these sets. We also use this assumption in the next section to get specific expressions for the upper bound on the secret communication rate.
To clarify how the encoding works, we start by considering two examples for relatively simple channels: first, the case where Alice is emulating a bit flip channel $\mathcal{N}^{BF}_{p}$ on the codeword, and second, the case where she is emulating the depolarizing channel. Finally we consider a more general error map $\mathcal{N}^{\otimes N}$. The message qubits are encoded into into the error syndromes of the codeword of the QECC she is using.
\subsection{The Bit Flip Channel}
Suppose that Eve believes the channel connecting Alice and Bob to be a bit flip channel, with a probability $p$ of error per qubit sent. (The actual physical channel is noiseless, as assumed above.) Alice sends a codeword of length $N$ to Bob, encoding some ``innocent'' covertext state $\rho_c$. The errors in the codewords that Alice sends to Bob should be binomially distributed: $pN$ is the mean number of errors of this distribution, and the variance is $(1-p)pN$. The total probability that there is an error of weight $w$ on the codeword should be
\begin{equation}
p_{k}={N \choose w}p^{w}(1-p)^{N-w} .
\label{eq:binomial_dist}
\end{equation}
There are
\[
{N \choose w} \equiv \frac{N!}{w! (N-w)!}
\]
such errors, all with equal probability $p^w (1-p)^{N-w}$.
If $N$ is large, then it is extremely likely that the number of bit flips will be a {\it typical} error---that is, an error of weight $w$ within a narrow range about the mean $pN$. Alice's encoding will make use of these typical errors. For each $w$ from $Np(1-\delta)$ to $Np(1+\delta)$, where $\sqrt{(1-p)/pN} \ll \delta \ll 1$, Alice chooses at random a set of $C_w$ possible error strings of weight $w$. (An {\it error string} of weight $w$ is a string of $N$ bits, with a 1 at every location with a bit flip and 0 at every location with no error.) This random choice is made using the shared secret key with Bob, so that Bob also knows which set of errors is being used to encode secret messages, but Eve (who does not share the key) could not know this.
Let these sets of error strings of weight $w$ be called $\{S_{w}\}$, and the set of all strings used in the encoding is
\begin{equation}
S=\bigcup\limits_{w}S_{w}.
\end{equation}
We sum up
\begin{equation}
C=\sum_{w=Np(1-\delta)}^{Np(1+\delta)} C_{w} = |S| .
\end{equation}
So the total number of strings in the set $S$ is $C$. This number $C$ is the total number of possible distinct secret messages that Alice can send to Bob (though she may also send {\it superpositions} of these messages). We assume all these messages to be equally likely. So the message encodes $M = \log_2 C$ bits (or qubits) of information.
Define the probability $q=1/C$. These error strings $S$ are typical strings (using the definition of weak typicality from information theory). Eve should not be suspicious at seeing such an error string, since it matches a probable result for the channel that she expects. For this encoding to be indistinguishable from the bit flip channel, the probability of the message being an error string of weight $w$ should equal the value from the distribution in Eq.~(\ref{eq:binomial_dist}) above. This means we want to satisfy
\begin{equation}
qC_{w} = \frac{C_{w}}{C} = p_{w} .
\end{equation}
Clearly we must have
\[
C_{w}\leq {N \choose w} ,
\]
for all $w$ in the typical range. This implies that:
\begin{align}
C_{w}p^{w}(1-p)^{N-w} \leq {N \choose w}p^{w}(1-p)^{N-w} &= C_{w} q , \nonumber \\
\Rightarrow p^{w}(1-p)^{N-w} &\leq q .
\label{eq:bitflipconstraint}
\end{align}
To communicate the maximum amount of information steganographically we want $C$ to be as large as possible, which means we want $q$ to be as small as possible. The constraint in Eq.~(\ref{eq:bitflipconstraint}) then gives us
\begin{equation}
q=p^{Np(1-\delta)}(1-p)^{N(1-p+p\delta)} .
\end{equation}
So Alice can send $M$ stego qubits to Bob, where
\begin{align}
M =& \log_{2}C = \log_{2}1/q \nonumber \\
=& N(-p\log_{2}p-(1-p)\log_{2}(1-p) \nonumber \\
&+\delta(p\log_{2}p-p\log_{2}(1-p))) \nonumber \\
=& N(h(p)-\delta p\log_{2}((1-p)/p)) \nonumber \\
\approx& Nh(p) ,
\end{align}
where $h(p)=-p\log_{2}p-(1-p)\log_{2}(1-p)$ is the entropy of the bit flip channel on one qubit. Therefore, with this encoding Alice can send almost $Nh(p)$ bits.
In \cite{shaw2011quantum} it is shown that the diamond norm distance between the channel $(\mathcal{N}_{p}^{BF})^{\otimes N}$ and Alice's encoding is exponentially small in $N$. This justifies the claim that this protocol will not arouse suspicion from Eve. In section III we use a slightly modified definition of secrecy that allows us to prove the converse bound on this rate of stego communication by information theoretic techniques. That means that this encoding is essentially optimal: the maximum rate of steganographic communication for a nondegenerate code in the case of the bit flip channel is $h(p)$.
\subsection{Depolarizing Channel}
Here we will consider the scenario where the channel Alice is emulating is the depolarizing channel. It turns out that due to the symmetric nature of the depolarizing channel the encoding looks quite similar to that of the bit flip channel. Recall that the depolarizing channel acting on a single qubit $\rho$ is given by
\begin{equation*}
\mathcal{N}^{DC}_{p}(\rho) = (1-p)\rho + (p/3)(X\rho X+Y\rho Y+Z \rho Z).
\end{equation*}
Applying this channel on $N$ qubits, the total probability of all errors with exactly $n_{1}$ $X$, $n_{2}$ $Y$, and $n_{3}$ $Z$ errors (and $n_{4} = N - n_1 - n_2 - n_3$ identity ``errors'') is
\begin{equation*}
p(n_{1},n_{2},n_{3},n_{4})=\frac{N!}{n_{1}!n_{2}!n_{3}!n_{4}!}(p/3)^{n_1+n_2+n_3}(1-p)^{n_4}.
\end{equation*}
Notice that instead of specifying $n_{1}, n_{2},$ and $n_{3}$ exactly, we can instead talk about errors with weight $w=n_{1}+n_{2}+n_{3}$. It follows by simple calculation that the total probability of all errors of weight $w$ is
\begin{equation*}
p(w)=3^w{N\choose w}(p/3)^w(1-p)^{N-w}={N \choose w}p^w(1-p)^{N-w},
\end{equation*}
which is just a binomial distribution in $w$. As in the bit flip case, we will need say what strings of errors are typical. There are a number of ways we could specify this, but for simplicity we will consider weights $w$ that lie between $Np(1-\delta)$ and $Np(1+\delta)$ for $\sqrt{(1-p)/pN}\ll\delta\ll 1$. The astute reader will notice that this set includes some errors that are not typical: for instance, it includes errors of weight $w$ where all (or most) of the errors are $X$'s and none (or few) are $Y$'s or $Z$'s. If such errors are used as codewords, they might make Eve suspicious. Still, the effect of this is not too large, because this set is still dominated by typical errors, and the probabilities of these strings are similar to the expected probabilities of atypical errors. With this definition of typicality, we can follow the exact same encoding given for the bit flip code using errors with weight $w$, except that the set of errors of weight $w$ is now of size
\[
\left(\begin{array}{c} N \\ w \end{array}\right) 3^w ,
\]
and errors of weight $w$ have probability $(p/3)^w (1-p)^{N-w}$. This then leads to the following encoding rate:
\begin{align}
M &= N(-p\log_{2}(p/3)-(1-p)\log_{2}(1-p) \nonumber \\
& +\delta(p\log_{2}(p/3)-p\log_{2}(1-p))) \nonumber \\
&= N(s(p)+\delta(p\log_{2}(p/3)-p\log_{2}(1-p)) \nonumber \\
&\approx Ns(p)
\end{align}
where we have defined $s(p)=-p\log_{2}(p/3)-(1-p)\log_{2}(1-p)$ to be the entropy of the depolarizing channel on one qubit.
\subsection{General Channels}
\subsubsection{Special case: random unitaries}
Consider a quantum channel acting on a single qubit of the form
\begin{equation}\label{eq:randomUnitaryChannel}
\mathcal{N}(\rho) = \sum_{i=1}^k p_{i} U_{i} \rho U_{i}^{\dagger} ,
\end{equation}
where the operators $U_i$ are all unitary, so $U_i U_i^\dagger = U_i^\dagger U_i = I$. The set of Kraus operators $\{\sqrt{p_i} U_i\}$ can be thought of as a set of possible single-qubit unitary errors $U_i$ that occur with probability $p_i$. Note that both the bit-flip and depolarizing channels are special cases of the random unitary channel, as is any Pauli channel. The channel acts on an $N$-qubit encoded state $\rho$ as $\mathcal{N}^{\otimes N}(\rho)$.
The total probability of all errors with $n_{1}$ $U_{1}$ errors, $n_{2}$ $U_{2}$ errors, and so forth, is given by the multinomial distribution:
\begin{equation}
p(n_{1},\ldots,n_{k})=\frac{N!}{n_{1}!\cdots n_{k}!}p_{1}^{n_{1}}\cdots p_{k}^{n_{k}}.
\end{equation}
Now consider weights $n_{j}$ in the range from $Np_{j}(1-\delta)$ to $Np_{j}(1+\delta)$, where $\delta$ is large enough that this set includes all the typical strings. (This definition can be modified, but for simplicity we stick with it in this paper.) Randomly choose $C_{n_{1},\ldots,n_{k}}$ strings with weights $n_1,n_2,\ldots,n_k$ in this range such that $n_{1}+\ldots+n_{k}=N$. As with the bit flip and depolarizing channels, let these sets of strings be called $S_{n_{1},\ldots,n_{k}}$ and let $S$ denote the union of all these sets of strings, which are a subset of the typical strings. For all weights $n_1,\ldots,n_k$ outside the typical set, we let $C_{n_{1},\ldots,n_{k}}=0$. The total number of strings in the set $S$ is $C$:
\begin{equation}
C = \sum_{n_1,\ldots,n_k} C_{n_{1},\ldots,n_{k}} .
\end{equation}
Defining $q \equiv 1/C$, we want to satisfy
\begin{equation}
C_{n_{1},\ldots,n_{k}}q=C_{n_{1},\ldots,n_{k}}/C=p(n_{1},\ldots,n_{k})
\end{equation}
for all weights $n_1,\ldots,n_k$ in the typical set, so that Eve does not become suspicious. Also, clearly $C_{n_{1},\ldots,n_{k}}$ must be less than $\frac{N!}{n_{1}!\cdots n_{k}!}$. This implies that:
\begin{align}
&C_{n_{1},\ldots,n_{k}}p_{1}^{n_{1}}\cdots p_{k}^{n_{k}}\leq\frac{N!}{n_{1}!\cdots n_{k}!}p_{1}^{n_{1}}\cdots p_{k}^{n_{k}} \nonumber \\
&C_{n_{1},\ldots,n_{k}}p_{1}^{n_{1}}\cdots p_{k}^{n_{k}}\leq C_{n_{1},\ldots,n_{k}}q \nonumber \\
&p_{1}^{n_{1}}\cdots p_{k}^{n_{k}}\leq q.
\end{align}
Notice that this time we cannot simply plug in the lower bounds of the sums for $n_{j}$, as we did for the depolarizing and bit flip channels, because we have the additional constraint that $n_{1}+\ldots+n_{k}=N$. However, the same general argument applies. Inside the set of typical weights, there is a string $\tilde{n}_1,\cdots,\tilde{n}_k$ with $|\tilde{n}_j/N - p_j| \le \delta p_j$ for all $j$, that maximizes the probability:
\begin{equation}
p_{\rm max} \equiv p_1^{\tilde{n}_1} p_2^{\tilde{n}_2} \cdots p_k^{\tilde{n}_k} .
\end{equation}
We can choose $q=p_{\rm max}$, and use this to put a bound on the number of stego qubits M Alice can send to Bob:
\begin{align}
M&=\log_{2}C =-\log_{2}(q) = -\log_2 p_{\rm max} \nonumber\\
&= -\tilde{n}_{1}\log_{2}(p_{1})-\ldots-\tilde{n}_{k}\log_{2}(p_{k}) \nonumber \\
&= N(-\frac{\tilde{n}_{1}}{N}\log_{2}(p_{1})-\ldots-\frac{\tilde{n}_{k}}{N}\log_{2}(p_{k})) \nonumber \\
&\ge N(1-\delta)(-\sum_{i=1}^{k}p_{i}\log_{2}(p_{i})) \nonumber\\
&= N(1-\delta)H(p_1,\ldots,p_k) .
\label{eq:randomUnitariesRate}
\end{align}
So in the limit of large $N$, we should approach a rate $H(p_1,\ldots,p_k)$ with this encoding.
\subsubsection{Encoding general channels across multiple code blocks}
This argument does not necessarily apply directly to a general quantum channel, since the probabilities of the different outcomes can be state dependent. However, we should be able to do a similar type of encoding for a general quantum channel $\mathcal{N}$ by encoding across multiple code blocks. Consider a general quantum channel acting on a single qubit as
\begin{equation}\label{eq:onequbitchannel}
\mathcal{N}(\rho) = \sum_{i=1}^k A_{i} \rho A_{i}^{\dagger} .
\end{equation}
The channel acts on an $N$-qubit encoded state $\rho$ as $\mathcal{N}^{\otimes N}(\rho)$, where we will let $N$ become large. For most states $\rho$, we can well approximate this $N$-qubit channel by a sum over the {\it typical} errors \cite{Klesse07,Klesse08},
\begin{equation}\label{eq:Nqubitchannel}
\mathcal{N}^{\otimes N}(\rho) \approx \sum_{\underline{i}\in \mathcal T} E_{\underline{i}} \rho E^\dagger_{\underline{i}},
\end{equation}
where $\rho$ is now the $N$-qubit codeword, the index is $\underline{i} = i_1i_2\ldots i_N$, the typical error $E_{\underline{i}}$ is
\begin{equation}
E_{\underline{i}} = A_{i_1} \otimes A_{i_2} \otimes \cdots \otimes A_{i_N} ,
\end{equation}
and $\mathcal{T}$ is the set of typical sequences $\underline{i}$ \cite{wilde2013quantum}.
We assume that the QECC Alice uses is one that can correct the typical errors of the channel. (Indeed, using a code that was not strong enough to correct the typical errors might well arouse Eve's suspicions.) We will also assume, for simplicity of this analysis, that the QECC is {\it nondegenerate}. This means that on a valid codeword in the QECC, the typical errors $E_{\underline{i}}$ all have distinct error syndromes, and act as unitaries that move the state to a distinct, orthogonal subspace labeled by $\underline{i}$. This means that error $E_{\underline{i}}$ occurs with a fixed probability $p_{\underline{i}}$ for all valid codewords of the QECC.
We can then essentially repeat the argument that leads to Eq.~(\ref{eq:randomUnitariesRate}), but now using the probabilities $p_{\underline{i}}$. Note that we now need to take two limits: the limit of many blocks, and also the limit where the individual blocks are large. For this argument to apply, we need to first go to the limit of many blocks, and then to the limit of large block size. In those limits, we can approach a rate
\begin{equation}\label{eq:effectiveEntropy}
- \frac{1}{N} \sum_{\underline{i}} p_{\underline{i}} \log_2 p_{\underline{i}} \equiv \bar{H} ,
\end{equation}
where $\bar{H}$ is an effective entropy per qubit from the channel.
Note that there are some ambiguities in making this argument. The Kraus map in Eq.~(\ref{eq:onequbitchannel}) is not unique. Choosing different sets of Kraus operators will lead to different sets of typical errors. However, these differences should not lead to significant changes to the effective entropy in the limit of large block size, so long as the code is nondegenerate on both sets of typical errors.
\subsection{Secret key consumption}
For the above encodings, how much secret key must be consumed? In general, we can assume that all the details of the encoding, etc., have been decided between Alice and Bob ahead of time. So in the protocol as described above, the only place where secret key is consumed is to pick the subsets of errors used in the encoding.
Let's consider the bit flip channel as a simple example. The possible messages are mapped onto a set of $C$ error syndromes, representing errors of weights $(1-\delta)Np \le w \le (1+\delta)Np$. For each error weight $w$ in that range, a subset of $C_w$ errors is chosen to represent possible messages. Alice and Bob can agree before the protocol begins to divide the set of errors of weight $w$ into $n_w$ nonoverlapping subsets of $C_w$ errors each, where
\begin{equation}
n_w = \left(\begin{array}{c} N \\ w \end{array}\right)/C_w
= \left(\frac{1-p}{p}\right)^{w-Np(1-\delta)} .
\end{equation}
(Since this is unlikely to be an exact integer, one must generally round down, which means that a small fraction of possible errors will be omitted. This will slightly reduce the match between the steganographic encoding and the noisy channel being simulated, but for large $N$ and $p\ll 1$ the difference will be small.)
For each transmitted block, Alice and Bob must randomly choose one of these $n_w$ subsets for each weight $w$ in the typical range. Choosing a subset requires $\log_2 n_w$ random bits, which are drawn from their shared key. However, since any given message is encoded as an error of some specific weight $w$, Alice and Bob can reuse the same secret key bits to choose the subset for each error weight $w$. So the number of key bits consumed to transmit one block is equal to the maximum value of $\log_2 n_w$ for $(1-\delta)Np \le w \le (1+\delta)Np$, which is
\begin{eqnarray}
K &=& \max_{Np(1-\delta) \le w \le Np(1+\delta)} \log_2 n_w \nonumber\\
&=& \max_{Np(1-\delta) \le w \le Np(1+\delta)} \log_2 \left(\frac{1-p}{p}\right)^{w-Np(1-\delta)} \nonumber\\
&=& (2Np\delta) \log_2 \left(\frac{1-p}{p}\right) .
\end{eqnarray}
How does this scale with $N$? Since this is a binomial distribution, $\delta$ will take the form
\begin{equation}
\delta = D \sqrt{\frac{1}{N} \left(\frac{1-p}{p}\right) } ,
\end{equation}
where $D$ is a fixed constant determining what fraction of all errors are included in the typical set. The key consumption therefore is
\begin{equation}
K = 2D \sqrt{N\left(\frac{1-p}{p}\right)} \log_2 \left(\frac{1-p}{p}\right) .
\end{equation}
The key consumption scales sublinearly with $N$, and asymptotically the key consumption rate goes to zero. While the details will vary, we expect this kind of sublinear scaling of $K$ with $N$ to be generic.
A few words more on secret key consumption are in order. In \cite{shaw2011quantum}, Shaw and Brun make a distinction between the {\it secrecy} and the {\it security} of a steganographic protocol. A steganographic protocol is {\it secret} if an eavesdropper without the secret key cannot distinguish between an encoded message being sent and the noisy channel being applied. It is {\it secure} if the eavesdropper cannot learn anything about the message, even if she knows that a message is begin sent.
Using a sublinear amount $K$ of shared secret key is sufficient to make the steganographic protocol secret, by this definition. However, it is {\it not} secure, in general. Since the number of qubits $M$ transmitted is typically larger than the number of secret key bits $K$ consumed, we would generically expect an eavesdropper to be able to learn on the order of $M-K$ bits of information about the message if she became aware of its existence.
This can be prevented by first encrypting the message before doing the steganographic encoding. Encryption requires $M$ bits of secret key in the case of a classical message (using a one-time pad), or $2M$ bits of secret key in the case of a quantum message (by twirling). In this case, the protocol is both secret {\it and} secure. However, there is a cost: the secret key is now consumed asymptotically at a linear rate.
\section{Secrecy, Reliability, and Bounds}\label{noiseless}
\subsection{The information processing task}
Here we consider the steganographic scenario as outlined above where Alice is using fake noise to hide her message from Eve, but the actual physical channel she is sending her information over is noiseless. We will consider the task known as {\it entanglement transmission}. This notion of quantum communication encompasses other quantum information-processing tasks such as mixed-state transmission, pure-state transmission, and entanglement generation. We follow closely the discussion of quantum communication in \cite{wilde2013quantum}.
\begin{figure}
\includegraphics[width=0.5\textwidth,height=0.5\textheight,keepaspectratio]{stegodrawingnoiseless.png}
\caption{The information processing task we consider for Alice sending $M$ stego qubits to Bob over a quantum channel (which is identity for the noiseless case). Alice encodes her message $M$ and an innocent covertext $\rho_c$ into a suitable quantum error-correcting code which has had typical errors applied to it, where the encoding depends on the secret key $k$. She sends this to Bob, who then decodes the message and covertext using his copy of the shared secret key $k$. Alice's message is entangled with a reference system $R$. The ability to transmit entanglement implies the ability to do general quantum communication.}
\label{fig:protocol}
\end{figure}
The information processing task we are considering is visualized in Figure \ref{fig:protocol}. Alice has a secret message of $M=\log_{2}|A_{1}|$ qubits, which is maximally entangled with a reference system $R$. She also prepares an innocent covertext $\rho_{c}$ which will be encoded into the $N$-qubit quantum error-correcting code. Let us first define her encoded state, dependent on the secret key element $k$:
\begin{equation}
\omega_{k,A'^{n}R}\equiv \mathcal{E}_{k,A_{1}C\rightarrow A^{'n}}(\rho_{c}\otimes\Phi_{A_{1}R}) .
\end{equation}
This dependence of the encoding on the secret key corresponds to choosing among the different sets of error strings $S$ in the protocols from the previous section. To someone (like Eve) who does not know the secret key $k$, the state is effectively
\begin{equation}
\omega_{A'^{n}R}\equiv\sum_{k} p_k \omega_{k,A'^{n}R} ,
\end{equation}
where $\omega_{A'^{n}R}$ is the state averaged over all possible values of the secret key $k$ with probabilities $p_k$. (We can choose this probability to be uniform for simplicity, $p_k = p$ for all $k$, if we so desire.)
What is a good way to guarantee secrecy from Eve? We propose the following {\em secrecy} condition:
\begin{equation}\label{eq:stegrlx}
\frac{1}{2}\|\Tr_{R}(\omega_{A'^{n}R})-
\mathcal{N}^{\otimes N}(V\rho_{c}V^{\dagger})\|_{1}\leq \delta
\end{equation}
where $\mathcal{N}$ is whatever channel Alice is emulating, $V$ is an isometry representing the encoding of the covertext into a suitably chosen codeword (one which can correct typical errors induced by the channel $\mathcal{N}$) and $\delta>0$ is some small parameter. What this condition says is that if Eve observes the quantum state, it will be effectively indistinguishable from an encoded covertext being sent through the noisy quantum channel $\mathcal{N}$.
We introduce another requirement which corresponds to a notion of {\em recoverability}. Once Bob receives the state, he applies his decoder $\mathcal{D}_{k,A'^{n}\rightarrow B_{1}C}$ to obtain the original $\rho_{c}\otimes\Phi_{B_{1}R}$. We can relax this by only requiring that the input states and output states are $\epsilon$ close, that is:
\begin{equation} \label{eq:afw}
\frac{1}{2}\|\mathcal{D}_{k,A'^{n}\rightarrow B_{1}C}(\omega_{k,A'^{n}R})-\rho_{c}\otimes\Phi_{B_{1}R}\|_{1}\leq\epsilon, \forall k
\end{equation}
where $\epsilon > 0$ is a small parameter.
\subsection{Upper bound on steganographic rate}
With these two assumptions of secrecy and recoverability, we can now put a bound on the number of qubits $M$ that can be sent reliably and stegonagraphically from Alice to Bob. Defining $\sigma_{E}\equiv \mathcal{N}^{\otimes N}(V\rho_{c}V^{\dagger})$ and applying the Fannes-Audeneart inequality to the secrecy condition we have:
\begin{equation}\label{eq:secretproof}
H(\Tr_{R}(\omega_{A^{'n}R}))\leq H(\sigma_{E})+\delta N+h_{2}(\delta)
\end{equation}
where $h_{2}$ is the binary entropy function. Furthermore, from the recoverability condition we have
\begin{align}\label{eq:reliableproof}
M&=\log|A_{1}|=I(R\rangle B_{1})_{\Phi} \nonumber \\
&\leq I(R\rangle B_{1})_{\mathcal{D}_{k}(\omega)}+ \epsilon N+(1+\epsilon)h_{2}(\epsilon/[1+\epsilon]) \nonumber \\
&\leq I(R\rangle A'^{n})_{\omega_{k}}+f(N,\epsilon) \nonumber \\
&\leq H(\Tr_{R}(\omega_{k,A^{'n}R}))+f(N,\epsilon).
\end{align}
The first equality follows from the fact that the coherent information of a maximally entangled state is just the logarithm of the dimension of one of the subsystems. The first inequality follows from the AFW inequality applied to \eqref{eq:afw}. The second inequality is the data processing inequality. The last inequality follows from the definition of the coherent information.
The concavity of entropy implies that
\begin{equation}
\sum_{k} p_k H(\omega_{k,A'^{n}}) \leq H\left(\sum_{k} p_k \omega_{k,A'^{n}}\right)
= H(\omega_{A'^{n}}) .
\label{eq:concavity}
\end{equation}
The encodings $\mathcal{E}_{k,A_{1}C\rightarrow A^{'n}}$ are isometries, which means that $H(\omega_{k,A'^{n}})$ has the same value for every $k$. We can therefore sum over the probabilities $p_k$ on the left-hand side of (\ref{eq:concavity}) to get
\begin{equation}
H(\Tr_{R}(\omega_{k,A^{'n}R}))\leq H(\Tr_{R}(\omega_{A^{'n}R})).
\end{equation}
Now putting \eqref{eq:secretproof} and \eqref{eq:reliableproof} together we arrive at our main result, which states that Alice can secretly and reliably send $M$ stego qubits to Bob, where $M$ is bounded above by
\begin{align}\label{eq:bound}
M&\leq H(\Tr_{R}(\omega_{RA^{'n}}))+f(N,\epsilon) \nonumber\\
&\leq H(\sigma_{E})+g(N,\delta)+f(N,\epsilon) ,
\end{align}
where $g(N,\delta)\equiv\delta N+h_{2}(\delta)$. Thus, if we can compute a maximum for $H(\mathcal{N}^{\otimes N}(\rho))$ when $\rho$ is pure (because $V$ is an isometric encoding and $\rho_{c}$ is pure), we have a tight upper bound on the number of qubits $M$ that can be sent steganographically over a noiseless quantum channel. (Of course, if the actual quantum channel is noisy, then this bound will in general be changed. This is the topic of future work.)
\subsection{Upper bounds for specific channels}
We will now apply our result \eqref{eq:bound} to the channels discussed in the previous section, where we make the implicit assumption that Alice is using a nondegenerate code. Though our result \eqref{eq:bound} is true in general, for a degenerate code the number of distinct error syndromes is smaller (depending on the code), and the bounds discussed here and achievable rates discussed in the previous section would be adjusted.
\subsubsection{The bit flip channel}
For the bit flip channel, i.e., $\mathcal{N}_{BF}(\rho)=(1-p)\rho+pX\rho X$, the maximum of $H(\mathcal{N}^{\otimes N}(\rho))$ over all $N$-qubit pure states $\rho$ is $Nh(p)$ where $h(p)=-p\log p - (1-p)\log (1-p)$ is the entropy of a single qubit sent through a bit flip channel. To prove this, consider some pure state $\rho=\ket{\psi}\bra{\psi}$. Then
\begin{equation}
\mathcal{N}_{BF}^{\otimes N}(\ket{\psi}\bra{\psi})=\sum_{s}p(s)X^{s}\ket{\psi}\bra{\psi}X^{s}
\end{equation}
where we are summing over all binary strings $s$ of length $N$; $X^{s}$ is the operator acting on $N$ qubits with an $X$ acting at every location where $s$ has a 1 and an $I$ where $s$ has a 0. The probability $p(s)$ is given by
\begin{equation}
p(s)= p^{w(s)}(1-p)^{(N-w(s))},
\end{equation}
where $w(s)$ is the weight of string $s$. The Shannon entropy of this distribution is $Nh(p)$, since it is a binomial distribution. The von~Neumann entropy is the minimum Shannon entropy over all possible ensemble decompositions of the given state, and it is not hard to check that it is achieved when $\ket{\psi}$ is a $Z$ eigenstate. Thus the encoding described in the previous section for steganography with an simulated bit flip channel is essentially optimal.
\subsubsection{More general channels}
Unfortunately, for a more general quantum channel $\mathcal{N}$ we may not know, in general, what $N$-qubit pure state $\rho$ maximizes $H(\mathcal{N}^{\otimes N}(\rho))$. However, we can still bound this quantity. First, consider a general quantum channel $\mathcal{N}$ that acts on an $N$ qubit pure state as follows:
\begin{equation}
\mathcal{N}^{\otimes N}(\rho)\approx\sum_{j}E_{j}\rho E_{j}^{\dagger}
\end{equation}
where $\{E_{j}\}$ is the set of typical errors associated with $N$ applications of the channel $\mathcal{N}$. Recall that we are choosing our isometric encoding to correct for typical errors of whatever channel $\mathcal{N}$ it is we are emulating. Though the set of correctable errors $\{E_{j}\}$ need not act like unitaries on the codespace, we can always find a set of correctable errors $\{\widetilde{E}_{j}\}_{j}$ that do \cite{nielsen2002quantum}. To see this, first consider the Knill-Laflamme condition:
\begin{equation}
\mathbb{P}E_{i}^{\dagger}E_{j}\mathbb{P}=\alpha_{ij}\mathbb{P}
\end{equation}
where $\mathbb{P}$ is the codespace projector and $\alpha$ is a Hermitian matrix. Thus, we can write $\widetilde{\alpha}=U^{\dagger}\alpha U$ where $U$ is a unitary matrix and $\widetilde{\alpha}$ is diagonal.
\begin{equation}
\widetilde{E}_{k}=\sum_{j}M_{jk}E_{k}
\end{equation}
where the unitary $M$ is chosen in such a way as to diagonalize $\alpha$. That is
\begin{align}
\mathbb{P}\widetilde{E}_{k}^{\dagger}\widetilde{E}_{l}\mathbb{P}&=\sum_{ij}M_{ik}^{*}M_{jl}\mathbb{P}E_{i}^{\dagger}E_{j}\mathbb{P} \nonumber \\
&= (\sum_{ij}M_{ik}^{*}\alpha_{ij}M_{jl})\mathbb{P} \nonumber \\
&=\widetilde{\alpha}_{kl}\mathbb{P} = \delta_{kl}\widetilde{\alpha}_{kk}\mathbb{P} .
\end{align}
Note that these errors $\{\widetilde{E}_{j}\}$ act unitarily on the codespace. So long as the Knill-Laflamme condition is satisfied, we can always diagonalize $\alpha$ in this way. Now going back to our expression for the channel action,
\begin{equation}
\sum_{j}E_{j}\rho E_{j}^{\dagger} = \sum_{k,l,j}M_{kj}M^{*}_{lj}\widetilde{E}_{k}\rho\widetilde{E}_{l}^{\dagger} = \sum_{k} \widetilde{E}_{k}\rho\widetilde{E}_{k}^{\dagger}.
\end{equation}
Because we have assumed that the typical errors are all correctable, and that the code is nondegenerate, the states $\widetilde{E}_{k}\rho\widetilde{E}_{k}^{\dagger}$ are all orthogonal to each other, and $\Tr\{\widetilde{E}_{k}\rho\widetilde{E}_{k}^{\dagger}\} = \alpha_{kk}$. The von~Neumann entropy is the Shannon entropy minimized over all possible decompositions, so the entropy of this state is clearly
\begin{equation}
H(\sigma_{E})=H(\mathcal{N}^{\otimes N}(V\rho_{c}V^{\dagger})) \leq -\sum_{k}\alpha_{kk}\log_{2}(\alpha_{kk}) .
\end{equation}
By \eqref{eq:bound} we have shown that the amount of steganographic communication allowed for a quantum channel $\mathcal{N}$ emulation is upper bounded by this quantity. Applying this to the general channel discussed in section II.C above, we see that this quantity is equal to $N\bar{H}$, where $\bar{H}$ is the effective entropy per qubit defined in Eq.~(\ref{eq:effectiveEntropy}). So this encoding approaches the maximum possible rate for the general channel, just as for the bit flip channel.
\section{Conclusions and Future Work}\label{conclusion}
Quantum steganography is the study of secret quantum communication between two parties, Alice and Bob. We have shown that Alice and Bob are able to communicate with each other secretly at a nonzero rate using a shared secret key, without arousing suspicion from a potential eavesdropper Eve. In this paper we gave explicit bounds on the number of stego qubits that Alice can send to Bob when Alice is simulating a general quantum channel $\mathcal{N}$ with her stego encoded message, as well as explicit encodings to that achieve these bounds, for the case when the actual physical channel is noiseless.
The obvious next question is what if the channel shared between Alice and Bob (as is generally the case) is noisy? There is reason to believe that so long as Eve has some ignorance about the actual physical channel, then Alice will still be able to communicate steganographically to Bob.
For instance, suppose the actual physical channel is a depolarizing channel $\mathcal{N}_{p}$ where $p$ is the depolarizing parameter and the channel that Eve expects is $\mathcal{N}_{p+\epsilon-4p\epsilon/3}$ for some small suitably chosen $\epsilon>0$. Then Alice can emulate a depolarizing channel $\mathcal{N}_{\epsilon}$ in such a way such that if Eve observes the state Alice is sending to Bob, it will look like an innocent encoded covertext passing through $N$ applications of a channel $\mathcal{N}_{p}\circ\mathcal{N}_{\epsilon}$ (where $N$ is the length of the codeword Alice is using). There should be elements of the encoding given in this paper that will generalize to the noisy case for general channels $\mathcal{N}$. This will certainly be an area of fruitful future study.
\section*{Acknowledgments}
Thanks to Mark Wilde and David Ding for helpful discussions. This research was supported in part by NSF Grants CCF-1421078 and QIS-1719778, and by an IBM Einstein Fellowship at the Institute for Advanced Study.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,213 |
Transfield and detention centres: It's no longer business as usual
opinion: Transfield and detention centres: It's no longer business as usual
The Drum
/ By Michael Bradley
Posted Wed 23 Sep 2015 at 11:39pm Wednesday 23 Sep 2015 at 11:39pm Wed 23 Sep 2015 at 11:39pm , updated Fri 25 Sep 2015 at 5:06am Friday 25 Sep 2015 at 5:06am Fri 25 Sep 2015 at 5:06am
A group called No Business In Abuse has been actively targeting Transfield's responsibility for what's been happening at Manus and Nauru. (AAP Image: Department of Immigration)
abc.net.au/news/bradley-transfield-and-detention-centres/6800448
Like many companies in the modern media age, Transfield may soon realise that doing bad things is bad business, and its operation of the Manus Island and Nauru detention centres will continue to face scrutiny, writes Michael Bradley.
The rules of engagement are changing. As with the political establishment and mainstream media, corporations are only slowly realising that business as usual isn't what it used to be.
The interesting point of intersection, between the institutions of the status quo and the uncontrollable new world of grass roots activism powered by digital media, is the assertion of a new morality into a landscape where it has been largely absent for several hundred years.
Consider these recent events on an increasing scale of gravity. A Brisbane JB HiFi store refused entry to a customer with Down syndrome, based on mistaken identity. The story went immediately viral and JB HiFi apologised and promised to review its policies. Do a Google search on JB HiFi today, and this story fills your screen.
7-Eleven is in all sorts at the moment, after revelations that it has been at least tolerating if not facilitating a systemic scam among its franchisees of underpaying their largely immigrant workforce by under-reporting their hours. The only constraint on that story blowing sky-high is the fear among employees of being deported for breaching their visa conditions if they blow the whistle.
US pharmaceutical company Turing recently bought the rights to a 62-year-old medicine that treats toxoplasmosis (a common side effect of HIV/AIDS) and immediately raised the retail price from $US13.50 to $US750 per pill. It took a day of social media outrage, including a Hilary Clinton tweet, for the company to backtrack and say it would be reducing the price again.
At the far end of the scale, the 78-year-old automotive giant Volkswagen has just admitted that it sold 11 million diesel cars worldwide with software designed specifically to cheat pollution emission tests, pursuant to what can only have been a deliberate scheme of fraudulent deception tacitly endorsed at the highest levels of the company.
Somewhere along the scale of corporate amorality sits Transfield, which operates the two offshore detention centres on Manus Island and Nauru for the Government. Transfield is desperately trying to fight off a new form of existential threat that goes well beyond the damage many companies have suffered at the hands of social media in recent years.
A group called No Business In Abuse* has been actively targeting Transfield's responsibility for what's been happening at Manus and Nauru through a campaign that includes "moving super funds away from harmful investments". NBIA's website says:
We'll make that rejection hit home, by drying up their potential investor and client market. Hundreds of us will start petitions targeting strategic businesses and institutions, asking them to pledge never to contract with companies like Transfield, unless they clean up their act.
As a public company listed on the stock exchange, Transfield is owned by traditionally disinterested investors, including super funds and other financial institutions.
NBIA seeks to harness the voice of "tens of thousands of individual Australians" to send a "loud and clear message to corporations like Transfield: being complicit in abuse has consequences".
Transfield's response has been assertive. It distributed a lengthy response to NBIA's allegations, noting its own commitment to respecting human rights while also noting that as a private company it is not bound by any international human rights instruments (such as the conventions on torture and the rights of children).
It says that investigations have shown no evidence to support the majority of allegations of human rights abuses in its centres, quotes approvingly the dissenting minority opinion of the recent Senate Select Committee inquiry and asserts that "much of (NBIA's) source data regarding conditions at Manus and Nauru is based on outdated public information, and is therefore incorrect".
For its part, NBIA points to successive findings by independent bodies that dreadful things are happening on Manus and Nauru. The United Nations High Commission for Refugees made such findings in 2013 and, in April this year, reported to the Senate inquiry that the centre at Nauru does not provide safe and humane conditions and that "the harsh conditions, lack of privacy for individuals, uncertainty regarding durable solutions remain largely unchanged". It maintained its view that no child should be transferred to or kept at Nauru.
The Senate Select Committee's majority report is damning. It found that the Nauru centre is not a safe environment for asylum seekers, particularly women, children and other vulnerable persons and that no children should be held there. It also found that the Immigration Department is not in a position to guarantee that any aspect of the Nauru centre is run well and that there is cause for ongoing concern about the performance and accountability of its contractors (Transfield).
Transfield is intent on pursuing an old school crisis management strategy: deny wrongdoing, assert absence of legal obligation while claiming pure motivation, point to gaps in the evidence and selectively quote the favourable bits, impugn the accusers' credibility and wait for the noise to subside, then get back to business.
In the end, what Transfield is banking on is that both the Coalition and ALP will continue to allow it to operate under the shroud of operational secrecy and help it limit public access to the truth, combined with the conspiracy of wilful ignorance that the Australian populace continues to share about the treatment of asylum seekers.
That's a tenuous form of security, because the reality of Manus and Nauru will continue to leak out and it is utterly appalling.
The genius of the corporate structure is that it allows all participants - directors, employees, investors (and investors in those investors) - to operate in a moral vacuum when it comes to the physical actions the entity undertakes. Those actions are no individual's responsibility, and so really dreadful things can be done without anyone at any point of the profit chain saying "stop, this is wrong". When the truth about Volkswagen and 7-Eleven comes out, we'll see that that's exactly what happened there.
Transfield's current experience is illustrating the vulnerability in this chain of amorality, which the new age of instantaneous democratised media is enabling. Corporations aren't familiar with that degree of externally applied pressure, but they'd better get used to it.
*Disclaimer: Marque Lawyers has provided legal advice to NBIA's partner organisation GetUp.
Editor's note (September 24, 2015): NBIA wishes to clarify that in the publication of this article Michael Bradley does not represent NBIA.
Editor's note (September 25, 2015): To clarify and better reflect the aims of NBIA, we have substituted some passages in the original article with quotations from the NBIA website.
Michael Bradley is the managing partner of Marque Lawyers, a Sydney law firm, and writes a weekly column for The Drum. Follow him on Twitter @marquelawyers.
Posted 23 Sep 2015 23 Sep 2015 Wed 23 Sep 2015 at 11:39pm , updated 25 Sep 2015 25 Sep 2015 Fri 25 Sep 2015 at 5:06am | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,129 |
Pierre Brantus, né Louis-Pierre Brantus le à Dijon et mort le dans la même ville, est un homme politique français.
Biographie
Résistant, membre du réseau Buckmester, il assure de dangereuses missions dont la prise de la forteresse de Langres. Chargé de mission auprès du commissaire de la République de Bourgogne-Franche-Comté, vice-président du comité régional de Libération, attaché parlementaire au cabinet de Jean Biondi, sous-secrétaire d'Etat à l'Intérieur, Pierre Brantus fait carrière dans la presse régionale. Il est trésorier-adjoint (1952-1954), trésorier (1954-1955), secrétaire général (1955-1971) du Syndicat national de la presse quotidienne régionale. Il est le directeur administratif du quotidien La Bourgogne républicaine, fondé en 1937 et dirigé par le député socialiste Jean Bouhey, puis son directeur général en 1957. Le journal se situe alors à gauche. La société qu'il dirige absorbe La République de Franche-Comté (de droite) en 1957 puis le Comtois, journal socialiste de Besançon, l'année suivante. Le , la Bourgogne républicaine devient La Bourgogne-Les Dépêches, puis bientôt Les dépêches du Centre-Est. Il fonde (1964) et préside la Société des presses nouvelles de l'Est, rachetée en 1973 par L'Est républicain. Sa société s'était rapprochée du journal nancéien dans les années 1960, Brantus devenant cogérant de la S.A.R.L. les Dépêches - l'Est républicain et directeur adjoint de l'Est républicain. Il reste président du conseil d'administration des Presses nouvelles de l'Est, en est brièvement le directeur général en 1975 puis il est désigné vice-président cette même année.
Il se tourne vers les responsabilités politiques à partir de 1972. Il est élu conseiller général de Montmirey-le-Château (Jura) en 1972, président du conseil général du Jura en 1980, sénateur (Union centriste) en 1983, succédant à Edgar Faure comme homme fort du département.
Républicain, centriste (CDP puis UDF-CDS), soutien de Raymond Barre en 1988, attaché à la liberté de la presse et à une philosophie humaniste et sociale, il fut longtemps le seul parlementaire "de droite" à être membre de la Ligue des droits de l'homme.
Détail des fonctions et des mandats
Mandat parlementaire
- : Sénateur du Jura
Notes et références
Voir aussi
Articles connexes
Canton de Montmirey-le-Château
Conseil général du Jura
Liste des sénateurs du département du Jura
Liens externes
Sa fiche sur le site du Sénat
Sénateur du département du Jura
Décès en septembre 1989
Naissance en octobre 1921
Président du conseil général du Jura
Titulaire de la médaille de la Résistance française avec rosette
Décès à 67 ans | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 513 |
The method used to repair a hole in a hollow-core door depends upon whether the door is painted or stained.
($8.95 - $175.00) : 903 matches. Find great deals on the latest styles of Hollow molded composite interior door. Compare prices & save money on Home Hardware.
A sheet molded compound (SMC) which is a fiberglass reinforced composite material, produced in a sheet format, for forming a door, is disclosed.
Available Molded in Matching Fleet Colors or "Ready to Paint" Composite Door Assemblies Equal Fit, ... Hole Forming Devices. Pipe Shaping Rings. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,924 |
They also have ornamental displays of perennial favorites: daylilies, roses, and irises. The Mordecai Children's Garden encourages hands-on exploration of soil and water and has stroller parking, picnic tables, and year-round programing. There are many gardens dedicated to the serenity of Japanese strolling gardens and bonsai, a South African garden with hardy plants from their steppe region, and a greenhouse overflowing with tropical and subtropical flowers.
While its Denver counterpart might be more popular, the Cheyenne Mountain Zoo in Colorado Springs is the most interesting Colorado zoo experience you'll find. It sits about 6,800ft (2,073m) above sea level and features exhibits built into the mountainside. You can also feed the giraffes, ride an open chairlift over the zoo, walk through an aviary of free-flying birds and expect plenty of animal encounters.
With over 750 animals of 170 species, the zoo is world-class, and contains the largest herd of reticulated giraffes in any zoo anywhere. A special feature of the zoo is the fact they allow visitors to hand-feed the giraffes, which are very tame. Hoping to inspire conservation action, the zoo has thoughtfully housed all of its animals in as natural an environment as possible so that visitors can understand the needs of each animal of each species.
Princess Reema returned to Saudi Arabia, where she focused on private sector initiatives and the empowerment of women in the Kingdom. In 2000, She co-founded Yibreen, a women's gym. From 2007 until 2015, Princess Reema was the Chief Executive Officer of Alfa International Company Limited – Harvey Nichols Riyadh, a multi-brand luxury retail company, where she collaborated with leading female recruitment agency, Glowork, and commissioned a study on Obstacles for Women in the Workplace, which set the tone for female inclusion in retail, and resulted in opening the first workplace nursery in a retail store, enabling more women to work while caring for children. In 2013, she founded Alf Khair, a social enterprise aimed at elevating the professional capital of Saudi women through a curriculum developed to enable financial self-sufficiency.
Located in Central Colorado, Bishop Castle has quickly become one of the most popular roadside attractions in the state. In 1959, fifteen-year-old Jim Bishop dropped out of school and purchased a 2.5 acre piece of land for $450. This piece of land was located alongside southern Colorado's San Isabel National Forest. In order to earn enough money to purchase the land, Bishop worked random side jobs and helped his father, Willard. Although Bishop funded the land purchase, his parent's legally owned the land since Bishop was only a teenager.
Visit any of the 5 visitors centers in the park; one is registered on the National Registry of Historic Places and was designed by the Frank Lloyd Wright School of Architecture. There is a variety of landscapes to explore, from mountains to mountain tundra, and a wide array of wildlife. Whether you come for a day and do a short hike, or stay and camp out to go on longer treks, the scenery will impress you. Don't miss the Arapaho National Forest or Indian Peaks Wilderness.
A registered National Natural Landmark just outside Colorado Springs, the Garden of the Gods Park is open year-round and offers stunning views of its 300ft (91m) sandstone rock formations, along with hiking, horseback riding and camping. The Visitor and Nature Center has all kinds of interactive exhibits. If you're looking for souvenirs, the Trading Post, which lies on the edge of the park, features artwork by local artists. Admission to the park is free.
The multi-tower buildings are still well preserved after eight hundred years, and seeing them is worth driving to the remote location on the Colorado/Utah border. Dogs are welcome on the hiking trails. The Square Tower Group has a small interpretative center, and rangers are available throughout the park to answer questions and give guidance. Hovenweep has a primitive 31-site campground that fills up on a first-come, first-served basis.
The Bishop Castle regularly hosts private special events throughout the year. Schools are welcome to use the Castle's facilities for educational purposes. Also, people are welcome to use the Bishop Castle as the location for their wedding ceremony. Although wedding ceremonies can be held at the Castle, people are not allowed to hold receptions or any other party similar to a reception at the Bishop Castle.
Make your Memorial Day memorable at Craig's Grand Olde West Days or MountainFilm in Telluride. June, July, August and September are perhaps the hardest hitting when it comes to sheer number of events. Every single weekend you'll have a handful of fun festivals to choose from. Splash around at FIBArk in June or the Colorado State Fair in Pueblo over Labor Day.
Vail is a tiny town at the foot of the Vail Mountains and nestled in the White River National Forest. The picturesque town is a prime ski resort destination in the winter, but also provides ample entertainment in the summer. Vail also has a creek that literally runs through town – follow the winding curves of Gore Creek on a walk some peaceful afternoon.
Still haven't packed enough into your Colorado vacation? Make time to see some of the great Colorado mountain attractions — tour a ghost town near Aspen, sail above the mountains in a balloon over Breckenridge, rock out at a music festival in Telluride, check out the nightlife scene in Vail and try your luck at a casino in Black Hawk, Central City or Cripple Creek. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,649 |
Q: Problem with the use of facet_nested for line plot I am trying to plot a line graph with facet nested (I often do that with bar plots). For some strange reason, one part of the plot is missing. I have worked around the code trying to add "group" to the aesthetics but it doesn't seem to work. Below is the graph I am ending up with
I am trying to plot a line graph with facet nested (I often do that with bar plots). For some strange reason, one part of the plot is missing. I have worked around the code trying to add "group" to the aesthetics but it doesn't seem to work. Below is the graph I am ending up with
data10 <- structure(list(Group = c("Visible", "Visible", "Visible", "Visible",
"Visible", "Visible", "Visible", "Visible", "Visible", "Visible",
"Visible", "Visible", "Remembered", "Remembered", "Remembered",
"Remembered", "Remembered", "Remembered", "Remembered", "Remembered",
"Remembered", "Remembered", "Remembered", "Remembered", "Visible",
"Visible", "Visible", "Visible", "Visible", "Visible", "Visible",
"Visible", "Visible", "Visible", "Visible", "Visible", "Remembered",
"Remembered", "Remembered", "Remembered", "Remembered", "Remembered",
"Remembered", "Remembered", "Remembered", "Remembered", "Remembered",
"Remembered"), Condition = c("CEN", "CEN", "CEN", "CEN", "IPS",
"IPS", "IPS", "IPS", "CTL", "CTL", "CTL", "CTL", "CEN", "CEN",
"CEN", "CEN", "IPS", "IPS", "IPS", "IPS", "CTL", "CTL", "CTL",
"CTL", "CEN", "CEN", "CEN", "CEN", "IPS", "IPS", "IPS", "IPS",
"CTL", "CTL", "CTL", "CTL", "CEN", "CEN", "CEN", "CEN", "IPS",
"IPS", "IPS", "IPS", "CTL", "CTL", "CTL", "CTL"), test1 = c("Pre-test",
"Pre-test", "Post-test", "Post-test", "Pre-test", "Pre-test",
"Post-test", "Post-test", "Pre-test", "Pre-test", "Post-test",
"Post-test", "Pre-test", "Pre-test", "Post-test", "Post-test",
"Pre-test", "Pre-test", "Post-test", "Post-test", "Pre-test",
"Pre-test", "Post-test", "Post-test", "Pre-test", "Pre-test",
"Post-test", "Post-test", "Pre-test", "Pre-test", "Post-test",
"Post-test", "Pre-test", "Pre-test", "Post-test", "Post-test",
"Pre-test", "Pre-test", "Post-test", "Post-test", "Pre-test",
"Pre-test", "Post-test", "Post-test", "Pre-test", "Pre-test",
"Post-test", "Post-test"), test = structure(c(1L, 1L, 2L, 2L,
1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L,
1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L,
1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L), .Label = c("Pre-test",
"Post-test"), class = "factor"), trial2 = c(1, 5, 9, 13, 1, 5,
9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1,
5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13, 1, 5, 9, 13,
1, 5, 9, 13), trial = c("Pre-1", "Pre-5", "Post-1", "Post-5",
"Pre-1", "Pre-5", "Post-1", "Post-5", "Pre-1", "Pre-5", "Post-1",
"Post-5", "Pre-1", "Pre-5", "Post-1", "Post-5", "Pre-1", "Pre-5",
"Post-1", "Post-5", "Pre-1", "Pre-5", "Post-1", "Post-5", "Pre-1",
"Pre-5", "Post-1", "Post-5", "Pre-1", "Pre-5", "Post-1", "Post-5",
"Pre-1", "Pre-5", "Post-1", "Post-5", "Pre-1", "Pre-5", "Post-1",
"Post-5", "Pre-1", "Pre-5", "Post-1", "Post-5", "Pre-1", "Pre-5",
"Post-1", "Post-5"), N = c(12, 10, 12, 11, 11, 9, 12, 12, 12,
10, 12, 12, 9, 11, 12, 12, 8, 12, 11, 12, 9, 9, 12, 12, NA, NA,
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
NA, NA, NA, NA, NA, NA), Variables = c("Eye reaction time", "Eye reaction time",
"Eye reaction time", "Eye reaction time", "Eye reaction time",
"Eye reaction time", "Eye reaction time", "Eye reaction time",
"Eye reaction time", "Eye reaction time", "Eye reaction time",
"Eye reaction time", "Eye reaction time", "Eye reaction time",
"Eye reaction time", "Eye reaction time", "Eye reaction time",
"Eye reaction time", "Eye reaction time", "Eye reaction time",
"Eye reaction time", "Eye reaction time", "Eye reaction time",
"Eye reaction time", "Hand reaction time", "Hand reaction time",
"Hand reaction time", "Hand reaction time", "Hand reaction time",
"Hand reaction time", "Hand reaction time", "Hand reaction time",
"Hand reaction time", "Hand reaction time", "Hand reaction time",
"Hand reaction time", "Hand reaction time", "Hand reaction time",
"Hand reaction time", "Hand reaction time", "Hand reaction time",
"Hand reaction time", "Hand reaction time", "Hand reaction time",
"Hand reaction time", "Hand reaction time", "Hand reaction time",
"Hand reaction time"), Eye_Rx = c(0.190333333, 0.213909091, 0.164583333,
0.2375, 0.24375, 0.215444444, 0.168916667, 0.259916667, 0.147333333,
0.277363636, 0.20425, 0.240833333, 0.189222222, 0.146727273,
0.111083333, 0.13225, 0.183375, 0.166583333, 0.115727273, 0.14,
0.15, 0.222222222, 0.206916667, 0.133083333, 0.423583333, 0.507636364,
0.374083333, 0.399166667, 0.45075, 0.401333333, 0.411583333,
0.459083333, 0.411166667, 0.433727273, 0.380333333, 0.4115, 0.411555556,
0.420636364, 0.294416667, 0.29875, 0.40425, 0.405833333, 0.330818182,
0.318, 0.353666667, 0.366555556, 0.338666667, 0.319333333), sd = c(0.113040084,
0.120666859, 0.093087894, 0.149627112, 0.164622502, 0.099557912,
0.058356987, 0.323980207, 0.094897393, 0.27273697, 0.199661589,
0.16341238, 0.143796538, 0.069264841, 0.03580873, 0.041850654,
0.155523023, 0.091204524, 0.065353027, 0.077994172, 0.098432718,
0.173048965, 0.179821861, 0.063938121, 0.138244486, 0.398595854,
0.107139211, 0.19240149, 0.296251467, 0.188482095, 0.183202054,
0.232274703, 0.155725769, 0.181612274, 0.135318032, 0.165268652,
0.169866941, 0.189299906, 0.071530614, 0.049045665, 0.117074762,
0.100927367, 0.090403339, 0.060077223, 0.073431941, 0.045735957,
0.208057393, 0.068832251), se = c(0.032631861, 0.036382427, 0.02687216,
0.043193627, 0.047522423, 0.033185971, 0.016846211, 0.09352503,
0.027394518, 0.08223329, 0.057637336, 0.047173091, 0.047932179,
0.020884135, 0.01033709, 0.012081243, 0.054985692, 0.026328478,
0.019704679, 0.022514978, 0.032810906, 0.057682988, 0.0519101,
0.018457346, 0.039907746, 0.120181172, 0.030928426, 0.055541526,
0.085520432, 0.062827365, 0.052885877, 0.067051931, 0.044954157,
0.054758161, 0.039062951, 0.04770895, 0.056622314, 0.057076069,
0.02064911, 0.014158264, 0.041392179, 0.029135221, 0.027257632,
0.0173428, 0.024477314, 0.015245319, 0.060060996, 0.019870159
), ci = c(0.071822243, 0.081065099, 0.059145226, 0.095068532,
0.104596148, 0.076526985, 0.03707826, 0.205847203, 0.060294927,
0.183227189, 0.126858921, 0.103827273, 0.110531804, 0.046532753,
0.022751782, 0.026590637, 0.130020501, 0.05794859, 0.043904761,
0.049555133, 0.075662085, 0.13301721, 0.114253359, 0.040624344,
0.087836356, 0.267780338, 0.068073007, 0.122246075, 0.188229202,
0.144880163, 0.116401031, 0.147580306, 0.098943433, 0.122008786,
0.085976975, 0.105006692, 0.13057129, 0.127173408, 0.045448384,
0.031162129, 0.097876951, 0.064126189, 0.060733789, 0.038171246,
0.056444787, 0.035155769, 0.132193361, 0.043733926)), row.names = c(NA,
-48L), spec = structure(list(cols = list(Group = structure(list(), class = c("collector_character",
"collector")), Condition = structure(list(), class = c("collector_character",
"collector")), test1 = structure(list(), class = c("collector_character",
"collector")), test = structure(list(), class = c("collector_double",
"collector")), trial2 = structure(list(), class = c("collector_double",
"collector")), trial = structure(list(), class = c("collector_character",
"collector")), N = structure(list(), class = c("collector_double",
"collector")), Variables = structure(list(), class = c("collector_character",
"collector")), Eye_Rx = structure(list(), class = c("collector_double",
"collector")), sd = structure(list(), class = c("collector_double",
"collector")), se = structure(list(), class = c("collector_double",
"collector")), ci = structure(list(), class = c("collector_double",
"collector"))), default = structure(list(), class = c("collector_guess",
"collector")), delim = ","), class = "col_spec"), class = c("spec_tbl_df",
"tbl_df", "tbl", "data.frame"))
library(tidyverse)
library(ggplot2)
library(ggthemes)
library(ggh4x)
p <- ggplot(data10, aes(x = trial2, y = Eye_Rx),group = test) +
geom_line(aes(color = Variables), lwd=1.2, show.legend = F) +
geom_ribbon(aes(ymin = Eye_Rx - 1.96 * se, ymax = Eye_Rx + 1.96 * se, fill = Variables), alpha = .6) +
scale_fill_manual(values = c("gray", "black"))+ scale_color_manual(values = c("black", "gray")) + facet_nested(Condition ~ Group + test)+ theme_bw() + xlab("Trial") + ylab("Hand and Eye Reaction time (s)") +
scale_x_continuous(limits = c(1,8), breaks = seq(1,8,1),labels = c("1", "2", "3", "4", "5", "6", "7", "8")) + theme(axis.text.x = element_text(size = 12,face="bold", angle = 90),#, angle = 10, hjust = .5, vjust = .5),
axis.text.y = element_text(size = 12, face = "bold"),
axis.title.y = element_text(vjust= 1.8, size = 18),
axis.title.x = element_text(vjust= -0.5, size = 18),
axis.title = element_text(face = "bold")) + theme(legend.position="top") +
guides(fill=guide_legend(title="")) + theme(legend.text=element_text(size=16),legend.title=element_text(size=14) ) +
theme(strip.text = element_text(face="bold", size=16))
p
A: You are excluding all of your post-test results because these all have x axis values of 9 or more. Remove the scale_x_continuous and add scales = "free_x" to facet_nested
ggplot(data10, aes(x = trial2, y = Eye_Rx),group = test) +
geom_line(aes(color = Variables), lwd = 1.2, show.legend = FALSE) +
geom_ribbon(aes(ymin = Eye_Rx - 1.96 * se, ymax = Eye_Rx + 1.96 * se,
fill = Variables), alpha = 0.6) +
scale_fill_manual(values = c("gray", "black"), name = NULL) +
scale_color_manual(values = c("black", "gray")) +
xlab("Trial") +
ylab("Hand and Eye Reaction time (s)") +
facet_nested(Condition ~ Group + test, scales = "free_x") +
theme_bw() +
theme(axis.text.x = element_text(size = 12,face="bold", angle = 90),
axis.text.y = element_text(size = 12, face = "bold"),
axis.title.y = element_text(vjust= 1.8, size = 18),
axis.title.x = element_text(vjust= -0.5, size = 18),
axis.title = element_text(face = "bold"),
legend.position = "top",
legend.text = element_text(size = 16),
legend.title = element_text(size = 14),
strip.text = element_text(face = "bold", size = 16))
A: Problem is with the x axis limits.
Modifying extending the limits give this:
ggplot(data10, aes(x = trial2, y = Eye_Rx),group = test) +
geom_line(aes(color = Variables), lwd=1.2, show.legend = F) +
geom_ribbon(aes(ymin = Eye_Rx - 1.96 * se, ymax = Eye_Rx + 1.96 * se, fill = Variables), alpha = .6) +
scale_fill_manual(values = c("gray", "black"))+
scale_color_manual(values = c("black", "gray")) +
facet_nested(Condition ~ Group + test, scales = "free")+ theme_bw() +
xlab("Trial") + ylab("Hand and Eye Reaction time (s)") +
scale_x_continuous(limits = c(1,13), breaks = seq(1,13,1),
labels = as.character(1:13)) +
theme(axis.text.x = element_text(size = 12,face="bold", angle = 90),#, angle = 10, hjust = .5, vjust = .5),
axis.text.y = element_text(size = 12, face = "bold"),
axis.title.y = element_text(vjust= 1.8, size = 18),
axis.title.x = element_text(vjust= -0.5, size = 18),
axis.title = element_text(face = "bold")) + theme(legend.position="top") +
guides(fill=guide_legend(title="")) +
theme(legend.text=element_text(size=16),legend.title=element_text(size=14) ) +
theme(strip.text = element_text(face="bold", size=16))
Removing the limits and making the scale = "free", gives this:
ggplot(data10, aes(x = trial2, y = Eye_Rx),group = test) +
geom_line(aes(color = Variables), lwd=1.2, show.legend = F) +
geom_ribbon(aes(ymin = Eye_Rx - 1.96 * se, ymax = Eye_Rx + 1.96 * se, fill = Variables), alpha = .6) +
scale_fill_manual(values = c("gray", "black"))+
scale_color_manual(values = c("black", "gray")) +
facet_nested(Condition ~ Group + test, scales = "free")+ theme_bw() +
xlab("Trial") + ylab("Hand and Eye Reaction time (s)") +
theme(axis.text.x = element_text(size = 12,face="bold", angle = 90),#, angle = 10, hjust = .5, vjust = .5),
axis.text.y = element_text(size = 12, face = "bold"),
axis.title.y = element_text(vjust= 1.8, size = 18),
axis.title.x = element_text(vjust= -0.5, size = 18),
axis.title = element_text(face = "bold")) + theme(legend.position="top") +
guides(fill=guide_legend(title="")) +
theme(legend.text=element_text(size=16),legend.title=element_text(size=14) ) +
theme(strip.text = element_text(face="bold", size=16))
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,030 |
Q: Cocos2d-x: possible to use HTML (UIWebView)? I'm trying out cocos2d-x and got to the point where I can build the Javascript samples for Android and run them inside a browser as well.
Now I want to create my own game, but coming from a HTML background, I'd rather use HTML tags with CSS than use Javascript to setup the user interface.
I've read about UIWebView which can display HTML-pages in an app, but I was wondering if anyone has ever done this in combination with Cocos2D-x ? And could this be transparent, to overlay a normal cocos2d-x screen in the app then?
If so, how could this be done?
A: You can use CCXWebview for that with Cocos2d-x. This extension is based on Cocos2d-x 2.0.4 and it seems to work also on Cocos2d-x 2.x with some modification.
However, if you want to use Cocos2d-x 3.0 for Android, you cannot use it because Cocos2d-x 3.0 uses NativeActivity, thus you cannot combine Android WebView on the Cocos2d-x screen.
EDITED
only problem is that I have little knowledge of Java or C++
... It would take me years to figure the Java and C++ things out :)
So why are you sticking to use Cocos2d-x??? Why don't you use Cocos2d-html5? It has same functionality as Cocos2d-x JavaScript binding and it uses HTML5 Canvas, so you can use DOM with it.
https://twitter.com/hyperandroid/status/311534580962295809 :
Cocoonjs can run Cocos2d-html5 games too.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,840 |
@interface SCMainWindow : NSWindow<NSDraggingDestination>
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,859 |
# Are You
Watching Closely?
Also in the series
William Rothman, editor, Cavell on Film
J. David Slocum, editor, Rebel Without a Cause
Joe McElhaney, The Death of Classical Cinema
Kirsten Moana Thompson, Apocalyptic Dread
Frances Gateward, editor, Seoul Searching
Michael Atkinson, editor, Exile Cinema
Paul S. Moore, Now Playing
Robin L. Murray and Joseph K. Heumann, Ecology and Popular Film
William Rothman, editor, Three Documentary Filmmakers
Sean Griffin, editor, Hetero
Jean-Michel Frodon, editor, Cinema and the Shoah
Carolyn Jess-Cooke and Constantine Verevis, editors, Second Takes
Matthew Solomon, editor, Fantastic Voyages of the Cinematic Imagination
R. Barton Palmer and David Boyd, editors, Hitchcock at the Source
William Rothman, Hitchcock: The Murderous Gaze, Second Edition
Joanna Hearne, Native Recognition
Marc Raymond, Hollywood's New Yorker
Steven Rybin and Will Scheibel, editors, Lonely Places, Dangerous Ground
Claire Perkins and Constantine Verevis, editors, B Is for Bad Cinema
Dominic Lennard, Bad Seeds and Holy Terrors
Rosie Thomas, Bombay before Bollywood
Scott M. MacDonald, Binghamton Babylon
Sudhir Mahadevan, A Very Old Machine
David Greven, Ghost Faces
James S. Williams, Encounters with Godard
William H. Epstein and R. Barton Palmer, editors, Invented Lives, Imagined Communities
Lee Carruthers, Doing Time
Rebecca Meyers, William Rothman, and Charles Warren, editors, Looking with Robert Gardner
Belinda Smaill, Regarding Life
Douglas McFarland and Wesley King, editors, John Huston as Adaptor
R. Barton Palmer, Homer B. Pettey, and Steven M. Sanders, editors, Hitchcock's Moral Gaze
Will Scheibel, American Stranger
Nenad Jovanovic, Brechtian Cinemas
Steven Rybin, Gestures of Love
Amy Rust, Passionate Detachments
# Are You
Watching Closely?
Cultural Paranoia, New Technologies,
and the Contemporary Hollywood
Misdirection Film
Seth Friedman
Cover image: From The Prestige. Credit: Newmarket/Photofest
Published by State University of New York Press, Albany
© 2017 State University of New York
All rights reserved
Printed in the United States of America
No part of this book may be used or reproduced in any manner whatsoever without written permission. No part of this book may be stored in a retrieval system or transmitted in any form or by any means including electronic, electrostatic, magnetic tape, mechanical, photocopying, recording, or otherwise without the prior permission in writing of the publisher.
For information, contact State University of New York Press, Albany, NY
www.sunypress.edu
Production, Eileen Nizer
Marketing, Anne M. Valentine
Library of Congress Cataloging-in-Publication Data
Names: Friedman, Seth, 1976– author.
Title: Are you watching closely? : cultural paranoia, new technologies, and the contemporary Hollywood misdirection film / by Seth Friedman.
Description: Albany : State University of New York Press, [2017] | Series: SUNY series, horizons of cinema | Includes bibliographical references and index.
Identifiers: LCCN 2016039673 (print) | LCCN 2016056195 (ebook) | ISBN 9781438465913 (hardcover : alk. paper) | ISBN 9781438465920 (ebook)
Subjects: LCSH: Motion pictures—Philosophy. | Space and time in motion pictures. | Narration (Rhetoric)
Classification: LCC PN1995 .F7445 2017 (print) | LCC PN1995 (ebook) | DDC 791.4301—dc23
LC record available at <https://lccn.loc.gov/2016039673>
10 9 8 7 6 5 4 3 2 1
For Leo and Clark Friedman
## Contents
List of Illustrations
Acknowledgments
Introduction
1Retrospective Issues: The Discursive Approach to Genre and the Misdirection Film
2The Truth Is Out There: Manufacturing Conspiratorial Narrative Coherence
3Constructing the (Im)perfect Cover: Masculine Masquerade and Narrative Agency
4Start Making Sense: Narrative Complexity, DVD, and Online Fandom
5The Masters of Misdirection: Branding M. Night Shyamalan and Christopher Nolan
6Genre Prestige: The Misdirection Film as Blockbuster and Middlebrow Art
Conclusion
Works Cited
Index
## Illustrations
Figure I.1.Young Elijah Price gets a surprise ending comic book gift, Unbreakable
Figure I.2.Olivia Wenscome and Robert Angier distract the audience, The Prestige
Figure I.3.Cole Sear divulges that he always sees dead people, The Sixth Sense
Figure 1.1.The unnamed narrator faints during the changeover, Fight Club
Figure 1.2.Marla Singer and the unnamed narrator hold hands at the end, Fight Club
Figure 1.3.The unnamed narrator imagines confronting Marla Singer, Fight Club
Figure 1.4.An advertisement displaying "Exodus 8:2" appears fleetingly, Magnolia
Figure 1.5.A plane contains one of many references in the prologue to 82, Magnolia
Figure 1.6.Claudia Wilson Gator breaks the fourth wall in the final shot, Magnolia
Figure 2.1.Michael Faraday falsely blamed by a television news report, Arlington Road
Figure 2.2.A subway advertisement alludes to the changeover, Jacob's Ladder
Figure 2.3.Heavy backlighting makes Louis appear angelic, Jacob's Ladder
Figure 2.4.Men in black, government agents abduct Jacob Singer, Jacob's Ladder
Figure 3.1.Elijah Price's emergency birth depicted in a mirror image, Unbreakable
Figure 3.2.David Dunn gives Kelly a woman's interest magazine, Unbreakable
Figure 3.3.M. Night Shyamalan's cameo as an accused drug dealer, Unbreakable
Figure 3.4.Roger "Verbal" Kint scans the bulletin board, The Usual Suspects
Figure 3.5.A flashback of Keyser Söze's rise to power, The Usual Suspects
Figure 4.1.Diane Selwyn makes arrangements with Joe at Winkie's, Mulholland Dr.
Figure 4.2.A barely perceptible "Hollywood is HELL" flyer, Mulholland Dr.
Figure 4.3.An Aunt Ruth double appears in the jitterbug sequence, Mulholland Dr.
Figure 4.4.Diane Selwyn and Woody Katz do a sensual audition, Mulholland Dr.
Figure 4.5.Leonard Shelby makes a note to tattoo Teddy's license plate, Memento
Figure 4.6.Teddy's license plate shown after Leonard Shelby's notation, Memento
Figure 4.7.Leonard Shelby replaces Sammy Jankis in a mental institution, Memento
Figure 4.8.Teddy's driver's license with an incorrect leap day expiration, Memento
Figure 4.9.Changed digits on Teddy's license plate, Memento
Figure 5.1.Titles from the opening of The Happening that set the scene
Figure 5.2.The Paris setting of the ending of The Happening
Figure 5.3.The Prestige's final shot reveals Robert Angier's facsimiles
Figure 6.1.The hotel room with impossible architecture, Inception
Figure 6.2.Alley walls close in on Dom Cobb, Inception
Figure 6.3.Teddy Daniels walks away after uttering his last line, Shutter Island
Figure 6.4.Evidence that Ashecliffe Hospital is run by the military, Shutter Island
Figure 6.5.Shutter Island's final shot suggests that there may be a second lighthouse
## Acknowledgments
As with Hollywood film production, virtually any academic book is a highly collaborative effort in which key contributors often do not receive the full credit that they deserve. Although devoting this section to thanking those who supported me might not jibe entirely with a book devoted partly to how innovation occurs amid standard genre, narrative, and formal conventions, I chose to leverage it traditionally because it is the least I can do to provide those backers with the recognition that they earned. This book would not have been possible without the valuable assistance of the following people.
This project began with the guidance of my MA thesis director, Matthew Solomon, who generously dedicated hours of his time to helping both to generate and shape the idea into a viable scholarly project. In fact, early work I wrote under his tutelage became an article in Journal of Film and Video 58.4 (Winter 2006): 16–28, portions of which appear in chapter 1. Matthew taught me how to be a professional academic and has graciously continued to mentor me. David Gerstner also developed that thesis into a rigorous piece of scholarship and remains a staunch supporter. Under the assured direction of Barbara Klinger, I transformed that thesis into a dissertation, some of which was derived from essays produced in her seminars, including one that was the source for an article in Genders 59 (Spring 2014) that is the basis of chapter 3. Barb's thorough and perceptive editorial suggestions consistently made my arguments more clear, concise, and persuasive. I am grateful that she lent so much of her seemingly endless energy and prodigious expertise to helping make the dissertation into a readily marketable book. I also want to thank the other prominent members of my dissertation committee—Joan Hawkins and Gregory Waller—for their insightful and influential feedback throughout the process.
Many of my peers, colleagues, and other professionals provided crucial comments on various iterations of the project that greatly improved it. Travis Vogan and Andrea Kelley both read numerous chapters of the manuscript and gave me sharp suggestions that strengthened a variety of key arguments. Additional peers from my graduate school community deserve special recognition for their generosity when this project was a dissertation and beyond, including Shelley Bradfield, Antonio Golán, Michael Lahey, James Paasche, and Justin Rawlins. During the dissertation phase, Shira Segal was a champion of my work as well as a loyal confidant and comrade in Joan's dissertation reading group. I also want to acknowledge the other members of that group throughout my tenure in it for their contributions. Joan again deserves special recognition for sacrificing many of her Saturdays to enhance the work of graduate students. At DePauw University, my colleagues in the Department of Communication and Theatre have graciously welcomed me into a group that is uncommonly supportive and compassionate. Finishing this book was a lot easier than it likely would have been elsewhere because I am fortunate to have found a home in which it is a genuine pleasure to work with each of my colleagues. Of those spectacular peers, Andrew Hayes, Kevin Howley, and Jonathan Nichols-Pethick were particularly instrumental to ensuring that DePauw became the place where I could complete this project with adequate support. Finally, thanks to Murray Pomerance for including me in the Horizons of Cinema series and for handing me over to Eileen Nizer, James Peltz, and Rafael Chaiken, who have been fantastic editors at SUNY Press. I also want to thank them for recruiting the anonymous readers of the manuscript, who offered astute feedback that bolstered the quality of the final product.
Finally, I want to thank my family and non-academic friends for their backing. Yona Belfort, Rich Caponetti, Steve Dorman, and Matt Morgan read papers on this subject in its earliest phases and have consistently believed in my abilities as a scholar. Jeremy Harmon gave me access to numerous films that I needed to use to complete the project. My parents, Jerry and Cheryl Friedman, and my brother, Justin Friedman, have been there for me every step of the way. My sons, Leo and Clark Friedman, provided the inspiration I needed to complete the project. Those two scoundrels immeasurably amplify the quality of everything I do each day.
## Introduction
SINCE THE EARLY 1990S, THERE HAS been a spate of Hollywood films that uncharacteristically inspire viewers to reinterpret them retrospectively. Films employing this narrative mode are hardly new in Hollywood and other cinema contexts. Numerous Hollywood and world cinema classics, such as The Cabinet of Dr. Caligari (1920), The Wizard of Oz (1939), and Citizen Kane (1941), contain a late revelation that encourages spectators to reassess the meaning of a majority of what has come before. As part of the recent upcropping of films, Hollywood also remade three of the most celebrated twist-ending films released before the 1990s: Diabolique (1955; 1996), Psycho (1960; 1998), and Planet of the Apes (1968; 2001). In addition to Diabolique (a remake of the French film, Les diaboliques), contemporary titles, like The Crying Game (1992, an Irish film) and Vanilla Sky (2001, a remake of Abre los Ojos, a 1997 Spanish film), indicate that these films have never been linked exclusively to the U.S. commercial film industry. Yet, between 1990 and 2010, Hollywood backed over 40 films that encourage viewers to reinterpret them retrospectively, making it the most fertile period for such films in history. Many of these films were commercial and critical successes, including The Sixth Sense (1999), A Beautiful Mind (2001), and Inception (2010), which each garnered significant box-office returns and considerable Academy Award attention. This book explores the reasons for this production trend. I examine these films in their cultural, industrial, and technological contexts to explain why they became unprecedentedly attractive to Hollywood producers and some audiences at the time.
Regardless of how they are packaged by the industry, these films constitute a genre defined partly by narrative structure. I use the term "misdirection" to describe them because it effectively captures how they provoke spectators to understand narrative information initially in one manner and subsequently comprehend it in drastically new ways. Despite this common element, they are not a clear-cut industrial genre. As with other genres, most notably film noir, no one set out to make a misdirection film per se because there is yet to be an agreed-upon label for them. The discursive evidence I chart throughout this book, however, shows that audiences, critics, and producers engage with misdirection films in a manner that differentiates them from other Hollywood fare. The ways in which these films are designed and packaged to prompt various groups to interact with them distinctly make them a viable cultural category set apart from other generic classifications.
## What is in a Name?: The Misdirection Film Genre in its Contexts
Existing scholarly literature begins to provide discursive evidence that demonstrates these films constitute a genre. A number of works have been recently published on increasing narrative complexity in commercial U.S. film and television, a few of which even grapple with some of the sociocultural, industrial, and technological circumstances contributing to its growing popularity. No book-length study and only a handful of articles, though, focus exclusively on misdirection films in relation to their contexts. Instead, scholars generally lump misdirection films into larger categories that encompass an array of contemporary, narratively complex media texts. In a rare essay that wrestles with these particular films directly, Cornelia Klecker deploys the term "mind-tricking narratives" to describe how misdirection films are constructed and interpreted. Like Klecker, I contend that the classification "complex storytelling," which Aristotle originally devised in Poetics, is too "vague" because it merely signifies that a film somehow "does not adhere to a classical narrative structure" (emphasis in original, 121). Other scholars have also created categories to distinguish specific storytelling developments in contemporary Hollywood cinema from the broader trend of increasing narrative complexity. Perhaps most influentially, in Convergence Culture, Henry Jenkins coined the concept of "transmedia storytelling" to identify the proliferation of contemporary U.S. films and television shows that migrate viewers across media by encouraging virtual communities to master their complexities. Similarly, David Bordwell and Barbara Klinger employ the term "puzzle film" in individual chapters of The Way Hollywood Tells It and Beyond the Multiplex, respectively, to describe contemporary Hollywood films that inspire spectators to gain deeper appreciations by engaging in repeated viewings. My definition of the misdirection film, however, does not include many of the devices that also prompt this activity, such as the creation of eminently quotable dialogue and elaborate fictional worlds, encompassed by these more amorphous categorizations.
As one sign of how influential the puzzle film moniker has become in Film Studies, Warren Buckland edited an anthology, entitled Puzzle Films, just three years after Bordwell and Klinger mobilized the term. A majority of essays in that collection are, like the preponderance of extant scholarship, almost exclusively preoccupied with issues of narrative construction and comprehension; however, Thomas Elsaesser's chapter on the "mind-game film" constitutes the most significant prior publication to connect these films to some of the cultural and technological conditions, including anxieties about the persistence of traditional ways of thinking and the ramifications of the advent of the World Wide Web, that I explore throughout the majority of this book. However, his categorization includes films that play games with either unwitting characters or both those characters and audiences by withholding important narrative information. In addition to conflating films that deceive actual audiences with ones in which only characters are tricked, his culturally attuned analysis underplays the importance of the industrial shifts contributing to their relative appeal to producers. My focus solely on films that inspire retrospective reinterpretations only by spectators, in contrast, isolates one of Hollywood's most successful responses to the ways that some viewers now commonly interact with its products. Misdirection, therefore, also provides an important historical dimension that links the genre's contemporary constituents to notable antecedents, setting up key comparisons and distinctions between the period under study and earlier moments in Hollywood.
There are a number of reasons why "misdirection" is more appropriate than the "puzzle film," the "mind-game film," "mind-tricking narratives," and other labels used to classify these films. Whereas such alternatives reduce these films to gimmicks and ostensibly trivial leisure time diversions, misdirection instead alludes expressly to "direction." It thus intimates how filmmakers working in the genre encourage initial misapprehensions of narrative information. Misdirection captures how these films are often created and promoted as contests of wits between filmmakers and audiences. In fact, the directors most closely associated with the misdirection film—M. Night Shyamalan and Christopher Nolan—have tried to bolster their reputations as self-styled masters of the genre through marketing and by incorporating references into the films themselves that alert audiences to the intellectual competition. Shyamalan, for instance, followed up Cole Sear's (Haley Joel Osment) humorous observation in The Sixth Sense that great storytelling needs "lots of twists" by having Elijah Price's mother (Charlayne Woodard) in Unbreakable (2000) tell her son that a comic book gift "has a surprise ending." Similarly, after the opening credits in Nolan's The Prestige (2006), a film about rival magicians, Alfred Borden (Christian Bale) asks in voiceover "Are you watching closely?," which is followed by many self-referential insinuations to its narrative structure, including John Cutter's (Michael Caine) exclamation that "a pretty assistant's the most effective form of misdirection." These meta-generic references are especially apt in a magic-themed film and serve as a telling evocation of the category as a whole because misdirection also invokes illusionism. Filmmakers both flaunt the trickery and conceal it. As Matthew Solomon documents in Disappearing Tricks, the historical links between magic and early cinema are underappreciated. Misdirection is also the appropriate moniker, then, because it indicates the persistent ways that magic and cinema relate, connoting the enduring connections between conjuring and film form and narrative.
A strong reliance on Hollywood principles is central to the particular brand of magic that most misdirection films employ, revealing a primary reason why they have already been of considerable interest to some film and media scholars. The release of an uncharacteristically high number of Hollywood films with similar unconventional narrative structures raises questions about the endurance of Hollywood's foundational storytelling and associated representational standards. Such issues make it valuable to grapple with questions associated with narrative and genre in relation to these films to link them to their contexts. Although connecting these films to their historical conditions of production and reception is my primary objective, identifying the properties that make them a distinct set that can be separated from other Hollywood fare first begins to provide the evidence as to why various groups might conceive of them as a genre.
Figure. I.1. Young Elijah Price's mother explains that her comic book gift contains a surprise ending in Unbreakable.
Figure. I.2. The Prestige's Robert Angier uses his assistant, Olivia Wenscome, to distract the audience from detecting the trick's secret.
Many assessments that already engage with these issues support the argument that, regardless of their ostensibly atypical properties, misdirection films buttress the storytelling and representational conventions that theorists, like Bordwell, claim have been dominant in Hollywood since the classical mode of narration calcified in the 1910s. As Bordwell explains in Narration in the Fiction Film, the "classical" Hollywood film's focus on "psychologically defined individuals who struggle to solve a clear-cut problem or attain specific goals" distinguishes it from alternatives (157). Formal devices in virtually all Hollywood films since the late 1910s have been compositionally motivated because they are subservient to narrative and used invisibly to forward a canonic story in which a typically white, heterosexual, male protagonist with consistent traits strives to overcome obstacles until all causal lines of action are tied up predictably and unambiguously. In spite of the misdirection film's unexpected or ambiguous conclusions, narrative causality is not typically revealed as being ultimately attributable to the random forces of chance. Rather, once the epiphany is exposed or its meaning is discovered, it usually shows, in conspiratorial fashion, that narrative causality can be reinterpreted as being driven by the actions of clearly identifiable agents, who appeared incapable of having such authority. This alternative explanation can provide consolation to viewers accustomed to Hollywood's standard storytelling and representational practices that generally support the status quo, indicating a major reason why they often resonate with mass audiences. Of course, the retrospective reinterpretations that misdirection films inspire are not always this clear-cut in relation to what "really" happened and the plurality of potential readings that they can prompt render proclamations about their cultural politics being monolithic as suspect. Yet, the atypical ways in which they generally encourage viewers to discover the actual "truth" distinguishes them from other Hollywood films and ultimately appeals to audiences seeking to make order out of chaos.
Regardless of this and other connections between these films and their contexts, almost all existing studies of them focus primarily, if not exclusively, on issues associated with narrative construction and comprehension to assess their relationship to the classical paradigm. Bordwell, for instance, uses these films to refute those who declare that the fall of the studio system has ushered in a postclassical era in Hollywood. Such arguments, he charges, typically focus too much on anomalies and ignore the continued dominance of the classical Hollywood film. Moreover, he claims that virtually all Hollywood films that seem to break with the classical mode of narration, including misdirection films, actually do nothing of the sort. In support of this assertion, Bordwell analyzes how the contemporary "puzzle film," keeps "one foot in the classical tradition" by providing "legible variants on well-entrenched strategies for presenting time, space, goal achievement, causal connection, and the like" (The Way 73, 75). As his examination of Memento (2000) reveals, Hollywood films that seemingly contain the most confounding narrative and formal innovations, paradoxically, typically rely most on classical devices to stay intelligible. Memento, therefore, hits spectators over the head with redundancy to orient them in time and space as well as remind them that the narrative unfolds backwards because it is compositionally motivated by the protagonist's short-term memory loss.
Although there is considerable research on how misdirection films operate narratively and are interpreted, there is surprisingly little interest in the reasons for their increasing appeal. Bordwell, for example, contends that to determine why Hollywood experienced a "narrative experimentation surge in the 1990s," the impulse "to look for some broad cultural change as the trigger" should be avoided (The Way 73). Moreover, even though he subsequently admits that Hollywood storytelling has recently been "enhanced by DVD," Bordwell only discusses the links between the development of new technologies and Hollywood's shifting narrative strategies fleetingly so that he can attempt to prove that the classical film still reigns supreme (The Way 103). Likewise, in his study of narrative in The Sixth Sense, Erlend Lavik initially remarks that "it is tempting to speculate that this boom in twist movies is related to the rise of the DVD"; however, he cuts off that line of inquiry to conduct a formalist analysis of its narrative structure (60). Yet, I contend that by examining these very sorts of issues, it is possible to determine how the contemporary misdirection film exemplifies the ways in which Hollywood production remains consistent in some regards and, importantly, shifts with the times in other fashions.
Rather than primarily investigate misdirection films in relation to the classical paradigm, after mapping their narrative properties and generic links at the outset of this book, I devote subsequent chapters to an assessment of how and why they epitomize a kind of narrative experimentation that has become a crucial facet of twenty-first-century audiovisual storytelling. This focus fills the gap in the scholarly literature on these films by highlighting their historical and cultural significance. Undoubtedly, as Jenkins, Klinger, and others argue, this spate of films is connected to the advent of new media technologies that make complex narratives designed to be watched repeatedly and dissected online attractive to an industry that depends exceedingly on post-theatrical markets. The development of new communication and film exhibition technologies has created profitable revenue streams that give industry executives financial incentive to back misdirection films. Neither changing industrial motives nor technological determinism, however, fully explains why these films have become more attractive.
There are also key cultural conditions contributing to the appeal of the contemporary Hollywood misdirection film. Most of these films, even those few that initially appear to be narratively incoherent, such as Magnolia (1999), Memento, and Mulholland Dr. (2001), can be reinterpreted along the lines of classical Hollywood storytelling, usually by making recourse to authorship. This is culturally relevant because although these films initially appear to interrogate traditional ways of thinking by seeming to support the discourses of relativity, subjectivity, and multiculturalism, the retrospective reinterpretations that they often inspire articulate a stronger reluctance to abandon familiar epistemologies, including a belief in absolute fact, faith in teleological narratives, as well as the notion that identity is static and biologically determined. In particular, although many misdirection films ostensibly present challenges to the social order, on further review they usually end up encouraging reinterpretations that reassure viewers that foundational American ideologies, such as white patriarchal capitalism, are still dominant. In addition to referencing the connections between these films and magic as well as the ways they prompt filmmakers and viewers to engage in cerebral competitions, the misdirection moniker expresses nostalgia for a bygone era that many people still want to exist. The term is thus also most appropriate because it references how these films can assuage growing cultural anxieties about the unknowability of the "truth" by diverting viewers away from the cognitive crises of relativity and subjectivity with the fantasy that it is possible to determine what "actually" occurred and who was "really" responsible for events.
That is not to say that all misdirection films uphold dominant ways of thinking indisputably or inspire retroactive readings that render their meanings absolute. As in virtually all Hollywood films, ideological contradictions abound in the genre's constituents, and they can prompt clashing interpretations that are often persuasively supported by textual and/or extratextual evidence to vastly different ends. Neither my own readings nor the ones produced by fans that I present throughout this book, then, are definitive. In fact, one of the reasons I devote so much space to both is to demonstrate the myriad and often compelling responses that these films can inspire, even when they contradict each other. Yet, like all readings, those that follow are contingent on the contexts that shape creative decisions and viewer comprehensions because they represent tendencies in artistic creation and audience interpretation that are framed by historically specific circumstances of production and reception.
My exploration of the genre's historical significance is thus driven by the "context-activated" theory of reception that Janet Staiger outlines in Interpreting Films, which rejects more traditionally employed "text-activated" or "reader-activated" alternatives. Rather than deem meaning as being determined by the author, the text, or the viewer, Staiger contends that "the interpretive event occurs at the intersection of multiple determinations," which means that "interpretation is contradictory and not coherent" (emphasis in original, 48). The film analyses that ensue highlight how all interpretations, including my own, are always contestable.
This does not imply, however, that reception studies disregards the importance of textual properties, is governed by relativism, or proves the futility of historical research because it is inexhaustible and subject to the whims of the critic. As Klinger acknowledges, "Without question, historical reception studies has a strong interpretive dimension" ("Film" 112). Although privileging context rather than texts, authors, or readers does not isolate a film's conclusive significance because it is still beholden to the researcher's interpretations, it productively means, as Klinger posits, "the aesthetic or political value of a film is no longer a matter of its intrinsic characteristics, but of the way those characteristics are deployed by various intertextual and historical forces" ("Film" 112). Examinations of how the films themselves mobilize or repress discourses in circulation when they are created and the various moments that they are consumed renders intertextuality compatible with reception studies. Textual analysis and reception studies are not irreconcilable and can coexist, then, provided that textual properties are considered just one discursive element in the complex meaning-making matrix. My own readings and the ones forwarded by fans negotiate this tension by situating these interpretations in the particular contexts that shape them. As with all comprehensions, my own exhuming of textual significance and my summaries of fan readings of these films are underscored by how conditions of production and reception influence meaning-making activities at given historical moments. To begin this contextual analysis, I thus turn to a discussion of how misdirection films are constructed and understood in relation to the dominant production logic of the time.
## Subsets of Misdirection: Defining the Changeover and the Master Key Films
In misdirection films the manner in which the viewer is encouraged to reinterpret narrative information retrospectively materializes in two primary ways. I use the terms "changeover" and "master key" to differentiate these two discernable, though not always mutually exclusive, narrative forms. Inspired by Fight Club's (1999) revelation in which the protagonist, in voiceover, uses the term to explain what is transpiring, the changeover is an incident that occurs within the narrative flow that forces a primary character and the spectator to question the validity of almost all that precedes its emergence. It has a recognizable lineage in Hollywood, from films like The Wizard of Oz, in which the changeover reveals that the Technicolor Oz sequences were a dream, to recent films, such as The Usual Suspects (1995), in which it exposes the fact that Verbal Kint (Kevin Spacey) evades the authorities and augments his criminal legend by constructing a cover story from the contents of a bulletin board. Changeover films explicitly signal that there is another way to reassess the meaning of a majority of narrative information by incorporating an explanatory sequence into the narrative itself.
In the master key film there is no single moment within the narrative that blatantly reveals that an alternative explanation exists. The master key film instead contains a subtext that, when its meaning is discovered, provides spectators with a different way to comprehend the meaning of a majority of narrative information. Mysterious objects or bizarre narrative occurrences remain unexplained by the master key film's conclusion. The existence of these enigmatic properties only alludes to the possibility that there is an alternative way to interpret the narrative significance of what has come before. However, once the meanings of clues related to these lingering ambiguities are discovered and understood, it becomes possible to reinterpret the significance of most narrative information. As with the changeover, this narrative form has appeared in Hollywood for decades, from studio-era films, like Citizen Kane, in which the belated identification of Rosebud may provide a totalizing explanation for the reasons for Charles Foster Kane's (Orson Welles) befuddling character traits, to contemporary titles, such as Magnolia (1999), in which the master key of Exodus 8:2 strongly suggests the significance of the film's seemingly inexplicable rain of frogs and its other puzzling elements. The master key, though, has a less discernable genealogy than the changeover because although the existence of an alternative narrative explanation can be obvious (Citizen Kane), there are instances in which it is never clearly present (Magnolia). In the latter case, it is only discovered if groups, like critics and audiences, unearth the secrets and communicate their findings to others persuasively enough to convince them that there is, in fact, a more compelling way to reassess narrative meaning.
Obviously, narrative devices like the changeover and the master key did not originate in cinema. The consistent use of similar narrative structures in print media makes it possible to use established terms from literary theory, such as a concealed "frame narrative" or an "allegory," to describe the ways that these films tell stories. As Shouhou Qi's PhD dissertation The Shift of Emphasis and the Reception of Surprise Ending Stories (1900–1941) reveals, the misdirection narrative appeared with its most prominence in print in the United States during the late nineteenth and early twentieth centuries. During that time, a handful of leading American authors, including Thomas Bailey Aldrich, Bret Harte, Frank Stockton, Ambrose Bierce, Richard Harding Davis, and, most famously, O. Henry, whose name is now synonymous with the ironic, twist ending, each published short stories, containing late revelations that encourage drastic retrospective reinterpretations of narrative information. In addition to illustrating the differences in the ways that the changeover and the master key film are constructed and interpreted, literary theory provides a framework for conceiving of the two devices as fundamentally similar. As Emma Kafalenos writes, renowned authors, like Edgar Allan Poe and Henry James, have long been considered masters of making meaning "functionally polyvalent" because the relevance of narrative information changes drastically in many of their works as readers "read, page by page, and acquire further information" (475). Both the changeover and the master key are functionally polyvalent narrative structures because they encourage reassessments of the significance of a majority of narrative information after the revelatory evidence is exposed or unearthed.
Literary theory is only a starting point for describing how misdirection films actually operate because it does not account for how they depend on particular cinematic techniques to render their narratives functionally polyvalent. The Sixth Sense exemplifies just how strongly many misdirection films rely on classical narrative and formal conventions to trick spectators into jumping to erroneous conclusions. To put it another way, The Sixth Sense typifies how these films often use classical principles as the magician's pretty assistant to distract viewers from discovering the secret. Up until the film's revelatory sequence, there is little reason for spectators accustomed to Hollywood standards to suspect that Malcolm Crowe (Bruce Willis) is dead. Instead, the quest narrative appears to center on the assistance that Malcolm provides to the tortured Cole Sear, who is cursed with the ability to see—hence the surname—dead people. Accordingly, spectators are led to believe that the narrative will be satisfactorily settled only when it builds to a climactic moment in which Malcolm helps the scared child finally cope with the spirits that haunt him and resolves the classical film's heterosexual coupling subplot that focuses on his attempts to reconcile with his wife, Hannah (Olivia Williams). This is exactly what seems to occur. Malcolm becomes Cole's pseudo-father as well as his mentor and appears to help the child overcome fears of his paranormal visions while trying to rekindle his marriage.
The exposure of the revelation, however, shows that narrative events are actually related to a much different causality. The changeover unexpectedly reveals that Malcolm did not survive the shooting that occurred at the film's outset. It exposes the fact that the narrative has not only focused on Malcolm's efforts to help Cole deal with his issues, but that Cole has also been helping Malcolm cope with a problem of his own. This causal line of action is not made explicit, though, because for most of the film the narrative appears to center primarily on Malcolm's efforts to convince Cole that the ghosts that haunt him are simply searching for closure, which, of course, remains significant upon reinterpretation because Malcolm is such a spirit himself. Consequently, at approximately the film's midpoint, when Cole famously mutters to Malcolm that he "see[s] dead people" everywhere and they do not realize that they are dead, it only becomes evident, in retrospect, that he is also referencing the film's hidden causal line of action. Ostensibly, his statement merely suggests he is finally opening up to Malcolm by providing him with the information that will help him conquer the demons. Yet, the changeover reveals that the confession has another meaning: Malcolm is unable to rest in peace until he comes to terms with his own death. Importantly, this duplicity operates at a formal level as well because the critical exchange between the two characters is filmed in standard shot/reverse shot manner that relies on quintessential techniques, like standard eye-line matches, making it seem as though it is simply a prototypical classical Hollywood conversation. In retrospect, though, the reasons for the camera focusing on Malcolm so exclusively when Cole divulges his ability to see ghosts is not only designed to display the doctor's reactions to the child's confessions. It also subtly reveals the doctor is dead because when Cole speaks about the ghosts that haunt him, the camera centers on Malcolm. In short, the changeover shows how classical storytelling and formal principles are deployed to trick viewers into drawing incorrect suppositions initially about narrative meaning.
Figure I.3. Cole Sear's perspective during a shot/reverse shot conversation with Dr. Malcolm Crowe in which the child divulges that he sees dead people in The Sixth Sense.
As with the changeover film, the revelatory evidence in the master key film dramatically alters the significance of narrative information. Pulp Fiction, for instance, can be classified as a master key film because of a particular alternative reading that has been popularized by fans, which postulates that Jules Winnfield (Samuel L. Jackson) and Vincent Vega (John Travolta) have been sent from God to retrieve a briefcase that contains Marcellus Wallace's (Ving Rhames) soul from the devil's henchmen. Of course, like some master key films, the film's narrative is largely comprehensible to most audiences without this insider knowledge. Many observers claim that such an alternative reading is absurd and unverifiable because of this fact. Such debates proliferate in virtual communities, such as on the fan website PulpFiction.com, in which there is a discussion board devoted exclusively to this issue entitled "The Briefcase & The Band Aid." On one hand, some contributors argue that the master key interpretation is partly verified by the enigmatic Band-Aid on the back of Wallace's neck because they claim the soul is removed from there by the devil. Furthermore, the briefcase is opened with the combination "666" and Vincent and Jules, the latter of who routinely quotes the Bible, are seemingly saved from a barrage of bullets by divine intervention when they attempt to recover it. Yet, on the other hand, some fans counter that the briefcase's exact contents are purposely never revealed because it is simply a quintessential MacGuffin. To support their position further, they cite numerous interviews in which Tarantino himself consistently maintains that what is in the briefcase is and should remain a mystery. This kind of disagreement about the correct way to understand Pulp Fiction's meaning is one of the primary reasons why it can be difficult to identify master key films definitively. Any film can be read figuratively since it always can be interpreted in a multiplicity of ways. A film's status as a master key film, then, can be challenging to validate because the presence of an alternative narrative explanation is often vehemently denied by those who believe that overzealous fans have discovered connections never intended by its makers.
The existence of such alternative narrative explanations that render a film's ostensible ambiguities narratively significant undoubtedly raises questions about the misdirection film's relationship to the classical paradigm. Although these films can be distinguished from standard Hollywood fare, most are more closely aligned with the classical film than they are with other cinematic traditions, such as experimental film or art cinema, in which Bordwell argues that "the tight causality of classical Hollywood construction is replaced by a more tenuous linking of events" (Narration 206). As with art cinema, I grant that viewers often rely on suppositions about authorial intent to explain the causal relationship of events in the misdirection film. In contrast to art cinema, however, misdirection films typically do not feature psychologically ambiguous characters who remain uncertain about even their own motives. Instead, misdirection films generally can be retrospectively reinterpreted hyper-classically in spite of their seemingly non-classical tendencies precisely because the revelation often exposes how an indisputable cause—someone or something concrete—has been secretly orchestrating events behind the scenes. The misdirection film represents Hollywood's concerted attempt to appeal to a niche market seeking innovation within relatively secure confines. The narrative and formal experiments contained in misdirection films, like innovations associated with genre, are ultimately attractive to some audiences because they promise to be at once something that is comfortably familiar as well as something that is new and different. These films exemplify how Hollywood has historically striven to capture audiences by making a subset of films that are at once classical and something more.
Although the revised interpretations that misdirection films inspire do not necessarily abandon Hollywood's storytelling and representational principles, their growing popularity suggests that some contemporary viewers are drawn to films that explicitly expose the constructedness of the classical film. Bordwell, then, may go too far by claiming that the misdirection film's innovations are almost always completely contained by the classical paradigm. He is right that the classical mode is still dominant and that virtually all contemporary Hollywood releases, including misdirection films, are rooted in the classical tradition; however, misdirection films also have a crucial non-classical element. Whereas the classical film is supposed to conceal its storytelling and representational mechanics to suture viewers into the narrative, the misdirection film typically shows that those very principles have been employed to distract viewers from discovering the truth, shattering the façade of Hollywood's invisible style and drawing attention directly to it as a construction, in retrospect. The critical awareness that this epiphany fosters renders the narrative itself spectacular. The historical prominence of such excess is one of the primary reasons that it is difficult to maintain that the classical narrative is, or has ever been, what most characterizes the Hollywood film. Even during the height of the studio-era, Elizabeth Cowie writes, producers aimed to obtain "multiple guarantees," secured "through other elements of the package, notably stars and high production values, but also sensational and spectacular elements" (182). Narrative is just one variable that Hollywood has considered in its production formula and the historical role that spectacular attributes, like excessive narrative itself, have played in attracting audiences, especially over the past few decades, needs to be accounted for more thoroughly.
That does not mean that misdirection films jettison the classical paradigm entirely. As with most Hollywood films concocted with an indie sensibility to cater to niche audiences, misdirection films often seem to be more unconventional than they actually are. This can allow misdirection films to uphold dominant ideologies covertly, as their meanings often do not become fully evident until after repeated viewings. The cultural politics of indie films can be similarly difficult to ascertain because their supposed alternative status may function to obscure their core messages. The connections between mainstream and indie films make it notoriously slippery to define the genre, a phenomenon that is only compounded by Hollywood's growing interest in the sector. Consequently, Michael Newman's contention that the indie film is best conceived of as "a cultural category," determined by "a cluster of interpretive strategies and expectations shared among" various groups rather than by "industrial criteria or formal and stylistic conventions" is a useful analog for my study of the misdirection film (Indie 11). Like Newman, I justify my generic grouping by charting the ways in which misdirection films provoke various groups to engage with them differently than mainstream fare, which is itself also always a constructed category subject to change based on historical circumstances. Using the broad indie label to classify these films, then, would present the same problem as the "narrative complexity" classification because it does not specifically identify how misdirection films can be distinguished from other contemporary cinematic forms that are also known for both challenging and upholding Hollywood conventions.
The way in which indies encourage interpretive activities that depart from the classical film, however, is a good starting point for comparing it to the misdirection film. In American Independent Cinema, Geoff King documents how developments in the late 1970s and 1980s, such as the explosion of film festivals, the success of the VCR, and Hollywood's growing blockbuster conservatism, created a larger audience for offbeat films, providing independent producers with more distribution opportunities, albeit on a limited basis. These conditions paved the way for a string of successful independently produced films that got the industry's attention, including Stranger Than Paradise (1984), She's Gotta Have It (1986), and sex, lies, and videotape (1989). Hollywood's subsequent, direct involvement in many films typically classified as indies suggests why King theorizes that indies are often conceptually distinguished by their tendency "to employ devices designed to deny, block, delay, or complicate the anticipated development of narrative, to reduce clarity or resolution and in some cases to increase narrative self-consciousness" (American 63). In spite of these attributes, King notes that a majority of indies are readily comprehensible to most audiences, situating them "somewhere between" Bordwell's "classical" and "art cinema" modes of narration, but ultimately rendering them closest in spirit to the standard Hollywood film (American 101). Like most films labeled as indies, misdirection films, many of which are also categorized as indies, largely adhere to Hollywood principles. The indie and the misdirection film, though, can also be differentiated from classical films by how they play with those very conventions. King's notion of the indie, therefore, relates to the misdirection film for two reasons. First, it demonstrates that Hollywood genres are determined by how films are perceived as linked to one another by various groups, including film scholars. Second, it suggests that misdirection films are part of a larger production trend, characterized by Hollywood's increasing willingness to back films that challenge some classical conventions at the same time that they uphold others.
Although Hollywood has growing faith in supporting narrative complexity, the conditions that have made the misdirection film more appealing to producers and audiences are not exactly the same as those that contributed to the concurrent rise in indies. Of course, the indie and misdirection film genres only represent a small fraction of Hollywood's output since the early 1990s. Misdirection films have coexisted with many other types of Hollywood films since then, a majority of which are more closely aligned with industry standards because they play far less with classical conventions. Like the indie-era identified by Newman, the periodization in this book is somewhat unsystematic; however, there was a quantifiable increase in the number of misdirection films distributed by Hollywood since roughly the same moment that he identifies as the beginning of the "Sundance-Miramax era" at the very end of the 1980s. This starting point also marks the time in which the most recognizable prototypes of the genre, films like The Usual Suspects, The Sixth Sense, and Inception, were released. As Newman theorizes, all periodization is arbitrary, but he also shows how such conditions make it as logical of a starting point for the study of these films as any alternative. In an industry that prioritizes profit over everything else, it is also important to link the selected period to trends in Hollywood that made it more appealing to produce these films. As with the indie, the misdirection film's growth is partly connected to the changing revenue streams resulting from technological developments that altered the ways that Hollywood films are distributed, exhibited, and experienced. As a result, the chosen timeframe also dovetails with the rise and fall of media and communication technologies that impacted film production strategies and reception practices, most notably the DVD player, which was at the height of its popularity during the misdirection film's peak and was in steady decline by the end of the period under study. The circumstances that made this epoch well-suited for films that encourage retrospective reinterpretations of narrative information are what warrant further examination.
At the same time, it is important to note that neither indies nor misdirection films are solely products of the contemporary moment. Narratively complex and prominent independent films were backed by Hollywood during the studio-era. Additionally, the industry has largely outsourced production to independents since it transformed into its current role as primarily a distributor and financer after the end of vertical integration. As Bordwell admits, "Hollywood has long been a stylized filmmaking tradition" because even at the height of the studio system, "Fritz Lang and Orson Welles" consistently "put formal problems at the center of their work" and that "[Alfred] Hitchcock is virtually the patron saint of young filmmakers who want to tinker with storytelling" (The Way 74). Welles, Lang, and Hitchcock indeed each made films in the studio-era, such as Citizen Kane, The Woman in the Window (1944), Beyond a Reasonable Doubt (1956), Stage Fright (1950), and Psycho, which encourage viewers to reinterpret a majority of narrative information retrospectively. The presence of these films reveals that misdirection films are not just a product of changing cultural, industrial, and technological contexts, as some audiences have long derived pleasure from films that challenge classical standards and producers found ways to make them successfully in previous periods.
Despite the historical persistence of the misdirection film, it has unquestionably become much more common of late. For Bruce Isaacs, such a production trend reveals that "narrative experimentation is no longer the privileged domain of the European art film but commonplace in American studio productions" (130). Yet, as Hitchcock's long struggle to make a commercially successful misdirection film suggests, there are key reasons why they were produced far less frequently prior to the 1990s. As I argued in "Misdirection in Fits and Starts," an array of evidence reveals that the director always aspired to make a successful misdirection film; however, after his belated entry into the genre with Stage Fright, Hitchcock was so reticent to direct another one because of that film's critical and commercial disappointment that he revealed Vertigo's (1958) big secret only to the audience approximately two-thirds of the way through the film instead of at the conclusion, the time that it is exposed in the source novel. Instead of its remarkable success being most attributable to narrative construction, then, the famously effective reception of Psycho, which similarly contains a changeover, stems from other primary factors. In particular, the marketing campaign leveraged how Alfred Hitchcock Presents (1955–1965) made the director into a star and colloquially reshaped his reputation as the "Master of Surprise," largely explaining why he shot the film to look like an extended episode of the show to capitalize on his fame. Finally, Psycho was exhibited and marketed in ways that optimized its surprises. Such radical tactics were necessary when the movie theater was virtually the only venue for viewing Hollywood films, showcasing the obstacles inhibiting the misdirection film's effectiveness at the time as well as the roles that promotion and exhibition can play in its successful reception.
Clearly, much has changed in Hollywood since the release of Psycho that has made misdirection films more attractive to the industry. New film exhibition and media communication technologies have been a boon for the industry in a number of ways. For starters, they have helped to bolster the reputations of films that otherwise likely would not have received much attention after they disappeared from theaters. As of August 2015, for instance, Fight Club is listed as the 10th most popular film of all time on the Internet Movie Database's (imdb.com) Top 250 list. Fight Club's canonical status is significant, considering that it generated an underwhelming $37 million at the domestic box office on its $63 million budget (imdb.com). Such box-office disappointment is not uncommon for misdirection films, as many struggle to recover negative costs during their theatrical runs. Yet, Fight Club's post-theatrical performance also indicates the potential rapid reversal of fortune associated with these films. Fight Club has since become renowned for being both one of the best DVDs available and among the best Hollywood films ever made. Entertainment Weekly, for example, put the film's two-disc special edition atop its 2001 list of "50 Essential DVDs" (ew.com). Total Film Magazine readers similarly ranked Fight Club as the second best film of all time on its 2006 Top 100 list (totalfilm.com). This swift appreciation has not only elevated Fight Club's reputation; it also transformed the film into a moneymaker. In April 2001, Variety reported that Fight Club grossed $55 million in home video revenues ten months after the two-disc special edition DVD became available (Bing). This small fragment of the film's post-theatrical revenues, which does not include subsequent earnings from home-video sales and rentals, epitomizes Hollywood's economic logic at a time when the theatrical take can have little impact on overall profitability.
A film's box-office performance is not now insignificant, however, as Hollywood films have a much greater chance of becoming hits in the aftermarket if they do well in U.S. theaters. Additionally, not all misdirection films only become canonized or profitable after their theatrical runs. To wit, Inception grossed nearly $300 million at the domestic box office and over $800 million theatrically worldwide, helping it immediately secure a spot near the top of the imdb.com 250 list, where it still resided in the lofty 14th position in August 2015 (imdb.com). Inception is the exception rather than the rule, though, as some of its success is attributable to its atypical blockbuster status. No other contemporary misdirection film comes close to its $160 million production budget and its estimated $100 million marketing expenditures (Fritz). Prior to Inception, The Sixth Sense was the template for the successful misdirection film, earning $293 million at the box office during its domestic theatrical run, which made it the seventh highest grossing film ever in the United States at the time (imdb.com). Unlike Inception and many other contemporary films that reap enormous profits, though, The Sixth Sense was an unanticipated sensation because it was not packaged as blockbuster fare. Similar to Inception, however, it was well-received by critics and arthouse audiences, suggesting that its particular narrative structure was the primary appeal for many viewers. Its changeover was so memorable that its high artistic reputation appears to be secure in a number of circles because the American Film Institute included it on its revised 2007 list of the Top 100 films ever made, and imdb.com voters ranked it as the 159th best film of all time on the Top 250 list, as of August 2015.
Although such theatrical profits are impressive, a film's financial performance in theaters only reveals so much about its cultural and economic worth. The domestic box-office take is now often just an indicator of the effectiveness of exorbitant marketing campaigns. More importantly, a Hollywood film's profitability and cultural legacy extend well beyond its run in U.S. theaters. The imdb.com rankings and other similar lists, likewise, have limited value in relation to what they express about the value that a culture places on a particular film. The results of the Top 250 list have probably been skewed by a number of factors, including the influence of preexisting critical discourses on voters and the relatively homogeneous demographic characteristics of participants on the site. However, even though lists such as these are not precise barometers of cultural tastes, they can reveal a great deal about what is considered superior by a particular interpretive community. For voters on imbd.com, films that provoke retrospective reinterpretations of narrative causality are among those deemed to have the greatest artistic merit. Many other recent films appearing on the imdb.com Top 250 list as of August, 2015, including Pulp Fiction (7th), The Usual Suspects (24th), Memento (44th), The Prestige (51st), A Beautiful Mind (148th), Shutter Island (2010, 192nd), and 12 Monkeys (1995, 205th), provide further support for this observation. The existence of this potentially lucrative niche audience begins to indicate why industry executives were increasingly willing to produce so many misdirection films during the period under study in spite of their often shaky box-office performances.
The unexpected commercial and critical success of The Sixth Sense played a key role in making misdirection films a big part of Hollywood's creative plans. However, the film's remarkably strong box-office performance and six Academy Award nominations were surprises even to industry insiders. The circumstances surrounding the film's release suggest that it was neither intended to be a cash cow nor a prestige product because it was modestly budgeted at $40 million and hit theaters on August 6, 1999, the tail end of the summer blockbuster season and before the beginning of the release period typically reserved for Oscar fare (imdb.com). Disney's lack of confidence in The Sixth Sense was perhaps best exemplified by the fact that the media conglomerate sold the rights to distribute the film to Spyglass Entertainment and kept only a small distribution fee for itself (Stewart 302). The unanticipated positive reception of The Sixth Sense contributed to Hollywood's subsequent greenlighting of many misdirection films, some of which were specifically designed to garner cultural and industrial cachet. A Beautiful Mind, for example, an adaptation of Sylvia Nasar's 1998 Pulitzer Prize-nominated novel of the same name, was up for eight Academy Awards and scored four of the most celebrated statues at the 2002 ceremony, including the only Best Picture win for a contemporary misdirection film. A Beautiful Mind was not an anomaly because Atonement, which similarly was adapted from a critically acclaimed novel, Ian McEwan's 2001 book of the same name, was nominated for seven Academy Awards in 2008, including Best Picture, Best Supporting Actress, and Best Adapted Screenplay (imdb.com). Misdirection films clearly became crucial components of the media conglomerates' portfolios in the 2000s, as best evidenced by blockbusters like Inception and prestige films like Shutter Island, directed by Martin Scorsese, arguably Hollywood's foremost auteur. Such production strategies suggest that they were not just unexpected hits by that time, as they were throughout the 1990s, thanks to the surprise success of films, like The Crying Game, Pulp Fiction, The Sixth Sense, and The Usual Suspects.
Perhaps nothing more effectively displays the growing faith that industry executives had in the economic potential of misdirection films than their increasing willingness to attach established A-list stars, such as Halle Berry (Perfect Stranger [2007]), Pierce Brosnan (Shattered [2007]), Nicholas Cage (Adaptation [2002]), Russell Crowe (A Beautiful Mind), Tom Cruise (Magnolia and Vanilla Sky), John Cusack (Identity [2003]), Robert De Niro (Hide and Seek [2005]), Leonardo DiCaprio (Inception and Shutter Island), Michael Douglas (The Game [1997]), Richard Gere (Primal Fear [1996]), Anthony Hopkins (The Human Stain [2003]), Nicole Kidman (The Others [2001] and The Human Stain), Ben Kingsley (Shutter Island), Sean Penn (The Game), Meryl Streep (Adaptation), Denzel Washington (Fallen [1998]), and Bruce Willis (Pulp Fiction, 12 Monkeys, The Sixth Sense, Unbreakable [2000], Lucky Number Slevin [2006], and Perfect Stranger), to these projects throughout the period. Misdirection films have also been produced with such regularity of late because they have provided Hollywood with many of its newest, bankable leading men and women. Actors, such as Christian Bale (American Psycho [2000], The Machinist [2004], and The Prestige), Edward Norton (Primal Fear, Fight Club, and The Illusionist [2006]), Guy Pearce (Memento), Brad Pitt (12 Monkeys and Fight Club), Kevin Spacey (The Usual Suspects), and Naomi Watts (Mulholland Dr.), established their esteemed critical reputations in large part by starring in one or more misdirection films early in their careers. Finally, a number of Hollywood's hottest young filmmakers during the period, including David Fincher (The Game and Fight Club), Gregory Hoblit (Primal Fear and Fallen), Nolan (Memento, The Prestige, and Inception), and Shyamalan (The Sixth Sense, Unbreakable, and The Village [2004]), helped to develop their authorial standing by directing multiple misdirection films. The genre, then, was on the minds of the majors and ultimately resonated with many viewers starting in the 1990s because shifting cultural, industrial, and technological circumstances made them more attractive to producers and some audiences.
## Chapter Outline
In subsequent chapters, I first differentiate misdirection films more thoroughly from affiliated classifications and further chart evidence to support the claim that these films constitute a verifiable genre. Consequently, I employ the discursive approach to genre to theorize and provide historical evidence to back my assertion that the misdirection film is a distinct category. As my discussion of the master key film already suggests, though, there are potential issues associated with using the discursive approach to genre because it can take years for the alternative interpretations of narrative information to originate, circulate, and calcify. Chapter 1, therefore, grapples with both the benefits and shortcomings of the discursive approach to genre. After distinguishing the misdirection film from some of its closest cinematic relatives, I examine the utterances associated with it to show that although the discursive approach is more culturally and historically attuned than traditional forms of genre study, dogmatic reliance on it can negatively impact how the genre's constituents are subsequently interpreted and evaluated.
I begin to forge, in greater detail, the connections between contemporary misdirection films and their cultural contexts in chapter 2. I determine the reasons why two overlapping narrative forms—misdirection films and conspiracy theories—have appeared with such regularity in the United States since the early 1990s. Irrespective of their subject matter, misdirection films prompt spectators to engage in interpretive behaviors that align with those employed by conspiracy theorists. Like misdirection films, conspiracy theories counter "official" explanations with an alternative account that is more satisfying than what was initially provided. Although conspiracy theorizing seems to challenge traditional ways of comprehending history, it also resembles misdirection films by relying on the same kind of causal reasoning as the narratives to which it is opposed by suggesting that everything can be understood according to a totalizing causal logic that can be traced back to the specific actions of historical agents. Not coincidentally, this ever-popular, American cultural pastime has flourished in the United States during a neoliberal-era characterized by the consolidation of corporate power and the diminishing agency of non-elites. Misdirection films frequently articulate these concerns about dwindling individual autonomy. This tendency is typified by the chapter's case studies—Jacob's Ladder and Arlington Road—which portray protagonists as victims of devious plots against them and also encourage viewers to reinterpret their meanings conspiratorially.
Chapter 3 continues to link the misdirection film to its cultural contexts and paranoid thinking by examining how its depictions of gender illustrate how it frequently relies on and upholds classical standards to work its deceptive magic as well as maintain the existing social order. Although the revelations in contemporary misdirection films typically show that seemingly feminized primary male characters are unexpectedly more powerful than their conventionally masculine protagonists, they usually do not suggest that multiple masculinities are a reality. Rather than demonstrating the progressive potential of the decoupling of masculinity from other aspects of identity, Unbreakable and The Usual Suspects exemplify how misdirection films often portray manhood regressively to uphold dominant ideologies about gender. The chapter details how these two films present disturbing fantasies of male masquerade in which men covertly maintain their authority by flaunting their purported fragilities. Even though their duplicitous narratives are well-suited to display masculinity as a construction, misdirection films like these instead confirm that gender performance is a skill men master to hide the male essence that "really" lies beneath the surface.
The narrative logic of these films begins to indicate why they are attractive to Hollywood's most coveted market: young, white, male viewers. Many core fans identify with the struggle and eventual triumph of seemingly disempowered male characters who surprisingly turn out to be primary causal agents. Chapter 4 explores the ways in which the online reception of misdirection films demonstrates how the industry successfully generates profit by catering to audiences that most commonly interact with its products obsessively in the digital age. I examine how Mulholland Dr. and Memento were produced and promoted to capitalize on this lucrative target market's propensity to engage in repeated, post-theatrical viewings, particularly on DVD, and discuss their meanings online. Their atypical complexities inspired an inordinate amount of interpretive work devoted to figuring out their "true" meanings in virtual communities. Interestingly, fans often cite the intentions of their almost always male creators to support their interpretations. In the end, these films are successful in the aftermarket largely because they satisfy a desire for mastery, a yearning often associated with young, tech-savvy, male film collectors who also consider themselves discerning Hollywood cinephiles.
As with any genre, certain creative personnel have become inextricably linked to it. Chapter 5 extends my industrial analysis by documenting the sharply contrasting career trajectories of the two filmmakers most closely connected to the contemporary Hollywood misdirection film: Shyamalan and Nolan. While Shyamalan's branding efforts in relation to the genre have, at least for the time being, derailed his once promising career, Nolan's connections to the misdirection film have helped make him into one of the industry's most valuable auteur commodities. In the wake of The Sixth Sense, Shyamalan jumped at the chance to market himself almost exclusively as the genre's preeminent director, a ploy that began to backfire with successive films and has now been abandoned, perhaps only temporarily, to reconstruct his floundering image. In contrast, Nolan's success as a marketable property is largely a consequence of promotional strategies that dovetail better with New Hollywood's industrial logic. Importantly, his ascent to the top of the misdirection film genre has not been ignored in advertising, but it never became the primary emphasis of marketing campaigns. This approach has proven much more effective for weathering the vicissitudes of taste and box-office volatility than the myopically focused one associated to Shyamalan. The comparative assessment of the two directors' changing reputations, then, expresses some of the perils of packaging misdirection films for consumption as well as the connections between authorship, genre, and industrial conditions in contemporary Hollywood cinema.
Although the industry's consistent willingness to attach A-list stars and up-and-coming directors to misdirection films since the 1990s illustrates Hollywood's continued faith in the genre, it was not until 2010 that it briefly reached fully elite status. Prior to this time, only a handful of misdirection films had been prestige products granted relatively large budgets. In 2010, however, the industry released two misdirection films—Inception and Shutter Island—that demonstrate its confidence in the genre had soared to new heights. Chapter 6 outlines how the creation and promotion of these two films highlight the importance of the misdirection film to the industry's larger strategies at the time. Interestingly, DiCaprio starred in both films, making his portrayal of the reconstruction of the broken man key to their success. The final chapter, therefore, ties the book's major arguments together by providing case studies that show how the genre had become optimal by this time for its cultural, industrial, and technological contexts.
Surprisingly, the misdirection film seems to have fizzled out temporarily after reaching its apotheosis in 2010. The Conclusion briefly explores the reasons for and results of this sudden and unexpected decline in generic output. Technological advancements are at the root of many of the most plausible culprits and explanations, as changes in home-video greatly impacted the industry's revenue streams. In the preceding years, the supremacy of the DVD was already being challenged seriously by other nontheatrical platforms, including Blu-ray, on-demand, and streaming video online. These developments paved the way for moving-image texts that employ the narrative mode to begin migrating more frequently to other media. The television industry, for instance, was able to replicate Hollywood's formula with DVD by adopting a viable publishing model for the first time in its history. As the success of shows, like Lost (2004–2010) and Heroes (2006–2010), reveals, these technological advancements enabled television producers to cash in on products that similarly respond to the same cultural anxieties and desires as the misdirection film. Such a tendency begins to suggest how and why the narrative mode has endured even though it has virtually disappeared, at least for now, from the silver screen.
The chapters that follow this introduction map the historical trajectory of the contemporary Hollywood misdirection film—from its high point, beginning in the early 1990s to its current downturn—by situating the genre in its cultural, industrial, and technological surround. My exploration ultimately demonstrates how, on one hand, these films have been fashioned in response to certain conditions that have remained stable in Hollywood, while, on the other hand, the ways they are constructed are a consequence of new circumstances that made their production and reception more favorable than ever before in the industry.
1
## Retrospective Issues
The Discursive Approach to Genre and the Misdirection Film
A lot of recent films seem unsatisfied unless they can add final scenes that redefine the reality of everything that has gone before; call it the Keyser Söze syndrome.
Roger Ebert, from his review of Fight Club
Nothing prepared me for Magnolia's conclusion, and for that I am grateful... Magnolia is admittedly not for everyone, but those who "get" the film are in for something that ranks as more of a cinematic experience than a mere movie.
James Berardinelli, from his review of Magnolia
DURING THE STUDIO ERA, IT WAS standard for the "A" picture to be part of a program that played on a continuous loop. Consequently, viewers were unaccustomed both to getting to theaters by precise start times and to experiencing the feature film uninterrupted from beginning to end. Historically specific promotional strategies were required to market the misdirection film effectively in that context because it was not well positioned for dominant movie-going practices at the time. As Joan Hawkins documents, to coincide with the release of Psycho (1960), Alfred Hitchcock virtually copied tactics from the marketing campaign for Les diaboliques (1955) by creating advertisements that instructed spectators to arrive before the film began and urged them not to spoil the ending (378). Thanks in part to Hitchcock's marketing ploys, exhibition practices grew more favorable for optimizing the misdirection film's narrative pleasures. It subsequently became routine for exhibitors to screen feature films at advertised start times. In spite of these new conditions, Hollywood did not back many misdirection films until the 1990s when cultural, industrial, and technological conditions all became more favorable for their production and reception. Not coincidentally, at the same time when these films exploded in popularity, the term "spoiler warning" became part of the common parlance to discourage viewers from ruining the primary pleasures associated with them in online forums or elsewhere.
This is the kind of discursive evidence that begins to demonstrate that misdirection films do, in fact, constitute a genre with a rich history that has changed over time. In this chapter, I chart the discourses associated with Fight Club (1999) and Magnolia (1999) to reveal how the particular ways in which various groups engage with misdirection films render them distinct from other Hollywood fare. Fight Club's critical reputation has grown immensely since its theatrical release largely because of its complex narrative structure; however, its changeover was typically cited as a weakness initially. As a result, it was identified as a constituent of the genre immediately and suffered commercially and critically as a result of being characterized as a clear-cut misdirection film that employed the changeover unsuccessfully. Fight Club's changeover, however, transformed into an asset in the post-theatrical market, vaulting it into the contemporary canon. Yet, much debate remains about the film's merits because its changeover's full significance is difficult to interpret definitively. In contrast, whereas Magnolia was initially received more favorably by some critics, it took some time for its status as a misdirection film to calcify because its master key first had to be unearthed and understood. This delayed discovery and reinterpretation of the film, likewise, ultimately improved its reputation. Similar to Fight Club, though, there is no consensus about the master key's impact on the film's gender commentary, making it challenging to determine Magnolia's cultural politics. Put simply, there is still extensive disagreement about both films' takes on gender because of how the meaning of all narrative information potentially changes dramatically in light of the revelation's significance.
Drastic reconsiderations of misdirection films' cultural relevance are common, particularly in relation to markers of identity, because of both the atypical ways they are constructed and viewers interact with them. Such deferred classifications and assessments reveal that reception can depend heavily on how these films are classified, discussed, and comprehended at distinct moments in time, suggesting why generic groupings and interpretations can enormously influence a film's reputation. The following analysis of both films, then, shows why the creation and persistence of generic classifications shape how constituent films are subsequently understood and evaluated.
## Theoretical Gag Order: The Drawbacks of the Discursive Approach to Genre
While terms associated with narrative surprise, like spoiler warnings, are frequently deployed in relation to misdirection films, they are not the only kinds of media texts that inspire groups to utter them. Such alerts are perhaps now most commonly used by television viewers, especially in a digital era in which shows are increasingly watched repeatedly via timeshifting technologies and discussed zealously in virtual communities. The advent of new devices and platforms, like DVD, DVR, and social media, has played a huge role in prompting the "forensic fandom" that Jason Mittell identifies as characterizing the reception of the narratively complex fictional programming that now pervades increasingly serialized American television ("Forensic"). As the success of shows, such as Alfred Hitchcock Presents (1955–1965) and The Twilight Zone (1959–1964), suggests, television has perhaps always been the ideal moving-image medium for misdirection narratives since its relatively short, episodic nature usually misleads audiences for far less time during a single viewing than the standard Hollywood film. In fact, Hitchcock once memorably quipped in TV Guide in 1957 that the audience acts "like grown-ups when they get something for free in their own homes" but "become children again when they have to pay" (qtd. in Kapsis 38). Although television has never actually been free, the perception that audiences do not pay for it can help foster greater acceptance of narrative experimentation on the medium. As Mittell and Jonathan Gray point out in their discussion of the reception of Lost (2004–2010), television fans are often willing to "give themselves over to creators to be manipulated and controlled through the storytelling process" and that, contrary to conventional wisdom, spoiling does not ruin the fun, but instead "make[s] a show that they love even more enjoyable." Conversely, discursive evidence indicates that most misdirection film fans agree with USA Today's Mike Clark who, in a review of The Sixth Sense (1999), claims its changeover should be preserved because "anybody who would divulge that deserves the kind of fate that would permit young Cole to see him walking around in blood" (10E).
Even though there may be similarities in how misdirection narratives are structured on film and television, the differences in the ways they are typically received begins to illustrate why it is appropriate to conceive of these films as constituting a distinct genre. Of course, spoiler warnings are also used for many other types of Hollywood films that do not fall in the misdirection genre, particularly those that are also loaded with narrative surprises or complexities. This suggests some of the drawbacks of solely relying on discursive evidence from user groups to determine generic categorizations because many films that do not inspire retrospective reinterpretations of all that has come before also are associated with these utterances. Like other humanistic methods, the discursive approach to genre is inexact and subjective because it has a strong qualitative dimension that cannot be precisely quantified. There is, for instance, no minimum threshold of utterances that determines if a given film should be classified in a genre. More importantly, the existence of such evidence is often a matter of happenstance to begin with, which leads to unsystematic results that can leave generic creation in the hands of those whose motives for executing the groupings vary widely.
Despite these methodological shortcomings, there are numerous reasons why the discursive approach is useful to theorists who strive to avoid traditional genre study's ahistorical pitfalls. Rick Altman's Film/Genre seminally illustrates how the discursive approach's culturally and historically attuned method considers film genres to be "defined by multiple codes, corresponding to the multiple groups who, by helping to define the genre, may be said to 'speak' the genre" (208). This summary shows how the discursive approach can free genre study from its static and reductive trappings by perceiving of genres as cultural categories always subject to reconstitution based on how user groups define them at distinct moments. Thus, I rely on Altman's semantic/syntactic/pragmatic model to highlight how the misdirection films' textual properties (semantics) and meanings (syntax) prompt groups of people to engage with them (pragmatics) in ways that separate them from other Hollywood fare.
Many scholars recently attempting to rescue genre theory from its ahistorical leanings also incorporate pragmatics to highlight how user groups, such as audiences, critics, exhibitors, and producers, contribute to perpetually fluctuating categorizations. James Naremore posits in More Than Night, for example, that film noir is best conceived of as "a loose, evolving system of arguments and readings that helps to shape commercial strategies and aesthetic ideologies" (11). Film noir is a touchstone category for genre theorists precisely because the term was coined ex post facto by French critics. Even though no one set out to make a film noir during its classical period since the genre had not been created yet, constituent films were distinct to those who classified them in the group retroactively. Regardless of how capricious or accurate any of these originating critics' categorizations are, their ramifications have been significant. The genre's conventions have been frequently mobilized by filmmakers since at least the late 1960s often to challenge some Hollywood standards and certain dominant ideologies. I grant that such accounts of film noir's history and legacy may overstate the genre's unconventionality and its ideological uniformity; however, its atypical characteristics were recognizable to those who initially identified it and their definitions of the genre have since influenced many producers. Genres, in short, are both out there and not out there. Definitive elements are always arbitrary and subject to change, but those very conceptions can instrumentally shape production trends for years to come.
Unfortunately, the way that the discursive approach to genre has developed discourages this critical intervention because it dissuades scholars from birthing new categories. According to most accounts, the discursive approach in Film Studies can be traced back to Andrew Tudor, who, in the 1970s, presciently argued that genre study is predicated on a self-fulfilling prophecy that adheres to a circular logic, whereby constituent texts are cherry-picked to exemplify the attributes already thought to distinguish a category. To counter this methodological shortcoming, Tudor instead views genres as "sets of cultural conventions" defined by what groups of people "collectively believe them to be" at given historical moments (139). This is the key principle that guides Mittell's influential, yet misguided, application of the discursive approach in Genre and Television, in which he urges scholars to "examine the cultural processes of generic discourse prior to examining the generic texts that have been traditionally viewed as identical to the genre itself" (emphasis in original, 16). Discovering utterances that reveal a film's generic identity first indeed mitigates Tudor's empiricist dilemma. This approach, though, is contingent on luck that becomes more likely with the luxury of retrospect. To avoid succumbing to ahistorical methods, scholars have to wait for others to make the generic connections to get the hard proof to group constituents accordingly. More disconcertingly, the self-fulfilling prophecy is still possible, as the majority of discursive evidence can be ignored in favor of atypical utterances, such as Ebert's pejorative account of Fight Club cited in this chapter's epigraph.
Although Mittell urges critics to attend to the extratextual universe first, he admits that categories "run through texts," raising the specters of textual analysis and intertextuality in discursive genre study (Genre 13). Yet, he also critiques Altman for adding pragmatics to account for the discursive surround as a mere addendum to his formative semantic/syntactic theory of genre, which, respectively, examines both a genre's recurrent textual elements and how those attributes are repeatedly deployed. Specifically, Mittell contends that "despite Altman's foregrounding of cultural processes, textual structure remains the centerpiece" rendering it incompatible with a focus on how "categories operate outside the bounds of the text" (Genre 16). Mittell, therefore, encourages a turn to textual evidence only after the requisite extratextual utterances are discovered, regardless of how random the rationale is for their inclusion in the first place. To take Fight Club as an example, according to this logic, I could mention the self-aware references to its duplicitous narrative as confirmers of its status as a misdirection film only because Ebert luckily connected it already to The Usual Suspects (1995). Consequently, the unnamed narrator's (Edward Norton) voiceover after discovering that Tyler Durden (Brad Pitt) is a product of his dissociative identity disorder, in which he says it is "a changeover, the movie goes on and nobody in the audience has any idea" now becomes harmonious with Mittell's conception of the discursive approach.
The potential pitfalls of Mittell's application of discursive genre theory are more clearly evinced by a master key film, like Magnolia, which was not immediately classifiable in the misdirection film genre because it only became one thanks to belated utterances by critics and fans. Unsurprisingly, I have discovered no initial reviews and promotional materials that definitively categorize it as such. This is because the reasons for and meanings of writer/director Paul Thomas Anderson's inclusion of a climactic rain of frogs were designed to mystify, at least initially. As with Berardinelli's review of Magnolia referenced in the epigraph, Ebert's review, which comes close to putting it in the misdirection genre, claims that the film's "threads converge, in one way or another, upon an event there is no way for the audience to anticipate. This event is not 'cheating,' as some critics have argued, because the prologue fully prepares the way for it, as do some subtle references to Exodus." Yet, he subsequently advises audiences to "Leave logic at the door" to appreciate the film fully. At best, then, reviewers could only speculate that Magnolia might be narratively coherent after repeated viewings, a critical trope that persists in reviews of some of Anderson's subsequent films also filled with seemingly eternal narrative ambiguities, especially The Master (2012). In his review of The Master, for instance, Colin Covert of the Minneapolis Star-Tribune epitomizes these suppositions by noting "Anderson's audacious films defy facile interpretation. Having seen it just once, I'm not sure I grasp it... I'm uncertain if the film's final scenes should be interpreted as dreams or reality." This discursive evidence merely suggests there could be a master key that unlocks the meaning of the film's many ambiguities, making it a stretch to call it a misdirection film only based on such speculation.
Figure 1.1. The unnamed narrator faints during Fight Club's changeover upon discovering that Tyler Durden is a manifestation of his dissociative identity disorder.
There are conceivably many instances for which no corroborating extratextual evidence exists for misdirection films prior to the publication of textual analysis that unearths their secrets. This presents a substantial challenge to identify potential constituents by using only pragmatics, which is why I adopt Altman's semantic/syntactic/pragmatic approach rather than heed Mittell's call for scholars to turn to textual properties only after first identifying the requisite extratextual evidence. This is partly because producers can and do initiate intertextual connections, which can be accounted for by semantics and syntax, that help to situate films in generic categories. Another key shortcoming of Mittell's approach is that the chances of discovering discursive affirmations of generic identity are more remote for many films released before new technologies democratized both film criticism and repeat viewings in post-theatrical settings. This comparative dearth of available evidence makes critical discourse the most likely repository of these generic utterances because, as Altman theorizes, "critics' desires to use regenrification as part of their critical arsenal" are "unpreventable" (82). If such a critical tendency is inevitable, I contend that it should be embraced rather than avoided in spite of its ahistorical drawbacks. As Altman notes, critics' generic inventions can have positive outcomes. Decades ago, for example, numerous feminist-inspired scholars reclassified some melodramas into the non-industrially recognized woman's film genre. Although this regrouping is unverifiable using the discursive approach, as existing utterances did not categorize them accordingly, the interventions of these scholars encouraged productive reconsiderations of these films in relation to patriarchy, Hollywood conventions, prevailing evaluations, and so on.
Historical distance, then, is the frill that permits theorists to map the discursive roots of critically generated genres, like the woman's film and film noir. Crucially, such originating utterances only exist in the first place because innovative scholars and critics created the labels and associated groupings based on semantic and syntactic evidence without waiting for others, like industry professionals, to do it for them. As Altman's rigorous historical research illustrates, these stories of generic initiation are not the exception because categories always calcify retroactively, regardless of who prompts the grouping. Now taken-for-granted monikers, such as the western and the musical, moved from first being adjectives associated with established categories to the nouns that ultimately denoted the genres themselves in industrial discourse (Altman 50–53). It is impossible to know, therefore, if or when a term will transform from being a modifier into the stable generic label itself. Although few media scholars have theorized how and why their own generic creations come into fruition, literary theorist Tzvetan Todorov tried to justify his origination efforts by distinguishing between "theoretical" and "historical" genres in his book, The Fantastic, a genre he invented that is characterized by the reader's hesitation between the uncanny and the marvelous, two related genres he also birthed. Fifteen years later, though, Todorov retracted his position by arguing that while it is "always possible" for individual critics to identify "a property common to two texts, and to put them together in a category," genre becomes "useful and operative" when "we agree to call genres only the classes of texts that have been historically perceived as such" (Genres 17). To salvage Todorov's useful differentiation, Steve Neale contends that media scholars should "distinguish theoretical genres from genres proper by renaming the former 'theoretical categories' " (43). In contrast, I argue that the term "theoretical genres" is appropriate because it's impossible to know if or when new terms and groupings will redraw previously agreed-upon boundaries.
Despite all of its problems, it is misguided to abandon the notion of genre entirely because it remains the primary way that groups, such as critics, exhibitors, producers, and spectators, relate Hollywood films to one another and differentiate them from each other. Marketing strategies for misdirection films begin to reveal why it is valuable to use Todorov's notion of theoretical genres rather than completely jettison the concept of genre. In particular, it shows how producers have capitalized on a growing awareness of these films as distinct in the minds of audiences, while, at the same time, accentuating their historical generic identities. These practices were exemplified by taglines associated with The Sixth Sense. Although one of the film's taglines, "Not every gift is a blessing," highlights its status as a supernatural thriller, other taglines, such as "Discover the secret of The Sixth Sense" and "Can you keep a secret?," foreground its memorable changeover (imdb.com). Similarly, The Usual Suspects was marketed as a crime drama with the tagline "Five Criminals. One Line Up. No Coincidence." Additional taglines, including "The truth is always in the last place you look" and "In a world where nothing is as it seems you have to look beyond...," though, more directly alert viewers that there will be significant narrative surprises (imdb.com). The impetus for this seemingly contradictory marketing strategy is twofold. On one hand, it maintains the secret by securing expectations in the conventions of historical genres that may not inspire retrospective reinterpretations of narrative information. On the other hand, it allows producers to advertise the films in a hybrid fashion, as belonging to historical genres as well as to a theoretical genre renowned for narrative unreliability that has not yet been industrially codified as such.
Theatrical trailers and television spots for misdirection films demonstrate a similar dual marketing approach. An advertisement that aired shortly after the release of The Usual Suspects clearly positions the film according to its historical and theoretical generic identities. The ad displays scenes from the film, as favorable excerpts from reviews are superimposed over the images. The first anecdote to appear is taken from Jack Kroll's Newsweek review, proclaiming it to be "The best crime movie of the 90s." This reaffirmation of the film's status in a historical genre is followed by an omniscient narrator's voiceover and snippets from other reviews that emphasize the presence of the changeover. The words "Twist... Twist... Twist," are extracted from Tom Christie's Details review and coupled with the narrator's statement that the film has a "twist and a twist and a twist." Immediately thereafter, the narrator announces that the film has "a whopper of an ending," as the same phrase from Janet Maslin's New York Times review concurrently appears onscreen. In short, even though the commercial begins by situating the film as a crime drama, it subsequently accentuates its status as a misdirection film.
A cursory examination of film reviews also indicates that critics use language to describe these films as part of a theoretical genre by discussing them in ways that differ from their historical generic identities. Ebert's aforementioned reference to The Usual Suspects and aversion to its legion of imitators undoubtedly connects films officially categorized in other genres according to a new criterion that identifies their unique narrative structures as the semantic element that binds them together. Critic Rob McKenzie makes similar observations in response to the same upcropping of films:
Nowadays, though, what used to be a surprise is like the toy at the bottom of the Cracker Jack Box; it's a surprise that is not a surprise, but if we don't get it, we feel ripped off. Not only are these twist endings almost inevitable, they've gotten a lot more twisted. What used to be a denouement—literally the untying of the knot—is now just as often a renouement. We can suspend our suspension of disbelief for the first 95% of the show because everything we need to believe is at the end. These films are like Enron's double bookkeeping: one story going on at the surface, the awful truth percolating unseen beneath. (SP 7)
These kinds of reactions exemplify how critics attempt to place labels, such as "Keyser Söze syndrome," "twist endings," and "renouement," on films that are industrially classified in other ways. Clearly, critics have written about these seemingly unrelated films in a manner that groups them together and distinguishes them from other Hollywood fare. The discourses surrounding these films demonstrate that various groups of people cluster otherwise unrelated Hollywood films together because of the particular narrative engagement that they demand from spectators.
Yet, as the spoiler warning issue suggests, discursive evidence alone is often not enough to distinguish misdirection films from others in closely affiliated genres that do not encourage the exact same viewer activities. Many other types of films have dramatic surprises at the end, but very few of these revelations also inspire spectators to reinterpret the meaning of virtually everything that has come before. If the discursive approach is coupled with more conventional genre analysis, then such issues can be redressed. Altman's semantic/syntactic/pragmatic model is thus appropriate because it combines discursive analysis with an examination of the films' textual properties and recurrent meanings. Continued attention to semantics and syntax offers the possibility of including otherwise neglected films in the genre, in turn, creating the requisite discursive evidence for subsequent scholars to justify sustained groupings in a historically sound fashion. The value of this approach can be best demonstrated by a brief discussion of how misdirection films are related to other genres with similar semantic and syntactic elements, but also have unique enough textual properties to distinguish them from these affiliated films.
## If You've Seen One, You Haven't Seen Them All: Differentiating the Misdirection Film
Hollywood has long depended on genre to niche market a relatively undifferentiated product line that largely adheres to classical storytelling and representational conventions. That is not to say that the industry's strategy is to promote generic purity. As Altman's historical analysis reveals, Hollywood usually downplays generic specificity in favor of hybridity in marketing campaigns. After all, the classical film's dual plot structure—the primary quest narrative and the heterosexual romance subplot—is engineered partly to appeal, respectively, to perceived masculine desires for action and to purportedly feminine wishes to see characters overcome romantic relationship struggles. Yet, a film's semantic genre elements, particularly when they are explicitly foreground from the outset, as they are in most classical films, can modify viewer expectations. For David Bordwell, generic motivation always has a potential bearing on the kinds of hypothesis forming activities that the spectator conducts when viewing classical Hollywood films. He contends that genre cues and constrains interpretive activities further than the classical film already does by limiting the narrative outcomes most likely to occur. For instance, he argues that most Hollywood films are clearly positioned as constituents of genres that, unlike the misdirection film, do not purposely mislead spectators about the meaning of most narrative information. Instead, viewer guesses about narrative causality are typically met in a highly predictable fashion because a majority of Hollywood films end when the protagonist's clearly defined goals are satisfactorily attained or denied, fulfilling expectations raised at the start and leaving no primary causal lines of action dangling permanently.
Bordwell acknowledges that some Hollywood genres contain narratives that intentionally fool spectators about the meaning of information. The whodunit film is just one prominent example of a genre in which spectators expect that crucial narrative information will be withheld. In the whodunit, a primary player is usually revealed to possess seemingly secure character traits that unexpectedly prove to be unstable by the conclusion. The genre conventions, therefore, encourage spectators to determine who is misleading them before he or she is unmasked as the culprit. In an attempt to explain why virtually all films of this ilk should still be considered classical in spite of these tendencies, he theorizes that the expectations raised by genre are what keep them from being non-classical. Bordwell maintains that the whodunit film is classical because its "overt play of narration and hypothesis forming is generically motivated," meaning that "we want uncertainty, we expect both characters and narration to try and deceive us, and we therefore erect specific sorts of first impressions, cautious provisional ones, based as much upon generic conventions as upon what we actually learn" (Classical 40). The con artist film is another prime example of a genre that is difficult to label as non-classical even though it induces both diegetic characters and viewers to interpret narrative information in a manner that ultimately proves to be incorrect, usually because of the exposure of a late revelation. I do not, therefore, include contemporaneous Hollywood con artist films, such as Catch Me if You Can (2002) and Matchstick Men (2003), or any of David Mamet's similarly themed films, like The Spanish Prisoner (1997) and Redbelt (2008), in the misdirection film genre precisely because, following Bordwell's logic, their narrative revelations expose elaborate ruses in accordance with the expectations raised from the beginning.
Misdirection films, by contrast, are often packaged as constituents of historical genres that do not alert audiences that they will be narratively unreliable: A Beautiful Mind (2001) is a biopic, Unbreakable (2000) is a superhero film, Atonement (2007) is a romantic drama, and so on. Of course, not all misdirection films are marketed in a manner that disguises the presence of a likely duplicitous narrative. However, misdirection films packaged as constituents of historical genres that are designed to mislead spectators, such as the mystery, detective, and thriller, also provoke them to reinterpret narrative information in a patently non-classical fashion. Misdirection films advertised as detective films, for example, typically do not abide by the same rules that traditionally govern the classical detective film. Like the whodunit, Bordwell argues that even though the Hollywood detective film often misleads viewers about the veracity of character motivation to prevent them from guessing its unexpected revelation, it still ultimately adheres to the rules that govern the classical film. Again, he relies on generic motivation as his primary defense for this argument. Bordwell claims that Hollywood detective films abide by the tenets of "fair play," a set of rules that became codified in detective literature, which imply that "the reader has as good a chance to discover the solution as the detective does" (Narration 67). As long as the viewer is made aware that there is a puzzle to solve and has a legitimate chance to figure it out before the explanation occurs, the detective film should still be considered classical because generic conventions compensate for its apparent departures from Hollywood's narrative and formal principles.
As David Richter's analysis of Fallen (1998) demonstrates, however, misdirection films packaged in the detective genre typically fool audiences precisely because they violate the tenet of fair play. Fallen centers on detective John Hobbes's (Denzel Washington) effort to hunt down and kill a murderous demon named Azazel, who has possessed a series of human hosts. The film begins in media res, as Hobbes explains, in voiceover, that what is being depicted is his brush with death. The reasons that he describes the event as such, though, do not become apparent until the end of the film. When the film returns to the opening scene at the conclusion, it finally starts to become clear that Hobbes previously described this moment as his near-death experience because it portrays the detective's attempt to destroy the demon. Specifically, he has concocted a plan to lure it to a deserted location to trick it into possessing him after he kills its current host and then ingests fast-acting poison to kill himself. Importantly, it already has been established that Azazel can only possess a new victim if its current host comes into direct contact with another living person, meaning that the demon should die after Hobbes commits suicide. Even though there are no other potential human hosts present, the demon does not perish after it enters Hobbes's rapidly dying body. Instead, it possesses a stray cat that inspects Hobbes as the poison takes effect. The revelation scene, as Richter explains, thus, invalidates the spectator's expectations of both "story logic and conventions of representation" (15). In terms of narrative, the established rules made it seem as though the demon could only possess human beings. As it relates to form, the presence of Hobbes's voice in the opening narration made it virtually impossible to guess that it was actually the demon describing its near death experience as it possessed the detective momentarily before it moved to its feline host. Although the film's genre immediately signals its narrative unreliability, it cannot be comprehended according to habitual standards because the revelation violates the spectator's expectations in such a way that it is almost impossible that anyone could have predicted the resolution before its exposure.
These examples reveal why an exploration of semantics, most notably, how the revelation transforms the meaning of information in ways that distinguish these films from similar genres, is a good starting point for identifying constituents that have not been discursively labeled accordingly, or for providing the corroborating evidence for films that have been already grouped as such by random utterances. According to Altman and other genre theorists, though, semantics should be combined with syntax partly to compensate for taxonomic genre theory's tendency to downplay film's relation to culture. Syntax attends to how semantic elements are recurrently deployed in connection to relevant cultural conditions by examining their thematic significance. This kind of genre study has been often referred to as the myth or ritual approach, which is exemplified by Thomas Schatz's Hollywood Genres. In that book, he theorizes that film genres remain salient as long as they provide satisfactory, imaginary resolutions to irreconcilable ideological oppositions in the broader cultural sphere. Hence the reason that the continued, formulaic deployment of semantic elements that characterize genres consistently appeals to audiences. The misdirection film, therefore, leverages its fundamental semantic element—the changeover and the master key—in relation, at least in part, to the spectator's desire to access the "truth" during an age in which its very existence has been increasingly challenged.
In subsequent years, Schatz and other adherents of the myth and ritual approach have been rightly criticized for relying on the method to achieve ahistorical ends. Although such a perspective can yield historically sound connections between films and their contexts, it has been typically deployed to show that genres always resolve underlying cultural tensions in the same fashion and evolve toward increasing self-consciousness, as they supposedly move from a nascent developmental phase to a self-referential stage of maturity. It is inaccurate, then, to contend that all contemporary misdirection films express the same ideological agenda in relation to the status of the "truth." Even though most misdirection films contain revelations that assuage fears about relativity, there are some that seem to revel in perpetual uncertainty. I also do not want to imply that the interpretations of any of the misdirection films presented in this book are absolute. My readings and the ones offered by fans that I rearticulate are often persuasively countered by alternative comprehensions because elements contained in many of these films, like eternal ambiguities, provoke a plurality of viable interpretations. Additionally, it is a mistake to claim that the genre has become increasingly self-reflexive over time, as there have always been varying degrees of intertextual references in the genre. In Arlington Road (1999), for instance, Oliver Lang (Tim Robbins) exclaims "I guess we're not in Kansas anymore, eh, Toto." Such an explicit reference to a highly recognizable misdirection antecedent—The Wizard of Oz (1939)—is significant because 1999 is the very year that these films began flooding the market and became culturally ubiquitous. Producers of Arlington Road had no idea that, just one month later, the release of The Sixth Sense would dominate the box office for weeks, spawn a legion of imitators, and contain dialogue that would become inescapable in popular culture. Self-reflexivity occurs at any point in a genre's development, especially because various groups, including scholars, can retroactively identify semantic and syntactic generic links, irrespective of filmmaker motives.
The impossibility of determining authorial intent begins to suggest why semantics and syntax are best combined with a pragmatic approach to genre. Since accessing the minds of filmmakers directly is a fantasy, an examination of discourses that circulate around and run through these films, including textual evidence itself, reveals how genres operate culturally. In American Film Cycles, Amanda Ann Klein provides a foundation for privileging pragmatics by arguing that "while film genres are primarily defined by the repetition of key images (their semantics) and themes (their syntax), film cycles are primarily defined by how they are used (their pragmatics)" (4). Although I agree with her that pragmatics should be paramount, I do not share her view that this applies only to cycles and not to genres. As Klein correctly notes, existing genre theory typically treats cycles "as messy structures in flux, poised either to become stable genres or to disappear quickly" (6). Herein lies the rub with the notion of the cycle. Since genres are a retroactive phenomenon, cycles are always on the precipice of turning into a genre. Labeling a set of films as a cycle is a precarious endeavor if the premise is based on the fact that it suddenly becomes a genre when groups of people notice that it reappears. Such logic renders the occasionally used term "transhistorical cycle" contradictory or nonsensical because once a set of films with similar semantic and syntactic properties, like misdirection films, returns and is discursively identified, it can be characterized as a genre. An examination of the discourses circulating within and outside of the text demonstrates how semantics and syntax are interpreted and activated by various groups to construct generic parameters at various historical moments.
Despite her insistence on emphasizing the unique properties of the cycle, many of Klein's central notions are applicable to my exploration of the misdirection film genre. In addition to foregrounding the importance of pragmatics, her work indicates how studying particular moments in a genre's history can yield precise findings about its relationship to its specific contexts. She contends that it is possible to "view film cycles as a mold placed over the zeitgeist, which, when pulled away reveals the contours, fissures, and complicated patterns of the contemporary moment" (Klein 20). A small slice of a genre's history can indeed reveal more micro-level information than an exhaustive genre study does about a set of films' relationships not only to concrete cultural circumstances, but also to more exact industrial and technological conditions. I focus only on contemporary misdirection films, then, because I intend to show how they have been constructed in direct response to particular cultural, industrial, and technological changes that impacted commercial film producers and audiences during a specific period in time. As Klein's study also highlights, such claims about the links between genres and their contexts are further strengthened by grounding them historically within the discursive surround. Consequently, the following examples epitomize how utterances related to this group of films circulate as well as how and why such discourses can significantly alter the ways in which constituents of the genre are discussed, evaluated, and understood for years to come.
## Obliterating the "Ideal" Man: Fight Club's Misunderstood Changeover
The story of the belated appreciation of Fight Club is already a legendary illustration of this phenomenon. It is correctly identified as a key moment in the history of post-theatrical exhibition because it was one of the first films to be completely reassessed as a consequence of its tremendously effective DVD release. Its delayed success in the aftermarket is partially attributable to the way that the film's two-disc collector's edition DVD (2000) was packaged for consumption more strategically than its theatrical release. Fight Club's theatrical marketing campaign told spectators little about the film itself. The promotional materials, for instance, centered as heavily on a pink bar of soap as they did the actors in the drama. Fight Club's taglines were also intentionally ambiguous. One positioned the film as being about "mischief, mayhem, and soap" and another claimed that it "works great even on bloodstains" (imdb.com). Although these advertisements subtly allude to the film's commentary on consumerism and masculinity in the United States, they most clearly obfuscate its narrative content. Such tactics were employed for two primary reasons. First, they functioned not to alienate Fight Club's intended audience—young, white, heterosexual men—by making the film's explicit critique of their behaviors implicit. Second, they cloaked the changeover entirely, a tactic that departs from conventional misdirection film advertising because it generally at least alludes to the presence of an alternate way to interpret the narrative.
By contrast, in addition to promoting its then virtually unprecedented array of special features, the back cover of the two-disc collector's edition DVD alerts audiences that the New York Times claims that Fight Club "just might require another viewing," blatantly signaling the presence of its changeover. Moreover, many of the excerpted quotations that pepper the booklet accompanying the DVD foreground the film's critiques of conventional masculinity. The final quote listed in the insert, for instance, is from Bret Easton Ellis, author of American Psycho, another renowned novel from the period that similarly critiques hegemonic masculinity and was adapted into a misdirection film, who claims that Fight Club both "rages against the hypocrisy of a society that continually promises us the impossible: fame, beauty, immorality, life without pain" and is a "dizzying take on the male fear of losing power." In sum, the film only found a core audience after both its duplicitous narrative and depiction of contemporary white, heterosexual, American male paranoia were featured prominently in its advertising.
The importance of the changeover in the film's aftermarket resurrection cannot be overstated. Fan discourse illustrates the impact of its narrative structure on its now lofty, but still controversial, reputation. Spectators have consistently expressed uncertainty about Fight Club's gender politics in the years since its theatrical release, spawning great disagreement online about the film's cultural merits, or lack thereof. A fan who posted on the film's "User Comments" page on the Internet Movie Database in April 2008, for instance, calls Fight Club "the greatest movie ever made" and declares that it provides "great insight into the universal male psychology" (imdb.com). Unfortunately, it is impossible to guess exactly what this contributor believes the film has to say about men because, as other participants on the site demonstrate, there is substantial disagreement about its final message in light of the changeover's meaning. For example, a presumably male viewer's response, accessed at the same time as the one above, reports that after seeing the film he "wanted to move into a broken house and get in touch with the primordial nature that has been silenced in men everywhere by years of materialism bullshit" (www.imdb.com). Conversely, another contemporaneous viewer's comment speculates that "the solution Tyler offers is horrible, but he's so charismatic that you'll hardly notice it" (www.imdb.com). On one hand, then, for some spectators, Tyler's character represents the remedy for dispossessed American men. On the other hand, some spectators interpret the film as ultimately lambasting the narrator's hyper-masculine alter-ego.
Reviewers were similarly divided about Fight Club's commentary on gender. Peter Rainer of New York magazine blasts the film for depicting "the squall of an essentially white-male generation that feels ruined by the privileges of women and a booming economy." Entertainment Weekly's Lisa Schwarzbaum similarly complains that the film "floats the idiotic premise that a modern-day onslaught of girly pop-cultural destinations (including IKEA and support groups) has resulted in a generation of spongy young men unable to express themselves as fully erect males." In contrast, Michael Wilmington of the Chicago Tribune claims that the film "satirizes and examines violence far more than exploiting it" because it is a "hilarious ride into the twisted recesses of the modern male psyche, with an amazing knife-twist surprise ending that some may compare to the ending of The Sixth Sense." In addition to praising the film's gender politics, Wilmington also champions its changeover, which, like Ebert, he connects directly to another prominent misdirection film. This is significant not only because it provides further discursive evidence of Fight Club's placement in the misdirection film genre. It also showcases the connections between how the changeover is interpreted and the way in which the film's larger take on masculinity is understood. Ebert's negative discussion of the film's changeover, for instance, bleeds into an unfavorable discussion of its portrayal of gender by characterizing it as "macho porn" that women "will instinctively see through" even though "men may get off on the testosterone rush." The discrepancy about the film's gender politics often hinges on how viewers comprehend the changeover's significance and its retroactive domino effect on the film's entire meaning, suggesting why there is a lot at stake in how it has been read in relation to genre and other constituents of the misdirection film category.
As these responses to the film indicate, Fight Club contains a complex narrative that is challenging to decipher initially and remains confounding on repeated viewings despite the fact that it has a changeover that ostensibly reveals its secrets unambiguously. The narrative centers on the reasons for and proposed remedies to the unnamed narrator's (who often refers to himself as "Jack," one of the pseudonyms he uses at support groups) malaise. The main cause for the film's polarized reception is the vastly different interpretations that persist even after the exposure of the changeover about what the film identifies as the culprits for and antidotes to the narrator's problems. Determining the film's takes on gender and sexuality, then, depend on how the changeover's significance is understood retrospectively. Most notably, considerable debate remains in virtual communities about the extent of the unnamed narrator's dissociative identity disorder, which Tyler is revealed to be a manifestation of during the film's memorable changeover sequence. Some fans speculate that, like the characters introduced and incorrectly presumed to be real in similarly themed misdirection films, such as A Beautiful Mind and Identity (2003), that a number of Fight Club's other characters, most prominently, Marla Singer (Helena Bonham Carter), may also be imagined, alternate personas. Such discussions are epitomized by the forum on the website, Movie & TV Stack Exchange.com, entitled "In Fight Club is Marla Singer a second figment of Jack's imagination?". Obviously, if Marla—the film's sole, primary female character—is only a fabricated product of the narrator's disorder, then interpretations of how she and the other characters are represented are likely to shift drastically.
Fight Club's narrator is initially depicted as a cubicle-inhabiting, corporate drone who works as a product recall cost appraiser for a large automaker and takes orders from a male boss who is infatuated with traditionally feminine concerns, like the color cornflower blue. The narrator's emasculating role demands that he help the company make huge profits by concealing the dangers associated with their vehicles. As the sequence in which the narrator's condo breathtakingly transforms into the pages of a furniture catalog illustrates, he seems to put up with the job to feed his insatiable hunger for consumer products. When his condo unexpectedly explodes and all of his possessions are incinerated, however, his values begin to change. The narrator's metamorphosis is guided by his decision to reach out for help to a mysterious acquaintance, named Tyler Durden, instead of Marla, the woman he loves to hate, after his condo is destroyed. When Tyler and the narrator subsequently meet at Lou's Tavern, it becomes apparent that the charismatic Tyler is trying to shepherd the narrator's masculine transformation by spewing cliché-ridden rants about the feminizing forces he deems responsible for their problems, such as the influences of consumer culture on traditional manhood.
Upon leaving the bar, Tyler alludes to a budding sexual tension between the two by asking the narrator to "cut the foreplay and just ask" if he can stay at his place. After the narrator finally makes the request, Tyler invites him to squat at his dilapidated house until he gets his life back together. In return for the favor, Tyler demands that the narrator "hit him as hard as he can," leading to their first fight. Surprisingly, the uptight narrator finds the sadomasochistic activity to be pleasurable because during their almost post-coital exchange, the narrator informs Tyler that "they should do this again sometime." When the two finally arrive at Tyler's house, however, it becomes clear that they will not be consummating their relationship with homosexual activity. As Melissa Iocco theorizes, the film's heterosexist tendencies are evident when Tyler shows him around the house because he only points out the location of the bathroom and stresses that they will stay in separate bedrooms. She keenly notes that his tour of the place reveals that "now that they are in a different and more personal environment and situation, their bodies and fluids should not mingle" (Iocco 52). A strictly homosocial relationship then develops between the two, eventually encouraging the narrator to emulate Tyler by giving up all of his possessions. The friends end up founding a bare-knuckled boxing organization, called "Fight Club," which meets in the basement of Lou's Tavern, to help other emasculated men reclaim their lost manhood. In this restricted setting, these seemingly feminized men also regain their virility by attaining sadomasochistic satisfaction, albeit in a violent and patently non-sexualized fashion.
Under Tyler's direction, Fight Club eventually "leaves the basement" and turns into a full-fledged rebellion, referred to as "Project Mayhem." Juxtaposed with the comparatively diverse Fight Club, Project Mayhem is comprised almost exclusively of young, white, and presumably heterosexual men, who are bent on annihilating the feminizing forces that they perceive to be the causes of their predicament. As Iocco contends, by taking this turn, the film ultimately seems to promote "homosocial and male bonding through violence and destruction" (50). Importantly, the film also shuts down the possibility that a homosexual relationship will ever develop between Tyler and the narrator. During their most sexually tinged scene in the house's bathroom, a naked Tyler tells the narrator what he believes is at the root of their problems. While taking a bath, he and the narrator share stories about how they were abandoned by their fathers, leading Tyler to speculate that "we're a generation of men raised by women, I'm wondering if another woman is really the answer to our problem." Thus, it seems that the burgeoning revolution should potentially strengthen the latent sexual bond between Tyler and the narrator by removing women from the picture, but it ultimately has the opposite effect.
For many critics and scholars, the presence of this kind of troubling dialogue that scapegoats women and fantasizes about eliminating them is a testament to Fight Club's misogynistic sensibility. In fact, the film has been condemned by a number of scholars, such as Terrell Carver, who disavows it for its purportedly reprehensible treatment of Marla Singer, whom he interprets as a metaphor for " 'woman,' very enigmatic and 'other' " who is "dangerous and unpredictable, sexually voracious in a totally ludicrous way and in sum the object in the most basic and stereotypical kind of male fantasy" (130). In relation to the film's complicated takes on capitalism and patriarchy, Henry Giroux also reads the film as having reactionary tendencies. He theorizes that Fight Club "is less interested in attacking the broader material relations of power and strategies of domination and exploitation associated with neoliberal capitalism than it is in rebelling against a consumerist culture that dissolves the bonds of male sociality and puts into place an enervating notion of male identity and agency" (henryagiroux.com). For Giroux, the film only draws attention to the ways that late capitalism and the associated triumph of consumerism have purportedly feminized many white, heterosexual, middle- and working-class men, meaning that it fails to demonstrate how neoliberalist economic policies have consolidated the power of elites and made it more difficult for traditionally subjugated groups to attain equality in economic, political, and social arenas.
Although Giroux's reading is admirable, it does not jibe with the ways in which Fight Club's ending encourages viewers ascribing to the "Marla is just another persona" explanation to reinterpret narrative information much differently. Like Carver, Giroux claims that Marla functions as a metonymical stand-in for the kind of women who some men would like to believe "exist to simultaneously make men unhappy and to service their sexual needs." Such a portrayal, Giroux argues, "reinscribes white heterosexuality within a dominant logic of stylized brutality and male bonding that appears predicated on the need to denigrate and wage war against all that is feminine." I grant that the film depicts how white, heterosexual, American men can overcome their emasculation; however, I contend that it does not urge them to accomplish this goal by blaming women for their problems and also laudably exposes the absurdity of hegemonic masculine gender performance in the process.
To ascribe a blatantly chauvinistic logic to Fight Club, critics have to ignore the full implications of the changeover, which is key to one of the most compelling retrospective reinterpretations of the film's meaning circulating in virtual communities: the "Marla is not real" explanation. Fight Club unexpectedly ends in classical Hollywood fashion with the romantic coupling of Marla and the narrator. Yet, this happy Hollywood ending, which seems incongruous with what has come before, is immediately thrown into question by the demolition of the offices of the major credit card companies that were the targets of Project Mayhem, which, in anti-capitalist fashion, erases the debt record. Additionally, shortly after the couple holds hands, the film cuts to an almost subliminal insertion of a close-up of a penis, an image that Tyler would typically splice into family films while working as a projectionist. These seemingly contradictory images strongly encourage viewers to scrutinize the film's positions on gender and sexuality by interrogating the veracity and meaning of the compulsory heterosexual coupling.
In addition to these yoked visual contradictions, the film's ending is rendered more ambiguous by the Pixies' "Where is my Mind?," which plays on the soundtrack as the towers collapse and the end credits roll. This music choice suggests that although it may seem that the protagonist's psychological state has become unequivocally clear because of the extended battle just depicted with an imaginary Tyler, it is important to remember that the narrative has been focalized through the perspective of an unreliable narrator throughout most of the film. Indeed, the film's final sequence, in which the action finally catches up with the narrator's opening voiceover, is the only time that narrative information is not filtered directly through the narrator. After all, the very first frame makes it clear that viewers are receiving narrative information through the mind of the unnamed narrator, as the creative opening credit sequence literally takes place inside of his brain. Once the opening credits end, the camera zooms out of the brain to showcase the character whose skull the camera was just inside. The narrator's voiceover then begins the film's prolonged flashback structure by hinting at the real relationship between Tyler and him. His narration informs spectators that he is constantly asked if he "knows Tyler Durden," "how that old saying of you only hurt the one you love works both ways," and that "[he] knows" about the details of Project Mayhem "because Tyler knows" about them. After a virtuosic, digitally enhanced tour of the foundations of multiple office buildings visually confirms Project Mayhem's destructive potential, the narrator concludes the opening voiceover by claiming that he "realizes that all of this, the gun, the bombs, the revolution, has got something to do with a girl named Marla Singer." Rather than interpret this statement as an expression of the film's misogyny, it is possible to read it as a commentary on the war occurring inside the narrator's mind because it is difficult to accept Marla as being any more real than Tyler.
Figure 1.2. Marla Singer and the unnamed narrator hold hands, as they watch the Project Mayhem-inspired demolition of credit card companies at the end of Fight Club.
The virtual impossibility of Marla's existence is suggested immediately by her character introduction. The narrator first encounters Marla at a meeting of a support group for testicular cancer survivors, a quintessential depiction of literal emasculation that would not likely accept a female member. Not coincidentally, the theme of castration recurs throughout Fight Club, as many male characters are threatened with having their balls cut off by others. Although the narrator never had testicular cancer, he decides to join the group because it provides him with a cathartic release that helps him cope with his insomnia. Marla, like the narrator, is a faker who visits the groups in order to ease her own maladies. Her obvious status as an imposter in this and other similar groups, such as the one composed of people suffering from tuberculosis in which she reports her chain "smoking doesn't go over at all," begins to signal the low probability of her physical existence.
The narrator then claims, in voiceover, that Marla is a problem because her presence in the groups ruins his ability to release his emotions, meaning that he again cannot sleep. When he follows Marla out the door presumably to confront her, the first visual clue that she might be simply a figment of his imagination appears. As the camera takes his point of view, an almost subliminal shot of Tyler, his other primary alter-ego, which reappears throughout the opening scenes, is superimposed on the image of her walking down the alley. The film then cuts to a fantasy scene from the narrator's perspective in which he challenges Marla about being a fraud. Her presence in his mind is subsequently reconfirmed when the narrator participates in a guided meditation seminar for cancer patients for a second time. As his voiceover informs viewers if he "had a tumor he would name it Marla," the camera cuts to what was previously depicted as his imagined place of serenity: a frozen cave inhabited by a talking penguin. When he enters the cave this time, though, he shockingly discovers that a smoking Marla has replaced the penguin.
Once the session ends, the angry narrator finally seeks out Marla to threaten her. When the two begin to exchange dialogue for the first time, it becomes clear that they at least think very similarly because they complete each other's sentences. After the narrator threatens to expose her, Marla surprisingly responds by claiming that she "saw him practicing telling her off," alluding to her privileged access to his mind because it can only reference the fantasy confrontation sequence that was depicted moments ago. To compromise, the two characters then begrudgingly attempt to split up the support group meeting schedule so that their paths never cross. Their discussion escalates into a disagreement because Marla claims to want to attend the meetings of both "the brain parasites" and "the organic brain dementia" groups, to which the unnamed narrator retorts she "can't have the whole brain!" The double-entendre-laden sequence ends with Marla standing in the middle of a busy city street in which, mysteriously, no cars have to swerve out of her way; this moment also hints at the revelation a final time, as their dialogue reminds viewers that the narrator's real name still has not been mentioned because he has gone by so many different aliases in support group meetings.
The alternate identity that initially becomes dominant in the narrator's mind, of course, is Tyler and not Marla. Tyler, like Marla, is a chain-smoking caricature; it is difficult to imagine he could be real as well because he is such an exaggeration of the masculine ideal. In spite of there being fleeting glimpses of Tyler early in the film, most explicitly as he passes the narrator on a moving walkway at the same time that the narrator's voiceover ruminates about the possibility of awakening "in a different time and a different place as a different person," his character is not formally introduced until the two serendipitously meet on an airplane. As with Marla, the actual relationship between the two characters is hinted at when the narrator tells Tyler that they "have the exact same briefcase." Their subsequent exchange reveals Tyler's self-assurance, sexual ambiguity, and anarchistic tendencies. After informing the narrator that he knows how to make explosives out of household materials, he complicates his masculinity and sexuality by deliberating about whether to give the narrator "the ass or the crotch" before exiting the row, suggesting both his traditional confidence and the budding sexual tension between the two.
Figure 1.3. Fight Club's unnamed narrator fantasizes about confronting Marla Singer to demand that she stop attending the same support groups.
Fight Club's memorable changeover sequence, the inspiration for my use of the term to describe the narrative tendency, leaves no doubt that Tyler's character should be interpreted as how the narrator initially imagines he would be if he was the ideal man. The narrator's growing awareness of his real relationship to Tyler prompts him to call Marla to ask if they have ever had sex, which the film previously revealed to be satisfying, but cartoonish and degrading to Marla. In fact, she informs the narrator in a subsequent conversation that this is one of Tyler's greatest talents. But, if both Tyler and Marla are not real, these ridiculous sex scenes become little more than the stuff of masturbatory fantasies. Marla's response on the phone sparks the apparitional appearance of Tyler, who chastises the narrator for violating their code of silence. Their exchange finally initiates the narrator's realization that he and Tyler are actually the same person. In classic changeover fashion, the film hammers viewers over the head with the revelatory information by presenting flashbacks of Tyler's previous acts of mischief and mayhem, only this time with the narrator in Tyler's place or with Tyler erased from the image entirely. To confirm his real identity further, Tyler then explains that he represents "all the ways" that the unnamed narrator "wishes he could be." Speaking not only to the narrator, but also to men in the audience, Tyler exclaims that he "looks like you want to look," "fuck[s] like you want to fuck," is "smart, capable, and, most importantly am free in all the ways that you are not." There was perhaps no better choice for this role than Brad Pitt, who had already won People magazine's "Sexiest Man Alive" award in 1995 and would be the first to be crowned with the solo title a second time in 2000 thanks largely to his chiseled appearance in Fight Club, to play the character who possesses the attributes that are the envy of many men. In comparison to the comparatively soft-bodied and plain-looking Edward Norton, there is little question which character spectators are supposed to identify as the ostensibly ideal man.
In contrast to critics who deem Fight Club as misogynistic for these reasons, however, the film can be read as progressive in relation to gender because of how what transpires after the changeover ultimately critiques Tyler's ideal masculinity, especially if Marla is also interpreted not to be real. Just moments before Marla and the narrator hold hands, the narrator destroys his hyper-masculine alter-ego with a self-inflicted gunshot that literally lobotomizes the Tyler portion of his brain to make room for the Marla persona. This act is necessary, as Tyler insinuates during the changeover sequence, because Marla knows too much about his real identity as a similar manifestation of the narrator's dissociative identity disorder. Consequently, the narrator's decision to obliterate his hyper-masculine persona in order to save Marla suggests that the film's gender politics are more radical than scholars like Carver and Giroux acknowledge. I grant that the narrator's decision to shoot himself is understood by the members of Project Mayhem, who may also be just a figment of his imagination, as a quintessential act of masculine strength. Yet, his decision to destroy Tyler and not Marla reveals that, as he tells Tyler, he does not "need [him] anymore" because his "eyes are open" to the destructive nature of his masculine violence and aggression. Put simply, he finally realizes that he has to allow his more feminine alter-ego to become the dominant persona to put an end to his discontent.
Although the narrator's choice to kill Tyler clearly expresses the unacceptability of this traditional form of masculinity, his act also allows him to pursue a conventional heterosexual romance with Marla, albeit perhaps only in his mind, which, in turn, demonstrates why critics are correct to claim that the film's cultural politics are contradictory and somewhat conservative, even if Marla is not real. His decision to be with Marla reiterates the film's heterosexism because the sexual tension between Tyler and the narrator posed such a threat to the patriarchal order that it needed to be destroyed and replaced with a more traditional romantic relationship. As the narrator assures Marla in the film's last line, although she met him "at a very strange time," the happy ending indicates that everything is now "going to be fine." In sum, Fight Club ultimately conveys mixed messages to viewers subscribing to virtually any reading, including the provocative "Marla is not real" interpretation, by rewarding its white, male protagonist with the literal woman of his dreams for abandoning his aberrant masculine identity in favor of a kinder and gentler, yet unquestionably heterosexual, masculine persona. The film's cultural status, therefore, is contingent on how groups interpret and judge the impact that the ambiguous changeover has on the rest of its meaning. What people make of the film, in the end, is overwhelmingly linked to the ways in which it is evaluated and discussed as a misdirection film and its relationships to other films in the genre.
## Atoning for the Sins of the Father: Unlocking the Meaning of Magnolia's Master Key
Like Fight Club, Magnolia's convoluted narrative is challenging to interpret definitively, even in retrospect, making it difficult to decipher its cultural messages. Magnolia's narrative ostensibly centers on a loosely connected series of events that depict the individual crises an array of characters—none of whom can be clearly identified as the protagonist—inhabiting suburban Los Angeles experience. Many of these characters interact little with each other and even though their lives sometimes intersect, they all seem to follow relatively distinct narrative trajectories—that is, until the film's climactic moment in which a rain of frogs affects them simultaneously and brings many of them into direct contact. Although the rain of frogs provides closure to all of the characters' predicaments, its meaning is never formally explained. Instead, the film's characters and spectators are seemingly left to ponder its causes and effects eternally.
Magnolia's three-hour-long and apparently non-classical narrative is largely a product of the atypical circumstances under which it was made. In an unusual move for the industry, New Line Cinema, a subsidiary of Time Warner, gave director Paul Thomas Anderson an almost unprecedented degree of artistic freedom on the project. Specifically, production head Michael De Luca famously told Anderson that he would have final cut on the film before the studio even knew anything about the script (Hirschberg SM 55). The uncharacteristic decision to give Anderson such authorial power was motivated by the critical and commercial success of Boogie Nights (1997), the director's second feature film and his first project for New Line, which was nominated for three of the most celebrated Academy Awards and won Golden Globes for Best Supporting Actor (Burt Reynolds) and Best Supporting Actress (Julianne Moore). However, although the film was ultimately profitable, it only brought in $26 million on its $15 million budget during its run in U.S. theaters (imdb.com). This is not a profit margin that typically warrants the decision to give a relatively unproven commodity such artistic latitude. New Line's gamble on Anderson's follow-up effort proved to be unsuccessful at the box office. Magnolia did not even make back its production costs during its domestic theatrical run, reaping only $22 million on its $37 million budget (imdb.com). As with its predecessor, though, the film was a hit with critics and garnered a number of prestigious accolades, including three Academy Award nominations and the Best Supporting Actor Golden Globe for Tom Cruise (imdb.com).
Like many other contemporary misdirection films that performed poorly in theaters, Magnolia's reputation has been bolstered by its run in post-theatrical markets. Upon its theatrical release, reviewers typically lauded the film's ambition, but chastised its narrative ineffectiveness. New York Times reviewer Janet Maslin typifies this reception by writing "It's astonishing to see a film begin this brilliantly only to torpedo itself in its final hour... as the desperate reach for some larger meaning begins, the sheer arbitrariness of [Anderson's] approach is laid bare." Although the seemingly incoherent rain of frogs was generally perceived to be the biggest narrative problem initially, the film's belated appreciation is largely attributable to a reinterpretation subsequently popularized by fans, which claims that the calamity is the key to understanding the film's meaning intelligibly. Importantly, a semantic/syntactic/pragmatic examination of the discourses associated with the film demonstrates that this reinterpretation of narrative information is not the sort that can be easily dismissed as baseless. In addition to reviewer and fan discourse, an analysis of textual properties, particularly the film's mise-en-scène (semantics), its display of Anderson's thematic preoccupations (syntax), and the way it was marketed (pragmatics), provides corroborating evidence to illustrate that it is a misdirection film because it suggests that the alternative meaning was deliberately buried for viewers and not just created out of thin air by overzealous fans.
The film's trailer is a logical start point for this analysis because even though it primarily advertises it as a melodrama by presenting many of the film's characters in emotionally over-wrought scenes, it also hints that it might be a misdirection film. After the preview depicts the last of its primary characters—police officer Jim Kurring (John C. Reilly)—kneeling in front of a cross, it inexplicably ends by cutting to an enigmatic shot of a frog, which is coupled with a voiceover narration that states, "And this will all make sense in the end." However, the preview may seem to be just disingenuous on first blush due to the film's ostensible narrative incoherence. In classical fashion, though, many of Magnolia's ardent fans claim that the catastrophe can be understood as being narratively related to the film's other events. One of the reasons that the film's reputation has grown considerably, then, is because, like most misdirection films, it can be reinterpreted retrospectively in a rather conventional manner by those in the know. Rather than understanding its "truth" as being unknowable or its events as being dictated by the random forces of chaos, according to this fan logic, those who "correctly" read between Magnolia's lines are able to determine what "actually" happened and why its most baffling events "really" occurred.
Although the rain of frogs literally comes out of the blue and appears to be completely unmotivated, many fans argue that it can be logically explained by the film's master key—the number 82. As Ebert speculates in his aforementioned review, events from the film's outset, especially in the seemingly unrelated prologue, consistently foreshadow its appearance. This reassessment of the film's meaning was ultimately buttressed by other critics' interpretations and Anderson's self-promotional tactics. Just a month after its December 1999 theatrical release and associated flurry of perplexed reviews, Mark Caro published an article in the Chicago Tribune in which he discussed with Anderson the many 82s and the direct references to Exodus scattered throughout the film's mise-en-scène, strongly suggesting that the number and its corresponding bible verse are somehow important to comprehending the narrative coherently. Perhaps most notable among these instances in the mise-en-scène occurs immediately before Kurring's windshield is about to be hit by the first of the falling frogs. He fleetingly passes a bus stop that contains a barely legible advertisement, which simply reads "Exodus 8:2." This subtle clue is only discernable on digital technologies, such as the DVD player, that allow viewers to pause the image pristinely and has become easier to distribute to other fans with the advent of the World Wide Web, particularly with the use of Web 2.0 platforms. The presence of this kind of explicit semantic detail indicates that the number references the Old Testament passage that tells of God's infliction of the ten plagues on the Egyptians as punishment for their persecution of the Jews. Exodus 8:1 reads "And the Lord spake unto Moses, Go unto Pharaoh, and say unto him, Thus saith the Lord, Let my people go, that they may serve me." Exodus 8:2 then reads "And if thou refuse to let them go, behold, I will smite all thy borders with frogs." Consequently, this evidence begins to offer spectators a way to interpret the strange catastrophe and all of the film's other ambiguous moments as actually being narratively relevant.
The film's extended prologue, for instance, seems to make little narrative sense until the master key is decoded. At first, it appears to be, at best, indirectly related to the rest of the film because it is only referenced again briefly at the conclusion and its meaning is never explicitly explained by subsequent events. A close analysis of the prologue's mise-en-scène, however, suggests that the scenes are about more than just a meditation on chance versus grand design. During the prologue's first scene, one of the men of Greenberry Hill, coincidentally either Joseph Green, Stanley Berry, or Daniel Hill, is hanged with the number 82 affixed to his shirt for the murder of family man Sir Edmund William Godfrey. In its second scene, Craig Hansen (Brad Hunt), the estranged father of four and the firefighting pilot who accidentally kills scuba diver, Delmer Derian (Patton Oswalt), flies a plane numbered 82 and is seen in a casino attacking a blackjack dealer, who is coincidentally Delmer, when he is upset by receiving an 8 instead of the 2 that he needs. The magnitude of the guilt from the act prompts Hansen to commit suicide. Finally, the bizarre shooting of 17-year-old Sydney Barringer (Chris O'Hara) is told during the 8:20 p.m. meeting of the American Association of Forensic Science. Sydney jumps from a rooftop ledge adorned with ropes coiled to form the number 82, only to be accidentally shot in the stomach by his mother as he passes his family's apartment #682 window just before he lands on a safety net that would have saved his life. As a result, Sydney's parents are arrested for turning the attempted suicide into a homicide. In short, the prologue is relevant to the rest of the film because it links the number 82 to divine punishments inflicted on people who destroy families.
Figure 1.4. A fleetingly observable advertisement that reveals the details of Magnolia's master key, which appears directly before Jim Kurring's car is hit by the first of the frogs.
If the number 82 is connected to the biblical passage, then the rain of frogs can indeed be understood as a modern-day punishment enacted by a vengeful God on the film's subsequent two primary father figures—Earl Partridge (Jason Robards) and Jimmy Gator (Phillip Baker Hall)—for ruining their families. Divine intervention disciplines these two animally named men for mistreating their children, as Partridge abandoned his son, Frank (Tom Cruise), and Gator sexually abused his daughter, Claudia (Melora Waters). Specifically, during the rain, a frog knocks a gun out of Gator's hand as he tries to commit suicide to escape facing the slow death of cancer alone and Partridge dies with Frank at his side when the frogs begin to fall before he can get a word out to fulfill his dying wish of reconciling with his son.
Figure 1.5. Craig Hansen's plane is emblazoned with one of the many explicit references to 82 in Magnolia's prologue.
The post-frog sequence ultimately reveals that the film's second generation of men, guys like Frank, need to embrace a new form of masculinity if they are ever to improve the situation their fathers created for subsequent generations. The sequence does this by focusing on the two characters from that cohort most closely linked to the film's two dying patriarchs: Frank is Earl's offspring and Jim shares more than just a name with Jimmy since he has become romantically involved with Claudia. Importantly, these two characters initially represented polar opposite masculinities, as Frank was a hyper-sexual misogynist and Jim was a lonely divorcee in search of a fulfilling heterosexual relationship. During the rain of frogs, however, Frank becomes the type of man that Jim is—a sensitive man who is not ashamed to reveal his emotions and care for women—as opposed to being like the destructive patriarchs represented by Partridge and Gator. Indeed, he cries uncontrollably at Earl's deathbed and is able to reconcile with his father even though Earl is unable to return the favor. He is then shown nurturing Linda Partridge (Julianne Moore), the stepmother he previously despised, in the hospital. Additionally, the catastrophic event encourages the film's only third-generation male character—wunderkind Stanley Spector (Jeremy Blackman)—to plead with his verbally abusive father to start "being nicer" to him. Finally, to confirm that Jim's behavior is what the film promotes as ideal, the once-wrathful God explicitly rewards the officer. As the frog aftermath scene concludes, Jim's gun, which he previously cried over losing, and which, as Joanne Clarke Dillman accurately contends, is "the phallic signifier par excellence," miraculously falls from the sky (144).
Significantly, such a reinterpretation of narrative meaning is plausible, at least in part, because fans can support it by making recourse to authorship. In addition to Anderson's own publicity efforts that began shortly after the film's release, a review of his oeuvre augments this reading's credibility. Each of Anderson's first six feature-length films, from Hard Eight (1996) to The Master (2012), centers on the terrible consequences that result when father figures desert their biological or adopted sons. Of course, this particular reassessment of Magnolia's meaning is anything but immediately evident because, instead of presenting a formal explanation for the rain of frogs, the film simply cuts to an intertitle that reads, "So Now Then." This intertitle is followed by a cut that briefly returns to the prologue, subtly alluding to its relationship to the master key. After highlights from the prologue replay, the voice-of-God narration then explains, "There are stories of coincidence and chance and intersections and strange things told... And it is in the humble opinion of this narrator that strange things happen all the time... And the book says we may be through with the past, but the past isn't through with us" (emphasis added). This offhanded reference to the Bible, which has been previously uttered by a number of the film's characters, potentially connects the rain of frogs to a divine plan. This insinuation, however, is not a blatant exposure of the hidden truth, as are the explanations provided by the changeover. In contrast, it is the kind of clue that signals that there might be a way to connect the events of the narrative according to a new master thread that attributes the events of the narrative to a specific causal agent, whom viewers never expected to be actually behind it all.
The film's open-ended conclusion, therefore, creates not only a likely response of frustration and befuddlement from viewers. It also has potential to inspire interpretations that contradict the master key reading outlined above. In fact, those who do not categorize it as a misdirection film often interpret the film to be, like Fight Club, ideologically conservative as it relates to gender. Dillman, for instance, observes that it is inaccurate and easy to think that the film places blame for the emasculation of most of the film's second-generation, white-male characters on familiar culprits. Initially, these men indeed appear to have been feminized by a number of developments, such as the intrusion of marginalized groups, like women and other minorities, into the workforce, the elimination of men from the reproductive realm, and the cultural acceptance of homosexuality. For starters, Phil Parma (Phillip Seymour Hoffman) is a male nurse who reveals in a conversation with Earl that, even though he is desperately trying, he does not have a girlfriend. Additionally, "Quiz Kid" Donnie Smith (William H. Macy) is openly gay, cannot find a partner with whom to share his love, and is fired from his job at an electronics store by his immigrant bosses. Finally, Jim has not been on a date since his divorce and garners less respect as a police officer than a female colleague who commandeers a crime scene investigation from him.
Not all of the younger male characters in the film, though, are as explicitly emasculated by contemporary cultural conditions as Phil, Donnie, and Jim. Frank is the mastermind behind "Search and Destroy," a workshop he devised to help men reclaim their lost traditional manhood. Frank's disconcertingly misogynistic performances at his seminars leave no doubt that he holds women accountable for the issues that men face. His frightening self-help sessions urge his followers to conquer their problems by copying his disgusting womanizing exploits. The circumstances leading to Frank's eventual character transformation, however, reveal that it is Earl (and not women, as he himself asserts and as some critics claim) who can be read as being most responsible for his predicament. Frank is hesitant to meet with his terminally ill father even after he learns that his death is imminent. He has justifiable hatred for Earl because he left Frank to tend to his dying mother by himself at a young age. Although it is odd that Frank eventually agrees to meet with Earl in the end, the rain of frogs rewards the act because his hated father dies before he can make peace with his son. According to this revised causal logic, it is men like Earl, and not the more traditionally identified scapegoats initially proffered by the film and by Frank himself, such as women, gays, and other minorities, who should be punished for the problems that Magnolia's second-generation, white-male characters experience.
Without knowledge of the meaning of the master key, it is easy to interpret the film's female characters, as Frank initially does, to be the ones largely responsible for the male struggle. The film's three primary, white, female characters—Linda, Rose Gator (Melinda Dillon), and Claudia—each indeed appear to stand in the way of the intergenerational maintenance of masculine authority. Magnolia's system of rewards and punishments, however, reveals that these women and the film's other female characters should not be held accountable. Instead, they too are ultimately depicted as the unfortunate victims of the outmoded, destructive behaviors of the film's traditional patriarchal oppressors. The characters who most blatantly expose the ways in which conventionally masculinist behaviors of men are no longer tenable are two African-American women: the disobedient apartment tenant, Marcie (Cleo King), and the journalist, Gwenovier (April Grace). Each of these women is strategically paired with Jim and Frank, respectively. As Dillman argues, the responses of both women to these men reveal how Jim and Frank's attempts to enact hegemonic masculinity are unsuccessful (149).
Early in the film, Jim is called to investigate a domestic disturbance in the apartment of Marcie, an imposing and unruly African-American woman, who practically disappears from the film after her memorable and extended introduction. Despite his best effort to assert his authority as a police officer, Marcie refuses to comply with Jim's demands. After tiring of her uncooperative antics, he attempts to handcuff her to a massive sofa in order to search her apartment. As he tries to administer the cuffs, Marcie defiantly slaps Jim numerous times. Once the cuffs are secured, she comically drags the sofa across the room in pursuit of him. Jim is obviously incapable of controlling this woman with his traditionally masculine performance. As Dillman summarizes, the film treats Marcie as a woman who "wreaks havoc on the social order" and "is an enigma that Jim and the film refuse to solve" (149). Marcie's uncontainable and inexplicable character shows how conventional strategies that white men employ to express their authority over women and other minorities are now ineffective.
Similarly, Gwenovier, who, unlike Marcie, is a professional career woman, brazenly challenges Frank's masculine power during an interview that he assumes is going to be a puff-piece. She confrontationally asks searing questions about concealed aspects of Frank's personal history. In response, Frank tries to evade her inquiries by heightening his hyper-masculinity. His tactics, though, do not faze the tenacious reporter. Like Jim, Frank cannot manage the situation that quickly escalates out of his control. All he can do in response, as Dillman claims, is to "stop speaking" (149). Gwenovier, who reveals that Frank's über-masculinity is a façade that has been fashioned to cover up his childhood emasculation, renders him impotent. In the end, Magnolia's African-American women are among the most powerful characters in the film because they are the ones who expose how white, heterosexual men can no longer rely on antiquated gender performances to maintain their authority.
The film's primary, white, female characters are also ultimately revealed not to blame for the men's problems, according to the reinterpretation inspired by the master key. Linda is a golddigger who married a much older man because she hoped to inherit his empire. Her intrusion into the Partridge family thus prevents the transfer of resources from Earl to Frank that would have sustained masculine authority. Indeed, she is initially so focused on keeping the money from Frank that she even assaults the passive Phil for attempting to fulfill Earl's dying wish to reunite with his son; however, Linda eventually falls in love with Earl and declares that she wants to renounce the inheritance. A number of powerful men, most notably Earl's attorney (Michael Murphy), stand in the way of her wishes. The guilt associated with her inability to remedy her selfish actions provokes Linda to try to take her own life, but unlike Craig Hansen from the prologue, whom God punishes by letting him kill himself, or Jimmy Gator, who is unable to escape his terminal cancer by committing suicide because of divine interference, she is saved by God. She is rescued by another narratively ambiguous African-American character—Dixon (Emmanuel Johnson)—a child rapper who refers to himself as a prophet in one of his verses. As a result of this divine intervention, she gets a new lease on life and forms the bond with her stepson, Frank, that her husband never got the chance to reestablish before his death.
The depiction of Rose, likewise, is easy to perceive as disturbing because she is portrayed as a dutiful wife to a man undeserving of that kind of support; however, she ultimately decides to leave Jimmy in the final stages of his cancer. Although her decision is warranted because she abandons her husband after he openly admits that he cheated on her with many women and practically confesses to molesting Claudia, her choice is surprising because she seems like she would stand by her man. Her characterization is further complicated by the fact that she neglected to care for her daughter's welfare by not stopping her husband's sexual abuse. Yet, Rose is not punished for her complicity and disloyalty because, like the other women in the film, there is little that she could have done to thwart the patriarch. Instead, she gets into a car accident during the rain of frogs, but is not hurt by the crash and is able to reach Claudia's apartment in time to comfort her during the event. Unlike Jimmy, then, God does not punish her for her transgressions. She ultimately comes out of the rain stronger because she is finally free from her husband and can help Claudia cope with the trauma inflicted by her monstrous father.
Jimmy's sexual abuse has provided Claudia with an inability to maintain a healthy, romantic relationship. As a result, she numbs her pain with a severe cocaine addiction. In fact, a strung-out Claudia tells Officer Jim on their first date that they should never talk to one another again even though they seem to be enjoying each other's company. By the end of the film, though, her character is also turned around by the rain of frogs. In the film's final scene, Claudia is in her bed as Aimee Mann's "Save Me" plays on the soundtrack. Barely audible under Mann's lyrics of "Come on and save me from the ranks of the freaks who suspect they could never love anyone," Jim can be heard attempting to convince Claudia that he will be a good companion. Instead of punishing her, then, the rain of frogs finally inspires her mother to recognize Jimmy's incest and rewards her with a relationship with Officer Jim, a compassionate man who is the exact opposite of her abusive father, perhaps putting a halt to her self-destructive habits. Surprisingly, the scene then concludes abruptly, with Claudia staring into the camera, breaking the fourth wall. After three harrowing hours, Magnolia's seemingly open narrative simply ends with Claudia's return of the gaze, challenging spectators to figure out how the film's narrative ambiguities might be able to be reinterpreted into a message about how the destructive actions of men threaten the future of humanity. Without knowledge of the way the master key changes the rest of the film's meaning and, therefore, of Magnolia's status as a misdirection film, such an interpretation is not possible. For those who lack that understanding or do not buy the associated reading, it is likely perceived as an indulgent and narratively incoherent film that contains troubling depictions of gender, instead of being an incendiary commentary on patriarchy and the negative consequences of traditional masculinity.
Magnolia and Fight Club exemplify what is at stake when user groups actively create and sustain genres. Identifying or failing to label constituents correctly can dramatically impact how films are interpreted and understood for years to come. Even when misdirection films are defined as such, determining their meanings, especially those that contain complicated narrative structures that resist being definitively reinterpreted with ease, can be a challenging endeavor. This phenomenon makes them particularly susceptible to formal and ideological critiques that may not hold up to closer scrutiny. Indeed, although it is easy to read both Fight Club and Magnolia as being narratively lazy or incoherent, a semantic/syntactic/pragmatic approach to genre reveals that their duplicitous narratives were strategically constructed to be byzantine, but perhaps ultimately decipherable. Consequently, whereas both films are often accused of being destructive to women and sending dangerous messages to men, a more careful analysis of the discourses running through and associated to them indicates that they may instead contain comparatively progressive representations of gender. Importantly, these retrospective reassessments are only possible after the full significance of the changeover or master key is understood. Comprehensions of their cultural politics change dramatically, in other words, once they become identified as constituents of the misdirection film genre and are read accordingly. In the end, it is imprudent to leave it to other groups of people, such as industry professionals, to create genres and classify films, as many who champion the discursive approach would have it, because such classifications can impact the way a film is evaluated for decades. Instead, scholars need to stop denying the inevitable genrefication impulse and embrace the power to birth potentially enduring and meaningful categories, regardless of the ahistorical drawbacks.
Figure 1.6. Claudia Wilson Gator breaks the fourth wall by staring directly into the camera in Magnolia's final shot.
2
## The Truth Is Out There
Manufacturing Conspiratorial Narrative Coherence
A new partnership of nations has begun, and we stand today at a unique and extraordinary moment. The crisis in the Persian Gulf, as grave as it is, also offers a rare opportunity to move toward an historic period of cooperation. Out of these troubled times, our fifth objective—a New World Order—can emerge: A new era—freer from the threat of terror, stronger in the pursuit of justice and more secure in the quest for peace. An era in which the nations of the world, east and west, north and south, can prosper and live in harmony.
—George H. W. Bush, to a joint session of Congress on September 11, 1991
This is a cover story, right?
—Martin Vail [Richard Gere] in Primal Fear [1996]
IN 1991, SHORTLY AFTER THE FIRST U.S.-led invasion of Iraq, President George H. W. Bush delivered a series of speeches in which he memorably called for the formation of a New World Order (NWO). The crisis in the Persian Gulf, he claimed, created a chance for the United States to lead a global power regime characterized by collective policing of international interests. As the first of the epigraphs above reveals, he once articulated his vision to a joint session of Congress on September 11, 1991, exactly ten years before the infamous al-Qaeda attacks on the World Trade Center and the Pentagon. Although Bush hoped that his NWO addresses would make it clear that he was directing a coalition of nations to ensure that principles such as freedom, justice, and the rule of law would reign supreme, his rhetoric was largely not received as he intended. As Harry West and Todd Sanders summarize, many instead believed that it outlined a conspiratorial plan to "subordinate the will of the American people to that of an unelected transnational bureaucracy and an international elite that might dictate its governing objectives" (3).
The paranoid reaction to Bush's addresses is incongruous, considering the drastic changes occurring around the globe at the time. The fall of the Berlin Wall in 1989 and the imminent crumbling of the Soviet Union meant that U.S. economic and military supremacy were now virtually unchallenged. The negative reaction to Bush's call for a New World Order is especially telling, then, because it exemplifies a kind of conspiracy theorizing that has been increasingly directed at the U.S. government of late. The dissemination of conspiracies about the government's suspected involvement in and motives for a number of events in recent years, ranging from the crash of TWA flight 800 and the unsolved murders of Tupac Shakur and the Notorious B.I.G. to the two invasions of Iraq and the targeted killing of Osama bin Laden, have been inescapable. Yet, the shockingly sinister and spectacularly catastrophic simultaneous hijacking of four commercial airliners by al-Qaeda operatives on September 11, 2001, was conspiracy theory's defining moment in the United States. In addition to paranoid accounts of the events of 9/11 that claim it was actually an inside job, the official explanation, as Ray Pratt argues, made it "the preeminent historical example of a real terrorist conspiracy theory" because it demonstrated beyond a doubt that "what we previously thought was paranoia just might have turned out to be a form of heightened awareness" ("Theorizing" 256). The massive death toll of 9/11 unquestionably made conspiracy theory all-too-real for many Americans who had the privilege of living in a nation that had never been subjected to a terrorist attack resulting in civilian casualties anywhere near that magnitude. The almost unbelievable details of the hijacking plot indeed gave credence to the proverb that "just because you're paranoid, doesn't mean they're not out to get you."
This chapter focuses on exploring why two overlapping discourses—misdirection films and conspiracy theories—have recently appeared with such regularity in the U.S. for many of the same reasons. According to Paul Silverstein, conspiracy theorizing is an interpretive practice that "prioritizes agency and fetishizes causality in making sense of everyday incoherence" (647). The misdirection film similarly ties up all potential loose ends by giving spectators an alternative account that, in retrospect, typically adheres to a classical narrative logic driven by individual character agency. Like many conspiracy theories, misdirection films provide viewers with a more attractive explanation of narrative information than what is initially supplied because they reveal that there is actually a more spectacular, yet still highly familiar way, to understand the relationship of events. Consumers of conspiracy theory and misdirection films alike derive pleasure from order being made out of chaos by those who see past the surface to draw the "correct" conclusions.
## The Popularization of a Great American Tradition: Agency Panic and Contemporary Conspiracy Theorizing
Conspiracy theory is a topic that has been ubiquitous in contemporary mass media representations and has long been popular in Hollywood. On television, for example, the Fox series, The X-Files (1993–2002), was hugely successful, grabbing "16 percent of all television viewers" at its peak in 1997 (qtd. in Hellinger 218). A number of recent conspiracy-themed documentary films also received significant media coverage. The most notable among these was Michael Moore's attempt to chronicle the "real" motives behind the second invasion of Iraq in Fahrenheit 9/11 (2004). Similarly, Nick Broomfield's Kurt and Courtney (1998) and Biggie and Tupac (2002) provided more sensational explanations for the mysterious deaths of three of the most recognizable popular musicians of the 1990s than what had been presented in mainstream news media. In addition, Loose Change (2007, 3rd Ed.), one of the most successful documentary films ever distributed primarily on the web at the time, suggests that 9/11 was an inside job. In mass-market literature, Dan Brown made the Catholic Church and the story of Jesus the subject of a grand conspiracy in The Da Vinci Code (2003). The book rapidly became an international sensation and was eventually adapted into a 2006 Hollywood film. Prior to the 1990s, Hollywood's conspiracy-themed film peak was in the 1970s, when it released films such as The Conversation (1974), The Parallax View (1974), and All the President's Men (1976) at the same time that the details of the Watergate scandal dominated the news headlines.
During the period under study, though, there has been a comparative explosion of conspiracy films that did not fall into the misdirection genre, such as JFK (1991), Bob Roberts (1992), Nixon (1995), Men in Black (1997), Wag the Dog (1998), Enemy of the State (1998), The X-Files (1998), Primary Colors (1999), The Contender (2000), The Manchurian Candidate (2004), Shooter (2007), The X-Files: I Want to Believe (2008), W. (2008), and the aptly titled Conspiracy Theory (1997). Yet, I focus this chapter exclusively on misdirection films for a couple of primary reasons. First, although recent conspiratorial-themed Hollywood films may similarly articulate a growing concern about the status of the "truth," they do not inspire viewers to reread the relationship of events in a paranoid fashion in the same way as misdirection films. Second, Pratt's Projecting Paranoia already does an admirable job of forging connections between the rise of conspiratorial-themed Hollywood films and changes in post-WWII U.S. culture. Pratt does not consider, however, how misdirection films, irrespective of whether or not they explicitly dramatize conspiracy, tap into similar fears and desires that have made conspiratorial-themed films appealing to some audiences since the early 1990s.
Just as conspiracy-themed films are hardly new, conspiracy theorizing is not solely a product of the contemporary moment. Consequently, a brief examination of its historical legacy in the U.S. will be useful for explaining why it has been both a persistent American cultural practice and the reasons that it recently has flourished in particular ways. Such an account also lays the foundation for revealing how the misdirection film's specific narrative structures assuage anxieties and fulfill desires in a manner that resembles conspiracy theorizing. As numerous critics point out, conspiratorial thinking has always been prevalent in the United States because the interpretive practice is endemic to American political culture. Of course, conspiracy theorizing is not uniquely American. As West and Sanders note, conspiracy theorizing "proliferates around the globe" and "may take on vastly different forms in different locales" (5). Yet, the manner in which it often materializes in the U.S. is connected to cultural, political, and historical circumstances. The United States, for starters, was founded on a healthy skepticism of centralized power because its democratic system of governance was created in direct response to the tyranny of British monarchical rule. A perpetual state of paranoia about the government should be expected, therefore, as U.S. citizens are supposed to question the motives of those in power to ensure that they do not become the unwitting subjects of an authoritarian regime. In spite of this civic responsibility, theorists, like Richard Hofstadter, have famously considered conspiracy theorizing to be a negative aspect of American political culture by arguing that it amounts to little more than "collective paranoid delusions" (5).
Some scholars have rightly taken issue with this aspect of Hofstadter's pejorative conception of conspiracy theorizing. Critics, like Fredric Jameson and Mark Fenster, counter that some contemporary iterations of the practice are anything but pathological because they reveal the limited options for most U.S. citizens to negotiate their relationship to authority in a neoliberal age. Fenster contends that conspiracy theories "ideologically address real structural inequities, constituting a response to a withering civil society and concentration in the ownership of means of production, which together leave the political subject without the ability to be recognized or achieve representation in the public realm" (67). For Fenster, the prevalence of conspiracy theorizing suggests that most U.S. citizens now have few plausible choices for opposing power. Such a perspective is similar to Jameson's notion of conspiracy theorizing as the "poor person's cognitive mapping," which he describes as the means on which the disenfranchised subject relies to cope with the overwhelmingly bureaucratic cultural logic of late capitalism ("Cognitive" 356). These scholars, then, contend that conspiracy theorizing is a justified response to the neoliberal economic and political policies dominant in the West since the Reagan-Thatcher revolution. In a neoliberal era dominated by corporate deregulation and consolidation, the gap between social classes has widened, the influence of big business has expanded both domestically and internationally, the power of labor unions has eroded, and so on. As a result, some paranoid sentiments, like those contemporary conspiracy theories often express, are appropriate in a milieu in which an increasingly small group of elites has gone to great lengths to secure power and resources.
Although these scholars posit that conspiracy theorizing can be a reasonable response to contemporary circumstances, they also criticize it for ultimately being apolitical. Fenster worries that conspiracy theory fails "to inform us how to move from the end of an uncovered plot to the beginning of a political movement" in which "we can begin to organize people in a world organized by complex divisions based on class, race, gender, sexuality, and other social antagonisms" (226). Conspiracy theory, at least as Fenster understands it, offers subscribers the opportunity neither to become politically engaged nor to change structural inequities. In contrast, Clare Birchall argues that an examination of conspiracy theorizing can be instructive to those who hope to conduct the kind of politically engaged scholarship that is fundamental to Cultural Studies. In particular, Birchall theorizes that Cultural Studies has "always involved itself in the nature of specialist and everyday knowledge and how we use them to produce, consume, and interpret the culture around us" (16). According to her conception, the Cultural Studies project, especially as it has been influenced by Michel Foucault, explores how dominant epistemologies are socially constructed. She depends on this perspective to theorize how certain interpretive practices, like conspiracy theorizing, are positioned as illegitimate in relation to more established ways of knowing. Before the findings of Copernicus, Galileo, and Kepler were accepted, for example, those who denied that the earth was the center of the universe were labeled heretics. Now, those who believe in a geocentric model are considered fanatics who refuse to acknowledge the more legitimate discourses of science. As Birchall argues, such examples show that "all knowledge is only ever 'theory' " and that "any transcendental truth claims rely on contingent strategies of legitimation" (73). As with the counter histories that Cultural Studies scholars often produce, conspiracy theorists discredit widely accepted accounts by challenging the veracity of those very explanations. Birchall, therefore, posits that conspiracy theorizing can ultimately be read as "affirming the cultural studies 'project'—as being endemic of cultural studies' openness to the question of what legitimate knowledge is" (85).
Accounts that contradict "official" narratives often concern those in power precisely because they potentially expose how those explanations are also highly constructed. As Daniel Hellinger and Dennis Judd document, America's ruling class, like "elites in every society," have always striven to "preserve their economic privilege and political power, and often engage in conspiracies for this purpose" (4). To support this bold statement, they contend that the flexibility of the U.S.'s democratic political system and the modernist rhetoric that it espouses are what have long served as both the cover story and means for the implementation of the agenda of those in power. According to Hellinger and Judd, for instance, the U.S. Constitution was not drafted by the people because a powerful group of elites instead conspired "in secret" and "beyond their legal mandate" to create a document much less democratic than alternatives that were actually drafted by the people at the time (4). Importantly, the often empty promises of equality and participation in government that the Constitution offers are still key factors in the dominance of elites. Throughout U.S. history, the decentralization of power the document purports has neither leveled the playing field nor provided the populace with clear explanations for the government's motives and actions. In fact, elites in the United States have consistently gone to extreme lengths to marginalize groups of people from the democratic process and have continually concealed the reasons for their tactics. Groups such as the illiterate, the poor, non-landowners, women, and African-Americans have been denied the right to vote in the U.S. precisely because of the threat that they posed to power. Similarly, with each passing year, evidence mounts that many politicians are more concerned with pleasing the benefactors who helped them get elected than they are with serving their constituents. Finally, since the Cold War, the instances in which those in power have blatantly violated the rights promised to U.S. citizens out of concern for "national security," exemplified by McCarthyism, the PATRIOT Act, as well as the NSA spying scandal, have become more frequent and spectacular.
For those in power, then, conspiracy theory can threaten one of the most important ways that authority is maintained. As Hellinger writes, conspiracy theories "cast suspicion on the transparency and legitimacy claims of actions undertaken by the police, military, and intelligence agencies, whose missions include actually undertaking conspiracies" (205). In contemporary democratic societies, institutions that serve and protect the people are often believed to operate in an open and honest fashion because they are designed to uphold fundamentally modernist ideals like truth, justice, freedom, and equality. As a result, when the supposedly irrational and paranoid claims of conspiracy theorists prove accurate, it can shake the very core of democratic society. In the aftermath of the Watergate scandal, for instance, it became impossible to deny that the government employs unauthorized surveillance techniques and creates elaborate cover-ups to maintain authority. Yet, as many scholars theorize, conspiracy theory's ultimate power lays not in its ability to reveal the "truth." Instead, it is a threat because it raises doubts about the veracity of official explanations. West and Sanders contend that conspiracy theories create "discourses that are constructed in contradistinction to the (also constructed) truths of transparency" (15). Silverstein, likewise, conceives of conspiracy theorizing as "a form of 'vernacular' knowledge production contrasting with overlapping 'official' modes of knowledge production that outlines an alternative 'truth regime' " (646). In short, conspiracy theories create oppositional accounts, which often suggest the likelihood that those constructing authoritative discourses also rely heavily on conspiratorial means to accomplish their objectives.
Clearly, it benefits those in power to make it seem as though the alternative explanations of conspiracy theorists are unfounded by referring to them as kooks, or other such derogatory labels, which can be accurate descriptions for some of its subscribers in certain instances. However, the real threat that conspiracy theorizing poses explains why, as West and Sanders argue, those "bound up with the notion of transparency expend a great deal of energy in attempts to paint Other ways of seeing with the brush of 'ignorance,' 'irrationality,' or 'superstition' " (12). Many conspiracy theorists are demeaned because it seems unreasonable to rely on the logic at a time in which power supposedly operates in the open and in the best interests of the people. Although conspiracy theorists have long been labeled as paranoiacs on the fringe, West and Sanders correctly contend that nonetheless "evidence suggests that a broad cross section of Americans today—traversing ethnic, gender, education, occupation, and other divides—give credence to at least some conspiracy theories" (4). In light of the recent wake of high-profile scandals that have exposed the reprehensible dealings of upper-echelon U.S. government and corporate officials, such as those involving executives at Enron, WorldCom, Tyco, and some of the world's preeminent financial service companies, it is understandable why such paranoid thinking is becoming more commonplace.
Virtually all discussions of contemporary conspiracy theorizing in the United States point to the assassination of President John F. Kennedy as a marker of a new period in its history. As Pratt argues, it was "an epistemological break" that caused "both the general populace and a significant subgroup of intellectuals to question their ability to know the truth rather than merely believe or sense it" (Projecting 221). Paradoxically, the more evidence that the Zapruder film revealed, the less the public was able to determine what "really" happened. The "truth" of the JFK assassination was also compounded by the mountain of information compiled by the Warren Commission in support of its unconvincing lone gunman theory, which, as Marita Sturken notes, most believed "constituted a cover-up and was at least part fiction" (72). The lack of closure to the assassination created, in Benedict Anderson's language, an "imagined community" of U.S. citizens, united by their simultaneous consumption of the same narratives that unsatisfactorily explained events. Significantly, for Anderson, this virtual camaraderie only became possible after the onset of modernity and the requisite development of print-capitalism. The newspaper, Anderson contends, was crucial to making a cohesive group out of disparate individuals because in the act of reading, the reader becomes "aware that the ceremony that he performs is being replicated simultaneously by thousands (or millions) of others of whose existence he is confident, yet of whose identity he has not the slightest notion" (35). Although Anderson fails to consider how oral traditions made this phenomenon possible in pre-modern societies, his thesis demonstrates that such practices have been amplified by the advent of subsequent mass media and communication technologies. Indeed, it is cliché for those old enough to remember the Kennedy assassination to report where they were when they learned about it on broadcast media. Similarly, many in that cohort would likely share their dissatisfaction with the "official" account after seeing the Zapruder film, Lee Harvey Oswald's murder on live television, and so on.
Television's development into the dominant mass medium has played a large role in the proliferation of similarly themed conspiracy theories in post-WWII U.S. culture. Since at least JFK's assassination in 1963, a number of seminal events in American history that received significant television news coverage, including the Vietnam War and subsequent misguided U.S.-led invasions, the Watergate scandal, as well as the similarly suspicious assassinations of Robert F. Kennedy, Martin Luther King, Jr., and Malcolm X, have either not been satisfyingly explained by official accounts or blatantly exposed the conspiratorial activities of those in power. Pratt theorizes that such events have conditioned American citizens to believe that "agencies of the U.S. government have the right—perhaps even the duty—to lie to protect national security" and that "even though they vote and pay taxes to support it, they do not have the right to know all the government's secrets" (Projecting 21–22). Unsurprisingly, not all U.S. citizens have taken kindly to this infantilization, which begins to explain why conspiracy theorizing spread widely in U.S. culture during the 1960s and 1970s and has only escalated since that time.
A casual search for conspiracy theories on the web indicates that conspiracy theorizing is not something that is only done by extremists. The Internet, like many technologies before it, has unquestionably made the distribution of conspiracy theories easier. Yet, the Internet is not solely responsible for the recent proliferation of conspiracy theories. During the 1980s and early 1990s, the distribution of conspiracy theories, as Birchall documents, was greatly "facilitated by technological developments that made personal computers, laser printers, and desktop publishing software more available" (35). These new technologies provided conspiracy theorists with a cost-effective way to disseminate messages through publications, such as zines, which have become easier to replicate on the web. Put simply, the Internet has dramatically intensified tendencies that predated its existence. The development of this new communication technology, in conjunction with the practice's omnipresence in popular culture, has, as Birchall claims, "ensured conspiracy theory a stable presence on the cultural scene" (38).
The persistence of the liberal democratic myth in the United States is the primary reason why conspiracy theory has always been and will likely continue to be central to American political culture. As Timothy Melley posits in Empire of Conspiracy, Americans are reared on the modernist principles of liberalism, which promotes free will, individual rights, and restrictions of government power. These discourses have long been the primary means by which most U.S. citizens conceive of their agency, revealing why there is strong suspicion of centralized government. The belief that the select few in power do not control the will of the people, however, has been strongly challenged recently by the exposure of the aforementioned conspiracies undertaken by the government and its conduits in big business as well as in surveillance and law enforcement. This nervous concern over the loss of individual sovereignty underpins much of the conspiratorial expression that has become more popular of late. Melley labels such paranoia as "agency panic," which he defines as "an intense anxiety about an apparent loss of autonomy or self-control—the conviction that one's actions are being controlled by someone else, that one has been 'constructed' by powerful external agents" (12). Many contemporary American conspiracy theorists, therefore, question the veracity of authoritative narratives by attributing what "really" happened to actual historical individuals and/or organizations with the power to dictate their actions. According to Melley, most post-WWII conspiracy theorizing is thus "a fundamentally conservative response—'conservative' in the sense that it conserves a traditional model of the self in spite of the obvious challenges that postwar technologies of communication and social organization pose to that model" (15).
## Postmodern Disguise: Misdirection Films and Hollywood Modernism
The premise that individual identity is most appropriately conceived of as a construct is often linked to the broader cultural logic of postmodernism because it contradicts the established notion that an "authentic" persona actually exists. Whereas modernist-inspired agency panic maintains that it is possible for those with the requisite interpretive acumen to locate a bounded self, postmodernist-inspired thought posits that it is a fantasy to identify a genuine persona. Scholars inspired by Jameson's writings on postmodernism, for instance, claim that schizophrenia is a prototypical condition of a mass-mediated era dominated by television and characterized by associated socially and culturally constructed identities. The much disputed, polysemic term "postmodernism" has been typically employed, as Robert Stam observes, to describe anything ranging from "a discursive/conceptual grid, a corpus of texts, a style or aesthetic, a paradigm shift, a prevailing sensibility" to "an epoch" (754). Crucially, these potential uses of the term are not mutually exclusive. As Stam points out, Jameson takes a "multidimensional approach which sees postmodernism as simultaneously a style, a discourse, and an epoch" (emphasis in original, 754). If postmodernism is deployed to name an indisputable change in multiple arenas, then I am hesitant to refer to the post-WWII period in the United States as the postmodern since neither contemporary American conspiracy theorizing nor recent Hollywood misdirection films constitute significant aesthetic, epistemological, or historical breaks from the conventional forms of expression with which they are most closely contrasted.
The ubiquity of the term "postmodernism" in scholarly and popular discourse, though, makes it difficult to ignore the concept completely. Consequently, I argue that postmodernism is, in Raymond Williams's terms, most appropriately conceived of as being "emergent" because it classifies "the new meanings and values, new practices, new relationships and kinds of relationships" that "are continually being created" by social formations at times in which existing dominant epistemologies and cultural practices still reign supreme (123). The tenets most commonly associated with postmodernism, such as relativity, multiculturalism, and the rejection of metanarratives, are not principal in a culture still characterized by well-established artistic canons, the competing discourses of religion and science, as well as firmly entrenched hierarchies of ability, class, gender, race, sexuality, and so on. Following Douglas Kellner's lead, I thus employ postmodernism, "as a placeholder, or semiotic marker, that indicates that there are new phenomena that require mapping or theorizing" (46). Postmodernism, therefore, describes emergent contextual shifts that have not yet superseded dominant ways of thinking. Although these forces have exerted some influence on aesthetics and culture, their impact is not substantial enough to signal that there has been an unequivocal transition into a new epoch.
My position on the postmodernist debate is influenced by scholars, like M. Keith Booker, who speculate that the post-WWII period in the U.S. represents a late phase of modernity rather than the start of the postmodern era. As with Booker, I grant that it is imprudent to ignore how a number of conditions resulting from and subsequent to WWII, including the ravaging of European lands, the dropping of atomic bombs on Hiroshima and Nagasaki, the eventual fall of the U.S.S.R., as well as the consequent international economic and military dominance of the United States, have all contributed to virtually unchallenged American power and the unfettered spread of global capitalism, which theorists like Jameson claim to be the defining characteristics of postmodernism. Booker persuasively theorizes, however, that these changes pale in comparison to "the radical transition from medieval Catholic hegemony to the emergent capitalism of early modernity" (30). Since the fall of feudalism and church rule in the West, in other words, there have been few, if any, economic, political, and cultural shifts that have altered the inexorable progress of modernity and its requisite industrialization, technological development, and establishment of liberalist governments that purport to serve the interests of the people.
The post-WWII period in the U.S., then, is best understood as a late phase of modernity that is still largely governed by its guiding principles. Hollywood misdirection films can be considered a product of this period because they also often seem to interrogate modern epistemologies on first blush; however, in the end, they rarely undermine that very way of comprehending the world completely. Yet, some scholars contend that these are precisely the kinds of films that demonstrate that Hollywood's production strategies have changed dramatically in response to postmodernity. In A World in Chaos, Tom Pollard and Carl Boggs argue that contemporary Hollywood films routinely challenge the modernist narrative and formal practices that dominate earlier periods. Interestingly, the authors offer a definition for modernist cinema that is similar to David Bordwell's notion of the classical film. They theorize that the ethos of "cinematic modernism" was "tied to the emergent studio system" and "set out to depict historic struggles between good and evil in a world where (typically white male) protagonists stood for a coherent, progressive set of values" in which "the hero (however flawed or tragic) was invested with the power to influence or transform society" (5–6). As with Bordwell, Boggs and Pollard contend that the portrayal of the protagonist's struggle to overcome seemingly insurmountable obstacles propels the typical golden age Hollywood film. They take this conception further, though, and theorize that in a majority of studio-era Hollywood films, the protagonist's objectives have strong ideological implications because they are typically tied to modernist notions of truth, justice, economic expansion, industrialization, nationalism, and so on.
Although Boggs and Pollard maintain that these kinds of narratives were dominant during the studio era, they admit that a few Hollywood films, such as misdirection films directed by Hitchcock, Lang, and Welles, challenged the principles of modernity at the time. However, they subsequently argue that the cultural and industrial changes precipitated by the U.S. victory in WWII and the ensuing breakup of the studio system have resulted in a sustained assault on the project of modernity. This cinematic shift is what many scholars identify as a tangible change in world cinema, a claim made perhaps most influentially by Gilles Deleuze in Cinema II, in which he argues that the classical Hollywood movement-image film has been supplanted by the time-image film in some non-U.S. cinemas after WWII. Boggs and Pollard, though, offer an important caveat to this conception because they note that although "postmodern" Hollywood films "build on a revulsion against tightly structured, formulaic, narrowly commercialized methods linked to the studio system," their ability to be made entirely "independently of corporate production has never been fully resolved or even confronted" (7). The latter half of this statement is particularly relevant to this book. Even though misdirection films encourage viewers to engage with them distinctly, they exist in a system in which a few media conglomerates still tightly control the industry. Boggs and Pollard thus theorize that this tension epitomizes an "age where the restless search for new epistemological and aesthetic paradigms—a search often taking its architects in the direction of chaos and even apocalypse—has infused the spirit of much contemporary filmmaking" and "entails a profound critique and rejection (but never full transcendence) of modernity" (7–9). Contemporary misdirection films similarly want to have it both ways. Although misdirection films appear to interrogate some of the classical film's basic narrative, formal, and ideological principles, I argue more strongly than Boggs and Pollard that they are not as radical as they initially appear. In retrospect, they instead can be read as usually remaining within the classical paradigm and often upholding dominant American ideologies.
Such a tendency aligns misdirection films culturally with recent explicitly conspiratorially themed Hollywood films. In Projecting Paranoia, Pratt relies heavily on Melley's conception of agency panic to argue that the films he studies often express a nervous concern that powerful forces behind the scenes nefariously dictate people's actions. As a result, Pratt claims that the films he analyzes "may be seen as unconscious reflections of state-supported repression of movements for human emancipation, or the belief among significant sectors of the public that their lives are no longer under their own control" (Projecting 2). Such a hypotheses is echoed by Patrick O'Donnell, who similarly posits that the presence of many conspiratorial-themed literary narratives during the brief historical period reveals that, "one knows that she is part of a series of orchestrated events over which she has no control, but knowing it confers a kind of legitimacy upon the knower; she can be manipulated, but she can't be fooled about being manipulated" (190). Hollywood's recent production trends demonstrate that films that depict powerful agents who conduct covert operations that go undetected by most U.S. citizens, such as the now iconic men in black, appeal to audiences. The popularity of these representations is noteworthy because even though they suggest that many spectators willingly admit that such agencies exist, they also indicate that most citizens acknowledge they are powerless to fight them. The contemporary misdirection film's success also reveals that many Hollywood spectators acknowledge that the classical film is a particular construction despite the fact that it still sutures them into its protagonist-driven narrative logic so effectively.
Importantly, then, Melley's notion of agency panic also accurately describes the existential plight of many of the misdirection film's protagonists. The heroes of these films often suffer from afflictions, such as dissociative identity disorder (Adaptation [2002], A Beautiful Mind [2001], Fight Club [1999], Identity [2003], and Shutter Island [2010]), that enable their actions to be controlled by other characters who know about and exploit their conditions to fulfill their own objectives. Moreover, even primary characters in misdirection films who do not have mental illnesses are often at the mercy of someone else's bidding, like those revealed to be the unwilling pawns in an evil mastermind's plot (Arlington Road [1999], Unbreakable [2000], The Usual Suspects [1995], etc.). Indeed, the revelatory information in almost every misdirection film exposes the fact that someone (Lucky Number Slevin [2006], The Village [2004], etc.) or something (whether it be ghosts in The Others [2001] and The Sixth Sense [1999], dreams in Jacob's Ladder [1990], Mulholland Dr. [2001], and Vanilla Sky [2001], or even deities in Magnolia [1999] and Pulp Fiction [1994]) has been controlling a main character's actions unbeknownst to both him or her and the audience. The epiphanies of these films clearly articulate a nervous concern over the authenticity of individual autonomy. As I demonstrate below, many misdirection films can thus be understood according to a hyper-classical narrative logic because they ultimately prioritize agency and fetishize causality—the key components of the definition that I use to understand conspiracy theorizing—in their alternative explanations of the "actual" reasons for and the "real" culprits behind otherwise unsatisfactorily explained events.
## Just Because You're Paranoid Doesn't Mean You Can Do Anything About It: Arlington Road and Agency Panic
Arlington Road offers perhaps the clearest example of why it is appropriate to examine contemporary misdirection films in relation to a specific kind of cultural paranoia; it depicts memorable conspiratorial plots in recent U.S. history and encourages audiences to reinterpret narrative information retrospectively in a conspiratorial fashion. The explicit combination of this theme and narrative structure in this particular film, though, was not entirely successful with audiences or critics. In comparison to other misdirection films released in 1999 (e.g., The Sixth Sense, Fight Club, and Magnolia), Arlington Road received little publicity, was completely ignored by the Academy of Motion Picture Arts and Sciences, and made little splash at the domestic box office. In fact, it netted approximately $24 million during its run in U.S. theaters, barely recuperating its modest $21.5 million budget (imdb.com). The film's narrative centers on Michael Faraday (Jeff Bridges), a George Washington University Political Science professor, who is experiencing difficulty coping with the tragic death of his FBI-agent wife, Leah Faraday (Laura Poe). Michael holds the Bureau responsible for her murder because she was killed during a mission based on faulty information. The film also depicts the struggle that develops between Faraday and his new neighbor, Oliver Lang (Tim Robbins), who initially seems to be little more than a prototypical suburban husband and father, but turns out to be a key member of a conspiratorial plot against the U.S. government that Michael tries to foil.
The casting of Robbins in one of the film's lead roles is noteworthy because it can be understood as one of the first moments in which a recognizable Hollywood star was unofficially typecast as a misdirection actor. He was likely considered for the part because of his performances in Jacob's Ladder and The Shawshank Redemption (1994), which although it is not a misdirection film, contains a major plot twist that Robbins helps to conceal with his performance. It is also significant that Spencer Treat Clark, who subsequently stars in Unbreakable, plays Michael's son, Grant Faraday, and that the film was scored by Angelo Badalamenti, who would later compose a similar eerie soundtrack for Mulholland Dr. The cast and crew, therefore, are peppered with names that demonstrate an increased industrial awareness that the misdirection film constitutes a genre replete with its own associated creative personnel.
The way that the film was marketed also suggests one of the reasons why misdirection films are often considered a genre in the popular imagination. Misdirection films contain narrative structures that inspire interpretive activities that both express paranoia about the concealment of the "truth" and a yearning to unearth it from its hiding place. Few contemporary misdirection films, though, make these shared cultural anxieties and desires explicit thematic concerns. As an exception to this trend, Arlington Road was promoted with taglines, such as "Your paranoia is real," "How well do you know your neighbor?," and "On July 9, terror hits home," prepping the audience to be on high alert. It is purely coincidental, of course, that the film was advertised in the rhetorical style of the George W. Bush administration and its War on Terror; however, such promotional ploys are noteworthy because they suggest that marketers recognized that recent events in U.S. history had made it such that many were willing to believe that, in the years leading up to September 11, 2001, it was plausible that they could be living amongst well-camouflaged terrorists.
Unlike many of Hollywood's post-9/11 terrorist-themed films released during the period under study, such as The Sum of All Fears (2002), Munich (2005), Syriana (2005), United 93 (2006), and World Trade Center (2006), though, Arlington Road's narrative was not inspired by the actions of international, radical extremist factions. Instead, its narrative was based largely on the tragic events associated with a number of domestic, right-wing militia groups, including those related to both Ruby Ridge's Randy Weaver and Timothy McVeigh, whose bombing of Oklahoma City's Murrah Federal Building was the deadliest terrorist attack in the United States prior to September 11, 2001. Significantly, as West and Sanders observe, in the aftermaths of both tragedies, Weaver and McVeigh publicly professed a "deep suspicion of power in the New World Order" (3). That is, the two were domestic terrorists in the 1990s whose behaviors were believed to be provoked partly by their profound distrust of the motives behind the U.S. government's recent actions.
Arlington Road presents thinly veiled fictional accounts of these significant events in 1990s U.S. history most explicitly during scenes that portray Faraday's course on American Terrorism. In classical fashion, Faraday's lectures are not narratively tangential because they provide key information about the protagonist's primary struggles. The scenes impart information about the circumstances of his wife's death and showcase his difficulty dealing with the loss. However, the scenes serve another important purpose: they reveal that Faraday is suspicious about the actions of the U.S. government and encourages viewers to identify with his justifiable paranoia. The first classroom scene opens with Faraday's assertion that the U.S. government has not lived up to the promises set forth in the Declaration of Independence and the Constitution. He informs his American Terrorism class that "Two hundred years ago a revolution was fought for certain beliefs like liberty, self-rule, self-reliance, and justice for all, and there are many in this country who feel we have not yet won that war." His lecture thus teaches his students that the modernist principles on which the country was founded are a charade. He subsequently points out that various groups have resorted to terrorism to protest the government's restrictions of their individual freedoms throughout U.S. history. Faraday then raises a series of questions about why such anti-government, terrorist acts have recently been on the rise. Echoing the sentiments of those puzzled by the paradoxical response to Bush's NWO speech, he asks "why, at a moment of social prosperity is the anti-government movement at its peak?" In response, Faraday articulates the growing sense of disenfranchisement in the U.S. by noting that "fewer and fewer of us are voting" and "more of us are joining the resistance." In short, the first classroom scene unfavorably portrays the U.S. government for limiting individual sovereignty and suggests that the growing opposition to its authority is understandable.
In the second classroom scene, Faraday offers a specific example of this anti-government resistance by presenting the tragic details associated with the demolition of the fictional Roosevelt Federal Building in St. Louis, Missouri, which are clearly based on the McVeigh bombing. Faraday begins by documenting what he deems to be the unconvincing "official" account. He reports that authorities concluded that a man, named Dean Scobee, acted alone in the attack. In particular, they claim that he committed the crime solely because he owed approximately $10,000 in back taxes to the IRS, whose offices were housed in the building. As Faraday challenges the veracity of these findings, a perturbed student interrupts the lecture by shouting, "Come on professor, the Feds did this whole investigation." An unimpressed Faraday counters that the "investigation didn't satisfy" him. He then puts another student on the spot by asking her to tell the class how she felt once the authorities identified Scobee. As Faraday surmises, her initial feelings of fear and her ultimate sense of security reveal why the "official" explanation can easily be dismissed as a cover story designed to provide the paranoid masses with a convenient scapegoat to pacify them. Faraday's second lecture articulates a compelling alternative account of the traumatic event fashioned in contradistinction to the "official" report, which he and many others believe to be completely unsatisfying.
For the final lecture scene, Faraday and his students are depicted outside of the classroom, at a field at Copper Creek, the location where Leah was killed. As Faraday begins recounting the details of the tragedy to his students, the film cuts to images of the event that are clearly coded as a flashback. In contrast to his description of the Scobee incident, it is significant that this event is presented in such a manner because classically trained spectators tend to expect flashbacks to be accurate. As Bordwell acknowledges in his discussion of Hitchcock's Stage Fright (1950), the director's first, unsuccessful attempt to make a misdirection film, it "is probably the canonic case of unreliable narration in classical cinema" because its opening flashback "turns out to have been the visual and auditory representation of a lie" (Narration 61). For Bordwell, that film failed with classical viewers because unless flashbacks are explicitly marked as potentially unreliable, convention dictates that they contain dependable information. Unlike his controversial account of the Scobee bombing, then, the Ruby Ridge inspired story depicted in the flashback is likely assumed to be factual. Of course, this is the case even though Faraday seems to be experiencing extreme psychological distress since his wife's death and harboring hatred for the FBI. It is subsequently revealed that the FBI mistakenly flagged Randy Weaver stand-in, Seaver Parsons, as a potentially dangerous domestic terrorist, who they believed was stockpiling weapons for a future act of violence. Specifically, during a surveillance mission at the Parsons' home, Leah and her colleagues are spotted by one of Parsons' sons, triggering a series of unfortunate events that end in a slaughter. As Faraday explains to his class, although it was true that Parsons was a renowned right-wing extremist, the FBI failed to realize that he "was a separatist, but not a terrorist," who was likely just legally trying to obtain merchandise to open a gun shop. In this scene, therefore, spectators become privy both to the kinds of covert operations that U.S. government agencies regularly conduct and the terrible consequences that can occur when incorrect causal connections are made as a result.
Arlington Road, however, is not a film that portrays conspiratorial acts as committed only by those in power. As the film's shocking changeover reveals, those who oppose the U.S. government's restriction of individual freedoms also regularly participate in conspiracies. The film's surprise conclusion shows, like other contemporary conspiratorial-themed Hollywood films, such as Conspiracy Theory, the remake of The Manchurian Candidate, and Shooter, that the consistent and seemingly unfathomable lone gunman theories produced by "official" accounts are fabrications. In Arlington Road, though, these theories are revealed as erroneous not because they have been manufactured by those in power to ease the anxieties of the public, as Faraday speculates in class in relation to the Scobee incident. Instead, they have been produced by those aiming to sabotage the government as a means to hoodwink the authorities. The film eventually exposes the fact that Faraday's seemingly perfect suburban neighbors—the Lang family—are actually key members of a militant, anti-government terrorist group that has played a crucial role in a number of horrific acts that have been incorrectly pinned on lone, rogue attackers. It is revealed, for instance, that the Langs lived in St. Louis at the time of the Roosevelt Federal Building bombing and that their son, Brady (Mason Gamble), was a member of the same Discovery Scout troop at which Dean Scobee volunteered. Unfortunately, by the time Faraday unearths the truth, it is too late because unbeknownst to both him and the audience, he has become the next unwitting participant in a new plot to destroy U.S. government agencies.
Most of the narrative thus focuses on Faraday's attempt to confirm his suspicions about his neighbors and convince others that his paranoia is justified. He constructs a conspiratorial account in which Oliver Lang is actually an alias for a convicted teenage pipe-bomber, named William Fenimore, who Faraday theorizes has sought revenge against the government since it unfairly seized his family's farm water source when he was a boy. Although no one else believes him for most of film, it is ultimately revealed that paranoia indeed can be a sense of heightened awareness because virtually all of Faraday's hypotheses eventually prove correct.
Oliver explicitly confirms many of Faraday's theories during the film's climactic chase sequence. When the two get into a physical altercation as Michael pursues a van (aptly emblazoned with the company name, Liberty), which he believes both contains Grant and is packed with explosives, Lang chastises Faraday for his obsessive desire to discover the truth by yelling, "You had to know, you couldn't leave your neighbor alone." It seems, then, that Faraday has become a legitimate threat to the terrorist organization because his conspiratorial account of Oliver's real identity may jeopardize the mission. He subsequently further validates Faraday's hypothesis by confessing that he is just "a messenger" and that "there's millions" like him "ready to take up arms" to make the government "pay for their sins." The dramatic confrontation between the film's protagonist and antagonist, therefore, appears to put an end to almost all of the film's primary causal lines of action. Now that Faraday has been proven right about his neighbor's involvement in a terrorist conspiracy, all that needs to happen for the film to conclude with a happy, Hollywood ending is both for him to stop the bombing and to save his son.
The film's ending, however, does anything but resolve the narrative in anticipated fashion. His dogged pursuit of the van ultimately takes him to the basement garage of the J. Edgar Hoover FBI building. Upon arrival, he convinces a group of skeptical FBI agents, led by Leah's ex-partner, Whitt Carver (Robert Gossett), to search the van. Shockingly, the van is empty and the trunk of Faraday's rental car is instead revealed to be packed with explosives. Consequently, it appears that the bomb was placed in the rental car when Michael was out of the vehicle for a brief period during his fight with Oliver, suggesting that their confrontation was staged by Lang's terrorist organization as part of the conspiratorial plan. The explosives are then detonated from a remote location by other members of Lang's team, destroying the offices of the FBI. Next, the film abruptly cuts to a television news report from the rubble in which a newscaster explains that "preliminary reports indicate the bombing was the working of" Faraday alone and that it has not yet been confirmed "if the bombing had anything to do with Faraday's wife, who was an FBI agent." The conspiratorial nature of the erroneous "official" account, then, is already being established by the initial media coverage of the event.
These inaccuracies are further codified in the news report by a series of anecdotes delivered by Faraday's former students, who claim that the professor was deeply disturbed by the circumstances surrounding his wife's death. Importantly, the final one of these statements is uttered by an ex-student of Faraday's, who was revealed during the bomb detonation sequence to belong to the same anti-government terrorist organization as the Langs. In the interview, she notes, "All I know is what he told me in his office after class—sweetheart, one day those men are going to pay, one day those men are going to burn." The news montage that explains the traumatic event indicates how supposedly transparent "authoritative" accounts are often just as constructed as conspiracy theories. It reveals how the "truth" is concealed by the organization behind the conspiratorial plot and its conduits in the media through a fetishization of causality and a prioritization of agency. Indeed, although spectators are now aware of what "actually" happened and who is "really" responsible for the traumatic event, the uninformed characters of the diegetic world are doomed to believe that one man acted alone to destroy the government institution that he blamed for his wife's unfortunate death.
Figure 2.1. A television newscaster presents an inaccurate report that attributes the bombing of the J. Edgar Hoover FBI building solely to Michael Faraday in Arlington Road.
The way that Arlington Road asks the audience to reread narrative causality in relation to how it is misunderstood by diegetic characters is especially relevant because it ultimately encourages viewers to engage in conspiratorial interpretive activities. Although it is true that most spectators likely never suspected that such an alternative explanation for narrative causality was possible, it is safe to assume that a majority of them are able to understand how the meaning of what has come before changes in light of the revelation. This is largely because the revised explanation of the causal relationship of events, like most conspiratorial accounts, adheres to a familiar narrative logic. Fenster even uses Bordwell's notion of the classical Hollywood film to conceptualize how those who engage in conspiracy theorizing routinely rely on many of the storytelling principles that are fundamental to its highly recognizable mode of narration. Citing one of Bordwell's sections of The Classical Hollywood Cinema he notes:
The "classical" conspiracy narrative attempts to unify seemingly disparate, globally significant elements and events within a singular plot, doing so through the traditional logic of conventional popular narratives, including "causality, consequence, psychological motivations, the drive toward overcoming obstacles and achieving goals. Character centered—i.e., personal or psychological—causality is the armature of the story." (Fenster 108)
The conspiratorial narrative, therefore, typically employs the same techniques as the classical film to explain narrative causality. In both cases, the events of the narrative can be fully understood as the workings of a single person or a group of individuals who, despite the many obstacles that arise, triumphantly attain the goals that were initially set forth.
Although the conventions of the classical narrative encourage spectators to believe that Arlington Road centers on Faraday's attempts to unmask the architects of the terrorist plot, the surprise conclusion, in hyper-classical fashion, reveals that the film only ends satisfactorily when the Langs are shown to accomplish their objectives. This is why the film concludes with scenes that depict both Oliver burning his files on Faraday and the Langs speculating about the details of their next mission. The existence of this surveillance evidence, in fact, indicates that Faraday had been explicitly selected for the gambit long before the Langs ever moved into the neighborhood. He was targeted as the perfect man to bomb the Hoover building because of the circumstances surrounding his wife's murder. Moreover, his scholarly fascination with radical extremist groups made him the ideal candidate to discover Oliver's secret identity, enabling them to trick Faraday into unknowingly delivering the bomb to the FBI's headquarters.
This logic only begins to suggest the deviousness and complexity of the Langs' plot. Faraday first meets his new neighbors, for instance, when he drives Brady to the emergency room after a purported fireworks accident. However, it is questionable whether he is even the Langs' son because during the changeover Grant is shown being cared for by other members of the conspiratorial cabal, suggesting that Brady could be an orphaned child of another victim from a prior plot. This seemingly outlandish explanation becomes even more plausible because Oliver is portrayed as an explosives expert, who will go to virtually any length, including severely injuring children, to lure Faraday into participating in the scheme. During his physical confrontation with Faraday, he even alludes to his willingness to harm kids if it will help him in the fight against the U.S. government by noting, "this is war Michael... in war children get killed." Admittedly, it seems far-fetched that such an intricate plot could ever really work. Viewers have to suspend disbelief to think that any group could arrange it so that a badly injured child wandering in the middle of a suburban neighborhood would be spotted by no one other than a lone passerby. Of course, Arlington Road was released shortly before the exposure of the revelations that al-Qaeda operatives took flight lessons in which they blatantly expressed no interest in learning how to land, were able to overtake four commercial airliners with little more than box-cutters, crashed planes into both towers of the World Trade Center and the Pentagon over the course of a one hour period without being stopped by the U.S. military, and so on. The existence of these kinds of overlapping discourses suggests why conspiracy theorizing has gained such cultural currency in the United States of late. Although the Langs' murderous scheme seems implausible, even for a Hollywood movie, a more diabolical, intricate, and horrifying conspiratorial plot dramatically unfolded in the U.S. just two years later.
It is the Langs, then, and not Faraday, who are ultimately Arlington Road's primary causal agents because they are revealed to be responsible for orchestrating the film's events. Such an epiphany clearly articulates the kind of agency panic that Melley argues underpins much contemporary conspiracy theorizing, as Faraday's actions are depicted as being completely manipulated by powerful forces beyond his control, even though he is an expert on the historical existence and practices of such terrorist organizations. In the end, the film's changeover leaves little doubt that, despite the fact Faraday was well aware of the Langs' nefarious objectives long before the destruction of the Hoover building ever occurred, he was incapable of stopping them from turning him into the perfect suicide bomber, making it a representation of contemporary conspiracy theorizing par excellence.
## Here Come the Men in Black: Jacob's Ladder and Paranoia as Heightened Awareness
Like Arlington Road, Jacob's Ladder is one of the few misdirection films to make justifiable paranoia an explicit thematic concern. Whereas Arlington Road foregrounds its focus on conspiratorial activities early and often, the presence of a devious plot orchestrated by the U.S. government does not become fully evident until the end of Jacob's Ladder. Specifically, the changeover reveals that narrative events have been a representation of the dying dream that a soldier, Jacob Singer (Tim Robbins), the film's protagonist, experiences while he unsuccessfully fights to stay alive after apparently suffering a mortal combat wound to his abdomen. During the film's ostensible climactic explanation before the changeover, a mysterious man, Michael Newman (Matt Craven), who trails Jacob throughout the dream, explains that he is a chemist who was forced by the military to manufacture an experimental, aggression-enhancing drug, nicknamed "The Ladder," to amplify soldiers' performance. Newman then notes that Jacob's platoon served as guinea pigs for the drug, which caused the soldiers to slaughter each other. This revelation is shocking because Jacob and viewers have been led to believe, since the film's opening bloody battle scene that a superimposed title classically indicates takes place in the "Mekong Delta" on "6 Oct. 1971," that the platoon was brutally ambushed by the Viet Cong. Although the changeover calls into question all that the dream shows, as it exposes how preceding events have been focalized through the mind of the comatose protagonist, a title card appears before the end credits roll, asserting that the U.S. military used a drug, colloquially referred to as BZ (quinuclidinyl benzilate), on test subjects during the Vietnam War, even though "The Pentagon denied the story." This epilogue suggests that many conspiratorial elements of the dream, such as the lawyer, Geary's (Jason Alexander) declaration that Jacob's platoon never served in Vietnam because they were "discharged on psychological grounds" after the military instead had them participate in clandestine "war games in Thailand," might, in fact, be true.
Jacob's Ladder's anti-Vietnam War message was part of a spate of Hollywood films, including more acclaimed titles, like Apocalypse Now (1979), Full Metal Jacket (1987), and Platoon (1986), that belatedly critiqued the controversial invasion. The popularity of the Vietnam War film cycle suggests that Jacob's Ladder's focus on the conflict had little to do with the filmmakers' struggles to get Hollywood backing. Instead, it was likely its duplicitous narrative structure that most made it seem potentially unprofitable because it took a decade until screenwriter and co-producer Bruce Joel Rubin's screenplay was finally released (Hartl). It was not until the now defunct Carolco, a fleetingly successful independent studio that also released another, more economically successful misdirection film that same year, Total Recall (1990), and collapsed as rapidly as it surprisingly rose to prominence, backed the project after the majors passed on it, that the film even went into production. Unlike some of Carolco's incredible successes in the early 1990s, most notably Terminator 2: Judgment Day (1991), Jacob's Ladder barely made back its production costs, earning approximately $26 million in U.S. theaters on its $25 million budget, where it was ultimately distributed by Sony subsidiary, TriStar Pictures (imdb.com). As with other misdirection films, then, it can be considered a Hollywood film from an industrial perspective because it was eventually distributed domestically by a media conglomerate even though it was produced independently.
These atypical production circumstances help to explain why the film epitomizes the textual properties of the contemporary misdirection film by mixing classical tendencies with non-classical elements. Aside from the aforementioned postscript, the only things that the viewer can accept as fact in light of the changeover is that a man named Jacob Singer died in a military infirmary tent after putting up a valiant struggle to stay alive while doctors tried to save him. The veracity of everything that precedes this moment, therefore, is uncharacteristically thrown into question by the changeover. In contrast to most misdirection films with a clear-cut changeover, Jacob's Ladder's revelation further complicates, rather than clarifies, what "actually" happened. Yet, director Adrian Lyne's decision to include the afterward about the military's purported experimental use of BZ strongly suggests the possibility that the dream narrative's explanatory sequence—Newman's admission that he concocted the drug surreptitiously administered to Jacob's platoon under duress—"really" occurred. Like Arlington Road, in retrospect, Jacob's Ladder is a film that quintessentially links the overlapping discourses of the misdirection narrative and conspiracy theorizing explicitly together by portraying a paranoid story of a protagonist who has been unwittingly manipulated by powerful forces beyond his control. In particular, even though Jacob, who is affectionately dubbed "the professor" by his platoon because he completed a PhD prior to going to Vietnam, is highly intelligent and is the only one of the crew who refuses to stop investigating the military's role in the incident after the other war veterans are coerced to drop the lawsuit, he is ultimately shown as being powerless to do anything about it. Viewers similarly realize at the end that they likely have been fooled despite the fact that the film uncharacteristically encourages them to interrogate the veracity of information consistently prior to the exposure of the changeover.
The lingering ambiguity after the changeover, then, is not the only aspect that renders Jacob's Ladder more non-classical than most other contemporary misdirection films. In comparison to many changeover films, which do not call the validity of events into question until the revelation, Jacob's Ladder constantly forces the viewer into a cognitive crisis by making it difficult to delineate between fantasy and reality. Such uncharacteristic ontological blurring transpires throughout the film, first occurring immediately after the opening battle scene in which spectators are led to believe that the strong marijuana that the group smokes before the attack is the reason they seem to go crazy during the bloodbath. Upon being bayonetted in the stomach by an unidentified attacker at the end of the opening combat scene, the film abruptly cuts to Jacob, now garbed in an U.S. Postal Service uniform, awakening on a New York City subway train with Albert Camus's classic, existential novel The Stranger in his hands. The edit initially appears to be a prototypical classical cut because Jacob grabs his abdomen as he wakes up startled, which helps orient the viewer in time and space by framing the preceding war scene as a dream. In retrospect, however, this dream's relationship to reality is complicated by the ways in which the changeover renders everything that comes before it functionally polyvalent.
Although the radical changes in Jacob's location and costume are initially jarring to the viewer, it subsequently becomes clear that the majority of the film is set in New York City after his discharge, ostensibly situating it in the "coming home" Vietnam War film subgenre. Constituent films, such as Coming Home (1978), The Deer Hunter (1978), First Blood (1982), and Born on the Fourth of July (1989), feature the trials and tribulations of returning veterans. Jacob's Ladder, likewise, appears to center on the post-traumatic stress that Jacob experiences after the war. Jacob's precarious mental state is the primary reason why the events that unfold throughout the New York City scenes are not initially understood to be the product of his dying dream. Viewers are instead led to believe that disturbing images may actually haunt him in his waking life in spite of his efforts to forget about the war. In fact, he tells his chiropractor, Louis (Danny Aiello), that he became a letter carrier rather than pursuing a faculty job upon returning from his service because "after 'Nam [he] didn't want to think anymore." Rather than interpret the entire New York City narrative as part of the dying dream, spectators are tricked into trying to determine if his terrifying visions are genuine or imaginings induced by postwar paranoia.
Yet, after Jacob awakes on the subway, the film slyly hints at the real explanation by focusing on subtle clues in the mise-en-scène that reference the dying dream. When Jacob comes to, the camera switches to his point of view as he gazes at two seemingly innocuous, but retrospectively narratively significant, advertisements. The first one claims that "New York may be a crazy town, but you'll never die of boredom," subtly alluding to the fact that Jacob is actually about to pass away at the end of the New York City dream. The second advertisement more cleverly reveals the film's big secrets, as it reads: "Hell. That's what life can be, doing drugs. But it doesn't have to be that way. Help is available, day or night." This ad is retroactively relevant because Jacob's dying dream about The Ladder centers on his visions of the demons that he thinks are trying to drag him and most of his fellow soldiers to hell even though no one, aside from most of his platoon-mates, believes they are real. Jacob's co-worker girlfriend in the dream, Jezebel (Elizabeth Peña), for instance, insists that the demons are actually just "winos and bag ladies" that populate the streets of New York City. Similarly, Rod (Anthony Alessandro), the only ex-platoon mate who is not haunted by the visions, asserts that his fellow vets are "all fucking paranoid" because "it was bad grass, that's all" and "there's no such things as fucking demons." Without knowledge of the changeover, both viewers and diegetic characters are encouraged to spend the dying dream narrative trying to determine if Jacob's visions are authentic or just delusions. As a result, the film's biggest clues about the presence of the changeover, such as when a palm reader tells Jacob that his lifeline indicates he is "already dead" and the many similar instances that hint that he is actually dying, are likely interpreted initially as potential manifestations of his post-traumatic stress-inspired hallucinations, instead of being read as evidence of the fact that he is about to pass away.
Figure 2.2. An advertisement in Jacob's Ladder alludes to its changeover that reveals Jacob Singer is having a drug-themed, dying dream in which demons try to drag him to hell.
The truth of the coming home narrative seems to be resolved when it is eventually revealed that Jacob both must acknowledge his impending death and stop holding onto his life's regrets for the demons to cease pursuing him. This finally happens after Newman's explanation about The Ladder in the dream, when Jacob returns to the Brooklyn apartment he shared with his family and realizes that Louis's Meister Eckhart-inspired advice of letting go of his earthly concerns will make the demons go away. In addition to lines of dialogue, like when Jacob asks Louis if "anyone has ever told him" that he looks like "an overgrown cherub," and formal choices, such as the heavy back-lighting that consistently makes Louis resemble a guardian angel, the narrative importance of the chiropractor and his advice are reiterated immediately prior to the changeover. Specifically, his earlier Eckhart-laden dialogue replays on the soundtrack right before the film cuts to a flashback, shot in home-video style, displaying key moments from what appears to be Jacob's actual family life. This formal choice is significant because it is not the first time that the aesthetic is used. The most notable time it was employed prior to this moment is during an earlier flashback within the dying dream, which reveals the tragic accident that killed Jacob's youngest son, Gabe (Macauly Culkin). Consequently, the stylistic change is significant because it is likely deployed to delineate these non-Vietnam scenes retroactively to signal their truth value in relation to the rest of the dying dream. After Jacob's life literally flashes in front of the viewer's eyes, he finally reunites with Gabe, who leads him up a staircase, symbolically indicating that his soul is ascending to heaven. Such a reading is reiterated formally, as the film cuts to a bright, white screen immediately before the shocking changeover occurs. In sum, the use of the home-video clips is designed to help spectators retrospectively distinguish fact from fiction in the dying dream narrative.
Figure 2.3. Jacob Singer's chiropractor, Louis, is depicted angelically largely because of heavy backlighting in Jacob's Ladder.
Throughout the rest of the pre-changeover sequences, the constant classically edited oscillations in and out of the dream make for often maddening transitions from what seems to be the subconscious imagination to waking life, but they also function consistently to fool the viewer by seemingly delineating fantasy from reality in clear-cut fashion. Retrospectively, though, it becomes virtually impossible to figure out what actually happened in Jacob's life because the changeover throws into question almost all that the dying dream depicts. As with the home-video footage, this is why the inclusion of the BZ postscript could be crucial to reinterpreting the film's meaning in a potentially coherent fashion. Indeed, its appearance at the film's conclusion after the changeover occurs suggests that Newman's revelation about the government's experimental use of BZ on Jacob's platoon has at least some merit. As a result, the film's most paranoid scenes, particularly those featuring U.S. government agents, become more believable, in retrospect, than many of the other elements of the dying dream. Most notably, it is Jacob's encounters with these mysterious men in black that potentially haunt spectators as being plausible after the changeover is revealed. Before he is able to escape, these intimidating, secret agents brutally kidnap Jacob, beat him, and attempt to kill him to stop him from investigating the military's actions. The now ubiquitous men in black in Hollywood cinema suggest that their actual existence has recently become highly plausible to U.S. audiences. Their important role in Jacob's Ladder's dying dream is thus a testament to the justifiable paranoia that pervades the viewing experience of the misdirection film. The popularity of such images illustrates that spectators and citizens alike readily acknowledge that they know they are now being controlled by powerful forces, such as men in black and Hollywood filmmakers, even though they are powerless to stop them. This awareness demonstrates why the retrospective reinterpretations the misdirection film inspires and much contemporary conspiracy theorizing about the abuses of formidable institutions, like the U.S. government, have resonated strongly with many people during the same historical moment.
Figure 2.4. Jacob's Ladder's men in black, government agents abduct Jacob Singer.
Contemporary misdirection films and some recent conspiracy theories are interrelated discourses that offer similar responses to their particular contexts. They are both narrative forms that have become popular of late because they appeal to a population that has grown increasingly skeptical of the project of modernity, but is not yet willing to abandon its foundational principles. Many contemporary conspiracy theories and misdirection films resonate with some audiences because they ultimately reveal, in accordance with the discourses of liberalism, that a select few individuals are really still in control of their lives. Although such alternative accounts should be applauded for exposing the constructedness of purportedly transparent "official" explanations, they paradoxically also usually adhere to the very narrative logic to which they are opposed. Most misdirection films indeed can be retrospectively reread in a hyper-classical fashion even though they initially seem to challenge some of Hollywood's most basic narrative and formal conventions. This tendency has strong cultural implications because many of these films, like the conspiracy theories they resemble, appeal to some viewers by transforming everyday incoherence into narratives that are both ultimately more conventional and supportive of dominant ideologies than they initially appear to be.
3
## Constructing the (Im)perfect Cover
Masculine Masquerade and Narrative Agency
SUCCESSFUL MISDIRECTION FILM ACTORS, like Tim Robbins, perform in ways that capitalize on the spectator's tendency to comprehend the causal relationship of narrative events as being attributable to the actions of a prototypical goal-oriented, male protagonist. Misdirection films prey on the spectator's propensity to associate narrative causal agency with men who behave in ways that are coded as authoritative and in control in relation to other characters. Such interpretations suggest that many audiences are inclined to consider some forms of manhood as ideal and active. It also demonstrates that many spectators are quick to judge other kinds of masculine performances as aberrant and passive. Gender performance, therefore, can function in a similar way that the classical Hollywood narrative does in the misdirection film. Indeed, although misdirection films challenge viewers to reconsider initial interpretations drastically, they also both rely on classical conventions to encourage audiences to draw incorrect causal suppositions and often can be reinterpreted according to a character-centered, causal logic that renders them hyper-classical, in retrospect. Similarly, gender performance in misdirection films generally leads audiences to believe that they understand the true personas of primary characters even though it is ultimately revealed that they jumped to incorrect conclusions about their real identities and narrative power.
The potential for misinterpretations of the links between gender performance and agency, of course, extend into the broader cultural sphere. Perhaps nothing demonstrates this phenomenon better than the aftermath of Osama bin Laden's targeted killing on May 2, 2011. His death raised serious concerns about the U.S. government's suppression of information related to the event, including photographic evidence confirming his death. It was peculiar, then, that among the first artifacts made public were videos showcasing the al-Qaeda leader studying and managing his media image. Even more bizarre was the subsequent decision to announce that a stash of heterosexual pornography was confiscated from his expensive Abottabad compound. These efforts were clearly an attempt to demystify bin Laden's legend as the pious, cave-dwelling commander of al-Qaeda. Such revelations suggest that government officials hoped his public persona would be reinterpreted as an intricate performance that concealed an "authentic" gendered identity aligned with hegemonic Western masculinity, characterized by a voracious appetite for wealth, fame, power, and women.
The event's fallout was also highlighted by news media accounts comparing it to a Hollywood film. This link was verified when Seal Team 6's successful raid became the climax of Zero Dark Thirty (2012). The escapade, though, appeals to Hollywood for reasons that extend beyond just the chance to reenact the operation. Zero Dark Thirty's central focus—its dogged heroine Maya's (Jessica Chastain) triumphant hunt for bin Laden—illustrates how Hollywood routinely distorts history by overemphasizing tales of exceptional individuals attaining lofty objectives. Such a tendency is unsurprising because whereas other cinematic traditions often do not present protagonist-driven quest narratives, Hollywood films have mass appeal largely because formal decisions are almost always subservient to its recognizable storytelling formula. In addition to the financial rewards linked to this canonical narrative and its associated invisible style, Hollywood benefits from it by mitigating political divisiveness. Its obsession with remarkable people overcoming seemingly insurmountable obstacles oversimplifies complex situations by boiling them down to Manichean battles between good protagonists and evil antagonists. Although this helps the industry avoid taking unambiguous stands on controversial issues, its recurrent practices have cultural ramifications. Classical protagonists, for instance, are disproportionately white, heterosexual men. Zero Dark Thirty thus notably features a female lead in a traditionally male role. How far it deviates from dominant ideology is debatable, however, as professional dedication results in alienation for Maya, who is denied even the joy of the compulsory heterosexual romance.
These examples show how mass media functions as an ideological state apparatus by engaging in the process of hegemonic negotiation. Media conglomerates do not portray culture monolithically because they both express and produce historically situated notions of identity. Yet, to maximize profits, producers generally play it safe by ultimately supporting conventional conceptions of cultural categories, like gender. In this chapter, I examine how a similar representation of masculinity in Unbreakable (2000) and The Usual Suspects (1995) epitomizes both how reactionary gender politics persist in Hollywood and the ways in which the misdirection film is particularly well-suited for surreptitiously maintaining male dominance. My argument thus extends the connections between misdirection films and conspiracy theorizing forged in the previous chapter by focusing specifically on how the genre's particular narrative machinations can effectively dovetail with expressions of contemporary white-male paranoia about the loss of cultural authority. Unbreakable seems to center on the role that African-American comic book dealer, Elijah Price (Samuel L. Jackson), who has a severe brittle bone disease, plays in the remasculinization of real-life superhero, David Dunn (Bruce Willis). Likewise, in The Usual Suspects, Verbal Kint (Kevin Spacey), an allegedly small-time crook purportedly afflicted with cerebral palsy, recounts how he helps reformed criminal legend, Dean Keaton (Gabriel Byrne), resurrect his outlaw glory. Both films' changeovers, however, show that their ostensible helper characters are actually archvillains who exploit their protagonist buddies. The following analysis of these two films illustrates how the misdirection film enables Hollywood to sustain traditional conceptions of manhood at a time when explicit mediations of hegemonic masculinity are received skeptically. Specifically, even though the changeovers reveal that seemingly feminized primary male characters are more powerful than conventionally masculine protagonists, they do not suggest that multiple masculinities are a reality. I ultimately contend that their apparent defiance of established male hierarchies instead supports dominant ideology by imagining that men's primal spirit endures behind a façade of aberrant masculinity.
## Masquerade Required: Preserving Male Essence in the Misdirection Film
Depictions of primary male characters concealing an authentic male identity under an emasculated cover represent a significant change in ideal manhood in Hollywood. These men are a far cry from the muscle-bound heroes, played by Arnold Schwarzenegger, Sylvester Stallone, and Bruce Willis, that characterized Hollywood in the 1980s. For Susan Jeffords, such "decisive, tough, aggressive, strong, and domineering" male protagonists were products of the times, as they intersected with Ronald Reagan's agenda to undo the policies of Jimmy Carter's administration, which were deemed "weak, defeatist, inactive, and feminine" (11). When Reagan's excessive masculine posturing no longer seemed as necessary in the immediate post-Cold War moment, she notes, sensitive family men began replacing hyper-masculine protagonists. This shift was typified by Schwarzenegger, who appeared in films blending his action hero persona with a domestic facet, including Kindergarten Cop (1990), True Lies (1992), and Junior (1994). Although such developments seem to embody vastly different masculine standards, Jeffords contends that they are actually "overlapping components of the Reagan Revolution," encompassing both "a strong militaristic foreign-policy position" and "a set of social values dependent on the centrality of fatherhood" (13). Her reading exemplifies how apparent transformations in gender representation back the same ideological project. How, then, do misdirection films, like Unbreakable and The Usual Suspects, extend this trend by articulating fantasies of continued male dominance that are appropriate for their contexts?
Discerning the gender politics of these two films is tricky partly because their atypical narrative structures effectively encourage spectators to draw incorrect conclusions, which can allow them to conceal their ideological messages more effectively and render their cultural expressions more ambiguous. The misdirection film's typical depiction of gender exemplifies how it frequently relies on classical standards to work its deceptive magic. As Psycho (1960 and 1998) and The Crying Game (1992) have famously shown, the misdirection film is perhaps more adept than any Hollywood genre at making audiences aware that it is easy to misconstrue markers of identity, such as gender and sexuality, exposing how viewers draw hasty conclusions about characters' relative narrative agency. Although most misdirection films do not prompt spectators to reevaluate a primary character's identity this drastically, many encourage audiences to understand gender as being unstable. This is especially true in relation to masculinity because numerous contemporary misdirection films stunningly reveal that male protagonists are victims of a fantasy or at the mercy of seemingly weaker male characters.
Even though these films contain surprise endings that illustrate that gender is constructed, they generally do not ultimately show, as Judith Butler seminally theorizes in Gender Trouble, that masculinity is entirely performative because "what we take to be an internal essence of gender is manufactured through a set of acts, posited through the gendered stylization of the body" (xv). Similarly, they do not demonstrate, as Jack Halberstam does in Female Masculinity, that "masculinity becomes legible where and when it leaves the white male middle-class body" since it can be mobilized by anyone, irrespective of ability, class, race, sex, and sexuality (2). Instead of demonstrating the progressive potential of decoupling gender from other markers of identity, Unbreakable and The Usual Suspects typify how misdirection films regressively depict gender performance as a way to conceal an antiquated male core that remains intact.
Although it is usually deployed in relation to women and femininity, the concept of masquerade relates to the kind of gendered deception male characters undertake in these two films. Like Steven Cohan in Masked Men, I thus use masquerade for its "theatrical rather than phallocentric implications," which is "in accordance with Butler's theorization of gender as 'performative' " (26). Masquerade is appropriate for analyzing mediated masculinity, as he contends, not because of Joan Riviere's psychoanalytically inspired conception of femininity as a way "to conceal a secreted theft of the phallus" (qtd. in Cohan 26). Instead, masquerade's theatrical dimension reveals how portrayals of masculine artifice can disrupt rigid notions of biologically determined gendered identity. Yet, as Jackie Stacey suggests in her analysis of Gattaca (1997), masquerade can be difficult to apply to representations of manhood since the "impossibility of masculinity" highlights "the more general façade of 'authentic' masculinity" (1862). I grant that such logic applies to a film, like Gattaca, in which an inauthentic perfect masculinity is the disguise because ideal masculinity is indeed shown to be unattainable when the copy and the original are both exposed as fabrications. Unbreakable and The Usual Suspects, by contrast, unveil a male essence behind an imperfect masculine cover. Masquerade, therefore, as Chris Holmlund notes, accounts for how films that reveal that an authentic masculinity exists below the surface "reinforce hegemonic power relations" by exhibiting "that there may be something underneath which is 'real' and/or 'normal' " (224).
The narrative fantasy of cloaked male quintessence appeals to a culture in which media representations of masculinity in crisis have become practically inescapable. Since the early 1990s, numerous Hollywood films have focused on the difficulties that white, heterosexual, American men, who perform their gender traditionally, are having maintaining authority. In 1999 and 2000 alone, for instance, as Philippa Gates observes, many films "centered on male protagonists in crisis," appearing "to indicate a broader social concern that at the turn of the new millennium masculinity was, indeed, in crisis" (46). Interestingly, of the films she cites—American Beauty (1999), Fight Club (1999), Magnolia (1999), The Sixth Sense (1999), American Psycho (2000), The Beach (2000), Memento (2000), and Unbreakable—all but American Beauty and The Beach are misdirection films, reiterating the narrative mode's suitability for expressing cultural fears and desires related to manhood. Of course, patriarchy is always in crisis because of perpetual threats to its supremacy. It is still valuable, though, according to Michael Kimmel, to analyze "the times when dominant masculinity is widely perceived to be under threat because there is a search for the timeless and eternal when old definitions no longer work and the new definitions are yet to be established" (3). Similarly, Nicola Rehling writes that "masculinity in crisis" troublingly "postulates a once stable, coherent, unified masculinity," making it valuable to identify "which particular forms of male insecurity are made manifest at specific historical junctures" (2–3).
Instead of abandoning the "crisis model" for examining masculinity, an exploration of the distinct historical periods in which traditionally accepted conceptions of manhood are widely perceived to be under threat can reveal the specificities of the paranoia directly related to the times. Although it is true that conceptions of "ideal" masculinity are always being contested, the topic of contemporary American men in crisis is particularly noteworthy because it has recently become so appealing to Hollywood producers and audiences. Susan Faludi's oft-referenced and controversial book, Stiffed, is particularly relevant in this regard because both the reasons it presents for the American male's purported fall from grace and the ways it offers to help him resurrect his authority are often vividly expressed by recent popular culture representations, including a number of misdirection films. According to Faludi, so many American men have become increasingly disillusioned because the institutions created by their predecessors to support their dominance have failed them. She theorizes that many of the changes precipitated by the U.S.'s emergence as a superpower after World War II, such as the shift from a production economy to a service economy as well as the gains made by the women's liberation, civil rights, and gay rights movements, have resulted in the depreciation of the roles that white, heterosexual men play in economic, political, and social spheres. Whereas their fathers were almost unequivocally considered the undisputed heads of their households, earned respectable livings in the production economy, and were revered as the authorities of culture, or so the story goes, contemporary American men do not have similar male-dominated institutional spaces to assert their traditional masculine control.
A cursory evaluation of the gendered makeup of corporate boardrooms, the military ranks, and the highest levels of government in the United States begins to raise doubts about the veracity of Faludi's claims. Regardless, Faludi's argument is relevant largely because of the ubiquity of contemporaneous mass media productions that echo her thesis that many men are now in an unenviable situation. She contends that American men are still expected to express their masculine control in the same manner as previous generations of men; however, they do not have the foundation necessary to display their manhood even though, paradoxically, their fathers came to power during a period of unprecedented abundance. Over the course of her research, Faludi eventually comes to the shocking conclusion that the younger men who were the subject of her quasi-ethnography had been forced to come "face-to-face with the collapse of some personal patrimony" because they almost inevitably suggested that behind all of their problems "lay their fathers' desertion" (596). The unfettered rise of consumerism and its corresponding mass media representations of "ideal" masculinity, she theorizes, also play crucial roles in the contemporary crisis in American manhood. Like the fifties housewife, who primarily acted as a domestic servant and the arbiter of consumption, Faludi claims that the "nineties man stripped of his connections to a wider world" has been "invited to fill the void with consumption and a gym-based display of his ultra-masculinity" (40). Unlike women, though, who have had men as the clear enemy oppressor, men have neither had a tangible nemesis nor a way to escape their predicament because there is no socially acceptable alternative to conceive of their gender.
For Faludi, the only way for men to rectify the situation is to find a way to regain their authority by "having authored something productive," which she admits has become increasingly difficult in a society where the kinds of contributions that most men now make in the service economy or on the high-tech warfront are much harder to quantify in the same terms as previous generations (86). Although Faludi ultimately argues that the task of men "is not, in the end, to figure out how to be masculine—rather their masculinity lies in figuring out how to become human," she never disputes that his "proper" role is as the active, and primary contributor to cultural, economic, and political arenas (607). I grant that she makes a valid point that men need to stop evaluating their worth based on how well they live up to gender standards that have long been codified as ideal in American culture. Her proposed solution, however, sustains, rather than redresses, the structural inequities that have kept white, heterosexual men on top and women and other minority groups on the bottom for centuries in the United States.
Irrespective of whether or not Faludi is ultimately right about either the culprits for or the appropriate responses to the contemporary crisis in American masculinity, Stiffed is particularly germane because, as Martin Fradley speculates, it has proven to be "symptomatic and representative of social perceptions and frustrations of many white, American men" (238). A number of Hollywood films that were released around the time of its publication indeed overlap with her fundamental claims about the status of American manhood, helping to produce cultural anxieties related to changing gender roles and relations in the United States. As J. Michael Clark observes, Tyler Durden (Brad Pitt) "virtually paraphrases Faludi" when he informs his fellow members of Fight Club that they have "no Great Depression, no Great War, no other dragon to battle heroically against, but that, instead, the 'great depression' is their own contemptuous lives lived so passively enslaved to consumerism" (67). These challenges prompted many men to respond in reactionary ways. In the 1990s, for example, there was a rise in men's groups, like the Mythopoetic Men's Movement, whose leader, Robert Bly, author of best-seller, Iron John, urged men to participate in wilderness retreats that enabled them to relocate their supposedly lost inner wild man.
Although the declarations that white, heterosexual, American men have recently been victimized by a series of unfavorable circumstances might be overblown, Faludi's claims are intriguing partly because many in this cohort have, in fact, fallen on harder times of late. The economic policies initiated by the Reagan and the George H. W. Bush administrations in the 1980s and early 1990s, strengthened by similar strategies enacted by subsequent U.S. presidents, had negative repercussions for many middle- and working-class Americans. A number of the developments associated with these neoliberal policies, as Kimmel documents, such as the "outsourcing of manufacturing jobs, plant closings, downsizing layoffs, cutbacks, and the gradual erosion of the safety net (health insurance, medical benefits, Social Security) instituted by the New Deal have ushered in a new era of 'social insecurity' " that is gendered because it "confounds men's sense of entitlement, their ability to be family providers and breadwinners" (218–19). The dramatic widening of the earning gap and the corresponding evisceration of the middle class since the early 1980s have made it increasingly challenging for many American males to continue fashioning themselves as self-made men. Moreover, the few mechanisms that were actually once deemed acceptable by many who subscribe to rugged individualism have been almost entirely eradicated. As Kimmel writes, the social programs that once "buttressed the self-made man's ability to be a successful breadwinner and provider—minimum wage, the GI bill, high wage employment, and unions—have eroded or disappeared" (220). Again, although these developments have diminished the economic standing of many white, heterosexual, American men, this group is still in a better position than minority groups to succeed. The real losses that many middle- and working-class white, American men have recently suffered, however, have caused confusion and anger for those who continue to struggle to demonstrate their cultural worth in a traditionally masculine fashion.
These paranoid responses convey worries about diminishing individual autonomy that jibe with Timothy Melley's aforementioned conception of agency panic. However, rather than address the culprits, such as the neoliberal policies that have consolidated wealth and bolstered corporate power, many disenchanted men instead blame familiar scapegoats, including big government, women, and other minority groups. Importantly, the perception that individual agency is dwindling has struck white, heterosexual men hardest because of their steadfast faith in rugged individualism and the American Dream, which, in spite of their meritorious myths, historically favor the privileged. In the 1990s, the destructive ramifications that could result from such reactionary fears were epitomized by the notorious cases of domestic terrorism committed by Timothy McVeigh and Ted Kaczynski, who, as Melley posits, believed that the supposedly "feminizing force" threatening their masculinity should be redressed by " 'regeneration through violence' " (14). Melley's reliance on Richard Slotkin's theory of regeneration through violence is significant because, as Slotkin argues, it was a crucial aspect of the frontier myth that "represented the redemption of American spirit or fortune as something to be achieved by playing through a scenario of separation, temporary regression to a more primitive or 'natural' state" (12). Centuries later, this logic endures for many American men hoping to find their mythical, primal manhood, as evidenced by groups like the Mythopoetic Men's Movement.
In contrast to the frontier era, there is now less unequivocal acceptance of men reverting to their supposed inner wild man. After the Reagan-era backlash against second wave feminism subsided, as David Greven documents, American masculinity "became aware of itself as both monolith and joke," resulting in growing Hollywood representations of a "post-Reagan New Man," that articulate a "split masculinity, which performed traditional roles of gendered identity while also acknowledging its ironic, meta-textual status" (16, emphasis in original). Brenton Malin, likewise, hypothesizes that this kind of dual portrayal characterized Bill Clinton's presidency and was embodied by a simultaneously dominant media representation of "conflicted masculinity that embraces and puts aside a variety of masculine stereotypes" (8). Until the Monica Lewinsky scandal calcified his budding reputation as a philanderer, for instance, Clinton was also consistently disparaged for being obsequious to his wife, Hillary Rodham Clinton. Such contradictions in masculinity, Malin argues, are palpable in many contemporaneous Hollywood films, featuring protagonists "salvaging the disturbing traditions from which the '90s man seemed to diverge" (10). Hollywood indeed released a number of Oscar Best Picture winners during the period under study, including Braveheart (1995), Jerry Maguire (1996), American Beauty, and Gladiator (2000), portraying broken protagonists who recapture their lost male spirit traditionally. In contrast to these depictions of explicit remasculinization, misdirection films often prey on the spectator's propensity to associate narrative agency with conventional protagonists by revealing that they are powerless at a time when other tactics are necessary to maintain male authority.
The substantial pressures on American men to prove their masculine prowess in a familiar manner without the same kind of supporting structures to do so has encouraged many men to seek alternative ways to counter their dwindling power. Throughout the 1990s, for example, some men turned their attention from displaying their competence at the workplace to showcasing their masculine proficiency in domestic affairs. As scholars like Jeffords argue, though, this turn to the home front did not represent a significant departure from the conservative agenda to maintain the patriarchal order. Although the home rapidly became a logical place for the dispossessed American man to reestablish his lost sense of worth, men did not necessarily change their approach to domestic affairs even though their role as the primary breadwinner was now often matched or exceeded by their female partners.
Many men stripped of power in the economic arena have thus sought to regain a foothold in the domestic sphere by reasserting their manhood as authoritative patriarchs. As a consequence, Kimmel argues that many men have focused on traditionally masculine activities, such as protecting their families and representing their family's interests in external affairs, which are "all valuable behaviors," but are "also behaviors that do not require that he ever set foot in his child's room" (237). This turn to the home front, then, is a classic case of hegemonic negotiation in action because it does not necessarily mean that men perform the child-rearing roles typically associated with women. Consequently, many men's movements in the 1990s attempted to determine the particular value that fathers, and not mothers, were thought to be well-positioned to provide to their families. Adrienne Burgess, author of Fatherhood Reclaimed, for example, cites the U.S. National Fatherhood Initiative as evidence that to help the family unit function most effectively, men must behave differently than women in the home because they should "contain emotion and be decisive" and should not be expressive or nurturing (27). For these kinds of activists, men need to assume a greater leadership role in the home because only they can provide the rational guidance that their purportedly over-emotional and irrational female partners are believed to be incapable of executing.
Not all proponents of the new family man, however, are opposed to men taking a more active role in parenting. For many throughout the 1990s, fatherhood was considered, as Stella Bruzzi claims, to be the primary "vehicle for teaching a man how to feel," making the "articulate, caring father the most valued male archetype" during the period (157). Unfortunately, though, the father often became highly prized in American culture not because of the actual contributions that he made around the house, but simply because he was actually there. In an era marked by the increasing presence of the single-parent household, the father who stayed and made time for his family, especially for his male offspring, became the prototype in some circles, such as the Mythopoetic Men's Movement. It became the job of dads to heal the "father wound" inflicted by a previous generation of men who had inexplicably deserted their families after promising so much to their sons. According to conservative thinkers, like Bly, this unfortunate trend began during the industrial revolution when men were encouraged to leave the home for extended periods to work in factory-style labor. The total disappearance of many men from the home in an era of post-1960s women's liberation only exacerbated this problem. As Kimmel documents, Bly and his followers expressed paranoia that this abandonment meant that many women "retained an incestuous dedication to their sons, excluding the father and keeping the boy dependent on her long after he needed to break away" (208).
Commercial U.S. films centering on young men who struggle to come to terms with the fathers who unexpectedly deserted them have been extremely popular for a number of decades, suggesting that Hollywood has played a key role in the ideological project to recapture masculine authority by turning to the home front. To wit, as Kimmel notes, some of the most commercially successful Hollywood films of the early blockbuster era—E.T.: The Extra-Terrestrial (1982), Close Encounters of the Third Kind (1977), and the films of the first Star Wars trilogy (1977, 1980, 1983)—use " 'healing the father wound' as both the motivation for and the ultimate result of the masculine quest" at the heart of their narratives (211). Indeed, both Steven Spielberg and George Lucas, arguably the most successful creative personnel working in Hollywood since the mid-1970s, have made veritable entertainment empires out of films that depict the efforts that young men make to establish relationships with the fathers who never taught them how to be a man. This paranoid drive to reassert patriarchal dominance by reclaiming fatherhood through often troubling means continues to be a central theme for some contemporary Hollywood filmmakers, including many of the misdirection film genre's most prominent directors.
## Unbreakable Masculinity: Reclaiming Men's "Rightful" Place
To coincide with the release of Signs (2002), M. Night Shyamalan's second film after his smash commercial hit The Sixth Sense, the August 5, 2002 edition of Newsweek featured a cover story on the director, touting him as "The Next Spielberg." The connection between the two filmmakers was logical for a number of reasons. To begin, just a few films into his Hollywood career, Shyamalan had already frequently professed his indebtedness to his filmmaking idol, such as numerous times in the cover story article by Jeff Giles in the aforementioned issue of Newsweek, and demonstrated a similar ability to create films that appealed to a family audience. Additionally, his films contain thematic preoccupations that align with Spielberg's favorite concerns. Most notably, Shyamalan's films are also obsessed with the influence of the meltdown of the American nuclear family on the lives of a generation of men and boys. The Sixth Sense, for example, centers on the plight of two male characters—one young (Cole Sear) and one middle-aged (Malcolm Crowe)—who both struggle to reconcile their small, splintering families. The two characters are only able to reconnect with their loved ones after developing a quasi-father-son relationship. Their bond finally enables Cole (Haley Joel Osment) to communicate openly with his mother (Toni Collette) about his problems and transforms him into a popular boy at school. Similarly, even though he is later revealed to be dead, the connection that Malcolm (Bruce Willis) forms with Cole allows him to reconcile with his wife (Olivia Williams) and provides him with the assurance that he is still an effective child psychologist.
Shyamalan already began to establish himself as a filmmaker in the auteurist mold before he made Signs, which also portrays a man's struggle to come to terms with death, albeit his wife's and not his own, to understand what it means to be a good father to his children in her absence, and to regain confidence in his chosen profession. Unbreakable, the film that Shyamalan made right before Signs also clearly showcases its direct connections to The Sixth Sense, its immediate predecessor. Unbreakable's trailer, for instance, advertises the film as another collaborative effort by the writer/director of The Sixth Sense and Bruce Willis. Elijah Price subsequently rhetorically asks the audience if they are "ready for the truth?," alluding to the presence of another memorable changeover.
Unbreakable's trailer could not be accused of false advertising because the film contained many of the same narrative and formal attributes as The Sixth Sense. It also included a changeover, the use of a particular color to signal danger (purple, in Unbreakable's case), and a fascination with the supernatural. Even more importantly, Unbreakable's narrative similarly centers on the existential angst of a dispossessed white, heterosexual, middle-aged, male character—David Dunn—who aims to reclaim his positions of authority in the home and on the job. Moreover, as in his previous film with Shyamalan, Willis's character must seek the help of marginalized people to reassert his agency. As with The Sixth Sense, it is not until David is truly willing to listen to the advice of a young boy, in this case his biological son, that he is able to discover his calling. Interestingly, David's son Joseph was played by Spencer Treat Clark, who also had a significant role in Arlington Road just a year earlier, suggesting that Willis was not the only actor cast for his ability to perform in the misdirection film genre. Shyamalan's familiar narrative scenario, then, appears once again in Unbreakable, and is actually taken a step further, in this instance, because David also needs to heed the advice of Elijah, a physically disabled African-American man, to overcome the obstacles that stand in his way.
The remarkable commercial success of The Sixth Sense established Shyamalan's reputation as an up-and-coming Hollywood writer/director, significantly influencing the ways that audiences received a follow-up effort that was clearly packaged in auteurist terms. Unbreakable opened to a lukewarm reception from both critics and audiences upon its release in 2000. The film grossed a respectable $95 million during its run in U.S. theaters on its healthy $75 million budget; yet its small profit margin was nowhere near the box-office gold generated by The Sixth Sense (imdb.com). Like audiences, film reviewers were divided about Unbreakable's merits. Many critics agreed with Eric Harrison of the Houston Chronicle, who claimed that "Shyamalan made the unfortunate, but predictable choice of trying to bottle The Sixth Sense's lightning and reuse it" (Houston 1). By contrast, many other critics agreed with the New York Post's Lou Lumenick, who wrote that "Shyamalan's dazzling reunion with Bruce Willis confirms that he's one of the most brilliant filmmakers working today" (47). Shyamalan himself also played a considerable part in encouraging these kinds of auteurist connections because he has made explicit efforts to establish his identity as the preeminent maker of misdirection films in many of his films since The Sixth Sense.
Another reason why Shyamalan has become widely referred to as an auteur is because of his penchant for working with a relatively consistent stable of actors. This anachronistic notion of authorship harks back to an era in which both actors and directors were forced to work under highly restrictive multi-picture contracts with a studio. Under the post-studio-era, package-unit system, in contrast, the teaming of celebrated directors with A-list actors on multiple films is instead often a marketing strategy because, as the industrial logic goes, there is a better chance to capture audiences if familiar names from successful collaborations are reunited on subsequent projects. Disney tried to repeat the success of The Sixth Sense by prominently featuring the pairing of Shyamalan and Bruce Willis in all forms of advertising in the hopes that Unbreakable would also attract both an arthouse and mainstream crowd. The re-teaming of Willis with Samuel L. Jackson was attractive to Disney for similar reasons: the success that the two actors had previously experienced together with both Pulp Fiction, another misdirection film with an indie sensibility, and Die Hard With a Vengeance (1995), the third installment in the highly successful action film franchise, could be leveraged to target these same audiences.
Although the Shyamalan/Willis and Willis/Jackson duos made sound economic sense, it was not merely a cynical marketing tactic. Shyamalan built his now fledgling brand in part by highlighting auteurist tendencies other than just the presence of an ending that forces spectators to reinterpret a majority of narrative information. In addition to the aforementioned thematic preoccupations, Shyamalan's films usually present faith in a divine plan, feature children or adolescents who see the world more clearly than adults do, are set in the Philadelphia vicinity, and so on. Among his stylistic tendencies are long takes, shots of reflected images, and the use of a mobile camera. The casting of Willis as the protagonist of Unbreakable can similarly be read in auteurist terms even though strong economic motives also undoubtedly drove the decision. In particular, Willis was an ideal candidate for the lead in Unbreakable because his secure position as a Hollywood action hero with limited acting ability could be used to the director's advantage, as it was in The Sixth Sense, to keep the audience from prematurely unearthing the changeover. Similarly, Jackson was a logical choice for Willis's ostensible sidekick because his familiar role as the prototypical "black buddy" in films, like Die Hard With a Vengeance, also effectively concealed his character's real motives.
Willis's performances in a number of profitable Hollywood action films released during the late 1980s and 1990s, including the first three films in the Die Hard series (1988, 1990, 1995), The Last Boy Scout (1991), and Armageddon (1998), helped to cement his reputation as a bad actor with great box-office appeal. Willis has subsequently become typecast as a wise-cracking, everyman hero in the mold of Die Hard's John McClane, who relies on his traditionally masculine attributes, such as physical prowess, cool under pressure, and mental toughness, to overcome seemingly insurmountable odds. Although his particular acting style and hyper-masculine onscreen identity have proven extremely popular with audiences, critics have remained largely unimpressed by his performances. Reviews of The Sixth Sense, for example, almost universally claimed that Willis had little to do with its success because child actor, Haley Joel Osment, was believed to have delivered the film's most impressive performance. Critic Rod Dreher, for example, claims that "Willis is OK in this movie, but it's not his picture anyway. This film belongs entirely to an 11-year-old kid named Haley Joel Osment" (43). Fellow reviewer Mick LaSalle speculates that Willis "has more screen time, but his role is reactive. Osment carries the drama" (emphasis added, C1). As these comments suggest, even though Willis was the star of a film that ended up being a critical and box-office sensation, his unfavorable reputation overshadowed the part that his acting played in its success.
Willis's perceived limited range as an actor and his inextricable connection to the Die Hard films, though, are key reasons why he became the quintessential misdirection film star. His familiar performance style from the Die Hard films is inseparable from his onscreen persona in any genre and has been almost universally reviled by film reviewers. Critic Stephen Holden, for example, pejoratively remarks that Willis wears the same smirk to suggest his sensitivity for Cole in The Sixth Sense as he does when "he is about to shoot someone in the face" (14). In his review of Unbreakable, James Berardinelli similarly complains that Willis's recognizable acting style is "too laconic" and showcases little "more than a glimmer of emotion." His performances are thus typically deprecated because they lack the kind of psychological depth that is most prized by the critical establishment. Traditional evaluations of Hollywood film performance such as these, however, neglect to account for how choices that actors like Willis make with their faces, bodies, and voices can inspire audiences to draw conclusions that extend beyond just character psychology.
In contrast to these typical assessments of Hollywood film acting, in Acting in the Cinema, James Naremore discusses the performer's ideolect, a term which he adapts from linguistics, as another way to evaluate "the actor's set of performing traits systematically highlighted in films." (4). As Andrew Higson also speculates, by approaching film performance in this way, the physical and vocal utterances of the performer can be conceived of "as a field of discourse" composed of "visual and aural signs" because in that context acting becomes "not the enactment of a coherent, psychologically complex character, but a montage of gestures (or 'gests')" that is "refined for the requirements of the shot" (154). Extending Higson's and Naremore's arguments, Willis's wooden acting style can be interpreted as entirely appropriate for the constraints of the misdirection film genre. His recognizable, monotone voice and limited facial expressions are effective for the misdirection film because they evoke an existing set of acting conventions with which the audience is familiar. Such choices are master "fabrications," which, in Frame Analysis, Erving Goffman defines as the "intentional effort of one or more individuals to manage an activity so that one or more others will be induced to have a false belief about what is going on" (83). Actors in misdirection films, then, must perform in a way that does not raise audience suspicions. Even more importantly, as Goffman theorizes, fabrications depend on audience expectations because they "require the use of a model, the use of something meaningful in terms of primary frameworks" (84). Effective fabricators keep the equivalent of a straight face by making their actions appear to be customary even though they are ultimately revealed to be anything but reliable. It is no coincidence, therefore, that Willis has been cast in a lead role in six contemporary misdirection films, more than any other Hollywood actor. His decision to perform these roles, as he would John McClane, is not out of sync with the requirements of the genre. In short, Willis was so often selected to star in these films not because of his ability to create psychologically complex characters, but because his stock techniques never signal to the audience that their interpretations of his character traits will be violated.
In Unbreakable, Willis's character struggle stems from his decision to become a security guard after abandoning a promising football career by faking an injury in order to marry his wife, Audrey Dunn (Robin Wright), who detests the violent sport. After miraculously surviving a train wreck, David reluctantly discovers how to reclaim his masculine prowess without jeopardizing his marriage. At the encouragement of Joseph and the enigmatic Elijah, who, thanks to faith in comic book lore believes that David was the sole survivor because he is really a superhero who is impervious to most injuries, he covertly learns to harness his superhuman strength and psychic ability to see the past crimes of the perpetrators he touches. As the generically inspired alliterative name, David Dunn, suggests, a clandestine superhero persona becomes the perfect alter-ego for a seemingly ordinary man, like Clark Kent and Peter Parker, whose spectacularly brutal talents must be kept secret from his pacifist wife.
Without knowledge of the changeover, this familiar narrative trajectory looks like just another example of how Hollywood imagines that white, heterosexual male dominance still exists even though hegemonic masculinity is less universally approved. After all, it ostensibly focuses on how the protagonist's closeted revival of his authentic male identity helps him overcome feminizing forces and restore his family. Additionally, its apparent remasculinization project seems disconcerting because of its racist implications. Heather Hicks, for example, documents that numerous critics jumped to the erroneous conclusion that Unbreakable contains the same dangerous racial ramifications as other contemporaneous Hollywood films, such as The Family Man (2000), The Green Mile (1999), and The Legend of Bagger Vance (2000), which also feature stereotypically mystical African-American friends, who, like Elijah, leverage their powers "toward helping and enlightening a white character" (28). As she contends, such critiques neglect how the changeover can alter comprehensions of narrative information. In contrast to conventionally altruistic magical African-American sidekicks, Elijah's objectives are eventually revealed to be anything but noble, if the narrative epiphany is interpreted as exposing a different motive for the antagonist's actions.
Had Unbreakable been exclusively about how David's superhero identity frees his suppressed male spirit, then it would have ended in standard classical fashion, as it appears to do when David rescues helpless children by killing their captor, reconciles with his wife, realizes his job protecting people is actually meaningful, and solidifies his bond with Joseph by covertly divulging his secret identity. Although all narrative lines of action are resolved satisfactorily, the end credits do not roll. The film instead cuts to an apparent epilogue in which David visits Elijah's store to thank him. Before expressing his gratitude, David meets Elijah's mother (Charlayne Woodard), who agrees with him that her son is a "miracle" for surviving accidents that should have "broken him," suggesting that Elijah is really the film's titular character. Once Elijah and David reunite and shake hands, the revelation shows that Elijah is indeed the film's primary causal agent. As is customary in many misdirection films, the changeover contains flashbacks exposing what actually happened. When the two finally touch for the first time, a bright light flashes and a loud screeching noise plays, signifying, as it has throughout, David's psychic ability to see the past illicit actions of touched subjects. Both David and the spectator simultaneously learn that Elijah is actually an archvillain, and not a benevolent helper, who has committed many terrorist acts, including David's train derailment, to find his adversary. The final scene can thus be reinterpreted as more than just an epilogue, as Elijah's desire to understand his brittle bone disease is really why he mentors David. As the successive reverse zoom-outs on both characters after they release hands show, they are most linked by Elijah's insistence on discovering a nemesis, like David, whose superhuman physical resilience renders the antagonist's opposite disorder meaningful. The narrative is only resolved classically, then, because David's belated understanding that he is a superhero confirms Elijah is an archvillain.
This revised explanation of narrative causality inspired by the revelation alters the meaning of almost all narrative information. Just as the final scene can be reinterpreted to be more than an epilogue, the film's opening scene can now be read as no longer simply being a tangential prologue. Although it is not immediately apparent that it is a flashback, a superimposed title indicates that the scene takes place in a Philadelphia-area shopping mall in 1961, decades before David and Elijah meet and perhaps not coincidentally the same year that Marvel Comics became a household name with the publication of the first issue of the Fantastic Four (DeFalco 84). The pre-opening credit scene depicts a flashback of the birth of baby Elijah, who is swaddled in a purple trimmed blanket (purple is linked to Elijah throughout the film) after suffering fractures during his emergency delivery in a department store dressing room. Significantly, virtually the entire scene is shot in a mirror image reflection, one of Shyamalan's formal signatures, which functions as an important visual motif in this film. In classical fashion, the technique is not just an artistic flourish because it subtly references both the paralleling of the two primary male characters and Elijah's hidden, archvillain alter-ego: "Mr. Glass." To reiterate the unstated relationship between the two, after the prologue, the film surprisingly cuts to an image of David aboard the soon-to-derail train, and not to a grown-up Elijah. The film's misdirection has already begun, in other words, as the edit leads spectators to identify with David by misleadingly positioning him, rather than Elijah, as the primary causal agent.
David's character is subsequently introduced classically because important information about his psychological traits is communicated rapidly. David first appears from an atypical angle that, on reverse shot, is revealed to be from a young girl's perspective. A purple-clad stranger, Kelly (Leslie Stefanson), then asks David if he is alone. David's affirmative response, which alludes to his isolation, prompts her to sit next to him. His sexual interest in Kelly is subsequently communicated nonverbally, as the camera gaze, mimicking the child's perspective, captures him removing his wedding ring. His attempted infidelity is inspired by his imminent separation from Audrey, which is later revealed to be driven by his admission that he keeps her and Joseph at an emotional distance. Rather than save the marriage, David has all but agreed to Audrey's wishes to take another security guard job in New York City and give up primary custody of his son. Although Joseph continues to admire David despite the impending separation, the film raises doubts about his parenting skills. For instance, when his injured son demands the school nurse call his father, David reports that Audrey "usually handles Joseph stuff" and asks if he has "to rub any smelly ointment" on him. David is on the verge of abandoning his family, then, because his conventional masculinity alienates him from his wife and kid. He thus begins to flirt with Kelly more aggressively in an awkward exchange always filmed from the child's point-of-view that never switches to the standard-shot/reverse-shot style of a classical conversation. In a matter of moments, David's introduction alerts viewers that the film contains non-classical and classical attributes as well as focuses on a man whose traditional masculinity is ruining his marriage and negatively influencing children.
Figure 3.1. A mirror-image shot of Dr. Mathison examining baby Elijah Price after his mother's emergency department store delivery in Unbreakable.
David's impropriety is further established when he offers Kelly a copy of a women's-interest magazine left onboard. Her unexpected response exposes his gender bias because she informs him she would prefer a discarded sports-themed magazine. Kelly clarifies her preference by noting that she is a sports agent traveling to meet a football prospect, making her the kind of woman who many men believe has encroached on their cultural authority by entering once all-male bastions, like the professional sports industries. David responds to her unexpected request in sexist fashion by joking that he wants to become a synchronized swimmer. He quickly retracts the misogynistic joke about the female-dominated sport by admitting that he is afraid of water, information that turns out to be crucial because it is later revealed to be his kryptonite. As the train passes through a tunnel, ensconcing David's placid face in shadows, he lies to her by claiming he dislikes football. As if it was not already clear that David's behavior is inappropriate, Kelly's embarrassing rejection of his advances confirms it. Like McClane, who at Die Hard's outset is revealed to be economically and socially inferior to his estranged wife, David's traditional masculine ways are outmoded and lead to embarrassing consequences when displayed explicitly. Such a reading of David's character was only amplified by offscreen events shortly before Unbreakable's release, as Willis's then-wife, Demi Moore, shockingly filed for divorce.
Figure 3.2. David Dunn attempts to give Kelly a woman's-interest magazine in Unbreakable.
If David's introduction is understood in relation to Willis's prototypical onscreen and changing offscreen personas, then it is clear why Unbreakable fools spectators into thinking that its classical ending will be David's discovery of a superhero identity that provides him with an acceptable way to reestablish his conventional masculinity, in secret. Such a reading is confirmed by Willis's performance because his relatively unexpressive acting style only seems to verify that he will again be playing a fallible action hero who uses his comedic wit, steady voice, and stoic facial expressions to combat and triumph over the now unfamiliar world that he inhabits. Like the viewer, though, it is David's proclivity to pigeonhole that most results in the surprise turn of events. In addition to misreading Elijah's and Kelly's true characters, David falsely accuses a man of South Asian descent, played by Shyamalan himself, of carrying drugs. As in The Sixth Sense, in which Shyamalan plays an archetypal Indian physician who misleads spectators by incorrectly diagnosing the situation, the director again uses his cameo not only to augment his burgeoning superstar image, but also to expose the audience's penchant for negative stereotyping. It is significant, then, that Elijah's disguise operates in relation to numerous markers of identity, extending beyond his race and the viewer's familiarity with Hollywood's interracial buddy film conventions, that are misconstrued as connoting weakness. His physical disability also helps him go undetected because it is misinterpreted as a flaw even though it is really the attribute that gives him the most strength by confirming he is David's foil. Indeed, the retroactive centrality of Elijah's disorder to his real identity counters cultural anxieties about disabled bodies being prisons for fully realized potential, as Vivian Sobchack theorizes in Carnal Thoughts, by transforming his supposedly debilitated body into one with "the transparent capacity for significant action and sensible meaning" (189).
Figure 3.3. Unbreakable director, M. Night Shyamalan, appears in a cameo as a stadium attendee whom David Dunn searches for drugs.
The importance of Elijah's disorder to his authentic identity is reiterated by a final twist that again links the two primary characters. Elijah acknowledges that he should have long realized he is a criminal mastermind because, like David, who consistently ignores his son's insights, he should have listened more closely to children, who called him Mr. Glass, the alter-ego he now presumably adopts. Importantly, it is Elijah's belief in the veracity of comic books that fuels his quest to find his superhero opposite who confirms his true identity, which justifies his permanent retreat out of adulthood. According to the film's logic, men like David and Elijah only fulfill their true promise if they are unencumbered by emasculating demands supposedly placed on contemporary adult males. Such an understanding, of course, comes at a cost for both characters, as Elijah resorts to mass murder to find David and reignites the violent fire extinguished by the hero's wife. Although the generically motivated concluding superimposed titles indicate that Elijah is sentenced to a psychiatric facility, he ultimately triumphs. Undoubtedly, Elijah's institutionalization marks him as a deviant pariah and renders him temporarily impotent; however, such places typically do not contain archvillains for long in the superhero genre. In fact, Shyamalan has consistently expressed his desire to make a sequel, featuring an actual showdown between David and an escaped Elijah, a rumor that Willis reconfirmed in a 2010 interview (Marshall). In the end, despite Elijah's capture, the villain wins by showing David that overcoming their malaise requires them to resurrect a stifled male essence free from feminizing constraints that is veiled by impaired alter-egos.
## Misreading The Usual Suspects: Feigning Fragility to Bolster Authority
Whereas Unbreakable depicts primary male characters as having to learn to deploy imperfect disguises to conceal their authentic identities, The Usual Suspects illustrates how masquerade is a powerful weapon for a man already certain of who he really is. The Usual Suspects was one of the first financially successful contemporary misdirection films, grossing over $23 million at the domestic box office on its $6 million budget (imdb.com). The film's complex narrative, which director Bryan Singer and screenwriter Christopher McQuarrie claim on their DVD commentary was partly inspired by John List's infamous 18-year disappearance after murdering his own family to shelter them from the shame of losing his job, was ultimately promoted as its primary draw; however, it initially scared off Hollywood. Singer and McQuarrie had to turn to Polygram Filmed Entertainment, a Dutch-owned company with ties to Universal Pictures, to finance and distribute the film theatrically (imdb.com). The gamble proved worthwhile because, in addition to its profitable theatrical run, Kevin Spacey won the Best Supporting Actor Oscar for his portrayal of Roger "Verbal" Kint and McQuarrie won the Best Original Screenplay Oscar. As with other contemporary misdirection films, a number of major players, including Columbia TriStar, MGM/UA, and Paramount subsequently distributed the film domestically and internationally in its various home video formats (imdb.com).
The film centers on U.S. Customs Agent David Kujan's (Chazz Palminteri) interrogation of Kint, who is about to post bail after being granted immunity by the District Attorney despite his role in the massacre associated with a purported $91 million cocaine deal at a San Pedro pier. Initially, the authorities and viewers suspect that Kint played a minor part in the crime because his riveting narration is revealed through a series of flashbacks from his perspective that accentuate his status as a crippled, small-time con artist. First, Kint's testimony to the D.A. reveals how a suspicious police lineup helped him team up with four seemingly more virile and accomplished crooks. Second, Kujan's interrogation of Kint in his friend's police station office depicts detailed flashbacks of the occurrences leading up to and during the events at the pier. In these flashbacks and the accompanying interrogation sequences, Kint presents himself as a weak lackey whose disabled body seems to inhibit his capacity to act authoritatively. Kujan and the viewer, therefore, are led to believe that Dean Keaton, the gang's most revered thug, is the film's primary causal agent. In standard heist film manner, Keaton is portrayed as the reluctant protagonist, who, despite his efforts to settle down by running a respectable business with his lawyer girlfriend, Edie Finnernan (Suzy Amis), possesses the traits to pull off the big job that will finally allow him to go legit. Classical standards are thus again deployed to trick viewers into thinking the alleged protagonist will inevitably prevail. Although Kint claims that Keaton is dead, Kujan refuses to believe it, leading audiences to presume that the con man is covering for his friend and co-conspirator. The interrogation, as a result, seems to be building to a climax in which Kujan finally gets Kint to admit that Keaton is really behind it all and escaped the law.
Kujan indeed hopes to use Kint's testimony to concoct his own furtive explanation to incriminate Keaton. As Kint effectively summarizes, Kujan's rigid theory demonstrates that "to a cop the explanation is always simple" because they just verify the suppositions they already believe. Kujan's construction of an alternative account, in other words, exemplifies how the authorities are depicted employing conspiratorial tactics to further their own agendas. To get Kint to participate in the interrogation, for instance, Kujan threatens to make up a story that, as part of his deal with the D.A., Kint ratted out Ruby Deemer, Kujan's most reliable, incarcerated informant. As Kujan also subsequently reports, during Keaton's stint with the NYPD, he was indicted seven times, including for multiple murder cases. In fact, Keaton was once involved in "New York's Finest Taxicab Service," a "ring of corrupt cops" handsomely compensated for chauffeuring smugglers. Keaton's insider-knowledge of this covert activity turns out to be valuable because he helps orchestrate a robbery of the Taxicab Service that also results in over 50 cops being busted. Ironically, the hit on the Taxicab Service only occurs because the police rely on unlawful means to rustle up the five criminals initially. As Kint claims, their "rights went right out the window" when the cops identified them as suspects. Kint's accusation that the authorities act unlawfully is verified by the police's interrogation of the five criminals when Keaton is punched in the face by a cop. Additionally, after Edie frees the five suspects, she reports that they were never officially charged.
Kujan similarly relies on underhanded tactics to pin the crime on his man. To get his information, Kujan challenges Kint to "convince" him that Keaton is dead by telling him "every last detail." In turn, Kint presents a byzantine account of events, eventually revealing that Keyser Söze, a legendary Turkish crime lord, is really the puppet master. Kint's recounting of the criminal scheme ultimately is believable, then, because it conspiratorially adheres to classical narrative conventions of causality and agency by attributing everything to the machinations of a powerful individual: Söze. Kujan, of course, is convinced that Keaton, and not the mythical Söze, is the mastermind. As a result of Kint's recollection, he concludes that Keaton is actually Söze. Consequently, when Kujan finally explains his theory to Kint about what really happened, the music on the soundtrack swells to a crescendo and the film frequently cuts to flashbacks that depict images from Kint's earlier testimony now taken out of their previous context. Kujan's acceptance of the totalizing plot of Keaton as the archcriminal seems logical, as events that ostensibly were initially unimportant to comprehending his "true" character now make Keaton look ruthless. In short, this scene appears to be the classical resolution in which the detective identifies the individual, whom he wanted to nail all along, as the real primary causal agent.
The film, though, does not actually uphold this conventional resolution by showing Keaton successfully fleeing with the money and Edie. Instead, its changeover provokes a new way to understand narrative causality. It stunningly shows that Kint is a master storyteller who knits—hence the anagrammed surname—the fictitious tale to escape the law and further his own legend as the mythic Söze. Upon Kint's release, Kujan and the audience simultaneously realize that the confession was fabricated because it is shockingly revealed that Kint both faked his cerebral palsy and used the contents of the interrogation room to create his contrived testimony. It is significant in this regard that Kint was briefly shown earlier scanning his surroundings when he first arrived in the police station office. The inclusion of such a seemingly narratively insignificant scene is crucial for two primary reasons. First, in classical fashion, it can be reinterpreted as anything but tangential because it becomes the most important scene for retrospectively reassessing the meaning of narrative information. Second, it helps to inoculate a film that is also packaged as a whodunit from complaints that it does not adhere to the tenet of "fair play," even though it is highly unlikely that viewers would be able to solve the mystery.
Figure 3.4. Roger "Verbal" Kint reads the bulletin board that becomes the source for his narrative immediately before David Kujan's interrogation begins in The Usual Suspects.
The changeover is depicted spectacularly, as the film cuts back and forth between Kujan's dumbfounded gaze, the objects that he observes in the office, and flashbacks of previous scenes from Kujan's explanation, which portrayed Keaton as the archcriminal, Söze, into whom Kint now transforms. Aural evidence also helps viewers make sense of what has really occurred, as earlier lines of dialogue are now associated with the objects captured by the camera's gaze. As Kujan stares at a bulletin board frame, for example, indicating its manufacture in Skokie, Illinois by the Quartet Corporation, Kint's offhanded remark that he once sang in a "barbershop quartet in Skokie, Illinois" replays. The new master thread of Kint as Söze reverses what both Kujan and spectators thought they knew about who really possessed narrative agency. Of course, once Kint's story is exposed as a fabrication, it becomes difficult to determine what, if anything, from his testimony is factual. However, although the revelation shows Kint's account is untrue, it leaves no question that he has been self-interestedly propelling narrative events all along.
Thus, it is Kint who is clearly revealed to be the film's master manipulator. Kint's brilliantly constructed narratives not only trick Kujan into confirming that Keaton is the perpetrator, they also bolster his own legend as a brazen criminal who is able to pull off seemingly impossible feats, perhaps suggesting he is even Söze himself. If he is also really Söze, then the methods he describes the crime lord deploying throughout also serve to amplify his reputation, as they are what enable him to convince the four other unsuspecting criminals to further his objectives. Specifically, Kobayashi (Pete Posthelwaite), Söze's purported attorney, gets the other felons to participate in the heist by presenting them with a mountain of surveillance evidence, revealing that Söze has the capability both to kill their loved ones and expose all of their previous wrongdoings. In fact, each of the criminals discovers that they once unknowingly stole from Söze, which is why they are being forced into his service at this time. Kint/Söze also expertly creates a number of cover stories to conceal his agenda and prowess. First, his convincing performance as a cripple lowers everyone's suspicions that he could be an archcriminal. Second, to distract his partners in crime and the authorities from his actual mission, Söze leads everyone to believe that the large sum of money at the pier is intended for a cocaine deal. As the film ultimately shows, though, the real aim of the mission is to kill Arturro Marquez (Castulo Guerra), who recently informed the authorities that he could identify Söze. Consequently, he is able to dupe the four other criminals into massacring practically everyone on the pier and the boat, allowing him to assassinate Marquez and steal $91 million that a group of rival Hungarians had brought to purchase the informant from a gang of Argentineans.
The self-serving reasons for Kint's storytelling agenda become most apparent retrospectively in relation to his earlier retelling of Söze's rise to power. Before beginning the tale about Söze, Kint strategically authenticates it by informing Kujan that "One story the guys told me, the story I believe, was from his days in Turkey." Kint's disclaimer triggers the Söze flashback, which, in contrast to the other recollections that are clearly framed as such but not delineated stylistically, is shot in a dreamlike fashion, obscuring the image. These quintessentially classical techniques alert spectators that the scene should be differentiated from the film's other flashbacks, rendering it potentially more believable in retrospect. After Kint finishes the story, he verifies it further by noting that few believe that Söze really exists. Kujan then asks, "Do you believe in him Verbal?," indicating that the fable has made him and the spectator let down their interrogative guards. Kint seizes the opportunity to persuade him even further by exclaiming, "Keaton always said I don't believe in God, but I'm afraid of him. Well, I believe in God and the only thing that scares me is Keyser Söze." The kind of believable details that Kint provides, therefore, encourages viewers to rely on the fable to reconstruct the narrative according to the revised logic of Söze as the primary causal agent.
Figure 3.5. The stylistically delineated flashback in The Usual Suspects that portrays Roger "Verbal" Kint's description of Keyser Söze's rise to power.
Kint's tale depicts Söze as a callous villain, who is especially fearsome because he commits horrific acts unfathomable to most other criminals. Specifically, during a raid on his home in which his wife is raped in front of his children, he mercilessly kills all but one member of a rival Hungarian gang as well as his own wife and kids. Söze's decisions to kill his family and spare one adversary are partly motivated by the hope that word about his exploits will spread. Accordingly, the spectator's revised understanding of Kint's true identity is inextricably linked to this legend. In a film virtually devoid of female characters and loaded with homoeroticism, doubts have been raised by Kint's feminized cover. In contrast to Unbreakable, by revealing that Kint's disorder was faked, The Usual Suspects does not depict disability as ultimately validating and enhancing masculine potential. Questions about Kint's real persona thus do not just evaporate along with his bogus cerebral palsy. To wit, after Kint is picked up outside the police station by his foreign and dandified associate, the man known as Kobayashi in his testimony, Kint smokes a cigarette effeminately. The film, then, does expose a few facts about Kint's authentic identity after revealing his lies, which could lead to a further interrogation of his manhood. The Söze legend, though, retrospectively secures his status as a former heterosexual family man who confects a pathetic façade to cover his ferocious male essence.
As Kint claims, it is Söze's ability to commit familicide that most defines his prowess and launches him to the top of the criminal underworld. Familicide, as Elizabeth Barnes argues, uses murder as "an expression of love as well as hatred" to enable "a man to (re)gain a sense of self-reliance (by eliminating his family) without abdicating his position as a devoted family man" (54). The horrific violence of familicide disturbingly allows offenders to free themselves from the perceived shackles of domestic obligations at the same time that it sustains their belief that they are fulfilling their familial duties by protecting their vulnerable kin from worse fates. This is exactly what happens for Söze, according to Kint, because his savagery saves his family from the consequences of his wife's rape and permits him to focus myopically on his criminal empire. Such tendencies, for Barnes, make familicide a distinctly male and characteristically American transgression. As her analysis of the prevalence of the crime and its literary representations in the immediate post-revolutionary United States demonstrates, during "a particular crisis in the history of U.S. masculinity, familicide perpetrators sought to exemplify manhood by asserting absolute sovereignty over their wives and children" (47). At a moment when American men were bent on distinguishing themselves from the British, the epidemic of patriarchs killing their own families to protect them from the embarrassment of having failed in a radically new economic context is especially telling. Söze's deeds are anything but foreign, then, as the actions of notorious American murderers, like List, reveal that familicide remains an ideal escape for men crushed by the pressure to provide for the families they so desperately want to protect.
The twisted fantasy of conflicted masculinity inherent in familicide relates to how manhood is represented in Unbreakable and The Usual Suspects. Both films demonstrate that men need to flee their emasculating predicaments by relying on elaborate disguises to hide a violent male core that enables them to assert control over their families. This veneer is necessary, the films imply, at a time when explicit displays of traditional masculinity are received with growing incredulity. In the end, these two films are troubling fantasies of male masquerade in which men secretly maintain their patriarchal supremacy by flaunting their purported fragilities. Although their duplicitous narrative structures are well-suited to reveal that gender is socially constructed, these films instead portray masculine performance as a way to conceal the "truth." They effectively counter pervasive paranoia about the loss of a male essence by showing how select men are capable of strategically protecting their power as well as reestablishing their authority in the family and beyond. In conspiratorial and classical fashion, the two films present narratives that privilege causality and agency to make order out of chaos. These films appeal to many spectators, therefore, by transforming everyday uncertainty into familiar causal narratives that support dominant ideologies, particularly the staunch belief that hegemonic masculinity endures and reigns supreme. Such a thematic preoccupation begins to suggest why they are attractive to viewers increasingly concerned about rediscovering who they "actually" are as well as reclaiming their "real" place in the home and in broader U.S. culture.
4
## Start Making Sense
Narrative Complexity, DVD, and Online Fandom
MANY YOUNG MEN HAVE EXPERIENCED hardships in recent years as they have struggled to overcome their perceived emasculation and maintain their dominance in the United States. A number of contemporary Hollywood misdirection films address these concerns by teaching young, white, heterosexual, American men how to adapt to a culture that has become increasingly intolerant of traditional ways that they have been encouraged to perform their gender. Rather than suggest that minority groups should be granted equality, however, the narrative solutions offered by these films typically give its target audience blueprints for maintaining authority in economic, political, and social spheres. Yet, because many misdirection films are difficult to interpret decisively, only those viewers who scrutinize the films most carefully are able to gain the privileged information that they believe their almost always male creators have hidden beneath the surface. In the end, although many misdirection films appear to revel in uncertainty, the work of devoted fans suggests that they can be made coherent by a high degree of interpretive labor, at least according to those who claim to have cracked the code definitively.
The presence of a way to make sense of seemingly ambiguous information renders most misdirection films hyper-classical because they can be reinterpreted according to a new narrative logic that is linked to the actions of a clearly defined causal agent who is almost inevitably male. The existence of this alternative explanation provides consolation to spectators who have grown accustomed to the narrative structure and relatively conservative ideological messages contained in an overwhelming majority of Hollywood films. Fans of these films thus often derive pleasure from a discovery of what they believe to be the filmmaker's "true" intentions.
The practice of turning films, which at first both seem to contain an unconventional narrative logic and depict a culture that has become increasingly unfamiliar, into ones that can be understood traditionally and coincide more closely with dominant ideologies, is one of the distinct joys that many viewers derive from watching misdirection films. Crucially, the spectator's ability to enjoy misdirection films in this manner has been greatly enhanced by the advent of new home-video and media communication technologies. In Beyond the Multiplex, her study of the impact that such developments have had on the production and reception of Hollywood films, Barbara Klinger contends that "familiarity" is one of the primary reasons why spectators revisit their favorite movies because it is "at once a central arena of satisfaction and the root of other functions and pleasures" of repeat viewing (152). Consequently, as Klinger argues, some recent Hollywood films seem to be designed specifically for consumption in the aftermarket since the familiarity offered by repetition "enables viewers to experience both comfort and mastery" not possible in one-time theatrical screenings (154). Many misdirection films, of course, are uniquely positioned to appeal to repeat viewers. It is only on subsequent viewings that spectators, who are initially dissatisfied with both their seemingly unconventional narratives and often confusing ideological messages, can potentially reread misdirection films coherently. In fact, once viewers have reinterpreted their narratives retrospectively, they are freer to focus on a number of different endeavors, such as gaining an even better understanding of narrative meaning, searching for evidence that was initially missed, ensuring that the trick was pulled fairly, more thoroughly comprehending the film's messages, and so on. Put simply, films that demand these kinds of viewing practices are tailor-made for audiences who use new media and communication technologies to gain a deeper appreciation of their complexities.
The misdirection film, then, encourages interpretation and viewing practices that differ from those typically enacted by the classical spectator. The ways that these films are usually comprehended challenge David Bordwell's conception of the classical film as a hermetically sealed entity that adheres to a recognizable narrative template and corresponding formal practices that make it easily understandable in one viewing. According to Bordwell, the classical narrative functions to retard the occurrence of the predictable conclusion because spectators derive pleasure from the way in which the protagonist overcomes a series of unknown variables, in the form of obstacles that stand in the way of his or her clear-cut objectives. Under this model, it is difficult to understand why spectators engage in multiple viewings of Hollywood films because once the narrative is consumed, it no longer effectively heightens the viewer's anticipation. Yet, misdirection films often cannot be interpreted classically in a single viewing and spectators generally do not enjoy them fully until they watch them repeatedly in post-theatrical settings.
I grant that it was difficult for Bordwell to contend with this development because in the years leading up to Narration in the Fiction Film's publication in 1985, earnings from the box office still outpaced those garnered from home-video. In fact, although viewers typically would venture to the cinema to see their favorite Hollywood films on multiple occasions and would await their re-release in theaters or on television, repeat viewing was a less common cultural practice at the time and, especially, in the decades before then. However, now that Hollywood's profits from home-video far exceed the box-office take, it is no longer tenable to equate Hollywood film primarily with the theatrical experience. Instead, as Derek Kompare theorizes, under these radically new conditions of commercial film exhibition, it is misguided to think of playback devices, such as the VCR and the DVD player, as "mere enhancements of media" because these new technologies, which are "designed on the premise of mediated repetition," are "reconceptions, profoundly altering our relationship with dominant media institutions, and with media culture in general" (Rerun 199). For instance, the ability to own and replay Hollywood films as well as to manipulate the narrative trajectory and the image were virtually inconceivable prior to the home-video age. These changes, though, have not only had a significant impact on the ways that spectators interact with commercially produced films, as Hollywood has also had to adapt its production strategies to respond in kind.
In this chapter, I examine how the highly complex and seemingly ambiguous narratives of two contemporary misdirection films—Mulholland Dr. and Memento—epitomize strategies that Hollywood has devised in response to these changing industrial and technological contexts. These two films are especially germane because their convoluted narratives and seemingly eternal ambiguities make them among the most byzantine Hollywood films ever released. This complexity has inspired an inordinate amount of interpretive work devoted to figuring out their "true" meanings in online communities. Such developments raise a series of questions about how the industry has encouraged spectators to engage with new media technologies that have significantly altered the manner in which Hollywood films are now commonly viewed, interpreted, and discussed. How do these two films demonstrate the ways in which the industry packages products for a niche audience that enjoys games of discovery and decipherment made possible by technologies, like the DVD player and the Internet? What tactics have producers recently deployed to entice a particular group of spectators to interact with films that require a comparatively high degree of interpretive labor? Who makes up this audience and why is it so attractive to Hollywood? What do these spectators hope to gain by watching misdirection films repeatedly on DVD and discussing them fervently online? What might discursive evidence on websites associated with these films tell us about how fans respond to the industry's practices?
## Hollywood in the Digital Era: Media Convergence and New Narrative Strategies
The recent proliferation of misdirection films exemplifies Hollywood's new economic logic amid changing industrial contexts because these films have been greenlit by media conglomerates even though their complex and often ambiguous narratives typically disappoint at the box office. Obviously, there have been enormously high-grossing misdirection films, like Inception (2010) and The Sixth Sense (1999), that are exceptions, as they both netted well over $200 million during their runs in domestic theaters; however, there are only a handful of other contemporary Hollywood misdirection films, such as A Beautiful Mind (1999), Planet of the Apes (2001), Shutter Island (2010), The Village (2004), and Vanilla Sky (2001), that have garnered as much as $100 million at the domestic box office. Yet, each of the misdirection films in the $100 million to $200 million club also had a substantial production budget that ranged from $60 million to $100 million, suggesting that their high revenues stemmed from expensive overhead costs that cut into profit margins (imdb.com). Although most contemporary Hollywood misdirection films have made at best only a modest splash at the box office, the theatrical performances of some smaller budgeted films, such as The Usual Suspects (1995) and Memento, are also of note because they greatly exceeded economic expectations. Importantly, the success of these films was attributable primarily to strong word of mouth and positive reviews because media conglomerates were initially hesitant to back them out of fear that their complex narratives would alienate audiences. Memento, for instance, earned $25 million at the domestic box office on a shoestring $5 million budget even though it was produced and distributed in the United States by Newmarket Films, an independent company that does not have the marketing resources to compete with the majors and their often astronomical advertising budgets (imdb.com).
In spite of their limited success at the box office, misdirection films typically perform well on DVD, which explains why Sony jumped at the chance to serve as Memento's distributor when it was released on home-video. Hence a reason why I classify misdirection films, such as Memento and Mulholland Dr., as Hollywood films even though they were produced independent of the U.S. commercial film industry. Although the ongoing nature of home-video sales as well as the studios' desire to conceal ancillary profits from parties seeking royalties and residuals make it difficult to locate exact figures on Memento's DVD revenues, its standing as of August 2015 as the 44th greatest film of all-time on the Internet Movie Database's Top 250 list suggests why Sony seized the opportunity to distribute the film on home-video. Its belated fan following is not out of the ordinary because a number of misdirection films that made little splash at the domestic box office are now consistently ranked among the greatest films ever made on lists created for popular cinema-related websites and entertainment-themed magazines. Drastic reassessments such as these indicate that many misdirection films do not find an audience, and do not become real moneymakers, until after they are released in post-theatrical markets.
Although earnings at the domestic box office continue to be the primary obsession of the U.S. entertainment-related news media, many scholars have demonstrated that theatrical receipts have long constituted only a fraction of a film's overall revenue. As Douglas Gomery asserts, in "little more than a decade after the 1976 introduction of the Betamax and the VHS alternative, rentals and sales of movies on tape surpassed the theatrical box-office take in the United States" (276). Even though the DVD player and the VCR were a boon for Hollywood, the industry actually struggled for years to reach a nontheatrical audience. As Peter Kramer summarizes, Hollywood has always striven to extend the life of its products beyond the silver screen: the "strong parallels between the integration of movies and domestic media technologies" that exist today were present at "the very beginnings of moving pictures in the late-nineteenth century" (13). In spite of these aspirations, industry executives have always been ambivalent about the development of moving-image exhibition technologies that supposedly threaten Hollywood's theatrical market. Fears relating to the loss of market share and the cannibalization of theatrical revenues have been present in the discourses surrounding every new platform for the nontheatrical exhibition of commercially produced films, including network television, cable television, satellite television, the Internet and, most significantly, home-video. The prevalence of such industry-induced hysteria is dubious, however, because it obscures the symbiotic relationships that have developed between the film industry and purportedly rival media industries since the start of broadcasting.
The well-publicized success of the DVD player, in particular, reveals that Hollywood is not primarily in the business of producing feature-length films for a theatrical audience. As Janet Wasko claims, DVD players became affordable to the middle class in the late 1990s, and by 2002 "the technology represented the fastest selling consumer electronic product ever" at that time, "having reached sales of 30 million units within five years" (133). Unlike VHS tapes, though, which initially flourished as a rental product and subsequently only experienced limited success in sell-through markets because Hollywood priced titles too high for most consumers to purchase thanks to concerns such as piracy, DVDs were immediately packaged for direct sale to consumers. The cheap cost to produce DVDs in large quantities, the difficulty to reproduce them illegally on DVD when the technology first appeared, their smaller packaging, and so on, instantly made them much more attractive to a sell-through than a rental market.
The advantages that DVDs provided to both consumers and producers all but eradicated the VCR and quickly made the DVD player the dominant home-video technology in the United States. Profits from DVDs rose steadily in the years after the technology first became widely available to the public in 1997. However, revenues from DVDs began their steady decline in 2007, slipping for the first time by "dropping three percent from 2006," yet still garnering a staggering "$23.4 billion" in "total sales and rentals," which "dwarfs Hollywood's $9.6 billion box-office total" for the same year (Snider, Life 1). Total sales of DVDs may have dipped slightly then, but the contemporaneous success of online DVD rental outlets, like Netflix.com, helped to stabilize the state of the DVD rental market, albeit precariously. The arrival of a number of new home viewing technologies such as Blu-ray (high definition DVD), video on-demand, DVR, and, particularly, streaming content online have now largely replaced the DVD, suggesting a major reason why the misdirection film genre went into a state of decline in the 2010s. Yet, the presence of all of these new technologies that purportedly threaten the theatrical feature-length film have only made the nontheatrical audience more important to Hollywood's bottomline.
Regardless of the continued importance of the domestic box-office for quickly paying off high interest loans and to a film's success in the aftermarket, the industry has adopted strategies to package the encounter with its films in each of its forms as distinct; the high sales revenue generated from DVDs in the late 1990s and early 2000s demonstrated that Hollywood also had success positioning post-theatrical interactions with its products as special. The marketing of home-video as a distinct form of entertainment is possible, as Gregory Waller hypothesizes, because "there is no 'film' apart from exhibition; we seek out, pay for, take pleasure or displeasure in the experience of a film—even if the film is shown in a 'home theater' rather than a public venue" (emphasis in original, 3). The industry has wisely understood that post-theatrical viewings, like the theatrical experience, are a unique recreational activity with a particular set of associated pleasures. Such tactics have become vital because Hollywood is working harder than ever to milk additional profits out of existing products to recover its investments. Klinger refers to this practice as "repurposing," which she defines as "the media industry's attempt to gain as much revenue as possible from a given property" by reselling it in ancillary markets (7).
The transfer of a litany of new, old, and otherwise unreleased Hollywood films to DVD was a temporary goldmine for an industry hoping to repurpose its products for nontheatrical audiences. Although clandestine accounting procedures made it difficult to obtain the precise amount that the media conglomerates pocketed from either the sale of a single ticket or post-theatrical products at the height of the DVD's popularity, Wasko reports that in 2002 a "studio head" claimed that they made an average of "$15 in profit on the sale of one DVD" (133). There was clearly great incentive for media conglomerates to encourage audiences to purchase films directly on DVD at the time because the money earned from those transactions greatly exceeded the take from individual theatrical admissions. Under these circumstances, only a niche audience had to be willing to buy a film on DVD to turn a profit from those that failed at the box office.
## Games of Discovery and Decipherment: Interpretive Practices in the Digital Age
DVD helped the film industry to expand a customer base that derives pleasure from viewing and, even more importantly, re-viewing Hollywood films in the home. Prior to the VCR's widespread adoption in the 1980s, film collecting was the province of a select few, who had both the economic resources and technological expertise necessary to acquire, maintain, and operate a library of Hollywood films on celluloid. In recent years, however, the film-collecting customer base created by the VCR was greatly expanded by the immediate status of the DVD as a sell-through product with the technological capacity to replicate many of the pleasures of the theatrical experience. Although the large and diverse market for DVDs has democratized personal ownership of Hollywood films, the industry initially targeted audiences historically proven to be the most profitable with products customized to their tastes when the technology was first available. Klinger demonstrates, for instance, that when the DVD was released in the late 1990s, demographic research revealed that middle- and upper-class white men were the earliest adopters of the new technology. Unsurprisingly, Hollywood genres typically coded as masculine were consistently among the best-selling DVDs at the time. These sales trends influenced Hollywood's production strategies for years, as "younger, well-to-do white men" were the "most important purchasers of DVD players," meaning this group had "great sway over which films are approved for production" during the period (Klinger 64).
The success of the DVD sell-through market encouraged producers to create films that target niche audiences like never before. Prior to the home-video age, Hollywood primarily made movies that appealed to broad audiences partly because the theater was practically the only venue for the consumption of its products. Hollywood's best interests are still served by making family friendly and politically safe films; however, the emergence of the sell-through, post-theatrical market decreased the financial risks associated with producing films that have greater potential of alienating certain audiences. The economic importance of this market, especially at the height of the DVD-era, led critics like Aaron Barlow to speculate that "the change in income sources is having a tremendous impact on how films are made" even though "few filmmakers are willing to admit it" (9). DVD technology enabled producers to repurpose their theatrical products in obvious and cost-effective ways by adding attractive extras, such as director commentaries, multilanguage soundtracks, deleted scenes, and behind the scenes featurettes that differentiate it from the theatrical version with minimal additional necessary investment.
The misdirection film is also well-positioned to appeal to viewers who routinely use DVD players and the Internet to maximize their enjoyment by engaging in practices, such as aesthetic appreciation and close narrative decipherment. Like Klinger, Henry Jenkins argues that films, including those that comprise The Matrix trilogy (1999, 2003, 2003), are now attractive to Hollywood producers because their transmedia narratives are "encyclopedic, containing a rich array of information that can be drilled, practiced, and mastered by devoted consumers" (Convergence 97). Although The Matrix films do not necessarily contain a moment that inspires viewers to reinterpret narrative information retrospectively, some audiences have been drawn to them partly because they offer pleasures that are similar to misdirection films. In fact, many of the hidden clues buried within the The Matrix trilogy are neither possible to unearth nor understand without the use of new media and communication technologies. As Jenkins writes, although some of the trilogy's secrets, "pop off the screen on first viewing," others "become clear only after you talk about the film with friends," and "still others... may require you to move through the film frame by frame on your DVD player" (Convergence 99). For fans willing to play the game, The Matrix films contain an " 'encyclopedic capacity,' " a term Jenkins borrows from Janet Murray, to describe the "new narrative forms" enabled by technologies that allow audiences to "seek information beyond the limits of the individual story" (Convergence 115–16).
For some scholars, the participatory activity inspired by many recent Hollywood films, like misdirection films, is evidence that spectators have become liberated from the authority of producers. Graeme Harper, for instance, notes that the arrival of the videotape meant that "a film need not be viewed at one time," and that it was thus "no longer even a teleological, or goal-directed art-form" (93–94). Similarly, Barlow contends that the popularity of "hang-out movies," watched repeatedly by groups in the home, goes "hand-in-hand with the viewing habits developed in the decades after the advent of the VCR, with the most important being that complete control the viewer assumes over the movie" (18).
In contrast, Jenkins counters that these freedoms have not necessarily liberated audiences from the authority of producers. Jenkins claims, for instance, that Lana and Lilly Wachowski, the directors of The Matrix films, "have positioned themselves as oracles" who possess, but are intentionally hesitant to reveal, the answers to the many enigmas contained in the trilogy (Convergence 99). The Matrix films demonstrate, for Jenkins, how filmmakers have created transmedia products with such detail that spectators, particularly younger viewers with substantial discretionary time and income, who are granted little expert authority in other realms, are encouraged to "become informational hunters and gatherers, taking pleasure in tracking down character backgrounds and plot points and making connections between different texts within the same franchise" (Convergence 129). The fictional world of The Matrix is so vast, complex, and riddled with mystery that one person alone cannot possibly tackle it. Yet, a core group of devoted fans eager to pool their resources has a much greater chance of being able to solve the Wachowskis' dense puzzle. In sum, The Matrix films are tailor-made for the digital age because they capitalize on "collective intelligence," a term that Jenkins adopts from Pierre Levy to describe "the ability of virtual communities to leverage the knowledge and expertise of their members, often through large-scale collaboration and deliberation" (Convergence 281).
Klinger similarly contends that spectators are still expected to follow the filmmakers' orders, even in nontheatrical contexts, if they hope to receive maximum pleasure from films that require repeated viewings and encourage virtual communities to examine their complexities. The spectator who unearths the secrets of these films, she argues, seems to become their omnipotent makers, "something of an authority—an intrepid explorer who has discovered a terra incognita and mapped every path" (Klinger 161). Like fans of The Matrix trilogy, however, these viewers do not necessarily reclaim power from producers. As Klinger posits, these films require viewer mastery that, in the discourses surrounding home-video technologies, "has often been inscribed within traditional associations of men and machines, white masculinity and technology" (250). Fans of misdirection films decode their secrets partly because they believe that their skilled detective work gives them privileged knowledge of the authorial intentions of their almost always male filmmakers. Rather than try to encourage a multiplicity of readings, misdirection films often demonstrate that producers hope to control and profit from the consumption of films nontheatrically by directing how they are interpreted in that context. Consequently, it is not surprising that misdirection films have been most appealing to an audience largely comprised of men, who both fashion themselves as discerning consumers of Hollywood films and use new technologies to gain a better understanding of their favorite titles. The online reception of misdirection films ultimately reveals that they have successfully attracted a modest, yet lucrative, target market that derives pleasure from obtaining insider knowledge. Specifically, the discovery of the films' "true" meanings and the transformation of seemingly unfavorable messages into ones that often support dominant ideologies provide these fans with a sense of superiority from films that, as box-office results suggest, some other viewers may find unappealing.
## Finding the Way on Mulholland Dr.: Willing Narrative Coherence and Closure
Although there is a strong correlation between the advent of new technologies and changing narrative strategies in Hollywood, these kinds of complex narratives are not solely a product of the digital age. There is perhaps no better example of this type of narrative experimentation in mainstream U.S. moving-image media prior to both the widespread adoption of the Internet and the advent of the DVD player than Twin Peaks (1990–1991), a television show co-created by Mark Frost and renowned Hollywood filmmaker, David Lynch. First broadcast by ABC in 1990, the show survived for only two seasons despite critical acclaim and a devoted fan following. Twin Peaks contained a remarkably complex narrative for network television, requiring multiple viewings and the shared expertise of a collective intelligence to try to decipher the show's core mystery of "Who Killed Laura Palmer?". As Jenkins observes in an essay about the discussions that the series generated on UseNet, a text-based computer networking platform that preceded the development of the contemporary World Wide Web, "Lynch's cryptic and idiosyncratic series seemed to invite the close scrutiny and intense speculation enabled by fans' access" to the VCR and the Internet ("Do" 54).
Twin Peaks indeed aired at a time in which both the VCR was the only playback device available to spectators who wanted to re-watch television programs repeatedly on their own schedules and in which the Internet was only in a rudimentary developmental stage. "It has been commonplace to remark upon the meteoric rise of the Web," Jeremy Butler writes, "but in 1991 it was essentially a text medium that seemed no more remarkable than other information distribution systems" (41). As he explains, the Internet did not really take off in the United States until the mid-1990s because before that time no browser was available that "could effectively handle images and position them on the screen" (42). It was not until the release of Netscape's Navigator in late 1994 and the subsequent development and dominance of Microsoft's Internet Explorer that the web became the medium that transformed the Internet into the enormously popular communication technology that it has since become. Although it is possible to speculate how the original Twin Peaks would fare with contemporary audiences, it is safe to assume that it failed, at least in part, because the technologies necessary to decode its narrative ambiguities were both relatively unsophisticated and available to only a select population.
The many viewers who did not or were unable to decipher the show's secrets thus likely echoed the primary complaints of critics, who, as Jenkins documents, grew impatient with Twin Peaks for dragging out the central enigma for too long. Even though a majority of middle- and upper-class Americans owned at least one VCR with time-shifting capabilities in the early 1990s, comparatively few U.S. households were connected to the Internet at the time. Fans attempting to crack the code, therefore, not only had to remember to record the show from week-to-week, but also did not have the ability to navigate or manipulate the show with the kind of precision that has been enabled by DVD technology. Additionally, although access to the Internet and the production of web content have been democratized greatly in recent years by the development of Web 2.0 as well as the sheer ubiquity and lower cost of personal computers with networking capability, it was initially available only to those technologically savvy few who possessed the substantial economic resources necessary to gain access. More specifically, the "digital divide," a term that entered the popular lexicon in the mid-1990s to describe the gap between those who had access to the Internet and those who did not, was tangibly raced, gendered, classed, and aged throughout the 1990s because young, white, middle- and upper-class men were most likely to possess the financial wealth and technological knowledge required to participate in the new communication technology. The show's deliberately convoluted narrative alienated many viewers and allowed only a select demographic to make the discoveries necessary to enjoy Twin Peaks in the manner that its creators seem to have intended.
These obstacles did not deter a core group, comprised of mostly men, from attempting to solve the dizzying puzzle at the center of the show. As Jenkins discusses, it is surprising that the show attracted a male audience because it was packaged as a nighttime soap opera. He also theorizes, however, that the show's unexpected success with men was attributable to the ways it encouraged interpretive practices that are typically gendered male. In comparison to groups of female fans of other cult television shows that Jenkins analyzes in Textual Poachers, who primarily "focus their interest on the elaboration of paradigmatic relationships, reading plot actions as shedding light on character psychology and motivations, the largely male fans in the Twin Peaks computer group focus on moments of character interaction as clues for resolving syntagmatic questions" (109). In particular, the male participants on the show's UseNet site rationalized their intense scrutiny of aspects of the show often considered the terrain of female fans, such as charting and imagining the details of romantic relationships, because the information was deemed essential for solving the puzzle. The show's dual generic status as both a soap opera and a mystery, as Jenkins claims, ultimately "provided the alt.tv.twinpeaks participants a space to examine the confusions of human interactions by translating them into technical problems requiring decoding" ("Do" 60).
The site's most active posters typically also relied extensively on their knowledge of Lynch's authorial tendencies both to support their devotion to the show and to validate their interpretations of its ambiguities. Although Frost shared the credit as creator, most fans attributed the show's complex narrative machinations solely to Lynch. The discourses of authorship effectively positioned Lynch as the show's primary creative force because of the authorial reputation that he already established, thanks to the critical success of films, such as Eraserhead (1977), The Elephant Man (1980), and Blue Velvet (1986). As Jenkins notes, fans on the Internet leveraged Lynch's auteur status to position him as a "master programmer" and "trickster author" who had engineered a sophisticated game of cat-and-mouse with loyal viewers ("Do" 61). Comments consistently made by the site's participants indeed revealed that one of the show's main sources of pleasure came from matching wits with the director. In particular, to display both their cultural capital and satisfaction with the show, a number of contributors routinely identified its many allusions to recognizable texts and Lynch's other works as well as offered their solutions to the complex narrative puzzle that they credited exclusively to him.
Although these core fans were not enough to sustain the show beyond a couple of seasons, in 1999, perhaps encouraged by substantial improvements in both Internet and home-video technologies, Lynch attempted to make another, similar foray into network television. He filmed a pilot for ABC for a show that was to be called Mulholland Dr., which, like Twin Peaks, was supposed to be propelled from week-to-week by an ongoing central mystery that remained perpetually unsolved. As Lynch explained to film critic, David Sterritt, the show never aired because network executives were reportedly unhappy with early versions of the pilot (15). Moreover, as they eventually did with Twin Peaks, which, in a March 1997 interview with Rolling Stone Lynch claimed had failed because he was forced to present revelatory information prematurely, network executives balked at the idea that the show would be driven by a never-ending mystery (Gilmore 41). As a result, the director subsequently accepted international funding to turn the pilot into a feature-length film that was picked up by Universal for theatrical distribution. Even though Mulholland Dr. eventually was developed for release in U.S. theaters, then, it was intended to be watched in nontheatrical venues from its inception. Lynch finally executed such a distribution strategy with his follow-up to Mulholland Dr., Inland Empire (2006), which contains many of the same narrative, formal, and thematic elements as its predecessor and was strategically shot using digital video so that it would be easy to disseminate widely on DVD after its extremely limited theatrical release.
In comparison to the success that it has experienced on DVD, the relatively poor performance of Mulholland Dr. during its run in theaters indicates that Lynch was correct to suspect that the film would fare better with a nontheatrical audience. However, its box-office disappointment is surprising because the discourses surrounding its domestic theatrical release were largely positive, suggesting that the film had a good chance to be commercially successful with some audiences. At the Cannes film festival in the summer of 2001, the film earned Lynch a share of the award for Best Director (the Coen brothers also won for The Man Who Wasn't There). Additionally, Mulholland Dr. was lauded by a number of America's most influential film critics in spite of its confusing narrative. Andrew Sarris, for instance, asserted that it is "one of the very few movies in which the pieces not only add up to much more than the whole, but supersede it with a series of (for the most part) fascinating fragments" (23). Similarly, Roger Ebert claimed to be "willing to forgive [Lynch] for Wild at Heart [1990] and even Lost Highway [1997]" because at last his "experiment doesn't shatter the test tubes... the less it makes sense, the more we can't stop watching it" (35). In particular, Ebert declared that Mulholland Dr. worked even though it contained virtually the same seemingly incoherent narrative structure as Lost Highway, which he bashed because he thought that he had been "jerked around" by a director who "knows how to put effective images on the screen, and how to use a soundtrack to create mood," but who does not seem to have "an idea, purpose, an overview, beyond the arbitrary manipulation of plot elements" (35).
Lynch's own comments prior to the film's theatrical release also advertised Mulholland Dr. as being more accessible than many of his other films. In fact, the director, who is notoriously unwilling to discuss the meaning of his films and maintains that some do not even contain coherent narratives, uncharacteristically claimed that there is a "correct" way to make sense of Mulholland Dr.'s narrative. In a 1997 interview for Rolling Stone that coincided with the release of Lost Highway, for example, Lynch complained that "every single element in a movie now has to be understood—and understood at the lowest common denominator," which, for him, is "a real shame, because there are so many places that people could go if they weren't corralled so tightly with those kinds of restraints" (qtd. in Gilmore 39). In contrast, after discussing Mulholland Dr. with Lynch at Cannes, Sterritt reported that the director informed him that the film "does tell a coherent and comprehensible story" and "though you may need multiple viewings to fit the pieces together, they'll form an elegant pattern if you ponder their perplexities long enough" (15). The positive reviews from critics and Lynch's own promotional efforts, however, were not enough to encourage Universal to release Mulholland Dr. widely in U.S. theaters. During its opening weekend in 2001, the film appeared on only 66 screens and, though that number grew to 247 screens the following weekend, it was never granted the kind of saturation booking that is typical of most Hollywood fare, contributing to its poor performance at the domestic box office. In the end, Mulholland Dr. recuperated less than half of its production costs during its short run in select U.S. theaters, bringing in approximately $7 million on a $15 million budget (imdb.com).
However, Mulholland Dr. has subsequently been elevated into the canon of contemporary Hollywood films even though it lost money at the box office. Critical praise and Lynch's Academy Award nomination for Best Director helped to fuel its delayed appreciation. Yet, its changing reputation has been bolstered most by how it was packaged for sale on DVD. The DVD version of Mulholland Dr. was designed with a number of features that encouraged viewers to watch it in a specific manner in post-theatrical settings. As with The Straight Story (1999), the original DVD of Mulholland Dr. contains no chapter stops, practically compelling spectators to watch it in a teleological fashion. The ploy forces repeat viewers to expend great energy to navigate the disc to unearth the meaning of its mysteries. Paradoxically, that version of the DVD also included an insert entitled "David Lynch's 10 Clues to Unlocking this Thriller," which read:
1) Pay particular attention in the beginning of the film; at least two clues are revealed before the credits. 2) Notice the appearances of the red lampshade. 3) Can you hear the title of the film that Adam Kesher is auditioning actresses for? Is it mentioned again? 4) An accident is a terrible event... notice the location of the accident. 5) Who gives a key, and why? 6) Notice the robe, the ashtray, the coffee cup. 7) What is felt, realized, and gathered at the club Silencio? 8) Did talent alone help Camilla? 9) Note the occurrences surrounding the man behind Winkies. 10) Where is Aunt Ruth?
Although no answers were offered to correspond to these hints, their presence strongly suggests that there is, in fact, a "correct" way to interpret the narrative coherently. It also signaled, however, that only those willing to engage in the interpretive work necessary to solve the puzzle would be rewarded with the privileged information that Lynch buried for their discovery.
The presence of an enigma to be solved eventually helped Mulholland Dr., like Twin Peaks before it, appeal unexpectedly to Hollywood's most desired target market, even though the film's storyline and thematic preoccupations are not well-positioned for success with young, male spectators. The film centers on Diane Selwyn (Naomi Watts), an aspiring female actress, who both struggles to succeed in the male-dominated world of Hollywood and copes with being rejected by her dream girl, Camilla Rhodes (Laura Elena Harring). A film that exposes the misogynistic tendencies of the American film industry and focuses on a lesbian character who longs for an unrequited romantic partnership is not the kind of topical focus that generally resonates with men. Of course, although one of the film's taglines—"a love story in the city of dreams"—wryly encapsulates its narrative thrust, its primary causal lines of action are anything but readily apparent on first viewing. Instead, as is suggested by its alternative tagline—"A woman in search of stardom. A woman in search of herself—in the city of dreams. A key to a mystery—lies somewhere on Mulholland Drive"—the film's narrative can only potentially be made coherent retrospectively with significant interpretive labor, initially rendering it non-classical.
Mulholland Dr. is difficult to interpret in classical fashion because it appears to contain a series of loosely related vignettes featuring characters who, like the primary players in the purportedly narratively incoherent Lost Highway, switch names and identities without explanation. The female leads played by Watts and Harring, for instance, are called Betty Elms and Rita, respectively, for most of the narrative and are not referred to as Diane Selwyn and Camilla Rhodes until late in the film. The film is also populated by bizarre and threatening characters, such as a powerful dwarf (Michael J. Anderson), a menacing cowboy (Lafayette Montgomery), psychotic brothers—Vincenzo (Dan Hedeya) and Luigi (Angelo Badalamenti) Castigliani—who finance Hollywood films, as well as a frightening homeless person (Bonnie Aarons), who lives behind the dumpster of a Winkie's diner. Although these characters appear to be important, they each turn up briefly, never exchange dialogue with the protagonist, and thus seem tangential. Additionally, many of the film's most memorable scenes, including those featuring the strange events at club Silenco, filmmaker Adam Kesher's (Justin Theroux) discovery of his wife's infidelity, as well as Luigi's terrifying display of revulsion with his espresso, function as ostensible narrative digressions. The film's peculiar characters and occurrences appear to render Mulholland Dr. non-classical because they seem to obfuscate narrative meaning, suggesting that they exist for something other than compositional motives.
The presence of a mystery to be solved, though, alerted viewers that the film's strange scenes and characterizations could each have narrative significance. Consequently, before moving on to my presentation of the many plausible readings of the film's narrative that have been generated in a particular online fan community, I briefly present the most readily available way to reinterpret it retrospectively in coherent fashion. In particular, many fans believe that Mulholland Dr.'s incomprehensible narrative actually depicts a causally related series of events because once its master key is discovered, the film can be understood as being divided into three distinct, but narratively related, sections. In fact, the most commonly held interpretation maintains that the film begins with a brief prologue before Diane falls asleep, is followed by a portrayal of her extended dream sequence, and ends with a series of post-dream fantasies and flashbacks that explain the source material for her dream. According to this reading, Diane's dream is fueled by her failed attempt to become a successful Hollywood actress. Diane's frustration inspires her to hire a hit man (Mark Pellegrino) at a Winkie's diner to kill Camilla, the woman she blames for her demise. This alternative explanation is largely derived from the information presented during the film's extended changeover, which is instigated by the master keys—literally a stylized blue key and its corresponding blue box. However, as opposed to the standard changeover film, Mulholland Dr. never blatantly signals to the audience that it is imparting revelatory information. Specifically, although the chic blue key appears to be important, its significance is never explicitly stated during or after the opening of the blue box.
The stylized blue key that eventually opens the box initially shows up early in the dream sequence shortly after Betty discovers a naked Rita in her beloved Aunt Ruth's (Maya Bond) apartment. Importantly, Betty, who is new to L.A., is only staying at the place temporarily because her aunt will be away for a short period to work on a film shoot in Canada. The key immediately seems as though it will be significant because it is one of the few pieces of evidence available to help Betty discover Rita's true identity, which is the film's ostensible quest narrative. Rita, who is experiencing amnesia after a freak accident in which a serendipitous car crash saves her from two men who threaten her at gunpoint in a limousine, cannot remember basic information, like her name (the inspiration for her alias comes from a Gilda [1946] poster, featuring Rita Hayworth) or why her purse contains a large stash of cash and the blue key.
The one thing that Rita initially remembers is that she was going to Mulholland Dr. when the accident occurred. Her memory is further jogged when she sees a waitress (Missy Crider) at Winkie's with the nametag "Diane," who looks a lot like Betty, prompting her to recall that the name Diane Selwyn is somehow important. In turn, Betty calls the mystery woman to see if she can provide any information about Rita. As she dials the phone, Betty offhandedly tells Rita that "it's strange to be calling yourself." When the two recognize that the voice on the outgoing message is not Rita's, Betty innocently suggests that "maybe that's your roommate." According to this particular reinterpretation of narrative information, Betty's seemingly tangential remark is the kind of evidence that becomes vital, in retrospect, because it hints that Betty and Diane may actually be the same person. Upon their return from the mysterious club Silencio later in the film, Betty inexplicably disappears and the key becomes relevant again when it opens the strange blue box that magically appeared in Betty's purse at the club. As Rita opens the box, the camera zooms into its darkness, slyly signaling, but never announcing, that a change is occurring. Shortly thereafter, the frightening cowboy reappears and signals that the dream is over by literally telling Betty that "it's time to wake up." Unlike the always groomed and peppy Betty, the woman (also played by Naomi Watts) who wakes up in the bed looks unkempt and gravely depressed. In sum, the opening of the box becomes the changeover because it subtly depicts the end of Diane's dream in which she both imagined herself as Betty and Camilla as Rita.
However, Lost on Mulholland Dr. (LOMD), an unofficial website created and maintained by fans of Mulholland Dr., demonstrates that some viewers are unconvinced that this particular rereading is the most persuasive. Consequently, they have leveraged their collective intelligence to resolve the seemingly eternal mysteries that it does not explain. On the site's "newcomer's guide," the contributors briefly summarize what they term the "classical interpretation" (the reading described above in which the opening of the blue box is deemed to be the changeover) because they consider it only "a useful starting point" for those interested in learning about other possible explanations. The site's participants, therefore, are devoted to discovering what they perceive to be alternative ways to reinterpret the film in a coherent fashion. Of course, there are a few posters to LOMD, like Dave H., who question the site's aims by wondering if the contributors are ultimately "reducing the beauty of this work of art by analysis." Yet, later in that same thread, he admits that he is "continually trying to figure it all out" even though he knows that this potentially ruins the film's pleasures. These contradictory statements exemplify participants' sentiments because although there is a desire to crack the code, LOMD's contributors consider the film's ambiguity to be a great asset.
While the anonymity of online posting makes it difficult to determine information about the identities of the site's primary contributors, it is safe to assume that a majority are male because the most active participants use avatars such as HarryTuttle, Alan Shaw, and richdubya that suggest that they are men. Such skewed gender representation on the site indicates that Jenkins is right to speculate that the presence of a mystery compensates for men's anxieties with a film that is not ostensibly targeted at them. It is possible to deduce more information about LOMD's creators from a link on the homepage that directs visitors to a list of the contributors' "recommended movies," which contains both their favorite films and films that they deem similar to Mulholland Dr. The lists are comprised primarily of narratively, formally, and thematically challenging films from a range of traditions, such as experimental, international art, and Hollywood cinemas. Unsurprisingly, Lynch's other films and a slew of contemporary misdirection films dominate both lists. This display of cultural capital is of note, then, because it signals that the site's creators consider themselves discerning cinephiles who possess exquisite taste in and an abundant knowledge about film.
LOMD's homepage also contains links to the site's primary sections, which each demonstrate that its contributors are highly devoted to solving Mulholland Dr.'s narrative puzzles. There is a link, for instance, to a page that features images associated with the film, such as photos and the corresponding addresses of primary shooting locations, snapshots from the making of the film and the television pilot, as well as still shots from the film deemed narratively significant. Additionally, there are pages that house sound clips from and related to the film, including songs from the soundtrack, key lines of dialogue, and interviews with creative personnel. Of most interest are the pages titled "theories" and "studies." The "theories" page provides detailed accounts of possible ways to interpret the meaning of the film. Similarly, the "studies" page offers explanations for the possible narrative relevance of just about every character, object, location, and event in the film.
On the "theories" page, visitors are invited to click on links to approximately 30 distinct ways to comprehend Mulholland Dr.'s narrative in a totalizing fashion. Although the content of the theories varies greatly, as is evidenced by their often outrageous titles, such as "The Abortion Theory," "Two Drug Trips," and "A Deal with the Devil," they each similarly aim to render the film classical by explaining how something other than the opening of the blue box can make the film's ambiguities narratively relevant. For starters, many posters claim that the post-dream scene in which Diane meets with the seedy man at Winkie's holds the key to comprehending the film coherently. According to a number of these contributors, those who ascribe to the "classical interpretation" incorrectly assume that the man she meets at the diner is a hit man. In the "Bribery Theory" explanation, for example, a poster argues that Diane is paying the unidentified man "to influence the casting of Camilla Rhodes in a film—starting her off on the road to stardom." This alternative reading is persuasive because it can be supported by the events that transpire in the dream. Its proponents posit that the blonde Camilla (Melissa George) in the dream is selected as the lead in the film within the film—The Sylvia North Story—because an array of shady businessmen, including the Castigliani Brothers and the dwarf, influence personnel decisions in Hollywood. Such a string of events implies that someone, presumably Diane, has arranged for it. Consequently, it is just as likely that her dream is inspired by the guilt associated with her decision to give him the money to buy Camilla's fame as it is that she paid him to kill her.
Figure 4.1. Diane Selwyn at Winkie's, making arrangements with Joe, whom most fans identify as a hit man, to do something to Camilla Rhodes in Mulholland Dr.
In the "Dying Dream/Afterlife Theory," three LOMD contributors, who adopt the usernames smapty, cuttingedgenyc, and Alfred Romo, also marshal convincing evidence to counter the "classical" interpretation. They believe that "the bulk of the film takes place after Diane has committed suicide." For these contributors, the key to understanding the film in this manner is revealed in the pre-opening credit sequence in which, through a P.O.V. shot, a character, presumably Diane, falls to the bed to begin the dream. Significantly, in this scene, the sound is highly muffled, potentially concealing a suicidal gunshot. Following this logic, the dream thus actually begins after Diane pulls the trigger because she falls on the same bed that, in the dream, contains a rotting corpse that resembles her. The meanings of many of the film's ambiguities and the ten clues, then, change dramatically from the "classical interpretation." To begin, the two elderly people—Irene (Jeanne Bates) and her companion (Dan Birnbaum)—depicted in the opening jitterbug sequence and later accompanying Diane when she first arrives in L.A., can now be understood as representing "Diane's guardian angels" because they reappear in her apartment just before she shoots herself. Similarly, club Silencio is transformed into a depiction of hell, which explains why the film ultimately returns there after Diane sees her own suicide occur. In fact, its status as hell explains why virtually all of the club's performers, such as the magician (Richard Green) and emcee (Geno Silva), who also appears as the manager of the seedy Park Hotel, are either dressed in red or have a demonic appearance. It also reveals why the red-clad Rebekah Del Rio apparently dies onstage. For those skeptical about this interpretation, the authors present a freeze-frame of an image that only appears for a fleeting moment immediately before Betty and Rita get into the cab that takes them to club Silencio, containing a flyer attached to a telephone poll that reads in small letters "Hollywood is" and in big letters "HELL." This plausible alternative explanation is thus bolstered by DVD technology because it enables viewers to control the image in such a fashion. Moreover, evidence to support this theory is effectively and easily communicated on the web, which allows users who possess even a little technological knowledge the ability to distribute text juxtaposed with images to a wide audience.
Like many other contributors to LOMD, the proponents of the "Dying Dream/Afterlife" theory also rely heavily on their knowledge of Lynch's authorial tendencies and their ability to spot intertextual references to buttress their interpretation. To wit, as Camilla treks toward downtown L.A. after she survives the attempt on her life in the back of the limo, the camera tilts up to reveal that she has reached Sunset Boulevard. As the authors of this theory note, that street name is also the title of a famous 1950 Hollywood film that is "narrated by a person who is already dead" that "tells us how he came to be killed." This is relevant to their reading because they assert it is a film that "Lynch has referred to as his favorite," suggesting why Mulholland Dr. contains so many "homages to Sunset Blvd." like "Lynch's use of the same car that Norma Desmond [Gloria Swanson] came to Paramount in." The authors cite similar evidence from Lynch's other works to validate their theory further. For instance, they contend that the cowboy has to be "a spirit" because when Adam meets him at his corral "the electricity in the lights overhead dimmer and flicker" and "in Twin Peaks, when a lodge spirit is nearby it alters the electrical charge of its surroundings." Most interestingly, to persuade readers that their interpretation is correct, they argue that "as an afterlife experience, Mulholland Dr. has NO unresolved issues." To discredit their opposition further, they also theorize that the "classical interpretation" is not plausible because "Lynch has never dealt directly with dreams. It's not his style." Put simply, the proponents of the "Dying Dream/Afterlife Theory" back their argument by claiming it explains everything and can be substantiated by the discourses of authorship.
Figure 4.2. An almost imperceptible "Hollywood is HELL" flyer is affixed to a pole, as Diane Selwyn and Camilla Rhodes hail a cab to take them to club Silencio in Mulholland Dr.
Many of the ambiguities that are seemingly clarified by this theory can be even more sensationally and compellingly explained by other hypotheses on LOMD. Like conspiracy theorists, contributors to LOMD participate in a game of hermeneutic one-upmanship to obtain the kind of narrative closure that the text itself does not provide. According to the writers of the "Dying Dream/Afterlife Theory," for example, the meaning of Aunt Ruth's character is the key to understanding the film. They argue that Aunt Ruth's apartment represents a portal into the afterlife. Since their explanation assumes that Diane commits suicide after ordering the hit on Camilla, they speculate that it makes sense that viewers see "Aunt Ruth leaving with suitcases as Camilla arrives" because it is subsequently revealed that the two characters actually died in that order. Moreover, this logic explains why "Diane arrives in the apartment AFTER Camilla/Rita." Such an interpretation also convincingly highlights why the changeover occurs where it does and the significance of what happens during it. Indeed, as Rita opens the blue box in Aunt Ruth's apartment after they return from club Silencio, Betty inexplicably vanishes and Aunt Ruth magically reappears in the apartment.
In contrast, though, as Aleksandra H., one of the site's only contributors adopting a female avatar counters on the "studies" page devoted to Aunt Ruth, Mulholland Dr. can be interpreted as depicting Aunt Ruth's, and not Diane's, dreams and flashbacks. For this contributor, a reading of the film as being about "a young girl who came from Canada to make a career in Hollywood," in the 1950s, as opposed to an aspiring Hollywood actress from the present, renders many of the "anachronistic touches" and ambiguities insufficiently explained by many of the other theories as narratively relevant. A reading of Aunt Ruth as the primary character explains why the first of three different aged, red-headed women who look a lot like her briefly appears as a competitor in the 1950s-inspired jitterbug sequence in which Diane is never shown dancing. In fact, as another contributor to the Aunt Ruth "studies" page named blu-ray posits, each of Aunt Ruth's doppelgängers can be read as representations of different phases of her Hollywood career: the jitterbug contestant is the pre-Hollywood Ruth, the casting agent who appears in Diane's steamy audition scene is Ruth in middle-age after she has given up acting, and the comparatively older, red-headed woman waiting for a limo is a depiction of Ruth living in the Sierra Bonita apartments in retirement. Additionally, if the story is being told from Aunt Ruth's perspective, The Sylvia North Story no longer appears to be a period piece. Instead, it becomes a film that actually was shot when Ruth arrived in Hollywood in the 1950s. Thus, it is also easy to explain why many of the characters use terms like "horse pucky" and "smart aleck," which makes it sound as though they exist in a 1950s sitcom.
According to this theory, Aunt Ruth's reappearance in the apartment when Betty disappears can be understood as the end of Ruth's naïve reimagining of how her experience as a film actress could have gone differently because Betty and Diane are actually representations of the two sides of her Hollywood story. Betty is a talented, young woman, who has no idea how things really work in Hollywood. Eventually, though, she turns into Diane, who, like the film's other elderly women characters—Coco (Ann Miller) and Louise Bonner (Lee Grant)—becomes a hardened, presumably out-of-work actress. Finally, it also becomes evident why Camilla draws her inspiration for her alter-ego from Aunt Ruth's poster of Rita Hayworth in Gilda: the film is widely renowned for being a quintessential studio-era depiction of the objectification of women. Moreover, Hayworth was at the height of her popularity in the 1940s and 1950s and famously dyed her hair red to conceal her Latina heritage. In short, this astute reinterpretation effectively explains many of the film's most puzzling moments both by imagining the details of a seemingly tangential character's personal history and drawing on a vast knowledge of Hollywood history.
Figure 4.3. One of the red-headed Aunt Ruth doppelgängers appears in Mulholland Dr.'s opening jitterbug sequence.
The remarkable explanation entitled "A Multi-Layered Analysis of Mulholland Dr.," posted by a contributor with the user name Alan Shaw reveals that the film has inspired the site's predominately male participants to engage in wild speculations about the personal histories of its characters to crack the code. Even though it is never explicitly stated in the film, Shaw posits, in a section of his analysis called "The Diane Selwyn Story," that Diane's parents died when she was a child. Consequently, she was forced her to live with her evil custodians—her grandmother Irene and her elderly male companion—in Deep River, Ontario. Diane's childhood was horrific, he theorizes, because the elderly man molested her. Worse still, her grandmother remained silent and demanded that Diane never speak of the abuse—hence, silencio. Fortunately, when Diane wins a jitterbug competition, she is able to try to fulfill her dream of becoming a Hollywood actress because it inspires her opportunistic custodians to give her the money she received from her recently deceased Aunt Ruth, who worked as a Hollywood casting agent.
Unfortunately, Diane's attempts to succeed in Hollywood are thwarted by consistent rejections. When her Aunt Ruth's money begins to run out, she is forced to become a waitress at Winkie's and moves into a room at the Park Hotel. To augment her meager income, Diane eventually becomes a prostitute, whose name gets placed in a pimp's black book, enabling her to move into the much nicer, yet still relatively humble, Sierra Bonita apartment complex. Such a reading is plausible, Shaw argues, "because her previous abuse as a child made her feel as though there was something inevitable about being treated like a commodity." Diane then tries out for the lead in The Sylvia North Story, a low-budget film produced by Wally Brown (James Karen), an old friend of Aunt Ruth's, which she believes is the perfect role for her because it depicts a woman in an illicit sexual relationship with an older man. However, as Shaw summarizes, "the role went to Camilla Rhodes" instead, who wowed everyone at the audition, including Diane, who told Camilla that she was impressed with her "ability to heat up what Diane thought was 'such a lame scene.' " Diane and Camilla subsequently develop a friendship after the fame-starved Camilla becomes enamored by Diane's flattery. The two eventually grow so close that they decide to move in together at Sierra Bonita and become lovers. Diane, who has been scarred by her male abusers, develops an unhealthy obsession for her female companion. Sadly, Camilla is not as committed to the relationship because she is only concerned with furthering her Hollywood career. After Camilla receives good reviews for her performance in The Sylvia North Story, she begins to be offered A-list parts, most notably in an Adam Kesher film. Camilla then agrees to get a desperate Diane some bit parts in her films, including Kesher's film, provided that she agrees to whore herself out to Hollywood executives like Luigi Castigliani, with whom Camilla routinely sleeps to secure her starring roles.
Camilla subsequently develops a romantic relationship with the director, which gives her financial stability and jeopardizes her relationship with Diane; however, even though Camilla moves out of the apartment, the two do not officially end their romance. To escape the trauma of once again being used, this time by a woman whom she loves deeply, Diane switches apartments with L. J. DeRosa (Johanna Stein), her neighbor at Sierra Bonita. Her attempt proves futile, though, because Camilla sadistically torments Diane by continually flaunting her promiscuity. Camilla's display of affection for another woman (the woman portrayed as the blonde Camilla in the dream) at a party celebrating her engagement to Adam becomes the final straw for the emotionally troubled Diane. As Shaw notes, "Diane had been able to hope that even if Camilla stayed with Adam" they "could still have an intimate relationship with each other;" however, "Camilla's kiss with another woman showed her that Camilla was not coming back to her." As a result, with the money she has earned from prostitution, Diane hires a hit man who promises to place a blue key behind a Winkie's dumpster after he kills Camilla. The sight of the blue key on her coffee table finally drives Diane over the edge. In turn, she falls into a drug-induced sleep and experiences the dream inspired by these traumatic events. Diane awakens from her dream more depressed than before and, presumably still under the influence of the drug, hallucinates the return of Irene and her companion. Horrified, she retreats to her bedroom, "the place," Shaw writes, "where her childhood abuse took place" and commits suicide.
"The Diane Selwyn Story" is only the start of his analysis because by hypothesizing that Diane was sexually abused by a number of perpetrators throughout her life, Shaw is able to close off a number of the film's most puzzling ambiguities and identify the narrative significance of the ten clues in a convincing fashion. Peppered throughout the "studies" pages, for instance, are his interpretations of how a variety of the film's mysterious characters, objects, and events can each be read as related to the master key of sexual abuse. He argues that the events at club Silencio portray Diane reliving her childhood sexual trauma. On the "studies" page devoted to the magician, he specifically theorizes that the magician's demonic performance, which culminates with both his almost imperceptible grunt and Diane's uncontrollable shaking in response, can be read as a coded representation of her childhood rape. Similarly, in his examination of Woody Katz (Chad Everett), Shaw speculates that his sexually charged audition with Diane can be understood as another time in which Diane's subconscious is contending with the horror of her abuse. He believes that Woody is closely related to Diane because of the scene's dialogue that suggests that he knows her parents well and because the actor, whose name has phallic connotations, prepares for the audition by saying "Dad's best friend goes to work." Moreover, evidence indicates that the liaison is both illegal and has occurred previously. During the scene, the two characters worry about going to jail if they are caught and Diane expresses remorse about the ongoing relationship. In sum, the sexual abuse theory provides an interpretive grid through which the narrative is made coherent by speculating about how unreferenced events from Diane's history shaped her Hollywood experience.
Figure 4.4. Mulholland Dr.'s Diane Selwyn and Woody Katz amplify the intimacy and sexual chemistry of their audition.
Shaw's highly imaginative reading of Mulholland Dr. also further showcases how LOMD's contributors attempt to convince visitors that they have definitively solved the puzzle by relying on a knowledge of Lynch's oeuvre, their ability to catch intertextual references, and the capabilities of DVD technology. On the page devoted to a summary of Shaw's sexual abuse theory, a number of stills from the film are posted as evidence that characters, such as Irene's elderly companion, the magician, and Woody Katz, are each representations of Diane's abusers. Similarly, to support the idea that Diane's abusive past is what leads her to work as a hooker, the page also contains a freeze-frame of the hit man interacting with a battered prostitute from the dream, who looks a lot like Diane. Additionally, the cryptic narrative logic of Twin Peaks, which also contained thematic undercurrents about the horrors of incest, is consistently cited as being relevant to understanding Mulholland Dr. as a nightmare about the trauma of childhood sexual abuse. Those who support the sexual abuse theory also correctly point out that Diane claims to hail from a town called Deep River, which happens to be the name of the apartment complex in Blue Velvet where Frank Booth (Dennis Hopper) violently rapes Dorothy Valens (Isabella Rossellini). To validate their claims further, the unnamed authors of this summary note that many historians have claimed that Rita Hayworth was likely molested by her father. Finally, according to the "studies" page devoted to a painting, entitled "The Beatrice Cenci," which hangs in Aunt Ruth's apartment, the famous portrait depicts a "Roman noblewoman" from the sixteenth century who supposedly "hired two hit men to kill her incestuous father."
The sexual abuse theory illustrates the lengths to which this fan community will go to obtain narrative closure. The "studies" pages, in fact, feature many other examples of the kind of interpretive labor that the film's release on DVD has inspired on the site, such as diagrams that depict both the numbers likely dialed on a rotary phone to call the dwarf and the layouts of both apartments at the Sierra Bonita complex. Additionally, there are English translations of the Spanish and French dialogue as well as the Chinese writing that appears in the film. There are also "studies" pages devoted to revealing the ways in which a number of films, including Sunset Blvd., Contempt (1963), and The Wizard of Oz are connected to Mulholland Dr. In fact, The Wizard of Oz, one of the most recognizable misdirection films in Hollywood history, has influenced Lynch throughout his career, especially in films like Wild at Heart and Blue Velvet. It clearly can be read as a narrative template for Mulholland Dr., according to the "classical interpretation." Like Dorothy's (Judy Garland) experiences in Oz, after the blue box is opened, the people, events, and objects that appear in the nightmare can be understood as distortions of crucial moments from Diane's waking life. To help support this reading of the film, the authors of The Wizard of Oz "studies" page not only list the extensive narrative similarities between the two films, they also juxtapose freeze-frame images of Mulholland Dr. with the Hollywood classic to demonstrate that shots from the former have been constructed as explicit homages to the latter. This impressive display of cultural capital again reveals that making such elaborate connections in a convincing manner has become feasible in an age in which DVD and Internet technologies allow this kind of visual evidence to be presented with relative ease.
LOMD may be the most dazzling example of how Mulholland Dr. has successfully encouraged fans to use new technologies to decipher its ambiguous narrative, but it was not the first page of this kind to appear on the web. In October 2001, shortly after the film premiered in U.S. theaters, Bill Wyman, Max Garrone, and Andy Klein published an article on Salon.com entitled "Everything you were afraid to ask about Mulholland Dr.," which was an early attempt to forward what LOMD participants term the "classical interpretation." Although the authors do an admirable job of making narrative sense of many of the film's most notable ambiguities, they ultimately acknowledge that their reading cannot explain all of its enigmas. This admission, in turn, inspired fans to submit their interpretations of the film to the editors, a number of which Salon.com published just a few days later in an article entitled "Whaddaya mean, 'We don't know about the box?'." Interestingly, scattered through these alternative readings are the kind of statements that suggest that audiences conceive of misdirection films as constituents of a distinct genre. One responder, for instance, theorizes that "the first two-thirds of the movie are Betty/Diane's fantasy, either masturbatory or upon dying (a sort of Jacob's Ladder cum Sixth Sense thing)." Humorously, another similarly themed response simply reads "Rosebud." Finally, a pleased reader commends the site for publishing the interpretation, because "This—and the Memento article" that Salon.com "published last summer" are the "type of film discussion" there should be "more of," provided that there are "the right kind of movies to discuss."
## Understanding Memento: Unreliable Memories and Untrustworthy Evidence
Salon.com readers were not the only ones to make these types of connections. The authors of the Salon.com article on Mulholland Dr. similarly note that "of recent American movies only Memento is remotely as challenging." Perhaps not surprisingly, then, in June 2001, just a few months before the Mulholland Dr. article appeared online, Salon.com published the aforementioned article on Memento. Written by Andy Klein, the article, entitled "Everything you wanted to know about Memento," aims to unravel the film's meaning. Klein justifies his decision to write the article about the film, which he initially compares to The Usual Suspects and The Sixth Sense, by eventually claiming that it is even more narratively complex than those predecessors. As he states, "Memento's puzzle cannot be undone with a single declarative sentence." Instead, as he summarizes, it is difficult to make sense of Memento's narrative because it is complicated by
an elegant but brain-knotting structure; by an exceedingly unreliable narrator through part of the film; by a postmodern self-referentiality that, unlike most empty examples of the form, thoroughly underscores the film's sobering thematic meditations on memory, knowledge and grief; and by a number of red herrings and misleading clues that seem designed either to distract the audience or to hint at a deeper, second layer of puzzle at work—or that may, on the other hand, simply suggest that, in some respects, the director bit off more than he could chew. (Klein)
Like Mulholland Dr., Memento is a film that contains such an uncharacteristically intricate narrative that spectators are encouraged to watch it on multiple occasions to sort out what "actually" happens. In the end, however, as Klein's comments also suggest, it may not be possible to determine Memento's "truth." Although it is highly unconventional for a Hollywood film to contain such narrative characteristics, some viewers cite the existence of these seemingly eternal ambiguities as a positive artistic attribute. Yet, many of these ardent fans are most devoted to solving these mysteries in a way that renders the film classical, in retrospect.
Even though it presents information in a highly unconventional manner, Memento does contain a seemingly straightforward narrative, making it necessary to explain why its events may not have the same retroactive significance. The noir-inspired plot is ostensibly driven by Leonard Shelby's (Guy Pearce) attempt to avenge his wife's rape and murder. In particular, although he shoots and kills a culprit during the attack, he remains convinced that a second assailant, named John G. or James G., remains at large. When Leonard shares the details of the event with Natalie (Carrie-Anne Moss), however, he tells her the authorities consider the case closed because they believe that there was only one intruder. Leonard's quest for justice is further complicated by the anterograde amnesia (short-term memory loss) that he purportedly incurred as a result of a head injury that he suffered while attempting to save his wife. Leonard's condition makes him an unreliable narrator because he claims not to be able to store any new information in his long-term memory since the incident, meaning that it is difficult to trust any information he divulges about himself. The presence of his memory loss also casts doubt on the veracity of his motives. Indeed, the film consistently hints that his objectives and even the severity of his condition may not be as clear-cut as they seem. After murdering Jimmy Grants (Larry Holden), for example, Leonard steals both his designer clothes and Jaguar, which also happens to have $200,000 cash in the trunk for a proposed drug deal. He subsequently kills Teddy (Joe Pantoliano), a purported cop, whose real name is supposedly John Edward Gammell, and who admits to have been manipulating Leonard ever since being assigned to the Shelbys' case. The all-too-coincidental benefits of both murders suggest that Leonard may both be out for more than vigilante justice and able to remember more than he claims.
Other factors trouble Leonard's traits or motives as being legitimate. First, the film contains an unorthodox narrative structure that replicates Leonard's memory loss and consequently disorients the spectator in time and space, making it difficult to determine the film's causal relationship of events. Specifically, a majority of scenes—those that play in color—unfold in reverse chronological order. Complicating things further, these backwards scenes alternate with black-and-white segments that move forward in time. Eventually, however, the two distinct, yet related, plotlines seamlessly meld together into a forward-moving color sequence late in the film when Leonard develops a Polaroid photo that depicts Jimmy's murder. Second, Teddy discloses much of the "truth" about the attack on Leonard's wife and what has happened since the incident. As with the film's other secondary characters, most notably Natalie, who seems to manipulate Leonard to kill Teddy, Teddy takes advantage of Leonard's condition to further his own objectives. The information that he provides in the explanatory sequence thus cannot be trusted because it may simply be designed to trick Leonard. Finally, even though the film contains a changeover that appears to explain the meaning of events fully, its validity is troubled by both Leonard's status as a highly untrustworthy narrator and the fact that the film itself contains an unreliable mise-en-scène.
Although Memento's meaning is complicated by its unconventional, nonchronological structure, theorists like Bordwell have demonstrated how the film's writer/director, Christopher Nolan, employs many techniques to help spectators comprehend it. The film contains a number of formal cues and narrative redundancies that indeed help spectators connect events in a classical fashion, such as the oscillation between color and black-and-white cinematography, Leonard's continuous reminders about the details of his condition, the parallel story about Sammy Jankis (Stephen Tobolowsky), who also suffers from anterograde amnesia and accidentally killed his diabetic wife by administering too many insulin shots, the use of dangling causes and dialogue hooks, as well as continuities and changes in makeup and costuming. Moreover, even though the film presents narrative information in an atypical fashion, its unconventional style and structure are tied directly to the subjectivity of the protagonist, rendering these decisions compositionally motivated. Although Bordwell grants that "closure operates retrospectively" in Memento, he ultimately claims that "the events still cohere through cause and effect," making its innovations contained by the classical paradigm (The Way 79).
Other critics argue, however, that even though many classical tactics are deployed to offset the film's complexities, it is still difficult to connect Memento's narrative events according to a definitive causal logic. For example, Klein, who admits to seeing Memento five times before penning his analysis, speculates that the film is difficult even for repeat viewers to understand because "its puzzles are so intriguing and so impenetrable at first viewing that filmgoers are almost forced to go back for a second look if they want to figure just what the hell is going on." To assist viewers perplexed by the film's unconventional structure, he presents a method for systematically charting the chronology of narrative events. First, he assigns letters in reverse order to each of the color scenes in the sequence in which they appear in the film, starting with "V" and ending with "A." Second, he uses numbers, ranging from "1" to "22," to correspond to the distinct black-and-white scenes in the chronological order in which they appear. Accordingly, the scene in which Leonard develops the Polaroid of Jimmy and the film transitions from black-and-white to color, should be labeled "22/A." "If you want to look at the story as it would actually transpire chronologically, rather than in the disjointed way that Nolan presents it" Klein writes, viewers should "reorder the events by starting with scenes 1–22 and ending with scenes A–V," which he exclaims will "be fun to do on DVD!" As with Mulholland Dr., Klein's provocative analysis of Memento inspired a slew of emails from readers, some of which Salon.com published on July 4, 2001. A number of these responses reveal that Klein was not the only one who believed that Memento was custom-made for DVD. One reader, for instance, tells Klein that "Like you, I am looking forward to Memento's release on DVD." Similarly, another reader admits to "being a relative babe in the woods, having only seen the movie once" and is "anxiously awaiting its release on DVD for further viewing."
In 2002, as if they had read Klein's article and the responses to it, Sony subsidiary Columbia TriStar gave viewers the opportunity to watch the film chronologically by releasing a two-disc limited edition of Memento on DVD, which was packaged to look like Leonard's psychiatric case file and even included a few doctored excerpts from his dossier. In addition to these ancillary materials and the disc containing the theatrical version of the film, the limited edition DVD was accompanied by a supplemental disc with a number of extras that provided new evidence for solving the mystery. Among these are the text of Jonathan Nolan's (Christopher Nolan's brother) short story on which the film was based, excerpts from Leonard's journal, and a transcript of the shooting script, replete with Christopher Nolan's handwritten notes. Of most interest, however, is the supplemental disc's hidden feature that enables viewers to watch the film re-cut in the exact chronological order detailed in Klein's article. Of course, it is not easy to find the Easter egg that enables this special feature because users must navigate a maddeningly convoluted menu comprised of mock psychiatric tests to access any of the features on either disc. To do something as simple as view the theatrical version, for instance, viewers first have to wait for a couple of sample psychiatric questions to disappear and then have to locate and select the term "watch" from a long list of words on its quasi-main menu. Fortunately, instructions for doing this are included in one of the inserts that accompany the limited edition DVD. In contrast, to access the reedited, chronological version of Memento, spectators must execute a complex series of commands, resembling the kind of interactive participation that is typically required of videogame players to activate cheats. In sum, the existence of the film's alternative edit on DVD and the collective intelligence that it likely takes to discover its presence suggests that producers both understood how difficult it is to decipher Memento and hoped to entice viewers by rewarding only devoted fans with tools for decoding its narrative secrets.
The presence of this Easter egg, in addition to many of the other extra features, implies that the film's creators hoped that sales would be bolstered by suggesting that there is indeed a "right" way to crack the code. Christopher Nolan's own comments about the film support this further. In his article, Klein claims that when Scott Timberg of the New Times of Los Angeles spoke to Nolan "about the film's outcome" just prior to its domestic theatrical release, the director talked about his interest in "ambiguity and subjectivity," but also claimed to know "the movie's Truth—who's good, who's bad, who can be trusted and who can't" and ultimately maintained "that close viewing will reveal all." Despite his declaration that there is a "correct" way to interpret the film, evidence suggests that Nolan, like Lynch, may have purposely created an unsolvable mystery. Moreover, as Lynch did with his ten clues for Mulholland Dr., the additional promotional materials that surrounded the film's various releases seem to have been designed with the intent of clouding narrative "truth." Such an apparently contradictory strategy makes perfect business sense because it helps to sustain a collective intelligence's attention for an extended period of time beyond the theatrical release by forcing fans to sort through both the film itself and the mountain of ancillary materials associated to it.
To market the theatrical release of Memento, for instance, cash-strapped Newmarket Films allowed Jonathan Nolan to create an innovative website that provides a critical backstory about Leonard and his wife that is never offered in the film. The official website, otnemem.com (cleverly, Memento spelled backwards), first greets visitors with the film's tagline "some memories are best forgotten." The tagline then disappears and is automatically replaced by a mock newspaper article, indicating that Leonard is the prime suspect in a murder investigation. Interestingly, the article also reports that Leonard escaped from a San Francisco Bay area mental institution in 1998. From the article, users are invited to click on a number of highlighted words that are hyperlinked to excerpts from Leonard's psychiatric dossier and his doctored copy of the police file, revealing information about his anterograde amnesia and what transpired since he was institutionalized. It notes that he was admitted to the mental hospital in late 1997, a number of months after the attack occurred. It also presents evidence that someone has been writing notes to Leonard in an attempt to warn him that another person may be manipulating him to kill the wrong individual to avenge his wife's death. These notes are what encourage Leonard both to tattoo important information to his body and to remember the tragic story of Sammy Jankis, whose all-too-coincidentally similar case Leonard had purportedly investigated when he worked in insurance prior to the attack. Additionally, evidence that helps Leonard eventually identify Teddy as the perpetrator, such as his driver's license and his license plate, is prominently featured on the site. It is ultimately difficult to determine what is factual on the site, however, because much of the information there is contradictory. For example, although a newspaper article reveals that Leonard's wife survived and that an accomplice is being sought by the authorities, hospital records suggest that she is dead and that Leonard fabricated the second perpetrator. Moreover, whereas some of his files indicate that he is incapable of making new memories, others suggest that his condition was improving before his escape.
Paradoxically, then, the insider information offered on the official website often makes it more challenging to determine the "truth" of many of the film's most troubling uncertainties, such as the actual severity of Leonard's condition, what happened to his wife after the attack, the real relationship between Leonard and Sammy Jankis, and who is responsible for his wife's murder. In fact, answers to these questions may not be knowable. There is never enough evidence to determine if Leonard is manipulating all of the other characters through a Keyser Sözesque fakery of his condition, or if other characters are actually taking advantage of his very real short-term memory loss. The answer may seem to be evident in the scene that reveals that the motel clerk, Burt (Mark Boone Junior), is exploiting Leonard's purported disability by charging him for unknowingly renting two rooms simultaneously at the Discount Inn. However, after Leonard exposes the scam, Burt jokingly reminds him that "he's not going to remember" the ordeal anyway, to which Leonard retorts, "you don't have to be that honest, Burt." Leonard's use of Burt's name during an exchange in which it is never otherwise mentioned is suspicious because Leonard always seems to have to check his Polaroids to remember anyone else's name. Furthermore, in the other scenes in which Leonard and Burt interact both face-to-face and on the phone, Leonard explicitly asks the motel clerk his name because he claims not to be able to remember meeting him before. Although I grant that Leonard's inexplicable recollection of Burt's name might only indicate the presence of a continuity error, it may also more sinisterly reveal that Leonard is willing to appear to be taken advantage of to perpetuate his elaborate ruse.
The ambiguity about Leonard's condition lingers eternally because, like the promotional material associated with the film, the film itself is loaded with inconsistent information. For starters, although it seems Teddy's long explanation about Leonard's past is a revelatory sequence that provides key information about what "really" happened and what Leonard's "true" motives are, it actually further convolutes narrative meaning. During his explanation, Teddy suggests that Leonard committed the acts that he continually attributes to Sammy Jankis. In contrast to what Leonard claims, Teddy contends that Leonard's wife survived the attack, she did not believe that his anterograde amnesia was real, she had diabetes, and that Leonard's work as an insurance fraud investigator revealed that Sammy was a con man who did not have a wife. Interestingly, as Leonard refutes the diabetes allegation, the film replays an earlier flashback in which Leonard pinched his wife's leg; however, this time he administers an insulin shot to the same spot. A few seconds later, though, as Leonard vehemently denies Teddy's claim, the film cuts back to the earlier flashback of him pinching her leg. The incongruous visual evidence in this scene demonstrates that the veracity of everything that Teddy has just exposed as well as the information that viewers both see and hear Leonard recall about his past should be interrogated.
The events that follow this explanation sequence, which comprise the changeover, further muddy the film's meaning even though they seem to offer a coherent explanation for Leonard's motives and actions. In voiceover, Leonard confirms Teddy's allegation that he makes up his own truths by purposefully doctoring the evidence. Consequently, after Teddy reveals that he has been using him, Leonard leaves clues to trick himself into believing that Teddy is really John G., which finally explains why he kills Teddy in the film's opening moments. Yet the scene also subtly alludes to the fact that there may be a major problem with accepting Teddy's assertion that Leonard actually suffers from anterograde amnesia and manipulates himself to remember things how he would like them to be. To catch John G., Leonard has devised a system in which he tattoos his body with what he deems as the most crucial clues to solving the case. As a result, he writes himself a note on an index card to tattoo Teddy's license plate as fact number six. The scene's first indication that his memory loss may not be as severe as he claims, then, is that he remembers this clue to be the sixth fact without first checking his existing tattoos. Even more troubling is how he also erroneously copies the plate number as "SG13 71U" (the final three digits should read 7IU and not 71U), which calls into question how he eventually determines that Teddy is the culprit. Importantly, though, the significance of his inaccurate transcription is only signaled visually because the film merely cuts back and forth twice between the index card and close-ups of the license plate. However, on a number of occasions earlier in the film, which are, of course, later in the sequence of narrative events, Leonard's incorrect notation of Teddy's license plate is further verified both aurally and visually. For instance, a tattoo artist administers a tattoo with two ones instead of an I, as Leonard misreads the digits on his leg as "SG13 7IU" in voiceover. The question that thus arises is how did Natalie help Leonard track down Teddy if she gave her contact at the DMV the incorrect plate number? On one hand, it can be assumed that Natalie simply fabricated all of the documents that lead Leonard to Teddy. On the other hand, it could also be inferred that the error is irrelevant because Leonard is a faker, who is fully aware that he plans to kill Teddy. In sum, these revelatory scenes present considerable uncertainty, rather than closure, in relation to Leonard's traits, motives, and actions.
Figure 4.5. Memento's Leonard Shelby writes himself a note, in which he fails to distinguish the number "1" from the letter "I," to tattoo Teddy's license plate number.
Figure 4.6. Shot of Teddy's actual license plate number that is juxtaposed with Leonard Shelby's notation of it to confirm his ambiguous transcription of the digits in Memento.
The unresolved issues that remain even after the explanation and changeover sequences transpire prompted fans to form knowledge communities on the web to solve Memento's puzzle. Perhaps the most notable of these sites is the now defunct Unofficial Christopher Nolan Web Site (UCNW). The site, originally created and maintained by Johannes Duckner, contains links to each of Nolan's films; however, the amount of information posted in relation to Memento and the comparatively high degree of discussion that the film generated on the site's message board (as of 2008, there were over 400 posts related to Memento, which dwarfed its closest competitor) indicates that the film was UCNW's main attraction when it was active. Additionally, the relatively few comments and pages related to The Prestige (2006) as well as the high number of postings that appeared in the years immediately following the release of Memento's DVD reveals that the site was at the height of its popularity in the early 2000s. When UCNW was in full operation, Duckner was its main contributor and the most active posters on the Memento discussion board adopted usernames such as Michael, Chad, and Larry, suggesting that an overwhelming majority of its participants were men. Like Mulholland Dr., therefore, the film successfully attracted a niche audience comprised primarily of male, tech-savvy cinephiles who largely derived pleasure from deciphering the film's mysteries. Memento's discussion board was thus dominated by threads that focus on decoding the film's most baffling enigmas, including Leonard's inexplicable recollection of Burt's name, the mystery of the license plate, and the "true" identity of Sammy Jankis.
Unsurprisingly, the UCNW pages created by Duckner devoted to Memento also contain a number of tools for solving the mystery, such as links to both Klein's Salon.com analysis and the reader responses to his essay, an FAQ about highly disputed plot points, interviews with creative personnel, promotional materials associated with the film, and still images from the film deemed narratively relevant. The Memento section of UCNW also includes Duckner's own stabs at making sense of some of the film's most puzzling ambiguities. Significantly, Duckner's "Memento Trivia: Memory is Treachery" page again demonstrates how misdirection films encourage fans to use the distinct capabilities of Internet and DVD technologies to communicate and validate their particular readings of narrative causality. To provide evidence that Sammy Jankis and Leonard are likely the same person, for instance, Duckner posts a series of still images from the scene in which Leonard describes the circumstances surrounding Sammy's wife's death to an unidentified caller, presumably Teddy, whom he speaks to throughout most of the black-and-white sequences. As Duckner reveals, when the flashback nears its conclusion, Sammy can be seen seated in a mental hospital, where he has been sent for accidentally killing his wife. His juxtaposition of two freeze-frames from the DVD shows that Leonard replaces Sammy in the same hospital chair for a fleeting instant right after a doctor passes between him and the camera. This visual evidence strongly indicates that Sammy may not be real. Moreover, it suggests that it was likely Leonard who was institutionalized for killing his own wife with insulin injections. As a number of contributors to the site's Memento message board similarly maintain, it can be argued, therefore, that Leonard is a con man who committed insurance fraud by pretending to have anterograde amnesia and subsequently killed his wife to keep up the charade. According to this master key, he was institutionalized after her death and has maintained the façade since escaping the mental hospital to continue to attain his objectives.
Figure 4.7. Leonard Shelby replaces Sammy Jankis in a mental hospital for a brief instant in Memento after a doctor passes between him and the camera.
Duckner presents other images likely captured from the DVD to encourage visitors to interrogate the validity of Leonard's condition even further. He questions the reliability of the information that leads Leonard to kill Teddy, for instance, by posting a freeze-frame of the photocopy of the driver's license that Natalie obtains from her contact at the DMV. As the site reveals, Teddy's driver's license expires on February 29, 2001, even though that was not a leap year, making it hard to believe that the document is authentic or that it actually helped Leonard identify Teddy as John G. As Duckner posits, "people could argue that this is a factual error," but "why would someone choose February 29th" and "not check if the year was really a leap year?"
It is Duckner's analysis of Teddy's license plate, however, that most persuasively demonstrates that it may ultimately be impossible to determine the "truth" about Leonard's condition. To explain the mystery of the plate, he initially posts a series of still images to show that Leonard erroneously transcribed and tattooed its digits. Taking things a step further, however, he also posts an image taken from much earlier in the film (later in the chain of narrative events) in which there is a brief glimpse of Teddy's rear license plate when Leonard and Teddy drive Dodd (Callum Keith Rennie) out of town. Inexplicably, the plate's digits now read SG13 71U (the I has now changed to a one to match Leonard's inaccurate tattoo!). To convince those who would argue that this is simply a continuity error, Duckner also posts scanned images of various iterations of the film's script, containing Nolan's handwritten notes, which confirm that the inconsistency was intentional. Consequently, Duckner's interpretation is predicated on the notion that the director purposely wanted spectators not to be able to trust the film's facts themselves because they have morphed to correspond to the whims of its unreliable protagonist. Following this logic, it once again becomes possible to believe that Leonard actually suffers from anterograde amnesia because even though he copied down the plate's digits inaccurately, the mise-en-scène has changed to render the transcription correct. Viewers are left with a conundrum because they too have to discern fact from fiction in a film in which there is no way to delineate between the two. Of course, by unearthing the purported intentions of the auteur, Duckner's reading suggests that the film's "truth" is ultimately knowable. His analysis of the license plate demonstrates that, like many of the contributors to LOMD, he believes that his particular interpretation should be accepted because it can be supported by both DVD technology and the discourses of authorship.
Figure 4.8. A copy of Teddy's driver's license that contains an impossible leap day expiration date in Memento.
Figure 4.9. Teddy's license plate appears with changed digits to correspond to Leonard's tattoo in Memento when the two drive Dodd out of town.
Even though some misdirection films contain narrative and formal attributes that appear to challenge classical conventions and are thus unappealing to many audiences, the economic motives for their increased production are clear. At a time in which DVD profits exceeded the box-office take, misdirection films, such as Memento and Mulholland Dr., were well-positioned to be repurposed for post-theatrical markets. The viewing and interpretive practices that these films inspire depart from the standard activities of the classical spectator because they practically demand that viewers both watch them countless times in post-theatrical venues and examine them in communal settings to appreciate them to their fullest. However, producers are also careful to assure audiences that these films can ultimately be read classically, even if that may not actually be the case. Such a strategy enabled these films to be marketed as being both novel and familiar. The complex narratives contained in these films are created for a transmedia viewing experience because they effectively migrate viewers across media, which differentiates them from most Hollywood fare. An examination of the comments associated with these films in online fan communities suggests spectators find misdirection films attractive precisely because they believe that it is possible to find definitive answers to all of their mysteries. Interestingly, fans most often make recourse to the discourses of authorship to validate their totalizing accounts of what "actually" happened and why events "really" occurred. Although a number of misdirection films contain non-classical tendencies, interpretive activity in online fan communities reveals that the primary pleasure many viewers derive from them comes from striving to render them classical by discovering how each of their ambiguities can be retrospectively read as being narratively relevant.
5
## The Masters of Misdirection
Branding M. Night Shyamalan and Christopher Nolan
AS THE ARDENT ONLINE ANALYSIS of Memento (2000) suggests, Christopher Nolan's authorial identity became associated with fervent fan activity in virtual communities at the same time that he began to become a marketable commodity. Despite the critical acclaim for Memento, there was no guarantee that Nolan would be able to parlay that film's success into a Hollywood career as a reliable auteur. This is partly because Memento was received as being dependent on the gambit of focalizing the narrative through an unreliable protagonist, who might be suffering from short-term memory loss. Roger Ebert, for example, claimed that the "device of telling his story backward, or sort of backward, is simply that—a device," leading him to report that, after re-watching the film, "greater understanding helped on the plot level, but didn't enrich the viewing experience." In addition to trying to avoid subsequently being associated with a narrative gimmick, Nolan had to contend with the fact that M. Night Shyamalan had already become recognized as being the misdirection film genre's preeminent director. By 2000, the remarkable performance of The Sixth Sense (1999) was leveraged to promote Unbreakable (2000) in auteurist terms as a similarly themed and narratively structured film from a Hollywood wunderkind. At the end of the decade, however, Nolan replaced Shyamalan atop of the genre with the release of Inception (2010), the most expensive and profitable Hollywood misdirection film made to date. How and why did such a rapid and unexpected turn of events occur?
By 2008, Shyamalan's once promising future in Hollywood had become a thing of the past. The critical and box-office disappointment of Lady in the Water (2006), the first Shyamalan film after The Sixth Sense to be a financial letdown, generating only $72 million worldwide theatrically on its $70 million budget, was a blow to the director's industrial standing (imdb.com). His next film, The Happening (2008), was almost universally panned by critics, cementing his declining reputation. Although it received terrible reviews, The Happening returned Shyamalan to box-office solvency, earning a respectable $63 million domestically and a healthy $163 million worldwide theatrically on its $48 million budget (Mendelson). Yet, studio executives could not bear continuing to advertise Shyamalan as an auteur in the same fashion after two consecutive films received such vitriol from critics. In fact, Shyamalan's two subsequent films—The Last Airbender (2010) and After Earth (2013)—were the first since the release of The Sixth Sense, when he was an unknown commodity, to be promoted with his name no longer featured as the primary draw in their marketing campaigns. In contrast, Christopher Nolan became a Hollywood superstar by 2008 largely because of the tremendous success of his first two installments in the most recent iteration of the Batman franchise, especially The Dark Knight (2008); however, Nolan did not achieve this lofty status merely by establishing a reputation as Hollywood's latest ruler of the blockbuster film. Instead, as Time critic Graeme McMillan summarizes, he has become renowned for making "smart, thought-provoking blockbusters" that do not simply adhere to the Hollywood formula. Nolan's reputation codified in this way, I argue, partly because he filled the void as the misdirection film genre's leading director in the wake of Shyamalan's critical misfires.
In this chapter, I chart the divergent career trajectories of Shyamalan and Nolan. Whereas Shyamalan's differentiation attempts have been disastrous, Nolan's continued linkage to the misdirection film has helped make him one of Hollywood's most powerful filmmakers. After The Sixth Sense and until The Last Airbender, marketing efforts attempted to distinguish Shyamalan's films and attract a fan following by promoting him as the master of the genre. His reputation began to decline precipitously, however, with The Village (2004), his third misdirection film, making it important to assess what transpired in relation to the film's release and in its aftermath. Nolan, however, has achieved great success, at least in part, by continuing to be tied to the genre. Shortly after the start of Shyamalan's demise, Nolan cemented his status as the heir apparent with The Prestige (2006), his second widely distributed misdirection film. The choices made on the film's production and promotion unabashedly announced his association with the genre, helping to transform him into its new dominant figure. A comparison of the discourses circulating around these two filmmakers, leading up to and since Nolan supplanted Shyamalan as the king of the misdirection film, reveals how closely genre and authorship are often intertwined in contemporary Hollywood. Consequently, I demonstrate the ways in which the misdirection film has been deployed, with varying effectiveness, to implement one of Hollywood's most reliable schemes for maximizing profit: manufacturing superstar auteurs to create brand-loyal enthusiasts. Such an analysis illustrates just how deeply authorial standing now depends on corresponding industrial strategies.
## Nice Package: Product Differentiation and Authorship in Contemporary Hollywood
Authorship has long been a notoriously contentious issue in Film Studies. The story of how directors came to be known as auteurs is exceedingly familiar. The director was initially championed as the auteur by a group of French critics writing for the legendary journal, Cahiers du Cinéma, in the late 1940s and 1950s. Incidentally, many of these critics were aspiring filmmakers themselves, who would become members of the influential French New Wave movement. Most of these critics, therefore, had a vested interest in giving the director greater artistic credence than other creative personnel, especially screenwriters, who theretofore typically garnered more significant credit for their creative contributions. Obviously, this intervention ultimately had dramatic ramifications in Hollywood and beyond because many directors are now thought of as the primary artistic force in spite of the highly collaborative nature of commercial filmmaking. Indeed, once these theories about film authorship were relayed by influential English-language critics in the 1960s, most notably Andrew Sarris, the idea of the director as the major creative force on a film gained broader traction. Thanks to critics, like Sarris, directors, particularly those working in the most restrictive industrial conditions, began to become evaluated and ranked artistically based on how their thematic preoccupations and stylistic tendencies were expressed across their output. When the Hollywood Renaissance was at its peak in the early 1970s, it started to become standard practice, rather than the exception, as it was in the studio-era, to advertise a film's director as a main attraction.
On first blush, this recognizable account of film authorship being a product of a series of critical interventions is logical. Although not inaccurate per se, it contains omissions about why the notion gained a foothold in Hollywood and film culture. Chief among these exclusions is an acknowledgment of how shifting industrial contexts in Hollywood buttressed the strategic decision to promote directors as auteurs. In the midst of a serious economic crisis, stemming from a steep decline in movie attendance in the 1960s because of a confluence of circumstances, including the impact of the Paramount Decree, the rise of television, mass suburbanization, and so on, Hollywood scrambled to find a remedy. The weakened financial status of the studios leading up to this time made them ripe for takeovers. A wave of mergers and acquisitions thus began at the moment that laid the foundation for the formation of the media conglomerates that now dominate the industry, catalyzing a revised approach to handling creative talent.
This structural change ushered in new business strategies. Gone was a time when Hollywood could rely on vertical integration and its associated (now illegal) practices, like block booking, to ensure the profitability of its films. Accordingly, it no longer made financial sense for the industry to maintain the overhead that it did during the studio era when it relied on multi-picture contracts with creative personnel to control content and costs. To mitigate risk, Hollywood eventually turned to the package-unit mode of production that now characterizes the industry in which creative personnel on individual films are assembled on an ad-hoc basis. This strategy helped resuscitate Hollywood's bottomline in the late 1960s and early 1970s. The production of a series of films, directed primarily by young, film-school educated, directors and targeted at an enormous youth audience, which generated substantial revenues in relation to their small budgets became an ideal, albeit temporary, solution to the industry's financial woes. For the first time in Hollywood history, it was economically advantageous to amplify, rather than conceal, the contributions of the director. Whereas the studios once largely obfuscated directors' impact to curtail their bargaining power in long-term contract negotiations, it now served the industry to promote filmmakers as key players in the package. Directors became valuable commodities for Hollywood, provided that they generated revenues to justify the increasingly exorbitant salaries that they subsequently demanded.
As Timothy Corrigan documents, this industrial shift changed the roles of the director in significant ways. Clearly, it affords executives with a convenient scapegoat when high profile films fall short of financial expectations. Such an impetus was exemplified by what happened to notable Hollywood Renaissance auteurs, such as Francis Ford Coppola and Michael Cimino, held liable by the industry after the artistic freedom they were granted led to purported self-indulgences that resulted in their films going substantially over budget and schedule or disappointing at the box office. It also allows Hollywood to market directors like other presold properties, including stars and genre. Since the classical mode of narration calcified in the late 1910s, the industry has used presold properties, most routinely familiar source material, stars, and genre, to differentiate a largely uniform product line. Although few filmmakers were leveraged to distinguish films during the studio-era, directors are now typically constructed as auteurs by the industry to market difference. As Corrigan writes, "institutional and commercial agencies now work, whatever the filmmaker's intention, to define auteurism as publicity or advertisement or as the dispersal of the control of the auteur into the total flow of television monitors" (50). This "commerce of auteurism," as Corrigan terms it, is Hollywood's modus operandi at a time in which it is economically expedient to transform directors into celebrities often on par with the star performers traditionally featured in marketing campaigns.
Even though it may seem that romantic notions of the uncompromising, artistic auteur would contradict the industry's motives in light of the fall of the Hollywood Renaissance and the rise of the blockbuster, promoting directors in this way, whatever their actual roles, still benefits Hollywood. In New Hollywood Cinema, Geoff King persuasively argues that there are overlooked industrial continuities that closely link together the Hollywood Renaissance and blockbuster-era, which began in earnest in the mid-1970s and has been the dominant mode of production since the 1980s. Hence, one of the reasons why the ostensibly dissimilar periods are often both referred to as representing the same transition from "Old Hollywood" to "New Hollywood." For King, the fall of the studio system and its replacement with the package-unit is the defining feature connecting the two seemingly dissimilar production trends. In both situations, the industry's primary agenda has been to employ methods to replicate the guaranteed return on investment from the studio-era by devising strategies that ensure profitability. After the end of vertical integration, Hollywood wanted a way to rig economic outcomes, like it did when block booking was standard practice. The temporary solution of the Hollywood Renaissance, then, was doomed to be short-lived, as the baby boom was destined to age and the industry always had its eye on recapturing the mass market, anyway. As media consolidation and the dominance of television subsequently accelerated, blockbuster production, saturation releases, astronomically growing marketing budgets, and the requisite economic logic of synergy became the norm because these tactics are well-suited for the new conglomerate landscape and present significant barriers to entry. Under this model, advertising directors as superstar auteurs serves as another presold property to bolster the big opening weekends necessary to pay off the high-interest loans that underscore perpetually escalating negative costs of the blockbuster model quickly, propel films to generate additional revenue streams in increasingly lucrative ancillary markets, and keep independent players out of the competition.
Of course, Hollywood made spectacle-laden, event films with high production values confected to attract mass audiences to theaters before the dawning of the blockbuster-era. What distinguishes the blockbuster age, though, is that this mode of production is now the central focus of the industry and not the exception. For Warren Buckland, this shift has instigated important changes in how the industry conceives of and packages film authorship. As with King, Buckland, in his study of Steven Spielberg and the blockbuster film, theorizes that the move to the package-unit mode of production is the key to comprehending the new roles of the director. Whereas critics traditionally evaluated auteurs retroactively based on how their artistic signatures were discernable in the films themselves during the studio age and its corresponding producer-unit mode of production, many scholars have rightly determined that authorship is now more accurately linked to how a director's image is constructed, marketed, and managed. Specifically, Buckland posits that the contemporary director "needs to become a power broker, a talent worker (which involves mastery of management skills), and must also create a brand image in order to gain positional advantage over the competition" (24). This is neither to say that textual properties, like thematic preoccupations and stylistic tendencies, are irrelevant in the blockbuster age nor was brand management of directors nonexistent in the studio-era. As scholars, like Robert Kapsis, have shown, there were anomalous directors, such as Alfred Hitchcock, who became marketable commodities before the fall of the studio system because of novel self-promotional efforts across media. Likewise, contemporary auteurs' recognizable textual practices are frequently mobilized, in promotional materials and in the films themselves, to reify their authorial status. The industry's ability to create and maintain a director's distinct brand has become vital to deploying authorship as a presold property that reliably boosts the theatrical take, creates barriers to entry, and sustains strong performance in the aftermarket.
The phenomenon of brand identity management in the blockbuster age is crucial for examining the vastly different fates of Shyamalan and Nolan in relation to the misdirection film. Up until the industry orchestrated a drastic change in Shyamalan's image in 2010 with the The Last Airbender, the director's films, from The Sixth Sense to The Happening, strongly exhibit textual continuities that coincide with traditional notions for assessing studio age authorship. By all of these accounts, even Shyamalan's pre-Airbender disappointments bear the hallmarks of an auteur with artistic integrity, rooted in consistent thematic preoccupations and technical acumen that results in distinctly recurrent stylistic tendencies. Yet, as the eventual decision to reconstruct Shyamalan's image shows, this uniformity across his output was not enough to maintain a brand that remained profitable. I contend, then, that the drastic transformation of his reputation is mostly attributable to the misguided promotional tactics that preceded the shift. Perhaps even more interestingly along these lines, is the heretofore failed reconstruction model devised for and by Shyamalan, which is modeled on Nolan's subsequent success in the misdirection film genre. In particular, the strategic decisions to have Shyamalan helm the first installment of a proposed franchise (The Last Airbender) and a big budget, sci-fi vehicle for Will Smith (After Earth) were attempts to disassociate the director from the misdirection film by having him prove his mettle in other, even more economically viable, blockbuster-era production trends. Such a strategy aligns with Nolan's ascent to superstardom, which is tied to the effective promotion of his aptitude in both the misdirection film genre and blockbuster production. In particular, after experiencing moderate financial success with melding the two in The Prestige, Nolan's subsequent genre hybrid, Inception, became the most economically successful misdirection film ever made. The following analysis of the authorial discourses surrounding and running through the works of these two filmmakers thus illustrates how the industry markets auteurs in relation to genre to maximize profits in the blockbuster age.
## A Surprising Authorial Twist: The Rise and Fall of M. Night Shyamalan
After his first two films, Praying with Anger (1992) and Wide Awake (1998), barely received theatrical distribution and netted next to nothing at the box office, the incredibly positive response to The Sixth Sense primed Shyamalan for authorial superstardom. The film received six Academy Award nominations (two of which, Best Director and Best Original Screenplay, were for Shyamalan himself), critical praise, became a cultural phenomenon, and, most importantly, was a box-office sensation. Although it might have been unreasonable to expect him to replicate those results, Unbreakable received generally good reviews, was widely interpreted as a continuation of the director's authorial vision, and turned a small profit at the domestic box office. Signs similarly garnered mostly favorable reviews and is Shyamalan's second highest grossing film to date, earning $228 in domestic theaters on its $72 million budget (imdb.com). The Village continued Shyamalan's profitable streak, bringing in $114 million domestically on a $60 million budget (imdb.com). Yet, it was his first film since before The Sixth Sense to be met with substantial critical scorn, precipitating a string of poor reviews on subsequent films and marking an important turning point in the director's career.
It is difficult to pinpoint why The Village had this impact on his reputation because it was profitable and not Shyamalan's first film to be branded in highly authorial terms. His name was previously sold as one of the primary attractions for both Unbreakable and Signs. As with Unbreakable, Shyamalan's name was featured in virtually all promotional materials for Signs. In contrast to Unbreakable, though, which evoked The Sixth Sense in conjunction with Shyamalan when it was uttered in the marketing campaign, the director had become familiar enough to advertise his second post-Sixth Sense film simply as "M. Night Shyamalan's Signs" in marketing materials, like the theatrical movie poster. The decision to rely on the exact same strategy with "M. Night Shyamalan's The Village," then, does not explain why that film experienced such a different critical fate than its immediate predecessors and initiated a sharp decline in the director's authorial standing. Whereas the favorable reception of Signs indicates that Disney's tactics to promote the director were well conceived, the critical backlash to The Village suggests that something elicited a very different reaction in response to his third misdirection film.
One change in the marketing of The Village is the amount of exposure the director himself received in comparison to his previous films. In addition to continuing to feature Shyamalan's name in promotional efforts, Disney capitalized on its television holdings to create a buzz for The Village. Just prior to the film's release in 2004, The Sixth Sense and Unbreakable were programmed on Disney subsidiary ABC in primetime specials. As Kim Owczarski summarizes, the network branded the event as "2 Days of Night," featuring "the presentation of The Sixth Sense on April 26 and the network television premiere of Unbreakable on May 3. Shyamalan hosted both airings by discussing his latest project and showing trailers and behind-the-scenes footage of The Village" (132). Clearly, Disney believed in the director's name and his now familiar persona to sell theatrical tickets as well as further enhance his authorial reputation.
These customary, synergistic marketing tactics were complemented by another, less routine promotional maneuver. Syfy television network (then Sci Fi), at the time a subsidiary of NBC Universal (now Comcast), aired a mockumentary entitled The Buried Secret of M. Night Shyamalan (2004) in the days leading up to the release of The Village. Importantly, the film was initially not packaged as a mockumentary. Although accounts of exactly what transpired during the making and marketing of the film differ, Shyamalan distanced himself from the mockumentary immediately before and after it appeared. Regardless of whether or not this was an elaborate publicity stunt, it had negative ramifications for Shyamalan's reputation. In the unlikely event that Shyamalan's trepidation was actually legitimate, there were justifiable reasons for his concern. The mockumentary portrays him in Orientalist fashion, as a mystical figure with strange powers, going as far as to suggest that his own biographical details inspired the supernatural elements of his three films released immediately before The Village. In particular, the film depicts Shyamalan's childhood as having parallels with primary characters in The Sixth Sense and Unbreakable. According to the film, the director, like Unbreakable's David Dunn (Bruce Willis), miraculously survived a drowning accident as a child (revealing that, as with Dunn and the aliens in Signs, water is his kryptonite) and developed the supernatural ability to communicate with the dead (as in The Sixth Sense), as a result. All of this would perhaps not be so bizarre if the mockumentary and the story surrounding its production had been handled differently. As part of its attempt to position the film as nonfiction, rather than as a mockumentary, Syfy "leaked" reports to the Associated Press of Shyamalan's purported desire to disassociate himself from it. Ultimately, Syfy's public relations moves crumbled right before the airing, as network representatives acknowledged that the film and its marketing campaign were a hoax, foreshadowing the kind of trouble that was to come for Shyamalan's branded identity after a subsequent series of similarly miscalculated promotional ploys (Taylor).
Of course, a little-seen mockumentary run on a niche cable channel is not primarily responsible for Shyamalan's rapidly declining marketability. It is indicative, however, of the kinds of ill-advised tactics that contributed to the radical transformation of his reputation. The strong box-office returns from The Village suggest that Disney's decision to present it explicitly as an "M. Night Shyamalan" film in virtually all promotional materials did little to hamper its initial box-office returns. If anything, its $50 million opening weekend box-office take suggests that the marketing campaign was initially effective, providing Disney with the quick revenues necessary to reimburse creditors and ensure post-theatrical success (imdb.com). Like many films in the blockbuster-era, in other words, its strong opening weekend performance was mostly attributable to successful advertising. The Village struggled subsequently, however, earning only an additional $16 million through its second weekend and just another $7 million at the end of its third weekend in theaters (imdb.com). Such an abrupt drop off at the box office often indicates that the expectations generated by the marketing campaign were not met by the film itself.
The theatrical trailer for The Village explicitly positions it as a product of Shyamalan's distinct imagination and implicitly connects the film to his earlier works. Although the primary actors are featured in the trailer's clips from the film itself, their names are never identified in the preview. Instead, the only name mentioned is the director's, as the trailer includes title cards that read "from writer and director M. Night Shyamalan" and "M. Night Shyamalan's The Village." Shyamalan's previous successes are also never explicitly uttered in the preview. However, the trailer evokes his association with both The Sixth Sense and Signs by affiliating it with the horror genre and alerting audiences that suspense will be generated by a mysterious, invading force that threatens primary characters. In contrast to most misdirection film marketing and Unbreakable's promotion, though, it never alludes to the changeover's presence. The decision to package it blatantly as a horror film aligns with the typical advertising strategies for the misdirection film by ostensibly packaging it in familiar genres that do not necessarily encourage retrospective reinterpretations of narrative information. Yet, the choice not to reference its status as a misdirection film even obliquely contradicts the genre's standard promotional tactics. This atypical strategy suggests that Shyamalan had become so closely linked to the genre by the time that The Village was released that marketing executives believed his name alone would be more than enough to signal to audiences that it is a misdirection film.
Rapidly declining box-office performance is not only a product of unmet expectations in relation to presold properties like genre. In addition to poor word of mouth that may have stemmed from the film failing to satisfy horror fans because of its PG-13 inspired lack of lurid material or it not being marketed as a misdirection film, negative reviews also strongly contributed to its steeply falling financial trajectory. Overall, reviews of The Village were mixed, but many who attacked the film were particularly venomous. Roger Ebert, for example, begins his review by describing the film as "a colossal miscalculation, a movie based on a premise that cannot support it, a premise so transparent it would be laughable were the movie not so deadly solemn." The Washington Post's Stephen Hunter notes that Shyamalan is "riding a one-trick pony and that poor pony is nearly dead" (C1). The one-trick pony that Hunter cites, of course, is Shyamalan's penchant for the changeover. This critique is not entirely accurate because Signs, Shyamalan's first film since before The Sixth Sense not to contain a changeover, had already interrupted the director's successive misdirection releases. Its unanticipated ending, in which the primary characters thwart an alien invasion, is affiliated with the misdirection film because it reveals how a divine plan imbues seemingly inconsequential events those characters previously experienced with great narrative meaning; however, although Signs' ending does inspire audiences to reconsider the narrative functionality of some information, it does not encourage viewers to reinterpret the significance of almost everything that precedes it in the way that the changeovers do in The Sixth Sense and Unbreakable. Although Signs represented a slight departure for Shyamalan, his decision to return to using a pure changeover in The Village was not well-received by critics who interpreted it as an ineffective gimmick from a director in need of fresh material. Reviews of The Village consistently show that critics were tiring of the changeover formula. Hunter epitomizes this by claiming that "Shyamalan really has to do some reconsidering. His surprises don't work anymore because we expect them. It was his obscurity, his lack of reputation that made the ending of The Sixth Sense so unforgettably jolting" (C1).
Yet, the consistent presence of significant narrative surprise is one aspect that should have helped rather than hindered Shyamalan's reputation as an auteur, according to traditional approaches. Indeed, it was employed as a quintessential and successful authorial calling card for Shyamalan on Unbreakable because the director's connection to the changeover had already became recognizable enough to sell his films almost on its own. I say almost because, as with Unbreakable, Shyamalan's name shared billing with Signs' star, Mel Gibson, in the marketing campaign. For The Village, however, virtually all promotional materials followed the logic of the trailer by featuring Shyamalan's name as the sole draw. This was not because the film was devoid of marketable actors, as it starred Sigourney Weaver, William Hurt, Joaquin Phoenix, and Adrien Brody, fresh off his Best Actor Oscar win for The Pianist (2002). None of these actors were cast as the film's lead, though. Instead, relative newcomer, Bryce Dallas Howard, Ron Howard's daughter, played the protagonist, Ivy Walker. Although Howard's first leading Hollywood role received heavy media coverage, her name and image were absent in most of the film's promotions. This decision was connected to fears about her bankability as an unknown actress in spite of her marketable genealogy. Rather than gamble on Howard or promote the ensemble, Disney exclusively foreground Shyamalan's name in The Village's marketing campaign, contributing to the direct backlash against the filmmaker after its negative reception.
Compounding matters was Shyamalan's choice to cast himself as the character who exposed the changeover to audiences in The Village. Shyamalan previously used the cameo in Hitchcockian fashion partly as a self-promotional maneuver to familiarize audiences with his image. He also deployed the device in cleverly self-reflexive ways to comment on his outsider status as a South Asian living in white, American culture and working in Hollywood, exemplified by his brief turns as an incompetent physician in The Sixth Sense and an alleged drug dealer in Unbreakable. These unlikable characters helped endear Shyamalan to audiences by making it seem as though he did not take himself too seriously and was unafraid to comment on the prejudice that South Asian Americans face, albeit tangentially. Shyamalan continued to develop his onscreen persona with Signs by casting himself as Ray Reddy, the man revealed to be responsible for the tragic and accidental death of the protagonist, Graham Hess's (Mel Gibson) wife, which leads to the existential crisis that provokes the film's classical quest narrative for the widower to regain his faith in his profession as a reverend. Although Shyamalan again plays an unlikable individual in Signs, his character is different than its immediate predecessors. Most significantly, his part in Signs is more than just a peripheral cameo because of Reddy's key narrative role. In particular, Reddy's apology to Hess for his part in his wife's accidental death and his hypothesis that the aliens are afraid of water help to tie all of the ostensibly random occurrences together at the end. This narratively significant role in a hit film helped make Shyamalan more recognizable than ever before, giving Disney the confidence to cultivate his growing authorial persona further to promote his next film. Such success not only resulted in more prominent use of his name in advertisements. It also encouraged Shyamalan to cast himself in similarly narratively significant roles in The Village and Lady in the Water.
In The Village, Shyamalan reverts to a smaller cameo because his brief appearance does not occur until near the end of the film, during its changeover sequence. Moreover, the director strategically chose not to feature his image prominently in front of the camera. Specifically, he plays an unfriendly park ranger who obliquely explains to a security guard surreptitiously helping Ivy, and, by extension to the viewer, what has really been happening. Their discussion reveals that Ivy and the members of her community actually exist in contemporary American society, but have just been living in secret in her family's Walker Wildlife Preserve, shielding them from outsiders. This changeover forces the spectator to realize that her ostensible pre-modern reality is actually a fabrication. It shows that the alternative community that her father, Edward Walker (William Hurt), and the other "elders" seeking a utopic refuge from violent American society created was deliberately constructed to shelter their offspring, explaining why the founders engage in farcical rituals to scare their children about fictional bogeymen in the surrounding woods.
The changeover, then, can be read as more than just a narrative gimmick because of its possible links to similar scare tactics enacted by the George W. Bush administration in the aftermath of 9/11. For some critics and fans, the changeover inspired a retrospective reinterpretation that connected it to contemporary U.S. culture in precisely this way, a thematic concern that Shyamalan began exploring in Signs, his first post-9/11 film, which also presents paranoid reactions to the threat of shadowy, invading others. Chicago Reader critic, Ben Sachs, for instance, posits that The Village's twist ending "recasts it as allegory" because the director described it as "his 'post-9/11 film,' " making the changeover "more provocative if read as an indirect critique of the Bush administration's war on terror." The changeover is thus significant not only for being precipitated by Shyamalan's character, but also because it moves the film into possible master key territory by potentially imbuing what precedes it with symbolic meaning related to the cultural context in which it was produced and received.
Shyamalan's formal choices during the changeover also have implications for his authorial brand because they suggest that he wanted to return to a more traditional cameo in The Village. To wit, he is shot from behind the head, reading The Philadelphia Inquirer, making him identifiable only through his voice, a fleeting reflection in a glass door, and his authorial links to his hometown. Although Shyamalan does not occupy as much screen time as he does in Signs, his role is even more narratively important than it is in that film, and especially than it is in The Sixth Sense and Unbreakable, in which his appearances serve a primary narrative function of comic relief for those in the know. This overlap between Shyamalan the filmmaker and Shyamalan the onscreen persona strongly underscored his connection to the changeover, which marketers for The Village hoped would be evoked exclusively by his name. In this instance, he was actually the one delivering the device to which he had become inextricably linked. Such a conflation between Shyamalan the character and Shyamalan the directorial master of the changeover provoked a shift in his reputation when the device was poorly received.
Mixed reviews for The Village were not enough, though, to prevent Buena Vista (a Disney subsidiary) from continuing to promote Shyamalan's association with the misdirection film in the aftermarket. The blurb on the back cover of the DVD version, for instance, links the director to other masters of the misdirection film by claiming that the film "ranks with the best of Hitchcock." It also advertises its array of special features, which includes clips starring "M. Night Shyamalan—That Reveal Clues To The Movie's Twists and Turns." Obviously, good overall box-office returns convinced executives that Shyamalan's authorial standing had been preserved despite some poor reviews since audiences seemed to respond favorably to the theatrical marketing campaign. In the end, The Village performed well globally, garnering an additional $144 million internationally in theaters, exceeding its respectable domestic take (boxofficemojo.com). The fact that Shyamalan's name was again used as a main attraction for his next film, Lady in the Water, is a testament to the confidence that remained in his authorial brand despite the often unfavorable critical reception of The Village.
If The Village signaled potential trouble for the Shyamalan brand, Lady in the Water unequivocally inflicted serious damage from which the director has yet to recover. As with The Village, Lady in the Water was marketed by positioning Shyamalan as the film's biggest star, though it represented a return to having the director share billing with other creative personnel. Although the theatrical trailer copies the tactics from The Village by explicitly mentioning Shayamalan's name only, print ads for Lady in the Water generally positioned the director as being part of a package with the film's lead performers. The theatrical movie poster, for instance, prominently depicts Bryce Dallas Howard's face, yet her and her costar Paul Giamatti's names are situated below the director's, as his name is featured atop of the poster in a larger font than the actors, advertising it as "A Film by M. Night Shyamalan." Howard's and Giamatti's names then appear below the image of Howard and above the film's title and tagline, which reads "Time is running out for a happy ending." This prototypical promotional artifact illustrates that there were minor changes in the marketing strategy after The Village because the director's penchant for unexpected endings and the stars were again more explicitly packaged as major selling points.
The return to the previously successful marketing campaign strategy for Shyamalan's films on Lady in the Water concealed a sordid production history that has become contemporary Hollywood legend. Shyamalan claims he initially got the idea for Lady in the Water based on a bedtime fairytale that he created for his daughters about what happens in their backyard pool, which he also turned into a children's book of the same name that was published in conjunction with the film's theatrical release (Sampson). Such a project seems ideal for Disney's family-oriented and synergistic logic. Yet, Lady in the Water was the director's first film since before The Sixth Sense not to be distributed by the media conglomerate. Disney had initially agreed to back the project; however, it was ultimately distributed by Warner Bros. after Shyamalan had a disagreement with executive, Nina Jacobson, his longtime advocate at the media conglomerate.
The details of the Shyamalan and Jacobson kerfuffle were chronicled in another unwise publicity ploy leading up to Lady in the Water's release that resembles Syfy's mockumentary for The Village. Sports Illustrated columnist Michael Bamberger penned an obsequious biography entitled, The Man Who Heard Voices: Or, How M. Night Shyamalan Risked His Career on a Fairy Tale, published just days before the film hit domestic theaters. In the book, Bamberger presents the circumstances that led to the film eventually being released by Warner Bros. instead of Disney. Purportedly out of concern for the script's confidentiality and because he insists on working out of Philadelphia instead of Los Angeles, Shyamalan had a personal assistant hand-deliver it to Jacobson in Hollywood. Shyamalan reportedly got angry when he learned that she was unable to read it immediately because of family obligations and then disputed her revision suggestions after she did review it. Even though Disney subsequently agreed to let him proceed with the production without making all of the edits, the director was upset enough to shop around the script. In light of his outstanding track record at the box office and eager to house the then-valuable Shyamalan brand, Warner Bros. agreed to back the film. Stunningly, Disney was so concerned about the situation that they subsequently fired Jacobson, even though her reservations would be confirmed by the film's box-office disappointment (Whipp U6).
One of Jacobson's biggest problems with the script was Shyamalan's ego-driven decision to cast himself in a more prominent role than he had in any of his films since before The Sixth Sense (Whipp U6). In Lady in the Water, Shyamalan plays Vick Ran, one of the apartment tenants of The Cove, a Philadelphia-area apartment complex, where a mystical creature named Story (Bryce Dallas Howard) magically appears. In turn, she relies on Vick and the other apartment denizens, most notably the complex's superintendent, Cleveland Heep (Paul Giamatti), to help her return to her native land. Like most of the apartment dwellers, Vick is initially reluctant to help. In Signs fashion, however, the unexpected ending reveals that he and the other Cove residents all live there for a fateful reason because they actually have unrealized special talents that they each need to deploy for her safe return. Lady in the Water, therefore, was Shyamalan's second film since The Sixth Sense not to contain a changeover because its big twists instead function like the ones in Signs, rendering allusions to the film's status as a misdirection film in promotional materials misleading. As it relates to Vick's hidden talent, specifically, he is portrayed as an aspiring author suffering from writer's block, whose book, Story reveals, needs to be written for her to return because it will eventually save the world, though, she claims he will be assassinated because of its ideas. Also disconcerting to Jacobson was Shyamalan's decision to depict a film critic, Harry Farber (Bob Balaban), as an idiot incapable of drawing correct interpretations about the world around him, which puts the film's other primary characters in grave danger (Whipp U6). As a consequence of his ineptitude, Farber is horrifically killed. Clearly, Shyamalan was not short on confidence after some critics dinged him for The Village. The decisions to portray Faber as he did and cast himself as a heroic martyr, who prevents the apocalypse with his unappreciated brilliance, indicate that the director hoped to highlight the inability of critics to understand his previous films fully.
Unsurprisingly, Lady in the Water was slammed by critics upon its release. The sometimes brutal disdain Shyamalan received for The Village pales in comparison to the almost universal disgust from reviewers about Lady in the Water. More than anything else, it was the director's decision to cast himself as the savior that made critics attack Shyamalan. Michael Booth of The Denver Post typifies this tendency by writing that "Shyamalan has sucked way too hard on the tailpipe of his fearsome publicity machine" and is "Blinded by his own aura." Slate's Dana Stevens is even more vicious, as she claims that "Lady in the Water marks Shyamalan's official leap off the deep end" and that the director "appears to have completely lost his marbles." Michael Atkinson of The Village Voice similarly wonders if Shyamalan "has lost his mind" and then rails against the director's narrative signature by noting that "It's beginning to chafe as a formula... Authorial vision is a non-issue, in the face of so much rootless, repetitive mumbo jumbo." For Mick LaSalle of the San Francisco Chronicle, the film was indisputably Shyamalan's alone, as he asserts that it "has the strengths and weaknesses of a one-man show," but ultimately notes that Shyamalan is misguided for making artistic originality a central theme "in a movie that's a dead ringer for [his] last two efforts." Finally, James Berardinelli of Reelviews declares that Lady in the Water is "the biggest misfire of M. Night Shyamalan's career, including his pre-Sixth Sense movies" and warns "For those who thought Shyamalan was stretching it in The Village, you ain't seen nothing yet." Although Berardinelli apparently ignores the big twists that unite all the characters by claiming that the film only contains "several minor misdirections," he does not believe that such a change will prevent "whatever luster still remains to Shyamalan's reputation" from being completely removed by Lady in the Water.
The overwhelmingly negative reaction from critics and box-office disappointment did not stop Fox from subsequently backing Shyamalan and packaging his next film, The Happening, in auteurist terms. Like his other post-Sixth Sense releases up until then, The Happening was marketed as a product of the director's distinct artistic vision. The theatrical trailer, for instance, contains a title card that explicitly marks it as "An M. Night Shyamalan Film." It subsequently presents title cards, though, that suggest marketers were growing weary of Shyamalan's bankability because it directly references his highest revenue generating films by identifying him as "Writer and Director of The Sixth Sense" and "Signs." Additionally, at the end of the trailer, star Mark Wahlberg's name is featured in a title card even though he had already been the centerpiece of the assorted film clips comprising the preview. Such decisions reveal that industry executives' faith in Shyamalan's ability to sell the film on his brand alone had waned because the preview for The Happening explicitly mentioned the name of the film's primary star for the first time since the trailer for Signs as well as linked Shyamalan directly to his biggest box-office hits, The Sixth Sense and Signs, instead of assuming his identity alone sufficed.
There was an even larger shift in the way The Happening was promoted differently than Shyamalan's preceding films. Likely in response to perceived issues with mismatched generic expectations, it was blatantly advertised as Shyamalan's first R-rated film in some promotional materials. Although the trailer does not accentuate this aspect explicitly, it is peppered with horrific images of death, such as people jumping off buildings, that are not only frightening because of their visceral impact, but also because they evoke footage of the World Trade Center suicides recorded during the 9/11 attacks. The film, therefore, was still packaged in auteurist terms despite the growing reservations about Shyamalan's fading brand. For starters, even though the film is ostensibly about nature taking revenge on humanity for destroying the planet, it marks a thematic return to The Village's allegorical representation of post-9/11 U.S. culture by consistently referencing the events of that day in its imagery and narrative. It centers on unearthing the reasons for the mystifying and terrifying mass deaths that begin on an otherwise mundane Tuesday (the same day of the week as the actual 9/11 events) morning in New York City, which are initially believed to be the result of a terrorist attack, and eventually lead the film's primary, Philadelphia-based characters to flee to rural Pennsylvania, the site of the final 9/11 plane crash. It also explicitly linked Shyamalan back to the horror film, the genre to which his previous box-office smashes had been most clearly connected. As with those films, The Happening was similarly marketed as a horror film that would generate suspense from an unknown force threatening its primary characters, with the additional promise that it would meet those expectations this time because its R-rating would enable Shyamalan to produce terror in a more gruesome manner than he did in The Village.
Figure 5.1. Superimposed title from opening scenes of The Happening that detail the time and place of the horrific mass suicides that stem from a yet-to-be-determined cause.
Reviewers did not respond in ways that suggest the marketing plan was effective. For many critics, The Happening represented a nadir in Shyamalan's career from which he might not recover. In his review, Time's Richard Corliss opines that "M. Night Shyamalan has lost the touch that made The Sixth Sense a suspense classic and his standing as a young master of creepiness in the grand Hitchcock tradition. He's just 37, but his best films are so far behind him, it's as if he's forgotten how he made them work." Kyle Smith of the New York Post is just as harsh, as he bashes the film and then situates it in auteurist terms by characterizing "the oeuvre of M. Night Shyamalan since The Sixth Sense" as "stupid ending, stupid ending, stupid ending and, in a change of pace with his last film Lady in the Water, stupid all the way through." Similarly, the Newark Star-Ledger's Stephen Whitty believes that The Happening denotes the end of Shyamalan's career as a bankable director. Echoing the sentiments of others about the film's ineffective marketing, Whitty claims that "The Happening was supposed to mark a fresh start" by "being almost defiantly rated R, as if to signal Shyamalan's decision to move away from subtler PG-13 films" and because "Shyamalan had been complaining he was tired of being known as the writer with the 'twist'—indicating that this story would try something riskier." Whitty ultimately notes that had "Shyamalan been serious about getting an R-rating, then he should have pushed the material in disturbing ways; if he wanted to eschew easy 'twists' then he should have embraced chaotic, even existential turns of the plot."
As these reviews demonstrate, The Happening was received by critics as anything but a departure from form for Shyamalan. His decision to pander to his detractors by incorporating more disturbing material to get an R-rating did little to change opinions about his continued inability to produce genuine fear in his audience. Although the film depicts numerous gruesome and imaginative suicides, especially in its opening sequences, reviewers still largely agreed with critics, like Whitty, who believed that it failed to generate sufficient terror. Additionally, his strategic choice not to incorporate a twist at the end that completely resolves all causal lines of action unexpectedly, regardless of if it is a changeover or not, did nothing to lessen his now detrimental connection to the misdirection film. Yet, it is inaccurate to claim that The Happening contains an ending that aligns it with the director's previous, post-The Sixth Sense films. In particular, just after it appears all causal lines of action have been satisfactorily resolved because the threats posed to characters have been thwarted, the film cuts to an eerily prescient concluding scene in relation to the actual spread of al-Qaeda and ISIS sponsored terrorism since 9/11, as the mass suicides that mysteriously ended in the United States have suddenly started to impact people in Paris. Rather than show what occurs next in Europe or around the globe, the film abruptly cuts to the end credits, making The Happening Shyamalan's most pessimistic and ambiguous film to date. Unlike the similarly gloomy Unbreakable, which also has a threat to humanity triumph in the end, there is no epilogue in The Happening to suggest that order is ultimately restored. In contrast to most reviewer accounts, then, The Happening actually represents a significant shift for Shyamalan because it was his first film to contain a concluding twist that amplified lingering narrative uncertainty instead of retrospectively resolving all causal lines of action classically.
The Happening thus seems to signal Shyamalan's effort to begin orchestrating a transformation of his image to salvage his brand. Undoubtedly, many of his auteurist signatures remain, such as his insistence on setting the film in the Philadelphia region, featuring an unknown menace that threatens the primary characters, casting himself in an unlikable cameo role (this time Shyamalan plays a man trying to court the protagonist's wife, but viewers only hear his voice), and so on. In spite of these similarities, his decisions to present more graphic images to elicit horror and abandon his signature twist ending that inspires a classical reinterpretation are notable divergences from the director's Sixth Sense-inspired filmmaking formula. The continuities outweighed the differences, however, as critics received it as anything but an actual tangible change of direction for Shyamalan.
Figure 5.2. The Happening's conclusion, depicting the spread of the mass suicides, which have mysteriously ended in the U.S., to Paris, where they inexplicably restart.
The remarkable shift engineered for the making and marketing of Shyamalan's next film, The Last Airbender, reveals that consistent critical derision mixed with poor authorial marketing strategies and the financial disaster of Lady in the Water had tarnished Shyamalan's reputation beyond repair in the minds of executives. Shyamalan was hired by Viacom's Paramount film division to make The Last Airbender because of the media conglomerate's synergistic desire to transform its subsidiary, Nickelodeon's, successful animated children's show, Avatar: The Last Airbender (2005–2008), into a film franchise. Although the project appeared to be largely unrelated to Shyamalan's previous films on the surface, there was potential to evoke his authorial brand in the marketing campaign because of previous efforts to promote Lady in the Water as a fairytale for family audiences. Of course, that film's failure made executives reticent to associate the director with it in advertisements for The Last Airbender. His name was still deemed marketable enough, however, to feature it in promotional materials, such as the theatrical movie poster, which positions The Last Airbender as an "M. Night Shyamalan Film," albeit in small font, much less prominently than for any of his other previous films after The Sixth Sense. The film's teaser trailer similarly highlights the director's involvement, as it packaged it as being "From M. Night Shyamalan" from that preview's opening shot.
The attachment of Shyamalan's name to the film in spite of his recent failures was a calculated gamble. The Last Airbender was initially conceived of as the first installment in a trilogy that was to be helmed by Shyamalan. Executives had so much faith in the proposed franchise that Shyamalan was given an $150 million production budget for The Last Airbender, his largest to date, and an additional $130 million was reportedly spent on the film's marketing (imdb.com). Although such a risk seems misguided in hindsight in light of the film's extremely poor critical reception, it made sense in relation to recent Hollywood trends. Despite being lambasted by critics and failing to recover its production costs at the domestic box office, the film ended up grossing over $300 million internationally, revealing why executives were willing to invest so much in the project and in Shyamalan (imdb.com). In particular, the industry is especially keen on developing big-budget, franchise films with international legs, adapted from presold properties with recognizable directors attached, such as Peter Jackson's critically acclaimed, box-office smash, The Lord of the Rings trilogy. Even more notable in this regard was the incredible performance of The Dark Knight, the second film in Nolan's Batman trilogy, which garnered over $1 billion worldwide at the box office (imdb.com). Considering the exploding popularity of Nolan's brand at the time, it is logical that executives attempted to capitalize on a similar marketing approach for Shyamalan by positioning him as a director also capable of making both cerebral, challenging fare, like misdirection films, and blockbuster franchises.
Ever conscious of his authorial reputation, Shyamalan contributed to these rebranding efforts in ways that made it seem as though he was not compromising his artistic integrity by transitioning to franchise production and the blockbuster formula. In numerous promotional statements, Shyamalan reiterated that he became aware of the Nickelodeon series on which the The Last Airbender is based because his daughters were fans of the show and the fairytale aspects were attractive to him. Additionally, in an interview with The Telegraph's Philip Horne, Shyamalan expands on the source material's appeal by noting that "There are Asian themes" and that it allowed him to depict aspects of his "life that [he] hasn't talked about," such as the "need for family," the ecological disasters that humanity is precipitating, and discovering "the reasons each of us are born." Of course, this self-promotional discourse is unreliable, as the thematic concerns that he claims are unique to The Last Airbender are featured in his preceding films. Such contradictory assertions indicate a central tension that helped damage Shyamalan's brand. On one hand, the director wants to sustain his reputation as a traditional auteur. Expressing frustration with the reception of his authorial intentions, Shyamalan told Aseem Chhabra of the Mumbai Mirror that he "just doesn't get it" and must be "speaking a different language" because he makes films "with great respect, integrity, and effort." Yet, on the other hand, he also tells Chhabra that if his "name is on top" of the movies "that means they are doomed."
Shyamalan's contradictory self-awareness suggests that he is fully cognizant of how the construction of his authorial reputation contributed to his demise. In other interviews surrounding The Last Airbender, he even identifies the culprit explicitly. Shyamalan told The Philadelphia Inquirer's Stephen Rea, for example, that "It's the presentation of the movies as author-driven... Here in the United States that comes with a stigma of hubris." Repeating these opinions, the director explained to Mark Naglazas of The Western Australian that he "has been cut down to size because American audiences are still not comfortable with the idea of the auteur" (TOD 6). These assertions suggest that the rebranding efforts orchestrated by and for Shyamalan in an attempt to repackage him as a franchise filmmaker did not go far enough to disassociate the director from his earlier releases and the authorial legend that calcified in response. As a consequence of keeping his name on the promotional materials and positioning the film as affiliated thematically with the rest of his oeuvre, audiences and critics received The Last Airbender as another disappointment from a director who had lost his touch and was only able to make misdirection films. Undoubtedly, The Last Airbender was constructed to represent a substantial shift for Shyamalan, as it is his first post-Sixth Sense film not to have significant plot twists, to be adapted from a presold property, set somewhere other than the Philadelphia region, and so on. These changes, however, were not enough to counter Shyamalan's status as a director incapable of anything other than the same tired, changeover formula.
Shyamalan's belated transition to blockbuster-style production collapsed after the originally proposed The Last Airbender sequels were shelved in favor of Sony's After Earth. After Earth continued the trend of the director getting bashed by critics even though his name went unmentioned in the marketing campaign, and it represented another departure from his post-Sixth Sense filmmaking template by embracing the blockbuster. After Earth also returned him to box-office disaster for the first time since Lady in the Water, as the film only garnered $60 million domestically on its $135 million production and estimated $100 million marketing budgets (imdb.com). Shyamalan shouldered most of the blame for After Earth's failure despite production history evidence that suggests the director had less creative clout than in the past. The film's treatment was written by its star, Will Smith, and Shyamalan claims that he agreed to make the movie when the actor "called him for his birthday," they "talked about his son Jaden's acting career," and he presented the director "a 45 second version of a story that just clicked" (Hiscock). Such circumstances are a testament to just how drastic the attempts to revise Shyamalan's reputation were at the time, as his comparatively limited artistic input on the production is a far cry from the kind of original screenplay project that established his industrial cachet as a coveted writer/director. After two successive disasters, the blockbuster rebranding strategy was abandoned. Shyamalan himself noted that "while there was a lot more action" in After Earth than any of his other films, he now wants "to try to make smaller movies based on his] experiences with this one" (Hiscock). Consequently, Shyamalan's next projects were his first foray into television, the current Fox series, Wayward Pines (2015– ), and The Visit (2015), a micro-budgeted $5 million film ([imdb.com). In the end, Shyamalan has been unable to undo the marketing tactics that inextricably connected him solely to the misdirection film, and executives are still trying to find a way to untangle him from that negative association.
## Authorial Hybridity: Christopher Nolan as Indie Auteur for the Blockbuster Age
In contrast with Shyamalan, Nolan has not become exclusively associated to the misdirection film. Although this is partially attributable to his desires to stay behind the camera instead of also appearing in front of it and eschew ill-conceived publicity stunts designed to augment his authorial legend, it is mostly a consequence of production and marketing strategies better suited for contemporary industrial contexts than those linked to Shyamalan. Rather than immediately capitalize on his connection to the misdirection film after Memento, Nolan delayed his return to the genre by directing two films more aligned with New Hollywood's focus on remakes and franchises: Insomnia (2002), an adaptation of a 1997 Norwegian film of the same name and Batman Begins, the inaugural film in his superhero trilogy as well as his first film co-financed by Syncopy, the production company started by the director and his wife, Emma Thomas. These decisions have helped both inoculate Nolan from the sorts of critiques that have plagued Shyamalan and made him the most renowned director of the misdirection film genre.
Whereas Shyamalan met expectations by immediately following-up The Sixth Sense with another project that directly mobilized that film's most distinctive qualities, many reviewers were surprised by Nolan's choice to direct more mainstream-style fare after he established a reputation as a narrative innovator with Following (1998) and Memento. Time's Richard Schickel exemplifies such reception by noting that Insomnia "does not tell its story backward" because directors are "allowed one gimmick that sensational per career" and that the film's narrative unexpectedly unfolds "rather conventionally, almost ploddingly." John Powers of LA Weekly agrees and also identifies a potential motive for the shift by claiming that fans of Nolan's first two features will likely be disappointed by this "conventional thriller without any of the time-jump shenanigans that gave Memento its special kick," but sees value in "his honorable, old-Hollywood knack for making entertainments geared to an intelligent audience." These typical responses to Insomnia appear to do anything but set up Nolan for the kind of authorial brand recognition constructed by and for Shyamalan in the wake of The Sixth Sense. Instead, Insomnia positioned Nolan as a budding studio-style director, willing to shed his arthouse reputation in favor of more customary fare. Industrially, the film established him as more than an indie director by earning generally positive reviews and netting a respectable $114 million global theatrical return on its modest $46 million budget (imdb.com). More importantly, it demonstrated that Nolan was capable of helming a more traditional Hollywood product, replete with A-level talent, including Al Pacino, Robin Williams, and Hilary Swank.
Nolan's post-Memento industrial savvy is best exhibited by his shrewd decision to use Syncopy to co-produce all of his films after Insomnia. As King documents in New Hollywood Cinema, the lots of auteurs during the late 1970s and 1980s is instructive for understanding how authorship actually functions in the blockbuster age. Building on the work of scholars like Jon Lewis and Corrigan, who both detail the reasons for Coppola's spectacular decline after his tremendous success in the early 1970s, King argues that the director's demise was most attributable to poorly conceived industrial strategies in relation to New Hollywood's changing economic logic. Specifically, he juxtaposes Coppola's short-lived Zoetrope Studio, as a haven for both filmmakers bent on preserving their artistic integrity and developing alternative distribution strategies, with directors, like George Lucas and Steven Spielberg, who founded production companies more congruent with Hollywood's financial motives and standard business practices. While Coppola's studio dovetailed with traditional notions of authorship being linked to consistent thematic preoccupations, stylistic tendencies, and artistic freedom, Lucas's Industrial Light and Magic concentrated on producing technical expertise, particularly special effects, for any Hollywood film, and Spielberg's Amblin Entertainment gave him the flexibility to serve as an executive producer on projects directed by other filmmakers as well as synergistically leverage his branded identity across a conglomerate's media holdings, especially television. Such tales of artistic success and failure relate directly to the fates of Nolan and Shyamalan decades later in an industry still governed by the same blockbuster economic logic. While Shyamalan has struggled to discard his seemingly unshakable, conventionally constructed authorial reputation and has used his own production company, Blinding Edge Pictures, only to co-finance films he directed, Nolan has thrived as an auteur and as a businessman adept at producing his own films and the works of others.
Although Syncopy was founded by Nolan and his wife in 2001, the company did not serve as a producer on Insomnia. It was not until the 2005 release of Batman Begins that the company co-produced its first film. After proving himself capable of helming studio-style production, Nolan further delayed his return to the misdirection film by instead next agreeing to direct an updated installment of the once-lucrative Batman franchise that had been temporarily abandoned after the box-office and critical disaster of Batman & Robin (1997), which failed to recuperate even its $125 million production budget at the domestic box office (imdb.com). The tenuous state of the Batman films and of superhero franchises in general at the time is important to stress now that they have become the centerpiece of contemporary blockbuster filmmaking. Although the success of misdirection veteran Bryan Singer's X-Men (2000) and X2 (2003) as well as the first two installments of Sam Raimi's Spiderman franchise in 2002 and 2004 signaled the potential for recently established auteurs to transition into the highly commercialized genre, the disappointment of Ang Lee's Hulk (2003) was cause for concern in relation to attempts to revive the severely damaged Batman brand. Perhaps this partially explains why Nolan received no guarantee from Time Warner subsidiary, Legendary Pictures, which co-produced the film, to make the requisite sequels that would revive the franchise after his first attempt. In fact, when asked by New York Times reporter Dave Itzkoff if he always had planned to follow up his Batman installments with films outside of the franchise, Nolan replied that "he's only ever done one film at a time" and initially "had no thoughts of doing a sequel at all" ("A Man"). Regardless of the veracity of these claims, the spacing between the Batman films indicates that there was no agreement for him to make additional films for the franchise immediately after Batman Begins.
In spite of any reservations that may have existed about Nolan's potential fit for reviving the Batman brand, the success of Insomnia and the creation of his own production company to co-finance the project gave Legendary Pictures enough confidence that he was ready to manage a blockbuster film that might ignite a resurrection of the franchise. They agreed to help back the $150 million budget for Batman Begins. The wager worked, as the film grossed $374 million globally in theaters (imdb.com). Although this revenue generation was impressive, it is nowhere near the enormous profit margins garnered by Nolan's two subsequent sequels in the franchise. The success of the sequels is partially attributable to savvy production strategies. Rather than copy the successful The Lord of the Rings trilogy's model, in which Jackson's three installments were unprecedentedly produced all at once and released in uninterrupted succession, the two misdirection films, The Prestige and Inception, that solidified Nolan as the genre's new master were interspersed between the Batman sequels. Moreover, as opposed to the standard production tactic set by other Indiewood directors, such as Steven Soderbergh and Richard Linklater, who oscillate between higher-budgeted classical fare and cheaper films with arthouse sensibilities, Nolan has leveraged his Hollywood standing since The Prestige to finance original projects with increasing negative costs, culminating most recently in Interstellar's huge $165 million production budget (imdb.com). This is notable because it highlights Nolan's unique status as a director renowned for his ability to blend narrative complexity with mass appeal, making him not exclusively linked to the misdirection film. Indeed, the most frequent criticism of his films—their extensive expository dialogue that caters to the masses—exemplifies Nolan's efforts to capture both arthouse viewers with narrative innovation and blockbuster fans alienated by radical experimentation or a filmmaking formula that is deemed too narrow for a wide audience.
Whereas virtually all promotional materials for Insomnia, such as the trailer and movie posters, explicitly emphasized that the film was "From the Director of Memento," references to Nolan were left off of a majority of the advertisements for Batman Begins. This runs counter to New Hollywood's tendency to highlight any element of the package that can bolster the theatrical take, which is especially salient considering that Nolan's two previous films were successful at the box office and generally liked by critics. Consequently, the conspicuous absence of his name is an indication that executives aimed to preserve the director's still tenuous reputation as an arthouse innovator and were unsure that he would be able to make the transition to blockbuster filmmaking without jeopardizing his status as a marketable, indie commodity. Despite these concerns, the trailer does feature some of the thematic preoccupations and stylistic tendencies that were already evident in Nolan's films, like his penchant for setting parts of his films in snowy mountainous regions, focusing on the links between obsession and revenge, as well as his protagonists' battles with past traumas that haunt them and their pining for lost women. Most importantly, however, the preview centers on the film's convoluted narrative structure. Perhaps the most novel aspect of Nolan's first installment in the franchise was his decision to use a complex, flashback-laden narrative structure that is atypical of blockbuster production. In classical fashion, this risky choice was made compositionally motivated because Batman Begins presents the origin story of Bruce Wayne's (Christian Bale) development of his superhero alter-ego, making its constant temporal shifts, such as those referencing his childhood, subservient to narrative demands. As it relates to his authorial standing, however, it was a cunning move by Nolan to maintain the burgeoning reputation he had cultivated as a narrative experimenter capable of satisfying a mass audience that can be easily alienated by departures from classical standards.
Reviews of Batman Begins reveal that the critical establishment largely received the film as an effective mix of blockbuster and arthouse qualities, which had great potential to spawn sequels. Lisa Schwarzbaum of Entertainment Weekly, for instance, writes that the film, "directed by indie-oriented storyteller Christopher Nolan (Memento) is a triumph—a confidently, original, engrossing interpretation... that announces, from the get-go, someone who knows what he is doing is running the show, and he's modestly unafraid to do something new." Schwarzbaum then ends the review by proclaiming that the world the director created "is a vertiginous time warp where only a risk-taking artist can navigate. Nolan ought to get back there soon and tell us what happens next." LaSalle concurs, noting that "the film adopts an elegant narrative strategy," which gives him confidence that even though "now that Batman has begun, the Batman movies will never end, at least for another 10 years. But maybe this time around they won't get so awful." As these reviews highlight, the film was discussed by most critics in authorial terms despite the fact that Nolan's name was left out of the marketing campaign. Although the promotional tactics might have been designed to protect Nolan's brand in case of failure, such a response helped to augment his budding reputation as much more than an indie experimenter and began to fuel anticipation for the director to make sequels in the franchise.
Rather than immediately work on the next Batman film after pleasing critics and audiences of Batman Begins with its complex narrative atypical of the blockbuster formula, Nolan returned to the misdirection film next with The Prestige. The director leveraged his consecutive post-Memento successes to convince Warner Bros. and Disney subsidiary, Touchstone Pictures, to co-finance The Prestige with Syncopy. Just a few years after the release of Memento, then, which had to be produced and distributed theatrically independent of Hollywood, the majors were now willing to back a project that directly evoked Nolan's breakthrough hit. Although the film's $40 million price tag is modest in relation to other contemporaneous Hollywood films, it dwarfed Memento's meager $5 million budget (imdb.com). Industry executives now had enough faith in the director to co-support a film with high potential to turn off audiences. Indeed, as The Prestige's sole theatrical tagline—"Are you watching closely?"—suggests, marketers were even willing to package the film's misdirection narrative explicitly as its defining and most spectacular element (imdb.com).
In contrast to the tagline, a majority of promotional materials for The Prestige accentuates its narrative structure as a primary, but not the exclusive, attraction. Instead of advertising Nolan as only connected to the misdirection film, the preview identifies him as "The Director of Memento and Batman Begins," firmly situating him in both the narratively complex, indie and blockbuster traditions. In addition, the trailer does not present Nolan as the film's only, or most prominent, star, as the names of its headlining performers—Christian Bale, Hugh Jackman, Michael Caine, and Scarlett Johansson—are all explicitly highlighted with individual title cards. The preview also markets generic hybridity by advertising it as a period drama, science fiction film, misdirection film, thriller, mystery, and so on. The clips from the film itself in the trailer additionally stress its link to the blockbuster model, as its most special-effects-laden facet—the CGI-enhanced, electricity generating, cloning machine—is heavily featured. Yet, it balances that spectacular element with arthouse attributes by emphasizing its narrative complexity because it contains key lines of dialogue that strongly reference the presence of the story's twists, secrets, and surprises. As scholars like Rick Altman and Steve Neale show, Hollywood's penchant for promoting films as constituents of multiple genres has always been the industry standard because it amplifies opportunities to capture a mass audience. King agrees that the industry's emphasis on genre hybridity is not new, but he grants that promoting genre mixing has become more pronounced for New Hollywood films. Like all other elements of the package, genre is an important component of what Richard Maltby calls "the commercial aesthetic," a kitchen-sink approach that Hollywood deploys to mitigate financial risk, which has only escalated since the fall of the studio system (14). Accordingly, genre hybridity, like authorship and stardom, is a significant and reliable presold property used to allay economic uncertainty in New Hollywood. Consequently, it is logical that the industry would try to position authorship similarly by constructing Nolan as a hybrid auteur, equally adept at blockbuster and indie-style filmmaking.
The foregrounding of The Prestige's particular narrative structure and its corresponding authorial mastermind did not stop at its theatrical marketing campaign. Nolan incorporated a number of overt diegetic references to its impending changeover to signal its status as a misdirection film and announce an intellectual competition with the audience from the film's opening frames. Like Shyamalan, who augmented his authorial reputation by alluding to his relationship with the misdirection film in Unbreakable's dialogue, Nolan used The Prestige's script as a means to cement his status as a preeminent maker of misdirection films. Immediately after the opening credits roll, magician, Alfred Borden (Christian Bale), reiterates in voiceover the tagline by asking viewers if they are "watching closely?" The film then cuts to a scene that depicts John Cutter (Michael Caine) performing a magic trick, as the stage engineer describes, in voiceover, how illusions work. His narration, parts of which are also featured in the trailer, first informs viewers that magicians commence their acts by requesting that the audience inspect props that appear to be mundane. He goes on to explain that the illusionist then "takes the ordinary something and makes it do something extraordinary." Subsequently, he claims that although audiences try to guess the secrets, they do not see through the ruse because they are "not really looking, don't want to know," and "want to be fooled." The film's opening blatantly warns viewers that they will be encouraged to draw incorrect conclusions by being lulled into relying on habitual forms of comprehension. However, it also alerts spectators that something remarkable will happen to violate preliminary interpretations. In sum, the film's first few minutes set expectations that the fun will derive from a spectacular narrative and the eventual exposure of an unforeseen revelation that drastically alters initial readings of narrative meaning.
The Prestige's metanarrative commentary, then, cleverly references how it will deceive and thrill viewers by both depending on and departing from Hollywood conventions. On one hand, the film is highly reliant on classical narrative and formal techniques to trick spectators into arriving at false causal suppositions about the significance of narrative information. On the other hand, its changeover reveals that an alternative explanation for narrative meaning has existed beneath the surface all along. At first, it appears that the film's narrative centers on how illusionist Robert Angier (Hugh Jackman) exacts revenge on Borden, whom he holds responsible for his wife's tragic on-stage death during their days as magicians' assistants. It focuses on Angier's efforts to develop his own version of "the transported man," his rival's most famous magic trick, in order to use it to frame Borden for his murder. After the opening voiceovers conclude, the narrative begins in media res, presenting scenes that appear to depict Angier's drowning and Borden's subsequent death sentencing for killing his rival. As a result, viewers expect that the ensuing flashbacks will showcase how Angier bested his nemesis. Up until the changeover, this is exactly what happens. The film presents the struggle between the two men, as they plot to ruin each other. Initially, it seems that Angier wins because he develops a version of the trick that is more popular than his adversary's is and destroys Borden's life in the process.
The changeover, though, unexpectedly reveals that Borden may actually get the better of Angier. Viewers shockingly discover that Borden's stunt only worked because the magician successfully kept its secret from everyone, including his nemesis. The sequence exposes the fact that Fallon, the magician's purported trusty stage engineer, is really a double, in disguise, who performed the trick in tandem with his doppelgänger. That is, although Borden convinced even his own family that he was just one person, he actually had a double, which he claims is a twin brother, with whom he took turns playing different roles on- and off-stage to give the appearance that the stunt was done by a single man. This is the reason that he was unable to maintain a healthy relationship with his closest confidants, including his wife, who commits suicide as a result of their troubled marriage. Even though the film hammers viewers over the head about the presence of the revelation from the beginning, then, it is hard to predict its exact contents. This is largely because critical information related to a primary character is either withheld or made to appear as though it has little narrative importance. For instance, characters consistently allude to the changeover by making what seem to be innocuous remarks about Borden's persona. In retrospect, though, ostensibly off-handed observations, such as the fact that a great magician always stays in character, "the transported man" could only be executed with a double, and that Borden often seems to have a split personality, take on great narrative significance. The epiphany is primarily effective, therefore, because viewers have been conditioned by the classical film to expect that main characters will possess a set of clear-cut traits that remain stable. As with many other changeover films, The Prestige demonstrates that a revelation that the protagonist is a much different person than viewers initially thought is particularly well-suited for an audience accustomed to classically constructed narratives.
Despite its presentation of a seemingly neat alternative explanation of narrative meaning, The Prestige's epiphany does not provide viewers with the kind of narrative closure that is customary in most changeover films. In fact, as with Memento, the more carefully the film is scrutinized in light of the changeover, the harder it is to determine what "really" happened. It is difficult to believe that Borden has a biological twin, for example, because Angier went to great lengths to clone himself in order to perform his own version of "the transported man." Moreover, Angier learns about the existence of the device that enables him to replicate himself after meeting with Nikola Tesla (David Bowie), the famous inventor with whom Borden previously collaborated and whose name is the cipher to his encrypted journal. It is thus possible that Borden used a similar apparatus to create facsimiles of himself before Angier. Additionally, it is hard to say definitively that Borden actually kills Angier in the end. After Borden fatally wounds Angier, the film's final shot reveals that his rival has produced multiple clones of himself, raising the possibility that other doubles may still exist. Put simply, the film's ending does not tie up all causal lines of action despite appearing to be an explanatory changeover.
At a time in which fans now routinely use digital media technologies to dissect and discuss their favorite films, misdirection films that inspire these interpretive activities have become more common in Hollywood than ever before. In the same year The Prestige was released, Hollywood distributed two more highly self-reflexive misdirection films: The Illusionist, another magician-themed film, and Lucky Number Slevin, in which there are similarly many diegetic references to the presence of its changeover. In addition, three Hollywood misdirection films—Perfect Stranger, Shattered, and Atonement—hit U.S. theaters in 2007. Interestingly, each of these films portrayed a female character as ultimately being the unexpected primary causal agent, indicating that producers had a growing awareness that contemporary misdirection films tend to center almost exclusively on the exploits of male characters and that a shift to focusing on women in the genre might represent an innovation to attract audiences tired of sameness. Executives thus believed that a potentially profitable audience recognized that films containing these particular narrative structures constituted a distinct genre with its own set of conventions. The decisions to begin The Prestige with a direct challenge to the viewer's interpretive acumen and to end it in an ambiguous manner, therefore, were calculated choices designed to appeal to viewers who enjoy participating in games of discovery and decipherment.
Figure 5.3. The final shot of The Prestige that suggests Robert Angier might still be alive by showing one of the many facsimiles killed during his act.
The Prestige was indeed targeted at spectators who derive pleasure from matching wits with a director already renowned for being a master of misdirection. This strategy worked. Critics were mixed, but many of them praised the film and applauded Nolan for his progress as a director. Booth epitomizes this critical discourse by noting that Nolan "first twisted Memento into a feverish dream," followed that with the "dark puzzle of Insomnia," and then his "most commercial success Batman Begins was disarmingly smart," which leads him to exclaim that "All those directing talents are in evidence with The Prestige." Although the film made little splash at the domestic box-office, netting a relatively modest $53 million, as of August 2015, it is ranked as the 51st best film of all time on the Internet Movie Database's Top 250 list. Such belated fan appreciation suggests that the film's reputation has been elevated by its run in post-theatrical markets. For Henry Jenkins, the increased production of films that require a comparatively high degree of interpretive labor indicates that they are "part of a corporate strategy to ensure viewer engagement with brands and franchises" across multiple media platforms (Convergence 56). Extending this logic into the realm of authorship, I contend that filmmakers like Nolan and Shyamalan are similarly branded commodities largely because of how they have become associated with the misdirection film. The presence of the director's name alone, then, is enough to differentiate these films from other Hollywood fare; it informs audiences that they should be prepared to perform interpretive activities that depart from habitual forms of comprehension simply because the directors are attached.
For Nolan, a key to his enduring appeal in relation to Shyamalan is how he has been consistently and effectively marketed as capable of combining tent-pole production with narratively complex fare that is more challenging than run-of-the-mill blockbusters. His success is linked to how he has never been exclusively associated to the misdirection film. Despite his string of indie and blockbuster hits, his name was still absent from the trailer for the The Dark Knight, which became the film that propelled him to the authorial stratosphere thanks to its gargantuan theatrical revenues and almost universal critical acclaim. By the time he completed the trilogy with the similarly lucrative The Dark Knight Rises (2012), marketers had enough confidence to package it in the preview as being "From Christopher Nolan" without referencing any of his other films. Nolan's Batman fame has even enabled the director to leverage his production company to back superhero franchises for other directors. Specifically, he served as executive producer for Man of Steel (2013), a second recent attempt to revive the moribund Superman franchise after Singer's Superman Returns (2006) failed to meet expectations at the box office and was largely panned by critics. Although Man of Steel received mixed reviews, it performed well at the box office, netting $115 million in its opening weekend and nearly $300 million during its run in U.S. theaters (imdb.com). Its big opening weekend take was partly attributable to how prominently Nolan was featured in the marketing campaign. Even though the film was directed by Zack Snyder, promotional materials, like the film's trailers, explicitly positioned the film as being from Snyder and Nolan, who was packaged as the "Director of The Dark Knight trilogy." Just a few years after The Prestige made him the leading director of the misdirection film, effective promotional strategies in relation to his blockbuster films transformed him into an authorial commodity that could also be mobilized on franchise films that he did not even direct.
The trailers for Inception are the most telling, though, about how the strategies used to promote Nolan as an auteur effectively respond to contemporary industrial contexts. Instead of positioning Nolan as the director of misdirection films and the Batman franchise, like was done for The Prestige, Inception's assorted previews advertise him only as the director of The Dark Knight. They also highlight the film's action-packed, spectacle-laden elements over its mind-bending narrative structure. Such a choice seems odd, considering that the film's complex narrative was one of its most defining attributes and a primary reason for Inception's massive profitability. Clearly, executives deemed it safer to activate Nolan's blockbuster persona than his misdirection identity because of the The Dark Knight's phenomenal performance. Yet, the final image of the trailers does allude to its status as misdirection because the film's title appears embedded in an intricate maze that mirrors Syncopy's logo.
As these emblems suggest, Nolan's link to the misdirection film may often be downplayed, but it has not yet had to be categorically denied like it recently has for Shyamalan. This is because Nolan has strategically avoided being pigeon-holed as only tied to the misdirection film genre. The same cannot be said for Shyamalan, whose career has suffered a significant setback from imprudent marketing tactics that ineffectively positioned him as the genre's top filmmaker. In contrast to Shyamalan's myopic use of Blinding Edge Pictures to back only his own authorial efforts, Nolan's decision to found his own production company early in his career has helped keep his image largely under his control and has made him just as associated with franchise filmmaking as he is with the misdirection film, enabling him to combine the two together in his films effectively as well as to produce blockbusters by other filmmakers. This crucial move has allowed Nolan to maintain a branded identity that aligns with New Hollywood's emphasis on authorship as a marketable commodity connected to other representative aspects of the package-unit logic, such as genre hybridity and a diverse production portfolio characterized by both blockbuster films as well as artistically innovative films with an indie sensibility, albeit within relatively familiar parameters.
6
## Genre Prestige
The Misdirection Film as Blockbuster and Middlebrow Art
THE DECISION TO MARKET INCEPTION (2010) primarily as a blockbuster, action film paid dividends at the box office. The film essentially matched The Sixth Sense's (1999) almost $300 million domestic, theatrical take and exceeded its predecessor's worldwide run in theaters by over $100 million, making it the highest grossing misdirection film ever released (imdb.com). Of course, this kind of data needs to be put fully into context to have significance because its comparative value is contingent on other factors, such as adjustments for inflation and who defines the genre's parameters. A review of the current evidence on Box Office Mojo (boxofficemojo.com), for instance, reveals some of the challenges associated with relying on these statistics to measure the economic merits of a genre's constituents. Among the many options for sorting all-time box-office figures, the website allows users to cull it according to genre. The site's genre label with the clearest connection to the misdirection film is the "mindbender" category. According to this classification logic, Inception indeed generated far more theatrical revenue than its closest competitor, almost tripling the category's second-place finisher: Shutter Island (2010). Its strong financial performance appears more remarkable than it might otherwise be because of the arbitrary choice not to include in the mindbender genre misdirection films that exceeded Shutter Island's theatrical take, including A Beautiful Mind (2001), Planet of the Apes (2001), and The Sixth Sense. These results reveal a potential ramification of random genre classifications. Such data suggests that 2010 was when the misdirection film was at its peak, as it marks the release of the two highest grossing films in the site's most closely affiliated genre.
A more careful review of the circumstances complicates the claim that 2010 was the misdirection film's definitive moment culturally and industrially. The genre's breakout year was 1999, when the modestly budgeted The Sixth Sense became a popular sensation as well as Fight Club and Magnolia received significant critical acclaim in some circles in spite of their box-office disappointment. To capitalize on this momentum, the industry rapidly backed more prestigious misdirection films than ever before, three of which landed in the top 20 grossing films of 2001: A Beautiful Mind, Planet of the Apes, and Vanilla Sky (boxofficemojo.com). Each of these films had a healthy production budget of at least $60 million and was directed, respectively, by an established Hollywood auteur—Ron Howard, Tim Burton, and Cameron Crowe—indicating the industry's growing willingness to allocate more resources to the genre than just a few years earlier when the misdirection film was primarily relegated to lower budgets and helmed by young directors striving to become branded commodities, like M. Night Shyamalan, Paul Thomas Anderson, and David Fincher.
The 2001 crop of films also represents a milestone for the genre because A Beautiful Mind captured the Best Picture Oscar, the only contemporary misdirection film to win it. Yet, as Box Office Mojo's failure to include any of these films in the mindbender category suggests, the 2001 slate of high profile misdirection films was not necessarily conceived of as primarily in connection with the genre. Instead, A Beautiful Mind was most prominently received as an esteemed biopic with some narrative surprises and Planet of the Apes and Vanilla Sky were largely considered part of a spate of recent Hollywood remakes. That is not to say that evidence of these films being constituents of the misdirection film genre is unavailable. It does suggest, however, that although the industry's faith in the genre had increased, it remained limited. This is why it mitigated risk by combining the misdirection film with Oscar bait content and source material that already proved successful elsewhere. In contrast, by 2010, the releases of Inception and Shutter Island illustrate that Hollywood briefly had enough faith in the genre to make artistically daring and expensive misdirection films designed to garner windfall profits and industrial cachet.
In this chapter, I examine how the production, marketing, and reception of Inception and Shutter Island reveal that the misdirection film had become an ideal genre for some of the broader conditions that most impacted Hollywood in the years leading up to the end of the first decade of the millennium. Specifically, my discussion of these two films shows that they exemplify the ways in which the genre was particularly well-suited for a number of the most significant cultural, industrial, and technological circumstances shaping the production and reception of Hollywood films at that historical moment. Rather than consider the misdirection film to be an anomalous genre in the industry's production agenda, my exploration of these two films highlights how it had developed into one of Hollywood's most reliable options for maximizing profits and amplifying its cultural capital for a short period of time before it quickly fell out of favor thereafter.
## Middlebrow Tastes: The Blockbuster Film and Award Show Accolades
Although the blockbuster film has been Hollywood's central focus since discovering it is ideal for the post-studio landscape and its associated package-unit system, media conglomerates have not been solely in the business of making expensive, spectacle-laden films constructed to lure audiences to theaters and synergistically optimize ancillary market revenues. Numerous scholars explain how and why Hollywood maintains a diverse production portfolio even though blockbusters are most likely to garner the biggest revenues in the domestic theatrical market and beyond. One of the issues that such critics must contend with first is the difficulty of identifying exactly what characterizes a blockbuster. Building on the work of discursive genre theorists like Rick Altman and James Naremore, Julian Stringer argues that the blockbuster is most appropriately conceived of as a cultural category, created and sustained by utterances that compare and contrast it to standard Hollywood fare, which is itself also always a constructed classification. As with any genre, the blockbuster is an unstable grouping subject to change based on variables, like when people acknowledge constituents and who labels them accordingly. Some films never designed to be, or packaged as, blockbusters get retroactively categorized as such by groups, like critics, because of their enormous box-office takes. A few of the top grossing titles of their respective years since 1990, including Forrest Gump (1994), Saving Private Ryan (1998), and American Sniper (2014), showcase how films not initially positioned as blockbusters can become ones based largely on how they unexpectedly dwarf modest box-office expectations. This phenomenon leads Stringer to claim that the blockbuster is a relative term, linked to "the money/spectacle nexus" as well as to a film's "size factor and bigness and exceptionality" (8). The blockbuster, therefore, is a genre that is entirely comparative. A film's status as one is predicated on how it is discursively differentiated from what is conceived of as run-of-the-mill Hollywood output by various groups in relation to an array of possible factors, such as budget, box-office take, spectacle, run-time, promotional ballyhoo, casting decisions, release date, synergistic tie-ins, and so on.
The haphazard and volatile classification logic of the blockbuster genre's constituents begins to demonstrate why the industry has never simply produced films confected with this sensibility since it became the most attractive option for New Hollywood's business model. If some films are only recognized as blockbusters retroactively and if it is a comparative term, then other kinds of films have to be backed by the industry for the blockbuster even to exist and for the genre's constituents to be so varied. Genre theory aside, there are other, more pragmatic reasons why the blockbuster did not turn into the only Hollywood product. In New Hollywood Cinema, Geoff King argues that, regardless of the central importance of the blockbuster to the media conglomerates' fiscal health, these companies maintain strategically diverse production portfolios for important reasons, including trying "to leave no opportunity for profit unexploited" and sustaining a positive "image of the studios, a matter of some significance given their potential vulnerability to federal regulation" (83). The industry's desires to generate better than anticipated revenues from lower-budgeted fare and to keep the patina of art to protect its remarkable history of self-censorship by avoiding the kind of government intervention that could result from the appearance of crass commercialism are strong incentives for Hollywood to continue varying its products. Blockbusters are often referred to as tent-pole films for precisely this reason: their synergistic revenue streams can prop up the fortunes of an entire media conglomerate, giving them the potential to offset losses that result from the other films it continues to back. Even though lucrative ancillary markets usually make blockbusters profitable in the long run, their huge negative costs also mean they are risky investments because of the initial hits they may take at the box office when high-interest loans need to be repaid rapidly. To allay the blockbuster's economic uncertainty, it behooves Hollywood to continue to churn out cheaper films that often turn modest profits and even can become box-office smashes.
For Thomas Schatz, New Hollywood's assorted production docket can be separated into "three classes of movie: the calculated blockbuster designed with the multimedia marketplace and franchise status in mind, the mainstream A-class star vehicle with sleeper hit potential, and the low cost independent feature targeted for a specific market and with little chance of anything more than 'cult film' status" ("The New" 35). Schatz prudently acknowledges that such divisions do not mean that Hollywood is akin to a caste system in which these three production categories are mutually exclusive or absolute. Yet, I argue even more strongly than he does that the blockbuster category should not be limited to films specifically constructed with synergistic revenue generation and the potential for franchising in mind. As his use of the ambiguous term "sleeper hit" to characterize a primary motive for the production of the second class of films indicates, there are films from his other categories that can go beyond just being unexpected hits and become blockbusters because of how groups, like audiences and critics, receive them as a consequence of the huge profits that they surprisingly garner at the box office.
What is most notable about the misdirection film in relation to New Hollywood's production logic is that its constituents have steadily migrated from the bottom- to the top-tier during the period under study. In the early 1990s, misdirection films were lower-budgeted and produced by independents, as evidenced by Jacob's Ladder (1990, Carolco), Total Recall (1990, Carolco), The Crying Game (1992, UK co-production), Pulp Fiction (1994, Miramax), and The Usual Suspects (1995, Polygram) (imdb.com). Always eager to capitalize on missed chances for profit, Hollywood first benefited from these films by distributing them more widely in their various formats. In the late 1990s, the industry moved to backing higher budgeted misdirection films, which mostly fall into Schatz's second class. Unsurprisingly, this began in a largely risk averse way, as Hollywood's initial attempts to support higher budgeted films in the genre were characterized by remakes: Diabolique (1996) and Psycho (1998). After The Sixth Sense shockingly became more than just a sleeper hit, the industry took the final step by making a misdirection film that was constructed with clear blockbuster aspirations from the outset: Fox's remake of Planet of the Apes, the first misdirection film with a nine-figure production budget, which was also both released in the midst of the summer blockbuster season and accompanied by video game tie-ins for a variety of popular console and handheld systems. Although the film performed well at the box office, garnering over $360 million in theatrical receipts globally, it received generally poor reviews and initial plans for a sequel were shelved (imdb.com). Yet, Fox's decision to resurrect the property a decade later with Rise of the Planet of the Apes (2011), Dawn of the Planet of Apes (2014), and additional forthcoming sequels indicates that executives were right to conceive of it as having franchise potential. As this belated revival suggests, the lackluster critical reception of the Planet of the Apes remake gave the industry pause about the genre's blockbuster possibilities, preventing it from returning to Schatz's top-tier for years.
In addition to the relative disappointment of Planet of the Apes, the incredible critical success of A Beautiful Mind helped cement the misdirection film in the A-list star vehicle category for most of the decade. The film's four prestigious Academy Award wins not only gave it enormous cultural capital. They also made the film into a moneymaker, as it ultimately took home $313 million at the worldwide box office on its $60 million budget (imdb.com). The importance of the Academy Awards for Hollywood's production calculus epitomizes why the blockbuster is most appropriately conceived of as a synecdoche for the industry's total output that conceals its actually more varied product mix. The industry indeed packages some of its films as designed primarily for something other than just pure entertainment. The Academy Awards provide a key way to recognize products that are heralded for other virtues, such as artistic expression and social commentary. Awards ceremonies, particularly the Oscars, serve the important function of publicly identifying the cultural value of these products, especially because the entertainment news media is otherwise fixated on box-office results. The industry is not opposed to this journalistic obsession, as it creates barriers to entry by making Hollywood films practically the exclusive focus in the public sphere as well as helps to propel these products to achieve greater success theatrically and in the aftermarket. To milk the most out of films that are not part of the blockbuster class, the industry relies heavily on award show accolades to create a buzz around selected constituents of mostly Schatz's second class of films. This crucial facet of the business further maximizes revenues on existing products, transforms creative personnel who win awards into more valuable presold properties in the future, and inoculates the industry from accusations that it is myopically focused on the bottomline.
Although films packaged as blockbusters generally fare poorly in the most prestigious Academy Award categories and are instead typically consigned to the technical awards, Gillian Roberts posits that Best Picture Oscar winners are often constructed in ways that actually have much in common with their seemingly distant big-budget, franchise relatives. This is because they are both primarily designed to appeal to a mass audience. As with the commercial aesthetic that drives blockbuster production, Roberts argues that Oscar winners usually possess qualities that exhibit Pierre "Bourdieu's characterization of middlebrow culture, namely as one that offers a negotiation between the accessibility of low culture and the prestige of high culture" (157). Bourdieu seminally theorized that culture industry production is characterized by an opposition to "intellectual art" because it is constructed to garner "investment profitability," which often results in appropriations of revered art forms to legitimize its status (126). Roberts's argument may seem to run counter to conventional wisdom because Oscar voters are typically chastised for being out of sync with mass taste; however, this is often not the case despite the box-office discrepancies that usually exist between Oscar winners and blockbusters. In addition to ignoring instances when indisputable blockbusters, like Titanic (1997) and The Lord of the Rings: The Return of the King (2003), dominated the ceremony, this false dichotomy fails to account for how frequently Best Picture winners are created for wide audiences. As Roberts contends, the Oscars provide "the impression of bringing legitimate culture within the reach of all by bestowing legitimacy on accessible cultural products" (157). As Best Picture winners from the period under study, like The Silence of the Lambs (1991), The Departed (2006), and No Country for Old Men (2007), illustrate, the Academy can bolster the cultural capital of films from familiar genres, such as the serial killer, gangster, and crime film, which are extremely popular, but are generally debased by the intelligentsia.
Most misdirection films coincide closely with a middlebrow cultural appeal that blends artistic legitimacy with mass audience accessibility, as they contain a combination of a reliance on classical conventions with narrative and formal innovations that challenge those very standards. This combination of low- and high-art tendencies is often cited by scholars as evidence of a postmodern turn in Hollywood cinema. Such contentions relate to Fredric Jameson's influential notion, articulated in Postmodernism, of the epoch being partly characterized by artistic production that increasingly exhibits the qualities of "pastiche," which indiscriminately samples disparate forms of expression, often resulting in "the effacement in them of the older (essentially high-modernist) frontier between high culture and so-called mass or commercial culture" (2). As I argued in chapter 2, although the mixing of high- and low-culture has recently intensified across various media and art forms, I conceive of the misdirection film as embodying a late phase of modernity, rather than of postmodernity, largely because its dependence on classical principles and general support of dominant ideologies exemplifies how it is most reliant on conventional aesthetic practices and entrenched epistemologies that have not yet been replaced by different ways of thinking or new modes of expression. Moreover, it is a stretch to claim that taste formations separating the elitist from the popular have collapsed to a point in which the two categories have become culturally indecipherable. If anything, allusions to high-art are most often deployed by Hollywood filmmakers to imbue their mass cultural productions with greater legitimacy precisely because such distinctions persist and film canons continue to be defined by traditional conceptions of artistic value.
The misdirection film's production trajectory illustrates Hollywood's enduring reticence to take radical artistic chances and instead highlights how it relies on familiar conventions. It is unsurprising, in an industry characterized by risk aversion and artistic conservatism, that it was only after a few lower-budgeted, independently produced misdirection films—The Crying Game, Pulp Fiction, and The Usual Suspects—won prestigious Academy Awards and subsequent representatives from the second class—Magnolia and The Sixth Sense—garnered numerous, non-technical Oscar nominations, that it finally won big in the genre in the most coveted categories with A Beautiful Mind. In addition to being well-suited for post-theatrical markets, the misdirection film thus served the industrial purpose of earning prestigious Oscar wins and nominations for ensuing films from Schatz's second-tier. The Oscar wins and nominations in non-technical categories from films, like Adaptation (2002) and Atonement (2007), which, thanks to its seven nominations became the second most decorated misdirection film by the Academy, are a testament to the industry's use of the genre to bolster its cultural capital during the 2000s.
The misdirection film's high critical acclaim and strong post-theatrical performance positioned the genre well for even more in the years leading up to 2010. Hollywood's backing of Inception and Shutter Island demonstrates that the misdirection film had become more than just a reliable genre for Schatz's second-tier at that moment. There was perhaps no better choice, therefore, of a director to appeal to middlebrow tastes than Martin Scorsese, who, by the 2000s, as Marc Raymond summarizes, "had come to represent the industry's best possible version of itself and the artistic quality it is capable of delivering" (201). At the time of Shutter Island's release, Scorsese had become Hollywood's crown jewel director, renowned for his image as the industry's foremost public intellectual, recent Oscar success, artistic integrity, and reliable, if often unspectacular, box-office returns. Scorsese's reputation for consistently directing aesthetically innovative films that do not alienate the masses made him ideal for helming an esteemed misdirection film because of its similar blend of those qualities.
As is the case under the package-unit model, the hiring of a director who has become a marketable commodity can serve as a valuable presold property. Scorsese's eventual participation in the project was critical to getting the film into production. The rights to Shutter Island's 2001 source novel of the same name, which was written by Dennis Lehane, who also penned the book that was adapted into Academy Award winner, Mystic River (2003), were optioned to Columbia Pictures in 2003; however, the project did not actually come into fruition until Scorsese and Leonardo DiCaprio subsequently signed on to direct and to star in it in 2007 (Fleming). Importantly, Scorsese and DiCaprio had established a track-record of working together on films with similar sensibilities that performed admirably at both the box office and the Academy Awards, including Gangs of New York (2002), The Aviator (2005), and The Departed, which finally earned one of the director's films the Best Picture and Best Director Oscars that many critics claimed he had been unfairly denied until then. As with Shutter Island, Inception was a gestating project that had been put on hold for a number of years. Nolan reportedly began toying with the idea before becoming a Hollywood filmmaker and originally pitched the project to Warner Bros. after directing Insomnia (2002), his first studio-style film. In contrast to Scorsese, however, who had already established a reputation as a filmmaker adept at making both critically acclaimed, arthouse films, like Mean Streets (1973), Taxi Driver (1976), and Raging Bull (1980), as well as commercially driven fare, like The Color of Money (1986) and the remake of Cape Fear (1991), Nolan had to prove his acumen in mainstream filmmaking before getting the greenlight to make Inception.
Although a name brand auteur might be enough to get the industry to back a film in Schatz's second-tier, the greater risks associated with blockbuster negative costs almost always require multiple presold properties to get them into production. Such a gamble is amplified for misdirection films because of their potential to turn off the mass audience. Both directors, then, relied heavily on stardom as an additional guarantee by casting DiCaprio, arguably Hollywood's most coveted leading man at the time, as their lead actor. After breaking into the industry as a teenager in the early 1990s, DiCaprio's star soared in popularity with Titanic, a blockbuster par excellence that shattered box-office highs and won a record-tying 11 Oscars. His performance, though, was not among the also record-tying 14 nominations that the Academy bestowed on Titanic, precipitating similar results on subsequent films that turned good profits and received Academy recognition, but long failed to net DiCaprio an acting Oscar (imdb.com). Despite being denied by the Academy for years, DiCaprio has become known for being the rare actor able to capture both arthouse and mainstream audiences. As Scott Bowles documents in his USA Today article on the actor's casting in Django Unchained (2013), DiCaprio has "a reputation for being one of the most selective actors in the industry," choosing to work with directors who, like Scorsese and Nolan, have proven capable of making films that are popular with wide audiences and have Oscar appeal, including Danny Boyle, James Cameron, Clint Eastwood, Baz Luhrman, Steven Spielberg, and Quentin Tarantino. This has not negatively impacted the actor at the box office, as DiCaprio's films have earned over $2 billion in domestic theaters, for an average of $97 million per film to date (boxofficemojo.com). Before finally winning his Oscar in 2016, DiCaprio was nominated for four Academy Awards as a performer, won three Best Actor Golden Globes, and helped the films in which he has been cast as the lead accumulate many coveted prizes (imdb.com). Clearly, DiCaprio has a reputation for starring in films that capture mass and arthouse audiences, situating him among the industry's top middlebrow performers.
The reasons for DiCaprio's appeal, however, extend beyond just savvy career choices. As with the success of the misdirection film more broadly, DiCaprio's stardom is also a product of the ways in which his persona is linked to salient cultural conditions. The protagonist that the actor typically portrays is a paragon contemporary of broken manhood. Such roles are epitomized by Inception and Shutter Island, in which he plays men who ostensibly struggle to regain their traditional masculinity after losing patriarchal control of their families because of the underhanded actions of their wives. These onscreen representations, which build on the doomed, romantic partner persona that DiCaprio developed in his star-making, melodramatic lead roles in Romeo + Juliet (1996) and Titanic, are emblematic of a culture in which there is a perceived crisis in hegemonic masculinity.
The connections between DiCaprio's image and anxieties about the persistence of traditional masculine dominance extend beyond his seminal film performances. As Richard Dyer argues in Heavenly Bodies, the star's persona is "made up of screen roles and obviously stage-managed public appearances, and also of images of the manufacture of that 'image' and of the real person who is the site or occasion of it" (7). This blend of textual and extratextual discourses means that the star is not associated with a single performance, as the image that calcifies supersedes a one-off role. Even more interestingly, Dyer contends that stars are attractive to audiences for the same reason that ritual genre theorists often cite for its appeal: the star's image often dovetails with dominant ideology by representing a satisfactory mollifying combination of cultural oppositions that are actually incompatible off screen. DiCaprio's popularity is linked to how his persona reconciles competing notions about appropriate forms of masculinity. According to his recent biographical legend, DiCaprio has become arguably Hollywood's most eligible bachelor, famous for his womanizing exploits. Perhaps most notoriously, the tabloids were headlined with news of the actor leaving a Miami club with 20 models after he and his then-girlfriend, Toni Garnn, reportedly split just weeks before the release of his most recent collaboration with Scorsese, The Wolf of Wall Street (2013), in which he is cast as real-life philanderer, Jordan Belfort (Saad). Those overlapping discourses, which position DiCaprio as an irresistible catch, express cultural desires for supporters of conventional masculinity about its endurance at a time when such notions of ideal manhood are less universally accepted.
Inception and Shutter Island resonate culturally not only because of how DiCaprio expresses their shared wishes about antiquated modes of masculinity remaining appropriate amid pervasive paranoia about changing gender roles and relations. They also depict the kind of agency panic about anxieties related to the loss of individual sovereignty that permeates many misdirection films. Both films contain changeovers that can be interpreted as depicting how their protagonists' free will is severely restricted by powerful forces beyond their control. As a consequence of how much these two films typify the misdirection film's connections to some of the most significant cultural, industrial, and technological conditions shaping Hollywood production and reception from 1990 to 2010, they are the ideal final case studies of this book. The following examinations of the ways in which they were interpreted by audiences as well as constructed and promoted by the industry, therefore, tie the book's arguments together by demonstrating how the misdirection film was specifically fashioned in response to its contexts.
## You Mustn't Be Afraid to Dream a Little Bigger: Inception and the Misdirection Film's Blockbuster Leap of Faith
A key aspect of Christopher Nolan's authorial lore that the director himself often repeats is that the decision not to make Inception as early in his career as he hoped turned out to be fortuitous. As Nolan explained in a 2010 interview with Dave Itzkoff of the New York Times, he "first pitched the film to the studio probably nine years ago, and he] wasn't ready to finish it. [He] needed more experience in making a big movie" ("A Man"). This is a primary reason why he directed the initial film of the Batman franchise first. Whether or not this is revisionist history that conceals lasting concerns Warner Bros. had about the project's economic potential, Nolan is right to position the film's timing as auspicious. Inception was a risky film to back because its complex narrative threatened to turn away some audiences. In contrast to the typically modestly budgeted films in the genre, though, it was a bigger gamble than any misdirection film before it, as its estimated $260 million negative costs put it in indisputable blockbuster territory in terms of budgetary size ([imdb.com). Moreover, whereas most films with that kind of gargantuan initial investment are greenlit because of their capacity to be franchised, tied to reliable presold source material, and capitalize on ancillary market revenues, Inception had few such opportunities, aside from likely exorbitant home-video sales and possible sequels, which have yet to be made. To alleviate risk, Warner Bros. chose to promote Inception's action-packed and spectacle-laden qualities most heavily, assemble a star-studded cast led by DiCaprio, as well as adhere to the template from The Prestige by highlighting generic hybridity, packaging it as a constituent of the action, blockbuster, heist, science fiction, and misdirection film genres. Most crucial to the production logic, however, was Nolan's development into a marketable auteur. Putting the production rationale in crude terms, President of Worldwide Marketing for Warner Bros., Sue Kroll, told the LA Times that "We don't have the brand equity that usually drives a big summer opening," but "Christopher Nolan as a brand is very powerful" (Fritz and Eller).
Nolan's status as the mastermind behind the Batman franchise was mobilized more prominently than any other aspect of his branded identity in Inception's promotional materials. This makes sense in light of the director's immediate predecessor to Inception, The Dark Knight, and its incredible performance at the box office, with critics, and at the Oscars, where it bucked the trend of blockbusters only netting technical Academy Awards by earning Heath Ledger a posthumous statue for Best Supporting Actor. Such a marketing strategy is logical because the film needed a wide audience to offset its high negative costs, which explains why Inception's affiliation with the blockbuster and its prototypical action-packed, spectacle-laden sequences was more central to its marketing than any other element in the package. Virtually all promotional materials, however, reference its links to the misdirection film, albeit often obliquely and fleetingly. For starters, the film's two taglines—"Your mind is the scene of the crime" and "The dream is real"—only allude to its convoluted narrative structure by foregrounding other generic links, including the heist and science fiction genres, as explicitly (imdb.com). Inception's trailers, which are dominated by clips of the film's most spectacular moments and images of its cast, also raise the specter of the misdirection film, but only briefly and implicitly. The teaser trailer, for example, bookends its featuring of Inception's blockbuster qualities with insinuations to the misdirection film by opening with extended shots of the spinning top, ostensibly the most important clue for deciphering its narrative mysteries, and ending with an image of the film's title in the middle of a gigantic maze that appears to be unsolvable. In short, although executives did not ignore the film's connection to the misdirection film in the marketing campaign, its blockbuster elements were the primary emphasis.
The strategic decision to position Inception as a blockbuster paid off at the domestic box office, where the film grossed approximately $63 million in its crucial opening weekend in mid-July during the blockbuster release season, allaying fears that its unfamiliar source material would turn off mass audiences (imdb.com). The film's initial success was buttressed by favorable reviews that highlighted its blockbuster status, but also lauded its misdirection film attributes. Elizabeth Weitzman of the New York Daily News typifies these responses by noting "Inception blends the blockbuster enormity of The Dark Knight with the indie insights of Memento to create an all-encompassing experience that makes other summer films seem mediocre." The Seattle Times reviewer Moira Macdonald makes practically the same assertion by claiming that the film merges "the twisty appeal of Memento with the cool chic of The Dark Knight," resulting in "the rare would-be blockbuster that demands close attention and surely would reward rewatching." Clearly, Nolan's reputation as a filmmaker with an uncommon history of pleasing both arthouse and wide audiences was enthusiastically activated by many reviewers despite the ploy to market Inception as a blockbuster film most conspicuously. In his review, Bill Goodykoontz of The Arizona Republic identifies why Nolan is able to blend indie and blockbuster filmmaking effectively. Evoking David Bordwell's theory that compositional motivation drives classical film form, Goodykoontz writes "The visuals are stunning, perhaps the most fully realized of any film... in this context it is not simply showing off for the sake of doing so, but a believable part of the story." Such hyperbole pervades Goodykoontz's astute review, in which he also acclaims DiCaprio's "trademark wounded man" performance and extols Nolan's construction of an "interlocking puzzle for the audience to figure out. (Don't worry, it's more fun than that sounds)." Similarly, praising the film's appeal to fans, Roger Ebert speculates that it "is sure to inspire truly endless analysis on the web."
These critics' predictions about the film's likelihood to generate fervent fandom online proved accurate, as Inception immediately became an Internet sensation and remains one of the most frequently discussed films on the web to this day. A Google search in August 2015, using the terms "Inception analysis," for example, yields over 46 million results, while the terms "The Dark Knight analysis," returns a still impressive, but comparatively measly, two million hits even though that film is Nolan's biggest box-office moneymaker. As with other misdirection films that initially spawned this sort of rabid fan activity online, such as Magnolia and Mulholland Dr. (2001), much of Inception's appeal to virtual communities can be linked to Nolan's decision to construct a narrative puzzle that resembles the ones he created in Memento and The Prestige because it similarly appears to be irresolvable. In contrast to the reception of those earlier misdirection films, particularly Memento, however, the advances of Web 2.0 technologies have made fan responses to Inception more common and diffuse than its generic predecessors. As a result, there is not a single online location housing the preponderance of Inception's virtual community activity, as there was for a film, like Mulholland Dr., because it was released when users needed coding skills to publish to the web, making the Lost on Mullholland Dr. (LOMD) website an unparalleled aggregator of that film's fan interpretations at the time.
Another key distinction between fan reception of Inception and earlier misdirection films is the specific kinds of interpretive activities it has inspired. In comparison to Nolan's previous misdirection films, most notably Memento, which provoked considerable online discussion dedicated to figuring out its "true" meaning, many fans of Inception begin from the premise that there is no conclusive solution to the film's central mystery of delineating dream from reality. For example, in an article on the Ropes of Silicon website, a once popular amateur movie blog, Brad Brevert challenges Michael Caine's earlier publicly stated assertion that he knows that the film's ending reveals it was not all just a dream by noting that it is unlikely that "we'll be hearing Chris Nolan explaining the ins and outs of Inception or confirming Caine's statement any time soon" because he would "expect to hear him talk about Inception's ending just as much" as he would envisage the director to provide "a final explanation for the existence of a certain tattoo in Memento." As this emblematic viewer response indicates, the discourse of authorship has strongly contributed to this shift in the reception of Nolan's films. Indeed, Nolan and DiCaprio alike both balked at opportunities to explain definitively what "really" happened in the immediate wake of Inception's release in their assorted interview and promotional appearances. Some of the director's fans thus continually make reference to his penchant for eternally lingering ambiguity and his tendency to avoid giving concrete answers about meanings of his previous misdirection films as evidence of why Inception's many enigmas cannot be explained definitively. Despite this admission, most fans still cannot resist attempting to offset the uncertainty associated with the film's truth being unknowable by deriving interpretations that resolve all of its mysteries.
Paradoxically, Nolan's own comments about the film's "true" meaning also have helped to fuel speculation that refutes claims that Inception is intentionally constructed to be eternally ambiguous. As he did with Memento, the director coyly leveraged publicity opportunities to encourage fans to search for an absolute answer even though he simultaneously claimed one might not actually exist. In an interview for Wired magazine published shortly after the film's theatrical release, Nolan responds to a question about the presence of an absolute explanation for what "actually" happened by claiming he does have a conclusive answer to the inquiry. Expanding on that response, he tells the interviewer that "I've always believed that if you make a film with ambiguity, it needs to be based on a true interpretation. If it's not, then it will somehow contradict itself and end up making the audience feel cheated. Ambiguity has to come from the inability of a character to know—and the alignment of the audience with that character" (Capps). In addition to suggesting to viewers that there might be a correct way to interpret Inception without giving it away, this assertion is significant for identifying a key characteristic of the misdirection film. It reiterates why these films should be differentiated from those in affiliated categories, such as Thomas Elsaesser's "mind-game film" genre, that also includes films in which only diegetic characters are fooled into jumping to erroneous conclusions about narrative information. In contrast, misdirection films center on tricking the spectator, irrespective of the fate of onscreen characters, even though both are often duped simultaneously. Additionally, Nolan's response is relevant for referencing how central focalization is to the misdirection film's particular brand of deceptive magic, as the viewer's strong identification with the classical film's prototypically goal-oriented protagonist, who surprisingly turns out to have less narrative agency than other characters, is the most common ploy for helping to pull off the sleight of hand.
Inception's convoluted narrative ostensibly focuses on Dom Cobb's (Leonardo DiCaprio) quest to return to his estranged children, whom he has been separated from since his then-wife, Mal (Marion Cotillard), supposedly framed him for her murder before killing herself. A mysterious magnate, Saito (Ken Watanabe), subsequently promises to exonerate Cobb if he is able to convince the recent heir of a competing energy conglomerate, Robert Fisher (Cillian Murphy), to break up his empire. Saito seeks out Cobb because of his reputation as the best dream extractor. That is, Cobb is a subconscious thief who specializes in entering the dreams of his marks and stealing their secrets. Despite his acumen in the arts of shared dreaming and subconscious burglary, Saito's request is more difficult than the standard job because it requires Cobb to plant an idea into the mind of his victim—hence inception—rather than to rob it. Complicating things further is Cobb's mysterious history with inception, which he apparently performed on Mal before her death, contributing to her psychological demise. Specifically, he convinced Mal that the two of them were stuck deep in a dream state and that they had to commit suicide to escape. Problems between the two ensue because Cobb is convinced that they are now awake in reality, while Mal believes they are still trapped in a dream and need to kill themselves again to return home. This creates the central retrospective interpretive conundrum for the viewer: who is correct about their ontological state? As a consequence of the guilt he feels from impacting Mal's death by incepting her with suicidal thoughts, she terrorizes his subconscious and invades any dream that he has or shares, jeopardizing Cobb's missions. Eager to complete the proverbial last job that will allow him to go legit, Cobb assembles, in quintessential heist film fashion, a team of elite accomplices, including Saito, the financer; Arthur (Joseph Gordon-Levitt), the planner; Eames (Tom Hardy), the forger; Yusef (Dileep Rao), the chemist; and Ariadne (Ellen Page), the architect. Together aboard a 747 from Sydney to Los Angeles, the team enters the minds of Fisher and each other in a multilayered, shared dreaming escapade, overcoming a series of daunting obstacles to accomplish their lofty objective, or so it seems.
All of this may sound like the stuff of standard classical Hollywood narrative, but it is not so simple or clear-cut throughout the film and, especially, after it ends. It is inaccurate, though, to label Inception as "non-classical" because one of the reasons that the film effectively appeals to a mass audience is its strong dependence on Hollywood principles in relation to narrative, form, and genre. The film's classical dual plot structure, which centers on Cobb's quest to return home and his attempt to reconcile with Mal, its compositionally motivated formal decisions that are often heavy-handedly accompanied by familiar classical cues, like dialogue hooks and narratively subservient parallel edits, as well as its reliance on recognizable genre conventions, all function to aid the Hollywood spectator in comprehending the byzantine story easier. Perhaps most notably in this regard is the film's bravura section in which Nolan cross-cuts between multiple levels of the dream and constantly returns to a van plunging off a bridge at the top-layer to anchor viewers in time and space, consistently reminding them how and why what is occurring in the vehicle is impacting the team members accordingly deeper down in the fantasy. Yet, as with other misdirection films that blur the ontological shifts between fantasy and reality, like Jacob's Ladder, Fight Club, and Mulholland Dr., it is difficult to decipher what "really" happens in Inception, even in retrospect, because although the film seems to uphold distinctions between dreaming and waking life, the line separating the two becomes increasingly indistinguishable on closer review.
The primary reason for the film's lingering ambiguity in relation to the difference between fantasy and reality is its changeover, which further complicates, rather than clarifies, what "actually" occurred. Specifically, after Cobb appears to achieve his goals of successfully incepting Fisher, letting go of Mal by forgiving himself for his role in her suicide, and reuniting with his children, the final shot lingers on a close-up of his purported totem—a top that he claims spins endlessly in the dream and only falls when he is awake—that he shockingly turns his back on before viewing its fate so he can instead grab his kids. After Cobb excitedly reunites with his children, the camera then moves to fixate its gaze on the spinning top, which wobbles, but does not actually topple, before the film cuts to the end credits. This edit means that neither Cobb nor the spectator can definitively determine from this sequence of events if he is dreaming this storybook resolution or if it is transpiring in his waking life, even though the preceding scenes have led the viewer to believe that he is no longer asleep.
Thanks to this uncertain conclusion, many fans contend that the top is a red herring, a MacGuffin that distracts viewers from all of the other clues planted throughout the film that may provide better evidence of if it is possible to distinguish between dream and reality. This interpretation has only been bolstered by Nolan's ensuing comments on the issue. To wit, in his 2015 commencement address at Princeton University, Nolan asserted that the film ends with Cobb "in his own subjective reality. He didn't really care anymore, and that makes a statement: perhaps, all levels of reality are valid" (Lee). Such an interpretation can be supported by other textual evidence in relation to the top that suggests it is simply a false clue. Like all totems, the top contains a secret that only its owner is supposed to know to prevent a shared dreamer from manipulating that person's grasp of reality. However, the top's status as Cobb's actual totem is troubled for a number of reasons. First, he informs Ariadne of it, rendering it no longer safe from her potential control. Second, he also reveals that it was really Mal's initially, which is troublesome because totems are not supposed to be handled by others. Finally, it behaves opposite of the way it needs to, as it acts normally in reality (eventually falls) and its special quality (endless spinning) happens in the dream. This last aspect is most crucial because a totem's secret property has to be unique only in waking life to force a dreamer to control it in expected ways in fantasy situations. A top that simply behaves normally in reality, therefore, is a useless totem because others would manipulate it to do the same exact thing in a dream, making it impossible for Cobb to use it to determine if he is actually awake.
As a result of the ambiguity that the open-ended changeover inspires, viewers are encouraged to watch the film repeatedly to decipher the mysteries that precede it, such as the puzzling details of Cobb's totem. Fans have thus competed in games of hermeneutic one-upmanship to convince each other that they have cracked the code most convincingly. A popular counter theory, for example, which is the official explanation that appears on the Inception Wiki page for "Dominick Cobb" and relies heavily on textual evidence, is that Cobb's wedding ring is his real totem because it only appears in the scenes coded as in the dream, meaning it is concealed from others in reality. Such activities have helped make the film a hit in the post-theatrical market. In terms of DVD and Blu-ray sales, for example, Inception has raked in an additional $160 million to date (the-numbers.com).
The presence of this and other eternal ambiguities that are central to determining the "true" meaning of narrative information is, as Bordwell argues, antithetical to a defining trait of the classical film, which is supposed to leave no primary causal lines of action unresolved at the conclusion. Instead, such an element potentially puts Inception closer to Bordwell's art cinema category from Narration in the Fiction Film in which narrative incomprehensibility as well as lead characters with uncertain traits and motives are more common. To lessen confusion that arises from the various interpretations that art cinema inspires, Bordwell contends that viewers typically make recourse to real life and authorship. More specifically, these ambiguities are often understood by viewers as existing because of the director's aims to express how chaos reigns off screen, distinguishing art cinema's narrative and associated formal properties from the classical film's valorization of protagonist-driven stories and corresponding representational principles that reassure audiences that the universe abides by a causal logic. Such atypical qualities leave Inception particularly open to multiple interpretations, making it a film that would appear likely to alienate classically trained spectators. Yet, it did not struggle with the mass audience partly because of the ways in which it was expertly constructed for middlebrow tastes by combining features typically reserved for "legitimate" art with more accessible elements that are characteristic of mass cultural products.
In addition to its heavy reliance on familiar genre conventions as well as classical narrative and formal devices, much of Inception's mass appeal is attributable to its use of a number of blockbuster standards. First, the film's huge production and marketing budgets are substantially bigger than they are for most Hollywood films. The film's cast is also filled with recognizable stars with broader international origins than is customary, as there are lead performers that hail from the United States, (Tom Berenger, DiCaprio, and Gordon-Levit), Canada (Page), Europe (Caine, Cotillard, Hardy, and Murphy), and are of Asian descent (Watanabe and Rao). The film's global appeal is also linked to location shooting decisions because it is set in exotic destinations, such as Japan, Mombasa, and Paris, that are, according to Itzkoff, actually an amalgamation of diverse production sites across the globe, including Alberta, London, Los Angeles, Paris, Tokyo, and Tangier ("The Man"). These extravagant casting and production decisions can only happen with blockbuster resources and are attractive to an industry that increasingly depends on courting the international market to recover escalating negative costs. It is the film's elaborate, action-packed, spectacle-laden set pieces, though, that situate it most securely in the blockbuster genre. More than anything else, it is these kinds of bombastic sequences that have colloquially come to define the blockbuster film.
Although staples of the blockbuster, like pyrotechnic-filled chase sequences, frequently are used to identify the genre's constituents, determining a film's status as such based on the amount of spectacle it contains is notoriously challenging. As King theorizes, this is tied to the ambiguity of the term "spectacle," which can refer to an array of attributes beyond special effects and similar elements most commonly cited as giving a film the quality. In fact, the exotic locales that serve as backdrops for many of Inception's most dazzling sequences are likely to be counted as spectacular by most observers. To assuage uncertainty associated with identifying this vague trait, King contends that in Hollywood spectacle is most appropriately conceived of as being excessive to the narrative, "as a source of distraction or interruption" from the classical film's primary objective of making its familiar storytelling format easily comprehensible by employing the corresponding invisible style (New Hollywood 179). Wisely, he acknowledges that Hollywood's emphasis on spectacle predates the fall of the studio system as well as grants that narrative and spectacle symbiotically coexist in New Hollywood. It is the latter part of this statement that most relates to how spectacle is effectively deployed in Inception. Nolan caters to his audience by making spectacle subservient to narrative, as a majority of the film's most astonishing sequences are clearly framed as a product of dream logic, regardless of how spectators reinterpret what actually happened. Consequently, many of the film's most spectacular moments are initially understood as being driven by storytelling demands and do not seem superfluous to the narrative. Even more notably in this regard, one of the most common ways to reinterpret Inception is that reality is never represented in the diegesis because the entire film actually solely depicts Cobb's extended dream. This popular reinterpretation of the film's meaning renders all of its spectacular moments as narratively explainable in retrospect since viewers are never actually privy to Cobb's waking life.
The theory that the entire film is a portrayal of Cobb's extended dream primarily hinges on a key aspect of Hans Zimmer's score, making the sound design also more narratively relevant and classically driven, in retrospect. Just prior to the opening shot, a musical theme that recurs throughout the film plays. As Itzkoff summarizes in an article for the New York Times ArtsBeat, many fans began to read this piece as a dramatically slowed down version of Edith Piaf's "La Vie en Rose" when a YouTube user posted a comparison of the two shortly after the film's release. Specifically, Itzkoff states "When the video's pseudonymous author, camiam321, plays the key musical cue from that score, two ominous blares from a brass section, followed by a slowed-down version of the Piaf song (which the Inception characters play at regular speed as a warning to wake up from a dream state), they sound nearly identical" ("Hans"). This evidence compellingly supports the "all a dream" interpretation because, prior to beginning their attempt to incept Fisher, Cobb's team agrees that Piaf's song will be the musical cue that they will play to signal that the kick has to be initiated to wake them from their strong, sedative-induced, multilayered dream state. Crucially, this theme replays at the very conclusion of the film to accompany the final segment of the end credits. In short, according to this explanation, everything that occurs during the film's blockbuster-length two hours and 28 minutes between these two musical cues is actually a representation of Cobb's extended dream while he struggles to stay asleep because he desperately wants the fantasy to continue forever, or is in the final stages of the dream before the kick takes full effect. Such an interpretation is bolstered by the rules conveyed by the dialogue because of how time expands in a dream, especially at multiple levels. Consequently, it is possible to understand the whole film as a portrayal of Cobb's dream, which occurs entirely in the few moments in which he hears the kick being played in reality.
This popular reinterpretation helps to explain the narrative significance of many of the film's most ambiguous and seemingly absurd elements. If it was all just a dream, for instance, then concerns about the veracity of Cobb's totem become irrelevant. It does not matter that it initially belonged to Mal or that others know its secret because it is just a projection of Cobb's subconscious. Additionally, it reveals why the flashback of Mal's suicide seems nonsensical, as she plunges from a window that is purportedly in their hotel room even though Cobb tries to persuade her not to jump from another window inside of that same room that inexplicably faces her directly. Similarly, the sequences ostensibly coded as depictions of Cobb's waking life that appear too good to be true, like the happy ending, or that are too far-fetched to be realistic, such as the fact that he enters situations in the midst of the action, can be explained as products of dream logic. Most notably in the impractical regard are the chase scenes in Mombasa, where Cobb is pursued mercilessly by agents who act like subconscious projections in dreams, nearly crushed by walls that defy physics by appearing to close in on him, and all-too-conveniently saved by Saito just in time. Accordingly, in retrospect, not even these spectacle-laden, action sequences are forced or narratively irrelevant because they are driven by story demands if Cobb is dreaming the whole time, including the sequences that appear to be coded as reality.
Figure 6.1. The impossible architecture of the hotel room in Inception from which Mal commits suicide.
In addition to explaining away Inception's most illogical properties, the extended dream theory helps to dispel many of the other primary critiques of the film. Chief among these disparagements are complaints about the amount of expository dialogue employed to explain the intricate rules of dreaming to the audience, which detractors claim leads to all of Cobb's team members being simplistic characters who lack psychological depth. David Denby of The New Yorker typifies this rhetoric by noting "Nolan is working on so many levels of representation at once that he has to lay in pages of dialogue just to explain what's going on." Although this is a fair critique of the film, regardless of how it is interpreted, it becomes a less severe issue if the entirety is understood to be an extended depiction of Cobb's dream. If the members of Cobb's team are just projections of his subconscious that represent different aspects of his personality, then their insufficient character complexity can be explained by the fact that they are supposed to be one-dimensional. As a result, Arthur becomes an embodiment of Cobb's rational side, Ariadne transforms into his creative spirit, and so on. This interpretation, then, can become a totalizing causal logic for comprehending all of the film's ambiguities and shortcomings in a way that resembles the paranoid logic of conspiracy theorists who rely on the discourses of causality and agency to understand more satisfactorily what otherwise remains insufficiently explained. In fact, many fans expand on this theory by claiming that the film is really an extended metaphor for Hollywood filmmaking and the difficulty of planting ideas in the audience's mind, with Cobb standing in for the director and his teammates representing Nolan's collaborators, like an executive producer (Saito), special effects designer (Yusef), and so on.
Figure 6.2. Alley walls appear to close in on Dom Cobb, as he evades being pursued in one of Inception's scenes ostensibly coded as not being part of a dream.
The interpretation that everything is just a dream also impacts the film's cultural implications. Considering the ways in which Inception was marketed and received according to such strong authorial terms, it is unsurprising that critics of Nolan's work often cite it as another example of his disconcerting gender politics. One of the director's most obvious thematic preoccupations is featuring heterosexual, male leads haunted by the tragic deaths of their love interests. Even more troubling to detractors in this regard is how these women cause their bereaved male partners to obsess dangerously about avenging their demise, a theme that is particularly acute in his two previous misdirection films: Memento and The Prestige. Yet, as with Memento, in which considerable ambiguity is raised about whether or not the purportedly dead wife that Leonard Shelby (Guy Pearce) pines for was actually murdered, an interpretation that Cobb dreamed the whole thing unsettles the explanation that Mal's suicide is the root of the protagonist's problems. Of course, if viewers instead think that Cobb is actually awake at the end, then Mal remains the film's unambiguous antagonist because her unwillingness to believe her husband about escaping limbo led to her framing him for murder and subsequently terrorizing his subconscious. In contrast, if the film is all really portraying Cobb's dream, then it becomes possible to interpret Mal's character in a very different fashion, as her theory about them still not being awake and stuck outside of reality might actually be correct.
Fans ascribing to the dream theory have used this as a basis for deriving some of the most outlandish interpretations of the film's "true" meaning. One explanation predicated on it all being a dream that has gained traction in some virtual communities is that Mal is actually the one performing inception on Cobb because he either unwittingly or intentionally refuses to come to terms with the fact that he is stuck in a fantasy. As its proponents contend, this interpretation is backed by the film's discussion about the rules of dreaming, as the limbo state that Cobb and Mal need to escape is the deepest dream level, meaning that it can only happen within another dream. Consequently, when the two commit suicide in limbo they would not have awoken to reality, but instead just entered a shallower level of the dream. Of course, it is possible that this is just a plot hole or a case of classical editorial intelligence in which superfluous information has been omitted for the purpose of privileging narrative propulsion. Nevertheless, some supporters of this reading maintain that Mal is trying to plant the idea of self-forgiveness into Cobb's mind. In particular, she is attempting to convince Cobb that he should not feel guilty for incepting her initially and that all will be well with their family when he finally awakens.
For subscribers of the Mal inception theory, there are many textual clues that can be leveraged to buttress its primary claims. For starters, they posit that Mal is conspiring with others to complete her objective. Cobb's father-in-law, Miles (Michael Caine), for instance, who never appears in the film's sequences ostensibly coded as part of the dream, is reinterpreted to be one of her primary collaborators. Hence, when Cobb conveniently appears in Miles's lecture hall and his father-in-law pleads with him to "come back to reality," he is actually communicating with his son-in-law in the dream state as a member of Mal's incepting team, rather than as simply a concerned family member trying to impart sage advice in reality. Additionally, this interpretation retroactively makes Mal's climactic disagreement with Cobb in limbo about who is right in relation to their ontological state the film's most important dialogue exchange. As Mal explains, it does seem coincidental that Cobb's purported reality is characterized by a persecution complex in which he is chased around the globe by mysterious agents who strikingly resemble the subconscious projections that attack dreamers. Put simply, this interpretation provides a conspiratorial, all-encompassing way to refute claims that Mal is the villain by transforming her into a character benevolently attempting to save her husband from his own guilty conscience.
Despite how such a reading helps to lessen the claims about Nolan's misogynistic tendencies, it does not completely absolve the film from having sexist sensibilities. Even if Mal, whose name in Spanish translates into "bad" in English, is actually a hero and not the antagonist, she is still secretly manipulating her husband's actions behind the scenes. Regardless of whether or not viewers interpret the film as a depiction of only an extended dream, then, Cobb remains the archetypal misdirection film protagonist by being portrayed as emasculated by powerful feminizing forces beyond his control. If it is all just a dream, then Cobb is, at best, trapped in a fantasy that he cannot or does not want to be liberated from or is, at worst, at the mercy of others, most likely his wife, who are incepting him with thoughts that are not his own. In contrast, if the distinctions between dream and reality can actually be sustained, then the film is a representation of his attempts to overcome the seemingly insurmountable obstacles that are attributable to his wife's misguided actions, which stem from her dimwitted inability to discern fantasy from reality, albeit thanks to Cobb's initial and unethical inception of her. Although the latter explanation provides Cobb with greater and more laudable agency than the former interpretation, it still positions him as a wounded man who must surmount Mal's imprudent decision to frame him for her murder because she hoped her suicide threat would also convince him to kill himself.
Such gender dynamics not only align Inception with the cultural politics of many misdirection films. They also overlap with the gendered representations contained in most blockbusters. One of the major issues that scholars, like King, have to contend with in their arguments that the seemingly incongruent Hollywood Renaissance and blockbuster-era are more alike than different is the oppositional cultural politics that characterize each production trend. As King details, it is tempting to link each moment to its corresponding zeitgeist because of the overarching political beliefs that are said to define the respective times. This line of thinking makes sense at a cursory level. Whereas the Hollywood Renaissance and its artistic innovation coincided with the countercultural revolution and is known for expressing largely progressive values, the shift to the blockbuster model and its more classically oriented filmmaking was contemporaneous with the Reagan-era and a turn to conservatism in the United States. King counters this simplistic account by highlighting the other contexts, particularly shifting industrial circumstances that influenced these developments as much as, if not more than, changing cultural conditions. He does grant, however, that the transition to blockbuster production did mean largely abandoning the experimentation of the Hollywood Renaissance in favor of a return to traditional classical principles, albeit not primarily because of broader political developments. This shift did have cultural repercussions, though. As King summarizes, since the fall of the studio system, "male audiences have been targeted more heavily" and "male-oriented genres have flourished," which "provides another explanation for the prominence of science fiction and the action film in the contemporary blockbuster economy" (New Hollywood 138). Of course, there are examples of constituent films, like Titanic, that performed exceptionally with female viewers, and not only men are attracted to the genres most affiliated with blockbuster production. What is clear, though, is that Inception, with its heavy science fiction and action film attributes as well as the casting of DiCaprio to appeal to female spectators enamored with his good looks was designed, at least in part, to correspond to the dominant production logic of the blockbuster-era, marking an important moment in the misdirection film's history.
## Is It Better to Make a Low- or High-Art Misdirection Film?: The Kafkaesque Genius of Shutter Island's Middlebrow Appeal
In contrast to Inception's clear status as a blockbuster, it is challenging to make a case for Shutter Island's inclusion in the category. The attachment of superstar branded commodities to the project, headlined by the reteaming of DiCaprio and Scorsese, certainly gave it blockbuster potential. Yet, the two's track record at the box office has not netted blockbuster revenues to date. In fact, Shutter Island is the duo's most profitable film so far in domestic theaters, garnering approximately $48 million over its estimated $80 million production costs (imdb.com). Such comparatively modest revenues and profit margin, however, are hardly blockbuster material. Even more notable, though, is the film's failure to match the Oscar attention heaped on all of the other collaborations between the two. Each other time the duo has teamed, it resulted in plentiful Academy Award nominations, including 10 for their first collaboration, Gangs of New York, 11 for The Aviator, five for The Wolf of Wall Street, and six for The Departed, which finally captured the biggest Oscars that had eluded Scorsese for decades (imdb.com). With zero Academy Award nominations, Shutter Island is obviously an outlier. These results cannot be blamed on its status as a misdirection film, however, as Inception was well-decorated at the 2011 Oscar ceremony, earning four wins in the technical categories and even garnering a rare nomination for both a blockbuster and misdirection film for Best Picture (imdb.com).
Instead, Shutter Island's anomalous result with the Academy of Motion Picture Arts and Sciences is largely attributable to its distribution strategy. All indications point to the film being initially designed to garner significant Oscar attention. In addition to it being the adaptation of Lehane's follow-up novel to Mystic River as well as the first reteaming of Scorsese and DiCaprio since The Departed, it was originally slotted for an autumn release, the time the media conglomerates reserve for their strongest Oscar hopefuls to give them momentum for the impending voting process. This distribution tactic has only grown more important economically, as post-theatrical revenues have increased because release window staggering schedules mean that home-video versions are first available near the time of the ceremony, bolstering the potential profits of nominees. Viacom's Paramount film division, however, shockingly decided to delay Shutter Island's initially announced October 2, 2009 domestic theatrical release until February 19, 2010. Paramount's CEO Brad Grey justified the decision by claiming that they made the move "because of the financial pressures associated with the downturn" (qtd. in Finke). It is improbable that the lingering impact of the 2008 U.S. financial crisis was the sole, or determining, factor in the shift. According to rumors, the delay was more likely attributable to DiCaprio's purported unavailability to travel for the international promotional junket and to concerns about rapidly declining profits that DVD sales started producing by then (Finke).
Precipitously declining DVD revenue is a liable culprit for Paramount's decision to alter its distribution plans. By 2010, the downturn in DVD purchases that began a few years earlier had plunged to unforeseen depths. Although the technology had run its course with consumers and consequently started to dip slightly in 2007, total revenues had already plummeted to less than half of what they were at that time by 2010, as DVD sales generated less than $5 billion for the industry in that year (Holden, Wade). This was a sobering statistic for Hollywood executives who had greenlit films due to their potential to perform exceptionally in the aftermarket, especially considering that the 2010 sales figures represented an over $3 billion drop from the nearly $8 billion revenue total of 2009, even though the industry released over 60 more titles on DVD in 2010 than it did in 2009 (Holden, Wade). The goal of constructing Shutter Island for the DVD market was central to its production logic, at least according to quotes attributed to Scorsese. James Gilligan, the film's psychiatric advisor, reported that "Scorsese said this film will make double the income because people will have to see it a second time to understand what happened the first time" (qtd. in Cox). Such discursive evidence indicates that the director was keenly aware of how changing revenue streams in the preceding years had altered Hollywood's production tactics. Although Scorsese was right that the industry still eagerly backed films for the DVD market when he joined the project in 2007, the sudden implosion of that line of revenue thereafter helps explain why Paramount unexpectedly postponed the release of a film that once seemed primed for optimal aftermarket performance. By 2010, Hollywood had practically abandoned DVD, preferring to privilege films that would do best on other post-theatrical platforms, such as on-demand and online streaming options, which encourage ephemeral rentals akin to the VCR-era, rather than the permanent purchases inspired by DVD.
Consequently, even though it received no Academy Award consideration, I contend that Shutter Island was conceived of as a prototypical prestige product at the outset. Had the film not been subject to industrial circumstances beyond the filmmakers' control, it likely would have been honored with numerous Oscar nominations, especially given the results of all other Scorsese and DiCaprio collaborations. Of course, this is speculative in spite of the corroborating evidence that suggests its poor fate with Oscar voters was virtually sealed when its release was pushed back to February since few films that hit theaters during the first months of the calendar year are ultimately recognized by the Academy. Yet, an examination of the ways in which the film was marketed and constructed illustrates that it was originally created with a middlebrow sensibility designed to maximize its profit and cachet.
Small distinctions between the first trailer made for Shutter Island and a subsequent one cut after the release date was delayed begin to highlight how high- and low-art elements were combined in the film. As is standard for New Hollywood advertising, the two trailers strongly foreground generic hybridity, as both accentuate its affiliations with film noir, the psychological thriller, horror, and the misdirection film. In the first trailer, the psychological thriller elements are prioritized in the presentation of the film's quest narrative, the film noir aspects are explicit in formal aspects, particularly in the mise-en-scène, while facets of horror and misdirection are peppered in, such as when a disturbingly disfigured prisoner stares at the camera and indirectly tells the spectator that "This is a game. You're a rat in a maze." What differentiates the second similar, but slightly re-edited, version of the preview, which advertises the belated February release date, is its greater emphasis on the film's horror properties. Images featuring gore absent from the initial trailer are present in the revised one, including shots of a character's face covered in blood. Additionally, the preview ends with a prototypical jump scare when a mysterious character leaps out of the darkness onto another one. These minor changes indicate that producers knowingly shifted their marketing approach to accentuate the film's more traditionally debased aspects after it was clear that it was now likely out of serious Oscar contention.
One reason why Shutter Island contains many horror facets and elements of other genres that are often degraded is because evidence suggests it was partly designed as Scorsese's extended homages to those films. Such properties are unsurprising because references to film history are one of the director's authorial calling cards, a fact made even more evident in his follow-up to Shutter Island, Hugo (2011), his love letter to Georges Méliès and early cinema, which returned Scorsese to familiar Academy Award fame by garnering 11 nominations and five wins (imdb.com). Framing the film for readers of a promotional interview the director did for TimeOut London magazine, Dave Calhoun describes Shutter Island as a "1950s-set thriller with Hitchcockian B-movie flavor." As this illustrates, the film was received by some critics as an amalgamation of low- and high-art elements. The combination of B-movie allusions with direct references to Hitchcock, a filmmaker once considered to be little more than a commercial entertainer, whose reputation has since transformed into that of a revered artist, epitomizes reviewers' reactions. For some critics, the B-movie traditions hampered the film. Joe Neumaier of the New York Daily News, for instance, complains about the script and DiCaprio's consequent performance by noting the actor "often simply mouths questions and waits for some B-movie tradition (the jabbering madman, the German doctor, the mystery woman) to provide answers."
Other reviewers, though, considered Shutter Island's many homages to film history to be a strength, especially if they could be linked to Hitchcock, arguably the most important progenitor of the contemporary misdirection film. In an article for The Guardian's website, entitled "Martin Scorsese: Master of the Hitchcock Tribute," Andrew Pulver writes that Scorsese "has taken the Hitchcock atmosphere of murderous insanity and run with it, shoehorning in one Hitchcock bit after another." He subsequently provides YouTube clip evidence of how Shutter Island references many of Hitchcock's films, including Psycho (1960), Spellbound (1945), Marnie (1964), Vertigo (1958), and Notorious (1946). There is a shot, for instance, that exactly replicates the one of the showerhead in Psycho and a scene of a run up a twisty, lighthouse staircase that evokes the treks up the bell tower stairs in Vertigo. The fact that Scorsese was thinking of Hitchcock while making the film is supported by his thematic preoccupations and the production history. As he told Calhoun, he "showed his colleagues The Wrong Man [1957]" to prepare for Shutter Island because "The main character in that is innocent but he feels guilty for who he is... I was raised a Catholic and I'm interested in that aspect of ourselves."
Shutter Island similarly focuses on a tormented protagonist, who is referred to as Edward "Teddy" Daniels or Andrew Laeddis (Leonardo DiCaprio), and has been committed to Ashecliffe mental institution for the criminally insane on Shutter Island. He is sentenced for his inability to cope with the guilt associated with his murder of his wife, who is called both Rachel Solando and Dolores Chanal (Michelle Williams), after he purportedly both denied her insanity and found she drowned their three children. Complicating things for the viewer is the fact that Teddy's (the name I use to refer to him as for clarity's sake) status as a patient is neither evident to him nor the spectator until the changeover, which is highly indebted to The Cabinet of Dr. Caligari's (1920) similar epiphany. The changeover exposes why it is difficult to know what "really" happened because it reveals that Teddy is the subject of Dr. Cawley's (Ben Kingsley) conspiratorial scheme. For the radical psychological experiment, the doctors pretend that Teddy is a U.S. Marshall, partnered with his primary psychiatrist, Dr. Sheehan (Mark Ruffalo), in disguise. Teddy is tricked into believing the two are investigating an escaped patient, Rachel Solando (Emily Mortimer). This is intended to make him realize he is living a lie to save him from a lobotomy, a ploy that initially seems to work. As Cawley explains during the changeover that leads to Teddy's cognitive crisis, for example, his and his wife's alter-egos are anagrams for the elaborate ruse. Like Inception, which also centers on DiCaprio's character's overcoming the guilt of his wife's apparent death, though, it is difficult to know what "actually" happened despite the revelation. Although Cawley seems to give a totalizing account during the changeover, its veracity depends on the viewer's beliefs about the doctors' motives and Teddy's mental state. This is because, as with Memento, the narrative is focalized through an unreliable protagonist with psychological issues, rendering him a potential pawn and his traits, such as his employment as a U.S. Marshall, open to scrutiny. Textual evidence that precedes the changeover bolsters this vagueness because Cawley goes to ethically dubious extremes to prove that his psychological methods are superior to alternatives he deems outmoded. It is also possible, however, that Cawley seems that way because the viewer learns about him through a delusional protagonist, or because it is just a cover for a more nefarious plot, making it hard to delineate fact from fiction.
Figure 6.3. Edward "Teddy" Daniels walks away from Dr. Sheehan in Shutter Island after uttering his ambiguous final line of dialogue.
Such ambiguity is compounded by what follows the changeover, as Teddy's last line of dialogue in which, after appearing to regress back into insanity, he waxes poetic about if it would be "worse to live as a monster, or to die as a good man?" This unexpected ending indicates that he is likely faking his relapse to submit to brain surgery, rather than either to continue to face what he now perceives to be true about his identity, or to fight the doctors futilely. Such a reading is supported by the film's ensuing, final shot of a lighthouse, the location where Teddy suspects the lobotomies happen. On cursory review, this implies that Teddy has consciously sabotaged Cawley's plan to convince the Board of Overseers and the doctor's supposed rival, Dr. Naehring (Max von Sydow), that his mental illness can be treated without brain surgery. Cawley informs Teddy during the changeover, though, that he relapsed after they had a previous breakthrough, suggesting that his current regression could be genuine. As Cox summarizes, this uncertainty has helped the film to be discussed intently by fans since its release. He also notes that, like Inception, Scorsese and DiCaprio have refused to provide further detail about the "correct" interpretation, only fueling more fan activity. This is exacerbated by the fact that Teddy's final line of dialogue is original to Laeta Kalogridis's screenplay, making Lehane's source novel a dead end. As Caine did in relation to Inception, however, Cox claims that Gilligan reported that the ending is unambiguous because it unequivocally reveals that Teddy was "too guilty to go on living," meaning that he is "going to vicariously commit suicide by handing [him]self over to the people who're going to lobotomise [him]." According to the film's psychiatric advisor, this reading is crucial because lobotomies are now threatening to return to the actual field. Like Cawley's explanation, then, Gilligan's declaration about his interpretation being definitive is questionable largely because of his vested interest in making the assertion.
Many spectators refuse to accept this simple explanation as absolute and instead leverage the film's lingering ambiguity to interpret it in numerous other ways, regardless of what Teddy's final rhetorical question and the concluding shot of the lighthouse insinuate. Again, recourse to authorship is a driving force in this regard because Scorsese has used eternal narrative uncertainty to probe the thin line separating sanity and insanity before, most famously in Taxi Driver. For most of these fans, Shutter Island's ending resembles Taxi Driver's unclear resolution of its protagonist's mental state and Inception's spinning top. The question of whether Teddy has regressed is thus irrelevant to them. Rather, these viewers begin from the premise that Teddy has consciously elected brain surgery and instead focus on the "real" reasons for that choice. These theories largely hinge on deciphering the doctors' objectives, rejecting Cawley's explanation during the changeover as truth. One such compelling interpretation is housed on an anonymous, amateur website, the Wistful Writer, which contains posts about the male author's (he uses male pronouns to describe himself) assorted interests, including film and literature. When I retrieved his "The Shutter Island Mystery" essay in August 2015, the site's listings of the month's top posts clearly revealed that this entry was its primary draw and had resonated with readers, as it had over 5,000 views, dwarfing the second-place finisher, which had under 400 hits. Importantly, he frames his sophisticated analysis as just one interpretation, claiming that he does not intend "to impose it on others." His reading, then, is predicated on the notion that the film's meaning cannot be understood definitively, a position that differs from standard misdirection film reception and typical readings of Shutter Island. Most participants on the film's Internet Movie Database "Message Board," for example, make absolute claims about the film's meaning, such as is if the protagonist is "actually" Teddy or Andrew. In contrast, this author reiterates the film's lasting ambiguity even in relation to his own account by noting "there are many issues with all interpretations of the story, including mine."
One aspect that makes the interpretation on the Wistful Writer's site compelling is its interesting overlaps with Inception. In particular, he argues that Cawley is anything but a benevolent psychiatrist because he is instead trying to plant false memories in Teddy's mind for malevolent purposes. Such a reading begins not only to explain the questionable ethics of Cawley and his associates, but it also imbues the film's many conspiratorial references to Ashecliffe actually being a government-funded facility run by the military to conduct clandestine tests with greater narrative significance. According to this interpretation, Ashecliffe is "really" comprised of doctors testing Nazi-inspired mind control experiments on patients to create more adept special soldiers to fight communism during the Cold War. Consequently, the seemingly paranoid beliefs that Teddy has and learns about what transpires on Shutter Island behind the scenes might not be delusions. That is not to say that all of his speculations are accurate, as even this theory's author grants that "maybe Teddy is just a conspiracy theorist obsessed with intelligence agencies." As he also notes, however, it is strange that the facility is policed by guards with military vehicles and guns, that an orderly does not dispute Teddy's claim that the Warden (Ted Levine) is "an ex-military prick," as well as that Cawley is renowned for his work with "Scotland Yard, MI5, and the OSS." This interpretation, then, aligns the film more closely with Arlington Road (1999) and, especially, with Jacob's Ladder than with any other misdirection film because of how it depicts agency panic about a protagonist made into a patsy by a powerful organization that transforms him into a warrior for their cause. Most importantly, like Jacob's Ladder, if Teddy is "actually" a test patient, it is difficult to determine what, if anything, is factual. This renders its mix of low- and high-art more narratively meaningful retrospectively because those elements can all be understood as fabrications for the ruse.
Figure 6.4. Visual evidence of Shutter Island's Ashecliffe Hospital being run and guarded by the U.S. military.
The theory forwarded by the author of the Wistful Writer site hinges primarily on challenging the claims that Cawley and his associates make about Teddy's "true" identity and his personal history by countering that they are "actually" part of the cover story. To wit, he contends that Teddy was never a U.S. Marshal, Andrew Laeddis (Elias Koteas) is a product of his dissociative identity disorder, and that he did not kill his wife for murdering their kids because they were actually childless. He instead posits that Teddy was committed to Ashecliffe for being a pyromaniac, who killed his wife when he burned their apartment building. To repress the crime, he creates the Laeddis alter-ego and constructs an alternative past. Thus, he is ideal for the conspiratorial ploy because he has already lost grasp of reality and his history. Hence the reason he identifies Laeddis as the "firebug" who killed his wife and why, when the manifestation of his dissociative identity disorder appears, he looks evil and deformed. As he also speculates, Andrew Laeddis seems much more like a fictional name than Edward Daniels. The contention that Teddy is "actually" an arsonist is supported by how he blows up Cawley's car with only his tie and a pebble, a feat the doctor seems to confirm during the changeover, as well as how he and Laeddis are shown in close-ups lighting matches the same way. In contrast, he reconstructs his Teddy identity as heroic, explaining why the doctors make him into a fake U.S. Marshall, complete with the film noir traits for the role play. Although this reading might be far-fetched, it explains ambiguities unresolved by the changeover's account. For starters, it highlights why Teddy is adamant that "it was the smoke that got her" in the fire that kills his wife and never mentions his children until Cawley prompts him, even though he says that "four people died" in the blaze. It also reveals why the doctors act unethically by having a nurse impersonate Rachel, patients submit to Teddy's interrogations, and Teddy fight other patients.
These inconsistencies are just the beginning because the interpretive possibilities created by such a theory potentially make all of Shutter Island's ambiguities narratively meaningful, in retrospect. Most significantly, they transform all of the B-film and high-art references from being potentially read as primarily artistic flourishes or authorial signatures into narratively relevant aspects. That is, these ostensibly outlandish elements become more plausible in relation to the story if they are reinterpreted as beholden to an evil mind-control scheme, as opposed to a compassionately curative plan. Indeed, the high- and low-art properties can now be understood as Hollywood-inspired creations that make the fakery more dramatic and believable for Teddy and the viewer, albeit often anachronistically, as the references to Hitchcock's post-1954 films demonstrate. When the viewer first sees Shutter Island through Teddy's eyes as he nears it on the boat he has been placed on, for example, the foreboding shot, accompanied by the ominous soundtrack, directly evoke Skull Island in the monster film, King Kong (1933). Likewise, the film noir qualities, such as the ubiquitous fedoras, trench coats, and chain-smoking, can be reinterpreted as conscious decisions by the scheming doctors that are even more related to that genre's convention of the protagonist's inescapable past haunting him because they manufactured that history. The conspiratorial story that Sheehan feeds Teddy in the creepy mausoleum also makes more sense if it is understood as designed by the doctors to spook their mark. Similarly, Naehring's amplification of his German heritage takes on greater meaning if it is comprehended as helping to convince Teddy about what "really" transpired during his military service and his associated memories of the liberation of Dachau. Obviously, the B-film references extend beyond these examples, but what is relevant is how the mind-control explanation retrospectively makes them more narratively significant for Teddy and the viewer.
The same is true of the high-art elements, which are most evident during Naehring's introduction. Teddy seamlessly exchanges German dialogue about concentration camps with Naehring, which is uncharacteristically not subtitled for classical viewers, encouraging them to search for a translation. Teddy also displays his knowledge of German culture by correcting Sheehan's likely purposeful misidentification of diegetic music as being composed by the Austrian Mahler, whose music would have been banned by the Nazis because he was a Jew, and not by the German Brahms, as his fake partner claims. These two seemingly benign moments are important retrospectively, according to the Wistful Writer, who argues that although Teddy was a soldier during WWII, it is difficult to know what he "actually" experienced. Cawley claims during the changeover that Teddy was "at the liberation of Dachau," but it is unknown if he killed any guards. Yet, during Teddy's flashback-inspired memory of Dachau, he murders German soldiers and makes the Kommandant suffer a slow death from a botched suicide. Crucially, then, the thoughts the doctors implant differ from Teddy's memories. The issue of Teddy's violence is frequently broached, most memorably during his disconcerting exchange with the sadistic Warden, who is costumed like an SS officer and unethically tries to incite the patient's ferocity. This is significant in light of the fact that the girl who haunts his memories from the mass of bodies at Dachau for not being able to save her becomes his daughter, Rachel Laeddis, in Cawley's revised explanation. As the Wistful Writer contends, however, she is the only of Teddy's supposed children who communicates with him in his delusions, raising doubt about him being a father at all. The doctors' alteration of the Dachau account, then, can be read as part of the effort to convince Teddy that his wife did not die in the fire he lit because he instead reverted to his "violent nature" after discovering she murdered their kids.
As the Rachel Solando (Patricia Clarkson), whom Teddy discovers hiding in the cave and provides much of the source material for the mind-control reinterpretation, such as how the patient has been surreptitiously fed psychotropic drugs by the doctors to instigate his delusions since the role play began, claims, this is the film's "Kafkaesque genius." Once Teddy is declared insane by the doctors, any of his attempts to refute their diagnosis can be used against him as further proof of his mental illness. Although she is almost certainly just a manifestation of Teddy's paranoid visions, her high-art reference to Franz Kafka's literature is perhaps the biggest clue to deciphering Shutter Island in this way. As the Kafkaesque allusion connotes, the author became renowned for expressing the confusion associated with the bureaucratic madness of modernity in his novels, especially The Trial. Like that book's protagonist, Josef K., who is unable to determine what he has been arrested for, or how to mount an appropriate defense in response, both Teddy and the viewer are left to ponder eternally what crimes he "actually" committed and what the doctors are "really" doing to him as a result. Again, such an intertextual connection only becomes possible if viewers accept that the changeover and what ensues, particularly Teddy's last line of dialogue, open more questions than provide definitive answers about the narrative's "true" meaning and have knowledge of Kafka's output.
This lingering ambiguity is further supported by the film's mise-en-scène, which, like Memento's, a film that it is also similar to, is itself unreliable. This becomes most evident during the scene when Teddy interrogates Mrs. Kearns (Robin Bartlett), in which she asks for a glass of water that appears to be there initially, disappears when she goes to drink it, and then reappears on the table when she sets it down. This is not the only instance when such visual unreliability occurs, as, for example, a liquor bottle initially seen in Teddy's wife's hand during a dream sequence disappears subsequently. Obviously, this impossibility can be attributed to the irrationality of dream logic; however, a similar discontinuity in relation to the lighthouse is less easy to explain. Although Deputy Warden McPherson (John Carroll Lynch) indicates that the lighthouse is a "sewage treatment facility," Teddy eventually believes that it is where the brain surgeries are conducted. When he finally arrives there, though, he discovers it is practically empty, aside from a small office where Cawley is waiting for him to provide the explanation that serves as the changeover. Yet, the lighthouse shown in the final shot does not exactly match the one he initially sees or visits because, as the Wistful Writer shows, it is not bordered by the same fence, raising the possibility that something underhanded is actually occurring in the alternate lighthouse that Teddy never actually finds. Cawley himself insinuates what might be occurring there when he taunts Teddy during the changeover by asking him "where are the Nazi experiments... the Satanic O.R.s?" The viewer, like Teddy, can never know because the insides of the possible second lighthouse remain forever undisclosed.
Figure 6.5. Shutter Island's final shot presents a seemingly distinct lighthouse from the previous one, as the long fence extending beyond the small one bordering it alone has vanished.
In contrast to the liquor bottle inconsistency, but like the lighthouse discrepancy, it is difficult to attribute the water glass issue to delusion, though, the retrospective knowledge that Teddy is experiencing psychological issues potentially explains its rationale in that way. How viewers ultimately interpret the scene's meaning, then, is again ultimately contingent on what they believe about the doctors' motives. Importantly, the scene initially seems to be a quintessential misdirection film moment because, in retrospect, it highlights how classical formal devices disguise the impending changeover at the same time that they flaunt it for viewer discovery. Specifically, when Teddy presses Mrs. Kearns about the purportedly absent Sheehan, the camera cuts numerous times, in prototypical shot/reverse shot fashion, to the psychiatrist's facial reactions, practically giving away a big secret, as both Teddy and the audience still believe he is also just a U.S. Marshall investigating the mystery. The way the water glass sequence is formally constructed, however, complicates this standard misdirection film reading. Although its illogical properties could be the product of a continuity error or an indication of Teddy's increasingly delusional perception, neither of those is the only possible, nor necessarily most convincing, explanations. This is because the full glass is clearly visible in Sheehan's hand, until the film cuts to a close-up of Mrs. Kearns imaginarily drinking it, before it cuts again to a birds-eye view of her placing the now empty cup on the table. What is most relevant about the three shots is that none of them are unquestionably framed as being from Teddy's perspective, though, the one with nothing in her hand can certainly be interpreted as portraying his vision; however, Sheehan is situated directly behind Teddy at that moment, making it as likely that the viewer could be getting the information from his perspective since he also frequently exchanges gazes with Mrs. Kearns during the discussion.
The point is that it is impossible to tell what "actually" happened or why it "really" occurred in that scene and in relation to the lighthouse, which makes them both fitting microcosms for how the viewer seems to be encouraged to interpret the entire film. The film's mise-en-scène, then, may ultimately be a red herring, too, suggesting that, more than anything else, Shutter Island is likely an unsolvable Kafkaesque narrative that does not provide a definitive answer about who Teddy is or what the doctors are doing to him. As many fans have pointed out, water and fire are deployed throughout the film constantly to signal the unreliability of Teddy's perceptions. In relation to water, there are many clues, in addition to the aforementioned glass, that suggest its significance, including the opening scene when Teddy splashes water on his face after expressing his dislike of it from getting sick on a boat in which the shackles that held him before the role play began are visible, the torrential rains that consistently obscure his ability to discern reality, and the water leaking from the fake revolver that he breaks during the changeover. Similarly, fire is also associated with his delusions, as the same roaring fire that demonically appears behind Naehring during his introduction rages behind the also sinister Laeddis, and a campfire is the only source of light when he encounters the cave-dwelling Rachel. Both elemental symbols, then, are not only linked to Teddy's unreliability; they are also connected to the two most plausible explanations given for his psychological instability: his inability to cope with his wife's death from his arson, or murdering her after she drowned their children. Either way, it is clear that Teddy is delusional and has been the subject of an elaborate role play. The ambiguity about what killed his wife, then, does little to change the viewer's conception of Teddy's condition and his lack of agency. Instead, it creates considerable uncertainty about why he is mentally ill and what the doctors are doing about it. That is, the film ultimately leaves it to the viewer to decide if Cawley conducted the role play in a last ditch effort to save Teddy from a lobotomy, or if the doctors conspired to manipulate the patient by drugging him and then subjecting him to brain surgery to complete his transformation.
Regardless of how the viewer interprets Shutter Island as a result of this lingering ambiguity, its connections to the broader thematic concerns and stylistic tendencies of Inception, particularly, and the misdirection film genre, generally, are clear. Whichever way it is read, it depicts a protagonist who lost his autonomy because of a traumatic event linked to his wife. His inability to exert traditional masculine control over his family caused him either to commit familicide by killing his wife because of his unhealthy obsession with fire or to murder her out of revenge for drowning their kids. This act resulted in him having to submit to the will of the doctors who now either benevolently or nefariously control his fate. As the last line of dialogue reveals, the only possible agency that remains for Teddy is faking his regression to save himself from guilt or giving in to the doctors' plans for brain surgery. Like the viewer of the misdirection film, then, Teddy knows he is being manipulated, but cannot do anything about it. This inability to know what the doctors are "really" doing is largely a product of storytelling and representational decisions to focalize the film through an unreliable protagonist, afflicted with "actual" or manufactured delusions. These indisputably paranoid visions make the film's expert combination of high- and low-art elements more narratively significant, in retrospect, because they can be read as products of Teddy's imaginings that are heightened either by the compassionate role play, or the doctors' drugging him as part of an evil plot to fulfill their sinister agenda. Such qualities align the film closely with Inception, which also presents an irresolvable puzzle that puts the viewer in the position of its broken protagonist who struggles to maintain a fading grasp on reality because of a tragedy related to his wife. That film's eternal uncertainty about the distinction between what is real and imaginary renders its many blockbuster elements more narratively relevant. Both films' similar presentation of the ontological blurring between reality and fantasy made them ripe for examination in virtual communities. These two films, therefore, are the contemporary misdirection film's apotheosis because they successfully attracted wide audiences by crystalizing the genre's formal tendencies and thematic concerns in Hollywood's most expensive and prestigious packages.
## Conclusion
The 2010 cultural and industrial zenith of the genre was short-lived, as comparatively few misdirection films have been released since then. Instead, narrative complexity has become more associated with commercial American television than it has with Hollywood film since at least that time, especially on premium cable. Although they were slower to seize the opportunity, television producers have also attempted to augment their bottomlines and authorial reputations by creating shows, which contain narratives that encourage viewers to watch them repeatedly and discuss them fervently in virtual communities. This evidence suggests that changing cultural, industrial, and technological conditions have similarly impacted the narrative strategies that have been deployed of late in commercial, moving-image media forms other than film. The proliferation of computer role-playing games since the 1990s, for instance, has been triggered by industrial motives and audience desires that resemble those that contributed to the rise in contemporary Hollywood misdirection films. Popular role-playing videogame franchises, such as the Final Fantasy series, which began as console games and have developed into massively multiplayer online games, similarly require gamers to devote considerable time and energy to solving their mysteries by sharing resources in virtual communities to unearth their secrets. My brief examination of recent narrative developments on television in this Conclusion, then, indicates potential areas for further study by showing that Hollywood was at the forefront of a trend that has subsequently become more common across the U.S. media industries.
Shows containing complex narratives that prompt audiences to watch them obsessively and go online to unravel their mysteries have indeed become more customary of late on U.S. television. Even a reality show that has become a cross-cultural phenomenon, like Survivor (2000– ), appeals to producers and spectators partly for reasons that are similar to those associated to the contemporary Hollywood misdirection film. Henry Jenkins contends that Survivor is "television for the Internet age—designed to be discussed, debated, predicted, and critiqued" (Convergence 25). As his analysis of the online message boards devoted to the show demonstrates, it was initially successful in attracting an ardent fan base dedicated to cracking its code. This transpired largely because the show coaxed viewers into guessing its most significant outcomes. Participants on these sites rely on evidence gleaned from the episodes themselves and their paratexts to draw conclusions before they air, such as who is going to be voted off the island each week, who makes it to the final four, and who will eventually be the winner. These hypotheses are difficult to make, however, because Mark Burnett, the U.S. version's executive producer, admits to monitoring message boards to edit the show in ways that provoke viewers to draw incorrect conclusions. Fan discourse, Jenkins writes, suggests that Burnett's efforts effectively positioned the show as a competition between producers and viewers, buttressed by "a belief that through a contest over information, some ultimate truth will emerge" (Convergence 43). These interpretive activities, therefore, resemble the behaviors of conspiracy theorists, who compete to derive superior alternative explanations. Like conspiracy theories, though, fan predictions are not necessarily intended to reveal the "truth." As Jenkins notes, viewers are generally disappointed by the premature exposure of secrets because they revel in continually battling Burnett. In sum, Survivor has appealed to a loyal audience who enjoys outwitting, outsmarting, and outlasting its purported mastermind, who strives to conceal the "truth."
The lingering ambiguity that remains unresolved in each episode of Survivor is not uncommon in commercial U.S. television programming. For decades, serialized television shows, which were once relegated primarily to daytime soap operas and primetime dramas, have contained ongoing storylines that extend for full seasons and beyond. As Michael Newman notes, however, this trend has migrated to other types of shows not traditionally affiliated with seriality. Newman claims, for instance, that Survivor and Arrested Development (2003–2006) exemplify how genres, like reality programming and the sitcom, now have narratives that are "thoroughly serialized" ("From Beats" 16). In contrast to the episodic format that once dominated fictional television programming, a wider variety of shows have recently relied on devices typically central to the serial, including open-ended storylines, long narrative arcs, and a sustained focus on character development. Survivor, for instance, helped inspire a litany of game-show-themed, reality programs that are highly invested in character and in which many narrative ambiguities remain unresolved after an episode or season's end. The growing popularity of the serial on television was perhaps best evidenced by the success of expensive dramas, like 24 (2001–2010), Lost (2004–2010), and Heroes (2006–2010), three of the most narratively complex shows ever to air on the traditional powerhouses at the time, that were uncharacteristically greenlit by historically risk averse network television executives. Of course, as I demonstrate in chapter 4, a show like Twin Peaks (1990–1991) indicates that programs requiring spectators to re-watch them and discuss their meanings in virtual communities to appreciate them deeply are not new. Since the early 2000s, however, shifting industrial and technological contexts have made these kinds of shows more attractive to producers and some spectators, suggesting that they are enticing in part because of how they respond similarly to the cultural paranoia that made the misdirection film appealing.
Although Twin Peaks lasted only two seasons, despite generating a small, yet loyal fan base, shows, such as 24, Lost, and Heroes, remained on the air for longer periods even though their intricate narratives alienated some viewers. This is largely because television, like the film industry, benefits from the additional revenue streams created by DVD and amplified by subsequent platforms, like direct-for-purchase streaming video. As Derek Kompare details in "Publishing Flow," the success of the DVD player changed the economic strategies of television executives. Prior to the VCR, it was difficult for viewers to re-watch their most beloved programs; audiences usually had to wait for reruns or hope their favorite shows would go into syndication to view them more times. Accordingly, television executives once solely aimed to create content that produced the greatest number of targeted viewers for advertisers because that was the industry's lone source of income. As Kompare documents, however, DVD was the first technology that allowed the television industry to adopt a "publishing" model in which it could sell its product directly to consumers, enabling the industry to turn profits from shows that built niche audiences at a slower rate. Of course, the VCR provided viewers the ability to record and replay programs at their leisure before the DVD player existed. However, the VCR was primarily embraced by consumers because it enabled them to view Hollywood films in the home. Additionally, the size, cost, and approximately two-hour capacity of the VHS tape made it virtually impossible for either retailers or consumers to devote the money and shelf-space necessary to obtain even a single season's worth of episodes. Consequently, most serialized programs were not transferred to VHS because their long narrative arcs did not translate well to the best-of format that was standard on the technology. The DVD, in contrast, could be hawked straight to consumers in a high quality, user-friendly, and comparatively cost-effective package.
Yet, the television industry did not capitalize on the DVD player when it became available to consumers in 1997 because it was initially perceived to be an extension of the VCR. During its first few years on the market, television executives remained unconvinced that there was the same kind of home-video customer base for their products as there was for Hollywood films. However, when the television DVD box set became available in 2000, it rapidly changed the industry's strategy. As Kompare writes, the DVD box set "brought television's home video practices more in line with film's and indicates how new technologies can prompt new uses and new practices while preserving old goals" ("Publishing" 338). It is still important for television producers to please advertisers because they remain the financial lifeblood of the industry. The additional revenue streams created by DVD technology and ensuing platforms, though, have undeniably altered the television industry's production tactics. In an age dominated by corporate consolidation and the logic of synergy, it has provided the media conglomerates that now control commercial U.S. television the chance to reap enormous profits from the sale of shows on DVD. As early as fall 2002, for example, "Fox chairman Peter Chernin reportedly claimed that television on DVD had already generated $100 million of revenue for his studio" (qtd. in Kompare "Publishing" 352). This staggering figure begins to illustrate why television executives eagerly backed shows that contain complex narratives that appealed to narrower, but loyal, audiences attracted to how those programs tap into desires and anxieties about the status of the "truth" in ways that resemble the misdirection film's expressions.
Interestingly, the first series ever to be manufactured in the box-set format was The X-Files (1993–2002), a narratively convoluted show that initially appealed to a cult audience, but eventually became one of the highest rated programs on television. As I argued in chapter 2, The X-Files can be read as being akin to the misdirection film even though it does not contain a narrative that inspires viewers to reinterpret a majority of narrative information retrospectively. More specifically, like the misdirection film, which encourages audiences to reread it conspiratorially, irrespective of its content, The X-Files engenders paranoia in viewers by forwarding a narrative that depicts an unending search for an elusive "truth" concealed from view by powerful agents. Even though a majority of its episodes aired before the DVD box set became available, then, its nine-season-long, serialized narrative appears tailor-made for the technology. Its focus on central narrative enigmas that seem to remain perpetually unsolved exemplifies the kind of storytelling tactics that now appear with greater regularity on commercial U.S. television because of changing cultural, industrial, and technological conditions. The success of The X-Files and other shows subsequently released in DVD box set form have revealed that a solid core of ardent fans can be enough to make a show appealing to producers even if it does not perform particularly well in the Nielsen ratings during its first few seasons. As Jason Mittell documents, for instance, shortly after Fox decided to end Family Guy (1999– ) because of low ratings, the show's DVD box sets "sold so well that Fox reversed its cancellation by returning the series to its lineup" (Television 424). Whereas television shows that appealed to only loyal fans were once viewed with skepticism by an industry that aimed for the highest ratings possible, programs targeted at potentially profitable and smaller target markets have become increasingly desirable at a time in which alternative revenue streams can now return hefty profits in the long run.
Just a few short years after the DVD box set became dominant, the television industry began to alter its narrative strategies in response to these new conditions of reception. Disney subsidiary ABC's Lost is perhaps the best example of the kind of storytelling that has become more common on television as a result of these changes. As Mittell and Jonathan Gray summarize, the interpretive activities that the show inspires demand that scholars reassess "what 'normal' narrative engagement might look like in the digital age." Lost depicts the adventures of a group of plane crash survivors, marooned on a mysterious island that is governed by supernatural forces. Its narrative is propelled by the perplexing events that occur on, or as a consequence of, the island. It thus centers on a search for answers to the island's most baffling enigmas, such as its specific location, the reasons that strange events consistently occur there, why the survivors have each ended up there, and so on. Consequently, each week fans were encouraged to try to solve the show's dizzying web of puzzles, which only became more elaborate and befuddling as the narrative unfurled in successive seasons.
Crucially, the information that is necessary to begin figuring out these mysteries was often presented during extended flashback, flashforward, and even flashsideway scenes. These sequences are usually of the utmost narrative importance because they typically inspire viewers to reinterpret the significance of what has come before. During season one, for instance, a flashback shockingly reveals that one of the most ostensibly virile survivors—John Locke (Terry O'Quinn)—was a paraplegic prior to the plane crash. As Locke's name epitomizes, the show also contains many high-culture references, as other characters are similarly inspired by famous philosophers, including Hume, Faraday, and Rousseau, coupled with low-culture staples, such as its soap opera inspired love triangles and supernatural, B-movie elements. Such a mix of references as well as its combination of artistic innovation and convention gives it a quintessential middlebrow sensibility similar to the one contained in Shutter Island (2010) and other misdirection films. This amalgamation of high- and low-culture facets helped bring the show great critical acclaim, as it won the 2006 Golden Globe for Best Television Drama and compiled 11 Emmy Award wins over its six-year run (imdb.com). In regard to its narrative innovations, after it is revealed that the island has restored Locke's ability to walk, it becomes possible to reassess the meaning of why previous episodes showed that he wants to remain there. This kind of drastic epiphany that requires retrospective reinterpretations occurs regularly. During season one, a number of other flashbacks also reveal that characters, such as Kate Austen (Evangeline Lilly) and James "Sawyer" Ford (Josh Holloway), similarly hope to stay on the island because of their unhappy pasts off of it. Complicating things further, these frequent temporal shifts are not always blatantly framed as such, making it more difficult than normal for television spectators to orient themselves in time and space as well as determine how the information imparted relates to larger narrative meaning. Moreover, the sequences are generally not explicitly referenced in subsequent episodes because they are designed to elicit suspense by giving loyal viewers privileged information about the obstacles facing the characters, at the same time that they give enough away not to alienate casual viewers more accustomed to the episodic series format.
The increasing prevalence of these once unconventional storytelling strategies on television have been inspired by changing viewing practices and the new revenue streams associated with them. Lost is well-suited for a time in which spectators routinely use new technologies, such as DVRs, on-demand, DVD players, and the Internet, to re-watch episodes of their favorite shows after they have initially aired. Additionally, as scholars like Jenkins argue, in an age of media convergence and synergy, fans now expect to be able to gain access to a wealth of extra information related to these narratively complex media texts on a variety of platforms, such as books, videogames, DVDs, and the web. The DVD box set for the second season of Lost, for instance, contains a chart entitled "Lost Connections" that maps the intricate web of relationships that exist between the characters. Similarly, ABC created "The Lost Experience," a website that both provided a space for fans to solve the show's mysteries and advertised the network's other programs. The decision to produce shows with narratives that strongly encourage viewers to engage in these interpretive activities, then, is backed by a dramatic change in the industry's economic logic, precipitated by shifting conditions of television reception.
The success of the DVD box set, though, does not mean that the Nielsen ratings became unimportant to the industry. Like the domestic theatrical runs of Hollywood films, television shows are likely to be most profitable in ancillary markets if they capture a mass audience when initially broadcast. The series 24, Lost, and Heroes each performed admirably in the Nielsen ratings, at least for some of their runs. However, their popularity generally dipped in the ratings after their maiden seasons, as their increasingly convoluted narratives in subsequent years turned off some spectators. The average number of viewers per episode during the third season of Heroes, for instance, was approximately 10 million, down substantially from the 14.3 million who watched the pilot. Likewise, ensuing season premieres of Lost never matched the 15.7 million viewers who tuned in for the show's highly publicized and expensive pilot in 2004 ("Season"). The atypically high production values of Lost's pilot are noteworthy because it was often received as having blockbuster film qualities. Stacey Abbott, for example, theorizes that the first episode, with its then astronomical $10 million production budget, as well as its extravagant marketing campaign and unprecedented Comic-Con premiere before it aired on television, gave it blockbuster attributes (13–14). Like Inception, these unconventional, blockbuster facets were palatable to fans largely because the show's fantasy attributes made them narratively subservient, in retrospect. Despite this retroactive classical structure, the audience for Lost dwindled to 13.4 million viewers for the first episode of season four in 2008 and 12.2 million viewers for the opening installment of season five ("Season"). Even Lost's much ballyhooed series finale attracted only 13.5 million viewers, according to the Nielsen ratings (Ross). Undoubtedly, the shrinking numbers for these shows are partly attributable to how they repel viewers who are unwilling or unable to put forth the effort necessary to decipher their narrative puzzles.
In spite of such discouraging trends in the ratings, shows with similarly complex narratives recently have been and continue to be key constituents of current television programming dockets. The tremendous success of subsequent programs, like Breaking Bad (2008–2013) and Mad Men (2007–2015), demonstrates that their predecessors were not anomalous on non-premium cable outlets. Undeniably, though, premium cable and streaming content providers have made those venues an ideal place for these kinds of shows because their lack of advertising-backed financing for original programming means that they require unconventional commitments from executives to survive on the networks or basic cable. As Mittell notes, in the face of falling audience numbers in 2007, ABC surprisingly agreed to producers' requests to extend Lost through six seasons, to an end date of 2010 (Television 266). Lost's atypical narrative structure practically demanded such a declaration. As my summary of Twin Peaks in chapter 4 exhibited, that show failed partly because critics and spectators grew concerned that its central mystery could never be resolved satisfactorily. Fans of Lost similarly expressed anxiety early in the show's run that its creators did not have a grand narrative plan. However, two of the show's most prominent creative personnel—Damon Lindelof and Carlton Cuse—made explicit efforts to refute those claims publicly. Specifically, in a press release, they noted that it was such a win to get the network to agree to sign on for six full seasons because they "always envisioned Lost as a show with a beginning, middle and end," which means that viewers "will now have the security of knowing that the story will play out as [they] intended" ("ABC").
It is a bit strange, however, that Lindelof and Cuse made this statement in spite of the credit they are often given for being Lost's primary showrunners. Although a number of individuals share the show's executive producer and writing credits, J. J. Abrams, who rose to fame as a writer/producer of Felicity (1998–2002) and the similarly narratively puzzling Alias (2001–2006), was typically cited in the popular press initially as the creative genius behind Lost. Abrams leveraged the critical success of Lost to become a marketable commodity in the commercial U.S. media industries. For example, Fringe (2008–2013), his subsequent narratively complex television project, featured the most expensive pilot episode ever to air on the medium at the time, eclipsing the exorbitant one produced for Lost (Schweitzer). Furthermore, his burgeoning reputation as an auteur gave Paramount enough faith in his brand name to make him the director of the $150 million first installment of the latest iteration of the Star Trek (2009) franchise, which also contains an unconventionally convoluted narrative that depicts time travel and alternate realities (imdb.com). Perhaps most notably, he was then given the chance to direct Star Wars: The Force Awakens (2015). As with promotional efforts for the misdirection film that foreground the director as the mastermind, the attachment of Abrams's name to Lost was important to its success because it originally provided fans a recognizable adversary with whom they could compete from week-to-week, even if other creative personnel ultimately became more visibly associated with orchestrating the narrative. It also helped Abrams construct an authorial reputation that, like Nolan, he has been able to leverage into a superstar image that mixes artistic innovation with blockbuster credibility as well as can be transported across genres and media.
The programming and production strategies related to Lost are thus relevant to this book for a number of reasons. For starters, they indicate that, in spite of popular discourses to the contrary, media industry executives do not necessarily still use traditional instruments to measure the success of their products. Now that many venues exist for a show to turn a profit after it is broadcast, the Nielsen ratings, like domestic box-office returns, no longer exert as much influence on the industry as they once did. The season two box set of Lost, for example, was the best-selling DVD on the market when it became available in September 2006, topping all Hollywood films released on DVD at the time (Arnold 18). Its strong performance in ancillary markets exemplifies why it is misguided to continue to think of the economic incentives that drive the U.S media industries in outdated terms. New media and communication technologies deeply influence the ways that many audiences now commonly interact with media texts, which, in turn, impact the production and promotional tactics of industry executives and creative personnel.
The success of contemporary Hollywood misdirection films and recent television shows that inspire analogous interpretive practices, though, cannot simply be explained by technological developments and the media industries' economic motives. New media technologies and changing industrial motives alone do not account for the specific ways that these similarly structured atypical narrative forms have come into fruition and why they have resonated strongly with some viewers. The kinds of duplicitous narratives that appear in these programs and misdirection films consistently tap into shared cultural anxieties and desires in relation to the status of the "truth." In particular, although they suggest that the "truth" is difficult to determine because it is hidden from view by powerful forces, they do not usually suggest that it is perpetually in flux. Instead, devoted spectators of most of these films and shows are ultimately assured that it is possible to know what "actually" occurred and who "really" made things happen. Of course, fans have to be up to the tasks of enduring the perpetual slew of convoluted narrative machinations and unearthing the secrets that they deem to be concealed by their makers to gain access to this privileged information. In misdirection films and these shows alike, then, a discovery of the "truth" is typically the ultimate payoff because it is usually either explicitly revealed, or perceived to have been deliberately buried for discovery by their creators.
Returning briefly to the Lost example to illustrate this phenomenon, the search for the "truth" is the show's central thematic concern. Not only did the show's creators deliver on their promise to dedicated fans willing to conduct the labor necessary to interpret the meaning that it would all make sense in the end, the existential search for "truth" was also depicted explicitly by the narrative, articulated most clearly by the ongoing thematic struggle for supremacy between faith and science. The characters closely linked to these competing philosophies shifted throughout the show's run; however, for a majority of it, Dr. Jack Shephard (Matthew Fox) represented science and faith was embodied by Locke. Although the cast of Lost is deliberately peppered with women and racial minorities to appeal to a diverse audience and inoculate the show from charges of racism and sexism, it is significant that Jack and John are the show's primary protagonists. Indeed, it does not take long before the survivors split into two separate camps on the island because they are drawn to the competing philosophies of their respective leaders. These two white, heterosexual men, therefore, are able to reclaim the traditional manhood on the island that they were previously stripped of off of it by being the individuals who are perceived as being capable of eventually bringing order out chaos.
As I discussed in chapter 3, then, Lost can be understood as expressing similar gender politics as many contemporary misdirection films because it depicts a culture in which purportedly emasculated male heroes are ultimately able to regain their masculine authority. In the case of Locke, for instance, the island revitalizes his manhood by enabling him to reuse his legs. The supernatural turn of events allows him to leverage his skills as a hunter to provide food for the other survivors, establishing him as a powerful figure on the island. Additionally, the show's complex narrative also often expressed nervous concern about the possibility that the actions of these two leaders may ultimately be irrelevant because a series of even more powerful white, heterosexual, male characters, including Benjamin Linus (Michael Emerson), Charles Whidmore (Alan Dale), Sawyer, and Jacob (Mark Pellegrino), were periodically shown to be the ones who may really be in control of the fates of the survivors. In the end, Lost did reveal that individual agency is restricted for Jack, the character who turned out to be the unequivocal protagonist, as the final changeover showed that he was dead and that the island was purgatory. Lost thus also expresses a panic about the viability of individual autonomy, which, as I claimed in chapter 2, has been a key factor driving the recent rise of conspiratorial narratives in both U.S. political and popular culture. As with the misdirection film, Lost's narrative is ultimately appealing to many viewers partly because it assures them that concepts associated with modernity, including the existence of absolute "truth" as well as the persistence of racial and gender hierarchies, have not been supplanted by tenets associated with postmodernity, such as relativity and multiculturalism.
As a result of these tendencies, the cultural implications of the contemporary Hollywood misdirection film will be generally disappointing to progressive critics who correctly argue that the industry still usually deploys its time-tested practices and conveys dominant ideologies to make its products attractive to a wide audience. Although it is true that many of these films are targeted at a smaller cohort than most Hollywood films once were, they are still constructed in ways that appeal to the market segment with the greatest profit potential. As I contended in chapter 4, contemporary misdirection films are often designed to capture a niche audience of predominantly young, white, male spectators, who derive pleasure from sorting out the clues that they believe have been laid out for their discovery by their male creators. Consequently, these films usually end up supporting traditional ways of thinking even though their unconventional narrative structures are particularly well-suited to challenge Hollywood's typically conservative agenda. Their duplicitous narratives, for instance, are uniquely positioned to expose the artificiality of narrative closure. Rather than attribute narrative causality to random or inexplicable forces, though, the alternative narrative explanation most often reveals that events were actually orchestrated by identifiable agents, who are almost inevitably male. Such retrospective reinterpretations appeal to audiences who have grown comfortable with Hollywood conventions. In the end, dedicated spectators are rewarded with the evidence necessary to transform films that initially appear to represent departures from the norm into ones that both adhere to the classical paradigm and express dominant American ideologies.
The presence of an alternative narrative explanation that renders these films comprehensible and comfortable to many viewers demonstrates why they exhibit the attributes that critics, like David Bordwell, identify as being embodied by the classical Hollywood film. The exposure or discovery of the revelatory information often makes these films hyper-classical because spectators can reinterpret their narratives according to a revised causality that is tied to the actions of clearly identifiable agents. Moreover, it is true that misdirection films rely heavily on Hollywood's generic, formal, ideological, and narrative conventions to work their deceptive magic. In fact, these very standards are what typically encourage viewers to draw the incorrect conclusions about the causal relationship of events. Although they are highly dependent on classical devices, however, it is difficult to claim that these films are simply constituents of that particular mode of narration. Even though fans often go to great lengths to try to make them narratively coherent, some of these films violate the most basic tenets of the classical film by lingering on perpetual ambiguity, presenting primary characters with indeterminate motives, making it impossible to distinguish fantasy from reality, and so on. These examples similarly respond to nervous concerns about the existence of absolute "truth," but amplify that anxiety by suggesting that it might be unknowable. This does not prevent most fans, though, from doggedly trying to find ways to reassess narrative meaning coherently. Regardless of how they respond to this broader and irreconcilable cultural conflict, misdirection films depart from the classical film by alerting audiences to their status as a construction, rendering the narrative and formal mechanics of the Hollywood film highly visible, in retrospect. Misdirection films, then, require multiple viewings and/or the shared knowledge of fans to be appreciated most deeply, which are interpretive behaviors that are not supposed to be expected of the classical spectator.
The constituent films of the genre suggest that Hollywood does not simply employ a simplistic production formula that has remained constant throughout its history. Rather, these films reveal that, when associated conditions are optimal, Hollywood is willing to design products that can be distinguished from its other fare precisely because they contain non-classical tendencies, at the same time that they continue to abide by some classical principles. The creation of a purely classically constructed narrative, therefore, is not and has never been Hollywood's primary concern. The misdirection film is a viable genre that discursive evidence reveals has existed throughout most of cinema history, but became more popular with many audiences and producers from 1990 to 2010 by effectively combining a specific blend of sameness and difference in response to its particular cultural, industrial, and technological contexts.
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## Index
Page numbers in italics indicate illustrations.
Abbott, Stacey,
Abrams, J. J.,
Abre los ojos,
Adaptation, , –,
After Earth, , ,
"agency panic," , , , , , ,
Aldrich, Thomas Bailey,
Alfred Hitchcock Presents (television series), ,
Alias (television series),
All the President's Men,
al-Qaeda, , , ,
Altman, Rick, –, , , ,
Amblin Entertainment,
American Beauty, –,
American Psycho,
American Sniper,
Anderson, Benedict,
Anderson, Paul Thomas, –, –,
Apocalypse Now,
Aristotle,
Arlington Road, , , –,
Arrested Development (television series),
arthouse cinema, –, –; indie films versus, ; misdirection films versus, –,
Atkinson, Michael,
Atonement, , , ,
auteurism, –; of Abrams, ; of Nolan, , –, , , , ; of Shyamalan, –, , –, –, ,
Aviator, The, ,
Badalamenti, Angelo,
Bale, Christian, ,
Bamberger, Michael, –
Barlow, Aaron, ,
Barnes, Elizabeth, –
Batman franchise, , , –, –, –
Beautiful Mind, A, , , –; Academy Awards for, –, , , ; marketing of, ; success of, , , –
Berardinelli, James, , , ,
Berry, Halle,
Betamax, , ,
Beyond a Reasonable Doubt,
Bierce, Ambrose,
bin Laden, Osama, ,
Birchall, Clare, –,
Blinding Edge Pictures, ,
blockbuster films, , –, –, ; conservatism of, ; as genre, –; middlebrow appeal of, –, , ; misdirection films as, , , , –, –, –; television series and, ,
Blue Velvet, ,
Bly, Robert, , ,
Boggs, Carl, –
Boogie Nights, –
Booker, M. Keith,
Booth, Michael, ,
Bordwell, David, –, , ; on classical Hollywood films, –, –, –, , ; on classical narrative functions, –, ; on Memento, , –; on "puzzle films," –, ,
Born on the Fourth of July,
Bourdieu, Pierre,
Boyle, Danny,
branding of directors, –, –
Braveheart,
Breaking Bad (television series),
Brevert, Brad,
Broomfield, Nick,
Brosnan, Pierce,
Brown, Dan,
Bruzzi, Stella,
Buckland, Warren, ,
Burgess, Adrienne,
Buried Secret of M. Night Shyamalan, The, –,
Burnett, Mark,
Burton, Tim,
Bush, George H. W., –, ,
Bush, George W., , ,
Butler, Jeremy,
Butler, Judith, –
Cabinet of Dr. Caligari, The, ,
Cage, Nicholas,
Cahiers du Cinéma,
Calhoun, Dave,
Cameron, James,
Camus, Albert,
Caro, Mark,
Carolco film studio,
Carter, Jimmy,
Carver, Terrell, ,
Catch Me if You Can,
censorship,
changeover films, –; Arlington Road as, , –; Fight Club as, , , –, , –; Jacob's Ladder as, –; semantic elements of, –, –; The Sixth Sense as, –, , ; Unbreakable as, , , –; The Usual Suspects as, , , –; The Wizard of Oz as, . See also individual movies
Chernin, Peter,
Chhabra, Aseem,
Christie, Tom,
Cimino, Michael,
Citizen Kane, , , –
Clark, J. Michael,
Clark, Mike,
Clark, Spencer Treat,
classical Hollywood films, Bordwell on, –, –, –, , ; Deleuze on, ; film noir versus, ; Inception as, , ; indie films versus, –; masculinity in, , ; misdirection films versus, , , , ; narrative functions of, –, ; new media and,
Clinton, Bill,
Clinton, Hillary Rodham,
Close Encounters of the Third Kind,
Cohan, Steven,
Cold War, , –, , ,
Color of Money, The,
Coming Home,
computer games, , , ,
conspiracy theories, –; "agency panic" and, , , ; misdirection films and, –, –, –, , ; about New World Order, , ; storytelling techniques of, –
Conspiracy Theory (film), ,
Constitution (U.S.), ,
Contender, The,
Conversation, The,
Coppola, Francis Ford, ,
Corliss, Richard,
Corrigan, Timothy, ,
counter histories,
Covert, Colin,
Cowie, Elizabeth,
Cox, David,
Crowe, Cameron,
Crowe, Russell,
Cruise, Tom, ,
Crying Game, The, , , , ,
cultural paranoia, , , –, ; in Arlington Road, ; about gender, , , ; in Jacob's Ladder, . See also conspiracy theories
Cusack, John,
Cuse, Carlton,
cycles. See film cycles
Dark Knight, The, , , , , –. See also Batman franchise
Davis, Richard Harding,
De Luca, Michael,
De Niro, Robert,
Declaration of Independence,
Deer Hunter, The,
Deleuze, Gilles,
Denby, David,
Departed, The, , ,
detective films, –
Diabolique, ,
DiCaprio, Leonardo, –; in Inception, , , –; in Shutter Island, , , , –,
Die Hard films, –,
Dillman, Joanne Clarke, –
Disney Corporation, ; Shyamalan and, , , –; Touchstone Pictures of,
dissociative identity disorder, –; in Fight Club, , , –; in Shutter Island,
Django Unchained,
Douglas, Michael,
Dreher, Rod,
Duckner, Johannes, –
DVD technology, –, , –, –; fan websites and, , ; Fight Club and, , ; Inception and, –; Klinger on, ; Magnolia and, ; Memento and, , –, –; Mulholland Dr. and, , , , , –; revenue decline from, ; television serials and, , –; VCR and, , , , , , , –; VHS and, , , ; The Village and,
Dyer, Richard,
Eastwood, Clint,
Ebert, Roger, ; on Fight Club, , , –, ; on Inception, ; on Magnolia, , ; on Memento, ; on Mulholland Dr., ; on The Village,
Eckhart, Johannes,
Ellis, Bret Easton,
Elsaesser, Thomas, ,
Enron Corporation,
E.T.: The Extra-Terrestrial,
experimental films, –,
Fahrenheit 9/11,
Fallen, , –
Faludi, Susan, –
familicide, –
Family Guy (television series),
Family Man, The,
fan websites, , , –, –; hermeneutic one-upmanship on, , ; for Inception, –, ; for Lost, ; for Memento, –, , ; for Mulholland Dr., –, ; for Pulp Fiction, –; for Shutter Island, –; for Twin Peaks, –. See also new media technologies
fatherhood, , , , –; in Magnolia, –; in The Sixth Sense, , ; in Unbreakable, ; in The Usual Suspects, –. See also masculinity
Felicity (television series),
Fenster, Mark, –, –
Fight Club, –, –, ; changeover in, , , –, , –; dissociative identity disorder in, , , –; Inception and, ; Magnolia and, –; marketing of, ; masculinity in, –, –, , , ; popularity of, –, ; stills from, , , ; success of, –, ; The Usual Suspects and, , , –
film cycles, ,
film noir, , –, , ,
Final Fantasy videogame,
Fincher, David, –,
First Blood,
Following,
Forrest Gump,
Foucault, Michel, –
Fradley, Martin,
French New Wave movement,
Fringe (television series),
Frost, Mark, –
Full Metal Jacket,
Game, The,
Gangs of New York, ,
Garnn, Toni,
Garrone, Max,
Gates, Philippa,
Gattaca,
gender, , , , ; race/class and, , , , , –; social construction of, –, , ; "woman's films" and, –. See also masculinity
genre, –; blockbuster films as, –; Bordwell on, –; differentiating of, –; discursive approach to, , –; film cycles and, ; hybrid, , , , ; middlebrow appeal and, –, , , –; "mindbender," –; niche marketing and, , ; self-reflexivity of, ; semantic/syntactic elements of, –, –; theorists of, –, ; Todorov on, . See also specific types
Gere, Richard,
Giamatti, Paul, ,
Gibson, Mel,
Giles, Jeff,
Gilligan, James, ,
Giroux, Henry, –,
Gladiator,
Goffman, Erving,
Gomery, Douglas,
Goodykoontz, Bill,
Gray, Jonathan, ,
Green Mile, The,
Greven, David,
Grey, Brad,
Halberstam, Jack, –
Happening, The, , , –, ,
Hard Eight,
Harper, Graeme,
Harrison, Eric,
Harte, Bret,
Hellinger, Daniel, ,
Henry, O. (William Sydney Porter),
Heroes (television series), , ,
Hicks, Heather,
Hide and Seek,
Higson, Andrew, –
Hitchcock, Alfred, –, , , ; Scorsese and, , ; television series of, ,
Hoblit, Gregory, –
Hofstadter, Richard,
Hollywood films. See classical Hollywood films
Hollywood Renaissance, –,
Holmlund, Chris,
Hopkins, Anthony,
Horne, Philip,
Howard, Bryce Dallas, , ,
Howard, Ron,
Hugo,
Hulk,
Human Stain, The,
Hunter, Stephen,
Identity, , –
Illusionist, The, –
Inception, , , , –; budget for, ; fan website for, –, ; marketing of, , , , –; masculinity in, , –; Shutter Island and, , ; stills from, , ; success of, , , , , –,
indie films, –, –, ; profit margins of,
Insomnia, –, ,
Interstellar,
Iocco, Melissa, –
Iraq, U.S. invasions of, –
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Itzkoff, Dave, , , –
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Jacobson, Nina, ,
James, Henry,
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Jerry Maguire,
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Kalogridis, Laeta,
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Kellner, Douglas,
Kennedy, John F., ,
Kennedy, Robert F.,
"Keyser Söze syndrome," , , –,
Kidman, Nicole,
Kimmel, Michael, , , ,
King, Geoff, , , , , ,
King, Martin Luther, Jr.,
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Klecker, Cornelia, ,
Klein, Amanda Ann,
Klein, Andy, , –
Klinger, Barbara, , , ; on "puzzle film," –; on repurposing, –
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Kroll, Sue,
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LaSalle, Mick, ,
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Lavik, Erlend, –
Ledger, Heath,
Lee, Ang,
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Lehane, Dennis,
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Luhrman, Baz,
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Lyne, Adrian,
Macdonald, Moira,
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Maslin, Janet, ,
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Mamet, David,
Manchurian Candidate, The, ,
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Memento, , , , –, ; Booth on, ; Bordwell on, , –; DVD of, , –, –; Ebert on, ; fan websites for, –, , ; Klein on, , –; marketing of, , , ; as master key film, ; as misdirection film, ; Mulholland Dr. and, , ; The Prestige and, ; stills from, –; success of, , –, ; The Usual Suspects and,
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Mittell, Jason, , –, ,
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Moore, Michael,
Mulholland Dr., , , , , –; box office sales for, ; DVD of, , , , , –; fan website for, –, ; Inception and, ; as master key film, –, ; Memento and, , ; as proposed TV series, ; stills from, , , , ; The Wizard of Oz and, –
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Newman, Michael, –,
Newmarket Films, –
Nickelodeon, ,
9/11. See September 11th attacks
Nixon, Richard, , ,
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Nolan, Christopher, –, –, –; as auteur, , –, , , , ; career of, , –, , , , –; on interpreting Inception, –, ; Scorsese and, ; unofficial website of, –. See also specific films
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Penn, Sean,
Perfect Stranger, ,
Piaf, Edith, –
Pitt, Brad, , . See also Fight Club
Planet of the Apes, , , , ,
Platoon,
Poe, Edgar Allan,
Pollard, Tom, –
postmodernism, , –,
Powers, John,
Pratt, Ray, , , , ,
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Primal Fear, ,
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Psycho, , , , , ,
Pulp Fiction, ; fan websites of, –; success of, ,
Pulver, Andrew,
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Rainer, Peter,
Raymond, Marc,
Rea, Stephen, –
Reagan, Ronald, , , , ,
Redbelt,
Rehling, Nicola,
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Richter, David, –
Riviere, Joan,
Robbins, Tim, , , ,
Roberts, Gillian, –
Romeo + Juliet,
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Ruby Ridge incident (1992), ,
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Schwarzbaum, Lisa, , –
Schwarzenegger, Arnold, –
Scorsese, Martin, , –, –
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| {
"redpajama_set_name": "RedPajamaBook"
} | 6,410 |
\section{Introduction}
\subsection{Potential Theory and polynomials}
The study of Approximation Theory in the complex plane and on the real line (by polynomials and rational functions) is deeply related to the Logarithmic Potential Theory (i.e., the study of subharmonic functions and Laplace operator); this is a classical and well established topic whose study goes back to Fekete, Leja, Szeg\"o, Walsh and many others.
These relations between Logarithmic Potential Theory and Approximation Theory spread among Markov, Bernstein and Nikolski type polynomial inequalities, asymptotic of optimal polynomial interpolation arrays and Fekete points, \emph{overconvergence phenomena} (i.e. uniformly convergent sequence of polynomials defining a holomorphic function in a larger open set) and its quantitative version, the Bernstein Walsh Theorem , asymptotic of orthogonal polynomials, random polynomials and random matrices. Moreover, most of such relations extend to the more general case of weighted polynomials and Logarithmic Potential Theory in presence of an external field. We refer to \cite{SaTo97,Sa10,StaTo92,WAL,Rans} and references therein for extensive treatments of this subjects.
More recently a non linear potential theory in multi dimensional complex spaces has been introduced and many analogies with the linear case have been shown, provided a suitable ''translation'' of the quantities that come into the play. \emph{Pluripotential Theory} (see for instance \cite{Kli,Klo05}) is the study of plurisubharmonic functions (i.e., functions which are subharmonic along each complex line) and the complex Monge Ampere operator; \cite{BeTa82}.
Though the lack of linearity makes this new theory much more difficult and requires to work with different tools, many connections with polynomial approximation has been extended to this multi dimensional framework; see \cite{LevSurRocky92,LevSur}. Indeed, polynomial inequalities in $\mathbb{C}^n$ are usually obtained by means of Pluripotential Theory, see for instance \cite{BaBi14,Ba92}, the Bernstein Walsh Theorem has been extended by Siciak to $\mathbb{C}^n$ \cite{Si81} and more general complex spaces by Zeriahi \cite{Ze91}. In his seminal work \cite{Za74I,Za74II,Za75}, Zaharjuta extended the equivalence between (a suitably re-defined version of) the Chebyshev Constant (i.e., the asymptotic of the uniform norms of monic polynomials) and the Transfinite Diameter (i.e., the asymptotic of the maximum of the Vandermonde determinant). Very recently, Berman Boucksom and Nystrom \cite{BeBo10,BeBoNy11} showed that Fekete points converge weak$^*$ to the \emph{pluripotential equilibrium measure} of the considered set in $\mathbb{C}^n$ and in much more general settings, this is a deep extension of the one dimensional case which can be obtained only by means of the weighted theory. The work of Berman and Boucksom stimulated different lines of research as $L^2$ theory and general orthogonal polynomials \cite{B97}, the study of multi-variate random polynomials \cite{ZeZe10,BlLe15,PrYeeAa15}, and the theory of sampling and interpolation arrays \cite{MaOr10,BeOr15,MaOr15}. More importantly from our point of view, the widely used heuristic that \emph{the "best" measure for producing uniform polynomial approximations by $L^2$ projection is the equilibrium measure} has been fully motivated and theoretically explained in \cite{BeBoNy11} also in its multivariate setting.
\vskip 0.5cm
The present work concerns, on one hand, to (partially) extend to the $\mathbb{C}^n$ case another \emph{connection between polynomials and Potential Theory}, on the other hand, to highlight how \emph{the polynomial $L^2$ approximation with respect to the equilibrium measure may be regarded as Fourier Analysis on a suitable Riemaniann manifold.} These ideas rest upon the relation between the Laplace Beltrami operator relative to the Baran metric and the orthogonal polynomials with respect to the pluripotential equilibrium measure.
We would like to introduce such relations starting by some examples that treat the instances of the interval $[-1,1]$ and the unit sphere.
\subsection{Two motivational examples}
\subsubsection{Chebyshev polynomials}
Chebyshev polynomials $T_n(x):=\arccos(n\cos x)$ are the orthogonal polynomials with respect to $\frac 1 {\pi\sqrt{1-x^2}}dx$, the equilibrium measure of the interval $[-1,1]$ as a subset of $\mathbb{C},$ i.e., the unique minimizer of the logarithmic potential $-\int \log|z-w|d\mu(z)d\mu(w)$ among all Borel probability measures $\mu$ on the interval $[-1,1]$. Another classical characterization of Chebyshev polynomials is given by the eigen-functions of the Sturm-Liouville eigenvalue problem
\begin{equation}
\label{STeigenproblem}
\begin{cases}\mathscr S[\phi](x):=(1-x^2)\phi''(x)-x\phi'(x)=-\lambda\phi(x),& x\in ]-1,1[\\
\phi'(x_0)=0,& x_0\in\{-1,1\}
\end{cases}.
\end{equation}
The set of eigenvalues turns out to be $\{n^2: n\in \mathbb{N}\}$ and $\mathscr S[T_n]=n^2 T_n.$
Instead, we re-write such an eigenvalue problem as
\begin{equation}\label{SLE}
\frac 1{\frac 1{\sqrt{1-x^2}}}\frac{d}{dx}\left(\frac 1{\sqrt{1-x^2}} (1-x^2) \phi'(x)\right)=-n^2\phi(x),\;\;x\in ]-1,1[.
\end{equation}
This apparently useless manipulation actually enlightens another property of Chebyshev polynomials. To explain this property, we first recall that the Laplace Beltrami operator relative to a metric $g$ can be written in local coordinates as
\begin{equation}\label{laplacebeltramidef}
\Delta_{LB}f=\frac 1{\sqrt {\det g}}\sum_{i=1}^n\partial_{x_i}\left(\sqrt{ \det g} \sum_j g^{i,j}\partial _{x_j}f \right),
\end{equation}
where $g^{i,j}$ are the components of the inverse of the matrix representing $g.$
Let us endow $]-1,1[$ with the Riemaniann metric $g(x):=\frac 1{1-x^2},$ we canonically obtain the Riemannian distance $d(x_0,x_1)=\int_{x_0}^{x_1}\frac 1 {\sqrt{1-x^2}}dx.$ Note that, up to a re-normalization, the resulting volume form is precisely the equilibrium measure of $[-1,1].$ If we plug $g(x):=\frac 1{1-x^2}$ in the expression \eqref{laplacebeltramidef} of the Laplace Beltrami operator, we obtain precisely the left hand side of \eqref{SLE}. In other words, we observe that:
\begin{itemize}
\item\emph{Chebyshev polynomials are eigenfunctions of the Laplace Beltrami operator with respect to the equilibrium measure of the interval.}
\end{itemize}
It is relevant to notice that the density of the equilibrium measure on $[-1,1]$ at $x$ is obtained as the normal (i.e., purely complex) derivative of the Green function of $\mathbb{C}\setminus [-1,1]$ with pole at infinity; see \cite[Ch II.1]{SaTo97}. This operation has a multidimensional counterpart (see \cite{Ba92}) that, under some assumptions, leads to the so called \emph{Baran metric} (\cite{BlBoLe12,BoLeVi12,Ba95}), see equation \eqref{baranmetricdef} below.
\subsubsection{Spherical harmonics}
We mention another relevant example of this relation between eigenfunctions of the Laplace Beltrami operator with respect to the metric defined by (pluri-)potential theory and the (pluripotential) equilibrium measure. In contrast with the case of Chebyshev polynomials, now we work in a multi dimensional setting and the flat euclidean space $\mathbb{C}^n$ is replaced by a complex manifold. A more detailed account of this example requires some preliminary notions in addition to the ones of Subsection \ref{subsectionPluripot}, thus we decided to present the explicit computations in Appendix \ref{appendix}, together with the needed recalls from pluripotential theory on algebraic varieties. At this stage we only sketch the results to underline the analogy with the case of Chebyshev polynomials.
Let us consider the unit sphere $\mathbb S^{n-1}\subset \mathbb{R}^n$ endowed it with the \emph{round metric} $g$ induced by the flat metric on $\mathbb{R}^n$ and denote by $\Delta$ the Laplace Beltrami operator on $\mathbb S^{n-1}.$ It is well known that \emph{spherical harmonics} are a dense orthogonal system of $L^2(\mathbb S^{n-1})$ which consists of polynomials that are eigenfunctions of $\Delta.$
Let us look at $\mathbb S^{n-1}$ as a compact subset of the \emph{complexified sphere} $\mathcal S^{n-1}:=\{z\in \mathbb{C}^n:\sum z_i^2 =1\}.$ By a fundamental result due to Sadullaev \cite{Sa82}, since $\mathcal S^{n-1}$ is a irreducible algebraic variety, one can relate ( see Appendix \ref{appendix}) the traces of polynomials on $\mathcal S^{n-1}$ to the \emph{pluripotential theory on the complex manifold} of $\mathcal S^{n-1}.$ On the other hand, due to Lemma \ref{differentiabilitylemma} below, we can define a smooth Riemaniann metric $g_{\mathbb S^{n-1}}$ on $\mathbb S^{n-1}$ suitably modifying the construction (see eq. \eqref{baranmetricdef}) of the Baran metric of convex real bodies. In particular such a definition is given by the generalization of the case of the real interval $[-1,1]$. Indeed, it turns out that $g_{\mathbb S^{n-1}}=g$ and its volume form is, up to a constant scaling factor, the pluripotential equilibrium measure (see equation \eqref{eqmsph} below) of $\mathbb S^{n-1},$ as compact subset of $\mathcal S^{n-1}.$ In other words
\begin{itemize}
\item\emph{the eigenfunctions of the Laplace Beltrami operator of $(\mathbb S^{n-1},g)$ are the orthogonal polynomials with respect to the pluripotential equilibrium measure of $\mathbb S^{n-1},$ seen as compact subset of $\mathcal S^{n-1};$} see Corollary \ref{corollarysphere}.
\end{itemize}
\subsection{Our results and conjecture}
The aim of the present paper is to present a conjecture on the extension to the $\mathbb{C}^n$ case of the relation between potential theory and certain Riemaniann structure that holds in the examples above. We support it by full proofs of all the few known instances fulfilling the required hypothesis, see Theorems \ref{ThBall} and \ref{ThSimplex} below.
\begin{conjecture}\label{Conjecture}
Let $\mathcal C$ denote either $\mathbb{C}^n$ or any irreducible algebraic sub-variety of it. Let $E\subset \mathcal C$ be a fat\footnote{This should be intended as the closure in $\mathcal C$ of the interior in $ \mathbb{R}^n\cap \mathcal C$ of $E$ equals to $E$ itself.} real compact set. Assume that the \emph{Baran metric} $\delta_E$ of $E$ is a Riemannian metric on $\interior_{\mathbb{R}^n\cap \mathcal C}E,$ then the orthonormal polynomials with respect to the pluripotential equilibrium measure $\mu_{E,\mathcal C}$ of $E$ in $\mathcal C$ are eigenfunctions of the Laplace Beltrami operator relative to the metric $\delta_E.$
\end{conjecture}
\begin{remark}
We stress that the orthogonal bases used in our proofs as well as most of their properties are already known in the framework of orthogonal polynomials (see \cite{DuXu01,DuXu14} and references therein). Moreover, our differential operators (i.e., Laplace Beltrami operators with respect to the metrics arising from Pluripotential Theory) turn out to be already studied in relation with certain symmetry groups \cite[Ch. 8]{DuXu14}, but they have not been related to any potential theoretic aspects before. More precisely, the Laplace Beltrami operator on the ball endowed with its Baran metric turns out to be the operator $\mathscr D_\mu$ in \cite[pg. 142]{DuXu01} with the parameter choice $\mu=0$. Instead, in the simplex case, $\Delta$ is precisely the operator defined in \cite[eq. 5.3.4]{DuXu01} (see equation \ref{differentialproperty} and Theorem \ref{simplexeigenfunctions} below) if we set (in the authors notation) $\kappa=(0,\dots,0)\in \mathbb{R}^{n+1}.$
\emph{Our goal is precisely to relate such families of functions and their properties to the Riemaniann structure that comes from Pluripotential Theory. }
\end{remark}
\begin{theorem}[Laplace Beltrami on the Baran Ball]\label{ThBall}
Let us denote by $\Delta$ the Laplace Beltrami operator of the Riemannian manifold $(B^n,g_{B^n})$ acting on
$$\mathscr C^2_b(B^n):=\left\{ u\in \mathscr C^2(B^n):\max_{|\alpha|\leq 2}\sup_{x\in B^n}|\partial^\alpha u(x)|<\infty\right\},$$
where $B^n:=\{x\in \mathbb{R}^n: |x|< 1\}$ and $g_{B^n}(x)$ is represented by the matrix \small
$$ G_{B^n}(x):=\left[\begin{array}{cccccc}
1+\frac{x_1^2}{1-\sum_{i=1}^nx_i^2}& \frac{x_1x_2}{1-\sum_{i=1}^nx_i^2} & \frac{x_1x_3}{1-\sum_{i=1}^nx_i^2}&\dots& \dots &\frac{x_1x_n}{1-\sum_{i=1}^nx_i^2}\\
\frac{x_1x_2}{1-\sum_{i=1}^nx_i^2} & 1+\frac{x_2^2}{1-\sum_{i=1}^nx_i^2}& \frac{x_2x_3}{1-\sum_{i=1}^nx_i^2}&\dots& \dots &\frac{x_2x_n}{1-\sum_{i=1}^nx_i^2}\\
\vdots & \vdots & \vdots&\vdots&\vdots&\vdots\\
\frac{x_nx_1}{1-\sum_{i=1}^nx_i^2}& \frac{x_nx_2}{1-\sum_{i=1}^nx_i^2} & \dots&\dots& \dots &1+\frac{x_n^2}{1-\sum_{i=1}^nx_i^2}
\end{array}\right].$$ \normalsize
The operator $\Delta$ is symmetric and unbounded, it
has discrete spectrum
$$\sigma(\Delta)=\{\lambda_s:=s(s+n-1): s\in \mathbb{N}\}$$
and the eigen-space of $\lambda_s$ is $\Span\{\phi_\alpha,|\alpha|=s\},$ where $\phi_\alpha$ (see Proposition \ref{orthogonalsystemball})
are orthonormal polynomials with respect to the pluripotential equilibrium measure
$$\mu_{B^n}:=\frac 1{\sqrt{1-|x|^2}}\chi_{B^n} \Vol_{\mathbb{R}^n}=\Vol_{g_{B^n}}.$$
Moreover, $\Delta$ can be closed to a self-adjoint operator $\mathcal D(\Delta)\rightarrow L^2(B^n,g_{B^n})$ (having the same spectrum), where
$$\mathcal D(\Delta):=\left\{u\in L^2(B^n,g_{B^n}): \sum_{s=0}^\infty\lambda_s^{2}\sum_{|\alpha|=s} |\hat u_{\alpha}|^2<\infty\right\}\subset H^1(B^n,g_{B^n})$$
and $\hat u_\alpha$ is the Fourier coefficient $\int_{B^n} u \frac{\phi_\alpha}{\|\phi_\alpha\|^2_{L^2(\mu_{B^n})}} d\mu_{B^n}.$
The operator $\Delta^{1/2}$ has domain
$$\mathcal D(\Delta^{1/2}):=\left\{u\in L^2(B^n,g_{B^n}): \sum_{s=0}^\infty\lambda_s\sum_{|\alpha|=s} |\hat u_{\alpha}|^2<\infty\right\}= H^1(B^n,g_{B^n})$$
\end{theorem}
For a precise definition of the Sobolev space $H^1(B^n,g_{B^n})$ see Subsection \ref{SubsectionSobolev} below.
\begin{theorem}[Laplace Beltrami on the Baran Simplex]\label{ThSimplex}
Let us denote by $\Delta$ the Laplace Beltrami operator on the Riemannian manifold $(S^n,g_{S^n})$, acting on
$$\mathscr C^2_b(S^n):=\left\{ u\in \mathscr C^2(S^n):\max_{|\alpha|\leq 2}\sup_{x\in S^n}|\partial^\alpha u(x)|<\infty\right\},$$
where $S^n:=\{x\in \mathbb{R}^n: \sum_{i=1}^n x_i < 1,x_i>0\,\forall i=1,2,\dots,n\}$ and $g_{S^n}(x)$ is represented by the matrix
$$G_{S^n}(x):=\left[\begin{array}{ccccc}
x_1^{-1} & 0&\dots& \dots &0\\
0& x_2^{-1}&0& \dots &0\\
\vdots & \vdots &\vdots&\vdots&\vdots\\
\vdots & \vdots &\vdots&\vdots&\vdots\\
0 & \dots&\dots& 0 &x_n^{-1}
\end{array}\right]+
\frac 1{1-\sum_{i=1}^n x_i}
\left[\begin{array}{ccccc}
1& 1&\dots& \dots &1\\
1& 1&\dots& \dots &1\\
1& 1&\dots& \dots &1\\
1& 1&\dots& \dots &1\\
1& 1&\dots& \dots &1
\end{array}\right].
$$
The operator $\Delta$ is symmetric and unbounded, it
has discrete spectrum
$$\sigma(\Delta)=\{\lambda_s:=s(s+\frac{n-1}{2}: s\in \mathbb{N}\}$$
and the eigen-space of $\lambda_s$ is $\Span\{\psi_\alpha,|\alpha|=s\},$ where $\psi_\alpha$ (see Proposition \ref{orthogonalsystemsymplex})
are orthonormal polynomials with respect to the pluripotential equilibrium measure of the simplex
$$\mu_{S^n}:=\frac 1{\sqrt{(1-\sum_{i=1}^nx_i)\prod_{i=1}^n x_i}}\chi_{S^n}(x)\Vol_{\mathbb{R}^n}=\Vol_{g_{S^n}}.$$
Moreover, $\Delta$ can be closed to a self-adjoint operator (still denoted by $\Delta$) $\mathcal D(\Delta)\rightarrow L^2(S^n,g_{S^n})$ (having the same spectrum) where
$$\mathcal D(\Delta):=\left\{u\in L^2(S^n,g_{S^n}): \sum_{s=0}^\infty\lambda_s^{2}\sum_{|\alpha|=s} |\hat u_{\alpha}|^2<\infty\right\}\subset H^1(S^n,g_{S^n})$$
and $\hat u_\alpha$ is the Fourier coefficient $\int_{B^n} u \frac{\psi_\alpha}{\|\psi_\alpha^2|_{L^2(\mu_{S^n})}} d\mu_{S^n}.$
The operator $\Delta^{1/2}$ has domain
$$\mathcal D(\Delta^{1/2}):=\left\{u\in L^2(S^n,g_{S^n}): \sum_{s=0}^\infty\lambda_s\sum_{|\alpha|=s} |\hat u_{\alpha}|^2<\infty\right\}= H^1(S^n,g_{S^n})$$
\end{theorem}
\begin{remark}[Refinement of Conjecture \ref{Conjecture}]
The computations performed in \cite{BoLeWa04}, \cite{BoLeWa08} and \cite{BlBoLe12} allow us to prove that Conjecture \ref{Conjecture} does hold for $E$ being the real ball and the real simplex. We remark that in this cases $\partial E$ is a real algebraic (possibly reduceble) set, thus one may add such an assumption to Conjecture \ref{Conjecture}.
\end{remark}
\begin{remark}\label{pullbacks}
In order to better understand how the Baran metrics of the ball and the simplex look like, it is worth to recall their special relation with certain portion of the sphere.
Let us denote by $(\mathbb H^n_+,g_{\mathbb H^n_+})$ the \emph{upper unit hemisphere}, i.e., the Riemaniann manifold which can be obtained by intersecting the unit sphere $\mathbb S^{n}$ (thought as a sub-manifold of $\mathbb \mathbb{R}^{n+1}$ endowed with the euclidean metric) with the positive half space $\{\xi\in \mathbb{R}^{n+1}: \xi_{n+1}>0\}.$ The map $\pi: \mathbb H^n_+\rightarrow B^n$, $\pi(\xi):=(\xi_1,\dots,\xi_n)$ clearly is a one-to-one $\mathscr C^\infty$ map of manifolds. Therefore we can define a metric $g$ on $B^n$ by means of the pull-back operator with respect to $F:=\pi^{-1}$:
$$g(v,w):=F^*g_{\mathbb H^n_+}(v,w)=g_{\mathbb H^n_+}(dF v,dF w), \forall v,w\in TB^n.$$
One can verify by direct computations that indeed $g\equiv g_{B^n}.$
Similarly, we can define the map $\Sqrt:S^n\rightarrow B^n\cap\{x\in \mathbb{R}^n: x_i>0,\forall i=1,\dots,n\}$, $\Sqrt(x):=(\sqrt{x_1},\sqrt{x_2},\dots,\sqrt{x_n}),$ and pull back by $\Sqrt$ on $S^n$ the Baran metric of the ball. Again this new metric indeed coincide with the Baran metric of the simplex.
In other words the maps $F$ and $sqrt$ are isometries of Riemaniann manifolds.
\end{remark}
Note that, since the manifolds $(B^n,g_{B^n})$ and $(S^n,g_{S^n})$ are isometric to certain portions of $\mathbb S^n$, the local differential and metric properties of this manifolds are the same of $\mathbb S^n.$
We recall that a Riemaniann manifold $(M,g)$ is termed Einstein when its metric tensor is a solution of the \emph{Einstein vacuum field equation}
\begin{equation}\label{vacuumfield}
\boldsymbol{\ric}=k g.
\end{equation}
Here
$$\boldsymbol{\ric}_{i,j}:=\sum_{l=1}^n (\partial_l \Gamma_{j,i}^l-\partial_j \Gamma^l_{l,i})\,+\,\sum_{l,k=1}^n( \Gamma^l_{l,k}\Gamma^k_{j,i}- \Gamma^l_{j,k}\Gamma^k_{l,i} )$$
is the Ricci tensor (written by means of the Christoffel symbols $\Gamma^i_{j,k}$) and $k>0.$
Since it is a well known fact that $(\mathbb S^n,g_{\mathbb S^n})$ is Einstein, we get the following proposition as a consequence of Remark \ref{pullbacks}.
\begin{proposition}
The unit ball and the unit simplex, endowed with their Baran metric respectively, are \emph{Einstein Manifolds}.
\end{proposition}
Since for all cases where the Baran metric is known to be Riemaniann it happens that it solves Equation \eqref{vacuumfield}, the following question naturally arises.
\begin{question}
Assume that $E$ is a Baran body in the sense of Definition \ref{defBB} below. Is it necessary for its Baran metric tensor to solve the Einstein vacuum field equation \eqref{vacuumfield}?
\end{question}
\begin{remark}
Recently, Zelditch \cite{Ze12} studied the spectral theory of the Laplace Beltrami operator on a real analytic Riemaniann manifold $M$ in connection with the Pluripotential Theory of the so called Bruhat-Whitney complexification $M_{\mathbb C}$ of $M.$ In particular, working under the assumption of ergodicity of the geodesic flow, \cite{Ze14,Ze07} present asymptotic results on the zero distribution of the eigenfunctions and series of functions with random Fourier coefficients. These results closely resemble the relation between the behaviour of zeros of orthogonal polynomials (or random polynomials) and the pluripotential equilibrium measure.
Even though our study is far to be as general as the context of the above references, in the author's opinion our result may be casted within this framework and offer concrete examples where explicit computations are performed. Indeed our Appendix \ref{appendix} exactly fits in the framework of \cite{Ze12}.
\end{remark}
The paper is structured as follows. In Section \ref{SecPreliminaries} we furnish all the required definitions from Pluripotential Theory, Operator Theory and Differential Geometry. In Section \ref{SecProofs} we prove Theorems \ref{ThBall} and \ref{ThSimplex}, giving a precise spectral characterization of the involved Sobolev spaces. Finally, in the Appendix \ref{appendix} it is shown how to define the Baran metric on the sphere and its equivalence with the standard round metric.
\subsection*{Acknowledgements}
The ideas of the present paper surfaced during the open problems session of the workshop \emph{Dolomites Research Week on Approximation} (\textsc{DRWA16}), held in Canazei (TN) Italy in September 2016. However, the content of the present paper has been deeply influenced by the note \cite{BoLeVi12} told by its second author during his visit at University of Padova in 2012. Therefore we would like to thank Norm Levenberg and the organizers of the conference and the Doctoral School of Mathematics of the University of Padova. Also, we would like to thank Prof. P.D. Lamberti for his helpfulness, Prof. M. Putti and M. Vianello for the useful discussions and the support they offered.
\section{Preliminaries and tools}\label{SecPreliminaries}
\subsection{The Pluripotential Theory framework}\label{subsectionPluripot}
Pluripotential Theory is the study of \emph{plurisubharmonic} functions, i.e., any upper-semicontinuous function $u:\Omega\rightarrow [-\infty, +\infty[$ being subharmonic along each one complex dimension affine variety in $\Omega\subseteq_{open}\mathbb{C}^n.$ We use the operators $d:=\partial+\bar\partial$ and $d^c:=i(-\partial +\bar\partial)$, where
\begin{equation*}
\partial:= \sum_{j=1}^n\frac{\partial}{\partial z_j}\cdot dz_j,\;\;\;\; \bar\partial:= \sum_{j=1}^n\frac{\partial}{\partial \bar z_j}\cdot d\bar z_j.
\end{equation*}
The operator $\ddc$ is sometimes referred as complex Laplacian and correspond with the usual Laplacian (up to a scaling factor) when $n=1.$
Since $\ddc$ is a linear operator, one can consider $\ddc u$ for a $L^1_{\text{loc}}$ function in the sense of currents (distribution on the space of differential forms) and it turns out that, for an upper-semicontinuous function $u$, $\ddc u\geq 0$ if and only iff $u$ is plurisubharmonic.
The \emph{complex Monge Ampere operator} $(\ddc{})^n$ is defined for $\mathscr C^2$ functions as
\begin{equation}
\label{MAdef}(\ddc{u})^n:=\ddc u \wedge \ddc u \wedge \dots \wedge \ddc u=c_n \det{(\ddc u)}d\Vol_{\mathbb{C}^n}.
\end{equation}
Clearly trying to define wedge products of factors of the type $\ddc u$ for any plurisubharmonic function $u$ leads to serious difficulties due to the lack of linearity. Bedford and Taylor \cite{BeTa82} showed that the definition of equation \eqref{MAdef} can be extended to any locally bounded plurisubharmonic function, being $(\ddc u)^n$ a positive Borel measure.
One may think to plurisubharmonic functions in $\mathbb{C}^n$ as ''the correct counterpart'' (see \cite[Preface]{Kli}) of subharmonic functions on $\mathbb{C}$, while harmonic functions should be replaced in this multi dimensional setting by \emph{maximal plurisubharmonic functions}, i.e., functions $u$ dominating on any subdomain $\Omega'$ any plurisubharmonic function $v$ such that $u\geq v$ on $\partial \Omega'.$ Locally bounded maximal plurisubharmonic functions satisfy $(\ddc u)^n=0.$
The multi dimensional counterpart of the Green function for the complement of a compact set $E$ is the \emph{pluricomplex Green function} (also known as Siciak-Zaharjuta extremal function) $V_E^*.$ Let $E\subset \mathbb{C}^n$ be a compact set, then we set
\begin{equation}\label{efdef}
\begin{split}
&V_E(\zeta):=\sup\{u(\zeta), u\in\mathcal L(\mathbb{C}^n), u|_E\leq 0\},\\
&V_E^*(z):=\limsup_{\zeta\to z}V_E(\zeta).
\end{split}
\end{equation}
Here $\mathcal L(\mathbb{C}^n)$ is the Lelong class of plurisubharmonic functions of logarithmic growth, i.e., $u(z)-\log|z|$ is bounded at infinity.
It is worth to recall that, as in the one dimensional case, due to \cite{Si81} (see also \cite{Kli}) we can express $V_E^*$ by means of polynomials $\wp(\mathbb{C}^n).$ That is
\begin{align*}
&V_E(\zeta)=\sup\left\{\frac 1{\deg p}\log^+|p(\zeta)|, p\in\wp(\mathbb{C}^n), \|p\|_E\leq 1\right\}.
\end{align*}
The function $V_E^*$ is either identically $+\infty$ or a locally bounded plurisubharmonic function on $\mathbb{C}^n$, maximal on $\mathbb{C}^n\setminus E$ (i.e., $\ddcn{V_E^*}$ is a positive Borel measure with support in $E$) having logarithmic growth at $\infty;$ if the latter case occurs we say that $E$ is \emph{non pluripolar}. In principle $V_E^*$ is only a upper semi-continuous function. When $V_E^*$ is continuous the compact set $E$ is said \emph{regular}. It is worth to recall that it turns out that $V_E^*$ is continuous if and only if $V_E^*$ identically vanishes on $E.$ We will treat only such a case in what follows.
For any non pluripolar compact set $E\subset \mathbb{C}^n$ the \emph{pluripotential equilibrium measure} of $E$ is defined as
\begin{equation}\label{pluripoteqmeasuredef}
\mu_E:=(\ddc V_E^*)^n,
\end{equation}
this is a Borel \emph{probability} measure supported on $E.$ We stress that, since $\mu_E(E)=1$ for any non pluripolar set \cite{BeTa82}, the total mass of the measures (and volume forms) that we are going to deal with is not important. We avoid to introduce normalizing constant in the metrics to keep the notation simple.
Let $E$ be a real convex body, Baran showed that in such a case
\begin{equation}\label{baranmetricdef}
\delta_E(x,v):=\limsup_{t\to 0^+}\frac {V_E^*(x+i t v)}{t}
\end{equation}
exists for any $x\in\interior E,$ $v\in \mathbb{R}^n.$ We refer to $\delta_E(x,v)$ as the \emph{Baran metric} of $E.$ We refer the reader to \cite{BoLeWa08} for a study on the connections among this metric, polynomials inequalities and polynomial sampling. The Baran metric defines in general a Finsler distance on $E$
$$d_E(x,y):=\inf\left\{\int_0^1 \delta_E(\gamma(s),\gamma'(s))ds, \gamma\in Lip([0,1],E),\gamma(0)=x,\gamma(1)=y\right\},$$
however it may happen that $\delta_E(x,v)$ is indeed Riemaniann, i.e.
$$\delta_E(x,v)=\sqrt{v^tG_E(x) v}$$
for a positive definite matrix $G_E(x).$ Note that $G_E(x)$ is then well defined by the \emph{parallelogram law}. More precisely we have
$$u^TG_E(x) v=\frac{\delta_E^2(x,u+v)-\delta_E^2(x,u-v)}{4}.$$
One of the possible motivation for the interest on the Baran metric comes from Approximation Theory. Indeed the \emph{Baran Inequality} (see \cite{Ba95,Ba92b} and \cite{BoLeVi12})
\begin{equation}\label{BaranInequality}
\begin{split}
&\frac{\left|\frac d{dt}p(\gamma((t))\right|}{\sqrt{1-p^2(\gamma(t))}}\leq \deg p\,\delta_E(\gamma(t),\gamma'(t)),\\
&\forall t\in[0,1], \gamma\in \mathscr C^1([0,1],E),p\in \wp^k(\mathbb{C}^n), \|p\|_E\leq 1,
\end{split}
\end{equation}
can be understood as a generalization of the classical Bernstein Inequality. For instance such inequality may be used to construct good sampling sets for polynomials, namely admissible meshes; see
\cite{CL08,Kr11,BoDeSoVi10,DePiSoVi15,Pi17}
We believe that the following definition is worth to be introduced.
\begin{definition}[Baran body]\label{defBB}
Let $\mathcal C$ denote either $\mathbb C^n$ or a irreducible algebraic variety of pure dimension $n,$ and let $\mathcal C_{\mathbb R}$ denote the real points of $\mathcal C.$ Let $E\subset \mathcal C_{\mathbb R}$ a compact fat\footnote{This mean that the closure in $\mathcal C_{\mathbb R}$ of the interior of $E$ in $\mathcal C_{\mathbb R}$ coincides with $E.$} non pluripolar set. If the Baran metric of $E$ is Riemaniann, then we term $E$ a \emph{Baran body}.
\end{definition}
In \cite{BoLeVi12, BoLeWa08}, the Baran metrics of the real ball, real simplex are computed (see Theorem \ref{ThBall} and Theorem \ref{ThSimplex} above), showing in particular that they are Baran bodies. To the best author's knowledge, these are all the known examples of Baran compact sets in $\mathbb{C}^n.$ We offer a further instance of a Baran compact in Appendix \ref{appendix}: the real sphere as subset of the complexified sphere.
\subsection{Differential operators and Sobolev spaces on a Riemaniann manifold}
\subsubsection{Differential operators}
We recall that a liner connnection on a vector bundle $\pi:E\rightarrow M$ (built on the differentiable manifold $M$) is an application (here $\mathcal E(M)$ is the space of smooth sections of the vector bundle $E$ and $\mathcal T(M)$ is the tangent bundle)
\begin{align*}
\nabla: &\mathcal T(M)\times \mathcal E(M) &\longrightarrow\;\;\;\;\;\;& \mathcal E(M)\\
&(X,V)\;\; &\longrightarrow\;\;\;\;\;\;& \nabla_X V
\end{align*}
such that it is $\mathscr C^\infty$-linear in $X$, $\mathbb{R}$-linear in $V,$ and for which holds the Liebnitz Rule $\nabla_X(f V)=V X(f)+ f\nabla_X(V),$ for any $f\in\mathscr C^\infty(M).$ In particular we have $\nabla_X f=X(f).$
Let $(M,g)$ be a (possibly non compact) Riemaniann manifold. It is well known that there exists a unique torsion-free linear connection on $\mathcal T(M)$ that is compatible with the metric $g$; namely the \emph{Levi-Civita connection}. Since we will deal only with such a connection we will still denote it by $\nabla.$ Indeed, the proof of the Levi Civita Theorem is fully constructive: the desired connection is expanded over a canonical basis and its coefficients, the \emph{Christoffel symbols} usually denoted by $\Gamma_{i,j}^k,$ are computed in terms the metric and its partial derivatives.
Note that, for a given $u\in \mathscr C^\infty(M),$ $\nabla u$ is a $(1,0)$ tensor field (i.e., point-wise it is a linear form) having the property that $( X,\nabla u)_g=\nabla_X u =X(u)$ and thus it can be written in local coordinates
$$\nabla u=\sum_j g^{i j}\frac{\partial u}{\partial x_j}dx_j.$$
Here $(\cdot ,\cdot)_g$ is the canonical duality induced by $g$ and $g^{ij}$ are the components of the matrix representing $g^{-1}.$ Hence it is convenient to define the tangent vector
$$(\grad u)_i :=\left(\sum_j g^{i j}\frac{\partial u}{\partial x_j}\right)_i,$$
namely the \emph{covariant gradient} of $u$, having the property that $( X,\nabla u)_g=\langle X,\grad u\rangle_g.$
The \emph{divergence} operator acting on $X\in \mathcal T(M)$ is defined by
$$\Div X:=\nabla \cdot X=\frac 1 {\sqrt{\det g}}\sum_i \partial_i( \sqrt{\det g} X^i).$$
Finally we can recall the definition of the \emph{Laplace Beltrami operator} $\Delta.$
\begin{equation}
\Delta u:=\Div(\grad u)=\frac 1 {\sqrt{\det g}}\sum_i \partial_i( \sqrt{\det g} (\grad u)_i).
\end{equation}
\subsubsection{Sobolev Spaces}\label{SubsectionSobolev}
Let $(M,g)$ be a Riemaniann manifold. Let us introduce on $\mathscr C^\infty(M)$ the norm
$$\|u\|_{1,2}:=\left( \int |u|^2 d\Vol_g\right)^{1/2}+ \left(\int |\grad u|^2d\Vol_g\right)^{1/2},$$
where $|\grad u|^2=\langle\grad u ,\grad u\rangle_g.$ Let us denote by $\mathscr C_{1,2}^\infty(M)$ the space $\{u\in \mathscr C^\infty(M), \|u\|_{1,2}<\infty\}.$
The \emph{Sobolev space} $H^1(M,g)$ is defined as the closure of $\mathscr C_{1,2}^\infty(M)$ with respect to $\|\cdot\|_{1,2}$ in the space of square integrable functions, also we introduce the space $H^1_0(M,g)$ defined as the closure of $\mathscr C^\infty_c(M)$ in the same norm. Note that in principle $H^1_0(M,g)\subseteq H^1(M,g).$
An important fact about Sobolev spaces and manifold is that the above two spaces may coincide, that is
\begin{equation}
H^1_0(M,g)\equiv H^1(M,g)\label{sobolevequivalence}
\end{equation}
\emph{Our interest on this phenomena is mainly due to the fact that the Laplace operator does not need to be complemented with boundary conditions in such a case.}
Indeed, $H^1_0(M,g)\equiv H^1(M,g)$ for any complete Riemaniann manifold $M$; see \cite[Th. 3.1]{He99}. We recall for the reader's convenience that a Riemaniann manifold $(M,g)$ is said to be complete if the metric space $(M,d_g)$ is complete, where \small
$$d_g(x,y):=\inf\left\{\int_0^1 \sqrt{\langle\gamma'(s),\gamma'(s)\rangle_{g(\gamma(s))}}ds, \gamma\in Lip([0,1],E),\gamma(0)=x,\gamma(1)=y\right\},$$
\normalsize
The Hopf-Rinow Theorem asserts that the completeness of $(M,g)$ is equivalent to the fact that any closed bounded subset of $M$ is compact.
We denote by $\mathscr C^\infty_b(M)$ the set uniformly bounded functions that have uniformly bounded partial derivatives of any order. Since for a complete manifold $\mathscr C^\infty_c(M)\subseteq \mathscr C^\infty_b(M)\subset H^1(M,g)$, it follows that for any complete manifold $(M,g)$, $\mathscr C^\infty_b(M)$ is dense in $H^1(M,g).$
Unfortunately, both $(B^n,g_{B^n})$ and $(S^n,g_{S^n})$ fail to be complete: it is very easy to construct a Cauchy sequence in $B^n$ not converging in $B^n.$ For instance take $\{x_k\}:=\cos{(2^{-k})}u$ for any unit vector $u\in \mathbb{R}^n$. Since $d(x_{k},x_{l})\leq 2^{-\min(k,l)},$ this is a Cauchy sequence, however $x_k\to u\notin B^n.$ Nevertheless, one may wonder weather equation \eqref{sobolevequivalence} holds true in this instances. This fact indeed depends on finer properties of the manifolds than completeness. Namely, Masamune \cite{Ma05,Ma99} showed that equality \eqref{sobolevequivalence} holds if and only if the metric completion of $M$ lies in the category of manifolds with almost polar boundary.
We recall that the Riemaniann manifold $(M\cup \Gamma, g)$ with boundary $\Gamma$ is said to have \emph{almost polar boundary} if the outer capacity $\Capa(\Gamma)$ of $\Gamma$ vanishes. Here we use the notation $\Capa(A)$ for the Sobolev (outer) capacity of the Borel subset $A$ of $M\cup \Gamma,$ where for any open subset $O$ of $M\cup \Gamma$ we set
\begin{equation*}
\Capa(O):=\inf\{\|u\|_{1,2}, u\in\mathscr C^\infty_c(M\cup \Gamma), 0\leq u\leq 1, u|_O\equiv 1\}
\end{equation*}
and for for any Borel subset $S$ we set
$$\Capa(A):=\inf\{\Capa(O), A\subset O\}.$$
It is clear that one can replace $\mathscr C^\infty_c(M\cup \Gamma)$ by $H^1_0(M\cup \Gamma,g)$ in the definition of $\Capa(O)$ obtaining an equivalent definition.
At this stage we can observe that $\partial B^n$ fails the \emph{sufficient} condition (see \cite[Th. 7]{Ma05}) to be polar
\begin{equation}\label{sufficientalmostpolar}
\liminf_{\epsilon\to 0^+}\frac{\log \Vol\left(\{x\in B^n: d(x,\partial B^n)<\epsilon \}\right)}{\log\epsilon}\geq 2.
\end{equation}
Here equality case is considered since $\partial B^n$ itself is a manifold (see \cite[Th. 7]{Ma05}).
Let us denote by $N_\epsilon$ the set $\{x\in B^n: d(x,\partial B^n)<\epsilon \}$, we have
$N_\epsilon=B^n \setminus (\cos \epsilon)\cdot B^n$, moreover
$$\Vol (N_\epsilon)=\pi \beta(1/2,n/2,1-(\cos \epsilon)^2).$$
Here $\beta(a,b,z)$ denotes the Incomplete Beta Function $\int_0^z t^{a-1}(1-t)^{b-1}dt.$
Hence
$$\frac{\Vol N_\epsilon}{\epsilon^2}\sim \frac{\Vol N_\epsilon}{1-(\cos \epsilon)^2} \frac{1-(\cos \epsilon)^2}{\epsilon^2}\sim 2\frac{\Vol N_\epsilon}{1-(\cos \epsilon)^2},\;\;\text{ as } \epsilon\to 0^+.$$
Note that
$$\liminf_{\epsilon\to 0^+}\frac{\Vol N_\epsilon}{1-(\cos \epsilon)^2}=\lim_{z\to 0^+}\frac{\beta(1/2,n/2,z)}{z}=\lim_{z\to 0^+}z^{-1/2}(1-z)^{n/2-1}=+\infty.$$
Thus we have $\liminf_{\epsilon\to 0^+}\frac{\Vol N_\epsilon}{\epsilon^2}=+\infty$ that in particular implies $\frac{\log \Vol N_\epsilon}{\log \epsilon}<2$ for any $\epsilon<\epsilon_0.$
Since the condition \eqref{sufficientalmostpolar} is not fulfilled by $\partial B^n$ nor $\partial S^n$ we wonder if the ball and the simplex, endowed with their Baran metrics, are not manifold with almost polar boundary. Indeed this is the case, as stated in the following proposition. However, since these conclusions are obtained as a consequence of Theorem \ref{ThBall} and Theorem \ref{ThSimplex} respectively, we cannot use them in the proof of such theorems.
\begin{proposition}\label{propnotalmostpolar}
The manifolds $(B^n,g_{B^n})$ and $(S^n,g_{S^n})$ are not manifold with almost polar boundary and
\begin{equation}
H^1(B^n,g_{B^n})\neq H^1_0(B^n,g_{B^n})\;,\;\;\;\;H^1(S^n,g_{S^n})\neq H^1_0(S^n,g_{S^n}).
\end{equation}
\end{proposition}
\begin{remark}
We warn the reader that $H^1(B^n,g_{B^n})\neq H^1_0(B^n,g_{B^n})$ does not imply in general that the eigenvalue problem $\Delta u =\lambda u$ is not well posed when we do not impose any boundary condition. The motivation depends on the following proposition which allows us to write the weak formulations \eqref{ballsymmetry} and \eqref{simplexsymmetry} of the Laplace Beltrami operator used in the proofs of Theorem \ref{ThBall} and \ref{ThSimplex} which is based on $\mathscr C^\infty_b$ functions (for which the boundary terms appearing in the integration by parts formulas we use vanish).
\end{remark}
\begin{proposition}\label{propdensity}
Let $(M,g)$ be $(B,g_B)$ or $(S,g_S)$. The space $\mathscr C^\infty_b(M)$ is dense in $\mathscr C^\infty_{1,2}(M)$ with respect to the norm $\|\cdot\|_{1,2}.$ Thus $\mathscr C^\infty_b(M)$ is dense in $H^1(M,g).$
\end{proposition}
Before proving Proposition \ref{propdensity} we need this two technical Lemmata whose proofs are omitted since it is sufficient to check the statements by easy direct computations.
\begin{lemma}[The inverse Baran metric of the ball]\label{Lemmaballinversemetric}
Let us denote by $G_{B^n}^{-1}$ the inverse of the matrix $G_{B^n}$ which represents the Baran metric of the $n$-dimensional ball. Then we have
\begin{equation}\label{ballinversemetric}
G_{B^n}^{-1}(x):=\left[\begin{array}{cccccc}
1- x_1^2& -x_1 x_2& -x_1 x_3&\dots&\dots&x_1 x_n\\
-x_2 x_1& 1-x_2^2 & -x_2 x_3&\dots&\dots&x_2 x_n\\
\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\
-x_n x_1&\dots & \dots&-x_n x_{n-2}&-x_n x_{n-1}&1- x_n^2
\end{array}\right].
\end{equation}
The matrix $G_{B^n}^{-1}(x)$ has eigenvalues $\{1,1-|x|^2\}$, where the eigen-space of $1$ is the tangent space at $x$ to the sphere of radius $|x|$ and centred at zero, while the eigen-space of $1-|x|^2$ is the Euclidean normal to such a sphere at $x.$
\end{lemma}
\begin{lemma}[The inverse Baran metric of the simplex]\label{lemmasimplexinversemetric}
Let us denote by $G_{S^n}^{-1}$ the inverse of the matrix $G_{S^n}$ which represents the Baran metric of the $n$-dimensional simplex. Then we have
\begin{equation}\label{simplexinversemetric}
G_{S^n}^{-1}(x):=\left[\begin{array}{ccccc}
(1- x_1)x_1& -x_1 x_2&\dots&\dots&x_1 x_n\\
-x_2 x_1& (1-x_2)x_2 &\dots&\dots&x_2 x_n\\
\vdots &\vdots &\vdots&\vdots &\vdots \\
-x_n x_1&\dots & \dots&-x_n x_{n-1}&(1- x_n)x_n
\end{array}\right].
\end{equation}
Moreover we have
\begin{equation}\label{relationinversegramian}
G_{S^n}^{-1}(x)= diag(\sqrt{x_1},\dots,\sqrt{x_n})\,G_{B^n}^{-1}(\sqrt{x_1},\dots,\sqrt{x_n})\,diag(\sqrt{x_1},\dots,\sqrt{x_n}).
\end{equation}
\end{lemma}
\begin{proof}[Proof of Proposition \ref{propdensity}]
Let us start by considering the case $M=B^n\subset \mathbb{R}^n.$ We denote by $\mathbb S^n$ the $n$ dimensional unit real sphere endowed with the standard \emph{round metric} $g_{\mathbb S^n}$ and we introduce the embedding map
$$E:\mathscr C^\infty_{1,2}(M)\rightarrow H_{\text{even}}^1(\mathbb S^n,g_{\mathbb S^n}),$$
where
\small
$$ E[f]\left(x_1,\dots,x_n,\pm \sqrt{1-\sum_{i=1}^n x_i^2}\right):=\frac 1 {\sqrt 2}\begin{cases} f(x_1,\dots,x_n)&,\sum_{i=1}^n x_i^2\neq 1\\ {\limsup}_{B\ni \xi\to x} f(\xi_1,\dots,\xi_n)&, \sum_{i=1}^n x_i^2= 1\end{cases}$$
\normalsize and \small
$$H_{\text{even}}^1(\mathbb S^n,g_{\mathbb S^n}):=\left\{g\in H^1(\mathbb S^n,g_{\mathbb S^n}),\;g(x_1,\dots,x_{n+1})=g(x_1,\dots,-x_{n+1})\, \Vol_{\mathbb S^n}-a.e.\right\}.$$
\normalsize We claim that $E$ is an \emph{isometry} of Hilbert spaces.
Before proving such a claim we stress that this would conclude the proof for the case of the ball. For, by standard mollification we can construct a sequence $\{\hat f_k\}$ of function in $\mathscr C^\infty(\mathbb S^n)$ converging to $E[f]$ in $H^1(\mathbb S^n,g_{\mathbb S^n}).$ To ensure that $\tilde f_k \in H_{\text{even}}^1(\mathbb S^n,g_{\mathbb S^n})$ we set $\tilde f_k:=(\hat f_k(x_1,\dots,x_{n+1})+\hat f_k(x_1,\dots,-x_{n+1}))/2.$
Finally define $\{f_k\}:=\{E^{-1}[\tilde f_k]\}$ and note that the claim above implies that $f_k\to f$ in $H^1(B^n,g_{B^n}).$
We stress that, while the injectivity of $E$ is trivial, one needs to notice that the global boundedness of $\tilde f_k$ together with its derivatives ensure that $E^{-1}[\tilde f_k]$ is a well defined element of $\mathscr C^\infty_{1,2}(M)$ which in particular is in $\mathscr C^\infty_b(M).$
Let us go back to prove that $E$ is an isometric embedding. For simplicity we work in the easy case of $n=2$, the general case can be proved in a completely equivalent way. Consider the spherical coordinates
$$\left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right)=\left(\begin{array}{c}\cos \theta \cos \phi\\ \cos \theta \sin \phi\\\sin \theta\end{array}\right).$$
We recall that the round metric represented in this coordinates is
$$g_{\mathbb S^2}(\theta, \phi):=\left[\begin{array}{cc}1 & 0 \\0 & \cos^2 \theta \end{array}\right]$$
and the corresponding volume form can be written $d \Vol_{\mathbb S^2}=\cos \theta d\theta d\phi.$ It follows that, for any $h\in H^1(\mathbb S^2,g_{\mathbb S^2})$ we have
$$\|h\|_{H^1(\mathbb S^2)}^2=\int_0^{2\pi}\int_{-\pi/2}^{\pi/2}\left(|h|^2+ |\partial_\theta h|^2+\frac{|\partial_\phi h|}{\cos^2\theta}\right)\cos \theta d\theta d\phi.$$
To compute $\|E[f]\|_{H^1(\mathbb S^2)}$ we perform the change of variables suggested by the first two components of the spherical coordinates, i.e.,
$$(x_1,x_2)\mapsto(\cos \theta \cos \phi, \cos \theta \sin \phi).$$
It is easy to verify by a direct computation that
\begin{align*}
&\|E[f]\|_{H^1(\mathbb S^2)}^2\\
=& \int_0^{2\pi}\int_{-\pi/2}^{\pi/2}\left(|E[f]|^2+ |\partial_\theta E[f]|^2+\frac{|\partial_\phi E[f]|}{\cos^2\theta}\right)\cos \theta d\theta d\phi\\
=& 2\int_0^{2\pi}\int_{0}^{\pi/2}\left(|E[f]|^2+ |\partial_\theta E[f]|^2+\frac{|\partial_\phi E[f]|}{\cos^2\theta}\right)\cos \theta d\theta d\phi\\
=&2 \int_{B^2} \left(\frac{|f|^2} 2 + (1-x_1^2-x_2^2) \frac{|\partial_n f|^2} 2 +\frac{|\partial_t f|^2} 2\right)\frac 1{\sqrt{1-x_1^2-x_2^2}}dx_1dx_2\\
=&\int_{B^2} \left(|f|^2 + |\grad f|_{g_{B^2}}^2\right) d\Vol_{g_{B^2}}\\
=&\|f\|_{H^1(B,g_{B^2})}^2.
\end{align*}
Let us now consider the case $M=S^n.$ We introduce the embedding map
$$F:\mathscr C^\infty_{1,2}( S^n,g_{S^n})\rightarrow V,$$
where
$$V:=\{f\in \mathscr C^\infty_{1,2}(B^n,g_{B^n}), f(\xi_1,\dots,\xi_j,\dots,\xi_n)=f(\xi_1,\dots,-\xi_j,\dots,\xi_n),\forall j\in \{1,\dots,d\}\}$$
and
$$ F[h]\left(\xi_1,\dots,\xi_n\right):= \frac 1 2 h(\xi_1^2,\dots,\xi_n^2).$$
Again if the closure of $F$ to $H^1(S^n,g_{S^n})$ is an isometric embedding we are done, since, for any given target function $h\in H^1(S^n, g_{S^n})$ we can pull back to $\mathscr C^\infty_b(S^n)$ any sequence of $\mathscr C^\infty_b(B^n)$ approximations to $F[h].$
To this aim, we introduce the partition $Q_1,\dots,Q_{2^n}$ of $[-1,1]^n$ given by the coordinates hyperplanes, we denote by $T:S^n\rightarrow B^n$ the map $(\xi_1,\dots,\xi_n)\mapsto (\xi_1^2,\dots,\xi_n^2)=x$ and we
notice that, for any $f\in \mathscr C^\infty (S^n)$, we have
$$\int_{B^n\cap Q_j} f\circ T d\Vol_{B^n}=\frac 1{2^n}\int_{S^n} f d\Vol_{S^n}.$$
Finally we compute
\begin{align*}
&\|F[h]\|_{H^1(B^n,g_{B^n})}\\
=&\sum_{j=1}^{2^n}\int_{B^n\cap Q_j} \left(|F[h](\xi)|^2 +|\grad F[h](\xi)|_{g_{B^n}}^2\right) d\Vol_{B^n}(\xi)\\
=&\frac 1 4\sum_{j=1}^{2^n}\int_{B^n\cap Q_j}\left(|h\circ T(\xi)|^2+Dh^t\circ T JT^t g_{B^n}^{-1} JT Dh\circ T(\xi)\right) d\Vol_{B^n}(\xi)\\
=&\frac 1 {4\cdot 2^n}\sum_{j=1}^{2^n}\int_{T^{-1}(B^n\cap Q_j)}\left(|h(x)|^2 +Dh^t (JT^t g_{B^n}^{-1} JT)\circ T^{-1} Dh(x)\right) d\Vol_{S^n}(x)\\
=&\frac 1 {4 \cdot 2^n}2^n\int_{T^{-1}(B^n\cap Q_1)}\left(|h(x)|^2 +Dh^t (JT^t g_{B^n}^{-1} JT)\circ T^{-1} Dh(x)\right) d\Vol_{S^n}(x).
\end{align*}
Since, due to equation \eqref{relationinversegramian},
$$(JT^t g_{B^n}^{-1} JT)\circ T^{-1}=4 diag(\xi)g_{B^n}^{-1}diag(\xi)\Big |_{\xi=\sqrt x}=4 g_{S^n}(x)$$
we conclude that $\|F[h]\|_{H^1(B^n,g_{B^n})}= \|h\|_{H^1(S^n,g_{S^n})}.$ In view of the above reasoning this concludes the proof.
\end{proof}
\subsection{Unbounded linear operators on Hilbert spaces, some tools.}
We need to recall some concepts from Operator Theory that allow a more precise and compact formulation of our results. A \emph{linear operator} on a Banach space $\mathscr B$ is a couple $(\mathcal D_{\mathscr B}(T),T)$, where $\mathcal D_{\mathscr B}(T)$ is a dense linear subspace of $\mathscr B$ and $T$ is a linear map $\mathcal D_{\mathscr B}(T)\rightarrow \mathscr B.$
Let $(\mathcal D_{\mathscr B}(T),T)$ be a linear operator. If for any sequence $\{f_n\}$ in $\mathcal D_{\mathscr B}(T)$ such that
\begin{itemize}
\item $\|f_n\to f\|_{\mathscr B}\to 0$ for some $f\in\mathscr B,$
\item there exists $g\in\mathscr B$ with $\|T f_n-g\|_{\mathscr B}\to 0$
\end{itemize}
it follows that $f\in \mathcal D_{\mathscr B}(T)$ and $Tf=g$, then the operator $T$ is said to be \emph{closed}.
If $\mathscr B$ is not finite dimensional, the notion of spectrum and set of eigenvalues are not coinciding. More precisely, we denote by $\sigma(T)$ the \emph{spectrum} of $T$
$$\sigma(T):=\{z\in \mathbb{C}: T-z\mathbb I \text{ is not invertible}\}.$$
Instead, $\lambda$ is an eigenvalue of $T$ if there exists an element $f\in \mathscr B$ such that $Tf=\lambda f.$
If an operator $T$ is not closed we may try to find and extension of it, i.e., $(\tilde T, \mathcal D_{\mathscr B}(\tilde T))$ such that $\mathcal D_{\mathscr B}(\tilde T)\supset \mathcal D_{\mathscr B}(T)$ and $\tilde Tf=Tf$ for any $f\in \mathcal D_{\mathscr B}(T).$ If we can find such an extension in the category of closed operators, then $T$ is said to be \emph{closable} and its minimal closed extension $\overline T$ is termed \emph{the closure} of $T.$
Now we replace the Banach space $\mathscr B$ by an Hilbert space $\mathscr H$, clearly the above terminologies are still well defined, since any Hilbert space is in particular Banach.
If for any $f, g\in \mathcal D_{\mathscr H}(T)$ we have $\langle Tf,g\rangle_{\mathscr H}=\langle f,Tg\rangle_{\mathscr H}$, then the operator $T$ is said to be \emph{symmetric}. It is a very useful fact that \emph{any symmetric operator is closable to a symmetric operator}. Again, if $\mathscr H$ is infinite dimensional, one must pay attention to the difference among symmetric and self-adjoint operators.
The adjoint $T^*$ of the operator $T$ is defined by the relation
$$\langle Tf,g\rangle_{\mathscr H}=\langle f,T^*g\rangle_{\mathscr H}, \forall f\in \mathcal D_{\mathscr H}(T), g\in \mathcal D_{\mathscr H}(T^*),$$
where
$$\mathcal D_{\mathscr H}(T^*):=\left\{g\in \mathscr H:\exists h\in \mathscr H\text{ such that }\langle Tf,g\rangle_{\mathscr H}=\langle f,h\rangle_{\mathscr H}, \forall f\in \mathcal D_{\mathscr H}(T) \right\}.$$
Clearly, we term $T$ self-adjoint when the two domains indeed coincide.
The proofs of our results, besides the explicit computations, rely on the following theorem which collects some classical results of Operator Theory; see for instance \cite[Ch. 1 and Ch. 4]{Da95}.
\begin{theorem}\label{ThOpTh}
Let $T$ be a linear non negative unbounded operator on the Hilbert space $(\mathscr H, \|\cdot\|)$ with domain $\mathcal D(T)$. Assume that
\begin{enumerate}[a)]
\item $T$ is symmetric,
\item It has discrete real spectrum $\sigma(T)=\{\lambda_j\}_{j\in \mathbb{N}}$ diverging to $+\infty.$
\end{enumerate}
Then
\begin{enumerate}[i)]
\item the closure $\bar T$ of $T$ is a self-adjoint unbounded operator (i.e., $T$ is essentially self-adjoint),
\item $\sigma( \bar T)=\sigma(T)$,
\item the domain of $\bar T$ is
\begin{equation}
\mathcal D(\bar T)=\{u\in \mathscr H:\;\sum_{j=1}^{\infty}\lambda_j^2|\hat u_j|^2<\infty\}
\end{equation}
\item the quadratic form
$$Q(u):=\langle T^{1/2}u,T^{1/2}u\rangle_{\mathscr H}$$
has domain
\begin{equation}
\mathcal D(Q)=\{u\in \mathscr H:\;\sum_{j=1}^{\infty}\lambda_j|\hat u_j|^2<\infty\}
\end{equation}
which is complete in the norm
$$|\|u\||:=\sqrt{Q(u)}+\|u\|_{\mathscr H}.$$
\end{enumerate}
\end{theorem}
\section{Proofs}\label{SecProofs}
The strategy of the proofs of Theorems \ref{ThBall} and \ref{ThSimplex} is to show that the conditions $a)$ and $b)$ of Theorem \ref{ThOpTh} holds for $T$ being the Laplace Beltrami operator (with respect to the considered metric), then to conclude applying Theorem \ref{ThOpTh}. This will be done by considering the weak formulation of the Laplace Beltrami operator and performing explicit computations on a suitable orthogonal system.
\subsection{Orthogonal polynomials in $L^2_{\mu_{B^n}}$}
The following family of orthogonal functions on the unit ball has been first introduced in the Approximation Theory framework, indeed the formula we will use is a special case of orthogonal polynomials for certain radial weight functions; see \cite[Ch. 5]{DuXu14}.
\begin{figure}[h]
\caption{The first ten basis functions $\phi_\alpha$ of Proposition \ref{orthogonalsystemball} generates $\wp^3(B^2).$ }
\label{basis}
\begin{tabular}{ccccc}
\includegraphics[scale=0.15]{polynomials00}&
\includegraphics[scale=0.15]{polynomials01}&
\includegraphics[scale=0.15]{polynomials02}&
\includegraphics[scale=0.15]{polynomials03}&
\includegraphics[scale=0.15]{polynomials11}\\
\includegraphics[scale=0.15]{polynomials12}&
\includegraphics[scale=0.15]{polynomials13}&
\includegraphics[scale=0.15]{polynomials22}&
\includegraphics[scale=0.15]{polynomials23}&
\includegraphics[scale=0.15]{polynomials33}
\end{tabular}
\end{figure}
\begin{proposition}[\cite{DuXu14}]\label{orthogonalsystemball}
Let us set for any $\alpha\in \mathbb{N}^{n}$\small
\begin{equation}
\phi_\alpha:=T_{\alpha_n}\left(\frac{x_n}{\sqrt{1-\sum_{k=1}^{n-1}x_k^2}} \right)\prod_{j=1}^{n-1}(1-\sum_{k=1}^{j-1} x_k^2)^{\alpha_j/2}C_{\alpha_j}^{\lambda_j}\left( \frac{x_j}{\sqrt{1-\sum_{k=1}^{j-1} x_k^2}} \right) ,
\end{equation}\normalsize
where $T_k$ is the Chebyshev polynomial of degree $k$, $\lambda_j:=\frac{n-j}2+\sum_{k=j+1}^n\alpha_k$ and $C^s_t$ denote the monic Gegenbauer polynomials of degree $t$ (i.e., $C^s_t:=J^{s-1/2,s-1/2}_t$ and $J_t^{\alpha,\beta}$ is the monic Jacobi polynomial).
The set $\{\phi_\alpha: \alpha\in \mathbb{N}^{n}\}$ is a dense orthogonal system in $L^2({B^n},g_{B^n})$ and
\begin{equation}\label{normsontheball}
\|\phi_\alpha\|^2_{L^2({B^n},g_{B^n})}=\|T_{\alpha_n}\|_{-1/2,-1/2}^2\prod_{j=1}^{n-1}\|C_{\alpha_j}^{\lambda_j}\|_{\alpha_j-1/2,\alpha_j-1/2}^2,
\end{equation}
where $\|f\|_{a,b}:=\left(\int_{-1}^1|f(t)|^2(1-t)^a(1+t)^bdt\right)^{1/2}.$
\end{proposition}
Note that the density of the linear subspace $\Span\{\phi_\alpha: \alpha\in \mathbb{N}^{n}\}$ in $H^1({B^n},g_{B^n})$ follows by Proposition \ref{propdensity}.
\subsection{Proof of Theorem \ref{ThBall}}
we \textbf{warn the reader} that we will denote throughout this section by $D f$ the \emph{Euclidean gradient of }$f.$
\begin{proof}[Proof of Theorem \ref{ThBall}]
We start showing that $\Delta$ acting on $\mathscr C^2_b(B^n)$ is a symmetric operator. Namely, for any $u,v\in \mathscr C^2_b(B^n),$ we have
\small
\begin{equation}\label{ballsymmetry}
\int_{B^n}u\Delta v d\Vol_{B^n}=-\int_{B^n}\langle \grad u,\grad v\rangle_{g_{B^n}}d\Vol_{B^n} =\int_{B^n}v\Delta u d\Vol_{B^n}.
\end{equation}
\normalsize
In order to prove this formula we perform two integrations by parts.
\begin{align*}
&-\int_{B^n}v\Delta u d\Vol_{B^n}=-\int_{B^n} \Div(\sqrt{\det g_{B^n}} G_{B^n}^{-1}D u)v dx\\
=&\lim_{r\to 1} -\int_{{B^n}_r} \Div(\sqrt{\det g_{B^n}} G_{B^n}^{-1}D u)v dx\\
=&\lim_{r\to 1} \left(\int_{{B^n}_r} D v^TG_{B^n}^{-1}D u \sqrt{\det g_{B^n}}dx-\int_{\partial {B^n}_r} \nu^T G_{B^n}^{-1}D u \sqrt{\det g_{B^n}}d\sigma\right)\\
=& \int_{B^n}\langle \grad u,\grad v\rangle_{g_{B^n}}d\Vol_{B^n}-\lim_{r\to 1}\int_{\partial {B^n}_r} \nu^T G_{B^n}^{-1}D u \sqrt{\det g_{B^n}}d\sigma\\
=& -\int_{B^n} \Div(\sqrt{\det g_{B^n}} G_{B^n}^{-1}D v)u dx+\\
&\;\;\;\;\;\;\;\; \lim_{r\to 1}\int_{\partial {B^n}_r} u\nu^T G_{B^n}^{-1}D v \sqrt{\det g_{B^n}}d\sigma -\lim_{r\to 1}\int_{\partial {B^n}_r} v \nu^T G_{B^n}^{-1}D u \sqrt{\det g_{B^n}}d\sigma\\
=&-\int_{B^n}u\Delta v d\Vol_{B^n}+ \lim_{r\to 1}\int_{\partial {B^n}_r} u\nu^T G_{B^n}^{-1}D v \sqrt{\det g_{B^n}}d\sigma\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; -\lim_{r\to 1}\int_{\partial {B^n}_r} v \nu^T G_{B^n}^{-1}D u \sqrt{\det g_{B^n}}d\sigma
\end{align*}
Here $\nu$ is the (euclidean) unit outward normal to $\partial {B^n}_r:=\{x\in \mathbb{R}^n:|x|\leq r\}.$
The proof of \eqref{ballsymmetry} is concluded if we show that
$$ \lim_{r\to 1}\int_{\partial {B^n}_r} v \nu^T G_{B^n}^{-1}D u \sqrt{\det g_{B^n}}d\sigma=0$$
for any $u,v\in \mathscr C^2_b({B^n}).$ For, simply observe (see Lemma \ref{Lemmaballinversemetric}) that $\nu$ is an eigenvector of $G_{B^n}^{-1}$ of eigenvalue $(\det g)^{-1}|_{|x|=r}=(1-|x|^2)$, thus we have
\begin{align*}
&\lim_{r\to 1}\int_{\partial B_r} v \nu^T G_B^{-1}D u \sqrt{\det g}d\sigma=\lim_{r\to 1} \sqrt{1-r^2}\int_{\partial B_r} v \partial_\nu u d\sigma\\
\leq&\lim_{r\to 1}\sqrt{1-r^2}C(u,f)=0.
\end{align*}
This shows that condition $a)$ of Theorem \ref{ThOpTh} holds for $\Delta.$ To conclude the proof we need to show that $b)$ holds as well, i.e., there exists a $L^2(B^n,g_{B^n})$-orthogonal system in $\mathscr C^2_b(B^n)$ dense in $L^2(B^n,g_{B^n})$ made of eigenfunctions of $\Delta$ such that the corresponding eigenvalues are a positive diverging sequence. We claim that such an orthogonal system is, indeed $\{\phi_\alpha, \alpha\in \mathbb{N}^n\},$ see Proposition \ref{orthogonalsystemball}.
For the sake of readability we present here the case $n=2$, which leads to slightly easier notation and computations with respect to the general one. However, all the elements of the proof of the general case are presented in such a simplified exposition. To easy the notation we denote $B^2$ by $B.$
The orthogonal basis of Proposition \ref{orthogonalsystemball} reads as
$$\phi_{s,k}(x,y):=(1-x^2)^{k/2}J_{s-k}^{k,k}(x) T_k\left( \frac y{\sqrt{1-x^2}}\right), 0\leq s\leq k\in \mathbb{N},$$
where we denoted by $J_m^{\alpha,\beta}$ the $m$-th Jacobi orthogonal polynomial with respect to $(1-x)^\alpha(1+x)^\beta.$ We need to verify
$$\langle -\Delta \phi_{s,k},\phi_{m,l}\rangle_{L^2(B,g_B)}=\lambda_{s,k}\delta_{s,m}\delta_{k,l}$$
Since $\phi_{s,k}$ are elements of $\mathscr C^{\infty}_b( B)$ we can use the above weak formulation \eqref{ballsymmetry} to get
\begin{align*}
&\langle -\Delta \phi_{s,k},\phi_{m,l}\rangle_{L^2(B,g_B)}=\int_BD \phi_{s,k}^T G_B^{-1}D \phi_{m,l}\sqrt{\det g_B}dxdy.
\end{align*}
Let us introduce a change of variables
$$(x,z)\mapsto \Psi(x,z):=(x,z\sqrt{1-x^2})=(x,y).$$
We denote by $J\psi$ the Jacobian matrix of $\Psi$ so we get
\begin{align*}
&\int_B D f_1^T G_B^{-1}D f_2 \sqrt{\det g}dx dy=\\
=&\int_{-1}^1\int_{-1}^1D(f_1\circ\Psi)^T J\Psi^{-T}G_B^{-1}J\Psi^{-1} D(f_2\circ \Psi) dx \frac {dz}{\sqrt{1-z^2}}\\
=&\int_{-1}^1\int_{-1}^1D(f_1\circ\Psi)^T \left[\begin{array}{cc} 1-x^2&0\\0&\frac{1-z^2}{1-x^2}\end{array}\right] D(f_2\circ \Psi) dx \frac {dz}{\sqrt{1-z^2}}.
\end{align*}
Note that not only $\Psi$ is a change of variables that diagonalizes $G_B^{-1}$, also it has the property of giving to the basis functions $\phi_{s,k}$ a tensor product structure. Indeed we have
$\phi_{s,k}\circ\Psi(x,z)=(1-x^2)^{k/2}J_{s-k}^{k,k}(x) T_k( z)$, thus
\begin{align*}
&\int_{-1}^1\int_{-1}^1D(\phi_{}s,k\circ\Psi)^T \left[\begin{array}{cc} 1-x^2&0\\0&\frac{1-z^2}{1-x^2}\end{array}\right] D(\phi_{m,l}\circ \Psi) dx \frac {dz}{\sqrt{1-z^2}}\\
=&\int_{-1}^{1}\partial_x[(1-x^2)^{k/2}J_{s-k}^{k,k}(x)] \partial_x[(1-x^2)^{l/2}J_{m-l}^{l,l}(x)](1-x^2)dx\;\cdot\\
&\;\;\;\;\;\;\;\int_{-1}^1 T_k(z)T_l(z)\frac {dz}{\sqrt{1-z^2}}\;+\\
&\;\;\;\int_{-1}^{1}(1-x^2)^{(k+l)/2-1}J_{s-k}^{k,k}(x)J_{m-l}^{l,l}(x)]dx\;\cdot\\
&\;\;\;\;\;\;\;\int_{-1}^1 \partial_zT_k(z)\partial_z T_l(z)\sqrt{1-z^2}dz
\end{align*}
It is well known that
$$\int_{-1}^1 T_k(z)T_l(z)\frac {dz}{\sqrt{1-z^2}}= 2^{\delta_k\delta_l}\pi/2\delta_{l,k}.$$
Also one has $T_k'=kU_{k-1},$ where $U_k$ are orthogonal Chebyshev ppolynomials of the second kind, i.e.,
$$\int_{-1}^1 U_k(z)U_l(z)\sqrt{1-z^2}dz= \pi/2\delta_{l,k}.$$
Using such orthogonality and differentiation relations in the above computation we get
\begin{align}
&\int_{-1}^{1}\partial_x[(1-x^2)^{k/2}J_{s-k}^{k,k}(x)] \partial_x[(1-x^2)^{l/2}J_{m-l}^{l,l}(x)](1-x^2)dx\;\cdot\notag \\
&\;\;\;\;\;\;\;\int_{-1}^1 T_k(z)T_l(z)\frac {dz}{\sqrt{1-z^2}}\;+\notag \\
&\;\;\;\;\;\;\;\int_{-1}^{1}(1-x^2)^{(k+l)/2-1}J_{s-k}^{k,k}(x)J_{m-l}^{l,l}(x)]dx\;\cdot\notag \\
&\;\;\;\;\;\;\;\int_{-1}^1 \partial_zT_k(z)\partial_z T_l(z)\sqrt{1-z^2}dz\notag \\
=&\frac\pi 2\delta_{l,k}\Big(\int_{-1}^{1}\partial_x[(1-x^2)^{k/2}J_{s-k}^{k,k}(x)] \partial_x[(1-x^2)^{k/2}J_{m-k}^{k,k}(x)](1-x^2)dx\;\cdot2^{\delta_k}\notag\\
&\;\;\;+k^2\int_{-1}^{1}(1-x^2)^{k-1}J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)dx\Big)\label{intermstep} .
\end{align}
Now we note that
\begin{align*}
&\int_{-1}^{1}\partial_x[(1-x^2)^{k/2}J_{s-k}^{k,k}(x)] \partial_x[(1-x^2)^{k/2}J_{m-k}^{k,k}(x)](1-x^2)dx\\
=&\int_{-1}^{1}\partial_x[J_{s-k}^{k,k}(x)]\partial_x[J_{m-k}^{k,k}(x)](1-x^2)^{k+1}dx\\
&\;\;\;\;+\int_{-1}^{1}-kx\partial_x[J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)](1-x^2)^{k}dx\\
&\;\;\;\;+k^2\int_{-1}^{1}x^2J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)(1-x^2)^{k-1}dx,
\end{align*}
integration by parts in the second term leads to
\begin{align*}
&\int_{-1}^{1}\partial_x[(1-x^2)^{k/2}J_{s-k}^{k,k}(x)] \partial_x[(1-x^2)^{k/2}J_{m-k}^{k,k}(x)](1-x^2)dx\\
=&\int_{-1}^{1}\partial_x[J_{s-k}^{k,k}(x)]\partial_x[J_{m-k}^{k,k}(x)](1-x^2)^{k+1}dx\\
&\;\;\;\;-2k^2\int_{-1}^{1}x^2J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)(1-x^2)^{k-1}dx\\
&\;\;\;\;+k\int_{-1}^{1}J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)(1-x^2)^{k}dx\\
&\;\;\;\;+k^2\int_{-1}^{1}x^2J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)(1-x^2)^{k-1}dx.
\end{align*}
We plug this last identity in \eqref{intermstep} so we get
\begin{align*}
&\langle -\Delta_B \phi_{s,k},\phi_{m,l}\rangle_{L^2(B,g_B)}\\
=&\frac\pi 2\delta_{l,k}2^{\delta_k}\Big( \int_{-1}^{1}\partial_x[J_{s-k}^{k,k}(x)]\partial_x[J_{m-k}^{k,k}(x)](1-x^2)^{k+1}dx\\
&\;\;\;\;+k^2\int_{-1}^{1}(1-x^2)J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)(1-x^2)^{k-1}\\
&\;\;\;\;+k\int_{-1}^{1}J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)(1-x^2)^{k}dx\Big)\\
=&\frac\pi 2\delta_{l,k}2^{\delta_k}\Big( \int_{-1}^{1}\partial_x[J_{s-k}^{k,k}(x)]\partial_x[J_{m-k}^{k,k}(x)](1-x^2)^{k+1}dx\\
&\;\;\;\;+k(k+1)\int_{-1}^{1}J_{s-k}^{k,k}(x)J_{m-k}^{k,k}(x)(1-x^2)^{k}dx\Big).
\end{align*}
The last term in the sum vanishes for any $m\neq s$, this follows from the orthogonality of Jacobi polynomials. When instead $m=s$ we have (see for instance \cite{DuXu01})
$$k(k+1)\int_{-1}^{1}(J_{s-k}^{k,k}(x)(x))^2(1-x^2)^{k}dx=\frac{k(k+1)2^{2k+1}(s!)^2}{(2s+1)(s+k)!(s-k)!}.$$
For the first term, we recall that $\frac d{dx}J_{s-k}^{k,k}=\frac{s+k+1}2J_{s-k-1}^{k+1,k+1}$, Hence, using again the orthogonality, we get
\begin{align*}
&\int_{-1}^{1}\partial_x[J_{s-k}^{k,k}(x)]\partial_x[J_{m-k}^{k,k}(x)](1-x^2)^{k+1}dx\\
=&\left(\frac{s+k+1}2\right)^2\int_{-1}^{1}(J_{s-k-1}^{k+1,k+1})^2(1-x^2)^{k+1}dx\\
=&(s+k+1) \frac{2^{2k+1}(s!)^2}{(2s+1)(s+k)!(s-k-1)!}.
\end{align*}
We finally computed
\begin{align*}
&\langle -\Delta_B \phi_{s,k},\phi_{m,l}\rangle_{L^2(B,g_B)}\\
=&\frac\pi 2\delta_{l,k}2^{\delta_k}\delta_{s,m}\frac{2^{2k+1}(s!)^2}{(2s+1)(s+k)!(s-k)!}\Big( k(k+1)+(s+k+1)(s-k)\Big)\\
=&s(s+1)\frac\pi 2\delta_{l,k}\delta_{s,m}2^{\delta_k}\frac{2^{2k+1}(s!)^2}{(2s+1)(s+k)!(s-k)!}\\
=&s(s+1)\|\phi_{s,k}\|_{L^2(B,g_B)}^2\delta_{l,k}\delta_{s,m}.
\end{align*}
Here the last line is due to Proposition \ref{orthogonalsystemball}.
\end{proof}
\subsection{Orthogonal polynomials in $L^2_{\mu_{S^n}}$}
\begin{proposition}[\cite{DuXu14}]\label{orthogonalsystemsymplex}
Let us set for any $\alpha\in \mathbb{N}^{n}$ and $x\in S^n$
\begin{equation}
\psi_\alpha(x):=\prod_{j=1}^n\left(1-\sum_{k=1}^{j-1}x_k \right)^{\alpha_j}J_{\alpha_j}^{a_j,-1/2}\left(\frac{2x_j}{1-\sum_{k=1}^{j-1}x_k}-1 \right),
\end{equation}\normalsize
where $J_m^{a,b}$ is the $m$-th Jacobi polynomial of parameters $a,b$ and
$$a_j:=2\sum_{k=1}^{\min(n,j+1)}\alpha_k+\frac{n-j-1}{2}.$$
The set $\{\psi_\alpha: \alpha\in \mathbb{N}^{n}\}$ is a dense orthogonal system in $L^2(S^n,g_{S^n}).$
\end{proposition}
This result (see Th. 8.2.2 in \cite{DuXu14}) plays a key role in our proof.
\begin{theorem}[\cite{DuXu14}]\label{simplexeigenfunctions}
Let us introduce the differential operator
\begin{equation}\label{differentialproperty}
\mathcal D f:=\sum_{i=1}^nx_i \partial_{i,i}^2 f-2\sum_{1\leq i<j\leq n}x_i x_j \partial_{i,j}^2 f +\frac 1 2\sum_{i=1}^n(1-(n+1)x_i)\partial_i f.
\end{equation}
Then we have
\begin{equation}
\mathcal D\psi_\alpha=|\alpha|\left(|\alpha|+\frac{n+1}2\right)\psi_\alpha.
\end{equation}
\end{theorem}
\subsection{Proof of Theorem \ref{ThSimplex}}
\begin{proof}[Proof of Theorem \ref{ThSimplex}]
Let us introduce, see Figure \ref{trianglefig}, the following notations for $\epsilon>0$
\begin{align*}
S^n_\epsilon &:=\left\{x\in S^n: x_i>\epsilon,(1-\sum_{k=1}^nx_i)>\epsilon\right\},\\
T^{n,0}_\epsilon&:=\left\{x\in \partial S^n_\epsilon: (1-\sum_{k=1}^nx_i)=\epsilon\right\},\\
T^{n,i}_\epsilon&:=\{x\in \partial S^n_\epsilon: x_i=\epsilon\},\;i=1,\dots, n.
\end{align*}
\begin{figure}
\begin{center}
\includegraphics[scale=0.1]{triangle}
\caption{Some notations used in the proof of Theorem \ref{ThSimplex}.}
\label{trianglefig}
\end{center}
\end{figure}
Also let $\nu_i$ be the Euclidean unit normal to $T^{n,i}_\epsilon$ (for any $\epsilon>0$). We note that $\partial S^n_\epsilon=\cup_{j=0}^n T^{n,i}_\epsilon.$
Following the first part of the proof of Theorem \ref{ThBall}, we show that $\Delta$ is a symmetric operator on the space $\mathscr C^\infty_b(S^n)$ which is dense (see Proposition \ref{propdensity}) in $H^1(S^n, g_{S^n}).$
To this aim we perform integration by parts twice. Let $u,v\in \mathscr C^\infty_b(S^n),$ then
\begin{align*}
&-\int_{S^n} v\Delta u d\Vol_{S^n}=-\int_{S^n} \Div(\sqrt{\det g_{S^n}} G_{S^n}^{-1}D u)v dx\\
=&\lim_{\epsilon\to 0^+} -\int_{S^n_\epsilon} \Div(\sqrt{\det g_{S^n}} G_{S^n}^{-1}D u)v dx\\
=&\lim_{\epsilon\to 0^+} \left(\int_{S^n_\epsilon} D v^TG_{S^n}^{-1}D u \sqrt{\det g_{S^n}}dx-\sum_{i=0}^n\int_{T^{n,i}_\epsilon} v\nu_i^T G_{S^n}^{-1}D u \sqrt{\det g_{S^n}}d\sigma\right)\\
=& \int_{S^n} \langle\grad u,\grad v\rangle_{g_{S^n}} d\Vol_{S^n}-\sum_{i=0}^n\lim_{\epsilon\to 0^+}\int_{T^{n,i}_\epsilon} \nu_i^T G_{S^n}^{-1}D u \sqrt{\det g_{S^n}}d\sigma\\
=&-\int_{S^n} v\Delta u d\Vol_{S^n}+\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \lim_{\epsilon\to 0^+}\int_{T^{n,i}_\epsilon} \left( u\nu_i^T G_{S^n}^{-1}D v-v\nu_i^T G_{S^n}^{-1}D u \right)\sqrt{\det g_{S^n}}d\sigma.
\end{align*}
Thus we need to prove that for any $u,v\in \mathscr C^\infty_b(S^n)$ and any $ i\in\{0,1,\dots, n\}$ we have
\begin{equation}
\lim_{\epsilon\to 0^+}\int_{T^{n,i}_\epsilon}u\nu_i^T G_{S^n}^{-1}D v\sqrt{\det g_{S^n}}d\sigma =0.\label{toshowsymmetry}
\end{equation}
For, it is sufficient to notice (using Lemma \ref{lemmasimplexinversemetric}) that for any $x\in T^{n,0}_\epsilon$
$$\nu_0^TG_{S^n}^{-1}\sqrt{\det g_{S^n}}=\sqrt{\frac \epsilon {\prod_{k=1}^nx_k}}(x_1,x_2,\dots,x_n)^T$$
and for any $x\in T^{n,i}_\epsilon$, $i=1,2,\dots,n$ \small
\begin{equation*}
\begin{split}
&\nu_i^TG_{S^n}^{-1}\sqrt{\det g_{S^n}}=\\
&\sqrt{\frac \epsilon {(1-\epsilon-\sum_{k=1,k\neq i}^nx_k)\prod_{k=1,k\neq i}^nx_k}}(x_1,x_2,\dots,x_{i-1},1-\epsilon,x_{i+1},\dots,x_n)^T.
\end{split}
\end{equation*}
\normalsize
Therefore we have
$$\left|\int_{T^{n,0}_\epsilon}u\nu_i^T G_{S^n}^{-1}D v\sqrt{\det g_{S^n}}d\sigma\right|\leq \sqrt\epsilon n\max_{S^n}(|Dv|_\infty |u|)\left\|\prod_{k=1}^n \sqrt{x_k}\right\|_{L^1(T^{n,0}_\epsilon)}\to 0$$
and, for any $i=1,2,\dots,n$
\begin{align*}
&\left|\int_{T^{n,i}_\epsilon}u\nu_i^T G_{S^n}^{-1}D v\sqrt{\det g_{S^n}}d\sigma\right|\\
\leq& \sqrt\epsilon n\max_{S^n}(|Dv|_\infty |u|)\left\| \left((1-\epsilon-\sum_{k=1,k\neq i}^nx_k)\prod_{k=1,k\neq i}^nx_k\right)^{-1/2}\right\|_{L^1(T^{n,i}_\epsilon)}\to 0
\end{align*}
and thus \eqref{toshowsymmetry} holds true. This shows that $\Delta$ is a symmetric operator on $\mathscr C^\infty_b(S^n),$ i.e., for any such $u$ and $v$
\small
\begin{equation}\label{simplexsymmetry}
\int_{S^n}u\Delta v d\Vol_{S^n}=-\int_{S^n}\langle \grad u,\grad v\rangle_{g_{S^n}}d\Vol_{S^n} =\int_{S^n}v\Delta u d\Vol_{S^n}.
\end{equation}
\normalsize
Now we want to show that $\Delta$ has discrete spectrum $\sigma(\Delta_S)=\{\lambda_s:=s(s+\frac{n-1}{2}): s\in \mathbb{N}\}$ and the eigen-space of $\lambda_s$ is $\Span\{\psi_\alpha,|\alpha|=s\}$ (see Proposition \ref{orthogonalsystemsymplex}).
Instead of proving this directly, we rely on the known properties of the basis $\{\psi_\alpha\}$, namely \eqref{differentialproperty}, and we simply show that for smooth functions
\begin{equation}\label{toshoweigen}
\Delta f=\mathcal D f,
\end{equation}
this allows us to characterize $\sigma(\Delta)$ due to Theorem \ref{simplexeigenfunctions}. Then we apply Theorem \ref{ThOpTh} and the thesis follows.
We introduce the notation $h(x):=(1-\sum_{k=1}^nx_k)\prod_{k=1}^n x_k.$ It is worth to note that
$$\sqrt{h(x)}\partial_i \frac{x_i}{\sqrt{h(x)}}=\frac{1-\sum_{k\neq i}x_k}{2(1-\sum_{k=1}^nx_k)}=\frac 1 2\left(1+\frac{x_i}{1-\sum_{k=1 }^nx_k} \right) .$$
For any smooth $f$ we have
\begin{align*}
&\Delta_{S^n}f\\
=&\sqrt{h(x)}\sum_{i=1}^n\partial_i\left(\frac {x_i}{\sqrt{h(x)}}(\partial_i f-\sum_{j=1}^nx_j\partial_j f) \right)\\
=&\sum_{i=1}^n\left\{ \sqrt{h(x)}\partial_i \frac{x_i}{\sqrt{h(x)}}(\partial_i f-\sum_{j=1}^nx_j\partial_j f)+x_i\partial_i(\partial_i f-\sum_{j=1}^nx_j\partial_j f)\right\}\\
=&\sum_{i=1}^n\left\{\frac 1 2\left(1+\frac{x_i}{1-\sum_{k=1 }^nx_k} \right)(\partial_i f-\sum_{j=1}^nx_j\partial_j f)+x_i\partial_i(\partial_i f-\sum_{j=1}^nx_j\partial_j f)\right\}\\
=&-\frac 1 2\sum_{j=1}^nx_j\partial_j f\cdot\sum_{i=1}^n\left(1+\frac{x_i}{1-\sum_{k=1}^nx_k} \right)+\\
&\;\;\;\;\;\frac 1 2 \sum_{i=1}^n\partial_i f+\frac 1{2(1-\sum_{k=1}^nx_k)}\sum_{i=1}^nx_i\partial_i f\;\;+\\
&\;\;\;\;\;\sum_{i=1}^n\left\{x_i \partial_i^2 f-x_i\sum_{j\neq i}x_j\partial_{i,j}^2f -x_i^2 \partial_i^2 f-x_i \partial_i f \right\}\\
=&\sum_{i=1}^nx_i(1-x_i)\partial^2_i f-2\sum_{1\leq j<i\leq n}x_ix_j\partial_{i,j}^2 f+\frac 1 2 \sum_{i=1}^n\partial_i f\\
&\;\;+\;\;\left(\sum_{i=1}^nx_i\partial_i f\right)\cdot\left\{-\sum_{i=1}^n\left(\frac 1 2+\frac{x_i}{2\left(1-\sum_{k=1 }^nx_k\right)}\right)+\frac{1}{2\left(1-\sum_{k=1}^nx_k\right)} -1\right\}\\
=&\sum_{i=1}^nx_i(1-x_i)\partial^2_i f-2\sum_{1\leq j<i\leq n}x_ix_j\partial_{i,j}^2 f+\frac 1 2 \sum_{i=1}^n\partial_i f\\
&\;\;+\;\;\left( \sum_{i=1}^n x_i\partial_i f\right)\cdot\left\{-\frac{n+2}{2} +\frac{-\sum_{i=1}^nx_i+1 }{2(1-\sum_{k=1}^nx_k)} \right\}\\
=&\sum_{i=1}^nx_i(1-x_i)\partial^2_i f-2\sum_{1\leq j<i\leq n}x_ix_j\partial_{i,j}^2 f+\frac 1 2 \sum_{i=1}^n\partial_i f\\
&=\sum_{i=1}^nx_i \partial_{i,i}^2 f-2\sum_{1\leq i<j\leq n}x_i x_j \partial_{i,j}^2 f +\frac 1 2\sum_{i=1}^n(1-(n+1)x_i)\partial_i f\\
=& \mathcal D f.
\end{align*}
\end{proof}
\subsection{Proof of Proposition \ref{propnotalmostpolar}}
Let us first recall a result of Masamune \cite[Th. 3]{Ma99} which the proof of Proposition \ref{propnotalmostpolar} relies on. Assume $(M,g)$ to be a compact Riemaniann manifold and let $\Sigma$ be a submanifold of $M,$ let us define $\Delta_M$ as the standard Laplace Beltrami operator acting on $\mathscr C^\infty_c(M\setminus \Sigma).$ Then
\begin{equation}\label{masamuneresult}
\Delta_M\text{ is essentially self-adjoint if and only if }\dim(M)-\dim(\Sigma)>3.
\end{equation}
\begin{proof}[Proof of Proposition \ref{propnotalmostpolar}]
Let $M:=S^n\subset \mathbb R^{n+1}$ and $\Sigma:=\{x\in M: x_{n+1}=0\}.$ Also introduce the notation $(x_1,x_2,\dots,x_n,x_{n+1})=(\xi,x_{n+1}).$
Let us assume by contradiction that $\mathscr C^\infty_c(B^n)$ is dense in $H^1(B^n,g_{B^n}).$
In view of the proof of Proposition \ref{propdensity} we have
\begin{equation}
\begin{split}
\mathscr H:=&\left(\mathscr C^\infty_c(B^n),\|\cdot\|_{1,2,g_{B^n}}\right){\leftrightarrows}_{\text{isometry}} \left(\mathscr C^\infty_{c,\text{even}}(M\setminus \Sigma),\|\cdot\|_{1,2,g_M}\right)=:\mathscr E_1.\\
&\left(\mathscr C^\infty_c(B^n),\|\cdot\|_{1,2,g_{B^n}}\right){\leftrightarrows}_{\text{isometry}} \left(\mathscr C^\infty_{c,\text{odd}}(M\setminus \Sigma),\|\cdot\|_{1,2,g_M}\right)=:\mathscr E_2.
\end{split}
\end{equation}
Here $\mathscr C^\infty_{c,\text{odd}}(M\setminus \Sigma)$ denotes the subspace
$$\left\{u\in \mathscr C^\infty_{c}(M\setminus \Sigma),g_{\mathbb S^n}),\;u(\xi,x_{n+1})=-u(\xi,-x_{n+1})\, \forall (\xi,x_{n+1})\in M\setminus \Sigma\right\}$$
and $\mathscr C^\infty_{c,\text{even}}(M\setminus \Sigma)$ is defined similarly. Note that, given $u\in \mathscr C^\infty_c(M\setminus \Sigma)$ we can define $u_{\text{even}}:=1/2(u(\xi,x_{n+1})+u(\xi,-x_{n+1}))\in \mathscr E_1$ and $u_{\text{odd}}:=1/2(u(\xi,x_{n+1})-u(\xi,-x_{n+1}))\in \mathscr E_2$ such that $u=u_{\text{even}}+u_{\text{odd}}$.
The assumption $\mathscr C^\infty_c(B^n)$ dense in $H^1(B^n,g_{B^n})$ together with Theorem \ref{ThBall} and the isometry property of the map $E$ in the proof of Proposition \ref{propdensity} implies that the Laplace Beltrami operator $\Delta_1$ acting on $\mathscr E_1$ and $\Delta_2$ acting on $\mathscr E_2$ are essentially self-adjoint. Moreover, since
$\Delta_M u=\Delta_1 u_{\text{even}}+ \Delta_2 u_{\text{odd}}$ for any $u\in \mathscr C^\infty_c(M\setminus \Sigma),$ it follows that $\Delta_M$ itself is essentially self-adjoint.
On the other hand, $\dim \Sigma= n-1$ and $\dim M=n$, this is in contrast with Masamune's result \eqref{masamuneresult} and thus $\mathscr C^\infty_c(B^n)$ can not be dense in $H^1(B^n,g_{B^n})$ and thus $H^1(B^n,g_{B^n})\neq H^1_0(B^n,g_{B^n}).$ Note that, in view of \cite[Th. 1]{Ma05}, this is equivalent to the fact that $B^n$ is not a manifold with almost polar boundary.
The proof for the simplex can be done in a equivalent way but using the map $F$ defined in the proof of Proposition \ref{propdensity} instead of the map $E.$
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,384 |
Has anyone been to the new outlet that just opened outside of Busan? On all the advertisements I saw around the city there were images of Forever 21 being there. But when I look at the website Forever 21 is not listed as a store. I just want to know if it is actually there or not! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,402 |
Michael Farrand Bennet (født 1964 i New Delhi i Indien) er en amerikansk politiker.
Bennet var kandidat til at blive Demokraternes præsidentkandidat til præsidentvalget i november 2020. Han havde problemer med at slå ordentligt igennem til vælgerskaren og trak sig derfor den 11. februar 2020.
Levned
Bennets far Douglas Bennet var da sønnen blev født rådgiver for den daværende amerikanske ambassadør i Indien, Chester B. Bowles, og boede derfor med sin kone Susanne Bennet i Indien. Senere arbejdede Bennet senior som rådgiver for den amerikanske vicepræsident Hubert H. Humphrey.
Michael Bennet voksede for det meste op i Washington, D.C.. Han viste politisk interesse og var i skoleferien bud i USAs kongres. Han tog sin uddannelse ved Wesleyan University og Yale University.
Bennets politiske karriere begyndte under Bill Clintons regeringstid, da han som knap 30 år gammel blev rådgiver til vicejustisminister Philip Heymann. Under Jamie Gorelick og Eric Holder fortsatte han i den funktion. Også faren Douglas arbejdede for Clinton, som viceudenrigsminister for internationale forbindelser.
I oktober 1997 blev Bennet gift med med statsadvokat Susan Daggett. De bosatte sig i 2001 i Denver (Colorado).
Michael Bennets blev i januar 2009, af guvernør Bill Ritter som efterfølger for Ken Salazar som demokratisk senator for Colorado. Salazar var blevet udpeget af Barack Obama til indenrigsminister.
Referencer
Eksterne henvisninger
Politikere fra USA
Personer fra Colorado
Medlemmer af USA's kongres for det demokratiske parti
Medlemmer af Senatet i USA | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,688 |
Q: Static Class vs Protected Constructor I Am getting a warning message in my class, like
Add a Protected constructor or the static keyword to the class declaration
Solution
The error is gone after I tried both the below ways,
static class without constructor
public static class Program {
}
Non static class with protected using constructor
public class Program
{
protected Program() { }
}
Question:
So What is the difference between Static Class vs Protected Constructor which is mentioned in my above solution? And which one is best to use?
A: A static class doesn't need an instance to access its members. A static class cannot have instance members (e.g. public int MyNumber; is not allowed on a static class because only static members are allowed on a static class). Both instance and static members are allowed on a non-static class though. A class with a protected constructor can only have an instance created by itself or something that inherits from it.
public class Program
{
protected Program()
{
// Do something.
}
public static Program Create()
{
// 100% Allowed.
return new Program();
}
public void DoSomething()
{
}
}
public static class AnotherClass
{
public static Program CreateProgram()
{
// Not allowed since Program's constructor is protected.
return new Program();
}
}
public class SubProgram : Program
{
protected SubProgram()
{
// Calls Program() then SubProgram().
}
public new static Program Create()
{
// return new Program(); // We would need to move the SubProgram class INSIDE the Program class in order for this line to work.
return new SubProgram();
}
}
Program.Create(); // Can be called since Create is public and static function.
Program.DoSomething() // Can't be called because an instance has not been instantiated.
var test = Program.Create();
test.DoSomething(); // Can be called since there is now an instance of Program (i.e. 'test').
AnotherClass.CreateProgram(); // Can't be called since Program's constructor is protected.
SubProgram.Create(); // Can be called since SubProgram inherits from Program.
As for performance, this distinction doesn't really have much to do with performance.
A: You probably only have static members in the class and the code analyser assumes that your intention is to not be able to create instances of the class so it is asking you to either make the class static
public static class Program {
//...static members
}
or put a protected/private constructor
public class Program {
protected Program { //OR private
}
//...static members
}
to prevent instances of that class from being initialized.
A static class is basically the same as a non-static class, but there is one difference: a static class cannot be instantiated.
Reference Static Classes and Static Class Members (C# Programming Guide)
The protected constructor means that only derived classes can call the constructor
and a private constructor wont allow any other classes to initialize the class with a private constructor
A: A static constructor is called when the class type is instantiated. The protected constructor is called when an instance of a class is created. The protected part means only classes that inherit the class can call it.
A: Static Constructor: Called once when the class type is instantiated and is used to initialize static members. Does not create an instance of the class.
Protected Constructor: A constructor that can be called only by the class or a class that inherits it.
The best practices for this is that you should have a static constructor for initializing static members and a protected constructor if you only want classes that inherit to be able to create an instance of your class. You can have both.
public class MyClass
{
static readonly long _someStaticMember;
private bool _param;
static MyClass()
{
//Do Some Logic
_someStaticMember = SomeValueCalculated;
}
protected MyClass(bool param)
{
_param = param;
}
}
public class ChildClass: MyClass
{
public ChildClass(bool param) : base(param);
}
public class NotChildClass
{
public MyClass someObject = new MyClass(true); //Will Fail
}
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"redpajama_set_name": "RedPajamaStackExchange"
} | 4,262 |
{"url":"https:\/\/mikesmathpage.wordpress.com\/2020\/07\/04\/3-ideas-involving-e-we-saw-this-week\/","text":"# 3 ideas involving e we saw this\u00a0week\n\nWe had a little surprise this week as we saw e show up three different times. Since each kid only saw one of the ideas, I thought reviewing all three this morning would make for a great math project.\n\nThe first thing I did was introduce and review the ideas. Those ideas are:\n\n(1) The proof that e is irrational,\n\n(2) This idea from a Nassim Taleb Twitter thread:\n\n(3) The idea in this twitter post from Sonia on Twitter:\n\nHere\u2019s the introduction to the ideas:\n\nFirst we talked about the proof that e is irrational. My younger son saw this idea as an exercise in the number theory book he\u2019s working through right now. The proof is accessible to kids, though a bit more difficult than some of the other proofs of irrationality the boys seen before:\n\nNext we moved to the idea in Nassim Taleb\u2019s tweet. The idea that $(1.01)^{365}$ and $e^{365\/100}$ are so close together is a really important idea from calculus and the general idea has many important applications:\n\nFinally, we looked at the tweet from Sonia and discussed the simplified mathematical problem in the tweet and the surprising relationship to e:\n\nI think these three ideas are fun ones for kids to see. The proof that e is irrational is something that I\u2019m pretty sure I didn\u2019t see until college, but is definitely accessible to kids. The other two ideas are really important ideas from calculus and probability and definitely worth exploring many times!","date":"2021-09-24 18:19:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 2, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5655642151832581, \"perplexity\": 733.3385511874272}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2021-39\/segments\/1631780057564.48\/warc\/CC-MAIN-20210924171348-20210924201348-00176.warc.gz\"}"} | null | null |
<?php
namespace RedKiteLabs\RedKiteCms\RedKiteCmsBundle\Core\ThemeChanger;
use RedKiteLabs\RedKiteCms\RedKiteCmsBundle\Core\Content\Template\TemplateManager;
use RedKiteLabs\RedKiteCms\RedKiteCmsBundle\Core\Repository\Factory\FactoryRepositoryInterface;
use RedKiteLabs\ThemeEngineBundle\Core\Theme\ThemeInterface;
/**
* ThemeChanger is deputed to change the website template
*
* @author RedKite Labs <webmaster@redkite-labs.com>
*/
class ThemeChanger
{
/** @var TemplateManager */
protected $templateManager;
/** @var FactoryRepositoryInterface */
protected $factoryRepository;
/** @var \RedKiteLabs\RedKiteCms\RedKiteCmsBundle\Core\Repository\Repository\LanguageRepositoryInterface */
protected $languagesRepository;
/** @var \RedKiteLabs\RedKiteCms\RedKiteCmsBundle\Core\Repository\Repository\PageRepositoryInterface */
protected $pagesRepository;
/**
* Constructor
*
* @param TemplateManager $templateManager
* @param FactoryRepositoryInterface $factoryRepository
*/
public function __construct(TemplateManager $templateManager, FactoryRepositoryInterface $factoryRepository)
{
$this->templateManager = $templateManager;
$this->factoryRepository = $factoryRepository;
$this->languagesRepository = $this->factoryRepository->createRepository('Language');
$this->pagesRepository = $this->factoryRepository->createRepository('Page');
}
/**
* Changes the current theme
*
* @param ThemeInterface $previousTheme
* @param ThemeInterface $theme
* @param string $path
* @param array $templatesMap
*/
public function change(ThemeInterface $previousTheme, ThemeInterface $theme, $path, array $templatesMap)
{
$this->saveThemeStructure($previousTheme, $path);
$this->changeTemplate($theme, $templatesMap);
}
/**
* Changes the website templates with the new ones provided into the $templatesMap
* array
*
* @param ThemeInterface $theme
* @param array $templatesMap
* @throws \Exception
*/
protected function changeTemplate(ThemeInterface $theme, array $templatesMap)
{
$ignoreRepeatedSlots = false;
foreach ($this->languagesRepository->activeLanguages() as $language) {
foreach ($this->pagesRepository->activePages() as $page) {
$templateName = $page->getTemplateName();
if ( ! array_key_exists($templateName, $templatesMap)) {
continue;
}
$page->setTemplateName($templatesMap[$templateName]);
$page->save();
$template = $theme->getTemplate($page->getTemplateName());
$this->templateManager
->refresh($theme->getThemeSlots(), $template);
$this->templateManager->populate($language->getId(), $page->getId(), $ignoreRepeatedSlots);
$ignoreRepeatedSlots = true;
}
}
}
/**
* Saves the current theme structure into a file
*
* @param ThemeInterface $theme
* @param string $themeStructureFile
* @throws \Exception
*/
protected function saveThemeStructure(ThemeInterface $theme, $themeStructureFile)
{
$templates = array();
foreach ($this->languagesRepository->activeLanguages() as $language) {
foreach ($this->pagesRepository->activePages() as $page) {
$key = $language->getId() . '-' . $page->getId();
$templates[$key] = $page->getTemplateName();
}
}
$themeName = $theme->getThemeName();
$currentTheme = array(
"Theme" => $themeName,
"Templates" => $templates,
);
file_put_contents($themeStructureFile, json_encode($currentTheme));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,262 |
Q: Parallel transport along meridian I'm trying to parallel transport a vector on a 2-sphere along a meridian, but I find something that is confusing me. Let's consider a vector at the north pole, or enough close to the north pole say $(\epsilon, \epsilon)$, with component ($v_{\theta}^0$, $v_{\phi}^0$) and let parallel transport it along the curve ${\phi}=0$. Just using the geodesic equation in polar coordinates I get equations
$$\frac{{\rm d}v_{\theta}}{{\rm d}\theta}=0\text{ and }\frac{{\rm d}v_{\phi}}{{\rm d}\theta}=-ctg(\theta)v_{\phi},$$
which can be integrated to
$$v_{\theta}={\rm const}\text{ and }v_{\phi}=v_{\phi}^0\frac{\sin{\theta^0}}{\sin\theta},$$
where $\theta^0=\epsilon$.
This result is irritating me because it means that parallel transporting the vector along the meridian would just change the $\phi$-component, while leaving the $\theta$-component unchanged. I expect that both components along a geodesic must be conserved because the angle to the tangent vector stays the same.
I found a review on parallel transport with the same results I get (formula 3.3 and 3.4 in "Parallel transport on a manifold", by Santiago Casas, 31.05.2011 - https://www.scribd.com/document/57524972/Parallel-Transport), but this is confusing me even more, because if it is the case for some rotation of the reference frame (that I cannot keep track of), then the final vector will have a different normalization than the initial one.
A: Got it! I think the results I get and the one on the paper I mention are correct: if you compute the norm of the vector at the beginning and end of the curve you get in both cases $||v||^2=r^2((v_{\theta}^0)^2+(v_{\phi}^0sin{\epsilon})^2)$, where $r$ is the radius of the sphere. I was just confused by the definition of scalar product in curved geometry. Although I still find difficult to imagine such a change of the only $\phi$-component of the vector by the time it is being parallel transported along a meridian, I believe this "kick" (in the sense of change of value of the $\phi$-component of the vector) is needed to keep constant the norm of the vector, because the reference frame is experiencing the same "kick" (in the sense of change of direction of $\hat{\phi}$-axis - see $g_{{\phi}{\phi}}$ component of the metric).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,217 |
<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<link href="style.css" type="text/css" rel="stylesheet">
<title>MOVZX—Move with Zero-Extend </title></head>
<body>
<h1>MOVZX—Move with Zero-Extend</h1>
<table>
<tr>
<th>Opcode</th>
<th>Instruction</th>
<th>Op/En</th>
<th>64-Bit Mode</th>
<th>Compat/Leg Mode</th>
<th>Description</th></tr>
<tr>
<td>0F B6 /<em>r</em></td>
<td>MOVZX <em>r16, r/m8</em></td>
<td>RM</td>
<td>Valid</td>
<td>Valid</td>
<td>Move byte to word with zero-extension.</td></tr>
<tr>
<td>0F B6 /<em>r</em></td>
<td>MOVZX <em>r32, r/m8</em></td>
<td>RM</td>
<td>Valid</td>
<td>Valid</td>
<td>Move byte to doubleword, zero-extension.</td></tr>
<tr>
<td>REX.W + 0F B6 /<em>r</em></td>
<td>MOVZX <em>r64, r/m8*</em></td>
<td>RM</td>
<td>Valid</td>
<td>N.E.</td>
<td>Move byte to quadword, zero-extension.</td></tr>
<tr>
<td>0F B7 /<em>r</em></td>
<td>MOVZX <em>r32, r/m16</em></td>
<td>RM</td>
<td>Valid</td>
<td>Valid</td>
<td>Move word to doubleword, zero-extension.</td></tr>
<tr>
<td>REX.W + 0F B7 /<em>r</em></td>
<td>MOVZX <em>r64, r/m16</em></td>
<td>RM</td>
<td>Valid</td>
<td>N.E.</td>
<td>Move word to quadword, zero-extension.</td></tr></table>
<p><strong>NOTES:</strong></p>
<p>*</p>
<p>In 64-bit mode, r/m8 can not be encoded to access the following byte registers if the REX prefix is used: AH, BH, CH, DH.</p>
<h3>Instruction Operand Encoding</h3>
<table>
<tr>
<td>Op/En</td>
<td>Operand 1</td>
<td>Operand 2</td>
<td>Operand 3</td>
<td>Operand 4</td></tr>
<tr>
<td>RM</td>
<td>ModRM:reg (w)</td>
<td>ModRM:r/m (r)</td>
<td>NA</td>
<td>NA</td></tr></table>
<h2>Description</h2>
<p>Copies the contents of the source operand (register or memory location) to the destination operand (register) and zero extends the value. The size of the converted value depends on the operand-size attribute.</p>
<p>In 64-bit mode, the instruction's default operation size is 32 bits. Use of the REX.R prefix permits access to addi-tional registers (R8-R15). Use of the REX.W prefix promotes operation to 64 bit operands. See the summary chart at the beginning of this section for encoding data and limits.</p>
<h2>Operation</h2>
<pre>DEST ← ZeroExtend(SRC);</pre>
<h2>Flags Affected</h2>
<p>None.</p>
<h2>Protected Mode Exceptions</h2>
<table class="exception-table">
<tr>
<td>#GP(0)</td>
<td>
<p>If a memory operand effective address is outside the CS, DS, ES, FS, or GS segment limit.</p>
<p>If the DS, ES, FS, or GS register contains a NULL segment selector.</p></td></tr>
<tr>
<td>#SS(0)</td>
<td>If a memory operand effective address is outside the SS segment limit.</td></tr>
<tr>
<td>#PF(fault-code)</td>
<td>If a page fault occurs.</td></tr>
<tr>
<td>#AC(0)</td>
<td>If alignment checking is enabled and an unaligned memory reference is made while the current privilege level is 3.</td></tr>
<tr>
<td>#UD</td>
<td>If the LOCK prefix is used.</td></tr></table>
<h2>Real-Address Mode Exceptions</h2>
<table class="exception-table">
<tr>
<td>#GP</td>
<td>If a memory operand effective address is outside the CS, DS, ES, FS, or GS segment limit.</td></tr>
<tr>
<td>#SS</td>
<td>If a memory operand effective address is outside the SS segment limit.</td></tr>
<tr>
<td>#UD</td>
<td>If the LOCK prefix is used.</td></tr></table>
<h2>Virtual-8086 Mode Exceptions</h2>
<table class="exception-table">
<tr>
<td>#GP(0)</td>
<td>If a memory operand effective address is outside the CS, DS, ES, FS, or GS segment limit.</td></tr>
<tr>
<td>#SS(0)</td>
<td>If a memory operand effective address is outside the SS segment limit.</td></tr>
<tr>
<td>#PF(fault-code)</td>
<td>If a page fault occurs.</td></tr>
<tr>
<td>#AC(0)</td>
<td>If alignment checking is enabled and an unaligned memory reference is made.</td></tr>
<tr>
<td>#UD</td>
<td>If the LOCK prefix is used.</td></tr></table>
<h2>Compatibility Mode Exceptions</h2>
<p>Same exceptions as in protected mode.</p>
<h2>64-Bit Mode Exceptions</h2>
<table class="exception-table">
<tr>
<td>#SS(0)</td>
<td>If a memory address referencing the SS segment is in a non-canonical form.</td></tr>
<tr>
<td>#GP(0)</td>
<td>If the memory address is in a non-canonical form.</td></tr>
<tr>
<td>#PF(fault-code)</td>
<td>If a page fault occurs.</td></tr>
<tr>
<td>#AC(0)</td>
<td>If alignment checking is enabled and an unaligned memory reference is made while the current privilege level is 3.</td></tr>
<tr>
<td>#UD</td>
<td>If the LOCK prefix is used.</td></tr></table></body></html> | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,752 |
In 2007, developers were hoping to build Idaho's first Whole Foods grocery story along with a 17-story tower of hotel space and condominiums. Just as the Great Recession was beginning to put a chokehold on the Treasure Valley economy, that proposal was downgraded to a 7-floor hotel and condos. Ultimately, developers opted to move forward with the grocery store—which opened its doors in Boise on Nov. 14, 2013—and put the hotel/condo concept on the back burner.
Now developers are set to move forward with a scaled-down proposal to construct a 5-story hotel and parking structure on a 1.7-acre site that fronts Myrtle Street immediately south of Whole Foods. When representatives of Eagle-based The Land Group go before the city of Boise Planning and Zoning Commission on Monday, Dec. 7, they'll describe plans for a 64,801-square-foot hotel building with an indoor pool on the ground floor and two-level parking garage south of Whole Foods.
Developers promise a hotel lobby space that "will be a pleasant change from the usual, providing a space more like your favorite coffee house than a traditional hotel lobby," according to internal documents. The ground floor also promises to include an outdoor lounge along Myrtle Street. Developers also promise the proposed hotel "will be affiliated with a national hotel flag" but don't mention what chain the hotel would be a part of.
Elevation plans for new hotel near Whole Foods. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,751 |
{"url":"http:\/\/math.stackexchange.com\/questions\/18853\/what-is-fu-v-in-mathbbqu-pm-v-pmfht-kt-in-mathbbzt-pm","text":"# What is $\\{f(u,v)\\in \\mathbb{Q}[u^{\\pm},v^{\\pm}]:f(ht,kt)\\in \\mathbb{Z}[t^{\\pm},1\/h,1\/k]\\forall h,k\\in \\mathbb{Z}\\backslash\\{0\\}\\}$?\n\nI'm not much good at even basic number theory, so this is really mysterious to me. I think the actual question might want to restrict to finite Laurent series, but I don't think it matters: I'm pretty sure what I want to do is figure out what each homogeneous component is allowed to look like. Moreover, I'm pretty sure that by plugging in $h=k=1$ I get that all the coefficients in nonnegative degrees need to be integral, so the first place to look is in degree $-1$.\n\nSo suppose we have \\begin{equation*} f(u,v)_{-1}=\\frac{a_1}{a_2 u}+\\frac{b_1}{b_2 v}. \\end{equation*} Then \\begin{equation*}f(ht,kt)_{-1}=\\frac{a_1}{a_2 ht}+\\frac{b_1}{b_2 kt} = \\frac{a_1 b_2 k+b_1 a_2 h}{a_2b_2hkt};\\end{equation*} for this to lie in $\\mathbb{Z}[t^{\\pm},1\/h,1\/k]$ we need $a_2b_2|a_1b_2k+b_1a_2 h$. I believe it's a basic fact that since this has to be true for all nonzero $h$ and $k$, we need that $a_2b_2|gcd(a_1b_2,b_1a_2)$. This implies that $a_2b_2|a_1b_2 \\Rightarrow a_2|a_1$ and similarly $b_2|b_1$. But assuming we started with our fractions in reduced form, this implies that $a_2=b_2=1$; that is, $f(u,v)_{-1}\\in \\mathbb{Z}[t^{\\pm}]$.\n\nSince I don't think I gained much insight from that calculation, I just tried to find restrictions on \\begin{equation*} f(u,v)_{-2}=\\frac{a_1}{a_2u^2}+\\frac{b_1}{b_2uv}+\\frac{c_1}{c_2v^2}. \\end{equation*} Now we have \\begin{equation*} f(ht,kt)_{-2} = \\frac{a_1}{a_2 h^2t^2}+\\frac{b_1}{b_2 hkt^2} + \\frac{c_1}{c_2 k^2t^2} \\end{equation*} \\begin{equation*} = \\frac{a_1b_2hkc_2k^2 + b_1a_2h^2c_2k^2 + c_1a_2h^2 b_2hk}{a_2b_2c_2h^3k^3t^2} \\end{equation*} \\begin{equation*} = \\frac{a_1b_2c_2k^2 + b_1a_2c_2hk + c_1a_2b_2h^2}{a_2b_2c_2h^2k^2t^2},\\end{equation*} and what we need is for $a_2b_2c_2|a_1b_2c_2k^2+b_1a_2c_2hk+c_1a_2b_2h^2$ (for all $h$ and $k$, of course). This is where I'm stuck. Reducing mod $a_2$ and recalling that $gcd(a_1,a_2)=1$ we should get that $a_2|b_2c_2k^2$, and similarly $b_2|a_2c_2hk$ and $c_2|a_2b_2h^2$. Putting $h=k=1$ this means that $a_2|b_2c_2$, $b_2|a_2c_2$, and $c_2|a_2b_2$. I tried breaking down these conditions into conditions on the prime factorizations but didn't get very far. And in any case, numbers that satisfy these need not satisfy the original relation. Help please????\n\nMotivation: In case you're still reading and you're wondering why I care about this specific calculation, it's because it's closely related to the \"$K$-theoretic homology of $K$-theory\"; precisely, it's the image of $K_*K\\rightarrow K_*K\\otimes \\mathbb{Q}$. This is important because for any (co)homology theory $E$, the $E$-cohomology of $E$ (written $E^*E$) is exactly the algebra of stable cohomology operations, and with some additional assumtions on $E$ (which are satisfied by $K$-theory), the $E$-homology of $E$ is the coalgebra of stable homology cooperations. Furthermore, the $e$-invariant is a homomorphism from the stable homotopy groups of spheres to a subquotient of $K_*K$!\n\n-\n\nI just came across a very pleasant algebro-geometric description of this ring $K_*K$, so I figure I'll record it here (a mere year and a half after I asked the question). My sense is that there isn't a better \"elementwise\" description, anyways.\n\nHere are a few background facts, each of which builds on the previous. $\\underline{\\mbox{Iso}}(G,H)$ denotes the scheme of isomorphisms between two group-schemes $G \\rightarrow X$ and $H \\rightarrow Y$; a $Z$-point is a pair of maps $(X \\xleftarrow{\\varphi} Z \\xrightarrow{\\psi} Y)$ along with an isomorphism $\\alpha:\\varphi^*G \\xrightarrow{\\sim} \\psi^*H$ of group-schemes over $Z$.\n\n1. For any line bundle $M$ over a scheme $S$, there is a canonical isomorphism $(\\mathbb{G}_m)_S \\cong \\underline{\\mbox{Aut}}(M)$: given an $S$-scheme $\\iota:\\mbox{spec }R \\rightarrow S$ and a point $(\\mathbb{G}_m)_S(R) = \\mbox{Hom}_S(\\mbox{spec }R,(\\mathbb{G}_m)_S)$ given by $\\lambda \\in R^\\times$, we obtain the automorphism $\\cdot \\lambda: \\iota^*M \\xrightarrow{\\sim} \\iota^*M$.\n2. The natural map $$\\underline{\\mbox{Aut}}((\\hat{\\mathbb{G}}_a)_\\mathbb{Q}) \\rightarrow \\underline{\\mbox{Aut}}(\\mbox{Lie}((\\hat{\\mathbb{G}}_a)_\\mathbb{Q}) \\cong (\\mathbb{G}_m)_\\mathbb{Q}$$ given by $f \\mapsto (df)_0$ is an isomorphism. If $R$ is a $\\mathbb{Q}$-algebra, then an $R$-point of the source is an element $f(x) \\in x\\cdot R[[x]]$ with $f'(0)\\in R^\\times$ and $f(x +_{(\\hat{\\mathbb{G}}_a)_R} y) = f(x) +_{(\\hat{\\mathbb{G}}_a)_R} f(y)$, but when you unwind this, having all denominators means that this must take the form $f(x)=\\lambda x$. Then $(df)_0$ is multiplication by $\\lambda$, which of course corresponds to the $R$-point of $(\\hat{\\mathbb{G}}_m)_\\mathbb{Q}$ picked out by $\\lambda$.\n3. There is an isomorphism $\\exp : (\\hat{\\mathbb{G}}_a)_\\mathbb{Q} \\xrightarrow{\\sim} (\\hat{\\mathbb{G}}_m)_\\mathbb{Q}$, which induces an isomorphism $\\underline{\\mbox{Aut}}((\\hat{\\mathbb{G}}_m)_\\mathbb{Q}) \\cong \\underline{\\mbox{Aut}}((\\hat{\\mathbb{G}}_a)_\\mathbb{Q}) \\cong (\\mathbb{G}_m)_\\mathbb{Q}$.\n4. For each $k \\in \\mathbb{Z} \\backslash \\{0\\}$, the endomorphism $[k]:\\hat{\\mathbb{G}}_m \\rightarrow \\hat{\\mathbb{G}}_m$ pulls back to an automorphism over $\\mathbb{Z}[1\/k]$, which yields a commutative diagram $$\\begin{array}{ccccccc} \\coprod_{k \\not= 0} \\mbox{spec }\\mathbb{Q} & \\xrightarrow{\\coprod [k]} & \\underline{\\mbox{Aut}}((\\hat{\\mathbb{G}}_m)_\\mathbb{Q}) & \\cong & \\underline{\\mbox{Aut}}((\\hat{\\mathbb{G}}_a)_\\mathbb{Q}) & \\cong & (\\hat{\\mathbb{G}}_m)_\\mathbb{Q} \\\\ \\downarrow & & \\downarrow \\\\ \\coprod_{k \\not= 0} \\mbox{spec }\\mathbb{Z}[1\/k] & \\xrightarrow{\\coprod [k]} & \\underline{\\mbox{Aut}}(\\hat{\\mathbb{G}}_m). \\end{array}$$ (The right-hand vertical map is hard to define diagramatically, but easy to define on functors of points once we observe that $|(\\mbox{spec }\\mathbb{Z})(R)|=1$ and $|(\\mbox{spec }\\mathbb{Q})(R)| \\leq 1$.)\n\nTheorem [Adams-Harris-Switzer '71, reinterpreted]: This diagram is a pushout in the category of affine schemes.\n\nThat is, the natural map $\\mathcal{O}(\\underline{\\mbox{Aut}}(\\hat{\\mathbb{G}}_m)) \\rightarrow \\mathbb{Q}[w,w^{-1}]$ (coming from the right-hand vertical map composed with the two isomorphisms) is injective, and its image consists of those Laurent polynomials $f(w)$ such that for all nonzero integers $k$, $f(k) \\in \\mathbb{Z}[1\/k]$.\n\nThe connection with K-theory is the following. Let $E$ and $F$ be two Landweber-exact spectra with formal group laws $f_E$ and $f_F$ (so $E_0X \\cong MUP_0X \\otimes_{\\eta_R,MUP_0,f_E} E_0 \\cong E_0 \\otimes_{f_E,MUP_0,\\eta_L} MUP_0 X$, naturally in X, and similarly for $F_0X$; here, $MUP$ is the periodified complex cobordism spectrum). Then, \\begin{eqnarray} E_0 F &=& MUP_0 F \\otimes_{\\eta_R,MUP_0,f_E} E_0 \\\\ &=& F_0 MUP \\otimes_{\\eta_R,MUP_0,f_E} E_0 \\\\ &=& F_0 \\otimes_{f_F,MUP_0,\\eta_L} MUP_0 MUP \\otimes_{\\eta_R,MUP_0,f_E} E_0. \\end{eqnarray} Thus, by Quillen's theorem, $\\mbox{spec }\\pi_0(E\\wedge F) = \\underline{\\mbox{Iso}}(G_E,G_F)$ (the scheme of isomorphisms between the associated formal groups). In the special case that $E=F=K$, we recover that $\\mbox{spec }K_0 K = \\underline{\\mbox{Aut}}(\\hat{\\mathbb{G}}_m)$.\n\nThe two-variate version in the original question arises when you consider $K_* K$ (with its natural map to $(K_\\mathbb{Q})_*K_\\mathbb{Q}=H\\mathbb{Q}P_* H\\mathbb{Q}P \\cong \\mathbb{Q}[u^\\pm,v^\\pm]$) instead of just $K_0 K$ (with its natural map to $(K_\\mathbb{Q})_0 K_\\mathbb{Q} = H\\mathbb{Q}_0 H\\mathbb{Q}P \\cong \\mathbb{Q}[w^\\pm]$).\n\n-","date":"2015-07-08 07:17:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.9912768006324768, \"perplexity\": 246.20068493335825}, \"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-2015-27\/segments\/1435376093097.69\/warc\/CC-MAIN-20150627033453-00211-ip-10-179-60-89.ec2.internal.warc.gz\"}"} | null | null |
Even Boomer Emmy-Award Winners Can't Get Hired | The Baby Boomer Generation -Trends, Research, Discussion, Comment for Baby Boomers.
We hate to keep harping on this (no, we don't), but here's yet another sad story of workers over 40 having trouble getting hired--the 'grays' as Hollywood calls them. If you happen to be looking for a job, you old gray you, there are some meaty statistics supporting the idea that employers should value (and hire) older workers. | {
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Decatur Central High School (DCHS) is a public high school in Indianapolis, Indiana, United States.
About
Decatur Central High School is part of the Metropolitan School District of Decatur Township. The school currently enrolls students from grades nine through twelve.
Small learning communities
Decatur Central has "small learning communities" with different learning styles. Students decide which SLC they would like to be entered in with a form explaining all the SLCs. This form is composed of a selection graph explaining the SLC they would like to be in, and a short essay area. In total there are five SLCs: Choice, Edge, Innovation, New Tech, and Q&I.
Demographics
Of Decatur Central's 1,632 students (in the 2007–08 school year), 83% were Caucasian, 10% were African American, 4% were Hispanic, 1% were Asian, and 2% were multiracial. 43% of students qualified for free lunches and 0% qualified for reduced price lunches.
Athletics
The mascot of Decatur Central is the Hawk; the school colors are Navy Blue and Varsity Gold.
Notable alumni
Aaron Gibson - former NFL offensive tackle for the Chicago Bears, Dallas Cowboys and Detroit Lions
Amy Cozad, Olympic diver in the 2016 Rio Olympics
Tommy Stevens - drafted by the New Orleans Saints in the seventh round with the 240th overall pick in the 2020 NFL Draft.
See also
List of high schools in Indiana
References
External links
Schools in Indianapolis
Public high schools in Indiana | {
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Whispered What Is an Element Chemistry Secrets
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The chemistry of an atom is dependent just on the number of electrons, which equals the amount of protons and is called the atomic number. The atomic mass of an isotope indicates the range of neutrons that are found within the heart of the atoms. As a consequence the amount of neutrons inside an atom can vary.
The outer area of the atom incorporates several electrons equal to the range of protons, making the standard atom electrically neutral. essay4me.org The outer field of the atom incorporates several electrons equal to the wide array of protons, making the conventional atom electrically neutral. It has several electrons equal to the variety of protons, making the normal atom electrically neutral.
It is simplest to learn by examples. For self-assessment, you will also locate a quiz with each lesson together with a chapter test. You will have a periodic table, the octet rule and a few physics concepts.
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Human interaction and compassion are key portions of the puzzle and cannot be discounted. Moreover, a master's degree will be able that will help you earn a higher salary. Acquiring theoretical knowledge doesn't have any appeal until finally pupils may just use it for sensible functions.
The atomic https://cuit.columbia.edu/content/security-and-privacy number is a significant idea of chemistry and quantum mechanics. Scandium does not have any biological role, but it's found in living organisms. With this type of a substantial region of study, it's not possible to comprehend everything about chemistry and very hard to summarize the field concisely.
The properties of elements in groups are alike in some respects to one another. It is a substance made from a couple of different elements that have been chemically joined. It is a substance made from two or three unique elements which have been chemically joined.
It's a substance composed of all of the identical kind of atom. The better part of the rest of the atom was simply empty space. Bear in mind that neutrons don't have any electric charge, so they don't influence the chemistry of an element.
The Hidden Treasure of What Is an Element Chemistry
The procedure for testing hardenability was proposed by means of a man named Grange. It will be related to the reason. There's no way a new, unstable element is going to get any uses as it deteriorates so quickly.
As a consequence of this effect, the radiation produced by the big bang would be expected to appear today as microwaves of only the type that were observed. If you're wondering precisely what canvas is made from, it's really quite fascinating. When it's the molecule is polar or non-polar may earn a difference in several ways.
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A molecule of a compound is created from a few elements. It is made from a couple elements. It has atoms of only one isotope, so every region of the substance has the specific bodily properties.
Again you will need to wax your rims with some very simple vehicle wax to stop extra oxidation or corrosion. The Greeks were the first to talk about the notion of an atom. The CAS index also has mixtures.
Managing pH is an essential portion of maintaining healthy water chemistry. It is a critical part of maintaining healthy water chemistry. They react in specific ways to make products that make fabrics, medicines and even human tissues.
The creation of a chemical includes not only its synthesis but in addition its purification to get rid of by-products and impurities involved with the synthesis. For example, the element sodium is made up of only sodium atoms. A chemical reaction is necessary.
In the existence of a spark, though, a small percent of the H2 molecules dissociate to form hydrogen atoms which are highly reactive. The positively charged particle is called proton. They are believed to be radioactive.
Furthermore, it regulates the entire quantity of water in the body. Another exceptional property of water is its capability to dissolve a huge collection of chemical substances. The individual elements of a mixture can be physically separated from one another.
So it's a wonderful green color. In general there's an increased quantity of substitution that occurs at higher temperature. The so called typical factors are observed in the first two rows.
Introducing What Is an Element Chemistry
Isotopes are a couple of types of an element. An atom is the fundamental unit of an element. For example, it may have a parameter controlling the sum of the string.
A standard example of a symbol that you use every day is an emoticon on your cellular phone. Nevertheless, it's now feasible to work out this number. Only numerous the Oracle-specific features are offered via the extension mechanism extended in JAXP.
What Is an Element Chemistry – Dead or Alive?
It requires the heated inside of a favorite oven. The so called typical components are observed in the first two rows. They are chemically the simplest substances and hence cannot be broken down using chemical reactions.
By contrast, it is a molecule that is made up of distinct kinds of atoms or several kinds of elements. By contrast, it's a molecule that is composed of distinct varieties of atoms or various kinds of elements. They have some of the same structural components.
Everything on Earth is made out of atoms. Experiments should be reproducible. They might also be classified into metals and non-metals.
Scientists that are employed inside this subject are called radiochemists. Atoms with over four shells are extremely rare in biological molecules. It's also called sustainable chemistry.
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मुख्यमंत्री ने अटलनगर (नया रायपुर) के राज्योत्सव स्थल का निरीक्षण कर
Just How People Develop A Effective Organization. Guidelines And Info About Accomplishment
Web Design fundamentals | {
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Should discussion of or at least promotion of certain political ideologies be off-limits? Particularly, should social media or even hosting platforms ban them even though theoretically as private companies they have the legal right to do so?
We seem to be heading toward a space where some people with any recognizable connection, however indirect, to the "alt-right" are supposed to be banned. OK, this is part of the backlash against Trump.
OK, some people think that if you talk about conservative principles with any objectivity, you're "supporting" white nationalism at least indirectly (You aren't). More seriously, if you talk openly online and then personally distance yourself from people whom you see as "second-handers" (Ayn Rand's term – later it was "moochers" and "looters") your speech automatically gets regarded as hate speech.
You could say that this debate whittles down to the idea that Hitler has to be considered worse than Stalin (or than Pol Pot, Mao, or Kim Jong Un).
The most extreme factions of the alt-right (as shown in the documentary "White Right", reviewed last night on my "cf" blog) do favor the establishment of ethno-states and the deportation or persons not belonging to the "correct" race or ethnicity. Short of that, there is this "blood and soil" populism. Is this a genuine political threat to minorities? This cannot happen in the United States because of various constitutional amendments (including 14th especially). But I do understand that in other parts of the world, where rights are in "laws" but not constitutionally guaranteed, it could be more of a threat (take Hungary, for example) – and these companies are global platforms.
Indeed, remember after Charlottesville, Trump was called out for not condemning the violence from the right more than that from the far left -- as if the most violent elements of Antifa can stay in circulation.
Today, the extreme right is more likely to become an existential threat to LGBT persons than the extreme left, but historically that has not generally been true in the past.
One argument against "free speech" online is that the practical context today is that "speech" tends to become the "dark money" of power over historically maligned groups – especially by race.
But if you stop free individual speech, you force people to take sides and join up in opposing groups to be heard at all. That indeed sounds like what the far Left today wants.
The far Left argues, for example, that free speech facilitates police profiling by race, or keeping PoC from getting to a level playing field. I've been asked if I've ever been targeted or beaten up for who I was – if I were, I'd want the legal system to give me some reparative protection. Well, it's complicated – yes, I was treated "unfairly" with the William and Mary expulsion of 1961, but I was privileged enough economically (as white) that I managed to outflank the whole thing and lead a fairly productive life as an individual, But not everyone has the chance to get that far as a "spectator".
So, I noticed that Westboro Baptist Church is still allowed to have its domain (by Cloudflare), and there is a marginal chance that sometime in my life, my life could be taken (or "sacrificed") because that site incites someone. I had visited Pulse a year before the shooting there. But I could have been there that night in 2016.
But I could have been at the Boston Marathon in 2013, or in Paris in November 2015. Or in the WTC on 9/11. Or in the Murrah building in OKC on 4/19/1995. You get the point.
I did one time cancel a volunteer assignment because it would have entailed driving my car into a particularly dangerous are in Washington DC. I have less exposure to unpredictable risk than less well off people – and this gets back to the skin-in-the-game idea.
It's also true that allowing homophobic speech could increase the political threat to my rights. In practice that isn't likely (even under Trump and Pence) because constitutional protections have gotten much stronger. Even so, I am reminded of comparable discussion about women's rights and Kavanaugh.
So it's a little mixed. But my own speech is worthless if I can't be objective and talk about anything in the news if it comes up – even the staging of Christchurch. I can't stay online if I have to yield to the idea that covering the ideology of someone who commits a horrific act only entices others to do the same in the future to get others to pay attention to them. Yet, that real-world potentiality does create a marginal, although very low probability risk, to my own safety, and of those connected to me -- an asymmetry or Black Swan that Taleb associates with his "skin in the game" moral theory. If it simply an inescapble consequence of mathematical logic. So, in the past, was the possibility that I could be drafted into combat.
(In the video, note what progression at 4:50).
Vice Motherboard (Joseph Cox and Jason Koebler) reports that Facebook now treats "white nationalism" and "white separatism" the same as "white supremacy". Perhaps an "ethno-state" would be, by definition, exclusionary and therefore supremacist (as a matter of intellectual logic). But Facebook was driven by influence from civil rights groups and probably left-leaning Academics. But as an intellectually principled matter, that would mean that pro-Israel support would be banned (although Israel is not "exclusionary" so it's hard to say). | {
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Holstebro Håndbold är en dansk damhandbollsförening som spelat i damehåndboldligaen sedan 2009. Klubben är en förening från 1990 av KFUM Holstebro och Holstebro HK, och har idag cirka 530 aktiva medlemmar, varav 480 är med i ungdomsavdelningen.
HH90 var moderklubb i TTH Holstebros damlag. HH90 är också en del av TTH Holstebros ungdomsavdelning, som hanterar U16- och U18-lagen i klubben. Från säsongen 2020-2021 kommer klubbens damligalag inte att vara anslutna till Team Tvis Holstebro längre. I framtiden kommer damlaget att spela under HH90 licens i damligan. Varje år i påsk arrangerar klubben Danmarks största ungdomstävling med 4000 deltagare med lag från Norge, Sverige, Tyskland, Nederländerna, Belgien och Danmark.
Det nya namnet för klubben Holstebro Håndbold publicerades den 14 april 2020.
2021-2022 Spelartruppen
Lagets nuvarande huvudtränare är Pether Krautmeyer, sedan 2016.
Meriter
Damlaget
EHF-cupmästare: 2013, 2015
Cupvinnarcupmästare: 2016
Spelare i urval
Ida Alstad (2013–2014)
Louise Bager Due (2013)
Linn Blohm (2014–2015)
Nycke Groot (2006–2011)
Nathalie Hagman (2014–2016)
Camilla Herrem (2015–2016)
Kristina Kristiansen (2007–2015)
Anna Loerper (2011–2013)
Ann Grete Nørgaard (2010–2015)
Jamina Roberts (2014–2016)
Silje Solberg (2014–2016)
Sandra Toft (2007–2014)
Externa länkar
Handbollsklubbar i Danmark
Sportklubbar bildade 2020
Holstebro kommun | {
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