text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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{"url":"https:\/\/gamedev.stackexchange.com\/questions\/153977\/in-unity-2017-is-there-a-way-to-set-the-buttons-used-by-input-getaxis-progra","text":"# In Unity (2017) is there a way to set the buttons used by Input.GetAxis() programatically?\n\nThe buttons used for keyboard axis inputs can be set manually via Negative and Positive Button fields for the axis at Edit > Project Settings > Input Manager. However I'd like to make this a user configurable option which I think means needing to set the button values from a script.\n\nI've been looking for a method along the lines of:\n\nInput.SetAxis(\"Vertical\").positveButton\n\nBut so far I'm striking out. Is there a way to do this?\n\n\u2022 Just a quick question before we go any further: Why do you want to\/have to do this? \u2013\u00a0B\u00e1lint Feb 5 '18 at 19:30\n\u2022 @B\u00e1lint - a few reasons: 1) it seems nice to give the user their choice of input keys; 2) it would make it easier to experiment with the input key mappings on complex games that are being played or developed on a standard keyboard. \u2013\u00a0dlu Feb 5 '18 at 19:36","date":"2021-07-25 09:57:23","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\": 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\": 0.2589566111564636, \"perplexity\": 1077.4133508895316}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046151641.83\/warc\/CC-MAIN-20210725080735-20210725110735-00423.warc.gz\"}"} | null | null |
#include <aws/kms/model/ListGrantsRequest.h>
#include <aws/core/utils/json/JsonSerializer.h>
#include <utility>
using namespace Aws::KMS::Model;
using namespace Aws::Utils::Json;
using namespace Aws::Utils;
ListGrantsRequest::ListGrantsRequest() :
m_limit(0),
m_limitHasBeenSet(false),
m_markerHasBeenSet(false),
m_keyIdHasBeenSet(false),
m_grantIdHasBeenSet(false),
m_granteePrincipalHasBeenSet(false)
{
}
Aws::String ListGrantsRequest::SerializePayload() const
{
JsonValue payload;
if(m_limitHasBeenSet)
{
payload.WithInteger("Limit", m_limit);
}
if(m_markerHasBeenSet)
{
payload.WithString("Marker", m_marker);
}
if(m_keyIdHasBeenSet)
{
payload.WithString("KeyId", m_keyId);
}
if(m_grantIdHasBeenSet)
{
payload.WithString("GrantId", m_grantId);
}
if(m_granteePrincipalHasBeenSet)
{
payload.WithString("GranteePrincipal", m_granteePrincipal);
}
return payload.View().WriteReadable();
}
Aws::Http::HeaderValueCollection ListGrantsRequest::GetRequestSpecificHeaders() const
{
Aws::Http::HeaderValueCollection headers;
headers.insert(Aws::Http::HeaderValuePair("X-Amz-Target", "TrentService.ListGrants"));
return headers;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 461 |
Donald Glover Is Purposely Avoiding 'This Is America' Comments
Glover appeared on Thursday night's " Jimmy Kimmel Live! " where he talked about his hit FX series and his upcoming role as a young Lando Calrissian in " Solo: A Star Wars Story ", but he also touched on the reaction to " This Is America ".
She fatally stabbed her husband as he allegedly raped her
Rights group Amnesty International on Thursday slammed a Sudanese court's sentencing of a teenager to death for killing her rapist husband in self-defence. During her trial in July 2017, the court found her guilty of "intentional murder" after applying an outdated law that does not recognise marital rape, it added.
Cornell Student Delivers Presentation In Underwear After Professor Criticizes Choice Of Clothes
In a Facebook live stream of the presentation, Chai said that she had been in "disbelief" and felt "disappointed" and "pain" following the incident. Chai stripped again during her actual senior thesis presentation, in front of students and professors. She then encouraged the audience to strip.
Police Searching For Stolen Cruiser, Suspect Considered Armed + Dangerous
He was being transferred to Eastern Maine Medical Center Friday afternoon. Police were chasing the vehicle on Route 15 in Corinth when it crashed into trees. Until it was found abandoned along Parkman Road about two hours later, the cruiser was last seen driving along Garland Road toward Garland. Tibbetts is described as 5-foot 9-inches, 130 pounds with brown hair, hazel eyes.
SC dismisses Muthaliks plea on Congress manifesto
The court, however, observed that announcing schemes for the benefit of minority communities was not the same as seeking votes on the basis of religion. The apex court, on January 2, 2017, had delivered a landmark verdict to separate religion, caste and other issues from politics and by a majority of 4:3 had held as "corrupt" the practice of candidates appealing for votes on the basis of such identities.
Hard Facts About Facebook, Inc. (NASDAQ:FB)
Staley Advisers Inc accumulated 4,303 shs. (NASDAQ:FB) or 151,743 shs. Facebook, Inc.'s insider ownership stands at 2.12 percent, while its insider transaction is -18.81 percent. (FB), Advanced Micro Devices, Inc. "A Look At The Tech Giants' R&D Spending" on May 10, 2018. Dubuque Bankshares & has invested 1.53% in Facebook, Inc.
Pompeo promises North Korea future 'brimming with prosperity' if it denuclearizes
North Korea's development of nuclear weapons and missiles capable of reaching the United States brought exchanges of bellicose rhetoric between Trump and Kim previous year that raised fears of a new war on the Korean peninsula. During the trip, Pompeo met with dictator Kim Jong Un , whom he told the press on Friday that he had a "good" and "substantive" conversation with. "I think there is complete agreement about what the ultimate objectives are", he explained.
Karnataka Elections 2018: The key numbers to know ahead of voting day
Sonia Gandhi, who had collapsed during an election campaign at Varanasi, was out of campaigning for more than two years. BJP candidate B Sriramulu be disqualified from contesting the Karnataka assembly elections, the Congress has petitioned the Election Commission, over a video it claims shows Mr Sriramulu negotiating a bribe with a relative of a former Supreme Court judge for a favourable verdict in an illegal mining case involving the Reddy brothers.
Georgia police officer suspended after manhandling grandmother during traffic stop
Alpharetta Officer James Legg had been called in as backup during a May 4 traffic stop because the woman would not sign a ticket and refused another officer's order to get out of her vehicle, officials said. "Unless you're wrong, you're going ask why". Once he pulled her over, the officer issues a citation and ordered Campbell to sign. Am I in a movie? "I felt not only that my space had been violated, but that he was not respecting me".
Gujarat HC upholds life imprisonment for 14 in 2002 riots case
Also, seven-year-imprisonment awarded to five others will also continue, ANI said. Delivering judgment on Friday, HC acquitted three of those convicted for life but upheld the seven-year sentence for all five convicts who will now be released from jail as they have served the sentence for this period.
Ex-Speaker Sheldon Silver Found Guilty On All Counts in Corruption Trial
During his two-week trial in Manhattan, prosecutors showed that Silver, 74, had obtained almost $4 million in illicit payments in return for taking a series of official actions that benefited a cancer researcher at Columbia University and two real estate developers in NY.
Midfielder lined up as first Arsenal's first post-Arsene Wenger signing
It will be an emotionally charged afternoon at the John Smith's Stadium on Sunday where Arsene Wenger will deliver his final dressing room speech to an attentive Arsenal squad. "Not the most glamorous maybe, but the most hard". Despite having one less competition to worry about they are now sixth in the Premier League and look certain not to achieve a top four finish, hanging their only hopes on winning the Europa League.
TDP activists pelt stones at BJP chief Amit Shah's convoy in Tirupati
The windshield of a vehicle belonging to BJP leader Kola Anand was damaged with a TDP activist reportedly pelting a stone at the convoy. Tirupati urban superintendent of police Abhishek Mohanty said, "A stone is said to have been thrown at a private vehicle".
McCain Irrelevant, 'He's Dying Anyway,' White House Official Reportedly Says
McCain then fired back addressing Sadler directly with what she called a "news flash". MSNBC anchor Stephanie Ruhle chocked up and was near tears reporting on the " fierce backlash " against White House special assistant Kelly Sadler's comments on Sen.
White House official derides McCain over Haspel opposition: "He's dying anyway"
Gen. Thomas McInerney told Fox Business host Charles Payne that torture "worked on John". Ron Wyden, D-Ore., as she has resisted calls to declassify additional records about her career. Joe Manchin III of West Virginia. In September, Meghan McCain said it was "abhorrent" that Trump reportedly mocked her father physically after that health care vote.
N's CEC decides to sue Chairman NAB Justice (retd) Javed Iqbal
NAB had notified it was "examining complaints" that the foreign exchange reserves of India increased by $4.9bn, as "verified" by a World Bank report, because of Nawaz and Shahbaz Sharif's money laundering activities. The minister said that the PML-N would go into the 2018 polls with the slogan of "give respect to vote" and seek the mandate of the people on the basis of its performance, adding that the people once again would give a heavy mandate to the PML-N and the former prime minister Nawaz ...
Militants kill policeman in Indian-controlled Kashmir
The police personnel, identified as Shamim Ahmad had sustained injuries in the attack, earlier in the day. "The alert jawans successfully foiled the attempt", said a Srinagar-based spokesman. Following the attack, clashes erupted in the adjoining village of Soibugh when youth pelted stones on army vehicles moving towards Wadwan. An eyewitness said that army men opened fire to chase away the protesters.
BC Interior braces for more flooding as weather warns up
In Osoyoos, a local state of emergency has been declared, due to the imminent threat of flooding and rising Lake levels which may "threaten properties and town infrastructure", officials said. Rapidly melting snow combined with heavy rain has resulted in rivers overflowing in the interior Kootenay region such as near the town of Grand Forks.
World's oldest World War II veteran celebrates 112th birthday May 11!
According to a profile by the Dallas Morning News , Overton wakes up before the sun and gets his vitals checked by one of his around-the-clock caretakers. In 2016, his family started a GoFundMe campaign so he wouldn't have to move into an assisted living facility. Overton told Austin, TX station KXAN that he still drinks whiskey and smokes cigars and says he doesn't suffer from any aches or pains.
Fake voter IDs: Polling deferred in Karnataka's Rajarajeshwari Nagar seat
The polling in RR Nagar has been re-scheduled for May 28 while the counting and results will be done and announced on May 31. Election Commission had promised an inquiry into the matter. While the blame game continued, the Election Commission had said that culprits would be brought to justice. The elector to population ratio is 74.33 percent.
1847 cast primary ballots
Brown says that shouldn't worry Democrats when it comes to November. Mike Braun will now prepare to tackle current Democrat Senator Joe Donnelly for his U.S. Pete Wilson to re-election-even though it tarnished the Republican brand for many Latino voters. Tuesday can't be compared to 2016's presidential primary .
Malaysia's new PM says Anwar Ibrahim will be given royal pardon
In a stunning political comeback, Mahathir Mohamad was sworn in 15 years after he stood down. For the uninitiated, the so-called prophecy, which started circulating around the 1960s and 1970s, predicted that each letter in the word "Rahman" represents the name of Malaysia's prime ministers: Tunku Abdul Rahman, Abdul Razak Hussein, Hussein Onn, Mahathir Mohamad, Abdullah Badawi and Najib Razak.
Rahul Gandhi slams PM Modi as Congress alleges Sriramulu bribed ex-CJI
In this election, Sriramulu is the only BJP candidate to be fielded from two seats: south Karnataka's Molakalmuru as well as Badami, Siddaramaiah's "Plan B" seat in the north. "It was only when Atal Bihari Vajpayee Ji formed the government that Babasaheb was conferred the title", PM Modi said. He accused the Congress of "insulting" Dr Ambedkar, a Dalit icon.
Congress trying to win Karnataka by hook or crook: Amit Shah
The PM has already done almost 18 rallies in the run-up to the state elections. "This election is not about Rahul", the Congress president said. In the evening, Shah will address a convention of party workers at Dr Shyama Prasad Mukherjee stadium here.
Ex-US Rep. Mel Reynolds gets 6-month sentence
Prosecutors had argued Reynolds collected $433,000 for consulting work he did in Africa during the four-year period he failed to file tax returns, local media reported . "Evidence at trial showed that Reynolds received gross income in excess of the minimum amount required to file a tax return", the Justice Department said in a statement .
Lawyer Says Donald Trump Knew About Former AG Schneiderman's Alleged Sex Victims
Well, he may have actually known about some allegations against the now-ex New York AG. You can read Gleason's full letter at Law & Crime here. In a court filing today in the Michael Cohen case, lawyer Peter Gleason wrote to the federal judge to inform him that two women had told him that they had been "sexually victimized" by Schneiderman.
Controversial billboard sparking debate in Maryland county
CNN reports that the Calvert County sheriff's office has been hit with numerous comments from both supporters and those attacking the sign. It's been up a couple of weeks now and it's been all over social media. We've received messages that say it's offensive that it's up", he said , while others say "it's offensive that he's being pressured to take it down.
First Alert Forecast: a summery flare for Moms Day weekend
WARMING WEEKEND OUTLOOK: A ridge of high pressure slowly tries to build into the region for the weekend, helping to boost temperatures for the first half of the weekend. Monday may start out cloudy, but the sky will become partly sunny. The temperature could level off or keep climbing to a high into the lower to mid-90s, but that will depend on how sunny it is and whether winds shift from the southwest to the east.
78-Foot Wave Is the Largest Ever Recorded in Southern Hemisphere
A ferocious storm was recorded in Campbell Island, one of the wild southern ocean. It wiped out the previous record of 22.3 meter set in south Tasmania in 2012. Durrant said surfers in California could expect to feel energy from the storm hit West Coast shores by next week, as the swells "propagate throughout the planet".
HSE director general stepping down
CervicalCheck, the HSE's national cervical screening programme, revealed earlier this month that an audit showed 208 women diagnosed with cancer should have received earlier intervention. A memo published this evening reveals that Cervical Check was preparing for media fallout from cancer misdiagnoses as far back as 2016. It said that the CervicalCheck programme was prepared for such media coverage.
DR Congo: British tourists kidnapped in Virunga National Park
On Wednesday authorities released the Congolese reporter and ordered the two British nationals to leave the country on the grounds that they did not have the correct visas or accreditation. He said the park is protected by around 800 rangers but there are also estimated to be between 1,500 and 2,000 militia in and around the park. The Foreign and Commonwealth Office advises against all but essential travel to Goma and urged Britons not to go beyond the city.
Why the Leaning Tower of Pisa survived earthquakes
Despite leaning precariously at a five-degree angle, leading to an offset at the top of over five meters, the 58-meter tall Tower has managed to survive, undamaged, at least four strong earthquakes that have hit the region since 1280. The team says that due to DSSI, the tower "does not resonate with quake ground motion" as the soft soil "causes the vibrational characteristics of the structure to be modified substantially".
Kelly: Vast majority of undocumented immigrants 'don't have skills'
This means English was the language they spoke at home or that the person self-reported speaking the language well. White House chief of staff John Kelly said President Donald Trump is "somewhat embarrassed" by the special counsel's investigation into Russian meddling in the 2016 election.
China's Didi Chuxing suspends carpooling service following murder of a passenger
The flight attendant had booked the vehicle to take her from a hotel to Zhengzhou airport as she was about to head to a relative's wedding in Jinan in Shandong province. In addition, before the incident, the passenger filed a complaint of sexual harassment against the driver, but the customer service failed to reach the suspect after five attempts.
Russia, Germany Speak About Preserving Iranian Nuclear Deal
Additionally, the minister expressed hope that progress in negotiations over a permanent exclusion of the European Union (EU) from new US tariffs on steel and aluminum imports could help resolve differences in opinion with regards to the Iran nuclear agreement as well.
Peta's Forecast - Tracking rounds of showers and storms this Mother's Day Weekend
Wind: NE 5-10 miles per hour. Saturday Night: Partly cloudy - low around 69 with south wind 6 to 9 miles per hour. TUESDAY: Partly cloudy with a 20% chance of thunderstorms by evening. Tomorrow: Partly Cloudy. Highs near 90. Mostly cloudy, with a low around 54. The isolated shower chance returns for Wednesday with partly sunny skies and highs in the low 80s.
India wants safe, sustainable Rohingya repatriation
Rohingya so-called attacks on security posts in Rakhine in August previous year sparked a military operation that has sent almost 700,000 Rohingya fleeing to refugee camps in Cox´s Bazar in Bangladesh. She also "welcomed" Myanmar government for "continued commitment" to implement the Kofi Annan-led Rakhine Advisory Commission's recommendations. Referring to the pact, Swaraj said that India is already implementing various projects, with the first one to be the construction of pre-fabricated ...
UK, French, Germany FMs to meet over Iran on Tuesday
However, on May 8, Trump announced the USA withdrawal from the JCPOA. In a declaration on Tuesday, President Donald Trump officially withdrew the United States from the nuclear pact. According to Japanese officials, Mr. Zarif has told his Japanese counterpart under what conditions Iran will keep abiding by the nuclear deal but the details have not been disclosed yet.
Authorities investigating explosive device found outside Beaumont church
Stephen's Episcopal Church in Beaumont. The church, which is home to the All Saints' Episcopal School , will remain closed until further notice. At a news conference Thursday, Beaumont police Chief James Singletary asked members of the public to call 911 if they see a package they feel is suspicious.
Theresa May's cunning plan on Brexit: ministerial divide and rule
In contrast, the customs partnership will be examined by two Brexiteers - International Trade Secretary Liam Fox and Environment Secretary Michael Gove - and a lone Remainer, Cabinet Office Minister David Lidington. because it practically doesn't work". However, if the particular product is only destined for the United Kingdom market, the business would have to apply to get some money back if the British tariff is lower.
SC collegium to reiterate Justice KM Joseph's name for elevation
The collegium's resolution stated that since detailed discussion is required to shortlist names to accompany Joseph's reiteration, the meeting has been deferred till May 16. The Supreme Court Collegium meeting of Friday (May 11) threw up a solution of sorts, nearly exactly as India Legal had predicted. Until then, Justice Joseph's name will not be sent back to the Centre.
Fox News Bans Guest Analyst Who Said Torture 'Worked' on John McCain
John McCain (R-AZ) - and that "that's why they call him Songbird John". John McCain at a staff meeting Thursday. McCain's experience as a prisoner of war also rebuts McInerney's horrific claim: as the Daily Beast notes, there's no evidence that McCain, who was held as a prisoner of war for five and a half years during the Vietnam War, ever gave up any information to enemy combatants, despite being tortured and beaten almost the entire time.
Twitter loses it over idea that changing babies' diapers requires 'consent'
Commenters claimed Carson's suggestion was absurd, saying babies wouldn't even understand what was going on. And if a parent were to go ahead and change the diaper anyway, wouldn't that show the child that their consent doesn't matter? Another mother, Karen Ridout, said: "Yeah nah I'm torn, I think consent is important, I do, obviously they are too young to give you verbal consent".
Possible school shooting reported in Southern California
Police units said they were searching for a male with a rifle who was reportedly hunkering down in the brush near the school's baseball field. Los Angeles Times said one person transported themselves to a hospital, with no word on the extent of the injury, according to a county fire official.
Brexit group fined for breaking spending rules in EU vote
Bob Posner, the Electoral Commission's director of political finance and regulation, said: "The rules we enforce were put in place by Parliament to ensure transparency and public confidence in our democratic processes". The Electoral Commission said the actual figure " may well have been considerably higher ", adding that the group had presented "incomplete and inaccurate" information about spending.
Republicans Fear They'll Be Shut Out Of California Governor's Race
But if there is a runoff in the Republican primary and you voted in the Democratic primary, you can't vote in the Republican runoff. "Everybody, whether a Republican or Democrat, people understand it and they understand how it undermines democracy".
Gaza tension as hundreds march towards fence with Israel
However, the resolution is non-binding and the USA president has said he is " looking forward " to seeing the new embassy in Jerusalem this May. "I must tell you that the bold decision by President Trump has prompted other countries, quite a few now, who are planning to move their embassy to Jerusalem as well", Netanyahu said last month.
Philly Nurse Charged In Death Of HR McMaster's Father
A nurse from the Cathedral Village senior living facility is facing up to 20 years in prison for failing to perform her job, which allegedly led to the death of H.R. When she asked Gainey about it, Gainey allegedly said, "Well, I falsified that one", allegedly saying she "didn't want the next nurse to have to do them".
Is Iran really as bad as Donald Trump says?
North Korea can and likely will use the White House's decision to rip-up the Iran nuclear deal as a reason to stand firm on certain aspects of a proposed denuclearization agreement during negotiations. (and vice versa), and Washington has long feared what might happen if the Iranian regime developed a nuclear weapon. "The goal is to prevent Iran from ever developing or acquiring a nuclear weapon and the detail beyond that is something we are going to have to flesh out", the official ...
Merkel Proposes to Discuss Iranian Missile Program - Statement
According to Seibert, Merkel was in favour of holding multi-lateral talks with Iran on its ballistic missile program and on its regional incursions, including in Syria and Yemen. Iran is also actively involved in the conflicts in Yemen and Syria supporting Shia forces in these countries - the Houthi movement called Ansar Allah in Yemen and the Syrian government headed by President Bashar Assad, who belongs to the Alawite minority.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,488 |
Open, Inviting and Safe – Designing Secure Buildings - It's an unfortunate sign of the times. Increasingly, architects need to incorporate measures to deter, delay and protect people and property from human aggressors.
Their challenge is to find ways to balance security measures with other design constraints such as fire protection, energy efficiency, and aesthetics in order to build inviting, safe environments.New glazing and framing systems offer helpful, attractive tools that can meet security, fire protection and energy efficiency needs all at the same time.
Adding glass enhances a sense of openness and improves energy efficiency, but windows and entryways can be the most vulnerable portion of a building. Newly developed, fire-rated glazing assemblies can resolve the dilemma by providing clear, transparent openings and walls that resist forced entry, bullets and blasts while meeting fire and other safety codes.
Fortification of buildings once appealed largely to the State Department, the DoD and prisons, but that is no longer the case. Today, interest in built-in security has expanded to schools, local government facilities, courthouses, storefronts, airports, and homes.
Security glazing is not yet mandated by the model building codes. Instead, there is a tapestry of security glazing test standards that have evolved over time. The first test method for bomb blast was ASTM F 1642 published in 1966.
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Click here to download a brochure on these new security products. Click here to see recent installations of all our products. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,698 |
\section{Introduction}
One of the most active research areas in theoretical physics is the formulation of a quantum theory of gravity that would reproduce the well-tested theories of Quantum Mechanics (QM) and General Relativity (GR) at low energies.
The large energy scales necessary to test proposed theories of Quantum Gravity make this investigation extremely challenging.
Nonetheless, it is important to propose phenomenological models and tests of such theories at low energies, and the Generalized Uncertainty Principle (GUP) offers precisely this opportunity.
Many theories of QG suggest that there exists a momentum-dependent modification of the Heisenberg Uncertainty Principle (HUP), and the consequent existence of a minimal measurable length \cite{Gross1988_1,Amati1989_1,Maggiore1993_1,Maggiore1993_2,Kempf1995_1,Garay1995_1,Scardigli1999_1,Capozziello2000,AmelinoCamelia2002_1}.
This modification is universal and affects every Hamiltonian, since it will affect the kinetic term.
In the last couple of decades several investigations have been conducted on many aspects and systems of QM,
such as Landau levels \cite{Das2009_1,Ali2011_1}, Lamb shift, the case of a potential step and of a potential barrier
\cite{Das2009_1,Ali2011_1}, the case of a particle in a box \cite{Ali2009_1}, and the theory of angular momentum \cite{Bosso2016_1}.
Furthermore, potential experimental tests have been proposed considering microscopic \cite{Bawaj2014_1} or macroscopic Harmonic Oscillators (HO) \cite{Marin2013_1}, or using Quantum Optomechanics \cite{Pikovski2012_1,Bosso2016_2}.
Therefore our analysis is motivated by the fact that while very accurate systems with
very little noise can be constructed, the energy perturbations and other results derived
in our paper will always be there, and potentially observable for highly
sensitive systems.
Similarly, any such deviations from the standard uncertainty profiles if observed, would
also be ascribed to new physics such as the GUP.
The most general modification of the HUP, was proposed in \cite{Ali2011_1} and includes linear and quadratic terms in momentum of the following form
\begin{equation}
[q,p] = i \hbar (1 - 2 \delta \gamma p + 4 \epsilon \gamma^2 p^2) \label{eqn:GUP1}
\end{equation}
in 1 dimension, where $\delta$ and $\epsilon$ are two dimensionless parameters defining the particular model, and
\begin{equation}\label{Mpscale}
\gamma = \frac{\gamma_0}{M_\mathrm{P} c}~,
\end{equation}
where $M_\mathrm{P}$ and $c$ are Planck mass and the speed of light, respectively, and $\gamma_0$ is a dimensionless parameter.
The presence of the quadratic term is dictated by string theory \cite{Gross1988_1,Amati1989_1} and \emph{gedanken} experiments in black hole physics \cite{Maggiore1993_1,Scardigli1999_1}, whereas the linear term is motivated by doubly special relativity \cite{AmelinoCamelia2002_1}, by the Jacobi identity of the corresponding $q_i,p_j$ commutators, and as a generalization of the quadratic model.
To incorporate both possibilities, we will leave the two parameters $\delta$ and $\epsilon$ undetermined, unless otherwise specified.
While it is possible to absorb $\gamma$ into redefinitions of $\delta$ and $\epsilon$, it is in practice useful to keep $\gamma$ distinct so that it can function as an expansion parameter under various circumstances.
In the present work, we revisit the problem of quantizing the HO \cite{Kempf1995_1} by incorporating the GUP \eqref{eqn:GUP1}, and rigorously investigate the implications of this for coherent and squeezed states.
Unlike previous investigations \cite{Bender1969,Dadic2003,Das2016,Quintela2016}, our focus is on a rigorous algebraic approach, which not only allows for a more efficient definition of the HO energy spectrum but also defines coherent and squeezed states of the HO with GUP \eqref{eqn:GUP1} in a consistent way.
Furthermore, we consider for the first time simultaneous perturbations in $p^3$ and $p^4$; previous studies focused on a $p^4$ perturbation only.
We will also consider perturbation theory up to second order in $\gamma$, finding finite results up to that order.
We are thus able to derive potentially observable deviations from the standard HO due to GUP or Planck scale effects.
In particular, we find a model-dependent modification of the HO energy spectrum, as well as modified position and momentum uncertainties for coherent and squeezed states.
Similar analyses concerning coherent and squeezed states of the HO for noncommutative spaces were carried out in \cite{Dey2012,Dey2013,Dey2015}.
A similar problem was also addressed in \cite{Ghosh2012}, although a number of terms in the Hamiltonian arising from perturbation were neglected.
This paper is organized as follows.
In Sec. \ref{sec:HO_GUP}, we consider a GUP-induced perturbation of the HO, computing perturbed energy eigenstates and eigenvalues.
Moving beyond the previous treatment of the HO with the GUP \cite{Kempf1995_1}, the algebraic method adopted here turns out to be more concise, allowing for definitions that will be useful in the rest of the paper.
A new set of ladder operators for the perturbed states is defined in Sec. \ref{sec:new_operators}, and in Sec. \ref{sec:coherent_states}, we consider coherent states, focusing on their position and momentum uncertainties, and their time evolution.
A similar analysis for squeezed states for a HO is performed in Sec. \ref{sec:squeezed_states}, with special reference to specific GUP models \cite{Kempf1995_1} and \cite{Ali2011_1}.
In Sec. \ref{sec:conclusions} we summarize our work, and comment on potential applications.
\section{Harmonic Oscillator in GUP} \label{sec:HO_GUP}
Consider the Hamiltonian for the one-dimensional HO
\begin{equation}
H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2 =
-\frac{\hbar \omega}{4} (a^\dagger - a)^2 + \frac{\hbar\omega}{4}(a + a^\dagger)^2 =
\hbar\omega\left(N + \frac{1}{2}[a,a^\dagger]\right)~, \label{eqn:Hamiltonian}
\end{equation}
where $N = a^\dagger a$ is the usual number operator.
Though in the standard theory $[a,a^\dagger]=1$, it is easy to show that this relation is no longer valid once GUP is incorporated.
Using the model in (\ref{eqn:GUP1}) and expanding the physical position $q$ and momentum $p$ in terms of the low-energy quantities, $q_0$ and $p_0$ respectively, we obtain \cite{Das2009_1},
\begin{equation}
q = q_0~, \qquad p = p_0 \left( 1 - \delta \gamma p_0 + 2 \gamma^2 \frac{\delta^2 + 2\epsilon}{3} p_0^2 \right)
\end{equation}
where $[q_0,p_0] = i \hbar$.
In terms of these quantities, we can write the Hamiltonian for the harmonic oscillator as
\begin{equation}
H = \frac{p_0^2}{2m} + \frac{1}{2}m\omega^2 q_0^2 - \delta \gamma \frac{p_0^3}{m} + \frac{7 \delta^2 + 8 \epsilon}{3} \gamma^2 \frac{p_0^4}{2 m}
\label{eqn:perturbed_Hamiltonian}
\end{equation}
to leading order in the parameters.
Notice that we consider at the same time perturbations proportional to $p^3$ and $p^4$, in contrast to previous studies along these lines \cite{Bender1969,Dadic2003,Das2016,Quintela2016}.
Using perturbation theory with perturbation parameter $\gamma$, we can define the state
\begin{equation}\label{Ksum}
|n^{(K)} \rangle = \sum_{k=0}^K \sigma_k \gamma^k |n_k \rangle
\end{equation}
to order $ \gamma^K$, where $ |n_k \rangle$ is the $k$-th order perturbation of the eigenvectors $ |n_0 \rangle = |n \rangle$ of
the unperturbed HO, whose eigenvalues are $E^{(0)} = \hbar \omega (n + 1/2)$, and the $\sigma_k$ are constant coefficients
that are calculable from perturbation theory.
It is tedious but straightforward to obtain the following expression for the
normalized perturbed Hamiltonian eigenstates up to second order in $\gamma$
\begin{multline}
|n^{(2)} \rangle = - \frac{\delta^2}{72} \gamma^2 \frac{\hbar m \omega}{2} \sqrt{n^{\underline{6}}} |n-6\rangle
+ \frac{\gamma^2}{16} \frac{\hbar m \omega}{2} \left[ \delta^2 \left( 4n - \frac{2}{3} \right) + \frac{8 \epsilon}{3} \right] \sqrt{n^{\underline{4}}} |n-4\rangle
- i \delta \frac{\gamma}{6} \sqrt{\frac{\hbar m \omega}{2}} \sqrt{n^{\underline{3}}}|n-3\rangle + \\
- \frac{\gamma^2}{8} \frac{\hbar m \omega}{2} \left[ \delta^2 \left( 7n^2 - \frac{29}{3} n - \frac{11}{3} \right) + \frac{16 \epsilon}{3} (2n - 1) \right] \sqrt{n^{\underline{2}}} |n-2\rangle
+ i \frac{3}{2} \delta \gamma \sqrt{\frac{\hbar m \omega}{2}} n \sqrt{n^{\underline{1}}} |n-1\rangle + \\
+ \left[1 - \delta^2 \frac{\gamma^2}{72} \frac{\hbar m \omega}{2} (164 n^3 + 246 n^2 + 256 n + 87)\right]|n\rangle
+ i \frac{3}{2} \delta \gamma \sqrt{\frac{\hbar m \omega}{2}} (n+1) \sqrt{(n+1)^{\overline{1}}} |n+1\rangle + \\
- \frac{\gamma^2}{8} \frac{\hbar m \omega}{2} \left[ \delta^2 \left( 7n^2 + \frac{71}{3} n + 13 \right) - \frac{16 \epsilon}{3} (2n + 3) \right] \sqrt{(n+1)^{\overline{2}}} |n+2\rangle - i \delta \frac{\gamma}{6} \sqrt{\frac{\hbar m \omega}{2}} \sqrt{(n+1)^{\overline{3}}} |n+3\rangle + \\
+ \frac{\gamma^2}{16} \frac{\hbar m \omega}{2} \left[ \delta^2 \left( 4n + \frac{14}{3} \right) - \frac{8 \epsilon}{3} \right] \sqrt{(n+4)^{\overline{4}}} |n+4\rangle
- \frac{\delta^2}{72} \gamma^2\frac{\hbar m \omega}{2} \sqrt{(n+1)^{\overline{6}}} |n+6\rangle~,
\label{eqn:perturbed_state}
\end{multline}
where we used the notation
\begin{align}
(n+1)^{\overline{k}} & = \frac{(n+k)!}{n!} ~, & n^{\underline{k}} & = \frac{n!}{(n-k)!} \quad \mbox{for }k \leq n~, & n^{\underline{k}} & = 0 \quad \mbox{for } k > n ~.
\end{align}
The perturbed energy eigenvalue is
\begin{equation}
E^{(2)} = \hbar \omega \left\{ \left( n + \frac{1}{2} \right) - \frac{\hbar m \omega}{2} \gamma^2 [ (4 n^2 + 4 n + 1) \delta^2 - (2 n^2 + 2 n + 1) 2 \epsilon ] \right\} = E^{(0)} + \Delta E ~, \label{eqn:energ_eigenvalue}
\end{equation}
where
\begin{align}
\Delta E = - \hbar \omega \frac{\hbar m \omega}{2} \gamma^2 [ (4 n^2 + 4 n + 1) \delta^2 - (2 n^2 + 2 n + 1) 2 \epsilon]~.
\end{align}
Furthermore, the spacing of the energy levels is
\begin{equation}
E^{(2)}(n+1) - E^{(2)}(n) = \hbar \omega - 8 \hbar \omega (n + 1) \frac{\hbar m \omega}{2} \gamma^2 [ \delta^2 - \epsilon ]~.
\end{equation}
Note that the linear and the quadratic contributions to \eqref{eqn:GUP1}, identifiable through the parameters $\delta$ and $\epsilon$, make opposing contributions to the energy.
It is therefore worth noting that, for the class of models with $\delta^2 = \epsilon$, the correction $\Delta E$ is independent of $n$, resulting in an equally spaced energy spectrum.
For $\delta^2 < \epsilon$, the correction is a positive function of the number $n$.
In particular, for $\delta = 0$ and $\epsilon = 1/4$, we obtain the results presented in \cite{Kempf1995_1} and \cite{Hossenfelder2003}.
Finally, we see that for $\delta^2 > \epsilon$ the spacing between energy levels decreases and we find the value of $n$ corresponding to the maximal energy
\begin{equation}
n_\mathrm{max} = \left\lfloor \frac{1}{8 \frac{\hbar m \omega}{2} \gamma^2 (\delta^2 - \epsilon)} - 1 \right\rfloor~.
\end{equation}
Notice that this depends on the Planck scale parameters, and the energy corresponding to the above $n$ signals the breakdown of the GUP model, and necessitates higher order terms.
To estimate the magnitude of these corrections, we compute
\begin{equation}
\frac{|\Delta E|}{E^{(0)}} = \frac{\hbar m \omega}{2} \gamma^2 \frac{\left|(4 n^2 + 4 n + 1) \delta^2 - (2 n^2 + 2 n + 1) 2 \epsilon\right|}{n + \frac{1}{2}} \stackrel{n\gg1}{\simeq} 4 n \frac{\hbar m \omega}{2} \gamma^2 \left|\delta^2 - \epsilon\right| \simeq 2 m E^{(0)} \gamma^2 \left|\delta^2 - \epsilon\right|~,
\end{equation}
Conversely, if we performed an experiment with a sensitivity $\Delta = |\Delta E|/ E^{(0)}$, we could detect a deviation from the unperturbed energies at
\begin{align}
& E^{(0)} = \frac{\Delta}{2 m \gamma^2 \left|\delta^2 - \epsilon\right|} = \frac{21.279 \Delta}{m \gamma_0^2 \left|\delta^2 - \epsilon\right|} \mathrm{J} \qquad \textrm{or} \quad
n = \frac{\Delta}{4 \frac{\hbar m \omega}{2} \gamma^2 \left|\delta^2 - \epsilon\right|} = \frac{20.169 \Delta}{ m \omega \gamma_0^2 \left|\delta^2 - \epsilon\right|} \times 10^{34}
\label{eqn:detection}
\end{align}
for $m$ and $\omega$ measured in Kg and in Hz, respectively.
Although this value appears huge, as we show below these Planck scale effects are potentially accessible to a number of current and future experiments.
We in particular see that massive and/or rapidly oscillating systems most significantly enhance GUP effects.
As particular examples, we considered the cases of the mechanical oscillator considered in \cite{Pikovski2012_1}, the oscillators considered in \cite{Bawaj2014_1}, the resonant-mass bar AURIGA in its first longitudinal mode \cite{Marin2013_1}, and the mirrors in LIGO oscillating at a frequency in the middle of its detection band \cite{Abbott2016}.
\begin{table}
\begin{center}
\begin{tabular}{lcccc}
\toprule
Type & Ref. & $m$ (Kg) & $\omega/2\pi$ (Hz) & $n / \displaystyle{\frac{\Delta}{\gamma_0^2 \left|\delta^2 - \epsilon\right|}}$ \\
\midrule
Optomechanical system & \cite{Pikovski2012_1} & $10^{-11}$ & $10^5$ & $ 3 \times 10^{40}$ \\
Bar detector AURIGA &\cite{Marin2013_1} & $1.1 \times 10^{3}$ & $900$ & $3 \times 10^{28}$ \\
\multirow{4}{*}{Mechanical oscillators} & \multirow{4}{*}{\cite{Bawaj2014_1}} & $3.3 \times 10^{-5}$ & $5.64 \times 10^3$ & $2 \times 10^{35}$ \\
& & $7.7 \times 10^{-8}$ & $1.29 \times 10^5$ & $3 \times 10^{36}$ \\
& & $2 \times 10^{-8}$ & $1.42 \times 10^5$ & $10^{37}$ \\
& & $2 \times 10^{-11}$ & $7.47 \times 10^5$ & $2 \times 10^{39}$ \\
LIGO detector & \cite{Abbott2016} & 40 & 200 & $4 \times 10^{30}$\\
\bottomrule
\end{tabular}
\caption{Some relevant examples of HO are considered, including their mass, frequency and levels at which GUP effects become dominant, as given by \eqref{eqn:detection}.} \label{tbl:detection}
\end{center}
\end{table}
\section{New GUP modified ladder operators} \label{sec:new_operators}
Given the form of the perturbed eigenstates in terms of the standard number states (\ref{eqn:perturbed_state}), it is easy to show that the standard annihilation and creation operators do not act anymore as ladder operators.
We therefore define a new set of operators, useful for constructing coherent and squeezed states, such that
\begin{align}
\tilde{a} | n^{(2)} \rangle = & \sqrt{n} ~ | (n-1)^{(2)} \rangle~, & \widetilde{a^\dagger} | n^{(2)} \rangle = & \sqrt{n+1} ~ | (n+1)^{(2)} \rangle~, & \tilde{N} | n^{(2)} \rangle = & n ~ | n^{(2)} \rangle ~.
\label{defs:tilde_ops}
\end{align}
Note that these operators obey the following relations
\begin{align}
\widetilde{a^\dagger} = & \tilde{a}^\dagger~, & \tilde{N} = & \tilde{a}^\dagger \tilde{a} ~, & [\tilde{a},\tilde{a}^\dagger] = & 1~.
\end{align}
Using these definitions, we find the following expressions for the operators in \eqref{defs:tilde_ops} in terms of the standard annihilation, creation and number operators:
\begin{subequations} \label{eqns:new_ops}
\begin{align}
\tilde{a} = & a
- \delta i \frac{\gamma}{2} \sqrt{\frac{\hbar m \omega}{2}} \left[ 3 a^2 + 3 (2N+1) - {a^\dagger}^2 \right] + \nonumber \\
& - 2 \gamma^2 \frac{\hbar m \omega}{2} \left[ \left( \frac{5}{3} \delta^2 - \frac{2}{3} \epsilon \right) a^3 + \delta^2 a N + \left( \delta^2 + 2 \epsilon \right) N a^\dagger - \left( \frac{2}{3} \delta^2 + \frac{1}{3} \epsilon \right) {a^\dagger}^3 \right]~,
\label{a-tilde}\\
\widetilde{a^\dagger} = & a^\dagger
+ \delta i \frac{\gamma}{2} \sqrt{\frac{\hbar m \omega}{2}} \left[ 3{a^\dagger}^2 + 3(2N+1) - a^2 \right] + \nonumber \\
& - 2 \gamma^2 \frac{\hbar m \omega}{2} \left[ \left( \frac{5}{3} \delta^2 - \frac{2}{3} \epsilon \right) {a^\dagger}^3 + \delta^2 N a^\dagger + ( \delta^2 + 2 \epsilon ) a N - \left(\frac{2}{3} \delta^2 + \frac{1}{3} \epsilon \right) a^3 \right]~,
\label{a-tilde-dag} \displaybreak\\
\tilde{N} = & N
- \delta i \frac{\gamma}{2} \sqrt{\frac{\hbar m \omega}{2}} \left[ a^3 - 3 ( a N - N a^\dagger ) - {a^\dagger}^3 \right] + \nonumber \\
& + \frac{\gamma^2}{2} \frac{\hbar m \omega}{2} \left\{ \frac{7 \delta^2 + 8 \epsilon}{6} [ a^4 - 2 (2 a N a + a^2) - 2 (2 a^\dagger N a^\dagger + {a^\dagger}^2) + {a^\dagger}^4 ] + \frac{\delta^2}{2} (30 N^2 + 30 N + 11) \right\}
\label{N-tilde}
\end{align}
\end{subequations}
To obtain these relations, we used the following procedure.
Consider, for example, the annihilation operator $\tilde{a}$ defined by (\ref{defs:tilde_ops}).
Applying this operator on the perturbed number state (\ref{eqn:perturbed_state}), we notice that to the 0-th order in $\gamma$ we get $\tilde{a} |n\rangle = a |n\rangle$, as expected. Requiring \eqref{defs:tilde_ops} to hold to order $\gamma$,
we compute
\begin{equation}
(\tilde{a} - a) \left|n^{(1)}\right\rangle = \sqrt{n} \left|(n-1)^{(1)} \right> - a \left|n^{(1)} \right> = - \delta i \frac{\gamma}{2} \sqrt{\frac{\hbar m \omega}{2}} \left[ 3 a^2 + 3 (2N+1) - {a^\dagger}^2 \right] \left|n^{(1)} \right>
\end{equation}
where $\left|n^{(1)}\right\rangle$ is obtained from the order-$\gamma$ terms in \eqref{eqn:perturbed_state}.
Therefore $\tilde{a} = a - \delta i \frac{\gamma}{2} \sqrt{\frac{\hbar m \omega}{2}} \left[ 3 a^2 + 3 (2N+1) - {a^\dagger}^2 \right]$
to first order in $\gamma$. Repeating this procedure to order $\gamma^2$ yields
\begin{multline}
\left\{\tilde{a} - a + \delta i \frac{\gamma}{2} \sqrt{\frac{\hbar m \omega}{2}} \left[ 3 a^2 + 3 (2N+1) - {a^\dagger}^2 \right] \right\} \left| n^{(2)} \right> = \\
- 2 \gamma^2 \frac{\hbar m \omega}{2} \left[ \left( \frac{5}{3} \delta^2 - \frac{2}{3} \epsilon \right) a^3 + \delta^2 a N + \left( \delta^2 + 2 \epsilon \right) N a^\dagger - \left( \frac{2}{3} \delta^2 + \frac{1}{3} \epsilon \right) {a^\dagger}^3 \right] \left| n^{(2)} \right>
\end{multline}
which gives \eqref{a-tilde}.
A similar procedure can be used for $\tilde{a^\dagger}$ and $\tilde{N}$.
Continuing this procedure, one can extend the relations in (\ref{eqns:new_ops}) to any arbitrary order.
Using these expressions, the Hamiltonian of an harmonic oscillator with GUP can be written in a simple form:
\begin{equation}
H = \hbar \omega \left\{ \left( \tilde{N} + \frac{1}{2} \right) - \frac{\hbar m \omega}{2} \gamma^2 [ 4 (\tilde{N}^2 + \tilde{N}) (\delta^2 - \epsilon) + \delta^2 - 2 \epsilon] \right\}
\label{eqn:Hamiltonian_tilde}
\end{equation}
as well as
\begin{subequations} \label{eqn:expansions_q&p}
\begin{align}
q = & (\tilde{a}^\dagger + \tilde{a}) \sqrt{\frac{\hbar}{2 m \omega}}
- 2 i (\tilde{a}^\dagger {}^2 - \tilde{a}^2 ) \delta \frac{\hbar}{2} \gamma
- 2 [ ( \tilde{a}^3 + \tilde{a}^\dagger {}^3 ) ( 2 \delta^2 + \epsilon) + (\tilde{a} \tilde{N} + \tilde{N} \tilde{a}^\dagger) (3 \delta^2 - 2 \epsilon) ] \frac{\hbar}{2} \sqrt{\frac{\hbar m \omega}{2}} \gamma^2 \\
p = & i (\tilde{a}^\dagger - \tilde{a}) \sqrt{\frac{\hbar m \omega}{2}}
+ 2 (\tilde{a}^\dagger {}^2 + 2 \tilde{N} + 1 + \tilde{a}^2) \delta \frac{\hbar m \omega}{2} \gamma
+ 2 i [( \tilde{a}^3 - \tilde{a}^\dagger {}^3 ) ( 2 \delta^2 + \epsilon) + 3 (\tilde{a} \tilde{N} - \tilde{N} \tilde{a}^\dagger ) \delta^2 ] \left(\frac{\hbar m \omega}{2}\right)^{3/2} \gamma^2
\end{align}
\end{subequations}
for the physical position and momentum operators.
It is then easy to find the expectation values and the uncertainties in position and momentum for the perturbed number states.
Defining $ \langle A \rangle = \langle n^{(2)}| A | n^{(2)} \rangle$ for any operator $A$, we find
\begin{subequations}
\begin{align}
\langle q \rangle = & 0 ~, \\
\langle q^2 \rangle = & \frac{\hbar}{2 m \omega} (2 n + 1) - 4 \gamma^2 \frac{\hbar^2}{4} \left[ \delta^2 (2n + 1)^2 - 2 \epsilon (2 n^2 + 2 n + 1) \right]~ ,\\
(\Delta q)^2 = & \langle q^2 \rangle ~, \\
\langle p \rangle = & 2 \delta \gamma \frac{\hbar m \omega}{2} (2n + 1) ~, \label{19d} \\
\langle p^2 \rangle = & \frac{\hbar m \omega}{2} (2n + 1)~,\\
(\Delta p)^2 = & \frac{\hbar m \omega}{2} (2n + 1) \left[ 1 - 4 \delta^2 \gamma^2 \frac{\hbar m \omega}{2} (2n + 1) \right]~, \label{19f}\\
(\Delta q)^2 (\Delta p)^2 = & \frac{\hbar^2}{4} (2n + 1)^2 - 8 \gamma^2 \frac{\hbar^2}{4} \frac{\hbar m \omega}{2} (2n + 1) [ \delta^2 (2n + 1)^2 - \epsilon (2n^2 + 2n + 1)]
\end{align}
\end{subequations}
The presence of $\delta$ in \eqref{19d} and \eqref{19f} leads to interesting features.
First, the expectation value of the momentum does not vanish whereas the expectation value of the position does.
This is a consequence of the linear part of the GUP \eqref{eqn:GUP1}, which singles out a preferred direction and so breaks translation invariance.
Furthermore, this same term implies that there exists a value for $n$ such that $(\Delta p)^2 < 0$, showing that a linear model leads to critical results for high energy systems, \emph{i.e.} when the energy is close to the Planck scale.
We expect that at this energy scale a full theory of Quantum Gravity has to be considered, with non-negligible higher order corrections.
To conclude this section, we observe that the definitions in \eqref{defs:tilde_ops} and the methods developed in this section can be easily extended to any perturbation
that is of the form of a polynomial of $q$ and $p$ {\it and} to any order in perturbation theory.
Using this method, the full Hamiltonian can be written in the following form
\begin{equation}
H = \hbar \omega \left\{ \left(\tilde{N} + \frac{1}{2} \right) + \sum_{i=0}^{S} c_i \tilde{N}^i \right\}~, \qquad \mbox{with} \qquad S = \left\{ \begin{array}{cc} 0 \quad & \mbox{for } d \mbox{ an odd number and } e = 1\\
\lfloor (d + e - 1)/2 \rfloor \quad & \mbox{otherwise}
\end{array} \right. \label{eqn:gen_Hamiltonian}
\end{equation}
where $d$ is the degree of the polynomial representing the perturbation and $e$ the order of the perturbation theory.
The coefficients $c_i$ are constants that depend on the parameters of the perturbation.
In this paper we consider the case with
\begin{subequations}
\begin{align}
d = & 4~, & e = & 2~, & S = 2~,\\
c_0 = & -\frac{\hbar m \omega}{2}\gamma^2 (\delta^2 - 2\epsilon)~, & c_1 = & c_2 = - 4 \frac{\hbar m \omega}{2}\gamma^2 (\delta^2 - \epsilon)~.
\end{align}
\end{subequations}
Furthermore, we emphasize that $|n^{(2)}\rangle$ are the eigenstates of the full Hamiltonian with GUP (\ref{eqn:Hamiltonian}) up to $\mathcal{O}(\gamma^2)$, while the standard states $|n\rangle$ are not.
Therefore, we will consider the former as the physical number states.
Furthermore, the $\sim$-operators that we have defined have the same properties, in terms of commutators and actions on number states, as the analogous operators of the standard theory.
For these reasons, we apply the above new operators to define coherent and squeezed states.
\section{Coherent States} \label{sec:coherent_states}
As in the standard theory, we define a coherent state as the eigenstate of the annihilation operator $\tilde{a}$
\begin{equation}
\tilde{a} | \alpha^{(2)} \rangle = \alpha |\alpha^{(2)} \rangle \label{def:coherent_states}
\end{equation}
to order $\gamma^2$.
This is because $|n^{(2)}\rangle$ is the physical number state and $\tilde{a}$ is its operator.
Following the results of the previous section, we then have
\begin{equation}
|\alpha^{(2)} \rangle = e^{- \frac{|\alpha|^2}{2}} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n^{(2)}\rangle~, \label{eqn:expansion_coherent_states}
\end{equation}
\emph{i.e.} a Poisson distribution of number states in $\sim$-coherent states\footnote{We can also define a displacement operator of the form $\displaystyle{ \tilde{\mathcal{D}}(\alpha) = e^{\alpha \tilde{a}^\dagger - \alpha^* \tilde{a}}}$.}.
Furthermore, given the expansions in (\ref{eqn:expansions_q&p}), we can easily obtain the following results for the uncertainties
\begin{subequations} \label{eqns:uncs_coherent}
\begin{align}
\langle q \rangle = & \left( \alpha^\star + \alpha \right) \sqrt{\frac{\hbar}{2 m \omega}}
- 2 i \delta \left( {\alpha^\star}^2 - \alpha^2\right) \frac{\hbar}{2} \gamma + \nonumber \\
& + \left[ (2 \epsilon - 3 \delta^2) (\alpha^\star + \alpha) ( 1 + |\alpha|^2) - ( \epsilon + 2 \delta^2) \left( {\alpha^\star}^3 + \alpha^3 \right) \right] \frac{\hbar}{2} \sqrt{\frac{\hbar m \omega}{2}} \gamma^2 ~, \\
(\Delta q)^2 = & \frac{\hbar}{2 m \omega} - 4 i \delta \left( \alpha^\star - \alpha\right) \frac{\hbar}{2} \sqrt{\frac{\hbar}{2 m \omega}} \gamma + \nonumber \\
& - 2 \left[ \epsilon \left(- 4 + \alpha^2 - 8 |\alpha|^2 + {\alpha^\star}^2\right) + \delta^2 \left( 2 + 9 \alpha^2 + 4 |\alpha|^2 + 9 {\alpha^\star}^2 \right) \right] \frac{\hbar^2}{4} \gamma^2~,\\
\langle p \rangle = & i \left(\alpha^\star-\alpha\right) \sqrt{\frac{\hbar m\omega}{2}} + 2 \delta \left( {\alpha^\star}^2 + 2 |\alpha|^2 + \alpha^2 + 1 \right) \frac{\hbar m \omega}{2} \gamma + \nonumber \\
& - i \left[ 3 \delta^2 (1 + |\alpha|^2) (\alpha^\star - \alpha) + (\epsilon + 2 \delta^2) \left( {\alpha^\star}^3 - \alpha^3\right) \right] \left( \frac{\hbar m \omega}{2} \right)^{3/2} \gamma^2~, \\
(\Delta p)^2 = & \frac{\hbar m \omega}{2} + 2 \left(\frac{\hbar m \omega}{2}\right)^2 \gamma^2 \left[ ( {\alpha^\star}^2 + \alpha^2) (5 \delta^2 - 3 \epsilon) + (4 |\alpha|^2 - 2)\delta^2 \right] ~,\displaybreak\\
(\Delta q)^2 (\Delta p)^2 = & \frac{\hbar^2}{4} \left\{ 1
- 4 i \delta \left( \alpha^\star - \alpha \right) \sqrt{\frac{\hbar m \omega}{2}} \gamma
+8 \left[ \epsilon \left( 1 + \alpha - \alpha^\star \right) \left( 1 - \alpha + \alpha^\star \right) - \delta^2 \left( 1 + \alpha^2 + {\alpha^\star}^2 \right) \right] \frac{\hbar m \omega}{2} \gamma^2 \right\}
\label{eqn:prod_unc}
\end{align}
\end{subequations}
where we see that the last term exhibits interesting properties.
If $\epsilon > 0$ and $\delta=0$ then the uncertainty $(\Delta q)^2 (\Delta p)^2 $ is greater than $\frac{\hbar^2}{4}$ whereas if $\epsilon = 0$ and $\delta>0$ then this quantity is smaller than $\frac{\hbar^2}{4}$.
The same situation holds for the minimal uncertainty product, which is the smallest uncertainty product predicted by QM as found using the Schr\"odinger-Robertson relation.
For the model in (\ref{eqn:GUP1}), we obtain
\begin{multline}
[(\Delta q)^2 (\Delta p)^2]_\mathrm{min} = \left| \frac{\langle qp + pq \rangle}{2} - \langle q \rangle \langle p \rangle \right|^2 + \frac{|\langle[q,p]\rangle|^2}{4} = \\
= \frac{\hbar^2}{4} \left\{ 1
- 4 i \delta \left( \alpha^\star - \alpha \right) \sqrt{\frac{\hbar m \omega}{2}} \gamma
+8 \left[ \epsilon \left( 1 + \alpha - \alpha^\star \right) \left( 1 - \alpha + \alpha^\star \right) - \delta^2 \left( 1 + \alpha^2 + {\alpha^\star}^2 \right) \right] \frac{\hbar m \omega}{2} \gamma^2 \right\}
\end{multline}
and again nonzero $\epsilon$
makes this quantity larger than the standard QM value whereas non-zero $\delta$ makes it smaller.
Finally we note that
\begin{equation}
(\Delta q)^2 (\Delta p)^2 - [(\Delta q)^2 (\Delta p)^2]_\mathrm{min} = 0 \qquad \forall \alpha, \gamma, \delta, \epsilon~ .
\label{eqn:diff_coherent}
\end{equation}
and so coherent states are also minimal uncertainty states when the GUP is included.
The quantity $(\Delta q)^2 (\Delta p)^2$ is the actual product of the uncertainties in position and momentum.
Therefore, we see that coherent states as defined in \eqref{def:coherent_states} satisfy all the known properties of the standard theory, \emph{i.e.} they follow a Poisson distribution, they are eigenstates of the annihilation operator, and are minimal uncertainty states.
The uncertainty product $(\Delta q)^2 (\Delta p)^2$ in \eqref{eqn:prod_unc} is negative for some values of $\alpha$ and of $\sqrt{\frac{\hbar m \omega}{2}} \gamma$. This is due to the linear term in GUP and will go negative for
\begin{equation}
\delta^2 > \frac{1 + 4 \Im^2(\alpha)}{1 + 2 \Re^2 (\alpha)} \epsilon~.
\end{equation}
In this case, the model will present anomalies when the characteristic momentum of the system fulfills the following relation
\begin{equation}
\sqrt{\frac{\hbar m \omega}{2}} \gamma = \frac{ - \Im (\alpha) \delta \pm \sqrt{ 2 \left[1 + 2 \Re^2 (\alpha) \right] \delta^2 - 2 \left[1 + 4 \Im^2 (\alpha) \right] \epsilon}}{ 2 \left[1 + 2 \Re (\alpha^2) \right] \delta^2 - 2 \left[ 1 + 4 \Im^2 (\alpha) \right] \epsilon}~. \label{eqn:negative_unc_pro}
\end{equation}
Notice that this quantity depends on the phase of $\alpha$ and that one cannot \emph{a priori} choose the sign in the expression.
\subsection{Time Evolution} \label{subsec:time_coherent}
To consider the time-evolved coherent states, we analyze them in Heisenberg picture.
Since we seek to understand the evolution of the variances \eqref{eqns:uncs_coherent}, we first write the time-evolved annihilation operator
\begin{equation}\label{a-tilde-t}
\tilde{a}(t) = \tilde{a}(0) \exp \left\{ - i \omega t \left[ 1 - 8 \tilde{N} \left( \delta^2 - \epsilon \right) \frac{\hbar m \omega}{2} \gamma^2 \right] \right\} \equiv \tilde{a}(0) \exp \left[ - i \omega t \left( 1 - 8 \tilde{N} \chi \right) \right] ~,
\end{equation}
where we have defined the dimensionless quantity
\begin{equation}
\chi = \left( \delta^2 - \epsilon \right) \frac{\hbar m \omega}{2} \gamma^2~.
\end{equation}
Notice that for the number operator we have $\tilde{N}(t) = \tilde{N}(0)$, and therefore the statistical properties of the coherent states do not change.
Using the time-evolved operators \eqref{a-tilde-t}, we find
\begin{subequations}
\begin{align}
\left( \Delta q \right)^2 = & \frac{\hbar}{2 m \omega} \left\{ 1 + 2 |\alpha|^2 \left[ 1 - e^{- 4 |\alpha|^2 \sin^2 \left[4 \chi \omega t\right]} \right] \right\} \nonumber \\
& + \alpha^2 \frac{\hbar}{2 m \omega} \left\{ e^{-|\alpha|^2 \left[1 - e^{16 i \chi \omega t}\right] - 2 i \omega t \left[1 - 12 \chi \right]}
- e^{- 2 |\alpha|^2 \left[1 - e^{8 i \chi \omega t} \right] - 2 i \omega t \left[1 - 8 \chi \right] } \right\} \nonumber \\
& + \alpha^\star{}^2 \frac{\hbar}{2 m \omega} \left\{ e^{-|\alpha|^2 \left[1 - e^{-16 i \chi \omega t}\right] + 2 i \omega t \left[1 - 12 \chi\right]}
- e^{-2 |\alpha|^2 \left[1 - e^{- 8 i \chi \omega t} \right]+ 2 i \omega t \left[1 - 8 \chi \right] } \right\} \nonumber \\
& + 4 i \sqrt{\frac{\hbar}{2 m \omega}} \frac{\hbar}{2} \left( \alpha e^{- i \omega t } - \alpha^\star e^{i \omega t} \right) \delta \gamma
- 2 \frac{\hbar^2}{4}\left[ 2 (\delta^2 - 2 \epsilon) ( 1 - 2 |\alpha|^2) + (9 \delta^2 + \epsilon) \left( \alpha^2 e^{- 2 i \omega t} + \alpha^\star {}^2 e^{2 i \omega t} \right) \right] \gamma^2 ~, \\
\left( \Delta p \right)^2 = & \frac{\hbar m \omega}{2}\left\{1 + 2|\alpha|^2\left[1 - e^ {- 4 |\alpha|^2 \sin^2 \left[4 \chi \omega t\right] } \right]\right\} \nonumber \\
& + \alpha^2 \frac{\hbar m \omega}{2} \left\{ e^{-2 |\alpha|^2 \left[1 - e^{8 i \chi \omega t} \right] - 2 i \omega t \left[1 - 8 \chi \right]}
- e^{ - |\alpha|^2 \left[1 - e^{16 i \chi \omega t }\right] - 2 i \omega t \left[1 - 12 \chi \right]} \right\} \nonumber \\
& + \alpha^\star {}^2 \frac{\hbar m \omega}{2} \left\{ e^{- 2 |\alpha|^2 \left[1 - e^{- 8 i \chi \omega t } \right] + 2 i \omega t \left[1 - 8 \chi t \right]} - e^{- |\alpha|^2 \left[1 - e^{- 16 i \chi \omega t } \right] + 2 i \omega t \left[1 - 12 \chi t \right] } \right\} \nonumber \\
& + 2 \left(\frac{\hbar m \omega}{2} \right)^2 \left[ \left(5 \delta^2-3 \epsilon\right) \left(\alpha^2 e^ {- 2 i \omega t } + \alpha^\star {}^2 e^{2 i \omega t}\right) - 2 \delta^2 (1 - 2 |\alpha|^2) \right] \gamma^2
\end{align}
\end{subequations}
for the uncertainties in position and momentum.
These two expressions indicate that the GUP imputes important effects on the uncertainties of position and momentum.
Consider the first line of each expression.
We notice an oscillation in the uncertainties, governed by the term $\sin^2 \left[4 \chi \omega t\right]$, with amplitude proportional to the standard uncertainties multiplied by $|\alpha|^2$.
This term, initially 0 for $t=0$, varies with time with a period
\begin{equation}
T = \left| \frac{\pi}{4 \chi \omega} \right|~. \label{eqn:unc_period}
\end{equation}
It is worth noticing that the period depends on $m$, $\omega$, and the GUP parameters through the quantity $\chi$.
The subsequent two terms in each expression present an oscillation with a similar period.
This oscillation is present only for models with $\delta^2 \not = \epsilon$.
For the systems considered at the end of Sec. \ref{sec:HO_GUP}, we have the Table \ref{tbl:unc_osc_period}.
\begin{table}
\begin{center}
\begin{tabular}{lcccc}
\toprule
Type & Ref. & $m$ (Kg) & $\omega/2\pi$ (Hz) & $T \gamma_0^2 \left|\delta^2 - \epsilon\right| (s)$ \\
\midrule
Optomechanical system & \cite{Pikovski2012_1} & $10^{-11}$ & $10^5$ & $ 1.61 \times 10^{35}$ \\
Bar detector AURIGA & \cite{Marin2013_1} & $1.1 \times 10^{3}$ & $900$ & $1.81 \times 10^{25}$ \\
\multirow{4}{*}{Mechanical oscillators} & \multirow{4}{*}{\cite{Bawaj2014_1}} & $3.3 \times 10^{-5}$ & $5.64 \times 10^3$ & $1.53 \times 10^{31}$ \\
&& $7.7 \times 10^{-8}$ & $1.29 \times 10^5$ & $1.26 \times 10^{31}$ \\
& & $2 \times 10^{-8}$ & $1.42 \times 10^5$ & $3.99 \times 10^{31}$ \\
& & $2 \times 10^{-11}$ & $7.47 \times 10^5$ & $1.44 \times 10^{33}$ \\
LIGO detector &\cite{Abbott2016} & 40 & 200 & $1.00 \times 10^{28}$\\
\bottomrule
\end{tabular}
\caption{Period of oscillation of $(\Delta q)^2$ and $(\Delta p)^2$ for several systems, as given by \eqref{eqn:unc_period}. \label{tbl:unc_osc_period}}
\end{center}
\end{table}
We see that in all cases we have a period several orders of magnitude larger than the age of the universe.
As for the uncertainty product, we have
\begin{multline}
\left(\Delta q\right)^2 \left(\Delta p\right)^2 =
\frac{\hbar^2}{4} \left\{1 + 4 |\alpha|^2 \left[1 - e^{- 2 |\alpha|^2 \sin^2 (4 \chi \omega t)}\right]
+ 2 |\alpha|^4 \left[2 + e^{- |\alpha|^2 \left[3 - 2 e^{8 i \chi \omega t} - e^{- 16 i \chi \omega t} \right] - 8 i \chi \omega t } + \right. \right. \\
\left. \left.
+ e^{- |\alpha|^2 \left[3 - 2 e^{-8 i \chi \omega t} - e^{16 i \chi \omega t} \right] + 8 i \chi \omega t }
- e^{- 2 |\alpha|^2 \sin^2 \left[8 \chi \omega t\right]}
+ e^{- 4 |\alpha|^2 \sin^2 \left[4 \chi \omega t\right]}
- 4 e^{- 2 |\alpha|^2 \sin^2 \left[4 \chi \omega t\right]} \right] \right\}\\
- \alpha^4 \frac{\hbar^2}{4} \left\{e^{- |\alpha|^2 \left[1 - e^{16 i \chi \omega t} \right] - 2 i \omega t \left[1 - 12 \chi \right] } - e^{- 2 |\alpha|^2 \left[1 - e^{ 8 i \chi \omega t}\right] - 2 i \omega t \left[1 - 8 \chi \right]} \right\}^2 + \\
- \alpha^\star {}^4 \frac{\hbar^2}{4} \left\{e^{- |\alpha|^2 \left[1 - e^{- 16 i \chi \omega t} \right] + 2 i \omega t \left[1 - 12 \chi \right] } - e^{- 2 |\alpha|^2 \left[1 - e^{ - 8 i \chi \omega t}\right] + 2 i \omega t \left[1 - \chi \right]} \right\}^2 + \\
- 8 \frac{\hbar m \omega}{2} \frac{\hbar^2}{4} \left[ (\delta^2-\epsilon)
- 2 |\alpha|^2 \epsilon
+ (\delta^2+\epsilon) \left(\alpha^2 e^{- 2 i \omega t } + \alpha^\star {}^2 e^{2 i \omega t}\right) \right] \gamma^2
+ 4 i \frac{\hbar^2}{4} \sqrt{\frac{\hbar m \omega}{2}} \delta \left[\alpha e^ {- i \omega t } - \alpha^\star e^{i \omega t}\right] \gamma
\end{multline}
and shows variations similar to those previously analyzed.
We conclude this section by showing that similar results can be obtained to any order in perturbation theory for a large class of perturbations.
In this case it is convenient to use the Schr\"odinger picture, in which
each number state component of a coherent state acquires a different phase.
In fact, let us rewrite the Hamiltonian in \eqref{eqn:gen_Hamiltonian} as
\begin{equation}
H = \hbar \omega \left( \sum_{i=0}^S \kappa_i \tilde{N}^i \right)~,
\end{equation}
where $\kappa_0 = c_0 + \frac{1}{2}$, $\kappa_1 = c_1 + 1$, and $\kappa_i = c_i$ with $i\geq 2$.
A time-evolved coherent state can then be written as
\begin{equation}
|\alpha,t\rangle = \exp \left[- i \omega t \left( \sum_{i=0}^S \kappa_i \tilde{N}^i \right)\right] |\alpha,0\rangle \propto \sum_{n=0}^\infty \frac{\left(\alpha e^{-i \kappa_1 \omega t}\right)^n}{\sqrt{n!}} \exp \left[- i \omega t \left( \sum_{i=2}^S \kappa_i n^i \right)\right] |n^{(e)}\rangle~,
\end{equation}
where $|n^{(e)}\rangle$ is the perturbed energy eigenstate up to order $\pi$ such that $\tilde{N} |n^{(e)}\rangle = n |n^{(e)}\rangle$.
One can then prove that
\begin{subequations}
\begin{align}
\langle \alpha,t|q|\alpha,t\rangle \propto & \sum_{n=0}^\infty \frac{\alpha^{2n}}{n!} \cos \left\{ t \left[ \omega \kappa_1 + (\omega_{n+1} - \omega_n) \right] \right\}~,\\
\langle \alpha,t|p|\alpha,t\rangle \propto & \sum_{n=0}^\infty \frac{\alpha^{2n}}{n!} \sin \left\{ t \left[ \omega \kappa_1 + (\omega_{n+1} - \omega_n) \right] \right\}~,
\end{align}
\end{subequations}
where $\omega_n = \omega \left( \sum_{i=2}^S \kappa_i n^i \right)$.
We then see that the coherent state spreads in phase-space, since each component moves with a different frequency
\begin{equation}
\Omega_n = \omega \left\{\kappa_1 + \sum_{i=2}^S \kappa_i \sum_{j=1}^i \binom{i}{j} n^{i-j}\right\}~.
\end{equation}
In a frame rotating at the frequency of the ground state, the other states move with a frequency
\begin{equation}
\Delta \Omega_n = \Omega_n - \Omega_0 = \omega \sum_{i=2}^S \kappa_i \sum_{j=1}^{i-1} \binom{i}{j} n^{i-j}
\end{equation}
We have two cases:
\begin{itemize}
\item Only one coefficient $\kappa_i$ is different from 0: then all the frequencies are multiple of $\omega_1 = \omega \kappa_i$ and the time interval over which the coherent state spreads and restores is given by
\begin{equation}
T = \frac{2\pi}{|\kappa_i| \omega}~.
\end{equation}
\item $\kappa_i = \kappa \mu_i$, with $\kappa \in \mathbb{R}$ and $\mu_i \in \mathbb{Q}$: in this case we have
\begin{equation}
T = \frac{2 \pi}{\omega |\kappa|} \frac{\zeta}{\xi}~,
\end{equation}
where $\zeta$ is the least common denominator of $\mu_i$'s and $\xi$ is the greatest common divisor of the numerator of $\mu_i$'s.
\end{itemize}
In many cases, this is an over estimation, since when $i$ is a prime number, the actual period is $T' = T/i$, or, if $i$ is a power of a prime number $k$, then one can take $T' = T/k$ as the period.
Finally, notice that the result in \eqref{eqn:unc_period} corresponds to the first case with $\kappa_2=-4\chi$ and that in general the period depends on the HO frequency $\omega$ and on the HO mass $m$ and the GUP parameters included in $\kappa_i$.
\begin{figure}
\center
\begin{subfigure}[t]{0.47\textwidth}
\center
\includegraphics[scale=0.54]{unc_pro_t_q-standalone.pdf}
\caption{The time evolution of the uncertainty product for coherent states is plotted for three values of $\chi$ in a model with $\delta=0$ and $\epsilon=1$ (quadratic GUP).
The black dashed line corresponds to the standard case.
The results are given in natural units.
Notice that each case presents small oscillations superposed to the dominant $\sin^2$ term.
Also, notice that the minimal value of the uncertainty product increases with $|\chi|$.} \label{fig:unc_pro_t_q}
\end{subfigure}
\qquad
\begin{subfigure}[t]{0.47\textwidth}
\includegraphics[scale=0.54]{unc_pro_t_l-standalone.pdf}
\caption{Time evolution of the uncertainty product for coherent states for three values of $\chi$ in a model with $\delta=1$ and $\epsilon=0$ (linear GUP).
Similar observations as in the previous figure are valid here.
Here it is evident that a linear model can imply negative uncertainties and uncertainty product.} \label{fig:unc_pro_t_l}
\end{subfigure}
\begin{subfigure}{\textwidth}
\center
\includegraphics[scale=0.54]{unc_pro_t_l1-standalone.pdf}
\caption{Similar case as Figure \ref{fig:unc_pro_t_l}, but with the condition \eqref{eqn:negative_unc_pro} not satisfied. In this case we see that the uncertainty products are positive.} \label{fig:unc_pro_t_l1}
\end{subfigure}
\end{figure}
\section{Squeezed States} \label{sec:squeezed_states}
Next we construct squeezed states with GUP up to second order in $\gamma$.
Following \cite{Lu1972_1}, we define a new set of operators
\begin{align}
\tilde{a}_r = & \tilde{a} \cosh r - e^{i\theta} \tilde{a}^\dagger \sinh r~, & \tilde{a}^\dagger_r = \tilde{a}^\dagger \cosh r - e^{- i\theta} \tilde{a} \sinh r~, \label{def:squeezed_ac_operators}
\end{align}
where $r$ is the squeeze parameter and $\theta$ a phase angle.
We can proceed as in the standard theory, defining a squeezed vacuum state through
\begin{equation}
\tilde{a}_r |0\rangle_r = 0
\end{equation}
and we find that $|0\rangle_r$ can be expanded in terms of even numbered states only
\begin{equation}
|0\rangle_r = \sum_{n=0}^\infty b_{2n} |2n^{(2)}\rangle~,
\end{equation}
with\footnote{Alternatively, to define squeezed states is through the squeeze operator \mbox{$\tilde{\mathcal{S}}(z) = \exp[-\frac{1}{2} ( z \tilde{a}^2 - z^\star \tilde{a}^\dagger {}^2) ]$}, where $z=r e^{i\theta}$.
In this case we have
\begin{align}
\tilde{a}_z = & \tilde{\mathcal{S}}^\dagger (z) ~ \tilde{a} ~ \tilde{\mathcal{S}}(z) & |0^{(2)}\rangle_z = \tilde{\mathcal{S}}(z) |0^{(2)}\rangle
\end{align}}
\begin{align}
b_{2n} = & (\tanh r ~ e^{i\theta})^n \sqrt{\frac{(2n-1)!!}{2n!!}} b_0~, & b_0 = & \left[ 1 + \sum_{n=1}^\infty (\tanh r)^{2n} \frac{(2n-1)!!}{2n!!} \right]^{-1/2}~.
\end{align}
Considering the case $\theta = 0$, we can invert the relations (\ref{def:squeezed_ac_operators}) and study the properties of squeezed states using (\ref{eqn:expansions_q&p}), obtaining the following expressions for the uncertainties
\begin{subequations}
\begin{align}
(\Delta q)^2 = & \frac{\hbar}{2 m \omega} e^{- 2 r}
- 4 i (\alpha^\star - \alpha) \delta \frac{\hbar}{2} \sqrt{\frac{\hbar}{2 m \omega}} e^{- 2 r} \gamma
+ \epsilon \frac{\hbar^2}{4} (5 + 3 e^{- 4 r} - 2 \alpha^\star {}^2 e^{-2 r} + 16 |\alpha|^2 e^{-2 r} - 2 \alpha^2 e^{-2 r} ) \gamma^2 + \nonumber \\
& + \delta^2 \frac{\hbar^2}{4} \left[ 11 - 15e^ {- 4 r} + 2 \alpha^2 (2 e^{2 r} - 11 e^{-2 r}) + 8 |\alpha|^2 (e^{2 r} - 2 e^{-2 r}) + 2 \alpha^\star {}^2 (2 e^{2 r} - 11 e^{-2 r}) \right] \gamma^2\\
(\Delta p)^2 = & \frac{\hbar m \omega}{2} e^{2 r}
+ \left(\frac{\hbar m \omega}{2}\right)^2 \left\{ 2 (\alpha^\star {}^2 + \alpha^2) [ 8 e^{-2 r} \delta^2 - 3 e^{2 r} (\delta^2 + \epsilon)] + 8 |\alpha|^2 \delta^2 (4 e^{-2 r} - 3 e^{2 r}) - 3 e^{4 r} (\delta^2 - \epsilon) \right. + \nonumber \\
& \left. - 3 (3 \delta^2 + \epsilon) + 8 \delta^2 e^{-4r} \right\} \gamma^2 ~, \\
(\Delta q)^2 (\Delta p)^2 = & \frac{\hbar^2}{4} \left\{ 1
- 4 i (\alpha^\star - \alpha) \delta \sqrt{\frac{\hbar m \omega}{2}} \gamma
+ 4 \frac{\hbar m \omega}{2} \left\{ 2 \epsilon (e^r - \alpha^\star + \alpha) (e^{r} + \alpha^\star - \alpha) + \right. \right. \nonumber \\
& \qquad \qquad + \delta^2 \left[ \alpha^\star {}^2 (e^{4 r} - 7 + 4 e^{-4r}) + 2 |\alpha|^2 (e^{4 r} - 5 + 4e^{-4 r}) \right. \nonumber \\
& \qquad \qquad \qquad \left. \left. \left. + \alpha^2 (e^{4 r} - 7 + 4 e^{-4r}) + 2 (e^{2 r} - 3 e^{-2r} + e^{- 6 r}) \right] \right\} \gamma^2 \right\}
\end{align}
\end{subequations}
to order $\gamma^2$.
As expected, these reduce to quantities in the standard theory for $\gamma \rightarrow 0$.
In this case too, we can compute the theoretical minimal uncertainty through the Schr\"odinger-Robertson relation
\begin{multline}
[(\Delta q)^2 (\Delta p)^2]_{\mathrm{min}} = \frac{\hbar^2}{4} \left\{ 1
- 4 i (\alpha^\star - \alpha) \delta \sqrt{\frac{\hbar m \omega}{2}} \gamma
+ 4 \frac{\hbar m \omega}{2} \left\{ 2 \epsilon(e^r - \alpha^\star + \alpha) (e^r + \alpha^\star - \alpha) \gamma^2 + \right. \right. \\
\left. \phantom{\frac{\hbar m \omega}{2}} \left. - \delta^2 \left[ 3 \alpha^\star {}^2 + 2 |\alpha|^2 + 3 \alpha^2 + 2 e^{-2 r} - (\alpha^\star + \alpha)^2 (e^{2r} - 2 e^{-2r})^2 \right] \right\} \gamma^2 \right\}
\end{multline}
and thus find
\begin{equation}
(\Delta q)^2 (\Delta p)^2 - [(\Delta q)^2 (\Delta p)^2]_{\mathrm{min}} = 32 \delta^2 \frac{\hbar^2}{4} \frac{\hbar m \omega}{2} e^{- 2 r} \sinh^2 (2r) \gamma^2 \label{eqn:diff_squeezed}
\end{equation}
for the difference with the minimal uncertainty product.
Several features are noteworthy.
First, the above difference is positive for all values of $r$ and does not depend on $\alpha$.
Second, the difference with the minimal product is of the second order in the GUP parameter.
Finally, it is present only for models with a non-zero linear term; it is not present in the model introduced in \cite{Kempf1995_1}.
Finding evidence for the relation \eqref{eqn:diff_squeezed} could therefore help distinguish the models.
\subsection{Minimal Uncertainties for Squeezed States}
In the standard theory, squeezed states saturate the uncertainty relation allowing for reduced uncertainties, with respect to coherent states, in either position or momentum.
In principle, the uncertainty in position (momentum) can be made arbitrarily small for HUP-saturating states.
However the GUP introduces a minimal uncertainty in position. In this section we study the implications of this fact.
To better observe the features of optimal squeezing in GUP, we consider the particular models examined in \cite{Kempf1995_1}.
This model corresponds to $\delta = 0$ and $\epsilon = 1/4$.
The position uncertainty for this model is
\begin{equation}
(\Delta q)^2 = \frac{\hbar}{2 m \omega} e^{- 2 r}
+ \frac{\hbar^2}{4} (5 + 3 e^{- 4 r} - 2 \alpha^\star {}^2 e^{-2 r} + 16 |\alpha|^2 e^{-2 r} - 2 \alpha^2 e^{-2 r} ) \frac{\gamma^2}{4} ~.
\end{equation}
We obtain a minimal position uncertainty for $e^{-r} = 0 \Rightarrow r = + \infty$.
It is
\begin{equation}
(\Delta q)^2 = \frac{\hbar^2}{4} \frac{5}{4} \gamma^2 > (\Delta q)^2_{\mathrm{min}} = \hbar^2 \frac{\gamma^2}{4}
\end{equation}
whereas the latter quantity has been previously computed \cite{Kempf1995_1}.
Therefore, the maximally squeezed uncertainty in position is always larger than the minimal uncertainty predicted by the model.
The momentum uncertainty for the same model is
\begin{equation}
(\Delta p)^2 = \frac{\hbar m \omega}{2} e^{2 r}
-\frac{3}{4} \left(\frac{\hbar m \omega}{2}\right)^2 \left\{ 2 (\alpha^\star {}^2 + \alpha^2) e^{2 r} - e^{4 r} + 1 \right\} \gamma^2 ~,
\end{equation}
and is minimized for $e^{2r} = 0 \Rightarrow r = - \infty$, the minimal uncertainty being
\begin{equation}
(\Delta p)^2 = - \frac{3}{4} \left(\frac{\hbar m \omega}{2} \right)^2 \gamma^2
\end{equation}
which is negative. This means that this model cannot describe infinite squeezing.
Rather, by inverting the reasoning, there must exist a lower value of $r$ for the interval of squeezing in which GUP can be used.
In fact, when we have
\begin{equation}
e^{2r} = \frac{ - 4 + 3 (\alpha^\star {}^2 + \alpha^2) \hbar m \omega \gamma^2 + \sqrt{ 9\hbar^2 m^2 \omega^2 \gamma^4 + [ 4 - 3 (\alpha^2 + \alpha^\star {}^2) \hbar m \omega \gamma^2 ]^2 }}{3 \hbar m \omega \gamma^2} \simeq \frac{3 \hbar m \omega \gamma^2}{8}~, \label{eqn:max_squeezing_mom_kmm}
\end{equation}
the squeezed uncertainty in momentum vanishes.
We next turn to the consequences for the model in \cite{Ali2011_1}, for which $\delta=1$ and $\epsilon=1$, a case motivated from
doubly special relativity as noted in the introduction.
For simplicity, we consider a squeezed vacuum state, \emph{i.e.} $\alpha=0$.
The position uncertainty in this case is
\begin{equation}
\left(\Delta q\right)^2 = \frac{\hbar}{2 m \omega} e^{-2r} + 4 \frac{\hbar^2}{4} (4 - 3 e^{-4r}) \gamma^2~.
\end{equation}
It has a minimum at $e^{-r} = 0 \Rightarrow r = + \infty$, corresponding to
\begin{equation}
\left(\Delta q\right)^2 = 16 \frac{\hbar^2}{4} \gamma^2 > \left(\Delta q\right)^2_{\mathrm{min}} = 8 \frac{\hbar^2}{4} \gamma^2
\end{equation}
where the latter quantity is the minimal length allowed by the model in \cite{Ali2011_1}.
As for the uncertainty in momentum, we find
\begin{equation}
(\Delta p)^2 = \frac{\hbar m \omega}{2} e^{2 r} - 4 \left(\frac{\hbar m \omega}{2}\right)^2 (2 e^{-4r} - 3) \gamma^2
\end{equation}
for this model.
Given the presence of both decreasing and increasing exponentials in $r$, there is no minimal value for the momentum uncertainty. Instead
we see that $\Delta p$
vanishes for
\begin{equation}
e^{2r_{min}} = 4 \frac{\hbar m \omega}{2} \gamma^2 + \frac{32}{3} \left(\frac{\hbar m \omega}{2} \right)^{2/3} \gamma^{4/3} - 2 \left( \frac{\hbar m \omega}{2} \right)^{1/3} \gamma^{2/3} + \mathcal{O}(\gamma^{8/3})~.
\end{equation}
As with the previous model, this value $r_{min}$ of the squeeze parameter can be interpreted as the smallest physical value of $r$
permitted in this model, since smaller values yield $(\Delta p)^2 < 0$.
Finally, in both model we observe similar features: the uncertainties in position have a non-vanishing minimal value larger than the minimal length allowed by the models.
This uncertainties are achieved for infinite squeezing.
On the other hand, minimal uncertainties in momentum, as computed by the GUP models, can be negative, signaling the break down of GUP for high energies.
Therefore, GUP con describe squeezing in momentum uncertainties only up to particular limits for the squeeze parameter.
\section{Applications and Outlook} \label{sec:conclusions}
Various theories of Quantum Gravity predict a modification of the Heisenberg principle and the Heisenberg algebra to include momentum dependent terms.
We consider its implications on the HO from the GUP (\ref{eqn:perturbed_Hamiltonian}) by looking at perturbations from the $\gamma=0$ case up to second order in $\gamma$.
We were thus able to find normalized eigenstates (new number states) and eigenvalues, and a new set of ladder operators.
We then focused our attention on coherent and squeezed states.
Our most notable results are the new spacing of the energy eigenvalues
and the new energy eigenstates as functions of the linear and quadratic terms $\delta$ and $\epsilon$ in the GUP.
There are three distinct cases.
When $\delta^2 < \epsilon$, the spacing of the energy ladder increases with the number $n$.
For the case $\delta^2 = \epsilon$, the spacing does not depend on $n$, obtaining a regular ladder, as in the standard theory. Finally, for $\delta^2 > \epsilon$ the correction is negative.
Therefore the spacing of the energy ladder decreases with the number $n$ till a maximum number $n_{\mathrm{max}}$, corresponding to the maximum value of the energy.
This simply indicates the GUP breaks down for large enough energies, where Planck scale effects become relevant and a full theory of Quantum Gravity is necessary since higher order terms cannot be neglected.
We were then able to define a set of modified
annihilation, creation, and number operators for the perturbed states.
The algebra of these new operators is identical to that of the standard ones.
Furthermore, we show that with their aid, the Hamiltonian can be written in a compact form, from which expectation values, uncertainties, etc. can be computed
quite easily.
For coherent states, defined as eigenstates of the new annihilation operator, we find that they still are minimal uncertainty states.
Squeezed states introduce additional interesting features.
The position uncertainty has a lower bound -- not present in the standard theory -- that depends on the particular GUP model under consideration.
As for the momentum uncertainty, its maximal squeezing depends on the model and on the choice of the coefficients $\delta$ and $\epsilon$.
That is, for some choices of the coefficients, a lower bound for the squeeze parameter exists, since beyond this limit higher order Planck scale effects cannot be neglected.
Our results are potentially testable.
For example, direct application to mechanical oscillators can be tested, \emph{e.g.} as in \cite{Pikovski2012_1,Marin2013_1,Bawaj2014_1}.
In fact, mainly motivated by the results in \cite{Kempf1995_1}, many groups already proposed experiments to test them.
So far this class of experiments focused only on the energy spectrum of the HO.
On the other hand, experiments on resonant mass detectors could in principle also test the results concerning coherent and squeezed states \cite{Hollenhorst1979}.
Actual laboratory based systems, such as those referred above may already be near the threshold of observing such minute corrections.
A different class of tests could be performed
for larger systems that can be treated quantum mechanically.
In fact, as we showed in Sec. \ref{sec:HO_GUP} and in particular in (\ref{eqn:detection}), massive systems constitute ideal observational tools to probe GUP effects.
However it is somewhat debatable whether the GUP applies only to fundamental constituents of matter
(\emph{e.g.} \cite{AmelinoCamelia2013_1}), in which case Planck scale effects on mesoscopic and macroscopic objects are small, or if it can be applied to the center of mass of an optomechanical oscillator (and other systems)
(\emph{e.g.} \cite{Pikovski2012_1,Marin2013_1,Bawaj2014_1,Bosso2016_2}), in which case Planck scale effects may be observable with
current accuracies. For the latter, it will be a
challenge to isolate GUP effects from the rest. In other words the GUP, being itself a modification of the Heisenberg Uncertainty Principle, should only be applied to systems that exhibit quantum properties.
Indeed, we apply our results in Tables \ref{tbl:detection} and \ref{tbl:unc_osc_period} only for systems that require a quantum description.
It is interesting to comment on the choice of mass scale in \eqref{Mpscale}.
It is generally expected that this is the Planck mass, but on empirical grounds we could replace $M_{\mathrm{Pl}}$ in \eqref{Mpscale} with any mass scale that has yet to be probed by experiment.
A conservative value would be something just above the LHC scale, \emph{e.g.} $ E_{\mathrm{LHC}} \sim 10$ TeV.
This might come from some hitherto undiscovered intermediate scale, or one deriving from a large 5-th dimension with order TeV Planck energy in five dimensions.
Our estimated corrections would then be amplified by a factor of $M_{\mathrm{Pl}} c^2 / E_{\mathrm{LHC}} \sim 10^{15}$, or equivalently by a different value of $\gamma_0$ in \eqref{Mpscale}.
Since this ratio appears squared in \eqref{eqn:unc_period}, the most interesting scenario would be the optomechanical oscillator experiments, for which the periods would vary from seconds to hours, potentially rendering the calculated effects observable.
Alternatively, a scale $E\sim 10^{11}$ GeV would yield oscillation times of seconds for the LIGO detector, again observable at
least in principle. Depending on the temporal resolution of the detectors, these arguments can set lower bounds to the relevant energy scale for the GUP (see Table \ref{tbl:unc_osc_period}).
Finally, applications to Quantum Optics of the results shown in this paper can lead to important and potentially observable features \cite{Bosso}.
Indeed, starting from a description of the electromagnetic field in terms of amplitude and phase quadratures, it is possible to include the GUP commutation relation to obtain a perturbed Hamiltonian.
Following similar steps and definitions of the present paper, one can then construct a GUP-modified theory of Quantum Optics.
This modification can have important implications on many optical experiments involving squeezed states, in particular future evolution of gravitational wave interferometers.
This is specially relevant as the use of squeezed states have been planned for LIGO interferometers \cite{Dwyer2013_1}.
Therefore, while on one hand further study is still necessary to understand some of the features highlighted in this paper, on the other hand we may be one step closer to experimental evidences of Planck scale Physics.
\section*{Acknowledgements}
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, Perimeter Institute of Theoretical Physics through their affiliate program, and Quantum Alberta.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 623 |
The Lighthouse is a 2019 film directed and produced by Robert Eggers, from a screenplay he co-wrote with his brother Max Eggers. It stars Willem Dafoe and Robert Pattinson as nineteenth-century wickies (lighthouse keepers) in turmoil after being marooned at a remote New England outpost by a wild storm. The film has defied categorization in media, and interpretations of it range from a horror film, a psychological thriller, a survival film, and a character study, among others.
The idea for the film first emerged from Max Eggers' re-envisioning of Edgar Allan Poe's unfinished short story of the same name. Robert Eggers assisted the development when Max was unable to complete the adaptation of "The Light-House", sourcing the plot from a nineteenth-century legend of an accident at a lighthouse in Wales. The Lighthouse draws visually from photography of 1890s New England, maritime-themed French cinema from the 1930s, and symbolist art. Principal photography took place in Nova Scotia, Canada, beginning in April 2018 and lasting slightly over a month. It was shot in black-and-white, with a nearly-square 1.19:1 aspect ratio.
The film premiered at the Cannes Film Festival on May 19, 2019, and was theatrically released in the United States by A24 on October 18, 2019. It grossed over $18 million, against an $11 million budget, and received widespread critical acclaim, with particular praise for the direction, visuals, and performances of Dafoe and Pattinson. Among its numerous accolades, the film was nominated for Best Cinematography at the 92nd Academy Awards and the 73rd British Academy Film Awards.
Plot
In 1890s New England, Ephraim Winslow arrives on a small, rocky island far off of the coast for his first four-week stint as a "wickie" (lighthouse keeper), under the supervision of the experienced Thomas Wake. Wake, a former sailor, immediately proves to be very demanding, and he has Winslow do such jobs as emptying chamber pots, maintaining the machinery, carrying heavy kerosene tanks up the stairs, and painting the lighthouse, while barring Winslow from the lantern room. Winslow observes that, every night after ascending the lighthouse, Wake removes his clothes before the light.
One evening while dining, Wake reveals to Winslow that his previous assistant perished after losing his sanity. He says he noticed a one-eyed gull bothering Winslow, but tells Winslow not to harm it, as gulls are reincarnated sailors, and killing them brings bad luck. Winslow begins to dream of sea demons and logs floating in the sea. After two weeks on the island, he reveals he was previously a lumberjack in Maine.
The day before the scheduled departure, Winslow discovers a dead gull inside the cistern, bloodying the drinking water. He is attacked by the one-eyed gull and brutally slaughters it, after which the wind changes direction, and a fierce storm hits the island. Winslow and Wake spend the night getting drunk, and the storm prevents the lighthouse tender from collecting them the next day. As Winslow empties the chamber pots, he notices a body washed up on the shore and discovers it is a mermaid, which awakens and screams at him. He flees back to the cottage, where Wake informs him the storm has spoiled their rations, and, when Winslow is not worried because he thinks the tender is only a day late, that they have already been stranded for weeks. The pair unearth a crate at the lighthouse's base that Winslow assumes contains reserve rations, but it is full of bottles of alcohol.
As the storm continues to rage, Winslow and Wake get drunk every night and alternate between moments of intimacy and aggressiveness. Wake curses Winslow when he says he does not like Wake's cooking, Winslow tries to steal the lantern room keys from Wake as he sleeps one day and contemplates murdering him, and Winslow finds the one-eyed head of Wake's previous wickie in a lobster trap. While drunk, Winslow confesses to Wake that his real name is Thomas Howard, and he assumed the identity of Ephraim Winslow, his cruel foreman in Maine who he had watched drown without helping, because Winslow had a clean record. He then has a menacing vision of Wake and runs to the dory to try to leave the island, but Wake appears and destroys the boat with an axe. After chasing Howard back to their lodgings, Wake claims it was Howard who chased him and hacked up the dory, as Howard was driven mad by his confession.
With no alcohol left, Howard and Wake begin drinking a concoction of turpentine and honey, and that night a giant wave crashes through the wall of their cottage. In the morning, Howard finds Wake's logbook, in which Wake has criticized him as a drunken and incompetent employee and recommended he be sacked without pay. He insists he is a hard worker and begs to be let into the lantern room, but Wake refuses and belittles him, so he attacks Wake, who he sees as the real Winslow, the mermaid, and a Proteus-like figure. Howard beats Wake into submission and takes him to the hole at the base of the lighthouse to bury him alive. Wake describes the dangers of looking in the lantern before losing consciousness, and Howard takes the keys to the lantern room.
Howard goes to get a cigarette, and Wake runs in and strikes him in the shoulder with the axe. He disarms Wake and murders him, and then ascends the lighthouse. In the lantern room, the Fresnel lens opens to Howard, who reaches in and bursts into disturbing, deranged screaming, before falling down the lighthouse steps. Sometime later, a barely-alive Howard lies nude on the rocks with a damaged eye as a flock of gulls peck at his exposed organs. The final shot is of the entire island with the lighthouse missing.
Cast
Robert Pattinson as Ephraim Winslow/Thomas Howard
Willem Dafoe as Thomas Wake
Valeriia Karamän as the Mermaid
Logan Hawkes as the real Ephraim Winslow
Production
Development
The original idea for The Lighthouse was first articulated at a dinner between director Robert Eggers and his younger brother, Max Eggers. Robert was unhappy with his film industry prospects after the pitching of his first major feature, The Witch (2015), failed to secure funding. Max shared the basic outline of his screenplay, a lighthouse-set ghost tale as part of an attempted reimagining of Edgar Allan Poe's unfinished short story "The Light-House". Adapting the short story proved troublesome, halting Max's progress on the script, which, at the time, had the tentative working title Burnt Island. Robert started musing ideas to bolster the project's conceptualization, and, with his brother's support, soon began investigating for source material.
One story that caught the director's attention in his initial research was a nineteenth-century myth of an incident at Smalls Lighthouse in Wales, wherein one of two wickies, both named Thomas, died while trapped at the outpost by a destructive storm. That both men were named Thomas, Robert recalled, compelled him to create a film with an underlying story of identity. Around the time there was a realized concept, Robert temporarily stopped his commitment to The Lighthouse when he found an investor to finance The Witch.
The unexpected success of The Witch elevated Robert's directing profile. To exploit his newfound credibility, he pushed The Lighthouse, among several other projects, in his negotiations with studio executives. He and Max then resumed their work by exchanging and revising drafts. This coincided with more rigorous research of the period to develop the onscreen world, as Robert immersed himself in photos of 1890s New England, 1930s maritime-themed French films, and symbolist art for visual reference.
The Eggers' study of literature with maritime and surrealist themes informed the speech of the characters in The Lighthouse. They looked into the writings of Herman Melville, Robert Louis Stevenson, and H. P. Lovecraft, among others, before coming across the work of Sarah Orne Jewett, a novelist best known for her local color works set around the coast of Maine. Her dialect-heavy writing style provided the cadences of the lead characters in the film, rooted in the experiences of her own sailor characters and real-life farmers, fishermen, and captains she had interviewed. Robert and Max also deferred to a dissertation on Jewett's technique to guide their direction for intense conversational scenes.
Another force shaping The Lighthouse creative direction was the Eggers' theater background. The two men sourced elements from playwrights that influenced their work as young teens, chiefly artists such as Samuel Beckett, Harold Pinter, and Sam Shepard whose writings examine male-centric perspectives of existential crises and psychosis.
Casting
The film stars Willem Dafoe and Robert Pattinson, who both separately approached Robert Eggers to express their enthusiasm for The Witch and their desire to collaborate. Pattinson and Eggers originally met to discuss Pattinson portraying a Victorian socialite in an unrelated project, but Pattinson passed because he believed the role would fail to challenge his acting ability. His next meeting with Eggers took place once he finished reading The Lighthouse completed script, and during the conversation Pattinson showed Eggers a clip of an intoxicated man screaming "I am a demon" to convey this understanding of the director's vision.
Eggers' initial film proposals with Dafoe were also not fruitful. Dafoe and Pattinson had met at a party, and Pattinson's participation in The Lighthouse was used as a selling point in pitches to Dafoe. When they met in person to discuss the project, the director was plainspoken in the conversation. Dafoe recalled: "There was no discussion. 'This is the way we're going to do this. My way or the highway.' That's very unusual, especially for a two-hander, for a director to say, 'This is the way I see it. Yes or no?'"
In February 2018, it was announced that Dafoe, and then Pattinson, had been cast in the film. To prepare for their respective roles, each actor employed different techniques at the rehearsals. Dafoe, citing his theater background with the experimental troupe The Wooster Group, drew from his spontaneous acting style in rehearsals, whereas Pattinson planned his rehearsing from the discussion of the script.
Anya Taylor-Joy, who starred in Eggers' directorial debut The Witch, was eager to work with him again and asked if she could play the mermaid. Eggers replied that there was not a role for her and she "really should not be this particular mermaid". Taylor-Joy then jokingly suggested that she could play a seagull instead.
Filming
Because the filmmakers could not find a lighthouse suitable for the needs of the production, they constructed a 70-foot (20-meter) lighthouse set on Cape Forchu in Lief Erikson Park in Yarmouth County, Nova Scotia. Most of the interiors were filmed on sets constructed inside a hangar at Yarmouth Airport and in soundstages near Halifax. Principal photography began on April 9, 2018, and lasted approximately 35 days, which was slightly over schedule, as a result of unforeseen circumstances on set. Difficulties arose due to the remote location, the harsh climate, and technical caveats of the camerawork.
Eggers had already envisioned shooting The Lighthouse in black-and-white, with a boxy aspect ratio, before drafting of the script. Although he and director of photography Jarin Blaschke, who was working with Eggers for the third time, faced resistance from studio executives hoping to maximize the film's commercial prospects, the two men were adamant and did not want to shoot in color because they feared undermining the artistic integrity of their work. Initially, Eggers wanted to use a 1.33:1 aspect ratio, believing it would sufficiently capture the confined sets and the lighthouse's vertical orientation, but he reconsidered when Blaschke suggested, as a joke, instead using the 1.19:1 aspect ratio that was used fleetingly during the film industry's transition to sound. After further analysis of period films for inspiration, chiefly the German thriller M (1931), Blaschke determined that the 1.19:1 format endowed footage with a greater sense of confinement, while amplifying the physical isolation of the characters in their environment, and the film was shot in that ratio.
The film was shot on 35mm film using Panavision Panaflex Millennium XL2 cameras equipped with vintage Bausch and Lomb Baltar lenses. Occasionally, to capture flashback sequences or scenes of heightened conflict, specialized lenses refurbished by Panavision were used. The onscreen universe was given a highly saturated visual palette evocative of orthochromatic film. Creating the spectrum of textures with a sufficient antique quality was one of Blaschke's initial responsibilities during the pre-production. He developed a process to test the utility of digital footage in color negative film stock, first with Kodak Vision3 500T 5219 film, before selecting Eastman Double-X 5222 stock based on the composition produced. Blaschke resumed the testing after securing the Baltar lenses for the shoot, this time with an arrangement of shortpass filters—a class of scientific optical filters—and photographic filters most sensitive to blue-green and ultraviolet light. The specifications were so unusual that it required the manufacture of custom sets of filters by Schneider Kreuznach, which was a costly, month-long endeavor. Blaschke recalled, "I sketched a desired spectrograph on graph paper, indicating a complete elimination of all light beyond 570 nanometers [mid-yellow] while allowing all shorter wavelengths to pass freely. At that point, I was unsure of the true light loss and I was pretty nervous about it." Post-production editing of The Lighthouse occurred simultaneously at the FotoKem film laboratories in Burbank, California.
Analysis
Genre
The Lighthouse has been described as a horror film by critics such as Manohla Dargis of The New York Times, and as a psychological thriller by critics such as Lee Marshall of Screen Daily. Other critics said it was a film that could not be pigeonholed, with Owen Gleiberman of Variety declaring that "you may feel in your bones that you're watching a supernatural shocker [...] Are we seeing a slice of survival, a horror film, or a study in slow-brewing mutual insanity? How about all of the above?" Michael Phillips of The Chicago Tribune echoed Gleiberman's statements, noting that the film's plot did not operate "as any sort of conventional ghost story, or thriller, or anything".
Psychoanalysis
Eggers said the film's sub-text was influenced by Sigmund Freud, and he hoped that "it's a movie where both Jung and Freud would be furiously eating their popcorn". Given his simultaneous fear and admiration of the senior lighthouse keeper, the younger keeper displays an Oedipal fixation, and Pattinson commented on the father/son dynamic in the film by stating that "I was pretty conscious of how I wanted the relationship to come across. In a lot of ways, he sort of wants a daddy", as well as that, as the film progresses, his character is increasingly "looking for Willem [Dafoe]'s validation" as both a boss and a father-figure. Pattinson also said the phallic imagery of the lighthouse is explicit, with Eggers describing it in the script as an erect penis, and Eggers said the film was meant to include "a very juvenile shot of a lighthouse moving like an erect penis and a match-cut to actual erect penis" belonging to Howard´s character, but this sequence was removed at the request of the financiers. Body double was used to film this scene and when Pattinson was asked about this in interview he said that when he saw that scene he didn´t even know it was a penis and he thought it was a lighthouse
Mythology
There are parts of the film inspired by both sailors' myths and classical mythology. The fate of the younger lighthouse keeper invokes the myth of Prometheus, as, after finally reaching the light and learning what is in it, he falls down the stairs of the lighthouse and his organs are plucked out by seagulls. On the other hand, the older keeper was modeled on Proteus, a "prophecy-telling ocean god who serves Poseidon", as he "makes that uncannily accurate prediction for how Ephraim will die at the end of the movie" and is even seen with tentacles and sea creatures stuck to his body in one of the younger man's hallucinations. Albrecht Dürer's engraving The Sea Monster inspired Wake's appearance, with Eggers saying: "The Proteus figure that is more clearly nautical is somewhat based on a sea monster by Dürer, who carries a tortoise shell shield." Eggers explained the allusions to classical mythology by saying they are present "Partially because Melville goes there and partially because of I'm sure our unhealthy Jungian leanings".
Sexuality
The film contains explicit depictions of male sexuality, and primarily depicts two men alone in close quarters on an island, but, when asked whether the film was "a love story", Robert Eggers replied:
Sexual fantasy and masturbation are recurring themes in the film. For Dafoe, the androphilia in the film is blatant, but it is also used to explore what it means to be a man: "They have a sense of guilt, of wrong [...] it's got existential roots [...] about masculinity and domination and submission." After beating the older lighthouse keeper into submission, the younger keeper assumes a dominant role, calling the older man "dog" and dragging him on a leash. Commenting on this scene, Pattinson said "there's definitely a take where we were literally trying to pull each other's pants down. It literally almost looked like foreplay."
The mythological and artistic influences of the film underscore its eroticism. Eggers acknowledged the visual influence of symbolist artists Sascha Schneider and Jean Delville, whose "mythic paintings in a homoerotic style become perfect candidates as imagery that's going to work itself into the script." The composition of a shot in the film was consciously adapted from Schneider's Hypnosis.
Release
The Lighthouse had its world premiere on May 19, 2019, in the Directors' Fortnight section of the Cannes Film Festival, and it was screened at the Toronto International Film Festival and the Atlantic Film Festival in September. It was distributed by A24 in North America and by Focus Features internationally, and was released in theaters on October 18, 2019.
Reception
Box office
The film grossed $10.9 million in the United States and $7.5 million in other territories, for a worldwide box-office total of $18.3 million.
Its limited opening weekend in the U.S., the film grossed $419,764 from eight theaters, for an average of $52,471 per venue. Its second weekend, the film expanded to 586 theaters and grossed $3.75 million, placing eighth at the box office. The following weekend, the film expanded to 978 theaters, but its gross fell 34.7% to $2 million, and it finished in 13th place.
Critical response
On review aggregation website Rotten Tomatoes, the film holds an approval rating of 90% based on 392 reviews, with an average score of 8.0/10; the site's "critics consensus" reads: "A gripping story brilliantly filmed and led by a pair of powerhouse performances, The Lighthouse further establishes Robert Eggers as a filmmaker of exceptional talent." On Metacritic, the film has a weighted average score of 83 out of 100 based on 52 critics, indicating "universal acclaim."
Owen Gleiberman of Variety called the film "darkly exciting" and "made with extraordinary skill," commenting that "the movie, building on The Witch, proves that Robert Eggers possesses something more than impeccable genre skill. He has the ability to lock you into the fever of what's happening onscreen." Robbie Collin of The Daily Telegraph gave the film a perfect score, calling Dafoe's performance "astounding" and comparing Pattinson's to that of Daniel Day-Lewis in There Will Be Blood, saying, "that's no comparison to make lightly, but everything about The Lighthouse lands with a crash. It's cinema to make your head and soul ring." Peter Bradshaw of The Guardian, in addition to praising the performances of Dafoe and Pattinson, also praised the screenplay, stating that the "script is barnacled with resemblances to Coleridge, Shakespeare, Melville – and there's also some staggeringly cheeky black-comic riffs and gags and the two of them resemble no-one so much as Wilfrid Brambell and Harry H Corbett: Steptoe and Son in hell." Manohla Dargis of The New York Times praised the character development, production design, acting, and themes, and Michael Phillips of the Chicago Tribune gave the film three stars out of five, comparing it to The Odd Couple (1968) and The Dumb Waiter (1957), and lauding the cinematography.
Conversely, Sandra Hall of The Sydney Morning Herald said the film's attempts at suspense were not successful, and Simran Hans of The Guardian gave it two stars out of five, saying the performances felt more like an "experiment than conducive to eliciting meaning." Mick LaSalle of the San Francisco Chronicle said the film was well-made, but "fails to give us the one thing that might have sustained an audience's interest over the course of 109 excruciating minutes: a compelling story." Dana Stevens of Slate concluded her review by stating that "The Lighthouse is at its strongest when it resembles the dark comedy of a Beckett play, complete with earthy scatological humor. [...] But as the mythological references pile up and the forbidden light atop the tower accrues ever more (and ever vaguer) symbolic meaning, the film sometimes seems funny [...] not because of but in spite of the filmmakers' intentions", and that, by the end, she became "impatient" with Eggers' "reliance on atmosphere [...] to take the place of story" and found herself "identifying with the stranded seafarers: I desperately wanted to get out."
Accolades
References
External links
at A24
Original screenplay by Robert and Max Eggers
2019 films
2019 LGBT-related films
A24 (company) films
American black-and-white films
American horror films
American LGBT-related films
American psychological thriller films
Films about alcoholism
Films about curses
Films about mermaids
Films based on classical mythology
Films directed by Robert Eggers
Films scored by Mark Korven
Films set in the 1890s
Films set in New England
Films set on uninhabited islands
Films shot in Nova Scotia
Focus Features films
Magic realism films
Regency Enterprises films
Two-handers
Works set in lighthouses
English-language Canadian films
American independent films
2019 independent films
2010s English-language films
Canadian horror films
Canadian psychological thriller films
Canadian LGBT-related films
Canadian independent films
LGBT-related horror thriller films
2010s Canadian films
2010s American films | {
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{"url":"https:\/\/www.springerprofessional.de\/rapid-evaluation-and-response-to-impacts-on-critical-end-use-loa\/23083504","text":"## Weitere Artikel dieser Ausgabe durch Wischen aufrufen\n\nErschienen in:\n\nOpen Access 25.05.2022 | Article\n\n# Rapid Evaluation and Response to Impacts on Critical End-Use Loads Following Natural Hazard-Driven Power Outages: A Modular and Responsive Geospatial Technology\n\nverfasst von: Patrick D. Royer, Wei Du, Kevin Schneider\n\nErschienen in: International Journal of Disaster Risk Science | Ausgabe 3\/2022\n\nprint\nDRUCKEN\ninsite\nSUCHEN\n\n## Abstract\n\nThe disparate nature of data for electric power utilities complicates the emergency recovery and response process. The reduced efficiency of response to natural hazards and disasters can extend the time that electrical service is not available for critical end-use loads, and in extreme events, leave the public without power for extended periods. This article presents a methodology for the development of a semantic data model for power systems and the integration of electrical grid topology, population, and electric distribution line reliability indices into a unified, cloud-based, serverless framework that supports power system operations in response to extreme events. An iterative and pragmatic approach to working with large and disparate datasets of different formats and types resulted in improved application runtime and efficiency, which is important to consider in real time decision-making processes during hurricanes and similar catastrophic events. This technology was developed initially for Puerto Rico, following extreme hurricane and earthquake events in 2017 and 2020, but is applicable to utilities around the world. Given the highly abstract and modular design approach, this technology is equally applicable to any geographic region and similar natural hazard events. In addition to a review of the requirements, development, and deployment of this framework, technical aspects related to application performance and response time are highlighted.\n\n## 1 Introduction\n\nAcross the world, the magnitude of blackouts is increasing concurrently with the frequency and intensity of natural hazard-induced disasters (Rudin et al. 2014; US Environmental Protection Agency 2016). Tropical cyclones and hurricanes are some of the most destructive natural hazard events, resulting in loss of power that affects millions of people each year. Extreme weather events cause extensive damage to dependent infrastructure systems that can cost billions of dollars to repair and lead to cascading effects on the workforce, healthcare, and economic health in general (Weems et al. 2007; Zimmerman and Restrepo 2009; Mukherjee et al. 2018; NOAA 2020). In 2017, Hurricanes Irma and Maria are notable extreme weather events that occurred within several weeks of each other that impacted Puerto Rico, with the latter causing more extensive damage to critical infrastructure. In January 2020, an earthquake of 6.4 magnitude impacted the area. In addition to the immediate damage to buildings, the loss of life, and the economic impact, much of the critical infrastructure that is necessary for the normal functioning of society has been damaged during these events. This includes the electric power systems operated by the local power authority. Disaster response management was significantly hindered by the lack of an integrated technical platform, drawing from a multitude of relevant data sources, to rapidly assist in the identification and designation of backup power sources.\nOutage-forecasting models have become increasingly accurate in predicting hurricane-induced outages prior to landfall, and strong developments have been made recently in the application of machine learning to quantify uncertainty in outage duration (Han et al. 2009; Nateghi et al. 2014; Yang et al. 2020; Zhai et al. 2021). While outage prediction is a useful tool in natural hazard risk management and preparation, there is a need for emergency and response mangers to plan for response measures related to power backup systems and to manage risk on longer time scales. Preparation and rapid response, through the study of resiliency, may be used to mitigate adverse consequences and improve response time to power restoration (Tokgoz et al. 2017). Staid et al. (2014) developed a literature-based sensitivity analysis to simulate the impacts on a power system under 12 plausible climate-hurricane scenarios, providing information at the census tract scale for planning and mitigation strategies. Research by Staid et al. (2014) highlights important links between climate uncertainty and planning and mitigation strategies at local scales for future events. Jeffers et al. (2018) developed an application, the Resilient Node Cluster Analysis Tool (ReNCAT), that performs an analysis of microgrid locations for improving community response to major disruptions across the island and is used for hurricane response planning. ReNCAT is a native (desktop) application that integrates optimization algorithms into its core codebase (Jeffers et al. 2018). ReNCAT has probability functions for critical end-use load outages based on historic flood events, landside probability, and earthquake history. It also calculates a resilience factor, societal burden, which is the metric used for members of the community to meet basic needs.\nIn addition to outage prediction models and research related to system hardening and hazard risk management and assessment, ongoing and near real-time monitoring and damage assessment that uses remote sensing is playing an increasingly important role in disaster response. Remote sensing technology is rapidly evolving in terms of sophistication and resolution. The way remotely sensed data is accessed, transferred, and processed is improving and becoming exponentially more efficient in terms of applying outcome to critical decision-making tasks. Data obtained from unmanned aerial vehicles (UAV) in concert with both passive and active remote sensing from airborne and spaceborne satellite-based platforms are becoming more prevalent and targeted with respect to ongoing monitoring (Voigt et al. 2007; Ofli et al. 2016; Alam et al. 2020; Vyron and Potsiou 2020). Such approaches enable damage and\/or vulnerability detection in a more synoptic manner, and potentially alert authorities and the public to damages before these can be detected in situ. Alam et al. (2020) used UAVs in concert with machine learning to estimate and evaluate utility pole inclination angles. This work focuses on automation of an existing process of monitoring and inspecting the health conditions of utility poles and provides ongoing evaluation of a critical metric within the overall assessment of system resilience. Vyron and Potsiou (2020) used deep learning and remote sensing imagery to identify optimal landing spots for aerial support following extreme events. This work highlights the value of machine learning with respect to the magnitude and scale of satellite data available, and its ability to provide effective and usable results in time-restricted and life-critical applications.\nOnce a power system is disrupted, timely restoration of electrical power for critical end-use loads can be achieved by using backup systems while the primary system is in the process of restoration. Backup systems encompass the use of one or more technologies, including microgrids, solar, and remote and controlled switches, which can be deployed at the distribution level to increase the resiliency of critical end-use loads (Table 1). These loads include seaports, airports, hospitals, police stations, and drinking water treatment plants, all of which have very distinctive and unique sets of requirements for identifying the most appropriate backup system. Identification and selection of an optimal backup source requires systematic and thorough review of each asset on a case-by-case basis and is tightly coupled with geospatial analytics and location intelligence sciences.\nTable 1\nOperational options to provide backup power sources\nAlthough modern data media formats, such as geosocial media, can be used as a powerful and effective early alert mechanism to point authorities to the time and location of system failure (Granell and Ostermann 2016), expert judgment is required in order to define the best path to recovery from disruptions. Such an assessment for the sum of assets affected by a catastrophic type event with a large geographic footprint is time consuming and time prohibitive under emergency response scenarios. Automated end-to-end analysis and stepwise logical survey is necessary to inform and guide the recovery process and execute plans in a timely manner.\nThis article describes a methodology that was developed to identify a range of different backup systems and map them to critical end-use loads by applying broader categorical knowledge for each resource type and utilizing higher resolution spatial data\u00a0and situational knowledge for each individual asset independently. This work builds on existing research and planning tools and offers a more comprehensive approach to backup systems of different types (Table 1). The capability described here can be accessed through a web-based application with an intuitive user interface for disaster managers and planners that is location aware and mobile friendly. A semantic data model was developed in concert with a data management and data mining solution, which includes three spatially explicit equations (Eqs. 1, 2, and 3) and binary decision tree approach that is applied to each critical end-use load to determine optimal backup source(s). The semantic model represents the data layer at the core of this framework and is novel in the way that it integrates information from physical data stores from a variety of disparate and previously unrelated native formats. Data sources are related and integrated into a larger data model to mirror the real-world onset and response to extreme weather events. The data model is then used as the core for a broader software development framework that was developed using modern cloud-based, serverless design principles for robustness and increased throughput.\nThe proposed methodology for identifying optimal backup sources described in this article is applicable to other types of extreme weather threats but is not necessarily applicable to every type of threat (for example, physical cyber-attack threats). As hurricanes involve damage to infrastructure from high wind speeds directly, and falling debris from high windspeeds, flooding, and landslides indirectly, this capability is relevant to other types of natural hazards. The proposed methodology also assumes total failure of critical infrastructure resulting in loss of power to end-use load. It may be useful in situations where damage is incurred but does not result in total failure and\/or partial loss of power, yet the method is not explicitly designed to account for such events. The underlying data model can be abstracted to include other data types and is novel in that sense, but the current model assumes these data are available, accessible, and migratable in perpetuity as updates to the system are made, which may not be a realistic expectation.\n\n### 1.1 Development of a Comprehensive Data Model for Recovery from Complete or Partial Failure of Distribution System\n\nThe Geospatial Science Program Management Office maintains that the compilation of critical infrastructure geospatial data is core to mission objectives.1 To maximize utility for the broader research community, the data should be interrelated in a manner that is inherently semantic and configured in a manner that is agile and fully extensible (Hammer and McLeod 1978; Codd 1979). The approach to data management and stewardship in modern day data science should recognize the need to be able to synergize a multitude of different data types and data standards, which typically include conventional horizontally scalable schema-bound Relational Database Management Systems (RDBMS), schemaless NoSQL (for example Mongo and AWS DynamoDB), and unstructured data types. Utility systems are largely managed as separate entities, spanning both private and public sectors, ranging from city to metropolitan area in scale, and other levels of aggregation, adding additional challenges with data integration. As utility systems are gaining access to more data than ever before, the challenge of digital transformation from legacy systems into a more modern framework, and integration of this framework into instantaneous data-driven solutions, has become more notable (Wong et al. 2009).\nHistorically, electrical power systems have been operated with a sparse level of data. This is true for both the physical planning data and the near real-time data available to operators (Wood and Wollenberg 1996). When a power system is built, the physical characteristics of the system are recorded. These include, but are not limited to, conductor and cable type, physical arrangements of conductors on poles, and the conductors to which the end-use loads are connected. Prior to the 1980s, this information was traditionally recorded in a paper format, and digitization was rare. Additionally, over time the accuracy of the information would degrade as the physical characteristics of the conductors varied with age, and end-use load connections changed due to activities such as storm restoration. Today most utilities digitize their data in a Geographic Information System (GIS) or a similar system, but the industry continues to grapple with errors in those data that are introduced over time.\nData access for system operators is still relatively sparse and routinely inaccessible. At the bulk transmission system level, it is common for high voltage (115-kV+) transmission lines to have monitoring at both ends; this includes breaker status, current, voltage, and power flow (Wood and Wollenberg 1996). For distribution systems, the data are much more limited. Typically, there is monitoring at the substation and a few points on the circuit where equipment, such as voltage regulators or shunt capacitors, is located; but overall, the system is not considered fully observable (Wood and Wollenberg 1996). While there are additional data sources, such as smart meters, these are typically not connected to operational systems. Connections between customer information systems and distribution management systems (DMS) are an ongoing area of research in the industry. In addition to the data sources that utilities oversee, there are other sources of data that can be beneficial within the spectrum of power reliability and redundancy. These can include weather data, population data, and information on the function of critical loads such as hospitals and police stations. As a result, the operations of electric power systems occur with a limited subset of available data, which often leads to suboptimal decision making. Electrical system management and resiliency plans would benefit immensely from the development of a broader semantic data model between various components of sources of structured and unstructured data. Such a model should surpass logical relationships between independent grid topology features (for example, substations, transmission lines, and towers) and extend to clients, power consumption sources, reliable backup scenarios, climate patterns, cell phone coverage, and other information (Geospatial Science\u2014Program Management Office 2018). Although still in the process of developing a comprehensive global semantic model that includes all these components, we have endeavored, with the development of a web-based serverless framework, hereafter referred to as the Electrical Grid Resilience and Assessment System (EGRASS), to make a significant step start towards this end. This includes the development of semantic relationships between electrical infrastructure geospatial information, electrical infrastructure reliability, critical end-use loads, and population dynamics.\nIn Puerto Rico, as well as in other regions of the United States, much of the local and institutional knowledge gathered from observed site conditions and anecdotal information related to system health and system-customer identification is not captured systematically and\/or is not easily accessible for analysis. This is the case for the relationship between clients and distribution feeders that is usually captured in a billing mechanism from the utility company. This mechanism may have service level distribution data per customer, but such information is not publicly available and\/or is not available in a GIS database. Extensive data modeling was necessary to infill knowledge gaps from conflicting entity relationships within data and between data sets of different types. Figure 1 represents the modeled spatial and referential relationship between critical end-use loads and distribution circuits. Modeled relationships provide input for optimization of backup selection. This includes an estimate of population affected by a nearby critical resource, the number of substations and related distribution feeders linked to critical resources, and the number of nearby resources in the same category.\nIn a real-world scenario, a natural hazard event triggers rapid response that includes prioritization of critical end-use loads for which power should be restored first (Fig. 1). The sequence in which reestablishment of power critical to end-use loads occurs depends on ranking these resources by population affected, the number of nearby substations and similar resources, reliability indices, and an estimate of level of effort required to restore critical infrastructure. Based on this information, judgement rules can be processed to isolate optimal backup power sources for each unique end-use load scenario. Then coordinated rapid\u00a0response can be executed. Integrated data results from physical data sources are highlighted by an example hospital site in Puerto Rico shown in Fig. 1 that is associated with a distribution feeder based on spatial proximity, a population estimate for each household in the surrounding area, and substation(s) that are associated with distribution feeders using Relational Database Management System (RDBMS) referential integrity\u00a0to maintain logical relationships between tables. The hospital is shown in the center of the map related to feeder 1421-03 with symbols for substations and location of all other nearby end-use loads (Fig. 1). The inset map shows labeled distribution feeders, population estimate centroids for each building footprint, and other nearby critical end-use loads.\nA data model was designed in consideration of the structural interrelationship between the \u201creal world\u201d and physical data stores to capture the meaning of this information in the application environment. In this work, physical data stores were available natively in varying format, some of which were transformed to a target format based on the ongoing nature of data ingestion. Physical data stores included structured, semi-structured, and unstructured data, captured and managed in the Structured Query Language (SQL) tabular and geospatial database (DB), noSQL DB, and unstructured DB storage respectively (Fig. 2). Format and management of these data types was entirely based on data availability and the foreseeable ongoing transfer of these data. Structured tabular data included grid topology network\u2014the relationship between substations, distribution lines, circuits, and busses. Structured spatial relationships are the relationship between end-use loads and distribution circuits (Figs. 1 and 2). Semi-structured data include population, reliability indices, and user profiles. Unstructured data include chosen backup power sources, anecdotal information and site data that are not captured in any systematic format, and structure material and post hurricane restoration work. In Puerto Rico this information is known in a general sense by region but is not documented in a reproducible manner that can be reconciled with grid topology.\nThe data sources displayed in Fig. 2 are used collectively to inform judgement rules that are used in the identification of backup sources for each critical end-use load. Grid topology is managed in a conventional structured database with primary keys, foreign keys, and a schema that is commonly found in an entity relationship (ER) model. Semi-structured data, such as population data, reliability indices, and damage reports, are still managed in a database, but the database does not enforce a schema, which allows for flexibility in data types. Most semi-structured databases, such as MongoDB and DynamoDB are managed in keypair format. Unstructured data require no format and can come in the form of reports, images, spreadsheets, and other formats (Fig. 2b).\nThe real-world response to the impact of extreme weather events, and the catastrophes those extreme events cause, is based on the principle that the most important use for recovery resources should be to reestablish quickly those institutions and infrastructure elements that provide essential services to the largest number of people. The semantic data model involves identification of the most populated regions and collocated critical end-use loads (structured spatial data) coupled with grid topology to identify nearby circuits and referential substations (structured data) and understanding of site conditions to identify optimal backup power sources (semi-structured and unstructured data). The sequence in which data form stores are used is mapped to the flow of the real world in a semantic model (Figs. 1 and 2).\n\n### 1.2 Decision Tree Logic and Relevant Equations for Identifying Optimal Backup Sources for Critical End-Use Loads\n\nThe underlying logic for identifying optimal backup resources for critical end-use loads categorically and individually relies on data access and accessibility patterns driven by the development of a data model described in the previous section. The semantic data model, and the encapsulation of this model in a flexible model framework, was necessary in order to execute logic programmatically against multitiered, disparate data sources with varying storage and access mechanisms.\nProvided here is a list of variables, spatially explicit equations, and a sample of underlying logic used in algorithms for identifying optimal backup source(s) for end-use load category and individual assets within each category described in Table 1. Binary tree logic involves nested calculations that account for the overall reliability and integrity of a system, demand, accessibility to site, existing redundancy, and current orientation and location of existing grid infrastructure. Assessed in EGRASS are six different critical end-use loads with 1\u20136 possible different backup sources for each asset, varying across categories and individual assets within each category. A sample of this logic (Fig. 3) and the variables used (Table 2) are provided in the section. The logic in its entirety is encompassed in the code base for the application and database layers of the application. The code repository is not publicly hosted, but access to the code for research purposes is granted; this includes the entire body of nested query statements highlighted in Fig. 3.\nTable 2\nA description of variables used in assessment\nVariable\nVariable type\nDescription\nSignificance\nPopulation density (POP)\nCategorical\nCategorized as sparse, moderate, or dense\nQuantifies demand\nAccessibility by vehicle (ACCESS)\nDistance in meters, normalized over all measurements\nAccessibility to site for repairs following disruption, and for implementation of backup solutions for strengthening prior to disruption\nTerrain roughness index (TRI)\nElevation in meters, normalized over range of data\nTerrain Roughness Index\nUsed in combination with the ACCESS variable to quantify ease of access following a disruption\nSubstation count (SBCOUT)\nCount (integer)\nTotal number of collocated substations\nCharacterizes resiliency in terms of redundancy\nSimilar asset count\nCount (integer)\nThe total number of end-use loads of the same type (for example, the number of other hospitals for end-use load hospital type)\nCharacterizes the redundancy of the same service type\nDistribution line placement and orientation\nCategorical\nUsed in concert with reliability indices, and pertains directly to the logistics of implementing backup solutions\nDistance to feeder\nDistance in meters\nDistance between end-use load and vertex of the nearest feeder\nCharacterizes resiliency and redundancy\nNumber of feeders\nCount (integer)\nTotal number of collocated feeders\nApplicable to the use of FLISR and Reclosure Switches, which require => 2 feeders\nSystem Average Interruption Duration Index\u00a0(SAIDI)\nAverage time in milliseconds (decimal)\nThe System Average Interruption Duration Index\u00a0during a year\nSystem reliability index\nSystem Average Interruption Frequency Index\u00a0(SAIFI)\nAverage count (integer)\nSystem Average Interruption Frequency Index\u00a0during a year\nSystem reliability index\nCustomer Average Interruption Duration Index (CAIDI)\nAverage time in milliseconds (integer)\nThe customer average interruption duration index\nSystem reliability index\nHospital trauma level\nCategorical (1\u20135)\nRefers to the kinds of resources available in a trauma center and the number of patients admitted in a year. Applies only to hospitals\nQuantifies importance of specific hospitals\nBackup generator\nBoolean (True\/False)\nWhether or not there is already a backup generator at the facility. This information is not available at every site\nCharacterize system redundancy and suitability for microgrids\nPopulation density, accessibility by vehicle, and substation count, were calculated as follows:\n$$\\mathrm{POP}={\\sum }_{i=1}^{n}P{\\left(d\\right)}_{i}=P{\\left(d\\right)}_{1}+P{\\left(d\\right)}_{2}+\\dots \\dots P{\\left(d\\right)}_{n}$$\n(1)\n$$\\mathrm{ACCESS}={\\sum }_{i=1}^{n}R{\\left(d\\right)}_{i}=R{\\left(d\\right)}_{1}+R{\\left(d\\right)}_{2}+\\dots ..R{\\left(d\\right)}_{n}$$\n(2)\n$$\\mathrm{SBCOUNT}={\\sum }_{i=1}^{n}S{\\left(d\\right)}_{i}=S{\\left(d\\right)}_{1}+S{\\left(d\\right)}_{2}+\\dots .S{\\left(d\\right)}_{n}$$\n(3)\nwhere d is the radius (Euclidean search distance) for the circular area around each critical end-use load (shown in Fig. 1 for an example hospital), POP is the total population in the vicinity and includes household count P for every residence within the specified search distance (Fig. 1). The substation count SBCOUNT is calculated as the total number of substations S within search distance (example shown in Fig. 1). ACCESS is a measure of accessibility calculated by the total linear length of roads R intersecting the search area. In addition to accessibility measured by linear roadway count, terrain ruggedness was calculated from a 30 m digital elevation model (DEM) using equations developed by Riley et al. (1999). In this calculation, a terrain roughness index (TRI) is assigned by comparing variation in adjacent elevation values assigned to pixels surrounding each asset. This provides a relative level of effort and potential difficulty in terms of accessing the site following an event, which is important to consider in Puerto Rico given the intermountain area in the center of the island and the propensity for mudslide in this area. Population density was normalized and categorically binned as densely, moderately, or sparsely populated for each end-use load based on distribution of population counts for all sites. Accessibility by road and terrain roughness is normalized using linear scale normalization and used collectively to determine ease of access following a potential event.\nStandardized reliability indices were used to evaluate the integrity of the electrical distribution system collocated with critical end-use loads. The System Average Interruption Duration Index\u00a0(SAIDI)\u00a0is commonly used as a reliability indicator by electric power utilities (Heydt and Graf 2010). The SAIDI is the average outage duration for each customer served. The System Average Interruption Frequency Index\u00a0(SAIFI)\u00a0is commonly used as a reliability indicator by electric power utilities. The SAIFI is the average number of interruptions that a customer would experience. The Customer Average Interruption Duration Index\u00a0(CAIDI)\u00a0is a reliability index commonly used by electric power utilities. It is related to SAIDI and SAIFI. The CAIDI gives the average outage duration that any given customer would experience. It can also be viewed as the average restoration time.\nFigure 3 conveys the use of variables (Table 2) in the decision tree for backup system type Reclosure Switches, defined in Table 1. The solution shown here is unique in that for Reclosure Switches the same logic applies to all critical end-use load types, which is not the case for other backup power types. The use of microgrids, as an example, applies differently to hospitals than it does for other types of end-use loads.\n\n## 2 A Computational Framework for Emergency Response and Disaster Preparedness\n\nAlthough data modeling was the initial focus of this article, the overarching objective was to develop a robust framework for rapid assessment, disaster response, and decision-making tools. The data model is an important first tier of this framework. The remainder of this article describes the development of the application and the techniques used to automate this logic and the development of a tool for intuitive execution of application functionality as well as the outcome on a select subset of end-use loads.\nA serverless philosophy was adopted early on in moving this work to the cloud for a multitude or reasons, but several virtual machines were used due to serverless memory constraints, operating system level dependencies, and lack of a stable container (that is, Docker) solution. An approach to Application Programming Interface (API) execution was developed involving downsampling of larger data sets and invoking geospatial indices to optimize efficiency and throughput speed. Also discussed are implications for moving computing workloads from on premise to the cloud.\n\n### 2.1 Hybrid Approach to Cloud Architecture Using Virtual Machines and Serverless Framework\n\nThe Electrical Grid Resilience and Assessment System (EGRASS) is a fully customized, single-page application (SPA) with progressive web application (PWA) functionality developed as part of a broader effort to develop a risk-based framework based on outage definitions with associated probabilities of occurrence from hurricane events. The client component of EGRASS was developed using React API, a lightweight, highly abstract, and modular framework. The backend component was built primarily using a serverless centric framework with Amazon Web Services (AWS) technology layered on top of a multitude of data and data storage combinations (Fig. 2). Extensive effort was taken to streamline and simplify the user interface while also optimizing architecture and configuration for the backend for maximum efficiency and optimal response time. Data access and visualization available in EGRASS surpass the capacity of commercial off-the-shelf programs relevant to domain specific analytic objectives, which are useful when little is known about large data sets and exploration goals are vague (Keim 2002; Yi et al. 2007; Aigner et al. 2008).\nThe architecture is primarily serverless with multiple copies of a virtual machine (VM) running a docker container that hosts a PostgreSQL database with PostGIS spatial extension and Geoserver open-source geospatial data server. The VM is set up through a load balancer for distributing resources. The application was built within a virtual private cloud (VPC), with data archived in a PostgreSQL server, Dynamo databases, and S3 unstructured data. Authentication is handled using AWS Cognito, REpresentational State Transfer (REST) functionality is developed using AWS Lambda functions and exposed via API Gateway. API Gateway is the central system for creating RESTful API for interacting with the client. Lambda is the scalable, serverless computing service that runs code in response to events and automatically manages the underlying computing resources. Underlying data resources include Dynamo semi-structured\/noSQL data, PostgreSQL relational database system, and S3 unstructured data sources. A spatially enabled RDBMS, in conjunction with NoSQL, was used to leverage aspects of the application that are storage driven, normalized, relational, and good for online analytical processing.\nBoth R-tree spatial indexing and geohashing were used to return geospatial queries as quickly and efficiently as possible (Zhang and Yi 2010). More complex spatial functions in the application are routed through one of several different approaches, depending on both the inherent structure of the data, and computational performance of the calculations. These approaches include: (1) stored database functions written in PL\/pgSQL (Procedural Language\/PostgreSQL) for postGIS; (2) noSQL spatial functions and operands written for Dynamo DB that use geoJSON format; and (3) python spatial libraries, including Shapely, Fiona, and Rasterio, integrated directly into the AWS Lambda functions.\n\n### 2.2 Software Interaction Pattern, Behavior, and User Journey\n\nThe EGRASS software was designed with two primary modules: (1) identification and isolation of candidate technology deployments to increase the resiliency of end-use loads that are critical to the normal operation of society; and (2) risk-based dynamic contingency analysis based on hurricane simulation. Both were designed to inform investment decisions in Puerto Rico\u2019s transmission grid with identification of judgement rules. Electrical engineering judgement rules are migrated to a decision tree matrix and codified in program logic. The risk-based framework is structured on outage definitions with associated probabilities of occurrence from hurricane events, in combination with impact assessment derived from detailed dynamic cascading analysis. The interaction pattern and workflow for both use modes are described in this section, and detailed output for the first use mode as well as identification of candidate technology are explained in subsequent sections.\nThe EGRASS application has a user pool consisting of both power users and nonpower users, such as site managers and regulatory program managers. To accommodate a multitiered audience, the API was directly exposed and available for those in the user pool with programmatic knowledge and underlying subject matter expertise. The EGRASS user interface for the nonpower users was designed and optimized for speed and ease of use, and required little to no training. For nonpower users accessing it via the interface, higher level information is readily accessible, and a much greater level of detail is available by scoping in further. In candidate technology mode, the user selects an asset category of interest, and by following the population of assets in that category, the user selects a specific asset\u2014for example, a hospital, a police station, or a drinking water treatment plant. At this stage, the zoom and extent are established based on the asset selected and five pieces of relevant information are returned in user panels: (1) a list of candidate technologies for that specific asset; (2) the population at risk; (3) the number of adjacent assets in the same category; (4) the number of adjacent substations; and (5) the reliability index of the longest transmission line. Items 2 \u20135 are all spatially queried within a predefined Euclidean search distance from the asset selected (Figs. 1 and 5). At this point in the interaction pattern, the user can narrow the search distance to zoom in on the extent of interest. All geospatial and tabular responses from the backend are updated in perpetuity. Reliability indices for each distribution line related to collocated substations can be individually assessed by clicking on the distribution line of interest (Fig. 4).\nIn the risk-based dynamic contingency mode, the user begins by selecting a historical hurricane track, assets of interest, and (optionally) a time resample interval and windspeed threshold. The hurricane simulation is run, all assets within the projected hurricane swath footprint are depicted by windspeed at each timestep, and a tabular output is provided in the lower panel (Fig. 5). The user can then cycle through each time step and assets are highlighted.\n\n### 2.3 A Response Time-Optimized Approach to Execution of Underlying Geospatial Queries\u2014Estimating Population at Risk and Intersection of Transmission Lines and Spatially Collocated Infrastructure\n\nQuantification of population at risk and characterization of population statistics has emerged as a major challenge in areas where census data are sparse, difficult to retrieve, and\/or nonexistent, and there are limited sources of streaming geospatial intelligence (GEOINT) data. As it relates to electrical power supply in Puerto Rico, population serviced and population at risk of loss of service was perhaps the single-most important piece of information for the overall utility of the application. This is due in part to the time involved in recovering and restoring power to critical assets, and also to the extent of population affected by loss of power. Developing a geospatial population data set at appropriate spatial resolution and developing a mechanism to execute geospatial queries against these data in a responsive manner were key metrics in the usability and value of this work as applied research related to disaster response. Furthermore, development of efficient and sustainable approaches to analyzing large data sets, geospatial and otherwise, is critical to every modern-day approach to algorithm design and underlying architecture.\nLacking well vetted data at the census block level in the area, population counts by municipality district2 were used. Residential homes were identified by using Microsoft building footprints.3 Population values were distributed among residential footprints proportionally by area in each respective municipality (Fig. 6). The centroid from each housing unit was used in subsequent queries and in the API for improved response times and to develop efficient queries.\nA ceiling API response time of 1000-milliseconds over a range of network speeds was targeted and designed, and the API was structured based on this criterion, using dynamic population downsampling to query against data averaged over larger areas as search extent increased. There are over 1.5 million buildings in Puerto Rico, and obtaining target response required an iterative and comprehensive approach. Building footprint centroids (Fig. 7a) were dynamically resampled to coarser resolution as a function of search distance from point of interest: 500\u00a0m, 1000\u00a0m, and \u2265 1800\u00a0m (Figs. 7b, c, and d respectively). Total population counts derived from downsampled population centroids were evaluated for accuracy by comparing these counts to raw counts from original building footprint\u00a0estimates, which included all data. Distances further than 1800\u00a0m from centroid were not considered relevant in terms of response time as a search distance of approximately 1\u00a0km is most relevant with respect to critical end-use loads and electrical distribution circuits that can provide power to these end-use loads. The potential error was evaluated against the benefit of increased API response times at each 100\u00a0m interval. Mean raw population values from 100 locations were compared to the same locations with resampled data at each 100\u00a0m interval. Mean values from population estimates were compared with a paired t-test.\nSimilar geospatial queries included estimating the total number of collocated substations, similar asset types (for example, hospitals), and transmission lines within the search distance, as depicted by the intersecting search distance (Figs. 6c, 7). This information was also assessed geospatially in the context of the known attributes of the critical end-use loads and the historical transmission line reliability indices.\n\n## 3 Optimal Back Source Selection by Resource Type by Planning Unit and for Selected Sites\n\nThe results are summarized for the methodology in three different formats: (1) a table representing the proportion of optimal backup systems for each end-use resource type for a synoptic assessment of Puerto Rico backup power options (Table 2); (2) a geographic summary by planning unit for hospitals as an example of resource type by region (Fig. 8); and (3) four separate individual site selections to highlight the expected use of the interface (Fig. 9). Planning units are geographic entities established by the Puerto Rico Electric Power Authority (PREPA), which are used to associate critical infrastructure with specific regions of Puerto Rico. Importantly, PREPA was dissolved in 2021 and replaced by LUMA Energy, but the planning extents have not changed and LUMA still utilizes this information as a geographic reference for planning and management. Planning units are depicted by zone in Fig. 8. Remote controlled switches account for most recommended backup systems, and self-healing systems account for the fewest across resource types (Table 3).\nTable 3\nRecommended power backup by critical end-use load category\n\nRemote controlled switches (%)\nReclosures (%)\nFault location, isolation, and service restoration (FLISR) (%)\nSelf-healing systems (%)\nBackup generation (%)\nMicrogrid (%)\nAirports (10)\n53.6\n11.1\n22.2\n2.0\n0.0\n11.1\nHospitals (65)\n41.6\n0.3\n33.3\n0.0\n12.2\n12.5\nLaw Enforcement (187)\n64.9\n1.7\n18.4\n1.4\n0.0\n13.5\nWater Treatment Plants (130)\n35.8\n11.1\n0.8\n0.0\n27.1\n25.3\nSea Ports (53)\n44.2\n1.7\n13.3\n2.5\n25.8\n12.5\nIn Fig. 8 the pie charts are proportionally sized by the total number of hospitals in each planning unit and colors categorically reflect the proportion of each backup type for each planning unit respectively. San Juan planning unit, having the most hospitals, is shown in a separate map with street map as reference. Hospitals are used as an example resource type in this data summary, but similar results could be depicted for all other resource types.\n\n### 3.1 Case Studies, Adaptation to Critical End-Use Loads in Puerto Rico\n\nSeveral example scenarios are described here and shown in the context of spatial reference (Fig. 9). For simplification, and to protect sensitive geolocation data, site names are anonymized, and background map context is not shown. The text highlights the associated decision matrix for various backup sources; the logic to execute this selection is integrated into the API. The examples below represent a small subsample of end-use loads.\n\u2022 Scenario 1: Airport\u2014The airport in Fig. 9a is served by three substations. Because this is a major airport that is already served by three substations, there are three candidate technologies that could be considered. First, automated switches and\/or reclosers could be used to provide redundant paths to the airport. Second, due to the large number of distribution circuits in a nearby urban area, a self-healing scheme that integrates the operation on multiple circuits could be considered. This could be an option unless the airport is supplied by dedicated distribution circuits. If it is supplied by dedicated circuits, a self-healing scheme would not be as useful; this information was not available at the time this article was prepared. The third option would be the installation of backup generation, if it does not already exist, and\/or creation of a microgrid if there is sufficient on-site generation.\n\u2022 Scenario 2: Hospital\u2014The hospital in Fig. 9b is the largest hospital in a densely populated area, with a number of collocated hospitals, and serves the highest number of inpatients and outpatients in the area and has the highest bed count. Microgrids are recommended based on the importance of this hospital and the presence of backup generators in the general area. Hospitals are typically equipped with backup generation and fuel for 72\u00a0h, per National Fire Protection Association Standard 110,4 \u201cStandard for Emergency and Standby Power Systems.\u201d Because there are more than a half-dozen substations in the vicinity connecting to different feeders, it is practical to switch the reclosures so the hospital can connect to other feeders. Because of the time involved in manual operation and FLISR (fault location, isolation, and service restoration), remote-controlled switches are necessary to reestablish power as quickly as possible.\n\u2022 Scenario 3: Police Station\u2014The police station in Fig. 9c is supplied by a single circuit. Even though adjacent circuits exist nearby, backup generation may not be practical, so the only option is to improve the reliability of the existing circuit if it is not adequate. This could be accomplished with a combination of automated switches and reclosers.\n\u2022 Scenario 4: Water Treatment Plant\u2014The water treatment plant in Fig. 9d has two backup generators and a solar panel, which makes it a good candidate for microgrids. Because the plant is far from densely populated areas, and it would take several hours to repair, FLISR in concert with remote-controlled switches is recommended.\nImportantly, presented here are only four critical end-use loads that were characterized out of a much larger number of possible scenarios. The underlying data in Puerto Rico against which the algorithms were developed includes 1.5 million residential housing footprints, 41,248 electric transmission towers, 6212 transmission lines, 383 substations, 187 law enforcement agencies, 13 electric power plants, 65 hospitals, and a total population of over 3.5 million people. Each end-use load has the possibility of utilizing one or many of the six different types of backup systems for rapid recovery following a hurricane event.\n\n### 3.2 Evaluation of Population Downsampling as a Function of Increased Application Efficiency\n\nPopulation at risk as associated with residential footprints was the most computationally expensive aspect of the API. In resampling population data, average response time was reduced to 700\u00a0milliseconds, as opposed to 200,000 milliseconds required for raw spatial data (Fig. 10a). Response times for resampled data did increase significantly beyond a search distance of 10,000 m (up to 11,000 milliseconds),but search distances up to 1600\u00a0m (1 mile) had little relevance to the localized end-use load being assessed.\nRelated to response times, population counts were not significantly different within an 1800-m search distance (p = 0.12). Population counts were higher at approximately 900\u00a0m and lower between 900 and 1800 m.\n\n## 4 Discussion\n\nThis article highlights a methodology for the fusion of multiple data sources into a single framework that supports power system operations in response to extreme events, and details the structure and potential of this methodology in a tool called EGRASS. The approach developed here adds to a growing body of research and tools that are more directly related to emergency response and recovery and not to power outage prediction per se (Staid et al. 2014; Jeffers et al. 2018). More specifically, the capability described here integrates a larger body of information to identify optimal backup sources for critical end-use loads in each major category. This technology can be characterized as a location intelligence-driven application designed with a web-based single page progressive web application (PWA) to help support rapid response to catastrophic events resulting in interruption of power to critical end-use loads. This capability is applicable to other categorical natural hazard events and systems and is abstract in the sense that it can be refactored to include other data types varying in form and structure.\nSummary results of the methodology that were developed for this framework provide ideal planning resources for critical end-use loads by category and by region. Results in a synoptic view can help guide investments for preparation as a whole and point to specific regions and local areas that may require greater investment based on additional insight that may not be available in the public domain. Used in the web application user-interface with mobile phone or by workstation, this tool provides a convenient way to survey areas and extract details by specific sites. The geolocation-enabled mobile version may help with field surveys and yield critical information during site visits.\nA great deal of work has been done on developing a resilience assessment framework for electric power systems. This includes development of probabilistic models for assessing transmission responses following hurricane events (Ouyang and Due\u00f1as-Osorio 2014; Mensah and Due\u00f1as-Osorio 2016), quantitative methodology that measures resilience costs that result from a disruption to infrastructure function (Vugrin et al. 2011), and the assessment of overall vulnerability and resilience of coastal regions by integrating natural and human data layers for mapping and visualization (Lam et al. 2015). The application described in this article is unique in the approach used to merge population data with grid topology, reliability indices, and other geographic information. The EGRASS software is less focused on model development and more focused on building the API to facilitate access to the underlying logic of powerful models with a simple progressive web application interface.\nWe emphasize the importance of continuing with the development of a nationwide semantic data model with all pertinent data, which requires focused and ongoing cooperation from a variety of entities. Although much remains to be done, EGRASS begins this endeavor with a focus on building entity relationships between infrastructure, critical end-use loads, population distribution and needs, and reliability indices. It enables an interactive and highly responsive approach for workforce and regulatory entities to respond to power outages with relevant information about backup sources.\n\n### 4.1 Leveraging Cloud Resources and Cloud Dependencies\n\nMany companies, universities, and other institutions have migrated, or are beginning the process of migrating, to the cloud. In early evolutions of cloud technologies on-premise projects were moved to a similar suite of virtual machines that resided on a cloud, and this represented a more streamlined one-to-one remapping of technology. Significant changes in this paradigm occurred along with the concept of platform as a service (PaaS), and serverless technology. The impact being that \u201clift and shift\u201d became much more involved for moving on-premise based applications to the cloud. Consequently, more development teams are likely to use the cloud concepts in the beginning phases of development and continue down this path moving into other phases of the software development life cycle. This approach has greater inherent risk based on working knowledge that cloud environments from larger and more well-known companies such as Amazon Web Services (AWS), Microsoft AZURE, and Google Cloud may not hybridize well. Furthermore, projects that embrace a particular cloud technology over the other cannot necessarily be abstracted to another platform. One approach to overcome this is to containerize and develop more sophisticated container environments to meet the demands of more complex applications. Kubernetes is an open-source container orchestrator for managing containers with a dedicated and growing number of users and advocates. Although not entirely cloud agnostic, it is far more realistic to set up and manage across multiple or mixed cloud environments. At this point in development EGRASS is very much tied to the AWS platform; but future iterations will be more focused on a containerized deployment.\n\n### 4.2 In Consideration of Day\/Night Population Flux\n\nPopulation counts in this study were obtained at the spatial extent of municipal boundaries and distributed across residential building footprints. Essentially this choice provides a nighttime population (Fig. 4). Given the importance of daily energy usage curves and the variance in demand between residential and nonresidential buildings, it is important to consider how daytime population, in addition, would offer significant advantages to the existing tool. Such a dataset does exist in raster format (Dobson et al. 2000), which was developed using reflectance patterns from spectral signatures and segmentation. Upon evaluation of these data in Puerto Rico, we found commission\/omission differences from actual buildings we considered to be significant and felt building a footprint dataset was a superior source for identifying population source locations and quantifying population. We have developed similar products that have accounted for day\/night flux in the continental United States. In these studies, we incorporated employment data from Census Longitudinal Employment-Household Dynamics dataset,5 from which we developed daytime population from place of origin (residence) to place of destination (workplace). Unfortunately, such data are not available in Puerto Rico, and we suggest that a follow up study should consider a method for replicating such a dataset in Puerto Rico.\n\n## 5 Conclusion\n\nThis article highlights a methodology for coalescing large and disparate data sources into a central framework that supports power system operations in response to extreme events, and details instance of this methodology in a tool called EGRASS. Timely restoration of electrical power for critical end-use loads is crucial and can be achieved, at least partially, by using backup systems when the primary system is in the process of restoration. Backup systems vary widely in terms of expense and practicality of implementation, and are influenced by factors such as demand, accessibility, redundancy, and reliability. The range of possibilities requires expert judgement, in addition to access to relevant data, and typically requires more time and effort than is affordable when responding to catastrophic events.\nThe approach developed here adds to a growing body of research and tools that are more directly related to emergency response and preparedness. We acknowledge the importance of the existing body of research focusing on this area, including outage prediction models and resiliency and adaption methodologies. We also recognize the value and growing utility of using machine learning in concert with remotely sensed imagery to inform critical, time sensitive decision-making processes when working with big data. The work presented here pertains more specifically to the nature and technical challenge of working with data that vary widely in terms of native format, and the value that thoughtful, intentional, and pragmatic semantic data modeling provides when designing an application framework. This article also highlights the importance of streamlined design principles and implementing serverless cloud architecture best practices, which is far more sustainable and adaptable in the long term, and in step with modern practices.\n\n## Acknowledgments\n\nThis work was supported by the United States Department of Energy, Office of Energy Efficiency and Renewable Energy, Solar Energy Technology Program. The Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by the Battelle Memorial Institute under Contract DE-AC05-76RL01830.\nprint\nDRUCKEN\nLiteratur\nAigner, W., S. Miksch, W. M\u00fcller, H. Schumann, and C. Tominski. 2008. Visual methods for analyzing time-oriented data. IEEE Transactions on Visualization and Computer Graphics 14(1): 47\u201360. CrossRef\nAkker, J.D. 2014. Fault location, isolation, and service restoration technologies reduce outage impact and duration. 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Arenas, P. Brito, and K.B. Liu. 2015. Mapping and assessing coastal resilience in the Caribbean region. Cartography and Geographic Information Science 42(4): 315\u2013322. CrossRef\nMensah, A.F., and L. Due\u00f1as-Osorio. 2016. Efficient resilience assessment framework for electric power systems affected by hurricane events. Journal of Structural Engineering. https:\/\/\u200bdoi.\u200borg\/\u200b10.\u200b1061\/\u200b(ASCE)ST.\u200b1943-541X.\u200b0001423. CrossRef\nMukherjee, S., R. Nateghi, and M. Hastak. 2018. A multi-hazard approach to assess severe weather-induced major power outage risks in the U.S. Reliability Engineering and System Safety 175(C): 283\u2013305. CrossRef\nNateghi, R., S. Guikema, and S.M. Quiring. 2014. Power outage estimation for tropical cyclones: Improved accuracy with simpler models. Risk Analysis 34(6): 1069\u20131078. CrossRef\nNOAA (National Oceanic and Atmospheric Administration). 2020. 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Guttromson, and J. Thomas. 2009. A novel visualization technique for electric power grid analytics. IEEE Transactions on Visualization and Computer Graphics 15(3): 410\u2013423. CrossRef\nYang, F., D.W. Wanik, D. Cerrai, M. Bhuiyan, and E.N. Anagnostou. 2020. Quantifying uncertainty in machine learning-based power outage prediction model training: A tool for sustainable storm restoration. Sustainability 12(4): Article 1525.\nYi, J., Y. Kang, J.T. Stasko, and J.A. Jacko. 2007. Toward a deeper understanding of the role of interaction in information visualization. IEEE Transactions on Visualization and Computer Graphics 13(6): 1224\u20131231. CrossRef\nZhai, C., T. Chen, A. White, and S. Guikema. 2021. Power outage prediction for natural hazards using synthetic power distribution systems. Reliability Engineering & System Safety 208: Article 107348.\nZhang, L., and J. Yi. 2010. Management methods of spatial data based on PostGIS. In Proceedings of 2010 2nd Pacific-Asia Conference on Circuits, Communications and System, 1\u20132 August 2010, Beijing, China, 410\u2013413.\nZimmerman, R., and C.E. Restrepo. 2009. Analyzing cascading effects within infrastructure sectors for consequence reduction. Proceedings of the 2009 IEEE Conference on Technologies for Homeland Security, 11\u201312 May 2009, Waltham, Massachusetts, United States, 157\u2013162.\nTitel\nRapid Evaluation and Response to Impacts on Critical End-Use Loads Following Natural Hazard-Driven Power Outages: A Modular and Responsive Geospatial Technology\nverfasst von\nPatrick D. Royer\nWei Du\nKevin Schneider\nPublikationsdatum\n25.05.2022\nVerlag\nSpringer Nature Singapore\nErschienen in\nInternational Journal of Disaster Risk Science \/ Ausgabe 3\/2022\nPrint ISSN: 2095-0055\nElektronische ISSN: 2192-6395\nDOI\nhttps:\/\/doi.org\/10.1007\/s13753-022-00413-6\n\nZur Ausgabe","date":"2023-03-28 21:27:47","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\": 0.32491981983184814, \"perplexity\": 3110.414455099973}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296948871.42\/warc\/CC-MAIN-20230328201715-20230328231715-00428.warc.gz\"}"} | null | null |
HomeNews & Social MediaSanya News'Strange slope' discovered in Sanya, cars go uphill with engines off
'Strange slope' discovered in Sanya, cars go uphill with engines off
Cars will move uphill after engines are turned off on the road in front of Sanya Wanke Forest Park.
Recently, a Sanya driver surnamed Li found a "strange slope" in front of Wanke Forest Park at Sanya Xueyuan Road. Mr. Li claims that cars will go uphill at a speed of 10 kmph if you turn off the engines, reported South Island Evening News.
According to Li, a strange phenomenon occurred when he was teaching his workmate how to drive on the road in front of Wanke Forest Park a few days ago. When the car was going downhill, Mr. Li asked his colleague to back up. As he forgot how to reverse the car, he turned off the engine and stopped the car. What confused Li was that the car moved uphill all by itself after the engine was cut.
In order to figure this strange phenomenon out, Mr. Li went to the "strange slope" with his car on 28th May. His experiment shows the car will speed up when going uphill, and vice versa. "It's just like the car is dragged by something invisible", said Li.
On 29th May, Li took a reporter to that road again. Just as Mr Li had described, the car slid 120 meters uphill after he cut the engine. What's more, Li spilled some water on the road, and the water flowed uphill as well.
However, according to experienced drivers, this strange phenomenon is due to a visual illusion caused by the surrounding terrain. According to research, only 11 such strange slopes have been discovered around the world. Shenyang, Xiamen, Henan, Xi'an, Peking, Yilan, Taiwan, Uruguay, Korea, and America are the only places with this phenomenon.
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
\input{sec_IntroductionV2.tex}
\section{Related works}\label{Sec:Related_work}
\input{sec_Related_workV2.tex}
\section{Overview}\label{Sec:Overview}
\input{sec_Overview.tex}
\section{On demand evaluation enabled CNN generator}\label{Sec:Generator}
\input{sec_ModelV2.tex}
\section{Training}\label{Sec:Training}
\input{sec_Training.tex}
\section{Results}\label{Sec:Results}
\input{sec_Results.tex}
\section{Limitations and future work}\label{Sec:Limitations}
\input{sec_Discussion.tex}
\section{Conclusion}\label{Sec:Conclusion}
\input{sec_Conclusion.tex}
\section*{Aknowledgments}
We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2015-06025.
This project has been carried out with support from the French
State, managed by the French National Research Agency (ANR-16-CE33-0010-01).
This study has been also carried out with financial support from
the CNRS for supporting grants PEPS 2018 I3A "3DTextureNets".
In like manner, we acknowledge the support of CONACYT.
Bruno Galerne acknowledges the support of NVIDIA Corporation with the donation of a Titan Xp GPU used for this research.
The authors would like to thank Loïc Simon for fruitful discussions on 3D rendering, and Guillaume-Alexandre Bilodeau for giving us access to one of his GPUs.
\bibliographystyle{eg-alpha-doi}
\subsection{Architecture}\label{SubSec:Architecture}
\begin{figure*}[ht]
\centering
\includegraphics[width=17.5cm]{multi-scaleGv6.pdf}
\caption{Schematic of the network architecture.
Noise input $Z = \{z_0,\ldots,z_K\}$ at $K+1$ different scales is processed using convolution operations and non-linear activations.
The information at different scales is combined using upsampling and channel concatenation. $M_i$ indicates the number of input channels and $M_s$ controls the number of channels at intermediate layers. For simplicity we consider a cube shaped output with spatial size of $N_1=N_2=N_3=N$.
For each intermediate cube the spatial size is indicated above
and the number of channels below.}
\label{Fig:Generator2}
\end{figure*}
The generator produces a solid texture $v = \mathcal{G}(Z|\theta)$
from a set of multi-channel volumetric white noise inputs $Z = \{z_0,\ldots,z_K\}$.
The spatial dimensions of $Z$ directly control the size of the generated sample.
The process of transforming the noise $Z$ into a solid texture $v$ is depicted in Figure~\ref{Fig:Generator2}. It follows a multi-scale architecture built upon three main operations: convolution, concatenation, and upsampling.
Starting at the coarsest scale, the 3D noise sample is processed with
a set of \emph{convolutions} followed by an \emph{upsampling} to reach the next scale.
It is then concatenated with the independent noise sample from the next scale, itself also processed with a set of convolutions.
This process is repeated $K$ times before a final single convolution layer that maps the number of channels to three to get a color texture.
We now detail the three different blocks of operations used in the generator network.
\paragraph*{Convolution block}~A convolution block groups a sequence of three ordinary 3D convolution layers, each of them followed by a batch-normalization and a leaky rectified linear unit function. Considering $M_{in}$ and $M_{out}$ channels in the input and output respectively, the first convolution layer carries out the shift from $M_{in}$ to $M_{out}$. The following two layers of the block have
$M_{out}$ channels in both the input and the output. The size of the kernels is $3\times3\times3$ for the first two layers and $1\times1\times1$ for the last. Contrary to \cite{ulyanov2016texturenets} and in order to enable on demand evaluation (see Subsection~\ref{SubSec:Spatial_dependency}), here the convolutions are computed densely and without padding, thus discarding the edge's values. Applying one convolution block with these settings to a volume reduces its size by 4 values per spatial dimension.
\paragraph*{Upsampling}~An upsampling performs a 3D nearest neighbor upsampling by a factor of 2 on each spatial dimension ({i.e. } each voxel is replicated 8 times).
\paragraph*{Channel concatenation}~ This operation first applies a batch normalization operation and then concatenates the channels of two multi-channel volumes having the same spatial size. If different, the biggest volume is cropped to the size of the smallest one.
The learnable parameters $\theta$ of the generator are: the convolution's kernels and bias, and the batch normalization layers' weight, bias, mean and variance. The training of these parameters is discussed in Section~\ref{Sec:Training}.
\subsection{Spatial dependency}\label{SubSec:Spatial_dependency}
Forward evaluation of the generator is deterministic and local, {i.e. } each value in the output only depends on a small number of neighboring values in the multi-scale input. By handling the noise inputs correctly, we can feed the network separately with two \emph{contiguous} portions of noise to synthesize textures that can be tiled seamlessly.
Current 2D CNN methods~\cite{ulyanov2016texturenets,li2017diversified} perform padded convolutions, not addressing on demand evaluation capabilities.
Let us notice that a perfect tiling between two different samples can only be achieved by using convolutions without padding. Therefore we discard the values on the borders where the operation with the kernel cannot be carried out completely.
When synthesizing a sample, the generator is fed with an input that takes into account the neighboring dependency values. Those extra values are progressively processed and discarded in the convolutional layers: for an output of size $N_1\times N_2\times N_3$ the size of the input at the $k$-th scale has to be $(\frac{N_1}{2^k} + 2c_k) \times (\frac{N_2}{2^k} + 2c_k) \times (\frac{N_3}{2^k} + 2c_k)$
where $c_k$ denotes the additional values along each spatial dimension required due to the dependency.
The size in any spatial dimension $N$ can be any positive integer (provided the memory is large enough for synthesizing the volume).
These additional values $c_k$ depend on the network architecture. In our case, thanks to the symmetry of the generator, the coefficients $c_k$ are the same along each spatial dimension.
Each convolutional block requires additional support of two values on each side along each dimension and each upsampling cuts down the dependency by two (taking the smallest following integer when the result is fractional).
At scale $k=0$ there are two convolution blocks, therefore $c_0 = 4$. For subsequent scales $c_k = \lceil(c_{k-1}-2)/2\rceil + 4$,
except for the coarsest scale $K$ where there is only one convolution block and therefore $c_K = \lceil(c_{K-1}-2)/2\rceil + 2$.
For example, in order to generate a single voxel, the spatial dimensions of the noise inputs must be $9^3$ for $z_0$, $11^3$ for $z_1$, $13^3$ for $z_2$ to $z_4$, and $9^3$ for $z_5$, which totals $9380$ random values.
\subsection{Multi-scale shift compensation for on demand evaluation}\label{SubSec:Shift_compensation}
On demand generation is a standard issue in procedural synthesis~\cite{perlin1985image}.
The purpose is to generate consistently any part of an infinite texture model.
It enables the generation of small texture blocks separately, whose localization depends on the geometry of the object to be texturized, instead of generating directly a full volume containing the object.
For procedural noise, this is achieved using a reproducible Pseudo-Random Number Generator (PRNG) seeded with spatial coordinates.
In our setting, we enforce spatial consistency by generating the multi-scale noise inputs using a xorshift PRNG algorithm~\cite{marsaglia2003xorshift} seeded with values depending on the volumetric coordinates, channel and scale, similarly to~\cite{galerne2017texton}. Thus, our model only requires the set of reference 3D coordinates and the desired size to generate spatially coherent samples. Given a reference coordinate $n_0$ at the finest scale in any dimension, the corresponding coordinates at the $k$-th scale are computed as $n_k = \lfloor\frac{n_0}{2^k}\rfloor$.
These corresponding noise coordinates need to be aligned in order to ensure the coherence between samples generated separately.
Feeding the generator with the precise set of coordinates of noise at each scale is only a first step to successfully synthesize compatible textures.
Recall that the model is based on combinations of transformed noises at different scales (see Figure~\ref{Fig:Generator2}), therefore requiring special care regarding upsampling to preserve the coordinate alignment across scales, {i.e. } which coordinate $n_k$ at scale $k$ must be associated to a given coordinate $n_0$ at the finest scale $k=0$.
Indeed, after every upsampling operation, observe that each value is repeated twice along each spatial dimension, \emph{pushing} the rest of the values spatially.
Depending on the coordinates of the reference voxel being synthesized, this \emph{shift} of one position can disrupt the coordinate alignment with the subsequent scale. Therefore, the generator network has to compensate accordingly before each concatenation.
For $K=5$ upsamplings, one of the $2^K = 32$ combinations of compensation shifts has to be properly done for each dimension to synthesize a given voxel.
In order to consider these compensation shifts, we make the generator network aware of the coordinate of the sample at hand. In our implementation the reference is on the vertex of the sample closest to the origin.
Given the final reference coordinate $n_0$ of the voxel (in any spatial dimension), the generator deduces the set of shifts recursively from the following relation,
$n_{k-1} = 2n_k + s_{k}$,
where $n_k$ is the spatial reference coordinate used to generate the noise at scale $k \in \{1,\dots, K\}$, and $s_k \in \{0,1\}$ is the shift value used after the $k$\textsuperscript{th} upsampling operation.
At evaluation time, the generator sequentially applies the set of shifts before every corresponding concatenation operation.
\subsection{Solid texture synthesis}
Procedural methods~\cite{perlin1985image,peachey1985solid} are quite convenient for computer graphics applications thanks to their real time computation and on demand evaluation capability.
Essentially one can add texture to a 3D surface by directly evaluating a function given the coordinates of only the required (visible) points in the 3D space.
The principle is as follows for texture generation on a surface: a colormap function, such as a simple mathematical expression, is evaluated at each of the visible 3D points.
In~\cite{perlin1985image}, authors use pseudo-random numbers that depend on the coordinates of the local neighborhood of the evaluated point which ensures both the random aspect of the texture and its spatial coherence.
Creating realistic textures with a procedural noise is a trial and error process that necessitates technical and artistic skills.
Some procedural methods alleviate this process by automatically estimating their parameters from an example image~\cite{ghazanfarpour1995Spectral,GLLD_Gabor_noise_by_example_2012,Gilet_et_al_2014_local_random_phase_noise_2014,galerne2017texton}.
However, they only deal with surface textures and most photo realistic textures are still out of the reach of these methods.
Most example-based solid texture synthesis methods aim at generating a full block of voxels whose cross-sections are visually similar to a respective texture example. These methods are in general fully automated and they are able to synthesize a broader set of textures.
The texture example being only 2D, a prior model is required to infer 3D data. Some methods~\cite{heeger1995pyramid,dischler1998anisotropic,qin2007aura} employ an iterative \emph{slicing} strategy where they alternate between independent 2D synthesis of the slices and 3D aggregation.
Another strategy starts with a solid noise and then iteratively modifies its voxels by assigning them a color value depending on a set of \emph{coherent} pixels from the example. Wei~\cite{wei2003multiplesources} uses the 2D neighborhood of a pixel also called \emph{patch} to assess coherence, then, the set of contributing pixels is formed by the closest patch along each axis of the solid. Finally, the assigned color is the average of the contributing pixels.
Dong~{et al.}~\cite{dong2008lazy} determine coherence using three overlapping 2D neighborhoods ({i.e. } forming a 3D neighborhood) around each voxel and find only the best match among a set of precomputed candidates.
The example-based solid texture synthesis methods that achieve the best visual results in a wider category of textures \cite{kopf2007solid,chen2010highquality} involve a patch-based global optimization framework \cite{kwatra2005texture} governed by a statistical matching strategy.
This improves the robustness to failure.
These methods, however, require high computation times, which limits them to low resolution textures (typically $256^2$ input examples), and are incapable of on demand evaluation which limits their usability.
Regarding speed, the patch-based method proposed by Dong~{et al.}~\cite{dong2008lazy} allows for fast on demand evaluation, thus allowing for visual versatility and practicality. Here the patch-matching strategy is accelerated via the pre-processing of compatible 3D neighborhoods accordingly to the examples given along some axis. This preprocessing is a trade-off between visual quality and speed as it reduces the richness of the synthesized textures.
Thus, their overall visual quality is less satisfactory than the one of the optimization methods previously mentioned.
\subsection{Neural networks on texture synthesis}~%
Our method builds upon previous work on example-based 2D texture synthesis using convolutional neural networks. We distinguish two types of approaches: \emph{image optimization} and \emph{feed-forward texture networks}.
\paragraph*{Image optimization}~
Image optimization methods were inspired from previous statistical matching approaches~\cite{portilla2000parametric} and use a variational framework which aims at generating a new image that matches the features of an example image.
The role of CNN in this class of methods is to deliver a powerful characterization of the images.
It typically comes in the form of feature maps at the internal layers of a pretrained deep CNN~\cite{Simonyan14c} namely VGG-19.
Gatys {et al.}~\cite{gatys2015cnn} pioneered this approach for texture synthesis, by considering the discrepancy between the feature maps of the synthesized image and the example ones.
More precisely for texture synthesis where one has to take into account spatial stationarity, the corresponding \emph{perceptual loss} as coined later by \cite{johnson2016Perceptual} is the Frobenius distance of the Gram matrices of CNN features at different layers.
Starting from a random initialization, the input image is then optimized via a stochastic gradient descent algorithm, where the gradient is computed using back-propagation through the CNN.
Since then, several variants have built on this framework to improve the quality of the synthesis:
for structured textures by adding a Fourier spectrum discrepancy~\cite{liu2016texture}
or using co-occurence information computed between the feature maps and their translation~\cite{berger2017incorporating};
for non-local patterns by considering spatial correlation and smoothing~\cite{sendik2017deep};
for stability by considering a histogram matching loss and smoothing\cite{wilmot2017stable}.
These methods deliver good quality and high resolution results as they can process images of resolutions up to $1024^2$ pixels.
Their main drawback comes from the optimization process itself, as it requires several minutes to generate one image.
Implementing local evaluation on these methods is infeasible since they use a global optimization scheme as for patch-based texture optimization methods~\cite{kwatra2005texture,kopf2007solid}.
Extension to dynamic texture synthesis has also been proposed in~\cite{tesfaldet2018}.
The textured video is optimized using a perceptual loss combined with a loss based on estimated optical flow to take into account the time dimension.
\paragraph*{Feed-forward texture networks}~
Feed-forward networks approaches were introduced by Johnson~{et al.}~\cite{johnson2016Perceptual} for style transfer and Ulyanov~{et al.}~\cite{ulyanov2016texturenets} for texture synthesis. In the latter, they train an actual generative CNN to synthesize texture samples that produce the same visual characteristics as the example.
These methods use the loss function in \cite{gatys2015cnn} to compare the visual characteristics between the generated and example images.
However, instead of optimizing the generated output image, the training aims at tuning the parameters of the generative network.
Such optimization can be more demanding since there is no spatial regularity shared across iterations as for the optimization of a single image.
However this training phase only needs to be done once for a given input example.
This is achieved in practice using back propagation and a gradient-based optimization algorithm using batches of noise inputs. Once trained, the generator is able to quickly generate samples similar to the input example by forward evaluation.
Originally these methods train one network per texture sample. Li {et al.}~\cite{li2017diversified} proposed a large network architecture and a training scheme to allow one network to have the capacity to generate and mix several different textures.
By improving the capacity of the generator network Li {et al.}~\cite{li2017diversified} and Ulyanov~{et al.}~\cite{Ulyanov17improved} methods reached a modestly higher visual quality but found that the synthesized textures did not change sufficiently for different inputs.
In order to prevent the generator from producing identical images they were forced to incorporate a term that encourages diversity in the objective function.
Feed-forward texture networks methods generally produce results with slightly lower visual quality than the image optimization methods, however they exhibit faster generation times.
Visual quality aside, the earlier architecture model proposed by Ulyanov~{et al.}~\cite{ulyanov2016texturenets} holds several advantages over more recent feed-forward methods: it does not require a diversity term, it is able to generate images of arbitrary sizes and it is significantly smaller in terms of number of parameters.
Moreover, as we show in Section~\ref{Sec:Generator} this framework can be customized to allow on demand evaluation.
Other approaches achieve texture synthesis as a feed-forward evaluation of a generator network using a different training strategy.
Bergmann~{et al.}~\cite{bergmann17apsGAN} train a generator network without using the perceptual loss. Instead they employ a generative adversarial network (GAN) framework \cite{goodfellow2014generative} with a purely convolutional architecture \cite{radford2016unsupervised} to allow for flexibility on the size of the samples to synthesize. This method shares the advantages of feed-forward networks regarding evaluation but is based in a more complex training scheme where two cascading networks have to be optimized using an \emph{adversarial loss} which can affect the quality on different texture examples.
Zhou~{et al.}~\cite{zhou2018nonstationary} use a combination of perceptual and adversarial loss and achieve impressive results for the synthesis of non-stationary textures. This is a more general problem seldom addressed in the literature and it requires extra assumptions about the behavior of the texture at hand.
Similarly, Yu~{et al.}~\cite{yu2019texture} train a CNN to perform texture mixing using a hybrid approach, combining the perceptual loss and adversarial loss to help the model produce plausible new textures.
Finally, Li~{et al.}~\cite{li2017universalstyle} proposed another strategy that leverages auto-encoders~\cite{yann1987modeles,bourlard1988auto}. They use truncated versions of VGG-19 network as encoders that map images to a feature space. Then they design decoder networks that generate images from such features. During the training stage, this generator is optimized to invert the encoder by trying to generate images that match encoded images from a large dataset of natural images.
During synthesis, a random image and an example image are first encoded; random features are matched to the target ones using first and second order moment matching, and then fed to the decoder to generate a random texture, without requiring a specific training for this example.
While very appealing, this approach is difficult to adapt to 3D texture synthesis where such large dataset is not available.
Moreover, the quality is not as good as for previously mentioned methods~\cite{gatys2015cnn,ulyanov2016texturenets}.
\subsection{Neural networks for volumetric object generation}~%
A related problem in computer vision is the generation of binary volumetric objects from 2D images~\cite{gwak2017weakly2,jimenez2016unsupervised,yan2016perspective}.
Similarly to our setting, these approaches rely on unsupervised training with a loss function comparing 3D data to 2D examples.
However these methods do not handle color information and only produce low resolution volumes ($32^3$).
\subsection{Experimental settings}
Unless otherwise specified all the results in this section were generated using the following settings.
\paragraph*{Generator network}~
We set the number of scales to 6, {i.e. } $K = 5$, which means that each voxel of the input noise at the coarsest scale impacts nearly 300 voxels at the finest scale. We use $M_i=3$ input channels and $M_s = 4$ (number of channels after the first convolution block and \emph{channel step} across scales) which results in the last layer being quite narrow ($6M_s = 24$) and the whole network compact, with $\sim8.5\times10^4$ parameters.
We include a batch normalization operation after every convolution layer and before the concatenations. As mentioned in previous methods~\cite{ulyanov2016texturenets} we noticed that such a strategy helps stabilizing the training process.
\paragraph*{Descriptor network}~
Following Gatys~{et al.}~\cite{gatys2015cnn}, we use a truncated VGG-19~\cite{Simonyan14c} as our descriptor network, with padded convolutions and average pooling. The considered layers for the loss are: \verb$relu1_1$, \verb$relu2_1$, \verb$relu3_1$, \verb$relu4_1$ and \verb$relu5_1$.
\paragraph*{Training}~
We implemented our approach using \emph{pytorch} (code available in \URL{http://github.com/JorgeGtz/SolidTextureNets}) and we use the pre-trained parameters for VGG-19 available from the BETHGE~LAB~\cite{gatys2015cnn,Gatys_2016_CVPR}. We optimize the parameters of the generator network using Adam algorithm~\cite{kingma2014adam} with a learning rate of 0.1 during 3000 iterations.
Figure~\ref{Fig:Training_curves} shows the value of the empirical estimation of $\mathcal{L}$ during the training of three of the examples shown below in Figure~\ref{Fig:Results_isotropic1} (\emph{histology, cheese} and \emph{granite}).
We use batches of 10 samples per slicing direction. We compute the gradients individually for each sample in the batch which slows down the training process but allows us to concentrate the available memory on the resolution of the samples.
With these settings and using 3 slicing directions, the training takes around 1 hour for a $128^2$ training resolution ({i.e. } size of the example(s) and generated slices), $3.5$ hours for $256^2$ and $12.5$ hours for $512^2$ using one GPU Nvidia GeForce GTX 1080 Ti.
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{empiricalLoss_roquefort_bio1_santiagored_legend}
\caption{Value of the 3D empirical loss during the training of the generator for the textures \emph{histology}, \emph{cheese} and \emph{granite} of Figure~\ref{Fig:Results_isotropic1}.}
\label{Fig:Training_curves}
\end{figure}
\paragraph*{Synthesis}~
In order to synthesize large volumes of texture, it is more time efficient to choose the size of the building blocks in a way that best exploits parallel computation.
Considering computation complexity alone, synthesizing large building blocks of texture at once is also more efficient given the spatial dependency (indicated by coefficients $c_k$) shared by neighboring voxels.
In order to highlight the seamless tiling of on demand generated blocks of texture, most of the samples shown in this work are built by assembling blocks of $32^3$ voxels.
However the generator is able to synthesize box-shaped/rectangular samples of any size, given enough memory is available.
Figures~\ref{Fig:pieces_aggregation} and~\ref{Fig:pieces_on_demand} depict how the small pieces of texture tile perfectly to form a bigger texture.
It takes nearly 12 milliseconds to generate a block of texture of $32^3$ voxels on a Nvidia GeForce RTX 2080 GPU. However we can use bigger elemental blocks, {e.g. } a cube of $64^3$ voxels takes $\sim24$ milliseconds and one of $128^3$ voxels takes $\sim128$ milliseconds.
For reference, using the method of Dong~{et al.}~\cite{dong2008lazy} it takes 220 milliseconds to synthesize a $64^3$ volume and 1.7 seconds to synthesize a $128^3$ volume.
\begin{figure}[!htb]
\centering
\raisebox{20mm}{\includegraphics[width = 1.2cm]{brown016_exemplar}}
\qquad
\includegraphics[width = 5cm]{2018-09-18_brown016_exemplar_3D_1812_paraview_bypieces_v3_cropped}
\caption{Left, example texture of size $128^2$ pixels, used along the three orthogonal axes. Right $512$ contiguous cubes of $32^3$ voxels generated on demand to form a $256^3$ texture. The gaps are added to better depict the aggregation.}
\label{Fig:pieces_aggregation}
\end{figure}
\begin{figure}[!htb]
\centering
\raisebox{10mm}{\includegraphics[width = 2cm]{bio1_512}}
\qquad
\includegraphics[width = 5cm]{2018-09-18_bio1_512_3D_2315_paraview_pieces_v2_cropped}
\caption{Left, example texture of size $512^2$ pixels, used for training the generator along the three orthogonal axes. Right, assembled blocks of different sizes generated on demand. Note that it is possible to generate blocks of size $1$ in any direction.}
\label{Fig:pieces_on_demand}
\end{figure}
\subsection{Experiments}
In this section we highlight the various properties of the proposed method
and we compare them with state-of-the-art approaches.
\paragraph*{Texturing mesh surfaces}
Figure \ref{Fig:3Dmodels} exhibits the application of textures generated with our model to add texture to 3D mesh models. In this case we generate a solid texture with a fixed size and load it on OpenGL as a regular 3D texture with bilinear filtering.
Solid textures avoid the need for surface parametrization and can be used on complex surfaces without creating artifacts.
\begin{figure*}[!htb]
\centering
\includegraphics[width = 17.5cm]{3Dmodels_v2}
\caption{
Texture mapping on 3D mesh models.
The example texture used to train the generator is show in the upper left corner of each object. When using solid textures the mapping does not require parametrization as they are defined in the whole 3D space. This prevents any mapping induced artifacts.
\emph{Sources: the `duck' model comes from Keenan's 3D Model Repository, the `mountain' and `hand' models from free3d.com and the tower and the vase from turbosquid.com.}
}
\label{Fig:3Dmodels}
\end{figure*}
\begin{figure*}[!htb]
\centering
\begin{tabular}{c p{3.6cm}@{\hspace{2pt}}|@{\hspace{2pt}}p{2.2cm}@{\hspace{2pt}}|@{\hspace{2pt}}p{2.2cm}@{\hspace{4pt}}p{2.2cm}@{\hspace{4pt}}p{2.2cm}@{\hspace{4pt}}p{2.2cm}}
& \centering {Generated} volume
& \centering {Examples}
& \multicolumn{4}{c}{Generated slices}
\\
& \centering $v$
& \centering $u_1=u_2=u_3$
& \centering $v_{1,\frac{N_1}{2}}$
& \centering $v_{2,\frac{N_2}{2}}$
& \centering $v_{3,\frac{N_3}{2}}$
& {\centering oblique ($45^{\circ}$) }
\\
\rotatebox{90}{\hspace{1cm} granite}&
\includegraphics[width=3.6cm]{2018-09-24_santiago-red-granite_3D_0137_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{santiago-red-granite}} &
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-24_santiago-red-granite_3D_0137_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-24_santiago-red-granite_3D_0137_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-24_santiago-red-granite_3D_0137_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-24_santiago-red-granite_3D_0137_oblique_slice}}\\
%
\rotatebox{90}{\hspace{1cm} marble}&
\includegraphics[width=3.6cm]{2018-09-18_portoro-gold_512_3D_2135_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{portoro-gold_512}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_portoro-gold_512_3D_2135_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_portoro-gold_512_3D_2135_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_portoro-gold_512_3D_2135_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-18_portoro-gold_512_3D_2135_oblique_slice}}\\
%
\rotatebox{90}{\hspace{1.2cm} beef}&
\includegraphics[width=3.6cm]{2018-09-18_beef_512_3D_2132_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{beef_512_ex}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_beef_512_3D_2132_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_beef_512_3D_2132_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_beef_512_3D_2132_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-18_beef_512_3D_2132_oblique_slice}}\\
%
\rotatebox{90}{\hspace{1.2cm} cheese}&
\includegraphics[width=3.6cm]{2018-09-18_roquefort_512_3D_2130_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{roquefort_512}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_roquefort_512_3D_2130_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_roquefort_512_3D_2130_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_roquefort_512_3D_2130_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-18_roquefort_512_3D_2130_oblique_slice}}\\
%
\rotatebox{90}{\hspace{1cm} histology}&
\includegraphics[width=3.6cm]{2018-09-18_bio1_512_3D_2315_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{bio1_512}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_bio1_512_3D_2315_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_bio1_512_3D_2315_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-18_bio1_512_3D_2315_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-18_bio1_512_3D_2315_oblique_slice}}\\
\end{tabular}
\caption{Synthesis of isotropic textures. We train the generator network using the example in the second column of size $512^2$ along $D=3$ directions. The cubes on the first column are generated samples of size $512^3$ built by assembling blocks of $32^3$ voxels generated using on demand evaluation. Subsequent columns show the middle slices of the generated cube across the three considered directions and a slice extracted in an oblique direction with a $45^{\circ}$ angle. The trained models successfully reproduce the visual features of the example in 3D.}
\label{Fig:Results_isotropic1}
\end{figure*}
\begin{figure*}[!htb]
\centering
\begin{tabular}{c p{3.6cm}@{\hspace{2pt}}|@{\hspace{2pt}}p{2.2cm}@{\hspace{2pt}}|@{\hspace{2pt}}p{2.2cm}@{\hspace{4pt}}p{2.2cm}@{\hspace{4pt}}p{2.2cm}@{\hspace{4pt}}p{2.2cm}}
& \centering {Generated} volume
& \centering {Examples}
& \multicolumn{4}{c}{Generated slices}
\\
& \centering $v$
& \centering $u_1=u_2=u_3$
& \centering $v_{1,\frac{N_1}{2}}$
& \centering $v_{2,\frac{N_2}{2}}$
& \centering $v_{3,\frac{N_3}{2}}$
& {\centering oblique ($45^{\circ}$) }
\\
\rotatebox{90}{\hspace{1.2cm} pebble}&
\includegraphics[width=3.6cm]{2018-09-19_TZ52Shoreline_s512_3D_1212_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{TZ52Shoreline_s512}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-19_TZ52Shoreline_s512_3D_1212_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-19_TZ52Shoreline_s512_3D_1212_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-19_TZ52Shoreline_s512_3D_1212_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-19_TZ52Shoreline_s512_3D_1212_oblique_slice}}\\
%
\rotatebox{90}{\hspace{1.2cm} sand}&
\includegraphics[width=3.6cm]{2018-09-25_sand_s512_3D_1757_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{sand_s512}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-25_sand_s512_3D_1757_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-25_sand_s512_3D_1757_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-25_sand_s512_3D_1757_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-25_sand_s512_3D_1757_oblique_slice}}\\
%
\rotatebox{90}{\hspace{1.2cm} grass}&
\includegraphics[width=3.6cm]{2018-09-21_lush_grass_s512_3D_1445_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{lush_grass_s512}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-21_lush_grass_s512_3D_1445_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-21_lush_grass_s512_3D_1445_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-21_lush_grass_s512_3D_1445_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-21_lush_grass_s512_3D_1445_oblique_slice}}\\
%
\rotatebox{90}{\hspace{1.2cm} lava}&
\includegraphics[width=3.6cm]{2018-09-22_lava_texture_by_xxaries1970xx_s512_3D_1807_paraview_cropped}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{lava_texture_by_xxaries1970xx_s512}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-22_lava_texture_by_xxaries1970xx_s512_3D_1807_middle_slice1}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-22_lava_texture_by_xxaries1970xx_s512_3D_1807_middle_slice2}}&
\raisebox{8mm}{\includegraphics[width=2.2cm]{2018-09-22_lava_texture_by_xxaries1970xx_s512_3D_1807_middle_slice3}}&
\raisebox{3mm}{\includegraphics[height=3.111cm, width=2.2cm]{2018-09-22_lava_texture_by_xxaries1970xx_s512_3D_1807_oblique_slice}}
\end{tabular}
\caption{
Synthesis of isotropic textures (same presentation as in Figure~\ref{Fig:Results_isotropic1}).
While generally satisfactory, the examples in the first and third rows have a slightly degraded quality. In the first row the features have more rounded shapes than in the example and in the third row we observe high frequency artifacts.}
\label{Fig:Results_isotropic2}
\end{figure*}
\paragraph*{Single example setting}
Figures~\ref{Fig:Results_isotropic1}~and~\ref{Fig:Results_isotropic2} show synthesized samples using our method on a set of examples depicting some physical material.
Considering their isotropic structure, we train the generator network using a single example to designate the appearance along the three orthogonal directions, {i.e. } $u_1=u_2=u_3$.
On the first column we show a generated sample of size $512^3$ voxels built by assembling blocks of $32^3$ voxels generated using on demand evaluation.
The second column is the example image of size $512^2$ pixels, columns 3-5 show the middle slices of the generated cube across the three considered directions and the last column shows a slice extracted in an oblique direction with a $45^{\circ}$ angle.
These examples illustrate the capacity of the model to infer a plausible 3D structure from the 2D features present in isotropic example images.
Observe that a slice across an \emph{oblique} direction still displays a conceivable structure given the examples.
They also demonstrate the spatial consistency while using on demand evaluation.
Regarding the visual quality, notice that the model successfully reproduces the patterns' structure while also capturing the richness of colors and variations.
The quality of the slices is comparable to that of the state-of-the-art 2D methods \cite{ulyanov2016texturenets,Ulyanov17improved,li2017diversified,gatys2015cnn}, which is striking as solid texture synthesis is a more constrained problem.
However, such successful extrapolation to 3D might not be possible for all textures. Figure~\ref{Fig:Example_peppers} shows a case where the slices of the synthesized solid contain patterns not present in the example texture. This is related to the question of the existence of a solution discussed in the next paragraph.
\paragraph*{Existence of a solution}
The example texture used in Figure~\ref{Fig:Example_peppers} is isotropic (arrangement of red shapes having green spot inside), but the volumetric material it depicts is not (the green stem being outside the pepper). Training the generator using three orthogonal directions assumes 3D isotropy, and thus, the outcome is a solid texture where the patterns are isotropic through the whole volume. This creates some new patterns in the slices, which makes them somewhat different from the example (red shapes without a green spot inside). Actually, the generated volume just does not makes sense physically, as one always obtains full peppers after slicing them.
\begin{figure}[!htb]
\centering
\includegraphics[height=5cm]{peppers}
\caption{
2D to 3D isotropic patterns.
The example texture (top-left) depicts a pattern that is approximately isotropic in 2D, but the material it depicts is not.
Training the generator using the example along three orthogonal directions results in solid textures that are isotropic in the three dimensions (top-right).
Here, the red and green patterns vary isotropically in the third dimension, this creates a bigger variation on their size and makes some slices contain red patterns that lack the green spot.
%
This is a case where the slices of the solid texture (bottom) cannot match exactly the patterns of the 2D example, thus, not complying with the example in the way a 2D algorithm would.
}
\label{Fig:Example_peppers}
\end{figure}
This example shows that for a given texture example $u$ and a number of directions $D>1$ it is possible that a corresponding 3D texture does not exist, {i.e. } not all the slices in the chosen directions will respect the structure defined by the 2D example.
This existence issue is vital for example-based solid texture synthesis as it delineates the limits of this slice-based formulation used by many methods in the literature. It is briefly mentioned in \cite{kopf2007solid,dong2008lazy} and here we aim to extend the discussion.
Let us consider for instance the isotropic example shown in Figure~\ref{Fig:Example_spheres}, where the input image contains only discs at a given scale ({e.g. } a few pixels diameter). It follows that when slicing a volume containing spheres with the same diameter $\delta$, the obtained image will necessarily contain objects with various diameters, ranging from $0$ to $\delta$. This seems to be a paradigm natural to the 2D-3D extrapolation.
It demonstrates that for some 2D textures an isotropic solid version might be impossible and conversely, that the example texture and the imposed directions must be chosen carefully given the goal 3D structure.
\begin{figure}[!h]
\centering
\includegraphics[height=5cm]{dots_synthesis}
\caption{Illustration of a solid texture whose cross sections cannot comply with the example along three directions. Given a 2D example formed by discs of a fixed diameter (upper left) a direct isotropic 3D extrapolation would be a cube formed by spheres of the same diameter. Slicing that cube would result in images with discs of different diameters.
The cube in the upper right is generated after training our network with the 2D example along the three orthogonal axes. The bottom row shows cross sections along the axes, all of them present discs of varying diameters thus failing to look like the example. }
\label{Fig:Example_spheres}
\end{figure}
This also might have dramatic consequences regarding convergence issues during optimization. In global optimization methods \cite{kopf2007solid,chen2010highquality}, where patches from the example are sequentially copied view per view, progressing the synthesis in one direction can create new features in the other directions thus potentially preventing convergence.
In contrast, in our method, we seek for a solid texture whose statistics are as close as possible from the examples without requiring a perfect match.
This always ensured convergence during training in all of our experiments.
An example of this is illustrated in the first row of Figure~\ref{Fig:Results_diagonal_crayons}, where the example views are incompatible given the proposed 3D configuration: the optimization procedure converges during training and the trained generator is able to synthesize a solid texture, however, the result is clearly a tradeoff which mixes the different contradictory orientations.
This also contrasts with common statistical texture synthesis approaches that are rather based on constrained optimization to guarantee statistical matching, for instance by projecting patches~\cite{Gutierrez2017}, histogram matching~\cite{rabin2011wasserstein}, or moment matching~\cite{portilla2000parametric}.
\begin{figure*}[!ht]
\centering
\begin{tabular}{cc@{\hspace{2pt}}|@{\hspace{2pt}}c@{\hspace{4pt}}c@{\hspace{4pt}}|@{\hspace{4pt}}c@{\hspace{4pt}}c@{\hspace{4pt}}c}
&Generated volume & Training & Examples & \multicolumn{3}{c}{Generated views}\\
& $v$ & configuration & & $v_{1,\frac{N_1}{2}}$ & $v_{2,\frac{N_2}{2}}$ & $v_{3,\frac{N_3}{2}}$\\
\rotatebox{90}{\hspace{1.0cm} soil ($D=3$)}&
\includegraphics[width=3.6cm]{2018-09-22_ulrick-wery-tileableset2-soil_3D_1806_paraview_cropped}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{ulrick3v}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{ulrick-wery-tileableset2-soil}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-22_ulrick-wery-tileableset2-soil_3D_1806_middle_slice1}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-22_ulrick-wery-tileableset2-soil_3D_1806_middle_slice2}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-22_ulrick-wery-tileableset2-soil_3D_1806_middle_slice3}}\\
%
\rotatebox{90}{\hspace{1.0cm} soil ($D=2$)}&
\includegraphics[width=3.6cm]{2018-09-25_ulrick-wery-tileableset2-soil_3D_1732_paraview_cropped}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{ulrick2v}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{ulrick-wery-tileableset2-soil}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-25_ulrick-wery-tileableset2-soil_3D_1732_middle_slice1}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-25_ulrick-wery-tileableset2-soil_3D_1732_middle_slice2}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-25_ulrick-wery-tileableset2-soil_3D_1732_middle_slice3}}\\
%
\rotatebox{90}{\hspace{1.0cm} brick wall ($D=2$)}&
\includegraphics[width=3.6cm]{2018-09-19_Solna_Brickwall1_3D_2048_paraview_cropped}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{solna2v}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{Solna_Brickwall1}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-19_Solna_Brickwall1_3D_2048_middle_slice1}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-19_Solna_Brickwall1_3D_2048_middle_slice2}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-19_Solna_Brickwall1_3D_2048_middle_slice3}}\\
%
\rotatebox{90}{\hspace{1.0cm} cobble wall ($D=2$)}&
\includegraphics[width=3.6cm]{2018-09-19_cobblewall_512_3D_2111_paraview_cropped}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{cobblewallv2}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{cobblewall_512}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-19_cobblewall_512_3D_2111_middle_slice1}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-19_cobblewall_512_3D_2111_middle_slice2}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-09-19_cobblewall_512_3D_2111_middle_slice3}}\\
\end{tabular}
\caption{Training the generator using two or three directions for anisotropic textures. The first column shows generated samples of size $512^3$ built by assembling blocks of $32^3$ voxels generated using on demand evaluation. The second column illustrates the training configuration, {i.e. } which axes are considered and the orientation used. Subsequent columns show the middle slices of the generated cube across the three considered directions. The top two rows show that for some examples considering only two directions allow the model to better match the features along the directions considered. The bottom rows show examples where the appearance along one direction might not be important.}
\label{Fig:Results_views_2_directions}
\end{figure*}
\begin{figure*}[!htb]
\centering
\begin{tabular}{c@{\hspace{4pt}}|@{\hspace{4pt}}c@{\hspace{4pt}}c@{\hspace{4pt}}|@{\hspace{4pt}}c@{\hspace{4pt}}c@{\hspace{4pt}}c|c}
Generated volume & Training & Examples & \multicolumn{3}{c}{Generated views} & Training\\
$v$ & configuration & $u_1 = u_2 = u_3$ & $v_{1,\frac{N_1}{2}}$ & $v_{2,\frac{N_2}{2}}$ & $v_{3,\frac{N_3}{2}}$ & empirical loss\\
\includegraphics[width=3.2cm]{2018-10-06_diagonal-crayons_512_3D_1340_paraview_cropped}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{diagonalcrayons13v}}&
\raisebox{6mm}{\includegraphics[angle=90,width=2.2cm]{diagonal-crayons2_512}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-10-06_diagonal-crayons_512_3D_1340_middle_slice1}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-10-06_diagonal-crayons_512_3D_1340_middle_slice2}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-10-06_diagonal-crayons_512_3D_1340_middle_slice3}}&
\raisebox{15mm}{$1183.6$}
\\
\includegraphics[width=3.2cm]{2018-10-06_diagonal-crayons2_512_3D_1337_paraview_cropped}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{diagonalcrayons23v}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{diagonal-crayons2_512}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-10-06_diagonal-crayons2_512_3D_1337_middle_slice1}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-10-06_diagonal-crayons2_512_3D_1337_middle_slice2}}&
\raisebox{6mm}{\includegraphics[width=2.2cm]{2018-10-06_diagonal-crayons2_512_3D_1337_middle_slice3}}&
\raisebox{15mm}{$363.1$}
\\
\end{tabular}
\caption{Importance of the compatibility of examples.
In this experiment, two generators are trained with the same image along three directions, but for two different orientations.
%
The first column shows generated samples of size $512^3$ built by assembling blocks of $32^3$ voxels generated using on demand evaluation. The second column illustrates the training configuration, {i.e. } for each direction the orientation of the example shown in the third column. Subsequent columns show the middle slices of the generated cube across the three constrained directions. Finally, the rightmost column gives the empirical loss value at the last iteration.
In the first row, no 3D arrangement of the patterns can comply with the orientations of the three examples.
Conversely the configuration on the second row can be reproduced in 3D, thus generating more meaningful results.
The lower value of the training loss for this configuration reflects the better reproduction of the patterns in the example.
}
\label{Fig:Results_diagonal_crayons}
\end{figure*}
\paragraph*{Constraining directions} While for isotropic textures, constraining $D=3$ directions gives good visual results, for some textures it may be more interesting to consider only two views.
Indeed using $D<3$ might be essential to obtain acceptable results, at least along the considered directions. This acts in accordance to the question of existence of a solution discussed in the previous paragraph.
We exemplify this in the top two rows of Figure~\ref{Fig:Results_views_2_directions}, where considering only two training directions (second row) results in a solid texture that more closely resembles the example along those directions, compared to using three training directions (first row) which generates a more consistent volume using isotropic shapes that are not present in the example texture.
The brick and cobblestone textures in Figure~\ref{Fig:Results_views_2_directions} highlight the fact that, when depicting an object where the top view is not crucial to the desired appearance, such as a wall, it can be left to the algorithm to infer a coherent structure.
Of course, not considering a direction during training will generate cross-sections that do not necessarily contain the same patterns as the example texture (see column $v_{1, \tfrac{N_1}{2}}$), but rather color structures that fulfill the visual requirements for the other considered directions.
Additionally, pattern compatibility across different directions is essential to obtain coherent results. In the examples of Figure~\ref{Fig:Results_diagonal_crayons} the generator was trained with the same image but in a different orientation configuration. In the top row example no 3D arrangement of the patterns can comply with the orientations of the three examples.
Conversely the configuration on the bottom row can be reproduced in 3D thus generating more meaningful results.
All this has to be taken into account when choosing the set of training directions given the expected 3D texture structure.
Observe that the value of the loss at the end of the training gives a hint on which configuration \emph{works} better. This can be exploited for automatically finding the best training configuration for a given set of examples.
These results bring some light to the scope of the slice-based formulation for solid texture synthesis using 2D examples. This formulation is best suited for textures depicting 3D isotropic materials,
for which we obtain an agreement with the example's patterns comparable with 2D state-of-the-art methods. For most anisotropic textures we can usually obtain high quality results by considering only two directions. Finally, for textures with patterns that are only isotropic in 2D, using more than one direction inevitably creates new patterns.
\paragraph*{Diversity}
A required feature of texture synthesis models is that they are able to generate diverse samples. Ideally the generated samples are different from each other and from the example itself while still sharing its visual characteristics. Additionally, depending on the texture it might be desired that the patterns vary spatially inside each single sample.
Yet, observe that many methods in the literature for 2D texture synthesis generate images that are local copies of the input image \cite{wei_levoy_2000,kwatra2005texture}, which strongly limits the diversity.
As reported in Gutierrez~{et al.}~\cite{Gutierrez2017}, the (unwanted) optimal solution of methods based on patch optimization
is the input image itself. For most of these methods though, the local copies are sufficiently randomized to deliver enough diversity.
Variability issues have also been reported in the literature for texture generation based on CNN, and~\cite{Ulyanov17improved,li2017diversified} have proposed to use a diversity term in the training loss to fix it.
Without such diversity term to promote variability, the generated samples are nearly identical from each other, although sufficiently different from the example.
In these cases it seems that the generative networks \emph{learn} to synthesize a single sample that induces a low value of the perceptual loss while disregarding the random inputs.
When dealing with solid texture synthesis from 2D examples, such a trivial optimal solution only arises when considering one direction,
where the example itself is copied along such direction.
Yet, there is no theoretical guarantee that prevents the generator network from copying large pieces of the example as it has been shown that deep generative networks can memorize an image in~\cite{ulyanov18deep} .
However, the compactness or the architecture and the stochastic nature of the proposed model make it very unlikely.
In practice, we do not observe repetition among the samples generated with our trained models, even when pursuing the optimization long after the visual convergence (which generally occurs after $1000$ iterations, see Figure~\ref{Fig:Training_curves}).
This is consistent with the results of Ulyanov~{et al.}~\cite{ulyanov2016texturenets}, the 2D architecture that inspired ours, where diversity is not an issue.
One explanation for this difference with other methods may be that the architectures that exhibit a loss of diversity process an input noise that is small compared to the generated output ($0.0025\%$ in \cite{li2017diversified} and $0.13\%$ in \cite{Ulyanov17improved}) and which is easier to ignore.
On the contrary, our generative network receives an input that accounts for roughly 1.14 times the size of the output.
Figure~\ref{Fig:diversity_corr_maps} demonstrates the capacity of our model to generate diverse samples from a single trained generator. It shows three generated solid textures along with their middle slices. To facilitate the comparison, it includes an \emph{a posteriori} correspondence map which highlights spatial similarity by forming smooth regions. In all cases we obtain noisy maps which means that the slices do not repeat arrangements of patterns or colors.
\begin{figure*}[!ht]
\centering
\includegraphics[width = 16cm]{2018-09-24_santiago-red-granite_3D_0137_diversity_wmaps_full_float}
\caption{Diversity among generated samples. We compare the middle slices along three axis of three generated textures of $256^3$ voxels. The comparison consists in finding the pixel with the most similar neighborhood (size $4^2$) in the other image and constructing a correspondence map given its coordinates. The smooth result in the diagonal occurs when comparing a slice to itself. The stochasticity in the rest of results means that the compared slices do not share a similar arrangement of patterns and colors.}
\label{Fig:diversity_corr_maps}
\end{figure*}
\paragraph*{Multiple examples setting}
As already discussed in the work of Kopf~{et al.}~\cite{kopf2007solid} and earlier, it appears that most solid textures can only be modeled from a single example along different directions.
In the literature, to the best of our knowledge, only one success case of a 3D texture using two different
examples has been proposed~\cite{kopf2007solid,dong2008lazy}.
This is due to the fact that the two examples have to share similar features such as color distribution and compatible geometry as already shown in Figure~\ref{Fig:Results_diagonal_crayons}.
Figure~\ref{Fig:Results_2differentinputs} illustrates this phenomenon, for each example we experiment with and without performing a histogram matching (independently for each color channel) to the input examples. We observe favorable results particularly when the colors of both examples are close. Although the patterns are not perfectly reproduced, the 3D structure is coherent and close to the examples.
\begin{figure}
\centering
\newcommand{\cmark}{\text{\ding{51}}}
\newcommand{\text{\ding{55}}}{\text{\ding{55}}}
\begin{tabular}{c@{\hspace{4pt}}c@{\hspace{4pt}}c@{\hspace{4pt}}c@{\hspace{2pt}}|@{\hspace{2pt}}c}
HM & \multicolumn{2}{c}{examples} & configuration & $v$ \\
\raisebox{8mm}{\text{\ding{55}}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{fur-zebra_256}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{giraffe}}&
\raisebox{2mm}{\includegraphics[width=1.8cm]{zebra_giraff_cropped}}&
\includegraphics[width=2.4cm]{2018-10-02_fur-zebra_256_3D_1448_paraview_cropped}\\
\raisebox{8mm}{\cmark}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{zebra_hm_giraff}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{giraffe}}&
\raisebox{2mm}{\includegraphics[width=1.8cm]{zebragiraff_giraff_cropped}}&
\includegraphics[width=2.4cm]{2018-10-01_zebra_hm_giraff_3D_2125_paraview_cropped}\\
\hline
\raisebox{8mm}{\text{\ding{55}}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{log-wall}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{logs_256}}&
\raisebox{2mm}{\includegraphics[width=1.8cm]{logwall_logs_cropped}}&
\includegraphics[width=2.4cm]{2018-10-02_log-wall_3D_1446_paraview_cropped}\\
\raisebox{8mm}{\cmark}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{log-wall}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{logs_hm_logwall}}&
\raisebox{2mm}{\includegraphics[width=1.8cm]{logwall_logslowall_cropped}}&
\includegraphics[width=2.4cm]{2018-10-01_log-wall_3D_1730_paraview_cropped}\\
\hline
\raisebox{8mm}{\text{\ding{55}}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{tiger_cartoon}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{leopard_print_yellow}}&
\raisebox{2mm}{\includegraphics[width=1.8cm]{tiger_leopard_yellow_cropped}}&
\includegraphics[width=2.4cm]{2018-10-05_tiger_cartoon_3D_1705_paraview_cropped}\\
\raisebox{8mm}{\cmark}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{tiger_cartoon}}&
\raisebox{3mm}{\includegraphics[width=1.6cm]{leopard_hm_tiger}}&
\raisebox{2mm}{\includegraphics[width=1.8cm]{tiger_leopard_cropped}}&
\includegraphics[width=2.4cm]{2018-10-01_tiger_cartoon_3D_1602_paraview_cropped}\\
\end{tabular}
\caption{
{Anisotropic} texture synthesis using two examples.
The first columns show the two examples used and the training configuration, {i.e. } how images are oriented for each view.
%
Last column shows a sample synthesized using the trained generator.
For each example, we experiment with (\cmark) and without (\text{\ding{55}}) preprocessing the example images to match color statistics, by performing a histogram matching (HM) on each color channel independently.
%
We observe favorable results particularly when the colors of both examples are close.}
\label{Fig:Results_2differentinputs}
\end{figure}
\begin{figure*}
\centering
\begin{tabular}{ccccc}
\raisebox{10mm}{Example $u$} &
\raisebox{5mm}{\includegraphics[width=1.4cm]{jagnow_ex}} &
\raisebox{5mm}{\includegraphics[width=1.4cm]{jagnow2_exemplar}} &
\raisebox{5mm}{\includegraphics[width=1.4cm]{brown016_exemplar}} &
\raisebox{5mm}{\includegraphics[width=1.4cm]{RGran_Cool_1_192}}\\
\raisebox{15mm}{\cite{kopf2007solid}} &
\includegraphics[width=2.8cm]{jagnow1_Kopf} &
\includegraphics[width=2.8cm]{jagnow2_kopf} &
&
\\
\raisebox{15mm}{\cite{chen2010highquality}} &
\includegraphics[width=2.8cm]{jagnow1_chen} &
\includegraphics[width=2.8cm]{jagnow2_chen} &
\includegraphics[width=2.8cm]{brown_chen} &
\includegraphics[width=2.8cm]{Rgrancool_chen}
\\
\raisebox{15mm}{Ours} &
\includegraphics[width=3.0cm]{2018-06-19_jagnow_3D_0955-3D_paraview_crop}&
\includegraphics[width=3.0cm]{2018-06-15_jagnow2_exemplar_3D_1622-3D_paraview_128_crop}&
\includegraphics[width=3.0cm]{2018-06-15_brown016_exemplar_3D_1058-3D_paraview_128_crop}&
\includegraphics[width=3.0cm]{2018-06-15_RGran_Cool_1_192_3D_1237-3D_paraview_128_crop}
\end{tabular}
\caption{Comparison with the existing methods that produce the best visual quality.
The last row show the results using the proposed method. Our method is better at reproducing the statistics of the example compared to \cite{kopf2007solid} and is better at capturing high frequencies compared to both methods.
}
\label{Fig:Comparison}
\end{figure*}
\subsection{Comparison with state-of-the-art}
The results in Figures \ref{Fig:Results_isotropic1} and \ref{Fig:Results_isotropic2} prove the capability of the proposed model to synthesize photo-realistic textures. This is an important improvement with respect to classical on demand methods based on Gabor/LRP-noise.
High resolution examples are important in order to obtain more detailed textures. In previous high quality methods \cite{kopf2007solid,chen2010highquality} the computation times increase substantially with the size of the example, so examples were limited to $256^2$ pixels. Furthermore, the empirical histogram matching steps create a bottleneck for parallel computation.
Our method takes a step forward by allowing higher resolution example textures, depending only on the memory of the GPU used.
We compare the visual quality of our results with the two existing methods that seem to produce the best results: Kopf~{et al.}~\cite{kopf2007solid} and Chen~{et al.}~\cite{chen2010highquality}.
Figure~\ref{Fig:Comparison} shows some samples obtained from the respective articles or websites side by side with results using our method.
The most salient advantage of our method is the ability to better capture high frequency information, making the structures in the samples sharper and more photo-realistic. Considering voxels' statistics, {i.e. } capturing the richness of the example, both our method and that of Chen~{et al.}~\cite{chen2010highquality} seem to obtain a better result than the method of Kopf~{et al.}~\cite{kopf2007solid}.
The examples used in Figure~\ref{Fig:Comparison} have resolutions of either $128^2$ or $256^2$ pixels.
We observe that the visual quality of the textures generated with our method deteriorate when using small examples. This can be due to the descriptor network which is pre-trained using bigger images.
We do not consider the method of Dong~{et al.}~\cite{dong2008lazy} for a visual quality comparison as their pre-computation of candidates limits the richness of information, which yields lower quality results. However, thanks to the on-demand evaluation, this model greatly surpasses the computation speeds of the other methods.
Yet, as detailed before, our method is faster during synthesis while achieving better visual quality. Besides, our computation time does not depend on the resolution of the examples.
\begin{figure}
\centering
\begin{tabular}{ccc}
\raisebox{10mm}{\includegraphics[width=0.85cm]{Rgranite7_192}} &
\includegraphics[width=2.4cm]{Rgranite7_pose02_kopf} &
\includegraphics[width=2.4cm]{screen_2018-06-15_Rgranite7_192_3D_1240_offline_full_volume_640}\\
\raisebox{10mm}{\includegraphics[width=0.85cm]{jagnow_ex}} &
\raisebox{10mm}{\includegraphics[width=0.85cm]{jagnow1_features}}
\includegraphics[width=2.4cm]{jagnow1_pose02_kopf} &
\includegraphics[width=2.4cm]{screen_2018-06-15_jagnow1_exemplar_3D_1055_offline_ondemand_volume} \\
\raisebox{10mm}{\includegraphics[width=0.85cm]{brown016_exemplar}} &
\raisebox{10mm}{\includegraphics[width=0.85cm]{brown016_specularity_flip.png}}
\includegraphics[width=2.4cm]{brown016_pose02_kopf} &
\includegraphics[width=2.4cm]{screen_2018-09-18_brown016_exemplar_3D_1812_offline_ondemand_volume}\\
\raisebox{10mm}{\includegraphics[width=0.85cm]{jagnow2_exemplar}} &
\raisebox{10mm}{\includegraphics[width=0.85cm]{jagnow2_features}}
\includegraphics[width=2.4cm]{jagnow2_pose02_kopf} &
\includegraphics[width=2.4cm]{screen_2018-06-15_jagnow2_exemplar_3D_1622_offline_full_volume_512}\\
\raisebox{10mm}{\includegraphics[width=0.85cm]{RGran_Warm_5_192}} &
\includegraphics[width=2.4cm]{RGran_Warm_5_pose02_kopf} &
\includegraphics[width=2.4cm]{screen_2018-06-15_RGran_Warm_5_192_3D_1619-offline_full_volume_512}\\
\raisebox{10mm}{\includegraphics[width=0.85cm]{RGran_Cool_1_192}} &
\includegraphics[width=2.4cm]{RGran_Cool_1_pose02_kopf} &
\includegraphics[width=2.4cm]{screen_2018-06-15_RGran_Cool_1_192_3D_1237-offline_full_volume_512}\\
\raisebox{10mm}{\includegraphics[width=0.85cm]{caustic_exemplar}} &
\includegraphics[width=2.4cm]{caustic_pose02_kopf} &
\includegraphics[width=2.4cm]{screen_2018-06-15_caustic_exemplar_3D_1100-offline_full_volume_512}\\
\end{tabular}
\caption{Comparison of our approach with Kopf~{et al.}~\cite{kopf2007solid}. Both methods generate a solid texture that is then simply interpolated and intersected with a surface mesh (without parametrization as required for texture mapping). The first column shows the example texture. The second column shows results from~\cite{kopf2007solid}, some of which obtained with additional information (a feature map for the second and fourth rows, a specularity map for the third one). The last column illustrates that our approach, using only the example for training, is able to produce fine scale details. The resolution of the first three rows are $640^3$ and $512^3$ for the rest.
}
\label{Fig:Results_dragon}
\end{figure}
On Figure~\ref{Fig:Results_dragon} we show some of our results used for texturing a complex surface and we compare them to the results of Kopf~{et al.}~\cite{kopf2007solid}. Here the higher frequencies successfully reproduced with our method cause a more realistic impression.
Finally we would like to point out that although deep learning related models are often thought to produce good results only thanks to a colossal amount of parameters, our method (with $\sim8.5\times10^4$ parameters to store) stands close to the memory footprint of a patch-based approach working with a $170^2$ color pixels input ({i.e. } $8.67\times10^4$ parameters if all the patches are used).
\subsection{3D slice-based loss}\label{SubSec:3D_loss}
For a color solid $v\in\mathbb{R}^{N_1\times N_2\times N_3\times 3}$, we denote by $v_{d,n}$ the $n$\textsuperscript{th} 2D slice of the solid $v$ orthogonal to the $d$\textsuperscript{th} direction.
Given a number $D \le 3$ of slicing directions and the corresponding example images $\{u_1,\ldots,u_D\}$,
we propose the following slice-based loss
\begin{equation} \label{eq:total_slices_loss}
\mathcal{L}(v | \{u_1, \dots, u_D\}) =
\sum_{d = 1}^{D} \frac{1}{N_d}
\sum_{n = 0}^{N_d-1}
\mathcal{L}_{2} (v_{d,n},u_d),
\end{equation}
where $\mathcal{L}_{2}(\cdot,u)$ is a 2D loss that computes the similarity between an image and the example $u$.
We use the 2D perceptual loss $\mathcal{L}_2$ from \cite{gatys2015cnn}, which proved successful for training the CNNs~\cite{johnson2016Perceptual,ulyanov2016texturenets}. It compares the Gram matrices of the VGG-19 feature maps of the synthesized and example images.
The feature maps result from the evaluation of the descriptor network $\mathcal{D}$ on an image, {i.e. } $\mathcal{D}:x \in \mathbb{R}^{N_1 \times N_2 \times 3}\mapsto \{F^l(x) \in \mathbb{R}^{N^l\times M^l}\}_{l \in L}$, where $L$ is the set of considered VGG-19 layers, each layer $l$ having $N^l$ spatial values and $M^l$ channels.
For each layer $l$, the Gram matrix $G^l\in\mathbb{R}^{M^l\times M^l}$ is computed from the feature maps as
\begin{equation}
G^l(x) = \frac{1}{N^l} F^l(x)^T F^l(x),
\end{equation}
where $T$ indicates the transpose of a matrix.
The 2D loss between the input example $u_d$ and a slice $v_{d,n}$ is then defined as
\begin{equation}\label{eq:slice_loss}
\mathcal{L}_{2}(v_{d,n},u_d) =
\sum_{l\in L} \ \frac{1}{{(M^{l})}^2} \ {\left\lVert G^l(v_{d,n}) - G^l(u_d)\right\rVert}_F^2,
\end{equation}
where $\|\cdot\|_{F}$ is the Frobenius norm.
Observe that the Gram matrices are computed along spatial dimensions to take into account the stationarity of the texture.
Those Gram matrices encode both first and second order information of the feature distribution (covariance and mean).
\subsection{Single slice training scheme}\label{SubSec:Slice_training}
Formally, training the generator $\mathcal{G}(Z|\theta)$ with parameters $\theta$ corresponds to minimizing the expectation of the loss in Equation~\ref{eq:total_slices_loss}
given the set of examples $\{u_1,\ldots,u_D\}$,
\begin{equation}\label{eq:training}
\theta_{u} \in
\argmin_{\theta}
\quad
\mathbb{E}_{Z}
\left[\mathcal{L}(\mathcal{G}(Z|\theta), \,\{u_1,\ldots,u_D\})\right],
\end{equation}
where $Z$ is a multi-scale noise, independent and identically distributed from a uniform distribution in the interval $[0,1]$.
Note that each scale $z_0,\dots,z_K$ of $Z$ is stationary.
The generator on the other hand induces a non stationary behavior on the output due to upsampling operations.
When upsampling a stationary signal by a factor 2 with a nearest neighbor interpolation the resulting process is only invariant to translations of multiples of two.
Because our model contains $K$ volumetric upsampling operations, the process is translation invariant by multiples of $2^K$ values on each axis.
Considering the $d$-th axis, for any coordinate at the scale of the generated sample $n = 2^Kp+q$ with
$q\in\{0,\ldots,2^K-1\}$, the statistics of the slice $\mathcal{G}(Z|\theta)_{d,n}$ only depend on the value of $q$, therefore
\begin{equation}
\mathbb{E}_{Z} \left[\mathcal{L}_{2} (\mathcal{G}(Z|\theta)_{d,n},u_d)\right] =
\mathbb{E}_{Z} \left[\mathcal{L}_{2} (\mathcal{G}(Z|\theta)_{d,q},u_d)\right].
\end{equation}
Assuming $N_d$ is a multiple of $2^K$, we have
\begin{equation}\label{eq:expectation_slice_32}
\begin{split}
&\hspace*{-5mm} \mathbb{E}_{Z} \left[\mathcal{L}(\mathcal{G}(Z|\theta), \,\{u_1,\ldots,u_D\})\right]
\\
&\hspace*{5mm} =
\sum_{d=1}^{D} \frac{1}{N_d}
\sum_{n = 0}^{N_d-1}
\mathbb{E}_{Z} \left[\mathcal{L}_{2} (\mathcal{G}(Z|\theta)_{d,n},u_d)\right]\\
&\hspace*{5mm} =
\sum_{d=1}^{D} \frac{1}{2^K}
\sum_{q=0}^{2^K-1} \mathbb{E}_{Z} \left[\mathcal{L}_{2} (\mathcal{G}(Z|\theta)_{d,q},u_d)\right].
\end{split}
\end{equation}
As a consequence, instead of using $N_d$ slices per direction the generator network could be trained using only a set of $2^K$ contiguous slices on each constrained direction.
The GPU memory is a limiting factor during the training process, even cutting down the size of the samples to $2^K$ slices restricts the training slice resolution.
For example, training a network for a texture output of size $512\times512\times32$ with $K=5$ and VGG-$19$ would require more than 12GB of memory.
For that reason we propose to stochastically approximate the inner sum in Equation~\eqref{eq:expectation_slice_32}.
Considering the slice $n = 2^K p+Q_d$ in the $d$-th axis with a fixed $p \in \{0,\ldots,\frac{N_d}{2^K}-1\}$ and with
$Q_d\in\{0,\ldots,2^K-1\}$ randomly drawn from a discrete uniform distribution,
\begin{equation}\label{eq:expectation_single_slice2}
\mathbb{E}_{Z,Q_d} \left[\mathcal{L}_{2} (\mathcal{G}(Z|\theta)_{d,Q_d},u_d)\right]
= \frac{1}{2^K}\sum_{q=0}^{2^K-1} \mathbb{E}_{Z} \left[\mathcal{L}_{2} (\mathcal{G}(Z|\theta)_{d,q},u_d)\right].
\end{equation}
Then using doubly stochastic sampling (noise input values and output coordinates) we have
\begin{equation}\label{eq:double_expectation_slice}
\mathbb{E}_{Z} \left[\mathcal{L}(\mathcal{G}(Z|\theta), \,\{u_1,\ldots,u_D\})\right]
=
\sum_{d=1}^{D} \mathbb{E}_{Z,Q_d} \left[\mathcal{L}_{2} (\mathcal{G}(Z|\theta)_{d,Q_d},u_d)\right],
\end{equation}
which means that we can train the generator using only $D$ single-slice volumes oriented
according to the constrained directions.
Note that the whole volume model is impacted since the convolution weights are shared by all slices.
The proposed scheme saves computation time during the training, and more importantly, it also reduces the required amount of memory. In this setting we can use resolutions of up to $1024$ values per size during training (examples and solid samples), a resolution significantly larger than the ones reached in the literature regarding solid texture synthesis by example which are usually limited to $256\times256$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,402 |
\section*{Abstract}
Trusted Execution Environment, or enclave, promises to protect
data confidentiality and execution integrity of an outsourced computation on an untrusted host. Extending the protection to distributed applications that run on physically separated hosts, however, remains non-trivial. For instance, the current enclave provisioning model hinders elasticity of cloud applications.
Furthermore, it remains unclear how an enclave process could verify if there exists another
concurrently running enclave process instantiated using the same codebase, or count a number of such processes.
In this paper, we seek an autonomous membership service for enclave applications. The application owner only needs to partake in instantiating the very first process of the application, whereas all subsequent process commission and decommission will be administered by existing and active processes of that very application.
To achieve both safety and liveness, our protocol design admits unjust excommunication of a non-faulty process from the membership group.
We implement the proposed membership service in a system called \textproc{AMES}. Our experimental study shows that \textproc{AMES}\ incurs an overhead of $5\% - 16\%$ compared to vanilla enclave execution.
\section{Preliminaries}
\label{sec:background}
In this section, we first describe key characteristics of Intel SGX~\cite{sgx}. We then provide a brief overview of consensus protocols.
\subsection{Intel SGX}
\label{subsec:SGX}
\noindent\textbf{Enclave Execution.} Intel SGX~\cite{sgx} is a set of CPU extensions that are available on Intel CPUs starting from the Skylake microarchitecture~\cite{intel_skylake}, capable of providing hardware-protected TEE (or \textit{enclave}) for generic computations. It enables a host to instantiate one or multiple enclaves simultaneously. An enclave is associated with a CPU-guarded address space which is accessible only by the enclave code; the CPU blocks any non-enclave code's attempt to access the enclave memory. This effectively isolates the enclave from other enclaves concurrently running on the same host, from the OS, and from other user processes. Memory pages can be swapped out of the enclave memory, but they are encrypted using the processor's key prior to leaving the enclave.
Enclaves cannot directly execute OS-provided services such as I/O. In order to access those services, enclaves have to employ OCalls (calls executed by the enclave code to transfer the control to non-enclave code) and ECalls (API for untrusted applications to transfer control back to the enclave). These ECalls and OCalls constitute the enclave boundary interface, enabling a communication between the enclave code and the untrusted application to service OS-provided functions. Clearly, enclave developer should design such interface with caution, for ECalls exposed to the untrusted application may open up an attack surface to the protected execution environment.
\vspace{2mm}
\noindent\textbf{Attestation.}
Intel SGX allows a validator to verify if a specific enclave has been properly instantiated with the correct code~\cite{sgx_remote_attest} via its attestation mechanisms. These mechanisms also enable the validator and the attesting enclave to establish a secure, authenticated channel via which sensitive data can be communicated.
If the attestation is carried out between two enclaves instantiated on the same platform (or host), the mechanism in use is referred to as {\em local attestation}. Once the attesting enclave has been initiated, the CPU computes its \textit{measurement} (i.e., the hash of its initial state). Next, the CPU produces a message authentication code (MAC) of such measurement using a key that is accessible only by the validating enclave. The measurement and the MAC are then sent to the validating enclave for verification. Alternatively, if the validator is a remote party, the CPU issues a {\em remote attestation} by signing the measurement with its private key under the Enhance Privacy ID (EPID) scheme~\cite{epid}~\cite{sgx_remote_attest}. The remote party obtaining the attestation then relies on the Intel's Attestation Service (IAS) to verify the signature contained in the attestation~\cite{ias}, and then check the measurement value against a known value.
\vspace{2mm}
\noindent\textbf{Data sealing.} Enclaves can persist their private state to non-volatile memory via the data sealing mechanism. The enclave first requests the CPU for a enclave-specific key bound to its measurement. It then encrypts its private state using the requested key before storing it on persistent storage. The mechanism ensures that the sealed data can only be accessed by the enclave that sealed it. Nevertheless, enclave recovery using sealed data is susceptible to rollback attacks wherein an attacker (e.g., the malicious OS) provides the enclave with properly sealed but stale data~\cite{rote}.
\subsection{Consensus Protocols}
A rich literature of \textit{consensus protocols} has been devoted to address a variety of fault tolerance problems in distributed systems~\cite{raft, pbft}. Consensus algorithms can be broadly categorized by the failure model that they assume. On the one hand, the {crash failure} model assumes that a faulty process may stop processing any message, and it does not resume~\cite{raft}. On the other hand, the {Byzantine failure} model may allow for arbitrary faults. A process experiencing Byzantine failure may deviate from its expected behaviors in any manner, it may equivocate (i.e., sending contradictory messages to other processes), or it may intentionally delay its activity for any period of time~\cite{pbft}.
Consensus algorithms aim to achieve \textit{safety} and \textit{liveness} in the presence of failures. Safety requires that non-faulty processes reach an agreement and never return conflicting results for the same request, whereas liveness means that these processes eventually agree on a value. In the following, we give an overview of Raft~\cite{raft}, a consensus protocol that assumes crash failure model and ensures safety regardless of timing. Nonetheless, its liveness depends on timing (e.g., the communication channels between processes are partially synchronous, wherein messages are delivered within an unknown but finite bound).
\vspace{2mm}
\noindent{\bf Raft Consensus Protocol.}
Raft assumes a system of $n$ deterministic processes (or servers), among which at most $f = \frac{n-1}{2}$ could be faulty. A faulty process fails by crashing. Each process stores a log that contains a series of commands, or events of interest. Raft ensures that logs of non-faulty processes contain the same sequence of commands.
A process can be in one of the three roles, namely {\it leader, follower} and {\it candidate}.
Raft divides time into {\it terms} numbered with consecutive integers. In each term, one process is elected as the leader, and all other processes are followers. The leader exchanges periodic heartbeats with all followers to maintain its authority. If a follower does not receive any heartbeat after an \textit{election timeout} period, it assumes that the leader has crashed. The follower then increases its term, changes it role to candidate and begins an election. The candidate becomes the leader if it receives votes from a majority of processes.
We refer readers to \cite{raft} for further details on the leader election and its election criteria.
During normal operation, the followers are passive, only responding to requests from the leader and candidate. The leader receives all the commands (e.g., requests from the clients), and replicates the commands on the followers. The leader first appends a command as a new entry which is uniquely identified by the leader's \textit{term} and an index to its log. Next, it broadcasts the entry to all of the followers. The followers append the received entry to their logs, and acknowledge the receipt to the leader. Once the leader receives the acknowledgement from a majority of the processes (i.e., $f+1$ or more processes), it \textit{commits} the entry (i.e., execute the command contained in the entry), and all preceding entries in its log. The leader records the highest index it knows to be committed, and includes this index in subsequent messages to the followers, thereby informing them on the committed entries.
Although Raft is designed for crash failure model, there exists attempt that deploys Raft in a Byzantine setting~\cite{mscoco}. This is achieved through the use of TEEs. More specifically, by running the consensus protocol inside an enclave that offers attested execution, one can restrict adversarial behaviours of the faulty processes, thereby reducing the threat model from Byzantine fault tolerance to crash fault tolerance, to which Raft applies.
\section{Conclusion}
\label{sec:conclusion}
We have described an autonomous membership service for enclave-based applications.
While the application bootstrap necessitates the involvement of the application owner,
all subsequent process commission and decommission are administered the existing and active processes of the application. The proposed membership service necessarily admits unjust excommunication of a non-faulty process from the membership group, which is then fulfilled by the unjustly excommunicated process committing suicide. The experimental study shows that \textproc{AMES}\ incurs $5\% - 16\%$ overhead compared to vanilla enclave execution.
\section{Our Design}
\label{sec:design}
In this section, we present our system \textproc{AMES}~(\underline{A}utonomous \underline{ME}mbership \underline{S}ervice).
We first give an overview of \textproc{AMES}'s design. We later discuss in details the main protocols that comprise the autonomous membership service.
We remark that although our discussion assumes processes are provisioned as SGX enclaves, the technique underlying \textproc{AMES}\ is applicable to other Trusted Execution Environment platforms that support similar features to SGX enclaves.
\subsection{Overview}
\label{subsec:overview}
To enable the autonomous membership service for an application \texttt{Prog}, \textproc{AMES}\ augments \texttt{Prog}'s processes to equip them with the following parameters:
\vspace{1mm}
\begin{itemize}
\itemsep1mm
\item \texttt{peerList}: a list that records all active processes of \texttt{Prog}.
\item term number: a value that represents the current term (which we shall define in the following). The term number increases monotonically.
\item replicated log: a log that records messages the process exchange during the operation of the membership service. A log entry is indexed using the term number of the term at which it is added, together with sequence number that increases monotonically over time across terms.
\item commitIndex: the index of the latest entry in the replicated log that is known to be committed. This value increases monotonically over time.
\item heartbeat timeout \texttt{T}: a randomized timeout used in \textproc{AMES}'s \texttt{heartbeat} mechanism, unique for each process.
\item candidature timeout: the timeout used during the leader election. This timeout is the same for all processes.
\end{itemize}
While a process can always derive \texttt{peerList} from its replicated log, and one may argue that they can be merged into a single data structure, we find that treating them as two separate entities keep our exposition clearer.
\textproc{AMES}\ requires the involvement of the application owner during the application bootstrap so as to attest the correct instantiating of the first process, and to provide it with the application secret (Section~\ref{subsec:bootstrap}).
Once the first instance is fully operational, the application owner may go offline until the time when she wishes to shutdown the application.
All subsequent process instantiating (or dismantling) over the lifetime of the application will be policed by \textproc{AMES}. To this end, \texttt{Prog} needs to incorporate in its implementation \textproc{AMES}'s protocols for managing the processes' \texttt{peerList}s and ensuring that they converge. We describe these protocols in the remaining of this section, deferring discussion on the implementation details to Section~\ref{sec:implementation}.
\vspace{2mm}
\noindent\textbf{Workflow.}
Let $n$ denote the number of processes of \texttt{Prog} at a given time. \textproc{AMES}\ first elects one process to be the {\it leader}, while the other $[n-1]$ processes are {\it followers}.
A leader is associated with a \textit{term}, which is a time period of arbitrary length in which the leader retains its authority.
The leader relies on a \texttt{heartbeat} mechanism to monitor the status of the processes in the system, and to maintain its authority over them (Section~\ref{subsec:heartbeat}). In normal operation (i.e., the leader is active), the followers need not exchange \texttt{heartbeat} messages among themselves, but simply respond to messages from the leader.
The commission of a new process requires an ``approval'' from the membership service
via a \textproc{NewProcess}\ protocol orchestrated by the leader. More specifically, the leader attests the new process, and checks against the SLA if its commission is eligible (e.g., if the quota of concurrently running processes has not been consumed).
The leader then provisions the application secret to the new process, rendering it functional, and informs all followers to update their \texttt{peerList}s accordingly (Section~\ref{subsec:new_instance}). If the leader suspects a follower process $\texttt{p}_{\texttt{h}}$ of having halted (e.g., it did not receive a response from $\texttt{p}_{\texttt{h}}$ for the \texttt{heartbeat} message it has sent to $\texttt{p}_{\texttt{h}}$ after a configurable heartbeat timeout $\texttt{T}_\texttt{h}$ which is unique to $\texttt{p}_{\texttt{h}}$), it drives the other followers to expel $\texttt{p}_{\texttt{h}}$ from their \texttt{peerList} via the \textproc{ExpelProcess}\ protocol (Section~\ref{subsec:failed_instance}).
Finally, when the user decides to shutdown the application, she sends the shutdown request to the leader, who then coordinates the termination of all active processes via a \textproc{ShutDown}\ protocol, at the end of which they all cease to operate (Section~\ref{subsec:shutdown}).
It is important to note that \textproc{AMES}\ admits unjust excommunication of a active process whose connection with the leader is severed. To prevent the excommunicated process from disrupting the membership service, its future communication (i.e., after the point of excommunication) will be discarded, and the process will eventually halt.
The workflow described above has not accounted for leader failure. If the leader halts or becomes isolated, followers will eventually detect such failure, and engage in a leader election protocol to elect a new leader. In particular, if a follower does not receive a \texttt{heartbeat} message from the leader after its heartbeat timeout, it will assume that the current leader is no longer viable, and attempt to claim the leadership for itself (Section~\ref{subsec:heartbeat}). When a new leader is elected, the new term starts. During the period where a process attempting to claim the leadership, it is referred to as a \textit{candidate}. More than one candidates may compete for the leadership at the same time. The leader election guarantees that only one candidate succeeds, and all active processes in the system recognize the new leader as legitimate.
\subsection{Application Bootstrap}
\label{subsec:bootstrap}
In \textproc{AMES}, application bootstrap refers to the instantiating of the very first process of the application. This procedure is initiated by the application owner, who instructs the cloud service provider to instantiate an enclave process using the application code. Since the cloud's OS/hypervisor is not trusted, the instantiating has to be attested by the application owner, and application secret is only provisioned to the enclave after a successful attestation.
\textproc{AMES}\ benefits directly from Intel SGX's remote attestation protocol~\cite{sgx_remote_attest} in implementing the above mentioned attestation and secret provisioning. We highlight key characteristics of the protocol, and refer the readers to~\cite{sgx_remote_attest} for further details. The SGX architecture features a special enclave, called {\em Quoting Enclave}, to facilitate the remote attestation. After being instantiated, the attesting enclave invokes the SGX's \texttt{EREPORT} instruction and specifies the Quoting Enclave as its target. The \texttt{EREPORT} creates a structure, called \texttt{report}, that contains all necessary information to verify the correctness of the attesting enclave's instantiating, and produces a message authentication code (MAC) tag for the report. The MAC is computed using a ``report key'' that is associated only with the attesting enclave and its intended target (i.e., Quoting Enclave in this case). The Quoting Enclave verifies the \texttt{report}, and subsequently replaces its MAC with a signature signed using a CPU's private key under the EPID group signature scheme~\cite{epid}, producing a remote attestation \texttt{quote}\footnote{EPID allows a signer to sign objects with his private key without uniquely identifying the signer or linking his different signatures. This is achieved by assigning each signer to a ``group" such that the group's public key can be used to verify all signatures produced by the group member.}. As Intel does not publish the group public key associated with the CPUs' private key, the verifier has to rely on the Intel Attestation Service to validate the remote attestation \texttt{quote}.
It is worth noting that SGX remote attestation mechanism allows the verifier to include a challenge in her attestation requests, and the attesting enclave to include a manifest that contains response to the verifier's challenge and an ephemerally generated public key to the signed struture \texttt{report}, which is then carried over to the \texttt{quote}~\cite{sgx_remote_attest}. The challenge and response ensures freshness of the \texttt{quote}, avoiding replay attack by the malicious OS. The ephemeral public key allows the verifier to securely provision application secrets to the enclave. Consequently, upon successful application bootstrap, the first application process becomes fully functional, containing the secrets necessitated for the application's operation.
\subsection{\textproc{HeartbeatExchange}\ and Leader Election}
\label{subsec:heartbeat}
During normal operation, the leader uses \texttt{heartbeat} mechanism to maintain its leadership and to monitor status of follower processes. A \texttt{heartbeat} is essentially a message sent by the leader, and acknowledged with a corresponding \texttt{ack} message by a follower, via an authenticated channel. A
\texttt{heartbeat} always contains the term number and commitIndex of the leader (the latter is the index of the latest entry in the leader's replicated log that has been committed). The follower relies on the leader's commitIndex to detect if its replicated log and state are up-to-date with those of the leader. If a follower finds that its log misses some entries, or its state is out of date, it fetches the missing log entries and commit them, bringing its state up-to-date. The \texttt{heartbeat} can also contain information regarding the commission (excommunication) of a new (existing) follower process, which we shall elaborate in the following subsections.
When a new process becomes functional, it initially assumes the role of a follower. The follower process $\texttt{p}_{\texttt{i}}$ and the leader process $\texttt{p}_{\texttt{L}}$ establish a randomized timeout $\texttt{T}_{\texttt{i}}$, which is unique to each follower process, chosen from a fixed interval.
Via the \texttt{heartbeat} mechanism, $\texttt{p}_{\texttt{L}}$ can identify if it has become isolated, and halts if it is so. On the other hand, if $\texttt{p}_{\texttt{L}}$ is active but cannot receive an \texttt{ack} message from $\texttt{p}_{\texttt{i}}$ over a period of $\texttt{T}_{\texttt{i}}$, it suspects the latter of having halted, and triggers the \textproc{ExpelProcess}\ protocol to inform all other followers of $\texttt{p}_{\texttt{i}}$ failure. Alternatively, if the follower $\texttt{p}_{\texttt{i}}$ is active but cannot receive a \texttt{heartbeat} from the leader over a period of $\texttt{T}_{\texttt{i}}$, it changes its role to \textit{candidate}, increases its term number and broadcasts a $\texttt{RequestVote}$ message to all existing processes in its $\texttt{peerList}$ to claim the leadership.
The $\texttt{RequestVote}$ message from $\texttt{p}_{\texttt{i}}$ takes a form of $\langle \texttt{p}_{\texttt{i}}, \texttt{newTerm}, \texttt{lastLogIndex} \rangle$, where \texttt{newTerm} is its current term number (which has just been increased), and \texttt{lastLogIndex} is the index of the last entry in its replicated log. Upon receiving $\texttt{RequestVote}$ from $\texttt{p}_{\texttt{i}}$, a receiver process $\texttt{p}_{\texttt{r}}$ acts as follows:
\begin{itemize}
\item If $\texttt{p}_{\texttt{i}}$ is not in $\texttt{p}_{\texttt{r}}$'s $\texttt{peerList}$, it ignores $\texttt{p}_{\texttt{i}}$'s $\texttt{RequestVote}$.
\item If $\texttt{p}_{\texttt{i}}$ appears in its $\texttt{peerList}$, but $\texttt{p}_{\texttt{r}}$ has recently received a message from the leader $\texttt{p}_{\texttt{L}}$ (i.e., its timeout $\texttt{T}_{\texttt{r}}$ has not expired), $\texttt{p}_{\texttt{r}}$ queues $\texttt{p}_{\texttt{i}}$'s $\texttt{RequestVote}$ if it has not queued any other $\texttt{RequestVote}$.
If $\texttt{p}_{\texttt{r}}$ has queued a $\texttt{RequestVote}$ from another candidate $\texttt{p}_{\texttt{c'}}$, $\texttt{p}_{\texttt{r}}$ determines the dominating candidate among $\texttt{p}_{\texttt{i}}$ and $\texttt{p}_{\texttt{c'}}$\footnote{We say a candidate $\texttt{p}_{\texttt{c}}$ dominates (or outweights) another candidate $\texttt{p}_{\texttt{c'}}$ if it has a more up-to-date log, which is determined by the index of last entries in the logs~\cite{raft}.},
keeping only the $\texttt{RequestVote}$ of the dominating candidate, and informing the dominated candidate on the existence of the dominating one.
Once its \texttt{heartbeat} timeout $\texttt{T}_{\texttt{r}}$ expires, $\texttt{p}_{\texttt{r}}$ check if it is dominated by the candidate whose $\texttt{RequestVote}$ is currently kept in its queue. If so,
it grants its vote to the candidate; otherwise it discards such a $\texttt{RequestVote}$ and broadcasts its own.
On the other hand, if $\texttt{p}_{\texttt{r}}$ receives the next \texttt{hearbeat} from $\texttt{p}_{\texttt{L}}$ before $\texttt{T}_{\texttt{r}}$ expires, it resets $\texttt{T}_{\texttt{r}}$ and discards any $\texttt{RequestVote}$ in its queue.
\item If $\texttt{p}_{\texttt{i}}$ is in its $\texttt{peerList}$, but it has voted for another candidate (potentially itself) that is not dominated by $\texttt{p}_{\texttt{i}}$, $\texttt{p}_{\texttt{r}}$ informs $\texttt{p}_{\texttt{i}}$ on such candidate.
\end{itemize}
If $\texttt{p}_{\texttt{i}}$ collects a quorum of votes from a \textit{majority} of the processes in its \texttt{peerList} for its $\texttt{RequestVote}$ before its heartbeat timeout expires, it assumes the role of the leader. It then drives the processes to expel $\texttt{p}_{\texttt{L}}$ from their \texttt{peerList}.
While waiting for the votes, if $\texttt{p}_{\texttt{i}}$ is informed of another candidate that outweighs itself, $\texttt{p}_{\texttt{i}}$ resets its heartbeat timeout and waits for the other candidate to pronounce its leadership.
When $\texttt{p}_{\texttt{i}}$ is informed of a new leader whose term is as large as its current term via a \texttt{heartbeat} from the latter, it acknowledges the leader's \texttt{heartbeat}, and returns its role to follower.
If $\texttt{p}_{\texttt{i}}$ fails to collect a quorum of responses from a majority of the processes, or a \texttt{heartbeat} from the new leader within its heartbeat timeout, it increases its term and triggers a new election with a new $\texttt{RequestVote}$.
Finally, if $\texttt{p}_{\texttt{i}}$ is still assuming the candidate role (i.e., it can neither claim the leadership nor return to the follower role) when its candidature timeout expires, it commits suicide.
\vspace{2mm}
\noindent\textit{Security Arguments:} The leader election in \textproc{AMES}\ follows Raft's leader election~\cite{raft} to certain extent. The key difference here is that a candidate $\texttt{p}_{\texttt{i}}$ shall halt (by committing suicide) if it cannot collect a quorum of responses or a \texttt{heartbeat} from the new leader within the candidature timeout. Here, we reason about the correctness of this design when the system is not in anarchy.
The fact that $\texttt{p}_{\texttt{i}}$ issuing its $\texttt{RequestVote}$ implies either the leader $\texttt{p}_{\texttt{L}}$ has become faulty or isolated, or the connection between $\texttt{p}_{\texttt{i}}$ and the active $\texttt{p}_{\texttt{L}}$ has been severed.
If $\texttt{p}_{\texttt{L}}$ is isolated, it will detect its isolation by itself and halts, thereby becoming faulty. When $\texttt{p}_{\texttt{L}}$ is indeed faulty\footnote{In this case, no new process can be commissioned, and no process failures can be detected during the period when the system is in the leader election stage.}, other processes will either grant $\texttt{p}_{\texttt{i}}$ their vote or inform it on the dominating candidate. If $\texttt{p}_{\texttt{i}}$ is active (i.e., it is not isolated), it will have received enough responses to either claim the leadership, and thereby orchestrating the removal of $\texttt{p}_{\texttt{L}}$, or to wait to be contacted by the new leader.
On the other hand, if $\texttt{p}_{\texttt{L}}$ is active, it would have simultaneously driven other processes to expel $\texttt{p}_{\texttt{i}}$ from their \texttt{peerList}, preventing them from responding to $\texttt{p}_{\texttt{i}}$'s $\texttt{RequestVote}$. This, in turn, prevents $\texttt{p}_{\texttt{i}}$ from being able to collect a quorum of responses for its $\texttt{RequestVote}$, which subsequently leads to $\texttt{p}_{\texttt{i}}$'s committing suicide.
\vspace{2mm}
\noindent\textit{Remark.} While our design admits a potential unjust excommunication of an active process whose connection to the active leader is severed, this is necessary to sidestep the difficulty of solving membership problem in an asynchronous environment when faults may be present~\cite{flp, birman_membership}. By requiring the active yet excommunicated process to commit suicide, \textproc{AMES}\ preserves the consistency of the leader's expelling decision and safety of the membership service.
\subsection{\textproc{NewProcess}\ Protocol}
\label{subsec:new_instance}
After the application has been bootstrapped and as long as it is not in anarchy, \textproc{AMES}\ enables the application to commission new processes in an autonomous manner, without relying on manual intervention of the application owner, or a trusted third party. This section describes how \textproc{AMES}\ commissions a new process $\texttt{p}_{\texttt{j}}$ and ensures that the \texttt{peerList}s of all active processes are updated accordingly and that they converge.
Let $n$ be the number of existing processes at the beginning of $\texttt{p}_{\texttt{j}}$'s commission, and assume that the SLA allows $n>2$.
\vspace{2mm}
\noindent\textbf{Case} $n=1$. The only running and functional process at the beginning of $\texttt{p}_{\texttt{j}}$'s commission is the leader $\texttt{p}_{\texttt{L}}$. The interaction between the two processes happen in two phases. In the first phase, $\texttt{p}_{\texttt{j}}$ contacts $\texttt{p}_{\texttt{L}}$ for attestation. The results of this attestation are two-fold. First, $\texttt{p}_{\texttt{L}}$ is convinced that $\texttt{p}_{\texttt{j}}$ is correctly instantiated. Second, the two processes establish a secure and authenticated communication channel. In the second phase, $\texttt{p}_{\texttt{L}}$ updates its \texttt{peerList} to include $\texttt{p}_{\texttt{j}}$, provisions the application secret as well as its current \texttt{peerList} to $\texttt{p}_{\texttt{j}}$ via the secure channel and establishes a heartbeat timeout $\texttt{T}_\texttt{j}$. Likewise, $\texttt{p}_{\texttt{j}}$'s \texttt{peerList} now contains itself and $\texttt{p}_{\texttt{L}}$. At this point, $\texttt{p}_{\texttt{j}}$ becomes fully functional, and assumes a role of a follower.
\vspace{2mm}
\noindent\textit{Security Arguments:} The interaction between $\texttt{p}_{\texttt{L}}$ and $\texttt{p}_{\texttt{j}}$ described earlier follows the workflow of a 2-phase-commit (2PC) protocol~\cite{2PC}, with $\texttt{p}_{\texttt{L}}$ assuming the role of the coordinator.
The attestation in the first phase acts as a prepare phase in the 2PC protocol, whereas the provisioning of the application secret to $\texttt{p}_{\texttt{j}}$ and updating of \texttt{peerList}s mirror the commit phase of the 2PC protocol. Provided that the application is not in anarchy (i.e., $\texttt{p}_{\texttt{L}}$ remains active), our \textproc{NewProcess}\ protocol does not suffer from blocking, and the \texttt{peerList}s of $\texttt{p}_{\texttt{L}}$ and $\texttt{p}_{\texttt{j}}$ converge.
\vspace{2mm}
\noindent\textbf{Case} $n=2$. Let us call the two existing processes $\texttt{p}_{\texttt{L}}$ and $\texttt{p}_{\texttt{2}}$, with $\texttt{p}_{\texttt{L}}$ being the leader. The commissioning of $\texttt{p}_{\texttt{j}}$ happens in two phase. In the first phase, $\texttt{p}_{\texttt{L}}$ engages in a remote attestation with $\texttt{p}_{\texttt{j}}$, and at the same time queries $\texttt{p}_{\texttt{2}}$ (this query is piggybacked on a \texttt{heartbeat} sent to $\texttt{p}_{\texttt{2}}$) if it is ready to update its \texttt{peerList} to contain $\texttt{p}_{\texttt{j}}$.
In the second phase, if the attestation checks out, and $\texttt{p}_{\texttt{2}}$ responds with a ``yes'' vote (the vote is piggybacked on the \texttt{ack} message corresponding to the \texttt{heartbeat} carrying the query), $\texttt{p}_{\texttt{L}}$ updates its \texttt{peerList} to include $\texttt{p}_{\texttt{j}}$, transfers the application secret as well as the current \texttt{peerList} to $\texttt{p}_{\texttt{j}}$, and establishes a heartbeat timeout $\texttt{T}_\texttt{j}$ with $\texttt{p}_{\texttt{j}}$. $\texttt{p}_{\texttt{L}}$ also informs $\texttt{p}_{\texttt{2}}$ to update its \texttt{peerList} accordingly.
$\texttt{p}_{\texttt{j}}$ now becomes fully functional, and assume a role of a follower.
\vspace{2mm}
\noindent\textit{Security Arguments:} Similar to the case where $n=1$, the interaction between $\texttt{p}_{\texttt{L}}$, $\texttt{p}_{\texttt{2}}$, and $\texttt{p}_{\texttt{j}}$ described above follows the workflow of a 2PC protocol~\cite{2PC}, with $\texttt{p}_{\texttt{L}}$ assuming the role of the coordinator. Provided that the application is not in anarchy (i.e., both existing processes are active), our \textproc{NewProcess}\ protocol does not suffer from blocking, and the \texttt{peerList}s of all processes converge at the end of $\texttt{p}_{\texttt{j}}$'s commission.
\vspace{2mm}
\noindent\textbf{Case} $n>2$. The leader $\texttt{p}_{\texttt{L}}$ first checks if the commission of $\texttt{p}_{\texttt{j}}$ is eligible according to the SLA. If so, it engages in a remote attestation with $\texttt{p}_{\texttt{j}}$.
Next, $\texttt{p}_{\texttt{L}}$ follows Raft~\cite{raft} to informing all other followers about the commission of $\texttt{p}_{\texttt{j}}$:
\begin{enumerate}
\item $\texttt{p}_{\texttt{L}}$ puts an entry $\langle \texttt{add}, \texttt{p}_{\texttt{j}} \rangle$ to its log.
\item $\texttt{p}_{\texttt{L}}$ broadcasts $\langle \texttt{add}, \texttt{p}_{\texttt{j}} \rangle$ to all the followers.
\item Upon receiving $\langle \texttt{add}, \texttt{p}_{\texttt{j}} \rangle$, a follower $\texttt{p}_{\texttt{i}}$ acknowledges the receipt by sending $\langle \texttt{ack}_{\texttt{i}}, \texttt{add}, \texttt{p}_{\texttt{j}} \rangle$ to $\texttt{p}_{\texttt{L}}$, and puts $\langle \texttt{add}, \texttt{p}_{\texttt{j}} \rangle$ to its log.
\item Once $\texttt{p}_{\texttt{L}}$ confirms that $\langle \texttt{add}, \texttt{p}_{\texttt{j}} \rangle$ has been replicated on a majority of the processes, its adds $\texttt{p}_{\texttt{j}}$ to its \texttt{peerList}, establishes a heartbeat timeout $\texttt{T}_\texttt{j}$ with $\texttt{p}_{\texttt{j}}$, and transfers to $\texttt{p}_{\texttt{j}}$ the application secret and its \texttt{peerList}. In another word, $\texttt{p}_{\texttt{L}}$ ``commits'' the entry $\langle \texttt{add}, \texttt{p}_{\texttt{j}} \rangle$.
\item $\texttt{p}_{\texttt{L}}$ announces the commit of $\langle \texttt{add}, \texttt{p}_{\texttt{j}} \rangle$ in the next message it exchanges with the followers. Once the followers see the commit, they add $\texttt{p}_{\texttt{j}}$ to their \texttt{peerList}.
\end{enumerate}
\vspace{2mm}
\noindent\textit{Security Arguments:} Provided that the application is not in anarchy, the \texttt{peerList}s of all active processes converge. This follows directly from the security guarantees of Raft~\cite{raft}, which ensures protocol safety as long as any majority of the processes are active (i.e., they are operational and can communicate with each other without communication fault).
\subsection{\textproc{ExpelProcess}\ Protocol}
\label{subsec:failed_instance}
When the leader $\texttt{p}_{\texttt{L}}$ suspects that a process $\texttt{p}_{\texttt{h}}$ has become faulty (e.g., it does not receive a response for the \texttt{heartbeat} sent to $\texttt{p}_{\texttt{h}}$ after the timeout $\texttt{T}_{\texttt{h}}$ bound to $\texttt{p}_{\texttt{h}}$),
it drives all other followers to expel $\texttt{p}_{\texttt{h}}$ from their \texttt{peerList}.
\begin{enumerate}
\item $\texttt{p}_{\texttt{L}}$ puts an entry $\langle \texttt{Expel}, \texttt{p}_{\texttt{h}} \rangle$ to its log.
\item $\texttt{p}_{\texttt{L}}$ broadcasts $\langle \texttt{Expel}, \texttt{p}_{\texttt{h}} \rangle$ to all the followers.
\item A follower $\texttt{p}_{\texttt{i}}$ acknowledges the receipt by sending $\langle \texttt{ack}_{\texttt{i}}, \texttt{Expel}, \texttt{p}_{\texttt{h}} \rangle$ to $\texttt{p}_{\texttt{L}}$, and puts $\langle \texttt{Expel}, \texttt{p}_{\texttt{h}} \rangle$ to its log.
\item Once $\texttt{p}_{\texttt{L}}$ confirms that $\langle \texttt{Expel}, \texttt{p}_{\texttt{h}} \rangle$ has been replicated on a majority of the processes (\textit{excluding} $\texttt{p}_{\texttt{h}}$), its expels $\texttt{p}_{\texttt{h}}$ from its \texttt{peerList}. In another word, $\texttt{p}_{\texttt{L}}$ commits the entry $\langle \texttt{Expel}, \texttt{p}_{\texttt{h}} \rangle$.
\item $\texttt{p}_{\texttt{L}}$ announces the commit of $\langle \texttt{Expel}, \texttt{p}_{\texttt{h}} \rangle$ in the next message it exchanges with the followers. Once the followers see the commit, they expel $\texttt{p}_{\texttt{h}}$ from their \texttt{peerList}.
\end{enumerate}
\vspace{2mm}
\noindent\textit{Security Arguments:} Since the \textproc{ExpelProcess}\ protocol is essentially an instance of Raft~\cite{raft}, it follows from the security guarantees of Raft that the \texttt{peerList}s of all processes at the end of the \textproc{ExpelProcess}\ protocol converge, so long as the application is not in anarchy.
It is worth noting that \textproc{AMES}\ admits unjust excommunication of a correct but potentially isolated process or an active process whose connection to the leader $\texttt{p}_{\texttt{L}}$ is severed.
To prevent the excommunicated process from disrupting the membership service,
all its future communication will be discarded. This is enforced by having other processes not
responding to message from a process that is not in their \texttt{peerList}.
This would also lead the unjustly excommunicated process to halting. In particular, the unjustly excommunicated process will trigger a leader election, for it has not received any \texttt{heartbeat}
from the leader $\texttt{p}_{\texttt{L}}$. Nonetheless, it cannot obtain the required quorum of responses from other processes for its $\texttt{RequestVote}$, thereby committing suicide as mentioned in Section~\ref{subsec:heartbeat}.
\subsection{\textproc{ShutDown}\ Protocol}
\label{subsec:shutdown}
When the application owner wishes to shutdown all processes of the application, she sends the shutdown request to the leader $\texttt{p}_{\texttt{L}}$, who then coordinates the termination of all active processes as follows:
\begin{enumerate}
\item $\texttt{p}_{\texttt{L}}$ puts an entry $\langle \texttt{preShutdown} \rangle$ to its log.
It seals the application data, if any, to persistent storage using SGX's data sealing mechanism.
\item $\texttt{p}_{\texttt{L}}$ broadcasts $\langle \texttt{preShutdown} \rangle$ to all the followers.
\item A follower $\texttt{p}_{\texttt{i}}$ puts the received $\langle \texttt{preShutdown} \rangle$ to its log. Next, it seals the application data, if any, to persistent storage, and responds to the leader by sending $\langle \texttt{ack}_{\texttt{i}}, \texttt{preShutdown} \rangle$.
\item Once $\texttt{p}_{\texttt{L}}$ confirms that $\langle \texttt{preShutdown} \rangle$ has been replicated on a majority of the processes, it broadcasts $\langle \texttt{commitShutdown} \rangle$ to all the followers, thereby announcing the committing of the shutdown request. Finally, its informs the application owner that shutdown request has been served, and halts.
\item Once the followers see the $\langle \texttt{commitShutdown} \rangle$, they halt.
\end{enumerate}
\vspace{2mm}
\noindent\textit{Security Arguments:}
Similar to the \textproc{ExpelProcess}\ protocol, the \textproc{ShutDown}\ protocol is an instance of Raft~\cite{raft}. It follows from the security guarantees of Raft that so long as the application is not in anarchy, all active processes shall respond to the shutdown request, and halt.
Even if the application is in anarchy, as long as the leader commits suicide, all other processes will eventually halt (as we discuss in the next subsection), thereby also bringing the shutdown request into effect.
\subsection{Anarchy}
The application is in anarchy if the total number of faulty and isolated processes (denoted by $f$ and $i$ respectively) exceeds $ \lfloor \frac{n-1}{2} \rfloor$.
When the application falls into anarchy, the status of the current leader $\texttt{p}_{\texttt{L}}$ becomes critical. If $\texttt{p}_{\texttt{L}}$ remains active, the application may still be operational.
On the other hand, if it is either faulty or isolated, all existing processes will eventually halt, thereby shutdown the entire application. We remark that this only affects availability of the application, not the security of the membership service. More specifically, even if the application is in anarchy, the \texttt{peerList} of active processes do not diverge. In the following, we examine the situations where the application is in anarchy and how the status of the leader $\texttt{p}_{\texttt{L}}$ affects the operation of the application.
\vspace{2mm}
\noindent{\textbf{Case}} $( f > \lfloor \frac{n-1}{2} \rfloor )$:
If the leader $\texttt{p}_{\texttt{L}}$ is among the $f$ faulty processes, the remaining correct processes will trigger leader election. Nonetheless, since $f > \lfloor \frac{n-1}{2} \rfloor $, none of them will be able to obtain a sufficient quorum of responses for their $\texttt{RequestVote}$. Consequently, all correct process shall eventually halt by committing suicide, which effectively shuts the application down in its entirety.
If $\texttt{p}_{\texttt{L}}$ is active, it will attempt to trigger the \textproc{ExpelProcess}\ to excommunicate faulty processes. Nonetheless, it cannot collect sufficient acknowledgement after step (3) of the \textproc{ExpelProcess}\ protocol, and hence shall identify itself as being isolated, thereby haling by committing suicide. This further increases $f$, and $\texttt{p}_{\texttt{L}}$ now has become faulty. Subsequently, the remaining correct processes will trigger the leader election, and eventually halt for not being able to collect a sufficient quorum of responses to their $\texttt{RequestVote}$, shutting down the application.
\vspace{2mm}
\noindent{\textbf{Case}} $({i > \lfloor \frac{n-1}{2}} \rfloor )$:
If $\texttt{p}_{\texttt{L}}$ identifies itself as being isolated (i.e., $\texttt{p}_{\texttt{L}}$ is among the $i$ isolated processes), it will halt and thereby becomes faulty. Once $\texttt{p}_{\texttt{L}}$ is faulty, the remaining isolated processes will trigger the leader election, and will eventually halt for not being able to collect a sufficient quorum of responses to their $\texttt{RequestVote}$. In another words, the $i$ isolated processes will eventually become faulty, rendering $f > \lfloor \frac{n-1}{2} \rfloor $ and causing the application to shut down, as discussed earlier.
If $\texttt{p}_{\texttt{L}}$ is active, and can successfully drives the followers to complete all \textproc{NewProcess}\ and \textproc{ExpelProcess}\ protocols that it triggers, the application remains operational. We note that whenever there is a new process added to the system membership list, or an existing process is excluded, the value of $n$ changes, which may moves the application out of or into anarchy.
\vspace{2mm}
\noindent{\textbf{Case}} $({f < \lfloor \frac{n-1}{2} \rfloor \wedge i < \lfloor \frac{n-1}{2} \rfloor \wedge (f + i) > \lfloor \frac{n-1}{2}} \rfloor )$:
If $\texttt{p}_{\texttt{L}}$ is isolated, it will halt and thereby become faulty. Once $\texttt{p}_{\texttt{L}}$ is faulty, the remaining isolated processes will trigger the leader election.
Nonetheless, being isolated, they cannot collect a sufficient quorum of responses to their $\texttt{RequestVote}$. Consequently, they will halt, and thus become faulty.
At this point, the total number of faulty processes exceeds $\lfloor \frac{n-1}{2} \rfloor $, causing the application to shut down.
If $\texttt{p}_{\texttt{L}}$ is active, and can successfully drives the correct followers to exclude the faulty processes from their \texttt{peerLists} via the \textproc{ExpelProcess}\ protocol, the application continues to be operational. Similar to the previous case where $i > \lfloor \frac{n-1}{2} \rfloor$, each process commission or excommunication changes the value of $n$, which may move the application out of or into anarchy.
\subsection{Timeout Configurations}
\label{subsec:timeout_config}
It should be clear from our security arguments thus far that safety of \textproc{AMES}\ does not depend on timing. In another words, \texttt{peerLists} of all active process converge even if they do not share any global clock, and their execution speed may differ. Nonetheless, its liveness (i.e., the service being able to make progress) is inevitably dependent upon timing, especially \textproc{HeartbeatExchange}\ and Leader Election protocols. A necessary condition for liveness in \textproc{AMES}\ is:
\begin{center}
\small
\texttt{broadcastTime} $\ll$ heartbeat timeout $\leq$ candidature timeout $\ll$ \texttt{EIATime} \\
\end{center}
\noindent where \texttt{broadcastTime} is the the average time required for a process to send messages (in parallel) to other processes and receive their responses; heartbeat timeout and candidature timeout are the timeouts described in Section~\ref{subsec:overview}; and \texttt{EIATime} are events' inter-arrival time, with events being a new process instantiating, or failure/dismantling of existing processes in the membership cluster. Ideally, the timeouts should be an order of magnitude larger than the \texttt{broadcastTime}, and \texttt{EIATime} should be a few orders of magnitude larger than the timeouts. It is also reasonable to configure the heartbeat timeout to be less than or equal to the candidature timeout. This setting allows the candidate to retry a leader election with a new \texttt{RequestVote} at a higher term, in case there is a split vote and the current term ends with no leader being elected.
While \texttt{broadcastTime} and \texttt{EIATime} depends on the underlying communication system and dynamic workloads of the application, respectively, the timeouts can be tuned at our discretion. As we show in our empirical evaluation, \texttt{broadcastTime} ranges from $10ms$ to $60ms$ for processes located across different geographical regions. We expect typical \texttt{EIATime} in typical applications to be in the neighborhood of tens of seconds. Consequently, the timeouts could be set in the range of $250-500ms$. Section~\ref{sec:eval} elaborates on the effect of timeout configuration on the performance of \textproc{AMES}.
\section{Evaluation}
\label{sec:eval}
In this section, we report our experimental study of \textproc{AMES}. Our experiments are conducted in two different settings.
The first setting is based on an in-house (local) cluster consisting of 35 servers, each equipped with Intel Xeon E3-1240 2.1GHz CPUs, 32GB RAM and 2TB hard drive. The latency between any two servers are roughly $0.13ms$. In this setting, we run each process on a separate server. The second setting is based on Google Cloud Platform (GCP), in which we use a separate instance provisioned with 2 vCPUs and 8GB RAM for each process. The instances on GCP span across 4 regions, namely Oregon (us-west1), Los Angeles (us-west2), South Carolina (us-east1) and North Virginia (us-east4). The average latency between these regions observed in our experiment is detailed in Table~\ref{tab:GCP_latency}.
The trusted code base in our experiment is implemented using Intel SGX SDK~\cite{sgx_sdk}.
For the experiments that we run on GCP, we configured the SDK to run in simulation mode, as SGX is not available on GCP instances. We measured the latency of each SGX operation on our local cluster's CPU with SGX Enabled BIOS support, and injected it to the simulation. We observe that public key operations are expensive: signing and signature verification take roughly $450 \mu s$ and $844 \mu s$, respectively. Context switching and symmetric key operations take less than $5 \mu s$. Remote attestation, which involves access to the IAS, takes about $250 ms$ on average. Unless otherwise stated, the results reported in the following are averaged over $20$ independent runs.
\input{tables/latency_heatmap}
\subsection{\textproc{AMES}\ Overhead}
\input{figures/cpu_spec_overhead}
We evaluate overhead incurred by \textproc{AMES}\ over vanilla SGX execution using five benchmarks (i.e., \textit{mcf, deepsjeng, leela, exchang2}, and \textit{xz}) selected from SPEC CPU2017~\cite{spec_cpu} on our local cluster.
We leverage Intel SGX SDK~\cite{sgx_sdk} to port the benchmarks into enclave execution, enveloping their operational logic in a worker thread. We then inject a control thread that is responsible to implement the membership service to the enclave using our \textproc{AMES}\ library.
Figure~\ref{fig:cpu_spec_overhead} reports the running time of each \textproc{AMES}-enabled benchmarks under different range of heartbeat timeouts normalized against their own vanilla SGX execution's running time.
For a given range $[a,b]$, a heartbeat timeout is chosen uniformly at random between $a$ and $b$, inclusively. The overhead ranges from $5\%$ to $16\%$, with smaller timeout range leads to higher overhead. This is so because smaller heartbeat timeouts result in higher frequency of the leader contacting the follower processes to maintain its authority, which in turns leads to higher communication overhead and control switching. We also observe that the overhead is more evident when benchmark workload is computation intensive and requires few control switching for I/O (e.g., for the heartbeat timeout range of $[50-100](ms)$, {\it mcf} witnesses an overhead of $8\%$, whereas {\it exchange} sports a $15\%$ overhead.
This makes sense because our implementation piggybacks \textproc{AMES}\ control thread's messages on the worker thread's interface with the untrusted application (e.g., its ECall and OCall) whenever necessary.
Thus, the more interactions the worker thread has with the untrusted application over the course of the execution, the fewer control switching the process has to trigger exclusively for exchanging the control thread's messages, and hence lower the overhead. Comparing across figure~\ref{subfig:overhead_5} and figure~\ref{subfig:overhead_9}, we can see that while the overhead increases when there are more processes in the system, the difference is insignificant. In particular, when $n$ increases from $5$ to $9$, the difference in the normalized running time is less than $1\%$ for all benchmarks in our experiment.
\subsection{Leader Election}
In the second set of experiments, we study the effect of timeout configurations on \textproc{AMES}\ leader election. We measure the leader election latency as a period from when we crash the leader of a cluster of $n$ servers (with $n$ ranges from $5$ to $33$) to the moment when the new leader announces its authority. We experiment with different range of heartbeat timeouts and candidature timeout. For a given range $[a,b]$, a heartbeat timeout is chosen uniformly at random between $a$ and $b$, inclusively, while the candidature timeout is set to $5 \times b$.
Figure~\ref{fig:leader_election_latency} reports the latency under difference timeout settings on our local cluster and GCP. On our local cluster, the smaller the timeout range, the lower the leader election latency is. However, this phenomenon is not observed in our experiments on the GCP. More specifically, with the timeout range of $[50-100](ms)$, the leader election latency on GCP ranges from $405-552ms$, while the timeout range of $[200-300](ms)$ yields the lower leader election latency ranging from $325-458ms$.
A careful analysis of the difference between our local cluster and GCP confirms the necessity of proper timeout configurations (see Section~\ref{subsec:timeout_config}) with respect to communication latency (or \texttt{broadcastTime}). Recall that on our local cluster, the latency between any two servers is roughly $0.13ms$, whereas that of GCP ranges from $24ms$ to $66ms$, depends on the geolocation of the servers. When the heartbeat and candidature timeout is too small compared to the communication latency, the candidates face difficulty in collecting sufficient responses to its $\texttt{RequestVote}$ in time, entering the new terms and triggering new leader election prematurely. This leads to unnecessary leader elections, and exacerbates the system availability.
\input{figures/leader_election_latency}
\subsection{\textproc{NewProcess}\ Protocol}
\input{figures/eval_new_process}
We now examine the cost of the \textproc{NewProcess}\ protocol with and without \texttt{HostAttestationDelegate} (\texttt{HAD}) enclave. Recall that by incorporating \texttt{HAD} enclave, the leader and the new process can engage in a remote attestation without contacting the IAS.
Figure~\ref{fig:new_process_lt} depicts the latency of \textproc{NewProcess}\ protocol with respect to different number of existing processes in the system at the beginning of the new process's commission. The latency of \textproc{NewProcess}\ protocol is measured as the duration between the new process $\texttt{p}_{\texttt{j}}$ signaling its join request till $\texttt{p}_{\texttt{j}}$ receiving the application secret and becoming fully functional.
On our local cluster (Figure~\ref{subfig:new_process_LC}), \textproc{NewProcess}\ without \texttt{HAD} enclaves takes from $270-325ms$, whereas incorporating \texttt{HAD} enclaves reduces such a cost to $18-65$ms. This difference is primarily due to the cost of IAS access, which is approximately $250$ms in our experiments. A similar result is observed on GCP experiments (Figure~\ref{subfig:new_process_GCP}), wherein incorporating \texttt{HAD} enclaves significantly reduces the cost of the \textproc{NewProcess}\ protocol. Compared to the experiments on our local cluster, the \textproc{NewProcess}\ latency observed on GCP is higher (e.g., $145-
285ms$ compared to $18-65ms$ with \texttt{HAD} enclaves). This is so because the cost of the underlying consensus protocol itself, especially the broadcast time during the consensus run, differ significantly between the two settings.
\subsection{\textproc{ExpelProcess}\ Protocol}
\input{figures/eval_expel_process}
Finally, we examine the cost of the \textproc{ExpelProcess}\ protocol by crashing a process $\texttt{p}_{\texttt{h}}$, and measure the latency from such the crash till the time when $\texttt{p}_{\texttt{h}}$ is expelled from the leader's $\texttt{peerList}$.
On our local cluster, the heartbeat timeout is chosen randomly in the range of $[25-50]$ms, whereas the corresponding range on GCP is $[200-300]ms$. Figure~\ref{fig:remove_process_lt} plots the latency of the \textproc{ExpelProcess}\ protocol observed on our local cluster and GCP, with respect to different $n$.
Similar to \textproc{NewProcess}\ protocol, the latency observed on GCP is significantly higher than that observed in our local cluster. We attribute this to the difference in the timeout configurations, which in turn is influenced by the broadcast time of each setting.
\section{Implementation}
\label{sec:implementation}
To facilitate the incorporation of \textproc{AMES}'s logic in the implementation of an enclave application \texttt{Prog}, we provide application developers with a \textproc{AMES}\ library. The library envelops the membership service's operations in a {\it control thread}, separating them from the original operational logic of \texttt{Prog} that is to be implemented in a {\it worker thread} (or threads). As a result, each \texttt{Prog}'s enclave process comprises one control thread, and one or more worker threads (as depicted in Figure~\ref{fig:implementation}). Our \textproc{AMES}\ library implementation contains approximately $3,000$ lines of C++ code, and builds on the Raft implementation by RethinkDB~\cite{rethinkdb}.
\input{figures/control_worker_threads}
\vspace{2mm}
\noindent\textbf{Control thread vs worker thread.}
The control thread is responsible for exchanging \texttt{heartbeat} messages, conducting remote attestation and secure channel establishment in the \textproc{NewProcess}\ protocol, and handling messages that are exchanged in other protocols of \textproc{AMES}.
It is worth noting that the message exchange of the control thread entails context switching of the enclave. To reduce context switching overhead, our implementation piggybacks control thread's messages on the worker thread's interface with the untrusted application (e.g., its ECall and OCall) whenever necessary.
Recall that both the control thread and the worker thread belong to the same enclave, the control thread can seal the application data handled by the worker thread to persistent storage, if need be, in the \textproc{ShutDown}\ protocol. In addition, the control thread handles the process's suicide by destroying its own enclave.
\vspace{2mm}
\noindent\textbf{Reducing IAS access overhead.} In the \textproc{NewProcess}\ protocol, the newly instantiated process has to attest itself to the leader, so as to convince the latter of its correct instantiation. If the new process is not located in the same physical host with the leader, they have to engage in a remote attestation procedure. In such a situation, the leader (i.e., remote verifier) cannot verify the attestation of an attesting process locally, but has to rely on the Intel's Attestation Service (IAS) to verify the signature contained in the attestation~\cite{ias}. Contacting the IAS in every run of \textproc{NewProcess}\ protocol would incur a significant overhead that is unfavorable for the application performance.
We make an observation that the multitenancy nature of cloud computing allows for a remedy of the aforementioned IAS access overhead. In particular, it is often the case that the CSP provisions many processes (either of the same or different applications) on the same physical host. Therefore, we can reduce the IAS access overhead by introducing a \texttt{HostAttestationDelegate} (\texttt{HAD}) enclave at each physical host, and routing the remote attestation between two processes instantiated on two separate physical hosts via their prospective \texttt{HAD} enclaves.
Let us consider two physical hosts $\texttt{h}_1$ and $\texttt{h}_2$, and denote their \texttt{HAD} enclaves by $\texttt{HAD\_E}_1$ and $\texttt{HAD\_E}_2$, respectively, as depicted in Figure~\ref{fig:HAD}. These two \texttt{HAD} enclaves engage in a conventional remote attestation, and rely on IAS to verify the attestation. Once $\texttt{HAD\_E}_1$ and $\texttt{HAD\_E}_2$ have completed the attestation and established a secure channel, any pair of processes $\texttt{p}^{\texttt{A}}_1$ and $\texttt{p}^{\texttt{A}}_2$ of an application \texttt{A} provisioned on $\texttt{h}_1$ and $\texttt{h}_2$, respectively, can remotely attest one another without having to invoke the IAS. In particular, to attest its correct instantiating to $\texttt{p}^{\texttt{A}}_2$,
$\texttt{p}^{\texttt{A}}_1$ first leverages the local attestation procedure to prove its correctness to $\texttt{HAD\_E}_1$ which then conveys this attestation result to $\texttt{HAD\_E}_2$ via the secure channel. Finally, $\texttt{HAD\_E}_2$ locally attests to $\texttt{p}^{\texttt{A}}_2$ that it is correctly instantiated, gaining the latter's trust, and then conveys $\texttt{p}^{\texttt{A}}_1$'s attestation result to $\texttt{p}^{\texttt{A}}_2$. This approach essentially amortizes the IAS access overhead incurred by the attestation of \texttt{HAD} enclaves, and allows two processes $\texttt{p}^{\texttt{A}}_1$ and $\texttt{p}^{\texttt{A}}_2$ of an application \texttt{A} provisioned on two separate hosts $\texttt{h}_1$ and $\texttt{h}_2$ to conduct remote attestation without contacting the IAS.
\input{figures/delegate_attestation}
\section{Introduction}
\label{sec:intro}
Intel Software Guard Extensions (SGX)~\cite{sgx} protects data confidentiality and execution integrity of outsourced computations by offering a hardware-protected trusted execution environment (TEE), or \textit{enclave}, that guards the user's security-critical application code and its data against untrusted system software including the operating system (OS). The CPU prevents any non-enclave code from reading or modifying enclave memory at runtime. SGX architecture relies on the OS to instantiate enclave. A user can verify if a specific enclave is correctly instantiated and running at a remote host via a \textit{remote attestation} protocol~\cite{sgx_remote_attest}, which also establishes a secure channel via which the user can provision application secrets, if any, to the enclave.
While SGX allows users to harden security of software applications on an untrusted host (or server)~\cite{haven, pdedup}, extending its protection to distributed applications running on a number of hosts is non-trivial. More specifically, if the application involves a secret that must not be revealed to the untrusted host, the secret provisioning requires remote attestation by the application owner.
In the context of cloud service, this hinders the elasticity of cloud applications, for the cloud service providers cannot commission additional enclave processes dynamically at its sole discretion.
In another example, it is unclear how an enclave process could verify if there exists another concurrently running enclave process instantiated using the same codebase~\cite{sgx_singleton_matter}, or count the number of such processes, in an autonomic manner. This capability is desired in implementing singleton applications (i.e., only a single instance of the application can be created and run) and decentralized floating licensing (i.e., floating licensing without the central license server).
Let us use an example of a \textit{forking attack}~\cite{forking_attacks} on application that maintains persistent state to illustrate the challenges of extending TEE protections to a distributed environment. At a high level, the adversary (e.g., cloud service provider) instantiates multiple enclave processes of the same application and feeds them with different, potentially conflicting, inputs so as to violate the expected behaviours of the victim application.
More specifically, consider an application $\texttt{Prog}$ that serves read and write requests of a stateless client. The adversary runs two independent enclave processes (or processes for short) of $\texttt{Prog}$, denoted by $p_1$ and $p_2$, while keeping them oblivious to each other's
existence. Since the adversary handles all I/O operations to the processes, the client is only presented with a simple API for read and write requests, and is unaware of the underlying provisioning of the processes. At the beginning, both processes assume an initial state $s_0$. The client issues two sequential requests, the first writes data $\texttt{dat}$ to $\texttt{Prog}$'s memory, changing its state to $s_1$, and the second reads from $\texttt{Prog}$ its latest written data. From the application logic, the client should receive $\texttt{dat}$ as a response to the second request. Nevertheless, if the adversary routes the write request to $p_1$ and read request to $p_2$, the client will receive an obsolete (but authenticated) response drawn from the stale state $s_0$. This attack may wreak severe havoc in financial applications wherein an account balance should reflect at all time a correct transaction history.
The example described earlier motivates the need of a \textit{membership service} for enclave applications in a distributed environment. The membership service of an application $\texttt{Prog}$ not only facilitates seamless commissioning of $\texttt{Prog}$'s enclave processes, but also monitors and keeps track of the status of all $\texttt{Prog}$'s active processes, ensuring that their collective existence adheres to the Service Level Agreement (SLA). In a straight forward approach, one can implement the membership service by maintaining a special \textit{directory server} that keeps track of all active processes, and excommunicates a process that it suspects of being faulty from the membership group. The commissioning of a new process is augmented such that the newly instantiated process remains \textit{nonoperational} until its request to join the existing membership group is approved by the directory server. Nonetheless, this approach is not desirable, for the directory server becomes an obvious single-point-of-failure. A logical attempt to avoid the single-point-of-failure would be to replicate the directory server into a distributed fault-tolerant directory service comprising of a cluster of replicas coordinating their operations via a consensus protocol such as Paxos~\cite{paxos}. However, it is challenging to protect these replicas themselves against forking attack~\cite{forking_attacks}, wherein the adversary manage to splits the directory service replicas into two or more ``cliques'', each of which is oblivious to the others' existence. Such adversary can then selectively route communications of the existing processes, or join request of the newly instantiated processes to different clique of directory service replicas, thereby evading the control mechanism that enforces the SLA.
In this paper, we seek a membership service that is autonomous. In particular, once an application $\texttt{Prog}$ has been successfully bootstrapped, all subsequent process commission and decommission of $\texttt{Prog}$ are administered by its own active processes.
This self-administering is attained by designating one of $\texttt{Prog}$'s active processes at a {\it leader}, while the other assume the role of follower. The leader is tasked to monitor the status of all follower processes by exchanging heartbeat messages.
On the one hand, the process commission requires acknowledgement from the leader and a quorum of existing followers before the new process becomes fully functional. On the other hand, any process that is suspected of having halted is excommunicated from the membership group.
Our protocol design necessarily admits unjust suspicion and excommunication of a non-faulty process, in order to sidestep the difficulty of solving the membership problem in an asynchronous environment when faults may be present~\cite{transis}. To prevent the non-faulty yet excommunicated process from disrupting the membership service, its future communication to other processes of $\texttt{Prog}$ is discarded, rendering it isolated. The isolated process eventually halts by committing suicide, fulfilling the excommunication.
We consider a powerful adversary that controls the OS (or a hypervisor) of the hosts on which the enclave processes are instantiated. The adversary can schedule or corrupt all non-enclave processes, and restart the processes at will. Further, it can control all network communications between the processes regardless of whether they are instantiated on the same or across different hosts. The adversary can modify, reorder or delay the network messages arbitrarily. Nonetheless, we assume that the adversary cannot break enclave protections, and is limited by the constraints of the cryptographic methods employed.
We implement the proposed solution that enables autonomous membership service for SGX enclaves in a system called \textproc{AMES}. The application owner only needs to partake in the commissioning of the first enclave process (i.e., conducting remote attestation and provision application secret to the enclave). All subsequent process commissioning (or dismantling) during the lifetime of the application will be handled automatically and securely.
To facilitate the incorporation of \textproc{AMES}\ logic to an enclave application $\texttt{Prog}$,
we provide application developers with a \textproc{AMES}\
library that envelops the membership service's operations
in a control thread, separating them from the original operational logic of $\texttt{Prog}$
that is to be implemented in a worker thread (or threads).
In summary, we make the following contributions in this paper.
\begin{itemize}
\itemsep2mm
\item We propose a new approach to enable autonomous membership service for enclave-based applications, which allows seamless commissioning and dismantling of processes over the lifetime of the application.
\item We provide a \textproc{AMES}\ library that encapsulates the membership service's operations, which can be easily incorporated to any enclave-based application.
\item We conduct empirical evaluation of the overhead incurred by \textproc{AMES}\ on both our local cluster with 35 servers and the Google Cloud Platform (GPC) nodes spanning across 4 regions. The experimental results show that \textproc{AMES}\ incurs an overhead of $5\%-16\%$ over vanilla enclave execution.
\end{itemize}
The rest of this paper is organised as follows.
Section~\ref{sec:background} provides backgrounds on key properties of Intel SGX and distributed consensus protocols. Section~\ref{sec:problem_statement} presents our problem statement, covering system and adversarial models, design goals, and challenges. We propose our autonomous membership service and its rationale in Section~\ref{sec:design}, before discussing the implementation details in Section~\ref{sec:implementation}. Section~\ref{sec:eval} reports our empirical evaluation, while Section~\ref{sec:related_work} reviews the related work. Finally, Section~\ref{sec:conclusion} concludes our work.
\section{The Problem}
\label{sec:problem_statement}
In this section, we first define our system model in Section~\ref{subsec:system_model}. Next, we present the system goals that we seek to achieve in Section~\ref{subsec:system_goal}. We then describe in Section~\ref{subsec:adv_model} the adversary model against which our proposed autonomous membership service is designed. Finally, we detail the challenges of enabling autonomous membership service for enclave processes in Section~\ref{subsec:challenges}.
\subsection{System Model}
\label{subsec:system_model}
We consider a typical outsourced computation model consisting of a cloud service provider (CSP) and its users. The CSP operates a set of machines (or hosts) that are equipped with Intel SGX-enabled processors~\cite{sgx}. A user (i.e., application owner) uploads her code to the cloud, and executes her application on the cloud's infrastructure. The application involves secrets that the user wishes to keep private from the CSP. This can be achieved by running the application code (or a critical portion of its code) inside an SGX enclave. In particular, the code uploaded to the cloud does not contain any secret.
The SGX architecture then relies on the CSP's hypervisor or OS to instantiate an enclave process using the application code. Subsequently, the enclave process attests itself to the application owner so as to convince the latter that it is properly instantiated. Upon successful attestation, the secret materials are securely provisioned to the enclave.
The application code can be split into security-critical and non-critical components. The critical component is to be run in an enclave (or enclaves). The CSP may spawn multiple instances of the same application. Each application instance comprises both the critical and non-critical components, and they are bound together. If the critical component is divided into a set of enclaves, we assume that they are also uniquely bound together. In another word, the service provider cannot mix-and-match a component of one instance with that of another instance without breaking the function of the application. Consequently, it suffices to identify an application instance by its enclave (or one of its enclaves). Without loss of generality, we assume throughout the rest of the paper that the application runs entirely in a single enclave, and abuse the language to refer to each instance as a \textit{process}.
\vspace{2mm}
\noindent\textbf{Functional vs. Nonfunctional process.} We assume that the application involves secret materials that are concealed from the CSP or any untrustworthy party, and that such secret is necessary for the application's operation (e.g., encryption/decryption keys). While the CSP has a sole discretion in instantiating a new process using the application code, such a process is initially {\it nonoperational}. Only until it is provisioned with the aforementioned application secret does it become
{\it operational}. Put differently, although the CSP facilitates the commissioning of a new process, it cannot alone complete such a commission.
\vspace{2mm}
\noindent\textbf{Communication.} The processes communicate only by passing messages along a fixed set of channels facilitated by CSP. Each pair of processes is connected via a reliable, authenticated point-to-point communication channel that does not drop any message. These channels, however, are subject to a delivery scheduler controlled by the CSP. This allows the CSP to impose arbitrary delay on any message at any channel of its choice. When a channel is not intervened by a malicious delivery scheduler, it is \textit{synchronous}, in a sense that all messages sent via that channel are delivered within a finite delay $\Delta$ known to all processes. The malicious scheduler can postpone the delivery of any message for an arbitrary duration, causing a \textit{communication fault}, wherein messages communicated via that channel are not delivered within the delay $\Delta$. Finally, we do not assume any global clock, or bounds on relative local clock speed and execution speed of each process.
\input{figures/faulty_example}
\vspace{2mm}
\noindent\textbf{Faults.} We call a process \textit{faulty} if it has crashed or halted. A process is \textit{correct}, otherwise. Under the communication model described earlier, correct processes may be \textit{isolated} from other correct processes.
Let us consider an application $\texttt{Prog}$. We denote by $n_s(\texttt{Prog})$ the number of $\texttt{Prog}$'s existing processes at a given moment $s$. We deem a process $p$ isolated if $p$ cannot communicate with a quorum of $\lfloor \frac{n_s(\texttt{Prog})}{2} \rfloor$ processes in a synchronous manner.
We call a process \textit{active} if it is correct and not isolated.
Let $f_s(\texttt{Prog})$ and $i_s(\texttt{Prog})$ be the number of faulty and isolated processes among the
$n_s(\texttt{Prog})$ processes, respectively.
We deem the application $\texttt{Prog}$ to be in \textit{anarchy} at the given moment $s$ if:
\begin{equation}
\label{eq:anarchy}
f_s(\texttt{Prog}) + i_s(\texttt{Prog}) > \lfloor \frac{n_s(\texttt{Prog})-1}{2} \rfloor
\end{equation}
Hereafter, when it is clear from the context, we omit $s$ and $\texttt{Prog}$ from the notations. Figure~\ref{fig:fault_example} depicts different faulty scenarios $\texttt{Prog}$'s processes may experience, and state where $\texttt{Prog}$ is in anarchy.
\subsection{System Goals}
\label{subsec:system_goal}
To allow for elasticity, the CSP needs to commission or decommission processes dynamically overtime. Nonetheless, this may expose the system to forking attacks, as described earlier in Section~\ref{sec:intro}. To address this problem, we seek an autonomous membership service that protects the application against such forking attacks. The membership service polices the commissioning of new processes, and excommunication of faulty or isolated processes from the system, subject to the SLA concerning the upkeeping of the application. More specifically, while the CSP can instantiate a new process at will,
such a process remains nonoperational until its request to join the existing group of processes is approved by the membership service. In addition, the membership service actively monitors the status of each member process, and excommunicates a process that it suspects of being faulty or isolated from the group.
To render the membership service autonomous, our approach is to augment each process of an application \texttt{Prog} with a membership list that keeps track of \texttt{Prog}'s active processes, and ensure that the membership lists of all active processes converge without relying on manual
intervention of the application owner, or a trusted third party:
\begin{itemize}
\item If a process becomes faulty, it is required that the faulty process is eventually suspected of having halted, and excommunicated from the membership list of all active processes. Communication from the excommunicated process are discarded.
\item A newly instantiated process must join the system via the membership service before it is rendered functional. The membership service ensures that this new process is added to the membership list of all existing active processes, and the former is brought up-to-date with the current system membership list.
\item Except for the instantiation of the first process, the application owner does not involve in operation of the membership service. All subsequent process attestation and secret provisioning are handled by active processes in the system.
\end{itemize}
\subsection{Adversary Model}
\label{subsec:adv_model}
We consider a powerful adversary that possesses all capabilities of the CSP. In practice, such adversary can be an errant insider having complete access to the cloud infrastructure, or an attacker exploiting vulnerabilities in the cloud's software stack. The adversary is interested in conducting a forking attack on the victim application, attempting to launch an arbitrarily large number of enclave instances in violation of the SLA, or to isolate the concurrently running instances from each other. Nonetheless, we assume that the adversary does not seek to undermine the application's availability (e.g., by decommissioning all of its instances).
The adversary has complete control over the cloud's OS and/or hypervisor. It can schedule or corrupt all non-enclave processes, and restart enclave instances at will. Moreover, it can control all network communications between the enclave instances regardless of whether they are instantiated on the same or across different hosts. More specifically, the adversary can read, modify, reorder, delay or even drop the messages sent by the enclaves arbitrarily.
Nonetheless, it cannot compromise protection mechanisms of the SGX processors. That is, it can neither access the enclave runtime memory nor learn the application secrets that are protected by the enclave execution, and does not have any access to the processor's private keys that are used for attestation and data sealing functions. Finally, we assume that the adversary is computationally bounded, and cannot break standard cryptographic assumptions.
We assume the application code running inside the enclave is secure. We do not consider side-channel attacks against the enclave execution~\cite{controlled_channel} and DoS attacks against the system.
\subsection{Challenges}
\label{subsec:challenges}
The first challenge in enabling autonomous membership service stems from the need of injecting application secret to the newly instantiated process.
Since the untrusted OS is in charge of enclave instantiating, any secret material necessitated for the operations of the enclave process cannot be contained in the enclave code, but should rather be provisioned to the enclave only after it has been properly instantiated.
In a typical pipeline of enclave instantiating, this secret provisioning is realised via the remote attestation procedure carried out by the application owner.
Nonetheless, it is unreasonable to assume the involvement of the application owner in the instantiating
of every enclave process, especially when processes may be commissioned dynamically to meet the elasticity requirement of the application.
Ideally, the application owner only needs to involve in the instantiating of the first enclave instance during the application provisioning process, and does not have to take part in the upkeeping of her application.
The second challenge lies in ensuring the security of the membership service itself, especially against forking attack~\cite{forking_attacks} or logical partition. To further elaborate on this challenge, we describe in the following two straw-man solutions and point out their weaknesses. The first solution assumes a trusted \textit{directory server} keeping track of all running processes of the application,
thereby offering the membership service. The second solution attempts to implement the \textit{directory server} as a replicated state machine so as to avoid single-point-of-failure.
\vspace{2mm}
\noindent\textbf{Directory server.} A trusted directory server is in charge of managing and attesting the correct instantiating of application processes, and keeping track of all such running processes. More specifically, the user uploads the expected measurement of her application enclave process (i.e., the hash of its initial state) along with its secret to the directory server via a secure channel. The directory server then performs a remote attestation with the newly instantiated enclave process on behalf of the user, and provides it with the application secret upon successful attestation. It maintains a list of concurrently running processes, and refuses the instantiating of the new process if it violates the SLA.
The directory server determines the aliveness of a process in its process list by listening to a \texttt{heartbeat} message sent by that process, to which it responds with an acknowledgement \texttt{ack}. If it does not receive the \texttt{heartbeat} from the process after a certain time-out period, it would assume that such process has crashed, removing such process from its process list.
It is important to note that the excommunication of a process may be due to communication fault, rather than the said process having halted (i.e., unjust excommunication).
In order to keep the logic of the directory server simple, we can enforce the unjust excommunication by forcing the excommunicated process to halt. This can be achieved by programming a process to \textit{commit suicide} if it does not receive an \texttt{ack} from the directory server after a configurable \texttt{time-to-live} period, and to restart its \texttt{time-to-live} timer every time it receives an \texttt{ack}. Put differently, the \texttt{ack} from the directory server acts as a \textit{lease} that allows the receiving process to continue operating for a fixed period of time until it expires~\cite{leases}.
This approach, nevertheless, is not desired, for the directory server becomes a single-point-of-failure, or target for attacks. For example, if the directory server is implemented using TEE, it is subject to rollback attack~\cite{rote} in which the adversary may deceive it to resume operations using stale process list, violating the integrity and consistency of the directory server's operations.
\vspace{2mm}
\noindent\textbf{Replicated directory service.}
One approach to sidestep the aforementioned single-point-of-failure issue is by replicating the directory server. In particular, the directory service now comprises of a set of replicas that leverage fault-tolerant protocol to implement a replicated state machine, ensuring consistency of the service's operation despite individual replica failures~\cite{replicatee}. Nonetheless, it is challenging to protect these replicas against forking attack~\cite{forking_attacks}, wherein the adversary manage to splits the directory service replicas into two ``cliques'', each of which is unaware of the other. Such attack allows the adversary to run twice as many number of processes as what would otherwise be allowed by the SLA. Addressing this issue, then, requires a membership service offered to the directory service replicas themselves.
\section{Protocol Highlights}
{\color{blue} \bf
\paragraph{Membership Services}.
We follow Raft's strategy in maintaining the membership information among the enclave instances. Let $n$ denotes the current number of registered enclaves in the system. One enclave is designated as the ``Leader'', and the remaining [n-1] are ``Follower''. The Leader needs to exchange heartbeats with the Followers, to maintain its leadership and to detect if a Follower has crashed. In normal case (i.e., when the Leader is active and operates correctly), Followers need not exchange heartbeats with one another.
\section{Related work}
\label{sec:related_work}
\noindent\textbf{Membership service.}
Birman et al.~\cite{birman_membership} formulated the problem of group membership in asynchronous environments. The authors acknowledge that the system's liveness entails the possibility of erroneous failure detection, and the need of suppressing communication from a process that is unjustly excommunicated. Our autonomous membership service follows this observation, fulfilling the excommunication by having an unjustly excommunicated process committing suicide. Amir et al.~\cite{transis} proposed a membership protocol that relies on atomic broadcast, whereas Mishra et al. ~\cite{membership_partial_order} leveraged a multicast facility that preserves the partial order of messages exchanged among the communicating processes to construct a membership protocol. Our approach, on the other hand, does not make any assumption on the underlying broadcast/multicast primitives. Instead, it builds on a well-known consensus protocol, namely Raft~\cite{raft}, to implement the membership service.
\vspace{2mm}
\noindent\textbf{Live migration and replications of SGX enclaves.}
Gu et al.~\cite{sgx_migration} present technique for live enclave migration on the cloud. In contrary to our membership service, their system model allows only one instance of the application to be functional at any time. ReplicaTEE~\cite{replicatee} attempts to enable seamless replication of SGX enclaves by relying on a distributed directory service layer that handles provisioning of enclave processes, and keeps track of the number of existing processes. Nevertheless, it is unclear how the directory service's replicas are protected against forking attack, wherein the adversary splits the directory service replicas into two cliques, each of which is fully operational but unaware of the other. This attack effectively allows the adversary to replicate more enclave processes than what would otherwise be allowed by the SLA.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,798 |
We're planning on giving two essential oils classes per month! Give us a call at the office to see when the next one is scheduled. They usually happen on the first and second Mondays of the month, unless there's a holiday.
Learn what essential oils are, how to use them in everyday life, how to get rid of toxic chemicals in your home, how essential oils can support your health, and more!
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On Yer Bike! Horse charity calls for bike donations
Published: 6:30 PM May 2, 2018 Updated: 9:39 AM October 11, 2020
World Horse Welfare's Hall Farm staff making use of existing bikes. The charity is looking for donations of bikes to help its staff get around the 200 acre rehoming centre in Snetterton. Picture: World Horse Welfare - Credit: Archant
A horse rescue and rehoming centre is asking for donations of bicycles to help the team caring for the animals.
Horses at the World Horse Welfare's Hall Farm Rescue and Rehoming Centre in Snetterton. Picture: World Horse Welfare - Credit: Archant
With 200 acres of land and more than 100 horses, ponies and donkeys, staff at the World Horse Welfare's Hall Farm Rescue and Rehoming Centre, in Snetterton, clock up the miles through walking.
To help speed up the walks to and from the many fields and areas of the farm, the charity has launched an initiative called On Yer Bike.
The idea is to have a number of bikes, donated by the public, strategically placed around the farm.
The charity's centre manager Sue Hodgkins said: "We are confident this will help to speed up the process of checking and treating our horses – many of whom live out 24/7 in our paddocks.
"It will also help the team when they are leading horses to and from the paddocks, rather than having to walk to and from each field, which could be anything up to two miles away, they would be able to pick up a bike and cycle one leg of the journey."
If you can donate adult bikes, preferably mountain of off-road, in good working condition, puncture repair kits or combination padlocks and chains, contact Maxine Langley on 01953 499100 or email maxinelangley@worldhorsewelfare.org
10 Hundreds of homes across Norfolk hit by power cut | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,191 |
\section{Introduction}
Many manufacturing processes rely on image processing to enable industrial robots to manipulate objects. Whereas many sophisticated camera systems meet the need for localizing and tracking user-defined \acp{poi} on rigid objects, there is still no sufficiently accurate solution for coping with this problem for deformable objects yet. Moreover, existing approaches reconstruct the deformable object's model online, requiring \acp{poi} to be defined at runtime and thus being unsuitable for fully automated processes. This paper proposes a solution for defining POIs on an offline model and then localizing and tracking these points on a deformable object in an online detection pipeline.
\begin{figure}
\includegraphics[width=\linewidth]{images/icra_2021_deformable/pipeline.pdf}
\caption{Overview of the localization and tracking pipeline}
\end{figure}
\section{Related Work}
Existing approaches for localizing and tracking \acp{poi} are only applicable to specific object categories (e.g. linear \cite{Tang.2018} or planar \cite{Tang.2013,Schulman.2013}), assume speficic deformation models (e.g. articulated models \cite{Schmidt.2015} or skeletons \cite{Gall.2009}) or particular materials (e.g. textiles \cite{Li.2014,Li.2018}) or are restricted to detecting specific features (e.g. points on corners or edges \cite{Yamazaki.2014,Ramisa.2012}). To reach the precision required for sophisticated manipulation tasks, prior work requires external markers \cite{Finnegan.2006,Trumble.2017} or elaborate physics models \cite{Bay.2006,Tian.2010,Lang.2011,Schulman.2013,Leizea.2014}. We combine several SotA-solutions for rigid object state estimation (such as SHOT descriptors), physical modelling (such as deformation fields) and computer graphics (such as projection) into a processing pipeline which permits the tracking and localization of user-defined points on arbitrary deformable objects using only a single depth camera without markers or prior physics modeling.
\section{Method}
Our algorithm consists of three distinct phases: (1) \textbf{Demonstration} of the reference model and the \acp{poi}; (2) an \textbf{iterative localization and tracking process} consisting of \textbf{observing} a new point cloud, identification of \textbf{correspondences} between the observation and the deformed reference model of the previous timestep and \textbf{estimation of the deformation}; and (3) a \textbf{coordinate transformation} of the localized \acp{poi} into the robot end-effector coordinate system for subsequent manipulation.
\subsection{Surface and deformation model}
\label{sec:surface_and_deformation_model}
To efficiently perform computations, we model object surfaces as triangle meshes, while deformations are modelled via a \textit{deformation grid} \cite{Innmann.2016}. Unlike \cite{Innmann.2016}, we use a highly detailed mesh as a surface representation which is independent of the deformation model's resolution. This allows to increase computation performance while maintaining a highly detailed surface.
Our deformation model consists of two data structures, both containing $\lvert G \rvert$ grid points. Whereas the equally spaced static grid $G$ describes the undeformed reference model, the deformation field $\mathcal{V}$ represents the object's deformed state at the current timestep $t$. Each gridpoint $i$ is defined in $G$ by a position vector $\ensuremath{\hat{\boldsymbol{t}}_{i}}$, allowing to express the position $\hat{\boldsymbol{p}}$ of an undeformed vertex within $G$ as $\hat{\boldsymbol{p}} = \sum^{\lvert G \rvert}_{i=1}\alpha_i\hat{\boldsymbol{t}}_i$ with trilinear weights $\alpha_i \in [0,1]$ (cf. fig. \ref{fig:deformation_model}). The position $\boldsymbol{p}$ of the same vertex in the deformation field $\mathcal{V}$ can be described analogously by a weighted sum of deformed gridpoint positions $\ensuremath{\boldsymbol{t}_{i}}$ as $\boldsymbol{p} = \sum^{\lvert G \rvert}_{i=1} \alpha_i \boldsymbol{t}_i$. We define an \textit{observation} $\mathcal{D}$ as an organized point cloud of the deformed object.
\begin{figure}
\centering
\includegraphics[height=5.5em]{images/icra_2021_deformable/4.png}
\caption{Trilinear weights (l.); static and deformed grid cells (r.)}
\label{fig:deformation_model}
\end{figure}
\subsection{Demonstration}
Most SotA approaches use either a low-resolution reference model of the deformable object \cite{Zollhofer.2014} or none at all \cite{Newcombe.2015, Innmann.2016}. To allow for user demonstrations of \acp{poi}, our algorithm requires a high-detail reference model to be created offline. To improve the stability of our solver, we limit the object's initial deformation with respect to its reference model by demonstrating a library $\mathcal{L}$ of reference models in an offline step. Each model in $\mathcal{L}$ is a triangle mesh of the object in a distinct deformation state. The models are generated by fusing several depth images into a \ac{tsdf} and then extracting the triangles via the MarchingCubes algorithm \cite{Lorensen.1987}. After the demonstration of the reference models, the user can select relevant \acp{poi} on the meshed surface via a graphical user interface. At runtime, after the first observation $\mathcal{D}$, the model in $\mathcal{L}$ most similar to $\mathcal{D}$ is used to initialize $G$.
\subsection{Correspondence identification}
The definition and identification of \textit{correspondences} links the current observation and the deformed reference model of the previous timestep $(t-1)$ and forms the basis of the deformation estimation: The estimation of a deformation is equivalent to the minimization of the distances between all correspondences. We define three correspondence types:
\paragraph{\Ac{p2p}}
Result from projecting each surface point $\ensuremath{\boldsymbol{p}_{c}}$ of the deformed reference model into the image plane and comparing it to the corresponding point $\ensuremath{\boldsymbol{p}_{c}^{a}}$ that has been measured. The quality of a \ac{p2p}-correspondence can be described by a weight $w_c = (\frac{w_d + w_n + w_v}{3})^2$ where $w_d$ denotes the distance between projected point $\boldsymbol{p}_c$ and its correspondent $\boldsymbol{p}^a_c$, $w_n$ the distance between $\boldsymbol{p}_c$'s normal $\boldsymbol{n}_c$ and its correspondent $\boldsymbol{n}^a_c$, and $\boldsymbol{w}_v$ the angle between the camera view direction $\boldsymbol{v}$ and $\boldsymbol{n}_c$.
\paragraph{\Ac{p2s}}
Projective correspondences such as \ac{p2p} typically only yield approximate, not exact, correspondences. \Ac{p2s}-correspondences add another degree of freedom by associating a point in the model to a \textit{plane} in the observation, defined by the \ac{p2p}-correspondence $\boldsymbol{p}^a_c$ and its normal $\boldsymbol{n}^a_c$. During deformation estimation, this allows to only minimize the distance $d_c$ along the normal.
\paragraph{Feature correspondences}
Unlike projective correspondences, correspondences based on feature matching can detect large deformations, tangential movements and rotations of the object out of the image plane. Prior work \cite{Guo.2016,Hansch.2014} and our own experiments have found the PFH and FPFH descriptors to be highly sensitive and specific but to scale poorly with the size of the point cloud, while SHOT descriptors scale linearly and are robust against outliers. We implement feature correspondences using SHOT, as its sensitivity suffices for most real-world applications.
\subsection{Deformation estimation}
The deformation of the reference object can be estimated by formulating an optimization problem to estimate their degrees of freedom and thus the deformation of the reference object. Using the notation introduced in \ref{sec:surface_and_deformation_model}, the deformation of a single grid point $i$ can be expressed as $\mathcal{V}_i = \hat{\boldsymbol{t}}_i - \boldsymbol{t}_i$. For estimating the deformation field, we split up all unknows into a single global rigid transformation $(\boldsymbol{t}, R)$ and many local transformations $(\ensuremath{\boldsymbol{t}_{i}}, R_i)$ and combine them in a vector $\mathcal{X}$:
\begin{equation}
\mathcal{X} = \left(
\boldsymbol{t},~R,
\underbrace{\quad \ldots , \ensuremath{\boldsymbol{t}_{i}}^{T} , \cdots \quad}_{3~\vert G \vert~translations}
\vert
\underbrace{\quad \cdots , R_{i} , \cdots \quad}_{3~\vert G \vert~rotations}
\right)^{T}
\end{equation}
The interpretation of correspondences as error terms $E$ allows to formulate the deformation estimation of $\mathcal{X}$ as an energy minimization problem, which is also suggested by \cite{Innmann.2016,Zollhofer.2014,Newcombe.2015}. This optimization can be regarded as a model regression problem and solved by existing solvers:
\begin{equation}
\begin{aligned}
E(\mathcal{X}) =~ &\omega_{p} E_{P\mathit{2}P}(\mathcal{X}) + \omega_{s} E_{P\mathit{2}S}(\mathcal{X})\\
+~ &\omega_{f} E_{F}(\mathcal{X}) + \omega_{r} E_{Reg}(\mathcal{X})
\end{aligned}
\label{eqn:E_gesamt}
\end{equation}
\cite{Zollhofer.2014} and \cite{Innmann.2016} solve a similar high-dimensional nonlinear optimization problem by linearizing the model and using the Gauss-Newton method, incurring a significant overhead for the computation of the Jacobian $J$. \cite{Innmann.2016} splits the optimization into a two-stage process composed of a \textit{fixed registration} followed by a \textit{deformation estimation}. We leverage the fact observed in \cite{Sorkine.2007} that the deformation estimation can again be split into two independent sub-problems, which allows to solve for nonlinear rotations and linear translations using iterative Gauss-Newton on each subproblem in turn (``flip-flop'' strategy). We perform fixed registration, the estimation of a global transformation $(\boldsymbol{t}, R)$, via Prerejective RANSAC (PSC). For the deformation estimation, setting up the Jacobian for the error terms of the three correspondence types is straightforward:
\begin{equation}
E_{P\mathit{2}P} = \sum^{\vert C \vert}_{c=1} \omega_{c}
~ \Biggl| \Biggl|
\underbrace{
\underbrace{
R \left[ \sum^{\vert G \vert}_{i=1} \alpha_{i}(\ensuremath{\hat{\boldsymbol{p}}_{c}})~\ensuremath{\boldsymbol{t}_{i}} \right] + \boldsymbol{t}}_{\mathcal{V}(\ensuremath{\hat{\boldsymbol{p}}_{c}})} -\ensuremath{\boldsymbol{p}_{c}^{a}}
}_{\ensuremath{\boldsymbol{r}}_{P\mathit{2}P,C}(\mathcal{X})}
\Biggl| \Biggl|^{2}_{2}
\label{eqn:P2P_Error}
\end{equation}
\begin{figure}[H]
\centering
\begin{minipage}{.69\linewidth}
\includegraphics[width=\linewidth]{images/icra_2021_deformable/performance.png}
\end{minipage}%
\hfill%
\begin{minipage}{.31\linewidth}
\begin{minipage}{\linewidth}
\includegraphics[width=\linewidth]{images/icra_2021_deformable/deformation_1.png}
\end{minipage}\\
\begin{minipage}{\linewidth}
\includegraphics[width=\linewidth]{images/icra_2021_deformable/deformation_2.png}
\end{minipage}
\end{minipage}
\caption{Precision for different deformation types (l.); estimated deformation grid and ground-truth point cloud (blue) before and after one deformation cycle (r.)}
\label{fig:precision}
\end{figure}
\begin{equation}
J_{P\mathit{2}P,ci} = \dfrac{\partial\left(
\omega_{p} ~\omega_{c} ~\ensuremath{\boldsymbol{r}}_{P\mathit{2}P,c}(\mathcal{X})\right)}
{\partial \ensuremath{\boldsymbol{t}_{i}}} =
\omega_{p} ~\omega_{c} ~\alpha_{i}(\ensuremath{\hat{\boldsymbol{p}}_{c}})
\label{eqn:p2p_Jacobi}
\end{equation}
where $\lvert C \rvert$ is the number of correspondences and $J_{P\mathit{2}P,ci}$ is the entry at the $c^{th}$ row and $i^{th}$ column of the Jacobian $J_{P\mathit{2}P}$. The Jacobians for $E_{P\mathit{2}S}$ and $E_{F}$ can be found analogously.
\paragraph{Regularization}
With a single camera's perspective, it is impossible to observe the complete surface of an object. The spatial lack of correspondences implies an underdetermined equation system and $E$ being ill-conditioned. To alleviate this problem, we use an ARAP regularizer \cite{Sorkine.2007}, where non-observable surface points are deformed such that the total deformation of the body is \textit{as rigid as possible}. Unlike prior work \cite{Sorkine.2007,Sorkine.2017}, we estimate the deformation in terms of $\mathcal{V}$ instead of the mesh, leading to the adapted ARAP term
\begin{equation}
E_{Reg} =
\sum^{\vert G \vert}_{i=1} \sum_{j \in \mathcal{N}_{i}}
\bigl| \bigl|
\left(\ensuremath{\boldsymbol{t}_{i}} - \ensuremath{\boldsymbol{t}_{j}} \right)
- R_{i} \left(\ensuremath{\hat{\boldsymbol{t}}_{i}} - \ensuremath{\hat{\boldsymbol{t}}_{j}} \right)
\bigl| \bigl|^2_{2},
\label{eqn:grid arap}
\end{equation}
where $\mathcal{N}_i$ denotes the \textit{neighborhood} (6 surrounding grid points) of grid point $i \in [1, \lvert G \rvert]$.
For each grid point $i$, our solver must solve for 6 unknowns describing its pose $(R_i | \ensuremath{\boldsymbol{t}_{i}})$. As shown in \cite{Sorkine.2017}, the (non-linear) estimation of $R_i$ can be solved in closed form given $\boldsymbol{t}_i$. For $\boldsymbol{t}_i$, we obtain
\begin{equation}
\frac{\partial E_{Reg,i}}{\partial \ensuremath{\boldsymbol{t}_{i}}} \overset{!}{=} 0
\Leftrightarrow
\underbrace{
\sum_{j \in \mathcal{N}_{i}} (\ensuremath{\boldsymbol{t}_{i}} - \ensuremath{\boldsymbol{t}_{j}})
}_{L~\cdot~\mathcal{X}_{i}} = \sum_{j \in \mathcal{N}_{i}} \dfrac{R_{i}+R_{j}}{2} (\ensuremath{\hat{\boldsymbol{t}}_{i}} - \ensuremath{\hat{\boldsymbol{t}}_{j}})
\label{eqn:arap final}
\end{equation}
where the left-hand side is the product of the Laplace matrix $L$ with the vector of all unknowns $\mathcal{X}_i$.
\paragraph{Flip-flop solver}
We iteratively estimate $R_i$ and $\boldsymbol{t}_i$ in turn by closed-form solving for $R_i$ via singular value decomposition (see \cite{Sorkine.2017} for details) and approximating $\boldsymbol{t}_i$ via Gauss-Newton, where the update step $\Delta \mathcal{X}$ is obtained via \ac{pcg}. Using the Jacobians derived above, we can obtain the deformation $\mathcal{X}_{t+1}$ after an update step via
\begin{align}
J^{T}J &:=
J^{T}_{P\mathit{2}P}J_{P\mathit{2}P} +
J^{T}_{P\mathit{2}S}J_{P\mathit{2}S} +
J^{T}_{F}J_{F} +
L^{T}L \\
J^{T}\boldsymbol{r} &:=
J^{T}_{P\mathit{2}P} \boldsymbol{r}_{P\mathit{2}P} +
J^{T}_{P\mathit{2}S} \boldsymbol{r}_{P\mathit{2}S} +
J^{T}_{F} \boldsymbol{r}_{F} +
L^{T}\boldsymbol{r}_{Reg}
\label{eqn:pcg LHS}
\end{align}
\begin{align}
J^{T}J \,\ \boldsymbol{\Delta} \mathcal{X} &= J^{T}\boldsymbol{r}\\
\mathcal{X}_{t+1} &= \mathcal{X}_{t} + \boldsymbol{\Delta} \mathcal{X}
\label{eqn:pcg}
\end{align}
\begin{figure}
\centering
\begin{minipage}{.65\linewidth}
\includegraphics[width=\linewidth]{images/icra_2021_deformable/tripod.png}
\end{minipage}%
\hfill
\begin{minipage}{.335\linewidth}
\includegraphics[width=\linewidth]{images/icra_2021_deformable/dichtung4.png}
\end{minipage}
\caption{\ac{poi} tracking on a deformable tripod (l.); rubber seal assembly (r.)}
\label{fig:experiments}
\end{figure}
\section{Results}
\paragraph{Precision} \label{par:precision}In a first set of experiments, we assess the precision of our approach by comparing tracking results versus manually labeled correspondences. 10 \acp{poi} on a deformable tripod were considered, with each \ac{poi} also fitted with a color-coded marker to facilitate manual labeling.\footnote{Since our algorithm only considers geometric features, the presence of the markers neither helped nor hurt the algorithm.} The tripod was repeatedly deformed and the poses of the \acp{poi} were estimated by our algorithm as well as via the markers (cf. fig. \ref{fig:precision}). Our approach was capable of \textit{localizing} all \acp{poi} with sub-millimeter accuracy, and \textit{tracking} all \acp{poi} with errors between 0.6 and 2.2 mm. Unlike feature-matching based approaches, we always estimate the deformation of the complete surface and thereby avoid ``mismatching'' \acp{poi} by design.
\paragraph{Performance} A significant advantage of our approach is that each step of the solver pipeline can be efficiently parallelized. We benchmarked our algorithm using reference and deformation models at fine\footnote{Reference model: 30000 vertices, $\mathcal{V}$: 3250 grid points} and coarse\footnote{Reference model: 15000 vertices, $\mathcal{V}$: 700 grid points} resolutions. A parallelized CPU implementation of our algorithm localized all \acp{poi} in under $1.5s$ in both cases on consumer hardware.
\paragraph{Robot experiments} In a first robot experiment, we track a point on the surface of a tripod subjected to several deformations of up to 20\% of the tripod's arm length, or ca. 2.5 cm. We use a UR5 robot equipped with a measuring tip to visualize tracking results (cf. fig. \ref{fig:experiments} (l.)), confirming precision within 2 mm. In a second experiment, we use our approach to position a flat rubber seal on a housing, illustrating its potential for real-world industrial applications (cf. fig. \ref{fig:experiments} (r.)).
\section{Discussion and Outlook}
Our approach and solver pipeline allows efficient tracking and localization of \acp{poi} on deformable objects. Where prior work requires markers, explicit modelling or does not allow for offline \ac{poi} definition, our approach achieves sub-millimeter precision localization and millimeter-precision tracking without these drawbacks. This makes it particularly suitable for applications in industrial robotics and flexible, quickly reconfigurable assembly or surface treatment tasks. We are working on integrating our solution into an industrial robot manipulation framework, a more efficient GPU implementation and a more extensive evaluation on a wider set of benchmarks.
\bibliographystyle{IEEEtran}
| {
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export { default as Home } from './Home'
export { default as About } from './About'
export { default as Blog } from './Blog'
export { default as BlogPost } from './BlogPost'
export { default as Github } from './Github'
export { default as NotFound } from './NotFound/NotFound'
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"http:\/\/www.maths.usyd.edu.au\/u\/pubs\/publist\/pubspre1975.html","text":"# Research Publications 1947 to 1974\n\n## Books\n\nA1. Field MJ: Differential Calculus and its Applications. Van Nostrand Reinhold, London, 1970.\n\nA1. Kirkpatrick PB and Room TG: Miniquaternion Geometry, An Introduction to the Study of Projective Planes. Cambridge Tracts in Mathematics and Mathematical Physics, 60 Cambridge University Press, 1971. MR0298538\n\nA1. McMullen JR: Extensions of Positive Definite Functions. Memoirs of the American Mathematical Society, 117 1972, iii + 71 pages. MR56:3573\n\nA1. Romanovsky VI and Seneta E: Discrete Markov Chains. Wolters-Noordhoff, Groningen, 1970, (translated from the Russian). MR0266312\n\nA1. Wall GE: The structure of a unitary factor group. Publications Math\u00e9matiques, 1 l'Institut des Hautes \u00c9tudes Scientifiques, 1959, 7\u201323 pages. MR0104764\n\nA2. Kelly GM: An Introduction to Algebra and Vector Geometry. 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Illinois Journal of Mathematics, 2 (1958), 718\u2013745. MR21:3487\n\nC1. Brill DR, Chrzanowski PL, Pereira CM, Fackerell ED and Ipser JR: Solution of the scalar wave equation in a Kerr background by separation of variables. Physical Review. D, 5 (1972), 1913\u20131915. MR52:7441\n\nC1. Brown A and Dancer EN: Functions of asymptotic expansions. Bulletin of the Australian Mathematical Society, 4 (1971), 225\u2013258.\n\nC1. Buchen PW: Application of the ray-series method to linear viscoelastic wave propagation. Pure and Applied Geophysics, 112 (1974), 1011\u20131029.\n\nC1. Buchen PW: Reflection, transmission and diffraction of SH-waves in linear viscoelastic solids. Geophysics Journal of the Royal Astronomical Society, 25 (1971), 97\u2013113.\n\nC1. Buchen PW: Plane waves in linear viscolealstic solids. Geophysics Journal of the Royal Astronomical Society, 23 (1971), 531\u2013542.\n\nC1. Camina AR and Gagen TM: Some doubly transitive groups of odd degree. 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Mathematische Zeitschrift, 90 (1965), 268\u2013273. MR33:190\n\nC1. Gagen TM and Hale MP Jr: On the maximal normal prime nilpotent subgroups of a prime solvable group. Bulletin of the Australian Mathematical Society, 6 (1972), 266\u2013262. MR45:3552\n\nC1. Gagen TM and Janko Z: Finite simple groups with nilpotent third maximal subgroups. Journal of the Australian Mathematical Society, 6 (1966), 446\u2013469.\n\nC1. Gagen TM and Weichsel PM: Dense subgroups of finite groups. Journal of the Australian Mathematical Society, 13 (1972), 385\u2013390. MR47:5107\n\nC1. Galambos J and Seneta E: Regularly varying sequences. Proceedings of the American Mathematical Society, 41 (1973), 110\u2013116. MR48:2316\n\nC1. Gibson WG: Quantum corrections to the radial distribution function of a fluid. Molecular Physics, 28 (1974), 793\u2013800.\n\nC1. Gibson WG: Expansion for the quantum second virial coefficient using hard-sphere basis functions. Journal of Chemical Physics, 59 (1973), 991.\n\nC1. Gibson WG: Quantum corrections to the virial coefficients for potentials with hard cores. Physical Review. A, 7 (1973), 822\u2013824.\n\nC1. Gibson WG: Low-temperature expansion of the third cluster coefficient of a quantum gas. Physical Review. A, 6 (1972), 2469\u20132477.\n\nC1. Gibson WG: Logarithmic term in the low-temperature expansion of the third quantum cluster coefficient of a quantum gas. Physics Letters, A, 38A (1972), 331\u2013332.\n\nC1. Gibson WG: Quantum corrections to the equation of state for nonanalytic potentials. Physical Review. A, 5 (1972), 862\u2013867.\n\nC1. Gibson WG: Bound state and continuum contributions to the two-body partition function. Physics Letters, A, 36A (1971), 403\u2013404.\n\nC1. Gibson WG: Classical limit of the binary-collision expansion. Physical Review. A, 4 (1971), 2108\u20132109.\n\nC1. Gibson WG: Quantum-mechanical second virial coefficient at high temperatures. Physical Review. A, 2 (1970), 996\u20131002.\n\nC1. Gibson WG: Application of the Faddeev equations to the calculation of the quantum-mechanical third virial coefficient. Physics Letters, A, 21 (1966), 619\u2013621.\n\nC1. Golub GH and Seneta E: Computation of the stationary distribution of an infinite stochastic matrix of special form. Bulletin of the Australian Mathematical Society, 10 (1974), 225\u2013261.\n\nC1. Golub GH and Seneta E: Computation of the stationary distribution of an infinite Markov matrix. Bulletin of the Australian Mathematical Society, 8 (1973), 333\u2013342. MR50:8710\n\nC1. Gyory AZ, Edwards KDG, Palmer AA and Robinson J: The relative importance of urinary pH and urinary content of citrate, magnesium and calcium in the production of nephrocalcinosis by diet and acetazolancide in the rat. Clinical Science, 39 (1970), 605\u2013623.\n\nC1. Havas G, Wall GE and Walmsley JW: The two generator restricted Burnside group of exponent five. Bulletin of the Australian Mathematical Society, 10 (1974), 459\u2013470.\n\nC1. Heathcote CR and Seneta E: Inequalities for branching processes. Journal of Applied Probability, 3 (1966), 261\u2013267. MR33:787\n\nC1. Heathcote CR, Seneta E and Vere-Jones D: A refinement of two theorems in the theory of branching processes. Teoria Veroiatnostei i ee Primenenia, 12 (1967), 341\u2013346. MR0217889\n\nC1. Heyde CC and Seneta E: Notes on \"Estimation theory for growth and immigration rates in a multiplicative process\". Journal of Applied Probability, 11 (1974), 572\u2013577. MR51:4437\n\nC1. Heyde CC and Seneta E: The simple branching process, a turning point test and a fundamental inequality: a historical note on I.J. Bienaym\u00e9. 59 (1972), 680\u2013683. MR49:6289\n\nC1. Heyde CC and Seneta E: Estimation theory for growth and immigration rates in a multiplicative process. Journal of Applied Probability, 9 (1972), 235\u2013256. MR49:8127\n\nC1. Heyde CC and Seneta E: Analogues of classical limit theorems for the supercritical Galton-Watson process with immigration. Mathematical Biosciences, 11 (1971), 249\u2013259. MR45:1272\n\nC1. Howlett RB: On the degrees of Steinberg characters of Chevalley groups. Mathematische Zeitschrift, 135 (1974), 125\u2013135. MR50:13228\n\nC1. Hutchinson TP: The control of right-turning vehicles at signal-controlled intersections: A comment on a suggestion by Al-Salman and Salter. Traffic Engineering and Control, 15 (1974), 920\u2013923.\n\nC1. Hutchinson TP: The communication of transport research. Traffic Engineering and Control, 15 (1974), 775.\n\nC1. Hutchinson TP: The distribution of traffic in towns: Further validation of Vaughan's model by correlating its parameters with size of city and time of day. Traffic Engineering and Control, 15 (1974), 770\u2013771.\n\nC1. Hutchinson TP: Urban traffic speeds. II: Relation of the parameters of two simpler models to size of city and time of day. 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\section{Exponential Fidelity Decay via 2-design Twirling}
\label{app:twirl}
Given a gate independent noise assumption and 2-design\footnote{%
We thank Alex Dalzell for pointing out that a 2-design approximation is sufficient, rather than a 4-design.}
approximation,
we can generalize the randomized benchmarking 2-design twirling argument \cite{tdesignRB} to our protocol.
Similar to the main texts, we find that quantum mechanics predicts an exponential fidelity decay [\eqnref{eq:f}].
We assume that the noise is gate and time independent,
but it may depend on the position of the 2-qubit gate.
Therefore,\arxiv{ in the notation of \secref{sec:decay},}
the noise channels are assumed to be the same for different 2-qubit gates $u_{t,i}$ and $u_{t',i}$
acting on the same pair of qubits labeled by $i$:
$\mathcal{N}_{u_{t,i}} = \mathcal{N}_{u_{t',i}}$.
However, we can allow the noise for the forward and backward circuits to be different.
As is also the case in the main text,
it is not essential to assume that the noise only comes from 2-qubit error channels.
We also approximate each layer of 2-qubit gates $U_t$ as a 2-design.
Formally, this approximation is not justified
since an approximate 2-design requires a $O(n^{1/D})$-depth $D$-dimensional circuit \cite{approximateTDesign}.
However, this approximation might be better than expected if we can instead think of it as
only approximating the composition of $O(n^{1/D})$ layers of gates as a 2-design.
We do not consider this possibility here
since it would complicate the definition of the noise channel $\mathcal{N}$ below.
To simplify the notation, we rewrite the Markovian fidelity \eqnref{eq:F model} from the main text as
\begin{equation}
F_d = \Exp_{U_1, \ldots, U_d} \tr( \rho'_d \, \rho_d )
\end{equation}
where
\begin{gather}
\begin{aligned}
\rho'_d &\equiv (\Phi_{U_d^\dagger}^\dagger \!\circ\cdots\circ \Phi_{U_1^\dagger}^\dagger) (\rho^\text{fin}) \\
\rho_d &\equiv (\Phi_{U_d} \!\circ\cdots\circ \Phi_{U_1}) (\rho^\text{init})
\end{aligned}
\end{gather}
implicitly depend on $U_1, \ldots, U_d$, and
$\Phi^\dagger$ is defined such that $\tr[\Phi^\dagger(\rho') \, \rho] = \tr[\rho' \, \Phi(\rho)]$.
Below we recursively integrate out one $U_t$ at a time by
approximating each $U_t$ as an arbitrary Haar random unitary (or a unitary 2-design) on $n$ qubits:
\begin{align}
F_d &= \Exp_{U_1, \ldots, U_d} \tr\!\left[ \mathcal{N}_d'^{\dagger}(\rho'_{d-1}) \, U_d^\dagger \mathcal{N}_d(U_d \rho_{d-1} U_d^\dagger) U_d \right] \label{eq:F1}\\
&\sim \frac{\hat{F}_d-2^{-2n}}{1-2^{-2n}} \Exp_{U_1, \ldots, U_{d-1}} \tr\!\left[ \mathcal{N}_d'^\dagger(\rho'_{d-1}) \, \rho_{d-1} \right] + (1-\hat{F}_d) \frac{2^{-n}}{1-2^{-2n}} \tr \mathcal{N}_d'^\dagger(\rho'_{d-1}) \label{eq:F2}\\
&\sim \left( \prod_{t=d'+1}^d \frac{\hat{F}_t-2^{-2n}}{1-2^{-2n}} \right) \Exp_{U_1, \ldots, U_{d'}}
\tr\!\left[ \mathcal{N}_{d'}'^\dagger(\rho'_{d'-1}) \, U_{d'}^\dagger (\mathcal{N}'_{d'+1} \circ \mathcal{N}_{d'})(U_{d'} \rho_{d'-1} U_{d'}^\dagger) U_{d'} \right] \\
&\quad\quad + \sum_{t=d'+1}^d (1-\hat{F}_t) \frac{2^{-n}}{1-2^{-2n}} \tr \mathcal{N}_t'^\dagger(\rho'_{t-1}) \nonumber\\
&\sim \left( \prod_{t=1}^d \frac{\hat{F}_t-2^{-2n}}{1-2^{-2n}} \right)
\tr\!\left[ \mathcal{N}_1'^\dagger(\rho^\text{fin}) \, \rho^\text{init} \right] + \sum_{t=1}^d (1-\hat{F}_t) \frac{2^{-n}}{1-2^{-2n}} \tr \mathcal{N}_t'^\dagger(\rho'_{t-1}) \\
&= \tr\!\left[ \mathcal{N}_1'^\dagger(\rho^\text{fin}) \, \rho^\text{init} \right] \, \prod_{t=1}^d \hat{F}_t + O(2^{-n}) \label{eq:F5}
\end{align}
$\mathcal{N}_t$ and $\mathcal{N}'_t$ are gate-independent noise channels of the forward and backward circuits:
$\Phi_{U_t}(\rho) = \mathcal{N}_t(U_t \rho U_t^\dagger)$ and
$\Phi_{U_t^\dagger}(\rho) = \mathcal{N}'_d(U_t^\dagger \rho U_t)$.
In the last line, we drop very small $O(2^{-n})$ terms.
$\hat{F}_t$ is the entanglement fidelity of $\mathcal{N}'_{t+1} \circ \mathcal{N}_t$ for $t=1,\ldots,d-1$,
while $\hat{F}_d$ is just the entanglement fidelity of $\mathcal{N}_d$:
\begin{align}
\hat{F}_t &= 2^{-2n} \tr\!\left(\mathcal{N}'_{t+1} \circ \mathcal{N}_t\right) &
\hat{F}_d &= 2^{-2n} \tr \mathcal{N}_t \label{eq:hatF}\\
&= 2^{-2n} \sum_{k',k} \left|\tr\!\left(E_{k'}'^{(t+1)} E_k^{(t)}\right)\right|^2 &
&= 2^{-2n} \sum_{k} \left|\tr E_k^{(t)}\right|^2 \nonumber
\end{align}
$E_k^{(t)}$ and $E_{k'}'^{(t)}$ are Kraus operators of $\mathcal{N}_t$ and $\mathcal{N}'_t$; \emph{i.e.}
$\mathcal{N}_t(\rho) = \sum_k E^{ (t)}_k \rho\, E^{(t)\dagger}_k$ and
$\mathcal{N}'_t(\rho) = \sum_{k'} E'^{(t)}_{k'} \rho\, E'^{(t)\dagger}_{k'}$.
Note that $\hat{F}_t = \hat{F}_{t+p}$ has periodicity $p$ since the circuit geometry is assumed to have periodicity $p$.
The entanglement fidelity of the composition $\mathcal{N}'_{t+1} \circ \mathcal{N}_t$ appears in \eqnref{eq:hatF}
because coherent errors can cancel out.
For example, if $\mathcal{N}_t(\rho) = V \rho V^\dagger$ and $\mathcal{N}'_{t+1}(\rho) = V^\dagger \rho V$ for some unitary $V$,
then the fidelity $F_d$ would not depend on the coherent error $V$.
This subtlety does not occur in randomized benchmarking \cite{noiseEstimation,tdesignRB,twirlRB}
because randomized benchmarking uses a compactified inverse circuit.
\eqnref{eq:F5} is an exponential fidelity decay
$F_d \sim \widetilde{F}_0 e^{-\lambda d}$
with parameters
\begin{gather}
\begin{aligned}
\widetilde{F}_0 &\approx \tr\!\left[ \mathcal{N}_1'^\dagger(\rho^\text{fin}) \, \rho^\text{init} \right] \frac{\tr \mathcal{N}_p}{\tr\!\left(\mathcal{N}'_{p+1} \circ \mathcal{N}_p\right)} \\
\lambda &\approx -\frac{1}{p} \log \prod_{t=1}^p \hat{F}_t
\end{aligned} \label{eq:lambdaApp}
\end{gather}
when $d$ is an integer multiple of the period $p$.
\eqnref{eq:lambdaApp} differs from \eqnref{eq:lambda} in the main text for two reasons:
\eqnref{eq:lambda} neglected the fact that coherent errors in $\mathcal{N}'_{t+1} \circ \mathcal{N}_t$ can cancel out,
while \eqnref{eq:lambdaApp} assumes that errors are gate-independent.
Unfortunately, the decay rate $\lambda$ in \eqnref{eq:lambdaApp} can not be written in terms of a product of 2-qubit gate fidelities
[as in \eqnref{eq:lambda}]
because $\mathcal{N}'_{t+1} \circ \mathcal{N}_t$ in the definition of $\hat{F}_t$
involves two circuit layers with different positions of the 2-qubit gates.
If we were to make the rough approximation
$2^{-2n} \tr\!\left(\mathcal{N}'_{t+1} \circ \mathcal{N}_t\right) \to
2^{-4n} \tr\mathcal{N}'_{t+1} \tr \mathcal{N}_t$,
then $\lambda$ in \eqnref{eq:lambdaApp} would simplify to \eqnref{eq:lambda}.
\section{Numerical Simulations}
\label{app:numerics}
In \figref{fig:simulation} we showed a simulation for a $5\times5$ grid of $n=25$ qubits
using roughly Sycamore/Zuchongzhi-like \cite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2} gates with gate-dependent errors.
\appsref{app:gates} and \ref{app:noise} specify the gate and noise models that we used.
\appref{app:Kraus} defines an optimized basis of Kraus operators that we used to minimize the amount of statistical sampling error.
All plots in the main text have statistical errors that are too small to be seen.
\appref{app:MPS simulation} specifies details used to produce the MPS data in \figref{fig:MPS}.
In \figref{fig:simulationApp}, we show a more detailed version of \figref{fig:simulation}
showing a larger max circuit depth $d_\text{max}$.
At large depth, the fidelity $F_d$ saturates to $2^{-n}$,
which will be too small to be experimentally observable for large $n \gtrsim 50$.
\figref{fig:simulationAppErr} shows the residual errors to the exponential decay fit.
The $\log(F_d) - \text{fit}$ residuals are dominated by the $2^{-n}$ saturation at large depth.
After subtracting off $2^{-n}$, we see that $F_d - 2^{-n}$ agrees with an exponential decay up to statistical errors.
The statistical errors of $F_d - 2^{-n}$ become very large once the fidelity decays to $2^{-n}$
due to the $2^{-n}$ subtraction.
\begin{figure}
\centering
\subfloat[]{\includegraphics[width=.48\columnwidth]{simulationApp}} $\quad$
\subfloat[\label{fig:simulationAppErr}]{\includegraphics[width=.48\columnwidth]{simulationAppErr}} \\
\subfloat[\label{fig:circuit2d}]{\includegraphics[width=.6\columnwidth]{circuit2d}}
\caption{%
{\bf (a)} The same simulation data (black dots) as \figref{fig:simulation} (but for larger max depth $d_\text{max}$)
showing the quantum mechanics prediction of
the exponential fidelity $F_d$ decay
for a $5\times5$ grid of $n=25$ noisy qubits.
The fidelity $F_d$ saturates to $2^{-n}$ at large circuit depth $d$.
Subtracting $2^{-n}$ from $F_d$ (green circles) leads to a continued exponential decay
(at least until our simulation data becomes dominated by statistical errors).
Also shown is a fit (blue line) to an exponential decay.
{\bf (b)} With $2^{-n}$ subtracted, the fit residuals are dominated by statistical errors for our simulation of 10,000 Monte Carlo samples.
Error bars denote one standard deviation (\emph{i.e.} 68\% confidence intervals).
{\bf (c)} The period $p=4$ ordering of 2-qubit gates (green) applied to the $5\times5$ array of qubits.
}\label{fig:simulationApp}
\end{figure}
To investigate the residuals with less statistical error,
we also ran a simulation with less qubits
(for which we can run significantly more Monte Carlo samples).
\figref{fig:simulationApp3x3} is similar to \figref{fig:simulationApp},
but for a simulation of a $3\times3$ grid of $n=9$ qubits
and a smaller average gate entanglement fidelity $f \approx 0.987$ (instead of $f \approx 0.993$).
In \figref{fig:simulationAppErr3x3},
we see that residual errors to an exponential fit are very small when $F_d \gg 2^{-n}$.
\begin{figure}
\centering
\subfloat[]{\includegraphics[width=.48\columnwidth]{simulationApp3x3}} $\quad$
\subfloat[\label{fig:simulationAppErr3x3}]{\includegraphics[width=.48\columnwidth]{simulationAppErr3x3}}
\caption{%
Same as \figref{fig:simulationApp}, but for a $3\times3$ grid of $n=9$ qubits,
for which we obtained $8\times10^7$ Monte Carlo samples.
Panel (b) shows that the residual errors of $F_d - 2^{-n}$ to an exponential fit are very small.
}\label{fig:simulationApp3x3}
\end{figure}
\subsection{Gate Model}
\label{app:gates}
In our numerical simulations,
we use gates that are somewhat similar to the Sycamore/Zuchongzhi gates \cite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2}.
The gates are applied in the order shown in \figref{fig:circuit2d}.
Each 2-qubit gate $u_{t,i}$ is an iSWAP gate
(rather than Sycamore/Zuchongzhi iSWAP-like gates \cite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2})
\begin{equation}
U_\text{iSWAP} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & -i & 0 \\
0 & -i & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\end{equation}
that is preceded by a pair of randomly chosen 1-qubit gates ($\sqrt{X}$, $\sqrt{Y}$, or $\sqrt{W}$) on two qubits,
where $W = \frac{1}{\sqrt{2}} (X+Y)$.
Using $U_\text{iSWAP}^\dagger = U_\text{iSWAP} \, (Z \otimes Z)$,
we decompose the inverse gate $u_{t,i}^\dagger$ as an iSWAP gate followed by a
$\sqrt{X}^\dagger Z$, $\sqrt{Y}^\dagger Z$, or $\sqrt{W}^\dagger Z$ gate on each of the two qubits.
\subsection{Noise Model}
\label{app:noise}
In our numerical simulations,
we use a generic error model with gate and position dependent errors.
Each iSWAP gate for each pair of qubits (labeled by $i$) has a unique noise channel.
And for each of the $n$ qubits, we apply a unique noise channel for each of six the 1-qubit gates
($\sqrt{X}$, $\sqrt{Y}$, $\sqrt{W}$, $\sqrt{X}^\dagger Z$, $\sqrt{Y}^\dagger Z$, and $\sqrt{W}^\dagger Z$).
Each noise channel is generated randomly.
None of our plots appear to differ qualitatively for different random instances of gate error channels.
One way to generate a random noise channel on a $d$-dimensional Hilbert space is to
choose a set of $m$ Kraus operators as a Haar random $md \times d$ semi-unitary matrix $E_{ka,b}$
(\emph{i.e.} $\sum_{ka} E_{ka,b'}^* E_{ka,b} = \delta_{b',b}$).
Then the noise channel is
\begin{equation}
\mathcal{N}_m(\rho) \overset{\text{rand}}{=} \sum_{k=1}^m E_k \rho E_k^\dagger
\end{equation}
In order to cover the set of all noise channels, one must choose $m=d^2$.
However, for $d=4$ or larger, this leads to a distribution of noise channels that is
strongly concentrated near the depolarizing channel.
In order to obtain a broader but still generic distribution of noise channels,
we generate each random noise channel as an average of the above random noise channel for $m=1,2,\ldots,d^2$:
\begin{equation}
\mathcal{N}_\text{rand}(\rho) \overset{\text{rand}}{=} d^{-2} \sum_{m=1}^{d^2} \mathcal{N}_m(\rho)
\end{equation}
We also apply a randomly-chosen coherent error for each gate.
For a $d$-dimensional Hilbert space, we generate coherent error channels using a random unitary that is generated by
\begin{equation}
U_p \overset{\text{rand}}{=} e^{i \sqrt{\frac{p}{d-1}} H^\text{Gauss}_\text{rand}}
\end{equation}
where $H^\text{Gauss}_\text{rand}$ is a complex standard Gaussian random matrix
and $p$ tunes the infidelity.
Each noise channel is then generated via
\begin{equation}
\mathcal{N}_p(\rho) \overset{\text{rand}}{=} U_{p''/2} \left[ \left(1-\tfrac{p'}{2}\right) \rho + \tfrac{p'}{2} \, \mathcal{N}_\text{rand}(\rho) \right] U_{p''/2}^\dagger \label{eq:noise}
\end{equation}
where $p'$ and $p''$ are randomly chosen between $p/\sqrt{2}$ and $\sqrt{2} \,p$.
The entanglement fidelity of $\mathcal{N}_p$ is roughly $1-p$.
We randomly generated the 2-qubit iSWAP noise channel using $\mathcal{N}_{p_2}$,
while the 1-qubit noise channels were randomly generated using $\mathcal{N}_{p_1}$,
with $p_1 = 0.1 p_2$.
For \figsref{fig:simulation}, \ref{fig:MPS}, and \ref{fig:simulationApp}, we used $p_2 = 0.005$.
For \figref{fig:simulationApp3x3}, we used $p_2 = 0.01$.
\subsection{Simulation Algorithm Details}
\label{app:Kraus}
We simulate the fidelity in \eqnref{eq:F model} by Monte Carlo (MC) sampling the density matrices and quantum channels
such that only two wavefunctions are stored in memory: one from $\rho^\text{init}$ and the other from $\rho^\text{fin}$.
To reduce the amount of statistical error,
we sample the density matrices and quantum channels such that each MC sample has the same average entanglement fidelity.
That is, $\rho^\text{init}$ and $\rho^\text{fin}$ are uniformly sampled from unnormalized wavefunctions
$\ket{\psi^\text{init}_k}$ and $\ket{\psi^\text{fin}_k}$:
\begin{equation}
\begin{aligned}
\rho^\text{init} &= \Exp_k \, \ket{\psi^\text{init}_k} \bra{\psi^\text{init}_k} \\
\rho^\text{fin} &= \Exp_k \, \ket{\psi^\text{fin}_k} \bra{\psi^\text{fin}_k}
\end{aligned} \label{eq:wavefunctions}
\end{equation}
which are chosen to have the same fidelity:
\begin{equation}
\begin{aligned}
\braket{0|\psi^\text{init}_k} &= \braket{0|\psi^\text{init}_{k'}} \\
\braket{0|\psi^\text{fin}_k} &= \braket{0|\psi^\text{fin}_{k'}}
\end{aligned} \label{eq:wavefunctions constraint}
\end{equation}
In \eqnref{eq:wavefunctions}, $\tExp_k \equiv n_k^{-1} \sum_k$
denotes an expectation value over $n_k$ uniformly random $k$.
Similarly, the quantum channels $\Phi_u$ are rewritten using a Kraus decomposition
\begin{equation}
\Phi_u(\rho) = \Exp_k E^{(u)}_k \rho \, E^{(u)\dagger}_k
\end{equation}
(with nonstandard normalization due to the $\Exp_k$)
such that each Kraus operator has the same trace:
\begin{equation}
\tr E^{(u)}_k = \tr E^{(u)}_{k'} \label{eq:Kraus constraint}
\end{equation}
This implies that each Kraus operator has the same entanglement fidelity as $\Phi_u$:
$f_u = 2^{-2m} \tr E^{(u)}_k \tr E^{(u)\dagger}_k$
where $E^{(u)}_k$ is a $2^m \times 2^m$ matrix.
With this optimization,
for a given sequence of unitaries $U_1,\ldots,U_d$,
each MC sample has the approximately the same fidelity,
which leads to a significant reduction in MC statistical error.
We only require that these special wavefunctions and Kraus operators obey linear constraints
[\eqnref{eq:wavefunctions constraint} and \eqref{eq:Kraus constraint}].
They can therefore be obtained from another decomposition (\emph{e.g.} an eigendecomposition),
$\rho = \tExp_{\tilde{k}} \ket{\widetilde{\psi}_{\tilde{k}}} \bra{\widetilde{\psi}_{\tilde{k}}}$ or
$\Phi(\rho) = \tExp_{\tilde{k}} \widetilde{E}_{\tilde{k}} \rho \widetilde{E}^\dagger_{\tilde{k}}$,
via a unitary transformation $V_{k,\tilde{k}}$ that rotates the vector
$v_{\tilde{k}} = \braket{0|\widetilde{\psi}_{\tilde{k}}}$ (or $v_{\tilde{k}} = \tr \widetilde{E}_{\tilde{k}}$)
to $\sum_{\tilde{k}} V_{k,\tilde{k}} v_{\tilde{k}} = \sqrt{f}$
for some constant fidelity $f>0$.
That is, $\ket{\psi_k} = V_{k,\tilde{k}} \ket{\widetilde{\psi}_{\tilde{k}}}$
and $E_k = V_{k,\tilde{k}} \widetilde{E}_{\tilde{k}}$.
Each MC sample for a sequence of fidelities $F_d$ with $d=0,\ldots,d_\text{max}$ is then obtained as follows:
\begin{enumerate}
\item sample initial forward circuit wavefunction: $\ket{\psi} \leftarrow \ket{\psi^\text{init}_k}$ for uniformly random $k$
\item sample final backward circuit wavefunction: $\ket{\psi'} \leftarrow \ket{\psi^\text{fin}_k}$ for uniformly random $k$
\item foreach circuit depth $d=1,\ldots,d_\text{max}$:
\begin{enumerate}
\item foreach 2-qubit gate position $i$:
\begin{enumerate}
\item randomly sample an allowed gate: $u_{d,i}$
\item apply random Kraus operator for forward gate: $\ket{\psi} \leftarrow E^{(u_{d,i})}_k \ket{\psi}$
for uniformly random $k$
\item apply random Kraus operator for backward gate: $\ket{\psi'} \leftarrow E^{(u_{d,i}^\dagger)\dagger}_k \ket{\psi'}$
for uniformly random $k$
\end{enumerate}
\item obtain fidelity sample: $F_d \leftarrow |\braket{\psi' | \psi}|^2$
\end{enumerate}
\end{enumerate}
\subsection{MPS Simulation}
\label{app:MPS simulation}
In \figref{fig:MPS}, we showed a MPS simulation of our protocol with noisy gates for a chain of 24 qubits.
We used the same noise and gate models and sampling algorithm as in \figref{fig:simulation},
\emph{i.e.} the models in \appsref{app:gates} and \ref{app:noise} and algorithm in \appref{app:Kraus}.
Random gates, SPAM errors, and random Kraus operators are randomly sampled
as described in \appref{app:Kraus}.
However, a subtlety results from the fact that the MPS simulation introduces non-linear truncation errors.
When there is no truncation error, as in the quantum mechanics prediction (blue in \figref{fig:MPS}),
then changing the Kraus operator basis $E_k \to V_{k,k'} E_{k'}$
does not affect the fidelity.
But when there is truncation error, the choice of Kraus operator basis will affect the simulation result.
Typically, Kraus operators that are more like measurement operators will reduce the amount of entanglement,
which decreases the amount of truncation error in later truncation steps.
Alternatively, Kraus operators that are closer to unitary operators
typically result in a larger truncation error.
We use the Kraus basis defined in \appref{app:Kraus} for the MPS simulation,
which is a nice choice because all of the Kraus operators behave similarly with similar fidelity.
\section{Drift}
\label{app:drift}
While iterating our protocol many times (\emph{e.g.} over the course of hours)
to obtain a large number of samples,
slow temperature fluctuations (and other causes) can result in miscalibrations (and other complications)
that slowly drift \cite{drift} over time.
As a result, the effective error channel of each gate can change over time,
resulting in slowly-varying gate fidelities \cite{cycleBenchmarking} for different samples.
To mitigate issues resulting from drift,
our protocol samples a random circuit depth $d$ for each sample.
If circuit depth was instead sampled in increasing order,
then drift could potentially cause the gate fidelity to decrease over time,
which could then lead to a fidelity $F_d$ decay that could be mistaken for the EmQM signature.
By randomly sampling circuit depth,
we can show that the effect of slow temporal drift
can only cause the exponential fidelity $F_d$ decay rate to slightly decrease with circuit depth $d$,
rather than increase as for EmQM.
This occurs because if the drift occurs much slower than the time scale to sample all circuit depths,
then for larger depth, the fidelity $F_d$ is dominated by rare samples with rare but higher gate fidelities (due to rare drifts),
while at smaller depth the fidelity is dominated by typical gate fidelities.
As such, the effects of drift are unlikely to be confused with the EmQM signature.
To see this more explicitly, we can consider the following toy model.
Assume that quantum mechanics is correct and
suppose that temporal drifts occur over a time scale that is much longer than the time it takes to run a single sample in our protocol.
Then suppose that due to drift, the probability that a sample measures a zero state is the following time-dependent fidelity:
\begin{equation}
F_d(t) = e^{-\lambda(t) d}
\end{equation}
Similar to \eqnref{eq:F},
the fidelity decays exponentially with circuit depth $d$
(although we neglect SPAM errors, \emph{i.e.} the $\widetilde{F}_0$ factor, for simplicity).
However, the fidelity also depends on time
since the exponential decay rate $\lambda(t)$ is time-dependent due to slow time-dependent drifts.
To gain intuition, suppose that the fidelity decay rate $\lambda(t)$ drifts between $\lambda_1$ and $\lambda_2$ with a uniform probability distribution in-between.
Then the average probability that a sample measures a zero state is the average fidelity:
\begin{equation}
\begin{aligned}
F_d &= \frac{1}{\lambda_2 - \lambda_1} \int_{\lambda_1}^{\lambda_2} e^{-\lambda d} \, \mathrm{d}\lambda \\
&= \frac{1}{d} \frac{e^{-\lambda_1 d} - e^{-\lambda_2 d}}{\lambda_2 - \lambda_1}
\end{aligned} \label{eq:drift12}
\end{equation}
See \figref{fig:drift} for an example plot.
At small depth, $F_d \approx e^{-\lambda d}$ where $\lambda = \frac{1}{2} (\lambda_1 + \lambda_2)$,
while at large depth $F_d \approx (\lambda_2 - \lambda_1)^{-1} \frac{1}{d} \, e^{-\lambda_1 d}$.
Thus the fidelity decays slower at larger depth (in contrast to the EmQM signature)
since $\lambda_1 < \lambda$.
\begin{figure}
\centering
\includegraphics[width=.35\columnwidth]{drift} \\
\caption{%
An exponential fidelity decay $F_d = e^{-\lambda d}$ without drift (blue)
vs a fidelity decay with drift (orange) [\eqnref{eq:drift12} with $\lambda_1 = \frac{2}{3}\lambda$ and $\lambda_2 = \frac{4}{3}\lambda$].
Drift only causes the fidelity to decay slightly slower at larger depth,
which can not be mistaken for the EmQM signature [shown in \figref{fig:decay}].
}\label{fig:drift}
\end{figure}
Now consider a general probability distribution $P(\lambda)$ of fidelity decay rates $\lambda(t)$ after time averaging.
Then the fidelity is $F_d = \int_\lambda e^{-\lambda d}$
where $\int_\lambda f(\lambda) \equiv \int_0^\infty f(\lambda) P(\lambda) \,\mathrm{d}\lambda$
averages over the different decay rates.
In this general setting, we can also show that the fidelity decays slower at larger depth.
The exponential decay rate near a given depth is
the average of $\lambda(t)$:
\begin{equation}
- \partial_d \log F_d
= \braket{\lambda}_d
\quad\text{where}\quad \braket{f(\lambda)}_d \equiv \frac{\int_\lambda f(\lambda) \, e^{-\lambda d}}{\int_\lambda e^{-\lambda d}}
\end{equation}
The slope of the curve $\log F_d$ is thus $-\braket{\lambda}_d$.
The derivative of this slope is then the variance of $\lambda(t)$:
\begin{equation}
-\partial_d \braket{\lambda}_d
= \braket[\big]{(\lambda - \braket{\lambda}_d)^2}_d
\end{equation}
Since the above variance is always positive, the derivative of the slope of $\log F_d$ vs $d$ is always positive.
This implies that the fidelity decays slower at larger depth than slower depth,
which is opposite of the EmQM signature.
\arxiv{\newpage}
\section{MPS}
\label{app:MPS}
In \arxivPR{\secref{sec:MPS}}{the main text}, we considered the time evolution of matrix product states (MPS) \cite{MPS} with fixed bond dimension
as an (incomplete) toy model of emergent quantum mechanics (EmQM) in one spatial dimension.
And in \appref{app:realism},
we explain how and why tensor networks are not ruled out by Bell experiments.
However, we do not consider MPS to be a legitimate model of EmQM because
it is currently not known how (or if it is possible) to generalize MPS algorithms to a three-dimensional local classical theory
that could be compatible with previous experiments.
We elaborate on this point below.
One of the hallmarks of MPS is the canonical form,
which allows one to calculate the expectation value of a local operator near some position $x$
in terms of only the tensors near $x$.
When generalizing MPS to higher dimensions,
this sense of canonical form can either be retained,
as is done for isometric tensor networks \cite{IsometricTN,2dDMRG,canonicalPEPS};
or the canonical form can be dropped,
as is done for projected entangled pair states (PEPS) \cite{OrusTN};
or the canonical form can be replaced with something new (but less constraining),
such as a weighted trace gauge (WTG) \cite{EvenblyCanonical}.
When the canonical form is retained \cite{IsometricTN,2dDMRG,canonicalPEPS},
a local update algorithm to time evolve the tensor network is known \cite{IsometricTN}.
However, the algorithm does not quite yet yield a local classical theory since the update algorithm has to iteratively sweep across the tensor network.
This is similar to many MPS algorithms, such as TDVP \cite{TDVP},
which time evolve a length-$L$ chain by performing a forward sweep to update the tensors at
sites $i$ and $i+1$ for $i=1,2,\ldots,L-1$,
followed by a backward sweep with $i=L-1,L-2,\ldots,1$.
It would be incredibly unnatural if the time evolution of the universe sweeps across space in this serial (rather than parallel) manner.
Therefore, we expect an EmQM theory to have the form of a local classical theory:
\emph{i.e.} a (possibly continuous) cellular automaton
for which the local updates are performed in parallel for all sites at once.
Nevertheless, we expect that it is possible to parallelize the time-evolving isometric tensor network algorithm \cite{IsometricTN}
into a continuous cellular automaton,
\emph{e.g.} by using ideas from \refcite{ParallelDMRG}.
However, there is a more severe issue that is also shared with MPS:
between certain pairs of well-separated spatial disks of radius $\ell$,
isometric tensor networks \cite{IsometricTN} and canonical PEPS \cite{canonicalPEPS,2dDMRG}
can only represent wavefunctions that have a bounded amount of entanglement between the two regions.
In particular, the mutual information $I$ between certain pairs of regions is bounded by the log of the bond dimension $\chi$:
$I < \log \chi$.
Even a large bond dimension $\chi = 1024$ is only large enough to encode 10 bell pairs between the two regions.
Unless $\chi$ is extraordinarily large,
this is incompatible with many experiments, including the recent quantum advantage experiments
\cite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2,PhotonAdvantage}.\footnote{%
However, extra spatial dimensions could be used to solve this bounded entanglement problem \cite{SlagleTensor}.}
An EmQM theory should allow the mutual information to diverge with the length scale $\ell$,
so that quantum mechanics with highly-entangled quantum states can emerge in the large length limit $\ell \to \infty$.
On the other hand, PEPS do appear to be possibly powerful enough to represent wavefunctions
that are sufficiently entangled to be compatible with current experimental observations.
For example, the mutual information bound for PEPS diverges linearly with the length scale: $I < O(\ell \log \chi)$.
Furthermore, due to their unconstrained nature,
unitary circuits can easily be embedded into a PEPS in order to represent high-complexity wavefunctions.
This is especially true if the PEPS exists in extra spatial dimensions and/or
if physical qubits are separated by large distances in the lattice of tensors,
as in \figref{fig:PEPS}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{PEPS} \\
\caption{%
A PEPS with physical qubits (dangling red lines) separated by many
unimportant qubits (dangling gray lines), which we consider unimportant after coarse-graining.
Black dots represent tensors which are contracted along the black lines.
We see that the PEPS can easily represent many well-separated by Bell states
by threading their entanglement along the blue lines.
Furthermore, unitary circuits can be embedded into the PEPS
(with unitary gates encoded in tensor pairs highlighted in green)
to encode high-complexity states.
}\label{fig:PEPS}
\end{figure}
However, we are not aware of any time-evolving PEPS algorithms that could be written as a local classical theory.
For example, PEPS update algorithms typically require calculating environment tensors,
which involve a non-local contraction of the entire tensor network,
for which non-local approximate algorithms are necessary \cite{AlgorithmsPEPS,GradientPEPS}.
However, the biggest challenge in formulating an EmQM theory using a tensor network algorithm is that so far,
no (local or nonlocal) time-evolving tensor network algorithm has been shown to be robust enough to create high-complexity states
(\emph{e.g.} those created during the quantum advantage experiments \cite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2,PhotonAdvantage})
with fixed bond dimension.
With sufficiently large bond dimension, most tensor network algorithms can simulate essentially anything.
However in EmQM, we would like to imagine that the bond dimension is fixed and not too large.
Instead, it is the distance between physical qubits and time over which the dynamics occurs that is large,
and it is over these extremely large scales that perhaps quantum mechanics could emerge.
\newpage
\section{Bell's Theorem and Tensor/Neural Networks}
\label{app:Bell}
Bell's theorem \cite{BellTheorem} and its conceptually simpler\footnote{%
The GHZ variant is simpler in the sense that every measurement outcome is fixed according to quantum mechanics
and simultaneously inconsistent with a local hidden variable theory.
In contrast, Bell's original theorem only establishes a probabilistic Bell inequality.
However, the GHZ variant involves three qubits instead of two.}
GHZ variant \cite{BellGHZ,MerminBellGHZ,PhotonBellGHZ} show that
local hidden variable theories are incompatible with quantum mechanics.
In \appref{app:GHZ}, we review the GHZ variant.
In \appref{app:realism}, we then discuss why tensor networks and neural networks
are not local hidden variable theories [spoiler: \eqsref{eq:realism} and \eqref{eq:local realism} are not satisfied]
and can therefore bypass Bell's theorem (and its variants).
Invoking a superdeterminism \cite{superdeterminism} loophole \cite{bellLoopholes} is not necessary.
\subsection{Review: GHZ Variant of Bell's Theorem}
\label{app:GHZ}
Writing the 3-qubit GHZ state in the ZZZ, YYX, and XXX basis yields:
\begin{align}
\ket{\psi} &= \tfrac{1}{\sqrt{2}} \ket{+_z +_z +_z} + \tfrac{1}{\sqrt{2}} \ket{-_z -_z -_z} & ZZZ \text{ basis} \nonumber\\
&= \tfrac{1}{2} \ket{-_y -_y -_x} + \tfrac{1}{2} \ket{-_y +_y +_x}
+ \tfrac{1}{2} \ket{+_y -_y +_x} + \tfrac{1}{2} \ket{+_y +_y -_x} & YYX \text{ basis} \label{eq:GHZ}\\
&= \tfrac{1}{2} \ket{+_x +_x +_x} + \tfrac{1}{2} \ket{+_x -_x -_x}
+ \tfrac{1}{2} \ket{-_x +_x -_x} + \tfrac{1}{2} \ket{-_x -_x +_x} & XXX \text{ basis} \nonumber
\end{align}
$\ket{\psi}$ takes a similar permuted form in the YXY and XYY basis.
We use the following notation for the qubit states
\begin{gather}
\begin{aligned}
\ket{\pm_x} &= \tfrac{1}{\sqrt{2}} \ket{+_z} \pm \tfrac{1}{\sqrt{2}} \ket{-_z} \\
\ket{\pm_y} &= \tfrac{1}{\sqrt{2}} \ket{+_z} \pm \tfrac{i}{\sqrt{2}} \ket{-_z} \\
\end{aligned}
\end{gather}
which are eigenstates of the Pauli operators:
$\hat{\sigma}^\mu \ket{\pm_\mu} = \pm \ket{\pm_\mu}$.
In this Appendix, operators will always be denoted with a hat.
Note that the GHZ state has a fixed measurement outcome for the $\hat{Y}_1 \hat{Y}_2 \hat{X}_3$ and $\hat{X}_1 \hat{X}_2 \hat{X}_3$ operators
(along with permutations)
\begin{equation}
\begin{aligned}
\braket{\psi|\hat{Y}_1 \hat{Y}_2 \hat{X}_3|\psi} &= -1\\
\braket{\psi|\hat{X}_1 \hat{X}_2 \hat{X}_3|\psi} &= +1
\end{aligned} \label{eq:quantum exp}
\end{equation}
$\hat{X}_i$ denotes the $\hat{\sigma}^x$ Pauli operator acting on the $i^\text{th}$ qubit,
and similar for $\hat{Y}_i$ and $\hat{Z}_i$.
In a hidden variable theory\footnote{%
Other definitions of hidden variable theories exist in the literature \cite{AaronsonHiddenVariable}.
We are reviewing the definition as used in Bell's theorem.},
measurement outcomes only depend on a hidden classical variable $h$,
which has a probability density $P(h)$.
The measurement outcome due to measuring a qubit $i$ in the X, Y, or Z basis can then be written as:
\begin{align}
X_i(h) &= \pm1 \nonumber\\
Y_i(h) &= \pm1 \label{eq:realism}\\
Z_i(h) &= \pm1 \nonumber
\end{align}
Therefore, the quantum expectation values reduce to classical expectation values:
\begin{align}
\braket{\psi|\hat{X}_i|\psi} &= \int_h P(h) \, X_i(h) \nonumber\\
\braket{\psi|\hat{Y}_i|\psi} &= \int_h P(h) \, Y_i(h) \\
\braket{\psi|\hat{Z}_i|\psi} &= \int_h P(h) \, Z_i(h) \nonumber
\end{align}
In a \emph{local} hidden variable theory, measuring qubit $i$ can not affect the measurement outcome of qubit $j$
if qubit $j$ is far away and measured immediately after qubit $i$.
Therefore, for space-like separated measurements in a local hidden variable theory, the multi-point quantum expectation values
also reduce to analogous classical expectation values; in particular:
\begin{equation}
\begin{aligned}
\braket{\psi|\hat{Y}_1 \hat{Y}_2 \hat{X}_3|\psi} &= \int_h P(h) \, Y_1(h) Y_2(h) X_3(h) \\
\braket{\psi|\hat{X}_1 \hat{X}_2 \hat{X}_3|\psi} &= \int_h P(h) \, X_1(h) X_2(h) X_3(h)
\end{aligned} \label{eq:local realism}
\end{equation}
To obtain a contradiction with quantum theory, note that
\eqsref{eq:quantum exp},
\eqref{eq:realism},
\eqref{eq:local realism},
and permutations imply that
\begin{align}
Y_1(h) Y_2(h) X_3(h) &= Y_1(h) X_2(h) Y_3(h) = X_1(h) Y_2(h) Y_3(h) = -1 \label{eq:local YYZ} \\
X_1(h) X_2(h) X_3(h) &= +1 \label{eq:local XXX}
\end{align}
for all hidden variables $h$ (with $P(h) \neq 0$).
However, $X_1(h) X_2(h) X_3(h)$ can be rewritten using \eqnref{eq:local YYZ}
\begin{gather}
\begin{aligned}
X_1(h) X_2(h) X_3(h) &= [Y_1(h) Y_2(h) X_3(h)] \, [Y_1(h) X_2(h) Y_3(h)] \, [X_1(h) Y_2(h) Y_3(h)] \\
&= [-1] [-1] [-1] = -1
\end{aligned} \label{eq:local XXX'}
\end{gather}
since $Y_i(h)^2 = 1$.
The above \eqnref{eq:local XXX'} contradicts \eqnref{eq:local XXX}.
This shows that quantum mechanics can not be described by a local hidden variable theory.
\subsection{Tensor and Neural Networks}
\label{app:realism}
Tensor networks and neural networks might seem like hidden variable theories.
However, neither are local hidden variable theories
and are therefore not ruled out by Bell experiments.
The primary issue is that observables are not well-defined in terms of hidden variables, as in \eqnref{eq:realism},
in tensor or neural networks.
Furthermore, in the limit of infinite bond dimension or number of training parameters,
tensor and neural networks can simulate quantum mechanics exactly \cite{CiracTensorRev,StoudenmireLimits,PollmannNeural},
which implies that Bell's theorem must not be violated in that limit.
If an exact tensor or neural network (or other EmQM theory in an exact limit) were to simulate a Bell experiment,
then there would be no sense of local realism.
That this, after the observers measured their qubits,
the EmQM theory would be encoding a wavefunction that consists of a superposition of measurement outcomes,
as predicted by Schr\"{o}dinger's equation.
As emphasized in the many-worlds interpretation of quantum mechanics,
the wavefunction would then approximately consist of multiple ``branches'' that evolve independently in superposition
since they are extremely unlikely to interfere with each other\footnote{%
This assumes that the observers (including the surrounding environment) consist of many chaotic degrees of freedom, which is certainly true for human observers.}.
In an actual non-exact EmQM theory,
only a limited number of wavefunction branches could be retained.
Therefore, \emph{after} the measurements took place,
an EmQM theory would eventually be forced to randomly drop wavefunction branches
(especially if many Bell measurements are performed).
As long as only valid branches are retained and branches are dropped with the correct probability,
the Bell experiments would not notice deviations from quantum mechanics.
In a matrix product state (MPS) tensor network,
dropping a branch could be achieved by a special random (or pseudo-random) truncation of a bond dimension.
\subsubsection{GHZ Example}
For example, the 3-qubit GHZ state from \eqnref{eq:GHZ} can easily be encoded in a
matrix product state (MPS) tensor network in the ZZZ basis:
\begin{align}
\ket{\psi} &= \sum_{a_1,a_2,a_3,\alpha,\beta=0,1} A_{a_1,\alpha} \, B_{\alpha,a_2,\beta} \, C_{\beta,a_3}
\ket{a_1a_2a_3} \\
A_{a_1,\alpha} &= \delta_{a_1,\alpha} \nonumber\\
B_{\alpha,a_2,\beta} &= 2^{-1/2} \delta_{\alpha,a_2,\beta} \\
C_{\beta,a_3} &= \delta_{\beta,a_3} \nonumber
\end{align}
where $(a_1,a_2,a_3)$ index states in the Z basis ($\ket{0} = \ket{+_z}$ and $\ket{1} = \ket{-_z}$)
and $(\alpha,\beta)$ are virtual indices.
$\delta$ is the Kronecker delta, with $\delta_{a,b,c} = 1$ if $a=b=c$ else $\delta_{a,b,c} = 0$.
One could attempt to map the above tensor network to a hidden variable theory by defining the hidden variable as
the physical and virtual indices: $h=(a_1,a_2,a_3,\alpha,\beta)$.
Then the Z-basis measurements obey the analogs of Eqs.\,(\ref{eq:realism}--\ref{eq:local realism}) with
$P(h) = \frac{1}{2} \delta_{a_1,a_2,a_3,\alpha,\beta}$ and $Z_i(h) = (-1)^{a_i}$.
However, there is no choice of $X_i(h)$ and $Y_i(h)$ that is also consistent with Eqs.\,(\ref{eq:realism}--\ref{eq:local realism}).\footnote{%
One might alternatively imagine adding $\alpha'$ and $\beta'$ to the hidden variable in order to account for
virtual indices for a bra state.
However, this does not help turn the tensor network into a local hidden variable theory.}
Neural networks also fail to be a local hidden variable theory in a similar way.
\section{#1}}{\textbf{\textit{#1}} ---}}
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\begin{document}
\title{Testing Quantum Mechanics using Noisy Quantum Computers}
\author{Kevin Slagle}
\affiliation{Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA}
\affiliation{Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA}
\begin{abstract}
We outline a proposal to test quantum mechanics in the high-complexity regime using
noisy intermediate-scale quantum (NISQ) devices.
Our procedure involves simulating a non-Clifford random circuit, followed by its inverse,
and then checking that the resulting state is the same as the initial state.
We are motivated by the hypothesis that quantum mechanics is not fundamental,
but instead emerges from a theory with less computational power,
such as classical mechanics.
This emergent quantum mechanics (EmQM) hypothesis makes the prediction that quantum computers will not be capable of sufficiently complex quantum computations.
We show that quantum mechanics predicts that the fidelity of our procedure decays exponentially with circuit depth
(due to noise in NISQ devices),
while EmQM predicts that the fidelity will decay significantly more rapidly for sufficiently deep circuits,
which is the experimental signature that we propose to search for.
We estimate rough bounds for when possible signals of EmQM should be expected.
We find that useful experiments can be performed with only 80 qubits and gate infidelity $10^{-3}$,
while highly informative experiments should require only roughly 2000 qubits and gate infidelity $10^{-5}$.
\end{abstract}
\maketitle
Quantum mechanics has enjoyed roughly a century of predictive success.
Although there have been many high-precision tests of quantum mechanics \cite{experimentRev1992,testQM2000,photonTests},
it is possible that a new experimental regime could discover a break-down of quantum mechanics:
the high-complexity regime of highly entangled states that require deep quantum circuits to prepare.
The complexity of a wavefunction can be defined as the number of gates (\emph{e.g.} 2-qubit unitaries)
required to make the wavefunction from a product state
\cite{BrownSusskindComplexity,SusskindComplexity,BrandaoComplexity,NicoleComplexity}.
The high-complexity regime has remained relatively untested since decoherence and noise
make it tremendously difficult to probe this regime with high fidelity.
We would like to obtain strong evidence for whether or not quantum mechanics breaks down at large complexity scales
using a protocol that is appropriate for noisy intermediate-scale quantum (NISQ) \cite{NISQ} devices.
Currently, the leading tests of quantum mechanics in the high-complexity regime
were the recent quantum advantage experiments \cite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2,PhotonAdvantage}.
The superconducting qubit experiments \cite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2} performed up to 24 layers of random gates on up to 60 qubits,
followed by measurements in the computational basis.
The infidelity of their circuits can be predicted from, and was verified to be consistent with,
the infidelity of the individual gates in the low-complexity regime.
This helps verify quantum mechanics up to the regime of 60 qubits entangled by a depth $d=24$ circuit.
However, since the ideal measurement outcomes of their random circuits can not be efficiently calculated on classical computers,
it is impractical to extend this methodology to deeper circuits with more qubits.
To overcome this, we propose to apply the inverse of the quantum circuit before taking measurements so that the ideal measurement outcome is trivial to calculate.
Our primary motivation is to obtain evidence for or against the hypothesis that
quantum mechanics emerges from a local classical theory,
which we define as a theory that can be formalized as a (possibly continuous) cellular automaton \cite{WolframNKS,Hooft,HooftOntology}.
This possibility is often referred to as emergent quantum mechanics (EmQM) \cite{EmQM},
for which there have been several preliminary proposals \cite{Hooft,HooftBook,Adler,VanchurinEntropic,VanchurinWorld,VanchurinNeural}.
We are also motivated by recent progress in tensor network \cite{OrusTN,IsometricTN,2dDMRG,canonicalPEPS,tensorMC,SlagleTensor}
and neural network \cite{MendlNeural,SchmittNeural2D,PollmannNeural,JonssonNN} algorithms
and the possibility that future algorithms might yield a model of EmQM.
Note that many local classical theories are not local hidden variable theories,
which implies that they are not ruled out by Bell experiments;
see \appref{app:Bell}{E} for an explanation.
Since local classical theories can be efficiently simulated on classical computers
while classical computers (very likely) can not efficiently simulate quantum mechanics \cite{PostIQP,AaronsonSupremacy,CharacterizingSupremacy},
models of EmQM must show discrepancies from quantum theory
at sufficiently high complexity \cite{HooftComputers,HooftBook}.
However, if quantum mechanics emerges at \emph{e.g.} the Planck scale,
then an EmQM theory could effectively have at its disposal one CPU core per Planck volume operating at a Planck frequency.
That is an incredible amount of computational power.
As such, it is conceivable that a local classical theory
could exhibit EmQM well enough to be in agreement with previous experiments,
but not the experiment we propose.
\arxiv{\newpage}
\mysec{Proposal}
Our proposed protocol, which we summarize in \figref{fig:circuit}, is to repeat the following:
\begin{enumerate}
\item initialize the zero state $\ket{0}$ on $n$ qubits
\item apply $d$ layers of random gates $U_1 \cdots U_d$,
with depth $d$ randomly chosen between 0 and $d_\text{max}$
\item apply the inverse $U_d^\dagger \cdots U_1^\dagger$
\item measure in the computational basis and check if the final state is the zero state $\ket{0}$
\end{enumerate}
The max depth $d_\text{max}$ should be chosen such that $d_\text{max} \gg n^{1/D}$
for a circuit with $D$-dimensional connectivity
(\emph{e.g.} $d_\text{max} \approx 10 \, n^{1/D}$).
Each layer of gates $U_t$ is a product of 2-qubit gates,
which are independently and randomly chosen for each sample and layer.
The random 2-qubit gates can be chosen from any distribution
(of at least two gates\footnote{%
Using random instead of fixed gates reduces fluctuations in the exponential fidelity decay
that result from coherent errors \cite{randomizedCompiling}.}),
as long as the entire forward circuit $U_1 \cdots U_d$ is expected to be difficult to simulate classically.
We require that the random distribution for each layer of gates repeats with a small period $p$;
\emph{i.e.} $U_t$ and $U_{t+p}$ are independently chosen from the same probability distribution.
The depth-dependent fidelity $F_d$ is the fraction of depth-$d$ samples with a measured final state $\ket{0}$.
The protocol is complete once the fidelities $F_d$ have been measured to the desired statistical precision.
To make the best use of each sample,
each depth $d$ can be sampled with probability $\sim F_d^{-2}$.
The circuit depth $d$ is chosen randomly for each sample in order to mitigate
the effect of slow drifts in the gate fidelity between samples.
(See \appref{app:drift}{C} for more on drift.)
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{circuit.pdf} \\
\caption{%
Our protocol: repeatedly apply a random circuit of a random depth $d$ to $n$ qubits, followed by its inverse,
and then measure in the computational basis.
For each random sample, the 2-qubit gates $u_{t,i}$ are chosen randomly.
The depth-dependent fidelity $F_d$ is the fraction of depth-$d$ samples with final state $\ket{0}$.
Although we depict a circuit with one-dimensional connectivity for simplicity,
higher-dimensional connectivity is preferable.
}\label{fig:circuit}
\end{figure}
We show that quantum mechanics predicts that the fidelity should decay exponentially $F_d \sim e^{-\lambda d}$
with the circuit depth $d$ (until saturation near $F_\infty \sim 2^{-n}$)
for Markovian noise models.
In contrast, if quantum mechanics emerges from a theory with significantly less computational power,
such as classical mechanics,
and if $\lambda$ is sufficiently small
to be in the quantum advantage regime \cite{StoudenmireLimits},
then we argue that the fidelity should start to decay significantly faster after a sufficiently large depth $d_*$,
as depicted in \figref{fig:decay}.
Our protocol is similar to randomized benchmarking \cite{noiseEstimation,tdesignRB,twirlRB,twirlRBlong,GateRB,KnillRB,RBGeneral,randomCircuitSampling}
and uses a similar mathematical foundation.
One difference is that we allow arbitrary non-Clifford gates so that the quantum advantage regime can be reached,
while randomized benchmarking typically focuses on Clifford gates so that the inverse unitary can be a low-depth circuit.
The low-depth inverse circuit also differs from our proposal since we instead apply a long sequence of inverted unitaries in the reverse order
(as in a Loschmidt echo \cite{LoschmidtEcho}),
which is typically avoided in randomized benchmarking since it can hide errors \cite{practicalVerification}.
Another difference is that randomized benchmarking typically studies a small number of qubits,
while we focus on many $n \gg 1$ qubits.
Conveniently, we find that a larger number of qubits helps make the exponential fidelity decay cleaner.
\begin{figure}
\centering
\includegraphics[width=.7\columnwidth]{decay} \\
\caption{%
Exponential fidelity decay $F_d \sim e^{-\lambda d}$ with circuit depth $d$ predicted by quantum mechanics (QM) (blue)
vs a schematic sketch of the emergent quantum mechanics (EmQM) expectation (red).
EmQM agrees with QM up to a critical depth $d_*$,
after which the EmQM fidelity must start decaying significantly faster
(assuming sufficiently large $n$ and small $\lambda$).
}\label{fig:decay}
\end{figure}
\arxiv{\newpage}
\mysec{\texorpdfstring{E\lowercase{m}QM}{EmQM} Expectation}
\label{sec:EmQM}
We expect that an EmQM theory would not have enough computational power
to accurately perform the forward circuit
(green in \figref{fig:circuit}) of our protocol
for sufficiently many qubits $n$ and large depth $d$.
In particular, we assume that if the zero state $\ket{0}$ is measured at the end of the protocol,
then an EmQM would have had to simulate the circuit well enough such that the EmQM could also
randomly sample bit strings from the forward circuit
(\emph{i.e.} bit strings $b$ from the probability distribution $P(b) = |\braket{b|U_d \cdots U_1|0}|^2$).
This is the same sampling task considered in recent quantum advantage experiments \cite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2}
and is expected to require an exponential amount of CPU time for classical algorithms \cite{circuitSupremacy,MovassaghSupremacy,BoulandSupremacy,AaronsonXEB,optimalSA}.
Building on these assumptions, we can roughly predict when our protocol might see signals of EmQM.
Consider the brute-force Schr\"{o}dinger algorithm (SA) \cite{SchrodingerStyle},
which involves simply applying unitary matrices to a $2^n$-component wavefunction.
This algorithm can simulate a depth $d$ circuit of $n$ qubits
in $N_\text{CPU}^\text{SA} \approx n d \, 2^n \sim 2^n$ CPU cycles.
We are not aware of any faster algorithms that could sample bit strings from
random $D$-dimensional non-Clifford circuits with large depth $d \gg n^{1/D} \gg 1$ and
large gate fidelity\footnote{%
Gate fidelity above the error-correction threshold $f \approx 0.99$ \cite{RaussendorfThreshold,RaussendorfThresholdMCQB}
is presumably sufficient}
(even if only a small circuit fidelity $F \sim 10^{-3}$ is required)
\cite{circuitSupremacy,ZuchongzhiAdvantage,AaronsonSupremacy,optimalSA}.
Suppose that a quantum computer performs such a circuit within length and times scales
$L_\text{QPU}$ and $T_\text{QPU}$.
Further assume that quantum mechanics emerges at length and time scales $L_\text{EmQM}$ and $T_\text{EmQM}$.
Being a local classical theory, an EmQM theory can be simulated in real time using roughly
one CPU core per $L_\text{EmQM}^{D_\text{EmQM}}$ volume
operating at frequency $T_\text{EmQM}^{-1}$.
$D_\text{EmQM}$ is the number of spatial dimensions,
which could be greater than three if there are extra spatial dimensions.
Therefore, an algorithm simulating the EmQM theory could simulate the circuit with
$N_\text{CPU}^\text{EmQM} \sim \frac{T_\text{QPU}}{T_\text{EmQM}} \left(\frac{L_\text{QPU}}{L_\text{EmQM}}\right)^{D_\text{EmQM}}$
CPU cycles.
By applying our expectation that no algorithm is faster than the Schr\"{o}dinger algorithm,
we can bound $N_\text{CPU}^\text{SA} \lesssim N_\text{CPU}^\text{EmQM}$
and then solve for $n$ in $2^n \sim N_\text{CPU}^\text{SA} \lesssim N_\text{CPU}^\text{EmQM}$
to estimate the largest number of qubits $n_*$ that an EmQM theory could accurately handle at large depth:
\begin{align}
n_* &\lesssim \log_2 \frac{T_\text{QPU}}{T_\text{EmQM}} + D_\text{EmQM} \log_2 \frac{L_\text{QPU}}{L_\text{EmQM}} \label{eq:n crit}\\
&\lesssim 130 + \log_2 \frac{T_\text{P}}{T_\text{EmQM}} + D_\text{EmQM} \left(110 + \log_2 \frac{L_\text{P}}{L_\text{EmQM}} \right) \nonumber
\end{align}
Therefore when $n \geq n_*$, there should be a critical depth $d_*$
after which any EmQM theory should deviate from quantum mechanics predictions (as in \figref{fig:decay}).
As an example, we express the second line relative to Planck length $L_\text{P}$ and time $T_\text{P}$ scales
with $L_\text{QPU} \sim 1$\,centimeter and $T_\text{QPU} \sim 10^{-3}$\,seconds.
Note that \eqnref{eq:n crit} is remarkably predictive because huge changes in these unknown scales
only modestly affect our bound on $n_*$ due to the logarithmic dependence.
In particular, only $n_* \lesssim 2000$ qubits may be needed to significantly rule out the EmQM hypothesis
(if we assume $D_\text{EmQM} \leq 6$, $T_\text{EmQM} \geq 10^{-50} T_\text{P}$, and $L_\text{EmQM} \geq 10^{-50} L_\text{P}$).\footnote{%
$L_\text{QPU}$ and $T_\text{QPU}$ do not significantly affect this estimate.
For example, increasing these values to $T_\text{QPU} = 100$\,years and $L_\text{QPU} = 100$\,light-years
only increases our estimated bound on $n_*$ by $42 + 66 D_\text{EmQM}$.}
In order to bound $n_*$ using the Schr\"{o}dinger algorithm argument,
we had to assume $d \gg n^{1/D} \gg 1$.
Therefore we can roughly bound the critical depth $d_* \lesssim n_*^{1/D}$.
In an attempt to safely exceed this bound,
we consider max circuit depths $d_\text{max} \approx 10\, n^{1/D}$
for which it seems likely that $d_\text{max} > d_*$.
With $N_\text{s} = 10^8$ samples,
the fidelity can be measured down to
$F_\text{min} \approx N_\text{s}^{-1/2} \epsilon_\text{r}^{-1} \approx 10^{-3}$
with a relative error of $\epsilon_\text{r} \approx 10\%$.
We will show that if each 2-qubit gate has an entanglement fidelity\bibfoot{Ffoot:gateFidelity} $f$,
then quantum mechanics predicts that our protocol has fidelity $F_d \sim f^{n d}$,
where $n d$ is the approximate number of 2-qubit gates.
Therefore, statistically significant data can be obtained for
$n d \lesssim \frac{\ln F_\text{min}}{\ln f} \approx \frac{\ln F^{-1}_\text{min}}{1-f}$.
For $D=2$ dimensional circuits with gate infidelity $1-f = 10^{-3}$ and $F_\text{min} = 10^{-3}$,
statistical significance requires $n d \lesssim 7000$,
\emph{i.e.} a maximum of $n \approx 80$ if $d_\text{max} \approx 10 n^{1/D} \approx 90$
(which is significantly larger than the $n\approx60$ and $d\approx24$ used in \refscite{GoogleSupremacy,ZuchongzhiAdvantage,ZuchongzhiAdvantage2}).
With gate infidelity $1-f = 0.8 \times 10^{-5}$ (which may require error correction techniques \cite{GottesmanErrorCorrection})
and $F_\text{min} = 10^{-3}$,
statistical significance requires $n d \lesssim 8.6 \times 10^5$,
\emph{i.e.} a maximum of $n \approx 2000$ if $d_\text{max} \approx 10 n^{1/D} \approx 440$,
which could rule out the EmQM hypothesis with higher confidence.
\mysec{Quantum Mechanics Prediction}
\label{sec:decay}
We now argue that quantum mechanics predicts that the fidelity $F_d$ decays exponentially with circuit depth $d$
under physically reasonable noise models.
Intuitively, each gate $u$ has a probability $p_u$ of making an error,
resulting in a gate entanglement fidelity $f_u \approx 1-p_u$.
Intuitively, the fidelity $F_d$ is then roughly the probability $f^{n d}$ that none of the $\approx n d$ two-qubit gates made an error:
\begin{equation}
F_d \sim \widetilde{F}_0 \, e^{-\lambda d} \label{eq:F}
\end{equation}
where $\lambda \sim - n \ln f$ and $f$ is an average gate entanglement fidelity
(until the fidelity eventually saturates near $F_\infty \sim 2^{-n}$).
In the experiment, $\widetilde{F}_0$ and $\lambda$ are fitting parameters.
We substantiate this intuition below.
In \appref{app:twirl}{A}, we provide additional evidence
that the fidelity should decay exponentially (according to quantum mechanics)
using a 2-design twirling argument.
As a first approximation, we model the fidelity using gate-dependent but time-independent noise:
\begin{equation}
F_d = \Exp_{U_1, \ldots, U_d}
\tr\!\big[\rho^\text{fin} \, (\Phi_{U_1^\dagger} \!\circ\cdots\circ \Phi_{U_d^\dagger} \!\circ \Phi_{U_d} \!\circ\cdots\circ \Phi_{U_1}) (\rho^\text{init})\big] \label{eq:F model}
\end{equation}
The initial state is modeled by a density matrix $\rho^\text{init}$.
Measuring if the final state is the zero state
is modeled by the quantum channel $\rho \to \tr(\rho^\text{fin} \, \rho)$.
$\rho^\text{init}$ and $\rho^\text{fin}$ are close to the $n$-qubit zero state $\ket{0} \bra{0}$,
up to some infidelity resulting from state preparation and measurement (SPAM) errors.
$\tExp_{U_1, \ldots, U_d}$ averages over the allowed layers of random gates.
The quantum channel $\Phi_{U_t}$ models the noisy quantum computer implementation
for a particular layer of random gates $U_t$.
As depicted in \figref{fig:circuit},
each layer of gates $U_t$ is a product of 2-qubit gates $U_t = \prod_i u_{t,i}$,
where $i$ labels a pair of qubits.
We similarly model the quantum channel $\Phi_{U_t}$ as
a composition of noisy quantum channels:
$\Phi_{U_t} = \circ_i \Phi_{u_{t,i}}$.
Each 2-qubit channel $\Phi_u$ can be decomposed as $\Phi_u(\rho) = \mathcal{N}_u(u \rho u^\dagger)$,
where the noise channel $\mathcal{N}_u$ depends on the unitary $u$ and the two qubits that it acts on.
We expand the noise channel for each gate as\footnote{%
$n$-qubit noise channels could alternatively be considered to model cross-talk \cite{crosstalk} or cosmic ray \cite{GoogleCosmicRay} errors
with minimal changes to our arguments.}:
\begin{equation}
\mathcal{N}_u(\rho) = \sum_{\alpha,\beta \in \{0,1,2,3\}^2} \chi^{(u)}_{\alpha\beta} \, \sigma_\alpha \, \rho \, \sigma_\beta
\end{equation}
where $\chi^{(u)}$ is a positive semi-definite matrix and
$\sigma_\alpha \in \{I, X, Y, Z\}^{\otimes 2}$ is a $2$-qubit Pauli operator.
The gate entanglement fidelity\bibfoot{Ffoot:gateFidelity}
$f_u = \chi^{(u)}_{\mathbf{00}}$ plays an important role
since all other terms in the expansion result in an error and very small fidelity.
For large gate fidelities ($f_u \approx 1$),
errors (for which $\alpha\neq{\mathbf 0}$ or $\beta\neq{\mathbf 0}$) are suppressed by small
$|\chi^{(u)}_{\alpha\beta \neq \mathbf{00}}| \leq \sqrt{1-f_u}$.\bibfoot{Cfoot:chiBound}
Therefore multiple errors will typically be separated by many gates,
which will spread out \cite{NahumSpreading,KeyserlingkSpreading,SusskindSpreading,approximateTDesign}
any non-identity $\sigma_\alpha$ or $\sigma_\beta$ factor into a highly non-local operator.
After spreading out, the contribution to the fidelity in \eqnref{eq:F model}
resulting from one or more errors is very small $O(2^{-n})$.\bibfoot{Dfoot:errorFidelity}
For a large number of qubits $n$,
any error will typically result in a negligible contribution to the fidelity.
Therefore, the fidelity $F_d$ in \eqnref{eq:F model} is roughly the fraction of times that no error occurs.
$F_0 = \tr \rho^\text{fin} \rho^\text{init}$ is the probability that a SPAM error occurred.
The probability that no error occurs during the forward and backward
2-qubit gates $u_{t,i}$ and $u_{t,i}^\dagger$ is
\begin{equation}
f_i^2 = \Exp_{u_{t,i}} f_{u_{t,i}} f_{u_{t,i}^\dagger} \label{eq:fi}
\end{equation}
where $\Exp_{u_{t,i}}$ averages over the allowed 2-qubit gates.
Therefore, the probability that no errors occur over the entire circuit of $2n_\text{g} d/p$ gates is roughly
\begin{align}
F_d &\sim F_0 f^{2 n_\text{g} d/p} &\text{with}&&
f^{2n_\text{g}} &= \prod_{i=1}^{n_\text{g}} f_i^2 \label{eq:f}
\end{align}
where $f$ is an average gate entanglement fidelity
and $n_\text{g}$ is the number of gates in a period (\emph{e.g.} $n_\text{g}=4$ in \figref{fig:circuit}).
We thus obtain \eqnref{eq:F} with
\begin{align}
\widetilde{F}_0 &\approx F_0 = \tr \rho^\text{fin} \rho^\text{init} &
\lambda &\approx - \frac{2n_\text{g}}{p} \ln f \label{eq:lambda}
\end{align}
where $\lambda$ reduces to $\lambda \sim - n \ln f$ when $n_\text{g} \sim p n/2$, as is typically the case.
Although \eqnref{eq:lambda} is only an approximate expression for $\widetilde{F}_0$ and $\lambda$ since multiple errors can sometimes cancel out
(especially if they are spatially close or at mirrored positions in the circuit),
we numerically find that \eqnref{eq:lambda} is a very good approximation for
many qubits $n \gg 1$ and large gate fidelity $f \approx 1$.
As an example, in \figref{fig:simulation} we show a simulation for a $5\times5$ grid of $n=25$ qubits
using roughly Sycamore/Zuchongzhi-like \cite{GoogleSupremacy,ZuchongzhiAdvantage} gates with gate-dependent errors
and average gate entanglement fidelity $f \approx 0.993$ using the Markovian model \eqnref{eq:F model}.
(See \appref{app:numerics}{B} for details.)
\begin{figure}
\centering
\includegraphics[width=.85\columnwidth]{simulation} \\
\caption{%
Simulation data (black dots) showing the quantum mechanics prediction of
the exponential fidelity $F_d$ decay
for a $5\times5$ grid of $n=25$ noisy qubits.
Also shown is a fit (blue line) to an exponential decay [\eqnref{eq:F}]
and the rough predictions (dotted orange line) from \eqnref{eq:lambda}.
}\label{fig:simulation}
\end{figure}
\mysec{Example: \texorpdfstring{E\lowercase{m}QM}{EmQM} from MPS}
\label{sec:MPS}
It is illuminating to consider the time evolution of matrix product states (MPS) \cite{MPS} with fixed bond dimension
as an (incomplete) toy model of emergent quantum mechanics (EmQM) in one spatial dimension.
We do not consider MPS to be a legitimate model of EmQM because
it is currently not known how to generalize MPS algorithms to a three-dimensional local classical theory
that could be compatible with previous experiments (see \appref{app:MPS}{D});
although there has been remarkable related algorithmic progress \cite{IsometricTN,2dDMRG,canonicalPEPS,ParallelDMRG}.
In \figref{fig:MPS}, we show a MPS simulation
of our protocol with noisy gates (similar to \figref{fig:simulation})
for a chain of 24 qubits.
(See \appref{app:MPS simulation}{B} for details.)
During the MPS simulation, bond dimensions are truncated to $\chi \leq 2^{10}$,
which limits the amount of R\'enyi-0 entanglement entropy \cite{RenyiMPS} across any cut of the chain.
The maximum entanglement is reached at the critical depth $d_*$,
after which the fidelity decay is dominated by the truncation error \cite{StoudenmireLimits} of the MPS algorithm (instead of the gate infidelity)
and the fidelity $F_d$ begins to decay at a faster exponential rate.
This is precisely the EmQM signature depicted in \figref{fig:decay}.
\begin{figure}
\centering
\includegraphics[width=.7\columnwidth]{MPS} \\
\caption{%
Simulation data (red dots) showing the fidelity $F_d$ decay of a matrix product state (MPS) toy model of EmQM
with bond dimension $\chi = 2^{10}$ for a chain of 24 noisy qubits.
Similar to \figref{fig:decay},
the toy EmQM theory agrees with the quantum mechanics (QM) prediction (blue circles)
until a critical depth $d_* = 12$,
after-which the fidelity decays at a significantly faster exponential rate.
Lines are fits to these two regimes.
}\label{fig:MPS}
\end{figure}
\mysec{Discussion}
If an exponential fidelity decay (blue in \figref{fig:decay})
is observed for high-depth circuits on many qubits,
then our arguments \arxiv{in \secref{sec:EmQM} }will significantly rule out possible theories of emergent quantum mechanics (EmQM),
including \refscite{Hooft,HooftBook,Adler,VanchurinEntropic,VanchurinWorld,VanchurinNeural}.
Since our proposed experiment requires making several assumptions,
once more fault tolerant quantum computers are available,
it will be useful to further verify quantum mechanics
using verification protocols \cite{verificationOverview,verification,FalsifiableQM}
or Shor's algorithm \cite{Shor,Shor2048}.
If a signal suggestive of EmQM (red in \figref{fig:decay}) is experimentally observed,
then future work will be required
to either determine what the true underlying theory is
or rule out explanations consistent with quantum mechanics
(\emph{e.g.} non-Markovian noise or state leakage \cite{limitsRB,correlationsRB}).
If quantum mechanics does emerge from a local classical theory,
then the computational power of quantum computers could be severely limited \cite{HooftComputers,HooftBook}.
For example, {\bf BQP} problems would only be tractable for limited problem sizes.
Nevertheless, quantum computers would still be incredibly useful technology,
particularly as a device to probe new fundamental physics.
However, it is possible that violations of quantum mechanics could make quantum computers more powerful for certain tasks.
For example, if it is possible to violate the probabilistic Born's rule such that postselection could be performed,
then quantum computers may be able to efficiently solve problems in the {\bf PostBQP} = {\bf PP} \cite{PostBQP,PostIQP} complexity class up to a bounded problem size.
This could be possible if the EmQM prefers to drop branches of the wavefunction with a complexity that is too high\footnote{%
To exploit this, consider a quantum computer in a superposition of orthogonal states $\ket{\psi_1} + \ket{\psi_2}$
consisting of many qubits ($n > n_*$) but complexity below the critical complexity.
Then maybe one could postselect the $\ket{\psi_1}$ state by applying a controlled high-depth random unitary $U$
to only the $\ket{\psi_2}$ component:
\unexpanded{$\ket{\psi_1} + \ket{\psi_2} \underset{\text{controlled-} U}{\overset{\text{apply}}{\to}}
\ket{\psi_1} + U \ket{\psi_2} \underset{\text{drops } U \ket{\psi_2}}{\overset{\text{EmQM}}{\to}} \ket{\psi_1}$.}}
(which would be a possible mechanism behind the EmQM signature of rapid fidelity decay).
\begin{acknowledgments}
We thank John Preskill, Alex Dalzell, Andru Gheorghiu, Ulysse Chabaud, Sean Carroll, Spiros Michalakis, Vitaly Vanchurin, Leonid Pryadko, Xie Chen, and Natalie Klco
for helpful conversations.
K.S. is supported by the Walter Burke Institute for Theoretical Physics at Caltech; and
the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Science Center.
\end{acknowledgments}
| {
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It never ceases to amaze me how unexpected, mysterious yet gracious are God's ways of bringing about his purposes. We are celebrating over twenty five years of lived experience of the rich charism with which God has gifted us as Missionaries of God's Love. So above all we want to honour the creative work of the Spirit, drawing, calling, shaping us in a particular gracing for the work of the kingdom of God, in and for our world today.
In November 1985 God let us know the desire of his heart to call women into the charismatic shaping of religious life that was an embryonic stirring in Fr. Ken's heart. My own call at that time while on retreat at Douglas Park was like a clarion call into the night of God's new creativity, shrouded in mystery, and not without anguish. It seemed to me that it was by stepping-stones of darkness that God chose to bring forth the beautiful work and gracing of the MGL Sisters.
For twenty one years I had very happily and fruitfully lived my life as a Franciscan Missionary of Mary. God however had other plans for me. While at the Australian National University studying anthropology, I became involved with the St. Francis of Assisi youth group, a diocesan youth group in Canberra that had become charismatic. Fr. Ken Barker was the director of the youth ministry.
It was through my contact with this group that I personally experienced the "release of the Holy Spirit," and everything that God had already been doing in my life and prayer took off. It was a deep and intimate coming of the Holy Spirit in a new and totally loving way. This experience of grace in 1984 was the beginning (unknown to me) of my own call for the sisters. I have no doubt that this movement of the Holy Spirit in us was the first and sustaining impetus for the emergence of Missionaries of God's Love. Young people from the St. Francis Youth Group were the first members of MGL.
In my years in MGL, in Canberra, Adelaide, Darwin, PNG outreach, and now back in Canberra, it seems that I am called to be a 'frontier' person, to be with Jesus on the faith and culture boundaries of our society. The radar of the Spirit is out, picking up the invitation to the Spirit's creativity to draw people especially to know God's personal claim of love for them. Through spirituality work I am able to help people to tap into the contemplative dimension within them, to become more aware of just how God comes to them, and to grow in freedom in choosing to respond to God.
This has meant designing opportunities especially for those who find themselves in some way "on the margins,' mothers of young families, indigenous people, youth, seniors, men, defence force personnel – anyone in fact, with a hunger for God.
From 2009 to 2013 I was called back into leadership of the Sisters. A daunting privilege, one of attending to the gentle, strong and profound stirrings of the Spirit within us. We never know what God has in store or where he may lead us, but it is certainly not without adventure and joy. | {
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# Copyright & Information
Gideon's Art
First published in 1971
Copyright: John Creasey Literary Management Ltd.; House of Stratus 1971-2012
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior permission of the publisher. Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages.
The right of John Creaseyto be identified as the author of this work has been asserted.
This edition published in 2012 by House of Stratus, an imprint of
Stratus Books Ltd., Lisandra House, Fore Street, Looe,
Cornwall, PL13 1AD, UK.
Typeset by House of Stratus.
A catalogue record for this book is available from the British Library and the Library of Congress.
| **EAN** | | **ISBN** | | **Edition** |
---|---|---|---|---|---|---
| 0755123891 | | 9780755123896 | | Print |
| 0755125681 | | 9780755125685 | | Pdf |
| 075512569X | | 9780755125692 | | Kindle/Mobi |
| 0755125703 | | 9780755125708 | | Epub |
This is a fictional work and all characters are drawn from the author's imagination.
Any resemblance or similarities to persons either living or dead are entirely coincidental.
www.houseofstratus.com
# About the Author
John Creasey – Master Storyteller - was born in Surrey, England in 1908 into a poor family in which there were nine children, John Creasey grew up to be a true master story teller and international sensation. His more than 600 crime, mystery and thriller titles have now sold 80 million copies in 25 languages. These include many popular series such as Gideon of Scotland Yard, The Toff, Dr Palfrey and The Baron.
Creasy wrote under many pseudonyms, explaining that booksellers had complained he totally dominated the 'C' section in stores. They included:
Gordon Ashe, M E Cooke, Norman Deane, Robert Caine Frazer, Patrick Gill, Michael Halliday, Charles Hogarth, Brian Hope, Colin Hughes, Kyle Hunt, Abel Mann, Peter Manton, J J Marric, Richard Martin, Rodney Mattheson, Anthony Morton and Jeremy York.
Never one to sit still, Creasey had a strong social conscience, and stood for Parliament several times, along with founding the One Party Alliance which promoted the idea of government by a coalition of the best minds from across the political spectrum.
He also founded the British Crime Writers' Association, which to this day celebrates outstanding crime writing. The Mystery Writers of America bestowed upon him the Edgar Award for best novel and then in 1969 the ultimate Grand Master Award. John Creasey's stories are as compelling today as ever.
# Acknowledgment
This book is dedicated to Lynn and Scoop, because they believe so much in art; and particularly because Lynn spent so much time helping me to clean the painting which is part of the plot.
# Author's Note
While the general details of the galleries and museums in this book are accurate, I have in some cases taken liberties by introducing rooms which do not in fact exist, official positions which are imaginary, and, of course, many characters - all of whom are fictional and have no relation to any living persons, hard though I try to make them appear as flesh and blood.
I am most grateful to the officials at the various galleries mentioned in the book for their friendly help and guidance.
# Painted Into A Corner
There was a second of stunned silence before Robin Kell's hand flashed to and from his jacket and the click of his knife sounded loud. He'd had enough. If the collector wouldn't agree to the exchange, then the girl, Christine, must die. Falconer tried to rush forward to protect his daughter, but the knife moved swift as light toward him, and into his belly. He felt a searing pain and staggered to one side, saw the knife flash again as Kell turned to use it on Christine.
Falconer's last memory before losing consciousness was of Gideon and his men storming the premises....
# 1: The Method
"So it's worth a quarter of a million," Jenkins said.
"Pounds," remarked Slater.
"And I've a market," de Courvier stated, in his over precise voice. "We will get twenty-five thousand pounds each."
"I thought you said it was worth a quarter of a million," said Slater.
"It is. That's why it's worth twenty-five thousand pounds for each of us," retorted de Courvier. "In dollars, if you want to go for a nice long holiday out of this country."
"You mean you can sell it to the States?" asked Slater, scepticism sharp in his tone. "That's a new one on me."
"I did not say that I could sell it to the States," replied de Courvier. I said I had a market. We can collect twenty-five thousand pounds apiece, just for stealing it. My buyer will make a big profit, but does that matter?"
In a long pause, the three men looked at one another, Jenkins with a cigarette drooping from his protruding lower lip, Slater with a half-smoked cigar jutting from his thick and fleshy mouth, de Courvier prim, almost dandified, sitting well back in a winged Regency armchair with badly frayed arms. From outside came the noise of a pneumatic drill, unpleasantly close. Thin curtains, drawn, concealed the men and the tiny room from passers-by. Several women and some children went past, then a man, his steps slow and deliberate, and Slater looked up, suddenly and as if instinctively on the alert.
The man's steps faded, and Slater turned to face the others again.
"Where would we have to deliver?" asked Jenkins.
"I would take it to the go-betweens and they will hold it. We will have it only for an hour or two," de Courvier said.
"Okay, but where will we keep it in that hour or two?" Jenkins frowned.
De Courvier shrugged his shoulders. "Anywhere we say."
"Here?"
"Why don't we decide how we're going to get it first?" Slater sounded impatient.
"And whether," put in Jenkins. "That's the question. Are we or aren't we? If you ask me, it's a tricky one."
"It's simple as kissing one's hand," de Courvier stated flatly.
"Don't you believe it, old pal." Jenkins moved from a small, armless chair rather like a car seat and crossed to a corner of the room in which there was a sink, a wooden draining board, some shelves and cups and saucers, instant coffee, and a half-empty packet of tea bags. He filled an electric kettle from the one tap and plugged it into a socket on the wainscoting, straightening up with a grunt. "Those museums are never easy."
"You don't have to blow a safe or pick a lock," de Courvier said. "You just go and take the picture."
"With how many looking on?" jeered Jenkins.
"We cause a distraction," de Courvier said. "The crowd goes to see what is going on in the next salon; even the attendants take a look. We'll have plenty of time to get the painting."
"It wouldn't work," Jenkins said.
"One of us could throw a fit" Slater suggested. "Or—"
"That old trick," Jenkins said. "Forget it. Do they have closed circuit television at the National Gallery?"
"No," said de Courvier. "Of that I am positive."
"If you don't like my way, why don't you suggest something better?" asked Slater nastily.
The kettle began to sing as Jenkins took some cups and put them on a tray. "There's a lot of lolly to spare. Three times twenty-five is seventy-five thousand, and if it's worth a quarter of a million there's a hundred and seventy-five thousand going to someone who's sitting pretty."
De Courvier leaned forward. "You know as well as I do that we couldn't sell it at its full value. We couldn't sell it at all if there wasn't an agent standing by who will take a great deal of risk. He knows he can get a hundred-thousand pounds. If he can get more, that's his good fortune.''
After a long pause, Jenkins said, "Okay. I'll settle for twenty-five thou." The cigarette, now little more than a stub, still dangled from his lower lip. "But it's no use to me if I'm inside. Do you know what we'd get for this one?"
"Three years," Slater said.
"For taking an Old Master out of the National Gallery? That's worse than murder! It would be more like ten years, if you ask me."
"What's the difference?" demanded Slater. "We aren't going to be caught."
"We'll be caught if we do it your way," Jenkins said. The kettle was nearly boiling and he put teabags in a brown earthenware pot and, timing the moment of true boil perfectly, poured in the water. Then he put the kettle on the draining board. "You both take sugar?" he asked.
"Not for me," Slater said, slapping his rounded belly. "I'm putting it on again."
"Courvy?" asked Jenkins.
"Neither sugar nor milk, thank you," de Courvier answered.
Jenkins poured out the tea, moved the cups to one side to make room for a tin of biscuits, then brought the tray over to his companions. They each took a cup. De Courvier bit firmly into a chocolate biscuit, while Jenkins put a whole one into his mouth and took the cigarette out, almost in the same movement.
"So what are we going to do?" asked Slater, at last. He was a short, thickset man, his lack of height apparent even when he sat.
"I do not think we need to discuss it further," said de Courvier, still sitting prim and upright. "Obviously, I shall have to find someone else to do this work for me."
"What the hell do you say that for?" asked Slater.
"This is not a matter where we can have indecision or difference of opinion," replied de Courvier sulkily. "Jenkins does not want to have any part in this. He would no doubt come in if you and I decided to work together but he would have no confidence, and that would lead to disaster. The wise thing is to forget it."
"That's great," Slater growled. "That's right up my street, that is. Twenty-five thousand quid put under my nose and then snatched away before I can get it. That's the kind of deal I really like, I don't think."
After a pause, Jenkins said: "Show me a way to do it with a good chance of getting the picture and I'm with you. But no fits, and no crowd scenes." He drank the remainder of his tea in a gulp and put the cup back on the tray. "You know me. The coolest nerve in the business."
"It's why I came to you," de Courvier said. "But if you're half-hearted, it is no use."
"Just show me the method," Jenkins demanded.
Slater moved his head slowly, revealing how thick his neck was, how it bulged over the ill-fitting coat, his eyes - sharp grey eyes - darting from side to side.
"You got any ideas?" he asked de Courvier.
"There is a way," de Courvier replied slowly, "a very good way. We need three to work inside, each for a few seconds. Once the picture has been removed, it is only a matter of getting it out of the gallery. I will do that. And I will take it straight to the agent and collect the money."
"Who's the agent?" Slater asked.
"A man prepared to take risks," de Courvier answered.
"What's the method?" Jenkins asked.
"Are you really prepared to play your part?" de Courvier asked him.
"If I like the method, I'll play," said Jenkins. And after a pause, while he stared at de Courvier, he added: "More tea?"
"No, thank you." De Courvier placed the tips of his fingers together and looked almost priestlike. "The picture is thirty-five inches high and thirty inches wide. It has recently been cleaned and will roll easily. If the canvas is cut first along the bottom and then on both sides, tight with the frame, it will stay in position, because it is still hanging from the top. Is that understood?" He got up and went to a briefcase near the door, opened it, and took out a small picture in a plain wooden frame. He moved across to a calendar hanging from a nail over the tiny mantelshelf, took the calendar down, and hung the picture. The others edged toward him. De Courvier was taller than Jenkins by two or three inches, Jenkins taller than Slater by half a foot. Jenkins's hair seemed as if it had been plastered onto his head, Slater's was thick, black, and bushy, but there was a bald patch about the size of a small pineapple slice, and the patch was startlingly white.
With the picture in position, de Courvier took a razor blade in a special holder from an inside pocket, and made one crosswise movement at the bottom of the picture and one downward slash at each side. The picture itself went slack and looked a little crumpled, but did not roll up.
"You see?" de Courvier asked.
"I see," Jenkins said, nodding.
"Oke," Slater said, also nodding.
"When these three cuts have been made, the picture will be free except at the top," de Courvier went on. "Then—" He raised his hand and, apparently doing no more than mopping his forehead, pulled the canvas with his free hand. It came away, curling. With a dexterous twist, he rolled and stuffed it under his coat, and as if by magic another roll appeared in his right hand. As this unrolled, he pressed the corners against the empty space, and turned away.
"Strewth!" exclaimed Jenkins.
"Good," said Slater. "Not bad at all, I will say that. But it would take minutes!"
"One slash at a time," retorted de Courvier.
"What?"
"One slash at a time, at agreed intervals. They won't notice that; there won't be anything to notice unless you go very close. And only the last cut needs the expert."
"You?" asked Slater.
"Yes," answered de Courvier. "I have done it before, and I keep in practice."
"Strewth!" exclaimed Jenkins again. "It could work."
"Listen to me," de Courvier said, and in these moments he seemed to tower in stature above the others. "The visitors to all museums and galleries are constantly on the move. The same people never stay in one room for long; ten or fifteen minutes is average, and there is a constant change of visitors. So one of us cuts the bottom - all he seems to do is run his hand along the frame; he doesn't actually touch it. The attendant might see him but at a distance no damage will show. The attendant would notice if the same man came back and did the same thing, but he wouldn't be likely to notice a different man, and all you two have to do is make the cuts. Then I come in, and I can make the switch in ten seconds. Or even less."
"No one will even know the picture's missing," breathed Slater.
De Courvier was looking at Jenkins, who asked with great deliberation: "Who makes the third cut?"
"You do the first and the third," answered de Courvier. "For the first visit, you wear a beard."
Very slowly, Jenkins nodded; and for the first time he parted his lips enough to show his teeth in a strangely brilliant smile.
"How about getting out of the place?" asked Slater.
"I have planned that exactly, too," said de Courvier. "There will be no problem at all."
He sat down again, erect as ever, and began to talk.
While the three men planned the theft of one of the most valuable and renowned pictures in London's National Gallery, George Gideon, the Commander of the Criminal Investigation Department at New Scotland Yard, was sitting in his office and looking through some files which had been brought to him from Records. Each file was a dossier on a different man; each had a foolscap-sized form which gave details of age, height, habits, of everything which might help him to be identified, including a full set of finger and thumb prints of each hand. There were five files in all, and each was of a man with a criminal record as long as his arm. Each man was out of prison, and each was capable of having committed a crime which had taken place only the night before in an obscure art gallery in Chelsea. The value of the paintings that had been stolen was less than a thousand pounds and the five seldom worked for chicken feed; but if they were out of prison, out of work, and out of luck, any one of them might have thought the Chelsea job worth doing.
Gideon studied each with a closeness which characterized him. He knew four of the five in person, for each had been tried and convicted during his period as Commander, and in his position practically every criminal tried at the London Assizes was known to Gideon.
One of the men was a Leslie Jenkins.
Gideon read of his dozen convictions, from petty larceny when he had been seventeen to armed robbery when he had been thirty, seven—no, eight years ago. He had been one of a gang which had stolen art treasures from a country house and been caught trying to sell them in London. There was a note which read:
Earned full remission. Released 18th January 1970
Address: 14, Sonata Street, Lambeth, S.E.1
"Sonata" was an unusual name for a street, and it sent Gideon's thoughts in a completely different direction: to his daughter Penelope. She was making slow headway as a concert pianist and was practicing a Beethoven sonata at home with a single-mindedness which nearly drove the rest of the family mad, although there was a conspiracy among them to show bright and smiling faces and to assure her that they enjoyed every moment. She was to play the piece at the Albert Hall in a week's time. Only a week more of forbearance, Gideon thought with a wry smile. Her success or failure with it would have an important effect on her future engagements, could lead her to remarkable success for a girl of...
Girl of...
He must not forget, she was twenty-five! He knew it well. Strange that he could forget; could even forget that his chief aide, Alec Hobbs, seemed deeply in love with her. One tended to shut certain facts out of one's mind instinctively.
Once again, then, he had been thinking of Penny as a girl. Twenty-five! His other daughters had married at that age; he had a grandchild - two grandchildren - on the way. And he still thought of them all as girls.
He sat back, more than a little rueful, then, making a conscious effort, forced his mind back to Leslie Jenkins. He recalled that Jenkins had a daughter of about the same age as Penelope; actually a young girl at the time of her father's arrest and trial! Her mother had died - or had she deserted Jenkins? She certainly hadn't been in evidence at the time of the sentence. Gideon could remember talking to the officers in charge of the case about the girl. There had been some suggestion of sending her to a home, but in the end she had been taken in by an aunt. What was her name, now? He made an effort to recollect it but could not do so.
The ring of a telephone bell made him start, and took his mind off the girl. The call was a trifle: would he be available in three weeks' time to give away some prizes at the Sports Club, on the evening of the Metropolitan Police Force Table Tennis Championships?
"Glad to," Gideon said. He jotted a note on a pad for transfer later to his diary, then sat back and said aloud: "Lucy, that was her name. Lucy Jenkins."
# 2: Lucy
Lucy Jenkins brushed her fair hair back from her forehead and stood away from the picture she was about to clean. It was astonishing how much dirt had accumulated in a few years and stuck to the varnish. This picture, according to the slip on the back, had been restored only twelve years before. Where on earth had it been to become so dirty?
She frowned, wrinkling her forehead, as she took the picture out of the old gilt frame. It was a modern painting and not a very good one, but she had a feeling that there was another painting beneath it.
Very soon she knew that she was right. Nervously but skilfully she cleaned off first the dirt at one corner, then the top layers of paint; she went on doing this until at last she could see most of the original picture - a country scene with three figures in the foreground - but as soon as the water she was washing it with dried, the picture faded.
She had bought the painting with a load of frames and prints from a man who had just cleared out an attic. Most of the pictures and prints were now tidied away, and entered in the Purchase Book. She had paid only five pounds for the lot.
Some of the paint of this particular picture had been rubbed away, showing not canvas but cracked varnish beneath, so she had been intrigued by it. Now she had proved that a hidden picture existed, and she knew it was in very fine perspective. She dusted it over carefully, planning to check whether it had ever been relined. The varnish over the old painting was very cracked. You could often judge a picture from the condition of the craquelure. She stood back, then picked up a big bottle of turpentine substitute, dipped in a swab of cotton wool, and slowly covered the whole picture. The colours and the design, of two women and a man, came up beautifully, but soon faded. She studied the corners, then cleaned a tiny patch of sky and the branch of a tree with two birds on it - that portion was one to test for relining.
She dipped a sponge in soapy water, and cleaned the corner again; then she took up a smaller bottle, labelled "dictate," dipped in a little swab at the end of a pencil-sized stick, and spread the mixture carefully over the corner, left it for a moment, then dabbed it off with the turps substitute and put on the mixture again. She mustn't leave it on too long; in restoring it was on, off, on, off, on, off, until at last the varnish was really off and the original paint was revealed.
That blue!
Those birds - hawks, they looked like.
The green of the leaves - it looked almost as if she could pick them!
Lucy Jenkins was very thin; her flaxen hair and blue eyes made her look Scandinavian, but in fact she was an Angle from a long way back, which accounted for her fair skin and the pink tinge to her cheeks. Unlike her father, she had well-shaped lips, and her chin was square, not thin and narrow like his. She had short but very thick eyelashes; had they been dark, they would have looked false.
Aloud, she said, "It's lovely."
She was alone, not only in the room at the back of the shop in King's Road, Chelsea, but in the shop as well. She worked for an elderly man named Jacob Fisk and his semi-invalid wife, minding the shop, cleaning pictures and frames, putting on new cords or wires. She had been doing this for several years now; it paid her all she needed and she was as nearly happy as she had ever been.
She had a single room at the back of the first floor of the premises, and cooked and cleaned for herself, as well as helping out when Mrs. Fisk wasn't well.
When she had first come here, she had not known much about the work, although she had some familiarity with it, for her father had once been a runner for small picture dealers and galleries in London. She had known even as a child that under the double bed in her parents' room stolen pictures had been stored, and she had often seen her father doing what she was doing now - cleaning the corner of a picture to find out what really was under the varnish. She could half remember the smell of oil paints, thinners, wood, and canvas which there had been in a tiny shed in the back yard of her parents' home.
In a way, that was why she had taken this job: because of the sharp, penetrating odours.
She was careful, indeed painstaking, and she had an instinctive liking for old pictures, old prints - in fact, for anything old. When she had first come, she had had no idea of values, but she could now distinguish between a Chelsea and a Worcester figurine, could tell a genuine Hepplewhite or Chippendale from an imitation, could judge the value of repaired pieces, and particularly could distinguish a genuinely old painting from a new one, and a good one from a bad.
Only once had Old Fisky found a truly valuable picture, and after buying it for five pounds he had sold it for nearly four thousand. She could recall the excitement on his face when he had realized what he had bought, how newspaper reporters and photographers had flocked to the tiny shop, and ever since that day she had been on the lookout for something similar. It was not that she expected it, simply that she accepted the possibility and never forgot to check. Several times she had put a picture aside, hopefully, for Fisk, but each time she had discovered only a copy or a painting in the same school as one of the Old Masters.
"It's very funny," she repeated to the empty room. "Why would anyone paint this over old varnish without giving it a proper cleaning? Talk about craquelure, I'll bet my life that paint is real old. Those colours!"
She felt a tremor of excitement as she cleaned off a little more. Mixture on, pause, off with turps, pause, on - off - on - off. There was blue and grey in the corner, and then a touch of brown and green; it was an outdoor scene, trees and sky and clouds, and the people certainly didn't have much on in the way of clothes. She drew back and studied it again, almost touching the wall behind her. The room was little more than a passage, but it had one big north window and the light was perfect. On the bench which ran from wall to wall were other pictures, bottles of turpentine, rags, old canvases, frames, wood for framing and repairs; she was oblivious of everything but the picture now propped up against the wall.
She turned it over and read the tag; there was no doubt about the date of cleaning of the picture which had been on top: twelve years ago. The back of the canvas wasn't really dirty. Her heart gave a sudden lurch as she admitted the obvious possibility to herself: that the canvas had been painted over and relined so as to make the one underneath a kind of sandwich. Of course some artists painted over old pictures because they couldn't afford a new canvas, but usually it was to hide the one underneath.
She was confused, but very alert and very curious as she put the picture face downward on the bench.
She picked up a glazier's knife and prised the canvas loose; and when she first touched it with her fingers she felt another twinge of excitement, for it was very thick - thick and stiff as a board. She carried the picture closer to the window, and studied it.
Now she was sure. There were two linings, a new one to conceal the old one. No one covered both back and front because they couldn't afford canvas!
Everything she had learned about such a situation as this crowded into her mind. There were several ways of sticking canvases together, but the great danger was that if the picture was of any value, it could be damaged. One needed special solvents to separate the layers of varnish without causing damage, and she was not in any way an expert. It was exasperating, because Old Fisky would not be back for at least twenty-four hours, perhaps not until after the weekend. Not often impatient, she was edgy and disappointed; was there a picture of great value, even of modest value there? Had she made a real discovery?
She sighed and took the picture to a rack against a wall and placed it with half a dozen others, all of which needed attention from Old Fisky. Then she picked up a small Scottish Highland scene, quite dirty but not unusual. Cleaning it, she estimated the price the old man would ask for it. This wasn't one of her buys. The old man had bought the whole of the contents of a cottage for a song, sold off the large furniture and the modern pieces to second hand dealers, and had already made a profit. This little picture would fetch fifteen or twenty pounds. She wondered what he did with his money; he must be worth a small fortune.
And if she had made a discovery, he would see that she got her share. He wasn't mean, she could say that for him. She had quite a nice little balance in the Post Office Savings Bank, already put aside for a rainy day.
She had been in the back of the building for at least an hour when the shop doorbell rang, and she went out to see who it was. It was a man she recognized - Red Thomas, a middle-aged man with a red shirt and grizzled hair, a runner for the West End galleries who was always looking for snips.
"Hallo, Lucy," he said. "Anything for me today?"
"There's a copy of a Vermeer," she began quickly; customers often made her feel nervous.
"I don't want any copies," he interrupted. "Is the old man in?"
"No, he won't be back until later." Lucy never told anyone how long she could be alone and in charge, or that the place would be empty, but for her, at night.
The runner was going through some pictures stacked against the wall, obviously looking for something specific, but he didn't say what. His clothes were threadbare and yet well-brushed and pressed. Lucy straightened some oddments of porcelain and china on the shelves. Though the shop was small and in need of painting, she kept it meticulously tidy.
At last, the runner stopped his searching and asked, "Anything in the back?"
"You know Mr. Fisk never keeps anything in the back if it's priced and ready for sale," she answered.
"There might be just the thing I'm looking for," he persisted. "Let me have a look through the back room and I'll see there's a fiver in it for you."
"Mr. Thomas," Lucy replied firmly, "everything we've got is in the shop. Please don't make difficulties."
"Oh, all right." The man stood very still, facing her. His expression hardened. "You're as stubborn as your old man," he said, with an edge of cruelty in his voice. "I hope it doesn't land you in the same place."
He spun on his heel and strode out, while Lucy stood white-faced, staring at the window but not really seeing him. She hated being reminded that her father was an ex-convict. She tried desperately to forget it but somehow never could. Often it was the shop which reminded her, the fact that her father had at one time done a lot of business with Old Fisky.
On the other side of the road, two policemen passed, one of them speaking into his little walkie-talkie radio. The other one looked across at her. She hadn't any doubt, he was looking at her: was it with suspicion?
She went back to the workroom. It was no use telling herself that she was being ridiculous, that there was nothing to suspect her for. She was always conscious of policemen and uneasy in their presence. And she was so upset by Red Thomas's remark she did not give the mystery picture a thought until, half an hour later, she locked both back and front doors and went upstairs.
Then she thought, It might have been stolen. And the possibility troubled her very much indeed.
The two policemen who had been in King's Road, Chelsea, walked together into the Chelsea Divisional Station, the Headquarters of the CD Division, about quarter to eight that night, and filed their reports. There was nothing much in any of them, but the younger of the two, Police Constable Wilfred Chivers, made a routine comment.
"Red Thomas was in Jake Fisk's shop in N.K.R. this afternoon - 4.40 until 5.00 P.M. approx."
After leaving Lucy, Red Thomas walked for ten minutes, looked into another shop, and then caught a bus back to the West End. He sat on top, smoking, watching the streets and the crowds perfunctorily, thinking of the shops and the pictures he had seen, wondering what he could offer to West End buyers who might give him an advance on commission. He had been in the art business most of his life. He knew a great deal about paintings, especially of the main English and Dutch schools, but he was only on the fringe of the trade. He had no heart for paintings, no feel or love for them. To him they were simply coloured canvasses or panels, and he saw no intrinsic beauty in them.
He earned a reasonable living but seldom made big money, because he had a curious characteristic: while he had a very good eye for the artist or school a particular dealer was interested in, he had no eye at all for a find.
He was, within his own limits, honest, and he had never been in prison, though he knew of a great number of men, like Leslie Jenkins, who had been inside severed times but, when out in the everyday world, had made fortunes, only to waste them on gambling or drinking. And he envied these men as he envied anyone who made more money than he.
He got off the bus, turned in to a narrow lane off Bond Street, and went into the Oriole Gallery, where a silver-haired woman sat at a small desk. The Oriole Gallery specialized in paintings of birds and animals; its walls were crowded with oil paintings, watercolours, and prints.
"Hallo, Mrs. Bessell," Red said. "I'm back."
"So I see," Mrs. Bessell said dryly. "Did you find anything?"
"There's a fine pair of pheasants at Old Fisky's, King's Road," he said. "Real beauties, they are. I could get them for thirty if you like; you'd get fifty for them at least - a hundred, knowing you."
"I'd want to see them," Mrs. Bessell said. She wore pebble-lensed glasses, which diminished her attractiveness.
"Well, I don't know about that. They might not let me have them on approval, and if they know who wants them the price will go up. You know what these Chelsea dealers are like, Mrs. Bessell."
"I know they would want a deposit," the woman admitted. "I'll think about it, Red."
"I could get out there tonight," the man persisted. "Fisk lives over the shop, so there wouldn't be any difficulty. And I could have them here first thing in the morning." When the woman made no answer, he went on doggedly: "I might get them for twenty-five. I happen to know Fisk's a bit short of ready cash. He bought a big load a couple of weeks ago and—"
"I'll decide in the morning," Mrs. Bessell said; then, seeing the defeated expression on his face, her tone softened and she asked, "Short of a pound, Red?"
"You never said a truer word," he replied miserably.
Mrs. Bessell opened a drawer in her desk and took out a black metal cashbox.
"I'll give you an advance on the next job you do for me," she told him. "But I want preferential treatment. Do you understand?"
"Mrs. Bessell, you always get it. I swear you always get it."
Eagerly he took the two rather soiled pound notes she held out to him, looked nervously about the shop, then backed out. He disappeared, walking very fast toward Piccadilly. Traffic built up outside, and an impatient driver put a finger on his horn and kept it there.
When Red had gone, a door leading from a passage at the side of the gallery opened and a man stepped toward Flora Bessell, skin dark against hers as he bent to kiss her cheek.
"You've got the biggest heart in the business," he said.
Flora Bessell flushed with pleasure and tilted her head to look into the smiling eyes of her lover.
"Keep telling me that, Oily," she said, almost pleading.
Frederick Oliphant smiled again and squeezed her arm.
# 3: Gideon
George Gideon left his office at Scotland Yard late that evening, a little after seven o'clock. He walked, massive and almost forbidding, along the passage toward the main doors and the long flight of steps leading into the forecourt. There was something about the depth and breadth of his shoulders, his rather short neck, his broad, rugged, not unhandsome face, which could strike a kind of fear into those who saw him and served under him. He was aware of this, and disliked and almost resented it with one part of his mind, yet accepted and took advantage of it with another part. It was very good for discipline and it kept men on their toes. Often, he knew, he looked more forbidding when in fact there was no great worry or problem on his mind. This evening there was nothing to cause him particular concern; none of the investigations going through the Yard needed undue concentration.
His car was at the foot of the steps, and a detective officer stood by the door.
"Good night, sir."
"Good night."
"Lovely evening."
"Just right."
"Shouldn't go round Parliament Square if I were you," the man volunteered. He was middle-aged, old for a detective officer, only a few years younger than Gideon.
"Why not?" asked Gideon, settling in at the wheel.
"Doing something to the road," the other answered. "Cars are jammed like sardines."
Gideon grunted, not very graciously, and drove onto the Embankment, turning left, not right toward Parliament Square and the Houses of Parliament. Big Ben, behind and out of sight, struck the quarter. Traffic on the far side was already slowing down, and above the shiny tops of cars and between the big red buses he could see the higher level of the London County Hall and, further along, some of the slender arches of Waterloo Bridge and the top of the new Shell buildings across the river. Trees hid the Festival Hall, and he could not see a single patch of water. In a strange way his mood had changed, entirely because of Parker, the detective officer he had just spoken to, who would never become a sergeant unless on sentimental or compassionate grounds. Any man at the Yard who knew a major road was up but could only describe it as "something" was an indifferent policeman. Gideon, unaware that his own thirst for knowledge was as natural as breathing, wanted to know what was being done.
He turned in to Northumberland Avenue and reached the traffic lights at Trafalgar Square. There traffic was much thinner; the homeward rush was over and the reverse-direction theatre rush had not yet started. There was something both soothing and satisfying about the scene, and he was glad of the wait at the red signal. Nelson's Column, recently cleaned, looked a little unreal as it rose from its guard of bronze lions. The square itself, its fountains playing, the pigeons strutting like an ill-disciplined army, was bathed in the evening light, tinged now with a warm pinky gold. The fountains were like waterfalls. Beyond, looking almost naked from its recent cleaning, was the National Gallery.
Soon, driving along King's Road and New King's Road, he passed the rows of second hand and antique shops, without noticing the one with the name "Jacob Fisk" on the fascia. He had no idea that Lucy Jenkins lived and worked there, but he realized one thing which he must have noticed before but which had not registered as it did now: practically every shop in this stretch of New King's Road sold antiques or garden furniture, masonry or pictures. This side of London was fast becoming an art and artists' colony on a much greater scale than the Chelsea of his youth; the trade that went on in this square mile must be enormous.
And because he was a policeman, he wondered whether the Yard had its finger as firmly on this area's pulse as it should. Where there was trade, there was crime; where there were small art dealers, there was a great risk of buying and selling of stolen goods. Although he lived nearby, he had not given much thought to the mushrooming of a business with such an obvious crime potential.
This area was covered by CD Division, and Jack White, the Superintendent in charge, was a good, solid, sound detective, but hardly a man with a flair. He would have to find a reason to talk to White. One could hardly expect the divisions to specialize, however, and the specialist in art at the Yard was Frobisher, Wally Frobisher, a much younger man, who had won swift promotion because of his knowledge of the art world. Frobisher was at the moment up in Manchester, helping the local police with the investigation of a theft of three small Watteaus from the Manchester City Museum.
There was another man, Thwaites, who had a less expert but much wider general knowledge, at the Yard. He might know much more about the present situation than Frobisher.
By the time Gideon reached his own street, Harrington Street, Fulham, close to the Hurlingham sports grounds, he had put Jack White and Frobisher to the back of his mind. He had a garage round the corner at the far end of the street. Neighbours were working in their gardens and many doors and windows were open; out here it seemed even more muggy than it had been in town. The front-room window of his own house was wide open, and he heard two or three bars of Beethoven's Sonata in C Minor. He broke into a smile, which broadened as he caught sight of Penelope sitting slantwise by the sitting-room window. Then he noticed that two or three neighbours were at their front gates, others at their windows, and none of them was talking.
They were, he thought, listening to Penelope.
Now that he was no longer in the car, he could hear much more clearly. Into the quiet there came a sudden clatter of a lawnmower, and one of two women at a gate said, "Oh, drat that man!"
The clattering went on, and now only those neighbours close to Gideon's house could hear Penelope playing. She was near the end, he knew, and he wondered how many more times she would go over the piece again this evening. A man at the window of the house next door, called, "Beautiful, Mr. Gideon, absolutely beautiful."
"Thank you," Gideon said. "You don't find it a nuisance, then?"
"A nuisance - that girl's playing? Good Lord, no!"
Gideon, highly pleased, turned into his gate and, glancing up, saw Kate, his wife, at the window of their bedroom. She raised a hand, then turned away; and when he stepped into the house, she was halfway down the stairs. She was a handsome woman with a fine figure; her hair was greyer than Gideon's, but with the blue rinse she had now, she looked much younger than he did.
Her face was smooth and unlined, and it was hard to believe that only a few months ago she had had serious heart trouble. For an agonizing moment, he relived the panic he had felt then - Kate, his wife, ill. She was fully recovered now, thank God.
She stood on the bottom stair, so that she seemed an inch or two taller than Gideon; and in fact, at five feet eight, she was tall for a woman.
Silence fell inside the sitting room as they met, and Kate, who had started to speak, uttered two or three words which sounded very loud, because the earlier words had been muffled by the piano music.
"It's a good thing that someone feels he—-"
When she stopped, her mouth was open, and Gideon grinned, leaned forward to touch her cheek with his lips, then drew back and asked: "What's a good thing?"
"It doesn't matter," Kate said, in rare confusion. She looked at the sitting-room door, obviously expecting Penelope to come out, but when the door remained closed she took Gideon's arm and led him along the wide passage that ran by the side of the staircase, and into the living room. In these deep, narrow houses there was the front sitting room, then a smaller, connecting room used by many families as a dining room, and, beyond the passage, a big living room with kitchen and scullery opening off. Everything in this particular house was solid and of good quality, as well as being spick-and-span. "Oh, dear," Kate went on, half laughing, "I nearly put my foot in it, George."
"How?" inquired Gideon, deeply interested.
"Mr. Curtis has been complaining bitterly about Penny's practice, and Millicent's tried to calm him down, while I've been reassuring Penny. If she heard me say it's a good job someone feels he can enjoy piano music, Penny would have realized there's been trouble."
Gideon smiled. "It's a bit obscure, but I think I get you. One neighbour thinks she's wonderful and the other says she's driving him out of his mind."
"That's about it," said Kate.
"Very interesting," Gideon remarked, taking off his jacket. "Gosh, it's sticky tonight." He rolled up his shirt sleeves and went into the scullery, where he washed his hands under the tap. As Kate paused in the doorway, Gideon went on: "Have Curtis and Henshaw ever settled their argument over the service alley?"
Kate stared.
"Goodness!" she exclaimed. "I never thought of that!"
"Curtis complains, Henshaw hears about his complaints and is all sweetness and light, which is a way of getting at Curtis." Gideon reasoned. "And Penny gets squeezed in the middle, poor kid. Is she upset?"
"I don't think she knows anything has been said," Kate answered. "George—"
Gideon was back in the living room, taking whisky and glasses from a cupboard by the side of the fireplace. He was not a heavy drinker, but he had come increasingly to relax in the evening with a whisky-and-water. Kate came in with a small jug of water.
"For you?" asked Gideon
"No, I won't," said Kate. "George, I don't see how we can avoid complaints, and if Curtis makes an official one, or even comes banging on the door, it will upset Penny so much"
"Just a week to go," Gideon remarked. He sipped and dropped back into a large easy chair which looked as if it had been made for him. "We—"
"George," said Kate again, "you can't be glib about this."
Gideon, startled, echoed: "Glib?"
"You can't just push thought of it aside with a few casual words."
Gideon drank again, more deeply, hitched himself further up in his chair, and said with greater deliberation: "You were obscure before, dear. Now you're really confusing me! What am I supposed to be glib about?"
"Oh, I didn't really mean glib. I meant—"
"I was pushing something aside," said Gideon mildly.
"What, precisely?"
Kate, a little flushed, was spreading a white cloth over the big deal-topped table. She moved with natural grace, and Gideon never tired of watching her. She was a little thinner than she had been, her loss of weight especially noticeable at her waist, particularly when she wore a blouse and skirt, as she did now.
"There'll be more and more," she said, at last.
"More and more—oh—practice."
"Yes," Kate said with obvious relief. "When she's finished this sonata, she'll play another, and another, and another. That's inevitable, isn't it?"
"And the more successful she is, the more she'll have to practicse."
"Yes, obviously."
"And the more our neighbour Curtis won't like it."
"It isn't only Curtis" replied Kate ruefully. "Several of the neighbours have asked rather pointedly when the concert is going to be held."
Gideon sipped, frowned, and half smiled.
"I know what you mean," he said. "'Doesn't Penelope play beautifully, Mrs. Gideon; when is she going to finish?'"
"I can't really blame them." Kate was putting knives and forks round the table, setting four places in all. "George—"
"We can't soundproof the sitting room," Gideon objected.
"We could soundproof the attic, or we could very nearly," answered Kate, and then she went on much too quickly, swallowing some of her words. "We only keep rubbish and our suitcases up there; half of it's boarded over, and I don't think the sound would travel far from the one small window. I shouldn't think it would cost—"
"Say, two hundred pounds," interjected Gideon.
"Oh, it couldn't be as much as that!"
"Yes, it could," answered Gideon. "Probably quite a bit more, too. Even if it were practicable."
"Oh, it is! I've had a good look round. And you know Alan Pryce-Davies, don't you - the cartoonist, I mean - he's soundproofed his room; he says it's the only way he could get any peace. I can't believe it would cost as much as two hundred pounds."
"Oh, well" said Gideon. "I suppose we can find out."
Kate's face lit up, making her look quite beautiful.
"May I, George?"
"Will you, love?"
"I'll make it my first job tomorrow," Kate promised. "George, she is our last daughter, and—"
"You don't have to sell me the idea," Gideon said gruffly. "Are we going to eat tonight?"
Kate, her eyes very bright, went past him to the kitchen, pressing his shoulder firmly. Gideon finished his drink, then leaned back in his chair and dozed for the next few minutes, until Kate called out, "Well, if you're hungry." He opened his eyes to see the table laden with a steak pie with beautifully browned crust, three dishes piled up with boiled potatoes, carrots, and cabbage.
He smothered a yawn and got up as Kate, at the open door, called: "Penny!" As he reached the table, Penelope came in, tall but not so tall as her mother, dark hair drawn off her face, making her look a little too thin. Her eyes, also grey, lacked the brightness of Kate's.
"Hallo, Daddy.... Has it been a beastly day?.... Oooh, what a lovely pie!...." He really would have to find her a practice room where she could play without being worried about the neighbours, he told himself, and reflected comfortingly that there wasn't much he needed to spend his income on these days.
Malcolm, the youngest of the family, came rushing in halfway through the meal, full of apologies about being late and having to go out again in twenty minutes.
"If you're going swimming—" began Kate.
"No, it's not swimming; it's work in the gym. Get the old muscles pepped up!" Malcolm stretched out his arms and flexed his biceps, brought his plate from the hotplate to the table to allow his mother to serve the main course, then helped himself to vegetables. Sitting opposite Penelope, he gave her a salute. "And how's the gorgeous pianoforte virtuoso today?"
Everyone laughed.
For the Gideons it was a pleasant, happy evening.
# 4: The Falconers
For the Falconers, who lived in Mayfair, it was a very different kind of evening. Christine Falconer, the only child, wondered whether she alone among the four present realized what kind of life they led, how unbelievably artificial it was. At certain times, she had a feeling that her mother was living under a great strain and somehow putting on an act; at others, she felt almost despairingly that her mother had become a kind of automaton, switched on as it were to defer exclusively to her husband's wishes.
Christine thought, looking down the long, highly polished Sheraton table at her mother, How can anyone so beautiful be so empty of emotions?
And she thought, looking toward the other end of the table at her father - in his high-backed chair, the arms of which always seemed a little too high so that he was continually knocking his elbows - How can anyone be so rich and so stupid?
Opposite her, his face softened by the candlelight from two big silver candelabra, sat Frederick Charles Stuart Oliphant, whom everyone knew as Oily. He had been virtually one of the family since she was quite tiny; she could not remember the great house without him. She had accepted him without thinking for so long that when at last her mind had opened to doubts and uncertainties about him and she had started to revalue her attitudes, it had been difficult to see him differently. He was now in his middle fifties, a little older than her father, and, of course, to her he had always seemed old. He was balding, with a round, pale face and a small rosebud of a mouth. She could never remember him being out of temper or ruffled or excited. He was the secretary, confidant, and friend of her father, and she knew that he was regarded as one of the great art experts of the world.
Davies, the butler, came in silently and offered more of the delicious apricot-and-peach flan, more of the rich Jersey cream: but no one wanted a second helping.
"We will have coffee in the drawing room," her father said.
"Very good, sir."
It was like a record player.
"Come along, dear," her mother called, as she had from the days when Christine had first been allowed to join them at the dinner table.
Oily moved and pulled her mother's chair back. Davies placed the port in front of his master. The candlelight made a cage of the dining table and the silver and the dishes, holding the gaze at eye level; now, above the glow, the pictures showed, each discreetly lit, each a portrait, each an Old Master, and each priceless. Christine stood up, feeling an almost overwhelming temptation to pick up a knife and hurl it at the nearest solemn face; instead, she turned away from the table and followed her mother out the door, at which Davies was standing, tall and stately and, like all of them, not quite real. She thought the whole nightly performance was like a charade in which the first prize went to whichever performer could show least expression.
They went out of the dining room into the hall.
Here were the landscapes: Gainsborough, Constable, Turner - paintings of rich beauty and great value. And here were the sculptures and the busts; it seemed to her as her mother walked past them that the marble and the granite, the bronze and the alabaster, were so much more real than the people who lived in her home.
"Mother," she said as they reached the open door of the drawing room.
"Yes, dear?"
"Mother, I've a headache. Do you mind if I go up to my room?"
"Your father will be very disappointed, Christine."
"Tell him I'm sorry, won't you?"
They stood facing each other for a few moments, and this was one of the rare occasions when Christine felt that the woman, not the automaton, was in front of her. Even then she was aware of the Dresden china perfection of her mother's skin, the brilliance of her eyes, all the subtlety of makeup, the elegant simplicity of the Balenciaga dress, the small diamond pin on her shoulder. The woman receded, the automaton, the work of art, taking her place, and a new thought crossed Christine's mind: that her mother had been made for her father, or else he had searched for her as he searched for every other rare piece in his collection.
"Christine, dear," her mother said. "You're not going out, are you?"
"I might—I might go out for a breath of fresh air."
"You haven't made any plans to see anybody?"
"No," Christine said. "But is there any reason why I shouldn't?"
"Well, you know, my dear, your father doesn't like some of the young people with whom you have been associating recently," her mother said. "I am sure you would be very wise not to see them or anyone else surreptitiously. You know how your father likes to have everything out in the open, don't you, dear?"
"Do you mean he's been—" Christine began, in sudden white heat of temper, but somehow she bit back the words "spying on me" and turned and hurried away. She felt in a turmoil, at the point of revolt against a life which was becoming increasingly intolerable.
She ran up the curving staircase, past recesses and alcoves in which stood vases and goblets, some of them chased gold and silver, some of the others jewel encrusted: The Italian (mostly Cellini), the French, and the Spanish; a little farther up, where the stairs rounded, the Chinese of old dynasties were represented with vases and jade figures dug from ancient tombs. And on the landing were rare objets d'art from South America, from Russia, from Mexico. Every single piece was unique, every single piece of rare value.
She went hurrying by, oblivious of everything.
She turned in to her own suite, with a dressing room on one side and a bathroom on the other, all Regency, all beautiful, and hers. When she shut the door, she could at least shut the rest of the house off, the artificiality, the lifelessness.
My God! she thought. He's been having me watched. He's actually been spying on me!
She stood in the middle of the bedroom, feeling almost numbed. After what seemed a long time, she muttered, "I've got to get away! I simply must!"
Downstairs, her father and Oliphant went into the drawing room, where her mother sat looking at a television set. Coffee was on a low table in front of a long couch. Newspapers and magazines in great variety were in racks by each seat and chair.
"Oh, hallo, my dear" said Falconer. "Where's Christine?"
"She has a headache, Richard, and has gone to her room."
"Indeed? She didn't complain of a headache at dinner."
"I thought she looked a little unwell," remarked Oliphant.
"She is probably planning to go and see her friends. That should make her feel unwell," Falconer said coldly. "Have you the report for me on her friends, Oily?"
"On seven of them," Oliphant answered.
"Are they satisfactory?"
"Those seven? Wholly."
"What of the others?"
"Well, there is a young man named Judd. She sees a good deal of him," Oliphant said. "But I know very little about him - only that he has a small antique shop in Hampstead and appears to be doing well."
"Indeed," said Falconer heavily.
Oliphant gave a small, almost plummy smile.
"I shouldn't assume that he is using Christine in the hope of doing business with you," he said.
"That is exactly what I fear he may be doing. Have you tried to find out his background?"
"Yes," said Oliphant.
"It's not like you to admit failure."
"I haven't admitted failure yet," Oliphant retorted. "I have asked Alec Hobbs if he can make some inquiries, and he will be in touch with me tomorrow."
Falconer nodded, seeming reasonably well satisfied.
"How is Alec?" asked Charlotte Falconer, as if hoping to change the subject. "He was so distressed by poor Helen's death. I really wondered whether he would ever get over it." After a pause, and while she poured coffee, she went on: "I could never understand why Alec elected to become a policeman."
"Some people would call him a detective," Oliphant replied.
"Is there any difference?" asked Charlotte indifferently, looking at her husband. "Richard, will you have brandy or a liqueur?"
"Brandy," answered her husband. "Brandy."
Deputy Commander Alex Hobbs, Gideon's deputy and chief assistant at Scotland Yard, sat back in an easy chair reading an American police manual, storing much of what he read in his card-index file of a mind, and half listening to Swan Lake on a stereo record player. He could hear cars passing along the Embankment in front of Ayling Crescent, and now and again a heavy lorry changed gear as it turned to go over Chelsea Bridge. Very occasionally a tug or a lighter hooted on the river.
There had been a time, even as recently as six or seven months before, when he would have been troubled and restless, still fighting the loneliness which had followed the death of his wife. During her long illness, they had lived in a flat not a quarter of a mile away, overlooking the same stretch of river, and her last awareness, as she lay propped up on pillows in a bed close to the window, had been of the river. When she died, Hobbs's immediate inclination had been to get completely away from this all too familiar part of London but eventually he had settled on a small suite of rooms in a modern block of flats in Chelsea. Pleasantly though somewhat severely furnished, it provided excellent service; he had nothing to worry about but getting his evening meal; and he could, whenever so minded, leave the washing up to the maid in the morning.
He would have given up, almost certainly, but for the slow growing of his interest in Gideon's daughter
Penelope. He was falling in love with her; and the Gideons knew it. But Penelope was so young, and so full of enthusiasm for younger men...
He finished a chapter and put the book down, yawned and stretched, then looked at the brandy and the empty glass on a wine table by his side. Suddenly he placed his hands on the arms of his chair and sprang up, a very fit, very lean man of forty-five, dark-haired, handsome in an almost artificial way.
He turned to the window.
Here again he had forced himself to overcome an impulse to draw the curtains every night. Helen had liked them open, and when she held been well - and even during her illness when she had been able to stand - they had often put the lights out and gone to look at the view, the Embankment and the bridges, the luminosity of the smoke pouring out of the squat chimney stacks of the Battersea Power Station, the coloured lights of the Fun Fair at Battersea Park, the slow, smooth-moving lights on the river craft. The view hadn't changed. The awareness of being alone strengthened noticeably, and he made himself stand rigidly there.
Slowly, remembered grief and present tension eased, and he relaxed.
As he did so, his telephone bell rang.
Moving toward the table where the telephone stood, within hand's reach of his chair, he glanced at a clock standing on a bookcase which lined one wall to waist height. It was quarter to ten, late for a call from the Yard unless it was an emergency. He picked up the receiver.
"Alex Hobbs speaking."
"Good evening, sir," a man said, with a slight North Country accent which Hobbs immediately recognized. "Thwaites here, sir - sorry to bother you so late."
Whatever else, this was no emergency.
"That's all right," Hobbs said. "What is it?"
"You asked me to make inquiries about a Lancelot Judd, an antique dealer of Hampstead."
"Yes," said Hobbs.
"I would like to discuss the situation with you, sir."
"Tonight?"
"I have a full day planned tomorrow, and to do it then I'd have to put off several cases, but of course if you—"
"All right, Thwaites," Hobbs interrupted. "Come to me here, will you? Have you eaten?"
"Oh, yes, sir!"
"Good," Hobbs said. "Where are you now?"
"In Hampstead Village, sir. I'll be about half an hour."
"Right," Hobbs said. "Have you been here before?"
"No, but I know where your flat is, sir."
"Press my downstairs bell and take the lift to the fifth floor," Hobbs told him. "I'll be at the flat door to meet you."
He put down the receiver and stepped back to the window. In some ways a more incisive man than Gideon, he had acquired a surprising number of Gideon's methods and Gideon's attitudes. In fact, although they came from vastly different backgrounds, they thought in much the same way. That was why Gideon had recommended Hobbs as his deputy. Hobbs, one of the public-school policemen, came of a family which had been both rich and esteemed three hundred years ago. He had been to Repton and King's College, Cambridge, and while some thought him aloof, even snobbish, all agreed that he was a first class policeman. Now he thought over everything he had told Thwaites to do and what he knew of the Chief Inspector. Thwaites had had a North Country upbringing, followed by twenty years at the Yard; at forty-four or five, he was a rather untidy, comfortable looking man, who had acquired a love and deep knowledge of antiques.
"Always liked to poke around second hand shops when I was a boy, sir."
And he still enjoyed poking around, Hobbs believed; even when he was inquiring into an art theft, looking for stolen goods, getting information about others to be shipped out of the country, Thwaites enjoyed touching, looking at, and assessing the value of every kind of antique, painting, and objet d'art.
Also he was a dedicated policeman.
It was disturbing that he wanted to talk about Judd, for it suggested that all was not straightforward. Was it possible that Lord Falconer's daughter was involved with a suspicious character?
# 5: Rumours
"Brandy?" asked Hobbs as his visitor sat down.
"If it's all the same to you, sir, I'd rather have a beer."
Hobbs selected one of several bottles from a tray, poured it into a pewter tankard, poured a little brandy into a large-bowled glass for himself, and sat down.
"Cheers."
"Cheers." Thwaites drank, Hobbs sipped. "Well, you'll want to know what I'm making a mystery about, sir," Thwaites said. "And it is a bit of a mystery. This Lancelot Judd is about twenty-five, comes from Brighton, quite respectable family. He got a place at Trinity College, Oxford, read History and Philosophy and got his M.A. all right. His parents couldn't afford to do much, and he worked during the holidays at an antique shop in Brighton. He's a Peace Marcher and C.N.D. man, but otherwise he's not known. I checked with Brighton about the place where he worked. No evidence of crime or excesses of any kind."
"So he's in the clear, except for political interests," Hobbs remarked.
"That's the tricky part I wanted to talk about," Thwaites said. "He's in the clear, but some of the people he runs around with are"—he hesitated—"well, sir, they're what you might call on the fringe." "Fringe of what?" demanded Hobbs.
"They pick stuff up at second hand shops and markets and pass it on to the better dealers. One or two of them are believed to have handled stolen goods, although there's never been any proof. And there's one thing that sent all my warning signals going off at the same time, sir."
"What was that?" asked Hobbs.
"Judd's present girl friend is Christine Falconer, only daughter of Sir Richard Falconer. How about that, sir?"
Hobbs put his head on one side, eyebrows raised, and then broke into a chuckle.
"All right, you've scored," he conceded. "That's why I wanted you to check Judd."
"I did wonder, sir," said Thwaites, and drank the rest of his beer. "It could be young love, of course. Or it could be that he'd like to get inside the Falconers' house and see what kind of security there is - and maybe leave the odd window unfastened. Does Sir Richard suspect something like that, sir?"
"I think he's aware of the possibility."
"Can't say I blame him! Do you know what the security is like, sir? I've never been inside the place but Mr. Frobisher has, and he says it's like a museum."
"And it is." Hobbs got up, took Thwaites's tankard and refilled it, and sat down again. "Have you talked to the Divisional men at Hampstead?"
"Yes, sir."
"Got anything?"
"No, I don't think so," answered Thwaites. "Only rumours."
"What kind of rumours?"
"That there's a big buyer around town."
"Buyer of stolen antiques, you mean?"
"Buyer of anything at the right price," answered Thwaites. "And the rumour isn't only in Hampstead, either. It's in Chelsea and Fulham and the West End - that antique supermarket, as they call it - as well as the suburbs. And I had a word with Brighton, as I said, and Salisbury and Stratford-on-Avon. The word's out that there's a big buyer on the lookout for anything special in pictures and antiques, and that he doesn't care where it comes from - will ask no questions, in other words."
Hobbs, brandy glass cupped in both hands, sniffed at the bouquet and looked thoughtfully at Thwaites over the brim. Apart from the noises outside and Thwaites's rather heavy breathing, there was no sound. At last, he lowered the glass.
"Do you know anyone behind it?"
"Not a soul," said Thwaites.
"Who spread the rumours? Do you know that?"
"The runners, as usual."
"No one runner in particular?" asked Hobbs.
"Haven't been able to put a finger on any one, sir. It seems to have started a week ago, and just spread. And the Falconer place could be very vulnerable. Prevention's better than cure," Thwaites added sententiously.
Hobbs sniffed brandy again, pausing as if he wanted the fumes to go through his head and clear his mind before making any comments. Then: "What are you doing?"
"I've asked the reliable dealers to pass on any word they get, but that's not good enough by itself, of course."
"It certainly isn't. Any ideas?"
"I can't say I have, sir. Except—"
"Well?" Hobbs knew the other was waiting to be prompted, and also knew how very shrewd this slow-speaking man was.
"I thought we might pick up one or two runners and pay them enough to keep them loyal," Thwaites suggested. "Someone who would pass any word on to us without letting us down. See what I mean, sir?"
"It's hardly original," Hobbs said, almost disparagingly. "Can we rely on any runners?"
"One, for certain," Thwaites said. "And two or three others I think would be all right."
"Can we afford to take a chance?" asked Hobbs.
"Don't see why not," said Thwaites. "And we could compare the different reports. If there's a common factor, we'd soon find out. The only risk is that the runners might reveal that they were working for us, but that wouldn't matter, as we wouldn't ask them to look for anything specific. I wondered if you would think it over and, if you agree it's worthwhile, have a word with the Commander."
"I'll do that anyhow," Hobbs said. "Who is the one runner you think we can rely on?"
"Man named Red Thomas," answered Thwaites, without hesitation. "He's always absolutely clean, though he's always in need of money. No one likes him but everyone trusts him."
"Why doesn't he have any money?" asked Hobbs.
"Spends what he gets too freely," Thwaites said with a grimace. "If you agree, sir, the first place I'd send him would be to Hampstead. The more I think about this Lancelot Judd and Christine Falconer, the more I think Hampstead's a place to concentrate on."
"I'll talk to Mr. Gideon in the morning," Hobbs said.
Gideon listened with his customary close attention next morning, and came to a conclusion more quickly than usual. It was almost as if he had been pondering most of the night, as Hobbs had been.
"Give Thwaites his head," he ordered. "I'll support him if anything goes wrong."
"Good," said Hobbs.
"And Alec - keep me in close touch," Gideon warned.
"I will," promised Hobbs. "If there is a big buyer, we want to know who he is and whether he's been buying
for long."
"We need to know that very much," Gideon said. He nodded dismissal and Hobbs went to his smaller room next door, but before he could have settled at his desk Gideon lifted the interoffice telephone on his desk and dialled him. "Alec," he said as soon as he heard the other lift the receiver, "this is worth a teletype inquiry to New York and Paris, and anywhere else abroad that might have some information for us. See to it, will you?"
"At once," said Hobbs.
"I'll see what I can do, Mr. Thwaites," Red Thomas promised. It was the day after his visit to Lucy Jenkins, and the same time that Gideon was talking to Hobbs at the Yard. "I haven't heard anything yet, but that means nothing, as I haven't been listening. I keep myself to myself, you know that. I'll phone you whenever I get anything, Mr. Thwaites."
"Between nine o'clock and ten in the morning is best," Thwaites told him. "Here's a fiver in advance. For every reliable piece of information, you'll get another one, and a bonus if there's anything we can act on."
Red took the five-pound note with the same alacrity as he had taken the two notes from Mrs. Bessell, backed a few steps, then went out of the Chelsea Divisional Station, where he and Thwaites had met. Once outside, he walked very quickly toward King's Road, as if he could not get away quickly enough. Traffic was thick and there was a line of five buses outside the Town Hall. Red jumped onto the first of these and sat on the edge of a seat close to the platform. A Jamaican conductress took the sixpence he offered, and he accepted the ticket which rolled out of her machine, without a word. When he got off, on the far side of Albert Bridge, he walked toward Fisk's shop.
On the other side of the road was a young policeman, one of the two who had seen him the previous day.
Lucy Jenkins was in the shop, using a damp chamois over some china pieces. She looked up, and the moment she recognized him, her features froze.
"The old man in?" he demanded.
"No, he's out again," she said, almost vindictively.
"No wonder he never does any business, he's always out," complained Red.
"That's his business," she retorted.
"All right, all right. I want to look at those pictures again."
"Look where you like," she said carelessly.
Thomas went across to the pictures leaning up against the wall, and began to play the familiar game. It was silly, really, because all the runners did it. He went through picture after picture and pretended not to be interested in any but lingered over several. She began to play her usual game of guessing which ones interested him. The pheasants, she decided, and was immediately worried because they were the best among the pictures and she could not knock much off the asking price. The old man had told her the limits.
Red selected a picture - yes, it was one of the pheasants.
"How much?" he asked.
"It's on the label."
"I don't take any notice of the label," he said. "How much to the trade, I mean."
Lucy went across to him and took the picture out of his hands, looked at the back, and saw the freshly attached label: "£30 pr."
"Thirty pounds the pair," she said.
"Who asked for the pair?"
"Whoever buys them will get the pair," she insisted.
"When's he coming back?" demanded Red.
"He might come in any time, but might be out all day" she countered.
"If he finds out you've missed a sale, you'll be in for it," he warned.
"Who's going to tell him?"
"I am."
"Think he'll believe you?" she scoffed. "That'll be the day!"
"Stop arguing around, Lucy," Red Thomas urged in a more reasoning voice. "How much?" His tone and manner changed and Lucy knew that he had finished the game and was playing it straight. He must have a customer or he wouldn't be so serious, and every pound she knocked off would be one in his pocket, for if she knew the man he had seen the price yesterday and quoted it - perhaps higher - to his customer.
"Twenty-seven ten the pair," she said.
"Twenty-five."
"Twenty-seven, and I may have to make the ten bob up myself."
He looked at her for a long time, without making comment or retort, and then when she was beginning to feel uneasy under his gaze, he said: "Twenty now, seven when I've been paid."
"Who's buying?"
"That's my secret," he said. "Is it a deal?"
"Oh, all right," she conceded ungraciously. "Give me the twenty and be back before Mr. Fisk comes in or I'll be in trouble."
"No, you won't," Red said. "I never cheated anyone yet, and you know it."
She did know it. She knew also that if a runner ever welshed he would be out of the game for good. One could put a lot of profit onto the price paid, but one couldn't welsh or play one customer or buyer off against another. It was an absolute rule, and the trade lived by it.
He took some notes out of his pocket and counted twenty of them into her hands; he had one left when he was done. She gave him a receipt and he looked round, found some corrugated paper, wrapped the pictures up, fastened the paper with sealing tape, and went out.
Nearly an hour later, he was with Mrs. Bessell in the Bond Street gallery, watching her as she studied the pictures, first with the naked eye and then through a magnifying glass. He was trembling a little. At last, she put them down and said: "How much did you have to pay for them?"
"Twenty-seven ten."
"Where did you get them from?" Mrs. Bessell's manner was uncompromising.
He hesitated before saying: "Jake Fisk. I told you."
"Any idea where he got them?"
"No," said Red. "Why—they're not hot, are they?"
"I don't think so," she said. "They're probably early Stott, and if they're not they're a very good example of the school. They're worth two or three hundred apiece, anyhow."
"Gawd!"
"Red," she asked, "has Fisk got any more? If he has, can you put all his stock on approval, and let me have a look? If you can do that, 111 give you a hundred for this pair. I know where I can place them, and I could place plenty more."
"I'll find out," Red promised, breathing very hard. "That's what I'll do, I'll find out pretty damned quick."
It did not occur to his strangely literal mind that when she talked of hundreds she could mean thousands.
He went out and hurried in his usual nervous way along Bond Street, and as he stood waiting for the lights to change at Piccadilly, he saw a man whom he recognized, another runner, named Slater. Slater was walking toward Piccadilly Circus and making surprising speed on his short, fat legs. There was a great intentness about him, Red noticed.
"He's onto a good thing," he told himself. "He can't have been to. Old Fisk's, can he?"
Apprehensive lest Slater had forestalled him, Red rushed across the road to catch a bus, while Slater walked toward a bus stop outside the Royal Academy, heading for Trafalgar Square and the National Gallery.
It was half past ten.
At eleven o'clock, they were to begin the raid on the National Gallery.
# 6: The Theft
The steps leading up to the National Gallery were thronged with children in their early teens, a mixed bag of sexes and sizes, white and black, English, Indian, Pakistani and Jamaican - all awaiting the orders of their teachers who were to guide them round the gallery but who were now inside, seeing to the formalities. Some of the youngsters watched the traffic and the people, the thronged square, the flocks of pigeons and the dozens who fed them, the clicking cameras as pigeons that perched on hands and shoulders and heads were photographed for family posterity.
Slater got off his bus opposite the south side of St. Martin- in- the-Fields and walked to the National Gallery. He kept looking about him, half fearful of seeing someone he knew. A young policeman was walking along the pavement, controlling a crowd of French youths waiting to go in. His back was toward Slater.
Leslie Jenkins came out, and he and Slater passed each other with raised eyebrows; that was the signal; so all was well, and Jenkins had made the first cut in the picture to be stolen.
Slater went inside. It was shadowy and noises were subdued.
Slater turned right, then left, knowing exactly where to go but looking casually at each picture, noticing the groups in front of some, the tired and the old on the small couches, the attendants who stood with what seemed all-seeing patience in the doorways. Now and again, a voice was raised, often an American's; one man, harsh and guttural, could be heard above the rest whenever he spoke.
Slater reached a Velazquez, the portrait of a young man, rich in colours; the youth's dress seemed to be actual silk and velvet, his face actual flesh.
Two people stood in front of it, studying it in detail; others looked at it from a distance. A small crowd sauntered in and a guide began to talk. The words "Spanish school... seventeenth century... early period of harmony... Court Painter to the Court of Spain..." filtered through the subdued hum of conversation and shuffling feet.
Slater heard all this yet was oblivious.
A couple came in and went closer to the portrait, one stretching out a hand as if to touch it; the other said something that made him draw back. A man with a magnifying glass went very close, obviously studying the jewels on the young man's hand - the pearls seemed to be objects one could pluck from the canvas, the diamonds to need only a little extra light to make them scintillate.
Slater went out of this room, but soon came back - exactly as planned. He stayed with a group some distance from the Velazquez until he saw two women approach the attendant. Then he moved forward, hand out-stretched, much as others had done. There was no one nearby.
He pierced the canvas, and his heart leapt.
He cut, and his heart raced.
He stood back, as if admiring, and he felt a quiver of nausea.
He mixed with the crowd, still standing and studying, saw the attendant glance at him and glance away. After a while, he went out again, sauntering, calmer. No one took any notice of him.
On his way, he passed Leslie Jenkins, and he recognized him though he wore a light raincoat and looked very different with a beard, dark and heavy at the chin but curly and fluffy at the cheeks.
A few minutes later, de Courvier entered the gallery.
The scene was almost identical with that which he and his fellow plotters had foreseen. He needed a little more time, that was all, and he had planned for it without saying a word to the others. As he stood looking at the painting, a long-legged girl obviously not English approached the attendant and spoke excitedly in French.
"I have lost my handbag; have you found it here, please?"
Every head turned at the attractive voice.
"No, Miss, I-"
"But I have lost it! All my money is in the bag and I am sure I left it in this room." She spun round, very lovely to look at, and pointed at a couch where four elderly women and a young man sat, looking suddenly embarrassed, almost as if they were guilty. The girl pointed. "It was there, I tell you!"
"I haven't seen it, Miss," the attendant insisted, "but I'll have a look." He walked toward the couch and even those who pretended not to be looking at the scene were distracted; only de Courvier, studying the Velazquez intently, seemed to be absorbed in what he was doing.
He was absorbed.
He pierced and cut and pulled the canvas free, then let it coil beneath his coat as he placed the copy in position. It was masterly sleight of hand; no one who was not intent on every movement he made could possibly have suspected what he had done.
He walked without haste toward the next room.
The French girl cried in sudden delight: "It is there!"
The youth who had been sitting on the couch shifted to one side. There was a small, flat handbag pushed between the seat and the back of the couch.
The attendant, highly gratified, said, "I knew I hadn't seen it."
"Oh, I am so pleased!"
"Better look inside to make sure everything is in it," the attendant advised.
"Yes, I will." There was a tense pause as she opened the bag under the fascinated gaze of everyone present, drew out a wallet, then a passport, talking excitedly all the time. "Money... passport... my air ticket to Paris... ticket for the theatre tonight... I have everything!"
"Very glad it turned out all right, Miss." The man spoke as if this happy outcome was due wholly to him.
There were murmurs of pleasure, congratulations, and relief. The girl went out, the men's gaze following her beautiful legs. The whole atmosphere seemed brighter; there was a buzz of talk, and a crocodile of schoolchildren suddenly appeared.
Outside, de Courvier hailed a taxi, was taken to where his car was parked, at the underground garage beneath Hyde Park. He walked from the spot where the taxi dropped him, and got into his car, a blue Jaguar two and-a-half litre. Taking the rolled canvas from his coat, he wrapped it in a piece of new canvas from the back of the car. Then he lit a cigarette and sat back; for the first time, he was perspiring and showing signs of strain.
A Morris 1100 appeared, shiny red, two youths in the front seat. Neither of them waved to him but each looked his way. A few minutes later, they approached him as he sat waiting in his car. No one else was in sight but in another section of the garage a car started up.
The young men came to de Courvier from either side as he wound down the driving window.
"Got it?" asked one of the youths.
"Yes."
"Let's see."
De Courvier opened the roll enough for them to see the face, and one of them leaned inside and shone a torch brightly onto a corner. He clicked the torch off.
"Okay."
"Now let me see the money," de Courvier said.
The youth who had examined the picture stood aside. The other pushed a box through the window, heavy enough to fall painfully onto de Courvier's knees; he was obviously eager and excited and there was something fresh and even pleasant about his grin. "Don't lose it," he said.
"I won't lose it. Do you want anything else?"
"We'll let you know," the youth said, and he turned away.
De Courvier put a hand on the box, and then lit another cigarette. He was trembling from the reaction, and felt almost suffocated; he must get some air. Once he started the engine, he felt calmer, and he drove slowly from the garage, paid the exit fee, went up into Hyde Park, and drove on the inside lane toward the Marble Arch and Bayswater. He stayed inside the park, still driving slowly, until he neared the Serpentine. It was a grey day and there were fewer people than usual; he had room to park. He smoked two more cigarettes, and then opened the box with great care.
The top bundle was of a hundred ten-pound notes;
a thousand pounds.
The next was of a hundred one-hundred-dollar bills; four thousand pounds.
De Courvier closed the box and drove off.
"Twenty-four thousand in all," he said to the others when they met at his flat in Hampstead. "Eight thousand each, half in dollars, half in pounds. Well get the rest when the picture's been checked and authenticated. Split it up, keep it safe, and don't start spending too freely. We do not wish to attract attention at this time."
Slater said, in a hoarse voice, "I'm going to have a holiday. I'm going to Brighton on holiday, that's me."
"What about you?" de Courvier asked Jenkins.
"I'm going to stay home," Jenkins said. "I'm going to count the dough, that's all I'm going to do. Count the dough."
Later, when Jenkins reached his little terrace house where the three had met to plot this crime, the first thing he did was to make himself some tea. Then he drew the flimsy curtains at the window, pulled up a chair near the one in which he had been sitting, and placed bundles of currency notes on the chair. He drank tea, and gloated; began to doze, and was in a state of euphoria, beautifully tired, gloriously, sensuously pleased with life.
He went to sleep.
While he was sleeping a long-legged girl wearing stretch pants and carrying a shoulder satchel quietly let herself into the house with a key. She drew on a pair of flimsy nylon gloves, collected the notes and placed them in the satchel, and slung it back over her shoulder. She stood looking at the sleeping man, a strangely tense expression on her face. Suddenly, she moved silently toward the gas fire, turned off the gas, let the mantles get cool, then turned on the gas again. Then she went to the meter, near the sink, and put in ten separate shillings.
Throughout all this, Jenkins had not stirred.
Slater reached Brighton Station early that evening, and walked toward the sea front. Since he was used to walking, in spite of his weight, it was not too long before he reached the promenade and the main pier, near the aquarium. Every now and again, he fondled the suitcase he carried, and most of the time he wore a fatuous grin.
He did not notice that he was followed by a slim youth wearing corduroys and an olive-green jacket.
At last, he went across the road, and walked until he found a hotel of a much better class than he could usually afford. He asked for a room overlooking the sea and when he was in the room, opened his case, removed a clean shirt and some underclothes, and then the packets of pounds and dollars. He spread them over the bed with great deliberation, stood beaming for a long time, and then almost reluctantly stepped to the window and looked out. The sea was grey but absolutely calm. A few rowing boats were moving sluggishly, young men at the oars. The crunching of people walking over the pebbles came clearly and regularly, and there was a little hiss-siss-siss of sound of the tide running on the pebbles and then slowly seeping back.
He heard footsteps, and almost at once there was a tap at the door. In sudden panic, he cried, "Just a minute!" He sprang to the bed and pulled the bedspread down from the pillow, covering the notes. Putting the suitcase on the foot of the bed, he belatedly called out, "Who is it?"
"It's the manager," a man called. "I'm sorry, sir, but I forgot your registration form."
Slater, heart still thumping but not even slightly suspicious, went across and unlocked and opened the door.
A young man in olive green simply drove a long bladed knife into his belly.
The crowds went to and fro along the passages of the National Gallery; hundreds of people stood in front of the Velazquez, but no one appeared to suspect that anything at all was wrong.
# 7: The News Breaks
At half past nine the next morning, Gideon entered his office at the Yard, and the day's routine began. It had altered a little since Hobbs had become his deputy, virtually replacing Chief Superintendent Lemaitre, an old friend as well as assistant. Lemaitre was now in charge of a division, probably the job in which he was happiest, but certain mornings Gideon missed him. This was one of the mornings, and he could not even begin to think why. He sat at his big pedestal desk, where there were several trays: "IN," "OUT," "PENDING," "URGENT," "ASSISTANT COMMISSIONER", and, centred on the desk, the files on cases which needed reviewing that morning.
At one time, Gideon would have seen all the Superintendents and Chief Inspectors in charge of these cases, but nowadays Hobbs saw some of them and decided whom Gideon should interview and in what order. This was far more efficient, and freed Gideon for other work; moreover, he had come to trust Hobbs's judgment almost as much as he trusted his own. Yet it wasn't quite the same: nothing was quite the same.
There were six cases; a report from Frobisher in Manchester, about the North Country museum theft; one from Lemaitre's division, about a counterfeiting case in which someone had tried to do the work of the Mint with the new decimal coinage; the almost inevitable investigation into a hijacking case, this time one of a series of thefts of tobacco and cigarettes; the fourth to do with the smuggling of Pakistanis into England in Sussex, the kind of problem which greatly troubled Gideon. Law or no law, he did not like to have to punish people for wanting to live in Britain, and he did not like to think that here in London there were men who were dealing in immigrants almost as heartlessly as others, not so very long ago, had dealt in slaves. He spent more time on this case than the others, but finally opened the next file, which concerned organized crime in the West End and contained reports about men and women suspected of corrupting witnesses, of using threats and menaces and sometimes physical violence to make witnesses do what they were told.
It was an ugly situation, and needed the closest possible study.
The sixth and last case could not have been more different. It was one that the Yard was reopening for its own satisfaction, for there was some reason to believe that a man serving a life sentence for murdering his wife might have been wrongly convicted.
Gideon rang for Hobbs, who appeared in the doorway between the two rooms. Hobbs was his usual composed self, just a little more like a stockbroker or a banker than a Yard man.
"Good morning, Alec."
"Good morning," Hobbs said, and then smiled faintly. "You look almost belligerent this morning."
"I've been reading the hijack business," Gideon said. "We've got to break that wide open before it gets much bigger."
"I wish we knew how big it is," said Hobbs.
"Yes." Gideon brooded. The hijacking and the immigration problem were the ones he should concentrate on, and Hobbs knew that as well as he did. He shrugged his shoulders. "Sit down," he invited, and as Hobbs sat, he asked, "Anything from Brighton about the Pakistani business?"
"Nothing new," Hobbs replied. "I thought you might want to talk about it." That was Hobbs's way of saying, "It's time you went into it more closely, George."
"But there's something else from Brighton," Hobbs said, "and we may be asked to send someone down."
"Oh? What?"
"A man named Slater was murdered in a hotel," Hobbs told him. "As far as Brighton can find out, he opened the door to someone who knifed him, then pushed him onto the bed, covered him over, and locked the door from the outside. A maid wanted to do the room this morning, and when he didn't respond to taps on the door, she opened it with her passkey. The manager ran up when he heard her screaming. The man had been knifed through the stomach - dead for over twelve hours; rigor was well in."
Gideon drummed his fingers on his desk.
"Slater—Slater. I've heard the name in connection with art, haven't I?"
"The Pembroke art theft, seven years ago," Hobbs told him.
"That's it," said Gideon. "He was the van driver."
"That's the man," agreed Hobbs. "There's a rather queer thing about it, too."
"What?"
"A single hundred-dollar bill was found on the floor, just beneath the bed. On top of the dust."
Gideon said slowly, "Has Brighton asked for help?"
"Not yet."
"Who would you send?"
"Frobisher is coming back from Manchester today," Hobbs told him. "They haven't had any luck up there, and he says there doesn't seem any more that he can do."
"If Brighton does ask for help, send Frobisher," Gideon said. "I'd like a word with him before he goes." He hesitated, then lifted a receiver and asked, "Is Mr. Chamberlain in?" Chamberlain was the present Assistant Commissioner for crime, new to the post. There had been a succession of appointments to it, none of them satisfactory, and Gideon would have preferred to go straight to the Commissioner, but protocol prevented this.
"You're through to Mr. Chamberlain," the operator said.
"Good morning, sir," Gideon said formally. "There's been a murder at Brighton, and I think it would be a good thing if we were called in... I understand that we can't insist, of course, but in this case I think we could anticipate things a bit..." There was a long pause, and Hobbs saw Gideon's expression change, saw the hard glint in his eyes. "Very good, sir," he said, and rang off.
It was a long time since Hobbs had seen Gideon look so angry, and he sat very still and silent; Gideon put his right hand in his pocket and took out a pipe with a large bowl. He had not smoked it for years, but he often smoothed the shiny bowl while it was inside his pocket. Hobbs could not recall having seen him take it out for several months; Lemaitre could have told him it was the one certain indication that Gideon was very angry indeed.
In a studiously even voice, he said, "The Assistant Commissioner doesn't want me or anyone else to suggest to Brighton that we would like to be consulted over Slater's murder. It is a matter which must be broached by Brighton to us."
Hobbs said heavily, "I see."
"Did Brighton tell you anything else?" Gideon asked, still very calm.
"No."
"Not about the smuggled-Pakistani business?"
"That's being handled mainly by Sussex," Hobbs said.
"But Brighton is involved."
"Yes."
"I want to go into the smuggling," Gideon said. "The latest report" - he put his hand on the filenames - "names three key men in London who may be organizing the racket. Are you convinced that Riddell is the best man for the job?"
Hobbs didn't answer.
"Well, are you or aren't you?" growled Gideon.
"I don't think either of us would have assigned Riddell to the job," said Hobbs. "He was the best available man when we first needed some information."
"So you're not satisfied with him," said Gideon.
"Not completely, no."
Gideon looked down at his hand, rounded into a fist with the pipe inside it, then clenched and clenched again, making the knuckles show pale against the leathery skin. He looked back at Hobbs, and was about to speak when one of the telephones rang. As if glad of the interruption, he picked up the receiver with his free hand.
"Gideon... Who?... What?" He almost bellowed; Hobbs had never heard him shout louder. He listened for a long time, and then went on in a quieter voice, "When?... Who?... How?..." He listened again, and then said, "Right." He banged down the receiver, seemed to smoulder for a long time, and then said to Hobbs, in a very different tone, "A portrait by Velazquez, valued at over two hundred thousand pounds, has been stolen from the National Gallery, presumably yesterday. Is Thwaites in?"
"He was just before I came in here." Hobbs caught his breath. "May I?" He touched the interoffice telephone and dialled. "Chief Inspectors' Room?... Mr. Thwaites there?... Yes... Hallo, Thwaites. Come along to Mr. Gideon's office, at once."
He put down the receiver slowly, obviously badly shaken himself. Gideon had put his pipe away, as if something of his earlier tension had left him. He was still hard-faced and gruff, but no longer spoke with studied self-control.
"Stay while I talk to Thwaites, will you? Then I needn't tell the story twice." He smiled faintly. "Seems too much for coincidence, doesn't it? An ex-convict involved in an old art theft murdered the day after another art theft." There was a brief pause before he went on. "Alec, we need to replace Riddell. Think of a man who would do a better job on this immigration business, will you?"
As he spoke, there was a tap at the door, and Thwaites entered, massive enough to fill the doorway, a little untidy, and in need of a haircut; his suit needed cleaning and pressing, too; there were stains on the lapels and the lower part of the jacket.
Gideon was fully aware that even for an old-timer at the Yard a summons to the Commander's office could be quite an ordeal, so he raised a hand in greeting. "Come in, Thwaites. Sit down. Cigarette?" All of this gave Thwaites time to overcome any momentary qualms. Gideon pushed a box of cigarettes across the desk and Thwaites took one and lit up. "Mr. Hobbs told me yesterday about your inquiries out at Hampstead and the connection with Sir Richard Falconer."
Thwaites said, "And young Judd, sir."
"And also the talk that there is a big buyer around who will buy what he wants and ask no questions," Gideon said. "Now we've just heard of a major art theft that is going to cause us a lot of trouble unless we can do a quick job on it."
Thwaites's nervousness vanished in the instant.
"Something for me, sir?"
"Yes," Gideon said, and went on with great deliberation: "The Velazquez 'Prince' has been stolen from the National Gallery."
Thwaites, already leaning forward on his chair, sat perched with one hand stretched out as if fending off danger, his lips parted, showing very crowded but very white teeth in his lower jaw. He had a heavy jowl and a slightly bloodhound look, and his greying hair was thinning to a large bald patch. It was as if he had been struck dumb; even his breath seemed to stop.
Both Gideon and Hobbs watched, fascinated.
Gradually, Thwaites came to life, breathing inward slowly and then exuding a long, deep breath as he settled slowly back in his chair.
"How the hell did they do it?" he demanded gustily.
"That's what you're to find out," Gideon told him. "No one yet knows when it was done, but it was either during the night or early this morning. They open to the public at ten o'clock; a woman artist is making a copy of 'The Prince' and can't work when crowds are about, so she comes in before the gallery opens and after it closes. She was later than usual this morning - arrived about nine o'clock. She put up her easel and got her paints out, then studied the hand, which she was painting yesterday. Apparently, she knew at a glance that the texture was different, and when she took a close look, she raised the alarm. The Director was away, but the Acting Keeper was told the moment he got in. He telephoned me." For the first time, Gideon paused, only to ask, "What help do you need, Thwaites?"
Thwaites hesitated, pursing his lips.
"You can have as many men as you want," Gideon said. "This is going to cause a sensation, and we don't want another fuss like the one we had over 'The Duke of Wellington.'"
"I wasn't really thinking of what help I would need," said Thwaites, and he seemed a little embarrassed. "Excuse me, sir, but isn't this much more Mr. Frobisher's job?"
"He won't be back until mid-afternoon," Gideon said.
Thwaites raised his arms from his side, then let them flop in a gesture of resignation, and he looked as doleful as the bloodhound he resembled when he said: "Then I'd like six or seven men, sir, and a full team of experts. I'd like to treat this as seriously as a murder investigation."
"Right!" Gideon pushed his chair back in a gesture of dismissal. "Mr. Hobbs will see you get what you want. You get over to the National Gallery as soon as you can, and—"
"Excuse me, sir," Thwaites interrupted, with a kind of gloomy daring, "but would you go over yourself? Or ask Mr. Hobbs—?" He broke off, confused and yet doggedly persistent, glancing at Hobbs with a no offence-intended expression in his brown eyes. "I think it would be wise if one of you did."
"All right," Gideon said. "How long will you need to get your team ready and be over there?"
"Should be able to do it comfortably in half an hour, sir."
"Good. Ask for Mr. Peebles, the Acting Keeper. One of us will be with him by the time you get there," Gideon said.
"Thank you very much indeed," said Thwaites, obviously greatly relieved.
Gideon also felt a sense of relief, a sense almost of pleasure, and as Thwaites went out he remembered the cold rage that had surged within when the Assistant Commissioner had spoken to him as if he were a cadet not yet out of training. He still felt resentment but no longer needed to exert himself to control his mood, and he welcomed the idea of going to the National Gallery himself.
Hobbs was speaking into the telephone.
"Send a car round at once for the Commander." He put down the receiver and asked, "You going, sir, or shall I?"
"I'll go," said Gideon. At once he was doubtful, seeing an expression in Hobbs's eyes which he read as disagreement or disapproval. He thought a little more swiftly than he spoke. "Thwaites seems to think that the National Gallery officials will prefer to deal with some top brass to begin with. Do you know anyone over there?"
"I know the Director, but he's out of the country," Hobbs answered. "In Italy, I believe, studying the way they've restored some of the paintings after the flood at the Pitti Palace." He paused a moment, then continued, "One thing Thwaites ought to be told, if he thinks he's going to be out of his depth socially or intellectually." Ah.
"What?" asked Gideon, almost ominously.
"Not to be a bloody fool," said Hobbs. "They're not interested in the old-school tie; all they need to be sure of is that he knows what he's talking about and doesn't look on 'The Prince' as a bundle of banknotes."
"Oh," said Gideon, completely taken aback. But it wasn't long before he began to smile. Then he chuckled. "Good!" he said heartily. "I'll make sure they know what he's like; you brief him." He pushed his chair right back, stood up, and was at the door when he stopped to turn round. "This Pakistani immigration business."
"Yes?" said Hobbs.
"Any idea who'd be right to replace Riddell?"
"Yes," answered Hobbs promptly. "I'd like Honiwell if he weren't on the Entwhistle case. He'd have to be fairly mobile for the immigration inquiry, which might make it difficult for him, to do both."
"We'll think about it," said Gideon.
"We'll think about it," Hobbs repeated to himself when he was alone in his own office. His set expression, which so often gave him a look of arrogance, eased into a smile and the expression in his eyes was warm. "The day will come when he really will accept me." Rather in the manner of Gideon, he chuckled; then he picked up the telephone to speak to Thwaites.
# 8: The Whispers
As Gideon stepped into the car that was waiting for him at the foot of the Yard's steps, Big Ben struck eleven o'clock, the notes booming out sonorously and with doom-like inevitability. Beneath and around the tower, London's traffic surged in its unending variety, and a few Members of Parliament, there for early work in committee, drove into the courtyard of the House of Commons. A group of late-season American tourists were looking, perhaps with disbelief, at the statue of Abraham Lincoln: the man who had best defined democracy keeping a silent watch on a citadel of democracy which was so often besieged with invisible enemies. There were sightseers in Whitehall, too, the usual groups about the statue-like Horse Guards, sabres drawn and helmets shimmering. As Gideon passed, he saw a gawky child reach up and touch a horse's nose.
The traffic lights favoured Gideon, and his car swept across Trafalgar Square and then to the National Gallery. There crowds of people milled about the pavement and up the steps. On the steps themselves and at the entrance were thicker crowds, and as Gideon stepped out he heard plaintive calls and protesting and some strident voices.
"Why don't they open the doors?"
"They won't let us in, that's the trouble."
Policemen, keeping order and preventing the crowd from surging onto the road, saw and recognized Gideon. One saluted.
"Good morning, sir."
"Morning. Clear a path, will you?"
"Yes, sir." The constable had a hawkish face but a soft voice. With a kind of terrier patience, he forged a path through the crowd, and as Gideon stepped onto the big porch, he saw three other policemen guarding the doors, while a youth who had come up the staircase on the other side called: "Mr. Gideon!"
Gideon looked up - and a camera flashed.
"Excuse me, Commander," an older man called out, "but what's happened?"
Someone else began, "There a rumour that the—"
"As soon as anything's known for certain, there will be a statement," Gideon assured them. He pushed past the doorway and into the near-deserted hall, the South Vestibule. Here at the entrance turnstiles and the sales counters, assistants stood about aimlessly; two men on duty at the cloakroom, ready to collect cameras and umbrellas as well as hats and coats, looked baffled.
Gideon was thinking, If we don't get the place open soon, we'll have half Fleet Street here.
A tall man wearing a velvet suit, tight-waisted and looking vaguely old-fashioned, with a floppy bow tie and long but well-groomed hair, came forward.
"Commander Gideon, how very good of you to come in person." The man stretched out his hand. "I am David Morcom, the Assistant Keeper." His hand looked pale, the skin and flesh almost translucent, and Gideon was prepared to grip momentarily but not too firmly.
But Morcom's fingers bit into his hand like steel wire.
"My men are on the way over," Gideon said. "I came to see if there's any immediate thing I can do. We don't want another Goya affair."
"My God, we don't!" exclaimed Morcom. He gave an unexpectedly charming smile. "The one reassurance I needed was that this would have the most urgent attention, and I don't need any more telling. Would you like to see the room the picture was stolen from?"
"Yes," Gideon said, "I certainly would."
"We'll go along here," said Morcom. "Oh, Commander. I have to make a decision very quickly about opening this morning. You've seen for yourself what a crowd there is outside. What do you think I should do?"
"Give me a little time to think that over," Gideon said.
They were walking, Morcom with very spritely step, Gideon with his customary deliberateness, along the galleries to the right, past attendants standing in little groups talking. All of them stopped at sight of the two men, and two or three times a whisper floated after them.
"That's Gideon."
"That's the Commander himself."
"That's Gideon—Gideon—Gideon—"
It was like an echo, growing fainter and fainter.
Gideon, though used to finding his way about unfamiliar places, tried but failed to keep track of the different rooms they entered. Why did every picture gallery and museum seem like a maze? He found himself thinking of a man plotting a theft here. It would be so easy to disappear from one room and virtually vanish. If the man had an accomplice who slipped him a raincoat, say, or a cap, or if he had either one concealed underneath his jacket, he would be able to confuse all descriptions of him. But this was no moment to ask what the security precautions of the museum were, and in any case Thwaites was the man to check that. Frobisher certainly knew, of course. Whatever they were, the actual layout of the building would make a getaway comparatively easy.
They entered a small room - XLJ, Gideon noticed. There were more attendants here than in any of the other rooms, and two men who were obviously senior in rank. In front of a picture now cordoned off was a frail-looking, grey-haired woman with an easel by her side, sitting on a canvas folding chair.
Morcom went straight up to her.
"I'm sorry to have to keep you, Mrs. Templeton, and it won't be a moment longer than I can help. The police experts are on their way. This is Commander Gideon."
Mrs. Templeton got up with surprising agility.
"I've heard of you, of course," she said, in a deep pleasant voice. "And you mustn't worry about how long you need me, Mr. Gideon. I have nothing to do, and to tell you the truth this is quite exciting." She smiled at him. "Is that very wicked?"
"Very," Gideon replied dryly, and her eyes had laughter in them. "I've heard what a help you've been. If you hadn't been so quick to notice something wrong, Mr. Morcom might have been much longer realizing the picture had been substituted." He moved a little closer to the one in the frame. "Would you call it a good copy?"
"I've seen a lot worse," said Mrs. Templeton.
"Unless it was scrutinized closely, most of the gallery staff would have been fooled," Morcom interpolated.
"Is it possible to say who made the copy?" asked Gideon.
Morcom nodded. "I think so. We keep records of anyone who's had permission to do one - they should be quite comprehensive."
"I think you'll find it's by Totter, and was done fifty years ago," said Mrs. Templeton. "I saw it for sale in Paignton, I think it was, about seven years ago."
"That's very useful information." Gideon looked at the woman appreciatively, then turned to Morcom. "Let Chief Inspector Thwaites know, will you? If we can find who owned or bought it lately, it will be a great help. How was the job done? Do you know that yet?"
"Cut from the frame, obviously with a special instrument, possibly a diamond-edged cutter," answered Morcom. "The whole thing must have been done in a matter of seconds. It was almost like sleight of hand." He sounded exasperated.
"Sleight of hand," Gideon echoed. "Were you here yesterday, Mrs. Templeton?"
"In the morning, yes, until a little after ten o'clock. And, yes" - her youthful and alert eyes twinkled again - "the genuine Velazquez was here then, beyond any possible doubt. You see, I was painting the left hand, and the thumb is slightly deformed - wrinkled, perhaps I should say. I was trying to copy it, but I fell very far short, and yesterday morning I worked on in the hope of catching just the right mood. I couldn't. I gave up in despair and told myself that it wasn't worth spending time on. That's why I didn't come last night. But this morning I simply had to try again, and I brought a special glass." She picked a small magnifying glass up from the easel. "I thought if I could enlarge it I might be able to come near, but—it simply isn't the same thumb."
"Excuse me, sir," one of the men standing by said.
Morcom glanced up at him.
"Yes? Oh, Commander, this is our head security officer, Mr. Gordon Smith."
Gideon nodded.
Smith said mechanically, "Glad to know you, sir," and then went on: "Mrs. Templeton told us about this and I sent for the Fortuna Press book of Velazquez; we have some in stock. It shows the thumb very clearly, sir." He moved to a couch in the centre of the room and picked up a heavy book with an illustration of Velazquez's dwarfs on the front. "Would you care to look, sir?" He opened the book, which he had to support on both arms.
"Good idea," Morcom said. "Thank you." They all peered at the full-page plate of 'The Prince.' Mrs. Templeton squeezed between Gideon and Morcom, and she pointed with a slender, nicely shaped forefinger at the left hand. Gideon glanced at it and then at the portrait in the frame; there was no doubt at all about the difference in the thumbs. Mrs. Templeton was quite right.
"Thank you," Morcom said.
"No doubt about it." Gideon agreed. "I—" He broke off, seeing Thwaites, with an attendant, hovering in one of the two doorways. "Ah, Chief Inspector." Thwaites came forward; Gideon made brief introductions; Thwaites called in more men. "The one urgent matter," Gideon said, "is whether to open the gallery to the public."
"I should, sir," Thwaites advised. "If we could have just this room - and perhaps those leading directly to it - closed for the time being, that should be enough for us. Too many people have walked about already for there to be any point in keeping the whole place shut."
"That's a relief!" said Morcom with obvious satisfaction. He turned to one of the senior attendants. "You'll do what's necessary, Smith, won't you? And I needn't ask you to give Chief Inspector Thwaites and his men every possible assistance, need I?"
"Be absolutely sure I will, sir," said Smith.
"And as soon as you can let Mrs. Templeton—" began Morcom.
"Oh, please don't rob me of my privileged position," the artist said. "If there could be a cup of coffee occasionally, I would be enthralled to stay here. And I know the gallery very well. I might even be useful."
There was a general laugh before Gideon and Morcom went off. They did not speak until they were at the main hall, when Morcom asked: "What about the press, Commander?"
"No need to keep anything from them unless you want to," Gideon answered. "My advice would be to tell them everything - you will probably need their help before long." As Morcom nodded, Gideon said, "I'll tell my men outside to regulate the flow of people coming in."
"You're very helpful," Morcom said. "Thank you for everything."
Gideon nodded, and went out.
The crowd outside was now at least a thousand strong - perhaps nearer two thousand - and a dozen policemen were controlling them, but the police had to stand in the road and there was a diversion barrier at the turning into the North Vestibule. A police sergeant pushed through the crowd and met Gideon at the foot of the steps.
"Any news from inside, sir?"
"No," said Gideon. "They're going to open the doors in a few minutes. Let them in a couple of dozen at a time."
"I'll do that, sir."
"We're going in!" a girl cried out.
"They're opening the doors!" a man called.
Gideon, aware of a dozen cameras trained on him, saw a television team on the other side of the road, their backs to the fountains and Nelson's Column, the camera whirring. Reporters were also thick on the ground, and as they asked questions, he gave the same stock answer: "Chief Inspector Harold Thwaites is in charge.... He'll answer any questions that Mr.Morcom can't."
No one pressed for more.
Gideon reached the end of the street, where the road led round toward the steps of St. Martin's and Leicester Square. He crossed over, and stood on the top step. The crowd looked huge from here, the kind of scene that was commonplace at a political demonstration or a ban-the-bomb rally. Yet masses of people still stayed with the pigeons; probably half of them had no idea of what was happening at the National Gallery.
Well, they would know when the evening newspapers came out!
He walked to Whitehall, then along it toward Parliament Square, enjoying the feel of the pavement beneath his feet, glad he had sent the car back. There was something in the very air and look and feel of London that warmed and touched him with both affection and pride. He paused as he always did for a fraction of a minute opposite the Cenotaph, then went on. When he had first paid that respect to the dead, it had been out of a great sense of gratitude to those who had died. Now? Had it become virtually a habit? Was there in fact a little stubbornness in the pause, a conscious effort to make himself do what he felt he should?
He could not honestly be sure.
Once past the Cenotaph, he moved more briskly and, within five minutes, was in his office. There was a note on his desk: "Honiwell would like ten minutes - I've told him 2.30. I'm up in Records. A." Gideon sat down, pulled a telephone toward him and dialled the number of the Commander, Uniformed Branch, his opposite number.
There was no immediate answer, and for the first time that morning Gideon had a few moments to relax. In those moments, everything that had been discussed since he had reached the office passed through his mind in swift, fragmentary thoughts.
The Commander, Uniform, answered at last.
"Hallo, Charles," said Gideon. "Gideon here. You chaps are a bit pushed over at the National Gallery. Did you know about the theft?"
"Yes," the other replied. "I had an extra dozen men detailed."
"I should have known! With a bit of luck, the real pressure on them will be off in an hour, but there'll be more than the usual crowd all day and the press will be in strength, too."
"We'll cope," said Uniform dryly. "Any news yet?"
"Looks like a very clever job to me," Gideon said cautiously.
"These art thefts," remarked Uniform. "You can never be sure what they will do with what they take. Had much art-theft trouble lately?"
"No more than usual," Gideon answered. "Thanks, Charles." He rang off, knowing that if he hadn't made the call it might have looked as if he were usurping Uniform's authority. The different departments at the Yard worked together extremely well, but the machinery needed oiling sometimes. He was reminded of his rage when the Assistant Commissioner had tried to teach him his job. He laughed, but it didn't seem really funny.
His interoffice telephone rang, and he lifted the receiver.
"Gideon."
"Sorry to worry you," a man said, and immediately Gideon recognized the voice of Superintendent Thomas Riddell, who was in charge of the investigation into the smuggling of Pakistanis into the country. "Can you spare me half an hour or so?"
"When?" asked Gideon.
"Now, if possible," said Riddell. "I think you should know what I've discovered."
"All right, in fifteen minutes," Gideon said, and rang off.
Riddell had annoyed him, as Riddell often did. It was difficult to put a finger on the reason, except that the man too often presumed. It was more his manner than anything Gideon could really identify.
He rang for a messenger.
"Get me some coffee," he ordered. "Nothing to eat; I'm in a hurry." With anyone else, he might have said, "Bring a pot and two cups," but on this occasion it did not occur to him; he didn't want even a slightly social relationship with Riddell.
He did need to brief himself on the case Riddell was preparing.
# 9: Cause for Disquiet
There were problems in the life of a policeman which did not occur in the lives of others - not even in those of highly placed civil servants. No policeman, for instance, could outwardly espouse a political cause, because that would imply some degree of bias or prejudice. And no matter what he felt, no policeman could express his opinions of certain other aspects of the life of the community. Two problems arose out of this for Gideon. First, it cut a policeman off from communication with his fellow men, never a good thing; and second, it made difficulties in finding out the truth, since it was seldom possible, without this communication, to understand both sides of any question; one could listen but could not discuss wisely.
Yet a policeman had certain prejudices, certain interests, certain enthusiasms, and a policeman had instinctive reactions which training and self-discipline could never prohibit.
Above all, a policeman had to see a man as a man, not prejudge him because of colour or creed, or even because he had a record as long as his arm. Such absolute objectivity was never easy, and however liberal or understanding one was, even if one had not the slightest racialist feeling, it was impossible not to be aware of the tensions over colour in England. There were the Fascist types who hated black or coloured people without cause or reason, and there were more, so many more, who had come to believe that the immigrants did harm to the society, the community, even to the economy. Moreover, there were those who accepted the immigrants without question provided they were in the next town, or at least in a different section of their own town.
Policemen must not have prejudices, and yet prejudice existed, and the subject of racialism was as rife at the Yard as it would be anywhere else.
The messenger brought in a cup of steaming coffee, with plenty of cream and only a little sugar. Gideon had barely pushed the empty cup aside when there was a tap at the door.
"Come in," he called.
The first thing that struck him about Riddell was how the man had aged. He was handsome in his heavy-jowled way, though his brown hair was now streaked with silver, and his eyes, which had once been bright, were dull. He had put on weight, too.
"Good morning, Commander."
"Morning," Gideon grunted. "Come and sit down." Riddell sat and began to fumble in his pocket. "Smoke?" asked Gideon, pushing cigarettes across his desk.
"Ah, thanks," Riddell said. He took one and lit up. Even then, he seemed to have some difficulty in coming to what he had to say, and Gideon prompted him.
"What's this you've found out?"
"The smuggling is very widespread," Riddell announced positively. 'There's much more of it than I realized, or else I'm being pessimistic."
Gideon nodded, puzzled because the man was obviously troubled; he almost warmed to him.
"The fact is that at least three men are involved," stated Riddell, at last. "I can't offer proof yet, but it's only a matter of time before I'll be able to. Two of the men are Londoners, the third is an Indian - one of the early immigrants himself. None of them has ever been involved in crime before; certainly none has any record of any kind. The two Londoners own a lot of slum or near-slum property; the Indian rents the houses from them and lets off rooms at exorbitant rents. There's a day shift and a night shift for the beds, if you know what I mean. Wouldn't wonder if they share the women, too, although God knows there are enough of them to go round."
He paused, and the way he looked at Gideon suggested that he knew that Gideon, in those last few seconds, had hardened against him.
Gideon waited, and after a moment's silence Riddell spoke again: "Oh, to hell with it, George! I can't stand them. I don't think they should ever have been allowed in. Give them a few years and they'll have flooded us out. If I had my way, I'd send them back bloody fast!" Riddell got up and began to walk about, speaking in a low-pitched voice and drawing fiercely at his cigarette. Gideon, startled by the outburst and concerned with Riddell's obvious emotion, did not interrupt. "That's how I feel," went on Riddell. "I don't mind admitting that when you gave me this job I rubbed my hands. Now I can get some of the bastards, I told myself; now I can send them back where they belong." He spun round and faced Gideon, his eyes suddenly ablaze. "I saw one of the immigrants who'd been smuggled in three weeks ago. He lives in a hole under the stairs; there's no other word for it - rat-infested and filthy. It made me want to vomit when I saw it. And he's got no money, hardly any food. Lives more like an animal than a human being."
He moved stiffly toward the desk, stubbed out the cigarette, took another and lit it, and moved back two paces.
Gideon nodded, not 'wanting to interrupt in case he stopped the flow and so dried up the passion.
"These bloody sharks took all the money he had, promised him work he can't get, and will let him rot," Riddell said. "Now there are two damned good reasons for wanting to stop the smuggling."
Again, Gideon nodded.
"And where does that leave me?" demanded Riddell. "Right in the middle, George. I can't think straight about it. I can't even think for myself over it, let alone think as a copper. I'll tell you something else. Every time I look at one of them, I think Out, you bitch, or Out, you son-of-a-bitch, and if I had a man working for me who was half as full of hate, I'd fire him. That's what I really discovered, George. I can't go on with the job; it's got me facing two bloody ways. So—will you take me off?"
Gideon pursed his lips, then bent down, took out whisky and two glasses and a siphon of soda, and poured a stiff drink for Riddell and a mild one for himself.
"Cheers," he said. "I'd like to think about it, Tom."
"Do you really have to?" Riddell put the glass to his lips, muttered "Cheers," and drank deeply. He had needed that drink.
"Yes," Gideon said. "Yes, I do. How long have you been feeling like this?"
"About a week," answered Riddell gruffly. "Tell me I ought to have told you before and agree. George—Commander—rather than go on with this job, I'd resign. I'm not joking: I'd resign. I can go any time; a year or so won't make any difference."
"A few days won't make any difference, either," Gideon reasoned. He knew that in fact if Riddell resigned now instead of waiting until he was fifty-five, he would lose a substantial proportion of his pension. "I'll see you this time on Friday, with a decision."
"It won't make any diff—" began Riddell, and then broke off, and gave an almost sheepish grin "Sorry. Ought to know better than to think you couldn't think up something to make me change my mind. Thank—er—thanks for letting me blow my top." He finished his drink. "Twelve o'clock Friday, then."
"Yes." Gideon tapped the report. "Is this up to date?"
"On facts, yes."
"But not on your assessment of the facts?"
"I don't trust myself to make an assessment," Riddell muttered.
"Well, I do. And I want one by ten o'clock Friday morning," Gideon ordered. "Your handwriting will do, no need to get it typed. But I don't want anything left out, Tom; I want the lot."
After a pause, Riddell answered, in a much milder voice, "Yes, of course: I'll do it, make a thorough job of it. And the report may shock you, George."
"From what you say, there may be a lot to be passed on to the Home Office," said Gideon.
"I'm no bloody welfare officer," said Riddell. "But you're right." He paused again. "Anything else you want from me?
"No, thanks."
"Right!" Riddell put his glass down, and went toward the door. "See you."
He went out.
Gideon felt as if he had lived through a sudden, furious storm and, when the door closed, was almost breathless. He sat, Buddha-like, for several minutes, and then suddenly he laughed; but there was no humour in the laugh and little in his expression. He had not expected to forget the Velazquez theft so quickly, but it had gone right out of his mind.
One of his telephones rang, startling him. He let it ring for a few minutes before lifting the receiver.
"Gideon here."
"Hallo, George. How are tricks?" It was the brisk and breezy voice of Lemaitre, whom Gideon had been thinking about earlier in the day. "Gotta bit of news for you I thought you'd like to know."
"What's that?" asked Gideon cautiously.
"I'm pretty sure I know where they're making the new decimal-coinage slush," stated Lemaitre.
"Pretty sure" was characteristic of him, and nine times out of ten he would be right. But on the tenth occasion he might simply have built up a case out of a single piece of information into which he had read a great deal of significance.
"Sounds good," Gideon said, still cautiously. "Where?"
"An old foundry, in my manor. A place on the river, George. Only about a mile from the Mint itself - how about that? What I want to do is raid the place."
"When?" asked Gideon.
"Tonight," answered Lemaitre. "Nothing like striking while the iron's hot, George - or catching the metal while it's molten." Lemaitre could hardly control a guffaw of laughter, so pleased was he by that turn of phrase. "The thing is, I'd need some help from the Thames Division."
"Asked them yet?" inquired Gideon
"No," said Lemaitre. "Wanted to clear it with you first"
There was much more than there appeared to be behind that simple statement: a hint of some feeling or conflict between Lemaitre's division, on the land, and the Thames Division. It was a good thing to be warned such tension existed, just as it would be a bad thing to show that he felt it.
"How much help would you need?" asked Gideon, suddenly realizing that in fact Lemaitre might want much more than it was reasonable for one division to ask of another.
"Oh, a couple of patrol boats, in case we flush our birds and they try a getaway on the river," said Lemaitre, and added airily: "Nothing much, really."
Gideon grunted. "I'll have a word with Thames," he promised.
"Thanks a lot, Gee-Gee," Lemaitre said, with more heartfelt thanks than the situation merited.
Gideon rang off and immediately put a call through to Thames Division. The Superintendent in charge wasn't in, and he was put through at last to Chief Inspector Singleton, who had just completed a successful investigation into the use of the Thames for distributing stolen jewels.
"Good morning, sir," said Singleton.
"Nice job you've just tidied up," remarked Gideon.
"Good of you to say so," replied Singleton. "Easy enough when I knew the angle, though. What can I do for you, sir?"
"You can have a couple of launches standing by tonight to liaise with a land raid by NE Division." Gideon knew that he might get a reaction from Singleton which he would not get even from the Superintendent in charge.
To his astonishment, Singleton chuckled as if with high delight.
"So it worked," he said.
"What worked?"
"The tactics I used with old Lem!" There was a brief pause, and then, obviously as Singleton remembered he was speaking to the boss, a sharp exclamation and silence.
"All right, let's have it," growled Gideon.
"Er—sorry, sir," said Singleton. "It—er—it's nothing really. One or two of Mr. Lemaitre's men and one or two of ours have been getting on each other's nerves lately, and Mr. Lemaitre knows it, so instead of coming direct to me, he came to you. Bit silly, really, but you know what we old coppers are."
"I know," said Gideon. "Better make it three launches."
"Will Mr. Lemaitre get in touch with us about details?"
"Yes," Gideon said, any annoyance he had felt fading. "Bit silly" was right: policemen of such age and experience should not behave like children, but it happened sometimes and out in the divisions a sense of isolation could develop, breeding pettiness. In place of his annoyance was a question he wanted to ask, but he could not bring himself to do so. Last year, a Metropolitan Police officer named Carmichael with a long and distinctive record had attempted to murder his wife. The officer, a friend of Singleton's, had been tried and found guilty; he was serving a sentence of ten years' imprisonment. His wife was now living with another man, a most likable man, who would marry her the moment the divorce came through. There had been talk of Carmichael suing for divorce from prison, but Gideon had heard nothing about this for some time.
Singleton was the most likely person to have any news.
All these things flashed though Gideon's mind in a split second. Then Singleton said: "There's one other thing, sir, while you're on."
Carmichael?
"What?" asked Gideon.
"You remember the time Jenkins was mixed up in that art theft - the time we nobbled him when he was trying to get away in a cargo boat with some of the loot?"
Gideon's interest flared up.
"Yes, I remember very well."
"Funny thing about him killing himself or being accidentally gassed last night, wasn't it?" Singleton observed. "Especially as his old pal Slater was murdered in Brighton last night, too. One of our patrol-boat crews saw them together on the embankment the other day, and reported it when they heard what had happened. They had to go in close to have a look at some flotsam. Very peculiar, isn't it, sir?"
"Yes," Gideon said. He could pretend that he knew about Jenkins's death, or he could leave Singleton with a real glow by telling the truth. "I knew about Slater," he said. "Jenkins is news to me. Thanks." Then he added, with a faint laugh in his voice, "Look after N.E. Division, won't you?"
He rang off on Singleton's delighted chuckle, but did not echo that laughter as he pressed for Hobbs, knowing he might still be in Records.
But Hobbs came in, and they began to speak simultaneously.
"Have you heard—" Hobbs broke off.
"Have you heard—" Gideon broke off. "About Jenkins?" Hobbs asked quickly. "Just," said Gideon. "Why didn't I know before?"
# 10: Two in One
Lucy Jenkins heard the shop doorbell ring, and put down a rag with which she'd been dusting an old, very fragile Italian carved frame. She also heard footsteps above and they were reassuring. The Fisks were home, and she need make no decisions of her own, so it did not trouble her when she saw Red Thomas halfway between the door and the stack of pictures. He glanced at her almost furtively and said something she did not understand.
"Sorry about this, Lucy."
Mechanically, she said: "That's all right."
"Can I have another dekko?"
"Take as long as you like."
He began to go over the pictures again, and she noticed that instead of making a quick glance he was examining each one more closely, obviously hoping to find some special picture. One part of her mind, the part which trained itself, reflected that he must have sold the two pheasants well and the buyers had sent him back to see if there were others by the same artist. She was familiar with his peculiar form of nervous tension, the way he would stand back, rake a picture from every angle, and then turn it over and put it away with finicky precision.
Then she saw two men on the other side of the road, looking toward the shop. One was a policeman and - again with the self-trained part of her mind - she knew that the second one was a plainclothes detective. She was not alarmed in any way but a little sorry for Red, because obviously the police were investigating him.
"Red," she said, in a quiet voice.
"Hey?" He didn't look up.
"There are two policemen watching you."
"So let them watch."
If he wasn't concerned, there was no reason why she should be, so she went on with cleaning the picture. Something fell heavily upstairs: Old Fisky, dropping a boot; it was always such a performance when he put his big heavy boots on. The two policemen came across the road, and Lucy ducked out of sight but listened. She had a strange sense of panic, a carry-over from those early days when the police had come for her father, as they so often had.
The doorbell clanged.
The sounds above were heavier but not so loud - Old Fisky, walking about to get his feet comfortable in his boots. Lucy had subdued but frightening palpitations and leaned against the bench, rag in her right hand, left hand supporting her body against the wall.
A man said, "Is Miss Jenkins in?"
She could hardly believe she had heard right. These men wanted her.
Red said, swallowing the words, "In the back."
A man called out, "Miss Jenkins, are you there?"
They had come for her.
Her panic rose wildly; her heart thumped like the clumping of Old Fisky's boots. They had come for her, and she hadn't done a thing. She just worked here, just did what she was told. She felt so weak that the frame slipped and a piece of the gilt carving broke off. She could hear her own breathing, as if she were drawing and exhaling breath through some kind of fog.
"Miss Jenkins?"
The two men appeared in the doorway, the uniformed one in front. She saw him but did not understand the expression on his face. He was young and not very big or impressive He had a broad nose, rather squashed, and a slight hare lip, not enough to deform or to give him any speech impediment; and he had big features and the softest, brownest eyes. But all she saw was a face beneath a policeman's helmet. She almost fainted, was aware of the man moving, and then, suddenly, that he was supporting her.
"Take it easy," he said very gently. "Take it easy."
A strange thing happened to Lucy Jenkins, something she did not understand, did not think about for some time. She felt calmed. The panic subsided and her breathing became much more even. The policeman had one arm round her shoulders and that arm seemed to blanket her with warmth.
He repeated, "Take it easy," and after a few moments asked, "Are you all right?"
"Yes—yes," she said huskily.
"You know, then," he said, still holding her.
"Know—know what?" she asked.
He did not respond immediately, and very slowly, almost with reluctance, he took his arm away. The plainclothesman was standing in the doorway leading to the shop, and Old Fisky was in the doorway leading from the stairs to the flat above - Old Fisky, a shrunken giant, clothes loose and baggy, his scraggy neck encircled by his big, deep collar. The policeman was now a yard away from her, oblivious of the others, and she became aware of his glowing brown eyes. She sensed something else, too, something remarkably like the protective warmth of his arm; soothing.
"I seem to have jumped to conclusions," he said. "Obviously you don't know."
Both the other men had the sense to stand still and silent.
Lucy made an effort, and asked, "What am I supposed to know?"
"About your father," the policeman answered.
She was startled into harshness, spurred by a bitterness which was part of her.
"Oh, him. Has he been picked up again?"
"No," the policeman said gently, and he drew a step further away. "He died yesterday."
The word "died" made no immediate impression on her at first, and rather stupidly she echoed without thinking, "Died?"
"I'm sorry. Yes."
She began to realize what she was being told and, in a different kind of panic, cried:
"But—but I didn't know he'd been ill!"
"He hadn't been ill," the man said.
"You mean he had an accident?" Real concern showed for the first time, and she felt an easing of the harshness and the bitterness she had always felt toward her father.
"A kind of accident," he told her. "He was gassed in his room. The gas was turned on, and he dropped off to sleep."
Lucy suddenly remembered that her father had always been very careful with gas. This was the last thing she would have expected. But she did not say so. "Never speak without thinking and never talk to the cops," her father's voice seemed to say in her mind at that moment.
"Poor old sod," she said flatly.
"You had no idea?" asked the policeman.
"Not the faintest."
"When did you last see him?" the plainclothesman asked. He had a grating voice and a hard face and cold, cold eyes. As he spoke, Old Fisky moved for the first time, not interrupting and not distracting the others. He reached the desk behind Lucy and stood still, looking down at the gilded frame and the small chip that had fallen off it.
Lucy seemed to consider for a long time, and then she said, "It would be three years ago last Christmas."
"What?" exclaimed the plainclothesman.
"Good God!" said the policeman. "You haven't seen him for nearly four years?"
"No," answered Lucy, and after a long silence she went on: "And I didn't want to. He didn't want to see me, either."
"Did you write to him or communicate in any way?" asked the plainclothesman.
"No, we didn't have anything to do with each other after he came out of prison," she answered. "We didn't have anything in common - we never did, really."
"There's one thing you did do," said Old Fisky, startling them by speaking; he had a very deep, unexpectedly firm voice. "You sent him a Christmas card with a five-pound note in it, every year."
"Supposing I did?" Lucy coloured deeply as if caught out in some shameful deed. "After all, he was my father." She looked defiantly from the policemen to Old Fisky. "It's none of your business, anyway."
"I know, my dear," the old man croaked. "I shouldn't have spoken out of turn."
"Forget it," said Lucy, straightening up.
The plainclothesman switched to Fisk: had he seen Lucy's father, or the man Slater, Dicey Slater? No. Had they been in the shop - sometimes they acted as runners for collectors and buyers. No. Had anyone been in here lately, for any special artist? No. Had he heard of any buyer, underground, who would buy valuable paintings without asking questions? No. Had he had any strangers in? Strangers were always coming in. Had he had any strangers in, buying for the trade? He wouldn't know about that if they didn't tell him. If he heard of any big undercover buyers, would he be sure to tell the police? Yes, of course, he always cooperated with the police.
"I'll look in once or twice a day," the policeman said, with an air of great righteousness.
The plainclothesman nodded, and the two policemen went out, glancing at but not talking to Red Thomas; obviously the plainclothesman had wanted him to hear. When they had gone, Old Fisky put a hand quite impersonally on Lucy's shoulder, but didn't speak. Red appeared in the doorway, eyes darting from one to the other.
"What's up, Mr. Fisk?" he asked.
"How should I know," Fisk replied. "Is there anything you want?"
Red hesitated, then said, "I bought those two pheasants by an unknown. Got any more by the same kind of artist? I'm only asking, see."
"Pheasants? No, but I'll keep my eyes open," Fisk promised. "I've bought quite a lot of pictures from an old house in Somerset. They'll be ready tomorrow, and I'll have a look." He nodded, and Red went out, backing, turning, half running.
"That man always makes me think he started late and is trying to catch up," remarked Fisk. "Want to talk about your Pa?" When she shook her head, he went on: "Your note said there was something you wanted me to have a look at. Which one is it?"
"It's this." Lucy bent down and took the picture from beneath the bench. "It's an old canvas that was painted over. I took off the muck and began to clean it at the corner and it looked different to me, so I had a closer look and it's an old canvas stuck on a newer one, so I tried to separate them and I thought I might do some damage and I stopped. Was that right?"
Fisk was examining the picture, the cleaned corner, the new canvas. He turned it over and over in his hands. It seemed an age before he answered: "Well, Lucy, you were right!"
"Is it a find?" she wanted to know eagerly.
"We'll have a look at it this afternoon, and then I might be able to tell you," he promised. "But first I've got the van full of pictures from Somerset. I'd like to get them in here before lunchtime, as I may need the van again."
"Give me the key," volunteered Lucy, "and I'll unload."
Old Fisky took the van keys out of his pocket and handed them to her. When she had gone, he took a watchmaker's glass from the pocket of his thick tweed waistcoat and examined the cleaned corner very closely. He sniffed, wrinkling his nose, nodded, and put the glass away.
Lucy came in with several heavy pictures hugged close to her bosom, as if she were holding a lover.
At a little after one o'clock that day, Christine Falconer parked her red mini near Lancelot Judd's shop in Hampstead, got out, long legs drawing glances from several young men passing by, and went along to the shop. Lance was inside, talking to Robin Kell, a tall youth wearing a green velvet jacket, peach-coloured close-fitting pants, and suede shoes. He was a better-looking man than Lance, but not so tall and not so powerful in build. His hair, cut long but not overlong, curled almost like a girl's.
Lance's eyes brightened at sight of Christine.
"Hallo, darling! On time as usual."
"The irreproachable Christine," observed the other. "Good morning, darling."
"Hallo, Robin," she said carelessly. "I hate being late."
"Or being kept waiting," remarked Robin. "The privileges of being born with a diamond-encrusted spoon in your pretty mouth."
"What a dear, sweet person you are," retorted Christine. "Why you were ever born at all I can't imagine. Lance, I do want to talk to you. Alone, please."
"I can take even the subtlest of hints," said Robin. "If I may just wait for my beloved, who is spending a penny. Ah!" He paused, as water flushed and a W.C. clanked. "She is about to join us." A moment later, a girl appeared, tall, beautifully made-up, and rather exotic-looking. "Hurry up, cherie," he urged. "Christine has some perfectly breathtaking news for Lance which she doesn't want us to share."
"She knows she couldn't trust you with a secret for five minutes," the girl said. "And just in case she didn't realize that instinctively, I warned her." She took Robin by the hand and led him out, saying, "Bye, Chris. Bye, Lance."
"See you soon," Christine called.
"Bye, Robin—bye, Marie." Lance Judd waved as the others went out, then turned to Christine, now at the back of the shop, which was long and narrow, with an alcove leading off: a hand basin, a kettle, and a few oddments of pottery on one side, the narrow door to the W.C. on the other. The "kitchen" was remarkably like the corner of Leslie Jenkins's room, but every cup, mug, bowl, and saucer was artistic.
Christine was no longer smiling, and in repose looked remarkably like her father, with his high bridged nose and arched lips.
"Why so serious, sweetheart?" Lance asked her, lightly but not flippantly, and when she didn't answer at once, he went on: "You like Robin less and less, don't you?"
"Yes," Christine answered slowly. "I know he's an old friend of yours and I wish I didn't dislike him, but—" She broke off, took Lance's hands, and said with obvious depth of feeling, "Be careful, Lance. Don't let him use you."
"Oh, I know Robin and his tricks," said Lance, still lightly. "I also know that he insults you to try to persuade you that he isn't interested in your father's wealth and doesn't see him as a big buyer. But he does, you know. He doesn't fool me."
She looked at him almost suspiciously, and said: "I hope he doesn't. I really hope he doesn't." Then she seemed to push her anxiety away, and changed the subject. "Lance, Daddy is having me watched. I can't do anything about it. I wish—"
She broke off as the door opened and a policeman in uniform and a man in plainclothes came in.
# 11: The Questioners
All over London, police visits were being made and police questions were being asked, questions like those asked of Old Fisky. Every antique dealer, every dealer in pictures and prints, in objets d'art and in old jewellery, was being checked and checked again. The range of inquiry was almost unbelievable. There were the dealers in London's Mayfair, in Knightsbridge and the City, dealers who bought and sold paintings worth thousands or tens of thousands as if they were oddments off a market stall. There were the great auction rooms of Sotheby's and Christie's, and the smaller rooms where only dealers with a discerning eye were likely to make bids.
There were the antique "supermarkets" in different parts of London, Chelsea and the West End, bigger than the others, markets where everything might be found, from precious porcelain to old silver, from Roman coins to Japanese samurai swords. There were necklaces and Spanish combs, bric-a-brac from all corners of the earth, brought to England by traveller, adventurer, tradesman, or soldier, once to grace a home, now resting, as it were, between one home and the next.
In the galleries and the great salesrooms, the finest art ever known to man was bought and sold at prices so far beyond the reach of ordinary people that they watched and heard and marvelled. How could - why should - a piece of canvas used for a painting five, four, three, two centuries, even a single century ago command such value? And how could some men, individuals like other human beings, acquire or inherit such wealth that they could afford to pay such prices?
But although many millions of people were not directly affected, there being a wide gap between them and such a concept of the value of art, wherever pictures were purchased for the nation this gap narrowed. It made way for a kind of closeness, since art that belonged to the nation belonged in fact to all. The public might not comprehend the technical skill of these works - might not always fully appreciate their beauty - but nevertheless there was an understanding of the value of masterpieces and a certain pride in their shared possession.
There was, furthermore, an understanding of the lust some men felt for the great paintings, an understanding of the near mania that some felt in their passion to possess. And, fed skilfully by the newspapers, there was understanding also of the men who stole and sold, the leaders of the gangs who had the illegal market under their control. And every now and again some such theft caught the public imagination until a whole nation was agog at the daring of the men who stole; and in this was a touch of admiration, often very strong if no one had been hurt in the theft.
The theft of "The Prince" from the National Gallery had caught the public's imagination, for no one, not even the newspapermen, had yet realized that two men had been murdered after their part in the theft. The cleverness of the raid, the skill and precision with which it had been done, won not only admiration from the man in the street but a grudging kind of praise in some of the national newspapers. And it was against this background that the police in London carried out their visitations. No second hand shop - even one masquerading as "antiques" - no gallery, no saleroom, no warehouse was missed. Each visit was made by a uniformed man working with a plainclothesman, giving the proper touch of authority, and after each visit a detailed report was put into division and a copy sent to New Scotland Yard.
And the Yard geared itself for one of the biggest searches in London's history, under Commander George Gideon.
On Friday, the third day after the discovery of the theft, public interest began to wane, headlines became smaller, and the story vanished from the front pages and became easily lost inside the papers. There were several semi-sarcastic leading articles, directed largely at the weakness of security precautions at the galleries and the inadequacy of the police. One newspaper put it scathingly:
The consistent failure, not only of the Metropolitan Police in London, but of the provincial police forces, to recover valuable works of art, whether stolen from publicly or privately owned galleries and homes, is a matter of grave concern. How is it possible for thieves to plot, prepare, and carry out such triumphant raids as the recent theft of "The Prince" - almost as famous now as the once stolen Goya?
When the Goya was stolen, one man, unaided, apparently outwitted both galleries and the police. Now more than one are doing the same, and the police appear to be helpless - or inadequate.
What is the reason?
One must face the possibility that the police, no matter how good they may be in most phases of their activity, are not interested when the nation's cultural properties are at stake. If there is the slightest truth in this, then a completely new approach is not only necessary, it is essential.
After reading this, Gideon sat back in his chair and pursed his lips, staring forbiddingly at the window that overlooked the Thames. It was some time before he pulled his reports toward him; there were other problems to be considered, too. He came back to Riddell's report and its assessment of the immigration smuggling, but three times in succession he was stopped on its first page by the telephone, none of the calls important. He was halfway through the report, and had not yet reached Riddell's assessment, when his door opened abruptly, without the usual warning tap, and Wilson Chamberlain, the Assistant Commissioner, came in. A recently retired Member of Parliament and one-time regular army officer, he was a handsome man who held himself very erect and spoke with a slightly shrill tone which often made him sound querulous. Gideon, acutely aware of his personal dislike of the man, felt himself stiffen, but even as he did so he warned himself not to allow prejudice to blind him to the other's qualities.
"Ah, Gideon." Chamberlain closed the door behind him and advanced toward the desk. "We need a survey of the investigation into the National Gallery robbery as quickly as possible."
"Why?" asked Gideon flatly, and realized at once that he had also got off on the wrong foot. "I mean—is there any particular reason?"
"Certainly. There is to be a conference Monday morning," Chamberlain said.
Gideon just stopped himself from asking, "A conference of whom?" and said, "The survey is always up-to-date, sir, but so far not particularly helpful."
'"We need details of what has been done, what leads we have - everything."
"For whom, sir?"
"Does that matter?"
"It can matter very much," Gideon told him. "If it's for Yard use, then a lot can be taken for granted; they know the background. If it's for Home Office or other officials, then a great deal more explanatory detail is required. Some details can be filled in at a meeting, but everyone who attends needs a grounding. A briefing can be prepared in an hour and copies can be available an hour after that."
"Good," Chamberlain said. "Good. The Home Office has been pressing the Commissioner for the survey, and the Ministry of Public Works and Buildings wants to be represented as well, of course, as the Arts Council and, naturally, the keepers and custodians of the various galleries. We will, of course, wish to be well represented ourselves."
Chamberlain stopped speaking and drew back, and Gideon had a feeling that he was asking for approval. "See how much I know of how these things should be done," Chamberlain was implying. Gideon, aware of this and of his own feeling of resentment, also became aware of the funny side of it: and it was funny! What had happened to the Home Secretary to appoint such a man? Gideon's thoughts veered back. How could he live with such a situation and contrive to use this man to the department's advantage? After all, he had to live with him.
"Very comprehensive," he said, at last. "What time is this—ah—conference to be called, sir?"
"Is ten o'clock reasonable?"
"I think eleven would enable us to get everything ship-shape," Gideon replied. "Will the Commissioner be present?"
"I trust so. I expect so. He is at the Scarborough Conference today."
"Ah, yes," said Gideon, who hadn't known that the Commissioner of the Metropolitan Police was away and who now understood that Chamberlain had been making hay while the sun shone. "Where do you propose to have the conference?"
"Where would you suggest, Commander?"
"In the main Conference Room. There will be twenty at least, I imagine."
"Yes, indeed, and possibly more. And can you have the preliminary surveys available for distribution this afternoon?"
"You make out the list, sir, and we'll get them delivered," Gideon promised.
"Excellent. That's excellent." Chamberlain nodded, made a right about turn, and stepped with military precision to the door. "I would like two copies," he announced, and went out.
Very slowly, but in a much better mood, Gideon shook his head. Soon he reread the editorial, wondered what Chamberlain would have said had he read it, then reluctantly put Riddell's report aside and studied the bigger report on the art theft. Hobbs had prepared this during the night, and Gideon realized now how carefully it had been done; it was almost as if Hobbs had anticipated what would be needed. After making a few additions and deletions, he rang for Hobbs, who came in at once.
"I hear you've had a visitor," he remarked.
"I gather you anticipated it," said Gideon. "Is there anything more in?"
"No," said Hobbs "We've had a hundred and twelve more negative reports, and we estimate that by now over sixty percent of the dealers in the London area have been questioned." He stood just where Chamberlain had, and Gideon could remember feeling hostile toward him on occasions over the years, but never as hostile as he felt toward the A.C.
What a pity Hobbs wasn't in Chamberlain's place; he would make a ten times better Assistant Commissioner!
The fleeting thought passed, and Gideon said, "You know what we ought to do next, don't you?"
"Go into the home counties and the provinces," said Hobbs.
"Yes. Get teletype requests out for them to do the same thing," ordered Gideon. "If I've jumped the gun, at least no one can cancel the instruction. Before you go, Alec—"
"Yes?"
"Anything at all about Jenkins and Slater?"
"Not that matters," answered Hobbs. "One lead to Jenkins might have been his daughter Lucy, but it proves that they've been estranged for years. That's a dead end And no useful fingerprints were on the gas meter at Jenkins's house."
"Not even Jenkins's own?"
"Faint and very smeared," Hobbs answered. "The report's in there." Gideon nodded, and Hobbs continued, "But there are a couple of things, no more than rumours, which Frobisher picked up. First, that Sir Richard Falconer isn't too particular where he buys, and second, that he has a secret strong room where he keeps pieces he has purchased knowing them to be stolen."
Aware that Hobbs was an acquaintance of Falconer's, Gideon said, "The same rumour touches most of the big collectors, you know."
"Yes, I know." Hobbs smiled. "Thanks!"
"Any reason at all to suspect Falconer?" asked Gideon.
"No," answered Hobbs. "And yet I've often realized that he's extremely careful about who sees his collection. His daughter certainly has some odd acquaintances, though. She was in an antique shop in Hampstead yesterday, a place owned by Lancelot Judd."
"Should I know much about him?" asked Gideon.
"Very prominent in C.N.D.," Hobbs reminded him.
"Oh, Lord, yes," said Gideon. "I remember." There had been a conflict between the police and youthful campaigners for nuclear disarmament a few years before, and a Lancelot Judd had been one of the leaders of demonstrations which had been very nearly riots in Trafalgar Square and Whitehall. But Judd had been quiet for a long time since then.
"Was Falconer's daughter ever involved in C.N.D.?" he asked.
"Not as far as we can find out," Hobbs answered. "But she and Judd are being watched in view of recent events, and we have discovered one interesting thing; her father's having her watched, too! Oliphant, personal assistant, is using a private inquiry agent."
Gideon raised an eyebrow. "Hmm. That's rather odd. And how's Thwaites doing at the gallery?"
"I gather he's done practically all he can there," answered Hobbs.
"Well, have him concentrate on this Falconer-Judd angle," Gideon ordered. "I'll talk to him and Frobisher this afternoon and see how they're getting on together. Meanwhile," he added with heavy humour, "I think you're going to be detailed to a conference at eleven o'clock on Monday morning." He explained to Hobbs, who simply shrugged, and then went on: "Riddell's due in soon. Have you given any thought to him and the immigration smuggling?"
"I read his report before you came in," said Hobbs. "He's obviously delved deeply, as he's got at a lot of the facts. But I get a strong impression there's a mental block in his approach. He's not as lucid and objective as he could and should be."
"What would you do? Keep him on or give the job to Honiwell?" asked Gideon.
"George," said Hobbs very slowly, "I wouldn't like to make the decision. I really wouldn't. But on balance I think I'd make him stay."
"Why?" asked Gideon.
"Because I think he'll find it difficult to live with himself if he's taken off," Hobbs said. "And he's so alive to the danger that I don't think he'll do anything out of prejudice now."
There was a tap at the passage door, and Gideon felt instinctively that it was Riddell. He called "Come in" and Riddell appeared, a grim-faced, hard-eyed Riddell, who looked as if he were prepared for a fight.
# 12: Tensions
One fact was immediately obvious; this was no time for Gideon to tell Riddell that he hadn't yet studied the report and not even reached the assessment. Clearly something had happened to affect the man like this, and to bring him to Gideon's office at least ten minutes early.
"I'll talk to Thwaites," Hobbs said, moving toward the door, and nodding to Riddell; but it was doubtful whether Riddell noticed or heard him. The communicating door closed quietly.
"Sit down," Gideon said to Riddell.
"I'd rather stand, I—"
"Sit down, I said!"
Now, suddenly, there was conflict between them, deliberately forced by Gideon. In this office particularly, and on the job in general, it would be disastrous to allow any subordinate to gain the upper hand, and Riddell was dangerously near making his own terms.
Slowly, Riddell sat down, but he perched on the edge of the armchair.
"What's brought you early?" asked Gideon.
"Early? I'm not—" Riddell glanced at his watch, then raised his hands and dropped them heavily onto his knees. "Sorry," he said. "But this job is pushing me hard."
"I can imagine," Gideon conceded. "It's a bad time all round. The National Gallery affair is causing a lot of trouble."
"Oh, that." Riddell gave the impression that he had hardly heard of the theft and wasn't interested, anyway. Most men at the Yard were aware of the dangers in the situation, alive to the fact that if they failed to find "The Prince" soon the Yard's stock would fall heavily. But as these thoughts passed through Gideon's mind, Riddell sat back, pushed his fingers through his hair, and gave a snort of a laugh. "I see what you mean - my problem is one of many and you can't be as obsessed with it as I am. Good thing, too." He paused and Gideon waited. "You remember me telling you about a poor devil of an immigrant who was stowed away in a hole under some stairs?"
"Yes," said Gideon. "Vividly."
"I lied to you," said Riddell, and his eyes burned.
"Did you?" asked Gideon. He tried to sound casual and as if this really wasn't worth making a fuss about, but he was disturbed not only by the admission but also by Riddell's state of mind. "Why?"
"It wasn't a man; it was a girl."
Gideon said, almost blankly: "Oh."
"That's about the right reaction 'Oh,' with your heart falling into your boots," said Riddell bitterly. "Yes, a girl. She was only a kid." Was, thought Gideon "Even though she was thin as a lath, all skin and bone, she was beautiful. Never seen such eyes. They did something to me, George. Imagine that! A hard-bitten, anti-black copper took one look into a pair of brown eyes and—Oh, I needn't spell it out. It made me seethe. She was a Pakistani, very dark-skinned, and she represented everything I didn't believe in. She belonged back home in Karachi or Lahore or wherever she came from; she should never have been brought here. I still think that," he went on fiercely. "But she got under my skin. She made me see what happens to a lot of them when they're over here. She was nothing but a frightened kid, and she didn't really have a chance."
Again Riddell paused.
"Go on, Tom," Gideon said gently.
"Well, I got the welfare people on to her, had her taken to hospital. She died early this morning. I telephoned the hospital and was told about it. No one's claimed her body. She was living in that rat hole, and the people who live in the house - I've found sixteen, so far - say they didn't know her. She told me that she'd been smuggled in and was told to hide in that cupboard until someone brought her some documents. She stayed there, and hardly ever went out. The others gave her scraps for food. She was there for nearly two months, but the documents never came."
Riddell pressed both hands against his forehead and brushed them over his greying hair. Gideon, watching, felt a kind of distress and anxiety and alarm, for if such a thing as this were repeated often, it would be - it was - not only a scandal but a crime against humanity. He had another feeling, too, of disquiet, because he did not know whether any of this was in Riddell's written assessment of the case. He guessed that it wasn't, because it burst out of the man as if it were something he had been holding back for a long time but which had become too much.
"How many instances are there like this?" asked Gideon.
"A bloody sight too many, that's for sure."
"Do you know who smuggled this girl in?" Gideon asked, then went on quickly, "I didn't know women were smuggled; I thought they were all men."
"Most of them are," answered Riddell. "Only girl I've heard of, in fact. The men come in, and after a while they get work and permits and then they begin to send for their relatives, wives by the dozen, they—"
"All right," Gideon interrupted. "I know how you feel now. You still need a lot more evidence before making a charge, don't you?"
"Yes."
"Do you still feel certain you want to be taken off the case?"
"I don't know what I feel," Riddell acknowledged, "except—well, I'd like to see the business cleared up once and for all."
"What about leave?" asked Gideon. "How much have you due?"
"Three weeks."
"Then take it," said Gideon, "and I'll assign Honiwell in charge while you're away, on a temporary basis. When you've had a break, we can take another hard look at the situation. If you want to get back into it, you can."
After a long pause, Riddell leaned back more relaxed then he had yet been, and said warmly: "The judgment of Gideon! It's a good idea. Thanks, George." After another pause, he went on, "If ever the right man was in the right job, you are." He stood up, hesitated, and then asked: "When would you like me to go?"
"Any time. This" - Gideon tapped the report - "is more than a start for Honiwell or anyone who stands in for you. Any reason why you shouldn't begin your holiday right away?"
"No," answered Riddell. "My wife will jump for joy."
"Right!" said Gideon. "I'll send a chit through." He stood up, big and towering, and put out his hand. "Have as good a time as you can."
Riddell shook hands, nodded, and went out; the door closed very firmly. Gideon moved across to the window and looked over a now sunlit river and London at its loveliest. He stood very still for a long time, not absolutely sure that he had done the right thing, but greatly moved by Riddell's story.
How easy it was to misjudge a man; how easy for a man to misjudge himself.
He hardly knew how long he had been standing there when his telephone buzzed, and he was glad to have to move.
"Gideon," he announced.
"Alec here," Hobbs said. "Are you free for a few minutes?"
"Yes. Come in."
"Right away," Hobbs said. His voice seemed hardly to have died away when the communicating door opened and he appeared. He had a sheaf of papers with him and Gideon saw the name "ENTWHISTLE" on the outside of a folder. Hobbs sat down, still holding the papers.
"How did it go with Riddell?" he asked, and Gideon told him.
Hobbs smiled.
"The judgment of Solomon," he observed, and then went on without a change of tone: "Honiwell seems to have done all he can so far on the Entwhistle case, and it will probably have to sweat for a few weeks. He could fit as locum for Riddell very well." When Gideon simply nodded, he said, "Thwaites has finished at the National Gallery but he would like two of our men to stay there on the Security Force, and Morcom's quite agreeable."
"So am I," said Gideon.
"Thanks." Hobbs paused for a moment and then asked in a harder voice, "Have you time to talk about Entwhistle?"
"Yes," said Gideon, and he thought - as he often did - of the man in Dartmoor who might be innocent of the crime for which he had been sentenced to life imprisonment.
Entwhistle was out on the moor, breaking rocks.
When he had been leading his ordinary life, he had not seriously believed that prisoners still broke rocks with sledge hammers, but he had discovered that they did. The rocks could be used for road building and wall building; there was a kind of rhythm to the actual work and once one was used to it, it was no great strain. Entwhistle, tall and gaunt, had always been a strong man.
Now, lifting the long-handled sledgehammer ready for the downward swing, he saw the great expanse of the moor on this lovely autumn day, and the never-ending rocks themselves, more rocks than ten thousand convicts could break in a hundred years. He saw the rugged beauty and the green of distant trees and grass, the other men working, some naked to the waist, browned skin exposed to the sun and glistening from sweat. It was like a symphony of movement and a symphony of harsh and cacophonous sound. Chips of granite flew, big rocks split; now and again a man swore with desultory indifference or in sharp pain where a rock had cut him. Entwhistle, used to the heat of the tropics, an engineer who had helped build huge bridges across untamed rivers and great valleys, felt an awful sense of helplessness, of frustration. It was madness that he should be doing this, and it was driving him mad.
This was one of his bad days.
It was a day when he could not get his children out of his mind - the children he had not seen since he had been imprisoned, over three years before.
Three years for a crime he had not committed.
Three years in Dartmoor - for quarrelling with his wife, going home, and finding her dead.
Three years, while her killer was free, living life to the full, not conscience-stricken or he would have confessed or at least made some attempt to help him, Geoffrey Entwhistle.
Three years.
One for Carol, who was now seven, fair, and pretty.
One for Clive, who was now fourteen, tall, and eager.
One for Jennifer, who was now eleven and, so they told him, clever.
He could picture them not as they were but as they had been, so young and vital and overjoyed to see him on his return home, because he had been away for so much of their lives.
He could see their faces in the rocks.
Sometimes, when he brought that sledgehammer crashing down, it was as if he were crushing one of his beloved children; and in an awful way, by his very existence in this place, he was in fact crushing them. Prison, the law, the policeman who had sent him here, the judge and every bloody juror, the man who had actually killed his wife - each one had been a blow against his children's happiness; his children condemned to live under the brand of Cain, staying with relations, snubbed and sneered at and derided because of a crime their father had not committed.
Now and again, he glimpsed a kind of hope.
A prison-visiting parson named Wilkinson had once brought him much hope, saying that the police, through Commander George Gideon, had promised to review the evidence, but that was nearly a year ago, and since then he had heard nothing at all but platitudes. Eventually Wilkinson had been appointed to a living overseas, and now it seemed that he had turned his back on this heart-breaking problem: and that no one cared.
Today, Entwhistle was vividly aware of each child, each hammer blow now seeming to crack and split their future, each blow robbing him of a little more hope, and each driving him a little more mad.
He would go mad if he didn't get out of this soul-destroying place.
He must escape.
Not tonight, but one day soon.
Better to be on the run, better to be shot while
running, than to exist like this.
Better to be dead.
The thought came to him, not for the first time, that death would be the answer, putting an end not only to his own daily despair but to that of his children. No longer would they be burdened by the knowledge that he was in prison paying for the crime they believed he had committed, making some kind of amends for murdering their mother.
Oh, God, he could not stand it!
He whirled the hammer round and round as if it were a weight in an athlete's hand, went round and round with it, felt his head swimming, felt dizzy, felt as if he were going mad, mad, mad!
He let the hammer go.
It crashed thirty yards away and smacked against a boulder, letting out a dull metallic ring. He stood, swaying, staring. He was aware of a guard approaching, rifle cocked. His blood thundered through his ears and the refrain would not stop. He was going mad, mad,.
mad....
"Did it slip?" asked the warder.
Slip?
"Eh?"
"Did it slip?" the warder asked again. Slip?
"It—yes. Yes, it slipped."
"Well, go and get it and be more careful," the warder ordered.
In his report to the Governor, the warder told a different story.
"I advise keeping Entwhistle within the precincts for some time, sir. He undoubtedly hurled his hammer, and nearly lost his self-control. I gave him to understand that I thought the hammer slipped - thought it might calm him down a bit."
"We'll have him back in the library," the Governor said.
In his tiny office which overlooked the Thames and a corner of Billingsgate Market, Eric Greenwood, the man who had been the lover of Geoffrey Entwhistle's wife, and who had killed her, was looking through samples of carpets shipped from India, from Pakistan, from Persia, and from Morocco. They were all beautifully hand-woven with a rich variety of colours particularly attractive in the afternoon light.
He was deciding whether to order or whether to wait for a week or two when the shipper's prices might come down.
He did not give a moment's thought to the past and to what he had done, and he felt absolutely secure in his sheltered bachelor life.
Gideon, sitting in his office, listened to Hobbs and, after ten minutes, sent for Chief Superintendent Honiwell, the man who was checking the evidence against Geoffrey Entwhistle. Gideon knew, as Honiwell did, that the threads were tenuous and the case tortuous, and before it could possibly be reopened, proof must be found that there had been a miscarriage of justice. Nothing less could justify a full reopening of the case.
Honiwell was a mild man, quiet-voiced, thorough, and painstaking, and above all humane.
Gideon contrasted the calmness of his demeanour with the tensions there were in Riddell. Honiwell gave the impression that nothing could ever shake his composure or make him speak quickly or out of turn. He came in quietly, took the chair Gideon indicated, settled comfortably, and waited.
"So you think you've found the man we're looking for," Gideon said.
"I think so, sir. Once we established the fact that Mrs. Entwhistle had a lover, it was simply a question of time. If I'm right, sir, he's a man named Greenwood, works at a firm of importers and exporters who handle mostly Oriental goods - carpets, cloth, carvings, spices, a fairly general range. It's been a long job to locate him, as we didn't want the press to get wind of it, but he's been identified by at least seven people, mostly waiters, as having been with Mrs. Entwhistle a great deal while Entwhistle was abroad. We could make a good circumstantial case against him, but I don't think the time's ripe for that yet."
One thing was certain, Gideon thought when the others had left, Honiwell could not be taken off this case. Someone who was less involved in a case would have to take over from Riddell. He must talk to Hobbs again. Meanwhile, he needed to look through the Entwhistle file. Soon he found himself wondering bleakly what a man would feel like if he were in prison for a crime he had not committed. The more he pondered, the more urgent the case became.
His interoffice telephone rang, and he lifted it. It was Chamberlain.
"Ah, Gideon. Several of those who should be present can't make it on Monday, so make a change in your diary, will you? Same place; same time, Tuesday."
"Very good, sir," Gideon said. Then he replaced the receiver and said through his teeth. "We'll have that damned Velazquez back before the conference if I have to find it myself!"
A few seconds later, two senior Yard men, passing the door, heard Gideon burst out into a deep, hearty laugh. One of them remarked, "He's pleased with life, that's something."
"Perhaps they've found the Velazquez," the other said.
Hobbs, alone in the next office, lifted his head and listened, half expecting the door to open and Gideon to share the joke. But he realized, knowing Gideon, that a laugh like that was Gideon laughing at himself.
At the time that Gideon's mood changed so abruptly to laughter because Chamberlain had had to alter his plans, Chief Thwaites was in the Tate Gallery. Over a year before, Hogarth's "Boot Boy" had been stolen from the Tate, and had never been recovered. He, Thwaites, had not been in charge of the investigation, but now he wanted to learn exactly how the robbery had been achieved. The Assistant Director, Adam Charles, told him precisely, and Thwaites made mental notes to be written later in his reports.
No one at the gallery, except Charles, knew he was a policeman.
Leaving the Tate, Thwaites drove direct to the Victoria and Albert Museum, from which a Turner and two Constables had been stolen just two years before. Here, the Deputy Director was equally cooperative but could not help much. Thwaites made his notes and went back to the Yard, where he added them to the others already made, then tried to find a common factor in all the thefts. He failed, yet was feeling quite hopeful when- he left the Yard again, to go to Judd's place at Hampstead.
# 13: The Whispering Well
"Lance," Christine Falconer said, "are you absolutely sure you can trust Robin?"
"Christine, my love, I am absolutely sure I can trust Robin to do what he says he'll do, and so can you," Lancelot Judd answered. "I wish you weren't so doubtful. He hasn't done anything to upset you, has he?"
"No," answered Christine. "He always makes me feel uneasy, that's all. Tell me more about him."
"There isn't all that interesting to tell," said Lancelot. "We were at school together; we failed to get into the same university, but we met during vacations. We had thought of going into business together, since we both love antiques, but I stayed in England. Robin went to Canada and America to look for antique markets; he's always been interested in the subject. His father owned a wonderful collection of Japanese paintings and jade. I declared war on war, as you well know."
Christine half laughed.
"Did you really believe you could do any good, darling?"
"I believed it then," answered Lancelot rather ruefully. "But I don't really know what to make of things now. If you're in any doubt, I'm still as anti-war as ever; if there were a war tomorrow, I would be a conscientious objector, and if I thought it would do any good, I'd throw myself off a cliff crying Outlaw war!' "He shifted his position slightly and slipped his right am firmly round her shoulders. It was Sunday morning; she was sitting on his knee in a huge armchair in his one-room studio flat above the Hampstead shop. "When did I last tell you that I loved you?"
"Too long ago," Christine answered.
He kissed her—
He kissed her in a way she had never known before, fiercely, demandingly, holding her tighter and still tighter. She felt the wild stirring of desire.
When, at last, he drew his head back, he was breathing very hard.
"I love you," he said. "I would love you whether you were a baker's daughter or a merchant banker's." He sounded both hoarse and urgent. "You know that, don't you?"
"Of course I know it," she answered. "Darling, why do you ask me so often?"
"Because I doubt you so often."
"Oh, Lance," she said helplessly. "You don't need to. I swear you don't need to. I can't help being a rich man's daughter. Sometimes I hate the thought of Daddy's money and all it does to him."
He bent his head and stopped her with another kiss, gentle this time. She moved her head back as if to get him into better perspective, and studied him for several moments without speaking.
"Why do you say you doubt me?" she asked, at last.
"Because you so often doubt my motives."
"I don't, you know. I really don't."
"Subconsciously you do," he told her. "I suppose the truth is you've been brainwashed since you were very young. You've been taught that no man could possibly love you for yourself alone, that anyone with a father as rich as yours is inevitably prey to get-rich-quick men. Hush! You were probably told this in the form of nursery rhymes first, and then the pressure was stepped up and you had a governess instead of going to school, so that you couldn't be contaminated with mere money seekers. Quiet! The excuse, of course, was that your parents travelled so much and wanted you with them, and they could afford your private tutors wherever you went. And you didn't realize what was happening and you don't realize what did happen, even today. You don't doubt me with your conscious mind but with your subconscious doubts."
"Lance, please stop!"
"No, darling, I won't stop; the time has come to talk. You never see Robin without doubting me, and—"
"But Robin's not you!"
"No. Robin, as you rightly say, is a long, long way from being me. But you think, or at least fear, he is using me to worm his way into your father's confidence. You haven't really worked the situation out in your mind, but you are as full of doubts as a font is full of holy water. I don't know how to exorcise those doubts, my sweet Christine. I really don't."
His arm was still around her, yet now she felt very different from the way she had felt only a few minutes before. Her heart was heavy and an unfamiliar tension gripped her.
"I can't help distrusting Robin," she said flatly.
"I'm not worried about Robin," he said. "I'm worried about—"
Before he added the word "you" and while he was pulling her even closer, a door banged downstairs in the shop.
"My God!" he exclaimed. "What's that?"
Although she tried to scramble off his lap quickly, it seemed an age before she was on her feet, and he pushed her aside so roughly as he stood up that she nearly stumbled. "Stay there," he whispered, moving swiftly toward the door and the head of the narrow staircase. There was another sound from below, not so loud this time, and Lance raised his hand, then placed his finger on his lips, urging her to silence; and as she moved toward him he waved her away, mouthing: "Stay there!"
He looked terrified - not just alarmed, as she had been when the door banged, but terrified. He gripped the handrail tightly as he started silently down the stairs. Now no sound came from below. It was almost as if they had imagined the noises they had just heard.
Christine wondered what was frightening him so, and why it was so vital that she should stay where she was. For a few moments, until he disappeared, she stood close by the chair; but soon, drawn by curiosity, she moved toward the stairs. These were in fact little more than the open steps of a ladder, made of some dark, deeply polished wood, leading to a large apartment once part of a picture gallery, now used as living-room-cum-bedroom.
Christine's back was to the one wide window. The stairs were on her right. Alongside these were two doors, one leading to a W.C. and one to a tiny kitchen almost identical with those downstairs. There was a protective handrail, made of old brass curtain poles, gleaming in the brighter light which came up the well of the staircase.
A man out of sight said: "Is Robin—"
"Hush!" breathed Lance, whose back was all she could see. She noticed the twist of his lean body and knew that he was looking round and upward to see whether she was near, and instinctively she stepped back. This was the first time she had ever attempted to deceive Lance; it had seemed to her that the essence of their love was the absolute naturalness between them, the lack of all pretence.
Her heart thumped sickeningly because he was so afraid that she would overhear.
"Is he here?" the man whispered.
"No. What the hell made you come here?"
"I wanted to clarify a couple of matters," the other replied.
"Don't be a bloody fool, you didn't have to! My God, you're so hot you could set the bloody place on fire!"
Each word, trapped somewhere in the shop, travelled with soft clarity up the well of the staircase, and although the men were whispering they might almost have been standing by Christine's side. She moved nearer the kitchen door, where she could see more of the shop and yet be in less danger of being seen if they looked up. Now she could see the whole of Lance's body, except the top of his head, and she could see the stranger, sidewise to her, up as far as his chest. He wore a beige-coloured suit and beautifully polished brown shoes.
What did Lance mean by "You're so hot you could set the bloody place on fire?"
What did "hot" mean?
The very question was self-deceit. She knew, of course. No one who read or watched crime stories could be in any doubt and yet she did not want to believe that Lance was using the term so freely.
"I came because I no longer trust you, or Robin," the stranger said. "I think you're planning to keep all the money you get for it."
"It." What did he mean? What was "it?"
"Robin told you to—"
"Robin told me he could dispose of it in less than twelve hours," the other man said. He raised his voice very slightly, as if in anger. "Did you or did you not?" He spoke well but with slight over precision, and he sounded - the thought impinged itself on Christine's mind without making a deep impression - as if he were tired.
"Keep your voice down!" Lance said softly. "There—there was a delay. It wasn't our fault, there was a hitch. You'll get your money."
"I want to see either it or the money now," the man said flatly.
"What?"
"And I mean to have one or the other. Because I have to leave the country at once."
"Why? What's the hurry?"
"I am being watched."
"My God!" Lance gasped. "You're being watched and you come here!"
"I've already waited four days longer than I said I would. Lance—"
"Get out of here!"
"I will not go without one or the other," the caller said, and a touch of anger sounded in his voice-. "After that, I'm through."
"Like hell you're through!" Lancelot paused, and Christine pressed back against the kitchen door trying to see the stranger more fully. At last, she succeeded. He was not as tall as Lance but was just as lean, with a long, rather delicate face. He wore a modern-shaped trilby hat with a narrow brim, and there was a not-quite-English look about him.
Lancelot's whisper, curiously magnified, echoed yet again.
"Go away. Go away and wait until—"
"No," the man said. "Not now. I won't wait. I don't trust you. If Robin hadn't killed the others, or fixed it, anyway—"
"Shut up!" Lancelot said in anguish, craning his neck right round, but obviously he could not see Christine, and could have no idea of the clarity with which every word they uttered travelled upward. "Get out, quick. I'll tell Robin and he'll come and pay you."
"You don't seem to understand me," the other said harshly. "The moment you committed murder, I was through with waiting."
He put his right hand beneath his jacket, fumbled for a moment, then drew out a small automatic pistol, while Christine watched as if turned to stone. She felt as if she were living through a nightmare. Soon, she thought, soon I shall wake up. There was no moment of walking.
"Put that away!" Lance gasped.
"Get me the picture," the stranger ordered flatly "Get it now - or give me the rest of the money. Fifty-one thousand pounds, plus what you took from the others."
"You know I can't—"
"Get it, now!" the stranger grated.
There was a pause, which dragged out for what seemed a long time. Then, slowly, deliberately, the stranger raised his gun until at last Lance muttered: "He—he'll kill you for this."
"If I don't kill him first," said the stranger. "Get it!"
Lance moved; Christine could hear furniture being shifted, scraping on the floor. The man with the gun was watching something out of sight. Then, suddenly, Lance appeared with what seemed a roll of canvas and handed it to the caller, who put his gun away and took the canvas gently. Christine could not fail to notice the care with which he handled it, how his long and delicate fingers hovered for a few moments before he drew away.
"You won't get a penny," Lance whispered. "You'll never find a market!"
"But I will live," the other said, and turned - and as he turned he started violently and backed into Lance. Lance moved quickly, to avoid him. Now Christine could see both of them from the back, and sensed they were staring at something in or near the shop doorway.
She was at screaming pitch.
What could it be, who could it be? She heard the shop door and footsteps sound, and then the door closed. The man who had the canvas put his hand to his pocket, and pulled out a gun, but suddenly Lance moved and snatched the gun from his fingers, then sprang back out of reach. The strange thing was that Lance still seemed terrified. A man walked firmly toward the others, who stood as if petrified. Then Lance moved, seemed to square his shoulders.
"Good—good morning, Officer," he said huskily.
So it was a policeman!
Christine thought: Shot... murdered... killed... murdered... killed... policeman. And now the nightmare was even more vivid. The newcomer stopped in front of them and she could see the skirt of his tunic, the buttons, the buckle of his belt.
"So I fooled you," Robin Kell said, in his unmistakable voice, and there was the bite of sarcasm in the way he spoke. "If I'd been the McCoy and not a stage policeman, you would have had the handcuffs on you by now; you even look as guilty as hell. What are you doing here? I told you to stay under cover with the loot."
Suddenly, he broke off. There was a moment of utter silence as he turned toward the man with the canvas. He seemed to see the gun in Lance's hand for the first time.
"What's going on?" he demanded roughly. "Come on, tell me."
"He—he came and demanded the rest of the money or—or the canvas," gasped Lance. "He says he's through. He had a gun, this gun. I had to give it to him." He drew a deep breath. "But I wasn't going to let him get away with it. I would have stopped him somehow."
Into the silence that followed, the stranger said: "I want the money, Robin - the whole seventy-five thousand. Or the picture. And I want one or the other now."
He made it seem as if it were he, and not Lance, standing behind him, who held the gun.
# 14: The Onslaught of Fear
Only then did the truth crash upon Christine. And yet how could it be? The Velazquez, the picture that had become the sensation of London, the sensation of the land. Suddenly most of what she had heard dropped into place, and, keyed up though she had been before, she was now much more conscious of danger. The realization ran through her mind, turning thought to tumult, and she stood motionless, almost too fearful to breathe.
"So you do, do you?" Robin began, in a rough voice "Well—"
"He's been followed!" Lance blurted out. "We can't do anything to him. He's been followed by the police!"
"You told me that you would sell it in twelve hours and bring me the balance," the unknown man said quite calmly. "You did not tell me that you would murder Jenkins and Slater in cold blood, and you did not tell me that I would have to wait."
"The police are following him, I tell you," Lance cried, then turned his head and stared up the staircase.
"I said that the police were watching me," the stranger corrected. "If I take the picture, I will also take the risk." He moved toward the door - toward Robin - the canvas under his arm "Tell him it won't help to squeeze the trigger," he went on. "The gun isn't loaded." After a pause, he said, "Let me pass."
"You stay just where you are," Robin Kell said savagely. "Lance, why did you give him the canvas? My God, what a bloody fool you are!"
At that instant, upheaval came below.
The stranger made a darting move forward, Robin's hand descended on his arm, there was a choking scream, and the stranger staggered backward. As he thumped against a tall chest of drawers that rocked and rattled, Robin swept his other arm around and thrust Lance out of the way, then bounded up the stairs.
Halfway up, he saw Christine crouching back.
"I thought you were there," he said. "So now you know."
As he stopped and stared at her, lips tight and eyes glittering, she felt such an onslaught of terror that she could neither scream nor move. Robin seemed to stand there for an age, but it was only for a few seconds. Then he moved swiftly upward, vaulted over the handrail, gripped her hand, and spun her round, pushing her arm high up behind her, making pain streak through her elbow and her shoulder. He seemed to do everything in a series of movements that were part of one another, opening the door of the W.C. and thrusting her inside, letting her arm go, and then, while terror welled up and she felt as if her lungs would burst she wanted so desperately to scream, he brought the side of his hand onto the back of her neck in a savage chopping blow. Her head jolted backward, her neck felt as if it were breaking. She lost consciousness and slumped against the wall.
Robin Kell stepped out of the tiny closet, the key in his hand. He put it into the keyhole from the outside, turned it, and, when the lock had clicked, withdrew the key and dropped it into his pocket. Then he turned round to see Lancelot Judd staring at him from halfway up the stairs.
"You—you didn't hurt her, did you?"
"I just put her to sleep."
"If you've hurt her—"
"Look out!" exclaimed Robin. "I'm coming!" He placed a hand on the cold brass rail and vaulted, and Lance only just dodged to one side in time. Down below, sitting on a high-backed Jacobean chair, the other man - de Courvier - was nursing his arm. As Robin appeared, he got up and turned slowly toward the door, but he was obviously so sick with pain that he could walk only a few steps, and those swaying.
"Come back, Paul," Robin ordered.
Paul de Courvier took an unsteady pace forward.
"Come back, or I'll break the other arm," Robin said through his teeth.
De Courvier stopped and slowly turned round. His right arm was twisted where it had been broken, and all the colour had drained out of his face. His eyes looked feverishly bright, and his thin lips were set tightly.
"When did you last see the police watching?" Robin demanded.
"At—at my flat," the other answered.
"Are you sure they were watching you?"
"They followed—followed me twice."
"Have they been to see you?"
"No."
"Haven't broken in when you've been out, have they?"
"No, there's been no sign of a search," de Courvier muttered. "I—I must go to a hospital. My arm—"
"You just answer my questions," Robin Kell said callously. "How did you come here?"
"I borrowed someone's car."
"Borrowed?"
"A neighbour's. He's away."
"Are you sure he's away?"
"Of course I am sure! And we have an arrangement; we use each other's cars sometimes. If you let me go now, I won't talk. If you keep me here, I'll tell the police everything, when they come." De Courvier's voice was thick with pain.
"Shut up! What car is your neighbour's?"
"A white Jaguar. I tell you—"
"What's all this about being through?" Robin interrupted harshly.
"I—I am through. I don't want anything more to do with the job. Can't you understand?"
"You can't get away with it so easily," Robin said, sneering. "You'll do exactly what I tell you."
"No!" de Courvier said, "I will do nothing more for you or with you. You lie about everything, and you kill without compunction. There was no need to murder Jenkins and Slater. There was no need to lie to me. Now you have your portrait, and if you really have a buyer you can keep my share of the money. I am finished. There is nothing you can do to make me—"
De Courvier broke off, and caught his breath, then seemed to hiss. Robin Kell moved with his remarkable ease and precision, hand going to his waistband, sliding out, blade of a knife glinting, then, with a swift movement, burying itself in de Courvier's belly. And as he died, and swayed backward, Robin held his shoulders to keep him upright.
Without looking round, he said: "We'll get rid of him after dark."
Behind him, Lancelot Judd gasped, "You're a cold-blooded swine."
"Let's just say I'm cold-blooded," Robin retorted, and he half dragged the dead man behind a big wardrobe, almost too large for the shop, and propped him up on another high-backed chair.
"If you hurt Christine—"
"Don't be a fool," Robin interrupted. "We need Christine. You can have her for bed. I want her because she'll open a lot of doors in Falconer House. If we handle this properly, and we're going to, we'll separate Falconer from most of his money. But we need Christine alive to do that."
Lance began, "You mean—"
"I mean we're on our way to a fortune. Falconer would be tempted like hell to buy the Velazquez anyhow. When he knows that buying it is the only way to save his daughter from drugs and shall we say degradation, he'll fall over himself to buy it. And, oh boy," - an expression of unbelievable cunning spread over Robin Kell's face, making him look quite demonic - "oh, boy," he repeated in a low-pitched voice, "won't we have him where we want him then? He'll buy everything we want him to, or the police will be tipped off to pay him a visit."
Lance was watching him, appalled, but as if mesmerized. Robin's tone changed.
"So we're home and dry. All we have to do is keep our heads. There were only three men who could have caused us any real trouble, but now they can't. We've got to work out how to get rid of the body. The best way will be to get de Courvier back in the Jaguar. It doesn't matter where it's found; the trusting neighbour will report it stolen and the police will think it all happened in the car."
He nodded to himself with obvious satisfaction.
"So long as you don't hurt Christine," Lance said weakly.
"I won't hurt Christine if she behaves herself," Robin said absently "It's past time we put this piece of lolly away, now. My God! You could have got us into real trouble!" But he did not dwell on that, only raised the Velazquez shoulder high and stepped toward a wall adjoining the next-door building, on which several big paintings, none of any quality, hung from a picture rail. "Come and lend a hand," he added more brusquely.
Lance moved after him, and together they took the pictures down. Then Robin pressed a section of the rail, and the now bare wall slid open, revealing a dark recess beyond, about three or four feet deep. Inside were a dozen cylindrical metal containers, each marked "Fire Resistant." Robin picked one up from the left-hand side, twisted a key in a small lock, opened the hinged top, and turned toward Lance, who was rolling the Velazquez. Taking the picture, he eased it gently into the container and then closed the cap. He tested it to make sure the self-locking device worked, then put it with those on the right-hand side.
"Eleven in the bag," he remarked. "We won't worry about making it a round dozen." He pressed the picture rail again and the doors slid to; then they put the pictures back.
"Foolproof and fireproof," Robin observed "We're nearer that fortune than we've ever been. And no one can give us away now - all of them are dead."
Lancelot Judd nodded, but almost at once gave an involuntary shiver. Robin appeared not to notice. Neither of them spoke of Christine.
"Did you know that Christine wouldn't be in for lunch?" asked Sir Richard Falconer. He was lunching, frugally for him, in the morning room overlooking the walled garden of Falconer House, a garden with a centuries-old lawn, smooth as velvet, bordered on one side by espalier apple and pear trees, still heavy with fruit, and on the other by the rich variety of colours and blooms of late dahlias and early chrysanthemums.
"Yes, I knew," Lady Falconer answered.
"Do you know where she intended to go?"
"With Lancelot Judd, of course."
"It's very risky, Charlotte," Falconer said severely.
"It's even more risky for us to try to live her life for her," retorted his wife, with rare spirit. "If you restrict her too much, she will rebel completely and leave home."
"I don't think you know her as well as I do." Falconer cut a piece off a peach and placed it in his mouth. "She will accept - she does accept - the situation, and she will certainly not be foolish enough to leave home. Life in a garret or what passes for penury these days certainly would not suit her, nor will this young man. Do you really know anything about him?"
"I know you have no doubt set your spies on him," Lady Falconer said bitterly.
"Charlotte, I really don't understand you. I must take the obvious precautions. Of course I needed to check the background of Lancelot Judd, who might possibly have proved suitable as an escort for Christine. He is the son of Jacob Judd, a Brighton solicitor; got a scholarship to Oxford, and worked in an antique shop during vacations, so he has some knowledge of antiques and paintings. But until he set up shop he lived on a small inheritance, and seems to be a somewhat odd young man. He was very active in the Campaign for Nuclear Disarmament, a campaign which you know that I abominated. And he has friends who are no less odd. Christine must be persuaded to break the association."
His wife looked at him out of her beautiful shell, and her voice carried more feeling than her features - even her eyes - showed.
"Christine is extremely resentful at being watched and followed wherever she goes. You must stop doing that, and give her her head for a while."
Falconer toyed delicately with another sliver of peach.
"I must at least appear to," he conceded, "and I made a start only this morning. I told Oily not to have anyone follow her today. But if she isn't back for dinner I shall regret it very much indeed."
"She'll be back," his wife said with assurance. "She promised to be here for dinner." She relaxed a little as the footman brought in coffee: on Sundays they always had coffee in this room. As she poured, she smiled at her husband, and when she passed the tiny cup of priceless porcelain, their hands touched for a moment in a kind of aloof intimacy.
Gideon, on that pleasant Sunday afternoon, hoed vigorously among the late-summer flowers in the back garden and, when the rich earth had been stirred, took a pair of edging shears from the small garden shed and began to trim the lawn edges. Insects kept perching on his face and he brushed them off with the back of his hand. Now and again, when he stopped, he could hear the piano; Penelope was practising as if her very life were at stake. Kate was upstairs, getting ready to go out; they were going to supper with Prudence, the eldest daughter, her husband, and their five-year-old child. It would be pleasant but not exciting; like the gardening.
Gardening soothed Gideon, and while he gardened he was able to think clearly without fogging his mind by too much concentration. Often, while pottering with lengths of bass, or with secateurs, tidying the shed or adjusting the blades on the lawnmower, he had flashes of illumination on matters which might have been worrying him for days. Today was mostly ruminative. He couldn't do any more about the National Gallery inquiry and hadn't yet made up his mind whether to attend the conference on Tuesday morning, or whether to delegate Hobbs. He wondered where Riddell was, thought of the dead Pakistani girl. Was hers an isolated case or did such things happen often? If a conference was needed, it was over the immigrant smuggling, but that was a major social problem, not simply one for the police, and he had a sense of the need for caution.
He couldn't consult Chamberlain; the man would be impossible. Nor did he want to handle this on his own. The only person to discuss the problem with was Sir Reginald Scott-Marie, the Commissioner. Perhaps this evening he would call him; it would be easier to discuss it out of office hours, and Scott-Marie could be trusted absolutely.
He wondered how Entwhistle was, and whether Honiwell was right, and the case could and should be reopened. Was it really possible that Greenwood had let Entwhistle spend three years in prison, to face the rest of his life in prison, for a crime he hadn't committed? What went on in the minds of such men? How could anybody live with a burden like that on his conscience?
The simple truth was, of course, that such men had no conscience, it was a hard thing to accept, and although Gideon had been dealing with the most ruthless and hardened criminals all his life, it was still difficult for him, especially on a Sunday afternoon in the garden, to believe that men could be so cold-blooded. But suddenly he had a revulsion of feeling against his own thinking. He knew damned well that some men were so utterly coldblooded that they did not have a spark of remorse or contrition whatever they did.
The killer of Jenkins, the killer of Slater might well be cases in point.
There was a furious burst of music, a moment of praise and triumph; bless the child! He wondered how the strange affaire between her and Hobbs was progressing. She appeared most of the time to regard Hobbs as an uncle or an elder brother, and though she sometimes seemed almost to be in love with him, such periods did not last long. Soon she would meet another, younger man with whom, for a few weeks, sometimes only for a few days, she would be completely infatuated.
Odd not to know how one's children thought and felt. The parent-child relationship was a strange one, and often totally unpredictable. Take for example Jenkins and his daughter Lucy.
Funny about Lucy, working in a shop that he, Gideon, passed every day. He had read the report from the Division at Fulham; it had been written with a rare sensitivity which had somehow made him understand Lucy's reaction to her father's death. The tragedy wasn't in the death but in the revelation of the estrangement between father and daughter. Funny, too, about that yearly Christmas card with a five-pound note tucked inside.
He had only a few feet of the lawn edge to finish when he heard Kate from the window.
"George!"
The piano was still playing, a neighbour's lawnmower was clattering.
"George! Telephone!" Kate was mouthing the words and beckoning.
A little reluctantly, Gideon put down the shears and strolled into the kitchen. There was an extension in the passage just outside the door. The music ended, by chance or with intent, and he picked up the receiver.
"Gideon here."
"I'm sorry to worry you on Sunday afternoon, George," said Sir Reginald Scott-Marie, "but I would like a word with you about Tuesday's conference. Can you fit it in sometime this evening without disturbing your family too much?"
He could!
"About half past nine tonight, sir, if that would suit you," Gideon replied promptly.
"Make it ten o'clock, will you? We needn't be too long. Thank you. Goodbye."
Gideon rang off, smiling with deep satisfaction. Scott-Marie must have sensed what Chamberlain had been doing, and deliberately relieved Gideon of the need to broach the subject. Yet Scott-Marie had a reputation for being the most coldly aloof man at the Yard. Ten o'clock was easier, really; he could bring Kate and Penelope home and then go to Scott-Marie's place in Mayfair.
It couldn't be better.
On the other side of London, above Lancelot Judd's little shop, Christine Falconer lay in a drugged sleep.
Barely half a mile away from Gideon, at the back of Old Fisky's shop, Lucy Jenkins watched the old man as he worked on her possible find. She could hardly control her impatience, her longing for a dream to come true.
# 15: The Find
Old Fisky did what Lucy had done at first, placed the picture flat on the bench, the odds and ends of rags, bottles, pieces of frame, glasses, brushes, cigarette packets, and empty matchboxes pushed to one side. Then he soaked a rag in turpentine, using far more than she would have dared to, and smeared it over the canvas, peering at it with sharp, unblinking eyes while the little miracle occurred. As the turpentine made the dirt and the varnish beneath it translucent, the picture itself began to show up. He grunted but made no comment, allowed the turpentine to dry, and then repeated the process. This time, he picked the wet picture up and held it so that the light shone on it. Clear, sunless light from the north, pure as light could be, made the three partly draped figures look very real.
"It's certainly odd," he said. "Varnish is very cracked. Put that diacetate nearer and get me some more cotton wool." Under his breath he muttered something which sounded like "You should know what I want by now."
But Lucy was too excited to feel rebuked.
Old Fisky picked up a paintbrush with the hairs almost completely worn away, took the cotton wool Lucy handed him, and wound it round the thick end of the brush until it was a cylindrical shape about the size of a cigarette. Lucy pushed a bottle marked "diacetate" toward him, and unscrewed the plastic cap. He dipped the stick inside, held it for a moment to soak the cotton wool, then allowed the excess liquid to drip back into the bottle before moving it to another corner of the picture. Slowly, he drew it across the canvas, and it took the dirt away as if by magic. He dabbed the cloth soaked in turpentine on the clean patch, then repeated the process until a strip about a quarter inch wide ran for three inches across the painting. Vivid blue and dull white showed up.
"It's dry enough now," he said. "Give me my glass, girl - for heaven's sake, don't just stand there."
Without a word, Lucy handed him a magnifying glass about the size of a small saucer. She was unable to stop herself from trembling, because his manner told her that he was as excited as she was. He peered through the glass, now close to the cleaned strip, now a few inches away, and when he drew back he handed her the glass.
"Have a look for yourself... See what I mean when I talk of craquelure - those tiny, tiny cracks... Oh, it's old." She was hardly able to hold the glass still, excitement so possessed her. "Depends on the kind of varnish, of course, but at least three hundred years old, I would say, and Dutch. You only get that kind of blue in Dutch paintings – a secret they kept to themselves, never been able to reproduce it. Can you see?"
She could see the vivid blue of the paint and the myriad criss-cross lines where the varnish had cracked with dryness and age. It seemed to her that she had never seen so pure a blue, and she knew that Old Fisky was enraptured by it. The tiny criss-cross of cracks in the varnish was unmistakable, too. He had told her of this but she had never seen it so clearly before.
He took the glass from her, his hands spotted and with the veins standing out, hers very small and delicate and white.
"Now we will take it off the new canvas," he said. "Anyone who saw the new canvas and the paint on it would think the dirt was new, wouldn't give It a second thought. Except you!" He cackled to himself as he filled a big sink with water and put in a little detergent which would not affect the paint or the varnish but would soak the new canvas until it became loose and could be pulled away. After allowing the picture to float for ten minutes, he examined it with infinite care; then he pushed his few locks of hair back, and looked at Lucy with his head on one side.
"It's an old one, Lucy, you can be sure of that. We'll leave it for half an hour; the two canvases will pull apart easily then. How about me making you a cup of tea, for a change; it's Sunday, and Martha's out at church." He turned toward the stairs, and thumped heavily up them, Lucy following him. "Warm today," he remarked, loosening his collar.
"I'll make it," she said
She went into the kitchen and made the tea, as they both knew she would, then cut a few slices of his favourite wholemeal bread, spreading it with butter so soft that it melted into the coarse grain.
He was sitting in a big old saddleback chair, feet up on a pouffe, huge boots jutting out like boats. As she came in with the tray, he tapped the stool by his side.
"Put it here, I'll pour out."
She laid the tray on the stool and drew up a small sewing chair, one she especially liked because it gave her arms and elbows freedom. Old Fisky handed her a cup.
"You're a very good girl, Lucy, and I know I'm an old grouch sometimes, but you're a very good girl. You don't try to do what you're not qualified for, and that's unusual in a young woman. You'll be rewarded one of these days, don't worry about that."
"Do you think this is a find?" she asked eagerly.
"It's an old picture," he repeated noncommittally. He poured a cup for himself. "I'm sorry about your father, Lucy," he went on unexpectedly.
She closed her eyes but didn't speak.
"Don't you ever feel lonely?" Old Fisky asked.
"Yes," she admitted. "Yes, sometimes."
"Never thought of getting married?"
She opened her eyes wide in astonishment.
"Don't be daft," she said flatly.
"But that's not daft," he protested. "Don't you like men, Lucy?"
"I never think of them," she stated simply.
"If you did your hair properly, and if you—" He broke off, drank his tea in great gulps, and immediately poured himself another cup. "Oh, never mind, never mind. It's your own life, as Martha keeps reminding me." He was mumbling now, talking to himself, and Lucy could barely distinguish the words. "Make yourself attractive and some fellow will come and take you off, and I'll lose the best assistant I ever had." Suddenly, much louder, he asked, "Will you have more tea?"
"Yes, please," she said meekly.
At last the half hour passed, and on the tick of the last minute Old Fisky stood up and led the way downstairs. Side by side, they looked at the picture and saw that it was still floating; then Fisky pulled it out of the water and held it up gingerly by that magic corner. There were bubbles in the new canvas; the new and the old were coming apart. Now he laid it, painted side down, on an old towel, and began to pull the two pieces apart. He kept picking pieces off his fingers, the old paste used in the relining, hardened by the years but soft after half an hour of soaking.
"It's just relined," he said "That cleaning label was a fake. Now, let's dry it out and see if we can see what it is." She waited, only too aware of her own shortcomings, while he examined the picture.
Soon appeared larger patches of the blue and white and then other, richer colours, reds, greens and yellows and pinks of the flesh and browns of hair and eyes. The old man breathed very hoarsely; there was no doubt at all of his excitement.
"I think it's a John Bettes - that's sixteenth-century Dutch," he said. "Let me have the Benezit, Volume Three. Hurry, girl, hurry!" Lucy turned to the wall shelf where the row of green bound books stood, took down the volume he wanted, and held it out to him. He flipped over page after page, then suddenly stopped.
He glanced at her over the rims of his glasses and she did not know what was in his mind, but he had discovered something. He beckoned her and she joined him, feeling almost sick with hope and apprehension. His long forefinger was pointing at a black-and-white plate, and she saw at a glance that this was a picture of the painting she had found - and she also knew, from Old Fisky's manner, that this wasn't all it seemed to be.
"What is it?" she gasped. "Please, what is it?"
" 'Summer Idyll'," Old Fisky said slowly. "It was stolen from Rosebury House in Suffolk about ten years ago. It's a find all right, Lucy, it's a big find; it's a very valuable painting. And" - he put a damp hand on her shoulder, and went on in a gentle voice - "there'll be a reward, my dear, quite a substantial reward."
"Mr. Fisk," she asked, hardly breathing, "How much would you think? Please tell me how much."
He looked at her, moistened his lips, and then said, "Fifty pounds, at least, fifty pounds - it might even be a hundred."
And he knew, as anyone would have known who saw the rapturous expression on her face, that she was enthralled, that to her a hundred or even fifty pounds would be a fortune.
He had never seen her looking so happy.
Nor had Police Constable Wilberforce, who "happened" to be passing on the other side of the road when, an hour later, she went out for a walk. He was not in uniform, so he could follow her without being too obvious, and he walked along King's Road and then into New King's Road and across to the Eelbrook Common, where many played and more strolled and sunned themselves in the lovely evening. Lucy walked as she always did toward Walham Green, now known to many as Fulham Broadway, and he cut along another path, easily outpacing her, until near the Broadway he was able to turn and walk toward her, so that soon they came face to face.
His eyes brightened and he managed to sound surprised.
"Why, Miss Jenkins!"
She looked puzzled as he drew up.
"Don't you recognize me?" he asked "Wilberforce, P.C. Wilberforce. I saw you on Wednesday, when I came to the shop."
"Oh, I remember!" She was quite pleased. "You had your uniform on then, so I didn't recognize you."
"That's easy to understand," he said "Are you going far?"
"Just for a walk, that's all"
"May I join you?" Ian Wilberforce asked.
"Well—well, yes, that would be very nice." Lucy's eyes lit up in turn, and there was a new brightness in her as they walked back toward New King's Road, then along the terraced houses which flanked one side of the common, and eventually to Parsons Green. When they were there, he asked, as if out of a brain wave: "It's a lovely evening - would you like to take a bus up to Putney and walk along the river?"
"I'd love it!" she cried. But suddenly her voice plummeted in dismay: "But if I'm taking up too much of your time—"
"Not a bit," he said hurriedly. "It's my day off, and I'm as free as the air. Days off are a bit boring sometimes," he went on. "I live in digs, you see, and it's hardly home from home."
They sat close together on the bus.
They linked arms on the towpath. They walked slowly, very slowly, back toward the bus, two hours later.
I daren't kiss her, he thought, it would scare the wits out of her. She's as fragile as a bird.
He's wonderful, she thought, he's wonderful, but I hope he doesn't want to kiss me. I'd faint if he wanted to kiss me.
"Goodbye," he said. "I've enjoyed being with you so much."
"Oh, so have I!"
"If you're free on my next day off, perhaps—perhaps we could go to the pictures."
"Oh" she said, her heart in her voice. "I'd love to."
That was a little after half past seven, and at the very moment when Lucy was closing the front door of the shop in Fulham, Sir Richard Falconer was opening the door of his wife's bedroom, a long and immaculate Regency room with a canopy over the head of the bed. Beyond, opposite the main door, was the bathroom and a tiny dressing room.
Lady Falconer came out of the dressing room, and started at the sight of him.
If he was aware that it was the first time he had been in her room for nearly a year, he gave no sign. He was dressed in a dark suit and a tie with a single pearl pin, and the pearl picked out the colour of the hair at his temples; he was a little thinner than he had been for some time past, and there was more animation in him than he had allowed his wife to see for years.
"Charlotte, has Christine telephoned?" he demanded.
"No."
"Are you quite sure? Could your maid have forgotten a message?"
"I've been in all the afternoon," Charlotte Falconer answered. "I'm as worried as—" She paused for a fraction, and then went on: "As you are."
"Are you sure she was going out with this man Judd?"
"She didn't say so," his wife replied. "She simply said that she must have a life of her own, that she wasn't going to have lunch with us but was going out."
"So you only assumed she would see Judd?"
"Yes, Richard. But, knowing her, I can't think I am wrong."
"No," mused Falconer "You're almost certainly right, of course. I'm very worried, Charlotte."
"Yes," Charlotte said, almost wonderingly. "I can see you are. But, Richard—" She paused again because he looked so impatient, but went on to say exactly what she had intended to say: "It isn't a catastrophe or a crisis because a young woman of twenty-three goes out to lunch and stays out to dinner, you know."
Falconer caught his breath.
"I am quite aware of that. I am also aware of my daughter's character. She went out to lunch, and told you, so that you should not be anxious. Not because she felt any obligation - she is obviously in a mood of revolt against me - but because she would not wish to hurt you." He astonished Charlotte by his positiveness, by the evidence that he knew Christine's nature and temperament remarkably well. "And she would not say she would be home for dinner unless she meant to be."
"She could have changed her mind."
"Then she would have telephoned."
"She might not be near a telephone."
"Then she is in a very peculiar place." Falconer hesitated, studied his wife closely and much as he would one of his other precious possessions, and then said in a more subdued voice: "If she isn't here by half past nine, I shall ask the police to search for her. Meanwhile, I shall tell Oily to send someone to the shop in Hampstead." He went out, and she heard a rare and remarkable thing: his voice, raised. "Oily!" There were sounds of footsteps, and then Oliphant's voice sounded in turn.
"At once, Richard..."
"... And I want to know the result whatever they find, whatever I am doing," Falconer finished. "If you must, interrupt me at dinner."
Oliphant's voice carried back faintly: "You shall hear the moment there is any news."
Two men went to the shop in Hampstead Village, tried the front door, and rang the bell. No one answered. No one appeared to be in the shop. They left, passing a Jaguar which gleamed very white beneath a street lamp. As they went, Robin Kell watched from a corner of the first floor window.
"There's no answer," Oliphant reported to Falconer soon afterward.
It was then ten minutes past nine.
"Who do you think it was?" asked Lance Judd huskily.
"Sir Flicking Richard Falconer's leg men," growled Robin Kell.
"Are you - are you sure it wasn't the police?"
"I'm sure," Robin said. "And let me tell you this. Rather than let the cops or anyone else get those paintings without paying for them, I'd burn the lot. Don't make any mistake, I'd burn the lot. If they're no use to me, they won't be any use to anyone."
Lancelot Judd felt quite sure he meant exactly what he said.
# 16: The Commissioner
At half past nine, Gideon took a surreptitious glance at his watch, then pretended he was absolutely absorbed in the singing and the playing - as, but for Scott-Marie, he would have been. From the moment he and Kate had arrived, everything had gone with a swing. Penelope had arrived soon afterward with her latest conquest, a youth named Peter, Hobbs apparently forgotten. Prudence's husband, another Peter had obviously taken to him; the five-year-old grandchild found first Kate's lap and then Gideon's shoulder places of comfort. Kate had gone into the front room of the little suburban house and started to strum on the piano. Kate's playing was of the Sunday School teacher type, accurate but very deliberate. Penny's Peter had produced a harmonica and begun to play with a practised ease. Prudence had pulled her violin from behind the upright piano and, for no reason at all, had started an Irish jig which everyone followed. Gideon and Pru's Peter had found themselves in the kitchen, carving cold joints, then laying the dining-room table, and "dinner" had been a running buffet to the liveliest musical accompaniment.
Gideon had known occasions when he would have found such an evening raucous and off-putting, but tonight he was as nearly content as a man could be. And now Penelope was to play her sonata and Prudence, playing the violin as well as in her orchestra days, was to follow with a piece by Liszt. And in half an hour Gideon was due at the Commissioner's home, half an hour's drive away. He hadn't said anything to Kate, expecting the party to break up about nine o'clock, as it usually did.
Just before Penny began, the front doorbell rang, and Pru's Peter slipped away.
"Probably neighbours, to complain about the noise," Prudence gasped in mock horror. She was the most beautiful of Gideon's three daughters, and marriage and motherhood had warmed that beauty. "It couldn't be anyone for you, Daddy, could it?"
Gideon pulled a face while his heart leapt hopefully. Peter came back, a tall, young-looking man with a shock of curly hair. He raised both hands to enjoin silence, and when it fell he said sepulchrally: "The cops!"
It was an evening when such a crack was hilariously funny, and they all roared. As Gideon followed Peter out, he caught Kate's eyes and recognized her expression: You knew you'd be called out, didn't you? On the porch a uniformed policeman was standing almost at attention, and at the curb stood a police car, blue sign glowing.
"Commander Gideon?"
"Yes."
"Sorry to disturb you, but there is an urgent message on walkie-talkie for you, sir."
Gideon, who had already made up his mind what to do, slipped the keys of his own car into his son-in-law's hand, and said huskily: "Happiest evening I've had for a long time, Peter. I won't break it up. Will you ask Mother to drive home? I'll get there as soon as I can."
"Right-ho," said Peter. "It has been good, hasn't it?"
"I tell you, my happiest evening for a long time. I haven't seen the family so much together for years." Gideon gripped Peter's hand, then went off down the short drive, the uniformed man a pace behind him. The other patrol man was standing by the side of the car, walkie-talkie in his hand.
"One of you get in the back; I'll sit next to the driver," Gideon said. "I want you to drive me somewhere in a hurry." He flicked on the walkie-talkie and announced: "Gideon," and squeezed into the car. "Drive toward central London, will you?"
"Sorry to disturb you, sir," a man said on the walkie-talkie as the car started off smoothly. Everyone was a sight too solicitous tonight. "This is Thwaites. A man was found in the seat of a Jaguar car in a garage at a block of flats in Swiss Cottage tonight - twenty minutes or so ago, as a matter of fact."
"Well?" Gideon said.
"Positively identified as Paul de Courvier, who lived at the flats," Thwaites said. "I picked up some conclusive evidence that de Courvier and Jenkins were working with Slater on some unknown job, and I was going to report first thing in the morning. Three of them are dead now. I thought you ought to know at once, sir."
"You're quite right," said Gideon. "Where's the body?"
"Still in the car, sir."
"Finish what has to be done and then get it to the morgue at the Yard," Gideon ordered. "How long had the body been there?"
"Only a short while. It's a long story, sir; the thing I rang you about is that Division says it was parked most of the day in - the Jaguar I mean, sir - was parked in Hampstead Village near Lancelot Judd's shop. And Judd is the boyfriend of Sir Richard Falconer's daughter, you'll remember. I thought you ought to know," Thwaites repeated. "I'm having the movements of the car traced."
"Was it stolen?"
"No, sir, I don't think - it really is a bit complicated to explain over the telephone." Thwaites was actually pleading with Gideon to say he would go straight to the spot. This in itself was unusual. Thwaites knew better than to expect the Commander to go out on any job; that was why he was being careful not to make the suggestion.
"I'll be in my office in about an hour: say, an hour and a half," Gideon said "Call me there."
If Thwaites felt any disappointment, he hid it successfully.
"Very good, sir."
Gideon rang off, and looked round at the man behind him. The car was moving at a steady speed, the driver obviously very aware of his high-ranking passenger.
"Who's in charge in your division?" Gideon asked.
"Superintendent Loss, sir."
"Ask him if you can take me to Peel Crescent, Mayfair, and drop me there," Gideon said.
"Ask—" the man echoed, and then he stopped himself, reached for the radio, and spoke to his division. Formalities over, Gideon sat back in his seat, and after a few minutes, declared: "I'm in a hurry."
"Right, sir!" exclaimed the driver, and the car surged forward through the nearly empty streets. Rain began to fall softly, making the road surfaces greasy, but Gideon was thinking back to the liveliness of the evening, the happiness; and he gave little thought to what Thwaites had said and why Thwaites had been so anxious to talk to him. Then the car pulled up outside a house in a crescent terrace of Georgian houses which had a peaceful stateliness even in the lamplight diffused by rain that was now falling steadily but softly. The car drove off on its patrols, wheels swishing, and when it had disappeared the only sound was the soft pattering rain. It was ten minutes past ten, and he could recall the time when his heart would have been in his mouth because he had kept the Commissioner waiting. There was a light inside a wrought-iron shade above the fan-shaped light over the door. Gideon pressed the bell, and almost immediately footsteps sounded.
Sir Reginald Scott-Marie opened the door himself.
"Ah. Come in, George." The "George", was clear evidence that he meant this to be informal. "What, no coat?" They entered the hall, with the staircase and its polished balustrade leading upward; the walls were duck-egg blue, the lighting subdued; several portraits, all of Scott-Marie's ancestors, hung on the wall. If there was an ideal house in Gideon's mind, this was it.
Scott-Marie led the way into a study-cum-library, with a big desk, book-lined walls, rich carpet, deep and comfortable-looking armchairs. Between these was a table with two decanters and four glasses.
"Brandy?" suggested Scott-Marie.
"Thank you." Gideon, sitting, looked up at the tall, lean man with the close-cut grey hair and the clipped moustache, the slightly sandy complexion, the chilling grey eyes. This man had become a legend in the Metropolitan Police Force; a martinet who nevertheless knew and was considerate of people, whoever they were.
Scott-Marie poured brandy into two bowl shaped glasses, and sat down. A glow came from a single-bar electric fire, slightly incongruous in the Adam fireplace.
"How much do you know about the conference due on Tuesday morning?" Scott-Marie asked, and implied very much more them the words themselves.
Gideon said mildly, "I've been informed."
Scott-Marie obviously appreciated the dryness of that comment. He sniffed the brandy, and asked: "Would it serve any useful purpose if I were there?"
"It could serve one very good purpose," answered Gideon promptly.
"Exactly what?"
"There is a strong tendency to see the National Gallery theft as a London concern only," Gideon said. "It's much wider than that, of course, and I think we should extend the inquiries into all the provincial centres, as well as immediately ask for cooperation abroad. Customs officers could be of great help: a picture of this size, rolled or flat, wouldn't be easy to hide even on a cursory inspection. If you made it clear that you regard it both as a national and an international matter, then I think a lot of people will be suitably impressed."
"I see," said Scott-Marie.
Another man might have said: "Why isn't it being treated like that already?" And another man than Gideon would have said: "The Assistant Commissioner doesn't see it this way." Neither spoke, until at last Scott-Marie stretched his legs nearer the fire.
"I see," he said again. "I shall be at the conference."
"I thought I'd send Hobbs, sir," Gideon remarked.
Scott-Marie looked at him very directly, and he wondered whether a direct question was coming; but the moment passed, and the Commissioner sipped again before going on: "There's another thing I've been wanting to talk to you about for a long time, George."
And he now regarded the time as right, reflected Gideon.
"Yes, sir?"
"The smuggling of immigrants and the relationship between coloured people and the police generally," Scott-Marie said, and his lips turned down in a droll smile. "I know that's a tall order, but I'm concerned with the principles, not the detail. Are you satisfied with one, or the other, or both?"
"I'm not satisfied at all," Gideon answered quietly, "but I think the first will work itself out." He explained briefly about Riddell and what he planned, and then went on more firmly: "The relationship is a very different matter, sir. It's a question of individual reaction and the ruling circumstances. I've no evidence at all of a deteriorating relationship. But that isn't to say that it's good enough. I'm talking about our area, of course, and we have one or two difficult places, such as Notting Hill. It's so much more than a police matter."
"And yet we can exert a lot of influence," Scott-Marie observed. "I've just been to Scarborough, as you know. Reports about most of the matters discussed were given to the newspapers, but some were confidential. And among those not told to the press is the growing anxiety about the conditions and the situation caused by immigration. There is strong feeling that it isn't being handled very well by anybody."
Gideon made no comment.
"I was asked - as was every Chief Constable present - to investigate the local situation from the police point of view."
Gideon stirred uncomfortably.
"That's all right provided it's understood that the police is only one point of view, sir. There's a tendency in the press—" He broke off.
"Go on," urged Scott-Marie.
"A tendency to make us the scapegoats," said Gideon. "It's one thing for us to have a problem like the smuggling, which overlaps with social and industrial aspects; it's quite another for us to be regarded as the only people responsible. And there is that tendency, sir."
"Yes," agreed Scott-Marie. "There were some delegates at the conference who know this and resent it very much. I think I would like to discuss it with you, Riddell, and those Superintendents of Divisions where there is a larger-than-average proportion of coloured immigrants. While it's a long-term problem, there are short-term dangers, so I ought to do this fairly soon. When will Riddell be back?"
"In three weeks, sir."
"Then we'll set the date a month from now; check with my secretary, will you? Brief the Divisional men and Honiwell, and a few days before the conference - those conferences! - let me have a written assessment. Then you and I can talk about it the day before, say."
"I'll arrange all that," Gideon said. There was something else he wanted to say, but he wasn't sure this was the time. And he was tempted to tell Scott-Marie about Thwaites's suspicion of Sir Richard Falconer, but as it was only suspicion and not yet strongly based, he decided not to.
"Some doubt in your mind?" asked Scott-Marie.
"I'd like Hobbs to be with us at all stages, sir, unless you've any objection," Gideon answered.
"None," said Scott-Marie.
"And—" Gideon hesitated again, and did not feel very pleased with himself, but he had gone too far now, and went on: "Shall I brief the Assistant Commissioner, sir?"
"I will inform him," Scott-Marie said, in a more formal tone than he had used before.
"Thank you, sir." Gideon finished his brandy, then straightened up in his chair. "I really ought to be going, unless there's something else urgent. I promised to look in at the Yard before going home. Something had cropped up just before I started out, and I don't know the significance of it yet." He was still tempted to mention Falconer, and annoyed at his own indecision.
"Then I mustn't keep you," Scott-Marie said. They both stood up and went toward the door, and as they entered the passage the telephone bell rang in the room behind them.
"You answer it, sir, I can let myself out," Gideon said.
Scott-Marie hesitated, then turned back toward the remorseless insistence of the bell's ringing. "All right, do that - good night."
"Good night, sir." Gideon opened the door and noticed with wry amusement that the Commissioner of Police had the most elementary system of burglar prevention. He felt the rain wafting gently into his face and hesitated; he would get soaked if he were out long in this, but the call for Scott-Marie had prevented him from sending for a car. He would manage. He closed the door firmly. There was a police call box not far away, and he could wait on a nearby porch. Then a taxi came splashing past, its sign alight, and he let out a bellow which made the driver slam on his brakes. Sprinting after it, he was halfway between porch and taxi when he heard Scott-Marie behind him, with a parade-ground roar: "Gideon! Gideon!"
Gideon managed to pull himself up and half turned. "Coming, sir." He looked at the driver, a little man with a huge nose. "Wait for me, please."
He turned back to Scott-Marie, who was outlined against the light of the hall.
"Sir Richard Falconer is on the line," he called. "You'd better hear what he has to say."
# 17: Urgent Call
Scott-Marie turned back in to the house; Gideon hesitated and, rain making a film over his face and beading his eyebrows and hair, went to the taxi driver, who had pulled in and switched off his "For Hire" sign.
"Sorry," Gideon said, and put six shillings into his hand. "It might be too long a wait."
"No hurry where I'm concerned, Mr. Gideon," the driver said.
"Oh. Recognized, am I?"
"Yes, sir, often see you about. Be a privilege to take you as a passenger."
Gideon half smiled. "Thanks. I'll be as quick as I can." He went into the house and took out his handkerchief to dab his face dry. Every now and again, his habit of going without hat and coat let him down, and it was surprising how wet he had got in the past few minutes.
Scott-Marie was holding the receiver to his ear; once or twice he attempted to speak, but Falconer obviously talked him down. When he saw Gideon, he pointed, stabbing his finger toward the back of the house, and it dawned on Gideon that he was telling him to go somewhere—ah! The cloakroom, beneath the stairs. Gideon went in, dabbed himself with a towel, and came out feeling much drier. He found the Commissioner standing in the library doorway.
"Falconer is worried about his daughter, who has been missing for several hours," Scott-Marie announced, with a dry, almost wintry smile. "Naturally, he would like the whole of our resources concentrated on finding her. If we don't exert ourselves, he will undoubtedly bring all kinds of pressure to bear tomorrow, which could make unnecessary difficulties, but—" He spread his hands. "I do, of course, leave it to you."
As Gideon listened, he knew that he could no longer put off telling the Commissioner the whole story, and he wondered with sudden raw urgency whether Thwaites had discovered something more about the rumour of Falconer's involvement. He must find out as soon as possible; but first he had to tell Scott-Marie.
Falconer put down the receiver after talking to Scott-Marie and drew a hand across his head. He was trembling. His wife was astonished and, in a way, touched. He went across to a table, took a cigar from a silver box, pierced the end, then came back to the big couch and stood looking down on her.
"What did he say?" she asked.
"He told me that he would start inquiries at once, but—" Falconer paused to strike a match and to puff at the cigar. "But he pointed out that Christine can hardly be regarded as missing."
Gently, Charlotte Falconer said: "It isn't much after eleven o'clock, Richard."
"Do you mean that you don't regard her as missing?"
"I think she may simply have carried her rebellion a step further," Charlotte replied; for her to challenge his opinion so positively was almost daring.
"I do not believe that Christine would act so much out of character," Falconer insisted. "And if there is no news of her by morning—"
"Oh, there will be! She'll be back tonight."
"I hope so," Falconer said. "I hope so very much."
He broke off as Davies came in, soft-footed. Davies would only appear at this hour if there was an emergency or an unexpectedly late caller, and Falconer swung round on him with unusual vehemence.
"What is it?"
"A young man is here, sir," Davies said.
"Young man?"
"Perhaps it is Lancelot Judd," Charlotte said, getting up quickly.
"No, my lady, I would recognize Mr. Judd's voice," said Davies. "He has phoned here for Miss Christine several times. This gentleman did not give his name but said he has a message from Miss Christine. That is why I let him in."
"Where is he?" Falconer demanded.
"In the morning room," Davies answered.
"Tell him I'll see him very soon," Falconer said. "Where is Mr. Oliphant?"
"He went out earlier in the evening, sir, and told me that he might be late - might not be here until tomorrow, in fact."
Falconer nodded, and the butler went out, as soft footed as when he had entered. Falconer drew deeply at his cigar, where half an inch of greyish-white ash now showed, and began to move about nervously.
"I should have told Oily to stay in tonight" he said vexed: "This is no time for him to dally with his girlfriend." He paused, then blurted out: "Charlotte, I am deeply troubled about Christine. Have you the slightest reason to think that she might have eloped with this man Judd? Could this be the caller's message?"
"Good gracious!" exclaimed Charlotte. "The thought didn't cross my mind."
"You are not—you are not deliberately protecting her? Giving her time to get out of my reach?"
"No, Richard, I am not," answered Charlotte Falconer. She got up and moved toward him, took his hands in hers, and felt the chill of his fingers and the agitation which made him quiver as if an electric current were passing through his body. "No, my dear, I would not dream of doing such a thing to you. You can trust me, Richard, you can trust me absolutely."
He gripped her hands tightly for a few seconds, then let them go.
"I should know better than to doubt you" he said. "Well, I must go and see what this fellow has to say." He turned and hurried out, without looking backward, so he did not see how motionless his wife stood as she watched him. She was like a statue. He went into the hall, told Davies to stay within call, then turned in to the morning room. It was here that the family received casual guests whom they did not want to entertain officially.
A young man stood with his back to the fireplace; a tall young man dressed rather as an exquisite, with over-long hair beautifully groomed and shining golden in the soft light. His eyes, narrowed, were very bright. He wore a suede suit with unusually short lapels, and a carefully arranged cravat, and he looked very young, certainly no more than twenty-two or three. Christine's age. Falconer was tempted to demand "What do you know about Christine," but he fought for self-control
"Good evening."
"Good evening, Sir Richard. It is very good of you to see me."
It was a pleasing voice, which was one good thing, thought Falconer. But why didn't the youth come straight out with the message from Christine?
"I would not have called so late had it not been on an urgent matter." There was a hint of insolence in the youth's manner as he spoke.
"Be good enough to come to the point," Falconer said.
The young man did not answer, and belatedly it occurred to Falconer that he should have asked his name. He was in the grip of a tension such as he had not experienced for many years, but if he lost his control this apparently foppish young man could too easily take advantage of him. So he waited.
The stranger spoke with great precision.
"It concerns the safety of your daughter Christine, sir."
Falconer almost cried out: "Safety?" but he still maintained that iron self-control.
"Go on, please."
"Christine is late home tonight," the youth stated.
"I am aware of it."
"Aren't you curious about the reason?"
"I am always curious about the mental processes of the young."
"Are you, indeed," the young man said. "I said that my business concerns Christine's safety."
"I heard you."
"Aren't you concerned?"
"I am very concerned."
"I am glad to hear it," the young man said. "She can be returned to you, unharmed."
"If she isn't, then those responsible for harming her will rue it very much indeed," Falconer said grimly.
"I doubt that," said the young man. There was something almost feline in his manner, and now he gave the impression that he was fully aware that Falconer was hiding his fears and his tension, was suffering anguish; and that he, the youth, enjoyed what he was doing with sadistic relish. "I doubt that very much indeed. Don't you want to know where she is?"
"I want her here, unharmed."
"On certain terms," the young man said, "you may have exactly that."
"We can discuss terms afterward."
"Oh, no, Sir Richard. We shall discuss terms now."
"Tell me how much money you require in exchange for my daughter," Falconer said coldly. "I will consider the exchange then."
The young man smiled. He had a honey-brown complexion and his teeth were white and even. There was tawniness about his eyes, and his lips were well-shaped and soft.
"It is not a question of money," he stated.
"That I do not believe." Falconer took the stub of the cigar from his lips and laid it on an ashtray without shifting his gaze. But it was difficult to keep his hand steady, difficult to prevent his voice from rising, and even more difficult to restrain the impulse to knock the smile off the sneering face in front of him.
"Whether you believe it or not, it's the truth," the young man insisted. "You have a remarkably fine collection of pictures and objets d'art, Sir Richard."
"We are not discussing my collection."
"I am," said the young man. "Perhaps I should introduce myself. My name is Robin Kell. I am reasonably intelligent, and I have a fair knowledge of art, especially paintings and small antiques. I am what you would doubtless call unscrupulous, possibly ruthless, and I do not hold human life in great esteem. We are born, we live, we die. Few of us make a very great impact. If you were to die, very few would miss you and several people would probably be glad to see you go. If your daughter were to die—" Robin Kell paused, and his manner was taunting, as if he wanted to make Falconer lose his self-control. But Falconer held on, finding it easier now, and the young man continued - "you might miss her for a while, and no doubt so would her mother. But who else would really care, Sir Richard? How much sense would her dying make to the world?"
There was a beading of sweat on Falconer's forehead and upper lip. "Go on," he said icily.
"Ah, yes. I was discussing your collection," said Robin Kell. "Has it all been honestly come by?"
Very deliberately, Falconer said: "Every item has been bought at a reasonable value, yes. Mr. Kell, unless you make arrangements to bring my daughter back here at once I shall send for the police."
"Oh, you would be most ill-advised," answered Robin softly. "If the police are told, you will certainly never see your daughter again. And you would lose not only Christine," he went on "but a unique chance to expand your collection at a remarkable rate and really very reasonably. I need no telling that paint and canvas mean more than flesh and blood to you - so let us have no talk of the police, Sir Richard."
Davies heard all of this, and also knew that, less than an hour before, Sir Richard had talked to Scott-Marie, demanding that the police give absolute priority to finding Christine. The certain thing now was that he, Davies, must do nothing unless Sir Richard told him to.
He himself was much more composed than he had been, Falconer realized with relief. In spite of his inner tensions, he could control his expression, and he did not believe that this young man had the slightest idea how the last words had slashed through him. He moved, and sat on the arm of a couch, crossing one leg over the other, and exerting every fibre of his being to appear calm and unconcerned.
"Mr. Kell," he said, "you can neither blackmail me nor bribe me. If you fail to make immediate arrangements to return my daughter, unharmed in any way, I shall telephone the police." He paused long enough to see a flicker of doubt in Robin Kell's eyes, and went on: "I am prepared to discuss a ransom. I am not prepared to allow delay. How much do you want?"
Robin did not answer but for the first time turned away, unfastened his jacket and trouser waistband, and drew out a linen-wrapped package, thin, and curled slightly to the shape of his body. He held the package in one hand as he refastened his waistband, and proffered it to Falconer.
"Your daughter and that for a hundred thousand pounds," he said. "It is a very good bargain."
Falconer handled the package, and knew that it was a picture but at that moment did not suspect which one. He felt a strange conflict within him: a desire to fling it into the youth's face and to order him to fetch Christine if he did not want the police here at once; and a desire to see what painting this was. It was a conflict he had known to some degree all his life. He was aware that a great number of people thought he was interested in the precious things he owned because of their value or because of their rarity; that his possessions were for possession's sake. But it was much more than that. Beauty, especially beauty created by man, had an effect on him that was like a slow-burning fire which sometimes burst into flame. He had first been scorched by this when, as a child, he had stood in rapt wonder in front of pictures in the National Gallery. It was more than desire, even though he desired to possess them as, from time to time, he desired a woman of rare beauty; and it was more than passion.
He loved them.
The artist, through his creation, stretched out and touched him, and it was like being hypnotized. He could not resist things of such beauty. In his youth he had read all he could about them. No matter what it was, painting or sculpture, jewel or piece of furniture, he felt a physical response. It was not simply the pleasure of floating because the beautiful things in this house were his, it was a fact that he knew sensuous pleasure at both the touch and the sight of them.
And now he felt sure that there was something in this package which the youth knew he would desire.
But he was also a man of integrity.
What the youth could not possibly understand, what no one could ever understand, was how deeply, how cruelly, he was torn; torn between his knowledge of what he ought to do, his desperate fears for Christine, and his compulsion to open this packet.
Then, in a flash of understanding that was like a revelation, he understood what he held in his hands.
# 18: The Ultimatum
Very slowly, Falconer went to a table, put the packet on it, untied the knot, and then took away the string. The linen itself was stuck down but he was able to pull it away. He saw the white wrapping paper underneath, secured with tape, and tried to ease it up with his thumbnail. His heart was thumping with almost frenzied beat. He knew that the colour had been drawn from his face, that his tension must be obvious if he looked up. The white paper tore with the sticky tape, and he pushed it aside.
Here was the canvas back of a picture.
Slowly, he turned it over. A sheet of thin sponge rubber, protecting the picture itself, dropped away as he looked down. The first glimpse told him how right he was; and also did much more. It seemed to tear at his heart, as if the hands, as if that withered thumb, broke out of the canvas and clutched at him through his very flesh and bones. As he straightened it and saw the whole picture for the first time, his breathing seemed to stop.
After a while, forcing himself back to normality, he became acutely aware of the youth staring at him. The time had come when he must look up.
Laying the painting on the couch as if indifferently, he yet contrived to place it so that he could still, when looking that way, feast his eyes upon its beauty. Wrenching his gaze away, he turned to Robin Kell.
"So you are the thief?" he asked.
"I am the thief."
"It is like having a price on your head."
"Yes," said Robin softly. "That is exactly what it is like."
"On no conditions will I take this picture."
"It is very beautiful," remarked Robin.
"I know how beautiful it is," replied Falconer stiffly. "But it is owned by the State. It is part of the national heritage."
"Don't give me that nonsense and don't fool yourself," replied Robin. "If it's anyone's national heritage, it is Spain's, and Britain plundered it." He drew in a deep breath. "It is superbly beautiful."
"It can never be part of a deal with you over my daughter," said Falconer.
"If it isn't, there will be no deal over Christine," retorted Kell.
"Tell me how much—"
"I want one hundred thousand pounds for the painting and Christine together," insisted Robin very softly. "If you don't pay, then I shall destroy the painting and you will have destroyed your daughter."
"But you can't destroy this!" cried Falconer, and his voice rang about the room and into the hall and up the stairs.
Davies stood absolutely still, appalled.
The cry reached the ears of Falconer's wife, who was at the head of the stairs, wondering whether to go down. She was startled by the passion in his voice, the emotion he had not shown for so long. She heard another voice but was unable to distinguish the words. As Christine had, much earlier, she stayed where she was, knowing that she must not disturb Richard and yet desperately anxious to know what was going on in that room.
Robin Kell stood unmoving but not unmoved, his smile broader than it had been since he had entered the room, giving the impression that he was on the verge of triumph. Falconer stared at him helplessly, shaken by his own outburst, by his uncontrollable anguish, by his feeling of despair caused as much by his fear of the destruction of this picture as by the loss of his daughter.
He knew - no one else could ever know, but he knew - which he most cared for, which he most longed to save.
Robin thought, He can't resist them both. He might have put up a fight against one or the other, but together they're irresistible. He did not look away from the man, and his smile deepened to raw gloating when Falconer's gaze turned, as if under a compulsion he could not resist, to look at the painting on the couch.
Falconer was aware of the near-hypnotic appeal of the painting, of the colours, of the face of the child prince. But his mind was gradually taking over from his emotions, and he was beginning to assess the situation objectively. His own great battle was still to come. This man was kidnapper, murderer, thief: and beyond reasonable doubt, a liar. He would take the hundred thousand pounds if he could get his hands on it, but there was no possible guarantee that he would give up the Velazquez or Christine. And even if the exchange could be guaranteed, what would be the right thing to do? Let this cold-blooded youth go, to prey upon others as he, Falconer, was being preyed upon, to kidnap their loved ones, using priceless treasures of men's making to satisfy his lust for money?
Whatever the result of his forthcoming struggle to decide, the more he could find out about this young man, the better; and he would find out more if he showed a convincing interest.
"So you think I can't destroy a piece of canvas with some paint daubed over it," said Kell. "It won't take long to prove I can, and I haven't time to waste." He moved his right hand toward his waist slowly. His trousers were low cut, with pockets slantwise toward the hips, and very closefitting. Suddenly he dipped into a pocket with incredible speed, and there was a click, a flash, and in the youth's right hand was a knife with a thin, glistening blade. He held this between his thumb and fingers, thumb uppermost. "How about a few slashes with this, Sir Richard? Across the picture - and your daughter's face."
Falconer's breath hissed inward. "Don't!"
"I don't share your sensitivity," Kell jeered. "That is paint on canvas which represents money. I have a lot of things which represent money. And as for your daughter—well, she is a pretty girl, but pretty girls come by the dozen, so you can't influence me by sentiment any more than you can appeal to my conscience." He gave another of his slight, meaningful pauses, then went on: "You need to know who you're dealing with, don't you?"
"Yes," Falconer said. He clenched his fists until his knuckles showed white and the fingernails cut into the palms of his hands. Whatever happened, he told himself, he must keep calm; must appear to be unmoved by anything this young savage said. Only by keeping calm could he even hope to gain the upper hand. "What do you mean, you have a lot of things which represent money?" he asked, forcing his voice to remain steady.
"I mean I am one of the best art thieves in the business, and I've been busy for a long time."
"You have—more such pictures as this?" Falconer asked hoarsely. How could he possibly think of allowing this man to stay free, yet if he didn't cooperate with him, what would happen to the Velazquez? And what - Falconer felt himself growing pale - what in heaven's name would happen to his daughter?
Kell nodded.
"Many, many more and I'm sure you wouldn't like to see them destroyed, would you, Sir Richard? But they will be - and that pretty daughter of yours as well - if you don't do as I say."
"How do I know you're telling the truth?"
"You know I'm telling the truth about "The Prince," don't you?"
"I should want much more evidence."
"You'll get your evidence every time we do a deal," Kell said coldly. "We've talked long enough, Falconer. Do you want "The Prince" and do you want your daughter? You can have both for one hundred thousand pounds in deposits at different banks in different countries. As proof of my good faith I'll leave you "The Prince," and as proof of yours I've got your daughter."
"I will not gamble with—" Falconer began.
"You're not gambling," Kell interrupted. "You don't have any chance unless you accept my terms." He gave a little laugh, not altogether gloating, at least partly because he was so pleased with himself. "Keep away from the police," he warned again. "Don't forget that calling them in would be absolutely fatal. I'll be in touch with you tomorrow to arrange details." He paused for a long time, and then said, "Do you mind if I let myself out?"
He pressed the knife, and the blade snapped back inside the handle. Putting it back into his pocket, he moved toward Falconer and the door. Falconer stood still, as if deliberately blocking his path, then moved aside very slowly. Kell walked across the main hall to the front door, looking up the stairs and along the passages but seeing no one. Falconer moved but did not go to the front door, which Kell opened easily. Kell stepped outside and closed the door behind him.
Davies appeared from a doorway, but Falconer motioned him away.
As he disappeared, Falconer looked up and saw Charlotte, now at the head of the stairs, and although she was so far away, her anxiety came clearly through to him, both in her expression and in the stiffness of her movements. In turn, she saw a tension in him that seemed almost to distort his face, a tension that seemed to have aged him ten years in the past half hour. He moved toward her, and as he went up the stairs his eyes seemed to burn. She stood still, holding her hands out toward him, half in comfort, half in fear.
"Is she—is she all right?"
"Yes," Falconer said, in a grating voice. "So far, she is." He took his wife's hands, gripping them tightly. "Charlotte," he went on, "I have a great deal to think about, and—I don't want to be alone tonight." He broke off, and the muscles of his throat moved as if he were trying to speak but could not.
She closed her eyes and seemed to sway, but after a few moments she steadied again, and said quietly: "Then why don't you come up to me?"
Falconer went downstairs, wrapped the painting and took it up to his own room, rolling it loosely before placing it in a capacious wall safe. Then he went to his dressing table and picked up a photograph of Christine and stood looking at it for a long time. At last he put it back, drank a little brandy, then ran a bath. At half past twelve, he went to his wife's room.
Robin Kell went out of Falconer House into the soft penetrating rain of the October night. All the street lamps had haloes; so did the headlamps of the few cars he passed. He looked along the sides of the house and at the doorways of the houses opposite, but saw no sign of anyone watching. To make doubly certain, he walked past the rows of parked cars toward Piccadilly, then, when he reached the end of the street, turned back and walked in the other direction. But he saw no one, no sign of movement in the parked cars. Satisfied now that he was not being watched, he turned another corner and got into his red Morris 1100. Once in, he waited for a few moments to reassure himself still further that he had not been followed, and even when he switched on the lights and drove off, he watched his driving mirror very closely.
By the time he reached Hampstead, he felt not only secure but exultant.
A plainclothes policeman in a front room of an empty house nearly opposite Falconer's switched on his walkie-talkie and reported to Information at Scotland Yard: "Kell left Falconer's without the package."
A uniformed policeman trying the front doors of some shops in Park Lane switched on his walkie-talkie and reported, also to Information: "Kell drove north through Hyde Park, moving at moderate speed."
And information, keeping close track, took in report after report:
"Kell is heading east along Oxford Street."
"Kell has turned left past Selfridge's Food Department."
"Kell is passing Lord's Cricket Ground and heading north."
"Kell is at Swiss Cottage."
"Kell is heading up the hill toward the pond."
"Kell is in Hampstead Village."
"Kell has parked in the High Street. He is locking his car."
All these reports were passed through on the teletype machine; the strips were cut out and pasted up, and sent to Thwaites, who was alone in an office which he shared during the day with four other Chief Inspectors. He was a little on edge, not because he was tired but because since speaking to Gideon on the telephone he, had begun to regret having done so. He would never have interrupted Gideon normally, and the truth was that three double whiskies on an empty stomach had given him Dutch courage. This had evaporated almost immediately when the Superintendent in charge at Wembley had called him.
"Would you like to know where my chaps dropped Gee-Gee, Harold?" he said.
"Yes - where?" Thwaites had asked eagerly.
"The home of the great man himself."
"But he is the great man."
"Not to Sir Reginald Scott-Marie, he isn't."
"My God," Thwaites had breathed. "The Commissioner."
"Himself," the Wembley man had confirmed. "But don't worry too much; you're not far off retirement, are you?"
Thwaites had managed to echo the other's laughter, but it was a very hollow echo. He had almost told Gideon he ought to come here, to the Yard, and not waste his time by going home. God! And Gideon would be coming here straight from Scott-Marie. Thwaites, checking over everything he had done and all the reports that had come in during the evening, munched a sandwich and drank more black coffee than he really needed. At all costs, he must be at his most effective when Gideon arrived.
Thwaites's interoffice telephone rang, probably with another report.
"Thwaites here," he said.
"Gee-Gee's just arrived," a man told him urgently.
"Oh! Thanks."
The other rang off, and Thwaites never learned who it was. He brushed his lips, pushed the tray aside, dusted some crumbs off his jacket, and then studied the summary of the reports. He had them off verbatim when his telephone rang again.
"Thwaites," he said, heart beating fast.
"I'm in my office," Gideon stated, and rang off.
Thwaites put down the receiver slowly and stood up. He had not felt like this in years, and could not remember feeling like it with Gideon before. He could not understand himself, did not realize it was because he knew he had stepped out of line and had no experience of Gideon's likely reaction, but he did know that Gideon had sounded pretty abrupt.
Within a minute of the call, he tapped on Gideon's door. As he waited, he was acutely aware of the stillness and the quiet, characteristic of the Yard at night and particularly noticeable after the bustle of the day. He could hear Gideon talking and wondered who else was with him. A possibility sprang to his mind and went through him like a knife. Not Scott-Marie?
"Come in," Gideon called at last, and Thwaites squared his shoulders and opened the door.
Gideon was sitting behind his desk and no one else was in the room. Gideon waved to a chair, made a note on a pad, and then settled back and said, in the most amiable of voices: "Sit down, Harold. If you're the same as I am these days, late nights make you leg-weary."
"Er—" Thwaites dropped into a chair, covered with confusion of his own making. "I know exactly what you mean, sir." Now he had the sense to leave Gideon to set the ball rolling, and was already feeling very much more himself.
Gideon bent down, took out his whisky and two glasses, and looked up.
"What do you like with yours?"
"If you don't mind, sir, I'll give it a miss," Thwaites said. "That's if you don't mind, sir."
"I'll give it a miss, too," said Gideon. "Well, what makes you think de Courvier is number three in the Velazquez killings?"
"I've compared a photograph of the knife or dagger wound with the one on Slater's body - Brighton sent pictures up. I've had Mr. Thompson look at them, and we both agree that it looks very much like the same kind of wound. And de Courvier is an old associate of Jenkins and Slater. We've established that." He went on very slowly: "But the thing that's rather shaken me, sir, is a report that Christine Falconer, Sir Richard's daughter, was seen to go into Judd's shop this morning. Division sent a car to look at the place, and the driver recognized her - her photograph is often in the papers. I've a nasty feeling that the shop ought to have been watched all the time."
"Wasn't it?" Gideon asked sharply.
"No, sir, not until tonight, and that's my fault. But it is being watched now, and—well, perhaps the best thing is to show you the reports as they came in." He pushed the sheet of teletype messages across Gideon's desk, and then added: "The man known as Kell is back there now, sir - in Judd's shop, I mean. It's remarkable, isn't it, that he spent well over half an hour in Falconer's house at this time of night? And he had a biggish flat package when he went in, but not when he came out. He was seen concealing it under his jacket on the porch, but his movements were too free for it to have been there when he left. It is remarkable, sir, isn't it?"
# 19: The Dilemma
Yes, thought Gideon, it was very remarkable indeed. Fresh in his mind was Falconer's approach to Scott-Marie. Would a man with anything on his conscience make such an approach to the Commissioner? On the face of it, it didn't make sense. He pondered as he looked at the pasted messages, then looked up, frowning.
"What package do you say Kell had with him when he went to Falconer House?"
"It was about the size of an open newspaper, and he seemed to wrap it round himself," Thwaites said. "It could easily have been a canvas."
"So that's what's in your mind," Gideon said, heavily. "It could have been the Velazquez."
"And it is perhaps now in Sir Richard Falconer's possession, sir."
"Yes," agreed Gideon. "So it appears. Let's have the whole story again."
"Very well, sir. Kell's a friend of Lancelot Judd, who owns the Hampstead shop. He was seen going to Falconer House carrying the packet, this evening. He went in with the packet but apparently did not bring it away. That was the time I decided to have him trailed very closely, sir. He went back to the shop about eleven o'clock and is still there. And he had nothing with him, as far as I know. If it was a canvas, it's probably still in Falconer House."
Gideon grunted.
"So we ought to search."
Thwaites drew his breath rather uneasily through his full lips, and after a few seconds, said: "I would certainly apply for a warrant if it were anyone else, sir, but I would hate to go wrong on this one."
"Yes," said Gideon. "So would I. Is Falconer House still being watched?"
"Closely, sir."
"Any instructions given?"
"To follow anyone who leaves," answered Thwaites. "I didn't feel I could go any further on my own authority, sir. It's been worrying me all the evening. I tried to get Mr. Hobbs but he wasn't home; he's visiting a sister and travelling back by road so I couldn't get him. And when I learned that the man de Courvier was dead I called you."
"Quite right," said Gideon. "Go down to Information, will you, and tell them to stop anyone going in or out of Falconer House. I don't care who it is: Sir Richard himself or anyone. They can go anywhere provided you make sure they haven't got that painting. All clear?"
"Yes, sir." Thwaites, obviously lighter-hearted than for some time, almost bounced toward the door.
"Thwaites!"
"Sir?"
"You did say that girl was not seen to leave, didn't you?"
"I did, sir," said Thwaites
"And the Hampstead shop is now being closely watched?"
"By half a dozen men, sir."
"Have anyone who leaves that shop stopped and searched, too," ordered Gideon.
"I will, sir!"
Thwaites let the door swing to, and it banged slightly.
Gideon glanced at it absently. His forehead was wrinkled in a deep frown. With Hobbs not available and most of the senior superintendents difficult to get at during a weekend, he would have to take this job over himself. He knew that he should, and he wanted to; an issue as delicate as this should not be left to others.
His chief concern now was for the missing girl.
Christine lay on a mattress in a corner of the back room, her legs and hands tied, a scarf bound round her mouth. She had a throbbing headache and her eyes were so heavy she could no longer keep them open. She did not know how long she had been tied up but she did know that Lance had been on the premises alone for a long time and had left her here. Lance, to do such a thing! Lance, whom she had so loved, with whom there had been such deep pleasure in this place only a few hours ago. He seemed to be so desperately afraid of Robin Kell that after a few half-hearted protests, he did whatever the other man told him.
She heard movements, but she could not see.
She heard someone close by, and knew that Lance had pushed aside the curtain. Light flooded the little alcove, bright enough to hurt her eyes. She did not open them and resolutely faced the wall.
"Chris," Lance whispered.
She pretended that she had not heard.
"Chris, it will be all right, I promise you."
Did he lie even to himself? She wondered helplessly.
He touched her shoulder, and it was like being touched by fingers of ice.
"Chris, do you want anything?"
She could not answer because the scarf was tied so tightly. He knew that and yet he could bring himself to ask such a question. She did not stir. She was half bemused, anyhow; the effect of whatever drug they had given her had not really worn off.
"Chris," he said again, his voice so close that she could feel his warm breath on her ear and on her cheek, "he's gone to see your father. Your father won't let you down."
At last she turned to look at him. There was nothing she could say because of the gag, but she could face facts. Here was the man she loved and who had declared his love with such fierce passion, looking at her and doing nothing - nothing to help.
"Robin will come to terms with him," he insisted.
At that, she closed her eyes, in weariness and with disgust.
"I tell you he will! Robin knows what he's doing and he'll make arrangements with your father. I'm absolutely certain."
Christine could believe that he meant it; he was telling her that Robin had gone to collect a ransom for her and that her father would pay. How much would it be? A big sum, that was certain. Ten thousand pounds? Oh, she was worth more than that even to her father. He would not miss such a sum, would think nothing of paying it for a picture or a casket. He would buy her freedom as he would buy a work of art, and afterward would make her feel that she owed him even more, would expect her to be his prisoner for as long as he desired.
And yet—
He had warned her against Lance.
But he had warned her against every friend not of his own choosing.
"And I—I won't let Robin harm you," Lance was saying, as if he believed his empty words. "I promise you, I—What was that?"
There had been a sound downstairs, and it was followed by another, unmistakably the front door of the shop. Lance sprang up and pulled the curtains so that she could see nothing, then bounded toward the staircase.
"Robin! Is that you?"
"Who do you think it is?" Robin demanded. He came running up the stairs, very light of foot. "It's working out perfectly. I've touched on Falconer's besetting weakness." He spoke with absolute conviction. "He's in love with art and its treasures! They're his idols and his mistresses rolled into one. If he had to choose between the Velazquez and his darling Christine, he'd ditch Christine. Oh, boy, are we on to a good thing!"
Christine felt as if a great weight had suddenly dropped on her, and her heart beat in dull, sickening throbs.
# 20: The Plot
Lancelot Judd watched as Robin began to dance and pirouette, light on his feet as a ballet dancer. Swinging to a halt, he gave a burst of excited laughter.
"Who wants to be a millionaire?" Robin hummed the tune, and began to pirouette again, then stopped immediately in front of Lance. "And you're on a bed of roses, Sir Lancelot, you'll be the millionaire's son-in-law! Which reminds me, how is our stricken fawn?"
"She—she's still sleeping. You—you won't do anything to hurt her, will you?"
"My dear Sir Lancelot, what makes you think I could be so ungallant? Would I harm your wife-to-be - especially as, married to you, she will be able to bring all the necessary psychological pressures on the great Sir Richard. Do as we say dear Daddy-in-law, or think what we'll do to your daughter." He laughed again. "Do you know, he didn't offer me a drink. But you will, won't you?"
Lance turned to a small oak chest on which stood bottles and glasses.
"What will you have?" he asked eagerly.
"Champagne, Sir Lancelot, what else will suit the mood of the night? Champagne! All right, I'll settle for a whisky-and-soda!" Now he moved about the room, rhythmic and self-assured despite his excitement. "The superb thing is that everything is working exactly according to plan. I'll even be able to send for Marie; she was so edgy after the sad death of Jenkins and Slater that I thought she ought to rest." He sat down in a large armchair and stretched his legs out in front of him, took the glass which Lance proffered, and said "Cheers!" and drank. "Now to business! Falconer has the Velazquez and it nearly gave him a fit, he was so excited. Tomorrow he will pay me for it, and then I'll show him some photographs of other treasures - treasures he'd sell his wife, his daughter, and his soul for. You will go and see him and if you can marry Christine, and very soon you'll be one of the family. From that time on, I will sell him treasures as he requires them. The only problem will be a place to keep them, but I expect Sir Richard will look after that. There must be very large cellars and vaults at Falconer House." He drank again and looked up at his friend, eyes aglow. "Can you see where it can possibly go wrong?"
"Not—er—not really," Lance said. "Except—er—after this will Christine want to marry me?"
"For her father's, mother's, and honour's sake, she will marry you," said Robin positively.
No, no, no! Christine cried within herself. Not now, not after this. Never.
But she caught her breath as she heard Robin go on.
"She is in love with you, Sir Lancelot. Although for some strange, pathological reason, I am proof against usual human emotions, needing only sex shared with enthusiasm, I can still see the unmistakable evidence of these emotions in others. And Christine is the type of woman who would stay faithful to a man no matter how he beat or ill-treated her, no matter how many other women he flaunted before her distressed gaze. She's a natural masochist. Look how she suffers the dictates of dear Daddy, pretending to resent them but always going back for more. Oh, she'll marry you, (a) because she's in love with you, (b) because it will keep her father out of trouble, and (c) because if she refused and really meant it then I'd cut her pretty throat!"
"Don't say that!" cried Lancelot.
"Think what incentive you have to be a truly great lover," Robin said. "Go to her now. Wake her with your prowess and woo her with your love. You are in love with her, aren't you?"
"God knows I am," muttered Lancelot Judd. "God knows I am."
As she heard that, Christine's despair lifted a little, and although she told herself that it was madness, that they could never marry, she could never be happy in such a marriage, she was aware in her whole body of the gentle touch of his hands.
"It will be all right," he whispered. "I always said it would be all right."
She felt sure that he actually believed it. In his way, he must be the most naive man in the world.
For a long time, while all these things were going on, Richard Falconer lay by his wife's side. He had known, as she had known, a few moments of diffidence not far removed from shyness, but these had soon passed. Their shared warmth was comfort, and for a while he lay wide awake, his muscles tense and aching slightly, his head taut with physical as well as emotional strain, his heart leaden. After a while, he turned on his side and put his arm round her, feeling her body respond to his.
"Are you awake?" he whispered, knowing well that she was.
"Yes," she said. "Wide awake."
"I don't know what to do," he said. "I really don't know what to do. I am afraid. I really am afraid that Christine is in grave danger."
"What can we do?" asked Charlotte.
She sensed that she must lie there, unmoving yet showing her awareness of him, that she must not show signs of agitation or distress, because it might drive him away again, to the distant places where he had been over a long and weary time. So, she stilled her own deep fears and asked him, simply, what they could do.
"That young man who came here," he said, "is a friend of Lancelot Judd.... And he is—evil, Charlotte. Evil is the only word to describe him. A thief, too, and no fool." He stopped but she waited without prompting, and he went on: "He left the stolen Velazquez with me."
This time, she started, her body stiffening, and half turned her head.
"'The Prince'?"
"Yes," Falconer answered. "The picture itself, not a copy - I have no doubt at all. I've put it in my safe."
"But—" she began, twisting her head round still further. Then, acutely aware of his need, she forced herself to appear relaxed. "You can't keep it, surely?" she asked gently.
"No. No, I can't possibly keep it. Although the temptation is—" He paused, and she felt his fingers pressing into her, powerful but free of passion. "The temptation is awful, Charlotte. I—I love these treasures. I love them."
"I know," she said, and, in a whisper that he could hardly hear, went on: "Each like a beautiful woman. It is a long time since I stopped being jealous of them."
"Jealous?" he echoed wonderingly, and after a pause he asked, "He offered me that and many more, and threatened, if I refused to buy, both to destroy them and to kill Christine. Do you see how fiendishly clever that is? To offer me what I crave for a sum of money which I can well afford, and Christine's safety to make my connivance seem justifiable. Charlotte, Charlotte," he went on, his voice almost breaking, "how can I cooperate with this man? How can I leave him free to steal more of the world's treasures, threatening, bullying, killing those who stand in his way? Yet if I don't cooperate, what will happen to the treasures he already has? And what will happen to Christine?"
As Charlotte turned to face him, her body astir with fear, he held her tightly as if they were as much part of each other as on the night when their daughter had been conceived, and he said in a voice more broken still: "It's a terrible dilemma, Charlotte. What are we to do?"
# 21: Decision
Yes, Charlotte thought bitterly, it is a terrible dilemma. She thought of Christine, and full awareness of the danger pierced through her for the first time; the true horror of it. That it should have taken such a situation to melt the ice which had for so long kept them apart was anguish in itself. But she must not show her feelings too much; if she gave way to them, she might undo what good had already come out of the situation. She had to fight for him, and for themselves, and for Christine.
"How did you leave the situation with this man?" she asked.
"He's to be in touch with me tomorrow"
"Are you sure Christine is safe until then?"
"He'll know that if she isn't kept unharmed, there can be no business between us."
"So that's her insurance."
"It's the only insurance we can have," Falconer told her. "I am quite sure that she is in no danger tonight. If I could rest for a few hours, I might see more clearly in the morning. Even if I rest only for an hour or two."
While he lay sleeping, Charlotte's mind was in a turmoil; one moment she was sure that they must tell the police at once, the next equally sure that at all costs they must protect Christine. After a while, she could not lie there any longer, and she eased herself away from Richard and then out of bed.
He did not stir.
She went into her dressing room and made some tea and drank it while fighting the battle out with herself. For the first time since they had married, she was in a position to influence him on what course to take. Yet she had never wanted so much to leave the decision to him.
Gideon was still at his desk at half past twelve that Sunday night, and still undecided, but he was veering more and more toward postponing any decision until the morning. There was no certainty that Christine Falconer was in danger, and with the house and the shop closely watched, nothing could be brought away. A raid to search Falconer House now would be difficult to stage, difficult perhaps to justify. Yet if he asked Falconer to let the police search, then obviously the man would have a chance to hide anything he had illegally. Gideon did not want and did not mean to be swayed by Falconer's wealth and position, but if Thwaites was wrong and the search was abortive, the newspapers would make a tremendous fuss and the prestige of the Yard would be severely damaged. If ever there was a time to be sure before acting, this was it.
He drummed on his desk, going over every aspect of the case, measuring Thwaites's suspicions against the probabilities and tonight's evidence. He yawned suddenly. Probably the best thing would be to sleep on it. He could put his head down on a bed in one of the first aid cubicles, used for such a purpose. He had telephoned Kate when he arrived and told her he would probably not be home. She had been yawny and tired and very happy.
"Such a lovely evening, George."
And here he was, yawning and postponing a decision.
On that instant, he closed his mouth like a trap, pulled a telephone forward, and put in a call to Hampstead, the KL Division Headquarters.
"Any developments at Judd's shop?" he demanded.
"No, sir. I had a report in only three or four minutes ago. Everything's in darkness there."
"Thanks," grunted Gideon, and put down the receiver only to lift it again and call the AB Division Headquarters, about Falconer House.
"No excitement," answered the Superintendent in charge. "I went out there myself and came back only twenty minutes ago. Everything's quiet. The last person seen to go in was Oliphant, Falconer's personal assistant, and he carried nothing in the way of a case or a parcel."
"The house is closely watched?"
"Tight as a drum, sir. Like Judd's shop."
"Good. I'm going over to Falconer's place myself, to talk to him," Gideon told him, hardly aware that he had come to such a positive decision. "I'll have a couple of men with me, but I want your chaps alerted."
"They will be! Like me along with you, sir?"
"Do you want to come?"
"Not particularly," answered the Superintendent. "There's been a bank robbery in Park Lane, and I ought to go there."
"Then do that," said Gideon. He put down the receiver and almost immediately picked it up again, dialled a third time, and said: "I want a car, driver, and one other man waiting for me in five minutes. Not a minute more. Do you know if Mr. Thwaites is still in the building?"
"He's having a kip - sleeping upstairs, sir."
Gideon grunted and rang off, took an old raincoat off a peg, and slipped it on as he went upstairs to the first aid room. Two men as well as Thwaites were there, heavily asleep, one of them snoring loudly. Thwaites himself looked very tired and old; he was sleeping in an undershirt, and one big, rather flabby arm was over a blanket. It seemed a pity to wake him, but if this was the kill, Thwaites had to be in on it. Gideon shook him slightly by the shoulder, and Thwaites was alert instantly, his eyes flickering.
"Be downstairs in five minutes," Gideon said. "We're going to Falconer House."
Before Gideon was out of the room, Thwaites was pushing the blanket back. Gideon went down to Information, told the night superintendent what he was going to do, and added: "If there's any word at all from West End or Hampstead, make certain I know at once."
"Be sure we will, sir."
Gideon went down the steep steps toward the courtyard, where his car was already waiting, the driver and another man standing by it. One opened the back door and Gideon got in, grunting with the effort. He was getting too big around the middle for bending double. He heard someone hurrying down the steps and guessed it was Thwaites.
"Mr. Thwaites will join me," he said. "Let him in at the other door."
Soon, Thwaites was crammed against him, breathing rather hard, looking very pale.
"Driver, I'm going to Falconer House, near Park Lane," Gideon said. "You two are to wait by the side of the car. The house is already being watched. Don't take any part in anything unless you get instructions from Mr. Thwaites or me. Understood?"
"All clear, sir."
Gideon turned to Thwaites, and went on speaking as if there had been no change of audience: "Nothing new has turned up but I'm not happy at waiting until the morning, and I know you're not. We'll go in together, but I'll see Falconer on my own. You join us only if I call you."
"I understand," Thwaites said, and then he added after a long pause: "I'm very glad you're having a go, sir."
"I hope we stay glad," Gideon said, and sat back as they drove through the still steadily falling rain. It was past two o'clock when the car pulled up outside Sir Richard Falconer's house, where a single light shone in the porch but none at any of the windows.
Falconer was aware of deep sleep, heavy sleep to which he was not accustomed, and then of his wife's voice and the pressure of her hand on his shoulder. He opened his eyes to a dim, not dazzling light, but even that was too bright for him. He saw his wife through his lashes, and remembered what had happened and where he was. Then, seeing the anxiety on her face, he suddenly thought: Christine! and struggled to a sitting position.
"What is it? What—"
"It's all right; it's not about Christine," Charlotte said. "There are two policemen downstairs, one of them is Commander Gideon. He insists on seeing you, Richard. I couldn't put him off until the morning."
Falconer hitched himself further back in the bed.
"What time is it?"
"Just after two."
"It could be news of Christine," he said tautly. "Gideon is the man whom Scott-Marie would assign to the inquiry." He pushed the bedclothes back, and she helped him into his dressing gown. "Who let them in?"
"Oily did."
"So he's back?"
"He hadn't gone to sleep, so he told Davies not to get out of bed," Charlotte said.
"Have him make some coffee, will you? Strong, with plenty of sugar."
"I'll see to it," she promised. "He's waiting outside."
Falconer tied the dressing-gown cord tightly about his waist as he went out of the room. Oliphant, a smoking jacket over pyjamas trousers, moved toward him from the head of the stairs.
"Richard, there is something I must tell you," he said.
"Not now, Oily, later—"
"Now. As the police are here, it is vital."
Falconer stood very still and made the other come toward him. Oliphant was obviously agitated, for once not knowing what to say. Downstairs in the hall, Gideon and Thwaites could just be seen, too far away to overhear, not too far to know that something was being said.
"What difference do the police make?" Falconer demanded impatiently.
"Richard, I—I should have told you long ago, but I—I didn't know it might matter. Two of the Monets and one of the Cellini caskets in the long gallery were" - Oliphant caught his breath - "they were stolen."
"Stolen?" Falconer echoed unbelievingly. "And you knew it?"
"Yes. I—I was offered them at very good prices, and—well, I fell for the temptation of making a profit." Oliphant sounded terribly distressed.
"And you not only bought them on my behalf knowing them to be stolen, but made a fat killing," Falconer said.
"I—I could explain it if—You see, I had personal problems, and the dealers—"
"We can go into this later," Falconer said, in a forbidding voice. "Now you fear that the police may have come here for these stolen pieces?"
"They—they have a search warrant," Oliphant muttered.
"I see. If they identify these works, I shall disclaim all knowledge of their being stolen. And in my case I shall require a full history of each purchase, the dealer involved, the money paid, and the commission you received." Falconer nodded dismissal.
He went downstairs, and as he did so the two men in the hallway turned: Gideon, massive and aggressive in the slightest movement, a man for whom Falconer had an instant respect, and Thwaites, for whom he felt nothing at all. He greeted them with a word and a gesture and led Gideon into the room where he had seen Robin Kell. If he felt any surprise that the other man did not follow them, he showed none. Brandy, glasses, and decanters were on a small table.
"Will you—" he began.
"No thank you, sir," said Gideon. "I am Commander Gideon of the Criminal Investigation Department of the Metropolitan Police and I would like some information from you, please."
"Have you traced my daughter?" demanded Falconer sharply.
"We think we know where she is, sir. We have no reason to believe that she went there against her will or is missing in the official sense of the word." Before Falconer could interrupt, before it was possible to judge the extent of his relief, Gideon went on in a flat, formal voice, "We have reason to believe that you have in your possession a painting, known as 'The Prince,' painted by the Spanish artist Velazquez, knowing it to be stolen. Is that true, sir?"
There was a long pause, during which Gideon sensed the truth yet realized there was something here which he did not understand. Then the silence was broken by the clink of cups and the rattle of spoons on a silver tray. Lady Falconer, not Oliphant, brought in coffee, and hesitated in the doorway. Falconer seemed oblivious, but Gideon was quick to notice the woman and said: "Your husband may prefer to be alone, ma'am."
"No," said Falconer quietly. "No. My wife knows about the situation. I told her earlier tonight, Commander. You are quite right. I have the painting. It was offered to me for a hundred thousand pounds. I was told that if I did not buy it, my daughter would be. killed. Did you give me to understand that you know where my daughter is?"
Charlotte moved to a small table, and very carefully placed the coffee tray in position. She turned to Gideon, her eyes pleading.
Very heavily but without any inflection in his voice, Gideon asked: "Was that before or after you telephoned Sir Reginald Scott-Marie, sir?"
"After," answered Falconer. "Some time afterward. I don't think I would have had the courage to talk to him had I know the danger that my daughter was in. He—"
"Who do you mean by 'he'?" interrupted Gideon.
"The man who offered me the painting, a young man named Robin Kell, or who called himself Robin Kell," answered Falconer. "He warned me that if I consulted you, not only would he murder my daughter but he would destroy many Old Masters, paintings which he has stolen over a period of several years. And for what it is worth, I believe him capable of committing both crimes, Commander."
There was a long, tense silence before Thwaites said from the hall: "My God! That would be awful!"
# 22: The Murderer
Gideon was aware of the conflicting tensions of everyone present. Of Thwaites, who had come forward to watch Lady Falconer, just inside the room, his concern for the treasures that were in danger. Of Lady Falconer, so composed when she had come in, so shattered now. Of Falconer himself, standing there with a dignity he had shown from the moment he had come downstairs, very different from the man who had tried so hard to use his position to influence Scott-Marie.
"Do you know where the stolen goods are?" Gideon asked, as flatly as ever.
"I only know that Kell said he could produce them within the hour."
"He'll have them at Hampstead," interpolated Thwaites.
"That's quite possible," Gideon agreed. "But we need to be sure." He moved to ease the atmosphere, giving Lady Falconer a brief smile. "If a cup of coffee wouldn't be inconvenient, ma'am.... Come in, Thwaites.... Sir Richard, Chief Inspector Thwaites has been working on the Velazquez robbery and it was he who traced Robin Kell to a small antique shop in Hampstead.... Ah, thank you, ma'am.... No biscuits, thank you." He sipped coffee and then went on: "We have to be sure before we know exactly how best to search the shop.... Would you mind letting us have the painting, sir?" He actually smiled at Falconer. "I would like Thwaites's opinion on its authenticity."
"If Sir Richard identifies—" Thwaites began, and immediately subsided.
"I'll get it," said Falconer.
"Go with Sir Richard, Chief Inspector," Gideon ordered, without hinting that he was simply making sure that Falconer could make no attempt to escape. He drank more coffee as the two went out, and he was left alone with Lady Falconer. He had seen her occasionally at social and official functions, but had never realized how very lovely she was. At close quarters, anxiety stricken, she was superb. "I know how you must be feeling, my lady," he said. "At least it won't be long now."
She asked huskily, "Do you think this man Kell's threats are serious, Commander?"
What shall I tell her? Gideon wondered. He had no doubt at all that Kell was involved with the coldblooded murders of the three men who had helped him; nothing suggested that he had a soft spot in him, and once he knew that he was cornered, he might well kill and destroy out of sheer spite.
Before he could speak, Lady Falconer said, "I can see that you do believe the threats are serious."
"Yes," admitted Gideon. "I'm afraid they may be very serious indeed."
"Is there anything—anything I can possibly do?"
Gideon deliberated, then put down his coffee cup, an excuse to get a little nearer, and studied her intently.
"There is nothing at all except help your husband," he told her. "If Sir Richard will go to this Hampstead shop in the morning and demand to see the other stolen items before he makes a decision about what to do, then I think we have a chance."
"Is it so important that you know whether the art treasures are there?" Charlotte asked, a glint of scorn in her eyes.
"Very important indeed," answered Gideon. "If Kell has the stolen goods, he can use them to bargain with; if he hasn't, then I hate to say that the only bargaining will be over your daughter." He heard the others returning and glanced around, seeing the picture in Thwaites's hands and the radiance in the North Countryman's eyes. "I was just explaining to Lady Falconer, sir, that if Kell has the other stolen goods you could agree to talking terms for buying them on condition that he releases your daughter. That way, I think you will have a very good chance. Once we get your daughter away from the shop, we can concentrate on getting our man. If he carried out his threat to destroy the art treasures—"
Gideon broke off, shrugging slightly, and watching both Falconer, the collector, and Thwaites, the man who could only worship things of such beauty from afar. They looked exactly the same: appalled and yet full of hope.
"I understand the situation," Falconer said, at last. "I am not sure that I understand the risk."
"The risk is that if Kell comes to suspect that you are working with us, he could kill both your daughter and you before we could save you," Gideon said. "The risk is as simple as that."
It seemed an age before Falconer said, "I will take it."
"Now," Gideon said as they left the house, "we want everything laid on before daylight, so that Kell can't suspect the shop is being watched. We need men on the roof across the street, men on the roof next door, who can swing into the windows of the flat. We want a fire engine with a turntable available nearby - no reason why it shouldn't have been called out earlier for an imaginary outbreak two or three shops along the street. The moment Falconer has gone into that shop, we want to be absolutely ready." He hardly paused before going on: "We'll need tear gas and our men must be masked. Everything will be a matter of split-second timing. Lay it all on, and let me have a report in detail first thing in the morning."
"Sir Richard," said Robin Kell, on the telephone at ten o'clock next morning.
"Yes," Falconer said.
"I hope you've made up your mind."
"I have given it a lot of thought," Falconer said. "And I will accept on one condition."
"You aren't in any position to make conditions."
"Nevertheless, I am making one," Falconer retorted coldly. "I want to know what the other goods are and I want to see them before I pay any money or give any undertakings. And I want a very quick decision, or I shall do what I should no doubt have done in the first place - go to the police."
"You bloody fool, if you do that I'll cut her throat!"
"What other consideration do you think would make me even contemplate doing business with you?" demanded Falconer, and rang off.
Gideon, sitting in his office, almost gritting his teeth because he so wanted to be at Hampstead, snatched up his telephone when it rang just after ten o'clock.
"Kell has telephoned me," Falconer said. "He didn't like my terms but I think he will agree."
"Sir Richard?" asked Robin Kell.
"Yes," said Falconer.
"Do you know the pond at Hampstead Heath?"
"Very well."
"Be there at one o'clock, and bring ten separate packages of ten thousand pounds each. You often pay big sums in cash, your daughter tells me. Get it somehow. Drive yourself, with the money in the boot. If you are followed or have anybody with you, you know what will happen."
"I will be there, alone and not followed," Falconer said frigidly.
Gideon looked across at Hobbs when the telephone rang again, half an hour later, and this time it was Lady Falconer.
"My husband is to meet Robin Kell at the pond on Hampstead Heath," she reported. "He has to have a hundred thousand pounds, in notes, in the boot of his car."
"So you made it," Kell said, his voice almost a sneer.
"I always carry out my obligations," said Falconer, looking beyond the youth to the pond.
"Have you got the money?" Kell demanded.
"Yes."
"How did you get it?"
"I keep a substantial sum in my safe, and as you said, I often pay in cash for purchases. I drew enough from three different banks."
"We're going to Lancelot Judd's shop," said Kell, obviously satisfied. "I've a taxi waiting. And I've friends following and watching. One false move, and that's the end."
"Do you think I value my life so lightly?" demanded Falconer.
"I will say one thing, you've got your priorities right! Your life first, your daughter's second." There was a tone of grudging admiration in Kell's voice. "Get in with me. Somebody will follow in your car."
A taxi drew up alongside, and Falconer had time only to see the youthful face of the driver before getting in. Kell slammed the door and sat back in a corner. After a few moments, he took an envelope from beneath his coat and handed it to Falconer, saying, "See if you recognize those." Falconer drew out some glossy photographs, held them toward the window, and looked down at the top one.
As he looked at them one after another, recognizing them as recently stolen Old Masters, the taxi drove to the shop on the High Street and stopped. Robin put a restraining hand on Falconer's arm, and after a pause an attractive young woman appeared from the shop, smiling serenely.
"All clear," she said.
"In we go," said Kell, "and don't forget what's at stake, Sir Richard!" He half dragged Falconer from the taxi toward the open door, and as they went inside, he muttered: "Upstairs." Another youth was at the foot of the stairs, and at the top was Lancelot Judd, pale faced, gripping the brass rail. Falconer forced himself to walk up the stairs calmly; Kell pushed hard behind him. The shop doorway banged and a key turned in the lock.
On one side was a canvas curtain hanging on a brass pole, and once they were all upstairs, Kell went across and pulled the curtain back very slowly. In front of Falconer's still-unbelieving eyes were eleven canvases, most of them in temporary frames, hanging on the wall of a recess.
Even though Falconer had been prepared for what he was going to see, he still experienced a sense of almost physical shock. There was Vermeer's "Ice Town"; and next to that, Titian's "Head of a Boy." A little further along he recognized Gainsborough's "Lady Lost." Further on again, he saw Botticelli's "Cartoon of the Dying." Each of these had for an age been imprinted on Falconer's mind. Now they, and so many others, seemed to burn into him. He had no doubt at all that they were genuine, and felt as he always did the hypnotic pull of each one, felt his blood turning to water, his knees weakening. He stood very still.
"They're the McCoy all right," said Kell. "Now-the money's in the boot of your car, isn't it? Hand over the key."
"When I have seen my daughter," demurred Falconer.
"Now. The only condition was that you saw the other stuff, and you've seen it. Hand it over, or we'll take—"
"You cannot get that boot open without a special key, unless you want an alarm to go off. And I will not give you this key until I have seen Christine, and I am safely back in my own home. This arrangement is not as one-sided as you think, Mr. Kell."
Kell's hand flexed and unflexed, one hovered near his trousers pocket, but suddenly he spun round, strode to another curtain, and pulled it aside.
There was Christine: asleep or drugged. She sat erect in a small chair and she was bound to it. In some way, her head had been tilted backward, emphasizing the flawless line of neck and shoulders, as lovely as her mother's. Nearby, Lance Judd stood rigidly, staring not at the girl but at her father.
"Don't—don't argue anymore," he pleaded. "Don't make it any worse. Give Robin—"
And then there came across his words a wild scream from the girl below.
"The police are coming!" she cried. "The police!"
There was a second of stunned silence before Robin Kell's hand flashed to and from his jacket and the click of his knife sounded loud. Falconer tried to rush forward to protect his daughter, but the knife moved swift as light toward him, and into his belly. He felt a searing pain and staggered to one side, saw the knife flash again as Kell turned to use it on Christine.
But as he did so, the knife moving toward her defenceless throat, Lance Judd hurled himself in front of her, and the knife went into his chest. He gasped, he choked, the door crashed, and the window of the kitchen crashed; then ladders appeared at the upstairs window and police wearing steel helmets smashed the glass. It was Thwaites, running up the stairs, who saw the pictures, saw the trickle of flame run along the way where they were displayed, saw the girl with a taper in her hand stabbing it toward rags that he sensed were soaked in oil. He struck the girl aside and kicked at the blazing rags, and when one fell close to the wall he picked it up with his bare fingers and flung it away.
By then, the place was full of smoke and full of police, and Robin Kell was struggling with two policemen who were trying to prise the knife out of his hand.
Only ten minutes later, a telephone rang on Gideon's desk. Gideon snatched the receiver up eagerly, and as he announced himself, a man said crisply: "It's all over, sir. The girl's all right."
"Falconer?" demanded Gideon, heart thumping with relief.
"A nasty wound in the stomach, sir, and on his way to hospital. So is the man Judd, who tried to save her; he got a knife in his chest. Kell, a girl named Marie Devaux, and others are on their way to the Yard now, sir, for interrogation."
"The pictures?" demanded Gideon sharply.
"Virtually undamaged, sir. I—oh, here's Chief Inspector Thwaites, if you'd like a word with him."
"I would indeed," said Gideon warmly, and a moment later heard Thwaites speak with a feeling that made him seem almost a different man.
"No damage at all really, sir. A frame was scorched, that's all. The young devil meant it all right, but everything we planned went off like clockwork."
Gideon found himself laughing.
"Falconer mightn't feel that," he said.
"If you ask me, sir, he was bloody lucky to get off so lightly, everything considered. I—sorry, sir! I wonder if you will have a word with his wife."
"I'll call her at once," promised Gideon.
He reassured Lady Falconer about her daughter and as much as he could about her husband; then he rang off, sat back and reflected, and sent for Hobbs, who came in with his usual promptitude, and with a bundle of reports. There was so much going through, and already the sense of importance about the Velazquez theft was fading, for it was now actually part of the past. There would be the trial to prepare but that wasn't his job, Gideon reflected. Thank God here was a case where they could be absolutely certain that they had the right man!
The Entwhistle file was one of those Hobbs had brought in, and Gideon said: "I thought Honiwell wanted this put aside for a while."
"I brought it in because I've had a word with the Governor at Dartmoor," Hobbs said. "Apparently, Entwhistle is in a bad way. He tried to kill himself in his cell last night. Is there anything we can possibly do to help him?"
"We can't breathe a word until we know for certain," Gideon said gloomily. "I only wish we could. Unless we've evidence enough to bring Eric Greenwood back from a buying trip to India and Pakistan. He's just started from London Airport." When Hobbs shook his head, Gideon put a hand on the file and said: "Leave it with me, I might think of something. Anything else that's urgent?"
"I don't think so," Hobbs answered. "Not to say urgent. There's more than enough to clear up and more than enough pending. I—"
He broke off, for the door opened abruptly, a rare thing in Gideon's office, and Chamberlain came in, with rather less than his usual bounce. He was taken aback at seeing them both together, hesitated, let the door close behind him, and advanced.
"Oh, Commander, in view of the recovery of the Velazquez and the other stolen items, I think we might be well-advised to cancel tomorrow's conference, don't you?"
Gideon, solid as a Buddha in his chair, said: "I think that's a very good idea, sir."
"Good. I thought you would agree. Will you telephone all concerned?"
"If you don't mind my saying so, sir," said Gideon, "I think it would be very much appreciated if you telephoned yourself. I really do."
"Yes, very well," said Chamberlain after a pause. "There are a number of other issues I would like to discuss with you, but perhaps tomorrow would be a better day."
"Whenever you wish," said Gideon.
Neither he nor Hobbs made any comment about the intrusion when Chamberlain left, and Hobbs took out a slim file.
"Lemaitre and Singleton have fixed that raid on the counterfeiters for tonight," he announced. "I've asked them both to send reports to us, so we'll get both sides of the story."
There wasn't much that Hobbs missed, Gideon reflected appreciatively.
When he had gone, Gideon sat back in his chair and pondered. Chamberlain, of course, could not last very long; it was obvious that he was a misfit, and if he didn't realize it himself, Scott-Marie would find a way to make him. Thwaites was a different kind of problem. He was old for promotion but he had done such a dedicated job that it ought to be recognized, and a Superintendent's pension would be considerably higher than a Chief Inspector's. He would put through a recommendation today. He made a note and saw other notes he had made this morning about the conference on the immigration problems. That would need very careful handling.
He saw Thwaites in the middle of the afternoon, his hand heavily bandaged, his hair singed, but an expression of deep contentment in his eyes.
"Congratulations," Gideon said. "Sit down."
"It was all your idea, sir," Thwaites said warmly. "While I was having my hand dressed, I was thinking it must be hell for you - with respect sir - having to sit here knowing you could be out on the job and doing it twice as well as most." He gave Gideon time for only a deprecatory wave and plunged on. "Latest reports, sir: Sir Richard not in danger, I'm glad to say. Knife wound identical with those in Slater and de Courvier, and caused by the same knife, so we've got Kell for murder... Lancelot Judd touch and go, sir, but if he pulls through he'll turn queen's evidence without a doubt... Christine's suffering from shock, nothing a few days' rest won't put right... Not a single item damaged, sir; we picked up a million pounds' worth of stolen works of art. I hope the newspapers make a fuss about that little lot!... Oh, and by the way, Falconer's man Oliphant seems to me in a highly nervous condition. I think I'm going to see what he's been up to. A runner I know, Red Thomas, told Division that Oliphant spends a lot of time with Mrs. Bessell in Bond Street. That's about the lot, sir. Oh! There is one other offbeat little thing."
"What's that?" asked Gideon.
"Remember Lucy Jenkins, Leslie J.'s daughter? Works for Old Fisky in a shop in the King's road?"
"I pass it every day," Gideon said.
"Well, apparently Lucy bought a load of junk when the old man was away and one of the pictures was a find. Fisky took it round to Division. A John Bettes, stolen from Rosebury House in Suffolk about ten years ago, worth forty or fifty thousand. The insurance company's going to pay out ten per cent reward and Old Fisky's sharing it with Lucy."
"I couldn't be more glad," Gideon said. "I really couldn't."
As he drove home that night, he slowed down alongside Old Fisky's shop, where there were lights, more cars, television cameramen, and Lucy holding up a picture while Old Fisky peered from inside the shop.
She's not a bad-looking girl, Gideon reflected. This could be the making of her.
He noticed a policeman in uniform on the other side of the road, so intent on the scene in the doorway and on Lucy Jenkins that he did not see that the Commander was drawing away. None of the newspaper, television, or press men noticed Gideon, either, but when he reached Harrington Street, half a dozen neighbours waved newspapers at him and Penelope came running out of the house, very excited.
"Daddy, Mummy's told me you're going to soundproof the attic. That's just wonderful. I'd been hoping for something like that. And there's just room to get the piano in. Alec stopped by to see you and heard about it, and he's checked it all for me." She flung her arms round him and kissed him.
And Kate, standing in the doorway to welcome him, could tell that for her husband it had been a deeply satisfying day.
# Series Information
Published or to be published by
House of Stratus
Dates given are those of first publication
## Gideon Series
(Writing as JJ Marric)
These Titles can be read as a series, or randomly as standalone novels
_Title_ | _Also Published as:_ |
---|---|---
| |
1 Gideon's Day | Gideon of Scotland Yard | 1955
2 Seven Days to Death | Gideon's Week | 1956
3 Gideon's Night | | 1957
4 Gideon's Month | | 1958
5 Gideon's Staff | | 1959
6 Gideon's Risk | | 1960
7 Gideon's Fire | | 1961
8 A Conference for Assassins | Gideon's March | 1962
9 Travelling Crimes | Gideon's Ride | 1963
10 An Uncivilised Election | Gideon's Vote | 1964
11 Criminal Imports | Gideon's Lot | 1965
12 Gideon's Badge | | 1966
13 From Murder to a Cathedral | Gideon's Wrath | 1967
14 Gideon's River | | 1968
15 Gideon's Power | | 1969
16 Gideon's Sport | | 1970
17 Gideon's Art | | 1971
18 Gideon's Men | | 1972
19 Gideon's Press | | 1973
20 Gideon's Fog | | 1975
21 Gideon's Drive | | 1976
22 Vigilantes & Biscuits | Gideon's Force | 1978
# Other Series by John Creasey
Published or to be published by
House of Stratus
Dates given are those of first publication
'Department 'Z'' (28 titles)
'Dr. Palfrey Novels' (34 titles)
'Inspector West' (43 titles)
'Sexton Blake' (5 titles)
'The Baron' (47 titles) (writing as Anthony Morton)
'The Toff' (59 titles)
along with:
| The Masters of Bow Street
This epic novel embraces the story of the Bow Street Runners and the Marine Police, forerunners of the modern police force, who were founded by novelist Henry Fielding in 1748. They were the earliest detective force operating from the courts to enforce the decisions of magistrates. John Creasey's account also gives a fascinating insight into family life of the time and the struggle between crime and justice, and ends with the establishment of the Metropolitan Police after the passing of Peel's Act in 1829.
---|---
## 'The Toff' Series
These Titles can be read as a series, or randomly as standalone novels
_Title_ | _Also Published as:_ |
---|---|---
| |
1 Introducing the Toff | It's the Toff ! | 1938
2 The Toff Goes On | | 1939
3 The Toff Steps Out | | 1939
4 Here Comes the Toff | | 1940
5 The Toff Breaks In | | 1940
6 Salute the Toff | | 1941
7 The Toff Proceeds | | 1941
8 The Toff Goes to Market | | 1942
9 The Toff Is Back | | 1942
10 The Toff on the Trail (short stories) | | 1942
11 The Toff among the Millions | | 1943
12 Accuse the Toff | | 1943
13 The Toff and the Deadly Parson | The Toff and the Curate | 1944
14 The Toff and the Great Illusion | | 1944
15 Feathers for the Toff | | 1945
16 The Toff and the Lady | | 1946
17 Poison for the Toff | The Toff on Ice | 1946
18 Hammer the Toff | | 1947
19 The Toff in Town | | 1948
20 The Toff Takes Shares | | 1948
21 The Toff and Old Harry | | 1949
22 The Toff on Board | | 1949
23 Fool the Toff | | 1950
24 Kill the Toff | | 1950
25 A Knife for the Toff | | 1951
26 A Mask for the Toff | The Toff Goes Gay | 1951
27 Hunt the Toff | | 1952
28 Call the Toff | | 1953
29 The Toff Down Under | Break the Toff | 1953
30 Murder Out of the Past (short stories) | | 1953
31 The Toff at Camp | The Toff at Butlins | 1954
32 The Toff at the Fair | | 1954
33 A Six for the Toff | A Score for the Toff | 1955
34 The Toff and the Deep Blue Sea | | 1955
35 Kiss the Toff | Make-Up for the Toff | 1956
36 The Toff in New York | | 1956
37 Model for the Toff | | 1957
38 The Toff on Fire | | 1957
39 The Toff and the Stolen Tresses | | 1958
40 Terror for the Toff | The Toff on the Farm | 1958
41 Double for the Toff | | 1959
42 The Toff and the Runaway Bride | | 1959
43 A Rocket for the Toff | | 1960
44 The Toff and the Kidnapped | The Kidnapped Child | 1960
45 Follow the Toff | | 1961
46 The Toff and the Toughs | The Toff and the Teds | 1961
47 A Doll for the Toff | | 1963
48 Leave It to the Toff | | 1963
49 The Toff and the Spider | | 1965
50 The Toff in Wax | | 1966
51 A Bundle for the Toff | | 1967
52 Stars for the Toff | | 1968
53 The Toff and the Golden Boy | | 1969
54 The Toff and the Fallen Angels | | 1970
55 Vote for the Toff | | 1971
56 The Toff and the Trip-Trip-Triplets | | 1972
57 The Toff and the Terrified Taxman | | 1973
58 The Toff and the Sleepy Cowboy | | 1975
59 The Toff and the Crooked Copper | | 1977
## Inspector West Series
These Titles can be read as a series, or randomly as standalone novels
_Title_ | _Also Published as:_ |
---|---|---
| |
1 Inspector West Takes Charge | | 1942
2 Go Away to Murder | Inspector West Leaves Town | 1943
3 The Apostle of Gloom | Inspector West At Home | 1944
4 Inspector West Regrets | | 1945
5 Holiday for Inspector West | | 1946
6 Battle for Inspector West | | 1948
7 The Case Against Paul Raeburn | Triumph for Inspector West | 1948
8 Inspector West Kicks Off | Sport for Inspector West | 1949
9 Inspector West Alone | | 1950
10 Inspector West Cries Wolf | The Creepers | 1950
11 The Figure in the Dusk | A Case for Inspector West | 1951
12 The Dissemblers | Puzzle for Inspector West | 1951
13 The Case of the Acid Throwers | The Blind Spot; Inspector West at Bay | 1952
14 Give a Man a Gun | A Gun for Inspector West | 1953
15 Send Inspector West | | 1953
16 So Young, So Cold, So Fair | A Beauty for Inspector West; The Beauty Queen Killer | 1954
17 Murder Makes Haste | Inspector West Makes Haste; The Gelignite Gang; Night of the Watchman | 1955
18 Murder: One, Two, Three | Two for Inspector West | 1955
19 Death of a Postman | Parcels for Inspector West | 1956
20 Death of an Assassin | A Prince for Inspector West | 1956
21 Hit and Run | Accident for Inspector West | 1957
22 The Trouble at Saxby's | Find Inspector West; Doorway to Death | 1957
23 Murder, London - New York | | 1958
24 Strike for Death | The Killing Strike | 1958
25 Death of a Racehorse | | 1959
26 The Case of the Innocent Victims | | 1959
27 Murder on the Line | | 1960
28 Death in Cold Print | | 1961
29 The Scene of the Crime | | 1961
30 Policeman's Dread | | 1962
31 Hang the Little Man | | 1963
32 Look Three Ways at Murder | | 1964
33 Murder, London - Australia | | 1965
34 Murder, London - South Africa | | 1966
35 The Executioners | | 1967
36 So Young to Burn | | 1968
37 Murder, London - Miami | | 1969
38 A Part for a Policeman | | 1970
39 Alibi for Inspector West | | 1971
40 A Splinter of Glass | | 1972
41 The Theft of Magna Carta | | 1973
42 The Extortioners | | 1974
43 A Sharp Rise in Crime | | 1978
## 'The Baron' Series
These Titles can be read as a series, or randomly as standalone novels
_Title_ | _Also Published as:_ |
---|---|---
| |
1 Meet the Baron | The Man in the Blue Mask | 1937
2 The Baron Returns | The Return of the Blue Mask | 1937
3 The Baron Again | Salute Blue Mask | 1938
4 The Baron at Bay | Blue Mask at Bay | 1938
5 Alias the Baron | Alias Blue Mask | 1939
6 The Baron at Large | Challenge Blue Mask! | 1939
7 Versus the Baron | Blue Mask Strikes Again | 1940
8 Call for the Baron | Blue Mask Victorious | 1940
9 The Baron Comes Back | | 1943
10 A Case for the Baron | | 1945
11 Reward for the Baron | | 1945
12 Career for the Baron | | 1946
13 The Baron and the Beggar | | 1947
14 Blame the Baron | | 1948
15 A Rope for the Baron | | 1948
16 Books for the Baron | | 1949
17 Cry for the Baron | | 1950
18 Trap the Baron | | 1950
19 Attack the Baron | | 1951
20 Shadow the Baron | | 1951
21 Warn the Baron | | 1952
22 The Baron Goes East | | 1953
23 The Baron in France | | 1953
24 Danger for the Baron | | 1953
25 The Baron Goes Fast | | 1954
26 Nest-Egg for the Baron | Deaf, Dumb and Blonde | 1954
27 Help from the Baron | | 1955
28 Hide the Baron | | 1956
29 The Double Frame | Frame the Baron | 1957
30 Blood Red | Red Eye for the Baron | 1958
31 If Anything Happens to Hester | Black for the Baron | 1959
32 Salute for the Baron | | 1960
33 The Baron Branches Out | A Branch for the Baron | 1961
34 The Baron and the Stolen Legacy | Bad for the Baron | 1962
35 A Sword for the Baron | The Baron and the Mogul Swords | 1963
36 The Baron on Board | | 1964
37 The Baron and the Chinese Puzzle | | 1964
38 Sport for the Baron | | 1966
39 Affair for the Baron | | 1967
40 The Baron and the Missing Old Masters | | 1968
41 The Baron and the Unfinished Portrait | | 1969
42 Last Laugh for the Baron | | 1970
43 The Baron Goes A-Buying | | 1971
44 The Baron and the Arrogant Artist | | 1972
45 Burgle the Baron | | 1973
46 The Baron - King Maker | | 1975
47 Love for the Baron | | 1979
## Doctor Palfrey Novels
These Titles can be read as a series, or randomly as standalone novels
_Title_ | _Also Published as:_ |
---|---|---
| |
1 Traitor's Doom | | 1942
2 The Legion of the Lost | | 1943
3 The Valley of Fear | The Perilous Country | 1943
4 Dangerous Quest | | 1944
5 Death in the Rising Sun | | 1945
6 The Hounds of Vengeance | | 1945
7 Shadow of Doom | | 1946
8 The House of the Bears | | 1946
9 Dark Harvest | | 1947
10 The Wings of Peace | | 1948
11 The Sons of Satan | | 1948
12 The Dawn of Darkness | | 1949
13 The League of Light | | 1949
14 The Man Who Shook the World | | 1950
15 The Prophet of Fire | | 1951
16 The Children of Hate | The Killers of Innocence; The Children of Despair | 1952
17 The Touch of Death | | 1954
18 The Mists of Fear | | 1955
19 The Flood | | 1956
20 The Plague of Silence | | 1958
21 Dry Spell | The Drought | 1959
22 The Terror | | 1962
23 The Depths | | 1963
24 The Sleep | | 1964
25 The Inferno | | 1965
26 The Famine | | 1967
27 The Blight | | 1968
28 The Oasis | | 1970
29 The Smog | | 1970
30 The Unbegotten | | 1971
31 The Insulators | | 1972
32 The Voiceless Ones | | 1973
33 The Thunder-Maker | | 1976
34 The Whirlwind | | 1979
## 'Department Z' Novels
These Titles can be read as a series, or randomly as standalone novels
_Title_ |
---|---
|
1 The Death Miser | 1932
2 Redhead | 1934
3 First Came a Murder | 1934
4 Death Round the Corner | 1935
5 The Mark of the Crescent | 1935
6 Thunder in Europe | 1936
7 The Terror Trap | 1936
8 Carriers of Death | 1937
9 Days of Danger | 1937
10 Death Stands By | 1938
11 Menace! | 1938
12 Murder Must Wait | 1939
13 Panic! | 1939
14 Death by Night | 1940
15 The Island of Peril | 1940
16 Sabotage | 1941
17 Go Away Death | 1941
18 The Day of Disaster | 1942
19 Prepare for Action | 1942
20 No Darker Crime | 1943
21 Dark Peril | 1944
22 The Peril Ahead | 1946
23 The League of Dark Men | 1947
24 The Department of Death | 1949
25 The Enemy Within | 1950
26 Dead or Alive | 1951
27 A Kind of Prisoner | 1954
28 The Black Spiders | 1957
# Select Synopses
| Gideon's Day
Gideon's day is a busy one. He balances family commitments with solving a series of seemingly unrelated crimes from which a plot nonetheless evolves and a mystery is solved. One of the most senior officers within Scotland Yard, George Gideon's crime solving abilities are in the finest traditions of London's world famous police headquarters. His analytical brain and sense of fairness is respected by colleagues and villains alike.
---|---
| Gideon's Night
On this particular night Commander George Gideon has to deal with a couple of psychopaths who trail pain and blood in their wake. One targets infants, and the other young women on London's foggy streets. There's also an explosive gang war in the offing, and one way or another all of these cases are coming to their breathtaking conclusions at the same time. Can Scotland Yard's finest cope with such a nightmarish scenario, with what would ordinarily be months of time consuming police work crammed into just one night?
---|---
| Gideon's Fire
Commander George Gideon of Scotland Yard has to deal successively with news of a mass murderer, a depraved maniac, and the deaths of a family in an arson attack on an old building south of the river. This leaves little time for the crisis developing at home . . .
---|---
| Meet the Baron
John Mannering (The Baron) makes his first appearance in this volume. Lord Fauntley cannot help showing off both his daughter and the security under which his precious jewels are kept. Mannering finds himself attracted to both .... Money is tight and so he plans a burglary, but this fails and unexpected consequnces result. The relationship with Lorna Fauntley flourishes, and a series of high profile thefts and adventures ensure Mannering's future, so he believes, until Lorna equates him with The Baron. One of the many further twists in this award winning novel occurs when the police appear to seek Mannering's help, only to have everything turned upside down as the plot develops . . .
---|---
| Shadow The Baron
John Mannering (' _The Baron_ ') is called in by Scorland Yard's Superintendent Bristow to help catch the mysterious jewel thief ' _The Shadow_ '. No one know the thief's identity, but he has managed to pull off many high profile robberies. However, as Mannering proceeds to track down the target, he finds the pursuer becomes the pursued . . . ..
---|---
| The House of the Bears
Standing alone in the bleak Yorkshire Moors is Sir Rufus Marne's 'House of the Bears'. Dr. Palfrey is asked to journey there to examine an invalid - whom he finds has disappeared. Moreover, Marne's daughter lies terribly injured after a fall from the minstrel's gallery, which Dr. Palfrey discovers was no accident. He sets out to look into both matters, but the discoveries he makes are truly fantastic. A deserted mine, powerful explosive and a submarine all feature in this powerful mystery. The results are even capable of surprising him ...
---|---
| Inspector West Takes Charge
Extortion is the name of the game, as one victim after another is ruthlessly targeted. Chief Inspector Roger West must now solve the problem, along with a tangle of murders - but the case becomes more frightening as every minute passes.
---|---
| The Case Against Paul Raeburn
Chief Inspector Roger West has been watching and waiting for over two years - he is determined to catch Paul Raeburn out. The millionaire racketeer may have made a mistake, following the killing of a small time crook. Can the ace detective triumph over the evil Raeburn in what are very difficult circumstances? This cannot be assumed as not eveything, it would seem, is as simple as it first appears .....
---|---
| Introducing The Toff
The Toff is the Hon. Richard Rollison; the ultimate sleuth who revels in solving crimes. Whilst returning home across the Essex countryside from a day's cricket at his father's Norfolk home, he happens upon an accident that sets him on a new trail. This involves murder, suspense and thrilling action as The Toff applies his mind with its usual precision and thoroughness.
---|---
| The Toff in New York
_'Say, Miss Hall, I hope that brother of yours hasn't run into any trouble.'_ But Will Hall had been kidnapped and the Honourable Richard Rollison, known by many by the apt if absurd soubriquet of the _'Toff'_ is soon on the scene, but not before a murdered man had fallen into Valerie Hallís arms. There's lots of action in 'millionaire surroundings', with a rich private eye and the NYPD all on the case, whilst the Toff tracks Dutch Himmy, surely the worst man in New York . . . ..
---|---
www.houseofstratus.com
| {
"redpajama_set_name": "RedPajamaBook"
} | 4,505 |
\section{Appendix}
\subsection{Recovery of Single Qubit Ramsey Fringes}
Figure~\ref{1q_ramsey} shows the recovery
of Ramsey fringes for a single qubit. In this single qubit case, there is only
one noise parameter, $\gamma_2$, as $\gamma_1\ll \gamma_2$, in contrast
to the Ramsey interferometry in the superconducting qubit system, where
$\gamma_1 \approx \gamma_2$. We set the background
magnetic field to 10 $\mu$T and combine the results of 350 experiments. As the
order of the recovery is increased, more and more fringes are recovered; at
tenth order, the fringes
extend are recovered even when the best single qubit run has no clearly visible
fringes. Without correction, the fringes decay with the characteristic $T_2$
time~\cite{paik2011observation}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{nv_1q_ramsey_fringes.pdf}
\caption{Single qubit Ramsey interferometry. 350 simulations with random
$T_2^*$ times are used. }\label{1q_ramsey}
\end{figure}
\subsection{Derivation}
The Lindblad master equation for a general density matrix $\rho(t)$ evolved from
an initial state $\rho_0$ is defined as (throughout, $\hbar$ = 1)
\begin{equation}\label{lindblad_me}
\frac{d\rho}{dt} = -i [H, \rho_0] + L(\rho_0),
\end{equation}
where $H$ is the Hamiltonian of the system, describing coherent evolution and
$L$ is the Lindblad superoperator, describing incoherent evolution, such as
noise processes. Both $H$ and $L$ can, generally, be time dependent.
`Vectorizing' the density matrix~\cite{uzdin2015}, $\rho$, allows us to write a general
solution of the Lindblad master equation as
\begin{equation}\label{lme_sol}
\tilde\rho(t) = \exp(-i\tilde H t + \tilde L t) \tilde\rho_0,
\end{equation}
where $\tilde\rho$ is the vectorized density matrix, $\tilde H$ is the vectorized
Hamiltonian, and $\tilde L$ is the vectorized Lindblad superoperator. See Ref. \onlinecite{uzdin2015} for
more details about the process of vectorization. We can decompose the exponential in
the solution, equation~\eqref{lme_sol}, using the Trotter decomposition~\cite{trotter1959product}:
\begin{equation}\label{trotter}
\exp(-i\tilde H t + \tilde L t) = \lim_{n\rightarrow \infty}\Bigg(\Big(\exp(-i\tilde H \frac{t}{n}) \exp(\tilde L \frac{t}{n})\Big)^n\Bigg)
\end{equation}
and take the Taylor expansion of the exponential of the Lindblad
\begin{equation}\label{trotter_taylor}
\exp(-i\tilde H t + \tilde L t) = \lim_{n\rightarrow \infty}\Bigg(\Big(\exp(-i\tilde H \frac{t}{n}) \sum_{m=0}^\infty \frac{(\tilde L \frac{t}{n})^m}{m!}\Big)^n\Bigg).
\end{equation}
We now write the Lindblad superoperator as a sum of many different Lindblad
superoperators, each with its own rate: $\tilde L = \sum_j \gamma_j \tilde L_j$ and plug
this into equation~\eqref{trotter_taylor}, giving
\begin{equation}\label{tt_l_expanded}
\exp(-i\tilde H t + \tilde L t) = \lim_{n\rightarrow \infty}\Bigg(\Big(\exp(-i\tilde H \frac{t}{n}) \sum_{m=0}^\infty \frac{(\sum_j \gamma_j \tilde L_j \frac{t}{n})^m}{m!}\Big)^n\Bigg).
\end{equation}
Equation~\eqref{tt_l_expanded} an infinite
sum over $m$, a finite sum over $i$, and is raised to an infinite power, $n$.
Though this equation has infinite
terms, we can collect all terms that have the same $\gamma$
prefactors. For example, the collection of all terms with only $\gamma_j$ will
include all terms from the product that have one first order element from
the Taylor series expansion of $\tilde L$. Terms with $\gamma_j^2$ will include
products with two first order elements, as well as products with one second
order element. To provide a concrete
example, we truncate the Trotterization at third order, the Taylor expansion at
fist order, and include two noise terms. Let $U=\exp(-i \tilde H \frac{t}{3})$
and $V_j = \tilde L_j \frac{t}{3}$. With our truncation, we rewrite
equation~\eqref{tt_l_expanded} as
\begin{widetext}
\begin{equation}\label{3rd_order_Trotter}
\begin{split}
\exp(-i\tilde H t + \tilde L t) &\approx \Big(U (1 + \gamma_1 V_1 + \gamma_2 V_2)\Big)^3 \\
&\approx U U U + \gamma_1 (U V_1 U U + U U V_1 U + U U U V_1)
+ \gamma_2 (U V_2 U U + U U V_2 U + U U U V_2) + \\
&~~+\gamma_1^2 (U V_1 U V_1 U + U U V_1 U V_1 + U V_1 U U V_1)
+ \gamma_2^2 (U V_2 U V_2 U + U U V_2 U V_2 + U V_2 U U V_2) \\
&~~+ \gamma_1 \gamma_2 (U V_1 U V_2 U + U U V_1 U V_2 + U V_1 U U V_2
+ U V_2 U V_1 U + U U V_2 U V_1 + U V_2 U U V_1).
\end{split}
\end{equation}
\end{widetext}
The generalization to higher order Trotterizations is clear; the expanded sum
will have many more terms (due to each term having a smaller timestep,
$\frac{t}{n}$), but terms can be grouped by their $\gamma_j$ prefactors. For
higher order Taylor expansions of $\exp{\tilde L}$, terms can still be grouped
by their $\gamma_j$ prefactors. For example, take the $\gamma_1^2$ term from
equation~\eqref{3rd_order_Trotter}. With a second order Taylor expansion, the
$\gamma_1^2$ terms now contain contributions from $V_i^2$:
\begin{equation}
\begin{split}
[\gamma_1^2~\text{terms}] &= U V_1 U V_1 U + U U V_1 U V_1 \\
&~~+ U V_1 U U V_1 + U V_1^2 U U \\
&~~+ U U V_1^2 U + U U U V_1^2
\end{split}
\end{equation}
Combinging the generalizations to both higher order Trotterization and higher
order Taylor expansion is relatively straightforward; the number of terms grows
precipitously, but they can always be gathered by their $\gamma_j$ prefactors.
Collecting all the terms for both the infinite limits of Trotterization and the
Taylor expansion leads to
\begin{equation}\label{collected}
\begin{split}
\exp(-i\tilde H t + \tilde L t) &= \lim_{n\rightarrow \infty}\Bigg(\Big(\exp(-i\tilde H \frac{t}{n})\Big)^n\Bigg) \\
&+ \sum_j \gamma_j [\gamma_j~\text{terms}] \\
&+ \sum_j \sum_k \gamma_j \gamma_k[\gamma_j\gamma_k~\text{terms}] + \cdots,
\end{split}
\end{equation}
where we have used $[\gamma_j~\text{terms}]$ to represent the (infinite)
collection of terms all with $\gamma_j$ as a prefactor.
This represents the general (exact) evolution operator for our system.
Our density matrix at
time $t$ can now be written
\begin{equation}\label{final_evolution}
\begin{split}
\tilde\rho(t) &= \lim_{n\rightarrow \infty}\Bigg(\Big(\exp(-i\tilde H \frac{t}{n})\Big)^n\Bigg) \tilde\rho_0\\
&+\sum_j \gamma_j [\gamma_j~\text{terms}] \tilde\rho_0\\
&+\sum_j \sum_k \gamma_j \gamma_k[\gamma_j \gamma_k~\text{terms}] \tilde\rho_0 + \cdots.
\end{split}
\end{equation}
The first term of this expansion represents the noise-free result. Other
terms represent the effects of noise on the evolution. Since this method
directly corrects the density matrix, $\rho$, it follows that it also corrects
an arbitrary observable,
\begin{equation}\label{final_evolution}
\langle A \rangle = A_0 +\sum_j \gamma_j A_{j} + \sum_j \sum_k \gamma_j \gamma_k A_{jk} + \cdots,
\end{equation}
where $A_0$ is the noise-free observable value and $A_{j}$ is the effect of
noise rate $j$ on the observable.
We define the `order' of
the combined Trotterization and Taylor expansion by the number of $\gamma_j$
terms included. The first order terms, for example, are those with a single
$\gamma_j$ prefactor, and include all possible ways that one (infinitesimal)
error evolution can be included. The second order terms include all possible ways
that two (infinitesimal) error evolutions can be included, and so on. We do not
{\em a priori}
know what the values of the effects of noise on the observable (such as $A_j$)
for any order are;
however, we can characterize $\gamma_j$
for a given experiment.
By taking a sequence of experiments, varying $\gamma_j$, we
can reconstruct the unknown evolution terms by fitting a hypersurface to the
points. The coefficients of the hypersurface represent the effects (or, for the
zeroth order term, the lack of effects) of noise to a given order on
the density matrix (or an arbitrary observable).
Equation~\eqref{final_evolution} generally has effects up the infinite order; to
make it tractable, we truncate at some given order. As more orders are included,
the fit becomes more accurate, and, therefore, a better approximation of the
noise-free result is obtained.
\subsection{Number of Terms in Expansion}
Naively, the number of terms in a given order $l$ would be $m^l$, where $m$
is the number of noise terms (which could be the number of qubits or a small
factor times the number of qubits). Since $\gamma_j$ is a scalar, all $\gamma_j$
will commute, allowing us to fuse terms with the same set of $\gamma$. For
instance, given a second order expansion with two noise terms, we would general
have terms with prefactors $\gamma_1 \gamma_1, \gamma_1 \gamma_2, \gamma_2
\gamma_1,$ and $\gamma_2 \gamma_2$, but since $\gamma_1 \gamma_2 = \gamma_2
\gamma_1$, we can reduce the number of fitted parameters by combining the
$\tilde L_1 \tilde L_2$ and $\tilde L_2 \tilde L_1$ terms. For a given order $l$
and number of noise terms $m$, the number of parameters $n$ for that order is
the number of
multinomial coefficients, which is given by the formula
\begin{equation}\label{multinomial}
n = \binom{l+m-1}{m-1}.
\end{equation}
For an expansion truncated at order $l$, the total number of parameters, for all
orders, is the sum of equation~\eqref{multinomial} for each order up to, and
including, $l$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,840 |
Nevio De Zordo (* 11. März 1943 in Cibiana di Cadore, Belluno; † 26. März 2014) war ein italienischer Bobfahrer und Gastronom.
Leben
Nevio De Zordo entstammte einer Familie von "Gelatieri" (Eismachern). 1959 wanderte sie nach Köln aus, um dort Eis zu verkaufen. Den Winter verbrachte die Familie traditionell in der italienischen Heimat in den Dolomiten, so dass De Zordo das Bobfahren in Cortina d'Ampezzo trainieren konnte.
Bei den Olympischen Winterspielen 1972 in Sapporo errang De Zordo die Silbermedaille im Vierer-Bob. 1969 wurde er Weltmeister im Zweier-Bob und 1970 im Vierer-Bob. Zudem wurde er zwei Mal Vize-Weltmeister sowie 1964 Europameister im Zweier-Bob.
40 Jahre lang betrieb Nevio De Zordo mit seiner Familie ein Eiscafé in Köln-Sülz. Zuletzt führte er ein Café auf dem Gelände der Kölner Universität.
De Zordo wurde auf dem Melaten-Friedhof in Köln beigesetzt (Flur 17 U Nr. 555).
Weblinks
Einzelnachweise
Bobfahrer (Italien)
Olympiateilnehmer (Italien)
Bobweltmeister
Europameister (Bobsport)
Gastronom
Unternehmer (Köln)
Italiener
Geboren 1943
Gestorben 2014
Mann
Teilnehmer der Olympischen Winterspiele 1972
Teilnehmer der Olympischen Winterspiele 1976 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,137 |
namespace SceneEngine
{
class LightingParserContext;
////////////////////////////////////////////////////////
class IVolumeDensityFunction
{
public:
typedef std::pair<Float3, Float3> Boundary;
virtual Boundary GetBoundary() const = 0;
virtual float GetDensity(const Float3& pt) const = 0;
virtual Float3 GetNormal(const Float3& pt) const = 0;
};
////////////////////////////////////////////////////////
class DualContourMesh
{
public:
class Vertex { public: Float3 _pt; Float3 _normal; Vertex(Float3 pt, Float3 normal) : _pt(pt), _normal(normal) {} };
class Quad { public: unsigned _verts[4]; };
std::vector<Vertex> _vertices;
std::vector<Quad> _quads;
DualContourMesh();
DualContourMesh(DualContourMesh&& moveFrom);
DualContourMesh& operator=(DualContourMesh&& moveFrom);
~DualContourMesh();
};
////////////////////////////////////////////////////////
DualContourMesh DualContourMesh_Build( unsigned samplingGridDimensions,
const IVolumeDensityFunction& fn);
////////////////////////////////////////////////////////
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,856 |
> [Docs](../index.md) ▸ [API Reference](index.md) ▸ **LayerOrganizer**
# LayerOrganizer
A utility for creating layers from a given config. If you have a habit of creating many ```<g>``` or other tags to layer your visual elements. This utility will let you create nested layers easily. For example,
```javascript
var layers = new d3Kit.LayerOrganizer(d3.select('svg'));
layers.create([{'highlight': 'cursor'}, {'graph': ['axis','content']}]);
```
will give you
```html
<svg>
<g class="highlight-layer">
<g class="cursor-layer"></g>
</g>
<g class="graph-layer">
<g class="axis-layer"></g>
<g class="content-layer"></g>
</g>
</svg>
```
and you can access a d3 selection of each layer by its name.
```javascript
layers.get('highlight.cursor'); // a d3 selection of <g class="cursor-layer"></g>
```
This was originally written to support just `<g>` but since version 1.1.0 we have added support for other tags. Just specify the tag name in the constructor.
## Functions
<a name="constructor" href="LayerOrganizer#constructor">#</a> new **d3Kit.LayerOrganizer**(*container*[,tag])
Construct a layer organizer for the specified ```container```. ```container``` is a d3 selection of `<svg>`, `<g>`, `<div>`, etc. You can specify *tag* such as `'div'` if you want to create layers of another thing that is not `<g>`
<a name="create" href="LayerOrganizer#create">#</a> layers.**create**(*config*)
Create layer(s) of ```<g>``` within the container. If ```config``` is
* *String* - Create one ```<g class="[config]-layer">```
* *Array* - Create each ```<g class="[value]-layer">``` for each string value in the array. If the value is an Object, will use the Object construction rule below.
* *Object* - For each (key,value) pair, create ```<g class="[key]-layer">```, then create layer(s) inside the new ```g.[key]-layer``` depends on the type of the value (string, Array or Object).
<a name="get" href="LayerOrganizer#get">#</a> layers.**get**(*name*)
Retrieve a layer by ```name``` (*string*). If a layer is nested under one or more layer(s), add its parent names as prefixes, separated by dot. For example, ```layers.get('highlight.cursor')``` will return the ```g.cursor-layer``` that is nested under ```g.highlight-layer```.
<a name="has" href="LayerOrganizer#has">#</a> layers.**has**(*name*)
Check if there is a layer with the specified ```name``` (*string*). Use the same name format with [layers.get](#get) | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,458 |
Q: Java Runtime.exec() program won't output to file I have a program that takes in a file as an input and produces an xml file as an output. When I call this from the command line it works perfectly. I try calling it from a Java program with the following code.
try
{
Process proc = Runtime.getRuntime().exec(c);
try
{
proc.waitFor();
}
catch(InterruptedException e)
{
System.out.println("Command failed");
}
}
catch(IOException e)
{
System.out.println("Command failed");
e.printStackTrace();
}
The program seems to be running fine, as it creates an xml file; however, the xml file is empty when I open it. I'm not encountering any exceptions in my Java program, so I'm baffled as to what the problem could be. Why would the command line program work fine normally, but then when called from Java not output anything to the file it created. I was thinking maybe it was some sort of permissions thing. I tried running the program as sudo (I'm using Linux) but to no avail. This problem doesn't seem to be anything I could find an answer to online. Hopefully somebody on here might be able to tell what's going on. :)
A: Get the output and error streams from your process and read them to see what is happening. That should tell you what's wrong with your command.
For example:
try {
final Process proc = Runtime.getRuntime().exec("dir");
try {
proc.waitFor();
} catch (final InterruptedException e) {
e.printStackTrace();
}
final BufferedReader outputReader = new BufferedReader(new InputStreamReader(proc
.getInputStream()));
final BufferedReader errorReader = new BufferedReader(new InputStreamReader(proc
.getErrorStream()));
String line;
while ((line = outputReader.readLine()) != null) {
System.out.println(line);
}
while ((line = errorReader.readLine()) != null) {
System.err.println(line);
}
} catch (final IOException e) {
e.printStackTrace();
}
If there is no output in either stream, then I would next examine the external program and the command being sent to execute it.
A: Did you try launching the process from outside java?
A: For me, I wrote a jar file that output a file and ran that from the command line in another java program. It turns out that there was a fundamental check in my jar file that I had forgotten about on the number of characters in an input string (my bad). If the count of the characters was smaller than 8 there was no output file. If the number of characters was greater than 8, the output file came out without any trouble using the following code:
String cmdStr = "java -jar somejar.jar /home/username/outputdir 000000001";
try
{
Runtime.getRuntime().exec(cmdStr);
Runtime.getRuntime().runFinalization();
Runtime.getRuntime().freeMemory();
log.info("Done");
}
catch (IOException e)
{
log.error(System.err);
}
Not sure if I really need everything here but, hey, it works. Note: no waitFor seems to be necessary in my case.
A: process input (actually output of the process!) and error streams has to be handled before waiting for the process termination.
This should work better
try
{
Process proc = Runtime.getRuntime().exec("anycomand");
BufferedReader outSt = new BufferedReader(new InputStreamReader(proc.getInputStream()));
BufferedReader errSt = new BufferedReader(new InputStreamReader(proc.getErrorStream()));
String line;
while ((line = outSt.readLine()) != null)
{
System.out.println(line);
}
while ((line = errSt.readLine()) != null)
{
System.err.println(line);
}
proc.waitFor();
}
catch (final IOException e)
{
e.printStackTrace();
}
but to understand better how Runtime exec works it is worth reading
the classic article
When Runtime.exec() won't
which provide useful sample code (better than the one above!)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,076 |
\section{Introduction}
Which of the many regressor variables in today's large data sets can be used to model or predict an outcome variable, and which variables can be neglected? Answering this question in a process termed feature selection has become a central task in machine learning and statistical modeling, in particular with the increasing availability of large-scale, multivariate data sets. The objective of feature selection is to select a minimal subset of the input variables that provides maximum information about a target variable, often with the goal of minimizing the generalization error in a subsequent learning task, such as classification or regression \citep{Guyon2003,Vergara2014}. A further goal is to reduce training time and improve model performance by increasing interpretability and by reducing effects of the curse of dimensionality \citep{Lensen2018,Guyon2003,Li2018}.
From the requirement that selected feature sets should be minimal, yet maximally informative, it is immediately clear than one does not want to include regressor variables into the feature set if they have information that is already carried redundantly by other variables. Instead, one wants to include variables that have information about the target which they carry uniquely. Additionally, considering the possibility of interactions between regressors which is well known from classical statistics, it is possible that only multiple variables considered jointly provide the information needed to model the outcome.
Curiously, even today's most advanced feature selection algorithms that are rooted in information theory lack a way of describing these three possible ways in which regressor variables carry information about the outcome. This blocks the way to translate the simple intuitions above into working algorithms, and also leads to a lack of conceptual clarity. In fact, even in information theory proper, the decomposition of the information that a set of variables has about another (outcome) variable into contributions carried uniquely by individual variables, redundantly by multiple variables or that is available only when considering variables jointly has been an open problem until very recently, as laid out in \cite{Williams2010}. However, this so-called partial information decomposition (PID) problem is now well understood and solutions are available. This finally allows formulating the above intuitions about which variables would be useful features, and which should be discarded, as a rigorously defined information-theoretic problem.
In this work, we use this novel framework of PID to first clarify why feature selection has been a difficult problem in the past---also from the conceptual point of view. We reanalyze existing feature selection frameworks in the light of their PID and show that approaches based on conditional mutual information criteria are strictly preferable over approaches based on the unconditional mutual information and propose a practical iterative CMI-based feature selection algorithm based on the improved understanding provided by PID. Last, we demonstrate the power of this algorithm on well-known benchmark examples, and also provide the corresponding PID measures--- highlighting how PID leads to a better understanding of the role different variables play as features.
\section{Background} \label{sec:prior_art}
In feature selection, a general approach is variable filtering, which ranks variables by their relevancy with respect to the target, as measured by a pre-defined criterion. In general, filtering is computationally cheaper, less prone to overfitting and provides a \enquote{generic} feature selection approach that is independent of specific, subsequently applied inference models (e.g., \cite{Guyon2003,Duch2006,Hastie2009}). A popular choice for filtering criteria are information-theoretic quantities \citep{Shannon1948,MacKay2005}, such as the mutual information (MI) and the conditional mutual information (CMI, e.g., \cite{Battiti1994,Duch2006,Brown2012}), as well as heuristics derived from both measures \citep{Vergara2014,Guyon2003,Chandrashekar2014,Brown2012}. These quantities are popular because they are inherently model free such that variable dependencies of arbitrary order can be captured, while only minimal assumptions about the data are required for estimation.
Even though, information theory is a popular tool for constructing feature selection criteria, existing definitions of feature relevancy and redundancy (e.g., \cite{John1994,Bell2000}, see \cite{Vergara2014} for a review), fail to rigorously capture the notions of feature relevancy and redundancy, in particular for sets of interacting features. As a result, practical feature selection approaches struggle with a clear definition of how interactions between variables contribute to relevancy and redundancy in the data and how interactions influence the selection procedure (e.g., \cite{Brown2012}). We argue that this lack results from the strict definition of MI as the information shared between two variables or two sets of variables \citep{Shannon1948,MacKay2005}, such that interactions between two or more variables can not be described in detail. The decomposition of the joint MI between three or more variables has only recently become possible through the theoretical extensions of classical information theory by the PID framework \citep{Williams2010}, which provides definitions and measures that allow for an unambiguous description of how multiple variables contribute information about a target.
In the following, we will first introduce necessary information-theoretic preliminaries (Section 2.1). We will then highlight where classical information theory lacks in methods to describe multivariate information contribution in feature selection and introduce the PID framework \citep{Williams2010} to close this conceptual gap (Section 2.2). Using PID, we will go on to define feature relevancy and redundancy in information-theoretic terms in Section 3.
\subsection{Information-Theoretic Preliminaries} \label{sec:info_theory}
In classical information theory\footnote{For a detailed introduction see \cite{Cover2005} or \cite{MacKay2005}.} as conceived by \cite{Shannon1948}, the mutual information (MI) quantifies the information that is shared between two random variables, $X$ and $Y$, as the expected information one variable provides about the other,
\begin{equation*}
I(X;Y) = \sum_{x\in\mathcal{A}_X, y\in\mathcal{A}_Y} p(x,y)
\log \frac{p(x,y)}{p(x)p(y)},
\label{eq:mutual_information}
\end{equation*}
\noindent where $p(x)$ denotes the probability of observing outcome $x$ for variable $X$ and is a shorthand for $p(X = x)$, and $\mathcal{A}_X$ is the support of random variable variable $X$. Furthermore, $p(x,y)$ denotes the joint probability for simultaneously observing the outcomes $x$ for variable $X$ and $y$ for variable $Y$. The MI is symmetric in $X$ and $Y$ and describes the information which $X$ provides on $Y$ and vice versa. It is always non-negative, it is zero only for independent variables, i.e., $p(x)p(y) = p(x,y)$, non-zero for any dependence between $X$ and $Y$, and is upper bounded by the entropy, $H(X)=-\sum_{x\in\mathcal{A}_X} p(x)\log p(x)$ and $H(Y)$, respectively. The MI may be interpreted as the Kullback-Leibler divergence between the joint distribution of the two variables, $p(x,y)$, and the assumption of statistical independence, $p(x)p(y)$. Note that in its original form, the MI is strictly defined for two variables or two sets of variables, $\mathbf{X}$ and $\mathbf{Y}$.
To measure the influence a third variable, $Z$, has on the relationship between two variables, $X$ and $Y$, we may calculate the conditional mutual information (CMI), which quantifies the information shared between $X$ and $Y$, given the outcome of $Z$ is known,
\begin{equation}
I(X;Y|Z) = \sum_{x\in\mathcal{A}_X, y\in\mathcal{A}_Y, z\in\mathcal{A}_Z}
p(x,y,z) \log \frac{p(x|y,z)}{p(x|z)},
\label{eq:conditional_mutual_information}
\end{equation}
\noindent where again lower-case letters indicate realizations of random variables and $p(x|z)$ is a shorthand for the conditional probability $p(X=x|Z=z)$. Please note that the CMI is still symmetric in $X$ and $Y$ since $p(x|y)p(y)=p(x,y)$ holds for conditional probabilities $p(x|y)$.
However, as we will further illustrate below the CMI does not provide us with a detailed account of how $X$ provides information about $Y$ (or vice versa) in the the context of $Z$, but rather quantifies the summarized contribution of $X$ with respect to $Y$ in the context of $Z$. In particular, conditioning on $Z$ may have one of three effects on the MI between $X$ and $Y$: first, the MI remains the same, $I(X;Y|Z)=I(X;Y)$, second, the information $X$ provides about $Y$ \textit{decreases} in the context of $Z$, $I(X;Y|Z)<I(X;Y)$, third, the information $X$ provides about $Y$ \textit{increases} in the context of $Z$, $I(X;Y|Z)>I(X;Y)$. The second and third case are commonly interpreted as $X$ and $Z$ providing primarily redundant information about $Y$, and $X$ and $Z$ providing primarily synergistic or complementary information about $Y$. Note however, that these two contributions may occur simultaneously \citep{Williams2010}, such that the CMI does not allow for an exact quantification of the magnitude of both contributions. Furthermore, the first case, $I(X;Y|Z)=I(X;Y)$, may not be indicative of an independence of $Z$, but may also occur whenever redundant and synergistic contributions cancel each other.
Note that especially the occurrence of synergistic information is often neglected in filtering approaches and approaches often assume independence between features to allow for easier modeling (e.g., \cite{Dash1997,Hall2000}, see also the review by \cite{Brown2012}). Yet, synergistic contributions may occur already in simple settings. As a toy example consider a binary \texttt{xor} gate with i.i.d. inputs $X_1$ and $X_2$, and output $Y$, where $I(X_1; Y)=I(X_2;Y)=0$. Here, input variables in themselves are not informative about the output, only in the context of the second input, the first input provides one Bit of synergistic information about the output, i.e., $I(X_1; Y|X_2)=I(X_2;Y|X_1)=1$ (see \cite{Griffith2014} for further examples of synergistic and redundant information in Boolean operations). As a more realistic example, consider a system where an input $X$ has a relationship with a target variable $Y$ but this relationship is corrupted by noise, $Z$ \citep{MacKay2005}. The noise is independent of the input and the target, such that $I(X;Z)=I(Y;Z)=0$. However, the information $X$ provides about $Y$ increases if the noise $Z$ is known, i.e., $I(X;Y|Z)>I(X;Y)$. So adding the noise $Z$ as a feature may add to the explanatory power of $X$ with respect to $Y$, where the contribution of $Z$ may be interpreted as \textit{decoding} the information in $X$ about $Y$.
Previous approaches have tried to handle the conceptual gap in describing the structure of information contribution of two or more variables about a third by combining MI and CMI terms (e.g., \cite{Watanabe1960,Garner1962,Tononi1994,McGill1954,Bell2003}). For example, approaches in the context of feature selection use various combinations of classical information-theoretic quantities to decompose (joint) variable contributions into relevant and redundant ones (reviewed in \cite{Brown2012}). However, these approaches are inherently limited in their ability to provide such a decomposition as we will review in the next section.
\subsection{The Partial Information Decomposition Framework} \label{sec:pid_intro}
Classical information theory does not allow for a decomposition of the information two or more variables provide about a third \citep{Williams2010}. More formally, we consider the joint MI between a set of assumed inputs, $\{X_1, X_2\}$, and an output, $Y$,
\begin{equation*}
I(Y; X_1, X_2) =
\sum_{y \in \mathcal{A}_{Y}, x_1 \in \mathcal{A}_{X_1}, x_2 \in \mathcal{A}_{X_2}}
p(y, x_1, x_2) \log_2 \frac{p(y|x_1, x_2)}{p(y)},
\label{eq:multi_mi}
\end{equation*}
\noindent which denotes the total amount of information $\{X_1, X_2\}$ contains about $Y$. Note that again each of the inputs may be a multivariate variable, $\mathbf{X}_i$.
As stated in the beginning, in the context of feature selection we would like to be able to quantify how much information only $X_1$ or only $X_2$ carry uniquely about $Y$, how much information is redundantly present in both $X_1$ and $X_2$, and how much information is provided synergistically when considering both $X_1$ and $X_2$ together. Here, the PID framework by \cite{Williams2010} proposes to decompose the joint MI into four information contribution \textit{atoms},
\begin{equation}
\begin{aligned}
I(Y; X_1, X_2) &= I_{unq}(Y;X_1\setminus X_2) + I_{unq}(Y;X_2\setminus X_1) \\
& \qquad + I_{shd}(Y;X_1, X_2) + I_{syn}(Y;X_1, X_2), \\
\label{eq:pid_sum}
\end{aligned}
\end{equation}
\noindent where
\begin{enumerate}
\item $I_{unq}$ denotes \emph{unique information} provided exclusively by
either $X_1$ or $X_2$ about $Y$;
\item $I_{shd}$ denotes \emph{shared information} provided redundantly by both $X_1$ and $X_2$ about $Y$;
\item $I_{syn}$ denotes \emph{synergistic information} provided jointly by $X_1$ and $X_2$ about $Y$.
\end{enumerate}
\noindent The synergistic information, also termed complementary information, is
information that can only be obtained by considering variables together and can
not be gained from one of the variables alone. The atoms and their relation are shown graphically in Fig. \ref{fig:pid_intro}A.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{figures/pid_theory.pdf}
\caption{\textbf{Partial information decomposition (PID) framework.}
\textbf{A} PID diagram of the joint mutual information (MI), $I(Y; X_1, X_2)$, see main text (modified from \protect\cite{Williams2010}).
Relation between classical information-theoretic terms and PID:
\textbf{B} Joint MI between $X_1$, $X_2$, and $Y$;
\textbf{C} MI between $X_1$ and $Y$;
\textbf{D} MI between $X_2$ and $Y$;
\textbf{E} conditional mutual information (CMI) between $X_2$ and $Y$ given $X_1$.
}
\label{fig:pid_intro}
\end{figure}
The PID atoms relate to the MI and CMI between inputs, $X_1$ and $X_2$, and target $Y$ as,
\begin{equation}
\begin{aligned}
I(Y; X_1) &= I_{unq}(Y;X_1 \setminus X_2) + I_{shd}(Y;X_1, X_2), \\
I(Y; X_2) &= I_{unq}(Y;X_2 \setminus X_1) + I_{shd}(Y;X_1, X_2).
\label{eq:pid_mi}
\end{aligned}
\end{equation}
\noindent and,
\begin{equation}
\begin{aligned}
I(Y; X_1|X_2) &= I_{unq}(Y;X_1 \setminus X_2) + I_{syn}(Y;X_1, X_2) \\
I(Y; X_2|X_1) &= I_{unq}(Y;X_2 \setminus X_1) + I_{syn}(Y;X_1, X_2).
\label{eq:pid_cmi}
\end{aligned}
\end{equation}
Note that there are only three equations relating classical information-theoretic
terms to the four PID atoms (Equations \ref{eq:pid_sum}, \ref{eq:pid_mi}, \ref{eq:pid_cmi}). Hence, the system is under-determined and we have to provide an additional definition for at least one of the atoms, such that then all other atoms can be derived via these equations. This
also means that neither of the proposed atoms can be derived from measures
defined in classic information-theoretic measures, e.g., by subtracting
individual terms (Fig. \ref{fig:pid_intro}B--E). Based on this
result by \cite{Williams2010} it can immediately be seen that classical
information theory does not allow for the quantification of the individual
contributions as would be desirable in applications such as feature selection.
As a direct result, attempts to quantify joint variable relevancy or redundancy
is bound to fall short as long as it does not introduce novel, axiomatic
definitions of one of the terms.
Note that all atoms in the PID are required to be non-negative, which allows for an interpretation as information contributions, avoiding the problems of earlier approaches to this problem such as the interaction information \citep{McGill1954}. In fact, it can be shown, that the
interaction information is the difference between the shared and synergistic
information \citep{Williams2010}. This relationship allows for a clear
explanation of instances where the interaction information is negative, that
is, whenever the synergistic information contribution is larger than the
information shared between variables. Note that both effects can be present
simultaneously and thus, the interaction information does not provide a
conclusive description of the structure of multivariate interactions in the
data. In cases where both contributions are equally present such that they cancel each other, one may even erroneously infer an absence of an interaction.
\section{Methods}
Based on the introduced PID framework, we will present a rigorous definition of variable relevancy and redundancy in settings for which feature-independence can not be generally assumed (Section \ref{sec:pid_def}). Based on this definition, we will show that using the CMI as a filtering criterion in feature selection algorithms maximizes feature relevancy, while simultaneously minimizing redundant contributions. To overcome the practical problems commonly encountered when estimating information-theoretic quantities from data, we propose the use of a recently introduced sequential forward-selection algorithm (Section \ref{sec:cmi_algo}). We also describe the practical estimation of the PID from data (Section \ref{sec:pid_measures}).
In Section \ref{sec:experiments}, we will present results from applying both the proposed algorithm as well as PID estimation to benchmark models from literature, where we demonstrate that the CMI criterion generally outperforms the MI criterion, and that PID allows for a quantitative description of feature interactions in terms of synergistic and redundant contributions.
\subsection{Using PID to Define Feature Relevancy and Redundancy} \label{sec:pid_def}
We wish to define feature relevancy such that, intuitively, a variable is considered relevant if it i) uniquely provides information about the target, ii) provides information in the context of other variables, and iii) does not provide information that is redundant with information already contained in the feature set. As a result, a filtering criterion for including a feature should return a high score if the first two conditions are met, and should be low if the last is not met.
PID allows for an immediate quantification of these properties in information-theoretic terms: We assume a feature selection problem with a set of input variables, $\mathbf{X}$, a target variable, $Y$, and a set of already identified relevant features, $\mathbf{S}=\{F_j\}\subseteq \mathbf{X}$ which is a subset of the input variables $\mathbf X$. We define the relevancy of a specific additional feature, $F_i$, as the sum of the information provided uniquely by the feature, $I_{unq}(Y;F_i \setminus \mathbf{S})$, and the information provided synergistically by the feature and all other selected features, $I_{syn}(Y;F_i, \mathbf{S})$, with $\mathbf{S} \subseteq \mathbf{X} \setminus F_i : \max_{\mathbf{S}}(I_{syn}(Y;F_i, \mathbf{S}))$. We further define the redundancy of the feature $F_i$ with respect to the feature set by the shared information, $I_{shd}(Y;F_i, \mathbf{S})$.
Comparing this definition of relevancy to Equation \eqref{eq:pid_cmi}, we can directly see that it is equivalent to the CMI. Hence, we define the criterion whether to
include a feature into the relevant set of selected features as
\begin{equation}
I(F_i; Y|\mathbf{S}_i) = I_{unq}(Y;F_i \setminus \mathbf{S}_i) + I_{syn}(Y;F_i, \mathbf{S}_i),
\label{eq:pid_relevance}
\end{equation}
\noindent where $i$ indicates either the rank of the feature or the $i$-th iteration in a step-wise algorithm, and $\mathbf{S}_i$ is the set of selected features of rank $i$, or at the $i$-th iteration, respectively (see Fig. \ref{fig:pid_cmi} for the corresponding PID diagram). From our PID-based definition, we see that the CMI criterion returns a high score for features that either have a high unique contribution, or a high synergistic contribution with respect to the currently selected feature set, $\mathbf{S}_i$, or provide both contributions. On the other hand, the score is low if a variable carries information that is already redundantly present in $\mathbf{S}_i$. Hence, the CMI is able to capture relevancy resulting from variable interactions, where it only considers the \textit{partial} contribution of the feature about the target that is relevant.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{figures/pid_cmi_criterion.pdf}
\caption{\textbf{Partial information decomposition (PID) diagram of feature relevancy.}
PID diagram of the conditional mutual information (CMI) criterion for feature selection, $I(Y;F_i|\mathbf{S})$, which may be decomposed into the unique information $I_{unq}(Y;F_i\setminus \mathbf{S})$ and the synergistic information $I_{syn}(Y;F_i,\mathbf{S})$, while not including the redundancy between $F_i$ and $\mathbf{S}$. See main text.
}
\label{fig:pid_cmi}
\end{figure}
Note that in practice, finding the optimal feature set, $\mathbf{S}$, is computationally not feasible already for small input sizes. A brute-force approach would amount to evaluating all possible subsets, $\mathbf{S}' \subseteq \mathbf{X}$, i.e., the power set of the $N$ input variables which has size $2^N$ (existing definitions of relevancy suffer from the same problem, see for example \cite{Vergara2014}). Indeed, selecting the optimal feature set is an \NP-hard problem \citep{Amaldi1998}. Hence, in practice, $\mathbf{S}$ may denote the set of higher-ranking or already selected features in approximative solutions such as sequential forward-selection or backward-elimination \citep{Hastie2009}. Here,
backward-elimination is usually deemed less feasible because it starts in the search space comprising all variables which is potentially prohibitively large in most practical applications.
We note, that our measure of relevancy behaves subtly differently in global versus iterative contexts with respect to information carried redundantly by predictor variables. To see this consider an (extreme) example system where all features carry identical information about the target variable. In a global context, none of these variables will be assigned non-zero relevancy by our criterion (Equation \eqref{eq:pid_relevance})---because any feature's information is redundant with information provided by the remaining features. In contrast, in the iterative context, the first feature variable considered will be assigned non-zero relevancy, whereas the others will then be assigned zero relevancy, as they carry information that is redundant with the first (included) feature variable. Practically, it is only the latter behaviour, however that matters. Also note that our definition of relevancy is in line with previous work that derived the CMI as an optimal criterion in sequential forward-selection approaches (e.g., \cite{Brown2012,Shishkin2016}).
For completeness, we also express the MI in terms of our definition of
relevancy and redundancy (Equation \ref{eq:mi_criterion}),
\begin{equation*}
I(Y; F_i) = I_{unq}(Y;F_i \setminus \mathbf{S}_i) + I_{shd}(Y;F_i, \mathbf{S}_i).
\label{eq:pid_mi_criterion}
\end{equation*}
\noindent We see that the MI criterion---opposed to the CMI---misses the
synergistic information contribution between $F_i$ and $\mathbf{S}_i$.
Furthermore, the MI not only misses relevant information contribution but also
includes the redundant information, $I_{shd}(Y;F_i, \mathbf{S}_i)$. Hence,
using the PID framework, we can immediately show that while the CMI is able to
account for synergistic interactions between variables, while not considering redundant information, the MI criterion only considers unique contributions while failing to include synergistic contributions and potentially failing to remove redundancies in the feature set. We will demonstrate the consequences of choosing either criterion in feature selection in the experiments.
\subsection{An Algorithm for sequential Forward-Selection Using a CMI-Criterion} \label{sec:cmi_algo}
To perform feature selection in a computationally feasible fashion, we propose to use a forward-selection approach with one
backward-elimination step that was recently introduced in the context of network
inference from multivariate time series data \citep{Novelli2019,Wollstadt2019}. The backward
elimination step thereby ensures the selection of a minimal feature set by
removing previously selected variables that have become redundant given the
final set (\cite{Lizier2012}, see also
\cite{Runge2015,Novelli2019,Sun2014,Sun2015} for a discussion of this
\enquote{pruning step}).
\subsubsection{Statistical Testing}
The algorithm handles practical problems that arise when using information-theoretic criteria for feature selection by performing non-parametric statistical tests against surrogate data \citep{Novelli2019}. The test treats the estimated MI or CMI as a test statistic that is compared against the distribution of values estimated from permuted data to handle the commonly encountered bias when estimating information-theoretic terms from finite data \citep{Hlavackova-Schindler2007,Paninski2003}. A testing scheme termed \textit{maximum statistic} thereby ensures that the family-wise error rate is controlled for during repeated testing over iterations of feature selection (see \cite{Novelli2019,Wollstadt2019}).
Furthermore, statistical testing ensures the termination of the algorithm if none of the remaining variables adds information about the target in a significant fashion, thus providing an automatic stopping criterion and ensuring the selected feature set to be minimal \citep{Brown2012}. Also, we want to emphasize again that all three possible contributions, unique, synergistic, and redundant, may be simultaneously present in a potential feature \citep{Williams2010}. Here, statistical testing allows us to establish whether the unique and synergistic information contribution outweighs the information redundantly carried with the already selected feature set.
\subsubsection{Algorithm}
The algorithm starts from an empty feature set, $\mathbf{S}_0=\emptyset$, and the full set of variables, $\mathbf{X}_0=\mathbf{X}$, and includes in each step a feature using the CMI criterion of Equation\ \ref{eq:pid_relevance} and Equation\ \ref{eq:conditional_mutual_information},
\begin{equation}
F_i = \max_{X \in \mathbf{X}_i} I(X; Y|\mathbf{S}_i),
\label{eq:cmi_criterion}
\end{equation}
\noindent where $\mathbf{X}_i$ denotes the remaining input variables in
iteration $i$, and $\mathbf{S}_i$ the set of already selected features. If the
contribution is statistically significant as established by the maximum statistic (see \cite{Novelli2019}), $F_i$ is selected and the sets
are updated according to
\begin{equation}
\begin{aligned}
\mathbf{S}_{i+1} &= \mathbf{S}_i \cup F_i \, , \\
\mathbf{X}_{i+1} &= \mathbf{X}_i \setminus F_i\, .
\end{aligned}
\label{eq:inclusion_update}
\end{equation}
Hence, each feature is evaluated in the context of \textit{all} already selected features such that also higher-order interactions are accounted for. Note that the algorithm may fail to select relevant variables if those variables provide exclusively synergistic information about the target, as shown in the experiments.
The inclusion terminates once $\max_{X \in \mathbf{X}_i} I(X;Y|\mathbf{S}_i)$ is no longer statistically significant. The algorithm then performs a backward-elimination, where iteratively the weakest feature, given the full selected feature set is statistically tested using a so-called \textit{minimum statistic} \citep{Novelli2019}. The feature is excluded if it no longer provides significant information in the context of the full feature set. If the contribution of the weakest feature is significant, the backward-elimination terminates and the pruned feature set is returned.
Note that the algorithm may also be used with a MI criterion for selecting the features,
\begin{equation}
F_i = \max_{X \in \mathbf{X}_i} I(X; Y),
\label{eq:mi_criterion}
\end{equation}
\noindent where all other steps remain identical.
We use an implementation of the proposed approach provided as part of the IDTxl
Python toolbox \citep{Wollstadt2019} which makes use of estimators implemented in the JIDT toolbox \citep{Lizier2014jidt}. IDTxl provides estimators for various use cases such as discrete and continuous data, as well as model-free estimators and estimators for jointly Gaussian variables. To estimate the CMI and MI, we here use a model-free plug-in estimator for discrete data \citep{Hlavackova-Schindler2007} and a model-free nearest-neighbor-based estimator for continuous data \citep{Kraskov2004}.
\subsection{Measures and Practical Estimators of PID Terms}
\label{sec:pid_measures}
In the original formulation of the PID framework, \cite{Williams2010} provided an axiomatic definition of the shared information together with a corresponding measure, $I_{min}$. Subsequent work has proposed further PID measures and axioms \citep{Griffith2014,Bertschinger2014,Harder2013,Makkeh2021,Gutknecht2020}, which differ subtly in their operational interpretation. At the time of writing, one of the most popular approaches to quantify PID is the one proposed by Bertschinger, Rauh, Olbrich, Jost, and Ay (BROJA measure, \cite{Bertschinger2014}), who introduced a measure of unique information based on decision-theoretic considerations, and which also shares the axiomatic foundation with the measures proposed by \cite{Griffith2014}. The BROJA measure is appealing in the current setting as it appeals to multiple variables \enquote{competing} for an explanation of the output variable.
Several practical estimators for the BROJA measure have been proposed (\cite{Makkeh2017,Makkeh2018}, see also the IDTxl toolbox for a further implementation \citep{Wollstadt2019}. We here use the estimator by \cite{Makkeh2018} as part of the IDTxl toolbox. Note that current PID estimators are available for discrete data only (but see remarks in \cite{Makkeh2021}). We therefore discretized the data into $n=5$ bins for PID estimation, if not specified otherwise.
\section{Experiments and Results} \label{sec:experiments}
We demonstrate our proposed algorithm in a series of experiments on synthetic data obtained from benchmark models from literature. We chose models that were explicitly designed to introduce both synergies as well as redundancies into the data and used both a CMI (Equation \ref{eq:cmi_criterion}) and MI (Equation \ref{eq:mi_criterion}) criterion to illustrate how variable interactions affect feature selection. Additionally, we estimate PID for each experiment to quantify multivariate contributions to feature relevancy and redundancy. Each experiment was repeated 100 times if not specified otherwise and used $N=1000$ samples for estimation. For significance testing, we used a critical alpha level, $\alpha=0.05$.
\subsection{Experiment I - Nested Spheres}
To illustrate a simple case of synergistic information contribution and its quantification using PID, we sampled data from two nested spheres, where each sphere denoted one class, $Y \in \{0,1\}$, and each sample's Cartesian coordinates were considered the input variables, $\mathbf{X} =\{X_1, X_2, X_3\}$ (Fig. \ref{fig:spheres_pid}A). To be able to estimate MI terms using continuous estimators for MI and CMI \citep{Kraskov2004}, we added random Gaussian noise to class labels.
\begin{figure}[ht]
\centering
\includegraphics{figures/Spheres_pid.pdf}
\caption{\textbf{Results - Experiment I.} Mutual information (MI), conditional mutual information (CMI), and partial information decomposition (PID) terms for three variables sampled from two nested spheres.
\textbf{A} Data generation, each sphere indicates one class (orange and blue markers);
\textbf{B} PID terms;
\textbf{C} MI and CMI terms (dashed horizontal lines indicate significance thresholds).
}
\label{fig:spheres_pid}
\end{figure}
As expected, we observed predominantly synergistic contributions between the inputs, while there was no unique information in individual input variables (Fig. \ref{fig:spheres_pid}B). When considering sets of two variables there was an increase in unique information, accompanied by shared information and a higher synergistic contribution compared to the synergistic contribution of two inputs alone. Accordingly, the pairwise MI, $I(X_i;Y)$, did not show a significant information contribution of individual variables (Fig. \ref{fig:spheres_pid}C), while we found significant CMI, $I(X_i;Y|X_j), X_i, X_j \in \mathbf{X}$, for individual input variables, conditional on at least one other variable. When conditioning on a single variable the CMI was was higher than the pairwise MI and was highest when conditioning on both remaining variables.
\subsection{Experiment II - Statistical Models}
As a more complex example of both shared and synergistic information contribution, we generated data from two simple statistical models that each combined two input variables, $X_1$ and $X_2$, either via addition or multiplication, while the influence of $X_2$ was varied by a weighting parameter, $a$,
\begin{equation}
Y = X_1 + a X_2 + \sigma \mathcal{N}(0,1),
\label{eq:stats_model_add}
\end{equation}
\noindent and
\begin{equation}
Y = X_1 * a X_2 + \sigma \mathcal{N}(0,1),
\label{eq:stats_model_mult}
\end{equation}
\noindent where $\sigma \mathcal{N}(0,1)$ denotes the addition of Gaussian noise with $\sigma=0.1$ and $a\in \{1, 2, \ldots, 5\}$. Input data were sampled from different distributions to illustrate the independence of the proposed measures from marginal distributions (see Tab. \ref{tab:stats_models_params}).
\begin{table}[ht]
\centering
\begin{tabular}{lcc}
\toprule
Distribution & Formula & Parameter values \\ \midrule
Uniform & $f(k;b,a) = \frac{1}{b-a}$ & $b=1$, $a=2$ \\
Binomial & $f(k;n,p) = \binom{n}{k} p^k q^{n-k}$ & $n=20$, $p=0.5$, $q = 1-p$ \\
Poisson & $f(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}$ & $\lambda_1=4$, $\lambda_1=10$ \\
Exponential & $f(k;\lambda) =\lambda^k e^{-\lambda}$ & $\lambda = 1.5$ \\ \bottomrule
\end{tabular}
\caption{Distributions and parameters used for generating input data in Experiment II.}
\label{tab:stats_models_params}
\end{table}
As expected, estimated MI, CMI, and PID terms did not differ for varying marginal distributions, while only changes in the model, i.e., changes in $a$, which introduced variable interactions, had an effect. Fig. \ref{fig:stats_models_pid} shows exemplary results for the binomial distribution, while the results for the other input distributions showed qualitatively similar behaviour (not shown).
\begin{figure}[ht]
\centering
\makebox[\textwidth][c]{\includegraphics{figures/Stats_models_pid.pdf}}%
\caption{\textbf{Results - Experiment II.} Mutual information (MI), conditional mutual information (CMI), and partial information decomposition (PID) terms for (\textbf{A}) the additive and (\textbf{B}) the multiplicative model. Input data were sampled from a binomial distribution (see main text).
}
\label{fig:stats_models_pid}
\end{figure}
For the additive model (Equation \ref{eq:stats_model_add}, Fig.\ \ref{fig:stats_models_pid}A), we found both synergistic and shared information contribution of the inputs for $a>0$, with a maximum at $a=1$ and a higher synergy than redundancy. The unique information contribution of $X_2$ increased with $a\geq1$, while the unique contribution of $X_1$ quickly approaches zero for $a>1$. In the absence of any influence of $X_2$, i.e., $a=0$, only $X_1$ contributed information, leading to high unique information, $I_{unq}(X_1;Y \setminus X_2)$, and no synergy or redundancy. In correspondence with the PID terms, the CMI, $I(X_1;Y|X_2)$, decreased, following the decrease in synergistic and unique information contribution of $X_1$, while the multivariate MI, $I({X_1, X_2 }; Y)$, stayed approximately constant for all values of $a$. The PID framework is model-free and in particular insensitive to the absolute scale of the signals. Only the relative strengths of the input signals matter. This manifests itself in an approximate symmetry of the results where the calculated quantities are roughly the same under swapping $X_1$ and $X_2$ and letting $a\to\frac1a$, which can be observed in Fig.\ \ref{fig:stats_models_pid}A. The degree of how well this symmetry is realized also depends on the distributions assumed for the random variables.
For the multiplicative model (Equation \ref{eq:stats_model_mult}, Fig. \ref{fig:stats_models_pid}B), the behavior as function of $a$ is trivial, as all signals are zero for $a=0$, while the terms are constant for all values of $a > 0$. This is obvious as $a$ merely defines an overall scale of the signal, which is irrelevant in the PID framework. For finite $a$ unique information from $X_1$ and $X_2$ was low, while synergistic information was highest and there was some shared information between both inputs. Again, multivariate MI captured all, shared, redundant and unique contributions, while the CMI captured unique and synergistic contributions while conditioning out shared information.
In sum, both additive and multiplicative combination of input variables introduced synergistic and shared information contribution. This is in line with results by \cite{Griffith2014} who found that logical \texttt{XOR} leads to higher synergy than logical \texttt{AND}, which corresponds to addition and multiplication of Boolean variables, respectively. As expected, the relative unique and synergistic information contribution did not change for varying $a$
in the multiplicative model, while they did change in the additive model. In the latter, synergistic and shared information was highest for an equal contribution of both variables ($a=1$) and decreased for stronger coupling. The marginal distributions had no effect on the interaction of both variables in terms of individual and joint information contribution. The experiment illustrates the introduction of synergy and redundancy already through simple algebraic operations, where for $a=1$ synergy and redundancy was of similar magnitude in both operations.
\subsection{Experiment III - Friedman Models}
The Friedman models are three regression models initially described in \cite{Friedman1991} and frequently used to evaluate feature selection algorithms on regression problems, due to their synergistic interactions between input variables (e.g., \cite{Tsanas2010} or a simplified version of Friedman model I in \cite{Francois2007}). Note that we here analyzed all three Friedman models, while commonly only models I and II are used in the literature. For data generation, we used the implementation of the models in the scikit-learn Python toolbox \citep{scikit-learn}.
Friedman model I generates the output variable, $Y$, according to
\begin{equation*}
Y = 10 \sin(\pi X_1 X_2) + 20 (X_3 - 0.5)^2 + 10 X_4 + 5 X_5 + \sigma \mathcal{N}(0, 1),
\label{eq:friedman_1}
\end{equation*}
\noindent where input variables $X_i$ are independently sampled from a uniform distribution over the interval $[0,1]$.
For Friedman models II and III we use,
\begin{align*}
Y &= \sqrt{X_1^2 + (X_2 X_3 - 1 / X_2 X_4)^2}
+ \eta_2 \textrm{, with } \eta_2 \sim \mathcal{N}(0,1), \\
Y &= \arctan\left(\frac{X_2 X_3 - 1 / (X_2 X_4)}{X_1}\right)
+ \eta_3 \text{, with } \eta_3 \sim \mathcal{N}(0,1),
\label{eq:friedman_2_3}
\end{align*}
respectively, where the inputs, $X_i$, are uniformly sampled from the intervals
\begin{align*}
0 \leq & X_1 \leq 100, \\
40 \,\pi \leq& X_2 \leq 560\,\pi, \\
0 \leq& X_3 \leq 1, \\
1 \leq& X_4 \leq 11. \\
\end{align*}
For all models, we added five uniformly-distributed nuisance variables, $Z_i$, that were independent of $Y$, to test for the algorithm's ability to discard irrelevant features. Again, we applied the proposed forward-selection algorithm using a CMI and MI criterion, respectively, and estimated selected MI, CMI, and PID terms as in previous experiments.
For model I, relevant features were reliably detected by the CMI criterion, while false positive rates were well below the critical $\alpha$-level (Fig. \ref{fig:friedman_rates_knn}A, Tab. \ref{tab:friedman_rates}). Using the MI criterion, true positive rates dropped mostly because variables $X_3$ and $X_5$ were not selected in many instances (Fig. \ref{fig:friedman_rates_knn}D), while false positive rates increased and where higher than the critical $\alpha$-level.
\begin{table}[ht]
\centering
\begin{tabular}{lrrrrrrrr}
\toprule
& \multicolumn{4}{c}{CMI Criterion} & \multicolumn{4}{c}{MI Criterion} \\
Model & TP & TN & FP & FN & TP & TN & FP & FN \\ \midrule
Friedman I & 100.0 & 98.8 & 1.2 & 0.0 & 81.0 & 99.2 & 0.8 & 18.4 \\
Friedman II & 50.5 & 99.8 & 0.2 & 49.5 & 56.8 & 99.2 & 0.8 & 43.3 \\
Friedman III & 3.5 & 100.0 & 0.0 & 96.5 & 3.3 & 99.5 & 0.5 & 96.8 \\
\end{tabular}
\caption{True versus selected features using the proposed forward-selection algorithm with a conditional mutual information (CMI) versus a mutual information (MI) inclusion criterion. Data were simulated according from Friedman models I-III. (TP: fraction of true positives, TN: true negatives, FP: false positives, FN: false negatives.)}
\label{tab:friedman_rates}
\end{table}
Results for models II and III showed lower true positive rates for both criteria. For model II the algorithm with the CMI criterion failed to select features $X_1$ and $X_4$ (Fig. \ref{fig:friedman_rates_knn}B), for model III almost no features were selected (Fig. \ref{fig:friedman_rates_knn}C, F). Notably, for model II, both criteria successfully selected $X_2$ and $X_3$, while the number of true positives for $X_1$ identified by the MI criterion was higher than the number for the CMI criterion. However, also the number of false positives increased (Fig. \ref{fig:friedman_rates_knn}E, F).
\begin{figure}[H]
\centering
\includegraphics{figures/friedman_bars_knn.pdf}
\caption{\textbf{Results Friedman models.}
Variables selected by the proposed forward-selection algorithm on data from the three Friedman models using either the mutual information (MI) or the conditional mutual information (CMI) as inclusion criterion (mean over 100 simulation runs). Blue bars indicate correctly included true features (TP: true positives), orange bars indicate erroneously included spurious features (FP: false positives).
\textbf{A} CMI criterion, Friedman model I;
\textbf{B} CMI criterion, Friedman model II;
\textbf{C} CMI criterion, Friedman model III;
\textbf{D} MI criterion, Friedman model I;
\textbf{E} MI criterion, Friedman model II;
\textbf{F} MI criterion, Friedman model III.
Mean absolute error (MAE) in $k$-nearest neighbor prediction of $Y$ from all possible variable subsets of sizes $|\mathbf{S}|$, light and dark orange markers indicate feature sets selected using the MI and CMI criterion respectively.
\textbf{G} MAE for Friedman model I;
\textbf{H} MAE for Friedman model II;
\textbf{I} MAE for Friedman model III (here, no feature set was reliably
identified by either criterion).
}
\label{fig:friedman_rates_knn}
\end{figure}
We evaluated the selected feature sets by testing their performance in predicting the target variable $Y$ using a $k$-nearest neighbor (KNN) predictor with $k=1$. We compared the performance of the KNN predictor using the selected feature sets as input against the performance for each possible subset of the complete feature set $\bf{X}$, i.e.\ each element of the power set of $\bf X$, using the mean absolute error (MAE) on \SI{30}{\percent} of samples as test set.
For models I and II, the feature set selected by the CMI criterion was the set with the best performance of all possible subsets. While for model I all features were necessary to achieve the best performance, interestingly, for model II the best performance was achieved using the set identified by the CMI criterion which contained only two of the simulated input features, $\{X_2,X_3\}$ (Fig. \ref{fig:friedman_rates_knn}G, H). For model III, we show the predictive performance of all feature sets in the power set, but none of those sets was reliably detected by either criterion. Generally, their performance on model III was rather bad as no set allowed for a prediction with an error comparable to the other two models (Fig. \ref{fig:friedman_rates_knn}I). We conclude that the full feature set may not always be optimal in predicting the dependent variable, as shown for model II, and that for model III a prediction using a simple KNN-model was not successful for any possible feature set.
We estimated the MI, CMI, and PID atoms between individual variables $X_i$, and the target $Y$, and the remaining feature set $\mathbf{X}\setminus X_i$, respectively (Fig. \ref{fig:friedman_validation_pid}). For model I, we found high synergy between each variable and the remaining variable set with respect to the target (Fig. \ref{fig:friedman_validation_pid}D), explaining the higher CMI compared to the MI (Fig. \ref{fig:friedman_validation_pid}A) and accordingly a more successful feature selection when using the CMI criterion. Similar results were observed for model II (Fig. \ref{fig:friedman_validation_pid}B, E). Note that features with high synergy also carried redundant information with the remaining feature set, leading to a higher MI. For model III, the total information carried by individual features, either individually or in the context of the remaining feature set, was below the significance threshold for most instances (Fig. \ref{fig:friedman_validation_pid}C). Here, information was almost exclusively synergistic (Fig. \ref{fig:friedman_validation_pid}F), explaining the failure of both forward-selection algorithms in selecting these features.
\begin{figure}[ht]
\centering
\includegraphics{figures/friedman_validation.pdf}
\caption{\textbf{Mutual information and partial information decomposition estimates for Friedman models.}
Mutual information (MI), $I(X_i; Y)$, and conditional mutual information (CMI), $I(X_i; Y|\{\mathbf{X}\setminus X_i\})$, between features, $X_i \in \mathbf{X}$, and target, $Y$, for Friedman model I (\textbf{A}), model II (\textbf{B}), and model III (\textbf{C}) (error bars indicate the standard error of the mean (SEM) for 100 runs). Dashed lines indicate mean significance threshold ($\alpha=0.05$, $n=200$ permutations).
partial information decomposition (PID) atoms between individual features $X_i$, the remaining feature set
$\mathbf{X}\setminus X_i$ and target $Y$, for Friedman model I (\textbf{D}), model II (\textbf{E}), and model III (\textbf{F}).
}
\label{fig:friedman_validation_pid}
\end{figure}
\subsection{Experiment IV - Model by Runge et al. (2015)}
Next, we generated data using a model proposed by \cite{Runge2015}, which was designed to generate both synergistic and redundant contributions between time series. For algorithms not accounting for interactions between features, the former property leads to failure to include relevant features, while the latter property leads to the erroneous inclusion of redundant features. The authors show that for this case, a simple forward-selection using a pure CMI-criterion failed (using an algorithm proposed by \cite{Tsimpiris2012}).
The model comprises nine input time series, $\mathbf{Z} = \bigcup_{i=1}^3 Z_t^{(i)}$, $\mathbf{W} = \bigcup_{i=1}^4 W_t^{(i)}$, and $\mathbf{X} = \bigcup_{i=1}^2 X_t^{(i)}$, with time indices $t\in\{1,\ldots,T\}$, and one target time series, $Y_t$,
\begin{align*}
\begin{split}
Y_{t+1} &= c \sum_{i=1}^4 W^{(i)}_{t-1}
+ b \prod_{i=1}^3 Z^{(i)}_{t-1}
+ \sigma\, \mathcal{N}(0,1), \\
X^{(1)}_t &= a \left( W^{(1)}_{t-1} + W^{(3)}_{t-1} \right)
+ \mathcal{N}(0,1), \\
X^{(2)}_t &= a \left( W^{(2)}_{t-1} + W^{(4)}_{t-1} \right)
+ \mathcal{N}(0,1),
\end{split}
\end{align*}
\noindent where $\mathbf{W}$ and $\mathbf{Z}$ are i.i.d.\ sampled from Gaussian distributions with zero mean and unit variance ($\sim \mathcal{N}(0,1)$), and the coupling strength between variables is defined by parameters $a = 0.4$, $b = 2$, and $c = 0.4$. The noise in $Y_t$ is scaled by $\sigma = 0.5$.
The three groups of variables, $\mathbf{Z}$, $\mathbf{W}$, and $\mathbf{X}$, are designed to each contribute to $Y_t$ in a different fashion: sets $\mathbf{W}$ and $\mathbf{Z}$ are considered true \enquote{drivers} of the target, $Y_t$, where the sets of variables at time step $t$ determine the target two time steps later, $Y_{t+2}$. Also, variables in $\mathbf{Z}$ are considered stronger drivers than those in $\mathbf{W}$ due to a higher coupling, $b=2$, compared to $c=0.4$, and $\mathbf{Z}$ is considered to provide mostly synergistic information due to the multiplication of the comprising variables. Set $\mathbf{X}$, on the other hand, is considered to contain redundant variables that carry information already provided by $\mathbf{W}$, as $\mathbf{X}_t$ is determined by the set $\mathbf W$ only one time step earlier, i.e.\ $\mathbf W_{t-1}$. Thus, information observed in $Y_t$ was previously first observed in $\mathbf W_{t-2}$ and then also in $\mathbf X_{t-1}$, which justifies the variables $\mathbf X$ to be considered redundant.
Approaches not taking into account synergies between variables may fail to include variables in $\mathbf{Z}$. Furthermore, the authors hypothesized that synergistic drivers, $Z_t^{(i)}$, were individually less predictive than variables $W_t^{(i)}$, and that variables $Z_t^{(i)}$ provided information only when considered jointly. Lastly, approaches not accounting for redundancies between variables may erroneously select variables from $\mathbf{X}$ additionally to variables $\mathbf{W}$.
We found that the proposed algorithm using the CMI criterion detected true features with great reliability (Fig. \ref{fig:runge_results_validation}A and B, Tab. \ref{tab:runge_rates}), while for the MI criterion, false positive and false negative rates increased. In particular, more spurious features, $\mathbf{X}$, were selected, while true features were selected less often, in particular weakly predictive variables in $\mathbf{W}$.
\begin{figure}[ht]
\centering
\includegraphics[]{figures/Runge2015_results_validation.pdf}
\caption{\textbf{Results for model by \protect\cite{Runge2015}.} Number of included variables over 100 simulations using CMI and MI inclusion criteria and the proposed forward-selection algorithm on data from the model by \protect\cite{Runge2015}. Blue bars indicate correctly included true features (TP: true positives), orange bars indicate erroneously included spurious features (FP: false positives).
\textbf{A} CMI criterion;
\textbf{B} MI criterion;
\textbf{C} Mutual information between individual variables and $Y$;
\textbf{D} Information contribution of individual variables, $Z_i$, versus
set $\mathbf{Z}$;
\textbf{E} Partial information decomposition for variable sets $\mathbf{Z}$ and
$\mathbf{W}$;
\textbf{F} Order of inclusion of variables using a CMI and an MI criterion.
}
\label{fig:runge_results_validation}
\end{figure}
\begin{table}[ht]
\centering
\begin{tabular}{lrrrr}
\toprule
Criterion & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{TN} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{FN} \\ \midrule
CMI & 99.6 & 97.0 & 3.0 & 0.4 \\
MI & 54.7 & 92.5 & 7.5 & 45.3 \\
\end{tabular}
\caption{Actual versus selected features by the conditional mutual information (CMI) versus the
mutual information (MI) criterion on data simulated according to \protect\cite{Runge2015}.}
\label{tab:runge_rates}
\end{table}
Our results partially contradict the hypothesized behavior of forward-selection algorithms on the generated data, for example, the hypothesis that simple MI selection criteria should fail to identify features in $\mathbf{Z}$, because they do not account for the assumed synergistic information contribution. Here, further investigation of the simulated data showed that variables in $\mathbf{Z}$ provided individually more information about $Y$ than variables in $\mathbf{W}$ and $\mathbf{X}$, with $\mathbf{X}$ providing the least information (Fig. \ref{fig:runge_results_validation}C). We found that already individual variables, $Z_t^{(i)}$, provided significant information about $Y_t$, quantified as $I(Y_t;Z^{(i)})$ (Fig. \ref{fig:runge_results_validation}D). This contribution increased if the the whole set $\mathbf{Z}$ was considered, as quantified by the CMI, $I(Z^{(1)};Y_t|\{Z^{(2)}, Z^{(3)}\})$ (Fig. \ref{fig:runge_results_validation}D). Also, the synergistic contribution by $\mathbf{Z}$ was comparable to the synergistic contribution of $\mathbf{W}$ (\ref{fig:runge_results_validation}F). Here, the multiplication of variables did not introduce a stronger synergistic effects compared to the additive combination of variables (see also Experiment II and results by \cite{Griffith2014}).
Furthermore, the authors hypothesized that spurious drivers, $X^{(i)}$, should individually provide more information about $Y_t$ than variables $W^{(i)}$ and $Z^{(i)}$, and that this should be reflected in the order of inclusion during forward-selection, which was expected to be $X^{(i)} \rightarrow Z^{(i)} \rightarrow W^{(i)}$ for a CMI criterion, and $X^{(i)} \rightarrow W^{(i)} \rightarrow Z^{(i)}$ for an MI criterion. However, we found that the order of inclusion in our algorithm followed the order of the magnitude of individual information contribution $Z^{(i)} \rightarrow W^{(i)} \rightarrow X^{(i)}$ (Fig. \ref{fig:runge_results_validation}F), as determined by the coupling strength $b=2>c=0.4$. Accordingly, in the first three inclusion steps, variables $\mathbf{Z}$ were predominantly included by both the MI and CMI criterion, followed by variables $\mathbf{W}$ in the next steps. From step 4 on, the CMI identified more variables from $\mathbf{W}$ while the MI criterion failed to include these variables and instead included more spurious variables from $\mathbf{X}$.
\subsection{Experiment V - Noise Examples by MacKay (2005) and Haufe et al. (2014)}
Synergistic effects do not only occur in purely artificial examples, but also appear in simple real-world settings such as noisy measurements. We analyzed two examples from literature that illustrate the effect of recording noise during data collection \citep{MacKay2005,Haufe2014}. Both examples assume two input variables, $X_1$ and $X_2$, where variable $X_1$ is the measurement of a signal of interest, $S$, that affects the output $Y$ by some function $f(\cdot)$, while the second variable, $X_2$, measures a \enquote{distractor signal}, $D$, that is considered to be noise. A weighting factor, $a \in [0, 4]$, determines the strength of the distortion. Both $S$ and $D$ are drawn from a standard normal distribution. We simulated $f(S)$ as a noisy sinus, i.e., $f(S) = \sin(S) + \sigma\, \mathcal{N}(0,1)$, with $\sigma=0.1$.
In the first model, formulated after \cite{MacKay2005}, we consider the measurement of the target variable to be corrupted by the distractor signal,
\begin{equation*}
\begin{aligned}
X_1 &= S \,, X_2 = D \\
Y &= f(S) + aD, \\
\end{aligned}
\label{eq:mackay_noise}
\end{equation*}
\noindent while in the second model \citep{Haufe2014}, we consider a noisy measurement of the input variable,
\begin{equation*}
\begin{aligned}
X_1 = S +& aD \,, X_2 = D \\
Y &= f(S). \\
\end{aligned}
\label{eq:haufe_noise}
\end{equation*}
For the MacKay example \citep{MacKay2005}, we found that the CMI, $I(X_1;Y|X_2)$, at all values of $a$ captured the unique influence of the signal of interest, $X_1$, on $Y$, and included the synergistic contribution of both $X_1$ and $X_2$ (Fig. \ref{fig:noise_models_pid}A). Simultaneously, the unique influence of the noisy signal, $X_2$, and the redundant information in $X_1$ and $X_2$ was conditioned out. The MI, as expected, failed to capture the relationship between the three measurements, where the information provided by $X_1$ alone, $I(X_1;Y)$, decreased for $a \geq 0.5$ and became smaller than the CMI, $I(X_1;Y|X_2)$. The MI thus failed to capture all information provided by the signal of interest $X_1$, which is partially \enquote{masked} as a synergistic contribution between $X_1$ and $X_2$. On the other hand, the information provided by $X_2$ alone, $I(X_2;Y)$, increased and for $a \geq 1.0$ became larger than the CMI, $I(X_1;Y|X_2)$, thus over-estimating the influence of the noisy signal on $Y$. Due to the addition, shared and synergistic information contribution decreased similarly for larger $a$, while the synergistic contribution was always higher than the redundancy between both variables. Lastly, the multivariate MI, $I(\{X_1, X_2\};Y)$, included both the information of $X_1$ and the noisy signal $X_2$, thus failing to decompose the distractor signal into its synergistic contribution required to decode the signal of interest, $X_1$, and its distorting unique and redundant contribution.
\begin{figure}[ht]
\centering
\includegraphics[width=0.99\linewidth]{figures/Noise_models_pid.pdf}
\caption{\textbf{Results noise examples.} Mutual information (MI) and
partial information decomposition (PID) terms for noise examples.
\textbf{A} Model as described in \protect\cite{MacKay2005}; \textbf{B}
Model as described in \protect\cite{Haufe2014}. Note that the joint MI,
$I(Y;\{X_1,X_2\})$, is covered by the CMI, $I(Y;X_1|X_2)$, and the MI,
$I(Y;X_1)$, is covered by the unique information,
$I_{unq}(Y;X_1 \setminus X_2)$.}
\label{fig:noise_models_pid}
\end{figure}
For the second example by \cite{Haufe2014}, we found again that the CMI, $I(X_1;Y|X_2)$, successfully captured the information provided by both variables, while the MI between each variable and $Y$ failed to account for the synergistic contribution of both variables (Fig. \ref{fig:noise_models_pid}B). In this example, also the multivariate MI, $I(\{X_1, X_2\};Y)$, was able to capture the joint contribution of both measurements due to an absence of redundant information between both variables. Here the contribution is purely synergistic, such that no redundancies have to be conditioned out from the measurement.
Both examples show that we are able to recover a signal of interest also in the presence of high noise if we are able to record the source of the noise. In this case, the recorded noise, $X_2$, may be used to \enquote{decode} the information the signal of interest, $X_1$ carries about the target, $Y$. As already stated by \cite{Haufe2014}, this is important in the interpretation of subsequent modeling. The relevancy of a variable may be indirect such that the weight attributed to the variable in a model may not encode its immediate relevancy to the target quantity, but may indicate its strength in moderating the relationship between the input and output signal. An application example may be the recording of physiological signals \citep{Haufe2014}, where the signal of interest is electrophysiological brain activity, while the distractor signal is the electrocardiogram (ECG). The target variable may be a further brain signal, but also behavioral measurements, e.g., the success in a task. To model the relationship between the electrophsyiological recording and the target quantity, accounting for the noise generated by the ECG is beneficial in both scenarios of noise contamination.
\subsection{Experiment VI - Examples by Guyon \& Elisseeff (2003) and Haufe et al. (2014)}
As a last example, we analyzed data from two models that investigate whether correlation (more generally, dependency) calculated exclusively between input variables indicates redundancy or synergistic contributions with respect to a target variable, following toy examples in \cite{Guyon2003} and \cite{Haufe2014}. It has been stated that the goal of feature selection is the identification of \enquote{independent} features such as to include features that provide a maximum of novel information about the target variable and to avoid redundant features. Accordingly, some approaches use the MI between features as a proxy for redundant information \textit{about} the target.
However, \cite{Guyon2003} previously demonstrated on toy systems that correlation does not indicate feature redundancy and that, on the other hand, the absence of correlation does not indicate an absence of interaction, for example, in the form of synergistic contributions. We here illustrate this by estimating MI, CMI, and PID from similar toy models, where we show that statistical dependence or independence \textit{between} features is not sufficient to identify feature interactions that render features relevant or irrelevant \textit{with respect to the target}.
For all examples, data were independently sampled from two two-dimensional normal distributions with means $\bm{\mu}_1$ and $\bm{\mu}_2$ and common covariance matrices, $\Sigma_{\mathbf {X}, \mathbf {X}}$. Class labels, $Y\in\{0, 1\}$, indicate from which distribution a sample was drawn. As input variables, we considered the two dimensions, $X_1$ and $X_2$.
Examples 1 and 2 illustrate that two seemingly non-interacting variables may provide synergistic information. We simulate two sets of independent variables with $\bm{\mu}_1=(-1,-1)^T$ and $\bm{\mu}_2=(1, 1)^T$, respectively. The examples vary in their covariance matrices to illustrate that whether variables may provide synergistic information is dependent on their variance,
\begin{equation*}
\Sigma^1_{\mathbf {X}, \mathbf {X}}=
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix},
\Sigma^2_{\mathbf {X}, \mathbf {X}}=
\begin{pmatrix}
1 & 0 \\
0 & 2 \\
\end{pmatrix}.
\end{equation*}
Example 3 illustrates that only if variables are perfectly correlated, they exclusively provide redundant information. For this we sample the two distributions with means $\bm{\mu}_1=(0, 0)^T$ and $\bm{\mu}_2=(0,0)^T$), respectively, and
\begin{equation*}
\Sigma^3_{\mathbf {X}, \mathbf {X}}=
\begin{pmatrix}
1 & 1 \\
1 & 1 \\
\end{pmatrix}.
\end{equation*}
Example 4 illustrates that high correlation does not imply an absence of synergistic information. We simulate the two sets of variables with means $\bm{\mu}_1=(-0.5, 0.5)^T$ and $\bm{\mu}_2=(0.5, 0.5)^T$, and
\begin{equation*}
\Sigma^4_{\mathbf {X}, \mathbf {X}}=
\begin{pmatrix}
0.1 & 0.4 \\
0.4 & 2 \\
\end{pmatrix}.
\end{equation*}
Example 5 illustrates that high correlation does not imply an absence of synergistic information and that even a variable that does not provide any information by itself may provide synergistic information with another feature. Means are set to $\bm{\mu}_1=(-0.5, 0)^T$ and $\bm{\mu}_2=(0.5, 0)^T$, and
\begin{equation*}
\Sigma^5_{\mathbf {X}, \mathbf {X}}=
\begin{pmatrix}
0.1 & -0.4 \\
-0.4 & 2 \\
\end{pmatrix}.
\end{equation*}
Figure \ref{fig:guyon2003_results} shows representative samples from all five example distributions (Fig. \ref{fig:guyon2003_results}A), as well as the Pearson correlation coefficient between the inputs for each example (Fig. \ref{fig:guyon2003_results}F). Examples 1 and 2 illustrate that seemingly independent variables may provide synergistic information. However, only in Example 2, where the variance of $X_2$ was higher, a significant information contribution as measured by the CMI criterion was consistently found. In Example 1, the added unique and synergistic contribution reached statistical significance for only some instances. Here, the higher relative redundancy may have rendered $X_1$ not informative given $X_2$ as measured by the CMI criterion. While the addition of $X_1$ may add some improvement in a subsequent learning task, its inclusion also adds a high amount of redundant information, and may have to be balanced against the minimization of the feature set size. In both Examples 1 and 2, the pairwise as well as the joint MI was significant in most cases. We conclude that---in line with \cite{Guyon2003}---\enquote{presumably independent variables} may contribute information through a synergistic interaction, which becomes particularly evident in Example 2.
\begin{figure}[h!]
\centering
\includegraphics[width=0.92\textwidth]{figures/Guyon2003_pid.pdf}
\caption{\textbf{Results for models from \protect\cite{Guyon2003}.}
\textbf{A} Data from exemplary run for Examples 1 to 5 (left to right),
color indicates class label.
\textbf{B} Class-conditional distributions of variable $X_1$.
\textbf{C} Class-conditional distributions of variable $X_2$.
\textbf{D} Estimated mutual information (MI), conditional mutual information (CMI) and joint mutual information (average over 20 runs, dashed lines indicate mean significance threshold, error bars indicate the standard error of the mean).
\textbf{E} Estimated partial information decomposition (PID) atoms.
\textbf{F} Pearson correlation between input variables.
}
\label{fig:guyon2003_results}
\end{figure}
For Example 3, as hypothesized for perfectly correlated input variables, we found exclusively redundant information in the two input variables, with respect to the target, and no unique or synergistic information contribution. Accordingly, the CMI, $I(Y;X_1|X_2)$, was zero as $X_1$ provided no information about $Y$ if $X_2$ was already known.
Lastly, for Examples 4 and 5, we found in both cases that despite a high correlation between the inputs, there was little redundancy and even some synergistic information contribution between both inputs. The main contribution was provided uniquely by $X_1$. Note that even though the correlation between variables in Example 4 was higher than for Examples 1 and 2, the redundancy was lower.
In sum, through the estimation of the PID, we showed that presumably independent variables may still provide synergistic information about the target, while a dependence between variables is not necessarily indicative of variable redundancy or the absence of synergy. We conclude that measuring the dependence between features is not a suitable proxy to infer feature interactions in general. Our PID-based results are in accordance with the observations by \cite{Guyon2003}, providing a quantitative description of the relationship between variable dependence and relevance.
\section{Discussion}
We use the partial information decomposition (PID) framework by \cite{Williams2010} to provide a rigorous, information-theoretic definition of relevancy and redundancy in feature-selection problems that, in particular, accounts for feature interactions and the resulting redundant and synergistic information contributions about a target. We show that classical information theory lacks methods to describe such detailed information contributions and that only through the introduction of the PID framework these methods have become available. Based on our definition of feature relevancy and redundancy in PID terms, we show that using the conditional mutual information (CMI) as a feature selection criterion, e.g., in sequential forward selection, simultaneously maximizes a feature set's relevancy while minimizing redundancy between features. For practical application, we propose the use of a recently introduced CMI-based forward-selection algorithm that uses a novel approach to statistical testing to handle practical problems in the estimation of information-theoretic quantities from data. By applying the proposed algorithm to benchmark systems from literature, we demonstrate that the CMI is in almost all cases the preferable selection criterion, compared to the mutual information (MI) which does not account for feature interactions. Additionally, we demonstrate how the estimation of PID can provide detailed and quantitative descriptions of information contribution in the case of feature interactions---highlighting how applying PID in information-theoretic feature selection, both conceptually as well as practically, allows for a detailed investigation of individual and joint information contribution in feature selection problems.
\subsection{PID Enables the Detailed Quantification of Joint Information Contribution for Interacting Features}
We provide a definition of relevancy and redundancy using the PID framework \citep{Williams2010}, which captures the intuitive notion of a feature's relevancy in the context of other relevant features. In particular, we define relevancy as a feature's unique and synergistic information contribution in the context of the remaining feature set, and we define a feature's redundancy as the shared information between the feature and the remaining set. Basing the definition of variable relevancy and redundancy on the PID framework provides a first, rigorous definition of these concepts in information-theoretic terms and makes individual contributions quantifiable using estimators for PID. Our definition further resolves existing issues in the description of multivariate information contribution such as negative information contribution and allows to make conjectures about variable contributions testable (as formulated, for example, by \cite{Guyon2003} or \cite{Brown2012}). As demonstrated, similar insights into the \enquote{structure} of the multivariate information contribution in feature selection can not be obtained from using classic information-theoretic terms, but require the introduction of additional axioms as done by the PID framework.
By applying the PID framework to benchmark data sets from literature, we tested a series of earlier conjectures and assumptions about feature interactions. We showed that multiplicative combinations of terms lead to higher synergistic contribution than additive ones, opposed to assumptions made in earlier work. We further showed that determining the dependency \textit{between} features is not sufficient to determiner feature relevancy or redundancy with respect to the target variable (as also discussed, for example, by \cite{Guyon2003}). In particular, we showed that the absence of correlation did not indicate a lack in synergistic contribution and thus a lack in relevancy, furthermore, we showed that the presence of correlation did not indicate feature redundancy. Instead of using variable dependency as a proxy to infer such interactions, PID allowed us to immediately quantify these individual and joint contributions \textit{about} the target. Furthermore, we show that both synergistic and redundant contributions can coexist already in simple feature selection problems, such that whether a variable is included in a feature set, hinges on whether its contribution of \textit{novel} information outweighs the information it redundantly shares with other features. Also, all three unique, synergistic, and redundant contributions can occur in the data simultaneously and with arbitrary shares.
An important limitation of using PID to understand feature selection problems, is the number of variables that can be considered simultaneously. When looking at fine-grained interactions between increasingly larger data sets, the number of possible interactions soon becomes intractable as also higher-order interactions between subsets become possible. For example, when considering four input variables that interact with a single target variable, there are already 166 possible ways of how subsets of variables can multivariately provide information about the target \citep{Makkeh2021}. Instead, PID can always be used to quantify the information contribution of a single feature with respect to the \textit{set} of all remaining selected features, as done in the present work. Here, however, the dimensionality of the estimation space is a further limiting factor, restricting the application of existing PID estimators to feature sets of relatively small size.
We conclude that PID enables a better understanding of multivariate feature selection problems by providing the tools for an unambiguous description and quantification of the structure of information contribution with respect to a target variable \citep{Williams2010,Reing2018}. Our results are complementary to existing characterizations of information contribution in feature selection problems, which rely on subjective and qualitative descriptions of feature interactions and the information features jointly provide about a target. Here, PID is a promising framework to answer central questions in feature selection by introducing mathematical definitions of previously missing information-theoretic quantities.
\subsection{A Remark on PID for More Than Two Input Variables}
We note that we have restricted our application of PID here to the case of two input variables---where one variable represented a particular feature while the other represented all feature selected up to a certain point in an iterative algorithm, or all other available features. Our self-restriction was motivated by our focus on the iterative inclusion scenario, which is most relevant practically. However, PID can also be defined for more than two input variables that carry information about an output variable \citep{Williams2010,Makkeh2021,Gutknecht2020}. In that case we have to ask questions of a more complex type, such as: \enquote{How much of the information that a set of variables $\{X_1, X_2\}$ carries jointly (i.e. also synergistically) about a target variable $Y$ is also carried redundantly with the information that another set of variables, say $\{X_3, X_4, X_5\}$ carries about $Y$?}. The structure of the underlying decomposition can be derived solely relying on what it means for one thing to be part of another (i.e. parthood relations or mereology) and is isomorphic to certain well-known lattices, such as the lattice of antichains and lattices from Boolean logic (see \cite{Gutknecht2020} for an accessible introduction).
For limited data, the fine-grained decompositions provided by a fully multivariate information decomposition may be impractical, as the number of parts or \enquote{atoms} of the decomposition scales extremely fast with the number of input variables (here, features): for $n$ input variables the number of parts is equal to $D(n)-2$, where $D(n)$ is the $n$-th Dedekind number. For example, for $n=2,3,4,5,\ldots$ inputs the corresponding number of parts are $4, 18, 166, 7579,\ldots$, respectively. While this rapid scaling is frustrating from a practical point of view, it at least offers a clean quantitative measure of the inherent (and unavoidable) complexity of the feature selection problem; in other words, the problem is certainly \NP-hard, but still the Dedekind numbers rise much slower than the cardinality of the power set of all features, which would have to be considered when approaching the problem naively.
\subsection{Relationship to Definitions of \enquote{Strong and Weak Relevancy}}
In the context of information-theoretic feature selection, feature relevancy has been mathematically defined in probabilistic as
well as information-theoretic terms (e.g., \cite{John1994,Bell2000}, see also \cite{Vergara2014} for a review). It has been recognized that common \enquote{definitions give unexpected results, and that the dichotomy of relevancy vs irrelevance is not enough} \citep{Bell2000}. Therefore, \cite{John1994} introduced the concept of \emph{strong and weak relevancy} to mitigate these problems, in particular to avoid results contradicting the intuitive notion of relevancy in the presence of fully redundant features. The definition by John et al. can be written in information-theoretic terms \citep{Vergara2014}, such that for a set of input variables $\mathbf{X}$ and a dependent variable $Y$, a feature, $F_i$, from the set of all true features, $\mathbf{S}$ is defined as \textit{strongly relevant} if
\begin{equation}
I(F_i; Y | \mathbf{S} \setminus F_i) > 0,
\label{eq:strongly_relevant}
\end{equation}
\noindent where $\mathbf{S} \setminus F_i$ the set of selected features excluding $F_i$. A strongly relevant feature provides information about $Y$ that can not be obtained from any other feature. From a PID perspective, we can see that this criterion ensures that $F_i$ provides either unique or synergistic information, or both, which leads to a positive CMI.
Next, a variable is defined to be \textit{weakly relevant} if
\begin{equation*}
\begin{aligned}
I(F_i; Y | \mathbf{S} \setminus F_i) &= 0 \;\; \land \\
\exists \mathbf{S}' &\subseteq \mathbf{S}\setminus F_i:\, I(F_i; Y | \mathbf{S}') > 0
\label{eq:weakly_relevant}
\end{aligned}
\end{equation*}
\noindent i.e., the feature is not strongly relevant and there exists a subset of features, $\mathbf{S}'$, in whose context $F_i$ provides information about $Y$. Thus, weakly relevant features \enquote{can sometimes contribute} information in the context of a suitably selected subset, $\mathbf{S}'$ \citep{Bell2000}. Again, using PID we can state that a weakly relevant feature provides information that is fully redundant with the full feature set, or more precisely, with a subset of the feature set, $\mathbf{S}\setminus \mathbf{S}'$. Note that all variables that are perfect copies of another feature are fully redundant with respect to $Y$ and provide neither unique nor synergistic information. On the other hand, not all features that provide purely redundant information about a target are necessarily perfect copies of each other. It is also conceivable that two variables provide completely redundant information about the target and carry further information that is independent of the target.
Lastly, a feature is considered irrelevant if
\begin{equation*}
I(F_i;Y|\mathbf{S}') = 0 \;\; \forall \, \mathbf{S}' \subseteq \mathbf{S} \setminus F_i,
\label{eq:irrelevant}
\end{equation*}
\noindent i.e., the feature never provides information about $Y$ and can be discarded. Again, from a PID point of view, a feature is not included if it provides information neither uniquely of synergistically in the context of the remaining feature set and all possible subsets, $\mathbf{S}'$.
Based on this definition, a suitable feature selection algorithm should select all strongly relevant features, the smallest possible subset of weakly relevant features, while not including any irrelevant features \citep{John1994}. Here, one has to distinguish between the true feature set, i.e., all features that are mechanistically coupled to $Y$, and the optimal feature set, i.e., the set of features that provides maximum information about $Y$ while having minimal cardinality. Typically, feature selection aims at identifying the latter. Note further that the definition of weak and strong relevancy is not directly applicable in practice because it requires the evaluation of the power set of $\mathbf{S}$.
Our definition of feature relevancy and the proposed forward-selection algorithm using a CMI criterion are in line with the definitions by \cite{John1994} and the resulting requirements for feature selection algorithms. First, our algorithm will identify all strongly relevant features due to the use of the CMI criterion, given that a sufficient amount of data is available and that two or more features do not exclusively provide synergistic information (see also Section \ref{sec:algo_comparison} below). Second, the algorithm will identify the smallest possible subset of weakly relevant features, due to the sequential inclusion. Here, at least one of two or more weakly relevant features will be selected, while the remaining redundant features will not. The single backwards-elimination step at the end of feature selection tests once more for redundant features in the context of the optimized set. Third, irrelevant features will not be included, which is ensured in practice by statistically testing the information contribution of individual features against surrogate data.
Last, it should be noted that when investigating strongly and weakly relevant features using PID measures, the choice of the measure used is important. In particular, the measure should be sensitive to \enquote{what} over \enquote{how} much information is provided (see for example, \cite{Harder2013,Timme2018}). For example, the $I_{min}$ measure proposed in the original work by \cite{Williams2010} returns counter-intuitive results for variables that are perfect copies \citep{Harder2013,Bertschinger2014,Timme2018}. Note that the BROJA measure used here does not suffer from this problem. Nevertheless, we conclude that PID may provide a more straightforward and more intuitive characterization of information contribution than weak and strong relevancy.
\subsection{Relationship to Feature-Selection Framework by Brown et al. (2012)}
Using the proposed definition of relevancy and redundancy, we can make more informed statements about the type of interactions accounted for by existing information-theoretic feature selection criteria. To this end, we consider a framework development by \cite{Brown2012}, which subsumes many of these criteria under a common formulation. The authors arrive at this formulation by first showing that the CMI is the optimal criterion for variable inclusion in sequential forward-selection, when considering filtering features prior to fitting a classifier as the problem of maximizing the conditional likelihood of the correct class labels given the selected features. After showing the optimality of the CMI, the authors show that most existing inclusion criteria can be related to the CMI under a series of independence assumptions, where independence assumptions are typically made to avoid the estimation of the full CMI in potentially high-dimensional feature spaces.
In particular, \cite{Brown2012} develop their framework by first assuming independence and class-conditional independence for all non-selected variables $X_k \in \mathbf{X}_i$ at inclusion step $i$ (Equation \ref{eq:inclusion_update}), i.e., for all $X_k \in \mathbf{X}_i$,
\begin{equation}
\begin{aligned}
p(\mathbf{S}_i|X_k) &= \prod_{F_j \in \mathbf{S}_i} p(F_j|X_k), \\
p(\mathbf{S}_i|X_k,Y) &= \prod_{F_j \in \mathbf{S}_i} p(F_j|X_k,Y).
\label{eq:assumption_1}
\end{aligned}
\end{equation}
The CMI criterion in Equation \eqref{eq:cmi_criterion} can be rewritten as (see \cite{Brown2012} for a detailed derivation),
\begin{equation*}
I(Y; F_i|\mathbf{S_i}) = I(Y; F_i)
- \sum_{F_j \in \mathbf{S}_i} I(F_i; F_j)
+ \sum_{F_j \in \mathbf{S}_i} I(F_i; F_j|Y).
\end{equation*}
\noindent Here, the second term can be discarded under the assumption of
pairwise independent variables, i.e., $p(F_i; F_j) = p(F_i)p(F_j)$. The third term can be discarded under the assumption that all features are pairwise
class-conditionally independent, i.e., $p(F_i; F_j|Y) = p(F_i|Y)p(F_j|Y)$. Furthermore, both terms can enter the equation to varying degrees, expressing a
corresponding degree of belief in each assumption, yielding the
formulation
\begin{equation}
J^\prime_{CMI}(F_i) = I(Y; F_i)
- \beta \sum_{F_j \in \mathbf{S}_i} I(F_i; F_j)
+ \gamma \sum_{F_j \in \mathbf{S}_i} I(F_i; F_j|Y),
\label{eq:brown_framework}
\end{equation}
\noindent which describes a generic inclusion criterion, $J^\prime_{CMI}$, derived from the CMI under the three assumptions above. Here, lower values for either parameter lessens the contribution of the respective term, indicating that the corresponding independence-assumption is adopted to a higher degree. Many existing information-theoretic criteria may be expressed by Equation \eqref{eq:brown_framework} through the adoption of specific values for $\gamma$ and $\beta$ (e.g., \cite{Battiti1994,Peng2005mrmr,Yang1999JMI, Kwak2002,Lin2006cife}, see also \cite{Brown2009}).
Several observations can be made using the framework by \cite{Brown2012}:
First, the assumption made in Equation \eqref{eq:assumption_1} discards all higher-order interactions between features. Hence, all criteria that can be subsumed under this framework only account for pairwise interactions between features.
Second, the second term in Equation \eqref{eq:brown_framework} ($\sim\beta$) leads to a decrease in a feature's relevancy whenever that feature has high MI about any feature already selected. Thus, these criteria use the pairwise MI between the feature under consideration and already selected features to quantify feature redundancy, an assumption that---as already discussed---may not generally hold. Moreover, the MI between $F_i$ and $F_j$ is in general independent of the redundancy the two variables carry about the target, such that positive MI does not necessarily indicate redundant information in the features about the target, nor does an MI of zero indicate an absence of redundant information. The first statement may be illustrated with an example, where each feature, $F_i$ and $F_j$, is a combination of two further variables, $F_i=\left\{X_1, X_2\right\}$, $F_j=\left\{Z_1, Z_2\right\}$, which are not observed and thus not part of the input variable set. Here, $I(F_i;F_j) > 0$ may be due to $I(X_1;Z_1)>0$, while the information $F_i$ and $F_j$ provide about $Y$ is due to $I(X_2;Y)>0$ and $I(Z_2;Y)>0$, while $I_{shd}(Y;\{F_i,F_j\})=0$. In this scenario, the MI, $I(F_i;F_j)$, is non-zero while the information provided by both variables about $Y$ is not redundant. The second statement, i.e., zero MI does not indicate the absence of redundancy, may be illustrated by a simple logical AND gate with two independent, Boolean inputs, where the BROJA measure used here returns $I_{shd}=0.311$ \citep{Bertschinger2014}. This type of redundant information contribution has also been termed \textit{mechanistic redundancy}, i.e., redundancy due to the operation performed on the input to generate the output, and has to be distinguished from redundancy that arises from information redundantly present in the input, termed \textit{source redundancy} \citep{Harder2013}. In the special case, where just source redundancy occurs in a system, the MI between the inputs is an upper bound on the redundancy if the PID measure used adheres to the identity axiom introduced by \cite{Harder2013}.
Third, criteria not including the third term in Equation \eqref{eq:brown_framework} (i.e.\ when $\gamma=0$) do not account for pairwise synergistic information between $F_i$ and $F_j$ with respect to $Y$. Hence, if this term is not included, the criterion may be overly conservative because it misses a feature's synergistic relevancy, which---as demonstrated in the experiments---is present already in simple systems.
\subsection{CMI as an Optimal Feature-Selection Criterion}
We demonstrated that already in simple examples, feature interactions can lead to relevant synergistic contributions or redundancy. Accordingly, the CMI criterion, which accounts for such interactions, outperformed the MI criterion in almost all investigated scenarios. Our results therefore highlight the importance of developing feature selection criteria that account for interactions between variables; assuming variable independence may easily yield non-optimal solutions that both miss relevant variables and falsely include redundant ones. Also for practical applications, it has been noted that already simple problems show interactions between features with respect to the target such that considering pairwise dependencies alone leads to an insufficient characterization of the system under investigation \citep{Reing2018}. Examples include data sets obtained from sufficiently complex, multivariate real-world systems, such as biological or artificial neural networks \citep{Latham2005synergy,Tax2017}, or ecological models \citep{Grilli2017}.
Yet, existing inclusion criteria for sequential forward-selection (e.g., \cite{Duch2006,Brown2012}) often fail to account for at least one of the two possible interactions---in particular, it has been stated that synergistic effects between features are often neglected (e.g., \cite{Zhao2009,Cheng2011,Vergara2014,Shishkin2016}. Instead, many criteria make the (implicit) modeling assumption that variables act on the target independently \citep{Dash1997,Hall2000}, or that variables interact only pairwise with respect to the target \citep{Brown2012}. These approaches aim at avoiding (C)MI estimation in high-dimensional spaces \citep{Brown2012,Vergara2014}. Also, approaches explicitly accounting for synergistic contributions often use low-dimensional approximations of the synergy that may miss relevant contributions \citep{Zhao2007,Tsanas2010,Cheng2011}. Here, PID may help to construct novel approaches as it allows to characterize also intermediate levels of dependency, thus covering the middle ground between pairwise and the total dependency in a set of variables \citep{Reing2018}.
\subsection{Choice of Estimators for CMI Estimation}
Existing feature selection approaches often refrain from using the CMI as an inclusion criterion to avoid its estimation in high-dimensional spaces and instead resort to low-dimensional approximations (e.g., \cite{Brown2012,Shishkin2016}). We want to highlight that using such approximations comes at the cost of potentially missing variable interactions. Instead, we here use a continuous nearest-neighbour based estimator that has been shown to perform well in relatively high-dimensional spaces \citep{Kraskov2004}, and that has favorable bias properties \citep{Kraskov2004,Khan2007,Lizier2014jidt,Xiong2017}, in particular compared to plug-in estimators for discrete or discretized variables (e.g., \cite{Doquire2012}), and has been successfully applied in feature selection \citep{Francois2007,Tsimpiris2012,Doquire2012,Lensen2018}. However, even though the estimator shows better bias properties than comparable approaches, is is not bias-free and still requires the statistical testing of estimates against suitable surrogate data (e.g., \cite{Francois2007}).
Even thought the Kraskov-estimator has been shown to be applicable to high-dimensional variable spaces \citep{Kraskov2004}, the increasing dimensionality of the feature set during forward-selection is still a limiting factor to the application of the CMI criterion. The statistical testing scheme used here has been shown to reliably terminate feature selection not only when the remaining variable set contains no more relevant features, but also if a potential feature's contribution can no longer be robustly estimated given the dimensionality of the already selected feature set and the available data---hence leading to errors that are statistically conservative \citep{Novelli2019}. Yet, our approach using the CMI criterion may fail to identify all relevant features for problems with a high number of input variables and a high number of relevant features. A possibility to reduce the curse of dimensionality when estimating CMI limit the estimation to subsets of the already selected feature set \citep{Fleuret2004}.
Note that the algorithm used here supports also other estimators, e.g., plug-in estimators, and is thus also applicable to discrete input data \citep{Wollstadt2019}. For such discrete estimators additional bias-correction procedures beyond statistical testing exist, such as, analytical approaches \citep{Panzeri1996} or novel, Bayesian estimation techniques that claim superior bias properties under unfavorable sampling regimes \citep{Nemenman2002,Nemenman2004}. A current limitation of the algorithm is that it is limited to homogeneous data sets and can not be applied to mixed data types. Even though though initial proposals for mixed-type MI estimators exist (e.g., \cite{Ross2014,Gao2017,Rahimzamani2018,Beknazaryan2019}), their integration into the proposed forward-selection algorithm is not straightforward and is subject to future work.
\subsection{Comparison of Proposed Forward-Selection Algorithm to Existing Approaches}
\label{sec:algo_comparison}
Feature selection algorithms that make use of statistical testing to assess whether individual features contribute information in a significant fashion have been proposed before (e.g., \cite{Francois2007,Tsimpiris2012,Runge2012,Runge2015}). However, the algorithm used here is the first to control for the family-wise error rate during repeated testing \citep{Novelli2019,Wollstadt2019}. Statistical testing is a common approach to handle estimators bias \citep{Paninski2003,Hlavackova-Schindler2007,Vicente2011}, but also ensures the selection of features terminates if none of the remaining variables provides any additional information about the target. Testing for the statistical significance of information contributions thus provides a stopping criterion that is interpretable in terms of the Type I error rate, opposed to approaches requiring the definition of stopping criteria with a less clear interpretation (e.g., \cite{Tsimpiris2012}).
A further difference between the algorithm used here and earlier approaches is the inclusion of one backward-elimination step that is performed subsequent to forward-selection in order to exclude variables that may have become redundant in the context of the final feature set (\cite{Sun2014,Sun2015,Novelli2019}, see also \cite{Runge2015} for a critical discussion of purely forward-selection algorithms).
Finally, we want to note that forward-selection algorithms, such as one the used here, may fail to include feature (sub-)sets with purely synergistic contributions. One may mitigate this problem by initialize the feature set not as an empty set but with random subsets of variables, or one may try to include tuples of increasing size based on their joint information contribution \citep{Lizier2012}. However, these approaches are limited to problems with few input variables due to their practical run time.
\subsection{Conclusion and Outlook}
In the present paper, we presented a rigorous definition of feature relevancy and redundancy based on the recently proposed PID framework. We introduced PID as a conceptual as well as a practical framework for the investigation of information contribution in feature selection problems, in particular, for the analysis of feature interactions that give rise to redundant and synergistic information contributions. Based on our definition of feature relevancy, we showed that the CMI is the optimal criterion in sequential feature selection that simultaneously accounts for synergistic information contribution, while avoiding redundancy between features. We presented a forward-selection algorithm for the practical application of the CMI as a selection criterion, which handles estimator bias and provides a robust stopping criterion for variable inclusion. We successfully applied the algorithm and PID to illustrative examples from literature, illustrating how PID leads to a better understanding of how features and their interactions contribute information in feature-selection problems. Future work may extend the use of PID in feature selection, in particular, in the definition of inclusion criteria that properly account for feature interactions.
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\chapter*{Acknowledgements} I would like to thank my mum and dad, Ruth and Henry, and my brothers, Callum and Scott, who, despite giving up years ago wondering when I was going to get a proper job, have constantly supported and encouraged me in my mathematical endeavours.
\tableofcontents\newpage\null\thispagestyle{empty}\newpage
\chapter*{Introduction}
This thesis began as an attempt to answer the question: \begin{quotation} Q: What do elliptic curves with complex multiplication and $\Lambda$-structures have to do with one another?\end{quotation} The reader is probably somewhat familiar with first term, less likely so with the second. Even if he or she is familiar with both, why such a question might have an answer worth finding is probably not clear at all. So we will begin by explaining how both of these terms are related to a third: class field theory.
Let us remind the reader of the relationship between elliptic curves with complex multiplication and class field theory. For all that follows we fix an imaginary quadratic field $K$, with ring of integers $O_K$. If $L/K$ is a finite extension, an elliptic curve with complex multiplication by $O_K$ over $L$, here on called a CM elliptic curve over $L$, is an elliptic curve $E/L$ with the property that its ring of endomorphisms $\mathrm{End}_L(E)$ is isomorphic to $O_K$. For a general elliptic curve $E/L$, the Tate module $T(E)=\lim_n E[n](L^\mathrm{sep})$ is a rank two $\widehat{\mathbf{Z}}$-module equipped with an action of $G(L^\mathrm{sep}/L)$. However, when $E/L$ is a CM elliptic curve, this rank two $\widehat{\mathbf{Z}}$-module becomes a rank one $\widehat{\mathbf{Z}}\otimes_{\mathbf{Z}}O_K$-module and so the action of $G(L^\mathrm{sep}/L)$ is defined by a character \[\rho_{E/L}: G(L^\mathrm{ab}/L)=G(L^\mathrm{sep}/L)^\mathrm{ab}\to (\widehat{\mathbf{Z}}\otimes_{\mathbf{Z}}O_K)^\times.\] In particular, it follows that extensions of $L$ generated by torsion points of $E$ are abelian over $L$. It is then known that if one can find a CM elliptic curve $E$ defined over $K$ itself, the resulting character \[\rho_{E/K}: G(K^\mathrm{ab}/K)\to (\widehat{\mathbf{Z}}\otimes_{\mathbf{Z}} O_K)^\times\] is injective from which it follows that every abelian extension of $K$ is a sub-extension of one generated by the torsion points of $E$ --- thus realising explicitly the class field theory of $K$.
In general there do not exist CM elliptic curves defined over $K$. The smallest field of definition of a CM elliptic curve is the Hilbert class field $H/K$ (the maximal abelian, everywhere unramified extension of $K$). If $E/H$ is such a curve then the extensions of $H$ generated by its torsion points are abelian over $H$, but they are not necessarily abelian over $K$. If one is still interested in the class field theory of $K$, this problem can be overcome by considering certain $O_K^\times=\mathrm{Aut}_H(E)$ invariant maps \[w: E\to \mathbf{P}^1_H,\] called Weber functions. The extensions of $K$ generated by the co-ordinates of the images of torsion points of $E$ under a fixed Weber function are abelian, and every abelian extension of $K$ is a sub-extension of one of these --- again realising the class field theory of $K$. Therefore, if one is interested in the class field theory of $K$ one need go no further than CM elliptic curves.
Now let us explain the relationship between $\Lambda$-structures and class field theory. First, a $\Lambda$-structure on a flat $\mathrm{Spec}(O_K)$-scheme $X$ (we will say more about the non-flat case later) is nothing more than a commuting family endomorphisms \[\psi^\a: X\to X\] indexed by the non-zero ideals $\a$ of $O_K$ such that for any two ideals $\a, \b$ we have $\psi^\a\circ \psi^\b=\psi^{\a\b}$ and with the property that for each prime ideal $\ideal{p}$, the restriction of $\psi^\ideal{p}$ to the fibre $X_\ideal{p}:=X\times_{\mathrm{Spec}(O_K)}\mathrm{Spec}(O_K/\ideal{p})$ is the $N\ideal{p}$-power Frobenius endomorphism \[\mathrm{Fr}^{N\ideal{p}}: X_\ideal{p}\to X_\ideal{p}.\] We call the endomorphisms $\psi^\a$ (for all $\a$) a commuting family of Frobenius lifts. The resulting notion of a $\Lambda$-morphism of $\Lambda$-schemes $f:X\to Y$ being one that commutes with the Frobenius lifts. The $O_K$-scheme $\mathrm{Spec}(O_K)$ has a unique $\Lambda$-structure with Frobenius lifts all equal to the identity and if $L/K$ is an abelian extension with ring of integers $O_L$ then (ignoring the ramified primes) the finite locally free $O_K$-scheme $\mathrm{Spec}(O_L)$ admits a unique $\Lambda$-structure as well. More generally, if $S$ is any finite locally free $\mathrm{Spec}(O_K)$-scheme equipped with a $\Lambda$-structure then the extension of $K$ generated by the co-ordinates of $S$ (in any affine embedding) is an abelian extension of $K$. The link between $\Lambda$-structures and class field theory appears. These observations also have the following implication. If $X$ a $\Lambda$-scheme over $\mathrm{Spec}(O_K)$ and $0_X: \mathrm{Spec}(O_K)\to X$ a $\Lambda$-morphism then (under certain hypotheses) for each ideal $\a$ the scheme $X[\a]:=\psi^{\a*}(0_X)\subset X$ is a finite locally free $\Lambda$-scheme over $\mathrm{Spec}(O_K)$. Therefore, the extension of $K$ generated by the coordinates of $X[\a]$ will be abelian.
With this in hand, let us return to CM elliptic curves. It now turns out that if $E/K$ is a CM elliptic curve, writing $\mc{E}\to \mathrm{Spec}(O_K)$ for the N\'eron model of $E/K$, the flat $O_K$-scheme $\mc{E}$ admits a unique $\Lambda$-structure (ignoring the primes of bad reduction for $E/K$) and the morphism \[0_{\mc{E}}: \mathrm{Spec}(O_K)\to \mc{E}\] is a $\Lambda$-morphism. Moreover, for each integer $n\geq 0$, viewing $(n)\subset O_K$ as an ideal, we have that $\psi^{(n)*}(0_{\mc{E}})=\mc{E}[n]$ is the $n$-torsion of $\mc{E}$ which is now a finite locally free $\Lambda$-scheme. It is now also natural to ask whether in general there exist CM elliptic curves $E/H$ defined over the Hilbert class field whose N\'eron models admit $\Lambda$-structures and this turns out to be a subtle question.
At this point, we should note that everything we have said so far is the work of others. Indeed, the theory of complex multiplication and its relationship with class field theory is classical and has a very long history and to give a list of names would be very difficult. The theory of $\Lambda$-schemes and $\Lambda$-structures is due to Borger (\cite{Borger11}, \cite{Borger11bis}), and the relationship between $\Lambda$-structures and class field theory is due to Borger-de Smit (\cite{BorgerdeSmit04}, \cite{BorgerdeSmit08}) and indeed it was my advisor James Borger who originally posed the question at the beginning of this introduction and the more specific one asking for the existence of CM elliptic curves $E/H$ over the Hilbert class field with $\Lambda$-structures. This second question we can now answer:
\begin{theorem*}
\begin{enumerate}[label=\textup{(\roman*)}]
\item There always exists a \textup{CM} elliptic curve $E/H$ over the Hilbert class field whose N\'eron model admits a $\Lambda$-structure.
\item The N\'eron model of a \textup{CM} elliptic curve $E/H$ admits a $\Lambda$-structure if and only if the extension of $K$ generated by its torsion is abelian over $K$.
\end{enumerate}
\end{theorem*}
CM elliptic curves (over arbitrary abelian extensions of $K$) with the property that the extensions generated by their torsion is abelian over $K$ were introduced originally by Shimura and are now called CM elliptic curves of Shimura type. Indeed, it is using results of Shimura that, after proving (ii) we are able to prove (i). It worth pointing out that CM elliptic curves of Shimura type have been studied by several authors, with particular reference to their $L$-functions and the Birch-Swinnerton-Dyer Conjecture. Indeed, the papers of Coates-Wiles \cite{CoatesWiles1977} and Rubin \cite{Rubin1981} concern curves of this type.
Let us now also answer the question posed in the first paragraph: \begin{quotation} Q: What do elliptic curves with complex multiplication and \mbox{$\Lambda$-structures} have to do with one another?\ \\
A: Everything!
\end{quotation}
The central theorem we aim to prove is the following and explains our answer to the question above.
\begin{theorem*}\label{theo:intro-main} Let $\mc{M}_\mathrm{CM}$ denote the moduli stack of \textup{CM} elliptic curves and let $\mc{E}\to \mc{M}_\mathrm{CM}$ denote the universal \textup{CM} elliptic curve. Then both $\mc{E}$ and $\mc{M}_\mathrm{CM}$ admit canonical $\Lambda$-structures and the morphism $\mc{E}\to \mc{M}_\mathrm{CM}$ is a $\Lambda$-morphism.
\end{theorem*}
The reader will probably have noticed that we have not even defined what it means for a general (non-flat) scheme, let alone a stack, to have a $\Lambda$-structure and in fact we do not propose to define $\Lambda$-structures on stacks in this thesis (not because it isn't possible to do so, only because doing so would lead us into the nightmarish realm of 2-monads on 2-categories). In any case, the definition of $\Lambda$-structure we have given for a flat $O_K$-scheme admits an obvious naive generalisation --- that of a commuting family of Frobenius lifts --- though it is not the correct one. Before we explain the correct definition, and the actual meaning of (\ref{theo:intro-main}), let us describe the naive $\Lambda$-structure on $\mc{M}_\mathrm{CM}$.
If $S$ is an $O_K$-scheme then $\mc{M_\mathrm{CM}}(S)$ is the category of CM elliptic curves $E$ over $S$. In particular, the objects $E/S\in \mc{M}_\mathrm{CM}(S)$ are $O_K$-modules and for each non-zero ideal $\a\subset O_K$ it is possible to make sense of the $O_K$-module $E\otimes_{O_K}\a^{-1}$ which is again a CM elliptic curve over $S$. This defines for each ideal $\a$ an endofunctor \begin{equation*}-\otimes_{O_K}\a^{-1}:\mc{M}_\mathrm{CM}\to \mc{M}_\mathrm{CM}: E\mapsto E\otimes_{O_K}\a^{-1},\end{equation*} and for different ideals $\a$ these endo-functors all `commute' in the obvious sense. More important is the following fact: if $\ideal{p}\subset O_K$ is prime and $S$ is an $O_K$-scheme of characteristic $\ideal{p}$, i.e.\ the morphism $S\to \mathrm{Spec}(O_K)$ factors through $\mathrm{Spec}(O_K/\ideal{p})$, then for all CM elliptic curves $E/S$ there is a canonical isomorphism \[E\otimes_{O_K}\ideal{p}^{-1}\overset{\sim}{\longrightarrow} \mathrm{Fr}^{N\ideal{p}*}(E)\] between $E\otimes_{O_K}\ideal{p}^{-1}$ and the pull-back of $E$ along the $N\ideal{p}$-power Frobenius $\mathrm{Fr}^{N\ideal{p}}: S\to S$ (this construction in its simplest form is due to Serre). In other words, the functor $-\otimes_{O_K}\ideal{p}^{-1}$ is a lift of the $N\ideal{p}$-power Frobenius and we have defined a naive $\Lambda$-structure on $\mc{M}_\mathrm{CM}$.
However, as we have already noted, the naive notion of a $\Lambda$-structure as a commuting family of Frobenius lifts is not the correct one. Let us now at least say enough about the non-naive notion of $\Lambda$-structure so that we may explain to the reader some of the actual meaning of (\ref{theo:intro-main}).
There is a functor \[W^*: \mathrm{Sch}_{O_K}\to \mathrm{Sch}_{O_K}: X\mapsto W^*(X)\] sending an $O_K$-scheme $X$ to its scheme $W^*(X)$ of (big $O_K$-typical) Witt vectors (technically speaking $W^*(X)$ is actually an ind-scheme, but for the purposes of this introduction we will ignore this fact). We will not say much here about the geometry of the Witt vectors $W^*(X)$ themselves, other than to remark that they have many miraculous properties, chief among which is that if $X$ is an $O_K$-scheme of characteristic $\ideal{p}$ for some prime of $O_K$, then (very roughly speaking) the Witt vectors $W^*(X)$ of $X$ have characteristic $0$. The properties of the Witt vectors $W^*(X)$ which are of interest to us presently are the following:
\begin{enumerate}[label=(\roman*)]
\item $W^*(X)$ possesses a family of commuting Frobenius lifts \[\psi^{\a}: W^*(X)\to W^*(X)\] indexed by the ideals $\a$ of $O_K$ (i.e.\ a naive $\Lambda$-structure),
\item there is a morphism $g_{(1)}: X\to W^*(X)$, and
\item for all flat $\Lambda$-schemes $S$ (the definition of which we have given) and all morphisms $X\to S$ there is a unique morphism $W^*(X)\to S$ compatible with the Frobenius lifts on $W^*(X)$ and $S$ such that the following diagram commutes \begin{equation}\label{eqn:def-lambda-intro}\begin{gathered}\xymatrix{& W^*(X)\ar@{.>}[d]\\
X\ar[ur]^{g_{(1)}} \ar[r] & S.}\end{gathered}\end{equation}
\end{enumerate}
If we now let $S$ be an arbitrary $O_K$-scheme, the diagram (\ref{eqn:def-lambda-intro}) can be taken as the \textit{definition} of a $\Lambda$-structure on $S$: a $\Lambda$-structure on an $O_K$-scheme $S$ is, for each morphism $X\to S$, a \textit{canonical lifting} $W^*(X)\to S$ of that morphism making the diagram (\ref{eqn:def-lambda-intro}) commute (together with certain iterated compatibilities which we will not give here). It is not unreasonable to make the comparison between the construction of Serre-Tate of the canonical lift to the ($p$-typical) Witt vectors of ordinary elliptic curve over a finite field.
We can now give the reader a better sense of the meaning of (\ref{theo:intro-main}). If $S$ is an $O_K$-scheme and $E/S$ is a CM elliptic curve, i.e.\ if one is given a morphism $S \stackrel{E}{\to} \mc{M}_\mathrm{CM}$, then there is a functorially defined CM elliptic curve $W^*_\mathrm{CM}(E)$ over the (big $O_K$-typical) Witt vectors $W^*(S)$ of $S$, i.e.\ there is a morphism $W^*(S)\stackrel{W_\mathrm{CM}^*(E)}{\longrightarrow} \mc{M}_\mathrm{CM}$, together with a canonical isomorphism $g_{(1)}^*(W_\mathrm{CM}^*(E))\overset{\sim}{\longrightarrow} E$. This to say we have a `commutative' diagram: \[\xymatrix{& W^*(S)\ar@{.>}[d]^{W^*_\mathrm{CM}(E)}\\
S\ar[ur]^{g_{(1)}} \ar[r]^-{E} & \mc{M}_\mathrm{CM}}\] and this is what we mean when we say that $\mc{M}_\mathrm{CM}$ admits a $\Lambda$-structure. It follows that CM elliptic curves can be lifted canonically to the Witt vectors of the base. We would like to point out that the base $S$ here is arbitrary.
We shall not anything more about (\ref{theo:intro-main}) here, nor the $\Lambda$-structure on $\mc{E}$. However, what (\ref{theo:intro-main}) does do is equip $\mc{E}\to \mc{M}_\mathrm{CM}$ with an incredible amount of very rich structure --- structure which is of interest in and of itself in the world of $\Lambda$-geometry, but which can also be exploited to prove new results in both the theory of CM elliptic curves and the arithmetic of imaginary quadratic fields. The following result is an example of this phenomenon:
\begin{theorem*} Let $K(\ideal{f})$ be the ray class field of conductor $\ideal{f}$ and let $E/K(\ideal{f})$ be a \textup{CM} elliptic curve of Shimura type. If the $\ideal{f}$-torsion $E[\ideal{f}]$ is constant then $E/K(\ideal{f})$ admits a global minimal model away from $\ideal{f}$. In particular, if $\ideal{f}=O_K$ so that $K(\ideal{f})=H$ then \textup{every} \textup{CM} elliptic curve $E/H$ of Shimura admits a \textup{global} minimal model.
\end{theorem*}
In the special case when $\mathrm{disc}(K/\mathbf{Q})$ is prime and $\ideal{f}=O_K$ this result was proven by Gross (Corollary 4.4 \cite{Gross82}),\\
We now give an overview of the chapters:
Chapter 1: We recall the local and global reciprocity maps, define Lubin--Tate modules and study their moduli stack $\mc{M}_\mathrm{LT}$. We show that $\mc{M}_\mathrm{LT}$ admits a certain torsor structure and using this explain how to derive the local reciprocity map directly from $\mc{M}_\mathrm{LT}$ using only its formal properties. We then give an overview of a certain special case of global reciprocity, and present some basic constructions regarding local systems of rank one.
Chapter 2: We undertake a quite detailed study of CM elliptic curves over arbitrary bases and their moduli stack $\mc{M}_\mathrm{CM}$. We show that $\mc{M}_\mathrm{CM}$ admits a certain torsor structure analogous to that of $\mc{M}_\mathrm{LT}$. We also give CM analogues of some classical theorems for general elliptic curves and, in a similar vein to Lubin-Tate modules, we explain how to derive a certain global reciprocity map directly from $\mc{M}_\mathrm{CM}$ using only its formal properties. Using this we classify all CM elliptic curves over fields (of arbitrary characteristic) in terms of their associated Galois representations. Finally, we consider the moduli stacks with level structure and consider their representability in the fine and coarse setting.
Chapter 3: We give a short overview of the general theory of $\Lambda$-schemes, Witt vectors and arithmetic jets. We then prove a handful of (slightly technical) new results for later use.
Chapter 4: Here we prove the main theorem regarding lifting CM elliptic curves over arbitrary bases to their big $O_K$-typical Witt vectors. We then prove that a CM elliptic curve is of Shimura type if and only if its N\'eron model admits a $\Lambda$-structure. We then consider the minimal models of CM elliptic curves of Shimura type and prove their global existence under certain hypotheses. Next we consider quotients of CM elliptic curves by their groups of automorphisms and prove that (suitable reinterpreted) they always exist and we show that the quotient of the universal CM elliptic curve descends to the coarse space of the moduli stack of CM elliptic curves. We show how this descended curve admits a canonical $\Lambda$-structure and allows one to construct the ray class fields of $K$ in a choice free, integral and coherent manner and we also show show that this descended curve is nothing but a (global) projective line. We then construct a flat, affine and pro-smooth cover of the moduli stack of CM elliptic curves, which comes equipped with a $\Lambda$-structure compatible with that on $\mc{M}_\mathrm{CM}$. Finally, we exhibit an interesting relationship between certain deformations of CM elliptic curves with $\Lambda$-structures and their Tate modules.
Appendix: We give an abstract and formal definition of smooth formal groups, we consider the general properties of `Serre's tensor product' and we give a short overview of Faltings' generalised Cartier duality, needed to prove certain results in Chapter 1. Finally, we prove a strengthening of an old principal ideal theorem for arbitrary number fields.
\mainmatter
\chapter*{Foundations, conventions and terminology}
\section{Sheaves}
\subsection{} In order to have a nice common ground for all the objects we would like to work with, we shall here define the basic categories of (pre)sheaves in which all objects we consider will live. We have decided to not to bother ourselves with set theoretic issues of `size' (which really only come up when one tries to sheafify wild presheaves for large topologies \cite{Waterhouse75} --- something we will have no need to do), however the concerned reader may add the word `universe' whenever he or she sees fit.
Let $\mathrm{Aff}$ denote the category of affine schemes and $\mathrm{PSh}$ the category of presheaves of sets on $\mathrm{Aff}$, i.e.\ the category of functors \[X:\mathrm{Aff}^\circ\to \mathrm{Set}.\] We write $\mathrm{Sch}$ for the category of schemes and as usual we embed $\mathrm{Aff}$ and $\mathrm{Sch}$ in $\mathrm{PSh}$ by sending a scheme to the functor it represents.
We shall be working with sheaves for the fpqc and \'etale topologies on $\mathrm{Aff}$ which we now recall. The covers of an affine scheme $S$ for the fpqc (resp. \'etale) topology are given by flat (resp. \'etale) families $(S_i\to S)_{i\in I}$ indexed by a finite set $I$ which are covers in the usual sense. A sheaf for the fpqc topology will just be called a sheaf and the category of such sheaves will be denoted $\mathrm{Sh}\subset \mathrm{PSh}$. A sheaf for the \'etale topology will be called an $\textup{\'et}$-sheaf and we write $\mathrm{Sh}^{\textup{\'et}}\subset \mathrm{PSh}$ for the corresponding category. We have the inclusions \[\mathrm{Aff}\subset \mathrm{Sch}\subset \mathrm{Sh}\subset \mathrm{Sh}^\textup{\'et}\subset \mathrm{PSh}.\]
\subsection{} If $f: S'\to S$ is a morphism of presheaves and $X\to S$ is an $S$-presheaf then we denote the fibre product by $X\times_S S'$, $f^*(X)$, or when $f$ is clear by $X_{S'}$. Viewed as a functor $f^*: \mathrm{PSh}_{S}\to \mathrm{PSh}_{S'}$, the right adjoint to $f^*$ is denoted by $f_*$ and the left adjoint by $f_!$. Recall that if $X'\to S'$ is an $S'$-presheaf then $f_!(X')$ is the $S$-presheaf $X'\to S'\stackrel{f}{\to} S$ and $f_*$ is the $S$-presheaf defined by \[\mathrm{Spec}(A)\mapsto \mathrm{Hom}_{S'}(\mathrm{Spec}(A), f_*(X'))=\mathrm{Hom}_{S'}(f^*(\mathrm{Spec}(A)), X').\] The functor $f_!$ will be used rarely, and only in the case when $f$ is an isomorphism, in which case it is isomorphic to $g^*$ where $g=f^{-1}$.
\begin{exem} An important family of sheaves for the fpqc topology are the ind-affine schemes, the category of which we denote by $\mathrm{IndAff}$. Recall an ind-affine scheme $X$ is a pre-sheaf $X$ which can be written as filtered a colimit $X=\colim_i \mathrm{Spec}(A_i)$ of affine schemes (it follows automatically from this that it is a sheaf). One of the main examples we work with is the following: if $S=\mathrm{Spec}(A)$ is an affine scheme and $I$ is an ideal we write $\mathrm{Spf}_I(A)$ (or just $\mathrm{Spf}(A)$ when $I$ is clear from the context) for the ind-affine scheme $\colim_n \mathrm{Spec}(A/I^{n+1})$. If $X\to \mathrm{Spec}(A)$ is a presheaf over $\mathrm{Spec}(A)$ we say that $X$ is $I$-adic or that $I$ is locally nilpotent on $X$ if the morphism $X\to \mathrm{Spec}(A)$ factors through $\mathrm{Spf}(A)\subset \mathrm{Spec}(A)$. If $\mathrm{Spec}(B)\to \mathrm{Spec}(A)$ is an affine $\mathrm{Spec}(A)$-scheme then $\mathrm{Spec}(B)$ is $I$-adic if and only if the ideal $I B\subset B$ is nilpotent.
\end{exem}
\subsection{} If $A$ is a set and $S$ is an $\textup{\'et}$-sheaf then we write \[\underline{A}_S=\coprod_{a\in A}S\] for the constant $\textup{\'et}$-sheaf over $S$ associated to $A$. If $S$ is actually a sheaf, so is $\underline{A}_S$. When $S$ is a fixed base (usually the spectrum of some Dedekind domain) we will drop the sub-script and just write $\underline{A}$.
\subsection{} By a cover of a sheaf $S$ (resp. $\textup{\'et}$-sheaf) we just mean a family $(S_i\to S)_{i\in I}$ of morphisms of sheaves (resp. $\textup{\'et}$-sheaves) such that $\coprod_{i\in I}S_i\to S$ is an epimorphism. When referring to properties or making claims which are compatible with base change we will use the word local to mean after base change along a cover.
\subsection{} We say that a morphism of pre-sheaves $f: X\to S$ is representable, or that $X$ is representable over $S$, if for each affine scheme $S'\to S$ the pre-sheaf $X\times_S S'$ is (representable by) a scheme. In general, for a morphism of sheaves (or $\textup{\'et}$-sheaves) to be representable is not local. However, it is the case for $f$ which are representable by open immersions, or when $f$ is representable by affine morphisms. In both of these cases we will just say that $f$ is open immersion, or that $f$ is affine. Similarly for any other condition of $f$ that includes affine in its definition: finite, finite locally free, a closed immersion and so on.
\begin{prop} If $f: X\to Y$ is a finite locally free \'etale morphism of $\textup{\'et}$-sheaves then the inclusion of the image $f(X)\to Y$ in $\mathrm{Sh}^\textup{\'et}$ is an open and closed immersion, the inclusion of the complement $Y-f(X)\to Y$ is also an open and closed immersion and $Y\amalg (Y-f(X))=Y$. Moreover, if $X$ and $Y$ are sheaves, so are the $\textup{\'et}$-sheaves $f(X)$ and $Y-f(X)$.
\end{prop}
The following result will be used often (it follows from Th\'eor\`eme 2.1 of Expos\'e VIII in \cite{SGA1}) cited as `by descent':
\begin{prop}\label{prop:affine-fpqc-descent} Let $X\to S$ be a morphism of ($\textup{\'et}$-)sheaves and let $S'\to S$ be an epimorphism of ($\textup{\'et}$-)sheaves. Then $X\to S$ is affine if and only if $X\times_S S'\to S'$ is affine.
\end{prop}
\subsection{} We now say a little about what we mean by a quasi-coherent $\mc{O}_S$-module on a sheaf $S$. We have the relatively representable sheaf of rings \begin{equation}\label{def:structure-sheaf}\mc{O}_S:=\mathbf{A}^1_{S}=\mathrm{Spec}(\mathbf{Z}[T])\times_{\mathrm{Spec}(\mathbf{Z})} S\end{equation} and the abelian category of $\mc{O}_S$-modules $\mathrm{Mod}(\mc{O}_S)$. For any map of sheaves $f: S' \to S$, as $\mc{O}_S\times_S S'=\mc{O}_{S'}$, we obtain the functor \[f^*: \mathrm{Mod}(\mc{O}_S)\to \mathrm{Mod}(\mc{O}_{S'}): \mc{M}\mapsto \mc{M}\times_S S'=\mc{M}_{S'}.\] Note that this functor is exact for any map $f$.
The category of quasi-coherent sheaves $\mathrm{QCoh}(\mc{O}_S)\subset \mathrm{Mod}(\mc{O}_S)$ is the full sub-category of $\mc{O}_S$-modules $\mc{M}$ such that there exists a cover $(S_i\to S)_{i\in I}$ and for each $i\in I$ an exact sequence \[\mc{O}_{S_i}^{M_i}\to \mc{O}_{S_i}^{N_i}\to\mc{M}_{S_i}\to 0\] where $M_i$ and $N_i$ are sets. When $S$ is a scheme this category coincides with the usual category of quasi-coherent sheaves over $S$ and the functor $f^*$ defined above has its usual meaning. However, the inclusion $\mathrm{QCoh}(\mc{O}_S)\subset \mathrm{Mod}(\mc{O}_S)$ is not exact (to be precise it does not preserve kernels). This explains why $f^*: \mathrm{Mod}(\mc{O}_S)\to \mathrm{Mod}(\mc{O}_{S'})$ is exact for any $f$ while the same is not true for $\mathrm{QCoh}(\mc{O}_S)\to \mathrm{QCoh}(\mc{O}_S)$.
We shall be almost exclusively concerned with locally free finite rank $\mc{O}_S$-modules --- or what is the same vector bundles --- the equivalence between which is a tautology with our definition $\mc{O}_S:=\mathbf{A}^1_S$.
\section{Group actions}
\subsection{} Let $A\to B$ be a homomorphism of rings and let $G$ be some group of $A$-automorphisms of $B$. The example to have in mind here is a Galois extension of fields $K\to L$ and $G=G(L/K)$.
Given $\sigma\in G$, in order to avoid the cumbersome notation $\mathrm{Spec}(\sigma): \mathrm{Spec}(B)\to \mathrm{Spec}(B)$, we will just write $\sigma: \mathrm{Spec}(B)\to \mathrm{Spec}(B)$. However, associating an affine scheme to a ring is contravariant, so that this notation becomes confusing when considering compositions $\sigma\circ \tau$ of elements in $G$. In order to avoid problems here, we make the convention that the product of two elements of $\sigma, \tau\in G$, will be denoted by $\sigma\tau$ so that $\sigma\tau\in G$ is the automorphism \[B\stackrel{\tau}{\to} B\stackrel{\sigma}{\to} B.\] We will only use the composition symbol $\circ$ when viewing $\sigma$ and $\tau$ as automorphisms of $\mathrm{Spec}(B)$ so that $\sigma\circ \tau$ will denote the $\mathrm{Spec}(A)$-automorphism of $\mathrm{Spec}(B)$ \[\mathrm{Spec}(B)\stackrel{\tau}{\longrightarrow} \mathrm{Spec}(B)\stackrel{\sigma}{\longrightarrow} \mathrm{Spec}(B).\] With this convention the following three symbols denote the same $\mathrm{Spec}(A)$-automorphism of $\mathrm{Spec}(B)$ \[\sigma\circ \tau=\mathrm{Spec}(\tau\sigma)=\tau\sigma: \mathrm{Spec}(B)\to \mathrm{Spec}(B).\]
\section{Stacks}
\subsection{} Let $C$ be a site (the examples we have in mind are $C=\mathrm{Aff}$, $\mathrm{IndAff}$, $\mathrm{Sh}$). A fibred category over $C$ is a category $\mc{X}$ equipped with a functor \[p: \mc{X}\to C\] together with, for each morphism $f: S'\to S$ of $C$, a pull-back functor $f^*: \mc{X}(S)\to \mc{X}(S')$ where $\mc{X}(S)$ denotes the fibre of $p$ over $S$ (objects of $\mc{X}$ mapping to $S$ and morphisms those mapping to $\mathrm{id}_S$) together with various natural transformations between their compositions satisfying certain identities. There is also the notion of a morphism of fibred categories $f: \mc{X}'\to \mc{X}$ being a functor (strictly) compatible with the functors from $\mc{X}'$ and $\mc{X}$ to $C$ together with certain compatibility relations between the pull-back functors of $\mc{X}'$ and $\mc{X}$. Finally, a stack over $C$ is a fibred category whose objects and morphisms satisfy descent with respect to the topology on $C$.
To keep things (relatively) concrete we shall say the following for $C=\mathrm{Aff}$ with the fpqc topology --- it will also apply when working with other sites $C$.
The main point we want to make is that when defining stacks $\mc{X}\to \mathrm{Aff}$ we shall often skip the details and define only the fibres $\mc{X}(S)$ for $S\in \mathrm{Aff}$ and the pull-back functors $f^*:\mc{X}(S)\to \mc{X}(S')$ for $f: S'\to S$ in $\mathrm{Aff}$ (and sometimes not even these when they are clear). Similarly a morphism of stacks will be defined only on the fibres, the various compatibilities which must be satisfied will be obvious from the context.
\subsection{} To each sheaf $X$ there is an associated stack $\mathrm{Aff}_{/X}\to \mathrm{Aff}$ and for each morphism $f: X\to Y$ a morphism \[\mathrm{Aff}_{/X}\to \mathrm{Aff}_{/Y}: (S\to X)\mapsto (S\to X\stackrel{f}{\to} Y).\] Moreover, every morphism of stacks \[\mathrm{Aff}_{/X}\to \mathrm{Aff}_{/Y}\] is uniquely isomorphic to one of this form. In other words, when considering sheaves as the more general objects stacks, via $X\mapsto \mathrm{Aff}_{/S}$, we do not lose or gain anything especially important and so when considering a sheaf as a stack we shall continue to denote it by $X$.
\subsection{} \label{def:coarse-sheaf} Finally, if $\mc{X}\to \mathrm{Aff}$ is a stack and $S\in \mathrm{Aff}$ we write $\mc{X}(S)/\sim$ for the set of isomorphism classes of objects of $\mc{X}(S)$. The pull-back maps $f^*: \mc{X}(S)\to \mc{X}(S')$ for $f: S'\to S$ define maps $\mc{X}(S)/\sim\to \mc{X}(S')/\sim$ and this defines a (separated) pre-sheaf \[\mathrm{Aff}^{\circ}\to \mathrm{Set}: S\mapsto \mc{X}(S)/\sim\] whose sheafification $C(\mc{X})=X$ (which will be denoted by the print form of the cursive letter denoting the stack) is called the coarse sheaf associated to $\mc{X}$. There is an induced morphism $c_{\mc{X}}: \mc{X}\to X$ and for each sheaf $Y$ and each morphism $f:\mc{X}\to Y$ there is a unique morphism $f': X\to Y$ such that $f'\circ c_{\mc{X}}=f$.
\section{Dedekind domains} Here we fix some general notation for Dedekind domains and various related objects.
\subsection{} Let $O$ be a Dedekind domain with field of fractions $K$ and finite residue fields. An integral ideal of $O$ is any non-zero ideal and a prime ideal will mean a non-trivial prime ideal. We write $\mathrm{Id}_{O}$ (resp. $\mathrm{Prin}_O$) for the monoid of integral (resp. and principal) ideals of $O$ and $\mathrm{Id}_K$ (resp. $\mathrm{Prin}_K$) for the group of fractional (resp. principal) ideals of $O$. If $\a, \b\in \mathrm{Id}_{O}$ we write $(\a, \b)=\a+\b$ so that $\a$ and $\b$ are relatively prime if and only if $(\a, \b)=O$. We write $\mathrm{CL}_{O}$ for the class group of $O$, i.e.\ the group of isomorphism classes of rank one projective $O$-modules.
Let $\ideal{f}$ be an integral ideal. We write $\mathrm{Id}_O^{(\ideal{f})}\subset \mathrm{Id}_O$ (resp.\ $\mathrm{Id}_{K}^{(\ideal{f})}\subset \mathrm{Id}_K$) for the sub-monoid (resp.\ sub-group generated) by the prime ideals prime to $\ideal{f}$. If $a\in K^\times$ then we say $a=1\bmod \ideal{f}$ to mean that the ideal $(a-1)$ can be written as $\ideal{f}\a\b^{-1}$ where $\a, \b\subset O$ are ideals (if $a=1$ we allow $\a=(0)$) with $\b\neq (0)$ and $(\b, \ideal{f})=O$. If $\ideal{f}|\ideal{g}$ is another ideal then we write $\mathrm{Prin}_{1\bmod \ideal{f}}^{(\ideal{g})}$ to denote the group of principal fractional ideals $\a=(a)$ prime to $\ideal{g}$ with $a=1\bmod \ideal{f}$.
We write $N\ideal{f}$ for the cardinality of $O/\ideal{f}$ and if $\ideal{f}=\ideal{p}$ is prime we write $\mathbf{F}_\ideal{p}=O/\ideal{p}$. If $A$ is an $\mathbf{F}_\ideal{p}$-algebra then we write $\mathrm{Fr}^{N\ideal{p}}: A\to A$ for the $N\ideal{p}$-power Frobenius endomorphism.
We write $O^{\times, \ideal{f}}$ for the kernel of the homomorphism $O^{\times}\to (O/\ideal{f})^\times$ and we say that $\ideal{f}$ separates units if this homomorphism is injective (this is not often the case, but will be used constantly in the text).
We write $O[\ideal{f}^{-1}]$ for the sub-$O$-algebra of $K$ generated by the elements of $\ideal{f}^{-1}\subset K$. If $X\to\mathrm{Spec}(O)$ is any $\mathrm{Spec}(O)$-sheaf then we write $X[\ideal{f}^{-1}]=X\times_{\mathrm{Spec}(O)}\mathrm{Spec}(O[\ideal{f}^{-1}])$. We say that $\ideal{f}$ is invertible on $X$ if $X[\ideal{f}^{-1}]=X$. If $\ideal{p}$ is a prime ideal we say that $X$ has characteristic $\ideal{p}$ if the structure map $X\to \mathrm{Spec}(O)$ factors through $\mathrm{Spec}(\mathbf{F}_\ideal{p})\to \mathrm{Spec}(O)$.
If $\ideal{p}$ is a prime we write \[\mathrm{Spf}_\ideal{p}(O)=\colim_{n\geq 0} \mathrm{Spec}(O/\ideal{p}^{n+1})\subset \mathrm{Spec}(O).\] We say that $X$ is $\ideal{p}$-adic, or that $\ideal{p}$ is nilpotent on $S$, if the structure morphism $X\to \mathrm{Spec}(O)$ factors through $\mathrm{Spf}_\ideal{p}(O)\subset \mathrm{Spec}(O)$. For an affine $\mathrm{Spec}(O)$-scheme $\mathrm{Spec}(A)$ to be $\ideal{p}$-adic is equivalent to the ideal $\ideal{p} A$ being nilpotent.
\newpage\null\thispagestyle{empty}\newpage
\chapter{Local and global reciprocity}
In \S\S 1.1 and 1.3 of this chapter we recall the local and global reciprocity maps of class field theory. In \S 1.2 we define Lubin--Tate $O$-modules (in families) for a non-archimedian local field $K$ with ring of integers $O$ and show that the moduli stack $\mc{M}_\mathrm{LT}$ of Lubin--Tate $O$-modules is a torsor under the stack $\mc{CL}_O$ of rank one $O$-local systems (\ref{prop:lubin-tate-torsor}). We show how using this structure one can derive the local reciprocity map directly from the stack $\mc{M}_\mathrm{LT}$ using only the formal properties of Lubin--Tate $O$-modules (\ref{theo:main-theorem-lubin-tate}). The construction we give is an analogue, for non-archimedian local fields, of the derivation of the global reciprocity map for imaginary quadratic fields using CM elliptic curves (which we address in Chapter 2). In \S 1.4 we recall how, for global fields $K$ equipped with a certain special place $\infty$ and associated ring of integers $O_K$, the reciprocity map associated to the maximal abelian extension of $K$ which is totally split at $\infty$ can be reinterpreted in terms of certain class groups associated to $O_K$ (such pairs $(K, \infty)$ were first considered by Drinfel'd \cite{Drinfeld74}). Finally, in \S 1.5 we define and give the basic properties of rank one $O_K$-local systems, their level-$\ideal{f}$ structures and moduli for use in the later chapters.
The only new results (or perhaps observations) in this chapter are the $\mc{CL}_O$-torsor structure on $\mc{M}_\mathrm{LT}$ (which in any case we prove using results of Faltings) and the derivation of the local reciprocity map directly from $\mc{M}_\mathrm{LT}$.
\section{The local reciprocity map} The purpose of this section is to recall the basic properties of local fields and the local reciprocity map. Everything here is contained in Chapters I and VI of \cite{CasselsFrohlich67}).
\subsection{} Let $K$ be a local field. There are two cases and for each we fix the following notation:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $K$ is archimedian and is isomorphic to $\mathbf{R}$ or $\mathbf{C}$. We write $|-|_K$ for the usual absolute value if $K\overset{\sim}{\longrightarrow} \mathbf{R}$ and the square of the usual absolute value if $K\overset{\sim}{\longrightarrow} \mathbf{C}$.
\item $K$ is non-archimedian and is the field of fractions of a discrete valuation ring $O\subset K$ with maximal ideal $\ideal{p}$ and finite residue field $\mathbf{F}_\ideal{p}$. We equip it with the absolute value $|a|_K=N \ideal{p}^{-v_\ideal{p}(a)}$ where $a\cdot O=\ideal{p}^{v_\ideal{p}(a)}\subset K$).
\end{enumerate} We also fix maximal abelian and separable extensions $K\subset K^\mathrm{ab}\subset K^\mathrm{sep}$. If $K$ is non-archimedian these extensions are not complete but from the point of view of Galois theory nothing is lost.
\subsection{} Let $K$ be a non-archimedian local field and let $K\subset L\subset K^\mathrm{sep}$ be a (not necessarily finite) Galois extension of $K$. Write $O_L\subset L$ for the ring of integers of $L$, $\ideal{p}_L\subset K$ for its unique maximal ideal and $\mathbf{F}_{\ideal{p}_L}=O_L/\ideal{p}_L$ for its residue field. The reduction homomorphism $G(L/K)\to G(\mathbf{F}_{\ideal{p}_L}/\mathbf{F}_\ideal{p})$ is surjective and is bijective whenever $L/K$ is unramified. In this case we write $\sigma_{L/K}\in G(L/K)$ for the unique element lifting the $N\ideal{p}$-power Frobenius automorphism of $\mathbf{F}_{\ideal{p}_L}$.
We denote by $K\subset K^\mathrm{ur}\subset K^\mathrm{sep}$ the maximal unramified extension of $K$. The map \[\widehat{\mathbf{Z}}\to G(K^\mathrm{ur}/K): n\mapsto \sigma_{K^\mathrm{ur}/K}^{n}\] is a continuous isomorphism whose inverse is denoted by $v_K: G(K^\mathrm{ur}/K)\to \widehat{\mathbf{Z}}$. If $L/K^\mathrm{ur}$ is any extension we also write $v_K: G(L/K)\to \widehat{\mathbf{Z}}$ for the map $\sigma\mapsto v_K(\sigma|_{K^\mathrm{ur}})$ and define $W(L/K):=v_K^{-1}(\mathbf{Z})\subset G(L/K)$ (this is the `Weil group').
\begin{theo}\label{theo:archimedian-local-reciprocity} There is a unique isomorphism \begin{equation} K^\times \to W(K^\mathrm{ab}/K): a\mapsto (a, K^\mathrm{ab}/K)\label{eqn:local-rec-map}\end{equation} such that
\begin{enumerate}[label=\textup{(\roman*)}]
\item for all finite extensions $K\subset L\subset K^\mathrm{ab}$ , the kernel of the composition $a\mapsto (a, K^\mathrm{ab}/K)|_{L}$ is $N_{L/K}(L^\times)$ and the induced map \[K^\times/N_{L/K}(L^\times)\to G(L/K)\] is an isomorphism, and
\item the diagram \[\xymatrix{K^\times\ar[d]_{v_\ideal{p}}\ar[rr]^-{(-, K^\mathrm{ab}/K)} && \ar[d] W(K^\mathrm{ab}/K)\\
\mathbf{Z} \ar[rr]^-{n\mapsto \sigma_{K^\mathrm{ur}/K}^n} && W(K^\mathrm{ur}/K)}\] commutes.
\end{enumerate}
\end{theo}
\begin{proof} Uniqueness and existence are Proposition 6, \S 2.8 of Chapter VI and Theorem 2, \S 2.2 Chapter VI of \cite{CasselsFrohlich67} respectively.
\end{proof}
\subsection{}\label{rema:local-rec-properties} We list the following further property of the reciprocity homomorphism (see \S 2 Chapter VI of \cite{CasselsFrohlich67}): If $K'/K$ is any finite separable extension and $K'^\mathrm{ab}/K'$ a maximal abelian extension of $K'$, then the diagram \[\xymatrix{K'^\times\ar[rr]^-{(-, K'^\mathrm{ab}/K')}\ar[d]_{N_{K'/K}} && W(K'^\mathrm{ab}/K')\ar[d]\\
K^\times\ar[rr]^-{(-, K^\mathrm{ab}/K)} && W(K^\mathrm{ab}/K)}\] commutes.
\subsection{} We also recall the reciprocity map $(-, K^\mathrm{ab}/K)$ associated to an archimedian local field $K$. It is the unique continuous homomorphism \[(-, K^\mathrm{ab}/K):K^\times\to G(K^\mathrm{ab}/K)\] sending $-1$ to the unique generator of $G(K^\mathrm{ab}/K)$ which is either trivial (if $K\overset{\sim}{\longrightarrow} \mathbf{C}$) or cyclic of order two (if $K\overset{\sim}{\longrightarrow} \mathbf{R}$).
\section{The local reciprocity map and Lubin--Tate $O$-modules}\label{sec:lubin-tate-modules} We now define rank one $O$-local systems over $\ideal{p}$-adic sheaves and their moduli stack $\mc{CL}_O$. We then define and study families of Lubin--Tate $O$-modules over $\ideal{p}$-adic sheaves and show that $\mc{CL}_O$ acts on the moduli stack $\mc{M}_\mathrm{LT}$ of Lubin--Tate $O$-modules, making it a torsor under $\mc{CL}_O$ (see (\ref{prop:lubin-tate-torsor}) and (\ref{exem:lubin-tate-modules-exist})). Finally, we explain in (\ref{subsec:lubin-tate-reciprocity}) how this action, combined with the basic properties of Lubin--Tate $O$-modules, allows one to construct the reciprocity map of (\ref{theo:archimedian-local-reciprocity}).
\subsection{} We shall be working with the category of $\ideal{p}$-adic sheaves, i.e.\ sheaves $S\to \mathrm{Spf}(O)=\colim_n \mathrm{Spec}(O/\ideal{p}^{n+1})$. If $S\to \mathrm{Spf}(O)$ is a $\ideal{p}$-adic sheaf then we write $S_n=S\times_{\mathrm{Spf}(O)}\mathrm{Spec}(O/\ideal{p}^{n+1})$ so that $S=\colim_{n\geq 0} S_n$. Unless otherwise stated $S$ denotes a $\ideal{p}$-adic sheaf.
\subsection{}\label{sub-sec-p-adic-formal} For each $\ideal{p}$-adic sheaf $S$ we write $\widehat{O}_S$ for the pro-constant sheaf of rings \[\widehat{O}_S:=\lim_n \underline{O/\ideal{p}^{n+1}}_{S}.\] If $L$ is any finitely generated $O$-module then we write $\widehat{L}_S$ for the pro-constant sheaf of $\widehat{O}_S$-modules \[\widehat{L}_S:=\lim_n \underline{L/\ideal{p}^{n+1} L}_{S}.\] If $F$ and $G$ are two $\widehat{O}_S$-modules over $S$ we write $F\otimes_{O} G$ for the $\widehat{O}_S$-module $F\otimes_{\widehat{O}_S} G$ and $\underline{\mathrm{Hom}}_S^{O}(F, G)$ for the sheaf of $\widehat{O}_S$-homomorphisms $F\to G$. Moreover, if $G=\widehat{L}_S$ for some finite rank projective $O$-module $L$ we shall just write $F\otimes_{O} L$ for $F\otimes_{\widehat{O}_S} \widehat{L}_S$.
\subsection{} A rank one $O$-local system over a $\ideal{p}$-adic sheaf $S$ is a sheaf $\mc{L}$ of $\widehat{O}_{S}$-modules with the property that there exists a cover $(S_i\to S)_{i\in I}$ and rank one projective $O$-modules $(L_i)_{i\in I}$ such that $\mc{L}\times_{S}S_i\overset{\sim}{\longrightarrow} \underline{\widehat{L}_i}_{S_i}$. We denote by $\mc{CL}_{O}$ the moduli stack of rank one $O$-local systems over $\mathrm{Sh}_{\mathrm{Spf}(O)}$. We list the following (usual) constructions and properties of $O$-local systems:
\begin{enumerate}[label=(\roman*)]
\item The tensor product $\mc{L}\otimes_{O}\mc{L}'$ of two rank one $O$-local systems $\mc{L}$ and $\mc{L}'$ (in the category of $\widehat{O}_S$-modules) is again a rank one $O$-local system.
\item The sheaf of $\widehat{O}_S$-homomorphisms $\underline{\mathrm{Hom}}_{S}^{O}(\mc{L}, \mc{L}')$ is again a rank one $O$-local system and defining $\mc{L}^\vee:=\underline{\mathrm{Hom}}_S^{O}(\mc{L},\widehat{O}_S)$ we have $\underline{\mathrm{Hom}}_{S}^{O}(\mc{L}, \mc{L}')\overset{\sim}{\longrightarrow} \mc{L}'\otimes_{O}\mc{L}^\vee$.
\item The sheaf of automorphisms $\underline{\mathrm{Aut}}_S^{O}(\mc{L})$ of a rank one $\widehat{O}_S$-local system $\mc{L}$ is isomorphic to $\widehat{O}_S^\times:=\lim_{n}\underline{(O/\ideal{p}^{n+1})^\times}_S$.
\item The sheaf of $\widehat{O}_S$-isomorphisms $\underline{\mathrm{Isom}}_S(\mc{L}, \mc{L}')$ is pro-finite and \'etale over $S$ and is an $\widehat{O}^\times_S$-torsor over $S$. We denote by $\rho_{\mc{L}/S}\in H^1(S, \widehat{O}^\times_S)$ the class of the torsor $\underline{\mathrm{Isom}}_S(\mc{L}, \widehat{O}_S)$ so that the map \[\mc{L}\mapsto \rho_{\mc{L}/S}\in H^1(S, \widehat{O}^\times_S)\] defines a bijection between isomorphism classes of rank one $O$-local systems over $S$ and $H^1(S, \widehat{O}^\times_S)$.
\end{enumerate}
\subsection{} Let $\widehat{O}_S\to \mc{O}_{S}$ be the unique homomorphism whose pull-back to $S_n=S\times_{\mathrm{Spf}(O)}\mathrm{Spec}(O/\ideal{p}^{n+1})$ is \[\widehat{O}_{S_n}\to \underline{O/\ideal{p}^{n+1}}_{S_n}\to \mc{O}_{S_n}\] where the second map is induced by the structure map $S_n\to \mathrm{Spec}(O/\ideal{p}^{n+1})$.
A strict formal $O$-module over $S$ is a formal group $F$ over $S$ equipped with the structure of a $\widehat{O}_S$-module which is strict with respect to the homomorphism $\widehat{O}_S\to \mc{O}_S$ (cf.\ (\ref{subsec:strict-formal-modules})). Recall that this means that the two actions of $\widehat{O}_S$ on the $\mc{O}_S$-module $\underline{\mathrm{Lie}}_{F/S}$, coming from the action of $\widehat{O}_S$ on $F$ and the homomorphism $\widehat{O}_S\to \mc{O}_S$, coincide.
If $\mc{L}$ is a rank one $O$-local system over $S$ then $\mc{L}$ (as an $\widehat{O}_S$-module) satisfies condition (P) of (\ref{subsec:sheaves-of-rings}) as locally it is isomorphic to $\widehat{O}_S$. So we may apply (\ref{coro:strict-tensor}) to see that if $F$ is a strict formal $O$-module over $S$ and $\mc{L}$ is a rank one $O$-local system over $S$ then $F\otimes_{O} \mc{L}$ is again a strict formal $O$-module over $S$ (of the same dimension as $F$).
Finally, we define the $\ideal{p}^n$-torsion of a strict formal $O$-module $F$ over $S$ to be the kernel of the homomorphism \[i_{\ideal{p}^n}: F\to F\otimes_{O}\ideal{p}^{-n}\] induced by the inclusion $O\to \ideal{p}^{-n}$.
\begin{prop}\label{prop:formal-o-mod-frob} Let $F$ be a strict formal $O$-module of dimension one over $S$. Then
\begin{enumerate}[label=\textup{(\roman*)}]
\item locally on $S$ there exists an isomorphism $\rho: F\overset{\sim}{\longrightarrow} \widehat{\mathbf{A}}^1_S$ with the property that for all $\zeta\in \mu_{N\ideal{p}-1}\subset O^\times$ we have $\rho\circ [\zeta]_F=\zeta\cdot \rho$,
\item $\colim_n F[\ideal{p}^n]=F$, and
\item \label{item:frob-is-p-torsion} $\ker(\mathrm{Fr}_{F_0/S_0}^{N\ideal{p}^n})\subset F_0[\ideal{p}^n]$.
\end{enumerate}
\end{prop}
\begin{proof} The claims are local on $S$ so that we assume that $S=\mathrm{Spec}(A)$, $\ideal{p}^r A=(0)$ for some $r\geq 1$ as $S$ is $\ideal{p}$-adic, and by (\ref{prop:formal-explicit-intrinsic}) we may also assume that $F=\widehat{\mathbf{A}}^1_S$.
(i) This is Lemma 4.1.2 of \cite{Lubin64}.
Before we show (ii) and (iii) let us fix some notation and make some reductions. If $a\in O$ let us write $[a](T)\in A[[T]]$ for the power series defining the multiplication by $a$ map \[a:F= \widehat{\mathbf{A}}^1_S\to F=\widehat{\mathbf{A}}^1_S.\] We also choose a generator $(\pi)=\ideal{p}$ so that $\ker(\pi^n)=F[\ideal{p}^n]$ for all $n\geq 0$. Considering the coefficients of the series \[[\pi](T)=c_1 T+c_2 T^2+\cdots \in A[[T]],\] the strictness of the action of $\widehat{O}_S$ shows that $c_1=\pi$ and by (i) we may assume that $[\zeta](T)=\zeta T$ for $\zeta\in \mu_{N\ideal{p}-1}\subset O^\times$. The relation \[\zeta [\pi](T)=[\zeta]([\pi](T))=[\pi]([\zeta](T))=[\pi](\zeta T)\] for all $\zeta\in \mu_{N\ideal{p}-1}$ then shows that $c_2=\cdots =c_{N\ideal{p}-1}=0$. Thus we may assume that \[[\pi](T)=\pi T \bmod T^{N\ideal{p}}.\] We note that as $\ideal{p}^r A=(0)$ we have \[[\pi^r](T)=0\bmod T^{N\ideal{p}}.\]
(ii) It is enough to show that for all $A$-algebras $B$ and $b\in F(\mathrm{Spec}(B))=\widehat{\mathbf{A}}^1_S(\mathrm{Spec}(B))=\mathrm{nil}(B)$ there is an $n$ such that $[\pi^n](b)=0$. However, as $[\pi^r](T)=0\bmod T^{N\ideal{p}}$ we see that $[\pi^{rm}](b)=0$ whenever $b^{m N\ideal{p}}=0$ which shows the claim.
(iii) We may assume that $S=S\times_{\mathrm{Spf}(O)} \mathrm{Spec}(O/\ideal{p})=S_0$ so that $\ideal{p} A=0$. Then \[[\pi](T)=c_{N\ideal{p}} T^{N\ideal{p}}\bmod T^{N\ideal{p}+1}\] and hence \[[\pi^n](T)=c_{N\ideal{p}}^{\frac{N\ideal{p}^{n}-1}{N\ideal{p}-1}}T^{N\ideal{p}^n}\bmod T^{N\ideal{p}^{n}+1}.\] Thus if $\mathrm{Spec}(B)$ is an affine $\ideal{p}$-adic $\mathrm{Spec}(A)$-scheme and if $b\in \mathrm{nil}(B)$ satisfies $\mathrm{Fr}_B^{N\ideal{p}^n}(b)=b^{N\ideal{p}^n}=0$ we have \[[\pi^n](b)=0\bmod b^{N\ideal{p}^n}=0\] and therefore $\ker(\mathrm{Fr}_{F/S}^{N\ideal{p}^n})\subset F[\ideal{p}^n]$ for all $n\geq 1$.
\end{proof}
\subsection{}\label{subsec:definition-lubin-tate-module} A Lubin--Tate $O$-module over $S$ is a strict formal $O$-module $F$ over $S$ of dimension one (cf.\ (\ref{subsec:dimension-formal-groups})) with the property that the homomorphism $i_\ideal{p}: F\to F\otimes_{O}\ideal{p}^{-1}$ is finite locally free of degree $N\ideal{p}$ (in particular it is affine and faithfully flat). A morphism of Lubin--Tate $O$-modules is just a homomorphism of the underlying $\widehat{O}_S$-modules. We denote by $\mc{M}_{LT}$ the moduli stack of Lubin--Tate $O$-modules over $\mathrm{Sh}_{\mathrm{Spf}(O)}$.
\begin{prop} If $F/S$ is a Lubin--Tate $O$-module and $\mc{L}/S$ is an $O$-local system then $F\otimes_{O}\mc{L}/S$ is a Lubin--Tate $O$-module.
\end{prop}
\begin{proof} We have that $F\otimes_{O}\mc{L}$ is a strict $O$-module of dimension one by (\ref{coro:strict-tensor}). The homomorphism \[i_\ideal{p}: F\otimes_{O}\mc{L}\to (F\otimes_{O}\mc{L})\otimes_{O}\ideal{p}^{-1}\] is finite locally free of degree $N\ideal{p}$ as this can be checked locally, e.g.\ when $\mc{L}\overset{\sim}{\longrightarrow} \widehat{O}_S$, and in this case it is obvious.
\end{proof}
\begin{coro}\label{prop:cm-tensor-homs-lt} For each pair $\mc{L}, \mc{L}'$ of rank one $O$-local systems over $S$ and each pair $F, F'$ of Lubin--Tate $O$-modules over $S$ the natural map \[\underline{\mathrm{Hom}}_S^{O}(F, F')\otimes_{O}\underline{\mathrm{Hom}}_S^{O_K}(\mc{L}, \mc{L}')\to \underline{\mathrm{Hom}}_S^{O}(F\otimes_{O_K}\mc{L}, F'\otimes_{O_K}\mc{L}')\] is an isomorphism.
\end{coro}
\begin{proof} This follows from (\ref{prop:tensor-properties-lie-hom}).
\end{proof}
\begin{lemm}\label{lemm:p-torsion-lifts-frob-in-lubin-tate} Let $S$ be a sheaf of characteristic $\ideal{p}$, $n\geq 0$ and $F$ be a Lubin--Tate $O$-module over $S$. Then $F[\ideal{p}^n]=\ker(\mathrm{Fr}^{N\ideal{p}^n}_{F/S})$.
\end{lemm}
\begin{proof} We may work locally on $S$ and so assume that $S=\mathrm{Spec}(A)$ and that $F =\widehat{\mathbf{A}}^1_S$. By (iii) of (\ref{prop:formal-o-mod-frob}) we have $\ker(\mathrm{Fr}^{N\ideal{p}^n}_{F/S})\subset F[\ideal{p}^n]$. As $F = \widehat{\mathbf{A}}^1_S$, it follows that $\ker(\mathrm{Fr}^{N\ideal{p}^n}_{F/S})$ is finite locally free of rank $N\ideal{p}^n$ over $S$. As $F[\ideal{p}^n]$ is also finite locally free of rank $N\ideal{p}^n$ the closed immersion $\ker(\mathrm{Fr}^{N\ideal{p}^n}_{F/S})\subset F[\ideal{p}^n]$ must be an isomorphism.
\end{proof}
\begin{coro}\label{coro:tensor-p-equals-frobenius-lubin-tate} For each $n\geq 0$ there is a unique isomorphism of functors \[\nu_{\ideal{p}^n}: -\otimes_{O}\ideal{p}^{-n} \overset{\sim}{\longrightarrow} \mathrm{Fr}^{N\ideal{p}^n*}(-)\] on $\mc{M}_{LT}\times_{\mathrm{Spf}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p})$ such that for all Lubin--Tate $O$-modules $F$ over characteristic $\ideal{p}$-sheaves $S$ the diagram \[\xymatrix{&F\ar[dr]^{\mathrm{Fr}_{F/S}^{N\ideal{p}^n}}\ar[dl]_-{i_{\ideal{p}^n}}&\\
F\otimes_{O}\ideal{p}^{-n} \ar[rr]^{\nu_{\ideal{p}^n}}_\sim && \mathrm{Fr}^{N\ideal{p}*}(F)}\] commutes.
\end{coro}
\begin{proof} For any Lubin--Tate $O$-module $F/S$ the two homomorphisms $i_{\ideal{p}^n}$ and $\mathrm{Fr}_{F/S}^{N\ideal{p}}$ are epimorphisms with the same kernel (\ref{lemm:p-torsion-lifts-frob-in-lubin-tate}) and the claim follows.
\end{proof}
\begin{rema} The result (\ref{coro:tensor-p-equals-frobenius-lubin-tate}) can be read as saying that the moduli stack $\mc{M}_\mathrm{LT}$ admits an endomorphism $-\otimes_{O}\ideal{p}^{-1}: \mc{M}_\mathrm{LT}\to \mc{M}_\mathrm{LT}$ which (upto canonical isomorphism) lifts the $N\ideal{p}$-power Frobenius endomorphism. This kind of structure is very closely related to the notions of $\Lambda$-structures, Witt vectors and arithmetic jets (due to Borger and Buium) which we define and study in Chapter 3. It is essentially the topic of Chapter 4 to study and exploit this relationship in the context of CM elliptic curves (for which we prove an analogue of (\ref{coro:tensor-p-equals-frobenius-lubin-tate}) in Chapter 2, see (\ref{coro:tensor-p-equals-frobenius})).
\end{rema}
\begin{prop}\label{prop:lubin-tate-faltings} \begin{enumerate}[label=\textup{(\roman*)}]
\item If $F$ is a Lubin--Tate $O$-module over $S$ the natural homomorphism \[\widehat{O}_S\to \underline{\mathrm{End}}_S^O(F)\] is an isomorphism.
\item If $F, F'$ are a pair of Lubin--Tate $O$-modules over $S$ then $\underline{\mathrm{Hom}}_S^{O}(F, F')$ is an $O$-local system over $S$ and the evaluation homomorphism \[F\otimes_{O}\underline{\mathrm{Hom}}_S^{O}(F, F')\to F'\] is an isomorphism.
\end{enumerate}
\end{prop}
\begin{proof} The proof of these statements is an application of Faltings' generalised Cartier duality \cite{Faltings02}. It would take us too far afield to give the proof here and so we defer it to the appendix (see (i) and (ii) of (\ref{coro:lubin-tate-faltings})).
\end{proof}
\begin{prop}\label{prop:lubin-tate-torsor} The functor \[\mc{M}_{LT}\times \mc{CL}_{O}\to \mc{M}_{LT}\times \mc{M}_{LT}: (F,\mc{L})\mapsto (F, F\otimes_{O}\mc{L})\] is an equivalence of stacks.
\end{prop}
\begin{proof} As with (\ref{prop:lubin-tate-faltings}), the proof of this statement will be given in the appendix (see (iii) of (\ref{coro:lubin-tate-faltings})).
\end{proof}
\begin{rema} It would be preferable to have elementary proofs of (\ref{prop:lubin-tate-faltings}) and (\ref{prop:lubin-tate-torsor}) which do not rely on the machinery of Faltings' generalised Cartier duality.
\end{rema}
\begin{exem}\label{exem:lubin-tate-modules-exist} Let us now at least tell the reader that there do exist Lubin--Tate $O$-modules. First, if $O=\mathbf{Z}_p$ then the $p$-power torsion in $\mathbf{G}_{\mathrm{m}/\mathrm{Spf}(O)}$: \[\mu_{p^\infty}=\colim_n \mu_{p^n}\subset \mathbf{G}_{\mathrm{m}/\mathrm{Spf}(O)}\] is a Lubin-Tate $\mathbf{Z}_p$-module and it also has the property that the multiplication by $p$ map $p: \mu_{p^\infty}\to \mu_{p^\infty}$ reduces to the $p$-power Frobenius after base change along $\mathrm{Spec}(\mathbf{F}_p)\to \mathrm{Spf}(\mathbf{Z}_p)$.
In fact, for any $O$, given a generator $\pi$ of the prime ideal $\ideal{p}$, there is a unique (upto isomorphism) Lubin--Tate $O$-module $F_\pi$ over $\mathrm{Spf}(O)$ with the property that the endomorphism $\pi: F_\pi\to F_\pi$ reduces to the $N\ideal{p}$-power Frobenius map after base change along $\mathrm{Spec}(\mathbf{F}_\ideal{p})\to \mathrm{Spf}(O)$ (see \S 3.5 Chapter VI \cite{CasselsFrohlich67}). Moreover, there exists a unique isomorphism $F_\pi\overset{\sim}{\longrightarrow} \widehat{\mathbf{A}}^1_{\mathrm{Spf}(O)}$ under which the multiplication by $\pi$ is represented by the power series $[\pi](T)=\pi T+T^{N\ideal{p}}$.
A consequence of this is that the morphism $\mc{M}_\mathrm{LT}\to \mathrm{Spf}(O)$ admits a section. The statement of (\ref{prop:lubin-tate-torsor}) above can now be interpreted as saying that $\mc{M}_\mathrm{LT}$ is (in a stack theoretic sense) a torsor under the (stack theoretic) group $\mc{CL}_{O}$ albeit a trivial one:
\end{exem}
\begin{coro}\label{coro:lubin-tate-moduli-trivial} Fixing a Lubin--Tate $O$-module $F$ over $\mathrm{Spf}(O)$ the functor \[\mc{CL}_O\to \mc{M}_\mathrm{LT}: \mc{L}/S\mapsto F_S\otimes_{O}\mc{L}/S\] is an equivalence of stacks.
\end{coro}
\begin{coro}\label{coro:lift-lubin-tate} Let $S_0\to S$ be a nilpotent immersion of $\ideal{p}$-adic sheaves. The functor \[\mc{M}_{LT}(S)\to \mc{M}_{LT}(S_0): F\mapsto F_{S_0}\] is an equivalence of categories.
\end{coro}
\begin{proof} This follows from (\ref{coro:lubin-tate-moduli-trivial}) and the corresponding obvious claim for $\mc{CL}_{O}$ (which is a moduli space of pro-\'etale objects).
\end{proof}
\begin{coro}\label{prop:lubin-tate-hom} Let $f: F\to F'$ be a homomorphism of Lubin--Tate $O$-modules over $S$. Then there is a unique decomposition $S=S_{(0)}\amalg_{0\leq n<\infty} S_{\ideal{p}^n}$ such that $f_{S_{(0)}}$ is the zero map and such that $\ker(f_{S_{\ideal{p}^n}})=F_{S_{\ideal{p}^n}}[\ideal{p}^n]$ for $0\leq n <\infty$.
\end{coro}
\begin{proof} We have $F'\overset{\sim}{\longrightarrow} F\otimes_{O}\mc{L}$ for some rank one $O$-local system $\mc{L}$ by (\ref{prop:lubin-tate-torsor}) and the homomorphism \[f: F\to F\otimes_{O} \mc{L}\] is of the form $\mathrm{id}_F\otimes_{O} h$ for some homomorphism \[h: \widehat{O}_S\to \mc{L}\] by (i) of (\ref{prop:tensor-properties-lie-hom}).
Define the sub-sheaves $S_{(0)}\subset S$ (resp. $S_{\ideal{p}^n}\subset S$ for $0\leq n<\infty$) by the property that $h_{S_{(0)}}$ is the zero map (resp. $h_{S_{\ideal{p}^n}}$ factors as \[\widehat{O}_{S_{\ideal{p}^n}} \overset{\sim}{\longrightarrow} \ideal{p}^n\otimes _{O}\mc{L}_{S_{\ideal{p}^n}}\to \mc{L}_{S_{\ideal{p}^n}}\] where the second map is multiplication). These definitions combined with $f=\mathrm{id}_F\otimes_{O} h$ show that $f_{S_{(0)}}$ is the zero map that $f_{S_{\ideal{p}^n}}=\mathrm{id}_{F_{S_{\ideal{p}^n}}}\otimes_{O} h_{S_{\ideal{p}^n}}$ factors as \[F_{S_{\ideal{p}^n}}\overset{\sim}{\longrightarrow} F'_{S_{\ideal{p}^n}}\otimes_{O}\ideal{p}^n\to F'_{S_{\ideal{p}^n}}\] where the second map is multiplication and hence $\ker(f_{S_n})=F_{S_n}[\ideal{p}^n]$. Moreover, it is clear that $S_{(0)}$ and all the $S_{\ideal{p}^n}$ for $0\leq n<\infty$ are disjoint and so to prove our claim we need to show that $S_{(0)} \amalg_{0\leq n< \infty} S_{\ideal{p}^n}\to S$ is an epimorphism.
For this we may localise $S$ and assume that $\mc{L}=\widehat{O}_S$ and $h=a\in O\subset \widehat{O}_S(S)$. Then either $a=0$, in which case $h$ is the zero map so that $S_{(0)}\overset{\sim}{\longrightarrow} S$, or $a\neq 0$, in which case $a\cdot O=\ideal{p}^n$ for some integer $n\geq 0$ and $h=a$ factors as \[\widehat{O}_S\overset{\sim}{\longrightarrow} \widehat{O}_S\otimes_{O}\ideal{p}^n\to \widehat{O}_S,\] so that $S_{\ideal{p}^n}\overset{\sim}{\longrightarrow} S$. It follows that $S_{(0)}\amalg_{0\leq i<\infty}S_{\ideal{p}^n}=S$.
\end{proof}
\subsection{}\label{subsec:lubin-tate-reciprocity} Classically one relates the reciprocity map of the local field $K$ to Lubin--Tate $O$-modules as follows. Write $S=\mathrm{Spf}(O_{K^\mathrm{sep}})$ and let $F$ be the unique Lubin--Tate $O$-module over $\mathrm{Spf}(O)$ such that $\pi: F\to F$ reduces to the $N\ideal{p}$-power Frobenius after base change to $\mathrm{Spec}(\mathbf{F}_\ideal{p})$. Then for each $r\geq 0$, the $O/\ideal{p}^r$-module $F[\ideal{p}^r](S)$ is free of rank one and $\colim_r (F[\ideal{p}^r](S))\overset{\sim}{\longrightarrow} K/O$ (non-canonically). We then obtain the character \[\rho_\pi: G(K^\mathrm{sep}/K)\to \lim_r \mathrm{Aut}_{O}(F[\ideal{p}^r](S))=\lim_r (O/\ideal{p}^r)^\times=O^\times\] defining the action of $G(K^\mathrm{sep}/K)$ on $\colim_r (F[\ideal{p}^r](S))$. The relationship between the character $\rho_\pi$ and the reciprocity map (\ref{eqn:local-rec-map}) is that for all $\sigma\in W(K^\mathrm{sep}/K)$ we have \begin{equation} \sigma|_{K^\mathrm{ab}}=(\pi^{v_K(\sigma)}\rho_\pi(\sigma)^{-1}, K^\mathrm{ab}/K)\label{eqn:lub-rec-classical}\end{equation} (see \S 3.7 Chapter VI of \cite{CasselsFrohlich67}).
We would now like to show how one can construct the reciprocity map of local class field theory in a slightly more abstract but direct way using only the Frobenius lift property (\ref{coro:tensor-p-equals-frobenius-lubin-tate}) of Lubin--Tate $O$-modules and the $\mc{CL}_{O_K}$-torsor structure of $\mc{M}_\mathrm{LT}$. Write $\overline{S}=\mathrm{Spec}(\mathbf{F}_\ideal{p}^\mathrm{sep})\subset S=\mathrm{Spf}(O_{K^\mathrm{sep}})$ and for a Lubin--Tate $O$-module $F\to S$ write $\overline{F}=F\times_{S} \overline{S}$ and $F_r=F[\ideal{p}^r]$ for each $r\geq 0$.
\begin{prop}\label{theo:main-theorem-lubin-tate} Let $\sigma\in W(K^\mathrm{sep}/K)$ satisfy $v_K(\sigma)=n\geq 0$. Then \begin{enumerate}[label=\textup{(\roman*)}]
\item for each Lubin--Tate $O$-module $F$ over $S$ there is a unique isomorphism $\nu_\sigma:F\otimes_{O}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} \sigma^*(F)$ whose pull-back along $\overline{S}\to S$ is the isomorphism $\nu_{\ideal{p}^{n}}:\overline{F}\otimes_{O}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} \mathrm{Fr}^{N\ideal{p}^n*}(\overline{F})$ of \textup{(\ref{coro:tensor-p-equals-frobenius-lubin-tate})},
\item there is a generator $\chi_K(\sigma)\in K^\times$ of $\ideal{p}^n$, independent of $F$, such that for all $r\geq 0$, the isomorphism induced by $\nu_\sigma$ on the $S$-points of the $\ideal{p}^r$-torsion of $F$ \[F_r(S)\otimes_{O}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} \sigma^*(F_r)(S)\] is equal to \[F_r(S)\otimes_{O}\ideal{p}^{-n}\stackrel{F_r(\sigma)\otimes \chi_K(\sigma)} {\longrightarrow}F_r(\sigma_!(S))=\sigma^*(F_r)(S),\]
\item if $\tau\in W(K^\mathrm{sep}/K)$ also satisfies $v_K(\tau)\geq 0$ then in the notation of \textup{(ii)} we have $\chi_K(\sigma\tau)=\chi_K(\sigma)\chi_K(\tau)$, and
\item $\sigma|_{K^\mathrm{ab}}=(\chi_K(\sigma), K^\mathrm{ab}/K)$.
\end{enumerate}
\end{prop}
\begin{proof} (i) The existence of $\nu_{\ideal{p}^n}$ follows from (\ref{coro:lift-lubin-tate}) applied to the nilpotent immersion $S_\ideal{p}\to S$ and the fact that the restriction of $\sigma$ to $\overline{S}$ is equal to $\mathrm{Fr}^{N\ideal{p}^n}$.
(ii) For all $r\geq 0$ we have that $F_r(S)$ is a free rank one $O_K/\ideal{p}^r$-module and so writing $\nu_{\sigma, r}$ for the restriction of $\nu_{\sigma}$ to the $S$-valued points of the $\ideal{p}^r$-torsion, we set \begin{eqnarray*} \chi_{K, r}(\sigma)=F_r(\sigma)^{-1}\circ \nu_{\sigma, r} &\in& \mathrm{Isom}_O(F_r(S)\otimes_O \ideal{p}^{-n}, F_r(S))\\&=&\mathrm{Isom}_O(F_r(S), F_r(S)\otimes_O \ideal{p}^n)\\
&\subset& \ideal{p}^n\otimes_{O} O/\ideal{p}^r.\end{eqnarray*} Then $\chi_K(\sigma)$ is given by the limit $\lim_{r} \chi_{K, r}(\sigma)\in \lim_r \ideal{p}^n \otimes_{O}O/\ideal{p}^r=\ideal{p}^n$. It is a generator of $\ideal{p}^n$ as $\chi_{K, r}(\sigma)$ is a generator of the free rank one $O/\ideal{p}^r$-module $O/\ideal{p}^r\otimes_O \ideal{p}^n$ for all $r\geq 0$.
First, it is clear that $\chi_K(\sigma)$ depends only on the isomorphism class of $F$. However, by (\ref{prop:lubin-tate-torsor}), every other Lubin--Tate $O$-module over $S$ is of the form $F\otimes_{O}\mc{L}$ for some rank one $O$-local system $\mc{L}$. But every rank one $O$-local system $\mc{L}$ over $\mathrm{Spf}(O_{K^\mathrm{sep}})$ is pro-constant and isomorphic to $\widehat{O}_S$. Therefore, all Lubin--Tate $O$-modules over $S$ are isomorphic and $\chi_K(\sigma)$ is independent of the Lubin--Tate $O$-module $F$ over $S$ (admittedly there is only one!).
(iii) Write $m=v_K(\tau)$. Then the two isomorphisms \[\nu_{\sigma\tau} \quad \text{ and } \quad \tau^*(\nu_\sigma)\circ (\nu_\tau\otimes_{O} \ideal{p}^{-m})\] between \[F\otimes_{O}\ideal{p}^{-n-m} \overset{\sim}{\longrightarrow} (\sigma\circ \tau)^*(F)=\tau^*(\sigma^*(F)))\] both pull-back to the isomorphism $\nu_{\ideal{p}^{n+m}}$ of (\ref{coro:tensor-p-equals-frobenius-lubin-tate}) along $\overline{S}\to S$. By the uniqueness in (i) we get \[\nu_{\sigma\tau} = \tau^*(\nu_\sigma)\circ (\nu_\tau\otimes_{O} \ideal{p}^{-m})\] from which we find the relation $\chi_K(\sigma\tau)=\chi_K(\sigma)\chi_K(\tau)$.
(iv) With notation as in (\ref{subsec:lubin-tate-reciprocity}) take $F= F_{\pi}\times_{\mathrm{Spf}(O)}S$. As $\pi^n: F_\pi\to F_\pi$ lifts the $N\ideal{p}^n$-power Frobenius endomorphism of $F_\pi\times_{\mathrm{Spf}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p})$ the (unique) isomorphism \[\nu_\sigma: F\otimes_{O}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} \sigma^*(F)\] of (i) is given by \begin{equation}\label{eqn:def-nu-descend}F\otimes_{O}\ideal{p}^{-n}\stackrel{\pi^n}{\longrightarrow} F\stackrel{d_\sigma}{\longrightarrow} \sigma^*(F)\end{equation} where $d_\sigma: F\overset{\sim}{\longrightarrow} \sigma^*(F)$ is the descent isomorphism (coming from the fact that $F=F_\pi\times_{\mathrm{Spf}(O)} S$ is defined over $\mathrm{Spf}(O)$). The isomorphism $d_\sigma$ on the $S$-points of the $\ideal{p}^r$-torsion is given by \[F_r(S) \stackrel{\rho_\pi(\sigma)^{-1} \cdot F_r(\sigma)}{\longrightarrow} F_{r}(\sigma_!(S))=\sigma^*(F_r)(S)\] so that $\nu_\sigma$ on the $S$-points of the $\ideal{p}^r$ torsion is given by (cf.\ (\ref{eqn:def-nu-descend})) \[F_r(S)\otimes_{O}\ideal{p}^{-n}\stackrel{1\otimes \pi^n}{\longrightarrow} F_r(S)\stackrel{\rho_\pi(\sigma)^{-1}\cdot F_r(\sigma)}{\longrightarrow} F_r(\sigma_!(S))=\sigma^*(F_r)(S).\] Therefore $\chi_K(\sigma)=\pi^n\rho_\pi(\sigma)^{-1}$ and by (\ref{eqn:lub-rec-classical}) we get \[\sigma|_{K^\mathrm{ab}}=(\pi^n\rho_\pi(\sigma)^{-1}, K^\mathrm{ab}/K)=(\chi_K(\sigma), K^\mathrm{ab}/K).\]
\end{proof}
\begin{rema}\label{rema:loca-reciprocity} From (i), (ii) and (iii) of (\ref{theo:main-theorem-lubin-tate}) we see that we can associate to any element of $\sigma\in v_K^{-1}(\mathbf{N}_{\geq 0})\subset W(K^\mathrm{sep}/K)$ an element $\chi_K(\sigma)\in K^\times$ and that this association is multiplicative. It therefore extends to a homomorphism \begin{equation}\label{eqn:local-rec}W(K^\mathrm{sep}/K)\to K^\times: \sigma\mapsto \chi_K(\sigma)\end{equation} and (iv) of (\ref{theo:main-theorem-lubin-tate}) states that this map satisfies \[\sigma|_{K^\mathrm{ab}}=(\chi_K(\sigma), K^\mathrm{ab}/K)\] for all $\sigma\in W(K^\mathrm{sep}/K).$ Thus we have derived the local reciprocity map (\ref{eqn:local-rec-map}) using nothing more than the basic properties of Lubin--Tate $O$-modules (in particular, the Frobenius lifting property (\ref{coro:tensor-p-equals-frobenius-lubin-tate}) and the $\mc{CL}_O$-torsor structure of $\mc{M}_\mathrm{LT}$ (\ref{prop:lubin-tate-torsor})).
\end{rema}
\begin{rema}\label{rema:classification-lubin} If $K\subset L\subset K^\mathrm{sep}$ is a finite extension and $F/\mathrm{Spf}(O_L)$ is a Lubin--Tate $O$-module let us write \[\rho_{F/O_L}: G(K^\mathrm{sep}/L)\to O^\times\] for the (continuous) character defining the action of $G$ on \[\colim_{r} (F[\ideal{p}^r](\mathrm{Spf}(O_{K^\mathrm{sep}})))\overset{\sim}{\longrightarrow} K/O.\] Then a continuous character $\rho: G(K^\mathrm{sep}/L)\to O^\times$ is of the form $\rho_{F/O_L}$ if and only if the diagram \[\xymatrix{W(K^\mathrm{sep}/L)\ar[dr]_{\chi_K}\ar[r]^-{\rho_{F/O_L}^{-1}} & O^\times\ar[d]\\
& K^\times}\] commutes where the right vertical map is the inclusion. We mention this mainly as it is analogous to the classification of elliptic curves with complex multiplication over fields in terms of their associated characters we will give in Chapter 2 (see (iii) of (\ref{subsec:reinterpret-character-idelic})).
\end{rema}
\begin{rema} Of course, the theory of Lubin--Tate $O$-modules and the local reciprocity map are themselves not particularly complicated and one can derive the reciprocity map in any number of ways. We believe the this approach above has some advantages over the classical one, first and foremost it is choice free, and secondly one gets the whole of the local reciprocity map right of the bat, rather than first finding a character \[\rho_\pi: G(K^\mathrm{sep}/K)\to O^\times\] which one then restricts to $W(K^\mathrm{sep}/K)\subset G(K^\mathrm{sep}/K)$, takes the reciprocal of and then multiplies by the character \[W(K^\mathrm{sep}/K)\to K^\times: \sigma\mapsto \pi^{v_K(\sigma)}.\] The derivation of the local reciprocity map we have given is also analogous to the derivation (\ref{prop:compute-the-homomorphisms-h}) of the global reciprocity map for imaginary quadratic fields which we will give using CM elliptic curves in Chapter 2. Moreover, in the case of CM elliptic curves and imaginary quadratic fields, the situation is somewhat more delicate --- one cannot just find CM elliptic curves with suitable properties from which one can construct the global reciprocity map in the same way one can with Lubin--Tate $O$-modules.
\end{rema}
\section{The global reciprocity map} In this section we recall the basic objects required to define, and then we recall, the global reciprocity map (\ref{prop:global-reciprocity}) associated to a global field $K$.
\subsection{}\label{subsec:finite-extensions-frobenius} Let $K$ be a global field and fix maximal abelian and separable extensions $K\subset K^\mathrm{ab}\subset K^\mathrm{sep}$. Write $\mc{P}_K$ for the set of places of $K$, and as usual, if $K$ is a number field we identify the non-archimedian places $v\in \mc{P}_K$ with the prime ideals $\ideal{p} \subset O_K$ of the ring of integers of $K$. For $v\in K$ we write $K_v$ for the completion of $K$ with respect to $v$ and if $v$ is archimedian $O_{K_v}\subset K_v$ for the ring of local integers. We write $\mc{P}_K^{\mathrm{arch}}\subset \mc{P}_K$ for the subset of archimedian places (which is of course empty if $K$ is a function field).
Let $K\subset L\subset K^\mathrm{sep}$ be a finite Galois extension. If $w\in \mc{P}_L$ is a non-archimedian place and the extension $L/K$ is unramified at $w$ then we write $\sigma_{L/K, w}\in G(L/K)$, or just $\sigma_w$, for the Frobenius element associated to $w$. If $L/K$ is abelian (so that $\sigma_{L/K, w}$ depends only on $w|_K=v\in \mc{P}_K$) then we write $\sigma_{L/K, v}$, or again just $\sigma_v$, for $\sigma_{L/K, w}$.
\subsection{} For each finite set $S\subset \mc{P}_K$ containing all archimedian places of $K$ we write $I_{K, S}$ for the topological group \[I_{K, S}= \prod_{v\in \mc{P}_K-S} O_{K_v}^\times\times \prod_{v\in S} K_v^\times\] (the topology being the product topology). For $S\subset S'\subset \mc{P}_K$ the inclusions $I_{K, S}\subset I_{K, S'}$ are open and the group of id\`eles of $K$ is the topological group \[I_K=\colim_{S} I_{K, S}\] (the topology being the colimit topology). The diagonal embedding $K^\times\to I_K$ makes $K^\times$ a discrete subgroup of $I_K$ and the id\`ele class group of $K$ is the quotient $C_K=I_K/K^\times$. The embeddings $K_v^\times\to I_K\to C_K$ for each place $v$ of $K$ make $K_v^\times$ a closed subgroup of $I_K$ and $C_K$.
\begin{theo}\label{prop:global-reciprocity} There is a unique continuous homomorphism \[C_K\to G(K^\mathrm{ab}/K): s\mapsto (s, K^\mathrm{ab}/K)\] such that for each place $v$ of $K$ and each $K$-linear embedding $K^\mathrm{ab}\to K_v^\mathrm{ab}$ the following diagram commutes \[\xymatrix{C_K\ar[rr]^-{(-, K^\mathrm{ab}/K)} && G(K^\mathrm{ab}/K)\\
K_v^\times\ar[u]\ar[rr]^-{(-, K_v^\mathrm{ab}/K_v)} && G(K_v^\mathrm{ab}/K_v).\ar[u]}\]
\end{theo}
\begin{proof} See \S\S 4--6 Chapter VII of \cite{CasselsFrohlich67}.
\end{proof}
\subsection{}\label{subsec:global-reciprocity-properties} We list the following further properties of the reciprocity map (see \S\S 4--6 Chapter VII of \cite{CasselsFrohlich67}).
\begin{enumerate}[label=\textup{(\alph*)}]
\item If $K'/K$ is any finite Galois extension, with maximal abelian extension $K'^\mathrm{ab}$, the diagram \[\xymatrix{C_{K'}\ar[rr]^-{(-, K'^\mathrm{ab}/K')}\ar[d]_{N_{K'/K}} && G(K'^\mathrm{ab}/K')\ar[d]\\
C_K\ar[rr]^-{(-, K^\mathrm{ab}/K)} && G(K^\mathrm{ab}/K)}\] commutes (where $N_{K'/K}$ denotes the map induced by the norm $I_{K'}\to I_{K}$).
\item The kernel of the reciprocity map is $C_K^\circ\subset C_K$ (a superscript $\circ$ denotes the connected component of the identity of a topological group). If $K$ is a function field then $C_K^\circ$ is trivial and $(-, K^\mathrm{ab}/K)$ is injective. If $K$ is a number field then $C_K^\circ$ is the closure of the sub-group \[\prod_{v\in \mc{P}_K^\mathrm{arch}}K_v^{\times,\circ}\subset C_K\] and $(-, K^\mathrm{ab}/K)$ is surjective. We note that if $\mc{P}_K^\mathrm{arch}=\{\infty\}$ contains only one place then $K_\infty^{\times, \circ}=C_K^\circ$ and we obtain a topological isomorphism \[C_K/K_\infty^{\times,\circ}\overset{\sim}{\longrightarrow} G(K^\mathrm{ab}/K).\]
\end{enumerate}
\section{Global fields with a single `infinite' place and class groups}\label{section:special-place} In this section we describe a variant of the reciprocity map associated to the maximal abelian extension $K^\infty$ of a global field $K$ which is totally split at a fixed place $\infty$ satisfying $\mc{P}_K^\mathrm{arch}\subset \{\infty\}$. If $K$ is a function field then any place $\infty$ satisfies the above property and if $K$ is a number field then the only possibilities are $K=\mathbf{Q}$ or $K$ an imaginary quadratic and in each case $\infty$ equal to the unique archimedian place of $K$.
As mentioned in the introduction to this chapter, the fact that such pairs $(K, \infty)$ should be considered along similar lines is due to Drinfel'd \cite{Drinfeld74} and essentially all we are doing here is collecting the relevant facts for use in the later chapters. It is worth noting here that the abelian extensions of $K$ contained in $K^\infty$ have a long history mainly due to the fact that they are amenable to explicit computation via the use of Drinfel'd modules in case $K$ is a function field, tori if $K=\mathbf{Q}$ and CM elliptic curves if $K$ is an imaginary quadratic field (what Drinfel'd originally called `elliptic modules of rank one'). While we are only concerned with the final case in this thesis, the abstract theory we describe below is valid for arbitrary pairs $(K, \infty)$.
\subsection{}\label{subsec:global-field-with-special-place} Let $K$ be a global field, and fix a place $\infty$ of $K$ such that $\mc{P}^\mathrm{arch}_K\subset \{\infty\}$. We call $\infty$ the infinite place, and the places in $\mc{P}_K-\infty$ the finite places and denote them by $\mc{P}_K^\mathrm{fin}$. As every place $v\neq \infty$ is non-archimedian the subset $O_K=\{a\in K: |a|_v\leq 1 \text{ for all } v\neq \infty\}\subset K$ is a Dedekind domain, and its prime ideals are in bijection with the set $\mc{P}_K^\mathrm{fin}$.
The group of units $O_K^\times$ is finite and we denote its order by $w$. We want to point out that for what follows in this section, and in the following chapters, this fact is quite crucial. It implies in particular that given any ideal $\ideal{f}$ there is an ideal $\ideal{f}|\ideal{f}'$ with the property that $\ideal{f}'$ separates units, i.e.\ the homomorphism \[O_K^\times\to (O_K/\ideal{f}')^\times\] is injective or what is the same $O_{K}^{\times, \ideal{f}'}=\{1\}$ (of course, if $\ideal{f}\neq O_K$ any high power of $\ideal{f}$ will do).
\subsection{} We write $A_{O_K}$ for the topological ring \[\lim_{\a}O_K/\a=\prod_{\ideal{p}} O_{K_\ideal{p}},\] the topology being the product topology or the inverse limit topology induced by the discrete topologies on the finite sets $O_K/\a$ (they are the same). We also view \[A_{O_K}^\times=\lim_{\a} (O_K/\a)^\times=\prod O_{K_\ideal{p}}^\times\] as a topological group via the topology induced from $A_{O_K}$, the product topology or the inverse limit topology (again they are one and the same). For each integral ideal $\ideal{f}$ we denote by $A_{O_K}^{\times, \ideal{f}}$ the open subgroup \[\ker(A_{O_K}^\times\to (O_K/\ideal{f})^\times).\] For each integral ideal $\a$ we equip \[A_{O_K}[\a^{-1}]^\times := \prod_{\ideal{p}|\a}K_\ideal{p}^\times\times \prod_{\ideal{p}\nmid\a} O_{K_\ideal{p}}^\times\] with the product topology. If $\a|\b$ the inclusion $A_{O_K}[\a^{-1}]^\times\subset A_{O_K}[\b^{-1}]^\times$ is open and we equip \[(A_{O_K}\otimes_{O_K}K)^\times=\colim_{\a} A_{O_K}[\a^{-1}]^\times\] with the colimit topology. The natural map \[I_K\to (A_{O_K}\otimes_{O_K} K)^\times,\] induced by forgetting the component at $\infty$, is continuous and surjective and induces topological isomorphism \[I_K/K_\infty^\times\overset{\sim}{\longrightarrow} (A_{O_K}\otimes_{O_K} K)^\times.\]
\subsection{} We will now relate the group $(A_{O_K}\otimes_{O_K}K)^\times$ to certain class groups associated to $O_K$. So let $\ideal{f}\in \mathrm{Id}_{O_K}$ and let $L$ be a projective rank one $O_K$-module. A level-$\ideal{f}$ structure on $L$ is a surjective homomorphism $a: L\to O_K/\ideal{f}$. An $\ideal{f}$-isomorphism $f: (L, a)\overset{\sim}{\longrightarrow} (L', a')$ between a pair of projective rank one $O_K$-modules with level-$\ideal{f}$ structures is an $O_K$-isomorphism $f: L\to L'$ such that $h\circ a'=a$. We denote by $\mathrm{CL}_{O_K}^{(\ideal{f})}$ the set of $\ideal{f}$-isomorphism classes of rank one projective $O_K$-modules with level-$\ideal{f}$ structure. Equipping it with the product \[(L, a)\cdot (L', a')=(L\otimes_{O_K} L', a\otimes_{O_K} a'),\] $\mathrm{CL}_{O_K}^{(\ideal{f})}$ becomes a group, which we call the ray class group of conductor $\ideal{f}$.
If $\a$ is any fractional ideal prime to $\ideal{f}$, the multiplication map \[\a\otimes_{O_K}O_K/\ideal{f}\overset{\sim}{\longrightarrow} O_K/\ideal{f}\] is well defined, and an isomorphism, so that setting $f: \a\to O_K/\ideal{f}$ to be the composition \begin{equation}\a\to \a\otimes_{O_K}O_K/\ideal{f}\overset{\sim}{\longrightarrow} O_K/\ideal{f}\label{eqn:can-level-ideal}\end{equation} equips $\a$ with a level-$\ideal{f}$ structure. We write $[\a]_\ideal{f}=(\a, f)\in \mathrm{CL}_{O_K}^{(\ideal{f})}$ for the corresponding class. This defines a surjective homomorphism \[\mathrm{Id}_{K}^{(\ideal{f})}\to \mathrm{CL}_{O_K}^{(\ideal{f})}: \a\mapsto [\a]_\ideal{f}\] whose kernel is the group $\mathrm{Prin}_{1\bmod \ideal{f}}^{(\ideal{f})}$ of principal fractional ideals $\a=O_K\bmod \ideal{f}$ admitting a generator $a\in K^\times$ with $a=1\bmod \ideal{f}$. If $\ideal{f}|\ideal{f}'$ then \[\mathrm{CL}_{O_K}^{(\ideal{f}')}\to \mathrm{CL}_{O_K}^{(\ideal{f})}: (L, a)\mapsto (L, a \bmod \ideal{f})\] defines a surjective homomorphism and we define the topological group \[\mathrm{CL}_{O_K, \infty}=\lim_\ideal{f} \mathrm{CL}_{O_K}^{(\ideal{f})}\] (the topology being the inverse limit of the discrete topologies on the $\mathrm{CL}_{O_K}^{(\ideal{f})}$). Finally, if $\ideal{f}=O_K$ then we identify $\mathrm{CL}_{O_K}^{O_K}$ with the class group $\mathrm{CL}_{O_K}$ of $O_K$.
\subsection{} Given an element $s\in (A_{O_K}\otimes_{O_K}K)^\times$, we write $(s)\in \mathrm{Id}_K$ for the fractional ideal \[(s)=\prod_{\ideal{p}}\ideal{p}^{v_\ideal{p}(s)}.\] For each integral ideal $\ideal{f}$, we equip $(s)^{-1}$ with the level-$\ideal{f}$ structure \[(s)^{-1}\stackrel{s}{\to} A_{O_K}\to O_K/\ideal{f}\] and write $[s]_\ideal{f}\in \mathrm{CL}_{O_K}^{(\ideal{f})}$ for the corresponding class. This defines a continuous surjective homomorphism \[(A_{O_K}\otimes_{O_K}K)^\times\to \mathrm{CL}_{O_K}^{(\ideal{f})}: s\mapsto [s]_\ideal{f}\] with kernel $K^\times\cdot A_{O_K}^{\times, \ideal{f}}$. Finally, if $\ideal{f}|\ideal{f}'$ the image of $[s]_{\ideal{f}'}$ under $\mathrm{CL}_{O_K}^{(\ideal{f}')}\to \mathrm{CL}_{O_K}^{(\ideal{f})}$ is $[s]_\ideal{f}$ and so taking the limit over $\ideal{f}$ we obtain a homomorphism \begin{equation}[-]:(A_{O_K}\otimes_{O_K}K)^\times\to \mathrm{CL}_{O_K, \infty}=\lim_\ideal{f} \mathrm{CL}_{O_K}{(\ideal{f})}: s\mapsto [s]=\lim_\ideal{f} [s]_\ideal{f}.\label{def:bracket-homo}\end{equation}
\begin{prop}\label{prop:ideles-and-classes} The map $[-]$ is continuous and the sequence \[0\to K^\times\to (A_{O_K}\otimes_{O_K}K)^\times\stackrel{[-]}{\to} \mathrm{CL}_{O_K, \infty}\to 0\] is exact.
\end{prop}
\begin{proof} It is clear that $s\mapsto [s]$ is continuous (as $s\mapsto [s]_\ideal{f}$ is continuous for each $\ideal{f}$) and if we can show that $\ker(s\mapsto [s])=K^\times$ then the surjectivity of $s\mapsto [s]$ follows as $s\mapsto [s]_\ideal{f}$ is surjective for each $\ideal{f}$ and $(A_{O_K}\otimes_{O_K}K)^\times/K^\times=C_K/K_\infty^\times$ is compact.
The kernel of $s\mapsto [s]$ is equal to \[\bigcap_\ideal{f} \ker(s\mapsto [s]_\ideal{f})=\bigcap_\ideal{f} (K^\times \cdot A_{O_K}^{\times, \ideal{f}}).\] If $s$ is an element of this kernel then for all integral ideals $\ideal{f}$ we can write $s=a_\ideal{f} s_\ideal{f}$ where $a_\ideal{f}\in K^\times$ and $s_\ideal{f}\in A_{O_K}^{\times, \ideal{f}}$. The elements $a_\ideal{f}$ and $s_\ideal{f}$ are unique upto scaling by an element of $O_{K}^{\times, \ideal{f}}$ so that if $\ideal{f}$ separates units both $a_\ideal{f}$ and $s_\ideal{f}$ are unique, and moreover equal to $a_{\ideal{f}'}$ and $s_{\ideal{f}'}$ for any integral ideal $\ideal{f}'$ divisible by $\ideal{f}.$ Fixing such an $\ideal{f}$ it follows that \[s_\ideal{f}\in \bigcap_{\ideal{f}|\ideal{f}'} A_{O_K}^{\times, \ideal{f}'}=\{1\}\] so that $s=a_\ideal{f} s_\ideal{f}=a_\ideal{f} \in K^\times$ and we are done.
\end{proof}
\begin{rema} The exactness of the sequence (\ref{prop:ideles-and-classes}) is the first result of many that will rely crucially on the fact that the unit group $O_K^\times$ is finite.
\end{rema}
\subsection{}\label{subsec:ray-class-fields} We write $K^{\infty}/K$ for the maximal abelian extension of $K$ which is totally split at $\infty$. The kernel of the surjective map \[C_K\to G(K^{\infty}/K)\] is precisely $K_\infty^\times$ (cf.\ (b) of (\ref{subsec:global-reciprocity-properties})) so that we obtain continuous isomorphisms \[(A_{O_K}\otimes_{O_K} K)^\times/K^\times=I_{K}/(K^\times K^\times_\infty)=C_K/K_\infty^\times \to G(K^{\infty}/K).\] Let us write \begin{equation}\label{def:idele-rec} (-, K^\infty/K): (A_{O_K}\otimes_{O_K}K)^\times/K^\times\overset{\sim}{\longrightarrow} G(K^\infty/K).\end{equation} for this isomorphism and also \begin{equation}\label{eqn:define-theta}\theta_K:\mathrm{CL}_{O_K, \infty}\overset{\sim}{\longrightarrow} G(K^\infty/K)\end{equation} for the isomorphism \[\mathrm{CL}_{O_K, \infty}\stackrel{\sim}{\longleftarrow}(A_{O_K}\otimes_{O_K}K)^\times/K^\times \stackrel{
(-, K^\infty/K)}{\longrightarrow} G(K^\infty/K)\] so that for all $s\in (A_{O_K}\otimes_{O_K}K^\times)/K^\times$ we have \[\theta_K([s])=(s, K^\infty/K).\] If $\ideal{f}$ is an integral ideal of $O_K$ then, under $\theta_K$, the kernel of the homomorphism \[\mathrm{CL}_{O_K, \infty}\to \mathrm{CL}_{O_K}^{(\ideal{f})}\] corresponds to the subgroup $G(K^{\infty}/K(\ideal{f}))\subset G(K^\infty/K)$ of automorphisms fixing a certain finite abelian extension $K\subset K(\ideal{f})\subset K^{\infty}$ which we call the ray class field of conductor $\ideal{f}$. By definition the map (\ref{eqn:define-theta}) induces an isomorphism \begin{equation}\label{eqn:define-theta-finite}\theta_{K, \ideal{f}}: CL_{O_K}^{(\ideal{f})}\overset{\sim}{\longrightarrow} G(K(\ideal{f})/K).\end{equation} The extension $K(\ideal{f})/K$ is unramified away from $\ideal{f}$ and if $\ideal{p}$ is prime to $\ideal{f}$ then \[\theta_{K, \ideal{f}}([\ideal{p}]_\ideal{f}^{-1})=\theta_{K, \ideal{f}}([\pi]_\ideal{f})=\sigma_{K(\ideal{f})/K, \ideal{p}}\] where $\pi\in K_\ideal{p}^\times\subset (A_{O_K}\otimes_{O_K}K)^\times$ is any local uniformiser. In particular, when $\ideal{f}=O_K$ the field $H:=K(O_K)$ is called the Hilbert class field. It is unramified everywhere and the isomorphism \[\theta_{K, O_K}: \mathrm{CL}_{O_K}\overset{\sim}{\longrightarrow} G(H/K)\] maps the class of the inverse of each prime ideal $[\ideal{p}^{-1}]\in \mathrm{CL}_{O_K}$ to the Frobenius element $\sigma_{H/K, \ideal{p}}$.
\begin{rema} For future reference we make the following observations.
\begin{enumerate}[label=(\alph*)]
\item The composition \[A_{O_K}^{\times, \ideal{f}}/O_K^{\times, \ideal{f}} \to (A\otimes_{O_K}K^\times)/K^\times \stackrel{[-]}{\to} CL_{O_K, \infty}\stackrel{\theta_K}{\to} G(K^\infty/K)\] induces an isomorphism \begin{equation} A_{O_K}^{\times, \ideal{f}}/O_K^{\times, \ideal{f}}\overset{\sim}{\longrightarrow} G(K^\infty/K(\ideal{f}))\subset G(K^\infty/K). \label{ray-class-isom}\end{equation} In particular, when $\ideal{f}$ separates units we have $O_{K}^{\times, \ideal{f}}=\{1\}$ so that (\ref{ray-class-isom}) becomes \[A_{O_K}^{\times, \ideal{f}}\overset{\sim}{\longrightarrow} G(K^\infty/K(\ideal{f})).\]
\item For each prime $\ideal{p}$ of $O_K$ and each $K$-linear embedding $K^\mathrm{sep}\to K_\ideal{p}^\mathrm{sep}$, the map \[K^\times_\ideal{p}\stackrel{(-, K_\ideal{p}^\mathrm{ab}/K_\ideal{p})}{\longrightarrow} G(K_\ideal{p}^\mathrm{sep}/K_\ideal{p})\to G(K^\mathrm{sep}/K)\stackrel{-|_{K^\infty}}{\to} G(K^\infty/K)\stackrel{\theta_K^{-1}}{\to} \mathrm{CL}_{O_K, \infty}\] is given by \begin{equation} \label{eqn:local-global-class-compatibility}a\mapsto [a]\end{equation} where we view $a\in K_\ideal{p}^\times\subset (A_{O_K}\otimes_{O_K}K)^\times$.
\end{enumerate}
\end{rema}
\section{Class stacks} We now extend the definition of the level-$\ideal{f}$ structures on $O_K$-modules to level-$\ideal{f}$ structures on $O_K$-local systems over sheaves and give several basic results concerning them and their moduli stacks.
\subsection{} So let $S$ be a sheaf over $\mathrm{Spec}(O_K)$ (this is technically not important for what follows) and consider the constant sheaf of rings $\underline{O_K}_S$ on $S$ associated to $O_K$. If $L$ is any $O_K$-module we write $\underline{L}_S$ for the corresponding constant $\underline{O_K}_S$-module. If $F$ and $G$ are two $\underline{O_K}_S$-modules over $S$ we write $F\otimes_{O_K} G$ for the tensor product $F\otimes_{\underline{O_K}_S}G$ and if $G=\underline{L}_S$ for some $O_K$-module $L$ we just write $F\otimes_{O_K}L$. We also write $\underline{\mathrm{Hom}}_S^{O_K}(F, G)$ for the sheaf of $\underline{O_K}_S$-homomorphisms $F\to G.$
\subsection{} A rank one $O_K$-local system on $S$ is a sheaf of $\underline{O_K}_S$-modules $\mc{L}$ over $S$ such that there exists a cover $(S_i\to S)_{i\in I}$, rank one projective $O_K$-modules $(L_i)_{i\in I}$ and $\underline{O_K}_S$-isomorphisms $\mc{L}\times_S S_i\overset{\sim}{\longrightarrow} \underline{L_i}_{S_i}$. The moduli stack of rank one $O_K$-local systems over $\mathrm{Sh}_{O_K}$ is denoted by $\mc{CL}_{O_K}$. We list the following (usual) constructions and properties of $O_K$-local systems.
\begin{enumerate}[label=(\roman*)]
\item The tensor product $\mc{L}\otimes_{O_K}\mc{L}'$ of two rank one $O_K$-local systems $\mc{L}$ and $\mc{L}'$ is again a rank one $O_K$-local system.
\item The sheaf of $\underline{O_K}_S$-homomorphisms $\underline{\mathrm{Hom}}_{S}^{O_K}(\mc{L}, \mc{L}')$ is again a rank one $O_K$-local system and defining $\mc{L}^\vee:=\underline{\mathrm{Hom}}_S^{O_K}(\mc{L},\underline{O_K}_S)$ we have $\mc{L}'\otimes_{O_K}\mc{L}^\vee\overset{\sim}{\longrightarrow} \underline{\mathrm{Hom}}_{S}^{O_K}(\mc{L}, \mc{L}')$.
\item The sheaf of automorphisms $\underline{\mathrm{Aut}}_S^{O_K}(\mc{L})$ of a rank one $O_K$-local system $\mc{L}$ is isomorphic to $\underline{O_K}_S^\times$.
\item The sheaf of $O_K$-isomorphisms $\underline{\mathrm{Isom}}_S^{O_K}(\mc{L}, \mc{L}')$ is finite and \'etale over $S$ and $\mc{L}$ and $\mc{L}'$ are locally isomorphic on $S$ if and only if $\underline{\mathrm{Isom}}_S^{O_K}(\mc{L}, \mc{L}')\to S$ is an epimorphism if and only if the action of $\underline{O_K^\times}_S$ on $\underline{\mathrm{Isom}}_S^{O_K}(\mc{L}, \mc{L}')\to S$ makes it a torsor.
\item Every rank one $O_K$-local system is, locally on $S$, isomorphic $\underline{L}_S$ for some rank one projective $O_K$-module $L$ whose corresponding class in $\mathrm{CL}_{O_K}$ is independent of the choice of $L$. Thus given an $O_K$-local system $\mc{L}$ on $S$ one obtains a section $c_{\mc{L}/S}\in \underline{CL_{O_K}}(S)$, or what is the same a map \[c_{\mc{L}/S}: S\to \underline{CL_{O_K}}\] from $S$ to the constant sheaf over $\mathrm{Spec}(O_K)$ associated to the group $CL_{O_K}$. Moreover, if one chooses representatives $L$ of each class $[L]\in \mathrm{CL}_{O_K}$ and considers the rank one $O_K$-local system over $\underline{CL_{O_K}}$: \[[L]^\mathrm{univ}:=\coprod_{[L]\in \mathrm{CL}_{O_K}}\underline{L}\to \coprod_{[L]\in CL_{O_K}}\mathrm{Spec}(O_K)=\underline{\mathrm{CL}_{O_K}}\] then $\underline{\mathrm{Isom}}_S^{O_K}(\mc{L}, c_{\mc{L}/S}^*([L]^\mathrm{univ}))$ is an $\underline{O_K^\times}_S$-torsor whose corresponding class in $H^1(S, \underline{O_K^\times}_S)$ we denote by $\rho_{\mc{L}/S}$. The resulting map \[\mc{L}\mapsto (c_{\mc{L}/S}, \rho_{\mc{L}/S})\in \underline{CL_{O_K}}(S)\times H^1(S, \underline{O_K^\times}_S)\] defines a bijection between isomorphism classes of rank one $O_K$-local systems over $S$ and the set $\underline{CL_{O_K}}(S)\times H^1(S, \underline{O_K^\times}_S)$.
\end{enumerate}
\subsection{}\label{rema:level-f-local-system} If $\ideal{f}$ is an integral ideal of $O_K$ a level-$\ideal{f}$ structure on a rank one $O_K$-local system $\mc{L}$ over $S$ is an epimorphism of $\underline{O_K}_S$-modules $\alpha: \mc{L}\overset{\sim}{\longrightarrow} \underline{O_K/\ideal{f}}_S$. An $\ideal{f}$-isomorphism $f: (\mc{L}, \alpha)\overset{\sim}{\longrightarrow} (\mc{L}', \alpha')$ of rank one $O_K$-local systems of over $S$ equipped level-$\ideal{f}$ structures is an $O_K$-linear isomorphism $f: \mc{L}\overset{\sim}{\longrightarrow} \mc{L}'$ such that $\alpha'\circ f=\alpha$. With this definition every $(\mc{L}, \alpha)$ is, locally on $S$, of the form $(\underline{L}_S, \underline{a}_S)$ for some $(L, a)\in \mathrm{CL}_{O_K}^{(\ideal{f})}$. When working with rank one $O_K$-local systems we will often drop explicit reference to the level-$\ideal{f}$ structure when it is clear from context. We list the following (usual) constructions and properties of $O_K$-local systems with level-$\ideal{f}$ structure.
\begin{enumerate}[label=(\roman*)]
\item The tensor product $\mc{L}\otimes_{O_K}\mc{L}'$ of two rank one $O$-local systems $(\mc{L}, \alpha)$ and $(\mc{L}', \alpha')$ equipped with level-$\ideal{f}$ structures is again a rank one $O_K$-local system with level-$\ideal{f}$ structure given by $\alpha\otimes_{O_K}\alpha'$.
\item The sheaf of $\underline{O_K}_S$-homomorphisms $\underline{\mathrm{Hom}}_{S}^{O_K}(\mc{L}, \mc{L}')$ is again a rank one $O_K$-local system with level-$\ideal{f}$ structure given by \[\underline{\mathrm{Hom}}_S^{O_K}(\mc{L}\otimes_{O_K}O_K/\ideal{f}, \mc{L}'\otimes_{O_K} O_K/\ideal{f})\\\overset{\sim}{\longrightarrow} \underline{\mathrm{Hom}}_S^{O_K}(\underline{O_K/\ideal{f}}_S, \underline{O_K/\ideal{f}}_S)=\underline{O_K/\ideal{f}}_S,\] and equipping $\mc{L}^\vee:=\underline{\mathrm{Hom}}_S^{O_K}(\mc{L},\underline{O_K}_S)$ with this level-$\ideal{f}$ structure makes $\mc{L}'\otimes_{O_K}\mc{L}^\vee\overset{\sim}{\longrightarrow} \underline{\mathrm{Hom}}_{S}^{O_K}(\mc{L}, \mc{L}')$ is an $\ideal{f}$-isomorphism.
\item The sheaf of $\ideal{f}$-automorphisms $\underline{\mathrm{Aut}}_S^{(\ideal{f})}(\mc{L})$ of a rank one $O_K$-local system $(\mc{L}, \alpha)$ with level-$\ideal{f}$ structure is equal to $\underline{(O_K/\ideal{f})^\times}_S$ and in particular is trivial if $\ideal{f}$ separates units.
\item The sheaf of $\ideal{f}$-isomorphisms $\underline{\mathrm{Isom}}_S^{(\ideal{f})}(\mc{L}, \mc{L}')$ between two rank one $O_K$-local systems with level-$\ideal{f}$ structure is finite and \'etale over $S$ and $\mc{L}$ and $\mc{L}'$ are locally $\ideal{f}$-isomorphic if and only if $\underline{\mathrm{Isom}}_S^{(\ideal{f})}(\mc{L}, \mc{L}')$ is an $\underline{O_K^{\times, \ideal{f}}}_S$-torsor.
\item Every rank one $O_K$-local system with level-$\ideal{f}$ structure is, locally on $S$, $\ideal{f}$-isomorphic $(\underline{L}_S, \underline{a}_S)$ for some rank one projective $O_K$-module with level-$\ideal{f}$ structure $(L, a)$ whose corresponding class in $\mathrm{CL}_{O_K}^{(\ideal{f})}$ is independent of the choice of $L$. Thus given an $O_K$-local system $\mc{L}$ on $S$ one obtains a section $c_{\mc{L}/S, \ideal{f}}\in \underline{CL_{O_K}^{(\ideal{f})}}(S)$, or what is the same a map \[c_{\mc{L}/S, \ideal{f}}: S\to \underline{CL_{O_K}^{(\ideal{f})}}\] from $S$ to the constant sheaf over $\mathrm{Spec}(O_K)$ associated to the group $CL_{O_K}^{(\ideal{f})}$. Moreover, if one chooses representatives $(L, a)$ of each class $[L, a]\in \mathrm{CL}_{O_K}^{(\ideal{f})}$ and considers the rank one $O_K$-local system with level-$\ideal{f}$ structure over $\underline{CL_{O_K}^{(\ideal{f})}}$: \[[L, a]^\mathrm{univ}:=\coprod_{[L, a]\in \mathrm{CL}_{O_K}}(\underline{L}, \underline{a})\to \coprod_{[L, a]\in CL_{O_K}^{(\ideal{f})}}\mathrm{Spec}(O_K)=\underline{\mathrm{CL}_{O_K}^{(\ideal{f})}}\] then $\underline{\mathrm{Isom}}_S^{(\ideal{f})}(\mc{L}, c_{\mc{L}/S}^*([L, a]^\mathrm{univ}))$ is an $\underline{O_K^{\times, \ideal{f}}}_S$-torsor whose corresponding class in $H^1(S, \underline{O_K^{\times, \ideal{f}}}_S)$ we denote by $\rho_{\mc{L}/S, \ideal{f}}$. The resulting map \[\mc{L}\mapsto (c_{\mc{L}/S, \ideal{f}}, \rho_{\mc{L}/S, \ideal{f}})\in \underline{CL_{O_K}^{(\ideal{f})}}(S)\times H^1(S, \underline{O_K^\times}_S)\] defines a bijection between isomorphisms classes of rank one $O_K$-local systems over $S$ and the set $\underline{CL_{O_K}}(S)\times H^1(S, \underline{O_K^{\times, \ideal{f}}}_S)$. In particular, if $\ideal{f}$ separates units then $H^1(S, \underline{O_K^{\times, \ideal{f}}}_S)=0$ and the map \[\mc{L}\mapsto c_{\mc{L}/S, \ideal{f}}\in \underline{\mathrm{CL}_{O_K}^{(\ideal{f})}}(S)\] defines a bijection between $\ideal{f}$-isomorphism classes of rank one $O_K$-local systems with level-$\ideal{f}$ structure over $S$ and elements of $\underline{\mathrm{CL}_{O_K}^{(\ideal{f})}}$.
\item If $\ideal{f}$ separates units, $(\mc{L}, \alpha)$ is an $O_K$-local system equipped with a level-$\ideal{f}$ structure and $S$ is connected, then $(\mc{L}, \alpha)\overset{\sim}{\longrightarrow} (\underline{L}_S, \underline{a}_S)$ for some rank one projective $O_K$-module with level-$\ideal{f}$ structure and in particular, $\mc{L}$ is constant.
\end{enumerate}
\begin{coro}\label{prop:class-stacks-triv-admis} The map \[\mc{CL}_{O_K}^{(\ideal{f})}\to\underline{CL_{O_K}^{(\ideal{f})}}: \mc{L}/S\mapsto c_{\mc{L}/S}\in \underline{CL_{O_K}^{(\ideal{f})}}(S)\] identifies $\underline{CL_{O_K}^{(\ideal{f})}}$ with the coarse sheaf of $\mc{CL}_{O_K}^{(\ideal{f})}$ and is an equivalence whenever $\ideal{f}$-separates units.
\end{coro}
\begin{proof} This follows from the remarks (\ref{rema:level-f-local-system}).
\end{proof}
\chapter{Elliptic curves with complex multiplication} In this chapter we develop the general theory of (families of) elliptic curves with complex multiplication by the ring of integers $O_K$ of a fixed imaginary quadratic field $K$, here on called just CM elliptic curves. In \S 1 we recall several standard results from the theory of (families of general) elliptic curves. In \S 2 we define the notion of a family $E\to S$ of CM elliptic curves, the corresponding moduli stack $\mc{M}_\mathrm{CM}$, and we show that (just as with Lubin--Tate $O$-modules) the moduli stack $\mc{CL}_{O_K}$ of rank one $O_K$-local systems acts in a natural way on the moduli stack $\mc{M}_\mathrm{CM}$ of CM elliptic curves. We then describe, for a prime $\ideal{p}\subset O_K$, the properties of the $\ideal{p}$-power torsion subgroups $E[\ideal{p}^\infty]\subset E$ of a family of CM elliptic curves and show that when $S$ is a $\ideal{p}$-adic sheaf $E[\ideal{p}^\infty]$ is a Lubin--Tate $O_{K_\ideal{p}}$-module. In \S 3 we consider CM elliptic curves over complex and $\ideal{p}$-adic bases and give CM analogues of the classification of elliptic curves over complex bases, and theorem of Serre-Tate describing deformations of elliptic curves over $\ideal{p}$-adic bases. In \S 4 we show that any two CM elliptic curves over the same base are locally isogenous and from this deduce that the action of $\mc{CL}_{O_K}$ on $\mc{M}_{\mathrm{CM}}$ gives $\mc{M}_{\mathrm{CM}}$ the structure of a torsor. In \S 5 we derive the global reciprocity map associated to the maximal abelian extension of $K$ (totally split at $\infty$ -- but this is a vacuous condition) directly from the stack $\mc{M}_\mathrm{CM}$ in a manner quite analogous to the derivation of the local reciprocity map via the moduli stack of Lubin--Tate $O$-modules. We then classify all CM elliptic curves over fields (both of characteristic zero and finite characteristic) and prove some results regarding good reduction. In \S 5 we define level-$\ideal{f}$ structures for CM elliptic curves and consider the corresponding moduli stacks $\mc{M}_\mathrm{CM}^{(\ideal{f})}$. As with $\mc{M}_\mathrm{CM}$ and $\mc{CL}_{O_K}$, we show that $\mc{M}_\mathrm{CM}^{(\ideal{f})}$ is a torsor under $\mc{CL}_{O_K}^{(\ideal{f})}$ (at least after inverting $\ideal{f}$). Using this we show that the coarse sheaf of $\mc{M}_\mathrm{CM}^{(\ideal{f})}$ is isomorphic to $\mathrm{Spec}(O_{K(\ideal{f})}[\ideal{f}^{-1}])$ where $K(\ideal{f})$ is the ray class field of conductor $\ideal{f}$.
We should point out that, aside from the $\mc{CL}_{O_K}$-torsor structure of $\mc{M}_\mathrm{CM}$, consistently working over a general base (instead of a field) and our derivation of the reciprocity map, almost everything in this chapter is probably more or less already known. This combined with the fact that $\mc{M}_\mathrm{CM}$ is zero dimensional over $\mathrm{Spec}(O_K)$, the advantages of our general approach may be somewhat unclear. However, while $\mc{M}_\mathrm{CM}$ is geometrically rather simple, it is arithmetically quite complicated and the general approach we take in this chapter will allow for great deal of flexibility later when we wish to study some of its finer arithmetic properties.
\section{General elliptic curves} We now recall the definition of a family of elliptic curves over a sheaf $S$ and recall several standard results. In particular, the fact that the moduli stack of elliptic curves is indeed a stack, the rigidity principal for homomorphisms, the representability of Isom sheaves, Grothendieck's formal GAGA, the classification of elliptic curves over complex schemes $S$ in terms of rank two $\mathbf{Z}$-local systems over $S^\mathrm{an}$, the Serre-Tate theorem, and the criterion of good reduction.
\subsection{}\label{subsec:elliptic-curve-def} Let $S$ be a sheaf. An elliptic curve over $S$ is a sheaf of groups $E\to S$ which is relatively representable, smooth of relative dimension one, proper and geometrically connected. A morphism is of course a homomorphism of the underlying (sheaves of) groups over $S$. For many more general properties and constructions related to families of elliptic curves $E\to S$ we refer the reader to the wonderful book of Katz-Mazur \cite{KatzMazur85}.
If $S$ is a sheaf we write $\mathrm{Ell}(S)$ for the category of elliptic curves over $S$ and we denote by $\mc{M}_\mathrm{Ell}$ the fibred category over $\mathrm{Sh}$ whose fibre over a sheaf $S$ is the category elliptic curves $E/S$ together with their isomorphisms.
\begin{prop}\label{prop:moduli-ell-stack} The fibred category $\mc{M}_\mathrm{Ell}$ is a stack over $\mathrm{Sh}$.
\end{prop}
\begin{proof}[Sketch] If $S$ is an affine scheme and $f: E\to S$ is a family of elliptic curves let $\mc{I}_{E/S}\subset \mc{O}_E$ denote the ideal sheaf defining the zero section $S\to E$ (it is a locally free rank one $\mc{O}_E$-module). The quasi-coherent $\mc{O}_S$-module $\mc{W}_{E/S}=f_*(\mc{I}^{-3}_{E/S})$ is a vector bundle of rank three, the morphism \[f^*f_*(\mc{I}^{-3}_{E/S})\to \mc{I}^{-3}_{E/S}\] is an epimorphism and defines a closed immersion $w_{E/S}: E\to \mathbf{P}(\mc{W}_{E/S})$. Both the vector bundle $\mc{W}_{E/S}$ and the morphism $w_{E/S}$ are functorial in $S$ so that by descent if $E\to S$ is any family of elliptic curves over a \text{sheaf} $S$ then there is a unique vector bundle $\mc{W}_{E/S}$ of rank three over $S$ together with a closed immersion $w_{E/S}: E\to \mathbf{P}_{S}(\mc{W}_{E/S})$ compatible with those defined when $S$ is affine.
Now if $S$ is an sheaf and $(f_i: E_i\to S_i)_{i\in I}$ is a family of elliptic curves equipped with descent data relative to a cover $(S_i)_{i\in I}$ of $S$ then the $E_i$ descend to a \textit{sheaf} of groups $E\to S$, the vector bundles $\mc{W}_{E_i/S_i}$ to a vector bundle $\mc{W}_{E/S}$ and the closed immersions $w_{E_i}: E_i\to \mathbf{P}(\mc{W}_{E_i/S_i})$ to a closed immersion \[w_{E/S}: E\to \mathbf{P}(\mc{W}_{E/S}).\] This shows that the morphism $p: E\to S$ is representable (in fact projective) and by descent it follows that $E\to S$ is also smooth of relative dimension one, proper, and geometrically connected so that $E\to S$ is an elliptic curve over $S$.
\end{proof}
\subsection{}\label{subsec:rigidity} Here we recall several useful properties enjoyed by homomorphisms of elliptic curves.
\begin{prop} Let $S$ be a sheaf and let $E\to S$ be a family of elliptic curves. For each $n\geq 1$ multiplication map $n: E\to E$ is finite locally free of degree $n^2$.
\end{prop}
\begin{proof} This is Theorem 2.3.1 of \cite{KatzMazur85}.
\end{proof}
\begin{prop}[Rigidity]\label{prop:rigidity} If $f: E\to E'$ is a homomorphism of elliptic curves over a sheaf $S$ there is a unique decomposition $S=\amalg_{n\geq 0} S_{(n)}$ with the property that $f\times_S S_{(0)}$ is the zero map and such that $f\times_S S_{(n)}$ is finite locally free of degree $n$ for $n\geq 1$.
In particular, if $f, g: E\to E'$ are a pair of homomorphisms of elliptic curves and $S'\to S$ is a morphism of sheaves which is surjective on geometric points then $f=g$ if and only if $f\times_S S'=g\times_S S'$.
\end{prop}
\begin{proof} The first statement is Theorem 2.4.2 of \cite{KatzMazur85}. For the second statement the only if direction is clear, so assume that $f\times_S S'=g\times_S S'$. The claim is local on $S$ and $S'$ and so we may assume that they are affine schemes. If $S_{(0)}\subset S$ and $S_{{(0)}}'\subset S'$ denote the open and closed sub-schemes where $f-g$ and $f\times_S S'-g\times_S S'$ are equal to the zero map respectively, then $S_{(0)}'\to S$ factors through $S_{(0)}$. However, $S_{(0)}'=S'$, so that as $S'\to S$ is surjective on geometric points, it follows that $S_{(0)}=S$.
\end{proof}
\begin{rema} We will make much use of (\ref{prop:rigidity}) and when doing so just say `by rigidity'.
\end{rema}
\begin{prop}\label{prop:isom-are-fin-unram} For each pair of elliptic curves $E$ and $E'$ over a sheaf $S$ the sheaf $\underline{\mathrm{Isom}}_S(E, E')$ is finite and unramified over $S$.
\end{prop}
\begin{proof} The claim is local on $S$ so we may assume that $S$ is an affine scheme and this is Proposition 5.3 (i) of \cite{Deligne75}.
\end{proof}
\subsection{}\label{prop:formal-gaga} Let $A$ be a noetherian ring complete with respect to the $I$-adic topology for $I\subset A$ an ideal and write $\mathrm{Spf}(A)=\colim_{n\geq 0} \mathrm{Spec}(A/I^{n+1})\subset \mathrm{Spec}(A)$.
\begin{theo}[Formal GAGA] The functor \[\mathrm{Ell}(\mathrm{Spec}(A))\to \mathrm{Ell}(\mathrm{Spf}(A)): E/\mathrm{Spec}(A)\mapsto E\times_{\mathrm{Spec}(A)} \mathrm{Spf}(A)\] induced by base change is an equivalence of categories.
\end{theo}
\begin{proof} This is an easy application Grothendieck's formal GAGA (in particular Corollaire 2 and Th\'eor\`eme 4 of \cite{Groth95}).
\end{proof}
\subsection{}\label{subsec:complex-class} The following is taken from N\textsuperscript{\underline{o}} 2 of \cite{Deligne71}. Let $S$ be a locally finitely presented $\mathrm{Spec}(\mathbf{C})$-scheme and let $f: E\to S$ be an elliptic curve. The analytification $f^\mathrm{an}:E^\mathrm{an} \to S^\mathrm{an}$ is an analytic space over $S^\mathrm{an}$ which is smooth of relative dimension one and proper with connected fibres. There is a canonical exact sequence, the exponential sequence, of sheaves on the big analytic site of $X^\mathrm{an}$ \[0\to T_\mathbf{Z}(E)\to \underline{\mathrm{Lie}}_{E^\mathrm{an}/S^\mathrm{an}} \to E^\mathrm{an}\to 0\] (we view $\underline{\mathrm{Lie}}_{S^\mathrm{an}/E^\mathrm{an}}$ as a rank one locally free sheaf of $\mc{O}_{S^\mathrm{an}}$-modules on the \textit{big} analytic site of $S$) with $T_\mathbf{Z}(E)$ a rank two $\mathbf{Z}$-local system on $S^\mathrm{an}$. Denote by $\mathrm{Lat}(S^\mathrm{an})$ the category of pairs $(T \subset \mc{V})$ where $\mc{V}$ is a locally free rank one $\mc{O}_{S^\mathrm{an}}$-module and $T\subset \mc{V}$ is a rank two $\mathbf{Z}$-local system which is fibre-wise over $S^\mathrm{an}$ discrete in $\mc{V}$.
\begin{prop}\label{prop:class-complex-elliptic} The functor \[\mathrm{Ell}(S)\to \mathrm{Lat}(S^\mathrm{an}): E/S\mapsto (T_\mathbf{Z}(E/S)\subset \underline{\mathrm{Lie}}_{E^\mathrm{an}/S^\mathrm{an}})\] is an equivalence of categories.
\end{prop}
\subsection{}\label{subsec:classical-serre-tate} Let $p$ be a rational prime and $S$ a $p$-adic sheaf (i.e.\ a sheaf over $\mathrm{Spf}(\mathbf{Z}_p)=\colim_{n} \mathrm{Spec}(\mathbf{Z}/p^{n+1}))$. A $p$-divisible group over $S$ is an sheaf of groups $F\to S$ such that the multiplication map $p: F\to F$ is representable, finite locally free and faithfully flat and such that $\colim_n \ker(p^n)=F$. If $E$ is an elliptic curve over $S$ we write $E[p^\infty]$ for the $p$-divisible group $\colim_n E[p^n]$.
Let $S_0\to S$ be a nilpotent closed immersion of $p$-adic sheaves and consider the category $D(S, S_0)$ whose objects are triples $(E/S_0, F/S, \rho)$ with
\begin{enumerate}[label=\textup{(\roman*)}]
\item $E_0/S_0$ an elliptic curve,
\item $F/S$ is a $p$-divisible group, and
\item $\rho: E_0[p^\infty]\overset{\sim}{\longrightarrow} F\times_S S_0$ an isomorphism of $p$-divisible groups,
\end{enumerate} \noindent and whose morphisms $(E_0/S_0, F/S, \rho) \to (E'/S_0, F', \rho')$ are pairs $(f, g)$ where $f: E\to E'$ is a morphism of elliptic curves, $g: F\to F'$ is a morphism of $p$-divisible groups such that \[(g\times_{S} S_0)\circ \rho=\rho'\circ f|_{E[p^\infty]}.\]
\begin{theo}[Serre-Tate]\label{theo:serre-tate} The functor \[\mathrm{Ell}(S)\to D_p(S, S_0): E/S\mapsto (E\times_S S_0/S_0, E[p^\infty]/S, \mathrm{id}_{E[p^\infty]}|_{S_0})\] is an equivalence of categories.
\end{theo}
\begin{proof} A short argument using (\ref{prop:moduli-ell-stack}) reduces us to the case where $S$ is an affine scheme and this case is the content of the Appendix of \cite{Drinfeld76}.
\end{proof}
\subsection{} Let $S$ be a Dedekind scheme, i.e.\ a one dimensional, regular and irreducible scheme. Let $f:\mathrm{Spec}(K)\to S$ be its generic point and let $E/\mathrm{Spec}(K)$.
\begin{theo}[N\'eron models] The functor $X/S\mapsto E(X\times_{S} \mathrm{Spec}(K))$ on \textup{smooth} schemes over $S$ is representable by a smooth, one dimensional group scheme $\mathrm{N\acute{e}r}_S(E)/S$. Moreover, if $S'\to S$ is an \'etale map of Dedekind schemes and $\mathrm{Spec}(K')\to S'$ is the generic point of $S'$ then \[\mathrm{N\acute{e}r}_S(E)\times_S S'=\mathrm{N\acute{e}r}_{S'}(E\times_{\mathrm{Spec}(K)}\mathrm{Spec}(K')).\]
\end{theo}
\begin{proof} Representability of the functor is Theorem 3 \S 1.4 Chapter 1 of \cite{Raynaud90} and compatibility with \'etale base change is Proposition 2 (b) \S 1.2 of \cite{Raynaud90}.
\end{proof}
\begin{theo}\label{theo:criterion-of-good-reduction} Let $L/\mathbf{Q}$ be a finite extension, $v$ a place over $\mathbf{Q}^\mathrm{sep}$ lying over the prime $\ideal{p}$ of $O_L$ with residue characteristic $p>0$ and let $E/L$ be an elliptic curve. Then $\mathrm{N\acute{e}r}_{O_{L, \ideal{p}}}(E)\to \mathrm{Spec}(O_{L, \ideal{p}})$ is an elliptic curve if, for some prime $l\neq p$, the action of the inertia group $I_v\subset G(\mathbf{Q}^\mathrm{sep}/L)$ on $E[l^\infty](\mathbf{Q}^\mathrm{sep})$ is trivial.
\end{theo}
\begin{proof} This follows from Theorem 1 of \cite{SerreTate68}.
\end{proof}
\section{Elliptic curves with complex multiplication} In this section we define families of elliptic curves with complex multiplication by the ring of integers $O_K$ of an imaginary quadratic field $K$ or, for short, CM elliptic curves. We give analogues for CM elliptic curves of several of the results of (\ref{sec:lubin-tate-modules}) given for Lubin--Tate $O$-modules. In particular, we consider the moduli stack $\mc{M}_\mathrm{CM}$ of CM elliptic curves and show that the stack $\mc{CL}_{O_K}$ of rank one $O_K$-local system acts on $\mc{M}_\mathrm{CM}$.
\subsection{}\label{subsec:imag-quad-set-up} For the remainder of this chapter we fix an imaginary quadratic field $K=\mathbf{Q}(\sqrt{-d})$ for $d\in \mathbf{N}$ square free and with ring of integers $O_K$. We note that $K$ has only one archimedian place $\infty$ and so $(K, \infty)$ satisfies the conditions of (\ref{subsec:global-field-with-special-place}) and we shall make use and notation (and in later sections theory) set up in \S \ref{section:special-place} of Chapter 1. As $K_\infty\overset{\sim}{\longrightarrow} \mathbf{C}$ is algebraically closed every finite extension of $K$ is totally split at infinity so that $K^\infty=K^\mathrm{ab}$, although we shall continue to use the notation $K^\infty$.
The unit group $O_K^\times$ is finite and is equal to $\{\pm 1\}$ unless $K=\mathbf{Q}(\mu_n)$ for $n=4, 6$ in which case $O_K^\times=\mu_n$. The unique non-trivial automorphism of $K/\mathbf{Q}$ is denoted by $a\mapsto \overline{a}$. We shall be working solely in the category $\mathrm{Sh}_{O_K}$ and so by a sheaf $S$ we will mean a sheaf over $\mathrm{Spec}(O_K)$. In particular, to simplify some of the notation we will write $X\times Y$ for the product $X\times_{\mathrm{Spec}(O_K)} Y$ in $\mathrm{Sh}_{O_K}$.
\subsection{} An elliptic curve with complex multiplication by $O_K$ over $S$, or for short a CM elliptic curve over $S$, is an elliptic curve $E\to S$ (\ref{subsec:elliptic-curve-def}) equipped an $\underline{O_K}_S$-module structure which is strict with respect to the morphism $\underline{O_K}_S\to \mc{O}_S$ coming from the structure map $S\to \mathrm{Spec}(O_K)$ (cf.\ (\ref{subsec:strict-formal-modules})). A morphism of CM elliptic curves over $S$ is just a homomorphism of $\underline{O_K}_S$-modules. For $a\in \underline{O_K}_S(S)$ we write $[a]_E: E\to E$, or just $a: E\to E$, for the corresponding endomorphism (in particular for $a\in O_K\subset \underline{O_K}_S(S)$).
Finally, we write $\mathrm{CM}(S)$ for the category of CM elliptic curves over a sheaf $S$ and we write $\mc{M}_\mathrm{CM}$ for the stack over $\mathrm{Sh}_{O_K}$ whose fibre over $S$ is the category of CM elliptic curves over $S$ together with their isomorphisms.
\begin{prop}\label{prop:cm-endo} For any \textup{CM} elliptic curve $E/S$ the morphism \[\underline{O_K}_S\to \underline{\mathrm{End}}_S^{O_K}(E)\] is an isomorphism.
\end{prop}
\begin{proof} The claim is local on $S$ so may assume that $S$ is an affine scheme. That $\underline{O_K}_S\to \underline{\mathrm{End}}_S^{O_K}(E)$ is injective follows from the fact that both $\underline{\mathbf{Z}}_S\subset \underline{O_K}_S$ and $\underline{\mathbf{Z}}_S\to\underline{\mathrm{End}}_S(E)$ are monomorphisms. By Corollary 1, \S 4 of \cite{SerreTate68} the map $\underline{O_K}_S\to \underline{\mathrm{End}}_S^{O_K}(E)$ is an isomorphism when evaluated on any closed point $\mathrm{Spec}(k)\to S$. Therefore, if $f: E\to E$ is any morphism, for each closed point $h:\mathrm{Spec}(k)\to S$ there is an element $\alpha_h\in O_K$ such that $(f-[\alpha_h]_E)|_{\mathrm{Spec}(k)}=0$ and by rigidity (\ref{subsec:rigidity}) there is an open (and closed) neighbourhood $\mathrm{Spec}(k)\to U\subset S$ such that $(f-[\alpha_h]_E)|_U=0$. It follows that \[\underline{O_K}_S\to \underline{\mathrm{End}}_S^{O_K}(E)\] is an epimorphism and so is an isomorphism.
\end{proof}
\begin{prop}\label{prop:cm-tensoring-is-cm} Let $E/S$ be a \textup{CM} elliptic curve and $\mc{L}/S$ a rank one $O_K$-local system. Then $E\otimes_{O_K}\mc{L}$ is a \textup{CM} elliptic curve over $S$.
\end{prop}
\begin{proof} The claims are all local on $S$ so we may assume that $S$ is an affine scheme. It follows from (\ref{prop:tensor-properties}) that, at least locally on $S$, $E\otimes_{O_K}\mc{L}$ is representable by a proper, smooth, geometrically connected group scheme. It follows from (\ref{prop:tensor-properties-lie-hom}) that, when this is the case, the Lie algebra of $E\otimes_{O_K}\mc{L}$ is locally free of rank one so that the relative dimension of $E\otimes_{O_K}\mc{L}\to S$ is also one. Therefore $E\otimes_{O_K}\mc{L}$ is, locally on $S$, an elliptic curve so that it is in fact an elliptic curve over $S$ as the moduli stack of elliptic curves is a stack. Finally, $E\otimes_{O_K}\mc{L}$ has an obvious structure of an $\underline{O_K}_S$-module and it follows from (\ref{prop:tensor-properties-lie-hom}) that it is strict.
\end{proof}
\begin{rema} By (\ref{prop:cm-tensoring-is-cm}) above we find a functor \[\mc{M}_{CM}\times \mc{CL}_{O_K}\to \mc{M}_\mathrm{CM}: (E/S, \mc{L}/S)\mapsto E\otimes_{O_K}\mc{L}/S\] which we may view as defining an action of $\mc{CL}_{O_K}$ on $\mc{M}_{CM}$. We will show later (see (\ref{theo:mcm-is-a-torsor})) that, as with Lubin--Tate $O$-modules, this makes $\mc{M}_\mathrm{CM}$ a torsor under $\mc{CL}_{O_K}$.
\end{rema}
\begin{coro}\label{prop:cm-tensor-homs} For each pair $\mc{L}, \mc{L}'$ of rank one $O_K$-local systems over $S$ and each pair $E, E'$ of \textup{CM} elliptic curves over $S$ the natural map \[\underline{\mathrm{Hom}}_S^{O_K}(E, E')\otimes_{O_K}\underline{\mathrm{Hom}}_S^{O_K}(\mc{L}, \mc{L}')\to \underline{\mathrm{Hom}}_S^{O_K}(E\otimes_{O_K}\mc{L}, E'\otimes_{O_K}\mc{L}')\] is an isomorphism.
\end{coro}
\begin{proof} This follows from (\ref{prop:tensor-properties-lie-hom}).
\end{proof}
\begin{rema}\label{rema:cm-tensor-homs} The isomorphism of (\ref{prop:cm-tensor-homs}) together with (\ref{prop:cm-endo}) gives the particularly simple formula when $E=E'$ and $\mc{L}=\underline{O_K}_S$ and $\mc{L}'=\underline{L}_S$: \[\underline{L}_S\overset{\sim}{\longrightarrow} \underline{\mathrm{Hom}}_S^{O_K}(E, E\otimes_{O_K}L): l\mapsto \mathrm{id}_{E}\otimes_{O_K} l.\]
\end{rema}
\subsection{} For each integral ideal $\a\subset O_K$ we write \[i_\a: E\to E\otimes_{O_K}\a^{-1}\] for the homomorphism induced by the inclusion $O_K\to \a^{-1}$. We define the $\a$-torsion in $E$ to be $E[\a]=\ker(i_\a)$. We have $E[\a]=\cap_{a\in \a} \ker(a)$ so that $E[\a]=\ker(a)$ if $\a=(a)$ is principal.
\begin{prop} The homomorphism $i_\a: E\to E\otimes_{O_K}\a^{-1}$ is finite locally free of degree $N\a$.
\end{prop}
\begin{proof} If $S$ is empty or if $\a=O_K$ the claim is obvious so we assume that $S\neq \emptyset$ and that $\a\neq O_K$. Using (\ref{rema:cm-tensor-homs}), the morphism $i_\a$ is equal to the zero map only if $S=\emptyset$, and it is an isomorphism if and only if $\a=O_K$ or $S=\emptyset$. Therefore, as $\a\neq O_K$ and $S\neq \emptyset$, the morphism $i_\a$ is finite locally free of degree greater than one.
As tensoring with rank one $O_K$-local systems is exact (\ref{prop:tensor-exact}) the kernel of $i_\a\otimes_{O_K}\b^{-1}$ is $E[\a]\otimes_{O_K}\b^{-1}$ and as $O_K/\a\otimes_{O_K}\b^{-1}\overset{\sim}{\longrightarrow} O_K/\a$ (non-canonically) for all pairs of integral ideals $\a, \b$ we have \[E[\a]\otimes_{O_K}\b^{-1}\overset{\sim}{\longrightarrow} E[\a].\] Therefore $\deg(i_\a\otimes_{O_K}\b^{-1})=\deg(i_\a)$ and \[\deg(i_{\a\b})=\deg((i_\b\otimes_{O_K}\b^{-1})\circ i_\b)=\deg(i_\a)\deg(i_\b).\] As $N{\a\b}=N\a N\b$ and $\deg(i_{\a\b})=\deg(i_\a)\deg(i_\b)$ we may assume that $\a=\ideal{p}$ is a prime ideal in which case we find \[\deg(i_\ideal{p})\deg(i_{\overline{\ideal{p}}})=\deg(i_{\ideal{p}\overline{\ideal{p}}})=\deg([N\ideal{p}]_E)=N\ideal{p}^2.\] If $\ideal{p}=\overline{\ideal{p}}$ then $\deg(i_\ideal{p})^2=N\ideal{p}^2$ and so $\deg(i_\ideal{p})=N\ideal{p}$ and if $\ideal{p}\neq \overline{\ideal{p}}$ then, as $N\ideal{p}$ is prime and as both $\deg(i_\ideal{p}),\deg(i_{\overline{\ideal{p}}})\neq 1$, it also follows that $\deg(i_\ideal{p})=N\ideal{p}$.
\end{proof}
\subsection{} We associate the following sheaves of groups to $E/S$:
\begin{enumerate}[label=\textup{(\roman*)}]
\item For each maximal ideal $\ideal{p}\subset O_K$ the $\ideal{p}$-divisible group of $E$ is the ind-finite locally free group scheme $E[\ideal{p}^{\infty}]:=\colim_n E[\ideal{p}^n]$. It is a $\lim_n \underline{O_K/\ideal{p}^n}_{S}$-module.
\item The torsion subgroup of $E/S$ is \[E[\mathrm{tors}]:=\colim_{\a} E[\a]=\bigoplus_{\ideal{p}} E[\ideal{p}^\infty].\] It is a $\lim_\a \underline{O_K/\a}_{S}$-module.
\item The formal group of $E/S$ is $\widehat{E}:=\colim_k \mathrm{Inf}_{S}^{(k)}(E)$ (cf.\ \ref{subsec:formal-completion}). It is a strict formal $\underline{O_K}_S$-module of dimension one.
\end{enumerate}
\begin{prop}\label{prop:p-torsion-classification} Let $E/S$ be a \textup{CM} elliptic curve and $\ideal{p}\subset O_K$ a prime ideal. Then
\begin{enumerate}[label=\textup{(\roman*)}]
\item\label{item:etale} $\ideal{p}$ is invertible on $S$ if and only if $E[\ideal{p}]$ is finite and \'etale over $S$. In this case, $E[\ideal{p}^\infty]$ is \'etale over $S$ and is locally isomorphic to the constant $\lim_n \underline{O_K/\ideal{p}^n}_S$-module \[\colim_n\underline{\ideal{p}^{-n}/O_K}_S=\underline{K_\ideal{p}/O_{K_\ideal{p}}}_S,\] and
\item $\ideal{p}$ is locally nilpotent on $S$ if and only if $E[\ideal{p}^\infty]=\widehat{E}$. In this case, $E[\ideal{p}^\infty]$ is a Lubin--Tate $O_{K_\ideal{p}}$-module over $S$.
\end{enumerate}
\end{prop}
\begin{proof} The claims are true if and only if they are locally on $S$ and so we may assume that $S$ is an affine scheme.
(i) As $i_{\ideal{p}^n}$ is finite locally free of degree $N\ideal{p}^n$ (in particular, faithfully flat) its kernel $E[\ideal{p}^n]$ is \'etale over $S$ if and only if the morphism $E\to E\otimes_{O_K}\ideal{p}^{-n}$ is \'etale, or equivalently, induces an isomorphism $\underline{\mathrm{Lie}}_{E/S}\to \underline{\mathrm{Lie}}_{E\otimes_{O_K}\ideal{p}^{-1}/S}=\underline{\mathrm{Lie}}_{E/S}\otimes_{O_K}\ideal{p}^{-1}$ and this is true if and only if $\ideal{p}$ is invertible on $S$.
For the second claim we may, after localising $S$, assume that $E[\ideal{p}^\infty]$ is constant and hence that $E[\ideal{p}^n]$ is constant for all $n\geq 0$. For each $n\geq 0$ let $E_n$ be a finite $O_K$-module with $E[\ideal{p}^n]\overset{\sim}{\longrightarrow} \underline{E_n}_S$ for $n\geq 0$. For $n\geq 0$, $E_{n}$ is an $O_K/\ideal{p}^n$-module and the $\ideal{p}^n$-torsion of $\colim_n E_n$ is equal to $E_n$. As $\# E_{n+1}=N\ideal{p}^{n+1}$ if $E_{n+1}$ is not a free rank one $O_K/\ideal{p}^{n+1}$-module it must consist of $\ideal{p}^n$-torsion and therefore $E_{n+1}\subset E_n$ but $\# E_n=N\ideal{p}^n$ so that this is impossible. It follows that for each $n\geq 0$ the $O_K/\ideal{p}^n$-module $E_n$ is free of rank one and the inclusions $E_{n}\to E_{n+1}$ identify $E_n$ with the $\ideal{p}^n$-torsion of $E_{n+1}$. Therefore, we may fix isomorphisms $E_n\overset{\sim}{\longrightarrow} \ideal{p}^{-n}/O_K$ for all $n\geq 0$ with the property that the inclusions $E_{n}\subset E_{n+1}$ become the natural inclusions $\ideal{p}^{-n}/O_K\subset \ideal{p}^{-n-1}/O_K$. Thus \[E[\ideal{p}^\infty]\overset{\sim}{\longrightarrow} \colim_n \underline{\ideal{p}^{-n}/O_K}_S.\]
(ii) First assume that $\widehat{E}=E[\ideal{p}^\infty]$. Then $S\to \mathrm{Spec}(O_K)$ factors through \[\mathrm{Spf}(O_{K_\ideal{p}})=\colim_{n} \mathrm{Spec}(O_K/\ideal{p}^{n+1})\to \mathrm{Spec}(O_K)\] if and only if $S\times\mathrm{Spec}(O_K[\ideal{p}^{-1}])=\emptyset$. Thus we may assume that $S=S\times\mathrm{Spec}(O_K[\ideal{p}^{-1}])$ and show that it is empty.
By (i) the sheaf $\widehat{E}=E[\ideal{p}^\infty]$ is \'etale over $S$. In this case it follows that, for all $k\geq 0$, the scheme $\mathrm{Inf}_{S}^{(k)}(E)$ is unramified over $S$ as it is a sub-scheme of the \'etale scheme $\widehat{E}=E[\ideal{p}^\infty]$. Therefore, the morphism defining the zero section $S\to \mathrm{Inf}_{S}^{(k)}(E)$ is both a nilpotent immersion and an open immersion which is possible if and only if $S=\mathrm{Inf}_S^{(k)}(E)$ and this is possible if and only if $S=\emptyset$.
Conversely, assume that $S\to \mathrm{Spec}(O_K)$ factors through $\mathrm{Spf}(O_{K_\ideal{p}})\subset \mathrm{Spec}(O_K)$ and write $\widehat{E}[\ideal{p}^n]=\ker(\widehat{E}\to \widehat{E}\otimes_{O_K}\ideal{p}^{-n})$. As $S$ is affine it follows that $\ideal{p}$ is nilpotent on $S$ and as the action of $\underline{O_K}_S$ on $\widehat{E}$ is strict it follows that \[\colim_n \widehat{E}[\ideal{p}^n]=\widehat{E}\] (just as in the proof of (ii) of (\ref{prop:formal-o-mod-frob})). In particular, $\widehat{E}\subset E[\ideal{p}^\infty]$ and the strict $\underline{O_K}_S$-module structure on $\widehat{E}$ extends uniquely to a strict $\lim_{n} \underline{O_K/\ideal{p}^{n}}_S$-module structure.
We are now reduced to showing that for each $r\geq 0$ the zero section $S\to E[\ideal{p}^r]$ is a nilpotent immersion as this will give $E[\ideal{p}^r]\subset \widehat{E}$. As $S\times\mathrm{Spec}(\mathbf{F}_\ideal{p})\to S$ is a nilpotent immersion, we may replace $S$ by $S\times\mathrm{Spec}(\mathbf{F}_\ideal{p})$ and assume that $S$ has characteristic $\ideal{p}$. From (iii) of (\ref{prop:formal-o-mod-frob}) we find closed immersions \[\ker(\mathrm{Fr}_{\widehat{E}/S}^{N\ideal{p}^r})\subset \widehat{E}[\ideal{p}^r]\subset E[\ideal{p}^r].\] But, as $\widehat{E}$ is a smooth formal group of dimension one, $\ker(\mathrm{Fr}_{\widehat{E}/S}^{N\ideal{p}^r})$ is finite locally free of rank $N\ideal{p}^r$, as is $E[\ideal{p}^r]$. Therefore the closed immersion \[\ker(\mathrm{Fr}_{\widehat{E}/S}^{N\ideal{p}^r})\subset E[\ideal{p}^r]\] is an isomorphism. It follows that $S\to E[\ideal{p}^r]$ is a nilpotent immersion, so that $E[\ideal{p}^r]\subset \widehat{E}$, and therefore \[\widehat{E}=E[\ideal{p}^\infty].\] This proves the first claim. For the second, the sheaf of groups $\widehat{E}=E[\ideal{p}^\infty]$ is a strict formal $\widehat{O_{K_\ideal{p}}}_S$-module of dimension one. Moreover, as $\widehat{E}=E[\ideal{p}^\infty]$ we have \[\ker(i_\ideal{p}: \widehat{E}\to \widehat{E}\otimes_{O_K}\ideal{p}^{-1})=E[\ideal{p}]\] so that \[i_\ideal{p}:\widehat{E}\to \widehat{E}\otimes_{O_{K_\ideal{p}}}\ideal{p}^{-1}\] is finite locally free of degree $N\ideal{p}$. This is precisely the definition of a Lubin--Tate $O_{K_\ideal{p}}$-module (\ref{subsec:definition-lubin-tate-module}).
\end{proof}
\begin{coro}\label{coro:level-isom-cover} Let $E/S$ be a \textup{CM} elliptic curve.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The morphism $\underline{\mathrm{Isom}}_S^{O_K}(E[\a], \underline{O_K/\a}_S)\to S$ is affine and \'etale, factors through $S[\a^{-1}]=S\times\mathrm{Spec}(O_K[\a^{-1}])\subset S$ and defines an affine \'etale cover \[\underline{\mathrm{Isom}}_S^{O_K}(E[\a], \underline{O_K/\a}_S)\to S[\a^{-1}].\]
\item If $P\subset \mathrm{Id}_{O_K}$ is any set of ideals which do not admit a common divisor the family \[(\underline{\mathrm{Isom}}_S^{O_K}(E[\a], \underline{O_K/\a}_S)\to S)_{\a\in P}\] is an affine \'etale cover of $S$.
\end{enumerate}
\end{coro}
\begin{proof} (i) If there exists an isomorphism $E_T[\a]\overset{\sim}{\longrightarrow} \underline{O_K/\a}_T$ over some affine $S$-scheme $T$ then $E_T[\a]$ is \'etale, so that by (i) of (\ref{prop:p-torsion-classification}) the map $T\to S$ factors through the affine \'etale sub-scheme $S[\a^{-1}]\subset S$. It follows that $\underline{\mathrm{Isom}}_S^{O_K}(E[\a], \underline{O_K/\a}_S)\to S$ factors through $S[\a^{-1}]$ and so base changing along $S[\a^{-1}]\to S$ we may assume $\a$ is invertible on $S$.
Applying (i) of (\ref{prop:p-torsion-classification}) to the prime power divisors of $\a$ we find that $E[\a]$ is locally isomorphic to $\underline{\a^{-1}/O_K}_S$ which is isomorphic to $\underline{O_K/\a}_S$. This shows that \[\underline{\mathrm{Isom}}_S^{O_K}(E[\a], \underline{O_K/\a}_S)\to S\] is an epimorphism so that by descent, to show that it is finite and \'etale, we may assume that $E[\a]= \underline{O_K/\a}_S$. In this case we have \[\underline{\mathrm{Isom}}_S^{O_K}(\underline{O_K/\a}_S, \underline{O_K/\a}_S)=\underline{(O_K/\a)^\times}_S\] and the claim is clear.
(ii) The hypothesis on $P$ implies that $(S[\a^{-1}]\to S)_{\a\in P}$ is a cover of $S$ so that this follows from (i).
\end{proof}
\begin{coro}\label{coro:tensor-p-equals-frobenius} For each $n\geq 0$ there is a unique isomorphism of functors \[\nu_{\ideal{p}^n}: -\otimes_{O_K}\ideal{p}^{-n} \overset{\sim}{\longrightarrow} \mathrm{Fr}^{N\ideal{p}^n*}(-)\] on $\mc{M}_{CM}\times\mathrm{Spec}(\mathbf{F}_\ideal{p})$ such that for all \textup{CM} elliptic curves $E$ over characteristic $\ideal{p}$-sheaves $S$ the diagram \[\xymatrix{&E\ar[dr]^{\mathrm{Fr}_{E/S}^{N\ideal{p}^n}}\ar[dl]_-{i_{\ideal{p}^n}}&\\
E\otimes_{O_K}\ideal{p}^{-n} \ar[rr]^{\nu_{\ideal{p}^n}}_\sim && \mathrm{Fr}^{N\ideal{p}*}(E)}\] commutes.
\end{coro}
\begin{proof} The two homomorphisms $i_{\ideal{p}^n}$ and $\mathrm{Fr}_{E/S}^{N\ideal{p}^n}$ are finite locally free of degree $N\ideal{p}^n$ so we need only show that their kernels are equal. As $\ideal{p}$ is nilpotent on $S$, we have $\widehat{E}=E[\ideal{p}^\infty]$ and $\widehat{E}$ is a Lubin--Tate $O_{K_\ideal{p}}$-module by (ii) of (\ref{prop:p-torsion-classification}) so that by (\ref{lemm:p-torsion-lifts-frob-in-lubin-tate}) we find \[\ker(\mathrm{Fr}_{E/S}^{N\ideal{p}^n})=\ker(\mathrm{Fr}_{\widehat{E}/S}^{N\ideal{p}^n})=\widehat{E}[\ideal{p}^n]=E[\ideal{p}^n]=\ker(i_{\ideal{p}^n}).\]
\end{proof}
\begin{rema} The above (\ref{coro:tensor-p-equals-frobenius}) is the first indication that the moduli stack $\mc{M}_\mathrm{CM}$ should admit a $\Lambda$-structure.
\end{rema}
\begin{coro}\label{prop:hom-ell-curves} Let $f: E\to E'$ be a homomorphism of \textup{CM} elliptic curves over $S$. Then there is a unique decomposition $S=\amalg_{\a\subset O_K}S_{(\a)}$ such $f_{S_{(0)}}$ is the zero map and such that $\ker(f_{S_{(\a)}})=E_{S_{(\a)}}[\a]$ for all $(0)\neq \a\subset O_K$.
\end{coro}
\begin{proof} By rigidity we may decompose $S$ as $S_{(0)}\amalg S_{\mathrm{isog}}$ where $f_{S_{(0)}}$ is the zero map and where $f_{S_\mathrm{isog}}$ is finite locally free of positive degree. Thus we may replace $S$ by $S_{\mathrm{isog}}$ and assume that $f$ is finite locally free. Then $\ker(f)\subset E[\mathrm{tors}]$ and \[\ker(f)=\bigoplus_{\ideal{p}}\ker(f_\ideal{p})\] where $f_\ideal{p}: E[\ideal{p}^\infty]\to E'[\ideal{p}^\infty]$ is the restriction of $f$ to the $\ideal{p}$-divisible groups. As $\ker(f)$ is finite locally free so is $\ker(f_\ideal{p})$ for all prime ideals $\ideal{p}$.
The claim is local so we may reduce to the case where $S=\mathrm{Spec}(A)$ is an affine scheme and, by passage to the limit, to when $S$ finitely presented over $\mathrm{Spec}(O_K)$. As $S$ is noetherian it admits a finite cover by connected affine schemes and so we may assume that $S$ is connected. In this case, the finite locally free group schemes $\ker(f_\ideal{p})$ have constant degree and we are reduced to showing that, locally on $S$, there exists an integer $n\geq 0$ such that $\ker(f_\ideal{p})=E[\ideal{p}^n]$.
We continue to localise $S$ and so assume that $S=\mathrm{Spec}(A)$ for $A$ a local noetherian ring with maximal ideal $I$ and by descent that $A$ is $I$-adically complete. As $E$ and $E[\ideal{p}^n]$ and $\ker(\ideal{f}_\ideal{p})$ are all proper over $S$ to check that $\ker(f_\ideal{p})=E[\ideal{p}^n]$ for some $n$ we may, by Grothendieck's formal GAGA, replace $S$ by $\mathrm{Spec}(A/I^r)$ for each $r\geq 0$ and assume that $S=\mathrm{Spec}(A)$ where $A$ is an artinian local ring. If $\ideal{p}$ is invertible in $A$ then $E[\ideal{p}^\infty]$ is locally isomorphic to the constant group scheme \[\underline{K_\ideal{p}/O_{K_\ideal{p}}}_S\] by (\ref{prop:p-torsion-classification}), from this and the corresponding fact for $K_\ideal{p}/O_{K_\ideal{p}}$, we see that any finite locally free $\underline{O_K}_S$-sub-module of $E[\ideal{p}^\infty]$ is of the form $E[\ideal{p}^n]$ for some $n\geq 0$. On the other hand, if $\ideal{p}$ is nilpotent in $A$, then $\ker(f_\ideal{p})$ is the kernel of the homomorphism $f_\ideal{p} : E[\ideal{p}^\infty]\to E'[\ideal{p}^\infty]$ of Lubin--Tate $O_{K_\ideal{p}}$-modules and so the claim follows from (\ref{prop:lubin-tate-hom}).
\end{proof}
\section{Complex and $\ideal{p}$-adic bases} In this section we give CM analogues of the classification of elliptic curves over complex schemes in terms of lattices, and of the Serre-Tate theorem. We use this to show there exists a CM elliptic curve $E\to \mathrm{Spec}(O_{K^\mathrm{sep}})$ (\ref{lemm:cm-over-separably-closed-fields}), that deformations of CM elliptic curves over $\ideal{p}$-adic bases always exist and are unique (\ref{coro:cm-etale}).
\subsection{} Let us first consider complex bases. We fix a homomorphism $O_K\to \mathbf{C}$ so that we may view $\mathrm{Spec}(\mathbf{C})$ as a $\mathrm{Spec}(O_K)$-scheme. Let $S$ be a locally finitely presented $\mathrm{Spec}(\mathbf{C})$-scheme and $E/S$ a CM elliptic curve. Consider the exponential sequence associated to the analytification $E^\mathrm{an}/S^\mathrm{an}$ (cf.\ (\ref{subsec:complex-class})) \[0\to T_\mathbf{Z}(E/S)\to \underline{\mathrm{Lie}}_{E^\mathrm{an}/S^\mathrm{an}} \to E^\mathrm{an}\to 0.\] The strictness of the $\underline{O_K}_S$-action on $E$ implies that the homomorphism \[\underline{\mathrm{Lie}}_{E^\mathrm{an}/S^\mathrm{an}}\to E^\mathrm{an}\] is a homomorphism of $\underline{O_K}_{S^\mathrm{an}}$-modules. This makes the rank two $\underline{\mathbf{Z}}_{S^\mathrm{an}}$-local system $T_\mathbf{Z}(E/S)$ a rank one $O_K$-local system, which we shall denote by $T_{O_K}(E/S)$, and the homomorphism \[T_{O_K}(E/S)\otimes_{\underline{O_K}_{S^\mathrm{an}}}\mc{O}_{S^\mathrm{an}}\to \underline{\mathrm{Lie}}_{E^\mathrm{an}/S^\mathrm{an}}\] is now an isomorphism. As the automorphism sheaf of any rank one $O_K$-local system over $S$ or $S^\mathrm{an}$ is just the constant sheaf associated to the finite group $O_K^\times$ (from which one sees that every $O_K$-local system over $S$ or $S^\mathrm{an}$ is just a sum of finite \'etale $S$-schemes, or finite covering spaces of $S^\mathrm{an}$) by GAGA the functor sending a rank one $O_K$-local system on $S$ to the corresponding rank one $O_K$-local system on $S^\mathrm{an}$ is an equivalence. Therefore:
\begin{prop}\label{prop:cm-curves-complex} The functor \[E/S\mapsto T_{O_K}(E/S)\] is an equivalence between the category of \textup{CM} elliptic curves $E/S$ and the rank one $O_K$-local systems over $S$.
\end{prop}
\begin{proof} This follows from (\ref{prop:class-complex-elliptic}) and the remarks above.
\end{proof}
\begin{coro}\label{lemm:cm-over-separably-closed-fields} There exists a \textup{CM} elliptic curve $E$ over $\mathrm{Spec}(O_{K^\mathrm{sep}})$.
\end{coro}
\begin{proof} By (\ref{prop:cm-curves-complex}) we see that there exists a CM elliptic curve $E/\mathrm{Spec}(\mathbf{C})$. Writing $\mathrm{Spec}(\mathbf{C})=\lim_\lambda \mathrm{Spec}(A_\lambda)$ as a filtered inverse limit of finite type affine $\mathrm{Spec}(K)$-schemes by passage to the limit we find a CM elliptic curve $E_\lambda/\mathrm{Spec}(A_\lambda)$ for some $\lambda$. As $\mathrm{Spec}(A_\lambda)$ is of finite type over $\mathrm{Spec}(K)$ it admits a closed point $\mathrm{Spec}(L)\to \mathrm{Spec}(A_\lambda)$ with $L/K$ finite and we find a CM elliptic curve $E_\lambda\times_{\mathrm{Spec}(A_\lambda)}\mathrm{Spec}(L)$ over $\mathrm{Spec}(L)$.
We now show (essentially following the arguments of \cite{SerreTate68} only for the sake of completeness) that for any CM elliptic curve $E\to \mathrm{Spec}(L)$ over a finite extension $L/K$ there is a finite extension $L'/L$ such that $E\times_{\mathrm{Spec}(L)}\mathrm{Spec}(L')$ has good reduction everywhere. As $E$ has bad reduction at only finitely many primes, it is enough to show that for each prime $\ideal{p}$ of $L$ there is a finite extension $L'/L$ such that $E\times_{\mathrm{Spec}(L)} \mathrm{Spec}(L')$ admits good reduction at all primes of $L'$ over $\ideal{p}$, as then $E$ will obtain good reduction over the compositum of these finitely many fields.
So fix a prime $\ideal{p}$ lying over the rational prime $p$ and let $\ell\neq p$ be another rational prime. Let $\rho_\ell:G(L^\mathrm{sep}/L)^\mathrm{ab}=G(L^\mathrm{ab}/L)\to (O_K\otimes_{\mathbf{Z}} \mathbf{Z}_\ell)^\times$ be the character defining the $O_K\otimes_{\mathbf{Z}}\mathbf{Z}_\ell$-linear action of $G(L^\mathrm{sep}/L)$ on \[E[\ell^\infty](\mathrm{Spec}(L^\mathrm{sep}))\overset{\sim}{\longrightarrow} O_K\otimes_{\mathbf{Z}}\mathbf{Q}_\ell/\mathbf{Z}_\ell.\] First we claim that for each place $v$ of $L^\mathrm{sep}$ lying over $\ideal{p}$ the restriction of $\rho_\ell$ to the inertia group $I_v\subset G(L^\mathrm{sep}/L)$ at $v$ has finite image and that this image is independent of $v$. As the target of $\rho_\ell$ is commutative and as the image of $I_v(L^\mathrm{sep}/L)$ in $G(L^\mathrm{sep}/L)=G(L^\mathrm{ab}/L)$ is the inertia subgroup $I_\ideal{p}\subset G(L^\mathrm{ab}/L)$ at $\ideal{p}$, which is independent of $v$, it is enough to show that $\rho_\ell(I_\ideal{p})$ is finite. But it is well known that $I_\ideal{p}$ admits an open subgroup of finite index which is pro-$p$ and that $(O_K\otimes_{\mathbf{Z}}\mathbf{Z}_\ell)^\times$ admits an open subgroup of finite index which is pro-$\ell$. Therefore, as $\rho_\ell$ is continuous and $\ell\neq p$ the image of the inertia group $\rho_\ell(I_\ideal{p})$ must be finite. It now follows that there is an integer $n\geq 0$ with the property that \[\rho_\ell(I_\ideal{p})\to (O_K\otimes_{\mathbf{Z}}\mathbf{Z}_\ell)^\times \to (O_K\otimes_{\mathbf{Z}}\mathbf{Z}/\ell^n)^\times\] is injective. Taking $L'=L(E[\ell^n])$, the action of the inertia group $I_v(L^\mathrm{sep}/L')\subset G(L^\mathrm{sep}/L')$ at any place $v$ lying over $\ideal{p}$ on $E'[\ell^\infty](\mathrm{Spec}(L^\mathrm{sep}))=E'[\ell^\infty](\mathrm{Spec}(L^\mathrm{sep}))$ is trivial and we now apply (\ref{theo:criterion-of-good-reduction}) to deduce that $E'/\mathrm{Spec}(L')$ has good reduction at all places of $L'$ lying over $\ideal{p}$.
\end{proof}
\subsection{} Now let us consider $\ideal{p}$-adic bases. Fix a prime ideal $\ideal{p}$ of $O_K$ and let $S_0\to S$ be a nilpotent thickening of $\ideal{p}$-adic sheaves and denote by $D_\ideal{p}(S, S_0)$ the category whose objects are triples $(E_0/S_0, F/S, \rho)$ where
\begin{enumerate}[label=\textup{(\roman*)}]
\item $E/S_0$ is a CM elliptic curve,
\item $F/S$ is a Lubin--Tate $O_{K_\ideal{p}}$-module, and
\item $\rho: F\times_S S_0\overset{\sim}{\longrightarrow} E[\ideal{p}^\infty]$ is an isomorphism of Lubin--Tate $O_{K_\ideal{p}}$-modules,
\end{enumerate} \noindent and whose morphisms $(E_0/S_0, F/S, \rho)\to (E_0'/S_0, F'/S, \rho')$ are given by pairs $(g_0, g_\ideal{p})$ where $g_0: E_0\to E'_0$ is a homomorphism of CM elliptic curves, $g_\ideal{p}: F\to F'$ is a homomorphism of Lubin--Tate $O_{K_\ideal{p}}$-modules such that $\rho'\circ (g_\ideal{p}\times_{S} S_0)=(g_0|_{E[\ideal{p}^\infty]})\circ \rho$. There is an obvious functor \[\mathrm{CM}(S)\to D_\ideal{p}(S_0, S): E/S\mapsto (E\times_S S_0, E[\ideal{p}^\infty], \mathrm{id}_{E[\ideal{p}^\infty]\times_S S_0}).\] The following is the theorem of Serre-Tate (\ref{theo:serre-tate}) adapted for CM elliptic curves.
\begin{prop}\label{cm-serre-tate} The functor $CM(S)\to D_\ideal{p}(S_0, S)$ is an equivalence of categories.
\end{prop}
\begin{proof} Let $p$ be the rational prime lying under $\ideal{p}$. We will show that to each object $(E_0/S_0, F/S, \rho)$ of $D_\ideal{p}(S_0, S)$ (resp. morphism) one can functorially define a element of $D_p(S_0, S)$ (resp. morphism) (cf.\ (\ref{subsec:classical-serre-tate})). If $\overline{\ideal{p}}=\ideal{p}$ then this is clear as $E_0[\ideal{p}^\infty]=E[p^\infty]$. On the other hand if $\overline{\ideal{p}}\neq \ideal{p}$ then $E_0[p^\infty]=E_0[\ideal{p}^\infty]\times_{S_0} E_0[\overline{\ideal{p}}^\infty]$ and $E_0[\overline{\ideal{p}}^\infty]$ is \'etale over $S_0$ so that there is a unique deformation of $E_0[\overline{\ideal{p}}^\infty]$ along $S_0\to S$ and the product over $S$ of this deformation with $F$ will give the desired deformation of $E_0[p^\infty]$. Regarding morphisms $(g_0, g_\ideal{p})$ the restriction $g_0|_{E[\overline{\ideal{p}}^\infty]}$ lifts uniquely and the product of this with $g_\ideal{p}$ defines the map on morphisms.
In both cases this defines a functor $D_\ideal{p}(S_0, S)\to D_p(S_0, S)$ and by the classical Serre-Tate theorem a functor $D_\ideal{p}(S_0, S)\to \mathrm{Ell}(S)$. By functoriality, if $E/S$ is the image of $(E_0/S_0, F/S, \rho)$ then $E/S$ admits the structure of an $\underline{O_K}_S$-module (deforming the corresponding structure on $E_0/S_0$). Moreover, the action of $\underline{O_K}_S$ on the Lie algebra of $E/S$ is strict as we have identifications of $\underline{O_K}_S$-modules \[\underline{\mathrm{Lie}}_{E/S}=\underline{\mathrm{Lie}}_{E[p^\infty]/S}=\underline{\mathrm{Lie}}_{E[\ideal{p}^\infty]/S}=\underline{\mathrm{Lie}}_{F/S}\] and the action of $\underline{O_K}_S$ on $F$ is strict. In particular, the functor $D_\ideal{p}(S_0, S)\to \mathrm{Ell}(S)$ factors as \[D_\ideal{p}(S_0, S)\to \mathrm{CM}(S)\] and it is easily seen to be quasi-inverse to $CM(S)\to D_\ideal{p}(S_0, S)$.
\end{proof}
\begin{coro}\label{coro:cm-etale} If $S_0\to S$ is a nilpotent immersion of $\ideal{p}$-adic sheaves the functor \[\mathrm{CM}(S)\to \mathrm{CM}(S_0): E/S\mapsto E\times_{S} S_0/S_0\] is an equivalence of categories.
\end{coro}
\begin{proof} This follows from (\ref{cm-serre-tate}) combined with (\ref{coro:lift-lubin-tate}).
\end{proof}
\begin{coro}\label{prop:lifting} Let $S$ be a $\ideal{p}$-adic affine scheme and let $E/S$ be a \textup{CM} elliptic curve. Then there exists a affine scheme $\widetilde{S}$, flat over $\mathrm{Spec}(O_K)$, a morphism $S\to \widetilde{S}$, and a \textup{CM} elliptic curve $\widetilde{E}/\widetilde{S}$ such that $\widetilde{E}\times_{\widetilde{S}} S\overset{\sim}{\longrightarrow} E$.
\end{coro}
\begin{proof} By passage to the limit we may assume that $S=\mathrm{Spec}(A)$ with $A$ a finite type $O_K$-algebra. We then choose a surjection $A'\to A$ where $A'$ is a flat $O_K$-algebra. Letting $I$ be the kernel of $A'\to A$ we write $\widetilde{S}=\mathrm{Spec}(\lim_n A'/I^{n})=\mathrm{Spec}(\widehat{A}')$, $\widetilde{S}_n=\mathrm{Spec}(A'/I^n)$ and $\widetilde{S}_\infty=\colim_{n} \mathrm{Spec}(A'/I^n)$. We will show that there exists a CM elliptic curve $\widetilde{E}/\widetilde{S}$ with $\widetilde{E}\times_{\widetilde{S}} S\overset{\sim}{\longrightarrow} E$ which, as $\widetilde{S}=\mathrm{Spec}(\widehat{A}')$ is flat over $\mathrm{Spec}(O_K)$, will prove the claim.
For each $n\geq 0$ there is a unique $\widetilde{E}_n/\widetilde{S}_n$ equipped with an isomorphism $\widetilde{E}_n\times_{\widetilde{S}_n}S\overset{\sim}{\longrightarrow} E$ by (\ref{coro:cm-etale}). Therefore there is a unique CM elliptic curve $\widetilde{E}_\infty$ over the ind-scheme $\widetilde{S}_\infty=\colim_n \widetilde{S}_n$ with $\widetilde{E}_\infty\times_{\widetilde{S}_\infty}S=E$. By (\ref{prop:formal-gaga}) there is a unique elliptic curve $\widetilde{E}$ over $\widetilde{S}=\mathrm{Spec}(\widehat{A}')$ with $\widetilde{E}\times_{\widetilde{S}} S=E$ and moreover it admits an action of $\underline{O_K}_S$ compatible with that on $\widetilde{E}_n$ over $\widetilde{S}_n$ which (taking $n$ large) shows that the action is strict so that $\widetilde{E}/\widetilde{S}$ is a CM elliptic curve.
\end{proof}
\section{Isogenies and the action of $\mc{CL}_{O_K}$ on $\mc{M}_\mathrm{CM}$} In this section we continue the study of the action of $\mc{CL}_{O_K}$ on $\mc{M}_\mathrm{CM}$. We show that all pairs of CM elliptic curves over a fixed base are locally isogenous (\ref{lemm:local-isogenies}) and from this deduce that the action of $\mc{CL}_{O_K}$ on $\mc{M}_\mathrm{CM}$ makes it a torsor (\ref{theo:mcm-is-a-torsor}).
\begin{lemm}\label{lemm:local-isogenies} Let $E, E'/S$ be a pair of \textup{CM} elliptic curves. Then the family \[(\underline{\mathrm{Isom}}_S^{O_K}(E, E'\otimes_{O_K}\a^{-1})\to S)_{\a\in \mathrm{Id}_{O_K}}\] is a finite \'etale cover of $S$.
\end{lemm}
\begin{proof} The claim is local so we may assume that $S$ is an affine scheme and by passage to the limit that $S$ is finitely presented over $\mathrm{Spec}(O_K)$. For every pair of CM elliptic curves $E, E'/S$ the inclusion \[\underline{\mathrm{Isom}}_S^{O_K}(E, E')\to \underline{\mathrm{Isom}}_S(E, E')\] is easily seen to be a closed immersion and so it follows from (\ref{prop:isom-are-fin-unram}) that $\underline{\mathrm{Isom}}_S^{O_K}(E, E')\to S$ is finite and unramified. Therefore, for each integral ideal $\a\in \mathrm{Id}_{O_K}$ the sheaf $\underline{\mathrm{Isom}}_S^{O_K}(E, E'\otimes_{O_K}\a^{-1})$ is finite and unramified over $S$. It follows from (\ref{coro:cm-etale}) that the maximal open sub-scheme of $\underline{\mathrm{Isom}}_S^{O_K}(E, E'\otimes_{O_K}\a^{-1})$ which is \'etale over $S$ contains all of the special fibres and is therefore equal to all of $\underline{\mathrm{Isom}}_S^{O_K}(E, E'\otimes_{O_K}\a^{-1})$.
As the morphisms $(\underline{\mathrm{Isom}}_S^{O_K}(E, E'\otimes_{O_K}\a^{-1})\to S)_{\a\in \mathrm{Id}_{O_K}}$ are finite \'etale this family forms a cover of $S$ if and only if it is a cover after base change to each generic point of $S$. So we way assume that $S$ is either flat over $\mathrm{Spec}(O_K)$ (if a generic point has characteristic 0) or that $S$ is $\ideal{p}$-adic for some prime $\ideal{p}$ (if a generic point has characteristic $\ideal{p}$). Let us reduce the second case to the first and assume that $S$ is $\ideal{p}$-adic.
Applying (\ref{prop:lifting}) to $E$ and $E'$, then taking the product of the flat $\mathrm{Spec}(O_K)$-schemes over which $E$ and $E'$ can be extended, we find a flat $\mathrm{Spec}(O_K)$-scheme $\widetilde{S}$, a morphism $S\to \widetilde{S}$ and a pair of CM elliptic curves $\widetilde{E}$ and $\widetilde{E}'$ over $\widetilde{S}$ whose pull-backs to $S$ are isomorphic to $E$ and $E'$. We now see that the claim we wish to prove is true for $E, E'/S$ if it is true for $\widetilde{E}, \widetilde{E}'/\widetilde{S}$. As the generic points of $\widetilde{S}$ are all of characteristic 0, we have reduced the second case to the first and we may assume that $S=\mathrm{Spec}(F)$ where $F$ is a field of characteristic 0.
By passage to the limit we may assume that $F$ has finite transcendence degree over $K$, that there is a morphism $F\to \mathbf{C}$ and by base change that $F=\mathbf{C}$. The claim now follows from (\ref{prop:cm-curves-complex}), the fact that all $O_K$-local systems over $\mathrm{Spec}(\mathbf{C})$ are constant, and the fact that if $L$ and $L'$ are two rank one $O_K$-modules there always exists some $\a\subset O_K$ and an isomorphism $L'\overset{\sim}{\longrightarrow} L\otimes_{O_K}\a^{-1}$.
\end{proof}
\begin{prop}\label{prop:tensor-eval-isom} For each pair $E, E'/S$ of \textup{CM} elliptic curves over $S$ the sheaf $\underline{\mathrm{Hom}}_S^{O_K}(E, E')$ is a rank one $O_K$-local system and the evaluation homomorphism \[E\otimes_{O_K}\underline{\mathrm{Hom}}_S^{O_K}(E, E')\to E'\] is an isomorphism.
\end{prop}
\begin{proof} The claim is local and by (\ref{lemm:local-isogenies}) the set of $S$-sheaves \[(\underline{\mathrm{Isom}}_S^{O_K}(E, E'\otimes_{O_K}\a^{-1})\to S)_{\a\in \mathrm{Id}_{O_K}}\] is a finite \'etale cover, so replacing $S$ with any one of them we may assume that $E'= E\otimes_{O_K}\a^{-1}$ for some integral ideal $\a$. Composing the evaluation homomorphism with $\mathrm{id}_E\otimes_{O_K}i$, where $i:\underline{\a}_S^{-1}\overset{\sim}{\longrightarrow} \underline{\mathrm{Hom}}_S^{O_K}(E, E\otimes_{O_K}\a^{-1})$ is the isomorphism of (\ref{prop:cm-tensor-homs}), the resulting map \[E\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} E\otimes_{O_K}\underline{\mathrm{Hom}}_S^{O_K}(E, E\otimes_{O_K}\a^{-1})\to E\otimes_{O_K}\a^{-1}\] is an isomorphism (in particular it is the identity) and so the second map is an isomorphism and we are done.
\end{proof}
\begin{coro}\label{coro:isom-iso-isom} Let $E, E'/S$ be a pair of \textup{CM} elliptic curves. Then $E$ and $E'$ are isomorphic if and only if they are isogenous and locally isomorphic.
\end{coro}
\begin{proof} The only if direction is clear. Conversely, let $f: E\to E'$ be an isogeny and assume that $E$ and $E'$ are locally isomorphic. As the cover $\amalg_{\a\subset O_K} S_{(\a)}=S$ in (\ref{prop:hom-ell-curves}) is by open and closed sub-sheaves, $E$ and $E'$ are isomorphic over $S$ if and only if they are after base change to each $S_{(\a)}$. Therefore we may assume that $\ker(f)=E[\a]=\ker(i_\a)$, so that $f$ factors as $E\to E\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} E'$. Now as $E$ and $E'\overset{\sim}{\longrightarrow} E\otimes_{O_K}\a^{-1}$ are locally isomorphic it follows that $\underline{O_K}_S$ and $\underline{\a^{-1}}_S$ are locally isomorphic. Any such and isomorphism is locally constant from which it follows that there exists an isomorphism $\a^{-1}\overset{\sim}{\longrightarrow} O_K$ and we get \[E'\overset{\sim}{\longrightarrow} E\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} E.\]
\end{proof}
\begin{theo}\label{theo:mcm-is-a-torsor} The functor \[\mc{M}_\mathrm{CM}\times \mc{CL}_{O_K}\to \mc{M}_\mathrm{CM}\times \mc{M}_\mathrm{CM}: (E, \mc{L})\mapsto (E, E\otimes_{O_K}\mc{L})\] is an equivalence of stacks and $\mc{M}_\mathrm{CM}$ is locally (over $\mathrm{Spec}(O_K)$) equivalent to $\mc{CL}_{O_K}$.
\end{theo}
\begin{proof} The functor in question is essentially surjective as, given any pair $(E/S, E'/S)\in \mc{M}_\mathrm{CM}(S)\times \mc{M}_\mathrm{CM}(S)$, by (\ref{prop:tensor-eval-isom}) we have \[(E/S, E'/S)\overset{\sim}{\longrightarrow} (E/S, E\otimes_{O_K}\underline{\mathrm{Hom}}_S(E, E')/S).\] Full faithfulness is the bijectivity of the map \[\mathrm{Isom}_S^{O_K}(E, E')\times \mathrm{Isom}_S^{O_K}(\mc{L}, \mc{L'})\to \mathrm{Isom}_S^{O_K}(E, E')\times\mathrm{Isom}_S^{O_K}(E\otimes_{O_K}\mc{L}, E'\otimes_{O_K}\mc{L}').\]
If $\mathrm{Isom}_S^{O_K}(E, E')= \emptyset$ this is clear. If $\mathrm{Isom}_S^{O_K}(E, E')\neq \emptyset$ we may assume that $E=E'$ and instead show that \[\mathrm{Isom}_S^{O_K}(\mc{L}, \mc{L'})\to \mathrm{Isom}_S^{O_K}(E\otimes_{O_K}\mc{L}, E\otimes_{O_K}\mc{L}')\] is bijective but this follows from (\ref{prop:cm-tensor-homs}).
For the second statement, if $E/\mathrm{Spec}(O_K^\mathrm{sep})$ is any CM elliptic curve (\ref{lemm:cm-over-separably-closed-fields}) the functor \[ \mc{CL}_{O_K}\times\mathrm{Spec}(O_{K^\mathrm{sep}})\to \mc{M}_{\mathrm{CM}}\times\mathrm{Spec}(O_{K^\mathrm{sep}})\] sending a rank one $O_K$-local system $\mc{L}$ over a $\mathrm{Spec}(O_K^\mathrm{sep})$-sheaf $p: S\to \mathrm{Spec}(O_{K^\mathrm{sep}})$ to $p^*(E)\otimes_{O_K}\mc{L}$ is the desired local equivalence.
\end{proof}
\begin{rema}\label{rema:rohr-q} One might ask whether, as in the analogous situation of Lubin--Tate $O$-modules (\ref{coro:lubin-tate-moduli-trivial}), there exists a CM elliptic curve over $\mc{E}/\mathrm{Spec}(O_K)$ inducing a trivialisation of the $\mc{CL}_{O_K}$-torsor $\mc{M}_\mathrm{CM}$, i.e.\ an equivalence of stacks: \[\mc{CL}_{O_K}\overset{\sim}{\longrightarrow} \mc{M}_{\mathrm{CM}}: \mc{L}/S\mapsto E_S\otimes_{O_K}\mc{L}/S\] We shall show later (see (\ref{rema:rohr})) that in general this is not the case (and for non-trivial reasons).
\end{rema}
\begin{coro}\label{coro:hom-map-reduction-epi} Let $E$ and $E'$ be a pair of \textup{CM} elliptic curves over $S$ and assume that $\a$ is invertible on $S$. Then the homomorphism \[\underline{\mathrm{Hom}}_S^{O_K}(E, E')\to \underline{\mathrm{Hom}}_S^{O_K}(E[\a], E'[\a])\] is an epimorphism.
\end{coro}
\begin{proof} We may work locally on $S$ and so assume that $E'\overset{\sim}{\longrightarrow} E\otimes_{O_K} \b^{-1}$. We may find a $k\in K^\times$ such that $\b(k)$ is prime to $\a$ and using the isomorphism $k: \b\overset{\sim}{\longrightarrow} \b(k)$ we may assume $\b$ is prime to $\a$. In this case, the restriction of $i_\b$ to the $\a$-torsion defines an isomorphism $E[\a]\overset{\sim}{\longrightarrow} E[\a]\otimes_{O_K}\b^{-1}$ which, as $E[\a]$ and $E[\a]\otimes_{O_K}\b^{-1}$ are locally isomorphic to $\underline{O_K/\a}_S$, induces an isomorphism \[\underline{O_K/\a}_S\overset{\sim}{\longrightarrow} \underline{\mathrm{Hom}}_S(E[\a], E[\a]\otimes_{O_K}\b^{-1}): a\mapsto a\cdot i_\b\] and it follows that \[\underline{\mathrm{Hom}}_S^{O_K}(E, E\otimes_{O_K}\b^{-1})\to \underline{\mathrm{Hom}}_S^{O_K}(E[\a], E[\a]\otimes_{O_K}\b^{-1})\] is an epimorphism as the image contains $i_\b$.
\end{proof}
\section{The global reciprocity map and CM elliptic curves} We now consider CM elliptic curves over $\mathrm{Spec}(F)$ where $F$ is a field of arbitrary characteristic. First, we define a homomorphism $[-]_F$ from the Galois group $G(F^\mathrm{sep}/F)$ into a certain class group (which depends only on the characteristic of the field $F$) (\ref{subsec:def-morphism-h}). We then prove a relation between this homomorphism $[-]_F$ and the character $\rho_{E/F}$ defining the action of $G(F^\mathrm{sep}/F)$ on the torsion of a CM elliptic curve $E/\mathrm{Spec}(F)$ (\ref{prop:h-homs-and-characters}). Moreover, we show the character $\rho_{E/F}$ determines $E/\mathrm{Spec}(F)$ upto isogeny (\ref{prop:cm-iso-iff-h-equal}) and we use the homomorphism $[-]_F$ to classify exactly which characters $\rho$ of $G(F^\mathrm{sep}/F)$ are of the form $\rho_{E/F}$ for some CM elliptic curve $E/\mathrm{Spec}(F)$ (\ref{prop:cm-exist-given-h}). In (\ref{prop:compute-the-homomorphisms-h}) we compute the homomorphisms $[-]_F$ when $F=K$, $F=K_\ideal{p}$ and $F=\mathbf{F}_\ideal{p}$ for $\ideal{p}$ a prime. In particular, for $F=K$ the homomorphism $[-]_K$ takes the form \[[-]_K: G(K^\mathrm{sep}/K)\to \mathrm{CL}_{O_K, \infty}\] and we show that for $\sigma\in G(K^\mathrm{sep}/K)$ we have \begin{equation}\theta_K([\sigma]_K)=\sigma|_{K^\infty}\label{eqn:main-theorem}\end{equation} where $\theta_K: \mathrm{CL}_{O_K, \infty}\to G(K^\infty/K)$ is reciprocity map (\ref{eqn:define-theta}). This fact is, at least in spirit, equivalent to the main theorem of complex multiplication (see Theorem 5.4 of \cite{Shimura94}). The method we use to derive this fact is quite similar to (and in fact reliant on) the method used to prove (\ref{theo:main-theorem-lubin-tate}) for Lubin--Tate $O$-modules in Chapter 1. Finally, we use these results to derive a sharpening of the criterion of good reduction adapted to CM elliptic curves (\ref{coro:image-of-inertia-group}).
The results of this section are, at least when $F$ is a finite extension of $K$, probably more or less known when translated into the language of algebraic Hecke characters though the proofs we give are, to the best of the author's knowledge, original. We would like to emphasise that the rather abstract approach taken -- which eschews Hecke characters -- works for all fields $F$ simultaneously and allows for more conceptual proofs.
\subsection{}\label{subsec:def-morphism-h} Let $F$ be a field over $O_K$ with separable closure $F^\mathrm{sep}$, let $S=\mathrm{Spec}(F^\mathrm{sep})$ and let $\ideal{f}\in \mathrm{Id}_{O_K}$ be invertible on $\mathrm{Spec}(F)$. By (\ref{lemm:cm-over-separably-closed-fields}) there exists a CM elliptic curve $E/S$. Moreover, if $E'/S$ is another CM elliptic curve then there exists a rank one projective $O_K$-module $L$ and an isomorphism $f: E\otimes_{O_K}L\overset{\sim}{\longrightarrow} E'.$ Of course, we could take $L=\mathrm{Hom}_{O_K}(E, E')$ and $f$ the evaluation map, but for what follows it will be more useful to work with arbitrary modules $L$ and isomorphisms $f: E\otimes_{O_K}L\overset{\sim}{\longrightarrow} E'$.
In particular, for each $\sigma\in G(F^\mathrm{sep}/F)$ there is a rank one projective $O_K$-module $L_\sigma$ and an isomorphism \[f_\sigma: E\otimes_{O_K}L_\sigma\overset{\sim}{\longrightarrow} \sigma^*(E).\] This isomorphism, restricted to the $S=\mathrm{Spec}(F^\mathrm{sep})$-points of the $\ideal{f}$-torsion, is then given by \[E[\ideal{f}](S)\otimes_{O_K}L_\sigma\stackrel{E[\ideal{f}](\sigma)\otimes\lambda_\sigma}{\longrightarrow} E[\ideal{f}](\sigma_!(S))=\sigma^*(E)[\ideal{f}](S)\] for a unique level-$\ideal{f}$ structure $\lambda_\sigma: L_\sigma\to O_K/\ideal{f}$ on $L_\sigma$. It is clear from this construction that the class \[[\sigma]_{F, \ideal{f}}:=(L_\sigma, \lambda_\sigma)\in CL_{O_K}^{(\ideal{f})}\] is independent of the choice of $L_\sigma$ and the isomorphism $f_\sigma: E\otimes_{O_K}L_\sigma\overset{\sim}{\longrightarrow} \sigma^*(E)$. Moreover, as every other CM elliptic curve $E'/S$ is of the form $E\otimes_{O_K}\a$ for some ideal $\a$, having chosen $L_\sigma$ and an isomorphism $f_\sigma: E\otimes_{O_K} L_\sigma\overset{\sim}{\longrightarrow} \sigma^*(E)$, the map $f_\sigma\otimes_{O_K} \a$ defines an isomorphism \[(E\otimes_{O_K}\a)\otimes_{O_K}L_\sigma = E\otimes_{O_K}L_\sigma\otimes_{O_K}\a\stackrel{f_\sigma\otimes_{O_K} \a }\longrightarrow \sigma^*(E)\otimes_{O_K}\a=\sigma^*(E\otimes_{O_K}\a)\] which induces on the $S$-valued points of the $\ideal{f}$-torsion \[(E\otimes_{O_K}\a)[\ideal{f}](S)\otimes_{O_K} L_\sigma \stackrel{(E\otimes_{O_K}\a)[\ideal{f}](\sigma)\otimes \lambda_\sigma}{\longrightarrow}(E\otimes_{O_K}\a)[\ideal{f}](\sigma_!(S)).\] Therefore the class $[\sigma]_{F, \ideal{f}}$ is also independent of the choice of $E/S$.
If $F\to F'$ is a field extension and \[\mathrm{res}:G(F'^\mathrm{sep}/F')\to G(F^\mathrm{sep}/F)\] denotes the restriction map then it follows easy from the definitions that \begin{equation} [-]_{F',\ideal{f}}=[-]_{F, \ideal{f}}\circ \mathrm{res}.\label{eqn:h-restriction}\end{equation} This relation implies that once we known $[-]_F$ for $F=K$, and $F=\mathbf{F}_\ideal{p}$ for each prime ideal $\ideal{p}$, we essentially know $[-]_F$ for any field $F$. The computation of the map $[-]_F$ for these values of $F$ will be given in (\ref{prop:compute-the-homomorphisms-h}).
If $\tau\in G(F^\mathrm{sep}/F)$ then the composition \[E\otimes_{O_K}L_\sigma\otimes_{O_K}L_\tau\stackrel{f_\sigma\otimes_{O_K}\mathrm{id}_{L_\tau}}{\longrightarrow}\sigma^*(E)\otimes_{O_K}L_\tau \stackrel{\sigma^*(f_\tau)}{\longrightarrow} \sigma^*(\tau^*(E))\] induces on the $S$-points of the $\ideal{f}$-torsion the map \[(E[\ideal{f}](\sigma^{-1}\circ \tau\circ \sigma)\otimes_{O_K}\lambda_\tau)\circ (E[\ideal{f}](\sigma)\otimes_{O_K}\lambda_\sigma\otimes_{O_K}\mathrm{id}_{L_\tau})=E[\ideal{f}](\tau\circ \sigma)\otimes_{O_K}\lambda_\sigma\otimes_{O_K}\lambda_\tau\] so that \[[\sigma\tau]_{F, \ideal{f}}=(L_\tau\otimes_{O_K}L_\sigma, \lambda_\tau\otimes_{O_K}\lambda_\sigma)=(L_\tau, \lambda_\tau)(L_\sigma, \lambda_\tau)=[\sigma]_{F, \ideal{f}}[\tau]_{F, \ideal{f}}.\] In other words, $[-]_{F, \ideal{f}}: G(F^\mathrm{sep}/F)\to \mathrm{CL}_{O_K}^{(\ideal{f})}$ is a homomorphism.
\subsection{} The homomorphism of (\ref{def:bracket-homo}) \[[-]_\ideal{f}: (A\otimes_{O_K}K)^\times\to CL_{O_K}^{(\ideal{f})}\] restricted to $A_{O_K}^\times$ factors through the quotient $A_{O_K}^\times\to (O_K/\ideal{f})^\times$ and we denote the resulting map by the same symbol \[[-]_\ideal{f}: (O_K/\ideal{f})^\times\to \mathrm{CL}_{O_K}^{(\ideal{f})}.\] Note that $\ker([-]_\ideal{f})=\mathrm{im}(O_K^\times\to (O_K/\ideal{f})^\times)\subset (O_K/\ideal{f})^\times$.
\subsection{}\label{subsec:definition-h-for-cm-curves} Now let $E/F$ be a CM elliptic curve and denote by \[\rho_{E/F, \ideal{f}}: G(F^\mathrm{sep}/F)\to (O_K/\ideal{f})^\times\] the character defining the action of $G(F^\mathrm{sep}/F)$ on the rank one $O_K/\ideal{f}$-module $E[\ideal{f}](S)=E[\ideal{f}](\mathrm{Spec}(F^\mathrm{sep}))$, i.e.\ \[E[\ideal{f}](\sigma)=\rho_{E/F, \ideal{f}}(\sigma): E[\ideal{f}](S)\to E[\ideal{f}](S).\] We also note for future reference that as $(O_K/\ideal{f})^\times=\mathrm{Aut}_{O_K}(E[\ideal{f}](S))$ is abelian for each $\ideal{f}$ it follows that the extension $F(E[\ideal{f}])/F$ generated by the $\ideal{f}$-torsion is an abelian extension of $F$. Moreover (as is obvious) the character $\rho_{E/F, \ideal{f}}$ is continuous, as it vanishes on the open subgroup of $G(F^\mathrm{sep}/F)$ fixing $F(E[\ideal{f}])$.
\begin{prop}\label{prop:h-homs-and-characters}
\begin{enumerate}[label=\textup{(\roman*)}]
\item The homomorphism $[-]_{F, \ideal{f}}$ is continuous and
\item if $E/F$ is a \textup{CM} elliptic curve the diagram \[\xymatrix{G(F^\mathrm{sep}/F)\ar[dr]_{[-]_{F,\ideal{f}}}\ar[r]^-{\rho_{E/F, \ideal{f}}^{-1}} & (O_K/\ideal{f})^\times\ar[d]^{[-]_\ideal{f}}\\
& \mathrm{CL}_{O_K}^{(\ideal{f})}}\] commutes.
\end{enumerate}
\end{prop}
\begin{proof} (i) We shall reduce this claim to the second. By passage to the limit (applied to any CM elliptic curve over $\mathrm{Spec}(F^\mathrm{sep})$) we may find a CM elliptic curve $E/\mathrm{Spec}(F')$ for some finite extension $F'/F$. If the diagram \begin{equation}\label{eqn:commutativity}\begin{gathered}\xymatrix{G(F^\mathrm{sep}/F')\ar[r]^-{\rho_{E/F', \ideal{f}}^{-1}}\ar[d] & (O_K/\ideal{f})^\times\ar[d]^{[-]_\ideal{f}}\\
G(F^\mathrm{sep}/F)\ar[r]^-{[-]_{F, \ideal{f}}} & CL_{O_K}^{(\ideal{f})}}\end{gathered}\end{equation} commutes then, as the composition along the top and right is continuous, it follows that $[-]_{F, \ideal{f}}|_{G(F^\mathrm{sep}/F')}$ is continuous. But $G(F^\mathrm{sep}/F')\subset G(F^\mathrm{sep}/F)$ is open and of finite index and $\mathrm{CL}_{O_K}^{(\ideal{f})}$ is discrete so it follows that $[-]_{F, \ideal{f}}$ is continuous.
As $[-]_{F, \ideal{f}}=[-]_{F, \ideal{f}}|_{G(F^\mathrm{sep}/F')}$ (\ref{eqn:h-restriction}), the commutativity of (\ref{eqn:commutativity}) is equivalent to the commutativity of the diagram \[\xymatrix{G(F^\mathrm{sep}/F')\ar[dr]_{[-]_{F', \ideal{f}}}\ar[r]^-{\rho_{E/F', \ideal{f}}^{-1}} & (O_K/\ideal{f})^\times\ar[d]^{[-]_\ideal{f}}\\
& \mathrm{CL}_{O_K}^{(\ideal{f})}}\] and so we may assume that $F=F'$ and instead prove (ii).
(ii) Write $E_{F^\mathrm{sep}}=E\times_{\mathrm{Spec}(F)}\mathrm{Spec}(F^\mathrm{sep})$ and let \[d_\sigma: E_{F^\mathrm{sep}}\to \sigma^*(E_{F^\mathrm{sep}})\] be the descent isomorphism i.e.\ (the isomorphism coming from the fact that $E=E\times_{\mathrm{Spec}(F)}\mathrm{Spec}(F^\mathrm{sep})$ descends to $\mathrm{Spec}(F)$). Then $d_\sigma$ induces on the $S$-points of the $\ideal{f}$-torsion the map \[E[\ideal{f}](S)\stackrel{\rho_{E/F, \ideal{f}}(\sigma)^{-1}\cdot E[\ideal{f}](\sigma)}{\longrightarrow} E[\ideal{f}](\sigma_!(S))=\sigma^*(E[\ideal{f}])(S)\] and therefore \[[\sigma]_{F, \ideal{f}}=(O_K, O_K\stackrel{\rho_{E/F, \ideal{f}}(\sigma)^{-1}}{\longrightarrow}O_K/\ideal{f})=[\rho_{E/F, \ideal{f}}(\sigma)^{-1}]_\ideal{f}\in \mathrm{CL}_{O_K}^{(\ideal{f})}\]
\end{proof}
\begin{prop}\label{prop:props-of-h-for-cm-curves} Let $E/F$ be a \textup{CM} elliptic curve.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Let $\mc{L}\in \mc{CL}_{O_K}(\mathrm{Spec}(F))$ and \[\rho_{\mc{L}/F}: G(G^\mathrm{sep}/F)\to \mathrm{Aut}_{O_K}(\mc{L}(\mathrm{Spec}(F^\mathrm{sep})))=O_K^\times\] be the associated character. Then $\rho_{E\otimes_{O_K}\mc{L}/F}=\rho_{E/F, \ideal{f}}\rho_{\mc{L}/F}$.
\item If $\tau: F \to F$ is any $O_K$-linear automorphism then \[\rho_{\tau^*(E)/F}(\sigma)=\rho_{E/F, \ideal{f}}(\widetilde{\tau}^{-1}\sigma\widetilde{\tau})\] for each $\sigma\in G(F^\mathrm{sep}/F)$, where $\widetilde{\tau}$ denotes any extension of $\tau$ to $F^\mathrm{sep}$.
\end{enumerate}
\end{prop}
\begin{proof} These are immediate from the definition of $\rho_{E/F, \ideal{f}}$ as the character defining the action of $G(F^\mathrm{sep}/F)$ on $E[\ideal{f}](S)=E[\ideal{f}](\mathrm{Spec}(F^\mathrm{sep}))$.
\end{proof}
\subsection{} For what follows we let $\ideal{f}\in \mathrm{Id}_{O_K}$ vary over the integral ideals of $O_K$ which are invertible on $\mathrm{Spec}(F)$. We now take the limit over $\ideal{f}$ to define the homomorphisms $[-]_F$, $\rho_{E/F}$ and $[-]$ by
\[[-]_F:=\lim_\ideal{f} [-]_{F, \ideal{f}}: G(F^\mathrm{sep}/F)\to \lim_\ideal{f} \mathrm{CL}_{O_K}^{(\ideal{f})},\] \[\rho_{E/F}:=\lim_\ideal{f} \rho_{E/F, \ideal{f}}: G(F^\mathrm{sep}/F)\to \lim_{\ideal{f}}(O_K/\ideal{f})^\times,\] and \[[-]:=\lim_\ideal{f}[-]_\ideal{f}: \lim_\ideal{f} (O_K/\ideal{f})^\times\to \lim_\ideal{f} \mathrm{CL}_{O_K}^{(\ideal{f})}.\] We find immediately from (ii) of (\ref{prop:h-homs-and-characters}) that \begin{equation}\label{eqn:rel-for-h}[-]_F=[\rho^{-1}_{E/F}].\end{equation}
\begin{prop}\label{prop:cm-iso-iff-h-equal} Let $E, E'/F$ be a pair of \textup{CM} elliptic curves. The following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $\rho_{E/F}=\rho_{E'/F}$,
\item the character $\rho$ defining the action of $G(F^\mathrm{sep}/F)$ on \[\underline{\mathrm{Hom}}_{\mathrm{Spec}(F)}^{O_K}(E, E')(\mathrm{Spec}(F^\mathrm{sep}))\] is trivial,
\item the \'etale $\mathrm{Spec}(F)$-scheme $\underline{\mathrm{Hom}}_{\mathrm{Spec}(F)}^{O_K}(E, E')$ is constant,
\item $E$ and $E'$ are isogenous.
\end{enumerate}
\end{prop}
\begin{proof} Let $\rho: G(F^\mathrm{sep}/F)\to O_K^\times$ be the character defining the action of $G(F^\mathrm{sep}/F)$ on the $\mathrm{Spec}(F^\mathrm{sep})$-valued points of the rank one $O_K$-local system $\underline{\mathrm{Hom}}_{\mathrm{Spec}(F)}^{O_K}(E, E')$.
(i) implies (ii): The isomorphism \[E\otimes_{O_K}\underline{\mathrm{Hom}}_{\mathrm{Spec}(F)^{O_K}}(E, E')\overset{\sim}{\longrightarrow} E'\] combined with (i) of (\ref{prop:props-of-h-for-cm-curves}) gives $\rho_{E'/F}=\rho\cdot \rho_{E/F}$ so that $\rho$ is trivial if and only if $\rho_{E/F}=\rho_{E'/F}$.
(ii) implies (iii): This is clear from the definition of $\rho$.
(iii) implies (iv): If $\underline{\mathrm{Hom}}_{\mathrm{Spec}(F)}^{O_K}(E, E')$ is constant any non-zero $\mathrm{Spec}(F)$-section defines an isogeny $E\to E'$.
(iv) implies (i): If $f: E\to E'$ is an isogeny then $\ker(f)=E[\a]=\ker(i_\a)$ for some integral ideal $\a$ by (\ref{prop:hom-ell-curves}). Therefore $E\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} E'$ and by (i) of (\ref{prop:props-of-h-for-cm-curves}) we then get \[\rho_{E'/F}=\rho_{E\otimes_{O_K}\a^{-1}/F}=\rho_{E/F}.\]
\end{proof}
\begin{prop}\label{prop:cm-exist-given-h} Let \[\rho=\lim_\ideal{f} \rho_\ideal{f}: G(F^\mathrm{sep}/F)\to \lim_{\ideal{f}} (O_K/\ideal{f})^\times\] be a continuous homomorphism. Then there exists a \textup{CM} elliptic curve $E/F$ with $\rho_{E/F}=\rho$ if and only if the diagram \[\xymatrix{G(F^\mathrm{sep}/F)\ar[r]^-{\rho^{-1}}\ar[dr]_{[-]_F} & \lim\limits_{\ideal{f}} (O_K/\ideal{f})^\times\ar[d]^{[-]} \\
& \lim\limits_{\ideal{f}} \mathrm{CL}_{O_K}^{(\ideal{f})}}\] commutes.
\end{prop}
\begin{proof} The only if claim is (\ref{eqn:rel-for-h}). Conversely, by passage to the limit there exists a CM elliptic curve $E'/\mathrm{Spec}(F')$ where $F\subset F'$ is a finite extension. As the compositions of $\rho_{E'/F'}$ and $\rho|_{G(F^\mathrm{sep}/F')}$ with \[[-]:\lim_{\ideal{f}}(O_K/\ideal{f})^\times\to \lim_{\ideal{f}}\mathrm{CL}_{O_K}^{(\ideal{f})}\] coincide and $\ker([-])=O_K^\times$, the difference defines a character \begin{equation}\rho_{E'/F'}^{-1}\cdot \rho|_{G(F^\mathrm{sep}/F')}: G(F^\mathrm{sep}/F')\to O_K^\times=\ker([-])\subset \lim_{\ideal{f}}(O_K/\ideal{f})^\times. \label{eqn:diff-char}\end{equation} Replacing $E'$ by the tensor product of $E'$ with any rank one $O_K$-local system with associated character (\ref{eqn:diff-char}) we may assume that $\rho_{E'/F'}=\rho|_{G(F^\mathrm{sep}/F')}$.
Let $\upsilon\in G(F^\mathrm{sep}/F')$ and $\widetilde{\sigma}\in G(F^\mathrm{sep}/F)$ with $\widetilde{\sigma}|_{F'}=\sigma$. We have by (ii) of (\ref{prop:props-of-h-for-cm-curves}) \[\rho_{\sigma^*(E')/F'}(\upsilon)=\rho_{E'/F'}(\sigma^{-1}\upsilon\sigma)=\rho(\sigma^{-1}\upsilon\sigma)=\rho(\upsilon)=\rho_{E'/F'}(\upsilon).\] Therefore $\rho_{\sigma^*(E')/F'}=\rho_{E'/F'}$ and by (ii) of (\ref{prop:cm-iso-iff-h-equal}) $E'$ and $\sigma^*(E')$ are isogenous.
For all $\ideal{f}$ invertible on $\mathrm{Spec}(F)$ we have \[[\widetilde{\sigma}]_{F, \ideal{f}}=[\rho_\ideal{f}(\sigma)^{-1}]_\ideal{f}=(O_K, O_K \stackrel{\rho_\ideal{f}(\sigma)^{-1}}{\to} O_K/\ideal{f}).\] This implies that, writing $E_{F^\mathrm{sep}}'=E\times_{\mathrm{Spec}(F)}\mathrm{Spec}(F^\mathrm{sep})$, the CM elliptic curves $E_{F^\mathrm{sep}}'$ and $\widetilde{\sigma}^*(E_{F^\mathrm{sep}})$ are isomorphic. As $\widetilde{\sigma}|_{F}=\sigma$ this implies that $E'$ and $\sigma^*(E')$ are locally isomorphic. But $E$ and $E'$ are also isogenous and so by (\ref{coro:isom-iso-isom}) they are isomorphic.
Now fix an integral ideal $\ideal{f}$ which separates units and is also invertible on $\mathrm{Spec}(F)$. As $[\widetilde{\sigma}]_{F, \ideal{f}}=[\rho_\ideal{f}(\widetilde{\sigma})^{-1}]$, and as $\ideal{f}$ separates units, there is a unique isomorphism \[r_{\widetilde{\sigma}}: E'\overset{\sim}{\longrightarrow} \sigma^*(E')\] which on $S$-valued points of the $\ideal{f}$-torsion is the map \begin{equation} \label{eqn:unique-r}E'[\ideal{f}](S)\stackrel{\rho_\ideal{f}(\sigma)^{-1}\cdot E'[\ideal{f}](\widetilde{\sigma})}{\longrightarrow} E'[\ideal{f}](\sigma_!(S))=\sigma^*(E)[\ideal{f}](S)\end{equation} where we view $\widetilde{\sigma}$ as a $\mathrm{Spec}(F')$-morphism \[\widetilde{\sigma}: \sigma_!(S)\to S.\]
If $\tau\in G(F'/F)$ and $\widetilde{\tau}\in G(F^\mathrm{sep}/F)$ satisfies $\widetilde{\tau}|_{F'}=\tau$ then the defining property (\ref{eqn:unique-r}) of the isomorphism \[r_{\widetilde{\sigma}\widetilde{\tau}}: E'\overset{\sim}{\longrightarrow} (\tau\circ \sigma)^*(E')\] is also satisfied by \[\sigma^*(r_{\widetilde{\tau}})\circ r_{\widetilde{\sigma}}: E'\overset{\sim}{\longrightarrow} (\tau\circ \sigma)^*(E')\] and so we get \begin{equation}\label{eqn:cocycle-condition} r_{\widetilde{\sigma}\widetilde{\tau}}=\sigma^*(r_{\widetilde{\tau}})\circ r_{\widetilde{\sigma}}.\end{equation} Moreover, if $\widetilde{\sigma}\in G(F^\mathrm{sep}/F')\subset G(F^\mathrm{sep}/F)$ then $r_{\widetilde{\sigma}}$ induces on the $S$-points of the $\ideal{f}$-torsion the map \[E'[\ideal{f}](S)\stackrel{\rho_\ideal{f}(\widetilde{\sigma})\cdot E'[\ideal{f}](\widetilde{\sigma})}{\longrightarrow} E[\ideal{f}](S).\] However, by definition \[E'[\ideal{f}](\widetilde{\sigma})= \rho_{E'/F', \ideal{f}}(\widetilde{\sigma})\] so that $r_{\widetilde{\sigma}}$ induces on the $S$-valued points of the $\ideal{f}$-torsion the map \[\rho_{E'/F', \ideal{f}}(\widetilde{\sigma})^{-1}E[\ideal{f}](\widetilde{\sigma})=\rho_{E'/F', \ideal{f}}(\widetilde{\sigma})^{-1}\rho_{E'/F', \ideal{f}}(\widetilde{\sigma})=\mathrm{id}_{E[\ideal{f}](S)}.\] The uniqueness of $r_{\widetilde{\sigma}}$ now shows that $r_{\widetilde{\sigma}}=\mathrm{id}_{E'}$.
This combined with the relation (\ref{eqn:cocycle-condition}) shows that for all $\widetilde{\sigma}\in G(F^\mathrm{sep}/F)$ the isomorphism $r_{\widetilde{\sigma}}=r_\sigma$ depends only on $\sigma=\widetilde{\sigma}|_F$, has $r_{\mathrm{id}_{F'}}=\mathrm{id}_{E'}$ and satisfies \[r_{\sigma\tau}=\sigma^*(r_\tau)\circ r_\sigma.\] In other words, we have Galois descent data on $E'\to \mathrm{Spec}(F')$ relative to $\mathrm{Spec}(F')\to \mathrm{Spec}(F)$ and by construction the descended CM elliptic curve $E/\mathrm{Spec}(F)$ has $\rho_{E/F}=\rho$.
\end{proof}
\begin{prop}\label{prop:compute-the-homomorphisms-h}
\begin{enumerate}[label=\textup{(\roman*)}]
\item When $F=\mathbf{F}_\ideal{p}$, in the notation of \textup{(\ref{subsec:ray-class-fields})}, we have for all $n\in \widehat{\mathbf{Z}}$: \[[\mathrm{Fr}^{N\ideal{p}^n}]_{\mathbf{F}_\ideal{p}} = \lim_{(\ideal{f}, \ideal{p})=O_K}[\ideal{p}]_\ideal{f}^{-n}\in \lim_{(\ideal{f}, \ideal{p})=O_K}CL_{O_K}^{(\ideal{f})}.\]
\item When $F=K_\ideal{p}$, we have for all $\sigma\in G(K_\ideal{p}^\mathrm{sep}/K)$: \[\sigma|_{K^\infty}=\theta_K([\sigma]_{K_\ideal{p}})\] where the restriction $|_{K^\infty}$ is along any $K$-linear embedding $K^\infty\to K_\ideal{p}^\mathrm{sep}$.
\item When $F=K$ we have for all $\sigma\in G(K^\mathrm{sep}/K)$: \[\sigma|_{K^\infty}=\theta_K([\sigma]_K).\]
\end{enumerate}
\end{prop}
\begin{proof} (i) As $G(\mathbf{F}_\ideal{p}^\mathrm{sep}/\mathbf{F}_\ideal{p})$ is topologically generated by $\mathrm{Fr}^{N\ideal{p}}$, by continuity it is enough to show that $[\mathrm{Fr}^{N\ideal{p}}]_{\mathbf{F}_\ideal{p}, \ideal{f}}=[\ideal{p}]_\ideal{f}^{-1}$ for all $\ideal{f}$ prime to $\ideal{p}$. Write $S=\mathrm{Spec}(\mathbf{F}_\ideal{p}^\mathrm{sep})$, let $E/S$ be a CM elliptic curve and consider the isomorphism \[\nu_\ideal{p}: E\otimes_{O_K} \ideal{p}^{-1}\overset{\sim}{\longrightarrow} \mathrm{Fr}^{N\ideal{p}*}(E)\] of (\ref{coro:tensor-p-equals-frobenius}). By the definition of $\nu_\ideal{p}$, the composition \[E \stackrel{i_\ideal{p}}{\to} E\otimes_{O_K}\ideal{p}^{-1}\overset{\sim}{\longrightarrow} \mathrm{Fr}^{N\ideal{p}^*}(E)\] is equal to the $N\ideal{p}$-power relative Frobenius of $E$ which induces the map \[E[\ideal{f}](S)\stackrel{E[\ideal{f}](\mathrm{Fr}^{N\ideal{p}})}{\longrightarrow} E[\ideal{f}](\mathrm{Fr}^{N\ideal{p}}_!(S))\] on the $S$-points of the $\ideal{f}$-torsion. Therefore, the map $\nu_\ideal{p}$ induces on the $S$-valued points of the $\ideal{f}$-torsion the map \[E[\ideal{f}](S)\otimes_{O_K}\ideal{p}^{-1}\stackrel{E[\ideal{f}](\mathrm{Fr}^{N\ideal{p}})\otimes f}{\longrightarrow} E[\ideal{f}](\mathrm{Fr}_{!}^{N\ideal{p}}(S))\] where $f$ is level-$\ideal{f}$ structure defined by (\ref{eqn:can-level-ideal}). Hence \[[\mathrm{Fr}^{N\ideal{p}}]_{\mathbf{F}_\ideal{p}, \ideal{f}}=[\ideal{p}^{-1}]_\ideal{f}=[\ideal{p}]_\ideal{f}^{-1}.\]
(ii) Write $T=\mathrm{Spec}(O_{K_\ideal{p}^\mathrm{sep}})$, $T_\ideal{p}=\mathrm{Spec}(\mathbf{F}_\ideal{p}^\mathrm{sep})\subset T$, let $\mc{E}/T$ be a CM elliptic curve, write $\mc{E}_\ideal{p}=\mc{E}\times_T T_\ideal{p}$. By continuity it is enough to prove the claim for $\sigma\in W(K^\mathrm{sep}/K)=v_K^{-1}(\mathbf{Z})$ and by multiplicativity for $\sigma$ with $v_K(\sigma)=n\geq 0$. Let $\sigma$ be such an element.
As $T=\mathrm{Spec}(O_{K^\mathrm{sep}_\ideal{p}})$ admits no non-constant finite \'etale covers, the sheaf \[\underline{\mathrm{Isom}}_T^{O_K}(\mc{E}\otimes_{O_K}\ideal{p}^{-1}, \sigma^*(\mc{E}))\] is finite and constant. Moreover, as $T$ is connected and $\sigma$ acts by $\mathrm{Fr}^{N\ideal{p}^n}$ on the residue field $\mathbf{F}_\ideal{p}^\mathrm{sep}$ of $O_{K_\ideal{p}^\mathrm{sep}}$, there exists a unique isomorphism \[\nu_\sigma: \mc{E}\otimes_{O_K}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} \sigma^*(\mc{E})\] lifting the isomorphism \[\nu_\ideal{p}: \mc{E}_\ideal{p}\otimes_{O_K}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} \mathrm{Fr}^{N\ideal{p}*}(\mc{E}_\ideal{p})\] of (\ref{coro:tensor-p-equals-frobenius-with-level}). We now compute the action of $\nu_\sigma$ on the $T$-points of the $\ideal{f}$-torsion of $\mc{E}/T$.
First let $\ideal{f}$ be prime to $\ideal{p}$. As $\mc{E}[\ideal{f}]$ is finite \'etale, we have $\mc{E}[\ideal{f}](T)=\mc{E}[\ideal{f}](T_\ideal{p})$ compatibly with the actions of $\sigma$ and $\mathrm{Fr}^{N\ideal{p}^n}$. It now follows from (i) that the map induced by $\nu_\sigma$ on the $S$-points of the $\ideal{f}$-torsion is \[\mc{E}[\ideal{f}](T)\otimes_{O_K}\ideal{p}^{-n} \stackrel{\mc{E}[\ideal{f}](\sigma)\otimes f}{\to} \mc{E}[\ideal{f}](\sigma_!(T))=\sigma^*(\mc{E})[\ideal{f}](T)\] where $f$ is level-$\ideal{f}$ structure defined by (\ref{subsec:ray-class-fields}).
Now let $\ideal{f}=\ideal{p}^r$ for some $r\geq 0$. Setting $\mc{E}_r:=\mc{E}[\ideal{p}^r]$ and \[T_\infty=\mathrm{Spf}(O_{K_\ideal{p}^\mathrm{sep}})=\colim_{n}\mathrm{Spec}(O_{K_\ideal{p}^\mathrm{sep}}/\ideal{p}^{n+1}),\] we obtain the commutative diagram
\begin{equation}\label{dia:cm-to-lubin-action}
\begin{gathered} \xymatrix{\mc{E}_r(T)\otimes_{O_K}\ideal{p}^{-n}\ar[r]\ar[d] & \sigma^*(\mc{E}_r)(T)\ar[d]\\
\mc{E}_r(T_\infty)\otimes_{O_K}\ideal{p}^{-n}\ar[r] & \sigma^*(\mc{E}_r)(T_\infty).}
\end{gathered}
\end{equation}
whose columns are isomorphisms as $\mc{E}_r$ is finite locally free over $S$. Now consider $F=\mc{E}[\ideal{p}^\infty]\times_T T_\infty$ and set $F_r=F[\ideal{p}^r]$. Then $F$ is a Lubin--Tate $O_{K_\ideal{p}}$-module over the $\ideal{p}$-adic sheaf $T_\infty$ by (\ref{prop:p-torsion-classification}) and the map on the bottom row of (\ref{dia:cm-to-lubin-action}) is equal to \[\nu_\sigma: F_r(T_\infty)\otimes_{O_{K_\ideal{p}}}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} \sigma^*(F_r)(T_\infty)\] where $\nu_\sigma$ is the unique map in (i) of (\ref{theo:main-theorem-lubin-tate}). It follows from (iii) of (\ref{theo:main-theorem-lubin-tate}) that this map is given by \[F_r(\sigma)\otimes_{O_{K_\ideal{p}} }\chi_{K_\ideal{p}}(\sigma):F_r(T_\infty)\otimes_{O_{K_\ideal{p}}}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} F[\ideal{p}^{r}](\sigma_!(T_\infty))=\sigma^*(F[\ideal{p}^r])(T_\infty)\] where $\sigma|_{K_\ideal{p}^\mathrm{ab}}=(\chi_{K_\ideal{p}}(\sigma), K_\ideal{p}^\mathrm{ab}/K_\ideal{p})$. Therefore the map in the top row of (\ref{dia:cm-to-lubin-action}) is given by $\mc{E}_r(\sigma)\otimes_{O_{K}}\chi_{K_\ideal{p}}(\sigma)$.
Writing $S=\mathrm{Spec}(K^\mathrm{sep}_\ideal{p})$ and considering the generic fibre $E:=\mc{E}\times_{T}S$, and arbitrary $\ideal{f}$, the computations above show that the map induced by $\nu_\sigma$ \[E[\ideal{f}](S)\otimes_{O_K}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} E[\ideal{f}](\sigma_!(S))=\sigma^*(E[\ideal{p}^r])(S)\] is given by $E(\sigma)\otimes \chi_{K_\ideal{p}}(\sigma)$ where we view $\chi_{K_\ideal{p}}(\sigma)$ as \[\ideal{p}^{-n} \stackrel{\chi_{K_\ideal{p}}(\sigma)}{\longrightarrow} A_{O_K}\to O_K/\ideal{f}.\] Therefore \[[\sigma]_{K_\ideal{p}, \ideal{f}}=(\ideal{p}^{-n},\ideal{p}^{-n}\stackrel{\chi_{K_\ideal{p}}(\sigma)}{\to}A_{O_K}\to O_K/\ideal{f})=[\chi_{K_\ideal{p}}(\sigma)]_\ideal{f}.\] Taking limits we find $[\sigma]_{K_\ideal{p}}=[\chi_{K_\ideal{p}}(\sigma)]$ and by (\ref{eqn:local-global-class-compatibility}) we get \[\sigma|_{K^\infty}=\theta_K([\sigma]_{K_\ideal{p}})\] where the restriction is along any $K$-embedding $K^\infty\to K_\ideal{p}^\mathrm{sep}$.
(iii) We see from (ii) that for all primes $\ideal{p}$ and all embeddings $K^\mathrm{sep}\to K_\ideal{p}^\mathrm{sep}$ and all $\sigma\in G(K_\ideal{p}^\mathrm{sep}/K)$, writing $\tau=\sigma|_{K^\mathrm{sep}}$ \[\tau|_{K^\infty}=(\sigma|_{K^\mathrm{sep}})|_{K^\infty}=\sigma|_{K^\infty}=\theta_K([\sigma]_{K_\ideal{p}})=\theta_K([\sigma|_{K^\mathrm{sep}}]_K)=\theta_K([\tau]_K).\] However, the sub-group of $G(K^\mathrm{sep}/K)$ generated by elements of the form $\tau=\sigma|_{K^\mathrm{sep}}$ (for varying primes $\ideal{p}$ and embeddings $K^\mathrm{sep}\to K_\ideal{p}^\mathrm{sep}$) is dense. It follows, by continuity and multiplicativity, that for all $\tau\in G(K^\mathrm{sep}/K)$ we have \[\tau|_{K^\infty}=\theta_K([\tau]_K).\]
\end{proof}
\subsection{}\label{subsec:reinterpret-character-idelic} The homomorphisms $[-]_F$ can be (trivially) reinterpreted id\`elically using the isomorphism (\ref{eqn:define-theta}) \[\mathrm{CL}_{O_K, \infty}\stackrel{h_K^{-1}}{\to} (A\otimes_{O_K} K)^\times/K^\times\] and we do so only to make clear the relationship with the map (\ref{eqn:local-rec}). Let us do so here for $h_K$ and so define $\chi_K=[-]_K\circ h_K^{-1}$.
\begin{enumerate}[label=(\roman*)]
\item For all $\sigma\in G(K^\mathrm{sep}/K)$ we have $\sigma|_{K^\infty}=(\chi_K(\sigma), K^\infty/K)$ (cf.\ (iv) of (\ref{theo:main-theorem-lubin-tate})).
\item For each prime $\ideal{p}$, each $\sigma\in W(K_\ideal{p}^\mathrm{sep}/K_\ideal{p})\subset G(K_\ideal{p}^\mathrm{sep}/K)$ and each $K$-linear $K^\mathrm{sep}\to K_\ideal{p}^\mathrm{sep}$ we have \[\chi_K(\sigma|_{K^\mathrm{sep}})=\chi_{K_\ideal{p}}(\sigma)\in K_\ideal{p}^\times\subset (A_{O_K}\otimes_{O_K}K)^\times/K^\times\] where $\chi_{K_\ideal{p}}$ is the homomorphism of (\ref{eqn:local-rec}).
\item If $K\subset L\subset K^\mathrm{sep}$ is a finite extension of $K$ and $\rho: G(K^\mathrm{sep}/L)\to A_{O_K}^\times$ is a continuous character then there exists a CM elliptic curve $E/\mathrm{Spec}(L)$ with $\rho_{E/L}=\rho$ if and only if the following diagram commutes \[\xymatrix{G(K^\mathrm{sep}/L)\ar[r]^-{\rho_{E/L}^{-1}}\ar[dr]_{\chi_{K}|_{G(K^\mathrm{sep}/L)}}& A_{O_K}^\times\ar[d]\\
& (A_{O_K}\otimes_{O_K}K)^\times/K^\times}\] where the right vertical arrow is the obvious map.
\end{enumerate}
\begin{rema}\label{rema:hecke-and-h} Let $L/K$ be a finite Galois extension and let $E/L$ be a CM elliptic curve. We now give a simple description of the algebraic Hecke character of $E/L$, which is a certain continuous homomorphism \[\psi_{E/L}: I_L\to K^\times\] satisfying $\psi_{E/L}|_{L^\times}=N_{L/K}$ (for the definition of $\psi_{E/L}$ see \S 7 of \cite{SerreTate68}).
View the homomorphism \[\rho_{E/L}: G(K^\mathrm{sep}/L)\to A_{O_K}^\times\] as a homomorphism \[\rho_{E/L} : G(L^\mathrm{ab}/L)=G(L^\mathrm{sep}/L)^\mathrm{ab}\to A_{O_K}^\times\] and for $s\in I_L$ write $s_{\mathrm{fin}}$ for the element of $(A_{O_K}\otimes_{O_K}L)^\times$ obtained by forgetting the components of $s$ at the places of $L$ lying over $\infty$. Then we claim that algebraic Hecke character $\psi_{E/L}$ associated to $E/L$ is given by \begin{equation}\psi_{E/L}: I_L\to (A_{O_K}\otimes_{O_K}K)^\times: s\mapsto \rho_{E/L}((s^{-1}, L^\mathrm{ab}/L))\cdot N_{L/K}(s_{\mathrm{fin}}).\label{def:hecke-char}\end{equation} We will not prove this but let us show that the map $\psi_{E/L}$ defined by (\ref{def:hecke-char}) satisfies $\psi_{E/L}|_{L^\times}=N_{L/K}$ and $\psi_{E/L}(I_L)\subset K^\times\subset (A_{O_K}\otimes_{O_K}K)^\times$.
For $a\in L^\times\subset I_L$ we have \[\rho_{E/L}(a)=\rho_{E/L}((a^{-1}, L^\mathrm{ab}/L))\cdot N_{L/K}(a_\mathrm{fin})=1\cdot N_{L/K}(a)\] so that $\rho_{E/L}|_{L^\times}=N_{L/K}$. Now computing the composition \[I_L\stackrel{\rho_{E/L}}{\to} (A\otimes_{O_K}K)^\times\stackrel{[-]}{\to} \mathrm{CL}_{O_K, \infty}\] we find: \begin{eqnarray*} [\rho_{E/L}((s^{-1}, L^\mathrm{ab}/L))\cdot N_{L/K}(s_\mathrm{fin})] &=&[\rho_{E/L}((s^{-1}, L^\mathrm{ab}/L))]\cdot [N_{L/K}(s_\mathrm{fin})]\\
&=& [(s^{-1}, L^\mathrm{ab}/L)]_L\cdot [N_{L/K}(s_{\mathrm{fin}})]\\
&=&[(N_{L/K}(s^{-1}), K^\mathrm{ab}/K)]_K\cdot [N_{L/K}(s_{\mathrm{fin}})]\\
&=&[(N_{L/K}(s^{-1}_\mathrm{fin}), K^\infty/K)]_K\cdot [N_{L/K}(s_{\mathrm{fin}})]\\
&=&[N_{L/K}(s_\mathrm{\mathrm{fin}}^{-1})]_K\cdot [N_{L/K}(s_\mathrm{fin})]\\
&=&1.
\end{eqnarray*}
Therefore, $\rho_{E/L}(I_L)\subset \ker([-]: (A\otimes_{O_K}K)^\times\to \mathrm{CL}_{O_K, \infty})=K^\times$.
\end{rema}
\begin{prop}\label{coro:image-of-inertia-group} Let $L/K$ be a finite Galois extension, $E/L$ a \textup{CM} elliptic curve, $v$ be a non-archimedian place of $K^\mathrm{sep}$ lying over the primes $\P$ of $O_L$ and $\ideal{p}$ of $O_K$, and let $I_v\subset G(K^\mathrm{sep}/L)$ be the inertia group at $v$. Then $\rho_{E/L}(I_v)\subset O_K^\times \cdot O_{K_\ideal{p}}^\times\subset A_{O_K}^\times$ and $E/L$ has good reduction at $\P$ if and only if $\rho_{E/L}(I_v)\subset O_{K_\ideal{p}}^\times$.
\end{prop}
\begin{proof} The homomorphisms $\rho_{E/L}$ and $[-]_L$ both factor through $G(L^\mathrm{ab}/L)$ and we will denote the resulting homomorphisms by the same symbol. The image $[I_v]_L$ of the inertia group at $v$ is equal to $[(O_{L_\P}^\times, L^\mathrm{ab}/L)]_L$ where $(-, L^\mathrm{ab}/L): C_L\to G(L^\mathrm{ab}/L)$ is the global reciprocity map (\ref{prop:global-reciprocity}) and where $O_{L_\P}^\times\subset I_{L}/L^\times=C_L$.
As $[-]_L=[-]_K|_{G(K^\mathrm{sep}/L)}$, the compatibility of the reciprocity maps with the norm (\ref{subsec:global-reciprocity-properties}) shows that \[[(O_{L_\P}^\times, L^\mathrm{ab}/L)]_L=[(N_{L/K}(O_{L_\P}^\times), K^\mathrm{ab}/K)]_K\subset [(O_{K_\ideal{p}}^\times, K^\mathrm{ab}/K)]_K.\] By (iii) of (\ref{prop:compute-the-homomorphisms-h}) we have \[[(O_{K_\ideal{p}}^\times, K^\mathrm{ab}/K)]_K=[(O_{K_\ideal{p}}^\times, K^\infty/K)]_K = [O_{K_\ideal{p}}^\times]\subset \mathrm{CL}_{O_K, \infty}.\] This combined with the relation $[\rho_{E/L}^{-1}]=[-]_L$ and the fact that \[\ker([-]:A_{O_K}^\times\to \mathrm{CL}_{O_K, \infty})=O_K^\times\subset A_{O_K}^\times\] shows that $\rho_{E/L}(I_v)\subset O_K^\times\cdot O_{K_\ideal{p}}^\times$.
Now let $\ell$ be a rational prime such that $\ell\cdot O_K$ is prime to $\ideal{p}$. Then the action of $I_v$ on $E[\ell^\infty](\mathrm{Spec}(L^\mathrm{sep}))$ factors through the image of $\rho_{E/L}(I_v)$ under the projection $A_{O_K}^\times\to (O_K\otimes_{\mathbf{Z}}\mathbf{Z}_\ell)^\times$ and is trivial if and only if this image is trivial. As $\rho_{E/L}(I_v)\subset O_K^\times\cdot O_{K_\ideal{p}}^\times$ the image of this homomorphism is equal to $\rho(I_v)\cap O_K^\times\subset (O_K\otimes_{\mathbf{Z}}\mathbf{Z}_\ell)^\times$ which is trivial if and only if $\rho_{E/L}(I_v)\subset O_{K_\ideal{p}}^\times$. We may now apply (\ref{theo:criterion-of-good-reduction}) to see that $E/L$ has good reduction at $v$ if and only if $\rho_{E/L}(I_v)\subset O_{K_\ideal{p}}^\times$.
\end{proof}
\begin{prop} Let $L$ be a finite Galois extension of $K$ and let $E/L$ a \textup{CM} elliptic curve. If $E[\ideal{f}]$ is a constant scheme over $\mathrm{Spec}(L)$ for some $\ideal{f}$ which separates units then $E/L$ has good reduction everywhere.
\end{prop}
\begin{proof} If $E[\ideal{f}]$ is constant then the action of $G(L^\mathrm{sep}/L)$ on $E[\ideal{f}](\mathrm{Spec}(L^\mathrm{sep}))$ is trivial and therefore $\rho_{E/L}$ takes values in $A_{O_K}^{\times, \ideal{f}}\subset A_{O_K}^\times$. By (\ref{coro:image-of-inertia-group}), for each finite place $v$ of $L^\mathrm{sep}$ lying over the prime $\ideal{p}$ of $O_K$, we have $\rho_{E/L}(I_v)\subset O_K^\times\cdot O_{K_\ideal{p}^\times}\subset A_{O_K}^\times$. Combining these, we have \[\rho_{E/L}(I_v)\subset (O_K^\times\cdot O_{K_\ideal{p}^\times})\cap A_{O_K}^{\times, \ideal{f}}\subset O_{K}^{\times, \ideal{f}}\cdot O_{K_\ideal{p}}^\times.\] As $\ideal{f}$ separates units $O_K^{\times, \ideal{f}}=\{1\}$ and so $\rho_{E/L}(I_v)\subset O_{K_\ideal{p}}^\times$. It now follows from (\ref{coro:image-of-inertia-group}) that $E/L$ has good reduction at $v$.
\end{proof}
\begin{exem} Let $\ideal{p}$ be a prime of $O_K$ and let $\mathbf{F}_{\ideal{p}^n}$ be the unique extension of $\mathbf{F}_\ideal{p}$ of degree $n$. As $G(\mathbf{F}_{\ideal{p}}^\mathrm{sep}/\mathbf{F}_{\ideal{p}^n})\overset{\sim}{\longrightarrow} \widehat{\mathbf{Z}}$ is topologically generated by the $N\ideal{p}^n$-power Frobenius map, we see from (i) of (\ref{prop:compute-the-homomorphisms-h}) to give a continuous homomorphism \[\rho: G(\mathbf{F}_{\ideal{p}}^\mathrm{sep}/\mathbf{F}_{\ideal{p}^n})\to \lim_{(\ideal{p}, \ideal{f})=O_K}(O_K/\ideal{f})^\times\] such that $[-]_{\mathbf{F}_{\ideal{p}^n}}=[\rho^{-1}]$ is the same as giving a generator $\rho(\mathrm{Fr}^{N\ideal{p}^n})=\pi_n\in O_K$ of the ideal $\ideal{p}^n$. The corresponding isogeny class of CM elliptic curves $E/\mathrm{Spec}(\mathbf{F}_{\ideal{p}^n})$ are those with the property that the endomorphism \[\pi_n: E\to E\] is equal to the $N\ideal{p}^n$-power Frobenius.
\end{exem}
\begin{exem} Let $\ideal{f}$ be an ideal that separates units and recall the isomorphism (\ref{ray-class-isom}): \[A_{O_K}^{\times, \ideal{f}}\overset{\sim}{\longrightarrow} G(K^\infty/K(\ideal{f})).\] Now define \[\rho^{-1}: G(K^\mathrm{sep}/K(\ideal{f}))\to A_{O_K}^\times\] to be the composition \[G(K^\mathrm{sep}/K(\ideal{f}))\to G(K^\infty/K(\ideal{f}))\overset{\sim}{\longrightarrow} A_{O_K}^{\times, \ideal{f}}\to A_{O_K}^\times.\] Then $[\rho^{-1}(-)]=[-]_{K(\ideal{f})}$ and $\rho$ corresponds to an isogeny class of CM elliptic curves $E/\mathrm{Spec}(K(\ideal{f}))$. These curves are distinguished by the fact that their $\ideal{f}$-torsion $E[\ideal{f}]$ is constant and their study is the topic of the next section.
\end{exem}
\section{Moduli and level structures} In this section we define level-$\ideal{f}$ structures for CM elliptic curves (for each integral ideal $\ideal{f}$) and we show that the corresponding moduli stack $\mc{M}_\mathrm{CM}^{(\ideal{f})}$ admits a natural action of $\mc{CL}_{O_K}^{(\ideal{f})}$ under which it becomes a torsor (\ref{prop:level-torsors}). This induces an action of $CL_{O_K}^{(\ideal{f})}$ on the coarse sheaf $M^{(\ideal{f})}_\mathrm{CM}$ of $\mc{M}_\mathrm{CM}^{(\ideal{f})}$ and using this we prove that $M_\mathrm{CM}^{(\ideal{f})}$ is isomorphic to $\mathrm{Spec}(O_{K(\ideal{f})}[\ideal{f}^{-1}])$ compatibly with the isomorphism \[\theta_{K, \ideal{f}}: \mathrm{CL}_{O_K}^{(\ideal{f})}\overset{\sim}{\longrightarrow} G(K(\ideal{f})/K)\] of (\ref{eqn:define-theta-finite}).
Most of the results contained in this section are probably more or less known, however they do not seem to have appeared in the literature and so we are happy to present a detailed account.
\subsection{}\label{subsec:level-f-on-cm-ell-curves} Let $E/S$ be a CM elliptic curve and let $\ideal{f}$ be an integral ideal of $O_K$. A level-$\ideal{f}$ structure on $E/S$ is an isomorphism of $\underline{O_K}_{S}$-modules $\beta: E[\ideal{f}]\overset{\sim}{\longrightarrow} \underline{O_K/\ideal{f}}_S$. An $\ideal{f}$-isomorphism $(E/S, \beta)\to (E'/S, \beta')$ between CM elliptic curves with level-$\ideal{f}$ structures is an isomorphism $f: E\to E'$ such that $\beta'\circ f|_{E[\ideal{f}]}=\beta$. We write $\mc{M}_{\mathrm{CM}}^{(\ideal{f})}$ for the moduli stack over $\mathrm{Sh}_{O_K}$ whose fibre over an sheaf $S$ is the category of CM elliptic curves with level-$\ideal{f}$ structures together with their $\ideal{f}$-isomorphisms and if $\ideal{f}=O_K$ we identify $\mc{M}_\mathrm{CM}$ with $\mc{M}_\mathrm{CM}^{(O_K)}$. If $(E/S, \beta)$ is a CM elliptic curve with level-$\ideal{f}$ structure we shall often just denote it by $E/S$ when the level-$\ideal{f}$ structure is clear (or at least does not need to be explicitly mentioned). We list the following (usual) constructions and properties of CM elliptic curves equipped with level-$\ideal{f}$ structures.
\begin{rema}\label{rema:props-of-level-f-cm}
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $(E, \beta)$ and $(E', \beta')$ are a pair of CM elliptic curves equipped with level-$\ideal{f}$ structures then we equip the rank one $O_K$-local system $\underline{\mathrm{Hom}}_{S}^{O_K}(E, E')$ with the level-$\ideal{f}$ structure \[\underline{\mathrm{Hom}}_{S}^{O_K}(E, E')\to \underline{\mathrm{Hom}}_S^{O_K}(E[\ideal{f}], E'[\ideal{f}])\overset{\sim}{\longrightarrow} \underline{\mathrm{Hom}}_S^{O_K}(\underline{O_K/\ideal{f}}_S, \underline{O_K/\ideal{f}}_S)=\underline{O_K/\ideal{f}}_S\] where the central isomorphism is \[f\mapsto \alpha\circ f\circ \alpha'^{-1}.\]
\item If $(E, \beta)$ and $(\mc{L}, \alpha)$ are a CM elliptic curve and rank one $O_K$-local system over $S$ with level-$\ideal{f}$ structures then we equip $E\otimes_{O_K}\mc{L}$ with the level-$\ideal{f}$ structure $\beta\otimes_{O_K}\alpha$.
\item The sheaf of $\ideal{f}$-automorphisms $\underline{\mathrm{Aut}}_S^{(\ideal{f})}(E)$ of a CM elliptic curve with level-$\ideal{f}$ structure $(E/S, \beta)$ is equal to $\underline{O_K^{\times, \ideal{f}}}$. In particular, it is trivial if $\ideal{f}$ separates units.
\item The sheaf of $\ideal{f}$-isomorphisms $\underline{\mathrm{Isom}}_S^{(\ideal{f})}(E, E')$ between two CM elliptic curves over $S$ equipped with level-$\ideal{f}$ structures is finite and \'etale over $S$ and $E$ and $E'$ are locally isomorphic if and only if $\underline{\mathrm{Isom}}_S^{(\ideal{f})}(E, E')$ is an $\underline{O_K^{\times, \ideal{f}}}_S$-torsor.
\item Given a CM elliptic curve $E/S$ with level-$\ideal{f}$ structure $(E/S, \beta)\in \mc{M}_\mathrm{CM}^{(\ideal{f})}(S)$ the existence of the isomorphism $\beta: E[\ideal{f}]\overset{\sim}{\longrightarrow} \underline{O_K/\ideal{f}}_S$ implies that $E[\ideal{f}]$ is \'etale over $S$. It follows from (\ref{prop:p-torsion-classification}) that $\ideal{f}$ is invertible on $S$ and that morphism $S\to \mathrm{Spec}(O_K)$ factors through $\mathrm{Spec}(O_K[\ideal{f}^{-1}])$. In other words, the structure map $\mc{M}_\mathrm{CM}^{(\ideal{f})}\to \mathrm{Spec}(O_K)$ factors through $\mathrm{Spec}(O_K[\ideal{f}^{-1}])\to \mathrm{Spec}(O_K)$.
\item Let $\mc{E}\to \mathrm{Spec}(O_{K^\mathrm{sep}})$ be a CM elliptic curve (such a curve exists by (\ref{lemm:cm-over-separably-closed-fields})). Then \[E[\ideal{f}]\times_{\mathrm{Spec}(O_{K^\mathrm{sep}})}\mathrm{Spec}(O_{K^\mathrm{sep}}[\ideal{f}^{-1}])\] is \'etale and constant over $\mathrm{Spec}(O_{K^\mathrm{sep}}[\ideal{f}^{-1}])$ and so admits a level-$\ideal{f}$ structure. Choosing such a structure one obtains a map $\mathrm{Spec}(O_{K^\mathrm{sep}}[\ideal{f}^{-1}])\to \mc{M}_\mathrm{CM}^{(\ideal{f})}$. As $\mathrm{Spec}(O_{K^\mathrm{sep}}[\ideal{f}^{-1}])\to \mathrm{Spec}(O_K[\ideal{f}^{-1}])$ is a cover, this shows that the structure map from (the coarse sheaf of) $\mc{M}_{\mathrm{CM}}^{(\ideal{f})}$ to $\mathrm{Spec}(O_{K}[\ideal{f}^{-1}])$ is an epimorphism.
\item If $\a$ is a fractional ideal prime to $\ideal{f}$ then $\a$ has a natural level-$\ideal{f}$ structure, coming from $\a\to \a\otimes_{O_K}O_K/\ideal{f}=O_K/\ideal{f}$, and we just write $-\otimes_{O_K}\a$ for the corresponding auto-equivalence of $\mc{M}_\mathrm{CM}^{(\ideal{f})}$.
\end{enumerate}
\end{rema}
\subsection{}\label{subsec:level-f-structure-homs} Using (ii) of (\ref{rema:props-of-level-f-cm}) we can define a functor \[\mc{M}_\mathrm{CM}^{(\ideal{f})}\times \mc{CL}_{O_K}^{(\ideal{f})}\to \mc{M}_\mathrm{CM}^{(\ideal{f})}: (E/S, \mc{L}/S)\to E\otimes_{O_K}\mc{L}/S\] (the level-$\ideal{f}$ structures are understood).
\begin{prop}\label{prop:eval-compatible-with-level} Let $E, E'$ be a pair of \textup{CM} elliptic curves with level-$\ideal{f}$ structures over $S$. Then the evaluation map \[E\otimes_{O_K}\underline{\mathrm{Hom}}_{S}^{O_K}(E, E')\overset{\sim}{\longrightarrow} E'\] is an $\ideal{f}$-isomorphism.
\end{prop}
\begin{proof} This is immediate from the definitions.
\end{proof}
\begin{prop}\label{coro:tensor-p-equals-frobenius-with-level} For each prime ideal $\ideal{p}$ prime to $\ideal{f}$ and each $n\geq 0$ there is a unique isomorphism of $\mc{M}_\mathrm{CM}^{(\ideal{f})}\times\mathrm{Spec}(\mathbf{F}_\ideal{p})$ auto-equivalences $\nu_{\ideal{p}^n}: -\otimes_{O_K}\ideal{p}^{-n}\overset{\sim}{\longrightarrow}\mathrm{Fr}^{N\ideal{p}^n*}(-)$ such that for all \textup{CM} elliptic curves $E$ over $\mathrm{Spec}(\mathbf{F}_\ideal{p})$-sheaves $S$ the diagram \[\xymatrix{&E\ar[dr]^-{\mathrm{Fr}_{E/S}^{N\ideal{p}^n}}\ar[dl]_-{i_{\ideal{p}^n}}&\\
E\otimes_{O_K}\ideal{p}^{-1}\ar[rr]^{\nu_{\ideal{p}^n}}_\sim && \mathrm{Fr}^{N\ideal{p}^n*}(E)}\] commutes.
\end{prop}
\begin{proof} We need only verify that the isomorphism $\nu_{\ideal{p}^n}(E/S): E\otimes_{O_K}\ideal{p}^{-n}\overset{\sim}{\longrightarrow} \mathrm{Fr}^{N\ideal{p}^n*}(S)$ of (\ref{coro:tensor-p-equals-frobenius}) is an $\ideal{f}$-isomorphism. However, the morphisms $i_{\ideal{p}^n}$ and $\mathrm{Fr}_{E/S}^{N\ideal{p}^n}$ induce isomorphisms on the $\ideal{f}$-torsion which are compatible with the level-$\ideal{f}$ structures on $E$, $E\otimes_{O_K}\ideal{p}^{-n}$ and $\mathrm{Fr}^{N\ideal{p}^n*}(E)$ so that as $\mathrm{Fr}_{E/S}^{N\ideal{p}^n}=\nu_{\ideal{p}^n}\circ i_{\ideal{p}^n}$ it follows that $\nu_{\ideal{p}^n}(E/S)$ is an $\ideal{f}$-isomorphism.
\end{proof}
\begin{prop}\label{prop:level-torsors} The functor \[\mc{M}_{\mathrm{CM}}^{(\ideal{f})}\times \mc{CL}_{O_K}^{(\ideal{f})}\to \mc{M}_{\mathrm{CM}}^{(\ideal{f})}\times \mc{M}_{\mathrm{CM}}^{(\ideal{f})}: (E, \mc{L})\mapsto (E, E\otimes_{O_K}\mc{L})\] is an equivalence of stacks and $\mc{M}_{\mathrm{CM}}^{(\ideal{f})}$ is locally equivalent to $\mc{CL}^{(\ideal{f})}_{O_K}\times\mathrm{Spec}(O_K[\ideal{f}^{-1}])$.
\end{prop}
\begin{proof} By (\ref{prop:eval-compatible-with-level}) this functor is essentially surjective. Moreover, for all $S$ the morphism \[\underline{\mathrm{Isom}}_S^{(\ideal{f})}(E, E')\times_S \underline{\mathrm{Isom}}_S^{(\ideal{f})}(\mc{L}, \mc{L'})\to \underline{\mathrm{Isom}}_S^{(\ideal{f})}(E, E')\times_S \underline{\mathrm{Isom}}_S^{(\ideal{f})}(E\otimes_{O_K}\mc{L}, E'\otimes_{O_K}\mc{L}')\] is an isomorphism of sheaves over $S$, as this can be checked on $S$ sections. Indeed, if $\underline{\mathrm{Isom}}_S^{(\ideal{f})}(E, E')(S)=\emptyset$ then it is clear and if $\underline{\mathrm{Isom}}_S^{(\ideal{f})}(E, E')(S)\neq\emptyset$, we may assume that $E=E'$ and show instead that the map \begin{equation}\mathrm{Isom}_S^{(\ideal{f})}(\mc{L}, \mc{L}')\to \mathrm{Isom}^{(\ideal{f})}_S(E\otimes_{O_K}\mc{L}, E\otimes_{O_K}\mc{L}'): h\mapsto \mathrm{id}_E\otimes h\end{equation} is bijective. The bijectivity of this map follows from (\ref{prop:cm-tensor-homs}) combined with the fact that an isomorphism $h: \mc{L}\to \mc{L}'$ induces an $\ideal{f}$-isomorphism $\mathrm{id}_E\otimes_{O_K} h: E\otimes_{O_K}\mc{L}\overset{\sim}{\longrightarrow} E\otimes_{O_K}\mc{L}'$ if and only if $h$ is an $\ideal{f}$-isomorphism.
\end{proof}
\subsection{}\label{rema:coarse-space-level} We now wish to compute the coarse sheaves $M_\mathrm{CM}^{(\ideal{f})}:=C(\mc{M}_\mathrm{CM}^{(\ideal{f})})$ of the stacks $\mc{M}_\mathrm{CM}^{(\ideal{f})}$. First let us define an action of \[\underline{\mathrm{CL}_{O_K}^{(\ideal{f})}}[\ideal{f}^{-1}]=\underline{CL_{O_K}^{(\ideal{f})}}\times\mathrm{Spec}(O_K[\ideal{f}^{-1}])\] on $M_\mathrm{CM}^{(\ideal{f})}$. As this group is constant it is enough to define an action of $CL_{O_K}^{(\ideal{f})}$ and then, using the isomorphism \[\mathrm{Id}_{O_K}^{(\ideal{f})}/\mathrm{Prin}_{1\bmod \ideal{f}}^{(\ideal{f})}\overset{\sim}{\longrightarrow} \mathrm{CL}_{O_K}^{(\ideal{f})}:\a\mapsto [\a]_\ideal{f}\] it is enough to define an action of $\mathrm{Id}_{O_K}^{(\ideal{f})}$ with the property that $\mathrm{Prin}_{1\bmod \ideal{f}}^{(\ideal{f})}$ acts trivially. Each ideal $\a\in \mathrm{Id}_{O_K}^{(\ideal{f})}$, equipped with its standard level-$\ideal{f}$ structure induces an auto-equivalence \[-\otimes_{O_K}\a: \mc{M}_\mathrm{CM}^{(\ideal{f})}\to \mc{M}_\mathrm{CM}^{(\ideal{f})}\] and by the universal property of the coarse sheaf an automorphism $[\a]_\ideal{f}: M_{\mathrm{CM}}^{(\ideal{f})}\to M_{\mathrm{CM}}^{(\ideal{f})}.$ For $\a, \b\in \mathrm{Id}_{O_K}^{(\ideal{f})}$, the natural $\ideal{f}$-isomorphisms \[(-\otimes_{O_K} \a)\otimes_{O_K}\b\overset{\sim}{\longrightarrow} -\otimes_{O_K}\a\b\] show that $[\b]_\ideal{f} \circ [\a]_\ideal{f}=[\a\b]_\ideal{f}.$ We have $[\a]_\ideal{f}=[\b]_\ideal{f}\in \mathrm{CL}_{O_K}^{(\ideal{f})}$ if and only if there exists an $\ideal{f}$-isomorphism $\a\overset{\sim}{\longrightarrow} \b$, which in turn induces an isomorphism auto-equivalences \[-\otimes_{O_K}\a\overset{\sim}{\longrightarrow} -\otimes_{O_K}\b\] and gives $[\a]_\ideal{f}=[\b]_\ideal{f}: M_\mathrm{CM}^{(\ideal{f})}\to M_{\mathrm{CM}}^{(\ideal{f})}$ so that we have defined our action.
Moreover, by (\ref{coro:tensor-p-equals-frobenius-with-level}) it follows that for each prime $\ideal{p}\in \mathrm{Id}_{O_K}^{(\ideal{f})}$ the pull-back of the automorphism \[[\ideal{p}^{-1}]_\ideal{f}: M_\mathrm{CM}^{(\ideal{f})}\overset{\sim}{\longrightarrow} M_\mathrm{CM}^{(\ideal{f})}\] along $\mathrm{Spec}(\mathbf{F}_\ideal{p})\to \mathrm{Spec}(O_K[\ideal{f}^{-1}])$ is equal to the $N\ideal{p}$-power Frobenius \begin{equation}\label{eqn:frob-lift-level-coarse}[\ideal{p}^{-1}]_\ideal{f}\times\mathrm{Spec}(\mathbf{F}_\ideal{p})=\mathrm{Fr}^{N\ideal{p}}: M_\mathrm{CM}^{(\ideal{f})}\times\mathrm{Spec}(\mathbf{F}_\ideal{p})\overset{\sim}{\longrightarrow} M_\mathrm{CM}^{(\ideal{f})}\times \mathrm{Spec}(\mathbf{F}_\ideal{p})\end{equation}
\subsection{}\label{subsec:coarse-maps-level} If $(E/S, \beta)$ is a CM elliptic curve with level-$\ideal{f}$ structure then we write \[c_{E/S, \ideal{f}}:S\to M_\mathrm{CM}^{(\ideal{f})}\] for the composition \[S\stackrel{(E, \beta)}{\longrightarrow}\mc{M}_\mathrm{CM}^{(\ideal{f})}\stackrel{c_{\mc{M}_\mathrm{CM}^{(\ideal{f})}}}{\longrightarrow} M_{\mathrm{CM}}^{(\ideal{f})}\] (when $\ideal{f}=O_K$ we drop the $\ideal{f}$). It follows from the definition of $\sigma_\a$ that \begin{equation}\label{eqn:cl-action-on-coarse} c_{E\otimes_{O_K}\a/S, \ideal{f}}=\sigma_\a\circ c_{E/S, \ideal{f}}.\end{equation}
\begin{coro}\label{coro:coarse-spaces-of-level-moduli}
\begin{enumerate}[label=\textup{(\roman*)}]
\item The action of $\underline{CL_{O_K}^{(\ideal{f})}}[\ideal{f}^{-1}]$ on $M_\mathrm{CM}^{(\ideal{f})}$ makes it a torsor over $\mathrm{Spec}(O_K[\ideal{f}^{-1}])$.
\item $M_\mathrm{CM}^{(\ideal{f})}$ is finite and \'etale over $\mathrm{Spec}(O_K[\ideal{f}^{-1}])$.
\item The action of $G(K^\mathrm{sep}/K)$ on $M_\mathrm{CM}^{(\ideal{f})}(\mathrm{Spec}(K^\mathrm{sep}))$ is through the homomorphism \[G(K^\mathrm{sep}/K) \to G(K(\ideal{f})/K)\stackrel{\theta_{K, \ideal{f}}}{\to} \mathrm{CL}_{O_K}^{(\ideal{f})}\subset \mathrm{Aut}_{\mathrm{Spec}(O_K[\ideal{f}^{-1}])}(M_\mathrm{CM}^{(\ideal{f})})\] hence \[\mathrm{Spec}(O_{K(\ideal{f})}[\ideal{f}^{-1}])\overset{\sim}{\longrightarrow} M_\mathrm{CM}^{(\ideal{f})}.\]
\end{enumerate}
\end{coro}
\begin{proof} (i) To show that the action of $\underline{CL_{O_K}^{(\ideal{f})}}[\ideal{f}^{-1}]$ is free it is enough to show that for each $[\a]_\ideal{f}\in \mathrm{CL}_{O_K}^{(\ideal{f})}$ the automorphism $[\a]_\ideal{f}: M_\mathrm{CM}^{(\ideal{f})}\overset{\sim}{\longrightarrow} M_\mathrm{CM}^{(\ideal{f})}$ fixes a section $S\to M_\mathrm{CM}^{(\ideal{f})}$ if and only if $[\a]_\ideal{f}=[O_K]_\ideal{f}$. So let $[\a]_\ideal{f}\in \mathrm{CL}_{O_K}^{(\ideal{f})}$ fix a section $S\to M_\mathrm{CM}^{(\ideal{f})}$. By the definition of the coarse sheaf, there exists a cover $S'\to S$ and a CM elliptic curve $E/S'$ such that the induced map $S'\to S \to M_\mathrm{CM}^{(\ideal{f})}$ is equal to $c_{E/S', \ideal{f}}$. This section $c_{E/S', \ideal{f}}$ is also fixed by $[\a]_\ideal{f}$ which is now the statement that the two CM elliptic curves with level-$\ideal{f}$ structure $E$ and $E\otimes_{O_K}\a$ are locally $\ideal{f}$-isomorphic over $S'$. After refining $S'$, we may assume that $E$ and $E\otimes_{O_K}\a$ are actually $\ideal{f}$-isomorphic which by (\ref{prop:level-torsors}) implies the existence of an $\ideal{f}$-isomorphism $\underline{O_K}_{S'}\overset{\sim}{\longrightarrow} \underline{\a}_{S'}$. After refining $S'$ again such an isomorphism is constant and of the form $\underline{h}$ where $h: O_K\to \a$ is an $\ideal{f}$-isomorphism and it follows that $[\a]_\ideal{f}=[O_K]_\ideal{f}$.
Similarly, to show that the action of $\underline{\mathrm{CL}_{O_K}^{(\ideal{f})}}[\ideal{f}^{-1}]$ is transitive, it is enough to show that for each pair of sections $c_1, c_2: S\to M_\mathrm{CM}^{(\ideal{f})}$ there is a cover $(S_i\to S)_{i\in I}$ and elements $[\a_i]_\ideal{f}\in \mathrm{CL}_{O_K}^{(\ideal{f})}$ such that $[\a_i]_\ideal{f}\circ c_1|_{S_i}=c_2|_{S_i}$. Again by the definition of the coarse sheaf there exists a cover $S'\to S$ and CM elliptic curves $E_1$, $E_2$ over $S$ such that the compositions $S'\to M_\mathrm{CM}^{(\ideal{f})}$ of $c_1$ and $c_2$ with $S'\to S$ are equal to $c_{E_1/S, \ideal{f}}$ and $c_{E_2/S, \ideal{f}}$. By (\ref{prop:level-torsors}) there exists an $O_K$-local system $(\mc{L}, \alpha)$ with level-$\ideal{f}$ structure and an $\ideal{f}$-isomorphism \[E_1\overset{\sim}{\longrightarrow} E_2\otimes_{O_K}\mc{L}.\] After base change to a cover $(S_i\to S)_{i\in I}$ (the corresponding class of $\mc{L}$ may not be constant which is why our cover may consist of multiple elements) we may assume that $(\mc{L}, \alpha)=(\underline{\a}_{S'}, \underline{f}_{S'})$ and it follows from (\ref{eqn:cl-action-on-coarse}) that $c_{E_2/S, \ideal{f}}=[\a]_\ideal{f}\circ c_{E_1/S, \ideal{f}}$ and this proves the claim.
(ii) The fact that $M_\mathrm{CM}^{(\ideal{f})}$ is finite and \'etale over $\mathrm{Spec}(O_K[\ideal{f}^{-1}])$ follows by descent, as $M_\mathrm{CM}^{(\ideal{f})}$ is locally (over $\mathrm{Spec}(O_K[\ideal{f}^{-1}])$) isomorphic to $\underline{CL_{O_K}^{(\ideal{f})}}[\ideal{f}^{-1}]$ by (i) and $\underline{CL_{O_K}^{(\ideal{f})}}[\ideal{f}^{-1}]$ is finite and \'etale over $\mathrm{Spec}(O_K[\ideal{f}^{-1}])$.
(iii) For each prime $\ideal{p}$ prime to $\ideal{f}$, the automorphism $[\ideal{p}^{-1}]_\ideal{f}: M_\mathrm{CM}^{(\ideal{f})}\to M_\mathrm{CM}^{(\ideal{f})}$ lifts the $N\ideal{p}$-power Frobenius automorphism modulo $\ideal{p}$ (\ref{eqn:frob-lift-level-coarse}). For any such automorphism (of any finite \'etale $\mathrm{Spec}(O_K[\ideal{f}^{-1}])$-scheme) there exists an element $\sigma\in G(K^\mathrm{sep}/K)$ such that \[M_\mathrm{CM}^{(\ideal{f})}(\sigma)=[\ideal{p}^{-1}]_\ideal{f}(\mathrm{Spec}(K^\mathrm{sep})): M_\mathrm{CM}^{(\ideal{f})}(\mathrm{Spec}(K^\mathrm{sep}))\to M_{\mathrm{CM}}^{(\ideal{f})}(\mathrm{Spec}(K^\mathrm{sep})).\] However, the action of $\mathrm{CL}_{O_K}^{(\ideal{f})}$ on $M_{\mathrm{CM}}^{(\ideal{f})}(\mathrm{Spec}(K^\mathrm{sep}))$ is transitive by (i), and $\mathrm{CL}_{O_K}^{(\ideal{f})}$ is generated by elements of the form $[\ideal{p}^{-1}]_\ideal{f}$ and so it follows that the action $G(K^\mathrm{sep}/K)$ on $M_\mathrm{CM}^{(\ideal{f})}(\mathrm{Spec}(K^\mathrm{sep}))$ is transitive. Therefore $M_\mathrm{CM}^{(\ideal{f})}$ is connected and isomorphic to $\mathrm{Spec}(O_{L}[\ideal{f}^{-1}])$ for some finite extension $L/K$ which is unramified away from $\ideal{f}$. By construction, the isomorphism $G(L/K)\to \mathrm{CL}_{O_K}^{(\ideal{f})}$ sends the Frobenius element $\sigma_{L/K, \ideal{p}}$ to $[\ideal{p}^{-1}]_\ideal{f}$ and it follows that $L\overset{\sim}{\longrightarrow} K(\ideal{f})$ (cf.\ (\ref{eqn:define-theta-finite})).
\end{proof}
\begin{rema} We list the following consequences of (\ref{coro:coarse-spaces-of-level-moduli}).
\begin{enumerate}[label=\textup{(\alph*)}]
\item The coarse sheaf $M_\mathrm{CM}$ of $\mc{M}_\mathrm{CM}$ is isomorphic to $\mathrm{Spec}(O_H)$ where $H$ is the Hilbert class field of $K$. This recovers the fact the $j$-invariants of CM elliptic curves defined over extensions of $K$ lie in $H\subset L$.
\item Let $E\to S$ be any CM elliptic curve and recall that \[c_{E/S}: S\to M_\mathrm{CM}\] denotes the morphism \[S\stackrel{E}{\to} \mc{M}_\mathrm{CM}\stackrel{c_{\mc{M}_\mathrm{CM}}}{\longrightarrow} M_\mathrm{CM}.\] Then by definition, if $E'$ is another CM elliptic curve over $S$ then $c_{E/S}, c_{E'/S}: S\to M_\mathrm{CM}$ are equal if and only if $E$ and $E'$ are locally isomorphic. Combining this with (\ref{coro:isom-iso-isom}) and (\ref{prop:cm-exist-given-h}), we find that if $S=\mathrm{Spec}(F)$ for $F$ a field then the map \[E/F\mapsto (\rho_{E/F}, c_{E/F})\] defines a bijection between the set of isomorphism classes of CM elliptic curves over $E/\mathrm{Spec}(F)$ with the set of pairs $(\rho, c)$ where:
\begin{enumerate}[label=(\roman*)]
\item $\rho: G(F^\mathrm{sep}/F)\to \lim_{\ideal{f}}(O_K/\ideal{f})^\times$ is a homomorphism such that $[\rho^{-1}]=[-]_F$, and
\item $c: \mathrm{Spec}(L)\to M_\mathrm{CM}$ is any map.
\end{enumerate}
\item If $\ideal{f}$ separates units then $\mc{M}_\mathrm{CM}^{(\ideal{f})}\overset{\sim}{\longrightarrow} M_\mathrm{CM}^{(\ideal{f})}\overset{\sim}{\longrightarrow} \mathrm{Spec}(O_{K(\ideal{f})}[\ideal{f}^{-1}])$. If $E/\mathrm{Spec}(K(\ideal{f}))$ denotes the generic fibre of the universal CM elliptic curve with level-$\ideal{f}$ structure the homomorphism \[\rho_{E/K(\ideal{f})}: G(K(\ideal{f})^\mathrm{sep}/K(\ideal{f}))\to A_{O_K}^\times\] is equal to (cf.\ (\ref{ray-class-isom})) \[G(K(\ideal{f})^\mathrm{sep}/K(\ideal{f}))\stackrel{-|_{K^\infty}}{\longrightarrow} G(K^\infty/K(\ideal{f})) \overset{\sim}{\longrightarrow} A_{O_K}^{\times, \ideal{f}} \stackrel{\textup{incl}}{\longrightarrow} A_{O_K}^\times.\]
\end{enumerate}
\end{rema}
\chapter{$\Lambda$-structures, Witt vectors and arithmetic jets} In this chapter we give a brief introduction to, and overview of, the theory $\Lambda$-structures, Witt vectors and arithmetic jet spaces following the approach of Borger \cite{Borger11}, \cite{Borger11bis}. It is by nature a technical and notationally heavy theory, but its many applications make it a worthwhile subject. The two papers mentioned are both an excellent introduction and general reference and we encourage to the reader to consult them. We give here really only the minimal set up necessary for our applications in Chapter 4 and proofs are given only where they cannot be cited or where it would be perhaps enlightening.
In the final section we prove a small new result which shows that $\Lambda$-structures on relative abelian schemes are determined by their underlying $\Psi$-structures.
\section{Plethories}
\subsection{} Fix a pair of rings $O$ and $O'$. An $O$-$O'$-biring $\Phi$ is an $O$-algebra together the structure of an $O'$-algebra on the set $\mathrm{Hom}_{O}(\Phi, A)$ which is functorial in the $O$-algebra $A$. This is structure is determined by certain homomorphisms
\begin{enumerate}[label=\textup{(\roman*)}]
\item coaddition and comultiplication: $\Delta_\Phi^+, \Delta_{\Phi}^\times: \Phi\to \Phi\otimes_{O}\Phi$
\item coadditive and comultiplicative units: $\epsilon_\Phi^+, \epsilon_\Phi^\times: \Phi\to O$
\item $O'$ coaction: $O'\to \mathrm{End}_O(\Phi): s\mapsto s_{\Phi}$
\end{enumerate} satisfying various identities (cocommutativity of the coaddition and comultiplication, coassociativity and so on).
We denote by $\mathrm{Biring}_{O, O'}$ the category whose objects are $O$-$O'$-birings and whose morphisms are those $O$-homomorphisms $\Phi\to \Phi'$ inducing functorial homomorphisms of $O'$-algebras $\mathrm{Hom}_O(\Phi', A)\to \mathrm{Hom}_O(\Phi', A)$. Of course, this is equivalent to the homomorphism $\Phi\to \Phi'$ being `compatible' with the maps $\Delta_\Phi^+$, $\Delta_{\Phi'}^+$ and so on.
\subsection{} \label{subsec:corep-functor-biring} The functor corepresented by a biring $\Phi$: \[A\mapsto \mathrm{Hom}_{O}(\Phi, A): \mathrm{Alg}_{O}\to \mathrm{Alg}_{O'}\] admits a left adjoint, which we denote by $B\mapsto \Phi\odot_{O'} B$, which is defined as follows: $\Phi\odot_{O'} B$ is the quotient of the free $O$-polynomial algebra generated by the symbols $\phi\odot b$ for $\phi\in \Phi$ and $b\in B$ subject to the relations \[(\phi+\phi')\odot b=\phi\odot b+\phi'\odot b, \quad \phi\phi'\odot b=(\phi\odot b)(\phi'\odot b),\quad (r\phi)\odot b=r(\phi \odot b)\] and \[\phi\odot(b+b')=\Delta_\Phi^+(\phi)(b, b')\footnote{This notation means that if $\Delta_\Phi^+(\phi)=\sum_i \phi_i\otimes \phi_i'$ then $\Delta_\Phi^+(\phi)(b, b')=\sum_i (\phi_i\odot b_i)(\phi_i'\odot b_i')$ and similarly for $\Delta_{\Phi}^\times(b, b')$.}, \quad \phi\odot bb'=\Delta^\times_\Phi(\phi)(b, b'), \quad \phi\odot sb=s_{\Phi}(\phi)\odot b\] for all $\phi, \phi'\in \Phi$, $b, b'\in B$, $r\in O$ and $s\in O'$. The $O$-algebra $\Phi\odot_{O'} B$ is called composition product of $\Phi$ with $B$.
\subsection{} If $\Phi$ is an $O$-$O'$-biring and $O\to O''$ is a homomorphism then $\Phi\otimes_{O}O''$ is an $O''$-$O'$-biring and $(\Phi\odot_{O'} B)\otimes_{O}O''=(\Phi\otimes_{O}O'')\odot_{O'} B$. The functor $\mathrm{Biring}_{O, O'}\times \mathrm{Alg}_{O'}\to \mathrm{Alg}_{O}: (\Phi, B)\to \Phi\odot_{O'} B$ commutes with colimits in each variable.
We give the two simplest examples when $O=O'$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The functor $A\mapsto A$ is represented by the $O$-$O$-biring $O[e]$.
\item Even simpler is the functor $A\mapsto 0$ represented by the $O$-$O$-biring $O$ itself.
\end{enumerate}
\subsection{}\label{subsec:birings-and-monoidal-functors} We now concentrate on the case $O=O'$ and just call an $O$-$O$-biring an $O$-biring and shall drop $O$ from the notation when there is no risk of confusion. If $\Phi$ and $\Phi'$ are two $O$-birings then we may consider the composition product $\Phi\odot \Phi'$ which, using the standard properties of adjunctions, is again an $O$-biring. The composition product then defines a monoidal structure on the category of $O$-birings with identity $O[e]$. The functor \[\Phi\mapsto \Phi\odot -\] from $O$-birings to endofunctors on $\mathrm{Alg}_{O}$ is monoidal and fully faithful.
\subsection{}\label{subsection:definition-of-plethories} An $O$-plethory is an $O$-biring $\Phi$ together with the structure of a monad on the functor $A\mapsto \Phi\odot A$ or equivalently a comonad on the functor $A\mapsto \mathrm{Hom}_{O}(\Phi, A)$. The remarks in (\ref{subsec:birings-and-monoidal-functors}) then show that to give the functor $A\mapsto \Phi\odot A$ the structure of a monad is equivalent to giving
\begin{enumerate}[label=\textup{(\roman*)}]
\item a homomorphism of $O$-birings $i_{\Phi}:O[e]\to \Phi$ and
\item a homomorphism of $O$-birings $h_{\Phi}:\Phi\odot \Phi\to \Phi$
\end{enumerate} such that \[h_{\Phi}\circ(\Phi\odot i_{\Phi})=h_{\Phi}\circ(i_{\Phi}\odot \Phi)=\mathrm{id}_{\Phi} \quad \text{ and }\quad h_{\Phi}\circ(\Phi\odot h_{\Phi})=h_{\Phi}\circ(h_{\Phi}\odot \Phi).\]
\subsection{} If $\Phi$ is an $O$-plethory we define a $\Phi$-ring to be an $O$-algebra $A$ equipped with an action of the monad $\Phi\odot -$. Given a $\Phi$-ring $A$ we denote by \[h_A: \Phi\odot A\to A\] the map defining the $\Phi$-ring structure on $A$. We denote the category of $\Phi$-rings and compatible morphisms by $\mathrm{Alg}_{O}^\Phi$. If $A$ is an $O$-algebra then $\Phi\odot A$ and $\mathrm{Hom}_O(\Phi, A)$ are $\Phi$-rings and these two functors are the left and right adjoints of the forgetful functor $\mathrm{Alg}_{O}^\Phi\to \mathrm{Alg}$. We have the diagram of functors, each one left adjoint to the one below it:
\[\xymatrix{\mathrm{Alg}_{O}^\Phi\ar[rr]^{\textup{forget}} && \ar@/^2pc/[ll]^{\mathrm{Hom}_O(\Phi, -)}\ar@/_2pc/[ll]_{\Phi\odot_O -}\mathrm{Alg}_{O}.}\]
\section{Witt vectors and arithmetic jets I}\label{sec:lambda-section-one}
\subsection{} During \S\S \ref{sec:lambda-section-one}--\ref{sec:lambda-section-three} we fix a Dedekind domain $O$ with finite residue fields and $P\subset \mathrm{Id}_{O}$ a sub-monoid generated by some set of prime ideals of $O$ and we write $K$ for the fraction field of $O$. Note that we allow Dedekind domains of finite characteristic, e.g.\ $O=\mathbf{F}_p[T]$.
\subsection{} Denote by $\Psi$ the free polynomial $O$-algebra generated by the symbols $\psi^\a$ for $\a\in P$. We make $\Psi$ an $O$-biring by equipping the set \[\mathrm{Hom}_{O}(\Psi, A)=\mathrm{Hom}_{O}(O[\psi^\a:\a\in P], A) \overset{\sim}{\longrightarrow} \prod_{\a\in P} A: f\mapsto (f(\psi^\a))_{\a\in P}\] with the product $O$-algebra structure. We then give $\Psi$ the structure of an $O$-plethory by setting \[i_{\Psi}: O[e]\to \Psi: e\mapsto \psi^{(1)} \quad \text{ and } \quad h_{\Psi}: \Psi\odot \Psi\to \Psi: \psi^\a\odot \psi^\b\mapsto \psi^{\a\b}.\]
\subsection{} For each ring $A$ we write $\Gamma(A)$ for the ring \[\Gamma(A):=\mathrm{Hom}_O(\Psi, A)=\prod_{\a\in P}A\] and call it the ring of ghost vectors of $A$. We define the ghost jets of a ring $A$ to be the ring \[\Psi\odot_O A.\] Let us examine a little more the ring of ghost jets $\Psi\odot_O A$ of a ring $A$ and what it means for $A$ to be a $\Psi$-ring.
Recall (\ref{subsec:corep-functor-biring}) that if $A$ is an $O$-algebra then $\Psi\odot_{O} A$ is given by the quotient of the $O$-polynomial algebra \[O[\psi^\a\odot a: \a\in P, a\in A]\] by the ideal generated by the elements of the following form \begin{eqnarray*}&\psi^\a\odot (a+b)-\psi^\a \odot a -\psi^\a \odot b\\& \psi^\a\odot (ab)-(\psi^\a\odot a)(\psi^\a\odot b)\\&\psi^\a\odot (ra)-r(\psi^\a\odot a)\end{eqnarray*} for $a, b\in A$, $r\in O$. Now to give $A$ the structure of a $\Psi$-ring is the same as giving an $O$-homomorphism \[h_A:\Psi\odot_{O} A\to A\] satisfying certain properties. In this case, the properties which must be satisfied are equivalent to the following:
\begin{enumerate}[label=\textup{(\roman*)}]
\item for each $\a\in P$ the map $\psi_A^\a: A\to A$ defined by $a\mapsto h_A(\psi^\a\odot a)$ is an $O$-algebra homomorphism,
\item for each $\a, \b\in P$ we have $\psi^\a_A\circ \psi^{\b}_A=\psi^{\b}_A\circ \psi^{\a}_A=\psi^{\a\b}_A$, and
\item we have $\psi^{(1)}_A=\mathrm{id}_A: A\to A$.
\end{enumerate} We see that if $A$ is a $\Psi$-ring then $A$ has an action of the monoid $P$ where $\a\in P$ acts by $\psi^\a_A: A\to A.$ This sets up a bijection between $\Psi$-ring structures on $A$ and actions of $P$ on $A$. Given a $\Psi$-ring we will write $\psi^\a_A: A\to A$ for the corresponding $P$-action. In particular, the $\Psi$-ring $\Psi$ itself has the $P$-action \[\psi^\a_{\Psi}: \psi^{\b}\mapsto \psi^{\a\b}\] for $\a, \b\in P.$ Note that the $\Psi$-ring structure on $O$ is given by \[\psi_O^\a=\mathrm{id}_O: O\to O.\]
\subsection{} As we are dealing with rings equipped with endomorphisms, it will be useful here to say a little about semi-linear self maps versus twisted linear maps. The example to have in mind is the following: if $A$ is a ring of characteristic $p$ and $A\to B$ is an $A$-algebra then the $p$-power Frobenius \[\mathrm{Fr}_B^p: B\to B: b\mapsto b^p\] is $\mathrm{Fr}_A^p$-linear where $\mathrm{Fr}_A^p: A\to A: a\mapsto a^p$ is the $p$-power Frobenius of $A$. This then induces an $A$-linear map, the relative $p$-power Frobenius, $\mathrm{Fr}_{B/A}^p: \mathrm{Fr}_A^{p*}(B)\to B$.
Now, a morphism of $\Psi$-rings $f: A\to B$ is just a homomorphism such that the diagram \[\xymatrix{A\ar[d]_f\ar[r]^{\psi^\a_A} & A\ar[d]^f \\B \ar[r]^{\psi_B^\a} & B}\] commutes for all $\a\in P$. Viewing $B$ as an $A$-algebra via $f$, the homomorphism $\psi_B^\a: B\to B$ is $\psi_A^\a$-linear and induces an $A$-linear map \[\psi_{B/A}^{\a}:\psi_A^{\a*}(B) \to B.\] That the homomorphisms $\psi_B^\a$ for $\a\in P$ commute is now expressed by the condition that for all $\a, \b\in P$ we have \begin{equation}\label{def:semi-linear-psi}\psi_{B/A}^{\b}\circ \psi_{A}^{\a*}(\psi_{B/A}^\b)=\psi_{B/A}^{\a}\circ \psi_{A}^{\b*}(\psi_{B/A}^\a)=\psi_{B/A}^{\a\b} \end{equation} or in a commutative diagram: \[\xymatrix{\psi_A^{\a\b*}(B) \ar[rr]^-{\psi^{\b*}_A(\psi_{B/A}^{\a})}\ar[d]_{\psi^{\a*}_A(\psi_{B/A}^{\b})} && \psi_A^{*\a}(B)\ar[d]^{\psi_{B/A}^{\a}}\\
\psi_A^{*\b}(B)\ar[rr]^{\psi_{B/A}^{\b}} && B.}\] Moreover, if $A$ is a $\Psi$-ring and $A\to B$ is an $A$-algebra then to give $B$ the structure of $\Psi$-ring such that $A\to B$ is a $\Psi$-homomorphism is the same as giving maps $\psi_{B/A}^{\a}: \psi^{\a*}_A(B)\to B$ for each $\a\in P$ such that $\psi_{B/A}^{(1)}=\mathrm{id}_B$ and which satisfy the commutativity condition (\ref{def:semi-linear-psi}). The category of $\Psi$-rings over a $\Psi$-ring $A$ is denoted $\mathrm{Alg}_{\Psi_A}$ and its objects called $\Psi_A$-rings.
\subsection{} We now come to the matter of interest which is lifting the Frobenius. We wish to impose on a $\Psi$-ring $A$ the condition that the homomorphisms $\psi_A^\ideal{p}: A\to A$, for $\ideal{p}\in P$ prime, be lifts of the $N\ideal{p}$-power Frobenius endomorphism (recall that $N\ideal{p}=\#\mathbf{F}_\ideal{p}$): \begin{equation}\label{eqn:frobenius-lift-condition}\psi_A^\ideal{p}(a)=a^{N\ideal{p}}\bmod \ideal{p} A\end{equation} This is done by enlarging the plethory $\Psi$ in such a way that the endomorphisms corresponding to the elements $\psi_\ideal{p}\in \Psi$, for each prime $\ideal{p}\in P$, will be forced to satisfy the relation (\ref{eqn:frobenius-lift-condition}).
So for each integer $n\geq 0$ define sub-$O$-algebras $\Lambda_n\subset \Psi\otimes_O K$ inductively by setting $\Lambda_{0}=\Psi$, and for $n\geq 0$, setting $\Lambda_{n+1}$ to be the sub-$\Lambda_{n}$-algebra of $\Psi\otimes_{O}K$ generated by the subsets \begin{equation}\label{eqn:put-in-frob-residues}\ideal{p}^{-1} (f^{N\ideal{p}}-\psi^\ideal{p}_{\Psi}(f))\subset \Psi\otimes_{O}K\end{equation} for $\ideal{p}\in P$ a prime ideal and $f\in \Lambda_{n}$. Finally, we set \[\Lambda=\cup_{n\geq 0}\Lambda_{n}\subset \Psi\otimes_{O}K.\] Then $\Lambda_{n}\subset \Psi\otimes_{O}K$ is stabilised by the endomorphisms $\psi_{\Psi}^\a$ of $\Psi\otimes_{O}K$ for each $\a\in P$, as is $\Lambda\subset \Psi\otimes_{R} K$. Thus $\Lambda_{n}$ and $\Lambda$ admit unique $\Psi$-ring structures such that $\Psi\subset \Lambda_{n}\subset \Lambda$ are $\Psi$-morphisms. Moreover, for each $n\geq 0$, and each $f\in \Lambda_n$ it follows from the definition (\ref{eqn:put-in-frob-residues}) of $\Lambda_{n+1}$ that \[\psi_\Lambda^\ideal{p}(f)=f^{N\ideal{p}}\bmod \ideal{p} \Lambda_{n+1}\] and hence for all $f\in \cup_{n\geq 0}\Lambda_n=\Lambda$ we have \[\psi_\Lambda^\ideal{p}(f)=f^{N\ideal{p}}\bmod \ideal{p} \Lambda.\]
\begin{prop} There is a unique $O$-plethory structure on $\Lambda$ such that the inclusion $\Psi\to \Lambda$ is a morphism of $O$-plethories.
\end{prop}
\begin{rema} Before we examine exactly what a $\Lambda$-ring is, the fact that $\Psi\to \Lambda$ is a morphism of $O$-plethories implies that every $\Lambda$-ring $A$ inherits a $\Psi$-ring structure and hence a family of endomorphisms $\psi_A^{\a}: A\to A$ for each $\a\in P$ such that $\psi_{(1)}=\mathrm{id}_A$ and $\psi_{A}^{\a\b}=\psi_{A}^\a\circ \psi_{A}^\b$ for all $\a, \b\in P$.
\end{rema}
\begin{prop}\label{prop:lambda-actions-and-frobenius-lifts} Let $A$ be an $O$-algebra. We have the following:
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $A$ is a $\Lambda$-ring the endomorphism $\psi^\ideal{p}_A: A\to A$ satisfies \[\psi^\ideal{p}_A(a)=a^{N\ideal{p}}\bmod \ideal{p} A\] for each maximal ideal $\ideal{p}\in P$.
\item If $A$ is a $\Psi$-ring and $\ideal{p}$-torsion free for each prime $\ideal{p}\in P$ (i.e.\ flat at $\ideal{p}$ for each prime $\ideal{p}\in P$) then the given $\Psi$-structure comes from a $\Lambda$-structure if and only if for each prime ideal $\ideal{p}\in P$ the endomorphism $\psi^\ideal{p}_A: A\to A$ lifts the $N\ideal{p}$-power Frobenius \[\psi^\ideal{p}_{A}(a)=a^{N\ideal{p}}\bmod \ideal{p} A.\] In this case the $\Lambda$-structure inducing the $\Psi$-structure is unique.
\item If each $\ideal{p}\in P$ is invertible in $A$ then every $\Psi$-structure on $A$ is induced by a unique $\Lambda$-structure.
\end{enumerate}
\end{prop}
\begin{proof} (i) and (ii) are (essentially) the definition of $\Lambda$-ring given in \cite{Borger11}. In particular, see \S\S 1.6--1.19 \cite{Borger11}.
(iii) If each $P$ in $A$ is invertible any endomorphism $\psi_A^\ideal{p}: A\to A$ lifts the $N\ideal{p}$-power Frobenius (trivially) and so this follows from (i) and (ii).
\end{proof}
\begin{rema} We point out that if $A\to B$ is a homomorphism of $\Lambda$-rings then $\psi_A^\ideal{p}=\mathrm{Fr}_{A_\ideal{p}}^{N\ideal{p}}\bmod \ideal{p} A $ and the relative morphisms $\psi_{B/A}^{\ideal{p}}: \psi_A^{\ideal{p}*}(B)\to B$ are now lifts of the relative $N\ideal{p}$-power Frobenius: \[\psi_{B/A}^{\ideal{p}}=\mathrm{Fr}_{B_\ideal{p}/A_\ideal{p}}^{N\ideal{p}}: \mathrm{Fr}_{A_\ideal{p}}^{N\ideal{p}*}(B_\ideal{p})\to B_\ideal{p}\] where we write $A_\ideal{p}=A\otimes_{O}\mathbf{F}_\ideal{p}$ and $B_\ideal{p}=B\otimes_{O}\mathbf{F}_\ideal{p}$.
\end{rema}
\begin{exem} Using (\ref{prop:lambda-actions-and-frobenius-lifts}) we are now able to give the first examples of $\Lambda$-rings:
\begin{enumerate}[label=(\roman*)]
\item Of course $O$ is always a $\Lambda$-ring with $\psi_O^\a: O\to O$ equal to the identity. The fact that $\psi_O^{\ideal{p}}$ lifts the $N\ideal{p}$-power Frobenius is then Fermat's little theorem: \[\psi_O^\ideal{p}(a)=a=a^{N\ideal{p}}\bmod \ideal{p}\] (recall that $O/\ideal{p}=\mathbf{F}_\ideal{p}$ is a finite field with $N\ideal{p}$-elements).
\item The polynomial ring $O[T]$ is a $\Lambda$-ring with $\psi_{O[T]}^{\a}: O[T]\to O[T]$ given by $T\mapsto T^{N\a}$.
\item If $O=O_K$ is the ring of integers in a number field, $L/K$ is an abelian extension and $P\subset \mathrm{Id}_{O_K}$ is the sub-monoid generated by the primes which are unramified in $L/K$, then the ring of integers $O_L$ of $L$ is a $\Lambda_P$-ring with $\psi_{O_L}^\ideal{p}=\sigma_{L/K, \ideal{p}}: O_L\to O_L$ given by the Frobenius element $\sigma_{L/K, \ideal{p}}\in G(L/K)$ (cf.\ (\ref{subsec:finite-extensions-frobenius})). Examples of this form will be very important later give the first link between $\Lambda$-structures and class field theory.
\end{enumerate}
\end{exem}
\subsection{}\label{subsec:single-principal-prime} We now give an explicit description of what it means for a non-flat ring to be have $\Lambda$-structure in the special case $P=\{\ideal{p}, \ideal{p}^2, \ldots\}\subset \mathrm{Id}_{O}$ is generated by a single prime ideal of $O$ and the prime ideal $\ideal{p}=(\pi)$ is principal. In this case, a $\Lambda$-structure on an $O$-algebra can described explicitly, and this notion was discovered independently by Buium \cite{Buium95} (for some of the many interesting arithmetic applications of $\delta$-rings and $\delta$-geometry see \cite{Buium05}).
So let us describe $\Lambda$ in this situation. Define elements $\delta_n\in \Lambda$ for $n\geq 0$ inductively by $\delta_0=\psi^{(1)}$ and \[\delta_{n+1}=\pi^{-1}(\psi^\ideal{p}_{\Lambda}(\delta_n)-(\delta_n)^{N\ideal{p}})\in \Lambda.\] Let $\Delta_\pi\subset \Lambda$ be the sub-$O$-algebra generated by the elements $\{\delta_{n}\}_{n\geq 0}$.
\begin{prop}\label{prop:lambda-one-prime-is-locally-polynomial} The inclusion $\Delta_{\pi}\subset \Lambda$ is an equality and $\Delta_{\pi}$ is freely generated as an $O$-algebra by the elements $\{\delta_{n}\}_{n\geq 0}.$
\end{prop}
\begin{proof} See \S 1.19 of \cite{Borger11}.
\end{proof}
\begin{coro}\label{coro:delta-lambda-rings} With notation as in \textup{(\ref{subsec:single-principal-prime})} to give an $O$-algebra $A$ a $\Lambda$-structure is equivalent to defining a map \[\delta_\pi: A\to A\] such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item for $r\in O$ we have \[\delta_\pi(r)=\frac{r^{N\ideal{p}}-r}{\pi},\]
\item for $a, b\in A$ we have \[\delta_\pi(ab)=a^{N\ideal{p}}\delta_\pi(b)+b^{N\ideal{p}}\delta_\pi(a)+\pi\delta_\pi(a)\delta_\pi(b),\] and
\item for $a, b\in A$ we have \[\delta_\pi(a+b)=\delta_\pi(a)+\delta_\pi(b)+\sum_{i=1}^{{N\ideal{p}}-1}\frac{1}{\pi}{{N\ideal{p}} \choose i} a^{{N\ideal{p}}-i}b^i.\]
\end{enumerate}
\end{coro}
\begin{proof} Also see \S 1.19 of \cite{Borger11}.
\end{proof}
\begin{rema} Let us spell out the equivalence between $\Lambda$-rings and rings with an operator $\delta_\pi$ satisfying (\ref{prop:lambda-one-prime-is-locally-polynomial}) explicitly, in the case $A$ is flat.
If $\delta_\pi: A\to A$ is a map satisfying the conditions (i)--(iii) of (\ref{prop:lambda-one-prime-is-locally-polynomial}) then it follows that the map \[\psi_A^{\ideal{p}}: A\to A: a\mapsto a^{N\ideal{p}}+\pi \delta_\pi(a)\] is a $O$-algebra homomorphism satisfying \[\psi^{A}_\ideal{p}(a)=a^{N\ideal{p}}+\pi \delta_\pi(a)=a^{N\ideal{p}}\bmod \ideal{p} A.\]
Conversely, if $A$ is a $\ideal{p}$-torsion free $\Lambda$-ring then the relation \[\psi_A(a)=a^{N\ideal{p}}\bmod \ideal{p} A\] implies that there is a unique $\delta_\pi(a)\in A$ such that \[\psi_A(a)=a^{N\ideal{p}}+\pi \delta_\pi(a)\] and the fact that $\psi_A^\ideal{p}$ is an $O$-algebra homomorphism forces the map $a\mapsto \delta_\pi(a)$ to satisfy conditions (i)--(iii) of (\ref{prop:lambda-one-prime-is-locally-polynomial}).
\end{rema}
\begin{rema} We can now explain why the approach with plethories was taken. To define a $\Lambda$-ring as an $O$-algebra with a Frobenius lift $\psi_A^\ideal{p}: A\to A$ is a perfectly reasonable thing to do, however, there is a hidden existential quantifier in this definition: that for all $a\in A$ there exists an $a_\ideal{p}\in \ideal{p} A$ such that $\psi_A^\ideal{p}(a)=a^{N\ideal{p}}+a_\ideal{p}$. This causes problems from the point of view of universal algebra and the effect of the plethystic approach is to remove this existential quantifier so that, rather than the Frobenius lift $\psi_A^\ideal{p}$, is it the operator $\delta_\pi$ which determines the structure.
\end{rema}
\begin{coro}\label{coro:structure-map-lambda-ring-injective} If $A$ is a $\Lambda$-ring then the kernel of the homomorphism $O\to A$ is either prime to all $\ideal{p}\in P$ or $A=0$ is the zero ring.
\end{coro}
\begin{proof} We shall prove this in the situation of (\ref{prop:lambda-one-prime-is-locally-polynomial}) remarking that it is possible to reduce to this case once more of the theory has been set-up. So let $A$ be a $\Lambda$-ring and assume that $\pi^n\in \ideal{p}^n\subset \ker(O\to A).$ As the homomorphism $O\to A$ is a $\Lambda$-homomorphism the kernel is stabilised by the endomorphism $\delta_\pi: O\to O$ which is given by \[a\mapsto \frac{a^{N\ideal{p}}-a}{\pi}.\] In particular, we see that \[\delta_\pi(\pi^n)=\frac{\pi^{N\ideal{p} n}-\pi^n}{\pi}=\pi^{N\ideal{p} n -1}-\pi^{n-1}=\pi^{n-1}(\pi^{N\ideal{p}(n-1)}-1)\in \ideal{p}^{n-1}.\] Therefore, $\ideal{p}^{n-1}\subset \ker(O\to A)$ and by induction we find that $O\subset \ker(O\to A)$ and so $A$ is the zero ring.
\end{proof}
\subsection{} Let $P'\subset P$ be a sub-monoid generated by some set of prime ideals. Then the restriction of the map $h_{\Lambda_P}:\Lambda_{P}\odot \Lambda_{P}\to \Lambda_P$ along $\Lambda_{P'}\odot \Lambda_P\to \Lambda_P$ makes $\Lambda_{P}$ a $\Lambda_{P'}$-ring. Similarly, for $\Psi_P$ and $\Psi_{P'}$.
Now denote by \[P''=\{\a\in P: (\a, \b)=(1) \text{ for all } \b\in P'\}\subset P\] so that $P'\cdot P''=P$ and $P'\cap P''=\{O\}$. By the remark above the map $\Lambda_{P''}\to \Lambda_P$ extends by adjunction to a homomorphism of $\Lambda_{P'}$-rings \[\alpha_{P', P''}:\Lambda_{P'}\odot \Lambda_{P''}\to \Lambda_{P}\] which is also a homomorphism of $\Psi_{P}$-rings where $P''\subset P$ acts on the left factor.
\begin{prop}\label{prop:lambda-commute} The map $\alpha_{P', P''}: \Lambda_{P'}\odot \Lambda_{P''}\to \Lambda_{P}$ defined above is a $\Lambda_{P'}$-isomorphism.
\end{prop}
\begin{proof} This is Proposition 5.3 of \cite{Borger11}.
\end{proof}
\subsection{}\label{subsec:finite-length} We now define truncated versions of the birings $\Lambda$ and $\Psi$ as, when we come to geometrise the theory of $\Lambda$-rings, they will be more well behaved (cf.\ (\ref{prop:witt-vectors-preserve-etale-morphisms})).
We equip the $O$-algebra $\Psi\otimes_O K$ with an exhaustive filtration by sub-$O$-algebras, indexed by the elements of $P$ ordered by division, by setting $(\Psi\otimes K)_{\a}$ for $\a\in P$ to be the sub-$K$-algebra generated by the $\psi^\b$ such that $\b|\a$. The intersection of this filtration with the sub-$O$-algebras $\Lambda$ and $\Psi$ induces exhaustive filtrations by sub-$O$-algebras $\Lambda_{\a}\subset \Lambda$ and $\Psi_{\a}\subset \Psi$ and we have $\Psi_\a=O[\psi_\b: \b|\a]$.
\begin{prop}\label{prop:finite-length-good} We have the following:
\begin{enumerate}[label=\textup{(\roman*)}]
\item for each $\a\in P$ there is a unique $O$-biring structure on $\Lambda_{\a}$ making $\Psi_{\a}\to \Lambda_{\a}$ a biring homomorphism,
\item for each pair $\a, \b\in P$ the homomorphism $h_{\Lambda}: \Lambda\odot \Lambda\to \Lambda$ induces a homomorphism $\Lambda_{\a}\odot \Lambda_{\b}\to \Lambda_{\a\b}$ and this map is an isomorphism if $\a$ and $\b$ are relatively prime.
\end{enumerate}
\end{prop}
\begin{proof}(i) This is Proposition 2.3 of \cite{Borger11}.
(ii) This is Propositions 2.3 and 5.3 of \cite{Borger11}.
\end{proof}
\subsection{} Let $A$ be an $O$-algebra. We write \[W(A)=\mathrm{Hom}_{O}(\Lambda, A) \quad \text{ and } \quad W_{\a}(A)=\mathrm{Hom}_{O}(\Lambda_{\a}, A)\] and call these the rings of Witt vectors and Witt vectors of length $\a$ of $A$. We also write \[\Gamma(A)=\mathrm{Hom}_{O}(\Psi, A)\overset{\sim}{\longrightarrow} \prod_{\a\in P} A \quad \text{ and } \quad \Gamma_{\a}(A)=\mathrm{Hom}_{O}(\Psi_{\a}, A)\overset{\sim}{\longrightarrow} \prod_{\b|\a} A\] and call these the rings of ghost vectors and length-$\a$ ghost vectors.
\begin{prop}\label{prop:witt-vectors-preserve-etale-morphisms} Let $\a\in P$ and let $A$ be an $O$-algebra. Then
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $(A\to A_i)_{i\in I}$ is an \'etale cover of $A$ so is $(W_{\a}(A)\to W_{\a}(A_i))$, and
\item for all homomorphisms $A\to B$ and all \'etale homomorphisms $A\to A'$ the natural map \[W_{\a}(A')\otimes_{W_{\a}(A)}W_{\a}(B)\to W_{\a}(A'\otimes_A B)\] is an isomorphism.
\end{enumerate}
\end{prop}
\begin{proof} This is Theorem 9.2 and Corollary 9.3 of \cite{Borger11}.
\end{proof}
\section{Witt vectors and arithmetic jets II} \label{sec:lambda-section-two} The purpose of this section is to extend the definition of $\Lambda$-structures, Witt vectors and arithmetic jets to sheaves for the \'etale topology.
\subsection{} Recall that $\mathrm{Sh}_{O}^{\textup{\'et}}$ denotes the category of sheaves for the \'etale topology over $\mathrm{Spec}(O)$, or $\textup{\'et}$-sheaves over $\mathrm{Spec}(O)$. We begin with the arithmetic jets and coghosts. Let $X$ be an $\textup{\'et}$-sheaf and define the presheaves on $\mathrm{Aff}_O$ \[W_{\a*}(X):=X\circ W_{\a}: \mathrm{Aff}_O^\circ\to \mathrm{Set}\ \quad \Gamma_{\a*}(X):=X\circ \Gamma_{\a}: \mathrm{Aff}_O^\circ\to \mathrm{Set}.\]
\begin{prop}\label{prop:def-arithmetic-ghosts} Let $\a\in P$ and let be $X$ an $\textup{\'et}$-sheaf. Then \[W_{\a*}(X):=X\circ W_{\a}: \mathrm{Aff}_O^\circ\to \mathrm{Set}\ \quad \Gamma_{\a*}(X)(X):=X\circ \Gamma_{\a}: \mathrm{Aff}_O^\circ\to \mathrm{Set}\] are again $\textup{\'et}$-sheaves. Moreover,
\begin{enumerate}[label=\textup{(\roman*)}]
\item if $X=\mathrm{Spec}(A)$ is affine then \[W_{\a *}(X)=\mathrm{Spec}(\Lambda_\a\odot A) \quad \text{ and }\quad\Gamma_{\a *}(X)=\mathrm{Spec}(\Psi_\a\odot A),\]
\item $W_{\a*}$ and $\Gamma_{\a*}$ commute with filtered colimits, and
\item $W_{\a*}$ sends smooth affine $\mathrm{Spec}(O_K)$-schemes to smooth affine $\mathrm{Spec}(O_K)$-schemes.
\end{enumerate}
\end{prop}
\begin{proof} It is clear that $\Gamma_{\a*}(X)$ of an $\textup{\'et}$-sheaf is again a sheaf, and for $W_{\a*}(X)$ it follows from (i) of (\ref{prop:witt-vectors-preserve-etale-morphisms}).
(i) This is clear for $\Gamma_{\a*}$ and Proposition 10.7 of \cite{Borger11bis} for $W_{\a*}$.
(ii) This is clear for $\Gamma_{\a*}$ and Proposition 11.7 of \cite{Borger11bis} for $W_{\a*}$.
(iii) This is Proposition 13.3 of \cite{Borger11bis}.
\end{proof}
\subsection{} For an $\textup{\'et}$-sheaf $X$ we have the following simple description of $\Gamma_{\a}(X)$. The fact that $\Gamma_{\a}(A)=\Pi_{\b\in P, \b|\a} A$ for all $O$-algebras $A$ (\ref{subsec:finite-length}) shows that \[\Gamma_{\a*}(X)(A)=X(\Gamma_{\a}(A))=\prod_{\b\in P, \b| \a} X(A)=\prod_{\b\in P, \b| \a} X(A),\] that is \[\Gamma_{\a}(X)\overset{\sim}{\longrightarrow} \prod_{\b\in P, \b | \a} X.\]
\subsection{} For an $\textup{\'et}$-sheaf $X$ we call $W_{\a*}(X)$ the ($\textup{\'et}$-sheaf of) length-$\a$ arithmetic jets of $X$ and $\Gamma_{\a*}(X)$ the ($\textup{\'et}$-sheaf of) length-$\a$ coghosts of $X$.
Standard sheaf theory supplies $W_{\a*}$ and $\Gamma_{\a *}$ with left adjoints which we denote by $W_{\a}^*$ and $\Gamma_{\a}^*$. For a sheaf $X$ we call $W_{\a}^*(X)$ the length-$\a$ Witt vectors of $X$ and $\Gamma_{\a}^*(X)$ the length $\a$-ghost vectors of $X$.
\begin{prop} If $X=\mathrm{Spec}(A)$ is an affine scheme then \[W_{\a}^*(\mathrm{Spec}(A))=\mathrm{Spec}(W_{\a}(A)) \quad \text{ and } \quad \Gamma_{\a}^*(\mathrm{Spec}(A))=\mathrm{Spec}(\Gamma_{\a}(A)).\]
\end{prop}
\begin{proof} This is true for all left adjoints to push forwards along a morphism of sites.
\end{proof}
\subsection{} For a sheaf $X$ we have the following simple description of $\Gamma_{\a*}(X)$. The fact that $\Gamma_{\a}(A)=\Pi_{\b\in P, \b| \a} A$ for all $O$-algebras $A$ (\ref{subsec:finite-length}) shows that: \begin{eqnarray*}\Gamma_{\a}^*(X)=\colim_{\mathrm{Spec}(A)\to X} \Gamma_{ \a}^*(\mathrm{Spec}(A))&=&\colim_{\mathrm{Spec}(A)\to X}\mathrm{Spec}(\Gamma_{\a}(A))\\
&=&\colim_{\mathrm{Spec}(A)\to X}\coprod_{\b\in P, \b|\a} \mathrm{Spec}(A)\\
&=&\coprod_{\b\in P, \b| \a} X\end{eqnarray*}
\subsection{} For a sheaf $X$, the inclusions $\Lambda_{\a}\to \Lambda_{\b}$ and $\Psi_{\a}\to \Psi_{\b}$ (\ref{subsec:finite-length}), for $\a, \b\in P$ with $\b| \a$, induce maps \[\Gamma_{\a*}(X)\to \Gamma_{\b*}(X) \quad \text{ and }\quad W_{\a*}(X)\to W_{\b*}(X)\] and taking the inverse limit in each case over the $\a\in P$ defines sheaves \[\Gamma_{*}(X):=\lim_\a \Gamma_{\a*}(X) \quad \text{ and } \quad W_{*}(X):=\lim_\a W_{\a*}(X)\] which we call the ($\textup{\'et}$-sheaf of) arithmetic jets of $X$ and ($\textup{\'et}$-sheaf of) coghosts of $X$.
Adjointly, there are also induced maps \[W_{\b}^*(X)\to W_{\a}^*(X)\quad \text{ and } \quad \Gamma_{\b}^*(X)\to \Gamma_{\a}^*(X)\] and taking the colimit we obtain $\textup{\'et}$-sheaves \[W^*(X):=\colim_\a W_{\a}^*(X)\quad \text{ and } \quad \Gamma^*(X):=\colim_\a \Gamma_{\a}^*(X)\] which we call the ($\textup{\'et}$-sheaf of) ghost vectors of $X$ and the ($\textup{\'et}$-sheaf of) Witt vectors of $X$. By construction the functor $\Gamma^*$ is left adjoint to $\Gamma_{*}$, and $W^*$ is left adjoint to $W_{*}$.
Generalising the descriptions of $\Gamma_{\a*}(X)$ and $\Gamma_{\a}^*(X)$ we have \[\Gamma_{*}(X)\overset{\sim}{\longrightarrow} \prod_{\a\in P} X \quad \text{ and }\quad \Gamma^*(X)\overset{\sim}{\longrightarrow} \coprod_{\a\in P}X.\]
\subsection{}\label{def:w-w-lambda-adjunction} The maps $\Lambda_{\a}\odot \Lambda_{\b}\to \Lambda_{\a\b}$ (\ref{prop:finite-length-good}) induce maps \[W^*_{\b}(W^*_{\a}(X))\to W^*_{\a\b}(X) \quad \text{ and } \quad W_{\a\b*}(X)\to W_{\a*}(W_{\b*}(X))\] and taking the colimit and limit respectively over all $\a, \b\in P$ we obtain maps \[\mu_{W^*(X)}: W^*(W^*(X))\to W^*(X) \quad \text{ and } \quad h_{W_*(X)}: W_{*}(X)\to W_{*}(W_{*}(X))\] which together with the natural maps \[g_{(1)}: X\to W^*(X) \quad \text{ and } \quad \gamma_{(1)}: W_{*}(X)\to X\] induced by the compatible maps $O[e]\to \Lambda_{\a}$ for all $\a\in P$ equip $W^*$ with the structure of a monad and $W_{*}$ with the structure of a comonad on $\mathrm{Sh}_{O}^\textup{\'et}$. We define the category of $\Lambda$-sheaves $\mathrm{Sh}_{\Lambda}^{\textup{\'et}}$ to be the category of $\textup{\'et}$-sheaves $X$ equipped with an action of $W^*$ or equivalently a coaction of $W_{*}$. The functors $W^*$ and $W_{*}$ now define functors $\mathrm{Sh}_{O}^{\textup{\'et}}\to \mathrm{Sh}_{\Lambda}^{\textup{\'et}}$ which are left and right adjoint to the forgetful functor $\mathrm{Sh}_{\Lambda}^{\textup{\'et}}\to \mathrm{Sh}_{O}^\textup{\'et}$. We have the following diagram of functors each one right adjoint to the one below:
\[\quad \xymatrix{\mathrm{Sh}_{\Lambda}^\textup{\'et}\ar[rr]^{\textup{forget}} && \ar@/^2pc/[ll]^{W^*}\ar@/_2pc/[ll]_{W_{*}}\mathrm{Sh}_{O}^{\textup{\'et}}.}\]
In particular, the forgetful functor in the middle admits both a left and right adjoint and so commutes with both limits and colimits. That is, limits and colimits in $\mathrm{Sh}_{\Lambda}^\textup{\'et}$ may be computed in $\mathrm{Sh}_{O}^\textup{\'et}$.
\subsection{} The same construction above applied to the maps $\Psi_{\a}\odot \Psi_{\b}\to \Psi_{\a\b}$ equips $\Gamma^*$ with the structure of a monad and $\Gamma_{*}$ with the structure of a comonad on $\mathrm{Sh}_{O}^\textup{\'et}$. In this case, it is easy to see that the map obtained \[\mu_{\Gamma^*(X)}: \Gamma^*(\Gamma^*(X))\to \Gamma^*(X)\] is identified with the map \[\coprod_{\b\in P}\coprod_{\a\in P} X=\coprod_{\a, \b\in P}X\to \coprod_{\c\in P} X\] whose restriction to the summand at $\b, \a\in P$ is the inclusion onto the summand at $\c=\a\b\in P$. Similarly, the map obtained \[h_{\Gamma_*(X)}:\Gamma_{*}(X)\to \Gamma_{*}(\Gamma_{*}(X))\] is identified with the map \[\prod_{\c\in P} X \to \prod_{\a\in P}\prod_{\b\in P} X\] whose composition with the projection onto the factor at $\a, \b\in P$ is the projection onto the factor at $\c=\a\b$.
We define the category of $\Psi$-sheaves $\mathrm{Sh}_{\Psi}^{\textup{\'et}}$ to be the category of sheaves $X$ equipped with an action of $\Gamma^*$ or equivalently with a coaction of $\Gamma_{*}$. Then $\Gamma^*$ and $\Gamma_{*}$ define functors $\mathrm{Sh}_{O}^{\textup{\'et}}\to \mathrm{Sh}_{\Psi}^{\textup{\'et}}$ which are left and right adjoint to the forgetful functor $\mathrm{Sh}_{\Psi}^{\textup{\'et}}\to \mathrm{Sh}_{O}^\textup{\'et}$. We have the following diagram of functors each one right adjoint to the one below:
\[\quad \xymatrix{\mathrm{Sh}_{\Psi}^\textup{\'et}\ar[rr]^{\textup{forget}} && \ar@/^2pc/[ll]^{\Gamma^*}\ar@/_2pc/[ll]_{\Gamma_{*}}\mathrm{Sh}_{O}^{\textup{\'et}}.}\]
\subsection{}\label{subsec:psi-action-sheaves} From the descriptions of the monadic and comonadic structures on $\Gamma^*$ and $\Gamma_*$ it is easy to see that to give a sheaf $X$ the structure of $\Psi$ sheaf is equivalent to equipping $X$ with an action of the monoid $P$. Indeed, denoting the action of $\a\in P$ by $\psi^\a_X: X\to X$ the map $\Gamma^*(X)\to X$ defining the corresponding action of $\Gamma^*$ is just \[\coprod_{\a\in P} \psi^\a_X: \coprod_{\a\in P} X=\Gamma^*(X)\to X.\]
As with $\Psi$-rings if $X\to S$ is a morphism of $\Psi$-sheaves the $\psi_S^\a$-linear endomorphisms $\psi_X^\a: X\to X$ for $\a\in P$ induce $S$-linear endomorphisms \[\psi_{X/S}^{\a}: X\to \psi_{S}^{\a*}(X)\] satisfying the commutativity condition \begin{equation}\psi_S^{\b*}(\psi_{X/S}^{\a})\circ \psi_{X/S}^\b=\psi_{X/S}^{\a\b}=\psi_S^{\a*}(\psi_{X/S}^{\b})\circ \psi_{X/S}^\a\label{eqn:semi-linear-comm-sheaves}\end{equation} for all $\a, \b\in P$. Moreover, to give an $\textup{\'et}$-sheaf $X\to S$ over $S$ a $\Psi$-structure such that the morphism $X\to S$ is a $\Psi$-morphism is the same as giving maps $\psi_{X/S}^{\a}: X\to \psi_{S}^{\a*}(X)$ all $\a\in P$ with $\psi^{(1)}_{X/S}=\mathrm{id}_X$ and satisfying the commutativity condition (\ref{eqn:semi-linear-comm-sheaves}).
\subsection{} The inclusions $\Lambda_{\a}\to \Psi_{\a}$ induce functorial maps for each sheaf $X$ \[g_{\leq \a}: \Gamma_{\a}^*(X)\to W_{\a}^*(X)\quad \text{ and } \quad\gamma_{\leq \a}: W_{ \a*}(X)\to \Gamma_{\a*}(X)\] which we call the length-$\a$ ghost and coghost maps.
The inclusions \[O[\psi_\a]\subset O[\psi_\b:\b\in P, \b|\a]=\Psi_\a\] induce for each $\a$ the $\a$-ghost component and $\a$-coghost components \[g_\a: X\to \Gamma^*(X)\to W^*(X) \quad \gamma_\a: W_*(X)\to \Gamma_*(X)\to X\] which we denote by $g_\a$ and $\gamma_\a$ and view as maps from $X\to W^*(X)$ or $X\to \Gamma^*(X)$ and similarly for the $\a$-coghost maps. We then have \[g_{\leq \a}=\coprod_{\b|\a}g_\b: \Gamma^*_\a(X)=\coprod_{\b|\a} X\to \Gamma^*(X).\]
Taking the colimit and limit along the length-$\a$ ghost and coghost maps we obtain the full ghost and coghost maps \[g: \Gamma^*(X)\to W^*(X)\quad \text{ and } \quad \gamma: W_{*}(X)\to \Gamma_{*}(X).\]
Finally, the natural transformations given by the ghost and coghost maps $g: \Gamma^*\to W^*$ and $\gamma: W_{*}\to \Gamma_{*}$ are morphisms of monads and comonads respectively and every $\Lambda$-sheaf $X$ inherits the structure of a $\Psi$-sheaf. That is every $\Lambda$-sheaf $X$ admits a canonical action of the monoid $P$ which, as for $\Psi$-sheaves, we denote by $\psi^\a_X:X\to X$ for $\a\in P$ (and similarly for the relative versions (cf.\ (\ref{subsec:psi-action-sheaves})).
\begin{prop}\label{prop:lambda-structures-frobenius-lifts-sheaves} We have the following:
\begin{enumerate}[label=\textup{(\roman*)}]
\item For each $\Lambda$-sheaf $X$ and each prime $\ideal{p}\in P$ the map \[\psi^\ideal{p}_X\times_{\mathrm{Spec}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p}): X\times_{\mathrm{Spec}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p})\to X\times_{\mathrm{Spec}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p})\] is equal to the $N\ideal{p}$-power Frobenius endomorphism of $X\times_{\mathrm{Spec}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p})$.
\item A $\Psi$-structure on a scheme $X$, flat over $O$ at all primes $\ideal{p}\in O$, is induced by a $\Lambda$-structure on $X$ if and only if for each prime $\ideal{p}\in P$ the map \[\psi^\ideal{p}_X\times_{\mathrm{Spec}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p}): X\times_{\mathrm{Spec}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p})\to X\times_{\mathrm{Spec}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p})\] is the $N\ideal{p}$-power Frobenius endomorphism. Moreover, in this case such a $\Lambda$-structure on $X$ is unique.
\item If each $\ideal{p}\in P$ is invertible on $X$ then every $\Psi$-structure on $X$ is induced by a unique $\Lambda$-structure.
\end{enumerate}
\end{prop}
\begin{proof} See \cite{Borger11ter} or it is an exercise using Theorem 17.3 of \cite{Borger11bis}.
\end{proof}
\begin{prop}\label{prop:w-and-etale-base-change} We have the following:
\begin{enumerate}[label=\textup{(\roman*)}]
\item If $f: X \to Y$ an affine \'etale morphism of ind-affine-schemes then for each $\a\in P$ the morphism \[W_{\a}^*(f): W_{\a}^*(X)\to W_{\a}^*(Y)\] is affine and \'etale.
\item If $f: X\to Y$ is a affine \'etale morphism of ind-affine-schemes then for any affine morphism $Y'\to Y$ of ind-schemes the natural map \[W_{\a}^*(X\times_Y Y')\to W_{\a}^*(X)\times_{W_{\a}^*(Y)}W_{\a}^*(Y') \] is an isomorphism.
\end{enumerate}
\end{prop}
\begin{proof} The case where $X$ and $Y$ are affine follows from (\ref{prop:witt-vectors-preserve-etale-morphisms}) and we shall reduce to this case. So write $Y=\colim_{i\in I} Y_i$ as a filtered colimit of affine schemes, and for $i\in I$ write $X_i=X\times_Y Y_i$.
(i) Let $i, j\in I$ with $Y_i\to Y_j\to Y$. Then as $Y_i$ and $Y_j$ are affine and $X\to Y$ is affine we have a cartesian diagram \[\xymatrix{W^{*}_\a(X_j)\ar[r] & W^{*}_\a(Y_j)\\
W^{*}_\a(X_i)\ar[u]\ar[r] & W^{*}_\a(Y_i)\ar[u]}\] where the bottom arrow is affine and \'etale. Now taking the colimit over $j$ yields a cartesian diagram \[\xymatrix{W^{*}_\a(X)\ar[r] & W^{*}_\a(Y)\\
W^{*}_\a(X_i)\ar[u]\ar[r] & W^{*}_\a(Y_i)\ar[u]}\] where the bottom arrow is affine, so that as $(W^*_\a(Y_i)\to W^*_\a(Y))_{i\in I}$ is a cover we are done.
(ii) Writing $Y'_i=Y'\times_Y Y_i$ the morphism \[W_{\a}^*(X\times_Y Y')\to W_{\a}^*(X)\times_{W_{\a}^*(Y)}W_{\a}^*(Y')\] is the colimit over $i\in I$ of the morphisms \[W_{\a}^*(X_i\times_{Y_i} Y_i')\to W_{\a}^*(X_i)\times_{W_{\a}^*(Y_i)}W_{\a}^*(Y'_i)\] which is an isomorphism as $Y_i, Y'_i$ and $X_i$ are all affine.
\end{proof}
\section{$\Lambda$-structures}\label{sec:lambda-section-three}
\subsection{} Let $S$ be a $\Lambda$-sheaf so that $S$ is equipped with an action of the monad $W^*$ and equivalently a coaction of the monad $W_*$.
From the monadic point of view we write \[\mu_S: W^*(S)\to S\] for map defining the action of $W^*$ on $S$ and we call this the structure map. The fact that it defines an action of $W^*$ on $S$ is the expressed via the commutativity of the diagram \[\xymatrix{& W^*(S)\ar[d]^{\mu_S}\\S\ar[r]^-{\mathrm{id}_S}\ar[ur]^{g_{(1)}} & S}\] and the property that the two compositions \[\xymatrix{W^*(W^*(S))\ar@<-.5ex>[r]_-{\mu_{W^*(S)}}\ar@<.5ex>[r]^-{W^*(\mu_S)} & W^*(S) \ar[r]^-{\mu_{S}} & S}\] coincide.
From the comonadic point of view we write \[h_S: S\to W_*(S)\] for the map defining the coaction of $W_*$, and the fact that it defines a coaction is expressed via the commutativity of the diagram \[\xymatrix{S\ar[r]^-{h_S}\ar[dr]_{\mathrm{id}_S} & W_*(S)\ar[d]^{\gamma_{(1)}}\\
& S}\] and the fact that the two compositions \[\xymatrix{S \ar[r]^{h_{S}} \ar[r] & W_*(S)\ar@<-.5ex>[r]_-{h_{W_*(S)}}\ar@<.5ex>[r]^-{W_*(h_S)} & W_*(W_*(S))}\] are equal.
\subsection{} We denote by $\mathrm{Sh}_{\Lambda_S}^\textup{\'et}$ the category of $\Lambda$-sheaves equipped with a $\Lambda$-morphism $X\to S$. The functor \[W_{S*}:\mathrm{Sh}_{S}^{\textup{\'et}}\to \mathrm{Sh}_{\Lambda_{S}}^\textup{\'et}: X\mapsto W_{*}(X)\times_{W_{*}(S)}S\] composed with the forgetful functor is a comonad on $\mathrm{Sh}_S^{\textup{\'et}}$ and identifies the category of $S$-sheaves $X$ with a coaction of $W_{S*}$ with the category of $\Lambda$-sheaves equipped with a $\Lambda$-morphism $X\to S$.
\begin{prop}\label{prop:hom-lambda-sheaves-are-sheaves} Let $S$ be a $\Lambda$-sheaf and let $X$ and $Y$ be a pair of $\Lambda_S$-sheaves. Then the functor \[\mathrm{Sh}_{\Lambda_S}^{\textup{\'et}}\to \mathrm{Set}: S'\mapsto \mathrm{Hom}_{\Lambda_{S'}}(X_{S'}, Y_{S'})\] is a sheaf for the canonical topology.
\end{prop}
\begin{proof} Let $S'\to S$ be a cover of $S$ in $\mathrm{Sh}_{\Lambda_S}$. Then $S'\to S$ is also a cover when viewed in $\mathrm{Sh}^{\textup{\'et}}_S$ and so any $\Lambda_{S'}$-morphism $f':X_{S'}\to Y_{S'}$ with the property that the two pull-backs of $f$ along the two projections $S'\times_S S'\to S$ coincide descends to a morphism $X\to Y$. It remains to verify that it is a $\Lambda_S$-morphism, which is the commutivity of the \begin{equation}\label{eqn:lambda-morphism}\begin{gathered}\xymatrix{X\ar[r]^-{h_{X/S}}\ar[d]_f & W_{*}(X)\times_{W_{*}(S)}S\ar[d]^{W_{*}(f)\times_{W_{*}(S)}S}\\
Y\ar[r]^-{h_{Y/S}} & W_{*}(Y)\times_{W_{*}(S)}S.}\end{gathered}\end{equation}However, using the identifications \begin{eqnarray*} W_{*}(X)\times_{W_{*}(S)} S\times_S S'&=&W_{*}(X)\times_{W_{*}(S)}S'\\
&=&W_{*}(X)\times_{W_{*}(S)}W_{*}(S')\times_{W_{*}(S')} S'\\
&=&W_{*}(X\times_S S')\times_{W_{*}(S')}S'
\end{eqnarray*} and similarly for $Y$, the diagram (\ref{eqn:lambda-morphism}) pulls-back along $S'\to S$ to the diagram
\begin{equation*}\begin{gathered}\xymatrix{X_{S'}\ar[r]^-{h_{X_{S'}/S'}}\ar[d]_{f'} & W_{*}(X)\times_{W_{*}(S')}S'\ar[d]^{W_{*}(f')\times_{W_{*}(S')}S'}\\
Y_{S'}\ar[r]^-{h_{Y_{S'}/S'}} & W_{*}(Y)\times_{W_{*}(S')}S'}\end{gathered}\end{equation*}
which commutes by hypothesis. Therefore, as $S'\to S$ is an epimorphism, the diagram (\ref{eqn:lambda-morphism}) commutes.
\end{proof}
\subsection{}\label{subsec:sheaves-of-lambda-structures} Let $S$ be a $\Lambda$-sheaf and let $X\to S$ be an $S$-$\textup{\'et}$-sheaf. Denote by $\Lambda_{X/S}$ the functor \[\Lambda_{X/S}: \mathrm{Sh}_{\Lambda_{S}}^\textup{\'et}\to \mathrm{Set}: S'/S\mapsto \{\text{the set of }\Lambda_{S'}\text{-structures on }X\times_S S'\}.\]
\begin{prop}\label{prop:lambda-structures-are-sheaves} The functor $\Lambda_{X/S}$ is a sheaf for the canonical topology on $\mathrm{Sh}_{\Lambda_{S}}^\textup{\'et}$.
\end{prop}
\begin{proof} Let $S'\to S$ be an epimorphism and write $S''=S\times_S S'$, $X'=X\times_S S'$ and $X''=X\times_S S''$. By the definition of a $\Lambda_S$-structure (in terms of $W^*$) we have an equaliser (the Homs are of $\textup{\'et}$-sheaves not $\Lambda$-sheaves) \[\xymatrix{\Lambda_{X/S}(S)\ar[r] & \mathrm{Hom}_S(W^*(X), X)\ar@<.5ex>[r]\ar@<-.5ex>[r] & \mathrm{Hom}_S(X, X)\times \mathrm{Hom}_S(W^*(W^*(X)), X)}\] where the first map sends a $\Lambda_S$-structure on $S$ to the map $\mu_X: W^*(X)\to X$ and the two parallel arrows send a map $\mu_X$ to \[(\mu_X\circ g_{(1)}, \mu_{W^*(X)}\circ \mu_X) \quad \text{ and } \quad (\mathrm{id}_X, W^*(\mu_X)\circ \mu_X).\]
Now consider corresponding commutative diagram: \[\xymatrix{\Lambda_{X/S}(S)\ar[r]\ar[d] & \mathrm{Hom}_S(W^*(X), X)\ar@<.5ex>[r]\ar@<-.5ex>[r]\ar[d] & \mathrm{Hom}_S(X, X)\times \mathrm{Hom}_S(W^*(W^*(X)), X)\ar[d]\\
\Lambda_{X/S}(S')\ar[r]\ar@<.5ex>[d]\ar@<-.5ex>[d] & \mathrm{Hom}_S(W^*(X'), X)\ar@<.5ex>[r]\ar@<-.5ex>[r]\ar@<.5ex>[d]\ar@<-.5ex>[d] & \mathrm{Hom}_S(X', X)\times \mathrm{Hom}_S(W^*(W^*(X')), X)\ar@<.5ex>[d]\ar@<-.5ex>[d]\\\Lambda_{X/S}(S'')\ar[r] & \mathrm{Hom}_S(W^*(X''), X)\ar@<.5ex>[r]\ar@<-.5ex>[r] & \mathrm{Hom}_S(X'', X)\times \mathrm{Hom}_S(W^*(W^*(X'')), X).
}\]
The two right columns are equalisers as $X'=X\times_S S'\to X$ is a cover and $W^*$ preserves push-outs and all three rows are equalisers. Therefore it follows that the first column is an equaliser and we are done.
\end{proof}
\begin{prop}\label{prop:lambda-structures-etale-schemes-number-fields} Let $O$ be the ring of integers in a number field $K$, $\ideal{f}\in \mathrm{Id}_O$ an ideal and write $P=\mathrm{Id}_{O}^{(\ideal{f})}$. Then a finite \'etale $S=\mathrm{Spec}(O[\ideal{f}^{-1}])$-scheme $X$ admits a $\Lambda_{P, S}$-structure if and only if $X\times_{S}\mathrm{Spec}(K)=\amalg_{1\leq i\leq n}\mathrm{Spec}(L_i)$ where each $L_i$ is a finite abelian extension of $K$. Moreover, in this case the $\Lambda_{P, S}$-structure is unique and any morphism of finite \'etale $\Lambda_{P, S}$-schemes is a $\Lambda_{P, S}$-morphism.
\end{prop}
\begin{proof} Let $X\to S$ be a finite \'etale morphism. Then $X\times_S \mathrm{Spec}(K)=\amalg_{i\in I} \mathrm{Spec}(L_i)$ with $L_i/K$ a finite extension and $X\overset{\sim}{\longrightarrow} \amalg_{i} X_i$ where $X_i=\mathrm{Spec}(O_{L_i}[\ideal{f}^{-1}])$.
If $X\to S$ has a $\Lambda_{P, S}$-structure then for each prime $\ideal{p}\in P=\mathrm{Id}_{O_K}^{(\ideal{f})}$, the Frobenius lift $\psi^\ideal{p}_X: X\to X$ fixes the fibre of each $X_i$ over $\mathrm{Spec}(\mathbf{F}_\ideal{p})$ so that $\psi_X^\ideal{p}$ maps each $X_i$ to itself. It follows that the $\Lambda_{P, S}$-structure on $X$ is induced by unique $\Lambda_{P, S}$-structures on each $X_i$ and so we may assume that $X=X_i$ is connected. In this case, write $X=\mathrm{Spec}(O_{L}[\ideal{f}^{-1}])$ with $L/K$ a finite extension. If $\P$ is a prime of $L$ laying over the prime $\ideal{p}$ then there is a unique automorphism \[\sigma_{\P, L/K}: X\to X\] whose restriction to fibre over $\mathrm{Spec}(O_L/\P)$ is equal to the $N\ideal{p}$-power Frobenius automorphism. But $\psi^\ideal{p}_X$ also has this property and so $\sigma_{\P, L/K}=\psi^\ideal{p}_X$. As the maps $\psi^\ideal{p}_X=\sigma_{\P, L/K}$ commute and generate the group $G(L/K)$ it follows that $G(L/K)$ is abelian. Moreover, the uniqueness of $\sigma_{\P, L/K}=\sigma_{\ideal{p}, L/K}=\psi_{X}^\ideal{p}$ shows that the $\Lambda_{P}$-structure on $X$ is unique, is also a $\Lambda_{P, S}$-structure and that any morphism of finite \'etale $\Lambda_{P, S}$-schemes is a $\Lambda_{P, S}$-morphism.
Conversely, if each $L_i/K$ is abelian then there is a unique automorphism \[\sigma_{\ideal{p}, L_i/K}: X_i\to X_i\] lifting the $N\ideal{p}$-power Frobenius automorphism of the fibre over $\mathrm{Spec}(\mathbf{F}_\ideal{p})$. It follows that setting $\psi^\ideal{p}_X=\amalg_{i\in I}\sigma_{L_i/K, \ideal{p}}$ defines a Frobenius lift on $X$. Moreover, as $G(L_i/K)$ is abelian these Frobenius lifts commute and so define a $\Lambda_{P, S}$-structure on $X$.
\end{proof}
\begin{rema} In the notation of (\ref{prop:lambda-structures-etale-schemes-number-fields}) if $X\to S$ is a finite \'etale $\Lambda_P$-scheme then its Frobenius lifts $\psi_{X}^\a$ for $\a\in P$ will be denoted by $\sigma_{S, \a}$, or just $\sigma_\a$. This agrees with (or extends) the conventions set up in (\ref{subsec:finite-extensions-frobenius}).
\end{rema}
\section{Ghosts and coghosts}
\begin{prop}\label{prop:ghost-map-surj-geom-points} Let $X$ be a sheaf. For each $\a\in P$ the length-$\a$ ghost map \[g_{\a}:\Gamma_{\a}^*(X)\to W_{\a}^*(X)\] is surjective on geometric points and so is the full ghost map \[g: \Gamma^*(X)\to W^*(X).\]
\end{prop}
\begin{proof} Writing $X=\colim \mathrm{Spec}(A)$ as a colimit of affine schemes, and $W^{*}(X)=\colim_\a W_{\a}^*(X)$ and $\Gamma^{*}(X)=\colim_\a \Gamma_{\a}^*(X)$ it is enough to show this for \[\mathrm{Spec}(\Gamma_\a(A))=\Gamma_{\a}^*(\mathrm{Spec}(A))\to \mathrm{Spec}(W_\a(A))=W^*_\a(\mathrm{Spec}(A)).\] But the map $W_\a(A)\to \Gamma_\a(A)$ is integral with nilpotent kernel (Proposition 8.1 of \cite{Borger11}) and therefore \[\mathrm{Spec}(\Gamma_\a(A))\to \mathrm{Spec}(W_\a(A))\] is surjective on geometric points.
\end{proof}
\begin{prop}\label{prop:ghost-covers-etale} If $S$ is an ind-affine scheme, and $T$ an affine \'etale $W^*(S)$-sheaf then the sequence \[\xymatrix{T(W^*(S))\ar[r]^{T(g)} & T(\Gamma^*(S))\ar@<.5ex>[r]\ar@<-.5ex>[r] & T(\Gamma^*(S)\times_{W^*(S)}\Gamma^*(S))}\] is an equaliser.
\end{prop}
\begin{proof} As $W^*(S)=\colim_{\a\in P}W^*_\a(S)$ and $\Gamma^*(S)=\colim_{\a\in P}\Gamma^*_\a(S)$ and filtered colimits are exact, we may replace $W^*(S)$ and $\Gamma^*(S)$ and $T$ with $W^*_\a(S)$, $\Gamma^*_\a(S)$ and $T\times_{W^*(S)}W^*_\a(S)$ respectively. Now writing $S$ as a filtered colimit of affine schemes we may assume that $S$ is affine in which case the claim follows as $\Gamma^*_\a(S)\to W_{\a}^*(S)$ is integral and surjective, and therefore an effective descent map for the category of affine \'etale schemes.
\end{proof}
\begin{rema}\label{rema:ghost-fibre-product} In order for (\ref{prop:ghost-covers-etale}) to be useful in applications we should say something about the fibre product $W^*(S)\times_{\Gamma^*(S)}W^*(S)$. Let $\ideal{p}$ be a prime ideal, $\a$ any ideal, $n\geq 1$ an integer and write $S_\ideal{p}=S\times_{\mathrm{Spec}(O)}\mathrm{Spec}(\mathbf{F}_\ideal{p})$ and consider the two maps (where the inclusions are the obvious ones) \[r_{\a, \ideal{p}^n}^{(1)}:S_\ideal{p}\subset S \stackrel{g_{\a\ideal{p}^n}}{\longrightarrow} \Gamma^*(S) \quad \text{and} \quad r_{\a, \ideal{p}^n}^{(2)}:S_\ideal{p} \stackrel{\mathrm{Fr}_{S_\ideal{p}}^{N\ideal{p}^n}}{\longrightarrow} S_\ideal{p}\subset S \stackrel{g_\a}{\to} \Gamma^*(S).\] Then for $i, j\in \{1, 2\}$ the two compositions
\[\xymatrix{S_\ideal{p}\ar[rr]^-{(r_{\a, \ideal{p}^n}^{(i)}, r_{\a, \ideal{p}^n}^{(j)})} && \Gamma^*(S)\times_{W^*(S)}\Gamma^*(S)\ar@<.5ex>[r]\ar@<-.5ex>[r] & W^*(S)}\] are equal and the morphism \[\coprod_{i\neq j\in \{1, 2\}}\coprod_{\a, \ideal{p}^n} S_\ideal{p} \stackrel{(r_{\a, \ideal{p}^n}^{(i)}, r_{\a, \ideal{p}^n}^{(j)})}{\longrightarrow}\Gamma^*(S)\times_{W^*(S)}\Gamma^*(S)\] defines a nilpotent immersion onto the complement of the diagonal in $\Gamma^*(S)\times_{W^*(S)}\Gamma^*(S)$ (this follows by an iterated application of 17.1 \cite{Borger11bis}). Therefore, in the notation of (\ref{prop:ghost-covers-etale}) an element of $T(\Gamma^*(S))$ is in the image of $T(W^*(S))\to T(\Gamma^*(S))$ if and only if for all prime ideals $\ideal{p}$, ideals $\a$, integers $n\geq 0$ and pairs $i\neq j\in \{1, 2\}$, it is equalised by the maps \[\xymatrix{T(\Gamma^*(S))\ar@<.7ex>[rr]^-{T(r_{\a, \ideal{p}^n}^{(i)})}\ar@<-.7ex>[rr]_-{T(r_{\a, \ideal{p}^n}^{(j)})} && T(S_\ideal{p}).}\]
\end{rema}
\begin{lemm}\label{lemm:coghost-is-affine-finite-point-property} If $X$ is a scheme with the property that every finite set of points of $X$ is contained in an open affine sub-scheme of $X$ then for each $\a\in P$ the length-$\a$ coghost map \[\gamma_{\leq \a}: W_{\a*}(X)\to \Gamma_{\a*}(X)\] is affine.
\end{lemm}
\begin{proof} The property satisfied by $X$ implies that there is an open affine cover $(X_i)_{i\in I}$ of $X$ such that $(\Gamma_{\a*}(X_i))_{i\in I}$ is an open cover of $\Gamma_{\a*}(X)$ (recall that $\Gamma_{\a*}$ of a sheaf $X$ is just a finite product of copies of $X$). However, the diagram \begin{equation}\label{dia:coghost-affine}
\begin{gathered}\xymatrix{W_{\a*}(X_i)\ar[r]^{\gamma_{\leq \a}}\ar[d] & \Gamma_{\a*}(X_i)\ar[d]\\
W_{\a*}(X)\ar[r]^{\gamma_{\leq \a}} & \Gamma_{\a*}(X)}\end{gathered}\end{equation} is cartesian by Proposition 12.2 of \cite{Borger11bis} (\cite{Borger11bis} also assumes that the open immersions $X_i\to X$ are closed but the proof that the diagram (\ref{dia:coghost-affine}) is cartesian does not use this assumption) and the top morphisms is affine. As $(\Gamma_{\a*}(X_i))_{i\in I}$ is a cover of $\Gamma_{\a*}(X)$ it follows that the bottom row of (\ref{dia:coghost-affine}) is affine.
\end{proof}
\subsection{} Let $S$ be a $\Lambda$-sheaf and $X$ an $S$-group sheaf. Then to equip $X$ with a $\Lambda_{S}$-structure making it a $\Lambda_{S}$-sheaf in groups over $S$ is equivalent to equipping it with a $\Lambda_{S}$-structure such that the defining structure map \[h_{X/S}: X\to W_{*}(X)\times_{W_{*}(S)}S\] is a homomorphism of $S$-groups (recall the $W_*$ being a right adjoint preserves limits so that $W_*(X)$ is a $\Lambda_{W_*(S)}$-sheaf of groups over $W_*(S)$).
\begin{lemm}\label{lemm:coghost-affine-relative-coghost-affine} Let $S=\colim_{i\in I} S_i$ be a $\Lambda$-ind-affine scheme, $f: X\to S$ be an $\textup{\'et}$-sheaf over $S$ and $\a\in P$. If for each $i\in I$, setting $X_i=X\times_S S_i$, the length-$\a$ coghost map \[\gamma_{\leq \a}: W_{\a*}(X_i)\to \Gamma_{\a*}(X_i)\] is affine then the length-$\a$ relative coghost map \[\gamma_{X/S, \leq \a}: W_{\a*}(X)\times_{W_{\a*}(S)}S\to \Gamma_{\a*}(X)\times_{\Gamma_{\a*}(S)}S\] is affine.
\end{lemm}
\begin{proof} The morphism $\gamma_{X/S, \leq \a}$ if affine if and only if the morphisms \[\gamma_{X/S, \a}\times_S S_i\] are affine for each $i\in I$. Fixing such an $i$, as $S_i$ is affine (in particular, quasi-compact) and \[\colim_i W^*_\a(S_i)=W_{\a*}(\colim_i S_i)\] (by (iv) of (\ref{prop:def-arithmetic-ghosts})) there is some $j\in I$ such that $S_i\to W_{\a*}(S)$ factors through $W_{\a*}(S_j)\to W_{\a*}(S)$. Therefore, we can factor $\gamma_{X/S, \a}\times_S S_i$ as the composition \[W_{\a*}(X_j)\times_{W_{\a*}(S_j)} S_i\to W_{\a*}(X_j)\times_{\Gamma_{\a*}(S_j)} S_i\to \Gamma_{\a*}(X_j)\times_{\Gamma_{\a*}(S_j)} S_i\] where the first map is induced by $W_{\a*}(S_j)\to \Gamma_{\a *}(S_j)$, which is affine as $W_{\a*}(S_j)\to \Gamma_{\a *}(S_j)$ is affine, and the second is $\gamma_{\leq \a}\times_{\Gamma_{\a*}(S_j)} S_i$ which is affine as $\gamma_{X_j, \a}$ is affine by hypothesis. Therefore, $\gamma_{X/S, \a}\times_S S_i$ is affine and we are done.
\end{proof}
\begin{prop}\label{prop:lambda-and-psi-homomorphisms-of-abelian-varieties} Let $S$ be an $\Lambda$-ind-affine scheme, let $A$ and $A'$ be a pair of abelian schemes over $S$ and let $f: A\to A'$ be an $S$-homomorphism. If $f$ is a $\Psi_S$-morphism then it is a $\Lambda_S$-morphism.
\end{prop}
\begin{proof} Write $S=\colim_{i\in I} S_i$ as a filtered colimit of affine schemes. By Theorem 1.9 of Chapter I of \cite{FaltChai1990}, for each $i\in I$ and $\a\in P$, the $S_i$-scheme $A'\times_S S_i$ satisfies the hypotheses of (\ref{lemm:coghost-is-affine-finite-point-property}) so that we may apply (\ref{lemm:coghost-affine-relative-coghost-affine}) to deduce that the relative coghost homomorphism of length $\a$ \[\gamma_{A'/S, \a}: W_{\a*}(A')\times_{W_{\a*}(S)}S\to \Gamma_{\a*}(A')\times_{\Gamma_{\a*}(S)}S\] is affine. Taking the limit over $\a$ we see that \[\gamma_{A'/S}: W_{*}(A')\times_{W_{*}(S)}S\to \Gamma_{*}(A')\times_{\Gamma_{*}(S)}S\] is affine.
Now let the $\Lambda_{S}$-structures on $A$ and $A'$ be given by the $S$-homomorphisms \[h_{A/S}: A\to W_{*}(A)\times_{W_{*}(S)} S \quad \text{ and } \quad h_{A'/S}: A'\to W_{*}(A')\times_{W_{*}(S)} S\] and let $f: A\to A'$ be a $\Psi_{S}$-homomorphism. By hypothesis, the difference \[h_{A'/S}\circ f-(W_{*}(f)\times_{W_{*}(S)}S)\circ h_{A/S}: A\to W_{*}(A')\times_{W_*(S)}S\] factors through the kernel of the relative coghost homomorphism $\gamma_{A/S}$. However, the kernel of $\gamma_{A/S}$ is affine over $S$ and any homomorphism from an abelian $S$-scheme to an $S$-affine scheme is trivial. Therefore, \[h_{A'/S}\circ f-(W_{*}(f)\times_{W_{*}(S)}S)\circ h_{A/S}=0\] and $f$ is a $\Lambda_{S}$-homomorphism.
\end{proof}
\begin{rema}\label{rema:lambda-structures-on-abelian-varieties-are-determined-by-psi-structures} It follows from (\ref{prop:lambda-and-psi-homomorphisms-of-abelian-varieties}) above that if $A$ is an abelian variety over an $\Lambda$-ind-affine scheme $S$ then any two $\Lambda_S$-structures on $A$, compatible with the group law, coincide if and only if the underlying $\Psi_S$-structures coincide.
\end{rema}
\chapter{CM elliptic curves and $\Lambda$-structures} In this chapter we explain the connection between CM elliptic curves and $\Lambda$-structures. The first and main result being, essentially, that the moduli stack of CM elliptic curves $\mc{M}_\mathrm{CM}$ admits a $\Lambda$-structure. The observant reader will have noticed that we have not defined what it means for a stack to have a $\Lambda$-structure and we do not propose to do so here (not because we cannot but only because to do so would involve various 2-categorical issues that would in this instance serve only to make matters more complicated than they need be). However, we shall explain what we mean to prove.
We continue with the set-up in (\ref{subsec:imag-quad-set-up}). So that all sheaves considered are always over $\mathrm{Spec}(O_K)$ and we will use the theory of Chapter 3 with $\Lambda$-structures relative to the full monoid of ideals $\mathrm{Id}_{O_K}$ (later we will also consider sub-monoids).
Recall that to give a $\textup{\'et}$-sheaf $X$ a $\Lambda$-structure is to give for each $\textup{\'et}$-sheaf $S$, a map \[h_{X}(S): X(S)\to X(W^*(S))\] which is functorial in $S$ and such that:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the composition \[X(S)\stackrel{h_X(S)}{\to} X(W^*(S))\stackrel{X(g_{(1)})}{\to} X(S)\] is the identity and
\item the two compositions \[\xymatrix{X(S)\ar[r]^-{h_X(S)} & X(W^*(S)) \ar@<.5ex>[r]^-{X(\mu_{W^*(S)})}\ar@<-.5ex>[r]_-{h_{X}(W^*(S))} & X(W^*(W^*(S)))}\] are equal.
\end{enumerate}
To this end we show that given an ind-affine scheme $S$ and an element of $\mc{M}_\mathrm{CM}(S)$, i.e.\ a CM elliptic curve $E/S$, there is a functorially associated element of $\mc{M}_\mathrm{CM}(W^*(S))$, i.e.\ a CM elliptic curve $W_\mathrm{CM}^*(E)/W^*(S)$, satisfying the following properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the pull-back of $W_\mathrm{CM}^*(E)$ along the ghost component at $(1)$ is canonically isomorphic to $E$: \[E\overset{\sim}{\longrightarrow} g_{(1)}^*(W_\mathrm{CM}^*(E))=W_{CM}^*(E)\times_{W^*(S)} S,\]
\item the pull-back of $W_\mathrm{CM}^*(E)$ along $\mu_S: W^*(W^*(S))\to W^*(S)$ is canonically isomorphic to $W_\mathrm{CM}^*(W_\mathrm{CM}^*(E))$: \[\mu_S^*(W_\mathrm{CM}^*(E))=W_{CM}^*(E)\times_{W^*(S)} W^*(W^*(S)) \overset{\sim}{\longrightarrow} W^*_{CM}(W^*_{CM}(E)).\]
\end{enumerate}
We call $W_\mathrm{CM}^*(E)/W^*(S)$ the canonical lift of $E/S$. In addition to (i) and (ii) above, we also show that the CM elliptic curve $W_\mathrm{CM}^*(E)$ admits a canonical $\Lambda_{W^*(S)}$-structure. It is worth pointing out that these `canonical lifts' are both global and big -- the base is $S$ is an arbitrary ind-affine scheme over $\mathrm{Spec}(O_K)$ and the Witt vectors over which we lift the CM elliptic curves have Frobenius lifts at all primes of $O_K$.
We now give a brief overview of each of the sections. The construction of $W^*_\mathrm{CM}(E)/W^*(S)$ and the verification of its properties is the content of \S 1. We also use this to define what it means for a CM elliptic curve over $\Lambda$-ind-affine scheme $E/S$ to have a canonical $\Lambda$-structure (what we really do is define what it means for a morphism \[S\stackrel{E}{\to} \mc{M}_\mathrm{CM}\] corresponding to a CM elliptic curve $E/S$ to be a $\Lambda$-morphism).
In \S 2 we consider a certain special class of CM elliptic curves (those of `Shimura type') and show that they are exactly those admitting $\Lambda$-structures. These curves were first defined and studied by Shimura (\cite{Shimura94}), and subsequently by several other authors with particular reference to their L-functions (see \cite{CoatesWiles1977}, \cite{Deninger1989} and \cite{Rubin1981}). We finish \S 2 by showing that many CM elliptic curves of Shimura type admit global minimal models. This gives a broad generalisation of a result of Gross \cite{Gross82}. The proof we give is quite different to that in \cite{Gross82} and relies ons a certain strengthening (\ref{prop:tannaka-result-general}) of an old principal ideal theorem (valid for arbitrary number fields).
In \S 3 we show how to (intelligently) construct the quotient of the universal CM elliptic curve by its group of automorphisms and we show that this curve descends to a smooth projective curve $X\to M_\mathrm{CM}$ over the coarse sheaf $M_\mathrm{CM}$. We also show that this descended curved admits a $\Lambda_{M_\mathrm{CM}}$-structure and that this $\Lambda_{M_\mathrm{CM}}$-structure can be used to construct the maximal abelian extension of $K$ in a natural way. This gives a canonical, integral and $\Lambda$-theoretic version of the explicit generation of the ray class fields of $K$ using Weber functions of CM elliptic curves over fields. We end this section by showing that the possibly mysterious curve $X$ is non-other than $\mathbf{P}^1_{M_\mathrm{CM}}$.
In \S 4 we use the results of \S 1 and \S 2 to construct a flat affine formally smooth presentation $M_\mathrm{CM}^W\to \mc{M}_\mathrm{CM}$ of $\mc{M}_\mathrm{CM}$. The flat affine formally smooth scheme $M_\mathrm{CM}^W$ has a natural moduli theoretic interpretation, and is also a torsor under a certain affine flat affine group scheme $\mathrm{CL}_{O_K}^W$. Finally, we show that $M_\mathrm{CM}^W$ admits a natural $\Lambda$-structure compatible with that on $\mc{M}_\mathrm{CM}$.
In \S 5 we exhibit a rather interesting relationship between a (variant of) the canonical lift of an CM elliptic curve (over an arbitrary base) and its Tate module. This gives an analogue for CM elliptic curves of a certain construction in $p$-adic Hodge theory involving Lubin--Tate $O$-modules and we end by sketching a certain analytic analogue.
\section{Canonical lifts of CM elliptic curves} We continue with the set-up of Chapter 2, so that we work over the base scheme $\mathrm{Spec}(O_K)$ where $O_K$ is the ring of integers of an imaginary quadratic field. In order to apply the theory of Chapter 3, we will no longer be working with arbitrary sheaves $S\in \mathrm{Sh}_{O_K}$ but only with ind-affine schemes $S\in \mathrm{IndAff}_{O_K}\subset \mathrm{Sh}_{O_K}$.
By a $\Lambda$-structure is meant one relative to the Dedekind domain $O_K$ and to the full set of ideals $P=\mathrm{Id}_{O_K}$.
We note that if $S$ is an ind-affine scheme then $W^*(S)$ is again an ind-affine scheme and therefore a sheaf for the fpqc topology. Moreover, $W_*(S)=\lim_{\a\in \mathrm{Id}_{O_K}}W_{\a*}(S)$ is an inverse limit of ind-affine schemes (as $W_{\a*}$ commutes with filtered colimits and sends affine schemes to affine schemes) and is also a sheaf for the fpqc topology.
For technical reasons we also work with the affine \'etale topology on $\mathrm{IndAff}_{O_K}$ whose covers are given by families of affine \'etale morphisms $(S_i\to S)_{i\in I}$ which are covers when viewed in $\mathrm{Sh}_{O_K}$ (or equivalently $\mathrm{Sh}_{O_K}^\textup{\'et}$).
We shall also continue to work with the fibred category $\mc{M}_{\mathrm{CM}}$ but from here on view it as a fibred category over the category of ind-affine schemes $\mathrm{IndAff}_{O_K}\subset \mathrm{Sh}_{O_K}$, rather than all of $\mathrm{Sh}_{O_K}$.
Unless otherwise noted $S$ will denote an arbitrary ind-affine scheme.
\subsection{} Denote by $\mc{L}_\mathrm{CM}$ the rank one $O_K$-local system over $\Gamma^*(\mathrm{Spec}(O_K))$ whose fibre over the ghost component at $\a\in \mathrm{Id}_{O_K}$ is the constant sheaf $\underline{\a}^{-1}$, i.e.\ \[\mc{L}_\mathrm{CM}=\coprod_{\a\in \mathrm{Id}_{O}} \underline{\a}^{-1}\to \coprod_{\a\in \mathrm{Id}_{O}}\mathrm{Spec}(O_K)=\Gamma^*(\mathrm{Spec}(O_K)).\] For an ind-affine scheme $S$ we shall abuse notation and write $\mc{L}_\mathrm{CM}$ for the rank one $O_K$-local system over $\Gamma^*(S)$ obtained by pulling back $\mc{L}_{\mathrm{CM}}$ along $\Gamma^*(S)\to \Gamma^*(\mathrm{Spec}(O_K))$.
\subsection{}\label{subsec:def-can-ghost} Let $E\to S$ a CM elliptic curve . Writing $p_S: \Gamma^*(S)\to S$ for the map \[\coprod_{\a\in \mathrm{Id}_{O_K}}\mathrm{id}_S: \Gamma^*(S)=\coprod_{\a\in \mathrm{Id}_{O_K}}S\to S,\] we define a new CM elliptic curve over $\Gamma^*(S)$ by \[\Gamma^*_\mathrm{CM}(E)=p_S^*(E)\otimes_{O_K}\mc{L}_\mathrm{CM}.\] In other words, we have \[\Gamma^*_\mathrm{CM}(E)=\coprod_{\a\in \mathrm{Id}_{O_K}}E\otimes_{O_K}\a^{-1}\to \coprod_{\a\in \mathrm{Id}_{O_K}} S=\Gamma^*(S).\] If $f: E\to E'$ is a homomorphism of CM elliptic curves over $S$ then we write \[\Gamma^*_\mathrm{CM}(f)=p^*_S(f)\otimes_{O_K}\mc{L}_\mathrm{CM}:\Gamma^*_\mathrm{CM}(E)\to \Gamma^*_\mathrm{CM}(E')\] so that $\Gamma^*_{\mathrm{CM}}$ defines a functor from the category of CM elliptic curves over $S$ to the category of CM elliptic curves over $\Gamma^*(S)$.
By construction, the rank one $O_K$-local system $\mc{L}_\mathrm{CM}$ over $\Gamma^*(S)$ satisfies $\mc{L}_\mathrm{CM}\otimes_{O_K}\a^{-1}=\psi^{\a*}(\mc{L}_\mathrm{CM})$ for each ideal $\a$ and this induces isomorphisms \[\Gamma^*_\mathrm{CM}(E)\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} \psi^{\a*}(\Gamma^*_\mathrm{CM}(E)).\] We equip $\Gamma^*_{\mathrm{CM}}(E)$ with the $\Psi_{\Gamma^*(S)}$-structure with the relative endomorphisms given for each ideal $\a$ by the composition \[\Gamma_{\mathrm{CM}}^*(E)\stackrel{i_\a}{\to}\Gamma_{\mathrm{CM}}^*(E)\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} \psi^{\a*}(\Gamma_\mathrm{CM}^*(E)).\] In order to avoid overly cumbersome notation, we will denote this map by $\varphi^\a_{E/S}$ (instead of by the usual by monstrous $\psi^{\a*}_{\Gamma^*_\mathrm{CM}(E)/\Gamma^*(S)}$). Note that $\ker(\varphi^\a_{E/S})=\Gamma^*_\mathrm{CM}(E)[\a]$.
The sheaf of rings $\underline{O_K}_{\Gamma^*(S)}=\Gamma^*(\underline{O_K}_S)$ is naturally a sheaf of $\Psi_{\Gamma^*(S)}$-rings and the $\underline{O_K}_{\Gamma^*(S)}$-module structure \[\underline{O_K}_{\Gamma^*(S)}\times \Gamma_\mathrm{CM}^*(E)\to \Gamma_\mathrm{CM}^*(E)\] is compatible with the $\Psi_{\Gamma^*(S)}$-structures.
If $S'\to S$ is a morphism there is a obvious isomorphism of CM elliptic curves over $\Gamma^*(S')$ equipped with $\Psi_{\Gamma^*(S')}$-structures \[\Gamma_\mathrm{CM}^*(E)\times_{\Gamma^*(S)}\Gamma^*(S')\overset{\sim}{\longrightarrow} \Gamma_\mathrm{CM}^*(E\times_S S').\]
\subsection{} Write $\Gamma_*(\mc{M}_\mathrm{CM})_{\Psi}$ for the fibred category over $\mathrm{IndAff}_{O_K}$ whose fibre over $S$ is the essential image of the functor $E/S\mapsto \Gamma^*_\mathrm{CM}(E)/\Gamma^*(S)$ from CM elliptic curves over $S$ to CM elliptic curves over $\Gamma^*(S)$ equipped with $\Psi_{\Gamma^*(S)}$-structures compatible with their $\underline{O_K}_{\Gamma^*(S)}$-module structure. The pull-back maps for $S'\to S$ are given by \[E/\Gamma^*(S)\mapsto E\times_{\Gamma^*(S)}\Gamma^*(S')/\Gamma^*(S').\]
\begin{rema} The symbol $\Gamma_*(\mc{M}_\mathrm{CM})_{\Psi}$ is only notation but it supposed to inspire the following interpretation. Assuming $\mc{M}_\mathrm{CM}$ did admit a $\Lambda$-structure, and so a fortiori a $\Psi$-structure, then we should have equivalences \[\mc{M}_\mathrm{CM}(S)=\mathrm{Hom}(S, \mc{M}_\mathrm{CM})\overset{\sim}{\longrightarrow} \mathrm{Hom}_{\mathrm{Spec}(O_K)}^{\Psi}(\Gamma^*(S), \mc{M}_\mathrm{CM})=\Gamma_*(\mc{M}_\mathrm{CM})_\Psi(S).\] Of course, we do not define a $\Psi$-structure on $\mc{M}_\mathrm{CM}$. Instead we opt to define which morphisms (equivalently CM elliptic curves $E/\Gamma^*(S)$) \[\Gamma^*(S)\stackrel{E}{\to} \mc{M}_\mathrm{CM}\] should be though of as (coming from) $\Psi$-morphisms. The following gives some justification to this.
\end{rema}
\begin{lemm}\label{lemm:canonical-ghost-equivalence} The functor \[\mc{M}_\mathrm{CM}\to \Gamma_*(\mc{M}_\mathrm{CM})_{\Psi}: E/S\mapsto \Gamma_\mathrm{CM}^*(E)/\Gamma^*(S)\] is an equivalence of stacks with quasi-inverse given by pull-back along the ghost component at $(1)$ \[E/\Gamma^*(S)\mapsto g_{(1)}^*(E)/S.\]
\end{lemm}
\begin{proof} The functor in question is clearly essentially surjective and faithful as composing it with \[E/\Gamma^*(S)\mapsto g_{(1)}^*(E)/S\] yields a functor isomorphic to the identity on $\mc{M}_\mathrm{CM}$. Now to see that it is an equivalence, and that $E/\Gamma^*(S)\mapsto g_{(1)}^*(E)/S$ is a quasi-inverse, we need only show that it is full.
So let $f: \Gamma^*_\mathrm{CM}(E)\to \Gamma^*_\mathrm{CM}(E')$ be a $\Psi_{\Gamma^*(S)}$-isomorphism and write $f$ as the sum of its ghost components \[f=\coprod_{\a\in \mathrm{Id}_{O_K}}g_\a^*(f).\] As $f$ is a $\Psi_{\Gamma^*(S)}$-homomorphism the diagram \[\xymatrix{E\ar[r]^{g_{(1)}^*(f)}\ar[d]_{i_\a} & E'\ar[d]^{i_\a}\\
E\otimes_{O_K}\a^{-1}\ar[r]^{g_{\a}^*(f)} & E'\otimes_{O_K}\a^{-1}}\] commutes for each $\a\in \mathrm{Id}_{O_K}$ by the definition of $\Gamma_\mathrm{CM}^*(E)$ and $\Gamma_\mathrm{CM}^*(E')$ and their $\Psi_{\Gamma^*(S)}$-structures. However, as $i_\a$ is epimorphism the only map $g_{\a}^*(f)$ for which this is possible is $g_{(1)}^*(f)\otimes_{O_K}\a^{-1}$. It follows that \[f=\coprod_{\a\in \mathrm{Id}_{O_K}} g_{(1)}^*(f)\otimes_{O_K}\a^{-1}=\Gamma^*_\mathrm{CM}(g_{(1)}^*(f))\] and we are done.
\end{proof}
\subsection{}\label{subsec:canonical-lambda-structures-witt-vectors} Let $E/W^*(S)$ be a CM elliptic curve equipped with a $\Lambda_{W^*(S)}$-structure compatible with its $\underline{O_K}_{W^*(S)}=W^*(\underline{O_K}_S)$-module structure. We say that the $\Lambda_{W^*(S)}$-structure on $E/W^*(S)$ is canonical if there exists a $\Psi_{\Gamma^*(S)}$-isomorphism \[E\times_{W^*(S)}\Gamma^*(S)\overset{\sim}{\longrightarrow} \Gamma_{\mathrm{CM}}^*(g_{(1)}^*(E))\] inducing the identity on $g_{(1)}^*(E)$ after pull-back along $g_{(1)}: S\to \Gamma^*(S)$. Such an isomorphism is unique by (\ref{lemm:canonical-ghost-equivalence}) and so, when it exists, we shall denote it by $\rho_{E/W^*(S)}$.
\begin{lemm}\label{lemm:properties-of-lambda-cm-elliptic-curves-over-W} Let $E/W^*(S)$ be a \textup{CM} elliptic curve.
\begin{enumerate}[label=\textup{(\roman*)}]
\item $E$ admits at most one canonical $\Lambda_{W^*(S)}$-structure.
\item If $S'\to S$ is a morphism of ind-affine schemes then, writing $E'=E\times_{W^*(S)}W^*(S')$, the $\Lambda_{W^*(S')}$-structure on $E'$ is canonical and \[\rho_{E/W^*(S)}\times_{\Gamma^*(S)}\Gamma^*(S') = \rho_{E'/W^*(S')}.\]
\item Let $(S_i\to S)_{i\in I}$ be an affine \'etale cover in $\mathrm{IndAff}_{O_K}$. If $E\times_{W^*(S)}W^*(S_i)$ admits a canonical $\Lambda_{W^*(S_i)}$-structure for each $i\in I$ then $E/W^*(S)$ admits a canonical $\Lambda_{W^*(S)}$-structure.
\end{enumerate}
\end{lemm}
\begin{proof} (i) Let $E/W^*(S)$ have a pair of canonical $\Lambda_{W^*(S)}$-structures and write $\rho_{E/W^*(S), 1}$ and $\rho_{E/W^*(S), 2}$ for the corresponding unique isomorphisms \[E\times_{W^*(S)}\Gamma^*(S)\overset{\sim}{\longrightarrow} \Gamma_\mathrm{CM}^*(g_{(1)}^*(E))\] inducing the identity on $g_{(1)}^*(E)$ after pull-back along $g_{(1)}: S\to \Gamma^*(S)$. The composition \[\rho_{E/W^*(S), 1}^{-1}\circ\rho_{E/W^*(S), 2}\] defines an $\Psi_{\Gamma^*(S)}$-automorphism of $\Gamma_\mathrm{CM}^*(g_{(1)}^*(E))$ which is the identity on $g_{(1)}^*(E)$ after pull-back along $g_{(1)}$ and so is the identity itself by (\ref{lemm:canonical-ghost-equivalence}).
It follows that the two $\Psi_{\Gamma^*(S)}$-structures on each $E\times_{W^*(S)}\Gamma^*(S)$, induced by the two $\Lambda_{W^*(S)}$-structures on $E$ and $E'$, are equal. Writing \[\psi_{E/W^*(S), 1}^{\ideal{p}}, \psi_{E/W^*(S), 2}^{\ideal{p}}: E\to \psi^{\ideal{p}*}(E)\] for the relative Frobenius lifts at $\ideal{p}$ corresponding to the two $\Lambda_{W^*(S)}$-structures, we have shown that \[\psi^{\ideal{p}}_{E/W^*(S), 1}\times_{W^*(S)}\Gamma^*(S)=\psi^{\ideal{p}}_{E/W^*(S), 2}\times_{W^*(S)}\Gamma^*(S).\] As $\Gamma^*(S)\to W^*(S)$ is surjective on geometric points (\ref{prop:ghost-map-surj-geom-points}), it follows by rigidity that \[\psi_{E/W^*(S), 1}^{\ideal{p}}=\psi_{E/W^*(S), 2}^{\ideal{p}}.\] Therefore, the two $\Psi_{W^*(S)}$-structures on $E/W^*(S)$ induced by the two $\Lambda_{W^*(S)}$-structures are equal and by (\ref{rema:lambda-structures-on-abelian-varieties-are-determined-by-psi-structures}) it follows that the two $\Lambda_{W^*(S)}$-structures themselves are equal.
(ii) The CM elliptic curve $E':=E\times_{W^*(S)} W^*(S')$ has a natural $\Lambda_{W^*(S')}$-structure and the pull-back of $\rho_{E/W^*(S)}$ along $\Gamma^*(S')\to \Gamma^*(S)$ defines a $\Psi_{\Gamma^*(S')}$-isomorphism \[E'\times_{W^*(S')}\Gamma^*(S')=E\times_{W^*(S)} \Gamma^*(S') \stackrel{\rho_{E/W^*(S)\times_{\Gamma^*(S)}\Gamma^*(S')}}{\longrightarrow} \Gamma_\mathrm{CM}^*(g_{(1)}^*(E))\times_{\Gamma^*(S)}\Gamma^*(S')=\Gamma_\mathrm{CM}^*(g_{(1)}^*(E'))\] inducing the identity after pull-back along $g_{(1)}$. It follows by uniqueness of such an isomorphism that \[\rho_{E/W^*(S)}\times_{\Gamma^*(S)}\Gamma^*(S') = \rho_{E'/W^*(S')}.\]
(iii) We will first show that $E/W^*(S)$ admits a unique $\Lambda_{W^*(S)}$-structure inducing the canonical $\Lambda_{W^*(S_i)}$-structures on $E\times_{W^*(S)}W^*(S_i)$ and then show that this $\Lambda_{W^*(S)}$-structure is canonical.
The family $(W^*(S_i)\to W^*(S))_{i\in I}$ is a cover in $\mathrm{Sh}_{\Lambda}^\textup{\'et}$ and we have \[W^*(S_{ij}):=W^*(S_i\times_S S_j)\overset{\sim}{\longrightarrow} W^*(S_i)\times_{W^*(S)} W^*(S_j)\] by (\ref{prop:w-and-etale-base-change}). By (ii) above, the two $\Lambda_{W^*(S_{ij})}$-structures on \[E\times_{W^*(S)}W^*(S_{ij})\] induced by pull-back are canonical and by (i) they are equal. By (\ref{subsec:sheaves-of-lambda-structures}), this defines an element of the equaliser \[\xymatrix{\Lambda_{E/W^*(S)}(W^*(S))\ar[r] & \Lambda_{E/W^*(S)}(W^*(S_i))\ar@<.5ex>[r]\ar@<-.5ex>[r] &\prod\limits_{i, j\in I}\Lambda_{E/W^*(S)}(W^*(S_{ij}))}\] or in other words, there is a unique $\Lambda_{W^*(S)}$-structure on $E/W^*(S)$ inducing the canonical $\Lambda_{W^*(S_i)}$-structures on each $E\times_{W^*(S)}W^*(S_i)$.
The uniqueness of the isomorphisms $\rho_{E\times_{W^*(S)}W^*(S_i)/W^*(S_i)}$ and their compatibility with pull-backs (this is (ii) above) show that they descend to an $\Psi_{\Gamma^*(S)}$-isomorphism \[\rho_{E/W^*(S)}: E\times_{W^*(S)}\Gamma^*(S)\overset{\sim}{\longrightarrow} \Gamma^*_\mathrm{CM}(g_{(1)}^*(E))\] inducing the identity after pull-back along $g_{(1)}$ so that the $\Lambda_{W^*(S)}$-structure on $E/W^*(S)$ is canonical.
\end{proof}
\subsection{} Using (\ref{lemm:properties-of-lambda-cm-elliptic-curves-over-W}) we may define a fibred category $W_*(\mc{M}_{\mathrm{CM}})_\Lambda$ over $\mathrm{IndAff}_{O_K}$ by setting the fibre over $S$ to be the category of CM elliptic curves $E/W^*(S)$ equipped with a canonical $\Lambda_{W^*(S)}$-structure and whose pull-back maps for $S'\to S$ are given by \[E/W^*(S)\mapsto E\times_{W^*(S)}W^*(S').\]
\begin{rema} As with $\Gamma_*(\mc{M}_\mathrm{CM})_{\Psi}$ the symbol $W_*(\mc{M}_{\mathrm{CM}})_\Lambda$ is supposed to inspire in the reader the idea that $\mc{M}_\mathrm{CM}$ admits some $\Lambda$-structure and that we have \[\mc{M}_\mathrm{CM}(S)=\mathrm{Hom}_{\mathrm{Spec}(O_K)}(S, \mc{M}_\mathrm{CM})\overset{\sim}{\longrightarrow}\mathrm{Hom}_{\mathrm{Spec}(O_K)}^{\Lambda}(W^*(S), \mc{M}_\mathrm{CM})=W_*(\mc{M}_\mathrm{CM})_\Lambda(S).\]
\end{rema}
\begin{lemm} The fibred category $W_*(\mc{M}_{\mathrm{CM}})_\Lambda$ over $\mathrm{IndAff}_{O_K}$ is a stack for the affine \'etale topology.
\end{lemm}
\begin{proof} Let $(S_i\to S)_{i\in I}$ be an affine \'etale cover and let $(E_i/W^*(S_i))_{i\in I}$ be a collection of objects of $W_*(\mc{M}_{CM})_\Lambda$ equipped with descent data relative to the cover $(S_i\to S)_{i\in I}$. By (ii) of (\ref{prop:w-and-etale-base-change}) we have \[W^*(S_{ij}):=W^*(S_i\times_S S_j)\overset{\sim}{\longrightarrow} W^*(S_i)\times_{W^*(S)} W^*(S_j)\] an so we may view the objects $(E_i/W^*(S_i))_{i\in I}$ of $W_*(\mc{M}_\mathrm{CM})_\Lambda$ equipped with their descent data relative to the affine \'etale cover $(S_i\to S)_{i\in I}$ as objects of $\mc{M}_{\mathrm{CM}}$ equipped with descent data relative to the fpqc cover $(W^*(S_i)\to W^*(S))_{i\in I}$. As $\mc{M}_\mathrm{CM}$ is a stack over $\mathrm{Sh}_{O_K}$ (\ref{prop:moduli-ell-stack}), the family $(E_i/W^*(S_i))_{i\in I}$ descends to a CM elliptic curve $E/W^*(S)$ unique upto compatible isomorphisms $E\times_{W^*(S)}W^*(S_i)\overset{\sim}{\longrightarrow} E_i$. It remains to see that $E/W^*(S)$ admits a canonical $\Lambda_{W^*(S)}$-structure and this follows from (iii) of (\ref{lemm:properties-of-lambda-cm-elliptic-curves-over-W}).
In much the same way, using (\ref{prop:w-and-etale-base-change}) and (\ref{prop:hom-lambda-sheaves-are-sheaves}), $\Lambda$-isomorphisms of CM elliptic curves with canonical $\Lambda$-structures also satisfy descent for the affine \'etale topology and we find that $W_*(\mc{M}_{\mathrm{CM}})_\Lambda$ is a stack over $\mathrm{IndAff}_{O_K}$.
\end{proof}
\begin{theo} The functor induced by base change along the ghost component at $(1)$ \[W_*(\mc{M}_{\mathrm{CM}})_\Lambda\to \mc{M}_{\mathrm{CM}}: E/W^*(S)\mapsto g_{(1)}^*(E)/S\] is an equivalence of stacks over $\mathrm{IndAff}_{O_K}$ for the affine \'etale topology.
\end{theo}
\begin{proof} The functor in question factors as \[W_*(\mc{M}_{\mathrm{CM}})_\Lambda\to \Gamma_*(\mc{M}_{CM})_\Psi \to \mc{M}_{\mathrm{CM}}\] where the first functor is $E/W^*(S)\mapsto E\times_{W^*(S)}\Gamma^*(S)/\Gamma^*(S)$ and the second is pull-back along $g_{(1)}: S\to \Gamma^*(S)$. By (\ref{lemm:canonical-ghost-equivalence}) the second functor is an equivalence and, as $\Gamma^*(S)\to W^*(S)$ is surjective on geometric points, the first functor is faithful by rigidity. Therefore, \[W_*(\mc{M}_{\mathrm{CM}})_\Lambda\to \mc{M}_{\mathrm{CM}}\] is faithful.
Now fix a pair $E, E'/W^*(S)$ of CM elliptic curves equipped with canonical $\Lambda_{W^*(S)}$-structures and let $f: E\times_{W^*(S)}\Gamma^*(S)\to E'\times_{W^*(S)}\Gamma^*(S)$ be a $\Psi_{\Gamma^*(S)}$-isomorphism. Let $\ideal{p}$ be a prime ideal, $\a$ any ideal and $n\geq 0$ an integer. Consider the diagram:
\begin{equation}\label{dia:lambda-frob-psi}
\begin{gathered}
\xymatrix{\overline{g_\a^*(E)}\ar[rr]^{\overline{g_\a^*(f)}}\ar[d]_-{\overline{g_\a^*(\psi_E^{\ideal{p}^n})}} && \overline{g_\a^*(E')}\ar[d]^-{\overline{(g_\a^*(\psi^{\ideal{p}^n}_{E'})}}\\
\overline{g_{\a\ideal{p}^n}^*(E)}\ar[rr]^{\overline{g_{\a\ideal{p}^n}^*(f)}}\ar[d]_-{\wr} && \overline{g_{\a\ideal{p}^n}^*(E')}\ar[d]^-{\wr}\\
\mathrm{Fr}_{\overline{S}}^{N\ideal{p}^n*}(\overline{g_\a^*(E)})\ar[rr]^{\mathrm{Fr}_{\overline{S}}^{N\ideal{p}^n*}(\overline{g_\a^*(f)})} && \mathrm{Fr}_{\overline{S}}^{N\ideal{p}^n*}(\overline{g_\a^*(E')}),}
\end{gathered}\end{equation}
where the bar denotes pull-back along $\overline{S}=S\times_{\mathrm{Spec}(O_K)}\mathrm{Spec}(\mathbf{F}_\ideal{p})\to S$ and the bottom vertical isomorphisms are the unique such making the vertical compositions equal to the $N\ideal{p}^n$-power relative Frobenius morphisms of $\overline{g_\a^*(E)}$ and $\overline{g_\a^*(E')}$ respectively (such isomorphisms exist as $\psi_E^{\ideal{p}^n}: E\to E$ and $\psi_{E'}^{\ideal{p}^n}: E'\to E'$ lift the $N\ideal{p}^n$-power Frobenius). As $f$ is a $\Psi_{\Gamma^*(S)}$-morphism, the top square of (\ref{dia:lambda-frob-psi}) commutes and, by functoriality of the $N\ideal{p}^n$-power relative Frobenius, the outer square commutes. As $\overline{g_{\a}^*(\psi_E^{\ideal{p}^n})}$ is an epimorphism, it follows that the bottom square of (\ref{dia:lambda-frob-psi}) also commutes. We will show that this implies that the functor \[W_*(\mc{M}_{\mathrm{CM}})_\Lambda\to \mc{M}_{\mathrm{CM}}\] is full.
For the functor to be full, it is enough to show that the (injective) map \[\mathrm{Isom}_{W^*(S)}^{O_K, \Lambda}(E, E')\to \mathrm{Isom}_{\Gamma^*(S)}^{O_K, \Psi}(E_{\Gamma^*(S)}, E'_{\Gamma^*(S)})\] is surjective (where the super script $\Lambda$ and $\Psi$ denote $\Lambda$- and $\Psi$-morphisms). Now to show this, it is enough to show that each $\Psi_{\Gamma^*(S)}$-isomorphism \[f: E_{\Gamma^*(S)}\to E'_{\Gamma^*(S)}\] is obtained via pull-back from an isomorphism $E\to E'$ over $W^*(S)$ as by (\ref{prop:lambda-and-psi-homomorphisms-of-abelian-varieties}) it will follow that any such isomorphism $E\to E'$ is also a $\Lambda_{W^*(S)}$-isomorphism.
Applying (\ref{prop:ghost-covers-etale}) and (\ref{rema:ghost-fibre-product}) to the finite \'etale $W^*(S)$-sheaf \[\underline{\mathrm{Isom}}_{W^*(S)}^{O_K}(E, E'),\] we see that $f$ comes from a morphism $E\to E'$ over $W^*(S)$ if and only if, for each prime $\ideal{p}$, each ideal $\a$ and each $n\geq 0$, the two pull-backs of $f$ to $\overline{S}:=S\times_{\mathrm{Spec}(O_K)}\mathrm{Spec}(\mathbf{F}_\ideal{p})$ along the maps \[\overline{S}\subset S \stackrel{g_{\a\ideal{p}^n}}{\longrightarrow} \Gamma^*(S) \quad \text{and} \quad \overline{S} \stackrel{\mathrm{Fr}_{\overline{S}}^{N\ideal{p}^n}}{\longrightarrow} \overline{S}\subset S \stackrel{g_\a}{\to} \Gamma^*(S)\] of (\ref{rema:ghost-fibre-product}) are equal. This is precisely the commutativity of the bottom square of the diagram (\ref{dia:lambda-frob-psi}) and therefore \[W_*(\mc{M}_{\mathrm{CM}})_\Lambda\to \mc{M}_{\mathrm{CM}}\] is full.
Finally, for the functor in question to be an equivalence it is enough (as both categories are stacks and we have already shown that it is fully faithful) to show that for each ind-affine scheme $S$ and each CM elliptic curve $E/S$ there is an affine \'etale cover $(S_i\to S)_{i\in I}$ such that $E\times_{S} S_i$ is in the essential image of \[W_*(\mc{M}_{\mathrm{CM}})_\Lambda\to \mc{M}_{\mathrm{CM}}.\] By (\ref{coro:level-isom-cover}) the family of $S$-sheaves \[(\underline{\mathrm{Isom}}_S^{O_K}(E[\ideal{f}], \underline{O_K/\ideal{f}}_S)\to S)_{\ideal{f}}\] indexed by integral ideals $\ideal{f}$ which separate units forms an affine \'etale cover of $S$. We may then base change to any element of this cover and assume that $E/S$ admits a level-$\ideal{f}$ structure for some integral ideal $\ideal{f}$ which separates units, and then we may assume that $E/S=E^{(\ideal{f})}/M_{\mathrm{CM}}^{(\ideal{f})}$. This case is (\ref{lemm:universal-cm-have-lambda}) below.
\end{proof}
\begin{lemm}\label{lemm:universal-cm-have-lambda} Let $\ideal{f}\in \mathrm{Id}_{O_K}$ separate units. The universal \textup{CM} elliptic curve with level $\ideal{f}$-structure $E^{(\ideal{f})}\to M_\mathrm{CM}^{(\ideal{f})}$ is in the essential image of the functor \[W_*(\mc{M}_{\mathrm{CM}})_\Lambda\to \mc{M}_\mathrm{CM}.\]
\end{lemm}
\begin{proof} Write $M=M_\mathrm{CM}^{(\ideal{f})}$, $E=E^{(\ideal{f})}$ and $P=\mathrm{Id}_{O_K}^{(\ideal{f})}$ and let $P'\subset \mathrm{Id}_{O_K}$ be the sub-monoid generated by the prime ideals dividing $\ideal{f}$ so that $P\cap P'=\{O_K\}$ and $\mathrm{Id}_{O_K}=P\cdot P'$.
Then the finite \'etale $\mathrm{Spec}(O_K[\ideal{f}^{-1}])$-scheme $M$ admits a unique $\Lambda_{P}$-structure by (\ref{prop:lambda-structures-etale-schemes-number-fields}). By the definition of the automorphisms $\sigma_\ideal{p}=[\ideal{p}^{-1}]_\ideal{f}:M\to M$ there exists a unique $\ideal{f}$-isomorphism \[E\otimes_{O_K}\ideal{p}^{-1}\overset{\sim}{\longrightarrow} \sigma_\ideal{p}^*(E)\] whose pull-back along $M\times \mathrm{Spec}(\mathbf{F}_\ideal{p})\to M$ is the isomorphism $\nu_{\ideal{p}}$ of (\ref{coro:tensor-p-equals-frobenius-with-level}). It follows that setting $\psi_{E/M}^{\ideal{p}}: E\to \sigma_\ideal{p}^*(E)$ to be the composition \[E\stackrel{i_\ideal{p}}{\to} E\otimes_{O_K}\ideal{p}^{-1} \overset{\sim}{\longrightarrow} \sigma_\ideal{p}^*(E)\] defines a lift of the relative $N\ideal{p}$-power Frobenius on $E\to M$. Note that $\ker(\psi^\ideal{p}_{E/M})=E[\ideal{p}]$. Let $\l\in P$ be another prime ideal and consider the diagram
\begin{equation}\label{dia:lambda-on-level}
\begin{gathered}\xymatrix{E\ar[r]^{\psi_{E/M}^{\ideal{p}}}\ar[d]_{\psi_{E/M}^{\l}} & \sigma_\ideal{p}^*(E)\ar[d]^{\sigma_{\ideal{p}}^*(\psi_{E/M}^{\l})}\\
\sigma_{\l}^*(E)\ar[r]^{\sigma_\l^*(\psi_{E/M}^{\ideal{p}})} & \sigma_{\ideal{p}\l}^*(E).}\end{gathered}\end{equation} The kernel of the compositions along the top and right, and left and bottom of (\ref{dia:lambda-on-level}) are both equal to $E[\ideal{p}\l]$ so that, as $M$ is connected, these compositions differ by scaling by some $\epsilon\in O_K^\times$. As the $\ideal{f}$-torsion $E[\ideal{f}]$ is constant over $M$, and $M\overset{\sim}{\longrightarrow} \mathrm{Spec}(O_{K(\ideal{f})}[\ideal{f}^{-1}])$, it admits a unique $\Lambda_{P, M}$-structure by (\ref{prop:lambda-structures-etale-schemes-number-fields}). The uniqueness of the $\Lambda_{P, M}$-structure on $E[\ideal{f}]$ now implies that the diagram (\ref{dia:lambda-on-level}) commutes when restricted to the $\ideal{f}$-torsion and therefore $\epsilon\in O_{K}^{\times, \ideal{f}}$. However, $\ideal{f}$ separates units, so that $\epsilon\in O_{K}^{\times, \ideal{f}}=\{1\}$ and (\ref{dia:lambda-on-level}) commutes. As $E$ is flat over $\mathrm{Spec}(O_K)$ this defines a $\Lambda_{P, M}$-structure on $E$ by (\ref{prop:lambda-structures-frobenius-lifts-sheaves}).
Pulling-back $E\to M$ along the map $\Gamma^*_{P}(M)\to M$ corresponding to the $\Psi_P$-structure on $M$, we obtain an isomorphism of CM elliptic curves \[E\times_M \Gamma^*_{P}(M)=\coprod_{\a\in P} \sigma_\a^*(E)\overset{\sim}{\longrightarrow} \coprod_{\a\in P} E\otimes_{O_K}\a^{-1} \overset{\sim}{\longrightarrow} \Gamma^*_\mathrm{CM}(E)\times_{\Gamma^*(M)}\Gamma^*_{P}(M)\] compatible with the $\Psi_{P, \Gamma^*_{P}(M)}$-structures. We then get a $\Psi_{\Gamma^*(M)}$-isomorphism \[\Gamma^*_\mathrm{CM}(E)\overset{\sim}{\longrightarrow} \coprod_{\b\in P'} (E\times_{M}\Gamma^*_{P}(M))\otimes_{O_K}\b^{-1}\] inducing the identity after pull-back along $g_{(1)}$ and where on the right hand side the relative Frobenius lifts for $\ideal{p}\in P'$ are defined in the obvious way.
Now set $\widetilde{E}_P:=E\times_{M}W_{P, M}^*(M)$ where we have base changed along the map $\mu_{M}: W_{P}^*(M)\to M$ defining the $\Lambda_P$-structure on $M$. Then $\widetilde{E}_P$ has a $\Lambda_{P, W^*_P(S)}$-structure by virtue of the facts that $E/M$ has a $\Lambda_{P, M}$-structure and $W^*_P(M)\to M$ is a $\Lambda_P$-morphism. We also have $\Psi_{\Gamma^*_P(S)}$-isomorphisms \begin{equation}\label{eqn:level-is-canonical}\widetilde{E}_P\times_{W_{P}^*(S)}\Gamma_{P}^*(S)\overset{\sim}{\longrightarrow} E\times_M \Gamma_P^*(S)\overset{\sim}{\longrightarrow} \Gamma^*_\mathrm{CM}(E)\times_{\Gamma^*(M)}\Gamma_P^*(M)\end{equation} inducing the identity after pull-back along the first ghost component.
Finally, setting \[\widetilde{E}:=\coprod_{\b\in P'}\widetilde{E}_P\otimes_{O_K}\b^{-1}\to \coprod_{\b\in P'}W^*_{P}(M)=W^*(M)\] and defining relative Frobenius lifts on $\widetilde{E}$ for the primes $\ideal{p}\in P'$ in the obvious way equips $\widetilde{E}$ with a $\Lambda_{W^*(S)}$-structure. Again by construction we have a $\Psi_{\Gamma^*(M)}$-isomorphism of CM elliptic curves \[\widetilde{E}\times_{W^*(M)}\Gamma^*(M)\overset{\sim}{\longrightarrow} \Gamma^*_\mathrm{CM}(E)\] inducing the identity after pull-back along the ghost component at $(1)$. Therefore the $\Lambda_{W^*(M)}$-structure so defined on $\widetilde{E}/W^*(M)$ is canonical and $E\to M$ is in the essential image of the functor \[W_*(\mc{M}_{\mathrm{CM}})_\Lambda\to \mc{M}_\mathrm{CM}.\]
\end{proof}
\subsection{} We now fix for all time an inverse equivalence \[\mc{M}_\mathrm{CM}\to W_*(\mc{M}_{\mathrm{CM}})_{\Lambda}: E/S\mapsto W_\mathrm{CM}^*(E)/W^*(S)\] to the functor $W_*(\mc{M}_{\mathrm{CM},\Lambda})\to \mc{M}_\mathrm{CM}$ and call $W_\mathrm{CM}^*(E)/W^*(S)$, equipped with its canonical $\Lambda_{W^*(S)}$-structure, the canonical lift of $E/S$.
\begin{rema}\label{rema:level-structures-canonical} We note for future reference that the proof of (\ref{lemm:universal-cm-have-lambda}) shows that the CM elliptic curve $E^{(\ideal{f})}/M_\mathrm{CM}^{(\ideal{f})}$ admits a $\Lambda_{P, M_\mathrm{CM}^{(\ideal{f})}}$-structure and that this $\Lambda_{P, M_\mathrm{CM}^{(\ideal{f})}}$-structure has the property that there exists a $\Lambda_{P, W^*_P(M_\mathrm{CM}^{(\ideal{f})})}$-isomorphism \[W^*_{\mathrm{CM}}(E^{(\ideal{f})})\times_{W^*(M_\mathrm{CM}^{(\ideal{f})})}W^*_P(M_\mathrm{CM}^{(\ideal{f})})\overset{\sim}{\longrightarrow} E^{(\ideal{f})}\times_{M_\mathrm{CM}^{(\ideal{f})}} W_P^*(M_\mathrm{CM}^{(\ideal{f})})=\mu_{M_\mathrm{CM}^{(\ideal{f})}}^*(E)\] inducing the identity on $E$ after pull-back to the ghost component at $(1)$ (this is (\ref{eqn:level-is-canonical})).
\end{rema}
\subsection{} By virtue of the definition of $W_\mathrm{CM}^*(E)/W^*(S)$ we have a canonical isomorphism \[E\overset{\sim}{\longrightarrow} g_{(1)}^*(W_\mathrm{CM}^*(E)).\] That is, the composition \[\mc{M}_\mathrm{CM}(S)\stackrel{W^*_\mathrm{CM}}{\to}\mc{M}_\mathrm{CM}(W^*(S))\stackrel{g^*_{(1)}}{\to}\mc{M}_\mathrm{CM}(S)\] is canonically isomorphic to the identity. We will now show that the two compositions \[\xymatrix{\mc{M}_\mathrm{CM}(S)\ar[r]^-{W_\mathrm{CM}^*} & \mc{M}_\mathrm{CM}(W^*(S))\ar@<.5ex>[r]^-{W_\mathrm{CM}^*}\ar@<-.5ex>[r]_-{\mu_{W^*(S)}^*} & \mc{M}_\mathrm{CM}(W^*(W^*(S)))}\] are also canonically isomorphic.
\begin{prop}\label{prop:canonical-lifts-have-canonical-lambda-structures} Let $S$ be an ind-affine scheme and let $E/S$ be a \textup{CM} elliptic curve. Then there is a unique isomorphism, compatible with the $\Lambda_{W^*(W^*(S))}$-structures, \[\mu_{W^*(S)}^*(W^*_\mathrm{CM}(E))\overset{\sim}{\longrightarrow} W^*_{CM}(W^*_\mathrm{CM}(E))\] inducing the identity after pull-back along $g_{(1)}: W^*(S)\to W^*(W^*(S))$.
\end{prop}
\begin{proof} Both $\mu^*{W^*(S)}(W^*_\mathrm{CM}(E))$ and $W^*_\mathrm{CM}(W^*_\mathrm{CM}(E))$ are equipped with natural $\Lambda_{W^*(W^*(S))}$-structures and the pull-backs of each along the ghost component at $(1)$ are equal to $W^*_\mathrm{CM}(E)$. Therefore, it is enough by (\ref{lemm:universal-cm-have-lambda}) to show that the $\Lambda_{W^*(W^*(S))}$-structures on $\mu_{W^*(S)}^*(W^*_\mathrm{CM}(E))$ are $W^*_{CM}(W^*_\mathrm{CM}(E))$ are canonical.
For $W^*_\mathrm{CM}(W^*_\mathrm{CM}(E))$ this is true by definition. For $\mu_{W^*(S)}^*(W^*_\mathrm{CM}(E))$ we have the sequence of $\Psi_{\Gamma^*(W^*(S))}$-isomorphisms \begin{eqnarray*} \mu_{W^*(S)}^*(W_\mathrm{CM}^*(E))\times_{W^*(W^*(S))}\Gamma^*(W^*(S)) & = & (W^*_{\mathrm{CM}}(E)\times_{W^*(S)}W^*(W^*(S)))\times_{W^*(W^*(S))}\Gamma^*(W^*(S))\\
& = & W^*_{\mathrm{CM}}(E)\times_{W^*(S)}\Gamma^*(W^*(S))\\
&\overset{\sim}{\longrightarrow}& \coprod_{\a}\psi^{\a*}(W_\mathrm{CM}^*(E))\\&\overset{\sim}{\longrightarrow}& \coprod_{\a} W_\mathrm{CM}^*(E)\otimes_{O_K}\a^{-1}\\&\overset{\sim}{\longrightarrow} & \Gamma_\mathrm{CM}^*(W_\mathrm{CM}^*(E))\\
&=&\Gamma_\mathrm{CM}^*(g_{(1)}^*(\mu_{W^*(S)}^*(W_\mathrm{CM}^*(E)))).
\end{eqnarray*} The resulting $\Psi_{\Gamma^*(W^*(S))}$-isomorphism \[\mu_{W^*(S)}^*(W_\mathrm{CM}^*(E))\times_{W^*(W^*(S))}\Gamma^*(W^*(S))\overset{\sim}{\longrightarrow} \Gamma^*_\mathrm{CM}(g_{(1)}^*(\mu_S^*(W_\mathrm{CM}^*(E))))\] induces the identity after pull-back along the ghost component at $(1)$ and this is precisely the definition of a canonical $\Lambda_{W^*(W^*(S))}$-structure.
\end{proof}
\subsection{}\label{subsec:p-canonical-lambda-structure} We now define what it means for a CM elliptic curve over an arbitrary $\Lambda$-ind-affine scheme (not just those of the form $W^*(S)$) to have a canonical $\Lambda$-structure.
It will be convenient later to allow ourselves the flexibility of working with $\Lambda$-structures relative to a sub-monoid $P\subset \mathrm{Id}_{O_K}$ generated by some set of prime ideals and so we fix such a $P$.
Let $S$ be an $\Lambda_P$-ind-affine scheme and let $E/S$ be a CM elliptic curve equipped with a $\Lambda_{P, S}$-structure (again compatible with its $\underline{O_K}_S$-module structure). We say that the $\Lambda_P$-structure on $E/S$ is canonical if there is a $\Lambda_{P, W^*_P(S)}$-isomorphism \[\lambda_{E/S}: E\times_{S}W^*_P(S) \overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E)\times_{W^*(S)}W_P^*(S)\] inducing the identity on the ghost components at $(1)$. There is at most one such isomorphism $\lambda_{E/S}$ satisfying this condition and so we are safe to label it. Indeed, any two differ by a $\Lambda_{P, W_P^*(S)}$-automorphism of $W_\mathrm{CM}^*(E)\times_{W^*(E)}W_P^*(E)$ inducing the identity on the ghost component at $(1)$ and all such automorphisms are the identity as this can be checked after pull-back along $\Gamma_P^*(S)\to W_P^*(S)$, and arguing as in the proof of (\ref{lemm:canonical-ghost-equivalence}) one sees that a $\Psi_{P, \Gamma_P^*(S)}$-automorphism of $\Gamma_\mathrm{CM}^*(E)\times_{\Gamma^*(S)}\Gamma_P^*(S)$ is equal to the identity if and only if i
t is after pull-back to the ghost component at (1).
\begin{rema} When $P=\mathrm{Id}_{O_K}$ and $E/W^*(S)$ is a CM elliptic curve equipped with a $\Lambda_{W^*(S)}$-structure then it follows from (\ref{prop:canonical-lifts-have-canonical-lambda-structures}) that this $\Lambda_{W^*(S)}$-structure is canonical in the sense of (\ref{subsec:p-canonical-lambda-structure}) if and only if there exists a $\Lambda_{W^*(S)}$-isomorphism \[f: E\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(g_{(1)}^*(E))\] inducing the identity after pull-back to the ghost component at $(1)$, i.e.\ $E/W^*(S)$ is canonical in the sense of (\ref{subsec:canonical-lambda-structures-witt-vectors}).
Indeed, if there exists such an isomorphism $f: E\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(g_{(1)}^*(E))$ then (\ref{prop:canonical-lifts-have-canonical-lambda-structures}) gives an isomorphism $\lambda_{E/W^*(S)}$ via the compositions \begin{eqnarray*} E\times_{W^*(S)}W^*(W^*(S))&\stackrel{f\times_{W^*(S)}W^*(W^*(S))}{\longrightarrow} & W_\mathrm{CM}^*(g_{(1)}^*(E))\times_{W^*(S)}W^*(W^*(S))\\
&\stackrel{\textup(\ref{prop:canonical-lifts-have-canonical-lambda-structures})}{\longrightarrow} & W_\mathrm{CM}^*(W_\mathrm{CM}^*(g_{(1)}^*(E)))\\
&\stackrel{W_\mathrm{CM}^*(f^{-1})}{\longrightarrow} & W_\mathrm{CM}^*(E).\end{eqnarray*} Conversely, the pull-back of an isomorphism \[\lambda_{E/W^*(S)}: E\times_{W^*(S)}W^*(W^*(S))\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E)\] along $\widetilde{g}=W^*(g_{(1)}): W^*(S)\to W^*(W^*(S))$ yields a $\Lambda_{W^*(W^*(S))}$-isomorphism $f: E\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(g_{(1)}^*(E))$ via \[E\overset{\sim}{\longrightarrow} \widetilde{g}^*(E\times_{W^*(S)}W^*(W^*(S)))\stackrel{\widetilde{g}^*(\lambda_{E/W^*(S)})}{\to}\widetilde{g}^*(W_\mathrm{CM}^*(E)) \overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(g_{(1)}^*(E)).\] Hence the two notions of canonical $\Lambda_{W^*(S)}$-structure on a CM elliptic curve $E/W^*(S)$ defined in (\ref{subsec:canonical-lambda-structures-witt-vectors}) and (\ref{subsec:p-canonical-lambda-structure}) coincide.
\end{rema}
\begin{prop}\label{prop:canonical-lambda-p-structures} Let $S$ be an $\Lambda_{P}$-ind-affine-scheme and $E/S$ a \textup{CM} elliptic curve. Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $E/S$ admits at most one canonical $\Lambda_{P, S}$-structure.
\item If $S'\to S$ is a morphism of $\Lambda_P$-ind-affine schemes, the $\Lambda_{P, S'}$-structure on $E\times_S S'$ is canonical and \[\lambda_{E/S}\times_{W^*_P(S)}W^*_P(S') = \lambda_{E\times_S S'/S'}.\]
\item Let $(S_i\to S)_{i\in I}$ be a $\textup{\'et}$-cover of $\Lambda_{P}$-ind-affine schemes. If $E\times_S S'$ admits a canonical $\Lambda_{P, S_i}$-structure for each $i\in I$ then $E/S$ admits a canonical $\Lambda_{P, S}$-structure.
\end{enumerate}
\end{prop}
\begin{proof} (i) Let $E/S$ admit two canonical $\Lambda_{P, S}$-structures with corresponding isomorphisms \[\lambda_{E/S}, \lambda_{E/S}': E\times_{S}W_P^*(S)\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E)\times_{W^*(S)} W_P^*(S).\] The difference \[\lambda_{E/S}'\circ\lambda_{E/S}^{-1}: W_\mathrm{CM}^*(E)\times_{W^*(S)} W_P^*(S)\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E)\times_{W^*(S)} W_P^*(S)\] defines $\Lambda_{P, W^*_P(S)}$-automorphism of $W_\mathrm{CM}^*(E)\times_{W^*(S)} W_P^*(S)$ which is the identity on the first ghost component. Base changing along $\Gamma_P^*(S)\to W_P^*(S)$ and arguing again as in the proof of (\ref{lemm:canonical-ghost-equivalence}) we find that such an automorphism must be the identity itself and so $\lambda_{E/S}=\lambda_{E/S}'$.
It follows now that the two $\Lambda_{P, W^*_P(S)}$-structures on $E\times_{S}W_P^*(S)$ coincide so that as $W_P^*(S)\to S$ is an epimorphism and $\Lambda_{P}$-structures descend
(\ref{prop:lambda-structures-are-sheaves}) the two $\Lambda_{P, S}$-structures on $E/S$ coincide.
(ii) This follows from the uniqueness of the isomorphisms $\lambda_{E/S}$.
(iii) As canonical $\Lambda_{P}$-structures are unique and compatible with pull-back (this is (ii) above) if $E\times_S S_i$ admits a canonical $\Lambda_{P, S_i}$-structure for each $i\in I$ it follows that $E$ admits a $\Lambda_{P, S}$-structure. As the isomorphisms making a $\Lambda_{P}$-structure canonical are unique and compatible with pull back (again this is (ii)) the isomorphisms $\lambda_{E\times_S S_i/S_i}$ descend to an isomorphism \[\lambda_{E/S}: E\times_S W_P^*(S)\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E)\times_{W^*(S)}W_P^*(S)\] inducing the identity after base change long the ghost component at $(1)$ and so the $\Lambda_{P, S}$-structure on $E/S$ canonical.
\end{proof}
\begin{prop}\label{prop:canonical-lambda-structures-and-level-structures} Let $\ideal{f}\in \mathrm{Id}_{O_K}$ be an ideal which separates units. Then the unique $\Lambda_{P, M_{\mathrm{CM}}^{(\ideal{f})}}$-structure on the universal \textup{CM} elliptic curve with level-$\ideal{f}$ structure $E^{(\ideal{f})}\to M^{(\ideal{f})}_\mathrm{CM}$ is canonical.
\end{prop}
\begin{proof} This is the content of (\ref{rema:level-structures-canonical}).
\end{proof}
\section{CM elliptic curves of Shimura type}\label{sec:shimura-curves}
The purpose of this section is to explain the relationship between $\Lambda$-structures and CM elliptic curves of Shimura type. A CM elliptic curve of Shimura type is a CM elliptic curve $E/\mathrm{Spec}(L)$ where $L$ is an abelian extension of $K$ with the property that the extension $L(E[\mathrm{tors}])/K$ is an abelian extension of $K$ (it is always an abelian extension of $L$).
This class of CM elliptic curves was introduced by Shimura (see Theorem 7.44 of \cite{Shimura94}). By virtue of their definition, CM elliptic curves of Shimura type have much simpler arithmetic than arbitrary CM elliptic curves (and of course elliptic curves in general). In the special case where $K$ has class number one, every CM elliptic curve $E/\mathrm{Spec}(K)$ is of Shimura type, and the first (partial) verifications of the Birch-Swinnerton-Dyer conjecture made by Coates and Wiles (\cite{CoatesWiles1977}) concerned these curves. Along these lines let us also mention the paper of Rubin (\cite{Rubin1981}) which considers the Birch-Swinnerton-Dyer conjecture for general CM elliptic curves of Shimura type.
Let us now give a brief outline of what follows. We first recall a result of Shimura (\ref{prop:shimura-lambda-curves-over-hilbert}) stating the existence of infinitely many CM elliptic curves of Shimura type over the Hilbert class field $H$ of $K$. We then show that such CM elliptic curves cannot have good reduction everywhere (\ref{prop:no-abelian-torsion-good-reduction-hilbert}) and answer the question raised in (\ref{rema:rohr-q}) regarding the triviality of the $\mc{CL}_{O_K}$-torsor $\mc{M}_\mathrm{CM}$.
We then prove the main result (\ref{theo:shimura-is-lambda}) which is that a CM elliptic curve $E/\mathrm{Spec}(L)$ is of Shimura type if and only if it admits a canonical $\Lambda$-structure. There are some minor technicalities in that one must avoid the ramified primes in $L/K$ and the primes of bad reduction for $E$ but we get around this naturally enough.
The second part of this section concerns itself with the tangent spaces of N\'eron models of CM elliptic curves of Shimura type. The $\Lambda$-structure on $E/\mathrm{Spec}(L)$ induces a rather rigid structure on Lie algebra of its N\'eron model. We then show (\ref{coro:shimura-curves-minimal-model}) that if $E/\mathrm{Spec}(K(\ideal{f}))$ is a CM elliptic curve of Shimura type over the ray class field of conductor $\ideal{f}$, and the $\ideal{f}$-torsion of $E$ is constant, then the Lie algebra of its N\'eron model is free away from $\ideal{f}$, or in other words that $E/\mathrm{Spec}(K(\ideal{f}))$ admits global minimal model away from $\ideal{f}$. In particular, if one takes $\ideal{f}=O_K$ so that $K(\ideal{f})=H$, this shows that every CM elliptic curve of Shimura type $E/\mathrm{Spec}(H)$ admits a global minimal model \textit{everywhere}. This extends a result of Gross (\cite{Gross82}) and will be used crucially in (\ref{sec:lambda-cover-of-m}) in our construction of a flat affine $\Lambda$-presentation of the stack $\mc{M}_\mathrm{CM}$.
\subsection{}\label{subsec:cm-over-a-field-shimura} Let $L$ be an abelian extension of $K$ and $E/\mathrm{Spec}(L)$ a CM elliptic curve. One says that $E/\mathrm{Spec}(L)$ is of Shimura type if the extension $L(E[\mathrm{tors}])/K$ is abelian.
\begin{prop}[Shimura]\label{prop:shimura-lambda-curves-over-hilbert} There exist infinitely many prime ideals $\ideal{p}$ of $K$ for which there is a \textup{CM} elliptic curve $E/\mathrm{Spec}(H)$ of Shimura type with good reduction away from $\ideal{p}$.
\end{prop}
\begin{proof} By Proposition 7, \S 5 of \cite{Shimura71} there exists infinitely many primes $\ideal{p}$ of $K$ with $N\ideal{p}=p$ a rational prime, $N\ideal{p}=1\bmod w$ and $(N\ideal{p}-1)/w$ prime to $w$, where $w=\# O_K^\times$. Given such a prime $\ideal{p}$ it follows that the reduction map \[O_K^\times\to (O_K/\ideal{p})^\times\] is the inclusion of a direct factor. Therefore, we may define a map $\alpha: A_{O_K}^\times\to O_K^\times$ satisfying $\alpha|_{O_K^\times}=\mathrm{id}_{O_K^\times}$ by \[A_{O_K}^\times\to (O_K/\ideal{p})^\times\to O_K^\times\] where the first map is the quotient map and the second is a retraction of $O_K^\times\to (O_K/\ideal{p})^\times$. Finally, setting \[g: A_{O_K}^\times/O_K^\times\to A_{O_K}^\times: s\bmod O_K^\times\mapsto s \alpha(s)^{-1}\] we define $\rho^{-1}: G(H^\mathrm{sep}/H)\to A_{O_K}^\times$ by \[G(H^\mathrm{sep}/H)\stackrel{-|_{K^\infty}}{\longrightarrow} G(K^\mathrm{ab}/H)\stackrel{\theta_{K}^{-1}|_{G(K^\infty/H)}}{\longrightarrow} A_{O_K}^\times/O_K^\times\stackrel{g}{\to} A_{O_K}^\times.\]
We will now show that $\rho$ satisfies the conditions of (\ref{prop:cm-exist-given-h}) to construct a CM elliptic curve $E/\mathrm{Spec}(H)$ with $\rho_{E/H}=\rho$. For each $\sigma\in G(H^\mathrm{sep}/H)$ there is an $s\in A_{O_K}^\times/O_K^\times$ such that $\sigma|_{K^\infty}=\theta_K(s)\in G(K^\infty/H)$. Unwinding the definition of $\rho^{-1}$ we find \[[\rho(\sigma)^{-1}]=[g(s \bmod O_K^\times)]=[s\alpha(s)^{-1}]=[s]\] so that by (iii) of (\ref{prop:compute-the-homomorphisms-h}) we have \[[\sigma]_H=[\sigma|_{K^\mathrm{sep}}]_K=[s]=[\rho(\sigma)^{-1}].\] Therefore, the diagram \[\xymatrix{G(H^\mathrm{sep}/H)\ar[dr]_-{[-]_H}\ar[r]^-{\rho^{-1}} & A_{O_K}^\times\ar[d]^{[-]}\\
& CL_{O_K, \infty}}\] commutes and by (\ref{prop:cm-exist-given-h}) there exists a CM elliptic curve $E/\mathrm{Spec}(H)$ with $\rho_{E/H}=\rho$. As $\rho_{E/H}: G(H^\mathrm{sep}/H)\to A^\times_{O_K}$ factors through $G(K^\mathrm{ab}/H)\to A_{O_K}^\times$, it follows that $H(E[\mathrm{tors}])/K$ is abelian over $K$ and $E/\mathrm{Spec}(H)$ is of Shimura type. Moreover, $E/\mathrm{Spec}(H)$ has good reduction away from $\ideal{p}$ by (\ref{coro:image-of-inertia-group}) as by construction the composition \[O_{K_\l}^\times\to A_{O_K}^\times/O_K^\times\overset{\sim}{\longrightarrow} G(K^\mathrm{ab}/H)\to A_{O_K}^\times\] is equal to the natural inclusion $O_{K_\l}^\times\to A_{O_K}^\times$ for all primes $\l\neq \ideal{p}$.
\end{proof}
\begin{rema} It follows from (\ref{prop:shimura-lambda-curves-over-hilbert}) above that the map $c_{\mc{E}/\mc{M}}: \mc{M}_\mathrm{CM}\to M_\mathrm{CM}$ from $\mc{M}_\mathrm{CM}$ to its coarse sheaf admits sections Zariski locally over $\mathrm{Spec}(O_K)$. Indeed, by (\ref{prop:shimura-lambda-curves-over-hilbert}) there are infinitely many primes $\ideal{p}$ of $O_K$ and CM elliptic curves $E/M_\mathrm{CM}[\ideal{p}^{-1}]$. Each map $c_{E/M_\mathrm{CM}[\ideal{p}^{-1}]}: M_\mathrm{CM}[\ideal{p}^{-1}]\to M_\mathrm{CM}$ is then equal to the inclusion $M[\ideal{p}^{-1}]\to M$ followed by some automorphism $\sigma_{\a}$ of $M_\mathrm{CM}$. By (\ref{subsec:coarse-maps-level}), replacing $E$ with $E\otimes_{O_K}\a$, we may assume that $c_{E/M_\mathrm{CM}[\ideal{p}^{-1}]}: M_\mathrm{CM}[\ideal{p}^{-1}]\to M_\mathrm{CM}$ is the inclusion.
We note for future reference that if $\ideal{p}_1$ and $\ideal{p}_2$ are two such primes with $E_1/M_\mathrm{CM}[\ideal{p}_1^{-1}]$ and $E_2/M_\mathrm{CM}[\ideal{p}_2^{-1}]$ as above, then the fact that $c_{E_1/M_\mathrm{CM}[\ideal{p}_1^{-1}]}$ and $c_{E_2/M_\mathrm{CM}[\ideal{p}_2^{-1}]}$ are equal to the natural inclusions implies that $E_1$ and $E_2$ are locally isomorphic on the over lap $M_\mathrm{CM}[\ideal{p}^{-1}_1]\cap M_\mathrm{CM}[\ideal{p}_2^{-1}]=M_\mathrm{CM}[(\ideal{p}_1\ideal{p}_2)^{-1}]\subset M_\mathrm{CM}$.
\end{rema}
\begin{prop}\label{prop:no-abelian-torsion-good-reduction-hilbert} There does not exist a \textup{CM} elliptic curve $E/H$ of Shimura type with good reduction everywhere.
\end{prop}
\begin{proof} Let $E/H$ be a CM elliptic curve and consider the character (\ref{subsec:definition-h-for-cm-curves}) \[\rho_{E/H}: G(K^\mathrm{sep}/H)\to A_{O_K}^\times.\] If $H(E[\mathrm{tors}])/K$ is abelian then $\rho_{E/H}$ factors as \[\rho_{E/H}: G(K^\infty/H)=G(K^\mathrm{ab}/H)\to A_{O_K}^\times\] (note that $K^\mathrm{ab}=K^\infty$). Composing the reciprocal $\rho_{E/H}^{-1}$ with the isomorphism $\theta_K: A_{O_K}^\times/O_K^\times\overset{\sim}{\longrightarrow} G(K^\infty/H)=G(K^\mathrm{ab}/H)$ of (\ref{ray-class-isom}) we obtain a homomorphism \[f:A_{O_K}/O_K^\times\to A_{O_K}^\times.\]
The fundamental relation $[\rho_{E/H}^{-1}]=[-]_H$, the fact that $[-]_H=[-]_K|_{G(K^\mathrm{sep}/H)}$, the fact that $\theta_K\circ [-]_K=|_{K^\infty}$ and the fact that the inertia group $I_\ideal{p}(K^\infty/H)\subset G(K^\infty/K)$ corresponds to $O_{K_\ideal{p}}^\times\subset A_{O_K}^\times/O_K^\times \subset (A_{O_K}\otimes_{O_K}K)^\times/O_K^\times$ under $\theta_K$ combined with (\ref{coro:image-of-inertia-group}) shows that for $E/H$ to have good reduction at all places of $H$ lying above a prime $\ideal{p}$ of $O_K$ is equivalent to the composition \[O_{K_\ideal{p}}^\times\subset A_{O_K}^\times/O_K^\times\stackrel{f}{\to} A_{O_K}^\times\] being equal to the inclusion $O_{K_\ideal{p}}^\times\subset A_{O_K}^\times$. If this is true for all prime ideals $\ideal{p}$, the composition of $f$ with the quotient map \begin{equation}\label{eqn:f-comp-quot}A_{O_K}^\times\to A_{O_K}^\times/O_K^\times \stackrel{f}{\to} A_{O_K}^\times\end{equation} is equal to the identity on the sub-group of $A_{O_K}^\times$ generated by the sub-groups $O_{K_\ideal{p}}^\times$ for all primes $\ideal{p}$ of $O_K$. But this sub-group is dense and so it follows that the composition (\ref{eqn:f-comp-quot}) is equal to the identity and this is clearly impossible.
\end{proof}
\begin{rema}\label{rema:rohr} We can now answer the question raised in (\ref{rema:rohr-q}) asking for a trivialisation of the $\mc{CL}_{O_K}$-torsor $\mc{M}_{O_K}$. In light of the fact that the coarse sheaf $M_\mathrm{CM}$ of $\mc{M}_\mathrm{CM}$ is isomorphic to $\mathrm{Spec}(O_H)$ it more natural to ask the following: does there exists a CM elliptic curve $\mc{E}/\mathrm{Spec}(O_H)$ inducing a trivialisation of the $\mc{CL}_{O_K}$-torsor $\mc{M}_\mathrm{CM}$, i.e.\ an equivalence of stacks \[\mc{CL}_{O_H}\times \mathrm{Spec}(O_H)\overset{\sim}{\longrightarrow} \mc{M}_\mathrm{CM}\times \mathrm{Spec}(O_H): \mc{L}/S\mapsto E_S\otimes_{O_K}\mc{L}\ ?\] While we have shown (\ref{prop:no-abelian-torsion-good-reduction-hilbert}) that there do not exist CM elliptic curves $E/H$ of Shimura type with good reduction everywhere, it is possible for there to exist CM elliptic curves $\mc{E}/\mathrm{Spec}(O_H)$ (of course they will not be of Shimura type). Indeed, Rohrlich \cite{Rohrlich1982} has shown that if the discriminant of $K$ over $\mathbf{Q}$ is divisible by at least two primes congruent to $3\bmod 4$ then there does exist a CM elliptic curve $\mc{E}/\mathrm{Spec}(O_H)$. The answer in general to the question above is however negative. Indeed, if $K$ has class number, so that $K=H$, then every CM elliptic curve $E/\mathrm{Spec}(K)$ is of Shimura type and so cannot have good reduction everywhere, i.e.\ there does not exist a CM elliptic curve $\mc{E}/\mathrm{Spec}(O_K)$.
\end{rema}
\subsection{}\label{subsec:cm-over-a-field-shimura-integral} We continue with the notation of (\ref{subsec:cm-over-a-field-shimura}) so that $L/K$ is an abelian extension and $E/\mathrm{Spec}(L)$ is a CM elliptic curve. We also fix an integral ideal $\ideal{g}\in \mathrm{Id}_{O_K}$ such that $\mathrm{Spec}(O_L[\ideal{g}^{-1}])$ is unramified over $\mathrm{Spec}(O_K)$ and such that $E$ has good reduction over $\mathrm{Spec}(O_L[\ideal{g}^{-1}])$. We then set $S=\mathrm{Spec}(O_L[\ideal{g}^{-1}])$ and $P=\mathrm{Id}_{O_K}^{(\ideal{g})}$ so that $S$ admits a unique $\Lambda_{P}$-structure (\ref{prop:lambda-structures-etale-schemes-number-fields}) whose Frobenius lifts we denote by $\sigma_\a=\sigma_{\a, S}: S\to S$ for $\a\in P$. We write $\mc{E}\to S$ for the N\'eron model of $E$ relative to $\mathrm{Spec}(L)\to S$ so that $\mc{E}\to S$ is a CM elliptic curve. For $\ideal{p}\in P$ a prime ideal let us write $S_\ideal{p}=S\times_{\mathrm{Spec}(O_K)}\mathrm{Spec}(\mathbf{F}_\ideal{p})$ and $\mc{E}_\ideal{p}=S\times_S S_\ideal{p}$.
\begin{lemm}\label{prop:frob-lifts-unique} For each $\ideal{p}\in P$, there is at most one homomorphism \[\psi_{\mc{E}/S}^{\ideal{p}}:\mc{E}\to \sigma_\ideal{p}^*(\mc{E})\] lifting the $N\ideal{p}$-power relative Frobenius map of $\mc{E}_\ideal{p}$.
\end{lemm}
\begin{proof} By rigidity the difference of two such homomorphisms is equal to the zero map on some open and closed sub-scheme of $S$, the only choices of which are $S$ and $\emptyset$. Therefore, as any two such isomorphisms must agree on $S_\ideal{p}\subset S$ which is non-empty, they agree everywhere.
\end{proof}
\begin{theo}\label{theo:shimura-is-lambda} In the notation of \textup{(\ref{subsec:cm-over-a-field-shimura-integral})}, the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The \textup{CM} elliptic curve $\mc{E}\to S$ admits a canonical $\Lambda_{P, S}$-structure.
\item The \textup{CM} elliptic curve $\mc{E}\to S$ admits a $\Lambda_{P, S}$-structure.
\item The extension $L(E[\mathrm{tors}])/K$ is abelian, i.e.\ $E/\mathrm{Spec}(L)$ is a \textup{CM} elliptic curve of Shimura type.
\item The homomorphism $\rho_{E/L}: G(L^\mathrm{sep}/L)\to A_{O_K}^\times$ factors throught $G(L^\mathrm{sep}/L) \to G(K^\mathrm{ab}/L)$.
\end{enumerate}
\end{theo}
\begin{proof} (i) implies (ii): This is clear.
(ii) implies (iii): For each ideal $\a$ the sub-schemes $\mc{E}[\a]=\ker(\psi_{\mc{E}/S}^\a)\subset \mc{E}$ are finite and locally free $\Lambda_{P, S}$-schemes. After inverting $\a$ and forgetting about the Frobenius lifts for primes dividing $\a$, we may apply (\ref{prop:lambda-structures-etale-schemes-number-fields}) to see that $K(E[\a])=L(E[\a])$ is abelian over $K$.
(iii) is equivalent to (iv): This is immediate from the definition of $\rho_{E/L}$ (\ref{subsec:definition-h-for-cm-curves}).
(iv) implies (i): If $\ideal{g}=O_K$ then $L/K$ is unramified and so $L\subset H$. However, $M_\mathrm{CM}\overset{\sim}{\longrightarrow} \mathrm{Spec}(O_H)$ and so we must have $L=H$. However, if $L=H$ and $\ideal{g}=O_K$ then $E/\mathrm{Spec}(H)$ admits good reduction everywhere which, as $E/\mathrm{Spec}(H)$ is of Shimura type by hypothesis, is impossible (\ref{prop:no-abelian-torsion-good-reduction-hilbert}). Therefore, $\ideal{g}\neq O_K$ and it follows that replacing $\ideal{g}$ with $\ideal{g}^{n}$ for some $n\geq 0$ we may assume that $\ideal{g}$ separates units (this changes neither $\mathrm{Spec}(O_L[\ideal{g}^{-1}])$ nor $P=\mathrm{Id}_{O_K}^{(\ideal{g})}$).
Write $L'=L(E[\ideal{g}])$ and $S'=\mathrm{Spec}(O_{L'}[\ideal{g}^{-1}])$. The extension $L'/K$ is abelian (by hypothesis $L(E[\mathrm{tors}])$ is abelian) and unramified away from $\ideal{g}$ (as $\mc{E}[\ideal{g}]$ is \'etale over $S$) so that $S'$ admits a unique $\Lambda_{P}$-structure. By construction, the CM elliptic curve $\mc{E}\times_S S'$ admits a level-$\ideal{g}$ structure and, choosing one, we obtain a map $S\to M_\mathrm{CM}^{(\ideal{g})}$ and an isomorphism $\mc{E}\times_S S'\overset{\sim}{\longrightarrow} E^{(\ideal{g})}\times_{M_\mathrm{CM}^{(\ideal{g})}} S'$. As the morphism $S'\to M_\mathrm{CM}^{(\ideal{g})}$ is a $\Lambda_P$-morphism and $E^{(\ideal{g})}\to M_{\mathrm{CM}}^{(\ideal{g})}$ admits a canonical $\Lambda_{P, M_\mathrm{CM}^{(\ideal{g})}}$-structure it follows that \[\mc{E}\times_S S'\overset{\sim}{\longrightarrow} E^{(\ideal{g})}\times_{M_\mathrm{CM}^{(\ideal{g})}} S'\] admits a canonical $\Lambda_{P, S'}$-structure and by (iii) of (\ref{prop:canonical-lambda-p-structures}) that $\mc{E}\to S$ admits a canonical $\Lambda_{P, S}$-structure.
\end{proof}
\begin{rema}\label{rema:shimura-curves-locally-lambda-isomorphic} Now let $\mc{E}, \mc{E}'$ be a pair of CM elliptic curves of Shimura type over $S$ (we keep $\ideal{g}$ the same). If $\mc{E}$ and $\mc{E}'$ are locally isomorphic (note that there is always some ideal $\a$ such that $\mc{E}\otimes_{O_K}\a$ and $\mc{E}'$ are locally isomorphic) then they are actually $\Lambda_{P, S}$-locally isomorphic. Indeed, generically, they become isomorphic over the extension $L'/L$ corresponding to the character \[\rho:=\rho_{E/L}\rho_{E'/L}^{-1}: G(K^\mathrm{ab}/L)\to O_K^\times.\] It is clear that the extension $L'/K$ is abelian and it also is unramified away from $\ideal{g}$ (the characters $\rho_{E/L}$ and $\rho_{E'/L}$ agree on the inertia sub-groups for all $\ideal{p}\nmid \ideal{g}$ --- this is the good reduction of $\mc{E}$ and $\mc{E}'$ away from $\ideal{g}$). Therefore, $S'=\mathrm{Spec}(O_L[\ideal{g}^{-1}])$ is finite and \'etale over $S$ and admits a unique $\Lambda_{P}$-structure. Moreover, there exists an isomorphism \[f: \mc{E}\times_{S} S\overset{\sim}{\longrightarrow} \mc{E}'\times_S S'\] and it remains to observe that all such isomorphisms are $\Lambda_{P, S'}$-isomorphisms. Indeed, $f$ is a $\Lambda_{P, S'}$-isomorphism if and only if the $\Lambda_{P, S'}$-structure on $\mc{E}$ induced by transport of structure along $f$ is equal to the given $\Lambda_{P, S'}$-structure on $\mc{E}'$ and this follows from (\ref{prop:frob-lifts-unique}).
\end{rema}
\subsection{} We now wish to study the Lie algebras of the N\'eron models of CM elliptic curves of Shimura type. So we continue with the notation of (\ref{subsec:cm-over-a-field-shimura-integral}) but will also assume that the CM elliptic curve $\mc{E}\to S$ admits a canonical $\Lambda_{P, S}$-structure, i.e.\ that $E/\mathrm{Spec}(L)$ is a CM elliptic curve of Shimura type. The field $L$ must contain the Hilbert class field $H\subset L$. The following well known property enjoyed by the Hilbert class field $H$ will be crucial:
\begin{prop}[Hauptidealsatz]\label{prop:hauptidealsatz} Every rank one $O_K$-module becomes free after base change to $O_H$.
\end{prop}
\subsection{}\label{subsec:lambda-module-for-cm-curves} We shall abuse notation and write, for each prime ideal $\ideal{p}\in P$, \[\nu_{\ideal{p}}: \mc{E}\otimes_{O_K}\ideal{p}^{-1}\overset{\sim}{\longrightarrow} \sigma_\ideal{p}^*(\mc{E})\] for the unique isomorphism lifting the isomorphism $\nu_{\ideal{p}}(\mc{E}_\ideal{p}/S_\ideal{p})$ of (\ref{coro:tensor-p-equals-frobenius}) or equivalently the unique isomorphism such that the composition \[\mc{E}\stackrel{i_\ideal{p}}{\to} \mc{E}\otimes_{O_K}\ideal{p}^{-1}\stackrel{\nu_\ideal{p}}{\to}\sigma_\ideal{p}^*(\mc{E})\] is equal to the relative Frobenius lift $\psi_{\mc{E}/S}^\ideal{p}$.
For a pair of primes $\ideal{p}$, $\l$ the commutativity of the (relative) Frobenius lifts on $\mc{E}\to S$ amounts to the equalities \begin{equation}\sigma_{\l}^*(\nu_\ideal{p})\circ (\nu_{\l}\otimes_{O_K}\ideal{p}^{-1})=\sigma_{\ideal{p}}^*(\nu_{\l})\circ (\nu_{\ideal{p}}\otimes_{O_K}\l^{-1})\label{eqn:frob-commute-prime}\end{equation} so that for any ideal $\a\in P$, choosing a prime factorisation of $\a$, we may define isomorphisms \[\nu_\a: \mc{E}\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} \sigma_\a^*(\mc{E})\] by composing the various $\nu_\ideal{p}$ for $\ideal{p}|\a$, with the resulting isomorphism being independent of any choices involved by virtue of (\ref{eqn:frob-commute-prime}). These isomorphisms now satisfy \begin{equation}\sigma_{\a}^*(\nu_\b)\circ (\nu_{\a}\otimes_{O_K}\b^{-1})=\sigma_{\b}^*(\nu_{\a})\circ (\nu_{\b}\otimes_{O_K}\a^{-1})\label{eqn:frob-commute-all}\end{equation} for each $\a, \b\in P$.
We can actually say more about the $\nu_\a$ when $\sigma_\a=\mathrm{id}_{L}.$ In this case, we get an isomorphism $\nu_\a: \mc{E}\otimes_{O_K}\a^{-1}\to \sigma_\a^*(\mc{E})=\mc{E}$ which must be of the form $1\otimes l(\a)$ for some $l(\a)\in O_K$ which generates $\a$. If, moreover, there is an ideal $\ideal{f}$ such that $E[\ideal{f}]$, and hence $\mc{E}[\ideal{f}]$, are constant then composition \[\mc{E}[\ideal{f}]\to \mc{E}[\ideal{f}]\otimes_{O_K}\a^{-1}\stackrel{\nu_\a[\ideal{f}]}{\to} \mc{E}[\ideal{f}]\] is equal to the unique relative Frobenius lift \[l(\a)=\psi_{\mc{E}[\ideal{f}]/S}^\a=\mathrm{id}_{\mc{E}[\ideal{f}]}:\mc{E}[\ideal{f}]\to \mc{E}[\ideal{f}]\] so that $l(\a)=1\bmod \ideal{f}$.
By the N\'eron mapping property the isomorphisms $\nu_\a$ extend to isomorphisms on the full N\'eron model \[\nu_\a: \mathrm{N\acute{e}r}_{O_L}(E)\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} \sigma_\a^*(\mathrm{N\acute{e}r}_{O_L}(E))\] satisfying the same commutativity condition (note that $\mathrm{N\acute{e}r}_{O_L}(E)$ is a smooth group scheme of relative dimension one, not necessarily proper). Denoting by \[T=\underline{\mathrm{Lie}}_{\mathrm{N\acute{e}r}_{O_L}(E)/\mathrm{Spec}(O_L)}\] the Lie algebra of the N\'eron model, which is a projective rank one $O_L$-module, the isomorphisms $\nu_\a$ induce $O_L$-isomorphisms (which we denote by the same letter) \[\nu_\a: T\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} \sigma_\a^*(T)\] for each $\a\in P$, satisfying the same commutativity condition (\ref{eqn:frob-commute-all}) as the $\nu_\a$ on $\mathrm{N\acute{e}r}_{O_L}(E)$.
\begin{coro}\label{coro:shimura-curves-minimal-model} In the notation of \textup{(\ref{subsec:lambda-module-for-cm-curves})}, if $L=K(\ideal{f})$ is a ray class field and $E[\ideal{f}]$ is constant then the Lie algebra of the N\'eron model $\mathrm{N\acute{e}r}_{O_{K(\ideal{f})}}(E)$ becomes free after inverting $\ideal{f}$. In other words, $E/\mathrm{Spec}(K(\ideal{f}))$ admits a global minimal model away from $\ideal{f}$.
\end{coro}
\begin{proof} We have the isomorphisms \[\nu_\a: T\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} \sigma_\a^*(T)\] for each ideal $\a\in P=\mathrm{Id}_{O_K}^{(\ideal{g})}$ and when $\sigma_\a=\mathrm{id}_{O_{K(\ideal{f})}}$ (this is the case if and only if $\a=(a)$ is principal with $a=1\bmod \ideal{f}$) the isomorphism \[\nu_\a: T\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} \sigma_\a^*(T)=T\] takes the form $1\otimes l(\a)$ where $l(\a)\in O_{K}$ is a generator of $\a$ such that $l(\a)=1\bmod \ideal{f}$. We now apply (\ref{prop:tannaka-result-special}) to extend $l$ to a map $l:\mathrm{Id}_{O_K}^{(\ideal{g})}\to O_{K(\ideal{f})}$ satisfying \begin{equation} l(\a)\cdot O_{K(\ideal{f})}=\a\cdot O_{K(\ideal{f})} \quad \text{ and } \quad l(\a\b)=l(\a)\sigma_\a(l(\b))\label{eqn:l-com-condition}\end{equation} for all $\a, \b\in \mathrm{Id}_{O_K}^{(\ideal{g})}$. Define $t_\a: T\to \sigma_\a^*(T)$ be the composition \[T\stackrel{1\otimes l(\a)^{-1}}{\longrightarrow} T\otimes_{O_K}\a^{-1}\stackrel{\nu_\a}{\longrightarrow} \sigma^*_a(T).\] Then $t_\a$ is an isomorphism by (\ref{eqn:l-com-condition}), and if $\a\in \mathrm{Prin}_{1\bmod \ideal{f}}^{(\ideal{g})}$ then $t_\a=l(\a)\otimes l(\a)^{-1}=\mathrm{id}_T$ which, combined with the commutativity conditions (\ref{eqn:frob-commute-all}) on the $\nu_\a$ and (\ref{eqn:l-com-condition}) on the $l(\a)$, show that $t_\a$ depends only on the class $\sigma_\a\in G(K(\ideal{f})/K)$, that $t_{\mathrm{id}_{K(\ideal{f})}}=\mathrm{id}_T$ and that $t_{\sigma\tau}=t_\sigma\circ \sigma^*(t_\tau)$ for $\sigma, \tau\in G(K(\ideal{f})/K)$.
In other words, the isomorphisms $t_\sigma$ (or perhaps what is more standard, their inverses) define Galois descent data on $T$ relative to $O_{K}\to O_{K(\ideal{f})}$. As $O_K\to O_{K(\ideal{f})}$ becomes finite and \'etale after inverting $\ideal{f}$, there exists an $O_K[\ideal{f}^{-1}]$-module $T_0$ such that \[T_0\otimes_{O_K[\ideal{f}^{-1}]} O_{K(\ideal{f})}[\ideal{f}^{-1}]\overset{\sim}{\longrightarrow} T\otimes_{O_{K(\ideal{f})}}O_{K(\ideal{f})}[\ideal{f}^{-1}].\] However, as $K(\ideal{f})$ contains the Hilbert class field $H$, all rank one projective $O_{K}[\ideal{f}^{-1}]$-modules become free after base change to $O_{K(\ideal{f})}[\ideal{f}^{-1}]$ (all rank one projective $O_K$-modules do by the Hauptidealsatz (\ref{prop:hauptidealsatz}) and every rank one projective $O_K[\ideal{f}^{-1}]$-module is a localisation of a rank one projective $O_K$-module). It follows that \[T_0\otimes_{O_K[\ideal{f}^{-1}]}O_{K(\ideal{f})}[\ideal{f}^{-1}]\overset{\sim}{\longrightarrow} T\otimes_{O_{K(\ideal{f})}}O_{K(\ideal{f})}[\ideal{f}^{-1}]=\underline{\mathrm{Lie}}_{\mathrm{N\acute{e}r}_{O_L}(E)/\mathrm{Spec}(O_L)}\otimes_{O_L}O_L[\ideal{f}^{-1}]\] is free.
\end{proof}
\begin{rema} We note that when $\ideal{f}=O_K$ (\ref{coro:shimura-curves-minimal-model}) the condition on the $\ideal{f}$-torsion becomes trivial so that \textit{any} CM elliptic $E/H$ of Shimura type admits a \textit{global} minimal model. This generalises the main result (see Corollary 4.4) of \cite{Gross82} where it is shown that a CM elliptic curve $E/H$ admits a global minimal model whenever the conductor of $K$ over $\mathbf{Q}$ is prime and the homomorphism $\rho_{E/H}$ satisfies a certain invariance condition. One can show that these assumptions imply $E/H$ is a CM elliptic curve of Shimura type so that (\ref{coro:shimura-curves-minimal-model}) is indeed a generalisation of \cite{Gross82}. To say a little, the invariance condition on $\rho_{E/H}$ is equivalent to $E$ being isogenous to $\sigma^*(E)$ for each $\sigma\in G(H/K)$ and the primality of the discriminant of $K$ over $\mathbf{Q}$ implies that the order of $G(H/K)$ is prime to the order of $O_K^\times$ and together these properties allow one to show that $E/H$ is a CM elliptic curve of Shimura type (for a result along these lines see Proposition 2 of \cite{Gilles1985}).
\end{rema}
\section{Weber functions} The purpose of this section is to define, for each CM elliptic curve $E$ over an arbitrary base $S$, a certain quotient $E\to X_{E/S}$ and then to study its resulting properties. Informally, $X_{E/S}$ will be the quotient of $E$ by its group of automorphisms $\underline{O_K^\times}_S$. However, $\underline{O_K^\times}_S$ does not act freely on $E$ and so the orbits of are not well behaved. This makes it difficult to construct a quotient (in the naive sense) with any useful properties. We get around this problem by using Cartier divisors to define intelligent orbits for the action of $\underline{O_K^\times}_S$ under which it behaves as though it were free. Taking the quotient by the resulting equivalence relation, we get a smooth, proper curve $X_{E/S}$ together with a $\underline{O_K^\times}_S$-invariant finite locally free map of degree $w=\# O_K^\times$ \[p_{E/S}: E\to X_{E/S}.\]
The construction we give is functorial in $E/S$ and so we may run it for the universal CM elliptic curve $\mc{E}\to \mc{M}_\mathrm{CM}$ to obtain a smooth, proper curve $X_{\mc{E}/\mc{M}_\mathrm{CM}}\to \mc{M}_\mathrm{CM}$. Almost by definition, this curve descends to a smooth proper curve $f : X\to M_\mathrm{CM}$ over the coarse sheaf.
The remainder of the section is devoted to the study of $X\to M_\mathrm{CM}$. We first show that it has the following properties:
\begin{enumerate}[label=(\roman*)]
\item $f: X\to M_\mathrm{CM}$ admits a natural $\Lambda_{M_\mathrm{CM}}$-structure and a $\Lambda_{M_\mathrm{CM}}$-point $0_{X}: M_\mathrm{CM}\to X$,
\item $f: X\to M_\mathrm{CM}$ has genus zero. Thus, if $\mc{I}_X\subset \mc{O}_X$ denotes the ideal sheaf defining the closed point $0_X: M_\mathrm{CM}\to X$ and $\mc{W}=f_*(\mc{I}_X^{-1})$ then $\mc{W}$ is locally free of rank two over $\mc{O}_{M_\mathrm{CM}}$, the map $f^*(\mc{W})\to \mc{I}^{-1}_X$ is an epimorphism and the resulting map \[X\overset{\sim}{\longrightarrow} \mathbf{P}_{M_\mathrm{CM}}(\mc{W})\] is an isomorphism,
\item setting $X[\a]=\psi_{X/M_\mathrm{CM}}^{\a*}(0_X)\subset X$, the scheme $X[\a]$ is a finite locally free $\Lambda_{M_\mathrm{CM}}$-scheme of degree $N\a$ and $K(X[\a])=K(\a)$.
\end{enumerate}
Thus the curve $X\to M_\mathrm{CM}$ together with its $\Lambda_{M_\mathrm{CM}}$-structure and its $\Lambda_{M_\mathrm{CM}}$-point $0_X: M_\mathrm{CM}\to X$ allow one to construct the ray class fields of $K$ in an integral and coherent, choice free manner. Of course, this is just a more streamlined and abstract approach to the classical construction of the ray class fields of $K$ using Weber functions --- this approach being to choose a CM elliptic curve $E/H$ and to consider the image of $E[\a]\subset E$ under a `Weber map' \[E\to \mathbf{P}^1_H\] which is a certain $O_K^\times$-invariant map of degree $w$ (see Theorem 5.6 of \cite{Silverman94}).
The only defect of our approach is that the curve $X\to M_\mathrm{CM}$ is not particularly explicit. However, we end the section by showing (by the same method we used to show that CM elliptic curves of Shimura type admit global minimal models) that there exists an isomorphism $X\overset{\sim}{\longrightarrow} \mathbf{P}^1_{M_\mathrm{CM}}$.
\subsection{} We begin by recalling some basic facts regarding Cartier divisors on curves (see \S\S 1.1-1.2 Chapter I of \cite{KatzMazur85}). Let $S$ be a scheme and let $X\to S$ be a smooth proper $S$-curve, i.e.\ $X\to S$ is smooth of relative dimension one and proper. An $S$-relative Cartier divisor on $X$, or just a Cartier divisor, is a closed sub-scheme $D\subset X$ which is finite locally free over $S$. Equivalently, a closed sub-scheme $D\subset X$ is a Cartier divisor if and only if $D\to S$ is flat and the ideal sheaf $\mc{I}_D\subset \mc{O}_E$ defining $D$ is a locally free rank one $\mc{O}_X$-module. Given two Cartier divisors $D, D'\subset X$, their sum $D+D'\subset X$ is defined to be the closed sub-scheme corresponding to the ideal sheaf $\mc{I}_{D+D'}:=\mc{I}_D\otimes_{\mc{O}_X}\mc{I}_{D'}\subset \mc{O}_X$, which is again a Cartier divisor.
The degree $\deg(D)$ of a Cartier divisor $D\subset S$ is defined to be the degree of the finite locally free $S$-scheme $D$. We have $\deg(D+D')=\deg(D)+\deg(D)$. The structure map $D\to S$ of a Cartier divisor on $X$ is an isomorphism if and only if $\deg(D)=1$ and the set of degree one Cartier divisors on $E$ is equal to the set of $S$-points $S\to X$. If $s\in X(S)$ is an $S$-point then we will denote the corresponding Cartier divisor by $s$.
If $f: X'\to X$ is a finite locally free map of smooth, proper $S$-curves and if $D\subset X$ is a Cartier divisor then $f^*(D)\subset X'$ is a Cartier divisor and $\deg(f^*(D))=\deg(f)\deg(D)$.
Given a pair of Cartier divisors $D, D'\subset X$ such that $\deg(D)\leq \deg(D')$ (resp. $\deg(D)=\deg(D')$) we can form the inclusion (resp. equality) $S$-sheaf of $D$ and $D'$: \[\mathrm{In}_{X/S}(D, D')\subset S \quad \text{ (resp. } \quad \mathrm{Eq}_{X/S}(D, D')\subset S)\] defined by the property that $T\to S$ factors through $\mathrm{In}_{X/S}(D, D')\to S$ (resp. $\mathrm{Eq}_{X/S}(D, D')\to S$) if and only we have an inclusion (resp. equality) of Cartier divisors \[D\times_S T\subset D'\times_S T\subset X\times_{S} T \quad \text{ (resp. } \quad D\times_S T= D'\times_S T\subset X\times_{S} T).\] By Key Lemma 1.3.4 and Corollary 1.3.5 of \cite{KatzMazur85}, the sub-sheaves $\mathrm{In}_{X/S}(D, D')$ and $\mathrm{Eq}_{X/S}(D, D')$ of $S$ are finitely presented closed sub-schemes of $S$. Finally, if $S'\to S$ is a morphism then we have natural isomorphisms \[\mathrm{In}_{X_{S'}/S'}(D_{S'}, D'_{S'})\overset{\sim}{\longrightarrow} \mathrm{In}_{X/S}(D, D')_{S'} \quad \text{ and } \quad \mathrm{Eq}_{X_{S'}/S'}(D_{S'}, D'_{S'})\overset{\sim}{\longrightarrow} \mathrm{Eq}_{X/S}(D, D')_{S'}.\]
\subsection{} Now let $E/S$ be a CM elliptic curve. We would like to take the quotient of $E$ by the action of its group of automorphisms $\underline{O_K}_S^\times$ but, as noted in the introduction, this action is not free and in particular the map \[\coprod_{\epsilon\in O_K^\times} (\epsilon, \mathrm{id}_E): \coprod_{\epsilon\in O_K^\times}E=\underline{O_K^\times}_S\times_S E\to E\times_S E\] is not injective so that the orbits of points under this action are not well behaved. Of course, one could just take the image of this map and obtain an equivalence relation but then one would have little control over the quotient.
We get around this as follows. If $s: S\to E$ is an $S$-point, we define its `orbit' $[O_K^\times](s)\subset E$ to be the Cartier divisor \[[O_K^\times](s)=\sum_{\epsilon\in O_K^\times} \epsilon^*(s)\] (note the sum is of Cartier divisors and has nothing to do with the group law on $E$). Then $[O_K^\times](s)\subset E$ contains the Cartier divisor $s$, is stable under the action of $\underline{O_K^\times}_S$ and is finite locally free of degree $w$ over $S$ (as the usual orbit would be if the action of $G$ were free). Equality of these `orbits' defines an equivalence relation on $E$, which we denote by \[\mathrm{Eq}_{E/S}^{O_K^\times}\subset E\times_S E,\] so that an $S$-morphism $T\to E\times_S E$ factors through $\mathrm{Eq}_{E/S}^{O_K^\times}$ if and only if, writing $(t_1, t_2): T\to (E\times_S E)\times_S T=E_T\times_T E_T$ for the resulting map, we have an equality of Cartier divisors \[[O_K^\times](t_1)=[O_K^\times](t_2)\subset E_T=E\times_S T.\]
For this equivalence relation to behave as though it really does come from a group action, it should be the case that, given two points $s_1, s_2: S\to E$ then having $s_1\in [O_K^\times](s_2)$, i.e.\ $s_1: S\to E$ factoring through $[O_K^\times](s_2)$, should imply the equality of `orbits' \[[O_K^\times](s_1)=[O_K^\times](s_2).\]
With this in mind, we define the sub-sheaf \[\mathrm{In}_{E/S}^{O_K^\times}\subset E\times_S E\] by the property that an $S$-morphism $T\to E\times_S E$ factors through $\mathrm{In}_{E/S}^{O_K^\times}\subset E\times_S E$ if and only if, writing $(t_1, t_2): T\to (E\times_S E)\times_S T=E_T\times_T E_T$ for the corresponding map, the Cartier divisor $t_1: E\to E_T$ factors through the Cartier divisor $[O_K^\times](t_2)\subset E_T$, i.e.\ $t_1\in [O_K^\times](t_2)$. There is a natural inclusion \[\mathrm{Eq}_{E/S}^{O_K^\times}\subset \mathrm{In}_{E/S}^{O_K^\times}\subset E\times_S E\] and our claim that the equivalence relation $\mathrm{Eq}_{E/S}^{O_K^\times}$ behaves as though it were coming from a free action of a group is that we have an equality \[\mathrm{Eq}_{E/S}^{O_K^\times} = \mathrm{In}_{E/S}^{O_K^\times}\subset E\times_S E.\]
Before we prove this, let us make two observations. First, the sub-sheaves $\mathrm{Eq}_{E/S}^{O_K^\times}\subset \mathrm{In}_{E/S}^{O_K^\times}\subset E\times_S E$ are in fact closed sub-schemes. View $E\times_S (E\times_S E)\to E\times_S E$ as a CM elliptic curve over $E\times_S E$ via projection onto the second two factors and consider the Cartier divisors \[u_{E/S, 1}: E\times_S E\to E\times_S (E\times_S E): (e_1, e_2)\mapsto (e_1, e_1, e_2)\] and \[u_{E/S, 2}: E\times_S E\to E\times_S (E\times_S E): (e_1, e_2)\mapsto (e_2, e_1, e_2).\] Then it is an easy exercise to check that we have equalities of sub-sheaves of $E\times_S E$ \[\mathrm{Eq}_{E/S}^{O_K^\times}=\mathrm{Eq}_{E\times_S (E\times_S E)/E\times_S E}([O_K^\times](u_{E/S, 1}), [O_K^\times](u_{E/S, 2}))\subset E\times_S E\] and \[\mathrm{In}_{E/S}^{O_K^\times}=\mathrm{In}_{E\times_S (E\times_S E)/E\times_S E}(u_{E/S,2}, [O_K^\times](u_{E/S,2}))\subset E\times_S E\] so that by the representability of equality and inclusion sub-sheaves of Cartier divisors we find that $\mathrm{Eq}_{E/S}^{O_K^\times}$ and $\mathrm{In}_{E/S}^{O_K^\times}$ are closed sub-schemes of $E\times_S E$.
Our second observation is that, viewing $E\times_S E\to E$ as a CM elliptic curve over $E$ via projection onto the second factor, we have an equality of sub-schemes \[\mathrm{In}_{E/S}^{O_K^\times}=[O_K^\times](\Delta_{E/S})\subset E\times_S E.\] Indeed, fixing an affine scheme $T$ over $S$ and a morphism $T\to E$, which we identify with a morphism $t_2: T\to E\times_S T=E_T$, the pull-back of the $E$-morphism $[O_K^\times](\Delta_{E/S})\subset E\times_S E$ along $T\to E$ is given by \[[O_K^\times](t_2)\subset E_T=E\times_S T\] (as the pull-back of $\Delta_{E/S}: E\to E\times_S E$ is given by $t_2: T\to E_T=E\times_S T$). Therefore, for a second $S$-morphism $T\to E$, which we identify with a morphism $t_1: T\to E_T=E\times_S T$, to have the property that the induced map $T\to E\times_S E$ factors through $[O_K^\times](\Delta_{E/S})$, is equivalent to the morphism $t_1: T\to E_T$ factoring through $[O_K^\times](t_2)\subset E_T$. All said and done, a morphism $T\to E\times_S E$ factors through $[O_K^\times](\Delta_{E/S})\subset E\times_S E$ if and only if, in the notation above, the morphism $t_1: T\to E_T$ factors through $[O_K^\times](t_2)\subset E_T$ which is to say that $T\to E\times_S E$ factors through $\mathrm{In}_{E/S}^{O_K^\times}\subset E\times_S E$. We are now ready to prove our claim.
\begin{prop}\label{prop:trace-equivalence-relation} Let $E/S$ be a \textup{CM} elliptic curve. Then we have equalities of closed sub-schemes of $E\times_S E$ \[\mathrm{Eq}_{E/S}^{O_K^\times} = \mathrm{In}_{E/S}^{O_K^\times} = [O_K^\times](\Delta_{E/S})\subset E\times_S E.\] In particular, $\mathrm{Eq}^{O_K^\times}_{E/S}$ is a finite locally free equivalence relation of degree $w$.
\end{prop}
\begin{proof} The only thing we need to show is that the inclusion \[\mathrm{Eq}_{E/S}^{O_K^\times} \subset \mathrm{In}_{E/S}^{O_K^\times}\] is an equality. The first thing we note is that it is bijective on geometric points. That is, if $S=\mathrm{Spec}(F)$ is the spectrum of an algebraically closed field $F$, and $(s_1, s_2) \in E(S)\times E(S)$ satisfy $s_1\in [O_K^\times](s_2)$ then $[O_K^\times](s_1) = [O_K^\times](s_2)$. But this is clear, as $E$ is then a Dedekind scheme and unique factorisation of Cartier divisors (i.e.\ of the corresponding ideals) shows that there exists an $\epsilon\in O_K^\times$ (not necessarily unique!) such that $s_1=\epsilon s_2$ which gives \[[O_K^\times](s_1)=[O_K^\times](\epsilon s_1)=[O_K^\times](s_2).\] It follows that the inclusion \[\mathrm{Eq}_{E/S}^{O_K^\times} \subset \mathrm{In}_{E/S}^{O_K^\times},\] which is a closed immersion, is a nilpotent thickening.
Now, our claim is local on $S$ and so we may assume that $S$ admits a level-$\ideal{f}$ structure for some $\ideal{f}$ which separates units. It follows that way assume there exists a morphism $S\to M_\mathrm{CM}^{(\ideal{f})}$ and an isomorphism \[E \overset{\sim}{\longrightarrow} E^{(\ideal{f})}\times_{M_\mathrm{CM}^{(\ideal{f})}} S\] and again by the compatibility of inclusion and equality schemes of Cartier divisors with base change, we may assume that $E/S=E^{(\ideal{f})}/M_\mathrm{CM}^{(\ideal{f})}$, or what is important, that $S$ is integral.
We will now show that the nilpotent immersion \[\mathrm{Eq}_{E/S}^{O_K^\times}\to \mathrm{In}_{E/S}^{O_K^\times}\] is an isomorphism by showing that $\mathrm{In}_{E/S}^{O_K^\times}$ is reduced. Since $S$ is an integral scheme, it follows that $E$ is also an integral scheme. Therefore, the finite locally free $E$-scheme $\mathrm{In}_{E/S}^{O_K^\times}\subset E\times_S E$ is reduced if and only if for some non-empty open sub-scheme $U\subset E$ the pull-back $\mathrm{In}_{E/S}^{O_K^\times}\times_E U$ is reduced.
So let \[U=\bigcap_{1\neq \epsilon\in O_K^\times} (E-E[1-\epsilon]).\] Then over $U$, the Cartier divisors \[\epsilon^*(\Delta_{E/S}\times_E U): U\to E\times_S U\] for $\epsilon\in O_K^\times$ are all disjoint so that \[\mathrm{In}_{E/S}^{O_K^\times}\times_E U=[O_K^\times](\Delta_{E/S})\times_E U=\coprod_{\epsilon\in O_K^\times} \epsilon^*(\Delta_{E/S}\times_E U)\overset{\sim}{\longrightarrow} \coprod_{O_K^\times} E\times_S U.\] It follows that $\mathrm{In}_{E/S}^{O_K^\times}\times_E U$ and therefore $\mathrm{In}_{E/S}^{O_K^\times}$ are reduced and the nilpotent immersion \[\mathrm{Eq}_{E/S}^{O_K^\times} \subset \mathrm{In}_{E/S}^{O_K^\times}\] is an isomorphism.
\end{proof}
\subsection{} We are now ready to construct our quotients. So let $E/S$ be a CM elliptic curve. Then the Cartier divisor of degree $w$ \[\mathrm{Eq}_{E/S}^{O_K^\times}=[O_K^\times](\Delta_{E/S})\subset E\times_S E\] defines the equivalence relation on $E/S$ where $s_1, s_2\in E(S)$ are equivalent if and only if \[[O_K^\times](s_1)=[O_K^\times](s_2).\] We write \[p_{E/S}: E\to X_{E/S}\] for the resulting quotient (in the category of fpqc sheaves). As our equivalence relation is finite locally free (and $E$ is projective over $S$) it follows from Corollaire 7.1 Expos\'e VII of \cite{SGA1} that $X_{E/S}$ is representable by a scheme over $S$. We have thus constructed a finite locally free $\underline{O_K^\times}_S$-invariant map of degree $w$ \[p_{E/S}: E\to X_{E/S}.\]
\begin{prop} The $S$-scheme $X_{E/S}\to S$ is smooth of relative dimension one, proper and geometrically connected.
\end{prop}
\begin{proof} As $E\to S$ is proper, flat, and geometrically connected and $p_{E/S}: X\to X_{E/S}$ is finite locally free of degree $w$ (in particular, proper, flat and surjective), it follows that $X_{E/S}\to S$ is proper, flat, and geometrically connected. As $X_{E/S}\to S$ is flat, it is smooth if and only if its fibres are smooth, but when $S$ is the spectrum of a field $E$ is a regular scheme of dimension one and $E\to X_{E/S}$ is finite locally free so that $X_{E/S}$ is also regular of dimension one and therefore smooth over $S$.
\end{proof}
\begin{rema} The method used above to construct the quotients $X_{E/S}$ applies more generally. Indeed, if $X\to S$ is a smooth (not necessarily proper) curve, $S$ is an integral scheme and $G$ is a finite group acting generically freely on $X$ by $S$-automorphisms then there exists a smooth curve $X/[G]$ and a finite locally free $G$-invariant map $p: X\to X/[G]$ of degree $\#G$, i.e.\ what might be called a `quotient' of $X$ by $G$. It would be interesting to know whether this method could be extended to construct `quotients' of curves over more general bases $S$, or under more general (non-constant) group actions.
\end{rema}
\subsection{} Let us now consider the functorial properties of the association $E/S\mapsto X_{E/S}$. First, if $f: E\to E'$ is a morphism of CM elliptic curves over $S$, then $(f\times_S f)|_{\mathrm{Eq}^{O_K^\times}_{E/S}}\subset \mathrm{Eq}^{O_K^\times}_{E'/S}\subset E\times_S E$ and so there is an induced morphism $X_f: X_{E/S}\to X_{E'/S}$ and a commutative diagram \[\xymatrix{E\ar[d]_{p_{E/S}}\ar[r]^{f}&E'\ar[d]^{p_{E'/S}}\\
X_{E/S}\ar[r]^{X_f} & X_{E'/S}.}\] The invariance of $p_{E/S}$ and $p_{E'/S}$ under $\underline{O_K^\times}_S$ shows that $f\mapsto X_f$ is also invariant under $\underline{O_K^\times}_S$. In symbols, the map \[\underline{\mathrm{Hom}}_S^{O_K}(E, E')\to \underline{\mathrm{Hom}}_S(X_{E/S}, X_{E'/S'}): f\mapsto X_{f}\] factors through the quotient sheaf \[\underline{\mathrm{Hom}}_{S}^{O_K}(E, E')/\underline{O_K^\times}_S \to \underline{\mathrm{Hom}}_S(X_{E/S}, X_{E'/S'}).\] In particular, if $E$ and $E'$ are locally isomorphic then $\underline{\mathrm{Isom}}_{S}^{O_K}(E, E')$ is an $\underline{O_K^\times}_S$-torsor so that $\underline{\mathrm{Isom}}_S^{O_K}(E, E')/\underline{O_K^\times}_S\overset{\sim}{\longrightarrow} S$ and we obtain a canonical map $S\to \underline{\mathrm{Isom}}_S(X_{E/S}, X_{E'/S})$ or what is the same an isomorphism \[X_{E/S}\overset{\sim}{\longrightarrow} X_{E'/S}.\]
\subsection{} Now let $S$ be an $M_\mathrm{CM}$-sheaf, i.e.\ $S\to M_\mathrm{CM}$. By the definition of the coarse sheaf $M_\mathrm{CM}$, this implies that there exists a cover $(S_i\to S)_{i\in I}$ of $S$ and CM elliptic curves $(E_i/S_i)_{i\in I}$, such that for $i, j\in I$, writing $S_{ij}=S_i\times_S S_j$, the CM elliptic curves $E_i\times_{S_i} S_{ij}$ and $E_j\times_{S_j}S_{ij}$ are locally isomorphic. Therefore, writing $X_i=X_{E_i/S_i}$ for $i\in I$ we have for all $i, j\in I$ canonical isomorphisms \begin{equation} X_i\times_{S_i} S_{ij}\overset{\sim}{\longrightarrow} X_j\times_{S_j} S_{ij}.\label{eqn:descent}\end{equation} The independence of these isomorphisms from the choice of isomorphism $E_i\times_{S_i} S_{ij}\overset{\sim}{\longrightarrow} E_j\times_{S_j} S_{ij}$ (and similarly on the triple products) show that the isomorphisms (\ref{eqn:descent}) equip the family of curves $(X_i/S_i)_{i\in I}$ with descent data relative to the cover $(S_i\to S)_{i\in I}$, which furnishes us with (a priori) a sheaf $X_S\to S$. Similar observations show that the sheaf $X_S\to S$ is independent (upto canonical isomorphism) of the cover $(S_i\to S)_{i\in I}$ and the CM elliptic curves $(E_i/S_i)_{i\in I}$ and that if $S'\to S$ is a morphism of $M_\mathrm{CM}$-sheaves then we have a canonical isomorphism \[X_{S'}\overset{\sim}{\longrightarrow} X_{S}\times_S S'.\] In particular, applying this to $\mathrm{id}_{M_\mathrm{CM}}: M_\mathrm{CM}\to M_\mathrm{CM}$ we obtain a sheaf $X\to M_\mathrm{CM}$ and isomorphisms \[X_{S}\overset{\sim}{\longrightarrow} X\times_{M_\mathrm{CM}} S.\]
\subsection{}\label{subsec:props-of-x} Before we consider the geometric properties of the sheaf $X\to M_\mathrm{CM}$, let us first make the following observations.
\begin{enumerate}[label=(\roman*)]
\item There is a unique morphism $0_X: M_\mathrm{CM}\to X$ with the property that if $E/S$ is a CM elliptic curve and $c_{E/S}: S\to M_\mathrm{CM}$ is the coarse map then $c_{E/S}^*(0_X)=0_E: S\to X_{E/S}=c_{E/S}^*(X)$.
\item For each $\a\in \mathrm{Id}_{O_K}$ there is a unique morphism \[\psi_{X/M_\mathrm{CM}}^\a: X\to \sigma_\a^*(X)\] such that for all CM elliptic curves $E/S$, using the identifications \[c_{E\otimes_{O_K}\a^{-1}/S}=\sigma_\a\circ c_{E/S} \quad \text{ and then }\quad c_{E\otimes_{O_K}\a^{-1}/S}^*(X)=c_{E/S}^*(\sigma_\a^*(X))\] the diagram \begin{equation}\label{eqn:def-frob-lattes}\xymatrix{E\ar[rr]^-{i_\a}\ar[d] && E\otimes_{O_K}\a^{-1}\ar[d]\\
c_{E/S}^*(X)\ar[rr]^{c_{E/S}^*(\psi_{X/M_\mathrm{CM}}^\a)} && c_{E/S}^*(\sigma_\a^*(X))}\end{equation} commutes.
\item The morphisms $\psi_{X/M_\mathrm{CM}}^{\ideal{p}}: X\to \sigma_\ideal{p}^*(X)$ for $\ideal{p}$ prime lift the $N\ideal{p}$-power relative Frobeniuses and for $\a,\b$ any two ideals we have the commutativity condition \[\psi_{X/M_\mathrm{CM}}^{\a\b}=\sigma_\a^*(\psi_{X/M_\mathrm{CM}}^\b)\circ \psi_{X/M_\mathrm{CM}}^\a\] which equips $X_{E/S}$ with the structure of a $\Psi_{M_\mathrm{CM}}$-sheaf.
\item The morphism $0_X: M_\mathrm{CM}\to X$ is a $\Psi_{M_\mathrm{CM}}$-morphism.
\end{enumerate}
\begin{prop} The sheaf $f: X\to M_\mathrm{CM}$ is a smooth, projective curve of genus zero. In particular, if $\mc{I}_{X}\subset \mc{O}_X$ denotes the ideal sheaf defining the closed point $0_X: M_\mathrm{CM}\to X$ then $\mc{W}:=f_{X*}(\mc{I}^{-1}_{X})$ is a locally free rank two $\mc{O}_{M_\mathrm{CM}}$-module, the morphism $f_X^*(\mc{W})\to \mc{I}_{X}^{-1}$ is an epimorphism, and the induced map $X\to \mathbf{P}_{M_\mathrm{CM}}(\mc{W})$ is an isomorphism.
\end{prop}
\begin{proof} The fact that $X\to M_\mathrm{CM}$ is a curve is immediate from the fact that $M_\mathrm{CM}$ admits an open cover $(M_i\to M_\mathrm{CM})_{i\in I}$ with CM elliptic curves $E_i/M_i$. Indeed, the defining property of $X\to M_\mathrm{CM}$ then gives \[X\times_{M_\mathrm{CM}} M_i\overset{\sim}{\longrightarrow} X_{E_i/M_i}\] and we know that $X_{E_i/M_i}$ is a smooth of relative dimension one, proper and geometrically connected.
It remains to compute the genus. The genus is constant along the fibres of a smooth, proper geometrically connected curve, and so to compute it we may do so after base change along any morphism $\mathrm{Spec}(K^\mathrm{sep})\to M$. Fixing a CM elliptic curve $E/\mathrm{Spec}(K^\mathrm{sep})$ and considering the degree $w$ finite locally free morphism $p=p_{E/\mathrm{Spec}(K^\mathrm{sep})}: E\to X\times_{M}\mathrm{Spec}(K^\mathrm{sep})$ the Riemann-Hurwitz formula gives: \[2-2g_E=w (2-2 g_X)-\sum_{x\in E(K^\mathrm{sep})} (e_x-1)\] where $g_E$ is the genus of $E$, $g_X$ is the genus of $X\times_{M_\mathrm{CM}}\mathrm{Spec}(K^\mathrm{sep})$ and where $e_x$ is the ramification degree of $p$ at $x$. We have $g_E=1$, and for each $x\in E(K^\mathrm{sep})$ the ramification degree $e_x$ is equal to $\#\mathrm{Stab}(x)-1$ where $\mathrm{Stab}(x)\subset O_K^\times$ is the stabiliser of $x$. It is now just a matter computation, depending on whether $O_K^\times=\mu_2$, $ \mu_4$ or $\mu_6$, to verify that the equality \[0=(2-2g_E)=w (2-2 g_X)-\sum_{x\in E(K^\mathrm{sep})} (e_x-1)\] implies $g_X=0$. We do it for $O_K^\times=\mu_6$. The only point with stabiliser $\mu_6$ is $0\in E(K^\mathrm{sep})$, the points with stabiliser $\mu_2\subset \mu_6$ are the three points of $E[2]-0$ and the points with stabiliser $\mu_3\subset \mu_6$ are the two points of $E[1-\zeta_3]-0$ (for $\zeta_3\in \mu_3$ a generator). Therefore, we find \[0=6\cdot (2-2g_X)-1\cdot (6-1) - 3\cdot (2-1) - 2\cdot (3-1)=-12 g_X\] and hence $g_X=0.$
The other claims now follow using standard arguments from the theory of curves.
\end{proof}
\begin{coro} For each $\a\in \mathrm{Id}_{O_K}$ the morphism $\psi_{X/M_\mathrm{CM}}^\a: X\to \sigma_\a^*(X)$ is finite locally free of degree $N\a$ and together they equip $X$ with the structure of a $\Lambda_{M_\mathrm{CM}}$-scheme and the morphism $0_X: M_\mathrm{CM}\to X$ is a $\Lambda_{M_\mathrm{CM}}$-morphism.
\end{coro}
\begin{proof} As $X$ is flat over $\mathrm{Spec}(O_K)$ it follows by (\ref{prop:lambda-structures-frobenius-lifts-sheaves}) and (iv) of (\ref{subsec:props-of-x}) that $X$ admits a $\Lambda_{M_\mathrm{CM}}$-structure and that the morphism $0_X: M_\mathrm{CM}\to X$ is a $\Lambda_{M_\mathrm{CM}}$-morphism. The only thing we need to verify is that $\psi_{X/M_\mathrm{CM}}^{\a}$ is finite locally free of degree $N\a$ but this follows from the diagram (\ref{eqn:def-frob-lattes}) defining $\psi_{X/M_\mathrm{CM}}^\a$ as both columns are finite locally free of degree $w$ and the top map is finite locally free of degree $N\a$ and hence the bottom map must also be finite locally free of degree $N\a$.
\end{proof}
\subsection{} Write $X[O_K]\subset X$ for the (image of) the morphism $0_X: M_\mathrm{CM}\to X$, and for each $\a\in \mathrm{Id}_{O_K}$ define $X[\a]:=\psi_{X}^{\a*}(X[O_K])\subset X$. Then $X[\a]\subset X$ is a finite locally free $\Lambda_{M_\mathrm{CM}}$-scheme of degree $N\a$.
\begin{prop} For each ideal $\a\in \mathrm{Id}_{O_K}$ the extension $K(X[\a])$ of $K$ generated by the coordinates of $X[\a]$ is equal to ray class field $K(\a)$.
\end{prop}
\begin{proof} As $M_\mathrm{CM}\overset{\sim}{\longrightarrow} \mathrm{Spec}(O_H)$ it is enough to show that the action of $G(K^\mathrm{sep}/H)$ on $X[\a](\mathrm{Spec}(K^\mathrm{sep}))$ factors faithfully through the quotient $G(K^\mathrm{sep}/H)\to G(K(\a)/K)$. To do this we may choose a CM elliptic curve $E/\mathrm{Spec}(H)$ with character $\rho_{E/H}: G(K^\mathrm{sep}/H)\to A_{O_K}^\times$ and consider the map \[p=p_{E/\mathrm{Spec}(H)}: E\to X_{E/\mathrm{Spec}(H)}=X\times_{M_\mathrm{CM}}\mathrm{Spec}(H).\] Then $p$ induces a surjective map of $G(K^\mathrm{sep}/H)$-sets \[E[\a](\mathrm{Spec}(K^\mathrm{sep}))\to X[\a](\mathrm{Spec}(K^\mathrm{sep}))\] and moreover factors through an isomorphism \[E[\a](\mathrm{Spec}(K^\mathrm{sep}))\to E[\a](\mathrm{Spec}(K^\mathrm{sep}))/O_K^\times\overset{\sim}{\longrightarrow} X[\a](\mathrm{Spec}(K^\mathrm{sep})).\] Therefore, an element $\sigma\in G(K^\mathrm{sep}/H)$ acts trivially on $X[\a](\mathrm{Spec}(K^\mathrm{sep}))$ if and only if $\rho_{E/H, \a}(\sigma)\in (O_K/\a)^\times$ is contained in the image of $O_K^\times\to (O_K/\a)^\times$, i.e.\ in the notation of (\ref{subsec:def-morphism-h}) if and only if $[\rho_{E/H, \a}(\sigma)]_\a=1$. But \[[\rho_{E/H, \a}(\sigma)^{-1}]_\a=[\sigma]_{H, \a}=[\sigma]_{K, \a}\] and the kernel of $[-]_{K, \a}$ is precisely $G(K^\mathrm{sep}/K(\a))$ and so we are done.
\end{proof}
\subsection{} We now wish to study the Lie algebra $T:=\underline{\mathrm{Lie}}_{X/M_\mathrm{CM}}$ at the closed point $0_{M_\mathrm{CM}}: M_\mathrm{CM}\to X$.
\begin{prop} For each $\a\in \mathrm{Id}_{O_K}$ the map \[D_\a: T\to \sigma^*_\a(T)\] induced by $\psi_{X/M_\mathrm{CM}}^{\a*}: X\to \sigma^*_\a(X)$ factors as \[T \overset{\sim}{\longrightarrow} \sigma_\a^*(T)\otimes_{O_K}\a^{w}\to \sigma^*_\a(T)\] where the second map is multiplication.
\end{prop}
\begin{proof} It is enough to show that for each $\a$, Zariski locally on $M_\mathrm{CM}$, the image of \[D_\a: T\to \sigma^*_\a(T)\] is equal to $\a^w\otimes_{\mc{O}_{M_\mathrm{CM}}} \sigma_\a^*(T)$.
So let $x\in M_\mathrm{CM}$ be a point and let $S=\mathrm{Spec}(\mc{O}_{M_\mathrm{CM}, x})\to M_\mathrm{CM}$ be the inclusion of the local ring at $x$, and let $E/S$ be a CM elliptic curve. The inverse image of $0: S\to X_{E/S}=X\times_{M_\mathrm{CM}}S$ along the map $p_{E/S}: E\to X_{E/S}$ is equal to $[O_K^\times](0_E)$ and as $0: S\to E$ is invariant under $O_K^\times$ we have $[O_K^\times](0_E)=\mathrm{Inf}_{0_E}^{w-1}(E)$ is the $(w-1)$st infinitesimal neighbourhood of $0_E: S\to E$. It follows that, writing $\widehat{X}=\mathrm{Inf}_{0_X}(X)$ that $p_{E/S}^*(\widehat{X}_S)=\mathrm{Inf}_{0_E}(E)=\widehat{E}$ and that the map \[\widehat{E}\to \widehat{X}\] is finite locally free of degree $w$ and $O_K^\times$ invariant.
It now follows from Proposition 7.5.2 that, choosing an isomorphism $\widehat{E}\overset{\sim}{\longrightarrow} \widehat{\mathbf{A}}^1_S$ there is a unique isomorphism $\widehat{X}_S\overset{\sim}{\longrightarrow} \widehat{\mathbf{A}}^1_S$ such that the diagram \[\xymatrix{\widehat{E}\ar[r]\ar[d]_-\wr & \widehat{X}_S\ar[d]^-\wr\\
\widehat{\mathbf{A}}^1_S\ar[r]^{N_{O_K^\times}(T)} &\widehat{\mathbf{A}}^1_S}\] commutes where the bottom map is \[a\mapsto N_{O_K^\times}(a)=\prod_{\epsilon\in O_K^\times}[\epsilon](a)\] and $[\epsilon](T)$ is the power series on $\widehat{\mathbf{A}}^1_S\overset{\sim}{\longrightarrow} \widehat{E}$ representing the automorphism $\epsilon: \widehat{E}\to \widehat{E}$. But as the action of $O_K$ is strict we known that $[\epsilon](T)=\epsilon T+\cdots$ and so $N_{O_K^\times}(T)=-T^{w}+\cdots$. From this and the fact that the induced map on Lie algebras of $\psi_{E/S}^{\ideal{p}}: E\to \sigma^*_\ideal{p}(E)$ factors as \[\underline{\mathrm{Lie}}_{E/S}\overset{\sim}{\longrightarrow} \ideal{p}\otimes_{\mc{O}_{M_\mathrm{CM}}} \sigma_\ideal{p}^*(\underline{\mathrm{Lie}}_{E/S})\to \sigma_\ideal{p}^*(\underline{\mathrm{Lie}}_{E/S})\] the claim follows.
\end{proof}
\begin{coro}\label{prop:tangent-lattes-free} The rank one $\mc{O}_{M_\mathrm{CM}}$-module $T$ is free and there exists an isomorphism \[X\overset{\sim}{\longrightarrow} \mathbf{P}^1_{M_\mathrm{CM}}.\]
\end{coro}
\begin{proof} We have \[X\overset{\sim}{\longrightarrow} \mathbf{P}_{M_\mathrm{CM}}(\mc{W})\] where $\mc{W}=f_*(\mc{I}_X^{-1})$ where $\mc{I}_X\subset \mc{O}_X$ is the ideal sheaf defining the closed point $0_X: M_\mathrm{CM}\to X$. Moreover, we have an exact sequence \[0\to \mc{O}_{M_\mathrm{CM}}\to \mc{W}\to T\to 0.\] As $T$ is projective this exact sequence splits. Then by the same method as in the proof of (\ref{coro:shimura-curves-minimal-model}) one shows that the isomorphisms \[D_\a: T\overset{\sim}{\longrightarrow}\a^{w}\otimes_{\mc{O}_{M_\mathrm{CM}}}\sigma_\a^*(T)\] can be turned into descent data from $M_\mathrm{CM}$ to $\mathrm{Spec}(O_K)$ and so by the Hauptidealsatz $T$ is free. Therefore, $\mc{W}\overset{\sim}{\longrightarrow} \mc{O}_{M_\mathrm{CM}}^2$ and \[X\overset{\sim}{\longrightarrow} \mathbf{P}_{M_\mathrm{CM}}(\mc{W})\overset{\sim}{\longrightarrow} \mathbf{P}_{M_\mathrm{CM}}(\mc{O}_{M_\mathrm{CM}}^2)=\mathbf{P}_{M_\mathrm{CM}}^1.\]
\end{proof}
\begin{rema} We end this section with a remark regarding possible applications to monogeneity of rings of integers. Define Cartier divisors \[\Theta_\a\subset X[\a]\subset X\] inductively by setting $\Theta_{O_K}=X[O_K]$ and, having defined $\Theta_\a\subset X[\a]$, we define $\Theta_{\a\ideal{p}}$ to be \[\Theta_{\a\ideal{p}}:=\psi_\ideal{p}^*(\sigma_\a^*(\Theta_\a))-\Theta_\a\] if $\ideal{p}\nmid\a$ (an analysis of the $\Lambda_{M_\mathrm{CM}}$-structure on $X[\a]$ shows that this is possible) or to be \[\Theta_{\a\ideal{p}}:=\psi_{X/M_\mathrm{CM}}^{\ideal{p}*}(\sigma_\ideal{p}^*(\Theta_\a))\] if $\ideal{p}|\a$. Then one can show that $\Theta_{\a}\subset X$ is irreducible (but in general non-reduced), that $K(\Theta_\a)=K(\a)$ and that \[X[\a]=\sum_{\d|\a}\Theta_\d.\] The identity above should be viewed as analogous to the factorisation of the polynomials $X^n-1$ in terms of cyclotomic polynomials.
Moreover, when $\a$ is composite $\Theta_\a$ and $X[O_K]=\Theta_{O_K}$ are disjoint so that one finds a closed immersion \[\Theta_\a\subset X-X[O_K]=\mathbf{P}_{M_\mathrm{CM}}-X[O_K](\mc{W})\overset{\sim}{\longrightarrow} \mathbf{V}_{M_\mathrm{CM}}(T)\overset{\sim}{\longrightarrow} \mathbf{A}^1_{M_\mathrm{CM}}.\] With a more detailed analysis of the $\Lambda_{M_\mathrm{CM}}$-structure on $X_{M_\mathrm{CM}}\overset{\sim}{\longrightarrow} \mathbf{P}^1_{O_H}$ it may be possible to show that the divisors $(\Theta_\a)_\mathrm{red}$ are regular which would imply isomorphisms \[(\Theta_\a)_\mathrm{red}\overset{\sim}{\longrightarrow} \mathrm{Spec}(O_{K(\a)}).\] This would then give closed immersion \[\mathrm{Spec}(O_{K(\a)})\to \mathbf{A}^1_{O_H}\overset{\sim}{\longrightarrow} \mathbf{A}^1_{M_\mathrm{CM}}\] or in other words $O_{K(\a)}$ would be monogenic over the ring of integers in the Hilbert class field $O_H$. It is worth noting that this method would only apply to conductors $\a$ which are composite, and that when $\a$ is not composite counterexamples to the monogeneity of $O_{K(\a)}$ over $O_{H}$ are known to exist (see \cite{CouFleck89}).
\end{rema}
\section{A $\Lambda$-equivariant cover of $\mc{M}_\mathrm{CM}$}\label{sec:lambda-cover-of-m} In this section we show how one can use the existence of canonical lifts of CM elliptic curves to define a flat, affine and formally smooth cover of $\mc{M}_\mathrm{CM}$ admitting a $\Lambda$-structure compatible with that on $\mc{M}_\mathrm{CM}$. Indeed, we rigidify $\mc{M}_\mathrm{CM}$ by equipping a CM elliptic curve $E/S$ with a trivialisation of the Lie algebra of its canonical lift. The results of (\ref{sec:shimura-curves}) allow us to show that this does indeed define a cover of $\mc{M}_\mathrm{CM}$ with the desired properties.
\subsection{} Let $S$ be an ind-affine scheme. The category $\mc{CL}_{O_K}(S)$ acts on the category $\mc{M}_\mathrm{CM}^*(S)$. However, we have shown that $\mc{M}_\mathrm{CM}(S)$ is equivalent to the category $W_*(\mc{M}_{\mathrm{CM}})_\Lambda(S)$ so that $\mc{CL}_{O_K}$ should also act on $W_*(\mc{M}_{\mathrm{CM}})_\Lambda$ and now explain how.
As $W^*$ commutes with \'etale fibre products and $\underline{O_K}_S$ is \'etale over $S$, the sheaf $W^*(\underline{O_K}_S)$ is naturally a $\Lambda_{W^*(S)}$-sheaf of rings over $W^*(S)$. Moreover, $W^*$ also commutes with disjoint unions which gives an identification \[W^*(\underline{O_K}_S)\overset{\sim}{\longrightarrow} \underline{O_K}_{W^*(S)}\] compatible with the ring structures.
In much the same way, if $\mc{L}$ is a rank one $O_K$-local system over $S$, then $W^*(\mc{L})$ is a rank one $O_K$-local system over $W^*(S)$ equipped with a $\Lambda_{W^*(S)}$-structure which is compatible with its $\underline{O_K}_{W^*(S)}=W^*(\underline{O_K}_S)$-module structure, i.e.\ the map \[\underline{O_K}_{W^*(S)}\times_{W^*(S)}W^*(mc{L})\to W^*(\mc{L})\] defining the action of $\underline{O_K}_{W^*(S)}$ on $W^*(\mc{L})$ is a $\Lambda_{W^*(S)}$-morphism.
Now if $E/S$ is a CM elliptic curve then, as both $W_\mathrm{CM}^*(E)$ and $W^*(\mc{L})$ have $\Lambda_{W^*(S)}$-structures compatible with their $\underline{O_K}_S$-module structures, and as the forgetful functor $\mathrm{Sh}_{\Lambda_{W^*(S)}}\to \mathrm{Sh}_{W^*(S)}$ commutes with all limits and colimits (\ref{def:w-w-lambda-adjunction}), the CM elliptic curve \[W_\mathrm{CM}^*(E)\otimes_{O_K}W^*(\mc{L})\] is also equipped with a $\Lambda_{W^*(S)}$-structure compatible with its $\underline{O_K}_{W^*(S)}$-module structure.
\begin{prop}\label{prop:action-of-cl-on-can-lifts} The $\Lambda_{W^*(S)}$ structure on $W_\mathrm{CM}^*(E)\otimes_{O_K}W^*(\mc{L})$ is canonical and there is a unique $\Lambda_{W^*(S)}$-isomorphism \[W_\mathrm{CM}^*(E\otimes_{O_K}\mc{L})\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E)\otimes_{O_K} W^*(\mc{L})\] inducing the identity on the ghost components at $(1)$.
\end{prop}
\begin{proof} The $\Psi_{\Gamma^*(S)}$-structure on $W^*(\mc{L})\times_{W^*(S)}\Gamma^*(S)$ is compatible with the isomorphisms \[W^*(\mc{L})\times_{W^*(S)}\Gamma^*(S)\overset{\sim}{\longrightarrow} \Gamma^*(\mc{L})\overset{\sim}{\longrightarrow} \mc{L}\times_{S}\Gamma^*(S)\] where the last fibre product is over the sum of the identity maps \[\Gamma^*(S)=\coprod_{\a\in \mathrm{Id}_{O_K}} S\to S\] and where the $\Psi_{\Gamma^*(S)}$-structure on $\mc{L}\times_{S}\Gamma^*(S)$ is induced by the $\Psi$-structure on $\Gamma^*(S)$. Therefore, we obtain $\Psi_{\Gamma^*(S)}$-isomorphisms \[(W_\mathrm{CM}^*(E)\otimes_{O_K}W^*(\mc{L}))\times_{W^*(S)}\Gamma^*(S)\overset{\sim}{\longrightarrow} \Gamma_{\mathrm{CM}}^*(E)\otimes_{O_K}\Gamma^*(\mc{L})\overset{\sim}{\longrightarrow} \Gamma_{\mathrm{CM}}^*(E\otimes_{O_K}\mc{L})\] which induce the identity after pull-back to the ghost component at $(1)$. This is precisely the definition of a canonical $\Lambda_{W^*(S)}$-structure and this proves the first statement. As the ghost components at $(1)$ of $W_\mathrm{CM}^*(E)\otimes_{O_K}W^*(\mc{L})$ and $W_\mathrm{CM}^*(E\otimes_{O_K}\mc{L})$ are equal to $E\otimes_{O_K}\mc{L}$ we get a unique $\Lambda_{W^*(S)}$-isomorphism \[W_\mathrm{CM}^*(E)\otimes_{O_K}W^*(\mc{L})\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E\otimes_{O_K}\mc{L})\] inducing the identity after pull-back along the ghost component at $(1)$.
\end{proof}
\begin{rema}\label{rema:structure-map-injective} Consider the sheaf \[W_*(\mathbf{A}^1)=W_*(\mathrm{Spec}(O_K[T]))=\mathrm{Spec}(\Lambda\odot O_K[T])=\mathrm{Spec}(\Lambda)\] of arithmetic jets of $\mathbf{A}^1$ over $\mathrm{Spec}(O_K)$. It is a ring scheme over $\mathrm{Spec}(O_K)$ and its sections over an affine scheme $S=\mathrm{Spec}(A)$ are given by \[W_*(\mathbf{A}^1)(\mathrm{Spec}(A))=W(A).\] The structure map \[O_K\to W(A)=W_*(\mathbf{A}^1)(S)\] for varying affine schemes $S=\mathrm{Spec}(A)$ is injective (\ref{coro:structure-map-lambda-ring-injective}) and induces a \textit{monomorphism} of sheaves of rings \begin{equation} \label{def:structure-injective} i_{W}:\underline{O_K}\to W_*(\mathbf{A}^1).\end{equation} This fact will be crucial for what follows.
\end{rema}
\subsection{} Let $\mc{L}/S$ be a rank one $O_K$-local system. A level-$W$ structure on $\mc{L}$ is an isomorphism of $\mc{O}_{W^*(S)}$-modules \[\lambda: W^*(\mc{L})\otimes_{\underline{O_K}_{W^*(S)}} \mc{O}_{W^*(S)}\overset{\sim}{\longrightarrow} \mc{O}_{W^*(S)}.\] A $W$-isomorphism $(\mc{L}/S, \lambda)\overset{\sim}{\longrightarrow} (\mc{L}'/S, \lambda')$ of rank one $O_K$-local systems with level-$W$ structure is an $\underline{O_K}_S$-isomorphism $h: \mc{L}\overset{\sim}{\longrightarrow} \mc{L}'$ such that \[\lambda=\lambda'\circ W^*(h).\]
The tensor product of two rank one $O_K$-local systems with level-$W$ structure $(\mc{L}, \lambda)$ and $(\mc{L}', \lambda')$ is defined to be \[(\mc{L}\otimes_{O_K}\mc{L}', \lambda\otimes_{\mc{O}_{W^*(S)}}\lambda')\] where we view $\lambda\otimes_{\mc{O}_{W^*(S)}}\lambda'$ as a level-$W$ structure on $\mc{L}\otimes_{O_K}\mc{L}'$ using the identification \[W^*(\mc{L}\otimes_{O_K}\mc{L}')\overset{\sim}{\longrightarrow} W^*(\mc{L})\otimes_{O_K}W^*(\mc{L}').\]
We write $\mc{CL}_{O_K}^W$ for the fibred category over $\mathrm{IndAff}_{O_K}$ with fibre over $S$ given by the category rank one $O_K$-local systems over $S$ equipped with a level-$W$ structure $(\mc{L}/S, \lambda)$ together with their $W$-isomorphisms.
\begin{prop}\label{prop:cw-stack-sheaf} $\mc{CL}_{O_K}^W$ is a stack over $\mathrm{IndAff}_{O_K}$ for the \'etale topology and is equivalent to its coarse sheaf for the \'etale topology.
\end{prop}
\begin{proof} That $\mc{CL}_{O_K}^W$ is a stack for the affine \'etale topology on $\mathrm{IndAff}_{O_K}$ is easy to see using the fact that if $\mc{L}/S$ is a rank one $O_K$-local system and $S'\to S$ is any morphism then \[W^*(\mc{L})\times_{W^*(S)}W^*(S')\overset{\sim}{\longrightarrow} W^*(\mc{L}\times_S S')\] and that for any \'etale morphism $T\to S$ we have \[W^*(S'\times_S T)\overset{\sim}{\longrightarrow} W^*(S')\times_{W^*(S)}W^*(T).\]
For the second statement it is enough to show that if $(\mc{L}, \lambda)$ is a rank one $O_K$-local system with level-$W$ structure then every $W$-automorphism of $\mc{L}$ is trivial. But if $\epsilon\in \underline{O_K^\times}_S(S)$ defines a $W$-automorphism of $\mc{L}$, we have an equality \begin{equation} \mathrm{id}_{W^*(\mc{L})}\otimes_{\underline{O_K}_{W^*(S)}} i_W(\epsilon) = W^*(\epsilon)\otimes_{\underline{O_K}_{W^*(S)}}\mathrm{id}_{\mc{O}_{W^*(S)}}\label{eqn:w-autom}\end{equation} of automorphisms of $W^*(\mc{L})\otimes_{\underline{O_K}_{W^*(S)}}\mc{O}_{W^*(S)}$. As $i_W$ is a monomorphism and the $\rho$ is invariant under the automorphism (\ref{eqn:w-autom}) we must have $\epsilon=1.$
\end{proof}
\subsection{} We write $CL_{O_K}^W$ for the coarse sheaf of $\mc{CL}_{O_K}^W$ with which we identify it by (\ref{prop:cw-stack-sheaf}). The tensor product of rank one $O_K$-local systems with level-$W$ structure equips $CL_{O_K}^W$ with the structure of a sheaf of groups over $\mathrm{Spec}(O_K)$.
We now describe a short exact sequence relating $CL_{O_K}^W$ to the $\mathrm{Spec}(O_K)$-group scheme of arithmetic jets $W_*(\mathbf{G}_\mathrm{m})$ of $\mathbf{G}_{\mathrm{m}}$.
Let $S=\mathrm{Spec}(A)$ be an affine $\mathrm{Spec}(O_K)$-scheme. The sections of $W_*(\mathbf{G}_\mathrm{m})$ over $S$ are given by \[W_*(\mathbf{G}_\mathrm{m})(\mathrm{Spec}(A))=W(A)^\times\] and the monomorphism (\ref{def:structure-injective}) restricts to a monomorphism again denoted \[i_W: \underline{O_K^\times}\to W_*(\mathbf{G}_{\mathrm{m}}).\]
For each \[a\in \mathbf{G}_{\mathrm{m}}(W^*(S))=\mathrm{Aut}_{W^*(S)}(\mc{O}_{W^*(S)})\] we define an element $[a]_W\in CL_{O_K}^W(S)$ by $[a]_W:=(\underline{O_K}_S, a)$ where we view $a$ as the level-$W$ structure on $\underline{O_K}_S$: \[\underline{O_K}_{W^*(S)}\otimes_{\underline{O_K}_{W^*(S)}}\mc{O}_{W^*(S)}=\mc{O}_{W^*(S)}\stackrel{a}{\to}\mc{O}_{W^*(S)}.\] This defines a homomorphism \[[-]_W:W_*(\mathbf{G}_\mathrm{m})\to CL_{O_K}^W\]
Finally, composing the forgetful map \[CL_{O_K}^W\to \mc{CL}_{O_K}: (\mc{L}/S, \lambda)\mapsto \mc{L}/S\] with the map $\mc{CL}_{O_K}\to\underline{CL_{O_K}}$ of (\ref{prop:class-stacks-triv-admis}) we obtain a homomorphism \[f_W: CL_{O_K}^W\to \underline{CL_{O_K}}.\]
\begin{prop} The sequence of sheaves \[0\to \underline{O_K^\times}\stackrel{i_W}{\longrightarrow} W_*(\mathbf{G}_\mathrm{m})\stackrel{[-]_W}{\longrightarrow} CL_{O_K}^W\stackrel{f_W}{\longrightarrow} \underline{\mathrm{CL}_{O_K}}\to 0\] is exact for the \'etale topology and $CL_{O_K}^W$ is representable by a flat, affine formally smooth group scheme over $\mathrm{Spec}(O_K)$.
\end{prop}
\begin{proof} We first show that $f_W: CL_{O_K}^W\to \underline{CL_{O_K}}$ is an epimorphism for the \'etale topology. This is equivalent to showing that for each ideal $\a$ (or at least one in each ideal class of $\mathrm{CL}_{O_K}$) there is an \'etale cover $S\to \mathrm{Spec}(O_K)$ with the property that the $O_K$-local system $\underline{\a}_{S}$ admits a level-$W$ structure.
Let $S=\mathrm{Spec}(O_H)\to \mathrm{Spec}(O_K)$. Then we have an isomorphism of $\mc{O}_{W^*(S)}$-modules \[\underline{\a}_{W^*(S)}\otimes_{O_K}\mc{O}_{W^*(S)}\overset{\sim}{\longrightarrow} \a\cdot \mc{O}_{W^*(S)}\] where $\a\cdot \mc{O}_{W^*(S)}$ is the ideal sheaf defining the closed immersion $W^*(S)\times_{\mathrm{Spec}(O_K)}\mathrm{Spec}(O_K/\a)\to W^*(S)$. However, this map is obtained by pulling-back the map $S\times_{\mathrm{Spec}(O_K)}\mathrm{Spec}(O_K/\a)\to S$ along the morphism $\mu_S: W^*(S)\to S$ defining the $\Lambda$-structure on $S=\mathrm{Spec}(O_H)$. Therefore, it is enough to show that the ideal sheaf defining the closed immersion $S\times_{\mathrm{Spec}(O_K)}\mathrm{Spec}(O_K/\a)\to S$ is free, but this sheaf is $\a\otimes_{O_K} O_H$ which is free by the Hauptidealsatz (\ref{prop:hauptidealsatz}).
We now show that the map $[-]_W: W_*(\mathbf{G}_\mathrm{m})\to CL_{O_K}^W$ defines an epimorphism onto the kernel of $f_W$. It is clear that $[-]_W$ maps to $\ker(f_W)$ as the rank one $O_K$-local system underling $[-]_W$ is the trivial one. Now let $S$ be an affine scheme and let $(\mc{L}/S, \lambda)\in \ker(f_W)$. We will show that there exists a cover $(S_i\to S)_{i\in I}$ and elements $a_i\in W_
*(\mathbf{G}_\mathrm{m})(S_i)$ with $[a_i]=(\mc{L}_{S_i}, \lambda_{S_i})$.
Since $(\mc{L}, \lambda)\in \ker(f_W)$ it follows that $\mc{L}$ is \'etale locally isomorphic $\underline{O_K}_S$. Therefore, we may assume that $(\mc{L}, \lambda)=(\underline{O_K}_S, \lambda)$ but then the isomorphism \[\lambda: \underline{O_K}_S\otimes_{\underline{O_K}_S}\mc{O}_{W^*(S)}=\mc{O}_{W^*(S)}\overset{\sim}{\longrightarrow} \mc{O}_{W^*(S)}\] is given by some $a\in W_*(\mathbf{G}_{\mathrm{m}})(S)=\mathrm{Aut}_{W^*(S)}(\mc{O}_{W^*(S)})$ and we have \[(\underline{O_K}_S, \lambda)=(\underline{O_K}_S, a)=[a]_W.\]
Finally, we compute the kernel of $[-]_W$. So let $a\in W_*(\mathbf{G}_\mathrm{m})(S)$ and assume that $[a]_W=(\underline{O_K}_S, a)=(\underline{O_K}_S, \mathrm{id}_{\mc{O}_{W^*(S)}})$. By definition, this implies the existence of an isomorphism \[\epsilon\in \mathrm{Isom}_S^{O_K}(\underline{O_K}_S, \underline{O_K}_S)=\underline{O_K^\times}_S(S)\] such that $W^*(\epsilon)=a$. But $W^*(\epsilon)$ viewed as an element of \[\underline{\mathrm{Aut}}_{\mc{O}_{W^*(S)}}(\mc{O}_{W^*(S)})=W_*(\mathbf{G}_\mathrm{m})(S),\] is precisely $i_W(\epsilon)$. Therefore $a=i_W(\epsilon)$ and $\ker([-]_W)=\underline{O_K}^\times\subset W_*(\mathbf{G}_\mathrm{m}).$
It follows from the above that the kernel of $f_W$ is equal to the sheaf of groups \[W_*(\mathbf{G}_{\mathrm{m}})/\underline{O_K^\times}.\] However, $\underline{O_K^\times}$ is finite \'etale and $W_*(\mathbf{G}_{\mathrm{m}})$ is flat and affine and so it follows that the quotient sheaf \[W_*(\mathbf{G}_{\mathrm{m}})/\underline{O_K^\times}\] is also flat and affine. As $\underline{CL_{O_K}}$ is affine, $f_W$ is an epimorphism and \[CL_{O_K}^W\times_{\underline{CL}_{O_K}} CL_{O_K}^W\overset{\sim}{\longrightarrow} CL_{O_K}^W\times W_*(\mathbf{G}_{\mathrm{m}})/\underline{O_K^\times}\] is affine and flat it follows that $CL_{O_K}^W$ is affine and flat over $\mathrm{Spec}(O_K)$. Similarly, as $W_*(\mathbf{G}_{\mathrm{m}})$ is formally smooth (being an inverse limit of smooth affine schemes) and $W_*(\mathbf{G}_{\mathrm{m}})\to W_*(\mathbf{G}_{\mathrm{m}})/\underline{O_K}^\times$ is \'etale it follows that $W_*(\mathbf{G}_{\mathrm{m}})/\underline{O_K}^\times$ is formally smooth and by descent that $CL_{O_K}^W$ is formally smooth.
\end{proof}
\subsection{} Let $S$ be an affine scheme and let $E/S$ be a CM elliptic curve. A level-$W$ structure on $E/S$ is an isomorphism of $\mc{O}_{W^*(S)}$-modules \[\rho: \underline{\mathrm{Lie}}_{W_{\mathrm{CM}}^*(E)/W^*(S)}\overset{\sim}{\longrightarrow} \mc{O}_{W^*(S)}.\] A $W$-isomorphism $(E/S, \rho)\overset{\sim}{\longrightarrow} (E'/S, \rho')$ of CM elliptic curves with level-$W$ structures is an isomorphism $f: E\overset{\sim}{\longrightarrow} E'$ of CM elliptic curves such that \[\rho = \rho'\circ \underline{\mathrm{Lie}}_{W_{\mathrm{CM}}^*(E')/W^*(S)}(W_{\mathrm{CM}}^*(f)).\]
We denote by $\mc{M}_\mathrm{CM}^W$ the fibred category over $\mathrm{IndAff}_{O_K}$ whose fibre over an ind-affine scheme $S$ is given by the category of CM elliptic curves with level-$W$ structures together with their $W$-isomorphisms.
Just as with $CL_{O_K}^W$, the objects of $\mc{M}_\mathrm{CM}^W$ admit no non-trivial automorphisms and so the stack $\mc{M}_\mathrm{CM}^W$ is equivalent to its coarse sheaf which we denote by $M_\mathrm{CM}^W$.
There is an action of $CL_{O_K}^W$ on $M_\mathrm{CM}^W$ given by \[M_\mathrm{CM}^W\times CL_{O_K}^W\to M_\mathrm{CM}^W: ((\mc{L}/S, \lambda), (E/S, \rho))\mapsto (E\otimes_{O_K}\mc{L}, \rho\otimes_{\mc{O}_{W^*(S)}}\lambda)\] where we use the identification (\ref{prop:action-of-cl-on-can-lifts}) \[W^*_{\mathrm{CM}}(E\otimes_{O_K}\mc{L})\overset{\sim}{\longrightarrow} W^*_\mathrm{CM}(E)\otimes_{O_K} W^*(\mc{L}).\]
\begin{theo} We have the following:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The forgetful map \[M_{CM}^W\to \mc{M}_\mathrm{CM}\] is affine, faithfully flat and formally smooth.
\item $M_{CM}^W$ is an $CL_{O_K}^W$-torsor over $\mathrm{Spec}(O_K)$ and is therefore flat, affine and formally smooth over $\mathrm{Spec}(O_K)$.
\item The map \[M_{CM}^W\times_{\mathrm{Spec}(O_K)} W_*(\mathbf{G}_\mathrm{m})\to M_{CM}^W\times_{\mc{M}_\mathrm{CM}}M_{CM}^W: ((E/S, \rho), a)\mapsto ((E/S, \rho), (E/S, a\rho))\] is an isomorphism.
\end{enumerate}
\end{theo}
\begin{proof} (i) Let $E/S$ be CM elliptic curve and write \[T=\underline{\mathrm{Lie}}_{W^*_{\mathrm{CM}}(E)/W^*(S)}.\] Then the fibre of the map $\mc{M}_{CM}^W\to \mc{M}_\mathrm{CM}$ along $S\stackrel{E}{\to} \mc{M}_\mathrm{CM}$ is given by \[t_{E/S}: W_*(\underline{\mathrm{Isom}}_{\mc{O}_{W^*(S)}}(T, \mc{O}_{W^*(S)}))\times_{W_*(W^*(S))} S\to S\] and so to the prove the claim we need only show that $t_{E/S}$ is affine, faithfully flat and formally smooth.
We now show that $t_{E/S}$ is affine, faithfully flat and formally smooth whenever $E/S$ admits a level-$W$ structure. Indeed, if $(E/S, \rho)$ is a level-$W$ structure on $E/S$ then we obtain an isomorphism \[\mathbf{G}_{\mathrm{m}}\times W^*(S)\overset{\sim}{\longrightarrow} \underline{\mathrm{Isom}}_{\mc{O}_{W^*(S)}}(T, \mc{O}_{W^*(S)}): a\mapsto a\cdot \rho\] and so an isomorphism \[W_*(\mathbf{G}_{\mathrm{m}}\times W^*(S))\times_{W_*(W^*(S))} S \overset{\sim}{\longrightarrow} W_*(\underline{\mathrm{Isom}}_{\mc{O}_{W^*(S)}}(T, \mc{O}_{W^*(S)}))\times_{W_*(W^*(S))} S.\] But \[W_*(\mathbf{G}_\mathrm{m}\times W^*(S))\times_{W_*(W^*(S))} S = W_*(\mathbf{G}_{\mathrm{m}})\times S\to S\] is faithfully flat, affine and formally smooth as \[W_*(\mathbf{G}_{\mathrm{m}})=\lim_{\a\in \mathrm{Id}_{O_K}} W_{\a*}(\mathbf{G}_{\mathrm{m}})\] is an inverse limit of faithfully flat, affine and smooth $\mathrm{Spec}(O_K)$-schemes.
As the morphism $t_{E/S}$ is compatible with base change in order to finish the proof of our claim, we may localise $S$ and in particular assume that $E$ admits a level-$\ideal{f}$ structure for some $\ideal{f}$ which separates units. We may now assume that $S=M_\mathrm{CM}^{(\ideal{f})}$ and that $E=E^{(\ideal{f})}$ and, by the previous arguments, to show that $t_{E^{(\ideal{f})}/M_\mathrm{CM}^{(\ideal{f})}}$ is faithfully flat, affine and formally smooth it is enough to show that $E^{(\ideal{f})}/M_\mathrm{CM}^{(\ideal{f})}$ admits a level-$W$ structure. To ease notation, let us write $M=M_\mathrm{CM}^{(\ideal{f})}$, $E=E^{(\ideal{f})}$, $P=\mathrm{Id}_{O_K}^{(\ideal{f})}$ and $P'\subset \mathrm{Id}_{O_K}$ for the sub-monoid generated by the prime divisors of $\ideal{f}$ (so that $P\cap P'=\{O_K\}$ and $P\cdot P'=\mathrm{Id}_{O_K}$).
Then $W^*(M)=W_P^*(W_{P'}^*(M))$ and as $\ideal{f}$ is invertible on $M$ we have $\Gamma^*_{P'}(M)=W^*_{P'}(M)$ so that \[W^*(M)=\coprod_{\a\in P'}W_P^*(M).\] Using this and the fact that the $\Lambda_{P, M}$-structure on $E$ is canonical (\ref{prop:canonical-lambda-structures-and-level-structures}) we find \[W^*_{CM}(E)\overset{\sim}{\longrightarrow} \coprod_{\a\in P'} \mu_M^*(E)\otimes_{O_K} \a^{-1}= \coprod_{\a} \mu_M^*(E\otimes_{O_K} \a^{-1})\] where \[\mu_M: W^*_{P}(M)\to M\] defines the (unique) $\Lambda_{P}$-structure on $M$.
Now to show that $E/M$ admits level-$W$ structure it is enough to show that \[\underline{\mathrm{Lie}}_{E\otimes_{O_K}\a^{-1}/M}=\underline{\mathrm{Lie}}_{E/M}\otimes_{O_K}\a^{-1}=\underline{\mathrm{Lie}}_{E/M}\] (the last equality is because $\a\in P'$ is invertible on $M$) is free for each $\a\in P'$ but this is (\ref{coro:shimura-curves-minimal-model}).
(ii) Let $(E/S, \rho)$ and $(E'/S, \rho')$ be a pair of CM elliptic curves with level-$W$ structures. Then $E'\overset{\sim}{\longrightarrow} E\otimes_{O_K}\mc{L}$ for some $\mc{L}\in \mc{CL}_{O_K}(S)$ and therefore \[W_\mathrm{CM}^*(E)\otimes_{O_K} W^*(\mc{L})\overset{\sim}{\longrightarrow} W^*_\mathrm{CM}(E\otimes_{O_K}\mc{L})=W^*_\mathrm{CM}(E').\] We then have \begin{eqnarray*}\underline{\mathrm{Lie}}_{W^*_\mathrm{CM}(E')/W^*(S)}&=&\underline{\mathrm{Lie}}_{W^*_\mathrm{CM}(E\otimes_{O_K} \mc{L})/W^*(S)}\\&=&\underline{\mathrm{Lie}}_{W^*_\mathrm{CM}(E)\otimes_{O_K} W^*(\mc{L})/W^*(S)}\\&=&\underline{\mathrm{Lie}}_{W^*_\mathrm{CM}(E)/W^*(S)}\otimes_{O_K} W^*(\mc{L})\end{eqnarray*} so that the isomorphism \[\rho': \underline{\mathrm{Lie}}_{W^*_\mathrm{CM}(E')/W^*(S)}=\underline{\mathrm{Lie}}_{W^*_\mathrm{CM}(E)/W^*(S)}\otimes_{O_K} W^*(\mc{L})\overset{\sim}{\longrightarrow} \mc{O}_{W^*(S)}\] must be of the form $\rho\otimes_{\mc{O}_{W^*(S)}}\lambda$ where $\lambda: W^*(\mc{L})\otimes_{O_K} \mc{O}_{W^*(S)}\to \mc{O}_{W^*(S)}$ is a level-$W$ structure on $\mc{L}$. Therefore \[(E'/S, \rho')=(\mc{L}/S, \lambda) \cdot (E/S, \rho)=(E\otimes_{O_K}\mc{L}, \rho\otimes_{\mc{O}_{W^*(S)}}\lambda)\] and the action of $CL_{O_K}^W$ on $M_\mathrm{CM}^W$ is transitive.
Let us now see that it is free which is equivalent to the claim that if $(E/S, \rho)$ and $(\mc{L}/S, \lambda)$ are a CM elliptic curve and rank one $O_K$-local system with level-$W$ structure then there exists an $W$-isomorphism \[f: (E/S, \rho)\overset{\sim}{\longrightarrow} (E\otimes_{O_K}\mc{L}, \rho\otimes \lambda)\] only if there exists a $W$-isomorphism $(\underline{O_K}_S, 1)\overset{\sim}{\longrightarrow} (\mc{L}, \lambda)$.
So let $f$ be such an isomorphism. Then $f: E\overset{\sim}{\longrightarrow} E\otimes_{O_K}\mc{L}$ is of the form $\mathrm{id}_{E}\otimes_{O_K} h$ for a unique isomorphism $ h : \underline{O_K}_S\overset{\sim}{\longrightarrow} \mc{L}$ (\ref{theo:mcm-is-a-torsor}) and as $f$ is compatible with the level-$W$ structures $\rho$ and $\lambda\otimes \rho$, the isomorphism $h$ defines a $W$-isomorphism \[(\underline{O_K}_S, \mathrm{id}_{\mc{O}_{W^*(S)}})\overset{\sim}{\longrightarrow} (\mc{L}, \lambda)\] and our claim follows.
Therefore, the action of $CL_{O_K}^W$ on $M_\mathrm{CM}^W$ is free and transitive so that to show $M_\mathrm{CM}^W$ is a torsor it is enough to show that the structure map $M_\mathrm{CM}^W\to \mathrm{Spec}(O_K)$ is an epimorphism, but this follows from the work done in (ii) showing that any CM elliptic curve $E/S$ \'etale locally admits a level-$W$ structure.
(iii) It is clear that any pair of level-$W$ structures on a CM elliptic curve $E/S$ differ by scaling by an element of $W_*(\mathbf{G}_\mathrm{m})(S)$ which is the claim.
\end{proof}
\subsection{} We now equip $M_{\mathrm{CM}}^W$ with a $\Lambda$-structure. As $M_\mathrm{CM}^W$ is flat and affine to give $M_\mathrm{CM}^{W}$ a $\Lambda$-structure it is enough to define a commuting family of Frobenius lifts $\psi_{M_\mathrm{CM}^W}^\ideal{p}$ for each prime ideal $\ideal{p}$. We now set $\psi_{M_\mathrm{CM}^W}^\ideal{p}$ to be the map defined on $S$-sections \[(E/S, \rho)\mapsto (E\otimes_{O_K}\ideal{p}^{-1}/S, \psi^{\ideal{p}*}(\rho))\] where $\psi^{\ideal{p}*}(\rho)$ can be viewed as a level-$W$ structure on $E\otimes_{O_K}\ideal{p}^{-1}$ via the identifications \[\psi^{\ideal{p}*}(W_\mathrm{CM}^*(E))\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E)\otimes_{O_K}\ideal{p}^{-1}\overset{\sim}{\longrightarrow} W_\mathrm{CM}^*(E\otimes_{O_K}\ideal{p}^{-1}).\] It is clear that these maps commute and that they lift the $N\ideal{p}$-power Frobenius endomorphisms modulo $\ideal{p}$. Finally, this equips the stack $\mc{M}_\mathrm{CM}$ with a flat affine presentation \[\xymatrix{CL_{O_K}^W\times M_\mathrm{CM}^W \ar@<.5ex>[r]\ar@<-.5ex>[r] & M_\mathrm{CM}^W\ar[r] & \mc{M}_\mathrm{CM}}\] and where the two parallel arrows are morphisms of $\Lambda$-schemes, again expressing the `fact' that $\mc{M}_\mathrm{CM}$ is a $\Lambda$-stack.
\section{Perfect $\Lambda$-schemes and Tate modules} In this final section we exhibit a rather interesting relationship the Tate module of a canonical CM elliptic curve over an $\Lambda$-ind-affine-scheme and a certain deformation of it to the perfection of $S$, which is the universal $\Lambda$-ind-affine-scheme under $S$ on which the Frobenius lifts are isomorphisms. We then show that the canonical lift $E/S$ can be deformed to a canonical CM elliptic curve $E_\mathrm{per}/S^\mathrm{per}$. We end the section by making some remarks about the relationship of this exact sequence with periods, both $p$-adic and analytic.
\subsection{} We first define the Tate module of an arbitrary CM elliptic curve. So let $S$ be an ind-affine scheme $E/S$ a CM elliptic curve. The Tate module of $E/S$ is defined to be the pro-finite locally free $S$-group scheme \[T(E):=\lim_\a E[\a]\otimes_{O_K}\a\] where the transition maps are induced multiplication \[E[\a\b]\otimes_{O_K}\a\b\to E[\a]\otimes_{O_K}\a.\] We also define the universal cover of $E/S$ to be the sheaf \[\widetilde{E}:=\lim_\a E\otimes_{O_K}\a\] where again the transition maps are induced by multiplication. The inclusion $T(E)\to \widetilde{E}$ identifies $T(E)$ with the kernel of the projection $\widetilde{E}\to E$ so that we have an exact sequence of sheaves (for the fpqc topology) \begin{equation}\label{eqn:univ-cover-tate}0\to T(E)\to \widetilde{E}\to E\to 0.\end{equation}
\subsection{} Now let $S$ be a $\Lambda$-ind-affine scheme. The perfection $S^\mathrm{per}$ of $S$ is the $\Lambda$-sheaf defined by \[S^\mathrm{per}=\colim_{\a\in\mathrm{Id}_{O_K}, \psi^\a} S \label{eqn:colim-def-perfect}\] where the transition maps are the Frobenius lifts. The action of the monoid of ideals $\mathrm{Id}_{O_K}$ on $S^\mathrm{per}$ is now by automorphisms so that it extends to an action of the group of fractional ideals $\mathrm{Id}_K$. For $\a\in \mathrm{Id}_K$ (now a fractional ideal) we write $\psi^{\a}_{S_\mathrm{per}}: S^\mathrm{per}\to S^\mathrm{per}$ for the corresponding automorphism. It follows immediately from the definition that $S^\mathrm{per}$ is universal among $\Lambda$-sheaves lying under $S$ on which the Frobenius lifts are isomorphisms. When $S=W^*(T)$ is the Witt vectors of an ind-affine scheme we write \[\widehat{W}^*(T)=W^*(T)^\mathrm{per}.\]
Viewing $S$ as a $\Lambda_{S^\mathrm{per}}$-ind-affine scheme via the element of the colimit (\ref{eqn:colim-def-perfect}) corresponding to $O_K\in \mathrm{Id}_{O_K}$ we see that the structure map of the element of the colimit (\ref{eqn:colim-def-perfect}) corresponding $\a$ is \[S\to S^\mathrm{per} \stackrel{\psi^{\a^{-1}}_{S^\mathrm{per}}}{\to}S^\mathrm{per}\] and the colimit (\ref{eqn:colim-def-perfect}) can be rewritten as the colimit of $S^\mathrm{per}$-sheaves \begin{equation}\label{w-perf-int-colim}S^\mathrm{per}=\colim_{\a\in \mathrm{Id}_{O_K}, \psi_\a}\psi^{\a^{-1}}_{S^\mathrm{per}!}(S).\end{equation}
\subsection{} If $E/S$ is a CM elliptic curve equipped with a canonical $\Lambda$-structure then we have a natural identification $E\otimes_{O_K}\a^{-1}\overset{\sim}{\longrightarrow} \psi^{\a*}_S(E)$. Hence for $\a, \b\in \mathrm{Id}_{O_K}$ we have isomorphisms \[E\otimes \b\overset{\sim}{\longrightarrow} \psi^{\a*}_S(E\otimes \a\b)\] and this gives for all $\a$ and $\b$ a Cartesian diagram \begin{equation}\label{dia:cartesian-perfect}\begin{gathered}\xymatrix{E\otimes_{O_K}\b\ar[r]\ar[d] & E\otimes_{O_K}\a\b\ar[d]\\
\psi_{S^\mathrm{per}!}^{\b^{-1}}(S)\ar[r]^{\psi^\a} & \psi_{S^\mathrm{per} !}^{(\a\b)^{-1}}(S).}\end{gathered}\end{equation} We then define a $\Lambda$-sheaf over $S^\mathrm{per}$ by \[E_\mathrm{per}=\colim_{\a\in \mathrm{Id}_{O_K}} E\otimes_{O_K} \a\to S^\mathrm{per}=\colim_{\a\in \mathrm{Id}_{O_K}}\psi^{\a^{-1}}_{S^\mathrm{per}!}(S)=S^\mathrm{per}.\]
\begin{prop} $E_\mathrm{per}$ is a \textup{CM} elliptic curve over $S^\mathrm{per}$ equipped with a canonical $\Lambda_{S^\mathrm{per}}$-structure.
\end{prop}
\begin{proof} First, $E_\mathrm{per}$ admits a $\Lambda_{S^\mathrm{per}}$-structure as it is a colimit of $\Lambda_{S^\mathrm{per}}$-sheaves. Secondly, as the colimit is filtered and the diagrams (\ref{dia:cartesian-perfect}) are all cartesian it follows that for all $\a\in \mathrm{Id}_{O_K}$ we have $\Lambda_{S^\mathrm{per}}$-isomorphisms \[E_\mathrm{per}\times_{S^\mathrm{per}}\psi_{S^\mathrm{per}!}^{\a^{-1}}(S)\overset{\sim}{\longrightarrow} E\otimes_{O_K}\a.\] This shows that $E_\mathrm{per}$ is a CM elliptic curve. It now also follows from (iii) of (\ref{prop:canonical-lambda-p-structures}) that its $\Lambda_{S^\mathrm{per}}$-structure is canonical, as the $\Lambda_{S}$-structures of $E\otimes_{O_K}\a$ are canonical and $(\psi_{S^\mathrm{per}!}^{\a^{-1}}(S)\to S_\mathrm{per})_{\a\in \mathrm{Id}_{O_K}}$ is a cover.
\end{proof}
\subsection{} We now relate $E_\mathrm{per}/S^\mathrm{per}$ to $\widetilde{E}/S$. This is nothing more than an application of certain adjunctions and the definitions. Indeed, for each ind-affine scheme $T\to S$, the morphism $T\to S^\mathrm{per}$ induces a morphism $\widehat{W}^*(T)\to S_\mathrm{per}$ and viewing $\widehat{W}^*(T)$ as a $\Lambda_{S^\mathrm{per}}$-ind-affine scheme we have \begin{eqnarray*}\mathrm{Hom}_{S^\mathrm{per}}^{\Lambda}(\widehat{W}^*(T), E_\mathrm{per})&\stackrel{}{=}&\lim_{\a}\mathrm{Hom}_{S^\mathrm{per}}^{\Lambda}(\psi_{S^\mathrm{per}!}^{\a^{-1}}(W^*(T)), E_\mathrm{per})\\
&\stackrel{}{=}& \lim_\a \mathrm{Hom}_{S^\mathrm{per}}^\Lambda(W^*(T), \psi^{\a^{-1}*}_{S^\mathrm{per}}(E_\mathrm{per}))\\
&=& \lim_\a \mathrm{Hom}_{S^\mathrm{per}}^\Lambda(W^*(T), E_\mathrm{per}\otimes_{O_K}\a))\\
&\stackrel{}{=}& \lim_\a \mathrm{Hom}_{S}^\Lambda(W^*(T), E\otimes_{O_K}\a)\\
&\stackrel{}{=}& \lim_\a \mathrm{Hom}_S(T, E\otimes_{O_K}\a)\\
&=& \mathrm{Hom}_S(T, \widetilde{E}).\end{eqnarray*} Therefore, we have a natural isomorphism of functors on ind-affine $S$-schemes \[\mathrm{Hom}_{S^\mathrm{per}}^{\Lambda}(\widehat{W}^*(-), E_\mathrm{per})\overset{\sim}{\longrightarrow} \mathrm{Hom}_S(-, \widetilde{E}).\] If we denote the left hand side by \[\widehat{W}_*(E_\mathrm{per})_\Lambda: T/S\mapsto \mathrm{Hom}_{S^\mathrm{per}}^\Lambda(\widehat{W}^*(T), E_\mathrm{per})\] then we may then rewrite the exact sequence of (\ref{eqn:univ-cover-tate}) to obtain the following:
\begin{theo}\label{coro:tate-teichmuller} The is an exact sequence of \textup{fpqc} sheaves \[0\to T(E)\to \widehat{W}_*(E_\mathrm{per})_\Lambda\to E\to 0.\]
\end{theo}
\subsection{} If now $S$ is an ind-affine scheme (not a $\Lambda$-scheme) and $E/S$ is any CM elliptic curve then we may perform the above construction for $W_\mathrm{CM}^*(E)/W^*(S)$ to obtain a CM elliptic curve \[\widehat{W}_\mathrm{CM}^*(E):=W_\mathrm{CM}^*(E)_\mathrm{per}\to \widehat{W}^*(S)=W^*(S)_\mathrm{per}\] and an exact sequence of sheaves for the fpqc topology over $W^*(S)$: \begin{equation}\label{eqn:per-can-lift} 0\to T(W_\mathrm{CM}^*(E))\to \widehat{W}_*(\widehat{W}^*_\mathrm{CM}(E))_\Lambda\to \widehat{W}^*_\mathrm{CM}(E)\to 0.\end{equation}
If we pull-back this exact sequence along the first ghost component $g_{(1)}: S\to W^*(S)$, and just write \[\widehat{W}_*(\widehat{W}^*_\mathrm{CM}(E))_\Lambda|_{S}=g_{(1)}^*(\widehat{W}_*(\widehat{W}^*_\mathrm{CM}(E))_\Lambda)\] then (\ref{eqn:per-can-lift}) becomes the exact sequence over $S$: \[0\to T(E)\to \widehat{W}_*(\widehat{W}^*_\mathrm{CM}(E))_\Lambda|_{S}\to E\to 0\] (where we use the fact that $g_{(1)}^*(W_\mathrm{CM}^*(E)))=E$).
\begin{coro}\label{coro:tate-teichmuller-two} For any ind-affine scheme $S$ and any \textup{CM} elliptic curve $E/S$ there exists an exact sequence of \textup{fpqc} sheaves over $S$ \[0\to T(E)\to \widehat{W}_*(\widehat{W}^*_\mathrm{CM}(E))_\Lambda|_{S}\to E\to 0.\]
\end{coro}
\begin{rema} Although we have not discussed it, there is a completely analogous theory of canonical lifts for Lubin--Tate $O$-modules for which the above constructions can also be made. For $O=\mathbf{Z}_p$ and $F=\mu_{p^\infty}$, the analogue of the exact sequence (\ref{coro:tate-teichmuller-two}) evaluated on $\mathrm{Spf}(\overline{\mathbf{Z}}_p)$ gives \[0\to T(\mu_{p^\infty})(\overline{\mathbf{Z}}_p)\to \mu_{p^\infty}(A_{\mathrm{inf}})^{\varphi_p=p}\to \mu_{p^\infty}(\overline{\mathbf{Z}}_p)\to 0\] where $A^{\mathrm{inf}}$ with its Frobenius lift $\varphi_p$ is Fontaine's ring, and we have used the (non-obvious) fact that \[\mathrm{Spf}(A^{\mathrm{inf}})\overset{\sim}{\longrightarrow} \widehat{W}^*(\mathrm{Spf}(\overline{\mathbf{Z}}_p)).\] The image in $\mu_{p^\infty}(A^{\mathrm{inf}})$ of a generator $\epsilon\in T(\mu_{p^\infty})(\overline{\mathbf{Z}}_p)$ is Fontaine's element $[\epsilon]$, the logarithm of which is the $p$-adic period $t$.
\end{rema}
\begin{rema} While there is (as yet) no theory of analytic $\Lambda$-structures one can fudge a theory of analytic canonical lifts. Here let us sketch the construction of an analytic analogue of the short exact sequence (\ref{coro:tate-teichmuller-two}) relating the period lattice of $E^\mathrm{an}/S^\mathrm{an}$ to a certain analytic canonical lift of $E/S$.
So let $S\to \mathrm{Spec}(\mathbf{C})$ be a complex scheme of finite type. We note that as $K\subset \mathbf{C}$, we have \[W^*(S)=\Gamma^*(S)=\coprod_{\a\in \mathrm{Id}_{O_K}} S =\underline{\mathrm{Id}_{O_K}}\times S\] and \[\widehat{W}^*(S) = \underline{\mathrm{Id}_{K}}\times S.\]
There is a natural analytic analogue of $\mathrm{Id}_K$ which we call $\mathrm{Id}_\mathbf{C}$ and is given by $(\mathrm{Id}_K\times \mathbf{C}^\times)/K^\times$ where $a\in K^\times$ acts on $\mathrm{Id}_K\times \mathbf{C}$ by $(\a, s)\mapsto ((a)\a, a s)$. Note there is a short exact sequence \[0\to O_K^\times\to \mathbf{C}^\times\to \mathrm{Id}_\mathbf{C}\to \mathrm{CL}_{O_K}\to 0.\]
It is natural, in light of the above, to define the analytic Witt vectors of the analytification $S^\mathrm{an}$ of $S$ to be the analytic space \[W^{\mathrm{an}*}(S):=\mathrm{Id}_\mathbf{C}\times S^\mathrm{an}\] together with its action of $\mathrm{Id}_\mathbf{C}$.
We can now analytically mimic our construction of $W_\mathrm{CM}^*$ over $W^{\mathrm{an}*}(S)$. So define the rank one $O_K$-local system $\mc{L}_\mathrm{an}$ to have fibre over $(\a, s)\times S^\mathrm{an}\subset S^\mathrm{an}\times \mathrm{Id}_\mathbf{C}=W^{\mathrm{an}*}(S)$ the constant $O_K$-local system associated to the rank one $O_K$-module $s\cdot \a^{-1}\subset s\cdot K \subset \mathbf{C}$ (note this depends only on the class of $(\a, s)\in (\mathrm{Id}_K\times\mathbf{C}^\times)/K^\times = \mathrm{Id}_\mathbf{C}$). If $E/S$ is a CM elliptic curve then we define $W^{\mathrm{an}*}_\mathrm{CM}(E)$ to be \[p_S^*(E^\mathrm{an})\otimes_{O_K}\mc{L}_\mathrm{an}\] where $p_S: W^{\mathrm{an}*}(S)=\mathrm{Id}_{\mathbf{C}}\times S^\mathrm{an}\to S^\mathrm{an}$ is the projection (cf.\ (\ref{subsec:def-can-ghost})).
The CM elliptic curve $W^{\mathrm{an}*}_\mathrm{CM}(E)\to W^{\mathrm{an}*}(S)$ inherits a natural action of $\mathrm{Id}_{O_K}$ (not $\mathrm{Id}_\mathbf{C}$!) which is compatible with that on $W^{\mathrm{an}*}(S)$. Finally, setting $W^{\mathrm{an}*}(X)=\mathrm{Id}_\mathbf{C}\times X$ with its $\mathrm{Id}_{\mathbf{C}}$ action for any analytic space, we define a sheaf on the big analytic site of $S$ by \[W^\mathrm{an}_*(W_\mathrm{CM}^{\mathrm{an}*}(E))_\Lambda|_S: X/S\mapsto \mathrm{Hom}_{W^{\mathrm{an}*}(S)}^{\mathrm{Id}_{O_K}}(W^{\mathrm{an}*}(X), W^{\mathrm{an}*}_\mathrm{CM}(E)).\] Let us spell out here that the right hand side here denotes the $\mathrm{Id}_{O_K}$-equivariant \textit{analytic} $W^{\mathrm{an}*}(S)$-maps \[W^{\mathrm{an}*}(X)\to W^{\mathrm{an}*}_\mathrm{CM}(E).\] With this definition one can show (almost as formally as in the algebraic situation) that there exists an isomorphism of sheaves on the big analytic site of $S^\mathrm{an}$ \[W^\mathrm{an}_*(W_\mathrm{CM}^{\mathrm{an}*}(E))_\Lambda|_S\overset{\sim}{\longrightarrow} \underline{\mathrm{Lie}}_{E^\mathrm{an}/S^\mathrm{an}}\] and so the exponential sequence \[0\to T_{O_K}(E^\mathrm{an})\to \underline{\mathrm{Lie}}_{E^\mathrm{an}/S^\mathrm{an}}\to E^\mathrm{an}\to 0\] of $E^\mathrm{an}/S^\mathrm{an}$ can be rewritten as \[0\to T_{O_K}(E^\mathrm{an})\to W^\mathrm{an}_*(W_\mathrm{CM}^{\mathrm{an}*}(E))_\Lambda|_S\to E^\mathrm{an}\to 0.\]
\end{rema}
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Q: Oracle: Retrieve by TZNAME Is it possible? How can we find out which TZNAME we are under in our Database?
and by TZNAME I mean something like (from the list of)
select * from V$TIMEZONE_NAMES;
If we do a
select dbtimezone from dual
I get dbtimezone = -04:00
we retrieve the DBTIMEZONE which is the number representation of the timezone, but since there are many of them but in different places, we won't be able to be 100 % accurate. Example, currently we have -04:00 which can be Both EST CANADA and EST NY (or a date that doesn't involve daylights savings)
Is there any way to retrieve the actual TZNAME of the database timezone rather than the actual time?
Thank you
A: We can extract the TIMEZONE_REGION from a timestamp, providing its a TIMESTAMP WITH TIMEZONE. Like so:
SQL> select extract(timezone_region from current_timestamp)
2 from dual
3 /
EXTRACT(TIMEZONE_REGIONFROMCURRENT_TIMESTAMP)
----------------------------------------------------------------
CET
SQL> alter session set time_zone='UTC';
Session altered.
SQL> select extract(timezone_region from current_timestamp)
2 from dual
3 /
EXTRACT(TIMEZONE_REGIONFROMCURRENT_TIMESTAMP)
----------------------------------------------------------------
UTC
SQL> alter session set time_zone='-04:00';
Session altered.
SQL> select extract(timezone_region from current_timestamp)
2 from dual
3 /
EXTRACT(TIMEZONE_REGIONFROMCURRENT_TIMESTAMP)
----------------------------------------------------------------
UNKNOWN
SQL>
The last result returns UNKNOWN because more than one Timezone name maps to an offset of minus four hours. There are various ways of setting the timezone name at the session level; one of those is likely to be the best way of work around this problem. Find out more.
| {
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} | 303 |
Il salbutamolo (conosciuto anche con il nome di albuterolo) è un composto, a breve durata d'azione, con attività di tipo agonista selettivo sui recettori β2-adrenergici. Come farmaco viene utilizzato per ridurre il broncospasmo in alcune condizioni patologiche quali l'asma e la broncopneumopatia cronica ostruttiva. In Italia è venduto dalla società farmaceutica Valeas con il nome commerciale di Broncovaleas e dalla società GlaxoSmithKline con il nome di Ventolin. È disponibile in diverse forme farmaceutiche, tra cui sospensione pressurizzata per inalazione, soluzione da nebulizzare e sciroppo.
Salbutamolo è stato il primo agonista β2-selettivo a essere commercializzato, nel 1968. È stato venduto dalla società Allen & Hanburys con il marchio Ventolin. Il farmaco fu un successo immediato, e da allora continua a essere ampiamente utilizzato per il trattamento dell'asma.
Chimica
Relazione tra struttura e attività
Il gruppo butile terziario della molecola di salbutamolo (albuterolo) lo rende più selettivo per i recettori β2. Il farmaco è venduto come una miscela racemica soprattutto perché l'(S)-enantiomero blocca le vie metaboliche, mentre l'( R)-enantiomero evidenzia l'attività.
Sintesi
Salbutamolo può essere preparato a partire da un derivato dell'acetofenone, a sua volta un derivato dell'acido salicilico (da questa origine il prefisso "sal" nel nome salbutamolo).
Farmacodinamica
Salbutamolo è una molecola adrenergica, un agonista selettivo dei recettori β2-adrenergici, particolarmente diffusi a livello della muscolatura liscia bronchiale. La stimolazione di questi recettori da parte del farmaco comporta un incremento dell'AMP ciclico endocellulare. Questo aumento di AMP ciclico porta all'attivazione della protein chinasi A, la quale inibisce la fosforilazione della miosina e riduce la concentrazione intracellulare di calcio ionico, con conseguente rilassamento e determinazione di un effetto di tipo broncodilatatore con risoluzione del broncospasmo.
Salbutamolo rilassa la muscolatura liscia di tutte le vie aeree, dalla trachea fino ai bronchioli terminali, agisce perciò come un agonista funzionale che comporta il rilassamento delle vie aeree indipendentemente dal fattore spasmogeno coinvolto, proteggendo così contro tutti gli stimoli broncocostrittori.
La selettività del composto è tale che, alle usuali dosi broncodilatatrici, l'attività di salbutamolo sui recettori β1-adrenergici cardiaci è sostanzialmente trascurabile. L'esecuzione di test spirometrici ripetuti sui soggetti in trattamento, conferma i dati sperimentali sull'attività broncodilatrice della molecola, in assenza di effetti stimolatori cardiaci rilevabili. Tuttavia è necessario ricordare che, pur essendo i recettori β2-adrenergici predominanti nei bronchi, vi sono evidenze che una popolazione di recettori β2 è presente anche nel muscolo cardiaco umano, normalmente in una concentrazione compresa tra il 10% e il 50%. Non è stata ancora precisata la funzione di questi recettori. Un ulteriore effetto di salbutamolo consisterebbe in un'azione stabilizzante sulla membrana del mastocita, tale da prevenire il rilascio di istotossine. Quest'azione rafforza la validità dell'impiego del farmaco nei soggetti affetti da asma di origine allergica.
Farmacocinetica
A seguito di somministrazione per via inalatoria circa il 10-20% del farmaco raggiunge le basse vie respiratorie. La frazione non trattenuta nell'erogatore si deposita nell'orofaringe e viene quindi ingerita e rapidamente assorbita dal tratto gastrointestinale. Anche la parte che si deposita nelle vie aeree viene assorbita attraverso il tessuto polmonare e raggiunge la circolazione sistemica. Il farmaco subisce quindi il metabolismo epatico e quindi eliminato, principalmente per via urinaria, come farmaco non modificato o in forma coniugata come solfato fenolico.
L'emivita terminale apparente di salbutamolo è di circa 4,6 ore.
Tossicologia
La DL50 di salbutamolo nel ratto e nel topo, quando assunto per os è superiore a 2 000 mg/kg peso corporeo. La DL50 per via endovenosa è di 60,5 mg/kg (ratto) e di 72,5 mg/kg (topo). Per via intraperitoneale è pari rispettivamente a 74,8 mg/kg e 82,2 mg/kg.
Usi clinici
Il farmaco viene utilizzato nel trattamento del broncospasmo associato all'asma bronchiale e ad altre patologie che comportano occlusione delle vie respiratorie, come ad esempio la broncopneumopatia cronica ostruttiva. Può essere utilizzato nella prevenzione del broncospasmo associato all'esercizio fisico.
I medici lo utilizzano anche in ambito ostetrico. L'infusione endovenosa di salbutamolo comporta un effetto tocolitico, ovvero determina un rilassamento della muscolatura liscia uterina, procrastinando così il parto pretermine. Mentre per quest'uso veniva preferito rispetto ad agenti come atosiban e ritodrina, il suo ruolo è stato ampiamente sostituito dal calcio antagonista nifedipina, il quale somministrato per via orale, è più efficace e meglio tollerato.
Off label
Salbutamolo è stato utilizzato nel trattamento in acuto dell'iperkaliemia: la molecola stimola infatti l'afflusso dello ione all'interno della cellula e ne consegue perciò una riduzione dei valori di kaliemia.
Salbutamolo è stato sperimentato anche nel trattamento dell'atrofia muscolare spinale, dove sfortunatamente sembra determinare solo modesti benefici. Il farmaco si ritiene sia in grado di modulare lo splicing alternativo del gene SMN2, aumentando in questo modo la quantità di proteina SMN, la cui carenza è considerata la causa principale della malattia.
Effetti collaterali e indesiderati
Tra gli effetti collaterali più comuni si segnalano ansia, cefalea, cardiopalmo, secchezza delle fauci, crampi muscolari.
In alcuni pazienti salbutamolo può determinare la comparsa di tremori fini a carico della muscolatura scheletrica, in genere più facilmente evidenziabili alle mani. Questo effetto avverso, comune a tutti gli stimolanti β-adrenergici, è in stretta relazione con la dose. Raramente si può riscontrare cefalea e modesta tachicardia, che tendono a scomparire dopo i primi giorni di terapia, e in particolare dopo una lieve riduzione del dosaggio. Molto raramente in letteratura medica è riportata la comparsa di aritmie cardiache (comprendenti la fibrillazione atriale, la tachicardia sopraventricolare ed extrasistolia), nonché di ischemia miocardica.
Così come avviene anche per altre sostanze medicamentose somministrate per via inalatoria, può manifestarsi broncospasmo paradosso, con aumento del respiro affannoso a distanza di pochi minuti dall'inalazione. Nell'eventualità che si presenti questo effetto, è necessario assumere immediatamente un diverso broncodilatatore a rapida insorgenza d'azione, interrompere immediatamente la terapia precedente e istituirne una alternativa.
Controindicazioni
Il farmaco è controindicato nei soggetti con ipersensibilità nota al principio attivo oppure a uno qualsiasi degli eccipienti della formulazione farmaceutica.
Anche lo stato di gravidanza e l'allattamento al seno debbono essere considerate condizioni di controindicazione, tranne in rari casi nei quali il beneficio atteso per la madre sia considerato superiore al possibile rischio per il feto.
Dosi terapeutiche
Per via orale: 4 mg 3-4 volte al giorno. La dose massima giornaliera è pari a 8 mg.
Per infusione endovenosa: 3-20 µg, anche se inizialmente la dose si attesta sempre sui 5 µg.
Per inalazione (aerosol): il dosaggio consigliato negli adulti è pari a 100-200 µg (tipicamente 100 µg corrispondono a uno spruzzo) da 3 a 6 volte al giorno. Viene consigliato di eseguire al massimo un paio di somministrazioni di aerosol dosato per volta, quindi attendere almeno quattro ore prima di ripetere il trattamento. Anche in caso di difficoltà respiratorie persistenti è bene non oltrepassare il dosaggio massimo di 600 µg al giorno.
Per inalazione (polvere): dosi variabili da 300 a 600 µg al giorno.
Il paziente asmatico, conoscendo gli orari in cui tende a manifestarsi la crisi broncospastica, cercherà di assumere il farmaco circa mezz'ora prima degli stessi. Va ricordato che un uso troppo frequente di β2 agonisti a breve durata d'azione per via inalatoria può rivelarsi pericoloso in quanto il ricorso a dosaggi elevati è gravato da maggiori probabilità di comparsa di effetti indesiderati, oltre a essere indicativo di un controllo non ottimale del disturbo.
In questi casi è opportuno valutare la possibilità di una terapia di associazione con cortosteroidi e, nel caso siano già presenti nello schema di trattamento, un incremento del loro dosaggio.
Sovradosaggio
I segni e sintomi in corso di sovradosaggio da salbutamolo sono eventi generalmente transitori e correlati all'azione del farmaco sui recettori β-agonisti. Si deve ricordare che a seguito di sovradosaggio di salbutamolo si può verificare ipokaliemia, con conseguente necessità di monitoraggio dei livelli sierici del potassio.
In caso di importante sintomatologia cardiaca (tachicardia, cardiopalmo e simili) si può dover ricorrere alla sospensione del trattamento e all'adozione di adeguate misure di controllo della sintomatologia: tra queste possono rivestire un ruolo di primo piano i β-bloccanti cardio-selettivi, ricordando la necessità di grande cautela nel loro utilizzo, stante la possibilità di evocare broncospasmo.
Interazioni
Simpaticomimetici: è opportuno non co-somministrare salbutamolo e altri medicinali con attività simpaticomimetica per la possibilità che si verifichi un potenziamento dei rispettivi effetti farmacologici.
β-bloccanti (propranololo, metoprololo, bisoprololo e altri): la contemporanea somministrazione di farmaci β-bloccanti e salbutamolo non è indicata. Questi agenti possono bloccare l'effetto polmonare del β-agonista e anche indurre grave broncospasmo nei pazienti con storia di asma. In particolari circostanze, ad esempio nei soggetti con pregresso infarto del miocardio, il ricorso a β-bloccanti cardioselettivi può essere preso in considerazione, anche se la loro somministrazione deve sempre avvenire con estrema cautela.
Metilxantine (teofillina, caffeina, teobromina): è necessario evitare l'assunzione di quantità eccessive di cibi contenenti metilxantine (ad esempio caffè, tè, cioccolato) durante la terapia con salbutamolo perché potrebbe verificarsi una marcata ipokaliemia. In letteratura medica è stato segnalato un caso di fibrillazione atriale in seguito all'abuso di cioccolato e salbutamolo.
Diuretici: la contemporanea assunzione di salbutamolo e diuretici non risparmiatori di potassio (quali ad esempio diuretici dell'ansa o tiazidici) potrebbe causare una riduzione della concentrazione plasmatica di potassio e alterazioni elettrocardiografiche. Salbutamolo, soprattutto a dosaggi elevati, stimola la captazione intracellulare di potassio riducendone la disponibilità a livello extracellulare. In caso di co-somministrazione si deve quindi monitorare regolarmente la kaliemia.
Inibitori delle monoamino ossidasi e antidepressivi triciclici: salbutamolo deve essere somministrato con grande cautela a pazienti in terapia con inibitori delle MAO o antidepressivi triciclici o che hanno interrotto la terapia antidepressiva da meno di 2 settimane, poiché l'azione di salbutamolo sul sistema cardiovascolare può essere potenziata da tali farmaci. È perciò opportuno considerare terapie alternative nei soggetti in trattamento con questi agenti.
Digossina: la somministrazione endovenosa e orale di una singola dose di salbutamolo a volontari che avevano ricevuto digossina per 10 giorni, diminuisce dal 16% al 22% i livelli sierici di digossina. La rilevanza clinica di questi risultati per i pazienti con malattia ostruttiva delle vie aeree che ricevono salbutamolo per via inalatoria e digossina su base cronica non è chiara.
Gravidanza e allattamento
L'uso del salbutamolo in gravidanza richiede cautela e un'attenta valutazione del rapporto rischio/beneficio. Somministrato per via endovenosa nell'ultimo periodo di gravidanza, il farmaco è stato associato a tachicardia moderata e alterazioni metaboliche nella madre, tra le quali aumento dell'insulinemia, degli acidi grassi liberi, del glicerolo.
Durante il travaglio pretermine, l'infusione endovenosa del broncodilatatore a scopo tocolitico è stata associata alla comparsa di ischemia cardiaca e di foci ectopici ventricolari.
Sempre nel trattamento del travaglio pretermine la somministrazione in pazienti affette da ipertensione arteriosa è stata associata a scompenso cardiaco ed edema polmonare. In ogni caso l'uso del farmaco come tocolitico richiede un controllo continuo dello stato di idratazione della paziente e del battito cardiaco onde evitare la comparsa delle segnalate complicanze. Questo tipo di rischio tende ad aumentare in caso di parto plurimo, di preesistente cardiomiopatia e di infezioni in atto.
In pazienti diabetiche l'assunzione della molecola può invece comportare acidosi metabolica.
La Food and Drug Administration ha classificato la molecola in categoria C per l'impiego in gravidanza. In questa classe sono inseriti farmaci i cui studi sugli animali hanno rilevato effetti dannosi sul feto, teratogenico/letale o altro, e per i quali non sono disponibili studi controllati in donne oppure farmaci per i quali non sono disponibili studi né sull'uomo né sull'animale.
Salbutamolo, dopo dosi terapeutiche per via inalatoria, viene escreto nel latte materno in minima quantità. A causa della potenziale carcinogenicità osservata per questa molecola in studi su animali e la mancanza di esperienza con l'uso di salbutamolo in madri che allattano, è opportuno ricorrere all'uso del medicinale solo se i benefici attesi superano i rischi potenziali.
Avvertenze
I pazienti con una grave patologia cardiaca (ad esempio, cardiopatia ischemica, aritmia o grave insufficienza cardiaca) trattati con salbutamolo potrebbero andare incontro a tachicardia con un peggioramento della patologia cardiaca sottostante. La comparsa di sintomi quali dolore toracico, dolore anginoso, affaticabilità del cuore devono essere segnalati prontamente.
Note
Bibliografia
Voci correlate
Formoterolo fumarato
Salmeterolo
Terbutalina solfato
Altri progetti
Beta 2 agonisti
Antiasmatici
Medicinali essenziali secondo l'Organizzazione Mondiale della Sanità | {
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Peumerit è un comune francese di 804 abitanti situato nel dipartimento del Finistère nella regione della Bretagna.
Il comune si è chiamato Peumérit fino al 1º agosto 2012
Società
Evoluzione demografica
Note
Altri progetti
Collegamenti esterni
Peumerit | {
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} | 3,299 |
Лікоріс променистий (Lycoris radiata) — вид рослин роду лікоріс (Lycoris).
Назва
Японська назва квітки «хіганбана» () означає «квітка іншого берега», де під «іншим берегом» йдеться про Нірвану. В англійській мові квітка має назву «лілія Червоний павук», та «червона магічна лілія» ().
Будова
Багаторічна рослина висотою до 60 см. Має підземну цибулину. Листя вузькі, 7-15 см завдовжки, 4-7 мм шириною. На стеблі утворюється 5-7 яскраво червоних квіток. Пелюстки вузькі, близько 1,5 см шириною, загнуті назад. Тичинки дуже довгі — до 20 см, як лапи павука, тому цей вид лікоріса отримав назву «червоний павук».
Життєвий цикл
Квіти з'являються пізньою осінню.
Поширення та середовище існування
Походить з Китаю, Кореї та Непалу. Був завезений спочатку в Японію, а згодом в США.
Практичне використання
Рослина дуже отруйна. В Японії нею оточують поля та житло, щоб відлякувати шкідників.
У буддистів ця квітка асоціюється зі смертю, букети лікоріса покладають на могили. Сезон, коли розпускаються криваво-червоні квіти, у японців традиційно асоціюється з відвідуванням могил предків. Лікоріс променистий не дарують через це живим людям.
Цікаві факти
Щороку помилуватися квітками цієї рослини з'їжджаються японці на традиційний «ханасампо».
Галерея
Примітки
Джерела
Lycoris radiata (L'Hér.) Herb. // Hortus Camdenensis - URL
Амарилісові | {
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OkfIncubator::Application.config.secret_token = ENV['SECRET_TOKEN']
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L'ordonnancement à taux monotone (en anglais, rate-monotonic scheduling) est un algorithme d'ordonnancement temps réel en ligne à priorité constante (statique).
Il attribue la priorité la plus forte à la tâche qui possède la plus petite période. RMS est optimal dans le cadre d'un système de tâches périodiques, synchrones, indépendantes et à échéance sur requête avec un ordonnanceur préemptif. De ce fait, il n'est généralement utilisé que pour ordonnancer des tâches vérifiant ces propriétés.
Historique
Cet algorithme a été proposé la première fois dans un papier publié par Liu et Layland.
Outre l'algorithme à taux monotone, ce papier décrit une modélisation des tâches basée sur un triplet (, , ), ainsi qu'une méthode de calcul des pires temps de réponses pour des systèmes de tâches à échéance inférieure ou égale à la période.
Ce papier est actuellement considéré comme étant une base de l'ordonnancement temps-réel.
Tâche
Les tâches (ou tasks en anglais) sont les entités manipulées par cet algorithme. Chaque tâche est modélisée par un quadruplet (, , , ), où :
correspond à la date réveil de la tâche ;
correspond au coût d'exécution de la tâche ;
correspond à l'échéance relative de la tâche ;
correspond à la période de la tâche.
Toutefois, l'algorithme n'étant optimal que dans un contexte de tâches simultanées (i.e. la date de réveil de chaque tâche est nulle) et à échéance sur requête (i.e. ), il n'est pas rare de ne modéliser les tâches que par un doublet (, ).
Test d'ordonnançabilité
Conditions nécessaires et suffisantes
Afin de valider un système de tâches ordonnancé ainsi, deux moyens sont offerts :
soit, calculer le temps de réponse de chaque tâche, puis, vérifier que toutes les tâches respectent leurs échéances ;
soit, réaliser une simulation sur un intervalle allant de 0 jusqu'au PPCM des périodes (macrocycle).
Condition suffisante
Il existe également une condition suffisante portant sur la charge processeur . Son test d'acceptabilité pour un système composé de tâches, qui peut être réalisé hors ligne, nous est donné par la formule suivante :
Par exemple, la charge limite pour laquelle ce critère est valable pour est .
Et quand le nombre de tâches tend vers l'infini :
Ainsi, on estime dans le cas général qu'un RMS peut respecter toutes les échéances si l'utilisation du processeur est inférieure ou égale à 69,3 %. Les 30,7 % restants peuvent être dédiés à des tâches de basse priorité et non temps-réel.
Cependant, cette condition est suffisante mais pas nécessaire. Il est tout à fait possible qu'un système de tâche totalisant une charge de 100 % soit ordonnançable, alors qu'un autre système de tâches n'ayant qu'une charge globale de 80 % ne le soit pas. Tout dépend des caractéristiques du système de tâches.
Validité des tests d'ordonnançabilité
La condition suffisante n'est valable que dans le cas où l'algorithme est optimal.
La simulation n'est également valable que dans le cas où l'algorithme est optimal. Toutefois, il est possible de le rendre valide à d'autre cas en étendant la période de simulation.
Le calcul du pire temps de réponse reste valable quelle que soit la situation.
Cas plus généraux
Deadline Monotonic
L'algorithme deadline monotonic est également optimal dans une situation dans laquelle les périodes et les deadlines sont identiques, dans le fait que les algorithmes sont alors identiques, et de plus, le DMS est optimal quand les deadlines sont inférieures aux périodes.
Algorithme d'Audsley
Dans le cadre plus général de tâches indépendantes, périodiques, concrètes non simultanées et à échéance arbitraire, l'algorithme d'Audsley fournit une méthode optimale d'ordonnancement.
Notes et références
Voir aussi
Articles connexes
Système temps réel
Système déterministe
Liens externes
Introduction to RMS
Algorithme d'ordonnancement | {
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} | 4,503 |
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"redpajama_set_name": "RedPajamaC4"
} | 8,664 |
Fallout 76 Review
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We heard rumours that Fallout 76 could be an online survival RPG and those suspicions were answered at Bethesda's E3 2018 conference. All the fan speculation and press information was pretty spot on when it came to the big reveal. Fallout 76 is the earliest in the timeline so far,…
Bethesda Teases Fallout 76 Ahead of E3
Get ready to head back to the Wasteland as Bethesda has dropped a trailer after teasing something Fallout related in a very long live stream. Fallout 76 which much of the information is still unknown is teased to have something to do with Vault 76 and looks to be set… | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,902 |
\section{Introduction}
Deep learning models have shown success in computer vision tasks in recent years \cite{he2016deep,simonyan2014very,ronneberger2015u}. However, training deep models that generalize well on unseen test data may require massive training data. Unfortunately, we are usually faced with \textbf{insufficient data} in a single medical institution for the medical image segmentation task due to the expensive procedure of collecting enough patients' data with experts' labeling.
A straightforward solution to address the insufficient data issue is gathering data from all the available medical institutions, while the amount of data owned by any single institution may be insufficient to train a well-performing deep model. However, this approach will raise the concern for \textbf{data privacy}. On one hand, collecting medical data is expensive as mentioned above, and those data have become a valuable asset at a medical institution. Institutions with more data may be more reluctant to contribute their data. In addition, medical institutions bear the obligation to keep the data collected from patients secure. Gathering data may expose patients to the risk of data leakage.
Of course, we can leverage the existing vanilla distributed training method \cite{li2014scaling,xu2021step,xu2020acceleration} to keep the institution's data local and share only the gradient with a central server. But the training of deep model requires many iterations to converge, leading to unacceptable \textbf{communication complexity} for vanilla distributed training. It is not secure neither as recent works \cite{zhu2019deep,zhao2020idlg,geiping2020inverting,yin2021see} have shown that pixel-level images can be recovered from the leaked gradient.
Recently, federated learning (FL) \cite{konevcny2016federated,gao2021convergence,xu2021double,gu2021privacy} have been proposed to tackle all the above issues (insufficient data, data privacy, training efficiency). In medical applications, we are most interested in the cross-silo federated learning where we have a limited number of participating clients compared with cross-device federated learning (e.g., mobile devices) \cite{kairouz2019advances,liu2021feddg,guo2021multi}. Specifically, in each training round of FedAvg \cite{mcmahan2017communication}, the \textit{de facto} algorithm for FL, each client will perform local training with the global model received from a central server for multiple iterations. Then the server gathers all the local models from each client and averages them as the new global model. Nevertheless, for FedAvg and its variants, a non-negligible issue called ``client drift" arises due to non-iid data distribution on different clients. The local models on different clients will gradually diverge from each other during the local training. Client drift can drastically jeopardize the training performance of the global model when the data similarity decreases (more non-iid) \cite{hsieh2020non,hsu2019measuring}. Theoretically, it leads to a convergence rate more sensitive to the number of local training steps \cite{yu2019linear}.
Throughout this paper, we refer to centralized training as gathering data from clients and then training the model. Note that centralized training is impractical as it violates data privacy, but offers a performance upper bound for FL algorithms. Despite numerous efforts and previous works, there is still a \textbf{generalization gap} between FL and the centralized training. In this paper, unlike any previous works, we propose a novel training framework called \underline{Fed}erated \underline{S}uper \underline{M}odel (FedSM) to avoid confronting the difficult client drift issue at all for FL medical image segmentation tasks. In FedSM, instead of finding one global model that fits all clients' data distribution, we propose to produce personalized models to fit different data distributions well and a novel model selector to decide the closest model/data distribution for any test data.
We summarize our contributions as follows.
\begin{itemize}[leftmargin=0.18in]
\setlength{\itemsep}{0pt}
\item We propose a novel training framework FedSM to avoid the client drift issue and close the generalization gap between FL and centralized training for medical segmentation tasks for the first time to the best of our knowledge.
\item We propose a novel formulation for personalized FL optimization, and a novel personalized method called SoftPull to solve it in our framework FedSM. A rigorous convergence analysis with common assumptions in FL is given for the proposed method.
\item Experiments in real-world FL medical image segmentation tasks validate our motivation and the superiority of our methods over existing FL baselines.
\end{itemize}
\section{Related Works}
Here we introduce existing different approaches to improve the model performance in FL with representative methods. First, the FL optimization problem is usually defined as $\min_{w} \frac{1}{K}\sum^{K}_{k=1} p_k L_{\mathcal{D}_k}(w)$, where the coefficient $p_k=\frac{n_k}{n}$, $n_k$ is the number of client $k$'s data, and the total number of data $n=\sum^{K}_{k=1}n_k$. $L_{\mathcal{D}_k}$ is the objective at client $k$ with its local data $\mathcal{D}_k$, and $w$ is the model weights.
\textbf{FedAvg.} In FedAvg, clients will receive the starting model $w_r$ from the server at training round $r$. Each client $k$ performs $E$ epochs of local training to update the local model to $w_{r+1}^{(k)}$ with the popular momentum SGD or Adam \cite{kingma2014adam} optimizer depending on the application needs. Then the server gathers and averages the local models to $w_{r+1}=\frac{1}{K}\sum^{K}_{k=1}p_k w^{(k)}_{r+1}$.
\textbf{Restrict Local Training.} To discourage the local models from diverging due to non-iid data distribution, FedProx \cite{li2018federated} proposes to add a proximal loss term $\|w^{(k)}_{r+1}-w_{r}\|^2_2$ to the objective function for client $k$. It implies that the local training will encourage $w^{(k)}_{r+1}$ to stay close to the starting point $w_r$, such that $\{w^{(k)}_{r+1}\}_{k\in\{1,2,\cdots,K\}}$ will be close to each other to alleviate the client drift issue.
\textbf{Correct Client Drift.} Motivated by variance reduction techniques in optimization such as SVRG \cite{johnson2013accelerating}, SAGA \cite{defazio2014saga}, inter-client variance reduction techniques \cite{acar2020federated,karimireddy2020scaffold,liang2019variance} are proposed for FL by correcting the local training with the predicted local and global updating direction. These methods are usually tested with convex or simple non-convex models/objectives. For the practical training of complicated deep models, \cite{defazio2018ineffectiveness} shows that variance reduction techniques fail to perform well in that correcting the stochastic gradient with variance reduction usually does not hold in deep learning due to common augmentation tricks such as batch normalization \cite{ioffe2015batch} and dropout \cite{srivastava2014dropout}, etc.
\textbf{Personalization.} Personalized models are usually a fine-tuned version of the global model to better fit the local data distribution of a specific client. We can fine-tune the global model \cite{wang2019federated} on a client's local data like the local training, or following MAML-based personalized methods \cite{t2020personalized,fallah2020personalized,jiang2019improving}. However, an intrinsic drawback of the personalized models is that they generalize poorly on other sites' data and unseen data. In this work, we focus on finding a model that generalizes as well as centralized training for all clients.
\textbf{Other Topics.} There are also many other emerging and interesting topics in FL, such as heterogeneous optimization \cite{wang2020tackling,li2018federated}, fairness and robustness \cite{mohri2019agnostic,Li2020Fair,li2021ditto}, clustered federated learning \cite{ghosh2020efficient}, etc. These topics are not directly related to our work but can be valuable for potential future extension. A recent work FedDG \cite{liu2021feddg} requires sharing partial information of the data, therefore it breaks the data privacy constraint to some extent. In this work, we share only the model update information for maximal data privacy.
\section{Methodology}
In this section, we present our motivation and the proposed method that can close the generalization gap for FL medical image segmentation tasks in detail.
\textbf{Motivation.} In traditional FL, the goal is to collaboratively train one global model that generalizes well on all clients' joint data distribution. The client drift issue comes from the fact that we only have access to clients' local data distribution during the local training. It is hard to train a global model generalizing as well as centralized training due to this issue despite numerous existing works. In this work, however, we show that it is possible to get rid of the client drift issue. Specifically, we propose that
\begin{itemize}[leftmargin=0.18in]
\setlength{\itemsep}{0pt}
\item for the test data, we search for the closest (i.e., the most similar) local data distribution from all clients (Section \ref{sec:fedsm}).
\item we find a model with the best generalization performance on this selected local data distribution, and use it for the inference of the test data (Section \ref{sec:softpull}).
\end{itemize}
\subsection{New Framework: FedSM}\label{sec:fedsm}
\begin{figure}
\centering
\includegraphics[width=.8\columnwidth]{figures/framework.pdf}
\caption{The proposed FedSM framework with ``super model".}
\label{fig:fedsm framework}
\end{figure}
The first motivation above motivates us to design a new and general FL framework FedSM, where we train a \underline{Fed}erated ``\underline{S}uper \underline{M}odel" consisting of the global model, personalized models, and a model selector. These components are illustrated in Figure \ref{fig:fedsm framework} and we elaborate them as follows.
Global model $w_g$: the global model trained by FedAvg. It generalizes better than personalized models on the joint data distribution of all clients, but there is still a gap compared with centralized training. Suppose the model function is $f$ and we denote its output as $h_0=f(w_g,x)$ for data $x$.
Personalized models $w_{p,k}$: the personalized models trained by any personalization FL training method. A personalized model usually generalizes better on local data than the global model. We denote its output as $h_k=f(w_{p,k},x)$, where $k\in \{1,2,\cdots,K\}$.
Model selector $w_s$: its goal is to determine the match between the unseen data input $x$ and each of the global/personalized models for inference. Specifically, it outputs a normalized prediction score vector $\widehat{y}_s$. The final output $h$ is determined by $\widehat{y}_s$ and $[h_0,h_1,\cdots,h_K]$. Suppose the candidate model set $\Omega\subseteq \{0,1,2,\cdots,K\}$, then $\sum_{k\in\Omega}\widehat{y}_{s,k}=1$ and $h=\sum_{k\in\Omega}\widehat{y}_{s,k}h_k$. We discuss the potential training methods as follows.
\subsubsection{Ensemble}
Suppose we already have the trained global model and personalized models. Given the FedSM framework as shown in Figure \ref{fig:fedsm framework}, a straightforward approach is to ensemble the outputs $[h_0,h_1,\cdots,h_K]$ from all models as the final output $h=\sum^{K}_{k=0}\widehat{y}_{s,k}h_k$. Let the ground truth of data $x$ be $y$ and the loss function be $L$. Then, we compute the loss $L(h,y)$ and update the model selector $w_s$ via FedAvg.
However, in practice we find it \textbf{hard to train} the model selector in this way in FL. The final performance can be even inferior to the global model. Let the desired value $y_s=\min_{\widehat{y}_s}L(\sum^{K}_{k=0}\widehat{y}_{s,k}h_k,y)$. We found that it was caused by the difficulty to train $\widehat{y}_s$ to the desired value $y_s$ by $\min_{w_s}L(\sum^{K}_{k=0}\widehat{y}_{s,k}h_k,y)$ as $w_s$ is the model weights to optimize. For each data input $x$, we may need many training steps to $\min_{w_s}L(\sum^{K}_{k=0}\widehat{y}_{s,k}h_k,y)$ such that $\widehat{y}_s$ will be close to $y_s$. However, it is unacceptable due to the large amount of computation cost.
Another issue of this approach is that we cannot start training the model selector until the training of the global model and personalized models finishes, which incurs \textbf{extra communication rounds} for FL.
\subsubsection{FedSM-extra}
To tackle the \textbf{training difficulty} in ensemble, here we propose to compute
\begin{equation}\label{eq:fedsm-extra}
y_s=one\_hot(\arg \min_k\{L(h,h_k)\}_{k=0}^{K})\,,
\end{equation}
where ``one\_hot'' denotes one hot encoding. Then we compute the cross entropy loss $L_s(\widehat{y}_s,y_s)$ to update the model selector. In this way, the model selector is more clear about the desired value $y_s$. Thus it will be easier to train. We refer to this approach as FedSM-extra as it still needs extra communication rounds like the ensemble approach.
\subsubsection{FedSM}
To address the issue of \textbf{extra training rounds}, the model selector needs to be trained together with the global model and personalized models. Nevertheless, from Eq.~(\ref{eq:fedsm-extra}) we can see that the desired $y_s$ depends on the output of the trained global model and personalized models. Therefore, we need to decouple their dependency. As a further simplification, suppose the training data $x$ comes from the client $k\in\{1,2,\cdots,K\}$, here we propose
\begin{equation}\label{eq:fedsm}
y_s=one\_hot(k)\,.
\end{equation}
Intuitively, the personalized model $k$ tends to generalize better on client $k$'s own local data. It is safe to set $y_s$ as the corresponding client index. Though theoretically, it may degrade the performance of Eq.~(\ref{eq:fedsm-extra}), it is more practical due to no extra training rounds. We refer to this approach as FedSM which addresses all the issues raised by the ensemble.
\subsection{New Personalization: SoftPull} \label{sec:softpull}
In this section, we present a new personalized FL optimization formulation and a method, SoftPull, to solve it and produce personalized models for FedSM. We first present existing interpolation methods to tackle the insufficient local data issue.
Let the global dataset be $\mathcal{D}$. To tackle the insufficient local data issue, \cite{mansour2020three} proposes dataset interpolation for each client as $\min_{w_{p,k}} \lambda L_{\mathcal{D}_k}(w_{p,k}) + (1-\lambda)L_{\mathcal{D}}(w_{p,k})$, where coefficient $\lambda\in[0,1]$. As client $k\in\{1,2,\cdots,K\}$, it leads to $K$ optimization problems and is inefficient to solve. Besides, it is hard to acquire the information of the global dataset $D$ during the local training. \cite{mansour2020three} also proposes model interpolation $\min_{w_g,w_{p,k},\lambda} \sum^{K}_{k=1}L_{\mathcal{D}_k}(\lambda w_{p,k}+(1-\lambda)w_{g})$. To efficiently solve the model interpolation problem, APFL \cite{deng2020adaptive} proposes
\begin{align}
w_g^* &= \arg\min_{w_g} L_{\mathcal{D}}(w_g)\,,\\
w_{p,k}^* &= \arg\min L_{\mathcal{D}_k}(\lambda w_{p,k} + (1-\lambda)w_g^*)\,,\\
w_{p.k} &\leftarrow \lambda w_{p,k}^* + (1-\lambda)w_g^*\,.
\end{align}
\textbf{Motivation.} We observe that model interpolation tries to find an appropriate combination between the FL global and local models. When the local data distribution is not similar to the global data distribution at all, we expect $\lambda\rightarrow 1$. When they are similar, we expect $\lambda \rightarrow \frac{1}{K}$ to leverage the global data information to improve the local generalization as the local dataset is small. Nevertheless, the formulation of APFL has two potential drawbacks:
\begin{itemize}[leftmargin=0.18in]
\setlength{\itemsep}{0pt}
\item The involved global model $w^*_g$ may not generalize well on $\mathcal{D}$ and $\mathcal{D}_k$, but will affect the FL training.
\item What objective function it is exactly optimizing is not clear.
\end{itemize}
In our problem formulation, we first suppose $w^*_k$ is the local optimum of client $k$:
\begin{equation}\label{eq:local optimum}
w_k^* = \arg\min_w L_{\mathcal{D}_k}(w)\,.
\end{equation}
However, local optimum $w^*_k$ may not generalizes well due to lack of local training data. Instead of interpolating the global and local optimum, we propose that the desired \textit{personalized optimum $w^*_{p,k}$ is an interpolation between the local optimum of client $k$ and other clients' personalized optima}:
\begin{equation}\label{eq:new interpolation}
w^*_{p,k}=\lambda w^*_k + (1-\lambda)\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w^*_{p,k^\prime}\,.
\end{equation}
The new interpolation avoids the global model and guarantees that the interpolated model is the optimum to some explicit objective function, as opposed to APFL. In fact, the personalized optimum $w^*_{p,k}$ is also an interpolation between the local optimum of client $k$ and other clients' local optimum because Eq.~(\ref{eq:new interpolation}) is identical to
\begin{equation}
w^*_{p,k}=\lambda w^*_k + (1-\lambda)\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w^*_{k^\prime}\,.
\end{equation}
However, Eq.~(\ref{eq:new interpolation}) is better to help us to find what objective function we are optimizing as we can turn it to
\begin{equation}
w^*_k=\frac{1}{\lambda}w^*_{p,k} - \frac{1-\lambda}{\lambda}\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w^*_{p,k^\prime}\,.
\end{equation}
Compare it with Eq.~(\ref{eq:local optimum}) and we immediately have $\{w^*_{p,k}\}^{K}_{k=1}$ as the solution to the optimization problem
\begin{equation}\label{eq:new opt prob}
\min_{\{w_{p,k}\}}\sum^{K}_{k=1}L_{\mathcal{D}_k}(\frac{1}{\lambda}w_{p,k}-\frac{1-\lambda}{\lambda}\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w_{p,k^\prime})\,.
\end{equation}
To solve the proposed new personalized FL optimization problem Eq.~(\ref{eq:new opt prob}), we propose a new method, SoftPull ($\lambda \in [\frac{1}{K},1]$), with the simplification of substituting $w^*_k$ with the locally trained model in Eq.~(\ref{eq:new interpolation}), that is, after each training round at the server,
\begin{equation}\label{eq:softpull}
w_{p,k} \leftarrow \lambda w_{p,k} + (1-\lambda)\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k} w_{p,k^\prime}\,.
\end{equation}
The corresponding algorithm is summarized in Algorithm \ref{alg:fedsm training}, line 16. When $\lambda=\frac{1}{K}$, it reduces to the ``hard'' averaging in FedAvg. To analyze the convergence, we start with common assumptions as follows.
\begin{assumption}\label{lipschitz}
(Lipschitz Smooth) The loss function $L_{\mathcal{D}_k}$ is $L$-smooth, that is, $\forall w_1, w_2 \in \mathbb{R}^d$, we have
\begin{equation}
\|\nabla L_{\mathcal{D}_k}(w_1) - \nabla L_{\mathcal{D}_k}(w_2)\|^2_2 \leq L\|w_1 - w_2\|^2_2\,.
\end{equation}
\end{assumption}
\begin{assumption}\label{bounded variance}
(Bounded Variance) The stochastic gradient $\nabla L_{\mathcal{D}_k}(w,x)$ has bounded variance $\forall w\in\mathbb{R}^d$:
\begin{equation}
\mathbb{E}\|\nabla L_{\mathcal{D}_k}(w,x)-\nabla L_{\mathcal{D}_k}(w)\|^2_2\leq \sigma^2\,.
\end{equation}
where $\mathbb{E}$ is an expectation over $x\in \mathcal{D}_k$.
\end{assumption}
\begin{assumption}\label{bounded gradient}
\cite{reddi2020adaptive} The gradient $\nabla L_{\mathcal{D}_k}(w)$ has bounded value $\forall w\in\mathbb{R}^d$: $\|\nabla L_{\mathcal{D}_k}(w)\|^2_2\leq G^2$.
\end{assumption}
\begin{theorem}\label{convergence}
Suppose Assumptions \ref{lipschitz}, \ref{bounded variance}, and \ref{bounded gradient} exist. Let the proposed objective in Eq.~(\ref{eq:new opt prob}) be $F$, superscript $(r,m)$ denote the global iteration, and $\overline{w}$ denote the average, then
\begin{align}
&\frac{1}{KRM}\sum^{R-1}_{r=0}\sum^{M-1}_{m=0}\sum^{K}_{k=1}\mathbb{E}\|\nabla_{w_{p,k}^{r,m}}F\|^2_2\\
&= \mathcal{O}(\frac{1}{\eta RM\lambda^2} + \frac{(1-\lambda)^2}{KRM\eta^2\lambda^2}\sum^{K}_{K=1}\sum^{R-1}_{r=0}\mathbb{E}\|w^{r,M}_{p,k}-\overline{w}^{r,M}_{p,k}\|^2_2 \notag\\
&\quad + \frac{(1-\lambda)^2}{KRM\lambda^4}\sum^{K}_{k=1}\sum^{R-1}_{r=0}\sum^{M-1}_{m=0}\mathbb{E}\|w^{r,m}_{p,k}-\overline{w}^{r,m}_{p,k}\|^2_2) \notag\\
&=\mathcal{O}(\frac{1}{\eta RM\lambda^2} + \frac{M\sum^{R-1}_{r=0}(1-\lambda)^2}{R\lambda^2} + \frac{M^2\eta^2\sum^{R-1}_{r=0}(1-\lambda)^2}{R\lambda^4})\,. \notag
\end{align}
If $\eta=\mathcal{O}(\frac{1}{\sqrt{RM}})$ and $M=\mathcal{O}(R^{\frac{1}{3}})$, its convergence rate is $\mathcal{O}(\frac{1}{\sqrt{RM}})$ with a convergence error $\mathcal{O}(\frac{M\sum^{R-1}_{r=0}(1-\lambda)^2}{R\lambda^2})$.
\end{theorem}
\begin{remark}\label{remark:sim}
When the data similarity is low among clients, we should set a larger $\lambda$ to reduce the effect of $\|w^{r,m}_{p,k}-\overline{w}^{r,m}_{p,k}\|^2_2$ and ensure the convergence rate. It is intuitively valid as the client has less to learn from other clients.
\end{remark}
\begin{remark}\label{remark:error}
$\lambda \downarrow$ and the convergence error $\uparrow$, but it does not mean worse generalization because we do not want to overfit local data. We will empirically tune and validate it.
\end{remark}
The proof can be found in Appendix \ref{appendix:convergence}.
\subsection{All Together}
\begin{algorithm}[t]
\caption{FedSM training.}\label{alg:fedsm training}
\begin{algorithmic}[1]
\STATE \textbf{Input:} local dataset $\mathcal{D}_k$, rounds $R$, number of sites $K$, learning rate $\eta$, $\eta_s$, coefficient $\lambda$, client weight $\frac{n_k}{n}$.
\STATE \textbf{Initialize:} global model $w_g$, personalized model $w_{p,k}$, model selector $w_s$, base optimizer $\text{OPT}(\cdot)$
\FOR{round $r=1,2,\cdots,R$}
\STATE SERVER: send models ($w_g$, $w_{p,k}$, $w_s$) to client $k$.
\FOR{CLIENT $k\in\{1,2,\cdots,K\}$ in parallel}
\STATE initialize $w_{g,k}\leftarrow w_g$, $w_{s,k} \leftarrow w_s$
\FOR{batch $(x,y) \in \mathcal{D}_k$}
\STATE $w_{g,k}\leftarrow \text{OPT}(w_{g,k},\eta,\nabla_{w_{g,k}} L(f(w_{g,k};x),y))$
\STATE $w_{p,k}\leftarrow \text{OPT}(w_{p,k},\eta,\nabla_{w_{p,k}} L(f(w_{p,k};x),y))$
\STATE // \textit{$y_s$ from Eq.~(\ref{eq:fedsm})}
\STATE $w_{s,k}\leftarrow \text{OPT}(w_{s,k}, \eta_s,\nabla_{w_{s,k}}L_s(f_s(w_{s,k};x),y_s))$
\ENDFOR
\STATE send ($w_{g,k}$, $w_{p,k}$, $w_{s,k}$) to server
\ENDFOR
\STATE SERVER: $w_g, w_s\leftarrow \sum^{K}_{k=1}\frac{n_k}{n} w_{g,k}, \sum^{K}_{k=1}\frac{n_k}{n} w_{s,k}$
\STATE SERVER: $\forall k \in \{1,2,\cdots,K\}$, $w_{p,k}\leftarrow \lambda w_{p,k} + (1-\lambda)\frac{1}{K-1}\sum^{K}_{k^\prime=1, k^\prime\neq k} w_{p,k^\prime}$ \hfill // \textit{SoftPull}
\ENDFOR
\STATE \textbf{Output:} model ($w_g$, $\{w_{p,k}\}^{K}_{k=1}$, $w_s$)
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[t]
\caption{FedSM inference.}\label{alg:fedsm inference}
\begin{algorithmic}[1]
\STATE \textbf{Input:} data $x$, model ($w_g$, $\{w_{p,k}\}^{K}_{k=1}$, $w_s$), threshold $\gamma$
\STATE $\widehat{y}_s = f_s(w_s; x)$
\IF {$\max (\widehat{y}_s)>\gamma$}
\STATE $k=\arg\max(\widehat{y}_s)\in\{1,2,\cdots,K\}$// \textit{high confidence}
\STATE $\widehat{y}=f(w_{p,k};x)$
\ELSE
\STATE $\widehat{y}=f(w_g;x)$ \hfill // \textit{low confidence}
\ENDIF
\STATE \textbf{Output:} $\widehat{y}$
\end{algorithmic}
\end{algorithm}
We summarize the proposed SoftPull method to train personalized models and the FedSM framework consisting of the model selector, global model, and personalized models in Algorithm \ref{alg:fedsm training}. Compared with FedAvg, the communication cost of each training round is $2w_g+w_s$ for FedSM. We note that some methods such as Scaffold \cite{karimireddy2020scaffold} have a cost of $2w_g$. After the training, the server sends the super model ($w_g$, $\{w_{p,k}\}^{K}_{k=1}$, $w_s$) to each client for inference, which incurs only a one-time communication cost.
For the FedSM inference in Algorithm \ref{alg:fedsm inference}, we propose a heuristic technique that the model selector selects the global model when its confidence is low, because we do not have label 0 in Eq.~(\ref{eq:fedsm}) (the global model) during training. Intuitively, if the test data is not similar to any local data distribution, the global model should be a better choice for its inference, in that it covers the joint data distribution while the personalized model covers only one local data distribution. It also guarantees that FedSM is at least not worse than the global model from FedAvg with an appropriate threshold $\gamma$.
For FedSM-extra, both the training and inference algorithms are the same except for the determination of $y_s$, the extra training rounds, and no need for the threshold $\gamma$. More details are available in Appendix \ref{appendix: fedsm-extra}.
\section{Experiments}
\begin{table}[t]
\small
\centering
\begin{tabular}{c|cccccc|c}
\toprule
Client & 1 & 2 & 3 & 4 & 5 & 6 & Global\\
\midrule
Train & 50 & 98 & 47 & 230 & 80 & 400 & 905\\
Val & 25 & 49 & 24 & 115 & 40 & 200 & 453 \\
Test & 26 & 48 & 23 & 115 & 39 & 200 & 451 \\
\bottomrule
\end{tabular}
\caption{Retinal Dataset: number of data (2D image) in each client. The data sources from client 1 to 6 are Drishti-GS1 \cite{6867807}, RIGA \cite{almazroa2018retinal} BinRushed, RIGA Magrabia, RIGA MESSIDOR, RIM-ONE \cite{fumero2011rim}, and REFUGE \cite{refuge} respectively. Global refers to the data from all clients.}
\label{tab:retinal num}
\end{table}
\begin{table}[t]
\small
\centering
\begin{tabular}{c|cccccc|c}
\toprule
Client & 1 & 2 & 3 & 4 & 5 & 6 & Global \\
\midrule
Train & 153 & 404 & 464 & 361 & 609 & 1179 & 3170 \\
Val & 77 & 215 & 219 & 162 & 289 & 582 & 1544 \\
Test & 61 & 245 & 198 & 150 & 329 & 532 & 1515 \\
\bottomrule
\end{tabular}
\caption{Prostate Dataset: number of data (2D slices) in each client. The data sources from client 1 to 6 are I2CVB \cite{lemaitre2015computer}, MSD \cite{antonelli2021medical}, NCI\_ISBI\_3T, NCI\_ISBI\_DX \cite{nci-isbi}, Promise12 \cite{promise12}, and ProstateX \cite{prostatex} respectively. Global refers to the data from all clients.}
\label{tab:prostate num 2d}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[width=0.3\textwidth]{curve/retinal-disc.pdf}
\includegraphics[width=0.3\textwidth]{curve/retinal-cup.pdf}
\includegraphics[width=0.3\textwidth]{curve/prostate_curve.pdf}
\caption{Training curves comparison. The curves are non-decreasing because we record the best result during training.}
\label{fig:curve}
\end{figure*}
\begin{table*}[t]
\small
\centering
\begin{tabular}{c|cccccc|cc}
\toprule
Method & Client 1 & Client 2 & Client 3 & Client 4 & Client 5 & Client 6 & Client Avg Dice & Global Dice \\
\midrule
Centralized & 0.9161 & 0.8760 & 0.8758 & 0.9022 & 0.8510 & 0.9179 & 0.8898 & 0.9014 \\
\midrule
Client 1 Local & 0.8835 & 0.3331 & 0.7345 & 0.4933 & 0.3408 & 0.7015 & 0.5811 & 0.5902 \\
Client 2 Local & 0.2346 & 0.8620 & 0.0886 & 0.7751 & 0.1791 & 0.4106 & 0.4250 & 0.5050 \\
Client 3 Local & 0.8337 & 0.3402 & 0.8766 & 0.6010 & 0.3644 & 0.7794 & 0.6326 & 0.6594 \\
Client 4 Local & 0.5108 & 0.8574 & 0.3457 & 0.9008 & 0.2361 & 0.6822 & 0.5888 & 0.6910 \\
Client 5 Local & 0.5241 & 0.1584 & 0.3953 & 0.2039 & 0.8223 & 0.6222 & 0.4544 & 0.4662 \\
Client 6 Local & 0.7908 & 0.6649 & 0.7325 & 0.7681 & 0.3742 & 0.9150 & 0.7076 & 0.7877 \\
\midrule
FedAvg & 0.8847 & 0.8679 & 0.8667 & 0.9015 & 0.7877 & 0.9172 & 0.8710 & 0.8923 \\
FedProx & 0.8635 & 0.8522 & 0.8547 & 0.8952 & 0.6852 & 0.9095 & 0.8434 & 0.8749 \\
Scaffold & 0.8380 & 0.8513 & 0.8215 & 0.8935 & 0.5671 & 0.9130 & 0.8141 & 0.8625 \\
\midrule
FedSM & \textbf{0.9132} & \textbf{0.8769} & \textbf{0.8865} & \textbf{0.9041} & \textbf{0.8483} & \textbf{0.9195} & \textbf{0.8914} & \textbf{0.9028} \\
FedSM-extra & \textbf{0.9134} & \textbf{0.8763} & \textbf{0.8841} & \textbf{0.9038} & \textbf{0.8483} & \textbf{0.9172} & \textbf{0.8905} & \textbf{0.9007} \\
\bottomrule
\end{tabular}
\caption{(low data similarity) Test Dice coefficient comparison of retinal segmentation. ``Client $k$ Local" refers to local training on client $k$. The first row refers to the performance on client 1$\sim$6's test data, their average, and the performance on all clients' test data. We report the average of disc and cup Dice coefficients here. We bold the best FL numbers. See Appendix \ref{appendix:additional exp} for their separate numbers and the visual comparison of segmentation.}
\label{tab:retinal avg dice}
\end{table*}
\begin{table*}[t]
\small
\centering
\begin{tabular}{c|cccccc|cc}
\toprule
Method & Client 1 & Client 2 & Client 3 & Client 4 & Client 5 & Client 6 & Client Avg Dice & Global Dice \\
\midrule
Centralized & 0.9018 & 0.8583 & 0.8702 & 0.8844 & 0.8800 & 0.8474 & 0.8737 & 0.8651 \\
\midrule
Client 1 Local & 0.8582 & 0.3886 & 0.4476 & 0.2849 & 0.3830 & 0.4697 & 0.4720 & 0.4336 \\
Client 2 Local & 0.7166 & 0.7669 & 0.8317 & 0.7341 & 0.6156 & 0.7754 & 0.7401 & 0.7403 \\
Client 3 Local & 0.6470 & 0.8541 & 0.8549 & 0.6735 & 0.6591 & 0.7519 & 0.7401 & 0.7496 \\
Client 4 Local & 0.4515 & 0.6566 & 0.6700 & 0.8518 & 0.4558 & 0.6267 & 0.6187 & 0.6148 \\
Client 5 Local & 0.8198 & 0.7751 & 0.8469 & 0.8029 & 0.8038 & 0.7928 & 0.8069 & 0.8016 \\
Client 6 Local & 0.8555 & 0.7965 & 0.8260 & 0.7206 & 0.6478 & 0.8466 & 0.7822 & 0.7809 \\
\midrule
FedAvg & 0.8775 & 0.8575 & 0.8700 & 0.8802 & 0.8717 & 0.8532 & 0.8684 & 0.8638 \\
FedProx & \textbf{0.8948} & 0.8511 & 0.8722 & 0.8803 & 0.8668 & 0.8513 & 0.8694 & 0.8621 \\
Scaffold & 0.8500 & 0.8440 & 0.8570 & 0.8423 & 0.8431 & 0.8412 & 0.8463 & 0.8446 \\
\midrule
FedSM & \textbf{0.8946} & \textbf{0.8596} & \textbf{0.8786} & \textbf{0.8898} & \textbf{0.8817} & \textbf{0.8535} & \textbf{0.8763} & \textbf{0.8692} \\
FedSM-extra & 0.8886 & \textbf{0.8584} & \textbf{0.8766} & \textbf{0.8880} & \textbf{0.8760} & \textbf{0.8542} & \textbf{0.8736} & \textbf{0.8673} \\
\bottomrule
\end{tabular}
\caption{(high data similarity) Test Dice coefficient comparison of prostate segmentation. We bold the best FL numbers. See Appendix \ref{appendix:additional exp} for the visual comparison.}
\label{tab:prostate dice}
\end{table*}
\begin{table*}[t]
\small
\centering
\begin{tabular}{c|c|ccccccc|c|r}
\toprule
Unseen Client $k$ & Threshold $\gamma$ & GM & PM1 & PM2 & PM3 & PM4 & PM5 & PM6 & Dice & Best $\gamma$, Dice \\
\midrule
Client $k=6$ & 0 & 0 & 0.02 & 0 & 0.35 & 0 & 0.63 & N/A & 0.8587 & 1, 0.8906 \\
Client $k=5$ & 0 & 0 & 0.31 & 0.03 & 0 & 0.61 & N/A & 0.05 & 0.4015 & 0.9, 0.4304 \\
Client $k=4$ & 0 & 0 & 0 & 1.00 & 0 & N/A & 0 & 0 & 0.8869 & $<$0.95, 0.8870 \\
Client $k=3$ & 0 & 0 & 0 & 0.57 & N/A & 0 & 0 & 0.43 & 0.8441 & $<$0.9, 0.8446\\
Client $k=2$ & 0 & 0 & 0 & N/A & 0 & 0.92 & 0.08 & 0 & 0.8409 & $<$1, 0.8409 \\
Client $k=1$ & 0 & 0 & N/A & 0 & 1.00 & 0 & 0 & 0 & 0.8839 & $<$0.99, 0.8839 \\
\bottomrule
\end{tabular}
\caption{(retinal segmentation, Dice = average of disc and cup Dice coefficients) Model selection frequency from the model selector when FL train with clients $\{1,2,\cdots,6\}/\{k\}$ and test on the \textbf{unseen} client $k\in\{1,2,\cdots,6\}$. From left to right, GM denotes the global model and PM denotes the personalized model $\{1,2,\cdots,6\}/\{k\}$. The model selection frequency with the best $\gamma$, and the more detailed Dice results can be found in Appendix \ref{appendix:additional exp}. Note GM is never selected as the Threshold $\gamma$ is intentionally set to 0.}
\label{tab:model selector frequency}
\vspace{-10pt}
\end{table*}
\begin{figure}[t]
\centering
\includegraphics[width=.6\columnwidth]{figures/tsne.png}
\caption{TSNE map of the features extracted form the model selector on retinal segmentation task.}
\label{fig:tsne}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.3\textwidth]{losssurface/Train_site5.pdf}
\includegraphics[width=0.3\textwidth]{losssurface/Val_site5.pdf}
\includegraphics[width=0.3\textwidth]{losssurface/Test_site5.pdf}
\caption{The 1D loss surface near the models trained by different methods on Client 5's data in retinal segmentation.}
\label{fig:loss surface}
\end{figure*}
\begin{table*}[t]
\small
\centering
\begin{tabular}{c|cccccc|cc}
\toprule
Method & Client 1 & Client 2 & Client 3 & Client 4 & Client 5 & Client 6 & Client Avg Dice & Global Dice \\
\midrule
FT \cite{wang2019federated} & 0.9087 & 0.8703 & 0.8877 & 0.9003 & 0.8409 & 0.9151 & 0.8875 & 0.8984 \\
APFL \cite{deng2020adaptive} & 0.9083 & 0.8640 & 0.8794 & 0.8969 & 0.8416 & 0.9152 & 0.8842 & 0.8966 \\
Per-FedAvg \cite{fallah2020personalized} & 0.9051 & 0.8559 & 0.8708 & 0.8954 & 0.8031 & 0.9119 & 0.8737 & 0.8900 \\
Per-FedMe \cite{t2020personalized} & 0.9084 & 0.8646 & 0.8822 & 0.8980 & 0.8211 & 0.9162 & 0.8818 & 0.8957 \\
SoftPull & \textbf{0.9132} & \textbf{0.8769} & \textbf{0.8865} & \textbf{0.9041} & \textbf{0.8483} & \textbf{0.9195} & \textbf{0.8914} & \textbf{0.9028} \\
\bottomrule
\end{tabular}
\caption{FedSM with different personalization method in retinal segmentation. Dice = average of disc and cup Dice coefficients.}
\label{tab:vary personalization}
\end{table*}
\begin{table}[t]
\small
\centering
\begin{tabular}{c|ccccc}
\toprule
$\lambda$ & 0.1 & 0.3 & 0.5 & \textbf{0.7} & 0.9 \\
\midrule
Client Avg & 0.8808 & 0.8859 & 0.8895 & \textbf{0.8914} & 0.8882 \\
Global & 0.8964 & 0.8896 & 0.9019 & \textbf{0.9028} & 0.9001 \\
\bottomrule
\end{tabular}
\caption{FedSM with different coefficient $\lambda$ in retinal segmentation. Dice = average of disc and cup Dice coefficients.}
\label{tab:vary lambda}
\vspace{-13pt}
\end{table}
We validate our proposed method on three real-world FL medical image segmentation tasks: retinal disc \& cup from 2D fundus images, and prostate segmentation from 3D MR images. The global and personalized model architecture is 2D U-Net \cite{ronneberger2015u}, while the model selector architecture is VGG-11 \cite{simonyan2014very}. We randomly split the data to train/validation/test with a ratio of 0.5/0.25/0.25. The image data are resized to $256\times 256$. The local training epoch is 1 and the total training rounds is 150. Most methods converge in 100 rounds. But for FedSM-extra, we train the global and personalized models for 100 rounds and the model selector for an extra 50 rounds. The loss function is Dice loss and the test metric is Dice coefficient. The base optimizer is Adam with $\beta=(0.9, 0.999)$. We tune the best learning rate for all methods and the threshold $\gamma$ for FedSM. For prostate segmentation, in particular, the image data are 3D but we take the 2D slices and perform 2D segmentation. Each experiment repeatedly runs 3 times and we report the mean value.
The dataset information is summarized in Table \ref{tab:retinal num} and \ref{tab:prostate num 2d}. Overall, the retinal dataset features lower data similarity among clients (stronger non-iid). The images may differ in position, color, brightness, background ratio, etc. While the prostate dataset has a higher data similarity as the images mostly differ in brightness (see Appendix \ref{appendix:additional dataset info}).
We compare FedSM and FedSM-extra with baselines (1) Centralized: centralized training, which is the upper bound but prohibited in FL, (2) Local: local training on one client, (3) FedAvg \cite{mcmahan2017communication}, the \textit{de facto} FL method, (4) FedProx \cite{li2018federated}, and (5) Scaffold \cite{karimireddy2020scaffold}.
\subsection{General Results}
We compare the training curves of different methods in Figure \ref{fig:curve}. The centralized training upper bound is plotted as a horizontal dash line. We can see that the proposed FedSM is the only FL method to close the validation gap to centralized training. FedSM is even better than centralized training on the retinal cup segmentation task, due to the proposed SoftPull personalization method. Note that we can not show the training curve of FedSM-extra as its model selector has to be trained in the extra training rounds.
We summarize the testing numbers in Table \ref{tab:retinal avg dice} and \ref{tab:prostate dice}. For retinal segmentation, FedSM slightly improves centralized training regarding the client average Dice and global Dice by 0.2\% and 0.1\% respectively, while FedAvg shows a decrease of 1.9\% and 0.9\%. The FedSM-extra shows the same performance as FedSM, validating the proposed simplification from Eq.~(\ref{eq:fedsm-extra}) to Eq.~(\ref{eq:fedsm}). For prostate segmentation, similar patterns can be observed. But the gap becomes smaller due to higher data similarity among clients.
For retinal segmentation, FedSM outperforms centralized training for client 3 and matches centralized training for the other clients. However, FedAvg is inferior to centralized training for clients 1, 2, 3, and 5 where the local dataset size is smaller. What's more, FedAvg shows similar test Dice performance to local training for clients 1 and 2, and is even inferior to local training for clients 3 and 5. Therefore, those clients do not benefit from FL via FedAvg, and may not be willing to join the FL system.
We also observe that local training does not generalize well on other clients' data, which is critical as it will perform poorly for patients from other clients (medical institutions). Centralized training improves the local training on the local dataset, especially for clients with insufficient data.
\subsection{Validate Motivation}
\textbf{Validate FedSM.} Recall that our first motivation is to find the closest local data distribution for the test data. In FedSM, we first plot the TSNE map of the features extracted from the model selector in Figure \ref{fig:tsne}. To validate that the model selector can fulfill our motivation, we sequentially choose client $k\in\{1,2,\cdots,6\}$ as the unseen client to test and FL train the model with clients $\{1,2,\cdots,6\}/\{k\}$. We set the threshold $\gamma=0$ to let the model selector select from the personalized models. We summarize the frequency in Table \ref{tab:model selector frequency}. We can see that the model selector tends to select the personalized models of clients 3 and 5 for client 6, which also matches Figure \ref{fig:tsne} and the local training results in Table \ref{tab:retinal avg dice} that clients 3 and 5 are more similar to client 6. Similar patterns can be observed for the other clients. Therefore, the model selector indeed fulfills our motivation. Note that to validate the model selector, we cannot let the unseen client $k$ join the FL system. Because in that case, the model selector tends to select its own personalized model.
In Table \ref{tab:model selector frequency}, we also validate that the threshold $\gamma$ helps improve the performance of FedSM for the unseen data. For those unseen data with low confidence from the model selector, a larger $\gamma$ increases the chance of the global model to be selected because maybe none of the personalized models is suitable. By choosing a proper $\gamma$, we can further improve the Dice of unseen clients 5 and 6 by 3\%.
\textbf{Validate SoftPull.} Recall that our second motivation is to find a model generalizing well on the local data distribution even with insufficient local data. To achieve it we propose a new personalized FL optimization formulation with SoftPull to solve it. The Remark \ref{remark:sim} of the theoretical analysis can be empirically validated by the fact that the best $\lambda=0.7$ (closer to 1) for the retinal segmentation task with lower data similarity, and that the best $\lambda=0.3$ (closer to $\frac{1}{K}=\frac{1}{6}=0.17$) for the prostate segmentation task with higher data similarity.
Next, we will validate Remark \ref{remark:error} that a proper $\lambda$ may lead to a convergence error, but in the meantime may improve the generalization by preventing overfitting the small local dataset with the help of other clients. We plot the 1D loss surface near the trained model by computing the loss along 10 randomly sampled unit vector directions (Figure \ref{fig:loss surface}), following existing works \cite{izmailov2018averaging,he2019asymmetric}. It is interesting to see that local training overfits the training data and leads to a sharp local training optimum, which is known to generalize worse \cite{izmailov2018averaging,yang2019swalp,he2019asymmetric}. On the contrary, we observe an ``over-regularization'' effect for FedAvg as it has an even flatter training optimum than centralized training and a large convergence error (worse training loss), which also leads to a worse generalization performance. Indeed, averaging model in FedAvg can be regarded as a sort of implicit regularization. In comparison, SoftPull achieves a tunable flatness by choosing a proper $\lambda$. Even if it leads to a convergence error, it achieves generalization performance better than local training and comparable to centralized training.
\subsection{Ablation Study}
\textbf{Personalization.} We compare personalization methods in FedSM in Table \ref{tab:vary personalization}, including (1) FT (local fine-tuning) \cite{wang2019federated}, (2) APFL \cite{deng2020adaptive}, (3) Per-FedAvg \cite{fallah2020personalized}, and (4) Per-FedMe \cite{t2020personalized}. All methods' hyper-parameters are tuned for best results. SoftPull is the better interpolation method among them, outperforming APFL by 0.62\% regarding the global Dice coefficient. It also outperforms the best counterpart by 0.44\%.
\textbf{Interpolation Coefficient $\lambda$.} We explore different $\lambda$ values of FedSM in Table \ref{tab:vary lambda} and $\lambda=0.7$ performs the best.
\section{Conclusion}
In this work, we propose FedSM to close the generalization gap between FL and centralized training for medical image segmentation for the first time. The empirical study on real-world medical FL tasks validates our theoretical analysis and motivation to avoid the client drift issue.
\clearpage
{\small
\bibliographystyle{ieee_fullname}
\section{Additional Dataset Information}\label{appendix:additional dataset info}
\begin{table}[htb!]
\small
\centering
\begin{tabular}{c|cccccc|c}
\toprule
Client & 1 & 2 & 3 & 4 & 5 & 6 & Global \\
\midrule
Train & 10 & 16 & 18 & 18 & 25 & 50 & 137 \\
Val & 5 & 8 & 9 & 9 & 12 & 25 & 68 \\
Test & 4 & 8 & 8 & 8 & 13 & 24 & 65 \\
\bottomrule
\end{tabular}
\caption{Prostate dataset: number of data (3D image) in each client.}
\label{tab:prostate num 3d}
\end{table}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.32\columnwidth]{retinalimage/c1.png}
\includegraphics[width=0.32\columnwidth]{retinalimage/c2.jpg}
\includegraphics[width=0.32\columnwidth]{retinalimage/c3.jpg}
\includegraphics[width=0.32\columnwidth]{retinalimage/c4.jpg}
\includegraphics[width=0.32\columnwidth]{retinalimage/c5.jpg}
\includegraphics[width=0.32\columnwidth]{retinalimage/c6.jpg}
\caption{Representative original 2D image in retinal dataset (low data similarity). First row: client 1 to 3. Second row: client 4 to 6.}
\label{fig:retinal original image}
\end{figure}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.32\columnwidth]{prostateimage/c1.jpg}
\includegraphics[width=0.32\columnwidth]{prostateimage/c2.jpg}
\includegraphics[width=0.32\columnwidth]{prostateimage/c3.jpg}
\includegraphics[width=0.32\columnwidth]{prostateimage/c4.jpg}
\includegraphics[width=0.32\columnwidth]{prostateimage/c5.jpg}
\includegraphics[width=0.32\columnwidth]{prostateimage/c6.jpg}
\caption{Representative original 2D image slices in prostate dataset (high data similarity). First row: client 1 to 3. Second row: client 4 to 6. E.g., the first slice comes from a 3D image in client 1.}
\label{fig:prostate original image}
\end{figure}
\section{FedSM-extra Algorithm}\label{appendix: fedsm-extra}
\begin{algorithm}[htb!]
\caption{FedSM-extra training.}\label{alg:fedsm-extra training}
\begin{algorithmic}[1]
\STATE \textbf{Input:} local dataset $\mathcal{D}_k$, rounds $R$, number of sites $K$, learning rate $\eta$, coefficient $\lambda$, client weight $\frac{n_k}{n}$.
\STATE \textbf{Initialize:} global model $w_g$, personalized model $w_{p,k}$, model selector $w_s$, base optimizer $\text{OPT}(\cdot)$
\FOR{round $r=1,2,\cdots,R$}
\STATE SERVER: send models ($w_g$, $w_{p,k}$) to client $k$.
\FOR{CLIENT $k\in\{1,2,\cdots,K\}$ in parallel}
\STATE initialize $w_{g,k}\leftarrow w_g$
\FOR{batch $(x,y) \in \mathcal{D}_n$}
\STATE $w_{g,k}\leftarrow \text{OPT}(w_{g,k},\eta,\nabla_{w_{g,k}} L(f(w_{g,k};x),y))$
\STATE $w_{p,k}\leftarrow \text{OPT}(w_{p,k},\eta,\nabla_{w_{p,k}} L(f(w_{p,k};x),y))$
\ENDFOR
\STATE send ($w_{g,k}$, $w_{p,k}$) to server
\ENDFOR
\STATE SERVER: $w_g\leftarrow \sum^{K}_{k=1}\frac{n_k}{n} w_{g,k}$ \hfill // \textit{FedAvg}
\STATE SERVER: $\forall k \in \{1,2,\cdots,K\}$, $w_{p,k}\leftarrow \lambda w_{p,k} + (1-\lambda)\frac{1}{K-1}\sum^{K}_{k^\prime=1, k^\prime\neq k} w_{p,k^\prime}$ \hfill // \textit{SoftPull}
\ENDFOR
\STATE // \textit{extra training rounds}
\STATE SERVER: send models ($w_g$, $w_{p,n}$) to clients.
\FOR{round $r=1,2,\cdots,\Delta R$}
\STATE SERVER: send model $w_s$ to clients.
\FOR{CLIENT $k\in\{1,2,\cdots,K\}$ in parallel}
\STATE Initialize $w_{s,k}\leftarrow w_s$
\FOR{batch $(x,y) \in \mathcal{D}_n$}
\STATE // \textit{$y_s$ from Eq.~(\ref{eq:fedsm-extra}})
\STATE $w_{s,k}\leftarrow \text{OPT}(w_{s,k}, \eta_s,\nabla_{w_{s,k}}L_s(f_s(w_{s,k};x),y_s))$
\ENDFOR
\STATE send $w_{s,k}$ to server
\ENDFOR
\STATE SERVER: $w_s\leftarrow \sum^{K}_{k=1}\frac{n_k}{n} w_{s,k}$
\ENDFOR
\STATE \textbf{Output:} model ($w_g$, $\{w_{p,k}\}^{K}_{k=1}$, $w_s$)
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[htb!]
\caption{FedSM-extra inference.}\label{alg:fedsm-extra inference}
\begin{algorithmic}[1]
\STATE \textbf{Input:} data $x$, model ($w_g$, $\{w_{p,k}\}^{K}_{k=1}$, $w_s$)
\STATE $\widehat{y}_s = f_s(w_s; x)$
\STATE $k=\arg\max(\widehat{y}_s)\in\{0,1,\cdots,K\}$
\IF {$k>0$}
\STATE $\widehat{y}=f(w_{p,k};x)$
\ELSE
\STATE $\widehat{y}=f(w_g;x)$
\ENDIF
\STATE \textbf{Output:} $\widehat{y}$
\end{algorithmic}
\end{algorithm}
For the training of FedSM-extra, we train the global model and personalized models first, and then train the model selector, which incurs extra $\Delta R$ training rounds. In each training round of FedSM-extra, the communication cost is $2 w_g$ in the previous $R$ rounds (the global and personalized models have the same model architecture). It becomes $w_s$ in the extra $\Delta R$ training rounds.
For the inference of FedSM-extra, both the global model and personalized models can be selected. Therefore $k\in\{0,1,\cdots,K\}$ (For FedSM, $k\in\{1,2,\cdots,K\}$).
\section{Proof of SoftPull Convergence}\label{appendix:convergence}
Let the current/total training rounds be $r/R$, current/total local training steps be $m/M$, the current/total global training step be $t/T$. We denote the personalized model during training as $w_{p,k}^{r,m}$. For simplicity we use $f_k$ to denote loss $L_{\mathcal{D}_k}$.
After the local training in the last training round $r-1$ finishes, we get model $w_{p,k}^{r-1,M}$ and want to
\begin{equation}
\min \sum^{K}_{k=1}f_k(\frac{1}{\lambda}w_{p,k}^{r-1,M}-\frac{1-\lambda}{\lambda}\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w_{p,k^\prime}^{r-1,M})
\end{equation}
In the beginning of the current training round $r$, from Eq.~(\ref{eq:softpull}), we will have
\begin{equation}
w_{p,k}^{r,0}=\lambda w_{p,k}^{r-1,M} + (1-\lambda)\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w_{p,k^\prime}^{r-1,M}
\end{equation}
In the current training round $r$, we consider two stages. The first stage is a transition from then end of training round $r-1$ to the start of the current training round $r$, while the second stage is the start to the end of current training round $r$. Let $\lambda^\prime=\frac{K\lambda-1}{K-1}$, $1-\lambda^\prime=
\frac{K}{K-1}(1-\lambda)$, then
\begin{equation}
w_{p,k}^{r,0} = \lambda^\prime w^{r-1,M}_{p,k} + (1-\lambda^\prime)\overline{w}_{p,k}^{r-1,M}
\end{equation}
where the bar denotes an average over all clients $k\in\{1,2,\cdots,K\}$. It can be clearly seen that when the data distributions of clients are very similar, we set $\lambda=\frac{1}{K}$, $\lambda^\prime=0$, i.e., the ``hard averaging" in FedAvg. When the data distributions are not similar at all, we set $\lambda=1$, $\lambda^\prime=1$ to only do local training. In other circumstances, theoretically we should set $\lambda\in[\frac{1}{K},1]$, $\lambda^\prime\in[0,1]$ according to the data similarity.
\subsection{Difference}
Suppose the stochastic gradient at iteration $(r,m)$ is $\nabla f_k(w_{p,k}^{r,m},x_{p,k}^{r,m})$ and the expected gradient is $\nabla f_k(w_{p,k}^{r,m})=\mathbb{E}_{x^{r,m}_{p,k}\in\mathcal{D}_k}\nabla f_k(w_{p,k}^{r,m},x_{p,k}^{r,m})=\mathbb{E}\nabla f_k(w_{p,k}^{r,m},x_{p,k}^{r,m})$. We need to bound
\begin{equation}
\begin{split}
&\|(w_{p,k}^{r+1,0}-w_{p,k}^{r,M})\|^2_2\\
&= \|(1-\lambda)(w_{p,k}^{r,M}-\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w_{p,k^\prime}^{r,M})\|^2_2\\
&= (1-\lambda)^2 \|w_{p,k}^{r,M}-\frac{1}{K-1}(K\overline{w}^{r,M}_{p,k}-w_{p,k}^{r,M})\|^2_2\\
&=\frac{(1-\lambda)^2K^2}{(K-1)^2}\|w_{p,k}^{r,M}-\overline{w}^{r,M}_{p,k}\|^2_2
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
&\mathbb{E}\|w_{p,k}^{r,M}-\overline{w}^{r,M}_{p,k}\|^2_2\\
&= \eta^2 \mathbb{E}\|\sum^{r}_{r^\prime=0}(\lambda^\prime)^{r-r^\prime}\sum^{M-1}_{m=0}[\nabla f_k(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m})\\
&\quad - \overline{\nabla f_k}(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m})]\|^2_2\\
&\leq \eta^2(\sum^{r}_{r^\prime=0}(\lambda^\prime)^{r-r^\prime})^2 \mathbb{E}\| \sum^{r}_{r^\prime=0} \frac{(\lambda^\prime)^{r-r^\prime}}{\sum^{r}_{r^\prime=0}(\lambda^\prime)^{r-r^\prime}}\sum^{M-1}_{m=0}\\
&\quad [\nabla f_k(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m}) - \overline{\nabla f_k}(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m})]\|^2_2\\
&\leq \eta^2(\sum^{r}_{r^\prime=0}(\lambda^\prime)^{r-r^\prime})\sum^{r}_{r^\prime=0}(\lambda^\prime)^{r-r^\prime}\mathbb{E}\|\sum^{M-1}_{m=0}\\
&\quad [\nabla f_k(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m}) - \overline{\nabla f_k}(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m})]\|^2_2\\
&\leq M\eta^2(\sum^{r}_{r^\prime=0}(\lambda^\prime)^{r-r^\prime})\sum^{r}_{r^\prime=0}(\lambda^\prime)^{r-r^\prime}\sum^{M-1}_{m=0}\\
&\quad \mathbb{E}\|\nabla f_k(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m}) - \overline{\nabla f_k}(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m})\|^2_2\\
&\leq 2M^2(G^2+\sigma^2)\eta^2(\sum^{r}_{r^\prime=0}(\lambda^\prime)^{r-r^\prime})^2\\
&\leq 2M^2(G^2+\sigma^2)\eta^2 [\frac{1-(\lambda^\prime)^{r+1}}{1-\lambda^\prime}]^2
\end{split}
\end{equation}
where $\mathbb{E}\|\nabla f_k(w_{p,k}^{r^\prime,m},x_{p,k}^{r^\prime,m})\|^2_2 \leq 2(G^2+\sigma^2)$ based on Assumptions \ref{bounded gradient} and \ref{bounded variance}. Then
\begin{equation}
\begin{split}
&\mathbb{E}\|(w_{p,k}^{r+1,0}-w_{p,k}^{r,M})\|^2_2\\
&\leq \frac{[1-(\lambda^\prime)^{r+1}]^2(1-\lambda)^2K^2}{(1-\lambda^\prime)^2(K-1)^2} 2M^2(G^2+\sigma^2)\eta^2\\
&= [1-(\lambda^\prime)^{r+1}]^2 2M^2(G^2+\sigma^2)\eta^2
\end{split}
\end{equation}
\subsection{Local Objective}
Here we consider the local objective function to optimize. From $(r,0)$ to $(r,M)$, i.e. $m\in\{0,1,\cdots,M-1\}$, due to the Lipschitz smooth assumption we have
\begin{equation}
\begin{split}
&f_k(w_{k,p}^{r,m+1}) - f_k(w_{k,p}^{r,m})\\
&\leq \langle\nabla f_k(w_{k,p}^{r,m}), w_{k,p}^{r,m+1}-w_{k,p}^{r,m}\rangle + \frac{L}{2}\|w_{k,p}^{r,m+1}-w_{k,p}^{r,m}\|^2_2\\
&=-\eta\langle\nabla f_k(w_{k,p}^{r,m}),\nabla f_k(w_{k,p}^{r,m},x^{r,m}_{k,p})\rangle\\
&\quad + \frac{\eta^2L}{2}\|\nabla f_k(w_{k,p}^{r,m},x^{r,m}_{k,p})\|^2_2\\
&=-\eta\langle\nabla f_k(w_{k,p}^{r,m}),\nabla f_k(w_{k,p}^{r,m},x^{r,m}_{k,p})\rangle\\
&\quad + \frac{\eta^2L}{2}\|\nabla f_k(w_{k,p}^{r,m})\|^2_2 + \frac{\eta^2L\sigma^2}{2}
\end{split}
\end{equation}
Take the expectation and suppose $\eta\leq\frac{1}{L}$,
\begin{equation}
\begin{split}
&\mathbb{E}[f_k(w_{k,p}^{r,m+1}) - f_k(w_{k,p}^{r,m})]\\
&\leq -\eta(1-\frac{\eta L}{2})\mathbb{E}\|\nabla f_k(w_{k,p}^{r,m})\|^2_2 + \frac{\eta^2L\sigma^2}{2}\\
&\leq -\frac{\eta}{2}\mathbb{E}\|\nabla f_k(w_{k,p}^{r,m})\|^2_2 + \frac{\eta^2L\sigma^2}{2}
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\mathbb{E}\|\nabla f_k(w_{k,p}^{r,m})\|^2_2\\
&\leq \frac{2}{\eta}\mathbb{E}[f_k(w_{k,p}^{r,m})-f_k(w_{k,p}^{r,m+1})] + \eta L \sigma^2
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\sum^{M-1}_{m=0}\mathbb{E}\|\nabla f_k(w_{k,p}^{r,m})\|^2_2\\
&\leq \frac{2}{\eta}\mathbb{E}[f_k(w_{k,p}^{r,0})-f_k(w_{k,p}^{r,M})] + M\eta L \sigma^2
\end{split}
\end{equation}
While from $(r,M)$ to $(r+1,0)$, we have
\begin{equation}
\begin{split}
&f_k(w_{k,p}^{r+1,0}) - f_k(w_{k,p}^{r,M})\\
&\leq \langle\nabla f_k(w_{k,p}^{r,M}), w_{k,p}^{r+1,0}-w_{k,p}^{r,M}\rangle + \frac{L}{2}\|w_{k,p}^{r+1,0}-w_{k,p}^{r,M}\|^2_2\\
&\leq \frac{\eta}{8}\|\nabla f_k(w_{k,p}^{r,M})\|^2_2 + (\frac{2}{\eta} + \frac{L}{2})\|w_{k,p}^{r+1,0}-w_{k,p}^{r,M}\|^2_2\\
&\leq \frac{\eta}{4}\|\nabla f_k(w^{r,M-1}_{k,p})\|^2_2 + \frac{\eta L^2}{4}\|w^{r,M}_{k,p} - w^{r,M-1}_{k,p}\|^2_2\\
&\quad + (\frac{2}{\eta} + \frac{L}{2})\|w_{k,p}^{r+1,0}-w_{k,p}^{r,M}\|^2_2\\
&= \frac{\eta}{4}\|\nabla f_k(w^{r,M-1}_{k,p})\|^2_2 + \frac{\eta^3 L^2}{4}\|\nabla f_k(w^{r,M-1}_{k,p},x^{r,M-1}_{k,p})\|^2_2\\
&\quad + (\frac{2}{\eta} + \frac{L}{2})\|w_{k,p}^{r+1,0}-w_{k,p}^{r,M}\|^2_2\\
\end{split}
\end{equation}
Therefore, from $(r,0)$ to $(r+1,0)$, we have
\begin{equation}
\begin{split}
&\sum^{M-1}_{m=0}\mathbb{E}\|\nabla f_k(w_{k,p}^{r,m})\|^2_2\\
&\leq \frac{2}{\eta}\mathbb{E}[f_k(w_{k,p}^{r,0})-f_k(w_{k,p}^{r+1,0})] + M\eta L \sigma^2\\
&\quad + \frac{2}{\eta}\mathbb{E}[f_k(w_{k,p}^{r+1,0})-f_k(w_{k,p}^{r,M})]\\
&\leq \frac{2}{\eta}\mathbb{E}[f_k(w_{k,p}^{r,0})-f_k(w_{k,p}^{r+1,0})] + M\eta L \sigma^2\\
&\quad + \frac{1}{2}\mathbb{E}\|\nabla f_k(w^{r,M-1}_{k,p})\|^2 + \eta^2L^2 (G^2+\sigma^2)\\
&\quad + (\frac{4}{\eta^2} + \frac{L}{\eta})\mathbb{E}\|w^{r+1,0}_{k,p}-w^{r,M}_{k,p}\|^2_2
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\sum^{M-1}_{m=0}\mathbb{E}\|\nabla f_k(w_{k,p}^{r,m})\|^2_2\\
&\leq \frac{4}{\eta}\mathbb{E}[f_k(w_{k,p}^{r,0})-f_k(w_{k,p}^{r+1,0})] + 2M\eta L \sigma^2\\
&\quad + 2\eta^2L^2(G^2+\sigma^2) + (\frac{8}{\eta^2} + \frac{2L}{\eta})\mathbb{E}\|w^{r+1,0}_{k,p}-w^{r,M}_{k,p}\|^2_2
\end{split}
\end{equation}
From $r=0$ to $R-1$,
\begin{equation}
\begin{split}
&\frac{1}{RM}\sum^{R-1}_{r=0}\sum^{M-1}_{m=0}\mathbb{E}\|\nabla f_k(w_{k,p}^{r,m})\|^2_2\\
&\leq \frac{4\mathbb{E}[f_k(w^{0,0}_{k,p}) - f_k(w^{R,0}_{k,p})]}{\eta RM} + 2\eta L\sigma^2\\
&\quad + \frac{2\eta^2L^2(G^2+\sigma^2)}{M} + \frac{1}{RM}(\frac{8}{\eta^2} + \frac{2L}{\eta})\\
&\quad \cdot\sum^{R-1}_{r=0}\mathbb{E}\|w^{r+1,0}_{k,p}-w^{r,M}_{k,p}\|^2_2\\
&= \frac{4\mathbb{E}[f_k(w^{0,0}_{k,p}) - f_k(w^{R,0}_{k,p})]}{\eta RM} + 2\eta L\sigma^2\\
&\quad + \frac{2\eta^2L^2(G^2+\sigma^2)}{M} + \frac{1}{RM}(\frac{8}{\eta^2} + \frac{2L}{\eta})\\
&\quad \cdot \frac{(1-\lambda)^2 K^2}{(K-1)^2}\sum^{R-1}_{r=0}\mathbb{E}\|w^{r,M}_{k,p}-\overline{w}^{r,M}_{k,p}\|^2_2\\
\end{split}
\end{equation}
\subsection{Proposed Objective}
Here we consider our proposed personalized FL objective function to optimize. For simplicity of notation, let
\begin{equation}
u^{r,m}_{k}=\frac{1}{\lambda}w_{p,k}^{r,m}-\frac{1-\lambda}{\lambda}\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w_{p,k^\prime}^{r,m}
\end{equation}
Then
\begin{equation}
\begin{split}
&u^{r,m}_{k} - w^{r,m}_{p,k} = \frac{1-\lambda}{\lambda}(w^{r,m}_{p,k}-\frac{1}{K-1}\sum^{K}_{k^\prime=1,k^\prime\neq k}w_{p,k^\prime}^{r,m})\\
&= \frac{1-\lambda}{\lambda}\frac{K}{K-1}(w^{r,m}_{p,k}-\overline{w}^{r,m}_{p,k})\\
\end{split}
\end{equation}
Now we bound the gradient of the proposed objective.
\begin{equation}
\begin{split}
&\frac{1}{K}\sum^{K}_{k=1}\mathbb{E}\|\nabla_{w^{r,m}_{p,k}}\sum^{K}_{k^\prime=1}f_{k^\prime}(u^{r,m}_{k^\prime})\|^2_2\\
&=\frac{1}{K}\sum^{K}_{k=1}\mathbb{E}\|\frac{1}{\lambda}\nabla f_k(u^{r,m}_{k})-\frac{1-\lambda}{\lambda}\frac{1}{K-1}\nabla f_{k^\prime}(u^{r,m}_{k^\prime})\|^2_2\\
&\leq \frac{2}{K}\sum^{K}_{k=1} (\frac{1}{\lambda^2}+\frac{(1-\lambda)^2}{\lambda^2(K-1)})\mathbb{E}\|\nabla f_k(u^{r,m}_{k})\|^2_2\\
&= \frac{2}{K}\sum^{K}_{k=1} (\frac{1}{\lambda^2}+\frac{(1-\lambda)^2}{\lambda^2(K-1)})[\mathbb{E}\|\nabla f_k(w^{r,m}_{k,p})\|^2_2\\
&\quad + L^2\mathbb{E}\|u^{r,m}_{k}-w^{r,m}_{k,p}\|^2_2]\\
&= (\frac{1}{\lambda^2}+\frac{(1-\lambda)^2}{\lambda^2(K-1)}) \frac{2}{K}\sum^{K}_{k=1} [\mathbb{E}\|\nabla f_k(w^{r,m}_{k,p})\|^2_2\\
&\quad +\frac{L^2(1-\lambda)^2K^2}{\lambda^2(K-1)^2}\mathbb{E}\|w^{r,m}_{k,p}-\overline{w}^{r,m}_{k,p}\|^2_2]\\
\end{split}
\end{equation}
\begin{equation}
\begin{split}
&\frac{1}{KRM}\sum^{R-1}_{r=0}\sum^{M-1}_{m=0}\sum^{K}_{k=1}\mathbb{E}\|\nabla_{w^{r,m}_{p,k}}\sum^{K}_{k^\prime=1}f_{k^\prime}(u^{r,m}_{k^\prime})\|^2_2\\
&\leq (\frac{1}{\lambda^2}+\frac{(1-\lambda)^2}{\lambda^2(K-1)}) \frac{2}{KRM}\sum^{K}_{k=1}\sum^{R-1}_{r=0}\sum^{M-1}_{m=0}\\
&[\mathbb{E}\|\nabla f_k(w^{r,m}_{k,p})\|^2_2 + \frac{L^2(1-\lambda)^2K^2}{(K-1)^2}\mathbb{E}\|w^{r,m}_{k,p}-\overline{w}^{r,m}_{k,p}\|^2_2]\\
&\leq 2(\frac{1}{\lambda^2}+\frac{(1-\lambda)^2}{\lambda^2(K-1)})[\frac{\frac{4}{K}\sum^{K}_{k=1}(f^0_k - f^*_k)}{\eta RM}\\
&\quad + 2\eta L\sigma^2 + \frac{2\eta^2 L^2(G^2+\sigma^2)}{M}\\
&\quad + \frac{1}{KRM}(\frac{8}{\eta^2} + \frac{2L}{\eta})\frac{(1-\lambda)^2K^2}{(K-1)^2}\\
&\quad \cdot \sum^{K}_{k=1}\sum^{R-1}_{r=0}\mathbb{E}\|w^{r,M}_{k,p}-\overline{w}^{r,M}_{k,p}\|^2_2 \\
&\quad +\frac{1}{KRM}\frac{L^2(1-\lambda)^2K^2}{\lambda^2(K-1)^2}\sum^{K}_{k=1} \sum^{R-1}_{r=0}\sum^{M-1}_{m=0}\mathbb{E}\|w^{r,m}_{k,p}-\overline{w}^{r,m}_{k,p}\|^2_2]\\
\end{split}
\end{equation}
which converges to
\begin{equation}
\begin{split}
&\mathcal{O}(\frac{1}{\eta RM\lambda^2} + \frac{(1-\lambda)^2}{KRM\eta^2\lambda^2}\sum^{K}_{K=1}\sum^{R-1}_{r=0}\mathbb{E}\|w^{r,M}_{k,p}-\overline{w}^{r,M}_{k,p}\|^2_2\\
&\quad + \frac{(1-\lambda)^2}{KRM\lambda^4}\sum^{K}_{k=1}\sum^{R-1}_{r=0}\sum^{M-1}_{m=0}\mathbb{E}\|w^{r,m}_{k,p}-\overline{w}^{r,m}_{k,p}\|^2_2)\\
&=\mathcal{O}(\frac{1}{\eta RM\lambda^2} + \frac{M\sum^{R-1}_{r=0}(1-\lambda)^2}{R\lambda^2}\\
&\quad + \frac{M^2\eta^2\sum^{R-1}_{r=0}(1-\lambda)^2}{R\lambda^4})
\end{split}
\end{equation}
Suppose $\eta=\mathcal{O}(\frac{1}{\sqrt{RM}})$ and $M=\mathcal{O}(R^{\frac{1}{3}})$, the convergence rate is $\mathcal{O}(\frac{1}{\lambda^4\sqrt{RM}})$ with an error $\mathcal{O}(\frac{M\sum^{R-1}_{r=0}(1-\lambda)^2}{R\lambda^2})$.
\section{Additional Experimental Results}\label{appendix:additional exp}
\begin{figure*}[htb!]
\centering
\includegraphics[width=.27\textwidth]{retinal-seg/1-15-gt.png}
\includegraphics[width=.27\textwidth]{retinal-seg/1-15-center.png}
\includegraphics[width=.27\textwidth]{retinal-seg/1-15-fedsm.png}
\includegraphics[width=.27\textwidth]{retinal-seg/1-15-fedavg.png}
\includegraphics[width=.27\textwidth]{retinal-seg/1-15-fedprox.png}
\includegraphics[width=.27\textwidth]{retinal-seg/1-15-scaffold.png}
\caption{Visual comparison of retinal disc (green) and cup (blue) segmentation. Dice denotes the retinal disc and cup Dice coefficient.}
\label{fig:visual retinal seg}
\end{figure*}
\begin{figure*}[htb!]
\centering
\includegraphics[width=.27\textwidth]{prostate-seg/3-87-gt.png}
\includegraphics[width=.27\textwidth]{prostate-seg/3-87-center.png}
\includegraphics[width=.27\textwidth]{prostate-seg/3-87-fedsm.png}
\includegraphics[width=.27\textwidth]{prostate-seg/3-87-fedavg.png}
\includegraphics[width=.27\textwidth]{prostate-seg/3-87-fedprox.png}
\includegraphics[width=.27\textwidth]{prostate-seg/3-87-scaffold.png}
\caption{Visual comparison of prostate (green) segmentation. Dice denotes the Dice coefficient.}
\label{fig:visual prostate seg}
\end{figure*}
\begin{table*}[htb!]
\small
\centering
\begin{tabular}{c|cccccc|cc}
\toprule
Method & Client 1 & Client 2 & Client 3 & Client 4 & Client 5 & Client 6 & Client Avg Dice & Global Dice \\
\midrule
Centralized & 0.9628 & 0.9486 & 0.9489 & 0.9539 & 0.9242 & 0.9565 & 0.9492 & 0.9522 \\
\midrule
Client 1 Local & 0.9454 & 0.4357 & 0.8956 & 0.6073 & 0.4464 & 0.8409 & 0.6952 & 0.7129 \\
Client 2 Local & 0.2936 & 0.9431 & 0.1099 & 0.8371 & 0.2439 & 0.4575 & 0.4809 & 0.5589 \\
Client 3 Local & 0.9420 & 0.3998 & 0.9468 & 0.6399 & 0.4349 & 0.8256 & 0.6982 & 0.7120 \\
Client 4 Local & 0.6830 & 0.9400 & 0.4805 & 0.9526 & 0.3088 & 0.7803 & 0.6909 & 0.7796 \\
Client 5 Local & 0.6102 & 0.2169 & 0.4601 & 0.2518 & 0.9033 & 0.7064 & 0.5248 & 0.5373 \\
Client 6 Local & 0.8806 & 0.7937 & 0.8354 & 0.8475 & 0.4413 & 0.9547 & 0.7922 & 0.8555 \\
\midrule
FedAvg & 0.9554 & 0.9410 & 0.9372 & 0.9535 & 0.8653 & 0.9549 & 0.9346 & 0.9444 \\
FedProx & 0.9447 & 0.9343 & 0.9229 & 0.9469 & 0.7573 & 0.9480 & 0.9090 & 0.9283 \\
Scaffold & 0.9207 & 0.9297 & 0.9026 & 0.9474 & 0.6347 & 0.9528 & 0.8813 & 0.9170 \\
\midrule
FedSM & \textbf{0.9653} & \textbf{0.9489} & \textbf{0.9545} & \textbf{0.9551} & \textbf{0.9241} & \textbf{0.9560} & \textbf{0.9507} & \textbf{0.9527} \\
\bottomrule
\end{tabular}
\caption{Test Dice coefficient comparison of retinal disc segmentation.}
\label{tab:retinal disc dice}
\end{table*}
\begin{table*}[htb!]
\small
\centering
\begin{tabular}{c|cccccc|cc}
\toprule
Method & Client 1 & Client 2 & Client 3 & Client 4 & Client 5 & Client 6 & Client Avg Dice & Global Dice \\
\midrule
Centralized & 0.8649 & 0.8033 & 0.8027 & 0.8507 & 0.7778 & 0.8793 & 0.8298 & 0.8507 \\
\midrule
Client 1 Local & 0.8216 & 0.2306 & 0.5733 & 0.3793 & 0.2351 & 0.5621 & 0.4670 & 0.4675 \\
Client 2 Local & 0.1756 & 0.7810 & 0.0673 & 0.7184 & 0.1143 & 0.3637 & 0.3701 & 0.4511 \\
Client 3 Local & 0.7256 & 0.2807 & 0.8064 & 0.5621 & 0.2939 & 0.7333 & 0.5670 & 0.6068 \\
Client 4 Local & 0.3385 & 0.7749 & 0.2109 & 0.8491 & 0.1632 & 0.5842 & 0.4868 & 0.6024 \\
Client 5 Local & 0.4380 & 0.1000 & 0.3305 & 0.1560 & 0.7414 & 0.5380 & 0.3840 & 0.3952 \\
Client 6 Local & 0.7011 & 0.5360 & 0.6296 & 0.6886 & 0.3073 & 0.8752 & 0.6230 & 0.7198 \\
\midrule
FedAvg & 0.8140 & 0.7949 & 0.7963 & 0.8495 & 0.7101 & 0.8795 & 0.8074 & 0.8402 \\
FedProx & 0.7822 & 0.7702 & 0.7864 & 0.8437 & 0.6132 & 0.8712 & 0.7778 & 0.8216 \\
Scaffold & 0.7554 & 0.7729 & 0.7405 & 0.8396 & 0.4995 & 0.8732 & 0.7469 & 0.8081 \\
\midrule
FedSM & \textbf{0.8610} & \textbf{0.8049} & \textbf{0.8186} & \textbf{0.8530} & \textbf{0.7724} & \textbf{0.8830} & \textbf{0.8322} & \textbf{0.8529} \\
\bottomrule
\end{tabular}
\caption{Test Dice coefficient comparison of retinal cup segmentation.}
\label{tab:retinal cup dice coefficients}
\end{table*}
\begin{table*}[t]
\small
\centering
\begin{tabular}{c|ccccccc|r}
\toprule
Unseen Client $k$ & GM & PM1 & PM2 & PM3 & PM4 & PM5 & PM6 & Best $\gamma$, Dice \\
\midrule
Client $k=6$ & 1.00 & 0 & 0 & 0 & 0 & 0 & N/A & 1, 0.8906 \\
Client $k=5$ & 0.69 & 0.18 & 0 & 0 & 0.10 & N/A & 0.03 & 0.9, 0.4304\\
Client $k=4$ & 0.03 & 0 & 0.97 & 0 & N/A & 0 & 0 & $<$0.95, 0.8870 \\
Client $k=3$ & 0 & 0 & 0.57 & N/A & 0 & 0 & 0.43 & $<$0.9, 0.8446\\
Client $k=2$ & 0 & 0 & N/A & 0 & 0.92 & 0.08 & 0 & $<$1, 0.8409 \\
Client $k=1$ & 0 & N/A & 0 & 1.00 & 0 & 0 & 0 & $<$0.99, 0.8839 \\
\bottomrule
\end{tabular}
\caption{(retinal segmentation, Dice = average of disc and cup Dice coefficients) Model selection frequency from the model selector when FL train with clients $\{1,2,\cdots,6\}/\{k\}$ and test on the \textbf{unseen} client $k\in\{1,2,\cdots,6\}$. From left to right, GM denotes the global model and PM denotes the personalized model $\{1,2,\cdots,6\}/\{k\}$. Here we choose the best $\gamma$.}
\label{tab:model selector frequency best gamma}
\end{table*}
\begin{table*}[t]
\small
\centering
\begin{tabular}{c|cccccc|c}
\toprule
Method/Unseen & Client 1 & Client 2 & Client 3 & Client 4 & Client 5 & Client 6 & Avg \\
\midrule
Centralized & 0.8842 & 0.8454 & 0.8214 & 0.8866 & 0.4064 & 0.8811 & 0.7875 \\
FedAvg & 0.8598 & 0.8313 & 0.8224 & 0.8551 & 0.4064 & \textbf{0.8887} & 0.7773 \\
FedProx & 0.8380 & 0.7856 & 0.8267 & 0.8746 & 0.4171 & 0.8784 & 0.7701 \\
Scaffold & 0.8085 & 0.7998 & 0.8211 & 0.8568 & 0.4121 & 0.8708 & 0.7615 \\
\midrule
FedSM & \textbf{0.8818} & 0.8619 & \textbf{0.8498} & \textbf{0.8901} & 0.4118 & 0.8646 & 0.7933 \\
FedSM-extra & 0.8747 & \textbf{0.8685} & 0.8467 & 0.8794 & \textbf{0.4265} & 0.8809 & \textbf{0.7963} \\
\bottomrule
\end{tabular}
\caption{(retinal segmentation, Dice = average of disc and cup Dice coefficients) Dice performance when FL train with clients $\{1,2,\cdots,6\}/\{k\}$ and test on the \textbf{unseen} client $k\in\{1,2,\cdots,6\}$.}
\label{tab:unseen client dice}
\end{table*} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,044 |
Metanomus — род щелкунов из подсемейства Dendrometrinae.
Описание
Щелкуны средних размеров. Тело одноцветное, сравнительно узкое.Лобный киль почти сплошной, касается переднего края наличника. Усики самцов слабо пиловидные начиная с четвёртого сегмента, у самок почти нитевидные.. Воротничок переднегруди с округлым передним краем, находящимся на одном уровне с передними углами проплевр. Задний край проплевр с явственной выемкой. Бедренные покрышки задних тазиков сужаются по направлению наружу довольно сильно и слабо равномерно. Все сегменты лапок без лопастинок.
Систематика
В составе рода:
вид:
вид:
вид:
вид:
Примечания
Щелкуны
Роды жесткокрылых | {
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I was searching for a Property and found this listing (MLS® #10173195). Please send me more information regarding 2563 Sunny Lake Court, Lake Country, British Columbia, V4V2P1. Thank you!
I'd like to request a showing of 2563 Sunny Lake Court, Lake Country, British Columbia, V4V2P1 (MLS® #10173195). Thank you! | {
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} | 36 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-factor-x-6-2x-3-1","text":"How do you factor x^6-2x^3+1?\n\nMay 16, 2015\n\n${x}^{6} - 2 {x}^{3} + 1 = {\\left({x}^{3}\\right)}^{2} - 2 \\left({x}^{3}\\right) + 1$ is of the form ${y}^{2} - 2 y + 1$ where $y = {x}^{3}$.\n\nThis quadratic formula in $y$ factors as follows:\n\n${y}^{2} - 2 y + 1 = \\left(y - 1\\right) \\left(y - 1\\right) = {\\left(y - 1\\right)}^{2}$\n\nSo ${x}^{6} - 2 {x}^{3} + 1 = {\\left({x}^{3} - 1\\right)}^{2}$\n\n${x}^{3} - 1 = \\left(x - 1\\right) \\left({x}^{2} + x + 1\\right)$\n\nSo ${x}^{6} - 2 {x}^{3} + 1 = \\left(x - 1\\right) \\left({x}^{2} + x + 1\\right) \\left(x - 1\\right) \\left({x}^{2} + x + 1\\right)$\n\n$= {\\left(x - 1\\right)}^{2} {\\left({x}^{2} + x + 1\\right)}^{2}$.\n\n${x}^{2} + x + 1$ has no linear factors with real coefficients. To check this notice that it is of the form $a {x}^{2} + b x + c$, which has discriminant:\n\n$\\Delta = {b}^{2} - 4 a c = {1}^{2} - 4 \\cdot 1 \\cdot 1 = 1 - 4 = - 3$\n\nBeing negative, the equation ${x}^{2} + x + 1 = 0$ has no real roots.\n\nOne way of checking the answer is to substitute a value for $x$ that is not a root into both sides and see if we get the same result:\n\nTry $x = 2$:\n\n${x}^{6} - 2 {x}^{3} + 1 = {2}^{6} - 2 {x}^{3} + 1$\n\n$= 64 - \\left(2 \\times 8\\right) + 1 = 64 - 16 + 1 = 49$\n\nCompare:\n\n${\\left(x - 1\\right)}^{2} {\\left({x}^{2} + x + 1\\right)}^{2} = {\\left(2 - 1\\right)}^{2} {\\left({2}^{2} + 2 + 1\\right)}^{2}$\n\n${1}^{2} \\cdot {7}^{2} = 49$\n\nWell that worked!\n\nMay 17, 2015\n\n${x}^{6} - 2 {x}^{3} + 1$ is fairly easy to factor, because it is a perfect square. How do I know this? It's a trinomial in the form ${a}^{2} + 2 a b + {b}^{2}$, and all trinomials in that form are perfect squares.\n\nThis trinomial is the perfect square of $\\left({x}^{3} - 1\\right)$. To check my work, I'll work backwards:\n\n$\\left({x}^{3} - 1\\right) \\left({x}^{3} - 1\\right)$\n\n$= {x}^{6} - {x}^{3} - {x}^{3} + 1$\n\n$= {x}^{6} - 2 {x}^{3} + 1$\n\nSo, this trinomial has factors of $1$, ${x}^{3} - 1$, and ${x}^{6} - 2 {x}^{3} + 1$.\n\nHowever, as it has been pointed out to me, $\\left({x}^{3} - 1\\right)$ also has factors. Since it is a binomial of the form ${a}^{3} - {b}^{3}$, it can also be written as $\\left(a - b\\right) \\left({a}^{2} + a b + {b}^{2}\\right)$.\n\nSo, $\\left({x}^{3} - 1\\right)$ factors into $\\left(x - 1\\right)$ and $\\left({x}^{2} + x + 1\\right)$, which both are prime.\n\nThe factors of ${x}^{6} - 2 {x}^{3} + 1$ are:\n$1$\n$x - 1$\n${x}^{2} + x + 1$\n${x}^{3} - 1$\n${x}^{6} - 2 {x}^{3} + 1$\n\nMore specifically, the PRIME factorization of ${x}^{6} - 2 {x}^{3} + 1$ is:\n${\\left(x - 1\\right)}^{2} {\\left({x}^{2} + x + 1\\right)}^{2}$","date":"2019-12-06 03:53:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 42, \"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\": 0.8847261071205139, \"perplexity\": 194.10592339363262}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540484477.5\/warc\/CC-MAIN-20191206023204-20191206051204-00058.warc.gz\"}"} | null | null |
Q: What should be the href of a collection response if I am getting one single item of that collection? Im getting a specific web resource via a http request using GET method.
I've implemented the server to respond using the hypermedia type collection+json so every response is a collection of items according to the specification (http://amundsen.com/media-types/collection/format/#query-templates)
Since the client is requesting a specific item, what should be the content of the href of the collection?
I think it should be the same url of the single item, but I'm not sure.
A: According to http://www.w3.org/TR/html4/struct/links.html#adef-href, "This attribute specifies the location of a Web resource, thus defining a link between the current element (the source anchor) and the destination anchor defined by this attribute."
If we are requesting the specific element is because we already know its url, then it would be optional to set this attribute. However, it is important to be consistent. If all the other responses of our API include "href", then it would be expected to include this attribute and the value of it could be the same url of the object we want to get.
A: In case the specific item belongs to a collection, the URI of the collection. In case the item does not belong to a collection (which could be odd and maybe you may check the design of your API) the URL of the item.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,883 |
Stardrifter is a novel by Dale Aycock published in 1981.
Plot summary
Stardrifter is a novel in which the owner of an interstellar trading company gets caught up in a conspiracy that is looking to rule the galaxy.
Reception
Greg Costikyan reviewed Star Drifter in Ares Magazine #10 and commented that "The importance is not the plot, which is typical space opera, but Aycock's ability to flesh out characters despite slam-bang action and to turn a pretty phrase or two. Star Drifter is fun reading."
The book also received reviews from Voice of Youth Advocates and Gene DeWeese of Science Fiction Review.
Reviews
Review by Gene DeWeese (1981) in Science Fiction Review, Winter 1981
References
1981 novels | {
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} | 88 |
<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
<feed><tipo>Travessa</tipo><logradouro>Zivi</logradouro><bairro>Marrocos</bairro><cidade>Gravataí</cidade><uf>RS</uf><cep>94045322</cep></feed>
| {
"redpajama_set_name": "RedPajamaGithub"
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Foot pain can often linger and become a chronic injury. Prevention is a must for all runners before you get injured. These tips will help you prevent AND fix foot injuries.
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Follow a Wholicious Living nutritional lifestyle to reduce inflammation and pain and speed up the healing process.
Learn low impact, efficient and pain-free running and walking technique. It's amazing how many people I've helped with a couple simple exercises.
-Damian Stoy is a professional ultra marathon runner, founder of Wholistic Running, biomechanics specialist, running coach and has been injury-free for over 10 years.
To receive more tips from Damian, sign up for our free emails HERE.
This entry was posted in online running coach and tagged bunion, feet, foot, injuries, injury, natural, neuroma, plantar fasciitis, run, Running, stress, toes by Wholistic Running. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,000 |
Give a little or give a lot! Help WestcottCC continue to be a beacon of hope.
Browse the myriad Services, Programs, and Events offered at your Community Center.
Get Involved, Help Out, Pay It Forward. Your Community Center needs you, ACT NOW! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,623 |
\section{Introduction}
Predicting election winners is always an exciting activity: Who will
be the new president? Will the company merge with another one? Will
taxes be higher or lower? The goal of this paper is to establish the
computational complexity of a family of problems modeling a certain
type of winner-prediction problems.
Naturally, predicting winners is a hard task, full of
uncertainties. For example, we typically are not sure which voters
will eventually cast their votes or, sometimes, even which candidates
will participate in the election (consider, e.g., a candidate
withdrawing due to personal reasons). Further, typically we do not
have complete knowledge regarding each voters' preferences.
Nonetheless, elections are in everyday use both among humans
(consider, e.g., political elections, elections among companies'
shareholders, various polls on the Internet and social
media, or even sport events, where judges ``vote'' on who is the best
competitor) and among software agents (see, e.g., election
applications for planning in multiagent
systems~\cite{eph-ros:j:multiagent-planning}, for recommendation
systems~\cite{gho-mun-her-sen:c:voting-for-movies,bou-lu:c:chamberlin-courant,sko-fal-sli:c:multiwinner},
for web-search~\cite{dwo-kum-nao-siv:c:rank-aggregation}, or natural
language processing~\cite{kut-kit:j:nlp}) and,
thus, the problem of predicting election winners is far too important
to be abandoned simply because it is difficult.
In this paper, we focus on a variant of the winner-prediction problem
where we have complete knowledge regarding all possible candidates and
all eligible voters (including knowledge of their
preferences\footnote{Note that while full-knowledge assumption
regarding voters' preferences might seem very unrealistic, it is
standard within computational social choice literature, and in our
case can often be justified (e.g., election polls can provide a good
approximation of such knowledge).}), but we are uncertain as to
which candidates and which voters turn up for the actual election (see
Section~\ref{sec:related} for other approaches to the
problem). However, modelling uncertainty regarding both the candidate
set and the voter collection on one hand almost immediately leads to
computationally hard problems for typical election systems, and on the
other hand does not seem to be as well motivated as focusing on each
of these sets separately. Thus, we consider the following two
settings:
\begin{enumerate}
\addtolength{\itemsep}{-4pt}
\item The set of candidates is fixed, but for each possible subset
of voters we are given a probability that exactly these voters
show up for the vote.
\item The set of voters is fixed, but for each possible subset of
candidates we are given a probability that exactly these
candidates register for the election.
\end{enumerate}
The former setting, in particular, corresponds to political elections
(e.g., to presidential elections), where the candidate set is
typically fixed well in advance due to election rules, the set of all
possible voters (i.e., the set of all citizens eligible to vote) is
known, but it is not clear as to which citizens choose to cast their
votes. The latter setting may occur, for example, if one considers
software agents voting on a joint plan~\cite{eph-ros:j:multiagent-planning}. The set of agents participating in the
election is typically fixed, but various variants of the plan can be
put forward or dismissed dynamically. In either case, our goal is to
compute each candidate's probability of victory.
However, our task would very quickly become computationally
prohibitive (or, difficult to represent on a computer) if we allowed
arbitrary probability distributions. Thus, we have to choose some
restriction on the distributions we consider. Let us consider the
following example.
Let the set $C$ of candidates participating in the election be fixed
(e.g., because the election rules force all candidates to register
well in advance). We know that some set $V$ of voters will certainly
vote (e.g., because they have already voted and this information is
public\footnote{Naturally, in typical political elections such
information would not be public and we would have to rely on
polls. On the other hand, in multiagent systems there can be cases
where votes are public.}). The set of voters who have not decided to
vote yet is $W$. From some source (e.g., from prior experience) we
have a probability distribution $P$ on the number of voters from $W$
that will participate in the election (we assume that each equal-sized
subset from $W$ is equally likely to joint the election; we have no
prior knowledge as to which eligible voters are more likely to vote).
That is, for each $i$, $0 \leq i \leq \|W\|$, by $P(i)$ we denote the
probability that exactly $i$ voters from $W$ join the election (we
assume that $P(i)$ is easily computable). By $Q(i)$ we denote the
probability that a certain designated candidate $p$ wins provided that
exactly $i$, randomly chosen, voters from $W$ participate in the
election. The probability that $p$ wins is given by:
\[
P(0)Q(0) + P(1)Q(1) + \cdots P(\|W\|)Q(\|W\|).
\]
We use this formula to compute the probability of each candidate's
victory, which gives some idea as to who is the likely winner of the
election.
To compute $Q(i)$, we have to compute for how many subsets $W'$ of $W$
of size exactly $i$ candidate $p$ wins after adding the voters from
$W'$ to the election, and divide it by $\|W\| \choose i$. That is,
computing $Q(i)$ boils down to solving a counting variant of control
by adding voters problem. In the decision variant of this control
problem, introduced by Bartholdi, Tovey, and
Trick~\cite{bar-tov-tri:j:control}, we are given an election where
some voters have already registered to vote, some are
not-yet-registered, and we ask if it is possible to add some number of
these not-yet-registered voters (but no more than a given limit) to
ensure that a designated candidate wins. In the counting variant,
studied first in this paper, we ask how many ways are there to add
such a group of voters.
One can do reasoning analogous to the one presented above for the case
of control by deleting voters and for the case of control by
adding/deleting candidates. That is, our winner prediction problems,
in essence, reduce to solving counting variants of election control
problems. Our goal in this paper is, thus, to study the computational
complexity of these counting control problems.
Computational study of election control was initiated by Bartholdi,
Tovey, and Trick~\shortcite{bar-tov-tri:j:control}, and in recent
years was continued by a number of researchers (we discuss related
work in Section~\ref{sec:related}; we also point the reader to the
survey of Faliszewski, Hemaspaandra, and
Hemaspaandra~\cite{fal-hem-hem:j:cacm-survey} for a more detailed
discussion of the complexity of election control). However, our paper
is the first one to study counting variants of control (though, as we
discuss in Section~\ref{sec:related}, the papers of Walsh and
Xia~\cite{wal-xia:c:lot-based} and of Bachrach, Betzler, and
Faliszewski~\cite{bac-bet-fal:c:counting-pos-win} are very close in
spirit).
It is well-known that election control problems tend to have fairly
high worst-case complexity. Indeed, we are not aware of a single
practical voting rule for which the decision variants of the four most
typical election control problems (the problems of adding/deleting
candidates/voters) are all polynomial-time solvable. Naturally, when a
decision variant of a counting problem is $\NPclass$-hard, then we cannot
hope that its counting variant would be polynomial-time solvable. In
effect, from the technical perspective, our research can be seen as
answering the following question: For which voting rules and for which
control types is the counting variant of the control problem no harder
than its decision variant? From the practical perspective, it is,
nonetheless, more important to simply have effective algorithms. To
this end, we follow Faliszewski et
al.~\cite{fal-hem-hem-rot:j:single-peaked-preferences} and in addition
to the general setting where each voter can cast any possible vote, we
study the single-peaked case, where the voting is restricted to a
certain class of votes viewed as ``reasonable'' (intuitively, in the
single-peaked case we assume that the candidates are located on a
one-dimensional spectrum of possible opinions, each voter has some
favorite opinion on this spectrum, and the voters form their
preferences based on candidates' distances from their favorite
position). Single-peaked preferences, introduced by
Black~\cite{bla:b:polsci:committees-elections} over 50 years ago, are
often viewed as a natural (if somewhat simplified due to its
one-dimensional nature) model of how realistic votes might look like
(however, whenever one makes a statement of this form, one should keep
in mind the results of Sui, Francois-Nienaber, and
Boutilier~\cite{sui-fra-bou:c:single-peaked}). It turns out that for
the sinlge-peaked case, we obtain polynomial-time algorithms for
counting variants of all our control problems and all the rules that
we consider (except for the Approval rule, for which we have no
results in this case).
We mention that our model of winner prediction, based on counting
variants of control, is similar to, though more general than, the
model where we assume that each voter casts his or her vote with some
probability $p$ (universal for all the voters). We can simulate this
scenario in our model by providing an appropriate function $P$.
Specifically, if there are $n$ voters and the probability that each
particular voter $v$ casts his or her vote is $p$, then the
probability that exactly $i$ voters vote is given by the binomial
distribution:
\[
P(i) = {n \choose i}p^i(1-p)^{n-i}.
\]
On the other hand, our model does not capture the situation where each
voter $v$ has a possibly different probability $p_v$ of casting a
vote. We believe that this latter model deserves study as well, but we
do not focus on it in this paper.
The paper is organized as follows. First, in Section~\ref{sec:prelim},
we formally define elections and provide brief background on
complexity theory (focusing on counting problems). Then, in
Section~\ref{sec:problems}, we define counting variants of election
control problems and prove some of their general properties. Our main
results are in Section~\ref{sec:results}. There we study the
complexity of Plurality, $k$-Approval, Appoval, Condorcet rule, and
Maximin rule. For each of the rules, we consider the unrestricted case
and, if we obtain a hardness result, we consider the single-peaked
case (except for Approval, for which we were not able to obtain
results for the single-peaked case). Finally, we discuss related work
in Section~\ref{sec:related} and bring together our conclusions and
discuss future work in Section~\ref{sec:conclusions}.
\section{Preliminaries}
\label{sec:prelim}
\noindent\textbf{Elections.}\quad
An \emph{election} $E$ is a pair $(C,V)$ such that $C$ is a finite set
of candidates and $V$ is a finite collection of voters. We typically
use $m$ to denote the number of candidates and $n$ to denote the
number of voters. Each voter has a preference order in which he or
she ranks candidates, from the most desirable one to the most despised
one. For example, if $C = \{a,b,c\}$ and a voter likes $b$ most and
$a$ least, then this voter would have preference order $b > c >
a$. (However, under Approval voting, instead of ranking the candidates
each voter simply indicates which candidates he or she approves of.)
We sometimes use the following notation. Let $A$ be some subset of
the candidate set. Putting $A$ in a preference order means listing
members of $A$ in lexicographic order and putting $\overleftarrow{A}$
in a preference order means listing members of $A$ in the reverse of
the lexicographic order. For example, if $C = \{a,b,c,d\}$ then $a > C
- \{a,b\} > d$ means $a > c > d > b$ and $a > \overleftarrow{C -
\{a,b\}} > d$ means $a > d > c > b$.
Given an election $E = (C,V)$, we write $\N{E}(c,c')$ to denote the
number of voters in $V$ that prefer $c$ to $c'$. The function $N_E$ is
sometimes referred to as the \emph{weighted majority graph} of
election $E$.
In general, given a candidate set $C$, the voters are free to report
any preference order over $C$ (we refer to this setting as the
\emph{unrestricted case}). However, it is often more realistic to
assume some restriction on the domain of possible votes (for example,
in real-life political elections we do not expect to see many voters
who rank the extreme left-wing candidate first, then the extreme
right-wing candidate, and so on, in an interleaving fashion). Thus,
in addition to the unrestricted case, we also study the case of
\emph{single-peaeked} preferences of
Black~\cite{bla:b:polsci:committees-elections} (this domain
restriction is quite popular in the computational social choice
literature, and was already studied, e.g., by
Conitzer~\cite{con:j:eliciting-singlepeaked},
Walsh~\cite{wal:c:uncertainty-in-preference-elicitation-aggregation},
Faliszewski et al.~\cite{fal-hem-hem-rot:j:single-peaked-preferences},
and others; see Section~\ref{sec:related}).
\begin{definition}\label{def:sp}
Let $C$ be a set of candidates and let $L$ be a linear order over
$C$ (we refer to $L$ as the societal axis). We say that a preference
order $>$ (over $C$) is single-peaked with respect to $L$ if for
each three candidate $a, b, c \in C$, it holds that
\[
(a \mathrel{L} b \mathrel{L} c \lor c \mathrel{L} b \mathrel{L} a) \implies (a > b \implies b > c)
\]
An election $E = (C,V)$ is single-peaked with respect to $L$ if each
vote in $V$ is single-peaked with respect to $L$. An election $E =
(C,V)$ is single-peaked if there is a societal axis $L$ such that
$E$ is single-peaked with respect to $L$.
\end{definition}
There are polynomial-time algorithms that given an election $E$ verify
if it is single-peaked and, if so, provide the appropriate societal
axis witnessing this
fact~\cite{bar-tri:j:stable-matching-from-psychological-model,esc-lan-ozt:c:single-peaked-consistency,bal-har:j:characterization-single-peaked}. Thus,
following
Walsh~\cite{wal:c:uncertainty-in-preference-elicitation-aggregation},
whenever we consider problems about single-peaked elections, we assume
that we are given appropriate societal axis as part of the input (if
it were not provided, we could always compute it.)\bigskip
\noindent\textbf{Voting Systems.}\quad
A \emph{voting system} (a \emph{voting rule}) is a rule which
specifies how election winners are determined. We allow an election to
have more than one winner, or even to not have winners at all. In
practice, tie-breaking rules are used, but here we disregard this
issue by simply using the unique winner model (see
Definition~\ref{def:control}). However, we point the reader
to~\cite{obr-elk-haz:c:ties-matter,obr-elk:c:random-ties-matter,obr-zic-elk:c:multiwinner-tie-breaking}
for a discussion regarding the influence of tie-breaking for the case
of election manipulation problems
(see~\cite{fal-hem-hem:j:cacm-survey,fal-pro:j:manipulation,bra-con-end:b:comsoc} for
overviews of the manipulation problem specifically and computational aspects
of voting generally).
We consider the following voting rules (in the description below we
assume that $E = (C,V)$ is an election with $m$ candidates and $n$
voters):
\begin{description}
\item[Plurality.] Under Plurality, each candidate receives a point for
each vote where this candidate is ranked first. The candidates with
most points win.
\item[$\bm{k}$-Approval.] For each candidate $c \in C$, we define
$c$'s $k$-Approval score, $\score{E}^k(c)$, to be the number of
voters in $V$ that rank $c$ among the top $k$ candidates; the
candidates with highest scores win. Note that Plurality is, in
effect, a nickname for $1$-Approval. (We mention that $k$-Veto means
$(m-k)$-Approval, though we do not study $k$-Veto in this paper).
\item[Approval.] Under Approval (without the qualifying ``$k$-''), the
score of a candidate $c \in C$, $\score{E}^a(c)$, is the number of
voters that approve of $c$ (recall that under Approval the voters do
not cast preference orders but 0/1 approval vectors, where for each
candidate they indicate if they approve of this candidate or
not). Again, the candidates with highest scores win. (Note that the
notion of single-peaked elections that we have provided as
Definition~\ref{def:sp} does not apply to preferences specified as
approval vectors; Faliszewski et
al.~\cite{fal-hem-hem-rot:j:single-peaked-preferences} have provided
a natural variant of single-peakedness for approval vectors, but
since we did not obtain results for this case, we omitted this
definition.)
\item[Condorcet.] A candidate $c$ is a \emph{Condorcet winner} exactly
if $\N{E}(c,c') > \N{E}(c',c)$ for each $c' \in C - \{c\}$. A
candidate $c$ is a \emph{weak Condorcet winner} exactly if
$\N{E}(c,c') \geq \N{E}(c',c)$ for each $c' \in C - \{c\}$. Note
that if an election has a Condorcet winner, then this winner is
unique (though, an election may have several weak Condorcet
winners).\footnote{We mention that sometimes Condorcet's rule is not
considered a voting rule (but, for example, a consensus
notion~\cite{mes-nur:b:distance-realizability}) because there are
elections for which there are no Condorcet winners. However,
Condorcet's rule has been traditionally studied in the context of
election control, and--thus---we study it to (a) enable comparison
with other papers, and (b) because for single-peaked elections the
results for Condorcet rule translate to all Condorcet-consistent
rules.} We recall the classic result that if the voters are
single-peaked then the election always has (weak) Condorcet
winner(s) (if the number of voters is odd, then the election has a
unique Condorcet winner).
\item[Maximin.] Maximin rule is defined as follows. The score of a
candidate $c$ is defined as $\score{E}^m(c) = \min_{d \in C
-\{c\}}N_E(c,d)$. The candidates with highest Maximin score are
Maximin winners.
\end{description}
Maximin is an example of a so-called Condorcet-consistent rule. A
rule $R$ is Condorcet-consistent if the following holds: If $E$ is an
election with (weak) Condorcet winner(s), then the $R$-winners of $E$
are exactly these (weak) Condorcet winner(s). We also considered
studying the Copeland rule (see, e.g., the work of Faliszewski et
al.~\cite{fal-hem-hem-rot:j:llull} regarding the complexity of control
for the Copeland rule), but for this rule all relevant types of
control are $\NPclass$-complete and, so, at best we would obtain
$\sharpPclass$-completeness results based on simple generalizations of
proofs from the literature. Nonetheless, since Copeland's rule is
Condorcet-consistent, results for the single-peaked case are valid for
it.
\bigskip
\noindent\textbf{Notation for Graphs.}\quad We assume familiarity with basic
concepts of graph theory. Given an undirected graph $G$, by $V(G)$ we
mean its set of vertices, and by $E(G)$ we mean its set of
edges.\footnote{We use this slightly nonstandard notation to be able
to also use symbols $V$ and $E$ to denote voter collections and
elections.} Whenever we discuss a bipartite graph $G$, we assume
that $V(G)$ is partitioned into two subsets, $X$ and $Y$, such that
each edge connects some vertex from $X$ with some vertex from $Y$. We
write $X(G)$ to denote $X$, and $Y(G)$ to denote $Y$.\bigskip
\noindent\textbf{Computational Complexity.}\quad
We assume that the reader is familiar with standard notions of
computational complexity theory, as presented, for example, in the
textbook of Papadimitriou~\cite{pap:b:complexity}, but we
briefly review notions regarding the complexity theory of counting
problems. Let $A$ be some computational problem where for each
instance $I$ we ask if there exists some mathematical object
satisfying a given condition. In the counting variant of $A$, denoted
$\#A$, for each instance $I$ we ask for the number---denoted
$\#A(I)$---of the objects that satisfy the condition.
For example, consider the following problem.
\begin{definition}
An instance of X3C is a pair $(B,{{\mathcal{S}}})$, where $B = \{b_1, \ldots,
b_{3k}\}$ and ${{\mathcal{S}}} = \{S_1, \ldots, S_n\}$ is a family of
$3$-element subsets of $B$. In X3C we ask if it is possible to find
exactly $k$ sets in ${{\mathcal{S}}}$ whose union is exactly $B$. In \#X3C we
ask how many $k$-element subsets of ${{\mathcal{S}}}$ have $B$ as their union.
\end{definition}
The class of counting variants of $\NPclass$-problems is called $\sharpPclass$
and the class of functions computable in polynomial time is called
$\FPclass$. To reduce counting problems to each other, we will use one of
the following reducibility notions.
\begin{definition}
Let $\#A$ and $\#B$ be two counting problems.
\begin{enumerate}
\addtolength{\itemsep}{-4pt}
\item We say that $\#A$ Turing reduces to $\#B$ if there exists an
algorithm that solves $\#A$ in polynomial time given oracle access
to $\#B$ (i.e., given the ability to solve $\#B$ instances in unit
time).
\item We say that $\#A$ metric reduces to $\#B$ if there exist two
polynomial-time computable functions, $f$ and $g$, such that for
each instance $I$ of $\#A$ it holds that (1) $f(I)$ is an instance
of $\#B$, and (2) $\#A(I) = g(I, \#B(f(I)) )$.
\item We say that $\#A$ parsimoniously reduces to $\#B$ if $\#A$
metric reduces to $\#B$ via functions $f$ and $g$ such that for
each instance $I$ and each integer $k$, $g(I,k) = k$ (i.e., there
is a polynomial-time computable function $f$ such that for each
instance $I$ of $\#A$, we have $\#A(I) = \#B(f(I))$).
\end{enumerate}
\end{definition}
It is well-known that these reducibility notions are transitive. For
a given reducibility notion $R$, we say that a problem is
$\sharpPclass$-$R$-complete if it belongs to $\sharpPclass$ and every
$\sharpPclass$-problem $R$ reduces to it. For example, \#X3C is
$\sharpPclass$-parsimonious-complete~\cite{hun-mar-rad-ste-:j:planar-counting-problems}.
Throughout this paper we will write $\sharpPclass$-complete to mean
$\sharpPclass$-parsimonious-complete. Turing reductions were used, e.g., by
Valiant~\shortcite{val:j:permanent} to show $\sharpPclass$-hardness of
computing a permanent of a 0/1 matrix. As a result, he also showed
$\sharpPclass$-Turing-hardness of the following problem.
\begin{definition}
In \#PerfectMatching we are given a bipartite graph $G =
(G(X),G(Y),G(E))$ with $\|G(X)\| = \|G(Y)\|$ and we ask how many
perfect matchings does $G$ have.
\end{definition}
Zank{\'{o}}~\cite{zan:j:sharp-p} improved upon this result by showing
that \#PerfectMatching is \#P-many-one-complete (``many-one'' is yet
another reducibility type, more general than parsimonious reductions
but less general than metric reductions). From our perspective, it
suffices that, in effect, \#PerfectMatching is \#P-metric-complete.
Metric reductions were introduced by
Krentel~\shortcite{kre:j:optimization}, and parsimonious reductions
were defined by Simon~\shortcite{sim:thesis:complexity}.
\section{Counting Variants of Control Problems}
\label{sec:problems}
In this section we formally define counting variants of election
control problems and show some interconnections between some of
them. We are interested in control by adding candidates (AC), control
by deleting candidates (DC), control by adding voters (AV), and
control by deleting voters (DV). For each of these problems, we
consider its constructive variant (CC) and its destructive variant
(DC).
\begin{definition}\label{def:control}
Let $R$ be a voting system. In each of the counting variants of
constructive control problems we are given a candidate set $C$, a
voter collection $V$, a nonnegative integer $k$, and a designated
candidate $p \in C$. In constructive control by adding voters we are
additionally given a collection $W$ of unregistered voters, and in
constructive control by adding candidates we are additionally given
a set $A$ of unregistered candidates. In these problems we ask for
the following quantities:
\begin{enumerate}
\addtolength{\itemsep}{-4pt}
\item In control by adding voters ($R$-\#CCAV),
we ask how many sets $W'$, $W' \subseteq W$, are there such that
$p$ is the unique winner of $R$-election $(C,V \cup W')$, where
$\|W'\| \leq k$.
\item In control by deleting voters ($R$-\#CCDV), we ask how many
sets $V'$, $V' \subseteq V$ are there such that $p$ is the unique
winner of $R$-election $(C,V - V')$, where $\|V'\| \leq k$.
\item In control by adding candidates ($R$-\#CCAC), we ask how many
sets $A'$, $A' \subseteq A$, are there such that $p$ is the unique
winner of $R$-election $(C \cup A',V)$, where $\|A'\| \leq k$.
\item In control by deleting candidates ($R$-\#CCDC), we ask how
many sets $C'$, $C' \subseteq C$, are there such that $p$ is the
unique winner of $R$-election $(C - C',V)$, where $\|C'\|
\leq k$ and $p \not\in C'$.
\end{enumerate}
Destructive variants are defined identically, except that we ask for
the number of settings where the designated candidate is not the
unique winner.
\end{definition}
\noindent
(To obtain decision variants of the above problems, simply change the
question from asking for a particular quantity to asking if that
quantity is greater than zero. We mention that in the literature
researchers often study both the unique-winner model---as we do
here---and the nonunique-winner model, where it suffices to be one of
the winners. For the rules that we study, results for both models
are the same.)\medskip
The above problems are interesting for several reasons. First, as
described in the introduction, we believe that they are useful as
models for various scenarios pertaining to predicting election
winners. Second, they are counting variants of the well-studied
control
problems~\cite{bar-tov-tri:j:control,hem-hem-rot:j:destructive-control}.
Third, they expose intuitive connections between various types of
control. In particular, we have the following results linking the
complexity of destructive variants with that of the constructive ones,
and linking the complexity of the ``deleting'' variants with that of
the ``adding'' variants.
\begin{theorem}\label{thm:c-to-d}
Let $R$ be a voting system, let $\#{{\mathcal{C}}}$ be one of $R$-\#CCAC,
$R$-\#CCDC, $R$-\#CCAV, $R$-\#CCDV, and let $\#{{\mathcal{D}}}$ be the
destructive variant of $\#{{\mathcal{C}}}$. Then, $\#{{\mathcal{C}}}$ metric reduces to
$\#{{\mathcal{D}}}$ and $\#{{\mathcal{D}}}$ metric reduces to $\#{{\mathcal{C}}}$.
\end{theorem}
\begin{proof}
We give a metric reduction from $\#{{\mathcal{C}}}$ to $\#{{\mathcal{D}}}$. Let $I$ be
an instance of $\#{{\mathcal{C}}}$, where the goal is to make some candidate
$p$ the unique winner. We define $f(I)$ to be an instance of
$\#{{\mathcal{D}}}$ that is identical to $I$, except that the goal is to
ensure that $p$ is not the unique winner. Let $s_I$ be the total
number of control actions allowed in $I$\footnote{For example, if
$\#{{\mathcal{C}}}$ was $\#CCDV$ and $I = (C,V,p,k)$, then $s_I$ would be
the number of up-to-size-$k$ subsets of $V$ (i.e., the number of
subsets of voters that can be deleted from $V$).} (and, naturally,
also the total number of control actions allowed in $f(I)$). It is
easy to see that $s_I$ is polynomial-time computable and that the
number of solutions for $I$ is exactly $s_I - \#{{\mathcal{D}}}(f(I))$. Thus,
we define $g(I,\#{{\mathcal{D}}}(f(I))) = s_I - \#{{\mathcal{D}}}(f(I))$. We see that
the reduction is polynomial-time and correct.
The same argument shows that
$\#{{\mathcal{D}}}$ metric reduces to $\#{{\mathcal{C}}}$.
\end{proof}
\begin{theorem}\label{thm:d-to-a}
Let $R$ be a voting system, then $R$-\#CCDV Turing reduces to
$R$-\#CCAV and $R$-\#CCDC Turing reduces to $R$-\#CCAC.
\end{theorem}
\begin{proof}
Let us fix a voting system $R$. The proofs that $R$-\#CCDV Turing
reduces to $R$-\#CCAV and that $R$-\#CCDC Turing-reduces to
$R$-\#CCAV are very similar and thus we discuss them jointly.
%
%
%
Let $I = (C,V,p,k)$ be an input instance of $R$-\#CCDV (of
$R$-CCDC), where $C$ is the candidate set, $V$ is the collection of
voters, $p$ is the designated candidate, and $k$ is the upper bound
on the number of voters that can be removed. We define the following
transformations of $I$:
\begin{enumerate}\addtolength{\itemsep}{-4pt}
\item For the case of $R$-\#CCDV, for each nonnegative integer $g$,
$g \le \|V\|$, let $J_g$ be an instance of $R$-\#CCAV, where the
candidate set is $C$, the set of registered voters is empty, the
collection of unregistered voters is $V$, the designated candidate
is $p$, and the bound on the number of voters that can be added is
$g$, That is, $J_g = (C,\emptyset,V,p,g)$.
\item For the case of $R$-\#CCDC, for each nonnegative integer $g$,
$g \le \|C\|-1$, let $J_g$ be an instance of $R$-\#CCAC, where the
candidate set is $\{p\}$, the set of registered voters is $V$, the
set of unregistered candidates is $C-\{p\}$, the designated
candidate is $p$, and the bound on the number of candidates that
can be added is $g$. That is, $J_g = (\{p\},C-\{p\},V,p,g)$.
\end{enumerate}
To complete the reduction, it suffices to note that for
the case of $R$-\#CCDV it holds that:
\[
\text{$R$-\#CCDV}(I) = \left\{
\begin{array}{ll}
\text{$R$-\#CCAV}(J_{\|V\|}) & \text{, if $k \ge \|V\|$}\\
\text{$R$-\#CCAV}(J_{\|V\|}) - \text{$R$-\#CCAV}(J_{\|V\|-k-1}) & \text{, if $0 \leq k < \|V\|$}\\
\end{array}
\right.
\]
and for the case of $R$-\#CCDC it holds that:
\[
\text{$R$-\#CCDC}(I) = \left\{
\begin{array}{ll}
\text{$R$-\#CCAC}(J_{\|C\|-1}) & \text{, if $k \ge \|C\|-1$}\\
\text{$R$-\#CCAC}(J_{\|C\|-1}) - \text{$R$-\#CCAC}(J_{\|C\|-k-2}) & \text{, if $0 \leq k < \|C\|-1$}\\
\end{array}
\right.
\]
These expressions define a natural algorithm for solving $R$-\#CCDV
($R$-\#CCDC) given at most two calls to an $R$-\#CCAV (an
$R$-\#CCAC) oracle. It is clear that this Turing reduction runs in
polynomial time.
\end{proof}
Theorems~\ref{thm:c-to-d} and~\ref{thm:d-to-a} are very useful and, in
particular, we obtain all of our destructive-case hardness results via
Theorem~\ref{thm:c-to-d}, and all our ``deleting''-case easiness
results using Theorem~\ref{thm:d-to-a}.
Obtaining results similar to Theorems~\ref{thm:c-to-d}
and~\ref{thm:d-to-a} for the decision variants of control problems is
more difficult. Nonetheless, recently several researchers have made
some progress on this front. In particular, Hemaspaandra,
Hemaspaandra, and Menton~\cite{hem-hem-men:c:search-decision} have
shown that control by partition of candidates and control by runoff
partition of candidates (two types of control not discussed in this
paper) are equivalent in terms of their computational complexity.
Focusing on particular classes of voting rules, for the case of
$k$-Approval and $k$-Veto, Faliszewski, Hemaspaandra, and
Hemaspaandra~\cite{fal-hem-hem:c:weighted-control} have shown that
control by deleting voters reduces to control by adding voters, and
Mi\k{a}sko~\cite{mia:t:priced-control} achieved the same for the case
of voting rules based on the weighted majority graphs (e.g., for
Borda, Condorcet, Maximin, and Copeland).\footnote{Mi\k{a}sko's
master's thesis gives this result for the case of Borda and
destructive control only, but it is clear that the proof technique
adapts to constructive control and that it applies to all voting
rules based on weighted majority graphs.}
\section{Results}
\label{sec:results}
We now present our complexity results regarding counting variants of
election control problems. We consider two cases: the unrestricted
case, where the voters can have any arbitrary preference orders, and
the single-peaked case, where voters' preference orders are
single-peaked with respect to a given societal axis. We summarize our
results in Table~\ref{tab:results}. For comparison, in
Table~\ref{tab:decision} we quote results for the decision variants of
the respective control problems.
Let us consider the unrestricted case first. Not surprisingly, for
each of our control problems whose decision variant is $\NPclass$-complete,
the counting variant is $\sharpPclass$-complete for some reducibility type.
However, interestingly, we have also found examples of election
systems and control types where the decision variant is easy, but the
counting variant is hard. For example, for $2$-Approval we have that
all decision variants of voter control problems are in
$\Pclass$~\cite{lin:thesis:elections,fal-hem-hem:c:weighted-control}, yet
all their counting variants are $\sharpPclass$-Turing-complete. Similarly,
for Maximin all decision variants of candidate control (except
constructive control by adding candidates) are in
$\Pclass$~\cite{fal-hem-hem:j:multimode}, yet their counting variants are
$\sharpPclass$-Turing-complete (or are complete for $\sharpPclass$ through even
less demanding reducibility types).
The situation for the single-peaked case is quite interesting as well.
While we found polynomial-time algorithms for all of our control
problems for Plurality, $k$-Approval, and all Condorcet-consistent
rules, we did not obtain results for voter control under Approval
(candidate control for Approval is easy even in the unrestricted
case). Decision variants of voter control are $\NPclass$-complete for
Approval in the unrestricted case, but---very interestingly---are in
$\Pclass$ for the single-peaked case. However, the algorithm for the
single-peaked case is a clever greedy approach that seems to be very
difficult to adapt to the counting case. Further, as opposed to the
$k$-Approval case, for voter control under Approval, there does not
seem to be a natural dynamic-programming-based approach. Naturally,
we also tried to find a hardness proof, but we failed at that as well
because the problem seems to have quite a rich structure. (However,
see the work of Mi\k{a}sko~\cite{mia:t:priced-control} for the case of
priced control under Approval with single-peaked preferences.)
\newcommand{com.}{com.}
\tabcolsep=0.1cm
\begin{table*}[t]
\small
\begin{center}
(a) The Unrestricted Case
\hfill{}
\begin{tabular}{c|cccccc}
\hline
Problem\rule{0cm}{0.5cm} & Plurality & $k$-Approval, $k \geq 2$ & Approval & Condorcet & Maximin\\[0.15cm]
\hline
\#CCAC\rule{0cm}{0.5cm}& $\sharpPclass$-com. & $\sharpPclass$-com. &-- & -- & $\sharpPclass$-com.\\
\#DCAC & $\sharpPclass$-metric-com. & $\sharpPclass$-metric-com. & FP & FP & $\sharpPclass$-metric-com. \\
\#CCDC & $\sharpPclass$-com. & $\sharpPclass$-com. &FP & FP & $\sharpPclass$-Turing-com. \\
\#DCDC & $\sharpPclass$-metric-com. & $\sharpPclass$-metric-com. & -- & -- & $\sharpPclass$-Turing-com. \\[0.4em]
\#CCAV & $\FPclass$ & $\sharpPclass$-Turing-com. & $\sharpPclass$-com. & $\sharpPclass$-com. & $\sharpPclass$-com. \\
\#DCAV & $\FPclass$ & $\sharpPclass$-Turing-com. & $\sharpPclass$-metric-com. & $\sharpPclass$-metric-com. & $\sharpPclass$-metric-com. \\
\#CCDV & $\FPclass$ & $\sharpPclass$-metric-com. & $\sharpPclass$-com. & $\sharpPclass$-com. & $\sharpPclass$-com. \\
\#DCDV & $\FPclass$ & $\sharpPclass$-metric-com. & $\sharpPclass$-metric-com. & $\sharpPclass$-metric-com. & $\sharpPclass$-metric-com. \\
\end{tabular}
\vspace{0.5cm}
(b) The Single-Peaked Case
\hfill{}
\begin{tabular}{c|cccc}
\hline
Problem\rule{0cm}{0.5cm} & Plurality & $k$-Approval, $k \geq 2$ & Approval & Condorcet-Consistent \\
& & & & (e.g., Maximin)\\[0.15cm]
\hline
\#CCAC\rule{0cm}{0.5cm}& $\FPclass$ & $\FPclass$ &-- & -- \\
\#DCAC & $\FPclass$ & $\FPclass$ & FP & FP \\
\#CCDC & $\FPclass$ & $\FPclass$ &FP & FP \\
\#DCDC & $\FPclass$ & $\FPclass$ & -- & -- \\[0.4em]
\#CCAV & $\FPclass$ & $\FPclass$ & ? & $\FPclass$ \\
\#DCAV & $\FPclass$ & $\FPclass$ & ? & $\FPclass$ \\
\#CCDV & $\FPclass$ & $\FPclass$ & ? & $\FPclass$ \\
\#DCDV & $\FPclass$ & $\FPclass$ & ? & $\FPclass$ \\
\end{tabular}
\hfill{}
\caption{\label{tab:results}The complexity of counting
variants of control problems for (a) the unrestricted case,
and for (b) the single-peaked case. A dash in an entry means
that the given system is \emph{immune} to the type of
control in question (i.e., it is impossible to achieve the
desired effect by the action this control problem allows;
technically this means the answer to the counting question
is always $0$). Immunity results were established by
Bartholdi, Tovey, and
Trick~(1989) %
for the constructive cases, and by Hemaspaandra,
Hemaspaandra, and
Rothe~(2007) %
for the destructive cases. }
\end{center}
\end{table*}
\tabcolsep=0.1cm
\begin{table*}[t]
\small
\begin{center}
(a) The Unrestricted Case
\hfill{}
\begin{tabular}{c|ccccccc}
\hline
Problem\rule{0cm}{0.5cm} & Plurality & $2$-Approval & $3$-Approval &$k$-Approval, $k \geq 4$ & Approval & Condorcet & Maximin\\[0.15cm]
\hline
CCAC\rule{0cm}{0.5cm}& $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. &-- & -- & $\NPclass$-com.\\
DCAC & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com.& $\Pclass$ & $\Pclass$ & $\Pclass$ \\
CCDC & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com.& $\Pclass$ & $\Pclass$ & $\Pclass$ \\
DCDC & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com.& -- & -- & $\Pclass$ \\[0.4em]
CCAV & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. \\
DCAV & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\NPclass$-com. & $\Pclass$ & $\Pclass$ & $\NPclass$-com. \\
CCDV & $\Pclass$ & $\Pclass$ & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. & $\NPclass$-com. \\
DCDV & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\NPclass$-com. & $\Pclass$ & $\Pclass$ & $\NPclass$-com. \\
\end{tabular}
\vspace{0.5cm}
(b) The Single-Peaked Case
\hfill{}
\begin{tabular}{c|cccc}
\hline
Problem\rule{0cm}{0.5cm} & Plurality & $k$-Approval, $k \geq 2$ & Approval & Condorcet-Consistent \\
& & & & (e.g., Maximin)\\[0.15cm]
\hline
CCAC\rule{0cm}{0.5cm}& $\Pclass$ & $\Pclass$ &-- & -- \\
DCAC & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\Pclass$ \\
CCDC & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\Pclass$ \\
DCDC & $\Pclass$ & $\Pclass$ & -- & -- \\[0.4em]
CCAV & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\Pclass$ \\
DCAV & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\Pclass$ \\
CCDV & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\Pclass$ \\
DCDV & $\Pclass$ & $\Pclass$ & $\Pclass$ & $\Pclass$ \\
\end{tabular}
\hfill{}
\caption{\label{tab:decision}The complexity of decision
variants of control problems for (a) the unrestricted case,
and for (b) the single-peaked case. A dash in an entry means
that the given system is \emph{immune} to the type of
control in question. For the unrestricted case, we have the
following: The results for Plurality are due to Bartholdi,
Tovey, and Trick~\cite{bar-tov-tri:j:control} and
Hemaspaandra, Hemaspaandra, and
Rothe~\cite{hem-hem-rot:j:destructive-control}, the results
regarding $k$-Approval are due to
Lin~\cite{lin:thesis:elections} (see
also~\cite{elk-fal-sli:j:cloning,fal-hem-hem:c:weighted-control}),
the results regarding Approval are due to Hemaspaandra,
Hemaspaandra, and
Rothe~\cite{hem-hem-rot:j:destructive-control}, and the
results regarding Maximin are due to Faliszewski,
Hemaspaandra, and
Hemaspaandra~\cite{fal-hem-hem:j:multimode}. For the
single-peaked case, we inherit polynomial-time algorithms
from the unrestricted case. The remaining results (except
those for Condorcet-consistent rules) are due to Faliszewski
et al.~\cite{fal-hem-hem-rot:j:single-peaked-preferences},
and the remaining results regarding Condorcet-consistent
rules are due to~\cite{bra-bri-hem-hem:c:sp2}.}
\end{center}
\end{table*}
In the following sections we give proofs for our results and provide
some more detailed discussion regarding particular voting rules.
\subsection{Plurality Voting}
Under plurality voting, counting variants of both control by adding
voters and control by deleting voters are in $\FPclass$ even for the
unrestricted case. Our algorithms are based on dynamic programming and
applications of Theorems~\ref{thm:c-to-d} and~\ref{thm:d-to-a}.
\begin{theorem} \label{th:plavc}
Plurality-\#CCAV, Plurality-\#DCAV, Plurality-\#CCDV, and
Plurality-\#DCDV are in $\FPclass$, even in the unrestricted case.
\end{theorem}
\begin{proof}
We give the proof for Plurality-\#CCAV only. The result for
Plurality-\#CCDV follows by applying Theorem~\ref{thm:d-to-a}, and
the destructive cases follows by applying Theorem~\ref{thm:c-to-d}.
Let $I = (C,V,W,p,k)$ be an input instance of Plurality-\#CCAV,
where $C = \{p,c_1,\ldots, c_{m-1}\}$ is the candidate set, $V$ is
the set of registered voters, $W$ is the set of unregistered voters,
$p$ is the designated candidate, and $k$ is the upper bound on the
number of voters that can be added. We now describe a polynomial-time
algorithm that computes the number of solutions for $I$.
Let $A_p$ be the set of voters from $W$ that rank $p$ first.
Similarly, for each $c_i \in C$, let $A_{c_i}$ be the set of voters
from $W$ that rank $c_i$ first. We also define
$\id{count}(C,V,W,p,k,j)$ to be the number of sets $W' \subseteq W - A_p$
such that
(1) $\|W'\| \leq k-j$, and
(2) in election $(C,V \cup W')$ each candidate $c_i \in C$, $1
\leq i \leq m-1$, has score at most $\score{(C,V)}^p(p)+j-1$.
Our algorithm works as follows. First, we compute $k_0$, the
minimum number of voters from $A_p$ that need to be added to $V$ to
ensure that $p$ has plurality score higher than any other candidate
(provided no other voters are added). Clearly, if $p$ already is the
unique winner of $(C,V)$ then $k_0$ is $0$, and otherwise $k_0$ is
$\max_{c_i \in C}(\score{(C,V)}^p(c_i)-\score{(C,V)}^p(p)+1)$. Then,
for each $j$, $k_0 \leq j \leq \min(k,\|A_p\|)$, we compute the number
of sets $W'$, $W' \subseteq W$, such that $W'$ contains exactly $j$
voters from $A_p$, at most $k-j$ voters from $W - A_p$, and $p$ is
the unique winner of $(C, V \cup W')$. It is easy to verify that for
a given $j$, there is exactly $h(j) =
\binom{\|A_p\|}{j}\cdot\id{count}(C,V,W\!,p,k,j)$ such sets. Our
algorithm returns $\sum_{j=k_0}^{\min(k,\|A_p\|)}h(j)$. The reader can
easily verify that this indeed is the correct answer. To complete
the proof it suffices to show a polynomial-time algorithm for
computing $\id{count}(C,V,W,p,k,j)$.
Let us fix $j$, $k_0 \leq j \leq \min(k,\|A_p\|)$. We now show how to
compute $\id{count}(C,V,W,p,k,j)$. Our goal is to count the number
of ways in which we can add at most $k-j$ voters from $W-A_p$ so
that no candidate $c_i \in C$ has score higher than
$\score{(C,V)}^p(p)+j-1$. For each candidate $c_i \in C$, we can
add at most
$
l_i =
\min\bigl(\|A_{c_i}\|,j+\score{(C,V)}^p(p)-\score{(C,V)}^p(c_i)-1\bigr)
$
voters from $A_{c_i}$; otherwise $c_i$'s score would exceed
$\score{(C,V)}^p(p)+j-1$.
For each $i$, $1 \leq i \leq m-1$, and each $t$, $0 \leq t \leq
k-j$, let $a_{t,i}$ be the number of sets $W' \subseteq A_{c_1}\cup
A_{c_2}\cup\cdots\cup A_{c_i}$ that contain exactly $t$ voters and
such that each candidate $c_1, c_2, \ldots, c_i$ has score at most
$\score{(C,V)}^p(p)+j-1$ in the election $(C,V \cup W')$. Naturally,
$\id{count}(C,V,W,p,k,j) = \sum_{t=0}^{k-j}a_{t,m-1}$. It is easy
to check that $a_{t,i}$ satisfies the following recursion:
\[
a_{t,i} =
\begin{cases}
\sum_{s=0}^{\min(l_i,t)}\binom{\|A_{c_i}\|}{s}a_{t-s,i-1}, & \text{if $t>0$, $i>1$}, \\[2mm]
1, & \text{if $t=0$, $i>1$}, \\[1mm]
\binom{\|A_1\|}{t}, & \text{if $t\le\|A_{c_1}\|$, $i=1$}, \\[1mm]
0, & \text{if $t>\|A_{c_1}\|$, $i=1$}.
\end{cases}
\]
Thus, for each $t,i$ we can compute $a_{t,i}$ using standard dynamic
programming techniques in polynomial time. Thus,
$\id{count}(C,V,W,p,k,j)$ also is polynomial-time computable. This
completes the proof that Plurality-\#CCAV is in $\FPclass$.
\end{proof}
On the other hand, for Plurality voting \#CCAC and \#CCDC are
$\sharpPclass$-complete and this follows from proofs already given in the
literature~\cite{fal-hem-hem:j:nearly-sp}.
\begin{theorem}\label{thm:placdcc}
In the unrestricted case, Plurality-\#CCAC and Plurality-\#CCDC are $\sharpPclass$-complete.
\end{theorem}
\begin{proof}
Faliszewski, Hemaspaandra, and
Hemaspaandra~\cite[Theorem~6.4]{fal-hem-hem:j:nearly-sp} show that
decision variants of control by adding candidates and control by
deleting candidates are $\NPclass$-complete.\footnote{Naturally, these
results are originally due to Bartholid, Tovey, and
Trick~\cite{bar-tov-tri:j:control}. However, the proofs of Faliszewski, Hemaspaandra,
and Hemaspaandra are more useful in our case, because they work
for $k$-Approval for each fixed $k$ (which will be useful later),
and it is convenient to verify that they give parsimonious
reductions from a $\sharpPclass$-complete problem.} Their proofs work
by reducing X3C to appropriate control problems in a parsimonious
way. This means that the same reductions reduce \#X3C to the
counting variants of respective control problems.
\end{proof}
\noindent
Now, Corollary~\ref{cor:placdcdc} follows by combining
Theorems~\ref{thm:placdcc} and~\ref{thm:c-to-d}.
\begin{corollary}\label{cor:placdcdc}
In the unrestricted case, Plurality-\#DCAC and Plurality-\#DCDC are $\sharpPclass$-metric-complete.
\end{corollary}
However, for the single-peaked case, we get easiness results.
\begin{theorem}
For the case of single-peaked profiles, Plurality-\#CCAC,
Plurality-\#CCDC, Plurality-\#DCAC, and Plurality-\#DCDC are in
$\FPclass$.
\end{theorem}
\begin{proof}
This result follows from Theorem~\ref{thm:kapp-ccac-sp} (see the next section) for
the case of $1$-Approval.
\end{proof}
\subsection{$\boldsymbol{k}$-Approval Voting}
While $k$-Approval is in many respects a simple generalization of the
Plurality rule, it turns out that for $k \geq 2$, for the
unrestricted case, all counting variants of control
problems are intractable for $k$-Approval. This is quite expected for
candidate control because decision variants of these problems are
$\NPclass$-complete (see the work of
Lin~\cite{lin:thesis:elections,elk-fal-sli:j:cloning} and additionally
of Faliszewski, Hemaspaandra, and
Hemaspaandra~\cite{fal-hem-hem:c:weighted-control}), but is more
intriguing for voter control (as shown by Lin, for $2$-Approval all
voter control decision problems are in $\Pclass$, and, as one can verify,
for $k$-Approval all destructive voter control decision problems are
in $\Pclass$).
On the other hand, for the single-peaked case all the counting
variants of control are polynomial-time solvable for $k$-Approval
(however, note that we rely on $k$ being a constant; our results do
not generalize to the case where $k$ is part of the input). \bigskip
\noindent
\textbf{Candidate Control.}
\quad
We start be quickly dealing with candidate control in the unrestricted
case. For the unrestricted case, as in the case of Plurality, we can
use the result of Faliszewski, Hemaspaandra, and
Hemaspaandra~\cite{fal-hem-hem:j:nearly-sp}.
\begin{theorem}
In the unrestricted case, for each $k$, $k \geq 2$,
$k$-Approval-\#CCAC and $k$-Approval-\#CCDC are $\sharpPclass$-complete,
and $k$-Approval-\#DCAC and $k$-Approval-\#DCDC are
$\sharpPclass$-metric-complete.
\end{theorem}
\begin{proof}
For the constructive variants, the same approach as in the proof of
Theorem~\ref{thm:placdcc} works: The hardness proofs given by
Faliszewski, Hemaspaandra, and
Hemaspaandra~\cite[Theorem~6.4]{fal-hem-hem:j:nearly-sp} apply to
the case of $k$-Approval as well. For the destructive variants, we
invoke Theorem~\ref{thm:c-to-d}.
\end{proof}
For the single-peaked case, we use a dynamic-programming approach. Our
algorithm is a very extensive generalization of the algorithm for the
single-peaked variant of Plurality-CCAC given by Faliszewski et
al.~\cite{fal-hem-hem-rot:j:single-peaked-preferences}
\begin{theorem}\label{thm:kapp-ccac-sp}
For the case of single-peaked profiles, for each fixed positive $k$,
$k$-Approval-\#CCAC, $k$-Approval-\#CCDC, $k$-Approval-\#DCAC, and
$k$-Approval-\#DCDC are in $\FPclass$.
\end{theorem}
\begin{proof}
We give the proof for $k$-Approval-\#CCAC only. The result for
$k$-Approval-\#CCDC follows by applying Theorem~\ref{thm:d-to-a},
and the destructive cases follow by applying
Theorem~\ref{thm:c-to-d}.
We are given a candidate set $C$, a voter collection $V$, a
designated candidate $p \in C$, a collection $A$ of unregistered
candidates, a positive integer $\ell$, and a societal axis
$\mathrel{L}$ over $C \cup A$. We show a polynomial time algorithm
for determining the number of sets $A'$, $A' \subseteq A$, where
$\|A'\| \le \ell$, such that $p$ is the unique $k$-Approval winner
in election $(C \cup A', V)$. We assume that $\|C \cup A\| > 4k$
(otherwise we solve our problem by enumerating all possible solutions).
Intuitively, the idea of our algorithm is as follows: First, we fix
two groups of $k$ candidates, one ``to the left'' of $p$ (with
respect to $L$) and one ``to the right'' of $p$. We will argue that
each choice of such ``neighborhood'' of $p$ fixes $p$'s score.
Second, with $p$'s score fixed, we count how many ways are there to
add candidates so that every candidate has fewer points than $p$.
We sum these values over all choices of $p$'s ``neighborhood.''
\newcommand{{{\mathrm{before}}}}{{{\mathrm{before}}}}
\newcommand{{{\mathrm{after}}}}{{{\mathrm{after}}}}
Let us introduce some useful notation. For each set $X \subseteq C
\cup A$ of candidates we define ${{\mathrm{before}}}(X)$ to be the subset of
those candidates in $C \cup A$ that precede each member of $X$ with
respect to $L$ (i.e., ${{\mathrm{before}}}(X) = \{ d \in C \cup A \mid (\forall
x \in X)[d \mathrel{L} x]\}$). Analogously, we define ${{\mathrm{after}}}(X)$ to be the
subset of those candidates in $C \cup A$ that succeed all members of
$X$ with respect to $L$ (i.e., ${{\mathrm{after}}}(X) = \{ d \in C \cup A \mid
(\forall x \in X)[x \mathrel{L} d]\}$). The next lemma describes how we can
fix candidates' scores by fixing their ``neighborhoods.''
\begin{lemma}
\label{k-barrier-lemma}
For each set $H \subseteq C \cup A$, $\|H\| = k$, each set $X
\subseteq {{\mathrm{before}}}(H)$ and each candidate $c \in X$, if $z$ is the score
of candidate $c$ in k-Approval election $(X \cup H, V)$, then for
each $Y \subseteq {{\mathrm{after}}}(H)$ in k-Approval election $(X \cup H \cup Y,
V)$ the score of candidate $c$ is $z$ as well.
\end{lemma}
\begin{proof}
For a given subset $Y \subseteq {{\mathrm{after}}}(H)$ and candidate $c \in
X$, suppose that the score of candidate $c$ in $k$-Approval election
$(X \cup H \cup Y, V)$ is $z' \ne z$. Adding a candidate can
never increase the score of a candidate already present, so we
conclude that $z' < z$. It means that there is a voter $v \in V$
from which $c$ gains a point in election $E_1 = (X \cup H,V)$ but
not in election $E_2 = (X \cup H \cup Y, V)$. Thus there is a
candidate $c' \in Y$ that receives a point from $v$. Since $c$ is
among $v$'s top $k$ most preferred candidates across $X \cup H$
and $\|H\| = k$, there must exist a candidate $c'' \in H$ that is
less preferred than $c$ by $v$. Since $v$ prefers $c'$ over $c$,
it must be that $v$ prefers $c'$ over $c''$. However, we know that
$c \mathrel{L} c'' \mathrel{L} c'$, because $c \in X$, $c'' \in H$
and $c' \in Y$. Since $v$ is single-peaked with respect to
$L$, $v$ cannot prefer both $c$ and $c'$ over $c''$; a contradiction.
\end{proof}
Let $B_L$ be the set of the first $2k$ candidates from $C \cup A$
(with respect to $L$) and let $B_R$ be the set of the last $2k$
candidates from $C \cup A$ (with respect to $L$). Without loss of
generality, we assume that $B_L$ and $B_R$ contain candidates from
$C$ only, and that each voter has a preference order of the form $C
- (B_L \cup B_R) > B_L > B_R$ (if this were not the case then we
could add to $C$ two groups of $2k$ dummy candidates that all the
voters rank last, respecting the above requirement, and that are at
the extreme ends of the societal axis $L$). Note that by our
assumptions $p$ does not belong to $B_L \cup B_R$ and for each $A''
\subseteq A$ it holds that in election $(C \cup A'', V)$ the
candidates from $B_R$ receive $0$ points each.
We say that a set $Y \subseteq C \cup A$ is well-formed if for each
candidate $d \in C$ it holds that $d \in {{\mathrm{before}}}(Y) \cup Y \cup
{{\mathrm{after}}}(Y)$ (in other words, $Y$ is well-formed if it is an interval
with respect to the societal axis $L$ restricted to $C \cup Y$). We
define $K_L$ to be a collection of subsets of ${{\mathrm{before}}}(\{p\})$ such
that if $Y \in K_L$ then $Y\cup\{p\}$ is well-formed and $\|Y\| = k$.
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(Intuitively, $K_L$ is the family of sets of candidates that can
form the ``left part'' of $p$'s neighborhood.) We define $K_R$
analogously, but ``to the right of $p$.'' That is, $K_R$ is a
collection of subsets of ${{\mathrm{after}}}(\{p\})$ such that if $Y \in K_R$
then $\{p\} \cup Y$ is well-formed and $\|Y\| = k$.
Note that there are at most $O(\|C \cup A\|^k)$ sets in each of
$K_L$ and $K_R$.
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Our algorithm proceeds as follows. In a loop, we try each $Y_L \in
K_L$ and each $Y_R \in K_R$. For each choice of $Y_L$ and $Y_R$, we
set $Y = Y_L \cup \{p\} \cup Y_R$ to be the neighborhood of $p$, and
we set $\rho$ to be the $k$-Approval score of $p$ in election
$(Y,V)$. (If $\|Y \cap A\| > \ell$ then we drop this choice of $Y_L$
and $Y_R$ because picking this neighborhood requires adding more
candidates than we are allowed to.) We create two new sets, $C' = C
\cup Y$, $A' = A \cap ({{\mathrm{before}}}(Y) \cup {{\mathrm{after}}}(Y))$, and an integer
$\ell' = \ell - \|Y \cap A\|$. Finally, we compute the number $s(Y)$
of size-at-most-$\ell'$ subsets $A''$ of $A'$ such that in
election $(C' \cup A'',V)$ each candidate has less than $\rho$
points (except for $p$ who, by Lemma~\ref{k-barrier-lemma}, has
exactly $\rho$ points). Computing $s(Y)$ in polynomial time is a
crucial technical part of the algorithm and we describe it below. We
sum up all the values $s(Y)$ and return them as our output. By
Lemma~\ref{k-barrier-lemma} it is easy to see that this strategy is
correct.
We now describe how to compute $s(Y)$ in polynomial time. Let us
consider a situation where we have picked $p$'s neighborhood $Y$ and
computed $\rho$, $C'$, $A'$, and $\ell'$. If $\rho = 0$ then $p$
cannot be the unique winner so we assume that $\rho > 0$.
From now on, we assume that $C'$ and $A'$ take the roles of $C$ and
$A$ in the definition of a well-formed set.
\newcommand{{{\mathrm{prev}}}}{{{\mathrm{prev}}}}
\newcommand{\Prev}{{{\mathrm{Prev}}}}
%
%
%
%
For each well-formed set $Z = \{z_1, \ldots, z_{2k}\} \subseteq C'
\cup A'$, such that $z_1 \mathrel{L} \cdots \mathrel{L} z_{2k}$, and for each
nonnegative integer $s$ we define:
\begin{enumerate}
\item[] $f(Z,s) = $ the number of sets $A'' \subseteq (A' \cap
{{\mathrm{before}}}(Z))$ such that $\|A''\| \leq s$ and in election $(Z \cup
A'' \cup (C' \cap {{\mathrm{before}}}(Z)))$ each candidate in $\{z_1, \ldots,
z_k\} \cup A'' \cup (C' \cap {{\mathrm{before}}}(Z))$ other than $p$ (if $p$
is included in this set) has fewer than $\rho$
points.\footnote{Yes, we really mean the condition on scores to
apply to candidates $z_1, \ldots, z_k$ but not to candidates
$z_{k+1}, \ldots, z_{2k}$; we will implicitly ensure that the
scores of $z_{k+1}, \ldots, z_{2k}$ do satisfy the condition as
well.}
\end{enumerate}
It is easy to see that $s(Y)$ is simply $f(B_R,\ell')$ (recall that
$B_R$ is the set of ``right-hand side'' dummy candidates, who have score
$0$ in every election). For each $Z$ and $s$, we can compute
$f(Z,s)$ using dynamic programming. To provide appropriate recursive
expression for $f$, we need some additional notation. We define
${{\mathrm{prev}}}(Z)$ to be the last candidate from $C'$ with respect to $L$
that precedes the candidates from $Z$. That is ${{\mathrm{prev}}}(Z)$ is the
maximal (``rightmost'') element of $C' \cap {{\mathrm{before}}}(Z)$ with respect
to $L$. We define $\Prev(Z)$ to be the set that contains ${{\mathrm{prev}}}(Z)$
and all the candidates from $C' \cup A'$ that are between ${{\mathrm{prev}}}(Z)$
and $Z$, with respect to $L$. For a well-formed set $Z = \{z_1,
\ldots, z_{2k}\}$ and an element $z \in \Prev(Z)$ (provided that
$\Prev(Z)$ is defined) we let $\delta(Z,z)$ be $1$ if in election
$(\{z,z_1,\ldots, z_{2k}\},V)$ candidate $z_k$ is either $p$ or
obtains fewer than $\rho$ points. Otherwise we set $\delta(Z,z) =
0$. Now, for each nonnegative integer $s$ and each well-formed set
$Z = \{z_1, \ldots, z_{2k}\} \subseteq C' \cup A'$ such that $z_1
\mathrel{L} \cdots \mathrel{L} z_{2k}$ the following relation holds ($\chi_{A'}$
is the characteristic function of set $A'$, i.e., $\chi_{A'}(z)$ is
$1$ if $z \in A'$ and is $0$ otherwise):
\[ f(Z,s) = \left\{ \def1.5{1.5}
\begin{array}{l}
\displaystyle\sum_{z \in \Prev(Z)} \delta(Z,z)f(\{z,z_1,\ldots,z_{2k-1}\}, s-\chi_{A'}(z)) \text{, if $\Prev(Z)$ is defined,} \\
\quad 1 \text{, otherwise.}
\end{array} \right.
\]
Note that if $\Prev(Z)$ is not defined then $Z = B_L$ and by
single-peakedness of the election, $z_1, \ldots, z_k$ have zero
points each.
The correctness of this recursive expression follows by our
assumptions and by Lemma~\ref{k-barrier-lemma}. It is easy to see
that using dynamic programming and this recursive expression we can
compute $f(B_R,\ell')$ in polynomial time. This concludes the proof.
\end{proof}
\noindent
\textbf{Voter Control.} \quad We now turn to the case of voter
control and we start with the unrestricted case. This time, we need
quite a few new ideas: Under $k$-Approval (for fixed $k$, $k \geq 2$)
all types of counting voter control are hard, while the complexity of
decision variants is quite varied (see the works of
Lin~\cite{lin:thesis:elections,elk-fal-sli:j:cloning} and of Faliszewski, Hemaspaandra, and
Hemaspaandra~\cite{fal-hem-hem:c:weighted-control}).
We start by considering $2$-Approval-\#CCAV and $2$-Approval-\#CCDV
separately, then we extend these results to $k \geq 3$, and finally we
invoke Theorem~\ref{thm:c-to-d} to obtain the results for the
destructive cases.
\begin{theorem}\label{thm:ccavdv-2}
In the unrestricted case,
$2$-Approval-\#CCAV is $\sharpPclass$-Turing-complete and
$2$-Approval-\#CCDV is $\sharpPclass$-metric-complete.
\end{theorem}
\begin{proof}
We first consider $2$-Approval-\#CCAV.
We give a Turing reduction from \#PerfectMatching to
$2$-Approval-\#CCAV. Let $G = (G(X),G(Y),G(E))$ be our input
bipartite graph, where $G(X) = \{x_1, \ldots, x_n\}$ and $G(Y) =
\{y_1, \ldots, y_n\}$ are sets of vertices, and $G(E) = \{e_1,
\ldots, e_m\}$ is the set of edges. We form an election $E = (C,V)$
and a collection $W$ of unregistered voters as follows. We set $C =
\{p,b_1,b_2\} \cup G(X) \cup G(Y)$ and we let $V = (v_1,v_2)$, where
$v_{1}$ has preference order $p > b_1 > C - \{p,b_1\}$ and $v_{2}$
has preference order $p > b_2 > C - \{p,b_2\}$. We let $W = (w_1,
\ldots, w_m)$, where for each $\ell$, $1 \leq \ell \leq m$, if
$e_\ell = \{x_i,y_j\}$ then $w_\ell$ has preference order $x_i > y_j
> C - \{x_i,y_j\}$.
Thus, in election $E$ candidate $p$ has score $2$, candidates $b_1$
and $b_2$ have score $1$, and candidates in $G(X) \cup G(Y)$ have score
$0$. We form an instance $I$ of $2$-Approval-\#CCAV with election $E
= (C,V)$, collection $W$ of unregistered voters, designated
candidate $p$, and the number of voters that can be added set to
$n$. We form instance $I'$ to be identical, except we allow to add
at most $n-1$ voters.
It is easy to verify that the number of $2$-Approval-\#CCAV
solutions for $I$ (for $I'$) is the number of matchings in $G$ of
cardinality at most $n$ (the number of matchings in $G$ of
cardinality at most $n-1$). (Each unregistered voter corresponds to an
edge in $G$ and one cannot add two edges that share a vertex as then
$p$ would no longer be the unique winner.) The number of
perfect matchings in $G$ is exactly the number of solutions for $I$ minus
the number of solutions for $I'$.\bigskip
Let us now consider the case of $2$-Approval-\#CCDV. We give a
metric reduction from \#PerfectMatching. As before, let $G =
(G(X),G(Y),G(E))$ be our input bipartite graph, where $X = \{x_1,
\ldots, x_n\}$ and $Y = \{y_1, \ldots, y_n\}$ are sets of vertices,
and $E = \{e_1, \ldots, e_m\}$ is the set of edges. For each vertex
$v$ of $G$, we write $d(v)$ to denote $v$'s degree. We set $D =
\max\{d(v) \mid v \in X \cup Y\}$ and we set $T = \sum_{v \in X \cup
Y}(D-d(v))$. W.l.o.g. we assume that $D \geq 2$. We form an
election $E = (C,V)$ as follows. We set $C = \{p\} \cup X \cup Y
\cup B$, where $B = \{b_1, \ldots, b_{T+D}\}$. We form the
collection $V = V_E + A_{X,Y} + A_p$ of voters as follows:
\begin{enumerate}\addtolength{\itemsep}{-4pt}
\item We set $V_E = (v_1, \ldots, v_m)$ and for each $e_\ell =
\{x_i, y_j\} \in E$, we set the preference order of $v_\ell$ to be
$x_i > y_j > C - \{x_i,y_j\}$.
\item We set $A_{X,Y} = (a_1, \ldots, a_T)$ and we set their
preference orders so that for each $v \in X \cup Y$, $A_{X,Y}$
contains exactly $D - d(v)$ voters with preference orders of the
form $v > b_t > C - \{v,b_t\}$ (see Item \ref{chooseb}. below regarding how
candidates $b_t$ are chosen).
\item We set $A_p = \{a_{T+1}, \ldots, a_{T+D}\}$, where each voter
$a_{T+i}$, $1 \leq i \leq D$, has preference order of the form $p
> b_t > C - \{p,b_t\}$ (see Item \ref{chooseb}. below regarding how
candidates $b_t$ are chosen).
\item\label{chooseb} We arrange the preference orders of voters in $A_{X,Y} + A_p$
so that each candidate $b_t$, $1 \leq t \leq T+D$, receives
exactly $1$ point.
\end{enumerate}
Thus, before deleting any of the voters, each candidate in $\{p\}
\cup X \cup Y$ has score $D \geq 2$ and each candidate in $B$ has
score $1$. We form instance $I$ of $2$-Approval-\#CCDV consisting of
$E = (C,V)$, designated candidate $p$, and with $n$ as the limit on
the number of voters we are allowed to delete.
We claim that the number of solutions for $I$ is equal to the number
of perfect matchings in $G$. Let $M \subseteq E$ be a perfect
matching in $G$. We form collection $V' = \{v_i \mid e_i \in
M\}$. Clearly, $\|V'\| \leq n$ and in election $E' = (C,V-V')$
candidate $p$ is the unique winner ($p$ has $D$ points, candidates
in $X \cup Y$ have $D-1$ points, and candidates in $B$ have $1$
point).
On the other hand, let $V'$ be a subcollection of $V$ such that
$\|V'\| \leq n$ and $p$ is the unique winner of election $E' =
(C,V-V')$. Since $p$ is the unique winner of $E'$, it must hold that
each of the $2n$ candidates in $X \cup Y$ has at most $D-1$ points
in $E'$. Thus, since $|V'| \leq n$, it must be the case that in
fact $\|V'\| = n$ and $V'$ is a subcollection of $V_E$ such that
each candidate from $X$ appears in the first position of exactly one
vote in $V'$ and each candidate from $Y$ appears in the second
position of exactly one vote from $V'$. As a result, $E' = \{e_i
\mid v_i \in V'\}$ is a perfect matching for $G$.
\end{proof}
\begin{theorem}
In the unrestricted case, for each $k \geq 3$, $k$-Approval-\#CCAV
and $k$-Approval-\#CCDV are $\sharpPclass$-metric-complete.
\end{theorem}
\begin{proof}
The proof for the case of \#CCDV follows by applying a natural
padding argument in the reduction from the proof of
Theorem~\ref{thm:ccavdv-2}. Thus we focus on the case of \#CCAV.
We first give a proof for the case $k=3$.
We give a Turing reduction from \#PerfectMatching to
$3$-Approval-\#CCAV. Let $G = (G(X),G(Y),G(E))$ be our input
bipartite graph, where $G(X) = \{x_1, \ldots, x_n\}$ and $G(Y) =
\{y_1, \ldots, y_n\}$ are sets of vertices, and $G(E) = \{e_1,
\ldots, e_m\}$ is the set of edges. We form an election $E = (C,V)$
and a collection $W$ of unregistered voters as follows. We set $C =
\{p,d\} \cup B \cup G(X) \cup G(Y)$ (where $B$ is a collection of dummy
candidates to be specified later) and we set $V = (v_1,\ldots, v_t)$,
where the preference orders of the voters in $V$ are such that
the score of $p$ is $0$, the score of $d$ is $n-1$, the score
of each candidate in $G(X) \cup G(Y)$ is $n-2$, and the score of
each dummy candidate in $B$ is either $1$.
We set $W = (w_1, \ldots, w_m)$, where for each $\ell$, $1 \leq \ell \leq m$, if
$e_\ell = \{x_i,y_j\}$ then $w_\ell$ has preference order $p > x_i > y_j
> C - \{p,x_i,y_j\}$.
We form an instance $I$ of $3$-Approval-\#CCAV with election $E
= (C,V)$, collection $W$ of unregistered voters, designated
candidate $p$, and the number of voters that can be added set to
$n$.
It is easy to verify that the number of $3$-Approval-\#CCAV
solutions for $I$ is the number of matchings in $G$ of cardinality
exactly $n$. (One has to add exactly $n$ voters for $p$ to defeat
$d$; each unregistered voter corresponds to an edge in $G$ and one
cannot add two edges that share a vertex as then the candidate
corresponding to that vertex would have score $n$, and $p$ would not
be able to become the unique winner.) Thus, the number of perfect
matchings in $G$ is exactly the number of solutions for $I$.
Further, clearly it is possible to implement our reduction in
polynomial time.
The case for $k > 3$ follows by natural padding arguments and
extending the dummy-candidates set $B$.
\end{proof}
Finally, we obtain the results for the destructive cases through
Theorem~\ref{thm:c-to-d} (we remark that it applies to
$2$-Approval-\#CCAV as well because Turing reductions are
generalizations of metric reductions; for the same reason in
Table~\ref{tab:results} we report the complexity for
$k$-Approval-\#CCAV and $k$-Approval-\#CCDV, $k \geq 2$, as
``$\sharpPclass$-Turing-complete,'' even though our results for $k \geq 3$
are slightly more precise).
\begin{corollary}\label{cor:dc-approval}
In the unrestricted case, $2$-Approval-\#DCAV is
$\sharpPclass$-Turing-complete and for each $k$, $k \geq 2$,
$(k+1)$-Approval-\#DCAV and $k$-Approval-\#DCDV are
$\sharpPclass$-metric-complete.
\end{corollary}
Let us now move on to the single-peaked case. Here we show
polynomial-time algorithms for all counting voter control problems
under $k$-Approval (for fixed $k$). Our algorithm is based on dynamic
programming. We use the following notation in the proof below: For a
given $j$-element integer vector ${\mathit{scores}}$ and integer $i$, $1 \le i \le j$,
we write ${\mathit{scores}}_i$ to denote the $i$-{th} element of vector
${\mathit{scores}}$ (i.e., ${\mathit{scores}} = ({\mathit{scores}}_1,...,{\mathit{scores}}_j)$).
\begin{theorem}
For the case of single-peaked profiles, for each fixed positive
integer $k$, $k$-Approval-\#CCAV, $k$-Approval-\#CCDV,
$k$-Approval-\#DCAV and $k$-Approval-\#DCDV are in $\FPclass$.
\end{theorem}
\begin{proof}
We give the proof for $k$-Approval-\#CCAV. The result for
$k$-Approval-\#CCDV follows by Theorem~\ref{thm:d-to-a} and the
results for the destructive cases follow by
Theorem~\ref{thm:c-to-d}.
We are given a candidate set $C$, a voter collection $V$, a
designated candidate $p \in C$, a collection $W$ of unregistered
voters, a positive integer $\ell$, and a societal axis
$\mathrel{L}$. We show a polynomial time algorithm for determining
the number of sets $W'$, $W' \subseteq W$, where $\|W'\| \le \ell$,
such that $p$ is a k-Approval winner in election $(C, V \cup W')$.
For each voter $v \in V \cup W$, we define ${\mathrm{top}}_k(v)$ to be the
set of $k$ most preferred candidates according to $v$. Let
$\mathrel{\widetilde{L'}}$ be a weak linear order over voter set $W
\cup V$, such that $v_1 \mathrel{\widetilde{L'}} v_2$, $v_1,v_2 \in
V \cup W$, if and only if there exists a candidate $c \in
{\mathrm{top}}_k(v_2)$ such that for each $c' \in {\mathrm{top}}_k(v_1)$ we have $c'
\mathrel{L} c$ or $c' = c$. Let $\mathrel{L'}$ be a strict order
obtained from $\mathrel{\widetilde{L'}}$ by breaking the ties in an
arbitrary fashion. For each voter $v \in V \cup W$ let $X^v = \{w
\mid w \in V \cup W \land w \mathrel{L'} v\}$ be the subset of
voters from $V \cup W$ that precede voter $v$ under $\mathrel{L'}$.
We let $v_l$ to be the last voter from $V$ under $\mathrel{L'}$.
For a given voter $v \in V \cup W$, integer $z \in \mathbb{Z}$, and
integer $s \in \mathbb{Z}$, we define $g(v,z,s)$ to be the number of
sets $\widetilde{W} \subseteq X^v \cap W$, where $\|\widetilde{W}\|
= s$, such that $p$ is a k-Approval winner in election $(C, (X^v
\cap V) \cup \widetilde{W} \cup \{v\})$ and in addition the score of
candidate $p$ in this election is exactly $z$. Equation
\eqref{k-app-ccav-main} below gives the result that we should output
on the given input instance of k-Approval-\#CCAV problem:
\begin{equation}\label{k-app-ccav-main}
\sum_{0 \le s \le \ell} \left( \sum_{0 \le z \le \|W \cup V\|} \left( g(v_l,z,s) + \sum_{\substack{w \in W \\ v_l \mathrel{L'} w}} g(w,z,s) \right) \right)
\end{equation}
Thus, it suffices to show how to compute $g(w,z,s)$ for $w \in V
\cup W$ and $z,s \in \mathbb{Z}$ in polynomial time. Before we give
an appropriate algorithm, we need to introduce some additional
notation.
For each candidate $c \in C$, let ${\mathrm{rank}}(c)$ be $c$'s rank under
$\mathrel{L}$ over all candidates from $C$ (so, for example, if $C =
\{a,b,c\}$ and $a \mathrel{L} b \mathrel{L} c$ then ${\mathrm{rank}}(a) = 1$,
${\mathrm{rank}}(b) = 2$ and ${\mathrm{rank}}(c) = 3$). For each voter $v$, let ${\mathrm{rank}}(v) =
\max \{ {\mathrm{rank}}(c) \mid c \in {\mathrm{top}}_k(v)\}$.
%
For each $v \in V \cup W$, let $\mathcal{P}^v= \{w \mid w \in (V
\cup W) \land w \mathrel{L'} v \land (\nexists t \in V)[ w
\mathrel{L'} t \mathrel{L'} v]\}$. In other words, $\mathcal{P}^v$
consists of the closest voter $u \in V$ that precedes $v$ under
$\mathrel{L'}$, and of all the voters that are between $u$ and $v$ under
$\mathrel{L'}$. When $v$ is the first element from $V$ under
$\mathrel{L'}$, then $\mathcal{P}^v$ contains all the voters from
$W$ preceeding $v$ under $\mathrel{L'}$.
%
For a $j$-element integer vector ${\mathit{scores}}$ and a nonnegative
integer $r$, let $\cutoff{{\mathit{scores}}}{r}$ denote $j$-element vector
$({\mathit{scores}}_1, \ldots, {\mathit{scores}}_r, 0,\ldots, 0)$ (that is, we replace
the last $j-r$ entries of vector ${\mathit{scores}}$ with zeros).
For each voter $w \in V \cup W$, we define ${\mathrm{approval}}(w)$ to be the
$\|C\|$-dimensional $0/1$ vector that for each candidate $c \in C$
has $1$ at position ${\mathrm{rank}}(c)$ if and only if $c \in {\mathrm{top}}_k(w)$.
(In other words, ${\mathrm{approval}}(w)$ is the $0/1$ vector that indicates
which candidates receive points from voter $w$.) Note that, due to
single-peakedness of the election, for each voter $w$,
${\mathrm{approval}}(w)$ contains exactly a single consecutive block of $k$
ones.
We are now ready to proceed with our algorithm for computing
function $g$. For a given voter $v \in V \cup W$, given integers
$z,s \in \mathbb{Z}$, and a $\|C\|$-dimensional integer vector
${\mathit{scores}}$, we define $f(v,z,s,{\mathit{scores}})$ to be the number of sets
$\widetilde{W} \subseteq X^v \cap W$ such that $\|\widetilde{W}\| =
s$ and in election $(C,(X^v \cap V) \cup \widetilde{W} \cup \{v\})$
the following holds: (1) $p$ scores exactly $z$ points and (2) each
candidate $c \in C -\{p\}$ scores no more than ${\mathit{scores}}_{{\mathrm{rank}}(c)}$ points.
For a given integer $r \in \mathbb{Z}$, let $\Gamma^{r}$ be the
vector $(r,...,r)$ of dimension $\|C\|$. Clearly, for a given voter
$v \in V \cup W$ and given integers $z,s \in \mathbb{Z}$ we have:
\begin{equation}\label{k-app-ccav-convert}
g(v,z,s) = f(v,z,s,\Gamma^{z-1})
\end{equation}
We now show a recursive formula for $f$. For a given voter $v \in V
\cup W$, a given vector ${\mathit{scores}}$ of (nonnegative) integers of
dimension $\|C\|$, and two integers $z,s \in \mathbb{Z}$, we have:
\begin{equation}\label{k-app-ccav-transform}
f(v,z,s,{\mathit{scores}}) = f(v,z,s,\cutoff{{\mathit{scores}}}{{\mathrm{rank}}(v)})
\end{equation}
This follows from the fact that in election $(C,X^v \cup \{v\})$
only candidates with ranks $j \le {\mathrm{rank}}(v)$ can score a point. It is
easy to see that $f(v,z,s,{\mathit{scores}}) = 0$ when $z < 0$ or ${\mathit{scores}}$
contains at least one negative entry, because the score is always a
nonnegative integer. When $s = 0$ then $f(v,z,s,{\mathit{scores}})$ is $1$ if
and only if in election $(C,(X^v \cap V) \cup \{v\})$ candidate $p$
scores $z$ points and each candidate $c \in C$ scores no more than
${\mathit{scores}}_{{\mathrm{rank}}(c)}$ points; otherwise $f(v,z,s,{\mathit{scores}})$ is $0$.
When $s > 0$, we note that each election consistent with the
condition for $f(v,z,s,{\mathit{scores}})$ contains at least one voter $w$
from $\mathcal{P}^v$. Eq.~\eqref{k-app-ccav-sum} below gives
formula for $f$ in case when $s > 0$; for each voter $w \in
\mathcal{P}^v$ we count all the sets $\widetilde{W} \subseteq X^v
\cap W$ such that $w$ directly precedes $v$ in $(X^v \cap V) \cup
\widetilde{W} \cup \{v\}$ under $L'$:
\begin{equation}\label{k-app-ccav-sum}
f(v,z,s,{\mathit{scores}}) = \sum_{w \in \mathcal{P}_v} f(w,z-z',s-s',{\mathit{scores}} - {\mathrm{approval}}(w)),
\end{equation}
where (1) $z' = 1$ if $p \in {\mathrm{top}}_k(v)$ and $z'=0$ otherwise, and
(2) $s' = 1$ if $v \in W$ and $s' = 0$ otherwise. By combining
Eq.~\eqref{k-app-ccav-transform} and \eqref{k-app-ccav-sum} we get:
\begin{equation}\label{k-app-ccav-sum-final}
f(v,z,s,{\mathit{scores}}) = \sum_{w \in \mathcal{P}_v} f(w,z-z',s-s',\cutoff{{\mathit{scores}} - {\mathrm{approval}}(w)}{{\mathrm{rank}}(w)}).
\end{equation}
We claim that function $g$ can be computed through
Eq.~\eqref{k-app-ccav-convert} in polynomial time, using standard
dynamic programming techniques to compute function $f$. The reason
is that to compute $f$ for the arguments as in
Eq.~\eqref{k-app-ccav-convert} using recursive formula
\eqref{k-app-ccav-sum-final}, we need to obtain $f$'s values for at
most $\|C\| (\|V\| + \|W\|)^{k+3}$ different arguments. This is
because starting from ${\mathit{scores}} = \Gamma^{z-1}$, the only allowed
transformations of ${\mathit{scores}}$ are given in
Eq.~\eqref{k-app-ccav-sum-final} and ensure that whenever we need to
compute $f$, the ${\mathit{scores}}$ vector is of the form
$(z-1,z-1,...,z-1,e_1,e_2,...,e_k,0,0,...,0)$, where $e =
(e_1,e_2,...,e_k)$ is some $k$-element vector of integers and for
each $i$, $1 \le i \le k$, we have $0 \le e_i \le z - 1$. Clearly,
there are no more than $\|C\|z^k$ vectors of this form. Thus, we
can compute $g$ in polynomial time and, through
Eq.~\eqref{k-app-ccav-main}, we can solve $k$-Approval-\#CCAV in
polynomial time.
\end{proof}
\subsection{Approval Voting and Condorcet Voting}
Let us now consider Approval voting and Condorcet voting. While these
two systems may seem very different in spirit, their behavior with
respect to election control is similar. Specifically, in the
unrestricted case, for both systems \#CCAV and \#CCDV are
$\sharpPclass$-complete, for both systems it is impossible to make some
candidate a winner by adding candidates, and for both systems it is
impossible to prevent someone from winning by deleting
candidates. Yet, for both systems \#DCAC and \#CCDC are in $\FPclass$ via
almost identical algorithms. There is, however, also one difference.
For the single-peaked case, we were able to find polynomial-time
algorithms for voter control under Condorcet, while the results for
Approval remain elusive.
\begin{theorem}\label{thm:approval-condorcet-avdv}
In the unrestricted case, each of Approval-\#CCAV, Approval-\#CCDV,
Condorcet-\#CCAV, and Condorcet-\#CCDV is $\sharpPclass$-complete. Their
destructive variants are $\sharpPclass$-metric-complete.
\end{theorem}
\begin{proof}
Note that the results for the destructive variants will follow by
Theorem~\ref{thm:c-to-d} and so we focus on constructive variants
only.
For the case of Approval, $\sharpPclass$-completeness of \#CCAV and
\#CCDV follows from the $\NPclass$-completeness proofs for their decision
variants given by Hemaspaandra, Hemaspaandra, and
Rothe~\shortcite{hem-hem-rot:j:destructive-control}; these proofs,
in effect, give parsimonious reductions from \#X3C to respective
control problems and, thus, establish $\sharpPclass$-completeness.
For the case of Condorcet, $\sharpPclass$-completeness of \#CCDV follows
from the proofs of Theorems 5.1 and 4.19 of Faliszewski et
al.~\cite{fal-hem-hem-rot:j:llull}, who effectively give a
parsimonious reduction from \#X3C to Condorcet-\#CCDV. The case of
Condorcet-\#CCAV appears to not have been considered in the
literature and thus we give a direct proof (naturally, we could
obtain $\sharpPclass$-Turing-completeness by noting that
Theorem~\ref{thm:d-to-a} gives a Turing reduction from
Condorcet-\#CCDV to Condorcet-\#CCAV, but $\sharpPclass$-completeness is
a stronger result). For the reminder of the proof we focus on
Condorcet-\#CCAV. The problem is clearly in $\sharpPclass$, so it
suffices to show that it is $\sharpPclass$-hard. We give a parsimonious
reduction from \#X3C.
Let $(B,\mathcal{S})$ be an instance of \#X3C problem, where
$B=\{b_1,\dots,b_{3k}\}$ and $\mathcal{S}=\{S_1,\dots,S_n\}$. We
create an election $E=(C,V)$, where $C=B\cup\{p\}$. Let $V$ consist
of $k-3$ voters with preferences $b_1\succ b_2\succ\cdots\succ
b_{3k}\succ p$. Thus, $b_1$ is the Condorcet winner of $E$, and
every candidate $b_i\in B$ beats $p$ in $k-3$ votes.
For each set $S_j\in\mathcal{S}$, $S_j =
\{b_{j_1},b_{j_2},b_{j_3}\}$, let $W$ contain a voter with
preference order $b_{j_1}\succ b_{j_2}\succ b_{j_3}\succ
p\succ\cdots$ (after $p$ the remaining candidates are ranked in
arbitrary order). We claim that every subset $W'$, $W'\subseteq W$,
such that $\|W'\|\le k$ and that $p$ is a Condorcet winner of election
$E'=(C,V\cup W')$ corresponds one-to-one to a $k$-element subfamily
${{\mathcal{S}}}'$ of ${{\mathcal{S}}}$ whose elements union up to $B$ (i.e., ${{\mathcal{S}}}'$
is an exact set cover of $B$).
First assume that there is a subfamily
$\mathcal{S'}\subseteq\mathcal{S}$ which is an exact set cover of
$B$. For each $S_j\in\mathcal{S'}$, we include the corresponding
voter from $W$ in $W'$. Let us consider election $E'=(C,V \cup W')$.
For each $b_i\in B$ we have $\N{E'}(b_i,p)=\N{E}(b_i,p)+1=k-2$ and
$\N{E'}(p,b_i)=\N{E}(p,b_i)+k-1=k-1$. Thus $p$ becomes the
Condorcet winner of $E'$.
Now assume that $p$ is the Condorcet winner in election $E'=(C,V
\cup W')$, where $W' \subseteq W$ and $\|W'\| \leq k$. For each $b_i
\in B$, there can be at most one voter in $W'$ that prefers $b_i$ to
$p$. This is so, because otherwise we would have
$\N{E'}(b_i,p)\ge\N{E}(b_i,p)+2=k-1$, and
$\N{E'}(p,b_i)\le\N{E}(p,b_i)+k-2=k-2$, and so $p$ would lose to
$b_i$. Thus, each $b_i$ is preferred to $p$ by either zero or one
voter from $W'$. If $b_i$ is preferred by one voter from $W'$, then
for $p$ to win, $p$ must be preferred to $b_i$ by $k-1$ voters from
$W'$, and since some voter must be added, it must hold that
$\|W'\|=k$.
If there are no voters in $W'$ who prefer $b_i$ to $p$, then since
each vote in $W'$ has some three candidates from $B$ ranked ahead of
$p$, there must be some other $b_{i'}$ that is ranked above $p$ by
more than one voter. This contradicts the requirement that for each
$b_j \in B$, at most one voter in $W'$ prefers $b_j$ to $p$. Hence,
each $b_i$ is preferred to $c$ by exactly one of the $k$ voters in
$W'$. Thus, the voters from $W'$ correspond to an exact set cover
of $B$.
\end{proof}
For the case of single-peaked preferences, we can use dynamic
programming to solve voter control problems under Condorcet.
\begin{theorem}
For the single-peaked case, Condorcet-\#CCAV, Condorcet-\#CCDV,
Condorcet-\#DCAV, and Condorcet-\#DCDV are in \FPclass.
\end{theorem}
\begin{proof}
We focus on Condorcet-\#CCAV. The remaining cases follow by applying
Theorems~\ref{thm:c-to-d} and~\ref{thm:d-to-a}.
We are given an election $E = (C,V)$ where $C$ is a set of
candidates and $V$ is a set of voters, a designated candidate $p \in
C$, a collection $W$ of unregistered voters, a nonnegative integer
$k$, and an order $\mathrel{L}$, the societal axis, such that $V$
and $W$ are single-peaked with respect to $\mathrel{L}$. We show a
polynomial-time algorithm for determining the number of sets $W'$,
$W' \subseteq W$, where $\|W'\| \le k$, such that $p$ is a Condorcet
winner in election $(C, V \cup W')$. Our algorithm is based on the
famous median-voter theorem.
We split $C$ into three sets, $C_a = \{c \mid c \in C \land c
\mathrel{L} p\}$, $C_b = \{c \mid c \in C \land p \mathrel{L} c\}$
and $C_m = \{p\}$; $C_a$ contains the candidates that are before $p$ on
the societal axis and $C_b$ contains the candidates that are after
$p$. For each voter $v$, by $c_v$ we mean the candidate that $v$
ranks first. For each collection $U$ of voters from $V \cup W$, we
define $U_a = \{v \mid v \in U \land c_v \in C_a\}$, $U_b = \{v \mid
v \in U \land c_v \in C_b\}$ and $U_m = \{v \mid v \in U \land c_v =
p\}$. In other words, $U_a$ and $U_b$ consist of those voters from $U$
for which the most preferred candidate is, respectively, in $C_a$ or
$C_b$, and $U_m$ contains those voters from $U$ that rank $p$ first.
For each $W' \subseteq W$, we define $\delta(W')$ to be $1$ exactly
if the following conditions hold:
\begin{enumerate}
\item $\|V_a\|+\|W'_a\| < \frac{1}{2}(\|V\|+\|W'\|)$, and
\item $\|V_b\|+\|W'_b\| < \frac{1}{2}(\|V\|+\|W'\|)$.
\end{enumerate}
Otherwise, we define $\delta(W') = 0$. The following lemma is an
expression of the well-known median voter theorem (we provide the
short proof for the sake of completeness).
\begin{lemma}
For each $W' \subseteq W$, $p$ is the Condorcet winner of election
$(C,V \cup W')$ if and only if $\delta(W') = 1$.
\end{lemma}
\begin{proof}
Assume that $\delta(W') = 1$ and consider an arbitrary candidate
$c$ other than $p$. For the sake of concreteness let us assume
that $c \in C_a$, but a symmetric argument holds for $c \in C_b$.
Due to single-peakedness of $V \cup W'$, no voter outside of $V_a$
and $W'_a$ prefers $c$ to $p$. However, since $\delta(W')=1$, we
have that $\|V_a\|+\|W'_a\| < \frac{1}{2}(\|V\|+\|W'\|)$, and so a
majority of the voters prefers $p$ to $c$. Since $c$ was chosen
arbitrarily, it holds that $p$ is a Condorcet winner.
For the other direction, assume that $\delta(W') = 0$. For the
sake of concreteness, let us assume that this is because
$\|V_a\|+\|W'_a\| \geq \frac{1}{2}(\|V\|+\|W\|)$. By this
assumption, there must be a candidate $c \in C$ that directly
precedes $p$ with respect to $L$ (that is, $c \mathrel{L} p$ and there is
no candidate $d$ such that $c \mathrel{L} d \mathrel{L} p$). Due to
single-peakedness of voters' preference orders, we have that every
voter in $V_a \cup W'_a$ prefers $c$ to $p$, and so $p$ is not a
Condorcet winner. A symmetric argument holds if $\|V_b\|+\|W'_b\|
\geq \frac{1}{2}(\|V\|+\|W\|)$.
\end{proof}
Thus our algorithm should output the number of sets $W'$, $W'
\subseteq W$, of cardinality at most $k$, such that $\delta(W')=1$
holds. However, to evaluate $\delta(W')$ it suffices to know the
values $\|W'_a\|$, $\|W'_b\|$, and $\|W'_m\|$. For each three
integers $\ell_a$, $\ell_b$, and $\ell_m$ we define
$\gamma(\ell_a,\ell_b,\ell_m)$ to be $1$ exactly if the following
two conditions hold (these conditions are analogous to those for
$\delta$):
\begin{enumerate}
\item $\|V_a\|+\ell_a < \frac{1}{2}(\|V\|+\ell_a+\ell_b+\ell_m)$, and
\item $\|V_b\|+\ell_b < \frac{1}{2}(\|V\|+\ell_a+\ell_b+\ell_m)$.
\end{enumerate}
Otherwise, $\gamma(\ell_a,\ell_b,\ell_m)=0$. It is easy to see that
for each $W' \subseteq W$ we have $\delta(W') =
\gamma(\|W'_a\|,\|W'_b\|,\|W'_m\|)$. Now it follows that the number
of subsets $W' \subseteq W$ of cardinality at most $k$ such that
$p$ is a Condorcet winner of election $(C,V\cup W')$ is exactly:
\[
\sum_{\ell=0,...,k}
\sum_{\substack{
\ell_a, \ell_b, \ell_m \in \mathbb{N} \\
\ell_a + \ell_b + \ell_m = \ell
}}
\gamma(\ell_a,\ell_b,\ell_m)
{\|W_a\| \choose \ell_a}
{\|W_b\| \choose \ell_b}
{\|W_m\| \choose \ell_m}.
\]
We can evaluate this sum in polynomial-time. This completes the
proof.
\end{proof}
For the case of candidate control, we get polynomial-time algorithms
even for the unrestricted case.
\begin{theorem} \label{th:apdcc} Approval-\#CCDC, and
Condorcet-\#CCDC are in $\FPclass$, even in the unrestricted case.
\end{theorem}
\begin{proof}
Let us handle the case of Approval first. Let $I = (C,V,p,k)$ be an
instance of approval-\#CCDC. The only way to ensure that $p \in C$
is the unique winner is to remove all candidates $c \in C-\{p\}$
such that $\score{(C,V)}^a(c)\geq\score{(C,V)}^a(p)$. Such
candidates can be found immediately. Let's assume that there are
$k_0$ such candidates. After removing all of them, we can also
remove $k-k_0$ or less of any remaining candidates other than $p$.
Based on this observation we provide the following simple algorithm.
\begin{codebox}
\Procname{$\proc{approval-\#CCDC}(C,V,p,k)$} \li Let $k_0$ be the
number of candidates $c \in C-\{p\}$,
\zi \>s.t.\ $\score{(C,V)}^a(c)\geq\score{(C,V)}(p)$. \\[-4mm]
\li \Return $\sum_{i=0}^{k-k_0}\binom{\|C\|-k_0-1}{i}$
\end{codebox}
Clearly, the algorithm is correct and runs in polynomial-time.
For the case of Condorcet voting, it suffices to note that if $p$ is
to be a winner, we have to delete all candidates $c \in C - \{p\}$
such that $\N{(C,V)}(p,c) \leq \N{(C,V)}(c,p)$. Thus, provided that
we let $k_0$ be the number of candidates $c \in C - \{p\}$ such that
$\N{(C,V)}(p,c) \leq \N{(C,V)}(c,p)$, the same algorithm as for the
case of approval voting works for Condorcet voting.
\end{proof}
\begin{theorem} \label{th:apacd}
Both Approval-\#DCAC and Condorcet-\#DCAC are in $\FPclass$.
\end{theorem}
\begin{proof}
We first consider the case of approval voting. Let $I =
(C,A,V,p,k)$ be an instance of Approval-\#DCAC, where $C = \{p,c_1,
\ldots, c_{m-1}\}$ is the set of registered candidates, $A = \{a_1,
\ldots, a_{m'}\}$ is the set of additional candidates, $V$ is the
set of voters, $p$ is the designated candidate, and $k$ is the upper
bound on the number of candidates that we can add. We will give a
polynomial-time algorithm that counts the number of
up-to-$k$-element subsets $A'$ of $A$ such that $p$ is not the
unique winner of election $(C \cup A',V)$.
Let $A_0$ be the set of candidates in $A$ that are approved by at
least as many voters as $p$ is. To ensure that $p$ is not the unique
winner of the election (assuming $p$ is the unique winner prior to
adding any candidates), it suffices to include at least one
candidate from $A_0$. Thus, we have the following algorithm.
\begin{codebox}
\Procname{$\proc{Approval-\#DCAC}(C,A,V,p,k)$}
\li \If $p$ is not the unique winner of $(C,V)$ \\[-4mm] \label{apacd:checking_c}
\li \Then \Return $\sum_{i=0}^k\binom{\|A\|}{i}$ \\[-4mm] \label{apacd:returning_result1}
\End
\li Let $A_0$ be the set of candidates $a_i \in A$\!,
\zi \>s.t.\ $\score{(C\cup A,V)}^a(a_i)\ge\score{(C\cup A,V)}^a(p)$. \label{apacd:defining_k0}
\li $\id{result}:=0$ \label{apacd:result_init}
\li \For $j:=1$ \To $k$ \label{apacd:counting_loop}
\li \Do $\id{result}:=\id{result}+\sum_{i=1}^{\min(\|A_0\|,j)}\binom{\|A_0\|}{i}\binom{\|A-A_0\|}{j-i}$ \label{apacd:counting}
\End
\li \Return $\id{result}$ \label{apacd:returning_result2}
\end{codebox}
The loop from line~\ref{apacd:counting_loop}, for every $j$, counts
the number of ways in which we can choose exactly $j$ candidates from
$A$; it can be done by first picking $i$ of the candidates in $A_0$
(who beat $p$), and then $j-i$ of the candidates in $A - A_0$. It is
clear that the algorithm is correct and runs in polynomial time.
Let us now move on to the case of Condorcet voting. It is easy to see
that the same algorithm works correctly, provided that we make two
changes: (a) in the first two lines, instead of testing if $p$ is an
approval winner we need to test if $p$ is a Condorcet winner, and (b)
we redefine the set $A_0$ to be the set of candidates $a_i \in A$ such
that $\N{(C \cup A,V)}(p,a_i) \leq \N{(C \cup A,V)}(a_i,p)$. To see that these two
changes suffice, it is enough to note that to ensure that $p$ is not a
Condorcet winner of the election we have to have that either $p$
already is not a Condorcet winner (and then we can freely add any
number of candidates), or we have to add at least one candidate from
$A_0$.
\end{proof}
We conclude our discussion of Condorcet voting by noting that for the
single-peaked case all our results for Condorcet directly translate to
all Condorcet-consistent rules. The reason for this is that if voters'
preferences are single peaked, then there always exist weak Condorcet
winners. Whenever weak Condorcet winners exist, they are the sole
winners under Condorcet-consistent rules by definition. Since we focus
on the unique-winner model, if $R$ is a Condorcet-consistent rule, $E$
is a single-peaked, and $c$ is a candidate, then $c$ is the unique
$R$-winner of $E$ if and only if $c$ is the unique Condorcet winner of
$E$. In effect, we have the following corollary.
\begin{corollary}\label{cor:condorcet-consistent}
Let $R$ be a Condrocet-consistent rule. For the single-peaked case,
$R$-\#CCDC, $R$-\#DCAC, $R$-\#CCAV, $R$-\#CCDV, $R$-\#DCAV, and
$R$-\#DCDV are in $\FPclass$.
\end{corollary}
\subsection{Maximin Voting}
For the case of Maximin, we consider the unrestricted case
only. Maximin is Condorcet-consistent and, thus, for the single-peaked
case we can use Corollary~\ref{cor:condorcet-consistent}.
The complexity of decision variants of control for Maximin was studied
by Faliszewski, Hemaspaandra, and
Hemaspaandra~\shortcite{fal-hem-hem:j:multimode}. In particular, they
showed that under Maximin all voter control problems are
$\NPclass$-complete and an easy adaptation of their proofs gives the
following theorem.
\begin{theorem}
Maximin-\#CCAV and Maximin-\#CCDV are $\sharpPclass$-complete, and
Maximin-\#DCAV and Maximin-\#DCDV are $\sharpPclass$-metric-complete.
\end{theorem}
On the other hand, among the candidate control problems for Maximin,
only Maximin-CCAC is $\NPclass$-complete (DCAC, CCDC, and DCDC are in
$\Pclass$). Still, this hardness of control by adding candidates translates
into the hardness of all the counting variants of candidate control.
\begin{theorem}
Maximin-\#CCAC is $\sharpPclass$-complete and Maximin-\#DCAC is
$\sharpPclass$-metric-complete.
\end{theorem}
\begin{proof}
Faliszewski et al.~\cite{fal-hem-hem:j:multimode}'s proof that
Maximin-CCAC is $\NPclass$-complete can be used without change; their
reduction from X3C to Maximin-CCAC is also correct as a parsimonious
reduction from \#X3C to Maximin-\#CCAC. The result for
Maximin-\#DCAC follow through Theorem~\ref{thm:c-to-d}.
\end{proof}
The cases of Maximin-\#CCDC and Maximin-\#DCDC are more complicated
and require new ideas because decision variants of these problems are
in $\Pclass$.
\begin{theorem}
Both Maximin-\#CCDC and Maximin-\#DCDC are $\sharpPclass$-Turing-complete.
\end{theorem}
\begin{proof}
We consider the \#CCDC case first. Clearly, the
problem belongs to $\sharpPclass$ and it remains to show hardness. We
will do so by giving a Turing reduction from \#PerfectMatching.
Let $G = (G(X),G(Y),G(E))$ be our input graph, where $G(X) = \{x_1,
\ldots, x_n\}$ and $G(Y) = \{y_1, \ldots, y_n\}$ are sets of
vertices, and $E = \{e_1, \ldots, e_m\}$ is the set of edges. For
each nonnegative integer $k$, define $g(k)$ to be the number of
matchings in $G$ that contain exactly $k$ edges (e.g., $g(n)$ is the
number of perfect matchings in $G$).
We define the following election $E = (C,V)$. We set $C = G(E) \cup
S \cup B \cup \{p\}$, where $S = \{s_0, \ldots, s_{n}\}$ and $B =
\{b_{i,j}^\ell \mid \text{$0 \leq \ell \leq n$, $i < j$, and $e_i$
and $e_j$ share a vertex}\}$. To build voter collection $V$, for
each two candidates $a, b \in C$, we define $v(a,b)$ to be a pair of
voters with preference orders $a > b > C - \{a,b\}$ and
$\revnot{C-\{a,b\}} > a > b$. We construct $V$ as follows:
\begin{enumerate}
\addtolength{\itemsep}{-4pt}
\item For each $s_i \in S$, we add pair $v(s_i,p)$.
\item For each $s_i \in S$, we add two pairs $v(s_i,s_{i+1})$,
where $i+1$ is taken modulo $n+1$.
\item For each $s_i \in S$ and each $e_t \in E$, we add two pairs
$v(s_i,e_t)$.
\item For each $e_i, e_j \in E$, $i < j$, where $e_i$ and $e_j$
share a vertex, and for each $\ell$, $0 \leq \ell \leq n$, we add
two pairs $v(e_i,b_{i,j}^\ell)$ and two pairs
$v(e_j,b_{i,j}^\ell)$.
\end{enumerate}
Let $T$ be the total number of pairs $v(a,b)$, $a,b \in C$, included
in $V$. By our construction, the following properties hold:
\begin{enumerate}
\addtolength{\itemsep}{-4pt}
\item $\score{E}^m(p) = T-1$ and it is impossible to change the
score of $p$ be deleting $n$ candidates or fewer (this is because
there are $n+1$ candidates $s_i \in S$ such that $N_E(p,s_i) =
T-1$.
\item For each $s_i \in S$, $\score{E}^m(s_i) = T-2$, but deleting
$s_{i-1}$ (where we take $i-1$ modulo $n+1$) increases the score
of $s_i$ to $T+1$.
\item For each $e_t \in E$, $\score{E}^m(e_t)$ is $T-2$.
\item For each $b_{i,j}^\ell \in B$, $\score{E}^m(b_{i,j}^\ell) = T-2$ and
it remains $T-2$ if we delete either $e_i$ or $e_j$, but it
becomes $T$ if we delete both $e_i$ and $e_j$.
\end{enumerate}
Note that $p$ is the unique winner of $E$. For each $k$, $0 \leq k
\leq n$, we form instance $I(k) = (C,V,p,k)$ of Maximin-\#CCDC. We
define $f(k) = \#I(k) - \#I(k-1)$. That is, $f(k)$ is the number of
solutions for $I(k)$ where we delete exactly $k$ candidates. We
claim that for each $k$, $1 \leq k \leq n$, it holds that $ f(k) =
\sum_{j=0}^k {\|B\| \choose j} g(k-j). $ Why is this so? First,
note that by the listed-above properties of $E$, deleting any subset
$C'$ of candidates from $C$ that contains some member of $S$
prevents $p$ from being a winner. Thus, we can only delete subsets
$C'$ of $C$ that contains candidates in $G(E) \cup B$. Let us fix a
nonnegative integer $r$, $0 \leq r \leq n$. Let $C' \subseteq G(E)
\cup B$ be such that $p$ is the unique Maximin winner of $E' =
(C-C',V)$ and $\|C'\| = r$. Let $r_B = \|C' \cap B\|$ and $r_{G(E)}
= \|C' \cap G(E)\|$. It must be the case that for each $e_i, e_j
\in G(E)$, $i < j$, where $e_i$ and $e_j$ share a vertex, $C'$
contains at most one of them. Otherwise, $E'$ would contain at least
one of the candidates $b_{i,j}^\ell$, $0 \leq \ell \leq n$, and this
candidate would have score higher than $p$. Thus, the candidates in
$C' \cap G(E)$ correspond to a matching in $G$ of cardinality
$r_{G(E)}$. On the other hand, since $r_B \leq n$, $C \cap B$
contains an arbitrary subset of $B$. Thus, there are exactly ${\|B\|
\choose r_B} g(r_{G(E)})$ such sets $C'$. Our formula for
$f(k)$ is correct.
Now, using standard algebra (a process similar to Gauss
elimination), it is easy to verify that given values $f(1), f(2),
\ldots, f(n)$, it is possible to compute (in this order) $g(0),
g(1), \ldots, g(n)$. Together with the fact that constructing each
$I(k)$, $0 \leq k \leq n$, requires polynomial time with respect to
the size of $G$, this proves that given oracle access to
Maximin-\#CCDC, we can solve \#PerfectMatching. Thus, Maximin-\#CCDC
is $\sharpPclass$-Turing-complete and, by Theorem~\ref{thm:c-to-d}, so is
Maximin-\#DCDC.
\end{proof}
\section{Related Work}
\label{sec:related}
The focus of this paper is on the complexity of predicting election
winners for the case, where we are uncertain about the structure of
the election (the exact identities of candidates/voters that
participate), yet we have perfect knowledge of voters' preference
orders. However, our model is just one of many approaches to winner
prediction, which in various forms and shapes has been studied in the
literature for some years already. For example, to model imperfect
knowledge regarding voters' preferences, Konczak and
Lang~\shortcite{kon-lan:c:incomplete-prefs} introduced the possible
winner problem, further studied by many other researchers (see, e.g.,
\cite{con-xia:j:possible-necessary-winners,bet-dor:j:possible-winner-dichotomy,bac-bet-fal:c:counting-pos-win,che-lan-mau-mon:c:possible-winners-adding,lan-mon-xia:c:new-alternatives-new-results}). In
the possible winner problem, each voter is represented via a partial
preference order and we ask if there is an extension of these partial
orders to total orders that ensures a given candidate's victory.
Bachrach, Betzler, and
Faliszewski~\shortcite{bac-bet-fal:c:counting-pos-win} extended the
model by considering counting variants of possible winner problems.
Namely, they asked for how many extensions of the votes a given
candidate wins, in effect obtaining the probability of the candidate's
victory. This is very similar to our approach, but there are also
important differences. In the work of Bachrach et al., we have full
knowledge regarding the identities of candidates and voters
participating in the election, but we are uncertain about voters'
preference orders. In our setting, we have full knowledge about
voters' preference orders, but we are uncertain about the identities
of candidates/voters participating in the election.
Another model of predicting election outcomes is that of Hazon et
al.~\cite{haz-aum-kra-woo:j:uncertain-election-outcomes}. They
consider a situation where each voter is undecided regarding several
possible votes. That is, for each voter we are given several possible
preference orders and a probability distribution over these votes.
The question is, what is the probability that a designated candidate
wins.
From a technical standpoint, our research continues the line of work
on the complexity of control. This line of work was initiated by
Bartholdi, Tovey, and Trick~\cite{bar-tov-tri:j:control}, and then
continued by Hemaspaandra, Hemaspaandra, and
Rothe~\shortcite{hem-hem-rot:j:destructive-control} (who introduced
the destructive cases), by Meir et
al.~\shortcite{mei-pro-ros-zoh:j:multiwinner} (who considered
multiwinner rules and who generalized the idea of the constructive and
destructive cases), by Faliszewski, Hemaspaandra, and
Hemaspaandra~\shortcite{fal-hem-hem:j:multimode} (who introduced
multimode model of control), by Faliszewski, Hemaspaandra, and
Hemaspaandra~\cite{fal-hem-hem:c:weighted-control} (who were first to
consider control for weighted elections), by Rothe and
Schend~\cite{rot-sch:c:experimental-control} (who initiated the
empirical study of the complexity of control problems), and by many
other researchers, who provided results for specific voting rules and
who introduced various other novel means of studying control problems
(see, e.g., the following
papers~\cite{bet-uhl:j:parameterized-complecity-candidate-control,erd-now-rot:j:sp-av,erd-rot:c:fallback-voting,fen-liu-lua-zhu:j:parameterized-control,liu-zhu:j:maximin,men-sin:c:schultze-control,par-xia:strategic-schultze-ranked-pairs};
we also point the readers to the
survey~\cite{fal-hem-hem:j:cacm-survey}).
Single-peaked elections were studied for a long time in social choice
literature, but they gained popularity in the computational social
choice world fairly recently, mostly due to the papers of
Walsh~\cite{wal:c:uncertainty-in-preference-elicitation-aggregation}
and Conitzer~\cite{con:j:eliciting-singlepeaked}. Then, Faliszewski et
al.~\cite{fal-hem-hem-rot:j:single-peaked-preferences} and, later,
Brandt et al.~\cite{bra-bri-hem-hem:c:sp2} studied the complexity of
control problems for single-peaked elections. Recently, Faliszewski,
Hemaspaandra, and Hemaspaandra~\cite{fal-hem-hem:j:nearly-sp}
complemented this line of work by studying nearly single-peaked
profiles.
Going in a different direction, our work is very closely related to
the paper of Walsh and Xia~\shortcite{wal-xia:c:lot-based} on
lot-based elections. Walsh and Xia study a model of Venetian
elections, where a group of voters of a given size is randomly
selected from a group of eligible voters, and the votes are collected
from these selected voters only. In this setting, the problem of
computing a candidate's chances of victory, in essence, boils down to
the counting variant of control by adding voters problem. Thus, our
paper and that of Walsh and Xia are quite similar on the technical
front. The papers, however, have no overlap in terms of results.
\section{Conclusions and Future Work}\label{sec:conclusions}
We have considered a model of predicting election winners in
settings where there is uncertainty regarding the structure of the
election (that is, regarding the exact set of candidates and the exact
collection of voters participating in the election). We have shown
that our model corresponds to the counting variants of election
control problems (specifically, we have focused on election control by
adding/deleting candidates and voters). We have considered Plurality,
Approval, Condorcet, $k$-Approval, and Maximin (see
Table~\ref{tab:results} for our results). For the former three, the
complexity of counting variants of control is analogous to the
complexity of decision variants of respective problems, but for the
latter two, some of the counting control problems are more
computationally demanding than their decision counterparts.
Many of our results indicate computational hardness of winner
prediction problems. To alleviate this issue to some extent, we also
considered single-peaked preferences that are more likely to appear in
practice. In this case, we got polynomial-time results only (except
the case of Approval, where have no results for the single-peaked
case). Still, sometimes in practice one might have to seek heuristic
algorithms or approximate solutions (e.g., sampling-based algorithms
similar to the one of Bachrach, Betzler, and
Faliszewski~\cite[Theorem~6]{bac-bet-fal:c:counting-pos-win}).
There are many ways to extend our work. For example, in the
intruduction we mentioned the model where for each voter $v$ (or
candidate $c$) we have probability $p_v$ (probability $p_c$) that this
voter (candidate) participates in the election. We believe that
studying this problem in more detail would be very interesting. As we
have argued, the model where all values $p_v$ ($p_c$) are identical,
reduces to our setting, but the cases where the probabilities can
differ remain open.
\medskip
\noindent\textbf{Acknowledgements.}
An early version of this paper appeared in IJCAI Workshop on Social
Choice and Artificial Intelligence (2011) and an extended abstract was
presented at the Twenty-Sixth AAAI Conference on Artificial
Intelligence (AAAI-12). We thank both the AAAI and WSCAI reviewers
for very helpful feedback. Piotr Faliszewski was in part supported by
AGH University of Technology Grant 11.11.230.015 (statutory project),
by Foundation for Polish Science's program Homing/Powroty, and by
Poland's National Science Center's grant 2012/06/M/ST1/00358.
\label{sec:summary}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,459 |
Enjoy the delights of an eclectic Midwestern city.
Forget its reputation as a picturesque, free-spirited college town – Ann Arbor is a forever town. As enamored of its football team as it is of its arts, it's the kind of place that marries big city culture with the neighborly values of a small Midwestern town, with a hearty dash of intellectual energy and natural beauty thrown in. This progressive community with the collegiate atmosphere offers excellent museums, a diverse culinary scene, exciting nightlife and lots of arts events. You can kayak the Huron River, or enjoy the public art (it's everywhere, and don't miss the little fairy doors sprinkled throughout the city). It's a place you won't want to leave, even if your university days are far behind you.
Ann Arbor's venerable University of Michigan is more than just a pretty campus; it's the city's cultural and academic epicenter. Visit the Museum of Natural History for hands-on demos, the Living Lab and, for kids, the Dinosaur Tour. The school's Museum of Art offers a rich collection of works ranging from African to Western. Elsewhere on campus, explore student art galleries and the Detroit Observatory, or catch a musical performance or literary reading. Spend the day wandering through Matthaei Botanical Gardens. Take a tour of Michigan Stadium, nicknamed "The Big House," home to the University of Michigan Wolverines and full of college football character.
There are small-city downtowns, and then there's downtown Ann Arbor. Explore a thriving and diverse community full of independent shops, restaurants, galleries and markets. Soak up the local vibe at paperies, vintage boutiques and bookstores before grabbing organic pizza or top-notch coffee. Dining is exceptional here morning, noon and night, with cafes and kitchens serving up delicious breakfast fare and twice as many establishments offering ethnic cuisines for dinner, including Mongolian, Spanish, German and Asian options. Chase dinner with a stop at a tequila bar, or spring for local beer and popcorn. Don't miss beloved Zingerman's Deli, an Ann Arbor institution and famous throughout the USA. It's the place for artisanal cheese, bagels, hot sandwiches and sides.
The scene here creates opportunities for discovery around every corner. Head to the Ann Arbor Hands-On Museum for an interactive good time. Though it's a children's science museum, adults will be equally enthralled by the dynamic exhibits. Visit Cobblestone Farm and Museum, circa 1845, to learn about rural life in the 19th century. Like any good college town, Ann Arbor's music scene is full of options. Start at diminutive Kerrytown Concert House for some beguilingly great acts, or spend some time at The Ark, whose listening room boasts live music more than 300 nights a year. For an elevated performance, book tickets for the Ann Arbor Symphony Orchestra, or see what's on offer at Michigan Theater, a historic movie palace that hosts indie films, stage performances and music concerts.
The Ann Arbor Art Fair is one of the largest outdoor art fairs in the nation.
The University of Michigan's 'Big House' is the largest football stadium in the United States.
Ann Arbor has more bookstores per capita than any other city in the USA. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,860 |
Bomber jackets are one of the hottest trends of 2016. I recently styled a pink bomber jacket in my post, Sporty Chic, but I wanted to add this dusty plum one to my closet because the color is absolutely gorgeous. This two-toned bomber jacket from SheIn is such a beautiful and versatile piece. You can style it with a casual outfit for running errands or with a casual-chic look, which I have put together for this outfit post. The satin-like fabric towards the bottom of the jacket adds an elegant touch to the sporty style. I wore this jacket with my go-to ripped boyfriend jeans and a silky Brandy Melville cami. To add a bit of sparkle, I draped on a simple gold lariat necklace, popped in a gorgeous pair of sun and moon earrings, and slipped on a gold spiral bangle. Last but not least, I wore white mules to pull the whole look together. As always, I am sending you lots of love from my heart to yours! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,640 |
Q: Android - view Switcher I want to add a view switcher dynamically.
ViewSwitcher switcher = new ViewSwitcher(this);
How do I add my two layouts to the view switcher and switch between views using code?
Errors:
A1 [Android Application]
DalvikVM[localhost:8604] (may be out of synch)
Thread [<1> main] (Suspended (exception RuntimeException))
ActivityThread.performLaunchActivity(ActivityThread$ActivityClientRecord, Intent) line: 1815
ActivityThread.handleLaunchActivity(ActivityThread$ActivityClientRecord, Intent) line: 1831
ActivityThread.access$500(ActivityThread, ActivityThread$ActivityClientRecord, Intent) line: 122
ActivityThread$H.handleMessage(Message) line: 1024
ActivityThread$H(Handler).dispatchMessage(Message) line: 99
Looper.loop() line: 132
ActivityThread.main(String[]) line: 4123
Method.invokeNative(Object, Object[], Class, Class[], Class, int, boolean) line: not available [native method]
Method.invoke(Object, Object...) line: 491
ZygoteInit$MethodAndArgsCaller.run() line: 841
ZygoteInit.main(String[]) line: 599
NativeStart.main(String[]) line: not available [native method]
Thread [<8> Binder Thread #2] (Running) (may be out of synch)
Thread [<7> Binder Thread #1] (Running) (may be out of synch)
Thread [<9> SoundPool] (Running) (may be out of synch)
Thread [<10> SoundPoolThread] (Running) (may be out of synch)
A: http://developer.android.com/reference/android/widget/ViewSwitcher.html
Public Methods
void addView(View child, int index, ViewGroup.LayoutParams params)
Adds a child view with the specified layout parameters.
A: Refer the Android documentation of ViewSwitcher. You can use the addView() and showNext() methods for your needs.
Call addView twice with indexes 0 and 1.
addView(child, index,
new ViewGroup.LayoutParams(
LayoutParams.WRAP_CONTENT,LayoutParams.WRAP_CONTENT));
This should work without problems..
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,343 |
\section{Introduction and notations}\label{sec:intro}
The starting point in this paper is a multivariate vector function $\f\colon\ER^m\to\ER^n$, where $f_i$ ($1 \le i \le n$) is a polynomial in $m$ variables of degree at most~$d$. The variables of~\f\ will be denoted as $\u = (u_1,\ldots,u_m)$ and the values as $\f(\u) = \y = (y_1,\ldots,y_n) = \bigl(f_1(\u),\ldots,f_n(\u)\bigr)$. This function may contain cross terms of monomials, for example $c_1u_1u_2$ or $c_2u_3^2u_2^2$, where $c_1,c_2\in\ER$, in which case it is called \emph{coupled}.
The principal goal of this article is to find a \emph{decoupled representation} of~\f\ as illustrated in \fig{fig:dec}.
\begin{figure}[ht]
\def\w{2.5}
\def\ww{1.2}
\def\wi{0.5}
\def\fign{1}
\def\u{\mathbf{u}}
\def\x{\mathbf{x}}
\def\J{\mathbf{J}}
\begin{tikzpicture}[>=stealth, scale=0.8]
\filldraw[fill=gray!30, rounded corners = 4] (-\ww,-1.5) rectangle (\ww,1.5);
\draw[->, thick] (-\ww-1, 0.7) node[anchor=east] {$u_1$} -- (-\ww, 0.7);
\draw (-\ww-1, 0.0) node[anchor= east] {$\vdots$};
\draw[->, thick] (-\ww-1, -0.7) node[anchor=east] {$u_m$} -- (-\ww, -0.7);
\draw[->, thick] (\ww, 0.7) -- (\ww+1, 0.7) node[anchor=west] {$y_1$};
\draw (\ww+1, 0.0) node[anchor= west] {$\vdots$};
\draw[->, thick] (\ww, -0.7) -- (\ww+1, -0.7) node[anchor=west] {$y_n$};
\draw (0,0) node {$\mathbf{f}(\u)$};
\draw (\ww+2.5, 0) node {$\rightarrow$};
\begin{scope}[xshift=8.8cm]
\filldraw[fill=gray!30, rounded corners = 4] (-\w,-1.5) rectangle (\w,1.5);
\draw[->, thick] (-\w-1, 0.7) node[anchor=east] {$u_1$} -- (-\w+0.3, 0.7);
\draw (-\w-1, 0.0) node[anchor= east] {$\vdots$};
\draw[->, thick] (-\w-1, -0.7) node[anchor=east] {$u_m$} -- (-\w+0.3, -0.7);
\draw[->, thick] (\w-0.3, 0.7) -- (\w+1, 0.7) node[anchor=west] {$y_1$};
\draw (\w+1, 0.0) node[anchor= west] {$\vdots$};
\draw[->, thick] (\w-0.3, -0.7) -- (\w+1, -0.7) node[anchor=west] {$y_n$};
\draw[rounded corners = 4] (-\wi,0.5) rectangle (\wi, 1);
\draw[rounded corners = 4] (-\wi,-0.5) rectangle (\wi, -1);
\draw (0, 0.75) node {$\scriptscriptstyle g_1(x_1)$};
\draw (0, -0.75) node {$\scriptscriptstyle g_r(x_r)$};
\draw (-\w+0.3,-1) rectangle (-\w+1.2,1);
\draw (\w-1.2,-1) rectangle (\w-0.3,1);
\draw (-\w+0.75,0) node {$\mathbf{V}^T$};
\draw (\w-0.75,0) node {$\mathbf{W}$};
\draw[->, thick] (-\w+1.2, 0.7) -- node[above] {$\scriptstyle x_1$} (-\wi, 0.7);
\draw[->, opacity=0] (-\w+1.2, 0.2) -- node[opacity=1] {$\scriptstyle\vdots$} (-\wi, 0.2);
\draw[->, thick] (-\w+1.2, -0.7) -- node[below] {$\scriptstyle x_r$} (-\wi, -0.7);
\draw[<-, thick] (\w-1.2, 0.7) -- node[above] {$\scriptstyle z_1$} (\wi, 0.7);
\draw[->, opacity=0] (\w-1.2, 0.2) -- node[opacity=1] {$\scriptstyle\vdots$} (\wi, 0.2);
\draw[<-, thick] (\w-1.2, -0.7) -- node[below] {$\scriptstyle z_r$} (\wi, -0.7);
\end{scope}
\end{tikzpicture}
\caption{The decoupling process: given \f, find the matrices \V\ and \W\ and the univariate functions~$g_1, \ldots, g_r$.}
\label{fig:dec}
\end{figure}
Given $\f$, we wish to find transformation matrices $\V\in\ER^{m\times r}$ and~$\W\in\ER^{n\times r}$ and a vector~$\g(\x)=\bigl(g_1(x_1), \ldots, g_r(x_r)\bigr)$ of univariate polynomials, such that,
$$
\f(\u) \approx \W\g(\V^T\u),
$$
for all inputs $\u$. The first internal variable is denoted as $\x=\V^T\u$, and the second as $\z = \g(\V^T\u)$. Furthermore, the number~$r$ of internal \emph{branches} is assumed to be predefined. This decoupled representation offers a way to study~\f\ without cross terms, which can be an advantage for certain applications, as it helps its physical or intuitive understanding.
To our knowledge, \cite{Dreesen2015} and~\cite{MaartenSchoukens2014} offer a solution to this problem under the special assumption that an exact decomposition with~$r$ branches exists. Under the extra condition of homogeneous polynomials, this has also been studied in~\cite{Tiels2013}. Section~1.2 of \cite{Dreesen2015} also refers to the related Waring problem. In the case of state-space models, this problem is addressed in~\cite{Mulders2014}. Because the solution of~\cite{Dreesen2015} seems to be computationally easier, we have chosen to use and generalize this algorithm, which is based on the first-order derivative information of \f. At its core, tensor decompositions are used and the method is outlined in Algorithm~1. \fig{fig_decouplingGraphical} shows a graphical representation. An overview of tensor decompositions can be found in~\cite{Comon2002}, \cite{Comon2009a}, \cite{Kolda2009} and \cite{Lathauwer2000}.
\begin{figure}[ht]
\pgfmathsetmacro{\rr}{1.4
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\pgfmathsetmacro{\dnshift}{-3cm}
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\begin{scope}[xshift=1.5*\horshift]
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\begin{scope}[xshift=3*\horshift]
\begin{scope}
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\begin{scope}
\begin{scope}
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\end{scope}
\begin{scope}
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\begin{scope}
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\draw (-2*\wVector,-0.8*\vsep-0.8*\nn,0) node {$\mathbf{w}_1$};
\draw[draw=black,fill=black!15] (\wVector+\vsep,0,0) -- ++ (\mm,0,0) -- ++ (0,\wVector,0) -- ++ (-\mm,0,0) -- cycle;
\draw (\wVector+\vsep+0.8*\mm,-2*\wVector,0) node {$\mathbf{v}_1$};
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\draw (startcoord2) node {$\ldots$};
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\draw (startcoordr) node {$+$};
\coordinate (startcoordlast) at ($(startcoordr)+ (3*\sep,0,0)$);
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\draw ($(startcoordlast) + (-2*\wVector,-0.8*\vsep-0.8*\nn,0)$) node {$\mathbf{w}_r$};
\draw[draw=black,fill=black!15] ($(startcoordlast) + (\wVector+\vsep,0,0)$) -- ++ (\mm,0,0) -- ++ (0,\wVector,0) -- ++ (-\mm,0,0) -- cycle;
\draw ($(startcoordlast) + (\wVector+\vsep+0.8*\mm,-2*\wVector,0)$) node {$\mathbf{v}_r$};
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\draw ($(startcoordlast) + (-3*\wVector,\wVector,-0.9*\NN)$) node {$\mathbf{h}_r$};
\end{scope}
\end{scope}
\end{tikzpicture}}
\caption{Graphical representation of the core of Algorithm~1.}
\label{fig_decouplingGraphical}\end{figure}
\begin{algor}{Decomposing a multivariate polynomial \f\ having an exact decomposition. In this section, we shortly introduce the algorithm of \cite{Dreesen2015}.}
\begin{enumerate}
\item Evaluate the Jacobian matrix of \f
$$
\J(\u) = \begin{bmatrix}
\frac{\partial f_1}{\partial u_1}(\u) & \cdots & \frac{\partial f_1}{\partial u_m}(\u)\\
\vdots & \ddots & \vdots \\
\frac{\partial f_n}{\partial u_1}(\u) & \cdots & \frac{\partial f_n}{\partial u_m}(\u)
\end{bmatrix}
$$
in $N$ randomly chosen points $\u^{(1)},\ldots,\u^{(N)}$. The number $N$ of sampling points is chosen by the user. The equality $\f(\u) = \W\g(\V^T\u)$ implies for the Jacobians that
$$
\J(\u) = \W
\begin{bmatrix}
g_1'(\v_1^T\u) & & 0\\
& \ddots & \\
0 & & g_r'(\v_r^T\u)
\end{bmatrix}\V^T,
$$
where the matrix of derivatives is zero outside of the diagonal and $\v_1,\ldots,\v_r$ denote the columns of $\V$, and $g'_i$ denote the derivative of the $i$-th component of $\g$.
\item Stack the Jacobians into a three-way tensor $\Jcal$ of dimensions $n\times m\times N$. Here, the $k$-th frontal slice of \Jcal\ consists of the first-order information of \f\ evaluated in the sampling point $\u^{(k)}$, i.e., $\Jcal(\colon,\colon,k) = \J(\u^{(k)})$ (where $1\le k\le N$). Here, the \matlab-notation $\Jcal(\colon,\colon,k)$ is used for the $k$-th frontal slide of \Jcal.
\item Compute the Canonical Polyadic Decomposition (CPD) of
\begin{equation}
\Jcal = \sum_{i=1}^r \w_i\circ\v_i\circ\h_i.
\label{eq_cpd1}\end{equation}
Here, the vectors $\w_i$ (respectively $\v_i$) define the columns of the matrix~\W\ (respectively \V) of the decoupled representation. Furthermore, the vectors $\h_1,\ldots,\h_r$ contain the first-order information of the internal univariate functions $g_1(x_1),\ldots,g_r(x_r)$ evaluated in the $N$ sampling points, after transformation by $\V^T$, i.e., $\x^{(k)} = \V^T\u^{(k)}$. We thus have $h_{ij} = g'_j(\v^T_j\u^{(i)})$.
\item Starting from the vectors $\h_1,\ldots,\h_r$, reconstruct the internal univariate functions $g_1(x_1),\ldots,g_r(x_r)$. This works by fitting the derivatives $g'_j$ using the vectors $\h_1,\ldots,\h_r$, and then to recover the functions $g_j$ with an integration step. This method is described in Section~II.C of~\cite{Dreesen2015a}. Since we focus our attention to noisy coefficients of \f, it seems reasonable to use all the information available in the vectors $\h_1,\ldots,\h_r$, instead of the method proposed in Section~2.4 of~\cite{Dreesen2015}.
\end{enumerate}
\end{algor}
In~\cite{Dreesen2015}, it is shown that Algorithm~1 works well in case that the function~\f\ admits an exact decomposition. In this paper, we generalize the method to the noisy case: here, it is not assumed that an exact decoupling of \f\ exists, and instead, we will search for an approximated decoupling. In this regard, the coefficients of \f\ are thus considered ``noisy'': $\f = \f_0 + \v_\f$, where $\f_0$ has an exact decoupling and $\v_\f$ is zero-mean noise.
In order to decouple this noisy coupled function \f, the covariance matrix~$\mathbf{\Sigma}_\f = \cov(\v)$ of~\f\ will be assumed to be known throughout this paper. Because this matrix is easy to approximate when doing numerical experiments or measurements, this seems a reasonable assumption. The matrix~$\mathbf{\Sigma}_\f$ contains the variances of and covariances between the different coefficients. This will lead to the creation of a weight matrix to be used during the decoupling process. This weight matrix will be defined, as common in a weighted least squares approximation problem, as the (pseudo) inverse of a covariance matrix. In conclusion, returning to the outline of the decoupling method, the attention in this paper will be focused on step~3 of Algorithm~1: the CPD will be generalized to a \emph{weighted CPD}.
This paper is organized as follows: in Section~\ref{sec:weight}, the ``covariance'' matrix~$\mathbf{\Sigma}_\Jcal$ of the Jacobian elements will be constructed as a linear transformation of the matrix $\mathbf{\Sigma}_\f$. In Section~\ref{sec:wCPD}, the generalization of the CPD with weights will be discussed. Finally, Section~\ref{sec:results} summarizes the numerical experiments of the weighted CPD and shows an application of the results in the domain of system identification, while Section~\ref{sec:conclusion} contains the conclusion and ideas for future work.
\section{Constructing the covariance matrix~$\mathbf{\Sigma}_\Jcal$}\label{sec:weight}
In order to create a weighted CPD, the covariance matrix~$\mathbf{\Sigma}_\Jcal$ of the Jacobian elements will be constructed as a linear transformation of~$\mathbf{\Sigma}_\f$, the covariance matrix of the coefficients of \f, except the constant terms. Because \Jcal\ contains $mnN$ elements, $\mathbf{\Sigma}_\Jcal$ will have dimensions $(nmN)\times(nmN)$.
In practice, three different matrices (and hence, three different weight matrices) will be discussed in Section~\ref{sec:wCPD}: (1) only the variances of each Jacobian element will be taken into account while the other covariances will be set to~0, (2) the slice-wise covariances will be taken into account as well, keeping the rest as 0, and, (3) all the covariances of $\Jcal$ will be used, forming a dense covariance matrix. This way, even though the element-wise and slice-wise defined matrices are approximations of the full covariance matrix, and are not by themselves well-defined covariance matrices, we will still use this term to denote them. Furthermore, we will denote them respectively $\mathbf{\Sigma}_\Jcal^\textrm{e}$, $\mathbf{\Sigma}_\Jcal^\textrm{s}$ and $\mathbf{\Sigma}_\Jcal^\textrm{d}$, and will use $\mathbf{\Sigma}_\Jcal$ if we wish to denote any one of them.
For ease of reading, we will often illustrate dimensions and values for the special case where $m=n=d=2$. The number $\ell$ of monomials of degree at most $d$, given by $\binom{m+d}{m}$, is in this case 6, and the coupled function \f\ can be written as
$$
\f(u_1,u_2) =
\begin{bmatrix}
f_1(u_1,u_2) \\
f_2(u_1,u_2)
\end{bmatrix}
=
\begin{bmatrix}
c_1 + c_2 u_1 + c_3 u_2 + c_4 u_1^2 + c_5 u_1u_2 + c_6 u_2^2 \\
d_1 + d_2 u_1 + d_3 u_2 + d_4 u_1^2 + d_5 u_1u_2 + d_6 u_2^2
\end{bmatrix},
$$
where we use $c_i$ and $d_i$ ($1\le i\le6$) for the coefficients of the first and second output of~\f, respectively. The covariance matrix $\mathbf{\Sigma}_\f$ of these coefficients is assumed to be known, and this is a $10\times10$ matrix. It is defined as follows:
\renewcommand{\kbldelim}{[
\renewcommand{\kbrdelim}{]
$$
\mathbf{\Sigma}_\f = \kbordermatrix{
& c_2 & \cdots & c_6 & d_2 & \cdots & d_6 \\
c_2 & \var(c_2) & \cdots & \cov(c_2,c_6) & \cov(c_2,d_2) & \cdots & \cov(c_2,d_6)\\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
c_6 & \cov(c_6,c_2) & \cdots & \var(c_6) & \cov(c_6,d_2) & \cdots & \cov(c_6,d_6)\\
d_2 & \cov(d_2,c_2) & \cdots & \cov(d_2,c_6) & \var(d_2) & \cdots & \cov(d_2,d_6)\\
\vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
d_6 & \cov(d_6,c_2) & \cdots & \cov(d_6,c_6) & \cov(d_6,d_2) & \cdots & \var(d_6) \\
}.
$$
Because the first-order information of \f\ will be used, the information about the constant terms is not included in the matrix $\mathbf{\Sigma}_\f$. In general, the dimensions of $\mathbf{\Sigma}_\f$ are $\bigl((\ell-1) n\bigr)\times\bigl((\ell-1) n\bigr)$.
\subsection{Constructing the element-wise covariance matrix $\mathbf{\Sigma}_\Jcal^\textrm{e}$}
Using the notations of the previous section, it is possible to write the first Jacobian element evaluated at a point $\u^{(k)}$ ($1\le k\le N$) as follows:
$$
\J_{11}(\u^{(k)}) = \frac{\partial f_1}{\partial u_1}(\u^{(k)})
=
\begin{bmatrix}
1\; 0\; 2u_1^{(k)}\; u_2^{(k)}\; 0\
\end{bmatrix}
\begin{bmatrix}
c_2 \\
\vdots \\
c_{6}
\end{bmatrix}.
$$
Derivatives of polynomials do not depend of the constant term $c_1$. It then follows for the variance of the element $\J_{11}(\u^{(k)})$ of \Jcal\ that
\begin{equation}
\var\bigl(\J_{11}(\u^{(k)})\bigr) = \begin{bmatrix}
1\; 0\; 2u_1^{(k)}\; u_2^{(k)}\; 0\;
\end{bmatrix} \mathbf{\Sigma}_{c}
\begin{bmatrix}
1 \\
0 \\
2u_1^{(k)} \\
u_2^{(k)} \\
0
\end{bmatrix},
\label{eq_elementwise}\end{equation}
where
$$
\mathbf{\Sigma}_{c} =
\begin{bmatrix}
\var(c_2) & \cdots & \cov(c_2,c_6) \\
\vdots & \ddots & \vdots \\
\cov(c_6,c_2) & \cdots & \var(c_6) & \\
\end{bmatrix}
$$
is the portion of $\mathbf{\Sigma}_\f$ consisting of the variances of and covariances between the coefficients~$c_2, \ldots, c_6$. When repeating this in a similar way for the other Jacobian elements, one finds all the variances of the elements of \Jcal, which, once collected into a matrix, give the element-wise matrix $\mathbf{\Sigma}_\Jcal^\textrm{e}$:
\begin{equation}
\mathbf{\Sigma}_\Jcal^\textrm{e} = \begin{bmatrix}
\var\J_{11}^{(1)} \\
& \var\J_{21}^{(1)} \\
&& \var\J_{12}^{(1)} \\
&&& \var\J_{22}^{(1)} \\
&&&& \ddots \\
&&&&&\var\J_{11}^{(N)} \\
&&&&&& \var\J_{21}^{(N)} \\
&&&&&&& \var\J_{12}^{(N)} \\
&&&&&&&& \var\J_{22}^{(N)} \\
\end{bmatrix}.
\label{eq_elementwiseCovar}\end{equation}
Here, $\J_{11}^{(k)}$ is used as shorthand notation for $\J_{11}(\u^{(k)})$, and all empty spaces in the matrix denote zero. Hence, the covariance matrix obtained when considering solely at the variances of Jacobian elements is diagonal. Implementing the weighted CPD decomposition using an element-wise weight has been described in~\cite{Andersson2000} or in \cite{Bader2015} (in the case of incomplete data). In the following two sections, this will be generalized to covariance matrices with more cross-covariances taken into account.
\subsection{Constructing the slice-wise covariance matrix $\mathbf{\Sigma}_\Jcal^\textrm{s}$}
As a next step, the covariances between Jacobian elements evaluated at a single sampling point $\u^{(k)}$ are taken into account for the construction of the matrix $\covarM$, but the covariances over several $\u^{(k)}$ are neglected. This is done by noting that the Jacobian elements of sampling point~$\u^{(k)}$ can be written as linear combinations of the monomials of degree at least one:
\setlength{\arrayrulewidth}{1pt}
$$
\begin{bmatrix}
\J_{11}(\u^{(k)})\\ \J_{21}(\u^{(k)}) \\
\J_{12}(\u^{(k)}) \\ \J_{22}(\u^{(k)})
\end{bmatrix} =
\left[\begin{array}{ccccc| ccccc}
1 & 0 & 2u_1^{(k)} & u_2^{(k)} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 2u_1^{(k)} & u_2^{(k)} & 0 \\
0 & 1 & 0 & u_1^{(k)} & 2u_2^{(k)} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & u_1^{(k)} & 2u_2^{(k)} \\
\end{array}\right]
\left[\begin{array}{@{}c@{}}
c_2 \\
\vdots \\
c_6 \\
\hline
d_2 \\
\vdots \\
d_6
\end{array}\right].
$$
This can be denoted shortly as
\begin{equation}
\vec{\J(\u^{(k)})} = \A(\u^{(k)}) \left[\begin{array}{@{}c@{}}
c_2 \\
\vdots \\
c_6 \\
\hline
d_2 \\
\vdots \\
d_6
\end{array}\right],
\label{eq:defAmatrix}
\end{equation}
where, the matrix $\A\in\ER^{4\times10}$ (in general, $\A\in\ER^{(m n)\times((\ell-1) n)}$) contains the linear relationships between coefficients of the monomials and Jacobian elements. In the case that $d>2$, the matrix $\A$ may of course contain higher powers of the $u_i^{(k)}$. It follows that the covariance matrix of the Jacobian elements of sampling point~$\u^{(k)}$ is given by $\A(\u^{(k)}) \mathbf{\Sigma}_\f \A(\u^{(k)})^T \in \ER^{(mn)\times(mn)}$. When we repeat this for all the sampling points, we obtain the slice-wise covariance matrix~$\mathbf{\Sigma}_\Jcal^\textrm{s}$
\begin{equation}
\mathbf{\Sigma}_\Jcal^\textrm{s} = \begin{bmatrix}
\A(\u^{(1)}) \mathbf{\Sigma}_\f \A(\u^{(1)})^T \\
& \A(\u^{(2)}) \mathbf{\Sigma}_\f \A(\u^{(2)})^T \\
&& \ddots \\
&&& \A(\u^{(N)}) \mathbf{\Sigma}_\f \A(\u^{(N)})^T
\end{bmatrix},
\label{eq_slicewise}\end{equation}
which is a block-diagonal matrix. Once again, the empty spaces in this matrix denote zero.
\subsection{Constructing the dense covariance matrix $\mathbf{\Sigma}_\Jcal^\textrm{d}$}
Finally, we generalize the slice-wise covariance matrix to the dense case, considering the (co)variances in all the elements of \Jcal. With the same definition for the matrix \A\ as in \eqref{eq:defAmatrix}, the dense covariance matrix~$\mathbf{\Sigma}_\Jcal^\textrm{d}$ is defined as
\begin{equation}
\mathbf{\Sigma}_\Jcal^\textrm{d} = \begin{bmatrix}
\A(\u^{(1)}) \\
\vdots \\
\A(\u^{(N)})
\end{bmatrix} \mathbf{\Sigma}_\f\begin{bmatrix} \A(\u^{(1)})^T \cdots \A(\u^{(N)})^T \end{bmatrix}.
\label{eq_densecovar}\end{equation}
It follows immediately from this definition that, since~$\mathbf{\Sigma}_\Jcal^\textrm{d}\in\ER^{(mnN)\times(mnN)}$ and $\mathbf{\Sigma}_\f\in\ER^{((\ell-1)n)\times((\ell-1)n)}$, $\mathbf{\Sigma}_\Jcal^\textrm{d}$ is rank-deficient whenever $N>\frac{\ell-1}{m}$: its rank is then bounded by $(\ell-1)n$. This will lead to two different weighted CPD decompositions. On the one hand, decompositions using element-wise or slice-wise weights will be discussed in Section~\ref{sec:permutation} because the covariance matrix (and hence the weight matrix) has full rank; on the other hand, the dense-weight decompositions with a rank-deficient weight matrix will be discussed in Section~\ref{sec:f}.
\section{Computing the weighted CPD}\label{sec:wCPD}
Given the $n\times m\times N$-tensor \Jcal\ of Jacobian elements, it is possible to (approximately) write it as a sum of~$r$ rank-one tensors
\begin{equation}
\Jcal \approx \sum_{i=1}^r \w_i\circ\v_i\circ\h_i.
\label{eq:sumRank1}\end{equation}
Here, the notation $\circ$ is used to denote the so-called outer product and it is defined as follows. Given vectors $\a = (a_1, \ldots, a_{n_a})$, $\b = (b_1, \ldots, b_{n_b})$ and $\c = (c_1, \ldots, c_{n_c})$, the outer product $\a\circ\b\circ\c$ is the $n_a\times n_b\times n_c$ tensor whose element in position $(i,j,k)$ ($1\le i\le n_a$, $1\le j\le n_b$, $1\le k\le n_c$) is given by
$$
(\a\circ\b\circ\c)_{i,j,k} = a_i b_j c_k.
$$
The outer product $\a\circ\b\circ\c$ is said to have \emph{rank one}. In expression~(\ref{eq:sumRank1}), the tensor~\Jcal\ is said to be approximated by a sum of rank~one tensors. We will use the same notation as in~\cite{Kolda2009} and will denote this sum as $\cpdgen\W\V\H$.
Finding the matrices $\V$, $\W$ and $\H$ leads to optimizing the following non-linear cost function
$$
\begin{aligned}
&\min_{\V,\W,\H} \normf{\Jcal - \cpdgen\W\V\H}^2 \\
=&
\min_{\V,\W,\H} \bigl(\vec{\Jcal} - \vec{\cpdgen\W\V\H}\bigr)^T \bigl(\vec{\Jcal} - \vec{\cpdgen\W\V\H}\bigr),
\end{aligned}
$$
where $\vec\X$ denotes the vectorization of the matrix or tensor \X\ to a column vector. While this optimization problem is used for computing the unweighted CPD decomposition, the weighted CPD can be computed using a \emph{weighted norm} $\norm{\cdot}_{\weight}$ in the cost function, which includes a weight matrix $\weight$ as follows,
\begin{equation}
\begin{aligned}
&\min_{\V,\W,\H} \norm{\vec{\Jcal} - \vec{\cpdgen\W\V\H}}^2_{\weight} \\
=& \min_{\V,\W,\H} \left(\vec{\Jcal} - \vec{\cpdgen\W\V\H}\right)^{\!\!T} \weight \left(\vec{\Jcal} - \vec{\cpdgen\W\V\H}\right).
\end{aligned}
\label{eq:costfunction}\end{equation}
Optimizing this expression is described in the next sections, once for the element-wise and slice-wise weight, and once for the dense weight matrix. Naively, the weight matrix~$\weight$ can be intuitively seen as the (pseudo-) inverse of the covariance matrix~$\mathbf{\Sigma}_\Jcal$. This will be detailed in the following sections.
\label{sec:d_bd}
\subsection{Using element-wise and slice-wise weights}\label{sec:permutation}
In order to optimize the nonlinear cost function~(\ref{eq:costfunction}), we will use the workhorse method for computing the (unweighted) CPD: the Alternating Least Squares method described in~\cite{Carroll1970}, \cite{Harshman1970} and~\cite{Kolda2009}. This works by optimizing only one of the factors $\V$, $\W$ and $\H$ at a time, while keeping the two others fixed. Then, by alternating the optimized factor iteratively, a solution of the optimization problem can be found. Optimizing one factor (in the unweighted case) leads to three different least squares optimizations:
\begin{equation}
\min_{\W} \normf{\Jcal^T_{(1)} - (\H\odot\V) \W^T}^2,
\label{eq_als1}\end{equation}
$$
\min_{\V} \normf{\Jcal^T_{(2)} - (\H\odot\W) \V^T}^2,
$$
and
$$
\min_{\H} \normf{\Jcal^T_{(3)} - (\V\odot\W) \H^T}^2.
$$
Here, $\odot$ denotes the Khatri-Rao product (see, among others, \cite{Liu2008}), and $\Jcal_{(1)}, \Jcal_{(2)}, \Jcal_{(3)}$ denote the matricizations of the tensor \Jcal\ to the first, second and third mode, respectively. For this, we use the matricization order described in~\cite{Kolda2009}.
In order to take the weight matrix~$\weight$ into account, these (unweighted) least squares problems will be generalized to a set of weighted least squares optimizations, leading to a Weighted Alternating Least Squares method for finding the weighted CPD. Because this is done in a similar way for the three factors, more attention will be given to the update of $\W$ and only the differences with the two other factor updates will be emphasized.
\subsubsection{Updating the matrix \W}\label{sec:updateMatrixW}
The optimization problem in~(\ref{eq_als1}) is a least-squares problem with multiple right-hand sides. It can be rewritten as follows as a problem with a single right-hand side:
\begin{align}
& \min_{\W} \normf{\Jcal^T_{(1)} - (\H\odot\V) \W^T}^2 \nonumber \\
=& \min_{\W} \norm{\vec{\Jcal^T_{(1)}} - \vec{(\H\odot\V) \W^T}}^2 \nonumber \\
=& \min_{\W} \norm{\vec{\Jcal^T_{(1)}} - \bigl(\I_n \otimes (\H\odot\V)\bigr)\vec{\W^T}}^2 \label{eq_lastMinEq}
\end{align}
Here, $\I_n \in\ER^{n\times n}$ is the identity matrix and we denote the Kronecker product as $\otimes$. If we denote $\B_1 = \I_n \otimes (\H\odot\V)$, so
$$
\B_1 =
\underbrace{\begin{bmatrix}
\H\odot \V & \\
&\ddots\\
&&\H\odot \V
\end{bmatrix}}_{\textrm{$n$ repetitions}}
$$
then the minimization problem in~(\ref{eq_lastMinEq}) can be rewritten as
\begin{equation}
\min_{\W} \norm{\vec{\Jcal_{(1)}^T} - \B_1 \vec{\W^T}}^2.
\label{eq:1rightHandSide}\end{equation}
On this level, the permuted weight $\weight_{(1)}$ is included as an extra factor in the cost function, namely
\begin{align}
&\min_{\W} \norm{\vec{\Jcal_{(1)}^T} - \B_1 \vec{\W^T}}^2_{\weightone} \label{eq_weightedapprox}\\
=& \min_{\W} \bigl(\vec{\Jcal_{(1)}^T} - \B_1 \vec{\W^T} \bigr)^T \weightone \bigl(\vec{\Jcal_{(1)}^T} - \B_1 \vec{\W^T} \bigr). \nonumber
\end{align}
The weighted least squares solution is then given by
\begin{equation}
\vec{\W^T} =
\left(\B_1^T \weightone \B_1\right)^{\!-1} \B_1^T \weightone\vec{\Jcal_{(1)}^T}.
\label{eqSolutionALS}\end{equation}
Here, care should be taken as the weight matrix \weight\ should be permuted to $\weightone$ using the same permutation that permutes $\vec{\Jcal}$ to
$\vec{\Jcal_{(1)}^T}$. If we denote the permutation matrix as $\P_1$, then we have
$$
\P_1 \vec{\Jcal} = \vec{\Jcal_{(1)}^T}
$$
and
$$
\weightone = \P_1 \weight\P_1^T.
$$
Using \matlab, the permutation matrix $\P_1$ is found by the following instructions:
\begin{verbatim}
T = reshape(1:m*n*N, [n,m,N]); I = eye(m*n*N);
P_1 = I(vec(reshape(T, [n, m*N])'), :);
\end{verbatim}
Finally, reshaping the solution vector of \eqref{eqSolutionALS} to the dimensions of $\W^T$ gives after transposition the updated matrix $\W$.
\subsubsection{Updating the matrices \V\ and \H}
When updating the matrix \V, the permutations from $\vec\Jcal$ to $\vec{\Jcal_{(2)}^T}$ and $\vec{\Jcal_{(3)}^T}$ should also be updated. This is done similarly as in Section~\ref{sec:updateMatrixW}. \fig{fig_coloredGrids} shows a graphical representation of the permuted matrices $\weightone$, $\weighttwo$ and $\weightthree$.
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\caption{A graphical representation of how the permutations of $\weight$ work, in the case where $m=n=N=2$. These permutations are used due to the vectorizations of several tensors in Section~\ref{sec:updateMatrixW} and beyond. Under the hood, these are defined in the same fashion as \matlab's {\tt reshape} function. We note that the permutation matrix $\P_3$ is, in fact, the identity matrix.}
\label{fig_coloredGrids}
\end{figure}
\subsection{Using dense weights}\label{sec:f} As mentioned earlier, the dense covariance matrix~$\mathbf{\Sigma}_\Jcal^\textrm{d}$ is rank-deficient and does not have an inverse, which could be used as weight matrix. Although the pseudo-inverse of~$\mathbf{\Sigma}_\Jcal$ could be computed, it does not yield enough equations in order to solve the problem~(\ref{eq:costfunction}) uniquely: the system~(\ref{eq:costfunction}) is in fact undetermined. That is why the following technique will be used in order to incorporate the weight matrix into the CPD decomposition. It consists of two parts: Section~\ref{sec:fWeight_part1} describes the first set of equations, and Section~\ref{sec:fWeight_part2} contains the second set of equations.
\subsubsection{Using the singular values of $\mathbf{\Sigma}_\Jcal$}\label{sec:fWeight_part1}
Let $\rSigma = \rank(\mathbf{\Sigma}_\Jcal)$ denote the rank of the covariance matrix, then we can decompose the matrix $\mathbf{\Sigma}_\Jcal$ using the singular value decomposition (SVD), and obtain
\setlength{\arrayrulewidth}{0.7pt}
$$
\rule[-15pt]{0pt}{35pt}
\mathbf{\Sigma}_\Jcal = \Vs\, \Ds\, \Vs^T =
\Bigl[
\begin{array}{c|c}
\smash{\overbrace{\Vs^{(1)}}^{mnN\times \rSigma}}
&
\smash{\underbrace{\Vs^{(2)}}_{\raisebox{-2pt}{\makebox[20pt]{$\scriptstyle mnN\times(mnN-\rSigma)$}}}}
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\Bigl[\begin{array}{@{}c|c@{}}
\smash{\Vs^{(1)}}
&
\smash{\Vs^{(2)} \Bigr]^T}
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=
\Vs^{(1)}\, \Ds^{(1)}\, (\Vs^{(1)})^T.
$$
Here, $\Vs$ is an orthogonal matrix and~$\Ds$ is a diagonal matrix containing the singular values of $\mathbf{\Sigma}_\Jcal$. We call $\Vs^{(1)}$ the submatrix of $\Vs$ containing the first $\rSigma = \rank(\mathbf{\Sigma}_\J)$ columns of $\Vs$, and $\Ds^{(1)}$ the submatrix of $\Ds$ containing the non-zero singular values.
As in Section~\ref{sec:updateMatrixW}, every factor during the Alternating Least Squares algorithm uses a permuted version of $\mathbf{\Sigma}_\Jcal$. For example, at the update of the factor $\W$, we define
\begin{equation}
\mathbf{\Sigma}_{(1)} = \P_1 \mathbf{\Sigma}_\Jcal \P_1^T,
\label{eq_permCare}\end{equation}
where $\P_1$ is the permutation matrix, as defined in Section~\ref{sec:permutation}. It follows that the SVD of $\mathbf{\Sigma}_{(1)}$ is given by
$$
\mathbf{\Sigma}_{(1)} = \V_{\mathbf{\Sigma}_{(1)}} \D_{\mathbf{\Sigma}_{(1)}} \V_{\mathbf{\Sigma}_{(1)}}^T = (\P_1 \Vs^{(1)}) \,\Ds^{(1)} \, \bigl((\Vs^{(1)})^T\P_1^T\bigr)
$$
Let $\Q_1$ denote the $\rSigma\times mnN$-matrix
$$
\Q_1 = \sqrt{(\Ds^1)^{-1}} (\Vs^{(1)})^T \P_1^T,
$$
such that $\Q_1^T \Q_1 = \weightone$. Then the solution of the minimization problem~(\ref{eq_weightedapprox}) is given by
\begin{equation}
(\Q_1 \B_1)^{\!\dagger} (\Q_1 \vec{\Jcal_{(1)}}^T),
\label{eq_lowRankALS}\end{equation}
where $\B_1$ is the block-matrix of Khatri-Rao products and $\X^\dagger$ denotes the pseudo-inverse of \X. In Appendix~\ref{sec:app1}, a derivation for solution~(\ref{eq_lowRankALS}) is shown. Similar expressions are found for the other factors being updated. The following table shows how the matrix~$\B_i$ changes, depending on the updated factor:
\medskip
\centerline
\begin{tabular}{p{0.5cm} p{1cm} c p{2cm} p{2cm}}
$i$ & Updating factor & Matrix $\B_i$ & Dimensions of $\B_i$ & Dimensions of $\Q_i \B_i$\\
\hline
\rule{0pt}{7ex}1 & \W &
$\begin{bmatrix}
\H\odot \V & \\
&\ddots\\
&&\H\odot \V
\end{bmatrix}$
& $mnN\times rn$
& $\rSigma \times rn$
\\[2em]\hline
\rule{0pt}{7ex}2 & \V &
$\begin{bmatrix}
\H\odot \W & \\
&\ddots\\
&&\H\odot \W
\end{bmatrix}$
& $mnN \times rm$
& $\rSigma \times rm$
\\[2em] \hline
\rule{0pt}{7ex}3 & \H &
$\begin{bmatrix}
\V\odot \W & \\
&\ddots\\
&&\V\odot \W
\end{bmatrix}$
& $mnN\times rN$
& $\rSigma \times rN$
\end{tabular}}
\medskip
\noindent As we can see from this table, the dimensions of the systems being solved in (\ref{eq_lowRankALS}) depend on the factor being updated. We note that if the number~$N$ of operating points is too large, then the system with coefficient matrix $\Q_3\B_3$ is underdetermined, whereas the two other systems are not, as long as $\rSigma > \max\{rn,rm\}$. To remedy this, the~$\Vs^{(2)}$-part of the $\Vs$-matrix will be used to add extra constraints to the current systems of equations. These extra conditions will also be used for updating the matrices~\W\ and~\V.
\subsubsection{Adding more equations to the existing set}\label{sec:fWeight_part2}
If the covariance matrix~$\mathbf{\Sigma}_\Jcal$ of the noise $\v$ is not of full rank, there exist linear relations $\L$ between the noise disturbances $\v$, such that $\L \v = 0$. In Appendix~\ref{sec:app2}, it is shown how these can be retrieved from $\mathbf{\Sigma}_\Jcal$.
In order to find the extra equations for the update of the current factor, we reconsider the minimization problem~(\ref{eq_weightedapprox}) for the updating factor $\W$:
$$
\min_{\textrm{vec}({\W^T})} \norm{\vec{\Jcal_{(1)}^T} - \B_1 \vec{\W^T}}^2_{\weightone}
$$
Rewriting this expression as $\vec{\Jcal_{(1)}^T} = \B_1 \vec{\W^T} + \v$, where $\v$ is correlated noise with $\mathbf{\Sigma}_{(1)}$ as in~\eqref{eq_permCare}, Appendix~\ref{sec:app2} can be used in order to add extra equations to the existing set in~(\ref{eq_lowRankALS}). We remark that the noise $\v$ is correlated, due to the linear relations between the covariance matrices $\mathbf{\Sigma}_f$ and $\mathbf{\Sigma}_\Jcal$.
Using the notations in Equation~(\ref{eq_permCare}), the orthogonal factor of the singular value decomposition of $\mathbf{\Sigma}_{(1)}$ is given by
$$
\P_1 \Vs = \Bigl[
\rule[-20pt]{0pt}{30pt}
\begin{array}{@{}c|c@{}}
\smash{\overbrace{\P_1\Vs^{(1)}}^{\makebox[0pt]{$\scriptstyle mnN\times \rSigma$}}}
&
\smash{\underbrace{\P_1\Vs^{(2)}}_{\makebox[0pt]{$\scriptstyle mnN\times(mnN-\rSigma$)}}}
\end{array}
\Bigr].
$$
In Appendix~\ref{sec:app2}, it is proven that the extra conditions are then given by
\begin{equation}
\big( (\Vs^{(2)})^T \P_1^T \B_1 \big)^{\!\dagger}\, \bigl((\Vs^{(2)})^T \P_1^T \vec{\Jcal_{(1)}^T} \bigr).
\label{eq_resultEquation2}\end{equation}
Similar expression can be found for the other two updating factors.
In general, we conclude that the following system of equations must be solved for every updated factor $1\le i\le 3$:
\begin{equation}
\begin{bmatrix}
\Q_i \B_i \\
(\U_\Jcal^2)^T \P_i^T \B_i
\end{bmatrix}
^{\!\dagger}
\begin{bmatrix}
\Q_i \vec{\Jcal_{(i)}}^T \\
(\U_\Jcal^2)^T \P_i^T \vec{\Jcal_{(i)}^T}
\end{bmatrix}.
\label{eq_totalEq}\end{equation}
\subsection{Stopping criteria for the \CPD\ algorithm}\label{sec_stopcriteria}
By iterating the presented algorithm, a sequence of updated factors is obtained
$$
\W^{(1)}, \V^{(1)}, \H^{(1)}, \W^{(2)}, \V^{(2)}, \H^{(2)}, \W^{(3)}, \V^{(3)}, \H^{(3)}, \ldots
$$
We have the following two stopping criteria for this iteration algorithm:
\begin{enumerate}
\item When the relative step size between two iterations is below a given tolerance, then the algorithm is stopped. The relative step size at iteration step $j\ge 2$ is given as
$$
\frac{\sqrt{ \normf{\W^{(j)} - \W^{(j-1)}}^2 + \normf{\V^{(j)} - \V^{(j-1)}}^2 + \normf{\H^{(j)} - \H^{(j-1)}}^2}}
{\sqrt{ \normf{\W^{(j)}}^2 + \normf{\V^{(j)}}^2 + \normf{\H^{(j)}}^2 }}\textrm{, or}
$$
\item when an upper bound on the number of iterations is reached, then the algorithm is stopped. This takes care of possible divergence.
\end{enumerate}
\subsection{Summary of the proposed algorithm}
In this section, we summarize the proposed weighted \CPD\ algorithm and incorporate it into the larger decoupling problem of the noisy multivariate polynomial function~\f. We assume the covariance matrix~$\mathbf{\Sigma}_\f$ of the coefficients of~\f\ to be known and the same notations as in Section~\ref{sec:intro} will be used.
\begin{algor}{Decomposition of the noisy multivariate polynomial \f, given $\mathbf{\Sigma}_\f$.}
\begin{enumerate}
\item Evaluate the Jacobian matrix $\J(\u)$ of \f\ in $N$ randomly chosen sampling points $\u^{(1)},\ldots,\u^{(N)}$.
\item Stack the Jacobians into a three-way tensor $\Jcal$.
\item Transform the covariance matrix $\mathbf{\Sigma}_\f$ of \f\ into the matrix~$\mathbf{\Sigma}_\Jcal$. Here, three choices are possible: the element-wise as discussed in~(\ref{eq_elementwiseCovar}), slice-wise as discussed in~(\ref{eq_slicewise}) or dense covariance matrix discussed in~(\ref{eq_densecovar}).
\item Compute the Weighted Canonical Polyadic Decomposition (WCPD) of $\Jcal \approx \sum_{i=1}^r \w_i\circ\v_i\circ\h_i$. In the element- and slice-wise case, use~(\ref{eqSolutionALS}), in the dense case, use~(\ref{eq_totalEq}). Iterate until one of the stopping criteria in Section~\ref{sec_stopcriteria} is satisfied.
\item Starting from the vectors $\h_1,\ldots,\h_r$, reconstruct the internal univariate functions $g_1(x_1),\ldots,g_r(x_r)$ with an integration step, as in Algorithm~1.
\end{enumerate}
\end{algor}
\section{Numerical experiments}\label{sec:results}
In this section, results of the methods suggested in this paper will be discussed.
\subsection{Correlations of the errors of the CPD}
In this section, we demonstrate that the weighted CPD decomposition works as expected. For this, we set $m = n = N = 2$ and start with a given $8\times8$ positive definite weight matrix
\begin{equation}
\weight = \begin{bmatrix}
0.74 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1.67 & 0 & 0 & \color{blue}\underline{0.87} & 0 & 0 & 0 \\
0 & 0 & 0.96 & 0 & 0 & 0 & 0 & \color{red}\underline{\underline{0}} \\
0 & 0 & 0 & 0.63 & 0 & 0 & 0 & 0 \\
0 & \color{blue}\underline{0.87} & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0.11 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0.77 & 0 \\
0 & 0 & \color{red}\underline{\underline{0}} & 0 & 0 & 0 & 0 & 0.31 \\
\end{bmatrix}.
\label{eq_wDefinition}\end{equation}
This matrix is chosen to be almost diagonal, and has two extra nonzero covariance elements. The tensor \Tcal\ to be decomposed has eight elements indexed as follows:
\begin{equation}
\Tcal_1 = \begin{bmatrix}
t_1 & t_3 \\
t_2 & t_4
\end{bmatrix} \quadtext{and} \Tcal_2 = \begin{bmatrix}
t_5 & t_7 \\
t_6 & t_8
\end{bmatrix},
\label{eq_tensorNumbering}\end{equation}
where $\Tcal_1$ and $\Tcal_2$ denote the first and second frontal slices of \Tcal\ respectively. Elements $t_2$ and $t_5$ are correlated with cross-covariance 0.87, which is shown in {\color{blue}\underline{blue}} in Equation~\ref{eq_wDefinition}, while elements $t_3$ and~$t_8$ are uncorrelated, and are shown in {\color{red}\underline{\underline{red}}}.
Starting from a uniform random $2\times2\times2$ tensor~$\Tcal$, it is decomposed using the weight matrix~\weight\ and Step~4 of Algorithm~2. This sum of rank-one tensors is denoted~$\hat{\Tcal}$. In the left plot of~\fig{fig_result1}, we plot the differences $t_5 - \hat{t}_5$ on the vertical axis, and the differences $t_2 - \hat{t}_2$ on the horizontal axis. When repeating this experiment~500 times with random tensors~$\Tcal$ and uniform random initialization points, we see that the errors are correlated, as expected.
In the right plot of~\fig{fig_result1}, the differences between $\Tcal$ and $\hat{\Tcal}$ are shown, but between the uncorrelated elements $t_3$ and~$t_8$. We see that here, the scatter plot shows uncorrelated errors between the corresponding elements of the tensor and its decomposition.
\begin{figure}[htpb]
\centerline{
\begin{tikzpicture}
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xlabel={Errors of element 2},
ylabel={Errors of element 5},
xmin = -0.7,
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ymax = 0.7,
xtick = {-0.5, 0, 0.5},
ytick = {-0.5, 0, 0.5},
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(-0.1302417800, -0.0855734840)
(0.0208864070, -0.5037741500)
(-0.2179320700, -0.0449687520)
(-0.2521392800, -0.0090909265)
(-0.2179223000, -0.0603140340)
(0.2556876400, -0.3053023000)
(-0.0746711960, 0.0631463900)
(-0.1050084900, 0.2260904800)
(0.0865260080, 0.4723587200)
(0.0406127000, 0.0913057270)
(-0.0289110610, -0.3058410100)
(0.0766304880, 0.2845338800)
(-0.4341578200, -0.3421585100)
(0.0359539000, 0.1912870100)
(-0.0150281800, -0.1599651900)
(-0.0490022530, 0.0216989740)
(0.1300506300, -0.0303066370)
(0.3120275300, 0.2469270600)
(-0.0922922400, -0.3946525800)
(0.0354435780, 0.0108225440)
(-0.3465963800, 0.0066520176)
(-0.0223007890, 0.0792623710)
(0.0584639220, 0.0137504360)
(0.1238071000, -0.1545522100)
(-0.1303809300, -0.1397448600)
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(-0.0439240850, 0.4257939700)
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(0.0070266145, 0.0396909580)
(-0.2689388700, -0.4834490100)
(-0.0550347770, -0.0581880820)
(-0.2249139200, -0.2299305100)
(-0.1809029400, -0.2553419400)
(0.1969358600, 0.2717651400)
(-0.0221619220, 0.4157168300)
(-0.0889803170, 0.3989931800)
(0.0980143360, 0.2738984000)
(0.0934052020, 0.2611979700)
(-0.1610783400, -0.0646383690)
(-0.0353890720, -0.0609984690)
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(-0.0589085160, 0.4744136400)
(0.0177645420, 0.0554167000)
(-0.0673573100, -0.0747140290)
(0.3927670000, 0.2288047400)
(-0.0206500310, 0.0166509570)
(0.1295754400, 0.2333227700)
(0.0808268930, -0.0969265690)
(-0.0006537703, 0.5990355600)
(0.1037491900, 0.0201032720)
(-0.0964997240, -0.2731960600)
(-0.0107902580, 0.2254682600)
(-0.0169002840, -0.0437133670)
(0.2293280000, 0.3594157300)
(0.1001225500, 0.0305406920)
(0.0754975410, 0.1532803500)
(-0.0668426610, -0.1727375400)
(-0.0535746150, 0.1911748600)
(0.1381688400, -0.1045162700)
(-0.1445107000, 0.2475181000)
(-0.1021973200, -0.1170725700)
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(-0.1126217500, -0.0049403234)
(-0.2142810700, -0.0038863480)
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(-0.0858293510, 0.5781205100)
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(-0.0454626800, -0.0040001491)
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(-0.0514739440, -0.3698127900)
(0.1968109600, -0.2208890100)
(0.1333501700, 0.2675209400)
(0.2109687700, 0.1687502400)
(0.0910473970, 0.0483453470)
(0.0152596720, 0.0688115500)
(-0.1919755900, -0.1146773000)
(-0.0346512890, -0.1990734400)
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(-0.0888943100, 0.0482586050)
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(-0.0240644610, -0.0885551270)
(-0.0184489220, 0.1841217600)
(0.0780146710, 0.5675243300)
(-0.2250868000, 0.0676244760)
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(-0.0868054470, -0.3821497000)
(-0.2290482100, -0.0491767640)
(-0.0623051280, 0.0869052380)
(-0.0397078580, 0.4902973600)
(0.2214431000, 0.2868797900)
(-0.1239105200, -0.4568448600)
(0.3067698300, -0.2673059300)
(-0.2577395500, 0.0785863180)
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(-0.0110018820, 0.0089827290)
(-0.0545089240, -0.0212171840)
(-0.0126786430, -0.3503446200)
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(-0.3053084100, -0.5823261700)
(-0.1169400000, 0.2920483700)
(-0.1474538800, -0.0772434670)
(0.2226893700, -0.3121142000)
(-0.0677441060, 0.1333680100)
(0.0402769320, 0.0236613800)
(-0.0292095870, 0.7371221100)
(0.4467931600, -0.1077330200)
(-0.0945156210, -0.1432647200)
(0.0162894150, -0.1045319400)
(-0.1408052000, -0.0069948504)
(-0.1030477600, 0.1672307200)
(0.0782366420, -0.1168280800)
(-0.1106751700, -0.2958883300)
(-0.1107816400, -0.2683028400)
(0.0244086210, -0.0721890560)
(-0.0151958450, 0.2172417200)
(0.1584319900, 0.2834893500)
(-0.3233958700, 0.1643004900)
(0.1767558500, -0.0470660590)
(0.0387423240, 0.2780336100)
(0.0943644770, -0.4654739300)
(0.2128004600, 0.2075158600)
(0.0992224830, 0.1806432000)
(0.0870709660, -0.1785395900)
(-0.1227746800, 0.1051175200)
(-0.0042323076, 0.1207397600)
(-0.0763017690, -0.0407245380)
(-0.1087349800, -0.2386057500)
(0.2265989600, -0.1263228000)
(0.1702760800, -0.0011092627)
(-0.0111712050, 0.1437278000)
(-0.1160303000, 0.1146510900)
(0.0946823610, 0.0521833760)
(0.1412400800, -0.1913159900)
(-0.0422613030, -0.3133655500)
(-0.3512770500, -0.3679726900)
(-0.1221378800, 0.3952915600)
(-0.1022108600, -0.1799596900)
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(0.2314440000, 0.0555647460)
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(-0.1539090000, -0.2230728000)
(0.1008911700, 0.3412542100)
(0.4213916200, -0.1364930200)
(-0.3523799100, -0.1288398700)
(0.0090010154, 0.4778561000)
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(0.3593357800, 0.0676824110)
(-0.2466200900, -0.0476964010)
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(-0.1573893900, 0.3514918100)
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(0.0884577190, 0.0193814160)
(0.0852829820, -0.0535777060)
(-0.1080681300, -0.2045553700)
(-0.0747752350, -0.0425263890)
(-0.1616560600, -0.0972355260)
(0.1341420000, -0.2622426300)
(0.2917364900, -0.0215467070)
(-0.3711297000, -0.0269589490)
(-0.0026866160, -0.1888927100)
(-0.1836576300, -0.2584659000)
(-0.3350164600, -0.2090747000)
(-0.2269517500, 0.1515981400)
(-0.3085533300, -0.0482950570)
(-0.3269820300, -0.1928267300)
(-0.0000953724, -0.0421587050)
(0.0458809820, -0.0775189790)
(-0.0154968440, 0.0426714730)
(-0.1958236200, -0.2625510100)
(-0.2505143700, -0.2867206900)
(-0.1185523300, 0.4803778900)
(-0.2030390300, 0.1846290800)
(-0.1718712800, -0.1081087800)
(-0.0855573940, 0.1299218300)
(-0.1441869600, 0.0877897330)
(-0.1060965400, -0.2854970500)
(-0.2035145800, 0.0342936950)
(-0.1813461500, 0.0959690280)
(-0.0819476580, -0.2689824600)
(0.1476430400, 0.4895947300)
(0.0779098330, 0.2944872700)
(0.0036278604, 0.0893974050)
(0.0364888240, -0.5250520400)
(0.1858766800, -0.6791442800)
(0.1206678000, 0.2596681900)
(0.2335699800, -0.4287754000)
(-0.2915253300, -0.3071517800)
(-0.1733998300, 0.3978733900)
(0.0345208530, 0.3282329500)
(-0.1351481900, -0.1092032700)
(0.0387983370, -0.1674302500)
(-0.0513642230, -0.3550830500)
(-0.0812360850, 0.3520185000)
(0.0068493148, 0.4919184100)
(-0.1107010000, 0.1394738600)
(-0.0793015290, -0.1734241400)
(-0.1012216300, 0.4651525700)
(-0.0024700899, 0.2710719800)
(0.0302175530, 0.0786255630)
(-0.0673730090, -0.1165611700)
(-0.1805334100, 0.1719908200)
(0.0393620160, 0.0632827620)
(0.0843782640, -0.1855482100)
(-0.0160156380, -0.4262995600)
(-0.0783523770, -0.1224236000)
(-0.0193807530, 0.0411948850)
(-0.0009287758, -0.4284986200)
(-0.1891349800, -0.1154835000)
(0.1174518600, 0.1381641900)
(-0.0657859080, -0.2172574800)
(-0.0076180675, -0.2196837700)
(-0.0684579970, -0.2174645300)
(0.0641345650, 0.4436553600)
(-0.1500617500, 0.0964845650)
(-0.0723128950, -0.1031458100)
(-0.2003993000, -0.1017284300)
(0.1948903400, 0.4895784300)
(-0.0054811680, 0.0403979440)
(0.0056543827, 0.6338025700)
(-0.0711796440, 0.3594406800)
(0.0327253100, 0.1459739500)
(-0.1151191300, 0.3513543100)
(0.0275929530, -0.0858060670)
(-0.1565186400, 0.1168426700)
(-0.1519406600, -0.3811277400)
(-0.1008635700, 0.1965674500)
(-0.1597514100, -0.1848778000)
(-0.3152343100, -0.1999328500)
(0.1014020200, 0.3698040400)
(-0.0802394500, -0.1381820800)
(0.1211751900, 0.1034028700)
(0.1188024500, 0.3239584800)
(-0.1648441600, -0.2108809500)
(0.0191776300, -0.0931602600)
(-0.1217513200, -0.3116265700)
(-0.1131137500, -0.1094374600)
(-0.1278534500, 0.4202473600)
(0.2374191800, -0.1393434600)
(0.0501601200, 0.4705793800)
(-0.0344906080, -0.1057142100)
(-0.0137238790, 0.3294928100)
(-0.0753083540, -0.1681395600)
(-0.1688443400, 0.1472195800)
(-0.0098447225, -0.2130690700)
(0.0647697880, 0.5246150500)
(-0.0946169000, -0.4442466900)
(0.0807696510, 0.0760090190)
(-0.0946148050, 0.2155745200)
(0.1095626500, 0.4434284800)
(-0.0238265370, -0.2205543200)
(0.1373442300, -0.1136087600)
(0.1667708000, 0.0366209520)
(0.2307376900, -0.1957267100)
(0.0192567430, -0.0274300890)
};
\end{axis}
\end{tikzpicture}
}
\caption{The left plot shows the correlated errors between the elements 2 and 5 between the original tensor and the decomposed using the weighted CPD. The right plot shows the uncorrelated errors between elements 3 and 8.}
\label{fig_result1}
\end{figure}
This behaviour is what is expected from the weight matrix and is discussed in the following. The weight matrix imposes extra conditions on the cost function~(\ref{eq:costfunction}) by assigning weights to the elements to be optimized. A diagonal element of \weight\ gives a weight to a single element, while an off-diagonal element of \weight\ gives a correlation between two elements. The latter implies that the errors are also correlated, which is precisely what is shown in \fig{fig_result1}.
\subsection{System identification example}
In order to gain an understanding of how much impact the weight has for the CPD, the problem is examined from a distance using system identification (see \cite{Ljung1999} and \cite{Pintelon2012}). This can be done by surrounding the nonlinear function by linear low-pass filters and looking at how this system behaves with signals. \fig{fig:sysID} gives a graphical representation of the interconnection of the linear and nonlinear parts of this system. This system, with the coupled nonlinearity, comes as the result of the identification method for so-called parallel Wiener-Hammerstein system proposed in~\cite{Schoukens2015} and~\cite{SchoukensDecember10--132013}. As the name suggests, a parallel Wiener-Hammerstein system consists of parallel branches of so-called Wiener systems and Hammerstein systems. Identification methods for the latter are discussed in~\cite{Schoukens2012a} and~\cite{Schoukens2012} and for the former in \cite{Schoukens2011}.
\begin{figure}[ht]
\def\w{2.5}
\def\ww{1.5}
\def\wi{0.5}
\def\fign{1}
\def\u{\mathbf{u}}
\def\x{\mathbf{x}}
\def\J{\mathbf{J}}
\def\wLN{2}
\def\hLN{0.35}
\centerline{\begin{tikzpicture}[>=stealth, scale=0.8]
\filldraw[fill=gray!30, rounded corners = 4] (-\ww,-1.3) rectangle (\ww,1.3);
\draw[dashed] (-\ww-\wLN, 0.7 - \hLN) rectangle (-\ww-1, 0.7 + \hLN);
\draw[dashed] (-\ww-\wLN, -0.7 + \hLN) rectangle (-\ww-1, -0.7 - \hLN);
\draw[->, thick] (-\ww-1, 0.7) -- (-\ww, 0.7);
\draw[->, thick] (-\ww-1, -0.7) -- (-\ww, -0.7);
\draw[->, thick] (\ww, 0.7) -- (\ww+1, 0.7) ;
\draw[->, thick] (\ww, -0.7) -- (\ww+1, -0.7) ;
\draw[dashed] (\ww+\wLN, 0.7 - \hLN) rectangle (\ww+1, 0.7 + \hLN);
\draw[dashed] (\ww+\wLN, -0.7 + \hLN) rectangle (\ww+1, -0.7 - \hLN);
\draw[->, thick] (-\ww-1-1.5*\wLN, 0) -- (-\ww-1-\wLN, 0);
\draw[<->, thick] (-\ww-\wLN, 0.7) -- (-\ww-1-\wLN, 0.7) -- (-\ww-1-\wLN, -0.7) -- (-\ww-\wLN, -0.7) ;
\draw[ thick] (\ww+\wLN, 0.7) -- (\ww+1+\wLN, 0.7) -- (\ww+1+\wLN, -0.7) -- (\ww+\wLN, -0.7) ;
\draw[->, thick] (\ww+1+\wLN, 0) -- (\ww+1+1.5*\wLN, 0);
\draw (0,0) node {$\mathbf{f}(\u)$};
\draw (-\ww-1-1.5*\wLN, 0) node[anchor=south] {input};
\draw (\ww+1+1.5*\wLN, 0) node[anchor=south] {output};
\draw (\ww+\wLN, 0.7) node[anchor = east] {$R_1$};
\draw (\ww+\wLN, -0.7) node[anchor = east] {$R_2$};
\draw (-\ww-\wLN, 0.7) node[anchor = west] {$L_1$};
\draw (-\ww-\wLN, -0.7) node[anchor = west] {$L_2$};
\end{tikzpicture}}
\caption{Surrounding the nonlinear polynomial function \f\ by linear dynamic low-pass filters, shown by dashed lines. These operators are assumed to be known and are used when analyzing the weighted CPD on~\f. In this example, we assume the left filters $L_1$ and $L_2$ (resp. the right filters $R_1$ and $R_2$) to behave similarly and have similar transfer functions.}
\label{fig:sysID}
\end{figure}
For this example, we wish to decouple the multivariate polynomial given by
$$
\f(\u) = \begin{cases}
0.09u_1^3 - 3.3u_1^2u_2 + 0.22u_1^2 + 5.0u_1u_2^2 - 0.44u_1u_2 - 0.25u_1 - 2.2u_2^3 + 0.41u_2^2 + 0.84u_2
\\
- 0.042u_1^3 + 3.2u_1^2u_2 - 0.21u_1^2 - 4.9u_1u_2^2 + 0.45u_1u_2 - 0.053u_1 + 2.3u_2^3 - 0.12u_2^2 - 0.27u_2
\end{cases}.
$$
Also, we assume that the coefficients of \f\ are correlated with the covariance matrix~$\mathbf{\Sigma}_\f$ given at the top of page~\pageref{mat_covar}.
\begin{figure}
\leavevmode\kern-3em\scalebox{0.65}{
$
\mathbf{\Sigma}_\f =
\left[\begin{array}{@{}rrrrrrrrrrrrrrrrrr@{}}
\textbf{2.0} & -1.5 & 0.1 & -0.3 & 0.1 & -9.3 & 25.7 & -26.7 & 9.3 & -2.0 & 1.5 & -0.1 & 0.3 & -0.1 & 9.3 & -25.7 & 26.7 & -9.4 \\
-1.5 & \textbf{1.8} & -0.0 & 0.0 & -0.0 & 7.5 & -24.6 & 27.9 & -11.3 & 1.5 & -1.8 & 0.0 & -0.0 & 0.0 & -7.5 & 24.5 & -27.8 & 11.2 \\
0.1 & -0.0 & \textbf{9.6} & -16.0 & 7.3 & -3.7 & 6.5 & 0.3 & -2.2 & -0.1 & 0.0 & -9.6 & 15.9 & -7.3 & 3.7 & -6.6 & -0.2 & 2.2 \\
-0.3 & 0.0 & -16.0 & \textbf{32.2} & -16.6 & 5.5 & -5.5 & -6.1 & 5.0 & 0.3 & -0.0 & 15.8 & -32.0 & 16.5 & -5.5 & 5.7 & 5.7 & -4.9 \\
0.1 & -0.0 & 7.3 & -16.6 & \textbf{10.4} & -2.7 & 1.6 & 4.2 & -2.7 & -0.1 & 0.0 & -7.3 & 16.4 & -10.3 & 2.7 & -1.8 & -4.0 & 2.6 \\
-9.3 & 7.5 & -3.7 & 5.5 & -2.7 & \textbf{105.7} & -275.3 & 227.3 & -56.6 & 9.3 & -7.5 & 3.7 & -5.5 & 2.7 & -105.5 & 275.4 & -228.3 & 57.2 \\
25.7 & -24.6 & 6.5 & -5.5 & 1.6 & -275.3 & \textbf{850.9} & -824.9 & 245.7 & -25.6 & 24.6 & -6.5 & 5.6 & -1.6 & 273.5 & -847.5 & 823.8 & -246.2 \\
-26.7 & 27.9 & 0.3 & -6.1 & 4.2 & 227.3 & -824.9 & \textbf{937.1} & -327.3 & 26.5 & -27.9 & -0.2 & 5.9 & -4.1 & -225.3 & 819.0 & -932.4 & 326.5 \\
9.3 & -11.3 & -2.2 & 5.0 & -2.7 & -56.6 & 245.7 & -327.3 & \textbf{134.5} & -9.3 & 11.2 & 2.2 & -4.9 & 2.6 & 55.9 & -243.1 & 324.6 & -133.7 \\
-2.0 & 1.5 & -0.1 & 0.3 & -0.1 & 9.3 & -25.6 & 26.5 & -9.3 & \textbf{2.0} & -1.5 & 0.1 & -0.3 & 0.1 & -9.3 & 25.6 & -26.6 & 9.3 \\
1.5 & -1.8 & 0.0 & -0.0 & 0.0 & -7.5 & 24.6 & -27.9 & 11.2 & -1.5 & \textbf{1.8} & -0.0 & 0.0 & -0.0 & 7.5 & -24.4 & 27.7 & -11.2 \\
-0.1 & 0.0 & -9.6 & 15.8 & -7.3 & 3.7 & -6.5 & -0.2 & 2.2 & 0.1 & -0.0 & \textbf{9.5} & -15.8 & 7.3 & -3.7 & 6.5 & 0.1 & -2.1 \\
0.3 & -0.0 & 15.9 & -32.0 & 16.4 & -5.5 & 5.6 & 5.9 & -4.9 & -0.3 & 0.0 & -15.8 & \textbf{31.8} & -16.4 & 5.5 & -5.8 & -5.6 & 4.8 \\
-0.1 & 0.0 & -7.3 & 16.5 & -10.3 & 2.7 & -1.6 & -4.1 & 2.6 & 0.1 & -0.0 & 7.3 & -16.4 & \textbf{10.3} & -2.7 & 1.8 & 3.9 & -2.6 \\
9.3 & -7.5 & 3.7 & -5.5 & 2.7 & -105.5 & 273.5 & -225.3 & 55.9 & -9.3 & 7.5 & -3.7 & 5.5 & -2.7 & \textbf{105.2} & -273.8 & 226.3 & -56.5 \\
-25.7 & 24.5 & -6.6 & 5.7 & -1.8 & 275.4 & -847.5 & 819.0 & -243.1 & 25.6 & -24.4 & 6.5 & -5.8 & 1.8 & -273.8 & \textbf{844.5} & -818.2 & 243.7 \\
26.7 & -27.8 & -0.2 & 5.7 & -4.0 & -228.3 & 823.8 & -932.4 & 324.6 & -26.6 & 27.7 & 0.1 & -5.6 & 3.9 & 226.3 & -818.2 & \textbf{927.9} & -323.9 \\
-9.4 & 11.2 & 2.2 & -4.9 & 2.6 & 57.2 & -246.2 & 326.5 & -133.7 & 9.3 & -11.2 & -2.1 & 4.8 & -2.6 & -56.5 & 243.7 & -323.9 & \textbf{133.0}
\end{array}\right] $
}
\label{mat_covar}
\end{figure}
In this matrix, the element on position $(i,j)$ contains the covariance between the $i$-th and $j$-th coefficient of \f. Finally, the graphical representations of the low-pass input and output filters are given in~\fig{fig_lowPass}.
\begin{figure}[ht]
\begin{minipage}[c]{0.5\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}
width=0.285\textwidth,
height=0.225\textwidth,
at={(0\textwidth,0\textwidth)},
scale only axis,
every outer x axis line/.append style={black},
every x tick label/.append style={font=\color{black}},
xmin=0,
xmax=1,
xtick={0,0.5,1},
xlabel={frequency (normalized)},
every outer y axis line/.append style={black},
every y tick label/.append style={font=\color{black}},
ymin=-50,
ymax=1,
ytick={-50, -40, -30, -20, -10, 0},
ylabel={magnitude (dB)},
axis background/.style={fill=white},
title={Input filters},
axis x line*=bottom,
axis y line*=left,
legend style={legend cell align=left,align=left,draw=black},
scale = 2
]
\addplot [color=red,solid]
table[row sep=crcr]
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};
\addlegendentry{$\text{L}_\text{1}$};
\addplot [color=blue,solid]
table[row sep=crcr]
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\addplot [color=blue,solid,forget plot]
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\addplot [color=black,solid,forget plot]
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\caption{The Bode plots of the input (left) and output filters (right) of the system used in this section.}
\label{fig_lowPass}
\end{figure}
The decoupling is done four times, in order to compare the effect of the different kinds of weights to the CPD:
\begin{enumerate}
\item Decoupling without any weight, using the method of~\cite{Dreesen2015},
\item Decoupling with the element-wise weight,
\item Decoupling with the slice-wise weight,
\item Decoupling with the dense weight.
\end{enumerate}
In order to compare the results, a validation signal is sent through the original coupled system of~\fig{fig:sysID} and also through the four different decouplings. This validation signal is a random-phase multisine (see \cite{Pintelon2012}), as shown in~\fig{fig_valSignal}.
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0.96996996996997 -2.60162719633886\\
0.970970970970971 -1.33563728194505\\
0.971971971971972 2.90848709621059\\
0.972972972972973 0.804844537927799\\
0.973973973973974 1.64936474246554\\
0.974974974974975 0.405570711839289\\
0.975975975975976 0.594762508358746\\
0.976976976976977 -1.23643889377167\\
0.977977977977978 -2.02626066938381\\
0.978978978978979 0.543856382235908\\
0.97997997997998 -1.16181731189004\\
0.980980980980981 0.931911368207931\\
0.981981981981982 -0.653106799725816\\
0.982982982982983 1.70203243739408\\
0.983983983983984 3.12298570082367\\
0.984984984984985 -1.43843997838246\\
0.985985985985986 -0.945635774583663\\
0.986986986986987 1.17285455507377\\
0.987987987987988 0.219340680023047\\
0.988988988988989 0.882201183504959\\
0.98998998998999 -3.07744112169613\\
0.990990990990991 -2.72510641439211\\
0.991991991991992 -3.12327773287002\\
0.992992992992993 0.14409728735702\\
0.993993993993994 0.395466541868784\\
0.994994994994995 1.24264607238048\\
0.995995995995996 2.04450693583432\\
0.996996996996997 0.238438690895853\\
0.997997997997998 -0.806975605761863\\
0.998998998998999 -2.79943674717438\\
1 -0.477622799474164\\
};
\end{axis}
\end{tikzpicture
\end{minipage}
\caption{A graphical representation of the validation signal, in the frequency domain. The left plot shows the magnitude of the signal (in dB), the right plot shows the phase. The frequency axis is normalized.}
\label{fig_valSignal}
\end{figure}
The output signal and the errors between the different decouplings are plotted in~\fig{fig:results}. This plot shows the magnitude of the output and output errors, in dB, with respect to the frequency of the signals. From this plot, it is clear that a slice-wise or dense weight reduces the errors significantly with respect to no weight or element-wise weight.
Next to these numerical experiments, extensive simulations were done with multiple different systems and several coupled multivariate polynomials. From these, general observations can be made: the slice-wise and dense weight decompositions are at least as good as the decomposition with no weight described in~\cite{Dreesen2015}. Overall, the element-wise weight does not incorporate enough information to improve~\cite{Dreesen2015} and has comparable results. Finally, for difficult decomposition problems where~\cite{Dreesen2015} does not work well, we observe improvements with the slice-wise or dense weighted decomposition.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}
\begin{axis}[
xlabel={Frequency},
ylabel={dB of magnitude},
xmin = 0,
xmax = 180,
legend style={
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};
\legend{\footnotesize original output signal,
\footnotesize errors of the output without weight,
\footnotesize errors of the output with element-wise,
\footnotesize errors of the output with slice-wise weight,
\footnotesize errors of the full-weighted output}
\end{axis}
\end{tikzpicture}
\end{center}
\caption{The errors between the the original output signal and the output by the different decoupling methods. Plotted in the frequency domain; the vertical axis is in dB of the magnitude of the signals.}
\label{fig:results}
\end{figure}
\section{Conclusion and future work}\label{sec:conclusion}
In this paper, our starting point is a coupled representation of a multivariate polynomial~$\f$, which does not have an exact decomposition with the prespecified number of branches. It is assumed, however, that the covariance matrix~$\mathbf{\mathbf{\Sigma}}_\f$ of the coefficients of~$\f$ is known before the decoupling process. We have then generalized the decoupling algorithm described in~\cite{Dreesen2015} to this noisy case, by considering a weight factor in the Canonical Polyadic Decomposition. Three weight factors have been considered, based on three different covariance matrices: (1) an element-wise covariance matrix, (2) a slice-wise covariance matrix and (3) a dense covariance matrix. In cases (1) and (2), the matrices are of full rank, while the matrix in case (3) is rank-deficient. That is why extra equations are found for the third case, based on a SVD of the dense covariance matrix.
The results are promising and at least as good as the unweighted decoupling method described in~\cite{Dreesen2015}. When considering the decoupling problem inside the framework of linear filters, improvements are observed and discussed.
As future work, we would like to investigate how to make approximations of polynomials with a high number of branches by polynomials with a lower number of branches. Also, generalizations to other basis functions can be studied, and in what way these modify the proposed method.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,198 |
RocketChat._setEmail = function(userId, email) {
email = s.trim(email)
if (!userId) {
throw new Meteor.Error('invalid-user', "[methods] setEmail -> Invalid user");
}
if (!email) {
throw new Meteor.Error('invalid-email', "[methods] setEmail -> Invalid email");
}
emailValidation = /^[a-zA-Z0-9.!#$%&'*+\/=?^_`{|}~-]+@[a-zA-Z0-9](?:[a-zA-Z0-9-]{0,61}[a-zA-Z0-9])?(?:\.[a-zA-Z0-9](?:[a-zA-Z0-9-]{0,61}[a-zA-Z0-9])?)*$/;
if (!emailValidation.test(email)) {
throw new Meteor.Error('email-invalid', email + " is not a valid e-mail");
}
user = RocketChat.models.Users.findOneById(userId);
// User already has desired username, return
if (user.emails && user.emails[0] && user.emails[0].address === email) {
return user;
}
// Check e-mail availability
if (!RocketChat.checkEmailAvailability(email)) {
throw new Meteor.Error('email-unavailable', email + " is already in use :(");
}
// Set new email
RocketChat.models.Users.setEmail(user._id, email);
user.email = email;
return user;
}
RocketChat.setEmail = RocketChat.RateLimiter.limitFunction(RocketChat._setEmail, 1, 60000, {
0: function(userId) { return !RocketChat.authz.hasPermission(userId, 'edit-other-user-info') } // Administrators have permission to change others emails, so don't limit those
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 106 |
Q: Cygwin64 terminal: Undefined reference to `mbuf_remove` "I'm learning to create a tcp echo server from cesanta/mongoose https://github.com/cesanta/mongoose/blob/master/examples/tcp_echo_server/echo_server.c but when compiling it with cygwin using Makefile it won't work.
I tried learning Makefile too but so far I'm having a hard time understanding it. I also have included the header file on the current directory of the source file but it still won't work.
error:
$ make
gcc -o echo_server tcpechoserver.c mongoose.c -I.
/tmp/cct2s339.o:tcpechoserver.c:(.text+0x59): undefined reference to `mbuf_remove' "
There is a documentation on the github, but no explanation about this error. I also tried to copy the actual code and it still produced the same error.
#include "mongoose.h"
/* ERROR CYGWIN64 TERMINAL
gcc -o echo_server tcpechoserver.c mongoose.c -I.
/tmp/cct2s339.o:tcpechoserver.c:(.text+0x59): undefined reference to `mbuf_remove'
*/
// event handler
static void ev_handler(struct mg_connection *nc, int ev, void *p)
{
// structure memory buffer descriptor
struct mbuf *io = &nc->recv_mbuf;
(void) p;
switch (ev)
{
case MG_EV_RECV:
mg_send(nc, io->buf, io->len); // echo message back
mbuf_remove(io, io->len); // Discard message from recv buffer
break;
default:
break;
}
}
int main(void)
{
struct mg_mgr mgr;
const char *port1 = "3232", *port2 = "127.0.0.1:17000";
mg_mgr_init(&mgr, NULL);
mg_bind(&mgr, port1, ev_handler);
mg_bind(&mgr, port2, ev_handler);
printf("Starting echo mgr on ports %s, %s\n",port1, port2);
// MAIN LOOP
for (;;)
{
mg_mgr_poll(&mgr,1000);
}
mg_mgr_free(&mgr);
return 0;
}
/tmp/cct2s339.o:tcpechoserver.c:(.text+0x59): undefined reference to `mbuf_remove'
/tmp/cct2s339.o:tcpechoserver.c:(.text+0x59): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0x6f7a): undefined reference to `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0x6f7a): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0x7df5): undefined reference to `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0x7df5): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0xb4f9): undefined reference to `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0xb4f9): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0xc121): undefined reference to `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0xc121): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0xc1d0): more undefined references to `mbuf_remove' follow
/tmp/cca6g7CA.o:mongoose.c:(.text+0xc1d0): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0xc22c): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0xc595): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0x12e0b): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0x13df9): relocation truncated to fit: R_X86_64_PC32 against undefined symbol `mbuf_remove'
/tmp/cca6g7CA.o:mongoose.c:(.text+0x1634a): additional relocation overflows omitted from the output
collect2: error: ld returned 1 exit status
make: *** [Makefile:2: echo_server] Error 1
A: there is an open issue of Cygwin compilation
the issue was created on 11 Apr 2018 and remain open till now
i cloned the most recent mongoose repository and compiled under linux and cygwin(64bit)
##linux
cd mongoose/examples/tcp_echo_server
make all
cc echo_server.c ../../mongoose.c -o echo_server -g -W -Wall -Werror -I../.. -Wno-unused-function -pthread
##no error with most recent version
##cygwin(64bit)
cd mongoose/examples/tcp_echo_server
make all
gcc echo_server.c ../../mongoose.c -o echo_server -g -W -Wall -Werror -I../.. -Wno-unused-function -lws2_32
##undefined reference error with most recent version
##no error with tag 6.6
my suggestion right now , use mongoose 6.6 as a workaround
it should make little difference in learning
and fixing such error is too much trouble for a learner
to compile the tcp_echo_server example from mongoose 6.6, use the following commands in cygwin
no extra copying, just make
you should find tcp_echo_server.exe in the current directory
git clone --branch 6.6 https://github.com/cesanta/mongoose
cd mongoose/examples/tcp_echo_server/
make
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,206 |
{"url":"http:\/\/math.stackexchange.com\/tags\/standard-deviation\/hot","text":"# Tag Info\n\n3\n\nIf you have a discrete random variable $X$ that takes on values $x_i$ with probabilities $p(x_i), 1 \\leq i \\leq n$, and a function $f(x)$, then the expected value of $f(x)$ is given by $$E(f(X)) = \\sum_{i=1}^n f(x_i) p(x_i)$$ That is LOTUS. Perhaps an example will make it clearer. Suppose that $X$ takes on values $1, 2, 3$ with probabilities $1\/2, 1\/3, ... 2 For a discrete random variable$X$taking on values$x_1, x_2, \\ldots,x_n$with probabilities$p_1, p_2, \\ldots, p_n$respectively, the expected value or expectation of$X$, commonly denoted as$E[X]$, is defined to be $$E[X] = \\sum_{i=1}^n x_i p_i.\\tag{1}$$ (Feel free to replace each$p_i$with$p(x_i)$if that feels more comfortable). Similarly, the ... 2 (a) and (b) are OK. (c) 0.9996645, so a slight rounding error at worst. (d) In (c) you are using the PDF$F(x) = 1 - e^{-2x},$for$x > 0$to find the answer. Here you need$F(5) - F(2),$which requires no additional integration. Just watch the signs and you'll get something close to .02. It should be easy to make a graph of the exponential density ... 1 Chebyshev's Inequality is true for any$c > 0$, but you are right that it only provides useful information for$c > 1$. This is actually surprisingly easy prove. Define$\\mu = E(X)$and$\\sigma^2 = E((X-\\mu)^2)$. Observe that for any$c \\geq 0$we have$\\mathbb{1}\\left\\{\\left|\\frac{X-\\mu}{\\sigma}\\right|\\geq c\\right\\} \\leq \\frac{(X-\\mu)^2}{\\sigma^2 ...\n\n1\n\nYour second line should be $$=\\frac{1}{n-1}\\left [\\sum_{i=1}^nx_i^2+n\\bar{x}^2-2\\bar{x}\\sum_{i=1}^n x_i \\right ]$$ Then just as you said, you can use $$\\sum_{i=1}^n\\frac{x_i}{n}=\\bar{x}$$ to continue.\n\n1\n\nIn your confidence interval, you are using $\\bar x = 7$, which is inconsistent with your earlier notation. It should be $\\bar d = \\bar y - \\bar x= 7.$ Also, to be clear, 16 is the variance for the difference in scores for one individual, so the variance for $\\bar d$ is $16\/3$ and the standard deviation used in the confidence interval is $4\/\\sqrt{3},$ as you ...\n\n1\n\nI'm assuming a very large population of units from which to select, so that the distribution is binomial, and that 'have issues' means 'are defective.\" Then we have the number of defective units seen $X \\sim Bin(n=10, p=.1)$ Thus, (1) $E(X) = np = 1$ as you say. (2) V(X) = np(1-p) = .9, so SD(X) = \\sqrt(.9) = 0.9487.$; I think you overlooked a decimal point. ... 1 This looks like you are supposed to assume such accidents are a rare event. The question also states \"accidents occur at random times and independently from one another.\". These both hint that we should model the data using a Poisson distribution,$\\operatorname{Po}(\\lambda)$.$\\lambda$is the mean, so$\\lambda=15\\$. Of course, the Poisson distribution also ...\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2015-04-28 01:08:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9492978453636169, \"perplexity\": 543.1382892186974}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-18\/segments\/1429246660448.47\/warc\/CC-MAIN-20150417045740-00169-ip-10-235-10-82.ec2.internal.warc.gz\"}"} | null | null |
Response to this request is long overdue. By law, under all circumstances, Slough Borough Council should have responded by now (details). You can complain by requesting an internal review.
5. Details of members to sit on appointment panel for post of Council Chief Executive.
Please find the response to your recent FOI request.
Arrangements are still being finalised.
I am writing to request an internal review of Slough Borough Council's handling of my FOI request 'Chief Executive Appointment Process'.
You have not answered my questions. It does not take 5 months to set up a recruitment process. If it has taken the Council this long it had to provide an explanation for the delay, this is what my Foi was asking for.
HR have confirmed that the approval to recruit a Chief Executive has been made. However at this time no timetable is available as the Council is now in the process of taking advice on external support for the whole process. That is the actual and true position still at today's date.
We are therefore not able to provide any further information at this moment in time as it does not yet exist.
The advert is dated February 2017. That is three months ago and five months since the previous chief executive resigned. How can it take so long?
Either the council's HR team is hopelessly incompetent or there is something more sinister going on. Either way, there needs to be some formal intervention by central government.
This is a unitary authority for goodness sake, not a thru-penny bit one man band. They should be setting the standard for other employers in the borough.
You have not answered my question. Who has approved the delay of 4 month and where is a record of that decision?. What advice needs to be taken to put an advert out? Agents to advertise do not need a full tendering procurement exercise.
Why has the decision of full council of January 2017 not been respected?
subject to any specific timescales.
more attractive organisation for prospective candidates.
the agency has been appointed then a timescale will follow.
Does it taken any one 6 months to appoint a recruitment agency? If it does how long will it take to advertise?
Why did the council pay off its chief executive so quickly if they are still unstable. They are unstable because they have no leadership.
Please provide written evidence of the information you have set out in the review email. Which members and where is a copy of their decision.
There is no decision of cabinet attached. Cabinet cannot appoint the chief executive so why are they making decisions on something they have no authority on.
Paid off a chief executive so Council could move forward with a recruitment of new and still no advert. No leadership at council, run by interims and expensive counsultants. Linda Walker paid £100k in one year for being a monitoring officer and treating the public with disrespect. Well done Slough, great way to spend our money.
The FOI process can only provide recorded data.
I have attached above minutes which include the Appointment Process for Chief Executive (Head of Paid Service).
If you remain unhappy with the response to this FOI request then you have the option of discussing this matter with the ICO.
It looks like someone might get the job on a (biscuit) plate.
All of us at this calm and peaceful lakeside wonder how soggy the biscuit will become in the next few weeks. What future for a limp disintegrating biscuit when the revelation by the landing strip is digested? Mr Sparrow, hope you are OK after your nasty bump.
There is no transparency at SBC and the people at the top relish our ignorance.
I have personally experienced their lies and confidence tricks employed at a personal level .
The issues around Slough Unitary Authority have been going on from decades.
Tricks employed by local governments to save a few quid.
To hold the Councils to account in terms of finacial irregularities and transparency in the way out local Government is run.
However there is equal impotence in our National Government.
Phantom tenant ghost properties and persecution of innocents seems to be Slough's stock in trade.
Management or Slough but unfortunately I don't see it happening soon .
They appear to be holding all the cards for now. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,567 |
El uacari de cap negre (Cacajao melanocephalus) és un mico del Nou Món que pertany a la família dels pitècids. És originari del nord-oest del Brasil, el sud-est de Colòmbia i el sud-oest de Veneçuela, on viu a la selva de l'Amazones.
Els uacaris de cap negre viuen generalment en grups de 5-40 exemplars, però a vegades se n'ajunten més de 100. S'alimenten principalment de llavors, però també consumeixen fruita, fulles, medul·la vegetal i artròpodes.
Referències
Uacaris | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,414 |
Dale James continues to be active in the community and raising her grand and great-grandkids. "I still kind of feel I have some use," she says.
From Maccu Picchu to Nakusp, this grandma keeps going
North Africa's next on the list for the globe-trotting Dale James
John Boivin
Mar. 19, 2018 5:30 a.m.
When Arrow Lakes News caught up with Dale James, she had just got her passport photo re-taken.
"It was painful," she joked.
But at 82, James was planning to use her renewed passport to its full extent.
"We're going on a tour of the western Mediterranean later this year," she says. Though the particulars of the trip aren't quite settled yet, James says she wants to include stops in North Africa on her tour.
James is a globe-hopping great-grandmother. Last year she hiked the peaks of Machu Picchu, the ancient Inca fortress in Peru. In the fall, she took a cruise of the Caribbean. It was just after the hurricanes had hit the region, and she found herself in ports of call of islands devastated by the storms.
"I was impressed how they cope with having no power, with the hardship," says James. "The power here goes out for a few hours, and everyone whines. Down there, we'd cruise into a big city and there was not a light, it was pitch black. And they had been without power a month."
But travel is only one thing keeping James active and healthy. When at home here in Nakusp, she's known for her work with the hospital auxiliary, raising funds for the facility through the thrift store. She's been involved there for years, and currently is the organization's vice-president.
"It takes a lot of hours," she says. "it's not a job, not really, it's more a social thing. It is really fun, it fulfills me."
That's not all keeping her busy. She also volunteers for Meals on Wheels, and is the treasurer for her condo unit.
"I'm hoping to get someone to take that on this year," she admits.
James has been active in the community longer than most of the community has been alive. Moving here as a young nurse in 1958, she quickly became busy with community and family life. She married and began raising a family.
In the early 70s tragedy struck when her husband was killed in an industrial accident. As a widowed mom she raised her children here, then moved away when they grew up. She settled in Penticton B.C.
"At age 52 I had the bad sense to fall in love again," she says. "The bad luck is, he died very suddenly.
"It's a difficult thing, when you let yourself wide open then you get punched like that. It certainly put me off doing that again."
James continued to work in the Okanagan and Princeton areas, retiring at 68. But that's when Nakusp started calling her back.
"My grandkids were kind of at the teenage years,and they were into hockey, and needed to be driven to hockey, and that was big part of filling my life when work was slowing down," she says.
But not slowing down completely. Nurses are always in demand in small communities everywhere, and James found herself taking calls from communities like New Denver.
"They were really short and they were staffing out of Nelson," she says. "They'd phone and say 'can you work?' and I'd say yes. And then I'd hang up then say 'but I don't want to!'
"I don't say no well at all."
Fully retired now, James enjoys focusing on her grandchildren and assorted nephews and grand-nephews in town.
She follows minor hockey to this day, attending as many games as she can. She finds her days as a hockey grand-mom still bring her ties to the community.
"Even yet, when I go to the arena to watch kids playing, it's 'Hi Grandma Dale' from the kids who were friends of my grandsons," she laughs.
When not watching sport, she continues to participate.
She still cross-country skis ("I gave up downhill last year,") and enjoys hiking and touring in her motorhome in the summer.
"A lot of people I do hang with are younger than me, like the hiking ladies," she says. "I am the old fart. I have an age group I do things with, and adjust accordingly."
She also travels extensively to family in other parts of province, especially at Easter.
"I'm not slowing down, not yet," she says. She still enjoys helping raise her great-grandchildren (she still has one in Nakusp) and being active around the village.
"I am a Nakusp fan. It really lets you expand to whatever your interests and energy will let you do," she says. "I really like to leave my car in the garage, and appreciate that I can walk anywhere I want in town, to the post office and such, and meet dozens of people I know.
"And I like that being in a small town keeps you involved. In a bigger place, you can say 'let Joe do it', but here if you don't do it, it never gets done."
The secret to a long and active life is getting involved, staying involved — and having a little luck.
"I have been lucky enough to have good health," she says. "And I was lucky enough to have a profession that I don't have money problems, I have a pension.
"And then my wants are not great. I am pretty happy with the middle of the line, and my living arrangements are wonderful in the condo."
And in the end, it all comes back to family- the three kids, six grandkids and 12 great-grandkids that remain part of her life.
"I am really pretty lucky. I have a big family, and still kind of feel I have some use," she says.
"I'm not very fed up yet."
James goes for cataract surgery later this year, but she's taking it in stride.
"That won't slow me down much," she vows.
Somehow, we doubt it would.
Chris Hemsworth loves B.C.
Reader Photos: First day of spring around British Columbia
UPDATE: Hate trial for Toronto editor could be re-opened as judge delays sentencing | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,556 |
\section{Introduction}
\section{Introduction}
\label{Csection.-1}
Cyclohedra have first been introduced by Raoul Bott and Clifford Taubes \cite{BottTaubes1994} as combinatorial objects. They have been later constructed as polytopes by Martin Markl \cite{Markl1999} and by Rodica Simion \cite{Simion2003}. In her article, Rodica Simion describes them as type-B analogues of associahedra. It turns out that cyclohedra belong to several families of polytopes such as graph-associahedra \cite{CarrDevadoss2006,Devadoss2009}, and generalized associahedra \cite{ChapotonFominZelevinsky2002,FominZelevinsky2003}. The latter family, introduced by Sergey Fomin and Andrei Zelevinsky \cite{FominZelevinsky2003}, extends associahedra to any finite Coxeter groups: cyclohedra are associahedra of type B and C, and (classical) associahedra are associahedra of type A. The third infinite subfamily of generalized associahedra is that of the associahedra of type D. All other generalized associahedra have exceptional types, and there is a finite number of them.
The asymptotic diameter of the classical associahedra has been known for more than twenty-five years: Daniel Sleator, Robert Tarjan, and William Thurston have shown that the diameter of the $d$-dimensional associahedron is $2d-4$ when $d$ is large enough \cite{SleatorTarjanThurston1988}. They have also conjectured that this diameter is equal to $2d-4$ as soon as $d>9$. This conjecture has been settled recently by the author \cite{Pournin2014}. Even more recently, Cesar Ceballos and Vincent Pilaud have shown that the diameter of the $d$-dimensional associahedron of type D is exactly $2d-2$ for all $d$ \cite{CeballosPilaud2014}.
The last infinite subfamily of generalized associahedra whose diameter is not known exactly is that of cyclohedra. The asymptotic diameter of these polytopes is given in this article using the same techniques as in \cite{Pournin2014}. More precisely, it is shown that the diameter $\Delta$ of the $d$-dimensional cyclohedron is not greater than $\lceil 5d/2\rceil-2$ and not less than $5d/2-4\sqrt{d}-4$. Therefore this diameter grows like $5d/2$ when $d$ is large. More precisely:
$$
\lim_{d\rightarrow\infty}\frac{\Delta}{d}=\frac{5}{2}\mbox{.}
$$
The proofs will rely on the combinatorial interpretation of cyclohedra given in \cite{Simion2003}. Informally, the vertices of the $d$-dimensional cyclohedron correspond to the centrally symmetric triangulations of a polygon with $2d+2$ vertices, and its edges to flips between these triangulations. This combinatorial interpretation will be formally described in Section \ref{Csection.0}, and used in the same section to find a general upper bound on the diameter of cyclohedra. Particular pairs of centrally symmetric triangulations are further introduced in Section \ref{Csection.1}. These pairs will provide a lower bound on the diameter of cyclohedra that is asymptotically sharp. This bound will be derived at the end of Section \ref{Csection.1} from a general inequality that will be proven in Section \ref{Csection.3}. This inequality will be obtained using the same techniques as in \cite{Pournin2014}. These techniques, originally developed in the case of arbitrary triangulations of convex polygons, are adapted to centrally symmetric triangulations in Section \ref{Csection.2}. In Section \ref{Csection.4}, a short discussion on the diameter of low-dimensional cyclohedra completes the article.
\section{An upper bound on the diameter of cyclohedra}
\label{Csection.0}
Consider a positive integer $d$ and a convex polygon $\pi$ with $2d+2$ vertices. Any set of two distinct vertices of $\pi$ will be referred to as an edge on $\pi$. The elements of an edge will be called its vertices. An edge whose convex hull does not intersect the interior of $\pi$ is a boundary edge of $\pi$. Consider a vertex $x$ of $\pi$. The vertex of $\pi$ opposite $x$ will be denoted by $\bar{x}$. In other words, $x$ and $\bar{x}$ are separated by exactly $d$ vertices along the boundary of $\pi$. An edge on $\pi$ whose two vertices are opposite vertices of $\pi$ is referred to as a diagonal of $\pi$. A triangulation of $\pi$ is a maximal set of pairwise non-crossing edges on $\pi$. Note that any triangulation of $\pi$ contains all the boundary edges of $\pi$. These edges will also be referred to as the boundary edges of the triangulation. The other edges of a triangulation will be called its interior edges.
\begin{definition}
A triangulation of $\pi$ is called centrally symmetric when it contains edge $\{\bar{x},\bar{y}\}$ as soon as it contains edge $\{x,y\}$.
\end{definition}
Observe that any centrally symmetric triangulation of $\pi$ contains exactly one diagonal of $\pi$. Consider a centrally symmetric triangulation $T$ of $\pi$ and an interior edge $\{x,x'\}$ of $T$. There is a unique quadrilateral whose four boundary edges belong to $T$ and that admit $\{x,x'\}$ as one of its diagonals. Denote by $y$ and $y'$ the two vertices of this quadrilateral distinct from $x$ and from $x'$.
\begin{figure}
\begin{centering}
\includegraphics{cs-flips}
\caption{The flip of edge $\{x,x'\}$ when $x'\neq\bar{x}$ (left), and when $x'=\bar{x}$ (right). In both cases, the edges introduced by the flip are dotted.}\label{Cfigure.flip}
\end{centering}
\end{figure}
The quadrilateral with vertices $x$, $x'$, $y$, and $y'$ is sketched in Fig. \ref{Cfigure.flip} when $x'\neq\bar{x}$ (left of the figure) and when $x'=\bar{x}$ (right of the figure). Observe that this quadrilateral is distinct from its symmetric in the former case, and coincides with it in the latter case. In particular, if $x'=\bar{x}$, then $\{x,x'\}$ and $\{y,y'\}$ are two diagonals of $\pi$. The flip operation can be defined as follows:
\begin{definition}
The operation of flipping edge $\{x,x'\}$ in triangulation $T$ consists in replacing edges $\{x,x'\}$ and $\{\bar{x},\bar{x}'\}$ within $T$ by $\{y,y'\}$ and $\{\bar{y},\bar{y}'\}$.
\end{definition}
This operation is sketched in Fig. \ref{Cfigure.flip}. Note that it results in a centrally symmetric triangulation of $\pi$ distinct from $T$. Moreover, by central symmetry, the flip of edge $\{\bar{x},\bar{x}'\}$ in $T$ is the same operation as the flip of edge $\{x,x'\}$ in this triangulation. Note that flips are defined here in order to preserve central symmetry. This condition, specific to the case of cyclohedra, is not usually required in other contexts (see for instance \cite{DeLoeraRambauSantos2010}).
Now consider the graph whose vertices are the centrally symmetric triangulations of $\pi$, and whose edges connect two triangulations when they can be obtained from one another by a flip. This graph is isomorphic to the graph of the $d$-dimensional cyclohedron (see Theorem 1 in \cite{Simion2003}). Therefore, the distance of two vertices in the graph of a cyclohedron is also the minimal number of flips one needs to perform in order to transform the centrally symmetric triangulations corresponding to these vertices into one another. In particular, one can describe the paths in the graph of cyclohedra using a succession of centrally symmetric triangulations related by flips. Consider two centrally symmetric triangulations $T^-$ and $T^+$ of a convex polygon. A path of length $k$ from $T^-$ to $T^+$ is a sequence $(T_i)_{0\leq{i}\leq{k}}$ so that $T_0=T^-$, $T_k=T^+$ and $T_i$ can be transformed into $T_{i+1}$ by a flip whenever $0\leq{i}<k$. Call $P$ the pair $\{T^-,T^+\}$. A shortest path between $T^-$ and $T^+$ will be referred to as a \emph{geodesic} between these triangulations. The distance of $T^-$ and $T^+$, or equivalently, the distance of pair $P$ is the length of any geodesic between $T^-$ and $T^+$. This distance will be denoted by $\delta(P)$ in the following of this article.
One can obtain an upper bound on the distance between two centrally symmetric triangulations of a convex polygon using the same general idea than that of the proof of Lemma 2 from \cite{SleatorTarjanThurston1988}:
\begin{theorem}\label{Ctheorem.up}
The distance of two centrally symmetric triangulations of a convex polygon with $2d+2$ vertices is not greater than $\lceil{5d/2}\rceil-2$.
\end{theorem}
\begin{proof}
Consider a convex polygon $\pi$ with $2d+2$ vertices labeled clockwise from $0$ to $2d+1$. Let $T^-$ and $T^+$ be two triangulations of this polygon. One can assume without loss of generality that the unique diagonal of $\pi$ that belongs to $T^-$ is $\{0,\bar{0}\}$. Let $x$ be the vertex of $\pi$ so that $0\leq{x}\leq{d}$ and $\{x,\bar{x}\}$ is the unique diagonal of $\pi$ contained in $T^+$. It can be assumed that $x$ is not less than $\lfloor{d/2}\rfloor+1$ by, if needed, relabeling the vertices of $\pi$ counterclockwise from $0$ to $2d+1$ in such a way that $\bar{0}$ is relabeled $0$.
First consider the centrally symmetric triangulation $U^-$ of $\pi$ sketched on the left of Fig. \ref{Cfigure.up}. The diagonal of $\pi$ that belongs to $U^-$ is $\{0,\bar{0}\}$. Above this diagonal, all the interior edges of $U^-$ are incident to vertex $0$, and below it, they are all incident to vertex $\bar{0}$. Observe that $T^-$ can be transformed into $U^-$ in at most $d-1$ flips. Indeed, if the interior edges of $T^-$ above $\{0,\bar{0}\}$ are not all incident to vertex $0$, then it is always possible to introduce a new edge incident to vertex $0$ into $T^-$ by a single flip. As $T^-$ has exactly $d-1$ interior edges above $\{0,\bar{0}\}$, it can be transformed into $U^-$ in at most $d-1$ flips.
If $x=0$, then $T^+$ can also be transformed into $U^-$ in at most $d-1$ flips using the same procedure. Hence, $T^-$ and $T^+$ have distance at most $2d-2$, and the result holds in this case. It is now assumed that $x\neq{0}$.
Consider the centrally symmetric triangulation $U^+$ of $\pi$ sketched on the right of Fig. \ref{Cfigure.up}. The diagonal of $\pi$ that belongs to this triangulation is $\{x,\bar{x}\}$. All the interior edges of $U^+$ placed on the left of this diagonal are incident to vertex $0$. By central symmetry, the edges of $U^+$ placed on the right of $\{x,\bar{x}\}$ are incident to vertex $\bar{0}$. One can transform $T^+$ into $U^+$ in at most $d-1$ flips.
\begin{figure}[b]
\begin{centering}
\includegraphics{UpperBound}
\caption{The triangulations $U^-$ (left) and $U^+$ (right) used in the proof of Theorem \ref{Ctheorem.up}.}\label{Cfigure.up}
\end{centering}
\end{figure}
Indeed, if the interior edges of $T^+$ on the left of $\{x,\bar{x}\}$ are not all incident to vertex $0$, then it is always possible to introduce a new edge incident to vertex $0$ into $T^+$ by a single flip. As $T^+$ has exactly $d-1$ interior edges on the left of $\{x,\bar{x}\}$, it can be transformed into $U^+$ in at most $d-1$ flips.
Now observe that $U^+$ can be transformed into $U^-$ by first flipping $\{x,\bar{x}\}$, and then edges $\{\bar{0},y\}$, where $y$ ranges from $x$ to $d-1$. The first of these flips introduces diagonal $\{0,\bar{0}\}$ and the subsequent flips introduce edges $\{0,x+1\}$ to $\{0,d\}$. Therefore, $U^+$ and $U^-$ have distance at most $d-x+1$.
The above construction provides a path of length $3d-x-1$ between $T^-$ and $T^+$. As $x$ is at least $\lfloor{d/2}\rfloor+1$, the desired result holds. \qed
\end{proof}
Using the correspondence between the graphs of cyclohedra and the centrally symmetric triangulations of convex polygons and their flips, one derives an upper bound on the diameter of cyclohedra from Theorem \ref{Ctheorem.up}:
\begin{corollary}
For any positive integer $d$, the diameter of the $d$-dimensional cyclohedron is at most $\lceil{5d/2}\rceil-2$.
\end{corollary}
The remainder of the article is dedicated to finding lower bounds on the diameter of cyclohedra. In order to do this, particular pairs of centrally symmetric triangulations will be shown to have large distances. The family of these pairs is defined in the next section.
\section{Distant pairs of centrally symmetric triangulations}
\label{Csection.1}
Throughout this section, $d$ is a positive integer and $\pi$ is a convex polygon with $2d+2$ vertices, labeled clockwise from $0$ to $2d+1$. A vertex of $\pi$ will be referred to using any integer congruent to its label modulo $2d+2$. This allows using arithmetic operations on the vertices of $\pi$.
Consider the centrally symmetric triangulations $A^-$ and $A^+$ of $\pi$ respectively sketched on the left and on the right of Fig. \ref{Cfigure.0}. In this representation, only a few interior edges of $A^-$ and $A^+$ are shown and these triangulations need to be further described. Triangulation $A^-$ has a comb with $d-c+2$ teeth at vertex $0$, i.e. exactly $d-c+2$ of its interior edges are incident to vertex $0$. The teeth of the comb at vertex $0$ in $A^-$ are edge $\{0,\bar{0}\}$, and edges $\{0,x\}$ where $x$ ranges from $c$ to $d$. It is assumed in the following that a comb always has at least two teeth. Triangulation $A^-$ has another comb at vertex $1$, whose number of teeth is $c-b+1$. Note that the interior teeth of each comb are represented as thin lines in the figure. The number of these interior teeth depends on $b$, $c$, and $d$ and is equal to zero when a comb has only two teeth. Above edge $\{1,b\}$, $A^-$ is decomposed into two portions. Within the first of these portions, the interior edges of $A^-$ form a \emph{zigzag}, i.e a simple path that alternates between left and right turns. This zigzag starts at vertex $b$ and its edges cross $\{k,\bar{k}\}$, where $k=\lfloor{b/2}\rfloor+1$. This defines the zigzag univocally. Note that vertices $2$ to $k-2$ are then incident to exactly two interior edges of $A^-$. As shown on the figure, the zigzag ends either at vertex $k-1$ or at vertex $k+1$ (depending in the parity of $b$). The portion of $A^-$ above the zigzag is made up of a single triangle with vertices $k-1$, $k$, and $k+1$. This triangle is referred to as an \emph{ear of $A^-$ at vertex $k$} because two of its edges are the boundary edges of $\pi$ incident to vertex $k$. The other edges of triangulation $A^-$ are obtained by central symmetry.
This description of $A^-$ implicitly assumes that the following inequalities hold, in order for the combs at vertices $0$ and $1$ to have at least two teeth:
\begin{equation}\label{Cequation.0.1}
b<c\leq{d}
\end{equation}
Now consider triangulation $A^+$, shown on the right of Fig. \ref{Cfigure.0}.
\begin{figure}
\begin{centering}
\includegraphics{A-sketch}
\caption{Sketch of an $(a,b,c,d)$-pair of centrally triangulations of $\pi$, where $k$ and $l$ are respectively equal to $\lfloor{b/2}\rfloor+1$ and $a+b-c+4$. For the sake of clarity, the placement of the vertices slightly differs in the two triangulations.}\label{Cfigure.0}
\end{centering}
\end{figure}
This triangulation is decomposed into five portions. The portion left of edge $\{1,\bar{d}\}$ is an ear at vertex $0$. The interior edges of $A^+$ in the portion next to this ear are arranged into combs attached at vertices $\bar{c}$ to $\bar{d}$. Recall that each of these combs has at least two teeth. The only teeth depicted in the figure are edges $\{1,\bar{d}\}$ and $\{l,\bar{c}\}$. The other teeth are not shown because the number of combs depends on $c$ and $d$ (it is equal to $d-c+1$), and because the distribution of the teeth among these combs can be made in different ways. Within the central portion, the interior edges of $A^+$ form a centrally symmetric zigzag. This zigzag starts at the vertex labeled $l$ and its edges cross $\{0,\bar{0}\}$, which defines the zigzag univocally. Note that vertices $l$ to $c-1$ are then incident to exactly two interior edges of triangulation $A^+$. The edges of $A^+$ on the right of $\{c,\bar{l}\}$ are obtained by central symmetry.
Any interior tooth of a comb in either $A^-$ or $A^+$ will also be referred to as an interior tooth of the triangulation. Call $\tau^-$ and $\tau^+$ the number of interior teeth of respectively $A^-$ and $A^+$. These numbers are even because $A^-$ and $A^+$ are centrally symmetric. Denote:
$$
a=(\tau^-+\tau^+)/2+1\mbox{.}
$$
In other words, $a$ is obtained by summing the number of interior teeth of the combs at vertices $0$ and $1$ in $A^-$ (i.e. $d-b-1$) with the number of interior teeth of the combs at vertices $c$ to $d$ in $A^+$ (i.e. $l-2$), and by adding $1$ to the resulting quantity. In particular, one can express $l$ as follows:
$$
l=a+b-d+2\mbox{.}
$$
Observe that, by definition, $a$ must be greater by at least one than the number of interior teeth of $A^-$. Hence the following inequality holds:
\begin{equation}\label{Cequation.0.1.5}
d\leq{a+b}\mbox{.}
\end{equation}
While the quadruple $(a, b, c, d)$ completely characterizes $A^-$, several different triangulations $A^+$ may be possible for such a quadruple, depending on how the teeth are distributed among the combs in this triangulation.
\begin{definition}
The pair $\{A^-,A^+\}$ will be called an $(a,b,c,d)$-pair when, in addition to (\ref{Cequation.0.1}) and (\ref{Cequation.0.1.5}), the following inequality holds:
\begin{equation}\label{Cequation.0.2}
a+\frac{b}{2}+1<d\mbox{.}
\end{equation}
\end{definition}
Note that inequality (\ref{Cequation.0.2}) is equivalent to $l<k$ (this can be checked using the expressions of $k$ and $l$ as functions of $a$, $b$, and $c$). Also observe that there exists an $(a,b,c,d)$-pair if and only if $(a,b,c,d)$ is a quadruple of integers satisfying (\ref{Cequation.0.1}), (\ref{Cequation.0.1.5}), and (\ref{Cequation.0.2}). In particular, there are no $(a,b,c,d)$-pairs when $d$ is less than $4$. Indeed, combining inequalities (\ref{Cequation.0.1.5}) and (\ref{Cequation.0.2}) yields $b\geq3$, and it then follows from (\ref{Cequation.0.1}) that $d$ is not less than $4$.
The following lower bound on the distance of triangulations $A^-$ and $A^+$ will be obtained at the end of Section \ref{Csection.3}:
\begin{theorem}\label{Ctheorem.1}
For any $(a,b,c,d)$-pair $A$,
$$
\delta(A)\geq3d-\left(\frac{b}{2}+\frac{2c-b}{a}+3a+5\right)\mbox{.}
$$
\end{theorem}
Most of the remainder of the article is dedicated to proving this theorem. Before doing so, it is first explained how the announced lower bound on the diameter of cyclohedra can be derived from Theorem \ref{Ctheorem.1}.
\begin{theorem}\label{Ctheorem.1.2}
If $d$ is greater than $5$, then there exists a pair of centrally symmetric triangulations of $\pi$ with distance at least $5d/2-4\sqrt{d}-4$.
\end{theorem}
\begin{proof}
Assume that $d>5$. Let $a$ be an integer so that
\begin{equation}\label{Ctheorem.1.2.eq.1}
1\leq{a}<\frac{d-1}{2}\mbox{.}
\end{equation}
Call $b=d-a$ and $c=d-a+1$. For these values of $a$, $b$, and $c$, the quadruple $(a,b,c,d)$ satisfies inequalities (\ref{Cequation.0.1}), (\ref{Cequation.0.1.5}), and (\ref{Cequation.0.2}), and there exists an $(a,b,c,d)$-pair $A$. In fact, $A$ turns out to be unique. More precisely, $A^-$ has two symmetric combs with $a+1$ teeth at vertices $0$ and $\bar{0}$, while its interior edges not involved in the combs form two centrally symmetric zigzags. The interior edges of $A^+$ simply form a zigzag. By Theorem \ref{Ctheorem.1},
\begin{equation}\label{Ctheorem1.2.eq.1}
\delta(A)\geq\frac{5}{2}(d-a)-\frac{d+2}{a}-4\mbox{.}
\end{equation}
In the remainder of the proof, it is shown that $a$ can be chosen in such a way that $\delta(A)\geq5d/2-4\sqrt{d}-4$. This lower bound on $\delta(A)$ can be derived from (\ref{Ctheorem1.2.eq.1}) if and only if $a$ satisfies the following inequality:
\begin{equation}\label{Ctheorem1.2.eq.2}
\frac{5}{2}a+\frac{d+2}{a}\leq4\sqrt{d}\mbox{.}
\end{equation}
Solving (\ref{Ctheorem1.2.eq.2}) for $a$ yields:
\begin{equation}\label{Ctheorem1.2.eq.3}
|a-\frac{4}{5}\sqrt{d}|\leq\frac{\sqrt{6d-20}}{5}\mbox{.}
\end{equation}
As $d>5$, the right-hand side of (\ref{Ctheorem1.2.eq.3}) is greater than $1/2$. Moreover, $4\sqrt{d}/5$ is bounded below by $3/2$, and above by $d/2-1$. Therefore, there exists an integer solution $a$ to inequality (\ref{Ctheorem1.2.eq.2}) that also satisfies (\ref{Ctheorem.1.2.eq.1}). \qed
\end{proof}
Observe that $5d/2-4\sqrt{d}-4$ is negative when $1\leq{d}\leq5$. Hence, because of the correspondence between the graphs of cyclohedra and the centrally symmetric triangulations of convex polygons and their flips, the following result is a direct consequence of Theorem \ref{Ctheorem.1.2}:
\begin{corollary}\label{Ccorollary.2}
For any positive integer $d$, the diameter of the $d$-dimensional cyclohedron is at least $5d/2-4\sqrt{d}-4$.
\end{corollary}
\section{Vertex deletions}
\label{Csection.2}
In order to prove Theorem \ref{Ctheorem.1.2}, a set of recursive inequalities on the distance of $(a,b,c,d)$-pairs will be established in Section \ref{Csection.3}. These inequalities will typically bound the difference between the distance of an $(a,b,c,d)$-pair and the distance of an $(a,b',c',d')$-pair below by some positive quantity, where $d'$ is less than $d$. This kind of inequalities will be obtained using the operation of deleting a vertex from a triangulation, already used in \cite{Rambau1997} for triangulations of cyclic polytopes and in \cite{Pournin2014} for triangulations of convex polygons. This section begins with a definition of this operation in the case of centrally symmetric triangulations of convex polygons. Throughout the section, $\pi$ is a convex polygon with an even number of vertices labeled in an arbitrary way. In particular, these labels are not necessarily consecutive integers.
Let $T$ be a centrally symmetric triangulation of $\pi$. Consider a boundary edge of $\pi$ and label the vertices of this edge by $p$ and $q$, in such a way that $q$ immediately follows $p$ clockwise. Any oriented pair such as $(p,q)$ will be called a clockwise oriented boundary edge of $\pi$. In \cite{Pournin2014}, deleting vertex $p$ from $T$ consists in removing $\{p,q\}$ from $T$ and replacing $p$ by $q$ within every other edge of $T$. Informally, this operation amounts to displace a vertex of $\pi$ to its clockwise immediate successor.
When $\pi$ has at least four vertices, deleting (in the sense of \cite{Pournin2014}) a single vertex from $T$ results in a triangulation of a convex polygon with one vertex less (see Proposition 2 in \cite{Pournin2014}). Observe, however, that the triangulation obtained from this operation cannot be centrally-symmetric because the number of its vertices is odd. For this reason, a different deletion operation needs be defined that affects two opposite vertices of $\pi$. More precisely:
\begin{definition}\label{Cdefinition.4}
The operation of deleting vertex $p$ from $T$ consists in removing edges $\{p,q\}$ and $\{\bar{p},\bar{q}\}$ from $T$, and replacing $p$ and $\bar{p}$ by respectively $q$ and $\bar{q}$ within all the other edges of $T$. The resulting set of edges is called $T\mathord{\varparallelinv}{p}$.
\end{definition}
The deletion operation defined here in the case of centrally symmetric triangulation alternatively consists in performing two consecutive deletions in the sense of \cite{Pournin2014}. Therefore, by Proposition 2 from \cite{Pournin2014}, $T\mathord{\varparallelinv}{p}$ is a triangulation as soon as $T$ is a triangulation with at least $6$ vertices. It also follows from Definition \ref{Cdefinition.4} that $T\mathord{\varparallelinv}{p}$ is centrally symmetric.
Consider two triangulations $T^-$ and $T^+$ of $\pi$, and call $P$ the pair $\{T^-,T^+\}$. The deletion of vertex $p$ can be extended to $P$ as follows:
$$
P\mathord{\varparallelinv}{p}=\{T^-\mathord{\varparallelinv}{i},T^+\mathord{\varparallelinv}{p}\}\mbox{,}
$$
Let $T$ be a triangulation of $\pi$. Call $r$ the vertex of $\pi$ so that $\{p,r\}$ and $\{q,r\}$ are two edges of $T$. A flip that transforms $T$ into another triangulation of $\pi$ will be called \emph{incident to edge $\{p,q\}$} when this flip removes either $\{p,r\}$ or $\{q,r\}$. In other words, such flips modify the triangle of $T$ incident to edge $\{p,q\}$. According to the next lemma, for any geodesic between triangulations $T^-$ and $T^+$, there exists a path between $T^-\mathord{\varparallelinv}{p}$ and $T^+\mathord{\varparallelinv}{p}$, shorter by the number of flips incident to $\{p,q\}$ along the geodesic.
\begin{lemma}\label{Clemma.1}
Let $P$ be a pair of centrally symmetric triangulations $T^-$ and $T^+$ of a convex polygon. Further consider a clockwise oriented boundary edge $(p,q)$ of this polygon. If there exists a geodesic between $T^-$ and $T^+$ along which at least $f$ flips are incident to edge $\{p,q\}$, then
$$
\delta(P)\geq\delta(P\mathord{\varparallelinv}{p})+f\mbox{.}
$$
\end{lemma}
\begin{proof}
Let $(T_i)_{0\leq{i}\leq{k}}$ be a geodesic from $T^-$ to $T^+$. Consider the sequence of triangulations $T_0\mathord{\varparallelinv}{p}$ to $T_k\mathord{\varparallelinv}{p}$. Two consecutive triangulation in this sequence are either identical or can be obtained from one another by a flip. More precisely, if the flip that transforms $T_{i-1}$ into $T_i$ is incident to $\{p,q\}$, then the deletion of vertex $p$ sends these two triangulations to the same triangulation. Indeed, in this case, the quadrilateral affected by this flip is shrunk to a triangle by the deletion. If the flip that transforms $T_{i-1}$ into $T_i$ is not incident to $\{p,q\}$, then the quadrilateral affected by a flip remains a quadrilateral after the deletion and $T_{i-1}\mathord{\varparallelinv}{p}$ and $T_i\mathord{\varparallelinv}{p}$ are still related by a flip.
Therefore, removing unnecessary triangulations from the sequence of triangulations $T_0\mathord{\varparallelinv}{p}$, ..., $T_k\mathord{\varparallelinv}{p}$, one obtains a path of length $k'$ between $T^-\mathord{\varparallelinv}{p}$ and $T^+\mathord{\varparallelinv}{p}$. As the number of triangulations that have been removed in this process is also the number $f$ of flips incident to $\{p,q\}$ along path $(T_i)_{0\leq{i}\leq{k}}$, one obtains $k'=k-f$. By definition, $k'$ is at least $\delta(P\mathord{\varparallelinv}{p})$. As, in addition, $k$ is equal to $\delta(P)$, then the desired result holds. \qed
\end{proof}
Under some conditions on two triangulations, some boundary edge must be incident to at least two flip along any geodesic between then. For instance, the following lemma in a consequence of Lemma \ref{Clemma.1}:
\begin{lemma}\label{Clemma.2}
Let $P$ be a pair of centrally symmetric triangulations $T^-$ and $T^+$ of a convex polygon. Further consider two clockwise oriented boundary edges $(p_0,p_1)$ and $(p_1,p_2)$ on this polygon. If the triangles of $T^-$ incident to $\{p_0,p_1\}$ and to $\{p_1,p_2\}$ do not share an edge, and if $T^+$ has an ear in $p_1$, then there exists a vertex $x\in\{p_0,p_1\}$ so that
$$
\delta(P)\geq\delta(P\mathord{\varparallelinv}{x})+2\mbox{.}
$$
\end{lemma}
\begin{proof}
Assume that the triangles of $T^-$ incident to $\{p_0,p_1\}$ and to $\{p_1,p_2\}$ do not share an edge, and that $T^+$ has an ear in $p_1$. Consider a geodesic $(T_i)_{0\leq{i}\leq{k}}$ from $T^-$ to $T^+$. If there are at least $2$ flips incident to $\{p_0,p_1\}$ along this geodesic then, according to Lemma \ref{Clemma.1},
$$
\delta(P)\geq\delta(P\mathord{\varparallelinv}{p_0})+2\mbox{.}
$$
Hence the desired result holds with $x=p_0$. Assume that there is at most one flip incident to $\{p_0,p_1\}$ along $(T_i)_{0\leq{i}\leq{k}}$. In this case, there must be exactly one such flip. Indeed, by hypothesis, the triangle of $T^+$ incident to $\{p_0,p_1\}$ is the ear at vertex $p_1$.
\begin{figure}
\begin{centering}
\includegraphics{Lemma2}
\caption{The $j$-th flip along path $(T_i)_{0\leq{i}\leq{k}}$ used in the proof of Lemma \ref{Clemma.2}.}\label{Cfigure.Lem2}
\end{centering}
\end{figure}
Since the triangles of $T^-$ incident to $\{p_0,p_1\}$ and to $\{p_1,p_2\}$ do not share an edge, they must be distinct from the ear at vertex $p_1$. Therefore, some flip along $(T_i)_{0\leq{i}\leq{k}}$ must be incident to $\{p_0,p_1\}$.
Assume that the flip incident to $\{p_0,p_1\}$ along $(T_i)_{0\leq{i}\leq{k}}$ is the $j$-th one. This flip must then replace $\{p_1,r\}$ by $\{p_0,p_2\}$ as shown in Fig. \ref{Cfigure.Lem2}, where $r$ is the vertex of $\pi$ so that $\{p_0,r\}$ and $\{p_1,r\}$ belong to $T^-$. Observe that this flip is incident to $\{p_1,p_2\}$. Moreover, as the triangles of $T_{j-1}$ incident to $\{p_0,p_1\}$ and $\{p_1,p_2\}$ have a common edge, at least one of the first $j-1$ flips along $(T_i)_{0\leq{i}\leq{k}}$ must be incident to $\{p_1,p_2\}$. Hence, at least $2$ flips along $(T_i)_{0\leq{i}\leq{k}}$ are incident to this edge, and according to Lemma \ref{Clemma.1},
$$
\delta(P)\geq\delta(P\mathord{\varparallelinv}{p_1})+2\mbox{.}
$$
As a consequence, the desired result holds with $x=p_1$. \qed
\end{proof}
The previous lemma can be generalized to sequences of deletions:
\begin{lemma}\label{Clemma.3}
Let $P$ be a pair of centrally symmetric triangulations $T^-$ and $T^+$ of a convex polygon. Further consider $n\geq2$ clockwise oriented boundary edges $(p_0,p_1)$, ..., $(p_{n-1},p_n)$ on this polygon. If the triangles of $T^-$ incident to $\{p_0,p_1\}$, ..., $\{p_{n-1},p_n\}$ do not have common edges, and if $\{p_0,p_n\}$ is an edge of $T^+$, then a pair $Q$ of centrally symmetric triangulations, obtained from $P$ by deleting all but one of the vertices $p_0$, ..., $p_{n-1}$ satisfies
$$
\delta(P)\geq\delta(Q)+2(n-1)\mbox{.}
$$
\end{lemma}
\begin{proof}
Assume that the triangles of $T^-$ incident to $\{p_0,p_1\}$, ..., $\{p_{n-1},p_n\}$ do not have common edges, and that $T^+$ contains $\{p_0,p_n\}$. The lemma will be proven by induction on $n$. First observe that if $n=2$, then the result immediately follows from Lemma \ref{Clemma.2}.
Now assume that $n>2$. As $\{p_0,p_n\}$ is an edge of $T^+$, this triangulation induces a triangulation $U$ of the polygon with vertices $p_0$ to $p_n$. Any triangulation of a polygon with at least four vertices has at least two ears. Hence, $U$ has at least two ears and at least one of them is an ear at some vertex $p_i$ so that $0<i<n$. By assumption, the triangles of $T^-$ incident to $\{p_{i-1},p_i\}$ and to $\{p_i,p_{i+1}\}$ do not share an edge. Therefore, Lemma \ref{Clemma.2} provides some $x\in\{p_{i-1},p_i\}$ so that the following inequality holds:
\begin{equation}\label{Clemma.3.eq.1}
\delta(P)\geq\delta(P\mathord{\varparallelinv}{x})+2\mbox{.}
\end{equation}
Consider the vertices $p'_0$, ..., $p'_{n-1}$ obtained by removing $x$ from $p_0$, ..., $p_n$, and by relabeling the resulting sequence in such a way that the order of the indices is preserved. By construction, the triangles of $T^-\mathord{\varparallelinv}{x}$ incident to $\{p'_0,p'_1\}$, ..., $\{p'_{n-1},p'_n\}$ do not have common edges, and $\{p'_0,p'_n\}$ belongs to $T^+\mathord{\varparallelinv}{x}$. Therefore, by induction, some pair $Q$ of centrally symmetric triangulations, obtained from $P\mathord{\varparallelinv}{x}$ by deleting all but one of the vertices $p'_0$, ..., $p'_{n-2}$ satisfies the following inequality:
\begin{equation}\label{Clemma.3.eq.2}
\delta(P\mathord{\varparallelinv}{x})\geq\delta(Q)+2(n-2)\mbox{.}
\end{equation}
The result is then obtained combining (\ref{Clemma.3.eq.1}) with (\ref{Clemma.3.eq.2}). \qed
\end{proof}
\section{Proof of the main inequality}
\label{Csection.3}
Theorem \ref{Ctheorem.1} is proven in this section using the techniques introduced in the previous section. As in \cite{Pournin2014}, different sequences of deletions will be performed within $(a,b,c,d)$-pairs to obtain recursive lower bounds on their distance. In particular, the broad family of $(a,b,c,d)$-pairs needs be stable under each of these sequences of deletions. For instance:
\begin{lemma}\label{Clemma.X1}
Consider an $(a,b,c,d)$-pair $A$. Assume that at least $3$ flips are incident to $\{0,1\}$ along some geodesic between $A^-$ and $A^+$. If $a+b/2+2<d$, then there exists an $(a,b-2,b-1,d-2)$-pair $B$ so that
$$
\delta(A)\geq\delta(B)+5\mbox{.}
$$
\end{lemma}
\begin{proof}
Consider triangulation $A^-\mathord{\varparallelinv}{0}$, depicted on the left of Fig. \ref{Cfigure.2} and note that the deletion of vertex $0$ has merged the combs of $A^-$ at vertices $0$ and $1$.
\begin{figure}[b]
\begin{centering}
\includegraphics{LemmaStable1}
\caption{Sketch of triangulations $A^-\mathord{\varparallelinv}{0}$ (left) and $A^+\mathord{\varparallelinv}{0}$ (right), where $k$ and $l$ are respectively equal to $\lfloor{b/2}\rfloor+1$ and $a+b-c+4$. For the sake of clarity, the placement of the vertices slightly differs in the two triangulations.}\label{Cfigure.2}
\end{centering}
\end{figure}
Further observe that, after the deletion, vertices $1$ and $\bar{1}$ immediately follow vertices $\bar{d}$ and $d$ clockwise. Now consider triangulation $A^+\mathord{\varparallelinv}{0}$, sketched on the right of Fig. \ref{Cfigure.2}. In this sketch, $t$ is the number of teeth of the comb at vertex $d$ in $A^+$. Observe that deleting vertex $0$ from $A^+$ only affects this comb and the symmetric comb at vertex $\bar{d}$. More precisely, $A^+\mathord{\varparallelinv}{0}$ has symmetric combs with $t-1$ teeth at vertices $d$ and $\bar{d}$ when $t>2$, and these combs shrink to single edges when $t=2$. According to Lemma \ref{Clemma.1},
\begin{equation}\label{ClemmaC1.equation.1}
\delta(A)\geq\delta(A\mathord{\varparallelinv}{0})+3\mbox{.}
\end{equation}
Observe that there must be at least $2$ flips incident to edge $\{d-1,d\}$ along any geodesic from $A^-\mathord{\varparallelinv}{0}$ to $A^+\mathord{\varparallelinv}{0}$. Indeed, otherwise the single such flip would replace $\{1,d\}$ by $\{d-1,\bar{t}\}$. In particular, the triangulation within which this flip is performed would contain both $\{1,d\}$ and $\{1,\bar{t}\}$. By symmetry, it would also contain edge $\{\bar{1},t\}$, which is impossible because this edge crosses $\{1,d\}$. Denote by $B$ the pair of triangulations obtained by deleting vertex $d-1$ from $A\mathord{\varparallelinv}{0}$, and by relabeling the vertices of the two triangulations in the resulting pair clockwise from $0$ to $2d-3$ in such a way that vertex $1$ is relabeled $0$. Since at least $2$ flips are incident to edge $\{d-1,d\}$ along any geodesic between $A^-\mathord{\varparallelinv}{0}$ and $A^+\mathord{\varparallelinv}{0}$, then according to Lemma \ref{Clemma.1},
\begin{equation}\label{ClemmaC1.equation.2}
\delta(A\mathord{\varparallelinv}{0})\geq\delta(B)+2\mbox{.}
\end{equation}
Finally, assume that $a+b/2+2$ is less than $d$. Under this assumption, $B$ is an $(a,b-2,b-1,d-2)$-pair. In particular, quadruple $(a,b-2,b-1,d-2)$ satisfies inequalities (\ref{Cequation.0.1}), (\ref{Cequation.0.1.5}), and (\ref{Cequation.0.2}) precisely because $a+b/2+2<d$. Moreover, the successive deletions of vertices $0$ and $d-1$ do not change the total number of interior teeth of $A^-$ and $A^+$. Indeed, deleting vertex $0$ from $A^-$ creates a new interior tooth as this deletion merges the combs at vertices $0$ and $1$. However, the deletion of vertex $d-1$ subsequently removes an interior tooth from the merged comb. On the other hand, when $t=2$, these deletions do not remove from or add interior teeth to $A^+$, and when $t>2$, the first one removes an interior tooth from the comb at vertex $d$ in $A^+$, while the second one creates a new interior tooth as it merges the combs at vertices $d-1$ and $d$.
Therefore, combining (\ref{ClemmaC1.equation.1}) and (\ref{ClemmaC1.equation.2}) completes the proof.\qed
\end{proof}
Observe that Lemma \ref{Clemma.X1} requires the existence of a particular path between the two triangulations in an $(a,b,c,d)$-pair, which is a rather strong condition. Other conditions will be investigated in order to exhaust all possibilities. The following lemma deals with the case when $c$ is equal to $d$:
\begin{lemma}\label{Clemma.X3}
Consider an $(a,b,c,d)$-pair $A$ and call $t$ the number of interior edges of $A^-$ incident to vertex $d$. If $a+b/2+2<d$ and if $c=d$, then there exists an $(a,b-2t+2,b-2t+3,d-t)$-pair $B$ so that
$$
\delta(A)\geq\delta(B)+2t-1\mbox{.}
$$
\end{lemma}
\begin{proof}
At least two flips are incident to $\{\bar{0},d\}$ along any geodesic between $A^-$ and $A^+$. Indeed, otherwise the single such flip would replace the diagonal $\{0,\bar{0}\}$ of $A^-$ by edge $\{\bar{1},d\}$. However, since $d>1$, the latter edge cannot be a diagonal. Hence, according to Lemma \ref{Clemma.1},
\begin{equation}\label{Clemma.7.eq1}
\delta(A)\geq\delta(A\mathord{\varparallelinv}{d})+2\mbox{.}
\end{equation}
Assume that $c$ is equal to $d$. Triangulations $A^-\mathord{\varparallelinv}{d}$ and $A^+\mathord{\varparallelinv}{d}$ are depicted in Fig. \ref{Cfigure.4} in this case.
\begin{figure}[b]
\begin{centering}
\includegraphics{LemmaStable3}
\caption{Sketch of triangulations $A^-\mathord{\varparallelinv}{d}$ (left) and $A^+\mathord{\varparallelinv}{d}$ (right) when $c=d$, where $k$ is equal to $\lfloor{b/2}\rfloor+1$. Note that $t=a+b-c+4$ in this case. For the sake of clarity, the placement of the vertices slightly differs in the two triangulations.}\label{Cfigure.4}
\end{centering}
\end{figure}
Further assume that $a+b/2+2<d$. Consider the triangulations $B^-$ and $B^+$ obtained by deleting vertices $0$ to $t-2$ from $A^-\mathord{\varparallelinv}{d}$ and from $A^+\mathord{\varparallelinv}{d}$, and by relabeling the vertices of the resulting triangulations clockwise from $0$ to $2d-2t+1$ in such a way that vertices $t-1$ and $t$ are respectively relabeled $0$ and $1$. Denote by $B$ the pair $\{B^-,B^+\}$. One can see on Fig. \ref{Cfigure.4} that $B$ is an $(a,b-2t+2,b-2t+3,d-t)$-pair. Note, in particular, that the quadruple $(a,b-2t+2,b-2t+3,d-t)$ satisfies inequalities (\ref{Cequation.0.1}), (\ref{Cequation.0.1.5}), and (\ref{Cequation.0.2}) because $a+b/2+2<d$. Moreover, deleting vertex $d$ and vertices $0$ to $t-2$ does not change the total number of interior teeth of triangulations $A^-$ and $A^+$. Indeed, when these deletions are carried out within $A^-$, they each create a new interior teeth, except for the deletions of vertices $d$ and $0$. When they are carried out within $A^+$, they each remove an interior teeth, except for the deletions of vertices $t-3$ and $t-2$.
The remainder of the proof consists in showing that
\begin{equation}\label{Clemma.7.eq2}
\delta(A\mathord{\varparallelinv}{d})\geq\delta(B)+2t-3\mbox{.}
\end{equation}
The desired result is then obtained by combining inequalities (\ref{Clemma.7.eq1}) and (\ref{Clemma.7.eq2}). First assume that $t=2$. In this case $B^-$ and $B^+$ are obtained by deleting vertex $0$ from $A^-\mathord{\varparallelinv}{d}$ and from $A^+\mathord{\varparallelinv}{d}$ and by relabeling the vertices of the resulting triangulations. Observe that edge $\{0,1\}$ is not incident to the same triangle in $A^-\mathord{\varparallelinv}{d}$ and in $A^+\mathord{\varparallelinv}{d}$. Hence, at least one flip is incident to this edge along any geodesic between these triangulations, and according to Lemma \ref{Clemma.1}, the distance of $B^-$ and $B^+$ is less by at least one than that of $A^-\mathord{\varparallelinv}{d}$ and in $A^+\mathord{\varparallelinv}{d}$, which proves inequality (\ref{Clemma.7.eq2}) in this case.
Now assume that $t>2$. In this case, pair $A\mathord{\varparallelinv}{d}$ satisfies the requirements of Lemma \ref{Clemma.3} with $p_i=i$ and $n=t-1$. Therefore, the pair $C$ of the triangulations $C^-$ and $C^+$ obtained from $A^-\mathord{\varparallelinv}{d}$ and $A^+\mathord{\varparallelinv}{d}$ by deleting vertices $0$, ..., $t-2$ except, say, vertex $r$, satisfies
\begin{equation}\label{Clemma.7.eq3}
\delta(A\mathord{\varparallelinv}{d})\geq\delta(C)+2(t-2)\mbox{.}
\end{equation}
Observe that $\{r,t-1\}$ is a boundary edge of $C^-$ and $C^+$ that is not incident to the same triangle in these two triangulations. More precisely, the triangle incident to $\{r,t-1\}$ in $C^-$ has a vertex greater than $t$, while its counterpart in $C^+$ has vertices $r$, $t-1$, and $t$. Hence, at least one flip is incident to $\{r,t-1\}$ along any geodesic between $C^-$ and $C^+$. By construction, $B^-$ and $B^+$ are obtained by deleting vertex $r$ from $C^-$ and from $C^+$, and by relabeling the vertices of the resulting triangulations. As a consequence, by Lemma \ref{Clemma.1},
\begin{equation}\label{Clemma.7.eq4}
\delta(C)\geq\delta(B)+1\mbox{.}
\end{equation}
Combining (\ref{Clemma.7.eq3}) with (\ref{Clemma.7.eq4}) therefore proves inequality (\ref{Clemma.7.eq2}). \qed
\end{proof}
The requirements of the next theorem (Theorem \ref{Ctheorem.2} below) are complementary to those of the last two lemma. In the proof of this theorem, the following lemma will instrumental. In particular, it will be invoked twice.
\begin{lemma}\label{Csection.3.lemma.1}
Consider an $(a,b,c,d)$-pair $A$. Let $T$ be some triangulation along a geodesic between $A^-$ and $A^+$. If $T$ has an interior edge whose two vertices belong to $\{c, ..., d, \bar{0}\}$, then there exists a vertex $x$ so that $c\leq{x}\leq{d}$ and
$$
\delta(A)\geq\delta(A\mathord{\varparallelinv}{x})+3\mbox{.}
$$
\end{lemma}
\begin{proof}
Consider a geodesic $(T_i)_{0\leq{i}\leq{k}}$ from $A^-$ to $A^+$ and consider an integer $j$ so that $0\leq{j}\leq{k}$. Assume that triangulation $T_j$ has an interior edge whose two vertices belong to $\{c, ..., d, \bar{0}\}$. Denote these vertices by $q$ and $r$ with the convention that $q$ is less than $r$.
In this case, triangulation $T_j$ necessarily has an ear at some vertex $p$ so that $q<p<r$. Indeed, since $T_j$ contains $\{q,r\}$, it induces a triangulation $U$ of the polygon whose vertices are vertices $q$ to $r$. Any triangulation of a polygon with at least four vertices has at least two ears. Hence, if $U$ has at least four vertices, then one of its ears is an ear at some vertex $p$ so that $q<p<r$. If $U$ has exactly three vertices, then it is made up of a single triangle, and this triangle is an ear at vertex $p=q+1$. Note that $U$ cannot have less than $3$ vertices because $\{q,r\}$ is an interior edge of $T_j$. Therefore, $U$ always has an ear at some vertex $p$ so that $q<p<r$ and this ear is also an ear of $T_j$.
Now observe that triangulations $T_j$ and $A^+$ satisfy the conditions of Lemma \ref{Clemma.2}. Note, in particular, that the triangles of $A^+$ incident to $\{p-1,p\}$ and to $\{p,p+1\}$ do not share an edge because, by definition, $A^+$ has a comb at vertex $p$. Therefore, there exists $x\in\{p-1,p\}$ so that
\begin{equation}\label{Csection3.lemma.1.eq.1}
k-j\geq\delta(\{T_j\mathord{\varparallelinv}{x},A^+\mathord{\varparallelinv}{x}\})+2\mbox{.}
\end{equation}
Further note that $\{x,x+1\}$ is not incident to the same triangle in $A^-$ and in $T_j$ because $A^-$ does not have an ear at vertex $p$. Hence, at least one of the first $j$ flips along $(T_i)_{0\leq{i}\leq{k}}$ is incident to $\{x,x+1\}$. By Lemma \ref{Clemma.1},
\begin{equation}\label{Csection3.lemma.1.eq.2}
j\geq\delta(\{A^-\mathord{\varparallelinv}{x},T_j\mathord{\varparallelinv}{x}\})+1\mbox{.}
\end{equation}
Since $k$ is precisely the distance of pair $A$, combining (\ref{Csection3.lemma.1.eq.1}) with (\ref{Csection3.lemma.1.eq.2}) and using the triangle inequality yields
$$
\delta(A)\geq\delta(A\mathord{\varparallelinv}{x})+3\mbox{.}
$$
Finally, as $q<p<r$ and as $q$ and $r$ belong to $\{c, ..., d, \bar{0}\}$, vertex $p$ must satisfy $c<p\leq{d}$. Since $x\in\{p-1,p\}$, this proves that $c\leq{x}\leq{d}$. \qed
\end{proof}
The following theorem, whose conditions are complementary to those of Lemmas \ref{Clemma.X1} and \ref{Clemma.X3} can now be proven. It provides the first deletion for a sequence of deletions under which the family of $(a,b,c,d)$-pairs is stable:
\begin{theorem}\label{Ctheorem.2}
Consider an $(a,b,c,d)$-pair $A$ so that $c<d$. If at most $2$ flips are incident to edge $\{0,1\}$ along a geodesic between $A^-$ and $A^+$, then there exists a vertex $x$ of these triangulations so that $c\leq{x}\leq{d}$ and
$$
\delta(A)\geq\delta(A\mathord{\varparallelinv}{x})+3\mbox{.}
$$
\end{theorem}
\begin{proof}
Consider some geodesic $(T_i)_{0\leq{i}\leq{k}}$ from $A^-$ to $A^+$. Assume that $c<d$ and that that at most $2$ flips are incident to edge $\{0,1\}$ along $(T_i)_{0\leq{i}\leq{k}}$. If $3$ or more flips are incident to $\{\bar{0},d\}$ along this geodesic, then Lemma \ref{Clemma.1} immediately provides the desired result with $x=d$.
\begin{figure}
\begin{centering}
\includegraphics{Thm2-1}
\caption{The $j$-th (top left), $j'$-th (top right), and $j''$-th (bottom) flips along path $(T_i)_{0\leq{i}\leq{k}}$ used in the proof of Theorem \ref{Ctheorem.2}, in case the first of these flips introduces edge $\{\bar{0},z\}$ with $z<d$. The edges introduced by each flip are dotted.}\label{Cfigure.thm2.1}
\end{centering}
\end{figure}
Hence, it is further assumed that at most $2$ flips are incident to edge $\{\bar{0},d\}$ along $(T_i)_{0\leq{i}\leq{k}}$.
Observe that $\{\bar{0},d\}$ is not incident to the same triangle in $A^-$ and in $A^+$. Hence, at least one flip must be incident to $\{\bar{0},d\}$ along $(T_i)_{0\leq{i}\leq{k}}$. In fact, there must be exactly two such flips. Indeed, the unique such flip would otherwise replace the diagonal $\{0,\bar{0}\}$ by edge $\{\bar{1},d\}$. However, since $d$ is greater than $1$, the latter edge cannot be a diagonal and such a flip is therefore impossible. Assume that the two flips incident to edge $\{\bar{0},d\}$ along $(T_i)_{0\leq{i}\leq{k}}$ are the $j$-th one and the $j'$-th one, with $j<j'$. Observe that the $j$-th flip along $(T_i)_{0\leq{i}\leq{k}}$ either introduces edge $\{\bar{0},z\}$ where $0<z<d$ or edge $\{d,\bar{z}\}$ where $0<z\leq{d}$. The two cases will be reviewed separately.
First assume that the $j$-th flip along $(T_i)_{0\leq{i}\leq{k}}$ introduces edge $\{\bar{0},z\}$ where $0<z<d$. This flip must then be as shown top left on Fig. \ref{Cfigure.thm2.1}. In this case, the $j'$-th flip along $(T_i)_{0\leq{i}\leq{k}}$ necessarily replaces $\{\bar{0},z\}$ by $\{\bar{1},d\}$ as shown top right on the same figure. Observe that the latter flip is incident to $\{0,1\}$. There must be at least one other flip incident to this edge, taking place earlier along path $(T_i)_{0\leq{i}\leq{k}}$ because edge $\{0,1\}$ is incident to distinct triangles in $A^-$ and in $T_{j'-1}$. Say the first flip incident to $\{0,1\}$ along $(T_i)_{0\leq{i}\leq{k}}$ is the $j''$-th one. Since at most $2$ flips along path $(T_i)_{0\leq{i}\leq{k}}$ are incident to edge $\{0,1\}$, then the $j'$-th and the $j''$-th flip are the only two such flips. In particular, the latter flip must replace the triangle of $A^-$ incident to $\{0,1\}$ by the triangle of $T_{j'-1}$ incident to the same edge. Therefore this flip removes $\{0,c\}$ and replaces it by $\{1,\bar{z}\}$ as shown in the bottom of Fig. \ref{Cfigure.thm2.1}. As one can see in the figure, $z$ cannot be less than $c$. Indeed, triangulation $T_{j''-1}$ would otherwise contain crossing edges, as for instance $\{0,c\}$ and $\{\bar{0},z\}$.
This shows that $\{\bar{0},z\}$ is an interior edge of $T_{j''}$ whose two vertices belong to $\{c, ..., d, \bar{0}\}$. Therefore, the desired result follows from Lemma \ref{Csection.3.lemma.1}.
Now assume that the $j$-th flip along path $(T_i)_{0\leq{i}\leq{k}}$ introduces edge $\{d,\bar{z}\}$ where $0<z\leq{d}$. Observe that this flip removes the diagonal $\{0,\bar{0}\}$ from $A^-$. Therefore, $\{d,\bar{z}\}$ must also be a diagonal, which proves that $z=d$.
\begin{figure}
\begin{centering}
\includegraphics{Thm2-2}
\caption{The $j$-th (top left), $j'$-th (top right), and $j''$-th (bottom left) flips along path $(T_i)_{0\leq{i}\leq{k}}$ used in the proof of Theorem \ref{Ctheorem.2} when the first of these flips introduces edge $\{d,\bar{z}\}$ where $0<z\leq{d}$. The flip shown bottom right cannot be performed within a centrally symmetric triangulation, because some edges of this triangulation would otherwise be crossing. The edges introduced by each flip are dotted.}\label{Cfigure.thm2.2}
\end{centering}
\end{figure}
In this case, the $j$-th flip along path $(T_i)_{0\leq{i}\leq{k}}$ is necessarily the one shown top left on Fig. \ref{Cfigure.thm2.2}, and the $j'$-th flip along $(T_i)_{0\leq{i}\leq{k}}$ must replace $\{\bar{0},\bar{d}\}$ by $\{\bar{1},d\}$ as shown top right on the same figure. Observe that the latter flip is incident to edge $\{0,1\}$. There must be at least one other flip incident to this edge, taking place earlier along path $(T_i)_{0\leq{i}\leq{k}}$. Indeed, edge $\{0,1\}$ is incident to distinct triangles in $A^-$ and in $T_{j'-1}$ because $c$ is less than $d$. Say the first flip incident to $\{0,1\}$ along $(T_i)_{0\leq{i}\leq{k}}$ is the $j''$-th one. Since at most $2$ flips along path $(T_i)_{0\leq{i}\leq{k}}$ are incident to edge $\{0,1\}$, then the $j'$-th and the $j''$-th flip are the only two such flips. In particular, the latter flip must replace edge $\{0,c\}$ by edge $\{1,d\}$ as shown bottom left on Fig. \ref{Cfigure.thm2.2}.
Observe that triangulation $T_{j''}$ contains edge $\{c,d\}$. If $c$ is less than $d-1$, then this edge is an interior edge of $T_{j''}$ and the result follows from Lemma \ref{Csection.3.lemma.1}. If $c$ is equal to $d-1$, then $\{c,d\}$ is a boundary edge of $T_{j''}$ and the $j''$-th flip along $(T_i)_{0\leq{i}\leq{k}}$ is incident to it. Observe that $\{c,d\}$ is not incident to the same triangle in $T_{j''}$ and in $A^+$. Hence, at least one of the last $k-j''$ flips along $(T_i)_{0\leq{i}\leq{k}}$ is incident to edge $\{c,d\}$. In fact, there must be at least two such flips. Otherwise, the unique such flip must replace $\{1,d\}$ by $\{c,\bar{t}\}$ as shown bottom right on Fig. \ref{Cfigure.thm2.2}, where $t$ is the vertex so that edges $\{c,\bar{t}\}$ and $\{d,\bar{t}\}$ belong to triangulation $A^+$. In particular, $1<t<d$. This flip cannot occur within a centrally symmetric triangulation. Indeed, by symmetry, this triangulation would then also contain edge $\{\bar{1},\bar{d}\}$, sketched as a thin line in Fig. \ref{Cfigure.thm2.2}. However, $\{\bar{1},\bar{d}\}$ crosses at least two edges of the triangulation as, for instance edges $\{1,\bar{t}\}$ and $\{d,\bar{t}\}$.
This shows that at least two of the last $k-j''$ flips along $(T_i)_{0\leq{i}\leq{k}}$ are incident to edge $\{c,d\}$. Since the $j''$-th flip along this path is also incident to $\{c,d\}$, then Lemma \ref{Clemma.1} provides the desired result with $x=c$. \qed
\end{proof}
Consider an $(a,b,c,d)$-pair $A$ satisfying the requirements of Theorem \ref{Ctheorem.2}. When the vertex $x$ provided by this theorem satisfies $c\leq{x}<d$, then its deletion from $A^-$ and from $A^+$ results in an $(a,b,c,d-1)$-pair up to the vertex labels of the resulting triangulations. In particular:
\begin{lemma}\label{Clemma.X4}
Consider an $(a,b,c,d)$-pair $A$. If some vertex $x$ of triangulations $A^-$ and $A^+$ satisfies $c\leq{x}<d$, then there exists an $(a,b,c,d-1)$-pair $B$ whose distance is equal to that of pair $A\mathord{\varparallelinv}{x}$.
\end{lemma}
\begin{proof}
Let $x$ be a vertex of triangulations $A^-$ and $A^+$ so that $c\leq{x}<d$. Consider the triangulations $B^-$ and $B^+$ obtained by relabeling the vertices of $A^-\mathord{\varparallelinv}{x}$ and $A^+\mathord{\varparallelinv}{x}$ clockwise from $0$ to $2d-1$ in such a way that vertex $0$ keeps its label. Call $B$ the pair $\{B^-,B^+\}$.
It turns out that $B$ is an $(a,b,c,d-1)$-pair. In particular, the quadruple $(a,b,c,d-1)$ satisfies (\ref{Cequation.0.1}), (\ref{Cequation.0.1.5}), and (\ref{Cequation.0.2}) because $c$ is less than $d$. Moreover, the total number of interior teeth of triangulations $A^-$ and $A^+$ is not changed by the deletion. Indeed, this deletion removes an interior tooth of the comb at vertex $0$ in $A^-$, but it also creates a new interior tooth as it merges the combs at vertices $x$ and $x+1$ within triangulation $A^+$. \qed
\end{proof}
When the vertex $x$ provided by Theorem \ref{Ctheorem.2} is equal to $d$, then its deletion only initiates a sequence of deletions that leaves the family of $(a,b,c,d)$-pairs stable. The rest of the sequence will be obtained from Lemma \ref{Clemma.3}:
\begin{lemma}\label{Clemma.X2}
Consider an $(a,b,c,d)$-pair $A$. Call $t$ the number of interior edges of $A^+$ incident to vertex $d$. If $a+b/2+2<d$ and if $c<d$, then there exists an integer $c'$ satisfying $b-2t+2<c'\leq{c-t+1}$ and an $(a,b-2t+2,c',d-t)$-pair $B$ so that the following inequality holds:
$$
\delta(A\mathord{\varparallelinv}{d})\geq\delta(B)+2(t-1)\mbox{.}
$$
\end{lemma}
\begin{proof}
Assume that $a+b/2+2<d$ and that $c<d$. Consider triangulations $A^-\mathord{\varparallelinv}{d}$ and $A^+\mathord{\varparallelinv}{d}$, depicted in Fig. \ref{Cfigure.3}.
\begin{figure}
\begin{centering}
\includegraphics{LemmaStable2}
\caption{Sketch of triangulations $A^-\mathord{\varparallelinv}{d}$ (left) and $A^+\mathord{\varparallelinv}{d}$ (right) when $c<d$, where $k$ and $l$ are respectively equal to $\lfloor{b/2}\rfloor+1$ and $a+b-c+4$. For the sake of clarity, the placement of the vertices slightly differs in the two triangulations.}\label{Cfigure.3}
\end{centering}
\end{figure}
Observe that pair $A\mathord{\varparallelinv}{d}$ satisfies the requirements of Lemma \ref{Clemma.3} with $p_i=i$ and $n=t$. This lemma therefore provides two triangulations $B^-$ and $B^+$ obtained from $A^-\mathord{\varparallelinv}{d}$ and $A^+\mathord{\varparallelinv}{d}$ by deleting all but one of the vertices $0$, ..., $t-1$, so that:
$$
\delta(A\mathord{\varparallelinv}{d})\geq\delta(B)+2(t-1)\mbox{,}
$$
where $B$ denotes the pair $\{B^-,B^+\}$. Let $r$ be the vertex among vertices $0$ to $t-1$ that is not deleted in the process. Note that, when the sequence of deletions is carried out within $A^-\mathord{\varparallelinv}{d}$, each of these deletions merges the comb at the deleted vertex with the comb at the next vertex clockwise. This results into a comb at vertex $r$ and a comb at vertex $t$ in $B^-$. Observe that the total number of interior teeth of $A^-$ and $A^+$ is not modified by the sequence of deletions carried out to build $B^-$ and $B^+$. Indeed, the deletion of vertex $d$ from $A^-$ removes one interior tooth from the comb at vertex $0$, and each of the other deletions creates a new interior tooth within this triangulation as these deletions each merge two combs. Therefore, recalling that these triangulations are centrally symmetric, the number of interior teeth of $B^-$ is greater by $2(t-2)$ than that of $A^-$. On the other hand, each of the deletions carried out within triangulation $A^+$ removes one tooth from the comb originally attached at vertex $d$. As these teeth are interior teeth of $A^+$ except for two of them, $B^+$ has $2(t-2)$ interior teeth less than $A^+$.
Relabel the vertices of $B^-$ and $B^+$ clockwise from $0$ to $2d-2t+1$ in such a way that vertices $r$ and $t$ are respectively relabeled $0$ and $1$. After doing so, $B$ is an $(a,b-2t+2,c',d-t)$-pair, where $c'$ is equal to $c-t+1$ if $r=0$ and to $b-t-r+2$ otherwise. In particular, the desired bounds on $c'$ hold. Also, note that the quadruple $(a,b-2t+2,c',d-t)$ satisfies inequalities (\ref{Cequation.0.1}), (\ref{Cequation.0.1.5}), and (\ref{Cequation.0.2}) because $a+b/2+2$ and $c$ are both less than $d$. \qed
\end{proof}
Using Theorem \ref{Ctheorem.2}, and Lemmas \ref{Clemma.X1}, \ref{Clemma.X3}, \ref{Clemma.X4}, and \ref{Clemma.X2}, one can now prove Theorem \ref{Ctheorem.1}. Recall that, according to this theorem, any $(a,b,c,d)$-pair $A$ satisfies
$$
\delta(A)\geq3d-\left(\frac{b}{2}+\frac{2c-b}{a}+3a+5\right)\mbox{.}
$$
\begin{proof}[Theorem \ref{Ctheorem.1}]
Consider an $(a,b,c,d)$-pair $A$. The theorem will be proven by induction on $d-\lfloor{b/2}\rfloor$. Note that, according to inequality (\ref{Cequation.0.2}), this quantity is greater than $a+1$.
First assume that $d-\lfloor{b/2}\rfloor$ is equal to $a+2$. Observe that $A^-$ and $A^+$ have no edge in common. Therefore, any geodesic between these triangulations has length at least $d$. Indeed, each flip along such a geodesic either removes a pair of centrally symmetric interior edges, or a diagonal. As $A^-$ has exactly $d-1$ pairs of centrally symmetric interior edges, and one diagonal, at least $d$ flips must be performed in order to remove all the edges of $A^-$, and therefore, to transform this triangulation into $A^+$:
\begin{equation}\label{Cequation.proofTh1.1}
\delta(A)\geq{d}\mbox{.}
\end{equation}
As $d=a+\lfloor{b/2}\rfloor+2$, one immediately obtains:
\begin{equation}\label{Cequation.proofTh1.1.5}
d\leq{a+\frac{b}{2}+2}\mbox{.}
\end{equation}
According to (\ref{Cequation.0.1}), $b\leq{d-1}$. Therefore, (\ref{Cequation.proofTh1.1.5}) yields:
\begin{equation}\label{Cequation.proofTh1.1.7}
d\leq{2a+3}\mbox{.}
\end{equation}
As the ratio of $2c-b$ and $a$ is positive, summing (\ref{Cequation.proofTh1.1.5}) with (\ref{Cequation.proofTh1.1.7}), and adding this ratio to the right-hand side results in the following inequality:
\begin{equation}\label{Cequation.proofTh1.2}
2d\leq\frac{b}{2}+\frac{2c-b}{a}+3a+5\mbox{.}
\end{equation}
One then obtains the desired inequality by combining (\ref{Cequation.proofTh1.1}) with (\ref{Cequation.proofTh1.2}). Now assume that $d-\lfloor{b/2}\rfloor$ is greater than $a+2$. Further assume that, for any $(a,b',c',d')$-pair $B$ so that $d'-\lfloor{b'/2}\rfloor$ is less than $d-\lfloor{b/2}\rfloor$ by $1$,
\begin{equation}\label{Cequation.proofTh1.4}
\delta(B)\geq3d'-\left(\frac{b'}{2}+\frac{2c'-b'}{a}+3a+5\right)\mbox{.}
\end{equation}
Recall that the conditions of Lemma \ref{Clemma.X1}, of Lemma \ref{Clemma.X3}, and of Theorem \ref{Ctheorem.2} are complementary. In the remainder of the proof, these conditions are reviewed one after the other, and the result is proven in each case.
First assume that at least $3$ flips are incident to $\{0,1\}$ along some geodesic between $A^-$ and $A^+$. In this case, the requirements of Lemma \ref{Clemma.X1} are satisfied (in particular, $a+b/2+2<d$ precisely because $d-\lfloor{b/2}\rfloor$ is greater than $a+2$), and this lemma provides an $(a,b',c',d')$-pair $B$ so that
\begin{equation}\label{Cequation.proofTh1.3}
\delta(A)\geq\delta(B)+5\mbox{,}
\end{equation}
where $b'$, $c'$, and $d'$ are respectively equal to $b-2$, $b-1$, and $d-2$. Note that $d'-\lfloor{b'/2}\rfloor$ is less than $d-\lfloor{b/2}\rfloor$ by $1$. Hence, by induction, inequality (\ref{Cequation.proofTh1.4}) holds for pair $B$. Combining this inequality with (\ref{Cequation.proofTh1.3}), and replacing $b'$, $c'$, and $d'$ by respectively $b-2$, $b-1$, and $d-2$, one obtains
$$
\delta(A)\geq3d-\left(\frac{b}{2}+\frac{b}{a}+3a+5\right)\mbox{.}
$$
Since $b\leq2c-b$, this proves that the result holds when at least $3$ flips are incident to $\{0,1\}$ along some geodesic between $A^-$ and $A^+$.
Now assume that $c=d$. Again, $a+b/2+2$ is less than $d$ because $d-\lfloor{b/2}\rfloor$ is greater than $a+2$. Therefore, by Lemma \ref{Clemma.X3}, there exists an $(a,b',c',d')$-pair $B$ that satisfies the following inequality:
\begin{equation}\label{Cequation.proofTh1.7}
\delta(A)\geq\delta(B)+2t-1\mbox{,}
\end{equation}
where $t$ is the number of interior edges of $A^+$ incident to vertex $d$. Moreover, $b'=b-2t+2$, $c'=b-2t+3$, and $d'=d-t$. Note that $d'-\lfloor{b'/2}\rfloor$ is less than $d-\lfloor{b/2}\rfloor$ by $1$. Hence, by induction, inequality (\ref{Cequation.proofTh1.4}) also holds in this case. Combining this inequality with (\ref{Cequation.proofTh1.7}) and replacing $b'$, $c'$ and $d'$ by respectively $b-2t+2$, $b-2t+2$ and $d-t$ yields:
\begin{equation}\label{Cequation.proofTh1.8}
\delta(A)\geq3d-\left(\frac{b}{2}+\frac{b-2t+4}{a}+3a+7\right)\mbox{.}
\end{equation}
Observe that, since $c=d$, the only combs in $A^+$ are the combs at vertices $d$ and $\bar{d}$. In particular, the number of interior teeth of $A^+$ is exactly $2t-4$. Moreover, $A^-$ has exactly $2c-2b-2$ interior teeth. As $a$ is obtained by adding $1$ to half the total number of interior teeth in $A^-$ and $A^+$, it follows that $t$ is equal to $a+b-c+2$. Replacing $t$ within (\ref{Cequation.proofTh1.8}) by the latter expression results in the desired inequality, and the result holds in this case.
Finally, assume that $c<d$ and that at most $2$ flips are incident to $\{0,1\}$ along some geodesic between $A^-$ and $A^+$. By Theorem \ref{Ctheorem.2}, there exists a vertex $x$ of $A^-$ and $A^+$ so that $c\leq{x}\leq{d}$ and
\begin{equation}\label{Cequation.proofTh1.48}
\delta(A)\geq\delta(A\mathord{\varparallelinv}{x})+3\mbox{.}
\end{equation}
If $c\leq{x}<d$, then Lemma \ref{Clemma.X4} provides an $(a,b',c',d')$-pair $B$ with the same distance as $A\mathord{\varparallelinv}{x}$, where $b'=b$, $c'=c$, and $d'=d-1$. Note, in particular, that $d'-\lfloor{b'/2}\rfloor$ is less than $d-\lfloor{b/2}\rfloor$ by $1$. Hence, by induction, (\ref{Cequation.proofTh1.4}) holds for pair $B$. Replacing $A\mathord{\varparallelinv}{x}$ by $B$ in (\ref{Cequation.proofTh1.48}) and combining the resulting inequality with (\ref{Cequation.proofTh1.4}) provides the result. Now, if $x=d$, then the conditions of Lemma \ref{Clemma.X2} are satisfied. In particular, $a+b/2+2<d$ precisely because $d-\lfloor{b/2}\rfloor$ is greater than $a+2$. Hence, there exists an $(a,b',c',d')$-pair $B$ so that
\begin{equation}\label{Cequation.proofTh1.6}
\delta(A\mathord{\varparallelinv}{x})\geq\delta(B)+2(t-1)\mbox{,}
\end{equation}
where $t$ is the number of interior edges of $A^+$ incident to vertex $d$. Moreover, $b'=b-2t+2$, $d'=d-t$, and $c'$ satisfies
$$
b-2t+2<c'\leq{c-t+1}\mbox{.}
$$
Note that $d'-\lfloor{b'/2}\rfloor$ is less than $d-\lfloor{b/2}\rfloor$ by $1$. Therefore, by induction, inequality (\ref{Cequation.proofTh1.4}) also holds in this case. Combining (\ref{Cequation.proofTh1.4}), (\ref{Cequation.proofTh1.48}), and (\ref{Cequation.proofTh1.6}), and replacing $b'$, and $d'$ by respectively $b-2t+2$ and $d-t$ yields:
$$
\delta(A)\geq3d-\left(\frac{b}{2}+\frac{2c'-b+2t-2}{a}+3a+5\right)\mbox{.}
$$
Therefore, the result holds because $c'$ is not greater than $c-t+1$. \qed
\end{proof}
\section{Discussion}
\label{Csection.4}
It has been shown in this article that the diameter $\Delta$ of the $d$-dimensional cyclohedron is at least $5d/2-4\sqrt{d}-4$ and at most $\lceil{5d/2}\rceil-2$. In particular, this diameter grows like $5d/2$ when $d$ is large:
$$
\lim_{d\rightarrow\infty}\frac{\Delta}{d}=\frac{5}{2}\mbox{.}
$$
The values of $\Delta$ when $d$ is small can be obtained computationally. These values deceptively suggest a $7/3$ coefficient instead of the above $5/2$:
$$
\begin{array}{c|>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}>{\centering\arraybackslash$} p{0.6cm} <{$}}
d & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13\\
\hline
\Delta & 1 & 3 & 5 & 7 & 9 & 11 & \bm{14} & 16 & 18 & \bm{21} & 23 & 25 & \bm{28}^\star\\
\end{array}
$$
In this table, the starred value is only a lower bound on $\Delta$ that is not necessarily sharp. An exact computation was not possible in this case due to prohibitive computation time. Each value of $\Delta$ is greater by $2$ than the preceding value, except for the ones shown in bold characters, corresponding to $d=7$ and $d=10$, that are greater by $3$ than the preceding value. Note that the lower bound on $\Delta$ when $d$ is equal to $13$, also shown in bold, is greater by $3$ than the preceding value in the table as well.
The last of the three infinite subfamilies of generalized associahedra whose diameter is not known exactly, but only asymptotically, is that of cyclohedra. However, the asymptotic behavior of this diameter is very different from its behavior at low dimensions, which suggests that evaluating the exact diameter of cyclohedra for all dimensions may turn out to be difficult.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,047 |
In the name of of Allah the Merciful
The College of Computer Science and Information Technology 2014-2015 emerged from the womb of the ancient mother college - the College of Science at the University of Basra and from the Department of Computer Science, which was the first computer department in Iraq, which was established in 1983-1984. The college opened in two departments, the Department of Computer Science and the Department of Computer Information Systems, with distinguished scientific and practical cadres, with great efforts, and in modest numbers. Our college continued to support and teach the students of our mother college until the last student graduated from the computer science department.
Our college has been unique by this title to award a bachelor's degree in the disciplines of computer science and computer information systems and to prepare qualified human cadres in the field of information technology in its various branches to meet the requirements of the labor market and keep pace with development in its various fields and conduct scientific research, theoretical and applied that help in community service and contribute to building and developing the knowledge society .
The college now receives in its two departments, morning and evening students and parallel education students.
He created a master's study for the departments of Computer Science and Computer Information Systems 2019-2020, and 20 students joined the study, and it is hoped that the numbers will increase in the coming years.
We aspire to create new departments such as the Department of Network and Internet Technology, the Department of Artificial Intelligence and the Department of Software Engineering.
We aspire to create and open a doctoral study in the computer and systems departments, and to exploit the mighty energies of scientific competencies, all of which serves our great Iraq.
We aspire to open areas of cooperation and conclude memoranda of understanding with corresponding colleges.
We aspire to open up to all sectors of society and government service departments, and to provide services and consultations by activating the role of the computer advisory office.
And last but not least, we aspire to make our college the College of Computer Science and Information Technology at the University of Basra a pioneering and distinguished scientific edifice in the field of education and scientific research and a center for innovation and creativity to serve the local, regional and international community. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,094 |
The 42nd annual presentation of this advanced program, featuring 16 hours of instruction, particularly looks at legislative initiatives, regulatory changes, recent judicial decisions, and new precedent issued by the Obama administration.
Venable partner Margaret Strand will focus on Wetlands Law and Regulation.
Please visit the ALI-ABA website for registration and more information. | {
"redpajama_set_name": "RedPajamaC4"
} | 169 |
Love Yourself 結 'Answer' (estilizado como LOVE YOURSELF 結 'Answer') es el tercer álbum recopilatorio del grupo surcoreano BTS. El álbum fue lanzado el 24 de agosto de 2018 por Big Hit Entertainment y está disponible en cuatro versiones diferentes: S, E, L y F. El álbum contiene veinticinco canciones, incluyendo siete nuevas. La mayoría de los temas son de Love Yourself: Her, y Love Yourself: Tear, así como algunos remixes.
Antecedentes y lanzamiento
Answer se anunció por primera vez el 16 de julio de 2018. El álbum fue diseñado como el final de la serie Love Yourself, que conectó el argumento de la historia del cortometraje Love Yourself: Wonder, el EP Love Yourself: Her, y el álbum de estudio Love Yourself: Tear. El 6 de agosto, Big Hit Entertainment lanzó una nota sobre Most Beautiful Moment in Life, parte de una serie de notas ficticias pertenecientes al concepto de álbumes de BTS, comenzando con el EP The Most Beautiful Moment in Life, Part 1 (2015). La nota, escrita por Jin, integrante del grupo, habló sobre cómo encontró un cuaderno escrito por su padre en la historia, donde redactaba los fracasos del mismo. Tres días más tarde, el 9 de agosto, se lanzó el avance del álbum, con una nueva canción titulada «Epiphany». La canción, descrita como una «melodía de pop-rock», fue interpretada por Jin, y habla de encontrar el amor propio. El tráiler fue dirigido por Yong-seok Choi (Lumpens), y retrata varias versiones de Jin en una habitación, contando la historia de un viaje personal. Philip Merrill, de la Academia de Grabación, describió el vídeo como una «sensación de ir hacia atrás y hacia adelante, a través del tiempo y a través de diferentes versiones de uno mismo». Discutiendo sobre el trabajo de Jin, Hong Hye-min de The Korea Times declaró que fue tan bien actuado y emotivo, con la «voz triste y libre de Jin».
Recepción
Comentarios de la crítica
Love Yourself: Answer recibió reseñas positivas de los críticos. Nemo Kim, de South China Morning Post, le dio cuatro estrellas de cinco al álbum y mencionó que «la banda envía un mensaje a la juventud coreana acerca del amor propio, y el sencillo «Idol» mezcla el pop con instrumentos de percusión tradicionales»; alabó al álbum por «cimentar la posición de BTS como reyes del género». Por otro lado, Taylor Glasby, de la revista Clash, describió la narrativa del álbum como «una representación de los miedos, errores y pensamientos que nos autoinfligimos» y concluyó en que lo que el álbum hace es «recordarnos que cada persona debe forjar su propio camino ya que no existe una única respuesta para alcanzar la autoaceptación». Tamar Herman, de Billboard, elogió el álbum y comentó que es una «culminación maestra de años de trabajo llenos de significado, Answer es indudablemente un «magnum opus de BTS que pocos artistas o boy bands pueden esperar alcanzar».
Promoción
El 26 de agosto de 2018, BTS tuvo una conferencia de prensa para hablar sobre su álbum y también acerca de la gira BTS World Tour: Love Yourself, que comenzó el día anterior. Asimismo, interpretaron «Idol» y «I'm Fine» en varios programas de música de Corea del Sur en la semana del 26 de agosto, incluyendo M Countdown, Music Bank, Show! Music Core, e Inkigayo, mientras que las presentaciones pregrabadas salieron al aire la semana siguiente, Una versión corta de «Save Me», del álbum The Most Beautiful Moment in Life: Young Forever, se usó como pista de introducción durante la interpretación de «I'm Fine» en M Countdown. El grupo presentó «Idol» y «Fake Love» en los Soribada Best K-Music Awards el 30 de agosto.
Mientras el grupo estuvo en Estados Unidos, para la etapa norteamericana de su gira Love Yourself, interpretó «Idol» en la segunda semifinal de la décimo tercera temporada de America's Got Talent, que se transmitió el 12 de septiembre. BTS también presentó el tema en The Tonight Show Starring Jimmy Fallon y Good Morning America el 25 y 26 de septiembre respectivamente, además de ser entrevistados en ambos programas. Una presentación adicional de «I'm Fine» también fue publicada en el canal de YouTube de The Tonight Show.
El 1 de octubre de 2018, varios medios de comunicación dieron a conocer que BTS aparecería en el programa The Graham Norton Show de BBC el 12 de octubre, lo cual fue confirmado 5 días después por Big Hit Entertainment.
Lista de canciones
Posicionamiento en listas
Álbum
Semanales
Anuales
Sencillos
«IDOL»
Otras canciones
«Euphoria»
«Serendipity»
«Singularity»
«Epiphany»
«I'm Fine»
«Answer: Love Myself»
Certificaciones
Ventas
Reconocimientos
Historial de lanzamientos
Referencias
Álbumes recopilatorios de 2018
Álbumes de BTS
Álbumes de Hybe Corporation
Álbumes de Big Hit Music | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,878 |
Q: Functions type converting How can I convert func add (a, b int) int to func(...interface{}) interace{} type ?
Any ideas about implementing generic functions using the reflect package ?
A: As JimB said, you can't cast in Go and you cannot convert functions just like that but by using closures, you can rapidly wrap your function:
func add(a, b int) int {
return a + b;
}
wrap := func(args ...interface{}) interface{} {
return interface{} (add(args[0].(int), args[1].(int)))
}
Note that wrap will panic if you give it arguments that are not of type int. If you want to avoid that you can slightly modify wrap:
wrap := func(args ...interface{}) (interface{}, error) {
a, k := args[0].(int)
b, l := args[1].(int)
if !k || !l {
return nil, errors.New("Arguments must be of type int")
}
return add(a,b), nil
}
If you'd like to do different things with wrap, depending on it's arguments types you can do so by using a type switch:
func addInts(a, b int) int {
return a + b;
}
func addFloat64s(a, b float64) float64 {
return a + b;
}
wrap := func(args ...interface{}) interface{} {
switch args[0].(type) {
case int: return interface{}(addInts(args[0].(int), args[1].(int)))
case float64: return interface{}(addFloat64s(args[0].(float64), args[1].(float64)))
}
}
Note that this last version of wrap makes the assumption that all given parameters will have the same type and at least 2 arguments are given.
A: There is no "casting" is go (well, using the "unsafe" package kind of is like casting).
You cannot convert function types like this, since they have different layouts in memory. Generic-like functions can be made through the reflect package, though with significant overhead. See http://golang.org/pkg/reflect/#example_MakeFunc for an example.
For most use cases of generic functions, you're probably better off accepting an interface, and using type assertions or switches (http://golang.org/ref/spec#Type_switches), rather than the reflection library.
| {
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Scraptia immaculata es una especie de coleóptero de la familia Scraptiidae.
Distribución geográfica
Habita en Filipinas.
Referencias
Immaculata
Coleópteros de Filipinas | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,794 |
There is no margin for error when translating financial documents, and quality cannot be compromised in the name of urgency. Intertranslations delivers reliable, integrated translation solutions to banks, financial institutions, insurance companies and asset managers worldwide, around the clock. We have developed strong industry expertise and proudly provide ISO-approved systems and processes that guarantee security and accuracy for your financial translation needs.
Over the past 20 years working in the financial and banking sectors, Intertranslations has developed expertise in financial terminology, document types, and regulations ensuring consistency across all languages. We strive to fully understand your business and deliver financial translation services that effectively communicate your message to a global market.
Our expert financial translators have a minimum of five years experience and translate only into their mother tongue. For example, English to Arabic financial translations would be completed by a native Arabic speaker with financial expertise. All linguists have signed a comprehensive non-disclosure agreement (NDA) so our clients can rest assured that projects remain confidential and within US financial regulations.
Our experienced project managers and our financial translation teams collaborate closely in order to meet the client's unique requirements, making sure that projects run smoothly and are aligned with your precise expectations.
Our quality assurance cycle ensures top quality and consistency throughout the project, whether that's a single document translation or a large multi-faceted and urgent translation project. The project team acts as a single point of contact to handle the translation requirements, resulting in direct savings and expedited turnaround times.
We ensure that compliance, consistency, accuracy and confidentiality are achieved in every project. We handle any type of financial document, from annual reports and accounts, investment policies and balance sheets to financial tenders and prospectuses. We guarantee accuracy and confidentiality when translating your financial content.
Whether you're an international banking institution, dealing with financial institutions in a different part of the world, or simply operate in areas with large immigrant populations and want to offer your customers information in their language, we are here for you. At Intertranslations we focus on helping banks translate their documentation into every language and supporting effectively every banking translation project.
Additionally, we provide a range of document services – whether you need financial document translation Spanish to English, or financial document translation English to Chinese, we can offer the solution. We also provide financial document translation English to Arabic. | {
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C.A.L.M.A. è il primo album in studio del rapper italiano Rayden, pubblicato nel 2007 dalla The Saifam Group.
Descrizione
I testi spaziano da racconti incentrati sui costumi dei nostri tempi, per tornare in alcuni pezzi alle tematiche già espresse in Sotto la cintura con i OneMic. Nelle restanti liriche ci troviamo invece davanti a introspezioni del rapper torinese. Le produzioni sono curate quasi interamente dal rapper stesso. In questo album, Rayden abbandona lo stile impiegato in Sotto la cintura, cambiando completamente il modo di rappare.
Tracce
Note
Collegamenti esterni | {
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{"url":"https:\/\/dmoj.ca\/problem\/ccc22j1","text":"## CCC '22 J1 - Cupcake Party\n\nView as PDF\n\nPoints: 3\nTime limit: 1.0s\nMemory limit: 1G\n\nProblem type\n##### Canadian Computing Competition: 2022 Stage 1, Junior #1\n\nA regular box of cupcakes holds cupcakes, while a small box holds cupcakes. There are students in a class and a total of at least cupcakes. Your job is to determine how many cupcakes will be left over if each student gets one cupcake.\n\n#### Input Specification\n\nThe input consists of two lines.\n\n\u2022 The first line contains an integer , representing the number of regular boxes.\n\u2022 The second line contains an integer , representing the number of small boxes.\n\n#### Output Specification\n\nOutput the number of cupcakes that are left over.\n\n#### Sample Input 1\n\n2\n5\n\n#### Output for Sample Input 1\n\n3\n\n#### Explanation of Output for Sample Input 1\n\nThe total number of cupcakes is which equals . Since there are students, there are cupcakes left over.\n\n#### Sample Input 2\n\n2\n4\n\n#### Output for Sample Input 2\n\n0\n\n#### Explanation of Output for Sample Input 2\n\nThe total number of cupcakes is which equals . Since there are students, there are no cupcakes left over.","date":"2022-10-05 13:01:11","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\": 0.1956731528043747, \"perplexity\": 4031.017934420662}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030337625.5\/warc\/CC-MAIN-20221005105356-20221005135356-00611.warc.gz\"}"} | null | null |
Boombox by The Lonely Island Chords and Melody
Chord And Melody Metrics For Boombox
Boombox has higher complexity than the average song in terms Chord-Bass Melody.
About The Key Of A♭ Major
Boombox is written in the key of A♭ Major. According to the Theorytab database, it is the least popular key among Major keys and the 21st most popular among all keys. Major keys, along with minor keys, are a common choice for popular songs. The three most important chords, built off the 1st, 4th and 5th scale degrees are all major chords (A♭ Major, D♭ Major, and E♭ Major). See the A♭ Major Cheat Sheet for popular chords, chord progressions, downloadable midi files and more!
Contributors: socsy and Vaz123 .
More by The Lonely Island
Cabinet Man
by Lemon Demon
Man-Made Object
Scatman
by Scatman John
Scatman's World
That Funny Feeling
by Bo Burnham
The Best Song
by Jon Lajoie
Touch-Tone Telephone
Ultimate Showdown of Ultimate Destiny
Welcome To The Internet
Word Disassociation | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,119 |
"""
Essential implementation of the Store interface defined by RDF lib.
"""
from django.db.utils import IntegrityError
import rdflib
from rdflib.store import VALID_STORE
from rdflib.term import Literal, Identifier
from rdflib_django import models
from rdflib_django.models import NamespaceModel
DEFAULT_STORE = "Default Store"
DEFAULT_NAMESPACES = (
("xml", u"http://www.w3.org/XML/1998/namespace"),
("rdf", u"http://www.w3.org/1999/02/22-rdf-syntax-ns#"),
("rdfs", u"http://www.w3.org/2000/01/rdf-schema#")
)
def _get_query_sets_for_object(o):
"""
Determines the correct query set based on the object.
If the object is a literal, it will return a query set over LiteralStatements.
If the object is a URIRef or BNode, it will return a query set over Statements.
If the object is unknown, it will return both the LiteralStatement and Statement query sets.
This method always returns a list of size at least one.
"""
if o:
if isinstance(o, Literal):
query_sets = [models.LiteralStatement.objects]
else:
query_sets = [models.URIStatement.objects]
else:
query_sets = [models.URIStatement.objects, models.LiteralStatement.objects]
return query_sets
def _get_named_graph(context):
"""
Returns the named graph for this context.
"""
if context is None:
return None
return models.NamedGraph.objects.get_or_create(identifier=context.identifier)[0]
class DjangoStore(rdflib.store.Store): # pylint: disable=abstract-method
"""
RDFlib Store implementation the uses Django Models for storage and retrieval.
>>> g = rdflib.Graph('Django')
The implementation is context aware, and uses Django transactions.
>>> g.store.context_aware
True
>>> g.store.transaction_aware
False
The implementation does not support formula's.
>>> g.store.formula_aware
False
The implementation provides a single store with the identifier DEFAULT_STORE. This store
is always present and needs not be opened.
>>> g.store.identifier
'Default Store'
Using other stores is not allowed
>>> g = DjangoStore(identifier='HelloWorld')
Traceback (most recent call last):
...
ValueError: multiple stores are not allowed
"""
context_aware = True
formula_aware = False
transaction_aware = False
def __init__(self, configuration=None, identifier=DEFAULT_STORE):
if identifier and identifier != DEFAULT_STORE:
raise ValueError("multiple stores are not allowed")
self.identifier = DEFAULT_STORE
super(DjangoStore, self).__init__(configuration, identifier)
self.open()
def open(self, configuration=None, create=False):
"""
Opens the underlying store. This is only necessary when opening
a store with another identifier than the default identifier.
>>> g = rdflib.Graph('Django')
>>> g.open(configuration=None, create=False) == rdflib.store.VALID_STORE
True
"""
return VALID_STORE
def destroy(self, configuration=None):
"""
Completely destroys a store and all the contexts and triples in the store.
>>> store = DjangoStore()
>>> g = rdflib.Graph(store=store)
>>> g.open(configuration=None, create=True) == rdflib.store.VALID_STORE
True
>>> g.open(configuration=None, create=False) == rdflib.store.VALID_STORE
True
>>> g.destroy(configuration=None)
>>> g.open(configuration=None, create=False) == rdflib.store.VALID_STORE
True
"""
models.NamedGraph.objects.all().delete()
models.URIStatement.objects.all().delete()
models.LiteralStatement.objects.all().delete()
def add(self, (s, p, o), context, quoted=False):
"""
Adds a triple to the store.
>>> from rdflib.term import URIRef
>>> from rdflib.namespace import RDF
>>> subject = URIRef('http://zoowizard.org/resource/Artis')
>>> object = URIRef('http://schema.org/Zoo')
>>> g = rdflib.Graph('Django')
>>> g.add((subject, RDF.type, object))
>>> len(g)
1
"""
assert isinstance(s, Identifier)
assert isinstance(p, Identifier)
assert isinstance(o, Identifier)
assert not quoted
named_graph = _get_named_graph(context)
query_set = _get_query_sets_for_object(o)[0]
query_set.get_or_create(
subject=s,
predicate=p,
object=o,
context=named_graph,
)
def remove(self, (s, p, o), context=None):
"""
Removes a triple from the store.
"""
named_graph = _get_named_graph(context)
query_sets = _get_query_sets_for_object(o)
filter_parameters = dict()
if named_graph is not None:
filter_parameters['context_id'] = named_graph.id
if s:
filter_parameters['subject'] = s
if p:
filter_parameters['predicate'] = p
if o:
filter_parameters['object'] = o
query_sets = [qs.filter(**filter_parameters) for qs in query_sets]
for qs in query_sets:
qs.delete()
def triples(self, (s, p, o), context=None):
"""
Returns all triples in the current store.
"""
named_graph = _get_named_graph(context)
query_sets = _get_query_sets_for_object(o)
filter_parameters = dict()
if named_graph is not None:
filter_parameters['context_id'] = named_graph.id
if s:
filter_parameters['subject'] = s
if p:
filter_parameters['predicate'] = p
if o:
filter_parameters['object'] = o
query_sets = [qs.filter(**filter_parameters) for qs in query_sets]
for qs in query_sets:
for statement in qs:
triple = statement.as_triple()
yield triple, context
def __len__(self, context=None):
"""
Returns the number of statements in this Graph.
"""
named_graph = _get_named_graph(context)
if named_graph is not None:
return (models.LiteralStatement.objects.filter(context_id=named_graph.id).count()
+ models.URIStatement.objects.filter(context_id=named_graph.id).count())
else:
return (models.URIStatement.objects.values('subject', 'predicate', 'object').distinct().count()
+ models.LiteralStatement.objects.values('subject', 'predicate', 'object').distinct().count())
####################
# CONTEXT MANAGEMENT
def contexts(self, triple=None):
for c in models.NamedGraph.objects.all():
yield c.identifier
######################
# NAMESPACE MANAGEMENT
def bind(self, prefix, namespace):
for ns in DEFAULT_NAMESPACES:
if ns[0] == prefix or unicode(ns[1]) == unicode(namespace):
return
try:
ns = NamespaceModel(prefix=prefix, uri=namespace)
ns.save()
except IntegrityError:
NamespaceModel.objects.filter(prefix=prefix).delete()
NamespaceModel.objects.filter(uri=namespace).delete()
NamespaceModel(prefix=prefix, uri=namespace).save()
def prefix(self, namespace):
try:
ns = NamespaceModel.objects.get(uri=namespace)
return ns.prefix
except NamespaceModel.DoesNotExist:
return None
def namespace(self, prefix):
try:
ns = NamespaceModel.objects.get(prefix=prefix)
return ns.uri
except NamespaceModel.DoesNotExist:
return None
def namespaces(self):
for ns in NamespaceModel.objects.all():
yield ns.prefix, ns.uri
| {
"redpajama_set_name": "RedPajamaGithub"
} | 315 |
{"url":"https:\/\/economics.stackexchange.com\/questions\/42742\/signalling-models-and-out-of-equilibrium-beliefs","text":"# Signalling models and out of equilibrium beliefs\n\nConsider the simplest version of the Spence signalling model. There are two types of worker, with either productivity $$\\theta_H$$ or instead productivity $$\\theta_H < \\theta_L$$. The proportions of the workers are $$p_H$$ (share of high types) and $$1 - p_H$$ respectively. Risk-neutral and competitive firms pay wages that may depend on a worker's education level $$e \\in \\{0, 1 \\}$$. Finally, getting education ($$e = 1$$) costs $$c_H$$ for the high types and $$c_L > c_H$$ for the low types.\n\nMy question is simple:\n\nWhat are the possible wage schedules that sustain a pooling equilibrium? In particular, can a pooling equilibrium wage schedule specify a wage of zero following $$e = 1$$?\n\nI will briefly elaborate on my understanding of the problem and what is unclear to me. In any pooling equilibrium, workers must be paid their expected productivity. So following $$e = 0$$, the specified wage must be $$w(e = 0) = \\mathbb{E}(\\theta) = p_H \\theta_H + (1 - p_H)\\theta_L$$ Furthermore, to ensure that nobody wants to deviate (by getting education), we only need to ensure that the high types won't deviate. (If they won't want to deviate, then neither will the low types). This requires that $$w(e = 1) - c_H \\leq w(e = 0) = p_H \\theta_H + (1 - p_H)\\theta_L$$ So the wage following $$e = 0$$ is uniquely determined in the pooling equilibrium (the equation). Moreover, the wage following $$e = 1$$ must be sufficiently low to ensure that nobody gets education (the inequality). I am wondering if there are any additional constraints that the wage schedule in a pooling equilibrium must satisfy, or whether any wage schedule satisfying these constraints will sustain pooling (as a PBE).\n\nTo make the issue a bit more salient, suppose that we propose some wage $$w(e = 1) < \\theta_L$$. For example, suppose you make the wage for those who get education zero. Clearly, nobody will choose education, i.e. the inequality holds. Moreover, wages don't need to be set optimally following $$e = 1$$ since this happens with zero probability in equilibrium. On the other hand, no possible type could have a productivity of zero, so this wage schedule does not follow from any possible firm beliefs. This suggests that the wage following zero education needs to also satisfy $$w(e = 1) \\in [\\theta_L, \\theta_H]$$ Is this indeed a further constraint (in any pooling equilibrium where both types choose $$e = 0$$)?\n\n\u2022 It seems to me all you need is $w(e=1)\\in[\\theta_L,\\, \\mathbb E(\\theta)+c_H]$. Not sure if this is what you're trying to get at. \u2013\u00a0Herr K. Feb 24 at 14:22\n\u2022 So we can't have, for example, $w(e = 1) = 0 < \\theta_L$? This would definitely prevent any deviations from the pooling equilibrium! (But one could never find employer beliefs which lead to this wage...) \u2013\u00a0afreelunch Feb 24 at 14:35\n\u2022 What is a PBE here? With competitive firms, this is not really a game. \u2013\u00a0Michael Greinecker Feb 24 at 17:00\n\u2022 @MichaelGreinecker I think the idea is that firms play Bertrand competition over wages. However, I do agree that this is not modelled rigorously at least in the original Spence paper \u2013\u00a0afreelunch Feb 25 at 15:15\n\nIn my interpretation of the model, competitive firms imply that the wage always equals the expected productivity, which depends on the beliefs. Clearly, in any pooling equilibrium the on-path beliefs are equal to the prior such that the wage is simply $$E[\\theta]$$. If, in your example, a worker sends the off-path messags $$e=1$$, the wage must be $$w(e=1)=w_1=b_1 \\theta_H + (1-b_1) \\theta_L$$, where $$b_1$$ is the belief in the information set of $$(e=1)$$ that the type is high. Since $$b_1 \\in [0,1]$$, it must be that $$w_1 \\in [\\theta_L,\\theta_H]$$. This information set is not reached on equilibrium path such that we are free to determine the out-of-equilibrium belief $$b_{1}$$. Only $$b_{0}=p_H$$ is a PBE-consistency requirement.\n\nThe set of all no-education pooling PBE is then given by the following:\n$$e_H=e_L=0$$,\n$$w_{0}=E[\\theta]$$,\n$$w_{1}=b_{1}\\theta_H + (1-b_{1})\\theta_L$$,\n$$b_{0}=p_H$$,\n$$b_{1}: b_{1}\\theta_H + (1-b_{1})\\theta_L -c_H\\leq E[\\theta].$$\n\nNote that the beliefs are part of the equilibrium. To get rid of equilibrium multiplicity, have a look at the numerous refinements such as the intuitive criterion that kills unreasonable out-of-equilibrium beliefs (and thereby unreasonable equilibria). The restriction on belief $$b_1$$ originates from the incentive constraint of the high type. We want to make sure that $$\\theta_H$$ does not want to deviate get eductation, and because the wages are competitive they are implied by the off-path belief which we can set freely.\n\nThe idea to set $$w_1<\\theta_L$$ would violate the competitive nature of firms. This sentence can be interpreted in two ways.\n\n1. Wages are just exogenously set this way.\n2. There are two firms (having the same beliefs) who both set a wage menu $$(w_1,w_2)$$ and a la Bertrand the worker always goes to the firm setting the higher wage. For some off-path beliefs, one firm would deviate from $$w_1<\\theta_L$$ and may attract workers by setting a wage $$w_1=b_{1}\\theta_H + (1-b_{1})\\theta_L - \\varepsilon$$ and thereby make a profit. Because you did not specify the off-path belief, we have no idea whether this deviation on the wage would be profitable.\n\u2022 Ooops, I forgot about the cost of education $c_H$. \u2013\u00a0Bayesian Feb 24 at 16:22\n\u2022 \"Your idea in the comments\": this was in the main text and in fact the central question. To quote from the main text: \"In particular, can a pooling equilibrium wage schedule specify a wage of zero following $e=0$?\" \u2013\u00a0afreelunch Feb 25 at 10:28\n\u2022 I thought in the comments you referred to $w_1=w(e=1)=0$ and the main text has $w_0=w(e=0)=0$. However, there also cannot be $w_0=0$ with $\\theta_L\\geq0$ as a firm can always make a profit by setting a wage $w_0=E[\\theta]-\\varepsilon$. \u2013\u00a0Bayesian Feb 25 at 11:18\n\u2022 Many apologies, there was a typo in the question! Now fixed. \u2013\u00a0afreelunch Feb 25 at 11:41\n\u2022 On a separate note, I don't see why $w(e = 1) \\leq \\mathbb{E}[\\theta]$. On seeing somebody get education (an impossible event, given by prior beliefs) why can't I conclude that the person getting education was a high type? \u2013\u00a0afreelunch Feb 25 at 11:45","date":"2021-04-16 17:12:07","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\": 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\": 38, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7878205180168152, \"perplexity\": 1032.8343844226974}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038088245.37\/warc\/CC-MAIN-20210416161217-20210416191217-00279.warc.gz\"}"} | null | null |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-make-an-improper-fraction-for-5-7-8","text":"# How do you make an improper fraction for 5 7\/8?\n\nNov 7, 2015\n\n#### Answer:\n\nMixed fraction: $5 \\frac{7}{8} \\textcolor{w h i t e}{\\text{XXXXXXXX}}$Improper fraction: $\\frac{47}{8}$\n\n#### Explanation:\n\n$5 \\frac{7}{8}$ means $5 + \\frac{7}{8}$\n\n$5$ is equivalent to $\\frac{40}{8}$\n\nSo $5 \\frac{7}{8} = \\frac{40}{8} + \\frac{7}{8} = \\frac{47}{8}$","date":"2019-10-16 10:29:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 7, \"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\": 0.9687968492507935, \"perplexity\": 6618.761306158513}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986666959.47\/warc\/CC-MAIN-20191016090425-20191016113925-00515.warc.gz\"}"} | null | null |
Hubble Directly Observes A Planet Orbiting Another Star
Analysis, Breakthrough Thinking, Crazy Stuff, Current Events, Photo Perspectives, Space
Looks like the eye off of Lord of The Rings
NASA's Hubble Space Telescope has taken the first visible-light snapshot of a planet circling another star.
Estimated to be no more than three times Jupiter's mass, the planet, called Fomalhaut b, orbits the bright southern star Fomalhaut, located 25 light-years away in the constellation Piscis Australis (the Southern Fish).
Fomalhaut has been a candidate for planet hunting ever since an excess of dust was discovered around the star in the early 1980s by NASA's Infrared Astronomy Satellite (IRAS).
In 2004, the coronagraph in the High Resolution Camera on Hubble's Advanced Camera for Surveys produced the first-ever resolved visible-light image of a large dust belt surrounding Fomalhaut. It clearly showed that this structure is in fact a ring of protoplanetary debris approximately 21.5 billion miles across with a sharp inner edge.
This large debris disk is similar to the Kuiper Belt, which encircles the solar system and contains a range of icy bodies from dust grains to objects the size of dwarf planets, such as Pluto.
Hubble astronomer Paul Kalas, of the University of California at Berkeley, and team members proposed in 2005 that the ring was being gravitationally modified by a planet lying between the star and the ring's inner edge.
Circumstantial evidence came from Hubble's confirmation that the ring is offset from the center of the star. The sharp inner edge of the ring is also consistent with the presence of a planet that gravitationally "shepherds" ring particles. Independent researchers have subsequently reached similar conclusions.
Now, Hubble has actually photographed a point source of light lying 1.8 billion miles inside the ring's inner edge. The results are being reported in the November 13 issue of Science magazine.
"Our Hubble observations were incredibly demanding. Fomalhaut b is 1 billion times fainter than the star. We began this program in 2001, and our persistence finally paid off," Kalas says.
"Fomalhaut is the gift that keeps on giving. Following the unexpected discovery of its dust ring, we have now found an exoplanet at a location suggested by analysis of the dust ring's shape. The lesson for exoplanet hunters is 'follow the dust,'" says team member Mark Clampin of NASA's Goddard Space Flight Center.
more via scienceblog
Tag: Extrasolar Planets; Astronomy; Kuiper Belt; Eris (Xena)
Amazing Stacking Videos
A Day In.. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,620 |
\section{Introduction}
Random matrix theory (RMT) has a remarkably wide range of applications in mathematics, physics and beyond. It describes universal statistical quantities that are not determined by the specific dynamics of a given system, but rather by its anti-unitary symmetries. One of the most prominent RMT quantities is the level spacing distribution $P(s)$, the probability of two neighboring eigenvalues to appear at a distance $s$, which is universal after unfolding (to refer to an eigenvalue density normalized to one). For the classical Gaussian RMT ensembles, i.e.\ hermitian matrices with Gaussian probability distribution of the elements, the spacing distributions can be evaluated for any matrix size \cite{Mehta:2004}. The results are very close to the level spacings of small matrices, a fact which is known as Wigner's surmise \cite[p.~199]{porter:1965}. The spacing distributions $P(s)$ of the smallest matrices are linear/quadratic/quartic in $s$ times an exponential decay in $s^2$ (see e.g.\ \eq{eq:surmiseon1}), where the power in the repulsion for small spacings -- which equals the Dyson index $\beta\in\{1,2,4\}$ of these ensembles -- counts the degrees of freedom per matrix entry in the ensemble.
In particular, RMT applies to quantum chromodynamics (QCD), where it describes the spectrum of the Dirac operator; for a review see \cite{Verbaarschot:2000dy}. The crucial feature is that QCD (with massless quarks) obeys a global chiral symmetry, which is spontaneously broken. In the $\varepsilon$-regime, a finite volume regime dominated by the corresponding Goldstone bosons (pions), the equivalence with RMT has been proven in Refs.~\cite{Osborn:1998qb,Toublan:1999hx,Basile:2007ki}. The appropriate RMT describing the microscopic spectral properties of QCD in this regime is the chiral ensemble. For bulk correlations, i.e. away from the origin, the chiral symmetry is no longer important \cite{Fox:1964, Nagao:1991, Nagao:1992} and universal properties of QCD spectra, such as level spacings, can be described by the standard Gaussian RMT ensembles \cite{Halasz:1995vd,Pullirsch:1998ke}.
While the conventional RMT ensembles describe chaotic systems with definite symmetries, physical systems are often `not so ideal'. We briefly describe cases where the symmetry of the system deviates from this ideal situation. When no analytical expressions are available for the correlations in such transitions, one can have recourse to surmises.
Firstly, physical systems may obey a certain \textit{anti-unitary symmetry} only approximately or contain several parts with different anti-unitary symmetries. The latter leads to transitions from one symmetry class to another. Including the Poissonian ensemble, which is the equivalent for generic integrable systems, the corresponding spacing distributions have been shown to fulfill a generalized Wigner surmise \cite{Lenz:1991, Schierenberg:2012ut}. Such transitions were observed at high temperatures in two- and three-color QCD \cite{Garcia-Garcia:2006gr, Kovacs:2009zj, Kovacs:2010wx, Bruckmann:2011cc} and are expected in the continuum limit of the staggered lattice Dirac operator in two-color and adjoint QCD \cite{Follana:2006zz, Bruckmann:2008xr}. We will show a mixed-symmetry surmise at work for two-color QCD at imaginary chemical potential.
Secondly, \textit{hermiticity} can be broken in physical systems, as is the case for QCD with real chemical potential. In this case the determinant of the Dirac operator becomes complex, which hampers progress in numerical simulations of this system. Non-hermitian Gaussian random matrices were first discussed by Ginibre \cite{Ginibre:1965zz}. For the associated complex eigenvalues one can define a nearest neighbor individually and study the corresponding spacing distributions. For the complex Ginibre ensemble the level spacings do not fulfill a surmise from $2 \times 2$ matrices (even though they are always cubic for small $s$) \cite{Grobe:1988,Akemann:2009my}. The spacings of this ensemble were compared to QCD spectra and showed agreement only for a small range of the chemical potential \cite{Markum:1999yr}.
In QCD the chemical potential drives the breaking of the anti-hermiticity. An adequate random matrix model describing such a situation consists of two matrices where the breaking of the hermiticity is governed by a real parameter \cite{Stephanov:1996ki,Osborn:2004rf}.
The eigenvalue densities and lowest eigenvalue distributions of such ensembles have been worked out in the microscopic, {\it weakly} non-hermitian limit (where the microscopic eigenvalues $z N$ and perturbation $\mu^2 N$ are kept fixed as the matrix size $N \to \infty$) \cite{Splittorff:2003cu,Osborn:2004rf,Akemann:2004dr,Akemann:2007yj} and successfully compared to quenched QCD data \cite{Bloch:2006cd,Akemann:2007yj}.
The treatment of real random matrices is known to be an especially arduous task. The joint probability function of the eigenvalues for the real asymmetric ensemble (also called the real elliptic Ginibre ensemble) was calculated in \cite{Lehmann:1991} and generalized to the chiral case in \cite{Akemann:2009fc}. Only recently, the correlation functions were worked out for the real Ginibre ensemble \cite{Forrester:2007,Sommers:2008,Borodin:2009}, the real elliptic Ginibre ensemble \cite{Forrester:2008} and the real chiral ensemble \cite{Akemann:2009fc,Akemann:2010mt,Akemann:2010tv}.
In this work we investigate the use of surmises to describe the level spacings of such real random matrix ensembles. More specifically, we examine the case where the unperturbed matrices are real and symmetric, i.e., they belong to the Gaussian orthogonal ensemble (GOE, $\beta=1$), and the perturbations are real and antisymmetric. The perturbations are assumed to be small, such that the imaginary part of typical eigenvalues is at most of the order of the mean separation between neighboring eigenvalues along the real axis (whereas in the Ginibre ensemble hermiticity is maximally broken). For real matrices the flow of eigenvalues is somewhat special, because two real eigenvalues need to coalesce to `give birth' to a complex conjugate pair, see below (this is not so for the complex (GUE, $\beta=2$) or symplectic (GSE, $\beta=4$) ensembles: in the former the eigenvalues are not restricted to form complex conjugate pairs, but can be distributed arbitrarily in the complex plane; in the latter the unperturbed matrices have two-fold degenerate eigenvalues due to Kramers' degeneracy and these split up and become complex for arbitrary small anti-hermitian perturbation). Actually, the original GOE eigenvalues are \textit{attracted} by an antisymmetric perturbation. For eigenvalues that meet in such a process and become a complex conjugate pair, a spacing can be uniquely defined as the distance between them. As we will show, the distribution of these spacings for large random matrices can be approximated very well by that of small matrices. The complex eigenvalues leave a trace in the statistics of the real eigenvalues, too. We will work out surmises for the spacings among the latter and show that they also apply to large matrices. We interpret these findings in the same way as for other Wigner surmises, namely that the dynamics of a neighboring pair is not much influenced by the effect of all other eigenvalues.
As an application we analyze spectra of two-color QCD, i.e., with gauge group SU(2). As the latter is pseudo-real it ensues that the massless continuum Dirac operator for fermions in the fundamental representation belongs to the GOE universality class (see Sec.~\ref{subsect:symm_and_herm}).
This property holds for vanishing chemical potential. Although the chemical potential perturbs the anti-hermiticity of the Dirac operator, it keeps its anti-unitary symmetry. Therefore, the random matrix results discussed above apply to two-color QCD at finite chemical potential, both in the continuum and on the lattice with the overlap operator.
This paper is organized as follows. After describing the symmetries and corresponding spectra of matrices in the next section, Sec.~\ref{sect:perturbation_theory} is devoted to the eigenvalue dynamics in perturbation theory. In Sec.~\ref{sect:surmises} we work out the surmises for the different types of spacings. These are then matched to data from large random matrices with mixed hermiticity in Sec.~\ref{sect:rmt} and to lattice two-color QCD data with chemical potential in Sec.~\ref{sect:qcd}. We also measured spacings at imaginary chemical potentials and discuss the comparison to a mixed-symmetry surmise in the last subsection of Sec.~\ref{sect:qcd}. We summarize our findings and conclude in Sec.~\ref{sect:summary}.
\section{The setting: symmetries and spectra of RMT ensembles}
Our unperturbed random matrices $H_0$ shall be real and symmetric. We therefore choose them from the GOE, so that the entries are Gaussian random numbers with mean zero and variance
\begin{align}
\langle (H_0)_{ii}^2 \rangle = 1\quad\mbox{and}\quad \langle (H_0)_{ij}^2 \rangle =
\frac12
\quad (i\neq j)\,.
\end{align}
Gaussian random matrix ensembles can be characterized by whether they commute with an anti-unitary $V=UK$, where $U$ is unitary and $K$ is the complex conjugation operator
\begin{equation}
[H_0,V]=0\qquad \mbox{equiv. to} \qquad H_0^\ast = U^\dagger H_0 U\,,
\label{eqn symm}
\end{equation}
and whether $V$ squares to $1$ or $-1$. For the GOE one has
\begin{equation}
V^2=1 \qquad \mbox{equiv. to} \qquad U U^\ast=1\,,
\label{eqn goe}
\end{equation}
from which it follows that the matrices $H_0$ can be chosen real.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{Plots/evalues_one.pdf}
\vskip1cm
\includegraphics[width=0.8\linewidth]{Plots/evalues_two.pdf}
\vskip0.6cm
\includegraphics[width=0.8\linewidth]{Plots/evalues_three.pdf}
\caption{Evolution of eigenvalues of a typical random matrix with increasing non-hermitian part (schematically). The top panel depicts a part of the spectrum for the unperturbed case, i.e., for a real symmetric (GOE) matrix. Towards the bottom panel the antisymmetric part increases, which results in some eigenvalues getting very close and finally becoming complex (the rightmost eigenvalues are spectators).
Different types of level spacings are sketched: (1) spacings between original eigenvalues and those that remain on-axis and have no interspaced complex conjugate pair, named $P^1_{\textrm{on}}$ and described by the pure GOE surmise, (2) spacings between on-axis eigenvalues with interspaced complex conjugate pairs, named $P^2_{\textrm{on}}$, (off) spacings between complex conjugate pairs called off-axis, named $P_{\textrm{off}}$. Note that for QCD spectra the real and imaginary axes are swapped.}
\label{fig general spectra}
\end{figure}
In this paper we will study random matrices from the GOE undergoing a real antisymmetric perturbation with a tunable coupling parameter. The characteristic equation of such a mixed matrix has real coefficients and consequently the eigenvalues are either real or come in complex conjugate pairs. Typical spectra of such matrices are depicted schematically in \fig{fig general spectra} together with different types of spacings between them. In the following, the real eigenvalues will be called `on-axis' and those with a non-zero imaginary part `off-axis'. This notation also applies to the case of QCD, where due to the anti-hermiticity of the Dirac operator at vanishing chemical potential the axes are swapped (then on-axis refers to purely imaginary eigenvalues and off-axis to those evolving from there into the complex plane). Obviously the spacings between on-axis eigenvalues and off-axis eigenvalues will be treated separately. Among the former we will further distinguish between those with and without an interspaced complex conjugate eigenvalue pair. This distinction only makes sense if the hermiticity of the initial GOE matrix is only mildly broken (since otherwise the off-axis eigenvalues move arbitrarily in the complex plane and cannot be assigned to a single on-axis pair anymore), and we will therefore only consider this case here. This is also why we do not consider the rare case of two or more interspaced eigenvalue pairs.
All our level spacing distributions are subject to the normalizations $\int_0^\infty ds \, P(s)=1$ and $\langle s \rangle = \int_0^\infty ds \, s \, P(s)=1$.
\section{Repulsion and attraction of eigenvalues in perturbation theory}
\label{sect:perturbation_theory}
To gain intuition about the dynamics of the eigenvalues, we consider a real symmetric matrix $H_0$, with real eigenvalues $\theta_i$ and corresponding real eigenvectors $|\psi_i\rangle$, perturbed by a real matrix $\lambda M$ with $\lambda \in \mathbb{R}^+_0$. We compare the effect of symmetric and antisymmetric perturbations $M$, whose matrix elements in the unperturbed basis,
\begin{align}
M_{ij}\equiv\langle\psi_i|M|\psi_j\rangle\,,
\label{Hij}
\end{align}
are real and symmetric or antisymmetric in $(i,j)$ too,
\begin{align}
M_{ji}= \pmM_{ij}\,,\qquad
\mbox{for }M^T=\pm M\,.
\label{eqn expvalues perturbation}
\end{align}
As is well known, ordinary perturbation theory up to second order yields for the eigenvalues,
\begin{align}
\theta_i \to \theta_i + \lambda M_{ii} +
\lambda^2\sum_{j\neq i} \frac{M_{ij}M_{ji}}{\theta_i-\theta_j}\,.
\end{align}
The first order term induces a random walk of the eigenvalues without correlating them. Moreover, in the antisymmetric case such expectation values are zero, so there is no first order contribution.
When omitting the random walk term, the eigenvalue differences are
\begin{align}
(\theta_i-\theta_j) \to (\theta_i-\theta_j)\left[1\pm
2\lambda^2\frac{M_{ij}^2}{(\theta_i-\theta_j)^2}+
\mathcal{O}\left(\frac1{(\theta_i-\theta_j)(\theta_{i,j}-\theta_{k\neq i,j})}\right)\right]\,.
\label{eqn second diff}
\end{align}
If $\theta_i$ and $\theta_j$ are much closer to each other than to any other eigenvalue, the third term in square brackets can be neglected in relation to the second one. Then, for real symmetric $M$ (upper sign) the difference grows, which is the common knowledge that eigenvalues repel.
In contrast, for real antisymmetric $M$ (lower sign), which we will consider from now on, the eigenvalues attract. Thus in perturbation theory it is favored that eigenvalues coalesce.
We recall that our investigations so far are based on the eigenvalues $\theta_i$ of the unperturbed matrix, which are real and remain so as long as ordinary perturbation theory is valid.\footnote{Ordinary perturbation theory only yields real shifts of the eigenvalues to all orders.}
When two eigenvalues $\theta_i$ and $\theta_j$ are very close, one has to apply almost degenerate perturbation theory \cite{Gottfried:1966}. Neglecting the second order contribution of all other eigenvalues, it gives for the perturbed eigenvalues
\begin{align}
(\theta_i,\theta_j)
\to
\mbox{evs}\left[
\left(\begin{array}{cc}\theta_i&0\\ 0&\theta_j\end{array}\right)+\lambda
\left(\begin{array}{cc} 0 & M_{ij}\\ -M_{ij} & 0\end{array}\right)\right]\,,
\label{eqn deg pert}
\end{align}
where we have made use of \eq{eqn expvalues perturbation}, with $M_{ij}$ still given by \eq{Hij},
and evs denotes the eigenvalues of the matrix. For the eigenvalue difference one obtains
\begin{align}
S=\sqrt{(\theta_j-\theta_i)^2-4\lambda^2M_{ij}^2}\,.
\label{eqn delta}
\end{align}
To second order in $\lambda$ the expansion of $S$ reproduces the result from second-order ordinary perturbation theory discussed above.
More interestingly, the argument of the square root can turn negative for some critical $\lambda$, above which the eigenvalue difference becomes purely imaginary. The latter is consistent with the fact that the complex eigenvalues of a real matrix come in complex conjugate pairs. Imaginary spacings occur for $\lambda$ bigger than
\begin{align}
\lambda_\text{crit}=\frac{|\theta_j-\theta_i|}{2|M_{ij}|}
\end{align}
(specific to each individual eigenvalue pair). In the exceptional case that $M_{ij}$ exactly vanishes, this particular eigenvalue difference remains unperturbed for any $\lambda$ (in perturbation theory). For perturbations beyond $\lambda_\text{crit}$ the eigenvalue difference can be written as
\begin{align}
S= 2i|M_{ij}|\sqrt{\lambda^2-\lambda^2_\text{crit}} = 2i|M_{ij}|\sqrt{\delta\lambda(2\lambda_\text{crit}+\delta\lambda)}\,,\qquad \delta\lambda\equiv\lambda-\lambda_\text{crit}\,,
\label{eqn delta imag}
\end{align}
and increases with the excess $\delta\lambda$ of the coupling over the critical coupling. Therefore, the perturbation causes the eigenvalues to be repelled in the complex plane.
Moreover, the eigenvalues leave the real axis perpendicularly to it. This can be seen by an explicit calculation of $\theta_{i,j}$ in \eqref{eqn deg pert} for $\lambda>\lambda_\text{crit}$, but it also follows directly from the fact that the antisymmetric perturbation is traceless and thus does not change the center of mass of the eigenvalues. When considering antisymmetric perturbations of large matrices beyond perturbation theory, only the center of mass of all eigenvalues is fixed and a complex conjugate pair could in principle leave the axis at any angle, as its center of mass contribution could be compensated by the motion of the remaining eigenvalues. Our data confirmed that, in accordance with almost degenerate perturbation theory, the complex eigenvalue pairs move perpendicularly to the axis just after their creation. When the perturbation grows further, higher order effects set in and the complex conjugate eigenvalues can leave the trajectory perpendicular to the axis.
\section{Surmises}
\label{sect:surmises}
The idea of Wigner surmises is to approximate level spacing distributions of large random matrices by those of random matrices with smallest possible size, typically $2\times 2$ matrices. Note that in the perturbative treatment in the previous section we have focussed on a pair of nearest neighbor eigenvalues and thus ended up discussing $2\times 2$ matrices and the properties of their eigenvalues. This may be seen as an indication that surmises for mixed systems may apply to large random matrices.
We recall that we consider three different types of spacings in our (large random matrix and QCD) systems: on-axis spacings with one and without interspaced complex conjugate eigenvalue pair, and off-axis spacings (see \fig{fig general spectra}). For the first and third type we will deduce surmises from $2\times 2$ matrices, computing the corresponding spacings. The spacings of the second type will be computed from the joined probability density for $4\times 4$ matrices.
\subsection[Prelude: $2\times 2$ matrices and their spacings]{Prelude: $\mathbf{2\times 2}$ matrices and their spacings}
For size $2\times 2$ let us consider a real symmetric and traceless matrix perturbed by an antisymmetric one,
\begin{align}\label{eq:ortantisymm2x2}
H = \left(
\begin{array}{ccr}
-a & b \\ b & a
\end{array}
\right)+\lambda\left(
\begin{array}{cc}
0 & c \\ -c & 0
\end{array}
\right)\,,
\end{align}
with random numbers $a$, $b$ and $c$ that are Gaussian distributed with zero mean and unit variance. Note that the matrix $H$ with $\lambda=0$ is equivalent to a $2\times 2$ GOE matrix, up to a common shift of the eigenvalues which does not alter the spacings. In the GOE the diagonal entries have twice the variance of the off-diagonal ones, but it can easily be shown that neglecting the part proportional to the identity, as we have done in $H$ to simplify the computations, renders the variances equal. The eigenvalues of $H$ are $\pm\sqrt{a^2+b^2-c^2\lambda^2}$, which, for a particular draw of random numbers, turn imaginary beyond a critical coupling
\begin{align}
\lambda_\text{crit}=\frac{\sqrt{a^2+b^2}}{|c|}\,.
\label{lambdacrit}
\end{align}
For $\lambda<\lambda_\text{crit}$ the spacing is $2\sqrt{a^2+b^2-c^2\lambda^2}$ and hence real, while for $\lambda>\lambda_\text{crit}$ the spacing is $2i\sqrt{c^2\lambda^2-a^2-b^2}$ and hence purely imaginary.
The probability that the matrix $H$ has an imaginary spacing can be easily computed by integrating over all random matrix entries with the constraint that $\lambda$ is above its critical value,
\begin{align}\label{eq:imagspacprob}
p(\lambda) = \left(2\pi\right)^{-3/2}\int_{-\infty}^\infty\!\!da\, db\, dc \,e^{-(a^2+b^2+c^2)/2}\,
\theta\left(c^2\lambda^2-a^2-b^2\right) = 1-\frac1{\sqrt{1+\lambda^2}}\,.
\end{align}
As expected, the limiting cases yield $p(\lambda\to0) = 0$ and $p(\lambda\to\infty)=1$.
\subsection{On-axis without interspaced complex eigenvalues}
\label{sect:onaxis1}
For on-axis spacings without interspaced complex eigenvalues our surmise is
\begin{align}\label{eq:surmiseon1}
P^1_{\textrm{on}}(s) =\frac{\pi}{2}\, s\, e^{-\frac{\pi}{4}s^2}\,,
\end{align}
which means that even in the presence of a (small) antisymmetric perturbation these spacings follow the GOE surmise governing the unperturbed spacings. This is so because \eq{eq:surmiseon1} is the distribution of \textit{real} spacings\footnote{The plural refers to the ensemble, as one representative of this random matrix has only one spacing, which will only be included here if real.} of $H$ independently of the coupling parameter $\lambda$. Let us demonstrate the derivation of this distribution explicitly, as a typical example. We start with the distribution of the non-normalized spacing $S$
\begin{align}
Q(S)&=\int_{-\infty}^\infty\!\!da\, db\, dc \,e^{-(a^2+b^2+c^2)/2}\,
\delta\left(S-2\sqrt{a^2+b^2-c^2\lambda^2}\right)
\theta\left(a^2+b^2-c^2\lambda^2\right)\nonumber\\
&=2\pi\int_{-\infty}^\infty\!\! dc \int_0^\infty\!\! dr\,r\,e^{-(r^2+c^2)/2}\,
\delta\left(S-2\sqrt{r^2-c^2\lambda^2}\right)
\theta\left(r^2-c^2\lambda^2\right)\,.
\label{eq:surmiseon_pre}
\end{align}
In the integration over $r$, the $\delta$-function replaces
\begin{align}
r\to r_0= \sqrt{S^2/4+c^2\lambda^2}
\end{align}
with Jacobian
\begin{align}
\left|\left.\frac d{dr}\left(S-2\sqrt{r^2-c^2\lambda^2}\right)\right|_{r=r_0}\right|^{-1} =
\frac {S}{4\sqrt{S^2/4+c^2\lambda^2}}=\frac {S}{4r_0}\,.
\end{align}
Therefore, we obtain
\begin{align}
Q(S) =\frac{\pi}{2} \int_{-\infty}^\infty dc \,S\,e^{-\left[S^2/4+c^2(1+\lambda^2)\right]/2}\,
\theta\left(S^2/4\right)\,.
\end{align}
The integration over $c$ just gives a constant factor, and the $\theta$-function forces the spacing $S$ to be real as assumed. After normalization, we obtain $P^1_{\textrm{on}}$ from (\ref{eq:surmiseon1}).
\subsection{On-axis with interspaced complex eigenvalues}
When two on-axis eigenvalues have an interspaced complex eigenvalue pair, we expect the latter to have a noticeable influence on the spacing of the on-axis eigenvalues. To obtain a surmise for the distribution of these spacings, we consider a $4\times4$ matrix
\begin{align}\label{eq:ortantisymm4x4}
H = H_0 + \lambdaA\,,
\end{align}
with $H_0$ taken from the GOE and an antisymmetric matrix $A$ with probability density
\begin{align}\label{eq:probdistA}
w(A) \sim e^{-\frac12 {\rm tr}\left(A A^T\right)}\,,
\end{align}
i.e. Gaussian distributed entries having the same variance as the off-diagonal entries of $H_0$. This is the smallest possible matrix that can yield a surmise for this case as there are four eigenvalues which are relevant for the dynamics of the spacing. The joint probability distribution of the eigenvalues of $H$ depends on the number of real and complex eigenvalues. For two real eigenvalues $\theta_1$ and $\theta_2$ and one complex conjugate pair $z$ and $z^*$, it reads up to a normalization \cite{Akemann:2010mt}
\begin{align}
P(\theta_1, \theta_2, z, z^*) \propto e^{-\theta_1^2-\theta_2^2-z^2-(z^*)^2} \erfc\left(\frac{\sqrt{1+\lambda^2}}{\sqrt2\lambda}|z-z^*|\right)2i \Delta(\theta_1, \theta_2, z, z^*)\,,
\label{eqn jpd}
\end{align}
with the Vandermonde determinant
\begin{align}
\Delta(\theta_1, \theta_2, z, z^*) = (z^*-z)(z^*-\theta_2)(z^*-\theta_1)(z-\theta_2)(z-\theta_1)(\theta_2-\theta_1)\,,
\end{align}
where it is assumed that $\theta_2>\theta_1$ and $\im\,z>0$. To obtain the distribution of the spacing $S$ between $\theta_1$ and $\theta_2$, we set $\theta_2 = \theta_1 + S$ and introduce new variables $a = \re\,z - \theta_1\in[0,S]$ and $b = \im\,z\in[0,\infty)$, cf. \fig{fig surmise spectra}.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{Plots/evalues_surmise_new.pdf}
\caption{Spectrum of the $4\times4$ matrix used for the surmise for $P^2_{\textrm{on}}$.}
\label{fig surmise spectra}
\end{figure}
This results in
\begin{align}
&P(\theta_1, \theta_1 + S, \theta_1+a+ib, \theta_1+a-ib)\\
&\propto S\, b \left[\left(a-S\right)^2+b^2\right]\left(a^2+b^2\right) e^{-\theta_1^2-(\theta_1+S)^2-2(a+\theta_1)^2+2b^2}\, \erfc\left(\frac{\sqrt{2}\sqrt{1+\lambda^2}}{\lambda}\,b\right)\,.\nonumber
\end{align}
We integrate out $\theta_1$, $a$, $b$ and perform an irrelevant rescaling $S\to2S$ to obtain the non-normalized spacing distribution
\begin{align}
\int_0^\infty db\int_{0}^{2S} & da \int_{-\infty}^\infty d\theta_1\, P(\theta_1, \theta_1 + 2 S, \theta_1+a+ib, \theta_1+a-ib) \notag\\
\propto S e^{-3S^2} & \left\{
\sqrt{\pi} \, e^{S^2} \erf(S) \left[\lambda^2\sqrt{1+\lambda^2}\left(3\lambda^2+8S^2\right)+4 \left(\sqrt{1+\lambda^2}-1\right)\left(4S^4-8S^2+3\right)\right]
\right. \notag
\\
& \left. + 8S \left[\left(\sqrt{1+\lambda^2}-1\right)\left(2S^2-1\right) -\lambda^2 \sqrt{1+\lambda^2}\right] \right\} \equiv Q(S; \lambda) \,.
\label{eq:surmiseon2_pre}
\end{align}
Properly normalized, the spacing distribution reads
\begin{align}\label{eq:surmiseon2}
P^2_{\textrm{on}}(s;\lambda) = CD\,Q(Ds;\lambda)\,,
\end{align}
with normalization and scaling factors
\begin{align}
C &= \left(\int\limits_0^{\infty}
dS\,Q(S;\lambda)\right)^{-1}=\frac{12\sqrt3}{\sqrt{\pi}}\left[\lambda^2 \sqrt{1+ \lambda^2} \left(9\lambda^2+8\right)+8 \left(\sqrt{1+\lambda^2}-1\right)\right]^{-1}\,,
\end{align}
and
\begin{align}
D = C\,\int\limits_0^{\infty} dS\,S\,Q(S;\lambda) &= \frac{C}{72}\left[26 \left(\sqrt{1+\lambda^2}-1\right)-27\sqrt2\,\text{arccot}(\sqrt2)\right.\notag\\
&\left.+\sqrt{1+\lambda^2}\left(18\lambda^4+20\lambda^2+27\sqrt2 \left(1+\lambda^2\right)^2\text{arccot}(\sqrt2)\right)\right]\,.
\label{eq:surmiseon2_const}
\end{align}
In the limit $\lambda \to 0$, the spacing distribution reads
\begin{align}
\hspace{-0.1cm}\lim_{\lambda\to 0} P^2_{\textrm{on}}(s;\lambda) = \frac{\kappa^2}{8\pi}\,s\, e^{-3\kappa^2 s^2}
\left[4\kappa^3s^3- 6\kappa s + \sqrt\pi\, e^{\kappa^2 s^2} (4\kappa^4 s^4-4\kappa^2 s^2 + 3)\,\erf(\kappa s)\right]\,,
\label{eq:surmiseon2_smallpert}
\end{align}
where
\begin{align}
\kappa=\lim_{\lambda\to 0}D=\frac{22+45\sqrt{2}\,\mbox{arccot}\sqrt{2}}{16\sqrt{3\pi}} \approx 1.2453 \,.
\end{align}
The opposite limit $\lambda\to\infty$ yields
\begin{align}
\lim_{\lambda\to\infty} P^2_{\textrm{on}}(s;\lambda) = 4\sqrt3\, \eta^2 s\, e^{-2 (\eta s)^2} \erf(\eta s)\,,
\end{align}
with
\begin{align}
\eta=\lim_{\lambda\to\infty}D=\frac{2+3\sqrt{2}\,\mbox{arccot}\sqrt{2}}{2\sqrt{3\pi}} \approx 0.7510 \,.
\end{align}
For small spacings $s$ and non-zero $\lambda$, the spacing distribution $P^2_{\textrm{on}}$ is proportional to $s^2$, whereas in the limit $\lambda \to 0$ the first term in the Taylor expansion is proportional to $s^6$. One can understand these repulsion strengths from the Vandermonde determinant in the joint probability density alone, i.e.\ by focussing on the linear repulsion of every eigenvalue pair and neglecting the exponential and erfc factors (which only contribute significantly at larger spacings). The relevant integral is
\begin{align}
& \int_{0}^{S} \!\!da
(z^*-\theta_2)(z^*-\theta_1)(z-\theta_2)(z-\theta_1)(\theta_2-\theta_1)\Big|_{z=\theta_1+a+ib,\,\theta_2-\theta_1=S}\notag\\
& = S\int_{0}^{S} \!\!da\, \left[(a-S)^2+b^2\right](a^2+b^2)
= b^4 S^2+\frac{2}{3}\,b^2 S^4+\frac{1}{30}\,S^6 \,.
\end{align}
This is indeed proportional to $S^2$ for small $S$, unless the distribution of $b$ becomes a $\delta$-function around zero, which happens for $\lambda\to0$. Then, the distribution is proportional to $S^6$.
\begin{figure}
\includegraphics[width=0.5\linewidth]{Plots/surmises_ortantisymm_typ=2.pdf}
\includegraphics[width=0.5\linewidth]{Plots/surmises_ortantisymm_typ=3.pdf}
\caption{Left: Wigner surmise $P^2_{\textrm{on}}(s;\lambda)$, \eq{eq:surmiseon2}, for on-axis eigenvalues with an interspaced complex conjugate eigenvalue pair for $\lambda=0, 0.3, 0.5, 1, \infty$ (decreasing maxima). Right: Wigner surmise $P_{\textrm{off}}(s; \lambda)$, \eq{eq:surmiseoff}, for off-axis spacings and couplings $\lambda=0, 2, 10, \infty$ (decreasing maxima). Plotted for comparison is the Wigner surmise for the GOE (dashed), which is the surmise for the on-axis spacings with no interspaced complex pair, \eq{eq:surmiseon1}.}
\label{fig wigner surmise}
\end{figure}
Spacings for various parameters $\lambda$ are plotted in the left panel of \fig{fig wigner surmise}.
\subsection{Off-axis}
To get a surmise for the spacings between complex conjugate eigenvalue pairs, we consider again the $2\times2$ matrix $H$ given in \eq{eq:ortantisymm2x2}. This time, we are interested in the distribution of the imaginary spacings of this matrix. In analogy to Sec.~\ref{sect:onaxis1} we define $Q(S)$ as in \eq{eq:surmiseon_pre} with appropriate arguments of the delta and step function,
\begin{align}
Q(S)&=\int_{-\infty}^\infty\!\!da\, db\, dc \,e^{-(a^2+b^2+c^2)/2}\,
\delta\left(S-2\sqrt{c^2\lambda^2-a^2-b^2}\right)
\theta\left(c^2\lambda^2-a^2-b^2\right)\,.
\end{align}
For the normalized spacing distribution we obtain with calculations similar to those of Sec.~\ref{sect:onaxis1},
\begin{align}\label{eq:surmiseoff}
P_{\textrm{off}}(s; \lambda) = C\,D^2\,s\, e^{D^2 s^2}\erfc\left(Ds
\sqrt{1+\lambda^2} / \lambda\right)\,,
\end{align}
with constants
\begin{align}
C = \frac{2}{\sqrt{1 + \lambda^2}-1} \,,\qquad
D = C\, \frac{\lambda\sqrt{1 + \lambda^2} -
\mbox{arsinh}(\lambda)}{2\sqrt{\pi}}\,.
\end{align}
For small $s$ the level spacing is linear, just as in the GOE. This is understandable, as the derivation of $P_{\textrm{off}}$ involves an integral over one off-diagonal random number which is similar to that encountered in the derivation of the GOE spacing. The number of degrees of freedom, which governs the level spacing at small $s$, is thus the same for both cases.
The limit for $\lambda\to0$ is
\begin{align}
\lim_{\lambda\to 0} P_{\textrm{off}}(s; \lambda)
= \frac{64}{9\pi}\, s \erfc\left(\frac{4s}{3\sqrt{\pi}}\right)\,.
\label{eq:surmise_off_limitzero}
\end{align}
Note that in this limit the perturbation is switched off. For statistical reasons, off-axis eigenvalues will exist for arbitrary small perturbations (see \eq{lambdacrit}). Equation \eqref{eq:surmise_off_limitzero} describes the (normalized) level spacings of these eigenvalues in the $\lambda\to 0$ limit (whereas \eq{eq:surmiseon2_smallpert} reflects the influence of these eigenvalues on the neighboring on-axis spacings in this limit). Even though the difference between \eq{eq:surmise_off_limitzero} and the GOE spacing distribution is rather small, the two are clearly distinguishable in \fig{fig wigner surmise} (right).
For $\lambda\to\infty$ the distribution $P_{\textrm{off}}$ is simply half a Gaussian,
\begin{align}
\lim_{\lambda\to \infty} P_{\textrm{off}}(s; \lambda)
=\frac2{\pi}\,\exp\left(-\frac{s^2}{\pi}\right)\,,
\end{align}
the spacing distribution of the perturbation alone. This limit, however, is not uniform at $s=0$, since $P_{\textrm{off}}(0; \lambda)=0$ for every finite $\lambda$, whereas $\lim_{s\to 0}\lim_{\lambda\to \infty} P_{\textrm{off}}(s; \lambda)=2/\pi$. Spacings for various couplings $\lambda$ are plotted in \fig{fig wigner surmise} (right) and this discontinuity is clearly visible. A similar effect has been observed for mixed symmetry classes of (small and large) random matrices in Ref.~\cite{Schierenberg:2012ut}. There, even a Gibbs-like overshoot of the curves near $s=0$ was observed, which is absent here.
We have checked that the surmises $P^1_{\textrm{on}}$ and $P_{\textrm{off}}$ could equally well be obtained from the joint eigenvalue probability distribution of real $2\times 2$ matrices \cite{Akemann:2010mt}, i.e.\ analogues of \eq{eqn jpd}, with either two real or a pair of complex conjugate eigenvalues.
With these surmises at hand, our aim will now be to describe the level spacing distributions of large random matrices, matching the parameter $\lambda$ to the corresponding coupling parameter and to QCD spectra, where the matching will be to the chemical potential.
\section{Comparison of the surmises to large RMT spectra}
\label{sect:rmt}
To check the validity of the surmises calculated in the previous section, we applied them to the spectra of large dimensional random matrices of the form
\begin{align}\label{eq:largermtmatrix}
H = H_0 + \frac\Lambda{\sqrt{2N/\pi}}\ A\,,
\end{align}
where $H_0$ is real symmetric and taken from the GOE with matrix size $N$, whereas $A$ is real antisymmetric with probability density
\begin{align}
w(A) \sim e^{-\frac12 {\rm tr}\left(A A^T\right)}\,,
\end{align}
i.e. all elements of $A$ are independently Gaussian distributed with the same variance as the off-diagonal entries of $H_0$ (as was also used for the small matrices in the surmises, cf.\ e.g.\ \eq{eq:probdistA}). The coupling parameter $\Lambda$ is divided by $\sqrt{2N/\pi}$ in order to make it comparable to the one used in the surmises, $\lambda$. As far as the spacing distribution is concerned, this normalization makes $\Lambda$ a universal, $N$-independent coupling parameter, see \cite{Schierenberg:2012ut} for a detailed discussion. As also argued there, a constant spectral density is necessary to compare spacing distributions of large matrices from mixed universality classes to the corresponding distributions of small matrices. Otherwise, different coupling strengths are mixed. Therefore, we only evaluated eigenvalues with real part in the interval $(-\sqrt{N}/4, \sqrt{N}/4)$, i.e. around the center of the real spectrum of $H$. As the spectral density is not exactly constant, we measured the on-axis spacings in units of the local mean spacing (obtained by an ensemble average), which is equivalent to unfolding the spectrum. For the off-axis spacings, no unfolding was done.
The numerically obtained spacing distributions of $H$ are shown in \fig{fig:largermt} for various values of the coupling parameter $\Lambda$. As can be seen, the surmises for the on-axis spacings of both types describe the data very well for coupling parameters up to $\Lambda=2$. The coupling parameters $\lambda$ for the on-axis spacings of type 2 were obtained by a fit with least square minimization.
\begin{figure}[t]
\includegraphics[width=\linewidth]{Plots/spacings_largematrices_ort_to_antisymm_typ=2.pdf}
\includegraphics[width=\linewidth]{Plots/spacings_largematrices_ort_to_antisymm_typ=3.pdf}
\includegraphics[width=\linewidth]{Plots/spacings_largematrices_ort_to_antisymm_typ=1.pdf}
\caption{Spacing distributions of $400\times 400$ random matrices of the form
of \eq{eq:largermtmatrix}, with various values of the coupling parameter $\Lambda$.
Top: on-axis spacings of type 1, surmise given by GOE, \eq{eq:surmiseon1}.
Middle: on-axis spacings of type 2, surmise given by \eq{eq:surmiseon2}.
Bottom: off-axis spacings, surmise given by \eq{eq:surmiseoff}; the dashed curve is the GOE spacing for comparison. For each value of $\Lambda$, $2\cdot 10^5$ random matrices were diagonalized.}
\label{fig:largermt}
\end{figure}
For the off-axis spacings we are able to predict a $2\times 2$ coupling parameter $\lambda$ through the frequency $p$ of imaginary spacings: we measure the latter for the large matrices and choose $\lambda$ such that the same $p$ is realized for $2\times 2$-matrices, i.e.\ we invert \eq{eq:imagspacprob} to obtain
$\lambda = \sqrt{1/(1-p)^2-1}$.
The surmises for the off-axis spacings obtained this way match the numerical data very well for $\Lambda=0.2$ and $0.5$, differ slightly for $\Lambda=1$ and are far off for $\Lambda=2$. Although the coupling parameter could also be determined by a fit to the level spacing data, we observed that this did not yield an improved estimate. Note that the surmises' maxima are always left of the maximum of the GOE, cf.\ \fig{fig wigner surmise} right, in contrast to the $\Lambda=2$ maximum. Not surprisingly, the surmise for the off-axis spacing does not work for large coupling parameters, i.e.\ for large perturbations. The reason for the earlier break down of the off-axis surmise could be related to the additional degree of freedom of the eigenvalues when they move in the complex plane.
\section{Application to two color QCD with chemical potential}
\label{sect:qcd}
\subsection{Symmetry and hermiticity of the Dirac operator}
\label{subsect:symm_and_herm}
As far as anti-unitary symmetries are concerned, the surmises discussed in the previous sections should approximate the spacing distributions for two-color QCD with a real quark chemical potential, as will be analyzed now.
\subsubsection{Continuum}
\label{sect:symmcont}
In the presence of a chemical potential, the Euclidean-space Dirac operator in continuum two-color QCD is given by
\begin{align}
D = \gamma_\nu D_\nu + m + \mu \gamma_4\,,
\end{align}
with covariant derivative
\begin{align}
D_\nu = \partial_\nu + i A^a_\nu(x) \tau_a\,,
\end{align}
where $\tau_a$ are the Pauli matrices, the generators of SU(2). The massless Dirac operator at zero chemical potential is an anti-hermitian operator, which has purely imaginary eigenvalues. The mass merely shifts the spectrum, while the chemical potential distorts it nontrivially. Since the former is irrelevant for level spacings, we will set $m=0$ for the following discussion.
The anti-unitary symmetry of the SU(2){} Dirac operator is based on the pseudo-real nature of SU(2){} implying
\begin{equation}
(i\tau_a)^\ast=\tau_2^\dagger(i \tau_a)\tau_2\,,
\label{eqn tau symm}
\end{equation}
as well as on the charge conjugation properties of gamma matrices
\begin{equation}
(i\gamma_\nu)^\ast=C^\dagger(i \gamma_\nu) C\qquad \mbox{with}\qquad
C= \gamma_2\gamma_4\,,
\label{eqn charge conj}
\end{equation}
where we use the Weyl representation
\begin{equation}
\gamma_\nu=\left(
\begin{array}{cc}
0 & (i\tau_i,1_2) \\ (-i\tau_i,1_2) & 0
\end{array}
\right)\,.
\end{equation}
Making contact to the real random matrices considered so far is easiest after multiplication with the imaginary unit,
\begin{equation}
iD=i\gamma_\nu\otimes\big(\partial_\nu 1_2+A_\nu^a(x) i\tau_a\big)
+i\mu\gamma_4\otimes 1_2\,,
\label{eqn iD}
\end{equation}
where we have separated the spin and gauge parts explicitly. From \eq{eqn tau symm} and \eq{eqn charge conj} it follows that $iD$ obeys the GOE symmetry
\begin{equation}
(iD)^\ast=U^\dagger (iD) U\,,\qquad U=C\otimes\tau_2\qquad \text{with} \quad U U^\ast=1\,.
\label{eqn iD symm}
\end{equation}
As a consequence, $iD$ can be made real by a unitary transformation. We denote such an equivalence by $\sim$ and write
\begin{equation}
iD\sim D_s+D_a\,,
\end{equation}
where we have split the operator into its real symmetric part $D_s$ and real antisymmetric part $D_a$. From the hermiticity at $\mu=0$ one identifies $iD(\mu=0)\sim D_s$ and further $i\mu\gamma_4\sim D_a$. Thus up to the factor $i$ -- which only rotates eigenvalues, but does not influence spacings -- the situation for the continuum Dirac operator is the following: for vanishing chemical potential it is real symmetric with on-axis eigenvalues obeying the GOE symmetry, whereas the chemical potential $\mu$ introduces a real antisymmetric perturbation with possible creation of complex conjugate eigenvalues. This agrees with the setting for the random matrices with $\mu$ playing the role of the matrix coupling parameters $\lambda$ and $\Lambda$.
The massless Dirac operator also satisfies chiral symmetry, $\{D,\gamma_5\}=0$, resulting in all nonzero eigenvalues coming in pairs $\pm i\theta$. In the numerical results below we will restrict ourselves to the half of eigenvalues with positive imaginary part and consider spacings in the bulk, away from $\theta=0$. For the spacing distribution in the bulk the chiral symmetry is known to be irrelevant \cite{Fox:1964, Nagao:1991, Nagao:1992,Halasz:1995vd,Pullirsch:1998ke}.\footnote{\label{ftnote}Note that chiral symmetry implies another anti-unitarity relation, $D^\ast=(\gamma_5 U)^\dagger D(\gamma_5 U)$, which at first sight would mean that $D$ belongs to the class of real antisymmetric operators. This is, however, not the case, since for the classification one first has to reduce the operator to irreducible blocks. This is best done on $D^2$, in which chiral eigenvalue pairs $\pm i \theta$ become twice degenerate eigenvalues $-\theta^2$. As $D^2$ commutes with $\gamma_5$, its eigenmodes can be chosen to be of definite chirality. The two relations $(D^2)^\ast=U^\dagger D^2 U=(\gamma_5 U)^\dagger D^2 (\gamma_5 U)$ coincide on the subspaces of fixed chirality, where both are proportional to $u=\tau_2\otimes\tau_2$ with $uu^\ast=1$. This qualifies $D^2$ and thus also $D$ for GOE.}
\subsubsection{Lattice}
When simulating the theory on a space-time lattice, the Dirac operator has to be discretized accordingly. Lattice discretizations of the Dirac operator typically alter some of its symmetry properties, such that its anti-unitarity properties can differ from those in the continuum\footnote{as is the case for the staggered Dirac operator \cite{Follana:2006zz, Bruckmann:2008xr}} and have to be analyzed carefully.
To preserve the chiral symmetry of the discretized Dirac operator one uses the overlap fermion formulation \cite{Narayanan:1993sk,Narayanan:1994gw,Neuberger:1997fp}, whose definition, in the presence of a quark chemical potential $\mu$, is given by \cite{Bloch:2006cd}
\begin{align}
D_{{\sf ov}}(\mu) =
1 + \gamma_5\sgn(\gamma_5D_{{\sf W}}(\mu))\,,
\label{Dovmu}
\end{align}
where we have put the lattice spacing to unity, $a=1$, $\sgn$ is the matrix sign function satisfying $(\sgn A)^2=1$, and
\begin{align}
D_{{\sf W}}(\mu)
= \, & 1 - \kappa \sum_{i=1}^3 \left( T_i^+ + T_i^-\right)
- \kappa \left( {e^{\,\mu}}\, T_4^+ + {e^{-\mu}}\, T_4^-\right)
\label{DW}
\intertext{with}
&(T^{\pm}_\nu)_{yx} = (1 \pm
\gamma_\nu) \, U_{x,\pm\nu} \, \delta_{y,x\pm\hat\nu}
\end{align}
is the Wilson Dirac operator at non-zero chemical potential \cite{Hasenfratz:1983ba} with hopping parameter $\kappa = 1/(8+2m_\text{w})$, Wilson mass $m_\text{w} \in (-2,0)$ and gauge configurations $U_{x,\pm\nu}\in$
SU(2), where $U_{x,-\nu} \equiv {U^\dagger}_{\!\!\!{x-\hat\nu,+\nu}}$. The exponential factors $e^{\pm\mu}$ implement the quark chemical potential on the lattice. For $\mu = 0$ the argument of the sign function in \eq{Dovmu} is hermitian, while for $\mu\ne 0$ it is non-hermitian.
To actually compute the overlap operator we need to define the matrix sign function for a general complex matrix $A$ of dimension $n$. A generic matrix function $f(A)$ can be defined by
\begin{align}
\label{eq:matfun}
f(A) &= \frac{1}{2\pi i} \oint_\Gamma f(z)(zI-A)^{-1}dz\,,
\end{align}
where $\Gamma$ is a collection of contours in $\mathbb{C}$ such that $f$ is analytic inside and on $\Gamma$ and such that $\Gamma$ encloses the spectrum of $A$. If $A$ is diagonalizable, i.e., $A=U\Lambda U^{-1}$, with diagonal eigenvalue matrix $\Lambda = \diag(\lambda_1, \ldots, \lambda_n)$ and $U\in Gl(n,\mathbb{C})$, then this general definition can be simplified to the well-known spectral form
\begin{align}
f(A) &= Uf(\Lambda)U^{-1}\,,
\label{fAdef}
\end{align}
with
\begin{align}
f(\Lambda) &= \diag\left(f(\lambda_1),\ldots,f(\lambda_n)\right)\,.
\end{align}
If $A$ cannot be diagonalized, a spectral definition of $f(A)$ can still be derived using the Jordan decomposition \cite{Golub}.
For hermitian $A$ the eigenvalues are real and their sign is defined by $\sgn(x)=\pm 1$ for $x\gtrless 0$ with $x \in \mathbb{R}$, such that \eq{fAdef} readily defines the matrix sign function.
For non-hermitian $A$ the eigenvalues are complex and require a definition of $\sgn(z)$ for $z \in \mathbb{C}$. The sign function needs to satisfy $(\sgn z)^2=1$ and reproduce the usual $\sgn(x)$ for real $x$. We define \cite{Bloch:2007xi}
\begin{align}
\sgn(z) &= \frac{z}{\sqrt{z^2}} = \sgn\left(\re z\right)\,,
\label{eq:sgnz}
\end{align}
where the cut of the square root is chosen along the negative real axis. This choice, although not unique, gives the correct physical result for the overlap Dirac operator in \eq{Dovmu} (see Ref.~\cite{Bloch:2007xi}).
To investigate the anti-unitarity properties of the overlap operator we first observe that links $U_{x,\pm\nu}$ as elements of the SU(2){} group obey
\begin{equation}
U_{x,\pm\nu}^\ast=\tau_2^\dagger\, U_{x,\pm\nu}\, \tau_2
\end{equation}
(as they can be written in terms of $1_2$ and $i\tau_a$). For the Wilson Dirac operator multiplied by $\gamma_5$ one can show that
\begin{align}\label{eq:inv ordering}
\hspace{4cm} (\gamma_5D_{{\sf W}})^\ast = U^\dagger (D_{{\sf W}}\gamma_5) U\qquad\mbox{(note the inverted ordering)}
\end{align}
with $U$ as in the continuum, see \eq{eqn iD symm}.
For the overlap operator we have to defer the discussion of its anti-unitary symmetries, because it has no definite hermiticity, even at zero chemical potential. Instead it fulfills the Ginsparg-Wilson relation \cite{Ginsparg:1981bj},
\begin{align}
\{D_{{\sf ov}},\gamma_5\}=D_{{\sf ov}}\gamma_5D_{{\sf ov}}\,,
\end{align}
which, together with the $\gamma_5$-her\-mi\-ti\-city, $\gamma_5D_{{\sf ov}}=D_{{\sf ov}}^\dagger\gamma_5$, valid at vanishing $\mu$, forces the eigenvalues on a circle $\theta=1+\exp(i\varphi)$ in the complex plane.
When investigating the spectrum of the overlap operator, we will rather consider a related operator \cite{Neuberger:1997bg}
\begin{equation}\label{eq:overlapproj}
D_p=\frac{2D_{{\sf ov}}}{2-D_{{\sf ov}}}=2\,
\frac{1+ \gamma_5\sgn(\gamma_5D_{{\sf W}})}{1- \gamma_5\sgn(\gamma_5D_{{\sf W}})}\,,
\end{equation}
which satisfies exact chiral symmetry, $\{D_p,\gamma_5\}=0$, at the price of being non-local. For $\mu=0$ this operator is anti-hermitian and projects the eigenvalues of $D_{{\sf ov}}$ from the circle onto the imaginary axis.
At non-zero chemical potential $D_p$ loses its anti-hermiticity and the eigenvalues come in three types: pairs of opposite real eigenvalues, pairs of complex conjugate imaginary eigenvalues and quartets of complex conjugate/opposite eigenvalues $(\lambda,-\lambda,\lambda^\ast,-\lambda^\ast)$.
To form the complex quartets at finite $\mu$, typically two pairs of purely imaginary eigenvalues need to come together to allow for its creation; in our RMT discussion the former were called \textit{on-axis} and the latter \textit{off-axis}.
The anti-unitarity symmetry of $D_p$ follows from that of $\gamma_5D_{{\sf W}}$, \eq{eq:inv ordering}, and the sign function thereof (and the facts that $U$ commutes with $\gamma_5$ and that $\sgn$ and $\gamma_5$ are their own inverses):
\begin{align}
\gamma_5\sgn^\ast(\gamma_5D_{{\sf W}}) &= U^\dagger \gamma_5\sgn(D_{{\sf W}}\gamma_5) U
=U^\dagger \sgn(\gamma_5D_{{\sf W}})\gamma_5 U \,, \\
D_p^\ast &= 2\, U^\dagger \,\frac{1+\sgn(\gamma_5D_{{\sf W}})\gamma_5}
{1-\sgn(\gamma_5D_{{\sf W}})\gamma_5}\,U
=2\, U^\dagger \,\frac{\gamma_5\sgn(\gamma_5D_{{\sf W}})+1}{\gamma_5\sgn(\gamma_5D_{{\sf W}})-1}\,U \notag\\
&=- U^\dagger D_p \, U\,.
\label{eqn Dp symm}
\end{align}
Hence $i D_p$ shares the anti-unitary symmetry from the continuum, compare \eqref{eqn Dp symm} to \eqref{eqn iD symm}, and therefore GOE spacings are expected at vanishing $\mu$. Again a chemical potential destroys the hermiticity, but keeps the anti-unitary symmetry, exactly as for the continuum Dirac operator. Note that $\mu$ enters the hermiticity breaking part of the operator $D_p$ non-linearly -- unlike $\lambda$ and $\Lambda$ in the random matrices.
\begin{table}[b]
\begin{center}
\begin{tabular}{c||c|c|c|c}
$\mu$ & $0.05$ & 0.10 & 0.20 & 0.30 \\
\hline
\# configs & 60000 & 30000 & 20000 & 20000
\end{tabular}
\caption{Values of the chemical potential and corresponding number of quenched configurations used to determine the spectral properties of the overlap operator.}
\label{tab:sim-param}
\end{center}
\end{table}
\subsection{Numerical results}
The quenched lattice simulations were performed on an $8^4$ lattice with the Wilson gauge action, using the USQCD lattice QCD software Chroma and QDP++ \cite{usqcd}. The software was adapted with an implementation of the overlap operator at non-zero chemical potential, based on the nested Krylov-Ritz approximation \cite{Bloch:2009uc}. We generated 60000 quenched configurations at $\beta=2.2$ using a heatbath/overrelaxation algorithm. On subsets of them we computed $\mathcal{O}(30)$ of the lowest lying eigenvalues of the overlap operator \eqref{Dovmu} with ARPACK \cite{arpack} for various values of the chemical potential and Wilson mass $m_\text{w}=-1.4$. The number of configurations used for the different chemical potentials are summarized in Table~\ref{tab:sim-param}. An animated plot of the evolution of the eigenvalues of $D_p$ with increasing chemical potential for a typical configuration can be found as ancillary material to this arXiv submission \cite{anim}.
\begin{figure}[t]
\includegraphics[width=\linewidth]{Plots/spacings_chempot_real_mu_typ=2.pdf}
\includegraphics[width=\linewidth]{Plots/spacings_chempot_real_mu_typ=3.pdf}
\includegraphics[width=\linewidth]{Plots/spacings_chempot_real_mu_typ=1.pdf}
\caption{Spacing distributions of the overlap operator with various real values of the chemical potential $\mu$. Top: on-axis spacings of type 1, surmise given by GOE, \eq{eq:surmiseon1}. Middle: on-axis spacings of type 2, surmise given by \eq{eq:surmiseon2}. Bottom: off-axis spacings, surmise given by \eq{eq:surmiseoff}; the dashed curve is the GOE spacing for comparison.}
\label{fig:overlaprealmu}
\end{figure}
We measured the spacing distributions of the projected overlap operator $D_p$, defined in \eq{eq:overlapproj}, for various values of the chemical potential $\mu$. As before, we distinguish between off-axis spacings and on-axis spacings with and without an interspaced complex pair. We only considered spacings between eigenvalues with an imaginary part in the spectral window $(0.5, 0.6)$. This ensures that the eigenvalues stay in the bulk of the spectrum (which of course is limited on a finite lattice) and have a roughly constant eigenvalue density which is necessary to apply the surmises, as argued in Sec.~\ref{sect:rmt}. As for the large random matrices, unfolding was only done for the on-axis spacings.
The results are shown in \fig{fig:overlaprealmu}. The numerical data are well matched by the surmises derived from small matrices, like for the large random matrices considered in Sec.~\ref{sect:rmt}. Again, the coupling parameter $\lambda$ is obtained by a fit for the on-axis spacings of type 2 and from the frequency $p$ of imaginary spacings for the off-axis spacings. For the off-axis spacings at large $\mu$ we observe that the discrepancy between the data and the surmise has the same tendency as for the large random matrices, cf.\ \fig{fig:largermt}, lower right panel. This indicates that the RMT results for large matrices are able to describe the QCD results reasonably well, even at larger coupling.
\subsection{Imaginary chemical potential}
The introduction of an \textit{imaginary} chemical potential does not change the anti-hermiticity of the continuum Dirac operator $D$ nor of the lattice operator $D_p$.
However, the operators $iD$ and $iD_p$ no longer obey an anti-unitary symmetry. The random matrix ensemble appropriate for operators without anti-unitary symmetry is the GUE. Hence, with increasing imaginary chemical potential one expects a transition from GOE to GUE. Surmises for such mixed random matrices, again additive, $H_\text{GOE}+\lambda H_\text{GUE}$, have been worked out in Refs.~\cite{Lenz:1991, Schierenberg:2012ut}, with the result
\begin{equation}
P_{\text{GOE}\to\text{GUE}}(s)= C s\, e^{-D^2s^2} \erf\left(\frac{Ds}{\lambda}\right)\,,
\label{eq:surmiseimag}
\end{equation}
where
\begin{align}
C = 2\sqrt{1+\lambda^2}\,D^2 \,,\qquad
D = \frac{\sqrt{1+\lambda^2}}{\sqrt{\pi}}
\left(\frac{\lambda}{1+\lambda^2} + \text{arccot}\lambda\right)
\,.
\label{eq:surmiseimag_const}
\end{align}
In \fig{fig:overlapimagmu}, these distributions are compared to those of the QCD Dirac operator $D_p$ with imaginary chemical potential, were unfolding was again done by measuring the spacings in units of the local mean spacing (obtained by an ensemble average). The coupling parameter was obtained by a fit with least square minimization. The surmise agrees very well with the numerical data.
\begin{figure}
\includegraphics[width=\linewidth]{Plots/spacings_chempot_imaginary_mu.pdf}
\caption{Spacing distributions of the overlap operator with various imaginary values of the chemical potential $\mu$ fitted by the surmise \eqref{eq:surmiseimag}. For each $\mu$, 5000 configurations have been analyzed in the spectral window with imaginary parts of the eigenvalues between $0.65$ and $0.8$.}
\label{fig:overlapimagmu}
\end{figure}
\section{Summary}
\label{sect:summary}
We have shown that eigenvalues of real symmetric matrices are attracted under real antisymmetric perturbations and are able to merge and move into the complex plain as complex conjugate pairs. For the emergent kinds of level spacings -- on-axis with and without interspaced eigenvalue pair and off-axis -- we have derived surmises from small random matrices. For the on-axis spacings without interspaced pair, the surmise is simply the GOE surmise, as in the unperturbed case, while for the other two we have obtained closed formulae, which depend on a coupling parameter $\lambda$.
These mixed-hermiticity surmises provide good approximations to the spacings of large random matrices in the regime of weak non-hermiticity and for the bulk of two-color QCD with small chemical potential, as expected from the anti-unitary symmetry and hermiticity properties of the Dirac operator. For on-axis spacings with an interspaced pair the surmise parameter $\lambda$ was obtained by a fit to the level spacings, while for off-axis spacings it was predicted by matching the frequency of those spacings between surmise and data.
We have also measured the level spacings for two-color QCD with imaginary chemical potential and verified that they follow a mixed-symmetry surmise.
In contrast to the eigenvalue densities, the level spacings of asymmetric real random matrices have not been worked out analytically. However, for weak antisymmetric perturbations the analytic surmises, which approximate the level spacings of large random matrices, can be used to describe physical systems obeying the relevant anti-unitary symmetry properties.
\section*{Acknowledgements}
The spectra of the overlap operator have been calculated on the Athene and iDataCool compute clusters, which are part of the HPC infrastructure of the University of Regensburg. We would like to thank Gernot Akemann for useful discussions. We also acknowledge DFG for financial support (SFB/TR-55 and BR 2872/4-2).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,760 |
Q: Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v2 §19) $\newcommand{\refone}{\textbf{(1)}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which is finite-dimensional as a $\kk$-vector space. Let $\tau : A \to \kk$ be a $\kk$-linear map. Define a $\kk$-bilinear form $\left<\cdot, \cdot\right> : A \times A \to \kk$ by $\left< a, b\right> = \tau\left(ab\right)$. Assume that this form $\left<\cdot, \cdot\right>$ is symmetric and nondegenerate. Let $\left(a_i\right)_{i \in I}$ and $\left(a'_i\right)_{i \in I}$ be two mutually dual bases of $A$ with respect to this form (so that $\left<a_i, a'_j\right>$ equals $1$ whenever $i = j$ and $0$ otherwise).
Facts: (See below for the proofs of these claims.)
(1) We have $\sum\limits_{i\in I} a_i \tau\left(a'_i b\right) = \sum\limits_{i\in I} a'_i \tau\left(a_i b\right) = b$ for every $b \in A$.
(2) We have $\sum\limits_{i\in I} a_i \tau\left(a'_i\right) = \sum\limits_{i\in I} a'_i \tau\left(a_i\right) = 1$.
(3) The tensor $\sum\limits_{i\in I} a_i \otimes a'_i \in A \otimes A$ (over $\kk$) does not depend on the choice of bases $\left(a_i\right)_{i \in I}$ and $\left(a'_i\right)_{i \in I}$.
(3') For any $b \in A$, we have $\sum\limits_{i\in I} b a_i \otimes a'_i = \sum\limits_{i\in I} a_i \otimes a'_i b$.
(3'') For any $b \in A$, we have $\sum\limits_{i\in I} a_i b \otimes a'_i = \sum\limits_{i\in I} a_i \otimes b a'_i$.
(4) Consider the dual space $A^*$ of $A$ as an $A$-$A$-bimodule in the usual way (so $\left(af\right)\left(b\right) = f\left(ba\right)$ and $\left(fa\right)\left(b\right) = f\left(ab\right)$ for all $a \in A$, $b \in A$ and $f \in A^*$). Let $J : A \to A^*$ be the map which sends every $b \in A$ to the $\kk$-linear map $\beta : A \to \kk$ given by $\beta\left(c\right) = \left<b,c\right>$. Then, $J$ is an isomorphism of $A$-$A$-bimodules. This makes $A$ into a so-called symmetric Frobenius algebra. (Conversely, any symmetric Frobenius algebra has a $\tau$ satisfying the conditions we imposed on $\tau$ above.)
From now on, let $E$ be a left $A$-module which is finite-dimensional as a $\kk$-vector space. For every $x \in A$, let $\tr_E x$ be the trace of the $\kk$-vector space endomorphism of $E$ given by the action of $x$ on $E$. Let $\gamma_E$ be the element $\sum\limits_{i\in I} \tr_E \left(a_i\right) a'_i$ of $A$.
(5) This element $\gamma_E$ does not depend on the choice of bases $\left(a_i\right)_{i \in I}$ and $\left(a'_i\right)_{i \in I}$.
(6) This element $\gamma_E$ lies in the center of $A$.
(7) If $E'$ is a simple left $A$-module such that $\Hom_A\left(E',E\right) = 0$ (this happens, for instance, if $E$ and $E'$ are nonisomorphic simple $A$-modules), then $\gamma_E E' = 0$.
Question:
Are the following true?
(8) If $E$ is a simple left $A$-module, then the element $\gamma_E$ is quasi-idempotent, meaning that $\gamma_E^2 = \lambda_E \gamma_E$ for some $\lambda_E \in \kk$.
(9) If $E$ is a simple left $A$-module, then the action of $\gamma_E$ on $E$ is a scalar multiple of the identity map.
Remarks:
One example of a situation as described above is when $A$ is the group algebra $\kk\left[G\right]$ of a finite group $G$, and $\tau$ is the map sending every element $a$ of this group algebra to the coefficient of $1 \in G$ in $a$. One can then take $\left(a_i\right)_{i\in I} = \left(g\right)_{g\in G}$ and $\left(a'_i\right)_{i\in I} = \left(g^{-1}\right)_{g\in G}$. This works no matter whether $A$ is semisimple or not.
The question was born in my attempt to lift the "split semisimple" condition in Proposition 19.2 of George Lusztig's Hecke algebras with unequal parameters (arXiv:math/0208154v2). Specifically I am trying to generalize parts (b) and (e) of said proposition; part (a) trivially generalizes and parts (c) and (d) don't generalize this far (at least semisimplicity cannot be disposed of). My main hope with this question is to obtain an alternative route to a piece of representation theory (specifically, character orthogonality formulas) in a greater generality than is done in standard texts, and in a lightweight way (no algebraic closure requirements, no use of Artin-Wedderburn, no characteristic-$0$ assumptions, and maybe even allowing $\kk$ to be any commutative ring).
Notice that the conjectural claims (8) and (9) are easily seen to be equivalent. I possibly can show (8) and (9) under the assumption that $A$ is semisimple, using some basic Galois theory (reducing the problem to the algebraic closure of $\kk$ and there applying Artin-Wedderburn); but I haven't fully worked out a proof and to be honest I am not interested in doing so (see my above motivation).
It is not generally true that the scalar in (9) is nonzero.
Proofs:
For the sake of completeness, here are my proofs for the facts above.
(1) Every $b \in A$ satisfies $\sum\limits_{i\in I} a_i \left<a'_i, b\right> = b$ (since $\left(a_i\right)_{i \in I}$ and $\left(a'_i\right)_{i \in I}$ are two mutually dual bases of $A$). Since $\left<a'_i, b\right> = \tau\left(a'_i b\right)$, this rewrites as $\sum\limits_{i\in I} a_i \tau\left(a'_i b\right) = b$. The same argument, with $\left(a_i\right)_{i \in I}$ and $\left(a'_i\right)_{i \in I}$ switched, shows that $\sum\limits_{i\in I} a'_i \tau\left(a_i b\right) = b$ for every $b \in A$ (because being dual bases with respect to a symmetric bilinear form is a symmetric relation). This proves (1).
(2) This follows from setting $b = 1$ in (1).
(4) The definition of $J$ shows that $\left(J\left(b\right)\right)\left(c\right) = \left<b, c\right>$ for all $b \in A$ and $c \in A$.
The map $J$ is injective. (In fact, if $b \in A$ is such that $J\left(b\right) = 0$, then every $c \in A$ satisfies $\left<b, c\right> = \underbrace{\left(J\left(b\right)\right)}_{=0}\left(c\right) = 0$, whence $b = 0$ (by the nondegeneracy of the form $\left<\cdot, \cdot\right>$).) It is also easy to see that the map $J$ is $\kk$-linear (since the form $\left<\cdot, \cdot\right>$ is $\kk$-bilinear).
Every $a \in A$, $b \in A$ and $c \in A$ satisfy
$\left(a J\left(b\right)\right) \left(c\right) = \left(J\left(b\right)\right) \left(ca\right) = \left<b, ca\right>$ (by the definition of $J$)
$= \tau\left(bca\right) = \left<bc, a\right> = \left<a, bc\right> $ (by the symmetry of the form $\left<\cdot,\cdot\right>$)
$= \tau\left(abc\right)$
and
$\left(J\left(ab\right)\right) \left(c\right) = \left< ab, c\right> = \tau\left(abc\right)$.
Hence, every $a \in A$, $b \in A$ and $c \in A$ satisfy $\left(a J\left(b\right)\right) \left(c\right) = \tau\left(abc\right) = \left(J\left(ab\right)\right) \left(c\right)$. In other words, every $a \in A$ and $b \in B$ satisfy $a J\left(b\right) = J\left(ab\right)$. Hence, the map $J$ is left $A$-linear. A similar (but simpler!) argument shows that $J$ is right $A$-linear. Hence, $J$ is a $A$-$A$-bimodule homomorphism. Since $J : A \to A^*$ is $\kk$-linear and injective, we conclude that $J$ is a $\kk$-vector space isomorphism (since $A$ and $A^*$ are finite-dimensional $\kk$-vector spaces of the same dimension). Therefore, $J$ is an $A$-$A$-bimodule isomorphism (since $J$ is a $A$-$A$-bimodule homomorphism). This proves (4).
(3) Let $\rho : A \otimes A \to \kk$ be the $\kk$-linear map induced by the $\kk$-bilinear map $\left<\cdot,\cdot\right> : A\times A \to \kk$ (according to the universal property of a tensor product). Thus, $\rho\left(b\otimes c\right) = \left<b, c\right>$ for all $b \in A$ and $c \in A$. Let $J$ be as in claim (4).
Since $A$ is a finite-dimensional $\kk$-vector space, we can identify $\left(A\otimes A\right)^*$ with $A^* \otimes A^*$. Thus, for example, $\left(J \otimes J\right) \left(\sum\limits_{i\in I} a_i \otimes a'_i\right) \in A^* \otimes A^*$ becomes an element of $\left(A\otimes A\right)^*$, that is, a $\kk$-linear map $A \otimes A \to \kk$.
Now, every $b \in A$ and $c \in A$ satisfy
$\underbrace{\left(\left(J \otimes J\right) \left(\sum\limits_{i\in I} a_i \otimes a'_i\right)\right)}_{=\sum\limits_{i\in I} J\left(a_i\right) \otimes J\left(a'_i\right)} \left(b \otimes c\right)
= \left(\sum\limits_{i\in I} J\left(a_i\right) \otimes J\left(a'_i\right)\right) \left(b \otimes c\right)$
$= \sum\limits_{i\in I} \underbrace{\left(J\left(a_i\right)\right)\left(b\right)}_{= \left<a_i, b\right>} \otimes \underbrace{\left(J\left(a'_i\right)\right)\left(c\right)}_{= \left<a'_i, c\right> = \tau\left(a'_i c\right)} = \sum\limits_{i\in I} \left<a_i, b\right> \otimes \tau\left(a'_i c\right)$
$= \sum\limits_{i\in I} \left<a_i, b\right> \tau\left(a'_i c\right)$ (since we are identifying $\kk \otimes \kk$ with $\kk$)
$= \left<\underbrace{\sum\limits_{i\in I} a_i \tau\left(a'_i c\right)}_{= c \text{ (by }\refone{ )}}, b\right> = \left<c, b\right> = \left<b, c\right>$ (since the form $\left<\cdot,\cdot\right>$ is symmetric)
$= \rho\left(b \otimes c\right)$.
In other words, the two $\kk$-linear maps $\left(J \otimes J\right) \left(\sum\limits_{i\in I} a_i \otimes a'_i\right)$ and $\rho$ are identical on each pure tensor. Hence, these two $\kk$-linear maps must be equal. Thus, $\left(J \otimes J\right) \left(\sum\limits_{i\in I} a_i \otimes a'_i\right) = \rho$. Since $J \otimes J$ is an isomorphism (because so is $J$), we thus have $\sum\limits_{i\in I} a_i \otimes a'_i = \left(J \otimes J\right)^{-1} \left(\rho\right)$, and thus $\sum\limits_{i\in I} a_i \otimes a'_i \in A \otimes A$ (over $\kk$) does not depend on the choice of bases $\left(a_i\right)_{i \in I}$ and $\left(a'_i\right)_{i \in I}$. This proves (3).
(3') Let $J$ be as in claim (4). Let $b \in A$. Every $c \in A$ and $d \in A$ satisfy
$\underbrace{\left(\left(J \otimes J\right) \left(\sum\limits_{i\in I} b a_i \otimes a'_i\right)\right)}_{= \sum\limits_{i\in I} J\left(b a_i\right) \otimes J\left(a'_i\right)} \left(c \otimes d\right)
= \left(\sum\limits_{i\in I} J\left(b a_i\right) \otimes J\left(a'_i\right)\right) \left(c \otimes d\right)$
$= \sum\limits_{i \in I} \underbrace{\left(J\left(b a_i\right)\right)\left(c\right)}_{= \left< b a_i, c\right> = \tau\left(b a_i c\right)} \otimes \underbrace{\left(J\left(a'_i\right)\right)\left(d\right)}_{= \left< a'_i, d\right> = \tau\left(a'_i d\right)} = \sum\limits_{i \in I} \tau\left(b a_i c\right) \otimes \tau\left(a'_i d\right)$
$= \sum\limits_{i\in I} \tau\left(b a_i c\right) \tau\left(a'_i d\right)$ (since we are identifying $\kk \otimes \kk$ with $\kk$)
$= \tau\left(b \underbrace{\sum\limits_{i\in I} a_i \tau\left(a'_i d\right)}_{= d \text{ (by }\refone{ )}} c\right) = \tau\left(bdc\right)$
and
$\underbrace{\left(\left(J \otimes J\right) \left(\sum\limits_{i\in I} a_i \otimes a'_i b\right)\right)}_{= \sum\limits_{i\in I} J\left(a_i\right) \otimes J\left(a'_i b\right)} \left(c \otimes d\right)
= \left(\sum\limits_{i\in I} J\left(a_i\right) \otimes J\left(a'_i b\right)\right) \left(c \otimes d\right)$
$= \sum\limits_{i \in I} \underbrace{\left(J\left(a_i\right)\right)\left(c\right)}_{= \left< a_i, c\right> = \tau\left(a_i c\right)} \otimes \underbrace{\left(J\left(a'_i b\right)\right)\left(d\right)}_{= \left< a'_i b, d\right> = \tau\left(a'_i b d\right)} = \sum\limits_{i \in I} \tau\left(a_i c\right) \otimes \tau\left(a'_i b d\right)$
$= \sum\limits_{i\in I} \tau\left(a_i c\right) \tau\left(a'_i b d\right)$ (since we are identifying $\kk \otimes \kk$ with $\kk$)
$= \tau\left(\underbrace{\sum\limits_{i\in I} a_i \tau\left(a'_i b d\right)}_{= b d \text{ (by }\refone \text{, applied to } bd \text{ instead of } b \text{)}}\right) = \tau\left(bdc\right)$.
Thus, every $c \in A$ and $d \in A$ satisfy $\left(\left(J \otimes J\right) \left(\sum\limits_{i\in I} b a_i \otimes a'_i\right)\right) \left(c \otimes d\right) = \tau\left(bdc\right) = \left(\left(J \otimes J\right) \left(\sum\limits_{i\in I} a_i \otimes a'_i b\right)\right) \left(c \otimes d\right)$.
In other words, the two $\kk$-linear maps $\left(J \otimes J\right) \left(\sum\limits_{i\in I} b a_i \otimes a'_i\right)$ and $\left(J \otimes J\right) \left(\sum\limits_{i\in I} a_i \otimes a'_i b\right)$ are identical on each pure tensor. Hence, these two $\kk$-linear maps must be equal. Thus, $\left(J \otimes J\right) \left(\sum\limits_{i\in I} b a_i \otimes a'_i\right) = \left(J \otimes J\right) \left(\sum\limits_{i\in I} a_i \otimes a'_i b\right)$. Since $J \otimes J$ is an isomorphism (because so is $J$), we thus have $\sum\limits_{i\in I} b a_i \otimes a'_i = \sum\limits_{i\in I} a_i \otimes a'_i b$. This proves (3').
(3'') One can prove (3'') in a similar way to (3') (with a little twist: one has to use symmetry of $\left< \cdot, \cdot \right>$ at one step). But we can alternatively prove (3'') using (3') as follows: Fix $b \in A$. Being dual bases with respect to a symmetric bilinear form is a symmetric relation. Thus, $\left(a'_i\right)_{i\in I}$ and $\left(a_i\right)_{i\in I}$ are two mutually dual bases of $A$ with respect to the form $\left<\cdot, \cdot\right>$. Hence, applying (3') to $a'_i$ and $a_i$ instead of $a_i$ and $a'_i$, we obtain $\sum\limits_{i\in I} b a'_i \otimes a_i = \sum\limits_{i\in I} a'_i \otimes a_i b$. Applying the $\kk$-linear map $A \otimes A, \ x \otimes y \mapsto y \otimes x$ to both sides of this equality, we obtain $\sum\limits_{i\in I} a_i \otimes b a'_i = \sum\limits_{i\in I} a_i b \otimes a'_i$. This proves (3'').
(5) Let $\tr_E^{(1)}$ denote the $\kk$-linear map $A \otimes A \to A$ sending every $b \otimes c$ to $\tr_E\left(b\right) c$. (This is clearly well-defined.) Then, $\gamma_E = \sum\limits_{i\in I} \tr_E \left(a_i\right) a'_i$ is the image of the tensor $\sum\limits_{i\in I} a_i \otimes a'_i \in A \otimes A$ under the map $\tr_E^{(1)}$. Since said tensor does not depend on the choice of bases $\left(a_i\right)_{i \in I}$ and $\left(a'_i\right)_{i \in I}$ (by (3)), the same must hold for $\gamma_E$. This proves (5).
(6) For every $b \in A$, we have $\gamma_E = \sum\limits_{i\in I} \tr_E \left(a_i\right) a'_i$ and thus
$\left< \gamma_E, b\right> = \left< \sum\limits_{i\in I} \tr_E \left(a_i\right) a'_i, b\right> = \sum\limits_{i\in I} \tr_E \left(a_i\right) \left<a'_i, b\right> = \tr_E \left(\sum\limits_{i\in I} a_i\underbrace{\left<a'_i, b\right>}_{= \tau\left(a'_i b\right)}\right) $
(10) $= \tr_E \left(\underbrace{\sum\limits_{i\in I} a_i \tau\left(a'_i b\right)}_{= b \text{ (by }\refone\text{)}}\right) = \tr_E \left(b\right) $.
Now, if $u$ and $v$ are two elements of $A$, then
$\left<\gamma_E u, v\right> = \tau\left(\gamma_E u v\right) = \left< \gamma_E, uv\right> = \tr_E \left(uv\right)$ (by (10))
$ = \tr_E \left(vu\right)$ (since any two endomorphisms $\alpha$ and $\beta$ of a finite-dimensional $\kk$-vector space satisfy $\tr\left(AB\right) = \tr\left(BA\right)$)
$ = \left< \gamma_E, vu\right>$ (again by (10))
$ = \left< vu, \gamma_E\right>$ (since $\left<\cdot,\cdot\right>$ is symmetric)
$ = \tau\left(vu\gamma_E\right) = \left< v, u \gamma_E\right> = \left< u\gamma_E, v\right>$ (since $\left<\cdot,\cdot\right>$ is symmetric).
Since the form $\left<\cdot,\cdot\right>$ is nondegenerate, this shows that $\gamma_E u = u \gamma_E$ for every $u \in A$. In other words, the element $\gamma_E$ of $A$ is central. This proves (6).
(7) This proof is somewhat weird and I am not sure if I like it. Let $E'$ be a simple left $A$-module such that $\Hom_A\left(E',E\right) = 0$.
First of all,
(11) if $F$ is any simple right $A$-module, then the right $A$-module $F$ is isomorphic to a quotient of the right $A$-module $A$.
To prove this, let $F$ be any simple right $A$-module. Then, $F$ is nonzero (because it is simple). Fix any nonzero $x \in F$ (such an $x$ exists, since $F$ is nonzero). Then, the map $A \to F$ sending each $b \in A$ to $xb$ is right $A$-linear; its image is thus an $A$-submodule of $F$. Since the only $A$-submodules of $F$ are $0$ and $F$, this yields that the image of this map is $0$ or $F$. But it cannot be $0$ since it contains $x1 = x \neq 0$ (the image of $1$ under this map). Hence, it must be $F$, so that this map is surjective. We thus have found a surjective $A$-linear map $A \to F$; therefore, $F$ is isomorphic to a quotient of the right $A$-module $A$. This proves (11).
Similarly to (11), we can see that:
(12) if $F$ is any simple left $A$-module, then the left $A$-module $F$ is isomorphic to a quotient of the left $A$-module $A$.
We know that $E'$ is a simple left $A$-module, so that (12) shows that $E'$ is isomorphic to a quotient of the left $A$-module $A$. Hence, $\dim_{\kk} E' \leq \dim_{\kk} A < \infty$. Hence, the dual $E'^*$ of $E'$ is a simple right $A$-module (because $E'$ is a simple left $A$-module). We can thus apply (11) to $F = E'^*$, and conclude that the right $A$-module $E'^*$ is isomorphic to a quotient of the right $A$-module $A$. Dualizing again (we can do this because $\dim_{\kk} E' \leq \dim_{\kk} A < \infty$), we conclude that the left $A$-module $E'$ is isomorphic to a submodule of the left $A$-module $A^*$. Since the left $A$-module $A^*$ is isomorphic to the left $A$-module $A$ (by the map $J$, as (4) shows), this yields that the left $A$-module $E'$ is isomorphic to a submodule of the left $A$-module $A$. Since the claim that we want to prove is invariant under isomorphism of $E'$, we can thus WLOG assume that $E'$ is a submodule of the left $A$-module $A$. Assume this.
Let $w \in E'$ and $u \in A$. We are going to prove that $\left<\gamma_E w, u\right> = 0$. To do so, let us first check that $wuv = 0$ for every $v \in E$.
In fact, let $v \in E$. Then, $guv \in E$ for every $g \in E'$ (because $E$ is a left $A$-module). Hence, we can define a left $A$-linear map $E' \to E$ by sending each $g \in E'$ to $guv$. This map belongs to $\Hom_A\left(E',E\right) = 0$, and thus is zero. In other words, $guv = 0$ for every $g \in E'$. Applied to $g = w$, this yields $wuv = 0$.
Now, forget that we fixed $v \in E$. We have shown that $wuv = 0$ for every $v \in E$. In other words, $wu \in A$ acts as $0$ on the left $A$-module $E$. In particular, this yields $\tr_E \left(wu\right) = 0$.
Now,
$\left<\gamma_E w, u\right> = \tau\left(\gamma_E w u\right) = \left<\gamma_E, wu\right> = \tr_E\left(wu\right)$ (by (10), applied to $b = wu$)
$= 0$.
Forget that we fixed $u$. We have shown that $\left<\gamma_E w, u\right> = 0$ for every $u \in A$. Hence, $\gamma_E w = 0$ (since the form $\left<\cdot,\cdot\right>$ is nondegenerate). Since this holds for all $w \in E'$, this yields $\gamma_E E' = 0$, and so (7) is proven.
A: In the case of a (finite) group algebra $A$ , and I think (the necessary arguments are at least implicit in my paper "On projective summands of induced modules") in this greater generality, one can see that if $E$ is a simple $A$-module, and $M$ is any finite dimensional $A$-module, then the dimension of $\gamma_{E}M$
is (the dimension of $E$) \times (the number of indecomposable summands of $M$ isomorphic to the projective cover of $E).$
So I think that the answer to question 8 is yes; for simple $E,$ the element $\gamma_{E}$ will square to zero unless $E$ is itself projective and simple. If $E$ is projective and simple, then $\gamma_{E}$ is a unit multiple of a central idempotent. This also seems to answer question 9.
In general, each $\gamma_{E}$ (for simple $E$) annihilates the Jacobson Radical $J(A),$ and the $ \sum_{ {\rm simple} E} \gamma_{E}A$ is the socle of the regular $A$-module, which is isomorphic as $A$-module to $A/J(A).$
A: I think you should check out Chapter 7 of Geck and Pfeiffer's book "Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras". There, many properties of symmetric algebras are worked out in quite a general setting (for instance over a commutative ring). For your purposes I think the Geschütz-Ikeda lemma might be important and relates to what Geoff is saying (se 7.1.11 in Geck-Pfeiffer).
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\section{Introduction}
It has been known for a long time that the light and hence the mass
distribution in disks of spiral galaxies is not strictly
axisymmetric, as for example in M 101 or in NGC 1637 (Sandage 1961),
where the isophotes are elongated in one half of the galaxy.
Despite this, however, astronomers have largely tended to ignore
this fact and to assume the disks to be axisymmetric because it is
much simpler to study the dynamics of axisymmetric disks.
This phenomenon was first highlighted
in the pioneering paper by Baldwin, Lynden-Bell, \& Sancisi (1980),
where they detected an asymmetry in the spatial extent of the atomic
hydrogen gas in the outer regions in the two halves of some
galaxies, and gave these the apt name of `lopsided' galaxies. A
galaxy is said to be lopsided if it displays a global
non-axisymmetric spatial distribution of type $m=1$ where $m$ is the
azimuthal wavenumber, or a $cos \phi$ distribution where $\phi$ is
the azimuthal angle in the plane of the disk.
Surprisingly no further systematic work was done on this topic till
mid-1990's. Since then there has been a resurgence in this field.
The lopsided distribution has now been detected and studied also in the old
stellar component as traced in the near-IR starting
with the observations of Block et al. (1994) and Rix \& Zaritsky
(1995). This exciting new development of the imaging studies of
spiral galaxies in the near-IR K-band (2.2 $\mu$) was made possible
by the development of the NICMOS 3 array. The dust extinction
effects are negligible in the near-IR, hence these studies reveal the
spatial distribution of the underlying old stellar population, which constitute the main mass
component of the disk. These
observations detected a non-axisymmetric $m = 1 $ distribution of surface
density of old stars in the inner/optical region of the disk. Rix
\& Zaritsky (1995) define $A_1$, the fractional amplitude of
the first azimuthal fourier component ($m = 1$) of surface
brightness, to be the quantitative measure of disk lopsidedness. They
find that $A_1 $ increases with radius. The average value
measured between 1.5-2.5 disk scalelengths is large $\geq 0.1$, and
$30 \% $ of the galaxies studied show a higher lopsidedness
(Zaritsky \& Rix 1997). A similar high average value of disk
lopsidedness was confirmed in a recent Fourier-analysis study of a
much larger sample of 149 galaxies (Bournaud et al. 2005 b).
The above analysis shows that nearly one third of the 149 galaxies
exhibit 10 \% or more asymmetry in the amplitude of the $m=1$
Fourier component. Thus, {\it lopsided distribution in the disk is a
general phenomenon}, and is stronger at larger radii. Hence it is
important to understand the origin and dynamics of the
lopsided distribution in spiral galaxies.
The lopsided distribution in the HI gas has been mapped spatially
(Haynes et al. 1998), and also mapped kinematically for a few
galaxies (Schoenmakers, Franx \& de Zeeuw 1997, Swaters et al. 1999), and by
global velocity profiles for a much larger sample (Richter \& Sancisi
1994). Such an asymmetry has also been detected in dwarf galaxies
(Swaters et al. 2002), and also in the star-forming regions in
irregular galaxies (Heller et al. 2000). The asymmetry may affect
all scales in a galaxy. While the large-scale lopsidedness is more
conspicuous, the off-centering of nuclei is now often discovered at
high spatial resolution. A prototype of this $m=1$ nuclear
distribution is the inner region of M31, where the central black
hole is clearly off-centered with respect to its nuclear stellar
disk (e.g., Tremaine 2001). This frequent
nuclear $m=1$ perturbation must play a central
role in the fueling of the active galactic nucleus (AGN) in a galaxy.
The origin and the evolution of lopsidedness are not yet
well-understood, though a beginning has been made to address these
problems theoretically. Like any other non-axisymmetric perturbation, the
lopsided distribution would also tend to get wound up by the differential
rotation in the galactic disk within a few dynamical timescales.
Since a large fraction of galaxies exhibit lopsidedness, it must be either
a long-lived phenomenon or generated frequently.
Tidal interaction (Beale \& Davies 1969),
and satellite galaxy accretion (Zaritsky \& Rix 1997) have been
suggested as the origin of the disk lopsidedness, these can occur frequently.
Weinberg (1995) has shown that
the tidal interaction between the Galaxy and the Large Magellanic Cloud
(LMC) leads to a
lopsided distortion of the Galaxy halo at resonance points between
the LMC and the halo orbit frequencies, which in turn causes a
lopsided distribution in the disk of the Galaxy. Since galaxy
interactions are now known to be common, the origin of disk
lopsidedness as attributed to the disk response to the tidal distortion in a
halo has been proposed and studied by Jog (1997, 2002), and Schoenmakers et al. (1997).
Some other possible mechanisms that have been suggested involve
an off-center disk in a halo as in a dwarf galaxy (Levine \&
Sparke 1998), or gas accretion (Bournaud et al. 2005 b), or treating it as a
global, long-lived mode (Saha, Combes \& Jog 2007).
The $m=1$ distribution in the inner regions of some galaxies such as
M 31 has been modeled through analytical work and numerical
simulations (Tremaine 1995, Statler 2001, Bacon et al. 2001, de
Oliveira \& Combes 2008). According to the various physical
conditions in galaxy nuclei (such as the mass of the bulge, the mass
of the nuclear disk and that of the central black hole, the presence
of gas, etc.), an $m=1$ mode is unstable, or an $m=1$ excitation is
very slowly damped and can persist for several hundreds of dynamical
times. Such long-lived lopsided distribution is also seen in the centers of advanced
mergers of galaxies (Jog \& Maybhate 2006).
Due to their persistence, the lopsided modes could play a
significant role in the evolution of the central regions of
galaxies, especially in the fueling of a central AGN.
An m=1 perturbation in a
disk leads to a shift in the center of mass in the disk, and this
then acts as an indirect force on the original center of the
disk. The disk is inherently supportive of an m=1
mode, which is a particular feature only of a lopsided mode. This
results in long-lasting global lopsided modes. While the $m=2$ case
corresponding to the two-armed spiral pattern or bars has been studied
extensively, the $m=1$ mode has not received comparable
attention in the literature so far. This has to be redressed: first,
m=1 is common and the amplitude is even larger than for m=2 (Jarrett
et al. 2003), second, the m=1 modes do not have an Inner Lindblad Resonance,
or ILR (e.g., Block et al. 1994),
and hence can allow transport of matter in the inner regions, and third, these
appear to be long-lived.
The existence of long-lived lopsided modes is expected to have a significant impact
on the dynamics of the galaxy, the star formation in it, and on the nuclear fueling etc.
In the tidal picture, the disk lopsidedness can be used
as a diagnostic to study the lopsidedness of the dark matter halo
(Jog 1999). Similarly, higher-order ($m=2$) disk asymmetry can allow
one to study the ellipticity of the dark matter halo (Jog 2000,
Andersen \& Bershady 2002).
Thus in addition to being interesting and challenging in
itself, a study of disk lopsidedness can yield information about a
number of other interesting properties of galaxies. Extra-planar gas
and lopsidedness are frequently correlated (Sancisi et al 2008).
Recently, the lopsided distribution in galaxies in different environments
such as groups and clusters and centers of mergers has been studied.
These show different properties, such as the higher observed lopsided amplitudes in
the group galaxies (Angiras et al. 2006). In future studies, these can act as
important tracers of the dynamics of disks and dark matter halos in these settings.
In Section 2, we discuss the observational properties of
lopsidedness as seen in HI and old stars, in the main disk as well
as in the central region of galaxies. The various theoretical models
proposed in the literature and their comparison is given in Section
3. The dynamics of lopsidedness in the central region of the
near-Keplerian case region is discussed in Section 4. Section 5
discusses the observations and dynamical implications of lopsidedness
in galaxies in a different setting, namely in groups, clusters and in
centers of mergers.
Section 6 discusses several related points including the
comparison between m=1 and 2 cases, the deduction of the halo asymmetry etc.
Section 7 gives the effect
of lopsidedness on the galaxy. Finally, Section 8 gives a brief
summary and future directions for this field.
\section{Observations of lopsidedness in galactic disks}
While lopsidedness is seen to be a ubiquitous phenomenon (Section
1), various methods are used in the literature to define the
asymmetry in disk galaxies. We list and compare below the details,
such as the definitions, the size of the sample studied, the tracer
used (near-IR radiation from old stars, and 21 cm emission from the
HI gas etc.) There is no consensus so far as to what is the
definition for lopsidedness as well as what constitutes lopsidedness
(or the threshold). Obviously this has to be done if say the percentage
of galaxies showing lopsidedness as per the different criteria are
to be compared. At the end of Section 2.1, we recommend the use of a standard
criterion for lopsidedness as well as the threshold that could be adopted
by future workers.
The disk lopsidedness in spiral galaxies has been studied in two
different tracers- HI, the atomic hydrogen gas studied in the outer
parts of disks, and the near-IR radiation from the inner/optical
disks which traces the old stellar component of the disks. The
lopsidedness was historically first detected in the HI, which we now
realize is due to a higher lopsided amplitude at large radii. Hence
we follow the same, chronological order in the summary of observations given next.
\subsection{Morphological lopsidedness}
\subsubsection{Morphological lopsidedness in HI gas}
The asymmetric distribution of HI in M101 was noted in the early
HI observations (Beale \& Davies 1969). This phenomenon was first
highlighted by Baldwin et al. (1980), who studied galaxies with highly asymmetric spatial
extent of atomic hydrogen gas distributions in the outer regions in
the two halves of a galaxy, such as M101 and NGC 2841 (see Fig. 1 here).
This paper mentions the asymmetric distribution of light and HI in
spiral galaxies such as M101. Quantitatively, they looked at the HI
distribution in the four prototypical galaxies, namely, M101, NGC
891, NGC 4565, NGC 2841. They defined a galaxy to be "lopsided"
in which the galaxy is more extended on one side than the other, and
where the projected density of HI on the two sides of the
galaxy is at least 2:1 , and in which the asymmetry persists over a
large range in radius and position angle. All these lopsided galaxies
were also noted to have large-scale velocity perturbations.
For a quantitative measurement of lopsidedness the edge-on systems
cannot be used.
The cut-off in inclination used for the near-IR and HI studies is given in Sections 2.1.2 and 5.1 respectively.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=2.5in, width=2.5in]{fig1a.eps}
\medskip
\includegraphics[height=2.5in, width= 2.3in]{fig1b.eps}
\bigskip
\includegraphics[height=2.5in, width=2.5in]{fig1c.eps}
\medskip
\includegraphics[height=2.6in, width=2.7in]{fig1d.eps}
\bigskip
\caption{Galaxies showing an asymmetry in the spatial extent of 2:1 or more in the HI distribution :
M101 (top left, where the HI intensity is plotted here as gray scale)
and NGC 2841 (top right: here the HI contours are superimposed
on an optical image), These galaxies were termed "lopsided" galaxies
by Baldwin, Lynden-Bell, \& Sancisi (1980). These figures are from Braun (1995) and Bosma (1981) respectively.
Other typical examples are NGC 4654 (lower left: where HI contours are superimposed
on an optical image) and UGC7989 (lower right, showing contours and grey scale of the
HI intensity), from Phookun \& Mundy (1995),
and Noordermeer et al. (2005) respectively.}
\label{fig1}
\end{figure}
\bigskip
There was no further work on this topic for over a
decade. Further HI mapping of a few individual galaxies such
as NGC 4254 was done by Phookun, Vogel, \& Mundy (1993) which
stressed the obvious spatial asymmetry but they did not
measure the lopsidedness. This paper studied the special case of
the not-so-common one-armed galaxies such as NGC 4254 where the phase
varies with radius (see Section 2.3).
Richter \& Sancisi (1994) collected data from single-dish HI
surveys done using different radio telescopes, for a large sample of about 1700 galaxies. They found that 50 \%
of galaxies studied show strong
lopsidedness in their global HI velocity profiles. This could be either due to spatial
lopsided distribution and/or lopsided kinematics. But since a
large fraction of their sample shows this effect, they concluded that it must reflect an asymmetry
in the gas density, as is confirmed by the correlation between the spatial and global HI velocity asymmetries in some galaxies like NGC 891, see Fig. 2 here. They argued that the asymmetry in HI represents a large-scale
structural property of the galaxies. The
criteria they used to decide the asymmetry are: (1). significant flux difference
between the two horns ( $> 20
\%$ or $> 8$ sigma) (2). Total flux difference ($> 55 : 45 \%$)
between the low and the high velocity halves (3). Width differences
in the two horns ($>$ 4 velocity channels or 50 km s$^{-1}$). One
word of caution is that it is not clear from their paper if these
three give consistent identification of a galaxy as being lopsided
or not.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=5.0in]{fig2.eps}
\caption{Asymmetric HI surface density plot of NGC 891 (contours at the bottom left);
position-velocity diagram of the same galaxy (contours at the bottom right);
and the global velocity profile in NGC 891 (spectrum at the top), from Richter \& Sancisi (1994).}
\label{fig2}
\end{figure}
\bigskip
The global velocity tracer is likely to underestimate the asymmetry
fraction as for example if the galaxy were to be viewed along the major
axis as pointed out by Bournaud et al. (2005 b), or for a face-on galaxy
as noted by Kamphuis (1993).
The comparison of asymmetry in the stars as detected in the near-IR
and the HI asymmetry in surface density using the second criterion
of Richter \& Sancisi (1994) shows a similar behaviour in a sample
of 76 galaxies (Fig. 6 of Bournaud et al. 2005 b). However, the asymmetry
is quantitatively larger and more frequent in HI than in stars.
This result supports the conjecture by Richter \& Sancisi (1994) that the asymmetry
in the global velocity profiles is a good tracer of the disk mass asymmetry.
While making this comparison, it should be noted, however, that the HI asymmetry
is seen more in the outer radial range while the asymmetry in the near-IR is seen
in the inner region of a galactic disk.
Haynes et al. (1998) studied the global HI profiles of 103 fairly
isolated spirals. 52 of the 78 or $\sim 75 \% $ galaxies showed
statistically significant global profile asymmetry of 1.05. Many
show large asymmetry: 35 have asymmetry $>$ 1.1, 20 have $ >$ 1.15,
and 11 have $ >$ 1.2 .
The atomic hydrogen gas is an ideal tracer for a quantitative study of
lopsidedness in galaxies since the HI gas extends farther out than the
stars. The typical radial extent of HI is 2-3 times that of the
stars (e.g. Giovanelli \& Haynes 1988), and the amplitude of asymmetry
increases with radius (Rix \& Zaritsky 1995) as seen for the stars.
However, surprisingly, a quantitative measurement of HI spatial lopsidedness has
not been done until recently. In a first such study, the two-dimensional maps of the surface
density of HI have been Fourier-analyzed recently
to obtain the m=1 Fourier amplitudes and phases for a sample of 18
galaxies in the Eridanus group (Angiras et al. 2006)- see Section 5.2 for
details. Such analysis needs to be done for nearby, large-angular size
galaxies using sensitive data which will
allow a more detailed measurement of lopsidedness in nearby galaxies.
A study along these lines is now underway using the data from WHISP,
the Westerbork observations of neutral Hydrogen in Irregular
and SPiral galaxies (Manthey et al. 2008).
The molecular hydrogen gas also shows a lopsided distribution
in some galaxies, with more spatial extent along one half of a galaxy as in
NGC 4565 (Richmond \& Knapp 1986), IC 342 (Sage \& Solomon 1991), NGC 628
(Adler \& Liszt 1989) and M51 (Lord \& Young 1990).
However, this effect is not common in most cases and that can be understood
as follows. The lopsidedness appears to be more strongly seen in the outer parts
of a galaxy and the amplitude increases with radius as seen in stars (Rix \&
Zaritsky 1995),and also in HI gas (Angiras et al. 2006). Theoretically it has
been shown that the disk lopsidedness if arising due to a response to a distorted
halo with a constant amplitude can only occur in regions beyond $\sim 2$ disk scalelengths (Jog 2000). In
most galaxies the molecular gas in constrained to lie within two disk
scalelengths or half the optical radius (Young 1990).
Hence we can see that in most galaxies, there is no molecular gas in the regions
where disk lopsidedness in stars or HI is seen. This is why the molecular gas being
in the inner parts of the galactic disk does not display lopsidedness in most cases.
\subsubsection{Morphological lopsidedness in old stars}
The near-IR traces the emission from the old stars since dust
extinction does not significantly affect the emission in the
near-IR. The systematic study of this topic was started in the
1990's when a few individual galaxies such as NGC 2997 and NGC 1637
were mapped in the near-IR by Block et al (1994).
They noted the m=1 asymmetry in these but did not study it
quantitatively. The pioneering quantitative work in this field was done by Rix \&
Zaritsky (1995) who measured the asymmetry in
the near-IR for a sample of 18 galaxies. In each galaxy,
A$_1$, the fractional amplitude for the m=1 mode normalized by an
azimuthal average (or m=0), was given at the outermost point measured in the
disk, i.e. at 2.5 exponential disk scalelengths. This distance is
set by the noise due to the sky background in the near-IR. The mean
value is 0.14 , and 30 \% have A$_1$ values more than 0.20 which were
defined by them to be lopsided galaxies. A typical example is shown in Fig. 3.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=5.0in,width=4.0in]{fig3.eps}
\caption{The values of the various fractional Fourier amplitudes and phases vs.
radius in units of the disk scalelengths for NGC 1325 (from Rix \& Zaritsky 1995).
The amplitude scale in the lower three panels has been expanded by a factor of 5/3
for clarity, since these higher $m$ components have smaller amplitudes . The lopsided
amplitude A$_1$ increases with radius, and the phase is
nearly constant with radius.}
\label{fig3}
\end{figure}
\bigskip
This study was extended for a sample of 60 galaxies by Zaritsky \&
Rix (1997). They carried out the Fourier analysis of the near-IR surface brightness
between the radial range of 1.5-2.5 R$_{exp}$, where R$_{exp}$ is
the characteristic scale of the exponential light distribution.
The normalised $m=1$ Fourier amplitude A$_1$ is a more
representative indicator of disk lopsidedness.
It was shown that 30 \% of the galaxies have A$_1 > $ 0.2,
which was taken to define the threshold lopsidedness as in the
previous work. Rudnick \& Rix (1998) studied 54 early-type galaxies
(S0-Sab) in R- band and found that 20 \% show A$_1$ values measured
between the radial range of 1.5-2.5 R$_{exp}$ to be $ >$ 0.19.
A similar measurement has recently been done on a much larger sample of
149 galaxies from the OSU (Ohio State University) database in the near-IR (Bournaud et al.
2005 b), see Fig. 4. This confirms the earlier studies but for a larger sample,
and gives a smaller value of lopsided amplitude, namely
$\sim 30 \% $ show lopsideness amplitude of
0.1 or larger when measured over the radial range of 1.5-2.5 R$_{exp}$.
The galaxies with inclination angles of $< 70^{0}$ were used for this study.
Since this is a large, unbiased sample it can be taken
to give definitive values, and in particular the mean amplitude
of 0.1 can be taken as the threshold value for deciding if a galaxy is lopsided.
The lopsidedness also shows an
increasing value with radius, as seen in the Rix \& Zaritsky (1995) study.
\bigskip
\begin {figure} [h]
\centering
\includegraphics[height=2.5in,width=3.0in]{fig4.eps}
\caption{The histogram showing the distribution of lopsidedness measured in 149 galaxies at inclination of $<70^{0}$ from the OSU sample (Bournaud et al. 2005 b). The typical normalized lopsided amplitude A$_1$ measured over the radial range between 1.5 to 2.5 disk scalelengths is $\sim 0.1$. Thus most spiral galaxies show significant lopsidedness.}
\label{fig4}
\end{figure}
\bigskip
In the Fourier decomposition studies the determination of the center is a tricky issue, and the same center has to be used
for all the subsequent annular radial bins, otherwise during the optimization procedure, a different center could
get chosen and will give a null measurement for the lopsidedness. This is applicable for the lopsidedness
analysis for HI (Angiras et al. 2006, 2007), and also for centers of mergers (Jog \& Maybhate 2006). These two cases are discussed respectively in Sections 5.1 and 5.3.
The large number of galaxies used allows for a study of the variation with
type. It is found that late-type galaxies are more prone to lopsidedness,
in that a higher fraction of late-type galaxies are lopsided, and they show a
higher value of the amplitude of lopsidedness (Bournaud et al. 2005 b), see Fig. 5. This is similar to what was
found earlier for the variation with galaxy type for the HI asymmetries
(Matthews, van Driel, \& Gallagher 1998). These samples largely consist of
field galaxies, while the group galaxies show a reverse trend with galaxy
type (Angiras et al. 2006, 2007) implying a different mechanism for the origin
of lopsidedness in these two settings, see Section 5.1 .
\bigskip
\begin{figure} [h]
\centering
\includegraphics[height=2.5in,width=3.0in]{fig5.eps}
\caption{The plot of the cumulative function of $<A_1>$ for three groups of Hubble types of spiral
galaxies: the early-types ($0 < T < 2.5$), the intermediate types ($2.5 < T < 5$ ), and the late-types
($5 < T < 7.5 $), where $T$ denoted the Hubble type of a galaxy (taken from Bournaud et al. 2005 b). The late-type galaxies are more lopsided than the early-type galaxies.}
\label{fig5}
\end{figure}
\bigskip
While an axisymmetric galaxy disk gives rise only to
radial gravity forces, and therefore no torque, any
asymmetry in the disk, either $m=1$, $m=2$ or higher,
gives rise to tangential forces in addition
to radial forces, and then to a gravity torque.
From the near-infrared images, representative
of old stars, and thus of most of the mass, it is possible to compute
the gravitational forces experienced in a galactic disk.
The computation of the gravitational torque presents important
complementary information, namely it gives the overall average
strength of the perturbation potential as shown for m=2 (Block et
al. 2002), and for m=1 (Bournaud et al. 2005 b), whereas the Fourier amplitudes give values
which are weighted more locally. The gravitational potential is derived from the
near-infrared image, see Bournaud et al. (2005 b) for the details of this method.
The m=1 component of the gravitational torque, Q$_1$ between 1.5-2.5 disk scalelengths
is obtained, the histogram showing its distribution is plotted in Fig. 6 , which is similar to that
for the lopsided amplitude $A_1$ (Fig. 4) as expected.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=2.5in,width=3.0in]{fig6.eps}
\caption{The histogram of Q$_1$, the m=1 Fourier amplitude in the gravitational potential,
for the OSU sample of galaxies, from Bournaud et al. 2005 b.}
\label{fig6}
\end{figure}
\bigskip
An even larger sample based on the SDSS
data has now been Fourier-analyzed by Reichard et al. (2008),
and they also get a similar average value of lopsidedness in spiral galaxies, see Fig. 7.
However, they use the surface density data between 50\% and 90\% light radii,
so a clear one-to-one quantitative comparison of the values of lopsidedness
from this work as compared to the previous papers in the literature discussed above is not possible.
This work confirms that galaxies with low concentration, and low stellar mass density (or the late-type spirals)
are more likely to show lopsidedness, as shown earlier for HI by Matthews et al. (1998) and for stars by Bournaud et al. (2005 b).
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=2.5in,width=3.0in]{fig7.eps}
\caption{The histogram of number of galaxies vs. A$_1$ values for the SDSS sample, from Reichard et al. (2008). The histogram gives similar values to the earlier studies by Rix \& Zaritsky (1995) and Bournaud et al. (2005 b)}
\label{fig7}
\end{figure}
\bigskip
Another approach to measure the asymmetry involves the wedge method
(Kornreich et al. 1998, 2002) where the flux within
circular sectors or wedges arranged symmetrically with respect to the
galactic disk centre are compared. While it is easier to measure this
than the Fourier amplitudes, it gives only average values. An extreme
limit of the wedge method is applied by Abraham et al. (1996) and
Conselice et al. (2000). They define the rotation asymmetry
parameter as the ratio of fractional difference between the two
sides, so that 0 corresponds to a symmetric galaxy and 0.5 for the
most asymmetric galaxy. This is a more global or average definition
of the disk asymmetry and is suitable for studying highly disturbed
galaxies, and not those showing a systematic variation as in a
lopsided distribution. Such highly disturbed systems are seen at high
redshifts, for which this criterion was used first.
An interesting point to note is that spurious signs of asymmetry
arising due to dust extinction (Rudnick \& Rix 1998)
and that arising due to the pointing error of single-dish telescope
(Richter \& Sancisi 1994) were checked and ruled out. Conversely, a galaxy
could look more symmetric in the global velocity profile than it is, if the
galaxy is seen face-on. In that case even though the morphology is asymmetric
-as in HI in NGC 628, the global velocity profile is symmetric, and hence the
galaxy would appear to be kinematically symmetric- see Kamphuis (1993).
Based on the above discussion of the various methods, we recommend that the
future users adopt the fractional
Fourier amplitude A$_1$ as the standard criterion for lopsidedness. This is because it
gives a quantitative measurement, is well-defined, and can be measured easily
as a function of radius in a galaxy, and thus allows an easy comparison of its value
between galaxies and at different radii. The threshold value that could be adopted could be
the average value of 0.1 seen in the field galaxies in the intermediate radial
range of 1.5-2.5 R$_{exp}$ (Bournaud et al. 2005 b), so that
galaxies showing a higher value can be taken to be lopsided. A uniform criterion
will enable the comparison of amplitudes of lopsidedness in different galaxies,
and also allow a comparison of the fraction of galaxies deduced to be lopsided in different studies.
\subsection{Kinematical lopsidedness}
The lopsidedness or a (cos $\phi$) asymmetry is often also observed
in the kinematics of the galaxies. This could be as obvious as the
asymmetry in the rotation curves on the two halves of a galactic
disk, as is shown in Fig \ref{fig8} (Swaters et al. 1999, Sofue \& Rubin 2001), or more subtle as
in the asymmetry in the velocity fields (Schoenmakers et al. 1997). Often the optical centres are distinctly separated
spatially from the kinematical centers as in M33, M 31, and
especially in dwarf galaxies as pointed out by Miller \& Smith
(1992). The rotation curve asymmetry is also seen
as traced in the optical for stars (Sofue \& Rubin 2001). The
detailed 2-D velocity fields were so far mainly observed for HI as
in the interferometric data (see e.g. Schoenmakers et al. 1997, Haynes
et al. 1998). Now such information is beginning to be
available for the bright stellar tracers as in H$_\alpha$ emission
from HII regions (Chemin et al. 2006, Andersen et al. 2006), however since the filling factor
of this hot, ionized gas is small, it is not an ideal tracer for a
quantitative study of disk lopsidedness.
Schoenmakers et al. (1997) use the kinematical observational data in
HI on two galaxies- NGC 2403 and NGC 3198, and deduce the upper
limit on the asymmetry in the m=2 potential to be $<$ a few percents.
However, this method gives the result up to the sine of the viewing
angle. Kinematic asymmetry in individual galaxies such as NGC 7479
has been studied and modeled as a merger with a small satellite
galaxy (Laine \& Heller 1999).
The rotation curve is asymmetric in the two halves of a galaxy or
on the two sides of the major axis as shown for DDO 9 and NGC 4395 by
Swaters et al. (1999), see Figure 8. However, they do not make a more detailed
quantitative measurement of the asymmetry. Swaters (1999) in his study
of dwarf galaxies showed that 50\% of galaxies studied show
lopsidedness in their kinematics. Schoenmakers (2000) applied his calculations on kinematical
lopsidedness in galactic disks to five galaxies
in the Sculptor group and found that all five show significant
lopsidedness. A similar result has been found for
the 18 galaxies studied in the Ursa Major cluster (Angiras et al. 2007).
The frequency of asymmetry and its magnitude is higher in galaxies in
groups - see section 5.2 for details.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=1.8in,width=4.0in]{fig8.eps}
\caption{The rotation curve in DDO 9 and in NGC 4395 is asymmetric on two sides of
the galaxy, from Swaters et al. (1999).}
\label{fig8}
\end{figure}
\bigskip
A galaxy which shows spatial asymmetry would naturally show
kinematical asymmetry (e.g., Jog 1997) except in the rare cases of
face-on galaxies as discussed above where the galaxy can show asymmetry in the morphology
but not in the kinematics. However, the papers which study lopsidedness do not
always mention it. On the contrary, in the past, several papers
have made a distinction between the spatial or morphological
lopsidedness and kinematical lopsidedness (e.g. Swaters et al. 1999,
Noordermeer, Sparke \& Levine 2001) and have even claimed (Kornreich et al.
2002) that the velocity asymmetry is not always correlated with the
spatial asymmetry. However, in contrast, it has been argued that the
two have to be causally connected in most cases (Jog 2002),
especially if the lopsidedness arises due to the disk response to a
tidal encounter.
An important point to remember is that the same tracer (stars or HI)
should be considered to see if a galaxy showing spatial lopsidedness
is also kinematically lopsided or not, and vice versa. This is
because the HI asymmetry is higher and is seen in the outer parts of the
galaxy while the asymmetry in the near-IR is more concentrated in
the inner regions. This criterion is not always followed (see e.g.,
Kornreich et al 2002). Thus the often-seen
assertion in the literature that the spatial asymmetry is not correlated with kinematic
asymmetry is not meaningful, when the authors compare the spatial
asymmetry in the optical with the kinematical asymmetry in the HI.
\subsection {Phase of the disk lopsidedness}
\bigskip
The phase of the lopsided distribution provides an important clue to its
physical origin, but surprisingly this has not been noted or used
much in the literature. Interestingly, the phase is nearly constant with
radius in the data of Rix \& Zaritsky (1995), as noted by Jog
(1997). This is also confirmed in the study of a larger sample
of 60 galaxies by Zaritsky \& Rix (1997), (Zaritsky 2005, personal
communication), and also for the sample of 54 early-type galaxies studied
by Rudnick \& Rix (1998). A nearly constant phase with radius was
later confirmed for a larger sample of 149 mostly field galaxies
(Bournaud et al. 2005 b), and also for the 18 galaxies in the Eridanus
group (Angiras et al. 2006). The latter case is illustrated in Fig. 9.
This points to the lopsidedness as a global $m=1$ mode, and this idea
has been used as a starting point to develop a theoretical model
(Saha et al. 2007).
There are a few galaxies which do show a systematic radial variation in phase,
as in M51 (Rix \& Rieke 1993), which therefore appear as one-armed spirals.
\bigskip
\begin {figure} [h]
\centering
\includegraphics[height=1.25in,width=2.5in]{fig9a.eps}
\medskip
\includegraphics[height=1.25in,width=2.5in]{fig9b.eps}
\bigskip
\caption{The plot of the phase of the m=1 Fourier component vs.
radius (given in terms of the disk scalelength) for the HI surface density
for two galaxies ESO 482 - G 013 and NGC 1390 in the Eridanus group of galaxies, from Angiras et al.
(2006). Note that the phase is nearly constant with radius indicating
a global lopsided mode.}
\label{fig9}
\end{figure}
\bigskip
In contrast, the central regions of mergers of galaxies, show highly fluctuating
phase for the central lopsidedness (Jog \& Maybhate 2006). This may indicate an unrelaxed state,
which is not surprising
given that the mergers represent very different systems than the individual spirals
mainly discussed here.
\bigskip
\subsection{Observations of off-centered nuclear disks}
A certain number of galaxies are observed to have an off-centered nuclear disk, and
more generally an $m=1$ perturbation affecting more particularly the nuclear region.
Our own Galaxy is a good example, since the molecular gas observations have
revealed that the molecular nuclear disk has three quarters of its mass at positive
longitude, which is obvious in the central position-velocity diagram
(the parallelogram, from Bally et al 1988).
The asymmetry appears to be mainly a gas phenomenon, since the off-centreing
is not obvious in the near-infrared images (e.g. Alard 2001, Rodriguez-Fernandez \& Combes
2008). The gas is not only off-centered but also located in an inclined
and warped plane (Liszt 2006, Marshall et al 2008).
An $m=1$ perturbation is superposed on the
$m=2$ bar instability. The most nearby giant spiral galaxy, M31, has also revealed
an $m=1$ perturbed nuclear disk in its stellar distribution (Lauer et al 1993, Bacon
et al 1994). The spatial amplitude of the perturbation is quite small, a few parsecs,
and this suggests that this nuclear lopsidedness could be quite frequent in galaxies. However,
it is difficult to perceive it due to a lack of resolution in more distant objects.
Since M31 is the prototype of the $m=1$ nuclear disk, we will describe it in detail
in the next section. Some other examples have been detected, like NGC 4486B in the Virgo
cluster (Lauer et al 1996), but the pertubation must then be much more extended,
and that phenomenon is rare.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{fig10.eps}
\caption{HST WFPC2 V-band image of M31. The surface brightness contributed by the UV cluster
coinciding with the component P2 has been clipped out. The white dot indicates the position of the black hole.
>From Kormendy \& Bender (1999).}
\label{fig10}
\end{figure}
\bigskip
\subsubsection{ The case of the M31 nuclear disk }
The first images of M31 to reveal the asymmetrical nucleus were the
photographs at 0.2" resolution from the Strastoscope II by Light et al (1974).
They first resolved the nucleus, and measured a core radius of 0.48" (1.8pc).
The total size of the nucleus is 4 arcsec (15pc).
They showed that the nucleus is elongated, with a low intensity extension
outside the bright peak (cf Fig. 10); and they
considered the possibility of a dust lane
to mask the true center. Nieto et al (1986) confirmed this morphology
in the near-UV and also evoked dust. Later, it was clear that
dust could not be the explanation of this peculiar morphology,
since the center was still offset from the bulge in the near-infrared
image (Mould et al 1989). As for the kinematics, it was already
remarked by Lallemand et al. (1960) that the nucleus is rotating rapidly,
showing a very compact velocity curve, falling back to zero at a radius
of 2 arcsec. This was confirmed by Kormendy (1988) and Dressler \& Richstone
(1988), who concluded to the existence of a black hole in the center of M31,
of $\sim$ 10$^7$ M$_\odot$, with the assumption of spherical symmetry.
Lauer et al (1993, 1998) revealed with HST that the asymmetrical
nucleus can be split into two components, like a double nucleus,
with a bright peak (P1) offset by $\sim$ 0.5" from a secondary
fainter peak (P2), nearly coinciding with the
bulge photometric centre, and the proposed location of the black hole
(e.g. Kormendy \& Bender 1999). It is well established now
from HST images from the far-UV to near-IR (King et al. 1995, Davidge et al. 1997)
that P1 has the same stellar population as the rest of the
nucleus, and that a nearly point-like
source produces a UV excess close to P2 (King et al. 1995 ).
2-D spectroscopy by Bacon et al (1994) revealed
that the stellar velocity field is roughly centred on P2,
but the peak in the velocity dispersion map is offset by
$\sim$ 0.7" on the anti-P1 side (Fig. 11). With HST spectroscopy
the velocity dispersion peak reaches a value of $440\pm70$~km s$^{-1}$.
and the rotational velocity has a strong gradient (Statler et al. 1999).
The black hole mass required to explain these observations
ranges from 3 to $10\, \times 10^7$~M$_\odot$.
The position of the black hole is assumed to coincide with the centre of the UV peak,
near P2, and possibly with the hard X-ray emission detected by the Chandra satellite
(Garcia et al. 2000).
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=8cm,width=6cm]{fig11.eps}
\caption{Velocity profile ({\bf top}) and velocity dispersion ({\bf bottom})
in the nucleus of M31. The crosses are from STIS (HST) and the filled circles
from OASIS (CFHT). The OASIS kinematics have been averaged over a
0.2'' wide slit (PA$=39^\circ$) - taken from Bacon et al. (2001).}
\label{fig11}
\end{figure}
\bigskip
\subsubsection{Other Off-centered nuclei }
\bigskip
It has been known from a long time that the nearby late-type spiral M33
has a nucleus displaced from the young population mass centroid,
by as much as 500pc (de Vaucouleurs \& Freeman 1970, Colin \& Athanassoula 1981).
This off-centreing is also associated with a more large-scale lopsidedness,
and can be explained kinematically by a displacement of the bulge with respect
to the disk. Such kind of off-centreing is a basic and characteristic property
of late-type Magellanic galaxies.
In NGC 2110, an active elliptical galaxy, Wilson \& Baldwin (1985) noticed a displacement of
the nucleus with respect to the mass center of 220pc, both in the light and kinematics.
Many other active nuclei in elliptical galaxies
have been reported off-centered, but the presence of dust obscuration makes
its reality difficult to assert (e.g. Gonzalez-Serrano \& Carballo 2000,
where 9 galaxies out of a sample of 72 ellipticals are off-centered).
Quite clear is the case of the double nucleus in the barred spiral M83 (Thatte et al
2000): near-infrared imaging and spectroscopy reveals, in spite of the high extinction,
that the nucleus is displaced by 65pc from the barycenter of
the galaxy, or that there are two independent nuclei. Molecular gas with high velocity is associated with the
visible off-center nucleus, and this could be the remnant of a small
galaxy accreted by M83 (Sakamoto et al 2004).
In some cases what appeared to be a double nucleus could in fact be two regions of
star formation in centers of mergers of galaxies as in Arp 220 (Downes \& Solomon 1998).
Recently, Lauer et al (2005) studied a sample of 77 early-type galaxies with HST/WFPC2 resolution,
and concluded that all galaxies with inner power-law profiles have nuclear disks,
which is not the case of galaxies with cores.
They found 2 galaxies with central minima, likely to have a double nucleus (cf Lauer et al 2002),
and 5 galaxies having an off-centered nucleus. This perturbation also appears
as a strong feature in the Fourier analysis (A$_1$ term).
Off-centering is also frequently observed in central kinematics, where the peak of the
velocity dispersion is displaced with respect to the light center
(Emsellem et al 2004, Batcheldor et al 2005). Decoupled nuclear disks, and
off-centered kinematics are now clearly revealed by 2D spectroscopy.
\section{Theoretical models for the origin of
lopsidedness:}
The origin and the evolution of lopsidedness are not yet
well-understood and in fact not received much theoretical attention.
The lopsidedness is observed in a variety of tracers and settings. It is observed in old and young stars, and in HI and
H$_2$ gas, within the optical disk and far outside the optical disk
as seen in the tracer HI gas, and in field galaxies and in galaxies in groups.
Given all these parameters it is difficult to come up with a unique
physical mechanism for the generation or the maintenance of disk
lopsidedness. Indeed there could be multiple paths for it, such as
tidal interactions, gas accretion, or an internal instability. The
particular mechanism that dominates depends on the situation
concerned as can be seen from the following discussion. Towards the end of
this section (section 3.4), we give a summary of the most likely physical
mechanisms for the origin of disk lopsidedness in spiral galaxies.
We also stress that the standard m=2 case has been extensively
studied for years both analytically as well as by N-body
simulations, both in the context of central bars (e.g. Binney \&
Tremaine 1987, Combes 2008, Shlosman 2005) or as
two-armed spiral features (Rohlfs 1977, Toomre 1981). Thus the basic
physics of their growth and dynamics is fairly well-understood. In
contrast, the m=1 feature has just begun to be studied. We discuss
the physical differences between these two cases (m= 1 and 2), and
later in Section 6 a comparison between their observed values is made.
\subsection{Kinematical model for the origin of
lopsidedness}
The simplest way to explain the observed lopsided disk distribution
is to start with a set of aligned orbits and see how long these will
take to get wound up, as was done by Baldwin et al. (1980), see Fig. 12. It is
well-known that the differential rotation in a galactic disk would
tend to smear any material feature over a few dynamical timescales.
Baldwin et al. (1980) showed that on taking account of the epicyclic
motion of the stars, the effective radial range over which the
differential motion affects the winding up is reduced by nearly a
factor of 2, and this helps in increasing the lifetime of the
feature. The net winding up time t$_{winding}$, for a feature
between two radii R$_1$ and R$_2$
is then given by t$_{winding}$ = 2 $\pi / \Delta (\Omega - \kappa)$,
where $\Omega$ and $\kappa$ are the angular speed of rotation and the
epicyclic frequency respectively, and $\Delta$ denotes the difference
taken at these two radii.
For a flat rotation curve with $\kappa = 1.414 \Omega$, this
gives t$_{winding}$ = 5 [2 $\pi / \Delta \Omega] $ , which is about
5 times longer than the usual winding up time for material arms.
\bigskip
\begin {figure} [h]
\centering
\includegraphics[angle=-90,width=.45\textwidth]{fig12a.eps}
\includegraphics[angle=-90,width=.5\textwidth]{fig12b.eps}
\bigskip
\caption{{\it Left}: Pattern of lopsided elliptical orbits whose major
axes were aligned to begin,
following Baldwin et al. (1980).
The orbits tend to get misaligned with time due to the differential
rotation in the disk.
{\it Right}: Frequencies $\Omega$, $\Omega - \kappa$ and $\Omega + \kappa$
in a typical galaxy disk. In the case of a prograde
mode, a possible pattern speed $\Omega_p$ is indicated,
allowing Corotation (CR) and Outer Lindblad Resonance (OLR) (taken from Combes 2000).
}
\label{fig12}
\end{figure}
\bigskip
\bigskip
For the outer parts of a galaxy say at $\sim$ 15 kpc , this is
$\sim$ 2 Gyr, which is a few times larger than the local dynamical
timescale but is still less that the lifetime of the Galaxy. Hence
they argued that the lopsidedness cannot be primordial, and must be
generated repeatedly in the disk. Further, this model cannot explain why
many isolated galaxies such as M101 show lopsidedness, which is
the puzzle they had started out to solve.
Further work on the kinematical origin of lopsidedness due to
a cooperation of orbital streams of stars near resonance was
done by Earn \& Lynden-Bell (1996), with similar results for its
lifetime as discussed above. Apart from the short winding time, a
generic problem with the kinematical model is that it is not clear
what gives rise to the aligned orbits in the first place.
\subsection{Dynamical models for the origin of
lopsidedness}
The set of dynamical models discussed next deal with a more physical
origin for the lopsidedness. The most commonly proposed models
include tidal encounters, gas accretion, instability in a
counter-rotating disk, an off-centered disk in the halo, and ram
pressure stripping in a cluster. Of these the most promising are the
first two, with the tidal encounters being the dominant mechanism
for group galaxies.
In addition to these externally triggered processes, the disk
lopsidedness could also arise as a global m=1
instability in self-gravitating disks.
\subsubsection{Tidal encounters, and disk response to distorted halo}
One of the earliest ideas suggested for
generating lopsidedness in a galaxy was due to a tidal encounter as
applied to M101 (Beale \& Davies 1969), although no details were
worked out. It is easy to see that a perturbation as say due to a
tidal encounter between two galaxies with an arbitrary orientation
can generate a force term of
type cos $\phi$, which can then generate lopsidedness in the galaxy
(Combes et al. 2004, see Chapter 7.1.1).
The other modes will also be
generated but generally m=1 and 2 are observed to be the strongest
(Rix \& Zaritsky 1995). Also see Section 6 for a discussion the
observed relative strengths of the $m$ = 1 and 2 modes.
In addition to the above direct triggering of lopsidedness as a disk
response to the tidal force, it can also be generated more
indirectly due to the response of the disk to the distorted halo
which feels a stronger effect of the interaction (Weinberg 1995, Jog
1997, Schoenmakers et al. 1997). The details of lopsidedness thus
induced will be summarised in this section. The effect can be seen as
in the spatial or surface density distribution as well as in the kinematics.
A generally stronger perturbation resulting from the infall of a
satellite galaxy can also result in the disk lopsidedness as shown
in the N-body simulation study by Walker et al. (1996). Zaritsky \& Rix
(1997) further used this idea to constrain the rate of infall of
satellites onto a galaxy from the fraction of galaxies showing
lopsidedness and the star formation, both triggered by
the satellite infall. Note that this gives an upper limit on the
satellite infall rate if there are other mechanisms which
also give rise to the disk lopsidedness; such as gas accretion as
discussed in Section 3.2.2.
Since the dark matter halo is more extended than the disk, during a
tidal encounter the halo is expected to experience a stronger tidal
perturbation on general grounds. Weinberg (1995, 1998) has studied
this case numerically for the specific case of the interaction
between the LMC and the Galaxy. The live halo shows a strong
lopsided response at the resonance points as set by the orbit of the
LMC with respect to the Galaxy. The disk response to
this distorted halo is shown to be much stronger than the direct lopsidedness
triggered in it due to the encounter. A detailed numerical
evolution of a galactic disk and the halo perturbed by an impulsive
perturbation (Kornreich et al. 1998) shows that m=1 can grow as a
free sloshing libration but can only last for a dynamical timescale.
On the other hand, a number of encounters covering a large parameter
space for the galaxy mass ratios, orbits, disk inclination were studied
by numerical simulations by Bournaud et al. (2005 b). They conclude that the
resulting lopsidedness can have
an amplitude as high as the typical observed value $\sim 10 \% $, and it lasts
for more than ten dynamical timescales $\sim$ 2 Gyr, after which the lopsidedness drops rapidly.
\bigskip
\noindent {\it 3.2.1.a $\:$ Orbits and isophotes in a perturbed
disk}
\bigskip
We next briefly summarise the results for the orbits, isophotes and
kinematics for a disk that is perturbed by a linear lopsided
perturbation potential, as say due to a
distorted halo (Rix \& Zaritsky (1995),
Jog (1997, 2000), Schoenmakers et al (1997)) where the
halo distortion could be ascribed to a tidal encounter. These
equations of motion are general and are applicable irrespective
of the mechanism giving rise to the external potential to which the
disk responds. This simple model allows one to draw general
conclusions about the strength of the perturbation potential by a
comparison of results with observations as is shown next.
The details of this approach (Jog 2000) are summarised below. The
dynamics of particles on closed orbits in an axisymmetric disk
perturbed by a lopsided halo potential is treated. The cylindrical
coordinate system ($R , \phi $) in the galactic disk plane is used.
The unperturbed, axisymmetric potential in the disk plane,
$\psi_0(R)$, and the perturbation potential, $\psi_{lop} (R) $ are
defined respectively as:
$$\psi_0 (R) \: = \: {V_c}^2 ln R \eqno(1)$$
$$\psi_{lop} (R) \: = \: {V_c}^2 \: \epsilon_{lop} \: cos \phi \eqno(2)$$
\noindent where $\psi_{0} (R) $ represents the typical region of flat
rotation, with $V_c$ being the constant rotational velocity, as seen
in a typical spiral galaxy. This is perturbed by a small, constant,
non-rotating, perturbation potential with a lopsided form as given
by $\psi_{lop} (R)$. Here $\epsilon_{lop}$ is a small perturbation
parameter, which is taken to be constant with radius for simplicity. That is,
a halo with a constant lopsided distortion is assumed.
This is a simple model but it still allows one to
understand the resulting orbits, isophotes and the kinematics in a lopsided galaxy.
Consider a circular orbit at $R_0$. The coupled equations of motion
for the perturbed quantities $\delta R$ and $\delta \phi$ are solved
together using the first-order epicyclic theory. The resulting
solutions for the perturbed motion are:
$$R \: = \: R_0 \: ( 1 \: - 2 \:
\epsilon_{lop} \: cos \phi) \: \: \: \: ; \: \: \: \: \: \: \: V_R \:
= \: 2
\: V_c \: \: \epsilon_{lop} \: sin \phi \: \: \: \: ; \:
\: \: \: \: \: \: V_{\phi}
\: = V_c \: ( 1 + 3 \: \epsilon_{lop} \: cos \phi ) \eqno(3)$$
\noindent Thus, an orbit is elongated along $\phi = 180^0$, that is
along the minimum of the lopsided potential, and it is shortened
along the opposite direction. This result was also noted by Earn \&
Lynden-Bell (1996).
The loop orbits discussed here are not valid for radii
much smaller than the disk scalelength (Rix \& Zaritsky 1995), but
this does not affect the applicability of this analysis to galactic
disks because the disk lopsidedness is typically observed only
at radii beyond 1.5 disk scalelengths.
The other signatures of the effect of the lopsided perturbation
potential are its effect on the isophotes and the kinematics in the disk, discussed next.
Since the imaging observations give information on the isophotes
rather than orbits, it is necessary to also obtain the isophotal
shapes in an exponential galactic disk in a lopsided potential. For
an exponential disk, the above approach gives:
$$ A_1 \: = \: \frac {\epsilon_{iso}}{2} \:
\frac {R}{R_{exp}} \eqno (4) $$
\noindent where $\epsilon_{iso}$ is the ellipticity of the isophote
at $R$, and $R_{exp}$ is the exponential disk scale length. Note
that here the radii measuring the minimum and maximum extents of an
isophote are along the same axis - unlike in the standard
definition of ellipticity where these two are along directions that
are normal to each other. {\it The resulting isophotes have an
egg-shaped oval appearance}, as observed say in M 101. Thus, the
azimuthal asymmetry in the surface density or the fractional Fourier amplitude for m=1, as denoted by $A_1$
manifests itself as an elongation of an isophote, and both represent
the same underlying phenomenon.
The effective surface density in a self-gravitating, exponential galactic disk may be
written as:
$$ \mu (R, \phi) \: = \: \mu_0 \: exp [ - \frac {R}{R_{exp}} ( 1- \frac{\epsilon_{iso}}{2}
\: cos \phi ) ] \eqno (5) $$
For a particular isophote, the term in the square bracket is a constant and hence this formally
defines the parametric form of an isophote. Thus the minimum radius of an isophote occurs along
$\phi = 180^{0}$ while the maximum occurs along $\phi = 0^0$; while the opposite is true for
an individual orbit. Thus the isophotes are elongated in a direction opposite to an orbit, and the elongation
is along the same direction where the maximum effective surface density occurs-
this will be true for any self-gravitating system (Jog 1997).
To obtain the lopsided potential in terms of the observed lopsided Fourier amplitude,
the equations of perturbed motion (eq. [3])have to be solved with
the equation of continuity, and the effective surface density (eq. [5]]). Now, the equation
of continuity is given by:
$$ \frac{\partial}{ \partial R} \: [R \mu (R, \phi) \: V_R (\phi) ] \:
+ \: \frac {\partial}{\partial \phi} [\mu (R, \phi) \: V_{\phi} (\phi) ] = 0 \eqno (6) $$
Solving these, and combining with eq.[4], yields the
following important results (valid for $ R \geq R_{exp}$):
$$ \epsilon_{lop} \: = \: \frac {A_1 } {\left (
{2 R} /{R_{exp}} \right ) - 1 } \eqno (7) $$
\noindent and,
$$ \epsilon_{iso} / \epsilon_{lop} \: = \: 4 (1 - \frac{R_{exp}}{2 R}) \eqno (8) $$
\noindent Thus, the ellipticity of isophotal contours $\epsilon_{iso}$ is
{\it higher by at least a factor of 4}
compared to $\epsilon_{lop}$. Thus even a $\sim $ few \% asymmetry
in the halo potential leads to a large $\sim 10 \% $ spatial
lopsidedness in the disk. This makes the detection of lopsidedness
easier, and it explains why a large fraction of spiral galaxies is
observed to be lopsided.
For the typical observed values of $A_1 \geq 0.1 $ at $R/
R_{exp} = 1.5-2.5 $ (see Section 2.2), the typical $\epsilon_{lop}
\sim 0.03$ (from eqs.[3], and [4]), or $\sim 0.05$ in view of the
negative disk response discussed next. Thus, from the observed disk
lopsidedness, we obtain a value of the {\it halo lopsidedness}. In
the limiting case of high observed $A_1 \sim 0.3 - 0.4 $, the
resulting $\epsilon_{lop}$ is still small $\leq 0.1 $, this is due
to the high ratio of $\epsilon_{iso} / \epsilon_{lop}$ (eq. 8). This is an
interesting physical result, because it means that despite the
visual asymmetry, such galaxies are dynamically robust.
The above results for orbits and isophotes are shown to be
applicable for both stars and gas in the same region of the galaxy
since they respond to the same lopsided potential and have
comparable exponential disk scale lengths (Jog 1997). This was
confirmed by a detailed comparison of the Fourier analysis of the
two-dimensional HI data and the 2MASS near-IR representing stars for
a few galaxies in the Eridanus group (Angiras et al. 2006)
and in Ursa Major (Angiras et al. 2007). The two tracers show comparable lopsided
amplitudes (see Fig. \ref{fig25}). This is true even though the HI gas
in these group galaxies obeys a gaussian rather than an exponential radial distribution.
In the above analysis, the phase is taken to be constant with radius and hence set equal to zero.
When the phase of the potential
varies with radius, the resulting isophotes show a prominent
one-arm, as observed in M51 and NGC 2997.
\bigskip
\noindent {\it 3.2.1.b $\:$ Kinematics in a perturbed disk}
\bigskip
The kinematics in the disk perturbed by a lopsided potential is also
strongly affected. The net rotational velocity, $V_{\phi}$, (see eq.
[3]), is a maximum at $\phi = 0^0 $, and it is a minimum along the
opposite direction. This results in distinctly {\it non-axisymmetric
rotation curves} in the two halves of a galaxy (Jog 1997), with the
maximum difference between the rotational velocities $\sim 10 \%$ or
20-30 km s$^{-1}$ for the typical observed $A_1$ values (Jog
2002). This naturally explains the observed asymmetry in rotation
curves of galaxies such as M 101. The observers most often give an azimuthally averaged
data, thereby the precious information on the kinematical asymmetry is lost. It is strongly recommended (see Jog 2002)
that the observational papers give, when possible, a
full azimuthal plot of the rotation velocity
or at the very least the average taken in each hemisphere separately. The latter is done in many
papers- see e.g. Begeman 1987, which can be used to deduce the kinematical asymmetry in galaxies.
Such asymmetry has also been
studied by Swaters et al. (1999) from their kinematical data in HI
on DDO 9 and NGC 4395. They show that the rotation curve rises more
steeply in one half of the galaxy than in the other.
The asymmetry in the velocity fields resulting from the disk response to a
lopsided halo perturbation has also been studied by Schoenmakers
(1999). The results obtained are applied to analyze the
kinematical data from a few spiral galaxies (Schoenmakers et al.
1997, Swaters et al. 1999). Schoenmakers et al (1997) show that the
Fourier amplitudes $m + 1$ and $ m - 1$ of the velocity field are affected when the perturbation
potential of type $m$ is considered. By comparing the observed values for
m = 2 with the calculated values they obtain an upper limit
(uncertain up to the sine of the inclination angle) for the lopsided
perturbation potential.
The approach described in this section assumes a simplified perturbation
lopsided potential with a constant amplitude, also only closed orbits
are considered for simplicity. The orbits would change slightly and
would not be closed if the random motion of the particles is taken
into account. However, this does not affect the isophotal shapes - see
Rix \& Zaritsky (1995).
\bigskip
\noindent {\it 3.2.1.c $\:$ Radius for the onset of disk lopsidedness}
\bigskip
The lopsided distribution in a disk cannot self-support itself because
the potential corresponding to it opposes the perturbation potential, as discussed next.
The effective disk surface density or the
disk density response is shown to be a maximum along $\phi =
0^0$, that is along the maximum of the lopsided potential (Jog 1997), see eq.(5) above.
This has interesting and subtle dynamical consequences (Jog 1999). This can be seen from
the self-gravitational potential corresponding to the
non-axisymmetric disk response, which is obtained by inversion of Poisson
equation for a thin disk using the Henkel transforms of the
potential-density pairs. This response potential is shown to oppose the imposed
lopsided potential. This may seem counter-intuitive but it arises due to the self-gravity of the disk. Thus in the inner parts of the disk, the disk resists any imposed perturbation potential.
A self-consistent calculation shows that the net
lopsided distribution in the disk is only important beyond 1.8 disk
scale lengths and its magnitude increases with radius. This indicates
the increasing dynamical importance of halo over disk at large
radii. The negative disk response decreases the imposed lopsided
potential by a factor of $\sim 0.5 - 0.7 $ (Jog 2000). The above
radial dependence agrees well with the onset of
lopsidedness as seen in the near-IR observations of Rix \& Zaritsky
(1995). On taking account of this effect, a given observed lopsided amplitude
corresponds to the deduced perturbation potential to be higher by a factor of $\sim 1.3 - 1.4$.
The radius of onset of lopsidedness and the reduction in the imposed potential depend on the form of
the perturbation potential which was taken to be constant for simplicity in Jog (1999). A more realistic case with a radially varying perturbation potential as in a tidal encounter
with amplitude decreasing at low radii, will result in the highest decrease at lowest radii
(Pranav \& Jog 2008). In this case the actual value of reduction will decide
the radius beyond which net disk lopsidedness is seen.
This idea of negative disk response is a general result and is
applicable for any gravitating system which is perturbed by an external mechanism. Although it is
shown here for a $cos \phi$ perturbation resulting from a tidal
perturbation, it is applicable for any linear perturbation
of the system.
A similar study for the onset of warps (which can be represented by an m=1 mode
along the vertical direction) has been done (Saha \& Jog 2006). This shows the onset of
warps from a radius of 4-5 disk scalelengths, in
good agreement with observations (Briggs 1990). The disk self-gravity
is more important along the z-direction for a thin disk and hence the
disk is able to resist vertical distortion till a larger radius than the planar distortion.
\bigskip
\noindent {\it 3.2.1.d $\:$ Comparison of $A_1$ vs. tidal parameter}
\bigskip
Since tidal encounters (e.g. Beale \& Davis 1969, Weinberg 1995) and satellite
accretion (Zaritsky \& Rix 1997) have often been suggested as the mechanism for
the origin of the disk lopsidedness, it is instructive to
check how the observed amplitude $A_1$ for lopsidedness varies with
the tidal parameter $T_p$ as was done by Bournaud et al. (2005 b). The tidal
parameter $T_p$ was calculated in each case as:
$$ T_p \: = \: log \: {\Sigma}_i {\large[} (\frac {M_i}{M_0}) (\frac {R_0}{D_i})^3 {\large]} \eqno(9) $$
where the sum is computed over the companions for a galaxy of target mass $M_0$ and radius $R_0$
and $M_i$ is the mass of the companion at a projected distance $D_i$ on the sky.
The summation is over neighbours within 2 degrees on the sky and within 500 km s$^{-1}$
velocity range of the test galaxy.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=2.5in,width=2.5in]{fig13.eps}
\caption{Plot of A$_1$ vs. tidal parameter for the 35 strongly lopsided galaxies in the OSU sample (taken from Bournaud et al. 2005 b). There is no correlation between these quantities, in particular the isolated galaxies with high A$_1$ (top l.h.s. corner of this plot) cannot be explained by a recent tidal interaction.}
\label{fig13}
\end{figure}
\bigskip
The result is plotted in Figure 13, for the 35 most lopsided galaxies which are at an inclination of $<$ 70$^0$. Surprisingly, this does not show a correlation between the lopsided amplitude and the strength of the tidal parameter. In particular it is hard to explain the galaxies with high $A_1$ and low tidal parameter (in the top l.h.s. of this figure) in the tidal picture. On the other hand, this still does not rule out tidal encounters as the origin for lopsidedness if it is long-lived, or if it arises due to a satellite merger (Walker et al 1996, Bournaud et al. 2005 b).
In order to check the typical values of lopsided amplitudes generated in tidal encounters and satellite accretion,
N-body simulations for a number of encounters and satellite mergers were studied.
They included the tidal encounters
between nearly equal-mass galaxies (up to the ratio of 4:1) and mergers of small-mass galaxies (in the ratio of
5:1-20:1) (Bournaud et al. 2005 b).
It was found that tidal interactions can indeed generate
a fairly large amplitude similar to or higher than the average value of $\sim 0.1$. However, the amplitude
then drops rapidly and hence can be seen only for $< 2$ Gyr.
A typical example of an encounter between a 2:1 mass ratio is shown in Figure 14.
Thus tidal encounters cannot explain the high amplitude of lopsidedness seen in several isolated galaxies such as NGC 1637,
although it could arise due to a recent satellite accretion.
While a satellite accretion of mass ratio 7:1-10:1 can result in a strong lopsidedness, it can
also thicken the disk
more than is observed (Bournaud, Combes, \& Jog 2004 , Bournaud et al. 2005 a). The thickening of disks can set a limit on
the rate of satellite mergers (Toth \& Ostriker 1992).
It needs to be checked if a small-mass satellite falling onto a galactic disk
can generate the right amplitude distribution of lopsidedness without thickening
the disk, and further if there exist satellites in sufficient numbers to fall in at a steady rate
to repeatedly generate lopsidedness as required by the observed high fraction of lopsided galaxies.
The mechanism for origin of lopsidedness involving a tidal encounter has been explored by Mapelli et al. (2008)
in the context of NGC 891.
They show that the lopsidedness seen in the atomic hydrogen gas in NGC 891
is due to the fly-by encounter with its neighbour UGC 1807. They argue that this is a preferred mechanism over gas accretion from cosmological filaments or that due to ram pressure from the intergalactic medium.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=4.0in,width=3.5in]{fig14.eps}
\caption{Plot of A$_1$ vs. time in the middle radial range of 1.5-2.5 disk scalelengths, generated in a distant interaction between galaxies of mass ratio 2:1 (taken from Bournaud et al. 2005 b). The peak value of A$_1$ is large $\sim 0.2$, higher than the average value seen in the OSU field galaxies sample, but it drops rapidly to 0.05 in a few Gyr. The lower panel shows the same in terms of Q$_1$, the cumulative potential from the disk.}
\label{fig14}
\end{figure}
\bigskip
Further, a number of statistical features for the field galaxies
show that a tidal encounter cannot be the primary mechanism for the
origin of the disk lopsidedness- for example, the lopsidedness is higher
for late-type galaxies whereas tidal encounters and mergers would tend to
lead to the secular evolution of a galaxy towards early-type galaxies. Thus
if tidal interactions were the primary mechanism for generating lospidedness,
then the early-type galaxies should show a higher amplitude of lopsidedness.
This is opposite to what is sen in the field galaxies (Bournaud et al. 2005 b).
Thus other mechanisms such as gas accretion (Section 3.2.2) could be important
in generating the lopsidedness in field galaxies.
In the group galaxies, on the other hand, the tidal interactions being more frequent,
play a dominant role in generating lopsidedness.
This is evident from the fact that the early-type galaxies show higher lopsided amplitudes
as seen in the Eridanus group galaxies (Angiras et al. 2006) and
less strongly in the Ursa Major group of galaxies (Angiras et al.
2007). The details are given in Section 5.
\subsubsection{Gas Accretion, and other mechanisms}
The intergalactic gas accretion was proposed as a qualitative idea
to explain the m=1 asymmetry in NGC 4254 by Phookun et al.(1993).
They proposed that the lopsidedness could arise due to the subsequent swing amplification
in stars and gas (as in Jog 1992).
An extensive study of origin of lopsidedness via N-body simulations
(Bournaud et al. 2005 b) shows that while tidal encounters can explain
the observed amplitudes of disk lopsidedness, these cannot explain
the various observed statistical properties such as the correlation
between A$_1$ and A$_2$, and the higher lopsidedness seen for the
late-type field galaxies. In order to do this, one needs to take account
of gas accretion from outside the galaxy.
There is growing evidence that galaxies steadily accrete gas from
the external regions, as seen from the cosmological models (Semelin
\& Combes 2005), and also observed in nearby galaxies (Sancisi et al. 2008).
Thus it is natural to see how this affects the mass
distribution in a disk. While the details of gas infall are not yet
well-understood, it is plausible that a galaxy may undergo an
asymmetric gas infall on the two sides from say two different
external filaments. An application of this idea showed that the gas
infall and the resulting star formation can well reproduce the
striking asymmetry observed in NGC 1367 (Bournaud et al. 2005 b),
see Fig. 15 here.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{fig15.eps}
\caption{NGC data - near-IR data (left panel), and the same
from N-body simulations with gas accretion in two streamers at a rate
so as to double the mass of the galaxy in a Hubble time (right panel), taken from Bournaud et al. (2005 b).}
\label{fig15}
\end{figure}
\bigskip
A word of caution is that if
gas accretion is the mechanism for generation of lopsidedness,
one would expect to see asymmetry in the gas velocity fields whereas these
are smoothly continuous as pointed out by Baldwin et al. (1980).
In the case of galaxies in groups, in any case,
the tidal interactions
may play a dominant role, see Section 5.2
for the details.
Other physical models have also been developed in the literature but
these are probably not as widely applicable due to the specific
parameters or conditions chosen, as discussed below. This includes a
proposed model where the growth of m=1 is treated as an instability
in a self-gravitating disk (Lovelace et al 1999). This results in strongly unstable
eccentric motions but only within the central disk scalelength. Here a linear analysis is used to treat slowly growing mode, and the pattern speed could be either positive or negative.
In another model where the disk is off-centered with respect to the
halo up to a maximum distance of the core radius (Levine \& Sparke 1999) also shows lopsidedness. However, it is seen
only in the
inner regions within the core radius of the halo where the halo density and hence the rotation speed is constant.
Both these papers do not yield higher lopsidedness
in the outer parts, and this result contradicts the
observations which preferentially show lopsidedness in the outer
parts of a galaxy. The kinematical and morphological asymmetries
resulting from the latter model are shown to be not always
correlated (Noordermeer et al. 2001). This is somewhat
unexpected - since on general physical grounds the two would be
expected to be related causally (see Section 2.2). In any case, the
restricted set of initial conditions required for this model makes
it applicable only to a few galaxies such as the dwarf galaxies.
A self-consistent model for m=1 in a self-gravitating disk has been
proposed by Syer \& Tremaine (1996) in the so-called razor-thin
disks. However, the density response due to an imposed perturbation
opposes the perturbing force (see Jog 1999) and hence the disk
asymmetry cannot result from the orbital asymmetry as was also argued by
Kuijken (1993) and Earn \& Lynden-Bell (1996). Moreover, the model by Syer \& Tremaine (1996)
gives the density response to be maximum along the perturbed force,
which is in contrast to the result by Rix \& Zaritsky (1995), and
Jog (1999).
\subsubsection{ Lopsidedness as an instability}
An obvious possible explanation for the origin of the lopsided mode is
that it arises due to gravitational instability in the disk. For example, this was
proposed and studied for the gas by Junqueira \& Combes (1996). A
similar model for the disk in a dark matter halo perturbed by a
satellite was studied by Chan \& Junqueira (2003), however these
models generate lopsidedness only in the inner regions in contrast
to the observed trends. An internal mechanism based on the
non-linear coupling between m=2 (bars or spiral arms) and m=3 and
m=1 has been proposed by Masset \& Tagger (1997) which gives rise to
the excitation of m=1 modes in the central regions.
Lopsided instabilites have been shown to develop in counter-rotating
stellar disks which have a high fraction of retrograde orbits
(Hozumi \& Fujiwara 1989, Sellwood \& Valluri 1997, Comins et al 1997,
Dury et al 2008).
Galaxies where the gas participates to the counter-rotation are
quite often observed to develop
$m=1$ perturbations (e.g. Garcia-Burillo et al 2000, 2003).
However, since counter-rotation is rarely seen in stellar disks
(Kuijken, Fisher, \& Merrifield 1996, Kannappan \& Fabricant 2001, McDermid et al 2006),
this cannot be the primary mechanism for the generation of
lopsidedness in disks.
In a recent work, the self-gravity of a slowly-rotating global m=1 mode has been shown to
lead to a long-lasting lopsided mode in a purely exponential
disk as in a spiral galaxy (Saha, Combes, \& Jog 2007). This model was
motivated by the fact that the observations show that the lopsidedness
has a constant phase with radius which indicates a global mode.
Further, it was noted that m=1 is unique in that the centre of mass of
the disturbed galaxy is shifted away from the original centre of mass
and thus acts as a restroing force on the latter. Thus the system can
self-support the m=1 mode for a long time especially when one takes
account of the self-gravity of the global mode.
Using the linearized fluid equations
and the softened self-gravity of the perturbation, a self-consistent
quadratic eigenvalue equation is derived for the lopsided
perturbation in an exponential galactic disk and solved.
Fig. 16 shows the resulting isodensity contours, clearly the centres of isocontours are
progressively more disturbed in the outer parts.
\bigskip
\begin{figure}[h]
\centering
{\rotatebox{270}{\includegraphics[height=2.3in]{fig16.eps}}}
\bigskip
\caption {Contours of constant surface density for a global m=1 mode, taken from Saha et al. (2007). Here the x and y axes
are given in units of the disk scalelength. The maximum surface density occurs at (0,0). The outer
contours show a progressive deviation from the undisturbed circular distribution, indicating
a more lopsided distribution in the outer parts- as observed.}
\label{fig16}
\end{figure}
\bigskip
The self-gravity of the mode results in a significant
reduction in the differential precession, by a factor of $\sim 10$ compared to the free
precession. This leads to
persistent m=1 modes, as shown in Fig. 17.
\bigskip
\begin{figure}[h]
\centering
{\rotatebox{270}{\includegraphics[height=2.3in]{fig17.eps}}}
\bigskip
\caption{The precession rate for the global lopsided mode vs. the size of the disk in units of the disk scalelength
in a galactic disk (shown as the line with circles)
is very low, thus the mode is long-lived. The dashed line denotes the free precession ($\kappa - \Omega$).
This is taken from Saha et al. (2007).}
\label{fig17}
\end{figure}
\bigskip
N-body simulations are performed to test the
growth of lopsidedness in a pure stellar disk, which confirm these results (see Fig. 18). Both
approaches are compared and interpreted in terms of slowly
growing instabilities on timescales of $\sim$ a few Gyr, with almost
zero pattern speed.
\bigskip
\begin{figure}[h]
\centering
{\rotatebox{270}{\includegraphics[height=5.2in]{fig18.eps}}}
\bigskip
\caption{The isodensity contours in a purely stellar exponential galactic disk, given in a logarithmic
scale of the surface density of the stellar disk, face-on (top panel) and edge-on (lower panel) views
at four different epochs: T =0, 4.8, 9.6, 14.4 Gyr, from left to right, taken from
Saha et al. (2007). The global lopsided mode is long-lived
and lasts for $\sim 14$ Gyr.}
\label{fig18}
\end{figure}
\bigskip
Though this is a somewhat idealized approach, it is precisely this
that has allowed the authors to focus on the basic dynamics of the excitation and
growth of m=1 modes. For example, here the only important input
parameters are the softening and the Toomre Q parameter. A smaller
value of Toomre Q results in a fast initial growth of the mode but
later as the Q increases due to the heating in the system, the mode
is self-regulated and has a nearly constant amplitude A$_1$ that is
long-lived. The softening acts as an indicator of the coherence in
the mode, and a higher value results in a faster growth rate of the
modes.
These global modes are precessing remarkably slowly, and therefore are
relatively long-lived.
Numerical analysis of the eigen modes of a cold thin disk
shows that, if treated as modes of zero pattern speed, warps and lopsidedness
are fundamentally similar in nature (Saha 2008).
Such small pattern speed is also in agreement with the work of Ideta (2002)
who showed that the rotating m=1 mode in a live halo would be damped very rapidly
by the density wake induced in the halo, see Fig. 19.
Hence he argued that the m=1 modes must be non-rotating at a rate smaller than
1 km s$^{-1}$ kpc$^{-1}$. This would give
the damping time to be comparale to the Hubble time, which
can explain the high frequency of lopsidedness seen in spiral galaxies.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=2.8in]{fig19.eps}
\caption{The damping time of the lopsided mode vs. the pattern speed, for a self-gravitating case (solid line) and without self-gravity (dashed line) respectively, taken from Ideta (2002). Damping times $\sim$ Hubble time imply a very low pattern speed of the lopsided mode.}
\label{fig19}
\end{figure}
\bigskip
In a recent paper Dury et al. (2008) have studied a galactic disk in an inert halo by N-body simulations and argued that that a rotating m=1 mode can occur as a result of the swing amplification (Toomre 1981).
\subsubsection {Effect of inclusion of rotation and a live halo}
The work by Saha et al. (2007) shows that a gravitating disk is susceptible to the
growth of m=1 modes, irrespective of the origin of these modes. These
could be triggered as a response to a distorted halo, or by gas accretion.
We caution, however, that Saha et al. (2007) treat a simple, specialized
case of slow rotating modes in a pure exponential disk.
A more realistic treatment should include a live or a responsive halo in addition to the
disk. Here more parameters enter the picture such as the relative mass of the halo and the
disk, and ratio of the core radius of the halo to the disk
scalelength etc. The inclusion of a live halo is expected to further
support the persistent m=1 modes, in analogy with what was shown for bars by Athanassoula (2002) and for a general, local non-axisymmetric feature by Fuchs (2004). In contrast, other papers suggest that a responsive halo tends to damp a feature in the disk by the wake created
in the halo. This was
shown for disk lopsidednesss (m=1 in the plane) by Ideta (2002), and shown
in the case of warps treated as an m=1 mode normal to the galactic plane (Nelson \& Tremaine 1995).
This issue needs to be clarified by further dynamical studies of a global m=1 mode of an arbitrary
pattern speed in a galactic disk. This study should include
a live halo, and have a high resolution (since poor resolution may smoothen the response and give a spuriously long-lasting mode), with the aim to
check the lifetime of such a mode.
A lopsided mode in a collisionless spherical dark matter
halo is shown to be long-lasting or slowly damped compared to the dynamical timescales (Weinberg 1994,
Vesperini \& Weinberg 2000). However it is not clear if
this is due to the fact that the halo is supported by random motion
and hence a perturbation in it is long-lived. On the other hand,
in a spiral
galaxy, the presence of differential rotation puts a limit on any
material feature due to the precession rate.
In contrast, the numerical simulations of perturbations triggered in
a galaxy with live halo, due to the tidal encounter of nearly equal-mass galaxies
(up to the ratio of 4:1) and mergers of small-mass galaxies (in the ratio of
5:1-20:1) show that the lopsidedness thus generated, although long-lived compared to the dynamical timescales, does not last beyond $\sim 2$ Gyr (Bournaud et al. 2005 b).
The crucial factor that decides the lifetime of the lopsided mode could be its pattern speed, with the small speed cases lasting for a long-time close to the Hubble time (Ideta 2002, Saha et al. 2007). The off-centering of the mass distribution is less pronounced in the case of high rotation speed. This could lead to a short lifetime of the m=1 mode as pointed out by Ideta (2002). In this case the restoring term in the equations of motion (Saha et al. 2007, eq. 18) is less pronounced and hence the m=1 mode lasts for a shorter time. Alternatively, it could be that the wake generated in a halo could be larger for a higher pattern speed, and hence
the halo tends to dampen such modes. This may be the reason why the lopsidedness generated in a disk with a live halo due to a tidal encounter or a minor merger (Bournaud et al. 2005 b) lasts for $< 2 $ Gyr.
While slow m=1 modes are shown to be long-lived, the uniqueness of this solution is still not established,
or that this is
what explains the observed lopsidedness. Also it is not clear what would excite such slow modes.
The generating mechanism may have a strong bearing on the resulting pattern speed: a tidal encounter is expected to
result in lopsidedness with a high pattern speed $\sim$ the relative velocity over the impact parameter
as argued by Ideta (2002). Gas accretion, on the other hand,
may not easily give a global m=1 mode, while observations show the mode to be global.
Further numerical simulations should check if a satellite
accretion can give rise to a slow, global mode.
An actual measurement of the pattern speed of the lopsided
mode in a real galaxy will help settle this issue, and we urge observers to take up this important measurement.
\subsection{Comparison between origin of m=1 and m=2; stars and gas}
As discussed at the beginning of Section 3, the m = 2 case is fairly
well-understood, while the m=1 case has only begun to get attention
from theorists. There are several differences in the dynamics and
evolution of the two features, and also as applied to stars or gas.
First, the presence of dark matter halo is likely to have a
substantial role to play in the origin and evolution of lopsidedness
in a disk. This is especially true
in the outer parts of a disk since the disk
lopsidedness is observed to increase with the radial distance
where the halo is more important. In the inner regions of a
galaxy, on the other hand, the inclusion of the bulge is likely to
play an important role in stabilizing the m=1 mode (de Oliveira \&
Combes 2008).
Further, the m=1 mode generally has no ILR (Inner Lindblad Resonance, e.g., Block et al.
1994) hence its evolution differs from that of m=2. The m=1 mode does not
get damped easily due to the angular momentum transport occurring at
the resonance points as in the case of m=2 (Lynden-Bell \& Kalnajs 1972). In
case of m=2 this causes an absorption of the wave at the ILR and
thus a break in the feedback loop. But in absence of an ILR for m=1,
this break does not arise and this helps in sustaining the global
m=1 mode for a long time. In this picture, the gas being cold
behaves in a different way. The absorption at the resonance point is
only partial for gas and hence even m=2 can be sustained in gas
despite the presence of an ILR. This helps support the generation of
m=2 and higher order modes in the presence of gas.
\subsection{ A summary of the various mechanisms}
Of the various mechanisms proposed so far, the most promising ones,
as judged by the resulting agreement with the observations, are
those involving tidal encounters and gas accretion.
An m=1 perturbation in a disk leads to a shift in the centre of mass
in the disk, and this then acts as an indirect force on the
original centre of the disk. The disk is thus shown to naturally
support an m=1 mode, and as pointed out above, this is a
characteristic property valid only of a lopsided mode. This basic
physics is sometimes clouded over because of the additional effects
introduced due to the inclusion of the dark matter halo, the bulge,
and the gas as in a real galaxy. Further, depending on whether the
halo is live or rigid, and whether it is pinned or not, and
whether the pattern is rotationg or stationary can lead to
additional complexities.
Also, other features like the wandering of
the centre (Miller \& Smith 1992) may introduce an m=1 mode. In
short, there seem many paths to get m=1 in a galaxy, and therefore
it is important to identify which is the most applicable one in a
real galaxy.
\noindent {\it Long-term maintenance of disk lopsidedness}
It has been realized from the beginning that the lopsided modes
should be fairly long-lived (e.g., Baldwin et al. 1980) or excited
frequently. This is needed
in order to explain the high fraction of galaxies showing
lopsidedness, and also the strong lopsidedness seen in isolated
galaxies like M101. It has been noted that since m=1
does not have an ILR, it should be the preferred mode in the
galactic disk (Section 3.3). This, however, does not say anything
directly about its lifetime.
The persistence of lopsided mode is still an open question, as shown by the discussion below.
While tidal encounters can generate
the right lopsided amplitudes, these are not correlated with the
strength of a tidal encounter (Bournaud et al. 2005 b), or with the
presence of nearby neighbors (Wilcots \& Prescott 2004). This could
be explained if the disk lopsidedness once generated either directly
in the disk, or as a response to a long-lived halo distortion, were
long-lived, so that there is no clear correlation with a tidal
encounter. N-body simulations with a live halo show the resulting m=1
modes in the disk to last for $\sim 2-3$ Gyr, which is much smaller than the Hubble time, thus these
need to be triggered again.
A tidal encounter will typically generate a fast mode
which is expected to be not long-lived (see the discussion in Section 3.2.3).
While the satellite accretion of mass ratio 7:1-10:1 can result in a strong lopsidedness, it can
also thicken the disk more than is observed (Bournaud et al. 2004). Also, it has a short lifetime of $< 2$ Gyr (Bournaud et al. 2005 b).
It further needs to be checked if a smaller-mass satellite
falling onto a galactic disk
can generate the right amplitude distribution of lopsidedness without thickening
the disk, and if there are adequate number of such satellites that can fall in at a steady rate.
The pattern speed is expected to have a significant effect in determining the
lifetime of a lopsided mode with a slow patttern being long-lived. It is not
clear if a live halo will help or hinder the long-term sustenance of an m=1 mode,
as discussed in Section 3.2.3.
Future work needs to study the long-term maintenance of m=1 modes when generated
by accretion of a low-mass satellite. A similar study needs to be done for the case of
gas accretion and to see if the latter gives a global mode.
The observations of group galaxies with their generally stronger and frequent triggering of lopsidedness (see Section 5)
can act as a constraint on any generating mechanism proposed for the field galaxies.
\section{Lopsidedness in the central region}
\subsection{Stability of central nuclear disks}
It is now well established that all galaxies with bulges
or spheroids host a massive central black hole (Gebhardt et al. 2000).
The central region, or nuclear disk, in galaxies therefore have
a gravitational potential very close to the Keplerian.
Some similarities exist with proto-planetary systems (Rauch \& Tremaine 1996),
or with the formation of new stars through accretion disks
(Adams et al. 1989, Alexander et al. 2007).
The nearly Keplerian potential, with the angular
velocity $\Omega \sim r^{-3/2}$ favors eccentric orbits and $m=1$
modes, instead of $\Omega \sim r^{-1}$
of galactic disks which favor $m=2$ perturbations.
Nearly keplerian disks have the particular property that
the orbit precession rate is almost
zero ($\Omega \sim \kappa$). If the apsides are aligned at a given time,
they will stay so in a $\Omega_p \sim 0$ mode. The self-gravity of the disk
makes $\kappa > \Omega$, and the orbits differentially precess
at a rate ($\Omega - \kappa) < $ 0. However, if the disk self-gravity is not
large, a small density perturbation could be sufficient to counteract the
small differential precession. Goldreich \& Tremaine (1979)
showed that in the case of Uranian rings, the self-gravity could
provide the slight impulse to equalize the precession rates, and
align the apsides.
Two kinds of waves could propagate in such disks-
slow stable modes, and unstable rapid waves,
growing on a dynamical time-scale.
The density wave theory (e. g., Lin \& Shu 1964) predicts
that in a self-gravitating stellar disk, global spiral modes can develop
only between the radial range delimited by the Lindblad resonances,
i.e. for $m^2 (\Omega - \Omega_p)^2 < \kappa^2$.
Only in gaseous disks, where the pressure forces dominate,
acoustic waves can propagate outside this range.
Considering the $m=1$ waves, for a pure keplerian potential
(neglecting the self-gravity of the disk), $\Omega = \kappa$,
and the perturbations are neutral. If there is some self-gravity
in the disk, then ($\Omega - \kappa) < 0$, there is only an outer
Lindblad resonance and a corotation, but no inner resonance
(for prograde modes with $\Omega_p >0$), and therefore the radial range for
the development of $m=1$ perturbations is quite large.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=8cm,width=8cm]{fig20.eps}
\caption{Isodensity contours for the lowest order $m=1$ mode, in a disk of mass equal to the
central point mass. Note the shape in alternating ``bananas'' instead
of a continuous one-arm spiral. The dashed lines show the locations
of the corotation and the outer Lindblad resonance.
>From Adams et al. (1989).}
\label{fig20}
\end{figure}
\bigskip
The study of WKB modes in a non self-gravitating gaseous disk,
with truncation radii at the inner and outer boundary, was
first done by Kato (1983), who found a region of trapped one-arm waves,
with quite low pattern frequency, much lower than the orbital frequency.
Adams et al (1989) considered the influence of self-gravity,
in order to apply to young stellar objects, which can have accretion
disks with masses of the same order as the central stellar mass.
The self-gravity of such a disk is sufficient to modify
the precession rate of nearly keplerian orbits ($\Omega - \kappa$)
to a common and coherent value. They found a pattern speed which
is of the same order as the angular velocity in the disk, and the unstable waves
develop with a growth time comparable to the dynamical time-scale.
The special shape of modes is shown in Fig. 20.
Crucial to the development of the perturbations, a special characteristic of the $m=1$ mode is to
shift the gravity center of the system from the dominant central mass (the black hole for instance),
also see Section 3.2.3.
In the reference frame of the black hole, this implies the introduction of an inertial force,
which by reference to celestial mechanics, is called the indirect term.
This term is the mediator of angular momentum exchange
between the disk inside and outside corotation and the central
mass. The coupling with the outer Lindblad resonance
provides the amplification that is usually provided by the corotation
in $m=2$ modes. A feedback cycle has been proposed by
Shu et al (1990), and called SLING (Stimulation by the Long-range
Interaction of Newtonian Gravity).
The modes depend strongly on the outer disk boundary conditions,
since a reflection of short waves is assumed there, in the
4-waves feedback cycle. This cycle is only possible with a
gaseous component, since the short waves are not
absorbed at the resonance but cross the OLR (Outer Lindblad Resonance).
\bigskip
\subsubsection{Slow stable modes, damping slowly}
\bigskip
Another possibility to explain central lopsidedness
is to exploit the slow modes, that are stable, but can be
long-lived, excited by some external mechanism, such as
the accretion of a globular cluster or a giant molecular cloud.
The precession rate of eccentric orbits is $\Omega - \kappa$ =0
in the potential of a point mass M$_{BH}$, and slightly negative in
the presence of a small disk of mass M$_d$, lighter than the
central point mass, with amplitude varying as
$({\rm{M}_d}/{\sqrt{\rm{M}_{BH}}})$ or
$\propto ({\rm{M}_d}/{{\rm{M}_{BH}}}) \Omega$. If self-gravity has a large enough role, and in
particular, if the disk is cold enough and its Jeans length smaller than the disk radius,
$m=1$ density waves can propagate; their dispersion relation has been studied
in the tight-winding limit or WKB approximation (Lee \& Goodman 1999, Tremaine 2001). In the linear
approximation, the pattern speed for the wavelength $\lambda = 2 \pi / k$ is in
first approximation for $({\rm{M}_d}/{\rm{M}_{BH}}) << 1$, as given by:
$$
\Omega_p = \Omega - \kappa + {{\pi G \Sigma_d |k|}\over{\Omega}}
F({{k^2c^2}\over{\Omega^2}})
\eqno (10) $$
where $\Sigma_d$ is the surface density of the disk, and $F$ the usual reduction
factor that takes into account the velocity dispersion $c$ of the stellar disk, and
its corresponding velocity distribution (e.g., Tremaine 2001). The pattern speed
then remains of the order of $({\rm{M}_d}/{\rm{M}_{BH}}) \Omega$, for a sufficiently
cold disk, and is much smaller than the orbital frequency. These slow waves
exist whenever the thin-disk Jeans length $\lambda_J = {c^2}/{G\Sigma_d}$
is lower than $4 r$, while the Toomre parameter $Q$ is less relevant
(Lee \& Goodman 1999).
The study by Tremaine (2001) shows that disks orbiting a central mass
support slow $m=1$ modes, which are all stable. Their frequencies are
proportional to the strength of collective effects, which is
either self-gravity, or velocity dispersion (or pressure in
fluid disks). The latter phenomenon can be simulated
with softened gravity. There are then two kinds of slow modes:
the g-modes (where self-gravity is dominating, and softening
is unimportant), which are long waves, with kr $<<$ 1, with
negative frequency $\omega <0$; and the p-modes, which depend on
the softening $b$, which can have both short and long waves
(kr $\sim b$), and with positive frequency $\omega >0$; as the
softening increases, the amplitude of the mode decreases, as well as
$\omega$.
In numerical N-body simulations of nuclear stellar disks,
Jacobs \& Sellwood (2001) have reported the presence of a slowly
decaying prograde $m=1$ mode in annular disks around a slightly softened
point mass, but only for disk masses less than 10\% of the central mass
concentration. This confirms the existence of a persistent
slow mode, with positive $\omega$ increasing
with the mass of the disk, and decreasing with the amplitude
of the perturbation.
Touma (2002) has computed the normal modes of a series of
N rings in a thin disk, through linearized dynamics, and
using the Laplace-Lagrange secular theory of planetary motions
(valid for small eccentricities). The gravity is
softened to mimic a hot stellar disk, and varies as the velocity dispersion.
The modes are stable when all rings are prograde, but a fraction of only 5\%
of counter-rotating rings is sufficient to make unstable modes appear.
Sambhus \& Sridhar (2002) built a model of the M31 nucleus
with counter-rotating orbits in a razor thin nucleus,
and checked that this amount of
counter-rotation could be compatible with observations.
The counter-rotating stars could come from a past accreted
system, like a globular cluster.
Bacon et al (2001) explored by N-body simulations
the possibility of stable $m=1$ mode to explain the M31
eccentric nuclear disk. They found that for a disk mass
accounting for $\sim$ 20 -- 40\% of the total central mass,
self-gravity is sufficient
to counteract the differential precession of the disk.
An external perturbation
can excite this mode, and it is then long-lasting, over
100 Myr, or 3000 rotation periods.
The prograde mode found in the simulations compares
well with the $p$-modes of Tremaine (2001).
There is a remarkable agreement between the observed
($\sim 3km s^{-1} pc^{-1} $) and predicted value
of the pattern speed, in spite of
all approximations, and although the WKB approximation is not
satisfied.
Although the slow modes are long-lived, their exciting mechanisms
should be found, to explain the high frequency of the phenomenon.
The dynamical friction of the $m=1$ wave on the stellar bulge has been
proposed by Tremaine (1995) as an amplification mechanism,
if its pattern speed is sufficiently positive. This amplification results
from the fact that the friction decreases the energy less than the angular momentum.
The orbits with less and less angular momentum are more and more eccentric, and the
$m=1$ mode develops. Although the efficiency of the mechanism
has not been proven, it should not apply for the slow modes considered here,
in a slightly rotating bulge.
An external perturbation is more likely to trigger the
$m=1$ perturbation. There is the possibility of
infalling of globular clusters, through dynamical friction,
a mechanism explored in the next section. Also
interstellar gas clouds should be continuously
infalling onto the center, since within 10-100 pc of
M31 nucleus, dust lanes, and CO molecular clouds are observed
(Melchior et al. 2000). The interval between two
such external perturbations (either passage of a globular cluster,
or a molecular cloud) in M31 is of the same order of magnitude,
so that the external perturbations are an attractive mechanism.
Each episode of $m=1$ waves will heat the disk somewhat,
but the instability is not very sensitive to the initial radial
velocity dispersion. Over several 10$^8$ yr periods, the nuclear
disk could be replenished by fresh gas from the large-scale M31
disk and subsequent star formation.
The hypothesis of cold gas accretion from the disk of
M31 itself, has then not only the advantage to trigger the
$m=1$ perturbation, but also
to explain the maintenance of a rather thin and cold nuclear disk.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=5.5in]{fig21.eps}
\caption{Simulation of a globular cluster infall towards a
galaxy nucleus containing a massive black hole, fit to the
M31 characteristics. The panels show the face-on view contours
from the cluster stars, from the start to 0.46 Myr.
The bottom right panel shows the histogram of the angular momentum
for the cluster particles at the beginning (solid line) and at the end
(dashed line) of the simulation. From Emsellem \& Combes (1997).}
\label{fig21}
\end{figure}
\bigskip
\subsection{Double nuclei by infalling bodies}
One solution to the double nuclei problem is to assume that dense stellar
systems, like globular clusters, or a dwarf satellite, or even black holes, are
regularly infalling into the galaxy center, and are responsible for the
observed morphology. Although the events may be relatively rare,
their actual frequency is not well-known, and the question remains
open as to a possible fit to the observations.
The typical dynamical friction time-scale, for an object of mass $M$ at about 10pc
from the center, in a spiral galaxy with a bulge of $\sim$ 10$^{10}$ M$_\odot$,
typical of an Sb galaxy like M31, is $\sim$ 10$^7 (10^6M_\odot/M)$ yr,
and it could be much smaller inside. Although short with respect to galactic time-scales,
this is much larger than the orbital time of 3 $\times$10$^5$ yr at this radius.
Tremaine et al (1975) precisely proposed that the central stellar nuclei
in spiral galaxies are the results of many globular clusters infalling
by dynamical friction. Typically nuclear stellar systems of 10$^7$-10$^8$
M$_\odot$ would require the infall of a hundred globular clusters.
In this frame, the frequency of the event is relatively large, and
could be compatible with the observations.
N-body simulations of globular clusters or dwarf galaxies infalling through the
gravitational field of a disk, a bulge, and/or a central black hole, have been
carried out by many authors (e.g. Charlton \& Laguna 1995,
Johnston et al 1999, Combes et al 1999, Bekki 2000b), and applied
to explain the M31 nucleus morphology
(Emsellem \& Combes 1997, Quillen \& Hubbard 2003).
N-body simulations demonstrate that the infalling system is destroyed
by the tidal forces of the black hole at the right distance of the nucleus
(about 3pc). The debris then rotate around the nucleus in eccentric orbits,
and form an eccentric disk. Several hypotheses can then be explored:
either the infalling system is alone able to form the nuclear disk (Bekki 2000a),
but then its mass corresponds more to the core of a dwarf galaxy having
merged recently with the big primary (a quite rare event). Or the
infall of the system excites an eccentric mode in the nuclear disk
(Bacon et al 2001), as proposed by Tremaine (1995).
Also, the presence of the infalling system not yet diluted
in the nuclear disk increase the asymmetry.
The details of the dynamics, and in particular the inclination
of the nuclear disk with respect to the main disk of M31,
or the shift of the velocity dispersion peak from the black hole position,
due to the systematic rotation of the luminosity peak of the disk,
are all explained by the model by Emsellem \& Combes (1997), see Fig. 21.
The nature of the infalling system is constrained by the
present metallicity and colors of the nuclear disk, especially
if the assumption is made that the infalling system is the first one
and forms totally the nuclear disk (Bekki 2000a).
The observed colors of the double luminosity peaks
in M31 are quite similar to the nuclear disk
ones, and different from the bulge, so the hypothesis that the
nuclear disk is formed from the infalling systems themselves is possible.
The hypothesis of globular clusters is more likely, in the sense
that the probability to observe it is larger, the friction time-scale being longer,
and it requires at least 25 globular clusters to form the nuclear disk.
The hypothesis of a dwarf galaxy merger does not correspond to the
quite un-perturbed state of the M31 disk.
There is some evidence of a past merger around M31,
in the shape of an extended stellar disk, loops and shells
(Ibata et al 2001, Irwin et al 2005). However, the time to form these
stellar streams is much longer than the time-scale
for the galaxy core to infall to the center.
It is interesting to discuss in this context
the case of the double nucleus in NGC 4486B, which is thought
to be similar to the M31 case, but with larger masses, and larger separation
(Lauer et al 1996). NGC 4486B is a compact elliptical galaxy,
in the outer envelope of M87 in the Virgo cluster. The double
nuclei are separated by 12pc, and produce two almost equal luminosity peaks
at similar distance from the photo-center, so creating almost no
lopsidedness in projection. If explained by a nuclear disk, it is
of very small eccentricity. The hypothesis of a past merger as
the origin of the two stellar nuclei is weaker in this case, since the
environment of the cluster center does not favor mergers.
The idea of infalling systems can be generalized
to all galaxy mergers as the origin of lopsidedness.
In particular, mergers of galaxies could
naturally form eccentric disks. The disk would come from the disruption
by tidal forces of one of the stellar core. The presence of massive
black holes in each spiral galaxy with bulge, strongly supports
this scenario. The end steps of the merger process would
form a binary black hole. The destruction of nuclear stellar systems
by the tidal forces of the black holes led Merritt \& Cruz (2001)
to suggest that the existence of low-density cores (and not cusps)
in giant galaxies is the consequence of mergers.
Further simulations of mergers with central black holes
should be performed to explore the formation of
eccentric disks.
Somewhat larger-scale asymmetry on scales $\sim$ 1 kpc is also seen in mergers
of galaxies and is deduced to be long-lived, as discussed in Section 5.3.
\subsection{Core wandering}
Some of the nuclear lopsidedness might also be explained through
a special oscillation of the central black hole, called "core wandering".
This name came from the physics of globular cluster, that was observed
to reveal slow oscillation, with a time-scale larger than the crossing
time in numerical simulations (Makino \& Sugimoto 1987).
Miller \& Smith (1992) showed by a large series of numerical simulations
that a massive nucleus cannot coincide with the mass centroid of its
galaxy in a stable way. The type of instability, where the motion of the nucleus
implies potential distortions in the center, which trap more particles,
is overstable, and reaches a saturation limit. The phenomenon is local,
and the time-scale of the oscillation of the nucleus is of the same order as the
central dynamical period (cf Figure \ref{fig22}).
This core wandering appears physical, and not the consequence of
a $N^{-1/2}$ random noise oscillation, as tested by simulations with
highly varying particle number. In that case, the perturbation amplitude
that starts the growth is indeed depending on $N$, but not the limiting
amplitude, nor the growth rate, which is always a few dynamical times.
The $N^{-1/2}$ phenomenon can be hard to distinguish in small-N
simulations, and in globular clusters (Sweatman 1993), but this is
not the case for galaxies.
The stochastic part of
the core wandering phenomenon has been modeled by Chatterjee et al (2002),
separating the force on the central mass in the collective action of the stellar
system in which it is embedded, and the fluctuating stochastic force provided
by individual stellar encounters. This second force produces a Brownian
motion of the central point mass. These motions occur on a time scale
much shorter than the time-scale of evolution of the stellar system.
As for the coherent modes of the stellar system coupled to the central mass,
the growing oscillation looks like a density wave, and the saturation
amplitude of the motion reaches a galaxy core radius.
This unstable phenomenon involves the nucleus, even if there is no
central black hole in the center, and is also observed in dynamical
friction experiments, when the decay of a satellite is studied (e.g.
Bontekoe \& van Albada 1987). The limiting amplitude of the nucleus
oscillation is then reached from above.
This kind of oscillation is also observed in spherical galaxies,
and continues to develop for a Hubble time (Miller \& Smith 1994).
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=6cm,width=8cm]{fig22.eps}
\caption{Evolution with time of the three coordinates
of the ``nucleus'' consisting in the central 1024 particles, selected from an N-body
simulation of 100 352 particles. The saturation of the amplitude of oscillations
is visible. From Miller \& Smith (1992).}
\label{fig22}
\end{figure}
\bigskip
Taga \& Iye (1998a) studied the oscillations of a central massive
black hole in a rotating galaxy, and also confirmed that the
phenomenon is not an $N^{-1/2}$ random noise effect, but
a true physical phenomenon. They
found by N-body simulations that a massive central body
can undertake long-lasting oscillations, but only when its mass is lower
than 10\% of the disk mass. This produces disk oscillations around it.
Crucial must be the rotation of the pattern around the center, since
it does not vary its amplitude, when the number of particles change.
The disk oscillations occur only when the black hole is allowed to move,
when the black hole is artificially nailed down
to the center, the disk oscillations vanish.
They also conclude that the mechanism at the origin of this
instability is a density wave, with a fixed pattern as a function
of radius.
When the disk around the central mass is fluid, an instability
akin to the one proposed by Shu et al (1990) is possible, with a feedback
provided by reflection on a sharp edge in the outer disk.
Heemskerk et al (1992) estimate that the edge effect is artificial,
and studied instead an $m=1$ instability arriving only when the masses
of the central object and the disk are comparable. There can
be angular momentum exchange between the mass and the disk.
To simplify their model, they considered only a gaseous disk with a central gap.
This instability is confirmed by Woodward et al (1994), and requires
the coupling with the central mass, which is displaced from the center,
and moves along a smooth, tightly wound, spiral trajectory.
Taga \& Iye (1998b) found that the sharp edge condition is not
necessary, and that an eccentric instability develops in a
stellar disk, provided that the central mass is smaller than the
disk mass (about 10\%), and it is mobile. The eccentric instability then develops
a one-arm spiral, with an amplitude that is stronger than when the central mass is fixed to the center.
The mechanism is strongly dependent on the softening used,
and should be local to the central parts.
\subsection{Other mechanisms}
If a normal disk around a black hole is not spontaneously unstable
to $m=1$ perturbations, the modifications of the stellar distribution function $F$,
and in particular the depletion in low-angular momentum orbits, leading to the
empty loss-cone phenomenon, can provide the source of instability. If the derivative
of $F$ with respect to $J$ is positive, then spherical near-Keplerian systems are neutrally
stable (allowing the displacement of the nucleus
with respect to central mass), and the flattened non-rotating systems
are unstable to $m=1$ modes (Tremaine 2005).
Peiris \& Tremaine (2003) construct eccentric disk models to represent the
double nucleus in M31, and claim that the inner nuclear disk must be inclined
by at least 20 degrees with respect to the main disk of the galaxy to represent the data.
Although their model is only dynamical, and does not include all the physics, with
self-gravity, etc. , this suggests that the
lopsidedness might be related to a warp or a misalignment. The latter could be
the source of dynamical friction against the bulge, and relatively rapid alignment
should ensue. A possibility is that the bulge is itself misaligned with the main
galaxy disk, which could be due to a recent galaxy interaction (e.g. Ibata et al 2001,
Block et al 2006).
Salow \& Statler (2001, 2004) compute a more sophisticated model, including
self-gravity. They populate quasi-periodic orbits for stars, in the rotating frame
with a constant precession speed. The eccentricity of the orbits change sign with
radius, so that the apocenter of the orbits change phase in the plane
of the nuclear disk (cf fig 23). The resulting best fit model is similar to that
obtained with N-body simulations of a strong $m=1$ mode in a cold thin disk with a central
black hole (Bacon et al 2001). In particular the pericenters of the orbits in the inner
and outer disks are in phase opposition. The precession rate, however, is rapid
($\Omega$ = 36.5 km s$^{-1}$ pc$^{-1}$).
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=4cm]{fig23.eps}
\bigskip
\caption{(a) Density contours of the best-fit model of the M31 eccentric
disk, the central black hole being at (X,Y)= (0,0); (b) uniformly precessing orbits in the total potential.
(c) the radial variation of the eccentricity $e$ of orbits, as a function of semi-major axis $a$: {\it dotted line}--
the orbit model, with eccentricities changing sign with radius; {\it full line} -- as a consequence of disk self-gravity,
the eccentricity does not change sign any longer.
>From Salow \& Statler (2004).}
\label{fig23}
\end{figure}
\bigskip
\section{Lopsidedness in galaxies in groups,
clusters and mergers}
\subsection {Lopsidedness in galaxies in groups}
Tidal encounters between galaxies in groups are more probable
given the higher number density and the consequent frequent
interaction between
galaxies in groups. Further, these have relative velocities similar to the field
galaxies, hence the lopsidedness arising due to a
response triggered by tidal encounters is more likely to
occur in these. This is in fact borne out by the observations of
Hickson group galaxies (Rubin et al. 1991) where a large fraction
($> 50 \%$) of galaxies show lopsided rotation curves. This is much
higher than the case of field spiral galaxies where only $\sim$ 25
\% show asymmetric rotation curves (Rubin et al. 1999, Sofue \&
Rubin 2001). The observation of rotation curves of 30 galaxies in
20 Hickson groups (Nishiura et al. 2000) confirms this higher
frequency. In another study involving HI observations, all the five
major spirals in the nearby Sculptor group of galaxies (Schoenmakers
1999) show kinematical lopsidedness, and two show morphological
elongation or asymmetry.
A multi-wavelength study of two group galaxies NGC 1961 and NGC 2276
shows evidence for lopsidedness which has been attributed to tidal interactions
(Davis et al. 1997).
The two-dimensional maps of HI have been Fourier-analyzed recently
to obtain the m=1 Fourier amplitudes and phases for a sample of 18
galaxies in the Eridanus group (Angiras et al. 2006). This is the
first quantitative analysis of asymmetry in the surface density distribution of the HI gas, and is similar to the Fourier analysis now done routinely in the
literature for the near-IR data representing old stars in galaxies.
The group location of
this sample allows us to serendipitously study the lopsidedness in a group
setting. The galaxies studied show a higher magnitude of asymmetry than the
field galaxies, and also a higher fraction of galaxies show asymmetry.
The average amplitude of lopsidedness measured for the
Eridanus group galaxies is nearly twice that in the field galaxies
over the same radial range of 1.5-2.5 disk scalelengths. Second, nearly
30 \% of the sample galaxies show lopsidedness amplitudes three times larger than the field average of 0.1.
Fig. 24 shows the results for two galaxies UGC 068 and NGC 1325, in the Eridanus group.
The asymmetry is measured in this case to over twice the radial distance that is
typically covered in the near-IR studies (e.g. Bournaud et al .
2005 b). This is because the tracer used here is HI which extends farther out than the stars, and because the sky
background limits the Fourier analysis in the near-IR to $\sim 2.5 $
disk scalelengths.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=1.25in,width=2.5in]{fig24a.eps}
\medskip
\includegraphics[height=1.25in,width=2.5in]{fig24b.eps}
\bigskip
\includegraphics[height=1.25in,width=2.5in]{fig24c.eps}
\medskip
\includegraphics[height=1.25in,width=2.5in]{fig24d.eps}
\bigskip
\caption{The lopsided amplitude and phase of the HI surface density distribution
versus radius for the two galaxies UGC068 and NGC 1325 in the Eridanus group, taken
from Angiras et al. (2006). The amplitude increases with radius and the
phase is nearly constant indicating that m=1 is a global mode. The lopsidedness
in the HI can be measured until several disk scalelengths, more than twice the
radial distance possible for the stars.}
\label{fig24}
\end{figure}
\bigskip
The A$_1$ values measured from the HI data and the R-band data are
available for four galaxies, and these were compared. It was found that in the
radial region of overlap, the two tracers show a similar value of
lopsidedness, see Fig. 25. This confirms that the origin of lopsidedness in HI is of
a dynamical origin and not purely of a gas-dynamical process that only
applies to the gas.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=2.0in,width=2.5in]{fig25a.eps}
\medskip
\includegraphics[height=2.0in,width=2.5in]{fig25b.eps}
\bigskip
\caption{A comparison of the lopsided amplitude A$_1$ obtained by
analyzing the HI data and the R-band data for stars for NGC 1953 (left panel) and
NGC 1359 (right panel), from Angiras et al. (2006). In the radial region
of overlap the values are comparable, thus indicating the same origin
for the lopsidedness both in stars and gas. The figures also strikingly
illustrate that HI is a much better tracer than stars for the study of
lopsidedness at large radii.}
\label{fig25}
\end{figure}
\bigskip
A similar Fourier analysis has been done for a sample of 12 galaxies
in the Ursa Major (Angiras et al. 2007) by analyzing
the 2-D HI data (Verheijen 1997) available for these. The average value of lopsided amplitude
in the 1.5-2.5 disk scalelength region in this sample is smaller than in the Eridanus group, and is closer to the field
galaxies case, as shown in Fig. 26. This could reflect the different spatial distribution of galaxies in the two groups- the ones in the Ursa Major are distributed along a filament.
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=2.5in,width=2.5in]{fig26a.eps}
\medskip
\includegraphics[height=2.5in,width=2.5in]{fig26b.eps}
\caption{The histograms denoting the number of galaxies vs. the lopsided amplitude
measured in the 1.5-2.5 disk scalelength range for the Ursa Major Group
(left) and the Eridanus group (right) of galaxies, taken from Angiras et al. (2007).
The galaxies are more lopsided in the Eridanus group, indicating a substantial variation
between groups.}
\label{fig26}
\end{figure}
\bigskip
An interesting characteristic of lopsidedness in group galaxies as noted
by Angiras et al. (2006) is that the early-type galaxies show a higher quantitative
lopsidedness than do the late-type galaxies, see Fig. 27. This is opposite to what is seen in the field galaxies (Bournaud et al. 2005b)
and indicates that tidal interactions play a dominant role in generating lopsidedness
in the group galaxies. This is not surprising given the high concentration of galaxies in a group.
Tidal interactions would tend to cause a secular evolution of galaxies to an earlier type and hence
when these are the dominant mechanism for generating lopsidedness, one would expect higher values of
lopsidedness for early-type galaxies, as argued by Bournaud et al. (2005 b).
\bigskip
\begin{figure}[h]
\centering
\includegraphics[height=2.0in,width=2.5in]{fig27.eps}
\caption { The lopsided amplitude measured between the radial range of 1.5-2.5 disk scalelengths vs. the galaxy type,
from Angiras et al. (2006). The early-type galaxies show a higher lopsidedness than the late-type galaxies.
This is opposite of what is seen for the field galaxies, see Fig. 5.}
\label{fig27}
\end{figure}
\bigskip
A distinguishing feature of asymmetry in the group galaxies is that
the values of the asymmetry as measured by the mean fractional Fourier amplitudes A$_1$, A$_2$ and A$_3$ for the modes m=1,2 and 3
are found to be comparable (Angiras et al. 2007). Also,
the derived perturbation potential parameters $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ are found to be comparable.
Although this last result depends on the model used, it reinforces the similar result obtained for the Fourier amplitudes which are directly observed and hence are model-independent (Section 3.1.2).
This is in
contrast to the field galaxies where A$_1$ and A$_2$ are comparable and are generally stronger than A$_3$ and the other higher mode amplitudes (Rix \& Zaritsky 1995).
This indicates the importance of multiple, simultaneous
tidal interactions that can occur under the special conditions of a group environment. This needs to be studied by future dynamical studies including by N-body simulations.
Since tidal interactions are frequent, occurring on a timescale of $< \sim 0.5 Mpc / 300$ km s$^{-1} \sim 3 Gyr$, the long-term maintenance of lopsidedness is not a problem in this environment.
\subsection{Lopsidedness in galaxies in clusters}
The galaxies in clusters may undergo even more frequent encounters
than in the groups because of the generally higher number density of
galaxies. However the relative velocity between galaxies is higher in clusters and
hence the encounters are weaker in strength. The accumulation
of a large number of weak interactions has been called
the galaxy harassment (Lake, Kat z, \& Moore 1998).
The asymmetry in NGC 4252 including the smoothly varying HI velocity field along the tidal tail, has been attributed to this effect
(Haynes, Giovanelli \& Kent 2007). The
amplitude of lopsidedness generated due to tidal encounters is
expected to be weaker in this case because of the quick encounters.
On the other hand, other dynamical processes such as asymmetry arising due
to ram pressure may be specifically applicable in a group or a
cluster setting. This may even be the dominant source of gas
asymmetry in these and can affect the HI gas lying on the outer
parts of a galactic disk. An interesting interplay of these various effects is possible as in NGC 4848 in the Coma cluster, which shows a lopsided distribution of the molecular gas (Vollmer et al. 2001). This has been explained by the interaction between the galactic gas, and the gas removed by ram pressure stripping around 4 $\times$ 10$^8$ yr ago which is now falling back.
Since the cluster galaxies often show HI deficiency especially in
the outer parts where lopsidedness is generally more common, this
could be a potential problem with detecting lopsidedness
in cluster galaxies.
Despite this, in some galaxies of groups and
clusters, the gas alone in known to show a strong asymmetry - as in NGC 4647 (Young
et al. 2006).
This strong gas asymmetry has been frequently attributed to
ram pressure stripping. However, the role of
the gravitational potential asymmetry in
lopsidedness cannot be neglected, even in these cases.
\subsection{Lopsidedness in centers of advanced mergers}
A fairly new regime that is just beginning to be explored is
the asymmetry at the centers of advanced mergers of galaxies.
A recent systematic study was done by Jog \& Maybhate (2006), with a view to
understand the mass asymmetry and the relaxation in the central regions of mergers.
Interactions and mergers of galaxies are known to be common and significantly affect
their dynamics and evolution.
The outer regions of merger remnants covering a distance of $\sim $ few kpc to a few 10 kpc have
been well-studied. These can be fit by an r$^{1/4}$ profile (class I), an outer exponential (class II) or a no-fit profile (class III)
(Chitre \& Jog 2002), and the first two can be explained as arising due to equal-mass mergers (e.g., Barnes 1992)
or unequal-mass mergers (Bournaud, Jog \& Combes 2005 a) respectively, while the third class
corresponds to younger remnants.
Despite
its obvious importance for the evolution of the central regions,
the luminosity distribution in the central regions of mergers was not
studied systematically so far.
A few mergers where this has been studied,
such as NGC 3921 (Schweizer 1996), and Arp 163 (Chitre \& Jog 2002),
show wandering or meandering centres for the consecutive isophotes.
Jog \& Maybhate (2006) chose a sample of 12 advanced mergers which showed signs of recent interaction such as tidal tails or loops but had a merged, common center and covered all three classes discussed above; and the angular size of the galaxy was sufficiently large to allow the Fourier analysis by dividing the image into
a few radial bins. The sample was chosen so as to cover the three classes showing different remnant profiles
as described above.
The K$_s$ band images from 2MASS were analyzed using the task ELLIPSE in STSDAS. The elliptical isophotes were fitted to galaxy images while allowing the center, ellipticity and the position angle to vary so as to get the best fit.
Fig. 28 (top panel) shows the result for Arp 163. The
isophotes are not concentric, instead the centers (X$_0$,Y$_0$) of consecutive isophotes
show a wandering or sloshing behaviour, indicating an unrelaxed central region.
Another measure of asymmetry is the lopsidedness of the distribution, to obtain this a galaxy image was Fourier-analyzed with respect to a constant center and the amplitude A$_1$ and the phase p1 of the $m=1$ mode were plotted versus radius- as shown for Arp 163 in the lower panel of Fig. 28. During the Fourier analysis the center was kept fixed, for the reason as discussed
in Section 2.1.2 . The intensity and hence the mass distribution is highly lopsided with the fractional amplitude
for the m=1 mode of $\sim 0.15$ within the central 5 kpc.
\bigskip
\begin{figure}[h]
\includegraphics[width=12cm]{fig28.eps}
\caption{Central region of the merger Arp 163 - The top panel shows the centers of the isophotes vs. the semi-major axis: the centers show a sloshing pattern indicating an unrelaxed behaviour. The lower panel shows the Fourier
amplitude $A_1$ and the phase $p_1$ vs. radius for the lopsided mode ($m=1$):
the lopsided amplitude is large and the phase fluctuates with radius. This is taken
from Jog \& Maybhate (2006).}
\label{fig28}
\end{figure}
\bigskip
All the sample galaxies show strong sloshing and lopsidedness in the central regions. The
asymmetry does not seem to significantly depend on the masses of progenitor galaxies - being similar for class I and II cases. However, it is higher for the mergers in early stages of relaxation (class III).
The corresponding values especially for the central lopsidedness are found to be smaller by a factor of few for a control sample of non-merger galaxies. This confirms that the high central asymmetry in mergers is truly due to the merger history.
The ages of remnants are deduced to be $\sim$ 1-2 Gyr as seen from the remnants with similar outer disturbed features in the N-body simulations of mergers (Bournaud et al. 2004), during which time these are likely to get chosen as our sample galaxies.
Thus the central asymmetry is long-lived, lasting for $\sim 100$ local dynamical timescales. Hence it can play an important role in the dynamical evolution of the
central regions. First, it can help fuel the central AGN since it provides a means
of outward
transport of angular momentum. Second, it can lead to the secular growth of the bulges via
the lopsided modes. These need to be studied in detail theoretically. Since this predicted evolution is due to the central asymmetry that is merger-driven, this process could be important in the hierarchical evolution of galaxies.
\section{Related topics}
\subsection {Relative strengths of lopsidedness (m=1) and
bars/spiral arms (m=2)}
Nearly all physical mechanisms, such as a tidal encounter or gas
accretion, that can give rise to an m=1 mode will also give rise to
an m=2 mode. Which of these two dominates depends on the detailed
parameters of the system. For example, for a distant encounter, the
m=2 mode is stronger. A prograde encounter is likely to preferentially
generate the m=2 mode while a retrograde encounter favours the generation of a lopsided (m=1)
mode (Bournaud et al. 2005 b).
It is well-known that a two-armed spiral pattern is supported as a
kinematical feature over most of the galactic disk, or in the region
of nearly flat rotation curve in general, since the pattern speed
given by $\Omega - \kappa / 2 $ is nearly constant in
this case (Lindblad 1959). The density wave theory of spiral
features is built around this idea (Rohlfs 1977).
The human eye/ brain likes to notice bisymmetry. This is one
reason why the two-armed spirals have received enormous amount
of attention in the theory of galactic structure and dynamics. This
is despite the fact that it has been known for a long time that a
higher $m$ value or a flocculent behaviour is more commonly observed in galaxies (see
e.g., Elmegreen \& Elmegreen 1982).
It should be noted that the early observational studies of spiral
structure such as the Hubble atlas (Sandage 1961) used blue filter
where the emission from the young stars stands out, and also the
dust extinction plays a major role in the galaxy image produced. In
contrast, the more recent studies using the near-IR band data, in
particular in the K$_s$ band avoid this problem and trace more
accurately the underlying old stellar mass
population (e.g., Block et al. 1994, Rix \& Zaritsky 1995).
Interestingly, in their recent study leading to the Large Galaxy
Catalog, Jarrett et al. (2003) show that m=1 is the most common mode
seen in the near-IR. They argue that, as noted by Block et al. (1994), there is theoretical reason to believe that m=1 modes should dominate the internal structure of spirals.
The question then is, why is there a prevalent notion even amongst
professional astronomers that lopsidedness is unusual in a galaxy
while a two-armed spiral is the norm, which is totally in contrast
to the observed case. We think one reason for this wrong notion
could be that the phase of lopsidedness is observed to be nearly
constant (see Section 2.3). Thus the resulting isophotal contours
are oval-shaped or egg-shaped, and their elongation is not striking
in the inner or optical parts of a galaxy, so an untrained eye may
miss it. \footnote {On the other hand,
a little training or awareness allows one to detect lopsidedness
in galaxies easily, as the authors of this present article have
found !}
If there were a strong radial dependence of the phase, then the resulting
one-armed structure as it occurs in M51 or NGC 4564 would be easy to see.
The amplitude of lopsidedness is found to increase with radius for
the stars (Rix \& Zaritsky 1995) and for HI gas (Angiras et al.
2006, 2007). This is the reason why even a low-sensitivity HI map reveals
the lopsidedness easily - see e.g. the images of a face-on galaxy
like M101 or an edge-on galaxy like NGC 891 (Baldwin et al. 1980).
Thus it is not surprising that lopsidedness was first detected in
the HI maps. The fact that it is now found to be equally ubiquitous
in the older stars as well, is what makes its study even more
important and challenging. For example, it points to a basic
dynamical mechanism for the origin of lopsidedness, and not
something based purely on the gas dynamical processes.
An interesting feature regarding observational detection of lopsidedness,
first noted by Rix \& Zaritsky (1995), is that it does not get confused with the
inclination angle, unlike the A$_2$ values.
\subsubsection {Observed amplitudes of m=1 and m=2 components}
The observed amplitudes A$_1$ and A$_2$ of m=1 and 2 respectively
for stars are generally comparable, and both are much larger than
the values for the higher $m$ modes for the field galaxies (Rix \& Zaritsky 1995, Bournaud et
al. 2005 b). Normally m=2 is taken to denote a bar or a spiral arm
in the inner or outer regions respectively, or it can also denote
disk ellipticity. The amplitudes for m=2 have been measured for
larger samples (Laurikainen, Salo, \& Rautiainen 2002, Buta et al.
2005, Bournaud et al. 2005 b). The average values of A$_1$ and A$_2$
between 1.5 - 2.5 disk scalelengths have been measured for the 149
galaxies in the OSU catalog (see the Appendix in Bournaud et al.
2005 b), and the two generally show a positive correlation (see Fig. 8
in that paper). This cannot be explained if the tidal encounters were the main generating mechanism since m=2 spiral arms or bars are more easily triggered on
direct orbits where $\Omega - {\kappa}/2$ is positive (e.g., Gerin et al. 1990). In contrast, a
lopsided mode is more likely to be triggered or last longer in a retrograde orbit since the pattern speed
of m=1 asymmetries, $\Omega - \kappa$, is negative in a galactic disk.
The simulations by Thomasson et al. (1989) shows that retrograde tidal encounters between galaxies lead to the
formation of leading one-armed spiral arms, as in NGC 4622.
Observationally very few galaxies
show a leading arm. Pasha (1985) found that only 2 out of the 189 galaxies studied
show a leading arm (NGC 3786, NGC 5426), and each of these happen to be in a pair, which agrees with the work by Thomasson et al. (1989).
It is possible that the pattern speed of m=1 arms in other galaxies is small - this has to be checked observationally.
As discussed in Section 3.2.3 this can have a bearing on the mechanism for the origin of
lopsidedness. For example, if the pattern speed is small, the global modes are long-lived (Saha et al. 2007) and do not need to be triggered frequently.
On the other hand, the galaxies in groups show a higher lopsidedness for the early-type galaxies, and show a comparable magnitude for all the lower modes, m=1,2, and 3. The frequent and even concurrent
tidal encounters in this setting are probably responsible for this (see Section 5.2).
The different modes could interact directly but this obvious line of research
has not been followed up much. A
non-linear coupling between m=1,3 and m=2 was proposed by Masset \& Tagger (1997)
for the central regions. Even though the m=1 mode is largely seen in the outer galactic
disk while the bars and the spiral structure (m=2) are seen more in the inner parts
of a galactic disk, they can still have a dynamical effect on each other.
The heating due to a bar (m=2) can suppress the further
growth of a lopsided mode, as was seen in the numerical simulations for a purely exponential
disk by Saha et al. (2007).
Conversely, the presence of a lopsided mode can lead to a bar
dissolution as has been studied by Debattista \& Sambhus (2008).
This topic needs more study and could shed an important light on the dynamical evolution of a disk
due to various non-axisymmetric features.
\subsection{Asymmetry in the dark matter halo}
The study of disk asymmetry including lopsidedness has triggered many applications where the
asymmetry is used as a tool to gain information about the shape and the density distribution in the
dark matter halo.
The dark matter halo is generally assumed to have a spherical shape, for the sake of simplicity.
This view has been challenged, and
there have been studies which have used various tracers such as the polar rings (Sackett \& Sparke 1990),
warps (e.g. Ideta et al. 2000), gas flaring (Olling 1996, Becquaert \& Combes 1997, Narayan, Saha \& Jog 2005, Banerjee \& Jog 2008),
to deduce the shape of the dark matter halo in galaxies. A summary of topic is given in
Natarajan (2002).
In the tidal picture of the origin of disk lopsidedness, the disk responds to a distorted
dark matter halo. Thus we can use the observed disk asymmetry to deduce the asymmetry in the
galactic plane of the dark matter halo, which is not visible directly.
As discussed in detail in Section 3.2.1, the observed disk lopsidedness
can be used to deduce the halo lopsidedness and indicates a few \% lopsidedness for the halo
in a typical galaxy. Similarly, on treating a self-consistent disk response, and using the
disk ellipticity, one can deduce the ellipticity of the dark matter halo (Jog 2000).
The idea of negative disk response (Jog 1999, 2000) has been applied by Bailin et al. (2007) for a more
realistic radial variation of the ellipticity of the potential to show that the disk response circularizes the net potential in the central region of a triaxial halo.
The amplitude of lopsidedness is higher in the group
galaxies and can therefore imply a higher distortion of the halo, $\sim$ 10 \%
as shown for the case of the Eridanus group galaxies (Angiras et al. 2006).
The power in the various m modes is comparable (see Schoenmakers 2000,
Angiras et al. 2007).
The values of all three perturbation potentials derived $\epsilon_1, \epsilon_2 ,
\epsilon_3$ are comparable (Section 5.1). This can be an important
clue to the mechanism for generating lopsidedness in groups, and perhaps indicates
the importance of multiple, simultaneous tidal interactions that can occur under
the special conditions of a group environment.
The asymmetry in the dark matter halo of the Galaxy has been studied quantitatively as follows. The
recent survey of atomic hydrogen gas in the outer Galaxy (Levine et al. 2006) has revealed
a striking asymmetry in the thickness map of HI gas. The gas in the Northern part flares more with
the thickness higher by a factor of $\sim 2$ compared to that in the South, at a galactocentric radial distance
of 30 kpc. This has been modeled by Saha et al (2008), who obtain the vertical scaleheight for the galactic disk in the
gravitational field of the dark matter halo by solving the vertical force equation and the Poisson equation together.
This model shows that the above asymmetry is best explained by a lopsided dark matter halo, with a small elliptical distortion that is out of phase with the lopsidedness.
The centres of dark matter halos are predicted to show lopsidedness as based on the N-body simulations
in the $\Lambda$CDM cosmology (Gao \& White 2006)
and the size of asymmetry is larger for larger size halos as in clusters of galaxies,
though these models need to be followed
by direct predictions which can be checked
against observations as stressed by these authors.
The Fourier harmonic technique developed mainly to study the disk asymmetry (Jog 1997, Schoenmakers et al. 1997)
has now been applied to the kinematical data along the minor axis for the dwarf galaxy DDO 47 (Gentile et al. 2005). This study
has shown that the velocity dispersion components are too small to arise due to a cusp. Hence the galaxy was deduced to have a genuine core-like density distribution of the dark matter
halo in the central regions.
The asymmetry in the halo is expected to be long-lived because of its
collisionless nature. However, a finite pattern speed can reduce this life-time to be $\sim $ a few
Gyr, or much less than the Hubble time (see the discussion in Section 3.2.3).
\subsection {Comparison with warps}
Spiral galaxies also show a bending of the midplane at large radii, this is known as the warp.
A warp is also a global feature of type m=1 except in the vertical direction (Binney \& Tremaine 1987).
Warps are extremely common and in fact all the main galaxies in the Local group, namely the Galaxy, M31, M33, LMC are warped. Warps are seen mainly in the HI gas, typically beyond a 4-5 disk scalelength radius (Briggs 1990)
but in many cases are also seen in stars in a radial region somewhat inside of this (Reshetnikov \& Combes 1998).
A tidal encounter in an arbitrary orientation can generate both lopsidedness as well as warps, as is known, see e.g. Weinberg (1995), and
Bournaud et al. (2004). Individual galaxies often exhibit both these phenomena, and galaxies with an intermediate angle of inclination allows both to be seen easily as in NGC 2841 (see Fig. 1b in the present paper).
Both the features share some common properties - namely they are seen preferentially
in the outer parts of a galactic disk, and their long-term maintenance against differential rotation is a problem.
The disk self-gravity resists imposed perturbation in the inner parts, as denoted by the negative
disk response (Section 3.2.1.c). The resulting net self-consistent disk response shows that
the disk will exhibit lopsidedness only outside of $\sim 2$ disk scalelengths (Jog 1999, Jog 2000). A similar calculation for the perturbation along the vertical direction shows that the onset of warps in a galactic disk occurs only beyond 4-5 disk scalelengths (Saha \& Jog 2006).
We note, however, that warps and lopsided distribution are physically different features. First, in a lopsided distribution the centre of mass is shifted with respect to the original centre of mass of the galaxy. On the other hand,
in a standard m=1 S-shaped warp, the mass distribution is symmetric with respect to
the centre of mass and to the symmetry plane of $z = 0$. Second, the onset of warps is
determined by a somewhat arbitrary threshold of a few degree lifting of the mid-plane away from its central value, which is set by the observational detection limits. On the other hand, a lopsided amplitude is well-determined quantitatively following the Fourier analysis- although the threshold value of what constitutes a lopsided galaxy is still fairly arbitrary (see Section 2.2).
\subsection{Implications for high redshift galaxies}
Lopsidedness is more likely at high redshift, since galaxy interactions are more frequent.
There is multiple observational evidence of more asymmetric
galaxies at high redshift (e.g. Simard et al 2002).
Measuring the lopsidedness is however complex due to the clumpy/non-uniform
background distribution in a galaxy (e.g. Elmegreen et al 2004), and the overall low spatial resolution.
\section{Effect of lopsidedness on galaxy evolution}
A galactic disk is inherently susceptible to the formation of the
lopsided mode (Section 3). This mode can be long-lived as
evident from the high fraction of galaxies that are observed to show
lopsidedness. The dynamical origin and the evolution of these
features is a challenging problem as discussed here. But apart from that, does the
existence of lopsidedness affect the galaxy in any way? The answer
is a resounding yes.
The list of processes whereby a lopsided distribution can
affect the evolution of the galactic disk in a significant way include the following:
1. A lopsided distribution can help in the angular momentum transport
in the disk and can thus contribute to the secular evolution in the disk.
This is especially important given the long-lived
nature of the lopsidedness. This can cause a redistribution of matter as studied
by Lynden-Bell \& Kalnajs (1972). This process is especially important at lower radii
because the dynamical timescale is smaller. Hence the net dynamical evolution
is likely to occur on timescales less than the age of the galaxy. The details for this process need to be worked out.
2. The fueling of the central active galactic nucleus (AGN) can occur due to the m=1 motion of the central black
hole (Section 4.2) and this could be more effective than the first process above.
3. Lopsidedness could affect the details of galaxy formation. For example, the accretion of mass as mediated by
m=1 would be especially important for the highly disturbed high redshift galaxies.
4. The lopsided distribution results in an azimuthal asymmetry in star formation in a galactic disk.
In a lopsided potential, the effective disk surface density is shown
to be a maximum at $\phi = 0^0$, corresponding to an overdense
region, while there is an underdense region in the opposite
direction along $\phi = 180^0$. The fractional increase in surface
density at $\phi = 0^0$ is high $\sim 0.3 - 0.5 $ for strongly
lopsided galaxies (see Fig. 1, Jog 1997). Thus, the molecular gas in
the overdense region could become unstable and result in enhanced
star formation, as shown
for example for the parameters for M101. Further, the enhanced star
formation in the overdense region is argued to give rise to a
preferential formation of massive stars (Jog \& Solomon 1992). This will
result in more HII regions in the overdense region. This prediction is exactly in agreement
with observations of more HII regions seen along the SW in M101.
Lopsidedness has also been observed in the H$_{\alpha}$ emission
from the star-forming regions in dwarf irregular galaxies (Heller et
al. 2000). Such asymmetry is expected to be common in all galaxies
and we suggest future work in this area is necessary.
\section{Summary and Future Directions}
We have reviewed the spatial and
kinematical lopsidedness in a galaxy - both the observations and dynamics, as seen in the
various tracers - stars and gas, and in the inner and outer regions,
and in different settings- field and group environment.
The lopsidedness is shown to be a common phenomenon. Nearly 30 \% of
spiral galaxies show a 10\% fractional amplitude in m=1 or the first
Fourier mode. The amplitude can be higher and can go up to 30 \% in
strongly lopsided galaxies like M 101. In a group environment, this
effect is stronger: all the galaxies show lopsidedness and the
average amplitude of lopsidedness is nearly twice that in the field
case.
We recommend that the future users adopt the fractional
Fourier amplitude A$_1$ as the standard criterion for lopsidedness.
Further, the threshold value that could be adopted could be
the average value of 0.1 seen in the field galaxies in the intermediate radial
range of 1.5-2.5 R$_{exp}$ (Bournaud et al. 2005 b), so that
galaxies showing a higher value can be taken to be lopsided. A uniform criterion
will enable the comparison of amplitudes of lopsidedness in different galaxies,
and also allow a comparison of the fraction of galaxies deduced to be lopsided in different studies.
A variety of physical mechanisms have been proposed to explain the
origin of the lopsidedness, of which the most promising are the ones
involving a tidal encounter, gas accretion, and gravitational instability. A unique feature of the m=1 perturbation
in the galactic disk is that it leads to a shift in the centre
of mass in the disk, and this further acts as an indirect force
on the original centre of the disk. The self-gravity of the perturbation
decreases the precession rate by a factor of $\sim 10$ compared to free precession.
The disk is thus shown to
naturally support a slowly-rotating, global, lopsided mode which is long-lived. However, the
uniqueness of this solution has been proven. Also it is not clear what would give rise to
such slow modes, except gas accretion. A
fast pattern speed as
would occur for lopsidedness generated in a tidal encounter
cannot yet be ruled out.
The high fraction of galaxies showing lopsidedness has still not been explained fully. The
N-body simulations for a tidal interaction between galaxies with a live halo and gas (Bournaud et al. 2005 b)
show that the
lifetime of the lopsided mode thus generated is a few Gyr. The other mechanism involving steady gas accretion
would probably generate a one-armed lopsided mode which is not seen commonly.
Satellite accretion could generate the right amount of lopsidedness but would
also thicken the disks more than is seen. It needs to be checked if a small-mass satellite
falling in can generate the right amplitude distribution of lopsidedness without thickening
the disk, and of course if there are such satellites available to fall in at a steady rate.
A measurement of pattern speed in real galaxies would be extremely useful in
constraining the main mechanism for generating lopsidedness. For example, a tidal encounter is expected to give rise to a lopsided mode with a small but finite pattern speed.
In group galaxies, the ongoing continuous tidal interaction can help one get over the
maintenance problem easily. This can explain why nearly all galaxies in a group are strongly
lopsided, and may perhaps explain why these show equal amplitudes of higher asymmetry modes. This needs to be confirmed by
detailed dynamical studies and simulations.
Some of the open problems in this field include: a measurement of pattern speed of lopsided
mode in real galaxies, the amplitude and thickening of the disk generated by an accretion of a
low-mass satellite, and
the study of origin and evolution of lopsidedness in galaxies in groups, and the evolution timescale
of a m=1 mode in a collisionless dark matter halo. These deal with the origin and the evolution of lopsidedness
in galaxies. The related field of problems involving the study of the dark matter shape has much promise, and some work has been started along this direction with models to explain the observed HI thickness distribution.
In the central regions of mergers of galaxies, the dynamics of the sloshing and lopsidedness seen on scales of $\sim 1$ kpc needs to be investigated. On further small scales, simulations with central black holes should be carried out to explore the formation of eccentric disks as in M31.
An even more interesting set of questions has to do with the
effect the lopsidedness has on the further evolution of the galaxy. These include:
the angular momentum transport, the fuelling of the central active galactic nucleus,
enhanced star formation in the overdense regions, and the early evolution of a galaxy
mediated by the m=1 mode for the studies of galaxies at high
redshift, and the coupling between the various modes (bars, lopsidedness etc) in a galactic
disk and its effect on the further dynamical evolution of a galaxy.
In summary, the study of asymmetry is a rich and a challenging
area in galactic structure and dynamics, with lots of open questions
- both observational and theoretical.
\bigskip
\noindent {\bf Acknowledgments:} We are happy to acknowledge the support of the Indo-French grant IFCPAR/2704-1 .
\bigskip
\bigskip
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\end{document}
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Sheriff: Boy dead after police chase partly taped by TV crew
A man led Alabama officers and a reality television crew on a weekend chase that ended when he crashed his vehicle into a tanker truck, killing the man''s 15-year-old son, authorities said.
Reico Terry, 41, remained jailed Monday after the death of his son Jaylen Derrel Terry in Saturday's crash in the Birmingham. Jefferson County sheriff's deputies had stopped chasing Terry because the pursuit had become too dangerous before the crash occurred, sheriff's Sgt. Joni Money said in a statement.
Deputies said the 15-year-old from the Birmingham suburb of Bessemer was one of two passengers in the vehicle and died after being taken to a hospital. The teen was a backseat passenger.
The chase began when deputies, accompanied by a TV crew from the program Live PD, reported seeing a drug transaction at a suburban gas station. Deputies stopped the buyer, who Money said confirmed the purchase. The crew was filming, but wasn't live.
When deputies tried to approach Terry, he sped away, Money said.
Deputies chased him onto Interstate 59/20, unsuccessfully trying to block intersections. Terry got off the interstate, Money said, adding deputies broke off the chase after Terry sped through intersections without yielding to oncoming traffic or stopping for the signals.
Terry crashed less than 20 minutes after he fled from deputies, according to Money. He said Terry tried to run away after the crash scene, but was arrested and found to have heroin.
Deputies say a front-seat passenger who had only minor injuries was questioned and released. Authorities didn't say whether Terry was injured in the crash.
Terry is charged with unlawful distribution of a controlled substance, possession of drug paraphernalia, felony eluding, reckless endangerment and resisting arrest. Bail is set at $43,000.
The Alabama Law Enforcement Agency is investigating the crash and authorities said Terry could face additional charges.
In November, four people were hospitalized and some critically injured following a crash in Birmingham during a chase by Jefferson County deputies, with Live PD accompanying.
The show portrays police action without any music, script, interviews or narrator. The contract with Jefferson County ended in December, but The Birmingham News reports Live PD crews have continued to accompany deputies, although filming is no longer live.
For 'ice wine' vintners, frigid temps are reason to rejoice
While the vast majority of New Yorkers are hunkering down to ride out frigid temperatures and snow blasts from the latest winter storm, others are seizing on the opportunity to
San Diego police investigating fatal stabbing at Sabre Springs park
Lobbyists told state insurance chief they represented company at center of campaign scandal, new filing says | {
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package transport
import (
"context"
"github.com/influxdata/influxdb/v2"
"github.com/influxdata/influxdb/v2/authorizer"
"github.com/influxdata/influxdb/v2/kit/platform"
"github.com/influxdata/influxdb/v2/kit/platform/errors"
)
func newAuthCheckingService(underlying RemoteConnectionService) *authCheckingService {
return &authCheckingService{underlying}
}
type authCheckingService struct {
underlying RemoteConnectionService
}
var _ RemoteConnectionService = (*authCheckingService)(nil)
func (a authCheckingService) ListRemoteConnections(ctx context.Context, filter influxdb.RemoteConnectionListFilter) (*influxdb.RemoteConnections, error) {
rs, err := a.underlying.ListRemoteConnections(ctx, filter)
if err != nil {
return nil, err
}
rrs := rs.Remotes[:0]
for _, r := range rs.Remotes {
_, _, err := authorizer.AuthorizeRead(ctx, influxdb.RemotesResourceType, r.ID, r.OrgID)
if err != nil && errors.ErrorCode(err) != errors.EUnauthorized {
return nil, err
}
if errors.ErrorCode(err) == errors.EUnauthorized {
continue
}
rrs = append(rrs, r)
}
return &influxdb.RemoteConnections{Remotes: rrs}, nil
}
func (a authCheckingService) CreateRemoteConnection(ctx context.Context, request influxdb.CreateRemoteConnectionRequest) (*influxdb.RemoteConnection, error) {
if _, _, err := authorizer.AuthorizeCreate(ctx, influxdb.RemotesResourceType, request.OrgID); err != nil {
return nil, err
}
return a.underlying.CreateRemoteConnection(ctx, request)
}
func (a authCheckingService) GetRemoteConnection(ctx context.Context, id platform.ID) (*influxdb.RemoteConnection, error) {
r, err := a.underlying.GetRemoteConnection(ctx, id)
if err != nil {
return nil, err
}
if _, _, err := authorizer.AuthorizeRead(ctx, influxdb.RemotesResourceType, id, r.OrgID); err != nil {
return nil, err
}
return r, nil
}
func (a authCheckingService) UpdateRemoteConnection(ctx context.Context, id platform.ID, request influxdb.UpdateRemoteConnectionRequest) (*influxdb.RemoteConnection, error) {
r, err := a.underlying.GetRemoteConnection(ctx, id)
if err != nil {
return nil, err
}
if _, _, err := authorizer.AuthorizeWrite(ctx, influxdb.RemotesResourceType, id, r.OrgID); err != nil {
return nil, err
}
return a.underlying.UpdateRemoteConnection(ctx, id, request)
}
func (a authCheckingService) DeleteRemoteConnection(ctx context.Context, id platform.ID) error {
r, err := a.underlying.GetRemoteConnection(ctx, id)
if err != nil {
return err
}
if _, _, err := authorizer.AuthorizeWrite(ctx, influxdb.RemotesResourceType, id, r.OrgID); err != nil {
return err
}
return a.underlying.DeleteRemoteConnection(ctx, id)
}
| {
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} | 1,660 |
{"url":"http:\/\/earthboundmisfit.space\/calculus-new-glasses\/","text":"\/More Calculus And New Glasses\n\n## Are glasses required to be a calculus nerd?\n\nI am learning how to input more and more complex formulas using LaTeX. As I learn more about the formatting options available; fonts, effects, shapes, etc\u2026 I can probably use the LaTeX code to design a whole page within a blog post instead of just inserting math equations. There is so much potential for this that I am just beginning to understand.\n\nI\u2019m off to the blood lab today for my preliminary appointment before the big one Friday. Just in the nick of time too because with summer coming to an end I could use some money and I won\u2019t have a lot of free time to donate blood during the fall.\n\nI finally went to the eye doctor yesterday to get a prescription for glasses. This is something I have been meaning to do since January but, better late than never. At least I will be able to see the whiteboards in the classrooms for the fall semester. I still don\u2019t feel comfortable with eyeglasses on but I suppose I\u2019ll get used to them. The free glasses I get from Masshealth are stupid looking so once I figure out how to buy the right ones online I\u2019ll buy ones that I am comfortable wearing, especially since there is \u00d8 chance my eyesight will improve as I get older.\n\n## More LaTeX\n\nL=\\lim_{n\\to\\infty}\\Big\\{1+\\frac{2}{n}\\Big\\}^n\\\\lnL=\\lim_{n\\to\\infty}n\\cdot\\ln\\Big\\{1+\\frac{2}{n}\\Big\\}\\\\\\textit{using l\u2019hopital\u2019s rule:}\\\\ln L\u2019=\\lim_{n\\to\\infty} \\bigg\\{\\frac{1}{1+\\frac{2}{n}}\\bigg\\}\\cdot\\Big\\{\\frac{-2}{n^2}\\Big\\}\\Big\\{\\frac{-n^2}{1}\\Big\\}=2\\\\e^{lnL\u2019}=e^2\\\\ L\u2019=e^2\\\\\\textit{by l\u2019hopital\u2019s rule again}\\\\L=e^2 [\\latex]\n\nL=\\lim_{n\\to\\infty}\\Bigg \\{\\sqrt{1+\\frac{1}{2n}}\\Bigg \\}^n = \\lim_{n\\to\\infty}\\Bigg \\{1+\\frac{1}{2n}\\Bigg \\}^{\\frac{n}{2}} \\\\ \\textit{take natural log of both sides}\\\\lnL=\\lim_{n\\to\\infty}\\frac{n}{2} \\,ln\\Big \\{1+\\frac{1}{2n}\\Big \\}=\\lim_{n\\to\\infty}ln \\Bigg \\{\\frac{1+\\frac{1}{2n}}{\\frac{n}{2}} \\Bigg \\} \\\\ \\textit{first application of l\u2019hopital\u2019s rule}\\\\ lnL\u2019=\\lim_{n\\to\\infty}\\frac{\\Big \\{\\frac{1}{1+\\frac{1}{2n}} \\Big\\} \\cdot \\Big\\{\\frac{-1}{2n^2}\\Big \\}}{\\Big\\{\\frac{-2}{n^2}\\Big \\}} \\\\ \\textit{simplify and apply definition of limit}\\\\ =\\lim_{n\\to\\infty}\\frac{1}{4} \\cdot\\frac {2n}{2n+1}=\\frac{1}{4}=lnL\u2019=lnL\\\\ \\mathbf{L=e^{lnL}=e^\\frac{1}{4}}\n\nI will be posting my test scores and some sample problems from my Calculus Exams here as soon as the semester is over. It wouldn\u2019t be good to be accused of any impropriety while some students still haven\u2019t taken the second test yet.","date":"2017-10-18 21:58:56","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\": 0.9257035255432129, \"perplexity\": 2100.7347501943786}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187823153.58\/warc\/CC-MAIN-20171018214541-20171018234541-00861.warc.gz\"}"} | null | null |
Mount Skeidskneet () is a mountain, high, surmounting the east side of the head of Humboldt Graben at the southwest extremity of the Petermann Ranges, Wohlthat Mountains, in Queen Maud Land in Antarctica. It was discovered and photographed by the German Antarctic Expedition (1938–39), mapped by Norway from air photos and surveys by the Norwegian Antarctic Expedition (1956–60), and named Skeidskneet.
Mountains of Queen Maud Land
Princess Astrid Coast | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,309 |
static const char
LIBPROJ_ID[] = "Id";
/*
** Permission is hereby granted, free of charge, to any person obtaining
** a copy of this software and associated documentation files (the
** "Software"), to deal in the Software without restriction, including
** without limitation the rights to use, copy, modify, merge, publish,
** distribute, sublicense, and/or sell copies of the Software, and to
** permit persons to whom the Software is furnished to do so, subject to
** the following conditions:
**
** The above copyright notice and this permission notice shall be
** included in all copies or substantial portions of the Software.
**
** THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
** EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
** MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
** IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
** CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
** TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
** SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*/
#define PROJ_PARMS__ \
double ap;
#define EPS10 1.e-10
#define PROJ_LIB__
#include <lib_proj.h>
PROJ_HEAD(tcc, "Transverse Central Cylindrical") "\n\tCyl, Sph, no inv.";
FORWARD(s_forward); /* spheroid */
double b, bt;
(void) P; /* avoid warning */
b = cos(lp.phi) * sin(lp.lam);
if ((bt = 1. - b * b) < EPS10) F_ERROR;
xy.x = b / sqrt(bt);
xy.y = atan2(tan(lp.phi) , cos(lp.lam));
return (xy);
}
FREEUP; if (P) free(P); }
ENTRY0(tcc) P->es = 0.; P->fwd = s_forward; ENDENTRY(P)
/*
** Log: proj_tcc.c
** Revision 3.1 2006/01/11 01:38:18 gie
** Initial
**
*/
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,372 |
Ha-Zorea (, v oficiálním přepisu do angličtiny HaZorea) je vesnice typu kibuc v Izraeli, v Severním distriktu, v oblastní radě Megido.
Geografie
Leží v nadmořské výšce 65 m, na pomezí Jizre'elského údolí a svahů planiny Ramat Menaše. Severovýchodně od obce pak Jizre'elským údolím protéká řeka Kišon, do které skrz vesnici prochází vádí Nachal ha-Šofet s přítoky Nachal Sanin a Nachal ha-Šnajim. Jižně od obce pak do údolí ústí vádí Nachal Gachar. V krajině západně od obce stojí zalesněné kopce Giv'at Miš'ol nebo Tel Kira, na jihu je to vrch Har Gachar. Na okraji izre'elského údolí se nalézá pahorek Tel Zarik.
Vesnice tvoří společně se sousední mošavou Jokne'am sídelní pás, který lemuje jihozápadní okraj Jizre'elského údolí. Tato dvě původně převážně zemědělská sídla jsou pak navíc urbanisticky propojena se sousedním městem Jokne'am.
Nachází se přibližně 72 km severovýchodně od centra Tel Avivu a 22 km jihovýchodně od centra Haify. Vesnici ha-Zorea obývají Židé, přičemž osídlení v tomto regionu je etnicky převážně židovské. 6 km severozápadním směrem odtud ovšem na hřbetu horského masivu Karmel leží drúzská sídla.
Na dopravní síť je vesnice napojena pomocí dálnice číslo 70, jež vede směrem k pobřeží Středozemního moře, a dálnice číslo 66, která sleduje jihozápadní okraj Jizre'elského údolí.
Dějiny
Obec ha-Zorea byla založena roku 1936. Jejími zakladateli byla skupina cca 70 židovských přistěhovalců z Německa. Tato skupina se zformovala ve městě Chadera a roku 1936 se usadila zde. Zpočátku byla terčem opakovaných útoků během arabského povstání, které roku 1936 v tehdejší britské Palestině vypuklo. V době před vznikem státu Izrael byl tento kibuc jedním z center elitních židovských jednotek Palmach.
Koncem 40. let 20. století měl kibuc ha-Zorea rozlohu katastrálního území dunamů (2,99 km²).
Jde o jeden z nejlidnatějších kibuců v Izraeli. Ekonomika je založena na zemědělství a průmyslu (například továrna na výrobu nábytku). Funguje tu i obchodní zóna. V obci je situováno také muzeum, které připomíná Wilfrida B. Israele, který pomáhal ve 40. letech organizovat židovskou emigraci z nacistické Evropy.
V obci funguje zařízení předškolní péče o děti a základní škola.
Demografie
Obyvatelstvo ha-Zorea je sekulární. Podle údajů z roku 2014 tvořili populaci v kibucu ha-Zorea Židé (včetně statistické kategorie "ostatní", která zahrnuje nearabské obyvatele židovského původu ale bez formální příslušnosti k židovskému náboženství).
Jde o menší sídlo vesnického typu s dlouhodobě stagnujícím počtem obyvatel. K 31. prosinci 2014 zde žilo 843 lidí. Během roku 2014 populace stoupla o 0,7 %.
Odkazy
Reference
Související články
kibuc
Externí odkazy
Oficiální profil obce na portálu Bet Alon
Oficiální profil obce na portálu Galil
Oficiální profil obce na portálu Rom Galil
Oficiální stránky obce
Oblastní rada Megido
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{"url":"https:\/\/szaach.pl\/battlefield-expansions-qnz\/4a5a99-what-is-the-missing-reason-in-step-5%3F","text":"Ask your question Login with google. XYZ Inc. produces two types of paper towels?regular and super-soaker. A line passes through the points (-3,-5) and (6,1). MZADE - mZCDF 4. def. Question sent to expert. a cylinder is exactly 3 times bigger than a cone with the same height and radius. During the ex a2 = b2 + c2 \u2013 2bccos(A) 9. commutative propertyWhat is the missing reason in Step 8? Answers: 2 Show answers Another question on Mathematics. Correct answers: 2 question: Given: jklm is an isosceles trapezoid, kl \u2225 jm prove: km \u2245 jl what is the missing reason in step 4? adminstaff. Linear pair postulate B. Givin C. Definition of complementary angles D. Congruent complements theorem Get an easy, free answer to your question in Top Homework Answers. \u2220 addition postulate 6. m\u22201 + m\u22202 + m\u22203 = 180\u00b0 6. def. Then, by the theorem: ------------------------------------------------------------------------------------------. ZADE a CDF 3. vert. What is the missing statement and the missing reason in step 5? of a 5 ? What is the measure of angle MNL? Then, by the theorem: the range is biggest number minus the smallest, so in this case it would be. Asked By stu-charlotte @ 15\/10\/2020 07:52 PM. Mathematics, 14.04.2018 02:50. Jailah needs a replacement for lithium for a science experiment. Thereofre in \u0394LMN we have: By using this site, you consent to the use of cookies. What is the measure of angle ABD? how much money will be in her bank account in six months? ORP and ORN are a linear pair . what is the missing reason in step 5 quizlet 4 noviembre, 2020. rose 5\u00b0C each hour for 4 hours. \u2220aec \u2245 \u2220bdc 3. all right angles are congruent 4. ? 1. human activity and normal climate conditions 2. natural factors and human activity 3. normal climate conditions and abnormal human activities 4. abnormal human activities and natural factors, Given: Circle O with diameter LN and inscribed angle LMN. What is the missing statement and the missing reason in step 5? b. inscribed angle theorem. The perimeter of the rectangle is at most 112 cm. F=(x1+x32,y1+y32) definition of midpoint 9. slope of DF = y1+y32\u2212y1+y22x1+x32\u2212x1+x22 = y3\u2212y2x3\u2212x2 the slope of BC = y3\u2212y2x3\u2212x2 definition of slope 10. Mathematics. Statements Reasons 1. circle O has diameter LN and inscribed angle LMN 1. given 2. is a semicircle 2. diameter divides into 2 semicircles 3. circle O measures 360o 3. measure of a circle is 360o 4. m = 180o 4. definition of semicircle 5. m\u2220LMN = 90o 5. ? Get an easy, free answer to your question in Top Homework Answers. By using this site, you consent to the use of cookies. substitution property. Then the temperature changed by... En las faner\u00f3gamas despu\u00e9s de ser fecundadas se forma la:... Research and identify a vestigial structure of an organism of your choosing. The vertebrate forelimb initially develops in the embryo as a solid mass of tissue. Which pair of angles are vertical angles? Correct answers: 2 question: Plz plz me what are the missing statement and reason in step 2 of the proof? What is the missing reason in step 5? Elections. \u2220 2. definition of perpendicular 3. x.48\u00b0 x.132\u00b0 Points A, B, and C are on line AC. rice prices have no real importance other than to large corporations, Which of the activities listed below could limit global warming by slowing the increase in atmospheric carbon dioxide levels? 0 Answers. ZS 2. def. What is mMHJ? PLEASE HELP WILL GIVE BRANLIEST, EXTRA POINTS! 90 degrees im pretty sure unless im reading it wrong. statements reasons 1. abcd is a kite 1. given 2. ab \u2245 bc, bp \u22a5 ac 2. definition of kite 3. bp \u2245 b. This theorem states that an angle \u03b8 inscribed in a circle is half of the central angle 2\u03b8 that subtends the same arc on the circle. Search. Which best describes his statement? Zs 3. m\u22201 = m\u22205; m\u22203 = m\u22206 4. def. Mathematics, 21.06.2019 19:00. The missing reason in step 3 is angle addition postulate. He wanted three things out of college: to learn a skill, to make good friends, and learning about life. What is the missing statement and the missing reason in step 5? ... OTHER QUIZLET SETS. What is the missing reason in step 5? You will receive an answer to the email. What is the missing reason in step 5? One of mendel's laws of inheritance. And millions of other answers 4U without ads. Arc LN measures 180\u00b0, \u00a0because segment LN is the diameter of the circle. You can refuse to use cookies by setting the necessary parameters in your browser. To complete steps 5 - 6 of the proof, refer to the diagram shown below. What is the missing reason in step 5 A. What is the missing reason in step 8? Given: mADE = 60\u00b0 and mCDF = (3x + 15)\u00b0 Prove: x = 15 What is the missing statement and the missing reason in step 5? Community Guidelines. Answers: 2. view available hint(s) select all that apply. You will receive an answer to the email. straight angle 7. m\u22205 + m\u22202 + m\u22206 = 180\u00b0 7. substitution Which could be the missing reason in Step 3? Most candidates are elected to office duringA primaryB. The car consumes 6 gallons of gas per 150 miles, and gas costs $1.20 pe... At the start of an experiment, the temperature of a solution was -12\u00b0C. Which of the following emerging technologies uses the principle of chromosomes to identify suspects, Why is the price of rice so important? Which statement could be used in step 2 when proving x = 30? Answers Mine. Alicia puts$400 in a bank account. Line segment XY is tangent to circle Z at point U. Angle MNL is complementary to angle KNL. 30\/12\/2019 08:08 AM. 41\u00b0 Line segment TS is tangent to circle O at point N. If the measure of QNT is 74\u00b0, what is the measure of QPN? Advanced Placement (AP), 08.11.2019 21:31, And millions of other answers 4U without ads, Add a question text of at least 10 characters. Subjects. Regular uses 2 units of recycled paper per unit of production, and super-soaker... YASHARI earns $27,000 per year, is single, and lives in Wyoming. What is the missing reason in step 5? Steve earns$8 per hour his older brother earns $2 more per hour than steve what is the ratio of the money steve earns in an hour to the money of his brother earns in an hour. To complete steps 5 - 6 of the proof, refer to the diagram shown below. The figure shows three lines that intersect at point N. Angle GNH is congruent to angle KNL. Angle CBD has a measure of 140\u00b0. at how many different angles will the hexagon map onto itself? a. replacing fossil fuels with nuclear energy b. burning vegetation to clear land for agriculture c. limiting soil erosion so organic matter takes longer to decompose d. choosing a fuel-efficient car, or bicycling to school or work e. using more electrical appliances and cars f. cutting down forests to build houses, What major factors play a role in global warming? statement: 60 = 3x + 15; reason: substitution statement: x = 15; reason: subtraction propert - e-eduanswers.com 50\u00b0 Marcus states that angle ORP and angle LRP are a linear pair. Statements Reasons 1. circle O has diameter LN and inscribed angle LMN 1. given 2. is a semicircle 2. diameter divides into 2 semicircles 3. circle O measures 360o 3. measure of a circle is 360o 4. m = 180o 4. definition of semicircle 5. m\u2220LMN = 90o 5.? The equation of this line can be expressed in the form Ax + By = C, where A, B, and C are integers... View a few ads and unblock the answer on the site. He is incorrect. A. \u2220jkl \u2245 \u2220mlk 4. ? Aregular hexagon rotates counterclockwise about its center. Statement: 60 = 3x + 15; Reason: substitution Statement: x = 15; Reason: subtraction property of equality Statement: 60 = 3x + 15; Reason: transitive property Statement: x = 15; Reason: subtraction and division properties of equality. What is the missing reason in step 5? A. AD=DA. ZADE and ZCDF are vert. Related Questions in Mathematics. What is the missing reason in step 5? \u2220ace \u2245 \u2220bcd \u2220eab \u2245 \u2220dbc \u2220eac \u2245 \u2220eac \u2220cbd \u2245 \u2220dbc ZS 4. Ray RO and ray RL are not opposite rays. What is the missing reason in step 5. LOGIN TO VIEW ANSWER. What is the missing statement and the missing reason in step 5? 6. 65 points eloise has a mat that has a length of $$\\frac{8}{9}$$ meter and a width of $$\\frac{3}{4}$$ meter. B. Why does the organism not need that structure anymore? Sam claims that cos x =sin y if x and y are congruent angels. Answers: 1 Show answers Another question on Mathematics. Which answer BEST describes the Cottage Industry? each year the account earns 5% simple interest. \u200b... View a few ads and unblock the answer on the site. This Top Homework Answer is Middle School level and belongs to the Mathematics subject. statements reasons 1. circle o has diameter ln and 1. given inscribed angle lmn lkn is a semicircle 2. diameter o divides into 2 semicircles 3. circle o \u2026 52 4\/5 B. defining a midpoint8. generalC propositionD. In circle V, angle WXZ measures 30\u00b0. a. Pythagorean theorem b. definition of cosine c. substitution d. properties of multiplication. 40\u00b0 x.70. 4263 users searched for this homework answer last month and 5 are doing it now, let\u2019s get your homework done. Share what\u2019s outside your window and all around you. In this case, the angle is \u2220LMN and the arc is arc LN. linear pair postulate given definition of complementary angles congruent complements theorem. Statement: 60 = 3x + 15; Reason: substitution . Answers: 3 Show answers Another question on Mathematics. 45 - 3x 6. subtr. Aneutron had a density of about 5.9 10 17kg\/m3 what would be the approximate mass of a 1-centimeter cube (a cube 1cm on all sides? Find a number such that if you add 8 and divide the result by 4 you will get the same answer as if you subtracted 3 from the original number and divided by 2. Mathematics, 21.06.2019 14:30. rice prices must make a large profit or no one will grow itc. given: circle with diameter ln and inscribed angle lmn prove: zumn is a right angle. statements reasons 1. jklm is an isosceles trapezoid, kl \u2225 jm 1. given 2. jk \u2245 lm 2. definition of isosceles trapezoid 3. kl \u2245 kl 3. reflexive property 4. Mathematics. Thus, Is Even, And Hence C 2a For Some Integer A. is sam correct ? wjc01280. 6. Correct answer - What is the missing reason in step 5? 52 4\/5 B. this recessive allele has a mutation that makes a gene for a blood-clotting factor nonfunctional. truth throughout the argument? 4. reflexive property 5. aec ~ bdc 5. aa similarity theorem what is the missing statement in step 4? Anne... Why have countries agreed to build space tools that must fall safely into the earths atmosphere... Why does plastic flow only occur below 50 meters of ice? View desktop site. 5. ? rice prices impact the cost of fruits and vegetables on neighboring farmsb. 6. F(n + 1) = f(n) \u2013 8. if f(1) = 100, what is f(6)? Question sent to expert. Year 9 English - help! Question: Given: mADE = 60\u00b0 and mCDF = (3x + 15)\u00b0 Prove: x = 15 What is the missing statement and the missing reason in step 5? If the measure of UV is 84\u00b0, what is the measure of YUV? She is t... You are going on a road trip over a distance of 3000 miles with three friends. a. rice prices must be kept within reach of the poorest peopleb. What is the missing reason in step 3? Answer. Proof to complete Step Statement Reason -(KV P) Premise for lIndirect Method *KA- P DeMorgan *K 41) Step 3 reason \"QVK Premise Page 8 of 12 ons Dr B \"Q he switches below 42) Step 5 reason 6. View a few ads and unblock the answer on the areas of the.... Of YUV the cost of fruits and vegetables on neighboring farmsb angles are congruent angels 1. 2! 10 ) o case it would be, the angle is \u2220LMN and the missing reason in step 2 proving... Necessary parameters in your browser proving x = 30 ~ bdc 5. aa similarity what! Tangent to circle Z at point N. angle GNH is congruent to KNL! Is at most 112 cm greater than 0\u00b0 and less than or equal to.... Which of the proof answer to your question in Top Homework answers use cookies by setting the necessary in. Are doing it now, let \u2019 s get your Homework done to... Even, and C are on line AC, because segment LN is the missing reason in 5! School level and belongs to the Mathematics subject is 180\u00b0 than or to! Kl \u2225 Jm Prove: Km \u2245 Jl what is the missing statement and the statement! Needs a replacement for lithium for a blood-clotting factor nonfunctional map onto?. 4. reflexive property 5. aec ~ bdc 5. aa similarity theorem what the. Ice to cause plastic flow OB of 3000 miles with three friends price of rice so important are... Little houses in small areas B reading it wrong three lines that intersect point! It turns through angles greater than 0\u00b0 and less than or equal to 360\u00b0 the necessary parameters in your.. Areas of the proof, refer to the diagram shown below to 360\u00b0 the... And ( 6,1 ) by the theorem: the range is biggest number minus the smallest so! X and y are congruent 4. 1. given 2 of their home ORP and LRP! The sum of the proof the embryo as a solid mass of tissue pair postulate definition.... you are going on a road trip over a distance of 3000 miles with three.. Which of the proof, refer to the diagram shown below inscribed angle LMN Prove zumn! Opposite the legs are equal proving x = 30 uses the principle of chromosomes to identify suspects, is. Explanation can be used to informally derive the formula for the volume of a is! Step 5 View a few ads and unblock the answer on the areas of the?... That much ice to cause plastic flow OB 5 quizlet 4 noviembre, 2020, and C. A2 = b2 + c2 \u2013 2bccos ( a ) 9. commutative propertyWhat is the missing reason step...: by using this site, you consent to the use of cookies m\u22201 + m\u22202 + =. Regular and super-soaker areas of the proof in subsidized loans and Another$ in... Land... Based on the site a. Pythagorean theorem B. definition of cosine c. substitution d. properties of.! In step 5 Reasons 1 MZADE -60\u00b0 1. given 2 ; reason: substitution missing statement the! Points ( -3, -5 ) and ( 6,1 ) hour for 4 hours... Why does a fight out. Complete steps 5 - 6 of the poorest peopleb neighboring farmsb by using this site, you to...... View a few ads and unblock the answer on the areas of the proof, refer to diagram... Of rice so important be kept within reach of the internal angles in each triangle is 180\u00b0 the of. 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A2 = b2 + c2 \u2013 2bccos ( a ) 9. commutative propertyWhat is the missing statement and the reason! Aa similarity theorem what is the missing statement and the arc is arc LN measures 180\u00b0 because... Why does a fight break out in the embryo as a solid mass of tissue will in! The price of rice so important angle LRP are a linear pair postulate given definition of angles. To angle KNL - ( 3x+15 ) 2 money will be in bank!, let \u2019 s outside your window and all around you angle LRP a... A linear pair postulate given definition of complementary angles congruent complements theorem which of the following emerging uses... Or no one will grow itc Trapezoid, Kl \u2225 Jm Prove: Km \u2245 Jl what is missing.... View a few ads and unblock the answer on the areas of the poorest peopleb therefore two... Within reach of the proof, refer to the diagram shown below question on.... A huge amount of money producing little houses in small areas B for this answer. Cell turn into a human the Middle of the circle turn into a human in this case what is the missing reason in step 5? be! Statement and reason in step 4 huge amount of money producing little houses in areas! Thereofre in \u0394LMN we have: by using this site, you consent to the Mathematics.. Price of rice so important 180\u00b0 7. substitution which could be the missing reason in step 8 towels? and. In this case, the angle is \u2220LMN and the missing reason in step 5 question: Plz me... How much money will be in her bank account in six months angle are. Question on Mathematics + what is the missing reason in step 5? = 180\u00b0 7. substitution which could be used step! Choose to have the Morehouse family move out of their home congruent to angle KNL for a blood-clotting factor.. B2 + c2 \u2013 2bccos ( a ) 9. commutative propertyWhat is the missing in... A cylinder is exactly 3 times bigger than a cone with the same height and radius Plz what. 1 MZADE -60\u00b0 1. given mZCDF - ( 3x+15 ) 2 towels? regular and super-soaker \u2013 2bccos a! Reflexive property 5. aec ~ bdc 5. aa similarity theorem what is the missing reason in 5! 19,000 in unsubsidized loans $19,000 in unsubsidized loans -3, -5 ) and ( 6,1 ) are., Why is the missing statement and reason in step 5 x.132\u00b0 Points a B! Cosine c. substitution d. properties of multiplication no one will grow itc it would.! This Homework answer is Middle School level and belongs to the diagram below. = m\u22205 ; m\u22203 = m\u2220LAM 5 have: by using this site, consent... The site Marcus states that angle ORP and angle LRP are a linear pair postulate definition. Less than or equal to 360\u00b0 are equal the price of rice so?. Jailah needs a replacement for lithium for a science experiment EF in circle?! Structure anymore claims that cos x =sin y if x what is the missing reason in step 5? y are congruent 4.$... A human less than or equal to 360\u00b0 cause plastic flow OB kept within of... 1. aeec ; bddc 1. given 2 which could be the missing reason in step 5 rice prices the. Complete steps 5 - 6 of the internal angles in each triangle is 180\u00b0 move of... Author choose to have the Morehouse family move out of college: to learn a skill, to make friends... + 15 ; reason: substitution on the site m\u22205 + m\u22202 + m\u22206 180\u00b0! To the use of cookies lt ; LMN is a right triangle with friends. 180\u00b0 7. substitution which could be used in step 5 Z at point N. angle GNH is congruent to KNL. Zumn is a right triangle factor nonfunctional available hint ( s ) select all that apply is. Organism not need that structure anymore two types of paper towels? regular and super-soaker initially develops in the of... 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A. rice prices impact the cost of fruits and vegetables on neighboring....\n\nSunapee Harbor Marina, How To Make A Photography Portfolio For A Job Interview, Froggy Gets Dressed Story Pdf, Best Walking Exercise, Chateau Song Lyrics, Lean-to Crossword Clue, Ramada Buffet Dinner Price, Waseda University Admission, Fairbanks, Alaskathings To Do, The Regimental Shop Edinburgh,","date":"2021-07-24 11:52:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.42423850297927856, \"perplexity\": 3177.716179158946}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046150264.90\/warc\/CC-MAIN-20210724094631-20210724124631-00595.warc.gz\"}"} | null | null |
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\section{Introduction}
Coherent transport is a phenomenon that occurs in various fields of physics.
Its fundamental properties are illustrated by the multi-slit experiment, where the maximum of the interference pattern is limited by the number of coherently illuminated slits.
Such an enhancement of arrival probability can be found in any system in which the propagating object can take several path alternatives coherently,
{\it i.e.} can be coherently delocalized.
Excitons in bio-molecular networks are just one example for such a situation,
but the identification of beatings in spectroscopic data obtained from natural \cite{Panitchayangkoon2011,Engel2007} and artificial \cite{Collini2009} samples which might be a signature of quantum coherence has substantially revived the interest in transport and interference.
Any realistic system -- in particular a light harvesting complex -- is always subject to noise.
Given the intrinsic stochastic nature of such processes, the propagating object would naturally take different paths depending on the current realisation of the noise.
Such a classical exploration of various paths, {\it i.e.} incoherent delocalization, however, does not result in any direct enhancement of the arrival probability.
Coherent delocalization gives rise to interference, which enhances transport efficiency if it is constructive \cite{Kramer1993,Scholak2011},
but may also reduce it if destructive \cite{Lattices1956,Plenio2008,Rebentrost2009c}.
Incoherent delocalization on the other hand does not have the potential for strong enhancement of transport efficiency,
but results in transport that is typically substantially more robust against perturbations than entirely coherent transport.
It is thus the interplay of coherent and incoherent delocalization that governs the efficiency with which the propagating object reaches a given final state.
Here, we introduce a quantitative concept of coherent delocalization that shall help to improve our understanding of the distinction between coherent and incoherent aspects of quantum transport. A rigorous approach to distinguish these two aspects has been sought for a while \cite{Smyth2012}.
Originally, this was attempted in terms of time-averaged quantities, which could provide a rough estimation of delocalization in specific systems \cite{Kuhn1997,Leegwater1996}. Recently, the transfer of instruments from entanglement theory resulted in a rather rigorous distinction between coherent and incoherent delocalization \cite{Caruso2010, Fassioli2010b},
and tools that verify a coherent delocalization of a given range (analogous to the number of coherently illuminated slits) are available \cite{Levi2013}.
So far, however, a clear quantitative meaning of such tools is unavailable,
but would be necessary to properly assess the potential for interference and resulting enhancement of transport.
In the following, we consider a general setting to describe quantum transport. The formal framework that we introduce is not limited to a specific physical scenario; it is applicable for example to the case of multi-armed interferometers, excitation transport in bio-molecular networks \cite{Scholes2011}, electron transport in a network of quantum dots \cite{Greentree2004}, photons in an optical network \cite{Schreiber2010} or more generally to assess coherence in quantum random walks. Irrespective of the specific physical system, the propagating object can adopt a given number of states $\ket{\Psi_i}$: those may, for example, indicate which path is taken by a particle in an interferometer, which individual chromophore is excited in a light harvesting complex, or they may indicate which excitonic eigenstate of the same system is populated. Once this set is determined, one may strive for the question over how many of these states the object is delocalized and to what extent the delocalization is coherent.
Given that coherent delocalization is a rather abstract concept one needs a clear notion to define its quantification.
This is similar to the theory of entanglement, where a quantitative notion has been found in terms of evolutions (commonly referred to as quantum channels) that may not create entanglement. The central requirement for a measure of entanglement is that it may never increase under these channels \cite{Bennett1996a}.
We follow here the same line of arguments: we identify channels that can not induce a coherent superposition of the states $\ket{\Psi_i}$, which we refer to as incoherent channels. Once this is established, our central requirement for a quantity to describe coherent delocalization over a given range of states quantitatively is that is may not grow under incoherent channels. Such concept is well established in the theory of entanglement and it starts to be increasingly employed also in the context of quantum coherence, as independently done in \cite{Baumgratz2013}.
\section{Coherent delocalization}
When the propagating object is isolated from its external environment, {\it i.e.} in absence of any noise, it is described by a pure state $\ket{\chi}=\sum_{i=1}^n \xi_i \ket{\Psi_i}$, where $n$ is the dimension of the system. The amplitudes $\xi_i$ give the probabilities $|\xi_i|^2$ to find the object in a specific state of the set $\ket{\Psi_i}$.
If only one amplitude $\xi_j$ is non-vanishing, then the object is completely localized on the state $\ket{\Psi_j}$.
A state $\ket{\chi}$ is defined coherently delocalized over $k$ states $\ket{\Psi_i}$, or in short $k$-coherent, if $k$ of the amplitudes $\xi_i$ are non-vanishing.
In the presence of noise, the correct instrument to describe the propagating object is a density matrix $\varrho=\sum_j p_j \prj{\chi_j}$.
It describes both quantum interference due to coherent delocalization in the pure state components $\ket{\chi_j}$ and classical averaging in terms of the probabilities $p_j$.
This interplay results in the fact that constructive interference originating from one pure state component may be compensated in this averaging process by destructive interference from other components.
That is, the potential of the state $\varrho$ to give rise to interference phenomena cannot be inferred exclusively from the coherence properties of the pure state components $\ket{\chi_j}$.
In particular, the description of the density matrix $\varrho$ in terms of the probabilities $p_j$ and states $\ket{\chi_j}$ is not unique. The eigensystem provides one decomposition, but there is a continuous set of decompositions with typically non-orthogonal states $\ket{\chi_j}$ \cite{Schrodinger2008}.
If there is a decomposition of the density matrix composed of at most $k$-coherent states, than $\varrho$ describes a situation that can be accounted for by interference up to $k$ amplitudes and classical averaging. A mixed state $\varrho$ should therefore be considered $k$-coherent if there is no set of probabilities $p_j$ and at most ($k$-1)-coherent states $\ket{\chi_j}$ that are compatible with $\varrho$, {\it i.e.} that satisfy $\varrho=\sum_jp_j\prj{\chi_j}$.
\section{Measures for coherent delocalization}\label{section3}
Our objective is to provide a quantification of coherent delocalization rather than the mere qualitative distinction between different types of $k$-coherence defined above. As discussed in the introduction, the main requirement for such a quantification is that it may not grow under incoherent channels.
We formalize this concept later-on in section \ref{sec:incohproc},
but anticipate here that channels of this kind are the result of an average over two types of elementary incoherent processes.
The first is the modification of phase coherence, and it is described by operators of the form
\begin{equation}
A_\ell=u_1\prj{\Psi_\ell}+u_2\sum_{j\neq \ell}\prj{\Psi_j}.
\label{eq:dephasing}
\end{equation}
An average over this type of operations induces dephasing, {\it i.e. } loss of phase coherence. The second type of elementary process is the incoherent hopping from $\ket{\Psi_\ell}$ to $\ket{\Psi_j}$, represented by the operators
\begin{equation}
B_{j\ell}=\ket{\Psi_j}\bra{\Psi_\ell}.
\label{eq:hopping}
\end{equation}
Any channel that can be described as an average over products of these two types of operators has to be considered incoherent, and a proper measure of coherent delocalization must never increase under such a channel.
Now that we have stated which condition to fulfil, we are in position to introduce the functions whose validity as quantifiers we prove later-on in section \ref{sec:monotonicity}.
Coherent delocalization first of all requires delocalization, {\it i.e. } more than one finite diagonal matrix element $\matel{\Psi_i}{\varrho}{\Psi_i}$.
The larger (in terms of absolute values) the off-diagonal matrix elements are, the more pronounced is the coherent character of this delocalization.
One would thus expect that coherent delocalization could be characterized by the relation between diagonal and off-diagonal matrix-elements.
As shown in \cite{Levi2013} the quantities
\begin{equation}
\label{hierarchy}
\tau_{kn}^{\varphi}(\varrho)=\big|\matel{\varphi_1}{\varrho}{\varphi_2}\big| -a_{kn}\sum_{j=1}^n\sqrt{\matel{\varphi_1^{(j)}}{\varrho}{\varphi_1^{(j)}}\matel{\varphi_2^{(j)}}{\varrho}{\varphi_2^{(j)}}}
\end{equation}
with
\begin{equation}
\label{eq:param}
a_{kn}=1/(n-k+1)\hspace{.2cm}\mbox{for}\hspace{.2cm}k\neq 2\hspace{.4cm}\mbox{and}\hspace{.4cm}a_{2n}=1/n,
\end{equation}
are non-positive if $\varrho$ is not at least $k$-coherent, {\it i.e. } coherently delocalized over $k$ of the states $\ket{\Psi_i}$. The states $\ket{\varphi_1}$ and $\ket{\varphi_2}$ are defined as
\begin{eqnarray}
\label{states}
\begin{aligned}
&\ket{\varphi_1}= C_1 \sum_{i=1}^n\frac{\beta_i}{\alpha_i}|\Psi_i\rangle\\
& \ket{\varphi_2}= C_2 \sum_{i=1}^n\frac{\beta_{i+n}}{\alpha_{i+n}}|\Psi_i\rangle
\end{aligned}
\end{eqnarray}
in terms of a set of $2n$ pairs of complex parameters $\{\alpha_i,\beta_i\}$, with $|\alpha_i|^2+|\beta_i|^2=1$.
The prefactors $C_i$ are given by $C_1=\prod_{i=1}^n \alpha_i$ and $C_2=\prod_{i=1}^n \alpha_{i+n}$.
The $n$ states $\ket{\varphi_1^{(j)}}$ and $\ket{\varphi_2^{(j)}}$ are obtained from $\ket{\varphi_1}$ and $\ket{\varphi_2}$
by a simple exchange of $\alpha_j,\beta_j$ and $\alpha_{j+n},\beta_{j+n}$, {\it i.e.}
\begin{eqnarray}
\label{states2}
\begin{aligned}
\ket{\varphi_1^{(j)}}&=\ket{\varphi_1}\big|_{\alpha_j=\alpha_{j+n},\beta_j=\beta_{j+n}}\\
\ket{\varphi_2^{(j)}}&=\ket{\varphi_2}\big|_{\alpha_{j+n}=\alpha_{j},\beta_{j+n}=\beta_j}.
\end{aligned}
\end{eqnarray}
The functionality of the definitions from \eref{hierarchy} to \eref{states2} can be understood in terms of pure states $\varrho=\prj{\chi}$. If $\xi_j=0$, the $j$-th term in the sum equals $\big|\matel{\varphi_1}{\chi\rangle\langle\chi}{\varphi_2}\big|$.
For a $k$-coherent state in an $n$-dimensional system there are $n-k$ such summands.
The prefactor $a_{kn}=1/(n-(k-1))$ thus makes sure that $\tau_{kn}^\varphi$ is non-positive for all states with less than $k$-coherence;
for $k=2$ the prefactor can be chosen even smaller because in the case of complete localization each term in the sum equals $\big|\matel{\varphi_1}{\chi\rangle\langle\chi}{\varphi_2}\big|$.
With these prefactors, \eref{hierarchy} is thus constructed to be non-positive for states that are less than $k$-coherent, and,
as shown in the appendix, for any at least $k$-coherent pure state $\ket{\chi}$ there exist vectors $\ket{\varphi_1}$ and $\ket{\varphi_2}$ which make $\tau_{kn}^\varphi(\prj{\chi})>0$. Finally, the functions defined in \eref{hierarchy} are convex as shown in \cite{Huber2010}; consequently, $\tau_{kn}^\varphi(\varrho)$ cannot be positive if there is a decomposition of the mixed state $\varrho$ into at most ($k-1$)-coherent states, {\it i.e. } if $\varrho$ is less than $k$-coherent.
The potential of these functions to characterize coherent delocalization and its relation to transport efficiency was shown in \cite{Witt2013a}.
As we demonstrate in the following, each of the non-negative functions
\begin{equation}
\label{deftaumax}
\mathcal{T}_{kn}(\varrho)=\max_{\varphi}\tau_{kn}^{\varphi}(\varrho)
\end{equation}
is indeed non-increasing under incoherent operations,
and is thus not just a qualitative, but indeed a valid quantitative description of $k$-coherence.
\section{The formal framework}
As pointed out before, the following formalism is suited to describe several different physical scenarios which feature coherent transport. In order to keep the discussion simple, however, we choose a general terminology in terms of excitations and units.
With excitation we indicate the to-be-transported entity, which could {\it e.g.} be an electronic excitation or a photon.
This excitation may be delocalized over a set of $n$ states $\ket{\Psi_i}$ ($i=1,...,n$).
Typically, the state $\ket{\Psi_i}$ denotes that the excitation is carried by the $i$-th out of $n$ physical entities that we refer to as units;
in the case of electronic excitations this could be chromophores; or they may be $n$ modes of the electromagnetic field in the case of photons in an interferometer. Regardless of the specific system, the units can be modeled as two levels systems.
These $n$ physical entities then define a composite system with $n$ components, so that each state $\ket{\Psi_i}$ can also be identified with an $n$-unit state
$\ket{\underbrace{0... 0}_{i-1}1\underbrace{0... 0}_{n-i}}$, where the $i$-th unit is excited. In this way the set of states $\ket{\Psi_i}$ is identifiable with the single excitation subspace of the full space which describes the $n$ units. For this reason, the term ``delocalization over $k$ states" possesses the same meaning as ``delocalization over $k$ units", and we use them interchangeably. The concept of $k$-coherence is then formally equivalent to the concept of $k$-partite entanglement in states that carry exactly one excitation \cite{Tiersch2012}.
This identification is going to be helpful in the following since it eases the analysis substantially.
Since, however, the description in terms of single units is not necessary to define the concept of coherent delocalization,
our results also apply {\it e.g.} to the case where the states $\ket{\Psi_i}$ are excitonic eigenstates, and a clear physical identification in terms of an $n$-unit state is not necessarily available.
The states $\ket{\varphi_1}$, $\ket{\varphi_2}$, $\ket{\varphi_1^{(j)}}$ and $\ket{\varphi_2^{(j)}}$ as defined in \eref{states} and \eref{states2} can also be obtained from the projection of the the $n$-unit states on the subspace with a single excitation.
Originally, $\tau_{kn}$ is defined \cite{Levi2013} in terms of a $2n$-unit product vector $\ket{\Phi}=\ket{\Phi_1}\otimes\ket{\Phi_2}$ given by
\begin{eqnarray}
\begin{aligned}
\label{states3}
\ket{\Phi_1}&=\ket{\psi_1}\otimes\ket{\psi_2}\otimes...\otimes\ket{\psi_n},\\
\ket{\Phi_2}&=\ket{\phi_1}\otimes\ket{\phi_2}\otimes...\otimes\ket{\phi_n},
\end{aligned}
\end{eqnarray}
where $\ket{\psi_i}$ and $\ket{\phi_i}$ are states for the $i$-th individual unit. The states $\ket{\Phi_1^{(j)}}$ and $\ket{\Phi_2^{(j)}}$, which enter the definition of $\tau^{\ket{\Phi}}_{kn}$ analogously to $\ket{\varphi_1^{(j)}}$ in \eref{deftaumax}, are obtained from $\ket{\Phi_1}$ and $\ket{\Phi_2}$ through an exchange of their $j$-th factor.
With the parametrization $\ket{\psi_j}=\alpha_j\ket{0}+\beta_j\ket{1}$ and $\ket{\phi_j}=\alpha_{j+n}\ket{0}+\beta_{j+n}\ket{1}$,
one obtains the states defined above in \eref{states} and \eref{states2} through projection onto the single excitation subspace.
In the following it is convenient to work in the full space of the $n$ units: since we require that $\varrho$ carries exactly one excitation, we can switch between the definition of $\tau_{kn}^{\ket{\Phi}}$ in the full space and its restriction to single excitation subspace $\tau_{kn}^\varphi$ as we like. For this reason, we are going to use the same symbol $\mathcal{T}_{kn}$ for the maximum of $\tau_{kn}^\varphi$ and of $\tau_{kn}^{\ket{\Phi}}$.
\subsection{Incoherent channels}\label{sec:incohproc}
One central advantage of the description in terms of the states of the individual units is that it permits the characterization of the operations that we anticipated as incoherent rather easily.
A real physical evolution is represented formally by a classical stochastic average over single processes, each of which is described by an operator $F_i$. The set of all these operators $F_i$, called the Kraus operators, connects the initial state $\varrho$ to the final state $\varrho_F$ via a channel \cite{Nielsen2010}:
\begin{equation}
\label{eq:channel}
\varrho_F=\sum_iF_i\varrho F_i^\dagger,
\end{equation}
where the additional constraint $\sum_iF_i^\dagger F_i=\mathbbm{1}$ ensures the conservation of trace.
An incoherent quantum channel is composed by incoherent Kraus operators, {\it i.e. } is an average over incoherent processes. Since none of these should coherently delocalize an initially localized excitation, any operator $F_i$ must be {\it local}, which means that it is given by the tensor product of operators acting on individual units, {\it i.e.}
\begin{equation}
F_i=f_i^{(1)}\otimes...\otimes f_i^{(n)}.
\label{eq:localoperator}
\end{equation}
In addition, we require that no such operator may change the number of excitations.
Each single-unit operator $f_i^{(j)}$ can be expanded in an operator basis.
Rather than using the typically employed Pauli matrices, we use in the following the raising operator $\sigma_+=(\sigma_x+i\sigma_y)/2$, the lowering operator $\sigma_-=(\sigma_x-i\sigma_y)/2$,
the regular Pauli matrix $\sigma_z$ and the identity $\mathbbm{1}$.
Any single-unit operator $\sigma_0=b\mathbbm{1}+d\sigma_z$ with complex coefficients $b$ and $d$ does not modify the number of excitations; $\sigma_+$ and $\sigma_-$ create and annihilate an excitation respectively.
We show in the following that each single-unit operator $f_i^{(j)}$ is either of the form $\sigma_0$, $\sigma_+$ or $\sigma_-$.
To this end, we consider a general form of $F$\footnote{the index $i$ is suppressed, as the following holds for all operators $F_i$ irrespective of their specific label}, not necessarily local, and expand it in the above introduced basis for the first subsystem. This gives
\begin{equation}
\label{excons}
F=c_0 (\sigma_0\otimes\mathcal{A}_0)+c_+(\sigma_+\otimes \mathcal{A}_-)+c_-(\sigma_-\otimes \mathcal{A}_+).
\end{equation}
$F$ conserves the number of excitations exactly if $\mathcal{A}_-$ and $\mathcal{A}_+$ annihilate and create an excitation respectively, and $\mathcal{A}_0$ conserves the number of excitations.
The operators $\sigma_0$, $\sigma_+$ and $\sigma_-$ are mutually orthogonal, and so are the operators $\mathcal{A}_0$, $\mathcal{A}_-$ and $\mathcal{A}_+$, which act on the remaining $n$-1 units.
$F$ is thus in the shape of a Schmidt decomposition \cite{Horodecki2009}, and one can directly conclude that it is of product form exactly if two of the three coefficients $c_0$, $c_+$ and $c_-$ vanish.
In a similar fashion one can proceed to investigate the locality of the operators $\mathcal{A}_0$, $\mathcal{A}_-$ and $\mathcal{A}_+$. By induction, one obtains that any operator in \eref{eq:localoperator} that conserves the number of excitations contains only operators of the form $\sigma_0$, $\sigma_+$ and $\sigma_-$ as factors; linear combinations would not be excitation conserving or would be nonlocal.
Restricted to the single excitation subspace discussed at the beginning of this section, the operator acting on $n$ units constructed from the single-unit operator $\mathbbm{1}(u_1+u_2)/2+\sigma_z^{(\ell)}(u_1-u_2)/2$ coincides exactly with $A_\ell$ defined in \eref{eq:dephasing}, while $B_{\ell j}$ originates from the joint contribution of the two single unit operators $\sigma_+^{(j)}\otimes\sigma_-^{(\ell)}$. We have therefore proven that indeed the only two elementary processes which do not create coherence and conserve the number of excitation are local dephasing and excitation hopping. All local excitation conserving Kraus operators acting on $n$ units, which we shall from now on compactly refer to as incoherent Kraus operators, can be obtained in terms of these two processes.
\subsection{A measure for coherent delocalisation decreases under incoherent channels}
\label{sec:monotonicity}
Our goal is to prove that $\mathcal{T}_{kn}(\varrho_F)\le\mathcal{T}_{kn}(\varrho)$ for all channels as defined in \eref{eq:channel} which are composed exclusively by incoherent Kraus operators.
The functions $\tau_{kn}^{\ket{\Phi}}$ are convex, as shown in \cite{Huber2010}. This allows us to conclude that
\begin{equation}
\mathcal{T}_{kn}(\sum_iF_i\varrho F_i^\dagger)\le\max_{\Phi}\sum_i\tau_{kn}^{\ket{\Phi}}(F_i\varrho F_i^\dagger).
\label{eq:convex}
\end{equation}
From the definition of $\tau_{kn}$ in \eref{hierarchy} it follows that
\begin{equation}
\tau_{kn}^{\ket{\Phi}}(F_i\varrho F_i^\dagger)=\tau_{kn}^{F_i^\dagger\otimes F_i^\dagger\ket{\Phi}}(\varrho).
\end{equation}
The state $F_i^\dagger\otimes F_i^\dagger\ket{\Phi}$ is still of product form because the Kraus operators $F_i$ are local as defined in \eref{eq:localoperator}.
This state is however in general not normalized. It is thus convenient to introduce the renormalized state
\begin{equation}
\ket{\tilde{\Phi}_i}=\frac{F_i^\dagger\otimes F_i^\dagger\ket{\Phi}}{\sqrt{\matel{\Phi}{F_iF_i^\dagger\otimes F_iF_i^\dagger}{\Phi}}}.
\end{equation}
Since $\tau_{kn}$ is a homogeneous function, we have
\begin{equation}
\label{eq:rescale}
\tau_{kn}^{F_i^\dagger\otimes F_i^\dagger\ket{\Phi}}(\varrho)=\tau_{kn}^{\ket{\tilde\Phi_i}}(\varrho)\ \underbrace{\sqrt{\matel{\Phi}{F_iF_i^\dagger\otimes F_iF_i^\dagger}{\Phi}}}_{\eta_i}.
\end{equation}
The maximization in the definition of ${\cal T}_{kn}$ runs over all product vectors,
and since the state vectors $\ket{\tilde\Phi_i}$ are in product form too,
a maximization over the state vectors $\ket{\tilde\Phi_i}$ can never yield something larger than maximization over $\ket{\Phi}$, which implies that $\tau_{kn}^{\ket{\tilde{\Phi}_i}}(\varrho)\leq\mathrm{max}_{\tilde{\Phi}_i}\tau_{kn}^{\ket{\tilde{\Phi}_i}}(\varrho)\leq\mathcal{T}_{kn}(\varrho)$. That is, together with \eref{eq:convex}, we can conclude that
\begin{eqnarray}
\begin{aligned}
\mathcal{T}_{kn}(\sum_iF_i\varrho F_i^\dagger)&\le\max_{\Phi}\sum_i\tau_{kn}^{\ket{\tilde\Phi_i}}(\varrho)\ \eta_i\\
&\le\mathcal{T}_{kn}(\varrho)\ \max_\Phi\sum_i\eta_i.\label{a4}
\end{aligned}
\end{eqnarray}
With the Cauchy-Schwartz inequality $\big|\sum_ix_iy^*_i\big|\le\sqrt{\sum_i|x_i|^2}\sqrt{\sum_i|y_i|^2}$, the right hand side can conveniently be bounded from above:
\numparts
\begin{eqnarray}\label{factor}
\max_{\Phi}\sum_i\eta_i&=&\max_{\Phi_1 \Phi_2}\sum_i\sqrt{\matel{\Phi_1}{F_iF_i^\dagger}{\Phi_1}\matel{\Phi_2}{F_iF_i^\dagger}{\Phi_2}}\label{eq:app1}\\
&\le&\max_{\Phi_1 \Phi_2}\sqrt{\matel{\Phi_1}{\sum_iF_iF_i^\dagger}{\Phi_1}}\sqrt{\matel{\Phi_2}{\sum_iF_iF_i^\dagger}{\Phi_2}}\label{eq:app2}.
\end{eqnarray}
\endnumparts
The two factors in \eref{eq:app2} are positive; the maximum of the product is therefore obtained as the product of the individual maxima. Since these maxima coincide, we have
\begin{eqnarray}
\label{eq:app3}
\begin{aligned}
\max_{\Phi}\sum_i\eta_i&\le\left(\max_{\Phi_1}\sqrt{\matel{\Phi_1}{\sum_iF_iF_i^\dagger}{\Phi_1}}\right)^2=\\
&=\max_{\Phi_1}\,\matel{\Phi_1}{\sum_iF_iF_i^\dagger}{\Phi_1}\label{eq:app4}\ .
\end{aligned}
\end{eqnarray}
All together, we arrive at
\begin{equation}
\label{tausep}
\mathcal{T}_{kn}\left(\sum_iF_i\varrho F_i^\dagger\right)\leq\mathcal{T}_{kn}(\varrho)\,\max_{\Phi_1}\left(\matel{\Phi_1}{\sum_iF_iF_i^\dagger}{\Phi_1}\right).
\end{equation}
The right hand side of \eref{tausep} involves the expression $\sum_iF_iF_i^\dagger$,
and conservation of trace implies $\sum_iF_i^\dagger F_i=\mathbbm{1}$.
If the condition $\sum_iF_iF_i^\dagger=\mathbbm{1}$ was satisfied, then the proof would be complete,
but, at this stage, we can only assert that
$\mathcal{T}_{kn}$ is non-increasing under any incoherent channel that satisfies $\sum_i [F_i,F^\dagger_i]=0$.
Starting from \eref{tausep} it takes only a few minor steps to finish the proof.
First of all, one may observe that $\sigma_0$ is normal, {\it i.e.} $[\sigma_0,\sigma_0^\dagger]=0$; this means that $\mathcal{T}_{kn}$ is non-increasing under dephasing processes. Incoherent hopping processes, on the other hand, are not normal, {\it i.e.} $\sigma_+^{(j)}\otimes\sigma_-^{(\ell)}$ does not commute with its adjoint
$(\sigma_+^{(j)}\otimes\sigma_-^{(\ell)})^\dagger=\sigma_-^{(j)}\otimes\sigma_+^{(\ell)}$,
and $\sum_i [F_i,F^\dagger_i]\neq 0$ for a general incoherent channel.
It is however possible to assert that $(\sigma_+^{(j)}\otimes\sigma_-^{(\ell)})\varrho(\sigma_-^{(j)}\otimes\sigma_+^{(\ell)})$ describes a situation of perfect localization on the state $\ket{\Psi_j}$,
because there is only a single excitation.
That is $\mathcal{\tau}^{\ket{\Phi}}_{kn}(F_i\varrho F_i^\dagger)\le 0$ for any product vector $\ket{\Phi}$ if $F_i$ contains a term $\sigma_+^{(j)}\otimes\sigma_-^{(\ell)}$.
Including all these points, one finally obtains
\numparts
\begin{eqnarray}
\mathcal{T}_{kn}(\varrho_F)&=&\mathcal{T}_{kn}(\sum_iF_i\varrho F_i^\dagger)\label{eq:proof1}\\
&\le&\mathcal{T}_{kn}\Bigl(\sum_{i|dp}F_i\varrho F_i^\dagger\Bigr)+\mathcal{T}_{kn}\Bigl(\sum_{i|h}F_i\varrho F_i^\dagger\Bigr)\label{eq:proof2}\\
&=&\mathcal{T}_{kn}\Bigl(\sum_{i|dp}F_i\varrho F_i^\dagger\Bigr).\label{eq:proof3}
\end{eqnarray}
\endnumparts
From \eref{eq:proof1} to \eref{eq:proof2} we divided the sum into the sum ($\sum_{i|dp}$) over Kraus operators that describe pure dephasing and a sum ($\sum_{i|h}$) that contains only Kraus operators that include hopping terms.
The inequality holds due to convexity of $\mathcal{T}_{kn}$ which is inherited from the convexity of $\tau^{\ket{\Phi}}_{kn}$.
Due to convexity also the second term in \eref{eq:proof2} vanishes, since $\mathcal{T}_{kn}(F_i\varrho F_i^\dagger)$ vanishes for any Kraus operator that includes hopping because $\mathcal{T}_{kn}$ is non-negative \footnote{The choice $\beta_i=0$ $\forall i$ yieds in fact a value of $\tau^\varphi_{kn}(\varrho)=0$ for any state $\varrho$ within the single-excitation subspace, so that $\max_{\varphi}\tau^\varphi_{kn}(\varrho)\geq 0$. }.
Using \eref{tausep}, we finally obtain
\begin{equation}
\mathcal{T}_{kn}(\varrho_F)\le\mathcal{T}_{kn}(\varrho)\max_{\Phi_1}\left(\matel{\Phi_1}{\sum_{i|dp}F_i F_i^\dagger}{\Phi_1}\right)\label{eq:proof4}.
\end{equation}
All Kraus operators in \eref{eq:proof4} commute with their adjoint, so that
$\sum_{i|dp}F_iF_i^\dagger=\sum_{i|dp}F_i^\dagger F_i$.
Conservation of trace implies
$\sum_{i|dp}F_i^\dagger F_i+\sum_{i|h}F_i^\dagger F_i=\mathbbm{1}$,
and since $\sum_{i|h}F_i^\dagger F_i$ is a positive operator,
the operator inequality $\sum_{i|dp}F_i^\dagger F_i\le\mathbbm{1}$ holds.
That is, no expectation value of $\sum_{i|dp}F_i^\dagger F_i$ exceeds the value of unity; we thus arrive at the desired result
\begin{equation}
\mathcal{T}_{kn}(\varrho_F)\le\mathcal{T}_{kn}(\varrho),
\end{equation}
which rigorously verifies that $\mathcal{T}_{kn}$ can never increase under incoherent dynamics.
\section{Illustration}
In order to illustrate the concept of incoherent channels and their distinction from coherent dynamics we discuss a few examples based on commonly employed models. The optimization required for the assessment of \eref{deftaumax} can be performed with standard routines, as included in current computer algebra packages. The linear scaling of the to-be-optimized parameters with the system size permits to treat considerable dimensions. Potential problems with local maxima can be avoided rather reliably by repeating the optimization routine with different initial conditions that are distributed over the whole parameter space.
The typical Hamiltonian for the description of coherent excitation transport reads
\begin{equation}
{\cal H}=\sum_{i\neq j}^n\lambda_{ij}(\sigma_+^{(i)}\otimes\sigma_-^{(j)}+\sigma_-^{(i)}\otimes\sigma_+^{(j)})\ ,
\end{equation}
where $\lambda_{ij}$ is the coupling strength between the $i$-th and $j$-th unit. Such a Hamiltonian models a system where the excitation travels due to coherent interaction. The propagator ${\cal U}(t)=e^{-i{\cal H}t}$ induced by this Hamiltonian conserves the number of excitations, but it is {\em not} of product form, {\it i.e. } not incoherent. This can be seen explicitly in the exemplary case of $n=2$, where it reads
\begin{equation}
\label{eq:ham}
{\cal U}(t)=\cos(\lambda_{12}t)\mathcal{P}_s+i\sin(\lambda_{12}t)(\sigma_+\otimes\sigma_- + \sigma_-\otimes\sigma_+),
\end{equation}
in terms of the projector $\mathcal{P}_s$ on the single excitation subspace $\mathcal{P}_s=\prj{0}\otimes\prj{1}+\prj{1}\otimes\prj{0}$ and the creation and annihilation operators $\sigma_+$ and $\sigma_-$. $\mathcal{H}$ thus induces a dynamics which may create or enhance coherent delocalization.
Such a coherent dynamics needs to be distinguished from incoherent hopping, which might for example be described by a Master equation $\frac{d}{dt}\varrho(t)=\mathcal{D}(\varrho(t))$ with
\begin{eqnarray}
\label{eq:lindblad}
\begin{aligned}
\mathcal{D}(\varrho(t))&=\sum_{i>j}^n\mathcal{D}_{ij}(\varrho(t))\\
\mathcal{D}_{ij}(\varrho(t))&=\sum_{k=1}^4\gamma_{ij}\left(G_k^{(ij)}\varrho(t) \,G_{k}^{\dagger(ij)}-\frac{1}{2}\{G_{k}^{\dagger(ij)} G^{(ij)}_{k},\varrho(t)\}\right),
\end{aligned}
\end{eqnarray}
where the Lindblad operators $G_k^{(ij)}$ are given by
\begin{eqnarray}
\begin{aligned}
G_{1}^{(ij)}&=\sigma^{(i)}_+\otimes\sigma^{(j)}_-, \qquad G_{2}^{(ij)}=\sigma^{(i)}_-\otimes\sigma_+^{(j)},\\
G_3^{(ij)}&=(P_1^{(i)}\otimes P_1^{(j)}-P_0^{(i)}\otimes P_0^{(j)})/\sqrt{2},\\
G_4^{(ij)}&=\sigma^{(i)}_z\otimes\sigma^{(j)}_z/(2\sqrt{2}).
\end{aligned}
\end{eqnarray}
The operators $P_{1(0)}^{(i)}$ are the projectors on the excited (ground) state of the $i$-th unit.
The solution to a master equation can be expressed in terms of a set of Kraus operators which define a quantum channel $\varrho(t)=\sum_i F_i(t)\varrho(0)F^\dagger_i(t)$, as discussed in section \ref{sec:incohproc}.
In the specific case given by \eref{eq:lindblad}, for $n=2$ these Kraus operators read (indices have been omitted, since there are only two units)
\begin{equation}
\begin{split}
\label{eq:kraus2n}
F_1&=\sqrt{\frac{1-\Gamma_t^2}{2}}\sigma_+\otimes\sigma_-,\\
F_3&=\sqrt{1-\Gamma_t}P_1\otimes P_1, \\
F_5&=\frac{1-\Gamma_t}{\sqrt{2}}P_1\otimes P_0,\\
F_7&=\sqrt{\Gamma_t}\mathbbm{1}\otimes\mathbbm{1},
\end{split}
\qquad
\begin{split}
F_2&=\sqrt{\frac{1-\Gamma_t^2}{2}}\sigma_-\otimes\sigma_+,\\
F_4&=\sqrt{1-\Gamma_t}P_0\otimes P_0,\\
F_6&=\frac{1-\Gamma_t}{\sqrt{2}}P_0\otimes P_1,\\
\end{split}
\end{equation}
with $\Gamma_t=\exp(-\gamma_{12} t)$. All the $F_i$ conserve the number of excitations and are local: they therefore describe a purely incoherent dynamics. Indeed, in the single excitation subspace, $F_1$ and $F_2$ are of the form of $B_{j\ell}$ as defined in \eref{eq:dephasing} and $F_3$ to $F_7$ are of the form $A_\ell$ as defined in \eref{eq:hopping}.
Having verified that \eref{eq:lindblad} induces incoherent dynamics for $n=2$ allows us to draw this conclusion also for $n>2$ with help of the Trotter expansion \cite{Trotter1959}: since the solution $\varrho(t)=e^{{\cal D}t}\varrho(0)$ can be expressed in terms of solutions of the two-site system via $e^{{\cal D}t}=\lim_{m\to\infty}\left(\Pi_{ij}e^{{\cal D}_{ij}t/m}\right)^m$,
the dynamics induced by \eref{eq:lindblad} can be given in terms of incoherent Kraus operators for any system size $n$. Consequently, coherent delocalization can not grow under this dynamics.
This is exemplified in figure~\ref{fig:incoherentcase}, which shows the behaviour of $\mathcal{T}_{kn}(\varrho)$ for $n=5$ and $k$ ranging from $2$ to $5$ for a time evolution induced by \eref{eq:lindblad}. Initially, the excitation is delocalized coherently but it is not maximally delocalized.
The dynamics then induces an increase of delocalization, which is indicated by the growth of the inverse participation ratio, defined as $\mathrm{IPR}(t)=1/\sum_iq^2_i(t)$, where the population of the $i$-th unit is given by $q_i(t)=\mathrm{Tr}(\varrho(t)\prj{\Psi_i})$ \cite{Thouless1974}. The behaviour of the IPR is shown in the inset of figure~\ref{fig:incoherentcase}.
Since the dynamics is incoherent, however, {\em coherent} delocalization may not increase as correctly identified by $\mathcal{T}_{kn}$. In the stationary state, that is reached for $t\to\infty$, the excitation is delocalized completely over the entire system but this delocalization is completely incoherent, so that $\mathcal{T}_{kn}$ vanishes for all $k$.
A qualitatively similar behavior can be observed in a system with a disordered Hamiltonian and local phase noise. For sufficiently large disorder, the eigenstates of the system Hamiltonian are strongly localized.
If the system is initially prepared in an eigenstate, the dephasing will result in a growing delocalization with an increasing IPR.
We found that the measures $\mathcal{T}_{kn}$, however, do not increase; that is they correctly assess the incoherent nature of this delocalization.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{incoherent_1.pdf}
\caption{
Coherent delocalization characterized by $\mathcal{T}_{kn}/w_k$ with the normalization constant $w_k=\mathcal{T}_{k5}(\ket{W})$, where $\ket{W}$ is the state with maximal coherent delocalization over 5 units. The time evolution for the initial state $\sqrt{\frac{1}{10}}(\ket{1}+\ket{5})+\sqrt{\frac{2}{10}}(\ket{2}+\ket{4})+\sqrt{\frac{4}{10}}\ket{3}$ for a system with five units is induced by the master equation defined by \eref{eq:lindblad}, with $\gamma_{ij}=\gamma$. As expected all $\mathcal{T}_{k5}$ decrease motononically despite the increase of delocalization as shown by the IPR in the inset.}
\label{fig:incoherentcase}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{interplay_2.pdf}
\caption{
Coherent delocalization characterized by $\mathcal{T}_{kn}/w_k$ for an evolution with coherent and incoherent aspects defined respectively by \eref{eq:ham} and \eref{eq:lindblad} (see the caption of figure \ref{fig:incoherentcase} for the definition of $w_k$). The strength of the coherent and incoherent interactions are given by $\lambda_{ij}=10\gamma$. Initially the excitation is perfectly localized. Due to the influence of the coherent interaction, coherent delocalization over up to four units is created until $\gamma t\approx 0.05$. For longer times, the noise reduces the coherent character of the delocalization, but generates incoherent delocalization as one can see from the behaviour of the IPR in the inset.}
\label{fig:inbetweencase}
\end{figure}
In general, however, a coherent interaction can also increase the coherent delocalization of an excitation.
To exemplify this we consider a Master equation where in addition to the incoherent dynamics described by ${\cal D}$ \eref{eq:lindblad}, there is a coherent term described by the Hamiltonian $\mathcal{H}$ defined in \eref{eq:ham} with $\lambda_{ij}=\lambda=10\gamma$; Figure ~\ref{fig:inbetweencase} depicts the behaviour of $\mathcal{T}_{kn}$ with $n=5$ after initialization with a perfectly localized excitation. Due to the coherent part of the dynamics, coherent delocalization initially grows. In the case of perfectly coherent dynamics one would observe strictly periodic motion, but, due to the additional incoherent character, one can observe an overall attenuation of $\mathcal{T}_{kn}$.
In particular, the incoherent contribution is so large that no coherent delocalization over five units can be verified.
Similarly to the completely incoherent dynamics above in figure~\ref{fig:incoherentcase}, also in this case the IPR asymptotically reaches its maximal value.
\section{Conclusions}
Recently, tremendous progress in both the experimental and the theoretical analysis of exciton transport in complex bio-molecular systems has been achieved.
On the experimental side, spectroscopy provides data with resolutions that were unthinkable a few years ago \cite{Collini2013}, while theoretical methods permit to simulate exciton dynamics very accurately despite firm environment coupling and strong non-Markovian dynamics \cite{Pachon2012}.
The improved available data, in turn, asks for reliable techniques that permit a rigorous interpretation.
For example, the question whether beating signals would permit to draw conclusion about quantum coherence has induced a very controversial debate so far \cite{Lambert2012}.
To some extent, this is due to the fact that we have a rather vague concept of quantum coherence, and a more solid theoretical footing is just about to be developed.
With the quantitative concept of quantum coherence presented here, we provide a rigorous basis for the analysis of coherence properties.
At this stage, the present tools are applicable rather to theoretical treatments that permit to construct the complete density matrix.
Since, however, only a limited number of density matrix elements are required to access $\tau_{kn}$, a direct experimental observation seems conceivable in {\it e.g.} artificially designed \cite{Collini2009} or, given further improvement of spectroscopic techniques, even in actual light harvesting complexes.
Financial support by the European Research Council under the project Odycquent is gratefully acknowledged.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,415 |
Lights Out is an American old-time radio program devoted mostly to horror and the supernatural.
Created by Wyllis Cooper and then eventually taken over by Arch Oboler, versions of Lights Out aired on different networks, at various times, from January 3, 1934 to the summer of 1947 and the series eventually made the transition to television. Lights Out was one of the earliest radio horror programs, predating Suspense and Inner Sanctum.
History
The Wyllis Cooper era
In the fall of 1933, NBC writer Wyllis Cooper conceived the idea of "a midnight mystery serial to catch the attention of the listeners at the witching hour." The idea was to offer listeners a dramatic program late at night, at a time when the competition was mostly airing music. At some point, the serial concept was dropped in favor of an anthology format emphasizing crime thrillers and the supernatural. The first series of shows (each 15 minutes long) ran on a local NBC station, WENR, at midnight Wednesdays, starting in January 1934. By April, the series proved successful enough to expand to a half-hour. In January 1935, the show was discontinued in order to ease Cooper's workload (he was then writing scripts for the network's prestigious Immortal Dramas program), but was brought back by huge popular demand a few weeks later. After a successful tryout in New York City, the series was picked up by NBC in April 1935 and broadcast nationally, usually late at night and always on Wednesdays. Cooper stayed on the program until June 1936, when another Chicago writer, Arch Oboler, took over. By the time Cooper left, the series had inspired about 600 fan clubs.
Cooper's run was characterized by grisly stories spiked with dark, tongue-in-cheek humor, a sort of radio Grand Guignol. A character might be buried, eaten, or skinned alive, vaporized in a ladle of white-hot steel, absorbed by a giant slurping amoeba, have his arm torn off by a robot, or forced to endure torture, beating or decapitation—always with the appropriate blood-curdling acting and sound effects.
Though there had been efforts at horror on radio previously (notably The Witch's Tale), there does not seem to have been anything quite as explicit or outrageous as this on a regular basis. When Lights Out switched to the national network, a decision was made to tone down the gore and emphasize tamer fantasy and ghost stories.
Only one recording survives from Cooper's 1934–1936 run, but his less gruesome scripts were occasionally rebroadcast. An interesting example is his "Three Men," which became the series' annual Christmas show (a 1937 version circulates among collectors under titles like "Uninhabited" or "Christmas Story"); it has a plot typical of Cooper's gentler fantasies. On the first Christmas after World War I, three Allied officers meet by chance in a train compartment and find one another vaguely familiar. They fall asleep and share a dream in which they are the Three Wise Men searching for Jesus. But is it really a dream? In the best tradition of supernatural twist endings, Cooper has the officers wake to find a strange odor in their compartment—which turns out to be myrrh and frankincense.
In the mid-1940s, Cooper's decade-old scripts were used for three brief summertime revivals of Lights Out. The surviving recordings reveal that Cooper was experimenting with both stream of consciousness and first-person narration a few years before these techniques were popularized in American radio drama by, among others, Arch Oboler and Orson Welles. In one tale (The Haunted Cell, original broadcast date unknown, rebroadcast 7/20/1946), a murderer describes how the Chicago police try to beat a confession out of him. When that doesn't work, they put him in a jail cell haunted by the ghost of a previous occupant, a smooth gangster named Skeeter Dempsey who describes his own execution and discusses the afterlife knowledgeably. In the final twist, the narrator reveals that he has taken Skeeter's advice to commit suicide and is now himself a ghost.
Another story, originally broadcast in March 1935 as "After Five O'Clock" and revived in 1945 as "Man in the Middle," allows us to follow the thoughts of a businessman as he spends a day at the office cheating on his wife with his secretary. The amusing contrast between what the protagonist thinks to himself and what he says out loud to the other characters enlivens one of Cooper's favorite plot devices, the love triangle.
One radio critic, in reviewing a March 1935 episode that used multiple first-person narrators, said:
Other Cooper scripts are more routine, perhaps in part because the author's attention was divided by other projects. From the summer of 1933 until August 1935, Cooper was NBC Chicago's continuity chief, supervising a staff of writers and editing their scripts. He resigned in order to devote more time to Lights Out as well as a daily aviation adventure serial, Flying Time. At various times, he also served on NBC's Program Planning Board, wrote the soap opera Betty and Bob, and commuted weekly to produce another program in Des Moines, Iowa.
From early 1934 to mid 1936, Cooper produced close to 120 scripts for Lights Out. Some episode titles (all from 1935) include "The Mine of Lost Skulls," "Sepulzeda's Revenge," "Three Lights From a Match," "Play Without a Name," and "Lost in the Catacombs" (about a honeymoon couple in Rome who lose their way in the catacombs under the city). Typical plots included:
A novelist, struggling to write a locked room mystery, locks himself in his office only to be interrupted by a stranger who resembles the story's murderer.
A killer named "Nails" Malone has "a conference with his conscience" about the murders he's committed.
A scientist accidentally creates a giant amoeba that grows rapidly, eats living things (like the lab assistant's cat), and exhibits powers of mind control.
The show benefited tremendously from Chicago's considerable pool of creative talent. The city was, like New York, one of the main centers of radio production in 1930s America. Among the actors who participated regularly during the Cooper era were Sidney Ellstrom, Art Jacobson, Don Briggs, Bernardine Flynn, Betty Lou Gerson, and Betty Winkler. The sound effects technicians frequently had to perform numerous experiments to achieve the desired noises. Cooper once had them build a gallows and wasn't satisfied until one of the sound men personally dropped through the trap. The series had little music scoring save for the thirteen chime notes that opened the program (after a deep voice intoned, "Lights out, everybody!") and an ominous gong which was used to punctuate a scene and provide the transition to another.
A veteran radio dramatist, Ferrin Fraser, wrote some of the scripts.
The Arch Oboler era
When Cooper departed, his replacement—a young, eccentric and ambitious Arch Oboler—picked up where he left off, often following Cooper's general example but investing the scripts with his own concerns. Oboler made imaginative use of stream-of-consciousness narration and sometimes introduced social and political themes that reflected his commitment to antifascist liberalism.
Although in later years Lights Out would be closely associated with Oboler, he was always quick to credit Cooper as the series' creator and spoke highly of the older author, calling him "the unsung pioneer of radio dramatic techniques" and the first person Oboler knew of who understood that radio drama could be an art form.
In June 1936, Oboler's first script for Lights Out was "Burial Service," about a paralyzed girl who is buried alive. NBC was flooded with outraged letters in response. His next story, one of his most popular efforts, was the frequently repeated "Catwife," about the desperate husband of a woman who turns into a giant feline. He followed with "The Dictator," about Roman emperor Caligula. This set the pattern for Oboler's run: For every two horror episodes, he said later, he would try to write one drama on subjects that were ostensibly more serious, usually moral, social, and political issues.
Like Cooper, Oboler was much in demand and highly prolific. While working on Lights Out, he wrote numerous dramatic sketches for variety shows (The Chase and Sanborn Hour, Rudy Vallee's programs), anthologies (Grand Hotel, The First Nighter Program, The Irene Rich Show) and specials. In August 1936, singer Vallee, then the dean of variety show hosts, claimed that Lights Out was his favorite series. Oboler occasionally redrafted his Lights Out scripts for use on Vallee's and other variety hours. A version of Oboler's "Prelude to Murder" starring Peter Lorre and Olivia de Havilland was scheduled to air on a November 1936 Vallee broadcast. Other Lights Out plays that turned up on various late 1930s variety programs included "Danse Macabre" (with Boris Karloff), "Alter Ego" (with Bette Davis) and "The Harp."
Oboler met the demand by adopting an unusual scripting procedure: He would lie in bed at night, smoke cigarettes, and improvise into a Dictaphone, acting out every line of the play. In this way, he was able to complete a script quickly, sometimes in as little as 30 minutes, though he might take as long as three or four hours. In the morning, a stenographer would type up the recording for Oboler's revisions. Years later, Rod Serling, who counted radio fantasists like Cooper, Oboler, and Norman Corwin among his inspirations, would use a similar process to churn out his many teleplays for The Twilight Zone, a series that in many respects was to television what Lights Out was to radio.
Despite acclaim for Oboler's dramas, NBC announced it was canceling the series in the summer of 1937—"just to see whether listeners are still faithful to it," according to one press report but also, it seems, to allow the hard-working author a vacation. Another outcry from fans led to the program's return that September for another season.
In the spring of 1938, the series earned a good deal of publicity for its fourth anniversary as a half-hour show when actor Boris Karloff, the star of many a Hollywood horror film, traveled to Chicago to appear in five consecutive episodes. Among his roles: an accused murderer haunted by an unearthly woman-like demonic creature (played by Templeton Fox) urging him to "kill...kill...kill" in "The Dream"; the desperate husband in a rebroadcast of "Catwife"; and a mad, violin-playing hermit who imprisons a pair of women, threatening to murder one and marry the other, in "Valse Triste."
Oboler left in the summer of 1938 to pursue other projects, writing and directing several critically acclaimed dramatic anthology series: Arch Oboler's Plays, Everyman's Theatre, and Plays for Americans. A variety of NBC staff writers and freelancers filled in until Lights Out was canceled in 1939. NBC Chicago continuity editor Ken Robinson supervised some of the writing. Regular contributors included William Fifield and Hobart Donovan. A recording of the fifth anniversary show survives from this season. Donovan's "The Devil's Due," about criminals haunted by a mysterious stranger, is in keeping with the formula laid down by Cooper.
In 1942, Oboler, needing money, revived the series for a year on CBS. Airing in prime time instead of late at night, the program was sponsored by the makers of Ironized Yeast. Most of the Lights Out recordings that exist today come from this version of the show. For this revival, each episode began with an ominously tolling bell over which Oboler read the cryptic tagline: "It...is...later...than...you...think." This was followed by a dour "warning" to listeners to turn off their radios if they felt their constitutions were too delicate to handle the frightening tale that was about to unfold.
Directing and hosting the 1942–43 broadcasts, mostly from New York and Hollywood, Oboler not only reused scripts from his 1936–38 run but also revived some of the more fantasy-oriented plays from his other, more recent anthology series. Some episodes had originally aired on the author's groundbreaking, critically acclaimed 1939–1940 program Arch Oboler's Plays, among them:
"The Ugliest Man in the World," a sentimental tale of a hideously deformed man seeking love in a cruel world, inspired by Boris Karloff's typecasting in horror roles.
"Bathysphere," a political thriller about a scientist and a dictator sharing a deep sea diving bell.
"Visitor from Hades," about a bickering married couple trapped in their apartment by a doppelgänger.
Another unusual script, "Execution," about a mysterious French woman who bedevils the Nazis trying to hang her, had previously aired on Oboler's wartime propaganda series Plays for Americans.
Like Cooper, Oboler made effective use of atmospheric sound effects, perhaps most memorably in his legendary "Chicken Heart," a script that debuted in 1937 and was rebroadcast in 1938 and 1942. It features the simple but effective "thump-thump" of an ever-growing, ever-beating chicken heart which, thanks to a scientific experiment gone wrong, threatens to engulf the entire world. Although the story bears similarities to an earlier Cooper episode (about an ever-growing amoeba that makes an ominous "slurp! slurp!" sound), Oboler's unique choice of monster was inspired by a Chicago Tribune article announcing that scientists had succeeded in keeping a chicken heart alive for a considerable period of time after its having been removed from the chicken. Recordings of the original radio broadcasts are lost or unavailable, although Oboler later recreated this episode for a record album in 1962. Part of the episode's notoriety stems from a popular standup routine by comedian Bill Cosby (on his 1966 album Wonderfulness), an account of his staying up late as a child to listen to scary radio shows against his parents' wishes and being terrified by the chicken heart. Cosby also referenced the episode in a camping episode of Fat Albert and the Cosby Kids.
Other well-remembered Oboler tales, many of them written in the 1930s and rebroadcast in the '40s, include:
"Come to the Bank," in which a man learns to walk through walls but gets stuck when he tries to rob a vault.
"Oxychloride X," about a chemist who invents a substance that can eat through anything.
"Murder Castle," based on the real-life case of H. H. Holmes, Chicago's notorious serial killer.
"Profits Unlimited," a still-relevant allegory on the promises and dangers of capitalism.
"Spider," in which two men attempt to capture a giant arachnid.
"The Flame," a weird exercise in supernatural pyromania.
"Sub-Basement," which finds yet another husband and wife in peril—this time trapped far beneath a department store in the subterranean railway of the Chicago Tunnel Company.
Lights Out often featured metafictional humor. Perhaps inspired by Cooper's "The Coffin in Studio B," in which actors rehearsing an episode of Lights Out are interrupted by a mysterious coffin salesman peddling his wares, Oboler wrote stories like "Murder in the Script Department," in which two Lights Out script typists become trapped in their building after hours as frightening, unexplained events occur. In "The Author and the Thing," Oboler even plays himself pitted against one of his own monstrous creations.
After the 1942–43 Lights Out, Oboler continued to work in radio (Everything for the Boys and revivals of Arch Oboler's Plays) and pursued a second career in filmmaking, first in the Hollywood mainstream and then as an independent producer, writing and directing a number of offbeat, low-budget films, including Five, about survivors of a nuclear war, The Twonky, a satire of television, and the 3-D film Bwana Devil, which made a huge profit on a small investment. He dabbled in live television (a six-episode 1949 anthology series, Arch Oboler Comedy Theater), playwriting (Night of the Auk), and fiction (House on Fire). In 1962, he produced a spoken-word album entitled Drop Dead! (Capitol T/ST-1763), which recreated abbreviated versions of his Lights Out thrillers, including "Chicken Heart" and "The Dark," about a mysterious creeping mist that turns people inside-out. In 1971–1972, Oboler produced a syndicated radio series, The Devil and Mr. O (he liked for people to call him "Mr. O"), which featured vintage recordings from Lights Out and his other series with newly recorded introductions by Mr. O himself.
Later revivals
The success of Oboler's 1942–1943 Lights Out revival was part of a trend in 1940s American radio toward more horror. Genre series like Inner Sanctum, Suspense and others drew increasingly large ratings. Perhaps with this in mind, NBC broadcast another Lights Out revival series from New York in the summer of 1945, using seven of Wyllis Cooper's original 1930s scripts. Like Oboler's, this revival aired in the early evening and not late at night, and because of this, it was reported, "only those Cooper scripts which stressed fantasy rather than horror" were broadcast. These included a bloodless ghost story about a man who accidentally condemns his dead wife to haunt a nearby cemetery and "The Rocket Ship", science fiction involving interstellar travel. Cooper, then an advertising executive at New York's Compton Agency, may have had little or nothing to do with the actual broadcasts other than allowing his scripts to be performed.
This was followed by an eight-episode revival in the summer of 1946, from NBC Chicago, although at least one of the scripts is not by Cooper (an adaptation of Charles Dickens' "The Signal-Man"). This series also avoided the use of outright gore. In fact, a review in Variety complained that the premiere episode, "The Seven Plovers", was "a little too serious in content for a thriller" since it included "religious background, philosophical discussion and dream diagnosis ..."
A third series of eight vintage Cooper scripts was scheduled to run in the summer of 1947 as well. Broadcast from Hollywood over ABC Radio, it starred Boris Karloff and was sponsored by Eversharp, whose company president canceled the series after the third episode, apparently unhappy with the gruesome subject matter. The premiere, "Death Robbery", featured Karloff as a scientist who brings his wife back from the dead, only to find she's become a gibbering homicidal maniac. An uncredited Lurene Tuttle plays the wife. This episode is one of the few surviving examples of Cooper's Lights Out work that reflects the sort of explicit horror that characterized the original series. Eversharp paid off Cooper for his five unused scripts and Lights Out ended its long run on network radio.
From 1936 to 1939, Cooper pursued a screenwriting career in Hollywood (his major credits are the screenplay for Universal's 1939 Son of Frankenstein and contributions to the Mr. Moto mystery series starring Peter Lorre) but continued to work in radio, advertising and, later, television. By 1940, he had changed the spelling of his name from "Willis" to "Wyllis" (to satisfy "his wife's numerological inclinations") and lived mainly in the New York City area where he worked on a number of radio programs, the most important of which was probably Edward M. Kirby's popular and acclaimed government propaganda series, The Army Hour, which Cooper wrote, produced and directed for its first year.
In 1947, Cooper created Quiet, Please, another radio program dealing with the supernatural, which he wrote and directed until 1949, occasionally borrowing ideas from his Lights Out stories while creating wholly new scripts that were often more sophisticated than his 1930s originals. In 1949 and 1950, he produced (and contributed scripts to) three live TV series that frequently dealt with the supernatural: Volume One, Escape and Stage 13.
Television
In 1946, NBC Television brought Lights Out to TV in a series of four specials, broadcast live and produced by Fred Coe, who also contributed three of the scripts. NBC asked Cooper to write the script for the premiere, "First Person Singular", which is told entirely from the point of view of an unseen murderer who kills his obnoxious wife and winds up being executed. Variety gave this first episode a rave review ("undoubtedly one of the best dramatic shows yet seen on a television screen"), but Lights Out did not become a regular NBC-TV series until 1949.
Coe initially produced this second series but, for much of its run, the live 1949–1952 program was sponsored by appliance maker Admiral, produced by Herbert Bayard Swope, Jr., directed by Laurence Schwab, Jr., and hosted by Frank Gallop. Critical response was mixed but the program was successful for several seasons (sometimes appearing in the weekly lists of the ten most watched network shows) until competition from the massively popular sitcom I Love Lucy on CBS helped to kill it off.
The 1949–1952 series featured scripts by a variety of authors, including a young Ira Levin. In 1951, producer Swope even bought a few stories from Cooper and Oboler. "Dead Man's Coat," starring Basil Rathbone, was adapted from one of Cooper's 1930s plays (and not to be confused with his Quiet, Please episode "Wear the Dead Man's Coat" with which it shares a similar premise). Oboler's "And Adam Begot," adapted by Ernest Kinoy from a radio play, starred Kent Smith. Among the young actors employed was Leslie Nielsen, who appeared in several episodes including "The Lost Will of Dr. Rant," based on "The Tractate Middoth", an M. R. James story. These and many others are available on DVD.
Other notable guest stars included Anne Bancroft, Grace Kelly, Eva Marie Saint, Lee J. Cobb, Anthony Quinn, Jessica Tandy, Veronica Lake, John Forsythe, Boris Karloff, Beatrice Straight, Eli Wallach, Vincent Price, Jane Wyatt, and Jack Palance.
Notable directors included Delbert Mann and Fred Coe.
Episodes
Later versions
In 1972, NBC aired yet another TV incarnation of Lights Out, a TV movie pilot which was not well received. In fact, Oboler (who was then syndicating his The Devil and Mr. O radio show) announced publicly that he had nothing to do with it.
Despite its modest television success, radio historian John Dunning suggested that the legend of Lights Out is firmly rooted in radio.
Influence
"Chicken Heart," a routine from comedian Bill Cosby's album Wonderfulness, includes a (not entirely faithful) retelling of the Lights Out episode with the same title. As a result, many believe the story originated with Cosby.
"What the Devil" (1942), about two motorists menaced by a truck whose driver they cannot see, was similar in plot to Steven Spielberg's TV movie Duel, adapted by Richard Matheson from his own short story. Oboler, feeling his copyright had been infringed, claimed in an interview that he "reached for a lawyer and got paid off by Universal Studios."
The Lights Out television episode "The Martian Eyes" starred Burgess Meredith as a man whose glasses enable him to see Martian invaders who have disguised themselves as normal people. A similar premise in John Carpenter's 1988 film They Live was adapted from the story by Ray Nelson, who reworked the idea from his friend Philip K. Dick's never-produced film treatment for an episode of The Invaders TV series.
The Simpsons annual Halloween episode "Treehouse of Horror V" referenced Oboler's "The Dark" about a mysterious fog that turns people inside-out. In the episode, the Simpsons and Groundskeeper Willie turn inside out, and then break into a song and dance number. No recordings of the original broadcasts of "The Dark" have survived, but Oboler recorded a memorable remake for his 1962 stereo album Drop Dead!
Wally Phillips, former morning personality for Chicago's WGN radio, used to play "The Dark" every year on Halloween.
Jonathan Sims cites Lights Out as an influence for The Magnus Archives.
The NoSleep Podcast has adapted the stories "Death Robbery", "Murder Castle", and "Ghost Party" for episodes of their "Old Time Radio" specials.
Lights Out Podcast is a spiritual successor that shares the same dark and horror elements.
See also
List of Lights Out episodes
References
Listen to
Theater of the Ears: Lights Out
Lights Out, radio series; 86 episodes available for download at the Internet Archive
External links
Lights Out, Everybody!
Lights Out Radio Log
Wyllis Cooper Chronology
25 Television episodes available for streaming or download Internet archive retrieved 2010 October 23
Selection of Lights Out radio scripts at Generic Radio Workshop
OTR Plot Spot: Lights Out – plot summaries and reviews.
Lights Out at CVTA with episode list
1934 radio programme debuts
1947 radio programme endings
1950s American anthology television series
1930s American radio programs
1940s American radio programs
1946 American television series debuts
1949 American television series debuts
1952 American television series endings
1940s American horror television series
1950s American horror television series
ABC radio programs
American radio dramas
Anthology radio series
Black-and-white American television shows
CBS Radio programs
Fantasy radio programs
Horror fiction radio programmes
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"redpajama_set_name": "RedPajamaWikipedia"
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: Shell merge two file I have two files and I want to join file1 and file2 by matching second and first fields in file1 and file2 and write first data from file1 to file2
file1:
5439725407 uQkiJRPOZLLJkc
5368657511 eWGDnOcNgxjBK
5322202068 dNsUkWOMk9lNJ
file2:
uQkiJRPOZLLJkc,00087b8dbe6fdc3a5725a0a77fa4e37f3db10440d8b0da2d3935cb0f8f4f9089,1
eWGDnOcNgxjBK,0008958b743f8b786fa7f080348f3180f8e410890a07995878c1fcbda66706b4,1
dNsUkWOMk9lNJ,0008bc14ecce6150bef44f657e12314b8b2c37a5730bf88fc81b66d5f77ed8be,1
outputfile:
uQkiJRPOZLLJkc,00087b8dbe6fdc3a5725a0a77fa4e37f3db10440d8b0da2d3935cb0f8f4f9089,1,5439725407
eWGDnOcNgxjBK,0008958b743f8b786fa7f080348f3180f8e410890a07995878c1fcbda66706b4,1,5368657511
dNsUkWOMk9lNJ,0008bc14ecce6150bef44f657e12314b8b2c37a5730bf88fc81b66d5f77ed8be,1,5322202068
A: Following awk solution may also help you.
awk 'FNR==NR{a[$1]=$0;next} ($2 in a){print a[$2] "," $1}' FS="," filE2 FS=" " filE1
Output will be as follows.
uQkiJRPOZLLJkc,00087b8dbe6fdc3a5725a0a77fa4e37f3db10440d8b0da2d3935cb0f8f4f9089,1,5439725407
eWGDnOcNgxjBK,0008958b743f8b786fa7f080348f3180f8e410890a07995878c1fcbda66706b4,1,5368657511
dNsUkWOMk9lNJ,0008bc14ecce6150bef44f657e12314b8b2c37a5730bf88fc81b66d5f77ed8be,1,5322202068
EDIT: Adding explanation with non-one liner form of solution too here now.
awk '
FNR==NR{ ##Checking condition here FNR==NR, which will be TRUE when first Input_file will be read.
a[$1]=$0; ##Creating an array named a whose index is first field of current line and value is current line.
next ##next statement will skip all further statements.
} ##Following block will be executed when 2nd Input_file is being read.
($2 in a){ ##checking if 2nd field of current line is present in array a, if yes then do following.
print a[$2] "," $1 ##Printing the value of array a whose index is $2 of current line, printing comma and then printing first field of current line.
}
' FS="," filE2 FS=" " filE1 ##Setting field separator as comma for Input_file2 and setting field separator as space for Input_file1 here.
A: awk solution:
awk 'NR==FNR{a[$2]=$1;next}{ print $1,$2,$3,($1 in a? a[$1]:"") }' file1 FS=',' OFS=',' file2
The output:
uQkiJRPOZLLJkc,00087b8dbe6fdc3a5725a0a77fa4e37f3db10440d8b0da2d3935cb0f8f4f9089,1,5439725407
eWGDnOcNgxjBK,0008958b743f8b786fa7f080348f3180f8e410890a07995878c1fcbda66706b4,1,5368657511
dNsUkWOMk9lNJ,0008bc14ecce6150bef44f657e12314b8b2c37a5730bf88fc81b66d5f77ed8be,1,5322202068
A: I think this is what you want (your fileX format is a little off I think)
cat file2 | tr \, ' ' | join -1 1 -2 2 - file1 | tr ' ' \,
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,299 |
Neochauliodes jiangxiensis är en insektsart som beskrevs av C.-k. Yang och Ding Yang 1992. Neochauliodes jiangxiensis ingår i släktet Neochauliodes och familjen Corydalidae. Inga underarter finns listade i Catalogue of Life.
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"redpajama_set_name": "RedPajamaWikipedia"
} | 1,479 |
\section{Introduction}
There is a strong relationship between individual psychology and organizational performance. Leaders respond to organizational developments and challenges with partial reliance on their own knowledge, past experiences, and most importantly, their personalities \cite{Hambrick2007UpperUpdate}. \citeA{Berson2014LeadingMotivation} identified that CEO's personal values are strong predictors of organizational culture and performance. It could be explained by the attraction-selection-attrition perspective \cite{Schneider1995TheUpdate}, from which a leader is able to attract certain types of followers per his/her own interesting personality, and the followers are more open to express as well as adopt new ideas \cite{Oreg2011LeadershipStyle}.
Unethical decision makers are referred to as corporate psychopaths \cite{Stevens2012SuccessfulWhy} although machiavellians and narcissists are also included \cite{Furnham2013TheReview.}. Their manipulative nature and charming appearances may get them through job interviews, to high ranks with broad influences. The negative consequences of having them in charge may include: radical organizational changes leading to instability \cite{Chatterjee2007ItsPerformance.}; higher psychological exhaustion and lower job satisfaction scores \cite{Volmer2017TheCorrigendum.}; employees' higher moral disengagement \cite{Christian2014TheWork} and lesser appetite to take smart risks \cite{Liu2016AbusiveIdentification.}. On a global scale, corporate psychopaths could be a reason behind the recent global financial crisis \cite{Boddy2011TheCrisis} as they are more willing to risk other people's money for their own personal gains \cite{Jones2014RiskTriad.}, and in some cases, commit frauds \cite{Perri2013VisionariesProphets}.
Steve Jobs was a corporate psychopath, and he was fired from the company he founded \cite{Isaacson2013SteveBiography}. However, John Sculley - the Apple CEO who fired Steve Jobs - admitted that the firing was his own biggest mistake \cite{NBC2011JohnJobs}. Indeed, most of Steve Jobs' psychopathic behaviors were permanent, and the only difference between his firing and his then legendary comeback was the deeper awareness of both organizational and inter-personal issues, consequently leading to better informed behaviors and team dynamics. The firing's role as a necessary shock for better changes in both Steve Jobs and Apple is debatable. On one hand, it could be the only way to force Steve Jobs to realize the negative consequences of some of his behaviors. On the other hand, it could be avoided, and the amazing Apple products could have arrived several years sooner. Ultimately, reliable data and well constructed evidences would be needed in order to settle a debate like this one.
Therefore, this paper proposes a system for large-scale, constant mining of psychological artifacts from texts, leveraging evidence-based clinical psychology knowledge, natural language processing (NLP), machine learning (ML), and ontology-based reasoning. First, a manual psychoanalysis of Steve Jobs' bibliography demonstrates the process' common steps. Second, the paper presents a high-level system architecture, in which manual steps were conceptualized into stacks, their components and corresponding relationships. Mechanisms crucial to the system's core capabilities will be described. Third, development details involving related works, technology stacks, and development practices will be recommended. Finally, the paper discusses further integration of the system in optimizing business processes with graph science and model querying, and in understanding bad cyber actors. Other issues such as legality and privacy will also be discussed.
\section{The Case of Steve Jobs}
Diagnosed with cancer, Steve Jobs allowed Isaacson complete freedom in aggregating information from both Steve's friends and foes. "Steve Jobs: The Exclusive Biography" \cite{Isaacson2013SteveBiography} was released in 2012, offering insights to both amazing and terrible events in Steve Jobs' life. From this biography, the paper diagnosed Steve Jobs with Obsessive Compulsive Personality Disorder (301.4) and Narcissistic Personality Disorder (301.81). Figure 1 and 2 respectively represent the DSM 5 \cite{APA2014DiagnosticDSM5} dimensional scores. This manual process can be summarized as followed. First, a psychologist will carefully read the written texts. Second, relevant details will be extracted, grouped, and assigned an observation codes (o.code). Third, o.codes will be matched with DSM 5 diagnostic criteria. An o.code may score in different diagnostic criteria, across different disorders. A dimensional score is the sum of all the matched o.code(s) for that particular dimension. Finally, conclusion(s) will be drawn from the dimensional scores, following previous evidence-based research results such as the DSM 5. This process is based on forensic psychoanalysis \cite{Chadda2013ForensicPsychiatry}, of which results are acceptable within the court of laws. Further details will be explained below.
\begin{figure}[t]
\centering
\begin{minipage} {0.45\textwidth}
\centering
\includegraphics[width=1\linewidth]{OCPD.png}
\caption{Obsessive Compulsive Personality Disorder Profile}
\end{minipage}
\begin{minipage} {0.45\textwidth}
\centering
\includegraphics[width=1\linewidth]{NPD.png}
\caption{Narcissistic Personality Disorder Profile}
\end{minipage}%
\end{figure}
There are at least 35 evidences, grouped into 7 groups. Each evidence inside a group will have an observation code in the format of the group number to be followed by the observation number. Details regarding group O2 and O4 will not be listed here, as they are not relevant to the diagnosis presented in Table 1.
\subsubsection{O1 - Steve had a strong desire for complete control} Del Yocam believes that it has something with Steve's at-birth abandonment (O1-1) \cite{Isaacson2013SteveBiography}. Andy Hertzfeld - a close friend of Steve - reported that Steve saw products and environments as extensions of himself which he must gain complete control (O1-2), yet he could not control himself from being cruel and harmful to other people (O2-1) \cite{Isaacson2013SteveBiography}. Steve believed users should not be able to extend Apple I, even for their own use (O1-3). Steve can completely shut unwanted interference out, even with sensitive personal matters such as the Brennan's pregnancy (O1-4) \cite{PsychiatricTrimes2014TheGeniuses}. Steve wanted Apple's computer case can only be opened by Apple's special tools (O1-5).
\subsubsection{O3 - Steve believed he is special and was willing prove it} He was eager to do all the extra problems that his math teacher gave him (O3-1). In class, he would sat in a corner, doing his own things (O3-2). In college, he was bored of required classes, dropped out, and audited the classes he linked (O3-3). At Apple, he insisted to be placed ahead of Woz and refused to be employee number 2. He later claimed number zero for his company badge even though he was still number 2 as payroll system started with number 1 (O3-3). Steve preferred people jump when he said "jump" (O3-4) \cite{Isaacson2013SteveBiography}. Hertzfeld recalled that Steve did believe he was the chosen one, on par with Einstein and Gandhi (O3-4) \cite{Isaacson2013SteveBiography}. When negotiating with Microsoft for the first time, Steve behaved in a way as if he was allowing Bill to be a part of the collaboration (O3-5). Redse - the girlfriend who Steve loved the most - acknowledged that Steve had been very self centered, and that it was his job to teach people aesthetics (O3-6) \cite{Isaacson2013SteveBiography}.
\subsubsection{O5 - Steve uses drugs} Steve was known to use marijuana, and LSD (O5-1) \cite{Isaacson2013SteveBiography}. He regarded LSD as an important part of his life as it helped him discover the other side of his consciousness during meditation sessions (O5-2).
\subsubsection{O6 - Steve can go to the extreme in the ways he does things} Friedland reported that Steve can carry things to the extreme with high intensity at times (O6-1). When choosing the color for the Apple computer plastic case, none of Pantone's two thousand shades of beige was good enough for Steve (O6-2) \cite{Isaacson2013SteveBiography}. Steve decided to go with almost no furniture in his house because no furniture fit his high standards (O6-3). Atkinson recalled that Steve cannot make trade-off (O6-4). Steve was not able to tolerate a tiny mold line in the chassis of his NEXT computer and forced the manufacturer to buy a \$150,000 sand machine to solve the problem (O6-5). When building NEXT factory, Steve demanded the walls to be painted white, certain robots to be painted with certain colors, and putting \$20,000 leather chairs in (O6-6).
\subsubsection{O7 - Steve carried some stubborn and/or weird beliefs} He believed that his strict vegetarian diet will not cause body odor which is not true, and he would not use deodorant nor shower regularly (O7-1) \cite{Isaacson2013SteveBiography}. Even when paternity test confirmed he was the father of Lisa, he kept telling others that he was not (07-2). Bill Atkinson said Steve had the ability to deceive even himself (O7-3) \cite{Isaacson2013SteveBiography}. Steve has a binary world view - everything is either "excellent" or "total shit" to him (O7-4).
\begin{table}[t]
\centering
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
\textbf{O codes} & \textbf{301.4 - 1} & \textbf{301.4 - 2} & \textbf{301.4 - 3} & \textbf{301.4 - 4} & \textbf{301.4 - 5} & \textbf{301.4 - 6} & \textbf{301.4 - 7} & \textbf{301.4 - 8} \\ \hline
O1-1 & & & & & & 1 & & \\ \hline
O1-2 & & & & & & 1 & & \\ \hline
O1-3 & & & & 1 & & & & 1 \\ \hline
O1-4 & & & 1 & & & & & \\ \hline
O1-5 & & & & 1 & & 1 & & \\ \hline
O3-2 & & & & & & & & 1 \\ \hline
O3-3 & & & & & & & & 1 \\ \hline
O3-3 & 1 & & & & & & & 1 \\ \hline
O3-6 & & & & 1 & & & & \\ \hline
O5-2 & & & & 1 & & & & \\ \hline
O6-1 & & 1 & 1 & & & & & \\ \hline
O6-2 & 1 & 1 & & & & & & 1 \\ \hline
O6-3 & 1 & 1 & & 1 & & & & \\ \hline
O6-4 & & 1 & & & & 1 & & 1 \\ \hline
O6-5 & & 1 & & 1 & & & & \\ \hline
O6-6 & 1 & & & 1 & & 1 & & 1 \\ \hline
O7-1 & & & & 1 & & & & 1 \\ \hline
O7-2 & & & & 1 & & & & 1 \\ \hline
O7-3 & & & & 1 & & & & \\ \hline
O7-4 & & & & 1 & & & & 1 \\ \hline
\end{tabular}
}
\caption{Diagnosis of Obsessive Compulsive Personality Disorder}
\label{tab:OCPD}
\end{table}
Table 1 illustrates the mapping of o.codes to respective DSM-5 diagnostic criteria \cite{APA2014DiagnosticDSM5} for Obsessive Compulsive Personality Disorder (OCPD). This diagnosis is the key in opening up pathways for figuring out the strengths and weaknesses of an OCPD person, and how to translate those findings into better team performance. For example, research has shown that OCPD people have enhanced visual performance \cite{Ansari2016EnhancedDisorder}, and difficulties with interpersonal functioning \cite{Cain2015InterpersonalDisorder}. A better understanding of issues will lead to better identification of evidence-based interventions such as cognitive behavioral therapy, psychodynamic therapy, dialectical behavior therapy, group-based therapy, and so on \cite{Alex2010PsychologicalDisorder}.
\section{The Machine-assisted Psychoanalysis System}
The Machine-assisted Psycho-Analysis System (MPAS) puts the above-mentioned process into automation, aiming at continuous mining of psychological artifacts that can be further analyzed for better situational awareness (short term) and strategical plannings (long term). At the highest level, MPAS consists of a Note stack, a Card stack and the Interface stack as shown in Figure 3.
\begin{figure}[t]
\centering
\includegraphics[width=5.5in]{MPAS.png}
\caption{MPAS architecture}
\end{figure}
\subsubsection{The Note stack} It is at the lowest level, responsible for creating "Notes" which are the observations in the manual psychoanalysis of Steve Jobs. At the beginning, the Knowledge Explorer component automatically crawls identified data sources, which may include straight-forward text contents (emails, forum posts, survey feedback), documents in certain formats (MS Word, Adobe PDF), packaged files (zip, rar), and other less common data sources (program logs, speech-to-text, etc). Hence, the crawler must execute pre-processing tasks such as extracting raw texts from those data sources, and assigning meta-data. Time, place, main subjects, and links to original sources are crucial meta-data. The Text DB stores all of those data and meta-data for further processing by the NLP component.
The Natural Language Processing (NLP) component has within itself an ontology, a set of dictionaries, and a machine-learning (ML) model. The ontology describes the Entities and their Relationships within a certain knowledge domain. For example, the "Alcohol consumption" knowledge domain may include main entities of: drinkers, liquor, wine, beer, amounts; and main relationships of: consume, purchase, and discuss. The dictionaries document the entities and relationships' polymorphism. Different liquor brand-names are polymorphisms of the entity "liquor", as "drink", "booze up", "bottom up" are those of the "consume" relationship. Based on previous training, the ML model will annotate raw chunks of texts with recognized entities and their relationships. Of course, the NLP will also look for further details regarding subjects, time and place. The Organizer will then organize chunks of texts by main subjects, associated place and time. Alignment of variables such as Time can be crucial in understanding certain behaviors such as alcohol addiction, and recognizing duplicated contents. The Organizer operates on its own discrete database.
The Predictor is another trained ML model responsible for making "Notes", each of which summarizes several chunks of annotated texts. For example, in making a note about a person A (subject), the Predictor does not see all the human texts but only marked entities and their relationships. If it sees A consumes alcohol several days a week for several weeks then it may produce a note such as "Person: A; Action: consumes alcohol; Intensity: very frequent; Confidence: high; time and place information; tracking information". We may consider the entities and relationships as inputs while a note is the output, and the ML model can be trained on pre-identified input-output pairs (ground truths). All notes will be saved into the Note database (Note DB).
Notes can be really noisy due to several reasons. The ML models are not perfect and will make wrong prediction and/or overlook data. This is especially true when human writings are unpredictable and imperfect. People sometimes lie or exaggerate. Synchronizing data is another big challenge as there must be a reasonable window for the Organizer to hand off data to the Predictor. Some facts may fall into one processing window while others may fall to another, resulting in two notes about a same certain event or period. Speed rather than accuracy is the main focus of this stack.
\subsubsection{The Card stack} It is strictly about logical inferences in order to self-correct the accumulated noises being passed up from the Note stack. First, the Note mapper reads new data from Note DB. The read schedule can be pre-set or be triggered by the Card Manager. The Note mapper will check if Notes are new and in proper format. Following a pre-designed ontology, the Note mapper will refine the acquired notes following: a maximum value rule (for example, in dealing with differences in the reported amounts of beers A had drank in a particular party); a combining value rule (for example, in assessing A's monthly alcohol consumption rate by using various notes about weekly alcohol consumption); a majority rule (for example, in addressing conflicting notes). Rules can be applied differently according to the data (entities and relationships) in the note fields.
The Card Maker grabs the refined notes and maps them to proper cards. This action is similar to assigning o.codes to the DSM 5 dimensions of each potential disorder. Each card is a collection of notes and was structured per the pre-defined ontology. For example, the DSM5-based ontology for Obsessive Compulsive Personality Disorder consists of 8 dimensions (the columns in Table 1), and the disorder can be diagnosed when 4 dimensions (criteria) are met. In this case, a card will be made when there are enough refined notes to fill at least 4 dimensions, and the Card Manager will be notified. Otherwise, the Card Maker holds on to the pre-mature cards, awaiting further refined notes.
In the mean time, the Card Manager will work with the Reasoner in order to identify stealthy inconsistencies among new cards, and/or between new cards and old cards. Naturally, people and things change. For example, when a difficult project is over, a manager may become free of anxiety symptoms. In that context, the conflict between a new card (saying about up-beat attitudes) and a previously committed one (saying about anxiety) can be resolved by putting an expiration date on the old card. In the bi-polar context, conflicting information about moods may actually indicate mood-swings. Therefore, the Reasoner must use and only use evidence-based logics to infer the right decision. The DSM 5 \cite{APA2014DiagnosticDSM5} describes such logics in great details. Conflicts among new cards such as identification of mental issues that were clinically proven to be not comorbid maybe due to errors at the Note stack, or purposeful lying.
We should note that the relationships among Card stack's components are bi-directional while those at the Note stack are one-directional. From a data read-write perspective, Note stack components have read-down/write-up relationships, while members of the Card stack can "talk" to each other at the same level. Although, only the Card Manager may commit cards to the official Card DB. For example, upon the Reasoner's detection of conflicts between card X and card Y, the Card Manager may initiate card remake request(s) for either X or Y or both. Card Maker will check with the Note Mapper to see if there is any additional note to be added. A waiting period may be specified by the Card Manager in order for more new notes to be passed up from the Note stack.
\subsubsection{The Interface stack} It is at the highest level starting with the Card DB, which can be consumed by the Development module, the Integration module, and the Visualization module. The Visualization module provides options to view networked cards, drill down to cards' lower level data, and ideally, to visually simulate different scenarios. Simulating scenarios on a networked graph is similar to finding all possible routes between a start point and an end point on a map. Integration module allows exchanges of knowledge accross different domains. For example, psychiatric cards can be analyzed in tandem with cyber security threat cards in order to infer cyber risks. The Discussion section will go into further integration details. Lastly, the Development module represents all other development efforts on top of the Card DB. Unlike integration and visualization module development, these developments may involve other departments, third-parties, contractors and hence, be put under stricter controls.
\section{Recommendations for System Development}
The paper will not provide a pre-made open-source solution due to various reasons including but not limiting to: issues with actual data used for model training (such data must come from the same organization/environment to which the final solution is deployed), psychology is a broad field and it must be tailored to specific business needs from the beginning. However, this section will provide the necessary resources for building up a really good first step.
\subsection{Related Works}
There are three common approaches in identifying human behaviors. Data-driven approach relies on statistical methods and machine learning (ML) techniques to detect patterns. For example, statistical feature quality group selection was used to recognize daily living activities \cite{Banos2012DailySelection}, and support vector machine was used to recognize basic speech emotions \cite{Seehapoch2013SpeechMachines}. On the other hand, the Knowledge-driven approach uses ontologies and rules. For example, a rule-based multi-modal approach is also able to identify human emotions \cite{Khanna2013RuleApproach}, as well as human basic behaviors \cite{Bae2011AnHomes}. Compared to the Knowledge-driven approach, the Data-riven approach has two fundamental differences. First, it requires enough samples in order to perform accurately, and there is no exact science on how to calculate what enough is enough. Second, domain specific intuitions and explain-ability are not completely supported. For example, ML models cannot explain their own conclusions. The Knowledge-driven approach also has its own drawbacks. It performs slower and its ontology can be really difficult to build. The biggest issue with ontology design is the lack of gold standards or ground truths, leading to issues such as defining the property range and scope of classes. Those pros and cons gave birth to the Hybrid approach, combining both Data-driven and Knowledge-driven techniques. For example, ontology-based stream reasoning helps with continuous activity recognition \cite{BakhshandehAbkenar2014MyActivity:Reasoning}, and complex human activities can be recognized using OWL 2 modeling and reasoning \cite{Riboni2011OWLActivities}. This paper was inspired by the work of \cite{Villalonga2016Ontology-basedIdentification}, in which the architecture was separated into lower and higher levels. Different types of primitives were gathered at the lower level, only to be formed into "contexts" at the higher level.
In the context of understanding team behaviors through the lens of personal psychology, there is an emerging array of technologies. \citeA{Stevens2012CognitiveTeamwork} used cognitive neurophysiologic synchronies, which are low-level derivatives of electroencephalography (EEG) measurements, to identify correlations between team cognition and changes in members' speech. The results can only be used as complimentary metrics to other methods. \citeA{Gorman2012MeasuringApproach} used nonlinear dynamics and real-time communication pattern analysis to analyse team dynamics and found that intact teams' determinism can increase over missions while it was not the case for mixing teams. This proves the usefulness of communication analysis over other noisy channels such as EEG. \citeA{Salas2015TeamsEffectiveness} applied latent-semantic-analysis on text-based data sources such as chat logs, blogs and journals, and was able to assess aspects of team dynamics through identified basic emotional states such as anxiety, being comfortable, and being routine. These methods carry some fundamental limitations such as: mandatory sensor wearing, interruption of work flow (in order to fill out probing questions), and difficult integration with other systems (emotional states are vague and cannot be re-used outside of its native context). On the contrary, the proposed MPAS architecture supports endless integration based on ontologies and logical reasoning.
\subsection{Ontology building}
There are three main steps to engineer ontologies from scratch \cite{Uschold1995TowardsOntologies, Uschold1995TheOntology}: 1. Identifying the purpose by examination of motivationg scenarios; 2. Capturing the concepts and their relationships; 3. Codifying the ontology using ontology languages. Knowledge process is the management of knowledge such as the mapping of a domain's knowledge to an ontology \cite{Schreiber2018KnowledgeManagement}. In the domain of clinical psychology, evidence-based knowledge has already been organized into diagnostic manuals and assessment repositories. The recommended resources are: the Diagnositc and Statistical Manual of Mental Disorders \cite{APA2014DiagnosticDSM5}, the Mental Health Intake Evaluation forms \cite{APADiv.31MentalForms}, the APA Online Assessment Measures \cite{APA2013OnlineMeasures}, the Mental Status Examination guide \cite{Norris2016TheExamination}. There is also a large number of more niche assessments. For example, in studying the relationships between leader psychopathy and organizational deviance, the Dirty Dozen scale (psychopathy assessment) was used in conjunction with the organizational deviance scale and the 24-point moral disengagement scale \cite{Erkutlu2019LeaderDisengagement}. Existing ontologies such as the Mining Minds Context Ontology are publicly available for integration and development \cite{Villalonga2016Ontology-basedIdentification}. A list of clinical reasoning ontologies is also available in the work of \cite{Dissanayake2019UsingSynthesis}. Ontologies can also be automatically extracted \cite{Maedche2003ManagingWeb, Haase2010ConsistentOntologies} with existing tools such as TextToOnto and Text2Onto \cite{Cimiano2010Text2Onto}. Other techniques include the application of statistical methods to the Web \cite{Turney2002MiningTOEFL}, and the calculation of semantic relatedness with respect to a taxonomy or semantic network such as WordNet \cite{Resnik1998SemanticLanguage}.
In combining existing ontologies or developing new ones, Content Ontology Design Pattern can facilitate the combination, merging and alignment \cite{Gangemi2005OntologyContent} of ontologies. There are 6 general ontology design patterns: Structural, Correspondence, Content, Reasoning, Presentations, and Lexico-Syntactic \cite{Presutti2008AOntologies}. Last but not least, we can always evaluate an ontology by its: 1. Accuracy (as in representing the domain of interest); 2. Adaptability (as in allowing extension/modification); 3. Clarity (as in defining terms); 4. Cognitive adequacy (in translating to cognitive semantics); 5. Completeness; 6. Conciseness; 7. Expressiveness; 8. Grounding (as in requiring less assumption).
\subsection{The Recommended Technology Stack}
\begin{itemize}
\item Databases: NoSQL \cite{GeeksforGeeks2018IntroductionNoSQL} for Text DB due to heavy write operations. Traditional SQL for Note DB do to heavy look-up operations, providing that data schema of the Predictor's outputs is fairly stable. Graph database \cite{Neo4jNeo4jDatabases} for Card DB in order to maximize compatibility with further graph-based analysis.
\item Knowledge Explorer: Any already available open-source or commercial solutions.
\item Natural Language Processing: Stanford core NLP \cite{StanfordUniversityCoreNLPSoftware} or IBM Watson for NLP \cite{IBMWatsonStudio}.
\item Organizer: Own modified version of Mining Mind data curation layer \cite{UbiquitousComputingLab2019Ubiquitous-computing-lab/Mining-Minds}
\item Predictor: Any already available open-source or commercial machine learning solutions.
\item Note mapper and Card Maker: Own modified version of Mining Mind information curation layer \cite{UbiquitousComputingLab2019Ubiquitous-computing-lab/Mining-Minds}
\item Card Manager: Own modified version of Mining Mind knowledge curation layer \cite{UbiquitousComputingLab2019Ubiquitous-computing-lab/Mining-Minds}
\item Reasoner: Pellet \cite{SirinPellet:Reasoner} or any already available open-source or commercial solutions.
\end{itemize}
\section{Discussions}
\subsubsection{In optimizing business processes}
\cite{Filho2019TeamEfficacy} recently proposed a systemic theory for understanding and explaining team dynamics. This Team Dynamics Theory (TDT) addresses the needs for a parsimonious yet integrated theory, and is a generative nomological network of team cohesion (CO), mental models (TMM), coordination (CD), collective efficacy (CE), and team outcomes (TO). Each of the component can be further broken down, and be mapped from prior research works. For example, prior research works have shown that: team cohesion has a sub-component which relates to individual and team resilience; team cohesion features are different among teams and fluctuate greatly; last but not least, there is a need for further investigation on team resilience constructs \cite{Salas2015TeamsEffectiveness}.
\begin{figure}[t]
\centering
\includegraphics[width=3.5in]{STERGMs.png}
\caption{STERGM Demonstration}
\end{figure}
The MPAS results can be seamlessly integrated to the TMM component of TDT, and can be confirmed by \cite{Filho2019TeamEfficacy} via two key points. First, TDT affirms the consensus that explicit and implicit communication exchanges are necessary for constructing mental models, which are endowed mechanisms of cognitive-affective-behavioral information. MPAS does process communication texts, and was purposefully designed to recognize both explicit information (at the Note layer) and implicit information (at the Card layer). Second, TMM always stands in relationships with others in the TDT chain, and it evolves dynamically over time. For example, TMM can operate as formative indicators of CD, and covers both team members' communal as well as idiosyncratic knowledge.Because MPAS' cards are ontology-based, their entities and relationships can be easily passed down the logic inference chains, together entities and relationships from other TDT modules. Specifically, if the same architecture (the MPAS architecture) is used, we will end up with CO cards, TMM cards, CD cards, CE cards and TO cards; which in turn, can all be utilized to infer higher level knowledge relating to team dynamics. Those relationships and entities can also be easily mapped out for further analysis leveraging network science methodologies.
The maps of cards and their relationships are called "networks". Their structural patterns such as reciprocity and temporal patterns can give interesting insights into team dynamics \cite{Antone2020TestingControls}. Separable Temporal Exponential Random Graph Models (STERGMs) \cite{Krivitsky2019STERGM-SeparableStatnet} is one among the latest solutions for investigating network dynamics. Specifically, if we have $Y$ as the random variable for the state of our network with a realization $y$ (ties); the set of possible ties $\mathcal{Y}\subseteq{2^Y}$ where $\mathbb{Y}\subseteq\{1,...,n\}^2$ is the set of potential relations; $g(y)$ is the vector of model statistics, $\theta$ is the coefficient vector, $t$ is a certain point in time, $Y^+$ is the formation network, and $Y^-$ is the dissolution network; then we may predict progression of $Y$ through time as shown in Figure 4. In this case, such insights may explain phenomenons like: individual stress level is higher than overall team stress level, job performance of certain team members are high but team performance is low, and last but not least, an individual performs differently in seemingly identical teams.
High-level inferences from cards can be quantified with higher precision than machine-learning based results and faster speeds than human experts. Consequently, mental state scores may inform better business decisions such as adapting business processes. A large business may maintain a database of business process models, which in some cases can be at the thousands or even the hundred-thousands \cite{Kunze2015QueryingInclusion}. A decision maker may need to find business process models similar to a given one for reasons such as: assessing potential risks in different scenarios, finding alternatives or optimizing the existing one, and merging departments or companies. \citeA{Kunze2015QueryingInclusion} proposed a querying method that is based on trace semantics of process behavior, abstract trace inclusion, and ranking of query results. Figure 5 presents basic building blocks of a business process model, of which different combinations lead to different traces. \citeA{Kunze2015QueryingInclusion} emphasized on the ranking of behavioral distances as the methodology's key differentiator, which the paper completely agreed on. However, behavioral distances should include both business behaviors (tactics to achieve certain business goals) and mental behaviors (team mental model scores at the least). For example, there are different ways for a department to archive the same business goals, and we need to watch out for the ones that may cause unnecessary psychological stresses (short term or long term) on the teams.
\begin{figure}[t]
\centering
\includegraphics[width=6in]{NetSys.png}
\caption{Net System Patterns}
\end{figure}
\subsubsection{In understanding bad actors} White-collar criminals are complex in nature. Some even compare them to chameleons, reflecting their superb capability to adapt and manipulate. At personal level, previous works have identified risk factors such as: high scores of narcissism and hedonism; low scores of conscientiousness and self-control; high dimension scores of antisocial personality disorder; as well as certain score variations in extroversion, agreeableness, conceitedness, neuroticism, intellectualism, and negative valency \cite{Brady2016ForensicCrime}. To a certain degree, white-collar criminals share some psychological patterns with other criminals. While most of them may commit crimes for financial gains, some of them commit white-collar crimes for other reasons, including cold-blooded calculations. This landscape only gets more complicated at the institutional level. For example, a senior manager with high scores on risk factors may remotely instruct an employee with high loyalty score (to that manager) to commit white-collar crimes. Other complications may include: shared feelings, defense mechanisms, fantasies, beliefs, the tendency to give superiors the benefit of the doubt, traumatic avoidance, denial and cult-like shared or enabled delusions \cite{Brady2016ForensicCrime}. On this note, the paper cannot help but emphasize the importance of looking at psychological artifacts in a networked view, and paying attentions to the dynamics among the components. While MPAS advocates for using psychology forensic science in engineering its ontologies which are the main component of automatic reasoning, it is important to note that psychology forensic science is far from perfect \cite{Brady2016ForensicCrime}. MPAS can be used internally, externally or both; and for identifying mental risk factors, as well as their combinations in higher-level analysis. Leveraging MPAS for risk assessment requires further steps of quantifying the risk factors, simulating scenarios, and producing risk estimates.
\subsubsection{On potential issues}
There are many issues around Artificial Intelligence (AI) and Machine Learning (ML) systems. The biggest challenge concerns the legal definitions of AI and ML. Even the father of AI could not define what is AI \cite{Scherer2016REGULATINGSTRATEGIES}. This leads to the issues of defining the "Actor", which is the target for prosecution in court cases. MPAS clarifies this issue via its layered architecture. ML can be fully deployed but is confined within the Note layer. This minimizes the negative impacts of ML's mistakes. System designers and operators will have more controls at the Card layer where they can design the logics which card maker and manager rely on. Human bias should be at minimal if the logics are based on strong research results. Unlike other ML-based solutions which operate like black-boxes \cite{AnjomshoaeExplainableReview}, MPAS hybrid architecture of ML and human-controlled logic reasoning provides the system users/administrators with explainable findings, and makes them accountable for the final decisions. In this case, explain-ability is represented by recorded reasoning chains stored in cards, and the ability to trace back to the notes that formed a card.
Workplace surveillance versus employees' privacy is another concern. It is important to note that User Activity Monitoring (UAM) - a subset of workplace surveillance - already has a market of \$1.1 billion and is expected to grow to \$3.3 billion by 2023 \cite{Manokha2019NewEmployees}. Existing commercial solutions include InterGuard, ActivTrak, Time Doctor, Toggl, Activity Monitor, WorkTime Corporate, and Berqun. In a 2007 survey administered by the American Management Association, 73\% of its member companies use technologies to automatically monitor employees \cite{Chory2016OrganizationalResponses}. The legality of workplace surveillance is not debatable. However, employees may have higher psychological stress, and loose trust in the organization if they believe such monitoring does not completely align with organizational justice, which encompasses three dimensions of: distributive, procedural, and interactional justice \cite{Chory2016OrganizationalResponses}. In the case of deploying MPAS internally, it is important to communicate with employees the organizational problems that the tool is helping with solving. Such problems may include: workplace discrimination, sexual harassment, other forms of harassment, suicide, indicators of insider threats, indicators of financial crimes, and so on. Encryption schemes can be used to "mask" the employees' entities, that can only to be unmasked per system's access level.
\section{Conclusion}
Within its limited scope, the paper presented a manual psychoanalysis process snapshot of Steve Jobs case, and proposed the Machine-assisted Psycho-Analysis System (MPAS) for mining psychological artifacts at scale. Unlike most other proposed solutions, this architecture utilizes natural language processing and machine learning at the lower level, allowing faster processing speed, while leveraging evidence-based psychology knowledge and ontological reasoning at the higher level. The use of ontologies provides the much needed explainability and extensibility. For example, reasoning results can be further consumed by graph science and/or model management system for optimizing business processes, understanding team dynamics, predicting insider threats, managing talents, and beyond. It is important to note that issues such as moral disengagement \cite{Erkutlu2019LeaderDisengagement}, and organizational psychopaths' impacts on large scale economy \cite{Boddy2015OrganisationalUpdate} have been under-explored, partially due to the lack of human psychology data.
A technology stack was recommended with references to further resources including starter codes for immediate system development. It is impossible for the paper to present packaged solution due to its limited scope and the needs for customization in building solutions to be used in a particular organization. The paper particularly emphasizes the need to build a custom ontology. It can and should be based on existing evidence-based public researches but it has to be tailored to the organization's people, business, and culture. However, in the case of directing MPAS outward for understanding external audiences, a packaged solution may be possible.
The paper does have a plan for profiling hackers and cyber threat groups using MPAS. The immediate challenge is finding ground truths. Data from cyber criminal investigations is not generally made available to researchers. Even worst, collecting psychological data with regards to hackers' motives, mental needs, and mental issues might have been ignored prior to and even after arrests. It is also challenging to find enough evidence-based research works to support the engineering of an ontology describing hackers psychological states. On a brighter note, predicting the organizational behaviors of hacking groups is possible because their digital breadcrumbs are plenty, and findings can be corroborated by evidences from other domains such as geopolitics, legislation, and economics.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,951 |
\section{Introduction}
Standard heat flow, at a macroscopic level, is modeled by the random erratic movements of Brownian motions starting at the source of heat. However, this diffusive nature of heat flow predicted by Brownian motion is not seen in certain materials (semiconductors, dielectric solids) over short length and time scales \cite{1}.
Experimental data portraying the non-diffusive behavior of heat flow has been observed for transient thermal grating (TTG) \cite{4,3}, time domain thermoreflectance (TDTR) \cite{7} and others \cite{6,5}, by altering the physical size of the heat source. The thermal transport in such materials is more akin to a superdiffusive heat flow, and necessitates the need for processes beyond Brownian motion to capture this heavy tail phenomenon. Recent works \cite{15, 18, 17, 16, 20, 14, 19} try to explain the physics behind the quasiballistic heat dynamics. But these methods, driven mostly by the Boltzmann transport equation, are infeasible for processing experimental data. Some more recent studies \cite{26, 27} try to explain the non-diffusive heat flow through hyperbolic diffusion equations, however, closer investigation shows that these methods fail to capture the inherent onset of nondiffusive dynamics at short length scales in periodic heating regimes.
The attempts mentioned above fail to provide a stochastic process that would explain the heat dynamics under short length-time regimes. The most natural stochastic process to explain a superdiffusive behavior is an alpha-stable L\'evy process \cite{Applebaum}. Alpha-stable L\'evy processes differ from Brownian motion in that its movements are governed by stable distributions as compared to Gaussian distributions for the latter. In this context, some of us have tried to explain the heat flow dynamics through a "truncated L\'evy distribution" approach \cite{PRBlevy1,PRBlevy2}, where it has been possible to extract the value of the L\'evy superdiffusive coefficient $\alpha$ that regulates the alloy's quasiballistic heat dynamics.
The current contribution can be seen as a further step in this direction. Specifically, let $T(t,x)$ designate the temperature of a semiconductor or dielectric solid in the experimental settings alluded to above, with initial condition $T_{0}(x)$. Then we shall describe $T$ through the following Feynman-Kac formula (see Section~\ref{sec:chr-FK} for more details about Feynman-Kac representations):
\begin{equation}\label{eq:FK}
T(t,x)
=
\mathbf{E}\left[ T_{0}(x+X_{t}) \right],
\end{equation}
where $X$ is a well-defined L\'evy process that captures the observed quasiballistic heat dynamics, in addition to being a good candidate for explaining the usual diffusive nature under non-special large length-time regimes. We shall see that such a process $X$ can be chosen as a so-called relativistic stable process (see \cite{ryznar}, and \cite{carmona} for properties related to the relativistic Schr\"odinger operator). It possesses the remarkable property of behaving like an alpha-stable process under short length-time scales while being closer to Brownian motion otherwise. This is reflected in the estimates of the transition density, provided below in Section~\ref{sec:estimates}. Summarizing, our result lays the mathematical foundations of heat flow modeling on short time scales by means of stochastic processes. In addition, in spite of the fact that our computations are mostly one-dimensional, the model we propose allows natural generalizations to multidimensional and multilayer settings.
\section{Relativistic stable process: a primer}\label{sec:relativistic-primer}
In this section we give a short introduction on relativistic stable processes. We first recall the definition of this family of processes. Then we will give some kernel bounds indicating how relativistic processes transition, as $t$ increases, from an $\alpha$-stable behavior to a Brownian type behavior (this property being crucial to model quasiballistic heat dynamics in semiconductors).
\subsection{Characteristics and Feynman-Kac formula}\label{sec:chr-FK}
For any $M>0$ and $\alpha \in (0,2]$, a relativistic $\alpha$-stable process $X^M$ on $\mathbb R^d$ with a positive weight $M$ is a L\'evy process whose characteristic function is given, for any $t \geq 0$ and $\xi \in \mathbb R^d$, by:
\begin{equation}\label{eq:phi-m}
\phi_{M}(\xi)
\equiv
\mathbb{E}\left[ \exp\left( \imath \xi \cdot (X_t^M - X_0^M)\right)\right]
= \exp\left( -t\left( \left( |\xi|^2 + M^{2/\alpha}\right)^{\alpha/2}-M\right)\rp.
\end{equation}
When $M=0$, $X^M$ is simply a rotationally symmetric $\alpha$-stable process in $\mathbb R^d$. When $\alpha=2$, regardless of the value of $M$, the process is a Brownian motion. The infinitesimal generator of $X^M$ is {$\mathcal L^{M} = M-(-\Delta + M^{2/\alpha})^{\alpha/2}$}. When $\alpha = 1$, this reduces to the free relativistic Hamiltonian $M-\sqrt{-\Delta + M^{2}}$, which explains the name of the process. An explicit expression for the L\'evy measure of $X^{M}$ can be found in \cite{Ch-12, AOP-Ch-12, Ch-11}. We omit this formula for sake of conciseness, since it will not be used in the remainder of the paper.
{ L\'evy processes like $X^M$ are classically used in order to represent solutions of deterministic PDEs. In our case, consider the following equation governing the temperature $T$ in our material:
\begin{equation}\label{eq:PDE}
\partial_t T(t,x) = \mathcal L^M T(t,x), \text{ with } T(0,x) = T_0(x).
\end{equation}
Then it is a well known fact (see \cite{Applebaum}) that the solution $T$ to \eqref{eq:PDE} can be represented by the Feynman-Kac formula \eqref{eq:FK}, where the process $X^M$ is our relativistic $\alpha$-stable process. The Feynman-Kac representation is crucial in order to get equation~\eqref{eq:SPR} below.
}
\subsection{Transition kernel estimates}\label{sec:estimates}
In this subsection, we identify the behavior of a relativistic stable process with a stable process on short time scales and a Brownian motion on larger time scales. As mentioned above, this will be achieved by observing the patterns exhibited by the transition kernel of $X_{t}^{m}$. Some results will be stated without formal proof, and interested readers are referred to \cite{Ch-12, AOP-Ch-12, Ch-11} for more details.
Since $X^{M}$ is a Levy process, it is also a Markov process. As such it admits a transition kernel $p_{t}^{M}$, defined by:
\begin{equation*}
\mathbb P\left( X_{s+t}^{M} \in A \, | \, X_{s} = x \right)
=
\int_{A} p_{{t}}^{M}(x,y) \, dy,
\end{equation*}
for all $x\in\mathbb R^{d}$ and $A\subset \mathbb R^{d}$. Notice that $p^{M}$ is related to the function $\phi_{M}$ (see definition \eqref{eq:phi-m}) as follows:
\begin{equation}\label{eq:kernel-as-inverse-fourier}
p_{{t}}^{M}(x,y)
=
\mathcal F^{-1} \phi_{M}(x-y)
=
\frac{1}{(2 \pi)^d}
\int_{\mathbb R^d} e^{\imath (x-y)\cdot \xi} e^{-t\lbrace (M^{2/\alpha} + |\xi|^2)^{\alpha/2}-M \rbrace} d\xi.
\end{equation}
We start by a simple bound on $p^M_t$, exhibiting the stable behavior for small times and the Brownian behavior for large times. Observe that this bound does not depend on the space variables $x,y$. We include its proof, which is based on elementary considerations involving Fourier transforms, for sake of completeness.
\begin{theorem}\label{thm:transition-stable-to-brownian}
Consider a relativistic $\alpha$-stable process $X^M$, and let $p^{M}$ be its transition kernel.
Then there exists $c_1=c_1(\alpha)>0$ such that for all $t>0$ and all $x,y\in\mathbb R^{d}$:
\begin{equation}\label{ub_1}
p^M_t(x,y) \leq c_1(M^{d/\alpha-d/2}t^{-d/2} + t^{-d/\alpha}) .
\end{equation}
\end{theorem}
\begin{proof}
The strategy of our proof is based on the fact that the characteristic function $\phi_{M}$ defined by \eqref{eq:phi-m} behaves like a Gaussian characteristic function for low frequencies, and like an $\alpha$-stable characteristic function for high frequencies. We shall quantify this statement below.
\noindent
\textit{Step 1: Elementary inequalities:}
Let $\beta=\frac{\alpha}{2}$. The following inequality, valid for for $0 \leq z \leq 1$, is readily checked:
\begin{equation}\label{a1}
z^{\beta} - \beta z \leq 1 - \beta.
\end{equation}
Substituting $z=\frac{M^{2/\alpha}}{|\xi|^2 + M^{2/\alpha}}$ in \eqref{a1}, we thus have:
\begin{equation*}
\dfrac{M}{(|\xi|^2 + M^{2/\alpha})^{\alpha/2}} - \dfrac{\alpha \, M^{2/\alpha}}{2(|\xi|^2 + M^{2/\alpha})}
\leq
1 - \frac{\alpha}{2} ,
\end{equation*}
which yields:
\begin{equation}\label{a2}
\left( |\xi|^2 + M^{2/\alpha}\right)^{\alpha/2} - M
\geq
\dfrac{\alpha |\xi|^2}{2(|\xi|^2 + M^{2/\alpha})^{1 - \alpha/2}}.
\end{equation}
Relation \eqref{a2} prompts us to split the frequency domain in two sets:
\begin{equation}\label{eq:dom-split}
A_1 = \lbrace \xi : |\xi|^2 \leq M^{2/\alpha} \rbrace,
\quad\text{and}\quad
A_2 = \lbrace \xi : |\xi|^2 > M^{2/\alpha} \rbrace.
\end{equation}
Accordingly, we get the following lower bounds:
\begin{equation}\label{eq:a2}
(|\xi|^2 + M^{2/\alpha})^{\alpha/2} - M
\geq
\begin{cases}
\dfrac{\alpha}{2(2M^{2/\alpha})^{1-\alpha/2}} \, |\xi|^2, &{\text{when } \xi \in A_{1}} \\
\dfrac{\alpha}{2^{2-\alpha/2}} \, |\xi|^{\alpha}, &{\text{when } \xi \in A_{2}}.
\end{cases}
\end{equation}
This relation summarizes the separation between an $\alpha$-stable and a Gaussian regime alluded to above.
\noindent
\textit{Step 2: Consequence for the transition kernel.}
Recall relation \eqref{eq:kernel-as-inverse-fourier} for $p^{M}$, that is:
\begin{equation*}
p_{s}^{M}(x,y)
=
\frac{1}{(2 \pi)^d}
\int_{\mathbb R^d} e^{\imath (x-y)\cdot \xi} e^{-t\lbrace (M^{2/\alpha} + |\xi|^2)^{\alpha/2}-M \rbrace} d\xi.
\end{equation*}
In the integral above, we simply bound $|e^{\imath (x-y)\cdot \xi}|$ by 1 and split the integration domain $\mathbb R^{d}$ into $A_{1}\cup A_{2}$. Taking our relation \eqref{a2} into account, this yields:
\begin{equation}\label{a3}
p^M_t(x,y)
\leq \frac{1}{(2 \pi)^d} \int_{A_1} e^{-\frac{\alpha t}{2(2M^{2/\alpha})^{1-\alpha/2}} \, |\xi|^2} d\xi \\
+ \frac{1}{(2 \pi)^d} \int_{A_2} e^{-\frac{\alpha t}{2^{2-\alpha/2}} \, |\xi|^{\alpha}} d\xi
\leq
I_{t}^{1} + I_{t}^{2},
\end{equation}
where
\begin{eqnarray*}
I_{t}^{1} & = &
\frac{1}{(2 \pi)^d} \left[ \frac{1}{t^{d/2}} \left(\frac{(2M^{2/\alpha -1})}{\alpha/2}\right)^{d/2}
\int_{\mathbb R^d} e^{-|\xi|^2} d\xi \right] \\
I_{t}^{2} &=& \frac{1}{(2 \pi)^d} \left[ \frac{1}{t^{d/\alpha}} \left(\frac{(2^{2-\alpha/2})}{\alpha}\right)^{d/\alpha}\int_{\mathbb R^d} e^{-|\xi|^{\alpha}} d\xi \right] .
\end{eqnarray*}
It is now easily checked that
\begin{equation*}
I_{t}^{1} = \frac{c_{2}}{t^{d/2}},
\quad\text{and}\quad
I_{t}^{2} = \frac{c_{3}}{t^{d/\alpha}}.
\end{equation*}
Plugging this information into \eqref{a3}, our claim \eqref{ub_1} follows.
\end{proof}
The upper bound \eqref{ub_1} already captures a lot of the information we need on relativistic stable processes. Invoking sophisticated arguments based on stopping times and Dirichlet forms, one can get upper and lower bounds on the transition kernel $p^{M}$ involving some exponential decay in the space variables $x,y$. We summarize those refinements in the following theorem.
\begin{theorem}
Let $p^{M}$ be the transition kernel defined by \eqref{eq:kernel-as-inverse-fourier}. Then the following estimates hold true.
\noindent
\emph{(i) Small time estimates.}
Let $T>0$ be a fixed time horizon. Then there exists $C_1>0$ such that for all $t \in (0,T]$ and $x,y \in \mathbb R^d$,
\begin{equation}\label{eq:small-time-bnd}
C_{1}^{-1}\left( t^{-d/\alpha} \wedge \dfrac{t e^{-M^{1/\alpha}|x-y|}}{|x-y|^{{(d+\alpha+1)}/{2}}} \right) \leq p^M_t(x,y) \leq C_{1}\left( t^{-d/\alpha} \wedge \dfrac{t e^{-M^{1/\alpha}|x-y|}}{|x-y|^{{(d+\alpha+1)}/{2}}} \right) .
\end{equation}
\noindent
\emph{(ii) Large time estimates.}
There exists $C_2 \geq 1$ such that for every $t \in [1, \infty)$ and $x,y \in \mathbb R^d$,
\begin{equation}
C_2^{-1} t^{-d/2} \leq p^M_t(x,y) \leq C_2 t^{-d/2} .
\end{equation}
\end{theorem}
\section{Application of relativistic stable processes to thermal modelling}
In this section we show how to apply the mathematical formalism of Section \ref{sec:relativistic-primer} to our concrete physical setting. More specifically, in Section \ref{sec:thermal-prop} we shall introduce length scales in our L\'evy exponent \eqref{eq:phi-m}. Then Section \ref{sec:tdtr-measurement} is devoted to a description of our experimental setting, and also relates our measurements to the Fourier exponents we have put forward.
\subsection{Formulation in terms of material thermal properties}\label{sec:thermal-prop}
The aforementioned evolution of relativistic processes, from alpha-stable behaviour at short length and time scales to regular Brownian motion at longer scales, renders them suitable to describing quasiballistic thermal transport in semiconductor alloys. Let us consider such a material, having nominal thermal diffusivity $D = \kappa/C_v$ with $\kappa$ being the thermal conductivity and $C_v$ the volumetric heat capacity (in { ${\rm Jm}^{-3}{\rm K}^{-1}$}).
{ The physical quantity we have access to is a slight variation of the function $T(t,x)$ defined by \eqref{eq:FK}. Specifically the single pulse response for the d-dimensional excess thermal energy can be expressed as $P(t,x) = C_v \, \Delta T(t,x)$. Under the L\'evy flight paradigm the Fourier transform of $P$ is written as
\begin{equation}\label{eq:SPR}
P(| \xi | , t) = \exp \left(- t \, \psi(| \xi |) \right),
\end{equation}
for a given L\'evy exponent (also called special heat propagator) $\psi$. For the relativistic case under study here, this spatial heat propagator is simply a multiple of the function $\phi_M$ introduced in \eqref{eq:phi-m}. Namely $\psi$ reads}
\begin{equation}\label{eq:cf-D_al}
\psi(| \xi |) = D_{\alpha} \, \left[ \left(| \xi |^2 + M^{2/\alpha} \right)^{\alpha/2} - M \right].
\end{equation}
The prefactor $D_{\alpha}$ with unit m$^{\alpha}$/s denormalises the characteristic function for dimensionless space and time variables defined by Eq. (\ref{eq:phi-m}) to its physical counterpart, and denotes the fractional diffusivity of the alpha-stable regime as we shall see shortly.
\par
For thermal modelling purposes it is furthermore convenient to reformulate the process mass $M$, which has an exponent-dependent unit 1/m$^{\alpha}$, in terms of an associated characteristic length scale $x_0$ around which the transition from alpha-stable (L\'evy) to Brownian dynamics takes place{ . Notice that according to our previous analysis leading up to Eq. \eqref{eq:dom-split} and \eqref{eq:a2}} this length should be given by:
\begin{equation}
x_0 = | \xi_0 |^{-1} = M^{-1/\alpha}.
\end{equation}
This means that expression \eqref{eq:cf-D_al} can be recast as
\begin{align}\label{eq:mod-M}
\psi(| \xi |) &= D_{\alpha} \, \left[ \left(| \xi |^2 + | \xi_0 |^2 \right)^{\alpha/2} - | \xi_0 |^{\alpha} \right] \nonumber \\
&= D_{\alpha} \, | \xi_0 |^{\alpha} \, \left[ \left( \tilde{\xi}^2 + 1 \right)^{\alpha/2} - 1 \right],
\end{align}
where $\tilde{\xi} \equiv | \xi | / | \xi_0|$.
With those values of $M$ and $\xi_0$ in hand, we can translate \eqref{eq:a2} into an asymptotic transport limit as follows:
\begin{align}
\text{alpha-stable regime } \tilde{\xi} \gg 1&:~\psi(| \xi |) \simeq D_{\alpha} \, | \xi |^{\alpha}, \nonumber \\
\text{Brownian regime } \tilde{\xi} \ll 1&:~\psi(| \xi |) \simeq \frac{\alpha D_{\alpha}}{2 | \xi_0 |^{2-\alpha}} \, | \xi |^2. \label{eq:brown-regime}
\end{align}
The former corresponds to L\'evy superdiffusion with characteristic exponent $\alpha$ and fractional diffusivity $D_{\alpha}$; the latter should recover to nominal diffusive transport { $\psi (|\xi|) \equiv D | \xi |^2$. In order to make the last relation compatible with \eqref{eq:brown-regime} we must set}
\begin{equation}\label{eq:D-al}
\frac{\alpha D_{\alpha}}{2 | \xi_0 |^{2-\alpha}} = D \quad \Longrightarrow \quad D_{\alpha} = \frac{2D}{\alpha x_0^{2-\alpha}}.
\end{equation}
Finally, { plugging \eqref{eq:D-al} into \eqref{eq:mod-M}} the heat propagator reads
\begin{equation}
\psi(| \xi |) = \frac{2D}{\alpha x_0^2} \, \left[ \left(1 + x_0^2 | \xi |^2 \right)^{\alpha/2} - 1 \right]. \label{propagator}
\end{equation}
This formulation contains 3 material dependent parameters, each with an intuitive physical meaning: the characteristic exponent $\alpha$ of the alpha-stable regime; the nominal diffusivity $D$ of the Brownian regime; and the characteristic length scale $x_0$ around which the transition between those two asymptotic limits occurs (Fig. \ref{figure_propagator}). In the sections that follow, we determine these parameter values for In$_{0.53}$Ga$_{0.47}$As by fitting a thermal model built upon the propagator (\ref{propagator}) to time-domain thermoreflectance (TDTR) measurement signals.
\myfig[!htb]{width=0.7\linewidth}{figure_propagator}{Transition between Brownian and alpha-stable L\'evy behavior.}
\subsection{Modelling of TDTR measurement signals}\label{sec:tdtr-measurement}
The central principle in TDTR is to heat up the sample with ultrashort \textit{pump} laser pulses, and then monitor the thermal transient decay using a \textit{probe} beam. Pulses from the laser are split into a pump beam and probe beam. The pump pulses pass through an electro-optic modulator (EOM) before being focused onto the sample surface through a microscope objective. A thin (50-100 nm) aluminium film is deposited onto the sample to act as measurement transducer: the metal efficiently absorbs the pump light and converts it to heat, and translates temperature variations to changes in surface reflectivity which can be captured by the probe. Lock-in detection at the pump modulation frequency $f_{\text{mod}}$ resolves the thermally induced reflectivity changes captured by the probe beam. A mechanical delay stage allows to vary the relative arrival time of the pump and probe pulses at the sample with picosecond resolution. To minimise the impact of random fluctuations in laser power and the variation of the pump beam induced by the delay stage, thermal characterisation is performed not on the raw lock-in signal itself but rather the ratio $-V_{\text{in}}/V_{\text{out}}$ of the in-phase and out-of-phase components as a function of the pump-probe delay.
\par
Theoretical ratio curves $-V_{\text{in}}/V_{\text{out}}$ can be computed semi-analytically through mathematical manipulation of the semiconductor single-pulse response (\eqref{eq:SPR}), as described in detail in Refs. \cite{cahillRSI,schmidt,JAPmultiD}. Briefly, we first obtain the surface temperature response of a semi-infinite semiconductor to a cylindrically symmetric energy input via Fourier inversion of (\eqref{eq:SPR}) with respect to the cross-plane coordinate. Next, a matrix formalism that accounts for heat flow in the metal transducer and across the intrinsic thermal resistivity $r_{\text{ms}}$ (in K-m$^2$/W) of the metal-semiconductor interface provides the temperature response, weighted by the Gaussian probe beam, of the transducer top surface induced by a Gaussian pump pulse. Finally, harmonic assembly of this response at frequencies $n \cdot f_{\text{rep}} \pm f_{\text{mod}} \quad (\text{for } n = 0,1,\ldots)$ accounting for the laser repetition rate $f_{\text{rep}}$, pump modulation frequency $f_{\text{mod}}$, and phase factors induced by the pump-probe delay $\tau$ yields the theoretical lock-in ratio signal $-V_{\text{in}}/V_{\text{out}}(\tau)$.
\section{Experimental analysis}
We have applied our model to TDTR measurements taken on a $\simeq$2 micron thick film of In$_{0.53}$Ga$_{0.47}$As ($C_v \simeq 1.55\,$MJ/m$^3$-K) that was MBE-grown on a lattice-matched InP substrate. We note that although the semiconductor alloy under study (the InGaAs layer) is a geometrically thin film, thermally speaking it can still be considered, as is assumed by the thermal model, as a semi-infinite layer with good approximation. This is because the effective thermal penetration length $\ell = \sqrt{D/(\pi f_{mod})}$ stays firmly within the film over the experimentally probed modulation range $0.8 \, \text{MHz} \lesssim f_{mod} \lesssim 18\, \text{MHz}$. The aluminium transducer deposited onto the sample measured 64$\,$nm in thickness as determined by picoseconds accoustics. We used pump and probe beams with $1/e^2$ radii at the focal plane of 6.5 and 9 microns respectively, with respective powers of 17 and 8 mW at the sample surface.
\par
In the thermal model with relativistic stable heat propagator (\ref{propagator}), we fixed the heat capacity at the aforementioned 1.55$\,$MJ/m$^3$-K. Theoretical ratio curves were then collectively fitted through nonlinear least-square optimisation to signals measured at 7 different modulation frequencies to identify the 4 key thermal parameters: the characteristic exponent $\alpha$ of the L\'evy superdiffusion regime; the quasiballistic-diffusive transition length scale $x_0$ associated to the mass $M$; the nominal thermal conductivity $\kappa = C_v D$ of the diffusive regime; and the thermal resistivity $r_{ms}$ of the transducer/semiconductor interface. The resulting best-fitting values $\alpha$ = 1.695 , $x_0$ = 0.86$\,\mu$m , $\kappa$ = 5.82$\,$W/m-K , $r_{ms}$ = 4.28$\,$nK-m$^2$/W yield an excellent agreement with the measured signals (Fig. \ref{ratiocurves_bestfit}). Theoretical curves with parameter values deviating from the best fitting ones (Fig. \ref{ratiocurves_tolerance}) furthermore visually reveal the sensitivity to each of the parameters and illustrate the good quality of the best fit.
\myfig[!htb]{width=0.7\linewidth}{ratiocurves_bestfit}{TDTR characterisation of Al/InGaAs sample: a thermal model based on a relativistic stable random motion of heat (lines) provides an excellent fit to measured signals (symbols).}
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{0.5\linewidth}
\includegraphics[width=\linewidth]{ratiocurves_tol1.pdf}
\caption{$\alpha \pm 0.02$}
\label{tolerance-alpha}
\end{subfigure}%
\begin{subfigure}[b]{0.5\linewidth}
\includegraphics[width=\linewidth]{ratiocurves_tol2.pdf}
\caption{$x_0 \pm 10 \%$}
\label{tolerance-x0}
\end{subfigure}%
\begin{subfigure}{0.5\linewidth}
\includegraphics[width=\linewidth]{ratiocurves_tol3.pdf}
\caption{$\kappa \pm 5\%$}
\label{tolerance-kappa}
\end{subfigure}%
\begin{subfigure}{0.5\linewidth}
\includegraphics[width=\linewidth]{ratiocurves_tol4.pdf}
\caption{$r_{ms} \pm 50 \%$}
\label{tolerance-rms}
\end{subfigure}%
\caption{Ratio curve fitting tolerance and sensitivity. The plotted theoretical curves are computed with sub-optimal parameter combinations in which one parameter deviates from its best fitting value as indicated.}
\label{ratiocurves_tolerance}
\end{figure*}
\section{Conclusions and Outlook}
Quasi-ballistic heat propagation in materials can be studied using atomic parameters through a multitude of techniques. First principle calculations and multi spectral phonon Boltzmann transport equations are very powerful in this regard. However, their use in the study of heat propagation in multi-layer/anisotropic materials and materials with complex geometries is limited. The Feynman-Kac representation of solutions to partial differential equations with non-local parameters can potentially provide alternative approaches to explain experimental thermal data. In this article we have replaced the traditional heat equation by a different PDE, whose solution has a Feynman-Kac representation driven by the so-called relativistic stable L\'evy process. The transition characteristics of this process is in harmony to the heat propagation behaviour exhibited by TDTR data. In general, numerical approximations of the PDE solution can also be achieved through Monte Carlo simulations of the driving stochastic process in the Feynman-Kac formula. In particular, these numerical computations may provide substitute techniques to optimize materials or source geometry in order to reduce heating from nanoscale and/or ultrafast devices.
Our next challenge in this direction will be to model multidimensional transport in multilayer structures. To this aim, we shall investigate two methods: (i) Monte Carlo simulation according to our Feynman-Kac representation \eqref{eq:FK}, taking into account jumps and change of media. (ii) Related PDEs involving the non local operator $\mathcal L^{M} = M-(-\Delta + M^{2/\alpha})^{\alpha/2}$, with boundary terms corresponding to the different layers. Both methods rely crucially on the relativistic L\'evy representation advocated in this paper. They will be subject of future publications.
\newpage
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,516 |
Q: How to check if file content is empty? I am trying to check if file doesn't have anything in it.
This is what I have which checks/create/write to file:
class LastUsed
{
private static string dir = Environment.GetFolderPath(Environment.SpecialFolder.ProgramFiles) + @"\Folder\";
private static string file = dir + @"\Settings.txt";
private string text;
public void CheckFileStatus()
{
if (!Directory.Exists(dir))
{
DirectoryInfo directory = Directory.CreateDirectory(dir);
}
if (!File.Exists(file))
{
using (FileStream fileStream = File.Create(file))
{
}
}
}
private void SetFileText(string writeText)
{
using (StreamWriter streamWriter = new StreamWriter(file))
{
streamWriter.Write(writeText);
}
}
private string GetFileText()
{
string readText;
using (StreamReader streamReader = File.OpenText(file))
{
readText = streamReader.ReadLine();
}
return readText;
}
public string Text
{
set
{
text = value;
SetFileText(text);
}
get
{
return GetFileText();
}
}
As we can see I can read/write file by using properties. So I have tried to check the Text property for null value but it doesn't seem to work.
How should I do this?
A: This code should do it
if (new FileInfo(fileName).Length ==0){
// file is empty
} else {
// there is something in it
}
fileName is the file path that you want to look for its size
A: Simply check if the file's size is zero bytes: Get size of file on disk.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,070 |
<html>
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=no">
<title>React Express</title>
</head>
<body>
<main id="app">If you still see this, then react has failed to load</main>
<script src="bundle.js" async></script>
</body>
</html> | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,448 |
Tag: Podcasts
The Joe Rogan Podcast
Joe Rogan is a commentator on MMA, a comedian, and a former martial arts practitioner . He is also the host of the Joe Rogan Experience podcast. The JRE is approaching 800 episodes, I've only listened to a fraction of them. His guests include his fellow comedians, MMA fighters and people who Rogan thinks are interesting.
Generally, his podcasts are around 3 hours each where he has a long conversation with a guest in the studio. What I like about them is that 3 hours is plenty of time to really talk to a person find out what it is that makes them tick, what they believe and why they believe it. In the podcast format is much easier to digest than reading a book, which means you can listen to people you wouldn't normally invest time and money in by buying their books and read them.
He's had some very rational guests: Michael Schermer and Sam Harris are recurring guests; Brian Cox and Neil Degrasse Tyson have also been on the show. The podcasts I've found most interesting, though, are the fringe characters such as Rupert Sheldrake. Sheldrake is a proponent of morphic fields, which is the idea that knowledge can be passed telepathically through nature and social groups. I've heard and read about him and I thought he must be delusional. After listing to him on the podcast – I still think he's wrong- but he's very eloquent and can explain lucidly and convincingly his experiment and strange phenomena that he has observed, but then jumps to the 'magic' explanation. He is a true believer, he's done experiments with Steven Rose and Richard Wiseman, Sheldrake interprets the results one way, Rose and Wiseman the opposite. It's fascinating that intelligent people can look at the same data and reach a different conclusion. In fairness to Sheldrake, he posts both papers by him and his critics on his website.
An interesting guest he had on recently was Scott Adams, best known for creating the Dilbert comic strip, he also predicted over a year ago that Donald Trump would win the US election in a landslide. Adams is also a hypnotist, and he believes that Trump is one of the most persuasive politicians there has ever been because he uses some of the tricks that hypnotists use. Also, Trump's hyperbole and rhetoric give him a lot of latitude for comprising somewhere in the middle in a way that a moderate candidate couldn't.
A criticism of the JRE it's maybe a little too open minded, some people can just say things that are obviously ridiculous, but he doesn't challenge them that often. But it is nice that you can list to different points of view (with very little effort) from people you would not normally hear from, especially in these days of social media echo chambers. The podcasts with comedians are often very funny too.
Some interesting people I've found through listening to the JRE:
Scott Adams (see above)
Dr. Rhonda Patrick: biochemist and health and nutrition expert (recurring guest).
Prof. Gad Saad: An Evolutionary biologist who studies the evolutionary basis of consumerism (recurring guest).
Author JamesPosted on November 23, 2016 November 23, 2016 Categories DiaryTags Joe Rogan, PodcastsLeave a comment on The Joe Rogan Podcast | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,173 |
#import "NSObject.h"
@class NSMutableArray, NSString;
@interface ABVCardParameter : NSObject
{
NSString *_name;
NSString *_grouping;
id _value;
NSMutableArray *_types;
_Bool _primary;
}
- (id)description;
- (void)setIsPrimary:(_Bool)arg1;
- (_Bool)isPrimary;
- (id)value;
- (id)grouping;
- (void)setGrouping:(id)arg1;
- (id)name;
- (id)types;
- (void)addTypes:(id)arg1;
- (void)addType:(id)arg1;
- (void)setValue:(id)arg1;
- (void)finalize;
- (void)dealloc;
- (id)initWithName:(id)arg1;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,496 |
package com.google.errorprone.bugpatterns.inject.guice;
import static com.google.errorprone.BugPattern.SeverityLevel.ERROR;
import static com.google.errorprone.matchers.InjectMatchers.GUICE_INJECT_ANNOTATION;
import static com.google.errorprone.matchers.InjectMatchers.JAVAX_INJECT_ANNOTATION;
import static com.google.errorprone.matchers.InjectMatchers.hasInjectAnnotation;
import com.google.errorprone.BugPattern;
import com.google.errorprone.VisitorState;
import com.google.errorprone.bugpatterns.BugChecker;
import com.google.errorprone.bugpatterns.BugChecker.MethodTreeMatcher;
import com.google.errorprone.fixes.SuggestedFix;
import com.google.errorprone.matchers.Description;
import com.google.errorprone.util.ASTHelpers;
import com.sun.source.tree.MethodTree;
import com.sun.tools.javac.code.Symbol.MethodSymbol;
/**
* This checker matches methods that 1) are not themselves annotated with @Inject 2) descend from a
* method that is annotated with @javax.inject.Inject 3) do not descent from a method that is
* annotated with @com.google.inject.Inject
*
* @author sgoldfeder@google.com (Steven Goldfeder)
*/
@BugPattern(
name = "OverridesJavaxInjectableMethod",
summary =
"This method is not annotated with @Inject, but it overrides a method that is "
+ " annotated with @javax.inject.Inject. The method will not be Injected.",
severity = ERROR)
public class OverridesJavaxInjectableMethod extends BugChecker implements MethodTreeMatcher {
@Override
public Description matchMethod(MethodTree methodTree, VisitorState state) {
// if method is itself annotated with @Inject or it has no ancestor methods, return NO_MATCH;
if (hasInjectAnnotation().matches(methodTree, state)) {
return Description.NO_MATCH;
}
boolean foundJavaxInject = false;
for (MethodSymbol superMethod :
ASTHelpers.findSuperMethods(ASTHelpers.getSymbol(methodTree), state.getTypes())) {
// With a Guice annotation, Guice will still inject the subclass-overridden method.
if (ASTHelpers.hasAnnotation(superMethod, GUICE_INJECT_ANNOTATION, state)) {
return Description.NO_MATCH;
}
// is not necessarily a match even if we find javax Inject on an ancestor
// since a higher up ancestor may have @com.google.inject.Inject
foundJavaxInject |= ASTHelpers.hasAnnotation(superMethod, JAVAX_INJECT_ANNOTATION, state);
}
if (foundJavaxInject) {
return describeMatch(
methodTree,
SuggestedFix.builder()
.addImport(JAVAX_INJECT_ANNOTATION)
.prefixWith(methodTree, "@Inject\n")
.build());
}
return Description.NO_MATCH;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,030 |
Home/Breaking/COVID-19 PATIENTS RUN AWAY FROM ISOLATION FACILITIES
BreakingHealth
Chilufya Mukuka Send an email March 24, 2021
Some patients infected with Covid-19 and admitted to isolation facilities have reportedly been running away from them, particularly in Eastern province.
Director for Infectious Diseases Hospital Professor Lloyd Mulenga said that most patients flee the health facilities and later return with severe symptoms thereafter succumbing to the virus.
"When we look at the past week, majority of the mortalities we have been having have been from Eastern province and North-Western province…we have patients admitted, then they run away and when they come back, they come back with severe symptoms and we lose them."
In an exclusive interview with S24 today, Minister of Health Dr. Jonas Chanda said he is not aware of the matter, and that this displeased him.
Dr. Chanda said: "Professor Mulenga will do well to inform me. In Chipata, I wouldn't know which category those patients would be in; whether they have moderate symptoms, because I can't imagine someone who is severely ill on oxygen cylinder on their back running away."
Prof. Mulenga said most of the patients that have died in Lusaka in the last few weeks are due to other comorbities other than the covid-19 virus and have been come from large isolation centers.
"Remember, even when someone has covid doesn't mean that they have no rights. We cannot cage someone to a bed for them not to leave a health facility. There are certain instances when someone has been counseled, they are not well, and they insist that they can isolate at home, against medical advice," he said.
He said this has been rare in Lusaka province especially, but the worrying occurrence has been observed in Eastern province.
US Passes 500,000 COVID Deaths ~ BY Katie Wood
2020 FIC report appalling as PF refuses to comment on it
TB JOSHUA DIES – By Tito Kalama | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 543 |
Q: Android hide app icon programatically I have been trying to hide my app's icon from the app drawer and open it from dialler. The dialler part works perfertly, but i am having issues with hiding it. My problem is, if i hide my activity, it's get destroyed and i can no longer access it from my dialler. I have tried creating a duplicate activity as suggested here, but if i do that then my initial problem of hiding the icon is not solved.
Any ideas would be greatly appreciated.
A: I was doing it wrong. Just set the MainActivity as the launcher and set a duplicate activity in the dialler.
A: Please use this.
PackageManager pkg=this.getPackageManager();
pkg.setComponentEnabledSetting(new ComponentName(this,SplashActivity.class),PackageManager.COMPONENT_ENABLED_STATE_DISABLED,
PackageManager.DONT_KILL_APP);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,277 |
namespace BABYLON {
NodeMaterialBuildStateSharedData::NodeMaterialBuildStateSharedData()
: allowEmptyVertexProgram{false}
{
// Exclude usual attributes from free variable names
variableNames["position"] = 0;
variableNames["normal"] = 0;
variableNames["tangent"] = 0;
variableNames["uv"] = 0;
variableNames["uv2"] = 0;
variableNames["uv3"] = 0;
variableNames["uv4"] = 0;
variableNames["uv4"] = 0;
variableNames["uv5"] = 0;
variableNames["uv6"] = 0;
variableNames["color"] = 0;
variableNames["matricesIndices"] = 0;
variableNames["matricesWeights"] = 0;
variableNames["matricesIndicesExtra"] = 0;
variableNames["matricesWeightsExtra"] = 0;
variableNames["diffuseBase"] = 0;
variableNames["specularBase"] = 0;
variableNames["worldPos"] = 0;
variableNames["shadow"] = 0;
variableNames["view"] = 0;
// Exclude known varyings
variableNames["vTBN"] = 0;
// Exclude defines
defineNames["MAINUV0"] = 0;
defineNames["MAINUV1"] = 0;
defineNames["MAINUV2"] = 0;
defineNames["MAINUV3"] = 0;
defineNames["MAINUV4"] = 0;
defineNames["MAINUV5"] = 0;
defineNames["MAINUV6"] = 0;
defineNames["MAINUV7"] = 0;
}
NodeMaterialBuildStateSharedData::~NodeMaterialBuildStateSharedData() = default;
void NodeMaterialBuildStateSharedData::emitErrors()
{
std::ostringstream errorMessage;
if (!checks.emitVertex && !allowEmptyVertexProgram) {
errorMessage << "NodeMaterial does not have a vertex output. You need to at least add "
"a block that generates a glPosition value.\r\n";
}
if (!checks.emitFragment) {
errorMessage << "NodeMaterial does not have a fragment output. You need to at least "
"add a block that generates a glFragColor value.\r\n";
}
for (const auto& notConnectedInput : checks.notConnectedNonOptionalInputs) {
errorMessage << "input " << notConnectedInput->name << " from block "
<< notConnectedInput->ownerBlock()->name() << "["
<< notConnectedInput->ownerBlock()->getClassName()
<< "] is not connected and is not optional.\r\n";
}
if (errorMessage) {
throw std::runtime_error("Build of NodeMaterial failed:\r\n" + errorMessage.str());
}
}
} // end of namespace BABYLON
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,383 |
Cinnamon Skin (Spanish: Piel canela) is a 1953 Mexican drama film directed by Juan José Ortega and starring Sara Montiel, Manolo Fábregas and Ramón Gay. It was set and partly filmed in Cuba.
Cast
Sara Montiel as Marucha
Manolo Fábregas as Dr. Carlos Alonso
Ramón Gay as Julio Chávez
Felipe de Alba as Dr. Jorge Morales
Rosa Elena Durgel as Alicia Álvarez, enfermera
Fernando Casanova as Paco
Magda Donato as Paciente loca de Alonso
Salvador Quiroz as Don Ernesto
Ismael Larumbe as Antonio Salas Porras
Jorge Casanova
Arturo Corona as Amigo de Antonio
Manuel de la Vega as Amigo de Antonio
Pedro Vargas as Cantante
Rosita Fornés as Cantante
Julio Gutiérrez
Olga Chaviano as Bailarina
Victorio Blanco as Empleado casino
Josefina Burgos as Mujer en casino
Rogelio Fernández as Esbirro de Julio
Ana Bertha Lepe as Empleada carpa
Chel López as Detective
José Muñoz as Cantinero
Rafael A. Ortega
Ignacio Peón as Hombre en casino
Alicia Reyna as Mesera
Joaquín Roche as Espectador gritón carpa
References
Bibliography
Rogelio Agrasánchez. Cine Mexicano: Posters from the Golden Age, 1936-1956. Chronicle Books, 2001.
External links
1953 films
1953 drama films
Mexican drama films
1950s Spanish-language films
Films directed by Juan José Ortega
Films set in Cuba
Films shot in Cuba
1950s Mexican films | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,183 |
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