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Biggest Earthquakes Near Alum Rock, California, United States
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Brno University of Technology (abbreviated: BUT; in Czech: Vysoké učení technické v Brně – Czech abbreviation: VUT) is a university located in Brno, Czech Republic. Being founded in 1899 and initially offering a single course in civil engineering, it grew to become a major technical Czech university with over 18,000 students enrolled at 8 faculties and 2 university institutes.
History
The Jesuits dominated university education in Moravia at the beginning of the 18th century as they controlled the University of Olomouc. The focus on theology and philosophy was not welcomed by the Moravian nobility. The nobility initiated the commencement of law education at the University of Olomouc in 1679. Later in 1725, the Moravian nobility enforced the establishment of the Academy of Nobility in Olomouc. Law and economy, mathematics, geometry, civil and military architecture, history, and geography were lectured there. As it aimed to promote knighthood also foreign languages, dance, swordsmanship, and equitation were taught there. The Academy was in Olomouc until 1847 when it was relocated to Brno, where it became the basis for what was later to become the University of Technology.
Due to the extinction of the University in Olomouc, no institution would provide an academic education in Moravia, and only one technical school, besides the German one, could not cover the lack of need, so the students mostly left to Prague, Vienna, or Kraków. Related to this situation the voices that called for the establishment of a university, but not a regional one in Olomouc but in the state capital – Brno, grew stronger. The Moravian Germans rejected the second Czech university and thus led to many quarrels (the settlement occurred after the collapse of the Austro-Hungarian Empire in 1919 by the establishment of Masaryk University). In September 1899 the disputes were solved by founding the Imperial Czech Technical University of Franz Joseph in Brno.
In the beginning, the university was settled in Augustine Street and had to make do with 4 professors and 47 students who could study only Civil Engineering. In the following year (1900) the teaching of the field of machine engineering was started and followed by cultural engineering (landscaping), electrical engineering, and chemical engineering. After World War I it was also possible to study architecture. In 1911 the university was moved into a newly built luxurious building in Veveří Street which is still used by the Faculty of Civil Engineering. In the interwar period, it was connected with the German Technical University and was renamed the Brno University of Technology. The school has used the name of the University of Technology of Dr. E. Benes for a short time.
After the German occupation of Czechoslovakia and installing the Protectorate of Bohemia and Moravia, all Czech universities, including the Czech Technical University, were forced to close (see International Students' Day). After the end of the war in 1945, the university was restored to its pre-war state, also taking over buildings of the German Technical University in Brno which was closed. After the war, the school was reopened under the older name the University of Technology of Dr. E. Benes. The school was ceased to exist in 1951; some parts were transferred to the newly established Military Technical Academy. The only faculties that provided civilian training remained the Faculty of Civil Engineering and Architecture Faculty, both under the name College of Engineering. Already in 1956, the university activity was gradually restored under the present name Brno University of Technology. Today's appearance was more or less stabilized in 1961.
After 1989 there was a reorganization of certain faculties and the emergence of many new ones. There was restored The Faculty of Chemistry (1992) and in addition to the technical fields, BUT focused on the economy (Faculty of Business founded in 1992) and arts (Faculty of Fine Arts, founded in 1993).
Another milestone is associated with the year 2000 when the BUT separates two faculties deployed in Zlín - Faculty of Technology and Faculty of Management and Economics - and thus established the Tomas Bata University.
The most recent significant organizational change is the splitting of the Faculty of Electrical Engineering and Faculty of Electrical Engineering and Computer Science at the Faculty of Electrical Engineering and Communication and the Faculty of Information Technology which occurred in 2002.
Most of the BUT buildings are now located in the area under Palacky Hill in the city district Kralovo Pole. There are the Faculty of Mechanical Engineering, Faculty of Business, Faculty of Chemistry, and a new building of the Faculty of Electrical Engineering and Communication as well as two college campuses.
Faculty of Information Technology is located in a former Carthusian monastery in Bozetechova Street and the new complex across the street. The Faculty of Civil Engineering is located in an extensively reconstructed building on Veveri Street. Faculty of Architecture is located on Porici Street, Faculty of Fine Arts on Udolni Street. BUT also uses the third college campus in Kounicova Street. Rector's Offices are located in a newly renovated Baroque building in Antoninska Street.
In more than 120 years BUT has matured into an internationally recognized institution offering education in a broad spectrum of fields ranging from technical and scientific disciplines through economics to the arts.
Authorities
The head of the university is the rector who is represented by five prorectors in various fields of activity. Financial matters of BUT are in the hands of the bursar, communication and promotion is the business of the public relations officer in cooperation with the Unit of external relations. Important documents and guidelines are discussed and approved by the Academic Senate which consisted of an employee and a student chamber. The scientific direction of BUT determines the Scientific Council, acting as BUT experts, other universities, and industry.
Each of the faculty is governed by the dean and his vice-deans. Also, the faculties have their academic senates which deal exclusively with the laws of the faculties. Similarly, the faculties have their scientific council. There are several student organizations in BUT, for historical reasons called the Student Unions. Each Faculty has its student chamber which represents students in the Academic Senate - students have the opportunity to participate in the management of their faculty.
Rectorate — prof. RNDr. Ing. Petr Štěpánek, CSc., dr. h. c.
The Academic Senate — current chairman of the senate is doc. Dr. Ing. Petr Hanáček.
Bursar — doc. Ing. Ladislav Janíček, Ph.D., MBA, LL.M. is the university bursar.
Faculties
Faculty of Civil Engineering
Faculty of Mechanical Engineering
Faculty of Chemistry
Faculty of Architecture
Faculty of Business and Management
Faculty of Electrical Engineering and Communication
Faculty of Fine Arts
Faculty of Information Technology
Faculty of Architecture (FA)
One of the oldest faculties of Brno University of Technology was established in 1919. During its existence the faculty was merged with the Faculty of civil Engineering. Currently provides training in architecture and urban design and has about three hundred students.
Faculty of Chemistry (FC)
Teaching Chemistry at BUT dates back to 1911 when there was established Department of Chemistry at the Czech Technical University. The development was interrupted in 1951 by converting BUT to the Military Technical Academy. The restoration of the teaching of chemistry was reached in 1992. Faculty realizes bachelor's, master's and doctoral degree programs in chemistry and food industry.
Faculty of Electrical Engineering and Communication (FEEC)
Studying at the faculty is focused on a wide range of scientific areas: control technology and robotics, biomedical engineering, power electronics and electrical engineering, electrotechnology and electronics, microelectronics, radioelectronics and teleinformatics.
Faculty of Information Technology (FIT)
Already in 1964 the department of automatic computers was established in the Faculty of Electrical Engineering, lately the institute of informatics detached which was transformed in 2002 into an independent faculty of information technology.
FIT campus is located in a former Carthusian monastery and the former estate.
The faculty consists of four institutes:
Department of Computer Systems
Department of Information Systems
Department of Intelligent Systems
Department of Computer Graphics and Multimedia
The faculty offers two basic levels of study - a three-year bachelor and two-year master's degree program. The third level is the doctoral degree program.
Faculty of Business and Management (FBM)
One of the youngest BUT faculty focuses on the economy and business fields of study. In 1992 the faculty was separated from the original Faculty of Mechanical Engineering.
Besides bachelor's, master's and doctoral programs in this field the faculty offers in cooperation with foreign universities postgraduate MBA studies. About 3 000 students study management, accounting, corporate finance, taxation and managerial science.
Faculty of Civil Engineering (FCE)
FCE is the oldest faculty of Brno University of Technology and the biggest one with its number of students – 6 500. In 1899 the university started its activities with this branch and that faculty was the only one which survived a violent change - transformation BUT into the Military Technical Academy in 1951.
Students can study in four programmes in Czech at this faculty:
Civil Engineering
Urban Engineering
Surveying and Cartography
Architecture
and one programme in English:
Civil Engineering
Faculty of Mechanical Engineering (FME)
The opening of the engineering department was in 1900 and herewith it is the second oldest faculty of the BUT. In the past there was taught energy branch from which subsequently became an independent Faculty of Electrical Engineering.
The faculty offers study of:
Engineering
Applied Sciences in Engineering
Mechanical Engineering
Production Systems
Industrial Design
Industrial Engineering
Machines and Equipment
Applied Natural Sciences
Physical and Materials Engineering
Metrology and Quality Assurance Testing
Faculty of Fine Arts (FFA)
On the contrary, one of the youngest BUT faculties is the Faculty of Fine Arts. In 1993 it was founded from the institute of fine arts which was based in the faculty of architecture a year earlier. As the number of students (3 thousands) it is the smallest faculty of current BUT.
Some of the study programs are:
painting
sculpture
graphics
graphic design
industrial design
conceptual tendencies
VMP (multimedia, video-performance).
University institutes
Institute of Forensic Engineering (IFE)
The aim of the institute is teaching forensic experts in master's degree programs - Venture engineering and Forensic engineering (expert engineering in transportation and real estate). It is possible to study The Forensic engineering program as a doctoral study. In preparation is the transformation of the institute to the separate department faculty with the temporary name the Faculty of Forensic Engineering.
Centre for Sports Activities (CESA)
In cooperation with the Faculty of Business CESA provides courses of study in Management of Physical Culture. Students can choose from more than 70 sports like basketball, swimming, athletics, golf or diving.
CEITEC Brno University of Technology (CEITEC BUT)
CEITEC is a centre for progressive innovative education which encourages scientific creativity in its research groups and a fostering of the principles of engagement. The institute offers a unique inter-institutional graduate programme, which in general is based on the CEITEC research programmes. Currently, one doctoral programme is available – Advanced Materials and Nanosciences.
Studies
The school is a holder of the European Commission ECTS certificates (European Credit Transfer System) - Label and DS (Diploma Supplement) Label for the period 2009-2013, reflecting the appreciation of quality of the university education in keeping with the principles of the Bologna Declaration. ECTS Label supports studying abroad at universities around the world. DS Label Certificate was awarded for the correct free award of the Diploma Supplement to all graduates of the principles of the European Commission.
Brno University of Technology offers its students:
295 study programs (of which 89 are accredited in foreign languages)
Participation in major international and scientific projects
Study at foreign and partner universities
More than 70 sports in 5 own specialized sports centers
Accommodation in halls of residences for the majority of applicants
9 libraries
Study programs:
Civil, Mechanical, Electrical and Forensic engineering
Information Technology
Chemistry
Economics and Management
Fine arts
Architecture
Cooperation with foreign institutions:
Framework agreements with 90 universities around the world
International educational research and development programs
Participation in EU programs for education:
CEEPUS
LLP / Erasmus
TEMPUS
Joint and double degree programs
Lifelong learning:
Is provided by faculties in their courses
MBA education (Master of Business Administration)
Lifelong Learning Institute:
provides counseling, information and organizational services
offers training and consultancy
organizes courses and seminars for seniors at the University of the Third Age
Science and Research
Research activities at the Brno University of Technology are realized in cooperation with national and international projects, programs, grants and research centers. BUT intensively cooperates with other universities and institutions, with the Academy of Sciences of the Czech Republic and private companies. The efforts of integration of teaching and scientific research are supported by application sector with which new curricula is prepared. Students can obtain a range of practical experience during their studies, thereby facilitating the selection of employment and competitiveness of BUT graduates. One of the BUT goals is to become a research university.
Main research areas
Environmental Technology
IT and communication technology
Aeronautical Engineering
Materials Engineering
Microelectronics and Nanotechnology
Studies for building and construction machinery
Advanced polymer and ceramic materials
Process and Chemical Engineering
Robotics and Artificial Intelligence
Sensing images and computer processing
Manufacturing technology
Research Centers
AdMaS - advanced building materials, engineering and technology
CEITEC - a center of excellence in biological sciences, materials and emerging technologies in cooperation of the Brno University of Technology and Masaryk University, Brno Mendel University, University of Veterinary and Pharmaceutical Sciences, Brno, Institute of Physics of Materials, Academy of Sciences and Veterinary Research Institute.
Centre for information, communication and biomedical technologies
Material Research Center in chemistry
Centre for Research and utilization of renewable energy
NETME Center - New Manufacturing Technology
Cooperation with industry
BUT cooperation with industry includes among others:
innovation and the preparation of new degree programs in collaboration with industry
direct cooperation in research and development companies in Czech Republic and abroad
personal participation of experts on education process
professional visits and internships
contact point for cooperation between enterprises and the BUT is Technology Transfer Office
Notable teachers
Bohuslav Fuchs, Major Czech modernist architect
Notable alumni
Mirek Topolánek, 7th Prime Minister of Czech Republic
Luděk Navara, Czech non-fictional author, publicist, scenarist and historian
Norbert Troller (1900-1984), artist and architect.
Radim Jančura, founder and CEO of Student Agency
Tomas Mikolov, Machine learning scientist
Notes and references
External links
Official website
History of the German Technical University in Brno
Article about a restoration of a former monastery in quarter Královo Pole, turned into a Centre of Application Technology
Central European Institute of Technology, CEITEC
MARABU, experimental aircraft
Universities in the Czech Republic
University
University
Educational institutions established in 1899
Buildings and structures in Brno
Engineering universities and colleges in the Czech Republic
1899 establishments in Austria-Hungary | {
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Kansas City Company Creates First-in-the-Nation PurpleStride Flagship Team
by Terra Hall — Apr 23, 2018
Members of PanCAN went to ScriptPro's headquarters to help kick off ScriptPro's participation as the first PurpleStride Flagship Team in the nation.
This May, Lindsey McDonald will slip on a purple T-shirt, lace up a pair of tennis shoes and hit the pavement for Kansas City's eighth PurpleStride walk to end pancreatic cancer.
Lindsey McDonald alongside her husband and children at PurpleStride Kansas City 2017.
While the 5K is timed for first, second and third place runners, in a way, McDonald and her team – which includes many of her colleagues – have already won the race.
"Community bonds and relationships with coworkers are important," McDonald said. "Those teambuilding relationships are what help you tackle tough problems at work. When you achieve a goal outside of the workplace, it will only help relationships inside the boardroom."
McDonald works at ScriptPro, a Kansas City-area company that provides robotics-enabled systems to optimize retail and ambulatory pharmacy services. ScriptPro just became the first PurpleStride Flagship Team in the country. As part of this partnership, the company dedicates itself to helping the Pancreatic Cancer Action Network (PanCAN) raise money and awareness to fight the world's toughest cancer.
McDonald's father-in-law, John McDonald, in front of his optical business.
"I'm so proud of my company for stepping up and leading the way," said McDonald, who has worked at ScriptPro for nearly two decades. "Being a part of something bigger is great for our employees. I'm interested to see how it shapes up this year and how it grows in the future."
Doctors diagnosed McDonald's father-in-law with pancreatic cancer in 2009. He passed away from the disease just four months later. That was a driving force for her when she joined PanCAN as a volunteer, starting the first PurpleStride Kansas City event in 2011, and when she became instrumental in making ScriptPro PanCAN's first PurpleStride Flagship Team.
"My father-in-law and his wife – all of their hopes and plans and all the things they waited so long to do were stolen," McDonald explained. "That story is not uncommon when you speak to people who have been affected by pancreatic cancer. I felt like I needed to do more."
McDonald and her family honoring her late father-in-law at Kansas City's inaugural PurpleStride event in 2011.
For her boss and ScriptPro owner Michael Coughlin, the feeling is mutual. ScriptPro employees, their family members and countless others have been directly affected by pancreatic cancer. He said that's why ScriptPro is motivated to help find a cure.
"It is wonderful to have the opportunity to lead the way with such a worthwhile effort," Coughlin said. "We are all really proud of the way our company and our people have responded and are making a difference. It reflects the dedication we have in all the things we do at ScriptPro to make the world better."
Read more about ScriptPro becoming the Pancreatic Cancer Action Network's first PurpleStride Flagship Team in the nation.
You too can help end pancreatic cancer and join PurpleStride Kansas City on Saturday, May 5 at Theis Park or register for an event near you at purplestride.org.
PurpleStride
7 Pancreatic Cancer Symptoms and Signs You Should Know
Alex Trebek Shares His Pancreatic Cancer Journey
7 Scientists Receive PanCAN Research Grants
8 Treatment Side Effects to Know
What Is Cancer Immunotherapy?
7 Questions to Ask About Pancreatic Cancer Treatment
St. Louis Cardinals Hall of Famer Bob Gibson Diagnosed with Pancreatic Cancer
5 New Members Join the Pancreatic Cancer Action Network's Survivor Council
10 Highlights You Made Happen
Comparing Pancreatic Tumor Tissue Types for Molecular Profiling
Mon – Fri, 7 a.m. – 5 p.m. PT | {
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Barathranthus es un género de arbustos perteneciente a la familia Loranthaceae. Comprende 16 especies descritas y de estas, solo 4 aceptadas.
Taxonomía
El género fue descrito por (Pieter Willem Korthals) Friedrich Anton Wilhelm Miquel y publicado en Flora van Nederlandsch Indië 1(1): 810, 834 en el año 1856.
Especies aceptadas
A continuación se brinda un listado de las especies del género Barathranthus aceptadas hasta noviembre de 2014, ordenadas alfabéticamente. Para cada una se indica el nombre binomial seguido del autor, abreviado según las convenciones y usos.
Barathranthus axanthus (Korth.) Miq.
Barathranthus mabioides (Thwaites) Danser
Barathranthus nodiflorus (Thwaites) Tiegh.
Barathranthus productus (King) Tiegh.
Referencias
Enlaces externos
http://gni.globalnames.org/name_strings?page=23&search_term=ns%3ABAR*
Loranthaceae | {
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\section{Introduction}
During islanded mode of operation of AC microgrid, grid-forming voltage source inverters (VSI) play an essential role by maintaining a stable voltage and frequency of the network and supplying uninterrupted power to the loads in the absence of a stiff distribution grid. Both centralized (\textit{master-slave}) and decentralized (\textit{droop-based}) \textit{Level-1} control layer architecture have been proposed in order to meet the uninterrupted power supply to the loads in the network via generating a voltage reference for the \textit{Level-0} control layer \cite{masterslave}. In \textit{Level-0} control layer, a voltage controller is required to be designed for each grid-forming VSI so that the output voltage waveform of each VSI tracks the respective voltage reference signal \cite{microgrid}.
\par Designing the voltage controller is essentially a multi-objective task. The design needs to primarily guarantee perfect reference tracking and good disturbance rejection which results in an output voltage with good regulation and low harmonic distortion in presence of various linear and non-linear loads. Additionally, the controllers are required to provide compensation to dynamic variations of the output current and improve the dynamic response \cite{yazdani}. It is also found that the unknown nature of the output loads can significantly alter the system behavior, especially during heavily loaded condition of VSI, that deteriorates the transient performance severely \cite{loaduncertainty}. The controller is required to perform all the aforementioned multi-objective tasks under output load uncertainties. As a result, requirement of robust performance of the voltage controller warrants an essential criterion during design stage.
\par Numerous voltage control strategies are proposed in the literature during past decades for grid-forming VSIs. Most of the earlier control techniques are based on linear control theory with a single voltage control-loop scheme \cite{guerrero1}. To further enhance the disturbance imposed by variations in input voltage and output current, a non-linear feed-forward loop is added in \cite{guerrero2}. However, single-loop classical controllers experience poor transient response and limited stability margins \cite{multiloop1}. To overcome these drawbacks, multiple-loop proportional-resonant (PR) and proportional-integral-differential (PID)-based classical control methods are proposed in \cite{multiloop1,multiloop2,multiloop3,multiloop4}. This common architecture has an internal current negative feedback loop, commanded by the error signal of the outer voltage regulation loop \cite{yazdani}. By introducing the inner current loop, the poles of the resonant $LC$ filter can be decoupled to compensate the non-idealities of the inductor. Moreover, it facilitates an inherent over-current protection of the VSI. However, having an inner-loop current controller requires more sensors that increases the overall cost \cite{prodanovic}. Moreover, classical controllers lack robustness in performance with output load uncertainties \cite{multiloop3}. In \cite{zhong1,zhong2,zhong3}, a lower order controller is designed using $\mathcal{H}_{\infty}$ methodology by removing the higher order part of the internal model using repetitive control technique. However, in these approaches load uncertainty is not taken into account. A robust tuning strategy for PID-based double-loop lag–lead compensator for VSI is proposed in \cite{multihinf1}. $\mathcal{H}_{\infty}$-based methodology is further explored in \cite{multihinf2,multihinf3,multihinf4,hinf2} in order to design an optimal controller for VSIs with linear and non-linear load disturbances. However, both these works inherently adapt the multi-loop structure for voltage controller and load uncertainty is not considered during design. In \cite{h2hinf1}, a sliding-mode control-based design for inner current loop and a mixed $\mathcal{H}_2/\mathcal{H}_{\infty}$-based optimal design for outer voltage loop is proposed. A very similar design is proposed in \cite{h2hinf2} where linear matrix inequality (LMI)-based linear quadratic regulator (LQR) for inner current loop and mixed $\mathcal{H}_2/\mathcal{H}_{\infty}$-based design for outer voltage loop are employed. Both these designs improve the transient response significantly in presence of uncertain filter parameters at the cost of added complexity in controllers. Moreover, uncertainty imposed by output loads is not taken into consideration while designing the controllers. \cite{hinf1} provides a robust optimal state-feedback controller based on the integration of optimal output regulation theory and back-stepping method where load is treated as a uncertain element. However, the impact of non-linear loads (various harmonic components) is not considered in this work. The primary contributions of this article are as follows:
\begin{enumerate}
\item A simple robust and optimal (single-loop, low-cost) voltage controller for \textit{Level-0} control layer is proposed that can address the aforementioned limitations along with taking into account the uncertainties in load variations.
\item It facilitates lower requirements of sensor measurements as opposed to other works in the literature without compromising in voltage regulation and disturbance rejection while guaranteeing robust performance.
\end{enumerate}
Time-domain MATLAB/SIMULINK-based simulation with low-cost controller are performed in the validation stage of the proposed control.
\begin{figure*}[t]
\centering
\subfloat[]{\includegraphics[scale=0.28,trim={18cm 2cm 14cm 3cm},clip]{circuit.pdf}%
\label{fig:circ}}~~~~~~~~
\subfloat[]{\includegraphics[scale=0.18,trim={0cm 0cm 0.0cm 0.0cm},clip]{GvZd1.pdf}%
\label{fig:GvZd}}
\caption{Description of the system with (a) the schematic of single-phase grid-forming VSI, (b) variation of plant dynamics model due to uncertain load.}
\label{fig:circuit}
\end{figure*}
\section{Problem Formulation}
\subsection{Architecture of Grid-forming VSI}
A single-phase grid-forming VSI terminated by a load, $Z_{Load}$, is shown in Fig.~\ref{fig:circuit}\subref{fig:circ}. The VSI is composed of a dc bus voltage, $V_{dc}$, four switching devices, $S_1$-$S_4$, and an $LC$ filter with $L_f$, $C_f$ as filter inductor and capacitor respectively. $R_f$ is the equivalent series resistance (ESR) of $L_f$ capturing parasitic element. The controller along with sinusoidal PWM switching technique generates the switching signals that result the output voltage. The dynamics of the VSI are described as:
\begin{align}
L_f \frac{d\langle i_{inv}\rangle}{dt} +R_f\langle i_{inv}\rangle &= \langle v_{inv}\rangle - \langle v_C\rangle, \label{eq1}\\
C_f \frac{d\langle v_C\rangle}{dt} &= \langle i_{inv}\rangle - \langle i_{grid}\rangle, \label{eq2}
\end{align}
where $\langle .\rangle$ signifies the average values of the corresponding variable over one switching cycle ($T_s$) \cite{yazdani}. Combining \eqref{eq1} and \eqref{eq2} and taking Laplace transformation, the open-loop output voltage dynamics of the grid-forming VSI is given as:
\begin{align}\label{eq3}
V_C(s) &= G_{inv}(s)V_{inv}(s) - G_{i}(s)I_{grid}(s),
\end{align}
where
\begin{align*}
G_{inv}(s) &= \dfrac{1}{L_f C_f s^2 +R_f C_f s +1},\\ G_{i}(s) &= \dfrac{L_f s+R_f}{L_f C_f s^2 +R_f C_f s +1}.
\end{align*}
\subsection{Load Characteristics and Modeling}
Characterizing the nature of the load is essential in the process of designing the controller. In this work, the load is modeled by a parallel combination of two components. First component is an admittance, $Y_L(s)$, with unknown $R$ and $L$ elements, and can be defined as:
\begin{align*}
Y_L(s):= [1+\Delta(s)]Y_L^N(s),
\end{align*}
where, $Y_L^N(s)$ is a frequency dependent weighting function, capturing the nominal behavior, and defined as:
\begin{align*}
Y_L^N(s):=\dfrac{1}{L_L^Ns+R_L^N}.
\end{align*}
A normalized dynamic LTI uncertainty, $\Delta(s)$, is defined to capture the uncertain behavior of load change such that
\begin{align*}
||\Delta(s)||_{\infty} \leq 1.
\end{align*}
The value of $L_L^N$ and $R_L^N$ can be selected as $R_L^N = V_{rated}^2/P_{rated}$ and $L_L^N = V_{rated}^2/(\omega_oQ_{rated})$, where $V_{rated}$, $P_{rated}$, $Q_{rated}$ and $\omega_o$ are the rated output voltage, active, reactive power of the grid-forming VSI and nominal frequency of the network respectively. Another component of loads is a parallel combination of current sources consisting of both fundamental and harmonic components \cite{loadmodel1} and defined as
\begin{align*}
I_d(s):=\sum_{h}I_{d,h}(s),~~ [~h~\text{is odd}~].
\end{align*}
Therefore, $Z_{Load}$ of Fig.~\ref{fig:circuit}\subref{fig:circ} can be characterized as
\begin{align}\label{eq4}
I_{grid}(s) &= Y_L(s)V_C(s) + I_d(s) \nonumber \\
&= [1+\Delta(s)]Y_L^N(s)V_C(s) + \sum_{h}I_{d,h}(s).
\end{align}
\begin{remark}
Multiplicative uncertainty in $Y_L(s)$ is used to shape the ``worst-case'' yet allowable output admittance of the grid-forming VSI. The exogenous disturbance signal, $I_d(s)$, captures the presence of non-linear loads as shown in Fig.~\ref{fig:control}\subref{control1}.
\end{remark}
\subsection{Impact of Load Uncertainty}
Combining \eqref{eq3} and \eqref{eq4}, system's dynamic equation becomes
\begin{align}\label{eq5}
V_C(s) = G_v^{\Delta}(s)V_{inv}(s) - Z_d^{\Delta}(s)I_d(s).
\end{align}
where,
\begin{align*}
G_v^{\Delta}(s) = \dfrac{G_{inv}(s)}{1+G_i(s)Y_L(s)},~~ Z_d^{\Delta}(s) = \dfrac{G_i(s)}{1+G_i(s)Y_L(s)}.
\end{align*}
Clearly uncertainty in $Y_L(s)$ imposes uncertainty in both $G_v^{\Delta}(s)$ and $Z_d^{\Delta}(s)$ as illustrated in Fig.~\ref{fig:circuit}\subref{fig:GvZd}. It can be observed that the unknown load across VSI causes significant change in the plant dynamics, especially the effective resonant frequency of the system. This deteriorates the robust performance of the controller severely as well as overall system stability \cite{loaduncertainty}.
\section{Proposed Solution}
\subsection{Objectives of the Designed Controller}
The objective is to design a feedback control law through controller, $C_{H_{\infty}}(s)$ as shown in Fig.~\ref{fig:control}\subref{control1}, which generates a control signal, $v_{inv}$, such that, i) $v_C$ tracks $v_{ref}$, ii) effects of $i_d$ on $v_C$ is attenuated, iii) $v_{inv}$ satisfies bandwidth limitations, and iv) $v_C$ recovers quickly after any sort of transients. Moreover, all aforementioned objectives need to be achieved with load uncertainty. In other words, it is necessary to provide the controller with enough robustness to deal with system uncertainty caused by load variation. These objectives are derived from acceptable standards on microgrid operation, IEEE Std-2030 \cite{ieee1}, and power quality, IEEE Std-519\cite{ieee2}.
\subsection{Design Procedure of the $\mathcal{H}_{\infty}$-based Controller}
$\mathcal{H}_{\infty}$-based controller design provides a framework for addressing multiple objectives. Here, the design is based on the system structure presented in Fig.~\ref{fig:control}\subref{control1} where user-defined weighting transfer functions, $W_S(s)$, $W_{CS}(s)$, $W_d(s)$ and $T_{des}(s)$, are selected based on the aforementioned requirements. The brief guidelines for designing the weighting transfer functions are provided below.
\begin{figure*}[t]
\centering
\subfloat[]{\includegraphics[scale=0.25,trim={5cm 1.0cm 8.5cm 2.5cm},clip]{control.pdf}%
\label{control1}}~~~~~~~~~
\subfloat[]{\includegraphics[scale=0.27,trim={19.5cm 5.5cm 19.0cm 3.5cm},clip]{genP.pdf}%
\label{control2}}
\caption{$\mathcal{H}_{\infty}$-based controller synthesis, (a) the closed-loop control system with selected weighting transfer functions, (b) general control configuration.}
\label{fig:control}
\end{figure*}
\subsubsection{Selection of $W_S(s)$}
To shape the sensitivity transfer function, the weighting function, $W_S(s)$ is introduced so that
\begin{itemize}
\item the tracking error at $\omega_o$ is extremely low,
\item the $LC$ resonance of VSI is actively damped,
\end{itemize}
$W_S(s)$ is modeled to have peaks at $\omega_o$ and $LC$ resonant frequency, $\omega_{res}$, with a overall gain $k_{S,1}$. Hence,
\begin{align*}
&W_S(s) \\
& = k_{S,1}\bigg[\dfrac{s^2+2k_{S,2}\zeta\omega_o s+\omega_o^2}{s^2+2\zeta\omega_o s+\omega_o^2}\bigg] \bigg[\dfrac{s^2+2k_{S,3}\zeta\omega_{res}s+\omega_{res}^2}{s^2+2\zeta\omega_{res}s+\omega_{res}^2}\bigg],
\end{align*}
where, $k_{S,2}$ and $k_{S,3}$ are selected to exhibit peaks and $\zeta$ takes care of the off-nominal system frequency.
\subsubsection{Selection of $W_{CS}(s)$}
$W_{CS}(s)$ is designed to suppress high-frequency control effort to shape the performance of $v_{inv}$. Hence, it is designed as high-pass filter with cut-off frequency at switching frequency and ascribed the form:
\begin{align*}
W_{CS}(s) = \dfrac{s+k_{CS,1}\omega_o}{s+k_{CS,2}\omega_o}, ~\text{where}~ k_{CS,1}<<k_{CS,2}.
\end{align*}
\subsubsection{Selection of $W_d(s)$}
$W_d(s)$ emphasizes the expected disturbances at $\omega_o$ and at different harmonic frequencies imposed by $i_d$ and emphasized by exogenous signal $\hat{i}_d$. It is based on the assumption that the load current comprises fundamental, $3^{rd}$, $5^{th}$, $7^{th}$, $9^{th}$, $11^{th}$ and $13^{th}$ harmonics. Hence, it is designed by a low-pass filter with peaks at selected frequencies. The resulting function is obtained as
\begin{align*}
W_d(s) = \prod_{h=1,3,\ldots,13}\dfrac{s^2+2k_{d,h}\zeta h\omega_o s+h^2\omega_o^2}{s^2+2\zeta h\omega_o s+h^2\omega_o^2},
\end{align*}
where, the values of $k_{d,h}$ are selected based on the THD standards mentioned in IEEE Std-519 \cite{ieee2}.
\subsubsection{Selection of $T_{des}(s)$}
To emphasize on the desired performance of the closed-loop system, a performance bound is designed with a typical first-order transfer function specified by the parameters $k_B$, $\omega_B^*$ in the following expression.
\begin{align*}
T_{des}(s) = k_B\dfrac{\omega_B^*}{s + \omega_B^*}.
\end{align*}
Clearly, the desired output response is decided based on the selection of $\omega_B^*$, which is determined by the requirement of speed of response. Desired bandwidth of at least $10$ times of $\omega_o$, decides the value of $\omega_{B}^*$ with $k_B$ as overall gain.
\begin{figure}[t]
\centering
\subfloat[]{\includegraphics[scale=0.25,trim={1.0cm 0.0cm 2.2cm 0cm},clip]{Wbode.pdf}%
\label{fig:weight}}
\subfloat[]{\includegraphics[scale=0.25,trim={1.0cm 0.0cm 2.2cm 0cm},clip]{GZmagbode.pdf}%
\label{fig:GZmag}}
\subfloat[]{\includegraphics[scale=0.25,trim={0.5cm 0.0cm 2.2cm 0.0cm},clip]{GZphasebode.pdf}%
\label{fig:GZphase}}
\caption{Bode plots of (a) magnitudes of weighting transfer functions, (b) magnitudes of $G(s)$ and $Z(s)$ and (c) phase of $G(s)$ and $Z(s)$.}
\label{fig:bode}
\end{figure}
\subsection{Problem Formulation and Resulting Optimal Controller}
The Bode plots of the selected weighting transfer functions in this work are shown in Fig.~\ref{fig:bode}\subref{fig:weight}. The $\mathcal{H}_{\infty}$ optimal problem is formulated to generate a feedback control law with controller, $C_{H_{\infty}}(s)$, stated as:
\begin{align}\label{eq10}
V_{inv}(s) = C_{H_{\infty}}(s)[V_{ref}(s)-V_C(s)].
\end{align}
As a result, the closed loop system equation can be found by combining \eqref{eq5} and \eqref{eq10} and can be written as
\begin{align}\label{eq11}
V_C(s) = G(s)V_{ref}(s) - Z(s)I_d(s),
\end{align}
where
\begin{align*}
G(s) = \dfrac{G_v^{\Delta}(s)C_{H_{\infty}}(s)}{1+G_v^{\Delta}(s)C_{H_{\infty}}(s)},~~ Z(s) = \dfrac{Z_d^{\Delta}(s)}{1+G_v^{\Delta}(s)C_{H_{\infty}}(s)}.
\end{align*}
Therefore, it is equivalent to state that the optimal controller is required to be designed such that
\begin{align*}
&G(s)|_{s=j\omega_o} \approx 1\angle 0^o,\\
&Z(s)|_{s=jh\omega_o} << 1 ~~[h=1, 3, 5, 7, 9, 11, 13].
\end{align*}
The control system of Fig.~\ref{fig:control}\subref{control1} can be realized as a general control configuration as shown in Fig.~\ref{fig:control}\subref{control2}\cite{robust1}. It has a generalized MIMO plant, $P$, containing nominal plant models defined as $G_v^{Nom}(s)$, $Z_d^{Nom}(s)$, $W_S(s)$, $W_{CS}(s)$, $W_d(s)$ and $T_{des}(s)$ with exogenous input signal $w:=\begin{bmatrix}v_{ref} & i_d\end{bmatrix}^\top$ and output signals $z:=\begin{bmatrix}z_S & z_{CS} & e_o\end{bmatrix}^\top$. The controller, $C_{H_{\infty}}(s)$ has input feedback signal, $e$, and output control signal, $v_{inv}$. The uncertainty function, $\Delta(s)$, with input signal, $z_{\Delta}$, and output, $w_{\Delta}$ is represented using upper linear fractional transformations \cite{robust1}. The goal is to synthesize the stabilizing controller, $C_{H_{\infty}}(s)$, that satisfies the following:
\begin{align}\label{eq12}
||T_{w\rightarrow z}||_{\infty} < 1.
\end{align}
\begin{remark}
Small-gain theorem states that the accomplishment of \eqref{eq12} together with $||\Delta(s)||_{\infty}\leq 1$ guarantees the robust stability of the closed-loop system.\cite{robust1}
\end{remark}
Using $\gamma$-iteration algorithm provided in \cite{robust1} (the state-space solution approach via solving two algebraic Riccati equations), the synthesis of optimal controller is achieved by means of \textit{hinfsyn} command of MATLAB toolbox. Usually $\mathcal{H}_{\infty}$-control algorithms produce controllers of higher order and model reduction becomes essential to design a low order implementable controller. It is achieved by using balanced truncation method removing modes faster than the switching frequency. The resulting controller, $C_{H_{\infty}}(s)$, is of the order of $17$ which is higher only by $3$ orders than that of conventional PR controller with up to $13^{th}$ order harmonic compensators. Fig.~\ref{fig:bode}\subref{fig:GZmag} and \ref{fig:bode}\subref{fig:GZphase} corroborate the accomplishment of the objectives by the resulting optimal controller. Moreover, the optimal controller along with the plant results $||T_{w\rightarrow z}||_{\infty}=0.77$ that makes the closed-loop system robust stable. Moreover, robust stability margin is verified by \textit{robstab} command of MATLAB.
\section{Results}
SIMULINK/MATLAB simulation studies are carried out for validation. The system parameters used in both the stages are tabulated in Table~\ref{table}. A combination of impedance type and current source type load models are used. However, for closer proximity with practical scenarios, a DC-drive motor load is also considered along with the others and the modelling is followed as mentioned in \cite{loadmodel1,loadmodel2}.
\renewcommand{\arraystretch}{1.2}
\begin{table}[t]
\centering
\caption{vsi parameters for matlab/simulink and chil simulation}
\label{table}
\begin{tabular}{|c|c|}
\hline
\textbf{VSI Parameter} & \textbf{Value} \\ \hline \hline
Rated RMS Output Voltage & $220$ (V) \\ \hline
Rated Output Frequency ($\omega_o$) & $2\pi 60$ (rad/s) \\ \hline
DC Link Voltage ($V_{dc}$) & $500$ (V) \\ \hline
Switching Frequency & $20$ (kHz) \\ \hline
$LC$ Filter Inductance ($L_f$) & $2$ (mH) \\ \hline
$LC$ Filter Capacitance ($C_f$) & $20$ ($\mu$F) \\ \hline
Rated Output Power & $2$ (kVA) \\ \hline
\end{tabular}
\end{table}
\begin{figure*}[t]
\centering
\subfloat[]{\includegraphics[scale=0.182,trim={1.0cm 1.0cm 0.0cm 1.0cm},clip]{vCHinf.pdf}%
\label{fig:vCHinf}}
\subfloat[]{\includegraphics[scale=0.182,trim={1.0cm 1.0cm 0.0cm 1.0cm},clip]{vCPR.pdf}%
\label{fig:vCPR}}
\caption{Simulation results, (a) waveform of reference voltage and output voltage with proposed controller, (b) with conventional PR-based controller\cite{yazdani}.}
\label{fig:sim}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[scale=0.185,trim={1cm 0.0cm 0.0cm 0.0cm},clip]{igrid.pdf}%
\caption{Simulation results of output load current waveform of VSI.}
\label{fig:igrid}
\end{figure}
\begin{table}[t]
\centering
\caption{reference tracking and thd performance comparison}
\label{THD}
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{\begin{tabular}[c]{@{}c@{}}Simulation\\ Study\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Controller\\ Type\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Magnitude \\ Error (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Phase \\ Error (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}THD \\ (\%)\end{tabular}} \\ \hline \hline
\multirow{2}{*}{SIMULINK} & $C_{H_{\infty}}(s)$-based & $\pm~0.74$ & $\pm~1.39$ & $0.41$ \\ \cline{2-5} & PR-based\cite{yazdani} & $\pm~1.41$ & $\pm~2.92$ & $1.39$ \\ \hline \hline CHIL & $C_{H_{\infty}}(s)$-based & $\pm~0.97$ & $\pm~1.89$ & $0.57$\\ \hline
\end{tabular}
\end{table}
\subsection{Simulation Results and Performance Comparison}
For the purpose of performance comparison, conventional multi-loop-based classical outer PR-based voltage controller and inner PI-based current controller is considered for the time-domain simulation. \cite{yazdani} provides an elaborated guidelines for designing the classical PR and PI based multi-loop voltage controller for grid-forming VSI with enough gain and phase margins. In this work, the guidelines are followed in order to possess a fair comparative study. PI-based inner loop is designed to have phase margin (PM) $\geq 60^{\circ}$ and gain margin (GM) $\geq 40$~dB with bandwidth $\geq 5$~kHz. Similarly, PR-based outer loop is designed to have PM $\geq 45^{\circ}$ and GM $ \geq 40$~dB with bandwidth $\geq 5$~kHz. The following sequence of events is used in the MATLAB/SIMULINK-based validation stage.
\begin{itemize}
\item VSI is in no-load condition until $t=0.05~s$.
\item At $t=0.05~s$, there is a transition from no-load to full-load and stays at full-load condition until $t=0.1~s$.
\item At $t=0.1~s$, reactive power drops to $0$ until $t=0.15~s$.
\item At $t=0.15~s$, both real and reactive power change.
\item At $t=0.2~s$, $v_{ref}$ drops and stays until $t=0.25~s$.
\item At $t=0.25~s$, VSI is switched to no-load condition.
\end{itemize}
The output voltage waveform of VSI, $v_C$, during the time-domain events with both proposed $\mathcal{H}_{\infty}$-based and conventional-multi-loop-based voltage controller are shown in Fig.~\ref{fig:sim}\subref{fig:vCHinf} and Fig.~\ref{fig:sim}\subref{fig:vCPR}. The enlarged portions in these figures are illustrating the transient behaviors during aforementioned situations. The results substantiate the fact that the proposed $\mathcal{H}_{\infty}$-based controller exhibits superior robustness in performance than the multi-loop controller with varying loading conditions. THD of $v_C$ with proposed controller is much smaller than the multi-loop controller with highly non-linear loads (output current is illustrated in Fig.~\ref{fig:igrid}). Finally, Table \ref{THD} corroborates the superiority of the proposed controller in reference tracking and disturbance rejection in compare with the classical multi-loop PR-based voltage controller at the cost of increasing the order of controller only by $3$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.33,trim={0.0cm 0.0cm 27cm 0.0cm},clip]{chil.pdf}%
\caption{OPAL-RT based hardware-in-the-loop simulation platform with Texas Instruments Delfino TMS320F28379D controller board.}
\label{fig:chilplatform}
\end{figure}
\subsection{CHIL Simulation Results}
The computational footprint of the proposed $\mathcal{H}_{\infty}$-based controller, while implemented in a real low-cost micro-controller board, is an essential check for further validating the performance. Therefore, controller hardware-in-the-loop (CHIL)-based simulation studies are also conducted on OPAL-RT real-time simulator with the proposed $\mathcal{H}_{\infty}$-based controller realized on a low-cost Texas-Instruments TMS28379D Delfino controller board as shown in Fig.~\ref{fig:chilplatform}. In similar way, four events are emulated in the OPAL-RT platform and are enlisted as:
\begin{itemize}
\item $\mathtt{CASE}$-$\mathtt{1}$: A sudden rise in VSI output current, $i_{grid}(t)$, at $t=158.05~s$ with unchanged reference signal, $v_{ref}(t)$.
\item $\mathtt{CASE}$-$\mathtt{2}$: A sudden fall in VSI output current, $i_{grid}(t)$, at $t=130.35~s$ with unchanged reference signal, $v_{ref}(t)$.
\item $\mathtt{CASE}$-$\mathtt{3}$: A sudden fall in output voltage reference signal, $v_{ref}(t)$, at $t=132.95~s$ with unchanged $i_{grid}(t)$.
\item $\mathtt{CASE}$-$\mathtt{4}$: A sudden rise in output voltage reference signal, $v_{ref}(t)$, at $t=166.05~s$ with unchanged $i_{grid}(t)$.
\end{itemize}
\begin{figure*}[t]
\centering
\subfloat[]{\includegraphics[scale=0.19,trim={0.0cm 6.5cm 5.0cm 1.0cm},clip]{chilvCHinf1.pdf}%
\label{fig:vCHinfchil1}}
\subfloat[]{\includegraphics[scale=0.19,trim={0.0cm 6.5cm 5.0cm 1.0cm},clip]{chilvCHinf2.pdf}%
\label{fig:vCHinfchil2}}
\subfloat[]{\includegraphics[scale=0.19,trim={0.0cm 6.5cm 5.0cm 1.0cm},clip]{chiligrid1.pdf}%
\label{fig:igridchil1}}
\subfloat[]{\includegraphics[scale=0.19,trim={0.0cm 6.5cm 5.0cm 1.0cm},clip]{chiligrid2.pdf}%
\label{fig:igridchil2}}
\caption{CHIL simulation results of output voltage waveform of VSI with (a) $\mathtt{CASE}$-$\mathtt{1}$: sudden rise in output current at $t=158.05~s$ and (b) $\mathtt{CASE}$-$\mathtt{2}$: sudden fall in output current at $t=130.35~s$ with proposed controller, (c) rise in output current in $\mathtt{CASE}$-$\mathtt{1}$, (d) fall in output current in $\mathtt{CASE}$-$\mathtt{2}$.}
\label{fig:chil1}
\end{figure*}
\begin{figure*}[t]
\centering
\subfloat[]{\includegraphics[scale=0.19,trim={0.0cm 6.5cm 5.0cm 1.0cm},clip]{chilvCHinf3.pdf}%
\label{fig:vCHinfchil3}}
\subfloat[]{\includegraphics[scale=0.19,trim={0.0cm 6.5cm 5.0cm 1.0cm},clip]{chilvCHinf4.pdf}%
\label{fig:vCHinfchil4}}
\subfloat[]{\includegraphics[scale=0.19,trim={0.0cm 6.5cm 5.0cm 1.0cm},clip]{chiligrid3.pdf}%
\label{fig:igridchil3}}
\subfloat[]{\includegraphics[scale=0.19,trim={0.0cm 6.5cm 5.0cm 1.0cm},clip]{chiligrid4.pdf}%
\label{fig:igridchil4}}
\caption{CHIL simulation results of output voltage waveform of VSI with (a) $\mathtt{CASE}$-$\mathtt{3}$: sudden fall in output voltage reference signal at $t=132.95~s$ and (b) $\mathtt{CASE}$-$\mathtt{4}$: sudden rise in output voltage reference signal at $t=166.05~s$ with proposed controller, (c) unchanged output current in $\mathtt{CASE}$-$\mathtt{3}$ and (d) in $\mathtt{CASE}$-$\mathtt{4}$.}
\label{fig:chil2}
\end{figure*}
The results for $\mathtt{CASE}$-$\mathtt{1}$ and $\mathtt{CASE}$-$\mathtt{2}$ are shown in Fig.~\ref{fig:chil1}. It is clearly observed that the output voltage waveforms of VSI, $v_C$, are insensitive to the sudden changes of $i_{grid}$ (shown in Fig.~\ref{fig:chil1}\subref{fig:igridchil1} and Fig.~\ref{fig:chil1}\subref{fig:igridchil2}) in both $\mathtt{CASE}$-$\mathtt{1}$ and $\mathtt{CASE}$-$\mathtt{2}$ as shown in Fig.~\ref{fig:chil1}\subref{fig:vCHinfchil1} and Fig.~\ref{fig:chil1}\subref{fig:vCHinfchil2}. Similarly, the results for $\mathtt{CASE}$-$\mathtt{3}$ and $\mathtt{CASE}$-$\mathtt{4}$ are shown in Fig.~\ref{fig:chil2}. It is observed that the output voltage waveforms of VSI, $v_C$, tracks the changes in reference voltage signal, $v_{ref}$, very quickly as shown in Fig.~\ref{fig:chil2}\subref{fig:vCHinfchil3} and Fig.~\ref{fig:chil2}\subref{fig:vCHinfchil4} for both $\mathtt{CASE}$-$\mathtt{3}$ and $\mathtt{CASE}$-$\mathtt{4}$. Moreover, due to constant-current type output loads terminated across the VSI, the output current waveform, $i_{grid}$ is unchanged as shown in Fig.~\ref{fig:chil2}\subref{fig:igridchil3} and Fig.~\ref{fig:chil2}\subref{fig:igridchil4}. The results shown in Fig.~\ref{fig:chil1} and Fig.~\ref{fig:chil2} clearly substantiate the fact that the dynamic performance of the proposed controller is less sensitive to transients. Moreover, Table~\ref{THD} shows that the reference tracking and harmonic rejection performance illustrated in CHIL results are close enough to the results from SIMULINK/MATLAB. However, due to practical and inevitable limitations like quantization error of ADC of the real controller and efficacy of discretization process, the results may differ slightly but under acceptable limit as evidenced in this case also. Along with the performances, this study also showcases the viability of the proposed $\mathcal{H}_{\infty}$-based controller in real low-cost control broads.
\section{Conclusion}
This article demonstrates the design and implementation of robust and optimal single-loop voltage controller for single-phase grid-forming VSI. The model uncertainty of VSI imposed by the unknown changing load is demonstrated and its impact on dynamic model of VSI is shown. $\mathcal{H}_{\infty}$-based controller design is introduced to address this issue and the required objectives for the optimal controller are discussed to formulate the optimization problem which leads to final optimal controller. A time-domain SIMULINK-based simulation study substantiates the fact that the resulting controller exhibits superior robustness in performance during varying loading of VSI that conventional multi-loop controller architecture. Moreover, OPAL-RT based CHIL simulations are conducted to verify the viability of the resulting controller.
\section*{Acknowledgment}
The authors acknowledge Advanced Research Projects Agency-Energy (ARPA-E) for supporting this research through the project titled ``Rapidly Viable Sustained Grid'' via grant no. DE-AR0001016.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,522 |
\section{Introduction}
By definition, a stable unsuperstable theory admits
a type that is not based on any finite subset of its domain.
\relax From this one sees that such a theory admits trees of
definable sets. That is, there is a sequence $\<\varphi_n(x,y):n\in\omega\>$
of formulas such that for any cardinal $\kappa$ there are
definable sets $\{\varphi_n(x,a_\nu):\nu\in {^{<\omega}\kappa}\}$
giving rise to $\kappa^{\aleph_0}$ partial types
$\{p_\mu:\mu\in {^\omega\kappa}\}$ where each $p_\mu$ forks over
$\{a_{\mu|k}:k<n\}$ for all $n\in\omega$. In~\cite{Shc}
the second author used these trees to count the number of uncountable models
or to find the maximal size of a family of pairwise nonembeddable
models of a fixed cardinality of any stable, unsuperstable theory.
However, for other combinatorial questions, such as computing
the Karp complexity of the class of uncountable models of such a theory,
the existence of these trees does not seem to be sufficient.
Here we prove that when the language is countable, any strictly
stable theory exhibits one of three more detailed nonstructural
properties. This trichotomy is used in \cite{LSh2}, but it is likely
to be used in other contexts as well. Two of the alternatives,
the Dimensional Order Property (DOP) or a theory being deep
appear in \cite{Shc} and are compatible with superstability.
The third alternative is new and is captured by the
following definition:
\begin{Definition} \label{witness}
{\em
An {\em abelian group witness to unsuperstability\/}
is a descending sequence $\<A_n:n\in\omega\>$ of abelian groups
with $[A_n:A_{n+1}]$ infinite for each $n$ such that
the intersection $A=\bigcap_n A_n$ is connected and whose
generic type is regular.
}
\end{Definition}
The existence of such a sequence readily contradicts superstability
as for any cardinal $\kappa$ one immediately obtains a tree
$\{C_\mu:\mu\in {^\omega\kappa}\}$ of cosets of $A$.
As well, with Theorems~\ref{ss} and \ref{rperp} we see that one
can frequently say more about the generic type of $A$.
This added information is used in \cite{LSh2}.
In order to establish these results, the bulk of the paper discusses
the notion of a {\em regular ideal\/} of formulas (see Definition~\ref{r}).
The origins of these ideas date back to Section V.4 of ~\cite{Shc}
and have been reworked and expanded in \cite{Hr1} and \cite{Pillay}.
Our notation is standard, and complies with either \cite{Pillay} or
\cite{Shc}. For a stable theory $T$ $\kappa_r(T)$
denotes the least regular cardinal $\kappa$ such that there is no
forking chain of length $\kappa$. Thus, a stable theory is superstable
if and only if $\kappa_r(T)=\aleph_0$ and $\kappa_r(T)=\aleph_1$
when $T$ is countable and strictly stable. We call a model
`a-saturated' (a-prime) in place of `${\bf F}^a_{\kappa_r(T)}$-saturated'
(${\bf F}^a_{\kappa_r(T)}$-prime).
{\bf Throughout the whole of this paper we assume `${\bf T=T^{{\rm eq}}}$.'}
That is, $T$ is a stable theory in a multi-sorted language,
${\frak C}$ is a large, saturated model of $T$, and the language $L$
is closed under the following operation:
If $E(\bar{x},\bar{y})$ is a definable
equivalence relation then there is a sort $U_E$ and a definable
surjection $f_E:{\frak C}^{lg(\bar{x})}\rightarrow U_E({\frak C})$ in the language
$L$. In particular, the set of sorts is closed under finite
products. Thus
any finite tuple of elements from varying sorts can be viewed as an
element of the product sort. With this identification,
every formula can be considered to have a single free variable.
As notation, $L({\frak C})$ denotes the set of formulas with parameters from
${\frak C}$ and for a specific sort $s$,
$L_s({\frak C})$ denotes the $L({\frak C})$-formulas
$\varphi(x)$ in which the free variable has sort $s$.
\section{Regular ideals}
\begin{Definition} {\em An {\em invariant ideal ${\cal ID}$\/} is a subset of
$L({\frak C})$ containing all algebraic formulas, closed under
automorphisms of ${\frak C}$, and for any sort $s$ and any $\varphi,\psi\in L_s({\frak C})$
\begin{enumerate}
\item If $\varphi,\psi\in{\cal ID}$ then $\varphi\vee\psi\in{\cal ID}$; and
\item If $\varphi\vdash\psi$ and $\psi\in{\cal ID}$, then $\varphi\in{\cal ID}$.
\end{enumerate}
A partial type $\Gamma$ (i.e., a subset of $L_s({\frak C})$ for some sort $s$)
is {\em ${\cal ID}$-small\/} if it entails some element of ${\cal ID}\cap L_s({\frak C})$.
}
\end{Definition}
Many times we will make use of the fact that formulas in ${\cal ID}$ may have
`hidden' parameters.
\begin{Lemma} \label{1} Let ${\cal ID}$ be any invariant ideal.
\begin{enumerate}
\item A complete type $p\in S(A)$ is ${\cal ID}$-small if and only if $p\cap{\cal ID}\neq\emptyset$.
\item For any $A$ and $a$, ${\rm stp}(a/A)$ is ${\cal ID}$-small if and only if ${\rm tp}(a/A)$ is
${\cal ID}$-small.
\item If $A\subseteq B$ and ${\rm tp}(a/B)$ does not fork over $A$, then
${\rm tp}(a/A)$ is ${\cal ID}$-small if and only if ${\rm tp}(b/A)$ is ${\cal ID}$-small.
\end{enumerate}
\end{Lemma}
{\bf Proof.}\quad (1) Right to left is immediate. For the converse, assume
$p$ entails $\psi\in{\cal ID}$. By compactness there is $\varphi\in p$ such
that $\varphi\vdash\psi$, hence $\varphi\in{\cal ID}$.
(2) Right to left is clear. If ${\rm stp}(a/A)$ entails $\psi(x,b)\in{\cal ID}$,
then by compactness and the finite equivalence relation theorem
there is an $A$-definable equivalence relation $E(x,y)$
with finitely many classes
such that ${\rm tp}(a/A)\cup\{E(x,c)\}\vdash\psi(x,b)$ for some $c$. Choose
$A$-automorphisms $\{\sigma_i:i<n\}$ of ${\frak C}$ such that
$\{E(x,\sigma_i(c)):i<n\}$ includes all the $E$-classes.
Since ${\cal ID}$ is an invariant ideal $\bigvee_{i<n}\psi(x,\sigma_i(b))\in{\cal ID}$
and ${\rm tp}(a/A)\vdash \bigvee_{i<n}\psi(x,\sigma_i(b))$.
(3) By (2) it suffices to prove this for strong types. Assume
${\rm stp}(a/B)$ is ${\cal ID}$-small. By (1) and (2), choose
$\psi(x,b)\in {\rm tp}(a/B)\cap{\cal ID}$. Choose $\{b_i:i\in\kappa(T)\}$ independent
over $A$, each having the same strong type over $A$ as $b$.
Since ${\cal ID}$ is invariant, $\psi(x,b_i)\in{\cal ID}$ for each $i$.
Furthermore,
since any $a'$ realizing ${\rm stp}(a/A)$ is independent with some $b_i$ over $A$,
$ab$ and $a'b_i$ realize the same strong type over $A$, hence
$\psi(a',b_i)$ holds. By compactness, there is a finite subset $F$
such that ${\rm stp}(a/A)\vdash\bigvee_{i\in F} \psi(x,b_i)$, so ${\rm stp}(a/A)$ is
${\cal ID}$-small.
\medskip
\begin{Definition} \label{r}
{\em
An invariant ideal ${\cal ID}$ is {\em regular\/} if, for all $L({\frak C})$-formulas
$\psi(y)$ and $\theta(x,y)$, {\bf IF} $\psi\in{\cal ID}$ and
$\theta(x,b)\in{\cal ID}$ for every $b\in\psi({\frak C})$ {\bf THEN}
$\exists y(\psi(y)\wedge\theta(x,y))\in{\cal ID}$.
}
\end{Definition}
We call a strong type ${\rm stp}(a/A)$ {\em ${\cal ID}$-internal\/} if there is
a set $B\supseteq A$ independent from $a$ over $A$, a $B$-definable
function $f$, and elements $\bar{c}$ such that ${\rm tp}(c/B)$ is ${\cal ID}$-small
for each $c\in\bar{c}$ and $a=f(\bar{c})$. The strong type ${\rm stp}(a/A)$
is {\em ${\cal ID}$-analyzable\/} if there is a finite sequence $\<a_i:i\le n\>$ from
${\rm dcl}(Aa)$ such that $a_n=a$ and ${\rm stp}(a_i/A\cup\{a_j:j<i\})$ is
${\cal ID}$-internal for each $i\le n$. Since ${\cal ID}$ is a collection of formulas,
this definition of analyzability is equivalent to the usual one, see e.g.,
\cite{Pillay}.
In order to iterate the defining property of a regular ideal, we need the
following notion, whose terminology is borrowed from \cite{HrSh}.
\begin{Definition} {\em A formula $\varphi(x,c)$
is {\em in ${\cal ID}$, provably over $B$\/} if there is
some $\theta(y)\in{\rm tp}(c/B)$ such that $\varphi(x,c')\in{\cal ID}$ for every
$c'$ realizing $\theta$.
}
\end{Definition}
\begin{Lemma} \label{iterate}
For all sets $B$ and every $n\in\omega$, if $\varphi(x,y_0,\dots,y_{n-1})$
is $B$-definable and $a,c_0,\dots,c_{n-1}$ satisfy:
\begin{enumerate}
\item ${\rm tp}(c_i/B)$ is ${\cal ID}$-small for each $i<n$;
\item $\varphi(x,c_0,\dots,c_{n-1})\in{\cal ID}$ provably over $B$; and
\item $\varphi(a,c_0,\dots,c_{n-1})$
\end{enumerate}
then ${\rm tp}(a/B)$ is ${\cal ID}$-small.
\end{Lemma}
{\bf Proof.}\quad Fix any set $B$. We argue by induction on $n$. If $n=0$ the formula
$\varphi(x)$ itself witnesses that ${\rm tp}(a/B)$ is ${\cal ID}$-small.
Assume the result holds for $n$ and fix a formula $\varphi(x,c_0,\dots,c_n)$
and $a,c_0,\dots,c_n$ as in the hypotheses. Choose a formula
$\theta(y_0,\dots,y_n)\in{\rm tp}(c_0\dots c_n/B)$ such that
$\varphi(x,c_0',\dots,c_n')\in{\cal ID}$ for all $c_0'\dots c_n'$ realizing $\theta$
and, using Lemma~\ref{1}, choose $\psi(y_n)\in{\rm tp}(c_n/B)\cap{\cal ID}$.
Let $\theta^*(y_0,\dots,y_n):=\theta(y_0,\dots,y_n)\wedge\psi(y_n)$,
$\theta'(y_0,\dots,y_{n-1}):=\exists y_n\theta^*$ and
$$\varphi'(x,y_0,\dots,y_{n-1}):=\exists y_n(\varphi(x,y_0,\dots,y_n)\wedge\theta^*(y_0,\dots,y_n))$$
We argue that Conditions (1)--(3) are satisfied by
$\varphi'$ and $a,c_0,\dots,c_{n-1}$.
Conditions (1) and (3) are clear. We claim that the formula
$\theta'$ witnesses that $\varphi'(x,c_0,\dots,c_{n-1})\in{\cal ID}$ provably
over $B$. Indeed, it is clear that $\theta'\in{\rm tp}(c_0\dots c_{n-1}/B)$,
so choose $c_0'\dots c_{n-1}'$ realizing $\theta'$.
Since $\psi\in{\cal ID}$ and $\theta^*(c_0',\dots,c_{n-1}',y_n)\vdash\psi$,
$\theta^*(c_0',\dots,c_{n-1}',y_n)\in{\cal ID}$. As well,
for any $c_n'$ such that $\theta^*(c_0',\dots,c_n')$ holds,
we have $\theta(c_0',\dots,c_n')$ holding as well,
so $\varphi(x,c_0',\dots,c_n')\in{\cal ID}$. Thus
$\varphi'(x,c_0',\dots,c_{n-1}')\in{\cal ID}$ since ${\cal ID}$ is a regular ideal.
\medskip
\begin{Proposition} \label{internal}
If ${\rm stp}(a/A)$ is ${\cal ID}$-internal, then
${\rm tp}(a/A)$ is ${\cal ID}$-small.
\end{Proposition}
{\bf Proof.}\quad Choose $B\supseteq A$ independent from $a$ over $A$,
a $B$-definable formula $\varphi(x,\bar{y})$, and
a tuple of elements
$\bar{c}$ such that each ${\rm tp}(c/B)$ is ${\cal ID}$-small for each $c\in\bar{c}$,
$\varphi(a,\bar{c})$ holds, and $\exists^{= 1}x\varphi(x,\bar{c})$.
But the formula $\varphi(x,\bar{c})\in{\cal ID}$ provably over $B$ via the formula
$\exists^{=1}x\varphi(x,\bar{y})$, so ${\rm tp}(a/B)$ is ${\cal ID}$-small
by Lemma~\ref{iterate}. That ${\rm tp}(a/A)$ is ${\cal ID}$-small
follows from Lemma~\ref{1}.
\medskip
The reader is cautioned that while ${\cal ID}$-internal types are
${\cal ID}$-small, this result does not extend to ${\cal ID}$-analyzable types.
In fact, the theory and type mentioned in Remark~8.1.6 of \cite{Pillay}
gives rise to an example of this. Much of the motivation of
this section, and in particular how it differs from
treatments in \cite{Hr1} and \cite{Pillay},
revolves around how we handle ${\cal ID}$-analyzable types that are
not ${\cal ID}$-small.
\begin{Definition} \label{foreign} {\em
A strong type $p$ is {\em foreign to ${\cal ID}$,\/} written
$p\perp{\cal ID}$, if $p\perp q$ for every ${\cal ID}$-small $q$.
}
\end{Definition}
\begin{Lemma} \label{charforeign}
The following are equivalent for any regular ideal ${\cal ID}$ and
any strong type $p$:
\begin{enumerate}
\item $p\perp {\cal ID}$;
\item $p\perp q$ for every ${\cal ID}$-internal strong type $q$;
\item $p\perp q$ for every ${\cal ID}$-analyzable strong type $q$;
\item If $p={\rm stp}(a/A)$ then there is no $a'\in{\rm dcl}(Aa)$
such that ${\rm tp}(a'/A)$ is ${\cal ID}$-small.
\end{enumerate}
\end{Lemma}
{\bf Proof.}\quad $(1)\Rightarrow (2)$ follows immediately from
Proposition~\ref{internal}. $(2)\Rightarrow(3)$
follows by induction on the length of the ${\cal ID}$-analysis,
using the fact that $p\perp {\rm tp}(b/B)$ and $p\perp {\rm tp}(a/Bb)$
implies $p\perp{\rm tp}(ab/B)$.
$(3)\Rightarrow(4)$ is trivial, and $(4)\Rightarrow(1)$
follows immediately from (say) Corollary~7.4.6 of \cite{Pillay}.
\medskip
The reader is cautioned that when the regular ideal is not
closed under ${\cal ID}$-analyzability, these definitions differ from
those in \cite{Pillay}.
\begin{Definition} {\em A partial type $\Gamma$ is {\em ${\cal ID}$-large\/}
if it is not ${\cal ID}$-small. $\Gamma$ is {\em ${\cal ID}$-minimal\/} if
it is ${\cal ID}$-large, but any forking extension of $\Gamma$
is ${\cal ID}$-small.
$\Gamma$ is {\em ${\cal ID}_\perp$-minimal\/} if it is ${\cal ID}$-large, but
any forking extension $\Gamma\cup\{\theta(x,c)\}$ is ${\cal ID}$-small whenever
${\rm stp}(c/{\rm dom}(\Gamma))\perp{\cal ID}$.
}
\end{Definition}
Clearly ${\cal ID}$-minimality implies ${\cal ID}_\perp$-minimality,
but one of the applications in Section~\ref{app} will use ${\cal ID}_\perp$-minimal
types that are not ${\cal ID}$-minimal.
\begin{Lemma} \label{regular} Let ${\cal ID}$ be any regular ideal.
If a strong type $p$ is both ${\cal ID}_\perp$-minimal and foreign to ${\cal ID}$,
then $p$ is regular.
\end{Lemma}
{\bf Proof.}\quad The point is that a counterexample to the regularity of
$p$ can be found within the set of realizations of $p$.
If $M$ is a-saturated and
$p={\rm tp}(a/M)$ is not regular then there are
a tuple
$\bar{c}=\<c_1,\dots,c_n\>$ realizing $p^{(n)}$
for some $n$ and a realization $b$ of $p$
such that ${\rm tp}(a/M\bar{c})$ forks over $M$,
${\rm tp}(b/M\bar{c})$ does not fork over $M$,
and ${\rm tp}(b/M\bar{c} a)$ forks over $M\bar{c}$.
Let $q={\rm tp}(a/M\bar{c})$ and
choose an $L(M)$-formula $\theta(x,\bar{c})\in q$ such that
$p\cup\{\theta(x,\bar{c})\}$ forks over $M$.
As $p\perp{\cal ID}$, $p^{(n)}\perp{\cal ID}$, so the ${\cal ID}_\perp$-minimality
of $p$ implies ${\rm tp}(a/M\bar{c})$ is ${\cal ID}$-small.
But, since $p$ is foreign
to ${\cal ID}$, ${\rm tp}(b/M\bar{c})$, which is a nonforking extension of $p$
would be orthogonal to $q$ by Lemma~\ref{charforeign}(2). In particular,
${\rm tp}(b/M\bar{c} a)$ would not fork over $M\bar{c}$.
\medskip
The following easy `transfer result' will be used in the
subsequent sections.
\begin{Lemma} \label{transfer}
Assume that $B$ is algebraically closed, $p={\rm tp}(a/B)$ is foreign
to ${\cal ID}$, $q={\rm tp}(b/B)$, and $b\in{\rm acl}(Ba)\setminus B$.
Then $q$ is foreign to ${\cal ID}$. If, in addition, $p$ is ${\cal ID}$-minimal
(${\cal ID}_\perp$-minimal) then $q$ is ${\cal ID}$-minimal (${\cal ID}_\perp$-minimal)
as well.
\end{Lemma}
{\bf Proof.}\quad If $q$ were not foreign to ${\cal ID}$, then by Lemma~\ref{charforeign}(4)
there is $c\in{\rm dcl}(Bb)\setminus B$ such that ${\rm tp}(c/B)$ is ${\cal ID}$-small.
Since ${\rm tp}(c/B)$ is not algebraic it is not orthogonal to $p$,
which, via Lemma~\ref{charforeign}(2), contradicts $p$ being foreign to ${\cal ID}$.
Thus $q\perp{\cal ID}$.
Next, suppose that $p$ is ${\cal ID}$-minimal. Since $p\not\perp q$ and $p\perp {\cal ID}$,
$q$ cannot be ${\cal ID}$-small. To see that $q$ is ${\cal ID}$-minimal, choose $C\supseteq B$
such that ${\rm tp}(b/C)$ forks over $B$. Then ${\rm tp}(a/C)$ forks over $B$, so ${\rm tp}(a/C)$
is ${\cal ID}$-small. Thus ${\rm tp}(b/C)$ is ${\cal ID}$-small by Lemma~\ref{iterate}.
\medskip
\section{Chains and witnessing groups}
Throughout this section ${\cal ID}$ always denotes a regular ideal.
\begin{Definition} {\em We say {\em $A$ is an ${\cal ID}$-subset of $B$,\/}
written $A\subseteq_{{\cal ID}} B$, if $A\subseteq B$ and ${\rm stp}(b/A)\perp{\cal ID}$
for every finite tuple $b$ from $B$.
When $M$ and $N$ are models we
write $M\preceq_{{\cal ID}} N$ when both $M\preceq N$ and
$M\subseteq_{{\cal ID}} N$.
A set $A$ is {\em ${\cal ID}$-full\/} if $A\subseteq_{{\cal ID}} M$ for some
(equivalently for every) a-prime model $M$ over $A$.
}
\end{Definition}
\begin{Lemma} \label{ext} Let ${\cal ID}$ be any regular ideal and assume $M$ is a-saturated.
\begin{enumerate}
\item If $M\preceq N$ are models then $M\preceq_{{\cal ID}} N$ if and only if
$\varphi(N)=\varphi(M)$ for all $\varphi\in L(M)\cap{\cal ID}$.
\item If $M\subseteq_{{\cal ID}} A$, then $M\preceq_{{\cal ID}} M[A]$, where
$M[A]$ is any a-prime model over $M\cup A$.
\end{enumerate}
\end{Lemma}
{\bf Proof.}\quad (1) First suppose $M\preceq_{{\cal ID}} N$ and choose $\varphi\in L(M)\cap{\cal ID}$.
If $c\in\varphi(N)$ then ${\rm tp}(c/N)$ is ${\cal ID}$-small. If ${\rm tp}(c/M)$ were not algebraic,
it would be nonorthogonal to an ${\cal ID}$-small type, contradicting ${\rm tp}(c/M)\perp{\cal ID}$.
So ${\rm tp}(c/M)$ is algebraic, hence $c\in\varphi(M)$. Conversely, if there
were $c\in N$ such that ${\rm tp}(c/M)\not\perp {\cal ID}$, then by Lemma~\ref{charforeign}(4)
there is $c'\in{\rm dcl}(Mc)\setminus M$ such that ${\rm tp}(c'/M)$ is ${\cal ID}$-small.
Then $\varphi(N)\neq \varphi(M)$ for any $\varphi\in{\rm tp}(c'/M)\cap{\cal ID}$.
(2) Recall that because $M$ is a-saturated,
$M[A]$ is dominated by $A$ over $M$.
Choose any tuple $c$ from $M[A]$. If ${\rm tp}(c/M)$ were not foreign to ${\cal ID}$,
then as $M$ is a-saturated, there is an ${\cal ID}$-small type $q\in S(M)$ such that
${\rm tp}(c/M)\not\perp q$, hence ${\rm tp}(c/M)$ is not almost orthogonal to $q$.
Since $c$ is dominated by $A$ over $M$, there is $a$ from $A$ such that
${\rm tp}(a/M)$ is not almost orthogonal to $q$, which contradicts $M\subseteq_{{\cal ID}} A$.
\medskip
\begin{Definition} {\em
A {\em saturated chain\/} is an elementary chain
$\<M_\alpha:\alpha<\delta\>$
of a-saturated models in which $M_{\alpha+1}$ realizes every complete
type over $M_\alpha$ for each $\alpha<\delta$.
An {\em ${\cal ID}$-chain\/} is a sequence
$\<M_\alpha:\alpha<\delta\>$ of a-saturated models
such that $M_\alpha\preceq_{{\cal ID}} M_\beta$ for all
$\alpha<\beta<\delta$ and $M_{\alpha+1}$
realizes every type over $M_\alpha$ foreign to ${\cal ID}$.
A chain (of either kind) is {\em ${\cal ID}$-full\/} if
the union
$\bigcup_{\alpha<\delta} M_\alpha$ is an ${\cal ID}$-full set.
}
\end{Definition}
In general, a saturated chain need not be ${\cal ID}$-full. However,
if ${\cal ID}$ is either
the ideal of algebraic formulas or superstable formulas (both of which are
regular),
then any a-saturated chain is ${\cal ID}$-full, since types are based on
finite sets. A more complete explanation of this is given in the proof of
Lemma~\ref{ri}.
By contrast, the
following Lemma demonstrates that ${\cal ID}$-chains are always ${\cal ID}$-full.
\begin{Lemma} \label{fullness}
Every ${\cal ID}$-chain is full. That is,
if $\<M_\alpha:\alpha<\delta\>$ is an ${\cal ID}$-chain, $\delta$ is
a nonzero limit ordinal, and $M_\delta$ is
a-prime over $\bigcup_{\alpha<\delta} M_\alpha$, then $M_\alpha\preceq_{{\cal ID}} M_\delta$
for all $\alpha<\delta$.
\end{Lemma}
{\bf Proof.}\quad By the characterization of $M\preceq_{{\cal ID}} N$
given by Lemma~\ref{ext}(1), the first sentence follows from the second.
So fix an ${\cal ID}$-chain $\<M_\alpha:\alpha<\delta\>$.
Let $N=\bigcup_{\alpha<\delta} M_\alpha$ and let $M_\delta$ be a-prime over $N$.
Fix any $\alpha<\delta$. Since $M_\alpha\subseteq_{{\cal ID}} M_\beta$ for all
$\alpha<\beta<\delta$, $M_\alpha\subseteq_{{\cal ID}} N$, so $M_\alpha\preceq_{{\cal ID}} M_\delta$
by Lemma~\ref{ext}(2).
\medskip
\begin{Definition} {\em A formula $\theta$ is {\em weakly ${\cal ID}$-minimal
(weakly ${\cal ID}_\perp$-minimal)\/}
if $\{\theta\}$ is ${\cal ID}$-minimal (${\cal ID}_\perp$-minimal).
}
\end{Definition}
We now state offer two complementary propositions.
The main point of both is that they produce regular types
that are `close' to a given regular ideal.
The advantage of (1) is that one obtains ${\cal ID}$-minimality
at the cost of requiring the chain to be ${\cal ID}$-full. In
(2) the fullness condition is automatically satisfied by Lemma~\ref{fullness},
but one only gets ${\cal ID}_\perp$-minimality.
\begin{Proposition} \label{split} Fix a regular ideal ${\cal ID}$, a countable,
stable theory $T$, and an ${\cal ID}$-large formula $\varphi$.
\begin{enumerate}
\item
{\bf Either} there is a weakly ${\cal ID}$-minimal formula
$\psi\vdash\varphi$
{\bf or} for every ${\cal ID}$-full saturated chain $\<M_n:n\in\omega\>$
with $\varphi\in L(M_0)$, there is an $\aleph_1$-isolated,
${\cal ID}$-minimal $p\in S(\bigcup_n M_n)$
with $\varphi\in p$ and $p\perp{\cal ID}$.
\item
{\bf Either} there is a weakly ${\cal ID}_\perp$-minimal formula
$\psi\vdash\varphi$
{\bf or} for every ${\cal ID}$-chain $\<M_n:n\in\omega\>$
with $\varphi\in L(M_0)$, there is an $\aleph_1$-isolated,
${\cal ID}_\perp$-minimal $p\in S(\bigcup_n M_n)$
with $\varphi\in p$ and $p\perp{\cal ID}$.
\end{enumerate}
Moreover, in either of the two `second cases' the type $p$ is regular.
\end{Proposition}
{\bf Proof.}\quad Assume that there is no weakly ${\cal ID}$-minimal $\psi\vdash\varphi$.
Fix
an ${\cal ID}$-full saturated chain $\<M_n:n\in\omega\>$
with $\varphi\in L(M_0)$, let $N=\bigcup_{n\in\omega} M_n$,
and let $M_\omega$ be $\aleph_1$-prime over $N$. Let
$\Delta_0\subseteq\Delta_1\subseteq\dots$ be finite sets of
formulas with $L=\bigcup_{n\in\omega} \Delta_n$.
We inductively construct a sequence $\<\varphi_n:n\in\omega\>$
of ${\cal ID}$-large formulas as follows:
Let $\varphi_0$ be our given $\varphi$.
Given $\varphi_n\vdash\varphi_0$ that is an ${\cal ID}$-large $L(M_n)$-formula
$$A_n=\{\psi\in L(M_{n+1}): \psi\vdash \varphi_n,
\ \psi\ \hbox{is}\ {\cal ID}\hbox{-large and
forks over}\ M_n\}.$$
As $M_{n+1}$ realizes every type over $M_n$ foreign to ${\cal ID}$
and $\varphi_n$ is not
weakly ${\cal ID}$-minimal, $A_n$ is nonempty.
Choose $\varphi_{n+1}\in A_n$ so as to minimize
$R(\psi,\Delta_n,2)$.
Let $\Gamma=\{\varphi_n:n\in\omega\}$. We first argue that
$\Gamma$ has a unique extension to a complete type in $S(N)$.
\medskip
{\bf Claim.} $\Gamma\vdash\neg\psi(x,b)$ for all
$\psi(x,b)\in {\cal ID}\cap L(N)$.
\medskip
{\bf Proof.}\quad
If the Claim were to fail, then $\Gamma\cup\{\psi(x,b)\}$
would be consistent,
hence would be realized in $M_\omega$, say by an element $c$.
As the chain is ${\cal ID}$-full, $c\in N$.
Choose $n$ such that
$b,c\in M_n$. But $\varphi_{n+1}$ was chosen to fork over
$M_n$, yet is realized in $M_n$, which is impossible.
\medskip
Now let $\psi(x,b)$ be any $L(N)$-formula.
Choose $n$ such that $\psi(x,y)\in\Delta_n$.
As $\varphi_{n+1}$ was chosen to be of minimal R(--,$\Delta_n,2)$-rank,
it is not possible for both $\varphi_{n+1}\wedge\psi(x,b)$ and
$\varphi_{n+1}\wedge\neg\psi(x,b)$ to be in $A_n$. As
${\cal ID}$ is an ideal, at least one of the two of them is ${\cal ID}$-large,
so is an element of $A_n$, thus the other one is ${\cal ID}$-small or
inconsistent. Using the Claim, either $\Gamma\vdash\psi(x,b)$
or $\Gamma\vdash\neg\psi(x,b)$.
Thus $\Gamma$ implies a complete type in $S(N)$, which we
call $p$.
By construction $p$ is $\aleph_1$-isolated
and is ${\cal ID}$-large by the Claim. Since $M_\omega$ is $\aleph_1$-saturated and $p$ is
$\aleph_1$-isolated, there is a realization $c$ of $p$ in $M_\omega$.
If $p$ were not foreign to ${\cal ID}$ then by Lemma~\ref{charforeign}(4) there would be
$c'\in{\rm dcl}(Nc)\setminus N$ with $c'/N$ ${\cal ID}$-small, directly
contradicting ${\cal ID}$-fullness.
It remains to show that any forking extension of $p$ is ${\cal ID}$-small.
let $\theta(x,a^*)$ be any $L({\frak C})$-formula such that $p\cup\theta(x,a^*)$
forks over $M_\omega$. Then for some $n$,
$\varphi_{n+1}\wedge\theta(x,a^*)$ $\Delta_n$-forks over $M_n$.
As $M_{n+1}$ realizes all types over $M_n$
there is $a'\in M_{n+1}$ such that ${\rm tp}(a'/M_n)={\rm tp}(a^*/M_n)$.
But then $\varphi_{n+1}\wedge\theta(x,a')$ $\Delta_n$-forks over
$M_n$, contradicting the
minimality of R(--,$\Delta_n,2)$ rank of $\varphi_{n+1}$.
As for (2) assume that there is no ${\cal ID}_\perp$-minimal formula
implying $\varphi$. Choose an ${\cal ID}$-chain $\<M_n:n\in\omega\>$,
which is automatically ${\cal ID}$-full by Lemma~\ref{fullness}.
The definition of $\{A_n\}_{n\in\omega}$ and the constructions of
$\Gamma$ and $p$ remain the same. All that is affected is that
in the final paragraph, as we only need to establish ${\cal ID}_\perp$-minimality,
one chooses a formula $\theta(x,a^*)$ with ${\rm tp}(a^*/N)\perp{\cal ID}$.
By Lemma~\ref{charforeign}(4) this implies ${\rm tp}(a^*/M_n)\perp{\cal ID}$
for all $n\in\omega$, so choosing $n$ as above, one obtains
$a'\in M_{n+1}$ satisfying ${\rm tp}(a'/M_n)={\rm tp}(a^*/M_n)$ and
a similar contradiction is obtained.
In both cases, the regularity of $p$ follows immediately from Lemma~\ref{regular}.
\medskip
Recall that a stable theory has {\em NDIDIP\/} if for every elementary
chain $\<M_n:n\in\omega\>$ of models, every type that is nonorthogonal
to some a-prime model over $\bigcup_{n\in\omega} M_n$ is nonorthogonal to some
$M_n$. Relationships between NDIDIP and NDOP are explored in \cite{LSh}.
\begin{Proposition} \label{group}
Fix a countable, stable theory $T$ with NDIDIP and a regular ideal ${\cal ID}$
such that the formula `$x=x$'$\not\in{\cal ID}$.
\begin{enumerate}
\item If there is an
an ${\cal ID}$-full, saturated chain
$\<M_n:n\in\omega\>$, but there is no weakly ${\cal ID}$-minimal formula then
there is an abelian group witness to unsuperstability, where in addition
the generic type of the intersection
is both ${\cal ID}$-minimal and foreign to ${\cal ID}$.
\item If there is no weakly ${\cal ID}_\perp$-minimal formula then
there is an abelian group witness to unsuperstability where the generic
type of the intersection is ${\cal ID}_\perp$-minimal and foreign to ${\cal ID}$.
\end{enumerate}
\end{Proposition}
{\bf Proof.}\quad (1) Fix an ${\cal ID}$-full, saturated chain $\<M_n:n\in\omega\>$ and let
$N=\bigcup_{n\in\omega} M_n$.
Using Proposition~\ref{split}(1)
choose $p\in S(N)$ to be $\aleph_1$-isolated, foreign to ${\cal ID}$, and ${\cal ID}$-minimal, hence
regular. Since $T$ has NDIDIP, $p\not\perp M_n$. Since $p$ is regular
and $M_n$ is a-saturated, by Claim~X~1.4 of \cite{Shc}
there is a regular type $r_0\in S(M_n)$ nonorthogonal to $p$.
Let $r$ denote the nonforking
extension of $r_0$ to $N$.
As $p$ and $r$ are nonorthogonal there is an integer
$m$ such that $p^{(m)}$ is not almost
orthogonal to $r^{(\omega)}$.
Since $p$ is $\aleph_1$-isolated and $M_n$ is
a-saturated, $Na$ is dominated by $N$ over $M_n$ for any $a$ realizing $p$.
Thus $p^{(1)}$ is not almost orthogonal to $r^{(\omega)}$ over $N$.
Choose $k\ge 1$ maximal such that
$p^{(k)}$ is almost orthogonal to $r^{(\omega)}$ over
$N$ and choose $\bar{c}$ realizing $p^{(k)}$. Let
$B={\rm acl}(N\bar{c})$ and choose a realization
$a$ of the nonforking extension of $p$ to $B$.
By Theorem~1 of \cite{Hr2}, there is $b\in{\rm dcl}(Ba)\setminus B$
and a type-definable, connected group
$A$ with a regular generic type $q$
(so $A$ is abelian by Poizat's theorem~\cite{Poizat})
and a definable regular, transitive action of $A$ on $p_1({\frak C})$, where
$p_1={\rm tp}(b/B)$. By Lemma~\ref{transfer} the type $p_1$ and hence $q$ are both
foreign to ${\cal ID}$ and ${\cal ID}$-minimal. By Theorem~2 of \cite{Hr3} there is
a definable supergroup $A_0\supseteq A$. By an easy compactness argument we may
assume $A_0$ is abelian as well. Furthermore, by iterating Theorem~2 of
\cite{Hr3} we obtain a descending sequence $\<A_n:n\in\omega\>$ of subgroups of $A_0$
with $A=\bigcap_{n\in\omega} A_n$.
Thus far we have not guaranteed that $A_{n+1}$ has infinite index in $A_n$.
In order to show that there is a subsequence of the $A_n$'s with this property
and thereby complete the proof of the Proposition,
it suffices to prove the following claim:
\medskip
{\bf Claim} For every $n\in\omega$ there is $m\ge n$ such that $[A_n:A_m]$ is infinite.
\medskip
{\bf Proof.}\quad By symmetry it suffices to show this for $n=0$. Assume that this were not the
case, i.e., that $[A_0,A_m]$ is finite for each $m$. Then $A$ has bounded index in $A_0$.
We will obtain a contradiction by showing that the definable set $A_0$ is
weakly ${\cal ID}$-minimal.
First, since $q$ is ${\cal ID}$-large, the formula defining $A_0$
is ${\cal ID}$-large as well.
Let $\varphi(x,e)$ be any forking extension of the formula defining $A_0$
and let $E\subseteq A_0$ be the set of realizations of $\varphi(x,e)$.
Let $\{C_i:i<2^\kappa\le 2^{\aleph_0}\}$
enumerate the $A$-cosets of $A_0$.
For each $i$, $E\cap C_i$ is a forking extension of $C_i$.
Since every $C_i$ is a translate of $A$ whose generic type is
${\cal ID}$-minimal, this implies that $E\cap C_i$ is ${\cal ID}$-small for
each $i$. Thus $\varphi(x,e)\in{\cal ID}$ by compactness (and the fact
that ${\cal ID}$ is an ideal). Thus, the formula defining $A_0$
is weakly ${\cal ID}$-minimal, contradiction.
The proof of (2) is identical, choosing an ${\cal ID}$-chain satisfying
the hypotheses and using Proposition~\ref{split}(2)
in place of \ref{split}(1).
\medskip
\section{Applications} \label{app}
Our first application gives a `trichotomy' for strictly stable
theories in a countable language. It uses the
ideal of superstable formulas. Let ${\bf R}^\infty$ denote the ideal of
\begin{Definition} \label{ss} {\em
${\bf R}^\infty$ denotes the ideal of
superstable formulas (i.e., all formulas $\varphi$ with $R^\infty(\varphi)<\infty$).
}
\end{Definition}
Equivalently, $\varphi\in{\bf R}^\infty$ if and only if for all cardinals
$\kappa\ge 2^{|T|}$, for any model $M$ of size $\kappa$
containing the parameters of $\varphi$, there are at most
$\kappa$ complete types over $M$ extending $\varphi$.
\begin{Lemma} \label{ri}
${\bf R}^\infty$ is a regular ideal,
any elementary chain $\<M_n:n\in\omega\>$ of a-saturated models is
${\bf R}^\infty$-full, and there are no weakly ${\bf R}^\infty$-minimal formulas.
\end{Lemma}
{\bf Proof.}\quad Invariance under automorphisms of ${\frak C}$ is clear and ${\bf R}^\infty$ being
an ideal follows by counting types. To show regularity,
choose $\psi(y)\in{\bf R}^\infty$ and $\theta(x,y)\in L({\frak C})$ such that
$\theta(x,b)\in{\bf R}^\infty$ for every $b$ realizing $\psi$.
Choose $\kappa\ge 2^{|T|}$ and a model $M$ of size $\kappa$
containing the hidden parameters of both $\psi$ and $\theta$.
Then there are at most $\kappa$ types $p(x,y)\in S(M)$ extending
$\theta(x,y)\wedge\psi(y)$, so the projection
$\exists y(\theta(x,y)\wedge\psi(y))\in{\bf R}^\infty$ as only $\kappa$
types $q(x)\in S(M)$ extend it.
To establish fullness, fix an elementary chain $\<M_n:n\in\omega\>$
of a-saturated models.
Let $N=\bigcup_{n\in\omega} M_n$ and choose an a-prime model $M_\omega$
over $N$. Because of Lemma~\ref{charforeign}(4), in order to show
that $N\subseteq_{{\cal ID}} M_\omega$ it suffices to show that
no element of
$c\in M_\omega\setminus N$ is ${\bf R}^\infty$-small. So choose
$c\in M_\omega$ such that ${\rm tp}(c/N)$ is ${\cal ID}$-small and we will show
that $c\in N$.
On one hand, since ${\rm tp}(c/N)$ contains a superstable formula
there is a finite $n$ such that ${\rm tp}(c/N)$ is based on $M_n$.
On the other hand, since $M_\omega$ is a-prime over $N$,
${\rm tp}(c/N)$ is a-isolated.
Thus ${\rm tp}(c/M_n)$ is a-isolated as well (see
e.g., Theorem~IV~4.3(1) of \cite{Shc}).
Since $M_n$ is a-saturated, this implies $c\in M_n\subseteq N$.
To show that there are no weakly ${\bf R}^\infty$-minimal formulas,
suppose that a formula $\varphi$ has the property that any forking
extension of $\varphi$ is ${\bf R}^\infty$-small.
We will show that $\varphi\in{\bf R}^\infty$ by counting types.
Fix a cardinal $\kappa\ge 2^{|T|}$ and a model $M$ of size $\kappa$
containing the parameters of $\varphi$. Let $M_0\preceq M$
have size $|T|$ that also contains the parameters containing $\varphi$.
It suffices to show that every $p\in S(M_0)$ extending $\varphi$
has at most $\kappa$ extensions to types in $S(M)$.
Clearly, there is a unique nonforking extension of $p$
and any forking extension of $p$ contains an $L(M)$-formula
witnessing the forking. Each such forking formula $\psi\in{\bf R}^\infty$,
so there are at most $\kappa$ $q\in S(M)$ extending $\psi$.
So, since the total number of $\psi\in L(M)$ is at most $\kappa$,
$p$ has at most $\kappa$ extensions to types in $S(M)$.
\medskip
\begin{Theorem}
Let $T$ be a stable, unsuperstable theory in a countable language.
Then at least one of the following three conditions occurs:
\begin{enumerate}
\item $T$ has the dimensional order property (DOP); or
\item $T$ has NDOP, but is deep (i.e., there is a sequence
$\<M_n:n\in\omega\>$ such that ${\rm tp}(M_{n+1}/M_n)\perp M_{n-1}$
for all $n\ge 1$); or
\item There is an abelian group witness to unsuperstability
(see Definition~\ref{witness})
in which the generic type of the intersection is both
${\bf R}^\infty$-minimal and foreign to ${\bf R}^\infty$.
\end{enumerate}
\end{Theorem}
{\bf Proof.}\quad
To begin, Corollary 1.12 of \cite{LSh} asserts
that any such theory $T$ has NDIDIP.
Since $T$ is not superstable the formula `$x=x$'$\not\in{\bf R}^\infty$.
As well, by Lemma~\ref{ri} there are no weakly
${\bf R}^\infty$-minimal formulas, so
Proposition~\ref{group}(1) asserts that an abelian group
witness to unsuperstability exists, whose generic type is
regular and
both ${\bf R}^\infty$-minimal and foreign to ${\bf R}^\infty$.
\medskip
Our second application comes from an attempt to solve the
`Main Gap for $\aleph_1$-saturated models.' As in the previous
theorem, the relevant setting is where a countable theory $T$
is stable, unsuperstable, with NDOP, and is shallow.
The main open question is whether, for such a theory
every nonalgebraic type $r$ is nonorthogonal to a regular type.
The following result sheds some light on this issue.
In order to analyze this problem, fix a nonalgebraic, stationary type $r$ over the
empty set. Let
$${\cal ID}_r=\{\varphi\in L({\frak C}):r\perp \varphi\}$$
Verifying that ${\cal ID}_r$ is an invariant ideal is straightforward.
To see that it is a regular ideal, fix $L({\frak C})$-formulas $\psi(y)\in{\cal ID}_r$
and $\theta(x,y)$ such that $\theta(x,b)\in{\cal ID}_r$ for every $b$ realizing
$\psi$. Choose an a-saturated model $M$ containing the parameters of
$\psi$ and $\theta$, pick a realization $c$ of the nonforking
extension of $r$ to $M$, and let $M[c]$ be
any a-prime model over $Mc$.
To show that $\varphi(x):=\exists y(\theta(x,y)\wedge\psi(y))\perp r$ it
suffices to prove that any realization of $\varphi$ in $M[c]$ is contained
in $M$. So choose any $a\in\varphi(M[c])$. Choose $b\in M[c]$
such that $\theta(a,b)\wedge \psi(b)$ holds. Since $r\perp\psi$, $b\in M$.
But then $\theta(x,b)$ is over $M$ and is $\perp r$, so $a\in M$ as well.
Thus ${\cal ID}_r$ is a regular ideal.
\begin{Theorem} \label{rperp} Assume that a countable
theory $T$ is stable, unsuperstable,
has NDOP, and is shallow. If a nonalgebraic,
stationary type $r$ is orthogonal to every regular
type, then there is an abelian group witness to unsuperstability in which the
generic type of the intersection $A=\bigcup_n A_n$ is both $({\cal ID}_r)_\perp$-minimal
and foreign to ${\cal ID}_r$.
\end{Theorem}
{\bf Proof.}\quad Fix such a type $r$. By naming constants we may assume that $r$ is over
the empty set. Note that any formula $\varphi\in r$ is not an element of ${\cal ID}_r$,
so `$x=x$'$\not\in{\cal ID}_r$.
\medskip
{\bf Claim.} There is no weakly $({\cal ID}_r)_\perp$-minimal formula.
\medskip
{\bf Proof.}\quad Assume that $\varphi$ were $({\cal ID}_r)_\perp$-minimal. We construct
a regular type $p\not\perp r$ as follows: Choose an a-saturated model
$M$ containing the parameters in $\varphi$, pick a realization $c$ of
the nonforking extension of $r$ to $M$, and choose
an a-prime model $M[c]$ over $Mc$. Since $\varphi$ is ${\cal ID}_r$-large
we can find an $a\in M[c]\setminus M$ realizing $\varphi$. Choose such an $a$
and let $p={\rm tp}(a/M)$. Clearly, $p\not\perp r$. To see that $p$ is regular,
first note that $p$ is $({\cal ID}_r)$-minimal
since $p$ is ${\cal ID}_r$-large and extends $\varphi$.
As well, $p$ is foreign to ${\cal ID}_r$, since if it were not, then by
Lemma~\ref{charforeign}(4) there would be $b\in{\rm dcl}(Ma)$ with ${\rm tp}(b/M)$
${\cal ID}_r$-small. But then $tp(c/Mb)$ would fork over $M$, implying that
$r$ is nonorthogonal to an ${\cal ID}_r$-small type, which is a contradiction.
So $p$ is $({\cal ID}_r)$-minimal and foreign to ${\cal ID}_r$, hence is regular
by Lemma~\ref{regular}.
\medskip
The theorem now follows immediately from Proposition~\ref{split}(2).
\medskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,969 |
Печера № 1 — геологічна пам'ятка природи місцевого значення. Об'єкт розташований на території Старобешівського району Донецької області, на північ від села Стила.
Площа — 0,5 га, статус отриманий у 1984 році.
Примітки
Джерела
Донбас заповідний. Науково-інформаційний довідник-атлас / за заг. ред. С.С. Куруленка, С.В. Третьякова. Видання друге, перероблене та доповнене. – Донецьк, Донецька філія Державного екологічного інституту Мінприроди України, 2008. – 168 с.
Заповедная природа Донбасса: Путеводитель / Сост. А.З. Дидова. - 2 изд., доп. - Донецк: Донбасс, 1987 - 168 с.
Геологічні пам'ятки природи Донецької області
Природоохоронні об'єкти, засновані 1984 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,499 |
<?php
namespace app\constants;
use Yii;
/**
*
* A News Type Class
* Author: Longbin
* Date: 2015/07/05
*
*/
class NewsType
{
const NEWS_TYPE_PAPER_ACCEPT = 1;
const NEWS_TYPE_PROJECT_INIT = 2;
const NEWS_TYPE_PEOPLE_JOIN = 3;
const NEWS_TYPE_PEOPLE_VISIT = 4;
const NEWS_TYPE_EVENT_HOST = 5;
public $id;
public $label;
public function __construct($id = null)
{
if ($id !== null) {
$this->id = $id;
$this->label = $this->getLabel($id);
}
}
public static function labels()
{
return [
self::NEWS_TYPE_PAPER_ACCEPT => Yii::t('main', 'Paper Acceptance'),
self::NEWS_TYPE_PROJECT_INIT => Yii::t('main', 'Project Initiation'),
self::NEWS_TYPE_PEOPLE_JOIN => Yii::t('main', 'People Join'),
self::NEWS_TYPE_PEOPLE_VISIT => Yii::t('main', 'People Visit'),
self::NEWS_TYPE_EVENT_HOST => Yii::t('main', 'Event Host'),
];
}
public static function getType($id)
{
$labels = self::labels();
return isset($labels[$id]) ? $labels[$id] : null;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,130 |
Оттомар Эллигер III (Ottomar Elliger III, 1703, Амстердам — 1735, Санкт-Петербург) — голландский рисовальщик и гравёр.
Один из сыновей и учеников Эллигера, Оттомара II (1666—1732), голландского живописца и гравёра, прибывшего в Санкт-Петербург для работы вИмператорской Санкт-Петербургской академии наук. Сам Оттомар Эллигер III с 1727 г. работал в Санкт-Петербурге, с 1728 г. — в Академии наук. Числился в конторе садовых дел. В качестве «гравера прошпектов и архитектуры» гравировал виды Петербурга, в том числе по рисункам Г. Марселиуса. Сочинял книжные иллюстрации. С 1731 г. руководил Гравировальной палатой, обучал русских гравёров: И. А. Соколова, Г. А. Качалова, рисовальщиков Ф. Маттарнови и М. И. Махаева.
Примечания
Литература
Власов В. Г. Стили в искусстве. В 3-х т. — СПб.: Кольна, Т. 3. Словарь имен, 1997. — С. 552—553.
Гравировальная палата Академии наук XVIII века: сборник документов / сост. М. А. Алексеева, Ю. А. Виноградов, Ю. А. Пятницкий. Л.: Наука, 1985. 293 с.
Записки Якоба Штелина об изящных искусствах в России: в 2-х т. / сост., пер. с нем., вступ. ст., предисл. к разделам и прим. К. В. Малиновского. М.: Искусство, 1990. Т. 1. 447 с.
Гравёры Нидерландов
Художники-педагоги
Художники Нидерландов | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,071 |
Alain Bonnet, más conocido como Alain Soral, es un ensayista, editor, instructor de boxeo francés relacionado con ideas extremas tanto de izquierda y derecha, llegando a ser calificado como «nacional-bolchevique». Tiene nacionalidad francesa y suiza.
Soral fue miembro activo del Partido Comunista Francés en la décadas de los 90s. Al mismo tiempo evolucionó en el mundo de los medios y el entretenimiento hasta principios de los 2000s. Soral abandonó el PCF, al que acusó de estar lleno de trotskistas encubiertos, y fue progresivamente cambiando su pensamiento hasta adoptar ideas consideradas de extrema derecha con tendencias acusadas de antisemitas. Durante estos años Soral pasa a militar en el Frente Nacional de Marine Le Pen y a desarrollar sus propios postulados. Soral afirma sentirse tanto nacionalista y conservador como de izquierdas, especialmente marxista, motivo por el cual ha sido calificado por numerosos medios de nacionalbolchevique francés. Durante estos años entabla amistad con el polémico humorista Dieudonné, llegando a convertirse en una especie de eminencia gris en sus discursos satíricos y actuaciones como comediante.
Progresivamente comienza alejarse del FN hasta abandonarlo en 2009. Tras esto, Soral, junto a exmiembros del Groupe union défense y disidentes del PCF, fundan un nuevo movimiento conocido como Égalité et Réconciliation. Dicha organización se describe a sí misma como nacionalista en lo moral y marxista en lo económico, autodefiniéndose como de «derecha» en los valores (patria, familia, tradición) e «izquierda» en lo económico (Socialismo, lucha de clases, anticapitalismo, marxismo). El pensamiento de la organización se encuentra influido tanto por autores nacionalrevolucionarios, pasando por los clásicos, tanto marxistas como fascistas.
E&R ha sido descrita por los observadores tanto como de nacionalistas de izquierda como de extrema derecha anti-capitalista, y a menudo se la suele relacionar y comparar con el nacionalbolchevismo. En marzo de 2011 funda Société à responsabilité limitée Culture pour tous, que incluye a la editorial Kontre Kulture. En 2008 fue condenado por acusaciones tales como "antisemitismo" e "incitación al odio, a la discriminación y a la violencia".
Biografía
Nació el 2 de octubre de 1958 en Aix-les-Bains (Francia). A mediados del 2013, el sitio web de la asociación que creó y preside, Egalité et Réconciliation («Igualdad y Reconciliación»), alcanzó el primer lugar en la audiencia de los blogs políticos. Su libro Comprendre l´Empire (Comprender el imperio), del año 2011, ha sido reimpreso más de diez veces, se han vendido más de 50.000 ejemplares (hasta el 2013) y ha llegado varias veces en el top 100 de ventas de Amazon; adicionalmente, ha sido traducido al ruso, coreano, e italiano. Todo eso, pese a que este libro nunca ha sido promocionado por la televisión o los periódicos.
Su progresiva irrupción en la política como un analista no invitado ni esperado lo ha expuesto a múltiples intentos de agresión por parte de milicias irregulares, sin que las autoridades judiciales sancionen a los agresores, con excepción del ataque del 5 de abril de 2013 por parte de un grupo de una decena de hombres enmascarados mientras cenaba en un restaurante; estas personas fueron filmadas por cámaras de seguridad y la justicia determinó su responsabilidad penal.
El popular comediante franco-camerunés Dieudonné declaró en una entrevista televisada en 2004 respecto de Alain Soral que «es importante preservar la visión de este tipo de artista, libre pensador, un poco trash, un poco punk, ahora que la expresión artística se ha vuelto completamente comercial»; desde entonces, acompaña activamente a Soral en su crítica aguda al oficialismo.
Manuel Valls, actual Primer Ministro de Francia, «en nombre de la República francesa, de la libertad de prensa y la democracia» condenó públicamente a las intervenciones de Soral a través de Internet, así como sus vínculos con Dieudonné.
Posiciones políticas
Alan Soral es conocido en Francia por su posición antisemita y antisionista, y uno de los fundadores de la organización política y blog de internet Igualdad y Reconciliación (Egalité et Réconciliation), que en principio estaba destinado a ser el nombre de un partido político que pensaba crear junto con Dieudonné, un comediante antisemita y creador del saludo "quennelle" (saludo nazi inverso). Alain Soral no dudó en tomarse una foto haciendo la "quennelle" en el Memorial del Holocausto de Berlín.
El domingo 7 de junio de 2009, figura como miembro de la Lista Antisionista que se presenta a las elecciones europeas, una curiosa mezcla de filonazis, gremialistas y propalestinos. En el portal de Internet de la agrupación aparecen fotos del presidente de Venezuela Hugo Chávez abrazando al presidente de Irán Mahmud Ahmadinejad y al presidente de Bolivia Evo Morales.
Es autor de la frase "Marx en Francia votaría al Frente Nacional". Su pensamiento es confuso de situar para los especialistas dado que sus ideas se sitúan entre la extrema derecha y la extrema izquierda.
Se reclama seguidor de Charles de Gaulle por su independencia ante los Estados Unidos. Defendió a Jean-Marie Le Pen, fundador del partido Front National (FN), y ahora retirado de la política. Ha manifestado su admiración por algunos jefes de Estado como Hugo Chávez, a quien ha visitado en Venezuela, el expresidente de Irán Mahmoud Ahmadinejad, el Presidente de Rusia Vladímir Putin o el líder del Hezbolá libanés, Hassan Nasrallah. También ha mostrado su apoyo a la organización fascista italiana CasaPound.
Condena por antisemitismo
En septiembre de 2019, el Tribunal Penal de Bobigny lo condenó a 24 meses de prisión de los cuales deberá cumplir 18 con prisión efectiva, por transmitir en línea un video de rap antisemita. El tribunal también dictaminó que Alain Soral elimine el clip de su sitio, bajo pena de una multa de 1.000 euros por cada día que demore en hacerlo.
Libros
Les Mouvements de mode expliqués aux parents («Los movimientos de moda explicados a los padres»), con Hector Obalk [Éric Walter] y Alexandre Pasche, Editorial Robert Laffont, 1984. Reeditado por France Loisirs y Le Livre de poche.
La création de mode. Comment comprendre, maîtriser et créer la mode («La creación de moda. Cómo comprender, dominar y crear la moda»), Editorial S.I.S., 1987.
Le Jour et la Nuit, ou la vie d'un vaurien («El día y la noche, o la vida de un bueno para nada»), Éditions Blanche, 1991.
Sociologie du dragueur («Sociología del ligón»), Éditions Blanche, 1996. Reeditado en 2004.
Vers la féminisation? («¿Hacia la feminización?»), Éditions Blanche, 1999. Reeditado en 2007 bajo el título Vers la féminisation? Pour comprendre l'arrivée des femmes au pouvoir («¿Hacia la feminización? Para comprender la llegada de las mujeres al poder»).
Jusqu'où va-t-on descendre? Abécédaire de la bêtise ambiante («¿Hasta dónde vamos a ir a parar? Abecedario de la idiotez reinante»), Éditions Blanche, 2002. Reediciones: Editorial Pocket en 2003 como Abécédaire de la bêtise ambiante y con el mismo título abreviado por Éditions Blanche en 2008, junto al siguiente.
Socrate à Saint-Tropez: texticules («Sócrates a Saint-Tropez: textículos»), Éditions Blanche, 2003. Reeditado junto al precedente como Abécédaire de la bêtise ambiante.
Misères du désir («Miserias del deseo»), Éditions Blanche, 2004.
CHUTe ! Éloge de la disgrâce («¡A callar! Elogio de la caída en desgracia»), Éditions Blanche, 2006.
Comprendre l'Empire. Demain la gouvernance globale ou la révolte des Nations («Comprender el Imperio. Mañana, la gobernanza global o la revuelta de las naciones, Éditions Blanche, 2011.
Chroniques d'avant-guerre («Crónicas de pre-guerra»), Éditions Blanche, 2012.
Dialogues désaccordés. Combat de Blancs dans un tunnel («Diálogos desacordados. Combate de blancos en un túnel») en colaboración con Éric Naulleau, Hugo, colección Blanche, 2013.
Referencias
Editores de Francia
Ensayistas de Francia del siglo XX
Ensayistas de Francia del siglo XXI
Miembros del Partido Comunista Francés
Antifeministas
Antisionistas
Antisemitismo en Francia
Antisemitas
Políticos del Frente Nacional (Francia)
Negadores del Holocausto
Nacional-revolucionario | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,700 |
Math art
Birthday Dominoes
(This post is dedicated to the most important Pisces birthday I know: EK.)
Dominoes is a well-known game that no one actually knows how to play. A much more accessible game is tiling dominoes: I give you a grid, and you tell me if you can cover the whole thing with dominoes.
For example, look at this 2x2 grid:
Easy! What about this 4x4 grid?
Also easy! What about a 3x3 grid?
It can't be done! Any grid you can cover with dominoes has to have an even number of squares, but a 3x3 grid has 9. We lose, through no fault of our own.
This pretty much solves grids. You can tile them with dominoes if and only if they have an even number of squares. But this gives us two natural questions to ask:
What if we used something other than dominoes?
What if we used something other than grids?
Something other than dominoes
Dominoes are pieces with two blocks "snapped" together. What if we used more than two blocks? This these exist, of course, and we call them triominoes, quadominoes, pentominoes, and in general, polyominoes.
There is only "one" domino—two blocks stuck at their ends—but there are three triominoes!
That pesky 3x3 grid that we couldn't tile with dominoes is a cinch with triominoes:
The most famous polyomino is the pentomino, which has five blocks stuck together. These are commonly used for fun in brain teasers, if you're into that sort of thing.
Something other than grids
The shape of the board is just as important as the number of spaces. For example, look at this "T" with four spaces:
Even though there are an even number of spaces, we can't tile this with dominoes! Here's a grid with 9 spaces that we cannot tile with triominoes:
In general, a board with $n$ spaces is only guaranteed to have a tiling of monomioes (pieces with 1 block) and an $n$-omio (a piece with $n$ blocks), both of which are kind of cheating. Board layout plays a big role.
I propose that we think about tiling the heart board, something I invented for just this occasion. Here are the first four heart boards:
It's a bit hard to make out the "grid" in these pictures, so here are the first two drawn by hand (kinda):
The number of blocks in the first four hearts is 10, 43, 96, and 169, respectively. The hearts follow a pattern that generalizes to arbitrary sizes. The $n$th heart board is the set of all $(x, y)$ such that
$0 \leq x < 2n$
$0 \leq y < 4n$
$x \leq y$
$y \leq 5n - x$
$y \leq x + 3n$,
and also the reflection of these points about the $y$-axis.
The $n$th heart board has exactly
\[10 n^2 + 3n - 3\]
spaces in it.
There are lots of questions to ask about this board, but let's settle for just one: When can the heart board be tiled by dominoes?
The first heart board cannot be tiled with dominoes. That's easy enough to see by hand because it only has 10 blocks:
The second heart board is much bigger at 43 blocks, but this is an odd size so no tiling by dominoes could exist. In fact, the general size $10 n^2 + 3n - 3$ is odd when $n$ is even, so only the "odd" heart boards could hope to be tiled by dominoes anyway.
So what about the third heart block, the one in the bottom left of the computer-generated image above? Can it be tiled using dominoes? It turns out that no, it cannot be. How do I know this? My computer told me. In fact, using the polyomino library, my computer told me more:
Heart board number
Domino tiling?
It seems like no heart boards can be tiled by dominoes. Is this true? Amazingly, yes! Not a single heart board can be tiled by dominoes.
The proof of this fact relies on a very simple observation: If you color the squares of an odd heart board in an alternating fashion, then then there are exactly two more squares of one color than the other. For example, look at the first heart board:
Every domino you put down must cover exactly one of each color. In the above picture, every domino covers one red and one blue tile. Once you place four dominoes, there are no blue tiles left, but two red tiles. We can't cover those pieces with dominoes! (This is what happened in our attempted tiling above.)
This pattern persists for every odd heart board. Here is a plot of the first four hearts again, now colored in this alternating way:
The first and third heart boards above (left column) have exactly two more red squares than blue squares. The second and fourth (right column) actually have a bigger difference in the number of squares, but we already knew that they couldn't be colored with dominoes.
This pattern is somewhat tricky to prove, but once you know the idea it's just calculations. See my math.SE question for details.
We've left lots of questions on the table that would be easy to answer. For example, what's the proportion of squares that are red versus blue in the even heart boards? A harder project: Let $L_n$ be a set of lines which intersect the square $[-n, n] \times [0, n]$ and each other. When is the region "inside" $L_n$ tileable with dominoes?
I don't know the answer to any of these offhand, but they sound fun. I hope that this delivery on my promise of math art taught everyone something new. Here's to much more in the future!
© 2023 Robert Dougherty-Bliss. Powered by Jekyll & AcademicPages, a fork of Minimal Mistakes. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,793 |
Sun buys Diba to attack set-top box market
Sun Microsystems is to buy Diba in a deal that takes it deep into the set-top box market.
By Martin Veitch | July 31, 1997 -- 15:21 GMT (08:21 PDT) | Topic: Hardware
Diba, is a two-year-old Silicon Valley pioneer in designing software and hardware for the world of "information appliances", futuristic devices such as handheld e-mail terminals, smart phones and mobile Web browsers. Many of the firm's staff previously worked for Oracle.
"We see this as very much a positive move," said Nigel Seed, VP of Europe for Diba. "The industry is consolidating very quickly with Microsoft buying WebTV and buying into [Internet telephony firm] Navitel, and Oracle buying Navio [a maker of Web browsers for small-format devices]. It was prettyy obvious that someone would come after us [and] Diba's work with consumer applications and Java is a wonderful combination. There is a huge market prospect but unless you find a very patient venture capitalist it's probably better to find a large, stable organisation."
Seed said that Diba's strength is in delivering a tight software architecture: "We produce small-footprint code. Code-bloat just drives up the price because of the processor, RAM, ROM and Flash memory you need."
Seed added that he expects the momentum for set-top boxes to quickly gather. "There's a huge pent-up demand for people to get on the Web and have e-mail and we expect to see set-top boxes based on our architecture available in the UK before Christmas."
The Diba-based systems will be based on a Motorola PowerPC and offer e-mail, Internet access and on-screen programming guides. Price is expected to duck £300.
More from Martin Veitch
Five years ago: IBM to ship 8.4Gb hard drive
Five years ago: IBM to push networked PC ticket
Five years ago: AOL calls flat-rate switch a success
Five years ago: Networks '97: MS presence gives UK Java Forum early rumpus | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 461 |
Der Coquitlam River ist ein 46 km langer rechter Nebenfluss des Fraser River im Südwesten der kanadischen Provinz British Columbia.
Der Coquitlam River entspringt 40 km nordöstlich von Vancouver in den Coquitlam Ranges, einem Gebirgszug der Pacific Ranges. Er fließt in überwiegend südlicher Richtung durch ein breites gletschergeformtes Tal und mündet nach 15 km in das obere nördliche Ende des Coquitlam Lake. Der Wasserspiegel des ehemals natürlichen Sees wird seit Anfang des 20. Jahrhunderts durch einen Staudamm erhöht. Das Wasser wird vom Coquitlam Lake zu Zwecken der Energiegewinnung über eine unterirdische 3,6 km lange Rohrleitung zum weiter westlich gelegenen Buntzen Lake geleitet. Dessen Wasser treibt die Turbinen zweier am Indian Arm gelegener Wasserkraftwerke an. Der 18 km lange Unterlauf des Coquitlam River erhält somit fast kein Wasser mehr aus dem See und wird heute hauptsächlich vom Or Creek, der knapp 2 km unterhalb des Staudamms in den Coquitlam River fließt, gespeist. Der Coquitlam River fließt zwischen den Städten Coquitlam im Westen und Port Coquitlam im Osten in südlicher Richtung und mündet schließlich einen Kilometer unterhalb des Pitt River in den Fraser River. 5,5 km oberhalb des Coquitlam Lake beträgt der mittlere Abfluss 6,69 m³/s.
Der Flussname "Coquitlam" leitet sich vermutlich von dem indianischen Wort für einen Fisch ab.
Weblinks
Einzelnachweise
Coquitlam
Fluss in den Coast Mountains | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,771 |
{"url":"https:\/\/support.bioconductor.org\/p\/118729\/","text":"1\n0\nEntering edit mode\nmodarzi \u25b4 10\n@modarzi-16296\nLast seen 7 months ago\n\nHi,\n\nI provide below query:\n\nquery <- GDCquery(project = \"TCGA-SARC\",sample.type = \"Primary solid Tumor\",\ndata.category = \"Transcriptome Profiling\",\ndata.type = \"Gene Expression Quantification\",workflow.type = \"HTSeq - FPKM-UQ\");\n\n\nGDCdownload(query, method= \"api\", directory = \"mydata\")\n\n\n\"<simpleWarning in file.create(to[okay]): cannot create file 'GDCdata\/TCGA-SKCM\/harmonized\/Transcriptome_Profiling\/Gene_Expression_Quantification\/3c9fe8ef-e394-4d7a-9189-de6ce2169c45\/50bbf24f-b914-4e53-98ee-0cf22b2d9f01.htseq.counts.gz', reason 'No such file or directory'>\n\n\nI appreciate if anybody share his\/ her comment with me.\n\nBest Regards,\n\n0\nEntering edit mode\n@tiago-chedraoui-silva-8877\nLast seen 9 months ago\nBrazil - University of S\u00e3o Paulo\/ Los A\u2026\n\nHello,\n\nI ran a small example and it worked. Probably the folder was not created in the system. If it is a windows OS, there is a limit of characters in the full path (I believe it is 256 characters), you might be able to check the path length it is trying to create with:\n\nstringr::strlength(file.path(getwd(),\"GDCdata\/TCGA-SKCM\/harmonized\/TranscriptomeProfiling\/GeneExpressionQuantification\/3c9fe8ef-e394-4d7a-9189-de6ce2169c45\/50bbf24f-b914-4e53-98ee-0cf22b2d9f01.htseq.counts.gz\"))\n\nBest regards, Tiago Chedraoui Silva\n\n0\nEntering edit mode\n\nHi,\n\nMy OS is windows 10. when I run getwd() I see below path:\n\n\"E:\/Biology_base\/RNA-seq\/GDC-TCGA-SARC\/Original_data\/SARC RNA-seq by TCGAbiolink Package-971215\"\n\n\nso when I run below code:\n\nGDCdownload(query)\n\n\nin \"SARC RNA-seq by TCGAbiolink Package-971215\" folder I see new folder by \"GDCdata\" and below path:\n\nGDCdata\\TCGA-SARC\\harmonized\\Transcriptome_Profiling\\Gene_Expression_Quantification\\0b4a4b61-ce8e-4fda-b0ee-4cc6ec3e2474\n\n\nbut in \"0b4a4b61-ce8e-4fda-b0ee-4cc6ec3e2474\" folder I can't find any file. Also I call stringr library but in that I couldn't find strlength(). I fould str_length()\n\nSo I request to help me and give your solution based on my explanation . Best Regards\n\n1\nEntering edit mode\n\nThe package is trying to create a file that has 267 characters (considering the full path) which more than the limit of 260 characters. You should either enable long paths ( https:\/\/www.howtogeek.com\/266621\/how-to-make-windows-10-accept-file-paths-over-260-characters\/), or reduce the name of same folders or work in the upper directories\n\nlike working inside this path: \"E:\/Biologybase\/RNA-seq\/GDC-TCGA-SARC\/Originaldata\/\"","date":"2021-06-15 12:06:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.24075070023536682, \"perplexity\": 9625.863168909778}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487621273.31\/warc\/CC-MAIN-20210615114909-20210615144909-00255.warc.gz\"}"} | null | null |
Lycaena gravenotata är en fjärilsart som beskrevs av Alexander Barrett Klots 1930. Lycaena gravenotata ingår i släktet Lycaena och familjen juvelvingar. Inga underarter finns listade i Catalogue of Life.
Källor
Juvelvingar
gravenotata | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,254 |
The killing outraged the entire state but what enraged people even more was that Davis had been in and out of prison his whole life and was still free...полностью>>
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In the years 1851-1855, Jean Baptiste Boussingault, French chemist, performed an experiment of plant growth in quartz and sand cultures with no soil....полностью>>
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Tile number three symbolizes the trinity; the "perfect" number, ten, was obtained by multiplying three times three, and adding one (which represented ...полностью>>
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Two hundred years ago, Thomas Robert Malthus, a British economist , wrote An Essay on the Principle of Population in which he argued that the world po...полностью>>
The Forgotten Chinese Holocaust Essay Research Paper
The Forgotten Chinese Holocaust Essay, Research Paper
Can you imagine your body being an object for experimentation while you?re still alive? That?s one of the things the Japanese did to the Chinese during the forgotten holocaust, the Chinese holocaust. Among the universal disputes between many countries, Japanese aggression on the Chinese was one of the worst events in history to ever take place. The Japanese also destroyed many cities of China. Specifically, they destroyed the city of NanJing by conducting mass bombings and remorseless killings. Other examples of Japanese horrific actions against the Chinese happened in a place called Unit 731.
During the 1920?s, NanJing only had a population of 250,000. However, during the 1930?s, the city was highly populated with over one million residents. This increase was a result of the Japanese occupation and countless refugees fleeing to the city from Manchuria and other Chinese areas to the east of NanJing. The city of NanJing was a safe city for the Chinese until Japanese forces advanced towards it from Shanghai on November 11th, 1937. The Japanese planes bombed the wealthy and more populated areas of the city. The most devastating bombing occurred on September 25th, 1937. Its targets were focused upon hospitals with a red cross on the roof, refugee camps, power plants, water works, and radio stations. About 500 bombs were dropped from 9:30 a.m. to 4:30 p.m., and as a result, there were over 600 civilian casualties. ?
On November 25th, Japanese forces attacked NanJing from three different directions. The Chinese city soon fell to the Japanese Imperial Army. As the Japanese entered the city, a massacre began which lasted six weeks. During that time, the Chinese were not simply murdered, but were humiliated, and tortured. The Japanese used unthinkable methods of murder. They chased the Chinese into the Yangtze River with machine guns, drowning them. They poured gasoline on people, shot them, and watched them flicker up. The Japanese cut the eyeballs out of men and then burned them while they were still alive.2 Some Chinese had their hearts cut out. Even babies were skewered and tossed into boiling water.
The Japanese soldiers showed to be heartless when they made games out of these atrocities and used the victims as toys. Japanese generals organized contests of who could kill the most Chinese. Whoever killed to most would be the winner. Sometimes the number of bodies reached as high as five hundred in a single contest.3 News reporters came and observed the barbaric competitions and the victors were praised back in Japan. Their generals encouraged the Japanese soldiers to rape whenever they please, and so they did. After which they killed off the women. The victims has their stomachs cut open or their breasts chopped off. ?Comfort women? were kept as sex slaves to serve the Japanese soldiers throughout the day. The Massacre of NanJing was therefore also known as the ?Rape of NanJing.?
The Japanese army finally left NanJing when the United States bombed Hiroshima and Nagasaki. After the six weeks of horror, NanJing was left in ruins. Nothing was left except the dead bodies that emitted an unbearable smell for miles around.
The Japanese started a secret ?research program? during and after World War II. The program was set up to develop weapons of biological warfare, including plague, anthrax, cholera, and a dozen other pathogens.4 Even after the Geneva Protocol was signed by 145 countries, including Japan, to ban all gas and biological weapons. By ?field testing,? Chinese cities were invaded by plague bombs dropped by the Japanese to see if they could start plague outbreaks?they did. The Japanese planes dropped plague-infected fleas over Ningbo in eastern China and over Changde in north central China. No accounts were found regarding how many died of this. The plague caused high fever, vomiting of blood, shivering, respiratory failure, and body pains that resulted in a dark purple body color. Three out of every 4 who contracted this disease died.5 The Japanese hoped to use the soon to be developed weapons on the United States. They proposed using balloon bombs to carry disease to America and they had a plan in the summer of 1945 to use kamikaze pilots to dump plague infected fleas on San Diego.6 And yet only the Chinese were largely effected by this plague.
In a place called Unit 731, a part of the research program, human experimentation on Chinese people took place. The Japanese army, which was then occupying a large chunk of China, evicted villages near the city of Harbin in Manchuria to make way for the headquarters of Unit 731. Head of Unit 731 was a man called Shiro Ishii. He was born in 1892 to a wealthy Japanese family. He grew up arrogant and had no regards for those who he considered lower than him. After the emperor approved his dream of building the unit, the horror began.
Unit 731 contained such jars with feet, heads, internal organs, all labeled. Medical researchers, doctors, dentists, technicians, and scientists all had part to do with this. Fifty different types of experiments were conducted in Unit 731. All chosen spontaneously. The researchers took the deliberately plague infected Chinese, who was not given a vaccine, and cut him open to see the effect of the disease to the man?s inside while he was still alive. The Chinese subjects used for the experiments were called marutas, or logs. A former medical worker in Unit 731 said he once saw a 6-foot high glass jar in which a western man was pickled in formaldehyde.7 The man was cut into two pieces, vertically. Medical researchers also locked up diseased prisoners with healthy ones, to see how various ailments would spread. The doctors even locked victims in pressure chambers to see how long the body would withstand before the eyes popped out of its sockets. Doctors even experimented on a three-day-old baby, measuring the temperature inside the infant?s middle finger.8 The needle was stuck in the finger to keep the baby?s fist from clenching and making the experiment easier. One experiment was to see which method was best to treat frostbite. Part of the unit held a freezing machine where they froze different body parts of the ?logs? and tried various ways to dehydrate it again.
The Chinese were used as dummies for other experiments outside of the unit. Tied to stakes in a pattern in a proving ground called Anda, the Chinese were used to see how effective the new technological weapons were. They wanted to know how many people the bomb would kill and the distance from the bomb the person was placed. Few survived that experiment so few Chinese had surgery, and those who died received autopsies. Ishii?s soldiers even went so low as to hand out chocolate candy laced with anthrax to starving Chinese children. Japanese troops also dropped cholera and typhoid cultures in wells and ponds, but the results were often counterproductive. In 1942, germ warfare specialists distributed dysentery, cholera and typhoid in Zhejiang Province in China, but Japanese soldiers themselves became ill and 1,700 died of the diseases.9
A recollection of a doctor, Dr. Ken Yuasa, says he remembers two Chinese men being brought in and stripped naked. Then he began practicing various types of surgery and when finished, the patients were killed with an injection. Dr. Yuasa said that the Chinese brought in for vivisections were used for practice and that they were routine among Japanese doctors working in China in the War.
Years after the NanJing Massacre, criminal trials were held. The Japanese that were not class A criminals were tried near the homes of their victims. However, the class A war criminals were tried at the Tokyo Trials in Tokyo. Twenty-eight men were persecuted and twenty-five were found guilty. Two of the three not found guilty died during trial and the third experienced a mental breakdown. Although Japanese criminals were charged and convicted, many Japanese citizens slowly developed a denial of the NanJing Massacre. Few Japanese civilians heard about the atrocities because of the Japanese control over the media, they heard only of the heroic war figures. In 1990, the Japanese began to officially deny the whole case with NanJing and stated that it was a lie. But, finally, in 1995 the Japanese Prime Minister Tomiichi Murayama gave the first clear and formal apology.
With all these actions taken upon the Chinese civilians, why was the head of Unit 731, Shiro Ishii, allowed to live peacefully until his death from throat cancer in 1959? It was partly because the Americans helped cover up the biological warfare program in exchange for the data collected by the experiments. A farmer who was a member of Unit 731 justified the reason for vivisection without anesthetic by saying that it might have had an effect on the body organs and blood vessels being examined. He even justified the use of children in the experiments. He said that the fathers of the children were probably spies. These attitudes contributed to a collective amnesia in Japan toward war atrocities.10
The NanJing Massacre and the experiments of Unit 731 could never be erased out of the past and should never be erased out of our minds. The cost of the breakthrough of new knowledge was borne by the victims of medical experiments. The apology of Murayama doesn?t include the apology of all, so rather than denying it, it should become excepted by Japan and a lesson for the rest of us in hopes that history won?t repeat itself. And yet still we can ask ourselves ?Can history repeat itself?? The answer is disappointing: There?s a possibility?because in war, you have to win.11 Seems like nothing else matters.
Holocaust Essay Research Paper Eleven million precious
Holocaust Essay, Research Paper Eleven million precious lives were lost during the Holocaust of ... a story made up by the Chinese, the "Nanjing Massacre never occurred" ... labeled the forgotten holocaust–not only forgotten, but denied by the Japanese ...
World War Ii 3 Essay Research Paper
World War Ii 3 Essay, Research Paper World War II and Beyond ... . Historians call it the Forgotten Holocaust. It is terrible that the Japanese military did ... 731, the dropping of the plague and anthrax on Chinese villages, and the Rape ...
Rape Of Nanking Essay Research Paper Although
... Of Nanking Essay, Research Paper Although many people thought that the World War II ... any resistance from the Chinese government. By the 1930s, the Japanese had launched ... Yasuhiko, entered into China and the forgotten holocaust had officially begun. About 50 ...
Japan At War Essay Research Paper Commentary
Japan At War Essay, Research Paper Commentary: Japan at WarThe ... reading experience. I myself as a Chinese who have never live through ... army did to us made the holocaust seem like a joke. I ... was more thought provoking than the Forgotten Soldier, which felt like ...
Nanking Essay Research Paper Chapter OneThe main
... from the soldiers of the Chinese army. The Japanese Journalists told us that the Japanese ... Nanking. This chapter she calls The Forgotten Holocaust: A Second Rape because people are ... takes us into the realm of World War Two China. The research done in ... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,453 |
import { Component, OnInit, Input } from '@angular/core';
import { Router, ActivatedRoute, Params } from '@angular/router';
import { isEmpty } from 'lodash';
import { RankingService } from '../shared/services/ranking.service';
import { SetOfResults } from '../shared/models/set-of-results.model';
import { FilterCriteria } from '../shared/models/filter-criteria.model';
@Component({
moduleId: module.id,
selector: 'agot-all-rankings',
templateUrl: 'all-rankings.component.html',
})
export class AllRankingsComponent implements OnInit {
@Input()
title: string;
@Input()
hideFilters: boolean = false;
@Input()
criteria: FilterCriteria;
results: SetOfResults;
loadingError: any = null;
isLoading: boolean;
constructor(private _route: ActivatedRoute,
private _router: Router,
private _RankingService: RankingService) {
}
ngOnInit() {
this._route.params
.map(this.setFiltering.bind(this))
.do(() => this.isLoading = true)
.switchMap((criteria: FilterCriteria) => this._RankingService.getRankings(criteria))
.subscribe(
(results) => {
this.loadingError = null;
this.results = results;
// TODO until loading states are sorted out
this.isLoading = false;
},
(err) => {
this.loadingError = err._body || err;
this.isLoading = false;
},
() => {
console.log('rankings component done');
this.isLoading = false;
}
);
}
onDateRangeChange(criteria: FilterCriteria) {
this.loadRankings(criteria);
}
loadRankings(criteria?: FilterCriteria) {
this._router.navigate(['/rankings', FilterCriteria.serialise(criteria)]);
}
private setFiltering(params: Params) {
const defaultFilter = this.criteria || new FilterCriteria();
return this.criteria = isEmpty(params) ?
defaultFilter :
FilterCriteria.deserialise(params, defaultFilter);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,087 |
Az Auchenoglanis a sugarasúszójú halak (Actinopterygii) osztályának harcsaalakúak (Siluriformes) rendjébe, ezen belül a Claroteidae családjába tartozó nem.
Tudnivalók
Az Auchenoglanis-fajok családjuknak az ősibb alakjait képviselik. Számos afrikai folyóban és tóban lelhetők fel. A nagy elterjedési területük a változatos étrendjüknek köszönhető. Táplálékaik között szerepelnek a rovarok, azok lárvái, puhatestűek, halivadékok, valamint magok és egyéb növényi eredetű törmelékek. E halnem fajait táplálkozási célokból halásszák, de akváriumokban is tarthatók.
Rendszerezés
A nembe az alábbi 2 élő faj tartozik:
Auchenoglanis biscutatus (É. Geoffroy Saint-Hilaire, 1809) - típusfaj
Auchenoglanis occidentalis (Valenciennes, 1840)
Csád nyugati részén előkerült a fosszilis Auchenoglanis soye Otero et al., 2007; ez a hal a miocén korban élhetett. Néhány egyéb fosszilis maradvány is ebbe a halnembe helyezhető, azonban faji szinten még nincsenek leírva.
Jegyzetek
Források
Auchenoglanis FishBase
Geerinckx, T. and E. Vreven, 2013. A re-evaluation of the species-level diversity within the catfish genus Auchenoglanis (Siluriformes: Claroteidae). J. Nat. Hist. 47(47-48):2979-3010.
Risch, L.M., 2003. Claroteidae. p. 60-96 In C. Lévêque, D. Paugy and G.G. Teugels (eds.) Faune des poissons d'eaux douce et saumâtres de l'Afrique de l'Ouest, Tome 2. Coll. Faune et Flore tropicales 40. Musée Royal de l'Afrique Centrale, Tervuren, Belgique, Museum National d'Histoire Naturalle, Paris, France and Institut de Recherche pour le Développement, Paris, France. 815 p.
Retzer, M.E., 2010. Taxonomy of Auchenoglanis Günther 1865 (Siluriformes: Auchenoglanididae). Zootaxa 2655:25-51.
Fordítás
Claroteidae
Halnemek | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,981 |
<page-layout [ready]="data$ | async">
<ng-container *ngIf="data">
<page-content [heading]="i18n.product.title.list"
[mobileHeading]="i18n.product.mobileTitle.list"
[headingActions]="headingActions$ | async">
<user-info [user]="data.user"></user-info>
<ng-container
*ngTemplateOutlet="products; context: {$implicit: data.userProducts, title: i18n.product.assigned.individual }">
</ng-container>
<ng-container
*ngTemplateOutlet="products; context: {$implicit: data.groupProducts, title: i18n.product.assigned.group, readOnly: true}">
</ng-container>
<ng-container
*ngTemplateOutlet="products; context: {$implicit: data.groupSetProducts, title: i18n.product.assigned.groupSet, readOnly: true}">
</ng-container>
</page-content>
</ng-container>
</page-layout>
<ng-template #products let-products let-title="title" let-readOnly="readOnly">
<ng-container *ngIf="!empty(products)">
<h2 class="mt-3 mb-2">{{ title }}</h2>
<ng-container *ngIf="layout.ltsm$ | async; else largeTable">
<div *ngFor="let prod of products" class="d-flex mb-2">
<div class="cell-main">{{ prod.name }}</div>
<div *ngIf="!readOnly && canRemove(prod)" class="ml-auto text-right">
<button type="button" [tooltip]="i18n.general.remove"
class="btn btn-icon"
(click)="remove(prod); $event.stopPropagation(); $event.preventDefault()">
<icon [icon]="SvgIcon.Trash"></icon>
</button>
</div>
</div>
</ng-container>
<ng-template #largeTable>
<table class="table table-hover small-height mb-0">
<thead>
<th width="40%" class="no-border">{{ i18n.general.name }}</th>
<th width="25%" class="no-border">{{ i18n.general.type }}</th>
<th width="25%" class="no-border">{{ i18n.product.userAccount }}
</th>
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| {
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} | 5,601 |
Stories that Inspire.
Victories to Celebrate.
Victor is taking advantage of his second chance at life
Victor was born in Mexico, but took off for the United States when he was just 16. "There was no work over there, so I came here in 2004," said Victor, who is a painter by trade. "When I started working, I also tried cocaine, and started using. I would work, get off work, and then use. I did that for five years."
Things changed when Victor had a daughter, now 12. He cleaned up his act, but eventually relapsed. "With my daughter's mom, I started using heroin and crystal meth," he said. "I was so depressed at the time. I would work and then party, do drugs, and just be irresponsible."
When Victor turned 28, he had a second daughter with a different woman. She is 4 years old now. "I was really battling my addiction then," he said. "I stopped working and I lived in the streets for a while. For two years. I was homeless and I had no hope, no nothing."
Victor said he would go to jail "all the time, many times. In two years, I went to jail at least 10 times," he said. "In jail, they gave me a chance to get sober and start doing things right. I also knew I'd get deported if I didn't quit my addiction."
Victor learned about the Mission and arrived in January. "I give thanks to God, first of all, for giving me another chance to be a normal person," he said. "I've been using drugs and in my addiction for half of my life. God give me this last chance to be a man, to mature. I want to do good. I've done enough partying in my life. I've thrown enough years of my life in the trash."
Victor, now 33, said he's always believed in God. He grew up in a Catholic family, although they didn't go to church. "I always knew there was a God somewhere. I knew that whenever I prayed, I believed God was there all the time. But right now, I believe more than ever before, and it's changing my life in a good way."
When things get tough, Victor thinks about his parents, who are still in Mexico. "I think about my mom and dad," he said. "They worry about me all the time. They really love me. My mom doesn't want to see me like this, in addiction.
"I'm so thankful to the Mission. It's a very nice program and they are very good people. They've been very kind. The whole program is good— everything: morning devotions, (volunteering) at the store, outreaches. I love to help people. I see people out there, and they need help. Life is hard on the streets. The Bible says to help each other."
Victor said he used to see his daughters, but hasn't recently. "Especially last year—I was homeless, really into drugs, on the streets," he said. "I don't want that life anymore. My daughters, I really love them. I want to be a part of their lives."
Now five months into the program, Victor is feeling good. "I love it. I'm pretty sure that I love myself now," Victor said with a laugh. "I don't want to be in addiction anymore. It's really bad, feeling like you are going nowhere, that you are going to end up dead or in jail. I don't want to be deported. Over there, life is really, really hard. But the main thing is, I want to change my life. And I want to be part of my daughters' lives.
"I'm going to do the best I can. I'm going to take this opportunity in a 100 percent positive way. Thanks to Jesus. He gave me another chance, and I want to take it."
Be Transformed
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Lives Changed
Looking forward to a new year at the Mission
Lives changed for good: Updates
← Drugs help Zach numb emotional wounds from his youth
VICTOR RECOMMITS TO THE PROGRAM, EXPERIENCES THE DIGNITY OF WORK → | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,034 |
CECIL, KARENL thru CECIL, KATHRYND
CECIL, KAREN L. (possibly her maiden name) had a baby named LAUREN MICHELLE HUDSON on 25 April 1974 in Jefferson County, Kentucky, United States of America.
Check the source file which is free. Also check Archives for KAREN L. CECIL.
CECIL, KAREN L. (possibly her maiden name) had a baby named ROBIN LORANE HUDSON on 10 February 1967 in Jefferson County, Kentucky, United States of America.
CECIL, KAREN L. (possibly her maiden name) had a baby named SHELLEY MARIE HUDSON on 7 February 1970 in Jefferson County, Kentucky, United States of America.
CECIL, KAREN L. married JEREMY L. HIGHER on 15 October 2011 using a marriage license issued in Hamilton County, Indiana, United States of America.
CECIL, KAREN L who was 18 (born ABT 1956) married 4 MAY 1974 in WICHITA COUNTY, TEXAS, U.S.A. a groom named VERNON L JR ROBERTS who was 18 (born ABT 1956).
Check the source file which is free. Also check Archives for KAREN L CECIL.
CECIL, KAREN L. (bride) of Fayette County, Kentucky, who was 21 years old, White, with no previous marriage, and WALTER M. CARPENTER of Fayette County, Kentucky, who was 25 years old, White, status 3 (Last marriage ended by divorce) with one previous marriage, had a wedding ceremony 20 November 1986 on a license issued in Fayette County, Kentucky, United States of America (Certificate number 40489) with the intention of residing in Fayette County, Kentucky.
CECIL, KAREN L. (bride) of Woodford County, Kentucky, who was 24 years old, White, with no previous marriage, and JOSEPH C. WILLETT of Jefferson County, Kentucky, who was 24 years old, White, status 1 (Never married) had a wedding ceremony 8 June 1991 on a license issued in Woodford County, Kentucky, United States of America (Certificate number 19665) with the intention of residing in Woodford County, Kentucky.
CECIL, KAREN L. was born 14 October 1946 to DORIS DAWKINS (possibly her maiden name) in Jefferson County, Kentucky, United States of America.
CECIL, KAREN L. was born 16 March 1960, received Social Security number 262-11-9368 (indicating Florida) and, Death Master File says, died 8 October 1994
CECIL, KAREN LEE was born 18 November 1957 in Missouri, United States of America. Special thanks to Reclaim the Records. Please consider donating to them.
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CECIL, Karen Lucille married in 1968 in Ohio, West Virginia, United States a groom named Kenneth William Castilow.
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CECIL, KAREN LYNN was born 12 January 1967 to MARY F. RAY (possibly her maiden name) in Fayette County, Kentucky, United States of America.
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CECIL, KAREN M. (possibly her maiden name) had a baby named GARRY HOWARD FOSTER on 12 April 1988 in Jefferson County, Kentucky, United States of America.
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CECIL, KAREN M. (possibly her maiden name) had a baby named JESSICA ANN FOSTER on 2 November 1982 in Jefferson County, Kentucky, United States of America.
CECIL, KAREN M. (bride) of Bullet County, Kentucky, who was 16 years old, White, with no previous marriage, and GARRY H. FOSTER of Bullet County, Kentucky, who was 18 years old, White, status 1 (Never married) had a wedding ceremony 15 April 1982 on a license issued in Bullet County, Kentucky, United States of America (Certificate number 03812) with the intention of residing in Bullet County, Kentucky.
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CECIL, KAREN MARIE was born 9 March 1966 to BETTY A. RHODES (possibly her maiden name) in Jefferson County, Kentucky, United States of America.
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CECIL, KAREN R. (possibly her maiden name) had a baby named ERICA ROSE CECIL on 13 September 1980 in Marion County, Kentucky, United States of America.
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CECIL, KAREN R. (possibly her maiden name) had a baby named JOHN VICTOR CECIL on 27 December 1997 in Marion County, Kentucky, United States of America.
CECIL, KAREN R who was 29 (born ABT 1958) married 13 NOV 1987 in BURNET COUNTY, TEXAS, U.S.A. a groom named PHIL R BOUZA who was 24 (born ABT 1963).
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CECIL, KAREN R. was born 27 December 1938, received Social Security number 305-42-2629 (indicating Indiana) and, Death Master File says, died 10 June 1993
CECIL, KAREN R. was born 28 November 1957 to MARY THOMPSON (possibly her maiden name) in Marion County, Kentucky, United States of America.
CECIL, KAREN S. (possibly her maiden name) had a baby named DAVID NICHOLAS BRYANT on 8 February 1983 in Jefferson County, Kentucky, United States of America.
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CECIL, KAREN S. (possibly her maiden name) had a baby named TRACEY RENEE BRYANT on 3 June 1985 in Jefferson County, Kentucky, United States of America.
CECIL, KAREN S. (bride) of Jefferson County, Kentucky, who was 23 years old, White, with no previous marriage, and DAVID C. BRYANT of Jefferson County, Kentucky, who was 23 years old, White, status 1 (Never married) had a wedding ceremony 2 August 1980 on a license issued in Jefferson County, Kentucky, United States of America (Certificate number 17529) with the intention of residing in Jefferson County, Kentucky.
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CECIL, KAREN S. was born 11 October 1956 to EVELYN SHIRLEY (possibly her maiden name) in Jefferson County, Kentucky, United States of America.
CECIL, KAREN W. (bride) of Marshall County, Kentucky, who was 32 years old, White, with one previous marriage, and BRIAN K. DENNIS of Marshall County, Kentucky, who was 26 years old, White, status 1 (Never married) had a wedding ceremony 3 May 1993 on a license issued in Marshall County, Kentucky, United States of America (Certificate number 13771) with the intention of residing in Marshall County, Kentucky.
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CECIL, KARIN was born 15 August 1883, received Social Security number 545-38-8307 (indicating California) and, Death Master File says, died November 1973
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CECIL, KARINA MICHELE and JOHN BOSCONCILLIO CUEVA completed their Douglas County (Nevada, United States of America) marriage license application number 201518 and, on 27 October 2002 had their wedding ceremony.
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CECIL, KARLA R who was 21 (born ABT 1984) married 2 MAY 2005 in JEFFERSON COUNTY, TEXAS, U.S.A. a groom named KEVIN A WEEMS who was 42 (born ABT 1963).
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CECIL, KARLINE was born 16 July 1951, received Social Security number 566-61-6055 (indicating California) and, Death Master File says, died 11 March 2001
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CECIL, KARRIE SHEA was born 17 March 1988 to ANGELA J. FAUGHT (possibly her maiden name) in Daviess County, Kentucky, United States of America.
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CECIL, KARRI STATEN and JEROME BERNARD SCHILDMEYER applied for a marriage license in Franklin County, Ohio, United States of America 6 June 1997, which license was issued 6 June 1997, which was valid until 5 August 1997, and which was returned 5 August 1997, indicating that a marriage happened 2 August 1997.
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CECIL, KASEY NACOLE was born 21 November 1986 to PATRICIA V. KNIGHT (possibly her maiden name) in Daviess County, Kentucky, United States of America.
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CECIL, KATE was born 27 August 1886, received Social Security number 223-62-7766 (indicating Virginia) and, Death Master File says, died March 1982
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CECIL, Kate (mother), and George Mays who was born in Tazewell Co., Va., had a baby boy, Edgar Cecil MAYS born 24 Jul 1876 in Tazewell Co., Va..
CECIL, Kate (mother), and George Mays, had a baby boy, Edgar Cecil MAYS born ABT 1877.
CECIL, Kate married in 1914 in Wetzel a groom named S F Dulaney.
CECIL, Kate married in 1914 in Wetzel, West Virginia, United States a groom named S F Dulaney.
CECIL, KATE C. was born 1 October 1921, received Social Security number 408-28-0432 (indicating Tennessee) and, Death Master File says, died 14 January 2006
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CECIL, KATELYN H. married JUSTIN M. WOOLEN on 10 May 2014 using a marriage license issued in Marion County, Indiana, United States of America.
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CECIL, KATELYNN SAMARA was born 29 April 1998 to PEGGY S. HENDRICKSON (possibly her maiden name) in Whitley County, Kentucky, United States of America.
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CECIL, KATE R. died 11 November 1916 at age 87 in Boyd County, Kentucky, United States of America.
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CECIL, KATHARINE was born 4 June 1892, received Social Security number 505-86-0962 (indicating Nebraska) and, Death Master File says, died September 1987
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CECIL, KATHARINE was born 24 August 1921, received Social Security number 405-20-7240 and, Death Master File says, died 16 February 2004
CECIL, KATHARINE I. was born 18 April 1898, received Social Security number 513-28-2248 (indicating Kansas) and, Death Master File says, died 9 November 1988
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CECIL, KATHERINE (possibly her maiden name) had a baby named BERNARO L. RITCHIE on 14 September 1956 in Nelson County, Kentucky, United States of America.
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CECIL, KATHERINE (possibly her maiden name) had a baby named DAVID R. RITCHIE on 18 February 1955 in Nelson County, Kentucky, United States of America.
CECIL, KATHERINE (possibly her maiden name) had a baby named THOMAS 1. SHORES on 14 August 1920 in Daviess County, Kentucky, United States of America.
CECIL, KATHERINE (possibly her maiden name) had a baby named THOMAS J. SHORES on 4 February 1918 in Daviess County, Kentucky, United States of America.
CECIL, KATHERINE (possibly her maiden name) had a baby named VIRGINIA A. WILLIAMS on 26 August 1928 in Boyd County, Kentucky, United States of America.
CECIL, KATHERINE was born 19 February 1926, received Social Security number 031-14-6683 (indicating Massachusetts) and, Death Master File says, died September 1974
CECIL, KATHERINE was born 25 October 1919, received Social Security number 560-20-2340 (indicating California) and, Death Master File says, died September 1984
CECIL, KATHERINE was born 28 September 1879, received Social Security number 229-60-1848 (indicating Virginia) and, Death Master File says, died January 1977
CECIL, KATHERINE who was 20 (born ABT 1946) married 28 JAN 1966 in CAMERON COUNTY, TEXAS, U.S.A. a groom named JOSEPH BENNETT VASQUEZ who was 24 (born ABT 1942).
CECIL, KATHERINE A who was 21 (born ABT 1976) married 9 AUG 1997 in DALLAS COUNTY, TEXAS, U.S.A. a groom named ANDREW P JARRETT who was 21 (born ABT 1976).
Check the source file which is free. Also check Archives for KATHERINE A CECIL.
CECIL, KATHERINE A. was born 19 June 1942, received Social Security number 356-32-8600 (indicating Illinois) and, Death Master File says, died 4 July 2012
CECIL, KATHERINE A. was born 23 February 1909, received Social Security number 504-12-9242 (indicating South Dakota) and, Death Master File says, died 15 August 1993
CECIL, KATHERINE ALEXANDRA was born 24 January 1996 in Missouri, United States of America. Special thanks to Reclaim the Records. Please consider donating to them.
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CECIL, KATHERINE C. (possibly her maiden name) had a baby named LISA CAROL RITCHIE on 17 November 1966 in Barren County, Kentucky, United States of America.
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CECIL, KATHERINE C. was born 15 February 1923, received Social Security number 051-18-3934 (indicating New York) and, Death Master File says, died 30 October 2002
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CECIL, KATHERINE C. was born 15 January 1932 to ANNA COY (possibly her maiden name) in Nelson County, Kentucky, United States of America.
CECIL, KATHERINE D. was born 19 February 1931 to BRIDGET BARTTEY (possibly her maiden name) in Jefferson County, Kentucky, United States of America.
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CECIL, KATHERINE F. was born 10 May 1955 to ANITA WESTFALL (possibly her maiden name) in Jefferson County, Kentucky, United States of America.
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CECIL, KATHERINE H. (possibly her maiden name) had a baby named HEATHER NICOLE CECIL on 9 July 1973 in Perry County, Kentucky, United States of America.
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CECIL, KATHERINE J who was 22 (born ABT 1970) married 14 FEB 1992 in NUECES COUNTY, TEXAS, U.S.A. a groom named JUAN M SANCHEZ who was 22 (born ABT 1970).
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CECIL, KATHERINE JANE was born 15 June 1954 in Missouri, United States of America. Special thanks to Reclaim the Records. Please consider donating to them.
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CECIL, KATHERINE L who was 24 (born ABT 1966) married 29 SEP 1990 in DALLAS COUNTY, TEXAS, U.S.A. a groom named STEPHEN P ORSAK who was 25 (born ABT 1965).
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CECIL, KATHERINE LOUISE was born 14 June 1985 to DONNA M. MUDD (possibly her maiden name) in Daviess County, Kentucky, United States of America.
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CECIL, KATHERINE LOUISE was born 26 January 1978 in Missouri, United States of America. Special thanks to Reclaim the Records. Please consider donating to them.
CECIL, KATHERINE M. was born 19 May 1920, received Social Security number 136-12-3528 (indicating New Jersey) and, Death Master File says, died 16 September 1996
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CECIL, Katherine Margaret married in 1946 in Wyoming, West Virginia, United States a groom named Robert Clayton Means.
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CECIL, KATHERINE R. was born 21 July 1912, received Social Security number 412-52-9994 (indicating Tennessee) and, Death Master File says, died 22 March 1995
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CECIL, KATHJO was born 20 December 1954, received Social Security number 527-98-9738 (indicating Arizona) and, Death Master File says, died 31 October 2004
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CECIL, KATHLEAN was born 7 April 1900, received Social Security number 234-40-1909 (indicating West Virginia) and, Death Master File says, died December 1978
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CECIL, KATHLEEN was born 9 September 1925, received Social Security number 235-24-0347 (indicating West Virginia) and, Death Master File says, died June 1980
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CECIL, KATHLEEN was born 31 January 1948 to PAULINE HUNTER (possibly her maiden name) in Menifee County, Kentucky, United States of America.
CECIL, KATHLEEN was born 5 September 1928 in Missouri, United States of America. Special thanks to Reclaim the Records. Please consider donating to them.
CECIL, Kathleen (mother), and R. Clayton Means who was born in Wva, had a baby boy, R. Clayton MEANS born in Mullens, Wva.
CECIL, Kathleen married in 1946 in Mercer, West Virginia, United States a groom named James P Little.
CECIL, KATHLEEN A. was born 3 December 1953, received Social Security number 035-34-4805 (indicating Rhode Island) and, Death Master File says, died 21 November 2012
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CECIL, KATHLEEN C. was born 12 May 1917, received Social Security number 539-01-7278 (indicating Washington) and, Death Master File says, died 2 March 2007
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CECIL, KATHLEEN D. was born 12 November 1956, received Social Security number 454-21-3991 (indicating Texas) and, Death Master File says, died 9 March 2004
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CECIL, KATHLEEN M. (possibly her maiden name) had a baby named JESSICA MARIE GARDNER on 16 November 1986 in Jefferson County, Kentucky, United States of America.
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CECIL, KATHLEEN M who was 23 (born ABT 1984) married 22 SEP 2007 in TARRANT COUNTY, TEXAS, U.S.A. a groom named STEVEN E SOTO who was 25 (born ABT 1982).
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CECIL, KATHLEEN M. was born 12 October 1957 to MARY SINGHISER (possibly her maiden name) in Jefferson County, Kentucky, United States of America.
CECIL, Kathleen Mae (mother) Michael Eugene CECIL born 19 Aug 1951 in Bluefield, W. Va..
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CECIL, Kathleen Marie married in 1952 in Raleigh, West Virginia, United States a groom named Clarence Raymond Cole.
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CECIL, KATHLEEN O. was born 9 December 1919, received Social Security number 288-05-1895 (indicating Ohio) and, Death Master File says, died 1 January 2004
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CECIL, KATHLEEN R. (possibly her maiden name) had a baby named JON THOMAS DONLON on 24 April 1983 in Jefferson County, Kentucky, United States of America.
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CECIL, KATHLEEN R. (possibly her maiden name) had a baby named KATIE ANN DONLON on 24 March 1980 in Jefferson County, Kentucky, United States of America.
CECIL, KATHLEEN R. (possibly her maiden name) had a baby named LAUREN MICHELLE DONLON on 8 March 1986 in Jefferson County, Kentucky, United States of America.
CECIL, KATHLEEN R. was born 4 November 1951 to HELEN CARRICO (possibly her maiden name) in Jefferson County, Kentucky, United States of America.
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CECIL, KATHLEN M. (bride) of Jefferson County, Kentucky, who was 25 years old, White, with no previous marriage, and JAMES C. GARDNER of Jefferson County, Kentucky, who was 33 years old, White, status 3 (Last marriage ended by divorce) with one previous marriage, had a wedding ceremony 14 October 1983 on a license issued in Jefferson County, Kentucky, United States of America (Certificate number 26998) with the intention of residing in Jefferson County, Kentucky.
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CECIL, KATHLEN R. (bride) of Jefferson County, Kentucky, who was 26 years old, White, with one previous marriage, and THOMAS M. DONLON of Jefferson County, Kentucky, who was 27 years old, White, status 1 (Never married) had a wedding ceremony 15 April 1978 on a license issued in Jefferson County, Kentucky, United States of America (Certificate number 06806) with the intention of residing in Oldham County, Kentucky.
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CECIL, KATHRYN was born 31 August 1907, received Social Security number 275-01-4908 (indicating Ohio) and, Death Master File says, died July 1965
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CECIL, KATHRYN A. was born 20 April 1950 to MARY KLUESNER (possibly her maiden name) in Jefferson County, Kentucky, United States of America.
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CECIL, KATHRYN A. of Jefferson County, Kentucky died 21 April 1950 at age U/1 in Jefferson County, Kentucky, United States of America.
CECIL, KATHRYN D. (possibly her maiden name) had a baby named KATHRYN JEAN LOGSDON on 24 November 1985 in Jefferson County, Kentucky, United States of America.
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CECIL, KATHRYN D. (bride) of Jefferson County, Kentucky, who was 19 years old, White, with no previous marriage, and RICKY L. LOGSDON of Jefferson County, Kentucky, who was 20 years old, White, status 1 (Never married) had a wedding ceremony 21 May 1980 on a license issued in Jefferson County, Kentucky, United States of America (Certificate number 10638) with the intention of residing in Jefferson County, Kentucky.
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CECIL, KATHRYN D. was born 30 November 1960 to NORMA GLASSCOCK (possibly her maiden name) in Jefferson County, Kentucky, United States of America. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,205 |
Co-op's official mascot Ko-suke is back with a new sticker set for you! Fun times are ensured whenever you use these adorable stickers in your chats. Friend Co-op Kyosai's official account to get them for free.
No matter how tough the situation, Ko-suke is ready to take on the challenge! Give your chats that extra boost with these special stickers. Friend Co-op Kyosai's official account to get them for free. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,747 |
Air Conditioning System Diagnostic Services | Cox Car Care, Inc.
Cox Car Care, Inc. provides Air Conditioning System Diagnostic services to Raleigh, NC, Cary, NC, Apex, NC, and other surrounding areas.
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Why Should You Have Air Conditioning System Diagnostic Services Performed at Cox Car Care, Inc.?
We proudly service the Air Conditioning System Diagnostic needs of customers in Raleigh, NC, Cary, NC, Apex, NC, and surrounding areas. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,422 |
<!DOCTYPE html>
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<head>
<title>Razorfish | Experiment Title</title>
<link rel="icon" href="favicon.ico" type="image/x-icon" />
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<script src="lib/stats.min.js"></script>
<script src="lib/dat.gui.min.js"></script>
<script src="lib/razor.min.js"></script>
<script src="lib/raf.js"></script>
</head>
<body>
<header>Experiment Title</header>
<canvas id="canvas"></canvas>
<script src="js/example.js"></script>
<script>
var example = new Example();
</script>
</body>
</html> | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,945 |
Eat, Sleep, Drink
OX Loves
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Home Love your area:
Knowledge, Perspectives, Eat, Sleep, Drink, Eat, Drink
Love your area:
Best of Witney
"Blanket makers also traditionally made mop heads and at one point it is believed that every Royal Navy ship had a Witney mop aboard."
Let your Home Earn its Keep:
From real ales to hay bales, via blankets and confectionary here are OX Magazine's best-loved Witney staples – the latest instalment in our celebration of Oxfordshire's many impeccable towns and villages.
Cogges Manor Farm
Cogges Manor Farm © Ric Mellis
Across the River Windrush stands Cogges Manor Farm. The house dates all the way back to the 13th century, while Cogges appears in the Domesday Book of 1086. You can even see the first owner of Cogges on a portion of the Bayeux tapestry. Cogges manor came under the purview of Henrys VII and VIII who then gave the land to the founder of Trinity College Oxford, Thomas Pope. Fast forward to the modern day and the grounds have been lovingly restored by volunteers into a working garden with buildings used for all manner of events. Check their website for details about pantos, carols and Christmas markets in December.
Church Lane | cogges.org.uk
Wychwood Brewery
'Afraid of the dark, Lagerboy?' – So reads the strapline to Wychwood Brewery's flagship ale, Hobgoblin. (A disgruntled reader of this ad made a complaint to the Advertising Standards Agency which was duly quashed.) This world-famous brewery offers two-hour guided tours that can be booked through their website and make excellent gifts for any ale fanatics in your life. See how the beer is concocted from its raw ingredients and sample the finished product at the tour's end.
Eagle Industrial Estate, The Crofts | wychwood.co.uk
The Shake Shop
Local sisters Kim and Debbie opened their incredibly popular little shop in 2011 and have been 'shaking up Witney' ever since. Dedicated to flying the flag for family-run, independent business, the shop has gone from strength to strength, expanding from their initial space to include a 50s diner. They have a smoothie/milkshake menu as long as your arm and a range of old-fashioned sweets sold by weight that is sure to resurrect pangs of nostalgia from anyone who used to spend their pocket money on penny sweets. They also run events raising money for local charities, and once made 270 milkshakes in their seventh birthday happy hour.
Market Square | shakeshop.co.uk
Witney Blanket Hall & Pie Shop
Witney has been synonymous with woollen blankets since The Middle Ages. Until 1908, the word 'Witney' was allowed to be used generally to denote quality and could be appended to products made up and down the country. Blanket makers also traditionally made mop heads and at one point it is believed that every Royal Navy ship had a Witney mop aboard. Witney Blanket Hall was built in 1721. For over 120 years, every blanket woven in the town passed through its doors to be weighed and measured. After passing through many different incarnations since the mid-19th century, the hall has reopened in a nod to Witney's heritage. On sale are blankets, throws and other woollen goods as well as tea, coffee, cakes and even their own Blanket Hall Beer.
High Street | www.cotswoldwoollenweavers.co.uk/blankethall
Greenway Antiques
Housed in a charming 17th century building, Greenway Antiques has been in Witney for 35 years. Independent and family-owned, the personally selected items change and evolve constantly, so there's always something to look out for on a visit. Given their personal engagement with their stock, there's no better place to get expert advice on a special antique. Covering the periods from the 17th to the mid-20th centuries, they carry items for the home and garden, furniture, decorative items and metalware, especially for the fireplace.
Corn Street | greenwayantiques.co.uk
Save a Life: Donate Blood
Myla's Story
Every 20 minutes someone in the UK is diagnosed with blood cancer and the register of stem cell donors – who are needed to save thousands of patients' lives – does not
A Band of Brothers
A Band of Brothers is a fast growing, community-based organisation established ten years ago, seeking to improve the lives of young men who are in crisis, by providing them with mentoring and
The Rise of the Small Museum
From hats to lawnmowers, Britain's speciality museums are going from strength to strength. With old industrial sites filling up with exhibitions of things like pens and coffins, as seen
Soul Fire Farm
"This Land is our Classroom"
Leah Penniman, author, activist and farmer, is committed to ending racism and injustice in our food system. She and her team at Soul Fire Farm in upstate New York are working to increase farmland
Subscribe to the OX Newsletter
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Site by Whitespace whitespace-dots | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,390 |
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<title>Form Test</title>
<link rel="stylesheet" href="//netdna.bootstrapcdn.com/bootstrap/3.0.0-wip/css/bootstrap.min.css" />
<link rel="stylesheet" href="//code.jquery.com/ui/1.10.4/themes/black-tie/jquery-ui.css" />
<style>
/* Form and inputs */
form {
width: 500px;
margin: 0 auto;
padding: 20px;
display: block;
}
input.form-control {
width: 375px;
}
button, input[type="submit"], .button {
margin-bottom: 8px;
}
/* While server is being requested */
form.validating-server-side {
background: #F2F2F2;
}
input.validating-server-side {
opacity: 0.5;
background: lightgoldenrodyellow;
}
/* modify inputs for password strength */
.password-strength input.form-control {
width: 375px;
margin-right: 4px;
display: inline;
}
.password-strength label {
display: block;
}
/* Checkboxes */
.form-group.check-boxes input {
margin-left: 10px;
}
span.help {
color: #999 !important;
}
/* Error container for form C */
#error-container div {
color: red;
line-height: 140%;
}
#error-container div:last-child {
padding-bottom: 10px;
}
</style>
</head>
<body>
<div>
<form action="" id="form-a" role="form">
<div class="form-group">
<label class="control-label" for="inline-suggestions">Inline suggestions</label>
<input name="inline suggestions" type="text" id="inline-suggestions" class="form-control" data-suggestions="Monkey, Horse, Fox, Tiger, Elephant" />
</div>
<div class="form-group">
<label class="control-label" for="country-suggestions">Country suggestions</label>
<input name="country suggestions" data-validation="country" type="text" id="country-suggestions" class="form-control" />
</div>
<div class="form-group">
<label class="control-label" for="country-suggestions">Swedish county suggestions</label>
<input name="Swedish county suggestion" data-validation="swecounty" type="text" id="swedish-county-suggestions" class="form-control" />
</div>
<div class="form-group">
<label class="control-label">Year</label>
<input name="birth" class="form-control"
data-validation="date"
data-validation-format="yyyy-mm-dd"
data-suggestions="2014-01-15,2014-01-16,2014-01-17" />
</div>
<div class="form-group">
<label class="control-label">Datepicker</label>
<input name="birth2" class="form-control"
data-validation="date"
data-validation-format="mm/dd/yyyy"
id="datepicker" />
</div>
<div class="form-group">
<label class="control-label">Number 0-10 (accepting floats with comma)</label>
<input name="floats" class="form-control"
data-validation="number"
data-validation-allowing="range[0;10], float"
data-validation-decimal-separator=","
/>
</div>
<div class="form-group password-strength">
<label class="control-label" for="password">Display password strength (only strong)</label>
<input name="password" type="password" id="password" class="form-control" data-validation="strength" data-validation-strength="3" />
</div>
<div class="form-group">
<label class="control-label">Alphanumeric and -_ and spaces</label>
<input name="alphanumeric with spaces" class="form-control" name="test" data-validation="alphanumeric" data-validation-allowing="-_ " />
</div>
<div class="form-group">
<label class="control-label">Alphanumeric only</label>
<input name="aplhanumeric only" class="form-control" name="test2" data-validation="alphanumeric" />
</div>
<div class="checkbox form-group">
<label>
<input name="checkbox" type="checkbox" data-validation="required" /> Must be checked
</label>
</div>
<div class="form-group">
<label class="control-label">Must choose one</label>
<br />
<input name="radio" type="radio" data-validation="required" value="1" /> A
<input name="radio" type="radio" value="1" /> B
<input name="radio" type="radio" value="1" /> C
<input name="radio" type="radio" value="1" /> D
</div>
<div class="form-group">
<label class="control-label">Even numbers only</label>
<input name="even numbers" class="form-control" name="test4" data-validation="even_number" />
</div>
<div class="form-group">
<label class="control-label">Make a choice</label>
<br />
<select name="choice" data-validation="required" data-validation-error-msg="Please make a choice">
<option value="">- - Choose - -</option>
<option>A</option>
<option>B</option>
<option>C</option>
<option>D</option>
</select>
</div>
<div class="form-group">
<label class="control-label">Text</label>
(<span id="max-len">20</span> chars left)<br />
<textarea id="text-area" class="form-control" name="some-text"></textarea>
</div>
<div class="form-group">
<label class="control-label">Server validation</label>
<input class="form-control" name="code" value="secret"
data-validation-help="The word is "secret""
data-validation="server"
data-validation-url="http://formvalidator.net/validate-email.php" />
</div>
<div class="form-group">
<label class="control-label">File validation</label>
<input type="file" name="some-file1" class="form-control"
data-validation="size mime required"
data-validation-size-error-msg="The file cant be larger than 400kb"
data-validation-error-msg="You must upload an image file (max 400 kb)"
data-validation-allowing="jpg, png, ico"
data-validation-max-size="400kb" />
</div>
<div class="form-group">
<label class="control-label">
Callback validation, set this value to "1" and
validation will fail
</label>
<input id="callback" class="form-control" />
</div>
<div class="form-group check-boxes">
<label>Checkbox group</label><br />
<label>
<input type="checkbox" name="box" value="1"
data-validation="checkbox_group"
data-validation-qty="1-2" /> 1
</label>
<label>
<input type="checkbox" name="box" value="2" /> 2
</label>
<label>
<input type="checkbox" name="box" value="3" /> 3
</label>
<label>
<input type="checkbox" name="box" value="4" /> 4
</label>
<label>
<input type="checkbox" name="box" value="5" /> 5
</label>
</div>
<p style="line-height: 200%">
<input type="submit" class="button">
<br />
<button class="button" type="button"
onclick="alert('From a is ' + ( $('#form-a').isValid({}, {}, false) ? 'VALID':'NOT VALID'));">
Test validation via js (<strong>without error messages</strong>)
</button>
<br />
<button class="button" type="button"
onclick="alert('From a is ' + ( $('#form-a').isValid() ? 'VALID':'NOT VALID'));">
Test validation via js (showing error messages)
</button>
<br />
<input type="reset" class="button">
</p>
</form>
<hr />
<form id="form-b">
<div class="form-group">
<label class="control-label">Test</label>
<input name="test" data-validation="number" type="text" />
</div>
<div class="form-group">
<label class="control-label">Password</label>
<input name="pass" data-validation="confirmation" type="password" />
</div>
<div class="form-group">
<label class="control-label">Password again</label>
<input name="pass_confirmation" type="password" />
</div>
<p>
<input type="submit" class="button">
<input type="reset" class="button">
</p>
</form>
<hr />
<form id="form-c">
<div class="form-group">
<label class="control-label">Country</label>
<input name="test" data-validation="country" data-validation-error-msg="No valid country given" />
</div>
<div class="form-group">
<label class="control-label">E-mail</label>
<input name="testmail" data-validation="email" data-validation-error-msg="E-mail is not valid" />
</div>
<div class="form-group">
<label class="control-label">Confirm e-mail</label>
<input name="test" data-validation="confirmation" data-validation-confirm="testmail" />
</div>
<div class="form-group">
<label class="control-label">Alphanumeric (will only be validated if the checkbox is checked)</label>
<input name="test2"
data-validation="alphanumeric"
data-validation-error-msg="Invalid..."
data-validation-if-checked="checker" />
<br />
<input type="checkbox" name="checker" />
</div>
<div id="error-container">
</div>
<p>
<input type="submit" class="button">
<input type="reset" class="button">
</p>
</form>
<hr />
<form id="form-d">
<h2>HTML5 attributes</h2>
<div class="form-group">
<label class="control-label">type="email"</label>
<input type="text" required="required" list="mejl" />
<datalist id="mejl">
<option value="Test">Test</option>
<option value="test2">test2</option>
<option value="test3">test3</option>
</datalist>
</div>
<div class="form-group">
<label class="control-label">type="url" (optional)</label>
<input type="url" />
</div>
<div class="form-group">
<label class="control-label">type="number"</label>
<input type="number" required="required" />
</div>
<div class="form-group">
<label class="control-label">type="number"</label>
<input type="number" required="required" maxlength="30" />
</div>
<div class="form-group">
<label class="control-label">type="number" range[-5;5]</label>
<input type="number" min="-5" max="5" required="required" />
</div>
<div class="form-group">
<label class="control-label">pattern="^([a-z]+)$"</label>
<input type="text" name="some-colorz" list="some-colorz" pattern="^([a-z]+)$" required="required" />
<datalist id="some-colorz" style="display: none">
<option value="Green">Green</option>
<option value="Blue">Blue</option>
<option value="Red">Red</option>
<option value="Black">Black</option>
<option value="White">White</option>
</datalist>
</div>
<p>
<input type="submit" class="button">
<input type="reset" class="button">
</p>
</form>
</div>
<script src="//ajax.googleapis.com/ajax/libs/jquery/1/jquery.min.js"></script>
<script src="//code.jquery.com/ui/1.10.4/jquery-ui.min.js"></script>
<script src="jquery.form-validator.js"></script>
<script>
(function($, window) {
var dev = '.dev'; //window.location.hash.indexOf('dev') > -1 ? '.dev' : '';
// setup datepicker
$("#datepicker").datepicker();
// Add a new validator
$.formUtils.addValidator({
name : 'even_number',
validatorFunction : function(value, $el, config, language, $form) {
return parseInt(value, 10) % 2 === 0;
},
borderColorOnError : '',
errorMessage : 'You have to give an even number',
errorMessageKey: 'badEvenNumber'
});
window.applyValidation = function(validateOnBlur, forms, messagePosition) {
if( !forms )
forms = 'form';
if( !messagePosition )
messagePosition = 'top';
$.validate({
form : forms,
language : {
requiredFields: 'Du måste bocka för denna'
},
validateOnBlur : validateOnBlur,
errorMessagePosition : messagePosition,
scrollToTopOnError : true,
borderColorOnError : 'purple',
modules : 'security'+dev+', location'+dev+', sweden'+dev+', html5'+dev+', file'+dev+', uk'+dev,
onModulesLoaded: function() {
$('#country-suggestions').suggestCountry();
$('#swedish-county-suggestions').suggestSwedishCounty();
$('#password').displayPasswordStrength();
},
onValidate : function($f) {
console.log('about to validate form '+$f.attr('id'));
var $callbackInput = $('#callback');
if( $callbackInput.val() == 1 ) {
return {
element : $callbackInput,
message : 'This validation was made in a callback'
};
}
},
onError : function($form) {
if( !$.formUtils.haltValidation ) {
alert('Invalid '+$form.attr('id'));
}
},
onSuccess : function($form) {
alert('Valid '+$form.attr('id'));
return false;
}
});
};
$('#text-area').restrictLength($('#max-len'));
window.applyValidation(true, '#form-a', 'top');
window.applyValidation(false, '#form-b', 'element');
window.applyValidation(true, '#form-c', $('#error-container'));
window.applyValidation(true, '#form-d', 'element');
// Load one module outside $.validate() even though you do not have to
$.formUtils.loadModules('date'+dev+'.js', false, false);
$('input')
.on('zbeforeValidation', function() {
console.log('About to validate input "'+this.name+'"');
})
.on('validationz', function(evt, isValid) {
var validationResult = '';
if( isValid === null ) {
validationResult = 'not validated';
} else if( isValid ) {
validationResult = 'VALID';
} else {
validationResult = 'INVALID';
}
console.log('Input '+this.name+' is '+validationResult);
});
})(jQuery, window);
</script>
<body>
</html> | {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"http:\/\/crypto.stackexchange.com\/questions\/8586\/in-ecdsa-how-many-field-operations-are-used-for-signature-verification\/8608","text":"In ECDSA, how many field operations are used for signature verification?\n\nI am wondering about the computational cost of ECDSA signature verification, in term of multiplications in the base field; and, as an aside, in term of (much cheaper) additions. To make things concrete, assume ECDSA in a field $GF(p)$ per FIPS 186-3 (which refers to ANSI X9.62:2005 (webstore)), using parameters as in appendix D.1.2.\n\nI'm interested first in the straightest standard-abiding method with a cost in the right ballpark; then in the most worthwhile refinements of that. I have identified (by name, but not quantitative effect, especially for the first one):\n\n\u2022 projective coordinate systems, said to allow replacing inversion in the base field (as used in straight point addition in the elliptic curve group) by fewer operations than standard inversion algorithms, e.g $x\\mapsto x^{p-2}\\bmod p$ (or the more efficient extended euclidian, but that does not quite match my \"multiplications in the base field\" evaluation criteria);\n\u2022 Shamir's trick, where $a*P+b*Q$ is computed by adding either $P$, $Q$ or $P+Q$ (on the elliptic curve group) when scanning the bits of integer multiplicands $a$ and $b$;\n\u2022 sliding window techniques when scanning the bits of a multiplicand.\n\nI'm not interested in the cost of verification of a certificate introducing the public key, or the cost of hashing; and only marginally by savings enabled by pre-computations, or techniques applicable only to $GF(2^n)$, or using the special form of the $p$ parameter (which allows speed-up of modular reduction $\\bmod p$, but leave the number of operations in the base field unchanged).\n\n-\nThe Ed25519 code (on a 255 bit curve) I'm using seems to need around 1500 squarings and 1500 additions to verify a signature. But it's 1) another signature algorithm 2) an edwards curve. | It uses Shamir's trick and some form of pre-computation, but no batch verification. \u2013\u00a0CodesInChaos Jun 5 '13 at 14:06\nI'm talking about operations in the underlying field. In this case multiplication or squaring modulo $2^{255}-19$. I didn't count additions etc. since those are cheap. \u2013\u00a0CodesInChaos Jun 5 '13 at 17:05\n@CodesInChaos: so that's 1500 squarings and 1500 multiplications in the base field. Makes sense. Thanks. \u2013\u00a0fgrieu Jun 5 '13 at 18:18\n\nUsually, the interleaving of wNAFs is faster than Shamir's trick for simultaneous point multiplication. Using interleaving with window sizes 5 (fixed) and 4 (random), the total number of operations is roughly (taken from Hankerson et al's Guide to ECC)\n\n$(0.37t + 3)A + (t + 1)D$\n\nwhere t is the bitlength of the scalars (which should be the same as the bitlength of the field size for standard curves), A is a mixed point addition (projective + affine) and D is a point doubling.\n\nFor standard prime curves with $a=-3$, using Jacobian coordinates, we have (from the EFD):\n\n$A = 7M + 4S$,\n\n$D = 3M + 5S$,\n\nwhere $M$ is a field multiplication and $S$ is a field squaring. Therefore,\n\n$M = (5.59t + 24)$,\n\n$S = (6.48t + 17)$.\n\nFor the 128-bit level of security, if I got everything right, $t=256, M = 1455, S = 1675$.\n\n-\nI'll try to cross-check this (which is much easier than coming with it in the first place). And that's pretty similar to the values in this comment. Thanks! \u2013\u00a0fgrieu Jun 6 '13 at 17:36","date":"2015-11-29 21:22:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.8360547423362732, \"perplexity\": 1413.97871218283}, \"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-2015-48\/segments\/1448398459875.44\/warc\/CC-MAIN-20151124205419-00077-ip-10-71-132-137.ec2.internal.warc.gz\"}"} | null | null |
Lederach is an unincorporated community in Lower Salford Township in Montgomery County, Pennsylvania, United States. Lederach is located at the intersection of Pennsylvania Route 113, Salfordville Road/Morris Road, Old Skippack Road, and Cross Road.
References
Unincorporated communities in Montgomery County, Pennsylvania
Unincorporated communities in Pennsylvania | {
"redpajama_set_name": "RedPajamaWikipedia"
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Q: Add integers to array using cin simultaneously in C++ Hellow guys i want to add nine integers to array at once time without pressing enter key in run time. please guys tell me how to add nine integers to array simultaneously in C++.
Thanks!
A: If you want to process each integer value right after its input in console is complete (e.g. in that a blank indicates that the next integer value shall begin), you are in a bad position.
The reason is that terminal input (beyond of what your C++ program can influence) often is buffered, and even cin might not receive any character until Enter or EOF is pressed in the terminal.
There may exist workarounds like conio.h or ncurses, but the are not standard and probably not worth the effort in your situation unless you really need to implement integer scanning for a production environment tightly connected to console input.
Try it out and compare input taken directly from console to input from a stream that is already "filled" with enough input:
int main() {
stringstream ss("12 34 56 78 90 10 11 12 13");
//istream &in = ss; // would output each integer immediately.
istream &in = cin; // will probably wait for enter before processing begins.
int value = 0;
for (int i=0; i<9; i++) {
if (! (in >> value))
break;
cout << value << "; ";
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,435 |
Q: How to append GET parameters to elastic-search when using tire I wish to append the parameter ?search_type=count to elastic-search.
curl -XGET 'http://127.0.0.1:9200/votes/_search?search_type=count' -d '
{
"facets" : {
"votes" : {
"terms" : {
"field" : "question_id"
}
}
}
}'
Something like that. How do I tell tire to append that search_type=count?
A: search_type=count queries is not supported yet in tire, workaround noted in github issue is to set size 0.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,559 |
\section{Introduction}
\lettrine{\textbf{T}}{he} frequency competition between the communication and radar systems will inevitably cause severe congestion within the finite spectrum of sub-6 GHz. To save RF resources, the sub-6 GHz spectrum allocated to radar systems can be made available for sharing between radar and wireless systems \cite{griffithsPROC2015}. Going beyond spectral co-existence, the joint radar-communication (JRC) systems integrate the complete hardware platform and the transmitted signal waveform of the two systems \cite{liuTCOM2020}. Recently the JRC systems have been advanced for hardware efficient architectures and employing low resolution converters \cite{aryanICC2021, dizdarICC2021} that are sought for in emerging fifth generation (5G) wireless communications \cite{aryanTGCN2021}.
The JRC systems are classified into three categories \cite{zhangJSTSP2021}. One is \emph{communication-centric} JRC which implements radar sensing as the secondary function of the existing communication system, e.g., \cite{strum2011}.
Second is \emph{radar-centric} JRC which integrates wireless communication as the secondary function of the existing radar system, e.g., \cite{eldarTSP2020}.
Third is \emph{joint JRC} that offers tunable trade-off between both the operations, e.g., \cite{aryanICC2021}. The design concept of the radar-centric JRC systems is to embed the communication information into the transmitted radar pulses and meanwhile maintain the sensing performance \cite{hanIET2013}. Furthermore, \cite{hassanienSPM2019} classifies the information embedding approaches for the radar-centric JRC systems into beam pattern modulation, index modulation (IM), and fast-time modulation.
The IM approach which utilizes the randomness and uniqueness of the radar system to convey the information outperforms the conventional methods with enhanced data throughput and robust error performance \cite{eldarTSP2020}. It aims to generate a comprehensive coding mechanism so that the blocks of information bits can be mapped accordingly to the indices of function blocks (communication codewords) within the JRC system \cite{clancyWCNC2015}. Example function blocks can cover the spatial position of the antennas in the multiple-input multiple-output (MIMO) system, the carrier frequency in the multiple-carriers system, and the type of waveform in the multi-waveform system. The work in \cite{eldarTSP2020_2} extends frequency agile radar \cite{axelsson2007} into the multi-carrier MIMO transmission.
\cite{eldarTSP2020} further extends this model for the JRC systems by exploiting the frequency and antenna agility. However, the exploitation of information embedding with more effective signal processing approaches and enhanced error performance has not been widely studied.
\emph{Contributions}: This paper implements an efficient IM based information embedding approach in JRC systems that utilizes the spectral and spatial randomness of the radar pulse signals. For that, the communication codewords are embedded in the transmitted radar pulses by selecting the corresponding indices of the carrier frequencies and antenna allocations. Firstly, we propose a novel codebook based minimum Euclidean distance (MED) maximization approach to select the best combinations of antennas and frequencies to form our signal constellation. We then further augment performance through a constellation randomization pre-scaling (CRPS) scheme for dual functionality transmission. We show the benefits of integrating the CRPS scheme followed by the codebook design that maximizes the signal-to-noise ratio (SNR) gain. Our numerical evaluation reveals an enhanced bit error rate (BER) for the proposed approaches when compared to the existing baseline.
\vspace{-2mm}
\section{System Model}
\label{sec:system}
We consider IM-JRC, which is implemented based on a multi-carrier MIMO radar (MCMR) \cite{eldarTSP2020_2} through IM. In MCMR, the collection of available carrier frequencies is as
$ F=\{f_\textrm{c}+m\Delta f | m\in (0,M-1)\}$,
where $M$ is the number of carrier frequencies available at the transmitter (TX), $f_\textrm{c}$ is the initial carrier frequency, and $\Delta f$ is the frequency step. Let $L_\textrm{R}$ be the number of the TX antennas, with $d=10 \frac{c}{f_\textrm{c}}$ being the spacing between the neighboring antenna elements and $c$
is the speed of light. Suppose that there are $N$ radar pulses repeatedly transmitted during each coherent processing interval of MCMR. Thus, the radar pulses will be transmitted within the time instance from $n T_\textrm{r}$ to $n T_\textrm{r}+T_\textrm{p}$, where $n \in (0,N-1)$.
During operation, MCMR randomly chooses a subset of $K$ carrier frequencies from $F$ for the $n$-th radar pulse as
$F_\textrm{n}= \{\Omega_\textrm{n,k} | k\in(0,K-1)\}$,
where $F_\textrm{n}\subset F$, and $\Omega_\textrm{n,k}$ is an individual selected carrier frequency. Subsequently, these frequencies are randomly allocated among the phased array antennas, so that each antenna utilizes a single carrier frequency from $F_\textrm{n}$. Let $f_\textrm{n,l}$ be the carrier frequency transmitted by the $l$-th antenna. If the same antenna element also happens to transmit at $\Omega_\textrm{n,k}$, then $\Omega_\textrm{n,k}=f_\textrm{n,l}$. In MCMR, the TX radar waveform is
\begin{equation}
\phi(f,t)= \text{rect}(t/T_\textrm{p})\text{exp}(j2\pi ft).
\end{equation}
In addition, the radar signal emitted from each antenna element should be multiplied with a weighting function to direct the antenna beam to the optimum angle $\theta$ \cite{eldarTSP2020}, given by
\begin{equation}
w_\textrm{l}(\theta)= \text{exp}(j2\pi f_\textrm{c} ld\text{sin}\theta/c).
\end{equation}
To formulate the transmitted signal vector $\mathbf{x}(n,t)\in \mathbb{C}^{L_\textrm{R}}$ of the entire antenna array for the $n$-th radar pulse, define $\mathbf{x}_\textrm{k}(n,t)$ as the subset of $\mathbf{x}(n,t)$ that transmits at frequency $\Omega_\textrm{n,k}$. Then the finalized transmission model of MCMR is
\begin{equation}\label{eq:tx_mcmr}
\mathbf{x}(n,t) \!\!=\!\! \sum_{k=0}^{K-1}\mathbf{x}_\textrm{k}(n,t)\!\!=\!\!\sum_{k=0}^{K-1}\mathbf{P}(n,k)\mathbf{w}(\theta)\phi(\Omega_\textrm{n,k},t\!-\!nT_\textrm{r}),
\end{equation}
where $\mathbf{P}(n,k)\in \{0,1\}^{L_\textrm{R}\times L_\textrm{R}}$ is the diagonal matrix containing the spatial agility information. It depends on the diagonal vector $\mathbf{p}(n,k)\in \{0,1\}^{L_\textrm{R}}$, whose $l$-th element is $1$ if $f_\textrm{n,l}=\Omega_\textrm{n,k}$, and is $0$ otherwise.
The parameter $\phi(\Omega_\textrm{n,k},t-nT_\textrm{r})$ denotes the transmitted waveform at time instance $t$, which carries the frequency agility information.
For IM-JRC, considering a single pulse transmission, i.e., $N=1$, we simplify notations as $\mathbf{x}(t)=\mathbf{x}(n,t),\mathbf{x}_\textrm{k}(t)=\mathbf{x}_\textrm{k} (n,t),\mathbf{P}_\textrm{k}=\mathbf{P}(n,k)$ and $\mathbf{w}_\textrm{k}=\mathbf{w}(\theta)$. At the initial stage of transmission, IM-JRC chooses a subset of $K$ carrier frequencies out of $F$. The set of all possible frequency selections $\zeta$ is
$\zeta=\{F^{(i)} | F^{(i)} \subset F,|F^{(i)}| =K,\,\, i=0,1,2,...\}$,
where $F^{(i)}$ is an individual selection of frequencies. The cardinality of $\zeta$ is
$|\zeta|=\frac{M!}{K!(M-K)!}$.
We define $L_\textrm{K}=L_\textrm{R}/K$, where $L_\textrm{K}$ is assumed to be an integer greater than $1$, i.e., each carrier frequency is transmitted by precisely $L_\textrm{K}$ antenna elements.
$P$ is the collection of all potential antenna allocations, given by
$P\!\!=\!\!\{\mathbf{P}_\textrm{k}^{(i)}, k\in(0,K\!-\!1) | \text{trace}(\mathbf{P}_\textrm{k}^{(i)})\!\!=\!L_\textrm{K}, \,\,i\!=\!0,1,2,..\}$,
where $\{\mathbf{P}_0^{(i)},...,\mathbf{P}_\textrm{{K-1}}^{(i)}\}$ represents one possible allocation pattern. The cardinality of $P$ is
$|P|=\frac{L_\textrm{R}!}{{(L_\textrm{K}!)}^K}$.
IM-JRC integrating the frequency and spatial index modulation
and following \cite{duhamel2010}, the maximum number of bits that can be embedded in a single radar pulse transmission is
\begin{equation}
\text{log}_2 |\zeta||P| = \text{log}_2 \frac{M!}{K!(M-K)!} + \text{log}_2 \frac{L_\textrm{R}!}{{(L_\textrm{K}!)}^K}.
\end{equation}
Assuming $L_\textrm{C}$ receiver (RX) antenna elements, we have
\begin{equation}
\mathbf{y}_\textrm{C} (t)=\mathbf{H}\mathbf{x}(t)+\mathbf{n}_\textrm{C} (t)=\sum_{k=0}^{K-1} \mathbf{H}\mathbf{x}_\textrm{k} (t)+\mathbf{n}_\textrm{C}(t),
\end{equation}
where $\mathbf{y}_\textrm{C}(t)\in \mathbb{C}^{L_\textrm{C}}$ represents the received JRC signal. $\mathbf{H} \in \mathbb{C}^{L_\textrm{C} \times L_\textrm{R}}$ is the Rayleigh flat fading channel.
$\mathbf{n}_\textrm{C}(t)\in \mathbb{C}^{L_\textrm{C}} $ is the additive white Gaussian noise (AWGN).
Before the sampling process, the received JRC signal $\mathbf{y}_\textrm{C}(t)$ is firstly down-converted by a factor of $e^{-j2\pi f_\textrm{c} t}$ to remove the high frequency component $f_\textrm{c}$. After that, IM-JRC employs the Nyquist sampling theorem and samples $\mathbf{y}_\textrm{C}(t)$ at the time instances of $iT_\textrm{s}$, where $T_\textrm{s}$ is defined as the sampling period and $i\in(0,\floor{T_\textrm{p}/T_\textrm{s}}$). Assuming the input bitstream is randomly and equally distributed,
the optimum Nyquist sampling period is given by $T_\textrm{s}=1/M\Delta f$.
Considering $L_\textrm{T}=\floor{T_\textrm{p}/T_\textrm{s}}+1$ as the number of sampled outputs per pulse and $c_\textrm{k}\in(0,M-1)$ being the frequency index corresponding to the selected carrier frequencies $\Omega_\textrm{k}$ which is
$c_\textrm{k}= (\Omega_\textrm{k}-f_\textrm{c})/\Delta f$.
Recall \eqref{eq:tx_mcmr}, and after sampling the finalized reception model of IM-JRC is
\begin{equation}
\mathbf{Y}_\textrm{C}=\sum_{k=0}^{K-1} \mathbf{H}\mathbf{P}_\textrm{k} \mathbf{w}_\textrm{k} \boldsymbol{\Phi}_{c_\textrm{k}}^T+\mathbf{N}_\textrm{C} ,
\end{equation}
where $\mathbf{Y}_\textrm{C}\in \mathbb{C}^{L_\textrm{C} \times L_\textrm{T}}$ and $\mathbf{N}_\textrm{C}\in\mathbb{C}^{L_\textrm{C} \times L_\textrm{T}}$ are the sampled RX signal,
and sampled AWGN signal, respectively. $\boldsymbol{\Phi}_{c_\textrm{k}}\in \mathbb{C}^{L_\textrm{T}}$ is the down-converted and sampled transmission waveform, computed as
$\boldsymbol{\Phi}_{c_\textrm{k}}={[1,e^{j2\pi c_\textrm{k} \Delta f T_\textrm{s}\times 1},...,e^{j2\pi c_\textrm{k} \Delta f T_\textrm{s}\times(L_\textrm{T}-1)}]}^T$.
It is also regarded as the baseband signal carrying the information of the frequency index $c_\textrm{k}$ of a single radar pulse.
\section{Proposed Codebook Design and CRPS Schemes}
In this section, we introduce the novel codebook design and CRPS schemes for IM-JRC.
Following \cite{eldarTSP2020}, for IM-JRC the maximum-likelihood (ML) decoder shows best BER performance in terms of estimating the frequency indices $\{c_\textrm{k}\}$ and antenna allocation patterns $\{\mathbf{P}_\textrm{k}\}$, which are previously embedded in the radar pulses through IM at the TX. The ML detection for IM-JRC is expressed as
\begin{equation}\label{eq:ml_estimation}
{\{{c}_\textrm{k},{\mathbf{P}}_\textrm{k}\}}_{k=0}^{K-1}
={_{\{{c}_\textrm{k},{\mathbf{P}}_\textrm{k}\}}^{\text{arg min}} \norm{\mathbf{Y}_\textrm{C}-\sum_{k=0}^{K-1} \mathbf{H}\mathbf{P}_\textrm{k}\mathbf{w}_k\boldsymbol{\Phi}_{c_\textrm{k}}^T}_F^2} ,
\end{equation}
where $\norm{.}_F$ denotes the Frobenius norm. Additionally, we assume that the RX has \emph{a priori} knowledge of $K$, the desired pointing angle $\theta$, and perfect channel state information (CSI).
\subsection{Codebook Design}
When the available codewords, i.e., combinations of antenna and frequency indices, are not equal to a power of $2$ \cite{duhamel2010}, IM-JRC only utilizes a subset of the available codewords for transmission. Defining the subset of codewords as valid codewords $V$, then
$|V|=2^B \le |\zeta||P|$,
where $B$ is the maximum number of bits that can be conveyed in an individual radar pulse. In general, the codebook design approach aims to locate the optimum $|V|$ valid codewords.
To define "optimum",
\cite{yangTWC2016} pointed out that the BER performance of the communication system is highly dependent on the MED between the neighbouring constellation symbols, i.e., valid codewords, where the MED is formulated as
\begin{equation}
\textrm{MED}=d_\textrm{min}=_{i\neq j}^\text{min}\norm{\mathbf{Y}_\textrm{i}-\mathbf{Y}_\textrm{j}}_F^2.
\end{equation}
Consequently, the designed codebook for IM-JRC should comprise the subset of valid codewords with maximum MED.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth, trim=36 100 55 0,clip]{Fig_1.pdf}
\caption{Illustration of Codebook Design Approach: (a) Original Codewords; (b) Designed Codebook.}
\vspace{-6mm}
\end{figure}
Fig. 1 demonstrates the proposed codebook design approach. In Fig. 1(a), there are $C=|\zeta||P|=6$ codewords available for transmission $\{\textrm{Y}_\textrm{0},...,\textrm{Y}_\textrm{5}\}$.
Hence, IM-JRC can convey a maximum of $\floor{\text{log}_2(C)}=2$ bits information with four valid codewords. The design philosophy is to locate the "worst" two codewords and make them idle. In Step 1, IM-JRC computes for the current MED and the corresponding neighbouring codewords, which are $\textrm{Y}_\textrm{0}$ and $\textrm{Y}_\textrm{1}$. After that, the system compares their second-minimum Euclidean distances with the remaining codewords. In this example, $d_{\textrm{Y}_0-\textrm{Y}_3}> d_{\textrm{Y}_1-\textrm{Y}_3}$; thus, the system keeps $\textrm{Y}_\textrm{0}$ and removes $\textrm{Y}_\textrm{1}$. In Step 2, IM-JRC repeats the procedure in Step 1, and it decides to keep $\textrm{Y}_\textrm{5}$ and make $\textrm{Y}_\textrm{3}$ idle. The remaining four valid codewords $\{\textrm{Y}_\textrm{0},\textrm{Y}_\textrm{2},\textrm{Y}_\textrm{4},\textrm{Y}_\textrm{5}\}$ comprise the finalized codebook design. The designed codebook maximizes the MED between the codewords, thereby reducing the probability of detection errors and enhancing the BER performance.
\vspace{-1mm}
\subsection{Constellation Randomization Pre-Scaling (CRPS)}
A constellation randomization approach is presented in \cite{masourosTVT2016, masourosCL2014} to improve the performance of spatial modulation in the MIMO communication systems by maximizing the system MED.
The operating principle is to randomize the original constellation diagram by multiplying codewords with a range of randomly generated transmit pre-scaling (TPS) factors.
We extend that technique for IM-JRC systems, termed as constellation randomization pre-scaling (CRPS).
Let $\mathbf{X}_\textrm{C}$ represent the original constellation diagram. At the beginning of CRPS, a number of normalized TPS factors are generated, given by $\boldsymbol{\alpha} = \{\boldsymbol{\alpha}_\textrm{d}|d \in (1,D)\}$, where $\boldsymbol{\alpha}$ is the set containing all the TPS factors with the cardinality of $D$. The parameter $\boldsymbol{\alpha}_\textrm{d} \in \mathbb{C}^{L_\textrm{R}}$ is one particular TPS factor, where each entry is randomized from the standard normal distribution, and $\mathbb{E}[\mathbf{\alpha}_\textrm{d}]=1$ where $\mathbb{E}[.]$ is the expectation operator. Defining $\mathbf{A}_\textrm{d} \in \mathbb{C}^{L_\textrm{R}\times L_\textrm{R}}$ as the diagonal matrix of $\boldsymbol{\alpha}_\textrm{d}$. Then the original constellation diagram is randomized by multiplying it with each of the TPS matrices, such as
$\mathbf{X}_\textrm{C-CRPS}\!\!=\!\!\mathbf{A}_\textrm{d} \mathbf{X}_\textrm{C}\!\!=\!\!\sum_{k=0}^{K-1} \mathbf{A}_\textrm{d} \mathbf{P}_\textrm{k} \mathbf{w}_\textrm{k} \boldsymbol{\Phi}_{c_\textrm{k}}^T$,
which is the collection of the randomized constellation diagrams. As shown in Fig. 2, the system MED varies accordingly with the randomization of the constellation diagram. Consequently, there are possibilities for the new MEDs to be greater than the original MED, where the possibilities increase proportionally with the number of TPS factors. Later, IM-JRC determines the optimal TPS factor which results in the maximum MED among the "pre-scaled" constellation symbols, hence the optimum BER performance, for which
\vspace{-4mm}
\begin{align}
\{\boldsymbol{\alpha}_\textrm{o},\mathbf{A}_\textrm{o}\}&\!\!=\!\!\text{arg} \max_d {_{\{c_{k_1},\mathbf{P}_{k_1}\}\neq\{c_{k_2},\mathbf{P}_{k_2}\}}^{\text{min}}}||\sum_{k=0}^{K-1} \mathbf{A}_\textrm{d} \mathbf{P}_{k_1} \mathbf{w}_\textrm{k} \boldsymbol{\Phi}_{c_{k_1}}^T \nonumber\\
&-\sum_{k=0}^{K-1}\mathbf{A}_\textrm{d}\mathbf{P}_{k_2} \mathbf{w}_\textrm{k} \boldsymbol{\Phi}_{c_{k_2}}^T||_F^2,
\end{align}
where $\{\boldsymbol{\alpha}_\textrm{o},\mathbf{A}_\textrm{o}\}$ are the optimal TPS factor and matrix, respectively. As shown in \cite{masourosTVT2016}, information of $\{\boldsymbol{\alpha}_\textrm{o},\mathbf{A}_\textrm{o}\}$ can be made available to the RX, or the RX can independently identify the TPS given an agreed codebook and CSI. Accordingly, the ML decoder in \eqref{eq:ml_estimation} is modified as
\begin{equation}
{\{{c}_\textrm{k},{\mathbf{P}}_\textrm{k}\}}_{k=0}^{K-1}={_{\{{c}_\textrm{k},{\mathbf{P}}_\textrm{k}\}}^{\text{arg min}} \norm{\mathbf{Y}_\textrm{C}-\sum_{k=0}^{K-1} \mathbf{H}\mathbf{A}_o \mathbf{P}_\textrm{k} \mathbf{w}_k\boldsymbol{\Phi}_{c_\textrm{k}}^T}_F^2}.
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth, trim=36 85 55 0,clip]{Fig_2.pdf}
\caption{Illustration of CRPS: (a) Original Constellation Symbols;
(b) Randomized Constellation Symbols.
\vspace{-2mm}
\end{figure}
\begin{table}
\centering
\def1{1}
\begin{tabular}{ |c|c|}
\hline
\textbf{Method} & \textbf{Complexity order} \\
\hline
IM-Codebook design & $[K L_\textrm{R}+(2K-1)L_\textrm{T}]L_\textrm{R} |\zeta||P|$ \\ & $+\frac{3L_\textrm{R} L_\textrm{T}+1}{2} \sum_{i=1}^Q(|\zeta||P|-i+1)(|\zeta||P|-i)$
\\ & $+[(L_\textrm{R}+3)L_\textrm{C} L_\textrm{T}+1](|\zeta||P|-Q)N$ \\
\hline
IM-CRPS approach & $[K L_\textrm{R}+(2K-1)L_\textrm{T}]L_\textrm{R} |\zeta||P|+[[L_\textrm{R}^2 L_\textrm{T}$\\ &$+((3L_\textrm{R} L_\textrm{T}+1)(|\zeta||P|-1))/2]|\zeta||P|+1]D$\\&$+[(L_\textrm{R}+3)L_\textrm{C} L_\textrm{T}+1]|\zeta||P|N$ \\
\hline
\end{tabular}
\caption{Computational complexity of the proposed schemes.}
\label{tab:Table1}
\vspace{-6mm}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth, trim=50 0 0 0,clip]{Fig_11.pdf}
\caption{BER performance of the proposed codebook design and CRPS schemes for $M=7$ and $L_\textrm{R}=6$.}
\vspace{-5mm}
\end{figure}
The overall computational complexity of IM with codebook design and CRPS scheme is provided in Table 1, where $Q$ is the number of codewords to be eliminated, expressed as $Q=|\zeta||P|-|V|=|\zeta||P|-2^B$ and $B=\floor{\text{log}_2|\zeta||P|}$. The complexity of the CRPS scheme also depends on $D$, which is the cardinality of the randomly generated TPS factors.
\section{Simulation Results}
\label{sec:simulation}
We set the system parameters as, unless stated otherwise, $M$ = 7, $f_\textrm{c}$ = 1.9 GHz, $\Delta f$ = 10 MHz, $L_\textrm{R}$ = 6, $\theta$ = $\frac{\pi}{4}$, $c = 3\times 10^8$ m/s, $T_\textrm{r}$ = 2 $\mu s$, $T_\textrm{p}$ = 1 $\mu s$, $K$ = 2, $L_\textrm{C}$ = 4, and $D$ = 100. Under this condition, the JRC model can convey a maximum of $8$-bits information during each radar pulse transmission. Additionally, the transmit signal power is normalized to unity. The BER results are averaged over $N=1\times10^5$ radar pulses.
Fig. 3 shows the BER performance of the proposed schemes and compare it with ML decoder baseline \cite{eldarTSP2020}.
The codebook design scheme slightly outperforms the ML decoder as it eliminates the idle communication codewords in the constellation diagram.
The CRPS scheme outperforms the codebook design approach as we can observe that this scheme meets the BER levels of $1\times10^{-3}$ and $1\times10^{-4}$ at the SNRs of -15.4 and -12.1 dB, giving the SNR gains of 2.5 and 2.1 dB, respectively.
Furthermore, in the case of codebook design followed by CRPS, the codebook design firstly eliminates 164 idle codewords and enlarges the temporal MED; after that, the CRPS scheme randomizes the remaining 256 valid codewords and maximizes the average symbol distance. However, this randomization inevitably decreases the system MED. For the case of CRPS followed by codebook design, the CRPS scheme randomizes the overall 420 communication codewords and maximizes the average symbol distance while leaving a set of lower symbol distances. In the following, these undesirable distances are handled by the codebook design approach during the elimination of the idle codewords. As a result, the finalized constellation diagram is more suitable for the communication transmission, which also accounts for the optimal BER performance, e.g., the corresponding SNR gains at the BER levels of $1\times10^{-3}$ and $1\times10^{-4}$ are both 4.4 dB.
The data rate being the maximum number of bits embedded per radar pulse, firstly we investigate in Fig. 4 the impact of varying $M$ on the BER performance of the optimal integrated CRPS and codebook design scheme. Following the Nyquist sampling period $T_\textrm{s}=\frac{1}{M \Delta f}$, it can be observed that when $M$ increases, the sampling period narrows, and the reliability of the decoding process is enhanced accordingly, which accounts for the BER performance enhancement of the IM-JRC model. This result also supports the trade-off between data rate and BER performance. We can also observe that the optimal proposed scheme outperforms ML decoder baseline for all the cases. Similarly Fig. 4 observes the BER performance with varying number of TX antennas $L_\textrm{R}$. The optimal integrated CRPS and codebook design approach again outperforms the ML decoder baseline for different $L_\textrm{R}$ values.
\begin{figure}[t]
\centering
\includegraphics[width=0.455\textwidth, trim=40 0 0 0,clip]{Fig_13_comb.pdf}
\caption{BER performance of the integrated CRPS and codebook design for varying $M$ ($L_\textrm{R}=6$) and varying $L_\textrm{R}$ ($M=7$).}
\vspace{-5mm}
\end{figure}
\section{Conclusion}
This paper proposes the novel and efficient codebook based MED maximization and CRPS schemes to employ IM in JRC systems with spectral and spatial agility. The proposed approaches outperform the state-of-the-art ML decoder baseline, and the integration of the CRPS scheme followed by the codebook design leads to the best optimal solution in terms of BER performance. In the specific case, employing CRPS followed by codebook design outputs the maximum SNR gain of around 4.4 dB when compared to the baseline approach.
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{"url":"https:\/\/joa.sh\/posts\/2016-08-02-free-monads.html","text":"# Free monads in category theory\n\nPart II of the series Free monads.\nAugust 2, 2016\n\n## Forgetting how to multiply\n\nIt\u2019s probably easiest to understand what a free monad is if we first understand forgetful functors1.\n\nIn category theory, a functor maps between categories, mapping objects to objects and morphisms to morphisms in a way that preserves compositionality2.\n\nA forgetful functor is just a functor that discards some of the structure or properties of the input category.\n\nFor example, unital rings have objects , where is a set, and are binary operations with identity elements respectively.\n\nLet\u2019s denote the category of all unital rings and their homomorphisms by , and the category of all non-unital rings and their homomorphisms with . We can now define a forgetful functor: , which just drops the multiplicative identity.\n\nSimilarly, we can define another forgetful functor , which maps from the category of rngs to the category of abelian groups. discards the multiplicative binary operation, simply mapping all morphisms of multiplication to morphisms of addition.\n\n## Forgetting monoids\n\nThe forgetful functor forgets ring multiplication. What happens if instead you forget addition? You get monoids! We can define monoids as the triple , where is a set, is an associative binary operation, and is the neutral element of that operation.\n\nThe forgetful functor maps from the category of rings to the category of monoids, , in which the objects are monoids, and the morphisms are monoid homomorphisms.\n\nMonoid homomorphisms map between monoids in a way that preserves their monoidal properties. Given , a monoid defined by , and , a monoid defined by , a function from to is a monoid homomorphism iff:\n\nit preserves compositionality3:\n\nand maps the identity element:\n\nTranslating into Haskell, if phi is a monoid homomorphism between monoid X to monoid Y, then:\n\nphi (mappend a b) == mappend (phi a) (phi b) -- (1)\n\nphi (mempty :: X) == mempty :: Y -- (2)\n\nFor example, we can define a monoid homomorphism that maps from the list monoid to the Sum monoid, the monoid formed from the natural numbers under addition:\n\nimport Data.Monoid\n\nlistToSum :: [a] -> Sum Int\nlistToSum = Sum . length\n\nIf it\u2019s too difficult (or we can\u2019t be bothered) to derive a formal proof, we can use QuickCheck to test properties of functions. Let\u2019s quickly check if listToSum is actually a monoid homomorphism:\n\nimport Test.QuickCheck\n\n-- (1)\nhomomorphism :: [()] -> [()] -> Bool\nhomomorphism a b =\nphi (mappend a b) == mappend (phi a) (phi b)\nwhere phi = listToSum\nghci> quickCheck homomorphism -- (1)\nOK, passed 100 tests.\n\nghci> listToSum (mempty :: [a]) == mempty :: Sum Int -- (2)\nTrue\n\nLet\u2019s forget some more things with yet another forgetful functor, 4.\n\nis a category where the objects are sets, and the arrows are just plain functions. So will map every monoid in to its underlying set, and every monoid homomorphism to a plain function.\n\nSum Int would just become Int, listToSum would just become length, mappend :: Sum a would map to (+), and so on. We forget that any of these things formed a monoid.\n\n## Natural Transformations\n\nMoving our discussion from forgetful functors to free constructions requires the concept of natural transformations. Recall that a functor must take all objects to , and all morphisms to , such that the following diagram commutes:\n\nThis diagram says that it doesn\u2019t matter if we start with , apply and then , or start with and instead apply and then - we always end up with . The functor has mapped between categories in a way that preserves the internal structure of the original category.\n\nA natural transformation is a similar sort of structure-preserving5 mapping, except instead of mapping between categories, it maps between functors.\n\nGiven functors , a natural transformation is a morphism between functors such that:\n\n1. For all , there exists a morphism, , where 6\n\n2. For every morphism , the following diagram- a naturality square- commutes:\n\nThis means we\u2019re making a rather strong claim about the properties of : is the same as !\n\nLet\u2019s consider two functors going in opposite directions, and .\n\nand aren\u2019t just any old functors though- they\u2019re equipped with a natural isomorphism:\n\nwhere the isomorphism is natural in and .\n\nSimply saying these hom-sets are naturally isomorphic is rather imprecise. We can pin down the naturality of by saying that certain natural transformations hold. But to define a natural transformation we need to define some functors!\n\nWe can define a natural transformation between these hom-functors from to , fixing :\n\nWe can use another notation to make their functorish nature more apparent:\n\nThese functors take every object to a hom-set of morphisms in , so it\u2019s perfectly valid to ask for a natural transformation between them:\n\nSo for every morphism , applying and then precomposing with is the same as precomposing with and then applying . That\u2019s naturality in . Naturality in is much the same, except we fix , and get functors from :\n\nfor all mophisms .\n\nWe can think of as a pair of hom-functors7 that take , and a pair of functors that take , such that each pair of functors creates a bijection between their corresponding sets, satisfying the above naturality conditions.\n\nWe describe this functorial relationship by saying that is left adjoint to , or .\n\n## Free monoids\n\nArmed with the ability to talk about the \u201cadjointness\u201d of functors, we can now examine what happens when we take to be a forgetful functor, when .\n\nIf is a forgetful functor that discards some information about its domain, must be able to \u201creconstruct\u201d enough to go from to . The left adjoint to a forgetful functor is always a free functor!\n\nReturning to our monoid example, if we take to be , the left adjoint to is the free functor .\n\nThis means there must be a natural isomorphism, , that creates a bijection between hom-sets of and , such that all functions to an underlying set of uniquely determines a monoid homomorphism that\u2019s natural in and :\n\nand vice-versa.\n\nHow could we construct so that the above conditions are met? Spoiler alert: we can just use List! Let\u2019s try to translate , and its inverse, into pseudo-Haskell8.\n\n-- Just delete the monoid constraint\nu :: Monoid m = m\n\nalpha :: (List a -> Monoid m) = (a -> u (Monoid m))\n\nalpha' :: (a -> u (Monoid m)) -> (List a -> Monoid m)\n\nNow we can translate this into actual Haskell9. Since u just removes the monoid constraint, we can substitute all instances of u (Monoid m) with simply m, and we can use the real list constructor and type constraint syntax:\n\nimport Data.Monoid\n\nalpha :: Monoid m => (a -> m) -> ([a] -> m)\nalpha g xs = mconcat $map g xs alpha' :: Monoid m => ([a] -> m) -> (a -> m) alpha' h x = h [x] To prove that alpha actually forms a natural isomorphism, we need to show that alpha . alpha' = id: -- Proof that alpha . alpha' = id alpha . alpha' -- eta expand = \\h x -> alpha (alpha' h) x -- substitute definition of alpha' = \\h x -> alpha h [x] -- substitute definition of alpha = \\h x -> mconcat (map h [x]) -- map f [x] = [f x] = \\h x -> mconcat ([h x]) -- mconcat [x] = x = \\h x -> h x -- eta-reduce = \\h = h -- definition of id = id and in the other direction, that alpha' . alpha = id: -- Proof that alpha' . alpha = id alpha' . alpha -- eta-expand = \\g xs -> alpha' (alpha g) xs -- substitute definition of alpha = \\g xs -> mconcat (map (alpha g) xs) -- eta-expand = \\g xs -> mconcat (map (\\x -> alpha g x) xs) -- substitute definition of alpha' = \\g xs -> mconcat (map (\\x -> g [x]) xs) -- map (f . g) = map f . map g = \\g xs -> mconcat (map g (map (\\x -> [x]) xs)) -- free theorem = \\g xs -> g (mconcat (map (\\x -> [x]) xs)) -- mconcat [[a],[b],[c]] = [a,b,c] = \\g xs -> g xs -- eta-reduce = \\g -> g -- definition of id = id So it follows that the list does indeed form a free monoid! Interestingly, what we\u2019ve already defined as alpha is just foldMap: alpha :: Monoid m => (a -> m) -> ([a] -> m) foldMap :: Monoid m => (a -> m) -> [a] -> m So in more Haskellish terms, we map each element of a list to a monoid, and then combine the results using the structure of that monoid. ghci> foldMap Product [2,4,6] Sum {getSum = 12} ghci> foldMap Product [2,4,6] Product {getProduct = 48} Of course, foldMap is really defining a monoid homomorphism, which means it should map the identity element: ghci> foldMap Product [] Product {getProduct = 1} ghci> foldMap Sum [] Product {getSum = 0} \u2026and preserve compositionality: homomorphism :: [Int] -> [Int] -> Bool homomorphism a b = phi (a ++ b) == phi a mappend phi b where phi = foldMap Sum ghci> quickCheck homomorphism OK, passed 100 tests. ## Free Monads Now let\u2019s take a look at free monads. Of course, monads are just monoids in the category of endofunctors, so we can apply what we\u2019ve already learned! A monad is an endofunctor equipped with natural transformations and , obeying the obvious10 axioms of identity and associativity. The category has monads as objects, and monad homomorphisms as arrows. We can define a forgetful functor, , that maps from the category of monads to the category of endofunctors. The category of endofunctors, , has endofunctors as objects and natural transformations as arrows, so should be a forgetful functor such that: \u2022 For every monad , will forget and , and just give us the underlying endofunctor . \u2022 For every monad homomorphism , will give us a natural transformation in . Now we can see what behaviour should have, when : \u2022 For every endofunctor , should be a monad. \u2022 For every natural transformation , should be a monad homomorphism. \u2022 The isomorphism should be natural in and . Again, this makes very strong claims about the behaviour of . It turns out that the following construction11 satisfies all these criteria: We\u2019re effectively asking for the existence of data Free f a = Pure a | Free (f (Free f a)) instance Functor f => Monad (Free f) where return a = Pure a Pure a >>= f = f a Free m >>= f = Free (fmap (>>= f) m) The monadic bind operation (>>=) can be defined in terms of \u201csubstitution followed by renormalization\u201d: Monad m => m a -> (a -> m b) -> m b m >>= f = join (fmap f m) In conventional monads, we substitute the a in our monad m a with m b, to get m m b, and then we renormalize with (which we call join in Haskell) to get m b. Free monads still perform the substitution of the underlying functor, but because Free type is defined recursively as Free (f (Free f a)), we effectively get for free by by sticking another layer of Free on top. It\u2019s a lossless process; everything you\u2019ve joined is retained12. In fact, Free looks suspiciously like List: data List a = Nil | Cons (List a) data Free f a = Pure a | Free (f (Free f a)) So Free is basically just a list of functors! When we defined the free monoid, the natural isomorphism constraint basically forced us into defining foldMap, which mapped each element of the list to a monoid, and then used the structure of that monoid to join the resulting elements: foldMap :: Monoid m => (a -> m) -> [a] -> m foldMap f xs = mconcat$ map f xs\n\nNow we\u2019re going to do the same for the free monad, by defining the natural transformation foldFree, and its inverse, foldFree':\n\nfoldFree :: (Functor f, Monad m) =>\n(forall a . f a -> m a) -> Free f a -> m a\nfoldFree' :: (Functor f, Monad m) =>\n(forall a . Free f a -> m a) -> f a -> m a\n\nDoing that is as simple as following the types:\n\nfoldFree _ (Pure x) = return x\nfoldFree phi (Free xs) = join $phi$ fmap (foldFree phi) xs\n\nfoldFree' psi = psi . Free . (fmap Return)\n\nProving that foldFree . foldFree' = id is left as an exercise for the reader.\n\n1. As far as I understand, there\u2019s no formal way of describing the \u201cforgetfulness\u201d of a functor.\n\n2. A functor \u201cpreserves compositionality\u201d if two morphisms of the input category compose to form a third morphism, such that the image of those two morphisms under the functor also compose to form the image of the third morphism.\n\n3. All homomorphisms have one constraint in common: they must preserve compositionality. We can be generalise the homomorphism constraint for any -ary operation; a function is a homomorphism between two algebraic structures of the same type if: for all\n\n4. Technically, in Haskell we\u2019d be mapping to the category , the category of Haskell types.\n\n5. It\u2019s actually exactly the same kind of structure preservation, because functors form a category where the objects are functors and the morphisms are natural transformations.\n\n6. We call \u201cthe component of at\n\n7. Category theory brings a great richness of perspectives, allowing us to think about relationships in whatever way that happen to suit whatever we\u2019re trying to talk about.\n\n8. I\u2019m not sure of a way to write a polymorphic function that \u201cforgets\u201d a monoid constraint, so we\u2019ll just wave our hands a bit until we get to Real Haskell.\n\n9. This might look a bit strange- even though we\u2019ve supposedly \u201cforgotten\u201d that m is a monoid in alpha', the type variable m is still bound by the monoid type constraint. We can cheat, though, and explicitly parameterize a forgetful function Monoid m => m -> b:\n\nalpha Monoid m=>(b->m)->(a->b)->([a]->m)\nalpha m g xs = mconcat $map (m . g) xs alpha' Monoid m=>(m->b)->([a]->m)->(a->b) alpha' m h x = m . h$ [x]\n10. Obvious to experienced category theorists, at least.\n\n11. The way this construction defines the bind operator results in a quadratic asymtotic complexity. Janis Voigtlander describes an approach for reducing the asymtotic complexity to linear in his paper Asymtotic Improvement of Computations over Free Monads.\n\n12. This gives an intuition for why any natural transformation between endofunctors, , can be fed to the free functor to form a monad homomorphism, . Just like the free monoid, we don\u2019t \u201cthrow away\u201d any information about the free construction beyond what\u2019s defined by the underlying category.","date":"2019-12-07 07:37:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9570573568344116, \"perplexity\": 1180.686526075872}, \"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-2019-51\/segments\/1575540496492.8\/warc\/CC-MAIN-20191207055244-20191207083244-00336.warc.gz\"}"} | null | null |
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Q: How a paper plane(rocket) flies in air? what is science behind it? What factors helps a paper plane to fly and can you explain how these factors help the plane to fly?
A: One commonly overlooked factor is how it is thrown. Modern air planes carry their own power while paper airplanes are typically thrown by hand. If it is thrown very quickly then the air flow likely to be turbulent. If thrown slowly we can model the air flow as laminar and predict its flight path more accurately.
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Sokoroni est une localité située dans le département de Koloko de la province du Kénédougou dans la région des Hauts-Bassins au Burkina Faso.
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Sokoroni est située à de Koloko et à de la frontière malienne sur la route menant à Sikasso.
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Sokoroni accueille un centre de santé et de promotion sociale (CSPS) tandis que le centre médical avec antenne chirurgicale (CMA) le plus proche se trouve à Orodara et que le centre hospitalier régional (CHR) est le CHU Souro-Sanon de Bobo-Dioulasso.
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\section{Introduction}
Automatic segmentation from multi-modal images is essential for clinical assessment, diagnosis and treatment planning \cite{ackaouy2020unsupervised,ouyang2019data}. Extensive literature has shown the effectiveness of convolutional neural networks in segmenting accurately cardiac structures \cite{li2021mdfa,painchaud2020cardiac}. Yet, without proper adaptation these models fail when deployed across modalities, new subjects and different clinical sites, due to a domain shift \cite{heimann2013learning} \textit{e.g.} between the modalities' appearance as in Fig.~\ref{fig1}. Designing models that can perform well across domains is key in medical applications where labels are scarce and expensive to obtain.
Semi-supervised and Unsupervised Domain Adaptation (UDA) approaches have been proposed to tackle the domain shift problem. The former assume a few labeled instances in the target domain can be used for joint-training with the source data \cite{puybareau2018left}. The more ambitious UDA strategies \cite{ackaouy2020unsupervised,chen2019synergistic,chen2020unsupervised,ouyang2019data,wu2020cf,yang2019domain} assume no labels are available for the target domain. The core idea of UDA is to go through an adaption phase using a non-linear mapping to find a common domain-invariant representation or a latent space $\mathcal{Z}$. The
domain shift in $\mathcal{Z}$ can be reduced by enforcing the two domains distributions to be closer via a certain
loss
(\textit{e.g.} Maximum Mean Discrepancy \cite{kumagai2019unsupervised}). Since $\mathcal{Z}$ is common to all domains who share the same label space, projected labeled source domain samples can be used to train a segmenter for all domains. In this paper, we deal with the problem of UDA for MR-CT cross-modality cardiac structure segmentation.
\begin{figure}[t]
\includegraphics[width=\textwidth]{figures/DomainShift.pdf}
\caption{The appearances of the cardiac structures look significantly different on MR and CT images. Both modalities share the same label space, yellow: left ventricle myocardium (LV-M), blue: left ventricle blood cavity (LV-B), red: left atrium blood cavity (LA-B), and green: ascending aorta (A-A). Bad prediction due to severe domain shift when no adaptation is considered. MM-WHS cardiac public database \cite{zhuang2016multi}.} \label{fig1}
\end{figure}
\textbf{Related Work.} Recent works on UDA for medical image segmentation rely on Generative Adversarial Networks \cite{chen2019synergistic,chen2020unsupervised,dou2018pnp,ouyang2019data,wu2020cf,yang2019domain} to translate the appearance from one modality to the other using multiple discriminators and a pixel-wise cycle consistency loss. Despite their success, they: i) suffer from instabilities \cite{arjovsky2017towards}, ii) rely on complex architectures with more than 95 million parameters, iii) are prone to model collapse \cite{liu2019spectral}, and iv) may generate images outside the actual target domain \cite{ackaouy2020unsupervised}. To alleviate some of these limitations, Ouyang~\textit{et.al.}~\cite{ouyang2019data} combined adversarial networks with VAEs~\cite{kingma2013auto}. They exploited the VAEs constraint imposed on the latent space to match a prior distribution and experimentally validated that it reduces the domain shift when used as a shared space across domains. To encourage appearance-invariance, \cite{ouyang2019data} deployed an adversarial loss guided by a cycle-consistency. The VAE model in \cite{ouyang2019data} is complex as equipped with three encoders, three decoders, a segmenter, and a domain classier. Its loss function has six trade-off hyper-parameters to tune. Recently, Optimal Transport (OT) theory \cite{villani2008optimal} jointly with deep learning methods was used by Ackaouy~\textit{et.al.}~\cite{ackaouy2020unsupervised} where a joint cost measure combining both the distances at feature space of a deep 3D-Unet between the samples and a loss function measuring the discrepancy at the output space between the two domains is proposed. A limitation of Seg-JDOT is that it employs image patches to enable the generation of a higher number of samples and to avoid curse of dimensionality when optimizing for the transport plan $\gamma$.
\textbf{Proposal.} We present a novel and lightweight domain-invariant variational -segmentation auto-encoder model. We use the latent space of a VAE that is constrained to follow a prior normal distribution as a common space similar to \cite{ouyang2019data} to reduce the domain shift. Then we exploit the geometry in $\mathcal{Z}$ for matching and aligning distances between probability distribution using OT theory by optimizing for a transport plan $\gamma$, similar to \cite{ackaouy2020unsupervised}, to further shrink the remaining domain shift. Different from \cite{ouyang2019data}, who maximized the image likelihood, we directly learn a semantic latent representation that maximizes the label likelihood. Our idea is that the prior normal distribution has a limited capacity to handle intensity and shape variations, but it can be efficiently exploited for modelling shapes alone. This claim is supported by \cite{painchaud2020cardiac} who use a VAE as a post-processor on the top of a U-Net output to convert the erroneous U-Net predictions to anatomical plausible outputs. Conversely, we simultaneously perform anatomical plausible segmentation and partial alignment of the label-conditional distributions. Also, different from \cite{ackaouy2020unsupervised}, i) we operate over the full image scale, ii) we bring the source and the target data closer to a normal distribution before solving for $\gamma$ to guarantee its convergence and iii) we do not require the alignment of the label-conditional distributions at the label space.
\textbf{\textit{Our main contributions are}:} i) We reduce the domain shift between the source and the target domain by projecting them into a shared semantic latent space which is regularized to follow a prior normal distribution; ii) We address the remaining shift by aligning latent vectors from both domains using a discrepancy measure based on OT theory; iii) Different from a typical VAE which forces the latent space to model image intensity variations, we concentrate the limited capacity of the prior normal distribution to model the shape of the segmentation masks; iv) Our model is lightweight with 1.7 million parameters and easy to adapt for other clinical application; v) We validate our model on the MM-WHS public dataset and outperform state of the art methods by a margin of $12.5\%$ dice score.
\section{Method}
Consider a labeled source domain dataset $\{{X}^{s},{Y}^{s}\}_{s=1}^N$ with $N$ images, and a target dataset $\{{X^t}\}_{t=1}^M$ with $M$ images, but with unknown labels ${Y}^{t}$. The goal of UDA is to build a common space for ${X}^{s},{X}^{t}$ while using the source labels ${Y}^{s}$ to guide a segmentation model to generalize across both domains. Here, we propose an Optimal Latent Vector Alignment (OLVA) method to learn a shared latent space that encodes all the structural information needed to generate image segmentation masks, regardless of the domain. We minimize the domain shift between the source and the target distributions in this latent space by pushing them close to a prior normal distribution with a VAE. An optimal transport discrepancy measure removes the remaining domain shift. Finally, a generative decoder guided by the source labels is trained to produce feasible semantic segmentation masks. A block diagram of the method is shown in Fig.~\ref{fig2}.
\begin{figure}[t]
\includegraphics[width=\textwidth]{figures/generaldiagram.pdf}
\caption{OLVA: a generative encoder enforces the shared latent vectors of both source and target domain to follow a prior normal distribution (a), further aligned through an OT plan $\gamma$ (b). A generative decoder is guided with the source domain labels to produce segmentation maps. We use t-distributed stochastic neighbor embedding (t-SNE)
to map the latent vectors to a
2D space for visualization purposes only.} \label{fig2}
\end{figure}
\subsection{VAEs for segmentation}
The goal of VAEs is to search for the best parameters $\phi^\ast,\theta^\ast$ in order to sample a latent variable $\boldsymbol{z}\sim q_{\phi^\ast}(\boldsymbol{z|x})$ whose
distribution can be relatively simple such as isotropic Gaussian distribution and to generate a new sample $\boldsymbol{\hat x}\sim p_{\theta^\ast}(\boldsymbol{z|x})$ as close as possible to the real observed data $\boldsymbol{x}$ such that $p_\theta^\ast(\boldsymbol{x|z})=p^\ast(x)$. The VAE loss formalized as in Eq.~(\ref{eq1}) enables an end-to-end training with a first term that maximize the marginal likelihood so that the generative model becomes better and a regularization term that minimize KL-divergence to better approximate $q_{\phi}(\boldsymbol{z|x})$ from the posterior $p_{\theta}(\boldsymbol{z})$. $\beta$ is a trade-off parameter between the two terms. Commonly, a prior model $p(\boldsymbol{z})$ is set to normal distribution $\mathcal{N}(0;I)$ and the re-parametrization trick is applied to facilitate the sampling process as $q_{\phi}(\boldsymbol{z|x})=\mathcal{N}(\mu_{\phi}(\boldsymbol{x}),\sigma_{\phi}(\boldsymbol{x})I)$. Thereby, $\mathcal{D}_{KL}$ become equivalent to $\frac{1}{2} \sum_{k=1}^{K} \left(1+\log(\sigma^{2}_{k}) - \mu^{2}_{k}- \sigma^{2}_{k}\right)$, $K$ is the latent space dimension and $-\log p_{\theta}(\boldsymbol{x}|\boldsymbol{z})$ is conveniently replaced by a reconstruction loss $||\boldsymbol{x}- p_{\theta}(\boldsymbol{x}|\boldsymbol{z})||^{2}$.
\begin{equation}
\mathcal{L}_{vae}(\phi,\theta;\boldsymbol{x}) =\underset{z \sim q_{\phi}}{\mathbb{E}} \left[-\log p_{\theta}(\boldsymbol{x|z}) \right] + \beta\mathcal{D}_{KL}(q_{\phi}(\boldsymbol{z|x})||p_{\theta}(\boldsymbol{z})),
\label{eq1}
\end{equation}
To use VAEs for segmentation, we let $\boldsymbol{z}$ represent directly the latent space of the segmentation masks, since modelling shape alone is less complex than shape and intensity together.
Moreover, VAEs lead to latent spaces that are continuous and structured (and not discrete as those of U-Net-like networks), facilitating interpolation. These choices will be determinant in producing valid segmentation masks that respect the anatomical variations of the source domain.
In practice, our decoder acts as a predictive generative model for the conditional label distribution $p_{\theta}(\boldsymbol{y|z})$ defined on the label space $\mathcal{Y}$. Using source domain data $\{X^s,Y^s\}$, our VAE segmentation loss is also guided by the soft dice loss $\mathcal{L}_{dice}^{s}$ to provide predictions of the segmentation maps as shown in Eq.~(\ref{eq2}.)
\begin{equation}
\mathcal{L}_{vae}^{s} = ||\boldsymbol{y}^{s} - p(\boldsymbol{y}^{s}|\boldsymbol{z}^{s})||^{2} +\beta\mathcal{D}_{KL}^{s}(q(\boldsymbol{z}^{s}|\boldsymbol{x}^{s})||p_{\theta}(\boldsymbol{z}^{s}))+ \mathcal{L}_{dice}^{s}(\boldsymbol{y}^{s},p(\boldsymbol{y}^{s}|\boldsymbol{z^{s}})).
\label{eq2}
\end{equation}
\subsection{Optimal Transport for Latent Vector Alignment}
To solve the domain adaptation problem within the segmentation task, we assume the existence of two distinct joint probability distributions $\mathcal{P}^s$ and $\mathcal{P}^t$ defined over a shared latent space $\mathcal{Z}$ and their marginal distributions ($\zeta^{s},\zeta^{t}$) are defined over $\Omega$ (the set of all probability measures).
Using $\mathcal{L}_{vae}^{s}(\phi,\theta;\boldsymbol{x}^{s},\boldsymbol{y}^{s})$, we regularize $\mathcal{P}^s$ to follow a normal distribution. To partially align $\mathcal{P}^s$ and $\mathcal{P}^t$, we argue that forcing both distribution to the same prior is beneficial. Therefore, we impose an additional cost function $\mathcal{D}_{KL}^{t}$ that measures the dissimilarity between latent vectors from the source $q(\boldsymbol{z}^s|\boldsymbol{x}^{s})$ and the target $q(\boldsymbol{z}^t|\boldsymbol{x}^{t})$ domains, with $\boldsymbol{z}^s, \boldsymbol{z}^t \in \mathcal{Z}$. Although necessary, this step is insufficient to completely align the two domains. The optimal transport (or Monge-Kantorovich) \cite{ackaouy2020unsupervised,OTCourty,gopalan2011domain,villani2008optimal} problem involves the matching of probability distributions defined over a geometric domain such as our latent semantic space. Here, using OT theory, we seek for a transportation plan matching distributions $\mathcal{P}^s$ and $\mathcal{P}^t$, which is equivalent to finding a probabilistic coupling, $\gamma_{0} \in \prod(\mathcal{P}^s,\mathcal{P}^t)$, as shown in Eq.~(\ref{eq3}). To simultaneously align the latent space through a coupling $\gamma_{0}$ while optimizing for $q(\boldsymbol{z|x})$, we adapt the Kantorovich OT formulation to the discrete case as in Eq.~(\ref{eq4}),
\begin{align}
\gamma_{0} = \argmin_{\prod(\mathcal{P}^s,\mathcal{P}^t)} \int_{\Omega \times \Omega} \mathcal{D}(q(\boldsymbol{z^s}|\boldsymbol{x^{s}});q(\boldsymbol{z^t}|\boldsymbol{x^{t}}))d\gamma(q(\boldsymbol{z^s}|\boldsymbol{x^{s}});q(\boldsymbol{z^t}|\boldsymbol{x^{t}})). \label{eq3}
\\
\min_{\gamma \in \prod, q(\boldsymbol{z}|\boldsymbol{x})
} \sum_{ij} \gamma_{ij}\mathcal{D}(q(\boldsymbol{z^s}|\boldsymbol{x^{s}});q(\boldsymbol{z^t}|\boldsymbol{x^{t}})) + \beta\mathcal{D}_{KL}^{t}(q(\boldsymbol{z}^{t}|\boldsymbol{x}^{t})||p_{\theta}(\boldsymbol{z^{t}})), \label{eq4}
\\
\min_{\gamma \in \prod, q(\boldsymbol{z}|\boldsymbol{x})
} \sum_{ij} \gamma_{ij}\mathcal{D}(q(\boldsymbol{z^s}|\boldsymbol{x^{s}});q(\boldsymbol{z}^{t}|\boldsymbol{x}^{t})) + \beta\mathcal{D}_{KL}^{t}(q(\boldsymbol{z}^{t}|\boldsymbol{x}^{t})||p_{\theta}(\boldsymbol{z}^{t})) + \mathcal{L}_{vae}^{s} \label{eq5}
\end{align}
\noindent where $\mathcal{D}(q(\boldsymbol{z^s}|\boldsymbol{x^{s}});q(\boldsymbol{z^t}|\boldsymbol{x^{t}})) = \alpha||q(\boldsymbol{z^s}|\boldsymbol{x^{s}}) - q(\boldsymbol{z^t}|\boldsymbol{x^{t}})||^{2}_{2}$ is the squared Euclidean distance and $\beta\mathcal{D}_{KL}^{t}$ regularizes the target distribution. The final objective of OLVA is formulated in Eq.~(\ref{eq5}), which optimizes jointly for: i) an embedding function $q(\boldsymbol{z}|\boldsymbol{x})$ that maps both the source and the target domain to a semantic latent space $\mathcal{Z}$ regularized to follow normal distribution; ii) a transportation matrix $\gamma$ that aligns similar semantic vectors $\boldsymbol{z}$ from both domains in the latent space; and iii) a predictive function $p(\boldsymbol{y}|\boldsymbol{z})$ for masks predictions.
\subsection{Learning OLVA}
With the formulation presented in in Eq.~(\ref{eq5}), our framework learns a common latent space that conveys aligned information for both the source and target domain. To solve Eq.~(\ref{eq5}), we use an alternating method ~\cite{OTCourty}. Therefore, we optimize $\gamma$, with fixed $q(\boldsymbol{z|x})$ and $p(\boldsymbol{y|z})$, which reduces to the problem in Eq.~(\ref{eq5}) to solving a classic OT problem with cost matrix $C_{i,j} = \alpha||q(\boldsymbol{z}^{s}|\boldsymbol{x}^{s})-q(\boldsymbol{z}^{t}|\boldsymbol{x}^{t})||^{2}_{2}$. Then, we optimize $q(\boldsymbol{z|x})$ and $p(\boldsymbol{y|z})$, with fixed $\gamma$, this turns the problem in Eq.~(\ref{eq5}) to a standard deep learning problem. Similar to Damodoran~\textit{et.al.}~\cite{damodaran2018deepjdot}, we solve the optimization problem with a stochastic approximation using mini-batches of size $m + n$ from the source and target domains respectively, which leads us to the optimization problem presented in Eq.~(\ref{eq6}). The stochastic approximation yields a computationally feasible solution for both the OT and VAE. The discrepancy measure and the KL-Divergence regularization are computed at the latent space layer, while the segmentation loss uses the output layer.
\begin{dmath}
\min_{q,p}\mathbb{E}\left[ \frac{1}{m}\sum_{i=1}^{m}\mathcal{L}_{dice}^{s}(\boldsymbol{y}^{s},p(\boldsymbol{y}^{s}|\boldsymbol{z}^{s})) + \frac{1}{m}\sum_{i=1}^{m}\beta\mathcal{D}_{KL}^{s} + \frac{1}{n}\sum_{i=1}^{n}\beta\mathcal{D}_{KL}^{t} + \frac{1}{m}\sum_{i=1}^{m}||\boldsymbol{y}^{s} - p(\boldsymbol{y}^{s}|\boldsymbol{z}^{s})||^{2} + \min_{\gamma \in \Gamma(\zeta^{s},\zeta^{t})} \sum_{i,j}^{m+n} \gamma_{i,j} \alpha|| q(\boldsymbol{z}^{s}|\boldsymbol{x}^{s}) - q(\boldsymbol{z}^{t}|\boldsymbol{x}^{t})||^{2}
\right]
\label{eq6}
\end{dmath}
\textbf{Architecture and implementation details:} OLVA accept batches containing $128$ source and $128$ target samples. The input dimension is $256\times256\times3$. The encoder is composed of five convolutional layers, with stride by 2 for down-sampling, and with a leaky rectified linear unit (lrelu) activation, with a leakage rate of $0.3$. The number of feature maps is successively $32,32,64,64,\text{and}~64$. The last convolutional is flattened and mapped using a linear fully connected layer into two vectors $(\mu,\sigma)$, each composed of $K=128$ features followed by a dropout of rate $0.3$. A latent vector $\boldsymbol{z}$ is generate as $\mu + \sigma \odot \boldsymbol{\epsilon}$, where $\boldsymbol{\epsilon}\sim\mathcal{N}(0,I)$ and given as an input to the decoder. The decoder is composed of five up-convolutional layers, with a lrelu activation, each composed of $64,64,32,32,\text{and}~4$ feature maps. The output layer with a sigmoid activation provides a mask of shape $256\times256\times4$. A learning rate of $0.0001$ is used with Adam optimizer. Using the validation set we experimentally tuned, $\alpha=10$ to focus more on the alignment loss and $\beta=0.1$. The total number of iterations is 10,000.
\section{Experiments and Results}
We use the public MM-WHS dataset \cite{zhuang2016multi} for cardiac segmentation consisting of $20$ MR ($\sim128$ slices) and $20$ CT ($\sim256$ slices) unpaired and multi-site images from $40$ patients. We followed the state-of-the-art data processing, domain adaptation protocol and evaluation metrics \cite{huo2018synseg,ouyang2019data,dou2018pnp,chen2019synergistic,chen2020unsupervised}. For data processing, we use the coronal view slices, cropped to $256 \times 256$ and normalized to zero mean and unit variance. To consider contextual information three adjacent slices ($256 \times 256 \times 3$) were stacked at the input and the middle slice label was used as the ground truth. Data augmentation included rotation, scaling, and affine transformations. A total of $11998$ MR and $9598$ sub-volumes were generated (each $256\times256\times3$). For domain adaptation, we randomly split each modality into training (16 subject) and testing (4 subjects). We use MR as a source domain, with $9599$ sub-volumes for training and $2399$ for validation. We set CT as a the target domain, with $8399$ sub-volumes for training and $1199$ for evaluation. We report the performance in terms of Dice Similarity Coefficient (DSC) and the Average Symmetric Surface Distance (ASSD).
\textbf{Experimental Settings and Results:} We consider four experimental settings. \emph{First}, we compare our model without optimal transport and trained with full supervision over the CT images (oracle VAE). We compare this setting against a U-Net~\cite{ronneberger2015u} to show how our VAE constrains the shape of the predictions to be valid. The oracle VAE also serves as an upper-bound baseline. \emph{Second}, to illustrate the domain shift problem, we consider the situation when no adaptation is performed, thereby, we train VAE-0 and U-Net-0 over MR images and evaluated them over CT images. \emph{Third}, we consider the SOA setting, in which 16 labeled and 16 unlabeled source and target sequences are used (OLVA-16). We compare this setting with four SOA methods for medical UDA: PnP-AdaNet \cite{dou2018pnp}, SIFA \cite{chen2019synergistic}, Synseg-net \cite{huo2018synseg} and Seg-DJOT \cite{ackaouy2020unsupervised}. We also compare to two SOA methods for natural image UDA: CycleGAN \cite{isola2017image} and AdaOutput \cite{tsai2018learning}. \emph{Fourth}, we consider a more ambitious scenario where we assume that only one unlabeled target sequence is available (OLVA-1). Therefore, we randomly draw one scan from the target set. To train OLVA-1 and to avoid overfitting, we fix all the model parameters and only the fully connected layer is retrained using loss evaluated at the latent space of Eq.~\ref{eq6}. We also perform an experiment where an auxiliary reconstruction task~\cite{duque2020spatio} is integrated (OLVA-R-1). The quantitative and qualitative results are presented in Table~\ref{TAB1} and Fig.~\ref{fig3}
\begin{figure}[b]
\includegraphics[width=0.9\textwidth]{figures/QualitativeResults.pdf}
\caption{Qualitative results of adaptation performances on segmentation.} \label{fig3}
\end{figure}
\begin{table}
\caption{Performances of UDA from MR to CT images under different settings. Postfix -0, -1 or -16 after names of each method indicate the number of unlabelled target scans used for training. We mark in bold the best results and we underline the second best. }
\small \setlength{\tabcolsep}{1.85pt}
\resizebox{0.85\linewidth}{!}{
\begin{tabular}{l *{5}{l} p{0.5\tabcolsep} *{5}{l}}
\toprule
& \multicolumn{5}{c}{\textbf{DSC Score}} && \multicolumn{5}{c}{\textbf{ASSD Score (mm)}} \\
\cmidrule(r){2-6} \cmidrule(l){8-12}
\textbf{Methods} & LV-M & LA-B & LV-B & A-A & \textbf{avg} & & LV-M & LA-B & LV-B & A-A & \textbf{avg} \\
\midrule
oracle U-Net & 0.83 & 0.89 & 0.92 & 0.93 & 0.89 && 0.38 & 0.39 & 0.28 & 0.31 & 0.34 \\
oracle VAE & \textbf{0.95} & \textbf{0.97} & \textbf{0.97} & \textbf{ 0.96} & \textbf{0.96} && \textbf{0.06} & \textbf{0.04} & \textbf{0.03 }& \textbf{0.05 }& \textbf{0.05} \\
\midrule
U-Net-0 & 0.10 & 0.27 & 0.02 & 0.24 & 0.15 && 36.0 & 19.4& 48.6 & 31.9 & 26.2\\
VAE-0 & \textbf{0.41} & \textbf{0.51} & \textbf{0.60} & \textbf{ 0.48} & \textbf{0.49} && \textbf{2.51} & \textbf{2.21}& \textbf{ 2.44 }& \textbf{2.95} & \textbf{2.53}\\
\midrule
\textbf{OLVA-16} & \textbf{0.79} & \textbf{0.87} & \textbf{0.88} & \textbf{0.88} & \textbf{0.85} && \textbf{0.56} & \textbf{0.53} & \textbf{0.37} & \textbf{0.45} & \textbf{0.31} \\
Seg-DJOT-16 & 0.57 & 0.60 & 0.57 & 0.62 & 0.59 && 3.64 & 3.62 & 3.85 & 5.20 & 4.07 \\
SIFA-16 & \underline{0.58} & 0.76 & \underline{0.76} & \underline{0.81} & \underline{0.73} && \underline{3.44} & 3.83 & 3.30 & 2.64 & {3.32}\\
Pnp-AdaNet-16 & 0.50 & \underline{0.77} & 0.60 & 0.79 & 0.66 && 10.2 & 4.04 & 8.60 & \underline{2.28} & 6.22 \\
SynSeg-Net-16 & 0.41 & 0.69 & 0.52 & 0.72 & 0.58 && 4.60 & 3.80 & 3.40 & 5.60 & 4.35 \\
AdaOutput-16 & 0.43 & 0.76 & 0.54 & 0.65 & 0.59 && 4.68 & \underline{2.89} & \underline{3.10} & 6.15 & 4.20\\
CycleGAN-16 & 0.28 & 0.75 & 0.52 & 0.73 & 0.57 && 4.85 & 6.20 & 3.92 & 5.54 & 5.30 \\
\midrule
OLVA-1 & 0.58 & 0.69 & 0.64 & \underline{0.67} & 0.64 && 2.10 & 1.95 & 1.85 & \textbf{2.30} & \underline{2.05} \\
OLVA-R-1 & \textbf{0.68} & \underline{0.70} & \underline{0.78} & 0.60 & \underline{0.69} && \textbf{1.89} & \textbf{1.88} & \textbf{1.51 }& \underline{2.43} & \textbf{1.92}\\
DECM-1 & \underline{0.60} & \textbf{0.78} & 0.71 & \textbf{0.78 }& \textbf{0.72} && 7.37 & 3.87& 6.44 &2.77& 5.11\\
Seg-DJOT-1 & 0.19 & 0.25 & 0.21 & 0.20 & 0.21 && 9.64 & 13.7 & 8.18 & 10.3 & 10.4 \\
SIFA-1 & 0.39 & 0.53 & \textbf{0.80} & 0.62 & 0.62 && 12.8 & 4.12 & 7.70 & 2.72 & 6.84 \\
Pnp-AdaNet-1 & 0.29 & 0.48 & 0.33 & 0.58 & 0.25 && 25.1 & 27.1 & 27.7 & 7.14 & 21.8\\
\bottomrule
\end{tabular} }\label{TAB1}
\end{table}
\textbf{Discussion:} Table~\ref{TAB1} shows that our supervised baseline method outperforms the U-Net, achieving a high DSC and more importantly producing valid cardiac shape predictions as seen in Fig.~\ref{fig3}, and as reflected by the ASSD score. When no adaptation is considered, VAE-0 achieves $49\%$ DSC, while U-Net achieved only $15\%$. As our VAE-0 pushes the latent semantic features to be close to normal distribution, it partially aligns the marginal distributions. In the UDA setting, OLVA-16 outperforms the SOA's best results by an additional $12.5\%$ in DSC! and having minimal erroneous prediction as seen in Fig.~\ref{fig3}, with average ASSD of $0.31~\text{mm}$. Considering the target data scarcity UDA Setting, OLVA-1 achieved the second best results after DECM-1 with an $8\%$ DSC difference. The results of OLVA-1 when the reconstruction auxiliary task is introduced (OLVA-R-1) reduce the gap to $5\%$ at the price of increasing the model complexity. The second-place is honorable, comparing the 1.7 million parameters of OLVA-1 with the more than 95 million parameters of DECM-1, and considering the quality of predictions as reflected by the ASSD. We also examine OLVA's performance when trained with randomly sampled 5 and 8 targets sequences. OLVA-5 achieves similar performance to DCEM-1, while OLVA-8 achieves better DSC score $79\%$. As for further ablation studies, we change the latent dimension to $64$, $256$ and $512$. With $K=64$, a degradation in the source domain performance was observed, yielding an average DSC score of $79\%$. With $K=256$, similar performances to $K=128$ is achieved. When $K=512$, a degradation in the performance over the source and the target domain is observed, leading to $69.6\%$ target DSC score for OLVA-16. This degradation is expected as optimizing for $\gamma$ requires a reasonable number of samples which grows with $K$'s dimensionality \cite{ackaouy2020unsupervised}.
\section{Conclusion}
To improve the applicability of deep learning model on new modality where it is expensive to acquire expert annotations, unsupervised domain adaptation represents a central solution. In this paper, we tackle the problem of unsupervised cross-modality medical image segmentation
with
a novel framework that jointly integrates VAE and OT theory to solve UDA problem. OLVA is a simple, efficient and lightweight model, which makes it practical to deploy in real-life without requiring a machine with huge computational resources. The usability of our method can be integrated within other learning regimes, for instance, a weakly-supervised model where sparse annotation of biomedical volumetric data are available and the aim would be to leverage the rest of the unlabeled data by matching them with the available labeled set. Future work will address the problem of building a general segmenter where the adaptation from one task to another is done with minimal task-specific information and to leverage other tasks information.
\bibliographystyle{splncs04}
\section{First Section}
\subsection{A Subsection Sample}
Please note that the first paragraph of a section or subsection is
not indented. The first paragraph that follows a table, figure,
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Subsequent paragraphs, however, are indented.
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The contribution should contain no more than four levels of
headings. Table~\ref{tab1} gives a summary of all heading levels.
\begin{table}
\caption{Table captions should be placed above the
tables.}\label{tab1}
\begin{tabular}{|l|l|l|}
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Heading level & Example & Font size and style\\
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Title (centered) & {\Large\bfseries Lecture Notes} & 14 point, bold\\
1st-level heading & {\large\bfseries 1 Introduction} & 12 point, bold\\
2nd-level heading & {\bfseries 2.1 Printing Area} & 10 point, bold\\
3rd-level heading & {\bfseries Run-in Heading in Bold.} Text follows & 10 point, bold\\
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schemas. Whenever possible, use vector graphics instead (see
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\begin{figure}
\includegraphics[width=\textwidth]{fig1.eps}
\caption{A figure caption is always placed below the illustration.
Please note that short captions are centered, while long ones are
justified by the macro package automatically.} \label{fig1}
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For citations of references, we prefer the use of square brackets
and consecutive numbers. Citations using labels or the author/year
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\begin{table}
\caption{Performances of UDA from MR to CT images under different setting.}\label{tab1}
\small \setlength{\tabcolsep}{1.85pt}
\begin{tabular}{l *{5}{l} p{0.5\tabcolsep} *{5}{l}}
\toprule
& \multicolumn{5}{c}{\textbf{DSC Score}} && \multicolumn{5}{c}{\textbf{ASSD Score (mm)}} \\
\cmidrule(r){2-6} \cmidrule(l){8-12}
\textbf{Methods} & LV-M & LA-B & LV-B & A-A & \textbf{avg} & & LV-M & LA-B & LV-B & A-A & \textbf{avg} \\
\midrule
& \multicolumn{10}{c}{\textbf{Supervised Setting}: 16 labeled $\boldsymbol{X}^{t}$ and 16 labeled $\boldsymbol{X}^{s}$} \\
\midrule
U-Net & 0.83 & 0.89 & 0.92 & 0.93 & 0.89 && 0.38 & 0.39 & 0.28 & 0.31 & 0.34 \\
VAE (ours) & \textbf{0.95} & \textbf{0.97} & \textbf{0.97} & \textbf{ 0.96} & \textbf{0.96} && \textbf{0.06} & \textbf{0.04} & \textbf{0.03 }& \textbf{0.05 }& \textbf{0.05} \\
\midrule
& \multicolumn{10}{c}{\textbf{No Adaptation}: 0 unlabeled $\boldsymbol{X}^{t}$ and 16 labeled $\boldsymbol{X}^{s}$} \\
\midrule
U-Net-0 & 0.10 & 0.27 & 0.02 & 0.24 & 0.15 && 36.0 & 19.4& 48.6 & 31.9 & 26.2\\
VAE-0 & \textbf{0.41} & \textbf{0.51} & \textbf{0.60} & \textbf{ 0.48} & \textbf{0.49} && \textbf{2.51} & \textbf{2.21}& \textbf{ 2.44 }& \textbf{2.95} & \textbf{2.53}\\
\midrule
& \multicolumn{10}{c}{\textbf{UDA Setting-16}: 16 unlabeled $\boldsymbol{X}^{t}$ and 16 labeled $\boldsymbol{X}^{s}$} \\
\midrule
\textbf{OLVA-16} & \textbf{0.79} & \textbf{0.87} & \textbf{0.88} & \textbf{0.88} & \textbf{0.85} && \textbf{0.56} & \textbf{0.53} & \textbf{0.37} & \textbf{0.45} & \textbf{0.31} \\
Seg-DJOT-16 & 0.57 & 0.60 & 0.57 & 0.62 & 0.59 && 3.64 & 3.62 & 3.85 & 5.20 & 4.07 \\
SIFA-16 & \underline{0.58} & 0.76 & \underline{0.76} & \underline{0.81} & \underline{0.73} && \underline{3.44} & 3.83 & 3.30 & 2.64 & {3.32}\\
Pnp-AdaNet-16 & 0.50 & \underline{0.77} & 0.60 & 0.79 & 0.66 && 10.2 & 4.04 & 8.60 & \underline{2.28} & 6.22 \\
SynSeg-Net-16 & 0.41 & 0.69 & 0.52 & 0.72 & 0.58 && 4.60 & 3.80 & 3.40 & 5.60 & 4.35 \\
AdaOutput-16 & 0.43 & 0.76 & 0.54 & 0.65 & 0.59 && 4.68 & \underline{2.89} & \underline{3.10} & 6.15 & 4.20\\
CycleGAN-16 & 0.28 & 0.75 & 0.52 & 0.73 & 0.57 && 4.85 & 6.20 & 3.92 & 5.54 & 5.30 \\
\midrule
& \multicolumn{10}{c}{\textbf{UDA Setting-1}: 1 unlabeled $\boldsymbol{X}^{t}$ and 16 labeled $\boldsymbol{X}^{s}$} \\
\midrule
OLVA-1 & 0.58 & 0.69 & 0.64 & \underline{0.67} & 0.64 && 2.10 & 1.95 & 1.85 & \textbf{2.30} & \underline{2.05} \\
OLVA-R-1 & \textbf{0.68} & \underline{0.70} & \underline{0.78} & 0.60 & \underline{0.69} && \textbf{1.89} & \textbf{1.88} & \textbf{1.51 }& \underline{2.43} & \textbf{1.92}\\
DECM-1 & \underline{0.60} & \textbf{0.78} & 0.71 & \textbf{0.78 }& \textbf{0.72} && 7.37 & 3.87& 6.44 &2.77& 5.11\\
Seg-DJOT-1 & 0.19 & 0.25 & 0.21 & 0.20 & 0.21 && 9.64 & 13.7 & 8.18 & 10.3 & 10.4 \\
SIFA-1 & 0.39 & 0.53 & \textbf{0.80} & 0.62 & 0.62 && 12.8 & 4.12 & 7.70 & 2.72 & 6.84 \\
Pnp-AdaNet-1 & 0.29 & 0.48 & 0.33 & 0.58 & 0.25 && 25.1 & 27.1 & 27.7 & 7.14 & 21.8\\
\bottomrule
\end{tabular} \label{TAB1}
\end{table}
\emph{Dataset:} to confirm the robustness of OLVA, we employ the Multi-Modality Whole Heart Segmentation Challenge 2017 dataset \cite{zhuang2016multi} for cardiac segmentation in 3D MR and 3D CT images. The dataset is composed of unpaired volumes of $20$ MRI ($\sim128$ slices) and $20$ CT ($\sim256$ slices) collected at different clinical sites from $40$ patients.
\emph{Data Processing:} recent studies \cite{ouyang2019data,dou2018pnp,chen2019synergistic,chen2020unsupervised} used the coronal view slices, cropped to $256 \times 256$ and normalized to zero mean and unit variance. The input is reformed to consider contextual information by stacking three adjacent slices ($256 \times 256 \times 3$) and the middle slice label were used as the ground truth. Data augmentation were applied: rotation, scaling, and affine transformations. A total of $11998\times256\times256\times3$ for MRI modality and $9598\times256\times256\times3$ for CT scan were generated.
\emph{Adaptation Setting and Protocol:} recent studies \cite{ouyang2019data,dou2018pnp,chen2019synergistic,chen2020unsupervised}, randomly split each modality of the data into training (16 subject) and testing (4 subjects). They used the MRI modality as a source domain with $9599\times256\times256\times3$ for training and $2399\times256\times256\times3$ for evaluation. The CT modality were set as a target domain with $8399\times256\times256\times3$ for training and $1199\times256\times256\times3$ for evaluation. We followed the same setting to compare our model performance with State-Of-the-Arts (SOA).
\emph{Evaluation Metrics:} we followed the common practice to quantitatively evaluate the segmentation performance. We report Dice Similarity Coefficient (DSC) and Average Symmetric Surface Distance (ASSD) for each label class. Within each setting, we mark in bold the best results and we underline the second best results.
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{"url":"https:\/\/www.physicsforums.com\/threads\/poorly-worded-question.50948\/","text":"Poorly worded question\n\nok here is the question exaclty- its worth 5 marks so i dont know what im missing..\n\nA car with a mass of 850kg is moving in a straight line at a constant speed of 110km\/h. It is brought to rest in 10.0s. What constant force is acting to stop the car?\n\nRelated Introductory Physics Homework Help News on Phys.org\n$$F_{avg}\\Delta t = m\\delta v$$\n\nLooks like an impulse question.\n\nThanks anyways guys but i got it. I think i was thinking it was asking what force as in friction or wind etc. was stopping it but its asking the force it would take to stop it in that time.\n\nDoc Al\nMentor\nIf you are not familiar with impulse\/momentum, you can always use kinematics to find the acceleration, then apply Newton's 2nd law to find the force.\n\ncepheid\nStaff Emeritus","date":"2020-01-24 22:41:23","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\": 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\": 0.4620160162448883, \"perplexity\": 613.4709605601497}, \"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-2020-05\/segments\/1579250626449.79\/warc\/CC-MAIN-20200124221147-20200125010147-00239.warc.gz\"}"} | null | null |
Q: basic line highcharts update using ajax I want to implement basic line highchart (http://www.highcharts.com/demo/line-basic)
using ajax. chart values update through ajax.
please help me how I do it
thanks
A: The highcharts website has an article about this here http://docs.highcharts.com/#preprocessing-live-data
This covers the basics of getting new data from the server using Ajax. If you want to add a new series to the chart, then you need to call
chart.addSeries(data)
Th is is documented here http://api.highcharts.com/highcharts#Chart.addSeries()
A: Please take look at example where data is loaded by ajax
http://www.highcharts.com/demo/line-ajax
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Emergency services rush in as woman is hit by car outside Boots in Sleaford
The patient was taken to Lincoln County
An ambulance rushed to the scene of the crash in Sleaford
Police and an ambulance rushed to the scene after a pedestrian was hit by a car.
The incident happened outside Boots on Southgate, Sleaford, on October 9 at around 8.25am.
The pedestrian was hit by a grey Suzuki car.
Police attend as road is blocked in Gainsborough after three cars crash near chip shop
A spokesperson for East Midlands Ambulance Service said: "We received a call at 08.24am on October 9 to an incident outside Boots on Southgate, Sleaford.
"The caller reported a road traffic collision involving a vehicle and a pedestrian.
"We sent a double crewed ambulance.
"One patient was taken to Lincoln County Hospital for further care."
The area where the crash happened (Image: Google Maps)
The incident happened at a busy junction on Sleaford's one way system.
Sleaford Town Councillor Ken Fernandes said he heard about the collision.
He said: "I hope the person recovers and they get well soon."
Woman dies after five-car crash at Lincoln shopping centre
He added that he would like to see the speed limit reduced in the town centre - although there is nothing to suggest speed was a factor in this incident.
He said: "The centre of town must be a 20 mile per hour speed limit.
"It's a 30 mile per hour zone but with the amount of people in the town centre it should be a 20 mile per hour zone during hours when children are out there and in the day.
"Lowering the speed limit lowers the damage caused."
East Midlands Ambulance Service | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,580 |
Q: How to query an object in an array embedded in mongodb? I am currently working on an application that takes control of Projects, which have Meetings and that these meetings have Participants.
I want to consult a Participant by his nomina field.
Structure for a project document object:
{
"id":"5c1b0616a0441f27f022bfdc",
"name":"Project Test",
"area":"Area",
"date":"2019-01-01",
"meetings":[
{
"id":"5c1b073d445707834699ce97",
"objetive":"Objetive",
"fecha":"2019-01-01",
"participants":[
{
"nomina":1,
"name":"Person 1",
"role":"Rol1",
"area":"area1",
"signature":null
},
{
"nomina":2,
"name":"Person 2",
"role":"rol 2",
"area":"área 2",
"signature":null
}
]
}
]
}
Expected behavior
I want to consult a Participant by nomina field knowing the id of the Project and also knowing the id of the Meeting.
Expected output
Having:
*
*id Project = 5c1b0616a0441f27f022bfdc
*id Meeting = 5c1b073d445707834699ce97
*nomina Participant = 1
It's expected that the query will return me:
{
"nomina":1,
"name":"Person 1",
"role":"Rol1",
"area":"area1",
"signature":null
}
A: For not so huge number of meetings in every document if you want to get the exact document stated, you can do this pipeline, it is straight forward:
db.collection.aggregate(
[
{
$match: {
id:"5c1b0616a0441f27f022bfdc"
}
}, {
$unwind: {
path : "$meetings"
}
},
{
$unwind: {
path : "$meetings.participants"
}
},
{
$match: {
"meetings.id":"5c1b073d445707834699ce97",
"meetings.participants.nomina":1
}
},
{
$replaceRoot: {
newRoot: "$meetings.participants"
}
}
]);
If you would have over thousands of elements in meetings then I'd suggest adding another match to meetings or grouping meetings and project IDs.
But if you just want to get the document containing what you want it is just a simple find query:
db.collection.find({id:"5c1b0616a0441f27f022bfdc","meetings.id":"5c1b073d445707834699ce97","meetings.participants.nomina":1 });
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,828 |
Q: Is there any way to make dialog's header and footer sticky in angular material When I scroll the content of Dialog, right top close and right bottom Ok button also getting scrolled.
I want to make those button fixed, not to scrolled so that any time I can close the dialog.
This my code
<div md-dialog-content>
<button class="close" mat-button (click)="onNoClick()">
<mat-icon>close</mat-icon>
</button>
//here my table content
</div>
<div mat-dialog-actions>
<button id="matbuttonClosedownSide" color="primary" mat-button [mat-dialog-close]="null">Ok</button>
</div>
code to open model
const dialogRef = this.dialog.open(myModalComponent, {
width: '80%',
height:'80%',
panelClass: 'my-dialog',
disableClose: true ,
data:this.data[index]
});
A: I had the same problem, I did it like follows:
.my-dialog {
.mat-dialog-container {
position: relative;
padding-bottom: 50px; // height of your sticky footer
}
.sticky-modal-footer {
position: absolute;
bottom: 0;
left: 0;
width: 100%;
border-top: 1px solid #efefef;
background-color: white;
padding: 10px 24px;
}
}
And then added the dialog buttons in a footer element:
<button class="close" matDialogClose>
<mat-icon>close</mat-icon>
</button>
<h1 mat-dialog-title>{{ modalTitle }}</h1>
<hr />
<mat-dialog-content> Dialog content </mat-dialog-content>
<footer class="sticky-modal-footer">
<button color="primary">OK</button>
</footer>
A: I ran into the same problem yesteday. After some research I found the mat-dialog-title and mat-dialog-actions directives. That did the trick for me:
<h3 mat-dialog-title>This is the sticky header</h3>
<mat-dialog-content>
<h3>Lorem Ipsum</h3>
<p>Lorem ipsum dolor sit amet...</p>
</mat-dialog-content>
<mat-dialog-actions align="start">
<button mat-button mat-dialog-close>Cancel</button>
<button mat-button [mat-dialog-close]="true">Accept</button>
</mat-dialog-actions>
You may need to fill the the <mat-dialog-content></mat-dialog-content> with more text to see the scrollbar. But it shows the basic idea: the header stays sticky above the scrollable content and the buttons stay sticky at the bottom.
My sample code is strongly inspired by this official Angular Material Dialog example: https://material.angular.io/components/dialog/examples#dialog-content
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,286 |
Q: KendoUI Treeview shows twisty even when has no children I have a kendoui treeview displaying some data obtained from a json datasource. The treeview is working with the exception that it displays the twisty for child items even when there are no child items.
I believe this is related to my json string but at this point I don't believe I can change it.
Here is the json string:
[{"Title":"Shared Documents","spriteCssClass":"folder","LastModified":"1/15/2013 10:42:20 AM","Items":[{"Title":"Folder 1","spriteCssClass":"folder","LastModified":"1/15/2013 10:42:20 AM","Items":[{"Title":"Subfolder 1","spriteCssClass":"folder","LastModified":"1/15/2013 10:41:52 AM","Items":[]},{"Title":"Test Tax Document.docx","spriteCssClass":"docx","LastModified":"1/15/2013 10:42:20 AM","Items":[]}]}]}]
I believe the problem is that the Items[] still exist even if there are no items.
Here is the code for my treeview...
var treeDS = new kendo.data.HierarchicalDataSource({
data: json,
schema: {
model: {
children: "Items"
}
}
});
var treeview = $("#CCA_DocLibTreeViewer_Tree").kendoTreeView({
template: "#= item.Title # - #= item.LastModified # <a href='\\#'>View</a>",
dataSource: treeDS,
dataTextField: ["Title", "Title"]
}).data("kendoTreeView");
Any thoughts on what I can do about this?
A: You are right, the question is that if it has Items no matter the length it assumes that has children.
The solution is either do not generate those empty Items or define treeDS as:
var treeDS = new kendo.data.HierarchicalDataSource({
data : json,
schema: {
model: {
children : "Items",
hasChildren: function (node) {
return (node.Items && node.Items.length > 0);
}
}
}
});
You can see that I have defined a hasChildren function that verifies that node.Items exists and its length is actually greater than 0.
You might see it running in JSFiddle here
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,927 |
Енергија ветра је енергија која потиче од снаге ветра. Представља конвенционалан обновљиви извор енергије, који се вековима користи за добијање механичке, а у новије време и електричне енергије. Међутим, производња електричне енергије из енергије ветра у већим количинама почела је тек после нафтне кризе 1973.
Енергија ветра се претвара у корисни облик енергије, електричну енергију, помоћу ветроелектрана. У класичним ветрењачама енергија ветра се претвара у механичку те се као таква директно користи за млевење житарица или пумпање воде. Крајем 2007. инсталирана снага ветроелектрана у свету била је 94,1 -{GW}-. Тренутно ветроелектране покривају тек 1% светских потреба за електричном енергијом, док у Данској та цифра износи 19%, Шпанији и Португалији 9%, Немачкој и Ирској 6% (подаци за 2007). Електричном енергијом из ветра ветроелектране снабдевају електро енергетску мрежу као што и појединачни ветроагрегати напајају изолована места. Ветар је богат, обновљив, лако доступан и чист извор енергије. Недостатак ветра ретко узрокује несавладиве проблеме када у малом уделу учествује у опскрби електричном енергијом, али при већем ослањању на ветар доводи до већих губитака.
Настанак
Настанак ветра је сложен процес. Како сунце неравномерно греје Земљу, полови примају мање сунчеве енергије него екватор. Поред тога, копно се брже гре и брже хлади од мора. Такво загрејавање покреће глобални атмосферски систем преноса топлоте с површине Земље према стратосфери која се понаша као виртуална таваница. Већина енергије таквог струјања ветра је на великим висинама где брзина ветра прелази и 160 km/h. Dio energije vjetra trenjem prelazi u difuznu toplinu kroz atmosferu i Zemljinu površinu. Predviđanja govore da je 72 TW energije vjetra iskoristivo u komercijalne svrhe. Treba napomenuti da ni teoretski ni praktično nije iskoristiva sva snaga vjetra.
Расподела брзине ветра
Ветар јако варира и средња вредност брзине за дату локацију није показатељ количине енергије коју ветроагрегат може произвести. Ипак, код предвиђања понашања ветра на одређеном месту, користе се подаци генерисани мерењима. Довољна је и мања промена локације да би јавиле велике промене у брзини ветра. Брзина ветра се мери и апроксимирамо Рејлијевом расподелом.
Како се велика количина енергије добија при већим брзинама ветра, доста енергије долази у краћим интервалима, односно на махове, као и ветар. Последица тога је да ветроелектране немају сталну снагу на излазу као што то имају нпр. термоелектране, те постројења која напајају ветроагрегати морају имати осигурану производњу електричне енергије и из неког другог извора. Сталност снаге код ветроелектрана би могао да осигура напредак у технологијама које се баве складиштењем енергије тако да се може користити енергија која је добијена за време јачег ветра онда када га нема.
Прикључак на мрежу
Најчешће се користе асинхрони генератори за ветроагрегате који захтевају реактивну снагу из мреже за побуду и стога садржавају кондензаторске батерије за њену компензацију. Различити типови ветроагрегата се понашају различито у случајевима поремећаја у електро енергетској мрежи тако да су преко потребна претходна испитивања и моделовања динамичних електромеханичких особина код нових ветроагрегата пре њиховог пуштања у погон. Постоје имплементације и са синхроним генератором, али такве нису често у примени, а асинхрони генератори с двостраним напајањем имају најпожељнија својства што се тиче спајања на мрежу.
Фактор оптерећења
Како је брзина вјетра промењива, годишња производња једног ветропарка није збир умножака називне снаге генератора и броја сати у години. Однос стварно произведене и теоријски највеће могуће произведене енергије назива се фактор оптерећења. Фактор оптерећења углавном износи 20 до 40% у најбољим случајевима.
За разлику од термоелектрана код којих на фактор оптерећења највише утиче цена горива и занемариво време за ремонт, код ветроелектрана фактор оптерећења зависи од непромјењивог својства ветра, и његове присутности. Што се тиче нуклеарних електрана, цена горива је изузетно ниска тако да фактор оптерећења досеже, па и прелази 90%.
Непредвидивост ветра
Електрична енергија добијена из енергије ветра варира из сата у сат, дневно и сезонски. Постоје и годишње варијације, али нису толико значајне. С обзиром на то може се краткорочно предвидети количина енергије која се може добити. Попут других извора електричне енергије, енергија ветра мора бити према одређеном распореду потрошње. Због тога се користе методе прогнозирања снаге ветра, али предвиђање износа добијене енергије из ветра није увек најпоузданија метода.
Производња и потрошња електричне енергије морају бити подеднаке како би мрежа остала једнолико оптерећена. Ова варијабилност може представљати изазов при спајању електричне енергије произведене ветром у мрежу. Интермитентност и непредвидива природа ветра повећавају трошкове за регулацију, подижу радну залиху, а при високој продорности могла би довести до повећања количине електричне енергије у систему што може проузроковати проблеме с преоптерећењем. Решење би било складиштење или повезивање мреже наизменичне струје високонапонским кабловима наизменичне струје. Енергија ветра може се заменити другим електранама у раздобљима слабог ветра. Мреже за пренос енергије већ сада се морају носити са застојима производње и дневним променама електричне потражње. Системи с великим капацитетом за енергију ветра би требали да имају више резерви (енергана које раде на мање од максималног оптерећења).
Реверзибилне хидроелектране или други облици складиштења енергије у мрежи могу похранити енергију добијену за време јаких ветрова и пустити је када је то потребно. Похрањена енергија повећава економску вредност енергије ветра, јер може заменити велике трошкове производње током највеће потражње. Потенцијални приход може премашити трошкове и губитке у похрани. Трошак складиштења може додати 25% на цену похрањене енергије ветра, али није предвиђено да се примењује на велики удео добијене енергије ветра. Динорвиг електрана је реверзибилна хидроелектрана од 2 -{GW}- у Велсу изједначава врхове потражње електричне енергије, омогућујући тако ефикаснији рад електрана које добављају електрицитет за базно оптерећење. Корисности од 75% и висока цена изградње таквих електрана нису проблем, јер је цена за рад тих електрана ниска и могућност смањења базне потрошње може смањити цену горива и укупне трошкове генерирања електричне енергије.
У неким регијама, вршна брзина ветра не може се поклопити с врхом потражње електричне енергије. У државама САЂа, Калифорнији и Тексасу, за време врућих летних дана брзине ветра су ниске, а потражња електричне енергије висока због масивног кориштења клима уређаја. Нека комунална предузећа субвенционирају куповину геотермалних топлотних пумпи својим корисницима, у сврху смањења потрошње електричне енергије током летних месеци чинећи кориштење клима уређаја и до 70% делотворнији. Друга могућност је да се међусобно распршена подручја повежу у тзв. "супермрежу" високоволтажних каблова за једносмерну струју.
У Великој Британији, потражња за електричном енергијом виша је зими него лети, пропорционално брзини ветра. Соларна енергија тежи да буде комплементарна енергији ветра. Подручја високог притиска ваздуха доносе ведро небо и слабије површинске ветрове, док су дани с нижим притиском ваздуха претежито ветровити и облачни. То значи да је соларне енергије обично највише лети, док је енергије ветра највише зими, те се тако интермитенција ветра и сунчеве енергије међусобно поништавају.
Као и код других извора и производња електричне енергије из ветроелектрана мора бити испланирана, али природа ветра то не омогућава, успркос помоћиметеорологије.
Инсталирана снага
Више је хиљада ветроагрегата у погону, укупно инсталиране снагe 73,904 -{MW}-, од чега је у Европи 65% (2006). Ветроелектране су имале најбржи раст од свих алтернативних извора енергије на почетку 21. века, капацитет им се више него учетворостручио од 2000. до 2006. 81% инсталиране снаге отпада на САД и Европу. Процене су да ће до 2010. бити инсталирано 160 -{GW}- снаге ветроагрегата с порастом од 21% годишње. Данска производи приближно једну петину електричне енергије ветроелектранама, што је чини земљом с највећим уделом ветроелектрана у властитој производњи. Она је значајни корисник и произвођач вјетротурбина. Немачка је водећи произвођач ветроелектрана, с уделом од 28% светске производње у 2006. и укупном производњом од 38,5 -{TWh}- у 2007. године (6,3% електричне енергије Немачке), а циљ јој је да до 2010. досегне производњу од 12,5% од укупне. Немачка има 18,600 ветротурбина, углавном на северу земље, укључујући и три највеће на свету (6-{MW}- и две по 5-{MW}-).
Кина је 2005. најавила изградњу ветропарка од 1000 -{MW}- у Хебеју до 2020. Циљ јој је био до исте године има и производњу од 20,000 -{MW}- из обновљивих извора. Сматра се да је од ветра на простору Кине могуће добити 253.000 -{MW}-.
Исплативост
Трендови
Према извештају Глобалног савета за енергију ветра, 2007. је инсталирано додатних 20 -{GW}- ветроелектрана, што је укупно инсталирану снагу довело на 94 -{GW}-. Гледано с привредног стајалишта, подручје производње електричне енергије из ветра је постало јако важно и финансијски интересантно на тржишту. Вредност уграђене опреме за ветроелектране у 2007. износи 25 милијарди €. Цена енергије из ветра 2004. је пала на једну петину цене из осамдесетих, а процена је да ће се пад наставити како расте масовна производња вишемегаватних ветротурбина. Како год, цена уградње је 2007. износила 1.300-{€/KW}-, што је више у поређењу с 2005. годином, када је износила 1.100 -{€/KW}-. Раст цене се објашњава великом потражњом за опремом, док је јако мало произвођача способно да произведе велике модерне турбине и носаче за ветроагрегате. На цену електричне енергије из ветро и хидроелектрана занемарив утицај има цена горива и јако мали утицај одржавање постројења, али су капитални трошкови значајни.
Теоретски потенцијал
Снага ветра у атмосфери је много већа од садашње светске потрошње. Најисцрпнија истраживања кажу да је укупна снага ветра на копну и близу обале 72 -{TW}-, што је еквивалентно 54 милијарде тона нафте годишње или пет пута више него што свет тренутно троши у било којем облику.
Директни трошкови
Многе потенцијалне локације ветроенергетских постројења су далеко од потрошачких центара, што повећава трошак због изградње нових мрежа за пренос електричне енергије. У неким подручјима то је зато што су јаки ветрови утицали на изградњу средишта даље од ветровитих подручја. Ветар који је некада био непријатан, данас је вредан извор енергије, без обзира на то што су се цивилизације настањивале у подручјима која су више заштићена од ветра.
Како су главни трошкови у добијању електричне енергије из енергије ветра заправо трошкови изградње, а не цена горива, просечна цена производње такве енергије зависи од израде и поправки електране. Гранична цена енергије након што је електрана изграђена износи мање од 1 цента по -{kWh}-.
Цене електричне енергије јако су регулисане широм света. Купци потписују дугорочне уговоре како би смањили ризик будућих флуктуација, осигуравајући тако стабилнији поврат новца за пројекте у стадијуму развоја. У таквим уговорима особа одговорна за рад система се обвезује на куповину енергије добијене ветром по фиксној цени за одређени период. Те цене се могу разликовати од цена енергије из других извора, па чак могу садржати и одређене субвенције.
Како је цена електричне енергије ствар тржишта, приходи су већи када се производња одвија у периодима више цене. Профитабилност ветроелектрана ће стога бити већа када се време њиховог рада подудара с тим периодима.
Историја
Људи користе енергију вјетра барем 5500 година, неки од примјера је да се чамац са једрима користи барем 5000 година и архитекти су користили управљан-ветар за природне вентилације још у античко доба. Коришћење вјетра да се обезбиједи механичка енергија је дошло негдје касније у антици.
У старој Персији, ветрењаче са вертикалном осовином, напола затворене (тако да вјетар потискује само једну половину ротора) и равним "једрима" се користе бар од 200. године нове ере.
Практичне вјетрењаче сличне конструкције су направљене у Авганистану у 7. вијеку. Са Блиског истока, идеја се проширила до Европе и вјетрењаче за млевење зрња у брашно или пумпање воде су забиљежене у 12. вијеку у Енглеској и Холандији.
До 19. вијека вјетрењаче су распрострањене по читавој Европи и донесене су и у Сјеверну Америку.
Крајем 19. вијека енергија вјетра се почела користити и за производњу електричне енергије (види ветроелектрана), али углавном у малим локалним постројењима до нафтне кризе 1973.
Послије кризе, долази у низу земаља до ужурбане активности за искориштење енергије вјетра за производњу струје. Са успонима и падовима, везаним углавном за раст и пад цијена нафте, развој се нарочито убрзава послије 2000. са непрекидним растом цијена нафте.
Развојне могућности
Енергија вјетра пружа велике могућности за даљи развој. При крају 2007. свјетски капацитет електрана на ветар је 94 -{GW}-, али то је и даље само 1% од укупне производње електричне енергије.
Земље које воде у производњи су:
Данска, 19% од укупне производње електричне енергије долази од ветра
Шпанија и Португал, 9%
Немачка и Ирска, 6%
Времена се ипак мењају. Производња електричне енергије из ветра се повећала пет пута од 2000. до 2007.
Производња је засад профитабилна и конкурентна по цени класичним изворима (хидроенергија, термоенергија, нуклеарна енергија) само у крајевима са већим брзинама ветра, као на обали мора и слично. Међутим са растом цена класичних енергената и са падом цена турбина на ветар, очекује се измјена овог односа у будућности.
Прорачун добијене снаге
Снага је пропорционална брзини вјетра, активној површини кракова ветрењаче и густини ваздуха. Прорачун искористиве снаге ветра је детаљније обрађен у чланку ветрењача.
Види још
Ветрењача
Обновљиви извори енергије
Референце
Спољашње везе
Енергија ветра са примерима
Канадска Асоцијација енергије ветра Нова технологија енергије ветра
Америчка Асоцијација енергије ветра
Енергија ветра и птице
Global Wind Energy Council (GWEC)
World Wind Energy Association (WWEA)
Dynamic Data Dashboard from the International Energy Agency
Current global map of wind power density
Обновљиви извори енергије | {
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An emotional video of a traumatised baby boy, covered in blood, in Syria clinging on to a nurse after his home was hit by an airstrike has been released.
The video was posted by the Syrian American Medical Society who reported that he had arrived at their hospital in Aleppo with his brother yesterday.
The medical situation in Aleppo is so dire that the only neurosurgeon left in the city performed brain surgery on the floor without anesthetics.
They wrote "This is what #ChildrenOfSyria endure everyday".
13 deaths were recording by the SAMS' medical personnel that day, and the hospital treated 40 wounded, the Daily Mail reports.
He was hit in an airstrike. We don't know whether it was the Russian or Syrian government.
They also revealed the shocking fact that in the east of the city, there are only 29 doctors left and a nurse said that 'Aleppo has been forgotten'.
Life in Aleppo has become very dire. No camera or no pen can describe what we're experiencing here.
Hospitals are overwhelmed. There's a severe shortage of medical supplies and specialized doctors. There's only one neurosurgeon, two general surgeons, and three orthopedic surgeons left in the besieged city.
The bombs have been so destructive that children cannot return to the underground schools.
Since Russian bombings began a year ago, thousands of civilians have been killed and at least 100,000 children remain trapped in the city alone.
SAMS operates 106 medical facilities to provide for citizens in need.
You can donate here to support them in providing this emergency care. | {
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{"url":"http:\/\/www.whxb.pku.edu.cn\/EN\/10.3866\/PKU.WHXB202008051","text":"Acta Phys. -Chim. Sin. \u203a\u203a 2021, Vol. 37 \u203a\u203a Issue (4): 2008051.\n\n\u2022 REVIEW \u2022\n\n### Mechanisms and Applications of Laser Action on Lead Halide Perovskites\n\nJiaxin Wang, Weili Shen, Jinning Hu, Jun Chen(), Xiaoming Li, Haibo Zeng()\n\n\u2022 Received:2020-08-19 Accepted:2020-09-11 Published:2020-09-16\n\u2022 Contact: Jun Chen,Haibo Zeng E-mail:chenjun@njust.edu.cn;zeng.haibo@njust.edu.cn\n\u2022 About author:Email:zeng.haibo@njust.edu.cn (H.Z.)\nEmail: chenjun@njust.edu.cn (J.C.)\n\u2022 Supported by:\nthe Natural Science Foundation of Jiangsu Province(BK20181296);the National Natural Science Foundation of China(11502116);the Fundamental Research Funds for the Central Universities(30919011253)\n\nAbstract:\n\nIn recent years, lead-halide perovskites, one of the most competitive material types in the field of semiconductors, has attracted widespread attention because of its easy preparation, low cost, and high performance. Lead-halide perovskites are a type of material with an ABX3 structure, in which A is an organic or inorganic monovalent cation, B is a divalent cation, and X is a halogen ion. Among them, the B-site ion and X-site ion form an octahedron, with the B-site ion occupying the center and the X-site ion located at the apex of the octahedron. This type of octahedron can undergo lattice changes such as rotation or tilt through the replacement of different halogen anions, which affects the material band gap. The octahedron is located in the center of a cube, which is composed of A-site ions. These structures constitute the basic unit of the perovskite. Compared with the widely used \u2161-\u2165 or \u2162-\u2164 semiconductor nanocrystalline materials, perovskite nanocrystals have great application potential owing to their superior optoelectronic performance. However, their stability problem restricts further development, making them unable to compete in commercial applications. Studies on the stability of perovskite materials began in 2009. It was discovered through experiments that perovskite materials would undergo irreversible degradation under the action of liquid polar solvents, which confirmed that humidity and air are important factors in perovskite degradation. With further research, the problem of illumination has also come to the surface. It was found through experiment that, when oxygen and humidity were excluded, the light condition could also have a certain negative impact on perovskite materials, and subsequently perform a certain repair effect. Research in this area can lay a foundation for the preparation of high-stability perovskite materials and devices, adjust the structure and performance of perovskite by lighting technology (especially laser irradiation), and expand its comprehensive application in the field of optoelectronics. This article focuses on the changes in perovskites under laser irradiation and the related applications. First, it reviews the unstable changes and micro-mechanisms that laser-irradiation induces in lead-halide perovskites, including accelerated degradation, repair of defects, segregation, phase transitions, and changes in the grain size. Second, based on these mechanisms, it explains how researchers have recently used laser-irradiation technology to control the performance of perovskite films and devices. In addition, it also introduces the application of the laser direct writing process in the fields of perovskite patterning and photoelectric detection. Finally, this paper summarizes the changes induced by the laser-irradiation illumination and applications of laser-irradiated lead-halide perovskites.\n\nMSC2000:\n\n\u2022 O644","date":"2021-02-28 21:45:37","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\": 0.2495918869972229, \"perplexity\": 4564.337345421293}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178361776.13\/warc\/CC-MAIN-20210228205741-20210228235741-00613.warc.gz\"}"} | null | null |
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\section{Introduction}
To decrease the power loss, we often design the shortest paths to connect with each components in electric integrated circuits.
Not only the electric loss but also any kinds of the cost are demanded to be minimized as far as possible in industrial products.
Such problems can be formulated into a more generic task to minimize or maximize a real single-valued function of multivariables, which is called optimization problems \cite{np,Opt}.
Several solvers for optimization problems have been invented in the context of the dynamical process in statistical physics \cite{SA,QA}.
In these methods, the system is driven to be trapped at the global minimum of the complicated valley structure.
In the present study, we approach this issue in a non-standard way with the recent progress in statistical mechanics.
We propose a method to measure the difference between the approximate result given by solvers of optimization problems and the true answer.
In practice, we do not always know the ground state energy, however the minimum value of the entropy is known trivially.
Therefore the entropy can be an indicator for the deviation of the approximate solution from the accurate answer.
To goal of our study, we extend the identity proposed by Adib, which is useful to estimate the entropy difference in changing the Hamiltonian in artificial dynamics \cite{isoje}.
This identity is inspired by the Jarzynski equality, which plays a key role to connect equilibrium states at beginning and end with a nonequilibrium process \cite{originalje}.
However the original formulation given by Adib is considered only for the isoenergy process, on which the energy is fixed to be a constant value, while the Hamiltonian changing.
We extend the identity of the entropy difference in the isoenergy process to an energy-controlled process \cite{mine}.
\section{Formulation}
In order to arbitrarily control the energy, we introduce an artificial field $\mathbf{F}=(\mathbf{F}_x, \mathbf{F}_p)$ to the Hamilton dynamics.
The equations of the modified dynamics are
\begin{equation}
\dot{\mathbf{x}}=\frac{\partial H}{\partial \mathbf{p}}+ \mathbf{F}_x(\mathbf{\Gamma}),\>\>\>
\dot{\mathbf{p}}=-\frac{\partial H}{\partial \mathbf{x}}+ \mathbf{F}_p(\mathbf{\Gamma}).
\label{eq:Hamilton dynamics}
\end{equation}
The above $\mathbf{\Gamma}=(\mathbf{x},\mathbf{p})$ describes a point on the phase space.
The energy follows an arbitrary function of time $E(t)$, while the Hamiltonian changing from $t=0$ to $\tau$, if we choose the functional form of $\mathbf{F}=(\mathbf{F}_x, \mathbf{F}_p)$ as
\begin{equation}
\mathbf{F}(\mathbf{\Gamma}) = \frac{\mathbf{X}}{\mathbf{X}\cdot \partial _{\mathbf{\Gamma}}H}\biggl( \frac{\rm d \it E}{\rm d \it t}-\frac{\partial H}{\partial t} \biggr).
\label{eq:energy reservoir}
\end{equation}
Here $\mathbf{X}$ is an arbitrary vector on the phase space satisfying $\mathbf{X}\cdot \partial _{\mathbf{\Gamma}}H\neq 0$.
>From equations (\ref{eq:Hamilton dynamics}) and (\ref{eq:energy reservoir}), we can easily confirm that the energy is accurately controlled as $E(t)$ since we have
\begin{equation}
\frac{\rm d \it H}{\rm d \it t}=\frac{\partial H}{\partial \mathbf{\Gamma}}\cdot \dot{\mathbf{\Gamma}}+\frac{\partial H}{\partial t}=\frac{\rm d \it E}{\rm d \it t}.
\label{eq:control}
\end{equation}
Under the special dynamics (\ref{eq:Hamilton dynamics}), the ensemble density $\rho _t(\mathbf{\Gamma})$ evolves following the Liouville equation:
\begin{equation}
\rho _t(\mathbf{\Gamma }_t) = \rho _0(\mathbf{\Gamma}_0) \rm e^{\it -t \overline{\rm \Lambda _{\it t}}(\mathbf{\Gamma}_t)},
\label{eq:Liouville eq}
\end{equation}
where
\begin{equation}
\overline{\Lambda _t}(\mathbf{\Gamma}_t) = \frac{1}{t}\int _0 ^t \rm d \it t' \rm \Lambda \it (\mathbf{\Gamma }_{t'}).
\label{eq:Lambda bar}
\end{equation}
Equation (\ref{eq:Lambda bar}) is the time average of the ``phase space compression factor" $\Lambda (\mathbf{\Gamma}_t) = \partial _{\mathbf{\Gamma}_t} \cdot \dot{\mathbf{\Gamma}_t}$ along the trajectory that connects $\mathbf{\Gamma}_0$ to $\mathbf{\Gamma}_t$ \cite{liouville}.
This factor appearing in the dynamics of the ensemble density $\rho _t(\mathbf{\Gamma}_t)$ will play the central role to estimate the entropy as shown below.
Similarly to the original Jarzynski equality, the system is assumed to be in an equilibrium state at the initial time $t=0$.
The distribution $\rho _0(\mathbf{\Gamma}_0) $ at the initial time is set to be the microcanonical distribution at $E = E(0)$:
\begin{equation}
\rho _0(\mathbf{\Gamma}_0) =\frac{\delta (H(\mathbf{\Gamma}_0)-E(0))}{\Omega _0},
\label{eq:inital}
\end{equation}
where $\Omega _0$ is the number of states at $t=0$.
Let us consider the average of $\rm e^{\tau\overline{\Lambda _{\tau}}}$ over all possible realizations from $t=0$ to $\tau$:
\begin{equation}
\langle \rm e^{\it \tau\overline{\Lambda _{\tau}}} \rangle = \int \rm d \it \mathbf{\Gamma}_{\rm \tau}\rho _{\tau}(\mathbf{\Gamma}_{\tau}) \rm e^{\tau \overline{\Lambda _{\tau}}} \\
= \int \rm d \mathbf{\Gamma}_{\tau} \frac{\delta (\it H(\mathbf{\Gamma} _{\rm 0}\it)-E(\rm 0))}{\Omega _0}.
\label{average}
\end{equation}
In the second transformation, equations (\ref{eq:Liouville eq}) and (\ref{eq:inital}) have been used.
Note that we control the energy of the system as $H(\mathbf{\Gamma}_t) = E(t)$, and we thus find
\begin{equation}
H(\mathbf{\Gamma}_t) -E(t)=H(\mathbf{\Gamma}_0) -E(0)
\label{eq:control}
\end{equation}
for any $t \in [0, \tau]$.
Therefore $H(\mathbf{\Gamma} _0)-E(0)$ in the Dirac delta function of equation (\ref{average}) can be replaced by $H(\mathbf{\Gamma} _{\tau})-E(\tau)$.
Equation (\ref{average}) can be reduced to
\begin{equation}
\langle \rm e^{ \tau\overline{\Lambda _{\tau}}} \rangle = \rm e^{ \Delta \it S}.
\label{eq:myje}
\end{equation}
Above $ \Delta S = \ln (\Omega _{\tau}/\Omega _{0})$ is the entropy difference between the equilibrium states at the different energy values $E(0)$ and $E(\tau)$.
Therefore we can obtain the entropy difference after the nonequilibrium procedure following the schedule of the energy $E(t)$.
If we estimate the entropy difference with such a naive method as directly calculating the entropy at the initial and final times, we have to investigate the entropy twice.
On the other hand, our formula (\ref{eq:myje}) gives the entropy difference by taking just a single average of $\rm e^{\tau \overline{\Lambda _{\tau}}}$.
This means that we can examine the entropy difference by the direct use of the resulting ensemble after the nonequilibrium process.
\section{Application to optimization problems}
We here show that our equality can be used for a quantitative estimation, which indicates how much an approximate solution differs from the true solution.
Let us consider an arbitrary potential energy $V(\mathbf{x})$ with continuous variables $\mathbf{x}$ which has no degeneracy at the ground state, and assume that a solution with the energy $\mathcal{E}$ has been obtained.
We consider the following Hamiltonian
\begin{equation}
H=\mathbf{p}^2+\frac{t}{\tau}V(\mathbf{x}),\label{eq:Hforopti}
\end{equation}
At the first stage of the dynamics, the system can visit all the locations since the potential energy is small.
By increasing the coefficient of the potential energy, the particle can recognize the energy barriers and be trapped into local minima.
Let us consider the schedule of the energy from $E(0) = \mathcal{E}'$ to $E(\tau) =\mathcal{E}$, where $\mathcal{E}'>\mathcal{E}$.
The entropy at the initial time $S(0)$ can be calculated easily since the system equals to non-interacting particles with the mass $m=1/2$ at $t=0$.
On the other hand, the entropy at the final time $S(\tau) = S(0)+\Delta S$ can be estimated by calculating $\Delta S$ from our identity.
The detailed protocol is described below.
First, we randomly choose an initial condition $( \mathbf{x}_0, \mathbf{p}_0)$ from the set $\{ (\mathbf{x}$, $\mathbf{p}) | H(\mathbf{x}, \mathbf{p}; t=0) = \mathcal{E}' \}$.
Note that the initial Hamiltonian depends only on $\mathbf{p}$.
We obtain the path toward a phase point with the lower energy following the energy-controlled dynamics (\ref{eq:Hamilton dynamics}) under the given initial condition.
Some initial conditions result in the divergence of the factor appearing on the right-hand side of equation (\ref{eq:energy reservoir}) when the system is trapped in a local minimum with $\partial_{\mathbf{\Gamma}_t}H=0$.
Such a divergence means that the energy can not decrease toward $\mathcal{E}$ and the system evolving from such initial conditions are unable to reach any points on the phase space at $t=\tau$.
Therefore such samples should be excluded in taking the average of $\exp\tau \overline{\Lambda _{\tau}}$.
Finally, we take the average of $\exp\tau \overline{ \Lambda _{\tau}}$ only in the case of the absence of such divergences, and we obtain $\Delta S$ appearing on the right-hand side of our equality (\ref{eq:myje}).
In the case of optimization problems, the value of the energy at the ground state cannot be known in advance.
On the other hand, the minimum value of the entropy is trivial as $-\infty$ in the case of classical systems.
Therefore $S(\tau) \to -\infty$ implies that the obtained solutions after the nonequilibrium process for the given potential energy are close to the minimum point.
In other words, estimating the difference of the entropy can be an indicator how much the approximate solution differs from the actual minimum.
The detailed estimation by use of a simple instance is given in the reference \cite{mine}.
Let us here emphasize another advantage in use of the entropy difference to measure the deviation from the true solution.
As described in figure 1, if $\Delta E =\mathcal{E} _2-\mathcal{E} _1$ takes a common value in both case, the entropy gap $\Delta S$ can become quite different, since the entropy rapidly decreases when the system breaks through local minima of the potential energy.
The energy cannot lead us to any information on the difference coming from the energy structure.
Therefore the entropy difference can be a good measure of the deviation from the true solution in optimization problems.
\begin{figure}[h]
\includegraphics[scale=0.4]{advantage3.eps}
\caption{Case that equal $\Delta E$ but different $\Delta S$.}
\label{f2}
\end{figure}
\section{Conclusion}
We have extended the Adib identity inspired by the Jarzynski equality to an energy-controlled system, and shown the possibility as the measure of the deviation of the approximate solution from the true answer.
However our study is still short of consideration, since the efficient dynamics to search for the ground state have not considered.
In addition we have trouble in numerical computation, since the value of the entropy of an approximate solution is compared with an infinite value in our protocol.
To solve these problems, we might establish the quantum version of our identity.
In the case of a quantum system, the entropy at the ground state is equal to $0$ which is clearly more suitable for the quantitative estimation than $- \infty$ of classical systems.
Furthermore, the tunneling effect can be useful, as quantum annealing, for searching for the global minimum of the complicated energy landscape as in spin glasses, with which most of the optimization problems are closely related \cite{sg}.
We also remark that the quantum version of our formulation can also open the possibility of quantum computation.
By use of the quantum degrees of freedom, we can implement our identity in the quantum computation \cite{annealing}.
The same techniques as in the literature would be available to suggest the algorithm to estimate the entropy as well as the free energy.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,094 |
// MirrorUploadDlg.cpp : implementation file
//
#include "stdafx.h"
#include "MirrorUpload.h"
#include "MirrorUploadDlg.h"
#include "EditProfile.h"
#ifdef _DEBUG
#undef THIS_FILE
static char THIS_FILE[] = __FILE__;
#endif
/////////////////////////////////////////////////////////////////////////////
// CMirrorUploadDlg dialog
CMirrorUploadDlg::CMirrorUploadDlg( CWnd* pParent /*=NULL*/)
: CDialog(CMirrorUploadDlg::IDD, pParent)
{
//{{AFX_DATA_INIT(CMirrorUploadDlg)
//}}AFX_DATA_INIT
// Note that LoadIcon does not require a subsequent DestroyIcon in Win32
m_hIcon = AfxGetApp()->LoadIcon(IDR_MAINFRAME);
}
CMirrorUploadDlg::~CMirrorUploadDlg()
{
CWinApp * pApp = AfxGetApp();
ASSERT( pApp != NULL );
// save all profiles, kill any running threads...
pApp->WriteProfileInt( _T("MirrorUpload"), _T("ProfileCount"), m_Profiles.size() );
for(int i=0;i<m_Profiles.size();i++)
{
CharString sProfileBase;
sProfileBase.format( "Profile%d", i );
if ( m_Profiles[i].pThread != NULL )
{
m_Profiles[i].pThread->kill();
delete m_Profiles[i].pThread;
m_Profiles[i].pThread = NULL;
}
pApp->WriteProfileString( _T("MirrorUpload"), CString( sProfileBase + "sName" ), CString( m_Profiles[i].sName) );
pApp->WriteProfileString( _T("MirrorUpload"), CString( sProfileBase + "sLogFile" ), CString( m_Profiles[i].sLogFile) );
pApp->WriteProfileString( _T("MirrorUpload"), CString( sProfileBase + "sPath" ), CString( m_Profiles[i].sPath) );
pApp->WriteProfileString( _T("MirrorUpload"), CString( sProfileBase + "sAddress" ), CString( m_Profiles[i].sAddress) );
pApp->WriteProfileInt( _T("MirrorUpload"), CString( sProfileBase + "nPort" ), m_Profiles[i].nPort );
pApp->WriteProfileString( _T("MirrorUpload"), CString( sProfileBase + "sUID" ), CString( m_Profiles[i].sUID) );
pApp->WriteProfileString( _T("MirrorUpload"), CString( sProfileBase + "sPW" ), CString( m_Profiles[i].sPW) );
}
}
void CMirrorUploadDlg::DoDataExchange(CDataExchange* pDX)
{
CDialog::DoDataExchange(pDX);
//{{AFX_DATA_MAP(CMirrorUploadDlg)
DDX_Control(pDX, IDC_LIST1, m_cProfileList);
//}}AFX_DATA_MAP
}
BEGIN_MESSAGE_MAP(CMirrorUploadDlg, CDialog)
//{{AFX_MSG_MAP(CMirrorUploadDlg)
ON_WM_TIMER()
ON_BN_CLICKED(IDC_BUTTON1, OnNewProfile)
ON_BN_CLICKED(IDC_BUTTON2, OnDeleteProfile)
ON_BN_CLICKED(IDC_BUTTON3, OnUpload)
ON_BN_CLICKED(IDC_BUTTON8, OnEditProfile)
ON_NOTIFY(NM_DBLCLK, IDC_LIST1, OnListOpen)
ON_BN_CLICKED(IDC_BUTTON7, OnOpenLog)
//}}AFX_MSG_MAP
END_MESSAGE_MAP()
/////////////////////////////////////////////////////////////////////////////
// CMirrorUploadDlg message handlers
BOOL CMirrorUploadDlg::OnInitDialog()
{
CDialog::OnInitDialog();
CWinApp * pApp = AfxGetApp();
ASSERT( pApp != NULL );
// load profiles from registry
int nProfileCount = pApp->GetProfileInt( _T("MirrorUpload"), _T("ProfileCount"), 0 );
for(int i=0;i<nProfileCount;i++)
{
CharString sProfileBase;
sProfileBase.format( "Profile%d", i );
Profile & profile = m_Profiles.push();
profile.sName = pApp->GetProfileString( _T("MirrorUpload"), CString( sProfileBase + "sName" ), _T("") );
profile.sLogFile = pApp->GetProfileString( _T("MirrorUpload"), CString( sProfileBase + "sLogFile" ), _T("") );
profile.sPath = pApp->GetProfileString( _T("MirrorUpload"), CString( sProfileBase + "sPath" ), _T("") );
profile.sAddress = pApp->GetProfileString( _T("MirrorUpload"), CString( sProfileBase + "sAddress" ), _T("") );
profile.nPort = pApp->GetProfileInt( _T("MirrorUpload"), CString( sProfileBase + "nPort" ), 9000 );
profile.sUID = pApp->GetProfileString( _T("MirrorUpload"), CString( sProfileBase + "sUID" ), _T("") );
profile.sPW = pApp->GetProfileString( _T("MirrorUpload"), CString( sProfileBase + "sPW" ), _T("") );
profile.pThread = NULL;
}
// now initialize our list control with the profiles
CRect rect;
m_cProfileList.GetClientRect( &rect );
m_cProfileList.SetExtendedStyle( LVS_EX_FULLROWSELECT );
int cWidth = rect.Width() / 5;
m_cProfileList.InsertColumn( 0, _T("Name"), LVCFMT_LEFT, cWidth * 2, 0 );
m_cProfileList.InsertColumn( 1, _T("Server"), LVCFMT_LEFT, cWidth * 2, 1 );
m_cProfileList.InsertColumn( 2, _T("Status"), LVCFMT_LEFT, cWidth, 2 );
m_cProfileList.InsertColumn( 3, _T("Path"), LVCFMT_LEFT, cWidth * 3, 3 );
m_cProfileList.InsertColumn( 4, _T("Log"), LVCFMT_LEFT, cWidth * 3, 4 );
// update our list control now..
OnTimer( 0 );
return TRUE; // return TRUE unless you set the focus to a control
}
void CMirrorUploadDlg::OnTimer(UINT nIDEvent)
{
bool bUploading = false;
// check the status of all profiles, and update the status field...
for(int i=0;i<m_Profiles.size();i++)
{
Profile & profile = m_Profiles[i];
if ( m_cProfileList.GetItemCount() <= i )
m_cProfileList.InsertItem( i, CString( profile.sName ) );
else
m_cProfileList.SetItemText( i, 0, CString( profile.sName ) );
m_cProfileList.SetItemText( i, 1, CString( CharString().format("%s:%d", profile.sAddress, profile.nPort)) );
m_cProfileList.SetItemText( i, 3, CString( profile.sPath ) );
m_cProfileList.SetItemText( i, 4, CString( profile.sLogFile ) );
if ( profile.pThread != NULL )
{
if ( profile.pThread->done() )
{
if ( profile.pThread->error() )
m_cProfileList.SetItemText( i, 2, _T("Error") );
else
m_cProfileList.SetItemText( i, 2, _T("Success") );
// remove the thread object now..
delete profile.pThread;
profile.pThread = NULL;
}
else
{
m_cProfileList.SetItemText( i, 2, _T("Uploading") );
bUploading = true;
}
}
}
// remove extra list items
while( m_cProfileList.GetItemCount() > m_Profiles.size() )
m_cProfileList.DeleteItem( m_cProfileList.GetItemCount()-1 );
if ( nIDEvent == 0x1 )
CDialog::OnTimer(nIDEvent);
if (! bUploading )
KillTimer( 0x1 ); // stop the updates if nothing is uploading...
}
//----------------------------------------------------------------------------
void CMirrorUploadDlg::OnNewProfile()
{
CEditProfile dialog;
if ( dialog.DoModal() == IDOK )
{
// create a new profile entry
Profile & profile = m_Profiles.push();
profile.sName = dialog.m_sName;
profile.sLogFile = dialog.m_sLogFile;
profile.sPath = dialog.m_sLocalPath;
profile.sAddress = dialog.m_sAddress;
profile.nPort = dialog.m_nPort;
profile.sUID = dialog.m_sUser;
profile.sPW = dialog.m_sPassword;
profile.pThread = NULL;
OnTimer( 0x0 );
}
}
void CMirrorUploadDlg::OnDeleteProfile()
{
Array< int > remove;
POSITION pos = m_cProfileList.GetFirstSelectedItemPosition();
while( pos )
{
int nItem = m_cProfileList.GetNextSelectedItem( pos );
Profile & profile = m_Profiles[ nItem ];
if ( MessageBox( CString( CharString().format("Confirm delete of profile %s?", profile.sName)), _T("Confirm Delete"), MB_YESNO) != IDYES )
continue;
remove.push( nItem );
}
for(int i=remove.size()-1;i>=0;i--)
{
Profile & profile = m_Profiles[ remove[i] ];
if ( profile.pThread != NULL )
{
profile.pThread->kill();
delete profile.pThread;
profile.pThread = NULL;
}
m_Profiles.remove( remove[i] );
}
OnTimer( 0x0 );
}
void CMirrorUploadDlg::OnUpload()
{
POSITION pos = m_cProfileList.GetFirstSelectedItemPosition();
while( pos )
{
int nItem = m_cProfileList.GetNextSelectedItem( pos );
Profile & profile = m_Profiles[ nItem ];
if ( profile.pThread != NULL )
continue; // upload already in progress
if ( MessageBox( CString(CharString().format("Please confirm upload from %s to %s:%d", profile.sPath, profile.sAddress, profile.nPort) ),
CString( profile.sName ), MB_YESNO) != IDYES )
continue; // skip
// start upload then...
profile.pThread = new UploadThread( profile.sPath, profile.sAddress,
profile.nPort, profile.sUID, profile.sPW );
profile.pThread->resume();
// start an update timer..
SetTimer( 0x1, 2500, NULL );
}
}
void CMirrorUploadDlg::OnEditProfile()
{
POSITION pos = m_cProfileList.GetFirstSelectedItemPosition();
while( pos )
{
int nItem = m_cProfileList.GetNextSelectedItem( pos );
Profile & profile = m_Profiles[ nItem ];
CEditProfile dialog;
dialog.m_sName = profile.sName;
dialog.m_sLogFile = profile.sLogFile;
dialog.m_sLocalPath = profile.sPath;
dialog.m_sAddress = profile.sAddress;
dialog.m_nPort = profile.nPort;
dialog.m_sUser = profile.sUID;
dialog.m_sConfirmPassword = dialog.m_sPassword = profile.sPW;
if ( dialog.DoModal() == IDOK )
{
// create a new profile entry
profile.sName = dialog.m_sName;
profile.sLogFile = dialog.m_sLogFile;
profile.sPath = dialog.m_sLocalPath;
profile.sAddress = dialog.m_sAddress;
profile.nPort = dialog.m_nPort;
profile.sUID = dialog.m_sUser;
profile.sPW = dialog.m_sPassword;
profile.pThread = NULL;
OnTimer( 0x0 );
}
}
}
void CMirrorUploadDlg::OnListOpen(NMHDR* pNMHDR, LRESULT* pResult)
{
*pResult = 0;
OnEditProfile();
}
void CMirrorUploadDlg::OnOpenLog()
{
POSITION pos = m_cProfileList.GetFirstSelectedItemPosition();
while( pos )
{
int nItem = m_cProfileList.GetNextSelectedItem( pos );
Profile & profile = m_Profiles[ nItem ];
ShellExecute( 0, _T("open"), CString( profile.sLogFile ), _T(""), _T(""), SW_NORMAL );
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 80 |
{"url":"https:\/\/mrcieu.github.io\/TwoSampleMR\/reference\/extract_instruments.html","text":"This function searches for GWAS significant SNPs (for a given p-value) for a specified set of outcomes. It then performs LD based clumping to return only independent significant associations.\n\nextract_instruments(\noutcomes,\np1 = 5e-08,\nclump = TRUE,\np2 = 5e-08,\nr2 = 0.001,\nkb = 10000,\naccess_token = ieugwasr::check_access_token(),\nforce_server = FALSE\n)\n\n## Arguments\n\noutcomes Array of outcome IDs (see available_outcomes). Significance threshold. The default is 5e-8. Logical; whether to clump results. The default is TRUE. Secondary clumping threshold. The default is 5e-8. Clumping r2 cut off. The default is 0.001. Clumping distance cutoff. The default is 10000. Google OAuth2 access token. Used to authenticate level of access to data. The default is ieugwasr::check_access_token(). Force the analysis to extract results from the server rather than the MRInstruments package.\n\ndata frame","date":"2021-05-07 14:03:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.34003308415412903, \"perplexity\": 10923.638512005506}, \"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\/1620243988793.99\/warc\/CC-MAIN-20210507120655-20210507150655-00484.warc.gz\"}"} | null | null |
{"url":"https:\/\/proofwiki.org\/wiki\/Edge_is_Minimum_Weight_Bridge_iff_in_All_Minimum_Spanning_Trees","text":"# Edge is Minimum Weight Bridge iff in All Minimum Spanning Trees\n\n## Theorem\n\nLet $G$ be an undirected network.\n\nLet every edge of $G$ have a unique weight.\n\nLet $e$ be an edge of $G$.\n\nThen $e$ is a bridge of minimum weight in $G$ if and only if $e$ belongs to every minimum spanning tree of $G$.\n\n## Proof\n\n### Necessary Condition\n\nAiming for a contradiction, suppose $e$ is a bridge of minimum weight that does not belong to some minimum spanning tree $Q$.\n\nLet $e$ be added to $Q$ to make $Q'$.\n\nThen $e$ forms part of a unique cycle $C$ in $Q$.\n\nThus there exists an edge $f \\in C$ such that $w \\left({Q}\\right) < w \\left({Q + e - f}\\right)$.\n\nThis contradicts the minimality of $Q$.\n\n$\\Box$","date":"2019-11-12 12:15:24","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\": 0.9312333464622498, \"perplexity\": 133.0506063270156}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496665521.72\/warc\/CC-MAIN-20191112101343-20191112125343-00506.warc.gz\"}"} | null | null |
\section*{1. Introduction}
The study of the exotic hadronic states -- the hidden-charm pentaquark resonances -- has received considerable
interest in recent years and is one of the most exciting topics of the nuclear and hadronic physics nowadays
after the discovery by the LHCb Collaboration pentaquark states $P_c^+(4380)$ and $P_c^+(4450)$ in the ${J/\psi}p$ invariant mass spectrum of the $\Lambda^0_b \to K^-({J/\psi}p)$ decays [1] and, especially, after the observation
by the Collaboration of three new narrow resonances $P_c^+(4312)$, $P_c^+(4440)$ and $P_c^+(4457)$ in these
decays [2], based on additional collected data and on an improved selection strategy, instead of initially
claimed $P_c^+(4380)$ and $P_c^+(4450)$ states.
Very recently, the LHCb Collaboration discovered a new narrow hidden-charm pentaquark denoted as
$P_{c}^+(4337)$ in the invariant mass spectrum of the ${J/\psi}p$ in the $B_s^0 \to {J/\psi}p{\bar p}$ decays [3].
On the other hand, the search for the LHCb pentaquarks $P_c^+(4312)$, $P_c^+(4440)$ and $P_c^+(4457)$ by the GlueX Collaboration in Hall D at JLab through a scan of the cross section of elastic reaction ${\gamma}p \to {J/\psi}p$ from threshold of 8.21 GeV and up to incident photon energy $E_{\gamma}=11.8$ GeV gave no evidence for them with present statistics and set the model-dependent upper limits on branching ratios of $P_c^+(4312) \to {J/\psi}p$, $P_c^+(4440) \to {J/\psi}p$ and $P_c^+(4457) \to {J/\psi}p$ decays of several percent [4]. The preliminary results from a factor of 10 more data on the $J/\psi$ photoproduction on a hydrogen target,
collected in the Hall C JLab E12-16-007 experiment (the so-called $J/\psi$-007 experiment),
also show no $P_c^+$ signal [5]. In this experiment the $e^+e^-$ pairs from the $J/\psi$ decays were detected
in coincidence using the two high momentum spectrometers of Hall C: the Super High Momentum Spectrometer
(SHMS) and High Momentum Spectrometer (HMS) for the electron and the positron, respectively.
In recent publications [6] and [7] the role, respectively, of the LHCb pentaquarks
$P_c^+(4450)$ and $P_c^+(4312)$, $P_c^+(4440)$, $P_c^+(4457)$ in charmonium photoproduction on protons and nuclei
has been investigated at near-threshold initial photon energies $E_{\gamma} \le 11$ GeV. Here, the description was
based on the consideration of the incoherent direct (${\gamma}N \to {J/\psi}N$) and two-step
(${\gamma}p \to P_c^+(4450) \to {J/\psi}p$ and ${\gamma}p \to P_c^+(4312) \to {J/\psi}p$,
${\gamma}p \to P_c^+(4440) \to {J/\psi}p$, ${\gamma}p \to P_c^+(4457) \to {J/\psi}p$) $J/\psi$ production
processes. As a measure for this role, the incident photon energy dependence of $J/\psi$ production
cross sections on protons and nuclei (excitation functions) has been adopted.
It was found that it is insignificant for the pentaquark resonances considered if branching ratios of their
decays to the ${J/\psi}p$ mode are less than a few percent, which is in line with the results of the JLab
experiments [4, 5].
In view of the aforementioned, to get a robust enough information for or against the existence of the
LHCb hidden-charm pentaquarks and to understand their better, it is crucial to investigate the possibility
of their observation by measuring not only the excitation functions for $J/\psi$ meson production
from photon-induced reactions on protons and nuclei at near-threshold photon energies, predicted in Refs. [6, 7],
but also the $J/\psi$ energy and momentum distributions in these reactions, not predicted in the previous
papers [6, 7]. Their prediction is the main aim of the present study.
In it, we consider the contribution of the $P_{c}^{+,0}(4312)$, $P_{c}^{+,0}(4337)$,
$P_{c}^{+,0}(4440)$ and $P_{c}^{+,0}(4457)$ resonances to near-threshold $J/\psi$ photoproduction
off protons and nuclei by adopting the Breit-Wigner shape for this contribution and by employing
the recent experimental data [4] on the total and differential cross sections of the ${\gamma}p \to {J/\psi}p$
process to estimate the background contribution.
The consideration is mainly based on the model, developed in Refs. [6, 7].
We present the predictions obtained within our present approach for the $J/\psi$ energy and momentum distributions
in ${\gamma}p$ as well as in ${\gamma}$$^{12}$C and ${\gamma}$$^{184}$W reactions at
near-threshold incident photon energies. These predictions may be useful in planning future
$J/\psi$ photoproduction experiments at the CEBAF facility.
\section*{2. Theoretical framework}
\subsection*{2.1. Direct non-resonant $J/\psi$ production processes}
At near-threshold photon beam energies below 11 GeV of our interest
\footnote{$^)$These energies are well within the present capabilities of the upgraded CEBAF facility
at JLab, which is providing an opportunity to study the observed [1--3] by the LHCb Collaboration
exotic hidden-charm pentaquark states in exclusive ${\gamma}p \to {J/\psi}p$ reactions in all
experimental Halls A, B, C, D [4, 5].}$^)$,
the following direct non-resonant elementary charmonium production processes with the lowest free production
threshold ($\approx$~8.21 GeV) contribute to the $J/\psi$ photoproduction on nuclei [6--8]:
\begin{equation}
\gamma+p \to J/\psi+p,
\end{equation}
\begin{equation}
\gamma+n \to J/\psi+n.
\end{equation}
The modification of the masses of the final high-momentum
$J/\psi$ meson and proton (see below) in the nuclear medium will be ignored in the present study.
Disregarding the absorption of incident photons in the energy range of interest as well as the $J/\psi$
meson quasielastic rescatterings on target nucleons [9], and describing
the charmonium final-state absorption in the nuclear matter by the
absorption cross section $\sigma_{{J/\psi}N}$
\footnote{$^)$For which we will use the value $\sigma_{{J/\psi}N}=3.5$ mb [6--10].}$^)$,
we represent the inclusive differential cross section for the production of $J/\psi$ mesons with the
momentum ${\bf p}_{J/\psi}$ on nuclei in the direct non-resonant processes (1), (2)
in the form [6--10]:
\begin{equation}
\frac{d\sigma_{{\gamma}A\to {J/\psi}X}^{({\rm dir})}({\bf p}_{\gamma},{\bf p}_{J/\psi})}
{d{\bf p}_{J/\psi}}=I_{V}[A,\sigma_{{J/\psi}N}]
\left<\frac{d\sigma_{{\gamma}p \to {J/\psi}p}({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}\right>_A,
\end{equation}
where
\begin{equation}
I_{V}[A,\sigma]=2{\pi}A\int\limits_{0}^{R}r_{\bot}dr_{\bot}
\int\limits_{-\sqrt{R^2-r_{\bot}^2}}^{\sqrt{R^2-r_{\bot}^2}}dz
\rho(\sqrt{r_{\bot}^2+z^2})
\exp{\left[-A{\sigma}\int\limits_{z}^{\sqrt{R^2-r_{\bot}^2}}\rho(\sqrt{r_{\bot}^2+x^2})dx\right]},
\end{equation}
\begin{equation}
\left<\frac{d\sigma_{{\gamma}p \to {J/\psi}p}({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}\right>_A=
\int\int
P_A({\bf p}_t,E)d{\bf p}_tdE
\left[\frac{d\sigma_{{\gamma}p \to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}\right]
\end{equation}
and
\begin{equation}
s^*=(E_{\gamma}+E_t)^2-({\bf p}_{\gamma}+{\bf p}_t)^2,
\end{equation}
\begin{equation}
E_t=M_A-\sqrt{(-{\bf p}_t)^2+(M_{A}-m_{p}+E)^{2}}.
\end{equation}
Here, $d\sigma_{{\gamma}p\to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi})/d{\bf p}_{J/\psi}$
is the off-shell differential cross section for the production of $J/\psi$ meson in process (1)
at the "in-medium" ${\gamma}p$ c.m. energy $\sqrt{s^*}$
\footnote{$^)$In Eq. (3), it is assumed that the cross sections for $J/\psi$ meson production in
${\gamma}p$ and ${\gamma}n$ interactions are equal to each other [6--8].}$^)$;
$\rho({\bf r})$ and $P_A({\bf p}_t,E)$ are normalized to unity the nucleon density and the nuclear
spectral function (they are given in Refs. [11, 12]) of target nucleus with mass number $A$,
having mass $M_{A}$ and radius $R$;
${\bf p}_{\gamma}$ and $E_{\gamma}$ are the momentum and
energy of the incident photon ($E_{\gamma}=|{\bf p}_{\gamma}|=p_{\gamma}$) in the laboratory system;
${\bf p}_{t}$ and $E$ are the momentum and binding energy of the intranuclear target proton,
participating in the reaction channel (1); $m_p$ is the free space proton mass.
Also, as previously in [6--8], we will suppose that the off-shell differential cross section\\
$d\sigma_{{\gamma}p\to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi})/d{\bf p}_{J/\psi}$
for $J/\psi$ production in process (1) is the same as the corresponding on-shell cross section
$d\sigma_{{\gamma}p\to {J/\psi}p}(\sqrt{s},{\bf p}_{J/\psi})/d{\bf p}_{J/\psi}$
determined for the off-shell kinematics of this process and in which the vacuum c.m. energy squared $s$,
given by the formula
\begin{equation}
s=W^2=(E_{\gamma}+m_p)^2-{\bf p}_{\gamma}^2=m_p^2+2m_pE_{\gamma},
\end{equation}
is replaced by the in-medium expression (6). The above off-shell differential cross section is then
(cf. [8, 10]):
\begin{equation}
\frac{d\sigma_{{\gamma}p \to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi})}
{d{\bf p}_{J/\psi}}=
\frac{\pi}{I_2(s^*,m_{J/\psi},m_{p})E_{J/\psi}}
\end{equation}
$$
\times
\frac{d\sigma_{{\gamma}p \to {J/\psi}{p}}(\sqrt{s^*},\theta_{J/\psi}^*)}{d{\bf \Omega}_{J/\psi}^*}
\frac{1}{(\omega+E_t)}\delta\left[\omega+E_t-\sqrt{m_{p}^2+({\bf Q}+{\bf p}_t)^2}\right],
$$
where
\begin{equation}
I_2(s^*,m_{J/\psi},m_{p})=\frac{\pi}{2}
\frac{\lambda(s^*,m_{J/\psi}^{2},m_{p}^{2})}{s^*},
\end{equation}
\begin{equation}
\lambda(x,y,z)=\sqrt{{\left[x-({\sqrt{y}}+{\sqrt{z}})^2\right]}{\left[x-
({\sqrt{y}}-{\sqrt{z}})^2\right]}},
\end{equation}
\begin{equation}
\omega=E_{\gamma}-E_{J/\psi}, \,\,\,\,{\bf Q}={\bf p}_{\gamma}-{\bf p}_{J/\psi},\,\,\,\,
E_{J/\psi}=\sqrt{m^2_{J/\psi}+{\bf p}_{J/\psi}^2}.
\end{equation}
Here, the off-shell c.m. charmonium angular distribution in reaction (1)
$d\sigma_{{\gamma}p \to {J/\psi}{p}}(\sqrt{s^*},\theta_{J/\psi}^*)/d{\bf \Omega}_{J/\psi}^*$
as a function of the $J/\psi$ production c.m. polar angle $\theta_{J/\psi}^*$ is given by [10]:
\begin{equation}
\frac{d\sigma_{{\gamma}p \to {J/\psi}p}(\sqrt{s^*},\theta^*_{J/\psi})}{d{\bf \Omega}_{J/\psi}^*}=
a{\rm e}^{b_{J/\psi}(t-t^+)}\sigma_{{\gamma}p \to {J/\psi}p}(\sqrt{s^*}),
\end{equation}
where $\sigma_{{\gamma}p \to {J/\psi}p}(\sqrt{s^*})$ is the off-shell total cross section
for $J/\psi$ meson production in this reaction and
\begin{equation}
t=m_{J/\psi}^2-2E^*_{\gamma}E^*_{J/\psi}+2p^*_{\gamma}p^*_{J/\psi}\cos{\theta^*_{J/\psi}},
\end{equation}
\begin{equation}
E_{\gamma}^*=p^{*}_{\gamma}, \,\,\,\,
E_{J/\psi}^*=\sqrt{m^2_{J/\psi}+p^{*2}_{J/\psi}},
\end{equation}
\begin{equation}
p_{\gamma}^*=\frac{1}{2\sqrt{s^*}}\lambda(s^*,0,E_{t}^2-p_t^2),
\end{equation}
\begin{equation}
p_{J/\psi}^*=\frac{1}{2\sqrt{s^*}}\lambda(s^*,m_{J/\psi}^{2},m_{p}^2),
\end{equation}
\begin{equation}
t^+=t(\cos{\theta^*_{J/\psi}}=1)=m_{J/\psi}^2-2E^*_{\gamma}E^*_{J/\psi}+2p^*_{\gamma}p^*_{J/\psi},
\end{equation}
\begin{equation}
t-t^+=2p^*_{\gamma}p^*_{J/\psi}(\cos{\theta^*_{J/\psi}}-1).
\end{equation}
The angle of charmonium production in the ${\gamma}p$ c.m. system, $\theta^*_{J/\psi}$, is expressed
through its production angle, $\theta_{J/\psi}$, in the laboratory system
($\cos{\theta_{J/\psi}}={\bf p}_{\gamma}{\bf p}_{J/\psi}/p_{\gamma}p_{J/\psi}$)
by means of equation [8, 10]:
\begin{equation}
\cos{\theta_{J/\psi}^*}=\frac{p_{\gamma}p_{J/\psi}\cos{\theta_{J/\psi}}+
(E_{\gamma}^*E_{J/\psi}^*-E_{\gamma}E_{J/\psi})}{p_{\gamma}^*p_{J/\psi}^*}.
\end{equation}
The condition of normalization
\begin{equation}
\int\limits_{4\pi}^{}a{\rm e}^{b_{J/\psi}(t-t^+)}d{\bf \Omega}_{J/\psi}^*=1
\end{equation}
gives for the parameter $a$ in Eq. (13) the following expression:
\begin{equation}
a=\frac{p^*_{\gamma}p^*_{J/\psi}b_{J/\psi}}{\pi}\left[1-{\rm e}^{-4p^*_{\gamma}p^*_{J/\psi}b_{J/\psi}}\right]^{-1}.
\end{equation}
Parameter $b_{J/\psi}$ in Eqs. (13), (21), (22) is an exponential $t$-slope of the differential cross section
of the reaction ${\gamma}p \to {J/\psi}p$ in the near-threshold energy region. It should be pointed out that
the differential cross section of this reaction was also very recently measured in the $J/\psi$-007 experiment [13]
as a function of the photon energy in the range of $9.1~{\rm GeV} \le E_{\gamma} \le 10.6~{\rm GeV}$ with the aim
to explore the impact of the collected data on the determination of the proton's gravitational form factors,
the proton-mass radius, and the contribution of the trace anomaly to the proton mass. Since the $t$-slope
$b_{J/\psi}$ was not determined in [13], we will adopt for it the GlueX result [4], namely:
$b_{J/\psi}$ $\approx$~1.67 GeV$^{-2}$. We will use this value in our calculations.
Now consider the off-shell total cross section
$\sigma_{{\gamma}p\to {J/\psi}p}(\sqrt{s^*})$ for $J/\psi$ production in process (1).
In line with the above-mentioned, it is the same as the vacuum cross section
$\sigma_{{\gamma}p \to {J/\psi}p}(\sqrt{s})$, in which the vacuum c.m. energy squared s,
defined by the formula (8), is replaced by the in-medium expression (6). For the vacuum total cross section
$\sigma_{{\gamma}p \to {J/\psi}p}(\sqrt{s})$ at near-threshold photon energies
we have used the following parametrization [7] of the available here experimental
information [4] on it from the GlueX experiment, based on the near-threshold predictions
of the two gluon and three gluon exchange model [14]:
\begin{equation}
\sigma_{{\gamma}p \to {J/\psi}p}({\sqrt{s}})= \sigma_{2g}({\sqrt{s}})+
\sigma_{3g}({\sqrt{s}}),
\end{equation}
where 2$g$ and 3$g$ exchanges cross sections $\sigma_{2g}({\sqrt{s}})$ and $\sigma_{3g}({\sqrt{s}})$
are given in Ref. [7] by formulas (7) and (8), respectively.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=16.0cm]{fig1gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The total cross section for the background reaction
${\gamma}p \to {J/\psi}p$ as a function of the photon energy $E_{\gamma}$.
Dashed and dotted-dashed curves are, respectively, calculations on the basis of the two gluon and
three gluon exchange model [14]. Solid curve is the incoherent sum of the above two calculations.
The GlueX experimental data are from Ref. [4].
The arrow indicates the threshold energy of 8.21 GeV
for direct non-resonant charmonium photoproduction off a free target proton at rest.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=16.0cm]{fig2gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) Plot of the allowed final $J/\psi$ meson and proton momenta
in the direct non-resonant ${\gamma}p \to {J/\psi}p$ reaction, occurring
in the laboratory system in the free space at initial photon energy of 9.44 GeV,
as functions of their production angles with respect to the photon beam direction in this system.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=16.0cm]{fig3gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The same as in Fig. 2, but for the initial photon energy of 10.12 GeV.}
\label{void}
\end{center}
\end{figure}
Fig. 1 shows that the GlueX near-threshold data are well fitted by only the combination (23) of the two gluon and
three gluon exchange cross sections and in the resonance incident photon energy range $\sim$ 9.5--10.0 GeV
the main contribution to the elastic $J/\psi$ production comes from the three gluon exchanges.
At the initial photon energies of interest the $J/\psi$ mesons are produced at small
laboratory polar angles (see below). Therefore, we will calculate the $J/\psi$ momentum
distributions from considered target nuclei
for the laboratory solid angle
${\Delta}{\bf \Omega}_{J/\psi}$=$0^{\circ} \le \theta_{J/\psi} \le 20^{\circ}$,
and $0 \le \varphi_{J/\psi} \le 2{\pi}$ (cf. [10]).
Then, integrating the differential cross section (3) over this solid angle,
we can represent the differential cross section for charmonium
production from the direct non-resonant processes (1) and (2) into this solid angle
as follows:
\begin{equation}
\frac{d\sigma_{{\gamma}A\to {J/\psi}X}^{({\rm dir})}
(p_{\gamma},p_{J/\psi})}{dp_{J/\psi}}=
\int\limits_{{\Delta}{\bf \Omega}_{J/\psi}}^{}d{\bf \Omega}_{J/\psi}
\frac{d\sigma_{{\gamma}A\to {J/\psi}X}^{({\rm dir})}
({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}p_{J/\psi}^2
\end{equation}
$$
=2{\pi}I_{V}[A,\sigma_{{J/\psi}N}]
\int\limits_{\cos20^{\circ}}^{1}d\cos{{\theta_{J/\psi}}}
\left<\frac{d\sigma_{{\gamma}p\to {J/\psi}{p}}(p_{\gamma},
p_{J/\psi},\theta_{J/\psi})}{dp_{J/\psi}d{\bf \Omega}_{J/\psi}}\right>_A.
$$
Before going further, we now consider, adopting the relativistic kinematics,
more simpler case of the free space production of $J/\psi$ mesons and protons
in the process ${\gamma}p \to {J/\psi}p$, proceeding on a free target proton being at rest,
to get an idea about their kinematic characteristics allowed in this process at incident
photon energies considered. The kinematics of two-body reaction with a threshold
(as in our present case) tell us that the laboratory polar $J/\psi$ and final proton
production angles $\theta_{J/\psi}$ and $\theta_{p}$ vary from 0 to a maximal values
$\theta^{\rm max}_{J/\psi}$ and $\theta^{\rm max}_{p}$, correspondingly, i.e.:
\begin{equation}
0 \le \theta_{J/\psi} \le \theta^{\rm max}_{J/\psi},
\end{equation}
\begin{equation}
0 \le \theta_{p} \le \theta^{\rm max}_{p};
\end{equation}
where
\begin{equation}
\theta^{\rm max}_{J/\psi}={\rm arcsin}[(\sqrt{s}p^{*}_{J/\psi})/(m_{J/\psi}p_{\gamma})],
\end{equation}
\begin{equation}
\theta^{\rm max}_{p}={\rm arcsin}[(\sqrt{s}p^{*}_{p})/(m_{p}p_{\gamma})].
\end{equation}
Here, the $J/\psi$ c.m. momentum $p^*_{J/\psi}$ is determined by Eq. (17),
in which the in-medium c.m. energy squared $s^*$ should be replaced by the
vacuum collision energy squared $s$, defined by the formula (8), and
$p^*_{p}$ is the final proton c.m. momentum. It is equal to the $J/\psi$ c.m. momentum $p^*_{J/\psi}$.
From Eqs. (27), (28) one can get, for example, that
\begin{equation}
\theta^{\rm max}_{J/\psi}=5.570^{\circ},\,\,\,\,\theta^{\rm max}_{p}=18.686^{\circ}
\end{equation}
at initial photon beam energy of $E_{\gamma}=9.44$ GeV and
\begin{equation}
\theta^{\rm max}_{J/\psi}=6.768^{\circ}, \,\,\,\,\theta^{\rm max}_{p}=22.892^{\circ}
\end{equation}
at photon energy of $E_{\gamma}=10.12$ GeV. Energy-momentum conservation in
the reaction (1), taking place in a vacuum, leads to two different solutions for the laboratory
$J/\psi$ meson and final proton momenta $p_{J/\psi}$ and $p_{p}$ at given laboratory polar production angles
$\theta_{J/\psi}$ and $\theta_{p}$, belonging, correspondingly, to the angular intervals (25) and (26):
\begin{equation}
p^{(1,2)}_{J/\psi}(\theta_{J/\psi})=
\frac{p_{\gamma}\sqrt{s}E^{*}_{J/\psi}\cos{\theta_{J/\psi}}\pm
(E_{\gamma}+m_p)\sqrt{s}\sqrt{p^{*2}_{J/\psi}-{\gamma^2_{\rm cm}}{v^2_{\rm cm}}m^2_{J/\psi}\sin^2{\theta_{J/\psi}}}}{(E_{\gamma}+m_p)^2-p^2_{\gamma}\cos^2{\theta_{J/\psi}}},
\end{equation}
\begin{equation}
p^{(1,2)}_{p}(\theta_{p})=
\frac{p_{\gamma}\sqrt{s}E^{*}_{p}\cos{\theta_{p}}\pm
(E_{\gamma}+m_p)\sqrt{s}\sqrt{p^{*2}_{p}-{\gamma^2_{\rm cm}}{v^2_{\rm cm}}m^2_{p}\sin^2{\theta_{p}}}}{(E_{\gamma}+m_p)^2-p^2_{\gamma}\cos^2{\theta_{p}}}.
\end{equation}
Here, ${\gamma_{\rm cm}}=(E_{\gamma}+m_p)/\sqrt{s}$, $v_{\rm cm}=p_{\gamma}/(E_{\gamma}+m_p)$,
the $J/\psi$ total c.m. energy $E^{*}_{J/\psi}$ is defined above by Eq. (15),
$E^{*}_{p}=\sqrt{m^2_{p}+p^{*2}_{p}}$ and
sign "+" in the numerators of Eqs. (31), (32)
corresponds to the first solutions $p^{(1)}_{J/\psi}$, $p^{(1)}_{p}$
and sign "-" - to the second ones $p^{(2)}_{J/\psi}$, $p^{(2)}_{p}$.
Looking at the expressions (31) and (32), one can come to the conclusion that the first solutions $p^{(1)}_{J/\psi}$ and
$p^{(1)}_{p}$ as well as the second ones $p^{(2)}_{J/\psi}$ and $p^{(2)}_{p}$
have different dependencies, respectively, on the production angles $\theta_{J/\psi}$ and $\theta_{p}$
within the angular intervals [0, $\theta_{J/\psi}^{\rm max}]$ and [0, $\theta_{p}^{\rm max}]$.
Namely, the former drop and the latter ones increase as the production angles
$\theta_{J/\psi}$ and $\theta_{p}$ increase in these intervals (cf. Figs. 2 and 3) and
\begin{equation}
p^{(1)}_{J/\psi}(\theta_{J/\psi}^{\rm max})=p^{(2)}_{J/\psi}(\theta_{J/\psi}^{\rm max})=
p_{J/\psi}(\theta_{J/\psi}^{\rm max}),
\end{equation}
\begin{equation}
p^{(1)}_{p}(\theta_{p}^{\rm max})=p^{(2)}_{p}(\theta_{p}^{\rm max})=
p_{p}(\theta_{p}^{\rm max}),
\end{equation}
where
\begin{equation}
p_{J/\psi}(\theta_{J/\psi}^{\rm max})=(p_{\gamma}m_{J/\psi}^2\cos{\theta_{J/\psi}^{\rm max}})/
(\sqrt{s}E^{*}_{J/\psi}),
\end{equation}
\begin{equation}
p_{p}(\theta_{p}^{\rm max})=(p_{\gamma}m_{p}^2\cos{\theta_{p}^{\rm max}})/
(\sqrt{s}E^{*}_{p}).
\end{equation}
According to Eqs. (35), (36), for $E_{\gamma}=9.44$ GeV we get then that
$p_{J/\psi}(\theta_{J/\psi}^{\rm max})=6.600$ GeV/c and $p_{p}(\theta_{p}^{\rm max})=1.593$ GeV/c.
For $E_{\gamma}=10.12$ GeV we obtain $p_{J/\psi}(\theta_{J/\psi}^{\rm max})=6.745$ GeV/c and
$p_{p}(\theta_{p}^{\rm max})=1.471$ GeV/c (cf. Figs. 2 and 3).
These figures show that the kinematically allowed charmonium laboratory momenta
and total energies in the direct non-resonant ${\gamma}p \to {J/\psi}p$ process, taking place
on the free target proton at rest, at given initial
photon energy vary within the following momentum and energy ranges:
\begin{equation}
p^{(2)}_{J/\psi}(0^{\circ}) \le p_{J/\psi} \le p^{(1)}_{J/\psi}(0^{\circ}),
\end{equation}
\begin{equation}
E^{(2)}_{J/\psi}(0^{\circ}) \le E_{J/\psi} \le E^{(1)}_{J/\psi}(0^{\circ}),
\end{equation}
where the quantities $p^{(1,2)}_{J/\psi}(0^{\circ})$ are defined above
by Eq. (31) and $E^{(1,2)}_{J/\psi}(0^{\circ})=\sqrt{m_{J/\psi}^2+[p^{(1,2)}_{J/\psi}(0^{\circ})]^2}$.
Finally, we calculate the $J/\psi$ energy distribution
$d\sigma_{{\gamma}p \to {J/\psi}{p}}[\sqrt{s},p_{J/\psi}]/ dE_{J/\psi}$
from the reaction ${\gamma}p \to {J/\psi}p$ within the kinematically allowed interval (38).
Integration of the more general differential cross section (9) over the angle $\theta_{J/\psi}$,
when this angle varies in the allowed angular region (25),
in the limits: ${\bf p}_t \to 0$, $E_t \to m_p$ and $s^* \to s$ yields:
\begin{equation}
\frac{d\sigma_{{\gamma}p \to {J/\psi}{p}}[\sqrt{s},p_{J/\psi}]}{dE_{J/\psi}}=
2\pi
\int\limits_{\cos{\theta_{J/\psi}^{\rm max}}}^{1}d\cos{\theta_{J/\psi}}p_{J/\psi}E_{J/\psi}
\frac{d\sigma_{{\gamma}p\to {J/\psi}p}[\sqrt{s},{\bf p}_{J/\psi}]}{d{\bf p}_{J/\psi}}=
\end{equation}
$$
=
\left(\frac{2{\pi}\sqrt{s}}{p_{\gamma}p^{*}_{J/\psi}}\right)
\frac{d\sigma_{{\gamma}p \to {J/\psi}{p}}[\sqrt{s},\theta_{J/\psi}^*(x_0)]}{d{\bf \Omega}_{J/\psi}^*}~{\rm for}
~E^{(2)}_{J/\psi}(0^{\circ}) \le E_{J/\psi} \le E^{(1)}_{J/\psi}(0^{\circ}),
$$
where
\begin{equation}
x_0=\frac{[p^2_{\gamma}+p^2_{J/\psi}+m^2_{p}-(\omega+m_p)^2]}{2p_{\gamma}p_{J/\psi}},\,\,\,\,\,
p_{J/\psi}=\sqrt{E^2_{J/\psi}-m^2_{J/\psi}}
\end{equation}
and the quantity $\cos{\theta_{J/\psi}^*(x_0)}$ is defined by Eq. (20), in which one has to perform the
replacement: $\cos{\theta_{J/\psi}} \to x_0$, and the photon and $J/\psi$ c.m. momenta $p^*_{\gamma}$
and $p^*_{J/\psi}$ are defined by formulas (16) and (17), correspondingly, in which
one needs also to make the substitutions: $E_t \to m_p$, $p_t \to 0$ and $s^* \to s$.
We will adopt the expression (39) for evaluating the free space $J/\psi$ energy distribution from
the direct process (1), proceeding on a proton at rest, for
incident photon beam resonant energies of 9.44, 9.554, 10.04 and 10.12 GeV (see below).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=16.0cm]{fig4gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The non-resonant total cross section $\sigma_1$ for the reaction
${\gamma}p \to {J/\psi}p$ (solid curve) and the incoherent sum (dotted curve) of it and the total
cross section $\sigma_2$ (short-dashed curve) for the
resonant $J/\psi$ production in the processes ${\gamma}p \to P^+_c(4312) \to {J/\psi}p$,
${\gamma}p \to P^+_c(4337) \to {J/\psi}p$, ${\gamma}p \to P^+_c(4440) \to {J/\psi}p$
and ${\gamma}p \to P^+_c(4457) \to {J/\psi}p$, calculated assuming that the resonances
$P^+_c(4312)$, $P^+_c(4337)$, $P^+_c(4440)$ and $P^+_c(4457)$ with the spin-parity quantum
numbers $J^P=(1/2)^-$, $J^P=(1/2)^-$, $J^P=(1/2)^-$ and $J^P=(3/2)^-$
decay to ${J/\psi}p$ with the lower allowed relative orbital angular momentum $L=0$
with all four branching fractions $Br[P^+_{ci} \to {J/\psi}p]=$1\%,
as functions of photon energy. The left and four right arrows indicate, correspondingly, the threshold energy
$E^{\rm th}_{\gamma}=8.21$ GeV for the reaction ${\gamma}p \to {J/\psi}p$ proceeding on a free target
proton being at rest and the resonant energies $E^{\rm R1}_{\gamma}=9.44$ GeV, $E^{\rm R2}_{\gamma}=9.554$ GeV,
$E^{\rm R3}_{\gamma}=10.04$ GeV and $E^{\rm R4}_{\gamma}=10.12$ GeV.}
\label{void}
\end{center}
\end{figure}
\subsection*{2.2. Two-step resonant $J/\psi$ production processes}
At photon energies below 11 GeV, incident photons can produce the observed [1--3] experimentally
non-strange charged $P^+_c(4312)$, $P^+_c(4337)$, $P^+_c(4440)$, $P^+_c(4457)$ pentaquark resonances
with quark structure $|P^+_c>=|uudc{\bar c}>$
and predicted [15], but non-observed yet their neutral isospin partners $P^0_c(4312)$, $P^0_c(4337)$,
$P^0_c(4440)$, $P^0_c(4457)$
\footnote{$^)$The minimal quark content of the $P_c^0$ states is $|P^0_c>=|uddc{\bar c}>$.
Following the observation of the narrow pentaquarks $P^+_c(4312)$, $P^+_c(4440)$ and $P^+_c(4457)$
by the LHCb Collaboration [1, 2], it was proposed to search for the $P_c^0$ states in
${\pi^-}p \to {J/\psi}n$ reaction [16].}$^)$
directly in the first inelastic collisions with intranuclear protons and neutrons
\footnote{$^)$We remind that, for example, the threshold (resonant) energies $E^{\rm R1}_{\gamma}$,
$E^{\rm R2}_{\gamma}$, $E^{\rm R3}_{\gamma}$ and $E^{\rm R4}_{\gamma}$ for the photoproduction
of the $P_c^+$ resonances with pole masses $M_{c1}^+=4311.9$ MeV, $M_{c2}^+=4337.0$ MeV, $M_{c3}^+=4440.3$ MeV
and $M_{c4}^+=4457.3$ MeV [2, 3] on a free target proton being at rest are $E^{\rm R1}_{\gamma}=9.44$ GeV, $E^{\rm R2}_{\gamma}=9.554$ GeV, $E^{\rm R3}_{\gamma}=10.04$ GeV and $E^{\rm R4}_{\gamma}=10.12$ GeV, respectively.}$^)$
:
\begin{eqnarray}
{\gamma}+p \to P^+_c(4312),\nonumber\\
{\gamma}+p \to P^+_c(4337),\nonumber\\
{\gamma}+p \to P^+_c(4440),\nonumber\\
{\gamma}+p \to P^+_c(4457);
\end{eqnarray}
\begin{eqnarray}
{\gamma}+n \to P^0_c(4312),\nonumber\\
{\gamma}+n \to P^0_c(4337),\nonumber\\
{\gamma}+n \to P^0_c(4440),\nonumber\\
{\gamma}+n \to P^0_c(4457).
\end{eqnarray}
Furthermore, the produced pentaquark resonances can decay into the final states ${J/\psi}p$
and ${J/\psi}n$, which will additionally contribute to the $J/\psi$ yield in the ($\gamma$,$J/\psi$) reactions
on protons and nuclei:
\begin{eqnarray}
P^+_c(4312) \to J/\psi+p,\nonumber\\
P^+_c(4337) \to J/\psi+p,\nonumber\\
P^+_c(4440) \to J/\psi+p,\nonumber\\
P^+_c(4457) \to J/\psi+p;
\end{eqnarray}
\begin{eqnarray}
P^0_c(4312) \to J/\psi+n,\nonumber\\
P^0_c(4337) \to J/\psi+n,\nonumber\\
P^0_c(4440) \to J/\psi+n,\nonumber\\
P^0_c(4457) \to J/\psi+n.
\end{eqnarray}
The branching ratios $Br[P^+_{ci} \to {J/\psi}p]$
\footnote{$^)$Here, $i=$1, 2, 3, 4 and $P^{+}_{c1}$, $P^{+}_{c2}$, $P^{+}_{c3}$ and $P^{+}_{c4}$
stand for $P^{+}_c(4312)$, $P^{+}_c(4337)$, $P^{+}_c(4440)$ and $P^{+}_c(4457)$, respectively.
Analogously, $P^{0}_{c1}$, $P^{0}_{c2}$, $P^{0}_{c3}$ and $P^{0}_{c4}$ will denote below
the $P^{0}_c(4312)$, $P^{0}_c(4337)$, $P^{0}_c(4440)$ and $P^{0}_c(4457)$ states.}$^)$
of the decays (43) have not been determined yet.
Model-dependent upper limits on branching fractions $Br[P^+_{c}(4312) \to {J/\psi}p]$,
$Br[P^+_{c}(4440) \to {J/\psi}p]$ and $Br[P^+_{c}(4457) \to {J/\psi}p]$ of several percent
were set by the GlueX Hall-D experiment [4] at JLab, having a moderate statistics (about 470 $J/\psi$ events).
Preliminary results from a factor of 10 more data (about 4000 $J/\psi$ events),
collected in the $J/\psi$--007 Hall-C experiment [17] at JLab as well, focused on the large $t$ region
\footnote{$^)$In which the rather flat resonant production of $J/\psi$ through the $P_c^+$
is expected to be enhanced relative to the suppressed here mostly forward diffractive production.}$^)$
in searching for the LHCb hidden-charm pentaquarks [1, 2], also observe no signals for them
and will set more stringent upper limits on the above branching fractions and on pentaquark-$J/\psi$
couplings. Based on the branching ratios and
fractions measured by the LHCb and GlueX Collaborations, the authors of Ref. [18] obtain that a lower
limit of $Br[P^+_{c} \to {J/\psi}p]$ is of the order of 0.05\% $\sim$ 0.5\%.
Taking into account these findings, we will adopt in our study for the four branching ratios
$Br[P^+_{ci}\to {J/\psi}p]$ of the decays (43) three following conservative options:
$Br[P^+_{ci} \to {J/\psi}p]=0.25$, 0.5 and 1\% ($i=1,2,3,4$), and in line with Ref. [15], will assume that
$Br[P^0_{ci} \to {J/\psi}n]=Br[P^+_{ci} \to {J/\psi}p]$. This will allow us
to get a better impression of the size of the effect of branching
fractions $Br[P^+_{ci} \to {J/\psi}p]$ and $Br[P^0_{ci} \to {J/\psi}n]$
on the resonant $J/\psi$ yield in ${\gamma}p \to {J/\psi}p$ as well as in
${\gamma}$$^{12}$C $\to {J/\psi}X$ and ${\gamma}$$^{184}$W $\to {J/\psi}X$ reactions.
Moreover, we will also suppose, analogously to [15], for the $P_{ci}^0$ states the same pole masses
$M_{ci}^0$ and total decay width $\Gamma_{ci}^0$ as those $M_{ci}^+$ and $\Gamma_{ci}^+$ for their
hidden-charm charged counterparts $P_{ci}^+$, i.e.: $M_{ci}^0=M_{ci}^+$ and
$\Gamma_{c1}^0=\Gamma_{c1}^+=9.8$ MeV, $\Gamma_{c2}^0=\Gamma_{c2}^+=29.0$ MeV,
$\Gamma_{c3}^0=\Gamma_{c3}^+=20.6$ MeV, $\Gamma_{c4}^0=\Gamma_{c4}^+=6.4$ MeV [2, 3].
In line with Refs. [6, 7, 19], we suppose that the in-medium spectral functions
$S_{ci}^+(\sqrt{s^*},\Gamma_{ci}^+)$ and $S_{ci}^0(\sqrt{s^*},\Gamma_{ci}^0)$
of the intermediate $P_{ci}^{+}$ and $P_{ci}^{0}$ resonances
are described by the non-relativistic Breit-Wigner distributions
\footnote{$^)$We ignore, for reasons of the simplification of calculations, the modification of the
$P_{ci}^{+}$ and $P_{ci}^{0}$ masses and total decay widths in the nuclear matter in our present study.}$^)$
:
\begin{equation}
S_{ci}^+(\sqrt{s^*},\Gamma_{ci}^+)=
\frac{1}{2\pi}\frac{\Gamma_{ci}^+}{(\sqrt{s^*}-M_{ci}^+)^2+({\Gamma}_{ci}^+)^{2}/4}
\end{equation}
and
\begin{equation}
S_{ci}^0(\sqrt{s^*},\Gamma_{ci}^0)
=\frac{1}{2\pi}\frac{\Gamma_{ci}^0}{(\sqrt{s^*}-M_{ci}^0)^2+({\Gamma}_{ci}^0)^{2}/4}.
\end{equation}
The in-medium total cross sections for production of these
resonances with the possible spin-parity quantum numbers $J^P=(1/2)^-$ for $P_{c1}^+$ and $P_{c1}^0$,
$J^P=(1/2)^-$ for $P_{c2}^+$ and $P_{c2}^0$, $J^P=(1/2)^-$ for $P_{c3}^+$ and $P_{c3}^0$, and
$J^P=(3/2)^-$ for $P_{c4}^+$ and $P_{c4}^0$
\footnote{$^)$Which might be assigned to them within
the hadronic molecular scenario for their internal structure (cf. [7, 15, 20--22]).}$^)$
in reactions (41), (42) can be determined, using the spectral functions (45), (46) and known
branching fractions $Br[P_{ci}^+ \to {\gamma}p]$ and $Br[P_{ci}^0 \to {\gamma}n]$ ($i=1$, 2, 3, 4),
as follows [6, 7, 19]:
\begin{equation}
\sigma_{{\gamma}p \to P_{ci}^+}(\sqrt{s^*},\Gamma_{ci}^+)=
f_{ci}\left(\frac{\pi}{p^*_{\gamma}}\right)^2
Br[P_{ci}^+ \to {\gamma}p]S_{ci}^+(\sqrt{s^*},\Gamma_{ci}^+)\Gamma_{ci}^+, \,\,i=1,2,3,4
\end{equation}
and
\begin{equation}
\sigma_{{\gamma}n \to P_{ci}^0}(\sqrt{s^*},\Gamma_{ci}^0)=
f_{ci}\left(\frac{\pi}{p^*_{\gamma}}\right)^2
Br[P_{ci}^0 \to {\gamma}n]S_{ci}^0(\sqrt{s^*},\Gamma_{ci}^0)\Gamma_{ci}^0, \,\,i=1,2,3,4.
\end{equation}
Here, the c.m. 3-momentum in the incoming ${\gamma}N$ channel, $p^*_{\gamma}$,
is defined above by Eq. (16)
\footnote{$^)$For simplicity, we assume that the neutron mass $m_n$ is equal to the proton mass $m_p$.}$^)$
and the ratios of the spin factors $f_{c1}=1$, $f_{c2}=1$,
$f_{c3}=1$, $f_{c4}=2$. Since we are mainly interested in the resonance $P_{c}$ region,
which is not far from the ${J/\psi}N$ production threshold, we suppose
in line with [7, 19, 23] that the hidden-charm pentaquarks $P_{ci}^+$ and
$P_{ci}^0$ decays to ${J/\psi}p$ and ${J/\psi}n$ modes
are dominated by the lowest partial waves with zero relative orbital angular momentum $L$.
In this case, adopting the vector-meson dominance model, one can obtain
that the branching ratios $Br[P_{ci}^0 \to {\gamma}n]$ and $Br[P_{ci}^+ \to {\gamma}p]$
are equal to each other (cf. [24])
\begin{equation}
Br[P_{ci}^0 \to {\gamma}n]=Br[P_{ci}^+ \to {\gamma}p]
\end{equation}
and the latter for $P^+_c(4312)$, $P^+_c(4440)$ and $P^+_c(4457)$
are expressed in the framework of this model via the branching fractions
$Br[P^+_c(4312) \to {J/\psi}p]$, $Br[P^+_c(4440) \to {J/\psi}p]$ and $Br[P^+_c(4457) \to {J/\psi}p]$
by formula (24) from Ref. [7] and within this model we get that
\begin{equation}
Br[P^+_c(4337) \to {\gamma}p]=1.48\cdot10^{-3}Br[P^+_c(4337) \to {J/\psi}p].
\end{equation}
Using Eqs. (47)--(49), we have
\begin{equation}
\sigma_{{\gamma}p \to P_{ci}^+}(\sqrt{s^*},\Gamma_{ci}^+)=\sigma_{{\gamma}n \to P_{ci}^0}(\sqrt{s^*},\Gamma_{ci}^0).
\end{equation}
According to Eq. (47), for example, the free total cross sections
$\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}p}(\sqrt{s},\Gamma_{ci}^+)$
for resonant charmonium production in the two-step processes (41)/(43), taking place on the target proton at rest,
can be represented as follows [6, 7]:
\begin{equation}
\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}p}(\sqrt{s},\Gamma_{ci}^+)=
\sigma_{{\gamma}p \to P_{ci}^+}(\sqrt{s},\Gamma_{ci}^+)\theta[\sqrt{s}-(m_{J/\psi}+m_{p})]
Br[P_{ci}^+ \to {J/\psi}p].
\end{equation}
Here, $\theta(x)$ is the step function and the c.m. 3-momentum in the incoming
${\gamma}p$ channel, $p^*_{\gamma}$, entering into Eq. (47), is determined above by the formula (16),
in which one has to make the replacements $E_t^2-p_t^2 \to m^2_p$ and $s^* \to s$.
In line with Eqs. (47) and (50), we see that these cross sections are proportional to $Br^2[P_{ci}^+ \to {J/\psi}p]$.
This fact enables us to evaluate upper limits on the branching fractions $Br[P_{c}^+(4312) \to {J/\psi}p]$,
$Br[P_{c}^+(4440) \to {J/\psi}p]$ and $Br[P_{c}^+(4457) \to {J/\psi}p]$, which are expected from
preliminary results of the JLab E12-16-007 experiment [17]. According to them, upper limits on the
cross sections (52) for $P_{c}^+(4312)$, $P_{c}^+(4440)$ and $P_{c}^+(4457)$ states almost an order of
magnitude below the respective GlueX limits [4]. With this and within the representation of Eq. (52), we
readily obtain the following relation between upper limits on the above branching fractions, which are
expected from the $J/\psi$-007 experiment, and those already available from the GlueX experiment:
$Br_{J/\psi-007}[P_{ci}^+ \to {J/\psi}p]\approx(1/\sqrt{10})Br_{\rm GlueX}[P_{ci}^+ \to {J/\psi}p]$ ($i=1$, 3, 4).
Model-dependent upper limits on the latter ratios of 4.6\%, 2.3\% and 3.8\% for
$P_{c}^+(4312)$, $P_{c}^+(4440)$ and $P_{c}^+(4457)$, assuming for each $P_{ci}^+$ spin-parity
combination $J^P=(3/2)^-$, were set by the GlueX Collaboration [4]. So that, following the above
relation, we get that $Br_{J/\psi-007}[P_{c}^+(4312) \to {J/\psi}p]\approx$1.46\%,
$Br_{J/\psi-007}[P_{c}^+(4440) \to {J/\psi}p]\approx$0.73\% and
$Br_{J/\psi-007}[P_{c}^+(4457) \to {J/\psi}p]\approx$1.20\%. This means that our choice of 1\%
for upper value of the branching ratios $Br[P_{ci}^+ \to {J/\psi}p]$ for all 4 states is quite
reasonable and justified.
Accounting for the fact that the most of the narrow $P_{ci}^+$ amd $P_{ci}^0$ resonances ($i=1$, 2, 3, 4),
having vacuum total decay widths in their rest frames of
9.8, 29.0, 20.6 and 6.4 MeV [2, 3], respectively, decay to ${J/\psi}p$ and ${J/\psi}n$
outside of the considered target nuclei [6] as well as
the results presented both in Refs. [6, 7, 10] and above by
Eqs. (3), (4), (51), (52), we can obtain the following expression for the $J/\psi$
inclusive differential cross section arising from the production and
decay of intermediate resonances $P_{ci}^+$ and $P_{ci}^0$ in ${\gamma}A$ reactions:
\begin{equation}
\frac{d\sigma_{{\gamma}A \to {J/\psi}X}^{({\rm sec})}({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}=
\frac{d\sigma_{{\gamma}A \to P_{ci}^+ \to {J/\psi}p}^{({\rm sec})}({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}
+
\frac{d\sigma_{{\gamma}A \to P_{ci}^0 \to {J/\psi}n}^{({\rm sec})}({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}
=
\end{equation}
$$
=
I_{V}[A,\sigma^{\rm in}_{{P_{c}}N}]
\left<\frac{d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}p}({\bf p}_{\gamma},{\bf p}_{J/\psi})}
{d{\bf p}_{J/\psi}}\right>_A, i=1, 2, 3, 4,
$$
where
\begin{equation}
\left<\frac{d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}p}({\bf p}_{\gamma},{\bf p}_{J/\psi})}
{d{\bf p}_{J/\psi}}\right>_A=
\int\int P_A({\bf p}_t,E)d{\bf p}_tdE
\left[\frac{d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi})}
{d{\bf p}_{J/\psi}}\right],
\end{equation}
and
\begin{equation}
\frac{d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi})}
{d{\bf p}_{J/\psi}}=\sigma_{{\gamma}p \to P_{ci}^+}(\sqrt{s^*},\Gamma_{ci}^+)
\theta[\sqrt{s^*}-(m_{J/\psi}+m_{p})]\times
\end{equation}
$$
\times
\frac{1}{\Gamma_{ci}^+(\sqrt{s^*},{\bf p}_{\gamma})}\int d{\bf p}_{p}
\frac{d\Gamma_{P_{ci}^+ \to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi},{\bf p}_{p})}
{d{\bf p}_{J/\psi}d{\bf p}_{p}},
$$
\begin{equation}
\frac{d\Gamma_{P_{ci}^+ \to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi},{\bf p}_{p})}
{d{\bf p}_{J/\psi}d{\bf p}_{p}}=\frac{1}{2E_{ci}^+}\frac{1}{2J+1}|M_{P_{ci}^+ \to {J/\psi}p}|^2
(2\pi)^4\delta(E_{ci}^+-E_{J/\psi}-E_{p})\times
\end{equation}
$$
\times
\delta({\bf p}_{ci}^+-{\bf p}_{J/\psi}-{\bf p}_{p})\frac{1}{(2\pi)^3{2E_{J/\psi}}}
\frac{1}{(2\pi)^3{2E_{p}}},
$$
\begin{equation}
\Gamma_{ci}^+(\sqrt{s^*},{\bf p}_{\gamma})=\Gamma_{ci}^+/\gamma_{ci}^+,
\end{equation}
\begin{equation}
E_{ci}^+=E_{\gamma}+E_t,\,\,\,\,\,{\bf p}_{ci}^+={\bf p}_{\gamma}+
{\bf p}_{t},\,\,\,\,\,\gamma_{ci}^+=E_{ci}^+/\sqrt{s^*}.
\end{equation}
Here, $E_{p}$ is the final proton total energy ($E_{p}=\sqrt{m^2_{p}+{\bf p}^2_{p}}$)
and $|M_{P_{ci}^+ \to {J/\psi}p}|^2$ is summarized over spin states of initial and final particles
matrix element squared describing the decays (43) for given $i$.
The quantity $I_{V}[A,\sigma^{\rm in}_{P_{c}N}]$ in Eq. (53) is defined above by Eq. (4), in which one
needs to make the substitution $\sigma \to \sigma^{\rm in}_{P_{c}N}$. Here the quantity
$\sigma^{\rm in}_{P_{c}N}$ denotes the inelastic total cross sections of the free $P_{c}N$ interaction.
Our estimates [6]
\footnote{$^)$These estimates also show that we can neglect quasielastic $P_{ci}^+{N}$ and
$P_{ci}^0{N}$ rescatterings in their way out of the target nucleus.}$^)$,
based on the ${J/\psi}p$ molecular scenario for the $P_{c}^+$ pentaquarks,
show that this quantity can be evaluated as $\sigma^{\rm in}_{P_{c}N} \approx 33.5$ mb. We will use
this value throughout our calculations.
In view of the aforesaid, the hidden-charm pentaquarks $P_{ci}^+$
(and $P_{ci}^0$) decays to ${J/\psi}p$ (and ${J/\psi}n$)
are dominated by the lowest partial $s$-waves with zero relative orbital angular momentum.
This implies that the matrix elements squared
$|M_{P_{ci}^+ \to {J/\psi}p}|^2$ (and $|M_{P_{ci}^0 \to {J/\psi}n}|^2$)
lead to an isotropic angular distributions of the
$P_{ci}^+ \to {J/\psi}p$ (and $P_{ci}^0 \to {J/\psi}n$)
decays for the considered spin-parity assignments of
the $P_{ci}^+$ (and $P_{ci}^0$) states. With this, we can readily
obtain the following relation between $|M_{P_{ci}^+ \to {J/\psi}p}|^2$ and the partial
width $\Gamma_{P_{ci}^+ \to {J/\psi}p}$ of the $P_{ci}^+ \to {J/\psi}p$ decay (cf. [10]):
\begin{equation}
\frac{1}{2J+1}\frac{|M_{P_{ci}^+ \to {J/\psi}p}|^2}{(2\pi)^2}=
\frac{2s^*}{\pi{p^*_{J/\psi}}}\Gamma_{P_{ci}^+ \to {J/\psi}p}.
\end{equation}
With it, we find for the expression (55) a more simpler form (cf. Eq. (9)):
\begin{equation}
\frac{d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}p}(\sqrt{s^*},{\bf p}_{J/\psi})}
{d{\bf p}_{J/\psi}}=\sigma_{{\gamma}p \to P_{ci}^+}(\sqrt{s^*},\Gamma_{ci}^+)
\theta[\sqrt{s^*}-(m_{J/\psi}+m_{p})]\times
\end{equation}
$$
\times
\frac{1}{I_2(s^*,m_{J/\psi},m_{p})}Br[P_{ci}^+ \to {J/\psi}p]
\frac{1}{4E_{J/\psi}}\frac{1}{(\omega+E_t)}
\delta\left[\omega+E_t-\sqrt{m_{p}^2+({\bf Q}+{\bf p}_t)^2}\right],
$$
where the quantities $\omega$ and ${\bf Q}$ are defined above by Eq. (12).
We will employ this expression in our calculations of the $J/\psi$ momentum distribution from the processes
(41)--(44) in ${\gamma}A$ reactions. Integrating the differential cross section (53)
over the angular range of ${\Delta}{\bf \Omega}_{J/\psi}$=$0^{\circ} \le \theta_{J/\psi} \le 20^{\circ}$,
$0 \le \varphi_{J/\psi} \le 2{\pi}$ of our interest,
we represent this distribution for given $i$ in this angular range in the following form (cf. Eq. (24)):
\begin{equation}
\frac{d\sigma_{{\gamma}A\to {J/\psi}X}^{({\rm sec})}
(p_{\gamma},p_{J/\psi})}{dp_{J/\psi}}=
\int\limits_{{\Delta}{\bf \Omega}_{J/\psi}}^{}d{\bf \Omega}_{J/\psi}
\frac{d\sigma_{{\gamma}A\to {J/\psi}X}^{({\rm sec})}
({\bf p}_{\gamma},{\bf p}_{J/\psi})}{d{\bf p}_{J/\psi}}p_{J/\psi}^2
\end{equation}
$$
=2{\pi}I_{V}[A,\sigma^{\rm in}_{{P_{c}}N}]
\int\limits_{\cos20^{\circ}}^{1}d\cos{{\theta_{J/\psi}}}
\left<\frac{d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}{p}}(p_{\gamma},
p_{J/\psi},\theta_{J/\psi})}{dp_{J/\psi}d{\bf \Omega}_{J/\psi}}\right>_A,\,\,\,i=1, 2, 3, 4.
$$
Before going to the next step, we calculate the free space resonant $J/\psi$ energy distribution
$d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}{p}}[\sqrt{s},p_{J/\psi}]/ dE_{J/\psi}$
from the two-step processes (41)/(43), proceeding on the free target proton at rest,
in addition to that from the
background ${\gamma}p \to {J/\psi}p$ reaction (cf. Eq. (39)).
The energy-momentum conservation in these precesses
leads to the conclusion that the kinematical characteristics
of $J/\psi$ mesons produced in them and in this reaction are the same at given incident photon energy.
The full on-shell differential cross section
$d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}{p}}[\sqrt{s},{\bf p}_{J/\psi}]/ d{\bf p}_{J/\psi}$
can be obtained from more general one (60) in the limits: ${\bf p}_t \to 0$,
$E_t \to m_p$ and $s^* \to s$. Its integration over the laboratory polar angle $\theta_{J/\psi}$,
when this angle belongs to the allowed angular interval (25), gives:
\begin{equation}
\frac{d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}{p}}[\sqrt{s},p_{J/\psi}]}{dE_{J/\psi}}=
2\pi
\int\limits_{\cos{\theta_{J/\psi}^{\rm max}}}^{1}d\cos{\theta_{J/\psi}}p_{J/\psi}E_{J/\psi}
\frac{d\sigma_{{\gamma}p \to P_{ci}^+ \to {J/\psi}p}[\sqrt{s},{\bf p}_{J/\psi}]}{d{\bf p}_{J/\psi}}=
\end{equation}
$$
=
\sigma_{{\gamma}p \to P_{ci}^+}(\sqrt{s},\Gamma_{ci}^+)\theta[\sqrt{s}-(m_{J/\psi}+m_{p})]\times
$$
$$
\times
\left(\frac{\sqrt{s}}{2p_{\gamma}p^{*}_{J/\psi}}\right)
Br[P_{ci}^+ \to {J/\psi}p]~{\rm for}
~E^{(2)}_{J/\psi}(0^{\circ}) \le E_{J/\psi} \le E^{(1)}_{J/\psi}(0^{\circ}).
$$
Eq. (62) shows that the free space $J/\psi$ energy distribution, which arises from the
production/decay chains (41)/(43), exhibits a completely flat behavior within the
allowed energy range (38).
\begin{figure}[!h]
\begin{center}
\includegraphics[width=16.0cm]{fig5gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The direct non-resonant $J/\psi$ energy distribution in the free space
elementary process ${\gamma}p \to {J/\psi}p$,
calculated in line with Eq. (39) at initial photon resonant energy of 9.44 GeV
in the laboratory system (solid curve). The resonant $J/\psi$ energy distributions in the two-step processes
${\gamma}p \to P_{c}^+(4312) \to {J/\psi}p$, ${\gamma}p \to P_{c}^+(4337) \to {J/\psi}p$,
${\gamma}p \to P_{c}^+(4440) \to {J/\psi}p$ and ${\gamma}p \to P_{c}^+(4457) \to {J/\psi}p$,
calculated in line with Eq. (62) at the same incident photon energy of 9.44 GeV
assuming that the resonances $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$ and $P_{c}^+(4457)$
with the spin-parity assignments $J^P=(1/2)^-$, $J^P=(1/2)^-$, $J^P=(1/2)^-$ and $J^P=(3/2)^-$, correspondingly,
all decay to the ${J/\psi}p$
with branching fractions 0.25\% (respectively, red dashed, blue dotted, dark cyan dashed-doted
and magenta dashed-dotted-dotted curves).
Incoherent sum of the direct non-resonant $J/\psi$ energy distribution and resonant ones, calculated supposing that the resonances $P_{c}^+(4312)$ and $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$, $P_{c}^+(4457)$
with the same spin-parity combinations all decay to the ${J/\psi}p$ with branching
fractions 0.25, 0.5 and 1\% (respectively, dark yellow short-dashed, wine short-dashed-dotted, olive dashed-dotted and navy short-dotted, pink dotted, royal dashed-dotted-dotted curves), all as functions of the total $J/\psi$ energy $E_{J/\psi}$ in the laboratory system.
The vertical dotted lines indicate the range of $J/\psi$ allowed energies in this system
for the considered direct non-resonant and resonant $J/\psi$ production off a free target proton at rest at
given initial photon resonant energy of 9.44 GeV.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=16.0cm]{fig6gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The direct non-resonant $J/\psi$ energy distribution in the free space
elementary process ${\gamma}p \to {J/\psi}p$,
calculated in line with Eq. (39) at initial photon resonant energy of 9.554 GeV
in the laboratory system (solid curve). The resonant $J/\psi$ energy distributions in the two-step processes
${\gamma}p \to P_{c}^+(4312) \to {J/\psi}p$, ${\gamma}p \to P_{c}^+(4337) \to {J/\psi}p$,
${\gamma}p \to P_{c}^+(4440) \to {J/\psi}p$ and ${\gamma}p \to P_{c}^+(4457) \to {J/\psi}p$,
calculated in line with Eq. (62) at the same incident photon energy of 9.554 GeV
assuming that the resonances $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$ and $P_{c}^+(4457)$
with the spin-parity assignments $J^P=(1/2)^-$, $J^P=(1/2)^-$, $J^P=(1/2)^-$ and $J^P=(3/2)^-$, correspondingly,
all decay to the ${J/\psi}p$
with branching fractions 0.25\% (respectively, red dashed, blue dotted, dark cyan dashed-doted
and magenta dashed-dotted-dotted curves).
Incoherent sum of the direct non-resonant $J/\psi$ energy distribution and resonant ones, calculated supposing that the resonances $P_{c}^+(4337)$ and $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$, $P_{c}^+(4457)$
with the same spin-parity combinations
all decay to the ${J/\psi}p$ with branching
fractions 0.25, 0.5 and 1\% (respectively, dark yellow short-dashed, wine short-dashed-dotted, olive dashed-dotted and navy short-dotted, pink dotted, royal dashed-dotted-dotted curves), all as functions of the total $J/\psi$ energy $E_{J/\psi}$ in the laboratory system.
The vertical dotted lines indicate the range of $J/\psi$ allowed energies in this system
for the considered direct non-resonant and resonant $J/\psi$ production off a free target proton at rest at
given initial photon resonant energy of 9.554 GeV.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=16.0cm]{fig7gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The direct non-resonant $J/\psi$ energy distribution in the free space
elementary process ${\gamma}p \to {J/\psi}p$,
calculated in line with Eq. (39) at initial photon resonant energy of 10.04 GeV
in the laboratory system (solid curve). The resonant $J/\psi$ energy distributions in the two-step processes
${\gamma}p \to P_{c}^+(4312) \to {J/\psi}p$, ${\gamma}p \to P_{c}^+(4337) \to {J/\psi}p$,
${\gamma}p \to P_{c}^+(4440) \to {J/\psi}p$ and ${\gamma}p \to P_{c}^+(4457) \to {J/\psi}p$,
calculated in line with Eq. (62) at the same incident photon energy of 10.04 GeV
assuming that the resonances $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$ and $P_{c}^+(4457)$
with the spin-parity assignments $J^P=(1/2)^-$, $J^P=(1/2)^-$, $J^P=(1/2)^-$ and $J^P=(3/2)^-$, correspondingly,
all decay to the ${J/\psi}p$
with branching fractions 0.25\% (respectively, red dashed, blue dotted, dark cyan dashed-doted
and magenta dashed-dotted-dotted curves).
Incoherent sum of the direct non-resonant $J/\psi$ energy distribution and resonant ones, calculated supposing that the resonances $P_{c}^+(4440)$ and $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$, $P_{c}^+(4457)$
with the same spin-parity combinations all decay to the ${J/\psi}p$ with branching
fractions 0.25, 0.5 and 1\% (respectively, dark yellow short-dashed, wine short-dashed-dotted, olive dashed-dotted and navy short-dotted, pink dotted, royal dashed-dotted-dotted curves), all as functions of the total $J/\psi$ energy $E_{J/\psi}$ in the laboratory system.
The vertical dotted lines indicate the range of $J/\psi$ allowed energies
in this system for the considered direct non-resonant and resonant $J/\psi$ production
off a free target proton at rest at given initial photon resonant energy of 10.04 GeV.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=16.0cm]{fig8gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The direct non-resonant $J/\psi$ energy distribution in the free space
elementary process ${\gamma}p \to {J/\psi}p$,
calculated in line with Eq. (39) at initial photon resonant energy of 10.12 GeV
in the laboratory system (solid curve). The resonant $J/\psi$ energy distributions in the two-step processes
${\gamma}p \to P_{c}^+(4312) \to {J/\psi}p$, ${\gamma}p \to P_{c}^+(4337) \to {J/\psi}p$,
${\gamma}p \to P_{c}^+(4440) \to {J/\psi}p$ and ${\gamma}p \to P_{c}^+(4457) \to {J/\psi}p$,
calculated in line with Eq. (62) at the same incident photon energy of 10.12 GeV
assuming that the resonances $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$ and $P_{c}^+(4457)$
with the spin-parity assignments $J^P=(1/2)^-$, $J^P=(1/2)^-$, $J^P=(1/2)^-$ and $J^P=(3/2)^-$, correspondingly,
all decay to the ${J/\psi}p$
with branching fractions 0.25\% (respectively, red dashed, blue dotted, dark cyan dashed-doted
and magenta dashed-dotted-dotted curves).
Incoherent sum of the direct non-resonant $J/\psi$ energy distribution and resonant ones, calculated supposing that the resonances $P_{c}^+(4457)$ and $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$, $P_{c}^+(4457)$
with the same spin-parity combinations all decay to the ${J/\psi}p$ with branching
fractions 0.25, 0.5 and 1\% (respectively, dark yellow short-dashed, wine short-dashed-dotted, olive dashed-dotted and navy short-dotted, pink dotted, royal dashed-dotted-dotted curves), all as functions of the total $J/\psi$ energy $E_{J/\psi}$ in the laboratory system.
The vertical dotted lines indicate the range of $J/\psi$ allowed energies in this system for the considered direct non-resonant and resonant $J/\psi$ production off a free target proton at rest at
given initial photon resonant energy of 10.12 GeV.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=16.0cm]{fig9gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The direct non-resonant momentum distribution
of $J/\psi$ mesons, produced in the reaction ${\gamma}{\rm ^{12}C} \to {J/\psi}X$
in the laboratory polar angular range of 0$^{\circ}$--20$^{\circ}$ and calculated in line with Eq. (24)
at initial photon resonant energy of 9.44 GeV in the laboratory system (solid curve).
The resonant momentum distributions of $J/\psi$ mesons, produced
in the two-step processes
${\gamma}p(n) \to P_{c}^+(4312)(P_{c}^0(4312)) \to {J/\psi}p(n)$,
${\gamma}p(n) \to P_{c}^+(4337)(P_{c}^0(4337)) \to {J/\psi}p(n)$,
${\gamma}p(n) \to P_{c}^+(4440)(P_{c}^0(4440)) \to {J/\psi}p(n)$ and
${\gamma}p(n) \to P_{c}^+(4457)(P_{c}^0(4457)) \to {J/\psi}p(n)$ and
calculated in line with Eq. (61) at the same incident photon energy of 9.44 GeV
assuming that the resonances $P_{c}^{+,0}(4312)$, $P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$
and $P_{c}^{+,0}(4457)$ with the spin-parity assignments
$J^P=(1/2)^-$, $J^P=(1/2)^-$, $J^P=(1/2)^-$ and $J^P=(3/2)^-$, correspondingly,
all decay to the ${J/\psi}p(n)$
with branching fractions 0.25\% (respectively, red dashed, blue dotted, dark cyan dashed-doted
and magenta dashed-dotted-dotted curves) and their incoherent sum (orange dotted curve).
Incoherent sum of the direct non-resonant $J/\psi$ momentum distribution and resonant ones,
calculated supposing that the resonances
$P_{c}^{+,0}(4312)$, $P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$, $P_{c}^{+,0}(4457)$
with the same spin-parity combinations all decay to the ${J/\psi}p(n)$ with branching
fractions 0.25, 0.5 and 1\% (respectively, green short-dashed, navy short-dotted and pink
short-dashed-dotted curves), all as functions of the $J/\psi$ momentum $p_{J/\psi}$ in the laboratory frame.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=16.0cm]{fig10gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The direct non-resonant momentum distribution
of $J/\psi$ mesons, produced in the reaction ${\gamma}{\rm ^{184}W} \to {J/\psi}X$
in the laboratory polar angular range of 0$^{\circ}$--20$^{\circ}$ and calculated in line with Eq. (24)
at initial photon resonant energy of 9.44 GeV in the laboratory system (solid curve).
The resonant momentum distributions of $J/\psi$ mesons, produced
in the two-step processes
${\gamma}p(n) \to P_{c}^+(4312)(P_{c}^0(4312)) \to {J/\psi}p(n)$,
${\gamma}p(n) \to P_{c}^+(4337)(P_{c}^0(4337)) \to {J/\psi}p(n)$,
${\gamma}p(n) \to P_{c}^+(4440)(P_{c}^0(4440)) \to {J/\psi}p(n)$ and
${\gamma}p(n) \to P_{c}^+(4457)(P_{c}^0(4457)) \to {J/\psi}p(n)$ and
calculated in line with Eq. (61) at the same incident photon energy of 9.44 GeV
assuming that the resonances $P_{c}^{+,0}(4312)$, $P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$
and $P_{c}^{+,0}(4457)$ with the spin-parity assignments
$J^P=(1/2)^-$, $J^P=(1/2)^-$, $J^P=(1/2)^-$ and $J^P=(3/2)^-$, correspondingly,
all decay to the ${J/\psi}p(n)$
with branching fractions 0.25\% (respectively, red dashed, blue dotted, dark cyan dashed-doted
and magenta dashed-dotted-dotted curves) and their incoherent sum (orange dotted curve).
Incoherent sum of the direct non-resonant $J/\psi$ momentum distribution and resonant ones,
calculated supposing that the resonances
$P_{c}^{+,0}(4312)$, $P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$, $P_{c}^{+,0}(4457)$
with the same spin-parity combinations all decay to the ${J/\psi}p(n)$ with branching
fractions 0.25, 0.5 and 1\% (respectively, green short-dashed, navy short-dotted and pink
short-dashed-dotted curves), all as functions of the $J/\psi$ momentum $p_{J/\psi}$ in the laboratory frame.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=16.0cm]{fig11gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The same as in Fig. 9, but for the initial photon energy of 10.12 GeV.}
\label{void}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=16.0cm]{fig12gampsiprot.pdf}
\vspace*{-2mm} \caption{(Color online) The same as in Fig. 10, but for the initial photon energy of 10.12 GeV.}
\label{void}
\end{center}
\end{figure}
\section*{3. Results}
The free space direct non-resonant $J/\psi$ production total cross section (23) (solid curve),
the total cross section for the resonant $J/\psi$ production in the processes (41)/(43)
determined on the basis of Eq. (52) for the considered spin-parity assignments of the hidden-charm resonances
$P_{ci}^+$ ($i=1$, 2, 3, 4) and for branching ratios $Br[P_{ci}^+ \to {J/\psi}p]=1$\% for all four
$P_{ci}^+$ states (short-dashed curve) and the combined (non-resonant plus resonant)
$J/\psi$ production total cross section (dotted curve) are presented in Fig. 4 as functions of photon energy.
It can be seen from this figure that the $P_{c}^{+}(4312)$ and $P_{c}^{+}(4337)$ as well as
$P_{c}^{+}(4440)$ and $P_{c}^{+}(4457)$ resonances exhibit itself as two narrow overlapping peaks, respectively,
at $E_{\gamma}=9.44$ and $E_{\gamma}=9.554$ GeV as well as at $E_{\gamma}=10.04$ and $E_{\gamma}=10.12$ GeV.
The strengths of these four peaks reach a value $\sim$ 0.1--0.2 nb. Whereas, the non-resonant contribution
in the resonance region is of about 1 nb. As a result, the combined total cross section of the reaction
${\gamma}p \to {J/\psi}p$ has no distinct peak structures, corresponding to the $P_{ci}^+$ states,
and it is practically not distinguished from that for the background reaction. If
$Br[P_{ci}^+ \to {J/\psi}p]=0.25$ and 0.5\%, then the resonant $J/\psi$ yield will be even more less than the
non-resonant one. This means that will be very hard to measure the $P_{ci}^+$ pentaquark states in
$J/\psi$ total photoproduction cross section on a proton target in the near-threshold energy region.
Evidently, to see their experimentally one needs to consider such observable, which is appreciably
sensitive to the $P_c^+$ signal in some region of the available phase space.
For example, the large $t$ region of the differential cross section $d\sigma/dt$
in the $J/\psi$-007 experiment [17], where the $t$-dependence of the background $J/\psi$ meson production
is suppressed while its resonant production is rather flat. This is also supported by the findings of Ref. [25],
where the photoproduction of initially claimed by the LHCb Collaboration hidden charm pentaquark states
$P_{c}^{+}(4380)$ and $P_{c}^{+}(4450)$ with the spin-parity assignments of $(3/2^-,5/2^+)$ or $(3/2^+,5/2^-)$,
respectively, on the proton target was considered by including the $t$-channel diffractive Pomeron exchanges
and the $s$-channel pentaquark productions. Here, by assuming that the pentaquark states decay into the ${J/\psi}p$
mode with fraction of 5\% was, in particular, shown that the contributions from the $P_c^+$ states calculated at
resonant c.m. energies $W=4.38$ GeV and 4.45 GeV for the two spin-parity combinations considered make the differential
cross section of the ${\gamma}p \to {J/\psi}p$ reaction strongly deviated from the diffractive one at off-forward
angles in the c.m. frame. This cross section indeed is rather flat at these angles and overestimates at them
significantly the contributions from the diffractive Pomeron exchanges. Furthermore, the predictions for the
differential cross section of the ${\gamma}p \to {J/\psi}p$ reaction, obtained in Ref. [26] within the approach
in which the Pomeron-exchange model with the parameters determined from fitting the available total cross section
data up to $W=300$ GeV is used to calculate the non-resonant amplitudes as well as the partial decay widths of
nucleon resonances with hidden charm, $N^*_{c{\bar c}}$, predicted by the considered meson-baryons ($MB$)
coupled-channel models to estimate the $N^*_{c{\bar c}} \to MB$ transition matrix elements and the vector-meson
dominance model to evaluate ${\gamma}p \to N^*_{c{\bar c}}$ as ${\gamma}p \to Vp \to N^*_{c{\bar c}}$ with
$V=\rho, \omega, J/\psi$ are adopted, demonstrate that the $N^*_{c{\bar c}}$ can be readily identified in the
near-threshold differential cross section of the ${\gamma}p \to {J/\psi}p$ process at large angles where the
contribution from Pomeron exchanges becomes insignificant. It should also be noted that an earlier prediction
of the differential cross section of this process, made in Ref. [27] at the resonant energy point $W=4.412$ GeV
by considering the non-resonant (${\gamma}p \to {J/\psi}p$) and resonant
(${\gamma}p \to N^*_{c{\bar c}}(4412) \to {J/\psi}p$) ${J/\psi}p$ photoproduction using, respectively, the two gluon
and three gluon exchange model [14] and vector-meson dominance model to generate vector mesons $\rho, \omega, J/\psi$
from photon which rescatter with the target proton to form intermediate hidden charmed nucleon resonance
$N^*_{c{\bar c}}(4412)$, shows as well that this cross section is quite weakly dependent on the c.m.s. $J/\psi$
production angle. It should be additionally pointed out that the feasibility of detecting the $P_{c}^{+}(4450)$
resonance with the spin-parity quantum numbers $J^P=3/2^-$ and $J^P=5/2^+$ in near-threshold $J/\psi$
photoproduction off protons in the CLAS12 experiment at JLab was also discussed in Ref. [23] in the framework
of a two-component model containing the directly produced resonance, diffractive background and accounting
for the experimental resolution effects. The contribution of the $P_{c}^{+}(4450)$ state, produced through
the vector-meson dominance mechanism, was parametrized using the Breit-Wigner ansatz and the non-resonant
contribution was described by the Pomeron exchanges. The fit of the available at that time data points for
differential cross section of the ${\gamma}p \to {J/\psi}p$ reaction, with $|t| \le~1.5$ GeV$^2$, covering
energy range from threshold to $E_{\gamma} \sim~120$ TeV in the lab frame, with this model showed that the upper
limits for branching ratio $Br[P_{c}^{+}(4450) \to {J/\psi}p]$ of the $P_{c}^{+}(4450)$ pentaquark range from
23\% to 30\% for $J=3/2$, depending on the experimental resolution, and from 8\% to 17\% for $J=5/2$.
These are essentially larger than those of several percent set later on by the GlueX Collaboration [4].
Finally, it is worth noting that the photoproduction of the $J/\psi$ off the proton near threshold was
studied in Ref. [28] using a novel final ${J/\psi}p$ production mechanism via the open charm $\Lambda_c^+{\bar D}^0$
and $\Lambda_c^+{\bar D}^{*0}$ intermediate states. The authors found that the existing experimental data [4]
on ${\gamma}p \to {J/\psi}p$ can be well described within the suggested mechanism. Moreover, they identified
a clear experimental signature for this mechanism: within it must be pronounced cusps at the
$\Lambda_c^+{\bar D}^0$ and $\Lambda_c^+{\bar D}^{*0}$ thresholds in the energy dependence of the total cross
section of the ${\gamma}p \to {J/\psi}p$ reaction, and found that the data [4] consistent with this feature
within their accuracy. One may hope that further measurements of the $J/\psi$ photoproduction off the proton at JLab
with higher statistics than GlueX will provide a deeper understanding of the $J/\psi$ photoproduction mechanism.
Taking into account the aforementioned, now we consider the $J/\psi$ energy distribution
from the considered ${\gamma}p \to {J/\psi}p$ elementary reaction.
The model developed by us allows to calculate the direct non-resonant $J/\psi$ energy distribution
from this reaction, the resonant ones from the production/decay chains (41)/(43),
proceeding on the free target proton being at rest.
They were calculated according to Eqs. (39), (62), respectively, for incident
photon resonant energies of 9.44, 9.554, 10.04 and 10.12 GeV.
The resonant $J/\psi$ energy distributions were determined for the considered spin-parity assignments
of the $P_{c}^+(4312)$, $P_{c}^+(4337)$, $P_{c}^+(4440)$, $P_{c}^+(4457)$ resonances for branching fractions
$Br[P_{ci}^+ \to {J/\psi}p]=$~0.25\% for all four states.
These dependencies, together with the incoherent sum of the non-resonant $J/\psi$ energy distribution
and resonant ones,
calculated assuming that all the resonances $P_{c}^+(4312)$ and $P_{ci}^+$ ($i=1$, 2, 3, 4),
$P_{c}^+(4337)$ and $P_{ci}^+$ ($i=1$, 2, 3, 4), $P_{c}^+(4440)$ and $P_{ci}^+$ ($i=1$, 2, 3, 4),
$P_{c}^+(4457)$ and $P_{ci}^+$ ($i=1$, 2, 3, 4) decay to the ${J/\psi}p$ mode
with three adopted options for the branching ratios $Br[P_{ci}^+ \to {J/\psi}p]$,
as functions of the $J/\psi$ total energy $E_{J/\psi}$ are shown, respectively, in Figs. 5, 6, 7, 8.
It is seen from these figures that the resonant $J/\psi$ production cross sections show a flat behavior
at all allowed energies $E_{J/\psi}$. Whereas the non-resonant cross section drops fastly as $E_{J/\psi}$
decreases. At incident photon resonant energies of 9.44, 9.554, 10.04 and 10.12 GeV of interest
its strength is essentially larger than those of the resonant $J/\psi$ production cross sections,
calculated for the value of the branching ratios $Br[P_{ci}^+ \to {J/\psi}p]=$~0.25\%
for "high" allowed $J/\psi$ total energies greater than $\approx$~7.25 GeV. Whereas at "low" $J/\psi$
total energies (below 7.25 GeV) and for each considered photon energy the contribution from the resonance
with the centroid at this energy, decaying to the ${J/\psi}p$ with the branching ratio of 0.25\%,
is much larger than the non-resonant one.
Thus, for instance, in this case for the $J/\psi$ mesons with total energy of 6.5 GeV
their resonant production cross section is enhanced compared
to the non-resonant one by sizeable factors of about 2.9, 3.6, 9.5 and 22.5 at initial photon energies of
9.44, 9.554, 10.04 and 10.12 GeV, respectively. Moreover, this contribution is also substantially larger
than those, arising from the decays of another three pentaquarks to the ${J/\psi}p$ channel with the
branching ratios $Br[P_{ci}^+ \to {J/\psi}p]=$~0.25\%, at the above-mentioned "low" $J/\psi$ total energies.
As a result, at each considered photon energy the $J/\psi$ meson combined energy distribution, deriving from the
direct $J/\psi$ meson production and from the decay of the pentaquark resonance located at this energy
to the ${J/\psi}p$ mode, reveals here a clear sensitivity to the adopted variations in the branching ratio
of this decay. Thus, for example, for the $J/\psi$ mesons with total energy of 6.5 GeV
and for the lowest considered incident photon energy of 9.44 GeV this $J/\psi$ combined distribution is enhanced
for the values of this ratio of 0.25, 0.5 and 1\% by notable factors of about 4.0, 12.5 and 46.8, respectively,
as compared to that from the directly produced $J/\psi$ mesons. And for the highest initial photon energy
of 10.12 GeV of our interest, at which the resonance $P_{c}^+(4457)$ appears as peak structure in the total
cross section of the exclusive reaction ${\gamma}p \to {J/\psi}p$, the analogous factors become much larger and
they are of about 23.5, 90.8 and 360.3, respectively. Furthermore, one can see that the above "partial"
combined energy distribution of the $J/\psi$ mesons is practically indistinguishable from their "total" combined
differential energy distribution, arising from the direct and resonant $J/\psi$ meson production via the
production/decay chains (41)/(43). This implies, on the one hand, that
the differences between the combined results, obtained by using a conservative value of the branching fractions
of the decays $P_{ci}^+ \to {J/\psi}p$ of 0.25\% and the non-resonant background, as well as between
the combined results, determined by employing the values of the branching ratios of these decays of
0.25 and 0.5\%, 0.5 and 1\%, are quite sizeable and experimentally measurable at "low" charmonium total
energies. On the other hand, at each incident
photon resonant energy considered the observation here of the specific hidden-charm LHCb pentaquark will be practically
not influenced by the presence of the another three hidden-charm pentaquark states and by the background reaction.
Since the $J/\psi$ production differential cross sections have a small absolute values $\sim$ 0.01--0.1 nb/GeV
at "low" $J/\psi$ total energies $E_{J/\psi}$, their measurement requires both high luminosities and large-acceptance detectors. Such measurement might be performed in the near future at the JLab in Hall A within the planned here high-statistics ($\sim$ 800k $J/\psi$ events in photoproduction) and high-precision E12-12-006 experiment using the SoLID detector [5, 17].
The momentum dependencies of the absolute non-resonant, resonant and combined $J/\psi$ meson differential
cross sections, correspondingly, from the direct (1), (2), two-step (41)/(43), (42)/(44) and direct plus two-step $J/\psi$ production processes in $\gamma$$^{12}$C and $\gamma$$^{184}$W interactions, calculated on the basis of
Eqs. (24), (61) for laboratory polar angles of 0$^{\circ}$--20$^{\circ}$ and for incident photon lowest resonant energy
of 9.44 GeV, are shown, respectively, in Figs. 9 and 10. The same as in these figures, but for initial highest photon
resonant energy of 10.12, is presented in Figs. 11 and 12. The resonant momentum differential cross
sections for the production of $J/\psi$ mesons in the two-step processes
${\gamma}p \to P_{ci}^+ \to {J/\psi}p$ and ${\gamma}n \to P_{ci}^0 \to {J/\psi}n$ ($i=1$, 2, 3, 4),
proceeding on the intranuclear nucleons of carbon and tungsten target nuclei, were obtained
for three employed values of the branching ratios $Br[P_{ci}^+ \to {J/\psi}p]$ and $Br[P_{ci}^0 \to {J/\psi}n]$.
It can be seen from these figures that the total contribution to the $J/\psi$ production on both these nuclei, coming
from the intermediate $P_{ci}^+$ and $P_{ci}^0$ states decaying to the ${J/\psi}p$ and ${J/\psi}n$
modes with branching fractions of 0.25\%, shows practically flat behavior, and it is significantly larger than that from the background processes (1), (2) in the "low"-momentum regions of 4.5--5.5 GeV/c
and 4.5--6 GeV/c for considered photon beam energies of 9.44 and 10.12 GeV, respectively.
As a result, in them the combined charmonium yield is completely governed by the presence of the $P_{ci}^+$ and $P_{ci}^0$ states in its production. Its strength is almost totally determined by the branching ratios $Br[P_{ci}^+ \to {J/\psi}p]$ and $Br[P_{ci}^0 \to {J/\psi}n]$ used in the calculations with a value, which is still large enough to be measured, as one may hope, at the CEBAF (cf. [13]), and which increases by a factor of about ten for both photon beam energies considered when going from carbon target nucleus to tungsten one
\footnote{$^)$It is interesting to note that the photoproduction of $J/\psi$-$^3$He bound state ([$^3$He]$_{J/\psi}$)
on a $^4$He target has been investigated in Ref. [29] using the impulse approximation, several
$\gamma+N \to J/\psi+N$ models based on the Pomeron-exchange and accounting for the pion-exchange mechanism
at low energies, and various $J/\psi$-nucleus potentials. The upper boundary of the predicted total cross
sections was found to be very small -- it is about 0.1--0.3 pb. The possibility of photoproduction of a six
quark-$J/\psi$ bound state ([$q^6$]$_{J/\psi}$) on the $^3$He target has been studied in Ref. [29] as
well. The upper boundary of the predicted total cross sections of $\gamma+^3{\rm He} \to [q^6]_{J/\psi}+N$
was obtained to be slightly larger than in the preceding case -- it is about 2--4 pb, depending on the model
of $\gamma+N \to J/\psi+N$ used in the calculations. These predictions may facilitate the planning of possible
measurements of [$^3$He]$_{J/\psi}$ and [$q^6$]$_{J/\psi}$ bound states at JLab.}$^)$
.
This leads to the well separated and experimentally distinguishable
differences between all combined calculations, corresponding to the adopted options for these ratios,
for both target nuclei and for both photon energies considered.
Since the $J/\psi$ meson production differential cross sections at photon beam energy of 9.44 GeV are larger
than those at the energy of 10.12 GeV by a factor of about 20 in the above "low"-momentum regions, their measurements
on light and especially on heavy nuclear targets at photon energies in the "low"-energy resonance region will open
an opportunity to determine accurately the above branching ratios -- at least to distinguish between their realistic
options of 0.25, 0.5 and 1\%. Such measurements could also be performed in the future at the JLab in the framework
of the proposed here E12-12-006 experiment [5, 17].
Accounting for the above considerations, we can conclude that the near-threshold $J/\psi$
energy and momentum distribution measurements in photon-induced reactions both on protons
and on nuclear targets will provide further evidence for the existence of the pentaquark
$P_{ci}^+$, $P_{ci}^0$ resonances, and will shed light on their decay rates to the channels ${J/\psi}p$ and ${J/\psi}n$.
\section*{4. Epilogue}
In this paper we studied the near-threshold $J/\psi$ meson photoproduction from protons and nuclei
by considering incoherent direct non-resonant
(${\gamma}p \to {J/\psi}p$, ${\gamma}n \to {J/\psi}n$) and two-step resonant
(${\gamma}p \to P_{ci}^+ \to {J/\psi}p$, ${\gamma}n \to P_{ci}^0 \to {J/\psi}n$, $i=1$, 2, 3, 4;
$P_{c1}^{+,0}=P_c^{+,0}(4312)$, $P_{c2}^{+,0}=P_c^{+,0}(4337)$, $P_{c3}^{+,0}=P_c^{+,0}(4440)$,
$P_{c4}^{+,0}=P_c^{+,0}(4457)$) charmonium production processes.
We have calculated the absolute excitation functions, energy and momentum distributions
for the non-resonant, resonant and for the combined (non-resonant plus resonant) production
of $J/\psi$ mesons on protons as well as, using the nuclear spectral function approach,
on carbon and tungsten target nuclei at near-threshold incident photon energies by assuming
the spin-parity assignments of the hidden-charm resonances $P_{c}^{+,0}(4312)$,
$P_{c}^{+,0}(4337)$, $P_{c}^{+,0}(4440)$ and $P_{c}^{+,0}(4457)$ as $J^P=(1/2)^-$, $J^P=(1/2)^-$, $J^P=(1/2)^-$
and $J^P=(3/2)^-$ within three different realistic scenarios for the branching ratios
of their decays to the ${J/\psi}p$ and ${J/\psi}n$ modes (0.25, 0.5 and 1\%).
It was shown that will be very hard to measure the $P_{ci}^+$ pentaquark states through the scan of the $J/\psi$
total photoproduction cross section on a proton target in the near-threshold energy region around the resonant
photon energies of 9.44, 9.554, 10.04 and 10.12 GeV if these branching ratios $\sim$ 1\% and less.
It was also demonstrated that at these photon beam energies the $J/\psi$ energy and momentum combined distributions considered reveal distinct sensitivity to the above scenarios, respectively, at "low" $J/\psi$ total energies and momenta, which implies that they may be an important tool to provide further evidence for the existence of the pentaquark $P_{ci}^+$ and $P_{ci}^0$ resonances and to get valuable information on their decay rates to the ${J/\psi}p$
and ${J/\psi}n$ final states. The measurements of these distributions could be performed in the near future
at the JLab in Hall A within the planned here high-statistics ($\sim$ 800k $J/\psi$ events in photoproduction) and high-precision E12-12-006 experiment using the SoLID detector.
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\section{Introduction}
\subsection{Motivation}
Instantons play a key role in the nonperturbative dynamics of
Yang-Mills theories, and indeed in a wide range of quantum mechanical
systems. One useful property of instantons is that they can allow a
semiclassical description where a full treatment is either difficult
or even ill defined, as in the case of gravity. At the other extreme,
in supersymmetric theories instantons are crucial in obtaining
exact results.
Within the programme of Euclidean Quantum Gravity,
multicentred gravitational instantons followed hotly on the tails of
their Yang-Mills counterparts \cite{Hawking:1976jb,
Gibbons:1979zt}. However, while the Gibbons-Hawking metrics have found a
surprising range of physical applications, their dynamical role within
quantum gravity remains unclear. One reason for this is that if the
instanton contains more than one centre, it is no longer
Asymptotically Euclidean ($\sim {{\Bbb R}}^4$) or Asymptotically Flat ($\sim
{{\Bbb R}}^3 \times S^1$). These are the
most natural asymptotics for infinite volume quantum gravity at
zero or finite temperature, respectively. In contrast, at constant
large radius the multicentred Gibbons-Hawking spaces tend to $S^1$
fibred over $S^2$ with increasingly high Chern number.
Said differently, the boundary conditions determine the
gravitational instanton topology. There is no sum over different
spacetime topologies for a fixed asymptotics. In this sense, the
Gibbons-Hawking spaces do not provide a semiclassical realisation
of spacetime foam.
It is therefore of interest to study gravitational theories in
which arbitrarily high instanton number is allowed with fixed
asymptotics. One example of such a theory is conformal gravity, in
which the Einstein-Hilbert term is replaced by the Weyl curvature
squared \cite{Strominger:1984zy,L91,Hartnoll:2004rv}. Despite some
rather attractive features of the gravitational instantons in this
theory, the physical status of the theory itself is uncertain due
to problems with higher derivative Lagrangians and unitarity.
In this paper we emphasise that Einstein-Maxwell theory also
admits regular multicentred instantons with arbitrarily
complicated topology for fixed asymptotics. These solutions have
essentially appeared before in the literature
\cite{Whitt:1984wk,Yuille:1987vw}. Various unsatisfactory
aspects of these previous treatments, for instance we have
preferred to use a Riemmanian Maxwell field that is real, have
lead us to carry out a systematic study {\it de novo}. We
furthermore extend our understanding of Einstein-Maxwell
gravitational instantons through discussions of uniqueness,
supersymmetry, moduli space metrics and lifts to five dimensions.
This final point may be of independent interest.
\subsection{Summary}
In Section 2 we present the instanton solutions. We detail the
possible asymptotics: ${{\Bbb R}}^4, {{\Bbb R}}^3 \times S^1$ and $AdS_2 \times S^2$,
and local versions thereof. We show that the solutions are half-BPS
when embedded into minimal ${\mathcal N} = 2$ supergravity and that they are
all the regular Riemannian half-BPS solutions. Finally, we evaluate
the action of the solutions. The Asymptotically Euclidean case is
found to only be well defined when a certain linear combination of the
charge and potential is fixed at infinity.
In section 3 we discuss the moduli space metric on the
Einstein-Maxwell instantons. We consider in some detail the
ambiguities involved in finding an inner product on the space of
metric fields. We show that there is a preferred inner product
which is inherited from the action and for which zero modes are
orthogonal to pure gauge modes.
Section 4 shows how the four dimensional instantons may be lifted
to solitons of five dimensional Einstein-Maxwell-Chern-Simons
theory, or minimal ${\mathcal N} = 2$ supergravity in five
dimensions. Generically the lifted solutions are either singular
or contain closed timelike curves. However, we find that one class
of solutions lift to regular, causal plane fronted wave spacetimes
with the fields localised in lumps orthogonal to the wave
propation. We call these `solitonic strings' as they do not have
an event horizon.
Section 5 briefly discusses the slow motion of the five
dimensional solitons. Unlike in the case of the Gibbons-Hawking
instantons and their lift to Kaluza-Klein monopoles, it seems that
there is not a direct connection between the four dimensional
instanton moduli space metric and the five dimensional soliton
slow motion moduli space metric in our case.
We end with a discussion of possible physical applications of
these multicentred Einstein-Maxwell instantons, and directions for
future work.
\section{The gravitational instantons}
\subsection{The solutions}
\label{sec:metric}
The gravitational instantons on a four dimensional manifold $M_4$
are solutions to the Einstein-Maxwell equations with Riemannian
signature
\begin{eqnarray}\label{eq:4deqns}
G_{a b} & = & 2 F_a{}^c F_{b c} - \frac{1}{2} g_{a b} F^{c d} F_{c d} \,, \\ \nonumber
\nabla_a F^{a b} & = & 0 \,.
\end{eqnarray}
The metric is given by
\begin{equation}
\label{IWmetric}
g^{(4)} = \frac{1}{U {\widetilde U}} (d\t + {\bf \w})^2 + U {\widetilde U} d{\bf x}^2
\,,
\end{equation}
where the functions $U,{\widetilde U}$ and the one form $\w$ depend on ${\bf
x}=(x, y, z)$ and satisfy
\begin{eqnarray}
\label{IWequations}
\nabla^2 U = \nabla^2 {\widetilde U} = 0 \,, \nonumber \\
\nabla \times \w = {\widetilde U} \nabla U - U \nabla {\widetilde U} \,.
\end{eqnarray}
We will work with four dimensional tangent space indices, $a,b,...$ and the vierbeins
\begin{equation}
e^4 = \frac{1}{(U {\widetilde U})^{1/2}} (d\t + {\bf \w}) \,, \qquad e^{i} =
(U {\widetilde U})^{1/2} dx^i \,.
\end{equation}
The electromagnetic field strength may now be written
\begin{eqnarray}\label{eq:fieldstrength}
F_{4 i} & = & \frac{1}{2} \pa_i \left[U^{-1} - {\widetilde U}^{-1}
\right] \,, \nonumber \\
F_{i j} & = & \frac{1}{2} \e_{ijk} \pa_k \left[U^{-1} + {\widetilde U}^{-1}
\right] \,,
\end{eqnarray}
where the derivatives are partial derivatives with respect to the
corresponding spacetime indices. One can check that this field
strength satisfies the Bianchi identities, and thus locally at least
we can write $F=dA$. Our expressions for the
field strength in Riemannian signature differ slightly from others in
the literature \cite{Whitt:1984wk,Yuille:1987vw} which were not real.
In particular the Riemannian Majumdar-Papapetrou metrics with
$U={\widetilde U}$ have purely magnetic field strength $F=-2\star_3 dU$.
These backgrounds were first found in the Lorentzian regime by Israel
and Wilson \cite{israelwilson} and by Perj\'es \cite{perjes} as a
stationary generalisation of the static Majumdar-Papapetrou multi
black hole solutions. However, it was shown by Hartle and Hawking that
all the non static solutions suffered from naked singularities
\cite{hartlehawking,Chrusciel:2005ve}.
With Riemannian signature however, regular solutions exist
\cite{Whitt:1984wk,Yuille:1987vw}. We can take
\begin{equation}\label{eq:sumpoles}
U = \frac{4\pi}{\b} + \sum_{m=1}^N \frac{a_m}{\mid {\bf x} - {\bf
x}_m \mid} \,,
\qquad {\widetilde U} = \frac{4\pi}{{\tilde \beta}} + \sum_{n=1}^{\widetilde N} \frac{{\tilde a}_n}{\mid {\bf x} -
{\bf {\tilde x}}_n \mid} \,,
\end{equation}
in these expressions $\b,{\tilde \beta},a_m,{\bf x}_m,{\tilde a}_n,{\bf
{\tilde x}}_n,N,{\widetilde N}$ are constants. For the signature to remain
$(+,+,+,+)$ throughout we can require $U,{\widetilde U} > 0$ which in turn
requires $a_m,{\tilde a}_n > 0$. From the explicit forms for $U$ and
${\widetilde U}$ in (\ref{eq:sumpoles}) we can write down explicit
expressions for the one forms ${\bf \w}$ and $A$, which so
far we have only defined implicitly. These are given in Appendix
A.
If there is at least one non coincident centre, ${\bf x}_m \neq
{\bf {\tilde x}}_n$, regularity requires that $\t$ is identified with
period $4\pi$ and that the constants satisfy the following
constraints at all the non-coincident centres
\begin{equation} \label{eq:constraints}
U({\bf {\tilde x}}_n) {\tilde a}_n = 1 \,, \qquad {\widetilde U}({\bf x}_m) a_m = 1 \,, \qquad \forall
m,n \,.
\end{equation}
Given the locations of the centres $\{ {\bf x}_n,{\bf {\tilde x}}_m\}$,
these constraints may be solved uniquely for the $\{a_n,{\tilde a}_m\}$
\cite{Yuille:1987vw}. When $\frac{4\pi}{\b} =
\frac{4\pi}{{\tilde \beta}} = 0$ the solution is only unique up to the overall
scaling
\begin{equation}\label{eq:scaling}
U \to e^{s} U \,, \qquad {\widetilde U} \to e^{-s} {\widetilde U}\,.
\end{equation}
In general this scaling leaves the metric invariant and induces a linear duality
transformation on the Maxwell field mapping solutions to solutions
\begin{equation}
{\bf E} \to \cosh s \, {\bf E} + \sinh s \, {\bf B} \,, \qquad
{\bf B} \to \sinh s \, {\bf E} + \cosh s \, {\bf B} \,.
\end{equation}
The rescaling does not leave the action and other properties of the
solutions invariant.
The constants $\b$ and ${\tilde \beta}$ determine the asymptotics of the
solution. There are three possibilities:
\begin{itemize}
\item The case $\frac{4\pi}{\b} = \frac{4\pi}{{\tilde \beta}} \neq 0$
gives an Asymptotically Locally Flat metric, tending to
an $S^1$ bundle over $S^2$ at infinity, with first Chern number $N-{\widetilde N}$.
Without loss of generality we have rescaled the harmonic functions
using (\ref{eq:scaling}) so that $\b = {\tilde \beta}$.
Equations (\ref{eq:constraints}) now imply that $\sum a_m - N = \sum
\tilde{a}_n - {\widetilde N}$. If $N={\widetilde N}$ the asymptotic bundle is trivial
and we obtain Asymptotically Flat ($\sim {{\Bbb R}}^3 \times S^1$) solutions.
\item The case $\frac{4\pi}{\b} = 0$, $\frac{4\pi}{{\tilde \beta}} = 1$
gives an Asymptotically Locally Euclidean metric, tending to ${{\Bbb R}}^4 /
{{\Bbb Z}}_{|N-{\widetilde N}|}$. We have used the rescaling (\ref{eq:scaling})
to set $\frac{4\pi}{{\tilde \beta}} = 1$ without loss of generality.
In this case the constraints (\ref{eq:constraints}) require that $\sum a_m =
N - {\widetilde N}$. Of course we can reverse the roles of $\b$ and ${\tilde \beta}$. If $N
= {\widetilde N} + 1$ the solution is Asymptotically Euclidean ($\sim {{\Bbb R}}^4$).
\item The case $\frac{4\pi}{\b} = \frac{4\pi}{{\tilde \beta}} = 0$ leads to
an Asymptotically Locally Robinson-Bertotti metric, tending to $AdS_2
\times S^2$ or $AdS_2/{{\Bbb Z}} \times S^2$. The former case only arises
if all of the centres are coincident, so that $U = {\widetilde U}$, and $\t$ need
not be made periodic. For both these
asymptotics, the constraints (\ref{eq:constraints}) require that $N =
{\widetilde N}$. We may further use the rescaling (\ref{eq:scaling}) to set
$\sum a_m = \sum \tilde{a}_n$.
\end{itemize}
As Riemannian solutions, the backgrounds are naturally thought of as
generalisations of the Gibbons-Hawking multicentre metrics which in
fact they include as the special case ${\widetilde U}=1$, albeit with an
additional antiselfdual Maxwell field. A crucial new aspect of the
Asymptotically Locally Euclidean (ALE) and Asymptotically Locally Flat (ALF)
Israel-Wilson-Perj\'es solutions is that when
\begin{equation}
N = {\widetilde N} \pm 1 \; \text{(for ALE)} \qquad \text{or} \qquad N = {\widetilde N} \; \text{(for
ALF)} \,,
\end{equation}
the fibration of the $\tau$ circle over $S^2$ at infinity is
trivial and the metrics do not require the ${{\Bbb Z}}_N$ identifications
at infinity that are needed in the Gibbons-Hawking case. The
spacetimes are therefore strictly Asymptotically Euclidean and
Asymptotically Flat respectively in these cases. The Euler number
is given by $\chi = N + {\widetilde N}-1$ in the ALF and ALE cases
\cite{Yuille:1987vw}. Thus the spaces
admit arbitrarily complicated topology, not restricted by the
asymptotic topology, and provide a semiclassical realisation of
spacetime foam in quantum Einstein-Maxwell theory.
The metric (\ref{IWmetric}) has vanishing scalar curvature. If
$U$ or ${\widetilde U}$ is constant then (\ref{IWmetric}) is Ricci flat,
and hyperK\"ahler. It is natural to ask whether any other
special choices of harmonic functions $U$ and ${\widetilde U}$ lead
to scalar flat K\"ahler metrics with a symmetry $\pa/\pa \t$
preserving the K\"ahler structure. Such metrics would be
conformally anti--self--dual and thus interesting in twistor theory.
The answer is negative.
From \cite{L91} any such metric is of the
form
\begin{equation}
g^{(4)}= \frac{1}{\cal W} (d\tau +\omega)^2 + {{\cal W}} h^{(3)}\,,
\end{equation}
where the metric $h^{(3)}$ on the three dimensional orbit
space of $\pa/\pa \t$, and the function ${\cal W}$ on this space
satisfies a coupled nonlinear system of PDEs. In the case that
$h^{(3)}$ is flat the equations reduce to
\begin{equation}
\nabla \times {\bf \w} =\nabla {\cal W}.
\end{equation}
Therefore ${\cal W}=U{\widetilde U}$ is harmonic, and then
(\ref{IWequations}) implies that
${\widetilde U}$ is a constant.
\subsection{Killing spinors}
The solutions have the further important property of admitting two
complex Killing spinors. These satisfy
\begin{equation}\label{eq:susy}
e^{\mu}{}_a \pa_\mu \e + \frac{1}{4} \left[\w^{b c}{}_a\, \G_{b c} + i F^{b
c}\, \G_{b c}\, \G_a \right] \e = 0 \,,
\end{equation}
where $\w^{b c}{}_a$ are the components of the the spin connection
one form $\w^{bc}$
defined by $de^b = \w^{b
c} \wedge e^{c}$. We use Greek letters $\mu,\nu,...$ to denote
Euclidean spacetime indices. Our gamma matrix conventions are
given in Appendix B, as is the spin connection for the background.
With these conventions one may solve the equation (\ref{eq:susy})
to find
\begin{equation}\label{eq:spinor}
\e =
\left( \begin{array}{c}
U^{-1/2} \e_0 \\
i {\widetilde U}^{-1/2} \e_0
\end{array}
\right) \,,
\end{equation}
where $\e_0$ is a constant two-component complex spinor: $\pa_\mu
\e_0 = 0$.
Within Einstein-Maxwell theory, the Killing spinors imply that the
solutions saturate a Bogomolny bound \cite{Gibbons:1982fy}. It is
also natural to view the solutions as half-BPS states of four
dimensional ${\mathcal{N}}=2$ supergravity
\cite{Ferrara:1976fu}. This theory has a complex
spin-$\frac{3}{2}$ Rarita-Schwinger field as well as the graviton
and photon. In fact, in a paper that anticipated current interest in
classifying supersymmetric solutions, Tod has shown that the
Lorentzian version of these solutions are all the supersymmetric
solutions to ${\mathcal{N}}=2$ supergravity with a timelike
Killing spinor \cite{Tod:1983pm}.
In the following subsection we shall repeat Tod's analysis in the
Riemannian case. As well as recovering the local form of the
metric, it will find that $|\nabla U^{-1}|$ and $|\nabla
{\widetilde U}^{-1}|$ are both bounded\footnote{This is stronger than
the Lorentzian result of \cite{Chrusciel:2005ve} where the separate
bounds cannot be established.}. Combined with a
result from analysis \cite{chrusciel_nad}, it will follow that (\ref{IWmetric}) together
with (\ref{eq:sumpoles}) is the most general regular supersymmetric
solution to minimal ${\mathcal{N}}=2$ supergravity. To put it differently, only
harmonic functions with a finite number of point sources lead to regular
metrics.
As usual, given Killing spinors $\e$ and $\eta$ we can build differential forms. In
particular, we have the one forms
\begin{equation}
V = \frac{1}{2} \bar \eta \G_a \e \, e^a \,, \qquad K = \frac{1}{2}
\bar \eta \G_5 \G_a \e \, e^a \,,
\end{equation}
and the two form
\begin{equation}
\Omega = - \frac{i}{2} \bar \eta \G_{ab} \e \, e^a \wedge e^b \,.
\end{equation}
In our representation of the Clifford algebra, given in the
appendix, all the gamma matrices are hermitian and therefore bar
simply denotes complex conjugation. From the Killing spinor
condition (\ref{eq:susy}) we have that
\begin{equation}
d \Omega = - 2 V \wedge F \,, \qquad d V = 0 \,, \qquad \nabla_{(a}
K_{b)} = 0 \,.
\end{equation}
With a little more work one can also show that
\begin{eqnarray}\label{eq:domega}
\nabla_a \Omega_{bc} & = & 2 V_a F_{bc} - 4 F_{a [b} V_{c]} + 4 V^d
F_{d [b} g_{c] a} \,, \nonumber \\
\nabla_a V_b & = & \frac{1}{4} F^{cd} \Omega_{cd} g_{ab} + F^{c}{}_{(a}
\Omega_{b) c} \,.
\end{eqnarray}
In fact there is more structure. The two form $\Omega$ can be split
into self dual and anti-self dual parts: $\Omega = \Omega^+ +
\Omega^-$. One can then show that $\Omega^+$ and $\Omega^-$ separately
satisfy the first equation in (\ref{eq:domega}) with $F$ replaced
by its self dual, $F^+$, and anti-self dual, $F^-$, parts
respectively.
Three important cases giving real forms are when $\eta = \e = \e^I$, for $I =
1,2,3$, which are defined by $\e_0$ in
(\ref{eq:spinor}) satisfying $\bar \e_0^I \t^J \e_0^I = \delta^{I
J}$ and $\bar \e_0^I \e_0^J = \delta^{I J}$. For these cases we find
\begin{equation}
V^I = d x^I \,, \qquad K = \frac{1}{U {\widetilde U}} \left(d\t + {\bf \w} \right) \,,
\end{equation}
and
\begin{equation}
\label{OmegaI}
\Omega^I = \left(U^{-1} - {\tilde U}^{-1} \right) e^4 \wedge
e^I + \frac{1}{2} \left(U^{-1} + {\tilde U}^{-1} \right)
\epsilon^{I j k} e^j \wedge e^k \,.
\end{equation}
Raising the index, the Killing vector is $K =
\pa/\pa \t$ as we should expect.
\subsection{Uniqueness of the solutions}
\label{killing_sp}
Here we show that the solution (\ref{IWmetric}),
(\ref{eq:fieldstrength}) with the harmonic functions described by
(\ref{eq:sumpoles}) and satisfying the constraints
(\ref{eq:constraints}) is the most general regular Einstein-Maxwell
instanton with a complex Killing spinor.
In this section it will be convenient to write the Dirac spinor
${\varepsilon}=(\alpha^A, \beta_{A'})$ as a pair of complex two-component
spinors. When dealing with these spinors we use the conventions
given in Appendix C. With positive signature, spinor conjugation
preserves the type of spinors. Thus if $\alpha_{A}=(p, q)$ we can
define $\hat{\alpha}_A=(\overline{q}, -\overline{p})$ so that
${\hat{\hat{\alpha}}}_A=-\alpha_A$. This hermitian conjugation
induces a positive inner product
\begin{equation}
\alpha_A\hat{\alpha}^A={\epsilon}_{AB}\alpha^B\hat{\alpha}^A=|p|^2+|q|^2 \,.
\end{equation}
We define the inner product on the primed spinors in the same way.
Here ${\epsilon}_{AB}$ and ${\epsilon}_{A'B'}$ are covariantly constant symplectic
forms with ${\epsilon}_{01}={\epsilon}_{0'1'}=1$. These are used to raise and lower spinor indices
according to $\alpha_B={\epsilon}_{AB}\alpha^A, \alpha^B={\epsilon}^{BA}\alpha_A$, and
similarly for primed spinors. In terms of our gamma matrices, $\hat \e
= \Gamma^{31} \bar \e$.
The Killing spinor equation (\ref{eq:susy}) becomes
\begin{equation}
\label{spinor_cond}
\nabla_{AA'}\alpha_B-i \sqrt{2}\phi_{AB}\beta_{A'}=0\,,
\qquad
\nabla_{AA'}\beta_{B'}+i\sqrt{2}\tilde{\phi}_{A'B'}\alpha_{A}=0\,,
\end{equation}
where the spinors $\phi$ and $\tilde{\phi}$ are symmetric in their respective
indices and give the anti-self dual and self dual parts of the electromagnetic
field
\begin{equation}
F_{ab}=\phi_{AB}{\epsilon}_{A'B'}+\tilde{\phi}_{A'B'}{\epsilon}_{AB}\,.
\end{equation}
Suppose that $\e = (\alpha^A, \beta_{A'})$ solves the Killing spinor
equation (\ref{spinor_cond}). Now we can reconstruct the spacetime
metric and Maxwell field.
Define
\begin{equation}
\label{definitions_UU}
U=(\alpha_A\hat{\alpha}^A)^{-1}, \qquad
{\widetilde U}=(\beta_{A'}\hat{\beta}^{A'})^{-1}.
\end{equation}
In our positive definite case, these two inverted functions do not
vanish unless $\a$ or $\b$ vanish.
In the Lorentzian case their possible vanishing leads
to plane wave spacetimes \cite{Tod:1983pm}. If $\a$ or $\b$ vanish
identically, we recover the Gibbons-Hawking solutions.
Now define a (complex) null tetrad
\begin{equation}
X_a=\alpha_A\beta_{A'}, \qquad \overline{X}_a=\hat{\alpha}_A\hat{\beta}_{A'},
\qquad Y_a=\alpha_A\hat{\beta}_{A'},\qquad \overline{Y}_a=-\hat{\alpha}_A{\beta}_{A'}\,.
\end{equation}
We can check that $\hat \e$ is also a solution to the Killing spinor
equation (\ref{spinor_cond}).
It therefore follows from (\ref{spinor_cond}) that $X_a, \overline{X}_a,
Y_a-\overline{Y}_a$ are gradients and that
$K_a= Y_a+\overline{Y}_a$ is a Killing vector.
Now define local coordinates
$(x, y, z, \tau)$ by
\begin{equation}
X=\frac{1}{\sqrt{2}}(dx+idy), \qquad (Y - \overline{Y})=
i \sqrt{2}dz, \qquad K^{a}\nabla_a=\sqrt{2}\frac{\partial}{\partial\tau}\,,
\end{equation}
where the form $X = X_a e^a = X_{A A'} e^{A A'}$ and similarly for
$Y,\overline{Y}$. The vector $K$ Lie derives
the spinors $(\alpha_A, \beta_{A'})$, implying that $U$ and ${\widetilde U}$ are
independent of $\t$.
The metric is now given by $ds^2 = {\epsilon}_{A B} {\epsilon}_{A' B'} e^{A A'} e^{B
B'}$. This expression may be evaluated by noting that from
(\ref{definitions_UU}) we have ${\epsilon}_{AB} = U (\a_A \hat \a_B - \a_B
\hat \a_A)$ and similarly for ${\epsilon}_{A'B'}$. Using the fact that from
the above definitions $K_aK^a=2(U{\widetilde U})^{-1}$, we find that the metric takes the form
(\ref{IWmetric}) for some one form ${\bf \omega}$. The next step is to
find ${\bf \omega}$.
The definitions of $U,{\widetilde U}$ and $K$ together with (\ref{spinor_cond}) imply
\begin{equation}
\label{Killing_relation}
\nabla_a K_b= i \sqrt{2} \left[ {{\widetilde U}}^{-1}\phi_{AB}{\epsilon}_{A'B'} +
U^{-1}\tilde{\phi}_{A'B'}{\epsilon}_{AB} \right] \,,
\end{equation}
and
\begin{equation}
\label{U_phi_relations}
\nabla_a U^{-1}= i\sqrt{2}\phi_{AB}K^{B}_{A'}, \qquad
\nabla_a {\widetilde U}^{-1}= -i\sqrt{2}\tilde{\phi}_{A'B'}K_{A}^{B'}\,.
\end{equation}
The formulae in (\ref{U_phi_relations}) may be inverted to find
expressions for $\phi_{AB}$ and $\tilde \phi_{A'B'}$, using $K_B^{A'} K^{B C'}
= \frac{1}{2} {\epsilon}^{A' C'} K_{D E'} K^{D E'}$. Substituting the result into
(\ref{Killing_relation}) yields the expression (\ref{IWequations}) for
$\nabla\times {\bf \omega}$.
Finally, differentiating the relations
(\ref{spinor_cond}) shows that the energy momentum tensor
is that of Einstein--Maxwell theory:
$T_{ab}=2\phi_{AB}\tilde{\phi}_{A'B'}$. The Maxwell equations
\begin{equation}
\nabla^{AA'}\phi_{AB}=0, \qquad \nabla^{AA'}\phi_{A'B'}=0 \,,
\end{equation}
now imply that $U$ and ${\widetilde U}$ are harmonic on ${{\Bbb R}}^3$. This completes
the local reconstruction of the solution from the Killing spinors.
So far everything has proceeded as in \cite{Tod:1983pm} with minor differences
in the reality conditions. The main difference arises in global regularity
considerations which lead us to consider the invariant
\begin{eqnarray}
F_{ab}F^{ab}&=&2(\phi_{AB}\phi^{AB}+\tilde{\phi}_{A'B'}\tilde{\phi}^{A'B'})
\nonumber\\
&=& |\nabla U^{-1}|^2+|\nabla {\widetilde U}^{-1}|^2,
\end{eqnarray}
where the norm of the gradients is taken with respect to the flat
metric on ${{\Bbb R}}^3$, and we have used (\ref{U_phi_relations}).
Regularity requires this invariant
be bounded. Therefore both $|\nabla U^{-1}|$ and $|\nabla {\widetilde U}^{-1}|$
must be bounded. The various boundary conditions we have described imply that
$U$ and ${\widetilde U}$ are regular as $|\bf x|\rightarrow \infty$. In particular,
they are both regular outside a ball $B_R$ of sufficiently large
radius $R$ in ${{\Bbb R}}^3$.
The coordinates $\{ {\bf x}, \tau\}$ cover ${{\Bbb R}}\times
({{\Bbb R}}^3 \setminus {\cal S})$, where ${\cal S}$ is the compact subset of $B_R$ on which
$U$ or ${\widetilde U}$ blow up.
A theorem from \cite{chrusciel_nad} can now be applied separately to
both harmonic functions to prove that ${\cal S}$ consists of a finite
number of points. In fact
\begin{equation}
\#{\cal S}<\mbox{max}\{|\nabla U^{-1}|, |\nabla {\widetilde U}^{-1}|\} |U(p)
+{\widetilde U}(p)|\;R+1,
\end{equation}
where $p$ is any point in $B_R$ which does not belong to ${\cal S}$.
This combined with the maximum principle shows
that (\ref{eq:sumpoles})
are the most general harmonic functions leading to regular metrics.
It also follows from (\ref{definitions_UU}) and the positivity
of the spinor inner product that $a_m$ and $\tilde{a}_n$ in
(\ref{eq:sumpoles}) are all non negative.
The spinors $\alpha_A, \beta_{A'}$ and their conjugates give a preferred
basis for the space $\Lambda^2(M)$ of two forms.
The anti-self dual two forms are given in terms of $\alpha_A$ by
\begin{equation}
\mbox{Re}(\alpha_A\alpha_B\epsilon_{A'B'}), \qquad
\mbox{Im}(\alpha_A\alpha_B\epsilon_{A'B'}), \qquad
i\alpha_{(A}\hat{\alpha}_{B)}\epsilon_{A'B'},
\end{equation}
and the self dual two forms are given in terms of $\beta_{A'}$ by analogous
expressions. The three two forms (\ref{OmegaI}) can be expressed in this
basis as
\begin{eqnarray}
\Omega^1+i\Omega^2&=&
-(\alpha_A\alpha_B\epsilon_{A'B'}+ \beta_{A'}\beta_{B'}\epsilon_{AB})\;
e^{AA'}\wedge e^{BB'},\nonumber\\
\Omega^3&=&
i(\beta_{(A'}\hat{\beta}_{B')}\epsilon_{AB}
-\alpha_{(A}\hat{\alpha}_{B)}\epsilon_{A'B'})\; e^{AA'}\wedge e^{BB'}.
\end{eqnarray}
The spinor expressions for (\ref{eq:domega}) can now be easily derived using
(\ref{spinor_cond}).
\subsection{Action of the instantons}
The contribution of instantons to physical processes is of course
weighted by their actions. Therefore it is important to evaluate
the actions of the spacetimes we are considering. Previous
computations on this subject should be approached with caution:
there are computational errors in
\cite{Whitt:1984wk} leading to unphysical results such as an action
unbounded from below, while in \cite{Yuille:1987vw} the Maxwell
contribution to the action is not considered. Both of these papers
also work with imaginary electric fields which leads to some
undesirable properties of the actions.
The Riemannian Einstein-Maxwell action, including the Gibbons-Hawking
boundary term, is
\begin{equation}\label{eq:plainaction}
S = - \int_{M_4} d^4x \sqrt{g^{(4)}} \left[ R^{(4)} - F_{a b} F^{a b} \right] - 2
\int_{\pa M_4}
d^3x \sqrt{\gamma} {\cal K} \,,
\end{equation}
where $\gamma$ is the induced metric on the boundary and ${\cal K}$ is the
trace of the extrinsic curvature of the boundary.
Evaluated on the Einstein-Maxwell instantons we are considering, one finds
\begin{equation}\label{eq:evalaction}
S = - 2 \pi \lim_{r \to \infty} \int_{S^2} d\Omega^2 r^2 \left[
\frac{(U + {\widetilde U})^2 \pa_r (U {\widetilde U})}{(U {\widetilde U})^2} + \frac{8}{r} \right] \,.
\end{equation}
Here we have introduced spherical polar coordinates $d{\bf x}^2 =
dr^2 + r^2 d\Omega^2$. The expression (\ref{eq:evalaction}) is
divergent and needs to be regularised by substracting off the action
of a reference geometry. This must be done separately for the
Asymptotically Locally
Flat, Euclidean and Robinson-Bertotti cases. We have assumed in
(\ref{eq:evalaction}) that $\t$ is identified with period $4\pi$.
The easiest case is Asymptotic Local Flatness, with $\b = {\tilde \beta} \neq 0$. Here the
background has simply $U = {\widetilde U} = \frac{4\pi}{\b}$, giving flat $S^1 \times
{{\Bbb R}}^3$ and a vanishing Maxwell field. One finds
\begin{equation}
\Delta S_{\text{ALF}} = 8 \pi \beta \left(\sum a_m + \sum {\tilde a}_n \right) \,.
\end{equation}
Recall that furthermore $\sum a_m = \sum {\tilde a}_n + N - {\widetilde N}$ in this
case.
The Asymptotically Locally Robinson-Bertotti case is also straightforward.
Here the background is the Robinson-Bertotti spacetime with $\t$
identified, $AdS_2 / {{\Bbb Z}} \times S^2$, supported by magnetic flux, that
is $U = {\widetilde U} = \frac{\sum a_m}{r} = \frac{\sum
{\tilde a}_n}{r}$. The regularised action turns out to vanish
\begin{equation}
\Delta S_{\text{ALRB}} = 0 \,.
\end{equation}
Now consider the Asymptotically Locally Euclidean case, with
$\frac{4\pi}{\b} = 0$ and $\frac{4\pi}{{\tilde \beta}}=1$. The required
background is Euclidean space with anti self dual Maxwell field, that
is $U = \frac{\sum a_m}{r}$ and ${\widetilde U} = 1$. Subtracting this
background regularises the gravitational action, but it does not
remove all the divergences from the Maxwell action. The divergence
of the regularised action tells us that we have not imposed the
correct boundary conditions for the Maxwell field with these
asymptotics.
The standard action (\ref{eq:plainaction}) is appropriate for fixing
the potential at infinity: $\delta A_a = 0$. Different boundary
conditions may be implemented by adding a boundary term to the
action. To obtain a finite action for ALE asymptotics we need to add a
boundary term that entirely cancels the bulk Maxwell action when
evaluated on solutions. The required term is
\begin{equation}\label{eq:maxwellboundary}
\left. S_{\text{ALE}} \right|_{\text{bdy.}} = 2 \int_{\pa M_4} d^3 x
\sqrt{\gamma} A^a F_{ab} n^b \,,
\end{equation}
where $n^b$ is a unit normal vector to the boundary. The resulting
boundary condition is
\begin{equation}\label{eq:boundarycondition}
A_a \delta (F^{a b} n_b) = \delta A_a F^{a b} n_b
\quad \text{on} \quad \pa M_4 \,.
\end{equation}
Physically this equation corresponds to keeping a certain linear combination of
the charge and potential fixed at infinity.
With the boundary term (\ref{eq:maxwellboundary}) added, the action is
found to be given by
\begin{equation}
\Delta S_{\text{ALE}} = 16 \pi^2 \sum {\tilde a}_n \,.
\end{equation}
At this moment, we do not have a physical understanding of why the
ALE instantons only contribute to processes in which the
particular boundary condition (\ref{eq:boundarycondition}) is
imposed.
\section{Instanton moduli space metric}
The analysis done in section (\ref{killing_sp}) has demonstrated
that the Einstein-Maxwell gravitational instantons with a Killing
spinor have $3(N+{\widetilde N})$ free parameters or moduli. The Euclidean
group in three dimensions can be used to fix six of these, except
in the case when $N + {\widetilde N} = 2$, in which case it only fixes five,
due to the axisymmetry. To obtain the moduli space one should also
quotient by the symmetric group $S_N \times S_{\widetilde N}$ acting on the
centres. Note that fixing the action then adds a further
constraint on the centres in the Asymptotically Locally Flat and
Euclidean cases.
While computation of the measure and metric on the moduli space of
Yang-Mills instantons is by now a highly developed field, the case
of gravitational instantons in four dimensions appears to have
been less systematically treated in the literature. In two
dimensions of course the measure plays a fundamental role in
string theory. Reflecting this state of affairs, we now give a
fairly general exposition of the formalism needed to compute
moduli space metrics for gravitational instantons in pure gravity
and Einstein-Maxwell theory.
\subsection{Inner products}
Let us recall the Yang-Mills procedure, but work with just the
$U(1)$ Maxwell case both for simplicity and because this is what we
need anyhow. One begins by writing down a natural ultralocal inner product
on the space of field perturbations. Strictly speaking it is an inner
product on the tangent bundle to the space of fields
\begin{equation}\label{eq:maxwellinner}
\langle \d A , \d A' \rangle = 2 \int_{M_4} d^4x \sqrt{g} g^{\mu \nu} \d
A_{\mu} \d A'_{\nu} \,.
\end{equation}
In this section it is appropriate to work with spacetime indices
$\mu,\nu\ldots$. One now restricts to considering only
perturbations that are orthogonal to pure gauge transformations.
Thus one requires
\begin{equation}
0 = \langle \d A , d \Omega \rangle = - 2 \int_{M_4} d^4x \sqrt{g}
g^{\mu \nu} \Omega \nabla_\mu \d A_\nu \,,
\end{equation}
for all $\Omega$. Therefore, perturbations must be considered in
Lorenz gauge
\begin{equation}\label{eq:lorentz}
\nabla_\mu \d A^{\mu} = 0 \,.
\end{equation}
Given this gauge, we can note that the inner product
(\ref{eq:maxwellinner}) should be thought of as coming from the
quadratic terms in the action. In particular, this
determines the normalisation. The quadratic action is
\begin{eqnarray}\label{eq:maxwellact}
S^{(2)}_{\d A} & = & 2 \int_{M_4} d^4x \sqrt{g} \left(
\nabla^\mu \d A^{\rho} \nabla_\mu
\d A_{\rho} - \nabla^\mu \d A^{\rho} \nabla_\rho \d A_{\mu} \right) \nonumber \\
& \rightarrow & - 2 \int_{M_4} d^4x \sqrt{g} g^{\rho
\sigma} \d A_{\rho} \nabla^2 \d A_{\sigma} + \text{non-derivative terms}\,.
\end{eqnarray}
Where the arrow denotes imposition of the Lorenz gauge. We can see
that the index structure of the gauge field is now that of the
inner product (\ref{eq:maxwellinner}). That is to say, the term in
the last line of (\ref{eq:maxwellact}) is just $-\langle
\d A, \nabla^2 \d A \rangle$, where $\nabla^2$ should be regarded as
an operator on $M_4$. In this way the inner
product is inherited from the action. The metric on the moduli
space is obtained by restricting the inner product
(\ref{eq:maxwellinner}) to zero modes. To summarise the logic: the
metric on the moduli space is inherited from the quadratic kinetic
terms in the action written in a specific gauge. However, that
gauge must simultaneously imply that field fluctuations are
orthogonal to pure gauge transformations.
We should note at this point that imposing orthogonality to gauge
transformations, with a consequent choice of gauge imposed, is not
completely essential. However, it does greatly simplify instanton
computations and gives a clear physical meaning to the moduli
space metric itself.
For the case of metric fluctuations, there is not a unique ultralocal
inner product with the correct symmetries. Rather we have the
family of de Witt metrics parametrised by $\lambda\in{{\Bbb R}}$
\begin{equation}
\langle \d g , \d g' \rangle_\lambda = \int_{M_4} d^4x G^{\mu \nu \rho
\sigma}_{(\lambda)} \d g_{\mu \nu} \d g'_{\rho \sigma} \,,
\end{equation}
where
\begin{equation}\label{eq:dewitt}
G^{\mu \nu \rho \sigma}_{(\lambda)} = \frac{1}{8} \sqrt{g}
\left[g^{\mu \rho} g^{\nu
\sigma} + g^{\mu \sigma} g^{\nu \rho} - 2 \lambda g^{\mu \nu} g^{\rho \sigma}
\right] \,.
\end{equation}
Thus $\lambda$ parametrises the possible inner products. The
metric is positive definite for $\lambda<1/4$ and non-degenerate
for $\lambda\neq 1/4$. In Appendix D we demonstrate that different
values of $\lambda$ indeed give non-equivalent inner products on
moduli space.
The de Witt metric with $\lambda=1$
also appears in Hamiltonian treatments of gravity.
This is not what we are doing here; the metric we want is on four
dimensional Riemannian geometries. In the case of pure gravity
there is a connection, as the four dimensional Euclidean theory
can be lifted to $4+1$ Einstein theory. The gravitational
instantons become Kaluza-Klein monopoles in five dimensions. In
this context the moduli space on the multicentred Gibbons-Hawking
spaces has been computed as the slow motion moduli space metric of
the Kaluza-Klein monopoles \cite{Ruback:1986ag}. We will describe
a lift of our solutions in a later section, but for the moment we
are pursuing a four dimensional treatment.
The ambiguity in the inner product translates into a choice of
gauge. Imposing orthogonality to pure gauge transformations now
requires
\begin{equation}
0 = \langle \d g , {\mathcal{L}}_{\xi} g \rangle_\lambda =
- \frac{1}{2} \int_{M_4} d^4x \sqrt{g}
\xi^{\mu} \left[\nabla^{\nu} \d g_{\mu \nu} - \lambda \nabla_{\mu} \d
g^{\nu}{}_{\nu} \right] \,.
\end{equation}
Here ${\mathcal{L}}$ is the Lie derivative. Therefore, metric
fluctuations must be considered in the gauge
\begin{equation}\label{eq:metgauge}
\nabla^{\nu} \d g_{\mu \nu} = \lambda \nabla_{\mu} \d
g^{\nu}{}_{\nu} \,.
\end{equation}
In Appendix D we discuss the extent to which the different choices
of $\lambda$ lead to isometric inner products. The result will
certainly not depend on $\lambda$ if all fluctuations are trace
free. All the gauges are equivalent in that case. Indeed, for
noncompact gravitational instantons, all normalisable zero modes
are trace free. This is not true for the compact gravitational
instanton, K3. However, we now need to check compatibility with
the quadratic kinetic terms in the action. The quadratic action,
only keeping track of derivative terms, is
\begin{eqnarray}\label{eq:quadraticgravity}
S^{2}_{\d g} & = & \frac{1}{4} \int_{M_4} d^4x \sqrt{g} \left(
\nabla^\mu \d g^{\rho \sigma} \nabla_\mu \d g_{\rho
\sigma} - \nabla^\mu \d g^\rho{}_\rho \nabla_\mu \d
g^\sigma{}_\sigma \right. \nonumber \\
& & \qquad \qquad \qquad \left. - 2 \nabla^\mu \d g^\rho{}_\mu \nabla^{\sigma} \d g_{\rho \sigma}
+ 2 \nabla^\mu \d g^\rho{}_\rho \nabla^{\sigma} \d g_{\mu \sigma} \right)\\
& \rightarrow & -\frac{1}{8} \int_{M_4} d^4x \sqrt{g}
\left[g^{\mu \rho} g^{\nu
\sigma} + g^{\mu \sigma} g^{\nu \rho} - 2 (1+2\lambda^2 - 2 \lambda) g^{\mu \nu} g^{\rho \sigma}
\right] \delta g_{\mu \nu} \nabla^2 \delta g_{\rho
\sigma}\,\nonumber,
\end{eqnarray}
where arrow denotes imposition of the gauge (\ref{eq:metgauge}).
Generically, this does not correspond to the de Witt
(\ref{eq:dewitt}) inner product which we started with. For
consistency, we now need to impose $2 \lambda^2 - 3 \lambda + 1
=0$. The two solutions to this equation are $\lambda = 1$ and
$\lambda = \frac{1}{2}$. These are in fact rather interesting
values. The first is that obtained from viewing the instanton
moduli space as the slow motion moduli space of 4+1 dimensional
Kaluza-Klein monopoles \cite{Ruback:1986ag}. The second
corresponds to de Donder gauge, perhaps the most natural gauge for
the theory, and was considered recently because gradient flow on
the space of metrics with this inner product is Ricci flow
\cite{Headrick:2006ti}.
It follows from the previous few paragraphs that for gravitational
instantons there are two preferred gauges, which correspond to
taking $\lambda = 1$ or $\lambda =
\frac{1}{2}$ in the de Witt metric. However, we are interested in
Einstein-Maxwell theory, so we furthermore need to take into
account the fact that the Maxwell field also transforms under
infinitessimal diffeomorphisms $A \to A + {\mathcal{L}}_{\xi} A$.
Orthogonality to such diffeomorphisms therefore requires
\begin{equation}\label{eq:bothgauge}
\langle \d g , {\mathcal{L}}_{\xi} g \rangle_\lambda + \langle \d A ,
{\mathcal{L}}_{\xi} A \rangle = 0 \,.
\end{equation}
Using the Lorenz condition on the gauge field perturbation
(\ref{eq:lorentz}) one finds that the orthogonality condition
(\ref{eq:bothgauge}) requires that the following gauge be
implemented for the moduli
\begin{equation}\label{eq:gandA}
\nabla^{\nu} \d g_{\mu \nu} - \lambda \nabla_{\mu} \d
g^{\nu}{}_{\nu} = - 4 \d A^\nu F_{\nu \mu} \,.
\end{equation}
Once again, we need to substitute this gauge choice into the
quadratic term of the action. This is similar to the case
of pure gravity (\ref{eq:quadraticgravity}) except that there are
two extra terms due to the right hand side of the gauge condition
(\ref{eq:gandA}). One of these does not involve any derivatives of
$\d A_\mu$ or $\d g_{\mu \nu}$ and so does not contribute to the
quadratic terms. However, the other term involves a single
derivative. This latter term is always present unless $\lambda =
\frac{1}{2}$, suggesting that this is the preferred gauge for
Einstein-Maxwell instantons.
\subsection{Towards the moduli space metric}
To find the moduli space metric we need to find the general
solution to the linearised Einstein-Maxwell equations satisfying
the gauge conditions (\ref{eq:lorentz}) and (\ref{eq:gandA}). Once
we have the solution, we should then evaluate the norm of the
fluctuations using the results of the previous section. Given that
we have the general solution at a nonlinear level, we can easily
solve the linearised Einstein-Maxwell system by perturbing the
full solutions. However, these solutions will not be in the
required gauge. Finding a gauge transformation to map the solution
into the correct gauge does not appear easy.
An alternative and more elegant approach is that employed in
\cite{Ruback:1986ag} to find the moduli space metric on the
Gibbons-Hawking gravitational instantons. This uses the existence
of $N$ closed self dual two forms on the background, $F^J$, as
well as the three self dual K\"ahler forms $\Omega^i$ to
write the metric fluctuation
\begin{equation}
\d g^{iJ}_{\mu\nu} = \Omega^{i\rho}{}_{(\mu} F^J_{\nu) \rho} \,.
\end{equation}
This perturbation solves the linearised Einstein
equations. Furthermore, it is transverse and tracefree and therefore
solves the gauge condition required for pure Einstein gravity.
Note that this approach combines supersymmetry,
which provides the three K\"ahler forms, and the topology of solution,
which has $b^+_2 = N$ and hence implies the existence of the closed self dual
forms $F^J$. Using these modes, \cite{Ruback:1986ag} shows that the
moduli space metric is given in terms of intersection matrix of the
Gibbons-Hawking background and is flat.
So far, we have not been able to adapt this argument to the
Einstein-Maxwell case in a way consistent with the gauge condition
(\ref{eq:gandA}). We hope that the framework presented in this
section will be a useful starting point for future work on the
moduli space metric.
\section{Lift to five dimensions}
\subsection{Lifting the solutions}
Recall the following feature of field theory instantons:
instantons in $D$ dimensions may be viewed as solitons in $(D+1)$
dimensions. Furthermore, the $L^2$ instanton metric coincides with
a natural Riemannian metric on the moduli space of solitons that
is induced from the kinetic term in the $(D+1)$ dimensional
action. This is interesting given the differing interpretations of
the metrics in each case. The metric is relevant at the classical
level in $(D+1)$ dimensions, as its geodesic motion approximates
the soliton dynamics in the nonrelativistic limit
\cite{Manton:1981mp}. However, in $D$ dimensions the metric is only
important in quantum field theory, where measures on solution
spaces are needed.
This procedure can also be applied to the 4 dimensional
Einstein-Maxwell gravitational instantons (\ref{IWmetric}) if it
is possible to lift them to Lorentzian metrics which are solitons
of some theory in higher dimensions. Of course the resulting
moduli space metric could depend on the choice of higher
dimensional theory. In this section we study one possible theory
in $(4+1)$ dimensions. The five dimensional metrics resulting from
the lift are interesting in their own right, and we clarify some
of their properties in this section. In the following section
\ref{Section_metric} we shall discuss the metric on the slow motion
moduli space of these solitons.
Einstein-Maxwell theory without a dilaton cannot be consistently
lifted to pure gravity in five dimensions\footnote{The need
for a dynamical scalar field was not originally
appreciated in the 1920s by Kaluza and Klein who set it to a constant.
This mistake was corrected more than 20 years latter by Jordan and Thiry.}.
However, Einstein-Maxwell
configurations may be lifted to solutions of five dimensional
Einstein-Maxwell theory with a Chern-Simons term. This lift is the
bosonic sector of the lift from ${\mathcal{N}}=2$ supergravity in
four dimensions to ${\mathcal{N}}=2$ supergravity in five
dimensions \cite{Chamseddine:1980sp,Lozano-Tellechea:2002pn}.
We are interested in lifting the four dimensional Riemannian theory to
a Lorentzian theory on a five dimensional manifold $M_5$.
The four dimensional action is
\begin{equation}
S_4 = \int d^4x \sqrt{g^{(4)}} \left[ R^{(4)} - F_{a b} F^{a b} \right] \,,
\end{equation}
with equations of motion given by (\ref{eq:4deqns}). The five
dimensional action is
\begin{equation}
\label{SEMCS}
S_5 = \int d^5x \sqrt{-g^{(5)}} \left[R^{(5)} - H_{\a \b} H^{\a \b} \right]
- \frac{8}{3\sqrt{3}} \int H \wedge H \wedge W \,,
\end{equation}
where $H = dW$ is the five dimensional Maxwell field. We use greek
indices ranging from $0$ to $4$ in five dimensions. The equations of
motion in five dimensions are
\begin{eqnarray}\label{eq:5deqns}
G_{\a \b} & = & 2 H_{\a}{}^{\g} H_{\b \g} - \frac{1}{2} g^{(5)}_{\a \b} H^{\g
\d} H_{\g \d} \,, \nonumber \\
d \star_5 H & = & - \frac{2}{\sqrt{3}} H \wedge H \,.
\end{eqnarray}
Given a solution, $g^{(4)}$ and $F=dA$, to the four dimensional
equations (\ref{eq:4deqns}), we may lift the solution to five dimensions as follows:
\begin{eqnarray}\label{eq:5dconfig}
g^{(5)} & = & g^{(4)} - (dt + \Phi)^2 \,, \nonumber \\
W & = & \frac{\sqrt{3}}{2} A \,,
\end{eqnarray}
where $\Phi$ is a one form determined by $g^{(4)}$ and $F$ through
\begin{equation}\label{eq:phieqn}
d \Phi = \star_4 F \,.
\end{equation}
One may then check that the five dimensional configuration
(\ref{eq:5dconfig}) solves the equations of motion
(\ref{eq:5deqns}). Note that solutions to (\ref{eq:phieqn}) exist
because $d \star_4 F = 0$ on shell. In our case we may solve for
$\Phi$ explicitly to find
\begin{equation}
\Phi = - \frac{1}{2} \left(U^{-1} + {\widetilde U}^{-1}\right) (d\t + {\bf \w}) + {\bf \chi} \,,
\end{equation}
where $\chi$ satisfies
\begin{equation}
\nabla \times {\bf \chi} = \frac{1}{2} \nabla \left(U - {\widetilde U} \right) \,.
\end{equation}
The supersymmetric solutions of ${\mathcal{N}}=2$ supergravity in five
dimensions have been classified \cite{Gauntlett:2002nw}. For the case
of a timelike Killing spinor the general solution is given as a $U(1)$
fibration over a four real dimensional hyperK\"ahler manifold.
It was shown in \cite{Gauntlett:2002nw} how the lift of the Lorentzian
Israel-Wilson-Perj\'es solutions to five dimensions could be expressed as a
fibration over the multicentred Gibbons-Hawking metrics \cite{Gibbons:1979zt}.
It turns out that the lift of the Riemannian
Israel-Wilson-Perj\'es solutions we are considering may also be
expressed as a fibration over the multicentred Gibbons-Hawking
metrics. The five dimensional metric (\ref{eq:5dconfig}) can be
written as follows\footnote{Writing the spacetime in the form
(\ref{eq:goodform}) locates the five dimensional solution in the
classification of \cite{Gauntlett:2002nw}. In section 3.7 of that
paper the general supersymmetric fibration over a Gibbons-Hawking
base with $\pa/\pa t$ a Killing vector is given in terms of three
harmonic functions. For our solution these correspond to $L = 2
{\widetilde U}$, $K = - {\widetilde U}$ and $M = - 2 {\widetilde U}$.}
\begin{equation}\label{eq:goodform}
g^{(5)} = - f^2 \left(d\t + \w' \right)^2 + f^{-1} g^{GH} \,,
\end{equation}
where the Gibbons-Hawking metric is
\begin{equation}
g^{GH} = V^{-1} \left(dt + \chi \right)^2 + V d{\bf x}^2 \,,
\end{equation}
with harmonic funtion
\begin{equation}
V = \frac{1}{2} \left(U - {\widetilde U} \right)\,.
\end{equation}
The remaining functions in the metric (\ref{eq:goodform}) are
\begin{equation}
f = \frac{V}{U {\widetilde U}} \,,
\end{equation}
and
\begin{equation}
\w' = {\bf \w} - \frac{1}{2 f^2} \left(U^{-1} + {\widetilde U}^{-1} \right)
\left(dt + {\bf \chi} \right) \,.
\end{equation}
Note that the hyperK\"ahler base itself is in general not regular,
even changing signature at points where $U = {\widetilde U}$. This is
perfectly compatible with regularity of the five dimensional
spacetime.
The case $U = {\widetilde U}$ is exceptional and cannot be written in the
form (\ref{eq:goodform}). Instead, these metrics have null
supersymmetry in five dimensions. The metric is\footnote{The
metric (\ref{eq:nullform}) falls within the classification of
\cite{Gauntlett:2002nw} for spacetimes with null supersymmetry by
setting their functions $H = - {\mathcal{F}}=U$ and
${\mathbf{a}}=0$.}
\begin{equation}\label{eq:nullform}
g^{(5)} = \frac{2 dt d\t}{U} - dt^2 + U^2 d{\bf x}^2 \,.
\end{equation}
\subsection{Regularity and causality}
The interesting points in the five dimensional metric are the
centres where $U \to \infty$ or ${\widetilde U} \to \infty$. In the four
dimensional Riemannian Israel-Wilson-Perj\'es solutions these can
always be made to be regular points \cite{Whitt:1984wk,
Yuille:1987vw} as we reviewed above. We need to re-examine the
regularity of the metric around these points and also check for the
possible occurrence of closed timelike curves.
Before zooming in on the centres note the following. Firstly, that
\begin{equation}
{g^{(5)}}_{\t\t} \equiv g^{(5)}\Big(\frac{\pa}{\pa \t}, \frac{\pa}{\pa \t}\Big)
= - \frac{(U-{\widetilde U})^2}{(2 U {\widetilde U})^2} < 0 \,,
\end{equation}
if $U \neq {\widetilde U}$. Therefore, to avoid closed timelike curves throughout
the five dimensional spacetime we must not identify $\t$.
Secondly, possible candidates for the location of horizons are where
the metric becomes degenerate
\begin{equation}\label{eq:horizons}
0 = {g^{(5)}}_{tt} {g^{(5)}}_{\t\t} - [{g^{(5)}}_{t\t}]^2 = -\frac{1}{U {\widetilde U}} \,.
\end{equation}
This occurs at the centres where $U$ or ${\widetilde U}$ diverge.
In order to understand the geometry near the centres,
there are three different cases we need to consider separately.
The first is that $U \to \infty$ while ${\widetilde U}$ remains finite. Using
polar coordinates $(r=\rho^2/4, \theta, \phi)$ centred on the point
${\bf x}_m$ and
requiring that $a_m {\widetilde U}({\bf x}_m) = 1$, the metric becomes
\begin{equation}
ds^2 = d\rho^2 + \frac{\rho^2}{4} \left[(d\t + \cos\theta d\phi)^2
+ d\Omega^2_{S^2} \right] - \left(dt - a_m d\t/2
\right)^2
\end{equation}
as $\rho \to 0$, with $d\Omega^2_{S^2}=d\theta^2+\sin^2\theta d\phi^2$.
The metric
may be made regular about this point if we identify $\t$ with period
$4\pi$. Unfortunately this introduces closed timelike curves as we
discussed. If we choose not to identify $\t$ we are left with
timelike naked singularities at the centres. We see that there is no
horizon at these points, but rather a (singular) origin of polar
coordinates. Therefore, metrics with this behaviour at the centres
cannot lift to causal, regular solitons in five dimensions.
The remaining two possibilities involve coincident centres where
both $U$ and ${\widetilde U}$ go to infinity, so that ${\bf x}_m = {\bf {\tilde x}}_m$. One
needs to treat separately the cases where $a_m = {\tilde a}_m$ and where
$a_m \neq {\tilde a}_m$. In the latter case we again find regularity at
the expense of closed timelike curves going out to infinity, or
alternatively naked singularities. This
leaves only the former case with $a_m = {\tilde a}_m$ for all $m$. That is,
${\widetilde U} = U + k$, with $k$ some constant.
By considering the asymptotic regime, one can see that in order to
obtain a sensible asymptotic geometry without closed timelike
curves, one requires that either both $U$ and ${\widetilde U}$ go to a
constant at infinity or they both go to zero. Rescaling the
harmonic functions and performing a duality rotation on the
Maxwell field, as we discussed in four dimensions above, implies
that without loss of generality $U={\widetilde U}$. We consider this case in
the following subsection.
\subsection{Multi solitonic strings}
The only lift that leads to a globally regular and
causal five dimensional spacetime is the case $U={\widetilde U}$, which corresponds to
the Euclidean Majumdar-Papapetrou metric in four dimensions.
The metric is (\ref{eq:nullform}), with a null Killing spinor.
Away from the centres, the spacetimes approach either
${{\Bbb R}}^{1,4}$ or $AdS_3 \times S^2$, with $U$ going to a constant or
zero at infinity, respectively.
With a rescaling of coordinates, the geometry near the centres
where $U \to \infty$ may be written
\begin{equation}\label{eq:ads}
ds^2 = a_m^2 \left[\frac{dr^2}{r^2} + 2 r dt d\t - dt^2 +
d\Omega^2_{S^2} \right] \,.
\end{equation}
Calculating the curvature shows that
this metric locally describes $AdS_3 \times S^2$. One might be tempted
to conclude that this represents the near horizon
geometry of an extremal black string in five dimensions. However, the
coordinates (\ref{eq:ads}) are a little unusual, the sign of
$dt^2$ differing from the metric of an extremal BTZ black hole
\cite{Banados:1992gq}. In particular, the Killing vector $\pa/\pa t$
is everywhere regular and timelike. This remains true in the full
spacetime (\ref{eq:nullform}). There is no horizon and the
degeneration of the metric at the centres is analogous to an origin of
polar coordinates.
The coordinates in (\ref{eq:ads}) may be mapped to Poincar\'e coordinates as follows
\begin{eqnarray}\label{eq:poincare}
Y & = & \frac{1}{r^{1/2}\cos \frac{t}{2}} \,,\nonumber \\
X & = & \frac{\t}{2} - \frac{1}{2} \left[\frac{1}{r}
- 1 \right] \tan \frac{t}{2} \,,
\nonumber \\
T & = & \frac{\t}{2} - \frac{1}{2} \left[\frac{1}{r} +
1 \right] \tan \frac{t}{2} \,,
\end{eqnarray}
so that the metric becomes
\begin{equation}
ds^2 = \frac{4 a_m^2}{Y^2} \left(-dT^2 + dX^2 + dY^2 \right) + a_m^2
d\Omega^2_{S^2} \,.
\end{equation}
There is no singularity at $t = \pm \pi$ as may be checked by
writing down the embedding of $AdS_3$ as a quadric in ${{\Bbb R}}^{2,2}$
in terms of these coordinates. The map (\ref{eq:poincare}) is
periodic in $t$. Taking $t$ with infinite range corresponds to
passing to the (causal) universal cover of $AdS_3$. There is no
need to identify $\t$ and therefore the spacetime is causal.
The metrics (\ref{eq:nullform}) give causal, regular
solutions to the five dimensional theory with an everywhere defined
timelike Killing vector. Writing the metric in the form
\begin{equation}
g^{(5)} = -(dt - d\t/U)^2 + \frac{d\t^2}{U^2} + U^2 d{\bf x}^2
\,,
\end{equation}
suggests that the spacetimes should be thought of as containing
$N$ parallel `solitonic strings'. The strings have worldvolumes in
the $t-\tau$ plane. There is a plane fronted wave
\cite{Gauntlett:2002nw} carrying momentum along the $\pa/\pa \t$
direction of the string. We call these plane fronted waves
solitonic strings to emphasise that the fields are localised along
strings and there are no horizons. The strings are magnetic
sources for the two form field strength
\begin{equation}
H = - \sqrt{3} \star_3 d U \,.
\end{equation}
This is possible because of the topologically nontrivial $S^2$ at
each centre (\ref{eq:ads}).
We end this subsection by remarking that any solution to
Einstein-Maxwell-Chern-Simons theory (\ref{eq:5deqns}) in $4+1$
dimensions can be lifted to a solution to 11 dimensional
supergravity given by the product metric of $g^{(5)}$ and a flat
metric on the six torus. The eleven dimensional four form is given
by $H\wedge\Omega_T$, where $\Omega_T$ is the K\"ahler form on the
torus. We have not pursued here an M theory interpretation of
these solutions.
\section{Slow motion in five dimensions}
\label{Section_metric}
An interesting feature of BPS solitons is the cancelation between
forces which makes static multi-soliton configurations possible.
This is clear for the 3+1 dimensional
Majumdar-Papapetrou multi black holes, where the electrostatic repulsion
is balanced by gravitational attraction. These black holes are in this sense
analogous to a nonrelativistic system of massive charged particles,
with the charge-to-mass ratio chosen to balance the Newtonian attraction and
Coulomb repulsion.
The nature of the forces in the, stationary but not static, 4+1
dimensional solution (\ref{eq:5dconfig}) is presumably more
complicated. We shall not study this problem here, and instead
focus on the scattering of slowly moving solitons. The question we
are interested in is whether there is a direct connection between
the metric on the moduli space of four dimensional instantons and
the metric on the moduli space describing slow motion of the 4+1
dimensional solitons. The metrics do coincide for pure gravity
instantons \cite{Ruback:1986ag}.
One can follow Manton's method for truncating the infinite number
of degrees of freedom of the gravitational field to the finite
dimensional moduli space ${\cal M}$ of solitons\footnote{In this
section we will refer to any of the solutions (\ref{eq:5dconfig})
as solitons, even if they are singular or contain closed timelike
curves. Part of our motivation is to compare with the moduli space
metric of four dimensional gravitational instantons
(\ref{IWmetric}) where everything is regular, even if $U \neq
{\widetilde U}$.}. This means that we shall be neglecting both gravitational
and electromagnetic radiation, and consider only velocity
dependent forces which perturbed solitons induce on each other. As
for the four dimensional instantons, the space ${\cal M}$ is not
the whole of ${{\Bbb R}}^{3(N+{\tilde N})}$. To obtain ${\cal M}$ we need
to quotient by the permutation group $S_N\times S_{\tilde N}$, and
the Euclidean group in three dimensions.
By considering the slow motion approximation to the initial value
formulation of 4+1 dimensional Einstein-Maxwell-Chern-Simons
(EMCS) theory, one can find the moduli space metric from the
effective action where the field degrees of freedom have been
integrated out. In the moduli space approximation the centres
become functions of $t$ and geodesic curves $\{{\bf x}_m(t), {\bf
\tilde{x}}_n(t)\}$ correspond to slow motion of a multi solitonic
string configuration.
The initial data for EMCS theory (\ref{SEMCS}) consists of a four
dimensional manifold $\Sigma$ together with a Riemannian metric
$\gamma_{\mu\nu}$, a symmetric tensor $K_{\mu\nu}$, a two form $B$
and a one form $E$. Given a metric $g^{(5)}$ and a one form
potential $W$ on $M_5$ we can perform a $4+1$ decomposition if
there exist a function $t$ whose gradient is everywhere timelike.
In this case $\Sigma$ is a level set of $t$, and we choose adapted
local coordinates $(t, x^a)$ such that the normal to $\Sigma$
takes the form
\begin{equation}
{\cal N}=N^{-1}(\pa_t-N^\mu\pa_\mu)\,,
\end{equation}
where $N$ and $N^\mu$ are the lapse function and the shift vector.
The spatial metric $\gamma_{\mu\nu}$ and the second fundamental
form $K_{\mu\nu}$ can now be read off from the formulae
\begin{eqnarray}\label{eq:ans1}
g^{(5)}&=&-N^2 dt^2+\gamma_{\mu\nu}(dx^\mu+N^\mu dt)(dx^\nu+N^\nu dt)\,, \nonumber\\
K_{\mu\nu}&=&\frac{1}{2}N^{-1}(\pa_t \gamma_{\mu\nu}-D_\mu
N_\nu-D_\nu N_\mu)\,,
\end{eqnarray}
where $D$ is the covariant derivative compatible with $\gamma$ on $\Sigma$.
We also decompose the one form $W$ and two form $H=dW$ as
\begin{equation}\label{eq:ans2}
W= W_0 N dt + W_\mu dx^\mu \,, \qquad H=E \wedge N dt + B\,.
\end{equation}
This last formula implies expressions for $E$ and $B$ as exterior
derivatives of the potentials $W_0$ and $W_\mu$.
The next step is to implement the 4+1 decomposition at the level of
the action. After neglecting a total derivative term, the following
action is obtained from substituting (\ref{eq:ans1}) and
(\ref{eq:ans2}) into the EMCS action (\ref{SEMCS})
\begin{eqnarray}\label{eq:decomaction}
S_{4+1} & = & \int d^4x dt N \sqrt{\gamma}
\left[R^{\gamma}+K_{\mu\nu}K^{\mu\nu}-K^2\right] \nonumber \\
& + & \int d^4x dt N \sqrt{\gamma} \left[2 E_\mu E^\mu - B_{\mu\nu} B^{\mu\nu} +
2 B_{\mu\nu} B_\rho{}^{\nu} \frac{N^\mu N^\rho}{N^2} \right] \nonumber \\
& - & \frac{8}{3\sqrt{3}} \int \left[W_0 B \wedge B - 2 B \wedge E
\wedge W_\mu dx^\mu \right] \wedge N dt \,.
\end{eqnarray}
Here $R^{\gamma}$ is the Ricci scalar of $\gamma$, $K=\gamma^{\mu
\nu}K_{\mu\nu}$, and all contractions use the metric $\gamma$. The three
lines come from the Einstein-Hilbert, Maxwell and Chern-Simons
terms in the action (\ref{SEMCS}), respectively. If we think of
the expression (\ref{eq:decomaction}) as an action for the fields
$\{\gamma_{\mu\nu},W_\mu,W_0,N_\mu,N\}$, then we see that the last
three of these appear without time derivatives. They are Lagrange
multipliers and impose the constraints of conservation of energy,
momentum and charge
\begin{equation}\label{eq:constraintx}
\frac{\d S_{4+1}}{\d N} = \frac{\d S_{4+1}}{\d N_\mu} = \frac{\d
S_{4+1}}{\d N W_0} =
0 \,.
\end{equation}
Arbitrary initial data will not evolve to a solution of the EMCS
theory. One needs to impose the constraint equations (\ref{eq:constraintx}).
To consider the slow motion dynamics of a perturbed stationary
solution, we allow the moduli to become time dependent and work to
first order in the velocities
\begin{equation}
v^J = \frac{d x^J}{dt},
\end{equation}
where we have used $x^J$ to denote a general modulus. This induces
a time dependence in the solution which to first order can be
written
\begin{equation}\label{eq:timederiv}
\frac{d \g_{\mu\nu}}{dt} = \d \g^J_{\mu\nu} v^J \,,
\qquad \frac{d W_{\mu}}{dt} = \frac{\sqrt{3}}{2} \d A^J_{\mu} v^J \,,
\end{equation}
where $\d \g^J_{\mu\nu}, \d A^J_{\mu}$ is the zero mode
corresponding to the modulus $x^J$. In general, simply allowing
the moduli to depend on time will not give a spacetime that solves
the constraint equations, even to first order in the velocities.
Instead, it will be necessary to add extra terms linear in the
velocities to the original solution. An early example of this
technique in gravity is the slow motion of Majumdar-Papepetrou
black holes \cite{Ferrell:1987gf}.
For the case of the Kaluza-Klein monopole lift of the
Gibbons-Hawking solutions, it turns out that it is sufficient to
simply promote the moduli to time dependent fields. The constraint
equations are automatically solved to first order in velocities
\cite{Ruback:1986ag}. This lies behind the simple identification
of the moduli space metrics in four and five dimensions. Let us
see whether the constraint equations are solved in our case.
To first order in velocities, the charge conservation and momentum
conservation constraints become
\begin{equation}\label{eq:conservation}
D^\mu (\d A_\mu^J/N) = 0 \,, \qquad D^\mu (\d\g_{\mu\nu}^J/N) =
D_\nu (\d \g^{J \mu}{}_{\mu}/N) \,.
\end{equation}
Here we used (\ref{eq:timederiv}). It is interesting to see that
these two constraints take the form of gauge conditions. They may
be imposed on the moduli fields and no extra terms are necessary.
Although these gauge conditions look similar to those encountered
in section 3.1, they are quite different. The choice of time
slicing is not the same. By comparing (\ref{eq:5dconfig}) and
(\ref{eq:ans1}) we see that $\g_{\mu\nu} = g_{\mu\nu} -
\Phi_{\mu} \Phi_{\nu}$. Working through the changes to the
covariant derivative shows that the charge conservation constraint
in (\ref{eq:conservation}), for instance, becomes
\begin{equation}
\nabla_\mu \left( \left[ (1-\Phi^2) g^{\mu \nu} + \Phi^\mu \Phi^\nu \right] \d A_\nu^J \right) = 0\, .
\end{equation}
Deriving this expression uses $1/N^2 = 1 - \Phi^2$. Here $\Phi^2$
is contracted with $g_{\mu \nu}$. A similar expression exists for
the momentum constraint. It is clear that this is not the Lorenz
gauge that we used for the instanton moduli space. As discussed,
the instanton moduli space metric is gauge dependent. This is the
first indication that there is not a direct connection between the
instanton and soliton moduli space metrics for our solutions.
A more significant problem arises from the Hamiltonian constraint.
To first order in velocities the constraint is
\begin{equation}\label{eq:hamiltonian}
\d g^J_{\mu\nu} D^\mu N^{\nu} - \d g^{J \mu}{}_\mu D^{\nu} N_{\nu}
= \frac{N}{\sqrt{\g}} \e^{\mu \nu \rho \sigma} F_{\mu \nu} \d
A^J_\rho A_{\sigma}\,.
\end{equation}
This is an algebraic relation between the various metric and
Maxwell field moduli. We might hope that (\ref{eq:hamiltonian}) is
solved for all moduli
for $\lambda=1$.
Unfortunately, it is
clear that this will not work. Notice that the Hamiltonian
constraint (\ref{eq:hamiltonian}) involves a symmetric derivative
of $N^\mu$. This translates into a symmetrised derivative of
$\Phi^\mu$. However, only the antisymmetrised derivative of
$\Phi^{\mu}$ can be expressed in terms of the four dimensional
fields via (\ref{eq:phieqn}). The Hamiltonian constraint will
require extra modes to be turned on for a consistent time
dependent solution.
The upshot of this section is therefore that, unlike in case of
Yang-Mills instantons or pure gravitational instantons, the slow
motion moduli space metric of the five dimensional soliton cannot
be directly reduced to the four dimensional instanton moduli space
metric. A full blooded computation of the backreaction of the
moduli velocities onto the spacetime is necessary.
\section{Discussion}
In this paper we have discussed various properties of multi-instanton
solutions of Euclidean Einstein-Maxwell theory. We have also shown how
these solutions may be lifted to `solitonic string' solutions of five dimensional Lorentzian
Einstein-Maxwell-Chern-Simons theory. There are roughly three types of
application for the solutions we have discussed. We hope that the present work
has provided a solid base for future investigations.
Firstly and perhaps most interestingly, given that the instantons only involve fields
that are observed to exist in nature, would be to understand the physical
effects mediated by these solutions. A well known example of the physical effect
of Euclidean Einstein-Maxwell theory is the bounce that describes the pair creation of
charged black holes in a sufficiently strong electromagnetic field \cite{Garfinkle:1990eq}.
One possible direction of study would be to ask whether the instantons tell us anything about the structure of the vacuum of Einstein-Maxwell theory, say as a function of temperature.
Secondly, it would be of interest to understand the role of these solutions as supersymmetric
building blocks within string and M theory. Either as higher dimensional supergravity instantons
\cite{deVroome:2006xu}, or as a component of Lorentzian compactification or brane solutions.
This would be analogous to the ubiquitous appearance of the Gibbons-Hawking metrics
in higher dimensions.
Thirdly, there are various mathematical aspects that we have not developed completely.
Some of these have physical consequences. It is important to understand the index theory
associated with the zero modes of the instantons. This will determine which correlators the instantons contribute to and also their effect on topological terms in the Lagrangian. Furthermore, we have not discussed determinants of quadratic fluctuations about the solutions. An interesting question is whether supersymmetry is sufficient in this case to force the one loop determinants to cancel.
On a slightly different note, a completely distinct set of Einstein-Maxwell instantons may be
constructed. LeBrun has found explict multicentred scalar flat K\"ahler metrics \cite{L91}.
These give solutions to Einstein-Maxwell theory with the field strength given by half the
K\"ahler form plus the Ricci form. It would interesting to study these solutions in more depth
and elucidate their relation, if any, with the solutions that we have discussed.
\section*{Acknowledgements}
We would like to thank David Berman, Roberto Emparan, Gary
Gibbons, Jan Gutowski, Matt Headrick, Hari Kunduri, James Lucietti, David
Mateos, Malcolm Perry, Simon Ross, Paul Tod and David Tong for
helpful comments at various points during this work.
This project was begun while SAH was supported by a research
fellowship from Clare College Cambridge. His research was
supported in part by the National Science Foundation under Grant
No. PHY99-07949.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,049 |
I had to buy a new printer the other day. The printer I wanted was like $200, but for some reason, I figured out that if I bought a printer/copier/scanner/faxer/coffee maker/clay oven/tennis racquet restringer, it's like $8.95. And I had to ask myself: why would it be that by paying less, I get more? How is it that the less I pay, the more I get? And I've figured out there's only one logical answer -- the giant, multinational, megacorporations really just want me to be happy. That's the only possible answer. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,310 |
using System;
using System.Diagnostics;
using System.Globalization;
using System.Linq;
namespace ClosedXML.Excel
{
internal class XLRangeAddress : IXLRangeAddress
{
#region Private fields
[DebuggerBrowsable(DebuggerBrowsableState.Never)] private XLAddress _firstAddress;
[DebuggerBrowsable(DebuggerBrowsableState.Never)] private XLAddress _lastAddress;
#endregion
#region Constructor
public XLRangeAddress(XLRangeAddress rangeAddress): this(rangeAddress.FirstAddress, rangeAddress.LastAddress)
{
}
public XLRangeAddress(XLAddress firstAddress, XLAddress lastAddress)
{
Worksheet = firstAddress.Worksheet;
FirstAddress = XLAddress.Create(firstAddress);
LastAddress = XLAddress.Create(lastAddress);
}
public XLRangeAddress(XLWorksheet worksheet, String rangeAddress)
{
string addressToUse = rangeAddress.Contains("!")
? rangeAddress.Substring(rangeAddress.IndexOf("!") + 1)
: rangeAddress;
string firstPart;
string secondPart;
if (addressToUse.Contains(':'))
{
var arrRange = addressToUse.Split(':');
firstPart = arrRange[0];
secondPart = arrRange[1];
}
else
{
firstPart = addressToUse;
secondPart = addressToUse;
}
if (XLHelper.IsValidA1Address(firstPart))
{
FirstAddress = XLAddress.Create(worksheet, firstPart);
LastAddress = XLAddress.Create(worksheet, secondPart);
}
else
{
firstPart = firstPart.Replace("$", String.Empty);
secondPart = secondPart.Replace("$", String.Empty);
if (char.IsDigit(firstPart[0]))
{
FirstAddress = XLAddress.Create(worksheet, "A" + firstPart);
LastAddress = XLAddress.Create(worksheet, XLHelper.MaxColumnLetter + secondPart);
}
else
{
FirstAddress = XLAddress.Create(worksheet, firstPart + "1");
LastAddress = XLAddress.Create(worksheet, secondPart + XLHelper.MaxRowNumber.ToInvariantString());
}
}
Worksheet = worksheet;
}
#endregion
#region Public properties
public XLWorksheet Worksheet { get; internal set; }
public XLAddress FirstAddress
{
get
{
if (IsInvalid)
throw new Exception("Range is invalid.");
return _firstAddress;
}
set { _firstAddress = value; }
}
public XLAddress LastAddress
{
get
{
if (IsInvalid)
throw new Exception("Range is an invalid state.");
return _lastAddress;
}
set { _lastAddress = value; }
}
IXLWorksheet IXLRangeAddress.Worksheet
{
get { return Worksheet; }
}
IXLAddress IXLRangeAddress.FirstAddress
{
[DebuggerStepThrough]
get { return FirstAddress; }
set { FirstAddress = value as XLAddress; }
}
IXLAddress IXLRangeAddress.LastAddress
{
[DebuggerStepThrough]
get { return LastAddress; }
set { LastAddress = value as XLAddress; }
}
public bool IsInvalid { get; set; }
#endregion
#region Public methods
public String ToStringRelative()
{
return ToStringRelative(false);
}
public String ToStringFixed()
{
return ToStringFixed(XLReferenceStyle.A1);
}
public String ToStringRelative(Boolean includeSheet)
{
if (includeSheet)
return String.Format("'{0}'!{1}:{2}",
Worksheet.Name,
_firstAddress.ToStringRelative(),
_lastAddress.ToStringRelative());
return _firstAddress.ToStringRelative() + ":" + _lastAddress.ToStringRelative();
}
public String ToStringFixed(XLReferenceStyle referenceStyle)
{
return ToStringFixed(referenceStyle, false);
}
public String ToStringFixed(XLReferenceStyle referenceStyle, Boolean includeSheet)
{
if (includeSheet)
return String.Format("'{0}'!{1}:{2}",
Worksheet.Name,
_firstAddress.ToStringFixed(referenceStyle),
_lastAddress.ToStringFixed(referenceStyle));
return _firstAddress.ToStringFixed(referenceStyle) + ":" + _lastAddress.ToStringFixed(referenceStyle);
}
public override string ToString()
{
return _firstAddress + ":" + _lastAddress;
}
public override bool Equals(object obj)
{
var other = (XLRangeAddress)obj;
return Worksheet.Equals(other.Worksheet)
&& FirstAddress.Equals(other.FirstAddress)
&& LastAddress.Equals(other.LastAddress);
}
public override int GetHashCode()
{
return
Worksheet.GetHashCode()
^ FirstAddress.GetHashCode()
^ LastAddress.GetHashCode();
}
#endregion
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,304 |
{"url":"https:\/\/www.impan.pl\/pl\/wydawnictwa\/czasopisma-i-serie-wydawnicze\/studia-mathematica\/all\/184\/2\/90366\/l-p-l-q-boundedness-of-analytic-families-of-fractional-integrals","text":"# Wydawnictwa \/ Czasopisma IMPAN \/ Studia Mathematica \/ Wszystkie zeszyty\n\n## $L^p$-$L^q$ boundedness of analytic families of fractional integrals\n\n### Tom 184 \/ 2008\n\nStudia Mathematica 184(2008), 153-174 MSC: Primary 42B20; Secondary 47B38, 44A35. DOI: 10.4064\/sm184-2-5\n\n#### Streszczenie\n\nWe consider a double analytic family of fractional integrals $S^{\\gamma,\\alpha}_{z}$ along the curve $t\\mapsto |t|^{\\alpha}$, introduced for $\\alpha =2$ by L. Grafakos in 1993 and defined by $$(S^{\\gamma,\\alpha}_{z}f)(x_1,x_2):= \\frac{1}{{\\mit\\Gamma}({z+1\\over2})}\\int\\int |u-1|^{z}\\psi(u-1) f(x_1-t,x_2- u|t|^{\\alpha}) \\,du\\, |t|^{\\gamma}\\,\\frac{dt}{t},$$ where $\\psi$ is a bump function on $\\mathbb R$ supported near the origin, $f\\in{\\cal C}^{\\infty}_{\\rm c} (\\mathbb R^2)$, $z,\\gamma\\in\\mathbb C$, $\\mathop{\\rm Re}\\nolimits \\gamma \\ge 0$, $\\alpha\\in\\mathbb R$, $\\alpha\\ge 2$.\n\nWe determine the set of all (${{1}\/{p}}, {{1}\/{q}},\\mathop{\\rm Re}\\nolimits z$) such that $S^{\\gamma,\\alpha}_{z}$ maps $L^p(\\mathbb R^2)$ to $L^q (\\mathbb R^2)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K^{i\\varrho,\\alpha}_{-1+i\\theta}$ is a product kernel on $\\mathbb R^2$, adapted to the curve $t\\mapsto |t|^{\\alpha}$; as a consequence, we show that the operator $S^{i\\varrho,\\alpha}_{-1+i\\theta}$, $\\theta, \\varrho \\in \\mathbb R$, is bounded on $L^p(\\mathbb R^2)$ for $1< p< \\infty$.\n\n#### Autorzy\n\n\u2022 Valentina CasarinoDipartimento di Matematica\nPolitecnico di Torino\nCorso Duca degli Abruzzi 24\n10129 Torino, Italy\ne-mail\n\u2022 Silvia SeccoDipartimento di Matematica\nPolitecnico di Torino\nCorso Duca degli Abruzzi 24\n10129 Torino, Italy\ne-mail\n\n## Przeszukaj wydawnictwa\n\nZbyt kr\u00f3tkie zapytanie. Wpisz co najmniej 4 znaki.\n\nOd\u015bwie\u017c obrazek","date":"2017-02-25 08:01:22","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\": 0.7336717247962952, \"perplexity\": 3048.879424125542}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501171670.52\/warc\/CC-MAIN-20170219104611-00405-ip-10-171-10-108.ec2.internal.warc.gz\"}"} | null | null |
/* -------------------------------------------------------------------------- */
/* Copyright 2002-2020, OpenNebula Project, OpenNebula Systems */
/* */
/* 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. */
/* -------------------------------------------------------------------------- */
define(function(require) {
/*
DEPENDENCIES
*/
// require('foundation.tab');
var BaseFormPanel = require("utils/form-panels/form-panel");
var Sunstone = require("sunstone");
var Locale = require("utils/locale");
var TemplateUtils = require("utils/template-utils");
var WizardFields = require("utils/wizard-fields");
var RoleTab = require("tabs/vmgroup-tab/utils/role-tab");
var AffinityRoleTab = require("tabs/vmgroup-tab/utils/affinity-role-tab");
var Notifier = require("utils/notifier");
var Utils = require("../utils/common");
/*
TEMPLATES
*/
var TemplateWizardHTML = require("hbs!./create/wizard");
var TemplateAdvancedHTML = require("hbs!./create/advanced");
/*
CONSTANTS
*/
var FORM_PANEL_ID = require("./create/formPanelId");
var TAB_ID = require("../tabId");
/*
CONSTRUCTOR
*/
function FormPanel() {
this.formPanelId = FORM_PANEL_ID;
this.tabId = TAB_ID;
this.affinity_role_tab = new AffinityRoleTab([]);
this.actions = {
"create": {
"title": Locale.tr("Create Virtual Machine Group"),
"buttonText": Locale.tr("Create"),
"resetButton": true
},
"update": {
"title": Locale.tr("Update Virtual Machine Group"),
"buttonText": Locale.tr("Update"),
"resetButton": false
}
};
BaseFormPanel.call(this);
}
FormPanel.FORM_PANEL_ID = FORM_PANEL_ID;
FormPanel.prototype = Object.create(BaseFormPanel.prototype);
FormPanel.prototype.constructor = FormPanel;
FormPanel.prototype.htmlWizard = _htmlWizard;
FormPanel.prototype.htmlAdvanced = _htmlAdvanced;
FormPanel.prototype.submitWizard = _submitWizard;
FormPanel.prototype.submitAdvanced = _submitAdvanced;
FormPanel.prototype.onShow = _onShow;
FormPanel.prototype.fill = _fill;
FormPanel.prototype.setup = _setup;
FormPanel.prototype.addRoleTab = _add_role_tab;
return FormPanel;
/*
FUNCTION DEFINITIONS
*/
function _htmlWizard() {
var opts = {
info: false,
select: true
};
return TemplateWizardHTML({
"affinity-role-tab": this.affinity_role_tab.html(),
"formPanelId": this.formPanelId
});
}
function _htmlAdvanced() {
return TemplateAdvancedHTML({formPanelId: this.formPanelId});
}
function _setup(context) {
this.roleTabObjects = {};
var that = this;
var roles_index = 0;
this.affinity_role_tab.setup(context);
// Fill parents table
// Each time a tab is clicked the table is filled with existing tabs (roles)
// Selected roles are kept
// TODO If the name of a role is changed and is selected, selection will be lost
$("#roles_tabs", context).on("click", "a", function() {
var tab_id = "#"+this.id+"Tab";
var str = "";
$(tab_id+" .parent_roles").hide();
var parent_role_available = false;
$("#roles_tabs_content #role_name", context).each(function(){
if ($(this).val() != "" && ($(this).val() != $(tab_id+" #role_name", context).val())) {
parent_role_available = true;
str += "<tr>\
<td style='width:10%'>\
<input class='check_item' type='checkbox' value='"+$(this).val()+"' id='"+$(this).val()+"'/>\
</td>\
<td>"+$(this).val()+"</td>\
</tr>";
}
});
if (parent_role_available) {
$(tab_id+" .parent_roles", context).show();
}
var selected_parents = [];
$(tab_id + " .parent_roles_body input:checked", context).each(function(){
selected_parents.push($(this).val());
});
$(tab_id + " .parent_roles_body", context).html(str);
$.each(selected_parents, function(){
$(tab_id + " .parent_roles_body #" + this, context).attr("checked", true);
});
});
$("#tf_btn_roles", context).bind("click", function(){
that.addRoleTab(roles_index, context);
roles_index++;
return false;
});
Foundation.reflow(context, "tabs");
// Add first role
$("#tf_btn_roles", context).trigger("click");
return false;
}
function _submitWizard(context) {
that = this;
var name = TemplateUtils.removeHTMLTags(WizardFields.retrieveInput($("#vm_group_name", context)));
var description = TemplateUtils.removeHTMLTags(WizardFields.retrieveInput($("#vm_group_description", context)));
var role = [];
$(".role_content", context).each(function() {
var role_id = $(this).attr("role_id");
role.push(that.roleTabObjects[role_id].retrieve($(this)));
});
var roles_affinity = this.affinity_role_tab.retrieve(context);
var vm_group_json = {
"NAME" : name,
"DESCRIPTION": description,
"ROLE" : role,
};
vm_group_json = $.extend(vm_group_json, roles_affinity);
if (this.action == "create") {
vm_group_json = {
"vm_group" : vm_group_json
};
Sunstone.runAction("VMGroup.create",JSON.parse(JSON.stringify(vm_group_json)));
return false;
} else if (this.action == "update") {
delete vm_group_json["NAME"];
Sunstone.runAction(
"VMGroup.update",
this.resourceId,
TemplateUtils.templateToString(vm_group_json));
return false;
}
}
function _submitAdvanced(context) {
if (this.action == "create") {
var template = $("textarea#template", context).val();
var vm_group_json = { vm_group: { vm_grp_raw: template } };
Sunstone.runAction("VMGroup.create", vm_group_json);
return false;
} else if (this.action == "update") {
var template_raw = $("textarea#template", context).val();
Sunstone.runAction("VMGroup.update_template", this.resourceId, template_raw);
return false;
}
}
function _onShow(context) {
var that = this;
$(".role_content", context).each(function() {
var role_id = $(this).attr("role_id");
that.roleTabObjects[role_id].onShow();
});
}
function _fill(context, element) {
$("#new_role", context)[0].parentElement.remove();
var that = this;
this.setHeader(element);
this.resourceId = element.ID;
$("#template", context).val(TemplateUtils.templateToString(element.TEMPLATE));
WizardFields.fillInput($("#vm_group_name",context), element.NAME);
$("#vm_group_name",context).prop("disabled", true);
WizardFields.fillInput($("#vm_group_description", context), element.TEMPLATE.DESCRIPTION );
//Remove row of roles
if (!Array.isArray(element.ROLES.ROLE)){
element.ROLES.ROLE = [element.ROLES.ROLE];
}
$.each(element.ROLES.ROLE, function(index, value){
var name = value.NAME;
if (name){
var html = "<option id='" + name + "' class='roles' value=" + name + "> " + name + "</option>";
$("#list_roles_select").append(html);
$("select #" + name).mousedown(function(e) {
e.preventDefault();
$(this).prop("selected", !$(this).prop("selected"));
return false;
});
}
});
this.affinity_role_tab.fill(context, element);
$("#btn_refresh_roles", context).remove();
$("#affinity", context).show();
}
function _add_role_tab(role_id, dialog) {
var that = this;
var html_role_id = "role" + role_id;
var role_tab = new RoleTab(html_role_id);
that.roleTabObjects[role_id] = role_tab;
//Append the new div containing the tab and add the tab to the list
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 33 |
An entrepreneur on a mission to make people from Birmingham and the Black Country loud and proud about their heritage, is looking for investors to take his business to the next level.
Donato Esposito is the founder of Bostin - a cult fashion business celebrating Birmingham and the Black Country with logos such as "Yummy Brummy", "Yummy Yammy" and, of course "Bostin", a Black Country word meaning "brilliant".
Mr Esposito registered Bostin as a trademark in 2002, after watching Black Country Rock, a musical by his brother - ex-Birmingham poet laureate Giovanni Esposito, aka "Spoz".
He said: "When they were using the word Bostin in the show it brought back fond memories of my childhood growing up in and around Birmingham.
"I started the brand because I wanted to promote a positive image about Birmingham and the Black Country.
"I have worked all over UK and Europe and some people often have a negative image of the West Midlands, so this is a way of shouting about what is good about our area ... or should I say Bostin!"
Since its launch, Bostin has developed a cult following amongst a number of young people in the city, and has featured in regional and national media worn by personalities such as BBC presenter Adrian Chiles and DJ Phil Upton.
The company - which is currently inter-net based - now has suppliers in the Black Country, Europe and the US and makes 20 per cent of its sales overseas.
Mr Esposito said: "We find that overseas orders predominantly come from ex pats or Australians who have lived in the region for a while and want a T-shirt as a memento. But there's so many reasons why people like the brand. Some people in the US have thought it's just a curious misspelling of Boston!"
Mr Esposito, who lives in Rubery and is also a freelance IT consultant, said he was now looking for a business mentor to help raise the profile of the brand and to secure a deal to supply it to retailers.
The 46-year-old, father of two, said: "I have been approached by a number of retail outlets who want to stock the Bostin range, but at the moment the mark-up is prohibitive.
"I need guidance more than anything - a mentor in the clothing industry.
"I know I have a talent for networking and my forte is getting the word out, but as far as clothing and distribution and manufacturing is concerned there is someone with more expertise out there that could develop the brand."
Mr Esposito said his ultimate aim would be for the brand to be instantly recognisable.
He said: "It's so simple - Bostin can be applied to anything from T-shirts to mobile phones. And it creates such a positive feeling, it's a really great word.
"Obviously I'd love to support myself and the kids. But it would be great if it could go bigger. If Richard Branson can do it with Virgin, why can't it be done with Bostin?" | {
"redpajama_set_name": "RedPajamaC4"
} | 2,818 |
\section{Introduction}
The classification and regression tree \citep[CART]{breiman1984classification} is among the most popular machine learning algorithms. It has apparent simplicity, visibility and intelligibility, and is thus widely used in data mining and analysis. The random forest \citep[RF]{breiman2001random}, as an ensemble method of CART, has arguably the most predictiveness for tabular data and is thus one of the most popular methods for machine learning.
However, it has been noticed by \citep{breiman2001random} from the time when CART was first proposed that using marginal variables as the splitting variable could cause problems in both theory and its numerical performance in classification and prediction. As a remedy for this disadvantage of CART, \citep[CART]{breiman1984classification} suggested using linear combination of the predictors as splitting variables. Later, the method became known as the oblique decision tree \cite[ODT]{Heath93inductionof} and has received much attention. ODT has better performance than CART, requiring fewer splits and therefore smaller trees than CART; the linear combination also makes ODT easy to interpret in data analysis. There are also various oblique random forests (ODRF) in the literature, including Forest-RC of \cite{breiman2001random}, Random Rotation Random Forest (RR-RF) of \cite{blaser2016random}, Canonical Correlation Forests (CCF) of \cite{rainforth2015canonical}, Random Projection Forests (RPFs) of \cite{lee2015fast} and Sparse Projection Oblique Random Forests of \cite{tomita2020sparse}. For these forests, the difference is on how to find the linear combination and can be regarded as different implementations of ODT or their random forests.
\subsection{A review of studies on statistical consistency}
Despite the popularity of CART and RF, its statistical consistency has posed a big theoretical problem for statisticians for a long time and is still far from being fully resolved. The difficulty for statistical consistency of CART or RF can be attributed to two issues. The first one is the limitation of the method itself, i.e., the use of only marginal variables as partitioning variables when building the tree of CART or RF. The second issue is the mathematical complexity in analyzing the relationship between different layers of the tree.
Early works on the consistency of CART or RF were mainly for the simplified version of the CART of RF. The most celebrated theoretical result is that of \cite{Breiman2001}, which offers an upper bound on the generalization error of forests in terms of correlation and strength of the individual trees. This was followed by \cite{breiman2004consistency} that focuses on a stylized version of the original algorithm. \cite{lin2006random} established lower bounds for nonadaptive forests (i.e., independent of the training set) via the connection between random forests and a particular
class of nearest neighbor predictors; see also \cite{biau2010layered}. In the past ten years, various theoretical developments, for example, \cite{biau2008consistency}, \cite{ishwaran2010consistency}, \cite{biau2012analysis}, \cite{genuer2012variance}, and \cite{zhu2015reinforcement}, have been made in analyzing consistency of simplified models. Recent attempts toward narrowing the gap between the theory and practice are \cite{denil2013consistency} who proves the first consistency result for
online random forests, and \cite{wager2014asymptotic} and \cite{mentch2014ensemble}
who study the asymptotic sampling distribution of forests.
The work of \cite{scornet2015consistency} is a milestone. They proved that CART-based RF is consistent in $L^2$ sense if the unknown underlying regression function is additive of the marginal variables. Following their proofs, \cite{syrgkanis2020estimation}, \cite{klusowski2021universal} and \cite{chi2020asymptotic} showed that RF is also consistent in the high-dimensional setting under different modelling assumptions. To be specific, \cite{klusowski2021universal} found there is a relationship between CART and greedy algorithms and further gives a consistency rate $(\ln n)^{-1}$, where $n$ is the sample size, for RF by using techniques in the greedy algorithms \citep{barron2008approximation}. \cite{chi2020asymptotic} improves his rate under an additional assumption called sufficient impurity decrease (SID) that includes the additive model as a special case under high-dimension setting. However, all above consistency results for CART or RF are based on very strong restrictions on $m(x)$, such as the additive model or the SID condition.
\subsection{Our contribution}
Note that the existing consistency results for decision trees or their corresponding random forests are either proved under very strong assumptions on the unknown ubderlying regression functions or are only for simplified versions of decision trees that are not practically used.
In view of this, our consistency results are novel. Our contributes are as follows.
\begin{itemize}
\item We establish the consistency results for ODT and ODRF for general regression functions as long as they are continuous. The results include those of CART or RF as their special cases or corollaries.
\item We introduce two ways for the "features bagging", leading to consistency for extended additive models or general regression with no assumption of model structure.
\item
We also refine the existing packages for ODRF according to the established theory. Intensive empirical study shows that our ODRF tends to has better performance than RF and other existing codes of decision forests.
\end{itemize}
During our study of ODT, \cite{cattaneo2022convergence} also presented some oracle properties of ODT. We summarize the three main differences between our results and those of Cattaneo et al. (2022) as follows. Firstly, oracle inequalities in \cite{cattaneo2022convergence} were intentionally designed for the high-dimensional regression and hold for functions whose Fourier transformations have finite first moments. Our study focuses on the consistency of ODT for fixed dimension regression and shows that ODT is consistent for more general regression functions, as long as they are continuous. Secondly, the number of layers $K_n$, which is unknown before the construction of ODT, is employed to control the growth of tree in \cite{cattaneo2022convergence}, while the number of terminal leaves $t_n$, which can be pre-defined in the algorithm, is used in our construction of ODT. In order to obtain the consistency of ODT \cite{cattaneo2022convergence} required $K_n$ must be $o(\log_2{n})$, which means the depth of ODT cannot be large. In other words, \cite{cattaneo2022convergence} can only guarantee consistencies for some special ODTs where $K_n$ must be much smaller than $\log_2{n}$. Note that for a given data, no one knows what the maximal depth of its ODT could be before performing the algorithm. Instead, this strong restriction is not required in our theory since we only need to assume $t_n=o(n)$ and the non-random $t_n$ can be indeed used as an input parameter to control the growth of ODT. Thirdly, we also introduce various random forests based on ODT and study their asymptotic properties whilst this part is missing in \cite{cattaneo2022convergence}.
\subsection{Organization of this paper}
The rest of this paper is organized according the following schema. In Section 2, we introduce notations used in the proofs and the Algorithm to describe how to construct ODT. Section 3 starts with the idea of our proofs, and gives our main results for the consistency of ODT. Section 4 extends the consistency results to ODRF. In section 5, we explain our implementation of the Algorithm, and compare its numerical performance with RF and other decision forests.
\section{Notation and Algorithm}
Suppose $Y$ is the response and $X\in [0,1]^p$ is the predictor. Our interest is to estimate the regression function $m(x)=\E(Y|X=x), x\in[0,1]^p$. Denote by $\D_n=\{(X_i,Y_i)\}_{i=1}^n$ the independent samples of $(X,Y)$. Let $ \Theta^p = \{ \theta: \theta \in \R^p \ \mbox{and } ||\theta||_2 = 1\}$ be the unit sphere in $\R^p$.
Denote by $A$ a node of ODT, which is a subset of $\R^p$, and denote its two daughters by $A^+_{\theta,s}=\{x\in A: \theta^Tx\le s\}$ and $A^-_{\theta,s}=\{x\in A: \theta^Tx> s\}$ satisfying $A=A^+_{\theta,s}\cup A^-_{\theta,s}$ if $A$ contains at least two data points.
Let $\bm{Y}=(Y_1,\ldots,Y_n)^\top$ be the response vector. For any node $A$, let $N(A):=Card(\{X_i\in A\})$ be the number of data points in $A$.
Let $\langle\bm{Y}-\bar{Y}_A, \bm{Y}+f\rangle_A:=\frac{1}{N(A)}\sum_{X_i\in A}{[(Y_i-\bar{Y}_A)\cdot (Y_i+f(X_i))]}$ and $\|\bm{Y}-f\|_A^2:=\frac{1}{N(A)}\sum_{X_i\in A}{(Y_i-f(X_i))^2}$, where $f: [0,1]^p\to\mathbb{R}$ and $\bar{Y}_A:=\frac{1}{N(A)}\sum_{X_i\in A}{Y_i}$ is the sample mean for data in $A$.
Define the impurity gain in regression problem \citep{breiman1984classification} by
\begin{equation}\label{impuritygainreg}
\Delta_A(\theta,s)= \|\bm{Y}-\bar{Y}_{A}\|_{A,n}^2-\left( P(A^+_{\theta,s})\|\bm{Y}-\bar{Y}_{A^+_{\theta,s}}\|_{A^+_{\theta,s},n}^2+P(A^-_{\theta,s})\|\bm{Y}-\bar{Y}_{A^-_{\theta,s}}\|_{A^-_{\theta,s},n}^2\right),
\end{equation}
where $P(A^+_{\theta,s})=N(A^+_{\theta,s})/N(A)$ and $P(A^-_{\theta,s})=N(A^-_{\theta,s})/N(A)$.
Let $t_n$ be the number of terminal leaves of an ODT, with $1\le t_n\le n$. The total variation of any univariate function $f(x), x\in [0,1]$ is denoted by $\|f\|_{TV}:= \sup_{\Gamma}{\sum_{j=0}^{n_{\Gamma-1}}|f(x_{j+1})-f(x_j)|}$, where the supremum runs over the set of all partitions $ \{\{x_0,\ldots, x_{n_\Gamma}\}: 0\le x_0<x_1<\cdots<x_{n_\Gamma}\leq 1\}$. If $f'\in L^1[0,1]$, we also have $\|f\|_{TV}=\int_0^1{|f'(x)|dx}$. We refer \cite{folland1999real} for more details about total variation. For any linear combination of ridge functions $g=\sum_{j=1}^J{g_j(\theta_j^T x)}, x\in [0,1]^p$, define its total variation by $\|g\|_{TV}:=\sum_{j=1}^J{\|g_{j}\|_{TV}}$.
\begin{figure}[ht!]
\centering
\resizebox{14cm}{!}
\begin{tikzpicture}
[node distance=1cm,
edge from parent/.style={blue,thick,draw},
edge from parent path={(\tikzparentnode) -| (\tikzchildnode)}]
\node (S01) [process] {$\mathbb{A}_0^1$};
\node (N01) [process,below of=S01] {};
\draw [arrow] (S01) -- (N01);
\node (N01) [process,below of=S01] {}
child {
node (S11) [process] {$\mathbb{A}_1^1$};
\node (N11) [process,below of=S11] {};
\draw [arrow] (S11) -- (N11);
\node (N11) [process,below of=S11] {}
child {
node (S21) [process] {$\mathbb{A}_2^1$};
\node (N21) [process,below of=S21] {};
\draw [arrow] (S21) -- (N21);
\node (N21) [process,below of=S21] {}
child {
node (S31) [process] {$\mathbb{A}_3^1$};
\node (S31) [process] {$\mathbb{A}_3^1$}
}
child [missing] {}
child {
node (S32) [process] {$\mathbb{A}_3^2$};
\node (S32) [process] {$\mathbb{A}_3^2$}
}
}
child [missing] {}
child [missing] {}
child {
node (S22) [process] {$\mathbb{A}_2^2$};
\node (N22) [process,below of=S22] {};
\draw [arrow] (S22) -- (N22);
\node (N22) [process,below of=S22] {}
child {
node (S33) [process] {$\mathbb{A}_3^3$};
\node (S33) [process] {$\mathbb{A}_3^3$}
}
}
}
child [missing] {}
child [missing] {}
child [missing] {}
child [missing] {}
child {
node (S12) [process] {$\mathbb{A}_1^2$};
\node (N12) [process,below of=S12] {};
\draw [arrow] (S12) -- (N12);
\node (N12) [process,below of=S12] {}
child {
node (S23) [process] {$\mathbb{A}_2^3$};
\node (N23) [process,below of=S23] {};
\draw [arrow] (S23) -- (N23);
\node (N23) [process,below of=S23] {}
child {
node (S34) [process] {$\mathbb{A}_3^4$};
\node (S34) [process] {$\mathbb{A}_3^4$}
}
child [missing] {}
child {
node (S35) [process] {$\mathbb{A}_3^5$};
\node (S35) [process] {$\mathbb{A}_3^5$}
}
}
child [missing] {}
child [missing] {}
child {
node (S24) [process] {$\mathbb{A}_2^4$};
\node (N24) [process,below of=S24] {};
\draw [arrow] (S24) -- (N24);
\node (N24) [process,below of=S24] {}
child {
node (S36) [process] {$\mathbb{A}_3^6$};
\node (S36) [process] {$\mathbb{A}_3^6$}
}
}
};
\end{tikzpicture}
}
\vspace{1cm}
\begin{tikzpicture}
[node distance=1cm,
edge from parent/.style={blue,thick,draw},
edge from parent path={(\tikzparentnode) -| (\tikzchildnode)}]
\node (input1) [process]{$\mathbb{A}_k^\tau$};
\node (decision1) [process,below of=input1] {$\theta_{\mathbb{A}_k^\tau} x \leq s_{\mathbb{A}_k^\tau} $};
\node (out1) [process,left of=decision1,xshift=-1.5cm,yshift=-1.5cm] {$\mathbb{A}_k^{\tau'}$};
\node (out2) [process,right of =decision1,xshift=1.5cm,yshift=-1.5cm] {$\mathbb{A}_k^{\tau''}$};
\draw [arrow] (input1) -- node {$(\theta_{\mathbb{A}_k^\tau}, s_{\mathbb{A}_k^\tau})=\mathop{\arg\max_{\theta\in\Theta^p, s\in\mathbb{R}}} {\Delta_{\mathbb{A}_k^\tau}(\theta,s)}$} (decision1);
\draw [arrow] (decision1) -| node {no} (out1);
\draw [arrow] (decision1) -| node {yes} (out2);
\end{tikzpicture}
\caption{This figure shows an example of $T_{\mathcal{D}_n,6,3}$, which has three layers $\L=3$ and $6$ leaves. To be specific, we have root node $\mathbb{A}_0^1$ in the layer 0, nodes $\mathbb{A}_1^1$ and $\mathbb{A}_1^2$ in the layer 1, nodes $\mathbb{A}_2^1, \mathbb{A}_2^2, \mathbb{A}_2^3, \mathbb{A}_2^4$ in the layer $\ell = 2$ and leaves $\mathbb{A}_3^1, \mathbb{A}_3^2, \mathbb{A}_3^3, \mathbb{A}_3^4, \mathbb{A}_3^5, \mathbb{A}_3^6, \mathbb{A}_3^7$ in the layer 3. Note that in this case $\mathbb{A}_2^2$ only contains one data point and can not be divided in further steps, which implies that $\mathbb{A}_2^2=\mathbb{A}_3^3$. It is also noteworthy that no matter how many data points in $\mathbb{A}_2^4$ we have $\mathbb{A}_2^4=\mathbb{A}_3^6$ because $t_n$ is preset to be 6. Finally, we have estimators $m_{n,2}(x)=\sum_{j=1}^4{\mathbb{I}(x\in \mathbb{A}_2^j)\cdot \bar{Y}_{\mathbb{A}_2^j}}$ and $m_{n,3}(x)=\sum_{j=1}^6{\mathbb{I}(x\in \mathbb{A}_3^j)\cdot \bar{Y}_{\mathbb{A}^j_3}}$ given data $\D_n$.}
\label{fig:decisiontree}
\end{figure}
Following \cite{breiman1984classification}, the best splitting criteria of each node $A$ that contains at least two data points is to choose $(\hat{\theta}_A,\hat{s}_A)$ which maximizes $\Delta_A(\theta,s)$ over $\mathbb{R}\times\Theta^p$.
Based on the best splitting, the construction of ODT in regression problem is shown in Algorithm \ref{Algorithm.ODTtreereg}.
For a regression tree, we call the root, i.e. original data, as layer 0, and the last layer, i.e. the layer that only contains leaves, $ \L $. Thus, the layer index $ \ell $ satisfies $ 0 \le \ell \le \L $.
Let $k_\ell$ be the number of leaves at layer $\ell$, whose corresponding nodes are denoted by $\{\mathbb{A}_l^j\}_{j=1}^{k_\ell}$. The estimator of $m$ by using nodes at layer $\ell$ is defined by
$$
m_{n,t_n,\ell}(x)=\sum_{j=1}^{k_\ell}{\mathbb{I}(x\in\mathbb{A}_\ell^j)\cdot \bar{Y}_{\mathbb{A}_\ell^j}}
$$
for each $ 0 \le \ell \le \L $. To simplify notation, $m_{n,t_n,\ell}(x)$ is sometimes abbreviated as $m_{n,\ell}(x)$ if no confusion arises. With above notations, the final estimator is $ m_{n,\L}(x)$ satisfying $k_\L=t_n$. As an example of a regression tree, as shown in Figure \ref{fig:decisiontree}, $ \L = 3 $ and $ t_n = 6$. Denote a regression tree with $t_n$ leaves by $ T_{D_n,t_n} $, which can also be written as $ T_{D_n,t_n, \L} $. For the proof of the consistency, we also need to consider a truncated tree at any layer $ \ell: 0 \le \ell \le \L$, denoted by $ T_{D_n,t_n, \ell} $. For example, tree $ T_{D_n,t_n,2} $ in Figure \ref{fig:decisiontree} has leaves $\{\mathbb{A}_2^1, \mathbb{A}_2^2, \mathbb{A}_2^3, \mathbb{A}_2^4\}$. Note that leaves and nodes are relative, and a node on a fully grown tree can become a leaf on a truncated tree.
\begin{algorithm}[h]\label{Algorithm.ODTtreereg}
\caption{Oblique Decision Tree in Regression}
\KwIn{Data $\D_n$ and pre-specified number of leaves $t_n$}
Set $\mathcal{P}_0=\{[0,1]^p\}$, the root node of the tree\;
For all $1\le \ell\le n$, set $\mathcal{P}_\ell=\varnothing$\;
Set $\text{layer}=0$ and $n_{\text{nodes}}=1$\;
\While{$n_{\text{nodes}} \in \{1, 2, ..., t_n\}$}{
\eIf{$\mathcal{P}_{\text{layer}}=\varnothing$}
{
$\text{layer}=\text{layer}+1$\;
}{
Let $A$ be the first node in $\mathcal{P}_{\text{layer}}$\;
\eIf{{\color{black}$A$ contains only one data point}}
{
$\mathcal{P}_{\text{layer}}\leftarrow \mathcal{P}_{\text{layer}}- \{A\}$\;
$\mathcal{P}_{\text{layer}+1}\leftarrow\mathcal{P}_{\text{layer}+1}\cup\{A\}$\;
}{
$(\hat{\theta}_A,\hat{s}_A)=\argmax_{\theta\in\Theta^p,s\in\mathbb{R}}{\Delta_A(\theta,s)}$ defined in \eqref{impuritygainreg}\;
Partition node $A$ into two daughter nodes: \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ $A^+_{\hat{\theta}_A,\hat{s}_A}=\{x\in A: \hat{\theta}_A^T\cdot x\le \hat{s}_A\}$ and $A^-_{\hat{\theta}_A,\hat{s}_A}=\{x\in A: \hat{\theta}_A^T\cdot x> \hat{s}_A\}$\;
$\mathcal{P}_{\text{layer}}\leftarrow \mathcal{P}_{\text{layer}}- \{A\}$\;
$\mathcal{P}_{\text{layer}+1}\leftarrow\mathcal{P}_{\text{layer}+1}\cup\{A^+_{\hat{\theta}_A,\hat{s}_A}\}\cup\{A^-_{\hat{\theta}_A,\hat{s}_A}\}$\;
$n_{\text{nodes}}=n_{\text{nodes}}+1$\;
}
}
}
\KwOut{Let $\{\mathbb{A}_\L^j\}_{j=1}^{t_n}$ be the set of leaves of above generated tree $T_{\mathcal{D}_n,t_n}$. Estimate $m(x)$ by $m_{n,\L}(x)=\sum_{j=1}^{t_n}{\mathbb{I}(x\in \mathbb{A}_\L^j)\cdot \bar{Y}_{\mathbb{A}_\L^j}}, x\in[0,1]^p$.}
\end{algorithm}
\section{Consistency of ODT}
Let us briefly describe our idea for the proof of consistency of ODT, i.e. the consistency of $m_{n,\L}(x)$ as an estimator of $m(x)=\E(Y|X=x), x\in[0,1]$. There are three main facts used in the proof. The first fact is the well-known universal approximation theorem; see, for example, \cite{cybenko1989approximation}. Under some mild conditions, function $ m(x) $ can be approximated by a sequence of ridge functions with additive structure, i.e.
\begin{equation}\label{ridgefunctionm}
m(x) = \sum_{j=1}^\infty g_j(\theta_j^T x),
\end{equation}
or for any $\epsilon>0$, there is a $ j_\epsilon\in\mathbb{Z}_+$ and
$
m_{\epsilon}(x) = \sum_{j=1}^{j_\epsilon} g_j(\theta_j^T x)
$
such that
$$
\sup_{x\in[0,1]^p}{|m(x)-m_{\epsilon}(x)|}\leq \epsilon.
$$
The second fact is that the directions searched in the ODT algorithm indeed can play the role of those in the universal approximation \eqref{ridgefunctionm} or those directions in $ m_{\epsilon}(x) $. The third fact is the consistency results of CART or RF for the additive model proved by \cite{scornet2015consistency} and that if $\theta_1, \theta_2, ... $ are known $
m_{\epsilon}(x) $ indeed has an additive structure.
The details of the proofs are given below. Let us start with several lemmas.
\begin{lemma}\label{Carteqqw}
Let $R(A):=\|\bm{Y}-\bar{Y}_A\|_A^2-\|\bm{Y}-g\|_A^2$ for any $g\in \mathcal{U}$, where $\mathcal{U}$ is a set of linear combinations of ridge functions:
$$
\mathcal{U}= \{\sum_{j=1}^J{g_j(\theta_j^Tx)}:\theta_j\in\Theta^p, g_j\in TV(\mathbb{R}), J\in\mathbb{Z}_+\}
$$
and $TV(\mathbb{R})$ consists of functions defined on $\mathbb{R}$ with bounded total variation (see \cite{folland1999real}). Let $A$ be an internal node of tree $T_{\mathcal{D}_n,t_n}$
which contains at least two data points. If $R(A)\ge 0$, then $\Delta_A(\hat{\theta}_A,\hat{s}_A)$ satisfies
$$
\Delta_A(\hat{\theta}_A,\hat{s}_A)\ge \frac{R^2(A)}{\|g\|_{TV}^2}.
$$
\end{lemma}
\begin{proof}
For each internal node $A$, we first introduce decision stump by
$$
\hat{Y}_A(x):=\frac{\mathbb{I}(x\in A^+_{\theta,s})P(A^-_{\theta,s})-\mathbb{I}(x\in A^-_{\theta,s})P(A^+_{\theta,s})}{\sqrt{P(A^+_{\theta,s})P(A^-_{\theta,s})}},
$$
obtained by a partition of $A$ with two daughters $A^+_{\theta,s}=\{x\in A: \theta^Tx\le s\}$ and $A^-_{\theta,s}=\{x\in A: \theta^Tx> s\}$. By some simple calculations, Lemma 3.1 in \cite{klusowski2021universal} shows that
\begin{equation}\label{Carteq}
\Delta_A(\theta,s)= |\langle \bm{Y}-\bar{Y}_A, \hat{Y}_A \rangle_A|^2
\end{equation}
when $\theta$ is parallel to any axis and $P(A^+_{\theta,s})P(A^-_{\theta,s})>0$. In fact, it is not difficult to prove that \eqref{Carteq} also holds for any $\theta\in\Theta^p$. Then, the remaining proof is similar to Lemma 7.1 in \cite{klusowski2021universal}.
\end{proof}
\begin{definition}[\cite{blumer1989learnability}]
Let $\mathcal{F}$ be a Boolean function class in which each $f:\mathcal{Z}\to\{0,1\}$ is binary-valued. The growth function of $\mathcal{F}$ is defined by
$$
\Pi_\mathcal{F}(m)=\max_{z_1,\ldots,z_m\in \mathcal{Z}}|\{(f(z_1),\ldots,f(z_m)):f\in\mathcal{F}\}|
$$
for each fixed integer $m\in\mathbb{Z}_+$.
\end{definition}
\begin{definition}[\cite{gyorfi2002distribution}] \label{coveringnumber}
Let $z_1,\ldots, z_n \in\mathbb{R}^p$ and $z_1^n=\{z_1,\ldots, z_n\}$. Let $\mathcal{H}$ be a class of functions $h:\mathbb{R}^p\to\mathbb{R}$. An $L_q$ $\epsilon$-cover of $\mathcal{H}$ on $z_1^n$ is a finite set of functions $h_1,\ldots, h_N: \mathbb{R}^p
\to \mathbb{R}$ satisfying
$$
\min_{1\leq j\leq N}{\left( \frac{1}{n}\sum_{i=1}^n{|h(z_i)-h_j(z_i)|^q}\right)^\frac{1}{q}}<\epsilon,\ \ \forall h\in \mathcal{H}.
$$
Then, the $L_q$ $\epsilon$-cover number of $\mathcal{H}$ on $z_1^n$, denoted by $\mathcal{N}_q(\epsilon,\mathcal{H},z_1^n)$, is the minimal size of an $L_q$ $\epsilon$-cover of $\mathcal{H}$ on $z_1^n$. If there exists no finite $L_q$ $\epsilon$-cover of $\mathcal{H}$, then the above cover number is defined as $\mathcal{N}_q(\epsilon,\mathcal{H},z_1^n)=\infty$.
\end{definition}
\begin{lemma}[\cite{gyorfi2002distribution}] \label{CoveringnumbervsVCd}
Let $\mathcal{H}$ be a class of functions $h: \mathbb{R}^p\to [0,B]$ with finite VC dimension $VC(\mathcal{H})\geq 2$. For any $B/4>\epsilon>0$, the cover number in Definition \ref{coveringnumber} satisfies
$$
\mathcal{N}_1(\epsilon,\mathcal{H},z_1^n)\leq 3\left( \frac{2eB}{\epsilon}\ln\left(\frac{3eB}{\epsilon} \right) \right)^{VC(\mathcal{H})}
$$
for all $z_1^n=\{z_1,\ldots,z_n\}, z_i\in \mathbb{R}^p.$
\end{lemma}
Let $\beta_n\asymp \ln{n}$ and $\hat{m}_{n,\L}(x)=\max{\{\min{\{m_{n,\L}(x),\beta_n\}},-\beta_n\}}$. Then,
$\hat{m}_{n,\L}(x) $ is a truncated version of $ m_{n,\L}(x) $. More generally, let $h:[0,1]^p\to\mathbb{R}$ be a function which is constant on each $\mathbb{A}_{\L}^j, j=1,\ldots, {t_n}$ in Algorithm \ref{Algorithm.ODTtreereg}.
Let $\mathcal{H}_{t_n}$ be the collection of all functions $ h $ defined above, and ${\mathcal{H}_{t_n}^{\beta_n}}=\{\max\{\min\{h,\beta_n\},-\beta_n\}: h\in\mathcal{H}_{t_n}\}$
\begin{lemma}[\cite{bagirov2009estimation}]\label{agirov2009estimation}
Assume
$
\E(e^{c\cdot Y^2})<\infty
$
for some $c>0$. Then, the truncated estimator $\hat{m}_{n,\L}(x)$ satisfies
\begin{equation}\label{Mainlemmaformulabeforeprun}
\begin{aligned}
\E_{\mathcal{D}_{n}} \int|\hat{m}_{n,\L}(x)-m(x)|^2d\mu(x)\leq & 2 \E_{\mathcal{D}_{n}}\left( \frac{1}{n}\sum_{i=1}^{n}|m_{n,\L}(X_i)-Y_i|^2-\frac{1}{n}\sum_{i=1}^{n}|m(X_i)-Y_i|^2\right)\\
&+\frac{c\ln^2 n}{n}\cdot\sup_{z_1^n}\ln\left(\mathcal{N}_1(1/(80n\beta_n),\mathcal{H}_{t_n}^{\beta_n}, z_1^n)\right)
\end{aligned}
\end{equation}
for some $c>0$, where $\mathcal{N}_1$ is the cover number given in Definition \ref{coveringnumber}, and $\mu$ is the distribution of $X$.
\end{lemma}
\begin{theorem}[Consistency of ODT before pruning]\label{consistencyodtreg}
Assume $ X \in [0,1]^p$ and
$
\E(e^{c\cdot Y^2})<\infty
$
for some $c>0$, and that $m(x)$ is a continuous function. If $t_n\to\infty$ and $t_n =o\left(\frac{n}{\ln^4{n}}\right)$, we have
$$
\E\left(\int{|m_{n,\L}(x)-m(x)|^2d\mu(x)}\right)\to 0,\ \text{as}\ n\to\infty.
$$
\end{theorem}
\begin{remark}
This theorem shows that ODT is mean squared consistent under very mild conditions. By comparing Theorem \ref{consistencyodtreg} with the consistency of CART given in \cite{scornet2015consistency}, we can conclude the advantages of ODT as follows. CART is consistent only when $m(x)$ has an additive structure of the predictors, while Theorem \ref{consistencyodtreg} guarantees the consistency of ODT for any smooth functions. In addition, Theorem \ref{consistencyodtreg} only needs very mild conditions on the distribution of $(X,Y)$.
\end{remark}
\begin{proof}
Steps of the proof are as follows. We first show that the truncated estimator $\hat{m}_{t_n}(x)$, defined above, is mean squared consistent by considering the two terms separately on the right hand side of Lemma \ref{agirov2009estimation}. The details are given in Part I and Part II respectively below. We will then prove that the un-truncated estimator $m_{t_n}(x)$ is also consistent in Part III.
\textbf{Part I:} Consider the first part of RHS of \eqref{Mainlemmaformulabeforeprun}. For the theoretical analysis, we introduce the following class of shallow neural networks
$$
\mathcal{G}_{J}=\Big\{\sum_{j=1}^{J} {c_j\cdot \sigma(\theta_j^Tx+d_j)}: \theta_j\in\Theta^p, c_j,d_j\in\mathbb{R}, \forall j\ge 1\Big\},
$$
where $\sigma(v)=e^v/(1+e^v)$ with $ v\in\mathbb{R}$.
Recall the definition of $T_{\mathcal{D}_n,\L}$, a truncated tree and $m_{n,\L}(x)$ is an estimator of $m(x)$ by taking averages of data in each terminal leafs of $T_{\mathcal{D}_n,\L}$, namely
$$
m_{n,\L}(x)=\sum_{j=1}^{k_\L}{\mathbb{I}(x\in \mathbb{A}_{\L}^j)\cdot \bar{Y}_{\mathbb{A}_{\L}^{j}} }.
$$
Let $\L_0:= \lfloor\log_2{t_n}\rfloor\le \L$. Note that $T_{\D_n,\ell}$ is fully grown for each $0\leq \ell\le \L_0$, namely $T_{\D_n,\ell}$ is generated recursively by splitting all leaves of previous $T_{\D_n,\ell-1}$ except those leaves which only contain one data point.
For any given $g\in\mathcal{G}_{J}$, we next prove
\begin{equation}\label{regpart1}
\|\bm{Y}-m_{n,\L_0}(X)\|_n^2- \|\bm{Y}-g(X)\|_n^2\le \frac{\|g\|^2_{TV}}{\lfloor\log_2{t_n}\rfloor+4}
\end{equation}
for any $t_n>1$, where $\|g\|_{TV}$ is the total variation of $g$ in Lemma \ref{Carteqqw}. Define the approximation error by $R_{\D_n,\ell}: =\|\bm{Y}-m_{n,\ell}(X)\|_n^2-\|\bm{Y}-g(X)\|_n^2$ for any $0\le \ell\le \L_0$. Without loss of generality, we can also assume $R_{\D_n,\L_0-1}\ge 0$ since $R_{\D_n,\L_0}\le R_{\D_n,\L_0-1}$ and \eqref{regpart1} holds obviously if $R_{\D_n,\L_0-1}< 0$. Similarly, define $R(A):=\|\bm{Y}-m_{t_n,\ell}(X)\|_A^2-\|\bm{Y}-g(X)\|_A^2$ for each $A\in \mathcal{O}_{\ell,1}:=\{\mathbb{A}_\ell^j\}_{j=1}^{k_\ell}$ which at least contains one data point. Then, for any $0\le \ell\le \L_0$ we have $$R_{\D_n,\ell}=\sum_{A\in\mathcal{O}_{\ell,1}}{w(A)R(A)},$$
where $w(A)=Card(A)/n$ is the proportion of data within $A$. Note that
\begin{align}
R_{\D_n,\L_0}= & R_{\D_n,\L_0-1}-\sum_{A\in\mathcal{O}_{\L_0-1,2}}{w(A)\Delta_A(\hat{\theta}_A,\hat{s}_A)} \label{dhasdgasdj}\\
\leq & R_{\D_n,\L_0-1}-\sum_{A\in\mathcal{O}_{\L_0-1,2}, R(A)> 0}{w(A)\Delta_A(\hat{\theta}_A,\hat{s}_A)} \nonumber\\
\le & R_{\D_n,\L_0-1}-\frac{1}{\|g\|_{TV}^2}\sum_{A\in\mathcal{O}_{\L_0-1,2}, R(A)> 0}{w(A)R^2(A)},\label{ygdai}
\end{align}
where $\mathcal{O}_{\L_0-1,2}\subseteq \mathcal{O}_{\L_0-1,1}$ is a collection of nodes which must contain at least two data points, and \eqref{dhasdgasdj} follows from the definition of impurity gain, and \eqref{ygdai} follows from Lemma \ref{Carteqqw}. Decompose $R_{\D_n,\L_0-1}$ into
\begin{align*}
R_{\D_n,\L_0-1}^+ := &\sum_{A\in\mathcal{O}_{\L_0-1,1} R(A)> 0}{w(A)R(A)},\\
\ R_{\D_n,\L_0-1}^-:=&\sum_{A\in\mathcal{O}_{\L_0-1,1}, R(A)\le 0}{w(A)R(A)}
\end{align*}
satisfying $R_{\D_n,\L_0-1}=R_{\D_n,\L_0-1}^++R_{\D_n,\L_0-1}^-$. By Jensen's inequality and $R(A)\le 0$ for any leaf $A$ of $T_{\D_n,\L_0-1}$ which contains only one data point, we have
\begin{equation}\label{dgiasydf}
\sum_{A\in\mathcal{O}_{\L_0-1,2}, R(A)> 0}{w(A)R^2(A)}\ge \left(\sum_{A\in\mathcal{O}_{\L_0-1,2}, R(A)> 0}{w(A)R(A)}\right)^2
=(R_{\D_n,\L_0-1}^+)^2.
\end{equation}
Then, combination of \eqref{ygdai} and \eqref{dgiasydf} implies that
\begin{align}
R_{\D_n,\L_0}&\le R_{\D_n,\L_0-1}- \frac{1}{\|g\|_{TV}^2}(R_{\D_n,\L_0-1}^+)^2\nonumber\\
&\le R_{\D_n,\L_0-1}- \frac{1}{\|g\|_{TV}^2} R_{\D_n,\L_0-1}^2, \label{dsfqwe}
\end{align}
where the last equation \eqref{dsfqwe} holds because $R_{\D_n,\L_0-1}^+>R_{\D_n,\L_0-1}$ and $R_{\D_n,\L_0-1}\ge 0$ by assumption. Again, following the same
arguments above, \eqref{dsfqwe} also implies
\begin{equation}\label{asdsa}
R_{\D_n,\ell}\le R_{\D_n,\ell-1}- \frac{1}{\|g\|_{TV}^2} R_{\D_n,\ell-1}^2
\end{equation}
for any integer $1\le \ell \le \L_0$. In conclusion, the fact that $R_{\D_n,1}\le R_{\D_n,0}- \frac{1}{\|g\|_{TV}^2} R_{\D_n,0}^2\le \|g\|_{TV}^2/{4}$ and \eqref{asdsa} imply that \eqref{regpart1} holds by mathematical induction.
Note that
$$
\|\bm{Y}-m_{n,\L}(X)\|_n^2\le \|\bm{Y}-m_{n,\L_0}(X)\|_n^2.
$$
Therefore, the first part on RHS of \eqref{Mainlemmaformulabeforeprun} satisfies, for any $g\in\mathcal{G}_J$,
\begin{align}
2 \E_{\mathcal{D}_{n}}\left( \frac{1}{n}\sum_{i=1}^{n}|m_{n,\L}(X_i)-Y_i|^2-\frac{1}{n}\sum_{i=1}^{n}|m(X_i)-Y_i|^2\right)
\le & 2\E_{\mathcal{D}_{n}}(\|\bm{Y}-m_{n,\L}(X)\|_n^2- \|\bm{Y}-g(X)\|_n^2)\nonumber\\
+&2\E_{\mathcal{D}_{n}}(\|\bm{Y}-g(X)\|_n^2-\|\bm{Y}-m(X)\|_n^2)\nonumber\\
\le& \frac{2\|g\|^2_{TV}}{\lfloor\log_2{t_n}\rfloor+4}+2\E(g(X)-m(X))^2\nonumber\\
\le & \frac{2\|g\|^2_{TV}}{\lfloor\log_2{t_n}\rfloor+4}+2\max_x{|g(x)-m(x)|^2}.\label{jlijl}
\end{align}
By \cite{pinkus1999approximation}, there is a series of $g_J\in\mathcal{G}_J, J\ge 1$ satisfying
\begin{equation}\label{iopipo}
\lim_{J\to\infty}{\max_x{|g_J(x)-m(x)|}}=0.
\end{equation}
In conclusion, \eqref{jlijl} and \eqref{iopipo} together imply that
\begin{equation}\label{djosijdisj}
2 \E_{\mathcal{D}_{n}}\left( \frac{1}{n}\sum_{i=1}^{n}|m_{n,\L}(X_i)-Y_i|^2-\frac{1}{n}\sum_{i=1}^{n}|m(X_i)-Y_i|^2\right)\to 0
\end{equation}
as $t_n\to\infty$, which completes the proof of \textbf{Part I}.
\textbf{Part II:} Now we consider the second part of the RHS of \eqref{Mainlemmaformulabeforeprun} by applying Lemma \ref{CoveringnumbervsVCd}. Recall that $\mathcal{H}_{t_n}$ consists of real functions $h:[0,1]^p\to\mathbb{R}$ that are constant on each $\mathbb{A}_{\L}^j, j=1,\ldots,t_n$ obtained in Algorithm \ref{Algorithm.ODTtreereg}. Define a Boolean class of functions :
$$
\mathcal{F}_{t_n}= \{sgn(f(x,y)): f(x,y)=h(x)-y, h\in\mathcal{H}_{t_n}\},
$$
where $sgn(v)=1$ if $v\ge 0$ and $sgn(v)=-1$ otherwise and $\mathcal{H}_{t_n}$ is defined below Lemma \ref{CoveringnumbervsVCd}. Recall that VC dimension of $\mathcal{F}_{t_n}$, denoted by $VC(\mathcal{F}_{t_n})$, is the largest integer $m\in\mathbb{Z}_+$ such that $2^m\leq\Pi_{\mathcal{F}_{t_n}}(m)$ holds
(see, for example, \cite{kosorok2008introduction}). Therefore, we next focus on bounding $\Pi_{\mathcal{F}_{t_n}}(m)$ for each positive integer $m\in\mathbb{Z}_+$. Let $z_1,\ldots,z_m\in\R^p$ be the series of points which maximize $\Pi_{\mathcal{F}_{t_n}}(m)$. Under the above notations, we have two observations as follows.
\begin{itemize}
\item For any $h_{t_n}\in\mathcal{H}_{t_n}$ that is constant on each $\mathbb{A}_{\L}^j, j=1,\ldots,t_n$, there is $h_{t_n-1}\in\mathcal{H}_{t_n-1}$ and a leaf $\mathbb{A}$ of corresponding $T_{\D_n,t_n-1}$ such that $\mathbb{A}= \mathbb{A}_{\L}^k\cup \mathbb{A}_{\L}^{k+1}$ for some $k$ and $h_{t_n-1}$ is constant on $\mathbb{A}_{\L}^j, j=1,\ldots,k-1,k+2,\ldots,t_n$ and $\mathbb{A}$.
\item All half-planes in $\mathbb{R}^p$ at most pick out
$\left({me}/{(p+1)}\right)^{p+1}$
subsets from $\{z_1,\ldots,z_m\}$ when $m\ge p+1$ (see, for example, \cite{kosorok2008introduction}), namely
$$
Card(\{ \{z_1,\ldots,z_m\}\cap\{x\in\R^p:\theta^T x\le s\}: \theta\in\Theta^p, s\in\mathbb{R} \})\leq \left({me}/{(p+1)}\right)^{p+1}.
$$
\end{itemize}
Based on above two facts, we can conclude
\begin{equation}\label{hunjhkn}
\Pi_{\mathcal{F}_{t_n}}(m)\le \Pi_{\mathcal{F}_{t_n-1}}(m)\cdot \left(\frac{me}{p+1}\right)^{p+1}.
\end{equation}
Then, combination of \eqref{hunjhkn} and $\Pi_{\mathcal{F}_{1}}(m)\le \left(\frac{me}{p+1}\right)^{p+1}$ implies that
\begin{equation}\label{fhskjfhsiu}
\Pi_{\mathcal{F}_{t_n}}(m)\le \left(\frac{me}{p+1}\right)^{t_n\cdot p+t_n}.
\end{equation}
Solving the inequality
$$
2^m\le \left(\frac{me}{p+1}\right)^{t_n\cdot p+t_n}
$$
by using the basic inequality $\ln x\leq\gamma\cdot x-\ln \gamma-1 $ with $ x,\gamma >0$ yields
\begin{equation}
VC(\mathcal{H}_{t_n})\le \frac{4}{\ln 2}\cdot p(t_n+1)\ln{\left(2p(t_n+1)\right)}\le c(p)\cdot t_n\ln(t_n),
\end{equation}
where constant $c(p)$ depends on $p$ only. Then, by Lemma \ref{CoveringnumbervsVCd} we have
\begin{align}
\mathcal{N}_1(1/(80n \beta_n),\mathcal{H}^{\beta_n}_{t_n},z_1^n) \leq & 3\left( \frac{4e \beta_n}{1/(80n \beta_n)}\ln\left(\frac{6e \beta_n}{1/(80n \beta_n)} \right) \right)^{VC({\mathcal{H}^{\beta_n}_{t_n}})}\nonumber \\
\leq & 3\left( \frac{4e \beta_n}{1/(80n \beta_n)}\ln\left(\frac{6e \beta_n}{1/(80n \beta_n)} \right) \right)^{VC({\mathcal{H}_{t_n}})}\nonumber \\
\leq & 3\left( \frac{4e \beta_n}{1/(80n \beta_n)}\ln\left(\frac{6e \beta_n}{1/(80n \beta_n)} \right) \right)^{c\cdot t_n \ln(t_n)}\nonumber \\
\leq & 3\left( 480e n \beta_n^2 \right)^{c\cdot t_n \ln(t_n)}\label{ConclusionPart2}
\end{align}
for any $z_1,\ldots,z_n\in\R^p$. Inequality \eqref{ConclusionPart2} implies that the second part of RHS of \eqref{Mainlemmaformulabeforeprun} satisfies
\begin{equation}\label{fsfsfz}
\frac{c\ln^2 n}{n}\cdot\sup_{x_1^n}\ln\left(\mathcal{N}_1(1/(80n\beta_n),\mathcal{H}_{t_n}^{\beta_n}, z_1^n)\right)\to 0,\ \text{as}\ n\to\infty,
\end{equation}
when $t_n =o\left(\frac{n}{\ln^4{n}}\right)$. This completes arguments of \textbf{Part II}.
\textbf{Part III}: By Lemma \ref{agirov2009estimation}, \eqref{djosijdisj} and \eqref{fsfsfz} imply that
\begin{equation}\label{chnosnolzm}
IV:=2\E_{\mathcal{D}_{n}} \int|\hat{m}_{n, \L}(x)-m(x)|^2d\mu(x)\to 0.
\end{equation}
Finally, we show that \eqref{chnosnolzm} also holds for the un-truncated estimator $m_{n,\L}(x)$. Note that
\begin{align}
\E_{\mathcal{D}_{n}} \int|m_{n, \L}(x)-m(x)|^2d\mu(x)\le & 2\E \int|\hat{m}_{n, \L}(x)-m_{n, \L}(x)|^2d\mu(x)+IV\nonumber\\
\le & 2\E \int|\hat{m}_{n, \L}(x)-m_{n, \L}(x)|^2\mathbb{I}(|m_{n, \L}(x)|>\beta_n)d\mu(x)+ IV\nonumber\\
\le & 2\E_{\D_n}\left(\E\left( |m_{n, \L}(X)|^2\cdot\mathbb{I}(|m_{n, \L}(X)|>\beta_n)|\D_n\right)\right)+ IV\nonumber\\
:= &V+IV. \label{fhuifhsdoui}
\end{align}
Since $\max_{x\in[0,1]^p}|m_{n, \L}(x)|\le \|m\|_{\infty}+\max_{1\le i\le n}{\epsilon_i} $, where $\epsilon_i=Y_i-m(X_i), 1\le i\le n$ are i.i.d. and share a common sub-Gaussian distribution, we have
\begin{align}
V\leq & 2\E_{\D_n}{\left[\left(2\|m\|_\infty^2+2\max_{1\le i\le n}{\epsilon_i^2}\right)\mathbb{I}\left(\max_{1\le i\le n}{\epsilon_i}\ge \beta_n-\|m\|_\infty\right)\right]}\nonumber\\
\le & 2\E_{\D_n}{\left[\left(2\|m\|_\infty^2+2\max_{1\le i\le n}{\epsilon_i^2}\right)\mathbb{I}\left(\max_{1\le i\le n}{\epsilon_i}\ge c\cdot\ln{n}\right)\right]}\nonumber\\
\le & 2\|m\|_\infty^2\cdot \P\left(\max_{1\le i\le n}{\epsilon_i}\ge c\cdot\ln{n}\right)+2\left[\E\left(\max_{1\le i\le n}{\epsilon_i^4}\right)\cdot \P\left(\max_{1\le i\le n}{\epsilon_i}\ge c\cdot\ln{n}\right)\right]^{\frac{1}{2}}.\label{fhoxufp}
\end{align}
Note that
\begin{align}
\P(\max_{1\leq i\leq n}|\epsilon_i|> c\cdot\ln{n})=&1- \P\left(\max_{1\leq i\leq n}|\epsilon_i|\leq c\cdot\ln{n}\right)\nonumber\\
=&1-\left[\P(|\epsilon_i|\leq c\cdot\ln{n})\right]^n
\leq 1-(1-c\cdot e^{-c\cdot \ln^2{n}})^n\nonumber \\
=& 1-e^{n\cdot \ln(1-c\cdot e^{-c\cdot \ln^2{n}})} \nonumber\\
\leq & -n\cdot \ln(1-c\cdot e^{-c\cdot \ln^2{n}})\label{opipo}\\
\leq & c\cdot n\cdot e^{-c\cdot \ln^2{n}},\label{duhsihrdksh}
\end{align}
where $\eqref{opipo}$ is obtained from the fact $1+v\le e^{v}, v\in\mathbb{R}$; and \eqref{duhsihrdksh} is due to the fact $\lim_{v\to 0}{\frac{\ln(1+v)}{v}}=1$. By \eqref{duhsihrdksh} and $\E\left(\max_{1\le i\le n}{\epsilon_i^4}\right)\le n\cdot \E(\epsilon_1^4)$, we have
\begin{equation}\label{opsdpz}
V\to 0.
\end{equation}
In conclusion, combination of \eqref{fhuifhsdoui}, \eqref{chnosnolzm} and \eqref{opsdpz} finishes arguments for \textbf{Part III}.
\end{proof}
For binary classification problem, where $Y$ only takes $0$ or $1$, we can still use ODT in which the Gini impurity is usually employed to divide each internal leaf defined by
$$
\Delta^c_A(\theta,s)=-\sum_{k=1}^2{P^2(k|A)}+P(A^+_{\theta,s})\sum_{k=1}^2{P^2(k|A^+_{\theta,s})}+P(A^-_{\theta,s})\sum_{k=1}^2{P^2(k|A^-_{\theta,s})},
$$
where $P(k|A)$ denotes the proportion of class $k$ in $A$. For the oblique classification tree, Algorithm \ref{Algorithm.ODTtreereg} still works after changing $\Delta_A(\theta,s)$ in \textbf{line 13} to $\Delta_A^c(\theta,s)$. Then by voting, the estimated class of input $x\in[0,1]^p$ is given by
\begin{equation}\label{hdfuioshjdfois}
\hat{C}_{n,t_n}(x)=
\left\{
\begin{array}{lc}
1, & m_{n,\L}(x) \geq 0.5, \\
0, & m_{n,\L}(x)< 0.5 .\\
\end{array}
\right.
\end{equation}
\begin{theorem}
The misclassification probability of classifier $\hat{C}_{n,t_n}(x)$ in \eqref{hdfuioshjdfois} satisfies
$$
\P(\hat{C}_{n,t_n}(X)\neq Y)-\inf_{f:[0,1]^p\to\{0,1\}}{\P\left(f(X)\neq Y\right)} \to 0
$$
under conditions in Theorem \ref{consistencyodtreg}.
\end{theorem}
\begin{proof}
By some calculations, it is easy to know
$$
\Delta^c_A(\theta,s)=2\Delta_A(\theta,s)
$$
if $Y$ only takes $0$ or $1$. Then, this theorem follows immediately from Theorem \ref{consistencyodtreg} and Theorem 1.1 in \cite{gyorfi2002distribution}.
\end{proof}
Pruning is important in the application of decision trees since it dramatically reduce the complexity of tree model and the variance of tree estimator. See, for example, \cite{breiman1984classification}. For regression tree, pruning is to select the best number of leaves by balancing squared loss function and a penalty:
$$
t_{n,r}^*=\argmin_{1\le \tau\le n}{\frac{1}{n}\sum_{i=1}^n{(Y_i-m_{n, \tau, \L}(X_i))^2}+\alpha_n\cdot \tau},
$$
where $\alpha_n>0$ is a given strength of penalty. Then, the pruned estimator in regression is given by
$$
m_{pru,n}(x):= m_{n, t_{n,r}^*, \L}(x).
$$
For classification tree, usually pruning is to select the best number of leaves by balancing $0-1$ loss and a penalty:
$$
t_{n,c}^*=\argmin_{1\le \tau\le n}{\frac{1}{n}\sum_{i=1}^n{\mathbb{I}(Y_i\neq \hat{C}_{n,\tau}(X_i))}+\alpha_n\cdot \tau},
$$
where $\alpha_n>0$ is a given strength of penalty and $\mathbb{I}(\cdot)$ is the indicator function. Then, the classifier after pruning is given by
$$
\hat{C}_{pru,n}(x):= \hat{C}_{n,t_{n,c}^*}(x).
$$
\begin{lemma}\label{afterprunlemmaa}
Let $\beta_n\asymp \ln{n}$. Assume
$
\E(e^{c\cdot Y^2})<\infty
$
for some $c>0$. Then, for any $1\le \tau\le n$ and $g_J\in \mathcal{G}_{J}$ we have
\begin{equation}\label{bSta4}
\begin{aligned}
\E_{\mathcal{D}_{n}}\left( \int|\hat{m}_{pru,n}(x)-m(x)|^2d\mu(x)\cdot\mathbb{I}(A_n)\right) \le &
2\E_{\mathcal{D}_{n}}\left(\left(\|m_{\tau}(X)-\bm{Y}\|_n^2-\|g_J(X)-\bm{Y}\|_n^2\right)\cdot \mathbb{I}(A_n)\right) \\
+& c\cdot\E\left(m(X)-g_J(X)\right)^2
+ 2\tau\cdot\alpha_n+c \cdot \frac{\ln^6{n}}{\alpha_n\cdot n} + c\cdot \frac{\ln n}{n},
\end{aligned}
\end{equation}
where $\hat{m}_{pru,n}(x)=\max{\{\min{\{m_{pru,n}(x),\beta_n\}},-\beta_n\}}$, and $A_n=\{\max_{1\le i\le n}{|Y_i|\le \beta_n}\}$, and $$
\mathcal{G}_{J}=\Big\{\sum_{j=1}^{J} {c_j\cdot \sigma(\theta_j^Tx+d_j)}: \theta_j\in\Theta^p, c_j,d_j\in\mathbb{R}, \forall j\ge 1\Big\}
$$
and $\sigma(v)=e^v/(1+e^v), v\in\mathbb{R}$.
\end{lemma}
\begin{proof}
The proof is similar to the proof of (6.23) in Lemma 1 of \cite{zhan2022ensemble}.
\end{proof}
\begin{theorem}[Consistency of ODT after pruning]\label{consistencyodtreg2}
Assume
$
\E(e^{c\cdot Y^2})<\infty,
$
for some $c>0$ and $m(x)$ is a continuous function. If $\alpha_n=o(1)$ and $\ln^6{n}/(\alpha_n\cdot n)=o(1)$, we have
$$
\E_{\D_n}\left(\int{|m_{pru,n}(x)-m(x)|^2d\mu(x)}\right)\to 0,\ \text{as}\ n\to\infty.
$$
\end{theorem}
\begin{proof}
Let $A_n=\{\max_{1\le i\le n}{|Y_i|\le \beta_n}\}$ and $\hat{m}_{pru}(x)=\max{\{\min{\{m_{pru}(x),\beta_n\}},-\beta_n\}}$, where $\beta_n\asymp \ln{n}$. Note that
\begin{align}
\E_{\D_n}\left(\int{|m_{pru,n}(x)-m(x)|^2d\mu(x)}\right)\le & 2\E_{\D_n}\left(\int{|m_{pru,n}(x)-\hat{m}_{pru,n}(x)|^2d\mu(x)}\right)\nonumber\\
&+2\E_{\D_n}\left(\int{|\hat{m}_{pru,n}(x)-m(x)|^2d\mu(x)}\cdot\mathbb{I}(A_n)\right)\nonumber\\
&+2\E_{\D_n}\left(\int{|\hat{m}_{pru,n}(x)-m(x)|^2d\mu(x)}\cdot\mathbb{I}(A_n^c)\right) \nonumber \\
:= & I+II+III.\label{guifshuik}
\end{align}
By similar arguments in proving $V\to 0$ in \textbf{Part III} of the proof of Theorem \ref{consistencyodtreg}, it is not difficult to show that
\begin{equation}\label{adhgauidh}
I\to 0.
\end{equation}
Next, we consider $II$. For any $\epsilon>0$, \eqref{iopipo} shows that there is $g_J\in\mathcal{G}_J$ for some large $J$ such that
\begin{equation}\label{jijipoq}
c\cdot\E\left(m(X)-g_J(X)\right)^2\le \frac{\epsilon}{3}.
\end{equation}
Fix above $g_J$ and let $\tau_n=1/\sqrt{\alpha_n}$ for each $n\in\mathbb{Z}_+$. Then, by \eqref{regpart1} we know there is a $N_1\in\mathbb{Z}_+$ such that \begin{equation}\label{reserrt1}
2\E\left((\|\bm{Y}-m_{\tau_n}(X)\|_n^2- \|\bm{Y}-g(X)\|_n^2)\cdot\mathbb{I}(A_n)\right)\le \frac{2\|g_J\|^2_{TV}}{\log_2{\tau_n}+4}\le\frac{\epsilon}{3}
\end{equation}
for all $n\ge N_1$. We can also find $N_2\in\mathbb{Z}_+$ such that
\begin{equation}\label{jdisjdfsp}
2\tau\cdot\alpha_n+c \cdot \frac{\ln^6{n}}{\alpha_n\cdot n} + c\cdot \frac{\ln n}{n}\le \frac{\epsilon}{3}
\end{equation}
for all $n\ge N_2$. By Lemma \ref{afterprunlemmaa}, the combination of \eqref{jijipoq} and \eqref{reserrt1} and \eqref{jdisjdfsp} shows that
\begin{equation}\label{huihdaskuhdku}
II\to 0.
\end{equation}
Finally, by $\|\hat{m}_{pru,n}\|_\infty\le\beta_n$ and $\|m\|_\infty<\infty$ and \eqref{duhsihrdksh}, it is easy to know
\begin{equation}\label{huohud}
III\to 0.
\end{equation}
In conclusion, combination of \eqref{guifshuik}, \eqref{adhgauidh}, \eqref{huihdaskuhdku} and \eqref{huohud} completes the proof.
\end{proof}
\section{Consistency of ODRF}
For ease of exposition, we introduce several notations that are often used in this part. Denote by $\mathcal{S}$ a (random) subset of $ \mathbf{\Omega}=\{ 1, ..., p\} $, and by $\mathcal{S}_\tau,\ \tau=1, 2, ...,$ a sequence of such subsets that may differ from one another. Let $ X_\S $ be a sub-vector of $ X $ consisting of coordinates indexed by elements of $\mathcal{S}$. Let $\mathbf{\Omega}=\cup_{j=1}^p\mathbf{\Omega}_j$, where $\mathbf{\Omega}_j$ is the collection of subsets with cardinality $j$. Let $q$ be a random number uniformly chosen from $\{1,\ldots,p\}$. Because given $q$ any element in $\mathbf{\Omega}_q$ will be selected with equal probability, $\mathbf{\Omega}$ is also a sample space equipped with probability
$$\P(\S)=\frac{1}{p}\cdot\frac{1}{\binom{n}{Card(\S)}}$$
for each $\S\subseteq \mathbf{\Omega}$. Thus, $ \S_\tau, \tau\geq 1 $, can be regarded as a (random) sample of $ \mathbf{\Omega} $. Let $\Xi_{t_n-1}=(\S_1,\ldots, \S_{t_n-1})$ with $t_n\geq 2$, which can be regarded as a random element in the product of probability spaces, $ \mathbf{\Omega} ^{\otimes t_n-1} $.
First, we introduce random ODT as follows. In Algorithm \ref{Algorithm.ODTtreereg}, we replace \textbf{lines 13-14} with following steps
\begin{itemize}
\item In the $\tau$-th division ($1\le \tau\le t_n-1$) of node $A$, randomly choose $\S_\tau\in\mathbf{\Omega}$ with $Card(\S_\tau)=q$;
\item $(\hat{\theta}_{A},\hat{s}_A)=\argmax_{\theta_{\S_\tau}\in\Theta^q,s\in\mathbb{R}}{\Delta_{A,q}({\theta}_{\S_\tau},s)}$, where $\Delta_{A,q}$ is defined in the same way as $\Delta_{A}$ except that only $\{(X_{i,\S_\tau},Y_i)\}_{i=1}^n$ are used in the calculation of $\Delta_{A,q}$;
\item Partition the node $A$ into $A^+_{\hat{\theta}_A,\hat{s}_A}=\{x\in A: \hat{\theta}_A^T\cdot x_{\S_\tau}\le \hat{s}_A\}$ and $A^-_{\hat{\theta}_A,\hat{s}_A}=\{x\in A: \hat{\theta}_A^T\cdot x_{\S_\tau}> \hat{s}_A\}$.
\end{itemize}
Denote the output estimator of Algorithm \ref{Algorithm.ODTtreereg} of the above randomization by $m^r_{n,t_n,\L}(x)$; the corresponding tree $T_{\mathcal{D}_n,t_n,\L}^r$ with $\L$ layers and $t_n$ leaves is named as random ODT. Note that $\L$ is a random variable depending on both $\D_n$ and $\{\S_\tau\}_{\tau=1}^{t_n-1}$. For the proof of the consistency, we also need to consider a truncated random tree at a particular layer $ \ell: 0 \le \ell \le \L $, denoted by $ T_{D_n,t_n,\L}^r $. For simplicity, we abbreviate $ T_{D_n,t_n,\ell}^r $ as $ T_{D_n,\ell}^r$ for each layer $\ell$ in the following context and similarly $m^r_{n,\L}(x)$ is an abbreviation of $m^r_{n,t_n,\L}(x)$ if there is no confusion. Adopting the idea of classical Random Forest \citep{breiman2001random}, our Random Forest based on ODT (ODRF) in regression is given by
\begin{equation}\label{ODTRandomforest}
m_{ODRF,n}(x)=\frac{1}{B}\sum_{b=1}^{B}{m^{r,b}_{n,t_n,\L}(x)},
\end{equation}
where $m^{r,b}_{n,t_n,\L}(x)$ is abbreviated as $m^{b}_{n,t_n,\L}(x)$ is the output estimator obtained by running above randomized Algorithm \ref{Algorithm.ODTtreereg} in the $b$-th time ($1\le b\le B$). It is noteworthy to highlight that ODRF is very similar to Forset-RC in \cite{breiman2001random} except that the later selects the best splitting plane with fixed $q$ variables from a number of finite randomly generated hyper-planes.
{\red{
\begin{lemma}\label{LemmaODTrandom}
Let $A$ be any internal node of random ODT $T_{\D_n,\L}^r$ in the $\tau$-th partition. Define $R(A):=\|\bm{Y}-\bar{Y}_A\|_A^2-\|\bar{Y}-g_q\|^2$ for any $g_q(v)=\sum_{k=1}^V{g_{k,q}(v)}$, where
$$
g_{k,q}(x)\in\mathcal{G}_{k,q,J}= \Bigg\{\sum_{j=1}^{J} {c_{k,q,j}\cdot \sigma(\theta_{\C_{k,q,j}}^Tx+d_{k,q,j})}: \C_{k,q,j}\in \mathbf{\Omega}_q, \theta_{\C_{k,q}}\in\Theta^{q}, c_{k,q,j},d_{k,q,j}\in\mathbb{R}, \forall j,k\ge 1\Bigg\},
$$
and $\sigma(v)=e^v/(1+e^v), v\in\mathbb{R}$. If $R(A)\ge 0$ given $q=q_0$, then we have
$$
\E_{\S_\tau}\left({\Delta_{A,q}(\hat{\theta}_{A},\hat{s}_A)}|\D_n,q=q_0\right)\geq c(p,q_0)\cdot \frac{R^2(A)}{\|g_{q_0}\|_{TV}^2\cdot V},
$$
where the expectation is taken over the random index $\S_\tau$ and $c(p,q_0)>0$.
\end{lemma}
}}
\begin{proof}
For ease of exposition, notations $A^+$ and $A^-$ are employed to denote $A^+_{\hat{\theta}_A,\hat{s}_A}$ and $A^-_{\hat{\theta}_A,\hat{s}_A}$ respectively when $q=q_0$. For each $1\le k\le L$, let $\Delta_{A}(\hat{\theta}_{A},\hat{s}_A, x_{\C_{k,q_0}})$ be the value of impurity gain in \eqref{impuritygainreg} when variables $x_{\C_{k,q_0}}$ are only used in the calculation of ${\Delta_{A}(\hat{\theta}_{A},\hat{s}_A)}$, namely only $x_{\C_{k,q_0}}$ are employed to divide the node $A$. Then, we have
\begin{equation}\label{huhadikl}
\E_{\S_\tau}\left({\Delta_{A,q}(\hat{\theta}_{A},\hat{s}_A)}|\D_n,q=q_0\right)\geq p_{q_0}\cdot\sum_{k=1}^V{\Delta_{A}(\hat{\theta}_{A},\hat{s}_A, x_{\C_{k,q_0}})}\geq p_{q_0}\cdot\max_{1\le k\le V}{\Delta_{A}(\hat{\theta}_{A},\hat{s}_A,x_{\C_{k,q_0}}}),
\end{equation}
where $p_{q_0}=1/p\cdot1/\binom{p}{q_0}$ is the probability of each ${\C_{k,q_0}}$. Let $g_{q_0}(x)=\sum_{k=1}^V{g_{k,q_0}(\theta_{\C_{k,q_0}}^Tx)}\in\mathcal{G}_{k,q_0, J}$, where $g_{k,q_0}(\theta_{\C_{k,q_0}}^Tx)= \sum_{j=1}^{J} {c_{k,q_0,j}\cdot \sigma(\theta_{\C_k}^Tx+d_{k,j})}, c_{k,q_0,j},d_{k,q_0,j}\in\mathbb{R}$. Define a series of weight functions of parameter $s\in [-1,1]$ by
$$
w_{k,q_0}(s)=\frac{|g'_{k,q_0}(s)|\sqrt{P(A^+|\C_{k,q_0})P(A^-|\C_{k,q_0})}}{\sum_{k=1}^V{\int_{-1}^{1}{|g'_{k,q_0}(s)|\sqrt{P(A^+|\C_{k,q_0})P(A^-|\C_{k,q_0})}}ds}},
$$
where $P(A^+|\C_{k,q_0})$ and $P(A^-|\C_{k,q_0})$ are proportions of data in $A^+$ and $A^-$ within $A$ respectively if the hyperplane $\theta^Tx_{\C_{k,q_0}}=s$ is used to divide $A$. Note that $w_{k,q_0}(s)\ge 0$ and $\int_{-1}^1{w_{k,q_0}(s)ds}\leq 1$. By following similar arguments in page 19 of \cite{klusowski2021universal}, it is not difficult to show that
\begin{equation}\label{pikojiq}
\max_{1\le k\le V}{\Delta_{A}(\hat{\theta}_{A},\hat{s}_A,x_{\C_{k,q_0}}})\geq \frac{|\langle \bm{Y}-\bar{Y}_A,g_{k,q_0}(\theta^T_{\C_{k,q_0}}x)\rangle_A|^2}{\left(\sum_{k=1}^V{\int_{-1}^{1}{|g'_{k,q_0}(s)|\sqrt{P(A^+|\C_{k,q_0})P(A^-|\C_{k,q_0})}}ds}\right)^2}
\end{equation}
for each $1\le k\le V$. By taking average on $|\langle \bm{Y}-\bar{Y}_A,g_{k,q_0}(\theta^T_{\C_{k,q_0}}x)\rangle_A|^2, 1\le k\le V$, \eqref{pikojiq} implies
\begin{equation}\label{bkdsahdlkh}
\max_{1\le k\le V}{\Delta_{A}(\hat{\theta}_{A},\hat{s}_A,x_{\C_{k,q_0}}})\geq \frac{|\langle \bm{Y}-\bar{Y}_A,g_{q_0}\rangle_A|^2}{V\cdot\left(\sum_{k=1}^V{\int_{-1}^{1}{|g'_{k,q_0}(s)|\sqrt{P(A^+|\C_{k,q_0})P(A^-|\C_{k,q_0})}}ds}\right)^2}.
\end{equation}
Note that
\begin{align}
\langle \bm{Y}-\bar{Y}_A,g_{q_0}\rangle_A &= \langle \bm{Y}-\bar{Y}_A,\bm{Y}\rangle_A-\langle \bm{Y}-\bar{Y}_A,\bm{Y}-g_{q_0}\rangle_A\nonumber\\
&\geq \|\bm{Y}-\bar{Y}_A\|_A^2- \|\bm{Y}-\bar{Y}_A\|_A\|\bm{Y}-g_{q_0}\|_A\nonumber\\
&\geq \|\bm{Y}-\bar{Y}_A\|_A^2-\frac{1}{2}\cdot \left(\|\bm{Y}-\bar{Y}_A\|_A\|^2+\|\bm{Y}-g_{q_0}\|_A^2\right),\label{lkpoakd}
\end{align}
where in the second line Cauchy-Schwarz inequality is applied. Therefore, \eqref{bkdsahdlkh} and \eqref{lkpoakd} and $R(A)\ge 0$ by the assumption of this lemma imply
\begin{equation}\label{eqeqa}
\max_{1\le k\le V}{\Delta_{A}(\hat{\theta}_{A},\hat{s}_A,x_{\C_{k,q_0}}})\geq \frac{R^2(A)}{4V\cdot\left(\sum_{k=1}^V{\int_{-1}^{1}{|g'_{k,q_0}(s)|\sqrt{P(A^+|\C_{k,q_0})P(A^-|\C_{k,q_0})}}ds}\right)^2}.
\end{equation}
Next we only need to consider how to bound the denominator of RHS of \eqref{eqeqa}. In fact, following similar arguments in page 20 of \cite{klusowski2021universal}, it is easy to get a bound by using total variation of each $g_{k,q_0}, 1\le k\le V$:
\begin{equation}\label{mdipajdmop}
\sum_{k=1}^V{\int_{-1}^{1}{|g'_{k,q_0}(s)|\sqrt{P(A^+|\C_{k,q_0})P(A^-|\C_{k,q_0})}}ds} \le \frac{1}{2}\cdot \|g_{q_0}\|_{TV}.
\end{equation}
Therefore, combination of \eqref{huhadikl} and \eqref{eqeqa} and \eqref{mdipajdmop} completes the proof.
\end{proof}
\begin{theorem}[Consistency of ODRF before pruning]\label{Theoremhusidhuioh}
Assume
$
\E(e^{c\cdot Y^2})<\infty,
$
where $c>0$ and $m(x), x\in[0,1]^p$ is continuous. If $t_n\to\infty$ and $t_n =o\left(\frac{n}{\ln^4{n}}\right)$, we have
$$
\E_{\mathcal{D}_{n},\Xi_{t_n-1}}\left(\int{|m_{ODRF,n}(x)-m(x)|^2d\mu(x)}\right)\to 0,\ \text{as}\ n\to\infty.
$$
\end{theorem}
\begin{remark}
This theorem shows that the random forest based on ODT is also consistent for the general model \eqref{Extendedmodel1} under the sub-Gauusian assumption of $Y$. Note that the classical Random Forest is only known to be consistent for additive models and strong conditions on the distribution of $X$ is also required in technique \citep{scornet2015consistency}.
\end{remark}
\begin{proof}
First, we prove the truncated estimator $\hat{m}_{n,\L}^r(x)=\max{\{\min{\{m_{n,\L}^r(x),\beta_n\}},-\beta_n\}}$, where $\beta_n\asymp \ln{n}$, is mean squared consistent under the conditions in Theorem \ref{Theoremhusidhuioh}. Following Lemma \ref{agirov2009estimation}, it is easy to show that
\begin{equation}\label{Mainlemmaformula1}
\begin{aligned}
\E_{\mathcal{D}_{n},\Xi_{t_n-1}} \int|\hat{m}^r_{t_n}(x)-m(x)|^2d\mu(x)\leq & 2\cdot \E_{\mathcal{D}_{n},\Xi_{t_n-1}}\left( \|m_{n,\L}^r(X)-Y\|_n^2-\|m(X)-Y\|_n^2\right)\\
&+\frac{c\ln^2 n}{n}\cdot\sup_{x_1^n}\ln\left(\mathcal{N}_1(1/(80n\beta_n),\mathcal{H}_{t_n}^{\beta_n}, z_1^n)\right)
\end{aligned}
\end{equation}
for some $c>0$, where ${\mathcal{H}_{t_n}^{\beta_n}}=\{\max\{\min\{h,\beta_n\},-\beta_n\}: h\in\mathcal{H}_{t_n}\}$ and $z_1^n\in \underbrace{\mathbb{R}^p\times\cdots\times \mathbb{R}^p}_n$; $\mathcal{H}_{t_n}$ is defined in Part II of the proof of Theorem \ref{consistencyodtreg}; $\mathcal{N}_1$ is the cover number in Definition \ref{coveringnumber}; and $\mu$ is the distribution of $X$. We prove this result by showing each part of RHS of \eqref{Mainlemmaformula1} converges to $0$ as $n$ goes to $\infty$, in a similar manner as Part I and Part II in the proof of Theorem \ref{consistencyodtreg}. Here, we only provide the proof of a key inequality:
\begin{equation}\label{regpart2dd}
\E_{\Xi_{t_n-1}}\left(\|\bm{Y}-m_{n,\L}^r(X)\|_n^2- \|\bm{Y}-g_p(X)\|_n^2\right)\le c(p)\cdot \frac{\|g_p\|^2_{TV}}{\lfloor\log_2{t_n}\rfloor+4}
\end{equation}
for any $t_n>1$ and real function $g_p\in\mathcal{G}_{1,p,J}$ presented in Lemma \ref{LemmaODTrandom} and some $c(p)>0$, which is analogous to \eqref{regpart1}.
Before the proof, recall the following facts:
\begin{itemize}
\item In the proof of \eqref{regpart2dd}, $\D_n$ is given and thus notation of conditional expectation or probability w.r.t. $\D_n$ is leaved out for simplicity.
\item For any internal node $A$ of $T^r_{\D_n,\L}$, $\Delta_{A,q}(\hat{\theta}_A,\hat{s}_A)$ only depends on $\S_\lambda$ for some $1\le\lambda\le t_n-1$ once data $\D_n$ is given.
\end{itemize}
Let us start prove \eqref{regpart2dd}. First, it is easy to observe following two facts:
\begin{itemize}
\item No matter the choice of each $\S_\tau,1\le \tau\le t_n-1$, the layer $\L$ of any random ODT $T_{\D_n,t_n}^r$ must be not smaller than $\L_0=\lfloor\log_2{t_n}\rfloor$;
\item Tree $T_{\D_n,\ell}^r$ is fully grown for each $0\leq \ell\le \L_0$, namely $T_{\D_n,\ell}^r$ is obtained recursively by splitting all leaves of $T_{\D_n,\ell-1}^r$ except for those which only contain one data point.
\end{itemize}
Recall that $\L$ is a random variable depending on both $\D_n$ and $\S_\tau, 1\le\tau\le t_n-1,$ and that $m_{n,\ell}^r(x)$ is the estimator obtained using leaves of $T_{\D_n,\ell}^r$. Given $\D_n$, define the expectation of $\|\bm{Y}-m_{\L}^r(X)\|_n^2$ over random $\theta$'s by
\begin{equation}\label{pokpok}
\E_{\ell}\left(\|\bm{Y}-m_{\L}^r(X)\|_n^2\right):=\sum_{T^r_{\D_n,\ell}} {\P_\theta{\left(T_{\D_n,\ell}^r\right)}}\cdot \|\bm{Y}-m_{n,\ell}^r(X)\|_n^2,
\end{equation}
where $\P_\theta{\left(T_{\D_n,\ell}^r\right)}$ is the probability of a realisation of $\S_\tau$s, each of which corresponds to a partition of an internal node of $T_{\D_n,\ell}$. By using the fact that $ \|\bm{Y}-m_{t_n}^r(X)\|_n^2$ is almost surely a decreasing sequence as $t_n$ increases, it is easy to check that
\begin{equation}\label{fuosjhnfliszjnkf}
\E_{\Xi_{t_n}}\left(\|\bm{Y}-m_{n,\L}^r(X)\|_n^2\right)\leq \E_{\L_0}\left(\|\bm{Y}-m_{n,\L_0}^r(X)\|_n^2\right).
\end{equation}
Therefore, we next only need to prove
\begin{equation}\label{dhzuadhnilq}
\E_{\L_0}\left(\|\bm{Y}-m_{\L_0}^r(X)\|_n^2\right)-\|\bm{Y}-g_p(X)\|_n^2\leq c(p)\cdot\frac{\|g_p\|^2_{TV}}{\lfloor\log_2{t_n}\rfloor+4}
\end{equation}
for some $c(p)>0$.
{ \red{First, we introduce some additional notations. Under the same notation in the proof of Theorem \ref{consistencyodtreg}, we define $R_{\D_n,\ell}: =\|\bm{Y}-m_{n,\ell}(X)\|_n^2-\|\bm{Y}-g(X)\|_n^2$ for each $1\le\ell\le \L_0$. Similarly, let $\mathcal{O}_{\ell-1,1}:=\{\mathbb{A}_{\ell-1}^j\}_{j=1}^{k_{\ell-1}}$ be leaves of $T^r_{\D_n,\ell-1}$ and $\mathcal{O}_{\ell-1,2}:=\{\mathbb{A}_{\ell-1}^{m_j}\}_{j=1}^{k'_{\ell-1}}\subseteq \mathcal{O}_{\ell-1,1}$ of which each node $\mathbb{A}_{\ell-1}^{m_j}$ must contain at least two data points. Let $\mathcal{Q}_{\ell-1,2}\subseteq\{\S_\tau\}_{\tau=1}^{t_n-1}$ which corresponds to each partition of each node in $\mathcal{O}_{\ell-1,2}$. Then, we have
\begin{equation}\label{fsjinflz}
R_{\D_n,\ell}= R_{\D_n,
\ell-1}-\sum_{A\in \mathcal{O}_{\ell-1,2}}{w(A)\cdot \Delta_{A,q}(\hat{\theta}_{A},\hat{s}_{A}}),
\end{equation}
where reacll $w(A)= \frac{1}{n}\cdot Card(\{X_i\in A: i=1,\ldots, n\})$.}} Given the tree $T^r_{\D_n,\ell-1}$, define the conditional expectation on $\mathcal{Q}_{\ell-1,2}$ by
\begin{equation}
\E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T^r_{\D_n,\ell-1} \right) := \sum_{m_{n,\ell}^r\ \text{is from } T^r_{\D_n,\ell}\ \text{generated by } T^r_{\D_n,\ell-1}}{\P_\theta{\left(T_{\D_n,\ell}^r\right)}\cdot \|\bm{Y}-m_{n,\ell}^r(X)\|_n^2}.\label{rtyrfht}
\end{equation}
Then, based on \eqref{fsjinflz} we also have
\begin{equation}\label{fjxisjflisdjflisj}
\E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T^r_{\D_n,\ell-1} \right) = R_{\D_n,
\ell-1}-\sum_{A\in \mathcal{O}_{\ell-1,2}}{w(A)\cdot \E_{\S^A}\left(
\Delta_{A,q}(\hat{\theta}_{A},\hat{s}_{A})
\right)},
\end{equation}
where random index $\S^A\in \{\S_\tau\}_{\tau=1}^{t_n-1}$ corresponds to the partition of $A$. Note that leaves $\mathcal{O}_{\ell-1,2}$ are not randomized once the tree $T^r_{\D_n,\ell-1}$ is given.
Now, we are ready to prove \eqref{dhzuadhnilq}. By \eqref{pokpok} and \eqref{rtyrfht}, it is easy to check
\begin{align}
\E_{\ell}\left(R_{\D_n,\ell}\right)&=\sum_{T^r_{\D_n,\ell}} {\P_\theta{\left(T_{\D_n,\ell}^r\right)}}\cdot \E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T_{\D_n,\ell-1} \right)\nonumber\\
&= \E_{\Xi_{t_n}}\left(\E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T^r_{\D_n,\ell-1} \right)\right),\label{pkoopkq}
\end{align}
which is similar to the law of iterated expectations. We highlight that both $T^r_{\D_n,\ell-1}$ and $\mathcal{Q}_{\ell-1,2}$ are random, so \eqref{pkoopkq} is not a trivial result from the classical law of iterated expectations. Then, from \eqref{fjxisjflisdjflisj} and Lemma \ref{LemmaODTrandom}, we have
\begin{align}
\E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T^r_{\D_n,\ell-1} \right) &= R_{\D_n,
\ell-1}-\sum_{A\in\O_{\ell-1,2}}{w(A)\cdot \E_{\S^A}\left(\Delta_{A,q}(\hat{\theta}_{A},\hat{s}_{A})\right)}
\nonumber \\
&\leq R_{\D_n,
\ell-1}-\sum_{A\in\O_{\ell-1,2}}{\frac{w(A)}{p}\cdot \E_{\S^A}\left(\Delta_{A,q}(\hat{\theta}_{A},\hat{s}_{A})|q=p\right)}
\nonumber \\
&\leq R_{\D_n,
\ell-1}-c(p)\cdot\sum_{A\in\O_{\ell-1,2}}{w(A)\cdot \frac{R^2(A)}{\|g_p\|_{TV}^2}}\nonumber\\
&\leq R_{\D_n,
\ell-1}-\frac{c(p)}{\|g_p\|_{TV}^2}\cdot\sum_{A\in\O_{\ell-1,2}:R(A)> 0}{w(A)\cdot R^2(A)},\label{koadk}
\end{align}
where $R(A)=\|\bm{Y}-\bar{Y}_{A}\|_{A}^2-\|\bm{Y}-g_p\|_{A}^2$ and $c(p)>0$. Now decompose $R_{\D_n,\ell-1}$ into two parts
$$R_{\D_n,\ell-1}^+:=\sum_{A\in\O_{\ell-1,1}:R(A)> 0}{w(A)R(A)},\ \ R_{\D_n,\ell-1}^-:=\sum_{A\in\O_{\ell-1,1}:R(A)\le 0}{w(A)R(A)}$$ satisfying $R_{\D_n,\ell-1}=R_{\D_n,\ell-1}^++R_{\D_n,\ell-1}^-$. By Jensen's inequality and the fact that $R(A)\le 0$ for any leaf $A$ of $T_{\D_n,\ell-1}$ which contains only one data point, we have
\begin{equation}\label{dgiasydfuip}
\sum_{A\in\O_{\ell-1,2}:R(A)> 0}{w(A)\cdot R^2(A)} \ge \left(\sum_{A\in\O_{\ell-1,2}:R(A)> 0}{w(A)\cdot R(A)}\right)^2=(R_{\D_n,\ell-1}^+)^2.
\end{equation}
Therefore, if $R_{\D_n,\ell-1}\ge 0$ then the combination of \eqref{koadk} and \eqref{dgiasydfuip} implies that
\begin{align}
\E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T^r_{\D_n,\ell-1} \right) &\le R_{\D_n,\ell-1}- \frac{c(p)}{\|g_p\|_{TV}^2}\cdot(R_{\D_n,\ell-1}^+)^2\nonumber\\
&\le R_{\D_n,\ell-1}- \frac{c(p)}{\|g_p\|_{TV}^2}\cdot R_{\D_n,\ell-1}^2, \label{dsfqwe23}
\end{align}
where \eqref{dsfqwe23} is from $R_{\D_n,\ell-1}^+>R_{\D_n,\ell-1}\ge 0$. In conclusion, \eqref{dsfqwe23} implies that
\begin{equation}\label{ojdfiasjdop}
\E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T^r_{\D_n,\ell-1} \right) \le R_{\D_n,\ell-1}- \frac{c(p)}{\|g\|_{TV}^2}\cdot \max{\{R_{\D_n,\ell-1},0\}}^2
\end{equation}
because $\E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T^r_{\D_n,\ell-1} \right) \le R_{\D_n,\ell-1}$ holds no matter what the sign of $R_{\D_n,\ell-1}$ is. By \eqref{pkoopkq} and Jensen's inequality again, taking expectation over $\Xi_{t_n}$ on both sides of \eqref{ojdfiasjdop} yields
\begin{align}
\E_{\ell}(R_{\D_n,\ell})&\le \E_{\ell-1}(R_{\D_n,\ell})-\frac{c(p)}{\|g_p\|_{TV}^2}\cdot \E_{t_n}\left(\max{\{R_{\D_n,\ell-1},0\}}^2\right)\nonumber\\
&\leq \E_{\ell-1}(R_{\D_n,\ell-1})-\frac{c(p)}{\|g_p\|_{TV}^2}\cdot [\E_{t_n}\left(\max{\{R_{\D_n,\ell-1},0\}}\right)]^2\label{jjoijkq}
\end{align}
for each $1\le\ell\le \L_0$. Finally, we complete the proof of \eqref{dhzuadhnilq} by considering the sign of $ \E_{\ell-1}(R_{\D_n,\ell-1})$ in the following two cases.
\textbf{Case 1:} There is an $ \ell_0 $, with $1\le\ell_0\le \L_0$, such that $\E_{\ell_0-1}{(R_{\D_n,\ell_0-1})}\le 0$. By checking \eqref{rtyrfht}, it is easy to know that
\begin{equation}\label{poqqasd}
\E_{\mathcal{Q}_{\ell-1,2}}\left(R_{\D_n,\ell}|T^r_{\D_n,\ell-1} \right) \le R_{\D_n,\ell-1}.
\end{equation}
By using \eqref{pkoopkq}, taking expectation over $\Xi_{t_n}$ on both sides of \eqref{poqqasd} implies that
\begin{equation*}
\E_{\ell}(R_{\D_n,\ell})\leq \E_{\ell-1}(R_{\D_n,\ell-1})
\end{equation*}
for all $1\le\ell\le \L_0$. Therefore, we have
\begin{equation}\label{poqaqasd}
\E_{\L_0}(R_{\D_n,\L_0})\le\E_{\ell_0-1}(R_{\D_n,\ell_0-1})\le c(p)\cdot\frac{\|g_p\|^2_{TV}}{\lfloor\log_2{t_n}\rfloor+4}
\end{equation}
by the assumption of this case.
\textbf{Case 2:} For each $1\le\ell\le \L_0$, we have $\E_{\ell-1}(R_{\D_n,\ell-1})> 0$. In this case, \eqref{jjoijkq} implies that
\begin{equation}\label{dgakdhkaa}
\E_{\ell}(R_{\D_n,\ell})\le \E_{\ell-1}(R_{\D_n,\ell-1})-\frac{c(p)}{\|g_p\|_{TV}^2}\cdot [\E_{\ell-1}\left({R_{\D_n,\ell-1}}\right)]^2
\end{equation}
for all $1\le\ell\le \L_0$ because $\E_{t_n}\left(\max{\{R_{\D_n,\ell-1},0\}}\right)\ge \E_{\ell-1}(R_{\D_n,\ell-1})>0$. When $\ell=1$, by \eqref{dgakdhkaa} it is easy to know that $\E_{1}(R_{\D_n,1})\le c(p)\cdot\frac{\|g_p\|_{TV}^2}{4}$. Using above initial condition and \eqref{dgakdhkaa} again, we also have
\begin{equation}\label{iaudfghbukajk122}
\E_{\L_0}{(R_{\D_n,\L_0})} \le c(p)\cdot \frac{\|g_p\|^2_{TV}}{\lfloor\log_2{t_n}\rfloor+4}
\end{equation}
by mathematical induction.
In conclusion, \eqref{dhzuadhnilq} follows \eqref{poqaqasd} and \eqref{iaudfghbukajk122}, which implies inequality \eqref{regpart2dd} is also true. By \eqref{Mainlemmaformula1} and following Parts I and II in the proof of Theorem \ref{Theoremhusidhuioh}, it is easy to know that $\hat{m}^r_{t_n}(x)$ is mean squared consistent. Then following the same arguments in Part III of that proof, we can also prove
\begin{equation}\label{ODTrandomfinal1beforeprun}
\E_{\mathcal{D}_{n},\Xi_{t_n-1}} \int|m^r_{n,\L}(x)-m(x)|^2d\mu(x)\to 0 \ \text{as}\ n\to 0.
\end{equation}
Finally, the application of Jensen's inequality
$$
\E_{\mathcal{D}_{n},\Xi_{t_n-1}} \int|m_{ODRF,n}(x)-m(x)|^2d\mu(x)\le \E_{\mathcal{D}_{n},\Xi_{t_n}} \int|m^r_{n,\L}(x)-m(x)|^2d\mu(x)
$$
shows that \eqref{ODTrandomfinal1beforeprun} also holds for the estimator $m_{ODRF,n}(x),x\in[0,1]^p$ obtained by ODRF.
\end{proof}
Next we consider Random Forest based on ODT (ODRF) after pruning. For each random tree indexed by $b, 1\le b\le B$, the estimator $m_{pru,b}^r(x), x\in[0,1]^p,$ corresponding to the pruned random ODT, is given by
$$
m_{pru,b}^r(x):= m_{t_{n,b}^*,b}^r(x),x\in[0,1]^p,
$$
where
$$
t_{n,b}^*=\argmin_{1\le \tau\le n}{\frac{1}{n}\sum_{i=1}^n{(Y_i-m^{b}_{n,\tau,\L}(X_i))^2}+\alpha_n\cdot \tau}
$$
and $\alpha_n>0$ is a pre-specified strength of penalty. Then, the estimator corresponding to the pruned ODRF is given by
$$
m_{ODRF,n}^{pru}(x):=\frac{1}{B}\sum_{b=1}^B{m_{pru,b}^r(x)}, x\in[0,1]^p,
$$
where $B\in\mathbb{Z}_+$ is the number of trees for the ensemble.
\begin{theorem}[Consistency of ODRF after pruning]\label{consistencyodtreg2forest}
Assume that the distribution of $Y$ satisfies
$
\E(e^{c\cdot Y^2})<\infty,
$
where $c>0$ and $m(x), x\in[0,1]^p$ follows model \eqref{Extendedmodel1} with each continuous $m_\tau(x), 1\le\tau\le V$. If $q\ge q_0$, $\alpha_n=o(1)$ and $\ln^6{n}/(\alpha_n\cdot n)=o(1)$, we have
$$
\E_{\D_n,\Xi_{t_n-1}}\left(\int{| m_{ODRF,n}^{pru}(x)-m(x)|^2d\mu(x)}\right)\to 0,\ \text{as}\ n\to\infty.
$$
\end{theorem}
\begin{proof}
The proof can be completed by following arguments in Theorem \ref{consistencyodtreg2} and Theorem \ref{Theoremhusidhuioh}.
\end{proof}
Note that the number of features in the linear combination is randomly selected from $1$ to $p$. Next, we introduce a simplified ODRF with fixed $q$ (ODRF$_q$), namely in each time $\theta_\tau$ is only uniformly chosen from $\bm{\Omega}_q$. Its corresponding tree is similarly named as random ODT$_q$. In this case, we show below that estimators $m_{ODRF_q,n}$ and $m_{ODRF_q,n}^{pru}$, corresponding to ODRF$_q$ before and after pruning respectively, are consistent for a extended additive model:
\begin{equation}\label{Extendedmodel1}
m(x)=\sum_{\tau=1}^V{m_\tau(x_{\C_\tau})},
\end{equation}
where $x_{\C_\tau}$ is defined similarly to above $x_{\A_\tau}$ and each $m_\tau$ is a function of only $q$ variables $x_{\C_\tau}$ with $ Card(\C_\tau) = q $. Note that $\C_\tau, 1\le \tau\le V,$ are fixed indexes and the maximum terms $ V $ in model \eqref{Extendedmodel1} is $ \binom{p}{q} $.
\begin{corollary}[Consistency of ODRF$_q$ before pruning]\label{Corollary1ds}
Assume
$
\E(e^{c\cdot Y^2})<\infty,
$
where $c>0$ and $m(x), x\in[0,1]^p$ follows model \eqref{Extendedmodel1} with each continuous $m_\tau(x), 1\le\tau\le V$. If $t_n\to\infty$ and $t_n =o\left(\frac{n}{\ln^4{n}}\right)$, we have
$$
\E_{\mathcal{D}_{n},\Xi_{t_n-1}}\left(\int{|m_{ODRF_q,n}(x)-m(x)|^2d\mu(x)}\right)\to 0,\ \text{as}\ n\to\infty.
$$
\end{corollary}
\begin{corollary}[Consistency of ODRF$_q$ after pruning] \label{Corollary2ds}
Assume
$
\E(e^{c\cdot Y^2})<\infty,
$
where $c>0$ and $m(x), x\in[0,1]^p$ follows model \eqref{Extendedmodel1} with each continuous $m_\tau(x), 1\le\tau\le V$. If $\alpha_n=o(1)$ and $\ln^6{n}/(\alpha_n\cdot n)=o(1)$, we have
$$
\E_{\D_n,\Xi_{t_n-1}}\left(\int{| m_{ODRF_q,n}^{pru}(x)-m(x)|^2d\mu(x)}\right)\to 0,\ \text{as}\ n\to\infty.
$$
\end{corollary}
\vspace{0.1cm}
As a summary of Theorems \ref{Theoremhusidhuioh} or \ref{consistencyodtreg2forest} and Corollaries \ref{Corollary1ds} and \ref{Corollary2ds}, if the subset size of the features $ q $ is randomly selected from 1 to p in each splitting, then the estimator is consistent for any continuous underlying regression function, if $ q $ is fixed, the estimator is consistent only if the underlying regression function has a special structure. As a special case, $q= 1$ corresponds to the traditional RF, the estimator is consistent for the additive models.
\section{Numerical Performance in Real Data} \label{secReal}
Many computer packages have been developed to implement ODT and ODRF, such as those developed by \cite{menze2011oblique}, \cite{blaser2016random} and \cite{tomita2020sparse}. However, none of them have become as popular as RF, probably because of the excessive computational time and the undesirable accuracy improvements over RF; see \cite{majumder2020ensembles}.
Note that the main difficulty in implementing ODT or ODRF is the estimation of the coefficient, $ \theta $, for the linear combinations, which is also one of the main differences amongst all the existing packages. The estimation methods of $ \theta $ include random projection, logistic regression, dimension reduction and many others. However,
our experiments suggest that these estimations actually make little difference in the results. In our calculation, instead of using one single projection or linear combination, we provide a number of $ \theta $s, each of which is for the projection of a set of randomly selected $ q $ predictors, and then use Gini impurity or residuals sum of squares to choose one combination as splitting variable and splitting point. This is similar to \cite{tomita2020sparse} where a number of random projections are provided from which one is selected. Also because of the low estimation efficiency of $ \theta $ as dimension $ q $ increases, we select $q $ randomly from 1 to $min([n^{0.5}], p) $ at each node. This selection of $ q $ satisfies the requirements of Theorem \ref{Theoremhusidhuioh} or \ref{consistencyodtreg2forest} for the consistency. Our code, called ODRF, is based on the above design of bagging. In our calculation, the logistic regression function is used to find $ \theta $ for each set of $ q $ predictors, but other options are also provided in the code.
Our ODRF will be compared with the following methods or packages: the Random Rotation Random Forest (RotRF) of \cite{blaser2016random} which randomly rotates the data prior to inducing each tree, the Sparse Projection Oblique Random Forests (SPORF) of \cite{tomita2020sparse} which simply uses the random projection method, and other methods that also use linear combinations as splitting variables including the method of \cite{silva2021projection}, denoted by \texttt{PPF}, and the method of \cite{menze2011oblique}, denoted by \texttt{ORF}. The comparison is also made with three axis-aligned popular methods, including Random Forest (RF), Generalized Random Forest (GRF) and extremely randomized trees (ERT), and the extreme gradient boosting (XGB). The following functions and packages in R are used for the calculations: \texttt{randomForest} \citep{breiman2001random} for RF, \texttt{regression\_forest} and \texttt{Classification\_forest} in package \texttt{grf} \citep{athey2019generalized} for GRF, \texttt{RLT} \citep{zhu2015reinforcement} for ERT, \texttt{xgboost} \citep{chen2016xgboost} for XGB. For all the R functions and packages, their default values of tuning parameters are used. Note that because \texttt{PPF} and \texttt{ORF} cannot be used for regression, we only report the classification results.
We use 20 real data sets with continuous responses and 20 data sets with binary categorical response (0 and 1) to demonstrate the performance of the above methods. The data are available at one of the following webpages (A) \url{https://archive.ics.uci.edu/ml/datasets}, (B) \url{https://github.com/twgr/ccfs/} and (C) \url{https://www.kaggle.com}.
If there are any missing values in a data, the corresponding samples are removed from the data. In the calculation, each predictor is scaled to [0, 1].
\begin{table}[h!]\small
\centering
\caption{Regression: average RPE based on 100 random partitions of each data set into training and test sets}\label{Table1
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{lrr|rrrr:rrr}
\hline
\multirow{2}[2]{*}{Dataset} & \multirow{2}[2]{*}{n} & \multirow{2}[2]{*}{p} & \multicolumn{4}{c:}{Axis-aligned} & \multicolumn{3}{c}{Oblique} \\
& & & RF & GRF & ERT & XGB & RotRF & SPORF & ODRF \\
\hline
Servo (B) & 166 & 4 & 0.287 & 0.268 & 0.632 & \textbf{\footnotesize 0.108} & 0.377 & 0.246 & 0.175 \\
Strike (B) & 624 & 6 & 0.844 & 0.793 & 0.790 & 1.197 & 0.797 & \textbf{\footnotesize 0.774} & 0.776 \\
Auto MPG (B) & 391 & 7 & 0.132 & 0.149 & 0.156 & 0.152 & 0.141 & 0.143 & \textbf{\footnotesize 0.127} \\
Low birth weight (B) & 188 & 9 & 0.397 & \textbf{\footnotesize 0.365} & 0.414 & 0.504 & 0.403 & 0.415 & 0.366 \\
Pharynx (B) & 192 & 12 & 0.398 & 0.388 & 0.557 & 0.353 & 0.541 & 0.503 & \textbf{\footnotesize 0.317} \\
Body fat (B) & 251 & 14 & 0.078 & 0.039 & 0.107 & 0.039 & 0.163 & 0.139 & \textbf{\footnotesize 0.034} \\
Paris housing price (C) & 10000 & 16 & 0.011 & \textbf{\footnotesize 0.000} & 0.035 & \textbf{\footnotesize 0.000} & 0.265 & 0.114 & \textbf{\footnotesize 0.000} \\
Parkinsons (A) & 5875 & 19 & 0.240 & 0.202 & 0.429 & \textbf{\footnotesize 0.068} & 0.456 & 0.362 & 0.208 \\
Auto 93 (B) & 81 & 22 & 0.402 & 0.460 & 0.369 & 0.491 & 0.438 & 0.421 & \textbf{\footnotesize 0.354} \\
Auto horsepower (B) & 159 & 24 & 0.114 & 0.198 & 0.118 & 0.147 & 0.216 & 0.166 & \textbf{\footnotesize 0.101} \\
Wave energy converters-Adelaide (C) & 71998 & 32 & 0.244 & 0.359 & 0.263 & 0.263 & 0.284 & 0.274 & \textbf{\footnotesize 0.238} \\
Baseball player statistics (C) & 4535 & 74 & 0.012 & 0.016 & 0.021 & \textbf{\footnotesize 0.001} & 0.293 & 0.165 & \textbf{\footnotesize 0.001} \\
Year prediction MSD (A) & 50000 & 90 & 0.820 & 0.873 & 0.823 & 0.927 & 0.817 & 0.814 & \textbf{\footnotesize 0.761} \\
Residential building-Sales (A) & 372 & 103 & 0.046 & 0.081 & 0.069 & \textbf{\footnotesize 0.018} & 0.196 & 0.157 & 0.019 \\
Residential building-Cost (A) & 372 & 103 & 0.064 & 0.101 & 0.081 & 0.050 & 0.138 & 0.118 & \textbf{\footnotesize 0.044} \\
Geographical original-Latitude (A) & 1059 & 116 & 0.737 & 0.831 & 0.759 & 0.822 & 0.792 & 0.773 & \textbf{\footnotesize 0.745} \\
Geographical original-Longitude (A) & 13063 & 249 & \textbf{\footnotesize 0.529} & 0.582 & 0.553 & 0.629 & 0.766 & 0.570 & 0.538 \\
Credit score (C) & 80000 & 259 & 0.111 & 0.151 & 0.111 & 0.113 & 0.251 & 0.124 & \textbf{\footnotesize 0.100} \\
CT slices (A) & 53500 & 380 & 0.049 & 0.123 & \textbf{\footnotesize 0.048} & 0.061 & 0.129 & 0.080 & 0.050 \\
UJIndoor-Longitude (A) & 19937 & 465 & 0.012 & 0.027 & 0.022 & 0.024 & 0.044 & 0.015 & \textbf{\footnotesize 0.010} \\
\hline
\multicolumn{3}{c|}{Average of RPE across all data sets} & 0.276 & 0.300 & 0.318 & 0.298 & 0.375 & 0.319 & 0.248 \\
\multicolumn{3}{c|}{no. of bests in 20 datasets} & 1 & 2 & 1 & 5 & 0 & 1 & 13 \\
\hline
\end{tabular
}
\end{table
For each data set, we randomly partition it into training set and test set. The training set consists of $n=\min(\lfloor 2N/3\rfloor,2000)$ randomly selected observations, where $N$ is the number of observations in the original data sets, and the remaining observations form the test set. For regression, the relative prediction error, defined as
$$RPE=\sum_{i\in \text{test set}}(\hat{y}_i-y_i)^2/\sum_{i\in \text{test set}}(\bar{y}_{\text{train}}-y_i)^2,$$
where $\bar{y}_{\text{train}}$ is naive predictions based on the average of $y$ in the training sets, is used to evaluate the performance of a method. For classification, the misclassification rate, defined as
$$MR=\sum_{i\in \text{test set}} 1(\hat{y}_i \neq y_i) /(N-n),$$
is used to assess the performance. For each data set, the random partition is repeated 100 times, and averages of the RPEs or MRs are calculated to compare different methods. The calculation results are listed in {Table \ref{Table1}} and {Table \ref{Table2}}. The smallest RPE or MR for each data set is highlighted in \textbf{bold} font.
\begin{table}[h!]\small
\centering
\caption{Classification: average MR (\%) based on 100 random partitions of each data set into training and test sets}\label{Table2
\centerline{ \hspace{-0.3cm}{ \small
\begin{tabular}{l@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}
|r@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}:r@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}r@{\hspace{0.5\tabcolsep}}r}
\hline
\multirow{2}[2]{*}{Dataset} & \multirow{2}[2]{*}{n} & \multirow{2}[2]{*}{p} & \multicolumn{4}{c:}{Axis-aligned} & \multicolumn{5}{c}{Oblique} \\
& & & RF & GRF & ERT & XGB & {\scriptsize Rot\footnotesize RF} & {\scriptsize SPO\footnotesize RF} & {\footnotesize PPF} &ORF& {\footnotesize ODRF}\\
\hline
MAGIC (B) & 19020 & 10 & 6.78 & 5.40 & \textbf{\footnotesize \small 4.98} & 10.93 & 6.64 & 6.27 & 9.51 & 7.56 & 5.38 \\
EEG eye state (A) & 14980 & 14 & 13.46 & 19.72 & 15.44 & 17.37 & 14.14 & 12.14 & 47.49 & \textbf{\footnotesize 8.03} & 9.06 \\
Diabetic retinopathy debrecen (A) & 1151 & 19 & 32.10 & 32.73 & 32.38 & 34.76 & 31.93 & 28.34 & 33.36 & 24.43 & \textbf{\footnotesize 23.68} \\
Parkinson multiple sound (C) & 1208 & 26 & 27.50 & 29.71 & 28.12 & 30.92 & 27.96 & 27.22 & 34.96 & 26.23 & \textbf{\footnotesize 25.67} \\
Pistachio (C) & 2148 & 28 & 10.31 & 11.76 & 10.45 & 11.09 & 9.19 & 9.26 & 10.85 & 7.39 & \textbf{\footnotesize 7.18} \\
Breast cancer (C) & 569 & 30 & 4.31 & 5.73 & 4.08 & 6.08 & 2.85 & 3.30 & 4.20 & \textbf{\footnotesize 2.81} & \textbf{\footnotesize 2.81} \\
Waveform (2) (B) & 5000 & 40 & 14.58 & 16.51 & 14.32 & 15.10 & 14.02 & 13.77 & 19.75 & 40.62 & \textbf{\footnotesize 13.71} \\
QSAR biodegradation (A) & 1055 & 41 & 13.34 & 15.34 & 13.12 & 15.75 & 13.22 & 13.03 & 17.48 & 12.94 & \textbf{\footnotesize 12.64} \\
Spambase (A) & 4601 & 57 & 5.34 & 7.36 & 5.36 & 6.74 & 7.54 & 5.19 & 10.60 & 7.37 & \textbf{\footnotesize 4.86} \\
Mice protein expression (A) & 1047 & 70 & 1.39 & 6.08 & 1.30 & 7.02 & 1.25 & 1.00 & 4.21 & \textbf{\footnotesize 0.60} & 0.95 \\
Ozone level detection (A) & 1847 & 72 & 6.17 & 6.83 & 6.44 & 7.22 & 6.21 & 6.16 & 19.69 & 6.06 & \textbf{\footnotesize 5.91} \\
Insurance company benchmark (A) & 5822 & 85 & 6.62 & \textbf{\footnotesize 5.97} & 6.16 & 9.20 & 6.19 & 6.19 & 6.62 & 6.97 & 6.42 \\
Company bankruptcy (C) & 6819 & 94 & \textbf{\footnotesize 3.08} & 3.21 & 3.09 & 3.64 & 3.19 & 3.13 & 3.11 & 3.18 & 3.13 \\
Hill valley (B) & 1211 & 100 & 40.83 & 45.68 & 42.16 & 37.81 & 11.21 & 1.30 & 29.23 & \textbf{\footnotesize 0.00} & \textbf{\footnotesize 0.00} \\
Hill valley noisy (B) & 1211 & 100 & 43.61 & 49.60 & 44.83 & 44.02 & 21.07 & 14.47 & 29.38 & 4.70 & \textbf{\footnotesize 3.83} \\
Musk (A) & 6598 & 166 & \textbf{\footnotesize 3.89} & 6.88 & 4.57 & 4.80 & 5.53 & 4.71 & 8.06 & 4.23 & 4.25 \\
ECG heartbeat categorization (C) & 14550 & 186 & 10.23 & 16.90 & 10.23 & 13.81 & 18.41 & 9.39 & 21.68 & 17.53 & \textbf{\footnotesize 8.76} \\
Arrhythmia (A) & 420 & 192 & \textbf{\footnotesize 19.19} & 23.10 & 21.32 & 24.45 & 24.10 & 20.96 & 23.00 & 23.14 & 21.42 \\
Financial indicators of US stocks (C) & 986 & 216 & 0.99 & 1.31 & 4.56 & \textbf{\footnotesize 0.04} & 14.31 & 2.40 & 1.98 & 12.11 & 0.15 \\
Madelon (A) & 2000 & 500 & 32.10 & 36.23 & \textbf{\footnotesize 29.22} & 30.26 & 41.25 & 38.29 & 41.83 & 43.99 & 39.93 \\
Human activity recognition (A) & 2633 & 561 & 0.06 & 0.14 & 0.12 & \textbf{\footnotesize 0.00} & 0.05 & 0.06 & 0.08 & \textbf{\footnotesize 0.00} & 0.06 \\
\hline
\multicolumn{3}{c|}{Average MR(\%) across all data sets} & 14.09 & 16.48 & 14.39 & 15.76 & 13.35 & 10.79 & 17.96 & 12.38 & 9.51 \\
\multicolumn{3}{c|}{no. of bests in 20 datasets} & 3 & 1 & 2 & 2 & 0 & 0 & 0 & 5 & 11 \\
\hline
\end{tabular
}}
\end{table
By comparing the prediction errors, both the RPE of regression and MR of classification, our ODRF is generally smaller than the other methods. Our ODRF is quite stable and attains the smallest RPE and MR in most data sets as listed in {Table \ref{Table1}} and {Table \ref{Table2}}. The advantages of ODRF is also verified by the fact that it has the smallest average of RPEs (or MSs) across all the data sets amongst all the methods. The numbers of data sets for which a method performs the best among the all the competitors, denoted by \textit{no. of bests}, also suggest ODRF's superiority over others including both marginal based forests and those with linear combinations as splitting variables.
Finally, we make a brief conclusion for the above experiments. Although many computer programs are developed for ODRF, but they don't show consistently better performance over RF and are not commonly received; see for example \cite{majumder2020ensembles}. We attribute this lack of improvement to the programming details and the choice of linear combinations for the splitting. After refining these issues and redesigning the bagging, all of our experiments, including many not reported here, can indeed produce a more significant overall improvement than RF. With this numerical improvement and theoretical guarantee of consistency, ODRF is expected to become more popular in the future.
\
\bibliographystyle{apa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,250 |
Q: How to manage user Onboarding Screen in ionic app How to Check when press the app icon that user is newly user for that app. and if that user is newly show onboard screen for this this user.
A: You can use localstorage to check if the app is opened for the first time and display an onboarding modal or page accordingly :
import { Storage } from '@ionic/storage';
constructor(private storage: Storage) { }
this.storage.get('firstOpening').then((firstOpening) => {
if (!firstOpening) {
//Open onboarding modal or navigate to page
}
});
And before leaving your onboarding page your can set the value in localstorage, so the page will not be displayed again :
this.storage.set("firstOpening", true);
A: At the initialization of the app you can do a check if the user has finished the onboarding. You can do the check by using the local storage / Ionic storage. I would suggest to do the check in app.component.ts.
Example:
app.component.ts
export class AppComponent {
constructor(
...
private router: Router,
private storageService: StorageService,
) {
this.initializeApp();
}
initializeApp() {
this.platform.ready().then(() => {
this.splashScreen.hide();
// Method to send user to onboarding pages
this.pushToAppOnboarding();
});
}
async pushToAppOnboarding() {
await this.storageService.getItem('onboarding').then((key) => {
// StorageService only takes strings. Thus, compare if the string equals 'true'.
if (key.value !== 'true') {
this.router.navigate(['/onboarding']);
}
});
}
}
onboarding.page.html
<ion-content>
<!-- place your onboarding content here. Use e.g. ion-slider -->
<ion-button (click)="finishAppOnboarding()">Lets Start</ion-button>
</ion-content>
onboarding.page.ts
constructor(
private router: Router,
private storageService: StorageService,
) { }
finishAppOnboarding() {
this.storageService.setItem('onboarding', 'true');
this.router.navigate(['/route-to-your-homepage']);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,768 |
{"url":"https:\/\/www.physicsforums.com\/threads\/what-is-absolute.2107\/","text":"# What is absolute?\n\nStaff Emeritus\nGold Member\nin your own words, define what you feel is absolute...\n\nVodka.\n\nMentat\nOriginally posted by Kerrie\nin your own words, define what you feel is absolute...\n\nDo you mean to define what it means to be absolute, or to define something that I personally believe to be absolute.\n\nwuliheron\nOriginally posted by Kerrie\nin your own words, define what you feel is absolute...\n\nAcceptance\n\nMentat\n\nOriginally posted by wuliheron\nAcceptance\n\nAcceptance of what?\n\nStaff Emeritus\nGold Member\nOriginally posted by LogicalAtheist\nVodka.\n\nOriginally posted by Kerrie\nin your own words, define what you feel is absolute...\n\nKnowledge of abstract objects (mental constructs) is absolute.\n\nwuliheron\n\nOriginally posted by Mentat\nAcceptance of what?\n\nAcceptance of course. :0)\n\nMentat\n\nOriginally posted by wuliheron\nAcceptance of course. :0)\n\nCool!\n\nIacchus32\nOriginally posted by Kerrie\nin your own words, define what you feel is absolute...\nPotentials? Where everything in between are just tendencies?\n\nabsolute is what this discussion isn't\n\nAcceptance of Absolute Vodka only...\n\nOriginally posted by wuliheron\nAcceptance of course. :0)\n\nGreetings !\nOriginally posted by Kerrie\nin your own words, define what you feel is absolute...\nAbsolute abstract systems.\nOriginally posted by wuliheron\nAcceptance\nCould you, please, prove even that is absolute ?\nOriginally posted by LogicalAtheist\nVodka.\n\nLive long and prosper.\n\nwuliheron\n\nOriginally posted by drag\n\nCould you, please, prove even that is absolute ?\n\nProving that acceptance of acceptance is absolute is a personal affair. You must prove it to yourself, the finger pointing at the moon is not to be confused with the moon itself.\n\nheusdens\nOriginally posted by Kerrie\nin your own words, define what you feel is absolute...\n\n\"Everything is a lie\"\n\nIacchus32\nOriginally posted by heusdens\n\"Everything is a lie\"\nOh, do you mean propaganda? ... Absolutely!\n\nPsychodelirium\nOriginally posted by Kerrie\nin your own words, define what you feel is absolute...\n\nAbsolute? What's that?\n\nMentat\n\nOriginally posted by heusdens\n\"Everything is a lie\"\n\nThat's the Liar's Paradox, and makes no rational sense, but I kind of agree (on a more broad, less specific, level).\n\nDissident Dan\nEverything that is is absolute. Things often are not true in absolutely all circumstances, but things that are true are absolutely true in their circumstances. It is absolutely true that there is existence outside of myself.\n\nwimms\nConcept of Time\/existence is absolute.\n\nIacchus32","date":"2022-12-09 12:43:47","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\": 0.85220867395401, \"perplexity\": 4621.459838235244}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446711396.19\/warc\/CC-MAIN-20221209112528-20221209142528-00796.warc.gz\"}"} | null | null |
Mike Cuddy reflects on Tech 20/20 through the years
on January 25 th, 2012 by Tom Ballard
Only a handful of individuals can claim to have been "officially" associated with Technology 2020 (or Tech 20/20 as it was recently rebranded) for its entire existence, and only one person can claim to have served as both a board Chair and CEO. That individual is Mike Cuddy, who has announced his retirement from the top leadership position at the end of the month.
Cuddy was on the organizing board in the early1990s and served as the second chair of the board of directors. He served continuously on the board until resigning in late 2008 to become CEO, succeeding Tom Rogers who had joined Oak Ridge National Laboratory (ORNL) earlier that year as Director of Industrial and Economic Development Partnerships.
In a recent interview with teknovation.biz, Cuddy reflected on his nearly 20-year history with Tech 20/20 and a 45-year career in Oak Ridge. He recalled the impetus for creating the organization, the initial strategy and early challenges, the growth years and, more recently, the impact of the multi-year recession.
"In many respects, it's déjà vu all over again," he said, noting that the Oak Ridge community is concerned today about economic challenges with federal funding just as it was 20 years ago.
In the late 1980s, Cuddy was a manager with Martin Marietta and later Lockheed Martin. He recalled that Oak Ridge was dealing with cutbacks such as closure of the Gaseous Diffusion Plant.
"Gene Joyce, a long-time Oak Ridge leader, said we need an information teleport," Cuddy said. Joyce pulled together a core group including Rogers, who was head of the Oak Ridge Chamber; Mayor Al Bissell and his son, Keith, then a member of the old Tennessee Public Service Commission; and Pete Craven, another long-time community leader.
"It was all about diversification of the local economy," Cuddy noted.
The core group recruited additional individuals from the region such as Alex Fischer with the former Tennessee's Resource Valley organization and Homer Fisher, a senior executive at UT. They secured funding from BellSouth to build a facility in Commerce Park in Oak Ridge. The BellSouth funding also provided five years of operating support.
Over the next decade, Tech 20/20 added programs and broadened its regional impact with Rogers as CEO and a diversified board of directors on which Cuddy has continuously served since its creation. (EDITOR'S NOTE: I served on the founding board, left the board for seven years and have since returned.)
"We took on the role of a pioneer," Cuddy said.
In his view, several key initiatives clearly made a difference. They included the emphasis on creating more investment capital in the region, launching the Digital Crossing (DC) shared services center in Downtown Knoxville, establishing the Information Technology Business Association that is now known as the Tennessee Valley Technology Council, launching the "Venture Forum" that has morphed into the "Entrepreneurial Imperative," and creating the Center for Entrepreneurial Growth (CEG).
Cuddy said Southeastern Community Capital (SCC), which was founded by Tech 20/20 and later spun-out, "is a real high water mark and credit" for the organization. SCC is now known as Pathway Lending with a $100 million loan fund.
Another wise decision was "investing in a data center in downtown Knoxville" that became known as the Digital Crossing. With grants and other investments, DC was launched and provided funding "that allowed the institution (Tech 20/20) to do things that would otherwise not have been possible," Cuddy said.
He also cited 1999 as a particularly significant year when the U.S. Department of Energy issued an RFP for the management of ORNL, and a partnership comprised of UT and Battelle Memorial Institute won. "This was really, really good for the region," Cuddy said, because of the role that UT-Battelle, LLC assigned to Tech 20/20 for technology commercialization.
"This was a real boost to our region," Cuddy said, because it led to the creation of the Center for Entrepreneurial Growth (CEG) to work with start-ups that licensed ORNL technologies. "Soon after that, UT followed suit" with a similar CEG program for its licensees.
Cuddy noted that the CEG expanded beyond the Oak Ridge-Knoxville region to help start similar programs in Chattanooga and Western North Carolina. The organization also manages a CEG program for the National Institute for Hometown Security in Somerset, KY.
Cuddy took over as CEO at a time when the nation's economic challenges were accelerating and Tech 20/20's nearly 15-year old facility was beginning to show its age. Investment capital was more challenging for start-ups to secure, and the Digital Crossing lost its best customer. In spite of these challenges, Cuddy believes that Tech 20/20 has weathered the storm and is poised for long-term success.
He cites the recent sale of the Digital Crossing which provides a $1.5 million cash reserve for Tech 20/20.
He talks about a strong relationship that he has developed with Oak Ridge City Manager Mark Watson and the planned renaming of the Tech 20/20 building as the "Oak Ridge Tech Commercialization Center" to highlight the importance of start-ups to the city's future.
He cites the key role that CEG is playing in the new East Tennessee Regional Accelerator Coalition (ETRAC), one of nine regional accelerators funded in part by grants from the Tennessee Department of Economic and Community Development.
He talks about a strong staff, although he regrets recent staff reductions that were necessary in light of the loss of cash flow from the Digital Crossing.
He cites the new partnership with Solidus called "Venture Incite" as a key foundation for success.
As to what the future holds for Cuddy, he said that he is still considering options, but he expects to undertake "projects for friends" and might even consider engaging with a start-up. In the meantime, he said, "It has been an honor for me to have worked with our outstanding Board of Directors and to have been a part of the Tech 20/20 family for the past 20 years."
UT's Fred Tompkins, 2012 board Chair, expressed the board's appreciation for Cuddy's years of service and his "nice accomplishments" during his tenure as CEO, and wished him well in his new pursuits.
Creative Agricultural Technologies still going strong after more than a decade
KPCB's co-founder says take big risks very carefully
CHINA UPDATE: More successful meetings for ABT
EDITOR'S NOTE: Apologies for the double emails
Latest Land Grant Films documentary premieres this Friday
Motivo raises $2.2 million seed round to expand staff | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,568 |
"The entire staff was excellent, compassionate, and thorough... I couldn't have hoped for a better handling of a stressful situation.".
"From the first appointment to the last, my experience could not have been better! Thank you all so much!".
"Dr. Vanderveen spent more time with me than any doctor in recent memory, and my wife and I never felt like she was rushed. She was down to earth in terminology and her manner.". | {
"redpajama_set_name": "RedPajamaC4"
} | 9,598 |
Q: How to read mouse up and mouse down events in linux? I used to work in windows and if in my C# wpf application I wanted to detect mouse up and mouse down , I did it using mouseup event.
Now I want to develop a simple application in C++ which detects mouse up and mouse down events in Linux . I have no idea about how to proceed , which is the best way and what libraries to use . Please guide me about how to go ahead .
A: Your question is just too broad... but I'll try anyway.
You can go with the device access level:
*
*You can read input events directly from /dev/input/*. It is not difficult, but your application will need root access, or else you'll have to change the permissions of the devices. The main advantage is that you can read the mouse without even create a connection with the X server.
*You can work as an X client:
a. You can use the X access directly, Xlib (not really recommended).
b. You can use a toolkit library, such as GTK+, Qt or WxWidgets, to name a few.
With option 2. you may have a hard time if you want to get the events that happen in windows from other applications. YMMV.
A: A really simple and quick solution may be libxdo.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,362 |
Cash and/or Credit Cards for purchases.
A lock so the bike can be left unattended while picking up items; an alternative is for seekers to go inside in shifts.
ID in case you're purchasing alcoholic beverages (picnic locations will allow discreet consumption).
Backpack or pannier if desired; helps Guide carry more items.
Lights (white facing front, red facing rear) when traveling after dark.
Guide will bring re-usable plastic cups for beverages, table cloth, pocket knife, bottle opener and napkins; you may want your own blanket if you've got room to carry.
At the end of the meal, you'll ride back to the starting location. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,532 |
Raul Galván (* 11. Jänner 2004 in Wien) ist ein österreichischer Fußballspieler uruguayischer Abstammung.
Karriere
Verein
Galván begann seine Karriere beim SK Rapid Wien, bei dem er ab der Saison 2018/19 auch sämtliche Altersstufen der Akademie durchlief. Im April 2022 erhielt er einen Jungprofivertrag bei den Wienern und rückte im Anschluss zur Saison 2022/23 in den Kader der zweiten Mannschaft Rapids. Sein Debüt in der 2. Liga gab er im August 2022, als er am sechsten Spieltag der Saison 2022/23 gegen den FC Blau-Weiß Linz in der 88. Minute für Pascal Fallmann eingewechselt wurde.
Nationalmannschaft
Galván spielte im September 2019 erstmals für eine österreichische Jugendnationalauswahl. Im Oktober 2020 absolvierte er gegen Slowenien sein einziges Spiel im U-17-Team. Im September 2021 spielte er zweimal für die U-18-Mannschaft.
Im Juli 2022 nahm er an einem Lehrgang des uruguayischen Fußballverbands für Auslandsuruguayer teil.
Weblinks
Einzelnachweise
Fußballspieler (SK Rapid Wien)
Österreicher
Geboren 2004
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,115 |
Christ said that His servants would fast. In Matthew 6 Jesus teaches by giving instructions on WHEN you give, WHEN you pray and WHEN you fast. Thus, in Jesus' teachings these three are duties for every Christian. But what is a fast? Is it starving yourself? Is it missing one meal – or more than one? How do I fast? How do I prepare to fast? How long should I fast for? All these and other questions are dealt with briefly in this document – A Quick Guide to Fasting – which I hope you find helpful.
Thank you Frans for this very practical and spiritual introduction to fasting. I hope we will develop this discilpline . It brings huge blessings indeed. | {
"redpajama_set_name": "RedPajamaC4"
} | 764 |
The European School Copenhagen (ESC) is an Accredited European School situated in the heart of the Carlsberg district of Copenhagen, Denmark. It caters to nursery, primary and secondary level students, leading to the European Baccalaureate as its secondary leaving qualification. The ESC is affiliated to the Sankt Annæ Gymnasium, a local Danish school, and is overseen by a joint management board of the two schools.
History
The plans for the school were presented by Copenhagen Municipality, Realdania, Novo Nordisk Fonden, Nordea-fonden and Industriens Fond. Welcoming its first students in 2014, the school was initially based out of a temporary location at the South Harbour School.
A competition for the design of the new school was won by BAM Danmark, Vilhelm Lauritzen Arkitekter, Nord Arkitekter and EKJ consulting engineers. The school moved into its new building on 25 October 2018 and it was officially inaugurated on 20 November 2018.
See also
Accredited European School
European Baccalaureate
European Schools
References
External links
Official website
Accredited European Schools
Sankt Annæ Gymnasium
International schools in Denmark
Primary schools in Copenhagen
Secondary schools in Copenhagen
Educational institutions established in 2014
2014 establishments in Denmark
School buildings completed in 2018 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,851 |
Anchorage Women's Clinic is a preferred provider with most commercial insurance plans and Medicaid. Please click here for a more comprehensive list of local plans. If you do not see yours, please contact us.
Anchorage Women's Clinic is committed to providing quality, affordable care. We have not raised our rates since 2014 and have always offered free cost estimates to patients.
In compliance with new State of Alaska laws and Senate Bill 105, we are posting the following list of health care services and undiscounted prices. Please note that these prices do not represent your actual cost for services. Costs depend on insurance specific contract rates or self-pay patient discounts.
If you would like an estimate of the anticipated charges for your non-emergency care, please let us know. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,128 |
\section{Introduction}
In response to the continued exponential growth of the data traffic in wireless networks \cite{5gwireless}, increasing efforts have been devoted to wireless communication technologies above the 100~GHz band, where abundant frequency resources exist \cite{nagatsuma2016advances}.
Four frequency bands with a total bandwidth of 137~GHz between 275~GHz and 450~GHz were identified for implementing land mobile and fixed service applications a few years ago \cite{WRC19_final}.
Data rates of 100~Gbit/s have been achieved at the frequency ranges by employing quadrature amplitude modulation (QAM) such as 16-QAM \cite{koenig2013wireless,nagatsuma2016real,chinni2018single, hamada2018300}.
A simple approach to further progress in data rates while keeping the same bandwidth occupancy is exploiting higher modulation orders.
So, higher power and lower white noise floor in carrier signals are demanded since a larger signal-to-noise ratio (SNR) in detected signals is required for that \cite{Chen2018Influence}.
There are roughly two generation and modulation/demodulation methods of high-frequency radio frequency (RF) waves with low phase noise: all-electronics and photonics-based approaches.
The former completes all processes of signal handling in electronic circuits.
It is one step ahead of the photonics-based counterpart regarding monolithic integrability, which is a key factor for mass production and reducing production costs.
However, multi-stage frequency multiplication and amplification processes required for producing high-frequency RF waves intrinsically degrade their phase noise and SNR, posing a limitation for the maximum data rate transmitted.
Modulation is usually carried out by mixing carrier and modulated signals.
On the other hand, a photonics-based approach is suitable for generating a lower-phase-noise wave.
It produces a high-frequency RF wave via heterodyne detection of optical lines with a fast photodiode (PD) \cite{nagatsuma2013terahertz}.
The phase noise is not limited by a frequency of a generated wave but by the relative phase noise of seed light, which can be controlled independently of the photo-mixing process.
In addition, it is advantageous that signal encoding is possible in the optical domain by using sophisticated modulation schemes in optical communication.
Pure electronic or optoelectronic devices carry out the signal reception \cite{harter2019wireless}.
On the other hand, systems to achieve ultra-low phase noise tend to be bulky \cite{li2019low}.
Also, the output power is limited by the maximum currents a PD can handle and the PD's responsivity.
A Kerr microresonator frequency comb~\cite{herr2014temporal,brasch2016photonic}, or microcomb, is a potential candidate to address the challenges in conventional photonics-based oscillators.
Its repetition frequency is typically located between 10~GHz and 1~THz, and the generation of RF waves with frequencies higher than 100~GHz has already been demonstrated via direct detection of the comb lines with ultra-fast PDs \cite{zhang2019terahertz, huang2017globally, tetsumoto2020300, wang2021towards, tetsumoto2021optically}.
There are three main advantages of using the microcomb in wireless communication.
(i) It can enhance detected modulation signal power thanks to constructive interference of multiple RF waves generated from multiple comb lines, as observed for carrier power in some studies~\cite{kuo2010spectral,wang2021towards}.
It will help to gain higher SNR out of the limited photocurrent of a PD.
(ii) The system can be more compact and simple than other photonics-based oscillators.
Aside from being on-chip, it will eliminate the need for some optical components required for other photonic systems (e.g., optical couplers and spectrum filters).
The large mode spacing allows a microcomb to be modulated and detected directly, as demonstrated in this study.
(iii) Its phase noise can be reduced significantly through stabilization to micro-wave references or optical fibers~\cite{zhang2019terahertz,tetsumoto2020300,kuse2022low}, optical frequency division \cite{tetsumoto2021optically}, dispersion engineering of microresonators \cite{stone2020harnessing}, or turning experimental parameters~\cite{yi2017single,tetsumoto2021effects}.
In fact, it has enabled demonstrating the record-low phase noise at 300~GHz \cite{tetsumoto2021optically}.
In this study, we demonstrate the unique features of a microcomb as a transmitter/receiver in wireless links.
The three advantages mentioned above are addressed, respectively, as follows.
The first one is demonstrated experimentally by employing power-equalized comb lines.
We observe more than 10~dB gain of modulation signal strength when multiple optical lines are detected compared to a two-line case.
We perform wireless communication experiments with a microcomb-based 300~GHz transmitter and receiver to show the second point.
As a result, successful transmission of up to 64-QAM signal is confirmed, whose limitation is not given by the microcomb.
The transmitter and receiver are stabilized to a common reference during the experiments, which corresponds to the condition that the third advantage is in effect.
This study will be a milestone in developing compact microcomb-based photonic millimeter- and terahertz-wave transceivers and help achieve higher data rates in future wireless communication systems.
\section{Operation principle}
We modulate the whole comb spectrum and detect it directly in this study.
By doing so, the detected modulation signal power is enhanced compared to two-line heterodyne detection under the same photocurrent.
This effect has been observed in previous studies for carrier signal power~\cite{kuo2010spectral,wang2021towards}.
\begin{figure*}[!ht]
\centering
\includegraphics[width=\linewidth]{fig1_rev2.pdf}
\caption{\footnotesize{ (\textbf{a}) Schematic illustration of operation principle. The upper half shows an experimental situation considered, and the lower half is the spectrum of signals at each stage. (\textbf{b}) Equalized comb lines from a 1~\% monitor port after the amplification. The power difference between combs is less than $\pm 0.5$~dB (dashed line). (\textbf{c}) Measured enhancement of detected modulation signal strength. The dashed line shows 0~dB of enhancement.}}
\label{fig1}
\end{figure*}
Power spectrum of photocurrent from a PD of $|I_\mathrm{PD}(\omega)|^2$ is given by,
\begin{equation}
|I_\mathrm{PD}(\omega)|^2 = |G(\omega ) \cdot P(\omega )|^2,
\label{eq1}
\end{equation}
where $\omega$ is angular frequency, $|G(\omega )|^2$ and $|P(\omega )|^2$ are power spectra of impulse response of a PD and an input wave, respectively.
So, the products of the PD output will be proportional to the square of the input wave power spectrum, assuming the PD's response is homogeneous and linear over the frequency of interest.
Figure~\ref{fig1}(a) illustrates the modulation/detection methods and spectra obtained at the respective stages.
The input is power-equalized $N$ comb lines with amplitude of $1/\sqrt{N}$ normalized so that total power becomes 1.
They are modulated with an intensity modulator (IM) with a modulation signal $\phi _\mathrm{RF} = M \sin{\Omega t }$, where $\Omega$ is an RF frequency, and $M$ is modulation depth.
There is no in-balance in the optical splitting and combining processes in the IM, and the light only on one arm experiences the modulation.
Then, the output of the modulator $E_\mathrm{out}$ is \cite{urick2015fundamentals},
\begin{eqnarray}
& E_\mathrm{out} &= \sum_{n=1}^{N} \frac{1}{\sqrt{N}}\left[ \cos{\omega _{n} t}
+ \sum_{k=-\infty}^{\infty} J_{k}(M)\sin\{(\omega _{n} + k\Omega )t \} \right] \nonumber \\
&\sim & \sum_{n=1}^{N} \frac{1}{\sqrt{N}}[ \cos{\omega _{n} t}
- J_{1}(M)\sin\{(\omega _{n} - \Omega )t \} + J_{0}(M)\sin{\omega _{n}t} + J_{1}(M)\sin\{(\omega _{n} + \Omega )t \} ] \nonumber \\
&=& \sum_{n=1}^{N} \frac{1}{\sqrt{N}} A_\mathrm{n},
\label{eq_opt}
\end{eqnarray}
where $n$ and $k$ are integers, $\omega _{n}$ is the angular frequency of $n$~th comb mode, $J_k$ is $k$~th-order Bessel function of the first kind, and $A_\mathrm{n}$ is the sum of the center and sideband modes around $n$-th optical line.
The factor $1/2$, which explains power splitting and combining, is omitted, and the phase difference between the two arms of $\pi/2$ is assumed for simplicity.
Higher order sidebands are neglected since the modulation depth $M$ is small, and $J_{-k}=(-1)^{k}J_{k}$ is employed.
The left bottom of Fig.~\ref{fig1}(a) depicts the optical spectrum of the comb lines at the IM output.
These optical lines are sent to a PD and generate a carrier and its sidebands (modulation tones) in the millimeter-wave domain through $I_\mathrm{PD} \propto |E_\mathrm{out}|^2$.
The millimeter-wave carrier with the angular repetition frequency $\omega _\mathrm{rep}=\omega _{n} - \omega _{n-1}$ and its sidebands are generated through the interference between adjacent optical lines and their sidebands of,
\begin{equation}
I_\mathrm{adj} = \frac{1}{N} \sum_{n=1}^{N-1} A_{n}A_{n+1}.
\label{eq_components1}
\end{equation}
\noindent
Each term of Eq.~\ref{eq_components1} gives the following components for the respective angular frequencies when the optical lines are phase-coherent:
\begin{eqnarray}
\omega _\mathrm{rep} &:& \frac{1}{2N} \{ 1 - J_{0}^{2}(M) - 2J _{1}^{2}(M) \} \cos {\omega _\mathrm{rep}t}, \nonumber \\
\omega _\mathrm{rep} - \Omega &:& -\frac{J_{1}(M)}{N} \sin\{ \left(\omega _\mathrm{rep} - \Omega \right)t\}, \nonumber \\
\omega _\mathrm{rep} + \Omega &:& \frac{J_{1}(M)}{N} \sin\{ \left(\omega _\mathrm{rep} + \Omega \right)t\}. \nonumber
\label{eq_components2}
\end{eqnarray}
\noindent
The resulting spectrum around the carrier is schematically shown in the center bottom of Fig.~\ref{fig1}(a), where higher-order sidebands are neglected because they are small in the weak modulation condition considered.
The amplitude of each frequency component is multiplied by $N-1$ owing to the contribution from each combination in Eq.~\ref{eq_components1}.
The envelope of the generated millimeter-wave can be captured with an RF detector, such as a Schottky barrier diode (SBD).
The modulation signal $P_\mathrm{mod} (N)$ with the angular frequency of $\Omega$ is detected in the microwave domain via the interaction between the carrier and the two sidebands, which is expressed as,
\begin{eqnarray}
P_\mathrm{mod} (N) &\propto & 2 \times \frac{N-1}{2N} \{ 1 - 2J _{1}(M)^{2} - J_{0}(M)^{2} \} \times J_{1}(M)\frac{N-1}{N} \times \frac{\sin{\Omega t}}{2} \nonumber \\
&=& \frac{1}{2} \left( \frac{N-1}{N} \right) ^{2} \{ 1 - 2J _{1}^{2}(M) - J_{0}^{2}(M) \} J_{1}(M) \sin{\Omega t}.
\label{eq_mod}
\end{eqnarray}
\noindent
Therefore, the enhancement factor of the modulation signal's power spectrum owing to the multiple comb lines can be expressed as follows:
\begin{equation}
\left| P_\mathrm{mod}(N) \right| ^2 \propto \left| \left( \frac{N-1}{N} \right) ^{2} \right| ^{2} = \left( \frac{N-1}{N} \right) ^{4} .
\label{eq_enhance}
\end{equation}
So, the modulation signal power will be enhanced by $|P_\mathrm{mod}(N)|^2/|P_\mathrm{mod}(2)|^2=16\left( \frac{N-1}{N}\right)^{4}$ compared to two-line cases, which will become 16 times (12~dB) when $N \to \infty$.
This number is from the 6~dB respective power enhancement of the carrier and sidebands in the millimeter-wave domain.
The key point is that each adjacent comb line pair produces an identical set of RF waves regarding amplitudes, frequencies, and phases at the detection.
Thus, the principle will be valid for modulation methods that gives the carrier and its sideband signals via photo-detection as long as the optical comb lines are phase-locked and the generated waves interfere constructively.
We demonstrate a simple experiment to confirm the effect.
We prepare microcomb lines with a 300~GHz frequency spacing and equalized power by using a waveshaper and an erbium-doped fiber amplifier (EDFA), as shown in Fig.~\ref{fig1}(b).
They are modulated by a 1.5~GHz sinusoidal wave with an IM and sent to an unitravelling-carrier photodiode (UTC-PD).
The envelope of the generated 300~GHz wave is detected with an SBD.
A dispersion compensating fiber (DCF, DCF-38 Thorlabs with $\sim $-38~ps/nm$\cdot$km) is inserted between the waveshaper and the UTC-PD to compensate for the dispersion effect.
We change the number of comb lines to inject, and record the change in the detected signal power of the modulation tone.
The photocurrent of the UTC-PD is kept at 0.5~mA for all measurements.
Figure~\ref{fig1}(c) shows the results measured with two different DCF lengths, where they are normalized by the power at $N=2$.
With the DCF of 9.5~m (red circle plots), the power of the detected modulation signal increases as the number of comb lines increases, and the trend is close to the theoretical expectation (black star plots).
The enhancement factor of as high as 10.9~dB is obtained with 7 comb lines.
On the other hand, the signal strength does not go up so high with increased numbers of comb lines but drops significantly with a 4.4~m DCF (blue triangular plots, at $N=5$), where the dispersion of the optical path is not compensated correctly.
This presents that the multiple comb line detection scheme can cause a reverse effect if the phase of each comb line is not appropriately aligned~\cite{wang2021towards}.
The small effect of the dispersion also can be seen in the plots for 9.5~m DCF at $N>9$.
\section{Microcomb generation, stabilization \& characterization}
Figure~\ref{fig2} depicts an experimental setup for microcomb generation and stabilization. Single sideband modulation is applied to continuous-wave (CW) light from an external cavity laser diode (ECL, Toptica CTL1550) with a single sideband modulator (SSBM), and the output is amplified with an EDFA.
The light is coupled with a ring resonator made of silicon nitride (SiN) after polarization alignment using a polarization controller (PC).
A soliton microcomb is excited by fast scanning the strong pump light with the SSBM \cite{briles2018interlocking, kuse2019control}.
The generated microcomb is polarized with another PC and a polarizer (PL). We employ polarization-maintaining fibers after here.
A small portion of the comb light is sampled for power and spectrum monitoring during the experiment.
The pump light is suppressed with a bandstop filter (BSF), and the output is split into two paths with a 90:10 optical coupler.
The 90~\% of the light is amplified with an EDFA and sent to a wireless link, explained later.
Figure~\ref{fig2}(b) shows the optical spectra of the generated microcomb and the filtered and amplified comb on the 90~\% port.
The bandwidth of the filtered comb spectrum is mainly limited by the BSF and EDFA gain bandwidths.
Pump-to-comb conversion efficiency is estimated to be $\sim 4$~\% from the ratio between the pump power and the total power of the other combs in the optical spectrum.
\begin{figure*}[!ht]
\centering
\includegraphics[width=\linewidth]{fig2_rev.pdf}
\caption{\footnotesize{ (\textbf{a}) Schematic drawing of the experimental setup for microcomb generation and stabilization. See text for details. Generated microcomb's (\textbf{b}) optical spectra, (\textbf{c}) phase noise, (\textbf{d}) frequency instability, and (\textbf{e}) relative intensity noise. The measurement floor in (e) (shaded curve) is obtained when no light is sent into the UTC-PD. (\textbf{f}) Result of the repetition frequency tuning.}}
\label{fig2}
\end{figure*}
The output from the 10~\% port is used to stabilize the microcomb through the method described in \cite{tetsumoto2020300}.
Although the suppression of the phase noise and frequency drift is not required for the communication experiments in this study based on self-homodyne detection, we implement this to show the microcomb's capability of being synchronized to a reference signal.
For the stabilization, sidebands of the comb lines are generated with two cascaded phase modulators (PMs) driven by a 10~GHz wave from a synthesizer (SYN), whose RF power is amplified to about 1~W with RF amplifiers (RF AMPs).
At around a center frequency of two comb lines, the 15~th sidebands from each comb are closely located thanks to the broad bandwidth of the generated electro-optic (EO) combs.
The spectrally overlapped frequency region is sampled with a bandpass filter (BPF), and the output is detected with a photodiode (PD).
The detected beat frequency $f_\mathrm{beat}$ is described as,
\begin{equation}
f_\mathrm{beat} = (f_\mathrm{n+1} - 15 f_\mathrm{10G}) - (f_\mathrm{n} + 15 f_\mathrm{10G}) = f_\mathrm{rep} - 30 f_\mathrm{10G}.
\label{eq3}
\end{equation}
where $f_\mathrm{n+1}$ and $f_\mathrm{n}$ are frequencies of $n+1$~th and $n$~th microcomb modes, and $f_\mathrm{10G}$ is frequency of the 10~GHz SYN.
So, the phase noise of the beat signal is,
\begin{equation}
\delta _{f_\mathrm{beat}} = \delta _{f_\mathrm{rep}} - 30 \delta _{f_\mathrm{10G}}.
\label{eq4}
\end{equation}
where we employ a notation of $\delta _{f}$ for the phase noise of a signal with frequency $f$.
We produce an error signal by mixing the beat signal with an RF reference and generate a control signal via a proportional integral derivative (PID) loop filter.
The control signal drives a voltage-controlled oscillator, whose output is applied to the frequency control channel of the SSBM.
Note that both the 10~GHz SYN and the reference RF wave in the system are synchronized to the same clock signal; thus, the microcomb is synchronized as well.
Figure~\ref{fig2}(c) presents the measured phase noise.
The free-running comb noise is suppressed to the calibrated noise level of the 10~GHz reference SYN in the locked condition.
The feedback bandwidth is set to slightly lower than the cross-point frequency of the free-running and reference noise of 100~kHz so that the noise after the stabilization follows the lower of them.
The phase noise at 10~kHz offset is -64~dBc/Hz (ignoring the spurious peak from the SYN, the noise level is $-75$~dBc/Hz).
The details of the phase noise measurement are shown in Supplementary Material.
Figure~\ref{fig2}(d) presents the frequency instability of the microcomb and the in-loop signal evaluated in terms of modified Allan deviation.
The obtained frequency instability at 1~second is $1.5\times 10^{-9}$ for the free-running microcomb and $1.9\times 10^{-15}$ for the in-loop signal, respectively.
The latter confirms that the microcomb is stabilized to the reference tightly.
In addition, we measure the relative intensity noise (RIN) of the generated 300~GHz wave as shown in Fig.~\ref{fig2}(e).
The number of comb lines and optical path dispersion are controlled in the same way as Fig.~\ref{fig1}(c), where $N=10$, the DCF of 9.5~m and the UTC-PD photocurrent of 6~mA are employed.
The RIN reaches a low noise floor of -124~dBc/Hz (grey dashed line) in both locked and unlocked conditions (black and red curves).
Also, we can use the stabilization scheme for fine-tuning the repetition frequency of the microcomb.
Equation~\ref{eq3} can be rewritten as,
\begin{equation}
f_\mathrm{rep} = 30 f_\mathrm{10G} + f_\mathrm{beat}.
\label{eq5}
\end{equation}
This indicates that $f_\mathrm{rep}$ can be tuned by controlling the reference RF wave frequency to which $f_\mathrm{beat}$ is stabilized. Note that changing $f_\mathrm{10G}$ will only shift $f_\mathrm{beat}$ not $f_\mathrm{rep}$.
To demonstrate this, we acquire the RF spectrum of the beat signal and observe its transition when we change the reference RF frequency in steps. Figure.~\ref{fig2}(f) shows the result.
The beat frequency follows the reference frequency linearly, and the tuning range of 26~MHz is obtained without losing the lock.
Though the frequency sweep is performed slowly (i.e., about 100~kHz/sec), which is limited by the data acquisition time for the measurement, the scanning speed can be faster as long as the phase lock loops can catch up.
\section{Wireless communication experiments}
\begin{figure*}[!ht]
\centering
\includegraphics[width=\linewidth]{fig3.pdf}
\caption{\footnotesize{(\textbf{a}) Schematic drawing of FPGA-based modulation/demodulation system. \textbf{b})-(\textbf{e}) Constellation diagrams for 4-QAM, 16-QAM, 64-QAM and 256-QAM modulation schemes, respectively, in FPGA direct link. Schematic drawing of wireless links based on (\textbf{f}) direct envelope detection and (\textbf{g}) coherent detection. See text for details. }}
\label{fig3}
\end{figure*}
Wireless communication systems in this study consist of a software part to produce and analyze digital signals and a hardware part to modulate, transmit and receive 300~GHz waves.
Digital signal processing (DSP) in the wireless system is performed with MATLAB \& Simulink (details are in Supplementary Material), and transmitter and receiver based on field programmable gate arrays (FPGAs) (USRP N210, Ettu) acting as interfaces between the software and the hardware parts.
Figure~\ref{fig3}(a) depicts the schematic of the system.
The host computer, which processes the Simulink program, the FPGA transmitter, and the receiver, are connected via Ethernet cables and hubs.
The transmitter and the receiver share the same clock signal as the microcomb.
The software program generates modulation signals encoded with either 4-QAM, 16-QAM, 64-QAM, or 256-QAM.
The center frequency and the symbol rate of the signal are set to 1.5~GHz and 200~kHz, respectively, where the rather low symbol rate is to avoid overrun and underrun in the software-hardware communication.
The transmitter is configured to generate the produced modulation signal as an arbitrary waveform generator.
The modulated signal is applied to an intensity modulator (IM) in the hardware system.
The in-phase and quadrature-phase components of the signal are superimposed in the RF wave domain.
The detected signal in the hardware is down-converted to baseband frequency, digitized in the FPGA receiver, and analyzed with the Simulink program in the host computer in real time. We evaluated the received signal's root mean square (RMS) error vector magnitude (EVM).
Figure~\ref{fig3}(b)-(e) shows the results of a back-to-back preliminary experiment, where the FPGA transmitter and the FPGA receiver are connected with a 1.5~m electric cable and a 10~dB attenuator.
Constellation points can be recognized clearly in 4-QAM, 16-QAM, and 64-QAM cases with EVM of 9.3~\%, 5.8~\%, and 7.2~\%, respectively, whereas 256-QAM constellation with EVM of 6.1~\% is not resolved well. These results pose the lowest levels of the reachable EVM in the following experiments.
We tested two types of wireless links.
One is based on direct envelope detection of the 300~GHz carrier with an SBD (Fig.~\ref{fig3}(f)).
The stabilized microcomb is modulated by the IM and detected with a UTC-PD, where a 300~GHz carrier is generated.
The 300~GHz wave is radiated into free space with a horn antenna, travels 0.1~m, is captured with another horn antenna, and is detected with the SBD.
The output from the detector is sent to the FPGA receiver.
The other system employs coherent mixing of two 300~GHz waves (Fig.~\ref{fig3}(g)).
The modulated wave is generated in the same way as the direct envelope detection.
Another wave for the down-conversion is produced by detecting the same microcomb before the modulation in this proof-of-concept study.
The generated two waves are mixed with a 300~GHz fundamental mixer, and the intermediate frequency (IF) signal is supplied to the FPGA receiver.
Although a homodyne detection system requires precise control of the relative phase of two waves detected in general, its fluctuation was slow enough in our experiment owing to the short path length after the fork, and no active control was implemented.
Note that we omit a DCF in the two setups since we can obtain sufficient SNR for the demonstration without it.
Figure~\ref{fig4}(a) presents a picture of the actual wireless link based on direct envelope detection. We employ horn antennas with a designed gain of 26~dBi. The transmittance is maximized by controlling the positions of the antennas with 3-axis mechanical stages. To check the transmission loss, we detect the 1.5~GHz sinusoidal wave modulation signal delivered by a 300~GHz carrier in the 0.1~m link and a 1~mm link. Figure~\ref{fig4}(b) shows the RF spectra of the detected signals. We observe about 13~dB reduction of the SNR through the 0.1~m free-space trip compared to the 1~mm link.
Figure~\ref{fig4}(c)-(f) displays the results of the wireless communication. The obtained EVM are 8.6~\%, 6.3~\%, 7.1~\%, and 6.4~\% for 4-QAM, 16-QAM, 64-QAM, and 256-QAM, respectively. So, no apparent degradation of signal quality is observed compared to the direct FPGA link experiment (Fig.~\ref{fig3}(b)-(e)).
The wireless link for the coherent detection is presented in Fig.~\ref{fig5}(a). Most components are the same as ones in the direct envelope detection system, but the SBD is replaced with a fundamental mixer, and another UTC-PD is attached to it to supply a 300~GHz wave to the LO port. Again, we demonstrate the transmission test and observe a 12~dB decrease in the SNR in the 0.1~m link. The acquired constellation diagrams are shown in Fig~\ref{fig5}(c)-(f). EVM of 8.8~\%, 6.2~\%, 6.5~\%, and 6.3~\% are obtained for 4QAM, 16QAM, 64QAM, and 256QAM, respectively, which are merely limited by the FPGA transmitter and receiver.
\begin{figure*}[!ht]
\centering
\includegraphics[width=\linewidth]{fig4.pdf}
\caption{\footnotesize{(\textbf{a}) A picture of the wireless link based on direct envelope detection. (\textbf{b}) RF spectra of detected 1.5~GHz modulated tones in transmission loss test. RBW is 1~kHz. (\textbf{c})-(\textbf{f}) Constellation diagrams for 4-QAM, 16-QAM, 64-QAM and 256-QAM modulation schemes, respectively, in direct envelope detection.}}
\label{fig4}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=\linewidth]{fig5.pdf}
\caption{\footnotesize{(\textbf{a}) A picture of the wireless link based on coherent detection. (\textbf{b}) RF spectra of detected 1.5~GHz modulated tones in transmission loss test. RBW is 1~kHz. (\textbf{c})-(\textbf{f}) Constellation diagrams for 4-QAM, 16-QAM, 64-QAM, and 256-QAM modulation schemes, respectively, in coherent detection.}}
\label{fig5}
\end{figure*}
Figure~\ref{fig6} summarises the results of the communication experiments.
EVM threshold to obtain bit error ratio (BER) of $4\times 10^{-3}$ is displayed by the black dashed line with star symbols as a measure of successful transmission \cite{shafik2006extended,chinni2018single}. With the EVM level, the BER will be decreased to $10^{-15}$ level by the forward error correction with 7~\% overhead \cite{chang2010forward}.
The obtained EVM levels are below the EVM limit in the experiment up to 64-QAM but slightly beyond the limit in the 256-QAM demonstration.
The limitation is given by the FPGA transmitter and receiver (phase noise is -80~dBc/Hz at 10~kHz offset for a 1.8~GHz signal according to their specification sheet).
This is convincing because the microcomb's phase noise is canceled out and we will see only modulation signal noise at the reception in the two wireless links demonstrated in this study.
\begin{figure*}[!ht]
\centering
\includegraphics[width=\linewidth]{fig6.pdf}
\caption{Summary of obtained EVM in wireless communication experiments}
\label{fig6}
\end{figure*}
\section{Discussion \& Outlook}
Here, we discuss microcomb's advantages for wireless communication and future perspective in more detail.
Our experiments demonstrate that a microcomb can be an oscillator in wireless links utilizing complex modulation formats.
The system is relatively simple compared to other photonics-based approaches because it starts from a single CW light, and no spectral filtering process is required, which allows comb lines to propagate in a shared path all the time.
This is, in particular, significant difference from photonics-based systems using conventional comb sources, which suffer from large loss due to splitting, filtering and combining processes and sometimes even require mechanisms to compensate fluctuation of path length difference \cite{nagatsuma2013terahertz}.
Also, the employed two architectures will benefit from respective merits as follows.
The direct envelope detection scheme does not require low-phase-noise carriers because the carrier noise will be canceled out at the detection, whereas its sensitivity is limited by a detector employed.
So, the critical parameters for this configuration are the noise property of RF sources in modulation/demodulation (e.g., arbitrary waveform generator for encoding, an oscillator for decoding), signal strength at a transmitter, and detector sensitivity.
A microcomb will contribute to improving the signal strength via the simultaneous modulation and detection of comb lines.
The coherent detection scheme will enhance detection sensitivity by a strong LO signal and give a larger bandwidth at the IF output in general.
However, detected SNR will be affected by the phase noise of both RF and LO signals injected into a mixer.
Also, the RF and LO signals need to have a small frequency drift or, ideally, share the common clock signals to help carrier frequency and phase synchronization in the DSP work properly.
Regarding the phase noise, a 300~GHz microcomb can be comparable (-100~dBc/Hz at 10~kHz) to an arbitrary waveform generator at 10~GHz used in a cutting-edge communication experiment \cite{hamada2018300} (-95~dBc/Hz at 10~kHz offset, Keysight M8196A \cite{keysight_M8196A}) through the optical frequency division technique~\cite{tetsumoto2021optically}.
So far, white phase noise floors of microcombs at 300~GHz are observed at around -120~dBc/Hz limited by shot noise and noise figure of electronic devices \cite{zhang2019terahertz,tetsumoto2021optically}. Note that this is still $>10$~dB lower than that of a typical electronic 300~GHz oscillator based on frequency multiplication \cite{dan2020superheterodyne}.
In addition, a microcomb oscillator can address the clocking problem and provide robust operation, as demonstrated in this study.
Two crucial challenges need to be addressed to realize low-cost and portable microcomb-based transceivers for future wireless communication: integrating the system and reducing power requirement.
Although a system based on the optical fiber connection like the one in this study can be packaged compactly, making it on-chip is more desirable for its miniaturization and mass production.
Such a platform will also mitigate dispersion effects, observed in Fig.~\ref{fig1}(e), thanks to much shorter path lengths.
Hybrid and heterogeneous integration \cite{liang2021recent,kaur2021hybrid} will be a key technology to achieve the goal since active components like lasers and media for nonlinear optical effects are usually made of different materials with different bandgap structures.
Recently, soliton generation has been demonstrated with InP/Si semiconductor lasers monolithically integrated with high $Q$ SiN resonators \cite{xiang2021laser}.
Regarding subsystems, the broadband EO comb we employed to stabilize the comb's repetition frequency may not be a suitable option when pursuing system integration as it requires high-power RF amplifiers.
Phase-locking the 300~GHz comb to a lower-frequency microcomb at the microwave domain can be an alternative way to down-convert and stabilize the comb frequency \cite{spencer2018optical,liu2020photonic}.
On-chip low half-wave voltage modulators may become a straightforward solution to relax the RF power requirement \cite{zhang2021integrated}.
The excitation power requirement for the microcomb generation has been lowered through progress in fabrication technologies and theoretical and experimental understanding of its generation physics.
For example, a soliton microcomb has been excited with 100-mW-level electrical power (1-mW-level optical power) by employing an ultrahigh $Q$ SiN ring resonator ($Q \sim 10^{7}$)\cite{stern2018battery}.
Pump-to-comb conversion efficiency is also an important measure to evaluate the energy cost.
It can be controlled via various parameters such as the resonator's size and loss and coupling between a waveguide and a resonator, where the efficiency close to 10~\% is predicted for a 300~GHz SiN resonator with intrinsic $Q$ in the order of $10^6$ under a highly over-coupled condition~\cite{jang2021conversion}.
To achieve the efficiency of $>10$~\%, using dark pulses or coupled resonator configurations can be effective approaches~\cite{xue2017microresonator,kim2019turn, xue2019super,helgason2022power}.
In summary, we demonstrated 300~GHz wireless communication links based on a microcomb for the first time.
The simultaneous modulation of the multiple microcomb lines enhanced the detected modulation signal strength by more than 10~dB compared to the two-line case.
We performed a transmission experiment of complex modulation format in two types of wireless links by using a microcomb stabilized to a reference clock.
No degradation of the signal quality is observed in the experiments, and the QAM signal as high order as 64 was transmitted successfully in both links.
The results show that a microcomb is a promising candidate as a photonic oscillator that boosts data capacity in future wireless communication.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,688 |
package com.hazelcast.spring.config;
import com.hazelcast.test.annotation.QuickTest;
import org.apache.maven.artifact.versioning.ComparableVersion;
import org.junit.Before;
import org.junit.Test;
import org.junit.experimental.categories.Category;
import org.junit.runner.RunWith;
import org.junit.runners.JUnit4;
import org.springframework.util.ResourceUtils;
import java.io.FileInputStream;
import java.util.Collection;
import java.util.HashSet;
import java.util.Properties;
import java.util.Set;
import java.util.regex.Matcher;
import java.util.regex.Pattern;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertTrue;
import static org.junit.Assert.fail;
/**
* A unit test which validates the spring.schemas file
*/
@RunWith(JUnit4.class)
@Category(QuickTest.class)
public class SpringSchemasValidityTest {
private Properties prop;
@Before
public void loadSpringSchemas() throws Exception {
prop = new Properties();
prop.load(new FileInputStream(ResourceUtils.getFile("classpath:META-INF/spring.schemas")));
}
@Test
public void testAllXSDsAvailable() throws Exception {
Collection<Object> values = prop.values();
Set<String> allXSDs = new HashSet<>();
for (Object o : values) {
allXSDs.add(o.toString());
}
for (String xsd : allXSDs) {
assertTrue(ResourceUtils.getFile("classpath:" + xsd).exists());
}
}
@Test
public void testUrlMatchesXSD() {
Set<String> names = prop.stringPropertyNames();
for (String name : names) {
if (name.endsWith("spring.xsd")) {
// perma link without version in file name
continue;
}
assertTrue(name.endsWith(prop.getProperty(name)));
}
}
@Test
public void testPermaLinkIsLatestVersion() {
Pattern pattern = Pattern.compile("hazelcast-spring-([0-9\\.]+)\\.xsd");
ComparableVersion latestVersion = null;
String latestVersionXsdFile = null;
for (Object o: prop.values()) {
String xsd = o.toString();
Matcher matcher = pattern.matcher(xsd);
assertTrue(matcher.matches());
String versionCode = matcher.group(1);
if (latestVersion == null) {
latestVersion = new ComparableVersion(versionCode);
latestVersionXsdFile = xsd;
} else {
ComparableVersion current = new ComparableVersion(versionCode);
if (current.compareTo(latestVersion) > 0) {
latestVersion = current;
latestVersionXsdFile = xsd;
}
}
}
String latestBySpringSchemas = prop.getProperty("https://www.hazelcast.com/schema/spring/hazelcast-spring.xsd");
assertEquals(latestVersionXsdFile, latestBySpringSchemas);
}
@Test
public void testBothHttpAndHttpsAvailable() {
Set<String> allURLs = prop.stringPropertyNames();
Set<String> http = new HashSet<>();
Set<String> https = new HashSet<>();
for (String url : allURLs) {
if (url.startsWith("https")) {
https.add(url);
} else if (url.startsWith("http")) {
http.add(url);
} else {
fail(url + " does not start with http or https");
}
}
assertEquals(http.size(), https.size());
for (String s : https) {
assertTrue(http.contains("http" + s.substring(5)));
}
for (String s : http) {
assertTrue(https.contains("https" + s.substring(4)));
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,552 |
Henri Dreyfus-Le Foyer, né le dans le et mort le dans le , est un professeur de philosophie français.
Son nom est resté célèbre parce qu'on a reproché à Jean-Paul Sartre, qui a été son successeur au lycée Condorcet, d'avoir profité de son éviction en raison du statut des Juifs édicté le par le régime de Vichy.
Biographie
Jeunesse et famille
Il naît en 1896 dans un milieu de la grande bourgeoisie juive parisienne, sous le nom d'Henri Nathan Dreyfus, fils d'Abraham Albert Dreyfus, négociant, et Marie Renée Loevel (sœur de l'écrivain Maurice Level et cousine germaine de Marcel Schwob), son épouse. Ses parents divorcent en 1906 et l'année suivante, sa mère se remarie avec l'avocat et homme politique (député de Paris en 1909-1910) Lucien Le Foyer. En vertu d'un jugement rendu par le Tribunal civil de la Seine le , Henri Dreyfus et son frère Pierre sont adoptés par leur beau-père, d'où l'adjonction du nom Le Foyer à leur patronyme de naissance.
Jean Daniel a affirmé qu'il était le petit-neveu du capitaine Dreyfus, mais cette erreur a été ensuite corrigée.
Carrière
Pendant la guerre, la propriété de la famille du député de Paris, Dreyfus-Le Foyer, ne fut pas confisquée et volée par les Allemands comme celles d'Alphonse Kann et des Goujon-Reinach, mais sauvée par la municipalité, qui mit tous leurs biens à l'abri.
Henri Dreyfus-Le Foyer, normalien (Ulm, 1919), agrégé de philosophie (1919), « médecin et philosophe, fut professeur au lycée Henri-IV ». Au cours d'études de médecine, il avait rédigé une thèse intitulée Le Vertige. Son frère Pierre Dreyfus-Le Foyer, chirurgien en pneumonectomie à l'hôpital Laennec, se réfugia en 1940 dans une clinique de Guéret (Creuse).
Il était professeur de philosophie en khâgne au lycée Condorcet à Paris en 1940 (il y enseignait depuis 1935) lorsque la France est envahie et occupée par l'armée allemande. Il obtint d'abord de Vichy le de quitter son poste à Paris pour être affecté « en repliement » au lycée Ampère à Lyon, en zone libre. En même temps, Ferdinand Alquié est nommé comme suppléant pour le remplacer à Paris au lycée Condorcet, en plus de son propre service maintenu comme professeur titulaire au lycée Rollin (aujourd'hui lycée Jacques-Decour). Peu après, Henri Dreyfus-Le Foyer reçoit du lycée Ampère la « circulaire concernant le statut des Israélites » puis un arrêté : « Monsieur Dreyfus-Le Foyer Henri, professeur de philosophie au lycée Condorcet, en repliement au lycée Ampère à Lyon, est admis à faire valoir ses droits à la retraite à dater du . […] Par suite de nécessités de service, il sera pourvu définitivement au remplacement de Monsieur Dreyfus-Le Foyer à partir de la même date ». À la rentrée 1941, c'est Jean-Paul Sartre, alors professeur au lycée Pasteur de Neuilly, qui obtient ce poste à Paris en classe préparatoire. Cet effet d'aubaine au détriment d'un Juif, qu'il ne pouvait ignorer, est l'objet d'une longue polémique depuis la fin du XX siècle.
Henri Dreyfus-Le Foyer s'installe alors à Lyon, puis dans le département des Hautes-Alpes, où il passera le reste de la guerre. À la fin du conflit, il est le médecin du maquis de Valgodemar (Hautes-Alpes).
Par la suite, il enseigne encore comme professeur de philosophie dans la khâgne du lycée Henri-IV à Paris (1955), aux côtés d'Henri Birault (khâgne), ainsi que de Maurice Savin et Étienne Borne (hypokhâgne).
En 1966, son Traité de philosophie générale reçoit le prix Broquette-Gonin de littérature, attribué par l'Académie française.
Il meurt en 1969 à Paris.
Œuvres
L'ouvrage Cours d'Algèbre. Livre I: calcul algébrique, 1926J attribué primitivement à H. Dreyfus-Le Foyer a été réalisé en réalité par Jacques Mayer Dreyfus-Lefoyer, ingénieur des arts et manufactures, promotion 22 B et frère de H. Dreyfus-Le Foyer.
Maurice Daurolle et H. Dreyfus-Le Foyer, Traité de dissertation philosophique, classe de philosophie et première supérieure, Paris, 1947. Réédition 1950, Delagrave.
Traité de philosophie générale, Paris, Armand Colin, 1965, collection U.
Bibliographie
Ingrid Galster, Sartre, Vichy et les intellectuels. L'Harmattan, 2001
Ingrid Galster, Sartre et les juifs. La Découverte, 2005
Ingrid Galster, « Sartre pendant l'Occupation. Réponse à une diffamation », in Commentaire, , été 2006,
Notes et références
Naissance en mars 1896
Naissance dans le 7e arrondissement de Paris
Décès en octobre 1969
Décès dans le 1er arrondissement de Paris
Décès à 73 ans
Lauréat du prix Broquette-Gonin (littérature)
Philosophe français du XXe siècle
Agrégé de philosophie
Élève de l'École normale supérieure
Enseignant au lycée Henri-IV
Enseignant au lycée Condorcet
Enseignant au collège-lycée Ampère | {
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To transfer data from USB to Android tablet is very very easy. But if you do not know its correct steps, you will waste your diamond time. For saving your diamond time, I have made this video for teaching the correct and simple steps.
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| {
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-find-the-derivative-for-x-4-3x-2-x-2","text":"# How do you find the derivative for (x^4 + 3x^2)\/(x^2)?\n\n##### 2 Answers\nJun 10, 2018\n\n$f ' \\left(x\\right) = 2 x$\n\n#### Explanation:\n\nWriting your term in the form\n\n$\\frac{{x}^{4} + 3 {x}^{2}}{x} ^ 2 = {x}^{4} \/ {x}^{2} + 3 {x}^{2} \/ {x}^{2} = {x}^{2} + 3$\nwe get\n\n$f ' \\left(x\\right) = 2 x$\n\nJun 10, 2018\n\n$2 x$\n\n#### Explanation:\n\n$\\frac{{x}^{4} + 3 {x}^{2}}{x} ^ 2 = {x}^{4} \/ {x}^{2} + \\frac{3 {x}^{2}}{x} ^ 2 = {x}^{2} + 3$\n\n$\\frac{d}{\\mathrm{dx}} \\left({x}^{2} + 3\\right) = 2 x$","date":"2021-01-27 08:19:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 6, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9851086139678955, \"perplexity\": 13575.802239466579}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 5, \"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-2021-04\/segments\/1610704821253.82\/warc\/CC-MAIN-20210127055122-20210127085122-00770.warc.gz\"}"} | null | null |
\section{Introduction}
These notes are concerned with the study of the following spaces.
\begin{blankdefinition}
The \emph{configuration space} of $k$ ordered points in the topological space $X$ is \[\mathrm{Conf}_k(X):=\{(x_1,\ldots, x_k)\in X^k: x_i\neq x_j\text{ if } i\neq j\},\] endowed with the subspace topology.\footnote{The reader is warned that, in the literature on configuration spaces, there are almost as many traditions of notation as there are references.} The \emph{unordered} configuration space is the quotient \[B_k(X):=\mathrm{Conf}_k(X)/\Sigma_k.\]
\end{blankdefinition}
Our goal is to attempt to understand the topology of these spaces, in the case of $X$ a manifold, through the lens of homotopy groups and (co)homology. Before beginning our study in earnest, we first mention a few reasons that configuration spaces are particularly interesting objects of study.
\subsection{Invariants}
The homotopy type of a fixed configuration space is a homeomorphism invariant of the background manifold, and these invariants tend to remember a rather large amount of information. A simple-minded example is provided by Euclidean spaces of different dimension; indeed, as we will see, there is a homotopy equivalence $B_2(\mathbb{R}^m)\simeq B_2(\mathbb{R}^n)$ if and only if $m=n$. In other words, configuration spaces are sensitive to the dimension of a manifold.
A somewhat more sophisticated example is provided by the fact that $B_2(T^2\setminus \mathrm{pt})\not\simeq B_2(\mathbb{R}^2\setminus S^0)$, which can be shown by a homology calculation. Note that $T^2\setminus \mathrm{pt}$ and $\mathbb{R}^2\setminus S^0$ have the same dimension and homotopy type, having $S^1\vee S^1$ as a common deformation retract. On the other hand, $(T^2\setminus\mathrm{pt})^+\cong T^2\not\simeq S^1\vee S^1\vee S^2\simeq (\mathbb{R}^2\setminus S^0)^+$, so we might conclude from this example that configuration spaces are sensitive to the \emph{proper} homotopy type of a manifold.
In order to discuss the most striking illustration of the sensitivity of configuration spaces, we recall that the \emph{Lens spaces} are a family of compact $3$-manifolds given by \[L(p,q):=S^3/C_p,\] where the cyclic group $C_p$ acts on $S^3\subseteq \mathbb{C}^2$ by multiplication by $(e^{2\pi i/p}, e^{2\pi iq/p})$. It is a classical theorem of Reidemeister that \begin{align*}
L(p,q_1)\simeq L(p, q_2)&\iff q_1q_2\equiv \pm n^2\mod p\\
L(p,q_1)\cong L(p,q_2)&\iff q_1\equiv \pm q_2^{\pm 1}\mod p.
\end{align*} In particular, $L(7,1)$ and $L(7,2)$ are homotopy equivalent but not homeomorphic, and, according to a theorem of Longoni--Salvatore \cite{LongoniSalvatore:CSNHI}, their configuration spaces distinguish them. Thus, configuration spaces are sensitive at least to the \emph{simple} homotopy type of a manifold.
\subsection{Braids}
A point moving in $B_k(\mathbb{R}^2)$ traces out $k$ different paths that weave among one another but can never overlap. For this reason, we think of the fundamental group $\pi_1(B_k(\mathbb{R}^2))$ as the group of geometric \emph{braids} on $k$ strands, with composition given by concatenation of braids. As we shall see, this braid group admits the remarkably simple presentation \[\pi_1(B_k(\mathbb{R}^2))\cong \langle \sigma_1,\ldots, \sigma_{k-1}\mid\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1},\, \sigma_i\sigma_j=\sigma_j\sigma_i \text{ if } |i-j|>1\rangle,\] due originally to Artin \cite{Artin:TB}. The combinatorial, algebraic, and geometric properties of these and related braid groups are of fundamental importance to a vast swath of mathematics that encompasses knot theory, mapping class groups \cite{Birman:BLMCG}, quantum groups \cite{Kassel:QG}, category theory \cite{JoyalStreet:BMC}, and motion planning \cite{Farber:CSRMPA}.
\subsection{Embeddings}
In the company of manifolds with trivialized tangent bundles, it is possible to speak of a \emph{framed embedding}, which is to say an embedding respecting the fixed trivialization, possibly up to a homotopy through bundle maps. The tangent bundle of any Euclidean space is canonically trivialized, and evaluation at the origin determines a homotopy equivalence \[\mathrm{Emb}^\mathrm{fr}(\amalg_k\mathbb{R}^n, \mathbb{R}^n)\xrightarrow{\sim} \mathrm{Conf}_k(\mathbb{R}^n).\] As a consequence of this observation and the fact that framed embeddings compose, we see that the collection $\{\mathrm{Conf}_k(\mathbb{R}^n)\}_{k\geq0}$ of homotopy types is equipped with hidden algebraic structure. This algebraic structure is that of an \emph{operad}, which goes by the name of $E_n$. This perspective has many important descendents, three of which we name.
\subsection{Iterated loop spaces} For any $X$, there is a collection of maps \[\mathrm{Emb}^\mathrm{fr}(\amalg_k\mathbb{R}^n,\mathbb{R}^n)\times (\Omega^nX)^k\to \Omega^n X,\] which arise from a variant of the Pontrjagin--Thom collapse construction. The compatibilities among these maps are summarized by saying that $\Omega^n X$ is an $E_n$-\emph{algebra}; in particular, the homology groups of the configuration spaces of $\mathbb{R}^n$ encode algebraic operations on $H_*(\Omega^nX)$. Remarkably, there is also a partial converse, due to May \cite{May:GILS}, which amounts to an algebraic classification of $n$-fold loop spaces.
\subsection{Factorization homology} The operad $E_n$ is defined in terms of embeddings among disjoint unions of Euclidean spaces, and so, after choosing a coordinate chart, we think of the structure of an $E_n$-algebra $A$ as being ``defined locally'' on a manifold $M$. Patching this local structure together across the elements of an atlas produces a manifold invariant, called the factorization homology of $M$ with coefficients in $A$ and denoted $\int_MA$, which can be thought of as a kind of space of configurations in $M$ labeled by elements of the algebra \cite{AyalaFrancis:FHTM}.
In the example of an $n$-fold loop space discussed above, the \emph{non-Abelian Poincar\'{e} duality} of Salvatore \cite{Salvatore:CSSl} and Lurie \cite{Lurie:HA} supplies the identification \[\int_M\Omega^nX\xrightarrow{\sim} \mathrm{Map}_c(M, X),\] as long as $X$ is $(n-1)$-connected. Later, we will encounter a precursor to this result in the configuration space models for mapping spaces introduced by McDuff \cite{McDuff:CSPNP,Boedigheimer:SSMS}.
In a sense, factorization homology is a method for using configuration spaces to probe manifolds, but it can also be used to study the configuration spaces themselves, reversing this flow of information \cite{Knudsen:BNSCSVFH}. In particular, a theorem of Ayala--Francis \cite{AyalaFrancis:FHTM} shows that configuration spaces enjoy a kind of Mayer--Vietoris property in the existence of a quasi-isomorphism \[C_*(B(M))\simeq C_*(B(M_1))\bigotimes^\mathbb{L}_{C_*(B(N\times\mathbb{R}))}C_*(B(M_2)),\] where $B(X):=\coprod_{k\geq0} B_k(X)$. The fundamental fact underlying this quasi-isomorphism is the contractibility of the unordered configuration spaces of $\mathbb{R}$.
\subsection{Embedding calculus} The embedding calculus of Goodwillie and Weiss \cite{Weiss:EPVIT,GoodwillieWeiss:EPVIT} produces a tower of approximations \[\xymatrix{
&\vdots\ar[d]\\
&T_2\mathrm{Emb}(M,N)\ar[d]\\
\mathrm{Emb}(M,N)\ar[uur]\ar[ur]\ar[r]&T_1\mathrm{Emb}(M,N),
}\] which can be thought of as \emph{algebraic} approximations, where algebra is construed in the operadic sense. Often these approximations become arbitrarily good---in particular, according to a theorem of Goodwillie--Klein \cite{GoodwillieKlein:MDSSE}, this occurs in codimension at least 3---so that one obtains a cofiltration of the space of embeddings. The layers of this cofiltration are described as spaces of sections of certain bundles over configuration spaces, so hard questions about embeddings can sometimes be translated, at least in principle, into softer questions about the ordinary topology of configuration spaces.
\subsection{What is in these notes?} Our guiding mathematical impulse is the basic imperative of the algebraic topologist. Configuration spaces are interesting; therefore, it is worthwhile to compute their homotopy groups and (co)homology.
From a more pedagogical standpoint, we have attempted to design a set of notes that might help to fulfill three goals of a graduate student hoping to enter the field: first, to become familiar with some of what is already known; second, to acquire a set of modern tools; and, third, to see how these tools might be applied in practice. For this reason, we have opted to explore a range of classical topics with a mix of classical and non-classical techniques, developing the latter along the way, with some old results being given new proofs. In particular, emphasis is placed on the use of hypercover techniques, which the author has found indispensable in his own work.
We now give a linear outline of the contents of these notes, including primary references.
\begin{enumerate}
\item[\S\ref{section:configuration spaces}] \emph{Configuration spaces and braids}. After exploring examples and developing basic properties, we exploit the Fadell--Neuwirth fibrations \cite{FadellNeuwirth:CS} to deduce a number of easy computations of homotopy groups. Afterward, we turn to the identification of $\pi_1(B_k(\mathbb{R}^2))$ with the braid group \cite{Artin:TB, FoxNeuwirth:BG}. We follow one of the approaches outlined in \cite{Birman:BLMCG}, reviewing the Reidemeister--Schreier method along the way from the topological viewpoint \cite{ZieschangVogtColdeway:SPDG}. We defer the proof of Fadell--Neuwirth, which will eventually be our first use of hypercover techniques.
\item[\S\ref{section:cohomology}] \emph{(Co)homology of $\mathrm{Conf}_k(\mathbb{R}^n)$}. The main result of this section is the computation of the integral cohomology ring of $\mathrm{Conf}_k(\mathbb{R}^n)$ \cite{Arnold:CRCBG, CohenLadaMay:HILS}. Our argument is essentially that of Cohen. As it turns out, the natural basis in homology is more convenient for some purposes, and we develop this viewpoint in detail following \cite{Sinha:HLDO}. We close by using these computations to dedue the rational homology of $B_k(\mathbb{R}^n)$.
\item[\S\ref{section:covering theorems}] \emph{Covering theorems}. We give a detailed development of the machinery of \v{C}ech covers, hypercovers, and complete covers following \cite{DuggerIsaksen:THAR}. The main result is that the weak homotopy type of a space may be recovered as the homotopy colimit over a complete cover---a topological basis, for example.
\item[\S\ref{section:deferred proofs}] \emph{Deferred proofs}. We use the covering techniques of the previous section, together with Quillen's Theorem B, to prove Fadell--Neuwirth, which was the main tool in \S\ref{section:configuration spaces}. We then use the same techniques to construct the Serre spectral sequence following \cite{Segal:CSSS}, from which we deduce the Leray--Hirsch theorem, which was used in \S\ref{section:cohomology}.
\item[\S\ref{section:mapping space models}] \emph{Mapping space models}. We introduce configuration spaces with labels in a pointed space \cite{McDuff:CSPNP, Boedigheimer:SSMS}, which we motivate as a type of homology theory for manifolds following \cite{AyalaFrancis:FHTM}. We give a new proof of the analogue of the exactness axiom for homology, namely the existence of a certain homotopy pullback square, using hypercovers and Quillen's Theorem B. Using scanning techniques following \cite{Boedigheimer:SSMS}, we show that labeled configuration spaces model mapping spaces, an analogue of Poincar\'{e} duality.
\item[\S\ref{section:homology from splittings}] \emph{Homology calculations from stable splittings}. The main result is the computation of the rational homology of $B_k(M)$ for odd-dimensional $M$ following \cite{BoedigheimerCohenTaylor:OHCS}. The key ingredient is to show that labeled configuration split stably \cite{Boedigheimer:SSMS, CohenMayTaylor:SCSC}.
\item[\S\ref{section:mod p cohomology}] \emph{Mod $p$ cohomology}. The main result is the computation of the mod $p$ cohomology of $B_p(\mathbb{R}^n)$. We follow the argument of \cite[III]{CohenLadaMay:HILS} in the large, giving simplified and corrected proofs of several key steps using the homology basis developed in \S\ref{section:cohomology}.
\item[\S\ref{section:lie algebra methods}] \emph{Postscript: Lie algebra methods}. We give an informal discussion of the main result of \cite{Knudsen:BNSCSVFH}, which expresses the rational homology of $B_k(M)$ for arbitrary $M$ as Lie algebra homology. We use this expression to give a proof of homological stability for configuration spaces.
\item[\ref{appendix:split simplicial spaces}] \emph{Split simplicial spaces}. In this appendix, following \cite{DuggerIsaksen:THAR}, we develop a criterion guaranteeing that degreewise weak equivalence between simplicial spaces induces a weak equivalence after geometric realization. We then verify that this criterion is satisfied in the cases of interest.
\item[\ref{section:homotopy colimits}] \emph{Homotopy colimits}. We motivate the bar construction model for the homotopy colimit and discuss its justification through the theory of derived functors and model categories \cite{Quillen:HA, Hirschhorn:MCL, Dugger:PHC}. We give a detailed proof that the bar construction is cofibrant in the projective model structure on functors. We discuss homotopy left Kan extensions and prove Quillen's Theorem B following \cite{GoerssJardine:SHT}.
\item[\ref{appendix:Spanier--Whitehead}] \emph{The Spanier--Whitehead category}. We discuss a very rudimentary form of stable homotopy theory, which is necessary for the theorem on stable splittings \cite{SpanierWhitehead:FAHT, DellAmbrogio:SWCAT}. We avoid the need to introduce the full machinery of spectra with the notion of a filtered stable weak equivalence.
\item[\ref{appendix:periodicity}] \emph{Tate cohomology and periodicity}. We review some facts about Tate cohomology \cite{CartanEilenberg:HA} and give a proof of the periodicity theorem used in \S\ref{section:mod p cohomology} following \cite{Swan:PPFG}.
\end{enumerate}
\subsection{What is not in these notes?} The title of these notes could---and someday, perhaps, will---be the title of a book, or of a book series. That being the case, there are vast and important swaths of the literature that we do not touch. What follows is a non-exhaustive list of omissions, in no particular order.
\begin{itemize}
\item Iterated loop space theory \cite{May:GILS, Segal:CSILS} and the computation of the mod $p$ homology of $B_k(\mathbb{R}^n)$ with all of the induced structure \cite[III]{CohenLadaMay:HILS}.
\item Factorization homology \cite{AyalaFrancis:FHTM} and embedding calculus \cite{Weiss:EPVIT}.
\item The Cohen--Taylor--Totaro spectral sequence \cite{Totaro:CSAV} and representation stability \cite{ChurchEllenbergFarb:FIMSRSG}.
\item The Fulton--MacPherson compactification \cite{FultonMacPherson:CCS,Sinha:MTCCS} and formality \cite{Kontsevich:OMDQ, LambrechtsVolic:FLNDO}.
\item The failure of homotopy invariance \cite{LongoniSalvatore:CSNHI}.
\item Configuration spaces of graphs \cite{Abrams:CSBGG, Ghrist:CSBGGR, AnDrummond-ColeKnudsen:SSGBG}.
\item Topological combinatorics \cite{Farber:TBPI, BlagojevichLueckZiegler:ETCS}.
\end{itemize}
The author hopes to remedy some of these omissions---as well as the unforgivable absence of pictures---in future versions of these notes.
\subsection{Prerequisites} Sections \ref{section:configuration spaces}-\ref{section:cohomology} of these notes should be accessible to anyone with a first course in algebraic topology \cite{Hatcher:AT}. In particular, facts about homology, cohomology, homotopy groups, covering spaces, and the like are used freely and without comment. In the remainder of the notes, particularly in \S\ref{section:covering theorems}, it will be helpful to have a working knowledge of basic category theory \cite{MacLane:CWM} and simplicial methods \cite{GoerssJardine:SHT}. We make some use of the Serre spectral sequence, which we review at the appropriate time, albeit briefly, and increasingly heavy use of homotopy colimits, which are discussed in some detail in Appendix \ref{section:homotopy colimits}. We make do with a very primitive form of stable homotopy theory, which is reviewed in Appendix \ref{appendix:Spanier--Whitehead}. No knowledge of spectra is assumed or used (although it never hurts).
\subsection{Conventions} For convenience, we take manifolds to be smooth of finite type. In dealing with graded vector spaces, we make use of the usual Koszul sign convention. For $V$ a graded vector space, $\mathrm{Sym}(V)=\bigoplus_{k\geq0}\mathrm{Sym}^k(V)$ is the tensor product of a polynomial algebra on the even part of $V$ with an exterior algebra on the odd part of $V$. We grade homologically and write $V[r]$ for the homological suspension of $V$---that is, the degree $n$ part of $V[r]$ is the degree $n-r$ part of $V$.
\subsection{Acknowledgments} These notes grew out of a course of the same name that I taught during the fall semester of 2017 at Harvard University. I am grateful to the members of the class for their engagement and to Harvard for the opportunity to design a course with complete freedom. Extra thanks are due to Sander Kupers for catching innumerable errors while they were still on the blackboard (if he missed any, we can blame him).
I wish to acknowledge a few key influences on the development of the course and notes. First and most important of these is the mathematics and metamathematics that I learned from my advisor, John Francis, without whom, I am certain, these notes would not exist. Second, the theory of hypercovers and complete covers developed by Dugger--Isaksen \cite{DuggerIsaksen:THAR} plays a central role here, as it has in most of what I have done as a mathematician. Third, the geometric description of the homology of configuration spaces in terms of planetary systems given by Sinha \cite{Sinha:HLDO} is not only very beautiful but also very useful, as it allows for greatly simplified arguments in some cases. Fourth, I would like to thank Haynes Miller for his sustained encouragement in the teaching of the course and the writing of these notes.
Finally, I thank everyone whose enthusiastic conversation has helped to foster my love for the ideas discussed here. In alphabetical order, with many omissions, this group includes Byung Hee An, Lauren Bandklayder, Mark Behrens, Lukas Brantner, Tom Church, Gabriel Drummond-Cole, Jordan Ellenberg, Elden Elmanto, Benson Farb, Ezra Getzler, Paul Goerss, Owen Gwilliam, Kathryn Hess, Sander Kupers, Pascal Lambrechts, Daniel L\"{u}tgehetmann, Jeremy Miller, Sam Nariman, Martin Palmer, Dan Petersen, Dev Sinha, Joel Specter, Alex Suciu, Hiro Tanaka, Nathalie Wahl, Brian Williams, Dylan Wilson, and Jesse Wolfson.
\section{Configuration spaces and braids}\label{section:configuration spaces}
\subsection{Basic examples} In order to begin to develop a feel for how configuration spaces behave, we begin with a few examples.
\begin{example}[$\varnothing$, $\mathbb{R}^0$]
It is usually best to begin with the trivial cases. In the case of the empty manifold, we have \[\mathrm{Conf}_k(\varnothing)=\begin{cases}
\mathrm{pt}&\quad k=0\\
\varnothing&\quad\text{ else,}
\end{cases}\] while \[\mathrm{Conf}_k(\mathbb{R}^0)=\begin{cases}
\mathrm{pt}&\quad k=0,1\\
\varnothing&\quad \text{else.}
\end{cases}\] Notice that $\mathrm{Conf}_0(X)$ is a singleton for any space $X$.
\end{example}
\begin{example}[$\mathbb{R}\cong (0,1)$]
From the definition $\mathrm{Conf}_2((0,1))=(0,1)^2\setminus\{(x,y):x=y\}$, it is clear that the ordered configuration space two points in $(0,1)$ is a disjoint union of two open 2-simplices, and that $\Sigma_2$ acts by permuting these components. In particular, $B_2((0,1))\cong \mathring{\Delta}^2\simeq \mathrm{pt}$. This description generalizes readily to higher $k$. Note that the natural orientation of $(0,1)$ induces a second ordering on the coordinates of any configuration, which is to say a permutation of $\{1,\ldots, k\}$, that any such permutation is possible, and that the assignment of configuration to permutation is locally constant by the intermediate value theorem. Thus, we have a $\Sigma_k$-equivariant bijection $\pi_0(\mathrm{Conf}_k((0,1)))\cong\Sigma_k$, and the unordered configuration space is naturally identified with the path component of the identity permutation. For this space, we define a map \begin{align*}
B_k((0,1))&\to\mathring{\Delta}^k:=\left\{(t_0,\ldots, t_k)\in \mathbb{R}^{k+1}: t_i>0,\, \sum_{i=0}^kt_i=1\right\}\\
\{x_1,\ldots, x_k\}&\mapsto (x_1, x_2-x_1, \ldots, 1-x_k),
\end{align*} where the set $\{x_1,\ldots, x_k\}$ is ordered so that $x_1<\cdots< x_k$. This map is a homeomorphism; in particular, the unordered configuration spaces of $\mathbb{R}$ are all contractible.
\end{example}
\begin{example}[$\mathbb{R}^n$]
In the configuration space of two points in $\mathbb{R}^n$, there are exactly three pieces of data---the direction, the center of mass, and the distance---and only one of these is not a contractible choice. Precisely, the map \begin{align*}\mathrm{Conf}_2(\mathbb{R}^n)&\to S^{n-1}\times\mathbb{R}_{>0}\times\mathbb{R}^n\\
(x_1,x_2)&\mapsto \left(\frac{x_2-x_1}{\|x_2-x_1\|}, \|x_2-x_1\|,\frac{x_1+x_2}{2}\right)
\end{align*} is a homeomorphism, and, in particular, the \emph{Gauss map} \[\mathrm{Conf}_2(\mathbb{R}^n)\to S^{n-1}\] given by the first coordinate of this homeomorphism is a homotopy equivalence. Since this map is also $\Sigma_2$-equivariant for the antipodal action on the target, we conclude that $B_2(\mathbb{R}^n)\simeq \mathbb{RP}^{n-1}$. The Gauss map will play a fundamental role in our later study; in particular, by pulling back a standard volume form, this map furnishes us with our first example of a nonzero higher dimensional class in the cohomology of configuration spaces.
We cannot be as explicit about $\mathrm{Conf}_k(\mathbb{R}^n)$ for higher $k$, but it should be noted that this space is an example of the complement of a \emph{hyperplane arrangement}, since \[\mathrm{Conf}_k(\mathbb{R}^n)=\mathbb{R}^{nk}\setminus \bigcup_{1\leq i<j\leq k}\{(x_1,\ldots, x_k)\in\mathbb{R}^{nk}: x_i=x_j\},\] which is an interesting and classical type of mathematical object in its own right with its own literature \cite{OrlikTerao:AH}.
\end{example}
\begin{example}[$S^1$] By definition $\mathrm{Conf}_2(S^1)$ is the $2$-torus with its diagonal removed, which is homeomorphic to a cylinder, and, by cutting and pasting, one can see directly that the corresponding unordered configuration space is the open M\"{o}bius band \cite{Tuffley:FSSS}. In the general case \cite[p. 292]{CrabbJames:FHT}, make the identification $S^1=\mathbb{R}/\mathbb{Z}$, and consider first the subspace $A\subseteq \mathrm{Conf}_k(S^1)$ of configurations $(x_1,\ldots, x_k)$ whose ordering coincides with the cyclic ordering induced by the standard orientation of $S^1$. For $1\leq i\leq k$, the difference $t_i=x_{i+1}-x_i\in (0,1)$ is well-defined, where we set $t_{k+1}=t_1+1$, and $\sum_{i=1}^kt_i=1$. Recording the normalized first coordinate and the $t_i$ defines a $C_k$-equivariant homeomorphism $A\cong S^1\times\mathring{\Delta}^{k-1}$, which induces a $\Sigma_k$-equivariant homeomorphism \[(S^1\times\mathring{\Delta}^{k-1})\times_{C_k}\Sigma_k\xrightarrow{\simeq} \mathrm{Conf}_k(S^1).\] In particular, $B_k(S^1)\simeq S^1/C_k\cong S^1$ for $k>0$.
\end{example}
\begin{example}[$S^n$]
In the case of two points in the $n$-sphere, as with Euclidean space, the choice of the direction in $S^n$ between the two points is the fundamental parameter. Precisely, the projection onto the first coordinate defines a map $\mathrm{Conf}_2(S^n)\to \mathrm{Conf}_1(S^n)$, and recording the unit tangent vector in the direction of the second coordinate produces a commuting diagram \[\xymatrix{
\mathrm{Conf}_2(S^n)\ar[d]\ar[r]&\mathrm{Sph}(TS^n)\ar[d]\\
\mathrm{Conf}_1(S^n)\ar@{=}[r]&S^n
}\] in which the top map is a homotopy equivalence. The projection map is a special case of a family of maps, the \emph{Fadell--Neuwirth fibrations} \cite{FadellNeuwirth:CS}, which relate different cardinalities of configuration space in any background manifold. The existence of this family of maps, which will be one of our most important tools in what follows, places very strong constraints on configuration spaces; for example, it ultimately implies that the $i$th Betti number of $B_k(M)$ is constant for large $k$ \cite{Church:HSCSM,RandalWilliams:HSUCS}. This phenomenon of \emph{homological stability}, and the corresponding \emph{representation stability} in the ordered case, has since become an active area of study in its own right \cite{Farb:RS}.
\end{example}
\subsection{First properties}
We begin our formal study of configuration spaces by cataloguing a few of their basic features. First, we note that, if $f:X\to Y$ is an injective continuous map, we have the $\Sigma_k$-equivariant factorization \[\xymatrix{
X^k\ar[rr]^-{f^k}&&Y^k\\
\mathrm{Conf}_k(X)\ar[u]\ar@{-->}[rr]^-{\mathrm{Conf}_k(f)}&&\mathrm{Conf}_k(Y)\ar[u]
}\] and thus an induced map $B_k(f):B_k(X)\to B_k(Y)$. This construction respects composition and identities by inspection, so we have functors \begin{align*}
\mathrm{Conf}_k:\mathcal{T}\mathrm{op}^{\mathrm{inj}}&\to \mathcal{T}\mathrm{op}^{\Sigma_k}\\
B_k:\mathcal{T}\mathrm{op}^{\mathrm{inj}}&\to \mathcal{T}\mathrm{op},
\end{align*} where $\mathcal{T}\mathrm{op}^\mathrm{inj}$ denotes the category of topological spaces and injective continuous maps, $\mathcal{T}\mathrm{op}^{\Sigma_k}$ the category of $\Sigma_k$-spaces and equivariant maps, and $\mathcal{T}\mathrm{op}$ the category of topological spaces.
\begin{proposition}
If $f:X\to Y$ is an open embedding, then $\mathrm{Conf}_k(f):\mathrm{Conf}_k(X)\to \mathrm{Conf}_k(Y)$ and $B_k(f):B_k(X)\to B_k(Y)$ are also open embeddings.
\end{proposition}
\begin{proof}
From the definition of the product topology, $f^k:X^k\to Y^k$ is an open embedding, and the first claim follows. The second claim follows from the first and the fact that $\pi:\mathrm{Conf}_k(X)\to B_k(X)$ is a quotient map.
\end{proof}
This functor also respects a certain class of weak equivalences, for which we now introduce some (very nonstandard) terminology.
\begin{definition}
Let $f,g:X\to Y$ be injective continuous maps.
\begin{enumerate}
\item We say that a map $H:X\times[0,1]\to Y$ is a \emph{monotopy} between $f$ and $g$ if it is a homotopy between $f$ and $g$ and if $H_t$ is injective for every $t\in [0,1]$.
\item We say that $f$ and $g$ are \emph{monotopic} if there is such an injectopy.
\item We say that an injective continuous map $f:X\to Y$ is an \emph{monotopy equivalence} if there is an injective continuous map $g:Y\to X$ such that $g\circ f$ is monotopic to $\id_X$ and $f\circ g$ is monotopic to $\id_Y$.
\end{enumerate}
\end{definition}
\begin{proposition}\label{prop:monotopy}
If $f$ and $g$ are monotopic, then $\mathrm{Conf}_k(f)$ and $\mathrm{Conf}_k(g)$ are homotopic (in fact monotopic).
\end{proposition}
\begin{proof}
In the solid commuting diagram
\[\xymatrix{
X^k\times[0,1]^k\ar@{=}[r]^-\sim& (X\times[0,1])^k\ar[r]^-{H^k}&Y^k\\\\
X^k\times[0,1]\ar[uurr]^-{H^\Delta}\ar[uu]^-{\id_{X^k}\times\Delta_k}\\
\mathrm{Conf}_k(X)\times[0,1]\ar@{-->}[rr]\ar[u]&&\mathrm{Conf}_k(Y),\ar[uuu]
}\] the diagonal composite is given by the formula \[H_t^\Delta(x_1,\ldots, x_k)=(H_t(x_1), \ldots, H_t(x_k)).\] By assumption, $H_t$ is injective for each $t\in[0,1]$, so the dashed filler exists. By construction, this map is injective for each $t$ and restricts to $\mathrm{Conf}_k(f)$ at $t=0$ and to $\mathrm{Conf}_k(g)$ at $t=1$.
\end{proof}
\begin{corollary}
If $f$ is a monotopy equivalence, then both $\mathrm{Conf}_k(f)$ and $B_k(f)$ are homotopy (in fact monotopy) equivalences.
\end{corollary}
\begin{proof}
The claim for $\mathrm{Conf}_k(f)$ is immediate from Proposition \ref{prop:monotopy}. For $B_k(f)$, we note that the homotopies constructed in the proof of that result are homotopies through $\Sigma_k$-equivariant maps, so they descend to the unordered configuration space.
\end{proof}
\begin{remark}
Another point of view on the previous two results is provided by the fact (which we will not prove here) that $\mathrm{Conf}_k$ and $B_k$ are \emph{enriched} functors, where the space of injective continuous maps from $X$ to $Y$ is given the subspace topology induced by the compact-open topology on $\mathrm{Map}(X,Y)$. Taking this fact for granted, these results follows immediately, since a monotopy is simply a path in this mapping space.
\end{remark}
The most important examples of monotopy equivalences will be isotopy equivalences of manifolds, as in the following example.
\begin{example}
If $M$ is a manifold with boundary, then $\partial M$ admits a collar neighborhood $U\cong \partial M\times (0,1]$. We define a map $r:M\to M$ by setting $r|_{\partial M\times(0,1]}(x,t)=(x,\frac{t}{2})$ and extending by the identity. This map is injective, and dilation defines monotopies from $r\circ i:\mathring{M}\to \mathring{M}$ and $i\circ r:M\to M$ to the respective identity maps. It follows that the induced map $\mathrm{Conf}_k(\mathring{M})\to \mathrm{Conf}_k(M)$ is a homotopy equivalence.
\end{example}
These functors also interact well with the operation of disjoint union.
\begin{proposition}
Let $X$ and $Y$ be topological spaces. The natural map \[\coprod_{i+j=k}\mathrm{Conf}_i(X)\times\mathrm{Conf}_j(Y)\times_{\Sigma_i\times\Sigma_j}\Sigma_k\to \mathrm{Conf}_k(X\amalg Y)\] is a $\Sigma_k$-equivariant homeomorphism. In particular, the natural map \[\coprod_{i+j=k}B_i(X)\times B_j(Y)\to B_k(X\times Y)\] is a homeomorphism.
\end{proposition}
\begin{proof}
From the definitions, the dashed filler exists in the commuting diagram
\[\xymatrix{\displaystyle \coprod_{i+j=k}X^i\times Y^j\times_{\Sigma_i\times\Sigma_j}\Sigma_k\ar[r]^-\simeq&(X\amalg Y)^k\\
\displaystyle\coprod_{i+j=k}\mathrm{Conf}_i(X)\times\mathrm{Conf}_j(Y)\times_{\Sigma_i\times\Sigma_j}\Sigma_k\ar@{-->}[r]\ar[u]& \mathrm{Conf}_k(X\amalg Y)\ar[u]}\] and is easily seen to be a bijection, which implies the first claim, since the vertical arrows are inclusions of subspaces. The second claim follows from the first after taking the quotient by the action of $\Sigma_k$.
\end{proof}
Thus, we may restrict attention to connected background spaces whenever it is convenient to do so.
Our next goal is to come to grips with the local structure of configuration spaces. We assume from now on that $X$ is locally path connected, and we fix a basis $\B$ for the topology of the space $X$ consisting of connected subsets. We define two partially ordered sets as follows.
\begin{enumerate}
\item We write $\B_k=\{U\subseteq X: U\cong \amalg_{i=1}^k U_i,\, U_i\in\B\}$, and we impose the order relation \[U\leq V\iff U\subseteq V \text{ and } \pi_0(U\subseteq V) \text{ is surjective}.\]
\item We write $\B_k^\Sigma=\{(U,\sigma): U\in \B_k, \,\sigma:\{1,\ldots, k\}\xrightarrow{\simeq}\pi_0(U)\}$, and we impose the order relation \[(U,\sigma)\leq (V,\tau)\iff U\leq V\text{ and } \tau=\sigma\circ\pi_0(U\subseteq V).\]
\end{enumerate} Denoting the poset of open subsets of a space $Y$ by $\op O(Y)$, there is an inclusion $\B_k^\Sigma\to \op O(\mathrm{Conf}_k(X))$ of posets defined by \[U\mapsto \mathrm{Conf}_k^0(U,\sigma):=\left\{(x_1\ldots, x_k)\in \mathrm{Conf}_k(U): x_i\in U_{\sigma(i)}\right\}\subseteq\mathrm{Conf}_k(X)\] and similarly an inclusion $\B_k\to \op O(B_k(X))$ defined by \[ U\mapsto B_k^0(U):=\left\{\{x_1,\ldots, x_k\}\in B_k(U): \{x_1,\ldots, x_k\}\cap U_i\neq \varnothing,\, 1\leq i\leq k\right\}\subseteq B_k(X).\] Note that these subsets are in fact open, since $U\subseteq X$ is open and configuration spaces respect open embeddings.
\begin{lemma}
For any $U\in \B_k$ and $\sigma:\{1,\ldots, k\}\xrightarrow{\simeq} \pi_0(U)$, there are canonical homeomorphisms \[B_k^0(U)\cong\mathrm{Conf}_k^0(U,\sigma)\cong\prod_{i=1}^kU_{\sigma(i)}.\]
\end{lemma}
\begin{proof}
It is easy to see from the definitions that the dashed fillers in the commuting diagram \[\xymatrix{X^k&\mathrm{Conf}_k(X)\ar[l]\ar[r]& B_k(X)\\
\displaystyle\prod_{i=1}^kU_{\sigma(i)}\ar[u]&\mathrm{Conf}_k^0(U,\sigma)\ar@{-->}[l]\ar@{-->}[r]\ar[u]&B_k^0(U)\ar[u]
}\] exist and are bijections. Since the lefthand map is the inclusion of a subspace and the righthand map is a quotient map, the claim follows.
\end{proof}
\begin{proposition}\label{prop:conf basis} Let $X$ be a locally path connected Hausdorff space and $\B$ a topological basis for $X$ consisting of connected subsets.
\begin{enumerate}
\item The collection $\{\mathrm{Conf}_k^0(U,\sigma): (U,\sigma)\in \B_k^\Sigma\}\subseteq \op O(\mathrm{Conf}_k(X))$ is a topological basis.
\item The collection $\{B_k^0(U): U\in \B_k\}\subseteq \op O(B_k(X))$ is a topological basis.
\end{enumerate}
\end{proposition}
\begin{proof}
By the definition of the product and subspace topologies, it will suffice for the first claim to show that, given $(x_1,\ldots, x_k)\in V\subseteq X^k$ such that \begin{itemize}
\item $V\cong \prod_{i=1}^k V_i$ for open subsets $x_i\in V_i\subseteq X$, and
\item $V\subseteq \mathrm{Conf}_k(X)$,
\end{itemize} there exists $(U,\sigma)\in \B_k^\Sigma$ with $(x_1,\ldots, x_k)\in \mathrm{Conf}_k^0(U,\sigma)\subseteq V$. Now, since $\B$ is a topological basis, we may find $U_i\in \B$ with $x_i\in U_i\subseteq V_i$. The second condition and the assumption that $X$ is Hausdorff imply that the $V_i$ are pairwise disjoint, so we may set $U=\coprod_{i=1}^kU_i$ and take $\sigma(i)=[U_i]$. With these choices \[(x_1,\ldots, x_k)\in \mathrm{Conf}_k^0(U,\sigma)\cong\prod_{i=1}^k U_i\subseteq \prod_{i=1}^k V_i=V,\] as desired.
The second claim follows from first, the fact that $\pi:\mathrm{Conf}_k(X)\to B_k(X)$ is a quotient map, and the fact that $\pi(\mathrm{Conf}_k^0(U,\sigma))=B_k^0(U)$ for every $\sigma$.
\end{proof}
These and related bases will be important for our later study, when we come to hypercover methods. For now, we draw the following consequences.
\begin{corollary}
Let $X$ be a locally path connected Hausdorff space. The projection $\pi:\mathrm{Conf}_k(X)\to B_k(X)$ is a covering space.
\end{corollary}
\begin{proof}
For $U\in \B_k$, we have $\Sigma_k$-equivariant identifications \[\pi^{-1}(B_k^0(U))=\bigcup_{\sigma:\{1,\ldots, k\}\cong \pi_0(U)} \mathrm{Conf}_k^0(U,\sigma)\cong B^0_k(U)\times\Sigma_k,\] where the second is induced by a choice of ordering of $\pi_0(U)$.
\end{proof}
As the example of the line with two origins shows, one cannot in general remove the hypothesis that $X$ be Hausdorff (we learned this example from Sander Kupers).
\begin{corollary}
If $M$ is an $n$-manifold, then $\mathrm{Conf}_k(M)$ and $B_k(M)$ are $nk$-manifolds.
\end{corollary}
\begin{proof}
We take $\B$ to be the set of Euclidean neighborhoods in $M$, in which case \[B_k^0(U)\cong \mathrm{Conf}_k^0\cong \mathbb{R}^{nk}\] for any $U\in \B_k$ and $\sigma:\{1,\ldots, k\}\xrightarrow{\simeq}\pi_0(U)$.
\end{proof}
\begin{exercise}
When is $B_k(M)$ orientable?
\end{exercise}
\subsection{Forgetting points}
From now on, unless otherwise specified, we take our background space to be a manifold $M$. In this case, we have access to a poweful tool relating configuration spaces of different cardinalities.
The starting point is the observation that the natural projections from the product factor through the configuration spaces as in the following commuting diagram: \[\xymatrix{\mathrm{Conf}_\ell(M)\ar@{-->}[d]\ar[r]&M^\ell\ar[d]&(x_1,\ldots, x_\ell)\ar@{|->}[d]\\
\mathrm{Conf}_k(M)\ar[r]&M^k&(x_1,\ldots, x_k)
}\] (we take the projection to be on the last $\ell-k$ coordinates for simplicity, but it is not necessary to make this restriction). Clearly, the fiber over a configuration $(x_1,\ldots, x_k)$ in the base is the configuration space $\mathrm{Conf}_{\ell-k}(M\setminus\{x_1,\ldots, x_k\})$. Our first theorem asserts that the situation is in fact much better than this.
\begin{recollection}
Recall that, if $f:X\to Y$ is a continuous map, the \emph{mapping path space} of $f$ is the space of paths in $Y$ out of the image of $f$. In other words, it is the pullback in the diagram \[\xymatrix{E_f\ar[r]\ar[d]&Y^{[0,1]}\ar[d]&p\ar@{|->}[d]\\
X\ar[r]^-f&Y&p(0).
}\] The inclusion $X\to E_f$ given by the constant paths is a homotopy equivalence, and evaluation at $1$ defines a map $\pi_f:E_f\to Y$, which is a fibration \cite[7.3]{May:CCAT}. The \emph{homotopy fiber} of $f$ is the fiber \[\mathrm{hofib}(f):=\pi_f^{-1}(y)\] of this fibration, where $y\in Y$ is some basepoint. The construction of $E_f$, and hence the homotopy fiber, is functorial, and we say that a diagram \[\xymatrix{
X\ar[d]_-f\ar[r]&X'\ar[d]^-{f'}\\
Y\ar[r]&Y'
}\] is \emph{homotopy Cartesian}, or a \emph{homotopy pullback square}, if the induced map $\mathrm{hofib}(f)\to \mathrm{hofib}(f')$ is a weak equivalence. The primary benefit of knowing that a square is homotopy Cartesian is the induced Mayer--Vietoris long exact sequence in homotopy groups.
\end{recollection}
\begin{theorem}[Fadell--Neuwirth \cite{FadellNeuwirth:CS}]\label{thm:Fadell--Neuwirth}
Let $M$ be a manifold and $0\leq k\leq \ell<\infty$. The diagram \[\xymatrix{
\mathrm{Conf}_{\ell-k}(M\setminus\{x_1,\ldots, x_k\})\ar[d]\ar[r]&\mathrm{Conf}_\ell(M)\ar[d]\\
(x_1,\ldots, x_k)\ar[r]&\mathrm{Conf}_k(M)
}\] is homotopy Cartesian.
\end{theorem}
\begin{remark}
In fact, Fadell--Neuwirth prove that this map is a locally trivial fiber bundle, and they give an identification of its structure group. We will not need this full statement, and our alternate proof of this weaker form will allow us to illustrate the efficacy of hypercover methods at a later point.
\end{remark}
The proof is a debt that we will return to pay after having developed a few more advanced homotopy theoretic techniques. For the time being, we concentrate on exploiting this result.
\begin{corollary}
If $M$ is a simply connected $n$-manifold with $n\geq3$, then $\mathrm{Conf}_k(M)$ is simply connected for every $k\geq0$. In particular, $\pi_1(B_k(M))\cong\Sigma_k$.
\end{corollary}
\begin{proof}
The case $k=0$ is trivial and the case $k=1$ is our assumption. The Van Kampen theorem and our assumption on $n$ imply that $M\setminus\{\mathrm{pt}\}$ is simply connected, so the first claim follows by induction using the exact sequence \[\xymatrix{\pi_1(M\setminus \{\mathrm{pt}\})\ar[r]&\pi_1(\mathrm{Conf}_{k}(M))\ar[r]&\pi_1(\mathrm{Conf}_{k-1}(M)).}\] The second claim follows from the observation that $\mathrm{Conf}_k(M)\to B_k(M)$ is a $\Sigma_k$-cover with simply connected total space and hence the universal cover.
\end{proof}
\begin{corollary}\label{cor:surface aspherical}
If $M$ is a connected surface different from $S^2$ or $\mathbb{RP}^2$, then $\mathrm{Conf}_k(M)$ is aspherical for every $k\geq0$. In particular, $B_k(M)$ is aspherical.
\end{corollary}
\begin{proof}
The case $k=0$ is obvious and the case $k=1$ follows from our assumption on $M$. This assumption further guarantees that $M\setminus\{\mathrm{pt}\}$ is also aspherical, so the first claim follows by induction using the exact sequence \[\xymatrix{\pi_i(M\setminus \{\mathrm{pt}\})\ar[r]&\pi_i(\mathrm{Conf}_{k}(M))\ar[r]&\pi_i(\mathrm{Conf}_{k-1}(M))}\] with $i\geq2$. The second claim follows from the first and the fact that $\pi:\mathrm{Conf}_k(M)\to B_k(M)$ is a covering space.
\end{proof}
In order to proceed further, it will be useful to have a criterion for splitting these exact sequences.
\begin{proposition}
If $M$ is the interior of a manifold with non-empty boundary, then the map $\pi_{k,\ell}:\mathrm{Conf}_\ell(M)\to \mathrm{Conf}_k(M)$ admits a section up to homotopy.
\end{proposition}
\begin{proof}
Write $M=\mathring{N}$, and fix a collar neighborhood $\partial N\subseteq U$ and an ordered set $\{x_{k+1},\cdots, x_\ell\}$ of distinct points in $U$. By retracting along the collar, we obtain an embedding $\varphi:M\to M$ that is isotopic to the identity and misses our chosen points. The assignment $(x_1,\ldots, x_k)\mapsto (\varphi(x_1),\ldots, \varphi(x_k), x_{k+1},\cdots, x_\ell)$ defines a continuous map $s:\mathrm{Conf}_k(M)\to \mathrm{Conf}_\ell(M)$ such that $\pi_{k,\ell}\circ s=\mathrm{Conf}_k(\varphi)\simeq \id_{\mathrm{Conf}_k(M)}$, since $\varphi$ is isotopic to the identity.
\end{proof}
\begin{corollary}
For $n\geq 3$, $k\geq0$, and $i\geq0$, there is an isomorphism \[\pi_i(\mathrm{Conf}_k(\mathbb{R}^n))\cong \prod_{j=1}^{k-1} \pi_i\left(\bigvee_jS^{n-1}\right).\]
\end{corollary}
\begin{proof}
For $k\in \{0,1\}$ the claim is obvious, as is the claim for $\pi_0$, and the claim for $\pi_1$ has already been established. In the generic case, we proceed by induction using the exact sequence \[\xymatrix{
\pi_{i+1}(\mathrm{Conf}_{k-1}(\mathbb{R}^n))\ar[r]& \pi_i(\mathbb{R}^n\setminus\{x_1,\ldots, x_{k-1}\})\ar[r]&\pi_i(\mathrm{Conf}_k(\mathbb{R}^n))\ar[r]&\pi_i(\mathrm{Conf}_{k-1}(\mathbb{R}^n)).
}\] The section up to homotopy constructed above induces a section at the level of homotopy groups, so the lefthand map is trivial and the righthand map is surjective. The result now follows from the homotopy equivalence $\mathbb{R}^n\setminus\{x_1,\ldots, x_{k-1}\}\simeq \bigvee_{k-1}S^{n-1}$ and the fact that all groups in sight are Abelian.
\end{proof}
The higher homotopy groups of bouquets of spheres being very complicated objects \cite{Hilton:HGUS}, this result is a striking contrast to the situation in dimension 2 as characterized in Corollary \ref{cor:surface aspherical}.
\begin{remark}
It should be noted that the product decomposition of the previous corollary is additive only. Viewed as shifted Lie algebra via the Whitehead bracket, $\pi_*(\mathrm{Conf}_k(\mathbb{R}^n))$ has a very rich structure---see \cite[II]{FadellHusseini:GTCS}.
\end{remark}
The following result is proved by essentially the same argument.
\begin{corollary}
The fundamental group of $\mathrm{Conf}_k(\mathbb{R}^2)$ is an iterated extension of free groups.
\end{corollary}
\subsection{Braid groups}
We turn our attention now to the fundamental group of the unordered configuration space $B_k(\mathbb{R}^2)$. We fix the basepoints $\{(2i,0)\}_{i=1}^k\in B_k(\mathbb{R}^2)$ and $(2,\ldots, 2k)\in \mathrm{Conf}_k(\mathbb{R}^2)$ and employ them implicitly throughout.
An element in $\pi_1(B_k(\mathbb{R}^2))$ is determined by the data of a permutation $\tau\in\Sigma_k$ and a path $p:[0,\pi]\to (\mathbb{R}^2)^k$ such that
\begin{enumerate}
\item $p(0)_r=(2r,0)$ for $1\leq r\leq k$,
\item $p(\pi)_r=(2\tau(r),0)$ for $1\leq r\leq k$, and
\item $p(t)_r\neq p(t)_{r'}$ for $1\leq r\neq r'\leq k$ and $t\in[0,\pi]$.
\end{enumerate} Composition is determined by composition of permutations and concatenation of paths; in particular, there is a canonical group homomorphism \[\pi_1(B_k(\mathbb{R}^2))\to \Sigma_k,\] which we will shortly see to be surjective. We have the following simple observation regarding this homomorphism.
\begin{lemma}
The subgroup $\pi_1(\mathrm{Conf}_k(\mathbb{R}^2))\leq \pi_1(B_k(\mathbb{R}^2))$ coincides with the kernel of the homomorphism $B_k\to \Sigma_k$.
\end{lemma}
\begin{proof}
It is obvious that $\pi_1(\mathrm{Conf}_k(\mathbb{R}^2))\leq \ker(B_k\to \Sigma_k)$, and both subgroups have the same index in $B_k$, since $\mathrm{Conf}_k(\mathbb{R}^2)\to B_k(\mathbb{R}^2)$ is $k!$-fold cover.
\end{proof}
To see verify surjectivity, we will exhibit elements $\sigma_i\in\pi_1(B_k(\mathbb{R}^2))$ lifting the respective transpositions $\tau_i=(i,i+1)$.
\begin{construction}
For $1\leq i\leq k$, define a path $p_i:[0,\pi]\to (\mathbb{R}^2)^k$ by the formula \[p_i(t)_r=\begin{cases}
(2r,0)&\quad r\notin\{i,i+1\}\\
c_{2i+1,1}(t+\pi)&\quad r=i\\
c_{2i+1,1}(t)&\quad r=i+1,
\end{cases}\] where $c_{a,b}(t)=(a+b\cos t,b\sin t)$ is the standard parametrization of the circle of radius $b$ centered at $(a,0)$.
The dashed factorization exists in the commuting diagram \[\xymatrix{[0,\pi]\ar[d]_-{p_i}\ar@{-->}[dr]\ar[r]&B_k(\mathbb{R}^2)\\
(\mathbb{R}^2)^k&\mathrm{Conf}_k(\mathbb{R}^2)\ar[l]\ar[u]_-\pi.
}\] Indeed, $\sin t=-\sin t$ if and only if $t\in \{0,\pi\}$, and in both of these cases we have $\cos t\neq -\cos t$. We write $\sigma_i$ for the homotopy class of the top horizontal map relative to $\{0,\pi\}$. Since $p_i(0)=((2,0),\ldots, (2i,0), (2i+2,0), \ldots, (2k,0))$ and $p_i(\pi)=((2,0),\ldots, (2i+2,0), (2i,0),\ldots, (2k,0))$, the class $\sigma_i$ defines an element of $\pi_1(B_k(\mathbb{R}^2))$.
\end{construction}
These elements satisfy some easy relations.
\begin{lemma}
If $|i-j|>1$, then $\sigma_i\sigma_j=\sigma_j\sigma_i$.
\end{lemma}
\begin{proof}
Without loss of generality, we may assume that $j>i$. Define $H:[0,\pi]^2\to (\mathbb{R}^2)^k$ by the formula \[H(s,t)_r=\begin{cases}
(2r,0)&\quad r\notin\{i,i+1,j,j+1\}\\
p_i(s)_r&\quad r\in\{i,i+1\}\\
p_j(t)_r&\quad r\in \{j,j+1\}.
\end{cases}\] For $r\in\{i,i+1\}$, the image of $H(s,t)_{r}$ is contained in a closed disk of radius $1$ around $(2i+1,0)$, and, for $r\in\{j,j+1\}$, in a closed disk of radius $1$ around $(2j+1,0)$. Since $|j-i|>1$, these sets are disjoint, and we conclude that $H$ factors through $\mathrm{Conf}_k(\mathbb{R}^2)$. The lemma now follows from the observation that $\sigma_j\sigma_i$ (resp. $\sigma_i\sigma_j$) is represented by the path obtained by composing $H$ with the counterclockwise (resp. clockwise) path around the boundary of $[0,\pi]^2$.
\end{proof}
\begin{lemma}
For $1\leq i\leq k$, $\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}$.
\end{lemma}
\begin{proof}
Define maps $q_1,q_2:[0,\pi]\to (\mathbb{R}^2)^k$ by the formulas \[
q_1(t)_r=\begin{cases}
(2r,0)&\quad r\notin\{i,i+1, i+2\}\\
c_{2i+2,2}(t+\pi)&\quad r=i\\
c_{2i+1,1}(2t)&\quad r=i+1\\
c_{2i+2,2}(t)&\quad r=i+2
\end{cases}
\]
\[
q_2(t)_r=\begin{cases}
(2r,0)&\quad r\notin\{i,i+1,i+2\}\\
\bar c_{2i+2,2}(t)&\quad r=i\\
\bar c_{2i+3,1}(2t+2\pi)&\quad r=i+1\\
\bar c_{2i+2,2}(t+\pi)&\quad r=i+2,
\end{cases}\] where a bar indicates reversal of parametrization. We claim that the concatenation of $q_1$ followed by $q_2$ represents $\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1}\sigma_i\sigma_{i+1}\sigma_i$. Assuming this claim, the lemma folllows, for the three nonconstant coordinate functions of this concatenation are nullhomotopic via homotopies with pairwise disjoint images.
To see why the claim is true, consider the representative $\tilde q$ of $\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1}\sigma_i\sigma_{i+1}\sigma_i$ that is given by concatenating the representatives $p_i$ and $p_{i+1}$ and their inverses. The path $\tilde q(t)_i$ first follows the lower half-circle of radius $1$ centered at $(2i+1,0)$, then follows the lower half-circle of radius $1$ centered at $(2i+3,0)$, then retraces these steps exactly. Thus, $\tilde q(t)_i$ is homotopic by a straight-line homotopy to a path that first follows the lower half-circle of radius 2 centered at $2i+2$ and then retraces this path, and the image of this homotopy away from $\tilde q(t)_i$ is disjoint from the images of $\tilde q(t)_r$ for $r\neq i$. In this way, we obtain a homotopy in the configuration space, and similar considerations apply to the $(i+2)$nd coordinate and the respective upper half-circles. Thus, $\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1}\sigma_i\sigma_{i+1}\sigma_i$ is represented by a path in which the $i$th and $(i+2)$nd coordinate trace and then retrace these larger half-circles while the $(i+1)$st coordinate traverses the circle of radius 1 centered at $2i+1$ followed by the circle of radius 1 centered at $2i+3$ in reverse. Since the concatenation of $q_1$ followed by $q_2$ is such a path, this establishes the claim.
\end{proof}
\begin{definition}[Artin \cite{Artin:TB}]
The \emph{braid group} on $k$ strands is the group $B_k$ defined by the presentation \[B_k=\left\langle \sigma_1,\ldots, \sigma_{k-1}\mid\sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1},\, \sigma_i\sigma_j=\sigma_j\sigma_i \text{ if } |i-j|>1\right\rangle.\]
\end{definition}
The previous two lemmas imply the existence of a canonical group homomorphism from the abstract group $B_k$ to $\pi_1(B_k(\mathbb{R}^2))$.
\begin{theorem}[Fox--Neuwirth \cite{FoxNeuwirth:BG}]\label{thm:fox-neuwirth}
The homomorphism $B_k\to \pi_1(B_k(\mathbb{R}^2))$ is an isomorphism.
\end{theorem}
Before proceeding with the proof, we first recall that the symmetric group on $k$ letters admits the Coxeter presentation \[\Sigma_k=\langle \tau_1,\ldots, \tau_k\mid \tau_i^2=1,\,(\tau_i\tau_{i+1})^3=1,\,(\tau_i\tau_j)^2=1\text{ if }|i-j|>1\rangle.\] It follows that the assignment $\sigma_i\mapsto \tau_i$ determines a group homomorphism $B_k\to \Sigma_k$.
\begin{definition}
The \emph{pure braid group} on $k$ strands is the subgroup $P_k=\ker(B_k\to \Sigma_k)\leq B_k$.
\end{definition}
\begin{proof}[Proof of Theorem \ref{thm:fox-neuwirth}]
Consider the following diagram of group homomorphisms: \[\xymatrix{
1\ar[r]&P_k\ar[r]\ar@{-->}[d]&B_k\ar[r]\ar[d]&\Sigma_k\ar@{=}[d]\ar[r]&1\\
1\ar[r]&\pi_1(\mathrm{Conf}_k(\mathbb{R}^2))\ar[r]&\pi_1(B_k(\mathbb{R}^2))\ar[r]&\Sigma_k\ar[r]&1.
}\] The top row is exact by definition and the bottom was shown to be exact above. The righthand square commutes by the construction of the $\sigma_i$, and it follows that the dashed factorization exists making the lefthand square commute. Thus, by the five lemma, it will suffice to verify that the induced map $P_k\to \pi_1(\mathrm{Conf}_k(\mathbb{R}^2))$ is an isomorphism for each $k$.
For this claim, we proceed by induction on $k$, the cases $k=0$ and $k=1$ being trivial. Consider the following diagram of group homomorphisms:
\[\xymatrix{
1\ar[r]&K\ar@{-->}[d]\ar[r]& P_k\ar[d]\ar@{-->}[r]& P_{k-1}\ar[d]^-{\mathrel{\rotatebox[origin=c]{-90}{$\simeq$}}}\ar[r]&1\\
1\ar[r]&\pi_1(\mathbb{R}^2\setminus \{x_1,\ldots, x_{k-1}\})\ar[r]&\pi_1(\mathrm{Conf}_k(\mathbb{R}^2))\ar[r]&\pi_1(\mathrm{Conf}_{k-1}(\mathbb{R}^2))\ar[r]&1.
}\] The middle and righthand vertical maps are the maps constructed above, the upper righthand horizontal map is defined by the inductive hypothesis and the requirement that the righthand square commute, and the subgroup $K\leq P_k$ is defined to be the kernel of this map. Since the top sequence is exact by definition and the bottom by Theorem \ref{thm:Fadell--Neuwirth} and the fact that the three spaces shown are all aspherical, the dashed vertical factorization exists making the lefthand square commute.
As we will see below in Lemma \ref{lem:kernel isomorphism}, $K$ is generated by $k-1$ elements, which map to a set of $k-1$ free generators under the lefthand vertical map, and assuming this fact, we may complete the proof. Indeed, these $k-1$ elements determine a homomorphism from the free group, which is a section of the map in question. This section is injective, since it is a section, as well as surjective, since its image contains a generating set. Since the section is an isomorphism, the map itself is so as well, and the five lemma implies the claim.
\end{proof}
In this way, we are led to the following question, which will next occupy our attention: given a presentation of a group $G$, how can we find a presentation for a subgroup $H\leq G$?
\subsection{The Reidemeister--Schreier method} We now discuss a technique from combinatorial group theory for producing presentations of subgroups---see \cite[2.3]{MagnusKarrassSolitar:CGT} for a standard account. We take the topological viewpoint expounded in \cite{ZieschangVogtColdeway:SPDG}.
\begin{definition}
A \emph{graph} is a 1-dimensional CW complex $\Gamma$ with at most countably many cells. The 0-cells of $\Gamma$ are its \emph{vertices} and the 1-cells its \emph{edges}. An \emph{edge path} is a sequence of oriented edges $\{e_i\}_{i=1}^n$ such that the head of $e_i$ is the tail of $e_{i+1}$ for $1\leq i<n$. An edge path is \emph{reduced} if it contains no subpath of the form $ee^{-1}$, where $e^{-1}$ denotes the edge $e$ with the opposite orientation. A \emph{tree} is a contractible graph. A \emph{spanning tree} for a graph $\Gamma$ is a subgraph $T\subseteq \Gamma$ such that $T$ is a tree and $T$ contains every vertex of $\Gamma$.
\end{definition}
\begin{lemma}
In a tree $T$, any pair of distinct vertices are connected by a unique reduced edge path.
\end{lemma}
\begin{proof}
After perhaps discarding redundant edges, the existence of two such edge paths produces an embedding of $S^1$ into $T$, a contradiction.
\end{proof}
\begin{lemma}
If $\Gamma$ is a connected graph, then $\Gamma$ admits a spanning tree.
\end{lemma}
\begin{proof}
Fix a vertex $v\in \Gamma$, set $T_0=\{v\}$, and define $T_i\subseteq \Gamma$ for $i>0$ recursively by adding to $T_{i-1}$ each vertex $w\in \Gamma$ that is separated from $T_{i-1}$ by a exactly one edge, together with one such edge for each such $w$. Then $T=\bigcup_{i=0}^\infty T_i$ is a spanning tree.
\end{proof}
This observation implies the following familiar consequence.
\begin{corollary}
If $\Gamma$ is a connected graph, $v\in \Gamma$ is a vertex, and $T\subseteq \Gamma$ is a spanning tree, then $\pi_1(\Gamma,v)$ is freely generated by a set of cardinality $|\pi_0(\Gamma\setminus T)|$.
\end{corollary}
\begin{proof}
Extend a contraction of $T$ to the vertex $v$ to obtain a deformation of $\Gamma$ onto a wedge of $|\pi_0(\Gamma\setminus T)|$-many circles, and apply the Van Kampen theorem.
\end{proof}
In particular, after fixing a vertex $v\in\Gamma$ and a spanning tree $T\subseteq\Gamma$, a set of free generators for $\pi_1(\Gamma, v)$ is given by the collection of loops given by concatenating an edge $e$ in $\Gamma\setminus T$ on either side with the unique reduced edge paths in $T$ from $v$ to the endpoints of $e$.
If the free group $F(S)$ on the set $S$ corresponds to the graph $\Gamma_S:=\bigvee_S S^1$, then a subgroup $H\leq F(S)$ corresponds to a covering space $\pi:\widetilde \Gamma\to \Gamma_S$. Since any covering space has unique lifting for maps with simply connected domains, a covering space of a graph is again a graph in a canonical way; in particular, we conclude the following classical fact.
\begin{corollary}[Nielsen--Scheier]
A subgroup of a free group is free.
\end{corollary}
We now consider the problem of finding generators for $H$. We begin by fixing a basepoint $\tilde v\in\widetilde\Gamma$ lying over the unique vertex $v\in \Gamma_S$ and a spanning tree $\tilde v\in T\subseteq \widetilde\Gamma$. The edges of $\widetilde\Gamma$ correspond via $\pi$ to edges of $\Gamma_S$, which is to say generators of $G$, and the vertices correspond bijectively to the set $F(S)/H$ of cosets. For any vertex $w\in T$, there is a unique reduced edge path in $T$ from $\tilde v$ to $w$, and the word corresponding to this edge path represents the coset corresponding to $w$. This edge path clearly has the property that each of its initial segments is again such a path.
\begin{definition}
A set $A=\{g_\ell\}_{\ell\in F(S)/H}$ of coset representatives is a \emph{Schreier set} for $H$ if, after writing each $g_\ell$ as a reduced word in $S$, every initial subword of $g_\ell$ is again an element of $A$.
\end{definition}
As a matter of notation, we write $g\mapsto \bar g$ for the composite $F(S)\to F(S)/H\cong A\to F(S)$.
\begin{theorem}[Reidemeister--Schreier, part I]
Let $H\leq F(S)$ be a subgroup and $A$ a Schreier set for $H$. The nontrivial elements of the form $g_\ell s(\overline{g_\ell s})^{-1}$ for $\ell\in F(S)/H$ and $s\in S$ freely generate $H$.
\end{theorem}
\begin{proof}
Fix a basepoint $\tilde v$. By path lifting, the Schreier set $A$ arises geometrically from a spanning tree $T$. The element $g_\ell s(\overline{g_\ell s})^{-1}$ corresponds to the path given by following the unique reduced edge path from $\tilde v$ to the vertex corresponding to $\ell$, crossing the edge $e$ corresponding to $s$, and then following the unique reduced edge path back to $\tilde v$. If $e$ lies in $T$, then $\overline{g_\ell s}= g_\ell s$, and the element is trivial; otherwise, we recognize $g_\ell s(\overline{g_\ell s})^{-1}$ as one of our previously established free generators for $H\cong\pi_1(\widetilde\Gamma, \tilde v)$.
\end{proof}
In order to accommodate the presence of relations, we extend our definition of a Schreier set as follows.
\begin{definition}
Let $G=\langle s\in S\mid r\in R\rangle$ be a group with a presentation and $H\leq G$ a subgroup. A set $A=\{g_\ell\}_{\ell\in G/H}$ of coset representatives is a \emph{Schreier set} for $H$ if it lifts to a Schreier set for $F(S)\times_GH\leq F(S)$.
\end{definition}
By the Van Kampen theorem, the group $G$ with the specified presentation is the fundamental group of the CW complex obtained by attaching 2-cells to $\Gamma_S$ according to the words $r\in R$. This cell structure lifts canonically to the covering space by attaching cells along the conjugates $g_\ell rg_\ell^{-1}$. Thus, we conclude the following generalization.
\begin{theorem}[Reidemeister--Schreier, part II]
Let $G=\langle s\in S\mid r\in R\rangle$ be a group with a presentation, $H\leq G$ a subgroup, and $A$ a Schreier set for $H$. Then $H$ is generated by the nontrivial elements $g_\ell s(\overline{g_\ell s})^{-1}$ for $\ell\in G/H$ and $s\in S$, with defining relations $g_\ell rg_\ell^{-1}$ for $\ell\in G/H$ and $r\in R$, written in the nontrivial generators.
\end{theorem}
\subsection{Back to braid groups} Before leveraging this result in completing the proof of Theorem \ref{thm:fox-neuwirth}, we pause to establish some notation. For $1\leq i<j\leq k$, we write $A_{ij}$ for the braid in which the $j$th strand winds once around the $i$th strand, passing in front of the intervening strands, and then returns to its starting position. In symbols, \[A_{ij}=\sigma_{j-1}\cdots\sigma_{i+1}\sigma_i^2\sigma_{i+1}^{-1}\cdots \sigma_{j-1}^{-1}.\] Note that the braids $A_{ij}$ are all pure.
We write $U_k\leq B_k$ for the subgroup generated by the elements $A_{ik}$ for $1\leq i<k$. It is easy to verify geometrically that $U_k\leq \ker(P_k\to P_{k-1})$ and that the induced homomorphism $U_k\to \pi_1(\mathbb{R}^2\setminus\{x_1,\ldots, x_{k-1}\})$ sends $\{A_{ik}\}_{i=1}^{k-1}$ to a set of free generators; in particular, $U_k$ is free on these generators. Our goal is to prove the following, which is the missing ingredient in the proof of Theorem \ref{thm:fox-neuwirth}.
\begin{lemma}\label{lem:kernel isomorphism}
The inclusion $U_k\leq \ker(P_k\to P_{k-1})$ is an isomorphism.
\end{lemma}
This result will be an easy consequence of the following.
\begin{proposition}\label{prop:pure semidirect}
There is an isomorphism $P_k\cong U_k\rtimes P_{k-1}$.
\end{proposition}
\begin{proof}[Proof of Lemma \ref{lem:kernel isomorphism}]
Inverting isomorphisms in the diagram
\[\xymatrix{
U_k\ltimes P_{k-1}\ar[d]_-\wr\ar[r]&P_{k-1}\ar@{=}[d]\\
P_k\ar[d]&P_{k-1}\ar[d]^-\wr\\
\pi_1(\mathrm{Conf}_k(\mathbb{R}^2))\ar[r]&\pi_1(\mathrm{Conf}_{k-1}(\mathbb{R}^2)),
}\] of group homomorphisms, we obtain two maps $P_k\to P_{k-1}$, the kernels of which are $U_k$ and the kernel in question; therefore, to conclude that these two coincide, it suffices to verify that the diagram commutes. By inspection of the element in homotopy corresponding to $A_{i k}$, it is clear that both composites annihilate $U_k$, so it suffices to verify that the composite \[P_{k-1}\to P_k\to \pi_1(\mathrm{Conf}_k(\mathbb{R}^2))\to \pi_1(\mathrm{Conf}_{k-1}(\mathbb{R}^2))\] is the canonical map. By the previous corollary and induction on $k$, $P_{k-1}$ is generated by $\{A_{ij}\}_{1\leq i<j\leq k-1}$, and the claim follows from the geometric interpretation of the $A_{ij}$.
\end{proof}
Our proof of Proposition \ref{prop:pure semidirect} will be somewhat indirect, proceeding through the intermediate group $D_k\leq B_k$ of braids that do not permute the last strand. In other words, $D_k$ is defined as the pullback in the diagram \[\xymatrix{
D_k\ar[r]\ar[d]&B_k\ar[d]\\
\Sigma_{k-1}\ar[r]&\Sigma_k.
}\] The proposition is an immediate consequence of the following result.
\begin{proposition}\label{prop:D semidirect}
There is an isomorphism $D_k\cong U_k\rtimes B_{k-1}$.
\end{proposition}
We prove this result by applying the Reidemeister--Schreier method to $D_k\leq B_k$. Define elements $g_\ell=\sigma_{k-1}\cdots \sigma_\ell$ for $1\leq \ell\leq k$.
\begin{lemma}\label{lem:schreier set}
The set $\{g_\ell\}_{\ell=1}^k$ is a Schreier set for $D_k$ in $B_k$.
\end{lemma}
\begin{proof}
Reading off the last entry of a permutation defines a bijection $\Sigma_k/\Sigma_{k-1}\cong \{1,\ldots, k\}$, and the last entry of the permutation $\tau_{k-1}\cdots \tau_\ell$ is $\ell$. Thus, the composite \[\{g_\ell\}_{\ell=1}^k\to B_k/D_k\to \Sigma_k/\Sigma_{k-1}\to \{1,\ldots, k\}\] is surjective and hence bijective. Since the rightmost two maps are also bijective, the first is as well, and we conclude that $\{g_\ell\}_{\ell=1}^k$ is a set of coset representatives for $D_k$ in $B_k$. Since any initial word in $g_\ell$ is obviously of the form $g_{\ell'}$ for some $\ell'$, the claim follows.
\end{proof}
In carrying out our computations below, we will need to know the following relations.
\begin{lemma}\label{lem:g sigma relations}
The following relations hold in $B_k$:\begin{align*}
&\sigma_i^{g_\ell}=\sigma_i, &1\leq i<\ell-1< k \\
&\sigma_i^{g_\ell}=\sigma_{i-1}, &1\leq \ell<i<k\\
&g_i\sigma_ig_{i+1}^{-1}=A_{ik}, &1\leq i< k\\
&g_i\sigma_{i-1}g_{i-1}^{-1}=1, &1<i\leq k.
\end{align*}
\end{lemma}
\begin{proof}
The third and fourth relations are obvious from the definitions, and the first is immediate from the commutativity relations in $B_k$. For the second, we have that \begin{align*}
g_\ell\sigma_ig_\ell^{-1}&=\sigma_{k-1}\cdots\sigma_\ell\sigma_i\sigma_\ell^{-1}\cdots \sigma_{k-1}^{-1}\\
&=\sigma_{k-1}\cdots \sigma_i\sigma_{i-1}\sigma_i\sigma_{i-2}\cdots \sigma_\ell\sigma_\ell^{-1}\cdots \sigma_{k-1}^{-1}\\
&=\sigma_{k-1}\cdots \sigma_i\sigma_{i-1}\sigma_i\sigma_{i-1}^{-1}\cdots \sigma_{k-1}^{-1}\\
&=\sigma_{k-1}\cdots \sigma_{i-1}\sigma_i\sigma_{i-1}\sigma_{i-1}^{-1}\cdots \sigma_{k-1}^{-1}\\
&=\sigma_{k-1}\cdots\sigma_{i+1}\sigma_{i-1}\sigma_{i+1}^{-1}\cdots\sigma_{k-1}^{-1}\\
&=\sigma_{j-1},
\end{align*} where the second equality follows from the commutativity relations, the third by cancellation, the fourth by the braid relations, the fifth by cancellation, and the last by commutativity.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:D semidirect}]
We have that, modulo $\Sigma_{k-1}$, \[\tau_{k-1}\cdots \tau_\ell\tau_i\equiv \begin{cases}
\tau_{k-1}\cdots \tau_\ell&\quad 1\leq i<\ell-1< k\text{ or }1\leq \ell<i< k\\
\tau_{k-1}\cdots \tau_{\ell-1}&\quad 1\leq \ell-1=i<k\\
\tau_{k-1}\cdots \tau_{\ell+1}&\quad 1\leq \ell=i<k.
\end{cases}\] Thus, we have \[\overline{g_\ell\sigma_i}=\begin{cases}
g_\ell&\quad1\leq i<\ell-1< k\text{ or }1\leq \ell<i< k\\
g_{\ell-1}&\quad 1\leq \ell-1=i<k\\
g_{\ell+1}&\quad 1\leq \ell=i<k,
\end{cases}\] and Lemma \ref{lem:g sigma relations} now implies that \begin{align*}g_\ell\sigma_{\ell-1}(\overline{g_\ell\sigma_{\ell-1}})^{-1}&=g_\ell\sigma_{\ell-1}g_{\ell-1}^{-1}=1\\
g_\ell\sigma_{\ell}(\overline{g_\ell\sigma_{\ell}})^{-1}&=g_\ell\sigma_\ell g_{\ell+1}^{-1}=A_{\ell k}\\
g_\ell\sigma_{i}(\overline{g_\ell\sigma_{i}})^{-1}&=\sigma_i^{g_\ell}=\begin{cases}
\sigma_i&\quad 1\leq i<\ell-1< k \\
\sigma_{i-1}&\quad 1\leq \ell<i<k.
\end{cases}
\end{align*}
We conclude by the Reidemeister--Schreier method that $D_k$ is generated by the collection $\{A_{\ell k}, \sigma_i: 1\leq \ell\leq k,\, 1\leq i< k-1\}$, which is to say by the subgroups $U_k$ and $B_{k-1}$.
In order to determine the relations, we conjugate the relations in $B_k$ by the $g_\ell$ and express the result in terms of our chosen generators using Lemma \ref{lem:g sigma relations}. We begin with the commutativity relations, for which there are six cases.
\begin{enumerate}
\item For $1\leq \ell<i<j-1< k-1$, \begin{align*}
[\sigma_i,\sigma_j]^{g_\ell}&= \sigma_i^{g_\ell}\sigma_j^{g_\ell}\sigma_i^{-g_\ell}\sigma_j^{-g_\ell}\\
&=[\sigma_{i-1},\sigma_{j-1}]
\end{align*}
\item For $1\leq i=\ell <j-1<k-1$, \begin{align*}
[\sigma_i,\sigma_j]^{g_\ell}&=g_\ell\sigma_ig_{\ell+1}^{-1}g_{\ell+1}\sigma_jg_{\ell+1}^{-1}g_{\ell+1}\sigma_i^{-1}g_{\ell}^{-1}g_\ell\sigma_j^{-1}g_{\ell}^{-1}\\
&=[A_{ik},\sigma_{j-1}].
\end{align*}
\item For $1\leq i=\ell-1<j-1<k-1$, \begin{align*}
[\sigma_i,\sigma_j]^{g_\ell}&=g_\ell\sigma_ig_{\ell-1}^{-1}g_{\ell-1}
\sigma_jg_{\ell-1}^{-1}g_{\ell-1}\sigma_i^{-1}g_\ell^{-1}g_\ell\sigma_j^{-1}g_\ell^{-1}\\
&=[1, \sigma_{j-1}]\\
&=1.
\end{align*}
\item For $1\leq i <\ell-1=j-1<k-1$, \begin{align*}
[\sigma_i,\sigma_j]^{g_\ell}&=g_\ell\sigma_ig_\ell^{-1}g_\ell\sigma_jg_{\ell+1}^{-1}g_{\ell+1}\sigma_i^{-1}g_{\ell+1}^{-1}g_{\ell+1}\sigma_j^{-1}g_\ell^{-1}\\
&=[\sigma_i,A_{jk}].
\end{align*}
\item For $1< i+1<j=\ell-1<k-1$, \begin{align*}
[\sigma_i,\sigma_j]^{g_\ell}&=g_\ell\sigma_ig_\ell^{-1}g_\ell\sigma_jg_{\ell-1}^{-1}g_{\ell-1}\sigma_i^{-1}g_{\ell-1}^{-1}g_{\ell-1}\sigma_j^{-1}g_\ell\\
&=[\sigma_i,1]\\
&=1.
\end{align*}
\item For $1< i+1<j<\ell-1<k-1$, \begin{align*}
[\sigma_i,\sigma_j]^{g_\ell}&=\sigma_i^{g_\ell}\sigma_j^{g_\ell}\sigma_i^{-g_\ell}\sigma_j^{-g_\ell}\\
&=[\sigma_i,\sigma_j].
\end{align*}
\end{enumerate} We turn now to the braid relations, for which there are five further cases.
\begin{enumerate}
\item[(7)] For $1\leq \ell<i< k-1$, \begin{align*}
(\sigma_i\sigma_{i+1}\sigma_i\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1})^{g_\ell}&=\sigma_i^{g_\ell}\sigma_{i+1}^{g_\ell}\sigma_i^{g_\ell}\sigma_{i+1}^{-g_\ell}\sigma_i^{-g_\ell}\sigma_{i+1}^{-g_\ell}\\
&=\sigma_{i-1}\sigma_{i}\sigma_{i-1}\sigma_{i}^{-1}\sigma_{i-1}^{-1}\sigma_{i}^{-1}.
\end{align*}
\item[(8)] For $1\leq \ell=i<k-1$, \begin{align*}
(\sigma_i\sigma_{i+1}\sigma_i\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1})^{g_\ell}&=g_\ell\sigma_ig_{\ell+1}^{-1}g_{\ell+1}\sigma_{i+1}g_{\ell+2}^{-1}g_{\ell+2}\sigma_ig_{\ell+2}^{-1}g_{\ell+2}\sigma_{i+1}^{-1}g_{\ell+1}^{-1}g_{\ell+1}\sigma_i^{-1}g_\ell^{-1}g_\ell\sigma_{i+1}^{-1}g_\ell^{-1}\\
&=A_{ik}A_{i+1,k}(A_{ik}A_{i+1,k})^{-\sigma_i}.
\end{align*}
\item[(9)] For $1\leq i=\ell-1\leq k-1$, \begin{align*}
(\sigma_i\sigma_{i+1}\sigma_i\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1})^{g_\ell}&=g_\ell\sigma_ig_{\ell-1}^{-1}g_{\ell-1}\sigma_{i+1}g_{\ell-1}^{-1}g_{\ell-1}\sigma_ig_{\ell}^{-1}g_{\ell}\sigma_{i+1}^{-1}g_{\ell+1}^{-1}g_{\ell+1}\sigma_i^{-1}g_{\ell+1}^{-1}g_{\ell+1}\sigma_{i+1}^{-1}g_\ell^{-1}\\
&=A_{ik}^{\sigma_i}A_{i+1,k}^{-1}.
\end{align*}
\item[(10)] For $1< i+1=l-1\leq k-1$, \begin{align*}
(\sigma_i\sigma_{i+1}\sigma_i\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1})^{g_\ell}&=g_\ell\sigma_ig_{\ell}^{-1}g_\ell \sigma_{i+1}g_{\ell-1}^{-1}g_{\ell-1}\sigma_ig_{\ell-2}^{-1}g_{\ell-2}\sigma_{i+1}^{-1}g_{\ell-2}^{-1}g_{\ell-2}\sigma_i^{-1}g_{\ell-1}^{-1}g_{\ell-1}\sigma_{i+1}^{-1}g_\ell^{-1}\\
&=[\sigma_i,1]\\
&=1.
\end{align*}
\item[(11)] For $1<i+1<\ell-1\leq k-1$, \begin{align*}
(\sigma_i\sigma_{i+1}\sigma_i\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1})^{g_\ell}&=\sigma_i^{g_\ell}\sigma_{i+1}^{g_\ell}\sigma_i^{g_\ell}\sigma_{i+1}^{-g_\ell}\sigma_i^{-g_\ell}\sigma_{i+1}^{-g_\ell}\\
&=\sigma_i\sigma_{i+1}\sigma_i\sigma_{i+1}^{-1}\sigma_i^{-1}\sigma_{i+1}^{-1}
\end{align*}
\end{enumerate}
Relations (3), (5), and (10) are vacuous; relations (1), (6), (7), and (11) yield the defining presentation for $B_{k-1}$; and relations (2), (4), (8), and (9) imply that $U_k$ is a normal subgroup. We recognize the resulting presentation as a presentation of a semidirect product of $B_{k-1}$ with the free group on the set $\{A_{\ell k}\}_{\ell=1}^k$.
\end{proof}
\section{(Co)homology of $\mathrm{Conf}_k(\mathbb{R}^n)$}\label{section:cohomology}
Our present goal is to understand the cohomology ring $H^*(\mathrm{Conf}_k(\mathbb{R}^n))$ with integer coefficients. Since the cases $n\in\{0,1\}$ are exceptional and easy, we assume throughout that $n\geq0$.
\subsection{Additive structure} We begin with the task of understanding the cohomology as a graded Abelian group.
\begin{definition}
Let $V$ be a degreewise finitely generated non-negatively graded Abelian group $V$. The \emph{Poincar\'{e} polynomial} of $V$ is the polynomial \[P(V)=\sum_{i\geq0}\mathrm{rk}(V_i)t^i.\] If $X$ is a space of finite type, the \emph{Poincar\'{e} polynomial} of $X$ is the $P(X):= P(H_*(X))$.
\end{definition}
The Poincar\'{e} polynomial for graded Abelian groups is additive under direct sum and multiplicative under tensor product, so the Poincar\'{e} polynomial for spaces is additive under disjoint union and, with appropriate torsion-freeness assumptions in place, multiplicative under Cartesian product.
\begin{theorem}[Leray--Hirsch]\label{thm:Leray--Hirsch}
Suppose that the diagram \[\xymatrix{
F\ar[r]\ar[d]&E\ar[d]\\
\mathrm{pt}\ar[r]&B
}\] is homotopy Cartesian and that \begin{enumerate}
\item $F$ and $B$ are path connected,
\item $H^*(F)$ is free Abelian,
\item $H^*(F)$ or $H^*(B)$ is of finite type, and
\item $H^*(E)\to H^*(F)$ is surjective.
\end{enumerate} There is an isomorphism $H^*(E)\cong H^*(B)\otimes H^*(F)$ of $H^*(B)$-modules. In particular, we have the equation \[P(E)=P(B)P(F).\]
\end{theorem}
The proof, which is premised on a few basic properties of the Serre spectral sequence, is deferred to a later point in the notes, at which we will discuss this tool in some detail.
In order to apply the Leray--Hirsch theorem, we must verify point (4). In doing so, we employ the \emph{Gauss maps} \begin{align*}
\mathrm{Conf}_k(\mathbb{R}^n)&\xrightarrow{\gamma_{ab}} S^{n-1}\\
(x_1,\ldots, x_k)&\mapsto \frac{x_b-x_a}{\|x_b-x_a\|}.
\end{align*}
\begin{lemma}\label{lem:cohomology surjective}
The natural map $H^*(\mathrm{Conf}_k(\mathbb{R}^n))\to H^*(\mathbb{R}^n\setminus\{x_1,\ldots, x_{k-1}\})$ is surjective.
\end{lemma}
\begin{proof}
We first construct a collection of maps $\varphi_a:S^{n-1}\to \mathbb{R}^{n}\setminus\{x_1,\ldots, x_{k-1}\}$ inducing a homotopy equivalence from the bouquet $\bigvee_{k-1}S^{n-1}$. We will then show that each composite \[\xymatrix{S^{n-1}\ar[r]^-{\varphi_a}&\mathbb{R}^n\setminus\{x_1,\ldots, x_{k-1}\}\subseteq \mathrm{Conf}_k(\mathbb{R}^n))\ar[r]^-{\gamma_{ka}}& S^{n-1}
}\] is the identity, implying that $H^*(\mathbb{R}^n\setminus\{x_1,\ldots, x_{k-1}\})$ is generated by classes of the form $\gamma_{ka}^*(\alpha)$ for $\alpha\in H^*(S^{n-1})$.
For the construction, we set $x_a=(3^a,0,\ldots, 0)$ for $1\leq a\leq k-1$ and define $\varphi_a:S^{n-1}\to\mathbb{R}^n\setminus\{x_1,\ldots, x_k\}$ by \[\varphi_a(v)_r=(3^a,0,\ldots, 0)+ 3^av\] where $v\in S^{n-1}$ is regarded as a unit vector in $\mathbb{R}^n$. Then $\varphi_a(-1,\ldots, 0)=(0,\ldots, 0)$ for $1\leq a\leq k-1$, and the induced map from $\bigvee_{k-1}S^{n-1}$ is clearly a homotopy equivalence. Finally, we have \[\gamma_{ka}\circ\varphi_a(v)=\frac{(3^a,0,\ldots, 0)+3^av-(3^a,0,\ldots, 0)}{\|(3^a,0,\ldots, 0)+3^av-(3^a,0,\ldots, 0)\|}=\frac{3^av}{\|3^av\|}=v.\]
\end{proof}
Write $\alpha_{ab}\in H^{n-1}(\mathrm{Conf}_k(\mathbb{R}^n))$ for the pullback along $\gamma_{ab}$ of the standard volume form on $S^{n-1}$. We are now able to give a complete additive description of the cohomology ring in terms of these generators.
\begin{corollary}
For any $k\geq0$, $H^*(\mathrm{Conf}_k(\mathbb{R}^n))$ is free with basis \[S_k=\{\alpha_{a_1b_1}\alpha_{a_2b_2}\cdots \alpha_{a_{m}b_m}: m\geq0,\,1\leq b_1<\cdots<b_m \leq k,\, a_\ell<b_\ell\}.\] In particular, the Poincar\'{e} polynomial is given by \[P(\mathrm{Conf}_k(\mathbb{R}^n))=\prod_{j=1}^{k-1}(1+jt^{n-1}).\]
\end{corollary}
\begin{proof}
Since $\mathbb{R}^n\setminus\{x_1,\ldots, x_{k-1}\}$ and $\mathrm{Conf}_{k-1}(\mathbb{R}^n)$ are path connected and the cohomology of the former is free, Theorem \ref{thm:Fadell--Neuwirth} and the Lemma \ref{lem:cohomology surjective} allow us to apply the Leray--Hirsch theorem. The first and third claim now follow by induction and the observation that $H^*(\mathbb{R}^n\setminus\{x_1,\ldots, x_{k-1}\})$ is free with Poincar\'{e} polynomial $1+(k-1)t^{n-1}$. For the third claim, we note that, by induction, the Leray--Hirsch theorem gives the additive isomorphism \[H^*(\mathrm{Conf}_k(\mathbb{R}^n))\cong \mathbb{Z}\langle S_{k-1}\rangle\otimes \mathbb{Z}\langle1,\, \alpha_{ak},\, 1\leq a\leq k-1\rangle,\] and it is easy to check that the map \begin{align*}
S_{k-1}\times \{1,\, \alpha_{ak},\,1\leq a\leq k-1\}\to S_k
\end{align*} given by concatenation on the right is a bijection.
\end{proof}
\subsection{Multiplicative structure}
In order to obtain a multiplicative description, we require information about the relations among the various $\alpha_{ab}$. We begin with a few easy but useful observations. Recall that $\Sigma_k$ has a right action on $\mathrm{Conf}_k(\mathbb{R}^n)$ given by \[(x_1,\ldots, x_k)\cdot \sigma=(x_{\sigma(1)},\ldots, x_{\sigma(k)}).\] This action induces a left action on cohomology.
\begin{proposition} The following relations hold in $H^*(\mathrm{Conf}_k(\mathbb{R}^n))$ for $1\leq a\neq b\leq k$.
\begin{enumerate}
\item $\alpha_{ab}=(-1)^n\alpha_{ba}$
\item $\alpha_{ab}^2=0$
\item $\sigma^*\alpha_{ab}=\alpha_{\sigma(a)\sigma(b)}$ for $\sigma\in\Sigma_k$.
\end{enumerate}
\end{proposition}
\begin{proof}
The first claim follows from the observation that the diagram \[\xymatrix{
&\mathrm{Conf}_2(\mathbb{R}^n)\ar[dl]_-{\gamma_{ab}}\ar[dr]^-{\gamma_{ba}}\\
S^{n-1}\ar[rr]^-{x\mapsto -x}&&S^{n-1}
}\] commutes, and that the degree of the antipodal map on $S^{n-1}$ is $(-1)^n$. The second follows form the fact that the volume form on $S^{n-1}$ squares to zero. For the third claim, we have the commuting diagram \[\xymatrix{
\mathrm{Conf}_k(\mathbb{R}^n)\ar[dr]_-{\gamma_{\sigma(a)\sigma(b)}}\ar[rr]^-\sigma&&\mathrm{Conf}_k(\mathbb{R}^n)\ar[dl]^-{\gamma_{ab}}\\
&S^{n-1}.
}\]
\end{proof}
We come now to the fundamental relation.
\begin{proposition}[Arnold relation]
For $ 1\leq a<b<c\leq k$, \[\alpha_{ab}\alpha_{bc}+\alpha_{bc}\alpha_{ca}+\alpha_{ca}\alpha_{ab}=0.\]
\end{proposition}
\begin{remark}
The Arnold relation holds without the assumption $a<b<c$. Indeed, the general form follows from the form given here and the antipodal relation $\alpha_{ab}=(-1)^n\alpha_{ba}$.
\end{remark}
We will discuss three proofs of this relation. For the time being, we concentrate on exploiting it. We write $\op A$ for the quotient of the free graded commutative algebra on generators $\{\alpha_{ab}\}_{1\leq a\neq b\leq k}$ by the ideal generated by $\{\alpha_{ab}+(-1)^{n+1}\alpha_{ba},\, \alpha_{ab}^2, \,\alpha_{ab}\alpha_{bc}+\alpha_{bc}\alpha_{ca}+\alpha_{ca}\alpha_{ab}\}$.
\begin{theorem}[Arnold \cite{Arnold:CRCBG}, Cohen \cite{CohenLadaMay:HILS}]
The natural map of graded commutative algebras \[\op A\to H^*(\mathrm{Conf}_k(\mathbb{R}^n))\] is an isomorphism.
\end{theorem}
\begin{proof}
The map is surjective, since its image contains a generating set, so it will suffice to show that the basis $S_k$ exhibited above, thought of as lying in $\op A$, spans. Indeed, it follows that $S_k$ is a basis for $\op A$, since any relation would map to a relation in $H^*(\mathrm{Conf}_k(\mathbb{R}^n))$, and this fact implies the claim.
To verify that $S_k$ spans, it suffices to show that the element $\alpha_{a_1b_1}\cdots\alpha_{a_m b_m}$ may be written as a linear combination of elements in $S_k$. By the antipodal relation and graded commutativity, we may assume that $a_\ell<b_\ell$ for $1\leq \ell\leq m$ and that $1\leq b_1\leq\cdots\leq b_m\leq k$. We proceed by downward induction on the largest value of $\ell$ such that $b_\ell=b_{\ell-1}=:b$. By the Arnold and antipodal relations, we have \begin{align*}
\alpha_{a_1b_1}\cdots\alpha_{a_{\ell-1}b}\alpha_{a_\ell b}\cdots\alpha_{a_m b_m}&=(-1)^{n+1}\alpha_{a_1b_1}\cdots(\alpha_{a_\ell a_{\ell-1}}\alpha_{a_{\ell-1}b}+\alpha_{ba_\ell}\alpha_{a_\ell a_{\ell-1}})\cdots\alpha_{a_mb_m}\\
&=(-1)^{n+1} \alpha_{a_1b_1}\cdots \alpha_{a_\ell a_{\ell-1}}\alpha_{a_{\ell-1}b}\cdots \alpha_{a_mb_m}\\&\qquad+(-1)^{2n+1+(n-1)^2}\alpha_{a_1b_1}\cdots \alpha_{a_\ell a_{\ell-1}}\alpha_{a_\ell b}\cdots \alpha_{a_mb_m}
\end{align*} After a second induction on the number of values of $\ell$ such that $b_\ell=b$, we may assume that $b_{l-2}<b$. Then, using our assumption that $a_\ell<b$ and $a_{\ell-1}<b$, these two monomials lie in the span of $S_k$ by the first induction.
\end{proof}
\subsection{The Arnold relation}
We now describe several approaches to proving the Arnold relation. The first reduction in all three cases is the observation that, using the projection $\mathrm{Conf}_k(\mathbb{R}^n)\to \mathrm{Conf}_3(\mathbb{R}^n)$ sending $(x_1,\ldots, x_k)$ to $(x_a,x_b,x_c)$, it suffices to verify the relation $\alpha_{12}\alpha_{23}+\alpha_{23}\alpha_{31}+\alpha_{31}\alpha_{12}=0$ in $H^*(\mathrm{Conf}_3(\mathbb{R}^n))$.
The first argument, due to Cohen \cite{CohenLadaMay:HILS}, is the most elementary, and we will be able to give a complete account, since it uses only techniques that we have already encountered.
\begin{proof}[Cohen's proof of the Arnold relation]
We have already seen that $H^{2n-2}(\mathrm{Conf}_3(\mathbb{R}^n))$ is free of rank 2 with basis $\{\alpha_{12}\alpha_{23}, \alpha_{31}\alpha_{12}\}$, where we have used the antipodal relation and graded commutativity to rearrange indices in the second case. Note that another basis for this module is $\{\alpha_{12}\alpha_{23},\alpha_{23}\alpha_{31}\}$, since the permutation $\binom{123}{321}$ interchanges this set with our known basis up to sign.
We conclude the existence of a relation \[x\alpha_{12}\alpha_{23}+y\alpha_{23}\alpha_{31}+z\alpha_{31}\alpha_{12}=0.\] Applying $\tau_{12}$ to this relation, we obtain the relation \begin{align*}
0&=x\alpha_{21}\alpha_{13}+y\alpha_{13}\alpha_{32}+z\alpha_{32}\alpha_{21}\\
&=(-1)^{(n-1)^2+2n}(x\alpha_{31}\alpha_{12}+y\alpha_{23}\alpha_{31}+z\alpha_{12}\alpha_{23}).
\end{align*} Canceling the sign and subtracting the result from the known relation yields \[(x-z)\alpha_{12}\alpha_{23}+(z-x)\alpha_{31}\alpha_{12}=0,\] whence $x=z$ by linear independence. Repeating the same process with $\tau_{23}$ shows that \[(x-y)\alpha_{12}\alpha_{23}+(y-x)\alpha_{23}\alpha_{31}=0,\] whence $x=y$ by linear independence. We conclude that the expression in question is $x$-torsion and therefore zero, since $H^*(\mathrm{Conf}_3(\mathbb{R}^n))$ is torsion-free.
\end{proof}
The original proof, due to Arnold \cite{Arnold:CRCBG}, is of a very different flavor, but is only valid in its original form in dimension 2.
\begin{proof}[Arnold's proof of the Arnold relation ($n=2$)]
Since there is no torsion, it suffices to prove the relation holds in cohomology with coefficients in $\mathbb{C}$. Make the identification $\mathbb{R}^2\cong\mathbb{C}$. The class $\alpha_{ab}$ is obtained by pulling back a standard generator of $H^1(S^1)$ along the composite \[\xymatrix{\mathrm{Conf}_k(\mathbb{C})\ar[r]^-{(z_a,z_b)}&\mathrm{Conf}_2(\mathbb{C})\ar[r]^-{z_2-z_1}&\mathbb{C}^\times\ar[r]^-{\frac{z}{\|z\|}}&S^1.
}\] A representative for this generator in $H^1(\mathbb{C}^\times)$ is given by the differential form $dz/z$, since \[\frac{1}{2\pi i}\int_{S^1}\frac{dz}{z}=1\] by the residue theorem. Therefore, we may represent $\alpha_{ab}$ by the differential form \[\omega_{ab}=\frac{dz_b-dz_a}{z_b-z_a}.\] The claim now follows from the easy observation that the differential forms $\omega_{ab}$ satisfy the Arnold relation.
\end{proof}
This line of argument actually yields the far stronger result of a quasi-isomorphism \[H^*(\mathrm{Conf}_k(\mathbb{C}))\xrightarrow{\sim} \Omega^*(\mathrm{Conf}_k(\mathbb{C});\mathbb{C})\] of differential graded algebras, where the cohomology is regarded as a chain complex with zero differential; in jargon, $\mathrm{Conf}_k(\mathbb{C})$ is \emph{formal}.
In higher dimensions, the corresponding differential forms satisfy the relation only up to a coboundary, i.e., we have the equation \[\omega_{12}\omega_{23}+\omega_{23}\omega_{31}+\omega_{31}\omega_{12}=d\beta.\] Roughly, the differential form $\beta$ is obtained by integrating the form $\omega_{14}\omega_{24}\omega_{34}$ along the fibers of the projection $\pi:\mathrm{Conf}_4(\mathbb{R}^n)\to \mathrm{Conf}_3(\mathbb{R}^n)$ onto the first three coordinates. To see why this might be the case, we imagine that a fiberwise version of Stokes' theorem should imply that the boundary of the fiberwise integral should be the fiberwise integral along the ``boundary'' of the fiber, which in turn should be a sum of four terms: the first three terms are the loci where $x_i=x_4$ for $1\leq i\leq 3$, and the fourth lies at infinity, where $x_4$ is very far away. We might imagine that the three terms in the Arnold relation arise from these first three terms and that the term at infinity vanishes.
Of course, the fiber of this projection is non-compact, so, in order to make this kind of reasoning precise, one must replace the configuration space $\mathrm{Conf}_k(\mathbb{R}^n)$ with its \emph{Fulton-MacPherson compactification} $\mathrm{Conf}_k[\mathbb{R}^n]$, which is defined as the closure of the image of $\mathrm{Conf}_k(\mathbb{R}^n)$ under the maps \[\mathrm{Conf}_k(\mathbb{R}^n)\to (\mathbb{R}^n)^k\times(S^{n-1})^{\binom{n}{2}}\times[0,\infty]^{\binom{n}{3}}\] given by the inclusion in the first factor, the Gauss maps $\gamma_{ab}$ for $1\leq a<b\leq k$ in the second, and the relative distance functions $\delta_{abc}(x_1,\ldots, x_k)=\frac{\|x_a-x_b\|}{\|x_a-x_c\|}$ for $1\leq a<b<c\leq k$ in the third---see the original references \cite{FultonMacPherson:CCS, AxelrodSinger:CSPT} or the detailed account \cite{Sinha:MTCCS}. It turns out that $\mathrm{Conf}_k[\mathbb{R}^n]$ is a manifold with corners on which the integration described above can actually be carried out.
Using this compactification, Kontsevich \cite{Kontsevich:OMDQ} was able to carry out an analogue of Arnold's program from above. The basic observation is that the construction of $\beta$ is an example of a more systematic method for generating differential forms from graphs, which, when pursued fully, yields a zig-zag of quasi-isomorphisms \[\xymatrix{H^*(\mathrm{Conf}_k(\mathbb{R}^n))& ?\ar[r]^-\sim\ar[l]_-\sim& \Omega^*(\mathrm{Conf}_k[\mathbb{R}^n]),}\] where the unspecified middle term is a certain ``graph complex.'' Thus, in higher dimensions, too, configuration spaces are formal. See \cite{LambrechtsVolic:FLNDO} for a detailed proof of the formality theorem.
\subsection{Planetary systems}
The third proof of the Arnold relation will proceed through a geometric, intersection-theoretic analysis following \cite{Sinha:HLDO}. In order to pursue this direction, we will need to understand something about the homology of $H_*(\mathrm{Conf}_k(\mathbb{R}^n))$. We begin by introducing a systematic method for generating homology classes.
\begin{definition}
Fix a subset $S\subseteq\{1,\ldots, k\}$.
\begin{enumerate}
\item An $S$-\emph{tree} $T$ is a pair of an ordering of $S$ and a binary parenthesization of $S$ with its ordering.
\item A $k$-\emph{forest} is an ordered partition $\{1,\ldots, k\}\cong \coprod_i S_i$ and an $S_i$-tree for each $i$.
\end{enumerate}
\end{definition}
\begin{example}
With $S=\{1,3,4,7,8\}\subseteq \{1,\ldots, 9\}$, the expression $((48)((17)3))$ is an $S$-tree.
\end{example}
\begin{example}
There is a unique $S$-tree with $S=\{i\}\subseteq\{1,\ldots, k\}$ given by the expression $i$.
\end{example}
The terminology is motivated by the observation that the data of an $S$-tree is equivalent to an isotopy class of planar tree $T$ with the following features:
\begin{enumerate}
\item $T$ has only univalent and trivalent vertices, called \emph{external} and \emph{internal}, respectively;
\item $T$ has a distinguished external \emph{root} vertex, and its other external vertices are \emph{leaves};
\item the leaves of $T$ are labeled by elements of $S$.
\end{enumerate}
The internal vertices in the geometric picture correspond bijectively to the pairs of matching open and closed parentheses in the combinatorial picture. We write $V(T)$ for the set of internal vertices of $T$. Note that an internal vertex lies on the path from the leaf $i$ to the root if and only if its parentheses enclose $i$; in this case, we write $v<i$. We define the \emph{height} $h(v)$ of a vertex $v$ to be the number of edges between $v$ and the root. In the combinatorial picture, the height of an internal vertex corresponds to the depth of the corresponding pair of parentheses.
Fix $S\subseteq \{1,\ldots, k\}$, and let $T$ be an $S$-tree. For each $i\in S$, define a map \begin{align*}
(S^{n-1})^{V(T)}&\xrightarrow{P_{T,i}} \mathbb{R}^n\\
(u_v)_{v\in V(T)}&\mapsto \displaystyle \sum_{v<i}(-1)^{\delta(i,v)}\epsilon^{h(v)}u_v,
\end{align*} where $\epsilon$ is a small positive real number, and $\delta(i,v)$ takes the value $1$ if the path from $i$ to the root passes through the left edge at $v$ and $0$ if the right edge.
\begin{lemma}
For $1\leq i\neq j\leq k$, $P_{T,i}\left((u_v)_{v\in V(T)}\right)\neq P_{T,j}\left((u_v)_{v\in V(T)} \right)$.
\end{lemma}
\begin{proof}
Let $w$ denote the highest internal vertex with $w<i$ and $w<j$, and assume without loss of generality that $\delta(j,w)=0$ and $\delta(i,w)=1$. Then, cancelling terms involving $v<w$, we have
\begin{align*}
P_{T,j}\left((u_v)_{v\in V(T)}\right)-P_{T,i}\left((u_v)_{v\in V(T)}\right)&=\displaystyle \sum_{v<j}(-1)^{\delta(j,v)}\epsilon^{h(v)}u_v-\displaystyle \sum_{v<i}(-1)^{\delta(i,v)}\epsilon^{h(v)}u_v\\
&=\epsilon^{h(w)}\left(2u_w+\epsilon(\cdots)\right).
\end{align*} For $\epsilon$ sufficiently small, this expression does not vanish.
\end{proof}
Thus, taking the coordinates of $\{1,\ldots, k\}\setminus S$ to be fixed at some large, distinct values, we obtain a map $P_T:(S^{n-1})^{V(T)}\to \mathrm{Conf}_k(\mathbb{R}^n)$. We refer to $P_T$ and to the image of the fundamental class under $P_T$ interchangeably as the \emph{planetary system} associated to $T$. A forest $F=\{T_i\}$ also defines a planetary system $P_F$ by taking products of translates of the planetary systems of its component trees.
\begin{definition}
A tree is \emph{tall} if it is of the form $(\cdots(i_1i_2)\cdots i_m)$ with $i_1$ minimal considered as a natural number. A forest is \emph{tall} if
\begin{enumerate}
\item each component tree is tall, and
\item the induced ordering on the minimal leaves of the component trees is the natural ordering.
\end{enumerate}
\end{definition}
\begin{proposition}
Planetary systems of tall trees form a basis for $H_{(k-1)(n-1)}(\mathrm{Conf}_k(\mathbb{R}^n))$.
\end{proposition}
\begin{proof}
From Leray--Hirsch and Fadell--Neuwirth, we know that $H^{(k-1)(n-1)}(\mathrm{Conf}_k(\mathbb{R}^n))$ is free Abelian of rank $(k-1)!$, so the homology group of interest has these properties as well. On the other hand, the set of tall trees on $\{1,\ldots, k\}$ is put into bijection with the set of permutations $\sigma\in \Sigma_{k}$ fixing $1$ by associating to $\sigma$ the tree \[T_\sigma=((\cdots(\sigma^{-1}(1)\sigma^{-1}(2))\cdots \sigma^{-1}(k))).\] Since this set has cardinality $(k-1)!$, we conclude that it suffices to show that the corresponding planetary systems are linearly independent. For this task, we define a map $\gamma_\sigma:\mathrm{Conf}_k(\mathbb{R}^n)\to (S^{n-1})^{k-1}$ by putting $\gamma_{\sigma^{-1}(i)\sigma^{-1}(i+1)}$ in the $i$th coordinate, and we set \[\alpha_\sigma=\gamma_\sigma^*(\mathrm{vol}_{S^{n-1}})=\alpha_{\sigma^{-1}(1)\sigma^{-1}(2)}\cdots\alpha_{\sigma^{-1}(k-1)\sigma^{-1}(k)}.\] Then the proof will be complete upon verifying that \[\left\langle P_{T_\sigma}, \alpha_{\tau}\right\rangle=\delta_{\sigma\tau}.\]
Write $\{v_1,\ldots, v_{k-1}\}$ for the vertices of $T_\sigma$, where $v_i$ is the unique internal vertex with $h(v_i)=k-i$. For $i<j$, we compute that $\gamma_{\sigma^{-1}(i)\sigma^{-1}(j)}(P_{T_\sigma}(u_{v_1},\ldots, u_{v_{k-1}}))$ is the unit vector in the direction of \begin{align*}
\left(\epsilon^{k-j+1}u_{v_{j-1}}-\sum_{\ell=j}^{k-1}\epsilon^{k-\ell}u_{v_\ell}\right)-\left(\epsilon^{k-i+1}u_{v_{i-1}}-\sum_{\ell=i}^{k-1}\epsilon^{k-\ell}u_{v_\ell}\right)
&=\epsilon^{k-j+1}\left(2u_{v_{j-1}}+\epsilon(\cdots)\right).
\end{align*} Letting $\epsilon$ tend to zero defines a homotopy between this map and the map \[(u_{v_1},\ldots, u_{v_{k-1}})\mapsto u_{v_{j-1}}.\] Therefore, $\gamma_\sigma\circ P_{T_\sigma}$ is homotopic to the identity, whence $\langle P_{T_\sigma},\alpha_\sigma\rangle=1$.
Assume now that $\tau\neq \sigma$. Then there is some $1<i<k$ such that $\sigma\tau^{-1}(i)$ is greater than both $\sigma\tau^{-1}(i-1)$ and $\sigma\tau^{-1}(i+1)$, for otherwise, using the fact that $\sigma\tau^{-1}(1)=1$, we conclude that $\sigma\tau^{-1}$ is order-preserving and hence the identity, a contradiction. But then, by the previous calculation, the two maps \begin{align*}\gamma_{\tau^{-1}(i-1)\tau^{-1}(i)}&=\gamma_{\sigma^{-1}(\sigma\tau^{-1}(i-1))\sigma^{-1}(\sigma\tau^{-1}(i))}\\ \gamma_{\tau^{-1}(i)\tau^{-1}(i+1)}&=\gamma_{\sigma^{-1}(\sigma\tau^{-1}(i))\sigma^{-1}(\sigma\tau^{-1}(i+1))}
\end{align*} differ by the antipodal map up to homotopy, so $\gamma_\tau$ factors through a submanifold of $(S^{n-1})^k$ of positive codimension. It follows that $\langle P_{T_\sigma},\alpha_\tau\rangle=0$.
\end{proof}
\begin{corollary}
Planetary systems of tall forests form a basis for $H_*(\mathrm{Conf}_k(\mathbb{R}^n))$.
\end{corollary}
\begin{proof}
Let $F=\{T_{\sigma_i}\}$ be a tall forest, where $T_{\sigma_i}$ has $k_i$ leaves. We apply the same reasoning to the diagram
\[\xymatrix{(S^{n-1})^{V(F)}\ar@{=}[d]\ar[r]^-{P_F}&\mathrm{Conf}_k(\mathbb{R}^n)\ar[r]& \displaystyle\prod_i \mathrm{Conf}_{k_i}(\mathbb{R}^n)\ar[r]^-{(\gamma_{\sigma_i})}&\displaystyle\prod_{i}(S^{n-1})^{V(T_i)}\\
\displaystyle\prod_i(S^{n-1})^{V(T_i)}\ar[urr]_-{(P_{T_{\sigma_i}})},
}\] which commutes up to homotopy.
\end{proof}
\begin{recollection}
One version of Poincar\'{e} duality for oriented, connected, boundaryless, possibly non-compact $n$-manifolds of finite type is the isomorphism \[\widetilde H_i(M^+)\cong H^{n-i}(M),\] where $M^+$ denotes the one-point compactification of $M$ and we reduce with respect to the point at infinity. In particular, such a manifold has a fundamental class $[M]\in \widetilde H^n(M^+)$, defined as the preimage of $1\in H^0(M)$ under this isomorphism. This duality can sometimes be interpreted geometrically.
\begin{enumerate}
\item If $N\subseteq M$ is a proper submanifold of dimension $r$ and $P\subseteq M$ is a compact submanifold of dimension $n-r$, we may contemplate the composite \[\xymatrix{
\widetilde H_r(N^+)\otimes H_{n-r}(P)\ar[r]&\widetilde H_r(M^+)\otimes H_{n-r}(M)\cong H^{r}(M)\otimes H_{n-r}(M)\ar[r]^-{\langle-,-\rangle}&\mathbb{Z}.
}\] (note that the existence of the first map uses the fact that $N$ is properly embedded). If $N$ and $P$ intersect transversely, then the value of this composite on $[N]\otimes [P]$ is the signed intersection number of $N$ and $P$.
\item Since cohomology is a ring, we may likewise contemplate the composite \[\xymatrix{
\widetilde H_r(N_1^+)\otimes \widetilde H_s(N_2^+)\ar[r]& H^{n-r}(M)\otimes H^{n-s}(M)\ar[r]^-\smile& H^{2n-r-s}(M)\cong \widetilde H^{r+s-n}(M^+),
}\] where $N_1$ and $N_2$ are proper submanifolds of dimension $r$ and $s$, respectively. If $N_1$ and $N_2$ intersect transversely, then the value of this composite on $[N_1]\otimes[N_2]$ is $[N_1\cap N_2]$.
\end{enumerate}
\end{recollection}
Now, consider the submanifold of $\mathrm{Conf}_3(\mathbb{R}^n)$ defined by requiring that $x_1$, $x_2$, and $x_3$ be collinear. This manifold has three connected components, and we let $C_a$ denote the component in which $x_a$ lies between $x_b$ and $x_c$. Then the map \begin{align*}
C_a&\to\mathbb{R}^n\times\mathbb{R}_{>0}\times\mathbb{R}_{>0}\times S^{n-1}\\
(x_1, x_2, x_3)&\mapsto \left(x_a,\, |x_b-x_a|,\, |x_c-x_a|, \,\frac{x_c-x_b}{|x_c-x_b|}\right)
\end{align*} is a homeomorphism. In particular, $\dim C_a=2n+1$. Note that $C_a$ is closed as a subspace of $\mathrm{Conf}_3(\mathbb{R}^n)$ and hence proper as a submanifold.
\begin{proof}[Sinha's proof of the Arnold relation]
Pushing forward $[C_1]$ and applying Poincar\'{e} duality as above, we obtain an element of $H^{n-1}(\mathrm{Conf}_3(\mathbb{R}^n))$. By our homology calculation, this class is determined by evaluating it on $P_{(12)}$ and $P_{(13)}$. These values are given by the respective intersection numbers with $C_1$, which are $\pm 1$ with opposite signs. Thus, with the appropriate choice of orientation, $C_1$ is Poincar\'{e} dual to $\alpha_{12}-\alpha_{13}$. Similar remarks apply to $C_2$, and, since $C_1\cap C_2=\varnothing$, we conclude that \begin{align*}
0&=(\alpha_{12}-\alpha_{13})(\alpha_{23}-\alpha_{21})\\
&=\alpha_{12}\alpha_{23}-\alpha_{12}\alpha_{21}-\alpha_{13}\alpha_{23}+\alpha_{13}\alpha_{21}\\
&=\alpha_{12}\alpha_{23}+(-1)^{n(n-1)}\alpha_{23}\alpha_{31}+(-1)^{2n}\alpha_{31}\alpha_{12}\\
&=\alpha_{12}\alpha_{23}+\alpha_{23}\alpha_{31}+\alpha_{31}\alpha_{12}.
\end{align*}
\end{proof}
\subsection{The Jacobi identity and little cubes}
The Arnold relation has its reflection in homology. For trees $T_1$ and $T_2$, we write $[T_1,T_2]$ for the tree obtained by grafting the roots of $T_1$ and $T_2$ to the leaves of $(12)$, in this order.
\begin{proposition}[Jacobi identity]\label{prop:jacobi}
The relation $[[T_1,T_2], T_3]+[[T_2,T_3], T_1]+[[T_3,T_1],T_2]=0$ holds in $H_*(\mathrm{Conf}_k(\mathbb{R}^n))$. More generally, if $R$ is any tree, then the trees resulting from grafting the roots of $[[T_1,T_2], T_3]$, $[[T_2,T_3], T_1]$, and $[[T_3,T_1],T_2]$ to any fixed leaf of $R$ sum to zero.
\end{proposition}
It is possible to give a geometric derivation of the Jacobi identity---see \cite{Sinha:HLDO}---but we will pursue an alternate route. We begin by observing that the most basic case of the identity, in which $T_1$, $T_2$, $T_3$, and $R$ are all trivial trees with no internal vertices, is essentially immediate from what we have already done.
\begin{proof}[Proof of Proposition \ref{prop:jacobi}, trivial case] We calculate that \begin{align*}
\left\langle ((23)1), \alpha_{12}\alpha_{23}\right\rangle&=\left\langle ((23)1), -\alpha_{23}\alpha_{31}-\alpha_{31}\alpha_{12}\right\rangle\\
&=-\left\langle((23)1), \alpha_{23}\alpha_{31}\right\rangle+(-1)^{1+2n+(n-1)^2}\left\langle ((23)1), \alpha_{21}\alpha_{13}\right\rangle\\
&=-\left\langle((13)2), \alpha_{13}\alpha_{32}\right\rangle+(-1)^n\left\langle((13)2), \alpha_{12}\alpha_{23}\right\rangle\\
&=-1,
\end{align*} where we have applied the permutation $\tau_{12}$ in going from the second to the third line, and the last equality follows from the perfect pairing between tall trees and the corresponding cohomology classes. A similar calculation shows that $\left\langle((23)1), \alpha_{31}\alpha_{12}\right\rangle=-1$, and it follows that \[((23)1)=-((31)2)-((12)3),\] as desired.
\end{proof}
The general form of the Jacobi identity follows from this basic case once we are assured that grafting of trees is linear. In order to see why this linearity might hold, we turn to an alternative model for the homotopy types of configuration spaces---for original references, see \cite{BoardmanVogt:HIASTS, May:GILS}.
\begin{definition}
A \emph{little $n$-cube} is an embedding $f:(0,1)^n\to (0,1)^n$ of the form $f(x)=Dx+b$, where $b\in(0,1)^n$ and $D$ is a diagonal matrix with positive eigenvalues.
\end{definition}
We write $\op C_n(k)$ for the space of $k$-tuples of little $n$-cubes with pairwise disjoint images, topologized either as a subspace of $\mathrm{Map}(\amalg_k (0,1)^n, (0,1)^n)$. Note that, since little cubes are closed under composition, we have a collection of maps of the form \[\op C_n(m)\times \op C_n(k_1)\times\cdots\op C_n(k_m)\to \op C_n(k)\] whenever $k_1+\cdots +k_m=k$. These maps furnish the collection $\{\op C_n(k)\}_{k\geq0}$ of spaces with the structure of an \emph{operad} \cite{May:GILS}, but we will not need to make use of the full strength of this notion.
\begin{proposition}
The map $\op C_n(k)\to \mathrm{Conf}_k((0,1)^n)\cong\mathrm{Conf}_k(\mathbb{R}^n)$ given by evaluation at $(1/2,\ldots, 1/2)$ is a homotopy equivalence.
\end{proposition}
\begin{proof}[Sketch proof]
A section of the map in question is defined by sending a configuration $x$ to the unique $k$-tuple of little cubes $(f_1,\ldots, f_k)$ with the following properties:
\begin{enumerate}
\item $f_i(1/2, \ldots, 1/2)=x_i$ for $1\leq i \leq k$;
\item all sides of each $f_i$ have equal length, and all $f_i$ have equal volume;
\item the images of the $f_i$ do not have pairwise disjoint closures.
\end{enumerate}
One checks that this map is continuous, so that we may view the configuration space as a subspace of $\op C_n(k)$. Scaling defines a deformation retraction onto this subspace.
\end{proof}
For further details, see \cite[4.8]{May:GILS}.
\begin{proof}[Proof of Proposition \ref{prop:jacobi}, general case]
By considering planetary systems of little cubes rather than configurations, one obtains the dashed lifts depicted in the diagram \[\xymatrix{
&\op C_n(k)\ar[d]\\
(S^{n-1})^{V(F)}\ar@{-->}[ur]^-{P_F^\Box}\ar[r]^-{P_F}&\mathrm{Conf}_k(\mathbb{R}^n).
}\] With these maps in hand, the combinatorics of grafting trees becomes the combinatorics of composing little cubes; that is, the tree $[[T_1, T_2], T_3]$ is the image of $\left( ((12)3), T_1, T_2, T_3\right)$ under the composition map \[\op C_n(3)\times \op C_n(3)\times \op C_n(k_1)\times \op C_n(k_2)\times\op C_n(k_3)\to \op C_n(k),\] and similar remarks pertain to grafting roots of trees onto a fixed leaf of a tree $R$. Thus, grafting, as the map induced on homology by a map of spaces, is linear, and the general identity follows from the basic case proven above.
\end{proof}
A similar argument as in our earlier cohomology calculation, using the Jacobi identity to rebracket forests into sums of long forests, proves the following.
\begin{theorem}[Cohen]
The graded Abelian group $H_*(\mathrm{Conf}_k(\mathbb{R}^n))$ is isomorphic to the quotient of the free Abelian group with basis the set of $k$-forests by the Jacobi relations and signed antisymmetry.
\end{theorem}
\begin{remark}
This isomorphism may be promoted to an isomorphism of the operad of graded Abelian groups given by the collection $\{H_*(\op C_n(k))\}_{k\geq0}$ with the operad controlling $(n-1)$-shifted Poisson algebras.
\end{remark}
\subsection{The unordered case}
We close this section with a calculation in the unordered case.
\begin{proposition}\label{prop:unordered rational}
For $k\geq2$ and $n\geq1$, there is an isomorphism \[
H_i(B_k(\mathbb{R}^n);\mathbb{Q})\cong\begin{cases}
\mathbb{Q}&\quad\text{if either $i=0$ or $i=n-1$ is odd}\\
0&\quad\text{otherwise.}
\end{cases}
\]
\end{proposition}
\begin{remark}
Note the vast difference in size and complexity between the rational homology of $B_k(\mathbb{R}^n)$ and that of $\mathrm{Conf}_k(\mathbb{R}^n)$. This disparity, which may at first seem surprising, is characteristic of the relationship between ordered and unordered configuration spaces in characteristic zero. In finite characteristic, as we will see, this relationship is reversed, and it is the homology in the unordered case that is by far more complex.
One obvious indicator of the rational difference between ordered and unordered is the fact that the $i$th Betti number of $\mathrm{Conf}_k(\mathbb{R}^n)$ tends to infinity with $k$, while that of $B_k(\mathbb{R}^n)$ quickly stabilizes to a fixed value. This observation is a simple example of the general phenomenon of \emph{homological stability} for configuration spaces of manifolds \cite{Church:HSCSM, RandalWilliams:HSUCS}. Although the Betti numbers in the ordered case do not stabilize, the analogous phenomenon of \emph{representation stability}, which takes the action of $\Sigma_k$ into account, does occur \cite{Farb:RS}.
\end{remark}
In making this calculation, we will use the following basic fact.
\begin{lemma}\label{lem:transfer}
Let $\pi: E\to B$ be a finite regular cover with deck group $G$. If $\mathbb{F}$ is a field in which $|G|$ is invertible, then the natural map \[\bar\pi_*:H_*(E; \mathbb{F})_G\to H_*(B;\mathbb{F})\] is an isomorphism.
\end{lemma}
This result is a consequence of the existence and basic properties of the \emph{transfer map}. Recall that the transfer is a wrong-way map on homology \[\mathrm{tr}:H_*(B)\to H_*(E)\] defined by sending a singular chain to the sum over its $|G|$ lifts to $E$, which is clearly a chain map. It is obvious from the definition that $\pi_*(\mathrm{tr}(\alpha))=|G|\alpha$.
\begin{proof}[Proof of Lemma \ref{lem:transfer}]
We claim that the composite \[\xymatrix{f:H_*(B;\mathbb{F})\ar[r]^-{\frac{1}{|G|}\mathrm{tr}}&H_*(E;\mathbb{F})\ar[r]&H_*(E;\mathbb{F})_{G}}\] is an inverse isomorphism to $\bar\pi_*$. Note that we have used the assumption that $|G|$ is invertible in $\mathbb{F}$ in defining $f$. In one direction, we compute that \[\bar\pi_*(f(\alpha))=\pi_*\left(\frac{1}{|G|}\mathrm{tr}(\alpha)\right)=\frac{1}{|G|}\pi_*(\mathrm{tr}(\alpha))=\alpha,\] and in the other we have \[f(\bar\pi_*([\beta]))=f(\pi_*(\beta))=\frac{1}{|G|}\left[\mathrm{tr}(\pi_*(\beta))\right]=\frac{1}{|G|}\left[\sum_{g\in G}g\cdot\beta\right]=\frac{1}{|G|}\left[\sum_{g\in G}\beta\right]=\beta.\]
\end{proof}
With the identification $H_*(B_k(\mathbb{R}^n);\mathbb{Q})\cong H_*(\mathrm{Conf}_k(\mathbb{R}^n);\mathbb{Q})_{\Sigma_k}$ in hand, we proceed by first identifying the coinvariants in top degree.
\begin{lemma}\label{lem:top homology}
For $k>1$, there is an isomorphism \[H_{(n-1)(k-1)}(\mathrm{Conf}_k(\mathbb{R}^n);\mathbb{Q})_{\Sigma_k}\cong\begin{cases}
\mathbb{Q}&\quad k=2 \text{ and $n$ even}\\
0&\quad\text{otherwise.}
\end{cases}\]
\end{lemma}
\begin{proof}
If $n$ is odd, then any tall tree $T$ is equal to the additive inverse of the tree obtained by switching the labels of the first two leaves of $T$. Since this operation may be achieved by the action of the symmetric group, it follows that $2[T]=0$ at the level of coinvariants, whence $[T]=0$. Since tall trees span the top homology, their images span the coinvariants, and the claim follows in this case.
Assume now that $n$ is even. If $k\geq3$, then the Jacobi identity applied to the bottom three leaves of a tall tree $T$ shows that $3[T]=0$, and so $[T]=0$, and we conclude as before. In the remaining case $k=2$, we note that $H_{n-1}(\mathrm{Conf}_2(\mathbb{R}^n))\cong\mathbb{Z}\langle P_{(12)}\rangle$, and that $\Sigma_2$ acts trivially.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:unordered rational}]
As a consequence of our description in terms of tall forests, we have the following calculation:
\begin{align*}
H_*(\mathrm{Conf}_k(\mathbb{R}^n))&\cong \bigoplus_{\text{partitions of [k]}}\bigotimes_i H_{(n-1)(k_i-1)}(\mathrm{Conf}_{k_i}(\mathbb{R}^n))\\
&\cong \bigoplus_{r\geq0}\left(\bigoplus_{k_1+\cdots+k_r=k}\bigotimes_{i=1}^rH_{(n-1)(k_i-1)}(\mathrm{Conf}_{k_i}(\mathbb{R}^n))\otimes_{\Sigma_{k_1}\times\cdots\times\Sigma_{k_r}}\mathbb{Z}[\Sigma_k]\right)_{\Sigma_r}\\
&\cong\bigoplus_{r\geq0}\left(\bigoplus_{k_1+\cdots+k_r=k}\bigotimes_{i=1}^rH_{(n-1)(k_i-1)}(\mathrm{Conf}_{k_i}(\mathbb{R}^n))\otimes\mathbb{Z}[\Sigma_k]\right)_{\Sigma_r\ltimes \Sigma_{k_1}\times\cdots\times\Sigma_{k_r}}.
\end{align*} Thus, tensoring with $\mathbb{Q}$, forming the $\Sigma_k$-coinvariants, and using that $k!$ is invertible, we find that \begin{align*}
H_*(B_k(\mathbb{R}^n);\mathbb{Q})&\cong \bigoplus_{r\geq0}\left(\bigoplus_{k_1+\cdots+k_r=k}\bigotimes_{i=1}^rH_{(n-1)(k_i-1)}(\mathrm{Conf}_{k_i}(\mathbb{R}^n);\mathbb{Q})\right)_{\Sigma_r\ltimes \Sigma_{k_1}\times\cdots\times\Sigma_{k_r}}\\
&\cong\bigoplus_{r\geq0}\left(\bigoplus_{k_1+\cdots+k_r=k}\bigotimes_{i=1}^rH_{(n-1)(k_i-1)}(\mathrm{Conf}_{k_i}(\mathbb{R}^n);\mathbb{Q})_{\Sigma_{k_i}}\right)_{\Sigma_r}.
\end{align*} The claim now follows easily from Lemma \ref{lem:top homology}, since the only nonvanishing terms up to the action of $\Sigma_r$ are $(k_1,\dots, k_m)=(1,\ldots, 1)$ and possibly $(k_1,\ldots, k_m)=(2,1,\ldots, 1)$.
\end{proof}
With a few more definitions in hand, this calculation may be packaged in a more succinct form.
\begin{definition}
A \emph{symmetric sequence} of graded Abelian groups is a collection $\{V(k)\}_{k\geq0}$ where $V(k)$ is a graded Abelian group equipped with an action of $\Sigma_k$.
\end{definition}
Thus, a symmetric sequence is equivalent to the data of a functor from the category $\Sigma$ of finite sets and bijections to graded Abelian groups. There is a notion of tensor product of symmetric sequences, which is given by the formula \begin{align*}(V\otimes W)(k)&=\bigoplus_{i+j=k}V(i)\otimes W(j)\otimes_{\Sigma_i\times\Sigma_j}\mathbb{Z}[\Sigma_k].
\end{align*} Defining a symmetric sequence by $H_*(\mathrm{Conf}(\mathbb{R}^n))(k)=H_*(\mathrm{Conf}_k(\mathbb{R}^n))$, we now recognize the identification \[H_*(\mathrm{Conf}(\mathbb{R}^n))\cong \mathrm{Sym}(H_\mathrm{top}(\mathrm{Conf}(\mathbb{R}^n)))\] with the symmetric algebra for this tensor product.
Now, a symmetric sequence $V$ determines a bigraded Abelian group $V_\Sigma$ by the formula \[V_\Sigma=\bigoplus_{k\geq0}V(k)_{\Sigma_k},\] and it is immediate from the formula that \[(V\otimes W)_\Sigma\cong V_\Sigma\otimes W_\Sigma.\] Thus, we have an isomorphism of bigraded vector spaces \begin{align*}
\textstyle\bigoplus_{k \geq0}H_*(B_k(\mathbb{R}^n);\mathbb{Q})&\cong H_*(\mathrm{Conf}(\mathbb{R}^n))_\Sigma\\
&\cong \mathrm{Sym}(H_\mathrm{top}(\mathrm{Conf}(\mathbb{R}^n)))_\Sigma\\
&\cong \mathrm{Sym}(H_\mathrm{top}(\mathrm{Conf}(\mathbb{R}^n))_\Sigma)\\
&\cong \mathrm{Sym}(\mathbb{Q}[0,1]\oplus \mathbb{Q}[n-1, 2]).
\end{align*}
\begin{remark}
From the operadic point of view, this bigraded Abelian group is the free shifted Poisson algebra on one generator.
\end{remark}
This calculation illustrates a valuable lesson, namely that configuration spaces tend to exhibit more structure when taken all together. This insight will be indispensable to us in our future investigations. Before pursuing this direction, however, we will need to invest in some new tools.
\section{Covering theorems}\label{section:covering theorems}
Having exploited it at length, our next long-term goal is to circle back and prove our version of the Fadell--Neuwirth theorem asserting that the diagram \[\xymatrix{
\mathrm{Conf}_{\ell-k}(M\setminus\{x_1,\ldots, x_k\})\ar[r]\ar[d]&\mathrm{Conf}_\ell(M)\ar[d]\\
(x_1,\ldots,x_k)\ar[r]&\mathrm{Conf}_k(M)
}\] is homotopy Cartesian. This type of statement is about the local homotopy type of the configuration space, while, through the topological basis that we exhibited in Proposition \ref{prop:conf basis}, we have fine control over the local topology. Of course, in some sense, everything about a space $X$ is determined by a basis, since $X$ can be reconstructed from the basis by gluing. On the other hand, as the following classic example illustrates, gluing is not a homotopically well-behaved operation.
\begin{example}
The two diagrams
\[\xymatrix{
S^{n-1}\times(0,1)\ar[d]\ar[r]&\mathring{D}^n\ar[d]&&S^{n-1}\ar[r]\ar[d]&\mathrm{pt}\ar[d]\\
\mathring{D}^n\ar[r]&S^n&&\mathrm{pt}\ar[r]&\mathrm{pt}
}\] are pushout squares, and there is a map from the left square to the right that is a homotopy equivalence on all but the bottom right corner.
\end{example}
This example illustrates that ordinary gluing, which is a colimit construction, is insufficient for the kind of homotopy theoretic questions we wish to pursue. The replacement will be the \emph{homotopy colimit}, which we review at length in Appendix \ref{section:homotopy colimits}.
The kind of result that we aim to prove is the following.
\begin{theorem}\label{thm:basis recovery}
Let $\mathcal{B}$ be a topological basis for $X$, regarded as a poset under inclusion and thereby as a category. The natural map \[\hocolim_{U\in \mathcal{B}}U\to X\] is a weak equivalence.
\end{theorem}
This result will be an immediate consequence of Theorem \ref{thm:complete cover recovery}. First, we explore the intermediary concepts of \v{C}ech covers and hypercovers, which are interesting in their own right. These tools all permit the reconstruction of the weak homotopy type of $X$ from various forms covering data. A general reference for this material is \cite{DuggerIsaksen:THAR}.
\subsection{\v{C}ech covers}
\begin{recollection}
We write $\Delta_+$ for the category of finite ordered sets and $\Delta\subseteq \Delta_+$ for the full subcategory of sets that are nonempty. Thus, up to isomorphism, the objects of $\Delta_+$ are the sets $[n]=\{0,\ldots, n\}$ for $n\geq-1$. A \emph{simplicial space} is a functor $\op X:\Delta^{op}\to \mathcal{T}\mathrm{op}$ (resp. \emph{augmented simplicial space}, $\Delta_+^{op}$).
Using the traditional notation $\op X_n:=\op X([n])$, we write $d_i:\op X_n\to \op X_{n-1}$ for map induced by the inclusion $[n-1]\to [n]$ that misses the element $i$ (the $i$th \emph{face map}), and $s_i:\op X_{n}\to \op X_{n+1}$ for the map induced by the surjection $[n+1]\to [n]$ that sends $i$ and $i+1$ to $i$ (the $i$th \emph{degeneracy}). If $\op X$ is augmented, we refer to the induced map $\op X_0\to \op X_{-1}$ as the \emph{augmentation}.
The \emph{geometric realization} of the simplicial space $\op X$ is the quotient \[|\op X|:=\faktor{\coprod_{n\geq0}\op X_n\times\Delta^n}{\sim}=\mathrm{coeq}\left(\coprod_{\Delta([\ell],[m])}\op X_m\times\Delta^\ell\rightrightarrows \coprod_{n\geq0} \op X_n\times\Delta^n\right),\] where the two arrows are given by the covariant functoriality of $\Delta^{(-)}$ and the contravariant functoriality of $\op X$, respectively. If $\op X$ is augmented, there results a canonical map $|\op X|\to \op X_{-1}$.
\end{recollection}
\begin{definition}
Let $\U=\{U_\alpha\}_{\alpha\in A}$ be an open cover of $X$. The \emph{\v{C}ech nerve} of $\U$ is the augmented simplicial space $\check{C}(\U):\Delta^{op}_+\to \mathcal{T}\mathrm{op}$ specified as follows.
\begin{enumerate}
\item In nonnegative simplicial degree, we have \[\check{C}(\U)_n=\coprod_{A^{n+1}}U_{\alpha_0}\cap\cdots\cap U_{\alpha_n},\] and $\check{C}(\U)_{-1}=X$.
\item The face maps $d_i$ is induced by the inclusions \[U_{\alpha_0}\cap\cdots\cap U_{\alpha_n}\subseteq U_{\alpha_0}\cap\cdots \widehat{U}_{\alpha_i}\cdots \cap U_{\alpha_n}.\]
\item The degeneracy $s_i$ is induced by the identifications \[U_{\alpha_0}\cap\cdots\cap U_{\alpha_n}=U_{\alpha_0}\cap \cdots\cap U_{\alpha_i}\cap U_{\alpha_i}\cap \cdots\cap U_{\alpha_n}\]
\item The augmentation is induced by the inclusions $U_\alpha\subseteq X$.
\end{enumerate}
\end{definition}
Applying $H_0$ to $\check{C}(\U)$ in each simplicial degree and taking the alternating sum of the face maps as a differential, we obtain the classical \emph{\v{C}ech complex} \[\cdots \to \bigoplus_{A^{n+1}}H_0(U_{\alpha_0}\cap \cdots \cap U_{\alpha_n})\to \cdots\to \bigoplus_AH_0(U_\alpha),\] which computes the homology of $X$ if $\U$ is a sufficiently good cover. In fact, this result can be strengthened to a recovery of the full weak homotopy type.
\begin{theorem}[Segal, Dugger-Isaksen]\label{thm:cech recovery}
For any topological space $X$ and any open cover $\U$ of $X$, the augmentation \[|\check{C}(\U)|\to X\] is a weak homotopy equivalence.
\end{theorem}
The proof will make use of a wonderful local-to-global principle. In establishing this principle, we will employ a little machinery, but see \cite[16.24]{Gray:HT} for a more elementary argument premised on subdivision.
\begin{proposition}\label{prop:mayer-vietoris}
Let $f:Y\to Z$ be a continuous map and $\U=\{U_\alpha\}_{\alpha\in A}$ an open cover of $Z$. If the induced map $f^{-1}(U_{\alpha_0}\cap\cdots\cap U_{\alpha_n})\to U_{\alpha_0}\cap\cdots\cap U_{\alpha_n}$ is a weak homotopy equivalence for every $(\alpha_0,\ldots, \alpha_n)\in A^{n+1}$, then $f$ is also a weak homotopy equivalence.
\end{proposition}
\begin{proof}
Suppose first that $\U=\{U,V\}$. Using the appropriate versions of the Van Kampen \cite{BrownRazekSalleh:VKTUNS} and Mayer-Vietoris \cite[5.13]{DavisKirk:LNAT} theorems, we conclude that $f$ induces isomorphisms on fundamental groupoids and on homology with arbitrary local coefficients, so the claim follows in this case from the Whitehead theorem \cite[6.71]{DavisKirk:LNAT}.
In the general case, we choose an ordinal $\lambda$ and a bijection $\varphi:\lambda\cong \U$ and proceed by transfinite induction, assuming that the claim is known for all covers with the cardinality of $\mu<\lambda$.
Suppose first that $\lambda$ is a successor ordinal. Setting $U=U_\lambda$ and $V=\bigcup_{\mu<\lambda}U_\mu$, it will suffice by the previous argument to verify that the restrictions of $f$ to $U$, to $V$, and to $U\cap V$ are all weak homotopy equivalences. The case of $U$ is a special case of our hypothesis, and the cases of $V$ and $U\cap V$ follow from the inductive assumption applied to the covers $\{U_\mu\}_{\mu<\lambda}$ and $\{U_\mu\cap U_\lambda\}_{\mu<\lambda}$, respectively, each of which is in bijection with $\lambda-1$ and satisfies the hypothesis of the proposition.
Suppose now that $\lambda$ is a limit ordinal. By compactness, any map $(D^{n+1}, S^n)\to (Z,Y)$ factors as in the solid commuting diagram
\[\xymatrix{
S^n\ar[d]\ar[d]\ar[r]&\ar[d]\displaystyle f^{-1}\left(\bigcup_{\mu<\mu_0}U_\mu\right)\ar[r]&Y\ar[d]^-f\\
D^{n+1}\ar[r]\ar@{-->}[ur]&\displaystyle\bigcup_{\mu\leq \mu_0}U_\mu\ar[r]& Z
}\] for some $\mu_0<\lambda$. The inductive hypothesis applied to the cover $\{U_\mu\}_{\mu\leq \mu_0}$ implies that the middle map is a weak homotopy equivalence, so the dashed lift exists making the upper triangle commute and the lower triangle commute up to homotopy. It follows that $\pi_n(f)=0$ for every $n\geq0$, and consideration of the long exact sequence in homotopy associated to $f$ completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:cech recovery}] For $(\alpha_0,\ldots, \alpha_n)\in A^{n+1}$, set $\U_{\alpha_0,\ldots, \alpha_n}=\{U_\alpha\cap U_{\alpha_0}\cap\cdots\cap U_{\alpha_n}\}_{\alpha\in A}$, and note that the diagram
\[\xymatrix{
|\check{C}(\U_{\alpha_0,\ldots, \alpha_n})|\ar[r]\ar[d]&|\check{C}(\U)|\ar[d]\\
U_{\alpha_0}\cap\cdots\cap U_{\alpha_n}\ar[r]& X
}\] is a pullback. Therefore, by Proposition \ref{prop:mayer-vietoris}, it suffices to prove the claim under the assumption that $X$ is a member of $\U$. In this case, $\check{C}(\U)$ has an extra degeneracy given by forming the intersection with $X$.
\end{proof}
\subsection{Hypercovers}
In light of Theorem \ref{thm:cech recovery}, it is natural to wonder what makes \v{C}ech nerves of covers special. The first step toward answering this question is to notice that \v{C}ech nerves may be completely characterized.
\begin{definition}
Let $f:Y\to Z$ be a continuous map. We say that $f$ is a \emph{covering map} if, up to homeomorphism, it is of the form $\coprod_{\alpha\in A} U_\alpha\to Z$ for some open cover $\U=\{U_\alpha\}_{\alpha\in A}$ of $Z$.
\end{definition}
\begin{definition}
We say that an augmented simplicial space $\op X$ is a \emph{\v{C}ech cover} if \begin{enumerate}
\item the augmentation $\op X_0\to \op X_{-1}$ is a covering map, and
\item for every $n>0$, the natural map \[\op X_n\to M_n(\op X):=\mathrm{eq}\left(\prod_{0\leq i\leq n}\op X_{n-1}\rightrightarrows\prod_{0\leq i<j\leq n}\op X_{n-2}\right)\] induced by the face maps is a homeomorphism.
\end{enumerate}
\end{definition}
The space $M_n(\op X)$ is called the $n$th \emph{matching space} of $\op X$.
\begin{example}
In low degrees, we have $M_0(\op X)=\op X_{-1}$ and $M_1(\op X)=\op X_0\times_{\op X_{-1}}\op X_0$.
\end{example}
\begin{exercise}
The \v{C}ech nerve of an open cover is a \v{C}ech cover, and, conversely, a \v{C}ech cover $\op X$ is the \v{C}ech nerve of some open cover $\U$---for example, we may take $\U$ to be the collection of connected components of $\op X_0$---but different open covers may have isomorphic \v{C}ech nerves.
\end{exercise}
Thus, a \v{C}ech cover is exactly what we obtain by deforming the constant simplicial object at $X$ by replacing the 0-simplices by a cover and filling in the rest of the object in the canonical way with matching objects. This observation identifies \v{C}ech covers as the first stage in an obvious hierarchy of covering notions.
\begin{definition}
We say that an augmented simplicial space $\op X:\Delta^{op}_+\to \mathcal{T}\mathrm{op}$ is a \emph{hypercover} if the canonical map $\op X_n\to M_n(\op X)$ is a covering map for all $n\geq0$. We say that the hypercover $\op X$ is \emph{bounded} if there is some $N$ such that this map is an isomorphism for $n>N$, the smallest such $N$ being the \emph{height} of $\op X$. If $\op X_{-1}=X$, then we say that $\op X$ is a hypercover of $X$.
\end{definition}
A hypercover of height 0 is precisely a \v{C}ech cover, while a hypercover of height $-1$ is isomorphic to the constant simplicial space with value $X$.
\begin{theorem}[Dugger-Isaksen]\label{thm:hypercover recovery}
For any topological space $X$, and any hypercover $\op X$ of $X$, the augmentation map \[|\op X|\to X\] is a weak homotopy equivalence.
\end{theorem}
The proof will make use of some formal machinery from the theory of simplicial spaces, which identifies the matching object $M_n(\op X)$ itself as the degree $n$ entry of a simplicial space. First, a few categorical reminders.
\begin{recollection}
Let $\op C$ be a category and $\D$ a category with small limits and colimits, and let $\iota:\op C_0\to \op C$ be a functor. Then the restriction functor $\iota^*:\op D^{\op C}\to \op D^{\op C_0}$ admits both a left and a right adjoint, the so-called \emph{Kan extension} functors, as depicted in the following diagram \[\doubleadjunct{\op D^{\op C_0}}{\op D^{\op C}}{\op D^{\op C_0}}{\iota_!}{\iota^*}{\iota^*}{\iota_*}.\] We may also use the notation $\mathrm{Lan}_\iota=\iota_!$ and $\mathrm{Ran}_\iota=\iota_*$ for the left and right Kan extensions, respectively. The value of the left Kan extension is given by the formula \[\iota_!F(C)\cong \colim\left((\op C_0\downarrow C)\to \op C\xrightarrow{F} \op D\right),\] where $(\op C_0\downarrow C)$ denotes the \emph{overcategory} whose objects are morphisms $f:C'\to C$ with $C'$ an object of $\op C_0$, and whose morphisms are commuting triangles \[\xymatrix{
C'\ar[rr]\ar[dr]&& C''\ar[dl]\\
&C.
}\] Dually, the right Kan extension $\iota_*F$ is computed as the corresponding limit over the undercategory $(C\downarrow\op C_0)$.
\end{recollection}
Taking $i:(\Delta^{op}_+)_{\leq n}\to \Delta^{op}_+$ to be the inclusion of the full subcategory of finite ordered sets of cardinality at most $n+1$, we obtain adjunctions \[\doubleadjunct{\mathcal{T}\mathrm{op}^{(\Delta^{op}_+)_{\leq n}}}{\mathcal{T}\mathrm{op}^{\Delta^{op}_+}}{\mathcal{T}\mathrm{op}^{(\Delta^{op}_+)_{\leq n}}}{\iota_!}{\iota^*}{\iota^*}{\iota_*}.\] We write $\tau_{\leq n}(\op X)=\iota^*\op X$, $\mathrm{sk}_n(\op X)=\iota_!\iota^*\op X$, and $\mathrm{cosk}_n(\op X)=\iota_*\iota^*\op X$ and refer to the $n$-\emph{truncation}, $n$-\emph{skeleton}, and $n$-\emph{coskeleton}, respectively, of $\op X$.
An exercise in the manipulation of limits shows that matching objects and coskeleta are related as \[M_n(\op X)\cong \mathrm{cosk}_{n-1}(\op X)_n,\] and the map $\op X_n\to M_n(\op X)$ discussed above is a component of the unit transformation of the appropriate adjunction. Note that $\mathrm{cosk}_{n-1}(\op X)_k\cong \op X_k$ for $k<n$, so the matching object is in some sense the primary measure of the difference between $\op X$ and its $n-1$-coskeleton.
We will need a few facts about the behavior of coskeleta of hypercovers. First, we note that, by elementary properties of Kan extensions, we have \[
\mathrm{cosk}_m(\mathrm{cosk}_n(\op X))\cong\begin{cases}
\mathrm{cosk}_n(\op X)&\quad m\geq n\\
\mathrm{cosk}_m(\op X)&\quad m\leq n
\end{cases}\] for any augmented simplicial space $\op X$. Second, we have the following pullback diagram for any $n$ and $m$
\[\xymatrix{
\mathrm{cosk}_{m}(\op X)_n\ar[d]\ar[r]&\displaystyle\prod_{[m]\subseteq [n]} \op X_m\ar[d]\\
\mathrm{cosk}_{m-1}(\op X)_n\ar[r]&\displaystyle\prod_{[m]\subseteq [n]} M_m(\op X),
}\] where the products are indexed on the set of injective order-preserving maps. This pullback square, which is dual to the inductive formation of skeleta by cell attachments, may be derived as an exercise in the manipulation of limits.
\begin{remark}\label{remark:mnemonic}
A mnemonic for this pullback diagram, which is rigorous at the level of simplicial sets and can be made rigorous for simplicial spaces with the correct interpretation of the symbols, is the following. By adjunction, we should think that \[\mathrm{cosk}_m(\op X)_n\cong\mathrm{Hom}(\Delta^n, \mathrm{cosk}_m(\op X))\cong \mathrm{Hom}(\tau_{\leq m}(\Delta^n), \tau_{\leq m}(\op X))\cong \mathrm{Hom}(\mathrm{sk}_m(\Delta^k), \op X),\] and there is a pushout diagram \[\xymatrix{
\displaystyle\coprod_{[m]\subseteq [n]}\partial \Delta^n\ar[r]\ar[d]&\mathrm{sk}_{m-1}(\Delta^n)\ar[d]\\
\displaystyle\coprod_{[m]\subseteq [n]}\Delta^n\ar[r]&\mathrm{sk}_m(\Delta^n).
}\]
\end{remark}
\begin{lemma}\label{lem:coskeleta}
Let $\op X$ be a hypercover.
\begin{enumerate}
\item The map $\op X\to \mathrm{cosk}_N(\op X)$ is an isomorphism if and only if $\op X$ is of height at most $N$.
\item For every $N\geq0$, $\mathrm{cosk}_{N-1}(\op X)$ is a hypercover, which is of height at most $N-1$ if $\op X$ is of height at most $N$.
\item If $\op X$ is of height at most $N$, then the map $\op X\to \mathrm{cosk}_{N-1}(\op X)$ is a degreewise covering map.
\end{enumerate}
\end{lemma}
\begin{proof}
For the first claim, assume first that $\op X$ is of height at most $N$. Since $\op X_n\cong\mathrm{cosk}_N(\op X)_n$ whenever $n\leq N$, it suffices to demonstrate the isomorphism for $n>N$. Since $\op X$ is of height at most $N$, the righthand map in the pullback diagram above is an isomorphism whenever $m>N$, so the lefthand map is an isomorphism in these cases as well. Thus, \[\op X_n\cong\mathrm{cosk}_n(\op X)_n\cong \mathrm{cosk}_{n-1}(\op X)_n\cong\cdots\cong \mathrm{cosk}_N(\op X)_n,\] as desired. On the other hand, suppose that $\op X\cong\mathrm{cosk}_N(\op X)$. Then for $m\geq N$, we have \[\mathrm{cosk}_m(\op X)\cong\mathrm{cosk}_m(\mathrm{cosk}_N(\op X))\cong \mathrm{cosk}_N(\op X)\cong \op X,\] which implies the claim after setting $m=n-1$ and evaluating at $n$.
For the second claim, it suffices by point (1) and the isomorphism $\mathrm{cosk}_{N-1}(\mathrm{cosk}_{N-1}(\op X))\cong\mathrm{cosk}_{N-1}(\op X)$ to show that $\mathrm{cosk}_{N-1}(\op X)$ is a hypercover. For $n\geq N$, we have \[M_n(\mathrm{cosk}_{N-1}(\op X))=\mathrm{cosk}_{n-1}(\mathrm{cosk}_{N-1}(\op X))_n\cong\mathrm{cosk}_{N-1}(\op X)_n,\] as desired, while, for $n\leq N$, we have the commuting diagram \[\xymatrix{
\mathrm{cosk}_{N-1}(\op X)_n\ar[r]&M_n(\mathrm{cosk}_{N-1}(\op X))=\mathrm{cosk}_{n-1}(\mathrm{cosk}_{N-1}(\op X))_n\\
\op X_n\ar[r]\ar@{=}[u]_-\wr&M_n(\op X)=\mathrm{cosk}_{n-1}(\op X)_n.\ar@{=}[u]^-\wr
}\] Since $\op X$ is a hypercover, the bottom map, and hence the top map, is a covering map, as desired.
The third claim is immediate for simplicial degree $n<N-1$. For $n=N-1$, we appeal to the pullback square above with $m=N$. Since $\op X$ is a hypercover, each of the maps $\op X_m\to M_m(\op X)$ is a covering map. Since covering maps are preserved under finite products and pullback, it follows that the lefthand map is a covering map, as desired.
\end{proof}
In the proof of Theorem \ref{thm:hypercover recovery}, we will thrice use that a degreewise weak homotopy equivalence between simplicial spaces induces a weak homotopy equivalence after geometric realization. This implication does not hold in general, but it does hold for a certain class of \emph{split} simplicial spaces, as shown in Appendix \ref{appendix:split simplicial spaces}, which includes the examples we need.
\begin{proof}[Proof of Theorem \ref{thm:hypercover recovery}]
We proceed by induction on the height $N$ of $\op X$, the base case $N=0$ being the case of a \v{C}ech cover. By Lemma \ref{lem:coskeleta}, the natural map $\op X\to \mathrm{cosk}_{N-1}(\op X)=:\op Y$ is a covering map in each degree, so we may extend this map to the horizontally augmented bisimplicial space \[\op W:=\left(\cdots\rightrightarrows\op X\times_{\op Y}\op X\rightrightarrows\op X\to \op Y\right)\] in which the $n$th row is the \v{C}ech nerve of the covering map $\op X_n\to \op Y_n$. By Theorem \ref{thm:cech recovery}, the map \[|\op W|_h\xrightarrow{\sim}\op Y\] is a degreewise weak homotopy equivalence, where $|-|_h$ denotes the geometric realization in the horizontal direction, and we conclude that \[|d^*\op W|\cong \big||\op W|_h\big|\xrightarrow{\sim}|\op Y|\xrightarrow{\sim} X,\] where $d:\Delta^{op}_+\to \Delta^{op}_+\times\Delta^{op}_+$ is the diagonal functor. Here the isomorphism follows from the general fact that the diagonal coincides with either iterated geometric realization \cite[p. 86]{Quillen:HAKTI}, the rightmost weak equivalence follows from the inductive hypothesis and the fact that $\op Y$ is a hypercover of height at most $N-1$, and the middle weak equivalence follows from Proposition \ref{prop:split criterion} in light of the fact that both $|\op W|_h$ and $\op Y$ are split by Corollaries \ref{cor:hypercovers are split} and \ref{cor:horizontal cech cover split}. Thus, in order to conclude the result for bounded $\op X$, it suffices to show that the natural map of simplicial spaces \[i:\op X\to d^*\op W=\left(\cdots\rightrightarrows \op X_1\times_{\op Y_1} \op X_1\rightrightarrows \op X_0\right)\] given by the diagonal in each degree, admits a retraction over the constant simplicial space with value $X$. Indeed, assuming this fact, it follows that the map $|\op X|\to X$ is a retract of a weak homotopy equivalence, and the claim follows.
To define the putative retraction $r:d^*\op W\to \op X\cong\mathrm{cosk}_{N}(\op X),$ it suffices by adjunction to exhibit a map $\bar r:\tau_{\leq N}(d^*\op W)\to \tau_{\leq N}(\op X)$, for which we take the isomorphism \[\bar r_n:\op X_n\times_{\op Y_n}\cdots\times_{\op Y_n}\op X_n=\op X_n\times_{\op X_n}\cdots\times_{\op X_n}\op X_n\cong\op X_n\] in degrees $n<N$. In degree $N$, we take the map \[\bar r_N:\op X_N\times_{\op Y_N}\cdots \times_{\op Y_N}\op X_N\to \op X_N\] to be any of the projections. One checks that, with these choices, $\bar r$ is a simplicial map, and that $\tau_{\leq N}(r\circ i)=\id_{\tau_{\leq N}(\op X)}$, implying the claim.
To conclude the claim for $\op X$ not necessarily bounded, it will suffice to show that the natural map \[\pi_n(|\op X|)\to \pi_n(|\mathrm{cosk}_{n+1}(\op X)|)\] is an isomorphism for every $n\geq0$, since $\mathrm{cosk}_{n+1}(\op X)$ is a bounded hypercover, so $\pi_n(|\mathrm{cosk}_{n+1}(\op X)|\cong\pi_n(X)$ by the argument above. For this claim, we note that \[|\op X| \xleftarrow{\sim}\big||\mathrm{Sing}(\op X)|_h\big|\cong |d^*\mathrm{Sing}(\op X)|,\] and similarly for $\mathrm{cosk}_{n+1}(\op X)$---these are the second and third cases in which we use that a degreewise weak equivalence induces a weak equivalence on realizations, for which we again invoke Proposition \ref{prop:split criterion} in light of Corollary \ref{cor:hypercovers are split} and Lemma \ref{lem:realization split}. From this equivalence, we conclude that \begin{align*}
\pi_n(|X|)&\cong \pi_n(|d^*\mathrm{Sing}(\op X)|)\\
&\cong\pi_n(|\mathrm{sk}_{n+1}(d^*\mathrm{Sing}(\op X))|)\\
&\cong \pi_n(|\mathrm{sk}_{n+1}(d^*\mathrm{Sing}(\mathrm{cosk}_{n+1}(\op X)))|)\\
&\cong \pi_n(|d^*\mathrm{Sing}(\mathrm{cosk}_{n+1}(\op X))|)\\
&\cong \pi_n(|\mathrm{cosk}_{n+1}(\op X)|),
\end{align*} where the second and fourth isomorphism follow as usual from the fact that $\pi_n(S^{m})=0$ for $n<m$, and the third follows from the fact that $\op X\to \mathrm{cosk}_{n+1}(\op X)$ is an isomorphism through simplicial degree $n+1$
\end{proof}
\subsection{Complete covers} We arrive now at the desired result.
\begin{definition}
We say that an open cover $\U$ of $X$ is \emph{complete} if $\U$ contains an open cover of $\bigcap_{U\in \U_0}U$ for every finite subset $\U_0\subseteq \U$.
\end{definition}
We regard $\U$ as a partially ordered set and thereby as a category.
\begin{theorem}[Dugger-Isaksen]\label{thm:complete cover recovery}
If $\U$ is a complete cover of $X$, then the natural map \[\hocolim_{U\in \U}U\to X\] is a weak equivalence.
\end{theorem}
\begin{remark}
We adopt the standard abuse of referring to the homotopy colimit as a space rather than an object of the homotopy category.
\end{remark}
Any open cover containing a basis for the topology of $X$ is complete, so Theorem \ref{thm:basis recovery} is a special case of this result.
The strategy of the proof is to reduce the result to Theorem \ref{thm:hypercover recovery} by relating the bar construction modeling the homotopy colimit in question to a certain hypercover.
\begin{construction}
We write $P_n$ for the set of nonempty subsets of $[n]$, regarded as partially ordered under inclusion and thereby as a category. Given a functor $F:I\to \mathcal{T}\mathrm{op}$, we write \[\mathrm{Bar}_n^\#(F)=\coprod_{f:P_n^{op}\to I}F(f([n]))\] and extend this to a simplicial space $\mathrm{Bar}_\bullet^\#(F)$ by letting the structure map associated to $h:[m]\to [n]$ be given by the restriction along the induced map $P_m^{op}\to P_n^{op}$. There is a canonical map $\mathrm{Bar}_\bullet(F)\to\mathrm{Bar}_\bullet^\#(F)$ of simplicial spaces induced by restriction along (the opposites of) the functors $P_n\to [n]$ sending a subset to its maximal element. Both simplicial spaces are naturally augmented over $\colim_IF$, and this map is compatible with the augmentations.
\end{construction}
Following Theorem \ref{thm:hypercover recovery}, the theorem will be an immediate consequence of the following two results.
\begin{proposition}\label{prop:subdivided comparison}
For any functor $F:I\to \mathcal{T}\mathrm{op}$, the induced map $|\mathrm{Bar}_\bullet(F)|\to |\mathrm{Bar}_\bullet^\#(F)|$ is a weak homotopy equivalence.
\end{proposition}
We will take up the proof of this proposition momentarily.
\begin{lemma}
Let $\U$ be an open cover and $F:\U\to \mathcal{T}\mathrm{op}$ the tautological functor. If $\U$ is complete, then the augmented simplicial space $\mathrm{Bar}_\bullet^\#(F)$ is a hypercover.
\end{lemma}
\begin{proof}
Writing $\overline P_n\subseteq P_n$ for the subcategory of proper nonempty subsets, we compute that \begin{align*}
M_n(\mathrm{Bar}_\bullet^\#(F))&=\mathrm{eq}\left(\prod_{S\subsetneq[n]}\coprod_{f:P_S^{op}\to \U}F(f(S))\rightrightarrows \prod_{S_1\subseteq S_2\subsetneq[n]}\coprod_{f:P_{S_1}^{op}\to \U}F(f(S_1))\right)\\
&\cong \mathrm{eq}\left(\coprod_{f:\overline{P}_n^{op}\to \U}\left(\prod_{S\in \overline P_n}F(f(S))\rightrightarrows\prod_{S_1\subseteq S_1\in \overline P_n}F(f(S)\right)\right)\\
&\cong \coprod_{f:\overline{P}_n^{op}\to \U}\bigcap_{S\in \overline P_n}F(f(S)),
\end{align*} and the canonical map to the matching object is the map \[\coprod_{f:P_n^{op}\to \U}F(f([n]))\to \coprod_{f:\overline{P}_n^{op}\to \U}\bigcap_{S\in \overline P_n}F(f(S))\] with components given by the inclusions $F(f([n]))\subseteq F(f(S))$ in $\U$ induced by the inclusions $S\subseteq [n]$ in $P_n$. Fixing $f:\overline P_n^{op}\to \U$, the inverse image under this map of the subspace $\bigcap_{S\in \overline P_n}F(f(S))$ is the disjoint union over all extensions of $f$ to $P_n^{op}$ of the value on $[n]$, which is to say the disjoint union of the elements of $\U$ contained in the intersection. Since the cover is complete, this collection of opens forms an open cover of the intersection and the claim follows.
\end{proof}
In order to prove Proposition \ref{prop:subdivided comparison}, we reinterpret the bar construction in a way that will generalize in parallel to the variant $\mathrm{Bar}_\bullet^\#(F)$.
\begin{definition}
Let $I$ be a category. The \emph{category of simplices} of $I$ is the category $\Delta^{op}I$ specified as follows.
\begin{enumerate}
\item An object of $\Delta^{op}I$ is a functor $f:[n]^{op}\to I$.
\item A morphism from $f:[n]^{op}\to I$ to $g:[m]^{op}\to I$ is a morphism $h:[m]\to [n]$ in $\Delta$ making the diagram \[\xymatrix{
[n]^{op}\ar[dr]_-{f}&&[m]^{op}\ar[dl]^-{g}\ar[ll]_-{h^{op}}\\
&I
}\] commute.
\item Composition is given by composition in $\Delta$.
\end{enumerate} The \emph{category of subdivided simplices} of $I$ is the category $\Delta_\#^{op}I$ with objects functors $f:P_n^{op}\to I$ and morphisms given as in $\Delta^{op}I$.
\end{definition}
\begin{remark}
The reader is warned that our terminology is nonstandard.
\end{remark}
\begin{remark}
In keeping track of the variance, it is helpful to think of the morphism from $f$ to $g$ as being given by the pullback functor $(h^{op})^*$.
\end{remark}
These categories come equipped with a number of functors, which we summarize in the following commuting diagram \[\xymatrix{
&I\\
\Delta^{op}I\ar[rr]^-\rho\ar[dr]_-\delta\ar[ur]^-\lambda&&\Delta^{op}_\#I\ar[ul]_{\lambda_\#}\ar[dl]^-{\delta_\#}\\
&\Delta^{op}
}\]
Here the functor $\delta$ records the domains of functors, and the functor $\lambda$ is given on objects by the formula \[\lambda\left(f:[n]^{op}\to I\right)=f(n).\] We use our choice of variance in extending $\lambda$ to a functor, for, since $h(m)\leq n$, there is a canonical map $n\to h(m)$ in $[n]^{op}$, and the functor $f$ provides a map $\lambda(f)=f(n)\to f(h(m))= g(m)=\lambda(g)$. Similarly, $\delta_\#(f:P_n^{op}\to I)=[n]$ and $\lambda_\#(f:P_n^{op}\to I)=f([n])$, and the functor $\rho$ is defined by the restrictions along the functors $P_n\to [n]$ mentioned above.
\begin{lemma}\label{lem:domain finality}
For every $n\geq0$, the natural functor $\delta^{-1}([n])\to (\delta\downarrow[n])$ is homotopy final, and similarly for $\delta_\#$.
\end{lemma}
\begin{proof}
We prove the first claim, the argument for the second being essentially identical. The overcategory $(\delta\downarrow[n])$ has objects the pairs of a functor $f:[m]^{op}\to I$ and a morphism $r:[n]\to [m]$ in $\Delta$, and a morphism is a commuting diagram \[\xymatrix{
&[n]^{op}\ar[dl]_-{r^{op}}\ar[dr]^-{(r')^{op}}\\
[m]^{op}\ar[dr]_-{f}&&[m']^{op}\ar[dl]^-{f'}\ar[ll]\\
&I.
}\] Note that the variance is such that this diagram represents a morphism from $(f, r)$ to $(f', r')$. Thus, denoting the natural inclusion by $\iota:\delta^{-1}([n])\to (\delta\downarrow[n])$, we see from the definitions that $\left((f,r)\downarrow\iota)\right)$ is the category of commuting diagrams of the form \[\xymatrix{
&[n]^{op}\ar[dl]_-{r^{op}}\ar@{=}[dr]\\
[m]^{op}\ar[dr]_-{f}&&[n]^{op}\ar[dl]\ar[ll]\\
&I.
}\] This category is isomorphic to the discrete category with one object, which is certainly contractible.
\end{proof}
\begin{corollary}\label{cor:bar kan extension}
There are natural degreewise weak equivalences \[\mathrm{hoLan}_\delta(\lambda^*F)\xrightarrow{\sim}\mathrm{Bar}_\bullet(F)\] and \[\mathrm{hoLan}_{\delta_\#}(\lambda_\#^*F)\xrightarrow{\sim}\mathrm{Bar}^\#_\bullet(F)\] of simplicial spaces.
\end{corollary}
\begin{proof}
We calculate that
\begin{align*}
\mathrm{hoLan}_\delta(\lambda^*F)_n&\simeq \hocolim\left((\delta\downarrow[n])\to \Delta^{op}I\xrightarrow{\lambda} I\xrightarrow{F}\mathcal{T}\mathrm{op}\right)\\
&\simeq \hocolim\left(\delta^{-1}([n])\to\Delta^{op}I\xrightarrow{\lambda} I\xrightarrow{F}\mathcal{T}\mathrm{op}\right)\\
&\simeq \coprod_{f:[n]^{op}\to I}F(f(n))\\
&\cong\coprod_{i_n\to \cdots \to i_0}F(i_n)\\
&=\mathrm{Bar}_n(F),
\end{align*} where Lemma \ref{lem:domain finality} and Proposition \ref{prop:hocolim facts}(1) are used in obtaining the second equivalence. The calculation in the subdivided case is essentially identical.
\end{proof}
The final ingredient that we will need is the following.
\begin{lemma}\label{lem:last value finality}
The functors $\lambda$ and $\lambda_\#$ are each homotopy final.
\end{lemma}
\begin{proof}
The claim for $\lambda$ will follow after verifying for each $i\in I$, first, that the category $\lambda^{-1}(i)$ has a final object, and, second, that the canonical functor $\iota:\lambda^{-1}(i)\to (i\downarrow\lambda)$ is homotopy initial. For the first claim, we note that the functor $[0]^{op}\to I$ with value $i$ is final in $\lambda^{-1}(i)$. For the second claim, we observe that an object of $(i\downarrow\lambda)$ is simply a composable tuple $i\to f(n)\to \cdots\to f(0)$, which determines a canonical functor $\bar f:[n+1]^{op}\to I$ such that $\lambda(\bar f)=i$, together with a universal map $\bar f\to f$. The argument for $\lambda_\#$ is essentially the same, the only difference being that we extend $f:P_n^{op}\to I$ to $\bar f:P_{n+1}^{op}\to I$ by the prescription \[
\bar f(S)=\begin{cases}
f(S)&\quad n+1\notin S\\
i&\quad n+1\in S.
\end{cases}
\]
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:subdivided comparison}]
The claim will follow after verifying that each of the marked arrows in the commuting diagram
\[\xymatrix{
\displaystyle\hocolim_IF\ar@{=}[d]&\displaystyle\hocolim_{\Delta^{op}I}\lambda^*F\ar[l]_-{(1)}\ar[d]\ar[r]^-{(3)}&\displaystyle\hocolim_{\Delta^{op}}\mathrm{hoLan}_\delta(\lambda^*F)\ar[d]\ar[r]^-{(5)}&|\mathrm{Bar}_\bullet(F)|\ar[d]\\
\displaystyle\hocolim_IF&\displaystyle\hocolim_{\Delta^{op}_\#I}\lambda_\#^*F\ar[l]_-{(2)}\ar[r]^-{(4)}&\displaystyle\hocolim_{\Delta^{op}}\mathrm{hoLan}_{\delta_\#}(\lambda_\#^*F)\ar[r]^-{(6)}&|\mathrm{Bar}^\#_\bullet(F)|
}\] is a weak equivalence. The first and second follow from Proposition \ref{prop:hocolim facts}(1) and Lemma \ref{lem:last value finality}, the third and fourth are formal, since left Kan extensions compose, and the fifth and sixth follow from Corollary \ref{cor:bar kan extension} and Proposition \ref{prop:hocolim facts}(2).
\end{proof}
\section{Deferred proofs}\label{section:deferred proofs}
\subsection{Fadell--Neuwirth fibrations}
We are now able to repay the first of our long outstanding debts, namely the proof of Theorem \ref{thm:Fadell--Neuwirth}, which asserts that the diagram
\[\xymatrix{
\mathrm{Conf}_{\ell-k}(M\setminus\{x_1,\ldots, x_k\})\ar[d]\ar[r]&\mathrm{Conf}_\ell(M)\ar[d]\\
(x_1,\ldots, x_k)\ar[r]&\mathrm{Conf}_k(M)
}\] is homotopy Cartesian. Recall that $\mathrm{Conf}_\ell(M)$ has a topological basis indexed by the partially ordered set \[\op B(M)_\ell^\Sigma=\left\{(U,\sigma):(\mathbb{R}^n)^{\amalg \ell}\cong U\subseteq M,\, \sigma:\{1,\ldots, \ell\}\xrightarrow{\simeq} \pi_0(U)\right\}\] consisting of the open sets $\mathrm{Conf}_\ell^0(U,\sigma)=\{x\in \mathrm{Conf}_\ell(M): x_i\in U_{\sigma(i)}\}.$ Thus, we have a functor \[\mathrm{Conf}_\ell^0:\op B(M)_\ell^\Sigma\to \mathcal{T}\mathrm{op}\] whose homotopy colimit is canonically equivalent to $\mathrm{Conf}_\ell(M)$, and similarly for $\mathrm{Conf}_k(M)$. We also have a functor $\pi:\op B(M)_\ell^\Sigma\to \op B(M)_k^\Sigma$ defined by \[\pi(U,\sigma)=\left(\coprod_{i=1}^kU_{\sigma(i)},\, \sigma|_{\{1,\ldots, k\}}\right)\] and a natural transformation fitting into the commuting digram \[\xymatrix{
\mathrm{Conf}_\ell^0\ar[d]\ar@{-->}[r]&\pi^*\mathrm{Conf}_k^0\ar[d]\\
\mathrm{Conf}_\ell(M)\ar[r]&\mathrm{Conf}_k(M),
}\] where the bottom map is the Fadell--Neuwirth map given by projection onto the first $k$ factors.
Now, since each $\mathrm{Conf}_\ell^0(U,\sigma)$ is contractible, we obtain the lefthand set of weak equivalences in the commuting diagram \[\xymatrix{
B(\op B(M)_\ell^\Sigma)\ar[d]_-{B\pi}&\hocolim_{\op B(M)_\ell^\Sigma}\mathrm{Conf}_\ell^0\ar[d]\ar[r]^-\sim\ar[l]_-\sim&\mathrm{Conf}_\ell(M)\ar[d]\\
B(\op B(M)_k^\Sigma)&\hocolim_{\op B(M)_k^\Sigma}\mathrm{Conf}_k^0\ar[r]^-\sim\ar[l]_-\sim&\mathrm{Conf}_k(M).
}\] Thus, understanding the homotopy fiber of the map rightmost map is tantamount to understanding the homotopy fiber of the map between classifying spaces induced by the functor $\pi$.
\begin{proof}[Proof of Theorem \ref{thm:Fadell--Neuwirth}]
We wish to apply Corollary \ref{cor:quillen b} to obtain the homotopy pullback square \[\xymatrix{
B((U,\sigma)\downarrow\pi)\ar[d]\ar[r]&B(\op B(M)_\ell^\Sigma)\ar[d]^-{B\pi}\\
\mathrm{pt}\ar[r]^-{(U,\sigma)}&B(\op B(M)_k^\Sigma).
}\] In order to verify that the hypotheses hold in this case, we note that $((U,\sigma)\downarrow\pi)$ is the category of $(W,\tau)\in \op B(M)_\ell^\Sigma$ such that $U_{\sigma(i)}\subseteq W_i$ for $1\leq i\leq k$. It is easy to see that the inclusion of the subcategory with $U_{\sigma(i)}=W_{\tau(i)}$ is homotopy initial, and this subcategory is isomorphic to $\op B(M\setminus U)_{\ell-k}^\Sigma$. Since homotopy initial functors induce weak equivalences on classifying spaces, we have the weak equivalences in the diagram \[\xymatrix{
B((U,\sigma)\downarrow\pi)&B(\op B(M\setminus U)_{\ell-k}^\Sigma)\ar[l]_-\sim&\hocolim_{\op B(M\setminus U)_{\ell-k}^\Sigma}\mathrm{Conf}_{\ell-k}^0\ar[l]_-\sim\ar[r]^-\sim&\mathrm{Conf}_{\ell-k}(M\setminus U)\\
B((U',\sigma')\downarrow\pi)\ar[u]&B(\op B(M\setminus U')_{\ell-k}^\Sigma)\ar[u]\ar[l]_-\sim&\hocolim_{\op B(M\setminus U')_{\ell-k}^\Sigma}\mathrm{Conf}_{\ell-k}^0\ar[u]\ar[l]_-\sim\ar[r]^-\sim&\mathrm{Conf}_{\ell-k}(M\setminus U')\ar[u]}\] for any $(U,\sigma)\leq (U',\sigma')$ in $\op B(M)_k^\Sigma$. Since $M\setminus U'\subseteq M\setminus U$ is a monotopy equivalence, the rightmost map is a weak equivalence, so all of the vertical arrows are weak equivalences. Therefore, by Corollary \ref{cor:quillen b} and what we have already shown, we have the homotopy pullback \[\xymatrix{
B(\op B(M\setminus U)_{\ell-k}^\Sigma)\ar[d]\ar[r]&B(\op B(M)_\ell^\Sigma)\ar[d]^-{B\pi}\\
\mathrm{pt}\ar[r]^-{(U,\sigma)}&B(\op B(M)_k^\Sigma).
}\] The proof is concluded upon noting that the inclusion \[\mathrm{Conf}_{\ell-k}(M\setminus U)\to \mathrm{Conf}_{\ell-k}(M\setminus \{x_1,\ldots, x_k\})\] is a weak equivalence for any $x_i\in U_{\sigma(i)}$.
\end{proof}
\subsection{Spectral sequences}
In proving the Leray--Hirsch theorem, and in much of what will follow, we will make use of the Serre spectral sequence. We begin with a few reminders on spectral sequences---for a general reference, see \cite{McCleary:UGSS}.
\begin{definition}
A (cohomological) \emph{spectral sequence} is a collection $\{E_r\}_{r\geq0}$ of bigraded $R$-modules, called the \emph{pages} of the spectral sequence, equipped with
\begin{enumerate}
\item differentials $d_r:E_r\to E_r$ of bidegree $(r,1-r)$, and
\item isomorphisms $H(E_r,d_r)\cong E_{r+1}$ of bigraded $R$-modules.
\end{enumerate}
\end{definition}
A \emph{map of spectral sequences} is a collection of bigraded chain maps $f_r:E_r\to \widetilde E_r$ compatible with these isomorphisms. We say that the spectral sequence $\{E_r\}$ is \emph{multiplicative} if each $E_r$ is equipped with the structure of a bigraded $R$-algebra for which $d_r$ is a bigraded derivation and the isomorphisms of (2) are algebra isomorphisms (note that this language is abusive, since multiplicativity is a structure rather than a property). Recall that $d_r$ is bigraded derivation if it satisfies the \emph{Leibniz rule} \[d_r(ab)=d_r(a)b+(-1)^{p+q}ad_r(b),\qquad |a|=(p,q).\]
\begin{remark}
Our statements and definitions are cohomological, since that is the nature of the application we have in mind, but the obvious dual notions are valid and often better behaved.
\end{remark}
In a typical situation, the $E_2$-page is something identifiable and relatively computable; each module $E_r^{p,q}$ is independent of $r$ for sufficiently large, and the identification of this common module $E_\infty^{p,q}$ is the goal; and the differentials are mysterious. We will not define $E_\infty$ or discuss convergence here---see \cite[3]{McCleary:UGSS} for details.
\begin{example}
If $(V,\partial,\delta)$ is a bicomplex satisfying mild boundedness conditions, there is a spectral sequence with $E_1\cong H(V,\partial)$ and \[E_2\cong H(H(V,\partial),\delta)\implies H(V,\partial+\delta).\] See \cite[2.4]{McCleary:UGSS} for a construction of this spectral sequence.
An alternative perspective on the spectral sequence of a bicomplex is offered by the \emph{homotopy transfer theorem} \cite{Vallette:AHO}. The starting observation is that the differential $\delta$ can be viewed as an algebraic structure on the chain complex $(V,\partial)$, in the form of an action of the dual numbers $k[\epsilon]/\epsilon^2$ (we work over a field $k$ for simplicity). After choosing representatives for homology and extending this choice to a chain deformation retraction of $(V,\partial)$ onto $(H(V,\partial),0)$, the homotopy transfer theorem endows $H(V,\partial)$ with the structure of a \emph{homotopy} $k[\epsilon]/\epsilon^2$-module, which amounts to the induced differential $\delta$ together with countably many higher operations $\{\delta_n\}$ satisfying certain relations (classically, this structure is known as a \emph{multicomplex}). These higher operations are direct analogues of the Massey products on the cohomology of a space, and they induce the differentials in the spectral sequence for $(V,\partial,\delta)$.
\end{example}
\begin{example}
If $X=\bigcup_{p\geq0} F_pX$ is a filtered space satisfying mild completeness conditions, there is a spectral sequence \[E_1^{p,q}\cong H^{p+q}(F_pX,F_{p-1}X)\implies H^{p+q}(X),\] i.e., there is an isomorphism of the $E_\infty$-page with the associated graded of the induced filtration on $H^*(X)$. The differential $d_1$ is the connecting homomorphism in the long exact sequence for the triple $(F_pX,F_{p-1}X, F_{p-2}X)$, and a filtration preserving map between spaces induces a map between the associated spectral sequences. The same construction goes through with integral cohomology replaced by an arbitrary cohomology theory. See \cite[2.2]{McCleary:UGSS} for details.
\end{example}
Applying this example to the skeletal filtration of the geometric realization of a simplicial space yields the following result. As a matter of notation, for a simplicial $R$-module $V$, we write $\mathrm{Alt}(V)$ for the associated chain complex of $R$-modules, whose differential is the alternating sum of the face maps of $V$.
\begin{corollary}[{\cite[5.1,\,5.3,\,5.4]{Segal:CSSS}}]\label{cor:simplicial spectral sequence}
Let $\op X:\Delta^{op}\to \mathcal{T}\mathrm{op}$ be a simplicial space and $\op H$ a cohomology theory. There is a spectral sequence \[E_2^{p,q}\cong H^p(\mathrm{Alt}(\op H^q(\op X)))\implies \op H^{p+q}(|\op X|),\] which is natural for simplicial maps and multiplicative if $\op H$ is.
\end{corollary}
Using this basic construction and our results on \v{C}ech nerves, we may associate a spectral sequence to a general map.
\begin{theorem}[Leray, Segal]\label{thm:spectral sequence of a map}
Let $f:X\to Y$ be a map between topological spaces with $Y$ paracompact and $\op H$ a cohomology theory. There is a spectral sequence \[E_2^{p,q}\cong H^p(Y; \op H^q(f^{-1}))\implies \op H^{p+q}(X),\] where $\op H^q(f^{-1})$ is the sheaf associated to the presheaf $U\mapsto\op H^q(f^{-1}(U))$. Moreover, this spectral sequence is natural on the arrow category and multiplicative if $\op H$ is.
\end{theorem}
\begin{proof}
For each open cover $\op U$ of $Y$, the collection $f^{-1}\op U=\{f^{-1}(V): V\in \U\}$ is an open cover of $X$, so we have a canonical weak homotopy equivalence \[|\check{C}(f^{-1}\op U)|\xrightarrow{\sim} X.\] We now apply Corollary \ref{cor:simplicial spectral sequence} to obtain a spectral sequence with \begin{align*}E_2^{p,q}&\cong H^p(\mathrm{Alt}(\op H^q(\check{C}(f^{-1}\op U))))\\
&= H^p\left(\cdots \to\bigoplus_{\op U^2}\op H^q(f^{-1}(V_0\cap V_1))\to \bigoplus_{\op U}H^q(f^{-1}(V))\right)\\
&\cong \check{H}(\op U;\op H^q(f^{-1}))\implies \op H^{p+q}(X),\end{align*} where $\check{H}$ denotes \v{C}ech cohomology. A refinement of open covers induces a map at the level of \v{C}ech nerves and therefore a map of spectral sequences, and, forming the colimit over the partially ordered set of open covers of $Y$, we obtain at last the spectral sequence \begin{align*}E_2^{p,q}&\cong \colim_{\op U}\check{H}^p(U;\op H^q(f^{-1}))\\
&\cong H^p(Y;\op H^q(f^{-1}))\implies \op H^{p+q}(X),
\end{align*} where the last isomorphism uses the assumption that $Y$ is paracompact.
\end{proof}
This spectral sequence specializes to two well-known spectral sequences.
\begin{corollary}[Atiyah-Hirzebruch]
Let $X$ be a paracompact space and $\op H$ a cohomology theory. There is a spectral sequence \[E_2^{p,q}\cong H^p(X; \op H^q(\mathrm{pt}))\implies \op H^{p+q}(X),\] which is natural and multiplicative if $\op H$ is.
\end{corollary}
\begin{proof}
We apply Theorem \ref{thm:spectral sequence of a map} to the map $\id_X$, in which case the sheaf in question is the constant sheaf with value $\op H^q(\mathrm{pt})$.
\end{proof}
The spectral sequence of greatest interest to us is the \emph{Serre spectral sequence} associated to a fibration \cite{Serre:HSEF}.
\begin{corollary}[Serre]\label{cor:serre ss}
Let $R$ be a ring and $\pi:E\to B$ a fibration with fiber $F$, and assume that $B$ is path-connected and paracompact. There is a spectral sequence \[E_2^{p,q}\cong H^p(Y; \underline{H^q(F; R)})\implies H^{p+q}(X;R),\] which is multiplicative and natural for maps of fibrations, where $\underline{H^q(F; R)}$ denotes the local coefficient system induced by the homotopy action of $\pi_1(B)$ on $F$.
\end{corollary}
\begin{remark}
Since an arbitrary map may be approximated by a fibration over a paracompact base, there is a Serre spectral sequence for an arbitrary map with connected target, which involves the homotopy fiber rather than the point-set fiber.
\end{remark}
\begin{corollary}\label{cor:ss of a cover}
Let $\pi:P\to X$ be a connected principal $G$-bundle. There is a spectral sequence \[E_2^{p,q}\cong H^p(G;H^q(P))\implies H^{p+q}(X),\] which is multiplicative and natural for maps of principal $G$-bundles.
\end{corollary}
\begin{proof} Forming the Borel construction on $P$, we obtain the homotopy pullback square \[\xymatrix{
P\ar[d]\ar[r]&EG\times_G X\ar[d]\\
\mathrm{pt}\ar[r]&BG,
}\] and, since $G$ acts freely on $X$, the natural map $EG\times_G P\to X$ is a weak equivalence. The claim follows from Corollary \ref{cor:serre ss} after identifying the group cohomology of $G$ with the twisted cohomology of $BG$.
\end{proof}
\subsection{The Leray--Hirsch theorem}
We turn now to the proof of the Leray--Hirsch theorem. We make use of the following basic observation, which identifies the \emph{edge maps} in the Serre spectral sequence.
\begin{lemma}\label{lem:edge maps}
Let $\{E_r\}$ be the spectral sequence for the fibration $F\xrightarrow{i} E\xrightarrow{\pi}B$. There is a commuting diagram \[\xymatrix{
H^q(E;R)\ar@{->>}[d]\ar[r]^-{i^*}&H^q(F;R)\ar@{=}[d]^-\wr\\
E_\infty^{0,q}\ar@{^{(}->}[r]&E_2^{0,q}
}\] for every $q\geq0$. Moreover, if $F$ is connected, then there is a commuting diagram \[\xymatrix{
H^p(B;R)\ar@{=}[d]_-{\wr}\ar[r]^-{\pi^*}&H^p(E;R)\\
E^{p,0}_2\ar@{->>}[r]&E^{p,0}_\infty\ar@{^{(}->}[u]
}\] for every $p\geq0$.
\end{lemma}
\begin{proof}
Both claims follow by naturality from the commuting diagram \[\xymatrix{
F\ar@{=}[d]\ar@{=}[r]&F\ar[d]^-i\ar[r]&\mathrm{pt}\ar[d]\\
F\ar[d]\ar[r]^-i&E\ar[r]^-\pi\ar[d]^-\pi&B\ar@{=}[d]\\
\mathrm{pt}\ar[r]&B\ar@{=}[r]&B,
}\] in which the bottom set of vertical arrows are all fibrations. The assumption that $F$ is connected permits the identification \[E_2^{p,0}\cong H^p(B;\underline{H^0(F;R)})\cong H^p(B;R).\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:Leray--Hirsch}]
The assumption that $B$ is connected permits the use of the Serre spectral sequence. The assumption that $F$ is connected and the surjectivity assumption together imply that the local system $\underline{H^q(F)}$ is trivial; indeed, the former implies that $\pi_1(E)$ surjects onto $\pi_1(B)$, so the action of $\pi_1(B)$ on $H^*(F)$ may be computed after lifting both to $E$. This action is trivial, since the action of $\pi_1(E)$ on $H^*(E)$ is so. The assumptions on the cohomology of $F$ and $B$ now permit us to apply the K\"{u}nneth theorem in cohomology to obtain the isomorphism \[E_2^{p,q}\cong H^p(B)\otimes H^q(F).\]
We claim that this spectral sequence collapses at $E_2$. To see why this is so, we first note that surjectivity of $i^*$ and Lemma \ref{lem:edge maps} imply the isomorphism $E_2^{0,q}\cong E_\infty^{0,q}$, so $d_r|_{E_r^{0,q}}=0$ for all $r\geq2$ and $q\geq0$. Assume for induction that we have established the isomorphism $E_2\cong E_r$, the base case being $r=2$. Since a bihomogeneous element in $E_r$ may be written as $a\otimes b$ for $a\in E_r^{p,0}$ and $b\in E_r^{0,1}$, it suffices by the Leibniz rule to show that $d_r(a)=0$ and that $d_r(b)=0$ separately. The latter equality was shown above, and the former holds for degree reasons.
Thus, we have an isomorphism of $H^*(B)$-modules between $H^*(B)\otimes H^*(F)$ and the associated graded for the induced filtration on $H^*(E)$. Since $H^*(F)$ is free Abelian, $i^*$ admits a section $s$, and the map \begin{align*}
H^*(B)\otimes H^*(F)&\to H^*(E)\\
a\otimes b&\mapsto s(a)\smile \pi^*(b)
\end{align*} is an isomorphism, since it becomes an isomorphism after passing to the associated graded.
\end{proof}
\section{Mapping space models}\label{section:mapping space models}
Our next goal is to explore the perhaps surprising connection between configuration spaces and mapping spaces, following the seminal work of McDuff \cite{McDuff:CSPNP} (see also \cite{Boedigheimer:SSMS}). A motivating idea due to Segal \cite{Segal:CSILS} is that of the \emph{electric field map} \[\coprod_{k\geq0}B_k(\mathbb{R}^n)\to \Omega^nS^n,\] which assigns to a configuration, viewed as a collection of point charges, the corresponding electric field. As a vector field, this electric field is a priori a map from $\mathbb{R}^n$ to itself; however, since it becomes infinite at the location of a point charge, and since it tends to zero at infinity, it naturally extends to a map between the respective one-point compactifications. Alternatively, we can understand this map as a kind of Pontrjagin-Thom construction, sending a configuration of $k$ points to the composite $S^n\to \vee_k S^n\to S^n$ of the Thom collapse map for the normal bundle of the configuration followed by the fold map.
The electric field map is not a homotopy equivalence; for example, the induced map on $\pi_0$ is the inclusion $\mathbb{N}\to \mathbb{Z}$. We would like to understand whether this failure can be rectified, as well as whether the same idea may be adapted to more general background manifolds and target spaces.
\begin{remark}
In a sense, this ``group completion'' discrepancy on connected component is the only obstruction to the map being an equivalence---see \cite{Segal:CSILS}.
\end{remark}
\subsection{Labeled configuration spaces}
We begin by identifying the type of combinatorics at play.
\begin{definition}
A \emph{pointed finite set} is a finite set $I$ together with a distinguished element, called the \emph{basepoint} and denoted $*$. We write $I^\circ$ for the pointed finite set $I\setminus \{*\}$. A map $f:I\to J$ is \emph{inert} if
\begin{enumerate}
\item $f(*)=*$, and
\item $f|_{f^{-1}(J^\circ)}$ is a bijection onto $J^\circ$.
\end{enumerate}
\end{definition}
We write $\mathrm{Inrt}$ for the category pointed finite sets and inert maps. Note that this category is isomorphic to the opposite of the category of finite sets and injective maps.
\begin{construction}
A manifold $M$ defines a functor from $\mathrm{Inrt}$ to $\mathcal{T}\mathrm{op}$ by sending $I$ to $\mathrm{Conf}_{I^\circ}(M)$ and the inert map $f:I\to J$ to the projection \[\xymatrix{
\mathrm{Conf}_{I^\circ}(M)\ar[d]\ar@{-->}[r]&\mathrm{Conf}_{\pi^{-1}(J^\circ)}(M)\ar[r]^-{\simeq}\ar[d]&\mathrm{Conf}_{J^\circ}(M)\ar[d]\\
M^{I^0}\ar[r]^-{\pi_f}&M^{\pi^{-1}(J^\circ)}\ar[r]^-\simeq&M^{J^\circ}
}\] given by the formula \[\pi_f\left((m_i)_{i\in I^\circ}\right)=(m_{f^{-1}(j)})_{j\in J^\circ}.\]
\end{construction}
\begin{construction}
A based space $(X,x_0)$ determines a functor from $\mathrm{Inrt}^{op}$ to $\mathcal{T}\mathrm{op}$ by sending $I$ to $\mathrm{Map}_*(I, X)\cong X^{I^\circ}$ and the inert map $f$ to the inclusion \[X^{J^\circ}\cong X^{f^{-1}(J^\circ)}\times \{x_0\}^{I^\circ\setminus f^{-1}(J^\circ)}\subseteq X^{f^{-1}(J^\circ)}\times X^{I^\circ\setminus f^{-1}(J^\circ)}\cong X^{I^\circ}.\]
\end{construction}
\begin{definition}
The \emph{configuration space of $M$ with labels in $X$} is the coequalizer \[\mathrm{Conf}_X(M)=\mathrm{coeq}\left(
\coprod_{J\to K}\mathrm{Conf}_{J^\circ}(M)\times X^{K^\circ}\rightrightarrows\coprod_{I}\mathrm{Conf}_{I^\circ}(M)\times X^{I^\circ}
\right),\] where the coproducts are indexed on the morphisms and objects of $\mathrm{Inrt}$, respectively.
\end{definition}
\begin{remark}
Note that this construction is sensible without the assumption that $M$ be a manifold.
\end{remark}
Thus, a point in $\mathrm{Conf}_X(M)$ is a finite formal sum $\sum m_ax_a$ with $m_a\in M$ distinct and $x_a\in X$, and the following relation holds \[\textstyle\sum m_ax_a\sim \sum m_ax_a+mx_0.\] We refer to the point $x_a$ as the \emph{label} of $m_a$. The topology is such that a point vanishes if its label moves to the basepoint of $X$; thus, if $x_k\to x_0$ in $X$, for example, then $mx_k\to \varnothing$ for any $m\in M$.
\begin{example}
For any $M$, $\mathrm{Conf}_\mathrm{pt}(M)=\{\varnothing\}$.
\end{example}
\begin{example}
For any $M$, there is a homeomorphism $\mathrm{Conf}_{S^0}(M)\cong \coprod_{k\geq0}B_k(M).$
\end{example}
We will also have use for a relative version of this construction. If $M_0\subseteq M$ is a closed subspace, we write $\mathrm{Conf}_X(M,M_0)$ for the quotient of $\mathrm{Conf}_X(M)$ by the further relation \[\textstyle\sum m_ax_a\sim \sum m_ax_a+mx,\qquad m\in M_0.\] We refer to this space as the labeled configuration space with \emph{annihilation in $M_0$}. In this space, a point vanishes also if it collides with the annihilation subspace $M_0$; thus, if $m_k\to m\in M_0$, for example, then $m_kx\to \varnothing$ for any $x\in X$.
\begin{example}
For any $M$ and $X$, $\mathrm{Conf}_X(M,M)=\{\varnothing\}$
\end{example}
\begin{example}
If either $X$ or $(M,M_0)$ is path connected, then so is $\mathrm{Conf}_X(M,M_0)$ (recall that a pair is path connected if the map on path components is surjective).
\end{example}
\begin{definition}
The \emph{support} of the configuration $\sum m_ax_a$ is the (finite) subset \[\textstyle\mathrm{Supp}\left(\sum m_ax_a\right)=\left\{m_a\mid m_a\notin M_0 \text{ and } x_a\neq x_0\right\}\subseteq M.\]
\end{definition}
The space $\mathrm{Conf}_X(M,M_0)$ is filtered by the closed subspaces \[\mathrm{Conf}_X(M,M_0)_{\leq k}:=\left\{\textstyle \sum m_ax_a\mid |\mathrm{Supp}(\sum m_ax_a)|\leq k\right\}.\] Moreover, both the successive quotients and the successive complements of this filtration are comprehensible, as \[\faktor{\mathrm{Conf}_X(M,M_0)_{\leq k}}{\mathrm{Conf}_X(M,M_0)_{\leq k-1}}\cong \mathrm{Conf}_k(M,M_0)\wedge_{\Sigma_k}X^{\wedge k},\] where $\mathrm{Conf}_k(M,M_0)$ is the quotient of $\mathrm{Conf}_k(M)$ by the subspace of configurations intersecting $M_0$ non-vacuously, while \[\mathrm{Conf}_X(M,M_0)_{\leq k}\setminus \mathrm{Conf}_X(M,M_0)_{\leq k-1}\cong \mathrm{Conf}_k(M\setminus M_0)\times_{\Sigma_k}(X\setminus x_0)^k.\]
\begin{example}
The filtration quotients of $\mathrm{Conf}_{S^r}(M)$ are given by the Thom spaces of the vector bundles $\mathrm{Conf}_k(M)\times_{\Sigma_k} \mathbb{R}^{rk}\to B_k(M),$ so, by the Thom isomorphism, we may compute the homology of the configuration spaces of $M$ from knowledge of this filtration. In fact, as we will show, this filtration splits at the level of homology, making this type of computation often feasible in practice.
Notice that, for $r>0$, $\mathrm{Conf}_{S^r}(M)$ is connected. In particular, the analogous electric charge map $\mathrm{Conf}_{S^r}(\mathbb{R}^n)\to \Omega^nS^{n+r}$ is a bijection on $\pi_0$, unlike in the case $r=0$ considered above. In fact, as we will show, this map is a homotopy equivalence.
\end{example}
We close this section by advancing the thesis that the construction $\mathrm{Conf}_X$ should be thought of as a kind of homology theory for manifolds (see \cite{AyalaFrancis:FHTM} for a detailed elaboration on this idea). It is easy to see that analogues of some of the Eilenberg-Steenrod axioms hold. For example, the construction is functorial in an obvious way for embeddings of pairs $(M, M_0)\to (N, N_0)$ and for baseed maps $X\to Y$; the map induced by an isotopy equivalence in the former case or a homotopy equivalence in the latter is a homotopy equivalence; we have a homeomorphism \[\mathrm{Conf}_X(M\amalg N, M_0\amalg N_0)\cong \mathrm{Conf}_X(M, M_0)\times\mathrm{Conf}_X(N, N_0),\] which is an analogue of the additivity axiom; and an analogue of excision is supplied by the homeomorphism \[\mathrm{Conf}_X(M\setminus U, M_0\setminus U)\xrightarrow{\simeq} \mathrm{Conf}_X(M,M_0)\] for $U\subseteq M_0$ open in $M$.
The basic building blocks of manifolds being disks rather than points, an appropriate analogue of the dimension axiom is supplied by the following result:
\begin{proposition}
Fix $X$ and $n\geq0$.
\begin{enumerate}
\item The inclusion $\mathrm{Conf}_X(D^n,\partial D^n)_{\leq 1}\subseteq \mathrm{Conf}_X(D^n,\partial D^n)$ is a weak homotopy equivalence, and
\item there is a homeomorphism $\mathrm{Conf}_X(D^n,\partial D^n)_{\leq 1}\cong \Sigma^nX$.
\end{enumerate}
\end{proposition}
\begin{proof}
For (1), we note that, by radial expansion, any pointed map from a compact space $K$ to $\mathrm{Conf}_X(D^n,\partial D^n)$ factors up to pointed homotopy through $\mathrm{Conf}_X(D^n,\partial D^n)_{\leq 1}$. For (2), we calculate that \begin{align*}
\mathrm{Conf}_X(D^n,\partial D^n)_{\leq 1}&\cong \frac{\mathrm{Conf}_0(D^n)\times X^0\amalg \mathrm{Conf}_1(D^n)\times X^1}{\sim}\\
&\cong \frac{(D^n\times X)_+}{(\partial D^n\times X)_+\cup (D^n\times\{x_0\})_+}\\
&\cong \frac{D^n_+\wedge X}{\partial D^n_+\wedge X}\\
&\cong S^n\wedge X,
\end{align*} as desired.
\end{proof}
Finally, we have the following analogue of exactness.
\begin{theorem}[McDuff]\label{thm:exactness}
Let $M_0\subseteq M$ be a closed submanifold of codimension 0, possibly with boundary. If $X$ is connected, then the diagram \[
\xymatrix{
\mathrm{Conf}_X(M_0)\ar[d]\ar[r]&\mathrm{Conf}_X(M)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(M,M_0)
}
\] induced by the maps $(M_0,\varnothing)\to (M,\varnothing)\to (M,M_0)$ is homotopy Cartesian.
\end{theorem}
We will take up the proof of this theorem presently. We conclude this section with an obvious question.
\begin{question} If $\mathrm{Conf}_X$ is a homology theory, what is Poincar\'{e} duality?
\end{question}
\subsection{Local topology} Our strategy in proving Theorem \ref{thm:exactness} will be to combine covering technology and Quillen's Theorem B as in our proof of Fadell--Neuwirth. In order to do so, we must come to grips with the local topology of $\mathrm{Conf}_X(M,M_0)$.
\begin{definition}
An open subset $U\subseteq M$ is \emph{pointed} with respect to $M_0$ if some union of components $U_*\subseteq U$ contains $M_0$ as an isotopy retract.
\end{definition}
\begin{remark}
The assumption that $M_0\subseteq M$ is a codimension 0 submanifold guarantees a ready supply of open sets in $M$ that are pointed with respect to $M_0$; indeed, we may take the union of $M_0$ with any tubular neighborhood of $\partial M_0$ in $M\setminus \mathring{M_0}$.
\end{remark}
Note that, if $U$ is pointed with respect to $M$, then $\pi_0(U, U_*)$ is canonically pointed by the class of $U_*$.
\begin{construction}
Fix $M_0\subseteq M$ and $x_0\in X$ as above. We define a category $\op B(M,M_0;X)$ by the following specifications.
\begin{enumerate}
\item An object of $\op B(M,M_0;X)$ is a pair $(U,\sigma)$ with $U\subseteq M$ a proper open subset pointed with respect to $M_0$ and $\sigma:\pi_0(U, U_*)\to\mathrm{Op}(X)$ a function with $x_0\in \sigma(U_*)$, such that, for $i\in \pi_0(U, U_*)^\circ$
\begin{enumerate}
\item $U_i\cong\mathbb{R}^n$,
\item $\sigma(U_i)\simeq \mathrm{pt}$, and
\item $\sigma(U_i)\cap \sigma(U_*)=\varnothing$.
\end{enumerate}
\item A morphism $(U,\sigma)\to (V,\tau)$ is an inert map $f:\pi_0(U, U_*)\to \pi_0(V, V_*)$ such that
\begin{enumerate}
\item $U_{i}\subseteq V_{f(i)}$ and $\sigma(U_{i})\subseteq \tau(V_{f(i)})$ for $i\notin f^{-1}(*)\setminus\{*\}$, and
\item either $U_i\subseteq V_*$ or $\sigma(U_i)\subseteq \tau(V_*)$ for $i\in f^{-1}(*)\setminus \{*\}$.
\end{enumerate}
\end{enumerate}
We write $\mathrm{Conf}_X^0(U,\sigma)\subseteq \mathrm{Conf}_X(M,M_0)$ for the subspace consisting of labeled configurations $\sum m_a x_a$ such that \begin{enumerate}
\item for each $i\in \pi_0(U)^\circ$, there is a unique $a$ with $m_a\in U_i$ and $x_a\in \sigma(U_i)$; and
\item otherwise, either $m_a\in U_*$ or $x_a\in \sigma(U_*)$.
\end{enumerate}
\end{construction}
We summarize the relevant facts about this category and these subspaces in a series of lemmas. First, a preliminary concept.
\begin{definition}
Let $\sum m_a x_a$ be a configuration lying in $\mathrm{Conf}_X^0(U,\sigma)$. We say that the point $m_{a}$ is \emph{essential} for $(U,\sigma)$ if $m_a\in U_i$ and $x_a \in \sigma(U_i)$ for some (necessarily unique) $i\in\pi_0(U,U_*)$.
\end{definition}
Equivalently, $m_a$ is essential\[\textstyle\sum_{a'\neq a}m_{a'}x_{a'}\notin \mathrm{Conf}_X^0(U,\sigma).\]
Recall that a category is a poset if and only if, first, each hom set contains at most one element, and, second, every isomorphism is an identity.
\begin{lemma}\label{lem:labeled poset}
The category $\op B(M,M_0;X)$ is a poset, and the inequality $(U,\sigma)\leq (V,\tau)$ holds if and only if the containment $\mathrm{Conf}_X^0(U,\sigma)\subseteq \mathrm{Conf}_X^0(V,\tau)$ holds.
\end{lemma}
\begin{proof}
Suppose that the inert maps $f$ and $g$ each determine a morphism $(U,\sigma)\to (V,\tau)$ in $\op B(M,M_0;X)$. Then there exists $i\in \pi_0(U,U_*)^\circ$ such that $f(i)=*\neq g(i)$; otherwise, $f$ and $g$ differ by a permutation of $\pi_0(V,V_*)$, and the conditions on morphisms in $\op B(M,M_0;X)$ force this permutation to be the identity. Now, since $f(i)=*$, either $U_i\subseteq V_*$ or $\sigma(U_i)\subseteq \tau(V_*)$; however, since $g(i)\neq *$, $U_i\subseteq V_{g(i)}$ and $\sigma(U_i)\subseteq \tau(V_{g(i)}$. It follows that either $V_{g(i)}\cap V_*\neq \varnothing$ or $\tau(V_{g(i)})\cap\tau(V_*)\neq \varnothing$, a contradiction. We conclude that hom sets in $\op B(M,M_0;X)$ have cardinality at most 1.
Now, if $(U,\sigma)\cong (V,\tau)$, then the associated inert map is also an isomorphism, which is to say a pointed bijection, so $U_i\subseteq V_{f(i)}\subseteq U_i$ and $\sigma(U_i)\subseteq\tau(V_{f(i)})\subseteq \sigma(U_i)$ for every $i\in \pi_0(U,U_*)$, and it follows that $(U,\sigma)=(V,\tau)$. Thus, $\op B(M,M_0;X)$ is a poset.
For the second claim, we verify the ``if'' implication, the converse being essentially obvious. Suppose, then, that $\mathrm{Conf}_X^0(U,\sigma)\subseteq \mathrm{Conf}_X^0(V,\tau)$, and choose $\sum m_ax_a\in\mathrm{Conf}_X^0(U,\sigma)$. We note that \begin{align*}
\pi_0(U,U_*)^\circ&\cong\left\{a\mid m_a\text{ is essential for $(U,\sigma)$}\right\}\\
&\supseteq\left\{a\mid m_a\text{ is essential for $(V,\tau)$}\right\}\\
&\cong\pi_0(V,V_*)^\circ,
\end{align*} where the inclusion follows from the definition of an essential point and our containment assumption. Since the data of an inert map is equivalent to that of an inclusion in the opposite direction, this observation defines an inert map $f:\pi_0(U,U_*)\to \pi_0(V,V_*)$. It is easy to see that two labeled configurations that may be joined by a path give rise to the same inert map; therefore, since $\mathrm{Conf}_X^0(U,\sigma)$ is path connected, $f$ is independent of the choice of $\sum m_ax_a$.
To see that $f$, so defined, witnesses the inequality $(U,\sigma)\leq (V,\tau)$, there are two points to verify.
\begin{enumerate}
\item For $i\in f^{-1}(\pi_0(V,V_*)^\circ)$, we note that $U_i\cap V_{f(i)}\neq \varnothing$, since there is some point $m_a$ that is essential both for $(U,\sigma)$ and for $(V,\tau)$. The existence of a path in $U_i$ from $m_a$ to a point lying outside of this intersection leads to a contradiction of the fact that $m_a$ is essential for $(V,\tau)$, so $U_i\subseteq V_{f(i)}$. On the other hand, since $U_*$ and $V_*$ both contain $M_0$ as an isotopy retract, we have the bijection $\pi_0(U_*)\cong\pi_0(M_0)\cong \pi_0(V_*)$. Thus, since $V_*\cap V_j=\varnothing$ for $j\neq*$, it follows that $U_*\cap V_j=\varnothing$ for $j\neq *$. Allowing a point to range over $U_*$ with a fixed label lying outside of $\tau(V_*)$ now shows that $U_*\subseteq V_*$.\\
\item For $*\neq i\in f^{-1}(*)$, if neither containment holds, then there exist $m\in U_i$ and $x\in \sigma(U_i)$ with $m\notin V_*$ and $x\notin \tau(V_*)$. Thus, there is a configuration of the form $\sum m_ax_a+mx\in \mathrm{Conf}_X^0(U,\sigma)$; however, since $m$ is not essential in $(V,\tau)$ by the definition of $f$, we must also have $\sum m_ax_a+mx\notin \mathrm{Conf}_X^0(V,\tau)$, contradicting our assumption.
\end{enumerate}
\end{proof}
\begin{lemma}\label{lem:labeled open}
Each $\mathrm{Conf}_X^0(U,\sigma)$ is open in $\mathrm{Conf}_X(M,M_0)$.
\end{lemma}
\begin{proof}
From the definitions, a subset of $\mathrm{Conf}_X(M, M_0)$ is open if and only if its preimage under each of the maps \[q_r:\mathrm{Conf}_r(M)\times X^r\to \mathrm{Conf}_X(M,M_0)\] is so. For $r<k:=|\pi_0(U)^\circ|$, we have $q^{-1}_r(\mathrm{Conf}_X^0(U,\sigma)=\varnothing$; otherwise, the inverse image is the $\Sigma_r$-orbit of the subspace \[\bigcup_{k+\ell+m=r} A_{k,\ell,m}\times\prod_{i=1}^k\sigma(U_i)\times X^\ell\times \sigma(U_*)^m,\] where $A_{k,\ell,m}\subseteq \mathrm{Conf}_r(M)$ is defined by requiring that each of the diamonds in the commuting diagram
\[\xymatrix{
&&A_{k,\ell,m}\ar[dr]\ar[dl]\\
&\bullet\ar[dl]\ar[dr]^-{(3)}&&\bullet\ar[dl]_-{(4)}\ar[dr]\\
\mathrm{Conf}^0_k(U\setminus U_*)\ar[dr]^-{(1)}&&\mathrm{Conf}_r(M)\ar[dr]\ar[dl]&&\mathrm{Conf}_\ell(U_*)\ar[dl]_-{(2)}\\
&\mathrm{Conf}_k(M)&&\mathrm{Conf}_\ell(M)
}\] be a pullback; in other words, $A_{k,\ell,m}$ is the subspace in which the first $k$ points lie $\pi_0$-surjectively in $U\setminus U_*$, the next $\ell$ points lie in $U_*$, and the remaining $m$ points are arbitrary. Since the first and second marked arrows are inclusions of open subspaces, so are the third and fourth marked arrows. Thus, $A_{k,\ell,m}$ is the intersection of two open subspaces and hence itself open, implying the claim.
\end{proof}
\begin{lemma}\label{lem:labeled complete cover}
The collection $\left\{\mathrm{Conf}_X^0(U,\sigma): (U,\sigma)\in B(M,M_0; X)\right\}$ is a complete cover of $\mathrm{Conf}_X(M,M_0)$.
\end{lemma}
\begin{proof}
We first verify that $\U$ is a cover. We begin by fixing $\sum m_ax_a\in \mathrm{Conf}_X(M,M_0)$ with $m_a\notin M_0$ and $x_a\neq x_0$ for all $a$. We now choose \begin{enumerate}
\item disjoint Euclidean neighborhoods $m_a\in U_a$;
\item a tubular neighborhood $M_0\subseteq U_*$ disjoint from $\bigcup_a U_a$;
\item contractible neighborhoods $x_a\in \sigma(U_a)$ disjoint from $x_0$; and
\item a contractible neighborhood $x_0\in \sigma(U_*)$ disjoint from $\bigcup_a \sigma(U_a)$.
\end{enumerate} Setting $U=U_*\cup\bigcup_a U_a$, we clearly have $\sum m_ax_a\in\mathrm{Conf}_X^0(U,\sigma)$.
For completeness, suppose that $\sum m_ax_a\in\bigcap_{r=1}^N\mathrm{Conf}_X^0(V_r,\tau_r)$. We make the same sequence of choices as in the previous step, while requiring that
\begin{enumerate}
\item if $m_a$ is contained in some $V_{r,i}$, then $U_a\subseteq \bigcap_{m_a\in V_{r,i}} V_{r,i}$;\item $U_*\subseteq \bigcap_{r=1}^N V_{r,*}$;
\item if $x_a$ is contained in some $\tau_r(V_{r,i})$, then $x_0\notin\sigma(U_a)\subseteq \bigcap_{x_a\in \tau_r(V_{r,i})}\tau_r(V_{r,i})$; and
\item $\sigma(U_a)\subseteq \bigcap_{r=1}^N \tau_r(V_{r,*})$.
\end{enumerate} We define a function $f:\pi_0(U,U_*)\to \pi_0(V_r, V_{r,*})$ by sending $U_a$ to the component $V_{r,i}$ such that $m_a\in V_{r,i}$, provided $m_a$ is essential for $(V_r,\tau_r)$, and to the basepoint otherwise. It is easy to check that $f$ is a well-defined inert map, and the proof is complete upon verifying that $f$ witnesses the inequality $(U,\sigma)\leq (V_r,\tau_r)$. There are two points to check.
\begin{enumerate}
\item If $f(a)\neq *$ or if $a=*$, then $U_a\subseteq V_{r,f(a)}$ and $\sigma(U_a)\subseteq \tau_r(V_{r,f(a)})$ by construction.
\item If $*\neq a\in f^{-1}(*)$, then $m_a$ is not essential for $(V_r, \tau_r)$, so either $m_a\in V_{r,*}$ or $x_a\in \tau_r(V_{r,*})$. By construction, then, either $U_a\subseteq V_{r,*}$ or $\sigma(U_a)\subseteq \tau_r(V_{r,*})$.
\end{enumerate}
\end{proof}
\begin{lemma}\label{lem:labeled contractible}
Each $\mathrm{Conf}_X^0(U, \sigma)$ is contractible.
\end{lemma}
\begin{proof}
A choice of contraction of $\sigma(U_*)$ onto $x_0$ defines a homotopy equivalence \[\mathrm{Conf}_X^0(U,\sigma)\simeq\prod_{\pi_0(U,U_*)^\circ}(U_i\times \sigma(U_i))\times \mathrm{Conf}_X(U_*, M_0).\] The second factor is contractible in view of the isotopy equivalence of pairs $(M_0, M_0)\xrightarrow{\sim} (U_*, M_0)$, and each $U_i$ and each $\sigma(U_i)$ is contractible by assumption.
\end{proof}
We write $q:\mathrm{Conf}_X(M)\to \mathrm{Conf}_X(M,M_0)$ for the map induced by $(M,\varnothing)\to (M,M_0)$.
\begin{lemma}\label{lem:labeled fiber}
For every $(U,\sigma)$, there is a homotopy equivalence $q^{-1}\mathrm{Conf}_X^0(U,\sigma)\simeq \mathrm{Conf}_X(M_0)$.
\end{lemma}
\begin{proof}
The inverse image $q^{-1}\mathrm{Conf}_X(U,\sigma)$ is the subspace of labeled configurations in $\mathrm{Conf}_X(M)$ with 1) exactly one essential point in each $U_i$, 2) all other points not lying in $U_*$ labeled by points of $\sigma(U_*)$, and 3) an arbitrary subconfiguration lying in $U_*$. Thus, a choice of contraction of $\sigma(U_*)$ onto $x_0$ defines a homotopy equivalence \[q^{-1}\mathrm{Conf}_X^0(U,\sigma)\simeq \prod_{\pi_0(U,U_*)^\circ}(U_i\times \sigma(U_i))\times\mathrm{Conf}_X(U_*).\] Since each $U_i$ and each $\sigma(U_i)$ is contractible, the claim follows from the isotopy equivalence $M_0\xrightarrow{\sim} U_*$.
\end{proof}
\begin{lemma}\label{lem:labeled locally constant}
If $X$ is connected, then the inclusion \[q^{-1}\mathrm{Conf}_X^0(U,\sigma)\to q^{-1}\mathrm{Conf}_X^0(V,\tau)\] is a homotopy equivalence whenever $(U,\sigma)\leq(V,\tau)$.
\end{lemma}
Before turning to the proof, we consider an example illustrating the necessity of the hypothesis on $X$.
\begin{example}
Set $M=D^n_6(0)$, $M_0=D_6^n(0)\setminus\mathring{D}_5^n(0)$, $X=S^0$. We define a pair $(U,\sigma)\leq (V,\tau)$ by setting \begin{align*}
U_1&=\mathring{D}^n_1(0)\\
U_2&=\mathring{D}_{1/3}(7/2,0,\ldots, 0)\\
U_*&=D^n_6(0)\setminus D_4^n(0)\\
V_1&=\mathring{D}_2^n(0)\\
V_*&=D^n_6(0)\setminus D_3^n(0).
\end{align*} Since $X=S^0$, $\sigma$ and $\tau$ are determined, and, since $U_1\subseteq V_1$ and $U_*\cup U_2\subseteq V_*$, the inert map $f:\{1,2,*\}\to \{1,*\}$ with $f(1)=1$ and $f(2)=*$ witnesses the claimed inequality. Now, in the commuting diagram
\[\xymatrix{
q^{-1}\mathrm{Conf}_{S^0}(U,\sigma)\ar[d]\ar@{=}[r]^-\sim& U_1\times U_2\times \mathrm{Conf}_{S^0}(U_*)\ar[d]&\{m_1\}\times\{m_2\}\times\displaystyle\coprod_{k\geq0}B_k(U_*)\ar[d]\ar[l]_-{\sim}\\
q^{-1}\mathrm{Conf}_{S^0}(V,\tau)\ar@{=}[r]^-\sim&V_1\times\mathrm{Conf}_{S^0}(V_*)&\{m_1\}\times\displaystyle\coprod_{k\geq0}B_k(V_*),\ar[l]_-{\sim}
}\] the point $\{m_1\}\times\{\varnothing\}$ does not lie in the image of the rightmost map; therefore, this map fails to surject on $\pi_0$.
\end{example}
\begin{definition}
Let $P$ be connected and $N\subsetneq P$ an isotopy retract. For $p\in P\setminus N$ and $x\neq x_0$, the \emph{stabilization map} with respect to $p$ and $x$ is the map \[\mathrm{Conf}_X(N)\cong \{px\}\times \mathrm{Conf}_X(N)\subseteq \mathrm{Conf}_X(D_{\epsilon}(p))\times\mathrm{Conf}_X(N)\to \mathrm{Conf}_X(M),\] where the rightmost map is the composite of the homeomorphism $\mathrm{Conf}_X(D_\epsilon(p))\times\mathrm{Conf}_X(N)\cong \mathrm{Conf}_X(D_\epsilon(p)\amalg N)$ with the structure map for the inclusion.
\end{definition}
Thus, in the homotopy category of spaces, there is a well-defined stabilization map $\mathrm{Conf}_X(P)\to \mathrm{Conf}_X(P)$ depending only on the class of $x$ in $\pi_0(X)$ and a choice of element in $\pi_0(P,N)$.
\begin{proposition}
Stabilization with respect to $p$ and $x$ is an equivalence if and only if $x$ lies in the path component of $x_0$.
\end{proposition}
\begin{proof}
If $x$ lies in the path component of $x_0$, then a path joining the two defines a homotopy from the stabilization map to the structure map for the isotopy equivalence $N\subseteq P$. Conversely, suppose that $X=X_1\amalg X_2$ with $x_0\in X_1$ and $x\in X_2$. The subspace \[\left\{{\textstyle\sum m_ax_a}\mid \{x_a\}\cap X_2\neq \varnothing\right\}\subset\mathrm{Conf}_X(P)\] is closed, open, proper, and contains the image of stabilization with respect to $p$ and $x$. It follows that stabilization does not surject on $\pi_0$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:labeled locally constant}]
Suppose that $f:\pi_0(U,U_*)\to \pi_0(V,V_*)$ witnesses the inequality $(U,\sigma)\leq(V,\tau)$, and write \[\pi_0(U,U_*)\cong *\sqcup f^{-1}(\pi_0(V,V_*)^\circ)\sqcup I\sqcup I',\] where $I=\{*\neq i\in f^{-1}(*)\mid\sigma(U_i)\subsetneq \tau(V_*)\}$ and $I'=\{*\neq i\in f^{-1}(*)\mid\sigma(U_i)\subseteq \tau(V_*)\}$. Choosing contractions of $\sigma(U_*)$ and $\tau(V_*)$ onto $x_0$ and points in $U_i\times\sigma(U_i)$ for $i\in\pi_0(U,U_*)^\circ$ induces the equivalences in the commuting diagram \[\xymatrix{
q^{-1}\mathrm{Conf}_X(U,\sigma)\ar[d]&\displaystyle\prod_I(U_i\times \sigma(U_i))\times\mathrm{Conf}_X(U_*)\ar[d]\ar[l]_-\sim&\mathrm{Conf}_X(U_*)\ar[l]_-\sim\ar[d]\\
q^{-1}\mathrm{Conf}_X(V,\tau)&\mathrm{Conf}_X(V_*)\ar[l]_-\sim&\mathrm{Conf}_X(V_*),\ar@{=}[l]
}\] where the righthand map is a composite of $|I|$ stabilization maps, each of which is an equivalence by our assumption on $X$.
\end{proof}
We now prove the theorem.
\begin{proof}[Proof of Theorem \ref{thm:exactness}]
Lemma \ref{lem:labeled poset} supplies the top arrow and Lemma \ref{lem:labeled open} the dashed filler in the commuting diagram \[\xymatrix{
\op B(M,M_0;X)\ar@{-->}[dr]\ar[rr]^-{\mathrm{Conf}_X^0}&&\mathcal{T}\mathrm{op}\\
&\mathrm{Op}(\mathrm{Conf}_X(M,M_0)).\ar[ur]
}\] By Lemma \ref{lem:labeled complete cover}, then, \[\mathrm{Conf}_X(M,M_0)\simeq \hocolim_{\op B(M,M_0;X)}\mathrm{Conf}_X^0\simeq B(\op B(M,M_0;X)),\] where the second equivalence follows from Lemma \ref{lem:labeled contractible}.
We similarly have the commuting diagram \[\xymatrix{
\op B(M,M_0;X)\ar@{-->}[dr]\ar[rr]^-{q^{-1}\mathrm{Conf}_X^0}&&\mathcal{T}\mathrm{op}\\
&\mathrm{Op}(\mathrm{Conf}_X(M)).\ar[ur]
}\] Lemma \ref{lem:labeled complete cover} implies that the collection $\left\{q^{-1}(\mathrm{Conf}_X^0(U,\sigma): (U,\sigma)\in \op B(M, M_0;X)\right\}$ is likewise a complete cover of $\mathrm{Conf}_X(M)$, so \[\mathrm{Conf}_X(M)\simeq \hocolim_{\op B(M, M_0;X)}q^{-1}\mathrm{Conf}^0_X.\] Since $X$ is connected, Lemma \ref{lem:labeled locally constant} and Corollary \ref{cor:hocolim quasifibration} grant that the diagram \[\xymatrix{
q^{-1}\mathrm{Conf}_X^0(U,\sigma)\ar[r]\ar[d]&\displaystyle\hocolim_{\op B(M,M_0;X)}q^{-1}\mathrm{Conf}_X^0\ar[d]\\
\mathrm{pt}\ar[r]^-{(U,\sigma)}&B(\op B(M,M_0;X))
}\] is homotopy Cartesian, and the desired conclusion follows from Lemma \ref{lem:labeled fiber} and the identifications already established.
\end{proof}
We close by pointing out that the same reasoning provides an analogue of the long exact sequence for a triple.
\begin{theorem}[McDuff]
Let $M_0$, $M_1$, and $M_0\cap M_1$ be closed submanifolds of codimension 0 in $M$. If either $X$ or $(M_0,M_0\cap M_1)$ is connected, then the diagram \[\xymatrix{
\mathrm{Conf}_X(M_0, M_0\cap M_1)\ar[r]\ar[d]&\mathrm{Conf}_X(M, M_1)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(M, M_0\cup M_1)
}\] is homotopy Cartesian.
\end{theorem}
\subsection{Duality for labeled configuration spaces}
We now disuss an analogue of Poincar\'{e} duality in the context of the ``homology theory'' of labeled configuration spaces. One approach to the classical isomorphism $H_*(M)\cong H_c^{n-*}(M)$ is through the following steps.
\begin{enumerate}
\item By taking a compact exhaustion, reduce to the statement $H_*(N)\cong H_c^{n-*}(N,\partial N)$ for compact manifolds with boundary.
\item Observe the local calculation $\mathbb{Z}\cong H_*(D^n)\cong H^{n-*}(D^n,\partial D^n)\cong \widetilde H^{n-*}(S^n)\cong\mathbb{Z}$.
\item Deduce the general case inductively using Mayer-Vietoris and/or exactness.
\end{enumerate}
Our approach will proceed along these same lines; first, however, we must identify the object that is to play the role of compactly supported cohomology.
\begin{recollection}
A Riemannian $n$-manifold $M$ has a principal right $O(n)$-bundle of \emph{orthonormal frames} \[\xymatrix{
\mathrm{Isom}(\mathbb{R}^n, T_pM)\ar[d]\ar[r]&\mathrm{Fr}_M\ar[d]\\
\{p\}\ar[r]&M,
}\] with fiber over $p$ the set of isometries from $\mathbb{R}^n$ with the standard inner product to the tangent fiber over $p$. The orthogonal group $O(n)$ acts on this space of isometries by precomposition. Note the $O(n)$-equivariant but non-canonical isomorphism $\mathrm{Isom}(\mathbb{R}^n, T_pM)\cong O(n)$.
\end{recollection}
We equip $M$ with a metric, whose features will be further specified in time.
\begin{construction}
Define a bundle $E_X=E_X(M)$ by \[E_X:=\mathrm{Fr}_M\times_{O(n)}\mathrm{Conf}_X(D^n,\partial D^n),\] where the action of $O(n)$ on $\mathrm{Conf}_X(D^n,\partial D^n)$ arises from the action by diffeomorphisms of $O(n)$ on the pair $(D^n,\partial D^n)$.
\end{construction}
The projection $\pi: E_X\to M$ has a canonical section defined by $s_0(p)=\varnothing$, and we may identify the fiber $\pi^{-1}(p)$ with $\mathrm{Conf}_X(D_1(T_p(M)), \partial D_1(T_pM))$. Moreover, the inclusion of the subbundle $\mathrm{Fr}_M\times_{O(n)}\mathrm{Conf}_X(D^n,\partial D^n)_{\leq 1}$ is a fiberwise homotopy equivalence, and, using the isomorphism $\mathrm{Conf}_X(D^n,\partial D^n)\cong\Sigma^n X$, which is $O(n)$-equivariant with $O(n)$ acting on the righthand side via the suspension coordiantes, we may identify this subbundle with the fiberwise smash product of $X$ with the fiberwise one point compactification of $TM$. Under this identification, the section $s_0$ is given by the fiberwise basepoint.
\begin{definition}
For a map $f:E\to B$, the \emph{space of sections} of $f$ over $A\subseteq B$ is the pullback in the diagram \[\xymatrix{
\Gamma(A;E)\ar[r]\ar[d]&\mathrm{Map}(A,E)\ar[d]^-{f_*}\\
\{A\subseteq B\}\ar[r]&\mathrm{Map}(A,B).
}\] Fixing a section $s_0$ we say that such a section $s$ has \emph{compact support} if $s|_{A\setminus K}= s_0|_{A\setminus K}$ for some compact subset $K\subseteq A$. We write $\Gamma_c(A;E)\subseteq\Gamma(A;E)$ for the subspace of compactly supported sections.
\end{definition}
Note that we have the identification \[\Gamma_c(A;E)\cong \colim_K \Gamma(A, A\setminus K;E),\] where the colimit is taken over the category of inclusions among compact subsets of $A$, and the space of relative sections $\Gamma(A,A_0;E)$ for $A_0\subseteq A$ is defined as the pullback in the diagram \[\xymatrix{
\Gamma(A,A_0;E)\ar[d]\ar[r]&\Gamma(A;E)\ar[d]^-{(A_0\subseteq A)^*}\\
\{s_0|_{A_0}\}\ar[r]&\Gamma(A_0;E).
}\]
The space $\Gamma_c(M;E_X)$ will play the role of compactly supported cohomology in our version of Poincar\'{e} duality. In order to make the comparison to the labeled configuration space $\mathrm{Conf}_X(M)$, we require a map.
\begin{recollection}
Given a Riemannian manifold $M$, a point $p\in M$, and a sufficiently small $\epsilon>0$, there is an \emph{exponential map} \[\exp_p^\epsilon:D_\epsilon(T_pM)\to M,\] which is an embedding.
\end{recollection}
We assume that $M$ is equipped with a metric such that $\exp_p^1$ is an embedding for all $p\in M$; for example, such a metric exists if $M$ is the interior of a compact manifold with boundary.
\begin{definition}
The \emph{scanning map} for $M$ and $X$ is the map \[s:\mathrm{Conf}_X(M)\to \Gamma(M;E_X)\] defined by letting $s\left(\sum m_a x_a\right)(p)$ be the image of $\sum m_ax_a$ under the composite map \begin{align*}\mathrm{Conf}_X(M)&\to\mathrm{Conf}_X(M,M\setminus \exp_p^1(\mathring{D}_1(T_pM)))\\
&\cong \mathrm{Conf}_X(\exp_p^1(D_1(T_pM)), \exp_p^1(\partial D_1(T_pM)))\\
&\cong \mathrm{Conf}_X(D_1(T_pM),\partial D_1(T_pM))\\
&\cong\pi^{-1}(p),
\end{align*} where the first map is induced by $(M,\varnothing)\to (M,M\setminus \exp_p^1(\mathring{D}_1(T_pM)))$, the second is excision, and the third is induced by the exponential map.
\end{definition}
Note that, since there are only finitely many $m_a$ with $x_a\neq x_0$, the section $s\left(\sum m_a x_a\right)$ always has compact support.
\begin{theorem}[McDuff]\label{thm:duality noncompact}
Scanning induces a weak equivalence \[\mathrm{Conf}_X(M)\to \Gamma_c(X;E_X)\] provided $X$ is connected.
\end{theorem}
As in our approach to classical Poincar\'{e} duality outlined above, we deduce this result from a version for compact manifolds with boundary $N$. In order to accommodate boundary into our scanning technique, we set $W=N\amalg_{\partial N}[0,1)\times\partial N$ and arrange as before that $\exp_p^1$ is an embedding for $p\in W$. Using a collar neighborhood of $\partial N$ in $N$, we choose an isotopy retract $N'\subseteq N$ such that $\exp_p^1(D_1(T_pN))\cap N'=\varnothing$ for $p\in \partial N\times[0,1)$, implying that the dashed filler exists in the commuting diagram \[\xymatrix{\mathrm{Conf}_X(N)\ar[r]^-{N\subseteq W}& \mathrm{Conf}_X(W) \ar[r]^-s& \Gamma(W;E_X)\\
\mathrm{Conf}_X(N')\ar[u]^-\wr\ar@{-->}[rr]&&\Gamma(W,\partial N\times[0,1);E_X)\cong \Gamma(N,\partial N;E_X).\ar[u]
}\]
\begin{theorem}[McDuff]\label{thm:duality with boundary}
Let $N$ be a compact manifold with boundary. Scanning induces a weak equivalence \[\mathrm{Conf}_X(N)\simeq \Gamma(N,\partial N;E_X)\] provided $X$ is connected.
\end{theorem}
Before turning to the proof of this result, we use it to deduce Theorem \ref{thm:duality noncompact}.
\begin{proof}[Proof of Theorem \ref{thm:duality noncompact}]
Realize $M$ as the interior of a compact manifold with boundary and choose a compact exhaustion $M=\colim_k N_k$ with each $N_k$ an isotopy retract along a collar neighborhood of the boundary. In the commuting diagram \[\xymatrix{
\mathrm{Conf}_X(M)\ar[r]&\Gamma_c(M;E_X)\\
\colim_k \mathrm{Conf}_X(N_k)\ar[u]\\
\colim_k\mathrm{Conf}_X(N_k'\ar[u])\ar[r]&\colim_k\Gamma(N_k,\partial N_k;E_X),\ar[uu]
}\] the upper left arrow is a homeomorphism; the lower left arrow is a obtained from a levelwise weak equivalence between diagrams of relatively $T_1$ inclusions, since $N_k'\subseteq N_k$ is an isotopy equivalence; the same holds for the bottom arrow by Theorem \ref{thm:duality with boundary}; and the righthand arrow is a homeomorphism, since $\Gamma(N_k,\partial N_k;E_X)\cong \Gamma(M,M\setminus N_k;E_X)$ and the diagram $\{N_k\}_{k\geq0}$ is final in the category of compact subsets of $M$. It follows that the top arrow is a weak equivalence, as desired.
\end{proof}
As in our proof schematic from before, our strategy in proving Theorem \ref{thm:duality with boundary} will be induction on a local calculation. We first clarify the nature of the induction in question.
\begin{recollection}
If $M$ is an $n$-manifold with boundary and $\varphi: \partial D^i\times D^{n-i}\to \partial M$ an embedding, then we obtain a new manifold with boundary $\overline M:= M\cup_\varphi (D^i\times D^{n-i})$, which we refer to as the result of \emph{attaching an $i$-handle} to $M$ along $\varphi$. A compact manifold with boundary may be built from the empty manifold by a finite sequence of handle attachments.
\end{recollection}
Our strategy, then, will be to proceed by induction on such a handle decomposition. Specifically, a handle attachment gives rise a homotopy pullback square \[\xymatrix{
\mathrm{Conf}_X(M)\ar[r]\ar[d]&\mathrm{Conf}_X(\overline M)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(D^i\times D^{n-i}, \partial D^i\times D^{n-i})
}\] provided $X$ is connected. Thus, if the theorem is known for $M$, then the theorem for $\overline M$ will follow by examining $\mathrm{Conf}_X(D^i\times D^{n-i}, \partial D^i\times D^{n-i})$. This logic leads us to formulate a third, relative version of the theorem.
\begin{theorem}[McDuff]\label{thm:duality relative}
Let $M$ be a compact manifold with boundary and $M_0\subseteq M$ a closed submanifold. Scanning induces a weak equivalence \[\mathrm{Conf}_X(M,M_0)\simeq \Gamma(M\setminus M_0, \partial M\setminus M_0; E_X)\] provided either $X$ or $(M,M_0)$ is connected.
\end{theorem}
We note first that it suffices to consider only those pairs $(M,M_0)$ with $M_0\subseteq \partial M$. Indeed, if $M_0\subseteq N$ is a closed tubular neighborhood, then \begin{align*}
\mathrm{Conf}_X(M,M_0)&\xrightarrow{\sim} \mathrm{Conf}_X(M, N)\\
&\cong \mathrm{Conf}_X(M\setminus \mathring{N}, \partial N),
\end{align*} and, similarly, \begin{align*}
\Gamma(M\setminus M_0, \partial M\setminus M_0)&\xrightarrow{\sim}\Gamma(M\setminus N, \partial M\setminus N)\\
&=\Gamma\left((M\setminus \mathring{N})\setminus \partial N, \partial (M\setminus \mathring{N})\setminus \partial N\right),
\end{align*} so the case of $(M,M_0)$ follows from the case of $(M\setminus \mathring{N}, \partial N)$.
In order to proceed, we must accommodate the relative case into our scanning setup.
\begin{construction} Assuming that $M_0\subseteq\partial M$, we write $\partial M=M_0\cup_{\partial M_1} M_1$, set $W=M\sqcup_{\partial M}\partial M\times[0,1)$, and let $M'$ be a retract along a collar of $M_1$ relative to $M_0$. Finally, choosing a tubular neighborhood $M_0\subseteq N\subseteq W$ such that $\exp_p^1(\mathring{D}_1(T_pM))\cap M_0=\varnothing$ for $p\notin N$, we obtain, up to homotopy, a scanning map
\[\mathrm{Conf}_X(M,M_0)\xleftarrow{\sim}\mathrm{Conf}_X(M', M_0)\to \Gamma(M\setminus N, \partial M\setminus N)\xleftarrow{\sim}\Gamma(M\setminus M_0, \partial M\setminus M_0),\] where the middle arrow is defined by the same formula as before.
\end{construction}
This construction is possible because $M_0\subseteq M\setminus \exp_p^1(\mathring{D}_1(T_pM))$ for $p\notin N$, so the required map \[\mathrm{Conf}_X(M, M_0)\to \mathrm{Conf}_X(M,M\setminus \exp_p^1(D_1(T_pM)))\cong \mathrm{Conf}_X(D_1(T_pM), \partial D_1(T_pM))\] is defined.
The key step in the proof of Theorem \ref{thm:duality relative} is the following result.
\begin{lemma}\label{lem:handle case}
For $0<i\leq n$ and any $X$, the conclusion of Theorme \ref{thm:duality relative} holds for the pair $(M,M_0)=(D^i\times D^{n-i}, \partial D^i\times D^{n-i})$. Moreover, if $X$ is connected, the conclusion also holds in the case $i=0$.
\end{lemma}
\begin{proof}
We proceed by downward induction on $i$. For the base case $i=n$, it suffices to show that the dashed restriction in the commuting diagram \[\xymatrix{
\mathrm{Conf}_X(D_1^n, \partial D_1^n)_{\leq 1}\ar[d]_-\wr\ar@{=}[r]^-\sim&\Sigma^nX\ar@{-->}[d] \\
\mathrm{Conf}_X(D_1^n,\partial D_1^n)\ar[r]&\Gamma(\mathring{D}_{1/2}^n;E_X)&\Gamma(\mathring{D}_{1}^n;E_X)\ar[l]_-\sim
}\] is a weak equivalence. For this claim, we note that radial expansion of the scanning neighborhood defines a homotopy from this restriction to the map $\Sigma^nX\to \Gamma(\mathring{D}^n_{1/2}; E_X)\cong \mathrm{Map}(\mathring{D}^n_{1/2}, \Sigma^nX)$ sending a point in $\Sigma^nX$ to the constant map to that point.
For the induction step, we write $\partial_\epsilon D^i$ for a closed collar neighborhood of $\partial D^i$, and we write $\partial_\epsilon D^i=\partial_\epsilon^+ D^i\cup \partial_\epsilon^- D^i$ with $\partial_\epsilon^+D^i\cap\partial_\epsilon^- D^i\cong \partial_\epsilon D^{i-1}\times D^1$. By exactnes, the diagram \[\xymatrix{\mathrm{Conf}_X(\partial^+_\epsilon D^i\times D^{n-i}, \partial_\epsilon D^{i-1}\times D^1\times D^{n-i})\ar[r]\ar[d]&\mathrm{Conf}_X(D^i\times D^{n-i}, \partial_\epsilon^- D^i\times D^{n-i})\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(D^i\times D^{n-i}, \partial_\epsilon D^i\times D^{n-i})
}\] is homotopy Cartesian provided $X$ is connected or $i>0$. The conclusion holds for the bottom right pair by induction, since \[\mathrm{Conf}_X(D^i\times D^{n-i}, \partial_\epsilon D^i\times D^{n-i})\simeq \mathrm{Conf}_X(D^i\times D^{n-i}, \partial D^i\times D^{n-i}),\] while for the upper right pair both the labeled configuration space and the section space are contractible. It follows that the conclusion also holds for the upper left pair, but \[\mathrm{Conf}_X(\partial^+_\epsilon D^i\times D^{n-i}, \partial_\epsilon D^{i-1}\times D^1\times D^{n-i})\cong \mathrm{Conf}_X(D^{i-1}\times D^{n-(i-1)}, \partial D^{i-1}\times D^{n-(i-1)}),\] so the claim follows.
\end{proof}
We now prove the theorem following \cite{Boedigheimer:SSMS}.
\begin{proof}[Proof of Theorem \ref{thm:duality relative}]
We first prove the claim in the special case $M_0=\partial M$. We proceed by induction on a handle decomposition of $M$, which we may take to involve no $n$-handles if no component of $M$ has empty boundary, which is the condition that $(M,\partial M)$ be connected. The base case of $M=D^n$ is known. For the induction step, we write $\overline M=M\cup_\varphi D^i\times D^{n-i}$ and assume the claim for $(M,\partial M)$. By exactness, the diagram \[\xymatrix{
\mathrm{Conf}_X(D^i\times D^{n-i}, D^i\times\partial D^{n-i})\ar[d]\ar[r]&\mathrm{Conf}_X(\overline M, \partial\overline M)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(\overline M, \partial \overline M\cup D^i\times D^{n-i})
}\] is homotopy Cartesian if $X$ is connected or if $i<n$. By excision, \[\mathrm{Conf}_X(\overline M,\partial \overline M\cup D^i\times D^{n-i})\cong\mathrm{Conf}_X(M,\partial M),\] so the claim is known for this pair by induction. Since it is also known for the upper left pair by Lemma \ref{lem:handle case}, the proof is complete in this case.
In the general case, we write $\partial M=M_0\cup_{\partial M_1} M_1$, set $\overline W=M\sqcup_{\partial M}\partial M\times[0,1]$, and note that the diagram \[\xymatrix{
\mathrm{Conf}_X(M_1\times[0,1], \partial M_1\times[0,1])\ar[r]\ar[d]&\mathrm{Conf}_X(\overline W, M_0\times[0,1])\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(\overline W, \partial M\times[0,1]).}\] is homotopy Cartesian if $X$ is connected or $(M_1,\partial M_1)$ is connected. Assuming one of these conditions to hold, excision and isotopy invariance show that \begin{align*}\mathrm{Conf}_X(\overline W, \partial M\times[0,1])&\cong\mathrm{Conf}_X(M, \partial M)\\
\mathrm{Conf}_X(\overline W, M_0\times[0,1])&\simeq \mathrm{Conf}_X(M, M_0),
\end{align*} so it suffices to prove the claim for the pair $(M_1\times[0,1],\partial M_1\times[0,1])$. To establish this last case, we use exactness once more, invoking our assumed connectivity of $X$ or $(M_1,\partial M_1)$, to obtain the homotopy pullback square \[\xymatrix{
\mathrm{Conf}_X(M_1\times[0,1], \partial M_1\times[0,1])\ar[d]\ar[r]&\mathrm{Conf}_X(M_1\times[0,2], \partial (M_1\times[0,2])\setminus \mathring{M_1}\times\{0\})\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(M_1\times [0,2], M_1\times[0,2]\setminus \mathring{M}_1\times(1,2)).
}\] The upper right entry and the corresponding section space are contractible, and the case of a manifold relative to its boundary shows that the theorem holds for the bottom right pair, since \[\mathrm{Conf}_X(M_1\times [0,2], M_1\times[0,2]\setminus \mathring{M}_1\times(1,2))\cong\mathrm{Conf}_X(M_1\times[1,2],\partial (M_1\times [1,2]))\] by excision. Thus, the proof is complete under the assumption that $X$ or $(M_1,\partial M_1)$ is connected.
In case $X$ and $(M_1,\partial M_1)$ are both disconnected, we let $N$ denote the union of the components of $\partial M$ not intersecting $M_0$, and we set $\widetilde M=M\cup_N M$. Exactness implies that the diagram \[\xymatrix{
\mathrm{Conf}_X(M,M_0)\ar[d]\ar[r]&\mathrm{Conf}_X(\widetilde M, M_0\sqcup M_0)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(\widetilde M, M\sqcup M_0)
}\] is homotopy Cartesian, since $(M,M_0)$ is connected by assumption. Since the theorem holds for the two righthand pairs by what has already been shown, it also holds for $(M,M_0)$.
\end{proof}
\subsection{Postmortem}Before moving on from the subject of Poincar\'{e} duality for labeled configuration spaces, we pause to give an informal discussion of several generalizations and continuations of this story.
\begin{perspective}
We have shown that, for connected $X$, there is a weak equivalence \[\mathrm{Conf}_X(D^n)\xrightarrow{\sim}\Omega^n\Sigma^nX.\] In fact, this equivalence may be upgraded to an equivalence of $E_n$-\emph{algebras} \cite{May:GILS}. Since the domain of this map is the free $E_n$-algebra on $X$ in the category of pointed spaces under Cartesian product, monadicity considerations show that the homotopy theory of connected $E_n$-algebras is equivalent to that of connected $n$-fold loop spaces. This equivalence can be further upgraded to an equivalence of the homotopy theory of all $n$-fold loop spaces with that of \emph{grouplike} $E_n$-algebras, i.e., those whose connected components form a group.
\end{perspective}
\begin{perspective}
For connected $X$, we have the equivalence \[\mathrm{Conf}_X(M)\xrightarrow{\sim}\Gamma_c(M;E_X)=\Gamma(M^+,\infty; E_X)\] as well as the dual equivalence \[\mathrm{Conf}_X(M^+,\infty):=\colim_K\mathrm{Conf}_X(K,\partial K)\xrightarrow{\sim}\colim_K\Gamma(\mathring{K};E_X)\cong \Gamma(M;E_X).\] Thus, the duality of the ``manifolds'' $M$ and $M^+$ mirrors the duality of the ``coefficient systems'' $\Omega^n\Sigma^nX$ and $\Sigma^nX$, a type of algebra and coalgebra, respectively, which is an algebraic phenomenon known as \emph{Koszual duality.} This observation has a generalization in terms of factorization (co)homology in the form of a ``Poincar\'{e}/Koszul duality'' map \[\int_Z A\to \int^{Z^\neg}B^nA,\] where $Z$ is a \emph{zero-pointed manifold} and $Z^\neg$ its \emph{negation}---in our case, $(M_+)^\neg=M^+$ and $(M^+)^\neg=M_+$. Like the scanning map it generalizes, this map is an equivalence under connectivity assumptions \cite{AyalaFrancis:PKD}.
\end{perspective}
\begin{perspective}
It is natural to wonder what can be said about the scanning map \[\mathrm{Conf}_X(M)\to \Gamma(M,\partial M; E_X)\] for disconnected $X$, particularly since our motivating example was the case $X=S^0$. The key to answering this question is the observation that, in the local case $M=D^n$, both the labeled configuration space and the section space carry the structure of a monoid up to homotopy---indeed, as we noted above, both are $E_n$-algebras---and the scanning map respects this structure. Like a group, a monoid has a classifying space, and, at least in the case of connected $X$, the homotopy pullback square \[\xymatrix{\mathrm{Conf}_X(D^n)\ar[d]\ar[r]&\mathrm{Conf}_X(D^{n-1}\times(D^1,\{1\}))\simeq\mathrm{pt}\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_X(D^{n-1}\times (D^1,\partial D^1))
}\] quaranteed by exactness shows that \[B\mathrm{Conf}_X(D^n)\simeq \mathrm{Conf}_X(D^{n-1}\times(D^1,\partial D^1))\overset{?}{\simeq} \mathrm{Conf}_{\mathrm{Conf}_X(D^1,\partial D^1)}(D^{n-1})\simeq\mathrm{Conf}_{\Sigma X}(D^{n-1}),\] and, in fact, this equivalence holds for any $X$ \cite[2.1]{Segal:CSILS}. Since $\Gamma(D^n, \partial D^n;E_X)\cong\Omega^n\Sigma^nX$ is grouplike, it follows from the group completion theorem \cite[1]{McDuffSegal:HFGCT} that we have the induced isomorphism \[H_*(\mathrm{Conf}_X(D^n))[\pi_0^{-1}]\cong H_*(\Gamma(D^n,\partial D^n;E_X)),\] giving a precise answer to the question of how far the scanning map is from a weak equivalence, at least in this local case. Note that $\pi_0$ acts on homology by stabilization, so, taking $X=S^0$, this isomorphism may be interpreted as the statement that \[\colim_k H_i(B_k(D^n))\cong H_i(\Omega^n_0\Sigma^n)\] for $i>0$, where $\Omega^n_0\Sigma^n\subseteq \Omega^n\Sigma^n$ is the degree zero component.
In the general case, assuming $\partial M\neq\varnothing$, we proceed by choosing a small disk $D$ sharing part of its boundary with $M$ and considering the diagram \[\xymatrix{
\mathrm{Conf}_X(D)\ar[r]\ar[d]& \Gamma(D,\partial D;E_X)\ar[d]\\
\mathrm{Conf}_X(M)\ar[r]\ar[d]_-q& \Gamma(M,\partial M;E_X)\ar[d]\\
\mathrm{Conf}_X(M,D)\ar[r]^-\sim&\Gamma(M\setminus D, \partial M\setminus D; E_X).
}\] Taking the homotopy colimit over stabilization maps corresponding to various components of $X$, the top arrow becomes an isomorphism on homology, and, while the map from $\mathrm{Conf}_X(D)$ to the homotopy fiber is not a weak equivalence, it does becomes a homology isomorphism in the limit.
One way to see this is by applying the following result, which is proved in exactly the same manner as Corollary \ref{cor:hocolim quasifibration}.
\begin{proposition}
If $F:I\to \mathrm{Top}$ is a functor sending each morphism in $I$ to a homology isomorphism, then the diagram \[\xymatrix{
F(i)\ar[r]\ar[d]&\hocolim_IF\ar[d]\\
\mathrm{pt}\ar[r]^-i&BI
}\] of spaces is homology Cartesian for every object $i\in I$.
\end{proposition}
In the scenario at hand, we apply this result to the functor \begin{align*}
\op B(M, D; X)&\to \mathcal{T}\mathrm{op}\\
(U,\sigma)&\mapsto(\hocolim q)^{-1}\mathrm{Conf}^0_X(U,\sigma).
\end{align*} The previous local group completion calculation shows that the hypothesis is satisfied. Finally, the Serre spectral sequence implies that scanning induces an isomorphism \[H_*(\hocolim \mathrm{Conf}_X(M))\cong H_*(\Gamma(M,\partial M; E_X)).\] In particular, the homology of $B_k(M)$ converges to that of a component of $\Gamma(M,\partial M; E_{S^0}),$ and, with more care, one can show that each $H_i(B_k(M))$ eventually stabilizes; one says that configuration spaces of open manifolds exhibit \emph{homological stability}. What is more surprising is that configuration spaces of closed manifolds also exhibit homological stability, at least rationally \cite{Church:HSCSM, RandalWilliams:HSUCS}.
\end{perspective}
\section{Homology calculations from stable splittings}\label{section:homology from splittings}
The goal of this section is the computation of the rational homology of the unordered configuration spaces of a large class of manifolds.
\begin{theorem}\label{thm:odd homology}
If $M$ is an odd dimensional compact manifold with boundary and $\mathbb{F}$ is a field of characteristic 0, then \[H_*(B_k(M))\cong \mathrm{Sym}^k\left(H_*(M)\right).\]
\end{theorem}
\begin{remark}
Since configuration spaces are isotopy invariant, this computation is valid for any odd-dimensional manifold of finite type.
\end{remark}
\begin{remark}
With slightly more care, it is not hard to show that this isomorphism is induced by the canonical inclusion $B_k(M)\to \mathrm{Sym}^k(M)$.
\end{remark}
\begin{remark}
Similar, but more complicated, statements are available for $\mathbb{F}_p$, and may be derived in essentially the same way, given the requisite knowledge of the mod $p$ homology of the configuration spaces of $\mathbb{R}^n$. In the case $p=2$, one can eliminate the requirement that $n$ be odd.
\end{remark}
Our plan is to exploit the natural filtration $\mathrm{Conf}_X(M)=\bigcup_{k\geq0}\mathrm{Conf}_X(M)_{\leq k}$, whose successive quotients are identified as
\[\faktor{\mathrm{Conf}_X(M)_{\leq k}}{\mathrm{Conf}_X(M)_{\leq k-1}}\cong \mathrm{Conf}_k(M)_+\wedge_{\Sigma_k}X^{\wedge k}.\] In the case $X=S^m$, these quotients are Thom spaces of vector bundles on the configuration spaces $B_k(M)$, so, by the Thom isomorphism, the primary obstruction to extracting the homology of $B_k(M)$ from the homology of $\mathrm{Conf}_{S^m}(M)\simeq \Gamma_c(M; E_{S^m})$ is the difference between $\mathrm{Conf}_{S^m}(M)$ and the graded space associated to the cardinality filtration. Happily, a result of \cite{CohenMayTaylor:SCSC,Boedigheimer:SSMS} guarantees that, at the level of homology, this difference is no difference at all.
\subsection{Stable splitting}\label{section:stable splittings}
In this section, we prove the following result. See Appendix \ref{appendix:Spanier--Whitehead} for an explanation of unfamiliar terms.
\begin{theorem}[B\"{o}digheimer, Cohen-May-Taylor]\label{thm:stable splitting}
There is a filtered stable weak equivalence \[\mathrm{Conf}_X(M,M_0)\xrightarrow{\sim_s}\bigvee_{k\geq1}\mathrm{Conf}_k(M,M_0)\wedge_{\Sigma_k}X^{\wedge k}\] for any manifold $M$, submanifold $M_0$, and pointed CW complex $X$.
\end{theorem}
Since filtered stable weak equivalences induce isomorphisms on homology by Corollary \ref{cor:filtered stable weak equivalence homology}, this result will be sufficient to allow us to proceed with our computation.
There are other lenses through which to view this result aside from this application to configuration spaces; indeed, one of its appealing features is that, through the scanning map, it allows information to flow equally in the other direction, implying stable splittings for many familiar mapping spaces.
\begin{example}[Snaith]
For $X$ connected, we have \[\Omega^n\Sigma^nX\xleftarrow{\sim} \mathrm{Conf}_X(D^n)\xrightarrow{\sim_s} \bigvee_{k\geq1}\mathrm{Conf}_k(D^n)_+\wedge_{\Sigma_k}X^{\wedge k}.\]
\end{example}
\begin{example}[Goodwillie]
For $X$ connected, we have \[\Lambda\Sigma X\xleftarrow{\sim} \mathrm{Conf}_X(S^1)\xrightarrow{\sim_s} \bigvee_{k\geq1}S^1_+\wedge_{C_k}X^{\wedge k}.\]
\end{example}
\begin{example}[\cite{Boedigheimer:SSMS}]
Let $K$ be a finite CW complex. For connected $X$, the mapping space $\mathrm{Map}(K,\Sigma^mX)$ splits stably for sufficiently large $m$. Indeed, choosing $M\simeq K$ with $M$ a parallelizable $m$-manifold, we have \begin{align*}
\mathrm{Map}(K,\Sigma^mX)&\simeq \mathrm{Map}(\mathring{M}, \Sigma^mX)\\
&\simeq\Gamma(\mathring{M}; E_X)\\
&\simeq \mathrm{Conf}_X(M,\partial M)\\
&\simeq_s \bigvee_{k\geq1}\mathrm{Conf}_k(M,\partial M)\wedge_{\Sigma_k} X^{\wedge k}.
\end{align*}
\end{example}
For the remainder of this section, we set $V_k=\bigvee_{i=1}^k\mathrm{Conf}_i(M,M_0)\wedge_{\Sigma_i}X^{\wedge i}$. To begin with, we seek to find a non-decreasing sequence $\{r_k\}_{k\geq1}$ of non-negative integers and a collection of maps $\Sigma^{r_k}\mathrm{Conf}_X(M,M_0)_{\leq k}\to \Sigma^{r_k} V_k$ making each of the diagrams \[\xymatrix{
\Sigma^{r_{k}}\mathrm{Conf}_X(M,M_0)_{\leq k}\ar[r]& \Sigma^{r_{k}} V_{k}\\
\Sigma^{r_{k}}\mathrm{Conf}_X(M,M_0)_{\leq k-1}\ar[u]\ar[r]& \Sigma^{r_{k}} V_{k-1}\ar[u]
}\] commute up to homotopy. Using the adjunction between loops and suspension and the fact that, by scanning, \[\Omega^r\Sigma^rA\simeq \mathrm{Conf}_A(\mathbb{R}^r)\implies \Omega^\infty\Sigma^\infty A\simeq \mathrm{Conf}_A(\mathbb{R}^\infty),\] it will suffice to produce the commuting diagram \[\xymatrix{
\mathrm{Conf}_X(M,M_0)\ar[r]^-\Phi&\mathrm{Conf}_V(\mathbb{R}^\infty)\\
\vdots\ar[u]&\vdots\ar[u]\\
\mathrm{Conf}_X(M,M_0)_{\leq k}\ar[u]\ar[r]^-{\Phi_k}&\mathrm{Conf}_{V_k}(\mathbb{R}^{r_k})\ar[u]\\
\mathrm{Conf}_X(M,M_0)_{\leq k-1}\ar[r]^-{\Phi_{k-1}}\ar[u]&\mathrm{Conf}_{V_{k-1}}(\mathbb{R}^{r_{k-1}})\ar[u]\\
\vdots\ar[u]&\vdots\ar[u]
}\]
\begin{construction}\label{construction:power set map}
Choose an embedding $\varphi:\coprod_{k\geq0}B_k(M)\to \mathbb{R}^\infty$ such that $\varphi|_{B_k(M)}$ factors through $\mathbb{R}^{r_k}$. We define $\Phi:\mathrm{Conf}_X(M,M_0)\to \mathrm{Conf}_V(\mathbb{R}^\infty)$ by the formula \[\Phi\left(\sum_{a\in A}m_ax_a\right)=\sum_{\varnothing\neq I\subseteq A}\varphi\left(\{m_a\}_{a\in I}\right)\cdot\left[\sum_{a\in I}m_ax_a\right]_{|I|},\] where $[-]_k:\mathrm{Conf}_X(M,M_0)_{\leq k}\to \mathrm{Conf}_k(M,M_0)\wedge_{\Sigma_k}X^{\wedge k}=V_k/V_{k-1}$ is the quotient map.
\end{construction}
\begin{lemma}
The map $\Phi$ is well-defined.
\end{lemma}
\begin{proof}
First, we note $\varphi(\{m_a\}_{a\in I}\neq \varphi(\{m_a\}_{a\in J}$ for $I\neq J$, so the set $\left\{\varphi(\{m_a\}_{a\in I})\right\}_{\varnothing\neq I\subseteq A}$ does in fact define an element of $B_{2^{|A|}-1}(\mathbb{R}^\infty)$. Indeed, if $I\neq J$, then $\{m_a\}_{a\in I}\neq \{m_a\}_{a\in J}$, since $m_a\neq m_b$ if $a\neq b$, and $\varphi$ is injective.
Second, we verify that the definition of $\Phi$ is independent of the choice of representative of a labeled configuration by an expression of the form $\sum_{a\in A}m_ax_a$. Indeed, any two representatives differ by a term of the form $m_bx_b$ with $m_b\in M_0$ or $x_b=x_0$, and $\left[\sum_{a\in I} m_ax_a\right]_{|I|}=0$ whenever $b\in I$; therefore, we have \begin{align*}\Phi\left(\sum_{a\in A}m_ax_a\right)&=\sum_{\varnothing\neq I\subseteq A\setminus \{b\}}\varphi\left(\{m_a\}_{a\in I}\right)\cdot\left[\sum_{a\in I}m_ax_a\right]_{|I|}+\sum_{b\in I\subseteq A}\varphi\left(\{m_a\}_{a\in I}\right)\cdot0\\&=\sum_{\varnothing\neq I\subseteq A\setminus \{b\}}\varphi\left(\{m_a\}_{a\in I}\right)\cdot\left[\sum_{a\in I}m_ax_a\right]_{|I|}\\
&=\Phi\left(\sum_{b\neq a\in A}m_ax_a\right),
\end{align*} as claimed.
\end{proof}
Note that, by construction, $\Phi|_{\mathrm{Conf}_k(M,M_0)_{\leq k}}$ factors through a map $\Phi_k:\mathrm{Conf}_k(M,M_0)_{\leq k}\to \mathrm{Conf}_{V_k}(\mathbb{R}^{r_k})$ as in the diagram displayed above.
\begin{lemma}\label{lem:phi compatibility}
For each $k\geq1$, the diagram \[\xymatrix{
\mathrm{Conf}_X(M,M_0)_{\leq k}\ar[d]_-{[-]_k}\ar[rr]^-{\Phi_k}&&\mathrm{Conf}_{V_k}(\mathbb{R}^{r_k})\ar[d]\\
V_k/V_{k-1}\ar@{=}[r]&\mathrm{Conf}_{V_k/V_{k-1}}(\mathbb{R}^0)\ar[r]&\mathrm{Conf}_{V_k/V_{k-1}}(\mathbb{R}^{r_k})
}\] commutes up to homotopy. In the case $k=1$, we adopt the convention that $V_0$ is the basepoint, so that $V_1/V_0=V_1$.
\end{lemma}
\begin{proof}
The clockwise composite is the map \[\sum_{a\in A}m_ax_a\mapsto \begin{cases}
\varphi\left(\{m_a\}_{a\in A}\right)\cdot\left[\sum_{a\in A}m_ax_a\right]_k&\quad |A|=k\text{ and }m_a\notin M_0,\, x_a\in x_0 \text{ for } a\in A\\
0&\quad\text{otherwise.}
\end{cases}\] Since $\varphi$ is homotopic to the constant map to the origin, the claim follows.
\end{proof}
The last ingredient is the following.
\begin{lemma}\label{lem:labeled cofibrations}
For any manifold $M$, submanifold $M_0$, and pointed CW complex $X$, the inclusion $\mathrm{Conf}_X(M,M_0)_{\leq k-1}\to \mathrm{Conf}_X(M,M_0)_{\leq k}$ is a Hurewicz cofibration.
\end{lemma}
Before proving this fact, we pause to deduce the theorem.
\begin{proof}[Proof of Theorem \ref{thm:stable splitting}]
By Lemma \ref{lem:phi compatibility}, we have the homotopy commutative diagram
\[\xymatrix{
\mathrm{Conf}_X(M,M_0)_{\leq 1}\ar@{=}[d]\ar[rr]^-{\Phi_1}&&\mathrm{Conf}_{V_1}(\mathbb{R}^{r_1})\ar@{=}[d]\\
V_1\ar@{=}[r]&\mathrm{Conf}_{V_1}(\mathbb{R}^0)\ar[r]&\mathrm{Conf}_{V_1}(\mathbb{R}^{r_1}),
}\] so the composite $\mathrm{Conf}_X(M,M_0)_{\leq 1}\to \mathrm{Conf}_{V_1}(\mathbb{R}^{r_1})\simeq \Omega^{r_1}\Sigma^{r_1}V_1$ is homotopic to the unit map. It follows that the induced map $\mathrm{Conf}_X(M,M_0)_{\leq 1}\to V_1$ in the Spanier--Whitehead category is the identity and in particular an isomorphism. Next, using the same lemma, together with the commutativity of the large diagram above, we find that the induced diagram \[\xymatrix{
\mathrm{Conf}_X(M,M_0)_{\leq k-1}\ar[d]\ar[r]&\mathrm{Conf}_X(M,M_0)_{\leq k}\ar[d]\ar[r]&V_k/V_{k-1}\ar@{=}[d]\\
V_{k-1}\ar[r]&V_k\ar[r]&V_k/V_{k-1}
}\] in the Spanier--Whitehead category commutes. The lefthand vertical arrow is an isomorphism by induction, so Lemma \ref{lem:five lemma} will allow us to conclude that the middle arrow is an isomorphism as long as we are assured that the top row is a cofiber sequence, which follows from Lemma \ref{lem:labeled cofibrations}. Thus, the adjoints of the maps $\mathrm{Conf}_X(M,M_0)_{\leq k}\xrightarrow{\Phi_k} \mathrm{Conf}_{V_k}(\mathbb{R}^{r_k})\simeq \Omega^{r_k}\Sigma^{r_k}V_k$ constitute the data of a filtered stable weak equivalence, as claimed.
\end{proof}
\begin{remark}
The proof of Theorem \ref{thm:stable splitting} given here is the classical one following \cite{Boedigheimer:SSMS} and \cite{CohenMayTaylor:SCSC}. More modern arguments have since become available. For one such agument, which uses hypercover-type techniques like those of \S\ref{section:covering theorems}, see \cite{Bandklayder:SSNPVNPD}.
\end{remark}
We turn now to the proof of Lemma \ref{lem:labeled cofibrations}. Recall that the condition for $A\to B$ to be a Hurewicz cofibration is the existence of a solution to all lifting problems of the form \[\xymatrix{
A\ar[d]\ar[r]&Y^I\ar[d]\\
B\ar@{-->}[ur]\ar[r]& Y,
}\] where $Y$ is an arbitrary topological space.
\begin{definition}
A pair of spaces $(B,A)$ is a \emph{neighborhood deformation retract} (NDR) pair if there exist maps $f:B\to I$ and $H:B\times I\to B$ such that \begin{enumerate}
\item $f^{-1}(0)=A$,
\item $H|_{A\times I}=\id_{A\times I}$, and
\item $H(b,1)\in A$ if $f(b)<1$.
\end{enumerate}
\end{definition}
We record the following standard facts about NDR pairs---see \cite[A]{May:GILS}, for example. As a matter of notation, given a subspace $A\subseteq B$, we write \[Z_k(A):=\left\{(x_1,\ldots, x_k)\in B^k: \{x_i\}_{i=1}^k\cap A\neq \varnothing\right\}\subseteq B^k.\]
\begin{proposition}\label{prop:NDR facts}
\begin{enumerate}
\item If $B$ is Hausdorff, then the inclusion of the closed subspace $A$ is a Hurewicz cofibration if and only if $(B,A)$ is an NDR pair.
\item If $(B,A)$ and $(B',A')$ are NDR pairs, then so is $(B\times B', B\times A'\cup A\times B')$.
\item If $(B,A)$ is an NDR pair, then $(B^k, Z_k(A))$ is a $\Sigma_k$-equivariant NDR pair.
\end{enumerate}
\end{proposition}
\begin{proof}[Proof of Lemma \ref{lem:labeled cofibrations}]
Since $M_0\subseteq M$ is a submanifold and $x_0\in X$ is a 0-cell, both $(M,M_0)$ and $(X,x_0)$ are NDR pairs. Thus, by Proposition \ref{prop:NDR facts}(3), $(M^k, Z_k(M_0))$ and $(X^k, x_0^k)$ are $\Sigma_k$-equivariant NDR pairs, and, by means of a tubular neighborhood, we may further arrange that the map $H_t:M^k\to M^k$ is injective for each $0\leq t<1$; therefore, $(\mathrm{Conf}_k(M), Z_k(M_0)\cap \mathrm{Conf}_k(M))$ is a $\Sigma_k$-equivariant NDR pair. Thus, by point (2), the pair \[\left(\mathrm{Conf}_k(M)\times X^k, \mathrm{Conf}_k(M)\times Z_k(x_0)\cup \left(Z_k(M_0)\cap \mathrm{Conf}_k(M)\right)\times X^k\right)\] is an NDR pair. It follows that the top arrow in the pushout diagram \[\xymatrix{
\mathrm{Conf}_k(M)\times_{\Sigma_k} Z_k(x_0)\cup \left(Z_k(M_0)\cap \mathrm{Conf}_k(M)\right)\times_{\Sigma_k} X^k\ar[r]\ar[d]&\mathrm{Conf}_k(M)\times_{\Sigma_k}X^k\ar[d]\\
\mathrm{Conf}_X(M,M_0)_{\leq k-1}\ar[r]&\mathrm{Conf}_X(M,M_0)_{\leq k}
}\] is the inclusion of an NDR pair and hence a Hurewicz cofibration, which implies that the bottom arrow is a Hurewicz cofibration as well.
\end{proof}
\subsection{Homology decomposition}
Having completed the proof of the theorem on stable splittings, our next task is to examine its implications for the homology of configuration spaces. Fix a field $\mathbb{F}$ and write $H_*=H_*(-;\mathbb{F})$. We will prove the following theorem of \cite{BoedigheimerCohenTaylor:OHCS}.
\begin{theorem}[B\"{o}digheimer-Cohen-Taylor]\label{thm:labeled conf homology}
Let $M$ be a compact manifold, possibly with boundary. For any $r>1$, there is an isomorphism of bigraded vector spaces \[H_*(\mathrm{Conf}_{S^r}(M))\cong \bigotimes_{i=0}^nH_*(\Omega^{n-i}S^{n+r})^{\otimes\dim H_i(M)}\]
provided $r+n$ is odd or $\mathbb{F}$ is of characteristic 2.
\end{theorem}
The auxiliary grading referred to in the statement is determined on the lefthand side by the isomorphism \[H_*(\mathrm{Conf}_{S^r}(M))\cong\bigoplus_{k\geq1} H_*(\mathrm{Conf}_k(M)_+\wedge_{\Sigma_k}S^{rk})\] and on the righthand side by the isomorphisms \begin{align*}H_*(\Omega^{n-i}S^{n+r})&\cong H_*(\mathrm{Conf}_{S^r}(D^{i}\times D^{n-i}, \partial D^i\times D^{n-i}))\\&\cong \bigoplus_{k\geq1}H_*(\mathrm{Conf}_k(S^i\wedge D^{n-i}_+, *)\wedge_{\Sigma_k}S^{rk})\end{align*} provided by scanning and stable splitting.
\begin{remark}
The obvious relative statement relating $\mathrm{Conf}_{S^r}(M,M_0)$ and $H_*(M,M_0)$ is also true, but we will not need to use it. For the necessary alterations in the argument, see \cite[3.5]{BoedigheimerCohenTaylor:OHCS}.
\end{remark}
\begin{remark}
By a different argument, the theorem can also be shown to hold in the case $r=1$---see \cite[4.5]{BoedigheimerCohenTaylor:OHCS}. The theorem is false for $r=0$.
\end{remark}
For the remainder of the lecture, we implicitly assume that the hypotheses of the theorem are satisfied.
The strategy will be to proceed by induction on a handle deccomposition of $M$ using exactness and the Serre spectral sequence. We may assume that $M$ is connected; then, in the base case $M=D^n$, we have $\mathrm{Conf}_{S^r}(D^n)\simeq\Omega^nS^{n+r},$ and the theorem holds without assumption on the parity of $r+n$ or the characteristic of the field. For the induction step, we write $\overline M=M\cup_\varphi D^i\times D^{n-i}$ with $i>0$ and consider the following two cases.
Assume first that $H_i(\overline M)$ surjects onto $H_i(\overline M,M)\cong \widetilde H_i(S^i)$. Since $r>0$, the sphere $S^r$ is connected, so exactness provides us with the homotopy pullback square
\[\xymatrix{\mathrm{Conf}_{S^r}(M)\ar[d]\ar[r]&\mathrm{Conf}_{S^r}(\overline M)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_{S^r}(\overline M, M).
}\] In the lower left corner, we have \begin{align*}
\mathrm{Conf}_{S^r}(\overline M, M)&\cong \mathrm{Conf}_{S^r}(D^i\times D^{n-i}, \partial D^i\times D^{n-i})\\
&\simeq \mathrm{Map}\left(\mathring{D}^i\times D^{n-i}, \mathring{D}^i\times \partial D^{n-i}, S^{n+r}\right)\\
&\simeq \Omega^{n-i}S^{n+r},
\end{align*} which is simply connected, since \[\pi_1(\Omega^{n-i}S^{n+r})\cong\pi_{n-i+1}(S^{n+r})=0\] for $n+r\geq n+1>n-i+1$. Thus, the action on the homology of the homotopy fiber is trivial. Since we are working over a field, we conclude that, in the homology Serre spectral sequence for this homotopy pullback, we have \begin{align*}
E^2&\cong H_*(\Omega^{n-i}S^{n+r})\otimes H_*(\mathrm{Conf}_{S^r}(M))\\
&\cong H_*(\Omega^{n-i}S^{n+r})\otimes\bigotimes_{j=0}^nH_*(\Omega^{n-j}S^{n+r})^{\otimes\dim H_j(M)}\\
&\cong \bigotimes_{i=0}^nH_*(\Omega^{n-i}S^{n+r})^{\otimes\dim H_i(\overline{M})},
\end{align*} where the second isomorphism uses the inductive hypothesis and the third our assumption on the homology of $\overline M$. Thus, in order to prove the theorem for $\overline M$, it will suffice to show that the Serre spectral sequence collapses at $E^2$.
Assume instead that the map $H_i(\overline M)\to H_i(\overline M, M)$ is trivial. Proceeding as before, and looping the resulting homotopy pullback once, we obtain the homotopy pullback square \[\xymatrix{
\Omega\,\mathrm{Conf}_{S^r}(\overline M, M)\ar[d]\ar[r]&\mathrm{Conf}_{S^r}(M)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_{S^r}(\overline M).
}\] We now use the assumption that $r>1$.
\begin{lemma}
For $r>1$, $\mathrm{Conf}_{S^r}(\overline M)$ is simply connected.
\end{lemma}
\begin{proof}
We again proceed by induction over a handle decomposition. Assuming the claim is known for $M$, the exact sequence
\[\cdots\to \pi_1(\mathrm{Conf}_{S^r}(M))\to \pi_1(\mathrm{Conf}_{S^r}(\overline M))\to \pi_1(\Omega^{n-i}S^{n+r})\to \cdots\] implies the claim for $\overline M$. For the base case, we note that \[\pi_1(\mathrm{Conf}_{S^r}(D^n))\cong \pi_1(\Omega^nS^{n+r})\cong \pi_{n+1}(S^{n+r})=0,\] since $r>1$.
\end{proof}
Thus, in this second case as well, the action on the homology of the homotopy fiber is trivial, and we have \begin{align*}
E^2\cong H_*(\mathrm{Conf}_{S^r}(\overline M))\otimes H_*(\Omega^{n-i+1}S^{n+r})\implies H_*(\mathrm{Conf}_{S^r}(M)\cong \bigotimes_{j=0}^nH_*(\Omega^{n-j}S^{n+r})^{\otimes\dim H_j(M)},
\end{align*} where the inductive hypothesis has been used in identifying the $E^\infty$ page. Thus, in order to prove the theorem for $\overline M$, it will again suffice to show that the Serre spectral sequence collapses at $E^2$.
Assuming the collapse of these spectral sequences, the final step in the proof of Theorem \ref{thm:labeled conf homology} is to identify the extra gradings on each side. In order to do so, we note that the maps \[\mathrm{Conf}_{S^r}(M)\to \mathrm{Conf}_{S^r}(\overline M)\to \mathrm{Conf}_{S^r}(\overline M,M)\] are compatible with the respective filtrations by cardinality, so this filtration gives rise to an third grading at the level of the Serre spectral sequence. Since this grading is precisely the auxiliary grading referred to in the statement of the theorem, and since the isomorphism in question was established by observing the spectral sequence to collapse, it follows that the isomorphism is compatible with this auxiliary grading.
In summary, we have reduced the theorem to the following result.
\begin{lemma}\label{lem:collapse lemma}
Let $\overline M=M\cup_\varphi D^i\times D^{n-i}$. \begin{enumerate}
\item If $H_i(\overline M)\to H_i(\overline M,M)$ is surjective, then the homological Serre spectral sequence for the fiber sequence \[\mathrm{Conf}_{S^r}(M)\to \mathrm{Conf}_{S^r}(\overline M)\to \mathrm{Conf}_{S^r}(\overline M,M)\] collapses at $E^2$.
\item If $H_i(\overline M)\to H_i(\overline M,M)$ is zero, then the homological Serre spectral sequence for the fiber sequence \[ \Omega\,\mathrm{Conf}_{S^r}(\overline M,M)\to\mathrm{Conf}_{S^r}(M)\to \mathrm{Conf}_{S^r}(\overline M)\] collapses at $E^2$.
\end{enumerate}
\end{lemma}
We will prove part (1) only, as the proof of part (2) is nearly identical---see \cite[3.4]{BoedigheimerCohenTaylor:OHCS} for full details. Our strategy in proving this result will be to relate the spectral sequence in question to a second spectral sequence, whose collapse will be easier to prove.
Recall from the proof of Theorem \ref{thm:stable splitting} that we have the map \[\eta_M:\mathrm{Conf}_{S^r}(M)\xrightarrow{\Phi}\mathrm{Conf}_V(\mathbb{R}^\infty)\to \mathrm{Conf}_{M_+\wedge S^r}(\mathbb{R}^\infty)\simeq \Omega^\infty\Sigma^\infty(M_+\wedge S^r),\] whose adjoint we might think of conceptually as comparing the ``homology theory'' of labeled configuration spaces to ordinary homology. In order to exploit this map, we require the following simple observation.
\begin{lemma}\label{lem:Q and fiber sequences}
If $A\to B\xrightarrow{f} C$ is a cofiber sequence, then \[\Omega^\infty\Sigma^\infty A\to \Omega^\infty\Sigma^\infty B\to \Omega^\infty\Sigma^\infty C\] is a fiber sequence.
\end{lemma}
\begin{proof}
From our previous observation that $\pi_i\circ \Omega^\infty\Sigma^\infty\cong\pi_i^s$, the long exact sequence in stable homotopy associated to a cofiber sequence, and the long exact sequence in homotopy associated to a fiber sequence, we obtain the commuting diagram \[\xymatrix{
\cdots\ar[r]&\pi_{i+1}(\Omega^\infty\Sigma^\infty C)\ar@{=}[d]\ar[r]&\pi_i(\Omega^\infty\Sigma^\infty A)\ar[d]\ar[r]& \pi_i(\Omega^\infty\Sigma^\infty B)\ar@{=}[d]\ar[r]&\cdots\\
\cdots\ar[r]&\pi_{i+1}(\Omega^\infty\Sigma^\infty C)\ar[r]&\pi_i(\mathrm{hofiber}\,f)\ar[r]&\pi_i(\Omega^\infty\Sigma^\infty B)\ar[r]&\cdots
}\] with exact rows. It follows from the five lemma that the canonical map from $\Omega^\infty\Sigma^\infty A$ to the homotopy fiber is a weak equivalence, as claimed.
\end{proof}
Therefore, since $M_+\to \overline M_+\to \overline M/M$ is a cofiber sequence, we obtain a map of fiber sequences as depicted in the following commuting diagram \[\xymatrix{
\mathrm{Conf}_{S^r}(M)\ar[d]\ar[r]&\Omega^\infty\Sigma^\infty(M_+\wedge S^r)\ar[d]\\
\mathrm{Conf}_{S^r}(\overline M)\ar[d]\ar[r]&\Omega^\infty\Sigma^\infty(\overline M_+\wedge S^r)\ar[d]\\
\mathrm{Conf}_{S^r}(\overline M,M)\ar[r]&\Omega^\infty\Sigma^\infty(\overline M/M\wedge S^r),
}\] and thereby a map at the level of Serre spectral sequences \[E(\mathrm{Conf})\to E(\Omega^\infty\Sigma^\infty).\] This map is the desired comparison map. The following lemma asserts that its target has the desired property.
\begin{lemma}\label{lem:surjective collapse}
If $H_*(\overline M)\to H_*(\overline M,M)$ is surjective, then the spectral sequence $E(\Omega^\infty\Sigma^\infty)$ collapses at $E^2$.
\end{lemma}
In order to prove this auxiliary collapse result, we make use of the following observation.
\begin{lemma}\label{lem:Q and surjections}
The functor $\Omega^\infty\Sigma^\infty$ preserves homology surjections.
\end{lemma}
This result, in turn, relies on the following basic fact.
\begin{lemma}\label{lem:conf classifying space}
The space $\mathrm{Conf}_k(\mathbb{R}^\infty)$ is contractible for every $k\geq0$.
\end{lemma}
\begin{proof}
We proceed by induction on $k$, the base case $k\in \{0,1\}$ being obvious. By Fadell--Neuwirth, we have a homotopy pullback square \[\xymatrix{\mathbb{R}^n\setminus\{x_1,\ldots, x_k\}\ar[d]\ar[r]&\mathrm{Conf}_{k+1}(\mathbb{R}^n)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_k(\mathbb{R}^n)
}\] for each $n\geq0$, and thus, in the limit, we obtain the homotopy pullback square \[\xymatrix{\mathbb{R}^\infty\setminus\{x_1,\ldots, x_k\}\ar[d]\ar[r]&\mathrm{Conf}_{k+1}(\mathbb{R}^\infty)\ar[d]\\
\mathrm{pt}\ar[r]&\mathrm{Conf}_k(\mathbb{R}^\infty).
}\] The space in the upper left corner is homotopy equivalent to $\bigvee_kS^\infty$, which is contractible, and the claim follows from the long exact sequence in homotopy.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:Q and surjections}]
We give the proof under the assumption that the source and target of the homology surjection are both connected, which is the only case that we will use. The general case may be treated in the same way with group completion techniques.
Let $f:X\to Y$ be a homology surjection between connected pointed spaces. We have the chain of isomorphisms \begin{align*}
H_*(\Omega^\infty\Sigma^\infty X)&\cong H_*(\mathrm{Conf}_X(\mathbb{R}^\infty))\\
&\cong \bigoplus_{k\geq0} H_*(\mathrm{Conf}_k(\mathbb{R}^\infty)_+\wedge_{\Sigma_k}X^{\wedge k})\\
&\cong \bigoplus_{k\geq0} H_*(\Sigma_k; \widetilde H(X)^{\otimes k}),
\end{align*} and similarly for $Y$, where the first isomorphism uses connectivity and scanning, the second uses stable splitting, and the third uses Lemma \ref{lem:conf classifying space}, the fact that the action of $\Sigma_k$ on $\mathrm{Conf}_k(\mathbb{R}^\infty)$ is contractible, and the K\"{u}nneth isomorphism. Since we are working over a field, the surjection $\widetilde H_*(X)\to \widetilde H_*(Y)$ splits, and this splitting induces an equivariant splitting of $\widetilde H_*(X)^{\otimes k}\to \widetilde H_*(Y)^{\otimes k}$ and hence a splitting at the level of group homology.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:surjective collapse}]
In the commuting diagram \[\xymatrix{
H_*(\Omega^\infty\Sigma^\infty(\overline M_+\wedge S^r))\ar@{->>}[d]\ar[r]&H_*(\Omega^\infty\Sigma^\infty(\overline M/M\wedge S^r))\ar@{=}[d]^-\wr\\
E^\infty_{*,0}\ar[r]&E^2_{*,0},
}\] the bottom arrow is an injection, since $E^\infty_{*,0}=\bigcap_{r}\ker d^r_{*,0}$ (recall that the differential $d^r$ in the homological Serre spectral sequence has bidegree $(-r,r-1)$). Since the quotient map $\overline M\to \overline M/M$ induces a surjection on homology by assumption, it does so after smashing with $S^r$; therefore, the top map is a surjection by Lemma \ref{lem:Q and surjections}. It follows that the bottom map is a surjection and therefore an isomorphism, and we conclude that $d_r|_{E^r_{*,0}}$ vanishes for every $r\geq2$.
Now, the functor $\Omega^\infty\Sigma^\infty$ takes values in homotopy associative and commutative $H$-spaces, so the spectral sequence in question is a spectral sequence of graded commutative algebras. By induction and Leibniz rule, it now follows that $d^r$ is identically zero for all $r\geq2$, as desired.
\end{proof}
The other relevant piece of information about this comparison is the following fact.
\begin{lemma}\label{lem:spectral sequence injective}
The map $E^2(\mathrm{Conf})\to E^2(\Omega^\infty\Sigma^\infty)$ is injective.
\end{lemma}
Before demonstrating this injectivity, we use it to prove the desire result collapse.
\begin{proof}[Proof of Lemma \ref{lem:collapse lemma}]
By induction with base case covered by Lemma \ref{lem:spectral sequence injective}, $E^2(\mathrm{Conf})\cong E^r(\mathrm{Conf})$ and $E^r(\mathrm{Conf})\to E^r(\Omega^\infty\Sigma^\infty)$ is injective. By Lemma \ref{lem:surjective collapse}, $d^r(x)$ becomes zero in $E^r(\Omega^\infty\Sigma)$ for any $r\geq2$ and $x\in E^r(\mathrm{Conf})$; therefore, we conclude that $d^r$ is identically zero, so $E^2(\mathrm{Conf})\cong E^{r+1}(\mathrm{Conf})$ and $E^{r+1}(\mathrm{Conf})\to E^{r+1}(\Omega^\infty\Sigma^\infty)$ is inejctive. Thus, the spectral sequence collapses at $E^2$.
\end{proof}
\subsection{Loop space calculations} We will give the proof of Lemma \ref{lem:spectral sequence injective} under the assumption that $r+n$ is odd and $\mathbb{F}$ has characteristic 0. The argument in finite characteristic proceeds along similar lines, leveraging a more complicated computation of the homology of iterated loop spaces carried out in \cite{CohenLadaMay:HILS}. The simpler computation that we will use is the following.
\begin{proposition}\label{prop:loop space odd sphere}
If $m$ is odd and $\mathbb{F}$ is of characteristic 0, then, for any $0\leq k<m$, there is an isomorphism of graded vector spaces \[\mathrm{Sym}_\mathbb{F}(\alpha)\xrightarrow{\simeq}H_*(\Omega^kS^m),\] where $\alpha$ is the image of the class of the identity under the composite map \[\pi_m(S^m)\cong \pi_{m-k}(\Omega^k S^m)\to H_*(\Omega^k S^m).\] Moreover, for $k>0$, this isomorphism is an isomorphism of algebras.
\end{proposition}
We will return to this calculation in the next lecture.
\begin{proof}[Proof of Lemma \ref{lem:spectral sequence injective}]
In light of the commuting diagram \[\xymatrix{
E^2(\mathrm{Conf})\ar[d]\ar@{=}[r]^-\sim&H_*(\mathrm{Conf}_{S^r}(\overline M,M))\otimes H_*(\mathrm{Conf}_{S^r}(M))\ar[d]^-{H_*(\eta_{(\overline M,M)})\otimes H_*(\eta_M)}\\
E^2(\Omega^\infty\Sigma^\infty)\ar@{=}[r]^-\sim&H_*(\Omega^\infty\Sigma^\infty(\overline M/M\wedge S^r))\otimes H_*(\Omega^\infty\Sigma^\infty(M_+\wedge S^r)),
}\] it suffices to show that $\eta_{(\overline M,M)}$ and $\eta_M$ are each injective on homology.
For the map $\eta_{(\overline M,M)}$, a brief consideration of the definition of the map $\Phi$, along the lines of earlier arguments, shows that the composite \[
\Omega^{n-i}\Sigma^{n-i}S^{r+i}\simeq \mathrm{Conf}_{S^r}(\overline M,M)\xrightarrow{\eta_{(\overline M,M)}} \mathrm{Conf}_{\overline M/M\wedge S^r}(\mathbb{R}^\infty)\simeq\Omega^\infty\Sigma^\infty S^{r+i}
\] is homotopic to the canonical inclusion. Thus, the claim in this case follows from Proposition \ref{prop:loop space odd sphere}, since \begin{align*}
H_*(\Omega^\infty\Sigma^\infty S^{r+i})&\cong \colim_\ell H_*(\Omega^\ell \Sigma^\ell S^{r+i})\\
&\cong \colim_s H_*(\Omega^{n-i+2s}\Sigma^{n-i+2s}S^{r+i})\\
&\cong \colim_s H_*(\Omega^{n-i+2s}S^{n+r+2s})\\
&\cong \colim_s\mathrm{Sym}_\mathbb{F}(\alpha_s)\\
&\cong\mathrm{Sym}_\mathbb{F}(\alpha_\infty),
\end{align*} where $|\alpha_s|=|\alpha_\infty|=r+i$, and we have used that $n+r+2s$ is odd.
The case of the map $\eta_M$ follows by induction on a handle decomposition, the base case of $D^n$ following as before from Proposition \ref{prop:loop space odd sphere}.
\end{proof}
Note that, in the characteristic 0 case, the injection $E^2(\mathrm{Conf})\to E^2(\Omega^\infty\Sigma^\infty)$ is in fact an isomorphism.
We abstract the general features of the calculation behind Proposition \ref{prop:loop space odd sphere} in the following two lemmas.
\begin{lemma}
Let $\mathbb{F}$ be any field and $\{E^r\}$ a first-quadrant, homological, multiplicative spectral sequence over $\mathbb{F}$. If\begin{enumerate}
\item $E^2_{*,*}\cong E^2_{*,0}\otimes E^2_{0,*}$,
\item $E^2_{*,0}\cong\wedge_\mathbb{F}(x)$ with $|x|$ odd, and
\item $E^\infty=E^\infty_{0,0}\cong\mathbb{F}$,
\end{enumerate} then $E^2_{0,*}\cong\mathbb{F}[y]$ with $|y|=|x|-1$.
\end{lemma}
\begin{proof}
We note first that $d^r|_{E_{*,0}}=0$ for $r<|x|$ by (1); therefore, by the Leibniz rule, $d^r\equiv0$ for $r<|x|$. Next, we note that $y:=d^{|x|}(x)\neq 0$, for otherwise $x$ is a permanent cycle, contradicting (3). By the Leibniz rule, we compute that \begin{align*}
d^{|x|}(x\otimes y^\ell)&=d^{|x|}(x)y^\ell+(-1)^{|x|}xd^{|x|}(y^\ell)\\
&=y^{\ell+1}.
\end{align*}
Note that $y^{\ell+1}\neq 0$ for each $\ell$, since otherwise $x\otimes y^\ell$ is a permanent cycle, and the result follows, since there can be no further differentials and hence no further elements in $E^2_{0,*}$.
\end{proof}
\begin{lemma}
Let $\mathbb{F}$ be a field of characteristic zero and $\{E^r\}$ a first-quadrant, homological, multiplicative spectral sequence over $\mathbb{F}$. If\begin{enumerate}
\item $E^2_{*,*}\cong E^2_{*,0}\otimes E^2_{0,*}$,
\item $E^2_{*,0}\cong \mathbb{F}[y]$ with $|y|$ even, and
\item $E^\infty=E^\infty_{0,0}\cong\mathbb{F}$,
\end{enumerate} then $E^2_{0,*}\cong\wedge_\mathbb{F}(x)$ with $|x|=|y|-1$.
\end{lemma}
\begin{proof}
As in the previous argument, we find that $d^r\equiv0$ for $r<|y|$, and that $x:=d^{|y|}(y)\neq0$. By the Leibniz rule and induction, we have that \begin{align*}
d^{|y|}(y^\ell)&=d^{|y|}(y)y^{\ell-1}+(-1)^{|y|} yd^{|y|}(y^{\ell-1})\\
&=\ell y^{\ell-1}\otimes x\\
&\neq 0,
\end{align*} since $\mathbb{F}$ has characteristic zero. Since there can be no further differentials, the claim follows.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:loop space odd sphere}]
We proceed by induction on $k$, the base case of $k=0$ being the observation that $H_*(S^m)\cong \wedge_\mathbb{F}(\alpha)$. For the induction step, we use the Serre spectral sequence for the homotopy pullback square \[\xymatrix{
\Omega^k S^m\ar[r]\ar[d]&P\,\Omega^{k-1}S^m\ar[d]\\
\mathrm{pt}\ar[r]&\Omega^{k-1}S^m,
}\] where $P$ denotes the path space functor. Since the path space is contractible, and since, for $k>1$, this spectral sequence is multiplicative, one of the two lemmas applies, depending on the parity of $k$. To identify the generator $\alpha$, it suffices to note that $\alpha\neq0$ by the Hurewicz theorem, and that $|\alpha|=m-k$.
In the case $k=1$, the spectral sequence is not multiplicative; rather, via naturality and the commutative diagram \[\xymatrix{
\Omega S^m\times\Omega S^m\ar[d]\ar[r]&\Omega S^m\ar[d]\\
\Omega S^m\times PS^m\ar[r]\ar[d]&PS^m\ar[d]\\
S^m\ar@{=}[r]&S^m
}\]it is a spectral sequence of $H_*(\Omega S^m)$-modules. This structure is sufficient to imply the necessary equation \[d^m(x\otimes y^\ell)=d^m(x)y^\ell=y^{\ell+1}\] in this case as well.
\end{proof}
\begin{example}
In finite characteristic $p$, the same calculation shows that $H_*(\Omega S^m)\cong \mathbb{F}[y]$. For $\Omega^2S^m$, however, the argument breaks down, for now\[d^{m-1}(y^p)=py^{p-1}=0,\] implying the existence of further elements \begin{align*}
Q^1(y)&:=d^{p(m-1)}(y^p)\\
\beta Q^1(y)&=d^{(p-1)(m-1)}(y^{p-1}\otimes x)
\end{align*} of degree $p(m-1)-1$ and $p(m-1)-2$, respectively. A systematic approach to these and other nontrivial higher differentials, and the resulting profusion of homology classes, is provided by the framework of homology operations for iterated loop spaces. The symbol $Q^1$ used above refers to a certain \emph{Dyer-Lashof} operation and the letter $\beta$ to the Bockstein operator, and it turns out that the homology of iterated loop spaces of spheres can be completely described in terms of composites of Dyer-Lashof operations and the Bockstein---see \cite{CohenLadaMay:HILS} for further details.
\end{example}
Our next move, having completed the proof of Theorem \ref{thm:labeled conf homology}, will be to exploit it in understanding the homology of ordinary configuration spaces.
\subsection{Odd dimensional homology} Our strategy in proving Theorem \ref{thm:odd homology} will be to reinterpret the two sides of the isomorphism of Theorem \ref{thm:labeled conf homology}, keeping track of the extra grading.
On the lefthand side of the isomorphism, as bigraded vector spaces, we have \begin{align*}H_*(\mathrm{Conf}_{S^r}(M))&\cong \mathbb{F}\oplus\widetilde H_*(\mathrm{Conf}_{S^r}(M))\\
&\cong \mathbb{F}\oplus \bigoplus_{k\geq1}\widetilde H_*\left(\mathrm{Conf}_k(M)_+\wedge_{\Sigma_k}S^{rk}\right)\\
&\cong\bigoplus_{k\geq0}\widetilde H_*\left(\mathrm{Conf}_k(M)_+\wedge_{\Sigma_k}S^{rk}\right)
\end{align*} by stable splitting. Writing $\pi_{k,r}$ for the natural projection \[\pi_{k,r}:\mathrm{Conf}_k(M)\times_{\Sigma_k}\mathbb{R}^{rk}\to B_k(M),\] we have the following observation.
\begin{lemma}\label{lem:thom space}
The map $\pi_{k,r}$ is a vector bundle of rank $rk$, and there is a pointed homeomorphism \[\mathrm{Th}(\pi_{k,r})\cong \mathrm{Conf}_k(M)_+\wedge_{\Sigma_k} S^{rk}.\]
\end{lemma}
\begin{proof}
For an open subset $U\subseteq M$ with $k$ connected components, each Euclidean, a choice of ordering $\tau:\{1,\ldots, k\}\cong \pi_0(U)$ gives rise to the commuting diagram
\[\xymatrix{\displaystyle\coprod_{\sigma:\{1,\ldots, k\}\cong \pi_0(U)}\prod_{i=1}^kU_{\sigma(i)}\times\mathbb{R}^{rk}\ar[d]&\displaystyle\coprod_{\sigma:\{1,\ldots, k\}\cong \pi_0(U)}\mathrm{Conf}_k^0(U,\sigma)\times\mathbb{R}^{rk}\ar[l]_-{\simeq}\ar[d]\ar[r]&\mathrm{Conf}_k(M)\times\mathbb{R}^{rk}\ar[d]\\
\displaystyle\prod_{i=1}^kU_{\tau(i)}\times\mathbb{R}^{rk}\ar[d]&\displaystyle\left(\coprod_{\sigma:\{1,\ldots, k\}\cong \pi_0(U)}\mathrm{Conf}_k^0(U,\sigma)\times\mathbb{R}^{rk}\right)_{\Sigma_k}\ar[d]\ar[l]_-\simeq\ar[r]&\mathrm{Conf}_k(M)\times_{\Sigma_k}\mathbb{R}^{rk}\ar[d]^-{\pi_{k,r}}\\
\displaystyle\prod_{i=1}^k U_{\tau(i)}&B_k^0(U)\ar[l]_-\simeq\ar[r]&B_k(M),
}\] in which the two righthand squares are pullback squares and the upper three vertical maps are projections to spaces of $\Sigma_k$-coinvariants. Since $B_k(M)$ is covered by open subsets of the form $B_k^0(Y)$, it follows that $\pi_{k,r}$ is locally trivial. Since addition and scalar multiplication in $\mathbb{R}^{rk}$, the linear structure of $\mathrm{Conf}_k(M)\times\mathbb{R}^{rk}$ descends to the quotient.
For the second claim, we have \begin{align*}
\mathrm{Th}(\pi_{k,r})&\cong D(\pi_{k,r})/S(\pi_{k,r})\\
&\cong \frac{\mathrm{Conf}_k(M)\times_{\Sigma^k}D^{rk}}{\mathrm{Conf}_k(M)\times_{\Sigma_k}S^{rk-1}}\\
&\cong \mathrm{Conf}_k(M)_+\wedge_{\Sigma_k}S^{rk}.
\end{align*}
\end{proof}
In order to apply the Thom isomorphism, we require the following result.
\begin{lemma}\label{lem:orientable}
If $r$ is even, then $\pi_{k,r}$ is orientable.
\end{lemma}
\begin{proof}
Since $r$ is even, any orientation of $\mathbb{R}^r$ induces a $\Sigma_k$-invariant orientation of $\mathbb{R}^{rk}$, whence of the trivial bundle $\mathrm{Conf}_k(M)\times \mathbb{R}^{rk}$. By $\Sigma_k$-invariance, this orientation descends to the quotient.
\end{proof}
\begin{remark}
Again, characterstic 2 is exceptional.
\end{remark}
\begin{corollary}\label{cor:lhs}
For $n$ odd and $r>1$ even, there is an isomorphism of bigraded vector spaces \[H_*(\mathrm{Conf}_{S^r}(M))\cong\bigoplus_{k\geq0} H_*(B_k(M))[rk].\]
\end{corollary}
As for the righthand side of the isomorphism of Theorem \ref{thm:labeled conf homology}, stable splitting gives us the following commuting diagram of isomorphisms \[\xymatrix{
H_*(\Omega^{n-i}S^{n+r})\ar@{=}[d]_-\wr& \displaystyle\bigoplus_{k\geq0}\widetilde H_*\left(\mathrm{Conf}_k(D^i\times D^{n-i}, \partial D^i\times D^{n-i})\wedge_{\Sigma_k}S^{rk}\right)
\ar[l]_-\simeq\ar@{-->}[d]\\
H_*(\Omega^{n-i}\Sigma^{n-i}S^{r+i})&\displaystyle\bigoplus_{k\geq0}\widetilde H_*\left(\mathrm{Conf}_k(\mathbb{R}^{n-i})_+\wedge_{\Sigma_k}S^{k(r+i)}\right).\ar[l]_-\simeq
&
}\] All four terms of this diagram are naturally bigraded, and the horizontal morphisms are compatible with these bigradings. It is possible, by direct consideration of the scanning map, to show that the vertical arrows are also compatible with the bigradings. Rather than proceed in this manner, however, we invoke the following result, which likewise assures us that we may work with the lower left and upper right corners interchangeably.
\begin{lemma}
For any manifold $N$ and $i,k\geq0$, there is a canonical, $\Sigma_k$-equivariant weak equivalence \[\Sigma^{ik}_+\mathrm{Conf}_k(N)\xrightarrow{\sim} \mathrm{Conf}_k\left(D^i\times N,\partial D^i\times N\right).\]
\end{lemma}
\begin{proof}
The map is defined via the inclusion \[\Sigma_+^{ik}\mathrm{Conf}_k(N)\cong \frac{(D^i)^k\times\mathrm{Conf}_k(N)}{\partial (D^i)^k\times \mathrm{Conf}_k(N)}\xrightarrow{\simeq} U\subseteq \mathrm{Conf}_k\left(D^i\times N, \partial D^i\times N\right)\] of the open subspace $U$ consisting of the basepoint together with all nontrivial configurations whose coordinates in $N$ are distinct. Letting $V$ denote the open subspace consisting of the basepoint together with all nontrivial configurations with at least two distinct coordinates in $D^i$, it is clear that $\mathrm{Conf}_k(D^i\times N,\partial D^i\times N)=U\cup V$. Now, both $V$ and $U\cap V$ are contractible by radial expansion, so the inclusion $i$ of $U$ induces the weak equivalences \begin{align*}
i^{-1}(U)&\xrightarrow{=} U\\
i^{-1}(V)=U\cap V&\xrightarrow{\sim} V\\
i^{-1}(U\cap V)&\xrightarrow{=}U\cap V.
\end{align*} It follows from Proposition \ref{prop:mayer-vietoris} that $i$ is a weak equivalence.
\end{proof}
The final ingredient in the proof will be the following calculation.
\begin{lemma}
For $n$ odd, $r>1$ even, $i\geq0$, and $\mathbb{F}$ of characteristic 0, \[\widetilde H_*\left(\mathrm{Conf}_k(\mathbb{R}^{n-i})_+\wedge_{\Sigma_k}S^{k(r+i)}\right)\cong\begin{cases}
\mathbb{F}[k(r+i)]&\quad i \text{ even or } k\in\{0,1\}\\
0&\quad \text{otherwise.}
\end{cases}\]
\end{lemma}
\begin{proof}
If $i$ is even, then $\pi_{r+i,k}$ is orientable, since $r$ is even, so the homology group in question is identified with \[H_*(B_k(\mathbb{R}^{n-i}))[k(r+i)]\cong\mathbb{F}[k(r+i)]\] by the Thom isomorphism, where we have used that $n-i$ is odd when $n$ is odd and $i$ is even. On the other hand, if $i$ is odd, then by the K\"{u}nneth theorem and the assumption on the charateristic of $\mathbb{F}$, we have \begin{align*}\widetilde H_*\left(\mathrm{Conf}_k(\mathbb{R}^{n-i})_+\wedge_{\Sigma_k}S^{k(r+i)}\right)
&\cong H_*(\mathrm{Conf}_k(\mathbb{R}^{n-i}))\otimes_{\Sigma_k}\widetilde H_*(S^{k(r+i)})\\
&\cong H_*(\mathrm{Conf}_k(\mathbb{R}^{n-i}))\otimes_{\Sigma_k}\mathbb{F}^{\mathrm{sgn}}[k(r+i)],
\end{align*} where $\mathbb{F}^\mathrm{sgn}$ denotes the sign representation of $\Sigma_k$. Earlier, we showed that the homology group $H_*(\mathrm{Conf}_k(\mathbb{R}^{n-i}))$ has a spanning set indexed by $k$-forests, where switching adjacent leaves of a tree introduces a sign $(-1)^\epsilon$, where $\epsilon$ is the degree of the antipodal map on $S^{n-i-1}$. Since $n$ and $i$ are both odd, it follows that $\epsilon=0$. After tensoring with the sign representation, it follows that any transposition in $\Sigma_k$ acts by $-1$. Thus, in this case, \[H_*(\mathrm{Conf}_k(\mathbb{R}^{n-i})\otimes_{\Sigma_k}\mathbb{F}^\mathrm{sgn}[k(r+i)]\cong\begin{cases}
\mathbb{Q}[k(r+i)]&\quad k\in\{0,1\}\\
0&\quad\text{otherwise,}
\end{cases}\] which completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:odd homology}]
Combining what we have done so far, we obtain the chain of isomorphisms of bigraded vector spaces \begin{align*}
\bigoplus_{k\geq0} H_*(B_k(M))[kr]&\cong \bigotimes_{i=0}^n\left(\bigoplus_{\ell\geq0}H_*\left(\mathrm{Conf}_\ell(\mathbb{R}^{n-i})_+\wedge_{\Sigma_\ell}S^{\ell(r+i)}\right)\right)^{\otimes\dim H_i(M)}\\
&\cong\bigotimes_{i\text{ even}}\left(\bigoplus_{\ell\geq0}\mathbb{F}[\ell(r+i)]\right)^{\otimes \dim H_i(M)}\otimes\bigotimes_{i\text{ odd}}\left(\bigoplus_{\ell\in\{0,1\}}\mathbb{F}[\ell(r+i)]\right)^{\otimes\dim H_i(M)}\\
&\cong \bigotimes_{i\text{ even}}\mathrm{Sym}(\mathbb{F}[r+i])^{\otimes \dim H_i(M)}\otimes\bigotimes_{i\text{ odd}}\mathrm{Sym}(\mathbb{F}[r+i])^{\otimes\dim H_i(M)}\\
&\cong\mathrm{Sym}(H_\mathrm{even}(M)[r])\otimes\mathrm{Sym}(H_\mathrm{odd}(M)[r])\\
&\cong \mathrm{Sym}\left(H_*(M)[r]\right),
\end{align*} where we have repeatedly used that $\mathrm{Sym}$ sends direct sums to tensor products. It follows that \[H_*(B_k(M))[kr]\cong \mathrm{Sym}^k(H_*(M))[kr],\] which completes the proof.
\end{proof}
\section{Mod $p$ cohomology}\label{section:mod p cohomology}
In the previous section, we expressed the mod $p$ (co)homology of the unordered configuration of a manifold of odd dimension in terms of the (co)homology of the spaces $\mathrm{Conf}_k(\mathbb{R}^n)_+\wedge_{\Sigma_k}S^{rk}$ for varying $k$, $n$, and $r$. Therefore, reformulating a little using the K\"{u}nneth isomorphism, we see that this computation amounts to essentially two computations of cohomology with local coefficients; indeed, we have \begin{align*}
H^*\left(\mathrm{Conf}_k(\mathbb{R}^n)_+\wedge_{\Sigma_k}S^{rk}\right)&\cong H\left(\mathrm{Map}_{\Sigma_k}\left(C_*(\mathrm{Conf}_k(\mathbb{R}^n)), \widetilde{C}^*(S^{rk})\right)\right)\\
&\cong H\left(\mathrm{Map}_{\Sigma_k}\left(C_*(\mathrm{Conf}_k(\mathbb{R}^n)), \mathbb{F}_p(r)\right)\right)[kr]\\
&=: H^*(B_k(\mathbb{R}^n);\mathbb{F}_p(r))[kr],
\end{align*} where $\mathbb{F}_p(r)$ is the $\Sigma_p$-module defined by \[
\mathbb{F}_p(r)=\begin{cases}\mathbb{F}_p^\mathrm{triv}&\quad r\text{ even}\\
\mathbb{F}_p^\mathrm{sgn}&\quad r\text{ odd}.
\end{cases}\] Our present goal is to describe the method behind this computation, which was first carried out in \cite[III]{CohenLadaMay:HILS}. For ease of exposition, we will focus on the case $r$ even, but the other case may be treated in an entirely parallel fashion. We also restrict to the case $k=p$, from which the other cases may be derived in a rather roundabout manner.
\subsection{Outline of argument} The strategy is to leverage our complete understanding of cohomology of the covering space $\mathrm{Conf}_k(\mathbb{R}^n)$ using the spectral sequence of Corollary \ref{cor:ss of a cover}. From the commutative diagram of covering spaces \[\xymatrix{
\mathrm{Conf}_p(\mathbb{R}^n)\ar[d]\ar@{=}[r]&\mathrm{Conf}_p(\mathbb{R}^n)\ar[d]\ar[r]&\mathrm{Conf}_p(\mathbb{R}^\infty)\ar[d]\\
\mathrm{Conf}_p(\mathbb{R}^n)_{C_p}\ar[r]&B_p(\mathbb{R}^n)\ar[r]&B_p(\mathbb{R}^\infty),
}\] we obtain the commutative diagram of fiber sequences \[\xymatrix{
\mathrm{Conf}_p(\mathbb{R}^n)\ar[d]\ar@{=}[r]&\mathrm{Conf}_p(\mathbb{R}^n)\ar[d]\ar[r]&\mathrm{pt}\ar[d]\\
\mathrm{Conf}_p(\mathbb{R}^n)_{C_p}\ar[d]\ar[r]&B_p(\mathbb{R}^n)\ar[d]\ar[r]&B\Sigma_p\ar@{=}[d]\\
BC_p\ar[r]&B \Sigma_p\ar@{=}[r]&B\Sigma_p,
}\] and hence the commuting diagram of spectral sequences \[\xymatrix{
H^s(\Sigma_p)\ar[d]\ar@{=}[r]&H^s(\Sigma_p)\ar[d]\\
H^s(\Sigma_p; H^t(\mathrm{Conf}_p(\mathbb{R}^n)))\ar[d]\ar@{=>}[r]&H^{s+t}(B_p(\mathbb{R}^n))\ar[d]\\
H^s(C_p;H^t(\mathrm{Conf}_p(\mathbb{R}^n)))\ar@{=>}[r]&H^{s+t}(\mathrm{Conf}_p(\mathbb{R}^n)_{C_p}).
}\] The utility of this maneuver lies in the following basic observation.
\begin{lemma}\label{lem:principal bundle transfer}
If $\pi:P\to X$ is a principal $G$-bundle and $H<G$ is a $p$-Sylow subgroup, then the induced map of spectral sequences is injective on $E_2$ and on $E_\infty$.
\end{lemma}
\begin{proof}
On $E_\infty$, the map is induced by the projection $\pi_H:P/H\to X$, which is a $[G:H]$-fold cover; therefore, the composite of $\pi^*_H$ followed by the transfer map is multiplication by $[G:H]$, which is injective, since $H$ is $p$-Sylow. On $E_2$, the map is induced by the map $BH\to BG$, which is also a $[G:H]$-fold cover, and a similar argument in cohomology with local coefficients applies.
\end{proof}
Thus, the spectral sequence of immediate interest is a summand of the spectral sequence obtained by restriction to $C_p$. Regarding this spectral sequence, we have the following
\begin{vanishingtheorem}
In the spectral sequence for the cover $\mathrm{Conf}_p(\mathbb{R}^n)\to \mathrm{Conf}_p(\mathbb{R}^n)_{C_p}$, we have $E_2^{s,t}=0$ if $s>0$ and $0<t<(n-1)(p-1)$.
\end{vanishingtheorem}
By Lemma \ref{lem:principal bundle transfer}, then, the same vanishing range holds for the spectral sequence of primary interest. Moreover, as we calculated earlier, $H^*(\mathrm{Conf}_p(\mathbb{R}^n))=0$ for $*>(n-1)(p-1)$, so the spectral sequence is concentrated in the $s=0$ column in degree at most $(n-1)(p-1)$ and in the two rows $t=0$ and $t=(n-1)(p-1)$.
As for the differentials, it follows from what has already been said that the only possible nonzero differentials are $d_r$ on the $s=0$ column for $2\leq r\leq (n-1)(p-1)$ and on the $t=(n-1)(p-1)$ column for $r=(n-1)(p-1)+1$. Moreover, we have the following simple, but crucial, observation:
\begin{lemma}\label{lem:high differential}
In the spectral sequence for the cover $\mathrm{Conf}_p(\mathbb{R}^n)\to B_p(\mathbb{R}^n)$, the differential $d_{(n-1)(p-1)+1}:E_{(n-1)(p-1)+1}^{s,(n-1)(p-1)}\to E_{(n-1)(p-1)+1}^{s+(n-1)(p-1)+1,0}$ is an isomorphism for $s>n+p-2$.
\end{lemma}
\begin{proof}
Assuming otherwise, it follows that $E_\infty^{s+(n-1)(p-1)+1,0}\neq 0$, and hence that $B_p(\mathbb{R}^n)$ has non-vanishing homology in some degree strictly greater than $(n+p-2)+(n-1)(p-1)+1=np$. Since $B_p(\mathbb{R}^n)$ is a manifold of dimension $np$, this is a contradiction.
\end{proof}
Together with the following classical calculation, this observation will allow us to populate much of the $t=(n-1)(p-1)$ row.
\begin{theorem}[Nakaoka]
There is a commuting diagram \[\xymatrix{
H^*(C_p)\ar@{=}[r]^-\sim&\wedge_{\mathbb{F}_p}[u]\otimes \mathbb{F}_p[\beta u]&u^{2(p-1)-1}\\
H^*(\Sigma_p)\ar@{=}[r]^-\sim\ar[u]&\wedge_{\mathbb{F}_p}[v]\otimes \mathbb{F}_p[\beta v]\ar[u]&v,\ar@{|->}[u]
}\] where $|u|=1$ and $\beta$ denotes the mod $p$ Bockstein.
\end{theorem}
In order to populate the remainder of this row, we make use of the following result, the proof of which is discussed in Appendix \ref{appendix:periodicity}.
\begin{periodicitytheorem}
Let $M$ be a $C_p$-module and $N$ a $\Sigma_p$-module.
\begin{enumerate}
\item Cup product with $\beta u$ induces an isomorphism \[H^s(C_p;M)\xrightarrow{\simeq} H^{s+2}(C_p; M).\]
\item Cup product with $\beta v$ induces an isomorphism \[H^s(\Sigma_p;N)\xrightarrow{\simeq} H^{s+2(p-1)}(\Sigma_p; N).\]
\end{enumerate}
\end{periodicitytheorem}
The final step is the identification of the $s=0$ column, which is simply the module of invariants for the action $\Sigma_p$. For simplicity, we do not state the analogous result for $p=3$.
\begin{invariantstheorem}
For any prime $p>3$,\[I:=H^*(\mathrm{Conf}_p(\mathbb{R}^n))^{\Sigma_p}\cong\begin{cases}
\wedge_{\mathbb{F}_p}(\alpha_{n-1})&\quad \text{$n$ even}\\
\mathbb{F}_p&\quad \text{$n$ odd}.
\end{cases}\]
\end{invariantstheorem}
Thus, the only possible differential aside from $d^{(n-1)(p-1)+1}$, whose effect is determined by Lemma \ref{lem:high differential} and periodicity, is $d^n$. After checking that this differential vanishes, the calculation follows.
\begin{theorem}[Cohen]
There is an isomorphism \[H^*(B_p(\mathbb{R}^n))\cong I\times_{\mathbb{F}_p} \frac{H^*(\Sigma_p)}{H^{>(n-1)(p-1)}(\Sigma_p)}.\]
\end{theorem}
In the remainder of this section, we now give this outline flesh.
\begin{warning}
This argument is the one given in the original reference \cite{CohenLadaMay:HILS}, but we adopt our own notational conventions and give different proofs of some result.
\end{warning}
\subsection{Invariants} We begin with a few sample calculations.
\begin{example}
For $p=3$, a basis for the degree $2n-2$ cohomology is given by the set $\left\{\alpha_{12}\alpha_{13}, \alpha_{12}\alpha_{23}\right\}$. We compute that \begin{align*}
\tau_{12}^*(\alpha_{12}\alpha_{13})&=(-1)^n\alpha_{12}\alpha_{23}\\
\tau_{23}^*(\alpha_{12}\alpha_{13})&=(-1)^{(n-1)^2}\alpha_{12}\alpha_{13}\\
\tau_{12}^*(\alpha_{12}\alpha_{23})&=(-1)^n\alpha_{12}\alpha_{13}\\
\tau_{23}^*(\alpha_{12}\alpha_{23})&=(-1)^{n+1}\alpha_{12}\alpha_{13}+(-1)^{(n-1)^2+1}\alpha_{12}\alpha_{23}.
\end{align*} Thus, if $\beta=\lambda_1\alpha_{12}\alpha_{13}+\lambda_2\alpha_{12}\alpha_{23}$ is a fixed point, we have the equations \begin{align*}
\lambda_1&=(-1)^n\lambda_2\\
\lambda_1&=(-1)^{(n-1)^2}\lambda_1+(-1)^{n+1}\lambda_2,
\end{align*} which imply $\beta=0$ unless $2\equiv(-1)^{(n-1)^2}\mod 3$, which occurs precisely when $n$ is even. This special case is the reason for the restriction $p>3$ in the statement of the theorem.
\end{example}
\begin{example}\label{example:fixed point}
Assuming that $n$ is even and setting $\alpha=\sum_{1\leq a<b\leq p}\alpha_{ab}$, we compute that \begin{align*}
\sigma^*\alpha&=\sum_{1\leq a<b\leq p}\alpha_{\sigma(a)\sigma(b)}\\
&=\sum_{\sigma(a)<\sigma(b)}\alpha_{\sigma(a)\sigma(b)}+(-1)^n\sum_{\sigma(b)<\sigma(a)}\alpha_{\sigma(a)\sigma(b)}\\
&=\alpha,
\end{align*} so the fixed point set is at least as large as claimed in the theorem above. Thus, it remains to show that $\alpha$ is the only possible nontrivial fixed point.
\end{example}
In order to prove the theorem, we pass from cohomology to homology in order to btain a more convenient basis. This passage is justified by the tensor/hom adunction; for a finite group $G$ and $G$-module $M$, we have \begin{align*}
H^*(G; M^\vee)&\cong \mathrm{Ext}^*_{\mathbb{F}[G]}\left(\mathbb{F}, M^\vee\right)\\
&\cong \mathrm{Ext}^*_{\mathbb{F}[G]}\left(\mathbb{F}, \mathrm{Ext}^*_\mathbb{F}\left(M,\mathbb{F}\right)\right)\\
&\cong \mathrm{Ext}^*_{\mathbb{F}}\left(\mathrm{Tor}_*^{\mathbb{F}[G]}\left(\mathbb{F}, M\right),\mathbb{F}\right)\\
&\cong H_*(G; M)^\vee.
\end{align*} In particular, in the case of interest, we have the isomorphisms \begin{align*}
H^*(\mathrm{Conf}_p(\mathbb{R}^n))^{\Sigma_p}&\cong \left(H_*(\mathrm{Conf}_p(\mathbb{R}^n))_{\Sigma_p}\right)^\vee\\
H^*(C_p;H^*(\mathrm{Conf}_p(\mathbb{R}^n)))&\cong H_*(C_p;H_*(\mathrm{Conf}_p(\mathbb{R}^n)))^\vee.
\end{align*}
Recall from \S\ref{section:cohomology} that $H_*(\mathrm{Conf}_p(\mathbb{R}^n))$ is spanned by classes indexed by $p$-forests modulo the Jacobi identity and graded antisymmetry, with a preferred basis given by the tall forests. The advantage of this perspective is that $\Sigma_p$ acts on a forest by permuting the leaves, which are independent of one another, whereas the indices of a monomial such as $\alpha_{ab}\alpha_{bc}\alpha_{ac}$ are far from independent.
The first of the theorems in question is now almost immediate; indeed, we essentially gave the argument previously when computing $H_*(B_k(\mathbb{R}^n);\mathbb{Q})$.
\begin{proof}[Proof of Invariants Theorem]
Let $\alpha$ be the class of a tall forest with at least three leaves; thus, $|\alpha|\geq 2(n-1)$. By the Jacobi identity, $3[\alpha]=0$ in $H_*(\mathrm{Conf}_p(\mathbb{R}^n))_{\Sigma_p}$; therefore, since $p>3$, the map from $H_*(\mathrm{Conf}_p(\mathbb{R}^n))$ to the module of coinvariants is zero, since it annihilates a basis. Since this map is also surjective, the claim follows in degree $2(n-1)$ and higher. The argument in degree $0$ is trivial. In degree $n-1$, there are two cases to consider. If $n$ is odd, then every basis element is equivalent to its negative in coinvariants, which must therefore be trivial. If $n$ is even, we note that $\Sigma_p$ acts transitively on our preferred homology basis, so $H_{n-1}(\mathrm{Conf}_p(\mathbb{R}^n))_{\Sigma_p}$ has dimension at most 1. Therefore, by Example \ref{example:fixed point}, the dimension is exactly 1.
\end{proof}
\begin{remark}
This argument differs from that given in \cite[III.9]{CohenLadaMay:HILS}. We were unable to justify some of the claims made in the course of that argument.
\end{remark}
\subsection{Vanishing}
We turn now to the vanishing theorem. As in the previous section, we may reformulate this result in homological terms as the assertion \[H_s(C_p; H_t(\mathrm{Conf}_p(\mathbb{R}^n))=0\] for $s>0$ and $0<t<(n-1)(p-1)$.
\begin{remark}
We emphasize that this theorem requires no restriction on $p$ (note that the case $p=2$ is vacuous).
\end{remark}
We will use the following vanishing criterion.
\begin{proposition}\label{prop:cyclic decomposition}
Let $V$ be a $C_p$-module over $\mathbb{F}$ and $\sigma\in C_p$ a fixed generator. If $V$ admits a decomposition of the form \[V\cong \bigoplus_{i=1}^{p}V_{\sigma^i}\] such that $\sigma\left(V_{\sigma^i}\right)\subseteq V_{\sigma^{i+1}}$, then $H_s(C_p;V)=0$ for $s>0$.
\end{proposition}
\begin{proof}
The trivial $C_p$-module $\mathbb{F}$ admits the so-called periodic resolution \[\cdots\to \mathbb{F}[C_p]\xrightarrow{N}\mathbb{F}[C_p]\xrightarrow{\sigma-1}\mathbb{F}[C_p]\xrightarrow{\epsilon}\mathbb{F},\] where $N=1+\sigma+\cdots+\sigma^{p-1}$ and $\epsilon$ denotes the augmention. Thus, the group homology of $V$ is computed by the complex \[\cdots \to V\xrightarrow{N} V\xrightarrow{\sigma-1} V,\] so it suffices to show that the inclusion $\mathrm{im}(N)\subseteq \ker(\sigma-1)$ is an equality.
Suppose that $\sigma(v)=v$. Our assumption on $V$ provides the unique decomposition \[\sum_{i=1}^pv_{\sigma^i}=v=\sigma(v)=\sum_{i=1}^p\sigma(v_{\sigma^i}).\] Since $\sigma(v_{\sigma^i})\in V_{\sigma^{i+1}}$, it follows by induction that $v_{\sigma^i}=v_\sigma$ for all $1\leq i\leq p$, so $v=N v_\sigma.$
\end{proof}
In order to apply this observation to our situation, we recall that a $p$-forest is simply an ordered partition of $\{1,\ldots, p\}$ together with a binary parenthesization of each block of the partition, and that changing the order of the partition introduces an overall sign. Since the Jacobi identity and antisymmetry do not change the partition of a forest, we have the direct sum decomposition \[H_*(\mathrm{Conf}_p(\mathbb{R}^n))\cong \bigoplus_{1\leq \ell\leq p}\bigoplus_{[\pi]\in \mathrm{Surj}(p,\ell)_{\Sigma_\ell}}F_{[\pi]},\] where $F_{[\pi]}$ denotes the subspace spanned by the forests with underlying unordered partition $[\pi]$. We now make three observations.
\begin{enumerate}
\item The degree 0 subspace is exactly the $\ell=p$ summand. Thus, we disregard this summand.
\item The degree $(n-1)(p-1)$ subspace is exactly the $\ell=1$ summand. Thus, we disregard this summand.
\item The symmetric group acts via the action on $\mathrm{Surj}(p,\ell)_{\Sigma_\ell}$ given by pre-composition. In particular, the $\ell$th summand above is closed under the action of $\Sigma_p$.
\end{enumerate}
We conclude that the Vanishing Theorem is equivalent to the claim that \[H_{s}\left(C_p;\bigoplus_{\mathrm{Surj}(\ell,p)_{\Sigma_\ell}}F_{[\pi]}\right)=0\] for $s>0$ and $1<\ell<p$. The essential observation in establishing this claim is the following.
\begin{lemma}\label{lem:partitions}
For any $1<\ell<p$ and $[\pi]\in \mathrm{Surj}(p,\ell)_{\Sigma_\ell}$, \[[\pi\circ\sigma]\neq[\pi].\]
\end{lemma}
\begin{proof}
There are numbers $1\leq i,j\leq \ell$ such that $|\pi^{-1}(i)|\neq |\pi^{-1}(j)|$; indeed, assuming otherwise implies that $\ell\mid p$, which contradicts our assumption that $\ell\notin\{1,p\}$. Now, if $\rho$ is a cyclic permutation taking any element of $\pi^{-1}(i)$ to $\pi^{-1}(j)$, then $[\pi]\neq [\pi\circ\rho]=[\pi\circ\sigma^i]$, which leads to a contradiction under the assumption that $[\pi\circ\sigma]=[\pi]$.
\end{proof}
\begin{proof}[Proof of Vanishing Theorem]
Since $F_{[\pi]}\cap F_{[\pi']}=0$ for $[\pi]\neq [\pi']$, Lemma \ref{lem:partitions} implies that $F_{[\pi]}\cap F_{[\pi\circ\sigma]}=0$ for any $1<\ell<p$ and $[\pi]\in \mathrm{Surj}(p,\ell)_{\Sigma_\ell}$. Thus, for fixed $[\pi]$, the submodule $\bigoplus_{i=1}^pF_{[\pi\circ\sigma^i]}$ satisfies the hypotheses of Proposition \ref{prop:cyclic decomposition}. By induction $\bigoplus_{\mathrm{Surj}(\ell,p)_{\Sigma_\ell}}F_{[\pi]}$ decomposes as a direct sum of submodules of this form. The proposition now implies the claim.
\end{proof}
\begin{remark}
The reader may compare the complexity of this argument in homology to the argument in cohomology of \cite[III.10]{CohenLadaMay:HILS}.
\end{remark}
\section{Postscript: Lie algebra methods}\label{section:lie algebra methods}
In this short final portion of these notes, we give an informal discussion of an approach to computing the rational homology of configuration spaces of general manifolds, which is premised on Lie algebras.
\subsection{Lie algebras and their homology} We begin with a few reminders. Fix a field $\mathbb{F}$ of characteristic zero.
\begin{definition}
A \emph{graded Lie algebra} is a graded vector space $\mathfrak{g}$ equipped with a map \[[-,-]:\mathfrak{g}^{\otimes 2}\to \mathfrak{g},\] called the \emph{Lie bracket} of $\mathfrak{g}$, satisfying the following two identities:
\begin{enumerate}
\item $[x,y]=(-1)^{|x||y|+1}[y,x]$
\item $(-1)^{|x||z|}\left[x,[y,z]\right]+(-1)^{|z||y|}\left[z,[x,y]\right]+(-1)^{|y||x|}\left[y,[z,x]\right]=0.$
\end{enumerate}
\end{definition}
\begin{example}
An ordinary Lie algebra may be viewed as a graded Lie algebra concentrated in degree 0.
\end{example}
\begin{example}
Given a graded vector space $V$, there is a free Lie algebra $\mathcal{L}(V)$ satisfying the obvious universal property. In particular, it follows easily from the definition that \[\mathcal{L}\left(v_r\right)=\begin{cases}
\mathbb{F}\langle v\rangle&\quad r\text{ even}\\
\mathbb{F}\langle v\rangle\oplus\mathbb{F}\langle [v,v]\rangle &\quad r\text{ odd.}
\end{cases}\]
\end{example}
\begin{example}
If $\mathfrak{g}$ is a graded Lie algebra and $A$ a graded commutative (possibly nonunital) algebra, then $A\otimes\mathfrak{g}$ is canonically a graded Lie algebra with bracket determined by the formula \[[\alpha\otimes x,\beta\otimes y]=(-1)^{|x||\beta|}\alpha\beta\otimes[x,y].\]
\end{example}
The first identity of the definition above, which is usually called \emph{graded antisymmetry}, ensures that the bracket descends to a map $\mathrm{Sym}^2\left(\mathfrak{g}[1]\right)[-1]\to \mathfrak{g}[1]$. The second identity, known as the \emph{Jacobi identity}, ensures that the appropriate composite \[\mathrm{Sym}^3\left(\mathfrak{g}[1]\right)[-2]\to \mathrm{Sym}^2\left(\mathfrak{g}[1]\right)[-1]\to \mathfrak{g}[1]\] is zero. We now systematize these observations.
\begin{recollection}
For a graded vector space $V$, write $\mathrm{Sym}(V)=\mathbb{F}[V_{\mathrm{even}}]\otimes\Lambda_\mathbb{F}[V_{\mathrm{odd}}]$. This object is a graded cocommutative coalgebra under the \emph{shuffle coproduct} \[\Delta(x_1\cdots x_k)=\sum_{i+j=k}\sum_{\Sigma_k/\Sigma_i\times\Sigma_j}\epsilon(\sigma; x_1,\ldots, x_k)\, x_{\sigma(1)}\cdots x_{\sigma(i)}\otimes x_{\sigma(i+1)}\cdots x_{\sigma(k)},\] where $\epsilon(-;x_1,\ldots, x_k):\Sigma_k\to \{\pm 1\}$ is the group homomorphism determined by the formula $\epsilon(\tau_j;x_1,\ldots, x_k)=(-1)^{|x_j||x_{j+1}|}$.
Recall that a \emph{coderivation} of a graded coalgebra $(C,\Delta)$ is a self-map $\delta$ satisfying the ``co-Leibniz rule'' $\Delta\circ\delta=(\id\otimes \delta+\delta\otimes \id)\circ\Delta$. Coderivations of $\mathrm{Sym}(V)$ of fixed degree $m$ decreasing word length by $n$ are in bijection with graded maps \[\mathrm{Sym}^{n+1}(V)\to V\] of degree $m$ \cite[22(a)]{FelixHalperinThomas:RHT}.
\end{recollection}
From this universal property and graded antisymmetry, we conclude that the bracket of a Lie algebra $\mathfrak{g}$ determines a coderivation $d_{[-,-]}=d$ of $\mathrm{Sym}\left(\mathfrak{g}[1]\right)$ of degree $-1$. Moreover, since the square of an odd coderivation is again a coderivation, the Jacobi identity implies that $d^2$ is the unique coderivation determined by 0, which, of course, is zero. Thus, the following definition is sensible.
\begin{definition}
Let $\mathfrak{g}$ be a graded Lie algebra. The \emph{Chevalley-Eilenberg complex} of $\mathfrak{g}$ is the chain complex \[\mathrm{CE}(\mathfrak{g}):=\left(\mathrm{Sym}(\mathfrak{g}[1]),\,d_{[-,-]}\right).\] The \emph{Lie algebra homology} of $\mathfrak{g}$ is \[H^\mathcal{L}(\mathfrak{g}):=H\left(\mathrm{CE}(\mathfrak{g})\right).\]
\end{definition}
\begin{remark}
The formula given above for the coproduct $\Delta$ on $\mathrm{Sym}(V)$ is determined by requiring that \begin{enumerate}
\item $\Delta(x)=1\otimes x+x\otimes 1$ and
\item $\Delta(xy)=\Delta(x)\Delta(y)$,
\end{enumerate}
or, in other words, by requiring that the elements of $V$ be primitive and that $\Delta$ be a map of algebras. Note, however, that the Chevalley-Eilenberg complex is a differential graded coalgebra but \emph{not} a differential graded algebra.
\end{remark}
\begin{remark}
The differential in the Chevalley-Eilenberg complex may be computed explicitly as \[d(x_1\cdots x_k)=\sum_{1\leq i<j\leq k}(-1)^{|x_i|} \epsilon(\sigma_{ij};x_1,\ldots, x_k)\, [x_i,x_j]\,x_1\cdots \hat x_i\cdots \hat x_j\cdots x_k,\] where $\sigma_{ij}$ is the unique $(2,k-2)$-shuffle sending $i$ to $1$ and $j$ to $2$.
\end{remark}
\begin{remark}
As the terminology suggests, there is an isomorphism \[H^\mathcal{L}(\mathfrak{g})\cong \mathrm{Tor}_*^{U(\mathfrak{g})}(\mathbb{F},\mathbb{F}),\] where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.
\end{remark}
\begin{remark}
If $\mathfrak{g}$ is the Lie algebra associated to a compact Lie group, we have a composite quasi-isomorphism \[\mathrm{CE}(\mathfrak{g})^\vee\xrightarrow{\simeq}\Omega(G)^G\xrightarrow{\sim}\Omega(G),\] where $\Omega(G)^G$ is the space of left-invariant differential forms on $G$ \cite{ChevalleyEilenberg:CTLGLA}.
\end{remark}
\subsection{Lie algebra homology and configuration spaces}
The relevance of Lie algebra homology from our point of view is the following result.
\begin{theorem}[{\cite{Knudsen:BNSCSVFH}}]
Let $M$ be an orientable $n$-manifold. There is an isomorphism \[\bigoplus_{k\geq0}H_*(B_k(M))\cong H^\mathcal{L}\left(\mathfrak{g}_M\right)\] of bigraded vector spaces, where $\mathfrak{g}_M=H_c^{-*}(M)\otimes \mathcal{L}(v_{n-1,1})$.
\end{theorem}
\begin{remark}
Explicitly, we have
\[\mathfrak{g}_M=\begin{cases}
H_c^{-*}(M)\otimes v&\quad n \text{ odd}\\
H_c^{-*}(M)\otimes v\oplus H_c^{-*}(M)\otimes[v,v]&\quad n\text{ even,}
\end{cases}\] where in the first case all brackets vanish and in the second the bracket is determined up to sign by the cup product.
\end{remark}
\begin{remark}
An analogous statement for nonorientable manifolds is also valid---see \cite{Knudsen:BNSCSVFH}.
\end{remark}
\begin{example}
If $n$ is odd, then the Lie bracket in $\mathfrak{g}_M$ is identically zero, so the differential in $\mathrm{CE}(\mathfrak{g}_M)$ vanishes. Thus, equating auxiliary gradings, we find in this case that \begin{align*}
H_*(B_k(M))\cong\mathrm{Sym}^k\left(H_c^{-*}(M)[n-1][1]\right)\cong \mathrm{Sym}^k\left(H_*(M)\right)
\end{align*} by Poincar\'{e} duality, recovering the computation of Theorem \ref{thm:odd homology}.
\end{example}
\begin{example}
Let $M_1=T^2\setminus \mathrm{pt}$ and $M_2=\mathbb{R}^2\setminus S^0$. Then, since $M_1^+\cong T^2$ and $M_2^+\simeq S^1\vee S^1\vee S^2$, we have \[H_c^{-*}(M_j)\cong\begin{cases}
\mathbb{F}\left\langle x_{-1}, y_{-1}, z_{-2}\mid xy=z\right\rangle&\quad j=1\\
\mathbb{F}\left\langle x_{-1}, y_{-1}, z_{-2}\right\rangle&\quad j=2.
\end{cases}\] Thus, as a bigraded vector space, we have \[\mathfrak{g}_{M_j}\cong\mathbb{F}\left\langle x_{0,1}, y_{0,1}, z_{-1,1}, \tilde x_{1,2}, \tilde y_{1,2}, \tilde z_{0,2}\right\rangle,\] and the bracket is determined by \[[x,y]=\begin{cases}
\tilde z&\quad j=1\\
0&\quad j=2.
\end{cases}\] It follows that the weight 2 subcomplex of the Chevalley-Eilenberg complex is given additively as \[\xymatrix{\mathbb{F}\left\langle \tilde x, \tilde y, xy\right\rangle \ar[rr]^-{xy\mapsto [x,y]}&& \mathbb{F}\left\langle xz, yz, \tilde z\right\rangle\ar[r]^-{0}& \mathbb{F}\left\langle z^2\right\rangle.}\] It follows that \[\dim H_2(B_2(M_j))=\begin{cases}
2&\quad j=1\\
3&\quad j=2.
\end{cases}\] In particular, $B_2(T^2\setminus \mathrm{pt})\not\simeq B_2(\mathbb{R}^2\setminus S^0)$.
\end{example}
\begin{remark}
This type of computation can be pushed much further; indeed, \cite{DrummondColeKnudsen:BNCSS} determines $H_i(B_k(\Sigma_g))$ for every degree $i$, cardinality $k$, and genus $g$. For example, \[\dim H_{101}(B_{100}(\Sigma_3))=28,449,499.\]
\end{remark}
\subsection{Homological stability}
In the previous section, we saw that the rational homology of unordered configuration spaces may be computed as Lie algebra homology. Our present goal is to leverage this information in order to understand some of the qualitative behavior of these homology groups. To begin, recall that, if $\partial M\neq \varnothing$, there is a stabilization map \[B_k(M)\to B_{k+1}(M)\] defined by inserting a point in a collar neighborhood of the boundary, and this map induces an isomorphism on integral homology \cite{McDuff:CSPNP}. Surprisingly, although the stabilization map may fail to exist, stability actually holds in general, at least rationally.
\begin{theorem}[Church]
Let $M$ be a connected $n$-manifold with $n>2$. There is a map \[H_i(B_{k+1}(M);\mathbb{Q})\to H_i(B_k(M);\mathbb{Q})\] that is an isomorphism for $i\leq k$.
\end{theorem}
Although our approach to this result will differ from \cite{Church:HSCSM}, it will be motivationally useful to recall the definition of the map used therein. The idea is that, although one does not have a stabilization map in general, there is always a map going the other direction, at least at the level of ordered configurations, namely the projection of the Fadell-Neuwirth fibration $\mathrm{Conf}_{k+1}(M)\to \mathrm{Conf}_k(M)$. This map is only $\Sigma_k$-equivariant and so fails to descend to the unordered configuration spaces, but this can be remedied by remembering that there are in fact $k+1$ different such projections. The desired map is then obtained as the dashed filler in the commuting diagram \[\xymatrix{
H_*(\mathrm{Conf}_{k+1}(M))\ar[d]\ar[r]&H_*(B_{k+1}(M))\ar@{-->}[dd]^-{\pi}\\
H_*(\mathrm{Conf}_k(M))^{\oplus k+1}\ar[d]_-\Sigma\\
H_*(\mathrm{Conf}_k(M))\ar[r]&H_*(B_k(M)),
}\] where the $i$th coordinate of the upper left map is the projection away from the $i$th coordinate.
The key to our argument is the observation that this map is a piece of a larger structure. Indeed, whenever $i+j=k$, we have a $\Sigma\times\Sigma_j$-equivariant map \begin{align*}
\mathrm{Conf}_k(M)&\to \mathrm{Conf}_i(M)\times\mathrm{Conf}_j(M)\\
(x_1,\ldots, x_k)&\mapsto \left(\left(x_1,\ldots, x_i\right), \left(x_{i+1},\ldots, x_k\right)\right),
\end{align*} which induces the commuting diagram \[\xymatrix{
H_*(\mathrm{Conf}_k(M))\ar[d]\ar[r]&H_*(\mathrm{Conf}_i(M))\otimes H_*(\mathrm{Conf}_j(M))\otimes_{\Sigma_i\times\Sigma_j}\Sigma_k\ar[d]\\
H_*(B_k(M))\ar@{-->}[r]&H_*(B_i(M))\otimes H_*(B_j(M)),
}\] which, after summing over $k$ and $i+j=k$, produces a map \[\Delta:H_*(B(M))\to H_*(B(M))\otimes H_*(B(M)),\] where $B(M):=\coprod_{k\geq0}B_k(M)$. This map is the comultiplication of a cocommutative coalgebra structure on $H_*(B(M))$. Morally speaking, this comultiplication is given by the formula \[\text{``}\Delta(x_1,\ldots, x_k)= \sum_{i+j=k}\sum_{\Sigma_k/\Sigma_i\times\Sigma_j}(x_{\sigma(1)},\ldots, x_{\sigma(i)})\otimes (x_{\sigma(i+1)}, \ldots, x_{\sigma(k)})\text{''}\]
From this point of view, Church's map $\pi$ is obtained by using the comultiplication to split points apart and then discarding all summands not of the form $i=1$ and $j=k-1$. This type of operation is familiar in the theory of coalgebras.
\begin{definition}
Let $(C,\Delta)$ be a differential graded coalgebra and $\lambda\in C^\vee$ an $r$-cocycle. The \emph{cap product} with $\lambda$ is the composite \[\lambda\frown(-):C\cong\mathbb{Q}\otimes C\xrightarrow{\lambda\otimes\Delta}C^\vee[r]\otimes C\otimes C\cong C^\vee\otimes C\otimes C[r]\xrightarrow{\langle-,-\rangle,\otimes \id_C}\mathbb{Q}[r]\otimes C\cong C[r].\]
\end{definition}
Explicitly, if $\Delta(c)=\sum_{i} c_i\otimes c_i'$, then \[\lambda\frown c=\sum_i\langle \lambda, c_i\rangle c_i'.\] Since $\lambda$ is assumed to be closed, $\lambda\frown(-)$ is a chain map and hence induces a map of the same name at the level of homology.
The moral of our discussion so far is that Church's map $\pi$ is given by the cap product with the unit in $H^0(M)$, which is dual to the homology class of a single point, since $M$ is connected.
Now, as we saw last time, $H_*(B(M))$ may be computed using the Chevalley-Eilenberg complex of the Lie algebra $\mathfrak{g}_M$, which is a cocommutative coalgebra whose comultiplication obeys a very similar formula to that given above. This observation leads us to formulate the following version of Church's theorem, which is the one that we will prove. Let $x\in H_c^{-*}(M)[n]\subseteq \mathfrak{g}_M[1]\subseteq \mathrm{CE}(\mathfrak{g}_M)$ denote the Poincar\'{e} dual of the class of a point in $M$, and let $\lambda$ be the dual functional to $x$. Finally, write $\mathrm{CE}(\mathfrak{g}_M)_k$ for the summand of the Chevalley-Eilenberg complex of weight $k$.
\begin{theorem}\label{thm:my stability}
Let $M$ be a connected $n$-manifold with $n>2$. Cap product with $\lambda$ induces an isomorphism \[H_i(\mathrm{CE}(\mathfrak{g}_M)_{k+1})\xrightarrow{\simeq} H_i(\mathrm{CE}(\mathfrak{g}_M)_{k})\] for $i\leq k$.
\end{theorem}
\begin{remark}
The same conclusion holds in the case $n=2$ with a slightly worse stable range. The theorem is false for trivial reasons for $n=1$ and vacuous for $n=0$.
\end{remark}
In fact, we will show that the chain level cap product is a chain isomorphism in a range. In order to do so, it will be useful to have a formula for the cap product.
\begin{lemma}
$\lambda\frown(-)=\frac{d}{dx}$.
\end{lemma}
\begin{proof}
The claim is equivalent to the claim that $\lambda\frown x^ry=rx^{r-1}y$ for every monomial $y$ not divisible by $x$. We compute that \begin{align*}
\Delta(x^ry)&=\Delta(x)^r\Delta(y)\\
&=(1\otimes x+x\otimes 1)^r\sum_j y_j\otimes y_j'\\
&=\sum_{i,j}\binom{r}{i}x^i y_j\otimes x^{r-i}y_j',
\end{align*} whence \[\lambda\frown x^ry=\sum_{i,j}\binom{r}{i}\langle \lambda, x^i y_j\rangle x^{r-i}y_j'\]Since $\lambda$ is the dual functional to $x$, the $(i,j)$ term of this term vanishes unless $i=1$ and $y_j$ is a scalar.
There are now two cases. If $y$ itself is a scalar, then $\Delta(y)=y\otimes y$, and the claim follows easily. If $y$ is not a scalar, then $\Delta(y)\equiv 1\otimes y+y\otimes1$ modulo elementary tensors in which neither factor is a scalar, and the claim again follows easily.
\end{proof}
\begin{corollary}
The chain map $\lambda\frown(-)$ is surjective with kernel spanned by the monomials not divisible by $x$.
\end{corollary}
Thus, it remains to determine which monomials are divisible by $x$. For simplicity, we assume $n>2$, but
\begin{lemma}
If $n>2$, then any nonzero monomial $y$ with $\mathrm{wt}(y)>|y|$ is divisible by $x$.
\end{lemma}
\begin{proof}
Write $y=y_1\cdots y_r$ with $y_j\in \mathfrak{g}_M[1]$. Since $\mathrm{wt}(y)>|y|$, we conclude that $\mathrm{wt}(y_j)>|y_j|$ for some $j$. Since $y_j\in \mathfrak{g}_M[1]$, the weight of $y_j$ is either 1 or 2, and we treat these cases separately.
If $\mathrm{wt}(y_j)=1$, then $y_j\in H_c^{-*}(M)[n]$, which is concentrated in degrees $0\leq *\leq n$, so the assumption $|y_j|<1$ implies that $y_j=0$. Since $M$ is connected, it follows that $y_j$ is a multiple of $x$.
If $\mathrm{wt}(y_j)+2$, then $y_j\in H_c^{-*}(M)[2n-1]$, which is concentrated in degrees $n-1\leq *\leq 2n-1$. Since $|y_j|<2$ and $n>2$, it follows that $y_j=0$, a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:my stability}]
What we have shown so far is that the chain map \[\mathrm{CE}(\mathfrak{g}_M)_{k+1}\to \mathrm{CE}(\mathfrak{g}_M)_k\] induced by $\lambda\frown(-)$ is surjective and an isomorphism through degree $k$. An easy exercise shows that any chain map with these properties induces an isomorphism in homology through degree $k$.
\end{proof}
\begin{appendix}
\section{Split simplicial spaces}\label{appendix:split simplicial spaces}
In this appendix, we develop a criterion guaranteeing that a degreewise weak homotopy equivalence of simplicial spaces induces a weak homotopy equivalence after geometric realization. We follow \cite{DuggerIsaksen:THAR}, but similar results may be found in \cite[11.15]{May:GILS} and \cite[A.5]{Segal:CCT}.
\subsection{Split simplicial spaces}\label{section:split}
\begin{definition}
A simplicial space $\op X$ is \emph{split} if there are subspaces $N_m(\op X)\subseteq \op X_m$ for each $m\geq0$, called the \emph{non-degenerate part} in degree $n$, such that the map \[\coprod_{[n]\twoheadrightarrow[m]}N_m(\op X)\to \op X_n\] induced by the degeneracies is a homeomorphism for every $n\geq0$.
\end{definition}
\begin{proposition}[Dugger-Isaksen]\label{prop:split criterion}
Let $f:\op X\to \op Y$ be a map between split simplicial spaces. If $f_n:\op X_n\to \op Y_n$ is a weak equivalence for every $n\geq0$, then $|f|$ is a weak equivalence.
\end{proposition}
The strategy of the proof is simple. First, we argue that $f$ induces a weak equivalence on geometric realizations of $n$-skeleta for every $n$; second, we argue that every element in homotopy of the full realization is captured by some skeleton. In order to put this plan into action, we need to have control over skeleta.
\begin{lemma}\label{lem:split pushout}
Let $\op X$ be a split simplicial space. The diagram \[\xymatrix{
N_n(\op X)\times\partial \Delta^n\ar[r]\ar[d]&|\mathrm{sk}_{n-1}(\op X)|\ar[d]\\
N_n(\op X)\times\Delta^n\ar[r]&|\mathrm{sk}_n(\op X)|
}\] is a pushout square.
\end{lemma}
\begin{proof}
Recall that the \emph{tensor} of a space $X$ with a simplicial space $\op Z$ is the simplicial space $(X\otimes\op Z)_n=X\times \op Z_n$, with simplicial structure maps induced by those of $\op Z$, together with the identity on $X$. Since geometric realization, as a left adjoint, preserves colimits, it suffices to produce a pushout square in simplicial spaces of the form
\[\xymatrix{
N_n(\op X)\otimes\partial \Delta^n\ar[r]\ar[d]&\mathrm{sk}_{n-1}(\op X)\ar[d]\\
N_n(\op X)\otimes\Delta^n\ar[r]&\mathrm{sk}_n(\op X),
}\] where we have indulged in the traditional abuse of using the same notation $\Delta^n$ for the representable simplicial set $\mathrm{Hom}_\Delta(-,[n])$ and its geometric realization, and similarly for $\partial\Delta^n$. To verify that this diagram is a pushout, it suffices to check in each level. Now, it is direct from the definitions that \[\mathrm{sk}_n(\op X)_m=\mathrm{sk}_{n-1}(\op X)_m\amalg\left(\coprod_{[m]\twoheadrightarrow[n]}N_n(\op X)\right),\] so $\mathrm{sk}_n(\op X)_m$ is the pushout of $\mathrm{sk}_{n-1}(\op X)_m$ and $N_n(\op X)\times \Delta^n_m$ over a coproduct of copies of $N_n(\op X)$ indexed by the set of maps $f:[m]\to [n]$ that fail to be surjective, which is exactly $\partial\Delta^n_m$.
\end{proof}
This fact will only be useful once we are assured that such pushouts are homotopically well-behaved. With regularity assumptions on the spaces involved, the following type of result is common knowledge, but in fact it holds in complete generality.
\begin{lemma}\label{lem:pushout invariant}
If $f:A\to A'$ and $g:B\to B'$ are weak homotopy equivalences, and if the front and back faces in the commuting diagram
\[\xymatrix{&A\times \partial\Delta^n\ar[rr]\ar[dd]_>>>>>>>{}|!{[d]}\hole\ar[dl]_-{f\times\id_{\partial\Delta^n}}&&B\ar[dd]\ar[dl]_-g\\
A'\times\partial\Delta^n\ar[rr]\ar[dd]&& B'\ar[dd]\\
&A\times\Delta^n\ar[dl]_{f\times\id_{\partial\Delta^n}}\ar[rr]^<<<<<<<<{}|!{[r]}\hole&&C\ar[dl]^-h
\\
A'\times\Delta^n\ar[rr]&&C'
}\] are pushout squares, then $h:C\to C'$ is a weak homotopy equivalence.
\end{lemma}
\begin{proof}
We cover $C$ by two open sets, the first being $U_1\times A\times D$, where $D\subseteq\mathring{\Delta}^n$ is a Euclidean neighborhood of the barycenter, and the second $U_2=B\coprod_{A\times\partial\Delta^n}(A\times P)$, where $P\subseteq \Delta^n$ is the complement of the barycenter. Similarly, we cover $C'$ by $U_1'$ and $U_2'$. Clearly, $h^{-1}(U_j')=U_j$ for $j\in\{0,1\}$. Consider the commuting diagrams \[\xymatrix{
U_1\ar[d]_-{h|_{U_1}}\ar[r]&A\ar[d]^-f&& U_2\ar[d]_-{h|_{U_2}}&B\ar[l]\ar[d]^-g&& U_1\cap U_2\ar[d]_-{h|_{U_1\cap U_2}}&A\times(D\cap P)\ar@{=}[l]_-{\simeq}\ar[d]^-{f\times \id_{D\cap P}}\\
U_1'\ar[r]&A'&& U_2'&\ar[l]B'&&U_1'\cap U_2'\ar@{=}[r]^-{\simeq}&A'\times(D\cap P),
}\] where the horizontal arrows in the leftmost diagram are the projections onto the first factor, and the horizontal arrows in the middle idagram are induced by the inclusion $\partial\Delta^n\subseteq P$. Both horizontal arrows in the leftmost diagram are homotopy equivalences, and $f$ is a weak homotopy equivalence by assumption; both horizontal arrows in the middle diagram are inclusions of deformation retracts, and $g$ is a weak homotopy equivalence by assumption; and $f\times\id_{D\cap P}$ is a weak homotopy equivalence by assumption. Thus, by two-out-of-three, all three restrictions of $h$ are weak homotopy equivalences, so $h$ itself is a weak homotopy equivalence.
\end{proof}
In verifying that elements in the homotopy groups of $|\op X|$ are all captured by skeleta, we must be assured that the inclusions among skeleta are not too pathological. This assurance takes the form of a relative separation axiom.
\begin{definition}
A subspace $A\subseteq B$ is \emph{relatively $T_1$} if any open set $U\subseteq A$ may be separated from any point $b\in B\setminus U$ by an open set $U\subseteq V\subseteq B$. An inclusion map is \emph{relatively $T_1$} if its image is so.
\end{definition}
This terminology is motivated by the observation that a space is $T_1$ if and only if each of its points is relatively $T_1$. Since finite intersections of open sets are open, we have the following immediate consequence:
\begin{lemma}
If $A\subseteq B$ is relatively $T_1$, then any open set $U\subseteq A$ may be separated from any finite subset of $B\setminus U$ by an open set $U\subseteq V\subseteq B$.
\end{lemma}
The importance of this notion for our purposes is the following result.
\begin{proposition}\label{prop:factor through colimit}
Let $Y_i\subseteq Y_{i+1}$ be a relatively $T_1$ inclusion for $i\geq 1$. If $K$ is compact, then any map $f:K\to \colim_\mathbb{N} Y_i$ factors through the inclusion of some $Y_i$.
\end{proposition}
\begin{proof}
If $f$ does not factor as claimed, then, without loss of generality, we may assume the existence of $x_i\in \mathrm{im}(f)\cap Y_i$ for each $i\geq1$. Recall that a subset of the colimit is open precisely when its intersection with each stage is open; thus, for each $j\geq1$, we may define an open subset $U_j\subseteq\colim_\mathbb{N} Y_i$ by the following prescription:
\begin{enumerate}
\item for $1\leq i<j$, set $U_{ij}=\varnothing$;
\item for $i=j$, set $U_{ij}=Y_j$;
\item for $i>j$, take $U_{ij}$ to be an open subset of $Y_i$ separating $U_{i-1,j}$ from the set $\{x_{j+1},\ldots, x_i\}$;
\item finally, set $U_j=\colim_\mathbb{N}U_{ij}$.
\end{enumerate}
Then $U_j\cap Y_i=U_{ij}$, so $U_j$ is an open subset the colimit, and, since $Y_j\subseteq U_{j}$, the collection $\{U_j\}_{j\in\mathbb{N}}$ is an open cover of $\colim_\mathbb{N}Y_i$. Since $K$ is compact, $\mathrm{im}(f)$ is compact, so it is contained in some finite subcover $\{U_{j_r}\}_{r=1}^N$. But, by construction, $U_{j_r}$ does not contain $x_i$ for $i>j_r$, so $\bigcup_{r=1}^N U_{j_r}$ does not contain $x_i$ for $i>\max\{j_r:1\leq r\leq N\}$, a contradiction.
\end{proof}
This fact will only be useful once we are able to identify relatively $T_1$ maps, a task that is made easier by the following observation.
\begin{lemma}\label{lem:stable under pushout}
Relatively $T_1$ inclusions are stable under finite products and pushouts along arbitrary continuous maps.
\end{lemma}
\begin{proof}
For the first claim, it suffices by induction to show that $A_1\times A_2\subseteq B_1\times B_2$ is relatively $T_1$ if each $A_j\subseteq B_j$ is so. Fix an open subset $U\subseteq A_1\times A_2$ and a point $(x_1,x_2)\in B_1\times B_2\setminus U$. By the definition of the product topology, we have $U=\bigcup_{i\in I}U_{i1}\times U_{i2}$ for open subsets $U_{ij}\subseteq A_j$. By our assumption on the inclusions of the $A_j$, we may find open subsets $U_{ij}\subseteq W_{ij}\subseteq B_j$ for each $i\in I$ such that $x_j\notin W_{ij}$. Then $U\subseteq W:=\bigcup_{i\in I}W_{i1}\times W_{i2}$ is open in $B_1\times B_2$, and $(x_1,x_2)\notin W$, as desired.
For the second claim, suppose that the diagram \[\xymatrix{
A\ar[r]^-f\ar[d]_-i&Y\ar[d]\\
B\ar[r]^-g&Z
}\] is a pushout square and that $i$ is a relatively $T_1$ inclusion. Fix an open subset $U\subseteq Y$ and a point $z\in Z\setminus U$ (here, in a small abuse, we identity $Y$ with its image in $Z$, since the pushout of an inclusion is an inclusion). There are two cases to consider.
Assume first that $z\in Y$. Since $f^{-1}(U)$ is open in $A$ and $i$ is an inclusion, there is an open subset $W\subseteq B$ with $W\cap A=f^{-1}(U)$, and $W\amalg_{f^{-1}(U)} U\subseteq Z$ is open. To see that $z$ is not contained in this subset, it suffices to show that $z\notin g(W)$, since $z\notin U$ by assumption. But $z\in Y$, and $Y\cap g(B)=f(A)$, so $Y\cap g(W)=f(W\cap A)=U$, and the claim follows.
On the other hand, suppose that $z\notin Y$; in particular, $z=g(b)$ for a unique $b\in B$. Then $b\notin i(f^{-1}(U))$ and $f^{-1}(U)\subseteq A$ is open, so, since $i$ is relatively $T_1$, there is an open subset $i(f^{-1}(U))\subseteq W\subseteq B$ with $b\notin W$. As before, $W\coprod_{f^{-1}(U)} U$ is open in $Z$ and clearly does not contain $z$.
\end{proof}
\begin{corollary}\label{cor:pushout is T1}
For any pushout diagram of the form\[\xymatrix{
A\times \partial \Delta^n\ar[r]\ar[d]_-{\id_A\times(\partial\Delta^n\subseteq\Delta^n)}&Y\ar[d]^-i\\
A\times \Delta^n\ar[r]& Z
}\] the inclusion $Y\to Z$ is relatively $T_1$.
\end{corollary}
Finally, we will need the following, essentially obvious, observation.
\begin{lemma}\label{lem:disjoint weak equivalence}
If $f:W\amalg X\to Y\amalg Z$ is a weak homotopy equivalence such that $f|_W$ factors through $Y$ as a weak homotopy equivalence, then $f|_X$ factors through $Z$ as a weak homotopy equivalence.
\end{lemma}
\begin{proof}
The claim that $f|_X$ factors through $Z$ is obvious after applying $\pi_0$ and considering the analogous claim for bijections of sets, since $\pi_0(f)$ is a bijection. The claim that this factorization is a weak homotopy equivalence follows in the same way after applying $\pi_n$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:split criterion}]
Fix $N_n(\op X)\subseteq \op X_n$ and $N_n(\op Y)\subseteq\op Y$ witnessing $\op X$ and $\op Y$ as split. We claim that the restriction of $f_n$ to $N_n(\op X)$ factors through $N_n(\op Y)$ as a weak homotopy equivalence for every $n\geq0$. Having established this, it will follow by induction and Lemmas \ref{lem:split pushout} and \ref{lem:pushout invariant} above that the induced map $|\mathrm{sk}_n(\op X)|\to |\mathrm{sk}_n(\op Y)|$ is a weak homotopy equivalence for every $n\geq0$.
To establish the claim, we proceed by induction on $n$, the base case following from our assumption, since $N_0(\op X)=\op X_0$ and similarly for $\op Y$. For the induction step, we note that the inductive assumption implies that the dashed filler in the diagram \[\xymatrix{
N_m(\op X)\ar@{-->}[d]\ar[r]&\op X_m\ar[d]^-{f_m}\ar[r]^-{s}&\op X_n\ar[d]^-{f_n}\\
N_m(\op Y)\ar[r]&\op Y_m\ar[r]^{s}&\op Y_n
}\] exists and is a weak homotopy equivalence for every $m<n$ and every degeneracy $s$. Thus, the dashed filler in the diagram \[\xymatrix{
\displaystyle\coprod_{[n]\twoheadrightarrow[m]\neq[n]}N_m(\op X)\ar[r]\ar@{-->}[d]&\op X_n\ar[d]^-{f_n}\\
\displaystyle\coprod_{[n]\twoheadrightarrow[m]\neq[n]}N_m(\op Y)\ar[r]&\op Y_n
}\] exists and is a weak homotopy equivalence. Since the righthand map is also a weak homotopy equivalence, the claim follows from Lemma \ref{lem:disjoint weak equivalence}.
Now, from Lemma \ref{lem:split pushout} and Corollary \ref{cor:pushout is T1}, it follows by induction that each of the inclusions $|\mathrm{sk}_n(\op X)|\to |\mathrm{sk}_{n+1}(\op X)|$ is relatively $T_1$, and similarly for $\op Y$. Since $|\op X|=\colim_\mathbb{N}|\mathrm{sk}_n(\op X)|$, and likewise for $\op Y$, we conclude from Proposition \ref{prop:factor through colimit} that any map $(D^{m}, S^{m-1})\to (|\op Y|,|\op X|)$ factors as in the solid commuting diagram \[\xymatrix{
S^{m-1}\ar[d]\ar[r]&|\mathrm{sk}_n(\op X)|\ar[d]\ar[r]&|\op X|\ar[d]^-{|f|}\ar[d]\\
D^m\ar@{-->}[ur]\ar[r]&|\mathrm{sk}_n(\op Y)|\ar[r]&|\op Y|.
}\] We have already shown the middle arrow to be a weak homotopy equivalence, so the dashed filler exists making the upper triangle commute and the lower triangle commute up to homotopy relative to $S^{m-1}$. Thus, $\pi_m(f)=0$ for every $m\geq0$, and the claim follows.
\end{proof}
\subsection{Some examples}\label{section:loose ends}
In this section, we identify a few classes of split simplicial spaces used in the main text. We begin by noting two easy consequences of the simplicial identities. First, for any simplicial space $\op X$, each degeneracy $s_i:\op X_{n-1}\to \op X_n$ is injective. Second, the intersection $s_i(\op X_{n-1})\cap s_j(\op X_{n-1})$ is contained in the union of the images of the various iterated degeneracies $\op X_{n-2}\to \op X_n$.
\begin{lemma}\label{lem:covering map split}
Let $f:\op X\to \op Y$ be a degreewise covering map. If $\op Y$ is split, then $\op X$ is split.
\end{lemma}
\begin{proof}
Setting $N_0(\op X)=\op X_0$, assume for $m<n$ that $N_m(\op X)\subseteq \op X_m$ has been constructed with the desired property. The degeneracies of $\op X$ induce a map \[s:\coprod_{\pi:[n]\twoheadrightarrow[m]\neq [n]}N_m(\op X)\to \op X_n,\] and the observations above imply that $s$ is injective. We claim that $s$ is a local homeomorphism and hence the inclusion of a collection of connected components, which is enough to imply the claim, for in this case we may take $N_n(\op X)=\op X_n\setminus\mathrm{im}(s)$. To see that $s$ is a local homeomorphism, we may restrict our attention to the component indexed by $\pi$, in which case it suffices to show that the top map in the commuting diagram \[\xymatrix{
\op X_n\ar[d]_-{f_n}& \op X_m\ar[l]_-{\pi^*}\ar[d]^-{f_m}\\
\op Y_n& \op Y_m\ar[l]^-{\pi^*}
}\] is a local homeomorphism. Since $\op Y$ is split, the bottom map is a local homeomorphism, and the righthand map, as a covering map, is a local homeomorphism. Thus, by commutativity, given $x\in \op X_m$, there is a connected open neighborhood $x\in U$ such that $(f_n\circ\pi^*)|_{U}$ is a homeomorphism onto its image. Since $U$ is connected and $f_n$ is a covering map, there is a subspace $V\subseteq \op X_n$ such that $\pi^*(U)\subseteq V$ and $f_n|_V$ is a homeomorphism onto its image. It follows that $\pi^*|_{U}$ is a homeomorphism onto its image, as desired.
\end{proof}
\begin{corollary}\label{cor:hypercovers are split}
If $\op X$ is a hypercover, then $\op X$ is split.
\end{corollary}
\begin{proof}
Since the condition of being split is a condition that, in each degree, involves only a truncation of the simplicial space in question, and since a hypercover $\op X$ coincides with some bounded hypercover through any simplicial degree, it suffices to assume that $\op X$ is bounded of height at most $N$. We proceed by induction on $N$, the base case being the observation that \v{C}ech covers are split, with degree $n$ non-degenerate part given by the union of the $(n+1)$-fold intersections of cover elements in which no two adjacent indices coincide. The inductive step follows from Lemma \ref{lem:covering map split}, since $\op X\to \mathrm{cosk}_{N-1}(\op X)$ is a degreewise covering map with split target by Lemma \ref{lem:coskeleta} and induction.
\end{proof}
We also have the following simple observation.
\begin{lemma}\label{lem:split levelwise}
If $f:\op X\to \op Y$ is a levelwise weak equivalence and $\op Y$ is split, then $\op X$ is split.
\end{lemma}
\begin{corollary}\label{cor:horizontal cech cover split}
Let $f:\op X\to \op Y$ be a degreewise covering map and $\op W$ the bisimplicial space obtained by forming the degreewise \v{C}ech nerve. If $\op Y$ is split, then $|\op W|_h$ is split.
\end{corollary}
Finally, we have the following observation, which is immediate from the fact that the singular functor and geometric realization each preserve coproducts.
\begin{lemma}\label{lem:realization split}
If $\op X$ is split, then the simplicial space $|\mathrm{Sing}(\op X)|$, obtained by applying the singular functor and geometric realization in each degree, is split.
\end{lemma}
\section{Homotopy colimits}\label{section:homotopy colimits}
\subsection{Bar construction}
Recall that, if $M$ and $N$ are differential graded modules over a ring $R$, then the homology of the relative tensor product $M\otimes_R N$ may fail to be invariant under quasi-isomorphisms in the factors. This defect can be eliminated by resolving $M$, say, by nice $R$-modules before computing the tensor product, and a canonical choice of such a resolution with free entries is given by the \emph{bar complex} \[\cdots\to M\otimes R^{\otimes n+1}\to \cdots\to M\otimes R\to M,\] where the differentials are defined as alternating sums of maps defined in terms of the multiplication in $R$ and the module structure of $M$.
In order to apply this intuition in more general contexts, we note that the bar complex arises from a simplicial $R$-module whose construction depended only on the existence of a free/forgetful adjunction relating differential graded $R$-modules to chain complexes. The corresponding simplicial \emph{monadic bar construction} is available whenever such adjunction data is present.
In our context, the role of the $R$-module $M$ is played by a functor $F:I\to \mathcal{T}\mathrm{op}$. Since a ring is roughly like a category with one object, we guess that the corresponding forgetful functor should be the functor that only remembers the values of a functor on objects. In other words, writing $I_0$ for the discrete category with the same objects as $I$, we should consider the forgetful functor \[\iota^*:\mathcal{T}\mathrm{op}^I\to \mathcal{T}\mathrm{op}^{I_0}\] given by restriction along the inclusion $\iota:I_0\to I$. This functor does admit a left adjoint $\iota_!$, given by left Kan extension. The standard formula for the left Kan extension gives us \begin{align*}
\iota_!\iota^*F(i)=\colim\left((\iota\downarrow i)\to I_0\xrightarrow{\iota} I\xrightarrow{F} \mathcal{T}\mathrm{op}\right)\cong \coprod_{i_1\to i}F(i_1),
\end{align*} and, more generally, \[(\iota_!\iota^*)^{n+1}F(i)\cong\coprod_{i_{n+1}\to \cdots \to i_1\to i}F(i_{n+1}).\] We obtain in this way a simplicial functor, whose geometric realization maps to $F$, which we wish to think of as a kind of resolution of $F$. Our guess, then, is that the homotopy colimit should be the colimit of this geometric realization. Since colimits commute with geometric realization, and since \[\colim_I(\iota_!\iota^*)^{n+1} F\cong \colim_{I_0}\iota^*(\iota_!\iota^*)^nF\cong \coprod_{i_0\in I}(\iota_!\iota^*)^nF(i_0)\cong \coprod_{i_n\to \cdots \to i_0}F(i_n),\] this guess may be rephrased in terms of the following object.
\begin{definition}
Let $F:I\to \mathcal{T}\mathrm{op}$ be a functor. The \emph{(simplicial) bar construction} on $F$ is the simplicial space $\mathrm{Bar}_\bullet(F)$ with \[\mathrm{Bar}_n(F)=\coprod_{i_n\to \cdots \to i_0}F(i_n)\] and with face and degeneracy maps given by the composition and identities in $I$, respectively.
\end{definition}
\begin{hypothesis}
Let $F:I\to \mathcal{T}\mathrm{op}$ be a functor. The homotopy colimit of $F$ is the space \[\hocolim_I F=|\mathrm{Bar}_\bullet(F)|.\]
\end{hypothesis}
The first check of this hypothesis is to verify the following.
\begin{proposition}
Let $F,G:I\to \mathcal{T}\mathrm{op}$ be functors and $\varphi:F\to G$ a natural transformation. If $\varphi(i)$ is a weak homotopy equivalence for every $i\in I$, then $|\mathrm{Bar}_\bullet(\varphi)|$ is also a weak homotopy equivalence.
\end{proposition}
\begin{proof}
It is easy to see that the simplicial space $\mathrm{Bar}_\bullet(F)$ is split, so the claim follows from Proposition \ref{prop:split criterion}.
\end{proof}
Since there is a natural map $|\mathrm{Bar}_\bullet(F)|\to \colim_I F$ supplied by the observation that \[\colim_IF\cong \mathrm{coeq}\left(\coprod_{i_1\to i_0}F(i_1)\rightrightarrows \coprod_{i_0}F(i_0)\right),\] we may summarize our progress so far as having exhibited one homotopy invariant approximation to the colimit. On the other hand, another homotopy invariant approximation to the colimit is the constant functor with value $\varnothing$! Why should we think that our proposed construction of the homotopy colimit is any better than this functor?
\subsection{Derived functors} We now formulate precisely what it means to be the best approximation by a homotopy invariant functor.
\begin{definition}
A \emph{category with weak equivalences} is a pair $(\op C, \mathrm{weq}(\op C))$ of a category and a collection of morphisms that contains the isomorphisms and has the property that if, in the commuting diagram \[\xymatrix{
A\ar[dr]\ar[rr]&&B\ar[dl]\\
&B
}\] in $\op C$, any two arrows lie in $\mathrm{weq}(\op C)$, then so does the third.
\end{definition}
The arrows in $\mathrm{weq}(\op C)$ are called \emph{weak equivalences}, and the closure property is called \emph{two-out-of-three}.
\begin{example}
Two cases of interest are the category of topological spaces with weak homotopy equivalences and the category of chain complexes with quasi-isomorphisms.
\end{example}
\begin{example}
If $(\op C, \mathrm{weq}(\op C))$ is a category with weak equivalences and $I$ is a category, then the functor category $\op C^I$ is again a category with weak equivalences when equipped with the \emph{pointwise} weak equivalences, i.e., a natural transformation is a weak equivalence if and only if each of its components is so.
\end{example}
\begin{definition}
Let $(\op C, \mathrm{weq}(\op C))$ be a category with weak equivalences. A \emph{homotopy category} for $\op C$ is a category $\mathrm{Ho}(\op C)$ equipped with a functor $\gamma=\gamma_{\op C}:\op C\to \mathrm{Ho}(\op C)$ with the following properties.
\begin{enumerate}
\item If $f\in\mathrm{weq}(\op C)$, then $\gamma(f)$ is an isomorphism.
\item Any functor $F:\op C\to \op D$ sending weak equivalences in $\op C$ to isomorphisms in $\op D$ factors uniquely through $\gamma$.
\end{enumerate}
\end{definition}
If $\op C$ has a homotopy category, then it is unique up to a unique equivalence of categories, so there is no harm in referring to \emph{the} homotopy category of $\op C$.
Often it will be the case, as with the colimit functor, that one is given a functor $F$ that does not send weak equivalences to isomorphisms. In this case, one can ask for the best approximation to $F$ by a functor having this property.
\begin{definition}
Let $(\op C, \mathrm{weq}(\op C))$ be a category with weak equivalences and $F:\op C\to \op D$ a functor. A \emph{left derived functor} of $F$ is a functor $\mathbb{L}F:\mathrm{Ho}(\op C)\to \op D$ equipped with a natural transformation $\mathbb{L}F\circ\gamma\to F$ that is final among functors $T:\mathrm{Ho}(\op C)\to \op D$ equpped with natural transformations $T\circ \gamma\to F$.
\end{definition}
Dually, we have the notion of a \emph{right} derived functor $\mathbb{R}F$. Note that, categorically speaking, the left derived functor $\mathbb{L}F$ is the right Kan extension of $F$ along $\gamma$.
\begin{definition}
Let $(\op C,\mathrm{weq}(\op C))$ be a category with weak equivalences, and assume that $\op C$ admits colimits indexed by $I$. The \emph{homotopy colimit} functor for $I$-shaped diagrams, if it exists, is the left derived functor of the composite $\gamma\circ\colim_I$, as depicted in the following digram: \[\xymatrix{
\op C^I\ar[d]_-\gamma\ar[rr]^-{\colim_I}&&\op C\ar[d]^-\gamma\\
\mathrm{Ho}(\op C^I)\ar[rr]^-{\hocolim_I}\ar@{=>}[urr]&&\mathrm{Ho}(\op C)
}\]
\end{definition}
One says that $\hocolim_I$ is the \emph{total left derived functor} of $\colim_I$.
Thus, the homotopy colimit is the closest approximation to the colimit by a homotopy invariant construction.
\subsection{Model structures} Our next task is to make this definition into something useable in practice. Our approach, following \cite{Quillen:HA}, will be to impose extra structure on our category, motivated by the structure observed ``in the wild'' in the homotopy theory of spaces, enabling us to make these abstract notions concrete. It should be emphasized, however, that this structure is scaffolding, and that the fundamental objects of interest are all at the level of the bare category with weak equivalences.
\begin{definition}
Let $(\op C, \mathrm{weq}(\op C))$ be a category with weak equivalences, and assume that $\op C$ has small limits and colimits. A \emph{model structure} on $\op C$ is a pair $(\mathrm{cof}(\op C),\mathrm{fib}(\op C))$ of classes of morphisms in $\op C$ satisfying the following axioms.
\begin{enumerate}
\item Both $\mathrm{cof}(\op C)$ and $\mathrm{fib}(\op C)$ are closed under retracts in the arrow category $\op C^{\Delta^1}$.
\item If $i\in\mathrm{cof}(\op C)$ and $p\in \mathrm{fib}(\op C)$, then the dashed filler exists in the commuting diagram \[\xymatrix{
A\ar[d]_-i\ar[r]&B\ar[d]^-p\\
C\ar@{-->}[ur]\ar[r]&D
}\] provided either $i$ or $p$ is a weak equivalence.
\item Any morphism $f:A\to B$ in $\op C$ may be factored as
\begin{enumerate}
\item $f=p\circ i$ with $i\in \mathrm{cof}(\op C)\cap \mathrm{weq}(\op C)$ and $p\in \mathrm{fib}(\op C)$ and as
\item $f=q\circ j$ with $j\in \mathrm{cof}(\op C)$ and $q\in \mathrm{fib}(\op C)\cap \mathrm{weq}(\op C)$.
\end{enumerate}
\end{enumerate}
A \emph{model category} is a category with weak equivalences equipped with a model structure. The morphisms in $\mathrm{cof}(\op C)$, $\mathrm{fib}(\op C)$, $\mathrm{cof}(\op C)\cap \mathrm{weq}(\op C)$, and $\mathrm{fib}(\op C)\cap \mathrm{weq}(\op C)$ are the \emph{cofibrations}, the \emph{fibrations}, the \emph{trivial cofibrations}, and the \emph{trivial fibrations}, respectively. An object is \emph{cofibrant} if the unique morphism from the initial object of $\op C$ is a cofibration (resp. \emph{fibrant}, final, fibration).
\end{definition}
In the situation of (2), we say that $i$ has the \emph{left lifting property} with respect to $p$, and that $p$ has the \emph{right lifting property} with respect to $i$.
\begin{exercise}[{See \cite[7]{Hirschhorn:MCL}}]
Derive the following consequences of the model category axioms.
\begin{enumerate}
\item Weak equivalences are closed under retracts in the arrow category.
\item (Trivial) cofibrations are closed under coproducts and pushouts along arbitrary morphisms.
\item (Trivial) fibrations are closed under products and pullbacks along arbitrary morphisms.
\item Cofibrations are exactly those morphisms with the left lifting property with respect to every trivial fibration, and similarly for the other classes of morphisms.
\end{enumerate}
\end{exercise}
From the point of view of our motivating questions, the benefit of the presence of a model structure on a category with weak equivalences is that it allows for explicit control over the homotopy category and derived functors.
\begin{proposition}[{\cite[8.3.5, 8.4.4]{Hirschhorn:MCL}}]
Let $\op C$ be a model category and $F:\op C\to \op D$ a functor.
\begin{enumerate}
\item The homotopy category $\mathrm{Ho}(\op C)$ exists.
\item If $F$ sends weak equivalences between cofibrant objects to isomorphisms in $\op D$, then $\mathbb{L} F$ exists, and the natural map $\mathbb{L}F(\gamma(C))\to \gamma(F(C))$ is an isomorphism for $C$ cofibrant (resp. fibrant, $\mathbb{R}F$).
\end{enumerate}
\end{proposition}
From what we have said so far, we draw the following strategy for working with homotopy colimits:
\begin{enumerate}
\item endow the functor category $\mathcal{T}\mathrm{op}^I$ with a model structure;
\item verify that, in this model structure, the total left derived functor of $\colim_I:\mathcal{T}\mathrm{op}^I\to \mathcal{T}\mathrm{op}$ exists; and
\item understand cofibrant replacement in this model structure.
\end{enumerate} With these steps accomplished, we may compute the homotopy colimit by cofibrantly replacing and computing the ordinary colimit.
We will be aided in accomplishing the second step of this strategy by the following result---see \cite[8.5.3, 8.5.18]{Hirschhorn:MCL}.
\begin{theorem}[Quillen]
Let $\op C$ and $\op D$ be model categories and \[\adjunct{\op C}{\op D}{F}{G}\] an adjunction. The following two conditions are equivalent: \begin{enumerate}
\item $F(\mathrm{cof}(\op C))\subseteq \mathrm{cof}(\op D)$ and $F(\mathrm{cof}(\op C)\cap \mathrm{weq}(\op C))\subseteq \mathrm{cof}(\op D)\cap \mathrm{weq}(\op D)$;
\item $G(\mathrm{fib}(\op D))\subseteq \mathrm{fib}(\op C)$ and $G(\mathrm{fib}(\op D)\cap \mathrm{weq}(\op D))\subseteq \mathrm{fib}(\op C)\cap \mathrm{weq}(\op C)$.
\end{enumerate} Moreover, if these conditions hold, there is an adjunction \[\adjunct{\mathrm{Ho}(\op C)}{\mathrm{Ho}(\op D).}{\mathbb{L}(\gamma_{\op D}\circ F)}{\mathbb{R}(\gamma_{\op C}\circ G)}\] In particular, both total derived functors exist.
\end{theorem}
Such an adjunction is commonly known as a \emph{Quillen adjunction} or \emph{Quillen pair}, and the functors $F$ and $G$ are left and right \emph{Quillen functors}, respectively.
Thus, if the putative model structure on $\mathcal{T}\mathrm{op}^I$ is chosen to be compatible with some fixed model structure on $\mathcal{T}\mathrm{op}$ in the sense that the diagonal functor $\mathcal{T}\mathrm{op}\to \mathcal{T}\mathrm{op}^I$ is a right Quillen functor, then the existence of the homotopy colimit will be assured.
We begin by specifying our choice of model structure on $\mathcal{T}\mathrm{op}$.
\begin{theorem}[Quillen]\label{thm:quillen model structure}
The following specifications define a model structure on the category $\mathcal{T}\mathrm{op}$ of topological spaces:
\begin{enumerate}
\item the weak equivalences are the weak homotopy equivalences;
\item the fibrations are the Serre fibrations;
\item the cofibrations are the retracts of relative cell complexes.
\end{enumerate}
\end{theorem}
\begin{proof}
We highlight only one aspect of the proof, namely the verification of the factorization axiom via the so-called \emph{small object argument}. For a detailed proof of the full result, see \cite{Hirschhorn:QMCTS}.
Let $f:X\to Y$ be any map, which we wish to factor as $f=p\circ i$ with $p$ a fibration and $i$ a trivial cofibration. We define $X_1$ as the pushout in the diagram \[\xymatrix{
\displaystyle\coprod\Lambda_k^n\ar[d]\ar[r]& X\ar[d]^-{i_1}\\
\displaystyle\coprod\Delta^n\ar[r]&X_1,
}\] where the coproduct in each case is indexed by the set \[\coprod_{n\geq0}\coprod_{0\leq k\leq n}\mathrm{Hom}(\Delta^n,Y)\times_{\mathrm{Hom}(\Lambda_k^n, Y)}\mathrm{Hom}(\Lambda_k^n, X).\] The simplicial set $\Lambda_k^n$ is the $k$th \emph{horn} of $\Delta^n$---see \cite[I.1]{GoerssJardine:SHT}. Setting $X_0=X$ and assuming $i_r:X_{r-1}\to X_r$ to have been defined, we apply the same procedure to the induced map $X_r\to Y$ to obtain $i_{r+1}:X_r\to X_{r+1}$, and we set $X_f=\colim_{r\geq0} X_r$. We claim that, in the commuting diagram \[\xymatrix{
X\ar[dr]_-f\ar[rr]^-i&& X_f\ar[dl]^-p\\
&Y,
}\] $i$ is a trivial cofibration and $p$ a fibration. The former claim involves finding a lift in the diagram \[\xymatrix{
X\ar[d]_-{i_1}\ar[d]\ar[rrr]&&&E\ar[ddd]^-q\\
X_1\ar[d]_-{i_2}\\
\vdots\ar[d]\\
X_f\ar[rrr]&&&B
}\] with $q$ a fibration, which may be accomplished inductively using the fact that each $i_r$ is a trivial cofibration, since each inclusion $\Lambda_k^n\subseteq \Delta^n$ is so. For the latter claim, we note that any map of pairs $(\Delta^n,\Lambda_i^n)\to (Y, X_f)$ factors as in the diagram \[\xymatrix{
\Lambda_i^n\ar[dd]\ar[r]&X_r\ar[d]_-{i_{r+1}}\ar[r]& X_f\ar[dd]^-p\\
&X_{r+1}\ar[ur]\ar[dr]\\
\Delta^n\ar@{-->}[ur]\ar[rr]&&Y,
}\] and the dashed filler exists by our definition of $X_{r+1}$ as the pushout.
In order to factor $f$ as a cofibration followed by a trivial fibration, we apply the same argument with horn inclusions replaced by the inclusion $\partial \Delta^n\subseteq \Delta^n$.
\end{proof}
\begin{remark}
Note that the factorization produced by the small object argument is even functorial.
\end{remark}
\subsection{Cofibrant generation} The only essential features of the category $\mathcal{T}\mathrm{op}$ used in the argument of Theorem \ref{thm:quillen model structure} was the existence of a collection $S=\{\Lambda_k^n\to \Delta^n\}$ of ``test maps'' for fibrations, and similarly for trivial fibrations, with the property that the domain of an element of $S$ is \emph{small} with respect to composites of iterated pushouts of members of $S$. These features can be axiomatized.
\begin{definition}
Let $\op C$ be a category and $S$ a set of morphisms in $\op C$. A morphism in $\op C$ is
\begin{enumerate}
\item $S$-\emph{injective} if it has the right lifting property with respect to every element of $S$;
\item an $S$-\emph{cofibration} if it has the left lifting property with respect to every $S$-injective;
\item a \emph{relative $S$-cell complex} if it is a composite of pushouts of elements of $S$.
\end{enumerate} We say that $S$ \emph{permits the small object argument} if the domains of elements of $S$ are small with respect to the collection of relative $S$-cell complexes.
\end{definition}
\begin{example}
Let $\op C=\mathcal{T}\mathrm{op}$. If $S$ is the collection of inclusions $\partial\Delta^n\subseteq\Delta^n$, then the $S$-injectives are the trivial fibrations, and the $S$-cofibrations are the cofibrations. On the other hand, if $S$ is the collection of inclusions $\Lambda_k^n\subseteq\Delta^n$, then the $S$-injectives are fibrations, and the $S$-cofibrations are the trivial cofibrations. Both of these collections permit the small object argument, since in both cases every element of $S$ is a relatively $T_1$ inclusion with compact domain.
\end{example}
Motivated by this example, we think of the elements of $S$ as ``generators'' for a set of cofibrations.
\begin{theorem}[Kan]
Let $(\op C, \mathrm{weq}(\op C))$ be a category with weak equivalences with $\mathrm{weq}(\op C)$ closed under retracts, and let $S_\mathrm{cof}$ and $S_\mathrm{triv}$ be sets of morphisms in $\op C$. Assume that \begin{enumerate}
\item $S_\mathrm{cof}$ and $S_\mathrm{triv}$ each permit the small object argument,
\item every $S_\mathrm{triv}$-cofibration is both an $S_\mathrm{cof}$-cofibration and a weak equivalence,
\item every $S_\mathrm{cof}$-injective is both an $S_\mathrm{triv}$-injective and a weak equivalence, and
\item one of the following conditions holds:
\begin{enumerate}
\item any map that is an $S_\mathrm{cof}$-cofibration and a weak equivalence is also an $S_\mathrm{triv}$-cofibration, or
\item any map that is an $S_\mathrm{triv}$-injective and a weak equivalence is also an $S_\mathrm{cof}$-injective.
\end{enumerate}
\end{enumerate}
Then $(\op C,\mathrm{weq}(\op C))$ extends to a model structure with cofibrations the $S_\mathrm{cof}$-cofibrations and fibrations the $S_\mathrm{triv}$-injectives.
\end{theorem}
\begin{proof}
Given $f:X\to Y$, we apply the same procedure as above to factor $f$ as $p \circ i$, where $p$ is an $S_\mathrm{triv}$-injective and $i$ is a relative $S_\mathrm{triv}$-cell complex. The latter is in particular an $S_\mathrm{triv}$-cofibration, so point (2) applies to show that this factorization is of the desired form. The dual argument, using point (3), furnishes the other factorization. Point (4) is used to demonstrate the lifting axiom. For a detailed account, see \cite[11.3.1]{Hirschhorn:MCL}.
\end{proof}
One can show that, in the situation of this theorem, the cofibrations are exactly the retracts of relative $S_\mathrm{cof}$-cell complexes, and similarly for trivial cofibrations and $S_\mathrm{triv}$---see \cite[10.5.22]{Hirschhorn:MCL}.
For obvious reasons, it is standard to refer to such collections $S_\mathrm{cof}$ and $S_\mathrm{triv}$ as \emph{generating} cofibrations and trivial cofibrations, respectively, and to refer to the resulting model category as \emph{cofibrantly generated}. One of the excellent features of cofibrantly generated model structures is that they are very portable.
\begin{corollary}
Let $\op C$ be a cofibrantly generated model category with generators $S_\mathrm{cof}$ and $S_\mathrm{triv}$, $\op D$ a category with small (co)limits, and \[\adjunct{\op C}{\op D}{F}{G}\] an adjunction. If \begin{enumerate}
\item $F(S_\mathrm{cof})$ and $F(S_\mathrm{triv})$ permit the small object argument, and
\item $G$ sends relative $F(S_\mathrm{triv})$-cell complexes to weak equivalences,
\end{enumerate} then the category with weak equivalences $(\op D, G^{-1}(\mathrm{weq}(\op C)))$ extends to model structure cofibrantly generated by $F(S_\mathrm{cof})$ and $F(S_\mathrm{triv})$. Moreover, $F$ and $G$ are a Quillen pair with respect to this model structure.
\end{corollary}
\begin{proof}
We apply the previous theorem. Prerequisitely, we note that $G^{-1}(\mathrm{weq}(\op C))$ satisfies two-out-of-three and is closed under retracts, and point (1) is true by assumption. For point (2), we note that $F(S_\mathrm{triv})$-cofibrations are retracts of relative $F(S_\mathrm{triv})$-cell complexes, which are sent to weak equivalences by assumption, so $F(S_\mathrm{triv})$-cofibrations are weak equivalences. To see that they are also $F(S_\mathrm{cof})$-cofibrations are also $F(S_\mathrm{triv})$-cofibrations, it suffices by definition to show that $F(S_\mathrm{cof})$-injectives are also $F(S_\mathrm{triv})$-injectives. For this, we note that the two lifting problems \[\xymatrix{
F(A)\ar[d]_-{F(i)}\ar[r]&B\ar[d]^-p&&A\ar[d]_-i\ar[r]&G(B)\ar[d]^-{G(p)}\\
F(C)\ar@{-->}[ur]\ar[r]&D&&C\ar@{-->}[ur]\ar[r]&G(D)
}\] are equivalent, and that $S_\mathrm{cof}$-injectives are also $S_\mathrm{triv}$-injectives.
The remaining assumptions are verified in a similar manner---see \cite[11.3.2]{Hirschhorn:MCL} for a detailed account.
\end{proof}
The resulting model structure on $\op D$ is called the \emph{transferred} model structure.
\subsection{Projective model structure} In our example of interest, we obtain the \emph{projective} model structure on functors:
\begin{corollary}[{\cite[11.6.1]{Hirschhorn:MCL}}]
Let $\op C$ be a cofibrantly generated model category and $I$ any small category. The following specifications define a model structure on the functor category $\op C^I$:
\begin{enumerate}
\item the weak equivalences are the pointwise weak equivalences;
\item the fibrations are the pointwise fibrations;
\item the cofibrations are the natural transformations with the left lifting property with respect to every pointwise fibration.
\end{enumerate}
\end{corollary}
Turning now to the question of cofibrant replacement, we write $\mathrm{Bar}_\bullet(F,-)$ for the augmented simplicial functor given in degree $n$ by \[\mathrm{Bar}_\bullet(F,-)=(\iota_!\iota^*)^{n+1}F(i)\cong\coprod_{I_0^{n+1}}F(i_{n+1})\times \mathrm{Hom}(i_{n+1}, i_n)\times\cdots\times\mathrm{Hom}(i_2,i_1)\times \mathrm{Hom}(i_1,-)\] and with the face and degeneracy maps induced by composition and insertion of identities, respectively. As we saw in the previous lecture, we have $\colim_I\mathrm{Bar}_\bullet(F,-)=\mathrm{Bar}_\bullet(F)$.
\begin{proposition}\label{prop:bar is cofibrant}
The augmentation $|\mathrm{Bar}_\bullet(F,-)|\to F$ is a pointwise weak homotopy equivalence. Moreover, if $F$ is pointwise cofibrant, then $|\mathrm{Bar}_\bullet(F,-)|$ is projective cofibrant.
\end{proposition}
\begin{proof}
Since $\iota^*$ reflects weak equivalences and commutes with colimits, it suffices to prove the first claim instead for the geometric realization of the augmented simplicial object $\iota^*\mathrm{Bar}_\bullet(F, -)$ in $\mathcal{T}\mathrm{op}^{I_0}$. But this augmented simplicial object has an extra degeneracy, which is given by the unit of the $(\iota_!,\iota^*)$-adjunction.
For the second claim, it will suffice to show that each of the maps $|\mathrm{sk}_{n-1}(\mathrm{Bar}_\bullet(F,-))|\to |\mathrm{sk}_n(\mathrm{Bar}_\bullet(F,-))|$ is a cofibration between cofibrant objects. We proceed by induction on $n$, the base case of the cofibrancy of the $0$-skeleton following from our assumption that $F$ is pointwise cofibrant. For the induction step, we write \[N_n(-)=\coprod_{I_0^{n+1}}F(i_{n+1})\times \mathrm{Hom}'(i_{n+1}, i_n)\times\cdots\times \mathrm{Hom}'(i_2,i_1)\times \mathrm{Hom}(i_1,-)\subseteq \mathrm{Bar}_\bullet(F,-),\] where \[\mathrm{Hom}'(i,j)=\begin{cases}
\mathrm{Hom}(i,j)&\quad i\neq j\\
\mathrm{Hom}(i,i)\setminus\{\id_i\}&\quad i=j.
\end{cases}
\] After evaluating at an object $i$, these spaces witness $\mathrm{Bar}_\bullet(F, i)$ as split, so we have the pushout square \[\xymatrix{
\partial\Delta^n\times N_n(i)\ar[r]\ar[d]&|\mathrm{sk}_{n-1}(\mathrm{Bar}_\bullet(F,i))|\ar[d]\\
\Delta^n \times N_n(i)\ar[r]&|\mathrm{sk}_{n}(\mathrm{Bar}_\bullet(F,i))|,
}\] which is moreover natural in $i$. Since the $(n-1)$-skeleton is known to be cofibrant by induction, it suffices to show that the map $\partial\Delta^n \times N_n(-)\to \Delta^n\times N_n(-)$ is a projective cofibration. For every $j\in I_0$, define a functor $N_n^{j}:I_0\to \mathcal{T}\mathrm{op}$ by \[N_n^{j}(i):=\begin{cases}
\displaystyle\coprod_{I_0^{n}}F(i_{n+1})\times \mathrm{Hom}'(i_{n+1}, i_{n})\times\cdots\times \mathrm{Hom}'(i_2,j)&\quad i=j\\
\varnothing&\quad\text{otherwise.}
\end{cases}\] Then we evidently have the commuting diagram \[\xymatrix{
\partial\Delta^n\times N_n(-)\ar[r]&\Delta^n\times N_n(-)\\
\displaystyle\coprod_{I_0}\iota_!\left(\partial\Delta^n\times N_n^{j}\right)\ar@{=}[u]^-\wr\ar[r]&\displaystyle\coprod_{I_0}\iota_!\left(\Delta^n\times N_n^{j}\right)\ar@{=}[u]_-\wr
}\] of functors, where the components of the bottom arrow are each induced by the inclusion $\partial \Delta^n\subseteq \Delta^n$. Since this map is a cofibration in $\mathcal{T}\mathrm{op}$, since products with cofibrant objects preserve cofibrations, since $\iota_!$ is a left Quillen functor, and since cofibrations are closed under coproducts, it follows that the bottom arrow, and hence the top arrow, in this diagram is a projective cofibration.
\end{proof}
\begin{corollary}
For any functor $F:I\to \mathcal{T}\mathrm{op}$, the canonical map $|\mathrm{Bar}_\bullet(F)|\to \colim_I F$ exhibits a homotopy colimit of $F$.
\end{corollary}
\begin{proof}
Since the values of the functor $\mathrm{Bar}_\bullet(-):\mathcal{T}\mathrm{op}^I\to \mathcal{T}\mathrm{op}^{\Delta^{op}}$ are all split, the functor $|\mathrm{Bar}_\bullet(-)|$ descends to a functor at the level of homotopy categories. Thus, from the existence of the canonical map to the colimit and the universal property of the derived functor, we obtain the commuting diagram
\[\xymatrix{
\gamma\left(|\mathrm{Bar}_\bullet(Q(F))|\right)\ar[d]\ar[r]&\gamma\left(|\mathrm{Bar}_\bullet(F)|\right)\ar[d]\\
\hocolim_IQ(F)\ar[r]&\hocolim_IF
}\] in $\mathrm{Ho}(\mathcal{T}\mathrm{op})$, where $Q:\mathcal{T}\mathrm{op} \to \mathcal{T}\mathrm{op}$ is any cofibrant replacement functor (for example, we may take $Q(X)$ to be the geometric realization of the singular simplicial set of $X$). The bottom map is an isomorphism, since weak equivalences in $\mathcal{T}\mathrm{op}^I$ are pointwise; the top map is a weak equivalence because both simplicial spaces are split; and the lefthand map is an isomorphism by Proposition \ref{prop:bar is cofibrant}. It follows that the righthand map is an isomorphism, as claimed.
\end{proof}
\begin{remark}
Most of what we have said carries over verbatim to the setting of a general cofibrantly generated simplicial model category, but the ability to forego pointwise cofibrant replacement seems to be an accident specific to $\mathcal{T}\mathrm{op}$, which is connected to the existence of the \emph{Str{\o}m model structure}---see \cite[A]{DuggerIsaksen:THAR} for further discussion.
\end{remark}
\begin{recollection}
To a category $I$, we may associate its \emph{nerve}, which is the simplicial set $NI$ given in degree $n$ by $NI_n=\mathrm{Fun}([n], I)$, the set of composable $n$-tuples of morphisms in $I$. The \emph{classifying space} of $I$ is the space $BI:=|NI|$, and we say that $I$ is \emph{contractible} if its classifying space is weakly contractible. We say that a functor $T:I\to J$ is \emph{homotopy final} if the overcategory $(j\downarrow T)$ is contractible for every object $j\in J$ (resp. \emph{homotopy initial}, $(T\downarrow j)$). Homotopy final functors induce weak homotopy equivalences on classifying spaces, and, since $BI\simeq BI^{op}$, the same holds for homotopy initial functors.
\end{recollection}
\begin{example}
A category with an initial or final object is contractible, as is a (co)filtered category.
\end{example}
We record the following standard facts about homotopy colimits.
\begin{proposition}[{\cite[6.7,\,20.3]{Dugger:PHC}}]\label{prop:hocolim facts}
Let $F:J\to \mathcal{T}\mathrm{op}$ be a functor.
\begin{enumerate}
\item If $T:I\to J$ is homotopy final, then the induced map \[\hocolim_I T^*F\to \hocolim_J F\] is a weak equivalence.
\item If $J=\Delta^{op}$ and $F$ is split, then \[\hocolim_{\Delta^{op}}F\simeq |F|.\]
\end{enumerate}
\end{proposition}
\subsection{Relative homotopy colimits} We will also have use for a relative version of the homotopy colimit. Note that, if $\lambda:I\to J$ is a functor, then the restriction $\lambda^*$ preserves fibrations and weak equivalences in the respective projective model structures, since both are pointwise. Thus, $(\lambda_!,\lambda^*)$ is a Quillen pair, and we may contemplate the \emph{homotopy left Kan extension} $\mathrm{hoLan}_\lambda:=\mathbb{L}\lambda_!$. In order to understand this functor, we recall that, from the commuting diagram of categories \[\xymatrix{
(\lambda\downarrow j)\ar[d]_-\mathrm{pt}\ar[r]^-{\pi_j} &I\ar[d]^-\lambda\\
\mathrm{pt} \ar[r]^-{\iota_j}&J,
}\] there is an induced \emph{base change isomorphism} \[\lambda_!F(j)=\iota_j^*\lambda_!F\xrightarrow{\simeq}\mathrm{pt}_!\pi_j^*F=\colim_{(\lambda\downarrow j)}\pi_j^*F.\] Our next results asserts that the analgous result holds for the homotopical versions of these functors.
\begin{corollary}
For any $\lambda:I\to J$ and $F:I\to \mathcal{T}\mathrm{op}$, there is a natural isomorphism \[\mathrm{hoLan}_\lambda F(j)\cong \hocolim_{(\lambda\downarrow j)}\pi_j^*F\] in $\mathrm{Ho}(\mathcal{T}\mathrm{op}^J)$.
\end{corollary}
\begin{proof}
We may assume that $F$ is objectwise cofibrant. In this case, by Proposition \ref{prop:bar is cofibrant}, we have the isomorphisms in $\mathrm{Ho}(\mathcal{T}\mathrm{op}^J)$ \[\mathrm{hoLan}_\lambda F(j)\cong \lambda_!|\mathrm{Bar}_\bullet(F, -)|\cong \colim_{(\lambda\downarrow j)}|\mathrm{Bar}_\bullet(F,\pi_j(-))|,\] whereas \[\hocolim_{(\lambda\downarrow j)}\pi_j^*F\cong \colim_{(\lambda\downarrow j)}|\mathrm{Bar}_\bullet(\pi_j^*F, -)|.\] By inspection, the simplicial functors $\mathrm{Bar}_\bullet(F,\pi_j(-))$ and $\mathrm{Bar}_\bullet(\pi_j^*F,-)$ are isomorphic, and the claim follows.
\end{proof}
\subsection{Quillen's Theorem B}
For any functor $F:I\to \mathcal{T}\mathrm{op}$, there is a natural map \[\hocolim_I F\to BI,\] which is induced on geometric realizations by the simplicial map $\mathrm{Bar}_\bullet(F)\to \mathrm{Bar}_\bullet(\underline \mathrm{pt})$ arising from the unique natural transformation $F\to \underline \mathrm{pt}$ (note that we have used the isomorphism $BI\cong BI^{op}$, since the latter bar construction is the nerve $NI^{op}$). Since the bar construction is split, this map is a weak equivalence whenever $F$ is pointwise contractible.
When $F$ is not pointwise contractible, it is often useful to be able to understand the homotopy fiber of this map. The result that will guide is in this task is due to Quillen and usually referred to as ``Theorem B.'' We follow the treatment of \cite[IV]{GoerssJardine:SHT}, beginning with the following preliminary result, which is interesting in its own right.
\begin{lemma}\label{lem:baby unstraightening}
If $F:I\to \mathrm{sSet}$ is a functor sending each morphism in $I$ to a weak equivalence, then the diagram \[\xymatrix{
F(i)\ar[r]\ar[d]&d^*\mathrm{Bar}_\bullet(F)\ar[d]\\
\Delta^0\ar[r]^-i&NI^{op}
}\] of simplicial sets is homotopy Cartesian for every object $i\in I$.
\end{lemma}
\begin{proof}
We will produce a diagram of the form
\[\xymatrix{
F(i)\ar[r]\ar[d]&K\times_{NI^{op}}d^*\mathrm{Bar}_\bullet(F)\ar[d]\ar[r]&\mathrm{Bar}_\bullet(F)\ar[d]\\
\Delta^0\ar[r]^-j& K\ar[r]^-p&NI^{op},
}\] in which $j$ is a trivial cofibration and $p$ a fibration, and we will show that the upper left map is a weak equivalence.
We take $K=\colim K_r$ to be the output of the small argument applied to $\Delta^0\to NI^{op}$; that is, $K_r$ is defined as the pushout in the diagram \[\xymatrix{
\displaystyle \coprod \Lambda_k^n\ar[dd]\ar[rr]&&K_{r-1}\ar[dl]\ar[dd]\\
&K_r\ar@{-->}[dr]\\
\displaystyle\coprod \Delta^n\ar[ur]\ar[rr]&& NI^{op},
}\] where $K_0=\Delta^0$. It now follows that each of the diagrams \[\xymatrix{
\displaystyle \coprod \Lambda_k^n\times_{NI^{op}}d^*\mathrm{Bar}_\bullet(F)\ar[d]\ar[r]&K_{r-1}\times_{NI^{op}}d^*\mathrm{Bar}_\bullet(F)\ar[d]\\
\displaystyle\coprod \Delta^n\times_{NI^{op}}d^*\mathrm{Bar}_\bullet(F)\ar[r]&K_r\times_{NI^{op}}d^*\mathrm{Bar}_\bullet(F),
}\] is a pushout, since colimits in $(\mathrm{sSet}\downarrow NI^{op})$ are computed in simplicial sets, and since the fiber product with $d^*\mathrm{Bar}_\bullet(F)$ admits a right adjoint on this category. Thus, since $K_0\times_{NI^{op}}d^*\mathrm{Bar}_\bullet(F)=F(i),$ it will suffice to show that each of the maps \[\Lambda_k^n\times_{NI^{op}}d^*\mathrm{Bar}_\bullet(F)\to \Delta^n\times_{NI^{op}}d^*\mathrm{Bar}_\bullet(F)\] is a weak equivalence. This map is obtained by applying the diagonal to the bottom map in the diagram \[\xymatrix{
\displaystyle\coprod_{(k_0\leq\cdots\leq k_r)\in (\Lambda_k^n)_r}F(\sigma(n))\ar[r]\ar[d]&\displaystyle\coprod_{(k_0\leq\cdots\leq k_r)\in (\Delta^n)_r}F(\sigma(n))\ar[d]\\
\displaystyle\coprod_{(k_0\leq\cdots\leq k_r)\in (\Lambda_k^n)_r}F(\sigma(k_r))\ar[r]&\displaystyle\coprod_{(k_0\leq\cdots\leq k_r)\in (\Delta^n)_r}F(\sigma(k_r))
}\] of bisimplicial sets, where $\sigma:[n]\to I^{op}$ is the given simplex of $NI^{op}$. The vertical maps, which are weak equivalences by our assumption on $F$, are supplied by the unique morphism to the final object $n\in [n]$, and the diagonal of the top map is the weak equivalence $\Lambda_k^n\times F(\sigma(n))\to \Delta^n\times F(\sigma(n))$. It follows by two-out-of-three that the bottom map becomes a weak equivalence after applying $d^*$, which was to be shown.
\end{proof}
\begin{corollary}\label{cor:hocolim quasifibration}
If $F:I\to \mathrm{Top}$ is a functor sending each morphism in $I$ to a weak equivalence, then the diagram \[\xymatrix{
F(i)\ar[r]\ar[d]&\hocolim_IF\ar[d]\\
\mathrm{pt}\ar[r]^-i&BI
}\] of spaces is homotopy Cartesian for every object $i\in I$.
\end{corollary}
\begin{proof}
Applying Lemma \ref{lem:baby unstraightening} to the functor $\mathrm{Sing}(F)$ yields the homotopy pullback square \[\xymatrix{
\mathrm{Sing}(F(i))\ar[d]\ar[r]&d^*\mathrm{Bar}_\bullet(\mathrm{Sing}(F))\ar[d]\\
\Delta^0\ar[r]^-i&NI^{op}.
}\] The claim now follows after applying geometric realization, since \[|d^*\mathrm{Bar}_\bullet(\mathrm{Sing}(F))|\cong \big||\mathrm{Bar}_\bullet(\mathrm{Sing}(F))|\big|\cong |\mathrm{Bar}_\bullet(|\mathrm{Sing}(F)|)|\simeq |\mathrm{Bar}_\bullet(F)|\simeq \hocolim_IF,\] and since geometric realization, as part of the Quillen equivalence between topological spaces and simplicial sets, preserves homotopy pullbacks.
\end{proof}
\begin{remark}
In case $I$ is the category associated to a group $G$, the statement becomes the familiar fact that the homotopy colimit of a $G$-space $X$, thought of as a functor, is given by the Borel construction on $X$, which forms a bundle over $BG$ with fiber $X$.
In general, this corollary encourages us to think of a functor as defining a sort of bundle over the classifying space with total space the homotopy colimit, fibers given by the functor itself, and, as we will see, space of sections given by the homotopy limit. Taking this kind of idea to its logical conclusion leads to the \emph{unstraightening} construction---see \cite[3.2]{Lurie:HTT}.
\end{remark}
\begin{corollary}[Quillen's Theorem B]\label{cor:quillen b}
Let $T:I\to J$ be a functor such that, for every morphism $\alpha:j\to j'$ in $J$, the map $B(j'\downarrow T)\to B(j\downarrow T)$ is a weak equivalence. The diagram \[\xymatrix{
B(j\downarrow T)\ar[r]\ar[d]&BI\ar[d]^-{BT}\\
\mathrm{pt}\ar[r]&BJ
}\] is homotopy Cartesian.
\end{corollary}
\begin{proof}
By Lemma \ref{lem:baby unstraightening}, it suffices to show that \[\hocolim_{J^{op}}B(-\downarrow T)\xrightarrow{\sim} BI.\] To see why this is so, we note that the lefthand side arises from the bisimplicial set given in bidegree $(m,n)$ by \begin{align*}
\coprod_{j_0\to\cdots \to j_n}N(j_n\downarrow T)_m&\cong \coprod_{j_0\to\cdots\to j_n\to T(i_0)\to \cdots \to T(i_m)}\mathrm{pt}\\
&\cong \coprod_{i_0\to \cdots \to i_m}N(J\downarrow T(i_0))_n.
\end{align*} Thus, \[\hocolim_{J^{op}}B(-\downarrow T)\simeq\hocolim_{I^{op}}B(J\downarrow T(-))\simeq \hocolim_{I^{op}}\underline{\mathrm{pt}}\simeq BI,\] where for the second equivalence we have used that $(J\downarrow T(i))$ is contractible for every $i\in I$, having the initial object $\id_i$.
\end{proof}
\section{The Spanier--Whitehead category}\label{appendix:Spanier--Whitehead}
\subsection{Stable homotopy}
We begin by recalling the following classical fact, which asserts that homotopy behaves like homology in a certain ``stable'' range.
\begin{theorem}[Homotopy excision]
Let $(B,A)$ be a $q$-connected CW pair. If $A$ is $p$-connected, then the map induced on $\pi_i$ by the map $(B,A)\to (B/A, *)$ is an isomorphism for $i\leq p+q$ and a surjection for $i=p+q+1$.
\end{theorem}
Recall that the suspension functor $A\mapsto \Sigma A$ is homotopy invariant if $A$ is a CW complex. Thus, we obtain in this case a \emph{suspension homomorphism}
\[\xymatrix{
\pi_i(A)=[S^i, A]_*\xrightarrow{\Sigma} [\Sigma S^i, \Sigma A]_*\cong [S^{i+1}, \Sigma A]_*=\pi_{i+1}(\Sigma A).
}\]
\begin{corollary}[Freudenthal suspension theorem]
If $A$ is a $p$-connected CW complex, then the suspension homomorphism \[\pi_i(A)\to \pi_{i+1}(\Sigma A)\] is an isomorphism for $i\leq 2p$ and a surjection for $i=2p+1$.
\end{corollary}
\begin{proof}
The suspension map coincides with the dashed map in the commuting diagram
\[\xymatrix{
\pi_{i+1}(CA)\ar[d]\ar[r]&\pi_{i+1}(CA,A)\ar[d]^-{(\star)}\ar[r]^-\simeq&\pi_i(A)\ar@{-->}[ddl]\\
\pi_{i+1}(SA)\ar@{=}[d]_-\wr\ar@{=}[r]&\pi_{i+1}(SA)\ar@{=}[d]^-\wr\\
\pi_{i+1}(\Sigma A)\ar@{=}[r]&\pi_{i+1}(\Sigma A)
}\] Since $A$ is $p$-connected by assumption, the pair $(CA,A)$ is $(p+1)$-connected, so the starred map is an isomorphism for $i+1\leq 2p+1$ and a surjection in the next degree, implying the claim.
\end{proof}
In light of this result, it is reasonable to make the following definition.
\begin{definition}
Let $A$ be a pointed CW complex. The $i$th \emph{stable homotopy group} of $A$ is \[\pi_i^s(A):=\colim_{r} \pi_{i+r}^s(\Sigma^rA)\cong \pi_{2i+2}(\Sigma^{i+2}A).\] A \emph{stable map} from $A$ to $B$ is a map $f:\Sigma^rA\to \Sigma^rB$ for some $r$, regarded as an element of $\colim_r[\Sigma^rA,\Sigma^rB]_*$.
\end{definition}
We may compose the stable maps $f:\Sigma^rA\to \Sigma^rB$ and $g:\Sigma^sB\to \Sigma^sC$ to obtain the stable map \[\Sigma^rg\circ\Sigma^s f:\Sigma^{r+s}A\to \Sigma^{r+s}C.\] Since an element of $\pi_i^s(A)$ is nothing other than a stable map from $S^i$ to $A$, it follows that stable maps induce morphisms at the level of stable homotopy groups.
\begin{definition}
A \emph{stable weak equivalence} is a stable map that induces an isomorphism on all stable homotopy groups.
\end{definition}
We shall use the notation $f:A\xrightarrow{\sim_s} B$ to indicate a stable weak equivalence.
\begin{recollection}
If $f:A\to B$ is a continuous map, the \emph{mapping cylinder} of $f$ is the pushout in the diagram \[\xymatrix{
A\ar[d]_-{\{1\}}\ar[r]^-f&B\ar[d]\\
A\times[0,1]\ar[r]& \mathrm{Cyl}(f).
}\] The \emph{mapping cone} of $f$ is the quotient \[C(f):=\frac{\mathrm{Cyl}(f)}{A\times\{0\}}.\] The diagram \[\xymatrix{
A\ar[d]\ar[r]^-f&B\ar[d]\\
\mathrm{pt}\ar[r]&C
}\] is a \emph{cofiber sequence} if the induced map \[C(f)\to B/f(A)\to C\] is a weak equivalence.
\end{recollection}
\begin{example}
If $f$ is a Hurewicz cofibration, then $A\xrightarrow{f} B\to B/f(A)$ is a cofiber sequence.
\end{example}
For our purposes, the key fact about stable weak equivalences is the following.
\begin{lemma}\label{lem:five lemma}
In the commuting diagram \[\xymatrix{
A\ar[r]\ar[d]_-{f_1}&B\ar[d]_-{f_2}\ar[r] &C\ar[d]_-{f_3}\\
A'\ar[r]&B'\ar[r]&C',
}\] if both rows are cofiber sequences and $f_1$ and $f_3$ are stable weak equivalences, then $f_2$ is also a stable weak equivalence.
\end{lemma}
It will be useful to have a means of producing stable maps.
\begin{lemma}\label{lem:stable maps adjunction}
Let $A$ be a finite CW complex. The set of stable maps from $A$ to $B$ is a in natural bijection with $[A,\,\Omega^\infty\Sigma^\infty B]_*$, where $\Omega^\infty\Sigma^\infty B:=\colim_r \Omega^r\Sigma^rB$.
\end{lemma}
\begin{proof}
By finiteness and adjunction, we have \begin{align*}
[A,\Omega^\infty\Sigma^\infty B]_*&=[A,\colim_r\Omega^r\Sigma^r B]_*\\
&\cong \colim_r[A, \Omega^r\Sigma^rB]_*\\
&\cong \colim_r[\Sigma^r A, \Sigma^rB]_*.
\end{align*}
\end{proof}
\subsection{Spanier--Whitehead category} In order to see why this lemma should be true, we locate the notion of stable weak equivalence within a convenient categorical context, which is a slight variant of that introduced in \cite{SpanierWhitehead:FAHT}.
\begin{definition}
The \emph{Spanier--Whitehead category} is the category $\mathcal{S}\mathrm{W}$ in which an object is a pair $(A,m)$ of a pointed CW complex and an integer, the set of morphisms are given by \[\mathcal{S}\mathrm{W}\left((A,m), (B,n)\right)=\colim_r \left[\Sigma^{r+m}A, \Sigma^{r+n}B\right]_*,\] where the colimit is taken over the set of natural numbers $r$ such that $r+m$ and $r+n$ are both nonnegative, and composition is defined in the same manner as composition of stable maps.
\end{definition}
We begin with a few basic observations on this category.
\begin{enumerate}
\item The full subcategory of objects of the form $(A,0)$ is the category stable maps between pointed CW complexes from the previous lecture. Note, however, that this subcategory is not closed under isomorphism in $\mathcal{S}\mathrm{W}$.
\item The assignment $A\mapsto (A,0)$ extends to a functor $\mathrm{Ho}(\mathcal{T}\mathrm{op}_*)\to \mathcal{S}\mathrm{W}$ fitting into the commuting diagram \[\xymatrix{
\mathrm{Ho}(\mathcal{T}\mathrm{op}_*)\ar[d]_-\Sigma\ar[r]& \mathcal{S}\mathrm{W}\ar[d]^-\Sigma&(A,m)\ar@{|->}[d]\\
\mathrm{Ho}(\mathcal{T}\mathrm{op}_*)\ar[r]&\mathcal{S}\mathrm{W}&(\Sigma A, m).
}\]
\item The class of the isomorphism \[\{\Sigma^m\Sigma A\cong\Sigma^{m+1}A\}\in [\Sigma^m\Sigma A,\Sigma^{m+1}A]_*\to\mathcal{S}\mathrm{W}\left((\Sigma A,m),(A,m+1)\right)\] defines a natural isomorphism $(\Sigma A, m)\cong (A,m+1)$, from which we conclude that $\Sigma:\mathcal{S}\mathrm{W}\to \mathcal{S}\mathrm{W}$ is an equivalence of categories with quasi-inverse $\Sigma^{-1}(A,m)=(A,m-1)$. In fact, the pair $(\mathcal{S}\mathrm{W},\Sigma)$ is universal with respect to this property in an appropriate sense---see \cite{DellAmbrogio:SWCAT}.
\item Any finite diagram in $\mathcal{S}\mathrm{W}$, after finitely many applications of the functor $\Sigma$, may be realized by a homotopy commutative diagram of CW complexes.
\item Since $(A,m)\cong(\Sigma A,m-1)$, the functor $\mathcal{S}\mathrm{W}((A,m),-)$ is naturally valued in groups; moreover, since $(A,m)\cong(\Sigma A,m-2)$, these groups are Abelian. This extra structure witnesses $\mathcal{S}\mathrm{W}$ as a \emph{preadditive} category, which is to say a category enriched in Abelian groups.
\item Fix $(A,m)$ and $(B,n)$, and let $N\geq 0$ be large enough so that $m+N$ and $n+N$ are both nonnegative. There is a natural chain of isomorphisms \begin{align*}
\mathcal{S}\mathrm{W}\left((\Sigma^{m+N}A\vee \Sigma^{n+N}B, -N), (C,p)\right)&= \colim_{r}\left[\Sigma^{r-N}(\Sigma^{m+N}A\vee\Sigma^{n+N}B), \Sigma^{p+r}C\right]_*\\
&\cong\colim_{r}\left[\Sigma^{m+r}A\vee \Sigma^{n+r}B, \Sigma^{p+r}C\right]_*\\
&\cong \colim_{r}\left(\left[\Sigma^{m+r}A, \Sigma^{p+r}C\right]_*\times \left[\Sigma^{n+r}B, \Sigma^{p+r}C\right]_*\right)\\
&\cong \colim_{r}\left[\Sigma^{m+r}A, \Sigma^{p+r}C\right]_*\times \colim_r\left[\Sigma^{n+r}B, \Sigma^{p+r}C\right]_*\\
&\cong \mathcal{S}\mathrm{W}\left((A,m), (C,p)\right)\times\mathcal{S}\mathrm{W}\left((B,n), (C,p)\right),
\end{align*} which exhibits a coproduct of $(A,m)$ and $(B,n)$. In the same way, we see that $\mathcal{S}\mathrm{W}$ has finite coproducts, and, since $\mathcal{S}\mathrm{W}$ is preadditive, finite biproducts \cite[VIII:2]{MacLane:CWM}; in other words, $\mathcal{S}\mathrm{W}$ is an \emph{additive} category.
\end{enumerate}
\subsection{Triangulated structure}
In fact, there is more structure to be uncovered.
\begin{definition}
Let $\op C$ be an additive category with an additive self-equivalence $\Sigma:\op C\to \op C$. A \emph{triangulation} of $\op C$ is a class $\op T$ of triples $(f,g,h)$ of morphisms of the form \[A\xrightarrow{f} B\xrightarrow{g} C\xrightarrow{h} \Sigma A,\] which satisfy the following axioms.
\begin{enumerate}
\item For every $A\in\op C$, $(\id_A,0,0)\in \op T$.
\item For each morphism $f$, there exists $(f,g,h)\in \op T$.
\item The class $\op T$ is closed under isomorphism.
\item If $(f,g,h)\in \op T$, then $(g,h,-\Sigma f)\in\op T$.
\item Given the solid commuting diagram \[\xymatrix{
A\ar@{=}[d]\ar[r]& B\ar[d]\ar[r]&C\ar@{-->}[d]\ar[r]&\Sigma A\ar@{=}[d]\\
A\ar[r]& B'\ar[d]\ar[r]&C'\ar@{-->}[d]\ar[r]&\Sigma A\\
&D\ar[d]\ar@{=}[r]&D\ar@{-->}[d]\\
&\Sigma B\ar[r]&\Sigma C
}\] in which the rows and lefthand column lie in $\op T$, the dashed fillers exist making the entire diagram commute, and the righthand column also lies in $\op T$.
\end{enumerate}
\end{definition}
\begin{remark}
Various equivalent combinations of axioms are possible. We follow \cite{May:ATC}.
\end{remark}
\begin{remark}
The elements of $\op T$ are typically referred to as \emph{distinguished triangles}, and the crucial fifth axiom is known variously as the ``octahedral axiom,'' for one of its visual representations, and ``Verdier's axiom.'' One way to understand this axiom is as enforcing a kind of third isomorphism theorem in $\op C$, i.e., that the ``quotient'' of $B'/A$ by $B/A$ should coincide with $B'/B$.
\end{remark}
We now seek to apply this formalism to the Spanier--Whitehead category.
\begin{definition}
A sequence $(A,m)\to (B,n)\to (C,p)$ in $\mathcal{S}\mathrm{W}$ is a \emph{cofiber sequence} if, after applying $\Sigma^N$ for some $N\in\mathbb{Z}$, it becomes isomorphic to the image of a cofiber sequence in $\mathrm{Ho}(\mathcal{T}\mathrm{op}_*)$.
\end{definition}
Recall that, for any map $f:A\to B$, we obtain a canonical map $C(f)\to \Sigma A$ by collapsing $B\subseteq C(f)$. Thus, any cofiber sequence in $\mathrm{Ho}(\mathcal{T}\mathrm{op}_*)$ extends canonically to a sequence of the form \[A\xrightarrow{f} B\xrightarrow{g} C\xrightarrow{h} \Sigma A.\]
We only sketch the proof of the following fundamental result in stable homotopy theory. For a detailed proof in an expanded context, see \cite[A.12]{Schwede:TTC}, for example.
\begin{theorem}[Puppe]
The collection of cofiber sequences is a triangulation of $\mathcal{S}\mathrm{W}$.
\end{theorem}
\begin{proof}[Sketch proof]
The first and third axioms are obvious, and the second follows from cellular approximation and the observation that the mapping cone of a cellular map between CW complexes is again a CW complex. After suspending, the fourth axiom follows from the standard fact that the rotation \[B\xrightarrow{g} C\xrightarrow{h} \Sigma A\xrightarrow{-\Sigma f} \Sigma B\] of a cofiber sequence is again a cofiber sequence \cite[8.4]{May:CCAT}. Finally, again after suspending, the fifth axiom reduces to checking that, given maps $f:A\to B$ and $f':B\to B'$, the natural sequence \[C(f)\to C(f'\circ f)\to C(f')\] is a cofiber sequence. After replacing $f$ and $f'$ by cofibrations, this claim follows from the homeomorphism \[\frac{B'/A}{B/A}\cong \frac{B'}{B}.\]
\end{proof}
\subsection{Consequences}
An important consequence of this structure is the following result, which is valid in any triangulated category---see \cite[13.4.2]{Stacks}, for example.
\begin{corollary}
For any $(A,m)\in \mathcal{S}\mathrm{W}$, the functors $\mathcal{S}\mathrm{W}\left((A,m),-\right)$ and $\mathcal{S}\mathrm{W}\left(-,(A,m)\right)$ each send cofiber sequences to long exact sequences of Abelian groups.
\end{corollary}
Since $\pi_i^s(A)=\mathcal{S}\mathrm{W}\left((S^0, i), (A,0)\right)$, Lemma \ref{lem:five lemma} now follows by the five lemma.
We also have the following appealing interpretation of stable weak equivalences.
\begin{corollary}\label{cor:finite stable weak equivalence}
If $A$ and $B$ are finite CW complexes and $f:(A,m)\to (B,n)$ induces an isomorphism \[\mathcal{S}\mathrm{W}\left((S^0,i),(A,m)\right)\xrightarrow{\simeq}\mathcal{S}\mathrm{W}\left((S^0,i),(B,n)\right)\] for every $i\in \mathbb{Z}$, then $f$ is an isomorphism. In particular, two finite CW complexes are stably weakly equivalent if and only if they become isomorphic in $\mathcal{S}\mathrm{W}$.
\end{corollary}
\begin{proof}
By induction on the skeletal filtration of $C$, using the fact that cofiber sequences induce long exact sequences, one shows that \[\mathcal{S}\mathrm{W}\left((C,p),(A,m)\right)\xrightarrow{\simeq}\mathcal{S}\mathrm{W}\left((C,p),(B,n)\right)\] for any finite CW complex $C$ and $p\in\mathbb{Z}$. The claim now follows from the Yoneda lemma for the subcategory of $\mathcal{S}\mathrm{W}$ on the finite CW complexes.
\end{proof}
\begin{corollary}\label{cor:stable weak equivalence homology}
Stable weak equivalences induce isomorphisms on homology.
\end{corollary}
\begin{proof}
By the suspension isomorphism, homology descends to a functor on $\mathcal{S}\mathrm{W}$. Since any functor sends isomorphisms to isomorphisms, the previous corollary implies the claim for stable weak equivalences between finite CW complexes. Since homology commutes with filtered colimits, the general case follows.
\end{proof}
\begin{remark}
It is a somewhat paradoxical fact that, although, as we have seen, the Spanier--Whitehead category effectively captures the stable phenomenon of homotopy behaving homologically, we in fact lose almost all Eilenberg-MacLane spaces upon passage to $\mathcal{S}\mathrm{W}$, since suspending destroys the property of being a $K(G,n)$. On the other hand, the Freudenthal suspension theorem guarantees that the homotopy groups remain correct in a range, and the map \[\Sigma K(G,n)\to K(G,n+1)\] adjoint to the weak equivalence $K(G,n)\xrightarrow{\sim} \Omega K(G,n+1)$ exhibits a sequence of objects in $\mathcal{S}\mathrm{W}$ that we might think should converge to the missing Eilenberg-MacLane object. This line of reasoning is one way to motivate the enlargement of the Spanier--Whitehead category to the full stable homotopy category or homotopy category of spectra---see \cite{Puppe:OSHC}, for example. Another motivation is taken up below.
\end{remark}
\subsection{Filtered stable weak equivalences} The purpose of this section is to address and partially remove the finiteness assumption in Corollary \ref{cor:finite stable weak equivalence}. We begin by noting that the assumption would have been unnecessary had we been assured that $(C,p)$ is the colimit in $\mathcal{S}\mathrm{W}$ of the objects $(\mathrm{sk}_k(C),p)$. Unfortunately, we have the computation \begin{align*}
\mathcal{S}\mathrm{W}\left((C,p), (A,m)\right)&=\textstyle{\colim_r}\left[\Sigma^{p+r}C, \Sigma^{m+r}A\right]_*\\
&\cong{\textstyle\colim_r}\,\pi_0\,{\textstyle\mathrm{Map}_*}\left(\Sigma^{p+r}C, \Sigma^{m+r}A\right)\\
&\cong{\textstyle\colim_r}\,\pi_0\left({\textstyle{\holim_k}}\,{\textstyle{\mathrm{Map}_*}}\left(\Sigma^{p+r}\mathrm{sk}_k(C),\Sigma^{m+r}A\right)\right),
\end{align*} and the formation of homotopy groups fails in general to commute with sequential homotopy limits; indeed, this failure is measured by the \emph{Milnor exact sequence} \[0\to \textstyle\lim^1_k\pi_{i+1}(X_k)\to \pi_i\left(\holim_k X_k\right)\to \lim_k\pi_i(X_k)\to0.\] To summarize the problem, the category $\mathcal{S}\mathrm{W}$ lacks certain filtered colimits that we might naively expect it to have.
In these notes, we adopt a somewhat ad hoc solution to this problem, which is nevertheless sufficient for our needs (but see Remark \ref{rmk:spectra} below). We make the following (non-standard) definition.
\begin{definition}
Let $X=\bigcup_{k\geq1} X_k$ and $Y=\bigcup_{k\geq1} Y_k$ be filtered pointed spaces. A \emph{filtered stable weak equivalence} is a collection of stable weak equivalences $f_k:X_k\xrightarrow{\sim_s} Y_k$ fitting into a commuting diagram \[\xymatrix{
X_{k-1}\ar[d]\ar[r]^-{f_{k-1}}&Y_{k-1}\ar[d]\\
X_k\ar[r]^-{f_k}&Y_k
}\] of stable maps.
\end{definition}
Since homology commutes with filtered colimits, we have the following consequence of Corollary \ref{cor:stable weak equivalence homology}.
\begin{corollary}\label{cor:filtered stable weak equivalence homology}
Filtered stable weak equivalences induce isomorphisms on homology.
\end{corollary}
\begin{remark}\label{rmk:spectra} A more principled solution to the problem of finiteness is provided by the category $\mathcal{S}\mathrm{p}$ of spectra, which may be constructed in many inequivalent ways, all of which yield the same homotopy category, typically called the \emph{stable homotopy category}. Using the subscript $\mathrm{fin}$ to indiate full subcategories spanned by finite spaces, the relationships among the various categories in question may be summarized in the following commuting diagram \[\xymatrix{
\mathcal{T}\mathrm{op}_*\ar[r]\ar[dd]_-{\Sigma^\infty}&\mathrm{Ho}(\mathcal{T}\mathrm{op}_*)\ar[d]&\mathrm{Ho}(\mathcal{T}\mathrm{op}_{*,\mathrm{fin}})\ar[l]\ar[d]\\
&\mathcal{S}\mathrm{W}\ar@{-->}[d]&\mathcal{S}\mathrm{W}_\mathrm{fin}\ar[l]\ar@{-->}[dl]\\
\mathcal{S}\mathrm{p}\ar[r]&\mathrm{Ho}(\mathcal{S}\mathrm{p}).
}\] The vertical dashed functor is badly behaved, being neither fully faithful nor essentially surjective, but the diagonal dashed functor is the inclusion of the full subcategory of compact objects in $\mathrm{Ho}(\mathcal{S}\mathrm{p})$; indeed, this fact may be used as one of a set of axioms characterizing the homotopy theory of spectra.
The category of spectra has many wonderful properties, but only two are directly relevant to our discussion: first, $\mathrm{Ho}(\mathcal{S}\mathrm{p})$ is again a triangulated category, so arguments in $\mathcal{S}\mathrm{W}$ translate directly to this new context; and $\Sigma^\infty C\cong\colim_k\Sigma^\infty \mathrm{sk}_k(C)$, repairing the deficiency of the Spanier--Whitehead category that hindered us before.
\end{remark}
\section{Tate cohomology and periodicity}\label{appendix:periodicity}
The goal of this appendix is to give an indication of the proof of the periodicity theorem used in the computation of the mod $p$ cohomology of $B_p(\mathbb{R}^n)$ for $p$ odd. The idea that we will pursue is that, if multiplication by the class in question is to be an isomorphism, then it should have an inverse, which we might imagine to be given by ``multiplication by an element of negative degree.'' Somewhat surprisingly, this idea is not nonsense, and the framework that gives it sense is the framework of \emph{Tate cohomology}, which we review briefly. For a more complete survey, see \cite[XII]{CartanEilenberg:HA}.
\subsection{Tate cohomology}
\begin{definition}
Let $G$ be a finite group and $V$ a $\mathbb{Z}[G]$-module. The \emph{norm map} for $V$ is the dashed filler in the commuting diagram \[\xymatrix{
V\ar[d]\ar[r]^-{\sum_{G} g}&V\\
H_0(G;V)\ar@{-->}[r]^-N& H^0(G;V).\ar[u]
}\]
\end{definition}
The norm map permits the definition of a cohomology theory combining ordinary group cohomology and group homology.
\begin{definition}
Let $G$ be a finite group and $V$ a $\mathbb{Z}[G]$-module. The \emph{Tate cohomology} of $G$ with coefficients in $V$ is \[
\hat H^n(G;V)=\begin{cases}
H^n(G;V)&\quad n>0\\
\mathrm{coker}\,N&\quad n=0\\
\ker N&\quad n=-1\\
H_{-n-1}(G;V)&\quad n<-1.
\end{cases}
\]
\end{definition}
We now summarize a few facts about Tate cohomology.
\begin{enumerate}
\item Tate cohomology is the cohomology associated to a certain type of resolution \cite[XII.3.2]{CartanEilenberg:HA}. By definition, a \emph{complete resolution} is a commuting diagram of free $\mathbb{Z}[G]$-modules \[\xymatrix{
\cdots\ar[r]&X_1\ar[r]^-{\delta_1}&X_0\ar[rr]^-{\delta_0}\ar[dr]_-{\epsilon}&&X_{-1}\ar[r]^-{\delta_{-1}}&X_{-2}\ar[r]&\cdots\\
&&&\mathbb{Z}\ar[ur]_-{\eta}
}\] in which the infinite row is exact, and Tate cohomology may be computed as \[\hat H^n(G;V)\cong H^n\left(\mathrm{Hom}_{\mathbb{Z}[G]}(X_\bullet, V)\right).\]
\item An injective group homomorphism induces a restriction on Tate cohomology \cite[XII.8]{CartanEilenberg:HA}. Tate cohomology is not functorial for general group homomorphisms.
\item Tate cohomology is multiplicative in the sense that there is a unique $C_2$-equivariant graded homomorphism \[\hat H(G;V_1)\otimes \hat H(G; V_2)\to \hat H(G;V_1\otimes V_2)\] extending the cup product in group cohomology and compatible with connecting homomorphisms \cite[XII.4]{CartanEilenberg:HA}. In particular, $\hat H(G;\mathbb{Z})$ is a graded commutative ring and $\hat H(G;V)$ is a module over this ring.
\item Tate cohomology enjoys a self-duality \cite[XII.6.6]{CartanEilenberg:HA} of the form \[\hat H^n(G;\mathbb{Z})\cong \mathrm{Hom}_\mathbb{Z}\left(\hat H^{-n}(G;\mathbb{Z}), C_{|G|}\right).\]
\end{enumerate}
\begin{remark}
According to (3), there are multiplication maps of the form \[H_m(BG)\otimes H_n(BG)\to H_{m+n+1}(BG),\] which may be interpreted from the point of view of singular chains in terms of the join of simplices \cite{Tene:PNTCFG}.
\end{remark}
\begin{example}
The periodic resolution for the cyclic group $C_k$ extends to a complete resolution \[\xymatrix{
\cdots\ar[r]&\mathbb{Z}[C_k]\ar[r]^-{\sigma-1}&\mathbb{Z}[C_k]\ar[rr]^-{\sum_{i=1}^k\sigma^i}\ar[dr]_-{\epsilon}&&\mathbb{Z}[C_k]\ar[r]^-{\sigma-1}&\mathbb{Z}[C_k]\ar[r]&\cdots\\
&&&\mathbb{Z}\ar[ur]_-{\sum_{i=1}^k\sigma^i}
}\] and it follows that \[
\hat H^n(C_k;\mathbb{Z})\cong\begin{cases}
C_k&\quad n \text{ even}\\
0&\quad n \text{ odd.}
\end{cases}
\] As for multiplicative structure, one can show that $\hat H(C_k;\mathbb{Z})\cong C_k[\beta,\beta^{-1}]$, where $\beta$ is the class represented by the 2-cocycle in $\mathrm{Hom}_{\mathbb{Z}[C_k]}\left(\mathbb{Z}[C_k], \mathbb{Z}\right)\cong \mathrm{Hom}_\mathbb{Z}\left(\mathbb{Z},\mathbb{Z}\right)$ corresponding to the identity \cite[XII.7]{CartanEilenberg:HA}.
\end{example}
\subsection{Periods}
The periodicity evident in the previous calculation is a special case, and also the source, of a more general phenomenon. For a $\mathbb{Z}[G]$-module $V$, write $\hat H(G;V)_p$ for the $p$-primary component of $\hat H(G;V)$ and $|G|_p$ for the largest power of $p$ dividing $G$.
\begin{proposition}
Multiplication by $\alpha\in \hat H^n(G;\mathbb{Z})$ induces an isomorphism on $\hat H(G;V)_p$ for every $\mathbb{Z}[G]$-module $V$ if and only if $\alpha$ has exact order $|G|_p$.
\end{proposition}
The idea of the argument is that such an element $\alpha$ determines a homomorphism $\hat H^n(G;\mathbb{Z})\to C_{|G|}$ and thus an element $\alpha^{-1}\in \hat H^{-n}(G;\mathbb{Z})$ by the duality referenced above, and this element will act as a multiplicative inverse to $\alpha$. For details, see \cite[XII.11.1, Exercise XII.11]{CartanEilenberg:HA}.
\begin{definition}
The $p$-\emph{period} of $G$, if it exists, is the least $n>0$ such that $\hat H^n(G;\mathbb{Z})$ contains an element of exact order $|G|_p$.
\end{definition}
\begin{remark}
Most groups do not have a $p$-period; indeed, if $p$ is odd, then $G$ has a $p$-period if and only if the $p$-Sylow subgroup of $G$ is cyclic \cite[Exercsie XII.11]{CartanEilenberg:HA}.
\end{remark}
\begin{example}
The cyclic group $C_{p^k}$ has $p$-period $2$.
\end{example}
The following theorem of \cite{Swan:PPFG} will allow us to determine the $p$-period of $\Sigma_p$. For a subgroup $H\leq G$, we write $C_G(H)$ and $N_G(H)$ for the centralizer and normalizer of $H$ in $G$, respectively.
\begin{theorem}[Swan]
Let $G$ be a finite group, $p$ an odd prime, and $H\leq G$ a $p$-Sylow subgroup. If $H$ is cyclic, then the $p$-period of $G$ is equal to $2\left|\frac{N_G(H)}{C_G(H)}\right|$.
\end{theorem}
Before discussing the proof of this theorem, we use it to prove the Periodicity Theorem. First, we recall a few standard facts about cyclic groups.
\begin{lemma} Fix $k\geq0$.
\begin{enumerate}
\item $C_{\Sigma_k}(C_k)=C_k$
\item $N_{\Sigma_k}(C_k)\cong C_k\rtimes \mathrm{Aut}(C_k)$
\item $\mathrm{Aut}(C_k)\cong C_k^\times$
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of Periodicity Theorem]
Since $V$ is defined over $\mathbb{F}_p$, we have $\hat H(G;V)_p=\hat H(G;V)$. Thus, the first claim is implied by our previous calculation of the $p$-period of $C_p$, while the second claim is implied by Swan's theorem and the calculation \[2\left|\frac{N_{\Sigma_p}(C_p)}{C_{\Sigma_p}(C_p)}\right|=2\frac{|C_p|\cdot|C_p^\times|}{|C_p|}=2(p-1).\]
\end{proof}
\subsection{Proof of Swan's theorem}
In order to deduce the Swan's theorem from our calculation in the case of a cyclic group, we will need to understand the relationship between the Tate cohomology of $G$ and that of its $p$-Sylow subgroup.
\begin{definition}
Let $H\leq G$ be a subgroup. An element $\alpha\in \hat H(H;V)$ is \emph{stable} if, for every $g\in G$, $\alpha$ equalizes the two homomorphisms \[\hat H(H; V)\to \hat H(H\cap H^g;V)\] induced by the inclusion $H\cap H^g\leq H$ and the injection $(-)^{g^{-1}}:H\cap H^g\to H$, respectively.
\end{definition}
A transfer argument, one proves the following \cite[XII.10.1]{CartanEilenberg:HA}.
\begin{proposition}
If $H\leq G$ is a $p$-Sylow subgroup, then, for every $\mathbb{Z}[G]$-module $V$, the natural map $\hat H(G;V)_p\to \hat H(H;V)$ is injective with image the stable classes.
\end{proposition}
Thus, in order to deduce the period of $G$ from the period of $H$, we must understand the stable classes.
\begin{lemma}
If the $p$-Sylow subgroup $H\leq G$ is Abelian, then $\alpha\in \hat H^*(H;\mathbb{Z})$ is stable if and only if $\alpha$ is fixed by $N_G(H)$.
\end{lemma}
\begin{proof}
The ``only if'' direction is obviously true without assumption on $H$, so assume that $\alpha$ is fixed by $N_G(H)$, and choose $g\in G$. Since $H$ is Abelian, both $H$ and $H^g$ are contained in $C_G(H\cap H^g)$, and, since $H$ is a $p$-Sylow subgroup of $G$, it follows that both are $p$-Sylow subgroups of this centralizer. Since all such are conjugate, it follows that $H=H^{tg}$ for some $t\in C_G(H\cap H^g)$, whence $tg\in N_G(H)$. We therefore have the commuting diagram \[\xymatrix{
H\cap H^g\ar@{=}[dd]_-{(-)^{t^{-1}}}\ar[rr]&&H\ar[dd]^-{(-)^{(tg)^{-1}}}\\\\
H\cap H^g\ar[rr]^-{(-)^{g^{-1}}}&&H,
}\] and the maps induced on Tate cohomology by the righthand map fixes $\alpha$ by assumption. It follows that $\alpha$ equalizes the two horizontal maps, as desired.
\end{proof}
\begin{lemma}
If the $p$-Sylow subgroup $H\leq G$ is cyclic, then $n$ is a multiple of the $p$-period of $G$ if and only if $n$ is even and $N_G(H)$ fixes $\hat H^n(H;\mathbb{Z})$ pointwise.
\end{lemma}
\begin{proof}
By assumption, the $p$-period of $H$ is $2$, so we may assume that $n$ is even \cite[XII.11.3]{CartanEilenberg:HA}. Then $n$ is a multiple of the $p$-period of $G$ if and only if $\hat H^n(G;\mathbb{Z})$ contains an element of exact order $|G|_p$. By our computation in the case of a cyclic group and the previous proposition, this statement is equivalent to the statement that the map $\hat H^n(G;\mathbb{Z})_p\to \hat H^n(H;\mathbb{Z})$ is an isomorphism, which is to say that every class in the target is stable. Since $H$ is in particular Abelian, the previous lemma shows that this condition is equivalent to the condition that $N_G(H)$ act trivially in degree $n$.
\end{proof}
\begin{proof}[Proof of Swan's theorem]
Suppose that $n$ is a multiple of the $p$-period of $G$; then, by the lemma, $n$ is even and every class in degree $n$ Tate cohomology is fixed by $N_G(H)$. Now, $N_G(H)$ acts on $H$ via the composite \[N_G(H)\twoheadrightarrow \frac{N_G(H)}{C_G(H)}\hookrightarrow \mathrm{Aut}(H)\cong H^\times.\] By our assumption on $H$, the target is cyclic, so the intermediate quotient is also cyclic, and the action on $\hat H^n(H;\mathbb{Z})$ is by multiplication by $r^{n/2}$ with $(r,p)=1$ \cite[Lemma 3]{Swan:PPFG}. Since this action is trivial, $n/2$ is a multiple of the order of the cyclic group $N_G(H)/C_G(H)$.
\end{proof}
\end{appendix}
\bibliographystyle{amsalpha}
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{"url":"http:\/\/blog.cdmansfield.com\/tag\/thermodynamics\/","text":"February 15, 2012 \u00a0Tagged with: \u00a0Comments Off on Useful Definitions and Identities for Thermodynamic Potentials\n\nThis is just a summary of many of the definitions and identities I find useful when working with the basic thermodynamics of a system or process. \u00a0It is not meant to be a comprehensive list.\n\n# Mathematical Identities\n\nUsing the reasoning behind the notation used for derivatives of thermodynamic potentials, there are some useful identities given the following prototypical system\n\nEq. 1) $\\displaystyle X=X\\left(Y,Z\\right)$\n\nThis equation represents a generic thermodynamic potential of an arbitrary system or process as a function of two implicitly extensive thermodynamic potentials. \u00a0Adopting the notation\n\nEq. 2) $\\displaystyle\\underline{X}=\\frac{X}{N}$\n\nTo denote the relationship between intensive variables, extensive variables, and the size of the system allows for the trivial conversion between system size dependent and system size agnostic thermodynamic relationships. \u00a0The total differential for the extensive form of the thermodynamic potential for the prototypical system is given by\n\nFebruary 14, 2012 \u00a0Tagged with: , ,\n\nWhen I took my process thermodynamics course as an undergraduate student, I was told to use what I had initially considered to be a completely redundant notation for partial derivatives for thermodynamic potentials. \u00a0The notation involved wrapping the partial derivative in a set of parentheses and noting which variables were \u201cheld constant\u201d as a subscript.\n\nEx. $\\displaystyle\\left(\\frac{\\partial P}{\\partial V}\\right)_{T,\\vec{N}}$\n\nThe derivation leading to this was woefully devoid of the mathematical basis for this apparently redundant notation. \u00a0Furthermore, so was the general literature on the subject, most of which consisted of fleeting introductions to their respective application. \u00a0As I played with the idea, it became clear why this notation was indeed very necessary. \u00a0It was only as I was solving a separate problem on my own that such notation yielded valuable information about the nature of the potentials being described.","date":"2018-08-19 11:15:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 3, \"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\": 0.8971807956695557, \"perplexity\": 754.0555469392045}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221215077.71\/warc\/CC-MAIN-20180819110157-20180819130157-00396.warc.gz\"}"} | null | null |
\section{Introduction}
Remote sensing technology has been advancing the measurement of massive amount of datasets for many geophysical processes. Statistical analysis for massive amount of data is challenging, since many geophysical processes evolve in space and time with complicated structures. The resulting data often exhibit nonstationary dependence structures. As remote sensing data are often collected by different satellite instruments over different footprints that have distinct shapes, orientations and sizes, these remote-sensing data are often noisy and incomplete with incompatible spatial supports and distinct measurement-error characteristics. In our real application, we focus on data collected from polar-orbiting satellites. A key characteristic of remote sensing data from polar-orbiting satellites is that these data are often collected at very high spatial resolution and relatively low temporal resolution. Spatial modeling for such data even at a single time point is computationally challenging, which leads to the well-known big ``$n$'' problem in spatial statistics \citep{Cressie2006, Cressie2008, Banerjee2008, Nychka2015, Ma2017}. In a spatio-temporal setting with more than one data sources, it is much more challenging to tackle this computational issue.
To analyze data from different satellite instruments, the resolution difference must be accounted for. There is a vast literature in spatial statistics to tackle the so-called change-of-support problem when statistical analysis is carried out with several data sources at different resolutions. Here, change of support (COS) refers to inference made at a resolution based on data from different scales \citep[e.g.,][]{Cressie1993, Cressie1996, Gelfand2001, Gotway2002}. A direct way to deal with the change-of-support problem is to represent the process at the block level as a stochastic integral of the process at the point level or areal-unit level. When data are obtained at different scales or resolutions, statistical inference for combining such data leads to the data-fusion problem. In spatial-statistics literature, data fusion has been approached in several different ways. \cite{Wikle2005} formulate a hierarchical Bayesian model that combines observations across different scales by assuming the same underlying true process. In a similar way, \cite{Fuentes2005} present an instance of Bayesian melding \citep{Poole2000} that assumes the underlying true process at the point level so that point-referenced observations of air pollution data and air-quality model output at the grid-cell level are linked to a same true process at the point level via measurement-error processes. The space-time extension of \cite{Fuentes2005} has been developed in \cite{Choi2009} to study the spatio-temporal association between mortality and pollution exposure to daily fine particulate matter (PM$_{2.5}$) based on point-referenced PM$_{2.5}$ and air quality model output. These models address the change-of-support problem explicitly, but their implementations require expensive computations, and hence their models are not suitable to directly analyze large datasets. Other different approaches for data fusion have been developed as well. For instance, \cite{McMillan2010} present a spatio-temporal model that combines point-referenced observations and numerical model output at the grid-cell level by assuming that the data process at point level is linked to the same underlying true process at the grid-cell level. Instead of assuming the same true underlying process for both observations and numerical model output, \cite{Berrocal2010, Berrocal2012datafusion} propose to regress the point-referenced observations over model output at the grid-cell level with spatial or spatio-temporal varying coefficients. These models have been applied to make inference based on point-referenced air quality observations and numerical model output at grid-cell level over the United States. In this paradigm, \cite{Sahu2010} also regress the point-referenced true process over numerical model output at the grid-cell level to predict chemical deposition in the eastern United States. These approaches do not address the change-of-support explicitly, and they require intensive computations to fit the model in a Bayesian framework. In remote sensing science, massive amount of data are often collected over space and time by satellite instruments, making these methods computationally intractable.
To tackle the massiveness of remote-sensing data, \cite{Nguyen2012} present the spatial data fusion methodology based on the spatial-random-effects model \citep{Cressie2006, Cressie2008}, where a single underlying true spatial process is assumed at the areal-unit level. \cite{Nguyen2014} further develop the spatio-temporal data fusion methodology based on the spatio-temporal-random-effects model \citep{Cressie2010, Kang2010, Katzfuss2011, Katzfuss2012}, where different underlying true spatio-temporal processes are assumed at the areal-unit level, and cross-dependence structures among different true processes are modeled through the spatio-temporal-random-effects model.
In this article, we propose a dynamic fused Gaussian process (DFGP) methodology for spatio-temporal data fusion to combine multiple datasets from different satellite instruments. As a generalization of \cite{Nguyen2014}, our DFGP methodology extends the spatial-only fused Gaussian process (FGP) in \cite{Ma2017} to a spatio-temporal setting. In particular, the FGP model extends the fixed rank kriging model \citep{Cressie2006, Cressie2008} by combining a low-rank representation with a general covariance matrix together with a graphical model with a sparse precision matrix. Based upon FGP, we take a dynamic-statistical approach to build the DFGP model under which the current state of the process of interest evolves from the previous state in a dynamic way. This hierarchical modeling approach has been adopted in many previous research \citep[e.g.,][]{Mardia1998, Wikle1998, Wikle1999, Berliner2000, Stroud2001, Wikle2001, Huang2002, Cressie2002, Xu2007}; see \cite{Cressie2011} for a comprehensive overview for spatio-temporal models. Our proposed DFGP falls into this paradigm, and extends the spatio-temporal-random-effects model \citep{Cressie2010, Kang2010, Katzfuss2011, Katzfuss2012, Zammit2017} with a more flexible covariance function.
The reminder of this article is organized as follows. Section~\ref{sec: data} introduces two different datasets from two satellite instruments onboard NASA's AQUA satellite. Section~\ref{sec: DFGP} presents the dynamic fused Gaussian process methodology in a data-fusion context. Kalman filtering and Kalman smoothing procedures are also derived. In Section~\ref{sec: SEM}, we give details on the stochastic expectation-maximization algorithm for parameter estimation in both filtering and smoothing procedures. In Section~\ref{sec: results}, we apply the DFGP methodology to analyze massive amount of sea surface temperature datasets, and make comparisons with other existing methods such as the spatio-temporal data fusion model in \cite{Nguyen2014}. Section~\ref{sec: conclusion} concludes with discussions and future research work.
\section{Data} \label{sec: data}
Sea surface temperature (SST) is a key climate and weather measurement, which plays a crucial role in understanding climate systems. Massive amount of SST datasets are collected from satellite instruments each day with the advances in new remote-sensing technology. For instance, the AQUA satellite launched on May 4, 2002 is a polar-orbiting satellite around the Earth, aiming at studying Earth's precipitation, evaporation, and cycling of water. The AQUA satellite carries two instruments: the Moderate Resolution Imaging Spectroradiometer (MODIS) and the Advanced Microwave Scanning Radiometer-Earth Observing System (AMSR-E). The MODIS instrument is an infrared radiometer with a ground swath width of 2,330 km, which is able to measure SST at fine spatial resolutions, but is unable to measure through cloud cover; the AMSR-E instrument is a microwave radiometer with a ground swath width of 1445 km, which is able to measure SST in all weather conditions except rain, but only at coarse spatial resolutions, though its quality is also subject to radio frequency interference. MODIS SST and AMSR-E SST data have been widely used for scientific research and operations \citep[e.g.,][]{Donlon2002, Carroll2006, Gentemann2014}, However, it is still challenging to combine these two different data products due to their different characteristics including different spatial resolutions. \textit{Heuristic} methods and empirical comparisons with in-situ observations are studied \citep[e.g.,][]{Guan2003, Kawai2006, Arai2013}. In this article, we propose a spatio-temporal statistical model to generate such high-resolution SST data products on a daily scale by combining MODIS SST and AMSR-E SST data in a rigorous way. The resulting high-resolution SST products can be critical and helpful for operational oceanography and numerical weather prediction.
In this study, we use daily daytime MODIS SST data at 9 km spatial resolution processed from the NASA Ocean Biology Processing Group Data Center (\url{oceancolor.gsfc.nasa.gov}), and daily daytime AMSR-E SST data at 25 km spatial resolution from \url{www.remss.com} from January 1 to 8 in the year 2010. These datasets have distinct error characteristics and are often sparse, irregular, and noisy with incompatible supports. Statistical methods for combining different sources of remote-sensing data will give much more accurate and reliable uncertainty analysis. The study region is chosen to be the tropical Pacific region between longitude $-30^{\circ}$ and $30^{\circ}$ and between latitude $120^{\circ}$ and $290^{\circ}$ from January 1 to 8 in the year 2010. Figure~\ref{fig: MODIS_AMSRE_day1} shows the MODIS SST and AMSR-E SST on January 1, 2010. The numbers of observations from MODIS instrument for each day are $n_1^{(1)} = 362,721, n_2^{(1)} = 398,662, n_3^{(1)}=409,445, n_4^{(1)}=385,490, n_5^{(1)}=425,541, n_6^{(1)}=415,869, n_7^{(1)}=416,721, n_8^{(1)}=415,467$, respectively, resulting in a total of 3,229,916 MODIS SST observations, where the subscript denotes the time step, and the superscript denotes the instrument with 1 for MODIS and 2 for AMSR-E. The numbers of observations from AMSR-E instrument for each day are $n_1^{(2)}=71,592, n_2^{(2)}=68,574, n_3^{(2)}=72,905, n_4^{(2)}=64,868, n_5^{(2)}=73,228, n_6^{(2)}=66778, n_7^{(2)}=71,431, n_8^{(2)}=67,245$, respectively, resulting in a total of 556,621 AMSR-E SST observations.
\begin{figure}[htbp]
\begin{center}
\makebox[\textwidth][c]{ \includegraphics[width=1.0\textwidth, height=0.4\textheight]{MODIS_AMSRE_day1.pdf}}
\caption{MODIS and AMSR-E SST data on January 1, 2010. The MODIS SST data are at 9 km resolution and the AMSR-E SST data are at 25 km resolution.}
\label{fig: MODIS_AMSRE_day1}
\end{center}
\end{figure}
\section{The Dynamic Fused Gaussian Process Model} \label{sec: DFGP}
For many physical spatio-temporal processes, it is quite natural that the process of interest cannot be observed directly, and we assume that the data process is a sum of a hidden process and a measurement-error process. Suppose we are interested in a real-valued spatio-temporal process $\{Y_t(\mathbf{s}): \mathbf{s} \in \mathcal{D}\subset \mathbb{R}^d,\, t\in \mathcal{T}\}$, where $\mathcal{D} \subset \mathbb{R}^d$ and $\mathcal{T}\equiv \{1, 2, \ldots, T\}$ for a positive integer $T$. The goal is to make fast statistical inference such as filtering and smoothing based on massive spatio-temporal datasets. In what follows, we present the dynamic fused Gaussian process (DFGP) model.
Suppose that the spatial domain $\mathcal{D}$ of interest is made up of $N$ non-overlapping, equal-areal, basic areal units $\mathcal{R}_i$ for $i=1, \ldots, N$, where each $\mathcal{R}_i$ is assumed to be associated with its centroid $\mathbf{s}_i$. These basic areal units represent the smallest spatial resolution at which prediction will be made, and they are called BAUs. In what follows, the discretized version of the domain $\mathcal{D}$ will be referred to as $\mathcal{R} \equiv \cup_{i=1}^N\{\mathcal{R}_i\}$ that is indexed by corresponding centroids $\{ \mathbf{s}_i: i=1, \ldots, N\}$. This discretization procedure has been used in many previous work \citep[e.g.,][]{Cressie2006, Cressie2008, Nguyen2012, Nguyen2014, Shi2017, Ma2017, Ma2018downscaling}. The data process is assumed to be observed at different spatial resolutions resulted from the same underlying true process $Y_t(\cdot)$ over these BAUs. This is a typical situation where multiple satellite instruments measure the same geophysical process at different resolutions in remote sensing science. Let $\mathbf{Z}_t^{(k)} \equiv (Z_t(\mathcal{A}_{t,1}^{(k)}), \dots, Z_t(\mathcal{A}_{t, n_t^{(k)}}^{(k)}))'$ be a vector of noisy version of the underlying true process $Y_t(\cdot)$ at the $k$-th spatial resolution over $n_t^{(k)}$ footprints $\{ \mathcal{A}_{t,i}^{(k)}: i=1, \ldots, n_t^{(k)}\}$ collected from the $k$-th satellite instrument at time $t$, where the quantity with superscript $k$ corresponds to that from the $k$-th satellite instrument. The total number of observations at time $t$ is denoted as $n_t = \sum_{k=1}^{k_0} n_t^{(k)}$ across all resolutions with $k_0$ being the number of instruments. The data process $Z_t(\cdot)$ over the $i$-th footprint $\mathcal{A}_{t,i}^{(k)}$ from $k$-th instrument at time $t$ is modeled as the true process $Y_t(\cdot)$ over footprint $\mathcal{A}_{t,i}^{(k)}$ plus a measurement-error process:
\begin{eqnarray}\label{eqn: data model in DFGP}
Z_t(\mathcal{A}_{t,i}^{(k)}) = Y_t(\mathcal{A}_{t,i}^{(k)}) + \epsilon_t(\mathcal{A}_{ t,i}^{(k)}), \quad \mathcal{A}_{ t,i}^{(k)} \subset \mathcal{D}; i=1, \ldots, n_t^{(k)}; k=1, \ldots, k_0
\end{eqnarray}
The true process $Y_t(\cdot)$ over the footprint $\mathcal{A}_{ t,i}^{(k)}$ is assumed to be the block average of the process $Y_t(\cdot)$ over the BAUs within the domain $\mathcal{A}_{ t,i}^{(k)}$:
\begin{eqnarray} \label{eqn: discrete sum in DFGP}
Y_t (\mathcal{A}_{t,i}^{(k)}) = \frac{1}{\sum_{\ell=1}^N I(\mathbf{s}_{\ell} \in \mathcal{A}_{t,i}^{(k)})} \sum_{\ell=1}^N I(\mathbf{s}_{\ell} \in \mathcal{A}_{t,i}^{(k)})\cdot Y(\mathbf{s}_{\ell}).
\end{eqnarray}
where $\mathbf{s}_{\ell}$ is the centroid of the BAU $\mathcal{R}_{\ell}$. The Eq.~\eqref{eqn: discrete sum in DFGP} is accounting for the change-of-support problem with BAUs assumed to have equal areas. This strategy has been widely taken in previous work \citep[e.g.,][]{Nguyen2012, Nguyen2014, Nguyen2017, Ma2018downscaling}.
The measurement-error process $\epsilon_t(\cdot)$ is assumed to be a Gaussian white noise term and $\{\epsilon_t(\mathcal{A}_{ t,i}^{(k)});$ $i=1, \ldots, n_t^{(k)}; k=1, \ldots, k_0; t=1, \ldots, T\}$ are assumed to be independent. It may have nonzero mean capturing the instrument bias, and has variance $\text{var}(\epsilon_t(\mathcal{A}_{ t,i}^{(k)}))= \sigma_{\epsilon_t, (k)}^{2} v(\mathcal{A}_{ t,i}^{(k)})>0$, where $v(\mathcal{A}_{ t,i}^{(k)})$ is known from validation data and instrument specification and allows for the possibility of nonconstant variance. The variance parameter $\sigma_{\epsilon_t, (k)}^{2}$ will be estimated via a maximum likelihood estimation procedure in Section~\ref{sec: SEM}. Under this assumption, the nugget variance parameters are both instrument-dependent and time-dependent. This allows great flexibility in modeling the measurement-error processes. The data model defined by Eq.~\eqref{eqn: data model in DFGP} and \eqref{eqn: discrete sum in DFGP} has been studied in many previous work \citep[e.g.,][]{Nguyen2012, Nguyen2014}.
As we are interested in the process $Y_t(\cdot)$ at the finest resolution defined by $N$ BAUs. Following \cite{Nguyen2012, Nguyen2014, Ma2018downscaling}, we define the process of interest $\{Y_t(\mathbf{s}): \mathbf{s} \in \mathcal{R}\}$ at BAU-level:
\begin{eqnarray}
Y_t(\mathbf{s}) &=& \mathbf{X}_t(\mathbf{s})' \boldsymbol \beta_t + \nu_t(\mathbf{s}) + \delta_t(\mathbf{s}), \label{eqn: model for Y}\\
\nu_t(\mathbf{s}) &=& \mathbf{S}_t(\mathbf{s})'\boldsymbol \eta_t, \label{eqn: model for nu} \\
\delta_t(\mathbf{s}) &=& \mathbf{B}_t(\mathbf{s})' \boldsymbol \xi_t, \label{eqn: model for delta}
\end{eqnarray}
where $\mathbf{X}_t(\cdot)\equiv (X_{t,1}(\cdot), \ldots, X_{t,p}(\cdot))'$ is a $p$-dimensional vector of covariates and $\boldsymbol \beta_t$ are corresponding unknown coefficients at time $t$; $\mathbf{S}_t(\cdot) \equiv (S_{t,1}(\cdot), \ldots, S_{t,r}(\cdot))'$ is an $r$-dimensional vector of basis functions at time $t$, and $\boldsymbol \eta_t$ is an $r$-dimensional random vector. The quantities $\mathbf{X}_t(\cdot), \mathbf{S}_t(\cdot), \mathbf{B}_t(\cdot)$ are defined at point level, and then are computed at BAU-level using Monte Carlo techniques following previous work \citep{Wikle2001, Gelfand2001, Fuentes2005, Katzfuss2011, Katzfuss2012}. More specifically, these quantifies at BAU-level can be approximated by Monte Carlo averages of them defined over 30 uniformly-distributed point locations within each BAU. In what follows, the pre-specified quantities defined at BAU-level are always obtained from their specification at point level via such Monte Carlo techniques.
Following \cite{Cressie2008}, we choose a bisquare basis function with the following form: $S_{t,i}(\mathbf{u}) = \{1 - (\|\mathbf{u} - \mathbf{c}_i\|/\ell_i)^2 \}^2\cdot\{I(\|\mathbf{u} - \mathbf{c}_i\| \leq \ell_i)\}$ for $i=1, \ldots, r$, where $\mathbf{c}_i$ is the center of the $i$-th bisquare basis function, and $\ell_i$ is the corresponding radius of the $i$-th bisquare basis function. In addition, we also assume that the number of basis functions is much smaller than the number of observations, i.e., $r\ll n_t$. The model in Eq~\eqref{eqn: model for nu} has a low-rank representation. $\mathbf{B}_t(\cdot)\equiv (B_{t,1}(\cdot), \ldots, B_{t, N}(\cdot))'$ is an $N$-dimensional vector of basis functions for the Markov random coefficients $\boldsymbol \xi_t$ at time $t$. $\boldsymbol \xi_t$ is an $N$-dimensional random vector defined on $N$ BAUs with Markov structure specified in Eq.~\eqref{eqn: model for xi}. Following \cite{Ma2017}, we choose a piecewise constant basis function for $B_{t,i}(\cdot)$ with the following form: $\mathbf{B}_{t,i}(\mathbf{u})=I(\mathbf{u} \in \mathcal{R}_i)$ for $i=1, \ldots, N$. The model in Eq.~\eqref{eqn: model for delta} has a high-rank representation, since $N\approx n_t$ or $N>n_t$. Notice that the quantities $\mathbf{X}_t(\cdot), \mathbf{S}_t(\cdot), \mathbf{B}_t(\cdot)$ defined at BAU-level lead to a model for the underlying true process $Y_t(\cdot)$ also defined at BAU-level. However, the data are collected at a resolution that is coarser than the resolution at which these BAUs are defined. The Eq.~\eqref{eqn: discrete sum in DFGP} links the process $Y_t(\cdot)$ at BAU-level to the resolution at which the data process is defined through the change-of-support property. In what follows, we give the model specification for $\boldsymbol \eta_t$ and $\boldsymbol \xi_t$.
Following \cite{Cressie2010}, we assume that the dynamical evolution of $\{\boldsymbol \eta_t: t=0, 1, \ldots, T\}$ follows a vector-autoregressive (VAR) model of order 1:
\begin{eqnarray} \label{eqn: evolution model}
\boldsymbol \eta_t \mid \boldsymbol \eta_{t-1}, \boldsymbol \eta_{t-2}, \ldots, \boldsymbol \eta_0 \sim \mathcal{N}_r(\mathbf{H}_t \boldsymbol \eta_{t-1}, \mathbf{U}_t),\, t=1, 2, \ldots, T,
\end{eqnarray}
with the initial state $\boldsymbol \eta_0 \sim \mathcal{N}_r(\mathbf{0},\, \mathbf{K}_0)$. The $r\times r$ matrix $\mathbf{H}_t$ and $r\times r$ matrix $\mathbf{U}_t$ are referred to as the propagation matrix and innovation covariance matrix, respectively.
The spatial-temporal process $\delta_t(\cdot)$ is linked to the random vector $\boldsymbol \xi_t$ through the link matrix $\mathbf{B}_t(\cdot)$. As the domain is partitioned into pairwise disjoint subregions $\{\mathcal{R}_i: i=1, \ldots, N\}$, we assume the following parsimonious spatial-temporal model for $\boldsymbol \xi_t=(\xi_t(\mathcal{R}_1), \ldots, \xi_t(\mathcal{R}_N))'$: for $i=1, \ldots, N$,
\begin{eqnarray} \label{eqn: model for xi}
\xi_t(\mathcal{R}_i) \mid \boldsymbol \xi_t^{-i} \sim \mathcal{N}(\gamma_{t} /e_{i+} \cdot \sum_{j\in \partial \mathcal{R}_i} e_{ij} \xi_t(\mathcal{R}_j),\, \tau_t^2/e_{i+}),
\end{eqnarray}
where $\boldsymbol \xi_t^{-i}\equiv (\xi_t(\mathcal{R}_1), \ldots, \xi_t(\mathcal{R}_{i-1}), \xi_t(\mathcal{R}_{i+1}), \ldots, \xi_t(\mathcal{R}_{N}))'$; $\gamma_{t}$ is the spatial dependence parameter at time $t$; $\mathbf{E}=(e_{ij})$ is an $N \times N$ adjacency matrix on the discretized domain $\mathcal{R}$ at BAU-level, see Chapter 6 of \cite{Cressie1993} for various specifications; $\partial \mathcal{R}_i$ is a set of indices corresponding to the neighbors of the BAU $\mathcal{R}_i$ that are defined by the spatial adjacency matrix $\mathbf{E}$; and $e_{i+}=\sum_{j=1}^N e_{ij}$ for $i=1, \ldots, N$. $\tau^2_t$ is the conditional marginal variance parameter at time $t$. From Eq.~\eqref{eqn: model for xi}, it is easy to derive that the joint conditional distribution of $\boldsymbol \xi_t$ for $t=1, \ldots, T$, is
\begin{eqnarray} \label{eqn: conditional dist for xi}
\boldsymbol \xi_t \sim \mathcal{N}(\mathbf{0},\, \mathbf{Q}_t^{-1}),
\end{eqnarray}
where $\mathbf{Q}_t \equiv \boldsymbol \boldsymbol \Delta^{-1} (\mathbf{I} - \gamma_{t} \mathbf{W})/\tau_t^2$; $\mathbf{W} \equiv\boldsymbol \boldsymbol \Delta \cdot \mathbf{E}$ is the $N\times N$ proximity matrix, and $\boldsymbol \Delta \equiv \text{diag}(1/ e_{1+}, \ldots, 1/e_{N+})$. This model is called a conditional autoregressive (CAR) model. The model for $\delta_t(\cdot)$ is a special case of the Gaussian graphical model (GGM). As discussed in \cite{Ma2017}, the model for $\delta_t(\cdot)$ can be constructed in a similar way as in \cite{Lindgren2011} and \cite{Nychka2015}. This will increase the flexibility of the model. Such implementation and demonstration is beyond the scope of this article. Although the model for $\delta_t(\cdot)$ does not incorporate dynamic evolution, the dynamic structure of the process $Y_t(\cdot)$ is inherited from the random vectors $\{\boldsymbol \eta_t: t=1, \ldots, T\}$.
Define the following quantities:
\begin{eqnarray}
X_{t, i}(\mathcal{A}_{t,j}^{(k)}) \equiv \frac{1}{\sum_{\ell=1}^N I(\mathbf{s}_{\ell} \in \mathcal{A}_{t,j}^{(k)})} \sum_{\ell=1}^N I(\mathbf{s}_{\ell} \in \mathcal{A}_{t,j}^{(k)})\cdot X_{t,i}(\mathbf{s}_{\ell}), \, i=1, \ldots, p, \\
S_{t, i}(\mathcal{A}_{t,j}^{(k)}) \equiv \frac{1}{\sum_{\ell=1}^N I(\mathbf{s}_{\ell} \in \mathcal{A}_{t,j}^{(k)})} \sum_{\ell=1}^N I(\mathbf{s}_{\ell} \in \mathcal{A}_{t,j}^{(k)})\cdot S_{t, i}(\mathbf{s}_{\ell}), \, i=1, \ldots, r, \\
B_{t,i}(\mathcal{A}_{t,j}^{(k)}) \equiv \frac{1}{\sum_{\ell=1}^N I(\mathbf{s}_{\ell} \in \mathcal{A}_{t,j}^{(k)})} \sum_{\ell=1}^N I(\mathbf{s}_{\ell} \in \mathcal{A}_{t,j}^{(k)})\cdot B_{t,i}(\mathbf{s}_{\ell}), \, i=1, \ldots, N,
\end{eqnarray}
where $t=1, \ldots, T$; $k=1, \ldots, k_0$; $j=1, \ldots n_t^{(k)}$. By combining Eq~\eqref{eqn: data model in DFGP} through Eq~\eqref{eqn: model for delta}, we have the following linear model at the $k$-th resolution:
\begin{eqnarray}
Z_t(\mathcal{A}_{t,i}^{(k)}) = \mathbf{X}_t(\mathcal{A}_{t,i}^{(k)})' \boldsymbol \beta_t + \mathbf{S}_t(\mathcal{A}_{t,i}^{(k)})' \boldsymbol \eta_t + \mathbf{B}_t(\mathcal{A}_{t,i}^{(k)})' \boldsymbol \xi_t + \boldsymbol \epsilon_t(\mathcal{A}_{t,i}^{(k)}).
\end{eqnarray}
Now stacking the above model over all footprints from the $k$-th instrument at time $t$ yields the following representation:
\begin{eqnarray}
\mathbf{Z}_t^{(k)} &=& \mathbf{X}_t^{(k)}\boldsymbol \beta_t + \mathbf{S}_t^{(k)} \boldsymbol \eta_t + \mathbf{B}_t^{(k)} \boldsymbol \xi_t + \boldsymbol \epsilon_t^{(k)}, \quad k=1, \ldots, k_0,
\end{eqnarray}
where $\mathbf{X}_t^{(k)}\equiv[\mathbf{X}_t(\mathcal{A}_{t,1}^{(k)}), \ldots, \mathbf{X}_t(\mathcal{A}_{t,n_t^{(k)}}^{(k)})]'$ is an $n_t^{(k)}$-by-$p$ matrix. $\mathbf{S}_t^{(k)}\equiv[\mathbf{S}_t(\mathcal{A}_{t,1}^{(k)}), \ldots, $ $\mathbf{S}_t(\mathcal{A}_{t,n_t^{(k)}}^{(k)})]'$ is an $n_t^{(k)}$-by-$r$ basis matrix in the low-rank component. $\mathbf{B}_t^{(k)}\equiv[\mathbf{B}_t(\mathcal{A}_{t,1}^{(k)}), \ldots, \mathbf{B}_t(\mathcal{A}_{t,n_t^{(k)}}^{(k)})]'$ is an $n_t^{(k)}$-by-$N$ basis matrix in the GGM component. $\boldsymbol \epsilon_t^{(k)}\equiv [\epsilon_t(\mathcal{A}_{t,1}^{(k)}), \ldots, \epsilon_t(\mathcal{A}_{t,n_t^{(k)}}^{(k)})]'$ is an $n_t^{(k)}$-dimensional random vector whose covariance matrix is $\sigma^2_{\epsilon_t, (k)} \mathbf{V}_{\epsilon_t, (k)}$ with $\mathbf{V}_{\epsilon_t, (k)} \equiv\text{diag}\{v(\mathcal{A}_{t,1}^{(k)}), $ $ \ldots, v(\mathcal{A}_{t,n_t^{(k)}}^{(k)})\}$.
Let $\mathbf{Z}_t \equiv [\mathbf{Z}_t^{(1)'}, \ldots, \mathbf{Z}_t^{(k_0)'}]'$ be an $n_t$-dimensional vector stacking all the observations together from $k_0$ satellite instruments at time $t$. Let $\mathbf{X}_t \equiv [\mathbf{X}_t^{(1)'}, \ldots, \mathbf{X}_t^{(k_0)'}]'$ be an $n_t$-by-$p$ matrix stacking all the covariates corresponding to all the observations at time $t$. Let $\mathbf{S}_t \equiv [\mathbf{S}_t^{(1)'}, \ldots, \mathbf{S}_t^{(k_0)'}]'$ be an $n_t$-by-$r$ basis matrix in the low-rank component. Let $\mathbf{B}_t \equiv [\mathbf{B}_t^{(1)'}, \ldots, \mathbf{B}_t^{(k_0)'}]'$ be an $n_t$-by-$N$ basis matrix in the GGM component. Then we have
\begin{eqnarray}
\mathbf{Z}_t = \mathbf{X}_t\boldsymbol \beta_t + \mathbf{S}_t \boldsymbol \eta_t + \mathbf{B}_t \boldsymbol \xi_t + \boldsymbol \epsilon_t,
\end{eqnarray}
where $\boldsymbol \epsilon_t\equiv (\boldsymbol \epsilon_t^{(1)'}, \ldots, \boldsymbol \epsilon_t^{(k_0)'})'$ is an $n_t$-dimensional random vector with covariance matrix $\mathbf{V}_t\equiv \text{diag}\{\sigma^2_{\epsilon_t, (1)} \mathbf{V}_{\epsilon_t, (1)}, \ldots, \sigma^2_{\epsilon_t, (k_0)} \mathbf{V}_{\epsilon_t, (k_0)}\}$.
\subsection{Kalman Filter and Kalman Smoother}
Suppose that inference is made on $Y_t(\mathbf{s}_0)$ for any centroid $\mathbf{s}_0$ of a BAU in $\mathcal{R}$ and any time $t=1, \ldots, T$. Let the set $\mathcal{R}^P$ be a collection of $m_t$ centroids where we want to make prediction of $Y_t(\cdot)$ at BAU-level for time $t$. So, the process vector of interest is
\begin{eqnarray}
\mathbf{Y}_t^P \equiv \mathbf{X}_t^P \boldsymbol \beta_t + \mathbf{S}_t^P \boldsymbol \eta_t + \boldsymbol \delta_t^P, \, t=1, \ldots, T,
\end{eqnarray}
where superscript $P$ denotes the quantities evaluated at the set $\mathcal{R}^P$ of prediction locations.
In what follows, sequential updates are given based on the hierarchical dynamical spatio-temporal process DFGP. The formulas are derived using Bayes' theorem in the context of dynamic spatio-temporal models described in \cite{Cressie2011}. To fix notation, we use $\mathbf{Z}_{1:u}\equiv [ \mathbf{Z}_1', \ldots, \mathbf{Z}_u']'$ to denote all the data collected from time $t=1$ to time $t=u$. For conditional expectations of $\boldsymbol \eta_t$ and $\boldsymbol \delta_t^P$ based on $\mathbf{Z}_{1:u}$, the following notations will be used: $\boldsymbol \eta_{t|u} \equiv E(\boldsymbol \eta_{t}\mid \mathbf{Z}_{1:u})$ and $\boldsymbol \delta_{t|u}^P\equiv E(\boldsymbol \delta_t^P \mid \mathbf{Z}_{1:u})$. The corresponding conditional covariance matrices will be denoted as $\mathbf{P}_{t|u}\equiv \mbox{\textrm{var}}(\boldsymbol \eta_t\mid \mathbf{Z}_{1:u})$ and $\mathbf{R}_{t|u}^P\equiv \mbox{\textrm{var}}(\boldsymbol \delta_t^P \mid \mathbf{Z}_{1:u})$, respectively.
Assuming initial states $\boldsymbol \eta_{0|0} \equiv \mathbf{0}$ and $\mathbf{P}_{0|0} \equiv \mathbf{K}_0$, the one-step ahead forecast distribution $[\boldsymbol \eta_t \mid \mathbf{Z}_{1:t-1}]$ is multivariate normal with mean $\boldsymbol \eta_{t|t-1}$ and covariance matrix $\mathbf{P}_{t|t-1}$ given as follows:
\begin{eqnarray} \label{eqn: one-step-ahead forecast}
\boldsymbol \eta_{t|t-1} &=& \mathbf{H}_t \boldsymbol \eta_{t-1|t-1}, \\
\mathbf{P}_{t|t-1} &=& \mathbf{H}_t \mathbf{P}_{t-1|t-1} \mathbf{H}_t' + \mathbf{U}_t,
\end{eqnarray}
where $\boldsymbol \eta_{t-1|t-1}$ is the conditional mean and $\mathbf{P}_{t-1|t-1}$ is the conditional covariance matrix for the filtering distribution $[\boldsymbol \eta_{t-1}|\mathbf{Z}_{1:t-1}]$.
The filtering distributions can be derived using Bayes' theorem: $[\boldsymbol \eta_t \mid \mathbf{Z}_{1:t}] \propto [\mathbf{Z}_t \mid \boldsymbol \eta_t] [\boldsymbol \eta_t \mid \mathbf{Z}_{1:t-1}]$ and $[\boldsymbol \delta_t^P \mid \mathbf{Z}_{1:t}] \propto [\mathbf{Z}_t \mid \boldsymbol \delta_t^P] [\boldsymbol \delta_t^P \mid \mathbf{Z}_{1:t-1}]$. As we assume Gaussianity for these random vectors, these filtering distributions also follow multivariate normal distributions. The filtering algorithm is proceeded sequentially for $t=1, \ldots, T$:
\begin{eqnarray} \label{eqn: filtering algorithm}
\boldsymbol \eta_{t|t} &=& \boldsymbol \eta_{t|t-1} + \mathbf{G}_t (\mathbf{Z}_t -\mathbf{X}_t\boldsymbol \beta_t-\mathbf{S}_t\boldsymbol \eta_{t|t-1}) \label{eqn: filter for eta}\\
\mathbf{P}_{t|t} &=& \mathbf{P}_{t|t-1} - \mathbf{G}_t \mathbf{S}_t \mathbf{P}_{t|t-1} \\
\boldsymbol \delta_{t|t}^P &=& \mathbf{B}_t^P \mathbf{Q}_t^{-1} \mathbf{B}_t'[\mathbf{S}_t\mathbf{P}_{t|t-1} \mathbf{S}_t' + \mathbf{D}_t^{-1}]^{-1} (\mathbf{Z}_t - \mathbf{X}_t\boldsymbol \beta_t - \mathbf{S}_t\boldsymbol \eta_{t|t-1}) \label{eqn: filter for delta}\\
\mathbf{R}_{t|t}^P &=& \mathbf{B}_t^P\mathbf{Q}_t^{-1}\mathbf{B}_t^{P'}
- \mathbf{B}_t^P \mathbf{Q}_t^{-1} \mathbf{B}_t' [\mathbf{S}_t\mathbf{P}_{t|t-1} \mathbf{S}_t' + \mathbf{D}_t^{-1}]^{-1} \mathbf{B}_t\mathbf{Q}_t^{-1}\mathbf{B}_t^{P'}
\end{eqnarray}
where $\mathbf{G}_t\equiv \mathbf{P}_{t|t-1} \mathbf{S}_t' (\mathbf{S}_t\mathbf{P}_{t|t-1} \mathbf{S}_t' + \mathbf{D}_t^{-1})^{-1} $ is the $r\times n_t$ Kalman gain matrix. $\mathbf{D}_t \equiv (\mathbf{B}_t \mathbf{Q}_t^{-1} \mathbf{B}_t' + \mathbf{V}_t)^{-1}$ is an $n_t$-by-$n_t$ matrix. $\mathbf{B}_t^P$ is the basis matrix in the GGM component, with $(i,j)$-th element being one if the $i$-th prediction location is the $j$-th BAU, and zero otherwise.
The smoothing distribution $[\boldsymbol \eta_t \mid \mathbf{Z}_{1:T}]$ can be written as
\begin{eqnarray*}
[\boldsymbol \eta_t \mid \mathbf{Z}_{1:T}] = \int [\boldsymbol \eta_t\mid \boldsymbol \eta_{t+1}, \mathbf{Z}_{1:T}] [\boldsymbol \eta_{t+1} \mid \mathbf{Z}_{1:T}] \,d \boldsymbol \eta_{t+1},
\end{eqnarray*}
where $[\boldsymbol \eta_t\mid \boldsymbol \eta_{t+1}, \mathbf{Z}_{1:T}] \propto [\boldsymbol \eta_{t+1} \mid \boldsymbol \eta_t] [\boldsymbol \eta_t \mid \mathbf{Z}_{1:t}]$ is derived using Bayes' theorem. The first term on the right-hand side is just the evolution distribution, and the second term on the right-hand side is the filtering distribution. Similar formulas can be derived for the smoothing distribution $[\boldsymbol \delta_t^P \mid \mathbf{Z}_{1:T}]$. The detailed derivation of these formulas can be found in Chapter 8 of \cite{Cressie2011}, where derivation is given for more general dynamic spatio-temporal models. Then the smoothing algorithm proceeds backwards in time for $t=T-1, T-2, \ldots, 1$:
\begin{eqnarray} \label{eqn: smoothing algorithm}
\boldsymbol \eta_{t|T} &=& \boldsymbol \eta_{t|t} + \mathbf{J}_t(\boldsymbol \eta_{t+1|T} - \boldsymbol \eta_{t+1|t}) \label{eqn: smoother for eta} \\
\mathbf{P}_{t|T} &=& \mathbf{P}_{t|t} + \mathbf{J}_t(\mathbf{P}_{t+1|T} - \mathbf{P}_{t+1|t}) \mathbf{J}_t' \\
\boldsymbol \delta_{t|T}^P &=& \boldsymbol \delta_{t|t}^P + \mathbf{M}_t(\boldsymbol \eta_{t+1|T} - \boldsymbol \eta_{t+1|t}) \label{eqn: smoother for delta} \\
\mathbf{R}_{t|T}^P &=& \mathbf{R}_{t|t}^P + \mathbf{M}_t(\mathbf{P}_{t+1|T} - \mathbf{P}_{t+1|t}) \mathbf{M}_t'
\end{eqnarray}
where
$\mathbf{J}_t \equiv \mathbf{P}_{t|t} \mathbf{H}_{t+1}' \mathbf{P}_{t+1|t}^{-1}$ and
$\mathbf{M}_t \equiv - \mathbf{B}_t^P \mathbf{Q}_t^{-1} \mathbf{B}_t' \mathbf{G}_t'\mathbf{H}_{t+1}' \mathbf{P}_{t+1|t}^{'-1}$.
The smoothing distribution of the initial state $\boldsymbol \eta_0$ has mean
$ \boldsymbol \eta_{0|T} = \boldsymbol \eta_{0|0} + \mathbf{J}_0(\boldsymbol \eta_{1|T} - \boldsymbol \eta_{1|t})$ and covariance matrix
$\mathbf{P}_{0|T} = \mathbf{P}_{0|0} + \mathbf{J}_0(\mathbf{P}_{1|T} - \mathbf{P}_{1|t}) \mathbf{J}_{0}'$.
The lag-1 cross-covariance term $\mathbf{P}_{t,t-1|T} \equiv \text{cov}(\boldsymbol \eta_t, \boldsymbol \eta_{t-1} \mid\mathbf{Z}_{1:T})$ is given by
\begin{eqnarray}
\mathbf{P}_{T,T-1|T} &=& (\mathbf{I}_r - \mathbf{G}_T \mathbf{S}_T') \mathbf{H}_T \mathbf{P}_{T-1|T-1} \\
\mathbf{P}_{t,t-1|T} &=& \mathbf{P}_{t|t} \mathbf{J}_{t-1}' + \mathbf{J}_t (\mathbf{P}_{t+1,t|T} - \mathbf{P}_{t|t}) \mathbf{J}_{t-1}', \, t=T-1, T-2, \ldots, 1.
\end{eqnarray}
\subsection{Filtering Distribution for Hidden Process $Y_t(\cdot)$}
For $t=1, \ldots, T$, the optimal filter of $\mathbf{Y}_{t}^P$ given the data $\mathbf{Z}_{1:t}$, denoted by $\mathbf{Y}_{t|t}^P$, is
\begin{eqnarray} \label{eqn: filtered predictor}
\mathbf{Y}_{t|t}^P \equiv E(\mathbf{Y}_t^P \mid \mathbf{Z}_{1:t}) = \mathbf{X}_t^P \boldsymbol \beta_t + \mathbf{S}_t^P \boldsymbol \eta_{t|t} + \boldsymbol \delta_{t|t}^P,
\end{eqnarray}
where $\boldsymbol \eta_{t|t}$ is given in Eq.~\eqref{eqn: filter for eta}, and $\boldsymbol \delta_{t|t}^P$ is given in Eq.~\eqref{eqn: filter for delta}. We call \eqref{eqn: filtered predictor} the \emph{Dynamic Fused Gaussian Process Filter (DFGPF)}. Its associated mean-squared-prediction-error covariance matrix is
\begin{eqnarray} \label{eqn: filter variance}
\boldsymbol \sigma^2_{t|t} \equiv E\{[\mathbf{Y}^P - \mathbf{Y}_{t|t}^P][\mathbf{Y}^P - \mathbf{Y}_{t|t}^P]'\}
= \mathbf{S}_t^P\mathbf{P}_{t|t} \mathbf{S}_t^{P'} + \mathbf{R}_{t|t}^{P} + \mathbf{S}_t^P \mathbf{C}_{t|t} + (\mathbf{S}_t^P \mathbf{C}_{t|t})',
\end{eqnarray}
where $\mathbf{C}_{t|t}\equiv \text{cov}(\boldsymbol \eta_t, \boldsymbol \delta_t^P \mid \mathbf{Z}_{1:t}) = -\mathbf{G}_t\mathbf{B}_t\mathbf{Q}_t^{-1}\mathbf{B}_t^{P'}$. We call the square root of the diagonal elements in $\boldsymbol \sigma^2_{t|t}$ the DFGPF standard errors.
\subsection{Smoothing Distribution for Hidden Process $Y_t(\cdot)$}
For $t=1, \ldots, T-1$, the optimal smoother of $\mathbf{Y}_t^P$ given the data $\mathbf{Z}_{1:T}$, denoted by $\mathbf{Y}_{t|T}^P$, is
\begin{eqnarray} \label{eqn: smoothed predictor}
\mathbf{Y}_{t|T}^P \equiv E(\mathbf{Y}_t^P \mid \mathbf{Z}_{1:T}) = \mathbf{X}_t^P \boldsymbol \beta_t + \mathbf{S}_t^P \boldsymbol \eta_{t|T} + \boldsymbol \delta_{t|T}^P,
\end{eqnarray}
where $\boldsymbol \eta_{t|T}$ is given in Eq.~\eqref{eqn: smoother for eta}, and $\boldsymbol \delta_{t|T}^P$ is given in Eq.~\eqref{eqn: smoother for delta}. We call \eqref{eqn: smoothed predictor} the \emph{Dynamic Fused Gaussian Process Smoother (DFGPS)}. Its associated mean-squared-prediction-error covariance matrix is
\begin{eqnarray}
\boldsymbol \sigma^2_{t|T} \equiv E\{[\mathbf{Y}^P - \mathbf{Y}_{t|T}^P][\mathbf{Y}^P - \mathbf{Y}_{t|T}^P]'\}
= \mathbf{S}_t^P\mathbf{P}_{t|T} \mathbf{S}_t^{P'} + \mathbf{R}_{t|T}^{P} + \mathbf{S}_t^{P} \mathbf{C}_{t|T} + (\mathbf{S}_t^{P} \mathbf{C}_{t|T})',
\end{eqnarray}
where $\mathbf{C}_{t|T}\equiv \text{cov}(\boldsymbol \eta_t, \boldsymbol \delta_t^P \mid \mathbf{Z}_{1:T})= -\mathbf{G}_t\mathbf{B}_t\mathbf{Q}_t^{-1}\mathbf{B}_t^{P'} + \mathbf{J}_t(\mathbf{P}_{t+1|T}-\mathbf{P}_{t+1|t})\mathbf{M}_t'$. We call the square root of the diagonal elements in $\boldsymbol \sigma^2_{t|T}$ the DFGPS standard errors.
\section{Maximum Likelihood Estimation via Stochastic EM Algorithm} \label{sec: SEM}
In what follows, we give a general derivation of the maximum likelihood estimation procedure for both filtering and smoothing methodology based on DFGP. Let $u$ be a generic time point. For the filtering-type estimator, the parameters will be estimated based on data $\mathbf{Z}_{1:u}$, where $u$ takes values from 2 to $T$, since data are collected from time point 1 to time point $T$. For the smoothing-type estimator, the parameters will be estimated based on the data $\mathbf{Z}_{1:u}$, where $u$ takes value $T$ only, since all the available data should be used in the smoothing-type methodology. To avoid identifiability issues, we assume that the propagation matrices $\{\mathbf{H}_t: t=1, \ldots, T\}$ and innovation matrices $\{\mathbf{U}_t: t=1, \ldots, T\}$ are time-invariant with common propagation matrix $\mathbf{H}$ and common innovation matrix $\mathbf{U}$. Let $\boldsymbol \theta \equiv \{ \boldsymbol \beta_1, \ldots, \boldsymbol \beta_u, \mathbf{K}_0, \mathbf{H}, \mathbf{U}, \tau^2_1, \ldots, \tau^2_u, \gamma_1, \ldots, \gamma_u\} \cup \{\sigma^2_{\epsilon_t, (k)}: t=1, \ldots, u; k=1, \ldots, k_0\}$ be a collection of model parameters up to time $u$. If the goal is to make filtering-type predictions, the letter $u$ denotes the current time at which predictions will be made, and parameters are estimated based on data up to current time $u$. If the goal is to make smoothing-type predictions, the letter $u$ denotes the time at which latest data are observed at time $T$. Smoothing-type predictions will be made at time $t=1, \ldots, T-1$. In what follows, we give an efficient parameter estimation procedure to ensure the scalability of the DFGP methodology for massive datasets.
\subsection{Likelihood Function}
Let $\boldsymbol \alpha_t = \mathbf{Z}_t - \mathbf{X}_t\boldsymbol \beta_t - \mathbf{S}_t \boldsymbol \eta_{t|t-1}$ be innovations for $t=1, \ldots, u$. These innovations are independently and normally distributed with mean zero and covariance matrix $\boldsymbol \Sigma_{t|t-1}=\mathbf{S}_t\mathbf{P}_{t|t-1}\mathbf{S}_t' + \mathbf{D}_t^{-1}$. Then, up to a constant, the negative twice marginal log-likelihood function is
\begin{eqnarray} \label{eqn: loglikelihood}
-2\ln L(\boldsymbol \theta) = -2 f(\boldsymbol \alpha_1, \ldots, \boldsymbol \alpha_u| \boldsymbol \theta) = \sum_{t=1}^u \ln |\boldsymbol \Sigma_{t|t-1}| + \sum_{t=1}^u \boldsymbol \alpha_t(\boldsymbol \theta)' \boldsymbol \Sigma_{t|t-1}(\boldsymbol \theta)^{-1} \boldsymbol \alpha_t(\boldsymbol \theta),
\end{eqnarray}
where $\boldsymbol \theta$ denotes model parameters. The inverse and log-determinant of $\boldsymbol \Sigma_{t|t-1}$ can be calculated as follows:
\begin{eqnarray} \label{eqn: matrix inversion}
\boldsymbol \Sigma_{t|t-1}^{-1} &=& \mathbf{D}_t - \mathbf{D}_t \mathbf{S}_t (\mathbf{P}_{t|t-1}^{-1} + \mathbf{S}_t' \mathbf{D}_t \mathbf{S}_t)^{-1} \mathbf{S}_t' \mathbf{D}_t \\
\ln |\boldsymbol \Sigma_{t|t-1}| &=& \ln |\mathbf{P}_{t|t-1}^{-1} + \mathbf{S}_t' \mathbf{D}_t \mathbf{S}_t| + \ln |\mathbf{P}_{t|t-1}| + \ln |\mathbf{D}_t^{-1}|,
\end{eqnarray}
where $\mathbf{D}_t=\mathbf{V}_t^{-1} -\mathbf{V}_t^{-1} \mathbf{B}_t (\mathbf{Q}_t + \mathbf{B}_t' \mathbf{V}_t^{-1} \mathbf{B}_t)^{-1} \mathbf{B}_t' \mathbf{V}_t^{-1}$ and $\ln |\mathbf{D}_t^{-1} | = \ln |\mathbf{Q}_t+\mathbf{B}_t'\mathbf{V}_t^{-1} \mathbf{B}_t| - \ln |\mathbf{Q}_t^{-1} | + \ln |\mathbf{V}_t|$.
To evaluate the negative twice marginal log-likelihood function in Eq.~\eqref{eqn: loglikelihood}, solving linear systems involving $\boldsymbol \Sigma_{t|t-1}$ is required. To solve $\boldsymbol \Sigma_{t|t-1}^{-1} \mathbf{T}$ for an $n_t$-dimensional vector or an $n_t$-by-$r$ matrix $\mathbf{T}$, one has to solve linear systems involving $r$-by-$r$ matrices and $N$-by-$N$ sparse matrix $\mathbf{Q}_t + \mathbf{B}_t' \mathbf{V}_t^{-1} \mathbf{B}_t$. The former requires $O(r^3)$ computational cost. If the sparse matrix $\mathbf{Q}_t + \mathbf{B}_t' \mathbf{V}_t^{-1} \mathbf{B}_t$ has bandwidth $p_0$ after appropriate reordering, the computational cost of the Cholesky decomposition for $\mathbf{Q}_t + \mathbf{B}_t' \mathbf{V}_t^{-1} \mathbf{B}_t$ is $O(N(p_0^2+3p_0))$. Linear systems involving $\mathbf{Q}_t + \mathbf{B}_t' \mathbf{V}_t^{-1} \mathbf{B}_t$ can be solved efficiently. The Cholesky decomposition of $\mathbf{Q}$ can be solved very efficiently with $O(N^{1.5})$ computational cost \cite[see][]{Ma2017}.
\subsection{Stochastic EM Algorithm}
The expectation-maximization (EM) algorithm introduced by \cite{Dempster1977} is a powerful method to solve maximum likelihood estimation problems iteratively, where the E-step is calculated exactly and then is followed by an M-step in each iteration. Instead of computing the conditional expectation exactly in the E-step, one can employ Monte Carlo algorithms to generate samples from the conditional distribution, and then replace the conditional expectation with an average of corresponding quantities evaluated at these samples. This estimation method is called the Monte Carlo EM algorithm \citep{Wei1990}. The success of the Monte Carlo EM algorithm relies on sufficiently large samples to approximate the E-step conditional expectations. A closely related modification of the EM algorithm described in \cite{Celeux1985} is known as the stochastic EM (SEM) algorithm, which substitutes the E-step with a single corresponding quantity evaluated with one random sample from the conditional distribution. The SEM algorithm is generally less computationally expensive than the Monte Carlo EM algorithm. It is robust with initial values and converges to a stationary distribution \cite[see][for details]{Celeux1985, Diebolt1996, Nielsen2000}. However, when computing resources are less constrained, one can employ the Monte Carlo EM algorithm estimate parameters in the DFGP methodology. In what follows, we give the SEM procedure to estimate parameters in the DFGP methodology.
In the EM algorithm of the DFGP model, we treat $\boldsymbol \eta_{0:u}$ and $\boldsymbol \xi_{1:u}$ as ``missing data''. Up to a constant, the negative twice complete-data log-likelihood is
\begin{eqnarray} \label{eqn: complete loglikelihood}
-2\ln L_c(\boldsymbol \theta) &=& -2\ln f(\mathbf{Z}_{1:u}, \boldsymbol \eta_{0:u}, \boldsymbol \xi_{1:u} | \boldsymbol \theta) \\ \nonumber
&=& \sum_{t=1}^u \left\{(\mathbf{Z}_t - \mathbf{X}_t \boldsymbol \beta_t - \mathbf{S}_t \boldsymbol \eta_t - \mathbf{B}_t \boldsymbol \xi_t)' \mathbf{V}_t^{-1} (\mathbf{Z}_t - \mathbf{X}_t \boldsymbol \beta_t - \mathbf{S}_t \boldsymbol \eta_t - \mathbf{B}_t \boldsymbol \xi_t) + \ln |\mathbf{V}_t|\right\} \\ \nonumber
&+& \sum_{t=1}^u \left\{(\boldsymbol \eta_t - \mathbf{H}\boldsymbol \eta_{t-1})' \mathbf{U}^{-1} (\boldsymbol \eta_t - \mathbf{H} \boldsymbol \eta_{t-1}) + \ln |\mathbf{U}|\right\} \\ \nonumber
&+& \ln |\mathbf{K}_0| + \boldsymbol \eta_0' \mathbf{K}_0^{-1} \boldsymbol \eta_0 + \sum_{t=1}^u\left\{ \boldsymbol \xi_t'\mathbf{Q}_t \boldsymbol \xi_t - \ln|\mathbf{Q}_t|\right\}.
\end{eqnarray}
Given the negative twice complete-data log-likelihood function in Eq.~\eqref{eqn: complete loglikelihood}, we now derive the $Q$-function in the EM algorithm first, and then present the derivation for the SEM algorithm. Consider the $(\ell+1)$th iteration in the EM algorithm. The E-step is to find conditional expectation of the complete-data log-likelihood for $\boldsymbol \theta=\boldsymbol \theta^{[\ell]}$ with respect to missing data, i.e., $Q(\boldsymbol \theta; \boldsymbol \theta^{[\ell]}) := E_{\boldsymbol \theta^{[\ell]}} [ -2\ln L_c(\boldsymbol \theta) \mid \mathbf{Z}_{1:u}]$.
In what follows, the following notations are used: $\boldsymbol \eta_{t|u}^{[\ell]} = E_{\boldsymbol \theta^{[\ell]}}(\boldsymbol \eta_t\mid \mathbf{Z}_{1:u}), \boldsymbol \delta_{t|u}^{[\ell]} = E_{\boldsymbol \theta^{[\ell]}} (\boldsymbol \delta_t\mid \mathbf{Z}_{1:u})$, $\mathbf{P}_{t|u}^{[l]} = \text{var}(\boldsymbol \eta_t\mid \mathbf{Z}_{1:u}, \boldsymbol \theta^{[\ell]}), \mathbf{R}_{t|u}^{[\ell]} = \text{var} (\boldsymbol \delta_t \mid \mathbf{Z}_{1:u}, \boldsymbol \theta^{[l]})$, $\mathbf{P}_{t,t-1|u}^{[\ell]} = \text{cov}(\boldsymbol \eta_t, \boldsymbol \eta_{t-1} \mid \mathbf{Z}_{1:u}, \boldsymbol \theta^{[\ell]})$ for all $t$.
In the EM algorithm, the actual prediction location $\mathcal{R}^P$ is the same as the observed locations, $\mathcal{R}^O$, for $t=1, \ldots, u$. We also define the quantities $\mathbf{K}_t^{[\ell+1]} \equiv \mathbf{P}_{t|u}^{[\ell]} + \boldsymbol \eta_{t|u}^{[\ell]}\boldsymbol \eta_{t|u}^{[\ell]'}$ and $\mathbf{L}_t^{[\ell+1]} \equiv \mathbf{P}_{t,t-1|u}^{[\ell]} + \boldsymbol \eta_{t|u}^{[\ell]} \boldsymbol \eta_{t-1|u}^{[\ell]'}$.
In the EM algorithm, these conditional expectations are computed exactly; while in the SEM algorithm, they are replaced by the complete log-likelihood function evaluated with a single sample from the posterior distribution $[\boldsymbol \eta_t, \boldsymbol \xi_t \mid \mathbf{Z}_{1:u}]$. To ensure the positive definiteness of matrices $\mathbf{K}_0^{[\ell]}$ and $\mathbf{U}_0^{[\ell]}$, we compute the conditional expectations involving them exactly. We only use samples from $[\boldsymbol \eta_t, \boldsymbol \xi_t \mid \mathbf{Z}_{1:u}, \boldsymbol \theta^{[\ell]}]$ to approximate the conditional expectations involving in $\boldsymbol \xi_t$. That is, we compute conditional expectations involving $\boldsymbol \eta_t$ explicitly, and we define $\boldsymbol \eta_{t|u}^{[\ell]} = E_{\boldsymbol \theta^{[\ell]}}(\boldsymbol \eta_t \mid \mathbf{Z}_{1:u})$. For expectations involving $\boldsymbol \xi_t$, we generate a sample from $[ \boldsymbol \eta_t, \boldsymbol \xi_t \mid \mathbf{Z}_{1:u}, \boldsymbol \theta^{[\ell]}]$, and use $\boldsymbol \xi_{t|u}^{[\ell]}$ to denote a sample for $\boldsymbol \xi_t$ from the distribution $[ \boldsymbol \eta_t, \boldsymbol \xi_t \mid \mathbf{Z}_{1:u}, \boldsymbol \theta^{[\ell]}]$. To generate a sample from the distribution $[ \boldsymbol \eta_t, \boldsymbol \xi_t \mid \mathbf{Z}_{1:u}, \boldsymbol \theta^{[\ell]}]$, we use the conditional simulation strategy in geostatistics to allow fast sampling procedure \citep[for details, see][Sec. 3.6.2]{Cressie1993}. The negative twice $Q_{SEM}(\cdot; \cdot)$ function in the SEM algorithm is
\begin{eqnarray*}
-2Q_{SEM}(\boldsymbol \theta; \boldsymbol \theta^{[\ell]})
&\equiv& \sum_{t=1}^u \left\{(\mathbf{Z}_t - \mathbf{X}_t \boldsymbol \beta_t - \mathbf{S}_t \boldsymbol \eta_{t|u}^{[\ell]} - \mathbf{B}_t \boldsymbol \xi_{t|u}^{[\ell]})' \mathbf{V}_t^{-1} (\mathbf{Z}_t - \mathbf{X}_t \boldsymbol \beta_t - \mathbf{S}_t \boldsymbol \eta_{t|u}^{[\ell]} - \mathbf{B}_t \boldsymbol \xi_{t|u}^{[\ell]}) \right\}\\
&+& u\ln |\mathbf{U}| + \sum_{t=1}^u tr\left\{ \mathbf{U}^{-1} (\mathbf{K}_t^{[\ell+1]} - \mathbf{H} \mathbf{L}_t^{[\ell+1]'} - \mathbf{L}_t^{[\ell+1]}\mathbf{H}' + \mathbf{H}\mathbf{K}_{t-1}^{[\ell+1]}\mathbf{H}') \right\} \\
&+& \ln |\mathbf{K}_0| + tr(\mathbf{K}_0 \mathbf{K}_0^{[\ell+1]}) + \sum_{t=1}^u \left\{ \boldsymbol \xi_{t|u}^{[\ell]'} \mathbf{Q}_t \boldsymbol \xi_{t|u}^{[\ell]} - \ln|\mathbf{Q}_t| + \ln |\mathbf{V}_t| \right\},
\end{eqnarray*}
where $\boldsymbol \xi_{t|u}^{[\ell]}$ is the sub-vector of a random sample from $[\boldsymbol \eta_t, \boldsymbol \xi_t \mid \mathbf{Z}_{1:u}, \boldsymbol \theta^{[\ell]}]$ corresponding to $\boldsymbol \xi_t$. We compute the expectations involving $\mathbf{U}$ and $\mathbf{K}_0$ exactly, since this will give a desirable property in their updating formulas that both of them are guaranteed to be positive definite in each iteration of the SEM algorithm.
In the M-step, this $Q_{SEM}$ function is maximized with respect to parameters $\boldsymbol \theta$, yielding the following formulas to update parameters for $t=1, \ldots, u$:
\begin{eqnarray*} \label{eqn: parameter updates}
\boldsymbol \beta_t^{[\ell+1]} &=& (\mathbf{X}_t' \mathbf{V}_t \mathbf{X}_t)^{-1} \mathbf{X}_t' \mathbf{V}_t^{-1} (\mathbf{Z}_t - \mathbf{S}_t \boldsymbol \eta_{t|u}^{[\ell]} - \mathbf{B}_t \boldsymbol \xi_{t|u}^{[\ell]}), \\ \nonumber
\sigma_{\epsilon_t, (k)}^{2 [\ell+1]} &=& (\mathbf{Z}_t^{(k)} - \mathbf{X}_t^{(k)}\boldsymbol \beta_{t}^{[\ell+1]}- \mathbf{S}_t^{(k)} \boldsymbol \eta_{t|u}^{[\ell]} - \mathbf{B}_t^{(k)} \boldsymbol \xi_{t|u}^{[\ell]})' \mathbf{V}_{\epsilon_t, (k)}^{-1} \\
&\cdot& (\mathbf{Z}_t^{(k)} - \mathbf{X}_t^{(k)}\boldsymbol \beta_{t}^{[\ell+1]}- \mathbf{S}_t^{(k)} \boldsymbol \eta_{t|u}^{[\ell]} - \mathbf{B}_t^{(k)} \boldsymbol \xi_{t|u}^{[\ell]})
\Big/ n_t^{(k)}, \\
\mathbf{K}_0^{[\ell+1]} &=& \mathbf{P}_{0|u}^{[\ell]} + \boldsymbol \eta_{0|u}^{[\ell]} \boldsymbol \eta_{0|u}^{[\ell]'}, \\
\mathbf{H}^{[\ell+1]} &=& \left(\sum_{t=1}^u \mathbf{L}_t^{[\ell+1]}\right) \left(\sum_{t=0}^{u-1} \mathbf{K}_t^{[\ell+1]}\right)^{-1}, \\
\mathbf{U}^{[\ell+1]} &=& \left(\sum_{t=1}^u \mathbf{K}_t^{[\ell+1]} - \mathbf{H}^{[\ell+1]} \sum_{t=1}^u \mathbf{L}_t^{[\ell+1]'}\right) \Big/u, \\
\tau_t^{2[\ell+1]} &=& \boldsymbol \xi_{t|u}^{[\ell]'} (\mathbf{I} - \gamma_t^{[\ell]}\mathbf{W}) \boldsymbol \xi_{t|u}^{[\ell]} \Big/ N,
\end{eqnarray*}
where the parameters $\beta_t, \sigma^2_{\epsilon_t, (k)}, \mathbf{K}_0, \mathbf{H}, \mathbf{U}$, and $\tau^2_t$ have closed-form updates. When the M-step is carried out w.r.t.~$\{\gamma_t: t=1, \ldots, u\}$, there is no explicit formula. However, it is straightforward to show that we can minimize the following function w.r.t.~these parameters:
\begin{eqnarray*}
g(\gamma_1, \ldots, \gamma_u) &=& \sum_{t=1}^u ( -\gamma_t\boldsymbol \xi_{t|u}^{[\ell]'} \mathbf{W} \boldsymbol \xi_{t|u}^{[\ell]}/\tau_t^{2[\ell]} - \ln|\mathbf{I} - \gamma_t\mathbf{W}|).
\end{eqnarray*}
To solve this nonlinear optimization problem, we use the interior-point method \citep[e.g.,][]{Byrd1999}. Notice that we can estimate all the parameters including the nugget variance parameters $\{ \sigma^2_{\epsilon_t, (k)}: t=1, \ldots, u; k=1, \ldots, k_0.\}$ in the SEM algorithm.
The SEM algorithm starts with certain initial values for parameters $\boldsymbol \theta$, and then these parameters are updated iteratively at each iteration. The initial values should be tuned to achieve better convergence results. Here, we give some practical suggestions. The initial values of regression coefficients can be set as the ordinary least square estimates. The initial values for $\{ \sigma^2_{\epsilon_t, (k)}: t=1, \ldots, u; k=1, \ldots, k_0.\}$ can be set as the parameter estimates by fitting empirical semivariograms near origin \citep[e.g.,][]{Kang2010}. The initial values for $\mathbf{K}_0$ and $\mathbf{U}$ can be set to the $r$-by-$r$ positive definite matrices with diagonal entries adjusted by the empirical variance of the data values $\{\mathbf{Z}_t: t=1, \ldots, u\}$. The initial values for $\tau^2_t$ can be set as a small portion (say 0.01) of the empirical variance of the data values $\{\mathbf{Z}_t: t=1, \ldots, u\}$. The initial values for $\mathbf{H}$ can be set as an identity matrix. After the initial values are specified, the SEM algorithm will update them in each iteration. The formulas to update all these parameters reveal that the positive definiteness of matrix $\mathbf{K}_0$ and $\mathbf{U}$ is guaranteed. To check the convergence of the SEM algorithm, we can monitor the change or relative change of log-likelihood function \eqref{eqn: loglikelihood} for a sufficient number of consecutive iterations. In addition, one can also monitoring the different of parameter values, i.e., $\| \boldsymbol \theta^{[\ell+1]} - \boldsymbol \theta^{[\ell]}\|$.
Although the procedure to obtain standard errors for the parameter estimates is not discussed here, these standard errors can be obtained in a certain way. For instance, one might use the bootstrap-sampling technique described in \cite{Stoffer1991} to compute the standard errors for these parameters estimates. Its detailed discussion is beyond the scope of this work. As in each iteration of the SEM algorithm, solving linear systems involving $r$-by-$r$ matrices and $N$-by-$N$ sparse matrices is required. We also need to carry out numerical optimization to update $\{\gamma_t: t=1, \ldots, u\}$, which requires the Cholesky decomposition for sparse matrices $\{\mathbf{Q}_t: t=1, \ldots, u\}$. These computations can be done efficiently but still require considerate amount of computing time.
When data are sparse, we can impose the time-invariant assumption for the nugget variance parameters $\{\sigma^2_{\epsilon_t, (k)}: t=1, \ldots, u; k=1, \ldots, k_0.\}$ so that stable estimates for these parameters can be obtained. Suppose that $\sigma^2_{\epsilon, (k)} \equiv \sigma^2_{\epsilon_1, (k)} = \ldots, = \sigma^2_{\epsilon_u, (k)}$. The formulas to update these nugget variance parameters are:
\begin{eqnarray*}
\sigma_{\epsilon, (k)}^{2 [\ell+1]} &=& \sum_{t=1}^u (\mathbf{Z}_t^{(k)} - \mathbf{X}_t^{(k)}\boldsymbol \beta_{t}^{[\ell+1]}- \mathbf{S}_t^{(k)} \boldsymbol \eta_{t|u}^{[\ell]} - \mathbf{B}_t^{(k)} \boldsymbol \xi_{t|u}^{[\ell]})' \mathbf{V}_{\epsilon_t, (k)}^{-1} \\
&\cdot & (\mathbf{Z}_t^{(k)} - \mathbf{X}_t^{(k)}\boldsymbol \beta_{t}^{[\ell+1]}- \mathbf{S}_t^{(k)} \boldsymbol \eta_{t|u}^{[\ell]} - \mathbf{B}_t^{(k)} \boldsymbol \xi_{t|u}^{[\ell]}) \bigg/ \left(\sum_{t=1}^u n_t^{(k)} \right).
\end{eqnarray*}
When data are available over a large number of time steps, i.e., $u$ is large, we can relax the time-invariant assumption for propagation matrices $\{\mathbf{H}_t: t=1, \ldots, u\}$, and innovation matrices $\{\mathbf{U}_t: t=1, \ldots, u\}$. Suppose that the propagation matrix and innovation matrix are constant over $b_0$ block time periods. Let $T_0\equiv 0$ and $T_{b_0}\equiv u$. We further assume that $\mathbf{H}_{1}=\ldots =\mathbf{H}_{T_1} \neq \mathbf{H}_{T_1+1} = \ldots =\mathbf{H}_{T_2} \neq \ldots =\mathbf{H}_{T_{b_0}}$ and $\mathbf{U}_{1}=\ldots =\mathbf{U}_{T_1} \neq \mathbf{U}_{T_1+1} = \ldots = \mathbf{U}_{T_2} \neq \ldots = \mathbf{U}_{T_{b_0}}$. For $b=1, \ldots, b_0$, the formulas to update $\mathbf{H}_{T_b}$ and $\mathbf{U}_{T_b}$ are
\begin{eqnarray*}
\mathbf{H}_{T_b}^{[\ell+1]} &=& \left(\sum_{t=T_{b-1}+1}^{T_{b}} \mathbf{L}_t^{[\ell+1]}\right) \left(\sum_{t=T_{b-1}+1}^{T_{b}-1} \mathbf{K}_t^{[\ell+1]}\right)^{-1}, \\
\mathbf{U}_{T_b}^{[\ell+1]} &=& \left(\sum_{t=T_{b-1}+1}^{T_{b}} \mathbf{K}_t^{[\ell+1]} - \mathbf{H}^{[\ell+1]}_t \sum_{t=T_{b-1}+1}^{T_b} \mathbf{L}_t^{[\ell+1]'}\right) \bigg/\left(T_b - T_{b-1}\right).
\end{eqnarray*}
\section{Results} \label{sec: results}
In what follows, we demonstrate DFGP methodology based on two SST datasets from MODIS and AMSR-E instruments. The tropical Pacific region is covered by $N=1,260,864$ grid cells at 9km resolution, which are used to define BAUs in this study. The CAR structure in the DFGP model is constructed based on first-order neighborhood structure on these BAUs. These two SST datasets have been bias-corrected to ensure that the mean zero assumption in the measurement-error process is valid in the DFGP methodology. Exploratory analysis suggests that the covariates $\mathbf{X}_t(\cdot) = [1, \text{latitude}(\cdot)$, $\text{latitude}(\cdot)^2]$ be included in the trend component for all time points. In Section~\ref{subsec: filtering CV} and Section~\ref{subsec: smoothing CV}, we demonstrate filtering and smoothing procedure of the DFGP methodology, respectively. Comparison of DFGP with with other existing methods is also made. In the following numerical illustrations, to assess the predictive performance for each method, we use the root-mean-sqaured-prediction error (RMSPE) to evaluate the accuracy of the predictions at held-out locations. We also use the continuous-rank-probability score \citep[CRPS;][]{Gneiting2007} to evaluate the quality of predictions for held-out locations, where small value of the CRPS indicates better prediction.
\subsection{Cross-Validation Study in the Filtering Methodology} \label{subsec: filtering CV}
We first carry out cross-validation to evaluate the filtering-type DFGP methodology and compare it with the spatio-temporal data fusion model in \cite{Nguyen2014}. In what follows, we refer to the spatio-temporal data fusion model in \cite{Nguyen2014} as \emph{Fixed Rank Filtering (FRF)}, and we refer to our filtering procedure of the DFGP methodology as \emph{DFGPF}. In FRF, we consider 99 equally-spaced basis functions at three different resolutions and 181 equally-space basis functions at four different resolutions with previous 99 basis functions included. As an additional comparison, we also implement a local kriging approach \citep[e.g.,][]{Haas1990, Vecchia1988, Kuusela2018}, which makes predictions based on observations in a moving window. This approach has been widely used in practice to make spatial/spatio-temporal predictions due to its computational efficiency and the usage of spatial/spatio-temporal varying covariance functions in a local moving window. In our implementation of the local kriging approach, we fit a spatio-temporal exponential covariance function model with its parameters estimated by the maximum likelihood method in a spatio-temporal moving window such that it contains 500 nearest observations. The exponential covariance function is chosen to be anisotropic in space and time, i.e., $C(h, u) = \sigma^2\exp\{-\sqrt{\frac{h^2}{\phi_s^2} + \frac{u^2}{\phi_t^2} } \} + \sigma^2_{\epsilon}I(h=0, u=0)$, where $h$ is the chordal distance in space and $u$ is the Euclidean distance in time. $\phi_s$ and $\phi_t$ are range parameters in space and time, $\sigma^2$ is the partial sill, and $\sigma^2_{\epsilon}$ is the nugget variance. This approach will be referred to as Local Kriging hereafter. Notice that when implementing the local kriging approach, we assume that each observation is associated with the centroid of the grid cell, and we ignore the resolution differences among each dataset. In the filtering context, the nearest observations are selected based on all training observations from $t=1, \ldots, 8$.
To set up the cross-validation exercise, we hold out MODIS SST in the block region between longitude $180^{\circ}$ and $190^{\circ}$ and between latitude $-20^{\circ}$ and $0^{\circ}$ for time point $t=2, \ldots, 8$; see Figure~\ref{fig: block region SST} for an example of the held-out region on January 8, 2018. This large contiguous region is used to test long-range prediction skills that are often required to make predictions for remote sensing data, because remote sensing data often have missing latitude bands when polar-orbiting satellite instruments collect the data around the Earth. In addition, we also hold out randomly sampled 10\% of remaining MODIS SST observations for time point $t=2, \ldots, 8$. These randomly held-out prediction locations will be used to test the short-range prediction skills. As our focus is to compare filtering-type predictions among Local Kriging, FRF and DFGPF, we only use the data up to time $t$ to estimate parameters and to make filtering predictions at held-out locations at time $t$, where $t=2, \ldots, 8$. Notice that we do not hold out observations at time $t=1$, since the filtering-type DFGP methodology reduces to the spatial-only FGP methodology, which has been studied in \cite{Ma2017} under various numerical examples.
Figure~\ref{fig: numerical comparison at different time points} shows the RMSPE and CRPS for FRF, DFGPF and Local Kriging at time point $t=2, \ldots, 8$. As we can see, DFGPF performs the best among all the three methods in terms of RMSPE and CRPS at all time points. Even though the number of basis functions in the low-rank component of FRF is almost twice as that in DFGPF, DFGPF still outperforms FRF in terms of RMSPE and CRPS at all time points, since the DFGPF model incorporates a more flexible covariance function than that in the FRF model. The Local Kriging approach gives better predictions than FRF. However, FRF may perform better than Local Kriging using adaptive basis functions; readers are referred to \cite{Ma2018downscaling} for details. In addition, we also tried in increase more basis functions, but this will pose numerical instabilities due to different contiguous regions at different time points. Basis function selection for such a spatio-temporal model is still an open problem. Methods in \cite{Tzeng2017} and \cite{Ma2018downscaling} can be extended to the spatio-temproral context to tackle this problem. This will be left for future research. DFGP outperforms Local Kriging at all tested time points in terms of RMSPE and CRPS. The reasons are as follows. First, Local Kriging fails to provide good predictions especially in large contiguous missing region, since it only uses local information without borrowing strength from distant observations. Second, Local Kriging uses a stationary exponential covariance function, which may not be flexible enough to capture nonstationary behavior of the underlying geophysical process. Third, Local Kriging does not address the change-of-support problem. It is shown in \cite{Ma2018downscaling} that ignoring the change-of-support problem can lead to unfavorable statistical inferential results.
Table~\ref{table: CV_SST} shows the average of numerical measure such as RMSPE and CRPS across all the seven tested time points as well as the total computing time (in hours) for parameter estimation and prediction on a 10-core machine with 20GB memory and Intel Xeon E5-2680 central processing unit. We see that DFGPF outperforms Local Kriging and FRF in terms of RMSPE and CRPS. For computing time, FRF is fastest, since FRF only needs to invert $r$-by-$r$ matrix and diagonal matrix. When $r$ is very small (e.g., $r=99, 181$), its computation can be very fast. Local Kriging is second fastest, since it only needs to solve small (e.g., 500-by-500) linear systems for every prediction location, and the parallel computing environment can be employed to facilitate computations.
DFGPF requires about 4 to 9 times, dependent on the number of basis functions in FRF, more computing time than FRF, since DFGPF not only needs to invert $r$-by-$r$ matrices but also needs to solve sparse linear systems for $N$-by-$N$ sparse matrices. Even though DFGPF requires more computing time, it can give very good predictive performance in a reasonable amount of time, since DFGPF is able to process a 8-day dataset with about 3.7 million observations in a time much less than one week.
\begin{figure}[htbp]
\begin{center}
\makebox[\textwidth][c]{ \includegraphics[width=1.0\textwidth, height=0.3\textheight]{MODIS_SST_holdout_T8_region.pdf}}
\caption{MODIS SST data are held out in the rectangular region on January 8, 2010. The delineated rectangular region is the held-out contiguous region to test long-range prediction skills.}
\label{fig: block region SST}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\makebox[\textwidth][c]{ \includegraphics[width=1.0\textwidth, height=0.2\textheight]{numerical_comparison_LFD_alldays.pdf}}
\caption{Numerical measures for predictions over all held-out locations based on Local Kriging, FRF, and DFGPF at $t=2, \ldots, 8$. The figure shows the RMSPEs at the left panel and CRPSs at the right panel for these methods, respectively. The asterisk represents the numerical measures based on Local Kriging; the plus sign represents the numerical measures based on FRF with $r=99$ basis functions; the cross sign represents the numerical measures based on FRF with $r=181$ basis functions; the circle sign represents the numerical measures based on DFGPF.}
\label{fig: numerical comparison at different time points}
\end{center}
\end{figure}
\subsection{Cross-Validation Study in the Smoothing Methodology} \label{subsec: smoothing CV}
DFGP not only allows filtering-type predictions, but also allows smoothing-type predictions. To demonstrate the smoothing methodology of DFGP, we compare it with the spatio-temporal data fusion model in \cite{Nguyen2014}. In what follows, we refer to our smoothing procedure of the DFGP methodology as \emph{DFGPS}, and refer to the spatio-temporal data fusion model in \cite{Nguyen2014} as \emph{Fixed Rank Smoothing (FRS)}. Notice that the methodology in FRS appeared first in \cite{Katzfuss2011} for a single data source. \cite{Nguyen2014} generalize this approach for multiple data sources. Similar to Section~\ref{subsec: filtering CV}, FRS is implemented with 99 equally-spaced basis functions at three different resolutions and 181 equally-spaced basis functions at four different resolutions. DFGPS is implemented with 99 basis functions that are the same as in FRS. As an additional comparison, we also implement Local Kriging. In the smoothing-type predictions, the nearest observations in Local Kriging are also selected based on all observations from January 1 to January 8, 2010. In the implementation of Local Kriging, we also chose 500 nearest observations. We also tried to increase the number of nearest observations, but improvement of prediction based on Local Kriging with more nearest observations is negligible and more computing resources are required in terms of computer memory and computing time.
To setup the cross-validation exercise, we hold out MODIS SST in the block region between longitude $180^{\circ}$ and $190^{\circ}$ and between latitude $-20^{\circ}$ and $0^{\circ}$ for time point $t=2, \ldots, 8$ as in Section~\ref{subsec: filtering CV}. Then 10\% of remaining MODIS SST are randomly held out to test the short-range prediction skills. Unlike the filtering context, the smoothing prediction of the process $Y_t(\cdot)$ for these held-out locations are obtained based on all remaining observations from both MODIS and AMSR-E instruments. Notice that the total number of observations that are predicted are much more in the smoothing context than those in the filtering context, since in the filtering context, we only evaluate prediction skills for $Y_t(\cdot)$ at current time $u$ based on data $\mathbf{Z}_{1:u}$, where $u=2, \ldots, 8$.
Figure~\ref{fig: numerical comparison in smoothing} shows the RMSPE and CRPS for held-out locations at time point $t=1, \ldots, 7$ based on FRS, DFGPS, Local Kriging. As we can see, the DFGPS model performs better than all the other models in terms of RMSPE and CRPS at all time points. This suggests that our proposed DFGP methodology can provide good smoothing-type predictions. Even though the number of basis functions in FRS is twice as that in the low-rank component of DFGPS, FRS still cannot outperform DFGPS. This is consistent with findings in the spatial-only context in \cite{Ma2017} and the filtering context in Section~\ref{subsec: filtering CV}. This is because the CAR structure introduces a spatial dependence structure that can capture the unexplained variation by the low-rank component. We do not further increase the number of basis functions, since we encountered numerical instability due to large portions of contiguous missing regions. The optimal selection of spatial basis functions in a spatio-temporal context is very challenging especially when data have different patterns of large contiguous missing region over time. DFGPS performs better than Local Kriging, which is expected with the same reasons given in Section~\ref{subsec: smoothing CV}.
Table~\ref{table: smoothing CV_SST} shows the average of RMSPE and CRPS across all seven time points as well as the total computing time (in hours) for parameter estimation and prediction on a 10-core machine with 20GB memory and Intel Xeon E5-2680 central processing unit. We see that DFGPS outperforms Local Kriging and FRS in terms of RMSPE and and CRPS. FRS is the fastest among all the methods due to small number of basis functions. The total computing time in FRS and DFGPS is smaller than those reported in FRF and DFGPF, since parameter estimation is only done once based on all available training data in the smoothing context; in contrast, parameter estimation and prediction are done individually for time $u=2, \ldots, 8$, based on data $\mathbf{Z}_{1:u}$ in the filtering context. As observations are held out at each time point in the smoothing procedure and only ``current'' observations are held out in the filtering procedure, more observations are held out in the smoothing procedure than those in the filtering procedure. In the implementation of Local Kriging, we only held out observations at each day, and predictions are made based on all remaining observations. DFGPS is faster than Local Kriging. However, more extensive parallelizations can be used to speed up computations in Local Kriging, it will require much more computing resources than FRS and DFGPS. DFGPS is slow compared to FRS, but it can provide very good inferential results in a reasonable amount of time, since we can obtain predictions at about 316,965 locations from all time points based on about 3.4 million training observations in a one-week time period.
\begin{figure}[htbp]
\begin{center}
\makebox[\textwidth][c]{ \includegraphics[width=1.0\textwidth, height=0.2\textheight]{numerical_comparison_LFD_smoothing_alldays.pdf}}
\caption{Numerical measures for predictions over all held-out locations based on Local Kriging, FRS, and DFGPS at $t=1, \ldots, 7$. The figure shows the RMSPEs at the left panel and CRPSs at the right panel for these methods, respectively. The asterisk represents the numerical measures based on Local Kriging; the plus sign represents the numerical measures based on FRS with $r=99$ basis functions; the cross sign represents the numerical measures based on FRS with $r=181$ basis functions; the circle sign represents the numerical measures based on DFGPS.}
\label{fig: numerical comparison in smoothing}
\end{center}
\end{figure}
\subsection{Filtering and Smoothing Predictions}
After carrying out cross-validation, we apply DFGPF to make filtering-type predictions for $t=2, \ldots, 8$, and apply DFGPS to make smoothing-type predictions for $t=1, \ldots, 7$. The SEM algorithm with different starting values is used to estimate parameters. It turns out that same parameters were obtained for a pre-specified small threshold after sufficient iterations. This also suggests that the SEM algorithm is robust to initial values as pointed out in \cite{Diebolt1996}. Figure~\ref{fig: predictions in DFGPF} shows the filtering-type predictions for $t= 4, 6, 8$ and associated standard errors, and Figure~\ref{fig: predictions in DFGPS} shows the smoothing-type predictions for $t=2, 4, 6$. As we can see, the resulting predictions are able to fill in the gaps by combining two sources of datasets. The associated prediction standard errors are also reasonable. We see that the predictions show larger uncertainties at locations where no SST data are collected than those at locations where SST data are available. As expected, DFGPS gives better predictions than DFGPF at same time points, since the smoothing methodology makes use of all the available observations. In Supplementary Materials, we also include two movies to show the filtering-type predictions for $t=2, 3, \ldots, 8$, and the smoothing-type predictions for $t=1, 2, \ldots, 7$.
\begin{figure}[htbp]
\begin{subfigure}{.95\textwidth}
\centering
\includegraphics[width=1.0\linewidth,height=0.3\textheight]{filtering_day4_center99_DFGP.pdf}
\label{fig: day4}
\end{subfigure}
\begin{subfigure}{.95\textwidth}
\centering
\includegraphics[width=1.0\linewidth,height=0.3\textheight]{filtering_day6_center99_DFGP.pdf}
\label{fig: day6}
\end{subfigure}
\begin{subfigure}{.95\textwidth}
\centering
\includegraphics[width=1.0\linewidth,height=0.3\textheight]{filtering_day8_center99_DFGP.pdf}
\label{fig: day8}
\end{subfigure}
\caption{DFGP filtering predictions and associated standard errors on January 4, 6, 8 in the year 2010 over the tropical Pacific ocean.}
\label{fig: predictions in DFGPF}
\end{figure}
\begin{figure}[htbp]
\begin{subfigure}{.95\textwidth}
\centering
\includegraphics[width=1.0\linewidth,height=0.3\textheight]{smoothing_day2_center99_DFGP.pdf}
\label{fig: day4}
\end{subfigure}
\begin{subfigure}{.95\textwidth}
\centering
\includegraphics[width=1.0\linewidth,height=0.3\textheight]{smoothing_day4_center99_DFGP.pdf}
\label{fig: day6}
\end{subfigure}
\begin{subfigure}{.95\textwidth}
\centering
\includegraphics[width=1.0\linewidth,height=0.3\textheight]{smoothing_day6_center99_DFGP.pdf}
\label{fig: day8}
\end{subfigure}
\caption{DFGP smoothing predictions and associated standard errors on January 2, 4, 6 in the year 2010 over the tropical Pacific ocean.}
\label{fig: predictions in DFGPS}
\end{figure}
\section{Discussion}\label{sec: conclusion}
In this article, we propose a dynamic fused Gaussian process model to allow both filtering and smoothing type predictions for massive remote sensing data. The parameters in DFGP are estimated via an efficient stochastic expectation-maximization algorithm. The DFGP methodology is demonstrated in a data-fusion context with multiple data sources at different spatial resolutions. We have applied our DFGP model to analyze massive amount of sea surface temperature data from MODIS and AMSR-E satellite instruments in both filtering and smoothing contexts. We found that DFGP gives better prediction results than the spatio-temporal data fusion model in \cite{Nguyen2014} in both filtering and smoothing contexts, even though more basis functions are incorporated. The DFGP methodology also gives much better prediction results than Local Kriging in both filtering and smoothing contexts.
Although DFGP requires more computational cost than Local Kriging, FRF and FRS, the computations in DFGP can be done efficiently with affordable computing resources, since a one-week dataset can be processed in much less than one week for about 3.7 million sea surface temperature observations. By borrowing strength across different time and instruments, DFGP is able to give good prediction results to fill in the gaps for massive amount of sea surface temperature data. A more compelling approach to compare DFGP might be \cite{Jurek2018}. However, it is not clear how this methodology can be extended to a data-fusion context for multiple data sources. The predictions are made at BAU-level, which is motivated by scientific study and available computing resources in practice. A statistical optimal way to choose the resolution of BAUs can be found in \cite{Bradley2017}, but a tradeoff has to be determined between available computing resources and statistical optimality.
The DFGP methodology assumes a single underlying true process, with the data process linked to this true process through different measurement-error processes. The resolution difference among each data process has been explicitly accounted for through the change-of-support property. When different underlying true processes are desired, one can extend the idea in \cite{Nguyen2014} to allow cross dependences among each underlying true process. In the DFGP methodology, the dynamic evolution is only exhibited in the low-rank component, which captures large-scale spatio-temporal variations. Future work might be introducing dynamic evolution structure in the graphical model component.
In this article, the DFGP methodology is demonstrated in a general context without any special structure imposed in the propagation matrix and innovation matrix. In practice, if physical knowledge of a geophysical process is available, this information can be incorporated in the DFGP methodology \cite[e.g.,][]{Wikle2001, Xu2007}. In fact, this will also help avoid potential non-identifiability issue because of over-parameterization in the model when data are sparse. To properly account for uncertainties in both parameter estimation and prediction, a fully Bayesian implementation of the DFGP methodology is recommended, but this will require much more computing resources for massive amount of spatio-temporal data especially for applications in remote sensing science. With high-performance computing facilities, the current DFGP methodology can be applied for much massive amount of data over much larger time period such as the work in \cite{Hoar2003}.
\section*{Acknowledgments}
This work was supported in part by an allocation of computing time from the Ohio Supercomputer Center.
Ma's research was partially supported by the National Science Foundation under Grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material do not necessarily reflect the views of the National Science Foundation. {Kang's research was partially supported by the Simons Foundation Collaboration Award (\#317298) and the Taft Research Center at the University of Cincinnati. We thank two anonymous reviewers and an associate editor for comments that greatly improved this work.
\bibliographystyle{apa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,758 |
\section{Introduction}
\label{sec:introduction}
Energy flows induced into magnetically dominated relativistic magnetospheres of compact objects are commonly modeled by numerical simulations in the force-free electrodynamics (FFE) limit. Fueled by the track record of observations in the era of multi-messenger astrophysics, current targets for such simulations include the magnetospheres of rapidly spinning black holes, spiraling neutron stars, magnetars, and pulsars. The tenuous, magnetically dominated atmosphere (magnetosphere) of pulsars is an active field of scientific interest. They fascinate both observers \citep[e.g.][]{Lorimer_1995MNRAS.273..411,Ransom_2005Sci...307..892,Abdo_2013ApJS..208...17,Jankowski_2018MNRAS.473.4436} and theorists \citep[e.g.][]{Kennel1984,Lyubarskii1996,Contopoulos1999,Contopoulos_2019MNRAS.482L..50,Goodwin_2004MNRAS.349..213,Timokhin2006,Timokhin_2013MNRAS.429...20, Petri_2020Univ....6...15}. With the remarkable progress in scientific computing, their rotating magnetosphere has captured designers of numerical methods that integrate FFE and magnetohydrodynamics (MHD) with ever improving accuracy \citep[e.g.][]{Komissarov_2006MNRAS.367...19,Spitkovsky2006,Tchekhovskoy2013,Parfrey2017,Carrasco_2020PhRvD.101f3017}.
Recently, particle-in-cell (PIC) simulations were able to resolve a broad range of scale separations and allow for unprecedented insight into the microphysics of pulsar magnetospheres across the global scale \citep[][]{Cerutti_2015MNRAS.448..606,Philippov2015,Guepin2020,Kalapotharakos_2018ApJ...857...44, Philippov_2018ApJ...855...94}. In this fascinating flurry of outcomes,
only few references scrutinised whether the results from \emph{ideal} plasma simulations are the best possible model for the pulsar magnetosphere that contains an inherently \emph{non-ideal} region, namely the equatorial current sheet (ECS) beyond the closed zone \citep{Contopoulos2016,Contopoulos_2019MNRAS.482L..50, Contopoulos2020}. Here, we study with rigorous technical depth how this non-ideal region can affect the global dynamics of the force-free aligned rotator magnetosphere - effectively serving as a blueprint for force-free magnetospheres of other compact objects.
\emph{Ideal} FFE evolve Maxwell's equations for the electromagnetic fields $\mathbf{E}$ and $\mathbf{B}$ while rigorously maintaining the force-free conditions $\mathbf{E}\cdot\mathbf{B}=0$ (equivalent to the no Ohmic heating condition $\mathbf{E}\cdot \mathbf{j}=0$, where $\mathbf{j}$ is the electric current) and $\mathbf{E}^2-\mathbf{B}^2<0$ (magnetic dominance). \emph{Non-ideal} force-free fields are those fields that allow for perturbations to the condition $\mathbf{E}\cdot\mathbf{B}=0$ by non-negligible electric fields $E_\parallel$ along the magnetic field $\mathbf{B}$ \citep[e.g.][]{Lyutikov_2003MNRAS.346..540}. Such fields were used in the literature to induce the concept of resistivity into FFE, either by specifically designed driving currents \citep[e.g.,][]{Alic2012} or by self-consistently modelled alterations to the current of ideal FFE \citep{Komissarov2004,Parfrey2017}.
In numerical models of FFE, it is common to enforce the preservation of the $\mathbf{E}\cdot\mathbf{B}=0$ and $\mathbf{E}^2-\mathbf{B}^2<0$ conditions algebraically by resetting the electric field \emph{instantaneously} wherever they are not fulfilled \citep[e.g.][]{Palenzuela2010,Mahlmann2020}.
Exploiting the specifics of our numerical methodology \citep{Mahlmann2020b}, which combines the instantaneous algebraic extraction of all non-ideal electric fields while evolving the charge continuity equation, we identified another mechanism that adds diffusivity to an ideal FFE scheme \citep{Mahlmann2020c}. Namely, the misalignment of electric fields $\mathbf{E}$ and charge density $\rho$ can significantly alter an FFE evolution. In this numerical survey, we further develop the findings obtained with an idealised setup that triggers tearing modes in force-free current sheets \citep{Mahlmann2020c} to the astrophysical relevant scenario of pulsar magnetospheres.
In the not uncontroversial realm of magnetospheric simulations in the force-free limit, we announced in \citet{Mahlmann2020b} that ambiguities in the standard reference of the force-free aligned rotator required further attention. In fact, \citet{Contopoulos2016} pointed out these ambiguities (\emph{trade secrets}) that arise when simulating magnetospheres with non-ideal regions, such as the pulsar and Wald magnetospheres, in the ideal plasma limit, as one finds in ideal MHD and FFE. A sensitivity for the dazzling amount of calibration that time-dependent numerical simulations require is rarely transmitted along with the visually appealing results themselves. This manuscript is an effort to bring transparency to the modeling of one astrophysical scenario that crosses the constraints set by FFE. It aims at enabling the reader to ask crucial questions when evaluating results from the simulations of magnetospheres and intends to place some landmarks for the development of future hybrid methods that are not restricted to the ideal regime.
This manuscript is organised as follows. In Sect.~\ref{sec:methodology}, we review the employed numerical methodology as well as the pulsar magnetosphere initial data (Sect.~\ref{sec:simulationsetup}). Sect.~\ref{sec:forcefreealigned} presents the outcome of the conducted simulations of a force-free aligned rotator. The results are grouped by different topics. First, we examine the dependency of the luminosity at the light cylinder (LC) on the method employed to preserve the consistency between the charge distribution and the currents (Sect.~\ref{sec:lcluminosity}). Section~\ref{sec:econservation} analyses the conservation of energy beyond the light cylinder. An array of ancillary high-resolution models yields additional insights into the subtleties of ideal FFE simulations in Sect.~\ref{sec:tradesecrets}. Specifically, we assess the role of the magnetic dominance condition (Sect.~\ref{sec:focuseddominance}), compare algebraic corrections of force-free violations to driving currents (Sect.~\ref{sec:drivingfocus}), and study the effect of resistivity models beyond the light cylinder (Sect.~\ref{sec:diffusivityfocus}). The discussion of Sect.~\ref{sec:discussion} includes views on the propagation of force-free violations (Sect.~\ref{sec:nonidealFF}), the diffusive time scales set by the employed hyperbolic/parabolic cleaning (Sect.~\ref{sec:cleaningscales}), and a general picture of diffusivity in force-free magnetospheres (Sect.~\ref{sec:diffusivitydiscuss}). We conclude this survey by summarizing the main takeaways of the presented results in Sect.~\ref{sec:conclusion}.
\section{Methodology}
\label{sec:methodology}
The aligned rotator problem has been studied vastly throughout the last 20 years and now appears to be a well-established test case for FFE codes. Even if it is likely a simplification of the more complex problem of a magnetic dipole that is misaligned with respect to the rotational axis of the pulsar, it still contains an element of special relevance for the overall problem, namely, the ECS. We approach this investigation by means of time-dependent FFE simulations performed with the numerical code presented in \cite{Mahlmann2020b,Mahlmann2020c}. Our method is an enhanced high-order conservative realization of FFE as introduced by \citet{Komissarov2004} and vastly benefits from the \textsc{Carpet} driver \citep{Goodale2002a,Schnetter2004} and its extension to spherical coordinates \citep{Mewes2018,Mewes2020} supported by the infrastructure of the \textsc{Einstein Toolkit}\footnote{\url{http://www.einsteintoolkit.org}}.
The scheme that we introduced in \cite{Mahlmann2020b} has since been compared to another force-free MHD method \citep[\textsc{BHAC},][]{Ripperda2019,Ripperda2019a} in the context of Alfvén wave interactions in the highly magnetised limit \citep{Ripperda2021}. It achieved a striking convergence of results across different methods.
Also in the context of the aligned force-free rotator \citep[cf. Sect. 5.2 in][]{Mahlmann2020b}, our FFE method reproduced the main characteristics of the pulsar magnetosphere: co-rotating magnetic field lines, a force-free closed zone in the wake of the Y-point, and an ECS.
However, \cite{Mahlmann2020b} observed a shift of the Y-point away from the light cylinder and a Poynting flux that was above the value which is treated as an established reference throughout the literature \citep[e.g.,][]{Spitkovsky2006, Tchekhovskoy2013, Etienne2017}. In the following subsections we briefly introduce the governing equations for the problem at hand, and the methods used to integrate them. We also provide the detailed numerical setup in Sect.~\ref{sec:numericalmethod}.
\subsection{Numerical method}
\label{sec:numericalmethod}
We employ the force-free scheme presented in \citet{Mahlmann2020b} to conduct 2D simulations of pulsar magnetosphere on a flat background without spacetime curvature \citep[cf. Sect. 5.2 in][]{Mahlmann2020b}. Ignoring general relativistic effects \citep[negligible for the global dynamics of the pulsar magnetosphere, especially far away from the neutron star surface, though very relevant, especially frame-dragging, for driving efficient pair production;][]{Philippov_2015ApJ...815L..19, Belyaev_2016ApJ...830..119, Gralla_2016ApJ...833..258, Philippov_2018ApJ...855...94}, the equations of FFE are the set of partial differential equations formed by the Maxwell equations together with the corresponding solenoidal constraint
($\nabla\cdot\mathbf{B}=0$) and the charge density, $\rho$ expressed as $\rho=\nabla\cdot\mathbf{E}$. These two equations are integrated in our code by employing the so-called hyperbolic/parabolic cleaning method \citep{Dedner2002,Komissarov2007MNRAS.382..995}, so that the former elliptic constrains become hyperbolic equations and form an \emph{augmented} system of equations:
\begin{align}
\partial_t \mathbf{B} &= -\nabla \times \mathbf{E}-c_\Psi^2\nabla\Psi\\
\partial_t \mathbf{E} &= \nabla \times \mathbf{B}+c_\Phi^2\nabla\Phi-\mathbf{j} \label{eq:Efield}\\
\partial_t \Psi &= -\nabla\cdot \mathbf{B}-\kappa_\Psi \Psi \label{eq:Psi}\\
\partial_t \Phi &= \nabla\cdot \mathbf{E}- \rho-\kappa_\Phi \Phi \label{eq:Phi}\\
\partial_t \rho &= -\nabla\cdot \mathbf{j} \label{eq:chargeconservation}
\end{align}
Our base numerical scheme employs the scalar potentials $\Phi$ and $\Psi$ to handle (numerical) errors to the constraints $\nabla\cdot\mathbf{E}=\rho$ and $\nabla \cdot\mathbf{B}=0$, respectively \citep[cf.][]{Komissarov_2006MNRAS.367...19,Mignone2010}. The action of the scalar potentials is controlled by a combination of damping constants, $\kappa_\Psi$ and $\kappa_\Phi$, as well as a set of advection speeds $c_\Psi$ and $c_\Phi$. We will refer to our default scheme as the charge conservative (CC) method, since it involves the charge conservation equation \eqref{eq:chargeconservation}, and guarantees that the charge distribution is consitent with the currents in the domain \citep{Komissarov_etal_2007MNRAS.374..415}.
A principal ingredient of our methodology is the current density as observed by the normal observer \citep[cf.][]{Mahlmann2020b,Mahlmann2020c}. It naturally splits into components
perpendicular and parallel to the magnetic field three-vector ($\mathbf{j}=\mathbf{j}_\perp + \mathbf{j}_\parallel$):
\begin{align}
\mathbf{j}_{\perp} &= \rho \frac{\mathbf{E} \times \mathbf{B}}{\mathbf{B}^2}, \label{eq:FFResCurrentPerpendicular1}\\
\mathbf{j}_\parallel &=
\frac{ \mathbf{B} \cdot(\nabla\times\mathbf{B}) -
\mathbf{E}\cdot(\nabla\times\mathbf{E}) + \kappa_I \: \mathbf{B}\cdot \mathbf{E}}{(1 + \kappa_I\eta)\,\mathbf{B}^2} \:\mathbf{B}
\label{eq:FFResCurrentPerpendicular}.
\end{align}
Here, $\kappa_I$ is the decay rate driving the electric field toward its target value $\mathbf{E}\rightarrow \eta\mathbf{j}$, and $\eta$ is a dissipation coefficient for the electric field that is parallel to the current. In the following sections, we specify whenever we extend the current of \emph{ideal} FFE ($\eta=\kappa_I=0$).
In a variation of the default numerical scheme, we ignore the charge conservation equation \eqref{eq:chargeconservation} and
compute the charge density appearing in the source terms by imposing $\rho=\nabla\cdot\mathbf{E}$. Specifically, the divergence of the electric field is computed from the cell-centered (volume averaged) electric field values in Eqs.~(\ref{eq:Efield}) and~(\ref{eq:FFResCurrentPerpendicular1}). We maintain the hyperbolic equation for the scalar potential $\Phi$ \eqref{eq:Phi} in order to dissipate and transport away any misalignment between charges and currents and preserve the charge conservation equation (\ref{eq:chargeconservation}) up to truncation error. We note that the $\nabla\cdot\mathbf{E}$ term appearing in Eq.~\eqref{eq:Phi} is computed as a numerical flux for the temporal update of $\Phi$ and, as such, it is obtained from the inter-cell values of the electric field. These interface values are monotonically reconstructed from cell-centred volume averaged values of the electric field. Hereafter, we will refer to this variation of the base scheme as the local charge reconstruction (LCR) method. We note that we employ a finite volume method where none of the variables is staggered off the cell centers \citep[differently from, e.g.][]{Spitkovsky2006, Mignone_2019MNRAS.486.4252}. Thus, the evaluation of $\nabla\cdot\mathbf{E}$ at each numerical cell is performed by employing a fourth order finite difference approximation based on a suitable number of neighboring cell values of $\mathbf{E}$ \citep[with an stencil similar to that used for the evaluation of $\mathbf{j}_\parallel$,][cf. Sect.~3.4]{Mahlmann2020}.
Finally, we also considered a second variation of the default numerical scheme that combines the two previous strategies into a \emph{hybrid} method (HCC hereafter). In the HCC algorithm we restrict the use of the LCR method to places where
the magnetic dominance condition is violated during any sub-step of the time integration, and the CC method is applied elsewhere. In practice, for the specific context of the aligned rotator magnetosphere, this limits the application of the LCR method to numerical cells affected by the ECS.
\subsection{Simulation setup}
\label{sec:simulationsetup}
As it is a common practice in the field, we employ a non-rotating dipole magnetosphere as initial data \citep[see Sect.~5.1 in][]{Mahlmann2020b}. The initially purely poloidal magnetic flux in the magnetosphere is then
\begin{align}
\mathbf{B}&=\mu\left(\frac{2\cos\theta}{r^3},\frac{\sin\theta}{r^4},0\right),\\
\left|\mathbf{B}\right|&\equiv B_{\rm d}\left(r,\theta\right)=\frac{\mu}{r^3}\left[3\cos^2\theta+1\right]^{1/2},
\label{eq:Bdipole}
\end{align}
where we scale the magnetic moment by the stellar radius, $\mu = r_*^{3/2}$ and the vector components are expressed in the orthogonal spherical basis for the coordinates $\{r,\theta,\phi\}$. An axisymmetric rotation is instantaneously switched on across the stellar surface, so that there is a transient period during which a torsional Alfvén wave propagates outwards throughout the magnetosphere. After this transient period, an \emph{approximately} steady state is reached
in the domain of extensions $r\times\theta=\left[r_*,751r_*\right]\times\left[0,\pi\right]$. Here, we use the stellar radius $r_*=13.67\,$km \citep[cf.][]{Mahlmann2019}. We evolve the magnetosphere in time for $t=8.35t_{\rm p}$, and $t=30.24t_{\rm p}$ for selected cases, where $t_{\rm p}=2\pi/\Omega_{\rm p}$ is the time of one pulsar revolution, and we choose $\Omega_{\rm p}\approx 646\,$Hz (equivalent to $\Omega_{\rm p}=0.02$ in the units of our numerical code). With this choice, the LC is located at distance $r_{\rm LC}\equiv c/\Omega_{\rm p}=5r_*$.
The study presented here evaluates numerical convergence for increasing resolution carefully, using combinations of the radial spacing $\Delta r=r_*/N_r$, $N_r\in\left[32,64,128\right]$, and angular spacing $\Delta \theta=\pi/N_\theta$, $N_\theta\in\left[100,200,400\right]$, respectively. The light-cylinder region is, thus, covered by a minimum of $N_{\rm LC}\in\left[160,320,640\right]$ radial mesh cells, while the total number of radial grid points exceeds this number by a large factor. We note that the outer boundary is sufficiently far away from the central object to avoid any feedback on the star itself. The inner boundary, located at the stellar surface, requires careful attention. To first order, we follow the treatment suggested by \citet{Parfrey2012}. We set the value of $B^r$ not only at the surface, but also within a thin layer $r_{\rm in} \le r\le r_*$. The remaining magnetic field components are treated differently. Namely, they are driven to their dipole values (Eq.~\ref{eq:Bdipole}) at some distance from the surface. In a thin boundary layer consisting of at least as many cells to be covered twice by the stencil of the chosen spatial reconstruction, $B^\theta$ and $B^\phi$ are evolved with the Maxwell equations. Deep inside the interior boundary, all deviations from the initial dipole fields are exponentially damped. Within the whole boundary layer, the electric field is obtained by assuming that it is purely inductive, i.e. \citep[cf.][]{Parfrey2012},
\begin{align}
\mathbf{E}=-\left[\mathbf{\Omega}_{\rm p}\times \mathbf{r}\right] \times\mathbf{B} .
\end{align}
This rather complicated boundary is needed to prevent spurious numerical behaviors at the stellar surface, which happen if all the magnetic field components are held fixed at $r_{\rm in} \le r \le r_*$. Likewise, this boundary acts equivalently to a conducting boundary in as much as it preserves the continuity of the electric field components parallel to the surface and the magnetic field perpendicular to it (i.e., no jumps in $B^r$, $E^\theta$ or $E^\phi$ develop at the stellar surface). It allows to relax the initial values of the electromagnetic field at the stellar surface to the approximate equilibrium values attained after a few rotational periods.
The electric charge inside the pulsar boundary layer is calculated in every sub-step of the time integration via $\rho=\nabla\cdot\mathbf{E}$. To ensure consistency at the inner boundary, we set $\Phi=0$ for $r<r_*$, while evolving $\Psi$ freely to evacuate errors to the solenoidal constraint on the magnetic field $\mathbf{B}$.
The evaluation of results presented throughout the following sections relies on the comparison of dimensionless quantities. Therefore, we normalise magnetic fields by the polar magnetic field $B_0=B_{\rm d}\left(r_*,0\right)=2\mu/r_*^3$. The charge density is normalised to the Goldreich-Julian charge density \citep{Goldreich_Julian_1969ApJ...157..869} at the polar cap, namely,
\begin{align}
\rho_0=\rho_{\rm GJ}\left(r_*,0\right)=-\frac{2\Omega_{\rm p} B_0}{c},
\end{align}
where $c$ is the speed of light ($c=1$ in the units of our code). Equally, electromagnetic currents will be normalised by the reference current density $j_0=|\rho_0c|$.
\begin{table}
\centering
\caption{Properties of the simulations corresponding to models described in Sect.~\ref{sec:forcefreealigned}. We provide the label of the respective model, the number of zones per stellar radius (in total, we have $\ge 750\times N_r$ radial zones) and in the interval $[0,\pi]$, the strategies to model the electric charge, the $\alpha$ parameter, the Y-point location, the luminosity at the LC, its relative decay up to a distance of $5r_{\rm LC}$, and the colatitude of the closed zone separatrix at the stellar surface. All models of this section evacuate force-free constraint violations by algebraic cropping of the electric fields.} \label{tab:mainmodels}
\begin{tabular}{P{0.2cm}P{1.2cm}P{0.4cm}P{0.5cm}P{0.5cm}P{0.7cm}P{0.8cm}P{0.4cm}}
\hline
\hline
& $N_r\times N_\theta$ & $\rho$ & $\alpha$ & $x_0$ & $L_{\rm Y}/L_0$ & $\Delta L/L_{\rm Y}$ & $\theta_{\rm c}$\\
\hline
\textbf{La} & $32\times 100$ & LCR & $72$ & $0.94$ & $0.97$ & $0.36$ & $33.2^\circ$\\%$0.58$ \\
\textbf{Lb} & $32\times 100$ & CC & 0.03 & $0.92$ & $1.03$ & $0.30$ & $33.8^\circ$\\%$0.59$ \\
\textbf{Lc} & $32\times 100$ & CC & $0.3$ & $0.88$ & $1.24$ & $0.08$ & $35.0^\circ$\\%$0.61$ \\
\textbf{Ld} & $32\times 100$ & CC & $2.9$ & $0.83$ & $1.67$ & $0.03$ & $36.7^\circ$\\%$0.64$ \\
\textbf{Le} & $32\times 100$ & CC & $9.3$ & $0.78$ & $1.95$ & $0.03$ & $37.8^\circ$\\%$0.66$ \\
\textbf{Lf} & $32\times 100$ & CC & $18$ & $0.71$ & $2.06$ & $0.05$ & $38.4^\circ$\\%$0.67$ \\
\textbf{Lg} & $32\times 100$ & CC & $37$ & $0.78$ & $2.12$ & $0.05$ & $38.4^\circ$\\%$0.67$ \\
\textbf{Lh} & $32\times 100$ & CC & $72$ & $0.77$ & $2.11$ & $0.04$ & $37.8^\circ$\\%$0.66$ \\
\textbf{Li} & $32\times 100$ & CC & $0.2$ & $0.89$ & $1.15$ & $0.10$ & $35.0^\circ$\\%$0.61$ \\
\textbf{Lj} & $32\times 100$ & CC & $0.6$ & $0.86$ & $1.45$ & $0.04$ & $36.1^\circ$\\%$0.63$ \\
\textbf{Lk} & $32\times 100$ & CC & $4.6$ & $0.79$ & $1.65$ & $0.04$ & $36.7^\circ$\\%$0.64$ \\
\textbf{Ll} & $32\times 100$ & CC & $19$ & $0.70$ & $2.21$ & $0.05$ & $38.4^\circ$\\%$0.67$ \\
\textbf{Lm} & $32\times 100$ & CC & $9.3$ & $0.78$ & $1.80$ & $0.06$ & $37.8^\circ$\\%$0.66$ \\
\textbf{Ln} & $32\times 100$ & CC & $37$ & $0.71$ & $2.28$ & $0.08$ & $39.0^\circ$\\%$0.68$ \\
\hline
\textbf{Ma} & $64\times 200$ & LCR & $36$ & $0.96$ & $1.01$ & $0.36$ & $32.7^\circ$\\%$0.57$ \\
\textbf{Mb} & $64\times 200$ & CC & 0.02 & $0.95$ & $1.06$ & $0.35$ & $33.2^\circ$\\%$0.58$ \\
\textbf{Mc} & $64\times 200$ & CC & $0.2$ & $0.91$ & $1.21$ & $0.06$ & $33.8^\circ$\\%$0.59$ \\
\textbf{Md} & $64\times 200$ & CC & $1.5$ & $0.86$ & $1.58$ & $0.03$ & $36.1^\circ$\\%$0.63$ \\
\textbf{Me} & $64\times 200$ & CC & $4.6$ & $0.80$ & $1.87$ & $0.04$ & $36.7^\circ$\\%$0.64$ \\
\textbf{Mf} & $64\times 200$ & CC & $9.3$ & $0.74$ & $1.95$ & $0.05$ & $37.2^\circ$\\%$0.65$ \\
\textbf{Mg} & $64\times 200$ & CC & $18$ & $0.76$ & $2.07$ & $0.04$ & $37.2^\circ$\\%$0.65$ \\
\textbf{Mh} & $64\times 200$ & CC & $36$ & $0.71$ & $2.20$ & $0.06$ & $37.8^\circ$\\%$0.66$ \\
\textbf{Mi} & $64\times 200$ & CC & $0.1$ & $0.92$ & $1.19$ & $0.07$ & $34.4^\circ$\\%$0.60$ \\
\textbf{Mj} & $64\times 200$ & CC & $0.3$ & $0.91$ & $1.43$ & $0.03$ & $35.0^\circ$\\%$0.61$ \\
\textbf{Mk} & $64\times 200$ & CC & $2.3$ & $0.87$ & $1.56$ & $0.02$ & $35.5^\circ$\\%$0.62$ \\
\textbf{Ml} & $64\times 200$ & CC & $9.2$ & $0.77$ & $2.17$ & $0.04$ & $37.8^\circ$\\%$0.66$ \\
\textbf{Mm} & $64\times 200$ & CC & $4.6$ & $0.83$ & $1.72$ & $0.04$ & $36.1^\circ$\\%$0.63$ \\
\textbf{Mn} & $64\times 200$ & CC & $19$ & $0.65$ & $2.37$ & $0.07$ & $39.0^\circ$\\%$0.68$ \\
\hline
\textbf{Ha} & $128\times 400$ & LCR & $18$ & $0.99$ & $0.96$ & $0.32$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Hb} & $128\times 400$ & CC & $0.1$ & $0.62$ & $1.17$ & $0.10$ & $33.8^\circ$\\%$0.59$ \\
\textbf{Hc} & $128\times 400$ & CC & $0.7$ & $0.81$ & $1.53$ & $0.01$ & $34.4^\circ$\\%$0.60$ \\
\textbf{Hd} & $128\times 400$ & CC & $2.3$ & $0.72$ & $1.83$ & $0.03$ & $35.5^\circ$\\%$0.62$ \\
\textbf{He} & $128\times 400$ & CC & $4.6$ & $0.68$ & $2.04$ & $0.04$ & $36.7^\circ$\\%$0.64$ \\
\textbf{Hf} & $128\times 400$ & CC & $9.2$ & $0.51$ & $2.24$ & $0.23$ & $36.7^\circ$\\%$0.64$ \\
\textbf{Hg} & $128\times 400$ & CC & $18$ & $0.41$ & $2.07$ & $0.21$ & $37.8^\circ$\\%$0.66$ \\
\textbf{Hh} & $128\times 400$ & CC & $0.4$ & $0.90$ & $1.37$ & $<0.01$ & $34.4^\circ$\\%$0.60$ \\
\textbf{Hi} & $128\times 400$ & CC & $1.5$ & $0.64$ & $1.91$ & $0.06$ & $36.1^\circ$\\%$0.63$ \\
\textbf{Hj} & $128\times 400$ & CC & $1.2$ & $0.84$ & $1.50$ & $0.01$ & $34.4^\circ$\\%$0.60$ \\
\textbf{Hk} & $128\times 400$ & CC & $4.6$ & $0.52$ & $2.20$ & $0.18$ & $37.2^\circ$\\%$0.65$ \\
\textbf{Hl} & $128\times 400$ & CC & $2.3$ & $0.85$ & $1.59$ & $0.02$ & $34.4^\circ$\\%$0.60$ \\
\textbf{Hm} & $128\times 400$ & CC & $9.3$ & $0.36$ & $2.42$ & $0.35$ & $40.7^\circ$\\%$0.71$ \\
\hline
\hline
\end{tabular}
\end{table}
\section{Force-free aligned rotator}
\label{sec:forcefreealigned}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{figures/figure1.jpg}
\vspace{-15pt}
\caption{Magnetic field component $B^\phi$ (normalised to $B_0$) and parallel force-free current $\mathbf{j}_\parallel/j_0$ for different values of the diffusivity parameter $\alpha\in\left[0.015,0.145,1.447,4.630,9.260,18.52,36.17\right]$ in the CC and in the LCR methods for a resolution of $N_r = 64$. In all cases, we employ $c_\Phi=1$, and for the run with the LCR method, we choose the extreme value $\alpha=36.17$. Black fieldlines are seeded at the same latitude on the stellar surface and may serve as a reference points for comparability. The light cylinder position is indicated by a solid black line, and an estimate of the Y-point location by a dashed black line.
}
\label{fig:FLDSJDENS}
\end{figure*}
In this section, we evaluate and interpret results from an extensive array of simulations (Tab.~\ref{tab:mainmodels}).
Specifically, Sects.~\ref{sec:econservation} and \ref{sec:lcluminosity}
present results from 41 simulations in the \emph{ideal} FFE limit, spanning different resolutions as well as a parameter range for $\kappa_\Phi$ and $c_\Phi$. Fig.~\ref{fig:FLDSJDENS} marks the starting point - and preceding motivation - of our exploration:
The FF equilibrium aligned rotator magnetosphere is vastly different when the LCR method is employed (top left panels), as opposed to a conservative evolution of the charge continuity equation in our CC scheme.
A useful dimensionless parameter to classify our results is
\begin{align}
\alpha=\frac{\kappa_\Phi}{c_\Phi}\Delta h,
\label{eq:alphadimensionless}
\end{align}
where $\Delta h=\text{min}\left[\Delta r, r \Delta\theta, r\sin\theta\Delta\phi\right]$. We provide a detailed interpretation of $\alpha$ throughout Sect.~\ref{sec:cleaningscales}. With an increasing damping coefficient $\kappa_\Phi$ (i.e. with increasing $\alpha$) of the hyperbolic/parabolic cleaning, the Y-point separating the closed zone from the equatorial current sheet moves closer to the central object and, thus, away from the LC. Equally, the amount of reconnecting field lines through the ECS notably decreases as $\alpha$ increases. The LRC method emerges as the \emph{most diffusive} limit of our parameter exploration. In comparable force-free simulations, \citet{Komissarov_2006MNRAS.367...19} finds results that are very similar to this limit. Observing significant reconnection beyond the Y-point, it is concluded that FFE has a tendency to facilitate the \textit{development of strongly dissipative current sheets}. Such solutions \textit{could only be relevant for magnetospheres with effective radiative cooling}. At the same time, one finds results in the literature where - without employing an explicit conservative evolution - the LCR method captures the equatorial current sheet of the pulsar magnetosphere rather well \citep{Spitkovsky2006,Etienne2017}. However, they employ different techniques to preserve the $\nabla\cdot\mathbf{E}=\rho$ and $\nabla\cdot\mathbf{B}=0$ constraints on the electromagnetic fields, based on either staggering the electromagnetic fields or employing the four-vector potential together with the energy-flux formulation of FFE \citep{McKinney2006}. We believe that scrutinizing this discrepancy up to its most finely granulated technical detail is crucial to understand the limits of FFE, how these limits affect astrophysical modeling, and how they can be overcome.
\subsection{Luminosity at the light cylinder}
\label{sec:lcluminosity}
The Poynting flux at the LC is the most commonly cited reference value for which the modeling of FF magnetospheres of aligned rotators is consulted. \citet{Timokhin2006} presents a thorough review of the FF steady pulsar magnetosphere, and we refer to the same reference luminosity \citep[see also][]{Gruzinov_2005PhRvL..94b1101},
\begin{align}
L_0 \approx \frac{\mu^2\Omega_p^4}{c^3}.
\end{align}
Previous results for time-dependent models show that the pulsar luminosity reached in the FF magnetosphere of an aligned pulsar \citep[with a Y-point located at the LC; note that equilibrium solutions with an inward-shifted Y-point exist][]{Goodwin_2004MNRAS.349..213,Contopoulos_2005A&A...442..579,Timokhin2006} is $L_{\rm LC}\equiv L(r=r_{\rm LC})=\left(1.0\pm 0.1\right)L_0$ \citep[e.g.][]{Komissarov_2006MNRAS.367...19,Spitkovsky2006,Tchekhovskoy2013}. In contrast, we shall argue that both the luminosity at the LC and the Y-point location depend on the (numerical) resistivity of the employed algorithm. Ultimately, this resistivity drives a slippage of the magnetic field lines at the Y-point as well as in the region of the ECS trailing it, and triggers their differential rotation in the magnetosphere. \cite{Contopoulos_2005A&A...442..579} explore the possibility that there is a differential rotational velocity of the open magnetic field lines to build solutions where the Y-point can be anywhere inside the LC. Figures~\ref{fig:Poynting} and~\ref{fig:PoyntingCSPEED} display the integrated Poynting flux through concentric spheres, as a function of distance from the central object. These plots show very different behavior beyond the LC (as we will discuss in Sect.~\ref{sec:econservation}). Indeed, we observe a transition towards the Poynting flux of the LCR method, and its dependence on distance from the star, for decreasing values of $\alpha$ in the set of data represented in Fig.~\ref{fig:Poynting}. The results shown for the CC method in Figs.~\ref{fig:Poynting} and~\ref{fig:PoyntingCSPEED} are obtained for different combinations of the cleaning parameters $\kappa_\Phi$ and $c_\Phi$ (the rest of the algorithmic elements in our numerical code are fixed). The broad range of luminosities spans between $\sim L_0$ (for the smallest value of $\kappa_\Phi=0.1$) and $\sim 2.3 L_0$ (for $\kappa_\Phi\ge 128$). There is a notable difference regarding the pulsar luminosity among different methods and within the CC method with distinct numerical parameters. As we argue below, the differences arise by the change in the location of the Y-point. Increasing $\kappa_\Phi$ and decreasing $c_\Phi$ (both changes yielding an increase of $\alpha$), thus, limiting the spread of numerical errors buffered in the cleaning potential $\Phi$, increases the luminosity at the LC. We observe a small growth of the total luminosity for $\kappa_\Phi>64$ when the numerical resolution is doubled. We will argue in Sects.~\ref{sec:tradesecrets} and \ref{sec:discussion} that this is because very large values of $\alpha$ also require very large numerical resolution to properly resolve the very fast damping of divergence cleaning errors in time. The LCR method corresponds to the limit in which errors to charge conservation spread through the domain without constraint. Hence, changes in the electric field on the ECS performed to restore the magnetic dominance condition are communicated \textit{instantaneously} all over the stencil of the discretization of the $\nabla$ operator in a single time iteration of the method (coupling as many as 12 zones around a given numerical cell, if a fourth order finite differences formula is employed).
Figure~\ref{fig:FLDSJDENS} lucidly illustrates that the Y-point moves away from the LC with increasing values of $\alpha$. A Y-point closer to the stellar surface boosts the electromagnetic luminosity of the pulsar \citep{Timokhin2006}. The amount of open magnetic field lines increases for a decreasing dimensionless distance $x_0\equiv r_{\rm Y}/r_{\rm LC}$ of the Y-point from the central object. This is directly related to the angular extension of the polar cap, which can be quantified by measuring the colatitude of the closed zone region (see below). In Fig.~\ref{fig:YpointLuminosity}, we present a large selection of simulated Y-point luminosities ($L_{\rm Y}$) vs. their estimated Y-point position, excluding results that did not yield equilibrium magnetospheres for very small or very large values of $\alpha$ (see discussion in Sect.~\ref{sec:discussion}). To approximate the Y-point location, we evaluate the drift velocity along the equator and assign $x_0$ to the position where the velocity is comparable to a small parameter, which equals the grid spacing $\Delta r$ in magnitude. We employ a similar approximation technique to the evaluation of the closed zone colatitude $\theta_{\rm c}$ (Tabs.~\ref{tab:mainmodels} and~\ref{tab:ancilmodels}), with a suitably strong decay at the transition to the co-rotation region. We find that, approximately, $\sin\theta_{\rm c}\sim (r_*/r_{\rm Y})^{1/2} \sim 0.5x_0^{-1/2}$. Roughly in line with the findings from \citet{Timokhin2006}, we measure a correlation between the Y-point location and luminosity. For large values of $\alpha$, associated to smaller values of $x_0$, the errors in these measurements - averaged over several pulsar revolutions - become larger. These errors are likely linked to degrading numerical accuracy induced by the increased stiffness of the augmentation equations (\ref{eq:Psi}) and (\ref{eq:Phi}) for large values of $\alpha$ (see Sect.~\ref{sec:discussion}).
We note that the correlation between $x_0$ and $L_{\rm Y}/L_0$ found numerically approaches the theoretical relation of \citet{Timokhin2006}, in which $L_{\rm Y}\propto x_0^{\lambda}$, with $\lambda = -2.065$, more so as the numerical resolution increases. Yet another interesting correlation derived from our results is that $L_{\rm Y}/L_0\approx 0.57(\sin\theta_{\rm c})^{0.19}$, which supports the claim that a larger luminosity at the Y-point correlates with a larger polar cap angle. Interpreting our findings, we cautiously suggest a possibility to obtain quasi-stationary magnetospheres in which the Y-point is not located at the LC but inside it. The location of the Y-point depends on the diffusivity at the ECS. Smaller diffusivity moves the Y-point inward and increases the outgoing Poynting flux, as we show in Fig.~\ref{fig:YpointLuminosity}.
The root of this dependency is the (magnetic fieldline) coupling between the fieldline footpoints at the stellar surface and part of the ECS (precisely, the part adjacent to the Y-point). The existence of such a region of coupling between the stellar surface and the ECS was suggested, e.g., by \citet{Contopoulos_2019MNRAS.482L..50} for stationary hybrid solutions combining FFE and a non-ideal ECS. Our time-dependent models reproduce many aspects of these equilibrium states. In the right panel of Fig.~\ref{fig:YpointLuminosity} we evaluate the specific dependency of the luminosity on the diffusion parameter $\alpha$. A power-law of the form $L_{\rm Y}/L_0\propto \alpha^{0.1}$ is found from a fit to our computed models. The parameter $\alpha$ controls the numerical diffusion of the electric field, if we assume that numerical diffusion mimics the physical one to some extent \citep[i.e. $\alpha\propto 1/\eta$, where $\eta$ is the resistivity, see discussion Sect.~\ref{sec:discussion} and][]{Mahlmann2020c}. Thus, our results suggest that the luminosity of the aligned rotator is inversely proportional to the resistivity at the ECS. Nevertheless, because of the small powerlaw index in the aforementioned $\alpha$-luminosity-relation, large changes in the resistivity are required to produce significant variations of the luminosity at the LC.
\subsection{Energy dissipation beyond the Y-point}
\label{sec:econservation}
\citet{Tchekhovskoy2013} hold a reliable benchmark for the convergence of pulsar magnetosphere modeling and its comparison between FFE and MHD. We emphasise the following property of their results: MHD models show a decay of the Poynting flux beyond the light cylinder that diminishes with higher resolution. However, their FFE models of the aligned rotator magnetosphere show $43\% - 50\%$ dissipation that converge to a stable amount of dissipation ($43\%$) for the highest resolution. We observe such behavior for the case of LCR, as it is marked by the thick red lines in Fig.~\ref{fig:Poynting}.
In all other cases using the CC method, the dissipation of Poynting flux along the radial direction is significantly less. This observation is also supported by
the field line images of Fig.~\ref{fig:FLDSJDENS}. There, field lines notably reconnect beyond the LC in the limit of low $\alpha$, while the cleaning of numerical errors shapes the ECS over longer distances. Stationary solutions of the force-free aligned rotator magnetosphere involving an ECS (where the FFE approximation does not hold) have been obtained \citep[e.g.][]{Contopoulos_2007A&A...466..301, Contopoulos_2014ApJ...781...46, Contopoulos2020}. The configurations built by \citet{Contopoulos2020} show that there is a region of the ECS that may extend beyond the LC, where magnetic field lines can still be closed. Our time-dependent models certainly reveal such a region, and its properties depend on the model parameters, which ultimately determine the level of (numerical) dissipation of the method. Small-scale structures in the equatorial plane (such as plasmoid-like formations) do not automatically appear by increasing resolution. Rather, it seems that an efficient damping of charge conservation errors allows them to emerge.
Remarkably, not only the smallest luminosity $L_{\rm LC}$ is attained by the LCR method, but, most importantly, also the largest relative decrease of the luminosity, $\Delta L/L_{\rm LC} = \left[L(r=5r_{\rm LC})- L_{\rm LC}\right]/L_{\rm LC}$ between the light cylinder and $r=5r_{\rm LC}$. Taking $\Delta L/L_{\rm LC}$ as a measure of the diffusivity of the algorithm, we conclude that the LCR method is more diffusive than the CC method. This conclusion is numerically robust, since duplicating the spatial resolution does not notably change the dissipation beyond the Y-point observed in the individual models.
Figs.~\ref{fig:Poynting} and~\ref{fig:PoyntingCSPEED} show a significant variability of the Poynting flux beyond the light cylinder for small values of $\kappa_\Phi$.
For most values of $\kappa_\Phi$, the Poynting flux beyond the LC is not monotonically decaying, but shows spikes associated to the ejection of plasmoid-like structures that move \emph{outwards} along the ECS (see Fig.~\ref{fig:FLDSJDENS}). The fact that these blobs of strong currents move outwards even if they are produced inside the LC is relevant: they do not contribute to the growth of the closed magnetospheric region. This finding is in contrast to \citet{Komissarov_2006MNRAS.367...19}, who claims that part of the plasmoids will move inwards, increasing the size of the closed magnetosphere with time and, hence, pushing the Y-point towards the LC. This assertion has also led \cite{Spitkovsky2005} and \cite{McKinney2005} to suggest that all configurations with $x_0<1$ would be unstable or transitory \citep[see also][]{Kalapotharakos_2009A&A...496..495,Yu2011,Parfrey_2012MNRAS.423.1416}. We do not notice such behavior here and, thus, our results suggest that closed magnetospheric configurations with a Y-point inside the LC may survive for many dynamical times.
In order to compute $\Delta L/L_{\rm LC}$ when it is spatially (and temporarily) variable (i.e., using the CC method with small values of $\alpha$), it is necessary to smooth out the data taking a suitable moving average. Larger values of $\kappa_\Phi$ or smaller advection speeds $c_\Phi$ (i.e., larger values of $\alpha$) yield magnetospheres with smaller Poynting flux dissipation beyond the LC. We note that the dissipation takes place at the ECS, where the force-free approximation is not strictly valid, specifically because the electric field strength becomes larger than the magnetic one and non-ideal electric fields with $\mathbf{E}\cdot\mathbf{B}\neq 0$ become dynamically important. Hence, the different amounts of relative dissipation are closely connected with the numerical handling of the force-free constraints and the propagation of errors from the regions where FFE is breached.
Extreme values of $\alpha$ suppress the hyperbolic/parabolic cleaning, letting the charge and the corresponding electric field divergence become misaligned over time. Nevertheless, very strong damping cleans errors very rapidly but effectively decouples the evolution equations of $\Phi$ from the underlying system of physical balance laws (in this case, Maxwell's equations), and renders the scalar potential dynamically negligible. As cases of intermediate $\alpha$ do not show this variability, we empirically demonstrate that there is an optimal range of values of $\kappa_\Phi$, corresponding to $\alpha\sim 1$. In this range, the CC method works very efficiently, effectively minimizing the dissipation at the ECS. We find strong evidence for a much better conservation of energy flux beyond the LC than what is quoted throughout the literature, where results seem to correspond to the most diffusive case of our parameter exploration, as to say the LRC method.
As a matter of fact, for the intermediate values of $\alpha\in\left[1.45, 4.63\right]$, the current $\mathbf{j}_\parallel$ is very efficiently suppressed in the equatorial region beyond the Y-point (Fig.~\ref{fig:FLDSJDENS}). Indeed, this is the reason which explains the significantly reduced dissipation beyond the LC in the models using the CC method and optimal values of $\alpha\sim 1$. For the smallest values of $\alpha$, the variability timescales are of the order of the polar cap light-crossing time, namely $\sim r_*\theta_{\rm c}/c\sim r_*/2c$ (see also the discussion in Sect.~\ref{sec:diffusivitydiscuss}).
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{figures/figure2.pdf}
\vspace{-6pt}
\caption{Poynting flux as a function of the distance from the central rotator. We present results after $16.7$ rotation periods for different damping constants $\kappa_\Phi$ (and a constant advection parameter $c_\Phi=1$) for different resolutions. The tests employing the LCR method are indicated by thick red lines.}
\label{fig:Poynting}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{figures/figure3.pdf}
\vspace{-6pt}
\caption{As Fig.~\ref{fig:FLDSJDENS} but for selected damping constants $\kappa_\Phi$ and different choices for the advection parameter $c_\Phi$.}
\label{fig:PoyntingCSPEED}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{figures/figure4.pdf}
\vspace{-6pt}
\caption{Dependence of the luminosity on the Y-point location and diffusion parameter $\alpha$. For the Y-point location (left panel), measurements from simulations at five different moments in time are averaged (standard deviation indicated in error bars). Fit functions are derived from for all models that find a reliable equilibrium (solid dots) and are represented by colored dashed lines. As a reference, we compare these results to the corresponding relation in \citet[][black line]{Timokhin2006}.
Errors to these fit functions are represented by the lightly shaded regions of the respective color. Outliers that do not find stable equilibria are denoted by colored circumferences. The right panel shows the normalised luminosity measured at the light cylinder, evaluated against the diffusion parameter $\alpha$. A fit function to the data is shown with a thick solid line. Uncertainties to the fit function are given as a shaded region around the central line.}
\label{fig:YpointLuminosity}
\end{figure*}
\section{Sources of dissipation}
\label{sec:tradesecrets}
In this part of the manuscript, we present an array of ancillary high-resolution simulations that will be key for assessing the role of numerical and/or phenomenological diffusivity in shaping the overall magnetospheric structure, and disclosing several \emph{trade secrets} \citep[cf.][]{Contopoulos2016} in the dynamical modeling of aligned rotator magnetospheres. In order to speed up the calculations at the higher numerical resolution, (i.e. with a grid spacing $\Delta r=r_*/128$, $\Delta\theta=\pi/400$) we will employ a radially re-scaled coordinate system for the remainder of this section. Specifically, for $r\gtrsim 3r_{\rm LC}$, the grid spacing increases by a factor of $a=1.001$ in each grid point along the radial direction. This helps us to reduce the number of points enclosed by the simulation domain while keeping sufficient resolution around the central object. Besides this change introduced for numerical convenience, we further replace the boundary layer described in Sect.~\ref{sec:simulationsetup} by perfect conductor boundary condition within the Riemann solver \citep[cf.][]{Munz2000}. Specifically, at the inter-cell face where the stellar surface is located, we set the following 'left' (L) state of the Riemann problem (corresponding to the interior of the star):
\begin{align}
\Psi_L &= \Psi_R\\
\Phi_L &= -\Phi_R\\
\mathbf{E}_L &= -\mathbf{E}_R + 2(\mathbf{E}_R \cdot \mathbf{\hat r}) \mathbf{\hat r}\\
\mathbf{B}_L &= +\mathbf{B}_R - 2(\mathbf{B}_R \cdot \mathbf{\hat r}) \mathbf{\hat r} ,
\label{eq:BCs}
\end{align}
where $\mathbf{\hat r}$ is the unit radial vector (normal to the stellar surface) and R denotes the respective 'right' state. The reason to consider different boundary conditions is the fact that we seek
stationary (or nearly stationary) magnetospheric configurations, in which case boundary conditions nearly completely determine the structure of the magnetosphere. Hence, using a variety of boundary conditions allows as to gauge their influence on the most salient features of our solutions, namely, the location of the Y-point and the closely related dissipation at the ECS. As we shall see, the boundary strategies used throughout this work produce consistent results. Table~\ref{tab:ancilmodels} shows a list of all the simulations that we describe in the following subsections.
\begin{table}
\centering
\caption{Properties of the set of high-resolution simulations corresponding to models described in Sect.~\ref{sec:tradesecrets}. We provide the label of the respective model, the strategies used to deal with violations of the force-free conditions, the strategies to model the electric charge, the phenomenological resistivity induced by a suitable current, the Y-point location, the luminosity at the LC, its relative decay up to a distance of $5r_{\rm LC}$, and the colatitude of the closed zone separatrix at the stellar surface. Models $\mathbf{Bg}$ to $\mathbf{Br}$ employ $\kappa_I=8$ in the parametrization of Eq.~(\ref{eq:FFResCurrentPerpendicular}).}
\label{tab:ancilmodels}
\begin{tabular}{P{0.2cm}P{1.2cm}P{0.4cm}P{0.5cm}P{0.5cm}P{0.7cm}P{0.8cm}P{0.4cm}}
\hline
\hline
& $\mathbf{E}\cdot\mathbf{B}$ & $\rho$ & $\alpha$ & $x_0$ & $L_{\rm Y}/L_0$ & $\Delta L/L_{\rm Y}$ & $\theta_{\rm pc}$\\
\hline
\textbf{Ba} & algebraic & LCR & $4.6$ & $1.04$ & $0.96$ & $0.29$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bb} & algebraic & CC & $4.6$ & $0.69$ & $2.10$ & $0.03$ & $37.2^\circ$\\%$0.65$ \\
\textbf{Bc} & algebraic & HCC & $4.6$ & $0.68$ & $2.04$ &$<0.01$ & $36.7^\circ$\\%$0.64$ \\
\textbf{Bd} & algebraic & LCR & $0.7$ & $1.02$ & $0.96$ & $0.32$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Be} & algebraic & CC & $0.7$ & $0.83$ & $1.52$ & $0.02$ & $34.4^\circ$\\%$0.60$ \\
\textbf{Bf} & algebraic & HCC & $0.7$ & $0.74$ & $1.49$ & $<0.01$ & $35.0^\circ$\\%$0.61$ \\
\hline
\textbf{Bg} & $\eta = 0.0$ & LCR & $4.6$ & $1.04$ & $0.95$ & $0.31$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bh} & $\eta = 0.0$ & HCC & $4.6$ & $0.81$ & $1.68$ & $<0.01$ & $34.4^\circ$\\%$0.60$ \\
\textbf{Bi} & $\eta = 0.0$ & LCR & $0.7$ & $1.04$ & $0.95$ & $0.30$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bj} & $\eta = 0.0$ & HCC & $0.7$ & $0.94$ & $1.11$ & $<0.01$ & $31.5^\circ$\\%$0.55$ \\
\hline
\textbf{Bk} & $\eta = 1.0$ & HCC & $0.7$ & $0.91$ & $1.15$ & $0.02$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bl} & $\eta = 10^{-1}$ & HCC & $0.7$ & $0.93$ & $1.10$ & $0.02$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bm} & $\eta = 10^{-2}$ & HCC & $0.7$ & $0.95$ & $1.10$ & $0.01$ & $30.9^\circ$\\%$0.54$ \\
\textbf{Bn} & $\eta = 10^{-3}$ & HCC & $0.7$ & $0.97$ & $1.10$ & $0.01$ & $30.9^\circ$\\%$0.54$ \\
\textbf{Bo} & $\eta = 1.0$ & CC & $0.7$ & $0.90$ & $1.30$ & $<0.01$ & $32.7^\circ$\\%$0.57$ \\
\textbf{Bp} & $\eta = 10^{-1}$ & CC & $0.7$ & $0.91$ & $1.29$ & $<0.01$ & $32.7^\circ$\\%$0.57$ \\
\textbf{Bq} & $\eta = 10^{-2}$ & CC & $0.7$ & $0.91$ & $1.28$ & $0.01$ & $32.7^\circ$\\%$0.57$ \\
\textbf{Br} & $\eta = 10^{-3}$ & CC & $0.7$ & $0.90$ & $1.29$ & $0.02$ & $32.7^\circ$\\%$0.57$ \\
\hline
\textbf{Bs} & algebraic & LCR & $0.0$ & $0.47$ & $1.34$ & $0.42$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bt} & algebraic & LCR & $0.1$ & $0.52$ & $1.03$ & $0.32$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bu} & algebraic & LCR & $2.3$ & $1.02$ & $0.96$ & $0.32$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bv} & algebraic & LCR & $9.3$ & $1.02$ & $0.96$ & $0.32$ & $31.5^\circ$\\%$0.55$ \\
\textbf{Bw} & algebraic & LCR & $18$ & $1.02$ & $0.96$ & $0.31$ & $31.5^\circ$\\%$0.55$ \\
\hline
\hline
\end{tabular}
\end{table}
\subsection{The role of the violations of FFE constraints}
\label{sec:focuseddominance}
\begin{figure*}
\centering
\includegraphics[width=0.98\textwidth]{figures/figure5.jpg}
\caption{Comparison of different treatments of the charge $\rho$ for some
configurations specified in Tab.~\ref{tab:ancilmodels}. We display the toroidal magnetic field in the left panel of each respective model. The corresponding right panels visualise the force-free violations accumulating during one full time-step of the FFE integration. We show the growth of non-ideal electric fields (absolute value) emerging transiently due to the violation of the $\mathbf{E}\cdot\mathbf{B}=0$ constraint by blue colors, and locations where the magnetic dominance condition is breached by yellow contours. The vertical solid and dashed black lines denote the position of the LC and of the Y-point, respectively.
}
\label{fig:FFCOND_COMPARE}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{figures/figure6.pdf}
\vspace{-6pt}
\caption{Dissipation via different channels of numerical diffusion as a function of the dimensionless parameter $\alpha$ (Eq.~\ref{eq:alphadimensionless}) in a quasi-equilibrium state at $t\approx 7.5t_p$. An estimate of the favorable range for $\alpha$ that minimises the dissipation by force-free violations is indicated as a light blue region. Different colors of the markers indicate different resolutions and treatment of charge density evolution. The shapes of the markers vary with different $c_\Phi$. We provide reference values of the respective dissipation channel for the LCR method of highest $\alpha$ as star-shaped data-points.}
\label{fig:QDissipation}
\end{figure}
We compare the final (quasi-)equilibrium of three different numerical treatments of the electric charge in the magnetosphere in models $(\mathbf{Ba})-(\mathbf{Bf})$. Since we have introduced changes in the boundary conditions and in the numerical grid, we confirm the results from the previous section (incorporating the aforementioned modifications with respect to models in Sect.~\ref{sec:forcefreealigned}). First, we use the LCR method throughout the entire magnetosphere for two distinct values of $\alpha$ (models $\mathbf{Ba}$ and $\mathbf{Bd}$). Second, we use the CC method in models $\mathbf{Bb}$ and $\mathbf{Be}$.
Finally, we apply the HCC method in models $\mathbf{Bc}$ and $\mathbf{Bf}$ using different values of $\alpha$.
Figure~\ref{fig:FFCOND_COMPARE} demonstrates that the cases where charge is not conservatively evolved in the entire domain (models $\mathbf{Ba}$ and $\mathbf{Bd}$) produce congruent results independent of the cleaning parameter $\alpha$. Such solutions have a large amount of reconnecting field lines beyond the light cylinder and a luminosity at $r_{\rm LC}$ that approaches $L_{\rm LC}/L_0\sim 1$. The small relative differences observed in the position of the Y-point ($\lesssim 2\%$) and the insignificant change in $L_{\rm LC}/L_0$ compared with the $\simeq 10\%$ relative difference in $\Delta L/L_{\rm LC}$ highlight the fact that the variation of $\alpha$ (by a factor $\approx 6.6$) mostly affects the dissipation dynamics of the ECS and its neighborhood.
When charge is evolved by a separate continuity equation, any misalignment of the electric field divergence and the charge density is cleaned out by the scalar potential $\Phi$. As we presented above (Sect.~\ref{sec:forcefreealigned}), altering the cleaning parameter $\alpha$ shifts the position of the Y-point and changes the Poynting flux at $r_{\rm LC}$. While most of the cases employing the CC method have a well-maintained current sheet beyond the Y-point (contrasting the reconnection in models $\mathbf{Ba}$ and $\mathbf{Bd}$), excessive cleaning (driven by large values of $\alpha$) can induce variability of the current sheet along the vertical direction.
This is obvious when comparing models $\mathbf{Bb}$ and $\mathbf{Be}$, but can also be observed for large values of $\alpha$ in Fig.~\ref{fig:FLDSJDENS}. As we see when comparing models $\mathbf{Bb}$ and $\mathbf{Bc}$, using the HCC scheme stabilises the current sheet on the equator, damping vertical displacements in the ECS. Inside the LC, models $\mathbf{Bc}$ and $\mathbf{Bf}$ only slightly differ from their CC counterparts. The location of the Y-point, and the luminosity differ less than $\simeq 2\%$ among HCC and CC models for $\alpha=0.7$, while for $\alpha=4.6$, HCC models show values of $x_0$ and $L_{\rm LC}/L_0$ in between of the ones of CC and LCR models (Tab.~\ref{tab:ancilmodels}). Beyond the LC, the differences between HCC and CC models are driven by insufficient damping, especially in the model with larger $\alpha$ ($\mathbf{Bb}$), where the ECS is distorted and the violations of the magnetic dominance condition are more patchy and intermittent along it. The fact that HCC models $\mathbf{Bc}/\mathbf{Bf}$ look more like $\mathbf{Bb}/\mathbf{Be}$ than like $\mathbf{Ba}/\mathbf{Bd}$, respectively, results from a subtle combination of two effects. On the one hand, the corrections to the electric fields after violations of the magnetic dominance condition are larger when using a local reconstruction of charge, either globally (LCR) or locally (HCC), than in CC models. This seems natural as the largest charges are created as a result of the sharp discontinuities of the electric field across the ECS, where LCR and HCC models undergo the same corrections in the charge density. On the other hand, the commonality in the (hyperbolic) evacuation of the errors triggered by violations of the FFE constraints at the ECS results in a greater similarity between HCC and CC models than to LCR models (the feedback of $\nabla\cdot\mathbf{E}$ on $\Psi$ is significantly reduced because of the explicit form in which $\rho$ is constrained in the LCR method; Sect.~\ref{sec:numericalmethod}). We can, thus, conclude that the large diffusivity in case of LCR method is primarily induced by corrections to the electric fields after violations of the magnetic dominance condition.
The (algebraic) correction of violations to the force-free conditions has different consequences for each of the constraints. Deviations from the $\mathbf{E}\cdot\mathbf{B}=0$ condition build up continuously and are distributed throughout the domain, though they are larger close to current sheets. We visualise this in the panels of Fig.~\ref{fig:FFCOND_COMPARE} that show the $\mathbf{E}\cdot\mathbf{B}$ errors normalised to the local magnetic field strength in a blue gradient. Violations include both very small inaccuracies that result from truncation errors of the algorithm, and strong non-ideal electric fields emerging, for example, at current sheets. In models employing the LCR method ($\mathbf{Ba}$ and $\mathbf{Bd}$), the hyperbolic part of the cleaning still operates to drive violations of the $\mathbf{E}\cdot\mathbf{B}$ constraint away from the ECS (we note a \emph{wave} pattern - concentric structures - apparently emerging from the equatorial plane at about $r\approx 1.2 r_{\rm LC}$ in the respective panels of Fig.~\ref{fig:FFCOND_COMPARE}). That pattern persists in models $\mathbf{Bb}$ and $\mathbf{Be}$, as well as in $\mathbf{Bc}$ and $\mathbf{Bf}$, where we employed the CC and the HCC method. However, in these cases, the violations of $\mathbf{E}\cdot\mathbf{B}$ can affect a larger region, especially for small values of $\alpha$. The mid-panel of Fig.~\ref{fig:QDissipation} shows that models employing the CC method systematically dissipate less energy by Ohmic processes (here approximated by the amount of current parallel to the electric field added up over the whole domain on a single timestep) than models using the LCR method for $\alpha\lesssim 5$.
The condition $\mathbf{E}^2-\mathbf{B}^2\leq 0$ is only relevant when significant non-ideal electric fields have built up, in the setup at hand at the ECS, as it is lucidly illustrated by the yellow contours in Fig.~\ref{fig:FFCOND_COMPARE}. It is a natural impulse to associate the high diffusivity observed for the setups using the LCR method to the violations of the magnetic dominance. Looking at the time evolution of the violation of the magnetic dominance constraint, we observe that in models implementing the LCR method everywhere, one finds regions of $\mathbf{E}^2>\mathbf{B}^2$ that consistently cover the ECS in the range $x_0\lesssim r/r_{\rm LC}\lesssim 1.7$ (and likely beyond). In contrast, models employing the CC and HCC methods display an intermittent set of spots of smaller extension, where $\mathbf{E}^2>\mathbf{B}^2$ in the equatorial region beyond the Y-point. The top panel of Fig.~\ref{fig:QDissipation} displays the electric energy subtracted from the whole domain during one iteration of the time-integrator. Contributions to this channel of diffusion result from mesh cells where the magnetic dominance constraint is breached. In practice, we calculate averages from several snapshots of the results to ensure that the quoted dissipation estimates have (more or less) stabilised. LCR models computed at the highest resolution ($N_r=128$; magenta squares) systematically dissipate more (the least about the same) energy by restoring the magnetic dominance condition than models computed with the CC method and the same resolution (dark green symbols) for $\alpha\lesssim 3$. Similarly to the electric energy dissipation in the correction of the $\mathbf{E}\cdot\mathbf{B}=0$ constraint, there is an interval $0.5 \lesssim \alpha\lesssim 3$, where dissipative losses induced by the restoration of the magnetic dominance condition are minimised.
In contrast to the trend found for the CC method, the dissipation triggered by violations of the FFE constraints is quite insensitive to $\alpha$ when evaluated for the LCC method (see magenta symbols in the upper and mid panels of Fig.~\ref{fig:QDissipation}). In order to understand this behavior, we have computed the electric energy dissipated by the hyperbolic/parabolic cleaning algorithm, which is given by $|c_\Phi^2\mathbf{E}\cdot\nabla \Phi|$ in the lower panel of Fig.~\ref{fig:QDissipation}. While the gradients of the cleaning potential $\Phi$ are large enough, increasing $\alpha$ (i.e. damping the divergence errors faster than shifting them away) reduces the diffusion through this channel, independent of the method used to evolve or reconstruct the charge density (LCR or CC). The dissipation through this channel is significantly larger than that driven by violations of the FFE constraints for $\alpha\lesssim 5$ and dominates the overall diffusion of electromagnetic energy in the magnetosphere. Above that value of $\alpha$, the total dissipated energy in the $|c_\Phi^2\mathbf{E}\cdot\nabla \Phi|$ channel is smaller than in the other two channels (for this, we compare the numerical values in the top and mid panels of Fig.~\ref{fig:QDissipation} to those in the bottom panel). From that point on, the cleaning algorithm in the CC method cannot efficiently evacuate and damp the errors induced in $\nabla\cdot\mathbf{E}$ by the restoration of the FFE constraints. Thus, the energy dissipation in models employing the CC method, becomes dominated by the violation of the FFE constraints above a certain value of $\alpha$. This reasoning substantiates our claim about the existence of an optimal interval for $0.7\lesssim\alpha\lesssim 3$, where the overall dissipation in the magnetosphere is smaller using the CC method than the LCR one. The insensitivity of the dissipation due to violations of the FFE constrains on $\alpha$ for LCR models (for a fixed numerical resolution) can be explained by the limited evacuation of FFE violations, e.g., from current sheets. As $\rho\approx\nabla\cdot\mathbf{E}$ (up to discretization errors), the right-hand-side of Eq.~\eqref{eq:Phi} is significantly smaller than in the models equipped with the CC method; the two mechanisms (cleaning of divergence errors and enforcing FFE violations) operate independently.
\subsubsection{Algebraic corrections versus driver currents}
\label{sec:drivingfocus}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{figures/figure7.jpg}
\vspace{-18pt}
\caption{Comparison of magnetospheric dynamics for different values of $\eta$ in the current given by Eq.~(\ref{eq:FFResCurrentPerpendicular}). From \textit{left} to \textit{right}, the phenomenological resistivity is increasing as $\eta\in\left\{10^{-3},10^{-2},10^{-1},1.0\right\}$. The background color denotes the (drift) velocity component $v^y$, as to say the velocity pointing perpendicular to the ECS. The separatrices where $v^y=0$ are highlighted by magenta contours. The top row comprises models implementing the HCC method, the bottom row those of fully conservative evolution of the charge density $\rho$ (CC models).}
\label{fig:ETA_COMPARE}
\end{figure*}
We have seen in the previous section that violations of the FFE constraints are fundamentally connected to the magnetospheric structure. It is, therefore, natural to assess whether different methods to restore the FFE constraints impact on the overall dissipation and topology of the magnetosphere.
Throughout the literature, various techniques are used to correct any deviations from the $\mathbf{E}\cdot\mathbf{B}=0$ condition. These techniques split up into the ones that apply algebraic resets of the electric field at each violation (as done in the models shown so far), and those that modify the Ohm's law encoded in a suitable current to drive the system into a force-free state \citep[cf.][and references therein]{Mahlmann2020b}. It is very much justified to suspect that all the variability in the luminosity identified in Sect.~\ref{sec:forcefreealigned} stems from this critical ingredient to force-free codes, rather than from their connection to charge conservation.
To dissect this subtle issue, we conduct simulations of the models $(\mathbf{Bg})-(\mathbf{Bh})$. Without algebraic resets of the $\mathbf{E}\cdot\mathbf{B}=0$ condition, we employ $\kappa_I=8$ in the force-free current presented in Eq.~(\ref{eq:FFResCurrentPerpendicular}). In this way, our method is comparable to the ones that employ driving currents, namely, a formulation of Ohm's law with a finite resistivity $\sigma_\parallel$ that acts along the direction of the magnetic field \citep[e.g.,][]{Alic2012,Komissarov2004}.
In Fig.~\ref{fig:FFCOND_COMPARE}, we contrast the magnetospheric equilibrium found when combining a suitable current to drive $\mathbf{E}\cdot\mathbf{B}\rightarrow 0$ for the LCR method (models $\mathbf{Bg}$ and $\mathbf{Bi}$), and for the HCC method (models $\mathbf{Bh}$ and $\mathbf{Bj}$). One straightforward observation is that a local reconstruction of charge remains the most diffusive limit in view of the much larger decrease of luminosity with distance measured by $\Delta L/L_{\rm LC}$ (Tab.~\ref{tab:ancilmodels}). For the HCC models $\mathbf{Bh}$ and $\mathbf{Bj}$, the luminosity decrease beyond the LC is as small as in the case in which algebraic cutbacks of the electric field enforce $\mathbf{E}\cdot\mathbf{B}=0$ (models $\mathbf{Bc}$ and $\mathbf{Bf}$). The luminosity of the HCC models that implement a driving current to enforce $\mathbf{E}\cdot\mathbf{B}\rightarrow 0$ decreases by $\sim 18\%-25\%$ with respect to models using algebraic resets of the electric field (models $\mathbf{c}$ and $\mathbf{f}$). This is a direct consequence of the larger value of $x_0$ in models $\mathbf{Bh}$ and $\mathbf{Bj}$, which place their Y-points closer to the LC. In contrast, models using the LCR method display only a small change of the luminosity ($\lesssim 1\%$) when comparing models with different strategies to enforce $\mathbf{E}\cdot\mathbf{B}=0$ (i.e. models $\mathbf{Ba}/\mathbf{Bd}$ vs models $\mathbf{Bg}/\mathbf{Bi}$). Thus, the smaller luminosity of models using the LCR method is not fully accounted for by the algebraic - rather harsh - corrections we apply to non-ideal electric fields.
\subsubsection{Diffusivity models beyond the light cylinder}
\label{sec:diffusivityfocus}
One technique that is often associated to the capacity of an FFE scheme to resolve current sheets is a finite resistivity induced by a suitably chosen Ohm's law \citep{Alic2012,Parfrey2017}. In \citet{Mahlmann2020c}, we explore the action of the current prescribed by Eq.~(\ref{eq:FFResCurrentPerpendicular}) during the development of 2D tearing modes. In this section, we review the impact of such phenomenological resistivities on the current sheet of the aligned rotator magnetosphere. To this purpose, we prescribe the following phenomenological resistivity:
\begin{align}
\eta=\eta_{\rm bg}+(\eta_{\rm d}-\eta_{\rm bg})\, \frac{1+\tanh(r_{\rm cyl}-r_{\rm LC})}{2} .
\end{align}
This resistivity becomes the driving value $\eta_{\rm d}$ for $r_{\rm cyl}>r_{\rm LC}$, where $r_{\rm cyl}=r\sin\theta$ is the cylindrical radius. In other words, inside the LC, all tests have the same (small) background resistivity $\eta_{\rm bg}$, where we use $\eta_{\rm bg}=10^{-5}\ll \eta_{\rm d}$.
In Fig.~\ref{fig:ETA_COMPARE}
we compare the magnetospheric evolution of different values of $\eta_{\rm d}$ for the HCC and CC models (\textbf{Bk})-(\textbf{Bn}) and (\textbf{Bo})-(\textbf{Br}), respectively. As we also denote in Tab.~\ref{tab:ancilmodels}, we observe - very much like in the previous section - that the Y-point is closer to the LC,
and the luminosity at the LC is lower in case of models using the HCC method. Regarding the dissipation of Poynting flux beyond the LC, the whole series of models (\textbf{Bk})-(\textbf{Br}) display similar and relatively low values of $\Delta L/L_{\rm LC}$, especially if we compare these values to the analogous ones obtained with the LCR method and $\eta=0$ (models \textbf{Bg} and \textbf{Bi}). For most of the phenomenological resistivities considered here (say $\eta\le 10^{-2}$), HCC models are more diffusive than CC models and this global measurement (of luminosity) manifests itself at the local level in a smoother velocity profile along the equatorial current sheet. Contrasting this, local reconnection events with large drift
velocities (normal to the ECS) become visible in Fig.~\ref{fig:ETA_COMPARE} (bottom row of panels) for the cases using the CC method (especially in models \textbf{q} and \textbf{r}). At the same time, the width of the resistive layer, in which we can identify an inflow (drift) velocity into the ECS, increases with larger $\eta_{\rm d}$.
\section{Discussion}
\label{sec:discussion}
\subsection{Action of non-ideal electric fields in FFE}
\label{sec:nonidealFF}
Our method extracts deviations from the ideal FFE condition $\mathbf{E}\cdot\mathbf{B}=0$ by the algebraic reset
\begin{align}
E^i\rightarrow E^k\left(\delta^i_{\hspace{4pt}k}-B_k\frac{B^i}{\mathbf{B}^2}\right).
\label{eq:DBcutback}
\end{align}
in each sub-step of the time-integrator. This surgical intervention instantaneously achieves (ideal) perpendicularity of electric and magnetic fields. At this point, the results presented in Sect.~\ref{sec:tradesecrets} suggest two different dynamical readjustments. First, a local reconstruction of charge adds an amount of charge into the domain that can be computed taking the divergence of Eq.~\eqref{eq:DBcutback}. By assuming a pointwise correction as well as $\nabla\cdot\mathbf{B}=0$, one obtains
\begin{align}
\rho\rightarrow\rho-\nabla\left[\frac{\mathbf{E}\cdot\mathbf{B}}{|\mathbf{B}|^2}\right]\cdot \mathbf{B}.
\label{eq:misalignedrecon}
\end{align}
As there is no discrepancy between the charge density $\rho$ and $\nabla\cdot\mathbf{E}$, the cleaning potential $\Phi$ reduces its role to a true scalar correction of (very small) numerical truncation errors. Non-ideal fields still alter the system of conservation laws by the loss of both, energy conservation, and charge conservation. The charge density is one constituent of the current that acts as a source of Ampère's law. A source or sink of it, due to the addition of charge when fixing the violation of the $\mathbf{E}\cdot\mathbf{B}=0$ constraint, is dynamically relevant and may have a global impact as it can be transported away from the numerical cells where it is initially generated.
Second, in a system that transports charge density in a fully conserved way, the correction in Eq.~\eqref{eq:DBcutback} does not induce local alterations of $\rho$. However, it induces a discrepancy $\mathcal{R}$ between $\rho$ and $\nabla\cdot\mathbf{E}$ that corresponds to the same amount that was identified above:
\begin{align}
\mathcal{R}\equiv\nabla\cdot\mathbf{E}-\rho\rightarrow\nabla\left[\frac{\mathbf{E}\cdot\mathbf{B}}{|\mathbf{B}|^2}\right]\cdot \mathbf{B}.
\label{eq:misalignedcons}
\end{align}
Such a mismatch between the divergence of electric fields and charge density will cause a change of the cleaning potential $\Phi$ that is not necessarily a small correction. Just as the current density, $\Phi$ acts as a source of Ampère's law. Furthermore, $\Phi$ is a correction to its conservative flux. With the same logic as above, it is, thus, dynamically relevant and with a potential impact beyond the places where charge corrections are induced due to the violation of the $\mathbf{E}\cdot\mathbf{B}=0$ constraint. With the similarity of Eqs.~(\ref{eq:misalignedrecon}) and~(\ref{eq:misalignedcons}) it is not surprising that the LCR method may be regarded, in some aspects, as a low $\alpha$ limit of the CC method (with only weak action of the scalar cleaning potential $\Phi$). Strikingly, the action of non-ideal electric fields on a global scale remains relevant even when replacing the algebraic corrections to force-free violations (Eq.~\ref{eq:DBcutback}) by the continuous action of a driving current as introduced in Eq.~(\ref{eq:FFResCurrentPerpendicular}). In this case, the non-ideal term
\begin{align}
\mathcal{S}_{\rm ni}=\kappa_I\frac{\mathbf{E}\cdot\mathbf{B}}{|\mathbf{B}|^2}\mathbf{B}=\kappa_I\mathbf{E}_\parallel
\label{eq:nonidealsource}
\end{align}
emerges a source of heating in the energy evolution equation \citep[][Eq.~2.73]{MahlmannPhD}. Its purpose is to continuously drive the electromagnetic field to a force-free state. The results presented in Sect.~\ref{sec:drivingfocus} allow two notable interpretations: i) the diffusion induced by Eq.~(\ref{eq:nonidealsource}) does not significantly change the dissipation of Poynting flux along the ECS (measured by $\Delta L/L_{\rm LC}$ obtained by applying Eq.~\ref{eq:DBcutback}); and ii) explicit driving currents on cell-centered meshes are not able to overcome the need of charge conservation in FFE. All the conducted simulations (including those employing driving currents) always reduce to the most diffusive limit whenever the LCR method is applied.
\subsection{Time-scales of the hyperbolic/parabolic cleaning}
\label{sec:cleaningscales}
Equations \eqref{eq:Efield} and \eqref{eq:Phi} can be manipulated to show that both, the scalar potential $\Phi$ and the difference parameter $\mathcal{R}$, obey the telegraph equation \citep{Komissarov2007MNRAS.382..995}
\begin{align}
&\partial_{tt}^2 \Phi + \kappa_\Phi \partial_t \Phi - c_\Phi^2\nabla^2 \Phi=0 \label{eq:telegraph1}\\
&\partial_{tt}^2 \mathcal{R} + \kappa_\Phi \partial_t \mathcal{R} - c_\Phi^2\nabla^2 \mathcal{R}=0.
\end{align}
Let $\tau$ and $l$ be the characteristic time and length scales of change of $\Phi$ (or $\mathcal{R}$), respectively. From Eq.~\eqref{eq:telegraph1}, approximating $\partial_t\Phi\approx \Phi/\tau$ and $\nabla\Phi\approx \Phi/l$, we obtain
\begin{align}
&\frac{\Phi}{\tau^2} + \kappa_\Phi \frac{\Phi}{\tau} - c_\Phi^2\frac{\Phi}{l^2}\approx 0 . \label{eq:telegraph2}
\end{align}
In the limit $\tau\ll 1/\kappa_\Phi$, Eq.~\eqref{eq:telegraph2} yields $\tau\approx l/c_\Phi\equiv\tau_a$, where $\tau_a$ has the meaning of an advection timescale for cleaning errors. In the complementary limit $\tau\gg 1/\kappa_\Phi$, one obtains $\tau\approx l^2 \kappa_\Phi/c_\Phi^2\equiv\tau_d$, where $\tau_d$ can be interpreted as the diffusion timescale for the cleaning of errors. The ratio between both time scales is precisely the parameter $\alpha$ defined in Eq.\,\eqref{eq:alphadimensionless}, i.e. $\alpha = \tau_d/\tau_a$.
As has been noted throughout the literature \citep[e.g.][]{Mignone2010,Mahlmann2019,Mahlmann2020c}, mostly in the context of cleaning errors to the $\nabla\cdot\mathbf{B}=0$ constraint, a careful calibration of the parameters controlling hyperbolic/parabolic cleaning is necessary. The same holds true for the cleaning of errors to the $\rho=\nabla\cdot\mathbf{E}$ constraint, and we conducted a thorough calibration for the simulations shown in this paper. One way of optimizing the cleaning parameter $\alpha$ is to evaluate the dissipation induced by the source terms to the energy evolution equation, as it was presented in \citet[][Eq.~2.73]{MahlmannPhD}. The relevant channels are dissipation by cleaning ($\propto c_\Phi^2\mathbf{E}\cdot\nabla\Phi$), as well as Ohmic heating fueled by non-ideal electric fields ($\propto\mathbf{E}\cdot\mathbf{J}$).
With the results in Fig.~\ref{fig:QDissipation} we find that employing the $\Phi$ cleaning with $\alpha\approx 1$ yields optimal results. This means that an optimal regime is reached when the timescales of dissipation ($\tau_d$) and advection ($\tau_a$) of numerical errors induced by the violation of FFE constraints (in the aligned force-free rotator, predominantly at the ECS) are of the same order. First, the dissipation by corrections of non-ideal electric fields is minimised across all models in this regime (light blue stripe). Second, the dissipation by the cleaning potential stabilises at a steady value across resolutions. This plateau roughly coincides with the blue shaded region and can be interpreted as a trade-off between excessively weak cleaning (inducing larger dissipation for small values of $\alpha$ because of the oscillatory behavior of the electric field after applying corrections to enforce the force-free regime) and the extreme case of over-damping that drives the scalar function $\Phi$ to zero very rapidly.
\subsection{Magnetospheric structure}
\label{sec:structure}
In Fig.~\ref{fig:CURRENT_ZOOM}, we display the structure of currents and charges in two selected regions of the magnetosphere. Specifically, we examine a location close to the polar cap and another one around the Y-point for models using the LCR method (model \textbf{Bd}) and CC method (model \textbf{Hc}). Both models have the same value of $\alpha=0.7$ and the same numerical resolution. The overall charge distribution looks qualitatively like the one obtained by previous force-free models \citep[e.g.][in axial symmetry or \citealt{Kalapotharakos_2009A&A...496..495, Kalapotharakos_2012ApJ...749....2} in three dimensions]{Parfrey_2012MNRAS.423.1416} and in a number of PIC simulations of aligned rotators \citep[e.g.,][]{Chen_2014ApJ...795L..22,Philippov2014,Cerutti2016,Brambilla_2018ApJ...858...81}. Negative charges fill the regions above the polar caps, while positive ones fill the closed magnetosphere up to the Y-point. The structure of the Y-point and of the ECS adjacent to it consists of a set of charge layers of alternating sign stacked vertically in model \textbf{Bd} (Fig.~\ref{fig:CURRENT_ZOOM}, bottom left panel). This layered structure is also observed in, e.g. \citet[][cf. their Fig.~16]{Parfrey_2012MNRAS.423.1416}. Along the equatorial region, a positively charged layer with a thickness $\sim 0.027r_{\rm LC}$ emerges, reproducing the results of a positive surface charge density along the ECS shown in \cite{Timokhin2006}. In model \textbf{Hc}, the layered charge structure is destroyed due to time-variable episodes of reconnection along the ECS. Still, the positively charged central layer is intertwined with regions of negative charge.
The (more) variable structure of the ECS in models using the CC method arises because of the existence of a finite time induced by restoring the stationary condition $\nabla\cdot\mathbf{E}=\rho$. This finite time is brought about by the (finite) propagation speed ($c_\Phi\sim c$) of the hyperbolic part of the equation controlling $\Phi$, and modulated by the (finite) diffusion timescale $\tau_d$ (see Sect.~\ref{sec:cleaningscales}). In the LCR method, this time is zero, as to say the restoration of $\nabla\cdot\mathbf{E}=\rho$ is instantaneous. However, the current does not follow the change in the charge density distribution instantaneously.
The directions of the poloidal current (displayed by the colored arrows in Fig.~\ref{fig:CURRENT_ZOOM}) shows a return current flowing along the ECS and continuing over the closed zone separatrix converging on the Y-point and extending up to the stellar surface. However, the structure of the current is not simple. Among currents flowing towards the stellar surface, we also find anti-parallel currents flowing away from the surface. We point out the interesting observation that above the polar caps a region with super Goldreigh-Julian charge density, i.e. $\rho>\rho_0$ (enclosed by the cyan dashed line in the upper panels of Fig.~\ref{fig:CURRENT_ZOOM}), emerges.
Under the assumption of stationarity, in a small layer around the ECS with a thickness $2h\ll r$, \cite{Contopoulos_2014ApJ...781...46} constructed a simple analytic model for the electromagnetic structure, where the electric field toroidal component is zero. However, in our
models, we find a time-dependent $|E_\phi(r,z,t)|\ne 0$ in places along the ECS (and $x>1$) where the magnetic dominance condition is breached. Indeed, the model using the CC method shows a quasi-periodic pattern of the form $E_\phi(r,z,t)\sim E_\phi(r,h)\sin\left[8\pi(x-t)\right] $.
For the model using the LCR method, the spatial frequency is about two times larger. The other two components of the electric field roughly follow the analytic model of \cite{Contopoulos_2014ApJ...781...46}, namely, they follow the relations $E_r(x,h)\approx -xB_z(x,z)$ and $E_z(x,h)\approx xB_r(x,z)$. Because of the presence of displacement currents and the (comparatively much smaller) contributions of the term $c_\Phi^2\nabla\Phi$ in Eq.~\eqref{eq:Efield}, the current density in the ECS is not given by $\mathbf{j}_{\rm steady}=\nabla\times\mathbf{B}$ as in the stationary analytic model of \cite{Contopoulos_2014ApJ...781...46}. This is lucidly represented in the bottom panels of Fig.~\ref{fig:CURRENT_ZOOM}, where the ratio $\Delta J/j_0=(|\mathbf{j}|-|\nabla\times\mathbf{B}|)/j_{0}$ differs from zero (indicating that the current density includes contributions other than $\mathbf{j}_{\rm steady}$) mostly in the vicinity of the ECS and also along the current sheets surrounding the closed magnetospheric region.
At this point, there is an interesting difference between models using LCR and CC methods, namely, models with a local charge reconstruction have a current smaller than $\mathbf{j}_{\rm steady}$ ($\Delta J/j_0<0$) along most of the current sheet. In contrast, the model \textbf{Hc} shows smaller average deviations along the ECS between $\mathbf{j}$ and $\mathbf{j}_{\rm steady}$, and surrounding it in intermediate patches beyond the LC. That finding is unexpected given the larger variability along the ECS encountered, in general, for CC models.
\subsection{Diffusivity in force-free magnetospheres}
\label{sec:diffusivitydiscuss}
\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{figures/figure8a.jpg}\\
\includegraphics[width=0.47\textwidth]{figures/figure8b.jpg}\\
\includegraphics[width=0.47\textwidth]{figures/figure8c.jpg}
\vspace{-6pt}
\caption{Charge and current structure in time-dependent aligned rotator models. Upper and mid panels: Charge ($\rho$; colored background, normalised to the absolute value of the Goldreich-Julian charge density, $|\rho_0|$) and current ($\mathbf{j}_\parallel$; arrows, normalised to $\rho_0 c$) structure of selected models (see panel legends). We visualise two different zooms into the magnetosphere, namely the polar cap region (top panels), and the Y-point vicinity (bottom panels). The direction of the current flow is indicated by arrows for regions where $j_\parallel/j_0>0.01$. The cyan dashed line denotes the location where $\rho=\rho_0$ (specifically, in between that line and the stellar surface, the charge density is larger than the Goldreich-Julian charge density). Bottom panels: Difference between the current $\mathbf{j}$ in our time-dependent models and the current in the steady state case $\mathbf{j}_{\rm steady}=\nabla\times\mathbf{B}$, normalised to the Goldreich-Julian current density $j_0$. We display the poloidal magnetic field lines; the vertical black line denotes the position of the LC.}
\label{fig:CURRENT_ZOOM}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{figures/figure9.jpg}
\vspace{-18pt}
\caption{Averaging of magnetospheric fields and Poynting fluxes during a time of $\Delta t=(7.5-6.8)t_{\rm p}=1.3t_{\rm p}$ for selected models ($\mathbf{Ba}$, $\mathbf{Be}$, and $\mathbf{Bb}$). Small variations beyond the light cylinder, especially those observed in Fig.~\ref{fig:Poynting} (shown again as thick, transparent lines in the background of the right panel) are compensated across timescale $\Delta t\gtrsim 1t_{\rm p}$.}
\label{fig:Averaging}
\end{figure*}
Resetting the charge density $\rho$ to the instantaneous value of $\nabla\cdot\mathbf{E}$ assures compatibility of electric fields and the corresponding charge distribution. However, removing significant fractions of the electric field from the domain for ensuring FFE conditions alters the charge distribution in the domain by an amount that is not necessarily of the order of the truncation error.
In our default (CC) scheme \citep{Mahlmann2020b}, the charge is evolved with a separate continuity equation. Changes to the electric field by the algebraic reset of force-free errors introduce a misalignment between the charge density and the divergence of electric fields. The degree of localization of this misalignment to inherently non force-free regions, such as current sheets, can be interpreted as a diffusive length scale. We control this localization in our hyperbolic/parabolic cleaning procedure with the parameter $\alpha$, that expresses the ratio of diffusion to advection timescales for the cleaning errors.
The LCR method turns out to be the most diffusive one, using as a measurement of the diffusivity the dissipation of Poynting flux beyond the LC. In the models resorting to the LCR method, the effect of violations to ideal FFE conditions spreads throughout the domain, with notable consequences for the global energetics. While the Y-point relaxes to a location close to the LC (cf. Fig.~\ref{fig:FFCOND_COMPARE}), field lines reconnect across the ECS and induce a notable dissipation of the outward transported energy ($\sim 30\%$). When augmented by a conservative evolution of the charge density with a suitable calibration of the divergence cleaning parameter $\alpha$, reconnection beyond the LC is greatly reduced - as is the dissipation of energy. At the same time, the Y-point is pushed away from the light cylinder (cf. Fig.~\ref{fig:FFCOND_COMPARE}), and the overall luminosity increases. Though the action of the cleaning potential $\Phi$ can become excessively large for $\alpha\gg 1$, the increase of the luminosity with the corresponding change of the Y-point location roughly follows the trend that is theoretically expected for small and intermediate values of $\alpha$ (cf. Fig.~\ref{fig:YpointLuminosity}).
We explore an interpretation of our results in light of the parameterization of the diffusivity beyond the LC in terms of the pair formation multiplicity $\kappa$ suggested by \cite{Contopoulos2020}. These authors show that smaller values of $\kappa$ eventuate in more dissipation at the ECS and, conversely, $\kappa\gg 1$ yields very little or no dissipation at all. When drawing a direct comparison to the results presented throughout this paper, the models endowed with the LCR method would correspond to values of $\kappa\sim 1$. Models employing our the CC scheme (our default), would compare to values $\kappa\gg 1$. The physical conditions in a typical pulsar magnetosphere tend to produce many pairs per Goldreich-Julian charge particle in the polar cap \citep[e.g.][and references therein]{Timokhin_2015ApJ...810..144,Contopoulos_2019MNRAS.482L..50}. Hence, in this regard, our CC models potentially reproduce the usual conditions met in actual rotating pulsars more closely, though the limitations set by the force-free regime naturally persist.
The panels of Fig.~\ref{fig:FFCOND_COMPARE} that show the location of violations of the FFE constraints illustrate two notable aspects: i) deviations from $\mathbf{E}\cdot\mathbf{B}=0$ are not a localised phenomenon. Emerging from genuinely non-ideal regions, such as current sheets, non-ideal fields can spread throughout the domain. ii) A change in the treatment of charge conservation not only minimises the dissipation induced by corrections of the $\mathbf{E}\cdot\mathbf{B}=0$ condition, but also changes the electromagnetic structure of the current sheet itself. Cases $\mathbf{Ba}/\mathbf{Bd}$ show a region with $\mathbf{E}^2-\mathbf{B}^2>0$ along the length of the current sheet. Contrasting this, such violations only occur at X-points in the current sheet of models $\mathbf{Bb}/\mathbf{Be}$. The acceleration of particles and the production of radiation demand the existence of an electric field component parallel to the magnetic field, namely that $\mathbf{E}\cdot\mathbf{B}\ne 0$. Hence, the structure of the regions of the magnetosphere, where the strongest violations of the $\mathbf{E}\cdot\mathbf{B}=0$ condition occur, are likely closely related to the production of pulsar radiation \citep[e.g.][]{Timokhin_2013MNRAS.429...20}. Looking at Fig.~\ref{fig:FFCOND_COMPARE}, these violations, although extended in the magnetosphere as stated above, are maximised in the vicinity of the Y-point, suggesting that the Y-point is the most important site for particle acceleration in the magnetosphere. This result is backed up by PIC simulations of, e.g. \cite{Chen_2014ApJ...795L..22} and gives support to the theoretical "ring-of-fire" model of \cite{Contopoulos_Stefanou_2019MNRAS.487..952}.
Violations to the ideal force-free conditions in combination with the transport of charge conservation errors originating during their correction (Sect.~\ref{sec:nonidealFF}) are the main driver of diffusivity in force-free aligned pulsar magnetospheres. The relative importance of the dissipation triggered by the enforcement of either the $\mathbf{E}\cdot\mathbf{B}=0$ or the $\mathbf{E}^2-\mathbf{B}^2<0$ conditions is similar across these two particular channels, as we can observe in the magnitudes of the electric energy lost within a given time step in Fig.~\ref{fig:QDissipation}. However, the magnitude of the dissipation triggered by each of them may depend on the order and frequency with which they are applied, as well as on the mesh and time-integrator \citep[cf.][]{Spitkovsky2006}.
Nevertheless, the dominant contribution to the dissipation along these channels stems from the ECS, where the magnetic dominance condition is chronically breached. We suggest that the magnetic dominance condition is really the origin of the differences between the various methods of dealing with the charge treatment as presented in this paper. In the HCC models using local charge reconstruction only in grid zones where $\mathbf{E}^2-\mathbf{B}^2>0$, the dissipation through this channel is minimised (see light green colored squares in the upper panel of Fig.~\ref{fig:QDissipation}), and the Ohmic dissipation is maximised (mid-panel of Fig.~\ref{fig:QDissipation}).
Since $\mathbf{E}^2-\mathbf{B}^2>0$ is only reached at the ECS, it is standing to reason that the specific restoration of the magnetic dominance constraint may induce global changes in the magnetospheric structure, its luminosity, and, certainly, on the amount of dissipation of Poynting flux beyond the LC.
The resistivity models used beyond the LC in Sect.~\ref{sec:diffusivityfocus} provide twofold insight. First, they allow us to estimate the numerically induced diffusivity $\eta_0$
across the ECS. The effect of $\eta_d$ will only become noticeable when the phenomenological resistivity is larger than the numerical diffusivity of the method. From Fig.~\ref{fig:ETA_COMPARE}
we can estimate that $\eta_0\lesssim 10^{-1}$. Second, and in line with the results presented in \citet{Mahlmann2020c}, a choice of $\eta_d\gtrsim 10^{-1}$ in the current presented in Eq.~(\ref{eq:FFResCurrentPerpendicular}) allows to properly model dynamics of the resistive layer of the ECS. Increasing $\eta_d$ gradually drives relatively large scale
inflows into the ECS (as traced by the drift velocity component perpendicular to it), effectively mimicking physical dynamics in the non-ideal region. As we established very competitive convergence of our high-order FFE method \citep{Mahlmann2020c}, we suggest that this relatively large value of $\eta_0$ is, indeed, induced by the non-ideal fields emerging in the ECS. In this context it becomes clear why several FFE methods need to employ special treatments of the ECS in the pulsar magnetosphere \citep{McKinney2006,Etienne2017}, namely, to reduce the extent of the diffusive regions by an ad-hoc prescription.
As stated in, e.g. \cite{Timokhin_2013MNRAS.429...20}, a necessary criterion imposed by observational constraints on the pulsar magnetosphere is the \emph{stationarity} in a statistical sense. In other words, any local fluctuations on timescales smaller than the LC light-crossing time $\tau_{\rm LC}=r_{\rm LC}/c$ (or more likely, over the rotational period of the pulsar $t_{\rm p}$) that average to a stationary state, may also account for the stability of the pulsar mean profiles and sharpness of the peaks in the spectra of gamma-ray pulsars. A very salient feature related to the treatment of the charge conservation equation in FFE is the obvious time-dependence of the magnetosphere, driven by episodes of magnetic reconnection along the ECS. However, the magnetospheres resulting from the CC method (including a suitable conservative treatment of the charge) are stationary if we average them out over timescales comparable to $\tau_{\rm LC}$. In order to support this statement, we show the time-averaged map of the toroidal magnetic field over an interval $\Delta t$ slightly larger than one pulsar rotational period in Fig.~\ref{fig:Averaging} (left panels). We find even stronger evidence than in the polar distribution of the toroidal field when evaluating the Poynting flux as a function of distance for the averaged models. It displays a radial dependence which radically smooths the spatial variability (Fig.~\ref{fig:Averaging}, right panel). We further probed the long-term stability of the representative models $\mathbf{Ba}$, $\mathbf{Be}$, and $\mathbf{Bb}$ by tracking their evolution during $\gtrsim 30$ rotational periods \citep{SupplementaryMediaA}. The models show stability over such time-scales in all the characteristic properties discussed in the previous section, especially regarding the Y-point location and pulsar luminosity.
\section{Conclusions}
\label{sec:conclusion}
In a deep exploration with our recently developed force-free code \citep{Mahlmann2020b,Mahlmann2020c}, we exploit the diffusion time-scale induced by hyperbolic/parabolic cleaning of charge conservation errors (Sect.~\ref{sec:cleaningscales}) to quantify an aspect that has not been systematically assessed so far in many FFE simulations, including our own. Namely, the \emph{global} imprint of \emph{local} violations to the force-free constraints. At the example of the force-free aligned rotator magnetosphere, we demonstrate that balancing the amount of damping and advection of charge conservation errors (encoded in the parameter $\alpha$) can alter the global structure of the simulated magnetosphere. Specifically, by decreasing the amount of numerical diffusion arising from violations to the FFE constraints, the Y-point moves away from the light cylinder while the outgoing Poynting flux increases by a factor of a few (Sect.~\ref{sec:forcefreealigned}).
In summary, our exploration clarifies several \emph{technical} aspects that should become central for the assessment of (global) FFE simulations. First, the localization of force-free violations to small regions in resistive layers, such as current sheets, are key to reduce the diffusivity that is induced into FFE by non-ideal electric fields or breaches of magnetic dominance (Sect.~\ref{sec:nonidealFF}). In our method, we achieve and control such a localization by combining a conservative evolution of charge density with a hyperbolic/parabolic cleaning of errors to Gauss' law. We suggest that $\alpha\lesssim 1$ is an optimal parameter for the minimization of the combined channels of numerical diffusion (Sect.~\ref{sec:focuseddominance}). Second, in the inherently non-ideal aligned rotator magnetosphere, the ECS is the
main source of numerical diffusion by inducing strong field gradients and violations to the FFE constraints. We identify a strong dependence of the dynamics on the specific treatment of violations to the magnetic dominance condition (Sect.~\ref{sec:diffusivitydiscuss}). Third, the extreme nature of algebraic corrections to the force-free conditions are not the main reason for the luminosity dependence of the global magnetosphere. Driving currents yield very similar results (Sect.~\ref{sec:drivingfocus}).
Finally, different treatments of force-free violations, especially at Y-points and current sheets, are likely to change the resistive time scales of the evolution and to have a notable impact on the equilibrium magnetospheres. We extend our analysis to so-called phenomenological resistivity models, where adapted driving currents mimic the development of resistive layers around current sheets (Sect.~\ref{sec:diffusivityfocus}).
\textit{FFE is a robust way to model energy flows in highly magnetised plasma}, as we find in the magnetospheres of many astrophysical objects. This statement can safely be extended to situations in which the global dynamics of field lines drives the transient appearance of inherently non-ideal regions, such as current sheets. Specifically, we argued this in the context of accretion of magnetic loops onto rapidly spinning BHs \citep{Mahlmann2020}, and the shearing of fields driven by interacting Alfvén waves \citep{Ripperda2021}. However, in situations where areas of genuine (physical) resistivity drive the global field line dynamics, employing FFE methods has to be carefully benchmarked. This work is, to our knowledge, the first extensive calibration of an FFE method for the specific application of astrophysical magnetospheres. We find the global flows of energy to be extremely sensitive to the treatment of FFE violations. Our results agree with comparable numerical surveys throughout the literature only in the limit of strong numerical diffusion induced by the ECS.
Finally, we suggest that scenarios like the aligned rotator should be handled with care when used as a standard test for FFE methods. The stability and magnitude of the Poynting flux beyond the light cylinder can be used as a primer for the diffusivity of the respective method. It could be argued that operating in the well-established, but ultimately limited, regimes of \emph{ideal} fluid approximations for the modeling of global scenarios that are affected or even driven by genuinely resistive effects will require more and more care in the future. The desire to overcome many orders of scale separation has to go hand-in-hand with a deep understanding of the diffusive properties of the employed numerical methods. Until we can transition into an era of multi-regime astro-plasma codes, we find it reassuring to have the limits of our FFE method laid out transparently.
\section*{Acknowledgements}
This work has been supported by the Spanish Ministry of Science, Education and Universities (PGC2018-095984-B-I00) and the Valencian Community (PROMETEU/2019/071). We furthermore thank for support from the COST Actions PHAROS CA16214 and GWverse CA16104. This manuscript relies vastly on high performance computing resources. They were provided with the \textit{LluisVives} machine at the Servei d'Informàtica of the \textit{Universitat de València} (financed by the FEDER funds for Scientific Infrastructures; IDIFEDER-2018-063) and extensively supplemented by allocations on the \textit{MareNostrum} and \textit{Tirant} supercomputers of the \textit{Spanish Supercomputing Network} (AECT-2021-1-0006, AECT-2021-1-0007).
\section*{Data Availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
\bibliographystyle{mnras}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 416 |
Q: copy directory to remote computer using powershell I'm a beginner and i tried to make a simple script to save our current scripts on other admin computer.
So:
$SAVE = Get-ADComputer -filter " name -notlike '* *mycomputer*' " -SearchBase 'OU=Supervision,OU=...'
Foreach ($S in $SAVE)
{
Copy-Item 'd:\doc\*' -destination '\\$S\d$\test'
}
I tried selecting the Name (adding a pipe after the get-adcomputer) but it fail and trying the line copy item without the for each work so i don't know where i went wrong.
If someone could point me the problem?
I tried Test-Path \\$A\D$ and it returned a true
Thank
A: i found a way to make it work: I deleted the " . I did not replace them and to make sure it would work created a new variable in the loop $tst = $S.IPv4Address and replaced the $S in the path name by $tst so now it's like that : Copy-Item 'd:\doc\*' -destination \\$tst\d$\test it seems to work.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,720 |
Corona Schröter – niemiecka śpiewaczka
Johann Hieronymus Schröter – niemiecki prawnik
Joseph Schröter – biolog
Martina Schröter – niemiecka wioślarka
Paweł Schroeter – polski lekarz | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,426 |
Q: Multiplying bytes with XOR So, before I get into my question. I tried search this but I am probably not wording it correctly to get any valid results. So the purpose is to use in the a AES 128-bit encryption program.
I need to multiply an unsigned char (which would be the hexadecimal value) by 2 or 3 and this would be an XOR operation. So basically, is there a way to do it without typing it out like this.
(SBOX[0] ^ SBOX[0]) ^ SBOX[0]
If I have to do it this way, each line is going to be fairly long but can be done I believe. It would be nice if there is an operator to just say 3 ^ SBOX[0].
A: If you're doing AES, then you're doing your arithmetic in a Galois Field (specifically GF(28)). Thus rules that you're used to for standard integers no longer hold.
In particular, whilst addition is XOR (in GF(2n)), multiplication isn't repeated addition. Your example shows why - multiplication by two would be x ^ x == 0 always.
The actual steps (in code) depend on the reducing polynomial of your Galois field (and in any case, deriving them is way beyond my ability nowadays). However, they're summarised in multiple places on the web. And in many case, these explanations specifically target the S-box MixColumns operation, e.g. Wikipedia.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,371 |
package org.apache.wink.server.internal.contexts;
import javax.ws.rs.core.HttpHeaders;
import javax.ws.rs.core.MediaType;
import javax.ws.rs.core.MultivaluedMap;
import org.apache.wink.common.RuntimeContext;
import org.apache.wink.common.internal.WinkConfiguration;
import org.apache.wink.common.internal.contexts.MediaTypeCharsetAdjuster;
import org.apache.wink.common.internal.runtime.RuntimeContextTLS;
import org.apache.wink.common.utils.ProviderUtils;
import org.apache.wink.server.internal.DeploymentConfiguration;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
public class ServerMediaTypeCharsetAdjuster implements MediaTypeCharsetAdjuster {
final private static ServerMediaTypeCharsetAdjuster instance =
new ServerMediaTypeCharsetAdjuster();
// enforce singleton
private ServerMediaTypeCharsetAdjuster() {
}
public static ServerMediaTypeCharsetAdjuster getInstance() {
return instance;
}
private static final Logger logger =
LoggerFactory
.getLogger(ServerMediaTypeCharsetAdjuster.class);
public MediaType setDefaultCharsetOnMediaTypeHeader(MultivaluedMap<String, Object> httpHeaders,
MediaType mediaType) {
logger.trace("setDefaultCharsetOnMediaTypeHeader({}, {}) entry", httpHeaders, mediaType); //$NON-NLS-1$
RuntimeContext context = RuntimeContextTLS.getRuntimeContext();
// we're on the server, so this is a safe cast
DeploymentConfiguration config = (DeploymentConfiguration)context.getAttribute(WinkConfiguration.class);
if (config.isDefaultResponseCharset() || config.isUseAcceptCharset()) {
if (httpHeaders != null && (httpHeaders.isEmpty() || httpHeaders
.get(HttpHeaders.CONTENT_TYPE) == null)) {
// only correct the MediaType if the MediaType was not explicitly
// set
logger.trace("Media Type not explicitly set on Response so going to correct charset parameter if necessary"); //$NON-NLS-1$
if (ProviderUtils.getCharsetOrNull(mediaType) == null) { //$NON-NLS-1$
try {
String charsetValue = "UTF-8"; //$NON-NLS-1$
if (config.isUseAcceptCharset()) {
// configuration says to inspect and use the Accept-Charset header to determine response charset
HttpHeaders requestHeaders = null;
if (context != null) {
requestHeaders = context.getHttpHeaders();
}
charsetValue = ProviderUtils.getCharset(mediaType, requestHeaders);
}
String newMediaTypeStr = mediaType.toString() + ";charset=" + charsetValue; //$NON-NLS-1$
mediaType = MediaType.valueOf(newMediaTypeStr);
httpHeaders.putSingle(HttpHeaders.CONTENT_TYPE, newMediaTypeStr);
logger.trace("Changed media type to be {} in Content-Type HttpHeader", newMediaTypeStr); //$NON-NLS-1$
} catch (Exception e) {
logger.trace("Caught exception while trying to set the charset", e); //$NON-NLS-1$
}
}
}
} else {
logger.trace("No default charset was applied to the response Content-Type header due to deployment configuration directive."); // $NON-NLS-1$ //$NON-NLS-1$
}
logger.trace("setDefaultCharsetOnMediaTypeHeader() exit returning {}", mediaType); //$NON-NLS-1$
return mediaType;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,649 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/257677\/perfect-separation-in-logistic-regression-and-data-transformation-can-it-help","text":"# Perfect separation in logistic regression and data transformation -> can it help?\n\nfirst of all, I am super happy that I found this great community. I am currently having trouble in my logistic regression analysis in that I get the error message display\n\nWarning message: glm.fit: fitted probabilities numerically 0 or 1 occurred\n\n\nI read a lot about the issues of perfect separation in this forum. A colleague told me that he is always using the log transformation of his data for his analysis (not logistic regression) and I noticed, that once I transformed the data, I won't get the error message anymore. Could this be a solution as well?\n\nFor background information, this is my data structure:\n\nmodel <- glm(formula = Customer.group ~ Price.Index , family = binomial())\n\n\nWith Customer.group being yes (1) or no (0) and Where Price.Index is a calculated measure for each customer consisting of the weighted sum of a price paid for a product in a category divided by the average price in this category. So it is not actual observed data, but a calculated measure.\n\n\u2022 I'm a bit puzzled by your dependent variables, if it is a sum of prices divided by an average price, is that not some continuous numerical response rather than a yes\/no (or some integer out of a larger integer total)? So why use logistic regression? \u2013\u00a0Bj\u00f6rn Jan 23 '17 at 13:22\n\u2022 I am rather surprised that your colleague thinks log transforming makes separation go away. Can you show us the result of your models with and without log transforming PriceIndex so we can see the difference? \u2013\u00a0mdewey Jan 23 '17 at 13:28\n\u2022 Hi Bj\u00f6rn, my dependent variable is \"Customer.Group\" being binary as yes\/no for indicating whether the customer is part of the group. My independent variable is the continuous price index which varies between ~0.1 and ~10 \u2013\u00a0Gabi Schneider Jan 23 '17 at 13:53\n\nmodel <- glm(Customer.Group ~ Price.Index, data=yourdata, family=\"binomial\") summary(model)\n\u2022 Hi Michael! Thanks so much! I tried this formula and I still get the error message with the non-transformed data. I also used the safeBinaryRegression package to assess if there is a perfect separation which was found in my data. Therefore, I used the brglm package to run a Firth regression (which was successul). But because I wasn't sure what diagnostic statistics and model fit estimates to use for Firth regression, I tried to run the glm model again this time with the log-transformed data. I am just wondering, if this procedure would be statistically correct or if it wouldn't make sense. \u2013\u00a0Gabi Schneider Jan 23 '17 at 13:46","date":"2019-04-21 22:18:07","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\": 0.5867331027984619, \"perplexity\": 989.0474209723415}, \"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-2019-18\/segments\/1555578532929.54\/warc\/CC-MAIN-20190421215917-20190422001917-00484.warc.gz\"}"} | null | null |
La Iglesia de San Pedro y San Pablo () es un edificio histórico propiedad del National Trust situada en la localidad de Muchelney, en el condado inglés de Somerset. Ha sido designado como un monumento clasificado de grado I. Tiene orígenes sajones, sin embargo, el edificio actual data en gran parte del siglo XV.
Historia
La iglesia, que se encuentra junto al sitio de la abadía de Muchelney y cerca del río Parrett, tiene un techo decorado con pinturas jacobeas de ángeles con los senos desnudos, cuya desnudez simboliza la pureza inocente. Fueron pintados a principios del siglo XVII.
La iglesia también contiene un organillo construido por Gray y Davison e instalado alrededor de 1835 a 1840. Es el último que se sabe que todavía está en la iglesia donde se instaló por primera vez y todavía funciona.
La iglesia cuenta con una planta dividida en tres naves y un presbiterio con una capilla corta a cada lado.
Tiene una torre Somerset de tres pisos, que data de alrededor de 1468, sostenida por pares de contrafuertes de esquina de altura completa. La torreta de escalera octogonal sureste conduce a una puerta exterior.
La parroquia es parte del beneficio del Ministerio del Equipo del Área de Langport dentro del decanato de Ilchester.
Referencias
Enlaces externos
Somerset
Arquitectura gótica de Inglaterra
Edificios listados de Grado I de Inglaterra | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,025 |
\section{Introduction}
Radio galaxies (RGs) are found from the cores to the extremities of galaxy clusters \citep[e.g.,][]{kale15,padovani16,Garon19}. Cluster RGs frequently appear significantly distorted from simple, bilateral, axial symmetry \citep[e.g.,][]{deGregory17,Garon19}, revealing non-axisymmetric environmental impacts. Sometimes the distortions can be attributed to galaxy motions relative to the cluster center. But, perhaps more revealing about cluster physics, many distortions are likely to reflect large-scale ICM flows and shocks; i.e., ``ICM weather'' related to cluster formation and evolution \citep[e.g.,][]{Bonafede14,Owen14,Shimwell14,vanWeeren17,Mandal18,WilberNov18}.
In order to improve understanding of the physics of these behaviors and associated observables, we have undertaken a broad-based study, primarily through simulations, but also including analytic modeling, analyzing dynamical RG-ICM interactions involving both steady winds through the life of the RG \citep[][]{jones16,ONeill19a} and shock impact on an existing RG \cite[][]{jones16,nolting19a,ONeill19b}. Most directly related to the present report, \cite{nolting19a} studied through simulations the interactions between cluster merger-strength ICM shocks and RG formed in a static medium when the incident shock normals are aligned with the axis of jets responsible for creating the RG. Here we consider the analogous interactions when the shock normals are orthogonal to the RG jet flows. \cite{nolting19a} pointed out that the evolution of a RG in response to a shock encounter has two successive components. The first component is associated with the abrupt change of conditions across the shock discontinuity, while the second component is a prolonged interaction with a post-shock wind whose properties are determined by the shock jump conditions. We will see in the present study that the same basic dynamical elements apply, independent of shock-RG orientation, However, some signature outcomes are sensitive to orientation. We also point to the \cite{ONeill19a} work analyzing in detail evolution of and emission from steady jets in a steady, orthogonal wind to form classical ``narrow angle tail'' (NAT) RG morphology.
\cite{nolting19a}, confirmed earlier studies demonstrating that shock impact on a low density cavity, such as a RG lobe, can transform the cavity into a ``doughnut-like'' ring vortex. This topological transformation, the most distinctive feature of a shock encounter with a lobed RG, results from shear induced by the enhanced post shock speed inside the lobe \citep[e.g.,][]{EnsslinBruggen02, PfrommerJones11}. In laboratory settings shocks in air striking helium bubbles have, for example, created analogous vortex rings \citep[e.g.,][]{Ranjan08}.
In the astrophysical context, rings of diffuse radio emission possibly related to shocked RG plasma have been discovered in, for example, Abell 2256 \citep{Owen14} and the Perseus cluster \citep{SibringdeBruyn98}. A distinct scenario related to this physics is the so-called ``radio phoenix.'' In the radio phoenix model aged cosmic ray electron (CRe) populations from expired AGN activity are overrun by an ICM shock wave \citep{Ensslin01, EnsslinBruggen02} and reaccelerated primarily by adiabatic compression to become luminous once again. Such objects could have complex morphologies as well as strongly curved, steep radio spectra \citep{vanWeeren19}. If, at the other extreme, the RG jets remain active through a shock encounter, so interact with the post-shock wind, RG-shock dynamics are considerably enriched, as already noted in \cite{nolting19a} for aligned shock-jet geometry and in \cite{ONeill19b} for shocked tailed RG. On the other hand, key signature behaviors that might be used to identify encounters generally and to constrain the conditions involved are yet to be established. Our further efforts aim to help fill that gap.
The remainder of this paper is organized as follows: Section \ref{sec:interactcartoon} outlines the underlying physics of the shock-RG encounter (\S \ref{subsec:cavities}), including vortex ring formation (\S \ref{subsec:VortRings}), and subsequent wind--jet interactions (\S \ref{subsec:bending}) when the wind velocity is transverse to ongoing jet flows. Section \ref{sec:methods} describes our simulation specifics, including numerical methods (\S \ref{subsec:numerics}) and details of our simulation setups (\S \ref{subsec:Setup}). In section \ref{sec:Discussion} we discuss the results of the simulations, while section \ref{sec:Summary} provides a brief summary of our findings.
\section{Outline of Orthogonal Shock--RG Interaction Dynamics}
\label{sec:interactcartoon}
The geometry of the problem we explore in this paper is illustrated in figure \ref{fig:orth-setup}. Specifically, a RG initially evolves in a homogeneous, stationary ICM prior to a shock encounter. The RG is formed, beginning at $t = 0$, by a pair of steady, oppositely directed jets that are identical except for the sign of the jet velocity. In the figure those jets are vertical. A plane shock whose normal is orthogonal to the jet axes first contacts the RG lobes at a time $t_i>0$ (from the left in the figure). Depending on the simulation, the jets may or may not remain active through the encounter. In one case jet activity is terminated long before the shock encounter to mimic a radio phoenix scenario.
\begin{figure*}
\centering
\includegraphics[scale=0.7]{OrthSetup.pdf}
\caption{Basic geometry of the orthogonal shock--RG encounter.}
\label{fig:orth-setup}
\end{figure*}
To describe the basic shock-RG encounter mechanics we need to specify several ICM, shock and RG properties and their relationships. In what follows properties associated with the unshocked ICM are identified by subscripts, `i', while properties of the post shock ICM wind, are marked by `w'. Properties of the RG cavities (= lobes) are identified by `c'. RG jet properties are designated by `j'. Where it is important to distinguish jet or cavity properties within the unshocked ICM from those same jet properties within the post shock wind, it is convenient to apply the distinct, hybrid labels, `ji' and `jw', or ``ci'' and ``cw'. It may also be useful up front to clarify that a feature or property is ``upwind'' of some second structure at a given time if an encounter between the two structures will occur in the future. Thus, in the current context, unshocked ICM material is upwind of the ICM shock, so in figure \ref{fig:orth-setup} to the right of the shock. Similarly, a vector in the post shock flow pointing ``upwind '' would point left in figure \ref{fig:orth-setup}.
We begin our outline with a characterization of the ICM shock transition. For this we need the incident shock Mach number, $\mathcal{M}_{si}$, along with the unshocked ICM density, $\rho_i$ and sound speed, $a_i$; that is, $\mathcal{M}_{si} = v_{si}/a_i$. The unshocked ICM pressure (assuming an adiabatic index, $\gamma = 5/3$) is, $P_i = (3/5)~\rho_i a_i^2$. Standard shock jump conditions give us properties of the post shock ICM wind; namely,
\begin{align}
\label{eq:jump-d}
\rho_w = \frac{4\mathcal{M}_{si}^2}{\mathcal{M}_{si}^2+3}\rho_i,\\
\label{eq:jump-p}
P_w = \frac{5M_{si}^2-1}{4}P_i,\\
\label{eq:jump-v}
|v_w| = \frac{3}{4}\frac{M_{si}^2-1}{M_{si}}a_{i},\\
\label{eq:jump-a}
a_w = \frac{\sqrt{(M_{si}^2 + 3)(5 M_{si}^2 - 1)}}{4 M_{si}} a_i,\\
\label{eq:windMach}
|M_w| = \frac{|v_w|}{a_w} = 3 \frac{M_{si}^2 - 1}{\sqrt{(M_{si}^2 + 3)(5M_{si}^2 - 1)}},
\end{align}
where the wind velocity, $v_w$, is measured in the frame of the unshocked ICM. Since our scenario involves a RG initially developing in a static ICM, we henceforth, unless otherwise stated, refer all velocities to the rest frame of the unshocked ICM (= the rest frame of the AGN/RG). In this study we carried out simulations involving two ICM shock strengths. Specifically, we considered $\mathcal{M}_{si} = 4$, for which $\rho_w/\rho_i = 3.37$, $P_w/P_i = 19.75$, $|v_w|/a_i = 2.81$, $a_w/a_i = 2.42$, and $|M_w| = 1.16$. For comparison, we also simulated one case with a weaker $\mathcal{M}_{si} = 2$ shock, leading to $\rho_w/\rho_i = 2.29$, $P_w/P_i = 4.75$, $|v_w|/a_i = 1.13$, $a_w/a_i = 1.44$, and $|M_w| = 0.78$. All our simulations reported here involve pre-shock ICM conditions with $\rho_i = 5\times 10^{-27}\rm{g/cm^3}$, $P_i = 1.33\times 10^{-11}~\rm{dyne/cm^2}$ and $a_i = 667~\rm{km/sec}$.
\subsection{Shock--Lobe Collisions}
\label{subsec:cavities}
Shock and post shock flow behaviors inside the RG cavities (lobes) are largely consequences of the large density contrast between the ICM and the cavities. Thus, to characterize this interaction we should specify $\rho_c$. For light jets, as in our simulated scenarios, we expect $\rho_c \la \rho_j \ll \rho_i$. Specifically, here we have used $\rho_j = 10^{-2} \rho_i$, and, indeed we find pre-shock cavity conditions with $\rho_c \la 10^{-2} \rho_i$. Such cavities generally reach at least rough pressure balance with their surroundings, and in our simulations we find $P_c \sim P_i$ before shock impact. Consequently, before shock impact $a_c \ga 10 a_i$.
A simple outline of shock-lobe interaction can be constructed from the fact that $a_c \gg a_i$. Detailed discussions can be found in \cite{PfrommerJones11} and references therein. Since the speed of the shock inside the cavity must satisfy $v_{sc} > a_c \gg a_i$, while in the scenario under discussion, $v_{si} = \mathcal{M}_{si} a_i \la\rm{a~few}~a_i$, the shock propagates more rapidly inside the cavity than in the surrounding ICM. Because the cavity is much hotter than the ICM, so that $a_c >> a_w$, the internal shock is considerably weaker than the incident shock; that is, $\mathcal{M}_{sc} = v_{sc}/a_c << \mathcal{M}_{si}$. Somewhat rarefied post shock ICM (wind) plasma, separated from cavity plasma by a contact discontinuity (CD), fills the cavity behind the shock at speeds $v_{CD} > v_w$. In the end, the cavity is crushed by this penetration. Coincidentally, the fast post shock penetration of ICM inside the cavity generates strong shear along the original cavity boundary. The result of these two developments is a topological transformation of the original cavity into a vortex ring whose axis aligns with the original shock normal. In the scenarios being examined here, there are two RG lobes being similarly transformed simultaneously. Thus, immediately after shock passage through the RG lobes there are two similar, coplanar vortex rings.
The simplicity of this outcome contrasts significantly with the outcome when the AGN jets and the shock normal align (or nearly align) as discussed in \cite{nolting19a}. For the latter geometry ICM-lobe encounters are sequential, rather than simultaneous. So, although vortex ring structures do develop, the flows, especially within the second, downwind lobe, are much more complicated than in the scenario outlined here. The events simulated in \cite{nolting19a} also all included continued active jets that were aligned (or nearly aligned) with the incident shock normal, which contributed further, distinctive behaviors to the dynamical evolution.
\subsection{Vortex Ring Dynamics}
\label{subsec:VortRings}
We return briefly to a basic discussion of what happens to the pair of vortex rings that emerge from the shock encounters under study in the present work. The full dynamics of vortex rings has been studied in depth analytically, in laboratory settings, and also numerically. Some useful and simple insights into the current situation come from such studies. In particular, a vortex line, or `filament,' can be shown to induce an associated velocity field in a relationship analogous to the Biot-Savart law of electromagnetism connecting a line of current to the encircling magnetic field. Specifically, a straight vortex line of infinite length and circulation, $\Gamma$, induces a velocity, $\delta v$, at a distance d given by
\begin{equation}
\delta v = \frac{\Gamma}{2\pi d}.
\label{eq:inducedVel}
\end{equation}
Conceptually, a vortex ring can be pictured as a vortex line connecting to itself, with opposite sides of the ring represented as counter-rotating, vortices. The electromagnetic analogy is a current loop, of course. Such counter-rotating vortices induce modifications in each other by equation \ref{eq:inducedVel} that project the vortex ring forward along its symmetry axis \citep[see, e.g.,][]{Leweke16}. When a vortex ring or filament is not circular, but possesses nonuniform curvature, these induction effects induce geometry changes. Where the curvature is highest, the induction effect is strongest. For instance, \cite{Hama62} showed that an initially parabolic vortex filament will result in a larger induced velocity at the vertex, causing it to lead the rest of the filament, which in turn alters the direction of the induced velocity at that point. The structure becomes 3 dimensional and the vertex acquires a vertical component in its induced velocity. In addition to self inducing a velocity forward along its axis, as a vortex ring propagates it is prone to entraining material from the surrounding medium, eventually slowing its propagation through the background medium (the post shock wind, in the present case) \citep{Maxworthy72}.
The same relationship leads multiple vortex rings to induce motions in each other. If two similar vortex rings propagate along parallel axes, as in this study, adjacent elements are counter-rotating vortices. But, the induced motion from this pair will be opposite to the induced motions from the top and bottom of a single vortex ring. This leads to a slowing of the motion of both vortex rings, with the slowing effect greatest at their nearest approach. This effectively attracts and tilts the rings towards each other. Lab experiments have verified this, demonstrating, as well, that ring pairs merge as the near edges touch. Since the vorticity in each ring at their nearest points is opposite, the net vorticity there vanishes, leading to a ``vortex reconnection event'' \citep{Oshima77}. Thus, the pair of vortex rings created by shock passage in our present scenario evolves into a single vortex ring roughly spanning the full extent of both RG lobes.
Finally, we point out that the vortex ring structures under discussion, once formed, are essentially isolated from the AGN itself, unless they come in contact with active jets. (This does not actually happen in our one simulation with sustained jet activity, $\bf{M_s4J}$ in Table \ref{table:tab1}, although with somewhat different jet dynamics, it could). The presence or absence of this interaction obviously impacts the evolution of the jets and their behaviors as synchrotron sources.
\subsection{Jet Propagation in the Post Shock Crosswind}
\label{subsec:bending}
If the RG jets remain active through the shock encounter (true in one of our simulations, $\bf{M_s4J}$), the post shock wind in the geometry under investigation induces a ram pressure-based force across each jet ($\sim \rho_w v_w^2/r_j$, with $r_j$ the jet radius) that deflects the jet's trajectory transversely. \cite{ONeill19a} examined in some detail jet trajectories for arbitrary relative orientations between the undisturbed jets and winds. So long as the jets are internally supersonic, the trajectories of steady jets can be expressed over a broad range of initial orientations with respect to a cross-wind in terms of a characteristic bending length, $\ell_b$, derived decades ago in the context of so-called ``narrow angle tail'' RG (NATs); \citep[][]{BegelmanReesBlandford,JonesOwen79}. In our present context the relation is
\begin{equation}
\ell_b= \frac{\rho_jv_j^2}{\rho_wv_w^2}~r_j.
\label{eq:ellb}
\end{equation}
\cite{ONeill19a} showed that long term jet/tail trajectories in steady winds are well-described as swept back tails with transverse displacements from their launch points of several $\ell_b$. In our simulation $\bf{M_s4J}$ $\ell_b \approx 4 r_j\sim 12$ kpc. The $\sim 40$ kpc lateral displacements for the jets from their launch points visible at late times in figure \ref{fig:orth3-PS-RHO} are consistent with this simple model, since the actual jet trajectories tend to be wider than the simple $\ell_b$ metric \citep[e.g.,][]{ONeill19a}.
We note that, so long as these jet trajectories do not intersect the vortex ring, the jets have no significant dynamical influence on the vortex ring, nor do they feed CRe or magnetic flux into the ring. We also note that the response of jets to the transverse winds encountered in this study is quite distinct from the response of a jet to a head or tail wind, as in the \cite{nolting19a} study \citep[see, also][and references therein]{jones16}.
\section{Simulation Specifics}
\label{sec:methods}
\subsection{Numerical Methods}
\label{subsec:numerics}
The simulations reported here used the Eulerian WOMBAT ideal 3D nonrelativistic MHD code described in \cite{PeteThesis} on a uniform, Cartesian grid employing an adiabatic equation of state with $\gamma = 5/3$. The simulations utilized the 2$^{nd}$ order TVD algorithm with constrained transport (CT) magnetic field evolution as in \cite{Ryu98}. Specific simulation setups are introduced in \S \ref{subsec:Setup} and listed in Table \ref{table:tab1}. While the AGN-launched jets in our simulations were magnetized as outlined below, the undisturbed ICM media in the simulations presented here were unmagnetized, allowing us to focus more directly on AGN-associated behaviors.
Bipolar jets in the simulations were created beginning at $t = 0$ within a ``jet launch cylinder'' of radius, $r_j$ and length $l_j$ within which a plasma of uniform density, $\rho_j$, and gas pressure, $P_j$ (so sound speed, $a_j = \sqrt{\gamma P_j/\rho_j}$), was maintained. A toroidal magnetic field, $B_{\phi} = B_0 (r/r_j)\hat{\phi}$ was also maintained within the jet launch cylinder. A characteristic ``plasma $\beta$'' parameter for the jets, reflecting the relative dynamical role of the jet magnetic field, is $\beta_{pj} = 8\pi P_j/B_0^2 = 75$ in the jets considered in this work. Thus, the magnetic pressures are subdominant to the gas pressure at the jet source. Aligned jet flows emerged from each end of the launch cylinder with velocity, $v_j$, along the cylinder axis, so with internal Mach number $\mathcal{M}_j = v_j/a_j$. The jet velocity, $v_j$, also changed sign midway along the cylinder length producing the bipolar jet symmetry. The launch cylinder was surrounded by a 2 zone, coaxial collar, within which properties transitioned to local ambient conditions. Jets were steady until a simulation-dependent time, $t_{j,off}$, after which they were cycled off.
Passive cosmic ray electrons (CRe) were injected into the simulations within the launched jets to enable computation of synthetic radio synchrotron emission properties of the simulated objects\footnote{Except for a negligible ICM population included to avoid numerical singularities in the CRe transport algorithm, all CRe were injected onto the computational domain via the jet launch cylinder.}. The CRe momentum distribution, $f(p)$, was tracked using the conservative, Eulerian ``coarse grained momentum volume transport'' CGMV algorithm in \cite{JonesKang05}. $f(p)$ spanned the range $10 \la p/(m_e c)\approx \Gamma_e\la 1.7\times 10^5$ (so, energies 5 MeV $\la E_{CRe} \approx \Gamma_e m_e c^2 \la$ 90 GeV) with uniform logarithmic momentum bins, $1\le k\le 8$. Inside a given momentum bin, $k$, $f(p) \propto p^{-q_{k}}$, with $q_k$ being bin dependent and evolving in time and space. $\Gamma_e$ represents CRe Lorentz factors.
At injection from the AGN source (= the jet launch cylinder), the CRe momentum distribution was a power law with $q = q_0 = 4.2$, over the full momentum range. This translates into a synchrotron spectral index, $\alpha = \alpha_0 = 0.6$ ($I_{\nu} \propto \nu^{-\alpha})$ using the conventional synchrotron-CRe spectral relation for extended power laws. The synchrotron emission, including spectra, reported here are computed numerically using $f(p)$ over the full momentum range specified above along with the standard synchrotron emissivity kernel for isotropic electrons in a local vector magnetic field $\vec{B}$ \citep[e.g.,][]{BlumenthalGould70B}. For our analysis below we calculated synthetic synchrotron emission at frequencies $150$ MHz $\leq \nu$ $\la 1$GHz. This emission, as it turns out, comes predominantly from regions with magnetic field strengths $\sim 1 \rightarrow\rm{few}~\mu$G, so mostly reflect CRe energies $\ga$ a few GeV ($\Gamma_e \sim 10^4$) (well inside our distribution).
We included adiabatic, as well as radiative (synchrotron and inverse Compton) CRe energy changes outside of shocks, along with test-particle diffusive shock (re)acceleration (DSA) at any shocks encountered. We did not include $2^{nd}$ order turbulent CRe reacceleration or CRe energy losses from Coulomb collisions with ambient plasma. The former depends on uncertain kinetic scale turbulence behaviors beyond the scope of this study, while the latter is most relevant for CRe with energies well below those responsible for the radio synchrotron emission computed in this work \citep[e.g.,][]{nolting19a}. CRe radiative losses combine synchrotron with inverse Compton (iC) scattered CMB radiation. The simulations reported here assumed a redshift, $z = 0.2$. The resulting radiative lifetime can be written
\begin{equation}
\tau_{rad} \approx 110 \frac{1}{\Gamma_{e4}\left[1+ B_{4.7}^2\right]}~\rm{Myr},
\end{equation}
where $\Gamma_{e4} = \Gamma_e/10^4$ and $B_{4.7} = B/(4.7\mu\rm{G})$. The first term in the denominator on the RHS reflects inverse Compton (iC) losses at z = 0.2, while the second represents synchrotron losses. Thus, we can see that for $\Gamma_e \sim 10^4$, of primary interest for the radio emission in this work, $\tau_{rad} \sim 100$ Myr, and that iC losses are predominant.
DSA of the CRe was implemented at shock passage by setting $q_{k,out} = \min(q_{k,in},3\sigma /(\sigma - 1))$ immediately post-shock, where $\sigma$ is the code-evaluated compression ratio of the shock.
This simple treatment is appropriate in the CRe energy range covered, since likely DSA acceleration times to those energies are much less than a typical time step in the simulations ($\Delta t \ga 10^4$ yr). Since our CRe have no dynamical impact, we treat the total CRe number density, $n_{CRe}$, as arbitrary. Consequently, while we compute meaningful synchrotron brightness, polarization and spectral distributions from our simulations, synchrotron intensity normalizations are arbitrary.
\subsection{Simulation Setups}
\label{subsec:Setup}
For this study we carried out four 3D MHD simulations (labeled $\bf{M_s4J}$, $\bf{M_s4}$, $\bf{M_s4Ph}$ and $\bf{M_s2Ph}$) of plane ICM shock impacts on symmetric, double-lobed RG formed prior to shock impact by light, bipolar AGN jets within a homogeneous, unmagnetized medium (see Table \ref{table:tab1}). While both the homogeneity and the lack of fields are significant simplifications from real cluster environments, we make these choices to simplify the interpretation of the outcomes of our simulations. Homogeneity of the medium helps isolate the dynamical effects of the particular interactions under study, without the influence of buoyancy effects and other nonuniformities. The lack of magnetic fields except those introduced by the jets help us understand the synchrotron emission we observe and how the jet fields evolve, without having to worry about how the ICM fields are interacting with those in the jet or contribute to the synchrotron emission. Dynamical studies in realistic, magnetized clusters with pressure and density profiles and static gravitational potential (not present in these simulations) are important and left to future work.
In each simulation, the incident ICM shock was oriented with its normal orthogonal to the symmetry axis of the RG (so orthogonal to the axis of the AGN jets that made the RG). The incident shock either had Mach number, $\mathcal{M}_{si} = 4$, reflected in the simulation label as $\bf{M_s4}$, or, in one case, Mach number $\mathcal{M}_{si} = 2$, reflected in the label as $\bf{M_s2}$.
\begin{deluxetable}{ccccccccc}
\tabletypesize{\footnotesize}
\tablewidth{0pt}
\tablecaption{Simulation Specifics\label{table:jetparams}}
\tablehead{
\colhead{Run} & \colhead{$M_{si}$} & \colhead{$P_{w}/P_i$} & \colhead{$v_{w}$} &\colhead{$x_{domain}$} &\colhead{$y_{domain}$} & \colhead{$z_{domain}$} & \colhead{$x_{jc}$} & \colhead{$t_{j,off}$} \\
\colhead{} & \colhead{} & \colhead{} & \colhead{($10^3$ km/sec)} & \colhead{kpc} & \colhead{kpc} & \colhead{(kpc) } & \colhead{(kpc)} & \colhead{(Myr)}
}
\startdata
$\bf{M_s4}J$ & 4.0 & 19.8 & 1.88 & $\pm$ 320 & $\pm$ 240 & $\pm$ 240 & -57 & N/A\\
$\bf{M_s4}$ & 4.0 & 19.8& 1.88 & $\pm$ 320 & $\pm$ 240 & $\pm$ 240 & -57 & 32\\
$\bf{M_s4Ph}$ & 4.0 & 19.8 & 1.88 & $\pm$ 208 & $\pm$ 240& $\pm$ 240 & -32 & 16\\
$\bf{M_s2Ph}$ & 2.0 & 4.75 & 0.75 & $\pm$ 160 & $\pm$ 240& $\pm$ 240 & -16 & 16\\
\enddata
\tablecomments{All simulations had: $\rho_i = 5\times 10^{-27}~\rm{g/cm^3}$, $P_i = 1.33\times 10^{-11}~\rm{dyne/cm^2}$, $a_i = 6.7\times 10^2~\rm{km/sec}$, $\rho_j = 10^{-2}\rho_i$, $P_{ji} = P_i$, $a_{j} = 10 a_i$, $v_j = 6.7\times 10^4~\rm{km/sec}$, $\mathcal{M}_j = 10$, $B_0 = 2.1\mu$ G, $\beta_{pj} = (8\pi P_{j})/B_0^2 = 75$, $r_j = 3~\rm{kpc}$, $l_j = 12$ kpc. All simulations employed uniform spatial grids with $\Delta x = \Delta y = \Delta z = 0.5$ kpc}
\label{table:tab1}
\end{deluxetable}
In one simulation, ${\bf{M_s4J}}$, the AGN jets remained steady throughout the simulation in order to explore dynamical relationships between the shock-induced vortex ring structures and the jets as they become deflected in the post shock wind, as well as to compare the relative synchrotron evolutions of the two dynamical components of the shocked RG. In addition, this allows us to look for distinctions between jet behaviors in this orthogonal shock context and the simple, steady cross wind studied in \cite{ONeill19a}. Simulation ${\bf{M_s4}}$ was identical to ${\bf{M4J}}$ except AGN jet activity ceased shortly after the shock first came into contact with the RG lobes (so no $\bf{J}$ in the simulation label). Since the RG prior to the shock interaction is identical in simulations ${\bf{M_s4J}}$ and ${\bf{M_s4}}$ we can look explicitly at roles of the jets in the post shock evolution of ${\bf{M_s4J}}$.
The other two simulations, ${\bf{M_s4Ph}}$ and ${\bf{M_s2Ph}}$, designed to simulate so-called ``radio Phoenix'' sources \citep[e.g.,][]{Ensslin01,Kempner04} (motivating the $\bf{Ph}$ in their labels), deactivated the AGN jets 89 Myr prior to first shock contact with RG evolution continuing in the interim. Recall from \S \ref{subsec:numerics} that 110 Myr represents a rough timescale for radiative energy losses by radio bright CRe, so that the CRe populations in those two simulations are significantly aged at shock impact. The only significant difference between the ${\bf{M_s4Ph}}$ and ${\bf{M_s2Ph}}$ simulations is the strength of the incident ICM shock.
In all the simulations the shock normal is along the $\hat{x}$ axis, so that $\vec{v}_w = v_w \hat{x}$. The jet launch cylinder is aligned to the $\hat{y}$ axis, with the center of the launch cylinder at rest with coordinates ($x_{jc}, 0, 0$), so centered in the y-z plane. As already noted, all the simulations involve pre shock ICM conditions with $\rho_i = 5\times 10^{-27}~\rm{g/cm^3}$, $P_i = 1.33\times 10^{-11}~\rm{dyne/cm^2}$, and with jet properties at launch, $\rho_j = 10^{-2} \rho_i$, $P_j = P_i$. The jets all had internal Mach numbers at launch, $\mathcal{M}_{ji} = 10$, so $v_j = 6.7\times 10^4~\rm{km/sec}$.
Table \ref{table:tab1} provides a summary of remaining key properties of each simulation. The first four table columns list the simulation label, the strength of the incident shock, $\mathcal{M}_{si}$, the resulting pressure jump across the incident shock and the post shock wind velocity. The dimensions of the computational domain are listed for each simulation in columns, 5-7, while $x_{jc}$ for each AGN jet is given in column 8. The final column lists the time during the simulated events when the jet launching is cycled `off,' or deactivated, $t_{j,off}$. In simulations $\bf{M_s4J}$ and $\bf{M_s4}$, first shock contact with the RG lobes takes place at $t = 19$ Myr, while in simulations $\bf{M_s4Ph}$ and $\bf{M_s2Ph}$ first shock contact takes place at $t = 105$ Myr. Again, in both the $\bf{M_s4J}$ and $\bf{M_s4}$ simulations AGN activity was steadily building the RG until at least 13 Myr after the shock first contact, while in the $\bf{M_s4Ph}$ and $\bf{M_s2Ph}$ simulations, jet activity ceased 89 Myr before any shock contact, leaving the RG to evolve passively during that interval.
\section{Discussion}
\label{sec:Discussion}
We now examine and compare the four simulations from Tables \ref{table:tab1}. All four of the simulations involved AGN jets with Mach number $M_{ji} = 10$, jet mass density, $\rho_j = 10^{-2}\rho_i$, and the characteristic magnetic field strength, $B_0 = 2.1 \mu$G. Each simulation involved an external ICM shock running over the structures generated by the RG jet, with three of the simulations having an ICM shock of Mach $M_{si}=4$, and the $\bf{M_s2Ph}$ simulation having $M_{si}=2$.
The simulations divide into two ``pairs,'' based on their properties and the motivations behind them. The $\bf{M_s4J}$ and $\bf{M_s4}$ simulations both involve Mach 4 ICM shock impact on lobed RG that had active AGN input at least until shock impact. They differ in whether the AGN jet remained constant throughout the simulation or were deactivated during shock impact on the RG. This difference allows us to explore the influence of the post shock jet flows on both the dynamics and observable emission of the post shock RG. In both cases the AGN activity means that CRe in the interaction are relatively fresh up at least to the time of shock impact.
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth3-PS-RHO-withlabels.pdf}
\caption{Volume renderings of the $\bf{M_s4J}$ at four times increasing top to bottom. The shock normal and jet axis are in the viewing plane. Shock impact on the RG begins soon after the top snapshot. Left: Jet mass fraction ($>30$\% visible). The location of the shock is outlined in dashed gray lines; Right: Log mass density spanning 3 decades in $\rho$, with key dynamical structures highlighted, including the ICM shock. Colors in all images follow the "CubeYF" colormap with "yellow" high and "purple" low. Images are rendered from a distance of 857 kpc from the RG.}
\label{fig:orth3-PS-RHO}
\end{figure*}
In contrast, the $\bf{M_s4Ph}$ and $\bf{M_s2Ph}$ simulations begin with a relatively short period of AGN jet activity (16 Myr), but, then the AGN jets deactivate and the RG lobe plasma is allowed to relax for 89 Myr before a shock impact. Of course, the CRe inside the RG lobes cool radiatively (and to a small degree adiabatically) in the interim. The intent was to investigate the ``radio phoenix'' scenario, in which fossil plasma from expired AGN is reactivated via ICM shocks. These two simulations differ only in the strength of the ICM shock incident on the lobe, with $M_{si}=4$ in the former case and $M_{si}=2$ in the later. This work extends the early simulation study of this scenario by \cite{EnsslinBruggen02}. There are two possibly significant distinctions in our approach, although both studies involved 3D MHD simulations of shock impact on low density cavities containing fossil CRe. The first difference is that in our simulations, the cavities formed dynamically in response to AGN jets, whereas \cite{EnsslinBruggen02} initialized their simulation with a static, spherical and uniform cavity with a discontinuous boundary. Dynamical cavities do not have uniform, static interiors, nor simple boundaries, even after substantial relaxation. This can, for example, influence the stability of the cavity boundary during shock passage, and, so impact expected vortex structures. The second distinction in the two simulation studies is that, while both followed evolution of passive CRe populations, our simulations allowed for the possibility of DSA, whereas \cite{EnsslinBruggen02} assumed it was absent. As it turns out neither of these distinctions is very significant, so that our results largely support the radio phoenix simulation results of \cite{EnsslinBruggen02}.
\subsection{Simulation $\bf{M_s4J}$: $M_{si} = 4.0$, $t_{j,off} =$N/A}
\label{subsec:Orth3}
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth3-synch-index.pdf}
\caption{Synchrotron images from $\bf{M_s4J}$ at the times in Figure \ref{fig:orth3-PS-RHO}. Resolution is 0.5 kpc. The AGN jet axis and shock normal are in the plane of the sky. Left: Linearly plotted 150 MHz intensity with arbitrary units. Right: 150/600 MHz spectral index, $\alpha_{150/600}$, for regions above 0.1\% of the peak intensity at 150 MHz. Spectral index scale is on the far right. At launch the jet synchrotron spectral index was $\alpha=0.6$. The location of the shock is outlined in dashed gray lines}
\label{fig:orth3-synch-index}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth3-BmagmG.pdf}
\caption{Volume renderings of the magnitude of the magnetic field in the $\bf{M_s4J}$ simulation at two of the times from figure \ref{fig:orth3-PS-RHO}, rendered from the same view point and orientation. The location of the shock is outlined in dashed gray lines.}
\label{fig:orth3-bmag}
\end{figure*}
The dynamical evolution of the $\bf{M_s4J}$ shock--RG interaction is shown in figure \ref{fig:orth3-PS-RHO}. The figure presents four snapshots of the volume-rendered\footnote{As viewed along the $\hat{z}$ axis at a distance roughly 857 kpc from the AGN.} jet mass fraction tracer (left panels) and logarithmic mass density (right panels) at: (1) $t =19$ Myr, just prior to RG--shock first contact (refer to Figure \ref{fig:orth-setup} for the geometry); (2) $t = 38$ Myr, after the cocoon has been shocked and the post-shock flow has begun to bend the jets; (3) $t = 104$ Myr, after the bent jets have penetrated through the vortex ring; and (4) $t = 202$ Myr, after the vortex ring had pulled inward toward the midplane and was mostly hidden by the jets and nascent NAT tails. In figure \ref{fig:orth3-PS-RHO} and all subsequent volume-renderings, the location of the shock in the ICM is outlined in dashed gray lines. We note that jet material leaving the AGN source after $t=32$ Myr, when the ICM shock passed the location of the jet source, is bent by the post shock wind and does not directly ``know'' about the ICM shock. The resulting NAT morphology does not explicitly require a shock, but is purely a result of the relative motion between the jet source and the medium. On the other hand, we point out below that the bent jets and the radio tails they produce ultimately reach the shock from downwind and modify it.
At $t=38$ Myr, the shock has propagated through the lobes of the RG. The jets have been obviously bent downwind by post shock ram pressure and are beginning to form what will become the tails of the future NAT. The previously planar shock has been modified during its passage through the RG lobes. In particular it has advanced ahead of the external, ICM shock in sections where it has intersected the low-density, high-sound-speed cocoon. Also visible at $t=38$ Myr is the beginning of the vortex ring structure formed from the remnants of the shocked cocoon material. Immediately after shock impact, it is still two distinct vortex rings originating from the two separate cocoons, with a small separation at the midpoint between the two remnant cocoons. The rings are elongated in the vertical direction because they trace the boundaries of the elongated cocoons prior to the shock impact. By $t=104$ Myr, the two parallel vortex rings have merged, as described in section \S \ref{subsec:VortRings}. The single ring structure is more apparent when rotated out of the plane of the sky, as in the left panel of figure \ref{fig:orth34-PS-rot60}.
Also by $t=104$ Myr, the jets have been bent completely downwind by the wind and a NAT structure has formed. In this construction we can roughly identify both jets, as coherent flows, and associated ``tails'', as somewhat more diffuse, blended flows with motions more or less aligned with the jets \citep[e.g.,][]{ONeill19a}. The tails, with embedded jets, can be seen to be passing through the vortex ring and advancing farther downwind. The impingement of the tail/jet structures on the shock from behind occurs because the downwind velocity of the tail plasma is actually greater than the post-shock wind speed. The vortex ring also advances downwind as a result of self-induction, as outlined previously, although the upwind advancement is less rapid than for the tails.
The downwind penetration by the tails, also pointed in the context of more traditional NAT formation by \cite{ONeill19a}, comes about quite simply as a result of the dynamics of tail formation. The physics is particularly straightforward when, as in this case, the launched jet velocities are orthogonal to the wind velocity. Then all of the downwind momentum in the deflected jets is necessarily extracted from the post shock wind. The tails include a mix of post shock ICM and jet plasma, so, again, all of their downwind momentum came from the post shock wind. Because mass densities in the tails are generally significantly less than in the post shock wind (see figure \ref{fig:orth3-PS-RHO}), the concentration of momentum flux in the tails leads to their enhanced velocities with respect to the wind. As long as a shock propagating into a medium at rest has Mach number $M_{s,i} \gtrsim 1.87$, the post-shock wind speed will be supersonic with respect to the pre-shock ICM sound speed. Therefore, as just noted, since the tails advance faster than the post-shock wind, they can overtake the external shock. In that case their progress could create effective bow shocks in advance of the external, ICM shock. By $t=104$ Myr this has occurred in the $\bf{M_s4J}$ simulation, and the visible shock surface in figure \ref{fig:orth3-PS-RHO} is a combination of the ICM shock and the bow shock from the tails.
By $t=202$ Myr, the vortex ring has pulled inward nearer to the jets, becoming difficult to distinguish in the renderings of jet mass fraction and density. The large curvature of the vortex structure near the top and bottom of the ring causes those locations to lead the rest of the ring slightly, in response to the increased induced velocity at that point (see equation \ref{eq:inducedVel}). This alters the direction of propagation of this section of the ring, adding a component in the direction toward the midplane between the jets, causing the ring to shrink in vertical size.
Radio synchrotron images with 0.5 kpc resolution are shown in figure \ref{fig:orth3-synch-index} at the same times as in figure \ref{fig:orth3-PS-RHO}. The AGN jets and the ICM shock normal are in the plane of the sky. Each image is constructed from integrated synchrotron emissivities along the line of sight. The left panels show the synchrotron brightness (arbitrary units) at 150MHz. The right panels show the radio spectral index, $\alpha_{150/600}$, with an intensity cut such that the image includes only pixels where the intensity at 150MHz is above 0.1\% of the peak intensity at 150 MHz at that time. As before, the location of the shock in the ICM is indicated by a dashed gray line. At $t=38$ Myr, the shock interaction causes brightening in the lobes as they are compressed, energizing the CRe and enhancing the magnetic field strength. Figure \ref{fig:orth3-bmag} shows volume renderings of the magnetic field strength from the $\bf{M_s4J}$ simulation at two times after the shock has impacted the RG. As a result of the shock impact, the magnetic fields in the remnant shocked lobes are compressed and amplified. This is greatest in the regions where the still active jets interact with the magnetic fields originally in the remnant lobes, relatively near the midpoint between the two.
By $t=104$ Myr, when the bending in the jets is well established, the magnetic field adjacent the jet launching cylinder is distorted by the shear associated with the post shock wind, amplifying the field and making it predominantly poloidal with respect to the jet axis. At launch the jets' magnetic field was purely toroidal. Also at $t=104$ Myr, the region where the two vortex rings converge into one ring shows a significant enhancement in the magnetic fields.
All of these regions of enhanced magnetic field strength show up significantly in the synchrotron images in figure \ref{fig:orth3-synch-index}. Indeed, the sensitivity of synchrotron emissivity to magnetic field is obvious in a comparison between the field strengths in figure \ref{fig:orth3-bmag} and the radio bright regions in figure \ref{fig:orth3-synch-index}.
By $t=104$ Myr, it becomes very difficult to see the vortex ring structure in the radio intensity images in contrast to the tails. There are two main reasons for this: first, CRe population contained in the vortex ring was deposited in the lobes prior to the shock impact, so it is an older population and has experienced substantially more cooling from inverse Compton and synchrotron losses. Second, as can be seen in figure \ref{fig:orth3-bmag}, the magnetic fields in the ring are generally weaker than in the tails. Overall, this means that in the presence of active jets, the emission from a vortex ring structure containing shocked lobe material will be subdominant, and the timescale over which the ring may be visible will be dependent on the cooling rate of the CRe.
In addition to the timescale for cooling being a limiting factor for the duration of vortex ring visibility, over time the dynamical evolution of the ring may also limit its visibility. In the $\bf{M_s4J}$ simulation, after the two vortex rings from the two lobes had merged, the resulting ring was highly elongated in the vertical direction. This more elliptical ring structure had high curvature at the top and bottom of the ring, resulting in higher self induced velocities at those points. This caused those parts of the ring to move forward downwind ahead of the rest of the ring, altered the geometry of the ring, and as a result, changed the direction of the induced velocity at those points to have a component toward the midplane. The end result is that the vertical extent of the ring decreases as it propagates. In our simulation, this limited the observability of the ring because the significantly radio brighter RG tails occupied the region interior to the ring, so as it decreased in vertical extent, it began to occupy the same region as the tails in projection, and became hidden. This dynamical situation is likely to occur in any elongated vortex ring, and if two vortex rings (from a pair of RG lobes) merge, they are likely to be elongated along the direction connecting the two previous ring centers. Whether or not the rings become hidden as they `shrink' will depend on the presence and detailed dynamics of any RG jets/tails.
The evolution of the CRe populations can also be seen in the spectral index images on the right of figure \ref{fig:orth3-synch-index}. At launch, the CRe in the jet have power law momentum spectra with $q_0 = 4.2$, so that the jet synchrotron spectrum is a power law with $\alpha \sim \alpha_0 = 0.6$. Some lobe material dominated by early jet activity displays slightly ``aged'', steeper spectra by the time of shock impact. In response to the shock passage, adiabatic compression energizes CRe and enhances field strength. Since the synchrotron intensity images are made at fixed frequency, the post shock emission comes from CRe that were previously lower energy, so that their radiative lifetimes were long compared to the expired time. Consequently, at this relatively early time, $t = 38$ Myr, there is little apparent spectral steepening in those populations. In contrast, at $t=104$ Myr, which now involves $t \sim \tau_{rad}$ for CRe of primary interest, the portion of the ring still bright enough to show up in the image has steepened to a spectral index, $\alpha \sim 1.0$. This is significantly steeper than the emission from the jet tails in the same region, since the latter contain plasma that only recently was launched in the jets. By $t=202$ Myr the vortex ring is no longer visible in the spectral index image, because, largely in response to radiative aging, the intensities used in determining $\alpha$ have fallen below the applied intensity cuts. Spectra displayed in the tails can be seen to steepen to $\alpha \gtrsim 1.4$ over a distance from the source of $\sim 400$ kpc. Those end tail portions represent CRe deposited largely during and soon after shock impact, so that $t \ga \tau_{rad}$ over much of the relevant CRe energy range.
While in this paper, we specifically did not set out to model any individual sources, but rather learn about the physics of a class of physical interactions in clusters, there are cases which bear resemblance to the radio images we produced of our simulations. One striking example worth noting here is the so-called ``Coma relic'' \citep[see, e.g.][]{Giovannini91}, in which radio galaxy jets are bent into a NAT which forms disrupted tails which lead to a bright steep spectrum feature transverse to the tails. This similarity in structure to the $\bf{M_s4J}$ case (see figure \ref{fig:orth3-synch-index}) could imply a similar dynamical origin. However, the nature of the shock associated with the Coma relic is a matter of ongoing investigation.
\subsection{Simulation $\bf{M_s4}$: $M_{si} = 4.0$, $t_{j,off} =32$ Myr}
\label{subsec:Orth4}
Figure \ref{fig:orth4-PS-RHO} shows volume renderings of the jet mass fraction (left) and the logarithmic mass density (right) from the $\bf{M_s4}$ simulation at times $t=38$ Myr and $t=104$ Myr. The $\bf{M_s4}$ simulation began as a restart of the $\bf{M_s4J}$ simulation from time $t=22$ Myr, but deactivated the AGN jet at $t=32$ Myr, approximately when the shock reached the jet launch cylinder. This distinction from $\bf{M_s}4J$, makes clearer the level of jet influence on evolution of the vortex rings and the shock front after its encounter with the RG, while also illuminating the role of fresh CRe injection by the jets as the dynamical structures evolve.
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth4-PS-RHO.pdf}
\caption{Volume renderings of the $\bf{M_s4}$ at two of the times from figure \ref{fig:orth3-PS-RHO}. The shock normal and jet axis are in the viewing plane. The jets deactivated shortly before the top snapshot. Left: Jet mass fraction ($>30$\% visible) with the location of the shock outlined in dashed gray lines; Right: Log mass density spanning 3 decades in $\rho$, with key dynamical structures highlighted, including the ICM shock. Images are rendered from a distance of 857 kpc from the RG.}
\label{fig:orth4-PS-RHO}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth4-synch.pdf}
\caption{Synchrotron images from $\bf{M_s4}$ at the times in Figure \ref{fig:orth4-PS-RHO}. Resolution is 0.5 kpc. The AGN jet axis and shock normal are in the plane of the sky. Left: Linearly plotted 150 MHz intensity with arbitrary units. Right: 150/600 MHz spectral index, $\alpha_{150/600}$, for regions above 0.5\% of the peak intensity at 150 MHz. Spectral index scale on the far right. At launch the jet spectral index was $\alpha=0.6$}
\label{fig:orth4-synch}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth3-4-0020-PS-rot60.pdf}
\caption{Volume renderings of the jet mass fraction from (Left) $\bf{M_s4J}$ and (Right) $\bf{M_s4}$ at 104 Myr. The view is rotated around the vertical axis by 60 degrees so the shock propagates into the page in order to highlight the ``ring'' structures produced. Images are rendered from a distance of 348 kpc from the RG.}
\label{fig:orth34-PS-rot60}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth34-Spectra.pdf}
\caption{Integrated spectral evolution of the $\bf{M_s4J}$ (left) and $\bf{M_s4}$ (right) simulations, in arbitrary flux units. Reference slopes of $\alpha=0.6$ and $\alpha=1.0$ are included. Shock impacts on the RGs begin at $t \sim 20$ Myr}
\label{fig:orth34-spectra}
\end{figure*}
As in the $\bf{M_s4J}$ simulation, the shock propagates relatively quickly through the low density cavity, moving ahead of the shock in the external medium. However, by time $t=104$ Myr, deviation from shock planarity has diminished significantly, in contrast to the behavior in the $\bf{M_s4J}$ simulation. This reinforces our conclusion that the significant deviations from shock planarity in the $\bf{M_s4J}$ simulation at this same time are more the result of added downwind momentum by the jet interacting with the shock than simply from the shock's interaction with the initial cavity. At time $t=104$ Myr in the $\bf{M_s4}$ simulation, a lower density (relative to the post-shock wind density) ``wake'' formed behind the jet launching cylinder, which for numerical reasons remained impenetrable, and connects to the vortex ring. The vortex ring itself formed in much the same way as in the $\bf{M_s4J}$ simulation. The cocoon (lobe) plasma became wrapped up into the shock-induced vortex rings developing along the peripheries of the cavities. The vortex ring then advanced at the same rate as in the $\bf{M_s4J}$ simulation. Based on this, we conclude that the vortex rings in the two simulations evolve mostly independent of the presence or absence of jets. This is due at least in part to the fact that the jets in this simulation are deflected into the interiors of the vortex rings, rather, than, for instance into the ring perimeters. Figure \ref{fig:orth34-PS-rot60} shows at time $t = 104$ Myr volume renderings illustrating the relationship between the jets and the vortex ring in the $\bf{M_s4J}$ simulation and the comparative vortex ring structure in the absence of the jets.
In figure \ref{fig:orth4-synch}, the synchrotron emission structure in the ring is visible. After the shock impact, at time $t=38$ Myr, the radio emissivity in the shocked cocoon was again enhanced as the CRe were energized and the fields amplified by compression. At time $t=104$ Myr, the visible parts of the ring are dominated by filamentary emission originating in magnetic flux tubes. The initially toroidal field topology that was dominant in the jet and in the cocoon prior to the shock interaction is stretched and folded into itself. As the vortex formed, the field was wrapped up around the vortex over an eddy time (the time it takes for the fluid to circle around the vortex core, $\sim 75$ Myr in this case). This structure cannot be seen in the $\bf{M_s4J}$ synchrotron images, because the emission from the tails dominate the vortex ring. This is because the tails are continuously refreshed with new CRe populations from the jet. Consequently, the tails generally contain younger CRe populations than those in the vortex ring. The latter is composed of aged CRe that filled the lobes before the shock interaction.
Additionally, more structure from the vortex ring can be seen in the spectral index maps on the right of figure \ref{fig:orth4-synch}, since the bright tails are absent. At $t=38$ Myr, the compressed material is again mostly near the injection index of $\alpha_0 = 0.6$, but near the midpoint between the lobes, the spectrum is steeper than in figure \ref{fig:orth3-synch-index}, since there are no jets to inject fresh CRe into this region. At $t=104$ Myr, the spectral index ranges over $0.7<\alpha < 1.4$, with much of the emission showing $\alpha \sim 1.0$. The brightest emission comes from those regions with higher field strength. Those regions generally produce emission with a flattened spectrum, because the higher fields imply the emission comes from lower energy CRe that have experienced less radiative cooling.
Figure \ref{fig:orth34-spectra} provides a summary of the spectral evolution of the integrated emission for both the $\bf{M_s4J}$ and $\bf{M_s4}$ simulations. The properties of both simulations are very similar at the two earliest times shown. However, at later times the intensities are greater and the spectra flatter with less curvature in the $\bf{M_s4J}$ simulation, reflecting the continued input of energy and CRe by the jets.
\subsection{Simulations $\bf{M_s4Ph}$: $M_{si} = 4.0$, $t_{j,off} =$16 Myr\\ and $\bf{M_s2Ph}$: $M_{si} = 2.0$, $t_{j,off} =$16 Myr}
\label{subsec:Orth56}
Each of the simulations in this pair began with a Mach 10 jet pair that was on for 16 Myr before deactivating. That activity inflated RG lobes, which resembled the early stages of those in the other simulated RGs, so similar to what is seen in the top panels of figures \ref{fig:orth3-PS-RHO} and \ref{fig:orth3-synch-index}. After jet energy input ceased the lobes relaxed towards pressure equilibrium with the ICM. From jet deactivation to shock impact about 89 Myr later, the cocoons were dynamically relatively quiet, although their bases did merge the structure into a single, connected, cocoon. (There was no buoyancy in this ICM, so the detached lobes did not move away from their source.) On the other hand, in the almost 90 Myr after jet inflow ceased, but before shock impact, the CRe in the cavities cooled significantly via radiation losses during that time. Those losses were dominated by inverse Compton scattering, so the cooling rate was almost constant. Had a gravitational potential been included, and thus buoyant effects been in play, adiabatic losses (as the lobes detached, rose, and expanded) would have contributed more substantially.
Of course, from shock impact forward, the evolution of both RG was dramatic. The principal distinction between the two simulation was the strength of the impacting ICM shock. In the $\bf{M_s4Ph}$ simulation the shock was Mach 4, while in the $\bf{M_s2Ph}$ the shock was Mach 2. Post shock dynamical evolution of the $\bf{M_s4Ph}$ simulation can be seen through volume renderings in figure \ref{fig:orth5-ps-rho}, with the jet mass fraction on the left, and the logarithmic mass density on the right. At $t=105$ Myr (slightly after the top panels in Figure \ref{fig:orth5-ps-rho}), the merged cocoon was impacted by the shock. At $t=230$ Myr, the expected vortex ring formed from the shocked cocoon can be observed. However, the jet mass fraction in the vortex is low ($\la 30\%$) due to substantial entrainment of ICM material.
The radio observable consequences of the shock interaction can be seen in figures \ref{fig:orth5-synch} and \ref{fig:orth56-spectra}. Prior to the shock impact, the radio emission at 150 MHz had faded dramatically due to the mentioned radiative cooling that makes this case into a radio phoenix scenario. Because this dimming is substantial, we display the radio intensity on a logarithmic scale spanning 3 decades in brightness, to better reveal the presence of the structures. After the shock passage, the brightness is substantially increased by adiabatic compression of the CRe as well as increased field strength in the cocoon. The radio spectrum also flattens because adiabatic CRe re-energization and magnetic field enhancement cause the emission in the observed band to be dominated by CRe previously at energies too low to radiate in this band, but also low enough to reduce their radiative losses (see the right panels of figure \ref{fig:orth5-synch}). Even 125 Myr after the shock impact there are regions of flatter emission ($\alpha_{150/600}\sim1.0$) than the situation immediately prior to the shock, when most of the cocoon exhibited spectral indices, $\alpha_{150/600}\sim 1.3$, with substantially steeper spectra at higher frequencies. This is also evident in the integrated spectra in figure \ref{fig:orth56-spectra}. The right panel shows the evolution of the $\bf{M_s4Ph}$ simulation, including the spectrum just before the jet is deactivated ($t = 13$ Myr) and at a time shortly after the shock has fully compressed the cocoon ($t = 164$ Myr). In the case of $\bf{M_s4Ph}$ the shock crossing time is about 25 Myr and ends around $t\sim 130$ Myr). The left panel shows for comparison the $\bf{M_s2Ph}$ spectral evolution with the weaker, $\mathcal{M}_s = 2$, shock . In this $\bf{M_s2Ph}$ case, the shock takes $\sim60$ Myr to fully compress the cocoon ($t \sim 165$ Myr). In both cases there is substantial brightening and flattening of the spectra following the shock interaction. This results mostly from the increase in magnetic field strength and adiabatic compression of the CRe, and not from any DSA, however. We examined the CRe momentum distributions directly and saw no evidence of flattening in the CRe spectra associated with DSA. Also, the radio spectra on the right in figure \ref{fig:orth56-spectra} for the $\bf{M_s4Ph}$ simulation is consistent with pure adiabatic compression of $10\pm2$\%. This is consistent with our observation that within the RG cocoons the shock strength is significantly reduced. Due to the lack of significant mixing between the ICM and the RG plasma prior to the shock impact, the cocoon is relatively homogeneous and the density is about 50-100 times less dense than the ICM. This leads to the shock becoming almost sonic with $M_s\gtrsim 1$. There are, however, some regions with $M_s\sim 2$ as it passes through the cocoon. As mentioned earlier, the results of the $\bf{M_s4Ph}$ and $\bf{M_s2Ph}$ simulations are consistent with analogous findings reported by \cite{EnsslinBruggen02}.
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth5-PS-RHO-2panel.pdf}
\caption{Volume renderings of the $\bf{M_s4Ph}$ at 98 Myr (top, right before the shock interaction) and 230 Myr (bottom). The shock normal and jet axis are in the viewing plane. Left: Jet mass fraction ($>30$\% visible) with the location of the shock outlined in dashed gray lines; Right: Log mass density spanning 3 decades in $\rho$, with key dynamical structures highlighted, including the ICM shock. Images are rendered from a distance of 410 kpc from the RG.}
\label{fig:orth5-ps-rho}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth5-LogSynch-Index-2Panel.pdf}
\caption{Synchrotron images from $\bf{M_s4Ph}$ at the times in Figure \ref{fig:orth5-ps-rho}. Resolution is 0.5 kpc. The AGN jet axis and shock normal are in the plane of the sky. Left: Logarithmic 150 MHz intensity spanning 3 decades in brightness. Right: 150/600 MHz spectral index, $\alpha_{150/600}$, for regions above 0.5\% of the peak intensity at 150 MHz. At both times, the shock is just out of the field of view, to the left(right) at $t=98(230)$ Myr. Spectral index scale on the far right. At launch the jet spectral index was $\alpha=0.6$}
\label{fig:orth5-synch}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.75\textwidth]{Orth56-spectra.pdf}
\caption{Integrated spectral evolution of the ``radio phoenix'' simulations in arbitrary flux units. Left: $\bf{M_s2Ph}$. Right: $\bf{M_s4Ph}$. In both simulations jet activity ceased at $t = 13$ Myr, while first shock contact was at $t = 112$ Myr. In each plot, the black line represents the time at which the shock has fully compressed the aged RG cocoon}
\label{fig:orth56-spectra}
\end{figure*}
\section{Summary}
\label{sec:Summary}
We have reported a 3D MHD study of the interactions between lobed radio galaxies initially at rest in a homogeneous ICM and plane ICM-strength shocks when the radio galaxy jet axis is orthogonal to the incident shock normal. These simulations included cases in which the radio jets remained active throughout the simulation, cases in which jet activity terminated during the interaction and cases in which the jet activity had ceased long enough before the shock impact, so to allow embedded relativistic electron populations to ``age'' radiatively before the encounter. This last case is designed as a probe of the so-called ``radio phoenix'' scenario that illuminates non-luminous fossil relativistic electron populations through shock encounters.
As in previous studies, these shocks, as they encounter low density RG lobes, propagate very rapidly through the lobes relative to the surroundings. This generates strong shear along the boundary between the lobes and the surrounding ICM. That causes each lobe to form a vortex ring in the shape of the projected cross section of the lobe from the perspective of the incident shock. Such vortex ring formation is the principal obvious signature of the shock encounter. In the cases studied here, where two similar lobes are impacted simultaneously by a shock, two co-planar rings form simultaneously. Due to their mutual induced motions, those two rings merge into a single ring as they propagate downwind behind the shock. The merged elongated rings acquire a velocity component toward the midplane through self induction at the high curvature top and bottom of the elongated ring as they propagate. In our simulations, this caused the ring to become hidden by the bright RG jets/tails as they began to overlap in projection.
If RG jets remain active following such a shock encounter, they are deflected by ram pressure from post-shock winds and form tails propagating downwind towards the shock. These tails extend downwind faster than the wind, even overtaking the shock. This can noticeably deform the shock surface.
Our simulations included the evolution of relativistic electrons introduced by the AGN jets, accounting for adiabatic, radiative and diffusive shock acceleration physics. From those results we computed synchrotron intensities and spectra, Because the shock strengths are strongly depressed inside the radio lobes, diffusive shock acceleration is not very important. On the other hand, as suggested in other studies, adiabatic compression of the relativistic electrons and amplification of magnetic fields during the shock encounter and lead to substantially enhanced synchrotron brightness, as well as spectral flattening and straightening. When the radio jets remain active we found that, because their relativistic electron populations are characteristically less aged, their emission mostly dominated emission from the remnants of the pre-impact radio galaxy. Our simulations of shock encounters with previously extinguished radio galaxy lobes produce results that are consistent with earlier studies of this scenario.
\acknowledgements
This work was supported at the University of Minnesota by NSF grant AST1714205 and by the Minnesota Supercomputing Institute. CN was supported by an NSF Graduate Fellowship under Grant 00039202 as well as with a travel grant through the School of Physics and Astronomy at the University of Minnesota. We thank numerous colleagues, but especially Larry Rudnick and Avery F. Garon for encouragement and feedback.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 742 |
from forms.haiku import HaikuForm
from forms.because import BecauseForm
from forms.acrostic import AcrosticForm
from forms.atoz import AtozForm
from forms.rhymingcouplet import RhymingCoupletForm
from forms.iambicpentameter import IambicPentameterForm
from forms.limerick import LimerickForm
from forms.alliterative import AlliterativeForm
from forms.ihate import IHateForm
from forms.roses import RosesForm
from forms.tanka import TankaForm
from forms.iambicpentameter_strict import IambicPentameterStrictForm
from forms.markov import MarkovForm
from forms.markov2 import Markov2Form
from forms.markovsounds import MarkovSoundsForm
from forms.all import AllForm
from forms.generate_dataset import GenerateDatasetForm
from forms.speedtest import SpeedtestForm
poem_forms = ["haiku","because","acrostic","atoz","couplet",
"iambicpentameter","limerick","alliterative", "ihate",
"roses","tanka","iambicpentameter_strict"]
tool_forms = ["all","generate_dataset","speedtest"]
other_forms = ["markov","markov2","markovsounds"]
def getForm(form):
## POEMS
if form==poem_forms[0]:
return HaikuForm()
elif form==poem_forms[1]:
return BecauseForm()
elif form==poem_forms[2]:
return AcrosticForm()
elif form==poem_forms[3]:
return AtozForm()
elif form==poem_forms[4]:
return RhymingCoupletForm()
elif form=="rhymingcouplet":
return RhymingCoupletForm()
elif form==poem_forms[5]:
return IambicPentameterForm()
elif form==poem_forms[6]:
return LimerickForm()
elif form==poem_forms[7]:
return AlliterativeForm()
elif form==poem_forms[8]:
return IHateForm()
elif form==poem_forms[9]:
return RosesForm()
elif form==poem_forms[10]:
return TankaForm()
elif form==poem_forms[11]:
return IambicPentameterStrictForm()
## TOOLS
elif form==tool_forms[0]:
return AllForm()
elif form==tool_forms[1]:
return GenerateDatasetForm()
elif form==tool_forms[2]:
return SpeedtestForm()
## OTHER THINGS
elif form==other_forms[0]:
return MarkovForm()
elif form==other_forms[1]:
return Markov2Form()
elif form==other_forms[2]:
return MarkovSoundsForm()
else:
return None
| {
"redpajama_set_name": "RedPajamaGithub"
} | 850 |
#pragma once
#include "Base/BaseTypes.h"
#include "FileSystem/XMLParserStatus.h"
namespace DAVA
{
class FilePath;
class XMLParserDelegate;
class XMLParser
{
public:
/** Parse xml data from specified file and delegate and return error information. */
static XMLParserStatus ParseFileEx(const FilePath& fileName, XMLParserDelegate* delegate);
/** Parse xml data from specified buffer and delegate and return error information. */
static XMLParserStatus ParseBytesEx(const char8* bytes, int32 length, XMLParserDelegate* delegate);
/** Parse xml data from specified string and delegate and return error information. */
static XMLParserStatus ParseStringEx(const String& str, XMLParserDelegate* delegate);
DAVA_DEPRECATED(static bool ParseFile(const FilePath& fileName, XMLParserDelegate* delegate));
DAVA_DEPRECATED(static bool ParseBytes(const unsigned char* bytes, int length, XMLParserDelegate* delegate));
};
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,336 |
from pybrain.supervised.trainers import BackpropTrainer
from pybrain.tools.shortcuts import buildNetwork
from pybrain.structure import TanhLayer
from pybrain.structure import LinearLayer
from pybrain.structure import SigmoidLayer
from pybrain.datasets import SupervisedDataSet
import matplotlib.pyplot as plt
from Tkinter import *
import numpy as np
n = buildNetwork(10,15,1, bias=True, hiddenclass=TanhLayer)
d = SupervisedDataSet(10,1)
X = np.random.randint(0, 2, (200, 10)).astype(np.float64)
def MSP(X):
weight = np.array([2./10.5, 2./10.5, 2./10.5, 1./10.5, 1./10.5, 0.5/10.5, 0.5/10.5, 0.5/10.5, 0.5/10.5, 0.5/10.5])
return np.dot(X, weight)
y = MSP(X)
for i in xrange(0,X.shape[0]):
d.addSample(X[i,:], y[i])
tol_max = 1e-3
max_iter = 200
trainer = BackpropTrainer(n, d, learningrate = 1e-3, momentum=0.9)
erroDpc = []
iter_t = 0
while max_iter>0:
tol = trainer.train()
erroDpc.append(tol)
max_iter -= 1
print 'erro: ', tol
if tol<=tol_max:
break
iter_t += 1
print 'Responda com 1 para sim e 0 para não'
print iter_t
q = np.zeros((1,10))
q[0,0] = int(raw_input('Você sente sede excessiva com frequência ?'))
q[0,1] = int(raw_input('Urina em grandes quantidades, cerca de 3,0 L por dia ?'))
q[0,2] = int(raw_input('Você tem reparado que está perdendo muito peso ultimamente ?'))
q[0,3] = int(raw_input('Você sente uma necessidade anormal de ingerir alimentos ?'))
q[0,4] = int(raw_input('Sua visão tem ficado turva ultimamente ?'))
q[0,5] = int(raw_input('Tem sentido tontura ultimamente ?'))
q[0,6] = int(raw_input('Tem se sentido muito cansaço ou fraqueza com frequência ?'))
q[0,7] = int(raw_input('Tem se sentido muitas dores de cabeça ?'))
q[0,8] = int(raw_input('Você fica doente com muita frequência ?'))
q[0,9] = int(raw_input('Você tem tido lesões nos membros inferiores que demoram a cicatrizar ?'))
print" Chance real", MSP(q)
print"predito", n.activate(q[0,:])
plt.plot(erroDpc)
plt.xlabel('Geracao')
plt.ylabel('Erro')
plt.title('Decaimento do erro')
plt.show() | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,160 |
While not as good as the Rapala Touch Screen Tourney Scale, this scale is smaller, making it a good option for storing in tackle bags and backpacks. It stores up to 8 fish weights, and has a very nice non puncturing grip for securely weighing fish.
After catching and weighing some bigger bass a few months ago, I realized it was time for a new digital scale. Not long after, I was scrolling Facebook when I saw a Pro bass fisherman with the new Rapala High Contrast Digital Scale. The scale had all the features of their popular handheld Tournament Scale, plus a fancy no-puncture lip grip. It looked sweet so I bought one.
Inverse Display – Rather than the typical black digits on grey background, Rapala implemented a black screen with light digits. The purpose is to make the numbers easier to see in harsh direct sunlight. This works well with polarized fishing sunglasses.
8 Weight Slots – Anglers can store up to 8 fish weights at a time. Weigh and save each fish during the day and the scale calculates the weight of a 5 fish limit. Use this with a culling system to speed up the process.
Fish Friendly Clamp – No more puncturing jaws or sliding hooks up gill plates. The plastic jaws securely clamp onto jaws and are much easier on fish.
Choice of Units – Change the units on your scale to match your tournament. Choose between pounds with ounces or pounds with decimals, or kilograms.
Memory Backup – The scale operates on two AAA batteries. Should the batteries die, there is an internal memory backup that saves the most recently stored weights until you can change the batteries.
The scale body fits nicely in the palm of the hand. The plastic is textured for good grip, and has a red rubber gasket that prevents slippage.
The grip clamp is only made of plastic, so I was interested to see how it feels in action.
The LCD screen is large enough to easily see the weights. The reverse image takes a little getting used to, but it turns out to work well in bright daylight. I wish the screen was large enough to see all the weights at once, like on the Rapala Tournament Scale. But this is a smaller, cheaper scale after all.
I also wish all the keys were slightly bigger, especially the directional keys. But, the keys work fine and they have a snappy click to them.
The first thing to do is set the units. Hold the power key for 2 seconds, then push the power key when the screen flashes to choose your units.
To store a weight, press the button with the Lock symbol. The weight will flash, then use the left and right keys to choose the memory slot, and push the center key to save. Use the same process to replace smaller weights over the day. You can use the down key and the scale will show you the MIN weight stored.
To clear all the saved weights, push the up and down key at the same time until all slots are flashing. Press the middle button and all weights will be cleared.
After a few weeks of testing, I think this is a decent digital fish scale. The weights have been accurate and consistent. I love the no puncture lip grip. It's fast, secure, and easier on the fish. The eight fish memory slots make it easy to track my livewell in a tournament and speed up the culling process.
This scale doesn't quite measure up to the Rapala Touch Screen Tourney Scale, but for the price is a good cheap digital fish scale for freshwater use. If you fish tournaments I would go with the Tourney Scale, but if you just want a good scale to track your success over a day, the Rapala High Contrast Digital Scale works great for a lower price. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,587 |
{"url":"http:\/\/www.koreascience.or.kr\/article\/ArticleFullRecord.jsp?cn=DBSHCJ_2007_v22n3_369","text":"ANOTHER METHOD FOR A KUMMER-TYPE TRANSFORMATION FOR A 2F2 HYPERGEOMETRIC FUNCTION\n\nTitle & Authors\nANOTHER METHOD FOR A KUMMER-TYPE TRANSFORMATION FOR A 2F2 HYPERGEOMETRIC FUNCTION\nChoi, June-Sang; Rathie, Arjun K.;\n\nAbstract\nVery recently, by employing an addition theorem for the con-fluent hypergeometric function, Paris has obtained a Kummer-type trans-formation for a $\\small{_2F_2(x)}$ hypergeometric function with general parameters in the form of a sum of $\\small{_2F_2(-x)}$ functions. The aim of this note is to derive his result without using the addition theorem.\nKeywords\ngeneralized hypergeometric function;Kummer`s first theorem for $\\small{_1F_1}$;Kummer-type transformation;addition theorem for $\\small{_1F_1}$;\nLanguage\nEnglish\nCited by\nReferences\n1.\nR. B. Paris, A Kummer-type transformation for $\\alpha_{2}F_{2}$ hypergeometric function, J. Comput. Appl. Math. 173 (2005), 379-382\n\n2.\nL. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, Cambridge, 1960\n\n3.\nL. J. Slater, neralized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966","date":"2018-08-16 13:43:51","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\": 4, \"\/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\": 0.7168218493461609, \"perplexity\": 2401.487123380556}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221211000.35\/warc\/CC-MAIN-20180816132758-20180816152758-00189.warc.gz\"}"} | null | null |
Q: Using std::decay with std::forward template<typename T>
struct test
{
std::string key;
T value;
template<typename U>
using decay = typename std::decay<U>::type;
template<typename U = decay<T>, typename US = decay<std::string>>
test(U&& _value, US&& _key) : value(std::forward<U>(_value)), key(std::forward<US>(_key)){}
};
I use decay like this in my almost every project. I want ask, if its a good use instead, of writing something like this :
test(T&& _value, std::string&& _key) : value(std::move(_value)), key(std::move(_key)){}
test(const T& _value, std::string&& _key) : value(_value), key(std::move(_key)){}
test(T&& _value, const std::string& _key) : value(std::move(_value)), key(_key){}
test(const T& _value, const std::string& _key) : value(_value), key(_key){}
A: You overthink this. You need just this:
template<typename T>
struct test
{
std::string key;
T value; // as a safety this could be replaced by:
// typename std::decay<T>::type value;
template<typename U, typename US>
test(U&& _value, US&& _key)
: value(std::forward<U>(_value))
, key(std::forward<US>(_key))
{}
};
This perfect forwarding will ensure that all constructors you have listed are available.
Looks like you do not understand what std::decay do, or when/how to use it.
Example: decay<std::string> is pointless, since this just represents std::string type, so you should write just std::string you do not have to do any conversion since you have full control over type passed to decay, you know this type doesn't contain reference or const since you have type this explicitly.
std::decay is useful to define a variable/filed of type which you could assign to. It strips references and constness, C-array converts to pointer, and ensures pointer to functions. See doc example.
Could you explain what was your plan to achieve with this default types for template parameters? I can't figure out what was your intention here.
A: If both values are actually kept inside your class, it is much simpler than that:
test(T value, std::string key)
: value(std::forward<T>(value)), key(std::move(key)) {}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,333 |
{"url":"http:\/\/www.cetatzeanul.ro\/rartr\/finding-angle-measures-between-intersecting-lines-worksheet-1857f8","text":"Here is a graphic preview for all of the Angles Worksheets.You can select different variables to customize these Angles Worksheets for your needs. Finding Angle measures between intersecting lines by Greg Lakey on Mar 22, 2018 This module deals with parallel, perpendicular and intersecting lines. You get the measure of angle AED is equal to 110 degrees. Nov 15, 2017 - Math Worksheets for Basic Geometry: Parallel, Perpendicular, Intersecting Also, recall that a straight angle is equal to 180\u00b0. Google Classroom Facebook Twitter. 7th lines and angles. For problems 9 \u2013 16, find the missing angles. Great resource! Angles in parrallel lines Presentation.A PowerPoint demonstrating angle properties for parallel lines. Using these challenging sheets, your child can practise identifying parallel (lines that run alongside one another and will never touch) and perpendicular lines (lines that meet at a 90\u00b0 angle). Theorem: The measure of the angle formed by 2 chords that intersect inside the circle is $$\\frac{1}{2}$$ the sum of the chords' intercepted arcs. Ask your question. 1) 2 1 3 4 m \u00d0 1 = m \u00d0 3 = m \u00d0 2 = m \u00d0 4 = Excellent resources to teach this topic- Thanks for sharing. Find the missing angle in the triangles and the covex quadrilaterals, worksheet #1. The Angles Worksheets are randomly created and will never repeat so you have an endless supply of quality Angles Worksheets to use in the classroom or at home. KS3 \/ GCSE Geometry. Distinguish between lines, line segments, rays, perpendicular lines, and parallel lines. Missing Measures: Intersecting Lines 1 Find the measures of the missing angles. To find the measure of an angle from two intersecting lines where you only know the slopes, you can use a method that involves finding the angle of inclination for each line. On the second page there is a short exercise with similar problems for the class to do themselves. 8. are angles that equal 180\u00b0. Jul 26, 2017 - A fun challenge for students! bachu 2 months ago 4. 4.7. SOLUTION: Here, ... Find the measure of the arc or angle indicated. Vertical angles are opposite angles formed by two intersecting lines. Diagram 1 . Practice: Finding angle measures between intersecting lines. 4. are 2 angles whose sum is 180\u00b0 and together form a straight line. So we got the exact result that we expected. Whether it is basic concepts like naming angles identifying the parts of an angle classifying angles measuring angles using a protractor or be it advanced like complementary and supplementary angles angles formed between intersecting lines or angles formed in 2d shapes we have them all covered for students. \u2022 \/2 and \/4 are both formed by the intersection of line j and line t. The angles are opposite of each other, so \/2 and \/4 are vertical angles. This homework \/ home learning pack contains a wealth of informative material that complements the task work later on in the resource. 7. are 2 adjacent angles that form a straight line. Finding angle measures between intersecting lines - Best diet to lose 10 pounds in 3 weeks, Use your knowledge about angles to find missing angle measures in various complex situations. Formula for angles and intercepted arcs of intersecting chords . Word Doc PDF. Take a look at our free worksheets! The angles that are formed at the intersection between this transversal line and the two parallel lines. Same-Side Interior Angles Angles in a triangle worksheets contain a multitude of pdfs to find the interior and exterior angles with measures offered as whole numbers and algebraic expressions. 1) 2) 130O 35O 3) 4) 120O 83O 5) 6) 141O 7) 8) 56O 170O. They have to find missing angle measures of various intersecting lines. Look at other angles to help you solve for the missing angle. Lines and Angles. How does this help my children\u2019s learning? Print Calculating the Angle Formed From Intersecting Lines Worksheet 1. ... We hope that the free math worksheets have been helpful. 3. are angles that have a shared vertex and a shared side. Assume that lines which appear tangent are tangent. Identify each triangle and classify it by both its sides and angles. Parallel and perpendicular lines are found all around us. Log in. Thank you. \u2022 \/1 and \/3, \/6 and \/8, and \/5 and \/7 are also vertical angles. I usually do this at the end of the unit and have students race to find all of the angle measures. And we could call that angle-- well, if we made some labels here, that would be D, this point, and then something else. If you're seeing this message, it means we're having trouble loading external resources on our website. Finding angle measures between intersecting lines Get the answers you need, now! 1. Angles between intersecting lines. So we could, first of all, start off with this angle right over here. Table of contents. Example: Find the measure of the arc or angle indicated. In other words, the measure of the larger angle is the sum of the measures of the two interior angles that make up the larger one. Bring into play the appropriate properties of these angles formed by intersecting lines and crack the pdf exercises on finding the unknown angle measures and solving variable equations involving angle measures. Angles of intersecting chords theorem. Use your knowledge about angles to find missing angle measures in various situations. free . An angle greater than 90\u00b0 but less than 180\u00b0 is called an obtuse angle. LINES AND ANGLES 91 An acute angle measures between 0\u00b0 and 90\u00b0, whereas a right angle is exactly equal to 90\u00b0. Angles with Parallel and Intersecting Lines. Vertical angles have the same measure. This simple worksheet is a good way to introduce\/review angles in parallel lines. Email. between intersecting lines, or angles formed in 2D shapes we have them all covered for students ... Angles Worksheets Find the value of angle x for each of the following triangles: a) x = degrees b) x = degrees c) x = degrees d) x = degrees. stone517 stone517 2 weeks ago Mathematics College +5 pts. An angle which is greater than 180\u00b0 but less than 360\u00b0 is called a reflex angle.Further, two angles whose sum is 90\u00b0 are It begins with diagrams of corresponding, alternate and allied (supplementary) angles, then there are some examples to work through with your class. Angles in Parallel Lines for KS3 development. Geometry Worksheets Angles Worksheets for Practice and Study. Or we could say the measure of angle AED plus the measure of angle CEA must be equal to 180 degrees. This free worksheet contains 10 assignments each with 24 questions with answers. Join now. Let this worksheet be the intersecting line between you and your pupils as it creates new angles for teaching the material of Angles in Parallel Lines! Exam help 29 featured 1 fun stuff. KS3 \/ GCSE Geometry. Some of them are absolutely free of cost! Cool graphics! annamak 2 months ago 5. 5. are lines that intersect at a right angle. Question 3 in the given fig. Angles in parrallel lines Presentation.A PowerPoint demonstrating angle properties for parallel lines. Find the missing angle for the intersecting lines, worksheet #2. Dollour1 5 days ago 5. Use your knowledge about angles to find missing angle measures in various situations. Example of one question: Watch bellow how to solve this example: circles-secant-tangent-angles-medium.pdf . Worksheets math grade 4 geometry. 1. top; Applet; Practice Probs ; Challenge Probs ; Angles formed by intersecting Chords. Download. This simple worksheet is a good way to introduce review angles in parallel lines. A variety of pdf exercises and word problems will help improve the skills of students in grade 3 through grade 8 to identify and differentiate between parallel, perpendicular and intersecting lines. In this lesson we\u2019ll look at angles whose sides intersect a circle in certain ways and how the measures of such angles are related to the measures of certain arcs of that circle. Log in. Properties of parallel lines properties of parallel lines q1. This is a step by step video tutorial on how to find the value of angles made by intersecting lines. Name : Score : Printable Math Worksheets @ www.mathworksheets4kids.com Intersecting Lines Find the unknown angles. Learn to apply the angle sum property and the exterior angle theorem, solve for 'x' to determine the indicated interior and exterior angles. Parallel & perpendicular lines. Word doc pdf find the missing angle for the intersecting lines worksheet 1. ... Find the missing angle for the intersecting lines, worksheet #1. So this angle right over here is 110 degrees. So you subtract 70 from both sides. We know the measure of CEA is 70 degrees. Join now. When two lines intersect and form 4 angles at the intersection, the two angles that are opposite each other are called \u201copposite angles\u201d or \u201cvertical angles\u201d and these vertical angles are \u201ccongruent\u201d \u2013 meaning they have the same shape and size. Word Doc PDF. But I'll just call it this angle right over here. We know it is 70 degrees. 6. are lines in a plane that intersect or cross. Word Doc PDF. Vertical Angles | Vertically Opposite Angles. How to find angle measures in figures of intersecting lines As we work through this lesson, remember that a chord of a circle is a line segment that has both of its endpoints on the circle. Reviews. Whether it is basic concepts like naming angles identifying the parts of an angle classifying angles measuring angles using a protractor or be it advanced like complementary and supplementary angles angles formed between intersecting lines or angles formed in 2d shapes we have them all covered for students. 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Cea is 70 degrees Math Worksheets @ www.mathworksheets4kids.com intersecting lines, and \/5 and \/7 are also angles.\n\nRoy Mustang Meets Edward, Walk Like A Bear Crossword Clue, Where Was The Epicenter Of The 1985 Mexico City Earthquake, Best All Terrain Knee Scooter 2020, Mitsubishi Heavy Industries Air Conditioner Remote Control Manual, Lost Lament Quest Steps, Alan Saffron Son,","date":"2021-04-12 01:36:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.36331838369369507, \"perplexity\": 1524.549647795464}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038065903.7\/warc\/CC-MAIN-20210411233715-20210412023715-00561.warc.gz\"}"} | null | null |
\section{Abstract}
Only astronomical observations can effectively probe in space-time the variability of the physical dimensionless constants such as the fine structure constant $\alpha$ and proton-to-electron mass ratio, $\mu$, which are related to fundamental forces of nature. Several theories beyond the Standard Model (SM) allow fundamental constants to vary, but they cannot make quantitative predictions so that only laboratory experiments and astronomical observations can show if this is the case or set the allowed bounds. At the moment of writing there are claims for a variability of both $\alpha$ and $\mu$ at 5 and 4 $\sigma$ of C.L., respectively, although for $\alpha$ they are contrasted by null results. The observations are challenging and a new spectrograph such as ESPRESSO at the combined incoherent focus of 4 VLT units (a potential 16 m equivalent telescope) will allow for a significant improvement in the precision measurement clearing up the controversy. If the variations will be confirmed, the implications are far reaching, revealing new physics beyond the SM and pointing a direction for GUTs theories. A most exciting possibility is that a variation of $\alpha$ is induced by quintessence through its coupling with the electromagnetic field. If this is the case an accurate measurement of the variability could provide a way for reconstructing the equation of state of Dark Energy \cite{a06}.
\section{Introduction}
The Standard Model (SM) of particle physics needs 26 dimensionless physical constants for the description of the natural world (\cite{t06}), of these few are directly related to the strength of fundamental forces. Among them the fine structure constant ($\alpha = {\rm e}^2/(\hbar c) $) and the proton-to-electron mass ratio, ($\mu = m_p/m_e$) are of particular interest for us since they can be measured accurately by astronomical observations of intervening absorption systems towards distant QSOs. The fine structure constant $\alpha$ is related to the strength of the electromagnetic force; $m_e$ is related to the vacuum expectation value of the Higgs
field, namely the scale of the weak nuclear force, and m$_p$ is related to the $\Lambda_{QCD}$ or the strong nuclear force, therefore $\mu$ is related to the ratio between the strong and weak nuclear forces.
A whatever small variability of these constants will produce a violation of the Weak Equivalence Principle (WEP) and would have far reaching implications revealing new physics beyond the Standard Model. In the Gev energy regime $\alpha$ has already been shown to vary, but at low energies laboratory measurements with cooled atomic clocks failed to detect variations at the fifteen decimal place. The most stringent laboratory value is $\dot{\alpha}/\alpha$ = (-2.7$\pm$3.9)$\cdot 10^{-16}$ yr$^{-1}$ \cite{p06}. Astronomy is providing some evidence for both $\alpha$ and $\mu$ variations, although the evidence for $\alpha$ has been contrasted by other groups.
The astronomical claims for a variability are at the level of 6 ppm, part-per-million, and are measured up to redshift 4, or 12 Gyr lookback time.
Several space-based missions as ACES, $\mu$SCOPE, STEP will soon improve existing laboratory bounds for WEP up to 6 orders of magnitude, and they should find violations if present claims of variability are correct under simple linear extrapolation. It is thus desirable that the astronomical community will be able to clear up the case before these accurate experiments will fly, but only astronomical observations can probe WEP non locally.
\subsection{Why constants should vary?}
Strings and multidimensional theories predict variable constants since the constants are defined in the whole multidimensional space and vary as extra dimensions are varying. The coupling between a scalar field
with the electromagnetic field gives also varying constants. The
required cosmological constant value is so small
that a quintessence is a likely candidate for Dark Energy. Thus varying constants could provide insights into the nature of dark energy and provide evidence for scalar fields \cite{ma07,f07}.
Avelino et al. (2006) have shown how a precise detection of the variability of a constant could be used for the reconstruction of the quintessence potential and of the equation of state of Dark Energy \cite{a06}.
If one constant is varying, then all the gauge and Yukawa couplings are also expected to vary. There is precise relation between the variation of $\alpha$ and $\mu$, but it depends on the context the unification is realized in. Thus, simultaneous measurements of the variability of $\alpha$ and $\mu$ at similar redshift will be a key discriminant of the several GUTs models \cite{d07}. Theoretical preferences are for a relative change between the $ \mu$ and $\alpha $ variations of $\le$ 50, but larger values are also possible, implying that the strong-coupling constant is running faster than $\alpha$ and therefore $\delta \mu$ should
be found to be larger then $\delta \alpha$.
\section{The observations}
Observations of the Werner and Lyman series of the molecular hydrogen in Damped Ly$\alpha$ galaxies (DLA) can be used to bound $\mu$ variations. The electron-vibro-rotational transitions have different dependence from the reduced mass and can be used to constrain a variability of $\mu$.
UVES observations of the DLA at z$_{abs}$ =3.0 towards QSO 0347-383 \cite{i05}, but see \cite{l02}, and of the DLA towards QSO 0405-443 have provided $\delta \mu
= (24 \pm 6)$ ppm, when the two systems are combined together \cite{r06}. The handful of systems investigated for this purpose reflects the difficulties of the measurement. There are few DLA showing H$_2$ and the restframe H$_2$ lines are at $\approx$ 1000 \AA\,, falling in the Lyman forest and requiring a z$_{abs} \ge$ 2 to be redshifted into the optical window. H$_{2}$ systems are extensively searched at the moment so that probably new observations will be available in the near future to verify these first findings.
Fine structure variability can be probed in the early universe through the primordial nucleosynthesis or through the CMB power spectrum but at the level of a few percent. The most effective way has been achieved through the analysis of metal lines of intervening absorption systems observed in the spectra of distant QSOs.
The energy levels of high mass nucleus are subject to relativistic corrections which are sensitive to the mass number.
These have been calculated for the most frequently observed resonance lines and constitute the popular Many-Multiplet method.
Murphy and collaborators \cite{mu04}
by comparing the redshift of several lines in a sample of 143 systems in the redshift interval 0.2$<z_{abs}<$4.2 found evidence for
$\Delta \alpha / \alpha$ = $(-5.7\pm 1.1)$ ppm. However, this evidence has been contrasted by two other groups which did not find
evidence for variability at the level claimed. Chand et al. found an average value of $(-0.6 \pm 0.6)$ ppm in a sample of 23 systems, while Levshakov and collaborators found $(-0.12 \pm 1.79)$ ppm and $(5.66 \pm 2.67)$ ppm in two systems at $z = 1.15$ and 1.84, respectively, and by using lines of Fe\,{\sc ii} only \cite{q04,l07}. What is the best methodology is currently under debate \cite{mu07a,mu07,s07,m07}.
\begin{figure}
\centering
\includegraphics[height=5cm]{fig3.eps}
\caption{Estimated accuracy in the position of an absorption line with EW =0.050 \AA\ and $b$= 2 km s$^{-1}$ as a function of the exposure time for ESPRESSO@4UT. On the figure legend other instrumental and observational parameters are given. }
\label{fig:1}
\end{figure}
\subsection{Would you like an ESPRESSO?}
These observations are challenging the instrumental performances of UVES-VLT or HIRES-Keck telescopes.
Measuring the variability of $\mu$ or $\alpha$ implies the measurement of a tiny variation of the position of one or few lines with respect to other reference lines.
It is not much different than revealing exoplanets, but with the limitations that only few lines can be used and QSO are much fainter than stellar sources.
The precision in the measure of a line position increases with the spectrograph resolving power till the intrinsic broadening of the metal lines is resolved, the signal-to-noise and with the decreasing of the pixel size
($\Delta \lambda$ $^{3/2}$, see \cite{b83} for a precise relation).
The ESPRESSO spectrograph described by L. Pasquini at this conference, both in the {\it Super-HARPS} or {\it Super-UVES} modes, holds the promise for one order of magnitude improvement compared to what presently achieved. Fig. 1 shows the accuracy which can be achieved in the photon limit approximation and accuracies of few 10 ms$^{-1}$ are reachable for single lines with relatively short exposures even for faint sources. An error of 30 m s$^{-1}$ corresponds to an error of 1 ppm for $\alpha$;
such an accuracy will be enough to resolve the present controversy and establish in a definitive way whether $\alpha $ or $\mu$ are varying as claimed. However, one important requirement is the improvement of the wavelength calibration, for instance with the LaserComb as discussed here by A. Manescau.
\section{Constants and Dark Energy}
Avelino and collaborators \cite{a06} have shown that the measurement of the behaviour of variations in $\alpha$ and $\mu$ with redshift can be used to infer the evolution of the scalar field and of the equation of state of the Dark Energy, not very differently from the reconstruction of the potential from the motion of a particle. Nelson Nunes kindly adapted their detailed analysis to a realistic set of observations which can be performed with ESPRESSO@4VLT. It is assumed that it has been possible to measure $\alpha$ and $\mu$ for a sample of 200 and 25 systems respectively and with an equal, for simplicity, accuracy of 1 ppm.
In the example case the scalar potential is taken as V($\phi$) = V$_0$(exp(10k$\phi$ + exp(0.1k$\phi$)), which is one of the simplest possible potential accounting for the accelerated expansion. Fig. 2 shows the Monte Carlo redshift distribution of the data with this scalar potential assuming that the variation of $\alpha$ {\bf is} -5 ppm at z=3 and that the two constants are mutually linked by a fix ratio of -6, as it is suggested by some of the observations. In Fig. 3 the red dotted line shows the assumed behaviour of the {\it w(z)} while the black continuos line shows its recovering through a fitting of the simulated data points with a polynomial of order m=3 (cfr \cite{a06} for details). The shaded regions show the 1 and 2 CL of the reconstruction, when both $\alpha$ and $\mu$ measurements have been considered. We emphasize that only few observations would clearly show if {\it w(z)} is an evolving function of z.
\begin{figure}
\centering
\includegraphics[height=5cm]{p4cl10k0.1Nc33a.eps}
\caption{Monte Carlo data set based on redshift
dependence of the scalar potential given in the text producing a $\Delta \alpha / \alpha$ =-5 ppm at z=3. Error bars are of 1 ppm for $\alpha$
and $\mu$ as expected with ESPRESSO@4VLT. }
\label{fig:2}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=5cm]{p4cl10k0.1Nc33c.eps}
\caption{Reconstruction of the equation of state and its error band. Dashed line represents the assumed dark energy and the solid line the reconstruction's best fit. Shaded regions show the 1 and 2 CL of the reconstruction}
\label{fig:2}
\end{figure}
\section{Conclusions}
Variability of physical constants is an important issue for physics
and only astronomy can probe this possibility for $\alpha$ and $\mu$ in the full space-time.
Present observations provide hints of variation for both constants but those for $\alpha$ have been contrasted by other investigations.
The ESPRESSO spectrograph presently conceived for the incoherent combined focus of the 4VLT would improve present accuracy by a significant factor
and therefore clarify the case. A confirmation of the variability would have far reaching implications revealing new physics beyond the SM, showing the right path for GUTs and possibly providing insights into the nature of Dark Energy.
If no variability is found, then the new more stringent bounds will be usefully combined with local space experiments for WEP violation.
Overall, this seems to be a great opportunity for the astronomical community and I hope that ESO will take advantage of it by considering the construction of the new high precision spectrograph at the incoherent combined focus of the 4 VLT units, a $\approx$ 16m equivalent telescope.
\section{Acknowledgements}
It is a pleasure to thank N. Nunes for adapting his simulations for ESPRESSO, all the ESPRESSO collaboration, in particular S. Levshakov and M. Murphy.
%
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