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King Solomon Hotel London
155/159 Golders Green rd, Golders Green, London NW11 9BX,
Enjoy a continental breakfast in the morning and return at the end of the day for a delicious Italian meal at 3 star King Solomon Hotel London.
Click here to book King Solomon Hotel London
King Solomon Hotel London offers affordable three star accommodation with a selection of 70 single, double, twin, triple and quad sized guestrooms. All of the rooms at King Solomon Hotel London have the convenience of private shower facilities and you will find a colour television, hairdryer, safe and a direct dial telephone inside too. You can also relax and make yourself a cup of tea or coffee and then sit down with your laptop to keep up with emails and surf the internet via the WiFi access.
London King Solomon Hotel is situated in North West London, close to local attractions such as Hampstead Heath and Brent Cross shopping centre. You can take a train on the London Underground from Golders Green and head to Central London and famous attractions such as Oxford Street and Leicester Square. Before heading out from London King Solomon Hotel you can enjoy the continental breakfast, and then return in the evening for a drink in the bar or a delicious Italian meal in the restaurant.
Where is King Solomon Hotel London Located?
King Solomon Hotel London is located in Golders Green, North West London Regional Map, Local Map
There are a number of universities easily accessible from King Solomon Hotel London, including the Hendon campus of Middlesex University, which is reachable via a five minute journey (2 stops on Northern Line), as well as University College London and Birkbeck University (8 stops). You can also to travel to Language Studies International Hampstead (1 stop).
The nearest underground station to London King Solomon Hotel, located in Zone 3, is Golders Green. From here you can take the Northern Line and travel in around twenty minutes to Leicester Square (10 stops). Once here you can enjoy the bright lights, bars, restaurants and cafes of Leicester Square, or walk to the shopping haven that is Oxford Street.
There are a number of popular local bars, cafes and restaurants within walking distance of King Solomon Hotel London, or you could pack up a picnic and head for Hampstead Heath. Shopping fans will love the close proximity of Brent Cross shopping centre too. Golders Green Underground Station is also close by, which will transport you into central London in no time at all.
King Solomon Hotel London Facilities
Direct Dial Phone
Wireless Internet Access In Rooms
Wake Up Service
Alternative accommodation in Golders Green
Golders Green Budget Hotels
All Golders Green accommodation
Universities, Colleges and Language Schools near to King Solomon Hotel London
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,715
|
In Colonial India, at a time of growing friction between the ruling British and the restless Indian populace, a Victorian woman and her young Tamil Indian servant defy convention, class, and heartbreak to investigate what is gained - and lost - by holding life still. Suggested by the life and work of photographic pioneer Julia Margaret Cameron, The Luminist filters 19th century Ceylon through the lens of an English woman, Catherine Colebrook and a 15 year old Tamil boy, Eligius Shourie. Left fatherless by soldiers, Eligius is brought as a servant to the Colebrooks' neglected estate. In the shadow of Catherine's obsession to arrest beauty - to select a moment from the thousands comprising her life in Ceylon and hold it apart from mere memory - Eligius transforms into her apprentice in the creation of the first haunting photographs in history.
Fledgling illustrator and Darcy fanatic Kay Ashton settles in the seaside town of Lyme to finish her book, The Illustrated Mr. Darcy, when a film company arrives to make a new adaptation of Persuasion. Kay is soon falling for the handsome bad boy actor playing Captain Wentworth, but it's the quiet screenwriter Adam Craig who has more in common with her beloved Mr. Darcy. Though still healing from a broken heart, Adam finds himself unexpectedly in love with Kay, but it will take more than good intentions to convince her that her real happy ending is with him.
The first time Julia Beckett saw Greywethers she was only five, but she knew at once that it was her house. Now, twenty-five years later, by some strange chance, she has just become the new owner of the sixteenth-century Wilshire farmhouse. But Julia soon begins to suspect that more than coincidence has brought her there. As if Greywethers were a porthal between worlds, she finds herself abruptly transported back in time. Stepping into seventeenth-century England, Julia becomes Mariana, a beautiful young woman struggling against danger and treachery, and battling a forbidden love for Richard de Mornay, handsome forebear of the present squire of Crofton Hall. Each time Julia travels back, she becomes more enthralled with the past, falling ever deeper in love with Richard...until one day she realizes Mariana's life threatens to eclipse her own--and that she must find a way to lay the past to rest, or risk losing a chance for love in her own time.
I enjoyed Mariana. Great Mailbox! Hope they all turn out to be good reads for you!
Enjoy all your new books. Mariana looks interesting, I loved Winter Sea but her.
Ooh, Susanna Kearsley! I just discovered her writing with The Rose Garden, a very enjoyable read. Have a great week, Laura, and enjoy your new books.
I often Dream of Mr. Darcy.
One of my best friends in all the world, who's not a blogger, told me about The Luminist last week. It's great to see it here; I think I'll need to get it!
Ooh that first book sounds really good. And I've been meaning to read Susanna Kearsley for a while. I look forward to your reviews!
I will read just about anything with Mr. Darcy in the title, that book looks really fun!
Ooh, The Luminist looks good! I'm also captivated by the cover and blurb for Mariana. It does look amazing!
Thanks for sharing, and for visiting my blog. Enjoy your reads!
We have Dreaming of Mr. Darcy in common. Can't wait to see what you think of it. Enjoy your new books!
Mariana has been on my wish list since I read (and loved) The Winter Sea. Enjoy!
Marianna is the one I will be looking out for.
Dreaming of Mr. Darcy caught my eye as I can never get enough of Darcy. Happy reading!
|
{
"redpajama_set_name": "RedPajamaC4"
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| 9,596
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Q: ld cannot find -lz in an empty environment even with LIBRARY_PATH, LD_LIBRARY_PATH and LD_PRELOAD I am trying to create a conda package that includes c code that have to compile with -lz. However, when the package is building, ld cannot find zlib even though I provide it with any paths possible.
As I understand, conda creates almost empty environment, and then fills it with necessary libraries and tools. It also installs zlib, so that there is zlib.h in $BUILD_PREFIX/include/ and libz.so, libz.a in $BUILD_PREFIX/lib.
Compilation itself looks like
$BUILD_PREFIX/bin/x86_64-conda_cos6-linux-gnu-cc -fPIC -g -Wall -O2 -Wc++-compat main.o -o <name> -L. -l<name> -lm -lz -lpthread
x86_64-conda_cos6-linux-gnu-cc is gcc version 7.3.0, and it calls ld defined here as $BUILD_PREFIX/bin/x86_64-conda_cos6-linux-gnu-ld. Then ld falls with an error cannot find -lz.
I tried using
export C_INCLUDE_PATH="$BUILD_PREFIX/include"
export LIBRARY_PATH="$BUILD_PREFIX/lib"
export LD_LIBRARY_PATH="$BUILD_PREFIX/lib"
export LD_PRELOAD="$BUILD_PREFIX/lib/libz.so"
in any combinations, but that did not work.
Are there any other ways to show ld path to the library?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,333
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Sign up for Updates and follow us on Facebook for the latest news and action in the struggle for climate action and justice!
"Why is a human rights organization working on environmental issues?" It was a question we used to get…a lot.
Global Exchange created popular toolkits, distributing thousands at teach-ins, rallies, and on campuses where we were working with large groups of student organizers. We created "No Blood for Oil" t-shirts which very pregnant co-founders Medea Benjamin and Kirsten sold at events and demos by laying them on their bellies to fundraise for campus groups.
As coalition-builders, Global Exchange helped to found United for Peace and Justice and CODEPINK Women for Peace, resurrecting the Gulf War slogan, No Blood for Oil.
In 1992 Nigerian writer, TV producer and environmental activist, Ken Saro Wiwa was imprisoned for leading a nonviolent campaign against petro giants like Royal Dutch Shell. The dirty extraction and highly explosive refinery operations were backed by the Nigerian military, threatening the very survival of the Ogoni people of the Niger Delta. Shockingly, troops waged a violent scorched-earth war against the Ogoni people for their protest, and despite a worldwide call for his release, Saro Wiwa was killed in 1995.
As environmental and human rights abuses continued, Global Exchange's Board of Directors President Walter Turner led our 2-week fact-finding delegation to the Niger Delta to investigate. As Walter recalls, "It was a dangerous place. Oil companies were acting with impunity, and we witnessed the constant gas flares and other daily violations against the Earth and the Ogoni people." The delegation also saw evidence of the California-based Chevron Corporation's participation in armed attacks against peaceful protestors, a local link to these frontline struggles that became the basis for our many years of activism supporting movement-building Justice for Nigeria work, and leading to our own campaign against Chevron.
We also set out to "put a human face on climate change." It was hard for everyday people to relate to something so intangible, yet our national addiction to fossil fuels impacted real places and people. We brought speakers to tour the U.S. from places like the sinking Maldives. Activist Elaine Alexie of the Gwichin First Nation in Alaska, whose traditional hunting and fishing lifestyle was being destroyed by warming temperatures, was one of the featured speakers.
And we took people to see the impacts on the ground through our Reality Tours program. We continued to visit these communities –in the Ecuador Amazon – impacted by the oil industry, raising their urgent call for justice. This work continues today.
It wasn't enough to simply ask people to drive less. The Jumpstart Ford campaign, in partnership with Rainforest Action Network (RAN) set its sights on pressuring decision makers at Ford Motor Company— the quintessential all-American corporation—to stop selling the dirtiest cars on Earth, live up to their reputation, and lead the industry in fuel efficiency.
Splashy actions at auto shows and "adopt a dealer" sit-ins at dealerships educated and activated the general public.
Our automaker focus naturally evolved from initially calling out only Ford Motor Company to declaring our "Freedom from Oil," bringing Canadian Mike Hudema to the Global Exchange family. Known for his book, "An Action a Day Keeps Global Capitalism Away," he launched colorful and creative actions to 'Separate Oil from State,' 'Save Hockey: Stop Climate Change,' and hijacking the 2006 Los Angeles auto show—interrupting the General Motors CEO's speech by asking him to sign a giant pledge sheet committing to make GM the most fuel efficient in the world by 2010. Although GM's Richard Wagoner refused to sign, the media was eager to listen.
We've joined with countless activists at Chevron's headquarters in San Ramon, CA. Antonia Juhasz, well-known author and activist, joined Global Exchange to take on Chevron as our new Energy Program director, coordinating Chevron-affected communities from Indonesia to the Niger Delta, Ecuador and here at home through The True Cost of Chevron network. Antonia led a lot of culture-jamming work mimicking the syrupy Chevron green-washed ads by infusing them with reality, and spearheading an Alternative Annual Report handed out to Chevron shareholders at the annual shareholders meetings.
The True Cost of Chevron network continues to support frontline communities affected by Chevron to raise their voices and call for accountability and action from this multi-billion dollar company.
In 2010, when BP's corporate negligence resulted in the Deepwater Horizon disaster in the Gulf Coast, we went there to investigate BP executives' and governmental claims that cleanup efforts were working. Antonia researched and wrote the book "Black Tide: The Devastating Impact of the Gulf Oil Spill," and toured the country with victims and the scant few scientists BP hadn't silenced.
In 2006, we began to look North, to Alberta, Canada's once pristine boreal forest, threatened by the dirtiest form of energy mining, the tar sands. From the Canadian perspective, the exploitation of the forest was directly linked to the U.S. addiction to oil, Canada's number one oil customer. Its impact on climate change was unparalleled. It was also wreaking destruction on First Nations homes and livelihoods and leaving an environmental scar so large as to be visible from space.
Under the direction of Global Exchange's next Executive Director, Carleen Pickard, the fight against the tar sands continued, becoming a powerful, national fight and movement against the Keystone XL Pipeline. The Keystone battle become one of the biggest political fights over energy in decades, and when President Obama refused to approve the permit one of the climate movement's greatest victories.
Now that is all at risk. President Trump opened the path for construction of both the Keystone XL and Dakota Access pipelines in his first 100 days in office. The fight continues.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 2,664
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The Wachbataillon (full name: Wachbataillon beim Bundesministerium der Verteidigung (WachBtl BMVg) (Guard Battalion at the Federal Ministry of Defence) is the German Bundeswehr's honour guard. The Wachbataillon number about 1,000 soldiers stationed in Berlin. It consists of seven active companies (see list below) and belongs to the Streitkräftebasis (Joint Service Support Command) of the Bundeswehr. The soldiers of the Wachbataillon often refer to themselves as Protter or Protokollsoldaten, meaning protocol soldiers.
Mission
The primary mission of the Wachbataillon is to perform the military honours for the German president, the German Chancellor, the Federal Minister of Defence and the Inspector General of the Bundeswehr during state visits or on comparable occasions.
The Wachbataillon executes the Großer Zapfenstreich ("Grand Tattoo") on special occasions (for example on the 50th anniversary of the Bundeswehr in front of the Reichstag in Berlin on the night of 26 October, 2005) or takes part in events like the ceremonial oath of the Bundeswehr ceremony, parades, state funerals, military tattoos and shows with its drill team which is the best trained special unit of the battalion.
A secondary mission is to perform (ceremonial) guard duty at the Ministry of Defence and other high-profile public places and to protect and guard members of the German government and the Ministry of Defence.
Another secondary mission is to secure and defend the alternate seat of the federal German government in conjunction with the Federal Police forces. Therefore, all soldiers of the Wachbataillon are trained as infantrymen and do regular exercises on military training areas (Truppenübungsplatz) in addition to their protocol duties.
Recruitment
Until recently, the Wachbataillon protocol- and security companies only admitted male personnel that had a body height ranging from up to , normal eyesight and body weight and are generally in good health. Wachbataillon personnel (especially protocol soldiers) still are not allowed to grow beards or moustaches as well as wearing glasses or being overweight (). These are internally known as the "three forbidden B's" ().
Since 2009, the 1./WachBtl BMVg staff- and supply company has admitted female personnel. In the for Karl-Theodor zu Guttenberg in March 2011, the first female soldier performed the role of torchbearer. Until the end of conscription in Germany in July 2011, about 80 percent of Wachbataillon's personnel were conscripts. The requirements for Wachbataillon personnel were relaxed to a body height ranging from up to , no visible visual aids and having a good general fitness. In March 2012 the first female officer took command of a platoon of the Wachbataillon.
The soldiers of the Wachbataillon are from all three German armed services (i.e. army, navy and air force) and have the uniforms from all forces at their disposal to be able to perform any drill mission on every occasion. Green berets are used for army uniform, the (dark) blue beret for air force and the navy wears traditionally the sailor suit (). The beret badge for army and air force uniform shows the letter "W" as an abbreviation for Wachbataillon.
Equipment
For reasons of tradition the Wachbataillon is the only unit of the Bundeswehr still using the Karabiner 98k bolt action rifle because the traditional Prussian rifle drills can be executed better with that historical weapon. In 1995 remaining swastikas and other Nazi-era markings were removed from these rifles, after criticism regarding the presence of such symbols on Wachbataillon kit by the SPD parliamentary party. As of 29 October 2011 women are allowed to participate in the drill training with the Karabiner 98k.
During normal duty the Wachbattalion uses the Heckler & Koch G36 assault rifle and other kit like the other units of the Bundeswehr. Since 2006 the 1./WachBtl BMVg is the last unit in the Bundeswehr using ten Feldhaubitze 105 mm guns in the role of salute guns.
Tradition
The Wachbataillon has been granted exemption of the Traditionserlass and is thus continuing the traditions of the 1st Foot Guards and the Infantry Regiment 9 Potsdam, the only unit in the Bundeswehr to officially have such ties. Veterans of these regiments are joined with active and former members of the Wachbataillon in the Semper Talis Union.
Structure
The Wachbataillon is stationed at the Julius Leber barracks in the centre of Berlin. Since 2001 the Wachbataillon is part of the Streitkräftebasis. The battalion is organized into seven active companies (four of them honour guard companies) and two inactive military reserve companies.
List of Wachbataillon's companies
See also
Beijing Garrison Honor Guard Battalion, China
Kremlin Regiment, Russia
Republican Guard, France
Representative Honor Guard Regiment of the Armed Forces, Poland
Friedrich Engels Guard Regiment, East Germany
SS-Verfügungstruppe, Nazi Germany
References
External links
Official information site about the Wachbataillon (German)
Semper Talis Union (German)
Protokollsoldat aus Leidenschaft - Zugführerin im Wachbataillon der Bundeswehr
Battalions of the Bundeswehr
Guards regiments of Germany
Military units and formations established in 1957
Guards of honour
Joint Support Service (Germany)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 8,689
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<!DOCTYPE html>
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|
{
"redpajama_set_name": "RedPajamaGithub"
}
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ASU Football: Freshmen play big role in upset of No. 16 Utah
The young guns played big this afternoon.
By Alec Henden Nov 3, 2018, 6:42pm MST
Share All sharing options for: ASU Football: Freshmen play big role in upset of No. 16 Utah
Nicole Hernandez/House of Sparky
Listening to both Arizona State head coach Herm Edwards and defensive coordinator Danny Gonzales all season they've preached the same storyline about their defense.
They're young.
ASU's opening night defensive unit featured just three players that had taken Division I snaps entering the season.
The unit has been much improved throughout the season, and have gone through a fair amount of growing pains. Seeing the process turn into a strong performance against No. 16 Utah is just that much sweeter.
Freshman linebacker Merlin Robertson and freshman Aashari Crosswell each collected their first career interceptions. Redshirt freshman linebacker Tyler Johnson recorded seven tackles.
"I can't say enough about our freshman guys," Edwards said. "Aashari got an interception, finally."
Crosswell's interception was a fairly accurate representation of his freshman season at safety. He was beat by the receiver on the route, was spinning in circles, but ended up in the right spot to make the play. Crosswell then proceded to run the ball back 47 yards, setting up ASU's second scoring drive.
"When the ball was in the air the sun was in my eyes and I couldn't see nothing," Crosswell said. "It was just a great play by me and it was great play by the people who was rushing for my first pick."
Crosswell received some good-natured jeering from his teammates for his comments, but he mentioned before the press conference that it was his first time in the setting.
Johnson was also impressed by the work of his teammate.
"We got pressure because he (Huntley) started to scramble a little bit and got nervous then he just chucked it down field," Johnson said. "I honestly didn't even think Aashari would catch it."
Crosswell was happy to make the play after dropping previous chances to intercept passes. The advice he received from senior quarterback Manny Wilkins was simple, enjoy the moment.
The defense held Utah to 20 total points and completely dominated the second half.
The third quarter defense was especially crucial. In their previous five games, ASU had allowed at least 14 points out of the break. They held the Utes to a field goal today.
It started with the big interception from Robertson. He's been a force patrolling the middle of this field for ASU and was in the right position to make the big play. It was a crucial catalyst to the defense's second half surge.
"We needed to play good in the third quarter," Edwards said. "Our home crowd helped us and then we got going again. Our offense found its way to make a couple big plays and then you find yourself in the lead and it's just a matter of managing the clock."
The players knew they had to prove to themselves that they could execute out of the break, and they were able to.
"In the recent games we have had trouble in the third quarter," Johnson said. "So coming out of halftime we need to stay warm and keep energy the fans helped us out a lot. We just kept pushing and pushing and the more turnovers we got and tackles for losses kept the energy up."
Gonzales was proud of the effort from his young players, and sees that they're beginning to make the plays they weren't earlier in the season.
"We're starting to mature," Gonzales said. "Our younger kids are starting to play better, they're starting to understand the concepts of what we're trying to do. They're playing hard and they're playing physical."
If ASU is going to capture the division title they will need to win out. While this seems unlikely, it is very much possible. And if the freshmen standouts continue to perform like they did against the Utes, the Sun Devils might be making an unexpected trip to the Bay Area at the end of November.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: forceChatter:feed is not working as expected I am trying to use forceChatter:feed from (https://developer.salesforce.com/docs/component-library/bundle/forceChatter:feed/documentation).
The below code is working fine and the result is as expected.
<aura:component controller="chatterController" implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,forceCommunity:availableForAllPageTypes,force:lightningQuickAction" access="global" >
<aura:handler name="init" action="{!c.doInit}" value="{!this}" access="global" />
<aura:attribute name="contactList" type="id" />
<forceChatter:feed type="Record" subjectId="a091y000000fXREAA2" /> //Passing Record Id directly
</aura:component>
Whereas if I pass the SubjectId with some attribute value, results are not getting displayed as expected.
<aura:component controller="chatterController" implements="force:appHostable,flexipage:availableForAllPageTypes,flexipage:availableForRecordHome,force:hasRecordId,forceCommunity:availableForAllPageTypes,force:lightningQuickAction" access="global" >
<aura:handler name="init" action="{!c.doInit}" value="{!this}" access="global" />
<aura:attribute name="contactList" type="id" />
<forceChatter:feed type="Record" subjectId="{!v.contactList}" /> //Passing Variable which has record-id(a091y000000fXREAA2) directly
</aura:component>
Not able to find what's the exact reason why the results are not displayed properly when I pass subjected with a variable which has the record Id.
A: I have found the answer to my question.
First, let me explain why the above thing is not working. As I am getting Id from the controller and in return, the Id is getting returned from the Apex controller and it's taking time as it is an asynchronous process and hence when the tag is getting rendered it is getting rendered as undefined.
Hence, to solve this issue, add this component in the controller and dynamically create it as follows:
$A.createComponent(
"forceChatter:feed", {
"context": "RECORD",
"subjectId": response.getReturnValue()
},
function(recordFeed) {
if (component.isValid()) {
var body = component.get("v.body");
body.push(recordFeed);
publisher.set("v.body", body);
}
});
This will solve your issue like a charm!
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
In the past years there has been an increasing interest in propulsion on small scales, both theoretically and experimentally. Interest to some natural modes of low-Reynolds-number locomotion has a long record in applied mathematics \cite{lighthill75,childress81} and underlying mechanisms of propulsion powered by flexible elastic filament \cite{taylor51,hancock53,beating}, rotating helical flagellum \cite{hancock53,rotating,AR07}, beating cilia \cite{GL92}, surface distortions of non-flagellated squirmers \cite{ESBM96,SS96} and some others are quite well understood. Artificial nature-inspired propellers, powered by either beating or rotating filaments \cite{exper} were recently fabricated and tested vs. the theoretical predictions; performance of such devices was also studied numerically via particle-based algorithms \cite{particle}.
Theoretical work on zero-Reynolds-number locomotion strategies (that are not necessarily biomimetic) for artificial micro-swimmers has attracted some attention quite recently and several modes of propulsion, such as three-link Purcell's swimmer \cite{3link_swim}, its ``symmetrized" version \cite{AR08} and generalized N-link swimmer \cite{dAK04}, three-sphere propeller \cite{3sph_swim}, swimmer propelled by arbitrary non-retractable cyclic shape strokes \cite{shape_stroke}, two-sphere ``pushmepullyou" \cite{pmpu}, surface treadmilling \cite{LKGA07} and surface tank-treading \cite{LK08} and others, were proposed and studied in details. A comprehensive review that provides the reader with state-of-the-art in low-Reynolds-number locomotion can be found in \cite{LP09}.
However, the aforementioned works address the propulsion through an unbounded Newtonian viscous liquid. Some bacterial cells are used to swim though complex and highly heterogeneous viscous environments rather than Newtonian viscous liquids, such as marine water. For instance, clinically important spirochetes navigate efficiently through dense extracellular matrix in host tissues and cross the blood-brain barrier \cite{spirochet,RL00}. Moreover, prospective design of artificial microrobots capable of propulsion through soft tissues, digestion tract, spinal canal, etc. relies on ability to efficiently navigate through complex heterogeneous environment. Some experimental observations of biological propulsion in gel-like polymer solutions cannot be explained in the framework of standard theories. The traditional theories for bacterial motion through purely viscous response predict that the swimming speed (at constant torque produced by flagellar motors) monotonically decreases with viscosity. Although the naive intuition suggests that, for instance, a rotating helical filament would move like a corkscrew when propelled through a very viscous liquid, this is actually never the case, no matter how large is the viscosity, as the zero-Reynolds-number swimming is purely geometric \cite{purcell77}. However, the swimming speed of Pseudomonas aeruginosa in polyvinylpyrollidone solutions increases with viscosity up to a certain point and thereafter decreases \cite{SD74}. Other flagellation types of externally flagellated bacteria (\eg Bacillus megaterium, Escherichia coli, Serratia marcescens, Sarcina ureae, Spirillum serpens and Thiospirillum jenense) also exhibit an increase in swimming speed in viscous solutions \cite{SD74}. The disagreement with traditional theories is more drastic for spirochetes, lacking external flagella, in which a helical or plane wave of the cell body moves backward yielding rolling of the cell body and forward propulsion. A remarkable feature of spirochete motility is that cells swim faster in a high viscosity gel-like media than they do in low-viscosity aqueous media \cite{KD75,spiro_swim}. The swimming speed of Leptospira interrogans monotonically increases with viscosity in medium supplemented with methylcellulose until the viscosity exceeds 300 cP \cite{KD75}. In low-viscosity, aqueous medium, T. denticola is observed to rotate without translating \cite{RLKNGS97}.
Berg and Turner \cite{BT79} investigated the effect of viscosity on propulsion of various microorganisms and suggested that the propulsion enhancement in \emph{in-vitro} experiments can be caused by loose and quasi-rigid networks formed by entangled linear polymer molecules such as methylcellulose. They argued that the enhanced propulsion is a result of flagellum pushing on the polymer network and, therefore, the propulsion resembles the motion of a screw boring through wood. Merely increasing the viscosity of the medium using a non-gel-forming sucrose polymer, such as Ficoll, did not enhanced motility, which indicated that propulsion enhancement may be dependent on viscoelastic rheology of the medium. However, recent findings of \cite{elastic}, who examined the effect of vicsoelasticity on propulsion for various rheological models of the liquid, showed that addition of elastic response to material constitutive equation does not yield the enhancement of propulsion. The propulsion through viscoelatsic liquid is always hindered (for the prescribed swimming gait), both velocity- and efficiency-wise, when compared to locomotion through viscous (Newtonian) solvent of the same viscosity.
The first attempt to explain the propulsion enhancement theoretically, following \cite{BT79}, was made in \cite{MK02} for externally flagellated bacteria and later extended in \cite{NAGM06} to spirochetes. They suggested that the local viscous resistance to the motion of the flagellum normal to its surface is considerably higher than that corresponding to the tangential motion (note that in purely viscous liquid the ratio of viscous resistances of a slender body is $\sim$2). Indeed, the normal motion of the helical threads yields pushing on polymer aggregates surrounding flagellum, while tangential motions yield no perturbation to the microstructure. Therefore, in gel-like media with a sufficient traction, the flagellum is propelled much like a corkscrew, without slippage. On the other hand, increase in the drag on the large bacterium head results in decay of the propulsion speed starting at some viscosity in agreement with previous experiments. In accord with these arguments is also the monotonic increase of the swimming speed of the organisms lacking a large passive head, such as spirochetes. When surrounded by a dense gel-like environment, the helical or flat waves of their body - depending on the spirochete species - get sufficient traction to propel the cell body through the medium like a corkscrew, i.e. without slippage. If traction is not sufficient, then the cell body slips against the external medium and the cell translates more slowly. Although the model proposed in \cite{MK02,NAGM06} shows limited qualitative agreement with experimental results, it involves two unknown phenomenological parameters -- ``apparent viscosities" of unclear physical origin.
The aim of the present study is to develop a more rigorous theoretical framework of propulsion through heterogeneous viscous media. In some sense, the working hypothesis behind this theory follows the original Berg and Turner's proposal \cite{BT79}, \ie the flagellum is pushing on the stationary matrix of obstacles, whereas the interaction is indirect, \ie mediated by a viscous solvent. The theory is based on solutions of averaged equations of viscous flow through random sparse array of obstructions such as fibers or spheres, mimicking heterogeneous gel-like polymer solutions where dense cores of the microgel particles (or chain-like aggregates in methylcellulose solutions \cite{gotlieb}) are surrounded by a viscous solvent containing dissolved polymer chains \cite{gel_struct}. We are interested to address the effect of the sparse network of stationary obstacles (\eg fibers or spherical cores) embedded into a viscous incompressible Newtonian solvent on propulsion of a force-(torque-)free swimmer. The properly averaged hydrodynamic fields (\ie pressure and velocity) in such media are governed by the \emph{effective Brinkman media} approximation \cite{brinkman47},
\begin{equation}
-\nabla p+\mu \Delta \te{u}=\mu \te{\kappa} \cdot \te{u}\:, \label{eq:brinkman}
\end{equation}
where $\te{u}$ is the average velocity field satisfying the continuity equation for the incompressible liquid, $\nabla\cdot\te{u}=0$, $p$ is the average pressure field and $\te{\kappa}$ is the tensorial damping coefficient. The essence of this mean field model is that, on average, the fluid in proximity to a stationary obstacle experiences a damping body force proportional to the local velocity accounting for the influence of the neighboring objects on the flow. By averaged quantities we mean ensemble averages over all possible arrangements of the surrounding obstructions about a ``test" obstacle. Far from the ``test" obstacle the velocity gradients are weak and the equation (\ref{eq:brinkman}) reduces to the differential form of Darcy's equation describing the flow through porous media and thus $\te{\kappa}$ is taken to be the Darcy resistance (or inverse of permeability). Spatially isotropic random matrix of obstacles can be described by a scalar resistance, $\kappa_{ij}=\alpha^2\:\delta_{ij}$ with $\delta_{ij}$ being the identity tensor. Thus, $\alpha^{-1}$ defines a new length scale related to the stationary obstacles (usually referred as ``screening" or shielding" length), so that for $r<\alpha^{-1}$ the velocity field is Stokesian, while for $r>\alpha^{-1}$ the velocity satisfies Darcy's equation. In particular, the fundamental solution of the singularly forced Brinkman equation (\ie flow driven by a point force $f_i$ acting at $\te{x}_0$) are identical to these corresponding to (Fourier-transformed) equations of oscillatory Stokes flow where the frequency parameter $\alpha^2=-\mathrm{i} \omega/\nu$ replaces the hydrodynamic resistance \cite{KK91}
\begin{equation}
u_i(\te{x})=\frac{1}{8\pi\mu}\: \mathcal{G}_{ij}(\te{x},\te{x}_0)\:f_i\:,\quad \mathcal{G}_{ij}=A\:\frac{\delta_{ij}}{r}+B\:\frac{\widehat{x}_i\:\widehat{x}_j}{r^3}\;, \label{eq:pointforce_brink}
\end{equation}
with
\begin{eqnarray}
A&=&2\re^{-\rho}\:(1+\rho^{-1}+\rho^{-2})-2\rho^{-2}\:, \nonumber \\
B&=&-2\re^{-\rho}\:(1+3\rho^{-1}+3\rho^{-2})+6\rho^{-2}\:,\nonumber
\end{eqnarray}
where $\widehat{\te{x}}=\te{x}-\te{x}_0$, $r=|\widehat{\te{x}}|$ and $\rho=\alpha r$. The fundamental solution for the pressure field corresponding to the Stokes flow \cite{KK91} remains unchanged. As expected, for $\alpha\rightarrow 0$ Stokes flow solution is recovered as $A,\:B\rightarrow 1$. Thus, the flow through random matrix of obstructions due to a point force decays like $r^{-3}$ (due to exponential screening of hydrodynamic interaction via $\re^{-\alpha r}$ in the above ``shielded Stokeslet" solution) vs. $r^{-1}$ decay of the classical point force (Stokeslet) solution in viscous incompressible liquid.
The dumping coefficient ${\te \kappa}$ in (\ref{eq:brinkman}) can be deduced theoretically for some representative obstacle arrangements by relating it to the total drag on the individual elements comprising the network \cite{SG68, Howells74}. The calculation is based on the fact that the mean resistance force per unit volume is equal to (minus) the number density of obstructions times the mean drag force $\te{F}_1$ exerted on a single obstruction in the uniform flow $\te{U}$ though a sparse matrix, $\nabla p=n \te{F}_1=\mu \alpha^2\te{U}$. For the simplest model system of sparse random matrix composed of stationary spherical obstructions of radii $a$, so that $\te{F}_1=6\pi \mu a \te{U}$, we arrive at $\alpha^2=9\phi/2a^2$, where $\phi$ is the volume fraction or concentration of the obstacles. Self-consistent calculations of the components of $\te{\kappa}$ corresponding to fibrous media for various spatial arrangement of long fibers, can be found in \cite{SG68}.
It should be noted that effective media approximation (\ref{eq:brinkman}) was originally derived in \cite{brinkman47} from heuristic arguments. However, it was shown later to provide the leading correction to the Stokes drag on a test sphere for random distribution of stationary spherical obstacles in a rigorous theory where the localized resistance due to neighboring obstacles was taken into account in the multi-particle expansion scheme for small concentration of obstacles \cite{Howells74}. The excellent agreement between the second-order (in concentration) multi-particle theory and the theory based on (\ref{eq:brinkman}) suggested that the latter may serve as an accurate approximation even in the case of moderate obstacle concentration. The numerical results based on Stokesian Dynamics suggested that (\ref{eq:brinkman}) accurately describes the flow through the random arrays of spheres with up to 30\% (by volume) fraction of obstructions \cite{DHSK95}.
In this work we consider several locomotion modes extensively studied in the past and relevant for propulsion of flagellated as well as non-flagellated organisms. First, we consider the G.~I. Taylor's undulating deformable sheet propagating small-amplitude traveling waves along its surface \cite{taylor51} and propelled through heterogeneous viscous medium whereas hydrodynamics is governed by (\ref{eq:brinkman}). The study of propulsion of non-flagellated swimmers powered by small-amplitude surface distortions (\ie squirmers) is greatly simplified via the use of Lorentz reciprocal theorem. Finally, we propose a local resistive force theory of propulsion through the effective media and test the predictions of the theory using rigorous numerical simulation of rotating helical filament through a sparse array of stationary spherical obstructions embedded within viscous Newtonian solvent.
\section{Propulsion of an undulating planar sheet}
\subsection{Transverse distortions}
The 2D analog of the flagellum-powered swimming is the infinite sheet along which the traveling waves are propagating. G.~I. Taylor was the first to consider this model problem in the limit of long-wavelength (or, alternatively, small-amplitude) waves \cite{taylor51}. The same model was considered in \cite{elastic} for modeling propulsion in viscoelastic liquid. The extension of the asymptotic small-amplitude analysis towards the case when the flow around the sheet is governed by (\ref{eq:brinkman}) is rather straightforward. We first consider purely normal mode of deformation, so that in the reference frame fixed with sheet the position of the material points follow
\begin{equation}
y_m=b \sin{(kx-\omega t)}\:,\qquad x_m=x\:,\label{eq:normal}
\end{equation}
where $b$ is the amplitude of modulation, $k$ -- the wavenumber and $\omega$ is the frequency of the wave. The traveling wave is propagating in the positive x-direction with the wave speed $c=\omega/k$ (the schematic of the problem is shown in Fig.~\ref{fig:schematic}).
\begin{figure}[t]
\includegraphics[scale=0.65]{fig1
\caption{(Color online) Schematic of the undulating sheet (red, solid) embedded in stationary random matrix of obstructions (blue dots). The sheet is propagating a traveling transverse wave in the positive $x$ direction with the wave speed $c=\omega/k$ and propelled to the left with velocity $U$. \label{fig:schematic}}
\end{figure}
Since the problem is 2D we recast the equation of motions (\ref{eq:brinkman}) and the no-slip boundary conditions in terms of the streamfunction, $\psi$.
The streamfunction is defined by $u=\partial_y \psi\:,v=-\partial_x \psi$, where $\te{u}=\{u,v\}$ so that the incompressibility condition $\nabla\cdot \te{u}=0$ is trivially satisfied.
Assuming that the arrangement of the obstructions is random and isotropic (random arrangement of fibers, aligned in all directions or spherical obstacles in a random arrangement), the damping coefficient $\kappa_{ij}$ in (\ref{eq:brinkman}) is diagonal and equal to, $\alpha^2 \delta_{ij}$, where $\alpha$ has the dimensions of inverse length. Then, applying curl on both sides of (\ref{eq:brinkman}) yields
\begin{equation}
\Delta \left(\Delta\: \psi-\alpha^2\:\psi\right)=0\:,\label{eq:brinkman1}
\end{equation}
where $\Delta\equiv\nabla^2$.
The no-slip boundary conditions for the velocity components at the \emph{deformed} sheet read
\begin{equation}
u=0\:,\qquad v=-b\omega\cos{(kx-\omega t)}\:.\label{eq:bc1}
\end{equation}
Obviously, purely transverse surface distortions do not preserve surface area, as $\nabla_s\cdot\te{u}\ne0$, where $\nabla_s=(\te{I}-\te{n}\bn)\cdot\nabla$ is the surface gradient operator, and cannot describe deformation of flexible but incompressible sheet. In the latter case, the velocity distribution is given by a combination of transverse and tangential distortions. In the leading approximation the 2D incompressibility condition ($\partial u_s/\partial s+\kappa u_n=0$, $s$ is the arc length measured in the direction of tangent unit vector $\te{s}$) yields $\partial_x u+y''v=0$, that leads to \cite{taylor51}
\begin{equation}
u=\frac{1}{4}b^2 k \omega \cos{(2kx-2\omega t)},\: v=-b\omega\cos{(kx-\omega t)}\:,\label{eq:bc1a}
\end{equation}
We will discuss the propulsion by tangential distortions separately in the next section.
We expand the velocity at the deformed sheet in terms of its value at the \emph{undistorted} surface (at time $t=0$, the choice of time is arbitrary) and choose the following characteristic scales: $k^{-1}$ for length, $b\omega$ for velocity and $(b\omega k)^{-1}$ for time. In terms of the scaled variables we obtain the following problem for the nondimensional streamfunction
\begin{eqnarray}
\Delta\left(\Delta-\alpha_*^2 \right)\psi=0\:, & \qquad y>0\:, & \label{eq:brinkman2}\\
\nabla \psi+\varepsilon \sin{x}\:\nabla \partial_y \psi=\cos{x}\: \te{e}_x\:, & \qquad y=0\:,& \label{eq:bc2}
\end{eqnarray}
where $\alpha_*=\alpha/k$ is the scaled resistance and $\varepsilon=bk\ll1$ is the scaled amplitude of the wave. We now expand the solution in series of powers of $\varepsilon$, $\psi=\psi_0+\varepsilon \psi_1+\ldots$. Submitting this expansion into (\ref{eq:brinkman2}-\ref{eq:bc2}) and matching terms of the same powers $\varepsilon$ we can construct the asymptotic solution. The net displacement will be determined by the $x$-velocity component of the flow far from the sheet,
\begin{equation}
u_\infty=\partial_y \psi\:,\quad y \rightarrow \infty\:.
\end{equation}
In the leading order $\mathcal{O}(\varepsilon^0)$ (\ref{eq:brinkman2}-\ref{eq:bc2}) produces
\begin{eqnarray}
\Delta\left(\Delta-\alpha_*^2 \right)\psi_0=0, &\qquad y>0, & \label{eq:brinkman3}\\
\nabla \psi_0=\cos{x}\: \te{e}_x, &\qquad y=0. &\label{eq:bc3}
\end{eqnarray}
The zeroth order solution is readily found as
\begin{equation}
\psi_0=\left(A_0 \re^{-\sqrt{1+\alpha_*^2}\:y}+B_0 \re^{-y} \right)\,\sin{x}\:,\label{eq:psi0}
\end{equation}
where the constants $A_0$ and $B_0$ are determined from (\ref{eq:bc3}),
\begin{equation}
A_0=-\frac{1}{\sqrt{1+\alpha_*^2}-1}\:,\quad B_0=\frac{\sqrt{1+\alpha_*^2}}{\sqrt{1+\alpha_*^2}-1}\:.\label{eq:A0B0}
\end{equation}
The propulsion velocity, as in the case of purely viscous liquid appears at the next order (because of symmetry $\varepsilon \rightarrow -\varepsilon$) as $\partial_y \psi_0\rightarrow 0$ as $y\rightarrow \infty$.
At $\mathcal{O}(\varepsilon)$ we obtain the following problem
\begin{eqnarray}
\Delta\left(\Delta-\alpha_*^2 \right)\psi_1=0\:, &\qquad y>0, &\label{eq:brinkman4}\\
\nabla \psi_1=-\sin{x} \nabla \partial_y \psi_0\:, &\qquad y=0. &\label{eq:bc4}
\end{eqnarray}
Substitution of the zeroth order solution (\ref{eq:psi0}) into (\ref{eq:bc4}) at $y=0$ yields
\begin{eqnarray}
\partial_y \psi_1&=&-\sin{x}\,\partial_{yy} \psi_0=\nonumber \\
&& -\left[(1+\alpha_*^2)A_0+B_0\right]\left(\frac{1}{2}-\frac{1}{2}\cos{2x}\right), \label{eq:bc4a} \\
\partial_x \psi_1&=&-\sin{x}\,\partial_{yx} \psi_0=0, \label{eq:bc4b}
\end{eqnarray}
as $\partial_y\psi_0=0$ at $y=0$. This suggests the form of the solution for $\psi_1$
\begin{equation}
\psi_1=C_1 y+\left(A_1 \re^{-\sqrt{4+\alpha_*^2}\:y}+B_1 \re^{-2y} \right)\,\cos{2x}\:.\label{eq:psi1}
\end{equation}
Substitution of (\ref{eq:psi1}) into (\ref{eq:bc4a}-\ref{eq:bc4b}) and using (\ref{eq:A0B0}) yields the unknown coefficients $A_1,\:B_1$ and $C_1$,
\[
A_1=-B_1=\frac{\sqrt{1+\alpha_*^2}\:\left(2+\sqrt{4+\alpha_*^2}\right)}{2\alpha_*^2},\; C_1=\frac{\sqrt{1+\alpha_*^2}}{2}\:
\]
The coefficient of the linear term, $C_1$, gives the velocity far from the undulating sheet,
\[
u_\infty=\left.\partial_y \psi_1\right|_{y\rightarrow\infty}=-\frac{1}{2}\left[(1+\alpha_*^2)A_0+B_0\right]=\frac{\sqrt{1+\alpha_*^2}}{2}.
\]
Thus, the sheet is propelled to the negative $x$ direction (propulsion velocity, $U$, is equal to minus $u_\infty$) as in the case of locomotion through viscous liquid without obstructions. Returning to the dimensional quantities, we obtain
\begin{equation}
u_\infty=\frac{(bk)^2 c}{2}\: \sqrt{1+(\alpha/k)^2}\:, \label{eq:Unormal}
\end{equation}
where $c=\omega/k$ stands for the speed of traveling wave. The classical Taylor's results is recovered at the limit $\alpha \rightarrow 0$. The propulsion velocity scaled with the wave speed is plotted in Fig.~\ref{fig:Unormal}. The crossover from Stokesian propulsion ($u_\infty=(bk)^2 c/2$) to that dominated by hydrodynamic resistance due to obstructions occurs at $\alpha_*\simeq 1$. In other words, propulsion powered by short wavelength surface distortions ($\alpha/k<1$) is not affected by the obstructions, while locomotion by long wavelength distortions (comparative to $\alpha^{-1}$) is enhanced, as the swimmer is experiencing the mean resistance due to stationary obstacles.
\begin{figure}[t]
\includegraphics[scale=0.85]{fig2
\caption{Propulsion velocity $u_\infty/(bk)^2 c$ vs. the scaled mean resistance due to stationary obstructions, $\alpha/k$ (log-linear plot). The crossover from Stokesian regime to the enhanced regime dominated by the network hydrodynamic resistance occurs when screening length $\alpha^{-1}$ becomes comparable to the wavelength $k^{-1}$. \label{fig:Unormal}}
\end{figure}
\subsection{Tangential distortions}
Let us consider now the effect of stationary obstacles on propulsion driven by purely tangential surface distortions (swimming gait of some non-flagellated microorganisms such as cyanobarcteria, \cite{ESBM96,SS96}). For this purpose we use the same model problem of the 2D sheet propagating traveling waves of compression/extension along its surface in the positive $x$ direction,
\begin{equation}
y_m=y\:,\quad x_m=x+b \sin{(kx-\omega t)}\:,\label{eq:tang}
\end{equation}
Similarly to the analysis of the transverse waves (Eqs. \ref{eq:brinkman2}-\ref{eq:bc2}), for purely tangential traveling waves we obtain the following problem in terms of the dimensionless stream function $\psi$ at the \emph{undeformed} surface:
\begin{eqnarray}
\Delta\left(\Delta-\alpha_*^2 \right)\psi=0, &\qquad y>0, & \label{eq:brinkman5}\\
\nabla \psi+\varepsilon \sin{x}\:\nabla \partial_x \psi=-\cos{x}\: \te{e}_y, &\qquad y=0. & \label{eq:bc5}
\end{eqnarray}
Substituting the asymptotic expansion $\psi=\psi_0+\varepsilon\psi_1+\ldots$ into (\ref{eq:brinkman5}-\ref{eq:bc5}) results at the
leading order $\mathcal{O}(\varepsilon^0)$ in the following boundary conditions
\begin{equation}
\nabla \psi_0=-\cos{x}\: \te{e}_y\:,\qquad y=0\:,\label{eq:bc6}
\end{equation}
while at any order $\psi_n$ should obey (\ref{eq:brinkman5}).
The zeroth order solution is
\begin{equation}
\psi_0=\left(A_0 \re^{-\sqrt{1+\alpha_*^2}\:y}+B_0 \re^{-y} \right)\,\cos{x}\:,\label{eq:psi00}
\end{equation}
where the constants are determined from by (\ref{eq:bc6}) and read
\[
A_0=-B_0=\frac{1}{\sqrt{1+\alpha_*^2}-1}\:.
\]
The net propulsion velocity is again restricted to the next order $\mathcal{O}(\varepsilon)$ as $\partial_y \psi_0\rightarrow 0$ as $y\rightarrow\infty$.
At $\mathcal{O}(\varepsilon)$ the boundary conditions for $\psi_1$ at $y=0$ read
\[
\nabla \psi_1=-\sin{x} \nabla \partial_x \psi_0\:,\qquad y=0\:.
\]
Substituting the coefficients $A_0,\,B_0$ we find that at $y=0$
\begin{eqnarray}
\partial_y\psi_1&=&-(\sqrt{1+\alpha_*^2}A_0+B_0)\sin^2{x}=-\left(\frac{1}{2}-\frac{1}{2}\cos{2x}\right), \nonumber \\
\partial_x\psi_1&=&(A_0+B_0) \cos{x}\sin{x}=0. \nonumber
\end{eqnarray}
These boundary conditions suggest the form of the solution for $\psi_1$ satisfying Eq.~\ref{eq:brinkman5}:
\[
\psi_1=-\frac{y}{2}+\left(A_1 \re^{-\sqrt{4+\alpha_*^2}\:y}+B_1 \re^{-2y} \right)\,\sin{2x}\:,
\]
and the remaining constants are readily found
\[
A_1=-B_1=-\frac{1}{2\sqrt{4+\alpha_*^2}-4}\:.
\]
The scaled fluid velocity at $y\rightarrow \infty$ is the same as for purely viscous liquid with viscosity $\mu$: $u_\infty=\partial_y \psi_1|_{(x,y \rightarrow\infty)}=-\varepsilon/2$, and being re-written in a dimensional form yields
\begin{equation}
u_\infty=-\frac{c (bk)^2}{2}\:,
\end{equation}
so the displacement is in the direction of wave propagation (positive $x$ direction). Therefore, the propulsion of the planer sheet powered by purely tangential small-amplitude surface distortions is not affected by presence of obstacles in the first approximation.
\subsection{Rate of viscous dissipation}
Since normal distortions are expected to facilitate propulsion, it is instructive to determine if such augmentation is costly in terms of power invested in swimming. The power expended by an arbitrary shaped organism and dissipated by viscosity is
\begin{equation}
{\cal P}=-\int_S (\te{\sigma}\cdot\te{n})\cdot \te{u}\: \rd S\:, \label{eq:dissp}
\end{equation}
If decompose the surface velocity as $\te{u}=\te{u}'+\te{U}$ where $\te{U}$ is the propulsion velocity of the swimmer and $\te{u}'$ is the velocity of the swimming stroke, and since $\te{\sigma}=\te{\sigma}'$ the integral in (\ref{eq:dissp}) remains unchanged when written in terms of $\te{u}'$ as the swimmer is force-free, $\int_S (\te{\sigma}\cdot\te{n})\:\rd S=0$, (we omit the prime hereafter for simplicity). Re-writing the dissipation integral as ${\cal P}=2\mu \int_V \te{E}\,\te{:}\,\te{E} \: \rd V$, where $\te{E}$ is the rate-of-strain tensor and expressing the product $\te{E} \te{:}\,\te{E}$ as $\sum \zeta_i \zeta_i+2 (\partial_i u_j) (\partial_j u_i)$, where $\te{\zeta}=\mbox{curl}\te{u}$ denotes vorticity, allows expressing ${\cal P}$ as \cite{SS96}
\begin{equation}
{\cal P}=\mu\int_V\te{\zeta}^2 \rd V-2\mu \int_S\:n_i\,(u_j\: \partial_j u_i) \:\rd S\:. \label{eq:dissp2}
\end{equation}
Here $V$ is the fluid volume surrounding the swimmer.
For purely tangential surface distortions the second term on r.h.s of (\ref{eq:dissp2}) can be re-written as $2\mu \int_S\:\te{u}^2 \kappa_s\:\rd S$, where $\kappa_s=-(\partial \te{s}/\partial s)\te{\cdot}\:\te{n}$ is the curvature of the surface along the direction of the flow \cite{SS96}, and it is zero for un undistorted planar sheet. For purely transverse surface distortions the surface integral on the r.h.s. of (\ref{eq:dissp2}) is equal, to the first approximation, to $2\mu \int_S U \,\partial_x v\: \rd S$, where $U \sim c \varepsilon^2$ and $\partial_x v=\mathcal{O}(c \varepsilon k)$ and, therefore, its contribution to the dissipation rate is restricted to $\mathcal{O}(\varepsilon^3 \mu c^2)$. Thus, for both swimming gaits (normal and tangential surface waves), the rate-of-work, to the leading approximation, is given by the volume integral in (\ref{eq:dissp2}) over squared vorticity, $\te{\zeta}^2=(\Delta\psi)^2$,
\begin{equation}
{\cal P}=\mu\int_V \:(\Delta\psi)^2 \rd V\:. \label{eq:dissp3}
\end{equation}
Substituting $\psi_0$ corresponding to purely normal (\ref{eq:psi0}) and purely tangential (\ref{eq:psi00}) gaits, respectively, and integrating over volume we find the rate of viscous dissipation to the leading approximation. It appears that to the leading approximation (with an error of $\mathcal{O}(\varepsilon)$) the dissipation rate (per unit area of the sheet) is the same for both swimming gaits and equal to
\begin{equation}
\mathcal{P}=\mu c^2 \varepsilon^2 k \:\frac{2 + \alpha_*^2 + 2\sqrt{1 + \alpha_*^2}}{4\:\sqrt{1 + \alpha_*^2}}\:, \label{eq:dissp4}
\end{equation}
where $\varepsilon=bk$. The ratio of rate-of-work (per unit area of the sheet) in propulsion through matrix of obstructions and that in unbounded viscous liquid ($\mathcal{P}_\mathrm{S}=\mu\: c^2 \varepsilon^2 k$ \cite{taylor51}, subscript `S' stands for ``Stokesian") $\mathcal{P}/\mathcal{P}_s \ge 1$ and it is a monotonically increasing function of $\alpha_*$. Again, the effect of the embedded obstruction matrix on rate of viscous dissipation is evident for distortions with wavelengths ($k^{-1}$) comparable to the screening length $\alpha^{-1}$.
The routine definition of swimming efficiency based on the ratio between the work invested in dragging the immobile swimmer and that expanded in swimming (Lighthill's efficiency \cite{lighthill75}) is not applicable for 2D swimmers as the drag force is not defined due to Stokes paradox. This may be considered as a mere issue of normalization \cite{LKGA07,LK08} and a natural measure for the propulsion efficiency of an undulating sheet can be given by the ratio
\[
\delta=\frac{\mu U^2 k}{\mathcal{P}}\:.
\]
Using the expression (\ref{eq:Unormal}) for the propulsion velocity by the normal distortion and (\ref{eq:dissp4}) we arrive at
\begin{equation}
\delta=\frac{\varepsilon^2\:(1 + \alpha_*^2)^{3/2}}{2 + \alpha_*^2 + 2\sqrt{1 + \alpha_*^2}}\:.\label{eq:eff}
\end{equation}
The propulsion efficiency $\delta/\varepsilon^2$ in (\ref{eq:eff}) is depicted in Fig.~\ref{fig:eff} as a function of $\alpha_*$.
It is readily seen that the propulsion through matrix of obstructions is more efficient that swimming in the unbounded viscous liquid with $\delta_S=\varepsilon^2/4$.
\begin{figure}[t]
\includegraphics[scale=0.85]{fig3
\caption{Propulsion efficiency of the oscillating sheet, $\delta/\varepsilon^2$ vs. the dimensionless resistance, $\alpha/k$ (log-log plot): transverse surface waves (solid line); tangential (extension/compression) surface waves (dashed line). \label{fig:eff}}
\end{figure}
We arrived at the improved propulsion efficiency for transverse distortions and retarded efficiency with longitudinal distortions. \emph{Therefore, propulsion through heterogenous viscous environment can be advantageous both speed-wise and efficiency-wise}. Although improved swimming speed is expected as a result of hydrodynamic interactions with obstacles for a prescribed swimming gait, the improved propulsion efficiency is rather surprising result. A similar occurrence of enhancement of dragging efficiency due to hydrodynamic interaction between a passive load and a micro-swimmer towing it was found in \cite{RL08}.
\section{Propulsion of non-flagellated squirmers by surface distortions}
We expect similar propulsion augmentation for non-flagellated swimmers (such as cyanobacteria \cite{ESBM96,SS96}) moving through the heterogeneous viscous media. In this case the analysis can be greatly simplified via the use of Lorentz reciprocity \cite{KK91}, that can be shown to hold for the effective media equations (\ref{eq:brinkman}) linear in $\te{u}$: for any two arbitrary solutions $(\te{u},\te{\sigma})$ and $(\widehat{\te{u}},\widehat{\te{\sigma}})$ decaying far from the swimmer:
\begin{equation}
\int_S (\widehat{\te{\sigma}}\cdot\te{n})\cdot \te{u}\: \rd S=\int_S (\te{\sigma}\cdot\te{n})\cdot \widehat{\te{u}}\: \rd S.\label{eq:recip}
\end{equation}
Here $S$ is the instantaneous surface of the swimmer, $\te{\sigma}\cdot\te{n}$ is the local drag force the fluid exerts on $S$. The aim of this section is to show that the propulsion of cyanobacteria through viscous liquid with embedded network of obstructions can be enhanced when locomotion is powered by traveling \emph{normal} distortions of the outer surface. Following \cite{SS96} we will consider asymptotically tractable case of nearly spherical squirmer of radius $a$ propelled by small amplitude surface waves. Substituting $\te{u}=\te{U}+\te{u}'$, where $\te{U}$ is the propulsion velocity and $\te{u}'$ is the surface distortion in the frame of reference fixed with the squirmer's center-of-mass, into (\ref{eq:recip}) and setting the net force on the swimmer to zero, we arrive at
\begin{equation}
\widehat{\te{F}}\cdot\te{U}\simeq-\int_S (\widehat{\te{\sigma}}\cdot\te{n})\cdot \te{u}'\:\rd S\:,\label{eq:recip1}
\end{equation}
where $\widehat{\te{\sigma}}$ is the stress field corresponding to the translation of the spherical squirmer when acted upon by an external force $\widehat{\te{F}}$. The equality in (\ref{eq:recip1}) is exact when the propulsion is governed by purely tangential distortions. For an auxiliary problem of towing the immobile spherical squirmer through viscous liquid with embedded sparse network of obstructions, we use the well-known results for the drag force
\[
\widehat{\te{F}}=6\pi\mu a \left(1+\alpha a+(\alpha a)^2/9\right) \te{U}\,
\]
and
\[
\left.\widehat{\te{\sigma}}\cdot \te{n}\right|_S=-\frac{3\mu}{2a}\: \te{U}\cdot \left[(1+\alpha a)\te{I}+\frac{(\alpha a)^2}{3}\:\te{n} \te{n} \right]\:.
\]
for the local traction on the immobile squirmer's surface (\eg \cite{Howells74}). The resulting equation for the propulsion speed in terms of surface motions reads
\begin{eqnarray}
\te{U} & \simeq &-\frac{1}{4\pi a^2 \left[1+\alpha a+(\alpha a)^2/9\right]}\times \nonumber \\
&& \:\int_S \left[(1+\alpha a) \te{u}'+\frac{(\alpha a)^2}{3} (\te{u}'\cdot\te{n})\,\te{n} \right]\,\rd S\:. \label{eq:U}
\end{eqnarray}
The propulsion speed of a spherical squirmer in an unbounded viscous liquid is recovered in the limit $\alpha a \rightarrow 0$ in accord with \cite{SS96}, $\te{U}_\mathrm{S} \simeq -(4\pi a^2)^{-1} \int_S \te{u}' \rd S$ (the subscript ``S" stands for ``Stokesian"). Decomposing the arbitrary surface velocity into the normal and the tangential component, $\te{u}'=(\te{I}-\te{n}\bn)\cdot\te{u}'+(\te{u}'\cdot\te{n})\,\te{n}$, allows Eq.~\ref{eq:U} to be re-written for either purely normal (superscript $n$) or purely tangential (superscript $s$) surface distortions as
\begin{equation}
\te{U}^{n,s}=\mathcal{F}^{n,s}(\alpha a)\,\te{U}_\mathrm{S}\:,\label{eq:U1}
\end{equation}
where $\mathcal{F}^{n,s}=\frac{\left\{1+\alpha a+(\alpha a)^2/3,\,\, 1+\alpha a\right\}}{1+\alpha a+(\alpha a)^2/9}$. Clearly, $1<\mathcal{F}^n<3$ and $0<\mathcal{F}^s<1$ for all values of $\alpha a>0$; the comparison of the propulsion velocity in a heterogeneous viscous medium and a purely viscous solvent for both kinds of surface motion (normal and tangential) is depicted in Fig.~\ref{fig:squirmer}.
\begin{figure}[t]
\includegraphics[scale=0.85]{fig4
\caption{Propulsion velocity of a spherical squirmer of radius $a$ moving through heterogeneous viscous medium as compared to Stokesian velocity of propulsion in unbounded viscous liquid, $U/U_\mathrm{S}$, vs. the dimensionless resistance, $\alpha a$: purely normal surface distortions (solid line); purely tangential surface distortions (dashed line) \label{fig:squirmer}}
\end{figure}
Thus, when both swimming gaits (normal and tangential waves) of a squirmer propelled through heterogeneous viscous environment are compared to propulsion through unbounded viscous liquid, we find that normal surface distortions enhance the locomotion, while motion powered by purely tangential waves is hindered.
\section{Propulsion of a rotating helical filament}
\subsection{Modified resistive force theory for propulsion through heterogeneous viscous media}
In this section we develop a modification of the resistive force theory (RFT) of the propulsion through effective heterogeneous viscous media powered by rotating helical flagellum. We assume again that small inertia-less flow in such media is governed by (\ref{eq:brinkman}) and we aim to find the propulsion velocity and rate-of-work of the rotating rigid helix as a function of its geometry (\ie pitch angle $\theta$ and the radius $r$, see Fig.~\ref{fig:helix_schematic}) and angular velocity $\omega$.
The result for the propulsion speed of the force-free \footnote{The torque is not zero, however, it is a simplified model of a flagellum of externally flagellated microorganisms, the torque required for rotation is provided by the ATP-powered machinery within the cell body; the interaction with the cell body is neglected for simplicity} slender helix rotating in Newtonian incompressible viscous liquid reads (\eg \cite{AR07,WN08})
\begin{equation}
\frac{U}{r \omega}= \frac{\sin{2 \theta}}{2\:(1+\sin^2{\theta})}\:,\label{eq:Uhelix}
\end{equation}
so that the ratio of velocities of a helix to a corkscrew is independent of viscosity and by (\ref{eq:Uhelix}): $U/(r\omega\cot{\theta}) \le 1/2$. Thus, in the unbounded Newtonian viscous liquid the helix needs at least two turns to progress over a distance of its threads. The rate-of-viscous dissipation is given by
\begin{equation}
\mathcal{P}=\frac{f_\perp L \: r^2 \omega^2}{1+\sin^2{\theta}}\:,\label{eq:dissp5}
\end{equation}
where $f_{\perp}$ and $f_{||}$ the force densities corresponding to transverse and axial motion of a filament, respectively; $f_{\perp}\approx 2f_{||}$ for purely viscous liquid.
We are interested to extend these results and derive the propulsion velocity and power as a function of the hydrodynamic resistance of the random matrix of obstacles $\alpha$. Let us consider propulsion speed of a single rigid helix that rotates with some constant angular speed $\omega$. Taking $\te{e}_3$ be the direction of the helical axis, the helix centerline is given by $\te{r}(s,t)=\{r\cos{(k s+\omega t)},\,r\sin{(k s+\omega t)},\,b s + U t\}$, where $k=2\pi/\ell$ with $\ell$ being a length of a single helical turn so $kr=\sin{\theta}$; $b=\cos{\theta}$ and $U$ is yet undetermined propulsion velocity. The basic assumption of the local RFT, is that the local force per unit length (force density) exerted on the slender filament is given by
\begin{equation}
\te{f}=f_{\perp}(\te{u}-(\te{u}\cdot\te{s})\,\te{s})+f_{||}(\te{u}\cdot\te{s})\te{s}\:,\label{eq:rft}
\end{equation}
where $\te{s}=\partial \te{r}/\partial s$ is the local tangent, $\te{u}=\partial \te{r}/\partial t$ is the local velocity and $f_{\perp}$ and $f_{||}$ the force densities corresponding to transverse and axial translation of a straight cylinder. For purely viscous liquid, $f_{\perp}=2f_{||}=4\pi\mu E$, with a small parameter $E=(\ln{2/\epsilon})^{-1}$ where $\epsilon=a/b\ll 1$ is an aspect ratio ($2b$ is the cylinder length and $2a$ is its diameter). The slenderness of the filament is controlled by the parameter $\kappa_f a\ll 1$, where $\kappa_f=|\partial^2 \te{r}/\partial s^2|=\sin^2{\theta}/r$ is the local curvature of the filament centerline.
\begin{figure}[t]
\includegraphics[scale=0.45]{fig5
\caption{(Color online) Schematics of a rigid helical filament: $2a$ is the diameter of the filament, $r$ is radius of the helix, $\theta$ is a pitch angle and a pitch is given by $p=2\pi r/\tan{\theta}$. \label{fig:helix_schematic}}
\end{figure}
We substitute the local velocity as $\te{u}=\te{u}_\perp+U\te{e}_3$ into (\ref{eq:rft}) and setting the total force in the $x_3$-direction to zero, we find that
\begin{eqnarray}
U\int_0^L \left[f_\perp+(f_{||}-f_\perp (\te{s}\cdot\te{e}_3)^2) \right] \rd s=\nonumber \\
\qquad -(f_{||}-f_\perp) \int_0^L (\te{s}\cdot\te{u}_\perp) (\te{s}\cdot\te{e}_3) \rd s\:.\label{eq:rft1}
\end{eqnarray}
Substituting $\xi=f_\perp/f_{||}$ and the expressions for the local tangent $\te{s}$ and $\te{u}_\perp$ into (\ref{eq:rft1}) one can derive the propulsion velocity of the force-free helical swimmer rotating around its central axis with angular velocity $\omega$ as a function of $\xi$:
\begin{equation}
\frac{U}{\omega r}=\frac{(\xi-1)\sin{2\theta}}{2\left[1+(\xi-1) \sin^2{\theta}\right]}\:.\label{eq:Uhelix2}
\end{equation}
The RFT result (\ref{eq:Uhelix}) is readily recovered from (\ref{eq:Uhelix2}) for $f_{\perp}=2\,f_{||}$ and $\xi=2$. Similarly, the rate-of-dissipation $\mathcal{P}=-\int_0^L \te{f} \cdot \te{u} \rd s$, can be found
\begin{equation}
\mathcal{P}=\frac{2 f_\perp L\: r^2 w^2}{1+\xi +(1-\xi ) \cos{2 \theta}}\:,\label{eq:power}
\end{equation}
that reduces to the known RFT result (\ref{eq:dissp5}) for $\xi=2$. Also, the propulsion efficiency can be defined as a ratio of power required to drag the helical filament along the $x_3$-axis with velocity $U$ and the power expanded in force-free swimming, $\delta=\te{F}\cdot \te{U}/\mathcal{P}$. The force required to tow the helix is easily obtained by integrating the local force $\te{f}$ in (\ref{eq:rft}) for $\te{u}=U\te{e}_3$: $\te{F}=U\:(f_\perp (1-\cos^2{\theta})+f_{||}\:\cos^2{\theta})\:L \te{e}_3$. Then, using $\te{F}$ and $\mathcal{P}$ yields an expression for propulsion efficiency as a function of the pitch angle and the ratio $\xi$:
\begin{equation}
\delta=\frac{(\xi-1)^2 \sin^2{2\theta}}{4\xi}\:.\label{eq:eff1}
\end{equation}
For unbounded viscous liquid $\xi=2$ and $\delta=\frac{1}{8}\sin^2{2\theta}$, yielding the optimal efficiency of 12.5\% at the pitch angle of $\theta=45^\circ$.
Now, let us solve the analogous problem for the force-free rotating helix propelled through a heterogeneous viscous medium. In this case (\ref{eq:rft1}) still holds, while the force densities $f_\perp$ and $f_{||}$ for a rigid cylinder of radius $a$ translating through medium with the effective resistance $\alpha$ are now functions of $\alpha a$ \cite{SG68},
\begin{equation}
\frac{f_\perp}{4\pi\mu}=\frac{1}{4}\;(\alpha a)^2+\alpha a\;\frac{K_1(\alpha a)}{K_0(\alpha a)}\:,\qquad \frac{f_{||}}{4\pi\mu}=\frac{1}{2}\:\alpha a\;\frac{K_1(\alpha a)}{K_0(\alpha a)}\:,\label{eq:fbrink}
\end{equation}
where $a$ is the radius of the filament and $K_p(x)$ are the modified Bessel functions of degree $p$. To determine the propulsion speed we only need to know the ratio of the resistance coefficients,
\begin{equation}
\xi=\frac{f_\perp}{f_{||}}=2+\frac{\alpha a}{2}\;\frac{K_0(\alpha a)}{K_1(\alpha a)}\ge 2\:. \label{eq:xi}
\end{equation}
Note that in the limit of vanishing matrix resistance $\alpha a\rightarrow 0$ the correct viscous limit, $\xi=2$, is recovered \footnote{However, the individual force densities $f_\perp$ and $f_{||}$ do not reduce to their corresponding Stokesian limiting values ($4\pi \mu E$ and $2\pi \mu E$, respectively) due to the fact that the limit $a\alpha \rightarrow 0$ is singular and requires special attention}. Substituting $\xi(\alpha a)$ into (\ref{eq:Uhelix2}) yields the expression for the propulsion velocity of slender helix rotating in viscous liquid with an embedded sparse matrix of obstacles. For vanishing resistance, $\xi \rightarrow 2$, and the propulsion velocity in (\ref{eq:Uhelix2}) tends to that in (\ref{eq:Uhelix}), as expected. In the opposite limit of large values of $\xi$ the motion resembles that of a corkscrew, $U=\omega r \cot{\theta}$, boring through solid without slip.
\begin{figure}[t]
\includegraphics[scale=0.85]{fig6a
\\
\includegraphics[scale=0.85]{fig6b
\caption{Propulsion of a rotating helical filament through heterogeneous viscous media: (\emph{a})
optimal velocity, $U/\omega r$, vs. the scaled resistance, $\alpha a$; (\emph{b}) optimal pitch angle $\theta=\frac{1}{2}\arccos\left(\frac{\xi-1}{\xi+1}\right)$ vs. the scaled resistance $\alpha a$.
\label{fig:helix_vel}}
\end{figure}
Optimizing swimming speed for prescribed rotation velocity, we find that swimming speed is maximized at the pitch angle $\theta=\frac{1}{2}\arccos\left(\frac{\xi-1}{\xi+1}\right)$, vanishing resistance yields the pitch angle $\theta=\frac{1}{2}\arccos{\left(\frac{1}{3}\right)}\simeq 35.26^\circ$ that maximizes the propulsion speed in purely viscous liquid, giving $U/\omega r \simeq 0.354$ in agreement with previous theories (\eg see \cite{AR07}). The optimal propulsion velocity of the rotating helix moving through heterogeneous viscous environment is depicted in Fig.~\ref{fig:helix_vel}\emph{a} as a function of dimensionless hydrodynamic resistance $\alpha a$. The optimal pitch angle, however, decreases with the increase in $\alpha a$ (see Fig.~\ref{fig:helix_vel}\emph{b}).
Lastly, the efficiency of propulsion can be found by substituting $\xi$ in (\ref{eq:xi}) into (\ref{eq:eff1}). The resulting dependence on the scaled hydrodynamic resistance $\alpha a$ and the pitch angle $\theta$ is depicted in Fig.~\ref{fig:helix_delta}. It can be readily seen that, again, the propulsion is advantageous not only speed-wise, but also efficiency-wise as $\delta$ grows with $\alpha a$. Interestingly, the optimal pitch angle remains 45$^\circ$ for all values of $\alpha a$ since $\delta \propto \sin^2{2\theta}$. The enhanced efficiency predicted by the theory is not sufficient, however, for maintaining the swimming speed for fixed torque applied to the filament. For a prescribed power expended in swimming, the swimming speed decays with $\alpha a$, which indicates that in addition to the proposed enhancement due to embedded matrix of obstacles, active control of motility by microorganisms may be involved.
\begin{figure}[t]
\includegraphics[scale=0.55]{fig7
\caption{(Color online) Propulsion efficiency $\delta$ (\ref{eq:eff1}) of a rotating helix in heterogeneous viscous media as a function of pitch angle $\theta$ (rad) and hydrodynamic resistance $\alpha a$. \label{fig:helix_delta}}
\end{figure}
\subsection{\label{sec:4b} Numerical simulations of propulsion through heterogeneous viscous medium}
To validate the theoretical predictions of the preceding section we implement the numerical scheme based on multipole expansion of the Lamb's spherical harmonic solution of Stokes equations \cite{Filippov00}. The filament is constructed from nearly touching rigid spheres (``shish-kebab" model) and the obstruction matrix is modeled as random sparse array of stationary spheres (see Fig.~\ref{fig:helix_spheres}). All $N$ spheres (composing the filament and the matrix) have the same radius $a$. The no-slip condition at the surface of all spheres is enforced rigorously via the use of direct transformation between solid spherical harmonics centered at origins of different spheres. The method yields a system of $\mathcal{O}( N \mathcal{L}^2)$ linear equations for the expansion coefficients and the accuracy of calculations is controlled by the number of spherical harmonics (i.e. truncation level), $\mathcal{L}$, retained in the series. The same approach was used in \cite{LK08,RL08} for modeling Purcell's toroidal swimmer.
\begin{figure}[t]
\includegraphics[scale=0.4]{fig8_lowres
\caption{(Color online) Illustration of a ``shish-kebab" filament propelled through heterogeneous viscous medium modeled as sparse matrix of stationary spheres immersed into Newtonian viscous liquid. \label{fig:helix_spheres}}
\end{figure}
The spheres composing the helical filament are partitioned along the backbone of the filament, $\te{r}(s)=\left\{r\cos{(k s)},\,r\sin{(k s)},\,b s\right\}$, so that the distance between centers of neighboring spheres is set to $2.02 a$. Here $kr=\sin{\theta}$ and $b=\cos{\theta}$ with $r$ and $\theta$ being a radius and a pitch angle of the helix, respectively. The motion of the $i$th sphere composing a rotating helix can be decomposed into translation and rotation about its center as $\te{V}_i=\te{U}_i+\tes{\Omega}_i\times \te{r}_i$ with $\te{U}_i=U\te{e}_3+\omega\te{e}_3 \times \te{R}_i$ and $\tes{\Omega}_i=\omega \te{e}_3$; here $\te{R}_i$ is a position vector to the $i$th sphere center in the fixed laboratory frame and $\te{r}_i$ is the radius vector with origin at the center of $i$th sphere, $\omega$ is the angular velocity of rotation and $U$ is the propulsion velocity along $x_3$-axis. The velocity of propulsion is determined by setting the net force exerted on rotating helix in the direction of propulsion to zero, $\sum_i {{F}_i}_3=0$, while translation and rotational velocities of the stationary obstacles are both set to zero. The rotation is powered by an external torque. The rate-of-work expended in propulsion of a rotating filament is found from
\begin{eqnarray}
\mathcal{P}&=&\sum \limits_{i=1}^{N_p} (-\te{U}_i\cdot\te{F}_i-\tes{\Omega}_i\cdot \te{T}_i)= \nonumber \\
&& -\sum \limits_{i=1}^{N_p} \omega (-{R_i}_2\:{F_i}_1+{R_i}_1\:{F_i}_2+ {T_i}_3)\:,\label{eq:power1}
\end{eqnarray}
where $\te{F}_i=\int_{\partial S_i} \te{\sigma\cdot}\te{n}\:\rd S$ and $\te{T}_i=\int_{\partial S_i} \te{r}_i\times (\te{\sigma\cdot}\te{n})\:\rd S$ are the hydrodynamic force and torque, respectively, exerted on $i$th sphere composing the filament.
We initially test the scheme for the case of rotating helix moving through unbounded viscous liquid without obstacles. The results of the calculation are presented in Fig.~\ref{fig:helix_pitch} for a helix composed of $30$ spheres with radii $r=2a,\,3a$ and $4a$ upon varying the pitch angle $\theta$ (empty symbols).
\begin{figure}[t]
\includegraphics[scale=0.65]{fig9
\caption{(Color online) Propulsion velocity $U/\omega r$ of a rotating helical filament built of $30$ spheres vs. the pitch angle $\theta$. The void symbols represent propulsion in unbounded viscous liquid: $r=2a$ ($\square$), $r=3a$ ($\bigcirc$) and $r=4a$ ($\bigtriangleup$); the full symbols stand for propulsion through viscous liquid with an embedded random matrix of 8\% (vol) stationary spherical obstacles: $r=2a$ ($\blacksquare$), $r=3a$ ($\bullet$) and $r=4a$ ($\blacktriangle$). The symbols stand for the mean value based on $20$ random configuration of obstacles; the error bars length is doubled mean standard deviation. The solid line is the theoretical result for propulsion in unbounded viscous liquid (\ref{eq:Uhelix2}) (with $\xi=1.52$ as found from simulations for a ``shish-kebab" rod with $N_p=30$, see Fig.~\ref{fig:shishkebab}) and the dashed curve is the theoretical prediction (\ref{eq:Uhelix2}) for propulsion through sparse random matrix of obstructions with $\alpha a=0.6$ (corresponding to a random matrix of spherical obstacles with $\phi=0.08$) for $\xi(\alpha a)$ given in Eq.~\ref{eq:xi} and corrected for the ``shish-kebab" shape of the filament. \label{fig:helix_pitch}}
\end{figure}
The results of the calculation of the scaled propulsion velocity $U/\omega r$ vs. $\theta$ for small pitch angles (i.e. for $\kappa_f a=(a/r)\:\sin^2{\theta}\ll 1$, where $\kappa_f$ is a local curvature of the filament centerline) are in a good agreement with the RFT result (solid line) corrected for a ``shish-kebab" shape of the helix. Actually, for such ``shish-kebab" filament variation of the shape occurs on a scale of the filament radius and not the length and therefore one should not expect the result of the local theory based on assumption of gentle variations of the shape, to hold in this case. However, this only changes the numerical coefficients in front of the resistance coefficients $f_\perp$ and $f_{||}$, leading to a slightly different ratio $f_\perp/f_{||}\approx 1.67$ at $\epsilon \rightarrow 0$ (instead of $\approx 2$ for slender particles with gradual variation of the geometry). The coefficient can be found numerically by computing the two force components, transverse and longitudinal, exerted on a ``shish-kebab" rod upon varying its length (\ie number of spheres composing the filament). The results of the calculation are presented in Fig.~\ref{fig:shishkebab} together with the best fit of the form $(c_1+c_2 \epsilon)/(c_3+c_4\epsilon)$. The theoretical prediction of the helix velocity in Fig.~\ref{fig:helix_pitch} (solid curve) is based on (\ref{eq:Uhelix2}) with $\xi=f_\perp/f_{||}=1.52$ corresponding to a straight ``shish-kebab" filament composed of $N_o=30$ spheres (see Fig.~\ref{fig:shishkebab}).
It can be readily seen that the agreement between the numerical (void symbols) and theoretical result is very good for pitch angles $\theta<0.3$ rad, as the data corresponding to different helix radii ($r=2a$, $3a$ and $4a$) collapse on the theoretical curve (solid line in Fig.~\ref{fig:helix_pitch}). For higher pitch angle the assumption of slenderness ($\kappa_f a \ll 1$) is violated and deviation from the theory is evident due to hydrodynamic interaction of the helix threads. Increase in the radius of the helix (for the same pitch angle) yields better agreement with the RFT prediction for both: viscous liquid and heterogeneous viscous media. The position of the maximum is observed for the pitch angle $\theta\simeq 0.6$ (rad) in agreement with theoretical predictions regardless of the total length and radius of the helix. Increasing the number of spheres composing the helix results in slightly increased propulsion velocity, while the increase in accuracy level, $\mathcal{L}$, slightly diminishes the speed. Note that for slender helices the calculations are rather accurate even for just two harmonics ($\mathcal{L}=2$) retained in the series, as the hydrodynamic interaction between treads is negligible.
\begin{figure}[t]
\includegraphics[scale=0.65]{fig10
\caption{Numerically calculated force ratio $f_{\perp}/f_{||}$ for a ``shish-kebab" straight rod of length $2b$ made of $N$ spheres of radii $a$ as a function of the aspect ratio $\epsilon=a/b$ (void symbols). The solid line stands for the best fit $(c_1+c_2 \epsilon)/(c_3+c_4\epsilon)$. \label{fig:shishkebab}}
\end{figure}
Next we consider propulsion through a sparse (8\% by vol) matrix of $N_o=30$ stationary spherical obstructions. Nonintersecting stationary spheres are arranged randomly in a rectangular box with its longer size oriented along the axis of the filament (see Fig.~\ref{fig:helix_spheres}). The size of the box is determined by the matrix density, i.e. volume fraction $\phi$ of the obstacles in the box and the slenderness of the filament. It should be realized, however, that the filament is propelled through a finite heterogeneous region surrounded by unbounded viscous liquid. However, the hydrodynamic disturbance from the rotating force-free filament in the heterogeneous effective media is expected to decay considerably faster due to ``shielded" interaction when compared to unbounded viscous liquid ($u=\emph{o}(1/r^3$), see Eq.~\ref{eq:pointforce_brink}), and thus the effect of the viscous domain outside the box is expected to be negligible. The propulsion velocity is determined from static rather than dynamic calculations in the same way as for filament rotating in purely viscous liquid. The value of the velocity is averaged over $20$ independent configurations and the results (full symbols) are shown in Fig.~\ref{fig:helix_pitch}. It can be readily seen that swimming through heterogeneous domain (see Fig.~\ref{fig:helix_pitch}) yields faster propulsion when compared to swimming in purely viscous liquid (for the prescribed swimming gait, \ie $\omega$). The agreement between the modified RFT (dashed curve) and the results of numerical simulations (full symbols) is very good. The theoretical prediction is based on (\ref{eq:Uhelix2}) with $\xi$ given by (\ref{eq:xi}) for $\alpha a=0.6$, corresponding to obstacle concentration of $\phi=0.08$ ($\alpha a=3\sqrt{\phi/2}$ for random array of spherical obstacles \cite{Howells74}), and multiplied by a factor of $0.83$ correcting for the ``shish-kebab" shape of the filament.
The concentration dependence on the propulsion speed is depicted in Fig.~\ref{fig:helix_conc} for a helix composed of $N=30$ spheres with $r=2a$, $\theta=0.3$ and $r=3a$, $\theta=0.5$ for concentration of spherical obstacles in a cell up to 8\% (vol). The monotonic increase in the propulsion speed $U/\omega r$ is evident and the agreement with the theoretical prediction (\ref{eq:Uhelix2}) is very good. The deviation from the theoretical prediction at small $\phi$ is probably due to the fact that derivation in \cite{SG68} leading to (\ref{eq:xi}) breaks down when the screening length and the filament length become comparable; this case requires a special consideration and will be addressed elsewhere.
\begin{figure}[t]
\includegraphics[scale=0.65]{fig11
\caption{Propulsion velocity of a rotating helical `shish-kebab" filament (as in Fig.\ref{fig:helix_spheres}) with $r=2a$, $\theta=0.3$ ($\square$) and $r=3a$, $\theta=0.5$ rad ($\vartriangle$) vs. concentration $\phi$ of stationary spherical obstructions, $\phi$. The solid curves are the modified RFT result (\ref{eq:Uhelix2}) with $\xi(\alpha a)$ given by (\ref{eq:xi}). Error bars show the doubled mean standard deviation of 20 random configurations. \label{fig:helix_conc}}
\end{figure}
The power, $\mathcal{P}$, expended in propulsion of the ``shish-kebab" filament is calculated via (\ref{eq:power1}). The corresponding hydrodynamic efficiency, $\delta$, is determined as $\delta=\mathcal{R}_{FU} U^2/\mathcal{P}$, where ${\cal R}_{FU}$ is the appropriate hydrodynamic resistance equal to the drag force on the (non-rotating) helix dragged along $x_3$-axis with $U=1$. The comparison of the numerical results for a helix composed of $50$ spheres and propelled through an unbounded viscous liquid, with the prediction of the RFT (\ref{eq:eff1}) corrected for the ``shish-kebab" shape (\ie $\xi=1.56$ corresponding to $N_p=50$) is depicted in Fig.~\ref{fig:helix_eff1} for several helix radii. We see that the theoretical prediction is less accurate than that for the propulsion speed (see Fig.~\ref{fig:helix_pitch}), and that the hydrodynamic interaction between treads at small $r$ diminishes the propulsion efficiency when compared to the theory. For $r=10\:a$ the agreement is good for pitch angles $<0.4$ rad. The position of the peak efficiency is slightly below the theoretical prediction of $\frac{\pi}{4}\approx 0.79$ rad and is around $0.7$ rad. Again, the increase in the radius of the helix (keeping helix radius fixed) results in a better agreement between the simulation results and the RFT prediction both for purely viscous liquid and heterogeneous media, as the slenderness parameter $\kappa_f a$ diminishes. The effect of the total length of the helix on propulsion efficiency is quite minor, a result for helix composed of $30$ spheres is very close to that of a longer helix with $N_p=50$ (see full squares vs. empty squares in Fig.~\ref{fig:helix_eff1}) for a wide range of the pitch angles.
\begin{figure}[t]
\includegraphics[scale=0.65]{fig12
\caption{(Color online) Efficiency of propulsion, $\delta$, of a rotating helical filament propelled through unbounded viscous liquid vs. the pitch angle $\theta$. Symbols stand for the results of the numerical calculation for a ``shish-kebab'' filament composed of $50$ spheres with: $r=6a$ ($\square$), $r=8a$ ($\circ$) and $r=10a$ ($\triangle$). Full squares ($\blacksquare$) are the results for a shorter helix with $N_p=30$ and $r=6a$. The continuous curve is a prediction of the RFT theory (\ref{eq:eff1}) with $\xi=1.56$ corresponding to the ``shish-kebab'' helix composed of $N_p=50$ particles. \label{fig:helix_eff1}}
\end{figure}
Next we consider propulsion efficiency through sparse random array of stationary spherical obstructions immersed into incompressible viscous liquid. Here we use helices composed of $30$ spheres and obstruction matrix composed of $N_o=30$ spheres with volume fraction of obstacles of 5\% (vol). The results for the swimming efficiency are shown in Fig.~\ref{fig:helix_eff2} as full symbols for $r=4a$($\blacksquare$) and $r=6a$ ($\bullet$). In accord with the prediction of the modified RFT (the dashed line), the helix is a more efficient propeller in the presence of obstacles: the maximum efficiency increases more than two folds. As for propulsion through purely viscous liquid (void symbols vs. solid line), the theoretical prediction based on effective media approximation overestimates the efficiency for large pitch angles, where hydrodynamic interaction of the treads is important, while the agreement is reasonably close for pitch angles $\theta<0.4$ rad. The RFT prediction was calculated from (\ref{eq:eff1}) using $\alpha a=0.474$ corresponding to obstacles' volume fraction of $\phi=0.05$, and $\xi$ multiplied by a constant factor of $0.83$ correcting for the ``shish-kebab" shape of the filament in the same way as done previously.
It should be stressed that no adjustable parameters are involved in comparison of the results of numerical simulations and the theory in Figs.~\ref{fig:helix_pitch}, \ref{fig:helix_conc}, \ref{fig:helix_eff1} and \ref{fig:helix_eff2} (besides the shape factor for the ``shish-kebab" filament that is estimated numerically).
\begin{figure}[t]
\includegraphics[scale=0.65]{fig13
\caption{(Color online) Efficiency of propulsion, $\delta$, of a rotating helical filament composed of $30$ spheres and propelled through viscous liquid with an embedded random matrix of 5\% (vol) stationary spherical obstacles, with vs. the pitch angle $\theta$. Full symbols stand for numerical results for propulsion heterogeneous viscous media with $5$\% (vol) of spherical obstacles embedded into viscous solvent: $r=4a$ ($\blacksquare$), $r=6a$ ($\bullet$); void symbols are the results of the numerical calculation for same filament, propelled through unbounded viscous liquid: $r=4a$ ($\square$), $r=6a$ ($\circ$). Error bars depict the doubled mean standard deviation based on 20 random configurations of the obstruction matrix. The curves stand for predictions of the RFT theory using (\ref{eq:eff1}) for the propulsion efficiency in both cases: unbounded incompressible viscous liquid for $\xi=1.52$, corresponding to a helix composed of $N_p=30$ spheres (solid curve); heterogeneous viscous media using $\xi$ as in (\ref{eq:xi}) with $\alpha a=0.474$ corresponding to obstruction volume fraction $\phi=0.05$ (dashed curve).
\label{fig:helix_eff2}}
\end{figure}
Lastly, we calculate the propulsion velocity of a helix scaled with the rotation speed controlled by the power invested in swimming (\ie torque applied to the filament), given by $\sqrt{\mathcal{P}/\mu L}$. The modified RFT theory (\ref{eq:Uhelix2},\ref{eq:power}) indicates that for a prescribed power the scaled propulsion speed, $U/\sqrt{\mathcal{P}/\mu L}$, should monotonically decay with the increase in $\alpha a$, while at $\alpha a \rightarrow 0$, this velocity grows unbounded as $f_\perp,\:f_{||}$ in (\ref{eq:fbrink}) vanishes (while their ratio $\xi$ remains finite). Note that $U/\sqrt{\mathcal{P}/\mu L}$ (in comparison to $U/\omega r$ and $\delta$ in (\ref{eq:Uhelix2}) and (\ref{eq:eff1}), respectively) is no longer a sole function of the ratio $\xi=f_\perp/f_{||}$, but also depends on the force density $f_\perp$ via $\mathcal{P}$. Obviously, as concentration of the obstacles tends to zero, it is expected that the force density $f_\perp$ should tend to a finite Stokesian value corresponding to an unbounded viscous liquid. This unphysical behavior near $\alpha a=0$ can be corrected by constructing a proper slender body approximation for the Brinkman equation (\ref{eq:brinkman}) and will be addressed elsewhere. However, the growth of $U/\sqrt{\mathcal{P}/\mu L}$ as $\alpha a$ diminishes indicates the propulsion speed can be enhanced for a fixed power. For this matter, we compute the scaled propulsion speed of the rotating helix composed of $30$ particles and the embedded matrix composed of $N_o=30$ spheres with the volume fraction of obstacles of 5\%. The results of the computation are provided in Fig.~\ref{fig:fixpower} vs. the pitch angle, $\theta$. The numerical results (symbols) demonstrate the propulsion augmentation (up to $\sim 18$ \% of the Stokesian velocity) for rotating helix with fixed rate-of-work. The continuous lines refer to the modified RFT theory corrected for the shape of the filament as before, and the agreement with the numerical results is quite good for small pitch angles. Since the scaled propulsion $U/\sqrt{\mathcal{P}/\mu L}$ decays with $\alpha a$ and in the limit of vanishing resistance should yield a Stokesian result, it is expected that there should be the optimal resistance $\alpha a$ maximizing the propulsion speed of the helix.
\begin{figure}[t]
\includegraphics[scale=0.65]{fig14
\caption{Dimensionless propulsion velocity, $U/\sqrt{\mathcal{P}/\mu L}$, of a rotating ``shish-kebab" helix of radius $r=4a$ and $N=30$, vs. the pitch angle $\theta$. Full symbols ($\square$) stand for a helix propelled through a unbounded viscous liquid of viscosity $\mu$ and empty symbols, ($\blacksquare$), correspond to the helix propelled through heterogenous medium modeled as 5\% (vol) random array of stationary spherical obstructions of radius $a$ embedded into a viscous solvent of viscosity $\mu$. The solid and the dashed lines stand for the RFT result of propulsion in purely viscous solvent and viscous liquid with an embedded matrix of obstacles, respectively. \label{fig:fixpower}}
\end{figure}
\section{Concluding remarks}
We have demonstrated that transverse strokes/motions can lead to enhancement of propulsion via the heterogeneous viscous environment, modeled as sparse matrix of stationary obstacles embedded into incompressible viscous liquid, for different swimming gaits. The tangential motions,however, do not offer such enhancement. The present theory is in accord with the qualitative argument made in \cite{BT79} suggesting that ``...efficiency of flagellar propulsion is probably enhanced by the gel-like structure" of the medium. Combining the local theory and rigorous numerical calculations we demonstrated that the propulsion can be enhanced both speed- and efficiency-wise. For instance, the translation speed of the rotating helical filament is increased not only for a prescribed rotation velocity but also for a fixed torque applied to the filament. These findings can provide the physical grounds for explaining experiments showing enhanced propulsion of externally flagellated bacteria \cite{SD74,BT79} in viscous polymer gels.
We should next address the question of applicability of the effective media approximation to biologically relevant cases. Originally, the theory was developed to described the resistance to the flow through such media, i.e. stationary arrays of obstructions (fibers or spheres), while we consider a propulsion of a swimmer though such media. While the equations of motions (\ref{eq:brinkman}) remain invariant under Galilean transformation (up to a uniform pressure gradient), there is still an issue related to ``permeability" of such matrix to a finite-size object propelled through it \cite{Broday02}. The spacing between obstacles that compose the matrix places a constrain on the relevant dimension of the object moving through it without altering its structure. Obviously, for high volume fraction of the obstructions the effect of the matrix effect is most certainly local and not described by the effective mean resistance as in (\ref{eq:brinkman}). The question to what extent the Brinkman's approximation holds when the structure of the matrix is distorted in response to swimmer's movement, is yet to be answered. We can estimate the range of formal validity of the theory using some simple geometric arguments. For instance, for fibrous media (\eg chain polymer aggregates in gels) where the fiber axes are completely randomly oriented the value of $\alpha$ is determined in a self-consistent manner as function of mean fiber content \cite{SG68},
\[
\alpha_*^2=4 \phi\:\left(\frac{1}{3}\:\alpha_*^2+\frac{5}{6}\alpha_*\: \frac{K_1(\alpha_*)}{K_0(\alpha_*)}\right)\:,
\]
where $\alpha_*=\alpha b$, $\phi=\lambda \pi b^2$ is the fiber volume fraction with $\lambda$ being the mean fiber length per unit volume and $b$ stands for the fiber radius. Therefore, for sparse fiber matrix, $\phi=0.001$ ($0.1$\% vol), we find that $\alpha_*=0.0304$. Thus, for the typical value of $a/b=50$ (\eg the radius of the spirochete cell helix is $\sim 0.05$ $\mu$m, the typical radius of polysaccharide chains in the extracellular domain is $0.5-2$ nm) the corresponding value of $\alpha a=\alpha_* (a/b)=1.52$ and the optimal velocity of propulsion is $U/\omega r\simeq 0.49$ which is an improvement of $\approx 40$\% comparative to the optimal speed of a helix propelled through viscous medium ($U/r\omega\simeq 0.35$). Assuming regular fiber spatial arrangements, one can estimate the average separation between fibers in the matrix $\Delta$ as
\[
\Delta/b \approx 2\left(\frac{\sqrt{3\pi}}{2}\:\phi^{-1/2}-1\right)\:.
\]
For $\phi=0.001$ this formula yields $\Delta/b\approx 100$, so that the mean spacing between fibers, $\Delta$, is sufficiently large to assume that the propelled helix is not distorting the matrix microstructure.
The situation is much more complicated when propulsion in gel-forming semi-dilute solutions of some polymers (such as methylcellulose) are considered. The typical ``mesh size" between polymer chain aggregates in these medium can be considerably higher than the above estimate based on uniform spatial arrangement, due to large concentration fluctuations typical for semi-dilute solutions of flexible and semi-flexible polymers \cite{DoiEdwards}. Therefore, the micro swimmer may navigate its way through polymer-lean region, taking advantage of the distributed hydrodynamic resistance due to clusters and/or aggregates in polymer-rich regions. The quantitative comparison of the present theory with the experimental results showing the significant increase of the propulsion velocity in gel-forming polymer solutions is difficult due to the complex microstructure of the gels, the local interaction of the propelled object with the polymer network, elastic response of the distorted network, etc. It should be further emphasized that the present work is not a quantitative description of propulsion in polymer gels, but it comes to qualitatively demonstrate the potential physical origin of propulsion enhancement due to screened hydrodynamic interaction with the embedded network.
The suggested numerical formalism of Sec.~\ref{sec:4b} can be readily extended to other cases of interest. If nearby obstacles are allowed to move in response to the propeller displacement when the certain stress threshold is exceeded, the rotating filament, for instance, could swim through arbitrary dense matrix of obstacles. Such approach can be applied for more realistic modeling of propulsion through viscoplastic materials exhibiting finite yield stress. A similar mechanism of locomotion enhancement is expected, as the swimmer is propelled through liquid-like region encapsulating the moving object, and it is ``pushing" against the unyielded solid bilk material. In the limit of small volume fractions of obstructions the effective media approximation (\ref{eq:brinkman}) should still hold, while in the other limiting case of freely suspended obstacles, the swimmer would experience hydrodynamic resistance controlled by the elevated effective viscosity, $\mu_\mathrm{eff}$, of the viscous suspension. Since zero-Reynolds-number propulsion through a viscous Newtonian liquid is purely geometric, propulsion velocity (for a prescribed swimming gait) is independent of the viscosity and no enhancement of locomotion is expected in this case. Adding elastic response of the distorted network of obstacles would allow more realistic modeling of propulsion in complex viscoelastic media such as viscous polymer gels.
\\
I would like to thank Oren Raz and Oded Kenneth for stimulating discussions on the subject. This work was supported by the Israel Science Foundation (Grant No. 923/07).
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{"url":"https:\/\/trueshelf.org\/multi-choices\/295\/jee-advanced-2016-paper-2-mathematics-question-40\/","text":"JEE Advanced 2016 Paper 2 Mathematics Question 40\n\nThe value of\u00a0$\\displaystyle\\int_{\\frac{-\\pi}{2}}^{\\frac{\\pi}{2}}\\frac{x^2\\text{cos}x}{1+e^x}dx$\u00a0is equal to\n\n0\n\n0\n\n0\n\n0\n\n0\n\n0\n\n0\n\n0\n\n0\n\n0","date":"2018-01-18 21:51:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.39657846093177795, \"perplexity\": 3882.245456621397}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084887621.26\/warc\/CC-MAIN-20180118210638-20180118230638-00320.warc.gz\"}"}
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Macroglossum glaucoptera är en fjärilsart som beskrevs av Butler 1875. Macroglossum glaucoptera ingår i släktet Macroglossum och familjen svärmare. Inga underarter finns listade i Catalogue of Life.
Källor
Svärmare
glaucoptera
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On behalf of Hastings Law Firm posted in Houston Medical Diagnosis Error Lawyer on Monday, September 7, 2015.
It is only natural for people in Texas with health problems to trust in the years of training and experience of medical professionals to resolve their issues. Unfortunately, patients are sometimes disillusioned by a doctor's failure to diagnose conditions, leaving them with additional expenses and in worse shape medically than before seeking treatment. A couple in another state recently filed a medical malpractice lawsuit against the practices of two doctors.
The complaint asserts that the wife suffered eye problems that were treated by the two defendants — a retina consultant and an eye surgeon — in Sept. 2013. It is alleged that the defendants failed to accurately diagnose giant cell arteritis, a condition in which the lining of the arteries becomes inflamed. Because of the claimed failure to diagnose the condition, proper treatment was not provided.
The condition can cause jaw pain, headaches, double vision, blurred vision and even blindness and stroke. The woman claims to have suffered a loss of vision that has caused significant financial and personal hardship. The plaintiffs alleged that the defendants beached a duty to exercise due care. They seek in excess of $50,000 for damages and loss of consortium, plus legal costs and interest.
Failure to diagnose a condition can adversely affect the quality of a patient's life. While a monetary judgment will not directly improve a victim's health, it may ease any financial burdens caused by the wrongful acts. As the top medical malpractice law firm in Houston, we are here to help. Texas residents who believe that substandard medical care is the cause for their health problems may pursue financial relief by filing a medical malpractice claim in a civil court. Most choose to utilize the services of an experienced medical malpractice attorneys to guide them through the legal proceedings.
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TechTalk offers interesting jobs and a great working environment.
When you start your job at TechTalk a buddy is helping you to get to know our company.
For professional questions a coach will be provided.
Every year, you have a personal budget for further education and training.
Our external training offer is constantly being developed and employees have the opportunity to attend these trainings.
We support and organize various community events.
On Fridays there is an exchange for example PizzaTalks, DevTnT, Dev-Meetings, Coding Dojo and Architectural, UX and Tester-Meetings.
Once a year, all employees attend a two-day TechTalk Teamevent.
Our teams meet regularly for exciting, sporting or culinary adventures.
An part of the sporty TechTalk calendar is the Business Run.
In addition, TechTalker also meet while running, cycling, skiing, etc.
In our modern equipped office there is a lot of light and a great view.
Drinks, fruits and sweets are available free of charge. Lunch is supported with 2 €. With our coffee machine you will quickly become a barista.
For a lunch break in the open air, the Danube park in front of the door and at the end of the day the deck chairs on the Copa Cagrana.
We always try to consider personal wishes for further development. An extra-occupational study course is therefore not uncommon at TechTalk.
In addition to maternity and paternity leave, we help our TechTalkers bring work and personal life together.
Once a year we offer a free health check.
Regular feedback from your colleagues support your personal development.
Once a year, the "TechTalk Award" honors outstanding achievements of your colleagues.
We are also happy to stay in touch with former colleagues and look forward to the joint meetings.
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COVID-19 Testing and Vaccinations at Pilgrim Baptist Church
December 28, 2021 December 29, 2021 The Positive Community Magazine
Part of Renowned Black Clergy Effort to Fight COVID-19
BY ANDREA FERGUSON-PETERSON AND REVEREND GLENN WILSON, SR
For more than a century, Pilgrim Baptist Church has stood as a beacon of hope and a symbol of resilience to Newark, New Jersey residents. The city of Newark is thriving and prevailing despite the devastation of COVID-19. To build on this progress, Pilgrim Baptist Church and nine other churches in Newark have joined a historic nationwide partnership with United Way of New York City for the Choose Healthy Life Initiative. This effort provides COVID-19 testing, vaccinations, vaccine awareness, and preventative health education.
Newark churches supporting The Choose Healthy Life Action Plan initiative include; Metropolitan Baptist Church; Bethany Baptist Church; City Hope Ministries; Greater Mt. Moriah Baptist Church; Jehovah-Jireh Praise and Worship Church Center; Mt. Calvary Missionary Baptist Church; Clear View Baptist Church; Pilgrim Baptist Church; Pleasant Grove Baptist Church and St. Marks Free Will Baptist Church.
Choose Healthy Life was founded by Debra FraserHowze as a sustainable, scalable, and transferable approach centered around the Black Church—the oldest and most trusted institution in the Black community—to address public health disparities. Choose Health Life is co-chaired by Rev. Al Sharpton and Rev. Calvin O. Butts III. Fraser-Howze said she handpicked the two pastors after working with them in the past on health equity issues. Reverend Al Sharpton, Reverend Calvin Butts III, and Reverend David Jefferson Sr., Esq., helped launch the effort in five major cities: Newark, Atlanta, Detroit, Washington D.C, and New York City.
Dr. Glenn B. Wilson, Sr., Pastor of Pilgrim Baptist Church, says "consistent testing and vaccinations are needed to be in place to save lives. As faith leaders, we along with the partner churches, have responsibility to guide community members through the testing, vaccination process, and serve as a COVID-19 education resource."
To help with our COVID-19 testing and vaccination efforts, with the following: Newark Urban League, Bessie Smith Community Food Distribution Outreach, Warriors Cycling New Jersey/New York, Newark Sarah Ward Pre-School, and Skips Tinnie Learners.
← Voting is a Right, Not a Privilege
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Rev. Jesse Williams' Video Message to The Positive Community Family
May 21, 2020 June 10, 2020 The Positive Community Magazine
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April 14, 2020 May 5, 2020 The Positive Community Magazine
NYSCAS Nursing Students Volunteer At Touro COVID-19 Vaccination Site
August 6, 2021 August 6, 2021 The Positive Community Magazine
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\section{Introduction}
In a series of recent papers~\cite{BL}, Bagger and Lambert (BL)
have constructed a three-dimensional, interacting superconformal
gauge theory of multiple M2-branes. The action is maximally
supersymmetric with 16 ordinary supersymmetries, and it has been
verified that the theory is indeed superconformal with 16 conformal
supercharges in \cite{Bandres:2008vf}. In the quest for the final
form of the theory, as usual, it was supersymmetry that provided
crucial guiding lights. The work was initiated as an attempt to
incorporate Basu and Harvey's generalized Nahm equation -which was a
proposal to describe M2-branes ending on an
M5-brane~\cite{Basu:2004ed}- in the full supersymmetric M2-brane
action. Their analysis revealed a novel algebraic structure, namely
the 3-algebra, which is also investigated independently by
Gustavsson~\cite{Gustavsson:2007vu}. Since the discovery, the
multiple M2-brane theory of Bagger and Lambert has attracted an
enormous degree of
attention~\cite{Ganor:2006ub,Ho:2007vk,Copland:2007by,Chen:2007ir,
MacConamhna:2007fw,Berman:2007bv,Chen:2007tt,Bandos:2008um,Gustavsson:2008dy,Mukhi:2008ux,
Berman:2008be,VanRaamsdonk:2008ft,Morozov:2008cb,Lambert:2008et,Gran:2008vi,Ho:2008bn,Gomis:2008cv,
Bergshoeff:2008cz,Hosomichi:2008qk,Papadopoulos:2008sk,Gauntlett:2008uf,Shimada:2008xy,Papadopoulos:2008gh,
Ho:2008nn,Gomis:2008uv,Benvenuti:2008bt,Ho:2008ei,Morozov:2008rc,Honma:2008un,Fuji:2008yj,
Ho:2008ve}. One might expect that, given this genuine superconformal
field theory, ${\cal M}$-theory is now about to unveil its mysterious and
fundamental features.
In the present paper, we set out to classify the BPS states, or the
BPS equations of the BL theory using a group theoretical
consideration. Apparently the theory of our interest has the Lorentz
group $\mbox{SO}(1,2)$ and the R-symmetry group $\mbox{SO}(8)$. Instead of
providing the full and thorough survey of possible BPS equations, we
focus mainly on two different types of BPS equations with different number
of supersymmetries, and classify them completely.
The first class is
completely Lorentz invariant, and the other is invariant under the
spatial rotation.
In the first type, the BPS equations are given
purely in terms of the three-algebra commutators and independent of the three-dimensional worldvolume
coordinates. Thus the corresponding nontrivial configurations possess infinite energy, typically corresponding to BPS objects of infinite size. Previously known analogous algebraic soultions include the longitudinal M5-brane in ${\cal M}$-theory matrix model which is realized in terms of Heisenberg algebra or large $N$ matrices \cite{Banks:1996nn}.
In the other type the equations are $\mbox{SO}(2)$ rotation invariant,
and the fields can be {\it time-dependent}.
A technical reason why we focus on the two classes is that in these cases, fully utilizing the $\mbox{SO}(8)$ triality we are able to classify the BPS equations completely.
In addition to the two classes, there is another
possibility to obtain third type of BPS equations \textit{via} simple tensor product. Namely one can obtain various generalizations of the
Nahm equations which are invariant under the boost $\mbox{SO}(1,1)\subset\mbox{SO}(1,2)$.
Our BPS equations manifest the division algebra structures:
octonion, quarternion or complex. In the paper we will mainly focus
on the BPS equations themselves. Our results hold for both the finite and infinite dimensional three-algebras.
Note however that the Lorentz invariant BPS equations can have nontrivial solutions only for infinite dimensional three-algebras. The specific solutions and the physical interpretation will be presented in a separate
publication \cite{preparation}.
The organization of the present paper is as follows. Sec.\ref{secPre}
is for preliminaries.
We first discuss the general features of the `supersymmetric projection matrices' and
review how to derive the corresponding BPS equations for a given projection matrix.
We also explain the relevant symmetries. Then we classify the projection matrices for
the $\mbox{SO}(1,2)$, $\mbox{SO}(2)$ and $\mbox{SO}(1,1)$ invariant equations.
Sec.\ref{secBPS} contains our
main results of the BPS equations.
Sec.\ref{secBPSSO12} classifies the $\mbox{SO}(1,2)$ invariant BPS equations preserving two,
four, six, eight, ten and twelve supersymmetries.\footnote{Note that in the present paper we focus on the sixteen ordinary supersymmetries
and not the sixteen conformal supersymmetries.
For the BPS equations preserving conformal supersymmetries in super Yang-Mills
we refer the readers to Ref.\cite{Park:2006kt}.} Sec.\ref{secBPSSO25} classifies the
$\mbox{SO}(2)$
invariant BPS equations preserving two, four, six and eight supersymmetries.
In Sec.\ref{secBPSSO11} we
discuss the $\mbox{SO}(1,1)$ invariant BPS equations which generalize the Nahm equations.
The final section, Sec.\ref{discussion} contains our results and discussions.
In Appendix we review the $\mbox{SO}(8)$ triality and its relation to octonions. \\
\textit{Note added}:
While this paper is being finished, Ref.\cite{KMlast} appears in
ArXiv which partially overlaps with our work, as it discusses the BPS equations
of the form: $D_{y}X_{I}=\textstyle{\frac{1}{3!}}{\cal C}_{IJKL}[X^{J},X^{K},X^{L}]$.
In the present paper, we explicitly spell the coefficients ${\cal C}_{IJKL}$ and classify
various BPS equations.\\
{}\\
\section{Preliminaries\label{secPre}}
The multiple M2-brane theory has 8 real scalar fields $X^I, I=1,2,\cdots,8$
and a 16 component Majorana spinor
$\Psi$.
The supersymmetry transformation of the fermions in the Bagger-Lambert theory
assumes the form:
\begin{equation}
\delta\Psi=\left(F_{\mu I}\Gamma^{\mu I}-\textstyle{\frac{1}{6}}F_{IJK}\Gamma^{IJK}\right)\varepsilon\,,
\label{BPS0}
\end{equation}
where all the variables are three-algebra valued and we set
\begin{equation}
\begin{array}{ll}
F_{\mu I}\equiv D_{\mu} X_{I}\,,~~~~&~~~~~F_{IJK}\equiv\left[X_{I},X_{J},X_{K}\right]\,.
\end{array}
\end{equation}
The bracket $[X_I,X_J,X_K]$ denotes the three-algebra product which is trilinear and totally
antisymmetric.
Note also that in contrast to the original convention \cite{BL} we let $I=1,2,\cdots,8$
and take $\mu\equiv 0,9,{10}$ directions as for the M2-brane worldvolume for convenience to present the BPS equations later,
\begin{equation}
\begin{array}{ccc}
x^{0}{\equiv t}\,,~~~~&~~~~x^{9}{\equiv x}\,,~~~~&~~~~x^{10}{\equiv y}\,.
\end{array}
\end{equation}
The supersymmetry parameter is real and subject to the $\mbox{SO}(1,2)$ projection condition:
\begin{equation}
\Gamma^{txy}\varepsilon=\varepsilon\,,
\label{chiral12}
\end{equation}
which is consistent with the opposite projection property, $\Gamma^{txy}\Psi=-\Psi$. Since the product of all the eleven-dimensional gamma matrices leads to the $32\times 32$ identity matrix $\Gamma^{txy123\cdots8}=1$, the above $\mbox{SO}(1,2)$ projection condition coincides with the chirality condition of $\mbox{SO}(8)$,
\begin{equation}
\Gamma^{123\cdots 8}\varepsilon=\varepsilon\,.
\label{chiral8}
\end{equation}
\subsection{Supersymmetry projection matrix - general}
In general for supersymmetric theories, the supersymmetry projection matrix $\Omega$ can be defined in terms of the commuting, real, orthonormal supersymmetry parameters $\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{N}$,
\begin{equation}
\begin{array}{ll}
\displaystyle{\Omega:=\sum_{i=1}^{N} \varepsilon_{i}\varepsilon^{\dagger}_{i}\,,}~~~~&~~~~\varepsilon_{i}^{\dagger}\varepsilon_{j}=\delta_{ij}\,,
\end{array}
\end{equation}
satisfying $\Omega^{\dagger}=\Omega^{2}=\Omega$. Here $N$ denotes the number of the preserved supersymmetries,
\begin{equation}
N={\rm Tr}\Omega\,.
\end{equation}
Naturally the eigenvalues of the projection matrices are either zero or one.\\
When the supersymmetry transformation of fermions takes the form $\delta\Psi={\cal F}\varepsilon$ where ${\cal F}$ denotes a bosonic quantity contracted with gamma matrices as in (\ref{BPS0}),
the general strategy to obtain the BPS equations is as follows~\cite{BPS68}:
\begin{enumerate}
\item \textit{Expand the projection matrix $\Omega$ in terms of the gamma matrix
product basis.}
\item \textit{Perform the matrix product ${\cal F}\Omega$ and reexpress it in terms of
the gamma matrix product basis.}
\item \textit{Read off the BPS equations from the coefficients of the linearly independent terms.}
\end{enumerate}
For example in the Euclidean four-dimensional minimal super Yang-Mills theory, we have two choices for the projection matrix $\Omega={{\textstyle\frac{1}{2}}}(1\pm\gamma^{1234})$, while ${\cal F}=F_{ij}\gamma^{ij}$. Consequently, noting $\gamma^{12}\Omega=\mp\gamma^{34}\Omega$ \textit{etc.}, we get $F_{ij}\gamma^{ij}\Omega=2(F\mp{\star\,F})_{i4}\gamma^{i4}\Omega$ such that the corresponding BPS equations are the well-known self-dual or anti-self-dual equations $F=\pm{\star\,F}$. In this way, the complete classifications of the BPS equations in six and eight-dimensional super Yang-Mills as well as the pp-wave M-theory matrix model~\cite{Berenstein:2002jq} have been carried out \cite{BPS68,Park:2002cba,Kim:2002zg}. \\
The present paper concerns the BPS equations of the Bagger-Lambert theory.
Since the eleven-dimensional spacetime admits Majorana spinors we can set all
the gamma matrices and the spinors to be real. In particular, the spatial gamma matrices are symmetric while the temporal gamma matrix is anti-symmetric.
Consequently, also from (\ref{chiral8}), the projection matrices of the Bagger-Lambert theory must satisfy
\begin{equation}
\begin{array}{lll}
\Omega=\Omega^{T}=\Omega^{\ast}\,,~~~~&~~~~\Omega=\Omega^{2}\,,~~~~&~~~~\Omega={\cal P}\Omega=\Omega{\cal P}\,,
\end{array}
\label{Omegacon}
\end{equation}
where ${\cal P}$ is the $\mbox{SO}(8)$ chiral projection matrix,
\begin{equation}
{\cal P}:={{\textstyle\frac{1}{2}}}(1+\Gamma^{123\cdots 8})\,.
\label{defcP}
\end{equation}
The most general form of such projection matrices reads
\begin{equation}
\Omega=\left[c+ \Upsilon_{4}+\Gamma^{x}(c^{\prime}+\Upsilon_{4}^{\prime})
+\Gamma^{y}(c^{\prime\prime}+\Upsilon_{4}^{\prime\prime})+\Gamma^{xy}\Upsilon_{2}\right]{\cal P}\,,
\label{GOME}
\end{equation}
where $c,c^{\prime},c^{\prime\prime}$ are constants,
$\Upsilon_{4},\Upsilon_{4}^{\prime}, \Upsilon_{4}^{\prime\prime}$ are foursome productions of the $\mbox{SO}(8)$ gamma matrices $\Gamma^{IJKL}$ contracted with self-dual four-forms, and $\Upsilon_{2}$ is a twosome production of the $\mbox{SO}(8)$ gamma matrices $\Gamma^{IJ}$ contracted with a two-form. All together, \textit{a priori}, there are ${3+3\times{{\textstyle\frac{1}{2}}}\left(\!\scriptsize{\begin{array}{c}8\\4\end{array}}\!\right)+\left(\!\scriptsize{\begin{array}{c}8\\2\end{array}}\!\right)=136}$ real parameters which must be determined by requiring the remaining condition $\Omega^{2}=\Omega$. The symmetry group $\mbox{SO}(1,2)\times\mbox{SO}(8)$ in the Bagger-Lambert theory may reduce the number of the free parameters, but is not big enough to transform all the free parameters, the two-form and the four-forms, into `canonical' forms. Note that the $\mbox{SO}(8)$ rotation may take only one of $\left\{\Upsilon_{4},\Upsilon_{4}^{\prime}, \Upsilon_{4}^{\prime\prime},\Upsilon_{2}\right\}$ into a canonical form. In our choice, the canonical form of a two-form reads
\begin{equation}
\Upsilon_{2}=a_{1}\Gamma^{12}+a_{2}\Gamma^{34}+a_{3}\Gamma^{56}+a_{4}\Gamma^{78}\,,
\end{equation}
while the canonical form of a self-dual four-form reads
\begin{equation}
\Upsilon_{4}=b_{1}{\cal E}_{1}+b_{2}{\cal E}_{2}+b_{3}{\cal E}_{3}+b_{4}{\cal E}_{4}+b_{5}{\cal E}_{5}+b_{6}{\cal E}_{6}+b_{7}{\cal E}_{7}\,,
\label{canon4}
\end{equation}
where we set
\begin{equation}
\begin{array}{cccc}
{\cal E}_{1}=\Gamma_{8127}{\cal P}\,,~&{\cal E}_{2}=\Gamma_{8163}{\cal P}\,,~&{\cal E}_{3}=\Gamma_{8246}{\cal P}\,,~& {\cal E}_{4}=\Gamma_{8347}{\cal P}\,,\\ {}&{}&{}&{}\\ {\cal E}_{5}=\Gamma_{8567}{\cal P}\,,~&{\cal E}_{6}=\Gamma_{8253}{\cal P}\,,~&
{\cal E}_{7}=\Gamma_{8154}{\cal P}\,.&{}
\end{array}
\label{Edef}
\end{equation}
The former is well known, while the latter is less familiar and we review it in Appendix~\ref{Appoct}. In (\ref{Edef}) the subscript spatial indices of the gamma matrices are organized such that the three indices after the common $8$ are identical to those of
the totally anti-symmetric octonionic structure constants~\cite{BPS68,Baez}:
\begin{equation}
\begin{array}{c}
~~e_{i}e_{j}=-\delta_{ij}+c_{ijk}\,e_{k}\,,~~~~~~~{i,j,k=1,2,\cdots,7\,,}\\ {}\\
1=c_{127}=c_{163}=c_{246}=c_{347}=c_{567}=c_{253}=c_{154}\,,~~~~~\mbox{others zero}\,.
\end{array}
\label{octconst}
\end{equation}
~\\
We say $\Omega$ is invariant under $\mbox{SO}(2)$ rotation invariant on $xy$-plane if
$~[\Gamma_{\!xy}\,,\,\Omega]=0$. When this holds, for a finite angle $\phi$ and rotation
$G=e^{\phi\Gamma_{\!xy}}$, from the equivalence
\begin{equation}
\begin{array}{lll}
{\cal F}\Omega=0~~&~~~\Longleftrightarrow~~~&~~G{\cal F}\Omega G^{-1}=G{\cal F} G^{-1}\Omega=0\,,
\end{array}
\end{equation}
we note that the corresponding BPS equations are, as a set, invariant under the rotation. Naturally this generalizes to an arbitrary subgroup of $\mbox{SO}(1,2){\times\mbox{SO}(8)}$.\\
In the present paper instead of attempting to solve for the most general
projection matrices, we restrict to the cases where $\Omega$ assumes the canonical form. Namely we focus on two types of the BPS equations and classify the corresponding BPS equations completely: one is the $\mbox{SO}(1,2)$ \textit{invariant cases}~\textit{i.e.}
\begin{equation}
{\Omega=\left(c+ \Upsilon_{4}\right){\cal P}\,,}
\label{SO12Omega}
\end{equation}
and the other is the $\mbox{SO}(2)^{\mathbf{5}}\equiv\mbox{SO}(2){\times\mbox{SO}(2)}{\times\mbox{SO}(2)}{\times\mbox{SO}(2)}{\times\mbox{SO}(2)}$ \textit{invariant cases}~\textit{i.e.}
\begin{equation}
{\Omega=\left(\mbox{constant}+\mbox{twosome~products~of~}\left\{\Gamma^{xy},\Gamma^{12},\Gamma^{34},
\Gamma^{56},\Gamma^{78}\right\}\right){\cal P}\,.}
\label{SO25Omega}
\end{equation}
Here $\mbox{SO}(1,2)$ and $\mbox{SO}(2)$ correspond to the M2 worldvolume Lorenz symmetry and the Cartan subgroup of the symmetry group $\mbox{SO}(1,2)\times\mbox{SO}(8)$ respectively. In addition, the former will easily generate various \textit{${\cal M}$-theoretic generalizations of the Nahm equations} which are invariant under $\mbox{SO}(1,1)\subset\mbox{SO}(1,2)$, as the corresponding projection matrices are of the form:
\begin{equation}
{\Omega=(1\pm\Gamma^{tx})\left(c+ \Upsilon_{4}\right){\cal P}\,.}
\label{SO11Omega}
\end{equation}
~\\
\subsection{$\mbox{SO}(1,2)$ invariant projection matrices}
The basic building blocks of all the possible $\mbox{SO}(1,2)$ invariant projection matrices are the following $N=2$ projection matrices~\cite{BPS68}:
\begin{equation}
\Omega=\textstyle{\frac{1}{8}}\left({\cal P}+{\alpha}_{1}{\alpha}_{2}{\cal E}_{1}+{\alpha}_{1}{\alpha}_{3}{\cal E}_{2}
+{\alpha}_{3}{\cal E}_{3}+{\alpha}_{2}{\cal E}_{4}+{\alpha}_{1}{\cal E}_{5}+{\alpha}_{1}{\alpha}_{2}{\alpha}_{3}{\cal E}_{6}
+{\alpha}_{2}{\alpha}_{3}{\cal E}_{7}\right)\,,
\label{SO12OG}
\end{equation}
where $\alpha_{1},\,\alpha_{2},\,\alpha_{3}$ are three independent signs,
\begin{equation}
\alpha_{1}^{2}=\alpha_{2}^{2}=\alpha_{3}^{2}=1\,.
\end{equation}
Three independent sign choices lead to eight possible combinations, hence eight $N=2$ projection matrices.
They are orthogonal to each other and complete, as summing all of them gives an identity. Namely they form an orthogonal basis for the $\mbox{SO}(1,2)$ invariant projection matrices. General $N=2k$ projection matrices can be straightforwardly obtained as a $k$ sum of the above eight $N=2$ projection matrices. Furthermore, from the $\mbox{SO}(8)$
triality, the ${8!}/[{k!(8{-k})!}]$ possibilities for the $k$ sum are all equivalent to each
other. The corresponding $N{=2k}$ BPS equations are $\mbox{SO}(1,2){\times\mbox{SO}(8{-k})}{\times\mbox{SO}(k)}$ invariant.
\subsection{$\mbox{SO}(2)$ invariant projection matrices}
The basic building blocks of all the possible $\mbox{SO}(2)$ invariant projection
matrices are the following $N=2$ projection matrices (see Appendix \ref{AppCartan}
for derivation):
\begin{equation}
\begin{array}{ll}
\Omega&=\textstyle{\frac{1}{8}}
\left[1+\Gamma^{xy}\!\left(\beta_{1}\Gamma^{12}+\beta_{2}\Gamma^{34}+\beta_{3}\Gamma^{56}+
\beta_{1}\beta_{2}\beta_{3}\Gamma^{78}\right)-\beta_{1}\beta_{2}\Gamma^{1234}
-\beta_{3}\beta_{1}\Gamma^{1256}-\beta_{2}\beta_{3}\Gamma^{1278}\right]\!{\cal P}\\
{}&{}\\
{}&=\textstyle{\frac{1}{8}}(1+\beta_{1}\Gamma^{xy12})(1+\beta_{2}\Gamma^{xy34})(1+\beta_{3}\Gamma^{xy56}){\cal P}\,,
\end{array}
\label{SO2OG}
\end{equation}
where $\beta_{1}$, $\beta_{2}$, $\beta_{3}$ denote three independent signs,
\begin{equation}
\beta_{1}^{2}=\beta_{2}^{2}=\beta_{3}^{2}=1\,.
\end{equation}
Eight possible $N=2$ projection matrices form an orthogonal basis for
the $\mbox{SO}(2)$ invariant projection matrices. General $N=2k$
projection matrices can be straightforwardly obtained as a $k$ sum of
the above eight $N=2$ projection matrices. However, if the sum contains a pair
of two opposite overall sign factors~\textit{e.g.} $\scriptstyle{(+++)}$ and
$\scriptstyle{(---)}$, the corresponding BPS configurations become $\mbox{SO}(1,2)$
invariant as $F_{\mu I}=0$ and the BPS equations reduce to those of $\mbox{SO}(1,2)$
invariant BPS equations. Excluding these cases, up to $\mbox{SO}(8)$ rotations,
there are five inequivalent $\mbox{SO}(2)$ invariant projection matrices as follows.
\begin{itemize}
\item $N=2$ $\,\mbox{SO}(2){\times\mbox{SU}(4)}$ invariant projection matrix, with the choice of $\scriptstyle{(\beta_{1},\beta_{2},\beta_{3})\,=\,(+++)}$,
\begin{equation}
\Omega=\textstyle{\frac{1}{8}}
\left[1+\Gamma^{xy}\!\left(\Gamma^{12}+\Gamma^{34}+\Gamma^{56}+\Gamma^{78}\right)-\Gamma^{1234}
-\Gamma^{1256}-\Gamma^{1278}\right]{\cal P}\,.
\label{SO251}
\end{equation}
\item $N=4$ $\,\mbox{SO}(2){\times\mbox{SU}(2)}{\times\mbox{SO}(4)}$ invariant projection matrix, with $\scriptstyle{(+++),(++-)}$,
\begin{equation}
\Omega=\textstyle{\frac{1}{4}}
\left[1+\Gamma^{xy}\!\left(\Gamma^{12}+\Gamma^{34}\right)-\Gamma^{1234}\right]{\cal P}\,.
\label{SO252}
\end{equation}
\item $N=6$ $\,\mbox{SO}(2){\times\mbox{SO}(2)}{\times\mbox{SU}(3)}$ invariant projection matrix, with $\scriptstyle{(+++),(++-),(+-+)}$,
\begin{equation}
\Omega=\textstyle{\frac{1}{8}}
\left[3+\Gamma^{xy}\!\left(3\Gamma^{12}+\Gamma^{34}+\Gamma^{56}-\Gamma^{78}\right)-\Gamma^{1234}
-\Gamma^{1256}+\Gamma^{1278}\right]{\cal P}\,.
\label{SO253}
\end{equation}
\item $N=8$ $\,\mbox{SO}(2){\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant projection matrix, with $\scriptstyle{(+++),(++-),(+-+),(+--)}$,
\begin{equation}
\Omega=\textstyle{\frac{1}{2}}(1+\Gamma^{xy12}){\cal P}\,.
\label{SO254}
\end{equation}
\item $N=8$ $\,\mbox{SO}(2){\times\mbox{SU}(4)}$ invariant projection matrix, with $\scriptstyle{(+++),(++-),(+-+),(-++)}$,
\begin{equation}
\Omega=\textstyle{\frac{1}{4}}
\left[2+\Gamma^{xy}\!\left(\Gamma^{12}+\Gamma^{34}+\Gamma^{56}-\Gamma^{78}\right)\right]{\cal P}\,.
\label{SO255}
\end{equation}
\end{itemize}
\subsection{$\mbox{SO}(1,1)$ invariant projection matrices}
For $\mbox{SO}(1,1)$ invariant projection matrices, we have the following $N=1$ projection matrices:
\begin{equation}
\Omega=\textstyle{\frac{1}{16}}\left(1+\alpha_{0}\Gamma^{tx}\right)
\left({\cal P}+{\alpha}_{1}{\alpha}_{2}{\cal E}_{1}+{\alpha}_{1}{\alpha}_{3}{\cal E}_{2}
+{\alpha}_{3}{\cal E}_{3}+{\alpha}_{2}{\cal E}_{4}+{\alpha}_{1}{\cal E}_{5}+{\alpha}_{1}{\alpha}_{2}{\alpha}_{3}{\cal E}_{6}
+{\alpha}_{2}{\alpha}_{3}{\cal E}_{7}\right)\,,
\label{SO11OG}
\end{equation}
where $\alpha_{0},\,\alpha_{1},\,\alpha_{2},\,\alpha_{3}$ are four independent signs,
\begin{equation}
\alpha_{0}^2=\alpha_{1}^{2}=\alpha_{2}^{2}=\alpha_{3}^{2}=1\,.
\end{equation}
Sixteen possible $N=1$ projection matrices form an orthogonal basis for the $\mbox{SO}(1,1)$ invariant projection matrices. Generic $N=k$ $\mbox{SO}(1,1)$ invariant projection matrices may be obtained straightforwardly as a $k$ sum of the above sixteen $N=1$ projection matrices. For each sum, we may decompose
\begin{equation}
\begin{array}{lll}
N=N_{+}+N_{-}\,,~~~~&~~~~N_{+}=n_{+}+n\,,~~~~&~~~~N_{-}=n_{-}+n\,,
\end{array}
\label{Npmdef}
\end{equation}
such that $N_{\pm}$ denotes the number of $N=1$ projection matrices in the sum whose $\alpha_{0}$ values are $\pm 1$, and $n$ counts the number of $N=1$ projection matrix pairs which have the same $\alpha_{1},\,\alpha_{2},\,\alpha_{3}$ values and opposite $\alpha_{0}$ signs. There are ${8!}/[{n_{+}!n_{-}!n!(8{-n_{+}}{-n_{-}}{-n})!}]$ possibilities for the sum which are all equivalent to another, thanks to the $\mbox{SO}(8)$ triality.
Furthermore, if $n$ is nontrivial~$n\neq 0$, then the BPS configurations become $\mbox{SO}(1,2)$ invariant as $F_{\mu I}=0$ and the number of the preserved supersymmetries is automatically increased from $n_{+}+n_{-}+2n$ to $2(n_{+}+n_{-}+n)$. In this case the BPS equations reduce to those of $\mbox{SO}(1,2)$ invariant BPS equations. Genuinely $\mbox{SO}(1,1)$ invariant BPS equations appear only when $n=0$. The corresponding $(N_{+},N_{-})$ BPS equations are then $\mbox{SO}(1,1){\times\mbox{SO}(N_{+})}{\times\mbox{SO}(N_{-})}{\times\mbox{SO}(8{-N_{+}}{-N_{-}})}$ invariant with the natural restriction $N_{+}{+N_{-}}\leq 8$.
\section{Classification of the BPS equations\label{secBPS}}
\subsection{$\mbox{SO}(1,2)$ invariant BPS equations\label{secBPSSO12}}
The generic $N=2$ $\mbox{SO}(1,2)$ invariant projection matrix~(\ref{SO12OG}) leads to the following
$N{=2}$ $\,\mbox{SO}(1,2){\times\mbox{SO}(7)}$ invariant BPS equations which involve three free sign factors $\,\alpha_{1}^{2}=\alpha_{2}^{2}=\alpha_{3}^{2}=1$:
\begin{equation}
\begin{array}{c}
F_{\mu I}=0\,,~~~~~~~~~~~\mu=t,x,y\,,~~~~~I=1,2,\cdots,8\,,
\label{FmuIzero}
\end{array}
\end{equation}
and
\begin{equation}
\begin{array}{l}
{\alpha_{1}\alpha_{2}}F_{278}+{\alpha_{2}\alpha_{3}}F_{548}+{\alpha_{3}\alpha_{1}}F_{638}
+\alpha_{1}F_{234}+\alpha_{2}F_{256}+\alpha_{3}F_{357}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{476}=0\,,\\
{}\\
{\alpha_{1}\alpha_{2}}F_{718}+{\alpha_{2}\alpha_{3}}F_{376}+{\alpha_{3}\alpha_{1}}F_{475}
+\alpha_{1}F_{143}+\alpha_{2}F_{165}+\alpha_{3}F_{468}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{538}=0\,,\\
{}\\
{\alpha_{1}\alpha_{2}}F_{456}+{\alpha_{2}\alpha_{3}}F_{267}+{\alpha_{3}\alpha_{1}}F_{168}
+\alpha_{1}F_{124}+\alpha_{2}F_{478}+\alpha_{3}F_{517}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{258}=0\,,\\
{}\\
{\alpha_{1}\alpha_{2}}F_{536}+{\alpha_{2}\alpha_{3}}F_{158}+{\alpha_{3}\alpha_{1}}F_{257}
+\alpha_{1}F_{132}+\alpha_{2}F_{738}+\alpha_{3}F_{628}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{167}=0\,,\\
{}\\
{\alpha_{1}\alpha_{2}}F_{346}+{\alpha_{2}\alpha_{3}}F_{418}+{\alpha_{3}\alpha_{1}}F_{427}
+\alpha_{1}F_{678}+\alpha_{2}F_{126}+\alpha_{3}F_{137}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{328}=0\,,\\
{}\\
{\alpha_{1}\alpha_{2}}F_{354}+{\alpha_{2}\alpha_{3}}F_{273}+{\alpha_{3}\alpha_{1}}F_{318}
+\alpha_{1}F_{758}+\alpha_{2}F_{152}+\alpha_{3}F_{248}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{174}=0\,,\\
{}\\
{\alpha_{1}\alpha_{2}}F_{128}+{\alpha_{2}\alpha_{3}}F_{236}+{\alpha_{3}\alpha_{1}}F_{245}
+\alpha_{1}F_{568}+\alpha_{2}F_{348}+\alpha_{3}F_{153}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{146}=0\,,\\
{}\\
{\alpha_{1}\alpha_{2}}F_{127}+{\alpha_{2}\alpha_{3}}F_{154}+{\alpha_{3}\alpha_{1}}F_{163}
+\alpha_{1}F_{567}+\alpha_{2}F_{347}+\alpha_{3}F_{246}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{253}=0\,.
\end{array}
\label{BPSMasterSO12}
\end{equation}
In particular, the $\mbox{SO}(1,2)$ invariance, the M2-brane worldvolume Lorentz symmetry, removes any worldvolume dependence, $D_{\mu}X_{I}=0$ for all $\mu$ and $I$. \\
The above set of BPS equations can be regarded as the master equations since any $N{=2k}$ BPS equations can be obtained by imposing $k$ copies of distinct $(\alpha_{1},\alpha_{2},\alpha_{3})$ choices. The corresponding $N{=2k}$ BPS equations are then $\mbox{SO}(1,2){\times\mbox{SO}(8{-k})}{\times\mbox{SO}(k)}$ invariant. We find for $N{=14}$ and $N{=16}$ the corresponding BPS equations are trivial, $\,F_{\mu I}=F_{IJK}=0$. Other nontrivial cases are as follows.
\subsubsection{$N=2$ $\,\mbox{SO}(1,2){\times\mbox{SO}(7)}$ invariant BPS equations - \textit{octonion}}
With the choice of ${(\alpha_{1},\alpha_{2},\alpha_{3})=(+++)}$,
the $N{=2}$ $\,\mbox{SO}(1,2){\times\mbox{SO}(7)}$ invariant BPS equations~(\ref{FmuIzero}), (\ref{BPSMasterSO12}) assume a compact form:
\begin{equation}
\begin{array}{ll}
F_{\mu I}=0\,,~~~~~&~~~~{\cal C}_{IJKL}F^{JKL}=0\,,
\end{array}
\label{SO12N2}
\end{equation}
where ${\cal C}_{IJKL}$ is a $\mbox{SO}(7)$ invariant four-form in eight dimensions, defined in terms of the octonionic structure constant (\ref{octconst}),
\begin{equation}
\begin{array}{lll}
{\cal C}_{ijk8}\equiv c_{ijk}\,,~~~~&~~~~{\cal C}_{ijkl}\equiv\textstyle{\frac{1}{6}}\epsilon_{pqrijkl}c_{pqr}~~~~~~~&\mbox{where}~~~~~~
1\leq i,j,k,l\leq 7\,.
\end{array}
\end{equation}
~\\
BPS states preserving $N{=2k}$ supersymmetries then satisfy $k$ copies of the $N{=2}$ BPS equations of different $\alpha$ choices. The corresponding $N{=2k}$ BPS equations are $\mbox{SO}(1,2){\times\mbox{SO}(k)}{\times\mbox{SO}(8{-k})}$ invariant, and involve $k$ different octonionic structures.
\subsubsection{$N=4$ $\,\mbox{SO}(1,2){\times\mbox{SO}(6)}{\times\mbox{SO}(2)}$ invariant BPS equations - \textit{complex}}
The $N{=4}$ $\,\mbox{SO}(1,2){\times\mbox{SO}(6)}{\times\mbox{SO}(2)}$ invariant BPS equations are, with $F_{\mu I}{=0}$,
\begin{equation}
\begin{array}{ll}
F_{IJK}{\cal J}^{JK}=0\,,~~~&~~~F_{IJK}=(1{\otimes{\cal J}}{\otimes{\cal J}}+{\cal J}{\otimes 1}{\otimes{\cal J}}+
{\cal J}{\otimes{\cal J}}{\otimes 1})_{IJK}{}^{LMN}F_{LMN}\,,
\end{array}
\label{SO12N4}
\end{equation}
where ${\cal J}$ is a complex structure ${\cal J}^{2}=-1$, ${\cal J}^{T}=-{\cal J}$ and hence $\mbox{SU}(4){\times\mbox{SO}(2)}$ invariant.\\
With the specific choice of $\alpha$'s as $\scriptstyle{(+++),(++-)}$, one gets
\begin{equation}
{{\textstyle\frac{1}{2}}}{\cal J}_{IJ}\Gamma^{IJ}=\Gamma^{12}+\Gamma^{34}+\Gamma^{56}+\Gamma^{78}\,.
\label{cJrep}
\end{equation}
{}\\
In terms of the corresponding holomorphic, anti-holomorphic coordinates $a,\bar{a}=1,2,3,4$ and the metric $\delta^{a\bar{a}}$, the above
$N{=4}$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(6)}{\times\mbox{SO}(2)}$ BPS equations~(\ref{SO12N4}) can be rewritten as
\begin{equation}
\begin{array}{ll}
F_{ab}{}^{b}=F_{\bar{a}b}{}^{b}=0\,,~~~~~&~~~~F_{abc}=F_{\bar{a}\bar{b}\bar{c}}=0\,.
\end{array}
\end{equation}
Namely $F_{(1,2)}$, $F_{(2,1)}$ are primitive and $F_{(3,0)}{=F_{(0,3)}}{=0}$. \\
We note that summing two $N{=2}$ projection matrices generates one complex structure. Hence in general,
summing $k>2$ of $N{=2}$ projection matrices will present \mbox{$\left(\!\scriptsize{\begin{array}{c}k\\2\end{array}}\!\right)$} number of complex structures to the corresponding $\mbox{SO}(1,2){\times\mbox{SO}(8{-k})}{\times\mbox{SO}(k)}$ invariant BPS equations. The ${{\textstyle\frac{1}{2}}} k(k-1)$ complex structures form singlets under $\mbox{SO}(8{-k})$ and are in the adjoint representation or $k$-dimensional two-form representation of $\mbox{SO}(k)$. In fact, they correspond to the generators of $\mbox{SO}(k)$. Nevertheless, the corresponding ${{\textstyle\frac{1}{2}}} k(k-1)$ number of complex structures are degenerate in the sense that distinct $[\frac{k+1}{2}]$ of them are sufficient to lead to the full $N{=2k}$ BPS equations.
\subsubsection{$N=6$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(5)}{\times\mbox{SO}(3)}$ invariant BPS equations - \textit{quarternion}}
The $N{=6}$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(5)}{\times\mbox{SO}(3)}$ invariant BPS equations are, with $F_{\mu I}{=0}$,
\begin{equation}
F_{IJK}{\cal J}_{p}^{JK}=0\,,~~~~~~~~p=1,2,3\,,
\label{SO12N6}
\end{equation}
where ${\cal J}_{1},{\cal J}_{2},{\cal J}_{3}$ are three distinct complex structures satisfying the quaternion relations:
\begin{equation}
{\cal J}_{1}^{2}={\cal J}_{2}^{2}={\cal J}_{3}^{2}={\cal J}_{1}{\cal J}_{2}{\cal J}_{3}=-1\,.
\end{equation}
It is worth to note that the remaining relation of (\ref{SO12N4}) \textit{i.e.} $F_{(3,0)}{=0}$ is fulfilled automatically for each complex structure. \\
With the specific choice of $\alpha$'s as $\scriptstyle{(+++),(++-),(+-+)}$, one gets
\begin{equation}
\begin{array}{l}
{{\textstyle\frac{1}{2}}}{\cal J}_{1}^{IJ}\Gamma_{IJ}=\Gamma^{12}+\Gamma^{34}+\Gamma^{56}+\Gamma^{78}\,,\\
{}\\
{{\textstyle\frac{1}{2}}}{\cal J}_{2}^{IJ}\Gamma_{IJ}=\Gamma^{14}+\Gamma^{23}+\Gamma^{58}+\Gamma^{67}\,,\\
{}\\
{{\textstyle\frac{1}{2}}}{\cal J}_{3}^{IJ}\Gamma_{IJ}=\Gamma^{13}+\Gamma^{42}+\Gamma^{57}+\Gamma^{86}\,.
\end{array}
\label{J3}
\end{equation}
~\\
Summing three $N{=2}$ projection matrices generates one quarternion structure. Hence in general,
summing ${k>3}$ of $N{=2}$ projection matrices will present \mbox{$\left(\!\scriptsize{\begin{array}{c}k\\3\end{array}}\!\right)$} number of quarternion structures to the corresponding $\mbox{SO}(1,2){\times\mbox{SO}(8{-k})}{\times\mbox{SO}(k)}$ invariant BPS equations. The \mbox{$\left(\!\scriptsize{\begin{array}{c}k\\3\end{array}}\!\right)$} quarternion structures are singlets under $\mbox{SO}(8{-k})$ and form a $k$-dimensional three-form representation of $\mbox{SO}(k)$. Nevertheless, the corresponding $\textstyle{\frac{1}{6}}k(k-1)(k-2)$ number of quarternion structures are degenerate in the sense that distinct $[\frac{k+2}{3}]$ of them are sufficient to give the full $N{=2k}$ BPS equations.
\subsubsection{$N=8$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations}
The $N{=8}$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations are, with $F_{\mu I}{=0}$,
\begin{equation}
F_{IJK}+{{\textstyle\frac{1}{2}}} F_{I}{}^{LM}{\cal T}_{JKLM}+{{\textstyle\frac{1}{2}}} F_{J}{}^{LM}{\cal T}_{KILM}+{{\textstyle\frac{1}{2}}} F_{K}{}^{LM}{\cal T}_{IJLM}=0\,,
\label{SO12N8}
\end{equation}
where ${\cal T}_{IJKL}$ is a $\mbox{SO}(4)\times\mbox{SO}(4)$ invariant self-dual four-form. With the specific choice of $\alpha$'s as $\scriptstyle{(+++),(++-),(+-+),(+--)}$, one gets
\begin{equation}
\textstyle{\frac{1}{4!}}{\cal T}_{IJKL}\Gamma^{IJKL}=\Gamma^{1234}+\Gamma^{5678}\,.
\label{Tsdff}
\end{equation}
~\\
Summing four $N{=2}$ projection matrices generates one self-dual four-form structure. Hence in general,
summing $k>4$ of $N{=2}$ projection matrices will present \mbox{$\left(\!\scriptsize{\begin{array}{c}k\\4\end{array}}\!\right)$} number of self-dual four-form structures to the corresponding $\mbox{SO}(1,2){\times\mbox{SO}(8{-k})}{\times\mbox{SO}(k)}$ invariant BPS equations. The \mbox{$\left(\!\scriptsize{\begin{array}{c}k\\4\end{array}}\!\right)$} self-dual four-form structures are singlets under $\mbox{SO}(8{-k})$ and form a $k$-dimensional four-form representation of $\mbox{SO}(k)$. Nevertheless, the corresponding $\textstyle{\frac{k!}{4!(k-4)!}}$ number of self-dual four-forms are degenerate in the sense that distinct $[\frac{k+3}{4}]$ of them are sufficient to give the full $N{=2k}$ BPS equations.
\subsubsection{$N=10$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(3)}{\times\mbox{SO}(5)}$ invariant BPS equations}
For $N{=10}$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(3)}{\times\mbox{SO}(5)}$ case there seems no novel structure to appear. One economic fashion to write the $N=10$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(3)}{\times\mbox{SO}(5)}$ invariant BPS equations is to employ a $\mbox{SO}(4)\times\mbox{SO}(4)$ invariant self-dual four-form and a complex structure: with $F_{\mu I}{=0}$,\footnote{
Alternatively we can express them in terms of two sets of \textit{either} $\mbox{SO}(4)\times\mbox{SO}(4)$ invariant self-dual four-forms one given by (\ref{Tsdff}) the other by $
{{\textstyle\frac{1}{2}}}({\Gamma_{1234}}{+\Gamma_{5678}}{+\Gamma_{1256}}{+\Gamma_{3478}}{+\Gamma_{1357}}{+\Gamma_{2468}}
{+\Gamma_{1467}}{+\Gamma_{2358}})$ \textit{\,or\,} quarternionic complex structures one by (\ref{J3}) and the other by
$\Gamma_{14}{+\Gamma_{85}}{+\Gamma_{76}}{+\Gamma_{23}}$, $\,\Gamma_{15}{+\Gamma_{48}}{+\Gamma_{73}}{+\Gamma_{62}}$,
$\,\Gamma_{18}{+\Gamma_{54}}{+\Gamma_{72}}{+\Gamma_{36}}$.}
\begin{equation}
\begin{array}{ll}
F_{IJK}+\textstyle{\frac{3}{2}}F_{[I}{}^{LM}{\cal T}_{JK]LM}=0\,,~~~~~&~~~~~
F_{IJK}{\cal J}^{JK}=0\,.
\label{SO12N10}
\end{array}
\end{equation}
The specific choice of $\alpha$'s as $\scriptstyle{(+++),(++-),(+-+),(+--),(-++)}$ gives
\begin{equation}
\begin{array}{ll}
\textstyle{\frac{1}{4!}}{\cal T}_{IJKL}\Gamma^{IJKL}=\Gamma^{1234}+\Gamma^{5678}\,,~~~~&~~~~
{{\textstyle\frac{1}{2}}}{\cal J}_{IJ}\Gamma^{IJ}=\Gamma^{18}-\Gamma^{27}+\Gamma^{36}-\Gamma^{45}\,.
\end{array}
\label{TJ}
\end{equation}
\subsubsection{$N=12$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant BPS equations}
The $N{=12}$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant BPS equations are,
with $F_{\mu I}{=0}$,\footnote{Of course, the above $N=12$ BPS equations can be obtained by imposing a pair of two distinct quarternionic BPS equations~(\ref{SO12N6}). There are ${{\textstyle\frac{1}{2}}}\!\left(\!\!\scriptsize{\begin{array}{c}6\\3\end{array}}\!\!\right)=10$ such pairs and any of them leads to the same $N=12$ BPS equations. For example we may choose one quarternion structure from (\ref{J3}) and the other by
$\Gamma^{12}{+\Gamma^{87}}{+\Gamma^{56}}{+\Gamma^{43}}$,
$~\Gamma^{17}{+\Gamma^{28}}{+\Gamma^{53}}{+\Gamma^{64}}$,
$~\Gamma^{18}{+\Gamma^{72}}{+\Gamma^{54}}{+\Gamma^{36}}$, corresponding to the $\alpha$ choices $\scriptstyle{(+++),(++-),(+-+)}$ and $\scriptstyle{(+--),(-++),(-+-)}$.}
\begin{equation}
F_{IJK}{\cal T}_{p}^{JK}=0\,,~~~~~~~~p=1,2,3,4,5,6\,,
\label{SO12N12}
\end{equation}
where ${\cal T}_{p}^{IJ}$'s are $\mbox{SO}(2)\times\mbox{SO}(6)$ covariant two-forms: fundamental under $\mbox{SO}(6)$ and singlet under $\mbox{SO}(2)$. With the specific choice of $\alpha$'s as $\scriptstyle{(+++),(++-),(+-+),(+--),(-++),(-+-)}$, one gets
\begin{equation}
\begin{array}{ll}
\textstyle{\frac{1}{2}}{\cal T}_{1}^{IJ}\Gamma_{IJ}=\Gamma^{14}+\Gamma^{23}\,,~~~~~&~~~~~
\textstyle{\frac{1}{2}}{\cal T}_{2}^{IJ}\Gamma_{IJ}=\Gamma^{67}+\Gamma^{58}\,,\\
{}&{}\\
\textstyle{\frac{1}{2}}{\cal T}_{3}^{IJ}\Gamma_{IJ}=\Gamma^{16}+\Gamma^{25}\,,~~~~~&~~~~~
\textstyle{\frac{1}{2}}{\cal T}_{4}^{IJ}\Gamma_{IJ}=\Gamma^{74}+\Gamma^{83}\,,\\
{}&{}\\
\textstyle{\frac{1}{2}}{\cal T}_{5}^{IJ}\Gamma_{IJ}=\Gamma^{17}+\Gamma^{28}\,,~~~~~&~~~~~
\textstyle{\frac{1}{2}}{\cal T}_{6}^{IJ}\Gamma_{IJ}=\Gamma^{35}+\Gamma^{46}\,.
\end{array}
\label{T6def}
\end{equation}
~\\
\subsection{$\mbox{SO}(2)$ invariant BPS equations\label{secBPSSO25}}
The generic $N=2$ projection matrix~(\ref{SO2OG}) leads to the following
$N{=2}$ $\,\mbox{SO}(2){\times\mbox{SU}(4)}$ invariant BPS equations which involve three free sign factors $\beta_{1}^{2}=\beta_{2}^{2}=\beta_{3}^{2}=1$:
\begin{equation}
\begin{array}{llll}
F_{x1}+\beta_{1}F_{y2}=0\,,~~~&~~~~F_{x3}+\beta_{2}F_{y4}=0\,,~~~&~~~~
F_{x5}+\beta_{3}F_{y6}=0\,,~~~&~~~~F_{x7}+\beta_{1}\beta_{2}\beta_{3}F_{y8}=0\,,\\
{}&{}&{}&{}\\
F_{x2}-\beta_{1}F_{y1}=0\,,~~~&~~~~F_{x4}-\beta_{2}F_{y3}=0\,,~~~&~~~~
F_{x6}-\beta_{3}F_{y5}=0\,,~~~&~~~~F_{x8}-\beta_{1}\beta_{2}\beta_{3}F_{y7}=0\,,
\end{array}
\label{SO25BPSeqMaster1}
\end{equation}
and
\begin{equation}
\begin{array}{ll}
{F_{t1}+\beta_{2}F_{134}+\beta_{3}F_{156}+\beta_{1}\beta_{2}\beta_{3}F_{178}=0\,,}~~~&~~~
{F_{135}-\beta_{1}\beta_{2}F_{245}-\beta_{2}\beta_{3}F_{146}-\beta_{3}\beta_{1}F_{236}=0\,,}\\
{}&{}\\
{F_{t2}+\beta_{2}F_{234}+\beta_{3}F_{256}+\beta_{1}\beta_{2}\beta_{3}F_{278}=0\,,}~~~&~~~
{F_{136}-\beta_{1}\beta_{2}F_{246}+\beta_{2}\beta_{3}F_{145}+\beta_{3}\beta_{1}F_{235}=0\,,}\\
{}&{}\\
{F_{t3}+\beta_{1}F_{312}+\beta_{3}F_{356}+\beta_{1}\beta_{2}\beta_{3}F_{378}=0\,,}~~&~~~
{F_{137}-\beta_{1}\beta_{2}F_{247}-\beta_{2}\beta_{3}F_{238}-\beta_{3}\beta_{1}F_{148}=0\,,}\\
{}&{}\\
{F_{t4}+\beta_{1}F_{412}+\beta_{3}F_{456}+\beta_{1}\beta_{2}\beta_{3}F_{478}=0\,,}~~&~~~
{F_{138}-\beta_{1}\beta_{2}F_{248}+\beta_{2}\beta_{3}F_{237}+\beta_{3}\beta_{1}F_{147}=0\,,}\\
{}&{}\\
{F_{t5}+\beta_{1}F_{512}+\beta_{2}F_{534}+\beta_{1}\beta_{2}\beta_{3}F_{578}=0\,,}~~&~~~
{F_{157}-\beta_{1}\beta_{2}F_{168}-\beta_{2}\beta_{3}F_{258}-\beta_{3}\beta_{1}F_{267}=0\,,}\\
{}&{}\\
{F_{t6}+\beta_{1}F_{612}+\beta_{2}F_{634}+\beta_{1}\beta_{2}\beta_{3}F_{678}=0\,,}~~&~~~
{F_{158}+\beta_{1}\beta_{2}F_{167}+\beta_{2}\beta_{3}F_{257}-\beta_{3}\beta_{1}F_{268}=0\,,}\\
{}&{}\\
{F_{t7}+\beta_{1}F_{712}+\beta_{2}F_{734}+\beta_{3}F_{756}=0\,,}~~&~~~
{F_{357}-\beta_{1}\beta_{2}F_{368}-\beta_{2}\beta_{3}F_{467}-\beta_{3}\beta_{1}F_{458}=0\,,}\\
{}&{}\\
{F_{t8}+\beta_{1}F_{812}+\beta_{2}F_{834}+\beta_{3}F_{856}=0\,,}~~&~~~
{F_{358}+\beta_{1}\beta_{2}F_{367}-\beta_{2}\beta_{3}F_{468}+\beta_{3}\beta_{1}F_{457}=0\,.}
\end{array}
\label{SO25BPSeqMaster2}
\end{equation}
The above set of BPS equations can be regarded as the master equations since any $N{=2k}$ $\mbox{SO}(2)^{\mathbf{5}}$ invariant BPS equations corresponding to the projection matrices (\ref{SO251}\,-\,\ref{SO255}) can be obtained by imposing $k$ copies of distinct $(\beta_{1},\beta_{2},\beta_{3})$ choices. We find,
among them, the $N=8$ $\,\mbox{SO}(2){\times\mbox{SU}(4)}$ invariant projection matrix~(\ref{SO255}) leads to the trivial BPS configuration $F_{\mu I}=F_{IJK}=0$. Other nontrivial cases are as follows.
\subsubsection{$N=2$ $\,\mbox{SO}(2){\times\mbox{SU}(4)}$ invariant BPS equations}
The $N=2$ $\,\mbox{SO}(2){\times\mbox{SU}(4)}$ invariant BPS equations corresponding to the projection matrix (\ref{SO251}) or the choice ${(\beta_{1},\beta_{2},\beta_{3})=(+++)}$ in (\ref{SO25BPSeqMaster1}) and (\ref{SO25BPSeqMaster2}) assume a compact form, up to Hermitian conjugation:
\begin{equation}
\begin{array}{lll}
F_{z\bar{a}}=0\,,~~~~~&~~~~~F_{ta}-iF_{ab}{}^{b}=0\,,~~~~~&~~~~~F_{abc}=0\,,
\end{array}
\end{equation}
provided we complexify the $\mbox{SO}(8)$ coordinates by the complex structure ${\Gamma_{12}}{+\Gamma_{34}}{+\Gamma_{56}}{+\Gamma_{78}}$, to introduce the holomorphic and anti-holomorphic variables $a,\bar{a}=1,2,3,4$ such that the metric is $\delta_{a\bar{a}}$ and
\begin{equation}
\begin{array}{ll}
\textstyle{D_{z}=\frac{1}{\sqrt{2}}}(D_{x}-iD_{y})\,,~~~~&~~~~
\textstyle{D_{\bar{z}}=\frac{1}{\sqrt{2}}}(D_{x}+iD_{y})\,,\\
{}&{}\\
F_{za}=\textstyle{\frac{1}{\sqrt{2}}}(D_{z}X_{{2a}{-1}}-i D_{z}X_{{2a}})\,,~~~~&~~~~
F_{z\bar{a}}=\textstyle{\frac{1}{\sqrt{2}}}(D_{z}X_{{2\bar{a}}{-1}}+iD_{z}X_{{2\bar{a}}})\,.
\end{array}
\end{equation}
\subsubsection{$N=4$ $\,\mbox{SO}(2){\times\mbox{SU}(2)}{\times\mbox{SO}(4)}$ invariant BPS equations}
The $N=4$ $\,\mbox{SO}(2){\times\mbox{SU}(2)}{\times\mbox{SO}(4)}$ invariant BPS equations corresponding to the projection matrix (\ref{SO252}) are, up to Hermitian conjugation,
\begin{equation}
\begin{array}{lllll}
F_{z\bar{a}}=0\,,~~~&~~~F_{zp}=0\,,~~~&~~~F_{pab}=0\,,~~~&~~~F_{tI}-iF_{Ia}{}^{a}=0\,,~~~&~~~
F_{Ipq}+{{\textstyle\frac{1}{2}}}\epsilon_{pqrs\,}F_{I}{}^{rs}=0\,,
\end{array}
\end{equation}
where $I=1,2,\cdots,8$, $~p,q,r,s=5,6,7,8$, $\epsilon_{pqrs}$ is a totally anti-symmetric tensor with $\epsilon_{5678}{=1}$ and $a,b,\bar{a}=1,2$ such that the $\mbox{SO}(4)\subset\mbox{SO}(8)$ coordinates are complexified by the complex structure ${\Gamma_{12}}{+\Gamma_{34}}$.
\subsubsection{$N=6$ $\,\mbox{SO}(2){\times\mbox{SO}(2)}{\times\mbox{SU}(3)}$ invariant BPS equations}
The $N=6$ $\,\mbox{SO}(2){\times\mbox{SO}(2)}{\times\mbox{SU}(3)}$ invariant BPS equations corresponding to the projection matrix (\ref{SO253}) are, up to Hermitian conjugation,
\begin{equation}
\begin{array}{llll}
F_{z\bar{\omega}}=0\,,~~~~&~~~~F_{za}=0\,,~~~~&~~~~F_{z\bar{a}}=0\,,~~~~&~~~~
F_{t\omega}-i\textstyle{\frac{1}{3}}F_{\omega a}{}^{a}=0\,,\\
{}&{}&{}&{}\\
F_{ta}-iF_{a\omega\bar{\omega}}=0\,,~~~~&~~~~
F_{\omega ab}=0\,,~~~~&~~~~F_{ab\bar{c}}=0\,,~~~~&~~~~F_{\omega a\bar{b}}-\textstyle{\frac{1}{3}}(F_{\omega c}{}^{c})\delta_{a\bar{b}}=0\,,
\end{array}
\end{equation}
where $a,\bar{a}=1,2,3$ such that we complexify the $\mbox{SO}(6)\subset\mbox{SO}(8)$ coordinates by the complex structure ${\Gamma_{34}}{+\Gamma_{56}}{+\Gamma_{87}}$ and also set separately for $\mbox{SO}(2)\subset\mbox{SO}(8)$,
\begin{equation}
\begin{array}{ll}
F_{z\omega}\equiv\textstyle{\frac{1}{\sqrt{2}}}(F_{z1}-iF_{z2})\,,~~~~&~~~~
F_{z\bar{\omega}}\equiv\textstyle{\frac{1}{\sqrt{2}}}(F_{z1}+iF_{z2})\,.
\end{array}
\label{omegacom}
\end{equation}
\subsubsection{$N=8$ $\,\mbox{SO}(2){\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant BPS equations}
The $N=8$ $\,\mbox{SO}(2){\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant BPS equations
corresponding to the projection matrix (\ref{SO254}) are, up to Hermitian conjugation,
\begin{equation}
\begin{array}{llll}
F_{z\bar{\omega}}=0\,,~~~~~&~~~~~F_{zp}=0\,,~~~~&~~~~
F_{tI}-iF_{I\omega\bar{\omega}}=0\,,~~~~&~~~~F_{Ipq}=0\,,
\end{array}
\label{SO2N8}
\end{equation}
where $I=1,2,\cdots,8$, $~p=3,4,5,6,7,8$ and we complexify the $\mbox{SO}(2)\subset\mbox{SO}(8)$ coordinates by the complex structure ${\Gamma_{12}}$ to employ (\ref{omegacom}).
\subsection{$\mbox{SO}(1,1)$ invariant BPS equations\label{secBPSSO11}}
The generic $N=1$ projection matrix~(\ref{SO11OG}) leads to the following
$N{=1}$ $\,\mbox{SO}(1,1){\times\mbox{SO}(7)}$ invariant BPS equations
which involve four free signs $\,\alpha_{0}^{2}=\alpha_{1}^{2}=\alpha_{2}^{2}=\alpha_{3}^{2}=1$:
\begin{equation}
\begin{array}{l}
F_{tI}-\alpha_{0}F_{xI}=0\,,~~~~~~~~~~~I=1,2,\cdots,8\,,\\
{}\\
\alpha_{0}F_{y1}-{\alpha_{1}\alpha_{2}}F_{278}-{\alpha_{2}\alpha_{3}}F_{548}-{\alpha_{3}\alpha_{1}}F_{638}
-\alpha_{1}F_{234}-\alpha_{2}F_{256}-\alpha_{3}F_{357}-{\alpha_{1}\alpha_{2}\alpha_{3}}F_{476}=0\,,\\
{}\\
\alpha_{0}F_{y2}-{\alpha_{1}\alpha_{2}}F_{718}-{\alpha_{2}\alpha_{3}}F_{376}-{\alpha_{3}\alpha_{1}}F_{475}
-\alpha_{1}F_{143}-\alpha_{2}F_{165}-\alpha_{3}F_{468}-{\alpha_{1}\alpha_{2}\alpha_{3}}F_{538}=0\,,\\
{}\\
\alpha_{0}F_{y3}-{\alpha_{1}\alpha_{2}}F_{456}-{\alpha_{2}\alpha_{3}}F_{267}-{\alpha_{3}\alpha_{1}}F_{168}
-\alpha_{1}F_{124}-\alpha_{2}F_{478}-\alpha_{3}F_{517}-{\alpha_{1}\alpha_{2}\alpha_{3}}F_{258}=0\,,\\
{}\\
\alpha_{0}F_{y4}-{\alpha_{1}\alpha_{2}}F_{536}-{\alpha_{2}\alpha_{3}}F_{158}-{\alpha_{3}\alpha_{1}}F_{257}
-\alpha_{1}F_{132}-\alpha_{2}F_{738}-\alpha_{3}F_{628}-{\alpha_{1}\alpha_{2}\alpha_{3}}F_{167}=0\,,\\
{}\\
\alpha_{0}F_{y5}-{\alpha_{1}\alpha_{2}}F_{346}-{\alpha_{2}\alpha_{3}}F_{418}-{\alpha_{3}\alpha_{1}}F_{427}
-\alpha_{1}F_{678}-\alpha_{2}F_{126}-\alpha_{3}F_{137}-{\alpha_{1}\alpha_{2}\alpha_{3}}F_{328}=0\,,\\
{}\\
\alpha_{0}F_{y6}-{\alpha_{1}\alpha_{2}}F_{354}-{\alpha_{2}\alpha_{3}}F_{273}-{\alpha_{3}\alpha_{1}}F_{318}
-\alpha_{1}F_{758}-\alpha_{2}F_{152}-\alpha_{3}F_{248}-{\alpha_{1}\alpha_{2}\alpha_{3}}F_{174}=0\,,\\
{}\\
\alpha_{0}F_{y7}-{\alpha_{1}\alpha_{2}}F_{128}-{\alpha_{2}\alpha_{3}}F_{236}-{\alpha_{3}\alpha_{1}}F_{245}
-\alpha_{1}F_{568}-\alpha_{2}F_{348}-\alpha_{3}F_{153}-{\alpha_{1}\alpha_{2}\alpha_{3}}F_{146}=0\,,\\
{}\\
\alpha_{0}F_{y8}+{\alpha_{1}\alpha_{2}}F_{127}+{\alpha_{2}\alpha_{3}}F_{154}+{\alpha_{3}\alpha_{1}}F_{163}
+\alpha_{1}F_{567}+\alpha_{2}F_{347}+\alpha_{3}F_{246}+{\alpha_{1}\alpha_{2}\alpha_{3}}F_{253}=0\,.
\end{array}
\label{SO11BPSeqMaster}
\end{equation}
The above set of BPS equations can be regarded as the master equations for generic
$\mbox{SO}(1,1)$ invariant BPS equations. One can classify the BPS equations according to the decomposition of the number of preserved supersymmetries as $(N_{+},N_{-})$ (\ref{Npmdef}). Among others, below we spell explicitly
$(N_{+},\,0\,)$ as well as $(N,N)$ BPS equations with $N_{+}=1,2,\cdots,7$, $~N=1,2,3,4$.
\subsubsection{$(N_{+},N_{-})=(1,0)$ $\,\mbox{SO}(1,1){\times\mbox{SO}(7)}$ invariant BPS equations - \textit{octonion}}
With the choice of ${(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})=(++++)}$,
the $(N_{+},N_{-}){=(1,0)}$ $\,\mbox{SO}(1,1){\times\mbox{SO}(7)}$ invariant BPS equations~(\ref{SO11BPSeqMaster}) assume a compact form:
\begin{equation}
\begin{array}{ll}
F_{tI}-F_{xI}=0\,,~~~~~&~~~~F_{y I}-\textstyle{\frac{1}{6}}{\cal C}_{IJKL}F^{JKL}=0\,,
\end{array}
\label{SO11N10}
\end{equation}
which generalizes the $N=2$ $\,\mbox{SO}(1,2){\times\mbox{SO}(7)}$ invariant BPS equations~(\ref{SO12N2}).
\subsubsection{$(N_{+},N_{-})=(2,0)$ $\,\mbox{SO}(1,1){\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant BPS equations - \textit{complex}}
The $(N_{+},N_{-})=(2,0)$ $\,\mbox{SO}(1,1){\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant BPS equations are, with $F_{tI}{-F_{xI}}{=0}$,
\begin{equation}
\begin{array}{ll}
{\cal J}^{IJ}F_{yJ}+{{\textstyle\frac{1}{2}}} F^{I}{}_{JK}{\cal J}^{JK}=0\,,~~~&~~~F_{IJK}=(1{\otimes{\cal J}}{\otimes{\cal J}}+{\cal J}{\otimes 1}{\otimes{\cal J}}+
{\cal J}{\otimes{\cal J}}{\otimes 1})_{IJK}{}^{LMN}F_{LMN}\,,
\end{array}
\label{SO11N20}
\end{equation}
which generalizes the $N=4$ $\,\mbox{SO}(1,2){\times\mbox{SO}(6)}{\times\mbox{SO}(2)}$ invariant BPS equations~(\ref{SO12N4}).
\subsubsection{$(N_{+},N_{-})=(3,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(3)}{\times\mbox{SO}(5)}$ invariant BPS equations - \textit{quarternion}}
The $(N_{+},N_{-})=(3,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(3)}{\times\mbox{SO}(5)}$ invariant BPS equations are, with $F_{tI}{-F_{xI}}{=0}$,
\begin{equation}
{\cal J}_{p}^{IJ}F_{yJ}+{{\textstyle\frac{1}{2}}} F^{I}{}_{JK}{\cal J}_{p}^{JK}=0\,,~~~~~~~~p=1,2,3\,,
\label{SO11N30}
\end{equation}
where ${\cal J}_{1},{\cal J}_{2},{\cal J}_{3}$ are three distinct complex structures satisfying the quaternion relations,
${\cal J}_{1}^{2}={\cal J}_{2}^{2}={\cal J}_{3}^{2}={\cal J}_{1}{\cal J}_{2}{\cal J}_{3}=-1$ (\ref{J3}).
It is worth to note that the remaining relation of (\ref{SO11N20})
$F_{(3,0)}=0$ is fulfilled automatically for each complex structure. Eq.(\ref{SO11N30}) generalizes the $N=6$ $\,\mbox{SO}(1,2){\times\mbox{SO}(5)}{\times\mbox{SO}(3)}$ invariant BPS equations~(\ref{SO12N6}).
\subsubsection{$(N_{+},N_{-})=(4,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations}
The $(N_{+},N_{-})=(4,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations are, with $F_{tI}{-F_{xI}}{=0}$,
\begin{equation}
{\cal T}_{IJKL}F_{y}{}^{L}+F_{IJK}+{{\textstyle\frac{1}{2}}} F_{I}{}^{LM}{\cal T}_{JKLM}+{{\textstyle\frac{1}{2}}} F_{J}{}^{LM}{\cal T}_{KILM}+{{\textstyle\frac{1}{2}}} F_{K}{}^{LM}{\cal T}_{IJLM}=0\,,
\label{SO11N40}
\end{equation}
where ${\cal T}_{IJKL}$ is a ${\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant self-dual four-form (\ref{Tsdff}). Eq.(\ref{SO11N40}) generalizes the $N=8$ $\,\mbox{SO}(1,2){\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations~(\ref{SO12N8}). Some mass deformations of the above BPS equations are studied in Ref.\cite{Hosomichi:2008qk}.
\subsubsection{$(N_{+},N_{-})=(5,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(5)}{\times\mbox{SO}(3)}$ invariant BPS equations}
The $(N_{+},N_{-})=(5,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(5)}{\times\mbox{SO}(3)}$ invariant BPS equations are,
with $F_{tI}{-F_{xI}}{=0}$,
\begin{equation}
\begin{array}{ll}
{\cal T}_{IJKL}F_{y}{}^{L}+F_{IJK}+\textstyle{\frac{3}{2}}F_{[I}{}^{LM}{\cal T}_{JK]LM}=0\,,~~~~~&~~~~~
{\cal J}^{IJ}F_{yJ}+{{\textstyle\frac{1}{2}}} F_{IJK}{\cal J}^{JK}=0\,,
\label{SO11N50}
\end{array}
\end{equation}
where ${\cal T}_{IJKL}$ and ${\cal J}^{IJ}$ are given in (\ref{TJ}). Eq.(\ref{SO11N50}) generalizes the $N=10$ $\,\mbox{SO}(1,2){\times\mbox{SO}(3)}{\times\mbox{SO}(5)}$ invariant BPS equations~(\ref{SO12N10}).
\subsubsection{$(N_{+},N_{-})=(6,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(6)}{\times\mbox{SO}(2)}$ invariant BPS equations}
The $(N_{+},N_{-})=(6,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(6)}{\times\mbox{SO}(2)}$ invariant BPS equations are, $F_{tI}{-F_{xI}}{=0}$,
\begin{equation}
{\cal T}_{p}^{IJ}F_{yJ}+{{\textstyle\frac{1}{2}}} F^{I}{}_{JK}{\cal T}_{p}^{JK}=0\,,~~~~~~~~p=1,2,3,4,5,6\,,
\label{SO11N60}
\end{equation}
where six of two-forms ${\cal T}_{p}$, $p=1,2,\cdots,6$ are given in (\ref{T6def}).
Eq.(\ref{SO11N60}) generalizes the $N=12$ $\,\mbox{SO}(1,2){\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant BPS equations~(\ref{SO12N12}).
\subsubsection{$(N_{+},N_{-})=(7,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(7)}$ invariant BPS equations}
The $(N_{+},N_{-})=(7,0)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(7)}$ invariant BPS equations are, with $F_{tI}{-F_{xI}}{=0}$,
\begin{equation}
{\cal T}_{p}^{IJ}F_{yJ}+{{\textstyle\frac{1}{2}}} F^{I}{}_{JK}{\cal T}_{p}^{JK}=0\,,~~~~~~~~p=1,2,3,4,5,6,7\,.
\label{SO11N70}
\end{equation}
Here we have seven of two-forms, six given by (\ref{T6def}) and last one by
\begin{equation}
{{\textstyle\frac{1}{2}}}{\cal T}_{7}^{IJ}\Gamma_{IJ}=\Gamma^{13}+\Gamma^{57}\,.
\end{equation}
They form a fundamental representation of $\mbox{SO}(7)$.\\
\subsubsection{$(N_{+},N_{-})=(1,1)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(6)}$ invariant BPS equations}
The $(N_{+},N_{-})=(1,1)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(6)}$ invariant BPS equations are, with $F_{tI}=F_{xI}=0$, best expressed in complex coordinates,
\begin{equation}
\begin{array}{ll}
F_{ab}{}^{b}=0\,,~~~~~&~~~~~F_{y\bar{a}}-\textstyle{\frac{1}{3}}\epsilon_{\bar{a}}{}^{bcd}F_{bcd}=0\,.
\label{SO11N11}
\end{array}
\end{equation}
\subsubsection{$(N_{+},N_{-})=(2,2)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(2)}{\times\mbox{SO}(2)}{\times\mbox{SO}(4)}$ invariant BPS equations}
The $(N_{+},N_{-})=(2,2)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(2)}{\times\mbox{SO}(2)}{\times\mbox{SO}(4)}$ invariant BPS equations are, with $F_{tI}=F_{xI}=0$,
\begin{equation}
(3{\cal J}^{[IJ}{\cal J}^{K]L}-{\cal T}^{IJKL})F_{yL}+F^{IJK}+\textstyle{\frac{3}{2}}F^{[I}{}_{LM}{\cal T}^{JK]LM}=0\,,
\label{SO11N22}
\end{equation}
where ${\cal J}^{IJ}$ is the complex structure of $\Gamma^{12}+\Gamma^{34}+\Gamma^{56}+\Gamma^{78}$ (\ref{cJrep}) and
${\cal T}^{IJKL}$ is the self-dual ${\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant four-form tensor of $\Gamma^{1234}+\Gamma^{5678}$ (\ref{Tsdff}).\\
\subsubsection{$(N_{+},N_{-})=(3,3)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(3)}{\times\mbox{SO}(3)}{\times\mbox{SO}(2)}$ invariant BPS equations}
We present the $(N_{+},N_{-})=(3,3)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(3)}{\times\mbox{SO}(3)}{\times\mbox{SO}(2)}$ invariant BPS equations with a pair of quarternion structures, one from (\ref{J3}) and the other from
$\Gamma^{12}{+\Gamma^{87}}{+\Gamma^{56}}{+\Gamma^{43}}$,
$~\Gamma^{17}{+\Gamma^{28}}{+\Gamma^{53}}{+\Gamma^{64}}$,
$~\Gamma^{18}{+\Gamma^{72}}{+\Gamma^{54}}{+\Gamma^{36}}$. With $F_{tI}=F_{xI}=0$ they are
\begin{equation}
\begin{array}{ll}
{\cal J}_{p}^{IJ}F_{yJ}+{{\textstyle\frac{1}{2}}} F^{I}{}_{JK}{\cal J}_{p}^{JK}=0\,,~~~~~&~~~~~\hat{{\cal J}}_{p}^{IJ}F_{yJ}-{{\textstyle\frac{1}{2}}} F^{I}{}_{JK}\hat{{\cal J}}_{p}^{JK}=0\,,~~~~~~~~~~p=1,2,3\,.
\label{SO11N33}
\end{array}
\end{equation}
{}\\
\subsubsection{$(N_{+},N_{-})=(4,4)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations}
The $(N_{+},N_{-})=(4,4)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations are, with $F_{tI}=F_{xI}=0$ , in terms of the self-dual ${\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant four-form tensor,
\begin{equation}
{\cal T}^{IJKL}F_{yL}+F^{IJK}=0\,.
\label{SO11N44}
\end{equation}
Especially among all the half BPS cases \textit{i.e.} $N_{+}+N_{-}=8$, only the case $(N_{+},N_{-})=(4,4)$ leads to the nontrivial BPS equations. \\
\section{Discussion\label{discussion}}
In this paper we studied and identified a number of BPS equations for the multiple M2-brane
theory proposed recently by Bagger and Lambert. We employed a method which had been
successfully applied to several analogous problems. One first constructs the basic projection matrices for the supersymmetry parameters, and then obtain the corresponding BPS equations. Our classifications are complete for $\mbox{SO}(1,2)$ as well as $\mbox{SO}(2)^{\mathbf{5}}$ invariant BPS equations, while may be not for $\mbox{SO}(1,1)$ invariant cases.
The BPS equations with different types and numbers of preserved supersymmetries are derived
in terms of the associated tensors which are invariant under the symmetry group
of the relevant BPS equations. In particular we derived three types of half BPS equations, which we recall:
\begin{itemize}
\item $N{=8}$ $\,{\mbox{SO}(1,2)}{\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations (\ref{SO12N8})
\begin{equation}
\begin{array}{ll}
F_{\mu I}=0\,,~~~~&~~~~
F_{IJK}+{{\textstyle\frac{1}{2}}} F_{I}{}^{LM}{\cal T}_{JKLM}+{{\textstyle\frac{1}{2}}} F_{J}{}^{LM}{\cal T}_{KILM}+{{\textstyle\frac{1}{2}}} F_{K}{}^{LM}{\cal T}_{IJLM}=0\,.
\end{array}
\end{equation}
\item $N=8$ $\,\mbox{SO}(2){\times\mbox{SO}(2)}{\times\mbox{SO}(6)}$ invariant BPS equations (\ref{SO2N8})
\begin{equation}
\begin{array}{llll}
F_{z\bar{\omega}}=0\,,~~~~~&~~~~~F_{zp}=0\,,~~~~&~~~~
F_{tI}-iF_{I\omega\bar{\omega}}=0\,,~~~~&~~~~F_{Ipq}=0\,,
\end{array}
\end{equation}
where $I=1,2,\cdots,8$, $~p=3,4,5,6,7,8$, and $\omega,\bar{\omega}$ are complex coordinates for $\mbox{SO}(2)\subset\mbox{SO}(8)$.
\item $(N_{+},N_{-})=(4,4)$ $\,{\mbox{SO}(1,1)}{\times\mbox{SO}(4)}{\times\mbox{SO}(4)}$ invariant BPS equations (\ref{SO11N44})
\begin{equation}
\begin{array}{ll}
F_{tI}=F_{xI}=0\,,~~~~~&~~~~~{\cal T}^{IJKL}F_{yL}+F^{IJK}=0\,.
\end{array}
\end{equation}
\end{itemize}
The BPS equations for different number of
supersymmetries exhibit the division algebra structures: octonion,
quarternion or complex. Let us take the Lorentz invariant type as examples.
For the least supersymmetric configurations preserving 1/8 supersymmetries, the relevant symmetry is $\mbox{SO}(1,2){\times\mbox{SO}(7)}$ and the
BPS equations can be elegantly written in terms of the invariant four-form which has close relation to octonions.
For 1/4-BPS
equations the symmetry is $\mbox{SO}(1,2){\times\mbox{SO}(6)}{\times\mbox{SO}(2)}$ and a complex structure appears. We next have 3/8 $\mbox{SO}(1,2){\times\mbox{SO}(5)}{\times\mbox{SO}(3)}$ invariant BPS equations, which are
naturally best expressed in terms of quarternions or hyper-K\"ahler structure. In addition, for 1/2-BPS equations we have the $\mbox{SO}(4)\times\mbox{SO}(4)$ invariant self-dual four-form structure. We have also
identified the exotic classes with more than 1/2 supersymmetry. Apparently the governing
symmetries include more than one hyper-K\"ahler structures, but we have not been
able to express the BPS equations in a succinct way. The true mathematical identity
of such systems certainly deserves more careful study.
The explicit solutions of the BPS equations will give the spectrum of supersymmetric
solitons in Bagger-Lambert theory. It is natural to ask the ${\cal M}$-theory interpretation
of such objects. The real scalar fields $X^{I}$ describe the locations of M2-branes in
the transverse $\mathbb{R}^{8}$. The spatial dependence of $X^{I}$ thus informs us on the shape
of M2-branes, or how they are embedded in the transverse
$\mathbb{R}^{8}$. Eq.(\ref{SO25BPSeqMaster1}) and the subsequent analysis clearly suggest
that the M2-brane worldvolume should occupy holomorphic curves, which is natural for
supersymmetry. Likewise, time-dependence of the scalar field obviously implies that
there is momentum along the particular direction. The three-algebra terms $F_{IJK}$
describe the truly ${\cal M}$-theoretic phenomena: polarization of multiple M2-branes into
M5-branes. Generically the BPS equations are given as various combinations of such basic
building blocks, and more detailed descriptions with explicit solutions will be reported
in a separate publication.\\
{}\\
{}\\
\section*{Acknowledgments}
We wish to thank Bum-Hoon Lee for discussions and encouragement.
This work is supported by the Center for Quantum Spacetime of Sogang
University with grant number R11 - 2005 - 021.
NK is partly supported by Korea Research Foundation Grant, No. KRF-2007-331-C00072.
The research of JHP is supported in part by the Korea Science and Engineering Foundation grant funded by the Korea government (R01-2007-000-20062-0).
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,963
|
Q: Are there any good explanations of kernel schedulers? I've recently begun wondering about kernel schedulers and whatnot. Is there any resource that provides an overview of commonly used kernel scheduling algorithm? The CFS scheduler has a lot of literature on its implementation, but I can't seem to find much along the lines of the queuing theory behind the algorithm.
A: This set of docs have been the most helpful for me by far although they don't talk about the queuing theory behind the algorithms.
A: Linux Kernel Scheduler Resources:
Inside the Linux scheduler
A short history of Linux schedulers
Completely Fair Scheduler (since 2.6.23)
Multiprocessing with the Completely Fair Scheduler
Real-Time Linux Kernel Scheduler
O(1) Scheduler (prior 2.6.23)
Linux Scheduler simulation
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,536
|
\section{#1}}
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\title{The Dynamics of Chern-Simons Vortices}
\author{Benjamin Collie and David Tong \\
Department of Applied Mathematics and Theoretical Physics, \\
University of Cambridge, UK\\
{\tt b.p.collie, d.tong@damtp.cam.ac.uk}}
\abstract{We study vortex dynamics in three-dimensional theories
with Chern-Simons interactions. The dynamics is governed by motion
on the moduli space ${\cal M}$ in the presence of a magnetic
field. For Abelian vortices, the magnetic field is shown to be the Ricci form
over ${\cal M}$; for non-Abelian vortices, it is the first Chern
character of a suitable index bundle. We derive these results by
integrating out massive fermions and following the fate of their
zero modes.}
\begin{document}
\pagestyle{plain} \setcounter{page}{1}
\newcounter{bean}
\baselineskip16pt
\section{Introduction}
The moduli space approximation provides an elegant description of
the low-energy behaviour of solitons \cite{manton}. Information
about soliton interactions is packaged in a simple geometric form
which has proven useful in extracting both the classical and
quantum dynamics of the system. In this paper we use the moduli
space approximation to study the motion of vortices in the
presence of Chern-Simons interactions \cite{cs}.
\para
For vortices in the Abelian-Higgs model in $d=2+1$ dimensions, a
moduli space ${\cal M}$ of solutions exists only when the
potential is tuned to critical coupling, meaning that the theory
lies on the borderline between Type I and Type II
superconductivity. For $k$ vortices, the moduli space has
dimension ${\rm dim}({\cal M})= 2k$, with the coordinates $X^a$,
$a=1,\ldots, 2k$, on ${\cal M}$ corresponding to the positions of
the vortices on the plane \cite{erick,taubes}. At low-energies,
the scattering of vortices can be described as geodesic motion on
${\cal M}$ with respect to a metric $g_{ab}$,
\begin{eqnarray} L_{\rm vortex} = \ft12
g_{ab}(X)\dot{X}^a\dot{X}^b\label{samols}\end{eqnarray}
Although the metric $g_{ab}$ is not known explicitly for $k\geq
2$, its properties have been well studied
\cite{samols,mansp,manchen}. Most notably, $g_{ab}$ is K\"ahler.
\para
One can ask how the dynamics of the vortices is affected by the
addition of a Chern-Simons interaction
\cite{csdyn,kimyeongold,kimyeong}. On general grounds, one expects
the low-energy dynamics of vortices to be governed by geodesic
motion on ${\cal M}$, now in the presence of a magnetic field
${\cal F}\in \Omega^2({\cal M})$. Locally we may write ${\cal
F}=d{\cal A}$ and the Lagrangian takes the form,
\begin{eqnarray} L_{\rm vortex} = \ft12 \tilde{g}_{ab}(X)\dot{X}^a\dot{X}^b -
\kappa {\cal A}_a(X)\dot{X}^a\label{right}\end{eqnarray}
where $\kappa$ is the coefficient of the Chern-Simons term in
three-dimensions. Working perturbatively in $\kappa$, Kim and Lee
found that to leading order $\tilde{g}_{ab}=g_{ab}$, while an
expression for ${\cal A}$ was given in terms of the profile
functions of the vortices \cite{kimyeong}. However, the geometric
meaning of ${\cal A}$ has remained mysterious. Here we remedy
this. We show that ${\cal F}$ is the Ricci form on ${\cal M}$.
\para
We further study the dynamics of non-Abelian $U(N)$ vortices
introduced in \cite{vib,auzzi} in the presence of Chern-Simons
interactions. In this case the moduli space has dimension
$\dim({\cal M})=2kN$ and the dynamics is again given by
\eqn{right}. We show that ${\cal F}$ is the first Chern character
of a particular index bundle over ${\cal M}$.
\para
The technique we use to derive these results is simple yet
indirect, and can be viewed as an application of the
Goldstone-Wilczek method \cite{gw,cw}. We make use of the
well-known fact that the Chern-Simons terms can be induced by
integrating out heavy fermions in three dimensions
\cite{redlich,agw}. We follow the fate of these fermions from the
perspective of the vortices. The fermi zero modes live in an index
bundle over ${\cal M}$ and we show that, as their mass becomes
large, they may be integrated out to reproduce the result
\eqn{right}. It is then simple to show that there is no further
contribution from non-zero modes. We recently employed this method
to derive the dynamics of instantons in five-dimensional Yang-Mills
Chern-Simons theories \cite{5dcs}.
\para
The plan of the paper is as follows: in Section 2 we introduce the
model of interest and describe its vortex solutions. It is a
$U(N)$ Yang-Mills theory, with Chern-Simons interactions, coupled
to matter fields. The Lagrangian admits ${\cal N}=2$ supersymmetry
and the vortices are BPS. In Section 3 we present our main
results, analyzing the impact on the vortex dynamics as fermions
are introduced, made heavy, and finally integrated out. Section 4
is devoted to two examples. In the first example, we study the
qualitative dynamics of two Abelian vortices and describe the
bound orbits. We also show that our technique correctly reproduces
the fractional statistics of Abelian vortices. The second example
concerns a single vortex in the $U(N)$ theory for which the moduli
space is ${\bf CP}^{N-1}$ and the appropriate magnetic field
${\cal F}$ is proportional to $\Omega$, the K\"ahler form. We also
show how to reproduce this magnetic field from a direct study of
the vortex equations in the moduli space approximation.
\section{The Vortex Equations}
The literature contains a veritable smorgasbord of
Chern-Simons models which admit vortex solutions. These include
Abelian theories with \cite{llm} and without \cite{hkp,jw,jlw} a
Maxwell term, non-Abelian theories \cite{klee,schap}, and theories
with non-relativistic kinetic terms for the matter fields
\cite{zhk,pij,jip,nonrel}. The properties of many of these models are
summarized in the excellent review \cite{dunne}.
\para
Our interest in this paper lies in a $U(N)$
Yang-Mills-Chern-Simons theory coupled to a real adjoint scalar
$\phi$ and $N_f$ scalars $q_i$, $i=1,\ldots,N_f$, each of which
transforms in the fundamental representation of the gauge group.
With suitable fermion content, the theory enjoys ${\cal N}=2$
supersymmetry (i.e. 4 four supercharges) which dictates the form
of the bosonic interactions:
\begin{eqnarray} {\cal L} &=& -\frac{1}{2e^2}{\rm Tr}\,F_{\mu\nu}F^{\mu\nu} -
\frac{\kappa}{4\pi}{\rm Tr}\,\epsilon^{\mu\nu\rho}(A_\mu \partial_\nu
A_\rho - \frac{2i}{3}A_\mu A_\nu A_\rho) +
\frac{1}{e^2}{\rm Tr}\,({\cal D}_\mu\phi)^2 \nonumber\\ && + |{\cal D}_\mu
q_i|^2 - q_i{}^{\!\dagger}\phi^2 q_i -
\frac{e^2}{4}{\rm Tr}(q_iq_i{}^{\!\dagger} -
\kappa\phi/2\pi-v^2)^2\label{lag}\end{eqnarray}
Notice that we have not considered separate Chern-Simons
coefficients for the $U(1)$ and $SU(N)$ parts of the gauge group,
but instead taken a specific combination in which they are
packaged together in $U(N)$. For $N\geq 2$, invariance of the
partition function under large gauge transformation requires that
$\kappa \in {\bf Z}$. For the Abelian theory, there is no such
constraint.
\para
To make contact with the other models on the market, it is
instructive to consider various limits of this Lagrangian.
\begin{itemize}
\item For $U(1)$ gauge group, the Lagrangian reduces to the
Maxwell-Chern-Simons-Higgs theory introduced in \cite{llm}.
\item When $\kappa=0$, the Lagrangian reduces to Yang-Mills theory
coupled to a number of fundamental scalar fields. This theory is
known to admit non-Abelian vortices, first introduced in
\cite{vib,auzzi} and since studied in some detail. (See, for
example, \cite{tasi,moduli,sy} for reviews). We will make much use of this limit.
\item When $e^2\rightarrow\infty$, the Yang-Mills term vanishes,
and the scalar field $\phi$ becomes auxiliary. Integrating out
$\phi$ reproduces the Chern-Simons-Higgs theory with sixth order
scalar potential, first introduced in the Abelian case in
\cite{hkp,jw,jlw}, and studied more recently in the non-Abelian
case in \cite{schap}. \end{itemize}
\para
Two important ground states of the theory are the unbroken phase and the Higgs phase.
The gauge symmetry is unbroken when the scalar fields take
the vacuum expectation values
\begin{eqnarray} \mbox{Unbroken Phase}:\ \ \ \ \ \phi^a_{\ b} = -\frac{2\pi
v^2}{\kappa}\delta^a_{\ b} \ \ \ ,\ \ \ q_i=0\label{unbroken}\end{eqnarray}
where $a,b=1,\ldots,N$ is the colour index. This state exists
regardless of the number $N_f$ of fundamental flavours. In
contrast, a ground state with fully broken gauge symmetry only
exists when $N_f \geq N$ and the rank $N_f$ term
$q_iq_i{}^{\!\dagger}$ in the potential can successfully cancel
the rank $N$ term $v^2$ (which comes with an implicit $N\times N$
unit matrix). For simplicity, in what follows we choose $N_f=N$.
There is then a unique ground state with fully broken gauge symmetry
given by
\begin{eqnarray} \mbox{Higgs Phase}:\ \ \ \ \ \ \ \ \ \phi =0 \ \ \ \ ,\ \ \
q_i^{\ a} = v\delta_i^{\ a}\label{HiggsPhase}\end{eqnarray}
In this vacuum, both the $U(N)$
gauge symmetry and the $SU(N)$ flavour symmetry which rotates the
Higgs fields $q_i$ are spontaneously broken. However, the diagonal
of the two survives: $U(N)_{\rm gauge}\times SU(N)_{\rm flavour}
\rightarrow SU(N)_{\rm diag}$.
The theory also has several ground states with partly broken gauge symmetry.
For each such state, the vacuum expectation values of the fields have some
diagonal entries equal to those in \eqn{unbroken} and the rest equal to those
in \eqn{HiggsPhase}. We will not consider these partly broken phases further.
\para
In the Higgs phase, the model admits topologically stable BPS
vortices. First order equations of motion may be derived using the
standard Bogomolnyi trick, and read
\begin{eqnarray} B = \frac{e^2}{2} (q_i q_i^\dagger - \kappa\phi/2\pi - v^2) \
\ ,\
\ \ {\cal D}_zq_i \equiv {\cal D}_1q_i-i{\cal D}_2q_i=0 \label{bog1}\\
E_\alpha + {\cal D}_\alpha\phi =0 \ \ \ ,\ \ \ {\cal D}_0\phi=0\ \
\ ,\ \ \ {\cal D}_0q_i+i\phi q_i=0\label{bog2}\end{eqnarray}
Here $B=F_{12}$ and $E_\alpha = F_{0\alpha}$. Note however that,
in contrast to vortices in $\kappa =0$ theories, it is not enough
to solve these first order equations alone: we must also solve
Gauss' law. This is most simply written in static gauge
$\partial_0=0$. Then the three equations in \eqn{bog2} may all be
solved by setting $A_0=\phi$, which is determined by Gauss' law
\begin{eqnarray} 2{\cal D}^2\phi +\frac{\kappa}{2\pi} e^2 B -
e^2\{\phi,q_iq_i^\dagger\}=0\label{gauss}\end{eqnarray}
Note that the presence of the Chern-Simons coupling ensures that
$\phi$ is sourced at the core of the vortex where $B\neq 0$. The
fact that the first order vortex equations \eqn{bog1} must be
supplemented by the second order equation \eqn{gauss} is what makes the study
of vortex dynamics somewhat more of a technical challenge in the
presence of a Chern-Simons interaction.
\para
Configurations satisfying \eqn{bog1} and \eqn{gauss} have energy,
\begin{eqnarray} E = -v^2 \int d^2x \ {\rm Tr}\,B = 2\pi v^2 k\end{eqnarray}
where $k\in {\bf Z}^+$ is the topological charge of the vortex. It
is expected that equations \eqn{bog1} and \eqn{gauss} enjoy a
moduli space of solutions of dimension $\dim({\cal M})=2kN$. This
is suggested by the counting of zero modes using index theorems
\cite{erick,hkp,vib}. However, to our knowledge the only rigorous
proof of this statement when $\kappa\neq 0$ holds in the
Abelian theory in the limit $e^2\rightarrow \infty$ \cite{wang}.
To some extent, the method we propose in the next section
circumvents this issue since our starting point will be the theory
with $\kappa =0$ where the existence of a moduli space has been
rigorously proven \cite{taubes}.
\para
Our theory has ${\cal N}=2$ supersymmetry. Yet so far we have not
mentioned the fermions. They consist of a single Dirac fermion
$\lambda$ in the adjoint representation of the gauge group (this
is the superpartner of $A_\mu$ and $\phi$) together with $N_f$
Dirac fermions $\psi_i$ in the fundamental representation (the
superpartners of $q_i$). In the background of the vortex, these
fermions carry zero modes. These zero modes will not be the focus of our
discussion in the next section, although one should remember that
they are present. Instead we will be interested in the zero modes
of some extra, supplementary, fermions that we now introduce.
\section{Integrating Out Fermions}
Our strategy in this section is to replace the Chern-Simons
interactions in the bosonic Lagrangian \eqn{lag} with something
that we understand better, namely fermions. To this end, we start
with the theory without Chern-Simons interactions by setting
$\kappa=0$ in \eqn{lag}. We now introduce $\tilde{N}$ chiral
multiplets $\tilde{Q}$, each transforming in the anti-fundamental
representation of the $U(N)$ gauge group. Each of these chiral
multiplets may be given a mass $m$ consistent with
supersymmetry\footnote{This mass term is not possible in $d=3+1$
dimensions, where it would break Lorentz invariance. It is allowed
in $d=2+1$, and was called a ``real mass" in \cite{ahiss} to
distinguish it from the more familiar complex mass that appears in
the superpotential. For the present purposes, the important point
is its effect on the fermions which is shown in the Dirac equation
\eqn{dirac}.}. In the limit $m\rightarrow \infty$, the chiral
multiplets may be happily integrated out. All of their effects
decouple, except for a remnant $U(N)$ Chern-Simons term, with
coefficient \cite{redlich,agw}
\begin{eqnarray} \kappa = - \frac{\tilde{N}}{2}\,{\rm sign}(m)\label{nkappa}\end{eqnarray}
Importantly, the $\kappa\phi$ term in the potential in \eqn{lag}
is the supersymmetric partner of the Chern-Simons term.
Supersymmetry is unbroken in this model (the Witten index is
non-vanishing), so when we integrate out the chiral multiplets $\tilde{Q}$,
the $\kappa\phi$ term must be generated together with the Chern-Simons term.
In fact, there is one further, related,
effect that is important: the scalar vev $v^2$ (which is a
Fayet-Iliopoulos parameter in the language of supersymmetry) picks
up a finite renormalization \cite{ahiss,memirror}:
\begin{eqnarray} v^2 \rightarrow v^2_{\rm eff} = v^2 + \frac{m \kappa}{2\pi} =
v^2 - \frac{\tilde{N}|m|}{4\pi}\label{veff}\end{eqnarray}
Notice that for suitably large $|m|$, we have $v^2 <0$, and the
theory exits the Higgs phase where the vortices live. If we wish
to stay in the Higgs phase, and keep the vortex mass fixed, we
must scale $v^2$ so that $v^2_{\rm eff}$ remains constant as
$m\rightarrow\infty$. If we perform such a scaling, we conclude
that the Chern-Simons theory \eqn{lag} is equivalent to the
Yang-Mills theory coupled to $\tilde{N}=2\kappa$ supplementary
massive chiral multiplets $\tilde{Q}$ in the limit $m\rightarrow
\pm \infty$.
\subsection{The Index Bundle of Fermi Zero Modes}
Let us now follow the effect of this procedure on the vortex
dynamics, focussing firstly on a Dirac fermion $\tilde{\psi}$ in
one of the chiral multiplets $\tilde{Q}$. The Dirac equation is
given by,
\begin{eqnarray} i\,\,{\raise.15ex\hbox{/}\mkern-12mu D} \tilde{\psi} - \tilde{\psi}\phi =
m\tilde{\psi}\label{dirac}\end{eqnarray}
We are interested in the solutions to this equation in the
background of the vortex. Since we are working in the theory with
$\kappa=0$, the bosonic fields of the vortex are solutions to
\begin{eqnarray} B=\frac{e^2}{2}(q_iq_i^\dagger - v^2)\ \ \ ,\ \ \ {\cal D}_z
q_i =0\ \ \ , \ \ \ A_0=\phi=0\label{veq}\end{eqnarray}
Of the full spectrum of solutions to the Dirac equation
\eqn{dirac} in the background of the vortex, only the zero modes
will prove important. We discuss these first, returning to the
non-zero modes shortly. We work with the basis of gamma matrices
$\gamma^\mu = (\sigma^3,i\sigma^2,-i\sigma^1)$. The zero modes
then take the form \cite{jr}
\begin{eqnarray} \tilde{\psi}(t,x_\alpha) = e^{-imt}
\left(\begin{array}{c} \tilde{\psi}_-(x_\alpha) \\
0\end{array}\right)\ \ \ {\rm or}\ \ \ \ \tilde{\psi}(t,x_\alpha)
= e^{+imt}
\left(\begin{array}{c} 0 \\
\tilde{\psi}_+(x_\alpha)\end{array}\right) \end{eqnarray}
where ${\cal D}_{\bar{z}}\tilde{\psi}_- = {\cal
D}_z\tilde{\psi}_+=0$. Standard index theorems state that the
equation ${\cal D}_{\bar{z}}\tilde{\psi}_-=0$ has $k$ solutions in
the background of the vortex, while ${\cal D}_z\tilde{\psi}_+=0$
has none\footnote{Recall that $\tilde{\psi}$ transforms in the
$\bar{\bf N}$ representation, while $q_i$ transforms in the ${\bf
N}$ representation -- this is responsible for the fact that ${\cal
D}_{\bar{z}}$ carries the zero modes in the background ${\cal
D}_zq_i=0$. More details on these fermi zero modes in the context
of vortex strings in related four-dimensional theories can be
found in \cite{het}.}. For example, in the Abelian case this
follows from the fact that there is no holomorphic line bundle of
negative degree.
\para
The space of zero modes of the Dirac equation defines a bundle
over the vortex moduli space ${\cal M}$, with fibre ${\bf C}^k$.
This is commonly referred to as the index bundle \cite{mansch,wy}.
As we move in moduli space by adiabatically changing the
background vortex configuration, the fermi zero modes undergo a
holonomy described by a Hermitian $u(k)$ connection $\omega$ over ${\cal
M}$. We denote the Grassmann-valued coordinates of the fibre as
$\xi^l$, $l=1,\ldots, k$. The low-energy dynamics of the vortex
should now be augmented to include these zero modes, described by
the kinetic terms\footnote{\label{footnote}There is an important
caveat here: the zero modes under discussion are non-normalizable;
they have a long-range $1/r$ tail, causing them to suffer from an
infra-red logarithmic divergence. In the context of
four-dimensional theories, there are several examples where
ignoring this fact, and treating these modes with kinetic terms of
the form \eqn{goodgood}, leads to quantitatively and qualitatively
correct physics \cite{vstring,qhet}. This approach has been
criticized in \cite{syok}. For the time being, we proceed by
ignoring this issue. However, in Section \ref{abstring} we will
present a slightly more involved construction that yields the same
answer, but doesn't suffer from this technical problem.}
\begin{eqnarray} L = \bar{\xi}^l(iD_t-m)\xi^l\label{goodgood}\end{eqnarray}
where the covariant derivative is defined by
\begin{eqnarray} D_t\xi^l = \partial_t\xi^l + i(\omega_a)^l_{\ m}\dot{X}^a\,
\xi^m\end{eqnarray}
Let us pause briefly to discuss how one should quantize these zero
modes. As usual, each complex fermionic zero mode gives rise to two states
--- occupied and unoccupied --- whose energy differs by $m$.
However, the question of the absolute ground state energy requires
us to resolve the usual ordering ambiguities. Comparison with the
renormalization of $v^2$ given in \eqn{veff} shows that a single
fermi zero mode should cause the mass of the vortex to shift by
$M_{\rm vortex} \rightarrow M_{\rm vortex} - |m|/2$. This strongly
suggests that we should take the ground state of the fermi zero
modes to have energy $-|m|/2$, and the excited state to have
energy $+|m|/2$. It would be interesting to understand better why
this choice of ordering is forced upon us.
\subsection{Integrating out the Index Bundle}
Throughout this discussion, we have been referring to the relevant
solutions of the Dirac equation as ``zero modes". This is a slight
misnomer because, as is clear from \eqn{goodgood}, they are
excited at a cost of energy equal to $|m|$. They become true zero
modes only in the $m\rightarrow 0$ limit which is, of course, to
be expected since they arose from fermions with mass $m$. However,
we are interested in the opposite limit $m\rightarrow \infty$. In
this limit, the effect of the fermi zero modes {\it almost}
decouples. They do not correct the metric $g_{ab}$. However, as we
now show, they do give rise to new Chern-Simons terms for the
moduli space dynamics.
\para
Integrating out the fermion $\xi$ in the path integral leads to
the ratio of determinants
\begin{eqnarray} \det\left(\frac{iD_t-m}{i\partial_t-m}\right)\end{eqnarray}
We can compute this ratio using standard methods. We work with
compact Euclidean time $\tau=it$, with periodicity
$\tau\in[0,\beta)$. We look for the eigenvalues $\lambda$ of the
operator
\begin{eqnarray} \left(-\partial_\tau - i\omega - m\right) \chi = \lambda\chi\end{eqnarray}
where $\omega=\omega_a\,\partial_\tau{X}^a$. The eigenfunctions
are subject to periodic boundary conditions $\chi(0)=\chi(\beta)$.
Solutions are given by the usual time-ordered product
\begin{eqnarray} \chi=e^{-(m+\lambda)\tau}\,V(\tau)\chi\ \ \ \ {\rm with}\ \ \
V(\tau) =
T\,\exp\left(-i\int_0^{\tau}d\tau^\prime\omega(\tau^\prime)\right)
\in U(k)\end{eqnarray}
Let us denote the eigenvalues of $V(\beta)$ as $e^{v_l}$,
$l=1,\ldots,k$. Then the periodicity requirement
$\chi(0)=\chi(\beta)$ means that the eigenvalues $\lambda$ are
given by
\begin{eqnarray} \lambda = \frac{2\pi i n+v_l}{\beta} -m\ \ \ \ \ \ \ n\in {\bf
Z},\ \ l=1,\ldots,k\end{eqnarray}
>From this we compute the ratio of determinants
\begin{eqnarray} \det\left(\frac{D_\tau+m}{\partial_\tau+m}\right) &=&\ \
\prod_{l=1}^k\prod_{n\in {\bf Z}}\left(\frac{2\pi i n/\beta +
v_l/\beta - m}{2\pi i n/\beta - m}\right) \nonumber\\ &=&\ \
\prod_{l=1}^k
\left(1-\frac{v_l}{m\beta}\right)\,\left(\frac{\sinh(\beta m/2 -
v_l/2)}{\sinh \beta m/2}\right) \nonumber\\
&\stackrel{\beta\rightarrow \infty}{\longrightarrow}&\ \
\exp\left(-\frac{1}{2}\,{\rm sign}(m)\sum_l v_l\right)\end{eqnarray}
where we assume that $\omega$ has compact support in taking the
limit in the last line. Translating back to Minkowski space, we
can write this as a contribution to the effective Lagrangian
involving the original $u(k)$ connection $\omega$.
\begin{eqnarray} L_{\rm eff} = \frac{1}{2}\,{\rm sign}(m)\,({\rm Tr}\,\omega_a) \dot{X}^a
\label{yes}\end{eqnarray}
This is the promised result. We see that, even in the limit
$m\rightarrow \infty$, the zero modes leave a remnant of their
existence by inducing an effective magnetic field ${\cal F}=d{\cal
A}$ on the moduli space, where ${\cal A} = {\rm Tr}\,{\omega}$ is
defined in terms of the connection on the index bundle. ${\cal F}$
is proportional to the first Chern character of the index bundle
while, conveniently enough, ${\cal A}$ is known as the
Chern-Simons 1-form. This is the worldline counterpart to the
statement that the parent three-dimensional fermions induce a
Chern-Simons term.
\para
The result \eqn{yes} holds for integrating out the zero modes
associated to a single chiral multiplet fermion. As we saw in
\eqn{nkappa}, we must integrate $\tilde{N}=2\kappa$ chiral
multiplets. Our final result is that the low-energy dynamics of
vortices in the Chern-Simons theory \eqn{lag} is given by,
\begin{eqnarray} L = \frac{1}{2}g_{ab}\dot{X}^a\dot{X}^b - \kappa {\cal A}_a
\dot{X}^a \label{final}\end{eqnarray}
\subsubsection*{Why Only Zero Modes Matter}
In deriving the Lagrangian \eqn{final}, we have integrated out
only the zero modes on the vortex worldline, while ignoring the
infinite tower of higher solutions to the Dirac equation. We now
show that this is consistent. The key point is that higher
excitations of fermions come in pairs, with energy $\pm E$:
\begin{eqnarray} \left(\begin{array}{cc} 0 & i{\cal D}_z \\ -i{\cal
D}_{\bar{z}} & 0 \end{array}\right)\left(\begin{array}{c}
\tilde{\psi}_- \\ \tilde{\psi}_+\end{array}\right) = E
\left(\begin{array}{c} \tilde{\psi}_- \\
\tilde{\psi}_+\end{array}\right) \ \ \Rightarrow\ \ \
\left(\begin{array}{cc} 0 & i{\cal D}_z \\ -i{\cal D}_{\bar{z}} &
0 \end{array}\right)\left(\begin{array}{c} \tilde{\psi}_- \\
-\tilde{\psi}_+\end{array}\right) = -E
\left(\begin{array}{c} \tilde{\psi}_- \\
-\tilde{\psi}_+\end{array}\right) \nonumber\end{eqnarray}
Contributions to the Chern-Simons term on the vortex
worldline cancel between each pair. To see this, we write
the general eigenfunction as $\tilde{\psi}^T =
(\tilde{\psi}_-\zeta_-, \tilde{\psi}_+\zeta_+)$ and promote
$\zeta_\pm$ to time-dependent Grassmann fields. The action for
these objects is schematically
\begin{eqnarray} L_{\rm non-zero-modes} = \bar{\zeta}_+(iD_t-m)\zeta_+ +
{\bar{\zeta}}_-(iD_t+m){\zeta}_- + E({\bar{\zeta}}_+{\zeta}_- +
\bar{\zeta}_-\zeta_+)\end{eqnarray}
which is schematic only in the sense that we have dropped overall
coefficients that arise from the overlap of the eigenfunctions.
Integrating out the non-zero modes now gives us a determinant of
the form,
\begin{eqnarray} \det\left(\begin{array}{cc} iD_t -m & E \\ E &
iD_t+m\end{array}\right)= \det(iD_t +\sqrt{m^2+E^2})\det(iD_t -
\sqrt{m^2 + E^2})\ \ \ \ \ \end{eqnarray}
We see that the effective mass of these objects is
$\pm\sqrt{m^2+E^2}$, leading to a cancellation due to the presence
of the ${\rm sign}(m)$ term in \eqn{yes}. In the limit $m\rightarrow
\infty$, these non-zero modes leave no trace of their existence on
the vortex dynamics.
\subsection{Abelian Vortices and the Tangent Bundle}
\label{abstring}
Our final answer \eqn{final} for the vortex dynamics is pleasingly
simple and geometrical. Yet it suffers from two drawbacks.
Firstly, we have no concrete expression for the index bundle and
its associated first Chern character. Secondly, as mentioned in
footnote \ref{footnote}, there is a technical subtlety due to the
non-normalizability of the zero modes. In this section we remedy
both of these issues for Abelian vortices. In Section \ref{doit}
we shall also remedy the problem of non-normalizability for
non-Abelian vortices.
\para
Our strategy is a slightly more refined version of that described
above. We again generate the Chern-Simons terms by integrating out
supplementary matter multiplets. The only thing that differs from
the previous discussion is the matter that we choose to integrate
out. Our starting point this time will be the Abelian Higgs model
with ${\cal N}=4$ supersymmetry (i.e. 8 supercharges). We set
$\kappa =0$ in \eqn{lag}, and introduce a neutral chiral multiplet
$A$, containing the Dirac fermion $\eta$, together with a single
chiral multiplet $\tilde{Q}$ of charge $-1$, containing the
fermion $\tilde{\psi}$. The extended supersymmetry requires that
these are coupled to the original chiral multiplet $Q$, containing the
scalar $q$, through the superpotential,
\begin{eqnarray} {\cal W} = \sqrt{2}\tilde{Q}AQ\label{w}\end{eqnarray}
The benefit of working in the ${\cal N}=4$ model is that the
geometry of the fermi zero modes is well understood. Indeed, in
the background of the Abelian vortex, the Dirac equations for
$\eta$ and $\tilde{\psi}$ reduce to\footnote{A recent
detailed discussion of these issues, with an explicit
demonstration of the relationship between fermionic and bosonic
zero modes, can be found in \cite{het}.},
\begin{eqnarray} i{\cal D}_{\bar z} \eta_- -
\sqrt{2}q^\dagger\tilde{\psi}^\dagger_- = 0\ \ \ ,\ \ \ -i{\cal
D}_z\tilde{\psi}^\dagger_--\sqrt{2}\eta_-q=0
\label{paul}\end{eqnarray}
The index theorem remains the same as before, and
these equations again have $k$ complex zero modes. However,
the presence of the coupling to $q$
--- which has a non-zero vacuum expectation value --- ensures that
the zero mode profiles are localized exponentially near the vortex
cores and are normalizable. This resolves the problem described in
footnote \ref{footnote}.
\para
Moreover, it can be shown that the $k$
fermi zero modes are proportional to the bosonic zero modes of the
vortex: they are related by the extended supersymmetry.
The upshot of this is that the fermi zero modes live
--- like their bosonic counterparts --- in the {\it tangent bundle}
over ${\cal M}$. The appropriate
covariant derivative for the $k$ Grassmann collective coordinates
$\xi$ is now,
\begin{eqnarray} (D_t\xi)^a = \partial_t\xi^a + \Gamma^a_{bc}\dot{Z}^b\xi^c\end{eqnarray}
where, in contrast to previous formulae, we have switched to
complex notation, defining the holomorphic coordinates $Z^a$,
$a=1,\ldots, k$ on a patch of the moduli space ${\cal M}$. The
$\Gamma^a_{bc}$ are the holomorphic components of the Levi-Civita
connection.
\para
The above is merely a review of well known results about fermi
zero modes of vortices in theories with ${\cal N}=4$
supersymmetry. As before, we now deform our theory by adding
a real mass $m$ for the chiral
multiplets $A$ and $\tilde{Q}$. We then integrate $A$ and $\tilde{Q}$
out. The multiplet $A$ is neutral and decouples in the $m\rightarrow
\infty$ limit. In contrast, $\tilde{Q}$ induces a Chern-Simons
interaction with coefficient $\kappa = -\ft12\, {\rm sign}(m)$.
\para
Integrating out the fermi zero modes on the worldline proceeds as
before. But, since the zero modes live in the tangent bundle,
locally we have $d{\cal A} = R$ where $R$ is the Ricci form. This
is defined in terms of the metric $g_{a\bar{b}}$ by
\begin{eqnarray} R = i\partial\bar{\partial}\ln\sqrt{g}\end{eqnarray}
In terms of local complex coordinates on ${\cal M}$, the vortex
dynamics becomes
\begin{eqnarray} L = g_{a\bar{b}}\dot{Z}^a\dot{\bar{Z}}{}^{\bar{b}} -
{\kappa}\left( {\cal A}_a\dot{Z}^a + \bar{\cal
A}_{\bar{a}}\dot{\bar{Z}}{}^{\bar{a}}\right)\label{evenmorefinal}\end{eqnarray}
where the complex Chern-Simons 1-form can be written locally as
\begin{eqnarray} {\cal A}_a = -\frac{i}{2}\,\frac{\partial}{\partial Z^a}\ln
\sqrt{g}\label{A}\end{eqnarray}
\subsection{Non-Abelian Vortices Revisited}
\label{doit}
The discussion in Section \ref{abstring} was solely for Abelian vortices. What goes
wrong if we try to repeat it for non-Abelian vortices? In order to
build the non-Abelian theory with ${\cal N}=4$ supersymmetry, we
must augment the $\kappa=0$ Lagrangian with $N$ chiral
multiplets $\tilde{Q}$ in the anti-fundamental representation, and
a single chiral multiplet $A$ in the adjoint representation.
Integrating out the $\tilde{Q}$ results in a $U(N)$ Chern-Simons
interaction of the type given in \eqn{lag}. However, integrating
out the adjoint multiplet $A$ contributes to the $SU(N)$
Chern-Simons term, but not the $U(1)$ Chern-Simons term. Thus the
mass deformed ${\cal N}=4$ theory does not yield the $U(N)$ ${\cal
N}=2$ theory of the form \eqn{lag}, but rather a theory with
different Chern-Simons coefficients for the $SU(N)$ and $U(1)$
parts of the gauge group.
\para
To make progress, we could instead augment the $\kappa=0$ Lagrangian
with $N$ chiral multiplets $\tilde{Q}$ in
the anti-fundamental representation, and a single neutral chiral
multiplet $A$. The theory no longer admits ${\cal N}=4$
supersymmetry, so we cannot use the above argument to show that
the zero modes live in the tangent bundle. Nonetheless, adding a
superpotential of the form \eqn{w} means that the Dirac equations
are once more of the form \eqn{paul}, and the fermi zero modes are
rendered normalizable. Thus, although we cannot show that the
magnetic field on the moduli space of non-Abelian vortices takes
the simple form \eqn{A}, any lingering worries caused by footnote
\ref{footnote} may now be left behind.
\section{Examples}
In this section, we illustrate our result with two examples. We
firstly examine the qualitative dynamics of two Abelian vortices
and show that the moduli space dynamics correctly captures their
fractional statistics. Secondly, we look at a single vortex in the
$U(N)$ theory, for which the internal moduli space is ${\bf
CP}^{N-1}$. We derive the dynamics both from the method described
in Section 3, and also from a direct moduli space computation.
\subsection{Two Abelian Vortices}
The relative dynamics of two Abelian vortices takes place in the
moduli space ${\cal M}\cong {\bf C}/{\bf Z}_2$. The metric is
given by
\begin{eqnarray} ds^2 = f^2(\sigma)(d\sigma^2 + \sigma^2d \theta^2)\end{eqnarray}
\EPSFIGURE{motion.eps,width=120pt}{The moduli space is a cone.}
\noindent where $\theta \in [0,\pi)$. Asymptotically, as $\sigma
\rightarrow \infty$, we have $f^2(\sigma)\rightarrow 1 +{\cal
O}(e^{-\sigma})$ \cite{samols,mansp} and the moduli space is a
cone with deficit angle $\pi$. Although the function $f(\sigma)$
is not known analytically, it can be shown that $f^2(\sigma) \sim
\sigma^2$ as $\sigma\rightarrow 0$, ensuring that the tip of the
cone is smooth. The moduli space is sketched in Figure 1, together
with an example of the motion which we will describe shortly.
\para
We work with the single valued holomorphic coordinate
$z=\sigma^2e^{2i\theta}$. Then the Chern-Simons 1-form \eqn{A} on the vortex
worldline is given by,
\begin{eqnarray} L_{CS} = -\kappa ({\cal A}\dot{z} + \bar{\cal A}\dot{\bar{z}})
= -\kappa \left(\frac{\sigma}{2}
\frac{\partial}{\partial\sigma}\log\,f^2 -
1\right)\dot{\theta}\label{twov}\end{eqnarray}
A similar expression, expressed in slightly different variables,
can be found in equation (85) of \cite{kimyeong}.
\para
Although the explicit function $f(\sigma)$ is not known, we may
still study the qualitative behaviour of vortices. The conserved
Noether charge associated to $\theta$ is given by
\begin{eqnarray} J = f^2\sigma^2\dot{\theta}
+\kappa\left(1-\frac{\sigma}{2}\frac{\partial\log
f^2}{\partial\sigma}\right)\label{j}\end{eqnarray}
As explained in \cite{kimyeong}, this differs from the angular
momentum of the two vortices by a constant. Meanwhile the
conserved Hamiltonian is
\begin{eqnarray} H = \frac{1}{2}f^2\dot{\sigma}^2 + V_{\rm eff}(\sigma)\end{eqnarray}
where the effective potential is due to the Chern-Simons term,
together with the usual angular momentum barrier,
\begin{eqnarray} V_{\rm eff}(\sigma) =
\frac{1}{2f^2\sigma^2}\left(J-\kappa+\frac{\kappa\sigma}{2}
\frac{\partial\log f^2}{\partial\sigma}\right)^2\end{eqnarray}
The classical scattering of vortices depends on the form of
$V_{\rm eff}$ which, in turn, depends on the relative values of
$\kappa$ and $J$. Let us fix $\kappa
>0$. On physical grounds, the form of the effective potential is shown in Figure 2:
\EPSFIGURE{veff.eps,width=380pt}{The effective potential for different values of $J$.}
\begin{itemize}
\item $J>\kappa$. In this regime, we have $\dot{\theta}>0$ and
$V_{\rm eff}$ is shown in Figure 2a. $V_{\rm eff}$ acts as an effective
angular momentum barrier and the scattering of vortices is not
qualitatively different from the case without a Chern-Simons term.
\item The regime $0 < J < \kappa$ is more interesting. The
effective potential is shown in Figure 2b. The root of the
effective potential corresponds to the static solution. We see
that, as emphasized in \cite{kimyeong}, static solutions with
different vortex separation $\sigma$ carry different angular
momentum $J$.
Small oscillations around the minimum of $V_{\rm eff}$ give rise
to bound orbits of vortices. From the expression \eqn{j}, we see
that $\dot{\theta}$ oscillates from negative to positive in such
orbits. The corresponding motion on the moduli space is drawn in
Figure 1. The two vortices trace Larmor circles, while orbiting one another.
This moduli space motion can be understood using a standard
argument involving adiabatic invariants: in the slowly varying
magnetic field, a particle drifts along
lines of constant field strength.
\item For $J<0$, we have $\dot{\theta}<0$. There are two distinct
shapes of $V_{\rm eff}$. For suitably small $|J|$, the effective
potential takes the form shown in Figure 2c. There are once again
bound orbits, including one at fixed $\sigma$. For $J\ll 0$, the
minimum of $V_{\rm eff}$ disappears and the potential once again
takes the shape of Figure 2a, with only scattering trajectories.
\end{itemize}
Before we move on, we also note that there is a simple quantum
effect that follows from \eqn{twov}. The first term vanishes as
$\sigma\rightarrow \infty$, while the second survives. This
ensures that as the particles orbit asymptotically, the
wavefunction picks up a phase $\exp(\pm i\pi\kappa)$. For $\kappa
\,{\raise.15ex\hbox{/}\mkern-12mu \!\in}{\bf Z}$, this endows the vortices with fractional statistics
in agreement with the analysis of
\cite{csdyn,kimyeongold,kimyeong}.
\subsection{One Non-Abelian Vortex}
For our second example, we examine a single vortex in $U(N)$. We
first review the dynamics of the vortex in the $\kappa =0$ case.
The vortex has an internal moduli space ${\cal M}\cong{\bf
CP}^{N-1}$, describing its orientation in colour and flavour
space \cite{vib,auzzi}. We introduce homogeneous coordinates on ${\cal M}$ by
starting with a solution $B_\star$ for the magnetic field of a
single Abelian vortex configuration. We can embed the Abelian
solution into a non-Abelian configuration by writing,
\begin{eqnarray} B^a_{\ b} =
\frac{B_\star}{r}\,\varphi^a\bar{\varphi}_b\label{namby}\end{eqnarray}
with a similar expression for the Higgs field which we will
describe in more detail in Section \ref{itsthis}. The coordinates
$\varphi_a \in {\bf C}$, $a=1,\ldots, N$, satisfy the constraint,
\begin{eqnarray} \sum_{a=1}^N |\varphi_a|^2 = r \label{constraint}\end{eqnarray}
where $r$ is a constant which is determined to be $r = 2\pi/e^2$
\cite{vib,auzzi,gsy}. The solutions \eqn{namby} are invariant
under the simultaneous rotation
\begin{eqnarray} \varphi_a\rightarrow e^{i\vartheta}\varphi_a\label{iden}\end{eqnarray}
The $\varphi_a$, subject to the constraint \eqn{constraint} and
identification \eqn{iden}, provide homogeneous coordinates on the
moduli space ${\cal M}\cong{\bf CP}^{N-1}$. The low-energy
dynamics of the vortex is described by a sigma-model on ${\cal M}$
endowed with the Fubini-Study metric and K\"ahler class $r$.
There is a simple way to impose the identification \eqn{iden} by
introducing an auxiliary gauge field $\alpha$ on the worldline.
The Lagrangian for the internal modes of the vortex takes the
form,
\begin{eqnarray} L_{vortex} = \sum_{a=1}^N |{\cal D}_t\varphi_a|^2\end{eqnarray}
where the degrees of freedom are subject to the constraint
\eqn{constraint}, and the covariant derivative is given by ${\cal
D}_t\varphi_a = \dot{\varphi}_a - i\alpha\varphi_a$.
\para
Let us now ask how this dynamics is altered by the presence of the
Chern-Simons term. The moduli space is compact and
the cohomology is generated by $\Omega$, the K\"ahler form. Thus
the first Chern character ${\cal F}$ of the
index bundle must be proportional to $\Omega$. We need only
determine the proportionality constant. In fact, this is simple to
achieve in the language introduced above. Let $\tilde{\psi}_\star$
denote the solution to the Abelian Dirac equation \eqn{dirac}.
Then the solution to the non-Abelian Dirac equation, with gauge
field given by \eqn{namby}, is
\begin{eqnarray} \tilde{\psi}^b = \tilde{\psi}_\star\ \xi\bar{\varphi}_b \end{eqnarray}
This is compatible with the symmetry \eqn{iden} if the Grassmann
collective coordinate $\xi$ is assigned charge,
\begin{eqnarray} \xi \rightarrow e^{i\vartheta}\xi\end{eqnarray}
This transformation rule determines the index
bundle, for it fixes the kinetic term of the Grassmann variable to
be given by the covariant derivative $D_t\xi =
\dot{\xi}-i\alpha\xi$. We may now take $m\rightarrow \infty$, and
integrate out $\xi$. The calculation is the same as that
described in Section 3, and yields
\begin{eqnarray} L_{1-vortex} = \sum_{a=1}^N|{\cal D}_t\varphi_a|^2 - \kappa
\alpha\end{eqnarray}
\subsubsection*{An Example of the Example}
For a single vortex in the $U(2)$ theory, the moduli space is
${\bf S}^2\cong{\bf CP}^1$. We now provide a more explicit
description of the dynamics in this case. The constraints
\eqn{constraint} are simply solved by
\begin{eqnarray} \varphi_1 = \sqrt{r} e^{i\psi-i\phi/2}\cos(\theta/2) \ \ \ ,\
\ \ \varphi_2 = \sqrt{r}
e^{i\psi+i\phi/2}\sin(\theta/2)\label{coords}\end{eqnarray}
where the angles take ranges $\psi\in [0,2\pi)$, $\phi\in
[0,2\pi)$ and $\theta\in[0,\pi)$. Expanding out the Lagrangian
gives
\begin{eqnarray} L_{vortex} =
r\cos^2(\theta/2)(\dot{\psi}-\dot{\phi}/2-\alpha)^2 + r
\sin^2(\theta/2)(\dot{\psi}+\dot{\phi}/2 - \alpha)^2 +
\frac{r}{4}\dot{\theta}{}^2 - \kappa\alpha\nonumber\end{eqnarray}
We now eliminate the gauge field $\alpha$ by its equation of
motion. Ignoring an overall constant term and treating total derivatives carefully, the resulting dynamics
is given by
\begin{eqnarray} L_{1-vortex} = \frac{r}{4}\left[\dot{\theta}{}^2+
\sin^2\theta\,\dot{\phi}{}^2\right] +
\frac{\kappa}{2}(\cos\theta-1)\dot{\phi}\label{mono}\end{eqnarray}
We recognize the first term as the familiar sigma-model on ${\bf S}^2$
with radius $R = \sqrt{r/2}$. The second term is the Dirac
monopole connection of strength $\kappa$, expressed in a form which
gives a well-defined potential everywhere except at the south pole.
\subsection{One Non-Abelian Vortex: Explicit Moduli Space Computation}
\label{itsthis}
In this final section, we show how to re-derive the Dirac monopole
connection \eqn{mono} from an explicit moduli space calculation. As
we shall see, the calculation requires that we take care with the
topology of the moduli space.
\para
Following \cite{kimyeong}, we work perturbatively both in the
velocity of the vortices, and in $\kappa$. Practically, this means
that we start with the Bogomolnyi equations with
$\kappa=0$,
\begin{eqnarray} B=\frac{e^2}{2}(q_iq_i^\dagger - v^2)\ \ \ ,\ \ \ {\cal D}_z q_i=0
\label{vortex}\end{eqnarray}
but with $\phi=A_0$ determined by Gauss' law
\eqn{gauss}\footnote{Since $\kappa \in {\bf Z}$, it does not seem
like a good candidate for perturbation theory. A more careful
study shows that $e^2\kappa^2/v^2 \ll 1$ is the small parameter.}.
Let's first quantify the price that we pay by working
perturbatively in $\kappa$. Since the Chern-Simons term clearly
plays a crucial role in this discussion, it is necessary to work
with the Lagrangian instead of the energy functional. We evaluate
the Lagrangian \eqn{lag} on the solution to the equations
\eqn{vortex} and \eqn{gauss}, with $\partial_0=0$. This gives
\begin{eqnarray} L = \int d^2x {\cal L} = -2\pi v^2 k -
\frac{e^2\kappa^2}{16\pi^2} \int\ d^2x {\rm Tr}\,\phi^2\end{eqnarray}
The last term is the correction to the Lagrangian due to the fact
that we chose to work with the $\kappa=0$ Bogomolnyi equations,
rather than the true equations \eqn{bog1} and \eqn{bog2}.
The mass of the configuration is
\begin{eqnarray} M_{\rm vortex} = 2\pi v^2 k \left( 1+ {\cal
O}\left(\frac{e^4\kappa^4}{v^4}\right)\right)\end{eqnarray}
The extra term is the price we pay for our approximation. At our
level of approximation, we neglect all terms of this order in
what follows.
\subsubsection*{Zero Modes}
Let us now turn to the dynamics of the system. Here we see the advantage
of our approximation, because we may deal with
the familiar vortex equations \eqn{vortex}. Denote the collective
coordinates of this system by $X^a$, with $a=1,\ldots 2kN$. The
zero modes of the solution are then given by differentiating, together
with a gauge transformation:
\begin{eqnarray} \delta_aA_\alpha = \frac{\partial A_\alpha}{\partial
X^a}-{\cal D}_\alpha w_a\ \ \ ,\ \ \ \delta_a q_i =
\frac{\partial{q_i}}{\partial X^a} - iw_aq_i\label{zeromodes1}\end{eqnarray}
The gauge transformation $w_a\in u(N)$ is dictated by the gauge fixing
condition,
\begin{eqnarray} {\cal D}_\alpha \delta_a A_\alpha =
-\frac{ie^2}{2}(\delta_aq_i\,q_i^\dagger -
q_i\delta_aq_i^\dagger)\label{cons}\end{eqnarray}
We next write $A_0=w+\phi$, where $w \equiv w_a\dot{X}^a$,
which ensures that the zero modes are related to the covariant time
derivatives as follows:
\begin{eqnarray} {\cal D}_0q_i = \delta_aq_i\dot{X}^a -i\phi q_i\ \ \ \ \ ,\ \
\ E_\alpha = \delta_aA_\alpha \dot{X}^a - {\cal D}_\alpha\phi\label{D5-4.20}\end{eqnarray}
The presence of the $\phi$ terms on the right-hand-side of these
equations is what distinguishes the Chern-Simons dynamics from the
case $\kappa=0$. Notice that in our approximation, we have not
needed to linearize the second order Gauss' law equation
\eqn{gauss} since the terms $({\cal D}_0\phi)^2$ are of order
$\kappa^2\dot{X}^2$ and may be safely ignored. Substituting into %
the Lagrangian \eqn{lag}, and making use of the constraint \eqn{cons}, we
derive an expression for the Lagrangian governing the dynamics of
the vortex,
\begin{eqnarray} L =
g_{ab}\dot{X}^a\dot{X}^b - 2\pi v^2 k
- \frac{\kappa}{4\pi}\int d^2x\ {\rm Tr}\, \left(2B w_a\dot{X}^a
-\epsilon_{\alpha\beta}A_\alpha\dot{A}_\beta\right) \label{kv}\end{eqnarray}
This generalizes the result derived in
\cite{csdyn,kimyeongold,kimyeong} to the non-Abelian case. The
first term in this expression is the usual metric on the vortex
moduli space, given by
\begin{eqnarray} g_{ab} = \int d^2x\ \left( \frac{1}{e^2} {\rm Tr}\,
\delta_aA_\alpha\delta_b A_\alpha +
\delta_{(a}q_i{}^{\!\dagger} \delta_{b)}q_i\right)\end{eqnarray}
The effect of the Chern-Simons interaction is shown in the last term
of \eqn{kv}, which is of order $\kappa\dot{X}$.
\subsubsection*{Non-Singular Gauge}
We now apply this formula to the simple case of a single vortex in
the $U(2)$ gauge theory. In this case, the moduli space is ${\bf
CP}^1$. Previous field theoretic studies of this system have
always employed singular gauge \cite{auzzi}, in which the Higgs
field $q_i$ has no winding at infinity. While this gauge is
perfectly adequate for studying the metric on moduli space (see,
for example \cite{gsy}), it hides the interesting topology of the
moduli space and is not suitable for studying the effect of the
Chern-Simons term.
We therefore first describe the collective coordinates of the
single $U(2)$ vortex in a gauge that does not suffer from singular
behaviour.
\para
Consider the $U(1)$ vortex equations \eqn{vortex}. We work in
polar coordinates on the spatial plane: $x_1= \rho \cos\chi$ and
$x_2=\rho\sin \chi$. Then the solution to the equations for the
$k=1$ vortex is given by
\begin{eqnarray} q = vq_\star(\rho) e^{i\chi}\ \ \ {\rm and}\ \ \ A_\chi =
1-f(\rho)\ \ \ ,\ \ \ A_\rho = 0 \end{eqnarray}
where the profile functions satisfy the ordinary differential
equations,
\begin{eqnarray} \rho q_\star^\prime = -fq_\star\ \ \ {\rm and}\ \ \
\frac{f^\prime}{\rho}=-\frac{e^2v^2}{2}(q_\star^2-1)\end{eqnarray}
subject to the boundary conditions $q_\star(\rho)\rightarrow 1,0$ and
$f(\rho)\rightarrow 0,1$ as $\rho \rightarrow +\infty, 0$. Given
these Abelian solutions, it is now a simple matter to embed them
into the fields of the $U(2)$ theory to arrive at new solutions.
There are two natural embeddings:
\begin{eqnarray} &i)&\ \ \ \ q_{(1)} = v\left(\begin{array}{cc} q_\star
e^{i\chi} & 0 \\ 0 & 1 \end{array}\right) \ \ \ ,\ \ \ A_\chi =
\left(\begin{array}{cc} (1-f) & \ 0 \\ 0 &\ 0
\end{array}\right) \ \ \ ,\ \ \ A_\rho = 0\label{q1}\\ &ii)&\ \ \ \
q_{(2)} = v\left(\begin{array}{cc} 1 & 0 \\ 0 & q_\star e^{i\chi}
\end{array}\right) \ \ \ ,\ \ \ A_\chi = \left(\begin{array}{cc}
0\ & 0 \\ 0\ & (1-f) \end{array}\right) \ \ \ ,\ \ \ A_\rho =
0 \label{q2}\end{eqnarray}
Here the rows and columns of the $q$ matrix correspond to colour
and flavour indices respectively. However, these embeddings are
not the only two. Given either of these solutions, one may act
upon it with a diagonal combination of the $SU(2)_{\rm flavour}$
symmetry and $SU(2)_{\rm gauge}$ symmetry of the model in such a
way that the diagonal structure of the vacuum remains invariant,
\begin{eqnarray} q \rightarrow U qV^\dagger\ \ \ ,\ \ \ A\rightarrow
UAU^\dagger - i (\partial U)\,U^\dagger\end{eqnarray}
where $V\in SU(2)_{\rm flavour}$ is a constant matrix, and
$U=U(\rho,\chi)\in SU(2)_{\rm gauge}$. In singular gauge, we would
impose the condition that $U\rightarrow V$ as $\rho\rightarrow
\infty$. However, the presence of the winding scalar field in
\eqn{q1} and \eqn{q2} means that cannot be quite right in the
present case. Indeed, the only transformation such that
$U\rightarrow V$ that is allowed is
$U=V=\tiny{\left(\begin{array}{cc} 0 & i
\\ i & 0
\end{array}\right)}$ which maps $q_{(1)}$ to $q_{(2)}$. For more
general transformations, $U$ must itself include some winding. The
necessary condition is not difficult to determine. For
$V=\tiny{\left(\begin{array}{cc} \hat{a}_1 & \hat{a}_2 \\
\hat{a}_3 & \hat{a}_4\end{array}\right)} \in SU(2)_{\rm flavour}$,
we require
\begin{eqnarray} U_{(1)}(\rho,\chi) = \left(\begin{array}{cc} a_1(\rho) &
a_2(\rho)e^{i\chi} \\ a_3(\rho)e^{-i\chi} &
a_4(\rho)\end{array}\right)\ \ \ {\rm or}\ \ \ U_{(2)}(\rho,\chi)
= \left(\begin{array}{cc} a_1(\rho) & a_2(\rho)e^{-i\chi} \\
a_3(\rho)e^{i\chi} & a_4(\rho)\end{array}\right) \ \ \ \ \ \ \
\label{u12}\end{eqnarray}
where the matrix $U_{(1)}$ is to be used for transformations away
from $q_{(1)}$, while the matrix $U_{(2)}$ is required for
transformations away from $q_{(2)}$. In both cases, the profile
functions in the gauge transformation satisfy the boundary
conditions $a_i(\rho)\rightarrow \hat{a}_i$ as $\rho \rightarrow
\infty$.
\para
Perhaps unsurprisingly, the picture that emerges is that two
patches are required to cover the moduli space. The solution
$q_{(1)}$ can be thought of as the north pole of ${\bf CP}^1$, and
combined gauge and flavour transformations given by $U_{(1)}$
cover nearly all the space, but cannot take us to $q_{(2)}$.
Similarly, $q_{(2)}$ is thought of as the south pole of the moduli
space and transformations using $U_{(2)}$ can reach the full
moduli space, except for the north pole.
\subsubsection*{Finding the Dirac Monopole Connection}
We now use these results to derive
the Dirac monopole connection on moduli space. Let's start with
the solution $q=q_{(1)}$. We look for zero modes corresponding to
a simultaneous $SU(2)$ gauge and flavour rotation, with parameters $\Omega$ and $\hat{\Omega}$ respectively. The zero modes are given
by
\begin{eqnarray} \delta q \equiv \delta_a q \dot{X}^a = i(\Omega q - q\hat{\Omega})\ \ \ , \ \ \ \delta A_\alpha \equiv \delta_a A_\alpha \dot{X}^a= {\cal D}_\alpha \Omega \label{zmo}\end{eqnarray}
The requirement that the vacuum remains invariant fixes $\Omega_\infty \equiv \lim_{\rho \rightarrow \infty} \Omega(\rho,\chi)$ in terms of $\hat{\Omega}$. The remaining freedom in $\Omega$ is fixed by the constraint \eqn{cons}, which now reads
\begin{eqnarray} {\cal D}^2\Omega =
\frac{e^2}{2}\left(\{\Omega,qq^\dagger\}-2q\hat{\Omega}q^\dagger\right)\label{omega}\end{eqnarray}
We demand that varying the fields with respect to the collective co-ordinates corresponds to the `large' part of the gauge and flavour rotation, with parameters $\Omega_\infty$ and $\hat{\Omega}$. This means that
\begin{eqnarray} \partial_0 q = \frac{\partial q}{\partial X^a} \dot{X}^a = i\left( \Omega_\infty q - q \hat{\Omega} \right)\ \ \ {\rm and} \ \ \ \partial_0 A_\alpha = \frac{\partial A_\alpha}{\partial X^a} \dot{X}^a = {\cal D}_\alpha \Omega_\infty \label{D6-4.31}\end{eqnarray}
To achieve this and satisfy \eqn{zmo}, we set $w = \Omega_\infty - \Omega$ in \eqn{zeromodes1}.
\para
We choose our flavour transformation to be $\hat{\Omega} =
\frac{\dot{\theta}}{2}\tiny{\left(\begin{array}{cc} 0 & 1 \\ 1 & 0
\end{array}\right)}\in su(2)_{\rm flavour}$, where we are using
the coordinates \eqn{coords}, and the factor of $\theta/2$ in this
expression follows directly from the same factor in \eqn{coords}.
Then \eqn{omega} is solved by \cite{gsy}
\begin{eqnarray} \Omega = \frac{\dot{\theta}}{2}\left(\begin{array}{cc} 0 & q_\star(\rho) e^{i\chi} \\
q_\star(\rho) e^{-i\chi} & 0 \end{array}\right)\label{431}\end{eqnarray}
where the boundary condition on $\Omega$ is inherited from
$\hat{\Omega}$. The asymptotic winding in \eqn{431} results from
working in non-singular gauge as in equation \eqn{u12}.
\para
To compute the terms in the low-energy dynamics of the Lagrangian, we substitute \eqn{D6-4.31} and $w = \Omega_\infty - \Omega$ into our moduli dynamics \eqn{kv} to get
\begin{eqnarray} L_{\rm CS} = -\frac{\kappa}{4\pi} \int d^2x \ {\rm Tr}\,\left( 2B(\Omega_\infty - \Omega) - \epsilon_{\alpha\beta}A_\alpha {\cal D}_\beta \Omega_\infty \right) \label{D6-4.33}\end{eqnarray}
Using \eqn{431} and \eqn{q1}, we see that $(\Omega_\infty - \Omega)$ and ${\cal D}_\rho \Omega_\infty$ are off-diagonal, while $B$ and $A_\chi$ are diagonal and $A_\rho$ is zero. Hence \eqn{D6-4.33} vanishes.
\para
\EPSFIGURE{cp1.eps,width=100pt}{\ }
However, we shouldn't be too hasty in concluding that the
Chern-Simons term has no effect on the vortex dynamics. We should
first compare with the expected Dirac monopole solution found in
Section 4.2. We have worked about the ``north pole" solution
\eqn{q1}. As discussed previously, this patch covers all but the
south pole of ${\bf CP}^1$. If we were to write the Dirac monopole
in these coordinates, the Dirac string would point along the
direction of the south pole. The corresponding term on the
worldline is given by
\begin{eqnarray} L_{\rm Dirac} = \frac{\kappa}{2}(\cos\theta-1)\dot{\phi}\end{eqnarray}
However, as shown in Figure 3, the calculation that we have just
done corresponds to moving downwards from the north pole. This is
equivalent to looking for a $\dot{\theta}$ term in the effective
action. It is not surprising that it gave a vanishing answer! Said
another way, there is always a coordinate choice so that a given
infinitesimal motion doesn't reveal a Dirac monopole connection in
the Lagrangian. We have made that coordinate choice above; moreover,
such a coordinate choice is always made implicitly if we work in singular gauge because this
gauge disguises the presence of the Dirac string.
\para
With this understanding of the topology of moduli space, it is a
simple matter to perform a calculation that does see the Dirac
monopole connection. Our first goal is to rotate the $q_{(1)}$
solution to a configuration corresponding to latitude $\theta$ on
the moduli space. This is done by a flavour rotation of the form,
\begin{eqnarray} V = \left(\begin{array}{cc} \cos(\theta/2) & i\sin(\theta/2)
\\ i\sin(\theta/2) & \cos(\theta/2)\end{array}\right) \in
SU(2)_{\rm flavour}\end{eqnarray}
together with a suitable gauge transformation $U_{(1)}$ with
boundary conditions given in \eqn{u12}. We now search for zero
modes around this new background. Our task is to solve for the
infinitesimal gauge transformation $\Omega$ satisfying
\eqn{omega}, subject to the appropriate
boundary condition. This boundary condition comes
from the requirement that the gauge transformation acts in the
longitudinal $\phi$ direction, and returns us to our starting point after $\phi$ has increased by $2\pi$. Using the coordinates
\eqn{coords}, we see that this can be achieved if we supplement our gauge and flavour transformations by $U(1)$ rotations corresponding to motion in the $\psi$ direction. An appropriate choice is
$\hat{\Omega} = \dot{\phi}\tiny{\left(\begin{array}{cc} 0 & 0 \\
0 & 1
\end{array}\right)}$. Since this is diagonal, we have $\Omega_\infty = \hat{\Omega}$ (see \eqn{u12}). Using the fact that $\partial_\alpha \Omega_\infty = 0$ and performing an integration by parts, we may write
\begin{eqnarray} \int d^2x\ {\rm Tr}\,\left(-\epsilon_{\alpha\beta}A_\alpha {\cal D}_\beta \Omega_\infty \right) = \int d^2x\ {\rm Tr}\,\left(-2B\Omega_\infty \right) \end{eqnarray}
Once we substitute this into the Lagrangian \eqn{D6-4.33}, we are left with
\begin{eqnarray} L_{\rm CS} = -\frac{\kappa}{4\pi} \int d^2x \ {\rm Tr}\,\left( -2B\Omega \right) \label{D6-LCS}\end{eqnarray}
\para
We may make use of the gauge covariance of \eqn{omega} to
translate the task of finding $\Omega$ into something equivalent: solving
\eqn{omega} in the background of the original vortex solution
\eqn{q1} now subject to the boundary condition arising from
\begin{eqnarray} V^\dagger \hat{\Omega} V= \dot{\phi}\,V^\dagger
\left(\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right)V =
\frac{\dot{\phi}}{2} \left(\begin{array}{cc} 1-\cos\theta & -i\sin\theta \\
i\sin\theta & 1+\cos\theta\end{array}\right)\end{eqnarray}
It is straightforward to show that the solution is given by
\begin{eqnarray} U^\dagger \Omega(\rho,\chi) U = \frac{\dot{\phi}}{2}
\left(\begin{array}{cc} 1-\cos\theta & -ie^{i\chi}q_\star(\rho)
\sin\theta \\ ie^{-i\chi} q_\star(\rho) \sin\theta &
1+\cos\theta\end{array}\right)\end{eqnarray}
We now substitute our results into the expression \eqn{D6-LCS}
arising from moduli space dynamics. Noting that the magnetic field associated with \eqn{q1} is given by $U^\dagger BU$, we have
\begin{eqnarray*} L_{\rm CS} &= &\frac{\kappa}{2\pi}\int d^2x\ {\rm Tr}\,U^\dagger\Omega U U^\dagger B U \\
&=&\frac{\kappa(1-\cos\theta) \dot{\phi}}{4\pi} \int d^2x\ {\rm Tr}\, B =
\frac{\kappa}{2} (\cos\theta-1) \dot{\phi} \end{eqnarray*}
This reproduces the Dirac monopole connection as claimed.
\setcounter{section}{0} \setcounter{equation}{0}
\renewcommand{\thesection}{\Alph{section}}
\section*{Acknowledgement}
We would like to thank Nick Dorey, Maciej Dunajski and Nick Manton
for many useful discussions. BC is supported by an STFC studentship.
DT is supported by the Royal Society.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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<h1 id="kubernetes-dashboard">[[full_name]] New And Episodic Issue Creators dashboard</h1>
<p>Links:</p>
<ul>
<li>New issues metric <a href="https://github.com/cncf/devstats/blob/master/metrics/shared/new_issues.sql" target="_blank">SQL file</a>.</li>
<li>Episodic issues metric <a href="https://github.com/cncf/devstats/blob/master/metrics/shared/episodic_issues.sql" target="_blank">SQL file</a>.</li>
<li>TSDB <a href="https://github.com/cncf/devstats/blob/master/metrics/kubernetes/metrics.yaml" target="_blank">series definition</a>. Search for <code>New and episodic issue</code></li>
<li>Grafana dashboard <a href="https://github.com/cncf/devstats/blob/master/grafana/dashboards/kubernetes/new-and-episodic-issue-creators.json" target="_blank">JSON</a>.</li>
</ul>
<h1 id="description">Description</h1>
<ul>
<li>This dashboard shows statistics about new and episodic issues and issue creators.</li>
<li>New issue creator is someone who haven't created any issue before given period.</li>
<li>New issue is an issue created by new issue creator</li>
<li>Episodic issue creator is someone who haven't created any issue in 3 months before given project and haven't created more than 12 issues overall.</li>
<li>Episodic issue is an issue created by episodic issue creator.</li>
<li>You can select single repository group or summary for all of them.</li>
<li>Selecting period (for example week) means that dashboard will calculate statistics in those periods.</li>
<li>See <a href="https://github.com/cncf/devstats/blob/master/docs/periods.md" target="_blank">here</a> for more informations about periods.</li>
<li>See <a href="https://github.com/cncf/devstats/blob/master/docs/repository_groups.md" target="_blank">here</a> for more informations about repository groups.</li>
</ul>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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\section{Introduction}
HB 21 (G89.0+4.7) is a large ($\sim120'\times90'$), middle-aged ($\sim5000-7000$ yr, \citealt{Lazendic(2006)ApJ_647_350,Byun(2006)ApJ_637_283}) supernova remnant (SNR) at a distance estimated to be from $\sim0.8$ kpc to $\sim1.7$ kpc \citep{Leahy(1987)MNRAS_228_907,Tatematsu(1990)A&A_237_189,Byun(2006)ApJ_637_283}.
Based on its indented, shell-like appearance in the radio and the existence of nearby giant molecular clouds, it is thought to be interacting with a molecular cloud \citep[cf.~Fig.~\ref{fig-obs};][]{Erkes(1969)AJ_74_840,Huang(1986)ApJ_309_804,Tatematsu(1990)A&A_237_189}.
The first direct evidence for this interaction was the detection of broad CO emission lines near the edge and the center of the remnant \citep{Koo(2001)ApJ_552_175,Byun(2006)ApJ_637_283}.
The existence of such an interaction was further supported by the suggestion that evaporation of the cloud might be responsible for the enhanced thermal X-rays seen in the central part of the remnant \citep{Leahy(1996)A&A_315_260}.
We performed infrared imaging observations toward two localized positions in HB 21, where the broad CO emission lines were observed (Fig.~\ref{fig-obs}), with two instruments: the InfraRed Camera \citep[IRC,][]{Onaka(2007)PASJ_59_S401s} aboard a Japanese satellite, \textit{AKARI}{} \citep{Murakami(2007)PASJ_59_S369s} and the Wide-field InfraRed Camera \citep[WIRC,][]{Wilson(2003)inproc} on the Palomar 5 m Hale telescope.
From the analysis of the northern part (``Cloud N'') data, we found that the mid-infrared diffuse features originated from shocked H$_2${} gas, with their excitation conditions well described by a thermal admixture of H$_2${} gas, whose infinitesimal H$_2${} column density has a power-law relation with the temperature $T$, d$N\sim T^{-b}dT$ \citep[][hereafter Paper I]{Shinn(2009)ApJ_693_1883}.
Such H$_2${} excitation conditions are consistent with the ``ankle-like'' energy level population diagram (i.e. a turn-up in population for higher energies, see Fig.~\ref{fig-pop}), hitherto observed at the shock-cloud interaction regions (cf. $\S$~1 of Paper I).
Here we present the analysis of the southern portion of HB 21 (``Cloud S''), following the method of Paper I.
The near- and mid-infrared images ($\sim2-13$ $\mu$m) we obtained show diffuse features around a shocked CO cloud.
We analyze them as emission lines of H$_2${} gas in statistical equilibrium.
We find the emission, as with the Cloud N case, to be well described with a power-law admixture model of thermal H$_2${} gas.
We then discuss these results with physical pictures of the shock-cloud interaction.
\section{Observations} \label{obs}
We observed two specific regions (Cloud N and Cloud S in Fig.~\ref{fig-obs}), where slow shocks ($\lesssim20$ km s$^{-1}$) propagate into clouds of $n(\textrm{H}_2)${} $\sim10^3$ cm$^{-3}${} \citep{Koo(2001)ApJ_552_175}, using two different instruments: IRC \citep{Onaka(2007)PASJ_59_S401s} aboard the \textit{AKARI}{} satellite and WIRC \citep{Wilson(2003)inproc} on the Palomar 5 m telescope.
The Cloud N data were analyzed in Paper I, and the Cloud S data are analyzed here.
Details on the observations and reduction of the IRC and WIRC data are described separately, below.
\subsection{\textit{AKARI}{} IRC observations} \label{obs-irc}
\textit{AKARI}{} is a satellite designed for both imaging and spectroscopy in the infrared \citep{Murakami(2007)PASJ_59_S369s}.
The IRC is one of \textit{AKARI}{}'s scientific instruments, which covers the wavelength range 2--30 $\mu$m{} and has a $\sim10'\times10'$ field-of-view for imaging.
The IRC pointed-imaging observations for Cloud S were performed on 2007 Jun 3rd towards (RA, Dec) = ($20^h46^m07.80^s$, $+50^\circ02'02.00''$) in J2000.
IRC comprises three channels (NIR, MIR-S, and MIR-L), each of which has three band-pass filters for imaging.
Among these, we employed four filters from NIR and MIR-S channels for the observations; the MIR-L channel was not used for observing Cloud S, due to lack of observing time.
Table \ref{tbl-obs} lists the wavelength coverage and the imaging resolutions ($\Gamma$), together with pixel sizes in each channel.
Data reduction was the same as for the Cloud N data (cf. Paper I), except for flat-fielding; we used a different MIR-S flat, since the dark pattern seen in the channel changed around 2007 Jan 7th\footnote{This is described in a note at http://www.ir.isas.jaxa.jp/AKARI/Observation/DataReduction/IRC/}.
We obtained the refined coadded image through the IRC Imaging Pipeline \citep[v.~20070104][]{Lorente(2007)man}.
Astrometric information was added to the coadded images, employing the 2MASS catalog \citep{Skrutskie(2006)AJ_131_1163s}, with a matching tolerance of 1.5 pixels.
The systematic errors ($\sim2-5\%$) of the calibration were included in the error estimation, as done for the Cloud N data.
Then, for the comparison between images from different bands, the pixel size was interpolated to $1''$ and the spatial resolution was smoothed to $\simeq7.43''$.
Point sources were removed applying the DAOPHOT package \citep{Stetson(1987)PASP_99_191} of IRAF making use of the simple-masking method.
The final images for Cloud S are displayed in Figure~\ref{fig-result}.
\subsection{Palomar WIRC H$_2${} observations} \label{obs-wirc}
The WIRC observations were taken together with those of Cloud N (cf. Paper I).
We carried out the H$_2$ $\upsilon=1\rightarrow0$ S(1){} 2.12 $\mu$m{} narrow-band filter imaging observation of Cloud S, centered at (RA, Dec) = (20h:46m:23.28s, $+49^\circ:54':16.72''$) in J2000, on the Palomar 5 m Hale telescope on 2005 August 29.
The WIRC is equipped with a Rockwell Science Hawaii II HgCdTe 2K infrared focal plane array, covering a $8.7'\times8.7'$ field of view with a $\sim0.25''$ pixel scale.
The WIRC field partially covered the IRC field, due to the mislocation of the WIRC observation center (cf. Fig.~\ref{fig-result}).
The data reduction was also the same as that of Cloud N data.
50 dithered images of 30 sec exposure were obtained.
We subtracted dark and sky background from each individual dithered frame and then divided by a normalized flat frame.
Finally, the dithered frames were combined to produce the final image.
Astrometry was obtained by matching the positions of 13 field point-sources with those of 2MASS catalog sources, and the positions agreed within $\sim0.3''$.
Flux calibration was also done using the 2MASS catalog.
We matched the magnitudes of 13 field point-sources with the corresponding $K_s$ magnitudes from the 2MASS catalog.
Their correlation coefficient was 0.9956, and their ratio, $M_{WIRC}/M_{K_s}$, was $1.026\pm0.011$.
The systematic error ($\sim14\%$) of the calibration was included in the error estimation.
Point sources were removed using DAOPHOT.
The full-width-at-half-maximum (FWHM) of these sources was found to be $\sim1.1''$.
\section{Results} \label{res}
The final images of IRC and WIRC are displayed in Figure~\ref{fig-result}, together with a $^{12}$CO $J=2\rightarrow1${} 230.583 GHz \citep{Koo(2001)ApJ_552_175} image for reference.
The peak positions of the shocked S1 and S2 clouds, where broad $^{12}$CO $J=2\rightarrow1${} lines were observed \citep{Koo(2001)ApJ_552_175}, are also indicated on the images.
\subsection{Morphology} \label{res-mor}
The IRC images (Fig.~\ref{fig-result}) show different features from band to band, as in Cloud N (Paper I).
The N3 and N4 images are dominated by point sources, while the S7 and S11 images show similar diffuse features.
They do not, however, look like bow shocks or planar shocks, unlike in Cloud N.
Around the cloud S1, common diffuse features are seen in all the IRC images, although they are faint in the N3 image.
However, there are no such diffuse features around the cloud S2.
This is in contrast with Cloud N, where the shocked CO clouds have corresponding diffuse features in the IRC S7 and S11 bands; this is more interesting considering that the cloud properties observed from CO emission lines are similar for both clouds, S1 and S2 \citep{Koo(2001)ApJ_552_175}.
Higher extinction toward the cloud S2 than the cloud S1 does not seem to be the reason for the absence of diffuse IRC features around the cloud S2, since very high column density $N(\textrm{H})$$\,\gtrsim10^{23}$ cm$^{-2}${} is required for the extinction to be effective at $\sim$ 10 $\mu$m{} \citep{Draine(2003)ARA&A_41_241}.
Recalling that H$_2${} emissions are the main source for the diffuse IRC features in the case of Cloud N (Paper I), this absence may be caused by the lack of H$_2${} gas around the cloud S2.
The dissociation of H$_2${} by hot gas ($\gtrsim10^6$ K) can be the reason, however it is unlikely since the X-ray emission is not strong around the cloud S2 \citep{Byun(2006)ApJ_637_283}.
The X-ray flux ($0.1-2.4$ keV) of HB 21 is $31.8\times10^{-10}$ erg s$^{-1}$ cm$^{-2}${} \citep{Leahy(1996)A&A_315_260}, and the cloud S2 locates $\sim10'$ away from the central X-ray emissions \citep{Byun(2006)ApJ_637_283}, which corresponds to a projected distance of $2-5$ pc.
With a hydrogen nuclei density of $10^4$ cm$^{-3}${} and an attenuating column density of $10^{20}-10^{22}$ cm$^{-2}$, this distance corresponds to an effective ionization parameter, Log $\xi_{eff}$, ranging from $-4$ to $-7$, sufficiently small so that the effects of X-rays are negligible \citep{Maloney(1996)ApJ_466_561}.
At the moment, it remains uncertain why such an absence of diffuse IRC features happens \emph{only} to the cloud S2.
A diffuse, looplike feature with a diameter of $\sim4'$ is seen in the northern portion of the S7 and S11 images, however, it does not seem to be related with the shocked CO clouds S1 and S2.
The $^{12}$CO $J=2\rightarrow1${} map shows a similar looplike feature at 9.4 km s$^{-1}${} \citep{Koo(2001)ApJ_552_175}.
Since the looplike feature looks similar in the IRC S7 and S11 bands, they may be generated by [Ar {\small II}]{} 6.99 $\mu$m{} and [Ne {\small II}]{} 12.8 $\mu$m{} emission lines, which are expected to show similar distributions in the shock-cloud interaction regions considering their ionization potentials \citep{Neufeld(2007)ApJ_664_890}; indeed, these lines have been frequently observed around SNRs \citep[e.g.~][]{Arendt(1999)ApJ_521_234,Oliva(1999)A&A_343_943,Reach(2002)ApJ_564_302,Neufeld(2007)ApJ_664_890}.
Thermal emission from warm dust ($\gtrsim100$ K) is another candidate.
However, it seems unlikely because hot gas ($\gtrsim10^6$ K)---the heat source for the warm dust---is not abundant around the cloud S1 \cite[Fig.~1 of][]{Byun(2006)ApJ_637_283}.
The WIRC H$_2$ $\upsilon=1\rightarrow0$ S(1){} image only covered a small portion of the field of the IRC images because of the mislocation, but does include the cloud S1 (cf. Fig.~\ref{fig-result}).
The H$_2${} image shows diffuse features around the cloud S1, similar to those seen in the IRC images.
The similarity is more easily recognizable in Figure \ref{fig-rgb}, which zooms into the area around the cloud S1.
The RGB color image is made with N4 (blue), S7 (green), and S11 (red).
Three elongated clumps, whose sizes are comparable with the FWHM of the IRC images ($\sim7.4''$), are apparent.
Overall, their colors are red-and-yellowish, although the southwestern part of the features shows a little bluish color.
The filamentary features seen in the WIRC image have a similar overall morphology to those seen in the RGB image.
They also surround the shocked CO cloud, S1 (cf.~Fig.~\ref{fig-result} and \ref{fig-rgb}).
This geometrical relationship thus suggests that the diffuse infrared features seen around the cloud S1 may also originate from shock excitation.
\subsection{Quantitative Infrared Characteristics of the Shocked Gas} \label{res-qua}
Since the cloud S2 shows no relevant feature in the IRC images and was not covered in the WIRC H$_2$ $\upsilon=1\rightarrow0$ S(1){} image, we analyzed the cloud S1 only.
To quantify the infrared characteristics of the cloud S1, we measured its intensity in the IRC and WIRC images.
The regions are outlined by the two concentric circles (see Fig.~\ref{fig-rgb}).
The inner circle is the source region and the surrounding annular shell is the background region.
To avoid any contamination during the measurement, possible point sources were excluded referring 2MASS point sources \citep{Skrutskie(2006)AJ_131_1163s}; they are indicated as white circles with a red slash on Figure \ref{fig-rgb}.
The White circles with black shadings on the IRC images (Fig.~\ref{fig-rgb}) are bright point sources masked out during the data reduction (cf.~$\S$~\ref{obs-irc}).
Also, the northern part of the IRC images was additionally excluded since the WIRC image does not fully cover this region.
The masked area is outlined by a white tetragon with a red slash.
Table \ref{tbl-result} lists the measured intensities together with the IRC colors, N4/S7 and S7/S11.
The IRC intensity is the strongest in S11, and decreases to shorter wavelength.
Comparing with Cloud N, the S7 and S11 intensities are greater by a factor of 2--3 in the cloud S1.
The colors are displayed as a point in Figure \ref{fig-ccd}.
The colors of the N2front in Cloud N are also displayed for comparison (cf.~section \ref{dis-ccd}).
The cloud S1 and N2front have similar colors.
The H$_2$ $\upsilon=1\rightarrow0$ S(1){} intensity was also extinction-corrected, as in Cloud N.
We calculated the extinction factor to be $\sim0.82$ ($A_V=1.8$ mag), derived from the extinction curve of ``Milky Way, $R_V=3.1$'' \citep{Weingartner(2001)ApJ_548_296,Draine(2003)ARA&A_41_241} with the foreground hydrogen nuclei column density, $N$(H)=$N$(H {\small I})+2$N$(H$_2$)$=(3.5\pm0.4)\times10^{21}$ cm$^{-2}$, towards the center of HB 21 \citep{Lee(2001)inproc}.
\section{Radiation Source of the Shock-Cloud Interaction Features Observed in the \textit{AKARI}{} IRC Bands} \label{irc-org}
To interpret the infrared intensities and colors (Table \ref{tbl-result}), we must identify the radiation source of the features we see in the shock-cloud interactions.
In Paper I, based on several arguments, we concluded that shocked H$_2${} gas was the most probable explanation for the interaction features observed in the IRC S7, S11, and L15 bands.
In the similar manner, we here attribute the infrared features seen around the cloud S1 in the IRC N4, S7, and S11 bands to shocked H$_2${} gas.
Firstly, the similarity between the features seen in the IRC images and the Palomar WIRC H$_2$ $\upsilon=1\rightarrow0$ S(1){} image is also seen in the cloud S1 case.
Although the IRC images are rather diffuse, they definitely show three elongated clumpy features, which similarly locate around the cloud S1 as in the H$_2$ $\upsilon=1\rightarrow0$ S(1){} image (Fig.~\ref{fig-rgb} and $\S$~\ref{res-mor}).
At least, this suggests that the features seen in the IRC bands arise partly from H$_2${} emissions.
Secondly, the same observational and theoretical arguments for H$_2${} emission, presented in Paper I, are valid over the IRC N4, S7, S11 bands ($\sim3-13$ $\mu$m): (1) only the H$_2${} emission lines belong to the ``lines of S and H$_2${}($J_{\textrm{\footnotesize{up}}}>2$)'' group can produce spatially similar features \citep{Neufeld(2007)ApJ_664_890}; (2) H$_2${} lines are the dominant emission from shocked molecular gas whose physical parameters are similar to the cloud S1 ($\upsilon_s$$=20$ km s$^{-1}$, $n(\textrm{H}_2)$$=10^4$ cm$^{-3}$; \citealt{Kaufman(1996)ApJ_456_611}).
We also considered other possible sources for the emission, presented in Paper I: fine structure ionic lines, thermal dust continuum, Polycyclic Aromatic Hydrocarbons (PAHs) bands, and synchrotron radiation.
Again, these do not seem likely either, as we discuss below.
\begin{itemize}
\item
Within the wavelength coverage of the IRC N4, S7, S11 bands, there are three strong ionic lines, [Fe {\small II}]{} 5.34 $\mu$m, [Ar {\small II}]{} 6.99 $\mu$m, and [Ne {\small II}]{} 12.8 $\mu$m, which have been observed in the shocked regions of SNRs \citep[e.g.~][]{Arendt(1999)ApJ_521_234,Oliva(1999)A&A_343_943,Reach(2002)ApJ_564_302,Neufeld(2007)ApJ_664_890}.
The ionization potentials of these ions are 7.9 eV (Fe$^{+}$), 15.8 eV (Ar$^{+}$), and 21.6 eV (Ne$^{+}$), respectively.
From their case study on four SNRs, \cite{Neufeld(2007)ApJ_664_890} showed that the ions in shock-cloud interaction regions have \emph{two} distinctive spatial distributions according to their ionization potential, $>13.6$ eV and $<13.6$ eV.
Thus, in principle, the three ionic lines ([Fe {\small II}], [Ar {\small II}], and [Ne {\small II}]) can generate similar features in the IRC N4, S7, and S11 band images.
However, the ions observed in the shock-cloud interaction regions have a low correlation with H$_2${} \citep{Neufeld(2007)ApJ_664_890}; indeed, such low correlations between [Fe {\small II}]{} and H$_2${} have been frequently observed around SNRs \citep[e.g.~][]{Oliva(1999)A&A_343_943,Koo(2007)ApJ_657_308,Lee(2009)ApJ_691_1042}.
For the case of cloud S1, which shows a good correlation between the diffuse IRC and the H$_2$ $\upsilon=1\rightarrow0$ S(1){} features (cf.~section \ref{res-mor}), it therefore seems that ionic lines from [Fe {\small II}], [Ar {\small II}], and [Ne {\small II}]{} are not responsible for the features we measured with AKARI.
\item
Thermal dust continuum is not likely responsible, either.
To produce a correlated features through the IRC N4, S7, and S11 bands, the dust temperature should be higher than 500 K.
In C-shocks, thought to be operating in the cloud S1, the dust temperature is below $\sim$50 K \citep{Draine(1983)ApJ_264_485}.
In addition, to heat up the dust over 500 K, there should be hot gas ($\gtrsim10^6$ K), well traced in X-rays, around the dust.
However, no significant hot gas exists around the cloud S1 \citep{Byun(2006)ApJ_637_283}.
\item
PAHs are another candidate since they are ubiquitous and have strong, broad band features at 3.3, 6.2, 7.7, 8.6, 11.2, 12.7, and 16.4 $\mu$m{} \citep{Tielens(2008)ARA&A_46_289}.
However, in the similar ways presented in Paper I, they are not likely to be the source for the diffuse features seen in the IRC bands.
PAHs are heated slowly and cool fast, thus it is hard to observe the shocked PAH emission above the background PAH emission \citep{Tielens(2008)ARA&A_46_289}; although one case was claimed to detect the shocked PAH emission \citep{Tappe(2006)ApJ_653_267}, hitherto, such emissions have not been observed in shocks \citep{vanDishoeck(2004)ARA&A_42_119,Tielens(2008)ARA&A_46_289}.
\item
Synchrotron radiation is unlikely, as well, since no obvious correlation between the $^{12}$CO $J=2\rightarrow1${} emission and the radio continuum was observed around the Cloud S \citep[section 4.2 of][]{Koo(2001)ApJ_552_175}.
\end{itemize}
\section{Comparison to Shock Models}
To interpret the infrared intensities and colors (Table \ref{tbl-result}), we must model the excitation of the features.
In Paper I, we concluded that shocked H$_2${} gas was the most probable explanation, based on several arguments.
Among these, the clearest was the similarity of the diffuse features seen in the IRC and the WIRC H$_2$ $\upsilon=1\rightarrow0$ S(1){} images.
Hence, here we now analyze the band intensities as emission lines from shocked H$_2${} gas, which we calculate by applying several different shock models.
These models are the same as in Paper I, except for the non-stationary model.
This has been excluded since some bright H$_2${} emission lines which fall into the IRC N4 band are not listed in the published models \citep[see Table 1 of][]{Flower(1999)MNRAS_308_271}.
Further descriptions of all these models are given in Paper I.
\subsection{C-Shock: Isothermal H$_2${} Gas} \label{cshock-iso}
It is known that the shocked H$_2${} gas behind a planar C-type shock can be approximated as an \emph{isothermal} and isobaric slab of gas \citep{Neufeld(2006)ApJ_649_816}, in view of the H$_2${} excitation diagrams predicted for such shocks \citep[e.g.][]{Kaufman(1996)ApJ_456_611,Wilgenbus(2000)A&A_356_1010}.
Hence, we first calculate the expected IRC colors from the emission lines of \emph{isothermal} H$_2${} gas.
Figure \ref{fig-ccd} displays the modeled IRC colors from isothermal H$_2${} gas as open circles ($\circ$).
Their trajectory moves from the lower-left corner to the upper-right corner as the temperature increases (i.e. becomes increasingly 'blue').
This is explained because pure rotational lines of H$_2$, which are dominant below a few 1000 K in the IRC bands, have shorter wavelengths for higher upper-levels.
As $n(\textrm{H}_2)${} increases, the populations are thermalized, approaching Local Thermodynamic Equilibrium (LTE).
As Figure \ref{fig-ccd} shows, isothermal H$_2${} gas can not explain the observed IRC colors with any combination of $n(\textrm{H}_2)${} and temperature.
The ortho-to-para ratio (OPR) was also varied from 0.5 to 5, since the OPR is expected to be different from 3.0 in the interstellar clouds \citep{Dalgarno(1973)ApL_14_77,Flower(1984)MNRAS_209_25,Lacy(1994)ApJ_428_L69} and even in shocked gas \citep{Timmermann(1998)ApJ_498_246,Wilgenbus(2000)A&A_356_1010}.
However, these variations are not able to reproduce the observed IRC colors (Fig.~\ref{fig-opr}).
The expected IRC colors at the \emph{same} temperature vary according to the adopted OPR; however, the \emph{locus} of the IRC colors do not differ much from the OPR=3.0 case, that is shown in Figure~\ref{fig-opr}.
A similar result was already found for the Cloud N, and it is consistent with the H$_2${} level populations displaying an \emph{ankle-like curve} (see Fig.~\ref{fig-pop}).
The critical density of an H$_2${} line transition increases as the energy level of the upper state increases \citep[cf.][]{LeBourlot(1999)MNRAS_305_802}; hence, isothermal H$_2${} gas can only produce either a \emph{straight line} (LTE) or a \emph{knee-like curve} (non-LTE, see Fig.~1 in Paper I) in the population diagram, neither of which are observed.
Therefore, the cloud S1 also has an ankle-like H$_2${} population.
This ankle-like population can be understood by the morphology of the diffuse H$_2$ $\upsilon=1\rightarrow0$ S(1){} features.
These features are filamentary and surround the shocked CO cloud (cf. Fig.~\ref{fig-result} and \ref{fig-rgb}).
Hence, if they are generated by shocks propagating into the cloud S1, a range of shock velocities are expected.
Since the postshock temperature of C-shock is proportional to the shock velocity as $\upsilon_s$$^{1.35}$ \citep{Neufeld(2006)ApJ_649_816}, a moderate difference in $\upsilon_s${} may result in a range of temperature in the shocked H$_2${} gas.
However, this explanation may not be valid, because the same population was also found for the N2front of Cloud N, whose appearance is so planar that little difference in shock speed is expected (Paper I).
\subsection{C-Shock: Power-law Distribution of H$_2${} Gas Temperature} \label{cshock-pow}
Figure \ref{fig-ccd} also displays the IRC colors calculated from the admixture model for H$_2${} gas, as filled circles ($\bullet$).
As can be seen, it can reproduce the observed IRC color ratios with an appropriate combination of $n(\textrm{H}_2)${} and $b$.
The derived parameters are listed in Table \ref{tbl-par}.
$N$(H$_{2}\,; T>100\,\textrm{K})${} is determined by scaling the modeled IRC intensity for the derived $n(\textrm{H}_2)${} and $b$ to meet the observed IRC intensity.
The detail contributions of H$_2${} line emission to IRC bands for these parameters are listed in Table~\ref{tbl-cont}.
The ``Weight'' column of Table~\ref{tbl-cont} lists the weighting factor for each line to the IRC band contribution.
For example, the S11 band intensity can be calculated as follows.
\begin{equation} \label{eq-cont}
\frac{I_{S11}(\textrm{H$_2$})}{\textrm{MJy sr$^{-1}$}}=\frac{0.610\,I[\textrm{H$_2$}\,S(2)]+0.921\,I[\textrm{H$_2$}\,S(3)]}{10^{-4}\,\textrm{erg s$^{-1}$ cm$^{-2}$ sr$^{-1}$}}
\end{equation}
As Table~\ref{tbl-cont} shows, the pure-rotational H$_2${} emission lines are dominant in all IRC bands.
Since the N4 band, in contrast to the L15 band used for Cloud N, is used for the color-color diagram of the S1 cloud (Fig.~\ref{fig-ccd}), the contribution from several higher-level emission lines, S(7)--S(11), was also included, while that from the S(1) 17 $\mu$m{} emission line was not applied (cf.~Table 4 of Paper I).
We here note that the model parameters for N2front derived from the N4/S7 vs. S7/S11 color-color diagram (Fig.~\ref{fig-ccd}), $b\sim4$ and $n(\textrm{H}_2)$$\sim4\times10^4$ cm$^{-3}$, are both \emph{larger} than those from the S7/S11 vs. S11/L15 diagram (Paper I), $b\sim3$ and $n(\textrm{H}_2)$$\sim2\times10^3$ cm$^{-3}$.
We discuss this issue further in section \ref{dis-ccd}.
In addition, since $^{12}$CO $v=1-0${} 4.6 $\mu m$ emission lines have been observed in shocked gas \citep[e.g.][]{Rosenthal(2000)A&A_356_705} and fall into the IRC N4 band coverage, we assessed their contribution to the band, for the power-law thermal admixture model with $b=4.0$ and $5.0$, referring the assessment of \cite{Neufeld(2008)ApJ_678_974}.
We used the CO vibrational energy state of \cite{Balakrishnan(2002)ApJ_568_443} and the CO vibrational transition rate of \cite{Chandra(1996)A&AS_117_557}.
We adopted the collisional rate coefficients for the excitation of CO vibrational transitions by H \citep{Balakrishnan(2002)ApJ_568_443} and by He \citep{Cecchi-Pestellini(2002)ApJ_571_1015}.
For excitation by H$_2$, we adopted the equations [7] and [8] of \cite{Thompson(1973)ApJ_181_1039} with the parameter $A$ of 68, the laboratory measurement of \cite{Millikan(1963)JChPh_39_3209}.
Unlike the excitation of H$_2${} gas, we included H as a collisional partner for the CO vibrational excitation since H excites CO vibrational levels ($\upsilon>0$) \emph{more} efficiently than He and H$_2$.
H excites the H$_2${} pure rotational levels ($\upsilon=0$) \emph{less} efficiently than He and H$_2${} \citep{LeBourlot(1999)MNRAS_305_802}, hence including H as a collisional partner in the excitation of H$_2${} gas makes negligible effects on the predicted IRC band intensity, where the pure rotational lines are dominant (cf. Table \ref{tbl-cont}).
Figure \ref{fig-cov} displays the results.
The fractional abundance of atomic hydrogen to molecular hydrogen, $N$(H {\small I})/$N(\textrm{H}_2)$, was varied as 0, 0.01, 0.1, and 1.0, while that of CO was fixed as $10^{-4}$.
As with the H$_2${} vibrational states, those of CO also have higher collisional coefficients for a collision with atomic hydrogen than with He or H$_2$; hence, Figure \ref{fig-cov} shows a sensitive dependence on the ratio, $N$(H {\small I})/$N(\textrm{H}_2)$.
In the range of $n(\textrm{H}_2)$=$10^3-10^6$ cm$^{-3}$, the contribution to the IRC N4 band is less than 0.1, hence negligible.
Furthermore, a robust simulation expects that $^{12}$CO $v=1-0${} emission lines are much weaker than those of H$_2${} in C-shocks of preshock H$_2${} densities $n(\textrm{H}_2)$$=10^4-10^6$ cm$^{-3}${} and shock velocities $\upsilon_s$=20--40 km s$^{-1}${} \citep{Kaufman(1996)ApJ_456_611}.
The derived parameters, except the power-law index $b$, are a little higher than those previously determined towards several SNRs, where interaction with nearby molecular clouds is occurring.
The density, $n(\textrm{H}_2)$=(3.9$_{-1.2}^{+2.1}$)$\times10^{4}$ cm$^{-3}$, is a few times higher than the value, derived from Large Velocity Gradient (LVG) analysis of CO data for HB 21, of $n(\textrm{H}_2)$$=7.0\times10^3$ cm$^{-3}${} by \cite{Koo(2001)ApJ_552_175}.
The column density we derived, $N$(H$_{2}\,; T>100\,\textrm{K})$=(2.8$_{-0.5}^{+0.2}$)$\times10^{21}$ cm$^{-2}$, is similarly higher than that derived towards shock-cloud interaction regions in four other SNRs (W 44, W 28, 3C 391, and IC 443), $N(\textrm{H}_2)${}$=(2.8-8.9)\times10^{20}$ cm$^{-2}$.
The latter values were determined from a two-temperature LTE fitting of pure-rotational H$_2${} spectra with varying OPRs \citep{Neufeld(2007)ApJ_664_890}.
Our derived $b$-value of 4.2$_{-0.1}^{+0.1}$ falls into the middle of the range, 3.0--6.0, found by \cite{Neufeld(2008)ApJ_678_974}.
These authors found the IRAC color ratios to be well explained with this range of power-law index $b$, analyzing \textit{Spitzer}{} IRAC observations towards the SNR IC 443.
From these three parameters derived---$n(\textrm{H}_2)$, $N$(H$_{2}\,; T>100\,\textrm{K})$, and $b$---we also determined the model prediction for the H$_2$ $\upsilon=1\rightarrow0$ S(1){} intensity, (1.5$_{-0.3}^{+0.5}$)$\times10^{-6}$ erg s$^{-1}$ cm$^{-2}$ sr$^{-1}$.
It is about a factor of four smaller than the observed value (see Table~\ref{tbl-result} and \ref{tbl-par}).
In contrast, for Cloud N the excess was found to be a factor of $17-33$ (Paper I).
We discuss these results further in $\S$\ref{discuss}.
Finally, we visualized the population state of the cloud S1, derived from the IRC color-color diagram, in Figure \ref{fig-pop} (left).
The pure-rotational levels which contribute to the IRC bands are designated with filled circles.
Also, the upper level of H$_2$ $\upsilon=1\rightarrow0$ S(1){} emission line is designated with a filled triangle; its population derived from the observed H$_2$ $\upsilon=1\rightarrow0$ S(1){} intensity, extinction corrected, is designated with a grey filled triangle with an error bar.
The diagram shows a severe deviation of $\upsilon>0$ levels from $\upsilon=0$ level, it thus seems that the two temperature LTE fitting, a model usually applied for shocked H$_2${} gas \citep[e.g.][]{Rho(2001)ApJ_547_885,Giannini(2006)A&A_459_821}, does not properly describe the population state of the cloud S1, even with varying OPR.
This is caused by the low $n(\textrm{H}_2)${} of the cloud S1, $\sim4\times10^4$ cm$^{-3}$, which is much lower than the critical densities for ro-vibrational H$_2${} lines, $\gtrsim10^8$ cm$^{-3}${} \citep{LeBourlot(1999)MNRAS_305_802}.
We also note that this type of $\upsilon>0$ states population may not be easily recognizable with near-infrared \emph{ground} observations, since only the population for a few lowest-$J$ levels of the $\upsilon>0$ states can be deduced due to the atmospheric absorption \citep[e.g.][]{Burton(1989)inproca,Giannini(2006)A&A_459_821}.
Hence, for an exact derivation of the H$_2${} level population, we must cover full range of $\sim2-30$ $\mu$m{} at once as in \cite{Rosenthal(2000)A&A_356_705}, and space observatories are ideal and mandatory in this sense.
\subsection{Partially Dissociative J-shock}
As Figure \ref{fig-other} shows, a partially dissociating jump-shock model does not reproduce the observed color of the cloud S1.
The observed color might appear to lie on a model extension to very high pressure, higher than $P=10^{11}$ cm$^{-3}$ K; however, this is implausible.
From their CO observations, \cite{Koo(2001)ApJ_552_175} derived $n(\textrm{H}_2)$$=7.0\times10^3$ cm$^{-3}${} and $\upsilon_s$$\lesssim20$km s$^{-1}${} for the cloud S1.
These give a postshock pressure of $\sim10^8$ cm$^{-3}$ K, which is more than $10^3$ times lower than the above limit.
A pressure enhancement can occur for the collision between molecular clumps and radiative shells of a remnant \citep[e.g.][]{Moorhouse(1991)MNRAS_253_662,Chevalier(1999)ApJ_511_798}; however, it is only about a factor of 20.
Insufficient cooling time cannot solve the disagreement between the observed and modeled colors, either.
If the postshock H$_2${} had not cooled as low as a few hundred K, then the modeled IRC colors move towards the upper-right direction in the color-color diagram (Fig.~\ref{fig-ccd}) to bluer colors.
This was not observed.
Overall, a partially dissociative J-shock does not seem to be a suitable model to explain the observed IRC colors.
\section{Discussion} \label{discuss}
The observed color ratios were only reproduced by the thermal admixture model, as was the case for Cloud N (Paper I).
Hence, we here discuss the derived parameters from this model, based on pictures for the shock-cloud interactions, as proposed in \cite{Neufeld(2008)ApJ_678_974} and in Paper I.
We also note here that we assumed the OPR=3.0 since no OPR information is available for the cloud S1; hence, the derived parameters can be changed according to the \emph{adopted} OPR value.
\subsection{Nature of Molecular Shocks Seen in the Infrared} \label{dis-nat}
Diffuse infrared features surround the shocked CO cloud S1 (see Fig.~\ref{fig-result} and \ref{fig-rgb}).
For instance, the H$_2$ $\upsilon=1\rightarrow0$ S(1){} image shows several filamentary features, as if excited by shocks propagating into the cloud core.
These may represent distinctive planar shocks, each with different speed.
From the previous observations toward SNR molecular shock regions, the H$_2${} level population diagram has been shown to have an ankle-like curve (cf.~Fig.~\ref{fig-pop}).
For planar C-shocks to explain such populations, it generally requires two components, whose shock velocities are $\sim10$ and $\sim30-50$ km s$^{-1}${}, with comparable amounts of $N(\textrm{H}_2)${} \citep[cf.~][]{Hewitt(2009)ApJ_694_1266}.
Hence, the filamentary features seen in the H$_2$ $\upsilon=1\rightarrow0$ S(1){} image may originate from such a mixture of planar C-shocks.
However, the possibility that the \emph{individual} filamentary feature bears the ankle-like population still remains, since such a property was seen in a very planar filamentary feature of Cloud N (N2front).
\cite{Neufeld(2008)ApJ_678_974} showed that the $b$ values they obtained, $\sim3.0-6.0$, can be explained by paraboloidal bow shocks, which are geometrical summations of planar C-shocks (see Fig.~\ref{fig-pic}).
In their picture, a paraboloidal bow shock, where H$_2${} survives the shock (i.e. T$\lesssim4,000$ K), has a power-law index $b\sim3.8$.
If some slower bow-shocks which do not reach 4000 K are then spatially averaged together, a value for $b$ of $\gtrsim3.8$ is generated.
The value $b$ for the cloud S1 was determined to be 4.2$_{-0.1}^{+0.1}$, which falls into the range derived for bow shocks, $b\gtrsim3.8$.
However, as discussed in Paper I, bow shocks should have been observed in the H$_2$ $\upsilon=1\rightarrow0$ S(1){} image, if any, since the expected shock width for a planar C-shock propagating into preshock gas of $n(\textrm{H}_2)${} $\sim10^4$ cm$^{-3}${} is $\sim10^{16}$ cm \citep{Draine(1983)ApJ_264_485}, comparable to the spatial resolution in the image, $\sim1.1''$ (cf.~$\S$~\ref{obs-wirc}) $\sim(1.3-2.8)\times10^{16}$ cm for the distance of $\sim0.8-1.7$ kpc \citep{Leahy(1987)MNRAS_228_907,Tatematsu(1990)A&A_237_189,Byun(2006)ApJ_637_283}.
This absence of bow shock features can be explained by the viewing angle.
If we consider the circular and filamentary appearance of the diffuse H$_2${} features around the cloud S1, it may be possible that a single paraboloidal bow-shock is being viewed along its symmetry axis, producing the circular feature seen in Figure \ref{fig-rgb}.
In this case, our result fit with the bow shock picture of \cite{Neufeld(2008)ApJ_678_974}, when seen face-on.
This bow shock picture has a difficulty in achieving a steady state for the shock, however.
It assumes a steady state planar C-shock at every point of the bow.
Through the C-type shock, the $n(\textrm{H}_2)${} can be increased up to a factor of ten \citep[e.g.][]{Timmermann(1998)ApJ_498_246,Wilgenbus(2000)A&A_356_1010}.
Thus, $n(\textrm{H}_2)${} at the upstream of the bow would be $\sim$ [$n(\textrm{H}_2)${} at downstream]$\times0.1\sim[4\times10^4]\times0.1\sim4\times10^3$ cm$^{-3}${} (cf.~Table~\ref{tbl-par} and Fig.~\ref{fig-pic}).
Also, the preshock gas velocity into the shock is known to be $\sim20$ km s$^{-1}${} from CO observations \citep{Koo(2001)ApJ_552_175}.
For these preshock density and shock velocity, the time required to achieve a steady shock is known to be $\sim10^4$ yr from the study on the early stage of shock generation \citep{Flower(1999)MNRAS_308_271}.
This time seems to be long for the bow shock around the cloud S1 to be in a steady state, considering the estimated age of HB 21, $\sim5000-7000$ yr \citep{Lazendic(2006)ApJ_647_350,Byun(2006)ApJ_637_283}, together with the location of the cloud S1 near the edge of the remnant (Fig.~\ref{fig-obs}); we here note that the remnant may be older than 5000-7000 yr, estimated at the distance of 0.8 kpc, since the distance is uncertain, $\sim0.8-1.7$ kpc \citep{Leahy(1987)MNRAS_228_907,Tatematsu(1990)A&A_237_189,Byun(2006)ApJ_637_283}.
In Paper I, we conjectured that a shocked clumpy interstellar medium (ISM) exists (cf.~Fig.~\ref{fig-pic}), based on the similar $b$ values of the N2front and N2clump regions and on the cyclodial (cuspy) feature seen in the N2clump region, together with numerical simulations \citep{Nakamura(2006)ApJS_164_477,Shin(2008)ApJ_680_336}.
If this picture also holds for the S1 cloud, the shocked clump must be unresolved in the WIRC H$_2$ $\upsilon=1\rightarrow0$ S(1){} image, since the H$_2${} features around this cloud do not show any cycloidal features.
However, even though this is the case, the absence of the wriggle expected for shock fronts propagating a clumpy ISM \citep[e.g.][]{Patnaude(2005)ApJ_633_240} still remains as an issue (cf.~Fig.~\ref{fig-rgb})---the wriggle is generated by shocks propagating further through a lower density medium, and vice versa.
This wriggle can be unresolved in the H$_2$ $\upsilon=1\rightarrow0$ S(1){} image if its scale is small enough ($\lesssim10^{16}$ cm).
However, it is uncertain whether the cycloidal feature would be maintained under such a small scale.
As noted in section \ref{res-qua}, the N4/S7 vs. S7/S11 colors of the cloud S1 and N2front are similar (cf.~Fig.~\ref{fig-ccd}).
This is intriguing considering that they are physically unrelated.
Their morphologies seen in H$_2$ $\upsilon=1\rightarrow0$ S(1){} images are also similar, i.e.~filamentary, although their sizes show a few factors of difference (cf.~ Paper I and Fig.~\ref{fig-rgb}).
These similarities suggest that the cloud S1 and N2front share similar shock conditions.
The interstellar ultraviolet radiation field may contribute to this similarity; however, a more robust study is required.
\subsection{H$_2$ $\upsilon=1\rightarrow0$ S(1){} intensity} \label{dis-h2s1}
We estimated the H$_2$ $\upsilon=1\rightarrow0$ S(1){} intensity of the cloud S1 for the derived model parameters---$n(\textrm{H}_2)$, $b$, and $N$(H$_{2}\,; T>100\,\textrm{K})$---from the mid-infrared IRC colors.
It is about four times weaker than the observed intensity (Table \ref{tbl-result} and \ref{tbl-par}).
This discrepancy is less severe than the Cloud N case, which shows a factor of 17--33 difference (Paper I).
However, the amount of excited gas, $N(\textrm{H$_2$};v=1,J=3)$, required to compensate for the difference is $\sim10^{14}$ cm$^{-2}${} in both cases (cf.~Fig.~\ref{fig-pop}).
In Paper I, we discussed two possible reasons for the discrepancy.
Firstly, the existence of additional H$_2${} gas, whose temperature and density are both high, but whose column density is low enough to have negligible effect on the mid-infrared line intensities.
For example, to compensate for a deficiency of $N(v=1,J=3)\sim10^{14}$ cm$^{-2}$, we need an additional amount of H$_2${} gas of $N(\textrm{H}_2)${} $\sim10^{16}$ cm$^{-2}${} in LTE with $T\sim2000$ K.
A compact, unresolved shocked cloud is a candidate for such additional H$_2${} gas.
The second explanation given was the omission of collisions with hydrogen atoms, which are effective in exciting the vibrational states of H$_2${} \citep[cf.~][]{Neufeld(2008)ApJ_678_974}.
The cross section for excitation of H$_2${} by H is several orders of magnitude greater for rovibrational transitions than it is for pure rotational transitions \citep[see Table~1 and Figure~1 in][]{LeBourlot(1999)MNRAS_305_802}.
Hence, with only a small fraction of H, $n(\textrm{H})${}/$n(\textrm{H}_2)${}$\sim0.025$, the rovibrational transition can be dominated by collisions with H, rather than with H$_2$, in the temperature range 300--4000 K.
Indeed, such a fraction of atomic gas is expected in interstellar clouds with $n(\textrm{H}_2)${} $\gtrsim10^3$ cm$^{-3}${} \cite[see Table~1 and Figure~1 in][]{Snow(2006)ARA&A_44_367}, as well as in theoretical models for shock waves that are fast enough to produce H$_2${} at temperatures of a few thousand K \citep[e.g.][]{Wilgenbus(2000)A&A_356_1010}.
\subsection{Limitation of the Thermal Admixture Model}
\subsubsection{H$_2${} Column Density $N(\textrm{H}_2)$}\label{dis-col}
In section \ref{cshock-pow}, we mentioned that the column density $N$(H$_{2}\,; T>100\,\textrm{K})${} of the cloud S1 is a few times higher than those of other SNRs (W 44, W 28, 3C 391, and IC 443).
The former is $N$(H$_{2}\,; T>100\,\textrm{K})$=(2.8$_{-0.5}^{+0.2}$)$\times10^{21}$ cm$^{-2}$, while the latter are $N(\textrm{H}_2)${}$=(2.8-8.9)\times10^{20}$ cm$^{-2}${} \citep{Neufeld(2007)ApJ_664_890}.
This seems unreasonable considering that the $n(\textrm{H}_2)${} of the cloud S1, (3.9$_{-1.2}^{+2.1}$)$\times10^{4}$ cm$^{-3}$, is lower than that of IC 443, $\sim10^7$ cm$^{-3}${} \citep{Neufeld(2008)ApJ_678_974}.
However, there is one important point to note before the comparison: those H$_2${} column densities are \emph{expectations} estimated using different methods.
The total column density of H$_2${} gas at a few 100 K is mainly determined by the values of $\upsilon=0$ $J=0,1$ levels, since other levels have much lower populations (cf.~Fig.~\ref{fig-pop}).
However, we cannot obtain the column densities of these levels directly from emission lines, because no transition to a lower state is allowed; therefore, we must estimate the column densities of $J=0,1$ levels from the observed column densities of $J>1$ levels.
The different methods used for this estimation causes the discrepancy between the cloud S1 and IC 443, mentioned above, as we discuss below.
The $N$(H$_{2}\,; T>100\,\textrm{K})${} of the cloud S1, (2.8$_{-0.5}^{+0.2}$)$\times10^{21}$ cm$^{-2}$, is estimated by applying the thermal admixture model to the observed IRC intensities, while the $N(\textrm{H}_2)${} of IC 443, $\sim5.0\times10^{20}$ cm$^{-2}$, was estimated with two temperature LTE fitting with varying the OPR \citep{Neufeld(2007)ApJ_664_890}.
If we estimate the $N(\textrm{H}_2)${} of the cloud S1 using the same two-temperature LTE fitting, we would obtain a lower values.
Figure \ref{fig-pop} (top-right) displays the result of such fitting.
The fitting was applied only to the H$_2${} levels whose emission lines contribute to the IRC bands (cf. Table \ref{tbl-cont}), and returned a column density of $\sim7.3\times10^{19}$ cm$^{-2}$.
This is about seven times less than IC 443, and now does not cause the discrepancy mentioned above.
Also, this column density is about \emph{40 times smaller} than the estimation using the thermal admixture model.
This difference stems from the fact that the thermal admixture model does not estimate the column densities of $J=0,1$ with a linear extrapolation in the population diagram as the LTE fitting does (cf.~Fig.~\ref{fig-pop}); it estimates the column densities with a curved population, defined by the equation $dN\sim T^{-b}dT$ (100 K $\le T \le$ 4000 K).
Such a difference is smaller when longer wavelength IRC bands are used for the column density estimation.
Figure \ref{fig-pop} (bottom-right) shows the results of the two temperature LTE fitting for the population of the N2front, determined using the IRC S7, S11, and L15 bands (Paper I); the result shows a column density of $\sim7.5\times10^{19}$ cm$^{-2}$, which is about \emph{3.3 times smaller} than the estimation using the thermal admixture model, $\sim2.5\times10^{20}$ cm$^{-2}${} (Paper I).
This trend is reasonable since the \emph{longer wavelength} IRC bands determine the level populations of \emph{lower-$J$} states in $\upsilon=0$, which results in a less severe extrapolation for the column densities, $N(J=0,1)$.
\subsubsection{H$_2${} Density $n(\textrm{H}_2)${} and power-law index $b$} \label{dis-ccd}
To compare the N4/S7 and S7/S11 colors of the cloud S1 with those of N2front in Cloud N (Fig.~\ref{fig-ccd}), we further determined the N4 intensity of N2front, which was not measured in Paper I because of strong point source contamination.
In order to remove this contamination, we additionally masked point-source dominated areas seen in the N4 band.
Table \ref{tbl-n2f} shows the measured intensities.
Since some areas are additionally masked unlike Paper I, the intensities of S7, S11, and L15 are a little different from those listed in Paper I; however, the colors S7/S11 and S11/L15 are almost unchanged (cf.~Fig.~7 in Paper I and Fig.~\ref{fig-n2f}).
Therefore, we think the point-source masking was done properly.
Figure \ref{fig-n2f} shows that the model parameters, $b$ and $n(\textrm{H}_2)$, obtained for N2front depends on which IRC bands are used for the color-color diagram.
The diagram of shorter wavelength bands (N4, S7, S11; Fig.~\ref{fig-n2f} leftmost) returns \emph{larger-b} and \emph{larger-$n(\textrm{H}_2)$} values than the diagram of longer wavelength bands (S7, S11, L15; Fig.~\ref{fig-n2f} rightmost), and the diagram of N4/S7 vs. S11/L15 (Fig.~\ref{fig-n2f} middle) returns the middle values of the former two diagrams'.
This inconsistency may be caused by the intrinsic property of shocked H$_2${} gas.
In other words, the whole level population of shocked H$_2${} gas may not be fully described by the power-law thermal admixture model with \emph{only one set} of $b$ and $n(\textrm{H}_2)$.
To check this possibility, in Figure \ref{fig-n2f}, we overplotted the IRC colors of Orion Molecular Cloud-1 (OMC-1), where extensive emission lines of shocked H$_2${} gas were observed over $2.5-30$ $\mu$m{} \citep{Rosenthal(2000)A&A_356_705}.
The OMC-1 emissions were adjusted to experience the same extinction with HB 21, to be placed in the model grid for N2front.
Interestingly, OMC-1 also shows the same trend for the model parameters, $b$ and $n(\textrm{H}_2)$, as the case for N2front.
This suggests that the variations of $b$ and $n(\textrm{H}_2)${} are needed for the thermal admixture model to describe the whole level population of shocked H$_2${} gas.
No significant variation of $b$ was seen in the SNR IC 443, where the thermal admixture model was applied first \citep{Neufeld(2008)ApJ_678_974}.
It may be caused by the narrow wavelength coverage of the bands they used (\textit{Spitzer}{} IRAC; $\sim3-8$ $\mu$m), which missed the longer wavelength information we used.
We here note that the model parameters---$b$ and $n(\textrm{H}_2)$---must be obtained from the \emph{same} band images for their comparison between different shocked regions, since the parameters are likely dependent on the wavelength.
\section{Conclusion}
We have observed a shock-cloud interaction region in the SNR HB 21 at near- and mid-infrared wavelengths, with the WIRC at the Palomar telescope and the IRC aboard the \textit{AKARI}{} satellite.
The IRC N4, S7, and S11 band images and the WIRC H$_2$ $\upsilon=1\rightarrow0$ S(1){} image reveal similar diffuse features, which surround the shocked CO cloud S1.
However, there are no infrared diffuse features seen around another shocked CO cloud S2.
Lack of shocked H$_2${} gas may cause this absence, but why it happens only for the cloud S2 is uncertain.
We found that the IRC colors of the cloud S1 are well explained by an admixture model of H$_2${} gas temperatures, whose infinitesimal column density varies as $dN\sim T^{-b}dT$.
Three physical parameters---$n(\textrm{H}_2)$, $b$, and $N$(H$_{2}\,; T>100\,\textrm{K})$---were derived from this thermal admixture model (cf.~Table~\ref{tbl-par}).
These can be understood with multiple planar C-shocks whose velocities are different.
Alternatively, the derived $b$ value ($\sim4.2$) can be understood through a bow shock picture, if we are looking at a single bow shock along the symmetry axis.
However, this picture has a difficulty in achieving a steady state.
A shocked clumpy ISM picture, conjectured in Paper I, remains as a possible explanation, but the absence of the wriggle, expected for shock fronts propagating a clumpy medium, in the filamentary features seen in H$_2$ $\upsilon=1\rightarrow0$ S(1){} image remains as an issue.
The model parameters, $b$ and $n(\textrm{H}_2)$, obtained for the cloud S1 and N2front (cf.~Fig.~\ref{fig-ccd}) are very similar, which means that these clouds share similar shock conditions.
We also compared the observed H$_2$ $\upsilon=1\rightarrow0$ S(1){} intensity to that predicted from the power-law admixture model.
It is about four times greater.
This excess might be caused by either an additional component of hot, dense H$_2${} gas (which has low total column density), or by the omission of collisions with hydrogen atoms in the power-law admixture model (which results in an under-prediction of the near-IR line intensity).
The limitation of the thermal admixture model is explored with respect to the derived model parameters.
The $N(\textrm{H}_2)${} estimation of the model shows a smaller difference with those of two temperature LTE fitting, when longer wavelength IRC bands are used for the determination of model parameters.
Investigating the infrared colors of N2front and OMC-1 in the four IRC bands (N4, S7, S11, and L15), we found that the thermal admixture model cannot describe the whole H$_2${} level population with only one set of $b$ and $n(\textrm{H}_2)$; the shorter wavelength bands returns higher-$b$ and higher-$n(\textrm{H}_2)$.
This tells we must use the same bands in determining the model parameters, for the comparisons of the shocked H$_2${} gas' properties.
\section*{Acknowledgments}
This work is based on observations with \textit{AKARI}, a JAXA project with the participation of ESA.
The authors thank all the members of the \textit{AKARI}{} project.
Also, the authors thank the referee for all the comments which make this paper clearer.
This work was supported by the Korea Science and Engineering Foundation (R01-2007-000-20336-0) and also through the KOSEF-NSERC Cooperative Program (F01-2007-000-10048-0).
This research has made use of SAOImage DS9, developed by Smithsonian Astrophysical Observatory \citep{Joye(2003)inproc}.
\bibliographystyle{G:/Work/Publication/bibtex/astronat/asr/elsart-harv}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Mobile PCI Express Module (MXM) ist eine Verbindungstechnologie für GPUs in Notebooks, basierend auf PCIe. Sie besteht aus einem Sockel, auf den passende MXM-Grafikmodule gesteckt werden. Dies erlaubt, im Gegensatz zu den sonst üblichen Lötverbindungen, ein Austausch des Grafikchips eines Notebooks.
Ursprünglich von Nvidia im Jahr 2004 vorgestellt, soll mit MXM ein industrieweiter, nicht-proprietärer Standard geschaffen werden. Über die Zeit wurde der Standard mehrmals revidiert, aktuell ist Version 3.1.
Technik
MXM 1 und 2
Die Karten werden in verschiedene Gruppen (nach Baugröße und Wärmeentwicklung) unterteilt:
55,3 mm Bohrungsabstand für Kühllösung (diagonal)
MXM-I (max. 18 W; 70 × 68 mm; 35 × 35 mm GPU)
MXM-II (max. 25 W; 73 × 78 mm; 35 × 35 mm GPU)
MXM-III (82 × 100 mm; 35 × 35 mm GPU)
MXM-HE (82 × 100 mm; 35 × 35 mm GPU)
60,5 mm Bohrungsabstand für Kühllösung (diagonal)
MXM-III (max. 35 W; 82 × 100 mm; 40 × 40 mm GPU)
MXM-HE (82 × 100 mm; 40 × 40 mm GPU)
Prinzipiell sind kleinere Module (mit derselben GPU-Größe) aufwärtskompatibel. Das bedeutet, ein MXM-I kann in ein für MXM-HE spezifiziertes Notebook verbaut werden – nicht aber ein MXM-III in ein MXM-II, da die entsprechende Abwärme bzw. Verlustleistung durch die vorhandene Kühllösung des Notebooks abgeführt werden muss.
Prinzipiell wurde der offene Standard (oft falsch als proprietäre Lösung bezeichnet) geschaffen, um Notebook-Herstellern ein schnelleres Umsteigen auf neue Grafikprozessoren zu ermöglichen.
MXM 3
Mit MXM 3 wurde der Aufbau der Module überarbeitet. Auch hier wird zwischen mehreren Größen unterschieden.
Typ-A (max. 55 W; 82 × 70 mm)
Typ-B (max. 200 W; 102 × 70 mm)
Module vom Typ-A sind dabei mechanisch kompatibel zum Typ-B-Sockel.
MXM 3.1 wurde 2012 vorgestellt. Nennenswert ist hier vor allem die hinzugekommene Unterstützung von PCIe 3.0.
Verbreitung
Nvidia legt viel Wert auf die Offenlegung der Informationen zu MXM. Es existieren auch MXM-Karten mit ATI-Grafikchips.
Anfang 2006 war die Verbreitung entsprechender Notebooks oder Grafiklösungen noch gering. Ende 2008 finden sich entsprechende Geräte aber in den Produktpaletten namhafter Hersteller, darunter Fujitsu-Siemens, Toshiba, Asus und Acer. Als erster Desktop-Computer verfügte der Apple iMac mit 24" Bildschirmdiagonale über einen MXM-Steckplatz, wobei die dort eingesetzte Karte jedoch nicht mit gängigen MXM-Karten tauschbar ist.
Bei Geräten mit MXM-Steckplatz besteht des Weiteren oftmals das Problem, dass kompatible Grafikkarten nur bei US-amerikanischen oder asiatischen Händlern ohne Funktionsgarantie bestellbar sind.
Weiter Verwendung
Qseven-Module nutzen den gleichen Steckverbinder wie MXM – SMARC-Module nutzen den gleichen Steckverbinder wie MXM 3. Diese beide Computer-on-Module sind nur mechanisch, jedoch nicht elektrisch zu MXM kompatibel.
Siehe auch
AXIOM (Advanced eXpress I/O Module)
Weblinks
MXM-Seite auf nVidia.com
Notebooks mit MXM-Modulen
Grafikkarte
Peripheriebus (intern)
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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package org.cyclops.integrateddynamics.core.evaluate.variable;
import net.minecraft.ChatFormatting;
import org.cyclops.cyclopscore.helper.Helpers;
import org.cyclops.integrateddynamics.api.evaluate.variable.IValue;
/**
* Wildcard value type
* @author rubensworks
*/
public class ValueTypeCategoryAny extends ValueTypeCategoryBase<IValue> {
public ValueTypeCategoryAny() {
super("any", Helpers.RGBToInt(240, 240, 240), ChatFormatting.RESET, IValue.class);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,418
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BH council allocates millions for recovery
Hotels, including the Beverly Wilshire, will benefit from the $3.4 million allocated to the Beverly Hills Conference and Visitors Bureau. (photo by Brica Wilcox/courtesy of the Rodeo Drive Committee)
Beverly Hills plans to continue to make big investments in tourism and hospitality to help restore the city's tax base, which was hampered by the pandemic.
On May 13, the City Council unanimously supported three separate funding requests related to helping the city's businesses and attracting new companies.
One of the contracts, which provides nearly $489,000 to the Beverly Hills Chamber of Commerce, was unanimously approved as part of the council's consent calendar. The other two contracts, awarding more than $158,000 to the Rodeo Drive Committee and more than $3.4 million to the Beverly Hills Conference and Visitors Bureau, were met with support from the council members, though they will return for formal adoption next month.
Laura Biery, the city's marketing and economic sustainability manager, said the chamber's contract was approved on May 13 "so they can get started right away."
"The way I view it is that this is an investment in the continued success of the city and that, in tough times like we've been in, it's important to continue to make that investment," Mayor Robert Wunderlich said.
Wunderlich noted that the recovery period has begun, but more work needs to be done.
"If you walk the streets, you can see that we've emerged [from the pandemic] strongly, but we have to stay focused on that," Wunderlich said.
Some of that work will be done on the internet by the Rodeo Drive Committee, which is working with the Los Angeles branch of the multinational public relations and marketing agency Bold. RDC President-elect Kathy Gohari and representatives of Bold advised the council that they plan to focus on five social media platforms: Instagram – the primary channel – plus Pinterest, Facebook, TikTok and Clubhouse.
Juliane Kringe, vice president of Bold, recommended "focusing on these five channels and doing it right as opposed to spreading it out" to other platforms like Twitter, Snapchat and YouTube.
"With all of these platforms, we can definitely cover all of the different generations and all of the different target groups," Kringe said.
The council members were unanimous in their support for the plan, which Councilman Lester Friedman noted "will certainly help our brand" as a city.
"I think the content that has been created has been great, and you have unanimous support from us. Keep up the good work," Wunderlich said.
The council members were similarly effusive in their praise for the BHCVB and Beverly Hills Chamber of Commerce, which laid out plans for a hopeful return to in-person events, which will bring back shoppers and hotel guests, aiding an industry that was especially hard-hit by the pandemic.
"I feel very strongly that we have the best CVB, the best team, out of anywhere – bar none – in the United States," Councilwoman Lili Bosse said.
Several groups aim to bring shoppers back to Rodeo Drive stores and other Beverly Hills businesses. (photo by Brica Wilcox/courtesy of the Rodeo Drive Committee)
Blair Schlecter, vice president of economic development and government affairs for the Beverly Hills Chamber of Commerce, said the immediate priority is to help local businesses get on the path to reclaiming their pre-pandemic success.
"Our plan for this next fiscal year is to ensure a quicker business recovery, which results in revenue for the community and the city, which helps support and enhance our already wonderful city services," said
The business groups said they hope to bring back conferences to the city, as well as bring city leaders back to New York City and on other trips to attract new businesses to the city.
"We think there's a lot of opportunities for business attraction," Schlecter said.
Beyond the next year, Councilman John Mirisch said it's important to consider the city's place as a destination for a significant time to come.
"This is about positioning us for the long term as a place where people want to go," Mirisch said.
One concern, however, was raised by Councilman Julian Gold, who pushed multiple times for metrics or other measures of success from the groups. Gold and Wunderlich supported the BHCVB, RDC and Beverly Hills Chamber of Commerce in finding a way to measure return on investment.
"I'm just questioning if there's something we can do that would be a more objective look at our performance," Gold said.
Despite his request for more information, Gold was clear that he joins his colleagues in strongly supporting the business groups and appreciates the work they do for the city.
"The chamber is really ground zero for our businesses and their return to health," Gold said.
Mixed-use project on La Brea survives appeals
Wilson selected as new WeHo city manager
Mary Wells elected as new BHUSD board president December 15, 2021
Rodeo Drive Committee elects members to board of directors October 28, 2021
Wunderlich optimistic about city's future October 14, 2021
COVID-19 mandate further complicates decisions in BH July 21, 2021
Gohari again to lead Rodeo Drive Committee July 1, 2021
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction
We start by introducing some notation relating to the order of certain elements modulo $n$. For every positive integer $n$ and every prime number $p$, we denote by $v_p(n)$ the \textit{$p$-adic valuation} of $n$, i.e. the greatest exponent $e\geq0$ for which $p^e$ divides $n$. The prime factorization of $n$ may then be written as
$$
n=\prod_{p\in\mathbb{P}}p^{v_p(n)},
$$
where $\mathbb{P}$ denotes the set of all prime numbers. We denote by $\rad(n)$ \textit{the radical of $n$}, i.e. the largest square-free divisor of $n$, namely
$$
\rad(n) = \prod_{\stackrel{p\in\mathbb{P}}{p|n}}p.
$$
\par For every positive integer $n$ and every integer $a$ coprime to $n$, we denote by $\mathcal{O}_n(a)$ \textit{the multiplicative order of $a$ modulo $n$}, i.e. the smallest positive integer $e$ such that $a^e\equiv1\pmod{n}$, namely
$$
\mathcal{O}_n(a) = \min\left\{e\in\mathbb{N}^*\ \middle|\ a^e\equiv1\pmod{n}\right\}
$$
and we denote by $\mathcal{R}_n(a)$ the \textit{multiplicative remainder} of $a$ modulo $n$, i.e. the multiple of $n$ defined by
$$
\mathcal{R}_n(a) = a^{\mathcal{O}_n(a)}-1.
$$
The multiplicative order of $a$ modulo $n$ also corresponds with the order of the element $\pi_n(a)$, where $\pi_n : \mathbb{Z} \relbar\joinrel\twoheadrightarrow \Z/n\Z$ is the canonical surjective morphism, in the multiplicative group ${\left(\Z/n\Z\right)}^*$, the group of units of $\Z/n\Z$. Note that $\mathcal{O}_n(a)$ divides $\varphi(n)$, $\varphi$ being the Euler's totient function.
\par For every positive integer $n$, we define and denote by $\alpha_n$ the arithmetic function
$$
\alpha_n :
\begin{array}[t]{ccl}
\mathbb{Z} & \longrightarrow & \mathbb{N}\\
a & \longmapsto & \left\{
\begin{array}{cl}
\mathcal{O}_n\left(a^n\right) & \text{for\ all}\ \gcd(a,n)=1,\\
0 & \text{otherwise},
\end{array}\right.
\end{array}
$$
where $\gcd(a,n)$ denotes the greatest common divisor of $a$ and $n$, with the convention that $\gcd(0,n)=n$. Observe that, for every $a$ coprime to $n$, the integer $\alpha_n(a)$ divides $\varphi(n)/\gcd(\varphi(n),n)$. This follows from the previous remark on $\mathcal{O}_n(a)$ and the equality $\alpha_n(a)=\mathcal{O}_n(a^n)=\mathcal{O}_n(a)/\gcd(\mathcal{O}_n(a),n)$.
\par For every positive integer $n$ and every integer $a$ coprime to $n$, we denote by $\mathcal{PO}_n(a)$ \textit{the projective multiplicative order of $a$ modulo $n$}, i.e. the smallest positive integer $e$ such that $a^e\equiv\pm1\pmod{n}$, namely
$$
\mathcal{PO}_n(a) = \min\left\{e\in\mathbb{N}^*\ \middle|\ a^e\equiv\pm1\pmod{n}\right\}.
$$
The projective multiplicative order of $a$ modulo $n$ also corresponds with the order of the element $\overline{\pi_n(a)}$ in the multiplicative quotient group ${\left(\Z/n\Z\right)}^*/\{-1,1\}$.
\par For every positive integer $n$, we define and denote by $\beta_n$ the arithmetic function
$$
\beta_n :
\begin{array}[t]{ccl}
\mathbb{Z} & \longrightarrow & \mathbb{N}\\
a & \longmapsto & \left\{
\begin{array}{cl}
\mathcal{PO}_n\left(a^n\right) & \text{for\ all}\ \gcd(a,n)=1,\\
0 & \text{otherwise}.
\end{array}\right.
\end{array}
$$
Observe that we have the alternative $\alpha_n=\beta_n$ or $\alpha_n=2\beta_n$.
\par In this paper, we study in detail these two arithmetic functions. In particular, we prove that, for every positive integers $n_1$ and $n_2$ such that
$$
\left\{
\begin{array}{lcl}
\rad(n_1)|n_2|n_1 & \text{if} & v_2(n_1)\leq1,\\
2\rad(n_1)|n_2|n_1& \text{if} & v_2(n_1)\geq2,
\end{array}
\right.
$$
the integer $\alpha_{n_1}(a)$ (respectively $\beta_{n_1}(a)$) divides $\alpha_{n_2}(a)$ (resp. $\beta_{n_2}(a)$), for every integer $a$. More precisely, we determine the exact relationship between the functions $\alpha_{n_1}$ and $\alpha_{n_2}$ and between $\beta_{n_1}$ and $\beta_{n_2}$. We prove that we have
$$
\alpha_{n_1}(a) = \frac{\alpha_{n_2}(a)}{\gcd\left(\alpha_{n_2}(a),\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right)}\ \text{for\ all}\ \gcd(a,n_1)=1
$$
in Theorem~\ref{thm21} of Section 2 and that we have
$$
\beta_{n_1}(a) = \frac{\beta_{n_2}(a)}{\gcd\left(\beta_{n_2}(a),\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right)}\ \text{for\ all}\ \gcd(a,n_1)=1
$$
in Theorem~\ref{thm31} of Section 3. Thus, for every integer $a$ coprime to $n$, the determination of $\alpha_n(a)$ is reduced to the computation of $\alpha_{\rad(n)}(a)$ and $\mathcal{R}_{\rad(n)}(a)$ if $v_2(n)\leq1$ and of $\alpha_{2\rad(n)}(a)$ and $\mathcal{R}_{2\rad(n)}(a)$ if $v_2(n)\geq2$. These theorems on the functions $\alpha_n$ and $\beta_n$ are derived from Theorem~\ref{thm2} of Section~2, which states that
$$
\mathcal{O}_{n_1}(a) = \mathcal{O}_{n_2}(a)\frac{n_1}{\gcd(n_1,\mathcal{R}_{n_2}(a))},
$$
for all integers $a$ coprime to $n_1$ and $n_2$. This result generalizes the following theorem of Nathanson which, in the above notation, states that for every odd prime number $p$ and for every positive integer $k$, we have the equality
$$
\mathcal{O}_{p^k}(a)=\mathcal{O}_p(a)\frac{p^k}{\gcd(p^k,\mathcal{R}_p(a))}
$$
for all integers $a$ not divisible by $p$.
\begin{nathanson}
Let $p$ be an odd prime, and let $a\neq\pm1$ be an integer not divisible by $p$. Let $d$ be the order of $a$ modulo $p$. Let $k_0$ be the largest integer such that $a^d\equiv1\pmod{p^{k_0}}$. Then the order of $a$ modulo $p^k$ is $d$ for $k=1,\ldots,k_0$ and $dp^{k-k_0}$ for $k\geq k_0$.
\end{nathanson}
For every finite sequence $S=\left(a_1,\ldots,a_m\right)$ of length $m\geq1$ in $\Z/n\Z$, we denote by $\Delta S$ the \textit{Steinhaus triangle} of $S$, that is the finite multiset of cardinality $\binom{m+1}{2}$ in $\Z/n\Z$ defined by
$$
\Delta S = \left\{ \sum_{k=0}^{i}{\binom{i}{k}a_{j+k}}\ \middle|\ 0\leq i\leq m-1\ ,\ 1\leq j\leq m-i \right\}.
$$
A finite sequence $S$ in $\Z/n\Z$ is said to be \textit{balanced} if each element of $\Z/n\Z$ occurs in its Steinhaus triangle $\Delta S$ with the same multiplicity. For instance, the sequence $\left(2,2,3,3\right)$ of length $4$ is balanced in $\mathbb{Z}/5\mathbb{Z}$. Indeed, as depicted in Figure~\ref{fig2}, its Steinhaus triangle is composed by each element of $\mathbb{Z}/5\mathbb{Z}$ occuring twice.
\begin{figure}[!h]
\centering
\begin{pspicture}(3,1.9485)
\pspolygon(0.375,1.9485)(2.625,1.9485)(1.5,0)
\rput(1.5,0.433){$0$}
\rput(1.25,0.866){$4$}
\rput(1.75,0.866){$1$}
\rput(1,1.299){$4$}
\rput(1.5,1.299){$0$}
\rput(2,1.299){$1$}
\rput(0.75,1.732){$2$}
\rput(1.25,1.732){$2$}
\rput(1.75,1.732){$3$}
\rput(2.25,1.732){$3$}
\end{pspicture}
\caption{\label{fig2}The Steinhaus triangle of a balanced sequence in $\mathbb{Z}/5\mathbb{Z}$}
\end{figure}
Note that, for a sequence $S$ of length $m\geq1$ in $\Z/n\Z$, a necessary condition on $m$ for $S$ to be balanced is that the integer $n$ divides the binomial coefficient $\binom{m+1}{2}$. In 1976, John C. Molluzzo \cite{Molluzzo1978} posed the problem to determine whether this necessary condition on $m$ is also sufficient to guarantee the existence of a balanced sequence. In \cite{Chappelon}, it was proved that, for each odd number $n$, \textit{there exists a balanced sequence of length $m$ for every $m\equiv0$ or $-1\pmod{\alpha_n(2)n}$ and for every $m\equiv0$ or $-1\pmod{\beta_n(2)n}$}. This was achieved by analyzing the Steinhaus triangles generated by arithmetic progressions. In particular, since $\beta_{3^k}(2)=1$ for all $k\geq1$, the above result implies a complete and positive solution of Molluzzo's Problem in $\Z/n\Z$ for all $n=3^k$.
\section{The arithmetic function $\alpha_n$
The table and the graphic depicted in Figure~\ref{fig3} give us the first values of $\alpha_n(a)$ for every positive integer $n$, $1\leq n\leq 20$, and for every integer $a$, $-20\leq a\leq 20$.
\begin{figure}[!p]
\centering
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n\backslash a$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ & $11$ & $12$ & $13$ & $14$ & $15$ & $16$ & $17$ & $18$ & $19$ & $20$\\
\hline
\hline
$1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$\\
\hline
$2$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$\\
\hline
$3$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$\\
\hline
$4$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$\\
\hline
$5$ & $1$ & $4$ & $4$ & $2$ & $0$ & $1$ & $4$ & $4$ & $2$ & $0$ & $1$ & $4$ & $4$ & $2$ & $0$ & $1$ & $4$ & $4$ & $2$ & $0$\\
\hline
$6$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$\\
\hline
$7$ & $1$ & $3$ & $6$ & $3$ & $6$ & $2$ & $0$ & $1$ & $3$ & $6$ & $3$ & $6$ & $2$ & $0$ & $1$ & $3$ & $6$ & $3$ & $6$ & $2$\\
\hline
$8$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$\\
\hline
$9$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$ & $0$ & $1$ & $2$\\
\hline
$10$ & $1$ & $0$ & $2$ & $0$ & $0$ & $0$ & $2$ & $0$ & $1$ & $0$ & $1$ & $0$ & $2$ & $0$ & $0$ & $0$ & $2$ & $0$ & $1$ & $0$\\
\hline
$11$ & $1$ & $10$ & $5$ & $5$ & $5$ & $10$ & $10$ & $10$ & $5$ & $2$ & $0$ & $1$ & $10$ & $5$ & $5$ & $5$ & $10$ & $10$ & $10$ & $5$\\
\hline
$12$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$\\
\hline
$13$ & $1$ & $12$ & $3$ & $6$ & $4$ & $12$ & $12$ & $4$ & $3$ & $6$ & $12$ & $2$ & $0$ & $1$ & $12$ & $3$ & $6$ & $4$ & $12$ & $12$\\
\hline
$14$ & $1$ & $0$ & $3$ & $0$ & $3$ & $0$ & $0$ & $0$ & $3$ & $0$ & $3$ & $0$ & $1$ & $0$ & $1$ & $0$ & $3$ & $0$ & $3$ & $0$\\
\hline
$15$ & $1$ & $4$ & $0$ & $2$ & $0$ & $0$ & $4$ & $4$ & $0$ & $0$ & $2$ & $0$ & $4$ & $2$ & $0$ & $1$ & $4$ & $0$ & $2$ & $0$\\
\hline
$16$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$\\
\hline
$17$ & $1$ & $8$ & $16$ & $4$ & $16$ & $16$ & $16$ & $8$ & $8$ & $16$ & $16$ & $16$ & $4$ & $16$ & $8$ & $2$ & $0$ & $1$ & $8$ & $16$\\
\hline
$18$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$\\
\hline
$19$ & $1$ & $18$ & $18$ & $9$ & $9$ & $9$ & $3$ & $6$ & $9$ & $18$ & $3$ & $6$ & $18$ & $18$ & $18$ & $9$ & $9$ & $2$ & $0$ & $1$\\
\hline
$20$ & $1$ & $0$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $1$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ & $0$\\
\hline
\end{tabular}\\[4ex]
\includegraphics[height=8cm]{alpha}
\caption{The first values of $\alpha_n(a)$}
\label{fig3}
\end{figure}
The positive integer $\alpha_n(a)$ seems to be difficult to determine. Indeed, there is no general formula known to compute the multiplicative order of an integer modulo $n$ but, however, we get the following helpful propositions.
\begin{lem}\label{lem11}
Let $n_1$ and $n_2$ be two positive integers such that $\rad(n_1)=\rad(n_2)$. Then, an integer $a$ is coprime to $n_1$ if, and only if, it is also coprime to $n_2$.
\end{lem}
\begin{proof}
This follows from the definition of the greatest common divisor of two integers and from the definition of the radical of an integer.
\end{proof}
\begin{prop}\label{prop2}
Let $n_1$ and $n_2$ be two positive integers such that $\rad(n_1)|n_2|n_1$. Then, for every integer $a$, the integer $\alpha_{n_1}(a)$ divides $\alpha_{n_2}(a)$.
\end{prop}
\begin{proof}
If $a$ is not coprime to $n_1$ and $n_2$, then, by definition of the functions $\alpha_{n_1}$ and $\alpha_{n_2}$ and by Lemma~\ref{lem11}, we have
$$
\alpha_{n_1}(a)=\alpha_{n_2}(a)=0.
$$
Suppose now that $a$ is coprime to $n_1$ and $n_2$. Let $p$ be a prime factor of $n_1$ such that $v_p(n_1)\geq2$. We shall show that $\alpha_{n_1}(a)$ divides $\alpha_{n_1/p}(a)$. By definition of $\alpha_{n_1/p}(a)$, there exists an integer $u$ such that
$$
a^{\alpha_{n_1/p}(a)\frac{n_1}{p}} = 1+u\frac{n_1}{p}.
$$
Therefore, by the binomial theorem, we have
$$
a^{\alpha_{n_1/p}(a)n_1} = {\left(a^{\alpha_{n_1/p}(a)\frac{n_1}{p}}\right)}^{p} = {\left(1+u\frac{n_1}{p}\right)}^{p} = 1+ un_1 + \sum_{k=2}^{p}{\binom{p}{k}u^k\left(\frac{n_1}{p}\right)^k}.
$$
Since $v_p(n_1)\geq2$, it follows that ${\left(n_1/p\right)}^k$ is divisible by $n_1$ for every integer $k\geq2$ and so
$$
a^{\alpha_{n_1/p}(a)n_1} \equiv 1 \pmod{n_1}.
$$
Hence $\alpha_{n_1}(a)$ divides $\alpha_{n_1/p}(a)$. This completes the proof.
\end{proof}
An exact relationship between $\alpha_{n_1}(a)$ and $\alpha_{n_2}(a)$, for every integer $a$ coprime to $n_1$ and $n_2$, is determined at the end of this section. We first settle the easy prime power case.
\begin{prop}\label{prop3}
Let $p$ be a prime number and $a$ be an integer. Then we have
$$
\alpha_{p^k}(a) = \mathcal{O}_p(a)
$$
for every positive integer $k$.
\end{prop}
\begin{proof}
Let $k$ be a positive integer. If $a$ is not coprime to $p$, then we have $\alpha_{p^k}(a)=\alpha_{p}(a)=0$. Suppose now that $a$ is coprime to $p$. By Proposition~\ref{prop2}, the integer $\alpha_{p^k}(a)$ divides $\alpha_{p}(a)$. It remains to prove that $\alpha_{p}(a)$ divides $\alpha_{p^k}(a)$. The congruence
$$
a^{\alpha_{p^k}(a)p^k} \equiv 1 \pmod{p^k}
$$
implies that
$$
a^{\alpha_{p^k}(a)p^k} \equiv 1 \pmod{p},
$$
and hence, by Fermat's little theorem, it follows that
$$
a^{\alpha_{p^k}(a)p} \equiv a^{\alpha_{p^k}(a)p^k} \equiv 1 \pmod{p}.
$$
Therefore $\alpha_{p}(a)$ divides $\alpha_{p^k}(a)$. Finally, we have
$$
\alpha_{p^k}(a) = \alpha_{p}(a) = \mathcal{O}_{p}(a^p) = \mathcal{O}_{p}(a).
$$
This completes the proof.
\end{proof}
\begin{rem}
If $p=2$, then, for every positive integer $k$, we obtain
$$
\alpha_{2^k}(a) = \mathcal{O}_{2}(a) = \left\{
\begin{array}{l}
0\ \text{for}\ a\ \text{even},\\
1\ \text{for}\ a\ \text{odd}.
\end{array}
\right.
$$
\end{rem}
\begin{prop}
Let $n_1$ and $n_2$ be two coprime numbers and $a$ be an integer. Then $\alpha_{n_1n_2}(a)$ divides $\lcm(\alpha_{n_1}(a),\alpha_{n_2}(a))$, the least common multiple of $\alpha_{n_1}(a)$ and $\alpha_{n_2}(a)$.
\end{prop}
\begin{proof}
If $\gcd(a,n_1n_2)\neq1$, then $\gcd(a,n_1)\neq1$ or $\gcd(a,n_2)\neq1$ and so
$$
\alpha_{n_1n_2}(a) = \lcm(\alpha_{n_1}(a),\alpha_{n_2}(a)) = 0.
$$
Suppose now that $\gcd(a,n_1n_2)=1$ and hence that the integers $a$, $n_1$ and $n_2$ are coprime pairwise. Let $i\in\{1,2\}$. The congruences
$$
a^{\alpha_{n_i}(a)n_i}\equiv1\pmod{n_i}
$$
imply that
$$
a^{\lcm(\alpha_{n_1}(a),\alpha_{n_2}(a))n_1n_2}\equiv1\pmod{n_i}.
$$
Therefore $\alpha_{n_1n_2}(a)$ divides $\lcm(\alpha_{n_1}(a),\alpha_{n_2}(a))$ by the Chinese remainder theorem.
\end{proof}
Let $n_1$ and $n_2$ be two positive integers such that
$$
\left\{
\begin{array}{lcl}
\rad(n_1)|n_2|n_1 & \text{if} & v_2(n_1)\leq1,\\
2\rad(n_1)|n_2|n_1& \text{if} & v_2(n_1)\geq2.
\end{array}
\right.
$$
By definition, we know that $\alpha_{n_1}(a)=\alpha_{n_2}(a)=0$ for every integer $a$ not coprime to $n_1$ and $n_2$. We end this section by determining the exact relationship between $\alpha_{n_1}(a)$ and $\alpha_{n_2}(a)$ for every integer $a$ coprime to $n_1$ and $n_2$.
\begin{thm}\label{thm21}
Let $n_1$ and $n_2$ be two positive integers such that
$$
\left\{
\begin{array}{lcl}
\rad(n_1)|n_2|n_1 & \text{if} & v_2(n_1)\leq1,\\
2\rad(n_1)|n_2|n_1& \text{if} & v_2(n_1)\geq2.
\end{array}
\right.
$$
Then, for every integer $a$ coprime to $n_1$ and $n_2$, we have
$$
\alpha_{n_1}(a) = \frac{\alpha_{n_2}(a)}{\gcd\left(\alpha_{n_2}(a),\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right)}
$$
\end{thm}
This result is a corollary of the following theorem.
\begin{thm}\label{thm2}
Let $n_1$ and $n_2$ be two positive integers such that
$$
\left\{
\begin{array}{lcl}
\rad(n_1)|n_2|n_1 & \text{if} & v_2(n_1)\leq1,\\
2\rad(n_1)|n_2|n_1& \text{if} & v_2(n_1)\geq2.
\end{array}
\right.
$$
Then, for every integer $a$ coprime to $n_1$ and $n_2$, we have
$$
\mathcal{O}_{n_1}(a) = \mathcal{O}_{n_2}(a)\frac{n_1}{\gcd(n_1,\mathcal{R}_{n_2}(a))}.
$$
\end{thm}
The proof of this theorem is based on the following lemma.
\begin{lem}\label{lem1}
Let $n$ be a positive integer and let $a$ be an integer coprime to $n$. Let $m$ be an integer such that $\rad(m)|\rad{n}$. Then, there exists an integer $u_m$, coprime to $n$ if $m$ is odd, or coprime to $n/2$ if $m$ is even, such that
$$
a^{\mathcal{O}_n(a)m} = 1 + u_m\mathcal{R}_n(a)m.
$$
\end{lem}
\begin{proof}
We distinguish different cases upon the parity of $m$. First, we prove the odd case by induction on $m$. If $m=1$, then, by definition of the integer $\mathcal{R}_n(a)$, we have
$$
a^{\mathcal{O}_n(a)} = 1 + \mathcal{R}_n(a).
$$
Therefore the assertion is true for $m=1$. Now, let $p$ be a prime factor of $m$ and suppose that the assertion is true for the odd number $m/p$, i.e. there exists an integer $u_{m/p}$, coprime to $n$, such that
$$
a^{\mathcal{O}_{n}(a)\frac{m}{p}} = 1 + u_{m/p}\mathcal{R}_n(a)\frac{m}{p}.
$$
Then, we obtain
$$
a^{\mathcal{O}_n(a)m}
\begin{array}[t]{l}
= \displaystyle\left(a^{\mathcal{O}_n(a)\frac{m}{p}}\right)^p = \left( 1 + u_{m/p}\mathcal{R}_n(a)\frac{m}{p} \right)^p\\[2ex]
= \displaystyle1 + u_{m/p}\mathcal{R}_n(a)m + \sum_{k=2}^{p-1}{\binom{p}{k}\left(u_{m/p}\mathcal{R}_n(a)\frac{m}{p}\right)^k} + \left(u_{m/p}\mathcal{R}_n(a)\frac{m}{p}\right)^p\\[2ex]
= \displaystyle 1 + \left( u_{m/p} + \sum_{k=2}^{p-1}{\frac{\binom{p}{k}}{p}{u_{m/p}}^{k}{\mathcal{R}_n(a)}^{k-1}{\left(\frac{m}{p}\right)}^{k-1}} + {u_{m/p}}^p\frac{{\mathcal{R}_n(a)}^{p-1}}{p}{\left(\frac{m}{p}\right)}^{p-1} \right)\mathcal{R}_n(a)m\\[2ex]
= \displaystyle 1 + u_m\mathcal{R}_n(a)m.
\end{array}
$$
Since $n$ divides $\mathcal{R}_n(a)$ which divides
$$
u_m-u_{m/p} = \sum_{k=2}^{p-1}{\frac{\binom{p}{k}}{p}{u_{m/p}}^{k}{\mathcal{R}_n(a)}^{k-1}{\left(\frac{m}{p}\right)}^{k-1}} + {u_{m/p}}^p\frac{{\mathcal{R}_n(a)}^{p-1}}{p}{\left(\frac{m}{p}\right)}^{p-1},
$$
it follows that $\gcd(u_m,n)=\gcd(u_{m/p},n)=1$. This completes the proof for the odd case. Suppose now that $n$ and $m$ are even. We proceed by induction on $v_2(m)$. Suppose that there exists an integer $u_{m/2}$, coprime to $n/2$, such that
$$
a^{\mathcal{O}_n(a)\frac{m}{2}} = 1 + u_{m/2}\mathcal{R}_n(a)\frac{m}{2}.
$$
We know that this statement is true for $v_2(m)=1$. Indeed, if $v_2(m)=1$, then $m/2$ is odd, this implying that there exists such an integer $u_{m/2}$, coprime to $n$ thus to $n/2$. Then, we obtain
$$
a^{\mathcal{O}_n(a)m}
\begin{array}[t]{l}
= \displaystyle \left(a^{\mathcal{O}_n(a)\frac{m}{2}}\right)^2 = \left( 1 + u_{m/2}\mathcal{R}_n(a)\frac{m}{2} \right)^2 = \displaystyle 1 + u_{m/2}\mathcal{R}_n(a)m + \left(u_{m/2}\mathcal{R}_n(a)\frac{m}{2}\right)^2\\[2ex]
= \displaystyle 1 + \left( u_{m/2} + {u_{m/2}}^2\frac{\mathcal{R}_n(a)}{2}\frac{m}{2} \right)\mathcal{R}_n(a)m = 1 + u_m\mathcal{R}_n(a)m.
\end{array}
$$
Since $n/2$ divides $\mathcal{R}_n(a)/2$ which divides $u_m-u_{m/2}$, it follows that $\gcd(u_m,n/2)=\gcd(u_{m/2},n/2)=1$. This completes the proof.
\end{proof}
We are now ready to prove Theorem~\ref{thm2}.
\begin{proof}[Proof of Theorem~\ref{thm2}]
The proof is by induction on the integer $n_1/n_2$. If $n_1=n_2$, then we have
$$
\frac{n_1}{\gcd(n_1,\mathcal{R}_{n_2}(a))} = \frac{n_1}{\gcd(n_1,\mathcal{R}_{n_1}(a))} = \frac{n_1}{n_1} = 1,
$$
since $\mathcal{R}_{n_1}(a)$ is divisible by $n_1$, and thus the statement is true. Let $p$ be a prime factor of $n_1$ and $n_2$ such that $n_2$ divides $n_1/p$ and suppose that
$$
\mathcal{O}_{n_1/p}(a) = \mathcal{O}_{n_2}(a)\frac{n_1/p}{\gcd(n_1/p,\mathcal{R}_{n_2}(a))}.
$$
First, the congruence
$$
a^{\mathcal{O}_{n_1}(a)} \equiv 1 \pmod{n_1}
$$
implies that
$$
a^{\mathcal{O}_{n_1}(a)} \equiv 1 \pmod{\frac{n_1}{p}}
$$
and so $\mathcal{O}_{n_1/p}(a)$ divides $\mathcal{O}_{n_1}(a)$. Now, we distinguish different cases upon $v_p(n_1)$.\\[2ex]
\textbf{1st case:} $v_p(n_1)\leq v_p\left(\mathcal{R}_{n_2}(a)\right)$.\\
Since $n_2$ divides $n_1/p$, it follows that $\mathcal{O}_{n_2}(a)$ divides $\mathcal{O}_{n_1/p}(a)$. Therefore $\mathcal{R}_{n_1/p}(a)$ is divisible by $\mathcal{R}_{n_2}(a)$ because we have
$$
\mathcal{R}_{n_1/p}(a)
\begin{array}[t]{l}
= \displaystyle a^{\mathcal{O}_{n_1/p}(a)}-1 = a^{\mathcal{O}_{n_2}(a)\frac{\mathcal{O}_{n_1/p}(a)}{\mathcal{O}_{n_2}(a)}}-1 = \left( a^{\mathcal{O}_{n_2}(a)}-1 \right) \sum_{k=0}^{\mathcal{O}_{n_2}(a)-1}{a^{\frac{k\mathcal{O}_{n_1/p}(a)}{\mathcal{O}_{n_2}(a)}}}\\
= \displaystyle\mathcal{R}_{n_2}(a)\sum_{k=0}^{\mathcal{O}_{n_2}(a)-1}{a^{\frac{k\mathcal{O}_{n_1/p}(a)}{\mathcal{O}_{n_2}(a)}}}.
\end{array}
$$
This leads to
$$
v_{p}(n_1)\leq v_p\left(\mathcal{R}_{n_2}(a)\right)\leq v_p\left(\mathcal{R}_{n_1/p}(a)\right).
$$
Therefore $\mathcal{R}_{n_1/p}(a)$ is divisible by $n_1$ and hence we have
$$
a^{\mathcal{O}_{n_1/p}(a)} = 1 + \mathcal{R}_{n_1/p}(a) \equiv 1 \pmod{n_1}.
$$
This implies that $\mathcal{O}_{n_1}(a)=\mathcal{O}_{n_1/p}(a)$. Moreover, the hypothesis $v_p(n_1)\leq v_p\left(\mathcal{R}_{n_2}(a)\right)$ implies that $\gcd(n_1/p,\mathcal{R}_{n_2}(a))=\gcd(n_1,\mathcal{R}_{n_2}(a))/p$. Finally, we obtain
$$
\mathcal{O}_{n_1}(a) = \mathcal{O}_{n_1/p}(a) = \mathcal{O}_{n_2}(a)\frac{n_1/p}{\gcd(n_1/p,\mathcal{R}_{n_2}(a))} = \mathcal{O}_{n_2}(a)\frac{n_1}{\gcd(n_1,\mathcal{R}_{n_2}(a))}.
$$\ \\
\textbf{2nd case:} $v_p(n_1)>v_p\left(\mathcal{R}_{n_2}(a)\right)$.\\
If $v_2(n_1)\leq1$, then $(n_1/p)/\gcd(n_1/p,\mathcal{R}_{n_2}(a))$ is odd. Otherwise, if $v_2(n_1)\geq2$, then $v_{2}(n_2)\geq2$ and every integer coprime to $n_2/2$ is also coprime to $n_2$. In both cases, $v_2(n_1)\leq 1$ or $v_2(n_1)\geq2$, we know, by Lemma~\ref{lem1}, that there exists an integer $u$, coprime to $n_2$, such that
$$
a^{\mathcal{O}_{n_1/p}(a)} = a^{\mathcal{O}_{n_2}(a)\frac{n_1/p}{\gcd(n_1/p,\mathcal{R}_{n_2}(a))}} = 1 + u\mathcal{R}_{n_2}(a)\frac{n_1/p}{\gcd(n_1/p,\mathcal{R}_{n_2}(a))} = 1 + u\frac{\mathcal{R}_{n_2}(a)}{\gcd(n_1/p,\mathcal{R}_{n_2}(a))}\frac{n_1}{p}.
$$
As $v_p\left(\mathcal{R}_{n_2}(a)\right)\leq v_p\left(n_1/p\right)$, it follows that $\mathcal{R}_{n_2}(a)/\gcd(n_1/p,\mathcal{R}_{n_2}(a))$ is coprime to $p$, and hence $\mathcal{O}_{n_1/p}(a)$ is a proper divisor of $\mathcal{O}_{n_1}(a)$ since
$$
a^{\mathcal{O}_{n_1/p}(a)} \not\equiv 1 \pmod{n_1}.
$$
Moreover, by Lemma~\ref{lem1} again, there exists an integer $u_p$ such that
$$
a^{\mathcal{O}_{n_1/p}(a)p} = 1 + u_p\mathcal{R}_{n_1/p}(a)p \equiv 1 \pmod{n_1}.
$$
This leads to
$$
\mathcal{O}_{n_1}(a) = \mathcal{O}_{n_1/p}(a)p = \mathcal{O}_{n_2}(a)\frac{n_1}{\gcd(n_1/p,\mathcal{R}_{n_2}(a))} = \mathcal{O}_{n_2}(a)\frac{n_1}{\gcd(n_1,\mathcal{R}_{n_2}(a))}.
$$
This completes the proof of Theorem~\ref{thm2}.
\end{proof}
We may view Theorem~\ref{thm2} as a generalization of Theorem~$3.6$ of \cite{Nathanson2000}, where $n_2=p$ is an odd prime number and $n_1=p^k$ for some positive integer $k$. Note that the conclusion of Theorem~\ref{thm2} fails in general in the case where $v_2(n_1)\geq2$ and $n_2=\rad(n_1)$. For instance, for $n_1=24=3\cdot2^3$, $n_2=6=3\cdot2$ and $a=7$, we obtain that $\mathcal{O}_{n_1}(a)=2$ while $\mathcal{O}_{n_2}(a)n_1/\gcd(n_1,\mathcal{R}_{n_2}(a))=24/\gcd(24,6)=4$.
We now turn to the proof of the main result of this paper.
\begin{proof}[Proof of Theorem~\ref{thm21}]
From Theorem~\ref{thm2}, we obtain
$$
\alpha_{n_1}(a) = \mathcal{O}_{n_1}(a^{n_1}) = \frac{\mathcal{O}_{n_1}(a)}{\gcd(\mathcal{O}_{n_1}(a),n_1)} = \frac{\mathcal{O}_{n_2}(a)\frac{n_1}{\gcd(n_1,\mathcal{R}_{n_2}(a))}}{\gcd\left(\mathcal{O}_{n_2}(a)\frac{n_1}{\gcd(n_1,\mathcal{R}_{n_2}(a))},n_1\right)} = \frac{\mathcal{O}_{n_2}(a)}{\gcd(\mathcal{O}_{n_2}(a),n_1,\mathcal{R}_{n_2}(a))}.
$$
Thus,
$$
\frac{\alpha_{n_2}(a)}{\alpha_{n_1}(a)}
\begin{array}[t]{l}
= \displaystyle\frac{\frac{\mathcal{O}_{n_2}(a)}{\gcd(\mathcal{O}_{n_2}(a),n_2)}}{\frac{\mathcal{O}_{n_2}(a)}{\gcd(\mathcal{O}_{n_2}(a),n_1,\mathcal{R}_{n_2}(a))}} = \frac{\gcd(\mathcal{O}_{n_2}(a),n_1,\mathcal{R}_{n_2}(a))}{\gcd(\mathcal{O}_{n_2}(a),n_2)}\\[4ex]
= \displaystyle\gcd\left(\frac{\mathcal{O}_{n_2}(a)}{\gcd(\mathcal{O}_{n_2}(a),n_2)},\frac{n_2}{\gcd(\mathcal{O}_{n_2}(a),n_2)}\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right).
\end{array}
$$
Finally, since we have
$$
\gcd\left(\frac{\mathcal{O}_{n_2}(a)}{\gcd(\mathcal{O}_{n_2}(a),n_2)},\frac{n_2}{\gcd(\mathcal{O}_{n_2}(a),n_2)}\right) = \frac{\gcd(\mathcal{O}_{n_2}(a),n_2)}{\gcd(\mathcal{O}_{n_2}(a),n_2)} = 1,
$$
it follows that
$$
\frac{\alpha_{n_2}(a)}{\alpha_{n_1}(a)} = \gcd\left(\frac{\mathcal{O}_{n_2}(a)}{\gcd(\mathcal{O}_{n_2}(a),n_2)},\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right) = \gcd\left(\alpha_{n_2}(a),\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right).
$$
\end{proof}
Thus, the determination of $\alpha_n$ is reduced to the case where $n$ is square-free.
\begin{cor}
Let $n$ be a positive integer such that $v_2(n)\leq 1$. Then, for every integer $a$, coprime to $n$, we have
$$
\alpha_n(a) = \frac{\alpha_{\rad(n)}(a)}{\gcd\left(\alpha_{\rad(n)}(a),\frac{\gcd(n,\mathcal{R}_{\rad(n)}(a))}{\rad(n)}\right)}.
$$
\end{cor}
\begin{cor}
Let $n$ be a positive integer such that $v_2(n)\geq 2$. Then, for every integer $a$, coprime to $n$, we have
$$
\alpha_n(a) = \frac{\alpha_{2\rad(n)}(a)}{\gcd\left(\alpha_{2\rad(n)}(a),\frac{\gcd(n,\mathcal{R}_{2\rad(n)}(a))}{2\rad(n)}\right)}.
$$
\end{cor}
\section{The arithmetic function $\beta_n$
First, we can observe that, by definition of the functions $\alpha_n$ and $\beta_n$, we have
$$
\alpha_n(a) = \beta_n(a) = 0
$$
for every integer $a$ not coprime to $n$ and
$$
\frac{\alpha_n(a)}{\beta_n(a)} \in \left\{1,2\right\}
$$
for every integer $a$ coprime to $n$. There is no general formula known to compute $\alpha_n(a)/\beta_n(a)$ but, however, we get the following proposition.
\begin{prop}\label{prop4}
Let $n_1$ and $n_2$ be two positive integers such that $\rad(n_1)=\rad(n_2)$. Let $a$ be an integer coprime to $n_1$ and $n_2$. If $v_{2}(n_1)\leq 1$, then we have
$$
\frac{\alpha_{n_1}(a)}{\beta_{n_1}(a)} = \frac{\alpha_{n_2}(a)}{\beta_{n_2}(a)}.
$$
If $v_2(n_1)\geq2$, then we have
$$
\alpha_{n_1}(a) = \beta_{n_1}(a).
$$
\end{prop}
\begin{proof}
Let $n_1$ be a positive integer such that $v_2(n_1)\leq 1$ and $a$ be an integer coprime to $n_1$. Let $p$ be an odd prime factor of $n_1$ such that $v_{p}(n_1)\geq2$. We will prove that
$$
\frac{\alpha_{n_1}(a)}{\beta_{n_1}(a)} = \frac{\alpha_{n_1/p}(a)}{\beta_{n_1/p}(a)}.
$$
If $\alpha_{n_1}(a) = 2\beta_{n_1}(a)$, then
$$
{a}^{\beta_{n_1}(a)n_1} \equiv -1 \pmod{n_1}
$$
and thus
$$
{a}^{\beta_{n_1}(a)p\frac{n_1}{p}} \equiv -1 \pmod{\frac{n_1}{p}}.
$$
This implies that $\alpha_{n_1/p}(a) = 2\beta_{n_1/p}(a)$. Conversely, if $\alpha_{n_1/p}(a) = 2\beta_{n_1/p}(a)$, then we have
$$
a^{\beta_{n_1/p}(a)\frac{n_1}{p}} \equiv -1 \pmod{\frac{n_1}{p}}.
$$
Since $v_p(n_1)\geq2$, it follows that
$$
a^{\beta_{n_1/p}(a)\frac{n_1}{p}} \equiv -1 \pmod{p}
$$
and thus
$$
{a}^{\beta_{n_1/p}(a)n_1} + 1 = 1 - {\left( - {a}^{\beta_{n_1/p}(a)\frac{n_1}{p}} \right)}^p = \left( 1 + {a}^{\beta_{n_1/p}(a)\frac{n_1}{p}} \right) \sum_{k=0}^{p-1}{{\left( - {a}^{\beta_{n_1/p}(a)\frac{n_1}{p}} \right)}^k} \equiv 0 \pmod{n_1}.
$$
This implies that $\alpha_{n_1}(a) = 2\beta_{n_1}(a)$. Therefore, by induction on $n_1/n_2$, we obtain that
$$
\frac{\alpha_{n_1}(a)}{\beta_{n_1}(a)} = \frac{\alpha_{n_2}(a)}{\beta_{n_2}(a)},
$$
for every integer $n_2$ such that $\rad(n_1)=\rad(n_2)$. Now, let $n_1$ be a positive integer such that $v_2(n_1)\geq2$ and $a$ be a non-zero integer. Suppose that we have $\alpha_{n_1}(a)=2\beta_{n_1}(a)$. Since
$$
{a}^{\beta_{n_1}(a)n_1} \equiv -1 \pmod{n_1}
$$
it follows that
$$
{\left({a}^{\beta_{n_1}(a)\frac{n_1}{4}}\right)}^{4} \equiv -1 \pmod{4}
$$
in contradiction with
$$
{\left({a}^{\beta_{n_1}(a)\frac{n_1}{4}}\right)}^{4} \equiv 1 \pmod{4}.
$$
\end{proof}
If $n$ is a prime power, then $\beta_n=\beta_{\rad(n)}$, in analogy with Proposition~\ref{prop3} for $\alpha_n$.
\begin{prop}
Let $p$ be a prime number and $a$ be an integer. Then we have
$$
\beta_{p^k}(a) = \beta_p(a)
$$
for every positive integer $k$.
\end{prop}
\begin{proof}
This result is trivial for every integer $a$ not coprime to $n$. Suppose now that $a$ is coprime to $n$. For $p=2$, then, by Proposition~\ref{prop4}, we have
$$
\beta_{2^k}(a) = \alpha_{2^k}(a) = 1
$$
for every positive integer $k$. For an odd prime number $p\geq3$, Proposition~\ref{prop4} and Proposition~\ref{prop3} lead to
$$
\beta_{p^{k}}(a) = \frac{\alpha_{p^k}(a)}{\alpha_p(a)}\beta_p(a) = \beta_p(a)
$$
for every positive integer $k$. This completes the proof.
\end{proof}
Let $n_1$ and $n_2$ be two positive integers such that
$$
\left\{
\begin{array}{lcl}
\rad(n_1)|n_2|n_1 & \text{if} & v_2(n_1)\leq1,\\
2\rad(n_1)|n_2|n_1& \text{if} & v_2(n_1)\geq2.
\end{array}
\right.
$$
It immediately follows that $\beta_{n_1}(a)=\beta_{n_2}(a)=0$ for every integer $a$ not coprime to $n_1$ and $n_2$. Finally, we determine the relationship between $\beta_{n_1}(a)$ and $\beta_{n_2}(a)$ for every integer $a$ coprime to $n_1$ and $n_2$.
\begin{thm}\label{thm31}
Let $n_1$ and $n_2$ be two positive integers such that
$$
\left\{
\begin{array}{lcl}
\rad(n_1)|n_2|n_1 & \text{if} & v_2(n_1)\leq1,\\
2\rad(n_1)|n_2|n_1& \text{if} & v_2(n_1)\geq2.
\end{array}
\right.
$$
Let $a$ be an integer coprime to $n_1$ and $n_2$. Then, we have
$$
\beta_{n_1}(a) = \frac{\beta_{n_2}(a)}{\gcd\left(\beta_{n_2}(a),\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right)}.
$$
\end{thm}
\begin{proof}
If $v_2(n_1)\leq 1$, then Theorem~\ref{thm21} and Proposition~\ref{prop4} lead to
$$
\frac{\beta_{n_2}(a)}{\beta_{n_1}(a)} = \frac{\alpha_{n_2}(a)}{\alpha_{n_1}(a)} = \gcd\left(\alpha_{n_2}(a),\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right).
$$
Since $v_2(n_2)=v_2(n_1)\leq1$, it follows that $\gcd(n_1,\mathcal{R}_{n_2}(a))/n_2$ is odd and hence, we have
$$
\frac{\beta_{n_2}(a)}{\beta_{n_1}(a)} = \gcd\left(\alpha_{n_2}(a),\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right) = \gcd\left(\beta_{n_2}(a),\frac{\gcd(n_1,\mathcal{R}_{n_2}(a))}{n_2}\right).
$$
If $v_{2}(n_1)\geq2$, then $\beta_{n_1}(a)=\alpha_{n_1}(a)$ and $\beta_{n_2}(a)=\alpha_{n_2}(a)$ by Proposition~\ref{prop4} and the result follows from Theorem~\ref{thm21}.
\end{proof}
Thus, as for $\alpha_n$, the determination of $\beta_n$ is reduced to the case where $n$ is square-free.
\begin{cor}
Let $n$ be a positive integer such that $v_2(n)\leq 1$. Then, for every integer $a$, coprime to $n$, we have
$$
\beta_n(a) = \frac{\beta_{\rad(n)}(a)}{\gcd\left(\beta_{\rad(n)}(a),\frac{\gcd(n,\mathcal{R}_{\rad(n)}(a))}{\rad(n)}\right)}.
$$
\end{cor}
\begin{cor}
Let $n$ be a positive integer such that $v_2(n)\geq 2$. Then, for every integer $a$, coprime to $n$, we have
$$
\beta_n(a) = \frac{\beta_{2\rad(n)}(a)}{\gcd\left(\beta_{2\rad(n)}(a),\frac{\gcd(n,\mathcal{R}_{2\rad(n)}(a))}{2\rad(n)}\right)}.
$$
\end{cor}
\section*{Acknowledgments
The author would like to thank Shalom Eliahou for his help in preparing this paper.
\nocite{*}
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,748
|
Copiapoa Britton & Rose – rodzaj sukulentów z rodziny kaktusowatych występujący w Chile. Rodzaj liczy 24 gatunki
. Gatunkiem typowym jest Copiapoa marginata (Salm-Dyck) Britton & Rose.
Systematyka
Pozycja systematyczna według APweb (aktualizowany system APG III z 2009)
Należy do rodziny kaktusowatych (Cactaceae) Juss., która jest jednym z kladów w obrębie rzędu goździkowców (Caryophyllales) i klasy roślin okrytonasiennych. W obrębie kaktusowatych należy do plemienia Notocacteae, podrodziny Cactoideae.
Pozycja w systemie Reveala (1993-1999)
Gromada okrytonasienne (Magnoliophyta Cronquist), podgromada Magnoliophytina Frohne & U. Jensen ex Reveal, klasa Rosopsida Batsch, podklasa goździkowe (Caryophyllidae Takht.), nadrząd Caryophyllanae Takht., rząd goździkowce (Caryophyllales Perleb), podrząd Cactineae Bessey in C.K. Adams, rodzina kaktusowate (Cactaceae Juss.), rodzaj Copiapoa Britton & Rose.
Gatunki
Copiapoa ahremephiana N.P.Taylor & G.J.Charles
Copiapoa alticostata F.Ritter
Copiapoa angustiflora Helmut Walter, G.J.Charles & Mächler
Copiapoa aphanes Mächler & Helmut Walter
Copiapoa calderana F.Ritter
Copiapoa cinerascens (Salm-Dyck) Britton & Rose
Copiapoa cinerea (Phil.) Britton & Rose
Copiapoa coquimbana (Karw. ex Rüpler) Britton & Rose
Copiapoa dealbata F.Ritter
Copiapoa decorticans N.P.Taylor & G.J.Charles
Copiapoa echinoides (Lem. ex Salm-Dyck) Britton & Rose
Copiapoa fiedleriana (K.Schum.) Backeb.
Copiapoa humilis (Phil.) Hutchison
Copiapoa hypogaea F.Ritter
Copiapoa krainziana F.Ritter
Copiapoa leonensis I.Schaub & Keim
Copiapoa longistaminea F.Ritter
Copiapoa marginata (Salm-Dyck) Britton & Rose
Copiapoa megarhiza Britton & Rose
Copiapoa montana F.Ritter
Copiapoa pendulina F.Ritter
Copiapoa rupestris F.Ritter
Copiapoa × scopa Doweld
Copiapoa serpentisulcata F.Ritter
Copiapoa solaris (F.Ritter) F.Ritter
Przypisy
Kaktusowate
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,130
|
Danika Wright does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.
Newly released analysis from Roy Morgan reaffirms that it is the lowest-income households that face the highest mortgage stress. And, contrary to what many might expect, the worst stress is not in Sydney and Melbourne, where property prices have hit record highs.
The Roy Morgan report estimates that 18.4% of Australian households are experiencing mortgage stress, a situation where over one-third of their income goes towards servicing a home loan.
Mortgage stress can lead to many complex social issues. It is considered one of the underlying causes of the Global Financial Crisis.
For many households affected by mortgage stress, defaulting is the last resort. Yet, as the mortgage-servicing pressure increases, so does mortgage risk.
Mortgage risk, the chance of a borrower defaulting, has increased to 83.2% for households earning under $60,000 per year. It is, however, virtually non-existent for households earning more than $150,000.
The Roy Morgan report highlights the importance of income, more so than house prices and borrowing costs, to mortgage stress. In fact, interest rates would need to more than double to match the impact of a loss of income on housing stress.
The previous peak in mortgage stress was in 2008-09, a period of high interest rates and bubble-like price growth in Sydney and Melbourne.
This time around, record low interest rates appear on the surface to be counter-balancing the default rate. Yet this is tied to a stagnation in income levels.
House prices and income levels moved in step until 2013. While house prices have continued to increase, household income levels have flattened since then, when the cash rate dropped to a historic low of 2.75%. The cash rate is now even lower at 1.50%, with further cuts forecast.
The troubling prediction from this is that mortgage stress among Australian households is set to remain high, despite the current low interest rates.
As home ownership concentrates among wealthier households, this report also shows that higher-income households are more resilient to increases in interest rates. This means, too, that home ownership increasingly requires a dual income.
The owner-occupied home is often referred to as the largest single asset that most households own. In countries like Australia and the USA, home ownership is promoted by government and linked to many aspects of future wellbeing. However, as recent HILDA analysis shows, owner-occupied households are becoming far less common.
The "Great Australian Dream" is expected to apply to only a minority of households next decade. With those in the already most marginalised parts of society most affected by mortgage stress, a change in the structures that incentivise home ownership is required to minimise the growing inequality gap.
The limitation with national averages is that pockets of pain are brushed over. The report drills into state-by-state analysis and metro vs regional comparisons.
While the largest mortgages across the country, averaging over $300,000, are in Sydney, mortgage stress is highest in Tasmania and South Australia.
Mortgage stress in Tasmania and South Australia sits well above the national average, as do their unemployment figures, 6.5% and 6.9% respectively, against a national average of 5.7%. Households in regional areas are also facing more acute mortgage stress than their city counterparts.
Regulators aren't taking any chances. With nearly $1 trillion in outstanding mortgage debt, double the pre-GFC level, the 2014 Financial Systems Inquiry identified that mortgages are now a significant systemic risk. In a recent speech on the prudential regulator's outlook, APRA general manager Heidi Richards stated that "the housing market now underpins our financial sector".
APRA has been tightening the lending standards of the big banks. Effective from July 1, the big banks have been required to apply higher "risk weightings" to residential mortgages. These determine the amount of capital held against assets to limit the likelihood of insolvency.
The silver lining to this otherwise depressing analysis is that the risks to financial stability are relatively low. Home ownership concentrated amongst wealthier households actually means there is a high degree of aggregate resilience to changes in future interest rates and incomes.
However, the report's focus is on current incomes. To brace for a true housing market downturn, the key will be monitoring employment and income statistics – unemployment rates as well as hours.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 965
|
Q: Android- how to decode html string with images to textview i want to know how to decode an html string containg images and text to a textview?
Example string:
String s="<p dir="ltr"><img src="ic_7.png"><img src="ic_4.png">byebye<img src="ic_13.png"><img src="ic_8.png">lol<img src="ic_9.png">asd<img src="ic_2.png"></p>"
I was using :
tv.setText(Html.fromHtml(s),TextView.BufferType.SPANNABLE);
But it doesn't work.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,131
|
Northport a step closer to information kiosk
The Village of Northport is located in the Town of Huntington on Suffolk County's North Shore. It was settled in 1656 and incorporated in 1894, the first area in Huntington to do so. (April 21, 2011) Credit: T.C. McCarthy
By MACKENZIE ISSLERmackenzie.issler@newsday.com September 2, 2013
The Village of Northport is a step closer to receiving county funding for an information kiosk that it wants to construct downtown, officials said.
The Suffolk County Downtown Revitalization Citizen's Advisory Board has approved the village's $15,500 application for grant funding, and now that project, along with several others, will go the county legislature for approval beginning in October, according to a statement from county Legis. William Spencer's office.
The plan is to construct an 8-foot by 7-foot kiosk building and two kiosk boards in Northport Park, near the entrance by the parking lot. They will provide information about the businesses and services available in the village, officials said.
"It is also a friendly, personal way of welcoming people, and that is the real spirit of Northport," Deputy Mayor Henry Tobin said.
Tobin said the Northport Chamber of Commerce has discussed putting up an information booth or kiosk for about 12 years.
Spencer (D-Centerport) said in the statement that the goal of the county's downtown revitalization program is to "create positive economic impacts on our communities through sales tax increases and job creation."
"Making Northport Village an even more attractive and vibrant place to visit, shop and enjoy family and friends is a smart way to not only improve our quality of life, but also to improve the economy of the county as a whole," Spencer said.
The county funds are awarded on a competitive grant basis using a merit-based scoring system. Applicants receive points for leveraging of additional funds, economic impact, reasonable expectation of completion, overall downtown improvement and proximity to downtown.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 8,696
|
{"url":"https:\/\/documen.tv\/question\/aqueous-hydrochloric-acid-hcl-will-react-with-solid-sodium-hydroide-naoh-to-produce-aqueous-sodi-24039794-83\/","text":"## Aqueous hydrochloric acid (HCL) will react with solid sodium hydroxide (NaOH) to produce aqueous sodium chloride (NaCl) and liquid water (H2\n\nQuestion\n\nAqueous hydrochloric acid (HCL) will react with solid sodium hydroxide (NaOH) to produce aqueous sodium chloride (NaCl) and liquid water (H2O). Suppose 4.0 g of hydrochloric acid is mixed with 7.96 g of sodium hydroxide. Calculate the maximum mass of sodium chloride that could be produced by the chemical reaction. Be sure your answer has the correct number of significant digits.\n\nin progress 0\n6 months 2021-07-20T21:53:13+00:00 1 Answers 3 views 0","date":"2022-07-03 14:07:42","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\": 0.8529565334320068, \"perplexity\": 5230.39351543987}, \"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-2022-27\/segments\/1656104244535.68\/warc\/CC-MAIN-20220703134535-20220703164535-00736.warc.gz\"}"}
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Home » Editor's Choice » Egyptian Actress Questioned Over Revealing Dress at Gala
Posted on December 5, 2018 December 6, 2018 by crimeandmoreworld - Copy Editing Desk
Egyptian Actress Questioned Over Revealing Dress at Gala
Rania Youssef Photo Courtesy: Twitter/ Raniah Youssef
It's a dress that has shaken Egypt and the uproar continues — prosecutors on Wednesday questioned actress Rania Youssef for at least four hours on accusations of public obscenity over a revealing dress she wore to a cinema gala last week, her lawyer said.
Youssef was allowed to go free after the questioning, pending the completion of the investigation, said the lawyer, Shaban Said.
But he added that she still faces trial on Jan. 12, a date set by court, and could face up to five years in prison, if convicted.
The initial complaint against the 45-year-old Youssef was filed by a group of lawyers with a reputation for moral vigilantism but they said they withdrew their complaint Tuesday.
The lawyers, Wahid al-Kilani, Hamido Jameel al-Prince, Amr Abdel Salam and Samir Sabry, said they decided to forego legal action after Youssef made a public statement.
Though she stopped short of an outright apology, Youssef said she did not mean to offend anyone with her long black dress, its see-through skirt revealing her legs in their entirety. She wore the dress last Thursday for the closing ceremony of this year's Cairo International Film Festival.
Images of Youssef in the dress were widely shared on social media in Muslim-majority Egypt, where ostensibly secular authorities often side with religious conservatives.
Her case prompted the country's Actors Guild to declare it intended to investigate and discipline actors who wore "inappropriate" attire during the opening and closing ceremonies of the weeklong film festival, arguing that they clashed with "traditions, values and ethics of the society."
A guild representative, Ayman Azab, attended Wednesday's questioning, Youssef's lawyer said.
Youssef said in a Facebook post that she may have misjudged how people would react to the dress.
"If I had known, I would not have worn this dress," she said. "I want to repeat my commitment to the values and ethics we have been raised by in Egyptian society."(VOA)
CategoriesEditor's Choice TagsEditor's Choice, Egyptian Actress, Egyptian Authority, Egyptian Judiciary, Egyptian Society, Moral Policing, Rania Youssef
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Dicyphonia är ett släkte av insekter. Dicyphonia ingår i familjen dvärgstritar.
Kladogram enligt Catalogue of Life:
Källor
Dvärgstritar
Dicyphonia
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Home > Publications/Tools > Using Wireless Handheld Computers to Seek Information at the Point of Care: An Evaluation by Clinicians.
Using Wireless Handheld Computers to Seek Information at the Point of Care: An Evaluation by Clinicians.
J Am Med Inform Assoc. 2007 Nov-Dec;14(6):807-15. Epub 2007 Aug 21.
Objective: To evaluate: (1) the effectiveness of wireless handheld computers for online information retrieval in clinical settings; (2) the role of MEDLINE in answering clinical questions raised at the point of care. Design: A prospective single-cohort study: accompanying medical teams on teaching rounds, five internal medicine residents used and evaluated MD on Tap, an application for handheld computers, to seek answers in real time to clinical questions arising at the point of care. Measurements: All transactions were stored by an intermediate server. Evaluators recorded clinical scenarios and questions, identified MEDLINE citations that answered the questions, and submitted daily and summative reports of their experience. A senior medical librarian corroborated the relevance of the selected citation to each scenario and question.Results: Evaluators answered 68% of 363 background and foreground clinical questions during rounding sessions using a variety of MD on Tap features in an average session length of less than four minutes. The evaluator, the number and quality of query terms, the total number of citations found for a query, and the use of auto-spellcheck significantly contributed to the probability of query success. Conclusion: Handheld computers with Internet access are useful tools for healthcare providers to access MEDLINE in real time. MEDLINE citations can answer specific clinical questions when several medical terms are used to form a query. The MD on Tap application is an effective interface to MEDLINE in clinical settings, allowing clinicians to quickly find relevant citations.
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There are not many runners who can honestly say that their running is never affected by mental or psychological factors. Almost everyone who competes or attempts to do their best at some important aspect of life has experienced both mental boosts and hindrances.
Doubts, confidence, courage and fears for example can have a significant impact on how well we perform. The more we understand the complex interrelationship between mind and body the more it becomes clear that in order to perform at our best physically we must properly prepare and use both body and mind.
Research and experience have proven that structured scientific mental training can help us learn to prepare and use our mind to improve our running. A key psychological ingredient needed to consistently run at our best is a winning attitude. In this context attitude is defined as those thoughts, feelings and emotions associated with a specific situation or performance. For example the thoughts and feelings we usually associate with the last three miles of a marathon is, for many of us, an overwhelming desire to get it over with! In general if we have doubts about our ability to cross the finish line within a certain time we have trouble doing so. On the other hand if we have confidence we tend to be much more likely to complete the race successfully.
There is a scientific basis for the commonly accepted axiom that confidence when you race leads to successful finishes. The muscles of the body involuntarily and instantaneously respond to thoughts, feelings and ideas. This phenomena is referred to as Ideomotor Activity. In practical terms what this means is that negative thoughts and emotions tend to have a negative effect on our muscles while positive thoughts tend to have a positive effect. In almost every situation of equal or nearly equal ability and preparation the faster runner is usually the one who maintains the most consistently positive attitude and pattern of thought.
Runners can and should learn to shape their attitude and thought pattern into one of predominantly positive and confident thinking. My next article will tell you how to play positive "mind games" that will begin to shape your thoughts and emotions and help you create and maintain a more dynamic and effective winning attitude.
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\section{Geometric setup}
We want to define some functors from commutative rings to sets
which are schemes, ending up with the quiver Grassmannian. These should turn out to be
useful to define morphisms between schemes naturally coming up in representation theory of quivers.
All of this section should be standard. We say that a scheme $X$ over some field $k$ is a variety
if it is separated, noetherian and of finite type over $k$. The main reference for the
geometric methods is \cite{DG}.
For a commutative ring $R$ and an $R$-module $P$ we say that $P$ has rank $r$ if for every prime
$\mathfrak{p} \in \Spec R$ we have that the localisation $P_\mathfrak{p}$ is free of rank $r$ as an $R_\mathfrak{p}$-module.
As a consequence $P$ is projective.
Let $d, e, r \in \mathbb{N}$. We define various schemes via their functors of points from commutative rings to sets. Let $R$ be a commutative
ring. We define
\begin{align*}
\Sch{Hom}(d,e) (R) \; &:= \Hom_R (R^d, R^e),\\
\Sch{Hom}(d,e)_r(R)\; & := \Set{ f\in \Sch{Hom}(d,e) (R) |
\begin{array}{l}
\Bild f \text{ is a direct summand}\\ \text{of }R^e \text{ of } \rank r
\end{array}},\\
\Sch{Inj}(d,d+e)(R)\; & := \Set{ f\in \Sch{Hom}(d,d+e) (R) | f \text{ is a split injection}}\\
& \; = \Sch{Hom}(d,d+e)_d(R)\\
\intertext{and}
\Sch{Surj}(d+e,e)(R) \; & := \Set{ g\in \Sch{Hom}(d+e,e) (R)| g \text{ is surjective} }\\
& \; = \Sch{Hom}(d+e,e)_e(R).
\end{align*}
We define the Grassmannian scheme via its functor of points
\[\Gr{d}{d+e} (R) := \Set{ P \subset R^{d+e} | P \text{ is a direct summand of }
\rank d}.\]
If $\seqv{d} = (d^0,\dots,d^\nu)$ is a sequence of integers, we define the flag
scheme $\OFl(\seqv{d}) \subset \prod \Gr{d^i}{d^\nu}$
as
\[\OFl(\seqv{d}) (R):= \Set{ (U^0, \dots, U^\nu) \in \prod_{i=0}^\nu \Gr{d^i}{d^\nu}(R) |
\begin{array}{c}
U^i \subset U^{i+1} \text{ for}\\ \text{all } 0 \le i < \nu
\end{array}}.\]
Let $n \in \mathbb{N}$. We define the group scheme $\GL_n$ via its functor of points
\[ \GL_n (R) := \GL(R^n). \]
\begin{lemma}
Let $d, e, r \in \mathbb{N}$ and $\seqv{d} = (d^0,\dots,d^\nu)$ a sequence of integers. Then
\begin{enumerate}
\item $\Sch{Hom}(d,e)$ is an affine space.
\item $\Sch{Hom}(d,e)_r$ is a locally closed subscheme of $\Sch{Hom}(d,d+e)$.
\item $\Sch{Inj}(d,d+e)$ is an open subscheme of $\Sch{Hom}(d,d+e)$.
\item $\Sch{Surj}(d+e,e)$ is an open subscheme of $\Sch{Hom}(d+e,e)$.
\item $\Gr{d}{d+e}$ and $\OFl(\seqv{d}) \subset \prod \Gr{d^i}{d^\nu}$
are projective schemes, smooth over $\mathbb{Z}$.
\item $\GL_n$ is a smooth group scheme over $\mathbb{Z}$.
\item The projection $\Sch{Inj}(d,d+e) \rightarrow \Gr{d}{d+e}$ sending each
$f$ to $\Bild(f)$ is a principal $\GL_d$-bundle, locally trivial in the
Zariski topology.
\end{enumerate}
\end{lemma}
\begin{proof}
For (1)--(4) and (7) see, for example, \cite{bill_homsgenrep}. For (5) and (6)
see \cite{DG}.
\end{proof}
Now we generalise these schemes to quivers.
For standard notations and results about quivers we refer the reader to
\cite{Ringel_integral}.
A quiver $Q=(Q_0, Q_1, s, t)$ is a directed graph with a set of vertices $Q_0$, a set of
arrows $Q_1$ and maps
$s,t \colon Q_1 \rightarrow Q_0$, sending an arrow to its starting respectively terminating vertex.
In particular, we write $\alpha \colon s(\alpha) \rightarrow t(\alpha)$ for an $\alpha \in Q_1$. A quiver $Q$ is finite
if $Q_0$ and $Q_1$ are finite sets. In the following, all quivers will be finite.
For a quiver $Q=(Q_0, Q_1, s ,t)$ we define $Q^{op} := (Q_0, Q_1, t, s)$ as the quiver with all arrows reversed.
A path of length $r\ge0$ in $Q$ is a sequence of arrows $\xi = \alpha_1 \alpha_2 \cdots \alpha_r$ such that
$t(\alpha_i) = s(\alpha_{i+1})$ for all $1 \le i < r$. We write $t(\xi) := t(\alpha_r)$ and $s(\xi):= s(\alpha_1)$.
For each $i \in Q_0$ there is the
trivial path $\epsilon_i$ of length zero starting and terminating in the vertex $i$. We identify the elements of $Q_0$ with
paths of length $0$. We denote by $Q(i,j)$ the
set of paths starting at the vertex $i$ and terminating at the vertex $j$.
If $R$ is a ring, then the path algebra $RQ$ has as basis the set of
paths and the multiplication $\xi \cdot \zeta$ is given by
the concatenation of paths if $t(\xi)=s(\zeta)$ or zero otherwise. In particular, the $\epsilon_i$ are pairwise orthogonal
idempotents of $RQ$, that is $\epsilon_i \epsilon_j = \delta_{ij} \epsilon_i$.
Let $Q_r$ denote the set of paths of length $r$. This extends the notation of vertices $Q_0$ and arrows $Q_1$. Then
\[ RQ = \bigoplus_{r\ge 0} RQ_r,\]
where $RQ_r$ is the free $R$-module with basis the elements of $Q_r$. By construction,
\[RQ_r RQ_s = RQ_{r+s},\]
thus
$RQ$ is an $\mathbb{N}$-graded $R$-algebra.
We define a partial order on
$\mathbb{Z} Q_0$ by $\dvec{d} = \sum_{i} d_i \epsilon_i \ge 0$ if and only if $d_i \ge 0$ for all $i\in Q_0$.
An element $\dvec{d} \in \mathbb{Z} Q_0$ is called a dimension vector.
We endow $\mathbb{Z} Q_0$ with a bilinear form $\bform{\cdot}{\cdot}_Q$ defined by
\[
\bform{\dvec{d}}{\dvec{e}}_Q := \sum_{i \in Q_0} d_i e_i - \sum_{\mathpalette\mathclapinternal{\alpha \colon i \rightarrow j \in Q_1}} d_i e_j.
\]
This form is generally called the Euler form or the Ringel form.
We also define its symmetrisation
\[
\sbform{\dvec{d}}{\dvec{e}}_Q :=\bform{\dvec{d}}{\dvec{e}}_Q + \bform{\dvec{e}}{\dvec{d}}_Q.
\]
Let $Q$ be a quiver and $k$ be a field. A $k$-representation
$M$ of $Q$ is given by finite dimensional $k$-vector spaces
$M_i$ for each $i\in Q_0$ and $k$-linear maps $M_\alpha \colon M_i \rightarrow M_j$ for each $\alpha \in Q_1$.
If $M$ and $N$ are $k$-representations of $Q$, then a morphism $f \colon M \rightarrow N$ is given by
$k$-linear maps
$f_i \colon M_i \rightarrow N_i$ for each $i \in Q_0$ such that the following diagram commutes
\[
\xymatrix{
M_i \ar[r]^{M_\alpha} \ar[d]_{f_i} & M_j \ar[d]^{f_j}\\
N_i \ar[r]_{N_\alpha} & N_j
}
\]
for all $\alpha \colon i \rightarrow j \in Q_1$. Denote by $\repK{Q}{k}$ the category of finite
dimensional $k$-representations of $Q$.
We will use the following notations for a $k$-algebra $\Lambda$ and two finite dimensional
$\Lambda$-modules $M$ and $N$:
\begin{itemize}
\item $(M,N)_\Lambda := \Hom_\Lambda(M,N)$,
\item $(M,N)^i_\Lambda := \Ext^i_\Lambda(M,N)$,
\item $[M,N]_\Lambda := \dim_k\Hom_\Lambda(M,N)$,
\item $[M,N]^i_\Lambda := \dim_k\Ext^i_\Lambda(M,N)$.
\end{itemize}
Note that $[M,N]^0 = [M,N]$ and $(M,N)^0 = (M,N)$.
If $Q$ is a quiver, we define $(M,N)_Q := (M,N)_{kQ}$ if $M$ and $N$ are $k$-representations and similarly for
the other notations. For dimension vectors $\dvec{d}$ and $\dvec{e}$ we denote
by $\hom_{kQ}(\dvec{d},\dvec{e})$ the minimal value of $[M,N]_Q$ for $M \in \RepK[Q]{\dvec{d}}{k}$ and
$N \in \RepK[Q]{\dvec{e}}{k}$ and similarly for $\ext^i_{kQ}(\dvec{d},\dvec{e})$. More generally, if
$\mathcal{A}$ and $\mathcal{B}$ are subsets of $\RepK[Q]{\dvec{d}}{k}$ and $\RepK[Q]{\dvec{e}}{k}$ respectively, then
$\hom(\mathcal{A}, \mathcal{B})$ and $\ext^i(\mathcal{A}, \mathcal{B})$ are defined analogously.
Whenever the algebra, the field or the quiver is clear from the context, we omit them from the notation.
If our algebra is hereditary we denote $\Ext^1$ by $\Ext$.
Let $M$ be a $k$-representation. A sequence of dimension vectors $\flvec{d}=(\dvec{d}^0, \dots, \dvec{d}^\nu)$
is called a filtration of $M$, or of $\dimve M$, if $\dvec{d}^i \le \dvec{d}^{i+1}$, $\dvec{d}^0 = 0$ and
$\dvec{d}^\nu = \dimve M$.
Let $Q$ be a quiver, $\dvec{d}$ and $\dvec{e}$ dimension vectors and
$\flvec{d} = (\dvec{d}^0, \dots, \dvec{d}^\nu)$ a filtration. By taking
fibre products we can generalise the previous constructions and define
$\Sch{Hom}(\dvec{d}, \dvec{e})$,
$\Sch{Inj}(\dvec{d},\dvec{d}+\dvec{e})$,
$\Sch{Surj}(\dvec{d}+\dvec{e},\dvec{e})$,
$\Gr{\dvec{d}}{\dvec{d}+\dvec{e}}$,
$\OFl(\flvec{d})$ and
$\GL_\dvec{d}$ pointwise. For all above schemes we can do base change
to any commutative ring $k$, and we will denote these schemes
by e.g. $\Sch{Hom}(\dvec{d}, \dvec{e})_k$.
\begin{definition}
Define the representation scheme
\[
\Rep[Q]{\dvec{d}} := \prod_{\alpha:i \rightarrow j} \Sch{Hom}(d_i,d_j).
\]
This is isomorphic to an affine space. Moreover, $\GL_\dvec{d}$ acts
by base change on $\Rep[Q]{\dvec{d}}$.
\end{definition}
\begin{remark}
For each scheme defined for a quiver $Q$ we often omit the index if $Q$ is obvious, e.g.
$\Rep{\dvec{d}} = \Rep[Q]{\dvec{d}}$.
\end{remark}
More generally, we have the module scheme.
\begin{definition}
Let $k$ be a field, $\Lambda$ a finitely generated $k$-algebra and
\[\rho \colon k\langle x_1, \dots, x_m\rangle \rightarrow \Lambda\]
a surjective map from the free associative algebra to $\Lambda$. The affine $k$-scheme $\MMod_\Lambda(d)$ is defined by
\[ \MMod_\Lambda(d) (R) =
\Set{ (M^1, \dots, M^m) \in (\End(R^d))^m | \begin{array}{c}f(M^1, \dots, M^m) = 0 \\ \forall\ f \in \Ker \rho\end{array}}.\]
There is a natural $\GL_d$-action on $\MMod_\Lambda(d)$ given by conjugation.
\end{definition}
\begin{remark}
We have that $\MMod_\Lambda(d)$ is isomorphic to the functor
\[
R \mapsto \Hom_{R-alg}(\Lambda \otimes R, \End(R^d)).
\]
Therefore, another choice of $\rho$ gives an isomorphic scheme.
\end{remark}
Let $\{\epsilon_i\}_{i\in I}$, $I$ some index set, be a complete set of (not necessarily primitive) orthogonal idempotents for $\Lambda$.
We say that a $\Lambda$-module $M$ has dimension vector $\dvec{d}=\sum d_i \epsilon_i \in \mathbb{N}^I$ with respect
to this idempotents, if $\dim M_i = \dim M \epsilon_i = d_i$.
The subscheme $\MMod_\Lambda(\dvec{d})$ consisting of $\Lambda$-modules of dimension vector $\dvec{d}$ is an open and closed
subscheme of $\MMod_\Lambda(d)$, therefore
\[ \MMod_\Lambda(d) = \coprod \MMod_\Lambda (\dvec{d}).\]
Let $Q$ be a quiver, $k$ a field, $d$ an integer and $\dvec{d}$ a dimension vector such
that $\sum d_i = d$. Then, after choosing an isomorphism
$\bigoplus k^{d_i} \rightarrow k^d$, there is a natural immersion
\[
\Rep[Q]{\dvec{d}} \rightarrow \MMod_{kQ}(\dvec{d}) \subset \MMod_{kQ}(d).
\]
Note that $\GL_\dvec{d} (k)$-orbits in $\Rep[Q]{\dvec{d}}(k)$ are in one-to-one
correspondence with isomorphism classes of $k$-representations of dimension vector $\dvec{d}$. Similarly,
$\GL_{d}(k)$-orbits in $\MMod_\Lambda(d)(k)$ are in one-to-one correspondence
with isomorphism class of $\Lambda$-modules of $k$-dimension $d$.
We define now quiver flags and quiver Grassmannians as varieties. Choose a $k$-representation
$M \in \Rep[Q]{\dvec{d} + \dvec{e}}(k)$.
We set
\[ \Gr[Q]{\dvec{d}}{M}(R) := \Set{ U \in \Gr{\dvec{d}}{\dvec{d} + \dvec{e}}_k (R) |
\begin{array}{c}
U \text{ is a subrepesentation of }\\
M \otimes_k R
\end{array}}
\]
for each $k$-algebra $R$. Then $\Gr[Q]{\dvec{d}}{M}$ is a closed subscheme of $\Gr{\dvec{d}}{\dvec{d}+\dvec{e}}_k$ and
therefore projective. In a similar way we obtain a closed subscheme
$\Fl[Q]{\flvec{d}}{M}$ of $\OFl(\flvec{d})_k$ for a filtration $\flvec{d}$ of $M$.
Let $\Lambda$ be a $k$-algebra and $d$, $e$ two integers.
For a $\Lambda$-module $M \in \MMod_\Lambda(d+e)$
we define
\[ \Gr[\Lambda]{d}{M}(R) := \Set{ U \in \Gr{d}{d + e}(R) | U \text{ is a submodule of }
M \otimes_k R}.
\]
Again, for some set of idempotents $\{\epsilon_i\}$
we define $\Gr[\Lambda]{\dvec{d}}{M}$ as the subscheme consisting of submodules having dimension vector $\dvec{d}$
and obtain
\[ \Gr[\Lambda]{d}{M} = \coprod \Gr[\Lambda]{\dvec{d}}{M},\]
each $\Gr[\Lambda]{\dvec{d}}{M}$ being open and closed in \Gr[\Lambda]{d}{M}.
Moreover, for $\Lambda = kQ$ we obtain that
\[ \Gr[Q]{\dvec{d}}{M} \cong \Gr[kQ]{\dvec{d}}{M} \]
via the immersion $\Rep[Q]{\dvec{d}} \rightarrow \MMod_{kQ}(\dvec{d})$.
We have the following well-known result.
\begin{proposition}
Let $X, Y$ be two schemes, $Y$ being irreducible.
Let $f \colon X \rightarrow Y$ be a morphism of schemes such that $f$ is open and for each $z \in Y$ the
fibre $f^{-1}(z)$ is irreducible. Then $X$ is irreducible.
\label{isch:propn:openirred}
\end{proposition}
\begin{proof}
Take $\emptyset \neq U, V \subset X$ and $U, V$ open. We need to show that $U \cap V$ is non-empty. We know that
$f(U)$ and $f(V)$ are non-empty and open in $Y$. Therefore, they intersect non-trivially. Let
$y \in f(U) \cap f(V)$.
By definition, we have
that $U \cap f^{-1}(y) \neq \emptyset$ and $V \cap f^{-1}(y) \neq \emptyset$ and both are open in $f^{-1}(y)$. By assumption,
the fibre $f^{-1}(y)$ is irreducible and therefore $U \cap V \cap f^{-1}(y) \neq \emptyset$ and we are done.
\end{proof}
\begin{remark}
The same is true for arbitrary topological spaces, since the proof relies purely on topology.
\end{remark}
\section{Tangent spaces}
Let $k$ be a field and $\Lambda$ a finitely generated $k$-algebra. We want to calculate the tangent
space at a point of $\Gr[\Lambda]{d}{M}$.
\begin{lemma}
Let $d, e \in \mathbb{N}$, $k$ a field, $\Lambda$ a finitely generated $k$-algebra and $M \in \MMod_\Lambda(d+e)(k)$.
Let $U \in \Gr[\Lambda]{d}{M}(K)$, for a field extension $K$ of $k$. Then
\[ T_U \Gr[\Lambda]{d}{M} \cong \Hom_{\Lambda\otimes_k K} (U, (M\otimes_k K)/U). \]
\end{lemma}
\begin{proof}
In this proof we use left modules since the notation is more convenient.
For the tangent space we use the definition of \cite[I, \S 4, no 4]{DG}.
By base change, we can assume that $K=k$. We prove the claim by doing a $k[\varepsilon]$-valued point calculation ($\varepsilon^2 = 0$).
The short exact sequence
\[
\xymatrix{ 0 \ar[r]& U \ar[r]^\iota & M \ar[r]^\pi \ar@/^/@{.>}[l]^p
& M/U \ar[r] \ar@/^/@{.>}[l]^j & 0}
\]
is split in the category of $k$-vector spaces, therefore there is a retraction $p$ of $\iota$ and a
section $j$ of $\pi$. We consider elements of $U$ as elements of $M$ via the inclusion $\iota$.
The map $k[\varepsilon] \rightarrow k$ given by $s + r \epsilon \mapsto s$ induces
a map $\theta\colon V \otimes k[\varepsilon] \rightarrow V$ for each $k$-vector space $V$.
For each homomorphism $f \in \Hom_\Lambda(U, M/U)$ we define
\[
S_f := \Set{ u + v \varepsilon | u \in U,\ \pi(v)=f(u) } \subset M \otimes k[\varepsilon].
\]
Note that $\theta(S_f) = U$.
We need to show that $S_f \in \Gr[\Lambda \otimes {k[\varepsilon]}]{d}{M \otimes k[\varepsilon]}$
and that every element $S$ of the Grassmannian with $\theta(S) = U$ arises in this way.
First, we show that $S_f$ is a $\Lambda \otimes k[\varepsilon]$-submodule. Let $u + v\varepsilon \in S_f$
and $r +s\varepsilon \in \Lambda \otimes k[\varepsilon]$. Then we have
\[ (r+s\varepsilon)(u+v\varepsilon)= ru + (rv + su)\varepsilon. \]
Since $\pi(v) = f(u)$, $u \in U$ and $f$ is a $\Lambda$-homomorphism we obtain that
\[ \pi(rv + su) = r\pi(v) = r f(u) = f(ru). \]
Therefore, $S_f$ is a $\Lambda \otimes k[\varepsilon]$ submodule.
Now we show that $S_f$ is, as a $k[\varepsilon]$-module, a summand of $M \otimes k[\varepsilon]$ of
rank $d$. Let $\tilde f\in \Hom_k(U, M):=j \circ f$
be a $k$-linear lift of $f$.Let
\begin{alignat*}{2}
\phi &\colon\quad& U \otimes k[\varepsilon] & \rightarrow M \otimes k[\varepsilon]\\
&& u + v\varepsilon & \mapsto u + (\tilde f (u) +v)\varepsilon.
\intertext{Obviously, $\phi$ is $k[\varepsilon]$-linear and $\Bild \phi = S_f$. Moreover,
$\phi$ is split with retraction}
\psi &\colon\quad& M \otimes k[\varepsilon] & \rightarrow U \otimes k[\varepsilon]\\
&& x + y\varepsilon & \mapsto p(x) + p(y-\tilde f\circ p(x))\varepsilon.
\end{alignat*}
Therefore, $S_f$ is a summand of $M \otimes k[\varepsilon]$ of rank $d$.
On the other hand, let $S \in \Gr[\Lambda \otimes {k[\varepsilon]}]{d}{M \otimes k[\varepsilon]}$
such that $\theta(S)=U$. Then, $U\varepsilon$ is a $k$-subspace of $S$ with
$\dim_k U\varepsilon = d$, therefore $\dim_k S/(U\varepsilon) = d$. The map sending
$u + v\varepsilon \in S$ to $u \in U$ is surjective, therefore the induced map from
$S/(U\varepsilon)$ is an isomorphism.
Hence, for each $u \in U$ there is a $v \in M$,
such that $u + v\varepsilon \in S$ and $v$ is unique up to a element in $U\varepsilon$.
Set $f(u) := \pi(v)$ and, by the discussion before, this does not depend on the choice of $v$
and we have that $S = S_f$. Moreover,
$f \in \Hom_{\Lambda}(U,M/U)$: Let $r \in \Lambda$, $u \in U$ and $v \in M$, such that $u+v\varepsilon \in S$.
Then, $ru + rv\varepsilon \in S$. By definition, $f(ru) = \pi(rv) = r\pi(v) = rf(u)$.
\end{proof}
\section{Tensor algebras}
We now want to calculate the tangent space at the quiver flag variety by using the previous result on
the tangent space at the module Grassmannian. For this we use tensor algebras, or more precisely
a generalisation of them.
Let $\Lambda_0$ be a ring and $\Lambda_1$ a $\Lambda_0$-bimodule. Define the tensor ring
$T(\Lambda_0, \Lambda_1)$ to be the $\mathbb{N}$-graded $\Lambda_0$-module
\[ \Lambda := \bigoplus_{r\ge 0} \Lambda_r, \quad \Lambda_r := \Lambda_1 \otimes_{\Lambda_0} \dotsm \otimes_{\Lambda_0} \Lambda_1 \: (r \text{ times}),\]
with multiplication given via the natural isomorphism $\Lambda_r \otimes_{\Lambda_0} \Lambda_s \cong \Lambda_{r+s}$. If $\lambda \in \Lambda_r$
is homogeneous, we write $\degr{ \lambda} = r$ for its degree.
The graded radical of $\Lambda$ is the ideal $\Lambda_+ := \bigoplus_{r \ge 1} \Lambda_r$. Note that
$\Lambda_+ \cong \Lambda_1 \otimes_{\Lambda_0} \Lambda$ as right $\Lambda$-modules.
\begin{lemma}
Let $R$ and $S$ be rings. Let $A_R$, $_R B_S$ and $C_S$ be modules over the corresponding rings. Assume that $_R B_S$ is $S$-projective
and $R$-flat. Then
\[ \Ext^n_S (A \otimes_R B, C) \cong \Ext^n_R (A, \Hom_S(B,C)).
\]
\end{lemma}
\begin{proof}
Choose a projective resolution $P_\bullet$ of $A_R$. The functor
$- \otimes_R B$ is exact, and for any projective module $P_R$ the functor
$\Hom_S(P \otimes_R B, -) \cong \Hom_R(P, -) \circ \Hom_S(B_S, -)$ is exact since $B_S$ is projective. Therefore,
$P_\bullet \otimes_R B$ gives a projective resolution of $A \otimes_R B_S$ as an $S$-module. We obtain
\begin{multline*}
\Ext^n_S (A \otimes_R B, C) = H^n \Hom_S(P_\bullet \otimes_R B, C) \cong \\
\cong H^n \Hom_R(P_\bullet, \Hom_S(B,C)) = \Ext^n_R (A, \Hom_S(B,C)).
\end{multline*}
\end{proof}
The following proof is the standard proof for showing that a ``classical'' tensor algebra, i.e.
a tensor algebra where $\Lambda_0$ is semisimple, is hereditary.
\begin{theorem}
Let $\Lambda$ be a tensor ring and $M \in \MMod \Lambda$.
Then there is a short exact sequence
\[ 0 \rightarrow M \otimes_{\Lambda_0} \Lambda_+ \xrightarrow{\delta_M} M \otimes_{\Lambda_0} \Lambda \xrightarrow{\epsilon_M} M \rightarrow 0,\]
where, for $m \in M$, $\lambda \in \Lambda$ and
$\mu \in \Lambda_1$,
\begin{align*}
\epsilon_M ( m \otimes \lambda)& := m \cdot \lambda,\\
\delta_M (m \otimes ( \mu \otimes \lambda) &:= m \otimes (\mu \otimes \lambda) - m \cdot \mu \otimes \lambda.
\end{align*}
Moreover, if $_{\Lambda_0} \Lambda_1$ is flat, then $\pd M \otimes_{\Lambda_0} \Lambda_+ \le \gldim \Lambda_0$ and
$\pd M \otimes_{\Lambda_0} \Lambda \le \gldim \Lambda_0$.
\end{theorem}
\begin{proof}
It is clear that $\epsilon_M$ is an epimorphism and that $\epsilon_M \delta_M = 0$. To see that $\delta_M$ is a monomorphism, we decompose
$M \otimes \Lambda = \bigoplus_{r\ge 0} M \otimes \Lambda_r$ and similarly $M \otimes \Lambda_+$ as $\Lambda_0$-modules.
Then $\delta_M$ restricts to
maps $M \otimes \Lambda_r \rightarrow (M \otimes \Lambda_r) \oplus (M \otimes \Lambda_{r-1})$ for each $r \ge 1$ and moreover acts as
the identity on the first component. In particular, if $\sum_{r=1}^t x_r \in \Ker(\delta_M)$ with $x_r \in M \otimes \Lambda_r$, then $x_t =0$.
Thus $\delta_M$ is injective.
Next we show that $M \otimes \Lambda = (M \otimes \Lambda_0) \oplus \Bild(\delta_M)$. Let $x = \sum_{r=0}^{t} x_r \in M \otimes \Lambda$.
We show that $x \in (M \otimes \Lambda_0) + \Bild(\delta_M)$ by induction on $t$. For $t=0$ this is trivial. Let $t \ge 1$. Then
$x_t \in M \otimes \Lambda_+$ and $x - \delta_M (x_t) = \sum_{r=0}^{t-1} x_r '$. So we are done by induction. If
$x \in \Bild(\delta_M) \cap M \otimes \Lambda_0$, then there is a $y \in M \otimes \Lambda_+$ such that $\delta_M (y) =x \in M \otimes \Lambda_0$.
But this is only the case if $y =0$ and therefore $x=0$.
Now let $_{\Lambda_0} \Lambda_1$ be flat. Then $_{\Lambda_0}\Lambda$ is flat. If $N$ is any $\Lambda_0$-module,
then $\pd_\Lambda N \otimes \Lambda \le \pd_{\Lambda_0} N$ since
\[\Ext^n_\Lambda (N \otimes \Lambda, L) \cong \Ext^n_{\Lambda_0} (N, \Hom_{\Lambda}(\Lambda, L)) = \Ext^n_{\Lambda_0} (N, L)\] for any
$L \in \MMod \Lambda$ by the lemma. Therefore, the second part follows since $\Lambda_+ = \Lambda_1 \otimes \Lambda$.
\end{proof}
Let $R$ be any $k$-algebra of finite global dimension and let $M$, $N$ be two $R$-modules. The Euler form of $M$ with $N$ is defined by
\[
\euf{M,N}_R := \sum_{i=0}^\infty \dim_k \Ext^i(M,N).
\]
Ringel showed that for a quiver $Q$ and two $k$-representations $M$ and $N$ we have that
\[
\euf{M,N}_Q = \euf{M,N}_{kQ}.
\]
Now let $Q$ be a quiver and $\Lambda_0 = R^{Q_0}$ for a fixed ring $R$. Let $\Lambda_1 := R^{Q_1}$ be the free $\Lambda_0$-bimodule
given by the arrows. The tensor ring $T(\Lambda_0, \Lambda_1)$ is then equal to $RQ$. Note that $\{\epsilon_i\}_{i\in Q_0}$ is
an orthogonal, complete set of idempotents. As before, for an $RQ$-module $M$ we denote
$M\epsilon_i$ by
$M_i$. As $R$-modules we have that $M = \bigoplus M_i$.
\begin{theorem}
\label{euf_quiverquiver}
Let $R$ be a $k$-algebra with $\gldim R = n < \infty$. Let $M$ and $N$ be two finite dimensional modules over $RQ$. Then the Euler form is given by
\[ \euf{M,N}_{RQ} = \sum_{i\in Q_0} \euf{M_i, N_i}_R - \sum_{\alpha\colon i \rightarrow j} \euf{M_i, N_j}_R. \]
\end{theorem}
\begin{proof}
Let $\Lambda_0 = R^{Q_0}$. There is a natural $\Lambda_0$-bimodule structure on $\Lambda_1 = R^{Q_1}$.
Let $\Lambda = T(\Lambda_0, \Lambda_1) = RQ$.
Let $0 \rightarrow M \otimes_{\Lambda_0} \Lambda_1 \otimes_{\Lambda_0} \Lambda \rightarrow M \otimes_{\Lambda_0} \Lambda \rightarrow M \rightarrow 0$
be the short exact sequence of the previous theorem. Apply $\Hom(-, N)$ to it and consider the long exact sequence
\[
\xymatrix@C=12pt{
0 \ar[r] & \Hom_\Lambda(M, N) \ar[r] & \Hom_\Lambda(M \otimes_{\Lambda_0} \Lambda , N) \ar[r]&
\Hom_\Lambda(M \otimes_{\Lambda_0} \Lambda_1 \otimes_{\Lambda_0} \Lambda, N) \ar `r[d] `[lll] `[dlll] `[dll] [dll]\\
& \Ext^1_\Lambda(M, N) \ar[r] &\Ext^1_\Lambda(M \otimes_{\Lambda_0} \Lambda , N) \ar[r]&
\Ext^1_\Lambda(M \otimes_{\Lambda_0} \Lambda_1 \otimes_{\Lambda_0} \Lambda, N) \\
&\cdots & & \\
\ar[r]& \Ext^n_\Lambda(M, N) \ar[r] &\Ext^n_\Lambda(M \otimes_{\Lambda_0} \Lambda , N) \ar[r]&
\Ext^n_\Lambda(M \otimes_{\Lambda_0} \Lambda_1 \otimes_{\Lambda_0} \Lambda, N) \\
\ar[r]& \Ext^{n+1}_\Lambda(M, N) \ar[r] & **[l]0.\qquad &}
\]
Here we obtain the last $0$ since the $\pd M \otimes \Lambda \le \gldim R$
by the previous theorem.
Now, since $ _{\Lambda_0} \Lambda_\Lambda$ is projective as a $\Lambda_0$-module and as a $\Lambda$-module, we obtain
\[\Ext^i_\Lambda(M \otimes_{\Lambda_0} \Lambda , N) \cong \Ext^i_{\Lambda_0}(M,N) \cong \bigoplus_{i \in Q_0} \Ext^i_R(M_i, N_i)\]
and
\[\Ext^i_{\Lambda}(M \otimes_{\Lambda_0} \Lambda_1 \otimes_{\Lambda_0} \Lambda, N) \cong \Ext^i_{\Lambda_0} (M \otimes_{\Lambda_0} \Lambda_1, N) \cong
\bigoplus _{\alpha \colon i \rightarrow j} \Ext^i_R(M_i, N_j). \]
This yields the claim.
\end{proof}
\begin{remark}
Note that, for a field $k$, we recover Ringel's result, namely that
\[
\sum_{i\in Q_0} \dim M_i \dim N_i - \sum_{\alpha\colon i \rightarrow j} \dim M_i \dim N_j = [M,N]_{kQ} - [M,N]^1_{kQ}
\]
for two $k$-representations $M$ and $N$ of a quiver $Q$.
\end{remark}
\section{Geometry of Quiver Flags}
Now we come back to quiver flags. Let $k$ be a field.
We can consider $\Rep[Q]{\dvec{d}}$ as an affine scheme over $k$ with the obvious functor of points. More
generally, we work in the category of schemes over $k$.
Fix a filtration
$\seqv{\dvec{d}}=(\dvec{d}^0 = 0\le \dvec{d}^1 \le \dots \le \dvec{d}^\nu)$.
Denote by $A_{\nu+1}$ the quiver having $\nu+1$ vertices numbered from $0$ to $\nu$ and arrows going from vertex $i$ to vertex $i+1$
for $0 \le i < \nu$. Let $\Lambda := (kQ) A_{\nu+1}$. Then $\Mod \Lambda$ is the category of sequences of $k$-representations of
$Q$ of length $\nu+1$ and
chain maps between them, i.e. a morphism between two modules
\[
\boldsymbol{M}= M^0 \rightarrow M^1 \rightarrow \dots \rightarrow M^\nu
\]
and
\[
\boldsymbol{N}= N^0 \rightarrow N^1 \rightarrow \dots \rightarrow N^\nu
\]
is given by a commutative diagram
\[
\begin{CD}
M^0 @>>> M^1 @>>> \cdots @>>> M^\nu\\
@VVV @VVV @. @VVV\\
N^0 @>>> N^1 @>>> \cdots @>>> N^\nu.
\end{CD}
\]
For $\Lambda$ it is easy to calculate the Euler form of two modules $\seqv{M}, \seqv{N} \in \Mod \Lambda$ by using
theorem $\ref{euf_quiverquiver}$:
\[ \euf{\seqv{M},\seqv{N}}_\Lambda = \sum_{i=0}^r \euf{M^i,N^i}_{kQ} - \sum_{i=0}^{r-1} \euf{M^i, N^{i+1}}_{kQ}. \]
We show that $\Gr[\Lambda]{\seqv{\dvec{d}}}{\seqv{M}} \cong \Fl[Q]{\seqv{\dvec{d}}}{M}$, where
$\seqv{M} = (M=M=\dots=M)$, and then use the previous
results to calculate the tangent space.
\begin{lemma}
Let $\flvec{d}$ be a filtration and
$M \in \Rep[Q]{\dvec{d}^\nu}(k)$.
Let
\[
\seqv{U} = (U^0, U^1, \dots, U^\nu) \in \Fl[Q]{\seqv{\dvec{d}}}{M}(K)
\]
for a field extension
$K$ of $k$. Then we have that
\[
T_\seqv{U} \Fl[Q]{\seqv{\dvec{d}}}{M} \cong \Hom_{\Lambda\otimes K}(\seqv{U}, (\seqv{M}\otimes K)/\seqv{U}),
\]
where $\Lambda = (kQ)A_{\nu + 1}$ and $\seqv{M} = (M=M=\dots=M)
\in \MMod_\Lambda (\dvec{d}^\nu,\dvec{d}^\nu, \dots, \dvec{d}^\nu)(k)$.
\end{lemma}
\begin{proof}
For a submodule
\[
\seqv{U}=(U^0 \rightarrow U^1 \rightarrow\dots\rightarrow U^\nu) \in \Gr[\Lambda]{\seqv{\dvec{d}}}{\seqv{M}}(R)
\]
we have automatically that the maps $U^i \rightarrow U^{i+1}$ are injections. Therefore,
such a submodule $\seqv{U}$ gives, in a natural way, rise to a flag
$\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}(R)$ and vice versa. This yields an isomorphism
$\Gr[\Lambda]{\seqv{\dvec{d}}}{\seqv{M}} \cong \Fl[Q]{\seqv{\dvec{d}}}{M}$. Since
$\Gr[\Lambda]{\seqv{\dvec{d}}}{\seqv{M}}$ is open in
$\Gr[\Lambda]{d}{\seqv{M}}$, where $d = \sum_{k,i} d^k_i$, we have that, for a point
$\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}(K)$,
\[ T_\seqv{U} \Fl[Q]{\seqv{\dvec{d}}}{M} \cong T_\seqv{U} \Gr[\Lambda]{\seqv{\dvec{d}}}{\seqv{M}}
= T_\seqv{U} \Gr[\Lambda]{d}{\seqv{M}} = \Hom_{\Lambda\otimes K}(\seqv{U}, (\seqv{M}\otimes K)/\seqv{U}). \]
\end{proof}
We define the closed subscheme $\RepFl[Q]{\seqv{\dvec{d}}}$ of
$\Rep[Q]{\dvec{d}^\nu} \times \OFl(\seqv{\dvec{d}})$ by its functor of points
\[ \RepFl[Q]{\seqv{\dvec{d}}}(R) := \Set{ (M, \seqv{U}) \in
\Rep[Q]{\dvec{d}^\nu}(R) \times \OFl(\flvec{d})(R) | \seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}}. \]
We have the following.
\begin{lemma}
Let $\flvec{d}$ be a filtration.
Consider the two natural projections from the fibre product restricted to $\RepFl[Q]{\seqv{\dvec{d}}}$.
\[
\xymatrix{
\RepFl[Q]{\seqv{\dvec{d}}} \ar[r]^{\pi_1} \ar[d]_{\pi_2}& \Rep[Q]{\dvec{d}^\nu}\\
\OFl(\flvec{d})&
}
\]
\label{lmm:geomflags:projtriv}
Then $\pi_1$ is projective and $\pi_2$ is a vector bundle of rank
\[
\sum_{k=1}^\nu \sum_{\alpha:i \rightarrow j} d^k_j (d_i^k - d_i^{k-1}).
\]
Therefore, $\RepFl[Q]{\seqv{\dvec{d}}}$ is smooth and irreducible of dimension
\[ \sum_{k=1}^{\nu-1} \euf{\dvec{d}^k, \dvec{d}^{k+1} - \dvec{d}^k}_Q + \dim \Rep[Q]{\dvec{d}^\nu}. \]
Finally, the (scheme-theoretic) image
$\mathcal{A}_\flvec{d}:=\pi_1(\RepFl[Q]{\seqv{\dvec{d}}})$ is a closed, irreducible
subvariety of $\Rep[Q]{\dvec{d}^\nu}$.
\end{lemma}
\newcommand{1.5ex}{1.5ex}
\tikzset{%
node style ge/.style={rectangle,minimum size=1.5ex}
}
\begin{proof}
$\pi_1$ is projective since it factors as a closed immersion into projective space times
$\ORep_{Q}$ followed by the projection to $\ORep_{Q}$.
For $\dvec{I}=(I_i)_{i \in Q_0}$, each $I_i \subset \{1,\dots,d^\nu_i\}$,
we set $W_\dvec{I}$ to be the
graded subspace
of $k^{\dvec{d}^\nu}$ with basis $\{e_j\}_{j \in I_i}$ in the $i$-th graded part $k^{d^\nu_i}$ and
$|\dvec{I}| := (|I_i|)_{i \in Q_0} \in \mathbb{N}^{Q_0}$.
For a sequence $\seqv{\dvec{I}} = (\dvec{I}^0, \dvec{I}^1, \dots, \dvec{I}^\nu)$ such that
$I_i^k \supset I_i^{k+1}$ and $|\dvec{I}^k|= \dvec{d}^\nu - \dvec{d}^k$ we set
\[
\seqv{W}_{\seqv{\dvec{I}}} := (W_{\dvec{I}^0}, \dots, W_{\dvec{I}^\nu})
\]
to be the decreasing sequence of
subspaces associated to $\seqv{\dvec{I}}$.
We show that $\pi_2$ is trivial over the open affine subset $U_{\seqv{\dvec{I}}}$ of
$\OFl(\seqv{\dvec{d}})$
given by
\[
U_{\seqv{\dvec{I}}} (R) := \Set{ \seqv{U} \in \OFl(\flvec{d})(R) |
U^k \oplus (W_{\dvec{I}^k} \otimes R) = R^{\dvec{d}^\nu}}.
\]
Without loss
of generality we assume $I^k_i = \{d^k_i+1, \dots, d^\nu_i\}$. Each element
$\seqv{U} \in U_\seqv{\dvec{I}}(R)$ is given uniquely by some matrices
$A_i^k \in \Mat_{(d^\nu_i-d^k_i)\times (d^k_i-d^k_{i-1})}$
such that
\[U^k_i = \Bild
\begin{array}{c@{}}\begin{tikzpicture}
\matrix (A) [matrix of math nodes,%
nodes = {node style ge},%
left delimiter = (,%
right delimiter = )]
{%
\id_{d^1_i} & 0 & \cdots & 0\\
\ & \id_{d^2_i-d^1_i} & \ddots & \vdots\\
\ & \hphantom{AB} & \ddots &0\\
\ & A^2_i & \ddots & \id_{d^{k}_i-d^{k-1}_i}\\
\hphantom{AB} & \hphantom{AB} & \cdots & A^k_i\\
};
\path ($ (A-2-1)!0.5!(A-4-1) $) node {$A^1_i$};
\draw (A-1-1.south west) rectangle (A-5-1.south east);
\draw ($ (A-2-2.south west)!0.3!(A-2-2.south) $) rectangle (A-5-2.south east);
\draw (A-5-4.north west) rectangle (A-5-4.south east);
\end{tikzpicture}\end{array}.\]
Let $V^k:=W_{(\{1,\dots,d^k_i\})}$. Let $X$ be the closed subscheme of
$\Rep[Q]{\dvec{d}^\nu}$
given by the functor of points
\[X(R) := \Set{ M \in \Rep[Q]{\dvec{d}^\nu} | V^k \otimes R \text{ is a subrepresentation of } M
\; \forall \; 0 \le k \le \nu}.\]
Note that $X$ is an affine space of dimension
\[\sum_{k=1}^\nu \sum_{\alpha:i \rightarrow j} d^k_j (d_i^k - d_i^{k-1}).\]
Let $g_\seqv{U}:= (g_i)_{i\in Q_0}$ where
\[g_i
\begin{array}{c@{}}
\begin{tikzpicture}
\matrix (A) [matrix of math nodes,%
nodes = {node style ge},%
left delimiter = (,%
right delimiter = )]
{%
\id_{d^1_i} & 0 & \cdots & 0 & 0\\
\hphantom{AB} & \id_{d^2_i-d^1_i} & \ddots & \vdots & 0\\
& \hphantom{AB}& \ddots & 0 & \vdots\\
\hphantom{AB} & A^2_i & \ddots & \id_{d^{\nu-1}_i-d^{\nu-2}_i} & 0\\
\hphantom{AB} &\hphantom{AB} & \cdots & A^{\nu-1}_i & \id_{d^{\nu}_i-d^{\nu-1}_i}\\
};
\path ($ (A-2-1)!0.5!(A-4-1) $) node {$A^1_i$};
\draw (A-1-1.south west) rectangle (A-5-1.south east);
\draw ($ (A-2-2.south west)!0.3!(A-2-2.south) $) rectangle (A-5-2.south east);
\draw (A-5-4.north west) rectangle (A-5-4.south east);
\end{tikzpicture}
\end{array}
\in \GL_{d^\nu_i}(k).\]
Then, the map from $X \times U_{\seqv{\dvec{I}}}$ to
$U_{\seqv{\dvec{I}}} \times_{\OFl} \RepFl[Q]{\flvec{d}}$ given by sending
$(M, \seqv{U})$ to $(g_\seqv{U} \cdot M, \seqv{U})$ is an isomorphism which induces
an isomorphism of vector spaces on the fibres.
Therefore, we have that $\pi_2$ is a vector bundle.
Finally, we prove the claim on dimension. Since $\OFl(\flvec{d})$ is smooth,
we have that
\begin{multline*}
\dim \OFl(\flvec{d}) =
\sum_{i \in Q_0} \sum_{k=1}^{\nu-1} \sum_{l=k+1}^\nu (d^k_i - d^{k-1}_{i}) (d^l_i - d^{l-1}_{i})\\
=\sum_{i \in Q_0} \left( \sum_{k=1}^{\nu-1} \sum_{l=k+1}^\nu d^k_i (d^l_i - d^{l-1}_{i}) -
\sum_{k=1}^{\nu-2} \sum_{l=k+2}^\nu d^{k}_{i} (d^l_i - d^{l-1}_{i}) \right)\\
=\sum_{i \in Q_0} \left( d^{\nu-1} (d^\nu_i - d^{\nu-1}_i) + \sum_{k=1}^{\nu-2} d^k_i (d^{k+1}_i - d^k_i) \right) =
\sum_{i \in Q_0} \left(\sum_{k=1}^{\nu-1} d^k_i (d^{k+1}_i - d^k_i) \right).
\end{multline*}
Since
$\RepFl[Q]{\flvec{d}}$ is smooth and $\pi_2$ is a vector bundle we obtain
\begin{multline*}
\dim \RepFl[Q]{\flvec{d}} = \dim \OFl(\flvec{d}) +
\sum_{\alpha \colon i \rightarrow j}\sum_{k=1}^\nu (d^k_i- d^{k-1}_{i}) d^{k}_j\\
=\sum_{i \in Q_0} \left(\sum_{k=1}^{\nu-1} d^k_i (d^{k+1}_i - d^k_i) \right) +
\sum_{\alpha \colon i \rightarrow j}\sum_{k=1}^\nu (d^k_i- d^{k-1}_{i}) d^{k}_j\\
=\sum_{k=1}^{\nu-1} \left( \sum_{i \in Q_0}d^k_i (d^{k+1}_i - d^k_i) +
\sum_{\alpha \colon i \rightarrow j} d^k_i (d^{k}_j-d^{k+1}_j)\right)
+ \sum_{\alpha \colon i \rightarrow j} d^\nu_i d^\nu_j\\
=\sum_{k=1}^{\nu -1} \euf{\dvec{d}^k, \dvec{d}^{k+1} - \dvec{d}^k}_Q + \dim \Rep[Q]{\dvec{d}^\nu}.
\end{multline*}
\end{proof}
\begin{remark}
Note that if $M$ is a $k$-valued point of $\mathcal{A}_\flvec{d}$ for $k$ not algebraically closed,
then $M$ does not necessarily have a flag of type $\flvec{d}$. This only becomes true after a finite
field extension.
\end{remark}
We now can give an estimate for the codimension of $\mathcal{A}_\flvec{d}$ in $\Rep[Q]{\dvec{d}^\nu}$.
For this we use Chevalley's theorem.
\begin{theorem}[Chevalley]
Let $k$ be a field, $X, Y$ irreducible $k$-schemes and $f \colon X \rightarrow Y$ a dominant morphism.
Then for every point $y \in Y$ and every point $x \in f^{-1}(y)$, the scheme theoretic fibre, we
have that
\[ \dim_x f^{-1}(y) \ge \dim X - \dim Y. \]
Moreover, on an open, non-empty subset of $X$ we have equality.
\end{theorem}
\begin{proof}
See \cite[\S 5, Proposition 5.6.5]{EGA4}.
\end{proof}
\begin{theorem}
Let $\flvec{d}$ be a filtration, $K$ a field extension of $k$, $\Lambda := (KQ)A_{\nu+1}$ and
$(M,\seqv{U}) \in \RepFl[Q]{\flvec{d}}(K)$. Let
$\seqv{M}=(M=\dots=M)$ as a $\Lambda$-module. Then we have that
\[ \codim \mathcal{A}_\flvec{d} \le \dim \Ext^1_{\Lambda}(\seqv{U}, \seqv{M}/\seqv{U})
\le \dim \Ext^1_{\Lambda}(\seqv{U}, \seqv{M})
\le \dim \Ext^1_{KQ}(M, M). \]
\label{them:codimflag}
\end{theorem}
\begin{proof}
Since $\dim \mathcal{A}_{\flvec{d}}$ is stable under flat base change we can
assume $k=K$.
Let $\seqv{V} := \seqv{M}/\seqv{U}$.
Then we have the following short exact sequence of $\Lambda$-modules:
\[
\begin{CD}
0:@.\quad @. 0 @>>> 0 @>>> \cdots @>>> 0\\
@VVV @. @VVV @VVV @. @VVV\\
\seqv{U}:@.\quad @. U^0 @>>> U^1 @>>> \cdots @>>> U^\nu\\
@VVV @. @VVV @VVV @. @VVV\\
\seqv{M}:@.\quad @. M @= M @= \cdots @= M\\
@VVV @. @VVV @VVV @. @VVV\\
\seqv{V}:@.\quad @. V^0 @>>> V^1 @>>> \cdots @>>> V^\nu\\
@VVV @. @VVV @VVV @. @VVV\\
0:@.\quad @. 0 @>>> 0 @>>> \cdots @>>> 0.\\
\end{CD}
\]
We already know that $\Hom_\Lambda(\seqv{U}, \seqv{V})$
is the tangent space of $\Fl{\flvec{d}}{M}$ at the point $\seqv{U}$.
Using Chevalley's theorem, we have that
\[\dim \Hom_\Lambda(\seqv{U}, \seqv{V}) \ge \dim_\seqv{U}\Fl{\flvec{d}}{M} \ge
\dim \RepFl[Q]{\flvec{d}} - \dim \mathcal{A}_\flvec{d} \]
and therefore
\[ \dim \mathcal{A}_\flvec{d} \ge \dim \RepFl[Q]{\flvec{d}} - \dim \Hom_\Lambda(\seqv{U}, \seqv{V}).\]
We now calculate
\begin{multline*}
\euf{\seqv{U}, \seqv{V}}_\Lambda = \sum_{k=0}^\nu \euf{U^k, V^k}_{Q} - \sum_{k=0}^{\nu-1} \euf{U^k, V^{k+1}}_{Q}\\
= \sum_{k=1}^{\nu-1} \euf{\dvec{d}^k, \dvec{d}^\nu - \dvec{d}^k}_{Q} - \sum_{k=1}^{\nu-1}
\euf{\dvec{d}^k, \dvec{d}^\nu - \dvec{d}^{k+1}}_{Q}
= \sum_{k=1}^{\nu-1} \euf{\dvec{d}^k, \dvec{d}^{k+1} - \dvec{d}^k}_{Q}.
\end{multline*}
Recall that
\[ \dim \RepFl[Q]{\seqv{\dvec{d}}} = \sum_{k=1}^{\nu-1}
\euf{\dvec{d}^k, \dvec{d}^{k+1} - \dvec{d}^k}_Q + \dim \Rep[Q]{\dvec{d}^\nu}. \]
In total
\begin{multline*}
\codim \mathcal{A}_\flvec{d} \le \dim \Rep{\dvec{d}} + \dim \Hom_\Lambda (\seqv{U}, \seqv{V})
- \dim \RepFl[Q]{\flvec{d}}\\
= \dim \Hom_\Lambda (\seqv{U}, \seqv{V}) - \sum_{k=1}^{\nu-1} \euf{\dvec{d}^k, \dvec{d}^{k+1} - \dvec{d}^k}_{Q}
=\dim \Hom_\Lambda (\seqv{U}, \seqv{V}) - \bform{\seqv{U}}{\seqv{V}}_\Lambda\\
= \dim \Ext^1_\Lambda(\seqv{U}, \seqv{V}) - \dim \Ext^2_\Lambda(\seqv{U}, \seqv{V}).
\end{multline*}
Here we have the last equality since $\gldim \Lambda \le 2$.
Since $\gldim kQ = 1$ and $P = P = \dots = P$ is projective in $\Mod \Lambda$
for every projective $P$ in $\Mod kQ$ we see that
$\pd_\Lambda \seqv{M} \le 1$ and similarly $\id_\Lambda \seqv{M} \le 1$. Consider, as before, the short exact sequence
\[ 0 \rightarrow \seqv{U} \rightarrow \seqv{M} \rightarrow \seqv{V} \rightarrow 0. \]
Since $\seqv{M}$ has projective dimension less than two we have that
$(\seqv{U}, \seqv{U})^2=0$ and that $(\seqv{U}, \seqv{V})^2 = 0$.
Applying $(-,\seqv{M})$ gives
a surjection $(\seqv{M},\seqv{M})^1 \rightarrow (\seqv{U}, \seqv{M})^1$.
Applying $(\seqv{U},-)$ yields a surjection $(\seqv{U}, \seqv{M})^1 \rightarrow (\seqv{U}, \seqv{V})^1$.
Hence the above result simplifies to
\[ \codim \mathcal{A}_\flvec{a} \le \dim \Ext^1(\seqv{U}, \seqv{V}) \le \dim \Ext^1(\seqv{U}, \seqv{M})
\le \dim \Ext^1(\seqv{M}, \seqv{M}). \]
Obviously, $\Ext^1_\Lambda(\seqv{M}, \seqv{M}) \cong \Ext^1_{kQ}(M,M)$ and the claim follows.
\end{proof}
\begin{remark}
Note that if the characteristic of $k$ is $0$, then, by generic smoothness,
there is a point $M \in \mathcal{A}_\flvec{d}$ and an $\seqv{U} \in \Fl{\flvec{d}}{M}$ such
that $\Fl{\flvec{d}}{M}$ is smooth in $\seqv{U}$ and
the value of $\dim \Ext^1_{\Lambda \otimes K}(\seqv{U}, (M\otimes K)/\seqv{U})$ is minimal. In this case we have that
\[ \codim \mathcal{A}_\flvec{d} = \dim \Ext^1_{\Lambda \otimes K}(\seqv{U}, (M\otimes K)/\seqv{U}).\]
\end{remark}
We also construct an additional vector bundle.
\begin{definition}
Let $\flvec{d}$ be a filtration.
Let $\Sch{Rep}_{Q,A_{\nu+1}}(\flvec{d})$ be the scheme given via its functor of points
\begin{multline*}
\Sch{Rep}_{Q,A_{\nu+1}}(\flvec{d})(R) := \\
\Set{ (\seqv{U}, \seqv{f}) \in \prod_{i=0}^\nu \Rep[Q]{\dvec{d}^i}(R) \times
\prod_{i=0}^{\nu-1} \Hom(\dvec{d}^i, \dvec{d}^{i+1})(R) | f^i \in \Hom_{RQ}(U^i, U^{i+1})}.
\end{multline*}
Let $\Sch{IRep}_{Q,A_{\nu+1}}(\flvec{d})$ be the open subscheme of
$\Sch{Rep}_{Q,A_{\nu+1}}(\flvec{d})$ given by its functor of points
\[
\Sch{IRep}_{Q,A_{\nu+1}}(\flvec{d})(R) := \Set{ (\seqv{U}, \seqv{f}) \in \Sch{Rep}_{Q,A_{\nu+1}}(\flvec{d})(R) |
f^i \in \Sch{Inj}(\dvec{d}^i, \dvec{d}^{i+1})(R)}.
\]
\end{definition}
\begin{remark}
Note that $\Sch{Rep}_{Q, A_{\nu +1}}(\flvec{d})(k)$ consists of sequences
of $k$-representations of $Q$. Therefore, these are modules over $(kQ)A_{\nu+1}$. Vice
versa, every $(kQ)A_{\nu+1}$-module of dimension vector $\flvec{d}$ is isomorphic to an element of
$\Sch{Rep}_{Q, A_{\nu +1}}(\flvec{d})(k)$.
We will often write $\seqv{U}$ instead of $(\seqv{U}, \seqv{f})$ for
an $(\seqv{U}, \seqv{f}) \in \Sch{Rep}_{Q, A_{\nu +1}}(\flvec{d})(R)$.
\end{remark}
\begin{lemma}
Let $\flvec{d}$ be a filtration.
Then the projection
\begin{alignat*}{2}
\pi&\colon\quad& \Sch{IRep}_{Q,A_{\nu+1}}(\flvec{d}) &\rightarrow
\prod_{i=0}^{\nu-1} \Sch{Inj}(\dvec{d}^i, \dvec{d}^{i+1})\\
\intertext{given by sending}
&&(\seqv{U}, \seqv{f}) &\mapsto \seqv{f}
\end{alignat*}
is a vector bundle and therefore flat.
In particular, $\Sch{IRep}_{Q,A_{\nu+1}}(\flvec{d})$ is smooth and irreducible.
\end{lemma}
\begin{proof}
The first part is analogously to lemma \ref{lmm:geomflags:projtriv}. Irreducibility
then follows by the fact that flat morphisms are open and proposition \ref{isch:propn:openirred}.
\end{proof}
Before we continue, we give the following easy lemma, stated by K. Bongartz
in \cite{Bongartz_singularities}, which gives
rise to a whole class of vector bundles.
\begin{lemma}
\label{lmm:geomflags:vecbun}
Let $X$ be a variety over a ground ring $k$. Let $m, n \in \mathbb{N}$ and
$f \colon X \rightarrow \Hom(m,n)_k$ a morphism. Then
for any $r\in \mathbb{N}$, the variety $X(r)$ given by the functor of points
\[ X(r)(R):= \Set{ x \in X(R) | f(x) \in\Hom(m,n)_{m-r}(R)} \]
is a locally closed subvariety of $X$. Moreover, the closed subvariety
\[
U_r(R):=\Set{ (x, v) \in X(r)(R)\times R^m | f(x)(v) = 0 }
\]
of $X(r) \times k^m$
is a sub
vector bundle
of rank $r$ over $X(r)$.
\end{lemma}
\begin{proof}
The claim follows easily by using Fitting ideals.
\end{proof}
\begin{example}
Let $(\seqv{M},\seqv{g}) \in \Rep[Q,A_{\nu+1}]{\flvec{e}}(k)$.
Let $\varphi \colon \Rep[Q,A_{\nu+1}]{\flvec{d}} \rightarrow \Hom(m,n)$
be the morphism given by
\[ (\seqv{U},\seqv{f}) \mapsto \left( \seqv{h}=(h_i^k) \mapsto
\left( (h^k_j U_\alpha^k - M_\alpha^k h^k_i)_{\substack{\alpha\colon i \rightarrow j\\0 \le k \le \nu}},
(h^{k+1}_i f^k_i - g^k_i h^k_i)_{\substack{i \in Q_0\\0 \le k \le \nu}} \right) \right)
\]
for every $(\seqv{U}, \seqv{f}) \in \Rep[Q,A_{\nu+1}]{\flvec{d}}(R)$,
where
\begin{align*}
m &= \sum_{i \in Q_0} \sum_{k=0}^{\nu} d_i^k e_i^k &
&\text{and}&
n &= \sum_{k=0}^\nu \sum_{\alpha \colon i \rightarrow j} d_i^k e_j^k +
\sum_{k=0}^{\nu-1} \sum_{i \in Q_0} d_i^k e_i^{k+1}.
\end{align*}
Then $\seqv{h} \in \ker \varphi (\seqv{U}, \seqv{f})$ if and only if
$\seqv{h} \in \Hom_{(R Q)A_{\nu+1}}(\seqv{U}, \seqv{M}\otimes R)$.
Set
\[
\RepHom[Q,A_{\nu+1}]{\flvec{d}, \seqv{M}}_r := U_r(R)
\]
from the previous lemma.
Note that elements $(\seqv{U}, \seqv{h}) \in \RepHom[Q,A_{\nu+1}]{\flvec{d}, \seqv{M}}_r(R)$
are all pairs consisting of a representation $\seqv{U} \in \Rep[Q,A_{\nu+1}]{\flvec{d}}(R)$
and a morphism $\seqv{h} \in \Hom_{(RQ)A_{\nu+1}}(\seqv{U}, \seqv{M})$ such that
$\rank \Hom_{(RQ)A_{\nu+1}}(\seqv{U}, \seqv{M})=r$.
The lemma yields that the projection
\begin{align*}
\RepHom[Q,A_{\nu+1}]{\flvec{d}, \seqv{M}}_r &\rightarrow \Rep[Q,A_{\nu+1}]{\flvec{d}}(r)\\
(\seqv{U}, \seqv{h}) &\mapsto \seqv{U}
\end{align*}
is a vector bundle of rank $r$.
It also stays a vector bundle if we restrict it to
the open subset $\Sch{IRep}_{Q,A_{\nu+1}}(\flvec{d})(r)$ of $\Rep[Q,A_{\nu+1}]{\flvec{d}}(r)$.
We denote the preimage under the projection to this variety by
$\Sch{IRepHom}_{Q, A_{\nu+1}}(\flvec{d},\seqv{M})_r$.\label{ex:geomflags:hombundle}
\end{example}
We obtain the following.
\begin{theorem}
Let $\flvec{d}$ be a filtration and $K$ a field extension of $k$.
\label{theom:geomflags:irred}
Assume that there is an $M \in \mathcal{A}_\flvec{d}(K)$ such that $\dim \Ext^1_{KQ}(M,M) =
\codim \mathcal{A}_\flvec{d}$. Then $\Fl[Q]{\flvec{d}}{M}$ is smooth over $K$ and
geometrically irreducible.
\end{theorem}
\begin{proof}
Smoothness is an immediate consequence of the last theorem, since we have that
$\dim T_{\seqv{U}}\Fl[Q]{\flvec{d}}{M}$ is constant and smaller or equal
to the dimension at each irreducible component living in $\seqv{U}$. Since $K$
is a field this implies smoothness. See \cite[I, \S 4, no 4]{DG}.
Now we prove irreducibility. By base change we can assume that $K$ is algebraically closed.
Consider all the following schemes as $K$-varieties.
We construct the following diagram of varieties.
\[
\xymatrix{ & \Sch{IRepHom}_{Q, A_{\nu+1}}(\flvec{d},\seqv{M})_r \ar@{->>}[d]^{\text{vector bundle}} &
\Sch{IRepInj}_{Q, A_{\nu +1}}(\flvec{d},\seqv{M})_r \ar@{_{(}->}[l]_{\text{open}} \ar@{->>}[d]\\
\Sch{IRep}_{Q,A_{\nu+1}}(\flvec{d}) & \ar@{_{(}->}[l]_{\text{open}} \Sch{IRep}_{Q,A_{\nu+1}}(\flvec{d})(r)
& \Fl[Q]{\flvec{d}}{M},\\
}
\]
$r$ being equal to $\euf{\flvec{d}, \seqv{M}}_\Lambda + [M,M]^1$.
Since open subvarieties and images of irreducible varieties are again irreducible and by application
of proposition \ref{isch:propn:openirred} we then obtain that $\Fl[Q]{\flvec{d}}{M}$ also is irreducible.
Consider the minimal value $r$ of $\dim\Hom(\seqv{U}, \seqv{M})$ for
$\seqv{U} \in \Sch{IRep}(\flvec{d})(K)$. Denote by
\begin{alignat*}{2}
\pi \colon&& \Sch{IRep}(\flvec{d}) &\rightarrow \Rep[Q]{\dvec{d}^\nu}\\
&& \seqv{U} &\mapsto U^\nu.
\end{alignat*}
Since $\mathcal{O}_M$ is open in $\mathcal{A}_\flvec{d}$ and $\Sch{IRep}$ is irreducible,
the intersection of the two open sets $\pi^{-1}(\mathcal{O}_M)$
and
$\Sch{IRep}(r)$ is non-empty. For all elements $\seqv{U}$ of $\pi^{-1}(\mathcal{O}_M)$
we have, by theorem \ref{them:codimflag}, that $[\seqv{U}, \seqv{M}]^1_\Lambda = [M,M]^1_Q$.
We already saw that $[\seqv{U}, \seqv{M}]^2 = 0$, therefore
\[
[\seqv{U}, \seqv{M}]_\Lambda = \euf{\seqv{U}, \seqv{M}}_\Lambda + [\seqv{U}, \seqv{M}]^1_\Lambda
= \euf{\flvec{d}, \seqv{M}}_\Lambda + [M,M]^1_Q.
\]
This means that the dimension of the homomorphism space is constant on $\pi^{-1}(\mathcal{O}_M)$ and
we obtain that $r=\euf{\flvec{d}, \seqv{M}}_\Lambda + [M,M]^1_Q$. Moreover, $\Sch{IRep}_r$
is irreducible as an open subset of $\Sch{IRep}$.
We then have that $\Sch{IRepHom}_{Q,A_{\nu+1}}(\flvec{d},\seqv{M})_r$
is irreducible, since it is a vector bundle on $\Sch{IRep}_r$ by example \ref{ex:geomflags:hombundle}.
Take the open subvariety $\Sch{IRepInj
(\flvec{d},\seqv{M})_r$
of $\Sch{IRepHom
(\flvec{d},\seqv{M})_r$ where the morphism to $\seqv{M}$ is injective.
It is irreducible as an open subset of an irreducible variety.
The projection
from this variety to $\Fl[Q]{\flvec{d}}{M}$ is surjective since
$\pi^{-1}(\mathcal{O}_M)$ is contained in $\Sch{IRep}(\flvec{d})(r)$, and therefore $\Fl[Q]{\flvec{d}}{M}$ is irreducible.
\end{proof}
We now want to interpret theorem \ref{theom:geomflags:irred} in terms of Hall numbers. Let $X_0$
be a variety defined over a finite field $\mathbb{F}_q$, where $q=p^n$ for a prime $n$. Denote
by $\overline{\mathbb{F}_q}$ the algebraic closure of $\mathbb{F}_q$ and by $X := X_0 \otimes \overline{\mathbb{F}_q}$
the variety obtained from $X_0$
by base change from $\mathbb{F}_q$ to $\overline{\mathbb{F}_q}$. Let $F$ be the Frobenius automorphism acting on $X$.
Denote by $H^i(X, \mathbb{Q}_\ell)$ the $\ell$-adic cohomology group with compact support for a prime $\ell \neq p$,
see for example \cite{Freitag_etaleweil}. Denote by $F^i$ the induced action of $F$ on
cohomology $H^i(X, \mathbb{Q}_\ell)$.
P. Deligne proved the following theorem.
\begin{theorem}[P. Deligne \cite{Deligne_weil2}, 3.3.9]
Let $X_0$ be a proper and smooth variety over $\mathbb{F}_q$.
For every $i$, the characteristic polynomial $\det( T \id - F^i, H^i(X, \mathbb{Q}_\ell))$
is a polynomial with coefficients in $\mathbb{Z}$, independent of $\ell$ ( $\ell \neq p)$.
The complex roots $\alpha$ of this polynomial have absolute value $\lvert \alpha \rvert = q^{\frac{i}{2}}$.
\end{theorem}
Moreover, the Lefschetz fixed point formula yields that
\[
\#X_0(\mathbb{F}_{q^n}) = \sum_{i \ge 0} (-1)^i \Tr( (F^i)^n, H^i(X, \mathbb{Q}_\ell)).
\]
Assume now that there is a polynomial $P \in \mathbb{Q}[t]$ such that, for each finite field extension
$L/\mathbb{F}_q$ we have that $\# X_0(L) = P(|L|)$, $|L|$ being the number of elements of the
finite field. We call $P$ the counting polynomial of $X_0$. Then we have the following.
\begin{theorem}
Let $X_0$ be proper and smooth over $\mathbb{F}_q$ with counting polynomial $P$.
Then odd cohomology of $X$ vanishes and
\[ P(t) = \sum_{i=0}^{\dim X_0} \dim H^{2i}(X, \mathbb{Q}_\ell) t^i.
\]
\end{theorem}
\begin{proof}
See \cite[Lemma A.1]{BillVandenBergh_absolutelyindec}.
\end{proof}
Assume now that $Y$ is a projective scheme over $\mathbb{Z}$ and set
$Y_k:=Y \otimes k$ for any field $k$. Note that for $Y$ $\ell$-adic
cohomology agrees with $\ell$-adic cohomology with compact support.
Assume furthermore that there is a counting polynomial $P \in \mathbb{Q}[t]$ such that,
for each finite field $k$, we have that $\# Y_k(k) = P(|k|)$.
By the base change theorem \cite[Theorem 1.6.1]{Freitag_etaleweil} we have
\[ H^i(Y_{\overline{\mathbb{Q}}}, \mathbb{Q}_\ell) \cong H^i (Y_\mathbb{C}, \mathbb{Q}_\ell). \]
By the comparison theorem \cite[Theorem 1.11.6]{Freitag_etaleweil} we have
\[ H^i(Y_\mathbb{C}, \mathbb{Q}_\ell) \cong H^i (Y_\mathbb{C}(\mathbb{C}), \mathbb{Q}_\ell),\]
where on the right hand side we consider the usual cohomology of the complex analytic manifold
attached to $Y_\mathbb{C}$.
Moreover, there is an open, dense subset $U$ of $\Spec \mathbb{Z}$ such that
$H^i (Y_{\overline{\kappa(v)}}, \mathbb{Q}_\ell) \cong H^i (Y_{\overline{\mathbb{Q}}}, \mathbb{Q}_\ell)$ for all
$v \in U$, where $\kappa(v)$ denotes the residue field at $v$. This means that for almost all primes $p$ we have that
\[H^i (Y_{\overline{\mathbb{F}_p}}, \mathbb{Q}_\ell) \cong H^i (Y_{\overline{\mathbb{Q}}}, \mathbb{Q}_\ell) \cong H^i(Y_\mathbb{C}(\mathbb{C}), \mathbb{Q}_\ell).\]
Therefore, if we know the Betti numbers of $Y_\mathbb{C}(\mathbb{C})$, then we know the coefficients
of the counting polynomial. In order to apply this to our situation we use
the following theorem of W. Crawley-Boevey \cite{Bill_rigidintegral}.
\begin{theorem}
\label{theom:geomflags:rigidreps}
Let $M$ be an $k$-representation without self-extensions. Then there is
a $\mathbb{Z}$-representation $N$ such that $M= N \otimes k$ and for
all fields $K$ we have that $\Ext (N \otimes K, N \otimes K) = 0$.
\end{theorem}
Putting all this together, we obtain the following.
\begin{theorem}
Assume that there is an $M \in \mathcal{A}_\flvec{d}(\mathbb{Q})$, being a direct sum of
exceptional representations, such that
\[
\dim \Ext^1_{\mathbb{Q} Q}(M,M) =
\codim \mathcal{A}_\flvec{d}.
\]
Let $N$ be a $\mathbb{Z}$-representation and $P \in \mathbb{Q}[t]$ a polynomial, such that
$N \otimes \mathbb{Q} \cong M$ and
$\# \Fl[Q]{\flvec{d}}{N \otimes \mathbb{F}_q} = P(q)$ for every prime power $q$. Then
$P(0) = 1$ and $P(1) = \chi\left( \Fl[Q]{\flvec{d}}{M \otimes \mathbb{C}}\right) > 0$.
Moreover, if $Q$ is Dynkin or extended Dynkin, then there is a representation $N$ and
a polynomial $P$ with the required properties.
\end{theorem}
\begin{proof}
Let $X := \Fl[Q]{\flvec{d}}{N}$ as a scheme
over $\mathbb{Z}$.
Using theorem \ref{theom:geomflags:irred} we
obtain that $X_k$ is smooth and irreducible for every field $k$. By the previous discussion
we have then that the
$i$-th coefficient of $P$ is exactly
$\dim H^{2i}(X_\mathbb{C}(\mathbb{C}), \mathbb{Q}_\ell)$ and that odd cohomology vanishes. Therefore,
\[
0<\sum\dim H^{2i}(X_\mathbb{C}(\mathbb{C}), \mathbb{Q}_\ell) = \chi\left( X_\mathbb{C} \right) = P(1).
\]
By irreducibility we have that
\[
P(0) = \dim H^{0}(X_\mathbb{C}(\mathbb{C}), \mathbb{Q}_\ell) = 1
\]
and this proves the first claims.
If $Q$ is Dynkin or extended Dynkin,
then let $N$ be the $\mathbb{Z}$-representation given by theorem \ref{theom:geomflags:rigidreps}. We have the
polynomial $P$ since we have Hall polynomials in both cases for exceptional representations
by results of Ringel \cite{Ringel_hallpolysforrepfiniteheralgs} and Hubery
\cite{Hubery_hallpolys}.
\end{proof}
Let $q$ be a prime power and $M$ an $\mathbb{F}_q$-representation of
a Dynkin quiver $Q$. If $\overline{\mathcal{O}_M}=\mathcal{A}_w$ for some word $w$ in simples, then
$F_w^M = 1 \mod q$. Therefore, in the Dynkin case, the generic coefficient
in the expression $u_w$ in the Hall algebra is equal to one modulo $q$. In section \ref{sec:reflhallnum}
we will show that even
more is true, namely that every non-zero coefficient in this expression is equal to one modulo $q$.
\section{A geometric version of BGP reflection functors}
In this chapter we want to define a geometric version of BGP reflection functors. First
we recall the definition of the usual BGP reflection functors as introduced by
Bern{\v{s}}te{\u\i}n, Gel$'$fand and Ponomarev
\cite{BernsteinGelfandPonomarev_coxeter}. Then we go on to define reflection functors on flags
of subrepresentations. In the remainder of this paper we fix a field $k$ and consider only
$k$-valued points of all varieties, if not otherwise stated.
Now we recall the definition of BGP reflection functors. The
main reference for this section is \cite{Ringel_integral}. For a nice
introduction see \cite{HKrause_repsofalgs}.
Let $Q$ be a quiver and $\dvec{d}$ be a dimension vector.
For each vertex $a \in Q_0$ we have the reflection
\begin{alignat*}{2}
\sigma_a &\colon\quad & \mathbb{Z} Q_0 & \rightarrow \mathbb{Z} Q_0\\
& & \dvec{d} &\mapsto \dvec{d} - \sbform{\dvec{d}}{\epsilon_a}_Q \epsilon_a.
\end{alignat*}
If $Q$ has no loop at $a$, one easily checks that $\sigma_a^2 \dvec{d} = \dvec{d}$.
We also define reflections on the quiver itself. The quiver $\sigma_a Q$ is
obtained from $Q$ by reversing all arrows ending or starting in $a$. If $\alpha\colon a \rightarrow j$
is an arrow in $Q_1$, then we call $\alpha^* : j \rightarrow a$ the arrow in the other direction and
analogously for $\alpha \colon i \rightarrow a$. We have that $(\sigma_a Q)_0 = Q_0$, therefore we can
regard $\sigma_a \dvec{d}$ as a dimension vector on $\sigma_a Q$.
Obviously, $\sigma_a^2 Q =Q$ if we identify $\alpha^{* *}$ with $\alpha$ (and we will do
this in the remainder).
If $a$ is a sink of $Q$, we define for each $k$-representation $M$ of $Q$ the homomorphism
\[
\phi^M_a \colon \bigoplus_{\alpha: j \rightarrow a} M_j \xrightarrow{\left(
\begin{smallmatrix}
M_\alpha
\end{smallmatrix}\right)} M_a.
\]
Dually, if $b$ is a source of $Q$, we define
\[
\phi^M_b \colon M_b \xrightarrow{\left(
\begin{smallmatrix}
M_\alpha
\end{smallmatrix}\right)} \bigoplus_{\alpha: b \rightarrow j} M_j .
\]
Note that $D\phi^{DM}_a = \phi^M_a$, where $D$ denotes the $k$-dual, and
that $d_a - \rank \phi^X_a = \dim \Hom(X, S_a)$ for $a$ a sink of $Q$ and
a representation $X$ of dimension vector $\dvec{d}$.
We define a pair of reflection functors
$S_a^+$ and $S_b^-$. To this end we fix a $k$-representation $M$ of $Q$ of dimension vector $\dvec{d}$.
If the vertex $a$ is a sink of $Q$ we construct
\[ S_a^+ \colon \repK{Q}{k} \rightarrow \repK{\sigma_a Q}{k} \]
as follows. We define $S_a^+ M := N$ by letting $N_i := M_i$ for a vertex $i \neq a$ and letting $N_a$ be
the kernel of the map $\Phi^M_a$. Denote by
$\iota \colon \Ker \phi^M_a \rightarrow \bigoplus M_j$ the canonical inclusion and by
$\pi_i \colon \bigoplus M_j \rightarrow M_i$ the canonical projection.
Then, for each
$\alpha: i \rightarrow a$ we let $N_{\alpha^*} \colon \Ker \phi^M_a \rightarrow M_i$
be
the composition $\pi_i \circ \iota$ making the following diagram commute
\[
\begin{CD}
0 @>>> \Ker \phi^M_a @>>\iota> \bigoplus M_j @>>> M_a\\
@. @VV{N_{\alpha^*}}V @VV{\pi_i}V @.\\
@. M_i @= M_i. \\
\end{CD}
\]
We obtain
a $k$-representation $N$ of $\sigma_a Q$. We call this representation $S^+_a M$.
This construction yields a functor $S^+_a$.
Let $s := d_a - \rank \phi^M_a$ be
the codimension
of $\Bild \phi^M_a$ in $M_a$.
We have that
\[e_a := \dim (S^+_a M)_a = \sum_{\alpha: i \rightarrow a} d_i - \rank \Phi^M_a =
d_a - (\dvec{d}, \epsilon_a) + s = d_a -s - (\dvec{d}- s \epsilon_a, \epsilon_a).\]
Therefore, $\dimv{ S^+_a M } = \sigma_a (\dvec{d} - s \epsilon_a) = \sigma_a (\dvec{d}) + s \epsilon_a$.
Dually, for a sink of $b$ of $Q$ we
obtain a functor $S^-_b$.
For a sink $a$ of $Q$ we have that $(S_a^-, S_a^+)$ is a pair of adjoint functors and
that $S_a^+$ is left exact and $S_a^-$ is right exact. There is a natural
monomorphism $\iota_{a,M} \colon S_a^- S_a^+ M \rightarrow M$ for $M \in \repK{Q}{k}$ and a natural epimorphism
$\pi_{a,N} \colon N \rightarrow S_a^+ S_a^- N$ for $N \in \repK{\sigma_a Q}{k}$.
We have the following lemma.
\begin{lemma}
Let $a$ be a sink and $X$ an indecomposable $k$-representation of $Q$. Then, the following are equivalent:
\begin{enumerate}
\item $X \ncong S_a$.
\item $S^+_a X$ is indecomposable.
\item $S^+_a X \neq 0$.
\item $S_a^- S_a^+ X \cong X$ via the natural inclusion.
\item The map $\Phi_a^X$ is an epimorphism.
\item $\sigma_a (\dvec{\dim} X) > 0$.
\item $\dvec{\dim} S^+_a X = \sigma_a (\dvec{\dim} X)$.
\end{enumerate}
\end{lemma}
An sequence $(i_1, \dots, i_r)$ of the vertices of $Q$ is called admissible,
if
$i_p$ is a sink in $\sigma_{i_{p-1}} \dotsm \sigma_{i_1} Q$ for each $1 \le p \le r$.
An admissible sequence is called admissible ordering if each vertex of $Q$ appears exactly once
in the sequence.
Note that there is an admissible ordering if and only if
$Q$ has no oriented cycles. Such a quiver is called acyclic. In the following we always assume that $Q$ is
acyclic.
For any admissible sequence of sinks $w=(i_1, \dots, i_r)$ define
\[S_w^+ := S_{i_r}^+ \circ \dots \circ S_{i_1}^+. \]
If $w=(i_1, \dots, i_n)$ is an admissible ordering of $Q$, we define the Coxeter functors
\begin{align*}
C^+ &:= S^+_{i_n} \circ \dots \circ S^+_{i_1} & C^- &:= S^-_{i_1} \circ \dots \circ S^-_{i_n}. \\
\end{align*}
Since both reverse each arrow of $Q$ exactly twice, these are endofunctors of $\repK{Q}{k}$. Neither functor
depends on the
choice of the admissible ordering.
A $k$-representation $P$ is projective if and only if $C^+ P =0$. Dually, a $k$-representation $I$
is injective if and only if $C^- I =0$.
An indecomposable $k$-representation $M$ of $Q$ is called preprojective if $(C^+)^r M=0$ for some $r\ge0$
and preinjective if $(C^-)^r M =0$ for some $r\ge0$. If $M$ is neither preprojective nor preinjective, then
$M$ is called regular.
An arbitrary $k$-representation is called preprojective (or preinjective or regular) if it is isomorphic to a direct sum of indecomposable
preprojective (or preinjective or regular) representations.
Let $M$ be a $k$-representation of $Q$ and $\flvec{d}$ a filtration of $\dimve M$.
We want to define reflections on a flag $\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}$. Let $a$ be
a sink and let $\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}$. Then, for each $i$, we have the following commutative diagram with exact rows.
\[
\begin{CD}
0 @>>> (S^+_a U^{i-1})_a @>>> \bigoplus U^{i-1}_j @>{\phi^{U^{i-1}}_a}>> \Bild \phi^{U^{i-1}}_a @>>> 0\\
@. @VVfV @VVV @VVgV @.\\
0 @>>> (S^+_a U^i)_a @>>> \bigoplus U^i_j @>>\phi^{U^i}_a>
\Bild \phi^{U^i}_a @>>> 0\\
\end{CD}
\]
By definition, $g$ and the map in the middle are injective. Therefore, $f$ is injective. This immediately yields that
$S^+_a \seqv{U}$ is a new quiver flag of $S^+_a M = S^+_a U^\nu$. The problem is that the
dimension of $S^+_a U^i$ is dependent on the rank of $\phi^{U^i}_a$.
This motivates the next definition. Recall that $d_a - \rank \phi^X_a = \dim \Hom(X, S_a)$
for a representation $X$ of dimension vector $\dvec{d}$.
\begin{definition}
Let $a$ be a sink, $\dvec{d}$ a dimension vector and $s$ an integer. Define
\[ \Rep[Q]{\dvec{d}}\lrang{a}^{s} :=
\Set{ M \in \Rep[Q]{\dvec{d}} | \dim \Hom(M, S_a) = s}.\]
Let $\seqv{\dvec{d}}$ be a $(\nu+1)$-tuple of dimension vectors
and
$\seqv{r} = (r^0, r^1, \dots , r^\nu)$ be a $(\nu+1)$-tuple of integers.
For each representation $M$ define
\[
\Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{r}} :=
\Set{ \seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M} |
\dim \Hom(U^i, S_a) = r^i}.
\]
Moreover, let
\[ \Rep[Q]{\dvec{d}}\lrang{a} := \Rep[Q]{\dvec{d}}\lrang{a}^0\] and
\[ \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a} := \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{0}}.\]
\end{definition}
\begin{remark}
Recall that a filtration $\flvec{d}$ of $\dimve M$ is a sequence of dimension
vectors such that $\dvec{d}^0=0$, $\dvec{d}^\nu = \dimve M$ and that $\dvec{d}^i \le \dvec{d}^{i+1}$.
In order to know a filtration $\flvec{d}$ it is enough to know the terms
$\dvec{d}^1, \dots, \dvec{d}^{\nu-1}$, since $\dvec{d}^0$ is always $0$ and
$\dvec{d}^\nu$ is always $\dimve M$. Therefore, we identify the $(\nu-1)$-tuple
$(\dvec{d}^1, \dots, \dvec{d}^{\nu-1})$ with the $(\nu+1)$-tuple $\flvec{d}$.
\end{remark}
\begin{example}
Let
\[Q =
\makeatletter%
\let\ASYencoding\f@encoding%
\let\ASYfamily\f@family%
\let\ASYseries\f@series%
\let\ASYshape\f@shape%
\makeatother%
\setlength{\unitlength}{1pt}
\includegraphics{geomreflfun-2_0.pdf}%
\definecolor{ASYcolor}{gray}{0.000000}\color{ASYcolor}
\fontsize{12.000000}{14.400000}\selectfont
\usefont{\ASYencoding}{\ASYfamily}{\ASYseries}{\ASYshape}%
\ASYalign(-53.064667,4.770035)(-0.500000,-0.500000){1.000000 0.000000 0.000000 1.000000}{$1$}
\ASYalign(-3.840845,4.770035)(-0.500000,-0.500000){1.000000 0.000000 0.000000 1.000000}{$2$}
\ASYalign(-28.452756,8.383535)(-0.500000,0.000000){1.000000 0.000000 0.000000 1.000000}{$\scriptstyle\alpha$}
.\]
Consider the representation $M$ given by $M_1 = M_2 = k^2$ and
$M_\alpha =\left(\begin{smallmatrix}
1 & 0\\
0 & 0\\
\end{smallmatrix}\right)$. We have that
$M \in \Rep[Q]{(2,2)}\lrang{2}^1$. Now consider flags of type $((0,0), (1,1), (2,2))$, i.e.
subrepresentations $N$ of dimension vector $(1,1)$. We need two injective linear maps
$f_1, f_2 \colon k^1 \rightarrow k^2$ making the following diagram commutative.
\[
\xymatrix{
k \ar[r]^{N_\alpha} \ar[d]_{f_1} & k \ar[d]^{f_2}\\
k^2 \ar[r]_{\left(\begin{smallmatrix}
1 & 0\\
0 & 0
\end{smallmatrix}\right)} & k^2
}
\]
We have the following situations.
\label{bsp:grass}
\begin{itemize}
\item $N \in \Gr[Q]{(1,1)}{M}\lrang{2}^1$: This means that $N_\alpha = 0$. Therefore,
we need that the image of $f_1$ is in
the kernel of $M_\alpha$, which is $1$-dimensional. Hence, a subrepresentation in
$\Gr[Q]{(1,1)}{M}\lrang{2}^1$ is given by $f_1 = \left( \begin{smallmatrix}
0\\
1
\end{smallmatrix}\right)$ and $f_2$ being an arbitrary inclusion. The point
$f_1 = f_2 = \left( \begin{smallmatrix}
0\\
1
\end{smallmatrix}\right)$ is special, since for this inclusion we have
that $M/N \cong S_1 \oplus S_2$ and otherwise $M/N \cong k \overset{1}{\rightarrow} k$.
\item $N \in \Gr[Q]{(1,1)}{M}\lrang{2}^0$: This means that $N_\alpha \neq 0$. Therefore,
we need that the image of $f_1$ is not in
the kernel of $M_\alpha$, which is $1$-dimensional. Hence, a subrepresentation in
$\Gr[Q]{(1,1)}{M}\lrang{2}^0$ is given by $f_1=f_2= \left( \begin{smallmatrix}
1\\
x
\end{smallmatrix}\right)$ for any $x \in k$.
\end{itemize}
The variety $\Gr[Q]{(1,1)}{M}$ consists therefore of two $\mathbb{P}^1_k$ glued together at one point.
Graphically,
\[
\Gr[Q]{(1,1)}{M} = {\color{red}\Gr[Q]{(1,1)}{M}\lrang{2}^1} \amalg {\color{blue}\Gr[Q]{(1,1)}{M}\lrang{2}^0} \cong
\begin{array}{c@{}}
\begin{tikzpicture}
\draw[blue] (1,0) circle (.5);
\draw[red] (0,0) circle (.5);
\filldraw[fill=red,red] (0.5,0) circle (0.025);
\end{tikzpicture}
\end{array}.
\]
Note that the Grassmannian is neither irreducible nor smooth.
\end{example}
In order to get rid of $\seqv{r}$ we define the following maps and then look at the fibres.
\begin{definition}
Let $a$ be a sink, $\dvec{d}$ a dimension vector and $M \in \Rep[Q]{\dvec{d}}\lrang{a}^s$ for an $s\in \mathbb{N}$.
We have that $M \cong M' \oplus S_a^s$ for some element $M' \in \Rep[Q]{\dvec{d} - s\epsilon_a}\lrang{a}$.
Without loss of generality we can assume that $M=M' \oplus S_a^s$ and we set
$\pi_a M := M'$. Obviously, $\pi_a M$ is unique up to isomorphism.
Now let $\flvec{d}$ be a filtration
and $\seqv{r}=(r^0, \dots, r^\nu)$ a $(\nu +1)$-tuple of integers.
Define
\begin{alignat*}{3}
\pi_a^{\seqv{r}} &\colon\quad& \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{r}} &\rightarrow
\Fl[Q]{\seqv{\dvec{d}} - \seqv{r} \epsilon_a}{\pi_a M}\lrang{a}\\
&& \seqv{U} &\mapsto \seqv{V} & & \text{where } V^i_j:=\begin{cases}
U^i_j & \text{ if } j \neq a,\\
\Bild \phi^{U^i}_a & \text{ if } j = a.
\end{cases}
\end{alignat*}
\label{def:flagtoredfib}
\end{definition}
\begin{remark}
Note that $\pi_a M$ is $\iota_{a,M} S^-_a S^+_a M$.
\end{remark}
\begin{example}
Coming back to example \ref{bsp:grass} we see that $\pi_2^1$ collapses $\Gr[Q]{(1,1)}{M}\lrang{2}^1$
to the point
\[\Gr[Q]{(1,0)}{k^2 \overset{\left(\begin{smallmatrix}
1 & 0\\
\end{smallmatrix}\right)}{\rightarrow} k}\lrang{2}.\]
The fibre of $\pi_2^1$ over this point is the vector space Grassmannian $\Gr{1}{k^2}$, being
isomorphic to $\mathbb{P}^1_k$.
\end{example}
We now introduce a little bit more notation. If $\seqv{d}$ is a sequence, then denote by
$\overleftarrow{\seqv{d}}$ the sequence given by $(\overleftarrow{\seqv{d}})^i = d^{\nu-i}$. Moreover,
we define the sequence $\seqv{e}$ by $e^i := d^\nu - d^{\nu-i}$.
Therefore, if $\seqv{\dvec{d}}$ is a filtration of $\dvec{d}^\nu$,
then $\seqv{\dvec{e}}$ is a filtration of $\dvec{d}^\nu$.
The fibre of the map $\pi_a^{\seqv{r}}$ is a set of the following type.
\begin{definition}
Let $\seqv{e} = (e^0, e^1, \dots, e^\nu)$ and $\seqv{r} = (r^0, r^1, \dots , r^\nu)$ be sequences of non-negative integers such that
$\seqv{e} + \overleftarrow{\seqv{r}}$ is a filtration.
Let $A_{\nu+1}$ be the quiver
\[
0 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow \dotsm \rightarrow \nu.
\]
Then define
\[
\xymatrix{
X^{\seqv{r}, \seqv{e}} := &
k^{e^{\nu}+r^0} \ar@{->>}[r] &k^{e^{\nu-1}+r^1} \ar@{->>}[r] & \cdots \ar@{->>}[r] & k^{e^1 + r^{\nu-1}}\ar@{->>}[r] & k^{e^0 + r^\nu}.
}
\]
\end{definition}
\begin{remark}
The $k$-representation $X^{\seqv{r}, \seqv{e}} \in \repK{A_{\nu +1}}{k}$ is injective and its
isomorphism class does not depend on the choice of the surjections.
\end{remark}
\begin{lemma}
Let $\seqv{e} = (e^0, e^1, \dots, e^\nu)$
and $\seqv{r} = (r^0, r^1, \dots , r^\nu)$ be sequences of non-negative integers such that
$\seqv{e} + \overleftarrow{\seqv{r}}$ is a filtration.
Then $X^{\seqv{r}, \seqv{e}}$ has
a subrepresentation of dimension vector $\seqv{r}$ if and only if $\seqv{e}$ is a filtration of $e^\nu$
\label{lmm:flagfib_not_empty}. Moreover, if $k$ is a finite field with $q$ elements, then
the number of $k$-subrepresentations is given by
\[
\# \Gr[A_{\nu+1}]{\seqv{r}}{X^{\seqv{r},\seqv{e}}} = \prod_{i=0}^\nu \qbinom{e^{\nu-i} - e^{\nu-i-1} + r^i}{r^i}_q.
\]
In particular, this number is equal to $1$ modulo $q$ if and only if the set of subrepresentations is non-empty.
\end{lemma}
\begin{proof}
We prove this by induction on $\nu$.
\begin{description}
\item[$\nu=0$]
There is a subspace of dimension $r^0$ of $k^{r^0 + e^0}$ if and only if $e^0 \ge 0$ and, for
$k$ a finite field of cardinality $q$, the number of
those is obviously $\qbinom{e^0 + r^0}{r^0}_q$.
\item[$\nu \ge 1$]
If $(U^0,U^1,U^2, \dots, U^\nu)$ is a subrepresentation of dimension vector $\seqv{r}$
of $X^{\seqv{r}, \seqv{e}}$,
then
$(U^1,U^2, \dots, U^\nu)$ is a subrepresentation of dimension vector $(r^1, r^2,\dots, r^\nu)$ of
$X^{(r^1, r^2, \dots, r^\nu), (e^0, e^1, \dots, e^{\nu-1})}$. Therefore, by induction,
$e^i \le e^{i+1}$ for $0 \le i < \nu-1$ and $0 \le e^{0}$. The preimage $V$ of $U^1$ under the surjection from $U^0$ has
dimension $r^1 + ( (e^{\nu} + r^0) - (e^{\nu-1} + r^1))= e^\nu - e^{\nu-1} + r^0$. Since $\seqv{U}$ is a subrepresentation, we must have that
$U^0 \subset V$. Therefore, $r^0 \le e^\nu - e^{\nu-1} + r^0$ or equivalently $e^{\nu -1} \le e^\nu$.
On the other hand, if $e^0 \ge 0$ and $e^i \le e^{i+1}$ for all $0 \le i < \nu$, then there is a subrepresentation
$(U^1,U^2, \dots, U^\nu)$ of $X^{(r^1, r^2, \dots, r^\nu), (e^0, e^1, \dots, e^{\nu-1})}$ of dimension vector $(r^1, r^2,\dots, r^\nu)$ by induction.
As before, the dimension of the preimage $V$ of $U^1$ under the surjection from $U^0$ has dimension
$e^\nu - e^{\nu-1} + r^0 \ge r^0$. If we choose now any subspace $U^0$ of dimension $r^0$ in $V$, then we obtain
a subrepresentation
of $X^{\seqv{r}, \seqv{e}}$ of dimension vector $\seqv{r}$.
If $k$ is a finite field of cardinality $q$, then, by induction, we have that
the number of subrepresentations
of $X^{(r^1, r^2, \dots, r^\nu), (e^0, e^1, \dots, e^{\nu-1})}$
of dimension vector $(r^1, r^2,\dots, r^\nu)$ is equal to
\[
\prod_{i=1}^\nu \qbinom{e^{\nu-i} - e^{\nu-i-1} + r^i}{r^i}_q.
\]
To complete such a subrepresentation to a subrepresentation of $X^{\seqv{r}, \seqv{e}}$ we
have to choose an $r^0$-dimensional subspace of an $(e^{\nu} - e^{\nu-1} + r^0)$-dimensional
space. Therefore, the number of subrepresentations is equal to
\[
\qbinom{e^{\nu} - e^{\nu-1} + r^0}{r^0}_q \prod_{i=1}^\nu \qbinom{e^{\nu-i} - e^{\nu-i-1} + r^i}{r^i}_q.
\]
This yields the claim.
\end{description}
\end{proof}
\begin{theorem}
Let $a$ be a sink of $Q$, $\flvec{d}$ a filtration, $\seqv{r}$ a $(\nu+1)$-tuple of non-negative integers
and $M \in \Rep[Q]{\dvec{d}^\nu}$. Then
\[\pi_a^{\seqv{r}} \colon
\Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{r}} \rightarrow
\Fl[Q]{\seqv{\dvec{d}} - \seqv{r} \epsilon_a}{\pi_a M}\lrang{a}\]
is surjective and
the fibre $(\pi_a^{\seqv{r}})^{-1}(\seqv{U})$
over any $\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}} - \seqv{r} \epsilon_a}{\pi_a M}\lrang{a}$
is isomorphic to $\Gr[A_\nu]{\seqv{r}}{X^{\seqv{r}, \seqv{\dvec{e}}_a}}$,
where $\dvec{e}^{\nu -i} := \dvec{d}^\nu - \dvec{d}^i$.
In particular the number
of points in the fibre only depends on $\seqv{r}$ and $\seqv{\dvec{d}}$ and not on $\seqv{U}$.
\label{theom:qgrass_to_red_fib}
\end{theorem}
\begin{proof}
Fix a flag $\seqv{V} \in \Fl[Q]{\seqv{\dvec{d}} - \seqv{r} \epsilon_a}{\pi_a M}\lrang{a}$. Let
$\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{r}}$. The flag $\seqv{U}$ is
in $(\pi_a^{\seqv{r}})^{-1}(\seqv{V})$ if and only if $U_j^i = V_j^i$ for all $j \neq a$
in which case $\Bild(\Phi^{U^i}_a) = V^i_a$.
Therefore, we only have to choose $U^i_a \subset M_a$ such that $V^i_a \subseteq U^i_a$,
$U^{i-1}_a \subset U^i_a$ and $\dim U^i_a = d^i_a$ for all $i$. This is the same as choosing
$\overline{U^{i}_a} \subset M_a/V^i_a$ such that $\theta^i(\overline{U^{i-1}_a}) \subset \overline{U^i_a}$ and
$\dim \overline{U^i_a} = r^i$ if we denote by $\theta^i \colon M_a/V^{i-1}_a \rightarrow M_a/V^i_a$ the
canonical projection. This is equivalent to finding a subrepresentation of
\[ M_a/V^0_a \rightarrow M_a/V^1_a \rightarrow \cdots \rightarrow M_a/V^\nu_a \]
of dimension vector $\seqv{r}$. All the maps in this representation are surjective since $\seqv{V}$
is a flag, therefore this representation
of $A_{\nu+1}$ is isomorphic to
$X^{\seqv{r}, \seqv{e}}$. Since $\flvec{d}$ is a filtration of $M$ we have that $\flvec{e}$ is a filtration.
Therefore, $\Gr[A_\nu]{\seqv{r}}{X^{\seqv{r}, \seqv{\dvec{e}}_a}}$ is non-empty by lemma \ref{lmm:flagfib_not_empty}
and $\pi_a^\seqv{r}$ is surjective.
\end{proof}
Now we are nearly ready to do reflections. The only thing left to define is what happens on a source. If $b$ is a source in
$Q$, then $b$ is a sink in $Q^{op}$, so we just dualise everything. Denote by $D:=\Hom_k(-,k)$.
\begin{definition}
Let $\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}$ and let $\dvec{e}^{\nu-i} = \dvec{d}^\nu - \dvec{d}^i$.
Then define
\begin{alignat*}{2}
\hat D &\colon\quad &
\Fl[Q]{\seqv{\dvec{d}}}{M}&\rightarrow
\Fl[Q^{op}]{\seqv{\dvec{e}}}{DM}\\
&&\seqv{U} & \mapsto (\hat D (\seqv{U}))^i := \ker( DM \rightarrow D(U^i)) = D(M/U^i).
\end{alignat*}
\end{definition}
\begin{remark}
Obviously, $\hat D ^2 \cong \id$ and the map $\hat D$ is an isomorphism of varieties.
\end{remark}
\begin{definition}
Let $b$ be a source, $\dvec{d}$ a dimension vector and $s$ an integer. Define
\[ \Rep[Q]{\dvec{d}}\lrang{b}^{s} :=
\Set{ M \in \Rep[Q]{\dvec{d}} | \dim \Hom(S_b, M) = s}.\]
Let $\seqv{\dvec{d}}$ be a $(\nu+1)$-tuple of dimension vectors
and
$\seqv{r} = (r^0, r^1, \dots , r^\nu)$ be a $(\nu+1)$-tuple of integers.
For each representation $M$ define
\[
\Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{b}^{\seqv{r}} :=
\Set{ \seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M} |
\dim \Hom(S_b, M/U^i) = r^i}.
\]
Moreover, let
\[ \Rep[Q]{\dvec{d}}\lrang{b} := \Rep[Q]{\dvec{d}}\lrang{b}^0\] and
\[ \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{b} := \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{b}^{\seqv{0}}.\]
\end{definition}
\begin{remark}
Note that $\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{b}^{\seqv{r}}$ if and only if
$\hat D \seqv{U} \in \Fl[Q^{op}]{\seqv{\dvec{e}}}{DM}\lrang{b}^{\seqv{r}}$.
\end{remark}
Now we state the main result on reflections.
\begin{theorem}
Let $a$ be a sink of $Q$, $\flvec{d}$ be a filtration and $M \in \Rep[Q]{\dvec{d}^\nu}\lrang{a}$.
The map
\begin{alignat*}{2}
S^+_a &\colon\quad & \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a} &\rightarrow
\Fl[\sigma_a Q]{\sigma_a \seqv{\dvec{d}}}{S^+_a M}\lrang{a}\\
&& \seqv{U} & \mapsto S^+_a \seqv{U}
\end{alignat*}
is an isomorphism of varieties with inverse $\hat D \circ S^+_a \circ \hat D = S^-_a$.
\label{theom:reflflagiso}
\end{theorem}
\begin{proof}
First, we show that $S^+_a \seqv{U}$ lies in the correct set. Let $\dvec{e}^{\nu-i} = \dvec{d}^\nu - \dvec{d}^i$.
For each $i$, we have the following commutative diagram with exact columns.
\[
\begin{CD}
@. 0 @. 0 @. 0 @. \\
@. @VVV @VVV @VVV @.\\
0 @>>> (S^+_a U^{i})_a @>>> \bigoplus\limits_{j \rightarrow a} U^{i}_j @>{\phi^{U^{i}}_a}>> U^{i}_a @>>> 0\\
@. @VVV @VVV @VVV @.\\
0 @>>> (S^+_a M)_a @>>> \bigoplus\limits_{j \rightarrow a} M_j @>\phi^{M}_a>> M_a @>>> 0\\
@. @VVV @VVV @VVV @.\\
0 @>>> (S^+_a M / S^+_a U^{i})_a @>>> \bigoplus\limits_{j \rightarrow a} (M/U^{i})_j @>\phi^{M/U^{i}}_a>> (M/U^{i})_a @>>> 0\\
@. @VVV @VVV @VVV @.\\
@. 0 @. 0 @. 0 @.
\end{CD}
\]
The two top rows are exact since $\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}$. By the snake lemma,
we have that the
bottom row is exact. Therefore, the map
\[ (S^+_a M / S^+_a U^{i})_a \rightarrow \bigoplus_{j\rightarrow a} (M/U^{i})_j \]
is injective and hence $S^+_a(M)/S^+_a(U^i) \in \Rep[\sigma_a Q]{\sigma_a(\dvec{e}^{\nu-i})}\lrang{a}$.
The diagram also yields that $\hat D \circ S^+_a \circ \hat D \circ S^+_a = \id$.
Since $S^+_a$ is a functor and all choices where natural, we have that $S^+_a$ gives
a natural transformation between the functors of points of these two varieties. Therefore,
it is a morphism of varieties.
Dually, $S^+_a \circ \hat D \circ S^+_a \circ \hat D = \id$. This concludes the proof.
\end{proof}
\section{Reflection functors and Hall numbers}
First we recall the definition of the Hall algebra.
The main references for this are Ringel \cite{Ringel_hallalgsandquantumgroups}
and
Hubery \cite{Hubery_ringelhall}.
Let $k$ be a finite field. Let $M, N, X \in \repK{Q}{k}$ be three
$k$-representations of $Q$. Then define
\[ F^X_{M N} := \#\Set{ U \le X | U \text{ subrepresentation}, U \cong N, X/U \cong M }. \]
This is a finite number.
Let $\mathcal{H}_k(Q)$ be the $\mathbb{Q}$-vector space with basis $u_{[X]}$ where $[X]$ is
the isomorphism class of $X$. For convenience we write $u_X$ instead of $u_{[X]}$. Define
\[
u_{[M]} \diamond u_{[N]} := \sum_{[X]} F^{X}_{M N} u_{[X]}.
\]
Then $(\mathcal{H}(\mathcal{A}), +, \diamond)$ is an associative
$\mathbb{Q}$-algebra with unit $1=u_0$, the Ringel-Hall algebra
or just Hall algebra. The composition algebra
is the subalgebra $\mathcal{C}_k(Q)$ of $\mathcal{H}_k(Q)$ generated by
the simple objects without self-extensions. Note that $\mathcal{H}_k(Q)$ and
$\mathcal{C}_k(Q)$ are naturally graded by dimension vector.
To each vertex $i \in Q_0$ there is a simple object $S_i$
given by $(S_i)_i = k$, $(S_i)_j = 0$ for $i \neq j$ and $(S_i)_\alpha = 0$ for all $\alpha \in Q_1$.
We set $u_i := u_{S_i}$ for each $i \in Q_0$. If $w = (i_1, \dots, i_r)$ is
a word in vertices of $Q$, we define
\[ u_w := u_{1} \diamond \dots \diamond u_{r}.\]
By definition, there are numbers $F_w^X$ for each $k$-representation $X$ of $Q$ such that
\[ u_w = \sum_{X} F_w^X u_X. \]
\label{sec:reflhallnum}
Let $k=\mathbb{F}_q$ be the finite field with $q$ elements and
$Q$ a quiver. Let $w=(i_r, \dots, i_1)$ be a word in vertices of $Q$. Define a filtration $\flvec{d}(w)$ by letting
\[\dvec{d}(w)^k := \sum_{j=1}^k \epsilon_{i_k}.\]
Then we obviously have $F_w^X = \# \Fl[Q]{\flvec{d}(w)}{X}$. Therefore, coefficients
in the Hall algebra are closely related to counting points of quiver
flags over finite fields. In the following, we will use reflection functors
to simplify the problem of counting the number of points modulo $q$. As an
application, we will show that for a preprojective or preinjective representation $X$
we have that $\# \Fl[Q]{\flvec{d}}{X} = 1 \mod q$ if $\Fl[Q]{\flvec{d}}{X}$ is non-empty.
\begin{lemma}
Let $a$ be a sink of $Q$, $k$ a field, $\flvec{d}$ a filtration and $M \in \RepK[Q]{\dvec{d}^\nu}{k}$.
Then
\[
\# \Fl[Q]{\seqv{\dvec{d}}}{M}
=\sum_{\seqv{r}\ge 0} \# \Gr[A_\nu]{\seqv{r}}{X^{\seqv{r}, \seqv{e}_a}}
\# \Fl[Q]{\seqv{\dvec{d}} - \seqv{r} \epsilon_a}{\pi_a M}\lrang{a}
\]
(on both sides we possibly have $\infty$).
Moreover, for each sequence of non-negative integers $\seqv{r}$, if
$
\Fl[Q]{\seqv{\dvec{d}} - \seqv{r} \epsilon_a}{\pi_a M}\lrang{a}
$
is non-empty,
then so is
$
\Gr[A_\nu]{\seqv{r}}{X^{\seqv{r}, \seqv{e}_a}}.
$
\label{reflflag:lmm:decompflag}
\end{lemma}
\begin{proof}
We have that
\[
\Fl[Q]{\seqv{\dvec{d}}}{M} = \coprod_{\seqv{r}\ge 0} \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{r}}.
\]
By theorem \ref{theom:qgrass_to_red_fib}, we have for each sequence of non-negative integers $\seqv{r}$ that
\[
\# \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{r}}
=
\# \Gr[A_\nu]{\seqv{r}}{X^{\seqv{r}, \seqv{e}_a}} \# \Fl[Q]{\seqv{\dvec{d}} - \seqv{r} \epsilon_a}{\pi_a M}\lrang{a}.
\]
By the same theorem we have that if
$\Fl[Q]{\seqv{\dvec{d}} - \seqv{r} \epsilon_a}{\pi_a M}\lrang{a}$ is non-empty, then so
is $\Gr[A_\nu]{\seqv{r}}{X^{\seqv{r}, \seqv{e}_a}}$.
\end{proof}
\begin{lemma}
Let $a$ be a sink of $Q$, $k$ a field, $\flvec{d}$ a filtration
and $M \in \RepK[Q]{\dvec{d}^\nu}{k}\lrang{a}^s$. Let $\seqv{r}_+= \seqv{r}_+(\flvec{d})$ be given as follows:
\begin{align*}
r^0_+ &:= 0; &\\
r^i_+ &:= \max\{0, (\sigma_a(\dvec{d}^{i-1} - \dvec{d}^i))_a + r^{i-1}_+\}& \text{for } 0 < i < \nu;\\
r^\nu_+ &:= s.
\end{align*}
Now let $\seqv{r}$ be a sequence of integers. If
$\Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{r}}$ is non-empty, then $\seqv{r} \ge \seqv{r}_+$.
\label{reflflag:lmm:rplusmin}
\end{lemma}
\begin{proof}
Let $\seqv{U} \in \Fl[Q]{\seqv{\dvec{d}}}{M}\lrang{a}^{\seqv{r}}$. By definition,
$\Hom(U^i, S_a) = r^i$. We have
\[r^i= \codim \Bild \Phi_a^{U^i} = \dim \ker \Phi_a^{U^i} + d^i_a - \sum_{j \rightarrow a} d^i_j
= \dim \ker \Phi_a^{U^i} - (\sigma_a \dvec{d}^i)_a.\]
We prove $r^i \ge r_+^i$ by induction on $i$. For $i=0$ the claim
is obviously true. Now let $0\le i \le \nu-2$. Obviously, $\dim \ker \Phi_a^{U^i} \le \dim \ker \Phi_a^{U^{i+1}}$.
Therefore,
\[ r^i_+ + (\sigma_a \dvec{d}^i)_a
\le r^i + (\sigma_a \dvec{d}^i)_a= \dim \ker \Phi_a^{U^i} \le \dim \ker \Phi_a^{U^{i+1}} =
r^{i+1} + (\sigma_a \dvec{d}^{i+1})_a.\]
Hence, $r^{i+1} \ge \max\{0,(\sigma_a(\dvec{d}^{i} - \dvec{d}^{i+1}))_a + r^i_+\} = r^{i+1}_+$.
For $r^\nu_+$ note that, by definition, $U^\nu = M$ and therefore
\[ r^\nu =\codim \Bild \Phi_a^{U^\nu} = \codim \Bild \Phi_a^{M} = s.\]
\end{proof}
\begin{remark}
Note that $\seqv{r}_+(\flvec{d}-\seqv{r}_+(\flvec{d}) \epsilon_a)=0$ since
\[
\sigma_a (\dvec{d}^i - \dvec{d}^{i-1})_a + r^i_+(\flvec{d}) - r^{i-1}_+(\flvec{d})
\ge r^i_+(\flvec{d}) - r^i_+(\flvec{d}) = 0.
\]
For any filtration $\flvec{d}$ of some representation $M$ it is enough to remember the terms
\[
(\dvec{d}^1, \dots, \dvec{d}^{\nu-1})
\]
since we always have $\dvec{d}^0 = 0$
and $\dvec{d}^\nu = \dimve M$. Note that the rule to construct $r_+^i$ for
$0<i<\nu$
depends neither on $\dvec{d}^0$ nor on $\dvec{d}^\nu$. Therefore, we
can define
\[S_a^+ \flvec{d} = S_a^+ (\dvec{d}^1, \dots, \dvec{d}^{\nu-1}) :=
(\sigma_a \dvec{d}^1 + r_+^1 \epsilon_a, \dots, \sigma_a \dvec{d}^{\nu-1} + r_+^{\nu-1}). \]
If $\flvec{d}$ is
a filtration of $M$, then $S_a^+ \flvec{d}$ is a filtration of $S_a^+ M$ if and only if
$(S_a^+ \flvec{d})^{\nu-1} \le \dimve S_a^+ M$.
\label{reflflag:rem:reflfilt}
\end{remark}
\begin{corollary}
Let $a$ be a sink, $k$ a field, $\flvec{d}$ a filtration and $M \in \RepK[Q]{\dvec{d}^\nu}{k}\lrang{a}^s$.
Then
\[
\# \Fl[Q]{\seqv{\dvec{d}}}{M}
=\sum_{\seqv{r}\ge 0} \# \Gr[A_\nu]{\seqv{r}+\seqv{r}_+}{X^{\seqv{r}+\seqv{r}_+, \seqv{e}_a}}
\# \Fl[\sigma_a Q]{S_a^+\seqv{\dvec{d}}+ \seqv{r}\epsilon_a}{S_a^+ M}\lrang{a}.
\]
In particular, if $k$ is a finite field of cardinality $q$, we have
\[
\# \Fl[Q]{\seqv{\dvec{d}}}{M}
=\sum_{\seqv{r}\ge 0}
\# \Fl[\sigma_a Q]{S_a^+\seqv{\dvec{d}}+\seqv{r}\epsilon_a}{S_a^+ M}\lrang{a} \mod q.
\]
\label{reflflag:cory:decompreflmodq}
\end{corollary}
\begin{proof}
By lemmas \ref{reflflag:lmm:decompflag} and \ref{reflflag:lmm:rplusmin} we obtain
that
\[
\# \Fl[Q]{\seqv{\dvec{d}}}{M}
=\sum_{\seqv{r}\ge 0} \# \Gr[A_\nu]{\seqv{r}+\seqv{r}_+}{X^{\seqv{r}+\seqv{r}_+, \seqv{e}_a}}
\# \Fl[Q]{\flvec{d} - (\seqv{r} + \seqv{r}_+)\epsilon_a}{\pi^s_a M}\lrang{a}.
\]
Note that $\sigma_a (\dvec{d}^i - (r^i+r^i_+)\epsilon_a) =
(S_a^+\flvec{d})^i + r^i \epsilon_a$ for all $0 < i < \nu$. Therefore, theorem \ref{theom:reflflagiso}
yields that
\[
\Fl[Q]{\flvec{d} - (\seqv{r} + \seqv{r}_+)\epsilon_a}{\pi^s_a M}\lrang{a} \cong
\Fl[\sigma_a Q]{S_a^+\flvec{d} + \seqv{r}\epsilon_a}{S_a^+ M}\lrang{a}.
\]
This proves the first claim.
Now let $k$ be a finite field of cardinality $q$. If
$\Gr[A_\nu]{\seqv{r}+\seqv{r}_+}{X^{\seqv{r}+\seqv{r}_+, \seqv{e}_a}}$
is non-empty, then its number is one modulo $q$ by lemma \ref{lmm:flagfib_not_empty}.
The second part of lemma \ref{reflflag:lmm:decompflag} yields that whenever
$\Fl[Q]{\flvec{d} - (\seqv{r} + \seqv{r}_+)\epsilon_a}{\pi^s_a M}\lrang{a}$
is non-empty, then $\Gr[A_\nu]{\seqv{r}+\seqv{r}_+}{X^{\seqv{r}+\seqv{r}_+, \seqv{e}_a}}$
is non-empty. This finishes the proof.
\end{proof}
We obtain the following.
\begin{theorem}
\label{theom:reflflagmodq}
Let $a$ be a sink, $k$ a field, $\flvec{d}$ a filtration and $M \in \RepK[Q]{\dvec{d}^\nu}{k}\lrang{a}^s$.
Then $\Fl[Q]{\seqv{\dvec{d}}}{M}$ is empty if and only if
$\Fl[\sigma_a Q]{S^+_a \flvec{d}}{S^+_a M}$ is.
Moreover, if $k=\mathbb{F}_q$ is a finite field, then
\[
\# \Fl[Q]{\seqv{\dvec{d}}}{M}=
\# \Fl[\sigma_a Q]{S^+_a \flvec{d}}{S^+_a M} \mod q.
\]
\end{theorem}
\begin{proof}
By corollary \ref{reflflag:cory:decompreflmodq} we obtain that
\[
\# \Fl[Q]{\seqv{\dvec{d}}}{M}
=\sum_{\seqv{r}\ge 0} \# \Gr[A_\nu]{\seqv{r}+\seqv{r}_+}{X^{\seqv{r}+\seqv{r}_+, \seqv{e}_a}}
\# \Fl[\sigma_a Q]{S_a^+\seqv{\dvec{d}}+ \seqv{r}\epsilon_a}{S_a^+ M}\lrang{a}.
\]
Note that $S_a^+ \flvec{d}$ is a filtration of $S_a^+M$ if and only if
$(S_a^+ \flvec{d})^{\nu-1} \le \dimve S_a^+ M$. Therefore, if $S_a^+ \flvec{d}$
is not a filtration of $S_a^+ M$, then
each $\Fl[\sigma_a Q]{S_a^+\seqv{\dvec{d}}+ \seqv{r}\epsilon_a}{S_a^+ M}\lrang{a}$
is empty for all $\seqv{r} \ge 0$ and hence, so is
$\Fl[Q]{\seqv{\dvec{d}}}{M}$. In this case, we also have that
$\Fl[\sigma_a Q]{S^+_a \flvec{d}}{S^+_a M}$ is empty. Both claims follow.
Assume now that $S_a^+ \flvec{d}$ is a filtration of $S_a^+M$.
We have that
\[\Fl[\sigma_a Q]{S^+_a \flvec{d}}{S^+_a M} \cong \Fl[\sigma_a Q^{op}]{\dimve S^+_a M - \rever{S^+_a \flvec{d}}}{DS^+_a M}\]
via $\hat D$. Let $\flvec{f}:=\dimve S^+_a M - \rever{S^+_a \flvec{d}}$. By using lemma
\ref{reflflag:lmm:decompflag} we obtain that
\[
\# \Fl[\sigma_a Q^{op}]{\flvec{f}}{DS^+_a M}
=\sum_{\seqv{r}\ge 0} \# \Gr[A_\nu]{\rever{\seqv{r}}}{X^{\rever{\seqv{r}}, (S^+_a\flvec{d})_a}}
\# \Fl[\sigma_a Q^{op}]{\flvec{f} - \rever{\seqv{r}} \epsilon_a}{D S_a^+ M}\lrang{a}.
\]
Moreover, by the same lemma we have for each $\seqv{r}\ge 0$ that if
$\Fl[\sigma_a Q^{op}]{\flvec{f} - \rever{\seqv{r}} \epsilon_a}{D S_a^+ M}\lrang{a}$ is
non-empty, then $\Gr[A_\nu]{\rever{\seqv{r}}}{X^{\rever{\seqv{r}}, (S^+_a\flvec{d})_a}}$ is non-empty.
Using $\hat D$ yields
\[
\Fl[\sigma_a Q^{op}]{\flvec{f} - \rever{\seqv{r}} \epsilon_a}{D S_a^+ M}\lrang{a}
\cong \Fl[\sigma_a Q]{S_a^+\flvec{d} + \seqv{r}\epsilon_a}{S_a^+ M}\lrang{a}.
\]
Combining these equalities, we have that
\[
\# \Fl[\sigma_a Q]{S^+_a \flvec{d}}{S^+_a M}
=\sum_{\seqv{r}\ge 0} \# \Gr[A_\nu]{\rever{\seqv{r}}}{X^{\rever{\seqv{r}}, (S^+_a\flvec{d})_a}}
\# \Fl[\sigma_a Q]{S_a^+\flvec{d} + \seqv{r}\epsilon_a}{S_a^+ M}\lrang{a}.
\]
Therefore, $\Fl[\sigma_a Q]{S^+_a \flvec{d}}{S^+_a M}$ is empty if and only if for all
$\seqv{r} \ge 0$ we have that the variety $\Fl[\sigma_a Q]{S_a^+\flvec{d} + \seqv{r}\epsilon_a}{S_a^+ M}\lrang{a}$
is empty. The same is true for $\Fl[Q]{\seqv{\flvec{d}}}{M}$ and this proves the first claim.
Now let $k$ be a finite field with $q$ elements. By the first part, if
$\Fl[\sigma_a Q]{S_a^+\flvec{d} + \seqv{r} \epsilon_a}{S_a^+ M}\lrang{a}$
is non-empty, then
$\Gr[A_\nu]{\rever{\seqv{r}}}{X^{\rever{\seqv{r}}, (S^+_a\flvec{d})_a}}$ is non-empty.
Therefore, lemma \ref{lmm:flagfib_not_empty} yields that
\[\# \Fl[\sigma_a Q]{S^+_a \flvec{d}}{S^+_a M}
=\sum_{\seqv{r}\ge 0}
\# \Fl[\sigma_a Q]{S_a^+\flvec{d} + \seqv{r}\epsilon_a}{S_a^+ M}\lrang{a} \mod q.\]
By corollary \ref{reflflag:cory:decompreflmodq} this is equal to
$\# \Fl[Q]{\flvec{d}}{M}$. This finishes the proof.
\end{proof}
\begin{remark}
The Coxeter functor $C^+$ is by definition the composition of reflection functors
associated to an admissible ordering
$(a_1, \dots, a_n)$ of $Q$. The action on a filtration, which we also denote by $C^+$, is
given by $C^+ \flvec{d} := S^+_{a_n} \dots S^+_{a_1} \flvec{d}$. It is not clear
that $C^+$ on a filtration does not depend on the choice of the admissible ordering.
\end{remark}
We immediately obtain the following.
\begin{corollary}
\label{reflflags:cory:preprojone}
Let $M$ be a preprojective $k$-representation and
let $\flvec{d}$ be a filtration of $\dimve M$. Take $r\ge0$ such that
$(C^+)^r M = 0$.
Then
$\Fl[Q]{\seqv{\dvec{d}}}{M}$ is non-empty if and only if
we have that $(C^+)^r \flvec{d} =0$ and that for every intermediate sequence $w$ of admissible sink reflections
$S^+_w \flvec{d}$ is a filtration of $S^+_w M$. In particular, this depends only
on the isomorphism class of $M$ and the filtration $\flvec{d}$, but not on the choice
of $M$ or the field $k$.
Moreover, if $k$ is a finite field with $q$ elements, then
$\Fl[Q]{\seqv{\dvec{d}}}{M}$ non-empty implies that
\[\# \Fl[Q]{\seqv{\dvec{d}}}{M} = 1 \mod q.\]
\end{corollary}
\begin{proof}
Using remark \ref{reflflag:rem:reflfilt} we obtain that for each reflection at a sink
$a$ of $Q$ we have that
$S^+_a \flvec{d}$ is again a filtration of $S^+_a M$ if and only if
$(S^+_a \flvec{d})^{\nu-1} \le \dimve S^+_a M$.
If this is not the case, then
the quiver flag is empty by theorem \ref{theom:reflflagmodq}.
Therefore, if the quiver flag is non-empty, then for every
intermediate sequence $w$ of admissible sink reflections
we have that $S^+_w \flvec{d}$ is a filtration of $S^+_w M$. We call this condition (*).
Assume that (*) holds. Iteratively applying
theorem \ref{theom:reflflagmodq} we have that
$\Fl[Q]{\flvec{d}}{M}$ is empty if and only if $\Fl[Q]{(C^+)^r\flvec{d}}{(C^+)^r M} = \Fl[Q]{(C^+)^r\flvec{d}}{0}$
is empty.
There is only
one filtration of the $0$ representation, namely $(0,0,\dots,0)$. Therefore,
$\Fl[Q]{\flvec{d}}{M}$ is non-empty if and only if $(C^+)^r\flvec{d} = 0$.
This proves the first part since we already have seen that if (*) does not hold, then
$\Fl[Q]{\flvec{d}}{M}$ is empty.
Assume now that $k$ is a finite field with $q$ elements. If (*) does not
hold, then the quiver flag is empty and the claim holds.
Assume therefore that (*) holds. As before, applying
theorem \ref{theom:reflflagmodq} yields that
\[ \#\Fl[Q]{\flvec{d}}{M} = \#\Fl[Q]{(C^+)^r\flvec{d}}{(C^+)^r M} = \#\Fl[Q]{(C^+)^r\flvec{d}}{0} \mod q.\]
There is only
one filtration of the zero representation, namely $(0,0,\dots,0)$, and
the number of flags of this type is obviously equal to one.
This concludes the proof.
\end{proof}
\section{Dynkin case}
In this section let $Q$ be
a Dynkin quiver. We first introduce the generic Hall algebra $\mathcal{H}_q(Q)$ and the
composition monoid $\mathcal{CM}(Q)$ and then use the results of the previous section to show that
$\mathcal{H}_0(Q) \cong \mathcal{CM}(Q)$, where the isomorphism is given by sending $u_i$ to
$S_i$.
Since $Q$ is Dynkin, every representation is preprojective. Moreover,
indecomposable representations of $Q$ are in bijection with
the set of positive roots $\Delta_+$ of the Lie algebra corresponding to the underlying Dynkin diagram.
An isomorphism class is therefore given by a function from $\Delta_+$ to $\mathbb{N}$ with finite support. We denote this
set by $\Phi$. For each element $\mu \in \Phi$ and each field $k$ we can choose a $k$-representation
$M(\mu, k)$ having isomorphism class $\mu$.
Hall polynomials exist
with respect to $\Phi$, as shown by Ringel \cite{Ringel_hallpolysforrepfiniteheralgs}.
More precisely, for $\mu, \nu, \xi \in \Phi$ there is a polynomial
$f^{\xi}_{\mu \nu} (q) \in \mathbb{Z}[q]$ such that for each finite field $k$ with $q_k$ elements
we have
\[ F^{M(\xi,k)}_{M(\mu,k) M(\nu,k)} = f^{\xi}_{\mu \nu} (q_k).\]
We define the generic Hall algebra $\mathcal{H}_q (Q)$ to be the free $\mathbb{Z}[q]$-module with basis
$\Set {u_{\alpha} | \alpha \in \Phi}$ and multiplication given by:
\[u_{\mu} \diamond u_{\nu} = \sum_{\xi} f^{\xi}_{\mu \nu} (q) u_{\xi}. \]
The generic composition algebra $\mathcal{C}_q(Q)$ is the subalgebra of $\mathcal{H}_q(Q)$ generated
by the simple representations without self-extensions, or more precisely their isomorphism
classes. For an acyclic quiver theses are exactly the $u_i$. If the quiver is fixed, then we often write $\mathcal{H}_q$ and
$\mathcal{C}_q$ instead of $\mathcal{H}_q(Q)$
and $\mathcal{C}_q(Q)$. Moreover, for convenience we identify for any representation $M \cong M(\alpha, k)$, $u_M$
with $u_\alpha$ and $f^{X}_{M N}$ with $f^{\xi}_{\mu \nu} (q)$.
When calculating Hall polynomials, certain quantum numbers appear. Let $R$ be some commutative ring
and let $q \in R$. Usually $R$ will be $\mathbb{Z}[q]$, the polynomial ring in one variable. We define for $r,n \in \mathbb{N}$,
$0 \le r \le n$:
\begin{align*}
[n]_q &:= 1 + q + \dots + q^{n-1} \\
[n]_q ! &:= \prod_{i=1}^{n} [i]_q \\
\qbinom{n}{r}_q &:= \frac{[n]_q !}{[r]_q ! [n-r]_q!}.
\end{align*}
Obviously, $\qbinom{n}{r}_0 = 1$.
Fix
an algebraically closed field $k$.
We say that a representation $M$ degenerates to $N$, $M \le_{\mathrm{deg}} N$, if
$\mathcal{O}_N \subseteq \overline{\mathcal{O}_M}$, where we take the closure in the Zariski topology.
For two arbitrary sets $U \subseteq \Rep{\dvec{d}}, V \subseteq \Rep{\dvec{e}}$ we define
\begin{align*}
\mathcal{E}(U,V) := \{ M \in \Rep{\dvec{d} + \dvec{e}} \: |& \: \exists \:A \in U, B \in V \
\text{and a short exact sequence } \\
& 0 \rightarrow B \rightarrow M \rightarrow A \rightarrow 0 \}.
\end{align*}
The multiplication on closed irreducible $\GL_\dvec{d}$-stable respectively $\GL_\dvec{e}$-stable
subvarieties $\mathcal{A} \subseteq \Rep{\dvec{d}},
\mathcal{B} \subseteq \Rep{\dvec{e}}$ is defined as:
\[ \mathcal{A} * \mathcal{B} := \mathcal{E}(\mathcal{A}, \mathcal{B}).\]
The set of closed irreducible subvarieties of nilpotent representations with
this multiplication is a monoid with unit $\Rep{\dvec{0}}$, the generic extension monoid $\mathcal{M}(Q)$. The composition monoid
$\mathcal{CM}(Q)$ is the submonoid
generated by the orbits of simple representations without self-extensions. All this is due to Reineke \cite{Reineke_monoid}.
For any word $w = (i_1, \dots, i_r)$ in vertices of $Q$ we define
$\mathcal{A}_w := \mathcal{O}_{S_1} * \dots * \mathcal{O}_{S_r}$. This is an element of $\mathcal{CM}(Q)$.
We can now use the machinery we developed in the previous section to prove
that, for $Q$ a Dynkin quiver, the generic composition algebra
specialised at $q=0$
and the composition monoid are isomorphic.
\begin{proposition}
\label{reflflags:propn:dynkinallone}
Let $X$ be a $k$-representation of $Q$ and $w$ a word in vertices of $Q$.
Then the condition that $X$
has a filtration of type $w$ only depends on $w$ and $[X] \in \Phi$ and not on the choice of $X$ or the field
$k$.
Moreover, we have that
\[ u_w = \sum_{[X] \in [\mathcal{A}_w]} u_{[X]} \in \mathcal{H}_0(Q). \]
\end{proposition}
\begin{proof}
Since all representations of $Q$ are preprojective, the first part of the
statement follows directly from corollary \ref{reflflags:cory:preprojone}.
Therefore, the sum in the second part is well-defined (i.e. the set
$[\mathcal{A}_w]$ does not depend on the field).
If $k$ is a finite field with $q$ elements, corollary \ref{reflflags:cory:preprojone} also yields that
\[ F_w^X = \# \Fl[Q]{\flvec{d}(w)}{X} = \begin{cases}
1 \mod q & \text{if } X \in \mathcal{A}_w,\\
0 & \text{else.}
\end{cases}
\]
Since $F_w^X= f_w^{[X]}(q)$ and we just showed that this is one modulo $q$ for all prime powers $q$
we have that
\[f_w^{[X]}(0) =\begin{cases}
1 & \text{if } [X] \in [\mathcal{A}_w],\\
0 & \text{else.}
\end{cases}
\]
This yields the claim.
\end{proof}
We obtain the main theorem for the Dynkin case.
\begin{theorem}
The map
\begin{alignat*}{2}
\Psi &\colon\quad & \mathbb{Q}\mathcal{M}(Q) & \rightarrow \mathcal{H}_0(Q)\\
&& \mathcal{A} & \mapsto \sum\limits_{[M] \in [\mathcal{A}]} u_{[M]}
\end{alignat*}
is an isomorphism of $\mathbb{Q}$-algebras.
\label{reflflags:theom:dynkin}
\end{theorem}
\begin{proof}
Note that for $Q$ Dynkin we have that $\mathcal{M}(Q) \cong \mathcal{CM}(Q)$
and $\mathcal{H}_q(Q) \cong \mathcal{C}_q(Q)$. Therefore,
for each $\mathcal{A} \in \mathcal{M}(Q)$ there is a word $w$ in
vertices of $Q$ such that $\mathcal{A} = \mathcal{A}_w$. In
the previous proposition we showed that the map sending $\mathcal{A}_w$ to
\[\Psi(\mathcal{A}_w) = \sum_{[M] \in [\mathcal{A}_w]} u_{[M]} = u_w\]
is well-defined. Therefore, $\Psi$ is a homomorphism, since
\[\Psi(\mathcal{A}_{w} * \mathcal{A}_{v}) = \Psi(\mathcal{A}_{wv}) =
u_{wv}=u_{w} \diamond u_v = \Psi(\mathcal{A}_{w}) \diamond \Psi(\mathcal{A}_{v}). \]
$\Psi$ is surjective since it is a homomorphism, and the generators
$u_i$ of $\mathcal{H}_0(Q)$ are in the image of $\Psi$. More precisely,
$\Psi(\mathcal{O}_{S_i}) = u_i$. Obviously, $\Psi$ is a graded morphism
of graded algebras. The dimension of the $\dvec{d}$-th graded part
of $\mathbb{Q}\mathcal{M}(Q)$ is the same as the dimension
of the $\dvec{d}$-th graded part of $\mathcal{H}_0(Q)$, namely the number of isomorphism classes of representations
of dimension vector $\dvec{d}$. Since each graded part is finite dimensional and
$\Psi$ is surjective, we have
that $\Psi$ is an isomorphism.
\end{proof}
\section{Further work}
In an upcoming paper, we show that if $Q$ is an acyclic extendend Dynkin quiver, then
\begin{alignat*}{2}
\Phi&\colon\quad& \mathcal{C}_0(Q) &\rightarrow \mathcal{CM}(Q)\\
&& u_i &\rightarrow S_i
\end{alignat*}
is a homomorphism and give generators for its
non-trivial kernel. We do this by a close analysis of the geometric version
of reflection functors and the result of a previous paper on the Hall algebra of an oriented cycle.
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Cardinals conundrum: What will be Carlos Martinez' role in the second half?
According to Derrick Goold, the Cardinals will spend some time over the All Star break assessing what to do with Carlos Martinez.
After optioning Martinez down to AAA on May 27th to stretch him out as a starter, the Cardinals recalled him on July 11th to give the team a fresh arm in the pen before the break. However, what was originally considered a temporary assignment may now become a more permanent one.
There are various factors that come into play here, the foremost being what makes the most sense, both for the team and for Martinez' development.
As GM John Mozeliak states:
"We have to assess what makes the most sense. He's electric. I definitely don't want him to go backward in terms of how much he has built up (as a starter). As we look at the next two or three weeks is there enough usage to justify keeping him here?"
In addition, Mike Matheny sums up the difficult decision like this:
"We do really like what he brings as a starter. For his development, it's better for him to be prepared to start. It's tough to weigh. We know he wants to be here. We like having him here. We think he can help us. But there is more that goes into the decision."
As Matheny says, there are other factors at play here as well.
For instance, a lot might depend on how Chris Carpenter fares in his rehab. He makes his first start tonight for AA Springfield and if he progresses, the need for Martinez as a starter decreases.
Also, will the Cardinals be able to find Martinez enough work? Prior to being moved into the starting rotation, Joe Kelly worked as the long reliever in the pen, the role that Martinez would likely take over. In the first two month of the season, Kelly only threw 18.2 innings, which would have put him on pace for about 55 innings. Hardly enough innings to allow a youngster like Martinez develop properly.
I think that the best option for the Cardinals and Martinez would be to send him back down to AAA to start. Although the Cardinals might not need another starter this year, they will certainly be looking for starters next year, with Jake Westbrook likely heading to free agency and the 5th starter spot still up in the air. Martinez could be one of the favorites to claim a starting role next year, but only if he's given time to properly develop this year.
However, the Cardinals may have already made their decision, as they optioned Keith Butler to AAA today. Butler, a righthander like Martinez, was unscored upon in his last 8 appearances, yet the Cards sent him down anyway, a possible indication that they have decided to keep Martinez.
In any case, I hope that if the Cardinals have decided to keep Martinez in the pen, it won't hamper his long-term development.
Labels: 2013 bullpen, 2013 rotation, Cardinals, Carlos Martinez, Chris Carpenter, Joe Kelly, Keith Butler
|
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| 7,974
|
How lucky are we that, in any capital city of Australia, we can walk down the street and be treated to the scents of South-East Asia, India, Japan, South America, France, Italy and more? With such a diverse food culture, with so much to offer, we are certainly spoiled for choice when it comes to eating out. Bring some of that flavour home with you with these outstanding recipes.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 4,081
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Anti-Gay Sect Leader Pleads Guilty for Murdering 4-Year-Old Boy and Adult Woman
Jadon Higganbothan, 4, (l) shot for allegedly being gay, and Antoinette McKoy, 28, (r) murdered for being unable to bear children.
Durham, North Carolina – The leader of an anti-gay sect has pleaded guilty to murder for killing a 4-year-old boy because he thought the toddler was gay, according to the Southern Poverty Law Center (SPLC). Peter Lucas Moses, 27, the leader of a polygamous group known as the "Black Hebrews," has agreed to testify against his mother, brother, and sister in order to avoid the death penalty for himself. He faces two life sentences for the murders of Jadon Higganbothan, 4, and Antoinette Yvonne McKoy, 28, if convicted of the crimes.
WRAL-TV reports that members of the Black Separatist cult addressed Moses as "Lord," and lived together in a house in Southeast Durham. In October 2010, because he believed he saw Higganbothan touch one of his sons "inapproriately" (the boy had allegedly spanked Moses' son on the bottom), he ordered the boy's mother to take him into the garage, where Moses shot the child in the head. The women in the group had arranged computer speakers in the garage to play the Lord's Prayer in Hebrew loudly enough to drown out the sound of the gunshot. Two months later, when Moses found out that his consort McKoy could not have children and had decided to escape the cult, he shot her to death in a bathroom of the house. On June 8, 2011, investigators found the bodies of Higganbothan and McKoy buried in trash bags in the basement of another house belonging to the sect. Moses' fingerprints were found on the tape used to secure the trash bags, and his handgun was proven to have been used in both murders.
The father of the little boy, Jamiel Higganbothan, told WRAL-TV News that he was furious the District Attorney had offered Moses a plea deal to save his life. "Me and my family wanted the death penalty," Higganbothan said after the deal was announced. Moses' brother, P. Leonard Moses, his sister, Sheila Moses, and his mother, Sheilda Harris, have been charged with accessories to the murder of Antoinetta McKoy. Jadon's mother, Vania Sisk, and two other women who lived with Peter Moses, Larhonda Renee Smith and Lavada Quinzetta Harris, have been charged with murder in McKoy's killing, and as accessories to the murder of the little boy.
The Black Hebrews, according to the SPLC, have roots going back to Black Separatist and Black Nationalist movements in the 19th century. They hold that they, not the Jews, are the true descendants of the Israelites in the Hebrew Bible. While most members of the modern movement in the United States are non-violent, a growing number of cells have become increasingly anti-Semitic, anti-gay, and prone to violence. They hold that modern Jews are imposters. These extremists also condemn whites for enslaving Blacks, and say that they are worthy of death or slavery because of it.
June 14, 2012 - Posted by unfinishedlives | anti-LGBT hate crime murder, Anti-Semitism, Black Hebrews, GLBTQ, gun violence, Hate Crimes, Heterosexism and homophobia, LGBTQ, Mistaken as LGBT, North Carolina, religious hate speech, religious intolerance | African Americans, anti-LGBT hate crime murder, Anti-Semitism, Black Hebrews, GLBTQ, gun violence, Hate Crimes, LGBTQ, Mistaken for being LGBT, North Carolina, religious hate speech, religious intolerance
|
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| 4,793
|
Lagos Government have began a massive hunt down of Okada riders across the state. The crackdown which began on Monday was directed by the state governor, Akinwunmi Ambode.
The governor had directed that Police and other agencies concerned to redouble efforts to ensure the law is complied with, assuring that the clampdown will be sustained vigorously on a daily basis.
But motorist complained that the crackdown is way beyond what was stipulated in the Lagos Traffic law that banned Okada in specific areas. They claimed the law only barred okada services on major state road and not inner road and streets as it is been currently enforced.
They said Okada riders that ply inner streets and lanes that are inaccessible to vehicles are also being arrested, thereby forcing untold hardship on Lagosians whose house are far from the major roads.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 3,518
|
To register or for more information, please call Cheryl at 403-201-6532.
I'm hanging onto summer for all it's worth, by teaching a workshop on painting fresh florals in oil & acrylics! And I'll be bringing in fresh flowers for everyone to paint from! The day will be a mix of learning by watching my demos, doing quick and easy exercises, and painting from your own floral bouquet. (You're also welcome to paint from your own reference photo.) We'll be talking about how to use color temperature, value, saturation, edge, volume and shape to create beautiful, fresh floral paintings. I'll supply one canvas, but you'll definitely want to bring in a couple more so you can get the most out of class.
This will be a small class, to ensure lots of one-on-one instruction time and relaxed atmosphere. Maximum number of students is 8. A supply list is available – I'll email it to you upon registration. We'll learn lots and have a great time doing it.
Coffee, tea and 'painting cookies' will be provided. Please either bring your own lunch, or aim to pick up a sandwich – Tim Horton's is a short drive down the road. To register, please call me at 403-201-6532. Cash, Cheque, MC and Visa welcome. I hope you can join us, and help us hang onto summer for one more weekend!
Previous Post Hittin' the Highway – Fall Art Classes in Cochrane!
|
{
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| 1,688
|
{"url":"http:\/\/mathhelpforum.com\/advanced-algebra\/72997-identity-element.html","text":"1. ## Identity Element\n\nI would appreciate any help given on this problem in abstract algebra. I spent an hour trying to figure it out but with no luck\n\nLet * be a binary operation on the nonempty set A. Prove that if A contains an identity element with respect to *, the identity element is unique. Prove using direct proof.\n\n2. Originally Posted by TitaniumX\nI would appreciate any help given on this problem in abstract algebra. I spent an hour trying to figure it out but with no luck","date":"2018-01-23 20:37:17","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\": 0.8160987496376038, \"perplexity\": 143.7239377351156}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084892238.78\/warc\/CC-MAIN-20180123191341-20180123211341-00510.warc.gz\"}"}
| null | null |
Felipe de Noircarmes (c 1530; Utrecht, 5 de marzo de 1574) militar al servicio de Carlos V y Felipe II de España. Él ganó notoriedad durante la represión de las insurrecciones calvinistas en los Países Bajos Españoles, por el Asedio de Valenciennes en 1566-7, la toma de Tournai el 2 de enero de 1567, y como miembro del Tribunal de los Tumultos al comienzo de la Guerra de los Ochenta Años. Él era estatúder del Condado de Henao desde 1566, y de Holanda, Zelanda y Utrecht de 1573 hasta su muerte.
Primeros años de vida
Noircarmes (como se le llama por lo general en la historiografía) cuyo nombre completo fue: Philippe René Nivelon Louis de Sainte-Aldegonde, Señor de Noircarmes era el hijo de Jean de Sainte-Aldegonde, descendiente de una antigua familia aristocrática de Saint-Omer y Marie de Rubempré Su padre había sido un chambelán de Carlos V (1538) y él mismo se menciona como una página de sus legajos.
Noircarmes casado Bona de Lannoy el 7 de septiembre de 1554. Tuvieron un hijo, Maximiliano-Lamoral, y una hija.
Referencias
Militares de la guerra de Flandes
Gobernadores de los Países Bajos Españoles
Estatúder
|
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| 8,652
|
Anaphes devillei är en stekelart som först beskrevs av Debauche 1948. Anaphes devillei ingår i släktet Anaphes och familjen dvärgsteklar. Inga underarter finns listade i Catalogue of Life.
Källor
Dvärgsteklar
devillei
|
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<td><a href="feathers/dragDrop/DragData.html" title="feathers.dragDrop.DragData">DragData</a></td>
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<td><a href="feathers/events/DragDropEvent.html" title="feathers.events.DragDropEvent">DragDropEvent</a></td>
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<td><a href="feathers/controls/Drawers.html" title="feathers.controls.Drawers">Drawers</a></td>
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<td><a href="feathers/controls/popups/DropDownPopUpContentManager.html" title="feathers.controls.popups.DropDownPopUpContentManager">DropDownPopUpContentManager</a></td>
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<!--Feathers Website | Feathers Documentation | Github Project | Support Forum<br/>Mon Nov 25 2013, 03:11 PM -08:00 -->
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,532
|
Q: How I set maxLength attribute in in HTML5? I want maxLength Validation in in HTML5.
I tried like this
<input type="number" name="trainer_mobile_no[]" maxlength="10" class="form-control" required="required"/>
A: You cannot set max length but max value for number input type something like this:
<input type="number" min="0" max="100">
A: You cannot set maxlength attr for number type, you can achieve this by returning keypress event false in jQuery.
$(document).ready(function () {
//called when key is pressed in textbox
$("#quantity").keypress(function (e) {
var maxlengthNumber = parseInt($('#quantity').attr('maxlength'));
var inputValueLength = $('#quantity').val().length + 1;
if (e.which != 8 && e.which != 0 && (e.which < 48 || e.which > 57)) {
return false;
}
if(maxlengthNumber < inputValueLength) {
return false;
}
});
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.7.2/jquery.min.js"></script>
Number : <input maxlength="8" type="number" name="quantity" id="quantity" />
A: you can add a max attribute that will specify the highest possible number that you may insert
<input type="number" max="999" />
if you add both a max and a min value you can specify the range of allowed values:
<input type="number" min="1" max="999" />
A: Use as
<input type="number" name="trainer_mobile_no[]" min="1" max="5" class="form-control" required="required">
A: There are two possibilities
<input type="number" min="1" max="999" />
and
if you want to check the length.
User will not be allowed to enter more than 4 digits
<input type="number" pattern="/^-?\d+\.?\d*$/" onKeyPress="if(this.value.length==5) return false;" />
Sample Code Given below.. run that
It will accept min 1 and max 999 numbers<br/>
<input type="number" min="1" max="999" />
<br/><br/>
and
<br/><br/>
if you want to check the length (max 5 is defined).<br/>
<input type="number" pattern="/^-?\d+\.?\d*$/" onKeyPress="if(this.value.length==5) return false;" />
A: you can use javascript in oninput like this:
<input type="number" name="contactNo" id="contactNo" class="form-control" placeholder="Enter Contact No" aria-label="contactNo" aria-describedby="basic-addon2" maxlength="10" data-rule-maxlength="10" oninput="javascript: if (this.value.length > this.maxLength) this.value = this.value.slice(0, this.maxLength); this.value = this.value.replace(/[^0-9.]/g, '').replace(/(\..*)\./g, '$1');">
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,369
|
{"url":"https:\/\/www.physicsforums.com\/threads\/find-the-area-enclosed-by-a-curve-and-two-lines.409165\/","text":"Homework Help: Find the area enclosed by a curve and two lines.\n\n1. Jun 10, 2010\n\neinsteinoid\n\n1. The problem statement, all variables and given\/known data\n\nFind the total area of the region enclosed by the curve x = y2\/3 and the lines x = y and y = -1.\n\n2. The attempt at a solution\n\nI graphed x = y2\/3 as y=x3\/2 and y=-x3\/2. Then I graphed the lines y=x and y=-1.\n\nI shaded the region in question, which appears off the cuff to have an area slightly larger than one. To find the area exactly I broke the problem up into two integration steps.\n\n1. $$\\int$$ x dx (from x=-1 to x=0) - $$\\int$$ -1 dx (in the same range of x=-1 to x=0)\n\n+\n\n2. $$\\int$$ -1 dx (from x=0 to x=1) - $$\\int$$ -x3\/2 dx\n\nI keep getting the whole area as 11\/10. This seems close enough, seeing as how the area appears to be slightly greater than 1. However, my textbook's answer glossary tells me that the correct answer is actually 6\/5. Can anyone tell me what i'm doing wrong?\n\nI attached a relevant graph (I did it in mspaint, lol, sorry). y=+x2\/3 is left out of the graph because it is irrelevant. The shaded region, i'm assuming, is the area in question:\n\nAttached Files:\n\n\u2022 area.jpg\nFile size:\n8.6 KB\nViews:\n150\nLast edited: Jun 10, 2010\n2. Jun 10, 2010\n\nTedjn\n\n3. Jun 10, 2010\n\neinsteinoid\n\nI was given the function x = y2\/3. This is a function of y but not a function of x. By that I mean every x value has more than 1 y value. In order for me to graph the equation as a function of x, I had to break it into two equations.\n\nGraphing x = y2\/3\n\nIs the same as graphing f(x)=x2\/3 and g(x)=-x3\/2 together.\n\nDoes this make sense?\n\n4. Jun 10, 2010\n\nStaff: Mentor\n\nIt's much simpler to use horizontal slices with typical area element given by $\\Delta A$ = (y2\/3 - y)$\\Delta y$, -1 <= y <= 1. Since the right and left endpoints of the typical area element don't change, a single integral will suffice.\n\n5. Jun 10, 2010\n\nTedjn\n\nWell, I get the same answer as you, but I don't trust my integration skills. If anyone else can confirm or point out the error, that would be good.\n\n6. Jun 10, 2010\n\neinsteinoid\n\nMathematically speaking, your response makes sense, Mark. If i integrate this:\n\n(y2\/3 - y)dy, -1<=y<=1\n\nI get 6\/5.\n\nAccording to the book, this is the correct answer, and for that I thank you! However, I'm now a little confused. I guess my new question is, what made you choose that integrand? And why does my choice of integrands not work?\n\n7. Jun 10, 2010\n\nTedjn\n\nOh, I see now the mistake we both made. There is a tiny sliver of area sandwiched between y = x and the upper half of the graph you left out.\n\n8. Jun 10, 2010\n\nStaff: Mentor\n\nIf you are given the bounding functions as x = g(y) and x = h(y), the most natural thing to try first is horizontal slices, because you don't have to convert both to their inverses. It still might be the case that you want to use vertical slices if the integration with horizontal slices turns out to be too much work or too difficult.\n\nI noticed that your graph was missing that bit of area in the first quadrant, but didn't mention it. Its area just happens to be 1\/10, which is why you were coming up with 11\/10 instead of 12\/10 = 6\/5.\n\n9. Jun 10, 2010\n\neinsteinoid\n\nI left out the curve in the first quadrant purposely because it didn't effect my shaded region in the 3rd\/4th quadrants. I guess it is safe to assume that the area I've shaded is incorrect, right?\n\nI can do the problem pretty easily now on paper but I'm confusing myself a little by trying to graph everything.\n\n10. Jun 10, 2010\n\nStaff: Mentor\n\nYes, the region in your graph was incorrect because it didn't include the part in the first quadrant. Your graph needs to be an accurate depiction of the region as described in the problem statement.\n\nIt's almost impossible to \"do the problem on paper\" unless you have a fairly accurate graph of the region you are going to integrate. These aren't separate, unrelated operations.\n\n11. Jun 11, 2010\n\neinsteinoid\n\nI was saying that I can calculate the area based on the integrand you provided, but was having trouble deducing that integrand myself. Anyways, thanks for your help.","date":"2018-06-20 23:33:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.7289614677429199, \"perplexity\": 673.0858514260356}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267863939.76\/warc\/CC-MAIN-20180620221657-20180621001657-00174.warc.gz\"}"}
| null | null |
Zigomar
by Léon Sazie
cover by Georges Vallée
"I'm accusing the one who signed these crimes with his initial… Z... I accuse Zigomar!"
Zigomar is a character dreamed up by Léon Sazie in 1909, two years before the now much more celebrated Fantômas. An evil, nefarious character, a criminal genius, Zigomar was so popular in his time that his picture could be found on bags of bread, pipes and matchboxes.
Masked, hooded, or in disguise, Zigomar constantly bedevils the law. The first of the masked super-criminals, he shares with Fantômas a taste for gratuitous, melodramatic crimes, imaginative atrocities (typhus-bearing mosquitoes being only one such example), murder, kidnapping, robbery, and torture. His inevitable escape from the clutches of the law, his perpetual evasion of justice, made him very popular with the public and he left his mark on the history of crime fiction.
Of Basque origin, Léon Sazie was born in Algeria in 1862 and died in an accident in Suresne near Paris in 1939. When he was still a child, his father committed suicide after being ruined in a bank fraud. Sazie eventually became a journalist, before turning to theater and, eventually, to serial fiction. He created Martin Numa, King of detectives, in 1908, and Zigomar a year later. He was also a brilliant fencer who fought several duels.
This volume, translated and introduced by Michael Shreve, contains a translation of the first of the six Zigomar novels, plus an introduction, bibliography and filmography (Zigomar was adapted three times for the screen in silent movie serials).
BOOK ONE : THE INVISIBLE MASTER (1909-10)
BOOK TWO: LIONS AND TIGERS (1910)
BOOK THREE: TIME FOR JUSTICE (1910)
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North Bristol NHS Trust
North Bristol NHS Trust (NBT) is a centre of excellence for health care in the South West region in a number of fields as well as one of the largest hospital trusts in the UK with an annual turnover of £544 million (including £33m related to the Community Children's Health Partnership). Sixty five per cent of our income for patient care comes from the three principal clinical commissioning groups of Bristol, North Somerset and South Gloucestershire. Most of the rest is received for specialist services commissioned by NHS England.
As the largest teaching Trust in the South West we have internationally renowned medical teams delivering incredible outcomes for our patients through the hard work and dedication of our 9,000 doctors, nurses, healthcare professionals and support staff.
For many years we have been at the forefront of new medical techniques and innovations, including brain and spinal surgery, joint replacement and the world's first radar breast imaging system, to name but a few.
We provide award-winning services and receive a significant number of referrals from other hospital Trusts for our wide range of specialist services. We treat over 300,000 patients a year, including over 6,300 births, and yet our philosophy is that each patient is treated with respect and dignity and, most important of all, as a person.
Our commitment is that each patient is treated with respect and dignity and, most important of all, as a person. We aim to provide "exceptional healthcare personally delivered" by offering services of exemplary quality, ensuring no unnecessary waits or delays, giving care in high quality facilities and having well trained and caring staff. Our vision is to be the provider of choice for patients needing our specialist care. We want to deliver innovative services with excellent clinical outcomes in the most appropriate setting for our patients. In May 2014 the vast majority of the Trust's inpatient services were centralised at Southmead Hospital Bristol when the new Brunel building opened its doors. Services from both Frenchay and Southmead hospitals were moved across during that month.
For 2015/16 the Trust Board signed up to focussing on three key strategic themes that were designed to address the challenges facing the Trust following the move. In addition, the Board wanted to focus on providing services for our patients which were safely delivered with an enhanced experience and made progress on meeting the constitutional standards. There was recognition of the need to have a stable workforce, with less dependency on agency staff, and for sustainable service delivery all within controlled finances. It was also important to conclude the major site redevelopment programme at Southmead.
The key themes of our plan in 2015/16 were:
*Delivering our core offer
*Being a healthy organisation
*Creating a future.
Westbury-on-Trym
BS10 5NB
Vacancy status: Closed
Ref: 339-CVP-PERI-Admin 1121 Vacancy ID: 3699364
Vaccination Administrator - Community Vaccination Team
Closed for applications on: 3-Dec-2021 00:00
Fixed term: 6 months
Full time - 37.5 hours per week (Up to 37.5 hours per week Monday to Saturday)
£20,330 - £21,777 per annum, pro rata.
Administration / Customer Service
Please note that if you apply for a position with this Trust, you may be contacted via Trac or via email. This includes invites for job interviews. We therefore recommend that you regularly check your Trac Account and email accounts.
Please note that this job advert will close as soon as sufficient applications have been received. So if you are interested, please apply for this vacancy as soon as you can.
The community vaccination team will work across local community settings within Bristol, North Somerset and South Gloucestershire to provide Covid immunisations to the local population.
The postholder will be required to travel to different community venues and work with local communities to deliver vaccinations and help maximise vaccination uptake. help!
Working to support the Vaccination teams your role will involve Front of house meet and greet patients, explanation of the vaccination process, co-ordination of patient information, booking 2nd vaccination dates, and maintaining patients' movement through the system.
We currently have vacancies across the Vaccination Programme:
Working hours will be on a Rota basis, Monday to Saturday between 9am and 7pm
Briefing and training on Covid-19 vaccines will be provided by our Staff Development Team.
If you apply for this vacancy and have not received a communication from this Trust within three weeks of the closing date, please assume that on this occasion your application has been unsuccessful.
Please note that this Trust does not reimburse travel expenses relating to interview attendance.
If you feel you meet the requirements of the Disability Act / Two Ticks scheme and require further support/advice, please contact us on tel 0117 414 1151.
This organisation is committed to safeguarding and promoting the welfare of children and young people and expects all staff and volunteers to share this commitment.
The successful applicant(s) will normally commence at the minimum of the scale unless they have previous NHS service at the same band. Progression through the scale is by annual increments.
Staff identify as 'At Risk' and who meet the essential shortlisting criteria of the post will be prioritised. Priority will be given to North Bristol NHS Trust employees in the first instance.
Please note that North Bristol NHS Trust has an arrangement with a number of organisations (please find a list below) whereby we confidentially share previous pre-employment information relating to your existing job in order to support an efficient recruitment process for candidates when offered a new position with us.
(University Hospitals Bristol and Weston NHS Foundation Trust, North Bristol NHS Trust, Bristol Community Health, Avon and Wiltshire Mental Health Partnership, North Somerset Community Partnership, NHS South Central and West Commissioning Support Unit and Bristol City Council)
Please note that stringent pre-employment checks are undertaken on all successful applicants prior to commencement in post. Please refer to the 'Information for Prospective Employees' attachment for information relating to DBS charges.
You must have appropriate UK professional registration.
The postholder will have access to vulnerable people in the course of their normal duties and as such this post is subject to the Rehabilitation of Offenders Act 1974 (Exceptions) Order 1975 (Amendment) (England and Wales) Order 2020 and as such it will be necessary for a submission for Disclosure to be made to the Disclosure and Barring Service to check for any previous criminal convictions.
Good communication skills.
Knowledge of prevoius admininstrator experience
Front of house experience
Job Description (PDF, 245.6KB)
Person Spec (PDF, 98.5KB)
Joanna Holmes
Recruitment Lead
recruitment.massvaccination@nbt.nhs.uk
No longer accepting applications
Sorry, this vacancy is no longer accepting applications.
You can search for similar jobs on the employer's job board, or visit our national jobs board Health Jobs UK.
Your data is being collected by North Bristol NHS Trust, whose privacy notice can be found here.
The data controller for this information is North Bristol NHS Trust. This application tracking system is provided by Civica UK Ltd (https://www.civica.com/en-gb/product-pages/trac/) as a data processor.
To make an enquiry, a request for your personal information held as part of this process, or to arrange for any mistakes to be corrected, you may contact either the team who are handling your application or the Data Protection Officer (helen.e.williamson@nbt.nhs.uk).
https://www.nbt.nhs.uk/about-us/information-governance/privacy-policy-data-protection
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Home Latest News Former Pentagon Chief Jim Mattis Said Trump Is Trying To Divide America
Former Pentagon Chief Jim Mattis Said Trump Is Trying To Divide America
Mattis was head of US Central Command when Obama fired him in 2013 over his hawkish views on Iran. It's also worthy of note that Mattis refused to criticize Trump publicly after he resigned, insisting the military must remain apolitical.
Former Pentagon chief Jim Mattis has finally commented on President Trump's leadership as he slammed the U.S. leader, on Wednesday, accusing him of trying to "divide" America and failing to provide "mature leadership" as the country staggers from days of protests over George Floyd's murder.
YOU MAY ALSO LIKE: Twitter Reacts To Melania Trump's Advice On 'Curfews' To American Citizen
Former Pentagon chief Jim Mattis resigned in December 2018 over Trump's decision to order a full troop withdrawal from Syria. Mattis has voiced his support for the demonstrators whose anti-racism rallies have shaken the country.
"Donald Trump is the first president in my lifetime who does not try to unite the American people — does not even pretend to try," Mattis wrote in a blistering statement posted online by The Atlantic.
"Instead, he tries to divide us," added the retired Marine general, who had previously argued it would be inappropriate for him to criticize a sitting president when he left office and was asked to make a comment.
"We are witnessing the consequences of three years without mature leadership," he stated.
Mattis described himself as "angry and appalled" after witnessing events of the last week, which saw Trump threaten a military crackdown on American citizens as nationwide protests turned violent in some cities.
The protest and agitation were sparked by the May 25 killing of George Floyd, a black man who suffocated beneath the knee of a white Minneapolis police officer, and whose agonizing death was filmed by passersby.
YOU MAY ALSO LIKE: U.S. Scrambles To Calm Unrest As Trump Faces Criticism For Violent Crackdown
Addressing the protester's demand, Jim Mattis wrote, "the protester's call for equal justice was a wholesome and unifying demand."
Mattis completely condemned Trump's decision to use the force to clear peaceful protesters from near the White House on Monday to allow 'him' to pose for photographs at a nearby damaged church, calling it an "abuse of executive authority."
The photo op has become a lightning rod for criticism of Trump's handling of the crisis, with religious leaders, politicians, and onlookers around the country expressing outrage.
"When I joined the military, some 50 years ago, I swore an oath to support and defend the Constitution," Mattis stated.
"Never did I dream that troops taking that same oath would be ordered under any circumstance to violate the Constitutional rights of their fellow citizens — much less to provide a bizarre photo op for the elected commander-in-chief, with military leadership standing alongside."
The U.S. President Trump has dismissed Mattis with a tweet, rehashing his claim that he "essentially" fired his Pentagon chief.
"Probably the only thing Barack Obama and I have in common is that we both had the honor of firing Jim Mattis, the world's most overrated General," the president wrote.
Mattis was head of US Central Command when Obama fired him in 2013 over his hawkish views on Iran. It's also worthy of note that Mattis refused to criticize Trump publicly after he resigned, insisting the military must remain apolitical.
Mattis Wednesday's statement is a signal that he no longer felt bound by his belief and respect for the president, as he called for solidarity with or without the president.
"We can unite without him, drawing on the strengths inherent in our civil society," Mattis wrote.
YOU MAY ALSO LIKE: Police Brutality Against Protesters Continue To Rise Across The Country
Mattis also seems not to be happy with the current Pentagon chief Mark Esper, although he didn't mention his name, but said, "We must reject any thinking of our cities as a 'battlespace' that our uniformed military is called upon to 'dominate.'"
The serving Pentagon chief Mark Esper came under heavy criticism after telling US governors on Monday that they should "dominate the battlespace" to end the protests, but had backtracked on Wednesday when he told reporters: "In retrospect, I would use different wording."
VIAAFP
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Q: Get the resource in the MainWindow in WPF I have added a Cursor file as a resource in my MainWindow.xaml, how can I access this resource from within a Border element that resides inside this window in code?
<Window.Resources>
<ResourceDictionary>
<FrameworkElement x:Key="OpenHand" Cursor="pack://application:,,,/Resources/openhand.cur"/>
</ResourceDictionary>
</Window.Resources>
<Grid x:Name="AppInterface">
// A Border is added here by code
// I want to be able to access the above resource from Border in code-behind
</Grid>
A: try this one :
var elementhand = Application.Current.MainWindow.Resources["OpenHand"] as FrameworkElement;
A: You can use Application.FindResource to find them by name.
FrameworkElement resource = Application.Current.FindResource("OpenHand");
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Everything Theatre An honest and unpretentious guide to the London theatre scene.
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Home » Reviews » Alternative » The Day of the Dead, Rich Mix – Review
The Day of the Dead, Rich Mix – Review
Author: Rachel Proctor in Alternative, Off West End, Reviews 23 November 2015 0 146 Views
Pros: An enchanting evening performed by experts in their field, full of beautiful words that lift you from your seat and take you elsewhere.
Cons: Every story might not be to your last, but don't worry: a different, fascinating story will be along in just a few minutes.
Pros: An enchanting evening performed by experts in their field, full of beautiful words that lift you from your seat and take you elsewhere. Cons: Every story might not be to your last, but don't worry: a different, fascinating story will be along in just a few minutes. This year marked the tenth anniversary of The Crick Crack Club's Day of the Dead. We were told that when the storytellers know they will be performing on this night, they start to compile and collect stories which weave in the threads of God, Death and the Devil. This evening is clearly…
Yet another winning evening from The Crick Crack Club - a joyous chance to revel in life and death.
This year marked the tenth anniversary of The Crick Crack Club's Day of the Dead. We were told that when the storytellers know they will be performing on this night, they start to compile and collect stories which weave in the threads of God, Death and the Devil. This evening is clearly an event on the Crick Crack Club's calendar, and well it should be. Drawing on the Mexican tradition, what unfolded was the perfect way to celebrate the nights drawing in and the death of the year.
I've come to worry a little bit about the use of the Mexican idea of the Day of the Dead in the last few years. It seems to me there is a real risk of cultural misappropriation, but this evening felt so honest and respectful, and had in it a real way of appreciating the rich history of storytelling handed down to us by the generations that have come before. In 2015, when we have such easy and fast access to the pre-digested narrative of rubbish films and trashy television, to sit with a group and listen as stories were unfolded felt soothing and natural.
We were treated to the narrative talents of three lovely storytellers, all coming from different storytelling traditions as well as drawing on others. Clare Muireann Murphy's beautiful lilting Irish voice was in stark contrast to the darkly humorous stories she told us, and she her first-rate comic physicality only served to enhance her storytelling. The best of this stories involved the attempts of three sisters to throughly humiliate their husbands, which they achieved valiantly, in incredibly creative fashions and to everyone's pleasure (except, I suspect, the poor husbands…)
Next up, was TUUP, born to Guyanese parents and raised in Acton. His stories narrated the lives of a hypochondriacal old woman, whose friends took revenge upon her over-dramatic ways, as well as narrating us through the grief on an old man who's wife had died, and, my personal favourite, the very funny story of one man's four day long party for his ancestors which was so loud and raucous it attracted the displeasure of God. What was most striking about TUUP's performance wasn't the humour, although he was very funny, or the drumming with which he kept his own pace up, but rather his apparent spontaneity. I can't tell, still, whether he was building these stories around a loose frame on the hoof, or he just had this remarkable ability to appear as if everything he was saying was coming to him then and there. Either way, his performance was some of the most skilful storytelling I've ever seen, and the wild passion with which he talked was thoroughly enthralling.
Our last storyteller was Xanthe Gresham. Most outstanding from her performance was the poem she had written for her grandmother, and which she performed with the help of the audience. In some ways it perfectly captured the evening: it somehow encompassed that odd mixture of emotions which descends when you recall the happy or funny moments you shared with someone you've subsequently lost. That simultaneous swell of grief and joy. Know the emotion I'm thinking of? That was this evening to a tee.
The thing that fascinates me about attending wonderful evenings of speaking, either rhythmically poetic spoken word nights or beautiful, referential storytelling, is the audience. Really, I should in no way be surprised by the amazing diversity that these evenings attract, after all the vast history and tradition of storytelling ranges across all cultures and ages. Yet sitting amongst the young and old, locals and people who had travelled far, first time attendees and loyal followers, I felt so happy that there has been a modern day revival of storytelling which has such broad ranging reach. Long may it continue – as long as it continues as brilliantly as this.
Performers: Clare Muireann Murphy, Xanthe Gresham and TUUP
Produced by: The Crick Crack Club
Booking Until: This was a one off event. To find out more about Crick Crack Club events visit http://www.crickcrackclub.com
Clare Muireann Murphy Crick Crack Club Day of the dead Tuup 2015-11-23
Rachel Proctor
Tagged with: Clare Muireann Murphy Crick Crack Club Day of the dead Tuup
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Next: 30-Second Shakespeare, Ivy Press – Book Review
About Rachel Proctor
Having intermittently been reviewing since the formation of ET, Rachel is currently taking a year off from working as a doctor to go back to University and study Medical Humanities, in an effort to basically do that English degree she didn't have a chance to do at medical school. It does mean there is plenty of time to get back into seeing loads of theatre in London, which she can basically pass off as further studying. She'll watch pretty much anything, with a penchant for an odd venue and anything with pretty lighting design.
The Maids / Hizmetçiler, The Hen and Chickens Theatre – Review
Luzia, Royal Albert Hall – Review
The CO-OP, White Bear Theatre – Review
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Trasimensko jezero (italijansko Lago Trasimeno [ˈlaːɡo traziˈmɛːno ], latinsko Trasumennus, etruščansko Tarśmina ) je jezero v pokrajini Perugia, v deželi Umbrija v Italiji na meji s Toskano. Jezero je južno od reke Pad in severno od bližnje reke Tibere, ima površino 128 km2 in je četrto po površini v Italiji. (Je nekoliko manjše od Komskega jezera) Samo dva manjša potoka se izlivata direktno v jezero in noben ne izteka. Nivo vode jezera znatno niha glede na količino padavin in sezonskih potreb mest, vasi in kmetij v bližini obale.
Jezero
Trasimeno je plitvo, blatno in bogato z ribami, vključno s ščukami, krapi in linji. V zadnjih 10 letih je bilo v povprečju globoko 5 metrov. Jezero Trasimeno je endoreično vodno telo; je zaprto jezero, ki sprejema vodo, vendar nima iztoka. Druga endoreična vodna telesa po svetu so na primer Kaspijsko jezero, Aralsko jezero, Veliko slano jezero v Utahu. Izhlapevanje lahko privede do kopičenja mineralov v vodi, kar ima za posledico slane razmere, zaradi česar so ta jezera občutljiva na pritiske zaradi onesnaženja.
Plitke vode so pomenile, da so uspevali malarijski komarji. Za boj proti malariji so bile v 1950-ih iz ZDA uvožene nekatere ribe, ki jedo ličinke komarjev. Te ribe so široko razpršene, nekatere pa živijo v jezerih blizu Trasimena. Čeprav pojedo ogromno ličink, je še vedno veliko komarjev in drugih žuželk.
Kakovost vode v jezeru je še vedno zelo dobra, kot je pokazala raziskava zaščite skupine Italia Nostra leta 2005. To naj bi bilo v veliki meri posledica majhne populacije in pomanjkanja velikih kmetij na tem območju.
Predlog za izčrpavanje jezera za reševanje težav z malarijo in globinskimi spremembami je bil zavrnjen. Konec 19. stoletja so spremembe nivoja rešili z izgradnjo prekopa v bližini San Feliciano. To je tudi zmanjšalo težavo z malarijo.
Izvor in zgodnja zgodovina
Pred tremi milijoni let je bilo v tem delu Umbrije plitvo morje. Depresija, ki je nastala zaradi geoloških prelomov, je omogočila oblikovanje današnjega Trasimenskega jezera.
Zgodovinsko je bilo Trasimeno znano kot Perugijsko jezero, saj je bilo pomembno za severozahodno Umbrijo in okrožje toskanska Chiana. V prazgodovini se je to jezero razširilo skoraj do Perugije. Trasimeno je mitološka figura, združena z Agillo, nimfo, rojeno v Agellu, danes griču na sredini med Perugijo in Trasimenom, prej otokom v jezeru.
Prva civilizacija, ki je naselila to območje, so bili Etruščani. Tri glavna etruščanska mesta – Perugia, Chiusi in Cortona - so oddaljena 20 kilometrov od jezera. Malo fizičnih dokazov je ostalo iz etruščanskega obdobja ali poznejših rimskih naselbin. Castiglione del Lago ima nekaj rimskih razvalin, njegove glavne ulice pa so strukturirane kot šahovnica v rimskem slogu.
Bitka pri Trasimenskem jezeru se je zgodila na severni obali jezera aprila 217 pr. n. št. med drugo punsko vojno. Natančna lokacija bitke ni znana, ker se je jezero razširilo naprej proti severu; bitka bi se lahko vodila med Cortono in Tuoro. V bližini Cortone je kraj, imenovan Ossaia, v italijanščini za 'kostnica'. Drug kraj glede bitke je kraj z imenom Sanguineto, katerega ime je povezano z italijanskim izrazom sangue, kar pomeni 'kri' ali, verjetno, 'krvavo mesto'.
Lokalno podnebje
Podnebje okoli jezera je precej toplo, zime so zmerne. Poletja so lahko zelo topla in vlažna, na splošno pa jezero moderira podnebje tako v hladnih kot toplih pogojih, ker tudi plitva voda daje zmerno termično vztrajnost. Od maja do septembra je temperatura dovolj visoka, da omogoča kopanje. Leta 1929 je hladna zima zamrznila celotno površino jezera in čez led so se lahko vozili z avtomobili. Hladne zime 1957, 1985 in 2002 so povzročile veliko škode na oljkah v bližini. Manj huda zamrznitev se je zgodila leta 1991. Glede na širino jezera je zamrznitev še vedno redek pojav.
Nivo vode
Nivo vode v jezeru je zelo odvisen od količine sezonskega dežja in se lahko iz leta v leto močno spreminja. Nivo vode je po navadi najnižje poleti (od septembra do oktobra), najvišje pa spomladi (aprila in maja).
Trasimeno ima na vzhodu visoke hribe, ki pomagajo ujeti dež in delno zaščititi jezero pred hladnimi vzhodnimi vetrovi. Večina vode v jezeru prihaja iz mreže potokov na zahodni strani jezera. Referenčna vodna gladina jezera je določena na 257,33 m nmv. Ta raven ustreza največji globini okoli 6 m. Načrtujejo se ukrepi za znižanje vodostaja le, če je ta višji od 257,60 m nmv, a ker je bila najvišja meja postavljena le občasno, je jezero tako visoko.
Po drugi svetovni vojni se je obala jezera umaknila za kilometer na zahodu (vzhodna obala je globlja in bolj strma). Leta 1958 je voda jezera dosegla najnižjo zabeleženo raven in sicer -2,63 m v primerjavi z referenčno gladino. Od leta 1958 se je raven ponovno povečala in je junija 1989 dosegla referenčno raven. Potem ko je gladina vode spet padla, se je leta 2003 obala umaknila za več kot 100 metrov, raven pa se je oktobra 2003 znižala na -1,85 m v primerjavi z referenčno vrednostjo. Od leta 2004 do poletja 2006 je bilo veliko dežja. V zadnjih 20 dneh avgusta 2005 je padlo 150 mm dežja, v preostalem delu leta pa več kot 700 mm. Na žalost je bilo v naslednjih petih letih dežja premalo (zlasti v letih 2006 in 2007 so bile zime zelo suhe, poletja vroča), zato je bil do oktobra 2012 vodostaj -1,51 m v primerjavi z referenčnim. Na srečo so bile zime od leta 2012 zelo deževne, zato se je gladina jezera postopoma povečevala, da je februarja 2014 dosegla referenčno raven. Aprila 2014 se je vodostaj še povečal in je dosegel 0,30 m nad referenčno vrednostjo .
Leta 2012 je bil odprt prekop iz akumulacijskega jezera Montedoglio v Toskani za oskrbo s kmetijstvom in obrežja mest in s tem odpravljena potreba po črpanju vode iz jezera.
Zaščita
Prebivalci občin okoli Trasimena in Umbrijci so uspešno zaščitili svoje jezero, katerega vode so primerne za kopanje in katerih doline in otoki so nedotaknjeno okolje. Leta 1995 je bil na celotni površini in obalah ustanovljen naravni park. Leta 2003 je bila okoli jezera odprta 50 km dolga kolesarska steza, ki turistom omogoča, da raziskujejo okolico. Obstajajo tudi pohodne poti, predvsem čez hribe na vzhodni strani.
V zadnjih dvajsetih letih so v jezero uvedli več tujih vrst (luzijanski rak (Procambarus clarkii), črni somič (Ameiurus melas), Psevdorazbora in navadna zlata ribica (Carassius auratus), ki ogrožajo avtohtono vodno življenje. Zlata ribica je bila prva tujerodna vrsta, ki je bila uvedena v jezero in danes predstavlja približno 50 % celotne prisotnosti rib.
Okolica jezera
Polovica Trasimena je obdana z griči, bogatimi z oljkami, ki so pomemben kmetijski vir. Na zahodni obali blizu Toskane so vinogradi, gojijo tudi sadje in zelenjavo. Hribi so precej nižji in podnebje toplejše. Monte Subasio v bližini Assisija je približno 70 km proti vzhodu in Monte Amiata, približno 70 km proti zahodu. Vegetacija vključuje borovce, vrbe in topole okoli obrežja.
Glavna naselja so Passignano sul Trasimeno, Tuoro, Monte del Lago, Torricella, San Feliciano, San Arcangelo, Castiglione del Lago in Borghetto. Castiglione del Lago ima najdaljšo obalo, saj je na edinem pomembnem jezerskem polotoku. To je bil morda otok, ki so ga Rimljani pridružili kopnemu.
Okoli jezera so stara majhna mesta in osamljeni gradovi, kot sta grad Zocco in stolp blizu Passignana. Monte del Lago je bil prvotno zgrajen za nadzor ceste od Trasimena do Perugie.
Letalstvo
Lokalno letališče Eleuteri v bližini Castiglione del Lago je bilo nekoč ena glavnih letalskih šol v Italiji z elegantnimi zgradbami, ki so jih poleti 1944 uničile nemške čete. To letališče je bilo nekoč skoraj tako veliko kot Castiglione. Blaga klima in popolna vidnost še vedno omogočajo uporabo tega letališča za letalske mitinge.
Na nekdanjem letališču obstaja socialno središče. Glavni vodni tok, hudournik Paganico, je letališče ločil od mesta.
Na nasprotni obali, v mestu Passignano, je bila tovarna letal SAI Ambrosini. To je zdaj opuščeno kot industrijsko središče, vendar se še vedno uporablja kot socialno središče. Ustanovljeno je bilo pred približno 80 leti, zgradbe še vedno obstajajo v bližini železniške postaje Passignano sul Trasimeno. To podjetje je izdelalo več vrst letal, ki jih je zasnoval ing. Sergio Stefanutti. Letala so bila testirana na letališču Eleuteri, le nekaj kilometrov stran od te tovarne. Eleuteri je bil uporabljen tudi kot testno središče za napredno letalo Ambrosini SS.4, ki je strmoglavilo na drugem poletu in projekt so opustili.
Povezave
Trasimeno je sorazmerno oddaljen od vseh večjih italijanskih mest, od katerih je najbližja Perugia. Pred več kot 100 leti je bila zgrajena zgodovinska železnica z glavno železniško postajo v Terontoli. Manjše železniške postaje so v Passignanu in Castiglione del Lago.
Zaradi povečanega prometa so pred približno 30 leti zgradili cesto čez Passignano v Perugijo. Ta cesta gre blizu severne in vzhodne obale Trasimena in do Perugije in Assisija. Obstajajo tudi številne manj pomembne ceste, na primer statal 75, zlasti na zahodni strani jezera. Avtocesta A1 poteka pet kilometrov zahodno od jezera.
Plovba
Za plovbo po jezeru veljajo strogi predpisi. Za celoten obod jezera Trasimeno je vzpostavljen zaščiten pas na razdalji 150 metrov od obale jezera in obale otokov. V zaščitenem pasu je plovba dovoljena samo za plovila, katerih največja dolžina je 9 metrov na vodni črti, ki jih poganja veslo ali jadro, pri največji hitrosti dveh vozlov. Izjeme veljajo za uradne čolne, ki jih poganja motor samo v vodi pred pristaniškimi območji ali dovoljenimi pristajalnimi mesti.
V pristanišču Passignano so tudi trajekti, 3 majhni, 2 srednje velika in dva velika (dve palubi), imenovana Perusia in Agilla II, s privezom v pristanišču Passignano in tudi dva bagra. Obstajajo pristanišča v Castiglione del Lago (pred kratkim popolnoma obnovljena), S. Arcangelo, S. Feliciano, Tuoro in več manjših sidrišč.
Otoki
V jezeru so trije otoki. Največji od teh otokov je Isola Polvese, skoraj 1 km2. Drugi največji, Isola Maggiore, je edini naseljen. Majhna ribiška vasica, ki je svoj višek dosegla v 14. stoletju, ima danes le okrog trideset prebivalcev. Večina zgradb, vključno z ruševinami frančiškanskega samostana, izvira iz 14. stoletja.
Maggiore je hribovit, medtem ko je Polvese ravninski in griči, Minore pa spominja na nagnjeno mizo. Minore je zdaj nenaseljen, v preteklosti pa je imel vas z več kot 500 prebivalci. Pred mnogimi stoletji je ob obali, v bližini samostana Olivetan, stal grad s peterokotno zgradbo. Grad še vedno ostaja, razvaline cerkve in samostana pa so skoraj popolnoma ohranjene, kljub opuščanju v 17. stoletju zaradi malarije. Malarijo so izkoreninili šele v 1950-ih. Pojavile so se tudi druge težave, saj so Trasimeno premagali Chiusi, Panicale, Perugia in Firence.
Firenške čete so v 17. stoletju porušile Polvese, naselje je začelo propadati, dokler do 19. stoletju ni obstajal le oskrbnik. Od številnih hiš ni ostalo nič.
Otok Minore v bližini Maggiora je zdaj nenaseljen, popolnoma ga pokriva lokalno rastje, razen majhnega sidrišča. V starih časih je prišlo do ločitve med obema skupnostima, saj je bil Polvese daleč stran od Maggiore-Minore. Po krajevnih zgodbah sta se obe skupnosti borili druga proti drugi, resnični problemi pa so bili v regionalnih silah, ki so se stoletja borile za to jezero.
Lokalno se je uporabljala ribolovna tehnika, imenovana Tuoro ali pesca da tuori. Ta zapleten sistem je bil sestavljen iz lesene pasti v vodi in krožne konstrukcije, ki drži mrežo okoli nje. Mreže so ujele ribe, ki so jih nato odpeljali v vas, da so jih posušili. Ta sistem je deloval ob visokem vodostaju, in je bil opuščen, ko je nivo padel. Pred nekaj leti je bil v bližini pristanišča otoka Polvese zasnovan navidezen tak sistem.
Trasimenski gradovi
Okoli Trasimenskega jezera so gradovi, mnogi so v središču majhnih naselij, drugi osamljeni in v razvalinah. Otoki Castiglione del Lago, Passignano, Magione, Maggiore in Polvese imajo gradove, medtem ko so grad Zocco, grad Montali in drugi na vrhovih hribov.
Grad Guglielmi na otoku Maggiore je bil zgrajen v poznem 19. stoletju na temeljih stare frančiškanske cerkve in je bil dolga leta priljubljen kraj. Do leta 1998 je bil še viden, nato so ga zaprli, ker je zgradba postala nevarno nestabilna. Zdaj ga obnavlja novi lastnik, vendar dela še zdaleč niso končana.
Med Monte del Lago in S. Feliciano je grad Zocco, porušen desetletja. Je v zasebni lasti, vendar ni vzdrževan. Je eden največjih gradov na tem območju in edini, ki ima v svojih stenah iz peščenjaka še vedno nedotaknjeno srednjeveško podobo. Pred leti je bil verjetno naseljena, saj je tam stavba, opremljena s TV anteno, zdaj pa je njen edini vhod zaprt. Najbolje ohranjeni deli sta vzhodna in južna stena, ki sta vse bolj ogroženi, ker se napake stene povečujejo. Preostali zidovi so večinoma porušeni ali propadli. Eden od južnih stolpov ima v sredini dve ogromni razpoki
Nagnjeni stolp Vernazzano (N43 13.210 E12 06.084), visok približno 20 metrov, se nagiba kot znameniti Poševni stolp v Pisi. Ta edinstven ostanek starodavnega gradu je bil zgrajen pred letom 1089, ko je družina Marchiones cel grad podarila samostanu Città di Castello. Leta 1202 je padel pod nadzor Perugie in to mesto je dobilo nadzor nad Severnim Trasimenom. Zgrajen je bil na Monte Castiglione, blizu dveh rek. Grad in okoliško naselje na Vernazzanu so v 15. stoletju in dve stoletji pozneje poškodovali močni potresi in popotresni sunki. Vodna erozija temeljev je povzročila, da se je stolp v 18. stoletju nagnil. Vernazzano je bil obnovljen v dolini, stran od tega mesta, ki je bilo učinkovito za teritorialni nadzor, vendar je bilo manj primerno za bivanje v. Nagnjeni stolp je bil zato zapuščen skoraj 300 let. Da bi se izognili propadu, so pred kratkim dodali jekleno armaturo. Stolp ni dobro znan, saj stoji stran od glavnih ulic. Viden je od daleč, vendar ni lahko dostopen.
Slike
Sklici
Zunanje povezave
Official website
Lake Trasimeno 2
Lake Trasimeno 3
Endoreična jezera
Jezera v Italiji
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: Reading dynamic routes I'm trying to create a dynamic routing system in my Node.js application using Express.js. Instead of a view, I'm using a Services layer in my MVC-like structure. I want to read the routes from a routes folder and dynamically create the routes based on the folder that contains routes for example student.js,teacther.js etc.
I've written the following code in my index.js file:
fs.readdir('./api/Public/Routes', (err, files) => {
if (err) throw err;
for (let file of files) {
const routeName = file.slice(0, -3);
console.log('routeName: ' + routeName);
console.log(3);
let routeFile = require(`./Public/Routes${routeName}`);
console.log(2);
app.use(`/${routeName}`, routeFile);
}
});
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
Correlation clustering is a concept of machine learning. It was introduced in Bansal et al. \cite{bbc}, which gives a good overview of the mathematical background as well. Let $G$ be a complete graph on $n$ vertices and label its edges with $+$ or $-$ depending on whether the endpoints have been deemed to be similar or different. Consider a partition of the vertices. Two edges are in conflict with respect to the partition if they belong to the same class, but are different, or they belong to different classes although they are similar. The ultimate goal of correlation clustering is to find a partition with minimal number of conflicts. The special feature of this clustering is that the number of clusters is not specified.
A typical application of correlation clustering is the classification of unknown topics of (scientific) papers. In this case the papers represent the nodes and two papers are considered to be similar, if one of them cite the other. The classes of an optimal clustering is then interpreted as the topics of the papers.
The number of partitions of $n$ vertices grows exponentially,
hence there is no chance for exhaustive search at computer experiments. Bansal et al. \cite{bbc} proved that to find an optimal clustering is NP-hard. They also presented and analyzed algorithms for approximate solutions of the problem. The correlation clustering can be treated as an optimization problem: minimize the number of conflicts by moving the elements between clusters. Hence one can use the traditional and new optimization algorithms to find near optimal partition.
Bak\'o and Aszal\'os \cite{ab} have implemented the traditional methods and invented some new ones.
Both above cited papers assumes nothing on the signing of the edges. It is natural to expect that the rule of the signing affects the optimal clustering. This expectation motivated the investigation of the present paper. To define a systematic signing it is straightforward to label the vertices by natural numbers and the sign between two vertices depends on a number theoretical property of the label of these vertices. As each pair of integers has greatest common divisor, it is natural to call two vertices $a\not=b$ similar if $\gcd(a,b)>1$ and different otherwise. In the sequel we prefer to use the words 'friend' and 'enemy' instead of 'similar' and 'different', respectively. Furthermore we refer to this graph as the network of integers. Note that the behavior of the $\gcd$ among the first $n$ positive integers has been investigated from many aspects; see e.g. a paper of Nymann, \cite{ny}.
Bak\'o and Aszal\'os \cite{ab} have made several experiments on the network of integers. They discovered that the classes of the near optimal clustering have regular structure. In the sequel denote $p_i$ the $i$-th prime, i.e., $p_1=2, p_2=3,\dots$. Set
$$
S_{i,n} = \{m \; : \; m\le n,\; p_i|m,\; p_j\nmid n\; (j< i)\}.
$$
In other words, $S_{i,n}$ is the set of integers at most $n$, which are divisible by $p_i$, but coprime to the smaller primes. Aszal\'os and Bak\'o found that
\begin{equation} \label{sejtes}
[2,n] \cap \ZZ = \bigcup_{j=1}^{\infty} S_{j,n}
\end{equation}
is highly likely to be an optimal correlation clustering for $n\le 500$. Notice that $S_{j,m}=\emptyset$ for all large enough $j$, i.e., the union on the right hand side is actually finite.
The aim of this paper is to show that for $n_0=3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23 = 111~546~435$ the decomposition \eqref{sejtes} is not optimal. We prove that the number of conflicts in
\begin{equation} \label{teny}
[2,n_0] \cap \ZZ = (S_{1,n_0}\cup\{n_0\}) \cup (S_{2,n_0}\setminus \{n_0\}) \bigcup_{j=3}^{\infty} S_{j,n}
\end{equation}
is less than in \eqref{sejtes} with $n=n_0$. Unfortunately, we are not able to verify that \eqref{teny} is optimal for $n<n_0$. However, we can prove that the natural greedy algorithm (Algorithm 1), presented in the next section, produces the clustering \eqref{sejtes} for $n< n_0$, but \eqref{teny} for $n= n_0$. Thus our result shads some light on the difficulty to find optimal clustering of large graphs.
Applying Algorithm~1 for the network of integers the results behave regularly until a certain large point, but then the regularity disappears. From our construction it will be clear that $n_0$ is the first, but not at all the last integer, which behaves irregularly. For example the numbers $3n_0, 5n_0, 9n_0, \dots$ are odd and are divisible by three, but adjoining them to $S_{1,n}$ causes less conflicts than adjoining them to $S_{2,n}$. Denote by $S_{i,n}^*$ the class, which contains $p_i$ and is produced by Algorithm~1. We have no idea whether these sets have some structure and what is their asymptotic behavior. For example does the limit
$$
\lim_{n \to \infty} \frac{|S_{1,n}^*|}{n}
$$
exist? Or is $\limsup_{n \to \infty} \frac{|S_{1,n}^*|}{n} = 1$ or is it smaller?
The paper is organized as follows. In the second section we present Algorithm~1 and the main theorem. The third section is devoted to the proof of combinatorial lemmata and in the last section we prove the theorem.
\section{Main result}
To find an optimal correlational clustering of a labeled network is an NP-hard problem. To find an approximation of the optimal solution, it is natural to use greedy algorithms. For the network of integers we use the following strategy. The optimal clustering for $\{2\}$ is itself. Assume that we have a partition of $N:=[2,n-1]\cap \ZZ$, then adjoin $n$ to that class, which causes the least new conflicts. The result is a locally optimal clustering, which is not necessarily optimal globally.
\begin{algorithm}
\caption{Natural greedy algorithm}
\begin{algorithmic}[1]
\Require{an integer $n\ge 2$}
\Ensure{a partition $\mathcal P$ of $N$}
\State ${\mathcal P}\gets \{\{2\}\}$;
\If {$n=2$} \textbf{return} $\mathcal P$ \EndIf
\State $m\gets 3$
\While{$m\le n$}
\State ${\mathcal P}_M\gets {\mathcal P}\cup \{\{m\}\}$
\State $M\gets \Call{conflicts}{ {\mathcal{P}}_M,m }$
\Comment{the number of conflicts with respect to the partition ${\mathcal P}_M$ caused by the pairs $(m,a),\ a<m$}
\State $C\gets\mbox{ number of classes in } {\mathcal P}$
\State $j\gets 1$
\While{$j\le C$}
\State $O \gets \Call{op}{ j,{\mathcal P}}$
\Comment{$OP(j,\mathcal P)$ denotes the $j$-th class in the partition $\mathcal P$.}
\State ${\mathcal P}_1\leftarrow {\mathcal P}\setminus \{O\}$
\State ${\mathcal P}_1\leftarrow {\mathcal P}_1 \cup \{O \cup \{m\}\}$
\State $M_1 \gets \Call{NuPair}{ {\mathcal P}_1, m}$
\Comment{ the number of pairs $(m,a)$ with $a<m$ causing a conflict in the partition ${\mathcal P}_1$}
\If{$M_1<M$}
\State $M\gets M_1$
\State ${\mathcal P}_M \gets {\mathcal P}_1$
\EndIf
\EndWhile
\EndWhile
\State \textbf{return} ${\mathcal P}_M$
\end{algorithmic}
\end{algorithm}
Starting with a partition of $\{2,\dots,n-1\}$ this algorithm establishes a partition of $\{2,\dots,n\}$ such that the conflicts caused by $n$ is minimal. The output of Algorithm~1 on the input $n$ is denoted by $G(n)$. It is a collection of disjoint sets, whose union is $[2,n] \cap \ZZ$. It is easy to see that
\begin{eqnarray*}
G(3) &=& \{\{2\},\{3\}\}\\
G(4) &=& \{\{2,4\},\{3\}\}\\
G(5) &=& \{\{2,4\},\{3\},\{5\}\}\\
G(6) &=& \{\{2,4,6\},\{3\},\{5\}\}\\
\dots\\
G(15) &=& \{\{2,4,6,8,10,12,14\},\{3,9,15\},\{5\},\{7\},\{11\},\{13\}\},
\end{eqnarray*}
moreover they are optimal as well.
Our main result is the following
\begin{theorem} \label{Lajos}
If $m<n_0=3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23 = 111~546~435$ then
\begin{equation}\label{egy}
G(m) = \bigcup_{j=1}^{\infty}S_{j,m}
\end{equation}
is true, but
$$
G(n_0) = (S_{1,n_0}\cup\{n_0\}) \cup (S_{2,n_0}\setminus \{n_0\})\bigcup_{j=3}^{\infty}S_{j,n_0}.
$$
\end{theorem}
\section{Auxiliary results}
To prove the main theorem we need some preparation. Throughout this paper the number of elements of a set $A$ will be denoted by $|A|$. In the first lemma we characterize that class of $G(n-1)$ to which Algorithm~1 adjoins $n$.
\begin{lemma}\label{hovategyuk}
Let $n>2$ be an integer. Write $G(n-1)=\{P_1,\dots, P_M\}$ and set $P_0=\emptyset$. For $1\le j\le M$ let
$$
E_{j,n} =\{m\;:\; m\in P_j, \gcd(m,n)=1\}
$$
and
$$
B_{j,n} =\{m\;:\; m\in P_j, \gcd(m,n)>1\}.
$$
Define $E_{0,n}=B_{0,n}=\emptyset$. Let $J$ be the smallest index for which $|B_{j,n}|-|E_{j,n}|$ $(j=0,\dots,M)$ is maximal. Then $G(n)=\{P'_0,\dots,P'_M\}$ such that
\[
P'_j =
\begin{cases}
P_j\cup\{n\},& \mbox{if}\; j=J,\\
P_j,& \mbox{otherwise}.
\end{cases}
\]
\end{lemma}
\begin{proof}
Let $K_j$ denote the number of new conflicts, which arise adjoining $n$ to $P_j$ $(j=0,\dots,M)$. Then
$$
K_j =|E_{j,n}|+\sum_{k=0 \atop k\not=j}^M |B_{k,n}|.
$$
Algorithm~1 adjoins $n$ to that $P_{\hat{J}}$ for which $K_{\hat{J}}$ is minimal and if there are more indices $j$ with minimal $K_j$ then $\hat{J}$ is minimal among them. This means that if $K_k\not=K_{\hat{J}}$ and $m\not=k$ then $K_k-K_{\hat{J}} \ge K_k-K_m$. This is equivalent to
$$
|E_{k,n}| + |B_{\hat{J},n}| - |E_{\hat{J},n}| - |B_{k,n}| \ge |E_{k,n}| + |B_{m,n}| - |E_{m,n}| - |B_{k,n}|
$$
and further to
$$
|B_{\hat{J},n}| - |E_{\hat{J},n}| \ge |B_{m,n}| - |E_{m,n}|.
$$
Thus $|B_{m,n}| - |E_{m,n}|$ $(m=0,\dots,M)$ assumes its maximal value at $m=\hat{J}$ and $\hat{J}$ is minimal among the indices with this property. Hence $J=\hat{J}$ and the lemma is proved.
\end{proof}
\begin{cor}\label{newcor} The following assertions are true.
\begin{itemize}
\item[(1)] If $n$ is even, then $n\in S_{1,n}$.
\item[(2)] If $n$ is a prime, then $\{n\}\in G(n)$.
\item[(3)] If the smallest prime factor of $n$ is $p_i$ and $n\in S_{j,n}$, then $j\leq i$.
\end{itemize}
\end{cor}
\begin{proof}
We start with noting that as we pointed out earlier, the assertions hold for $n\le 10$. That is, we have
$$
G(n-1) = \{S_1,\dots,S_M\},
$$
where, for simplicity, we set $S_j = S_{j,n-1}$ $(j=1,\dots,M)$. Put $S_0=\emptyset$.
(1) If $n$ is even then $B_{1,n}=S_1$, thus $|B_{1,n}| = n/2 -1$. If $2\le j\le M$ then $B_{j,n}\subseteq S_j$, thus $|B_{j,n}|\le [(n-1)/p_j] < n/3$. As $n/2-1 > n/3$ for $n\ge 8$ we have
$$
|B_{1,n}|-|E_{1,n}| > |B_{j,n}|-|E_{j,n}|\ \ (j=2,\dots,M).
$$
Hence Algorithm~1 adjoins $n$ to $S_1$, i.e., to the class of even numbers.
(2) If $n=2$ then (2) holds by Step~1. of Algorithms~1. Let $n$ be an odd prime, $G(n-1)=\{S_{1},\dots,S_{M}\}$ and $S_0=\emptyset$. By Lemma \ref{hovategyuk} Algorithm~1 adjoins $n$ to that $S_J$ for which $|B_{j,n}|-|E_{j,n}|$ $(j=0,\dots,M)$ is maximal.
Since $n$ is a prime, $B_{j,n}=\emptyset$ for all $j=1,\dots,M$. Thus $|B_{j,n}|-|E_{j,n}|<0$ $(j =1,\dots,M)$, but $|B_{0,n}|-|E_{0,n}|=0$. Hence $n$ will be adjoined to the empty set, i.e. it will form alone a class in $G(n)$.
(3) We may assume that $n$ is odd and composite. Let $n=q_1^{\alpha_1}\cdots q_t^{\alpha_t}$, where $q_1<\dots<q_t$ are odd primes and $\alpha_1,\dots,\alpha_t$ are positive integers. We obviously have $\{q_1,\dots,q_t\}\subseteq \{p_1,\dots,p_M\}$.
Assume that $q_1 = p_i$. Then every elements of $S_i$ is divisible by $q_1$. Thus $B_{i,n}=S_i$ and $E_{i,n}= \emptyset$, hence $|B_{i,n}|-|E_{i,n}|= |S_i|$.
If $j>i$ then $|B_{j,n}|-|E_{j,n}|\le |S_j|\le|S_i|$. Thus, by Lemma \ref{hovategyuk}, if $n$ will be adjoined to $S_j$ then $j\le i$.
\end{proof}
The next lemma describes a simple, but useful property of the integer part function.
\begin{lemma} \label{egeszresz}
Let $q_1,\dots,q_t$ be pairwise different odd primes, $\alpha_1,\dots,\alpha_t$ positive integers. Let $u$ be a positive integer coprime to $q_i$ $(i=1\dots,t)$ and $n=q_1^{\alpha_1}\cdots q_t^{\alpha_t}$. If $\{i_1,\dots,i_k\}\subseteq \{1,\dots,t\}$ then
\begin{equation} \label{egyegeszresz}
\left[ \frac{n-1}{u q_{i_1}\cdots q_{i_k}} \right] = \left[\frac{\frac{n}{q_{i_1}\cdots q_{i_k}}-1} {u} \right].
\end{equation}
In particular, if $u=2$ then
\begin{equation} \label{ketegeszresz}
\left[ \frac{n-1}{2 q_{i_1}\cdots q_{i_k}} \right] = \frac{n-q_{i_1}\cdots q_{i_k}}{2 p_{i_1}\cdots q_{i_k}} = \frac{n}{2 q_{i_1}\cdots q_{i_k}} - \frac12.
\end{equation}
\end{lemma}
\begin{proof} We have
$$
\left[ \frac{n-1}{u q_{i_1}\cdots q_{i_k}} \right] = \frac{n-m}{u q_{i_1}\cdots q_{i_k}},
$$
where $m$ is the smallest positive integer such that the fraction on the right hand side is an integer. As $q_{i_j}|n$, we must have $q_{i_j}|m$ $(j=1,\dots,k)$ as well, which implies $q_{i_1}\cdots q_{i_k} | m$. This proves \eqref{egyegeszresz}.
If $u=2$ then $m = q_{i_1}\cdots q_{i_k}$ is the smallest positive integer with the required property, because $n-m$ is even.
\end{proof}
The next lemma gives a good approximation for the size of $S_{i,u}$.
\begin{lemma} \label{Sjelemszam}
Let $u$ be an odd integer. Then we have $|S_{1,u}| = \frac{u-1}{2}$. Further, if $p_i$ is an odd prime, then
$$
\left||S_{i,u}|- \frac{u}{p_i} \prod_{\ell=1}^{i-1} \left(1-\frac1{p_{\ell}}\right)\right| \le 2^{i-2}.
$$
\end{lemma}
\begin{proof}
The first statement is obvious. To prove the second one we start with the identity
\begin{eqnarray*}
S_{i,u} &=& \{m\;:\; m\le u,\; p_i|m,\; p_{\ell}\nmid m\; \mbox{for all} \; 1\le \ell < i\}\\
&=& \{m\;:\; m\le u, p_i|m\} \setminus \bigcup_{\ell=1}^{i-1} \{m\;:\; m\le u, p_i\cdot p_{\ell} | m\}.
\end{eqnarray*}
In the remaining of the proof we assume that the elements of the occurring sets are at most $u$. The sieve formula implies
\begin{eqnarray*}
|S_{i,u}| &=& |\{m\;:\; p_i|m\}|\\&
-&\sum_{\ell=1}^{i-1} (-1)^{\ell-1} \sum_{1\le i_1<\dots < i_{\ell}< i}|\{m\;: \; p_i\cdot p_{i_1}|m\}\cap\dots \cap \{m\;:\; p_i\cdot p_{i_{\ell}}|m\}|\\
&=& |\{m\;:\; p_i|m\}| + \sum_{\ell=1}^{i-1} (-1)^{\ell} \sum_{1\le i_1<\dots < i_{\ell}< i}|\{m\;:\; p_i\cdot p_{i_1}\cdots p_{i_{\ell}}|m\}|\\
&=& \sum_{\ell=0}^{i-1} (-1)^{\ell} \sum_{1\le i_1<\dots < i_{\ell}< i}|\{m\;:\; p_i\cdot p_{i_1}\cdots p_{i_{\ell}}|m\}|.
\end{eqnarray*}
Thus
\begin{equation} \label{haromegeszresz}
|S_{i,u}| = \sum_{\ell=0}^{i-1} (-1)^{\ell} \sum_{1\le i_1<\dots < i_{\ell}< i} \left[\frac{u}{p_i\cdot p_{i_1}\cdots p_{i_{\ell}}}\right].
\end{equation}
Using $x-1< [x]\le x$ we obtain
$$
- \sum_{\ell=0\atop {\ell \text{is even}} }^{i-1} {i-1 \choose \ell}\le |S_{i,u}| - \frac{u}{p_i} \prod_{\ell=0}^{i-1} \left(1-\frac1{p_{\ell}}\right) < \sum_{\ell=0\atop {\ell \text{is odd}} }^{i-1} {i-1 \choose \ell}.
$$
As
$$
\sum_{\ell=0\atop {\ell \text{is even}}}^{i-1} {i-1 \choose \ell} = \sum_{\ell=0\atop {\ell \text{is odd}} }^{i-1} {i-1 \choose \ell} = 2^{i-2},
$$
the lemma is proved.
\end{proof}
In the next lemma we prove an estimation for $|B_{j,n}|-|E_{j,n}|$, where
$$
B_{j,n} = \{m\; : \; m \in S_{j,n-1},\; \gcd(m,n)>1\}
$$
and
$$
E_{j,n} = \{m\; : \; m \in S_{j,n-1},\; \gcd(m,n)=1\}.
$$
The elements of $B_{j,n}$ and $E_{j,n}$ are the friends and enemies of $n$ in $S_{j,n-1}$, respectively.
\begin{lemma} \label{Bjn}
Let $q_1<\dots<q_t$ be odd primes, $\alpha_1,\dots,\alpha_t$ positive integers and $n=q_1^{\alpha_1}\cdots q_t^{\alpha_t}$. Let $j\ge2$ be such that $p_j<q_1$. Then
$$
\left||B_{j,n}|-|E_{j,n}| - \frac{n-1}{p_j} \prod_{\ell=1}^{j-1}\left(1-\frac1{p_{\ell}} \right) \left( 1- 2\prod_{k=1}^t\left(1-\frac1{q_k}\right)\right)\right| \le 2^{t+j-2}.
$$
\end{lemma}
\begin{proof}
As $|B_{j,n}|+|E_{j,n}| = |S_{j,n-1}|$ we have
\begin{equation} \label{Bjnegy}
|B_{j,n}|-|E_{j,n}| = 2 |B_{j,n}| - |S_{j,n-1}| = |S_{j,n-1}| - 2(|S_{j,n-1}| - |B_{j,n}|).
\end{equation}
For $|S_{j,n-1}|$ we can use the estimations of Lemma \ref{Sjelemszam}, thus we have to deal only with the second summand. As $p_j<q$ for all prime factors $q$ of $n$, we have
$$
B_{j,n} = \bigcup_{\ell=1}^t \{m\;:\; m\in S_{j,n-1}, q_{\ell}|m\}.
$$
Using again the sieve formula we get
\begin{eqnarray*}
|B_{j,n}| &=& \sum_{\ell=1}^t (-1)^{\ell-1} \sum_{1\le j_1<\dots<j_{\ell}\le t} \left|\bigcap_{k=1}^{\ell}\{m\;:\; m\in S_{j,n-1}, q_{j_k}|m\}\right|\\
&=& \sum_{\ell=1}^t (-1)^{\ell-1} \sum_{1\le j_1<\dots<j_{\ell}\le t} |\{m\;:\; m\in S_{j,n-1}, q_{j_1}\cdots q_{j_{\ell}}|m\}|.
\end{eqnarray*}
Set $U_{j,\ell}(q_{j_1},\dots,q_{j_{\ell}}) = U_{j,\ell} = \{m\;:\; m\in S_{j,n-1}, q_{j_1}\cdots q_{j_{\ell}}|m\}.$ Then
$$
U_{j,\ell} = \{m\;:\; m\le n-1,p_j\cdot q_{j_1}\cdots q_{j_{\ell}}|m\}\setminus \bigcup_{i=1}^{j-1}\{m\;:\; p_i p_j q_{j_1}\cdots q_{j_{\ell}}|m\}.
$$
We can compute the number of elements of $U_{j,\ell}$ by using the sieve formula.
\begin{eqnarray*}
|U_{j,\ell}| &=& \left[\frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}}\right]- \sum_{i=1}^{j-1} (-1)^{i-1} \sum_{1\le h_1<\dots<h_i< j} \left[\frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}p_{h_1}\cdots p_{h_i}}\right]\\
&=& \sum_{i=0}^{j-1} (-1)^{i-1} \sum_{1\le h_1<\dots<h_i< j} \left[\frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}p_{h_1}\cdots p_{h_i}}\right].
\end{eqnarray*}
Combining these formulae we get
$$
|B_{j,n}| = \sum_{\ell=1}^t (-1)^{\ell-1} \sum_{1\le j_1<\dots<j_{\ell}\le t} \sum_{i=0}^{j-1} (-1)^{i} \sum_{1\le h_1<\dots<h_i< j} \left[\frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}p_{h_1}\cdots p_{h_i}}\right].
$$
The last formula together with \eqref{haromegeszresz} implies
$$
|S_{j,n-1}|-|B_{j,n}| = \sum_{\ell=0}^t (-1)^{\ell} \sum_{1\le j_1<\dots<j_{\ell}\le t} \sum_{i=0}^{j-1} (-1)^{i} \sum_{1\le h_1<\dots<h_i< j} \left[\frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}p_{h_1}\cdots p_{h_i}}\right].
$$
Changing the order of the summation we get
$$
|S_{j,n-1}|-|B_{j,n}| = \sum_{i=0}^{j-1} (-1)^{i} \sum_{1\le h_1<\dots<h_i< j} \sum_{\ell=0}^t (-1)^{\ell} \sum_{1\le j_1<\dots<j_{\ell}\le t} \left[\frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}p_{h_1}\cdots p_{h_i}}\right].
$$
With Lemma \ref{egeszresz} we get
$$
-1<\left[\frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}p_{h_1}\cdots p_{h_i}}\right]- \frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}p_{h_1}\cdots p_{h_i}}\le 0.
$$
Thus
$$
\left|\sum_{\ell=0}^t (-1)^{\ell} \sum_{1\le j_1<\dots<j_{\ell}\le t} \left[\frac{n-1}{p_j\cdot q_{j_1}\cdots q_{j_{\ell}}p_{h_1}\cdots p_{h_i}}\right]- \frac{n-1}{p_j\cdot p_{h_1}\cdots p_{h_i}}\prod_{k=1}^t\left(1-\frac1{q_k}\right)\right| \le 2^{t-1}
$$
and
$$
|S_{j,n-1}|-|B_{j,n}| = \frac{n-1}{p_j}\prod_{k=1}^t\left(1-\frac1{q_k}\right) \prod_{\ell=1}^{j-1}\left(1-\frac1{p_{\ell}}\right) + C,
$$
where the constant $C$ is at most $2^{t+j-3}$. This estimate together with Lemma \ref{Sjelemszam} and \eqref{Bjnegy} gives
$$
|B_{j,n}| - |E_{j,n}|= \frac{n-1}{p_j} \prod_{\ell=1}^{j-1}\left(1-\frac1{p_{\ell}} \right)- 2 \frac{n-1}{p_j}\prod_{k=1}^t\left(1-\frac1{q_k}\right) \prod_{\ell=1}^{j-1}\left(1-\frac1{p_{\ell}}\right) + C_1,
$$
where $|C_1| \le 2^{t+j-3} + 2^{j-2}< 2^{t+j-2}$.
\end{proof}
The next lemma plays a key role in the proof of Theorem \ref{Lajos}. In contrast to the classes of odd numbers, it is possible to give the exact values of the difference of the number of friends and enemies in the class of even numbers.
\begin{lemma} \label{fontos}
Let $n= q_1^{\alpha_1}\cdots q_t^{\alpha_t}$ with $q_1<\dots< q_t$ odd primes and $\alpha_1,\dots, \alpha_t$ positive integers. Then
\begin{eqnarray*}
|E_{1,n}| &=& \frac{\varphi(n)}{2} = \frac{n}2 \left(1-\frac1{q_1}\right)\cdots \left(1-\frac1{q_t}\right),\\
|B_{1,n}| &=& \frac{n-1}2 - |E_{1,n}|.
\end{eqnarray*}
\end{lemma}
\begin{proof}
The statement could be proved by repeating the proof of Lemma \ref{Bjn} and using Lemma \ref{egeszresz}. However, there is a much more direct and simple way, which we present.
Let $A_1=\{h\;:\; 1\le h\le \frac{n-1}2, \gcd(h,n)=1\}$ and $A_2=\{h\;:\; \frac{n+1}2 \le h< n, \gcd(h,n)=1\}$. Then $A_1$ and $A_2$ are disjoint and their union is $A=\{h\;:\; 1\le h < n, \gcd(h,n)=1\}$. Plainly $|A| = \varphi(n)$. The mapping $\psi\;:\; h\mapsto n-h$ is bijective between $A_1$ and $A_2$. Moreover, $\psi(h)$ is odd if and only if $h$ is even. Thus the number of even positive integers, which are coprime to $n$ is $\varphi(n)/2$. As $E_{1,n}$ is exactly the set of even integers, less than and coprime to $n$, the proof is complete.
\end{proof}
\section{Proof of Theorem \ref{Lajos}}
Despite of the lengthy preparation, the proof of Theorem \ref{Lajos} is complicated. First we confirm the cases where $n$ is odd, $3\mid n$, and prove the second assertion. The hard part is to prove that \eqref{egy} is true for $n<n_0$. This is done by a combination of comparison of the estimations of Lemmata \ref{Sjelemszam} and \ref{Bjn}, some computer search and finally application of a tool from prime number theory.
\subsection{The cases $n$ is odd with $3\mid n$, and proving the second assertion}
Let $n> 2$ be an integer and assume that \eqref{egy} holds for all $m< n$. Suppose further that $k=2$, i.e. the smallest prime factor of $n$ is $q_1=3$. Then by Lemma \ref{fontos} we have $|B_{1,n}|-|E_{1,n}|=\frac{n-1}2 - \varphi(n)$. A simple computation shows that $|S_2| = \frac{n-3}6$, which is a bit stronger then the statement of Lemma \ref{Sjelemszam}.
By Lemma \ref{hovategyuk}, $n$ is adjoined to $S_1$ precisely when $|S_2|< |B_{1,n}|-|E_{1,n}|$. This inequality implies $\frac{n-3}6 < \frac{n-1}2 - \varphi(n)$, which is equivalent to $\varphi(n)<\frac{n}3$. Using the explicit form of $\varphi(n)$ and dividing by $n$ we get $\left(1-\frac13\right)\cdot \left(1-\frac1{q_2}\right)\cdots \left(1-\frac1{q_t}\right)<\frac13$. Thus Algorithm~1 adjoins $n$ to $S_1$ if and only if
\begin{equation} \label{ketto}
\left(1-\frac1{q_2}\right)\cdots \left(1-\frac1{q_t}\right)<\frac12.
\end{equation}
Inequality \eqref{ketto} is independent from the exponents $\alpha_1,\dots,\alpha_t$, thus $n$ is minimal with this property if all exponents are one. For fixed $t$ the number $n$ is minimal if $q_1,\dots,q_t$ are consecutive odd primes starting with $q_1=3$. A simple computation shows that the smallest $n$ which is divisible by $3$ and satisfies \eqref{ketto} is $n_0$.
\begin{rem} An analogous computation shows that $n_1=5\cdot p_4\cdots p_{14} = 2~180~460~221~945~005$ is a candidate to be the smallest odd integer, which is not divisible by $3$ and is adjoined to $S_1$. However, as $n_1$ is much larger than $n_0$ and many odd integers between $n_0$ and $n_1$, e.g. $3n_0, 5n_0, 9n_0,\dots$ are adjoined to $S_1$ we are not sure that for example $n_1'=5\cdot p_4\cdots p_{13}$ will belong to $S_1$ or to $S_3$.
\end{rem}
\subsection{Numbers with middle sized factors}
Let $n= q_1^{\alpha_1}\cdots q_t^{\alpha_t}<n_0$ be such that $p_i=q_1<q_2\dots<q_t$. For $i\le 2$ Theorem \ref{Lajos} is already proved. Assume that $i\ge 3$. By Lemma~\ref{hovategyuk} Algorithm~1 adjoins $n$ to $S_{i,n-1}$ if and only if
\begin{equation} \label{feltetel}
|B_{j,n}| - |E_{j,n}| < |S_{i,n-1}|
\end{equation}
holds for all $1\le j<i$. By Lemmata \ref{Sjelemszam}, \ref{Bjn} we have
$$
|S_{i,n-1}| \ge \frac{n-1}{p_i} \prod_{\ell=1}^{i-1} \left(1-\frac1{p_{\ell}}\right) - 2^{i-2}
$$
and
$$
|B_{j,n}| - |E_{j,n}| \le \frac{n-1}{p_j} \prod_{\ell=1}^{j-1}\left(1-\frac1{p_{\ell}} \right) \left(1- 2\prod_{k=1}^t\left(1-\frac1{q_k}\right)\right)+ 2^{t+j-2}.
$$
Thus if
$$
\frac{n-1}{p_j} \prod_{\ell=1}^{j-1}\left(1-\frac1{p_{\ell}} \right) \left(1- 2\prod_{k=1}^t\left(1-\frac1{q_k}\right)\right)+ 2^{t+j-2} \le \frac{n-1}{p_i} \prod_{\ell=1}^{i-1} \left(1-\frac1{p_{\ell}}\right) - 2^{i-2},
$$
then \eqref{feltetel} holds, too. The last inequality is equivalent to
$$
\frac{n-1}{p_i} \prod_{\ell=1}^{i-1} \left(1-\frac1{p_{\ell}}\right) + \frac{n-1}{p_j} \prod_{\ell=1}^{j-1}\left(1-\frac1{p_{\ell}} \right) \left( 2\prod_{k=1}^t\left(1-\frac1{q_k}\right)-1 \right) \ge 2^{t+j-2} + 2^{i-2}.
$$
For fixed $t$ and $i$ the product $\prod_{k=1}^t\left(1-\frac1{q_k}\right)$ assumes its smallest value if the $q_k$-s are the $t$ consecutive primes starting with $p_i$. Thus if $n_1=n_1(i,j,t)$ denotes the smallest $n$, which satisfies
$$
n \ge 1+ (2^{t+j-2} + 2^{i-2})/T,
$$
where
$$
T = \frac{1}{p_i} \prod_{\ell=1}^{i-1} \left(1-\frac1{p_{\ell}}\right) + \frac{1}{p_j} \prod_{\ell=1}^{j-1}\left(1-\frac1{p_{\ell}} \right) \left( 2\prod_{k=1}^t\left(1-\frac1{q_k}\right)-1 \right)
$$
then \eqref{feltetel} holds for all $n\ge n_1$ having exactly $t$ prime factors, from which the smallest is $p_i$.
We computed $n_1(i,j,t)$ for all triplets $(i,j,t)$ with $3\le i \le 20, 3\le j\le i-1, 1\le t\le t_i$, where $t_i$ is the largest $t$ such that $\prod_{k=0}^{t-1} p_{i+k} \le n_0$. Remark that $n_1(i,j,t)>n_0$ for $i\ge 20$. By our computation $n_1(i,j,t)$ is a monotone increasing function of $j$ for fixed $(i,t)$, thus we displayed in Table \ref{T2} only the values $n_1(i,i-1,t)$.
\begin{table}
\begin{tabular}{|c|c|c|c|c|}
\hline
$i=3, p=5$& $i=4, p=7$& $i=5,p=11$& $i=6,p=13$& $i=7,p=17$ \\
\hline
$t, n1$ & $t, n1$ & $t, n1$ & $t, n1$ & $t, n1$ \\
\hline
$1,24$ &$1,93$ & $1,308$ &$1,953$& $1,2521$ \\ \hline
$2, 46$ & $2, 159$ & $2,514$ & $2, 1533$ & $2, 4033$ \\ \hline
$3, {\it 92}$ &$3, {\it 297}$ & $3, {\it 933}$ &$3, {\it 2720}$ & $3, {\it 7091}$ \\ \hline
$4, {\it 196}$ &$4, {\it 583}$ & $4, {\it 1813}$ &$4, {\it 5157}$ & $4, {\it 13318}$ \\ \hline
$5, {\it 422}$ &$5, {\it 1194}$ & $5, {\it 3654}$ &$5, {\it 10151}$ & $5, {\it 26186}$ \\ \hline
$6, {\it 929}$ &$6, {\it 2480}$ & $6, {\it 7471}$ &$6, {\it 20484}$ & \\ \hline
$7$, {\it 2044} && & & \\ \hline \hline
$i=8,p=19$ & $i=9, p=23$& $i=10,p=29$& $i=11,p=31$& $i=12,p=37$\\
\hline
$t, n1$ & $t, n1$ & $t, n1$ & $t, n1$ & $t, n1$ \\
\hline
$1,6531$ &$1,15889$ & $1,40751$ &$1,98726$& $1,228806$ \\ \hline
$2, 10285$ & $2, 24799$ & $2,63466$ & $2, 152425$ & $2, 352739$ \\ \hline
$3, 17815$ &$3, 42901$ & $3, 109166$ &$3, 260747$ & $3, 603486$ \\ \hline
$4, {\it 33246}$ &$4, {\it 79673}$ & $4, {\it 202185}$ &$4, {\it 481157}$ & $4, {\it 1112639}$ \\ \hline
$5, {\it 64728}$ &$5, {\it 154809}$ & $5, {\it 392470}$ &$5, {\it 929762}$ & \\ \hline
\hline
$i=13,p=41$ & $i=14, p=43$& $i=15,p=47$& $i=16,p=53$& $i=17,p=59$\\
\hline
$t, n1$ & $t, n1$ & $t, n1$ & $t, n1$ & $t, n1$ \\
\hline
$1,542016$ &$1,1198905$ & $1,2623122$ &$1,5937759$& $1,13554766$ \\ \hline
$2, 833706$ & $2, 1838933$ & $2,4014787$ & $2, 9071489$ & $2, 20693167$ \\ \hline
$3, 1421830$ &$3, 3126106$ & $3, 6813569$ &$3, 15389987$ & $3, 35045873$ \\ \hline
$4, {\it 2612025}$ &$4, {\it 5727822}$ & $4, 12481252$ &$4, 28153619$ & $4, 64044517$ \\ \hline
\hline
$i=18,p=61$ & $i=19, p=67$& & & \\
\hline
$t, n1$ & $t, n1$ & & & \\
\hline
$1,29627101$ &$1,64068095$ & && \\ \hline
$2, 45131528$ & $2, 97553073$ & & & \\ \hline
$3, 76322166$ & & & & \\ \hline
\end{tabular}\\[1.5 ex]
\caption{\label{T2}}
\end{table}
If $n_1(i,i-1,t)> \prod_{k=0}^{t-1}p_{i+k}$, which appeares often, then \eqref{feltetel} cannot hold for the pair $(i,t)$. These values are displayed in italics in Table \ref{T2}.
\subsection{Verification of the statement for numbers with one and two prime factors}
The following lemma verifies Theorem \ref{Lajos} if $n$ is a prime power.
\begin{lemma} \label{primhatvany}
Let $p=p_i\ge 3$ be a prime. If $p\le 67$ and $p^{\alpha}<n_0, \alpha>0$ then $p^{\alpha}\in S_{i,n}$. Generally, if $\alpha\le 4$ then $n=p^{\alpha} \in S_{i,n}$.
\end{lemma}
\begin{proof}
If $\alpha=1$ then as for all $j<i$ $S_{i,n-1}=B_{j,n}=\emptyset$ and $|N_{j,n}|>0$ the statement is true. The statement is valid for $p=3$ too because the smallest integer divisible by three, which lands in $S_1$ is $n_0$.
Let $\alpha>1$ and $j<i$. Then we have
\begin{equation}\label{baratok}
B_{j,p^{\alpha}}= \bigcup_{k=1}^{\alpha-1}p^k E_{j,p^{\alpha-k}}.
\end{equation}
If $m\in B_{j,p^{\alpha}}$, then either $m$ is divisible only by the first power of $p$, thus $m/p\in E_{j,p^{\alpha-1}}$ or $m$ is divisible by a higher power of $p$, in which case $m/p\in B_{j,p^{\alpha-1}}$, i.e.
$$
B_{j,p^{\alpha}} = pE_{j,p^{\alpha-1}} \cup p B_{j,p^{\alpha-1}}.
$$
Using this identity we get the proof of \eqref{baratok} by induction.
{\it Case 1, $\alpha=2$.} Then $|B_{j,p^{2}}| = |E_{j,p}|$. By Tchebishev's theorem there exists at least one prime $q$ with $p/p_j< q < p$. Hence $p_jq\in E_{j,p^{2}} \setminus E_{j,p}$, which implies $|E_{j,p^{2}}| > |B_{j,p^{2}}| = |E_{j,p}|$. Thus $p^2 \in S_{i,n}$, in particular $7^2\in S_{4,n}, 11^2\in S_{5,n}$. These facts together with the entries $(i,t)=(3,1), (4,1)$ of Table \ref{T2} show that the assertion is true for $p=5,7$.
{\it Case 2, $\alpha=3$.} Then there exists a prime $q$ with $p^2/p_j< q < p^2$, i.e. $p_jq\notin E_{j,p^{2}}$. If $m\in E_{j,p}$ then $qm \le qp <p^3$, thus
$$
|E_{j,p^{3}}| \ge |B_{j,p^{3}}| = |E_{j,p^{2}}| + |E_{j,p}|,
$$
and this case is proved. By Table \ref{T2} the assertion is true for $p\le 19$.
{\it Case 3, $\alpha=4$.} Identity \eqref{baratok} implies
$$
|B_{j,p^{4}}| = |E_{j,p^{3}}| + |E_{j,p^{2}}| + |E_{j,p}|.
$$
We have plainly $E_{j,p^{2}} = E_{j,p} \cup E_{j,(p,p^2/p_j]} \cup E_{j,(p^2/p_j,p^2]}$, where the second and third sets include all elements of $E_{j,p^{2}}$, which belong to the interval $(p,p^2/p_j]$ and $(p^2/p_j, p^2]$ respectively. As $p_j\ge 5$ there exist at least two different primes $q_1,q_2$ with $p^3/p_j<q_1,q_2<p^3$, i.e.
$$q_kE_{j,p} \cap E_{j,p^{3}} =q_1E_{j,p} \cap q_2E_{j,p} = \emptyset, k=1,2.$$
Further, there exist primes $q_3,q_4$ with $p^2 < q_3 < 2p^2$ and $p^2/2 < q_4 < p^2$. The sets $q_kE_{j,p}, k=1,2, q_3E_{j,(p,p^2/p_j]}$ and $q_4 E_{j,(p^2/p_j,p^2]}$ are by the construction pairwise disjoint. If $m \in q_3E_{j,(p,p^2/p_j]} \cap q_4 E_{j,(p^2/p_j,p^2]}$ then $m$ is divisible by the pairwise different primes $q_3,q_4,p_j$, thus $m\ge p_jq_3q_4> p_j p^4/2>p^4$, which is a contradiction.
\end{proof}
The next lemma verifies Theorem \ref{Lajos} for integers, which have two different prime divisors and the smaller is at most $53$.
\begin{lemma} \label{ketprim}
Let $p=p_i\ge 3$ and $q>p$ be primes. If $p\le 53$ and $p^{\alpha}q^{\beta}<n_0, \alpha,\beta>0$ then $p^{\alpha}q^{\beta}\in S_{i,n}$. Generally, if $q < p^3$ then $n=pq \in S_{i,n}$.
\end{lemma}
\begin{proof}
The idea of the proof is similar to the proof of Lemma \ref{primhatvany}. The assertion is true for $i=3$ because the smallest integer divisible by three and not lying in $S_{2,n}$ is $n_0$. We start the proof of the general case with the identity
\begin{equation} \label{ketprimazonossag}
B_{j,pq} = qE_{j,p} \cup p(E_{j,q} \cup B_{j,q}) = qE_{j,p} \cup pE_{j,q} \cup p^2 E_{j,\lfloor q/p \rfloor} \cup p^2 B_{j,\lfloor q/p^2 \rfloor} .
\end{equation}
If $p^2>q$ then $B_{j,\lfloor q/p^2 \rfloor} = \emptyset$ and we get
$$
B_{j,pq} = qE_{j,p} \cup pE_{j,q} \cup p^2 E_{j,\lfloor q/p \rfloor}.
$$
There exist primes $r_1,r_2$ with $q/2<r_1<q$ and $p^2/4<r_2<p^2, r_2\not=q,r_1$. Then
$$
E_{j,pq} \supset r_1E_{j,p} \cup E_{j,q} \cup r_2 E_{j,\lfloor q/p \rfloor}
$$
and the sets on the right hand side are disjoint. Thus if $q<p^2$ then $pq\in S_{i,pq}$. Combining this with Table \ref{T2}, implies the assertion for $p\le 17$.
If $p^2<q<p^3$ then \eqref{ketprimazonossag} implies
$$
B_{j,pq} = qE_{j,p} \cup pE_{j,q} \cup p^2 E_{j,\lfloor q/p \rfloor} \cup p^3 E_{j,\lfloor q/p^2 \rfloor}.
$$
If $q\le 4p^2$ then we have $E_{j,\lfloor q/p^2 \rfloor}= \emptyset$ and the argument of the last case works here too. The other extreme case $q>p^3/2$ also needs a separate analysis. Hence we assume $4 p^2< q\le p^3/2$.
We have $E_{j,\lfloor q/p \rfloor} = E_{j,p} \cup E_{j,(p,\lfloor q/p \rfloor]}$, where $E_{j,(p,\lfloor q/p \rfloor]}$ denotes the set of those elements of $E_{j,\lfloor q/p \rfloor}$, which belong to the interval $(p,\lfloor q/p \rfloor]$.
There exist primes $r_k, k=1,2,3,4$ such that $r_1,r_2\in (q/4,q), r_1\not= r_2, \;r_3\in (p^2/2,p^2), r_4\in (p^3/16,p^3), r_4\not= q,r_1,r_2$. Plainly $r_3<r_1,r_2,r_4$. Then
$$
E_{j,pq} \supset E_{j,q} \cup r_{1}E_{j,p} \cup r_{2}E_{j,p} \cup r_{3}E_{j,(p,\lfloor q/p \rfloor]} \cup r_4 E_{j,\lfloor q/p^2 \rfloor}
$$
and the sets on the right hand side are disjoint. The first claim holds because of the choice of the size of the $r_k$'s. If $m\in r_i E_{j,p}, i=1,2$ then $r_ip_j|m$, thus $m\ge r_ip_j> qp_j/4>q$, hence $m \notin E_{j,q}$. A similar argument shows that $E_{j,q} \cap r_4 E_{j,\lfloor q/p^2 \rfloor} = \emptyset$. If $m \in E_{j,(p,\lfloor q/p \rfloor]}$ then $mr_3> p p^2/2 \ge q$, i.e. $E_{j,q} \cap r_3 E_{j,(p,\lfloor q/p \rfloor]} = \emptyset$, which finishes the proof of this case.
There remains to treat the case $q> p^3/2.$ We have also $E_{j,\lfloor q/p \rfloor} = E_{j,2p} \cup E_{j,(2p,\lfloor q/p \rfloor]}$. There exist primes $r_k, k=1,2,3,4$ such that $r_1,r_2\in (q/8,q/2), r_3\in (p^2/2,p^2), r_4\in (p^3/16,p^3), r_4\not= q,r_1,r_2$. Plainly we also have $r_3<r_1,r_2,r_4$. Then
$$
E_{j,pq} \supset E_{j,q} \cup r_{1}E_{j,p} \cup r_{2}E_{j,2p} \cup r_{3}E_{j,(2p,\lfloor q/p \rfloor]} \cup r_4 E_{j,\lfloor q/p^2 \rfloor}
$$
and the sets on the right hand side are disjoint. The proof is the same except that if $m \in r_{3}E_{j,(2p,\lfloor q/p \rfloor]}$ then $mr_3> 2p p^2/2 >q$. The proof of the lemma is complete.
\end{proof}
\subsection{Numbers with three prime factors}
Up to now we proved Theorem \ref{Lajos} for integers with at most two prime divisors, such that the smaller is at most $53$. Unfortunately we did not found any meaningful generalization of Lemma \ref{ketprim} to integers with three prime divisors. By Table 1 the smallest prime factor of such a candidate is at least $19$. For each prime $29\le p \le 43$ we computed all integers, which are divisible by $p$, lie below the bound given in Table 1 and have three different prime divisors, which are at least $p$. Their number, $n(p)$ is given in Table \ref{T3}.
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$p$& 19& 23& 29& 31 & 37 & 41 & 43 \\
\hline
$n(p)$ & 4 & 18 & 65 & 216 & 513 & 1302 & 3097 \\
\hline
\end{tabular}\\[1.5 ex]
\caption{\label{T3}}
\end{table}
Fix $p=p_i$. For each candidates $n$ we computed $|B_{i-1,n}|-|E_{i-1,n}|$. For this purpose we used a variant of the wheel algorithm, see e.g. \cite{w}. In our case this listed efficiently the elements of $S_{i-1,n}$ because we know than they are divisible by $p_i$ and, on the other hand, relative prime to $2,3,5,7$. For each $m$ produced by the wheel algorithm we computed the $\gcd(m, \prod_{j=5}^{i-2}j)$. If this is not one, then $m$ does not belong to $S_{i-1,n}$, otherwise we added one to the counter of $|E_{i-1,n}|$ or $|B_{i-1,n}|$ according as $\gcd(m,n)=1$ or not. We found in each case that $|B_{i-1,n}|-|E_{i-1,n}|<0$, which means $n$ cannot be adjoined to $S_{i-1,n}$. The total computational time was some minutes on a notebook.
\begin{rem}The extension of the computation for larger values of $p$ is very time consuming. We performed the same procedure for $p=67$, in which case the number of candidates is $90338$. The total computational time on the same computer took about one day. The application of tools of the prime number theory saved lot of computer time and is much more elegant.
\end{rem}
\subsection{Numbers with large prime factors}
In the previous sections we proved Theorem \ref{Lajos} for integers such that their smallest prime factor is at most $47$.
\begin{proposition} Let $n_0=111546435$. Assume that for any $n$ with $n<n_0$ having a prime factor $\leq 37$, the partition is the expected one, i.e. \eqref{sejtes} holds. Suppose that $n<n_0$, such that $n$ is composed of primes $\geq 41$. Then $n$ belongs to the class $S_{i,n}$, where $p_i$ is the smallest prime divisor of $n$.
\end{proposition}
To prove the Proposition, we need the following lemma.
\begin{lemma}
\label{lemnew1}
Let $x\geq n_0^{2/3}$ and $t\geq 41$ be real numbers. Then we have
$$
\pi(x)-\pi(\sqrt{x})>18\pi(x/t)+56.
$$
\end{lemma}
\begin{proof} By a classical result of Rosser and Schoenfeld \cite{rs} we have
$$
{\frac{x}{\log x}}\left(1+{\frac{1}{2\log x}}\right)<\pi(x)<
{\frac{x}{\log x}}\left(1+{\frac{3}{2\log x}}\right)
$$
for $x\geq 59$. Hence the statement easily follows by a simple calculation.
\end{proof}
\begin{proof}[Proof of the Proposition] First observe that $n$ cannot have more than four prime divisors, counted with multiplicity. Write $n=q_1\cdots q_I$ with $2\leq I\leq 4$, $41\leq q_1\leq \dots \leq q_I$. Take an arbitrary prime $p_\ell$ with $37\leq p_\ell < q_1$. We show that $n$ cannot be put into the class $S_{\ell,n}$. For this it is sufficient to show that $n$ has more enemies in $S_{\ell,n}$ than friends.
First observe that the elements of $S_{\ell,n}$ have at most four prime divisors, counted with multiplicity. Hence the friends of $n$ in this class have the form
$$
p_\ell q_ir_1r_2,\ p_\ell q_iq_jr_1,\ p_\ell^2 q_ir_1,\ p_\ell q_ir_1,
$$
$$
p_\ell q_i,\ p_\ell q_iq_j,\ p_\ell q_iq_jq_k,\ p_\ell^2 q_i,\ p_\ell^2 q_iq_j,\ p_\ell^3q_i
$$
where $r_1$ and $r_2$ are primes $>p_\ell$ distinct from
$q_1,\dots,q_I$, and the indices $i,j,k$ may be equal.
Observe that if $p_\ell q_ir_1r_2$ is a friend of $n$, then
$$
p_\ell r_1r_2,p_\ell^2 r_1r_2,p_\ell^2 r_1,p_\ell^3 r_1\in S_{\ell,n}
$$
are distinct enemies of $n$. We show that the number of other friends of $n$ is smaller than the number of enemies of $n$ in $S_{\ell,n}$ of the form $p_\ell q$ where $q$ is a prime greater than $p_\ell$. For this first note that the remaining friends of $n$ are of the form $qp_\ell P$ or $p_\ell P$, where $P$ is a product of a power of $p_\ell$ and of $q_i$-s, and $q$ is a prime $>p_\ell$.
The number of friends of the form $p_\ell P$ is at most $52$, since $P$ can be
$$
q_i,\ p_\ell q_i,\ p_\ell^2q_i,\ q_i^2,\ p_\ell q_i^2,\ q_iq_j,\ p_\ell q_iq_j,\ q_i^3,\ q_i^2q_j,\ q_iq_j^2,\ q_iq_jq_k
$$
with $1\leq i<j<k\leq 4$, while for the friends of the form $qp_\ell P$ we have $q\leq n_0/p_\ell q_1$, and here $P$ may take at most the $18$ values
$$
q_i,\ \ \ p_\ell q_i,\ \ \ q_i^2,\ \ \ q_iq_j\ \ \ (1\leq i<j\leq 4).
$$
Hence the number of friends of $n$ in $S_{\ell,n}$ of the latter form is at most $18\pi(n_0/p_\ell q_1)$. On the other hand, the number of enemies of $n$ in $S_{\ell,n}$ of the form $qp_\ell$ is at least $\pi(n_0/p_\ell)-\pi(p_\ell)-4$.
Set $x:=n_0/p_\ell q_1$ and $t:=q_1$. If we further suppose that $p_\ell\leq \sqrt[3]{n_0}$, then by Lemma \ref{lemnew1} we have
$$
\pi(n_0/p_\ell)-\pi(p_\ell)\geq \pi(n_0/p_\ell)-\pi(\sqrt{n_0/p_\ell})
>18\pi(n_0/p_\ell q_1)+56
$$
and the statement follows.
So we are left with the case $p_\ell>\sqrt[3]{n_0}$. Then obviously $n=q_1q_2$ with $p_\ell<q_1\leq q_2$. However, then obviously, $n$ has the only friends
$$
p_\ell q_1,\ \ \ p_\ell q_2
$$
in $S_{\ell,n}$. Since $p_\ell$ and $p_\ell^2$ are enemies of $n$ in $S_{\ell,n}$, and all the elements of $S_{i,n}$ with $p_i=q_1$ are friends of $n$, the statement also follows in this case.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 7,348
|
Meteor.startup(function(){
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var user = Meteor.user();
return (user === null || typeof user !== "undefined");
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{
"redpajama_set_name": "RedPajamaGithub"
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| 8,101
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{"url":"https:\/\/cirq.readthedocs.io\/en\/latest\/generated\/cirq.Operation.html","text":"# cirq.Operation\u00b6\n\nclass cirq.Operation[source]\n\nAn effect applied to a collection of qubits.\n\nThe most common kind of Operation is a GateOperation, which separates its\neffect into a qubit-independent Gate and the qubits it should be applied to.\n__init__()\n\nInitialize self. See help(type(self)) for accurate signature.","date":"2020-04-08 05:36:43","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\": 0.295574426651001, \"perplexity\": 7746.022890380736}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371810617.95\/warc\/CC-MAIN-20200408041431-20200408071931-00273.warc.gz\"}"}
| null | null |
Usually you can find them on torrent (quality can be questionable read comments before download) Project Gutenberg is one site where you can download free ebooks in different formats like pdf, epub kindle text Page on gutenberg.org Edit: There... Sci-Hub is a website that provides free access to millions of paywalled and open-access research papers and books. Sci-Hub obtains paywalled papers by presenting compromised university usernames and passwords to the publishers' websites via the universities' proxy websites.
Sci-Hub and LibGen: what if� why not? Open Access is becoming the predominant form of getting access to peer reviewed articles. Many new non-traditional tools (institutional repositories, social media, and peer to peer sites) are available out there to retrieve the full-text of peer reviewed articles.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 4,001
|
package cucumber.runtime.java.guice.integration;
import cucumber.api.junit.Cucumber;
import org.junit.runner.RunWith;
/**
* The Cucumber integration tests use a mixture of annotation and module binding to demostrate both techniques.
* The step definition classes are all bound in scenario scope using the @ScenarioScoped annotation.
* The test object classes are bound using cucumber.runtime.java.guice.loadguicemodule.YourModuleClass.
*/
@RunWith(Cucumber.class)
public class RunCukesTest {
}
|
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"redpajama_set_name": "RedPajamaGithub"
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| 3,889
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{"url":"http:\/\/clay6.com\/qa\/26354\/if-sigma-x-sigma-y-and-x-y-are-related-by-u-x-y-v-x-y-then-cov-u-v-is","text":"Browse Questions\n\n# If $\\sigma_x= \\sigma_y$ and $x,y$ are related by $u=x+y\\: v=x-y$ then $cov (u,v)$ is\n\n$\\begin {array} {1 1} (A)\\;-1 & \\quad (B)\\;0 \\\\ (C)\\;1 & \\quad (D)\\;cov\\: (x,y) \\end {array}$\n\nCan you answer this question?\n\nGiven\n$u = x+y$\n$v = x-y$\n$\\Rightarrow \\overline u = \\overline x+\\overline y$\n$\\overline v=\\overline x-\\overline y$\n$u - \\overline u=(x-\\overline x)+(y-\\overline y)$\n$v-\\overline v=(x-\\overline x)-(y-\\overline y)$\n$(u-\\overline u)(v-\\overline v)=(x-\\overline x)^2-(y-\\overline y)^2$\n$\\Rightarrow \\large\\frac{1}{n} \\Sigma ( u-\\overline u)(v-\\overline v) = \\large\\frac{1}{n} \\Sigma (x-\\overline x)^2-\\large\\frac{1}{x} \\Sigma (y-\\overline y)^2$\n$= \\sigma_x^2-\\sigma_y^2=0$\n$\\Rightarrow cov (u,v)=0$\nAns : (B)\nanswered Jan 31, 2014","date":"2017-01-19 19:15:31","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9623491168022156, \"perplexity\": 1537.5146052161774}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560280730.27\/warc\/CC-MAIN-20170116095120-00034-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
Improved characterization of high-energy \mbox{$p$-$p$}\ collisions may extend our understanding of QCD, provide a more accurate reference for heavy ion (\mbox{A-A}) collisions and support LHC searches for physics beyond the Standard Model. While a major effort has been devoted to \mbox{$p$-$p$}\ measurements and theoretical analysis some issues remain unclear, specifically the production and manifestations of low-energy jets near the kinematic limits of jet fragment production~\cite{pptheory,fragevo}. Given the steeply-falling spectrum of scattered partons near mid rapidity, {\em minimum-bias} jets are dominated by (mini)jets near an effective spectrum lower bound at 3 GeV, those that are least understood~\cite{anomalous}.
In this study we employ the systematics of 1D $p_t$\ spectra from 200 GeV \mbox{$p$-$p$}\ collisions to derive a two-component model (TCM) for 2D {\em trigger-associated} (TA) correlations. Based on the 2D TCM we propose to extract jet-related {\em hard components} (fragment-pair distributions) from measured TA correlations and determine the kinematic limits for \mbox{$p$-$p$}\ minimum-bias jet production.
Extraction of isolated jets from \mbox{$e^+$-$e^-$}\ and \mbox{$e$-$p$}\ collisions has provided accurate determination of in-vacuum jet properties for specific jet energies. Nonperturbative fragmentation functions (FFs) have been measured down to small hadron momenta~\cite{opal,tasso,eeprd} and have been parametrized simply and precisely over a large jet energy range (2 to 200 GeV)~\cite{ppprd}.
Jet systematics in elementary \mbox{$p$-$p$}\ and composite \mbox{A-A}\ collisions are less certain. Event-wise jet reconstruction in \mbox{$p$-$p$}\ or \mbox{$p$-$\bar p$}\ collisions, emphasizing high-energy jets, must contend with a nonjet background or {\em underlying event} (UE). Separation of (di)jets from the UE background is typically accomplished with a {\em jet finder} based on a numerical algorithm~\cite{ua1,cdf}. Such algorithms may introduce biases in the inferred jet energy and other jet properties, such as the mean fragment number, fragment momentum distribution and angular distribution.
For \mbox{A-A}\ collisions trigger-associated ``dihadron'' 1D azimuth and 2D angular correlations provide a combinatoric jet analysis method for the high-density \mbox{A-A}\ environment~\cite{starhipt}. High-$p_t$\ {\em trigger} hadrons defined by restrictive $p_t$\ cuts estimate a jet thrust (parton momentum) axis. Other $p_t$\ cuts define a class of {\em associated} hadrons based on strong assumptions about jet structure. A combinatoric background is then subtracted based on a physical model designated by the expression ``zero yield at minimum'' (ZYAM). Strong biases in inferred jet yields and angular structure may result from such methods~\cite{tzyam}.
An alternative approach is based on {\em minimum-bias} (MB) 1D spectrum and 2D two-particle correlation analysis. No a priori assumptions are imposed on jet structure, and no special $p_t$\ cuts are applied (except an effective low-$p_t$\ cutoff reflecting a limited detector $p_t$\ acceptance). Measured event-averaged jet contributions to $p_t$\ spectra and two-particle correlations emphasize minimum-bias (mainly low-energy) jets. Jet-related spectrum hard components and {\em symmetrized} angular and $p_t \times p_t$ correlations have been measured for \mbox{$p$-$p$}\ and \mbox{A-A}\ collisions over a range of collision systems~\cite{ppprd,hardspec,fragevo,porter2,porter3,axialci,anomalous}.
In the present study we extend the MB program. We derive a method to isolate the 2D trigger-associated (TA) correlation structure of minimum-bias jets from \mbox{$p$-$p$}\ collisions by analogy with a two-component subtraction method developed for 1D $p_t$ spectrum analysis~\cite{ppprd}. For the 1D analysis a jet-related hard component or {\em fragment distribution} was revealed with quantitative correspondence to pQCD predictions~\cite{fragevo}. In the present study we develop a method to isolate a hard component from {\em asymmetric} (conditional) 2D TA correlations that may be related quantitatively to fragmentation function (FF) data from \mbox{$e^+$-$e^-$}\ collisions and to a predicted pQCD parton spectrum.
This article is arranged as follows:
Sec.~\ref{anal} summarizes several jet-structure analysis methods.
Sec.~\ref{2comp} reviews two-component models for hadron yields, spectra and two-particle correlations.
Sec.~\ref{trig} presents a 1D two-component model for trigger spectra.
Sec.~\ref{tata} develops a 2D TCM for trigger-associated correlations.
Sec.~\ref{isolate} illustrates how jet-related hard components can be isolated from TA data.
Sec.~\ref{syserr} estimates systematic uncertainties.
Secs.~\ref{disc} and~\ref{summ} present Discussion and Summary.
\section{Analysis methods} \label{anal}
In this section we review analysis methods directed to separating jet structure from backgrounds in high-energy nuclear collisions. Jet structure can be inferred by several methods: event-wise 2D jet reconstruction~\cite{opal,cdf}, isolation of nominal jet-related angular correlations using model-dependent $p_t$ cuts and combinatoric-background subtraction~\cite{starhipt} and identification of MB jet fragment distributions within correlations and spectra~\cite{ppprd,hardspec,porter2,porter3}.
In each case systematic biases may arise from limitations in the method and unjustified assumptions.
\subsection{Event-wise jet reconstruction}
Event-wise jet reconstruction with a jet-finder algorithm is intended to isolate jet fragments from nonjet hadrons. Because of the complexity of the fluctuating fragmentation cascade jet isolation on $(\eta,\phi)$ is biased at some level. If the jet (parton) energy is the object of measurement systematic bias may be reduced to a relatively low level, since most of the parton energy is carried by a few higher-momentum fragments located near the jet thrust axis (parton momentum). If the number and properties of low-momentum fragments (the majority) are the object systematic biases may be relatively large, since those hadrons may emerge farther in angle from the thrust axis and be excluded by a jet finder~\cite{pptheory}.
Underlying-event studies are complementary to event-wise jet reconstruction. By assumption UE analysis addresses the nonjet background---what is {\em not} the triggered dijet~\cite{rick,cdf}. Any low-momentum (and large-angle) jet-related hadrons excluded from the dijet by a jet-finder algorithm may be assigned (incorrectly) to the UE. Inferred UE multiplicities, angular correlations and $p_t$\ distributions could then be misleading~\cite{pptheory}.
\subsection{Combinatoric dihadron analysis and ZYAM}
Jet structure on azimuth inferred from so-called dihadron correlation analysis relies on strong assumptions about background structure and jet fragment distributions. Background estimates invoke measurements of azimuth quadrupole amplitude $v_2$ conventionally associated with ``elliptic flow''~\cite{2004}. But jet structure can contribute strongly to $v_2$ as a ``nonflow'' bias (e.g., the $m=2$ Fourier or quadrupole component of a same-side 2D jet peak narrow on 1D azimuth)~\cite{gluequad,tzyam,multipoles}. And the assumption of zero correlation amplitude at the minimum is inconsistent with substantial same-side and away-side peak overlaps that are expected {\em and observed} for such correlations~\cite{porter2,porter3}. ZYAM background subtraction can then result in underestimated and distorted (severely biased) jet structure~\cite{tzyam,multipoles}. It is further assumed that (a) only ``high-$p_t$'' hadrons should be associated with pQCD jets and (b) any hadrons below 2 GeV/c must emerge from a thermalized bulk medium in \mbox{A-A}\ collisions~\cite{starbulk,starhipt}. ``Trigger-associated'' $p_t$ cuts based on such assumptions can further distort and diminish inferred jet structure.
\subsection{p-p collision charge-multiplicity classes}
The charge-multiplicity $n_{ch}$ dependence of spectrum and correlation data from \mbox{$p$-$p$}\ collisions provides essential information on collision dynamics and particle production mechanisms. Distinct correlation components are observed to scale quite differently with $n_{ch}$, permitting accurate separation of the individual components without imposing an a priori physical model. Measured systematic properties of the separate components can then be related to candidate collision mechanisms.
Charge multiplicity $n_{ch}$ or its {\em soft component} $n_s$ (defined below) is an analog to \mbox{A-A}\ centrality measures $N_{part}$ and $N_{bin}$ (Glauber parameters). But the relevant \mbox{$p$-$p$}\ ``centrality'' parameter may be ``depth'' on nucleon momentum fraction $x$ rather than \mbox{$p$-$p$}\ impact parameter $b$~\cite{pptheory}.
Anticipating future experimental analysis we define seven multiplicity classes ($n_{ch}$ bins, corresponding to fractions of the \mbox{$p$-$p$}\ total cross section) over a statistically-significant interval, with bin means $\bar n_{ch}/ \Delta \eta = 1.7, 3.4, 5.5, 7.6, 10, 13.7, 18.7$.
\subsection{Single-particle $\bf y_t$ spectra}
Although transverse momentum $p_t$ is directly measured by particle detectors, {\em transverse rapidity} $y_t \equiv \ln[(m_t + p_t)/m_h]$ (with transverse mass $m_t = \sqrt{p_t^2 + m_h^2}$ and $m_h = m_\pi$ as the default for unidentified hadrons) visualizes important spectrum structure equally well at smaller and larger $y_t$. In semilog plots on $y_t$ the ``power-law'' data trend arising from the underlying pQCD parton spectrum appears as a straight line at larger $y_t$.
Analysis of the multiplicity dependence of $y_t$\ spectrum shapes from 200 GeV \mbox{$p$-$p$}\ collisions revealed two fixed functional forms---later identified as soft and hard components---scaling approximately as $n_{ch}$ and $n_{ch}^2$~\cite{ppprd}. From subsequent analysis and theory comparisons one component (soft) was associated with projectile nucleon fragmentation proportional to soft multiplicity component $n_s$, and the other component (hard) was associated with dijet production proportional to $n_s^2$~\cite{fragevo}. The combination defines the two-component model for \mbox{$p$-$p$}\ spectra.
\subsection{Symmetrized two-particle correlations} \label{symmcorr}
Symmetrized combinatoric pairs on two-particle momentum space $(\vec p_1,\vec p_2)$ can be factorized with minimal information loss into pair distributions on 2D transverse rapidity space $(y_{t1},y_{t2})$ or $y_t \times y_t$ and on 4D angle space $(\eta_1,\phi_1,\eta_2,\phi_2)$.
Minimum-bias correlations (no $y_t$\ cuts) have been studied extensively for \mbox{$p$-$p$}~\cite{porter2,porter3} and \mbox{Au-Au}~\cite{axialci,anomalous} collisions. The quantitative connection between jet-related angular correlations and the $y_t$ spectrum hard component has been demonstrated in Ref.~\cite{jetspec}.
If within some angular acceptance $(\Delta \eta,\Delta \phi)$ the correlation structure is invariant on pair mean angle (e.g., on $\eta_\Sigma = \eta_1 + \eta_2$), 4D angular correlations can be {\em projected by averaging} onto difference variables (e.g., $\eta_\Delta = \eta_1 - \eta_2$) without loss of information to form {\em angular autocorrelations}~\cite{axialcd,inverse}. The 2D subspace ($\eta_\Delta,\phi_\Delta$) then retains all correlation information and can be visualized.
The azimuth pair acceptance can be separated into same-side (SS, $|\phi_\Delta| < \pi/2$) and away-side (AS, $|\phi_\Delta| > \pi/2$) regions.
Features in angular correlations can be modeled by simple functional forms including 1D and 2D Gaussians and sinusoids. Sinusoids $\cos(m\phi_\Delta)$ on azimuth can be characterized as {\em cylindrical multipoles} with pole number $2m$, e.g., dipole, quadrupole and sextupole for $m = 1,~2,~3$.
Angular correlations on $(\eta_\Delta,\phi_\Delta)$ formed from all combinatoric pairs but excluding ``self'' pairs---particles numerically paired with themselves---are complementary to correlations on transverse rapidity $y_t\times y_t$. The pair distribution is symmetrized about the main diagonal on $y_t \times y_t$. $y_t$\ acceptance [1,4.5] corresponding to $p_t$\ interval [0.15,6] GeV/c is consistent with typical detector $p_t$ acceptance and data volumes (statistics limitations at larger $p_t$).
Two-particle correlations can be obtained for individual charge-pair types: like-sign (LS), unlike-sign (US), charge-independent (CI = LS $+$ US) and charge-dependent (CD = LS $-$ US). {\em Intra}-jet correlations (same-side jet cone) are dominated by the US combination. {\em Inter}-jet correlations (back-to-back jet pairs) show no charge dependence (CD = 0). Bose-Einstein correlations are observed only for the LS combination. The nonjet azimuth quadrupole conventionally associated with ``elliptic flow'' shows no significant charge dependence.
\subsection{Conditional trigger-associated correlations}
Minimum-bias {\em conditional} (asymmetric) TA correlations on $(y_{t,assoc},y_{t,trig})$ (with $y_{t,assoc} < y_{t,trig}$ as defined below) are related to, but not equivalent to, symmetrized correlations on $y_t \times y_t$. In either case all nontrivial combinatoric pairs from all events in a given $n_{ch}$ class appear---no particles are rejected by $p_t$ cuts. But there are quantitative differences in the two structures: Unsymmetrized TA correlations retain additional correlation information, and the TA hard component is more directly comparable to pQCD parton spectrum predictions and measured fragmentation functions.
TA correlations are formed as follows: All events in a given $n_{ch}$ class are separated into ``trigger'' classes ($y_{t,trig}$ or $y_{tt}$ bins) based on the greatest hadron $y_t$ in each event (trigger particle). The single trigger hadron for each event is assumed (with some probability) to be the proxy for a scattered parton. The spectrum for $n_{ch}-1$ {\em associated} hadrons (some may be jet fragments) distributed on $y_{t,assoc}$ or $y_{ta}$ is accumulated for each event in the trigger class, with trigger particles excluded (no self pairs). The resulting 2D TA distribution denoted by $F(y_{ta},y_{tt},n_{ch})$ can be factorized (according to Bayes' theorem) into a trigger spectrum $T(y_{tt},n_{ch})$ (comparable to a pQCD parton spectrum) and a 2D ensemble of distributions $A(y_{ta}:y_{tt},n_{ch})$: associated-hadron spectra conditional on specific trigger $y_{tt}$ values (comparable to an ensemble of fragmentation functions conditional on parton energy).
We employ the two-component model of 1D single-particle $p_t$ spectra derived from 200 GeV \mbox{$p$-$p$}\ collisions to define a TCM for 2D TA correlations. By subtracting a 2D soft-component model from TA data we intend to isolate a TA hard component that may be compared quantitatively with pQCD parton spectra and fragmentation function systematics. The method may also be applied to \mbox{A-A}\ collisions to study modified parton fragmentation to jets in more-central collisions.
\section{two-component model} \label{2comp}
The two-component (soft+hard) model~\cite{kn} has been applied to analyses of \mbox{$p$-$p$}~\cite{ppprd,pptheory} and \mbox{Au-Au}~\cite{hardspec,fragevo} collisions. The soft component is attributed to participant-nucleon dissociation
and the hard component to large-angle-scattered parton fragmentation to jets. The relation of $y_t$\ spectrum hard-component structure to pQCD theory was established in Ref.~\cite{fragevo}.
It is also possible to decompose minimum-bias correlation structure according to the TCM. Soft and hard correlation components in more-peripheral \mbox{Au-Au}\ collisions scale $\propto N_{part}$ (participant nucleons) and $\propto N_{bin}$ (\mbox{N-N}\ binary collisions) respectively~\cite{anomalous,nov2}. Corresponding multiplicity scaling with soft multiplicity $n_s$ for \mbox{$p$-$p$}\ collisions (projectile nucleon fragments $\propto n_s$, dijet production $\propto n_s^2$) is discussed in Ref.~\cite{pptheory}. Identification of the 1D spectrum and 2D correlation hard components with jets in \mbox{$p$-$p$}\ and more-peripheral \mbox{Au-Au}\ collisions is well supported by data systematics and comparisons with pQCD~\cite{ppprd,fragevo,anomalous,jetspec}.
\subsection{Soft and hard event types and multiplicities} \label{tcmmult}
\mbox{$p$-$p$}\ collisions can be separated into soft and hard {\em event types}. By definition hard events include at least one minimum-bias jet within the angular acceptance $(\Delta \eta,\Delta \phi)$. Soft events include no jet structure within the acceptance. Soft and hard event types should be distinguished from soft and hard {\em components} of spectrum and correlation structure from individual events or ensemble averages.
Each $n_{ch}$ event class with $N_{evt}(n_{ch})$ events includes $N_s$ soft and $N_h$ hard events. Soft $n_s$ and hard $n_h$ multiplicity components averaged over the event ensemble are related by $n_s + n_h = n_{ch}$. Reference~\cite{ppprd} reported that \mbox{$p$-$p$}\ hard-component multiplicity $n_h$ scales approximately as $n_h = \alpha\, n_{ch}^2$, implying a preliminary soft-component definition $n_s = n_{ch} - \alpha\, n_{ch}^2$. A more accurate hard-component trend is $n_h = \alpha\, n_{s}^2$ with $\alpha \approx 0.006$ (for acceptance $\Delta \eta = 1$).
The observed mean dijet number $n_j$ within acceptance $\Delta \eta$ varies with soft-component multiplicity $n_s$ as $n_j = 0.015\, \Delta \eta\, (n_s / 2.5\Delta \eta)^2$ scaled from non-single-diffractive (NSD) \mbox{$p$-$p$}\ collisions~\cite{ppprd}. Poisson probabilities for soft and hard events are then $P_s(n_s) = N_s/N_{evt} = \exp(-n_j)$ and $P_h(n_s) = N_h/N_{evt} = 1- P_s(n_s)$. For soft events $n_s'' = n_{ch}$ and for hard events $n_s' + n_h' = n_{ch}$, defining those symbols. The several $n_{ch}$ components then satisfy the following relations: (i) $n_{ch} = n_s + n_h = P_s n''_{s} + P_h (n_s' + n_h')$, (ii) $n_s = P_s n_{ch} + P_h n_s'$ and $n_h = P_h n_h' = n_j \bar n_{ch,j}$, (iii)
$n_s' = n_s - (P_s/P_h) n_h = n_s - P_s n_h' \approx n_s - P_s \bar n_{ch,j}$, where $\bar n_{ch,j}$ is the ensemble-mean dijet fragment multiplicity within $\Delta \eta$. For smaller event multiplicities $P_h \approx n_j \propto n_s^2$.
\subsection{p-p single-particle $\bf y_t$ spectra} \label{ppspecc}
Distinct soft and hard components of 1D $y_t$ spectra from \mbox{$p$-$p$}\ collisions are observed to have fixed forms on $y_t$, but their relative amplitudes vary with $n_{ch}$ (or $n_s$). The two-component model of $y_t$ spectra $\rho(y_t,n_{ch}) = \rho_0( n_{ch}) F_0(y_t, n_{ch})$ from \mbox{$p$-$p$}\ collisions conditional on measured $ n_{ch}$ integrated within some angular acceptance $(\Delta \eta,\Delta \phi)$ is described by~\cite{ppprd}
\begin{eqnarray} \label{ppspec}
\rho_0( n_{ch}) F_0(y_t, n_{ch}) &=& S(y_t,n_{ch}) + H(y_t,n_{ch}) \\ \nonumber
&=& \rho_s( n_{ch}) S_0(y_t) + \rho_{h}( n_{ch}) H_0(y_t),~~~
\end{eqnarray}
where $\rho_0 = n_{ch} / \Delta \eta \Delta \phi$ is the $y_t$-integral single-particle (SP) angular density, and $\rho_s = n_s / \Delta \eta \Delta \phi$ and $\rho_h = n_h / \Delta \eta \Delta \phi$ are corresponding soft and hard charge angular densities.
The inferred spectrum soft and hard model components [unit integral $S_0(y_t)$ and $H_0(y_t)$ functions, see App.~\ref{tcmmodelfunc}] are independent of $n_{ch}$. Soft-component model $S_0(y_t)$ is defined as the limiting form of the normalized spectra appearing in Fig.~\ref{ppspectra} (left) as $n_s \rightarrow 0$. Hard-component model $H_0(y_t)$ represents data hard components $H(y_{t},n_{ch})$ obtained from spectrum data by subtracting the soft-component model and is predicted by measured parton fragmentation functions (from \mbox{$p$-$p$}\ collisions) folded with a minimum-bias pQCD parton (dijet) spectrum integrating to $\sigma_{dijet} \approx 2.5$ mb~\cite{fragevo}.
Figure~\ref{ppspectra} (left) shows $y_t$ spectra for several multiplicity classes (thin solid curves). Each spectrum is normalized by a corresponding soft multiplicity $n_s$ inferred iteratively (Sec.~\ref{tcmmult}). The bold dash-dotted curves in the left panel represent soft component $S_0(y_t)$. The plotted spectra are uncorrected for detector inefficiencies. The $y_t$-averaged inefficiency cancels in the normalization ratio. A $y_t$-dependent inefficiency function deviating from unity at lower $y_t$ and included in the $S_0$ model to relate corrected and uncorrected spectra is indicated by the ratio of the two dash-dotted curves.
\begin{figure}[h]
\includegraphics[width=1.65in,height=1.6in]{ppcms110a}
\includegraphics[width=1.65in,height=1.6in]{ppcms110b}
\caption{\label{ppspectra}
Left: Normalized single-particle spectra (thin solid curves) from seven multiplicity $n_{ch}$ classes of 200 GeV \mbox{$p$-$p$}\ collisions. The dash-dotted curves are inferred soft component $S_0$ for uncorrected (lower) and efficiency-corrected (upper) spectra.
Right: Hard components (thin solid curves) extracted from spectra in the left panel according to the vertical axis label. The bold solid curve is the jet-related hard-component model function $H_0$ with coefficient $\alpha = 0.006$. The dash-dotted curve is soft component $S_0$ scaled by $\Delta \eta / n_{s}$ for the highest multiplicity class (7).
}
\end{figure}
Figure~\ref{ppspectra} (right) shows the {\em measured} hard components $H(y_t,n_{ch})$ (thin solid curves) inferred by subtracting the soft-component model $S_0(y_t)$ from each normalized data spectrum in the left panel and normalized by soft density $n_s / \Delta \eta$ with acceptance $\Delta \eta = 2$. Data for the higher $n_{ch}$ classes coincide with TCM function $H_0(y_t)$ (bold dashed curve) within data uncertainties except for the lowest $y_t$ points, but the data for two $n_{ch}$ classes fall significantly below the majority. $H(y_t)$ for the first $n_{ch}$ bin is reduced by factor 3 (and the next bin by a smaller factor) from the principal trend, as noted in Ref.~\cite{ppprd}.
The observed systematic deviations are included in the TA TCM defined below.
The dashed curve is $\alpha H_0(y_t)$. Hard-component model $H_0(y_t)$ is a Gaussian {\em with power-law tail} as introduced in Ref.~\cite{hardspec} and described in App.~\ref{tcmmodelfunc}. A preliminary value $\alpha = 0.005 \pm 0.0015$ was reported in Ref.~\cite{ppprd}. The relation of $\alpha$ to dijet production is discussed below.
The TCM for 1D $y_t$\ spectra reveals minimum-bias jet structure (hard component) in quantitative agreement with pQCD predictions~\cite{ppprd,hardspec,fragevo,jetspec}.
The dash-dotted curve in the right panel is soft component $S_0$ scaled according to the $y$-axis label for the highest multiplicity class. Direct comparison with the ensemble of hard components indicates that the TCM hard component is accurately determined down to $y_t \approx 2$ by subtraction of model $S_0$, especially for larger $n_{ch}$.
Measured SP spectra integrated over some angular acceptance can also be decomposed in terms of soft and hard event types as
\begin{eqnarray} \label{spevtype}
\frac{d n_{ch}}{y_t dy_t} &=& P_s(n_{ch}) S_s(y_t,n_{ch}) \\ \nonumber
&+& P_h(n_{ch}) [S_h(y_t,n_{ch}) + H_h(y_t,n_{ch})]
\end{eqnarray}
with corresponding TCM model elements $S_s \rightarrow n_{ch} S_0$, $S_h \rightarrow n_s' S_0$ and $H_h \rightarrow n_h' H_0$, and with $n_h' \approx \bar n_{ch,j}$ for smaller $n_{ch}$.
\subsection{p-p minimum-bias dijet production}
Equation~(\ref{ppspec}) integrated over some angular acceptance $(\Delta \eta,\Delta \phi)$ becomes
\begin{eqnarray} \label{ppspec2}
\frac{d n_{ch}}{y_t dy_t} &=& n_s S_0(y_t) + n_h H_0(y_t).
\end{eqnarray}
The measured data trend in Fig.~\ref{ppspectra} (right) implies $n_h/n_s = \alpha\, n_s/\Delta \eta$ or
\begin{eqnarray} \label{freq}
\frac{n_h}{\Delta \eta} &=& 0.006 \pm 0.001 \left( \frac{n_s}{\Delta \eta}\right)^2 \\ \nonumber
&\equiv& f(n_{ch}) \bar n_{ch,j}(\Delta \eta),
\end{eqnarray}
where the second line represents the jet hypothesis from Ref.~\cite{ppprd} and defines dijet frequency $f = dn_j/ d\eta$ (dijet number per event per unit $\eta$) with mean dijet fragment multiplicity $\bar n_{ch,j}$.
That result can be compared with a pQCD prediction of the dijet frequency for non-single-diffractive (NSD) \mbox{$p$-$p$}\ collisions. From the trend in Eq.~(\ref{freq}) we have
\begin{eqnarray}
f(n_{ch}) &=& f_{NSD} \frac{n_{s}^2}{n_{s,NSD}^2} =\frac{f_{NSD}}{(n_{s,NSD}/\Delta \eta)^2} \left(\frac{n_s}{\Delta \eta}\right)^2 \\ \nonumber
f_{NSD} &=& \frac{0.006 \times 2.5^2}{\bar n_{ch,j}(\Delta \eta)} \approx 0.015,
\end{eqnarray}
with $n_s / \Delta \eta \approx 2.5$ for NSD \mbox{$p$-$p$}\ collisions and $\bar n_{ch,j}(\Delta \eta) \approx 2.5$~\cite{eeprd}. The value $f_{NSD} \approx 0.015$ inferred from the \mbox{$p$-$p$}\ spectrum $n_{ch}$ dependence is consistent with a pQCD prediction 0.015 based on a parton spectrum cutoff near 3 GeV corresponding to dijet total cross section 2.5 mb~\cite{fragevo}. Thus, we obtain from a TCM analysis of 1D $y_t$\ spectra the dijet $\eta$ density $f = dn_j/d\eta$ as a function of $n_s$.
The minimum-bias dijet rate determined directly from \mbox{$p$-$p$}\ $y_t$\ spectra can be contrasted with Monte Carlo parametrizations~\cite{hijing,pythia,herwig,herwig2}. Dijet production in \mbox{A-A}\ collisions modeled by \textsc{hijing} (based on the \textsc{pythia} model of \mbox{$p$-$p$}\ collisions) is determined by an effective \mbox{$p$-$p$}\ dijet total cross section $\sigma_{dijet} \approx 25$ mb~\cite{liwang}, ten times the value inferred from measured $y_t$\ spectra~\cite{fragevo}.
\textsc{hijing} predictions for \mbox{Au-Au}\ hadron production disagree strongly with data trends, prompting some to reject the TCM for heavy ion collisions~\cite{review}.
Ironically, while the \textsc{hijing} {\em implementation} of the TCM overpredicts low-energy scattered partons it also underpredicts corresponding jet angular correlations compared to data~\cite{anomalous}. The \textsc{hijing} fragmentation model for low-energy partons appears to proceed by partons (gluons) $\rightarrow$ single hadrons, not correlated charge-neutral hadron pairs as observed for instance in \mbox{$p$-$p$}\ collisions~\cite{porter2,porter3}. In contrast, measured \mbox{$p$-$p$}\ dijet production is fully consistent with TCM centrality trends for jet-related structure in more-peripheral \mbox{Au-Au}\ collisions~\cite{anomalous}.
\subsection{In-vacuum jet phenomenology}
A primary goal of TCM analysis is direct comparison between measured spectrum and correlation hard components and pQCD jet systematics. The main source of isolated (in-vacuum) jet properties is \mbox{$e$-$p$}\ and \mbox{$e^+$-$e^-$}\ data from the HERA and LEP, both dijet fragment multiplicities~\cite{jetmult} and fragment momentum distributions or {\em fragmentation functions} (FFs)~\cite{tasso,opal}. FF systematics are conveniently plotted on the space $(y,y_{max})$, where $y \equiv \ln[(E+p)/m_\pi]$ and $y_{max} \equiv \ln(Q/m_\pi)$ with dijet energy $Q = 2E_{jet}$. Such FF ensembles are conditional distributions $D(y:y_{max})$ on hadron fragment rapidity $y$ given some value of parton rapidity $y_{max}$. Conditional hard components isolated from TA correlations as in the present study should be directly comparable.
Figure~\ref{qcd} (left) shows a parametrization of measured FFs~\cite{eeprd} from the lowest observed jet energies ($\approx 2$ GeV or $y_{max} \approx 3.3$ from Ref.~\cite{cleo}) to the highest ($\approx 100$ GeV or $y_{max} \approx 8$ from LEP). The lower bound on $y$ is determined by \mbox{$p$-$\bar p$}\ FF data~\cite{fragevo}. The parametrization has the simple form $D(y:y_{max}) = 2n_{ch}(y_{max}) \beta(u;p,q)$, where $u \approx y/y_{max}$ is a scaled rapidity, $\beta(u;p,q)$ is the unit-normal beta distribution, $n_{ch}(y_{max})$ is determined by parton energy conservation within a jet, and parameters $(p,q)$ are nearly constant over a large parton energy interval. That simple parametrization describes all quark or gluon $\rightarrow$ unidentified hadron FFs over a large energy interval to the uncertainty limits of the FF data.
\begin{figure}[h]
\includegraphics[width=1.65in,height=1.65in]{aleph11alogz}
\includegraphics[width=1.65in,height=1.65in]{aleph11cpp}
\caption{\label{qcd}
Left: Parametrization of the fragmentation-function (FF) ensemble for unidentified hadrons from light (uds) quarks or gluons over the parton energy range 2-200 GeV ($y_{max} \in$ [3.3,8]), from Ref.~\cite{eeprd}.
Right: The same FF ensemble folded with a perturbative QCD parton spectrum on $y_{max}$ as in Ref.~\cite{fragevo}. The z axis is logarithmic in both cases.
}
\end{figure}
Figure~\ref{qcd} (right) shows the \mbox{$e^+$-$e^-$}\ FF ensemble in the left panel multiplied by a pQCD parton (power-law) spectrum on $y_{max}$~\cite{fragevo}. The most-probable jet (minijet) energy is $E_{jet} \approx 3$ GeV ($y_{max} \approx$ 3.75) consistent with measured jet systematics~\cite{ua1,anomalous,fragevo}. That distribution is the argument of a pQCD folding integral (factorization, Bayes' theorem). Projection onto the vertical axis (integration over the parton spectrum on $y_{max}$) produces a {\em fragment distribution}---a pQCD prediction for the measured TCM spectrum hard component $H(y_t)$~\cite{fragevo}. The hard components of 2D TA correlations may be compared quantitatively with the FF data in Fig.~\ref{qcd}. Such comparisons should establish the kinematic limits for jet manifestations in high-energy nuclear collisions.
\subsection{Symmetrized two-particle correlations}
Correlation structures on $(\eta_\Delta,\phi_\Delta)$ and $y_t \times y_t$ from 200 GeV NSD \mbox{$p$-$p$}\ collisions~\cite{porter2,porter3} are consistent with extrapolation of the centrality systematics of angular correlations from \mbox{Au-Au}\ collisions~\cite{anomalous}. Both angular correlations and correlations on transverse rapidity are described accurately by the TCM (aside from small-amplitude structures representing Bose-Einstein correlations and gamma conversion to \mbox{$e^+$-$e^-$}\ pairs).
The soft component of 2D angular correlations is represented by a narrow 1D Gaussian on $\eta_\Delta$. Soft-component pairs are exclusively the unlike-sign (US) charge combination consistent with projectile-proton dissociation (mainly to gluons near mid-rapidity), and the r.m.s.\ peak width on $\eta_\Delta$ is $\sigma_{soft} \approx 0.5$~\cite{porter2}.
The hard-component 2D peak on $y_t \times y_t$ corresponds approximately (when projected onto 1D $y_t$) to the 1D SP spectrum hard component $H_0(y_t)$ of Eq.~(\ref{ppspec}). The hard component of 2D angular correlations (two features) is consistent with expectations for intra-jet correlations (SS 2D peak, ``jet cone'') and inter-jet correlations (AS 1D peak on azimuth, back-to-back jets). The volume of the SS 2D peak corresponds quantitatively to the hard component of the total hadron yield inferred from $y_t$ spectrum data and to pQCD calculations~\cite{jetspec}. Measured \mbox{$p$-$p$}\ angular correlations~\cite{porter2} show only semiquantitative agreement with the \textsc{pythia} Monte Carlo~\cite{pythia}.
\subsection{Conditional trigger-associated correlations}
Single-particle spectrum hard components (jet fragment distributions) are 1D projections of more-complex extended objects (dijets). We seek to isolate 2D trigger-associated hard components that reveal further details of jet structure, especially the analog between TA conditional distributions on space $(y_{ta},y_{tt})$ inferred from \mbox{$p$-$p$}\ collisions and parton-fragment conditional distributions on space $(y,y_{max})$ inferred from \mbox{$e^+$-$e^-$}\ collisions~\cite{eeprd,fragevo}.
The TA pair distribution for given $n_{ch}$ class is denoted by $F(y_{ta},y_{tt},n_{ch})$. To extract a 2D hard component from TA data we require a 2D soft-component model. Equation~(\ref{ppspec}) is a TCM for SP $y_t$\ spectra from which the 1D hard component $H(y_t)$ was isolated. Deriving a TCM for the 2D TA distribution is an exercise in compound probabilities and random sampling from multiple parent distributions. Trigger and associated particles are approximately random samples from fixed spectrum components $S_0(y_t)$ and $H_0(y_t)$ in Eq.~(\ref{ppspec}).
We assume that the soft component of the TA TCM is factorizable (uncorrelated samples). We further assume, only for purposes of illustration in this study, that the 2D hard-component {\em model} is also factorizable. We expect that the TA {\em data} hard component derived by subtracting the soft-component model is not factorizable in general.
We factorize the 2D TA distribution for given $n_{ch}$ class according to Bayes' theorem as
\begin{eqnarray} \label{ta2comp}
F(y_{ta},y_{tt}) &=& T(y_{tt}) A(y_{ta}:y_{tt}),
\end{eqnarray}
where $T(y_{tt})$ is a {\em trigger spectrum} and $A(y_{ta}:y_{tt})$ is an ensemble of conditional {\em associated-particle spectra} given a trigger at $y_{tt}$. We first obtain a TCM for the 1D trigger spectrum and then develop the 2D TA TCM as a Cartesian product.
\section{One-dimensional Trigger spectra} \label{trig}
Trigger particles arise from (sample) soft or hard events, and for hard events arise from either the soft or hard spectrum component. For a given trigger particle the remainder of an event with multiplicity $n_{ch}$ is constrained to $n_{ch} - 1$ associated particles. We now combine information from Refs.~\cite{ppprd,fragevo} on dijet frequency and the two-component 1D $y_t$\ spectrum model to derive a 1D TCM for trigger spectrum $T(y_{tt},n_{ch})$. The total multiplicity in the angular acceptance is $n_{ch} = \Delta \eta\, dn_{ch}/d\eta$.
For a sequence of {\em independent} trials (collision events), each including $n_{ch}$ samples from a fixed parent spectrum $F_x(y_t)$ ($x$ denotes soft $s$ or hard $h$ event type), we sort events according to the maximum sample value $y_{tt}$ in each event.
Parent distributions are denoted by $F_s(y_{tt}) = S_0(y_{tt})$ for soft events and $F_h(y_{tt},n_{ch}) = p_s' (n_{ch})S_0(y_{tt}) + p_h'(n_{ch}) H_0(y_{tt})$ for hard events, where $p_x' = n_x' / n_{ch}$.
We then define the trigger spectrum
\begin{eqnarray} \label{trigspec}
T(y_{tt},n_{ch}) &\equiv& \frac{1}{N_{evt}(n_{ch})} \frac{dn_{trig}}{y_{tt}dy_{tt}} \\ \nonumber
&& \hspace{-.3in} = P_s(n_{ch})T_{s0}(n_{ch})G_s(y_{tt},n_{ch})\, n_{ch} F_s(y_{tt}) \\ \nonumber
&& \hspace{-.27in} + P_h(n_{ch})T_{h0}(n_{ch})G_h(y_{tt},n_{ch})\, n_{ch} F_h(y_{tt},n_{ch}) \\ \nonumber
&& \hspace{-.3in} = P_s(n_{ch})T_s(y_{tt},n_{ch}) + P_h(n_{ch})T_h(y_{tt},n_{ch}).
\end{eqnarray}
In each term factor $F_x(y_{tt},n_{ch})$ is the probability of a $y_{tt}$ sample from the parent spectrum, and factor $G_x(y_{tt},n_{ch})$ is the probability of a void (no samples) above the given $y_{tt}$. The sum gives the overall density of trigger particles at $y_{tt}$ from either event type.
The {\em void probability} $G_x(y_{tt},n_{ch})$ is defined as follows: For events with trigger at $y_{tt}$ no sample should appear with $y_t > y_{tt}$ (void interval). The event-wise spectrum integral {\em above} $y_{tt}$ within acceptance $\Delta \eta$ is
\begin{eqnarray} \label{nsum}
n_{x\Sigma}(y_{tt},n_{ch})
&=& \int _{y_{tt}}^\infty dy_t y_t n_{ch} F_x(y_{t})
\end{eqnarray}
separately for unit-normal spectra $F_x(y_t)$ from soft or hard events. The void probability for event type $x$ is $G_x(y_{tt},n_{ch}) = \exp[- \kappa n_{x\Sigma}(y_{tt},n_{ch})]$, where $O(1)$ factor $\kappa$ may account for non-Poisson (e.g., jet) correlations. $O(1)$ coefficients $T_{x0}(n_{ch})$ are defined such that the product $T_x \equiv T_{x0}(n_{ch}) G_x\, n_{ch} F_x$ is unit normal
\begin{eqnarray} \label{g0yt}
\int_0^\infty dy_{tt} y_{tt} T_x(y_{tt},n_{ch}) &=& 1
\end{eqnarray}
(the number of trigger particles from any event is 1). Small deviations from $T_{x0} = 1$ may arise from incomplete $y_t$ acceptance and particle detection inefficiencies.
\begin{figure}[h]
\includegraphics[width=3.3in]{ppcms118ano}
\put(-20,105) {\bf (d)}
\put(-20,232) {\bf (b)}
\put(-140,105) {\bf (c)}
\put(-140,232) {\bf (a)}
\caption{\label{trigspec2}
Two-component models (TCM) of trigger spectra for four multiplicity classes of 200 GeV \mbox{$p$-$p$}\ collisions generated by Eq.~(\ref{trigspec}). Trigger spectra for soft events (no jets, dotted), hard events (at least one jet, dash-dotted) and total (solid).
}
\end{figure}
Figure~\ref{trigspec2} shows trigger spectra $T(y_{tt},n_{ch})$ obtained from Eq.~(\ref{trigspec}) (solid) and the soft and hard components $P_s T_{s}$ (dotted) and $P_h T_{h}$ (dash-dotted) vs trigger rapidity for four multiplicity classes. TCM model parameters for $S_0(y_{tt})$ and $H_0(y_{tt})$ are those from the spectrum analysis in Ref.~\cite{ppprd} with no changes (see App.~\ref{tcmmodelfunc}). The spectrum mode shifts to larger $y_{tt}$ values with increasing $n_{ch}$.
\begin{figure}[h]
\includegraphics[width=3.3in]{ppcms118b}
\put(-22,85) {\bf (d)}
\put(-22,230) {\bf (b)}
\put(-142,85) {\bf (c)}
\put(-142,230) {\bf (a)}
\caption{\label{trigprobs}
TCM trigger spectrum ratios for four multiplicity classes of 200 GeV \mbox{$p$-$p$}\ collisions. Ratios for soft events (dotted curves), hard events (dash-dotted curves) and hard components of hard events (dashed curves). Soft/hard event-type crossings occur at 6, 2, 1 and 0.75 GeV/c respectively for the four classes.
}
\end{figure}
Figure~\ref{trigprobs} shows {\em trigger fractions} $R_{s}$, $R_{h}$ and $R_{hh}$ vs trigger rapidity, where $ R_{x} = P_x T_{x} / T$ is the ratio of the corresponding $T_{x}$ term in Eq.~(\ref{trigspec}) to the full trigger spectrum $T(y_{tt},n_{ch})$. The bold dash-dotted curve is $R_h$ and the bold dotted curve is $R_s$. For higher-multiplicity events and above $p_t \approx 1$ GeV/c ($y_t \approx 2.7$) trigger particles represented by $R_{hh}$ (hard components of hard events) arise mainly from the 1D $y_t$\ spectrum hard component associated with jets. Those hadrons are the only proper parton proxies. Although the {\em fraction} of soft triggers for larger $n_{ch}$ still dominates at lower $y_t$\ the absolute number of soft triggers is negligible. Ratios $R_x$ and the corresponding spectra $T_x$ from Eq.~(\ref{trigspec}) play a central role in the definition of the TCM for 2D TA correlations.
\section{Trigger-associated 2D model} \label{tata}
We now combine trigger spectrum elements from Eq.~(\ref{trigspec}) with conditional distributions based on the TCM for 1D $y_t$\ spectra and certain {\em marginal constraints} (described below and in App.~\ref{marg}) to define the 2D TCM $F(y_{ta},y_{tt},n_{ch})$ for TA correlations.
\subsection{TCM TA distribution $\bf F(y_{ta},y_{tt})$}
We assume that for a given trigger the associated particles are sampled from parent distributions similar to those for the 1D SP spectrum TCM subject to additional marginal constraints described below. For multiplicity class $n_{ch}$ the total number of triggers is $N_{evt}(n_{ch})$, the associated-particle number per event is $n_{ch} - 1$, and the total trigger-associated pair number for the given $n_{ch}$ class (excluding self pairs) is $N_{evt}(n_{ch})(n_{ch}-1)$.
We assume that the expressions for soft and hard event types are linearly independent, and event types occur with probabilities $P_x(n_{ch})$ as defined previously. We then have $F(y_{ta},y_{tt}) = P_s F_s(y_{ta},y_{tt}) + P_h F_h(y_{ta},y_{tt})$. According to Bayes' theorem the probability of a TA pair $F_x(y_{ta},y_{tt})$ can be written as the product of trigger probability $T_x(y_{tt},n_{ch})$ and $A_x(y_{ta}:y_{tt},n_{ch})$, the conditional probability of an associated particle at $y_{ta}$ given a trigger at $y_{tt}$.
$A_x(y_{ta}:y_{tt})$ is based on the $F_x(y_t)$ from the 1D SP TCM but is set to zero above $y_{tt}$ (void).
Combining the conditional probabilities with the corresponding trigger-spectrum components we obtain a 2D TCM for unit-normal $F(y_{ta},y_{tt},n_{ch})$
\begin{eqnarray} \label{tadist}
F(y_{ta},y_{tt}, n_{ch})
&=& \frac{1}{N_{evt}(n_{ch})(n_{ch}-1)}\frac{d^2n_{ch}}{y_{tt}dy_{tt}y_{ta}dy_{ta}}~~ \\ \nonumber
&=& P_s (n_{ch}) T_s(y_{tt},n_{ch})A_s(y_{ta}:y_{tt},n_{ch}) \\ \nonumber
&+& P_h (n_{ch}) T_h(y_{tt},n_{ch}) A_h(y_{ta}:y_{tt},n_{ch}),
\end{eqnarray}
where $A_s(y_{ta}:y_{tt},n_{ch}) = S_0''(y_{ta}:y_{tt},n_{ch})$ for soft and $A_h(y_{ta}:y_{tt},n_{ch}) = p_s'(n_{ch}) S_0'(y_{ta}:y_{tt},n_{ch}) + p_h'(n_{ch}) H_0'(y_{ta}:y_{tt},n_{ch})$ for hard event types. Primes indicate that the conditional probabilities may deviate from the corresponding 1D SP spectrum models because of imposed marginal constraints. Modifications to the SP spectrum components are represented by $O(1)$ weight functions $D_x(y_{tt})$ and $E_x(y_{ta})$ as described in App.~\ref{marg}. Note that for associated samples $p_x' = n_x'/(n_{ch}-1)$.
\begin{figure}[h]
\includegraphics[width=3.3in]{ppcms117anew}
\put(-20,105) {\bf (d)}
\put(-20,232) {\bf (b)}
\put(-140,105) {\bf (c)}
\put(-140,232) {\bf (a)}
\caption{\label{spec1a}
Two-component model for trigger-associated correlations from four multiplicity classes of 200 GeV \mbox{$p$-$p$}\ collisions generated by Eq.~(\ref{tadist}).
}
\end{figure}
Figure~\ref{spec1a} shows the predicted TCM $F(y_{ta},y_{tt},n_{ch})$ simulating measured 2D TA correlations for four \mbox{$p$-$p$}\ multiplicity classes. The rapidity interval $y_t \in [1,4.5]$ is divided into 25 histogram bins consistent with previous $y_t \times y_t$ data analysis~\cite{porter2,porter3}. The hard components of measured 2D TA distributions should relate to the jet systematics of Fig.~\ref{qcd}. The soft components of the TCM defined by Eq.~(\ref{tadist}) may be used to isolate TA data hard components for such comparisons. We now consider details of the conditional distributions $A_x(y_{ta}:y_{tt},n_{ch})$.
\subsection{TCM marginal constraints}
The 2D TA distribution $F(y_{ta},y_{tt},n_{ch})$ must integrate over $y_{ta}$ to the unit-normal trigger spectrum $T(y_{tt},n_{ch})$ and over $y_{tt}$ to the unit-normal SP spectrum $F(y_{ta},n_{ch})$ (minus the trigger spectrum if self pairs are excluded). Those marginal constraints modulate the conditional distributions $A_x(y_{ta}:y_{tt})$ in the 2D TA TCM and must be met separately for soft and hard events. We seek the most general 2D model functions that satisfy the constraints.
For event-type $x$ a sum over all events (triggers) should return the 1D parent spectrum $F_x(y_t)$, but we exclude self pairs (triggers) from the 2D TA distribution. Since the trigger self pairs appear along the main diagonal they are described on $y_{ta}$ by the same function $T_x(y_{ta},n_{ch})$. The complement is the unit-normal marginal spectrum of associated particles $A_x(y_{ta})$ defined for given $n_{ch}$ by
\begin{eqnarray}
(n_{ch}-1)A_x(y_{ta}) =n_{ch} F_x(y_{ta}) - T_x(y_{ta}).
\end{eqnarray}
The following must be true for independent sampling from SP parent distribution $F_x(y_t)$ for given $n_{ch}$
\begin{eqnarray} \label{f0yt}
A_x(y_{ta}) &=& \int_{y_{ta}}^\infty dy_{tt} y_{tt} T_x(y_{tt}) A_x(y_{ta}:y_{tt}).
\end{eqnarray}
Each associated-particle conditional probability should be unit normal on $y_{ta}$ for any $y_{tt}$
\begin{eqnarray} \label{f0yt2}
\int_0^{y_{tt}} dy_{ta} y_{ta} A_x(y_{ta}:y_{tt}) &=& 1,
\end{eqnarray}
so that integration of Eq.~(\ref{f0yt}) over $y_{ta}$ returns Eq.~(\ref{g0yt}).
Equations~(\ref{f0yt}) and (\ref{f0yt2}) define the required marginal constraints on the 2D conditional probability distributions.
Conditional distributions $A_x(y_{ta}:y_{tt})$ can be obtained by an iterative process. We refer to histogram $A_x(y_{ta}:y_{tt})$ as having rows indexed by $y_{ta}$ and columns indexed by $y_{tt}$. To form the initial approximation to $A_x(y_{ta}:y_{tt})$ 1D distribution $F_x(y_{ta})$ from the SP TCM is set to zero above $y_{tt}$ for each $y_{tt}$. Equation~(\ref{f0yt2}) is imposed for each $y_{tt}$ to renormalize the corresponding column. Equation~(\ref{f0yt}) is then imposed for each $y_{ta}$ to renormalize the corresponding row, which may change the column normalizations. The resulting alteration of SP model functions $F_x$ is represented by weight functions $D_x$ and $E_x$. Iteration converges rapidly and a single pass provides sufficient accuracy within the kinematic region on $(y_{ta},y_{tt})$ relevant to the TA hard component. Further details are presented in App.~\ref{marg}.
\begin{figure}[h]
\includegraphics[width=1.65in]{ppcms117e}
\includegraphics[width=1.65in]{ppcms117f}
\caption{\label{spec3aa}
Left: Soft-event conditional distribution $A_s(y_{ta}:y_{tt}) = S_0''(y_{ta}:y_{tt})$ showing the effects of weight factor $D_s(y_{tt})$ at smaller $y_{tt}$ as described in App.~\ref{marg}. The z axis is logarithmic.
Right: Hard-event hard component conditional distribution $H_0'(y_{ta}:y_{tt},n_{ch})$ showing the effects of weight function $D_h(y_{tt})$ at smaller $y_{tt}$. The z axis is linear.
}
\end{figure}
Figure~\ref{spec3aa} illustrates the effects of marginal constraints on $S_0''(y_{ta}:y_{tt})$ (left) and $H_0'(y_{ta}:y_{tt})$ (right). The initial $A_x(y_{ta}:y_{tt})$ forms follow the SP parent spectra $F_x(y_{t})$ on $y_{ta}$ and are uniform on $y_{tt}$ subject to the condition $y_{ta} < y_{tt}$. The constraint of Eq.~(\ref{f0yt2}) represented by $D_x(y_{tt})$ causes elevated amplitudes at smaller $y_{tt}$ due to the reduced acceptance on $y_{ta}$. The constraint of Eq.~(\ref{f0yt}) represented by $E_x(y_{ta})$ might alter amplitudes at larger $y_{ta}$ relative to the SP parents. Any renormalization on $y_{ta}$ could then disturb the normalization on $y_{tt}$ requiring multiple iterations for convergence, but we observe that such effects are negligible. Systematic uncertainties for the renormalizations are discussed in Sec.~\ref{syserr}.
\subsection{TCM conditional distribution $\bf A(y_{ta}:y_{tt})$}
The full TA distribution can be factorized according to Bayes' theorem to define a complementary conditional distribution.
We divide $F(y_{ta},y_{tt},n_{nch})$ from Eq.~(\ref{tadist}) by trigger spectrum $ T(y_{tt},n_{ch})$ from Eq.~(\ref{trigspec}) to define a TCM for measured $A(y_{ta}:y_{tt},n_{ch})$. The result is an ensemble of unit-normal conditional spectra
\begin{eqnarray} \label{ftrat}
A(y_{ta}:y_{tt},n_{ch}) &=& \frac{1}{(n_{ch}-1)} \frac{dn_{ch}(y_{ta}:y_{tt},n_{ch})}{y_{ta}dy_{ta}} \\ \nonumber
&=& R_{s}(y_{tt},n_{ch}) A_s(y_{ta}:y_{tt},n_{ch}) \\ \nonumber
&+& R_{h}(y_{tt},n_{ch}) A_h(y_{ta}:y_{tt},n_{ch}).
\end{eqnarray}
\begin{figure}[h]
\includegraphics[width=3.3in]{ppcms117b}
\put(-85,110) {\bf (d)}
\put(-85,237) {\bf (b)}
\put(-205,110) {\bf (c)}
\put(-205,237) {\bf (a)}
\caption{\label{spec3a}
Two-component model for trigger-associated conditional spectra from four multiplicity classes of 200 GeV \mbox{$p$-$p$}\ collisions generated by Eq.~(\ref{ftrat}). The jet-related hard-component contribution at upper right increases with $n_{ch}$. The z axis is logarithmic.
}
\end{figure}
Figure~\ref{spec3a} shows TA histograms from Fig.~\ref{spec1a} divided by trigger spectra from Eq.~(\ref{trigspec}) to obtain TCM TA histograms $A(y_{ta}:y_{tt},n_{ch})$. As noted, the increase in amplitude with decreasing $y_{tt}$ for smaller $y_{tt}$ occurs because each conditional distribution must be unit normal on $y_{ta}$ but extends over a decreasing interval $y_{ta} \in[1,y_{tt}]$.
\section{Isolating the TA hard component} \label{isolate}
Conditional ratio $A(y_{ta}:y_{tt})$ obtained directly from measured TA histograms $F$ divided by measured trigger spectra $T$ is model independent and should provide the least-biased route to sought-after jet-related structure. Isolation of the hard component requires subtraction of a soft-component model (part of the TA TCM).
\subsection{Hard components of $\bf A(y_{ta}:y_{tt},n_{ch})$}
The 2D TA histograms in the previous section can be constructed from measured particle data. We wish to isolate a measured TA hard component $H_h(y_{ta}:y_{tt},n_{ch})$ by subtracting a 2D soft-component reference according to the method described in Ref.~\cite{ppprd}. Solving Eq.~(\ref{ftrat}) for the hard component of {\em hard events} we obtain
\begin{eqnarray} \label{hardcomp}
H_h'(y_{ta}:y_{tt})/(n_{ch}-1) &=& [A - R_s S''_0]/R_h - p'_s S'_0~~
\end{eqnarray}
where for real data $A_s(y_{ta}:y_{tt}) \equiv S_0''(y_{ta}:y_{tt})$ and $ A_h(y_{ta}:y_{tt}) \equiv p'_s S_0' + H_h'(y_{ta}:y_{tt})/(n_{ch}-1)$. The primes on the $F_x$ recall the effects of marginal constraints on the TCM (App.~\ref{marg}). Subscript $h$ denotes a component from hard events as in Eq.~(\ref{spevtype}). For this model exercise we expect the factorization result $H_h'(y_{ta}:y_{tt},n_{ch}) \rightarrow n_{h}' H'_0(y_{ta}:y_{tt})$.
\begin{figure}[h]
\includegraphics[width=3.3in]{ppcms117c}
\put(-85,110) {\bf (d)}
\put(-85,237) {\bf (b)}
\put(-205,110) {\bf (c)}
\put(-205,237) {\bf (a)}
\caption{\label{hardcol1}
Hard components of trigger-associated conditional spectra from four multiplicity classes of 200 GeV \mbox{$p$-$p$}\ collisions in the form $H_h'(y_{ta}:y_{tt})/n_{h}'$ derived according to Eq.~(\ref{hardcomp}). The procedure returns the hard component $H'_0(y_{ta}:y_{tt})$ of the TCM approximately independent of $n_{ch}$ as expected. The z axis is linear.
}
\end{figure}
Figure~\ref{hardcol1} confirms that model hard component $H_0'(y_{ta}:y_{tt})$ is returned by soft-component subtraction applied to the simulated data in Fig.~\ref{spec3a} for each $n_{ch}$ class (indicating minor variation of weight function $D_h$ with $n_{ch}$ as in Fig.~\ref{specc4}, upper right). The TA TCM assumes factorization for the hard component. For real data we expect to encounter the primary goal of 2D TA analysis: nontrivial jet-related correlation structure in $H_h(y_{ta}:y_{tt},n_{ch})$ reflecting low-energy parton fragmentation systematics.
We can compare a limiting case of the 2D TA system with the 1D SP spectrum analysis. For larger $y_{tt}$ and $n_{ch}$ we expect $R_s \ll R_h \approx 1$ (see Fig.~\ref{trigprobs}) and
\begin{eqnarray}
H_h'(y_{ta}:y_{tt}) \hspace{-.05in} &\approx&\hspace{-.05in} (n_{ch}-1) A(y_{ta}:y_{tt}) \hspace{-.01in} - \hspace{-.01in} n_s' S_0'(y_{ta}:y_{tt}),~~~~~
\end{eqnarray}
equivalent to the SP spectrum problem described in Sec.~\ref{ppspecc}, with $P_s \ll P_h \approx 1$.
For ensemble-averaged SP spectra we observe $H_h(y_t,n_{ch}) \approx n_h'(n_{ch}) H_0(y_t)$ with $n_h'(n_{ch}) \approx \bar n_{ch,j}$ -- factorization of $H_h(y_t,n_{ch})$. For measured TA data, jet-related $H_h(y_{ta}:y_{tt},n_{ch})$ ($H_h'$ corrected for constraint distortions) may not factorize. The 2D structure should correspond to Fig.~\ref{qcd} and may reveal further details of minimum-bias jets in p-p collisions.
\subsection{Hard components of $\bf F(y_{ta},y_{tt},n_{ch})$}
\begin{figure}[h]
\includegraphics[width=3.3in]{ppcms117d}
\put(-85,110) {\bf (d)}
\put(-85,237) {\bf (b)}
\put(-205,110) {\bf (c)}
\put(-205,237) {\bf (a)}
\caption{\label{hardcol2a}
Hard components of minimum-bias trigger-associated correlations from Fig.~\ref{spec1a} for four multiplicity classes of 200 GeV \mbox{$p$-$p$}\ collisions. These distributions were reconstructed by combining hard components $H'(y_{ta}:y_{tt})$ from Fig.~\ref{hardcol1} with model trigger-spectrum hard components $T_h(y_{tt})$ from Eq.~(\ref{trigspec}). The z axis is logarithmic.
}
\end{figure}
Figure~\ref{hardcol2a} shows the product $T_h(y_{tt},n_{ch}) H_h'(y_{ta}:y_{tt},n_{ch})/n_h'$ representing the form of the (factorized) hard component of TA distributions $F(y_{ta},y_{tt},n_{ch})$ in Fig.~\ref{spec1a}. The distribution mode on $y_{tt}$ moves to larger $y_{tt}$ with increasing $n_{ch}$. For large multiplicities the mode approaches 3 GeV ($y_{tt} \approx 3.75$), the observed most-probable parton energy. The mode for lower multiplicities is closer to $y_t = 2.7$ observed for minimum-bias symmetrized $y_t \times y_t$ data from 200 GeV \mbox{$p$-$p$}\ collisions~\cite{porter2,porter3}. In this simulation exercise the distributions extend below $y_{ta} = 2$. We expect systematic uncertainties in inferred hard components for data to be large below that point.
\section{Systematic uncertainties} \label{syserr}
The object of proposed trigger-associated correlation analysis is extraction of a jet-related hard component that can be compared with measured in-vacuum jet properties and a predicted pQCD parton spectrum. The accuracy of the inferred TA hard component depends on possible limitations of the analyzed data and the validity of both the overall TA two-component description and of the defined soft-component model function.
\subsection{Validity of the trigger-associated TCM}
The predicted TA TCM is based on the measured TCM for single-particle spectra ($S_0$ and $H_0$ defined in App.~\ref{tcmmodelfunc}) which has been verified in comparisons with spectrum data at the few-percent level~\cite{ppprd}. The structure of the TA model is $F = P_s T_s S''_0 + P_h T_h (p_s' S'_0 + p_h' H'_0)$. The $n_{ch}$ trends from SP spectra determine soft and hard particle probabilities $p_x'$ with few-percent accuracy. Modifications to the SP model components denoted by primes are discussed in App.~\ref{marg} and should not contribute significantly to uncertainties in the inferred TA hard component.
The probabilities $P_x$ of soft and hard events depend on the inference of jet frequency $f$ from spectrum data, which in turn depends on interpretation of the SP spectrum hard component as a jet fragment distribution. That interpretation has been confirmed quantitatively in Ref.~\cite{fragevo}.
For a given event type the corresponding TA distribution $F_x(y_{ta},y_{tt})$ is factorized according to Bayes' theorem to a product of trigger spectrum and associated-particle conditional spectra. Trigger spectra are defined by product $T_x = G_x n_{ch} F_x$ reflecting the probability that a given sample (the trigger) occurs at $y_{tt}$ and that no other sample occurs above that point (the void). The predicted $T(y_{tt},n_{ch})$ can be compared directly with measured trigger spectra as a critical test of the compound-probability analysis that is the basis for the TA TCM.
\subsection{TA TCM model elements}
The TA hard component must be isolated from TA correlation data by subtracting a 2D soft-component model with contributions from both soft and hard event types. Elements of the 2D soft component are expressed as $T_s S_0''$ (soft events) and $T_h p_s' S_0'$ (hard events). We assume that each element is factorizable (no correlations between trigger and associated particles, both from soft processes).
The TA hard-component model is introduced in this study only as a place holder to verify the subtraction procedure -- that the hard component included in the TCM emerges from subtraction with only expected small distortions due to marginal constraints. The TA hard-component model is factorized and thus cannot represent the complex fragmentation conditional systematics that appear in Fig.~\ref{qcd} and are expected for TA real data. Jet-related correlations are the main objective of TA analysis.
\subsection{Common-mode uncertainty reduction}
By inferring the TA hard component from ratio $A = F / T$ certain common-mode reductions in systematic bias may be achieved. Conditional associated-particle spectra have the form $A = R_s S_0'' + R_h (p_s' S_0' + H'/n_{ch})$, where $F$, $T$ and $A$ are obtained directly from TA data. We expect the hard-component structure to be slowly varying with $n_{ch}$ (the jet formation process is approximately independent of \mbox{$p$-$p$}\ multiplicity). For larger $n_{ch}$ and for $y_{tt} > 2.5$ ($p_t > 0.8$ GeV/c) $R_s \ll R_h \approx 1$. Uncertainty in the inferred hard component $H'$ then arises only from $p_s'$, which is accurately known, from the forms of $S_0''$ and $S_0'$ which are determined from measured SP spectrum structure and from marginal constraints reviewed in App.~\ref{marg}.
\subsection{Data hard component inferred by subtraction}
Based on results for SP spectrum data as in Fig.~\ref{ppspectra} we expect the jet-related hard component to be resolvable (systematic uncertainties $< 10$\%) only for $y_t > 2$~\cite{ppprd}. Uncertainties contributed by the soft-component model fall sharply above that point since the model is rapidly decreasing while the hard component is increasing toward a maximum near $y_t = 2.7$ ($p_t = 1$ GeV/c). The relevant kinematic region for the 2D TA hard component is then defined by both $y_{ta}$ and $y_{tt} > 2$ ($p_t > 0.5$ GeV/c). We then summarize uncertainties relevant to that region.
The soft-component model is assumed to be factorized (no correlations). Soft-component correlations are observed in \mbox{$p$-$p$}\ data, but only below $y_t = 2$~\cite{porter2}, and so should not affect isolation of the TA hard component within the relevant kinematic region.
The soft-component model is distorted by marginal constraints (mainly the projection onto $y_{tt}$ represented by $D_x$). However, the effect is significant only below $y_{tt} = 2.5$ (Fig.~\ref{specc4}, upper panels) and can be corrected to a few percent. Similarly, distortions of the soft component on $y_{ta}$ should be less than 20\% (Fig.~\ref{specc4}, lower panels) for any $y_{ta}$. We also note from the same figure that any corrections to the inferred hard component [$H_h'(y_{ta}:y_{tt}) \rightarrow H_h(y_{ta}:Y_{tt})$] in response to marginal-constraint distortions would be $< 15$\%, with few-percent accuracy of corrected data.
The $n_{ch}$ dependence of corrected $H_h(y_{ta}:y_{tt},n_{ch})$ could provide an important check on the overall method. As noted, the hard component may represent universal parton fragmentation to low-energy jets which should be approximately independent of the \mbox{$p$-$p$}\ underlying event.
\section{Discussion} \label{disc}
In this study we have derived a two-component model of trigger-associated correlations for \mbox{$p$-$p$}\ collisions. Subtraction of the model soft components from TA data should reveal a 2D hard component that may establish the kinematic limits of jet production in \mbox{$p$-$p$}\ (and possibly \mbox{A-A}) collisions. We discuss the relation between \mbox{$p$-$p$}\ TA correlations and other aspects of nuclear collisions.
\subsection{Relation to previous MB correlation analysis}
Trigger-associated correlation analysis is an extension of previous measurements of minimum-bias angular and $y_t \times y_t$ correlations~\cite{porter2,porter3,anomalous} to develop a more complete description of minimum-bias jets in nuclear collisions. From correlation studies of \mbox{$p$-$p$}\ collisions we observe that the soft component of angular correlations is restricted to $y_t < 2$ ($p_t < 0.5$ GeV/c) and appears only for unlike-sign (US) charge pairs. Similarly, the jet-related same-side correlation peak is dominated by US pairs if $y_t < 4$ ($p_t < 4$ GeV/c). Both trends are consistent with local charge conservation expected during fragmentation of low-energy partons (gluons) to hadrons at the end of any fragmentation cascade. This study establishes the basic algebraic relations. TA analysis of real data may address separately the various charge combination LS, US, CI and CD (Sec.~\ref{symmcorr}) to confirm correspondence of the TA hard component with parton fragmentation. The dependence of TA structure on azimuth relative to the trigger direction can also be studied to confirm a jet interpretation of certain MB angular correlation structure.
\subsection{pQCD and parton fragmentation functions}
We have asserted that the TA hard component may be compared in some sense to Fig.~\ref{qcd}. In Fig.~\ref{qcd} (left) the normalization for conditional distribution $D(y:y_{max})$ at each $y_{max}$ is the corresponding dijet multiplicity $2n_{ch,j}(y_{max})$~\cite{eeprd}, the ``associated particle'' (fragment) multiplicity emerging from a pair of (trigger) partons. TA associated-particle spectra $A_h(y_{ta}:y_{tt},n_{ch})$ from hard events include soft and hard components. The normalization is the same associated-particle multiplicity $n_{ch}-1$ for all $y_{tt}$ conditions, but the fraction of $n_{ch} - 1$ from the soft component of hard events is unrelated to the dijet. For larger $n_{ch}$ soft component $n_s$ should dominate $n_{ch}$ and the $n_{ch} - 1$ constraint on the total associated-particle spectra may have only a small effect on (bias) hard component $H_h(y_{ta}:y_{tt},n_{ch})$. Thus, $D(y:y_{max})$ and $H_h(y_{ta}:y_{tt},n_{ch})$ may be directly comparable.
In Fig.~\ref{qcd} (right) the FF ensemble $D(y:y_{max})$ is combined with a calculated parton spectrum. The corresponding result for TA correlations is shown in Fig.~\ref{hardcol2a}. It may be possible to relate the hard component $T_{hh}$ of the hadron trigger spectrum to the pQCD parton spectrum and FFs using the compound-probability methods employed in this study to predict the trigger-hadron spectrum, similar to the analysis of Ref.~\cite{fragevo}
\subsection{Conventional spectrum and correlation analysis}
Nominally jet-related spectrum structure and angular correlations have been studied extensively in \mbox{A-A}\ collisions at RHIC in the search for jet modifications in a dense QCD medium. Conjectured modifications include high-$p_t$\ jet suppression (spectra)~\cite{starraa} and low-energy jet (minijet) thermalization in an opaque medium (ZYAM analysis of azimuth correlations)~\cite{starzyam}. Conventional analysis invokes restrictive trigger-associated $p_t$\ cuts based on questionable assumptions about jet structure, including the assumption that only ``high-$p_t$'' hadrons can be associated with jets.
By establishing the actual kinematic boundaries for jet fragment production the proposed TA analysis may counter some assumptions that support conventional data analysis. Whereas it is commonly assumed that hadrons below $p_t = 2$ GeV/c emerge from a thermalized bulk medium the TA hard component may confirm that a substantial fraction of hadrons in that $p_t$\ interval are part of a significant jet-correlated contribution. Imposing restrictive $p_t$\ cuts in ZYAM analysis then may {\em exclude most jet fragments} from nominal jet analysis~\cite{tzyam}. Interpreting spectrum structure in that interval as determined by bulk medium properties (e.g., radial flow) may erroneously assign a flow interpretation to jet structure.
\subsection{p-p underlying-event studies}
In Refs.~\cite{cdfue,cmsue} measurements of $N_\perp$ (hadron yield within the azimuth {\em transverse region} or TR centered at $\pi/2$ relative to the trigger) vs trigger condition $p_{t,trig}$ and $N_\perp(p_t)$ spectra are employed to characterize the UE. The TR is expected to exclude contributions from the triggered dijet. Extrapolation of the $N_\perp$ spectrum to $p_t = 0$ is interpreted to indicate an excess yield within the TR relative to that expected for a beam-beam contribution (projectile dissociation or soft component). Substantial increases of $N_\perp$ with higher $p_{t,trig}$ values relative to the minimum-bias or non-single diffractive (NSD) multiplicity are also interpreted to reveal novel contributions to the UE, including {\em multiple parton interactions} (MPI).
Measurements of MB dijet properties~\cite{porter2,porter3} indicate that jet-related SS and AS peaks strongly overlap on azimuth, contradicting UE assumptions about exclusion of the triggered dijet from the TR~\cite{pptheory} and assumptions common to ZYAM analysis about no significant SS and AS peak overlap~\cite{tzyam}. TA analysis applied to limited intervals on azimuth relative to the trigger (``toward'' and ``away'' regions as well as the TR defined for UE analysis) may confirm a substantial triggered dijet contribution to the TR and the momentum structure of that contribution.
\subsection{QCD Monte Carlos for p-p collisions}
Monte Carlo (MC) simulations commonly employed to describe \mbox{$p$-$p$}\ conditions (e.g., \textsc{pythia} and \textsc{herwig}) invoke critical physical assumptions, including a fixed projectile-proton parton distribution function, a scattered-parton spectrum with assumed lower bound, a parton fragmentation model and the eikonal model for collisions of composite projectiles. One outcome of such MCs is the prediction of MPI as a substantial contribution to the UE (within the TR). Such predictions may be questioned.
The effective dijet total cross section implied by the parton spectrum lower bound assumed for a \mbox{$p$-$p$}\ Monte Carlo may exceed measured MB dijet production in \mbox{$p$-$p$}\ collisions by a factor 10 or more~\cite{liwang}. But jet-related correlations predicted by the same Monte Carlo may fall well below those actually observed in \mbox{$p$-$p$}\ and peripheral \mbox{A-A}\ collisions~\cite{anomalous}, casting doubt on the MC hadronization model. MPI contributions attributed to $N_\perp$ data from some UE analysis may actually be part of the triggered dijet that should be expected based on measured MB jet properties. Improved understanding of \mbox{$p$-$p$}\ collisions may result from detailed comparisons of MCs with dijet rates inferred directly from $p_t$\ spectra, with measured MB angular and $y_t \times y_t$ correlations and with results from TA analysis as proposed in the present study.
\section{Summary} \label{summ}
We have derived a two-component (soft+hard) model (TCM) for 2D trigger-associated (TA) correlations from 200 GeV \mbox{$p$-$p$}\ collisions. The model is based on a TCM for 1D single-particle (SP) $y_t$\ spectra. The elements of the 1D spectrum model are combined as compound probabilities to construct the 2D TA model.
TA correlations are constructed as averages of pair distributions from single events where the highest $p_t$\ or $y_t$\ (transverse rapidity) particle (trigger) in each event is combined with all other particles (associates) to form TA pairs. By subtracting the soft component of the TA TCM from TA correlation data we extract the TA hard component, which should be dominated by jet-related structure.
The projection of the TA TCM onto trigger rapidity $y_{t,trig}$ is the trigger spectrum $T({y_{tt}})$. A trigger spectrum can also be formed directly from data and compared with the TCM prediction. Projection of the TA TCM onto associated rapidity $y_{t,assoc}$ should return the SP associated spectrum $A(y_{ta})$ (SP spectrum without trigger particles, which are excluded as self pairs).
A trigger hadron from a \mbox{$p$-$p$}\ collision acts (with some probability) as proxy for the leading parton of a jet. Trigger-associated hadron correlations then include jet correlations that emulate parton-fragment correlations. TA conditional hard component $H_h(y_{ta}:y_{tt})$ may be compared directly with measured fragmentation functions $D(y:y_{max})$, where $y$ is the hadron fragment rapidity and $y_{max}$ is the parton rapidity. Such comparisons should establish kinematic limits on parton energy and fragment momentum for jet production in \mbox{$p$-$p$}\ collisions.
TA correlations can also be constructed for restricted azimuth intervals relative to the trigger momentum. Of special interest is the transverse region (TR), an azimuth interval bracketing $\pi/2$ relative to the trigger direction. Underlying event (UE) analysis assumes that a triggered dijet is confined within jet cones at 0 and $\pi$ and should not contribute to the TR. The TA hard component extracted from the TR may challenge that assumption.
The TA TCM established in this study can be applied to both \mbox{$p$-$p$}\ and \mbox{A-A}\ data. The TA hard component may provide new insights into jet production from nuclear collisions, especially modified jet formation in more-central \mbox{A-A}\ collisions. Application to Monte Carlo data may test basic assumptions invoked by QCD models, including the structure of the scattered-parton spectrum and the frequency of multiple parton interactions (MPI).
This work was supported in part by the Office of Science of the U.S.\ DOE under grant DE-FG03-97ER41020.
\begin{appendix}
\section{Marginal Constraints} \label{marg}
According to the two-component model of high-energy nuclear collisions hadron production arises mainly from projectile-nucleon dissociation (soft) or large-angle-scattered parton fragmentation (hard). The TCM for SP spectra assumes that all final-state hadrons are sampled from single-particle model spectra for soft or hard events, and the latter from soft or hard spectrum components. The TCM 2D TA distribution must project to trigger and associated 1D marginal spectra that are consistent with SP spectrum structure. Model construction is based on $F_x(y_t,n_{ch})$, the unit-normal SP spectrum model for event type $x$ corresponding to spectrum data from multiplicity class $n_{ch}$. The goal is TCM reference $F_x(y_{ta},y_{tt},n_{ch})$, the most general model for trigger-associated correlations consistent with marginal constraints and a factorization assumption representing minimal TA correlations.
\subsection{Marginal spectra}
$T_x(y_{tt},n_{ch})$ is the per-event marginal trigger-particle spectrum integrating to one trigger particle per event. The total charged-particle number within the angular acceptance is $n_{ch} = \Delta \eta\, dn_{ch}/d\eta$. The trigger spectrum is derived from the unit-normal SP spectrum $F_x(y_t,n_{ch})$ by $T_x(y_{tt},n_{ch}) = G_x(y_{tt},n_{ch})\, n_{ch} F_x(y_{tt},n_{ch})$, where $G_x(y_{tt},n_{ch})$ is the void probability that no samples appear above $y_{tt}$ [defined by Eqs.~(\ref{nsum}) and (\ref{g0yt})].
Trigger particles may appear on the main diagonal on $(y_{ta},y_{tt})$ as a self-pair contribution to TA correlations. The trigger spectrum then has the same form in projections onto $y_{ta}$ and $y_{tt}$. Self pairs are excluded from the present analysis.
The per-event marginal associated-particle spectrum is denoted by $A_x(y_{ta})$. The SP spectrum for each $n_{ch}$ class is the sum of trigger and associated spectra
\begin{eqnarray}
n_{ch} F_x(y_{ta}) &=& T_x(y_{ta}) + (n_{ch} - 1)A_x(y_{ta}),
\end{eqnarray}
leading to an expression for associated-particle spectra
\begin{eqnarray} \label{axdef}
(n_{ch}-1) A_x(y_{ta}) &=& [1 - G_x(y_{ta})]\, n_{ch} F_x(y_{ta}).
\end{eqnarray}
\subsection{TCM marginal constraints}
2D TA correlations can be factorized according to Bayes' theorem as $F_x(y_{ta},y_{tt}) = T_x(y_{tt})\, A_x(y_{ta}:y_{tt})$. We seek the most general form for conditional spectra $ A_x(y_{ta}:y_{tt})$ consistent with imposed constraints. The definition $A_x(y_{ta}:y_{tt}) = D_x(y_{tt}) E_x(y_{ta})F_x(y_{ta})$ (for $y_{ta} < y_{tt}$) assumes SP distributions $F_x(y_{ta})$ as the initial case. $O(1)$ weight functions $D_x(y_{tt})$ and $E_x(y_{ta})$ (with initial values 1) represent the effect of marginal constraints as described below.
The first constraint is defined by projecting $F_x(y_{ta},y_{tt})$ onto $y_{tt}$ to obtain $T_x(y_{tt})$. Assuming initial values $E_x = 1$ and canceling common factor $T_x(y_{tt})$ gives
\begin{eqnarray} \label{dx}
D_x(y_{tt}) \int_0^{y_{tt}} dy_{ta} y_{ta} F_x(y_{ta}) &=& 1,
\end{eqnarray}
which defines weight function $D_x(y_{tt}) \geq 1$. That constraint is equivalent to requiring that any event in multiplicity class $n_{ch}$ contains $n_{ch}-1$ associated particles.
The second constraint is defined by projecting $F_x(y_{ta},y_{tt})$ onto $y_{ta}$ to obtain marginal spectrum $A_x(y_{ta})$, effectively the trigger-weighted average of conditional associated spectra. The uncorrected projection onto $y_{ta}$ is
\begin{eqnarray} \label{margint}
I_x(y_{ta})
&=& F_x(y_{ta}) \int_{y_{ta}}^{\infty} dy_{tt} y_{tt} D_x(y_{tt}) T_x(y_{tt}).
\end{eqnarray}
Combining Eqs.~(\ref{axdef}) and (\ref{margint}) in the form $A_x = E_x I_x$ given $D_x(y_{tt})$ from Eq.~(\ref{dx}) then determines weight functions $E_x(y_{ta}) \approx 1$. The combined weight functions define the primed associated-particle spectra referred to in the text as $A_s = S_0'' = D_s E_s S_0$ and $A_h = p_s' S_0' + p_h' H_0' = D_h E_h (p_s' S_0 + p_h' H_0)$ (for $y_{ta} < y_{tt}$).
The marginal constraints simplify for certain limiting cases. $D_x(y_{tt})$ is $\gg 1$ for $y_{tt}$ small and $\approx 1$ for $y_{tt}$ large.
$G_x(y_{t}) \rightarrow 1$ for $y_{t}$ large and $1 - G_x(y_{t}) \rightarrow 1$ for $y_{t}$ small.
For $y_{ta}$ large (and therefore $y_{tt}$ large) Eq.~(\ref{margint}) becomes
\begin{eqnarray} \label{runint}
I_x(y_{ta})
&\approx & F_x(y_{ta}) D_{x0} \int_{y_{ta}}^{\infty} dy_{tt} y_{tt} n_{ch} F_x(y_{tt}) \\ \nonumber
&\approx & -\ln[G_x(y_{ta})] D_{x0} F_x(y_{ta})
\end{eqnarray}
where $D_{x0} \approx 1$ is the limiting value of $D_x$ for large $y_{tt}$. The first line follows from $G_x \approx 1$, and the second line follows from $n_{x\Sigma} = -\ln(G_x) \approx 1 - G_x$ defined by Eq.~(\ref{nsum}). That result is consistent with $A_x(y_{ta})$ from Eq.~(\ref{axdef}) for $y_{ta}$ large.
For $y_{ta}$ small Eq.~(\ref{margint}) becomes
\begin{eqnarray}
I_x(y_{ta}) &\approx& F_x(y_{ta}) \int_{y_{ta}}^{\infty} dy_{tt} y_{tt}D_x(y_{tt}) T_x(y_{tt}) \\ \nonumber
&\approx& \langle D_x(y_{ta})\rangle F_x(y_{ta})
\end{eqnarray}
also consistent with Eq.~(\ref{axdef}) for $y_{ta}$ small.
For the following we use simulation data to compare marginal associated-particle spectra $A_x(y_{ta})$ as defined by Eq.~(\ref{axdef}) with projection integrals $I_x(y_{ta})$ defined by Eq.~(\ref{margint}), separately for soft and hard events and for four $n_{ch}$ classes.
\subsection{Simulation results}
\begin{figure}[h]
\includegraphics[width=3.3in]{ppcms119a}
\put(-20,105) {\bf (d)}
\put(-20,232) {\bf (b)}
\put(-140,105) {\bf (c)}
\put(-140,232) {\bf (a)}
\caption{\label{specc3}
a), (b): Predicted marginal associated-particle spectra $A_x(y_{ta},n_{ch})$ (dashed curves) as defined by Eq.~(\ref{axdef}). Limiting cases are represented by $-\ln[G_x(y_{ta},n_{ch})] F_x(y_{ta},n_{ch})$ (thin solid curves) for larger $y_{ta}$ and SP model functions $F_x(y_{ta},n_{ch})$ (dotted curves) for smaller $y_{ta}$.
(c), (d): Running integrals $I_x(y_{ta},n_{ch})$ (dashed curves) as defined by Eq.~(\ref{margint}). The limiting cases (thin solid and dotted curves) are the same as for the upper panels.
}
\end{figure}
Figure~\ref{specc3} (a), (b) shows marginal associated-particle spectra $A_x(y_{ta},n_{ch})$
(dashed curves) defined by Eq.~(\ref{axdef}) compared to limiting cases $F_x(y_{ta})$ (for smaller $y_{ta}$, dotted curves) and $-\ln[G_x(y_{ta})]F_x(y_{ta})$ (for larger $y_{ta}$, thin solid curves).
Figure~\ref{specc3} (c), (d) shows running integrals $I_x(y_{ta})$ for four $n_{ch}$ classes (dashed curves) defined by Eq.~(\ref{margint}) compared to the same limiting cases. The lowest dashed curves are for the lowest $n_{ch}$ class.
The $F_s$ for all soft events are the same (by construction), whereas the $F_h$ for hard events depend on $n_{ch}$. Because all hard events include at least one jet those events with the smallest $n_{ch}$ have the hardest spectra (uppermost dotted curves in right panels). It is the {\em fraction of hard events} in any $n_{ch}$ class that increases with $n_{ch}$, thus making the observed ensemble-averaged spectra harder.
\begin{figure}[h]
\includegraphics[width=3.3in]{ppcms119b}
\put(-22,105) {\bf (d)}
\put(-22,232) {\bf (b)}
\put(-142,105) {\bf (c)}
\put(-142,232) {\bf (a)}
\caption{\label{specc4}
(a), (b): Weight functions $D_x(y_{tt})$ defined by Eq.~(\ref{dx}). The separate curve in the right panel corresponds to the lowest $n_{ch}$ bin.
(c), (d): Weight functions $E_x(y_{tt})$ defined by the vertical axis label and Eqs.~(\ref{axdef}) and (\ref{margint}).
}
\end{figure}
Figure~\ref{specc4} (a), (b) shows $D_x(y_{tt})$ inferred from Eq.~(\ref{dx}). The limiting values for larger $y_{tt}$ denoted by $D_{x0}$ reflect the effects of integration over incomplete $y_t$ acceptance and possible detector inefficiency (for real data). The data integration on $y_{tt}$ or $y_{ta}$ extends over $[1,4.5]$, not [$0,\infty$] and thus excludes a significant fraction $O(10\%)$ of the SP spectrum soft component.
Figure~\ref{specc4} (c), (d) shows $E_x(y_{ta})$ inferred from the combination of Eqs.~(\ref{axdef}) and~(\ref{margint}). The deviations from unity are small. The exceptional curves in the right panels correspond to the smallest $n_{ch}$ class and therefore the hardest spectrum.
Because both $E_x$ factors are close to unity for smaller $y_{ta}$ they would not significantly change the $D_x$ inferred from Eq.~(\ref{dx}). A single normalization iteration is then sufficient.
Within the interval $y_{ta} < 2.5$ where the soft-component model is relevant for isolation of a data hard component $E_s $ remains within 10\% of unity.
The hard-component weight functions $E_h$ for all but the lowest $n_{ch}$ class do not deviate more than 15\% from unity, suggesting that corrections $H' \rightarrow H$ for marginal distortions may not be required. These results confirm the accuracy of numerical integration over several decades and reveal that the TA TCM is self-consistent to about 10\%.
\section{TCM model functions} \label{tcmmodelfunc}
We summarize the TCM single-particle spectrum model functions that provide the basis for this 2D trigger-associated analysis. The soft-component model is a limiting case of measured spectra. The hard-component model is quantitatively consistent with a pQCD parton spectrum folded with measured jet fragmentation functions to describe a parton fragment distribution~\cite{fragevo}.
\subsection{SP spectrum soft and hard model functions}
The unit-integral functions for the two-component model (TCM) of $m_t$ or $y_t$ spectra used in this analysis are defined in Refs.~\cite{ppprd,hardspec}. For 200 GeV \mbox{$p$-$p$}\ collisions the soft-component model (L\'evy distribution on $m_t$) is
\begin{eqnarray}
S_0(y_t) = \frac{20.6}{[1 + (m_t - m_\pi)/{nT}]^n}
\end{eqnarray}
with $m_t = m_\pi \cosh(y_t)$, $n = 12.8$ and $T = 0.145$ GeV. The Gaussian form of the hard-component model is
\begin{eqnarray}
H_0(y_t) = 0.33 \exp\{-(y_t - y_{t0})^2/2\sigma_{y_t}^2\}
\end{eqnarray}
on $y_t$, with $y_{t0} = 2.67$ ($p_t \approx 1$ GeV/c) and $\sigma_{y_t} = 0.445$. The coefficients (determined by the unit-integral condition) depend on the specific model parameters.
\subsection{Constructing a power-law tail} \label{power}
In Ref.~\cite{hardspec} hard-component model $H_0(y_t)$ is generalized to a Gaussian with added power-law tail to accommodate the underlying parton energy spectrum. The power-law trend $\exp(-q\, y_t)$ appears as a straight line when $\ln(H_0)$ is plotted vs $\ln p_t$ or vs $y_t \sim \ln(p_t)$. Parameter $q$ is the power-law exponent on $y_t$ (different from that on $p_t$). $H_0(y_t)$ is constructed as a Gaussian function with power-law tail as follows. For given Gaussian parameters the Gaussian trend transitions to the power-law trend (slopes equal) at $y_t- y_{t0} = q \sigma_{y_t}^2 $ where the exponent of $H_0(y_t)$ is $q^2 \sigma_{y_t}^2/2$. The exponent function below that point is $(y_t - y_{t0})^2/2\sigma_{y_t}^2$ and above that point is $q(y_t - y_{t0}) - q^2 \sigma_{y_t}^2/2$. The required function $H_0(y_t)$ is obtained by exponentiating those functions within the specified $y_t$ intervals. For the TCM parameters used in this analysis (including $q \approx 5.5$) the transition to power-law tail occurs near $y_t = 3.75$ (see Fig.~\ref{ppspectra}, right).
In Ref.~\cite{fragevo} a pQCD-calculated minimum-bias fragment distribution derived from measured jet fragmentation functions corresponds well with the Gaussian+tail model of the spectrum hard component except below 0.5 GeV/c ($y_t = 2$) where the Gaussian model drops below the pQCD calculation. Within that same low-$p_t$ interval systematic uncertainties in the inferred spectrum hard component and pQCD prediction are relatively large.
\end{appendix}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,713
|
#ifndef MIDIClientProxy_h
#define MIDIClientProxy_h
#include "modules/webmidi/MIDIClient.h"
#include "platform/heap/Handle.h"
namespace blink {
class MIDIAccessInitializer;
}
namespace blink {
class WebMIDIClient;
class MIDIClientProxy FINAL : public blink::MIDIClient {
public:
static PassOwnPtr<MIDIClientProxy> create(WebMIDIClient* client)
{
return adoptPtr(new MIDIClientProxy(client));
}
// blink::MIDIClient
virtual void requestSysexPermission(blink::MIDIAccessInitializer*) OVERRIDE;
virtual void cancelSysexPermissionRequest(blink::MIDIAccessInitializer*) OVERRIDE;
private:
explicit MIDIClientProxy(WebMIDIClient*);
WebMIDIClient* m_client;
};
} // namespace blink
#endif // MIDIClientProxy_h
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,128
|
Q: Select by time range and segment results by 1 hour I need help with the following requirement:
I have the following table:
id | status | created_at | closed_at
1 'OPEN' '2019-05-08T12:30:24Z' null
2 'CLOSED' '2019-05-08T12:50:22Z' '2019-05-08T13:05:53Z'
3 'CLOSED' '2019-05-08T13:20:00Z' '2019-05-08T13:40:12Z'
4 'CLOSED' '2019-05-08T13:55:47Z' '2019-05-08T14:05:36Z'
5 'OPEN' '2019-05-08T14:15:57Z' null
6 'CLOSED' '2019-05-08T14:30:29Z' '2019-05-08T14:40:00Z'
7 'CLOSED' '2019-05-08T14:55:38Z' '2019-05-08T15:05:51Z'
For time range
created_at='2019-05-08T13:00:00Z'
closed_at ='2019-05-08T15:00:00Z'
The output should be:
timestamp | id
2019-05-08 13:00:00 1,2,3,4
2019-05-08 14:00:00 1,4,5,6,7
Notice that 1 exists at 13:00 and at 14:00 because it's created_at starts at 12:30 and it doesn't have a closed_at (null) so it "lives" throughout the entire time range.
The closest I could come up with is the following:
SELECT TIMESTAMP WITH TIME ZONE 'epoch' +
INTERVAL '1 second' * round(extract('epoch' from created_at) / 3600) * 3600 as created_at,
id
FROM issues
WHERE (created_at BETWEEN '2019-05-08T13:00:00Z' AND '2019-05-08T15:00:00Z') OR
(closed_at BETWEEN '2019-05-08T13:00:00Z' AND '2019-05-08T15:00:00Z') OR
(status='OPEN')
GROUP BY
created_at,id
ORDER BY created_at;
which gives
"2019-05-08 13:00:00+00" "3"
"2019-05-08 13:00:00+00" "2"
"2019-05-08 13:00:00+00" "1"
"2019-05-08 14:00:00+00" "4"
"2019-05-08 14:00:00+00" "5"
"2019-05-08 15:00:00+00" "6"
"2019-05-08 15:00:00+00" "7"
Is that even possible to do from a select query?
A: Assuming you meant created_at = '2019-05-08T13:00:00Z', you should be able to do this pretty easily with tstzranges.
WITH time_ranges as
(SELECT tstzrange(t, t + '1 hour'::interval, '[]') as t_range
FROM generate_series('2019-05-08T13:00:00Z'::timestamptz,
'2019-05-08T14:00:00Z'::timestamptz,
'1 hour'::interval) g(t))
,
test_values AS (
SELECT * from (values
(1, '2019-05-08T12:30:24Z'::timestamptz, null::timestamptz),
(2, '2019-05-08T12:50:22Z', '2019-05-08T13:05:53Z'),
(3, '2019-05-08T13:20:00Z', '2019-05-08T13:40:12Z'),
(4, '2019-05-08T13:55:47Z', '2019-05-08T14:05:36Z'),
(5, '2019-05-08T14:15:57Z', null),
(6, '2019-05-08T14:30:29Z', '2019-05-08T14:40:00Z'),
(7, '2019-05-08T14:55:38Z', '2019-05-08T15:05:51Z')
) v(id, created_at, closed_at)
)
select lower(t_range), string_agg(id::text, ',' ORDER BY id)
FROM time_ranges
JOIN test_values on tstzrange(created_at, closed_at, '[]') && t_range
GROUP BY lower(t_range);
lower | string_agg
------------------------+------------
2019-05-08 13:00:00+00 | 1,2,3,4
2019-05-08 14:00:00+00 | 1,4,5,6,7
(2 rows)
The only somewhat tricky part is joining on the overlap between the tstzrange from created_at to closed_at and the given hours. You don't need the status column at all.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,638
|
\section{}
The large applied potential of spin-orbit-torques for magnetic random access memory has stimulated intensive interest in investigating spin orbit coupling (SOC) in heavy metals such as Pt and Ta~\cite{ref01,ref02,ref03,ref04,ref05,ref06,ref07,ref08,ref09,ref10,ref11}. Their spin Hall angle ($\thsh$), spin diffusion length ($\ls$) and spin relaxation time ($\ts$), which influence switching efficiency are important parameters for determining their effectiveness, but especially the latter two are experimentally hard to assess. Accurate determination of $\ts$ could also help to identify the spin relaxation mechanisms~\cite{ref12}. Though $\thsh$ and $\ls$ have been measured by spin pumping~\cite{ref13,ref14,ref15,ref16,ref17} and 2nd harmonic Hall measurement~\cite{ref18,ref19,ref20}, $\ts$ of Pt and Ta is rarely reported. In principle, $\ts=\ls^2/D$, with $D$ being the diffusion constant, which is also difficult to determine independently.
Electron spin resonance (ESR) has been a standard technique to measure the spin relaxation time of bulk light metals~\cite{ref21}. However, it is not suitable for ultrathin films~\cite{ref22,ref23}. In addition, Elezzabi et al.~\cite{ref24} developed a time-resolved optical technique to directly measure the spin relaxation process in Au to be $\tsau=(45\pm5)$ ps. However, this method is not suitable for heavy metals such as Pt, Ta and W with short $\ts$ \cite{ref25}. Recently, Dyakonov\cite{ref26} theoretically, then V\'{e}lez et al.~\cite{ref27} and Wu et al.~\cite{ref28} experimentally demonstrated a so-called Hanle magnetoresistance (MR) effect in Pt and Ta: a spin accumulation at the sample boundaries caused by the spin Hall effect is dephased by a magnetic field via the Hanle effect, which results in an additional positive MR. This electrical method can be applied to estimate $\ts$ from the magnetic field dependence\cite{ref27,ref28}. Using this approach $\tspt=$1.9 ps was determined for Pt/\ce{SiO2} and 0.61 ps for Pt/YIG~\cite{ref28}.
In fact, spin injection experiments in nonlocal spin valves~\cite{ref29,ref30,ref31,ref32,ref33,ref34,ref35} and 3-terminal geometries~\cite{ref36,ref37,ref38,ref39,ref40} are both powerful tools in measuring $\ts$ in metals and semiconductors. In these experiments, ferromagnetic layer (FM)/tunnel barrier/nonmagnetic layer (NM) junctions are adopted to both inject a non-equilibrium spin accumulation and simultaneously determine their magnitude. These measurement were used to determine spin relaxation times in a wide variety of materials, e.g., $\tssi$=55 - 285 ps for heavily doped silicon~\cite{ref40}; $\tsg >$ 1 ns for graphene/BN~\cite{ref41}; $\tsal$=110 ps for aluminum~\cite{ref29}, $\tscu$=22 ps for copper~\cite{ref42} and $\tsau$=45 ps for gold~\cite{ref32}.
However, it is impractical to apply these spin injection experiments to measure $\ts$ in heavy metals with strong SOC for at least two reasons. First, $\ls$ in this case is so short (about several nanometers) that the preparation of nonlocal spin valves with comparable dimensions is beyond current lithography capabilities. Second, the real contact resistance is $r=\rc + \rsi$, where $\rsi$ and $\rc$ are the contact resistance induced by spin injection (SI) and the original contact resistance without $\rsi$, respectively. Here $\rsi$ equals to $\rn\rc/(\rn+\rc)$ and the spin resistance in the NM layer $\rn$ is defined as $\roun\lsn$. $\roun$ and $\lsn$ are the resistivity and spin relaxation length of NM, respectively. For this discussion we ignore the influence of spin resistance in FM on $\rsi$ due to the small values of $\ls$ in FM. Because $\rn\ll\rc$ for metals, $r\approx\rc+\rn$. As one increases a field perpendicular to the spin polarization in the NM, the spin accumulation dephases, resulting in a vanishing $\rn$ due to the Hanle effect. This gives rise to a $MR\equiv[r(H\rightarrow\infty)-r(0)]/r(0)=-\rn/(\rc+\rn)\approx-\rn/\rc<0$. This negative spin-injection-induced MR (SIMR) has been utilized in 3-terminal geometries to measure $\ts$ in semiconductors~\cite{ref36,ref37,ref38,ref39} but is negligible in metallic systems, since $\rn\ll\rc$ by several orders of magnitude. Besides, $\rc$ can also exhibit a field dependence due to SOC in FM/Barrier/NM junctions~\cite{ref43,ref43.1}. This so-called tunneling anisotropy MR (TAMR)~\cite{ref44} further complicates the analysis.
In this Letter, we will show that even with a 3-terminal geometry, SIMR can be clearly observed by 2nd harmonic voltage measurements, since TAMR only dominates the 1st harmonic voltages. We adopted this method to determine $\ts$ in Pt and Ta and also their corresponding temperature dependences.
First we discuss the basic concept of these measurements. The tunneling conductance $\gc=1/\rc$ is composed by counterparts for opposite spin channels, $\gc=\gcup+\gcdn$. Here we have already neglected $\rn$ in the contact resistance due to the fact that $\rn\ll\rc$. Spin injection into the NM or spin extraction from NM induces a non-equilibrium spin accumulation $\mu_\mathrm{N}$ in NM, which increases or decreases Fermi levels of opposite spin channels. This can further lead to a change of $\gc$ by $\triangle\gc=\frac{\dif\gcup}{\dif E}\mu_\mathrm{N}-\frac{\dif\gcdn}{\dif E}\mu_\mathrm{N}=\frac{\dif(\gcup-\gcdn)}{\dif E}\mu_\mathrm{N}$. The spin accumulation is given by $\mu_\mathrm{N}=p\rn j$, where $p$ and $j$ are the tunneling spin polarization and current density across the junction~\cite{ref45}. Thus $\triangle\gc=\alpha p\rn j$ with $\alpha\equiv\frac{\dif(\gcup-\gcdn)}{\dif E}$. The voltage across the junction $v=\rc j$ is then
\begin{equation}
v=\frac{1}{(g_\mathrm{C,0}+\triangle\gc)}j\approx ( \frac{1}{g_\mathrm{C,0}}-\frac{\triangle\gc}{g_\mathrm{C,0}^2}) j=\frac{1}{g_\mathrm{C,0}}j-\frac{\alpha p\rn }{g_\mathrm{C,0}^2}j^2
\end{equation}
Here $g_\mathrm{C,0}$ is the contact conductance at zero current, or $v=r_\mathrm{C,0}j-\alpha p\rn r_\mathrm{C,0}^2 j^2$ with $r_\mathrm{C,0}$ being the contact resistance at zero current. Note that $r_\mathrm{C,0}$ does not contain SIMR. Assuming that $r_\mathrm{C,0}=r_\mathrm{C,00}(1+\mathrm{TAMR})$ and $\rn=r_\mathrm{N,0}(1+\mathrm{SIMR})$, results in $v=r_\mathrm{C,00}(1+\mathrm{TAMR})j-\alpha p r_\mathrm{N,0} r_\mathrm{C,00}^2\mathrm{(1+SIMR)(1+TAMR)}^2 j^2$, where $r_\mathrm{C,00}$ and $r_\mathrm{N,0}$ are the contact resistance and spin resistance at $H=0$ and $j=0$, respectively. This equation can be further reduced considering TAMR$\ll$1 and SIMR$\ll$1:
\begin{widetext}
\begin{equation}
v\approx r_\mathrm{C,00}(1+\mathrm{TAMR})j-\alpha p r_\mathrm{N,0} r_\mathrm{C,00}^2(1+\mathrm{SIMR+2TAMR})j^2
\end{equation}
\end{widetext}
In practice, an AC current $j=j_0\sin(\omega t)$ satisfying $\triangle\gc<g_\mathrm{C,0}/10$ was selected to make the above Taylor expansion reasonable. Thus $\f= r_\mathrm{C,00}(1+\mathrm{TAMR})j_0$ has no explicit dependence on SIMR while $\ff=\frac{1}{2} \alpha p r_\mathrm{N,0} r_\mathrm{C,00}^2(1+\mathrm{SIMR+2TAMR})j_0^2$ has a dependence on both SIMR and 2TAMR. They also differ in phase by 90$^\circ$. We would expect that TAMR dominates in $\f$ while SIMR becomes comparable to the TAMR and thus observable in $\ff$ as shown in the following experiments.
\begin{figure}[thb!]
\includegraphics[width=9cm]{fig1.jpg}%
\caption{\label{fig1}(Color online) (a) and (b) Magnetic moment $m$ vs $H$ curve of Ta/ MgO/CoFeB and Pt/ MgO/CoFeB film. (c) Schematic of heavy metal/ MgO/\ce{Co40Fe40B20} junctions. Top electrode 1 and bottom electrodes 2, 3 and 4 are on opposite sides of 40-nm MgAlO$_x$ around the tunnel junction area. (d) The ISHE measurement setup applying an AC current between E1 and E3 and detecting the voltage between E2 and E4 with preamplifier and lock-in amplifier. (e) and (f) 1st harmonic ISHE voltage of Ta/ MgO/CoFeB and Pt/ MgO/CoFeB. High temperature (orange circle) or low temperature (blue) data are shown together for the Ta and Pt stacks, respectively. The current amplitude is 100 $\mathrm{\mu A}$ for Ta and 500 $\mathrm{\mu A}$ for Pt. Opposite field dependencies (e) and (f) indicate different signs of $\thsh$ of Ta and Pt.}
\end{figure}
Stacks of \ce{SiO2}//Ta(10) or Pt(10)/MgO(2)/\ce{Co40Fe40B20}(4)/Ta(5)/Ru(7) (thickness in nm) provided by Singulus Technologies AG were deposited via magnetron-sputtering and then post-annealed with a magnetic field of 1 $\mathrm{T}$ along the $x$-axis at 300 $^o$C for 1 hour to induce an easy axis along the $x$-axis. $M$-$H$ curves acquired by vibrating sample magnetometer (Microsense) showed in-plane magnetic anisotropy for both Ta/MgO/CoFeB and Pt/MgO/CoFeB stacks [Fig. 1(a) and (b)]. The anisotropy field of each sample is about 15 kOe along the $z$-axis, while the easy axis is along the $x$-axis. $H_x$ smaller than 1 kOe is sufficient to align the magnetization along the easy axis.
The extended films were then processed into magnetic tunneling junctions by ultraviolet lithography and argon ion etching. The junctions had one top electrode (E1) and three bottom ones (E2, E3 and E4) [Fig. 1(c) and (d)]. The size of the junctions was 6 $\mathrm{\mu m}\times$6 $\mathrm{\mu m}$. Ta/MgO/CoFeB or Pt/MgO/CoFeB junctions were surrounded by MgAlO$_x$ for protection and also for isolating E1 from the remaining electrodes. Magnetotransport properties were measured in a physical property measurement system (Quantum Design-9T). To measure the inverse spin Hall effect (ISHE) of the bottom electrodes, an AC current with sine wave and $f=\omega/\pi=8.7$ Hz was applied between E1 and E3 using a Keithley 6221 and the 1st harmonic voltage $V\w$ between E2 and E4 was firstly pre-amplified (Stanford Research, SR560) and then picked up by a lock-in amplifier (SR830) [Fig. 1(d)].
In this setup, spin-polarized current was perpendicularly injected from the FM to the NM layer. Their spin orientation was along the $x$-axis at $|H_x|>$500 Oe. Then a voltage in the open circuit can be detected along the $y$-axis due to the ISHE. The field dependences of the 1st harmonic voltage $V\w^\mathrm{ISHE}$ between E2 and E4 in Ta and Pt junctions are illustrated in Fig. 1(e) and (f). The sign of $V\w^\mathrm{ISHE}$ reverses as expected with reversed sign of $H_x$. $V\w^\mathrm{ISHE}$ has opposite signs in the Ta and Pt due to their opposite $\thsh$~\cite{ref47,ref48}, which indicates successful spin injection into the bottom heavy metal layer. Similar ISHE behaviors in both junctions have also been observed near room temperature. The maximum $V\w^\mathrm{ISHE}/j_0$ of Ta and Pt junctions is about 1 $\umo$ and 0.1 $\umo$ at 300 K, which is in the same order of magnitude as in Ref. ~\cite{ref46}.
\begin{figure}[thb!]
\includegraphics[width=9cm]{fig2.jpg}%
\caption{\label{fig2}(Color online) TMR obtained from 1st harmonic voltage with the 3-terminal (3T) measurement setup applying AC currents between E1 and E3 and detecting the voltages between E1 and E4 in the inset at high temperature (a) 300 K for Ta/MgO/CoFeB, (b) 250 K for Pt/MgO/CoFeB or low temperature 10 K for (c) Ta/MgO/CoFeB or (d) Pt/MgO/CoFeB. The external field is either in plane along $x$-axis (black square) or out of plane along $z$-axis (red circle). The currents are identical as in Fig. 1. (e) and (f), 100 $\mathrm{\mu A}$ for Ta/MgO/CoFeB [(a) or (c)] and 500 $\mathrm{\mu A}$ for Pt/MgO/CoFeB [(b) or (d)].}
\end{figure}
3-terminal MR measurements are further performed on both Ta and Pt junctions. We have first detected the 1st harmonic voltage $V\w^\mathrm{3T}$ between E1 and E4 with an AC current applied between E1 and E3 [inset of Fig. 2(a)]. TMR$\w$ is defined as [$V\w^\mathrm{3T}(H)$-$V\w^\mathrm{3T}(0)$]/$V\w^\mathrm{3T}(0)$ and its field dependences is shown in Fig. 2(a)-(d). The MR originates from the tunneling junction instead of the anisotropy magnetoresistance (AMR) of the CoFeB layer. Direct measurements of AMR of the Ta and Pt showed negligible field dependence in the 1st harmonic measurements. AMR only appears in the DC measurement, whose value is negligibly small, only less than 0.05$\%$ at 10 K\cite{ref48.1}. Except Thus, the TMR is mainly attributed to anisotropic tunneling magnetoresistance (TAMR) of the CoFeB/MgO/heavy metal junctions, and we use TAMR instead of TMR in the following analysis.
At high temperature, TAMR$\w^z$ first quadratically increases as $H_z$ increases from zero in both Ta and Pt junctions [Fig.2(a) and (b)] and later gradually saturates at 0.20$\%$ for Ta and 0.14$\%$ for Pt junction as $H_z$ approaches 15 kOe which is also the anisotropy field of the CoFeB layer. Further increasing $H_z$ leads to a MR reduction for both junctions. When $H_x$ is applied, TAMR$\w^x$ increases only by about 0.01$\%$ and then decreases gradually toward the negative MR. Note that TAMR$\w^z$ is much larger than TAMR$\w^x$. $H_z$ aligns the magnetization from in-plane to out-of-plane, which subsequently changes the density of state of the interfacial FM layer via SOC and results in TAMR as predicted theoretically~\cite{ref44,ref49}. The phenomenon TAMR$\w^z>$TAMR$\w^x$ is consistent with Ref.\cite{ref50}, since $H_x$ keeps the magnetization along the easy axis, and consequently TAMR$\w^x$ varies little.
Similar behaviors are also observed at 10 K, except for larger saturation fields and slightly larger TAMR$\w^z$ values [Fig. 2(c) and (d)]. The negative MR ,which depends on applied field instead of magnetization, is also observed at 10 K. Its origin is still unknown and beyond the scope of this study. The only remarkable difference between 10 K and high temperature is that a small negative MR (about -0.014$\%$) appears at low $H_z$ in the Ta junction [Fig.2(c)]. This negative MR exhibits a similar field dependence as the Hanle-effect-induced SIMR discussed below. Thus we attribute it to spin injection into Ta. This SIMR$\w$ should have been negligibly small due to the fact $\rn\ll\rc$. In fact, it turns out to be unobservable in the Pt junction or at high temperatures.
It might be possible that inhomogeneities of the MgO layer result in a significant reduction of the effective tunneling area and smaller $\rc$ in the Ta junction. This may lead to a reemerging of SIMR$\w$ although SIMR$\w$ is still one order smaller than TAMR$\w^z$. Inhomogeneous current distribution due to the resistance of the nonmagnetic layer within the junction area could reduce the measured tunneling resistance below the real tunneling resistance by about 10.8$\%$ and 4.5$\%$ for Ta and Pt junctions respectively due to device geometry as well as inhomogeneous current distribution within the junction~\cite{ref50.1,ref50.2}. However, this would not affect the injected spins and their dephasing process in the heavy metal layers. Therefore, this resistance adjustment would not physically influence the field dependence of the TAMR and the SIMR effects which is the basis of estimating the spin relaxation times.
$V\ww^\mathrm{3T}$ was detected in the same setup as shown in the inset of Fig. 2(a). The only difference is that the 2nd harmonic voltage with 90$^\circ$ phase shift was measured with the lock-in amplifier. As shown in Eq. (2), SIMR should be comparable to TAMR within a factor of 2 for the 2nd harmonic signal. Thus this method renders Hanle and inverted Hanle effect signals induced by SIMR detectable even in the presence of a TAMR background (Fig.3).
\begin{figure}[thb!]
\includegraphics[width=9cm]{fig3.jpg}%
\caption{\label{fig3}(Color online) 2nd harmonic voltage with the 3-terminal (3T) measurement setup for Ta/MgO/CoFeB at (a) 300 K or (b) 10 K, and for Pt/MgO/CoFeB at (c) 250 K or (d) 10 K. The magnetic field was applied along the $x$-axis (black square) for inverted Hanle measurement or the $z$-axis (red circle) for Hanle measurement.}
\end{figure}
The field dependence of $V\ww^\mathrm{3T}$ at 300 K or 250 K for Ta and Pt junction is shown in Fig. 3(a) and (b). For small $H_z$, the magnetization is still aligned along the easy axis. An AC current injects (extracts) spins into (from) NM and leads to a non-equilibrium spin accumulation, which conversely influences tunneling resistance and contributes an additional $V\ww$. A vertical $H_z$ can dephase the spin accumulation via the Hanle effect and therefore diminish the additional $V\ww$, leading to a negative MR with a Lorentzian shape in the 2nd harmonic signal. This Hanle dephasing is the same as established by Silsbee~\cite{ref51} for DC measurement. It is worth noting that TAMR$\ww$ and SIMR$\ww$ contribute to a positive and negative MR, respectively. In addition, TAMR$\ww$ has a $H_z^2$ dependence at low field according to our results in Fig. 2, while SIMR$\ww$ exhibits a Lorentzian-shape dependence. By fitting $V\ww$ vs. $H_z$ curves with a Lorentzian function plus a $H_z^2$ function, we can obtain a spin relaxation time $\ts=e/(mB_0)$ with the electron charge $e$, electron mass $m$ and $B_0$ being the half width at half maximum of the Lorentzian fitting. $\ts$ is (7.8$\pm$1.6) ps at 300 K and (13.1$\pm$0.6) ps at 10 K for Ta [Fig.3(a) and (b)]. By further increasing $H_z$ beyond 10 kOe, $V\ww^\mathrm{3T}$ increases due to both tilting of magnetization and the concomitant TAMR contribution.
In contrast, $H_x$ avoids dephasing of the spin polarization along $x$, and therefore extends spin relaxation process and finally causes a positive MR in small field. This picture accounts for the inverted Hanle effect~\cite{ref40}. A similar positive SIMR also occurs for the 2nd harmonic signal (Fig.3). Besides, $V\ww$ exhibits a $H_{z/x}$ dependence at high fields, especially at 10 K, but the origin of this field dependence is unclear at this point. The Hanle signal in Fig.3 (c) and (d) results in $\ts$ (5.0$\pm$1.5) ps at 250 K and (7.3$\pm$0.6) ps at 10 K for Pt. The inverted Hanle SIMR shows similar behavior for Ta. More than 4 devices are measured to estimate the $\ts$ for each type of stacks. The data for the other devices are attached in the Supplementary Information\cite{ref48.1}.
\begin{figure}[thb!]
\includegraphics[width=9cm]{fig4.jpg}%
\caption{\label{fig4}(Color online) Temperature dependence of the 2nd harmonic voltage of Hanle measurements for (a) Ta/MgO/CoFeB and (b) Pt/MgO/CoFeB from 10 K to 300 K. And temperature dependence of spin relaxation time (c) for Ta/MgO/CoFeB and (d) for Pt/MgO/CoFeB acquired via fitting the data with a Lorentzian curve plus a parabolic function for the TAMR correction applied in different field ranges $\pm$13 Oe (red triangle), $\pm$14 Oe (olive square) and $\pm$15 Oe (black pentagon) or without the parabolic function fitting (blue circle). Inset in (d) shows that $\ts\rho$ remains nearly constant from 300 K to 10 K for all fitting ranges.}
\end{figure}
In order to investigate the temperature ($T$) dependence of $\ts$, we have conducted the 2nd harmonic SIMR measurement in a Hanle geometry at different temperatures [Fig.4(a) and (b)]. As $T$ decreases from 300 K to 10 K, the Hanle-effect-induced $\triangle V\ww$ grows significantly by nearly one order of magnitude. In order to examine whether the field range for selecting the data affects $B_0$, we have tried different ranges ($\pm$13 kOe, $\pm$14 kOe and $\pm$15 kOe) for the fitting. The $T$ dependence is basically the same for different fitting ranges. Their variance is less than 2 ps for both materials. Taking the $\pm$14 kOe fitting range, $\ts$ in Ta gradually decays from (13.1$\pm$0.6) ps at 10 K to (7.8$\pm$1.6) ps at 300 K. In contrast, if the TAMR correction is ignored in the fitting $\ts$ stays at 20 ps below 150 K and then decays to 14 ps at 300 K. These values are not only 50$\%$ higher than those with TAMR correction but also exhibits an unreasonable $T$ dependence. Thus the TAMR correction is indispensable. $\ts$ of Pt and Ta is about 10 ps or below. These values are 1-3 orders smaller than $\ts$ in light metals or semiconductors, consistent with the trend that elements with larger atomic number have stronger SOC. $\tspt$ is about half of $\tsta$ at all temperatures in our experiment and much smaller than $\tsau$ of 45 ps. Here $\tspt$=(5.0$\pm$1.5) ps at 250 K is about twice of 1.9 ps measured by Hanle MR, which might be caused by lower resistivity in the former Pt and different film thickness in the two experiments. In our experiment, $\roupt$=24.4$\uuocm$ at 300 K, while it is 58$\uuocm$ in Ref.~\cite{ref28}. $\ts \rho$ appears to be a constant for these two samples. The $T$ dependence of $\roupt$ is also measured. For resistivity measurement, the top structure MgO/CFB/capping layers in the Pt/MgO/CFB stacks is etched away. $\roupt$ decreases weakly with decreasing temperature and $\ts \rho$ in Pt is nearly a constant from 300 K to 10 K for all the fitting ranges [inset in Fig.4(d)]. The momentum relaxation time $\tp$ is inversely proportional to $\rho$. Thus $\ts/\tp$ is also a constant, which indicates that the spin relaxation in Pt is governed by Elliott-Yafet mechanism ~\cite{ref12}. We also applied a THz technique~\cite{ref52} to directly measure momentum relaxation time and resistivity of Pt with 30 nm thickness, which gives $\tp$=(5$\pm$3) fs and $\roupt$=16$\uuocm$ at 300 K. Assuming that $\tp$ is proportional to 1/$\roupt$, $\tp$ in Pt/MgO/CFB is thus around 2.7 fs. Therefore the spin flip probability of each scattering $\tp/\ts$ is around 7$\times$10$^{-4}$ for Pt at 300 K.
Our $\routa$ is about 342$\uuocm$ at 300 K, much larger than those reported for the resistivity of $\alpha$-phase and even $\beta$-phase Ta~\cite{ref53,ref54}, which might be due to oxidation of Ta after the top structure is etched. Therefore $\routa$ vs. $T$ is not used here for examining the spin relaxation mechanism.
In conclusion, TAMR$\w$ dominates the 1st harmonic 3-terminal MR measurement while SIMR$\ww$ becomes significant compared to the TAMR$\ww$ background and turns out to be much easier measured in the 2nd than in the 1st harmonic signal. This renders conventional 3-terminal FM/barrier/NM devices suitable for directly measuring the spin relaxation time $\ts$ of heavy metals without complications from proximity effects~\cite{ref55,ref56,ref57,ref58} that occur, when the heavy metal is in direct contact with a ferromagnet. ISHE is also observed, which proves successful spin injection into Ta and Pt. By fitting Hanle curves with a Lorentzian function plus a parabolic TAMR background, we have obtained $\ts$ of Ta and Pt. The $\ts$ for both materials exhibits a small increase from 300 K to 10 K, such that $\ts$ is about (7.8$\pm$1.6) ps and (5.0$\pm$1.5) ps for Ta and Pt at high temperature while it is about (13.1$\pm$0.6) ps and (7.3$\pm$0.6) ps at 10 K, respectively. Since $\ts \rho$ stays constant at all temperatures, the spin relaxation in Pt seems to be dominated by the Elliott-Yafet mechanism. This experimental approach provides an electrical manner to directly quantify spin relaxation time of heavy metals, which have been elusive from conventional SIMR or optical measurements. Furthermore, there is no physical limitation for this method to be generalized to other light metals and semiconductors.
\section{acknowledgments}
\begin{acknowledgments}
This work was supported by the 863 Plan Project of Ministry of Science and Technology (MOST) (Grant No. 2014AA032904), the MOST National Key Scientific Instrument and Equipment Development Projects [Grant No.2011YQ120053], the National Natural Science Foundation of China (NSFC) [Grant No. 11434014, 51229101, 11404382] and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (CAS) [Grant No.XDB07030200]. The work from Axel Hoffmann contributing to experiment conception and data analysis was also supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. X. M. Liu and Z. M. Jin have contributed to THz measurement. The annealed raw films were provided by Singulus Technologies AG.
\end{acknowledgments}
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"redpajama_set_name": "RedPajamaArXiv"
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import { shallowEqualKeys } from "core/utils"
export default function() {
return {
fn: { shallowEqualKeys }
}
}
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"redpajama_set_name": "RedPajamaGithub"
}
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Q: Spring Boot application stuck at: Initializing Spring DispatcherServlet 'dispatcherServlet' I am upgrading a Spring Boot application from version 1.5.6 to 2.1.1. When I start the application, it gets stuck at this line:
INFO: Initializing Spring DispatcherServlet 'dispatcherServlet'
When I hit this URL: http://localhost:8888/actuator/health, I get {"status":"UP"}
Also when I hit this URL: http://localhost:8888/swagger-ui.html, I see the Swagger UI.
But my main application doesn't start. Any idea why it's stuck?
A: You have just to exclude security auto configuration.
Just add on your main SpringbootApplication
@SpringBootApplication(exclude = SecurityAutoConfiguration.class)
A: I was facing the same issue. Below code was already a part of my application's main class.
@SpringBootApplication(exclude = SecurityAutoConfiguration.class)
I could not see any logs saying that application has started or anything as such. I had logging level to DEBUG.
logging:
level:
root: ERROR
org:
springframework: DEBUG
But when I tried hitting my APIs from localhost, it could get expected response. The application was running on port 8080 in my case.
A: I know its old.
I was stuck similarly when I tried to move from SpringBoot 1.5 to 2.
What fixed it for me was when I added
@SpringBootApplication(exclude = { SecurityAutoConfiguration.class })
based on differences in springboot2 and this springboo2 security tutorial . Spring 2 migration guide -> 2
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{"url":"https:\/\/www.infoq.com\/articles\/Automated-Builds-2\/","text":"Facilitating the spread of knowledge and innovation in professional software development\n\nContribute\n\n### Choose your language\n\nInfoQ Homepage Articles Automated Builds: How to Get Started\n\n# Automated Builds: How to Get Started\n\n## Introduction\n\nThe first part of this series discussed some of the benefits of automating your build and deployment processes. There are many reasons you may want to do this - to allow your developers to focus on core business instead of administration, to reduce the potential for human error, to reduce the time spent on deployment, and a variety of others. Whatever your motivations are, automating your build process is always the right answer.\n\nIn this article, we will take a common example of a corporate web application for a fictional financial institution, and walk through fully automating their build process.\n\n## Case Description\n\nOur company is 3rd National Bank, a local financial institution. Our online banking application consists of the front-end web application (ASP.NET); a RESTful service (WebAPI) for connecting from mobile applications; a series of internal services (WCF) which use a traditional domain-driven design to separate out business logic, domain objects, and database access (Entity Framework); and a SQL Server database.\n\nThe software team uses Mercurial as their source control system, and delivers features regularly, using a feature-branch strategy - a branch is created for each new feature or bug, and once tested, the code is merged into the main line for release.\n\nCurrently all of the build and deployment steps are done manually by the software team, causing developers to spend several hours every week maintaining their repositories and servers instead of writing code. We\u2019re trying to change that, and automate as much of the process as possible.\n\n## Build Scripts\n\nBuild scripts are the first step toward automating your build. These scripts come in all shapes and sizes \u2013 they can be shell or batch scripts, XML-based, or written in a custom or an existing programming language; they can be auto-generated or hand-coded; or they can be totally hidden inside an IDE. Your process may even combine multiple techniques. In this example, we'll use NAnt as our build script engine.\n\nIn our environment, we have separate Visual Studio solutions for the front-end web application, the external service, and the internal service application, and a database solution for the SQL database. We\u2019ll create a single master build script file, 3rdNational.Master.build, which looks something like this:\n\n<project name=\"3rd National Bank\" default=\"build\">\u00a0\u00a0\u00a0<target name=\"build\">\u00a0\u00a0\u00a0\u00a0\u00a0<call target=\"createStagingEnvironment\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<nant buildfile=\"BankDB\/BankDB.build\" target=\"build\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<nant buildfile=\"ServiceLayer\/ServiceLayer.build\" target=\"build\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<nant buildfile=\"OnlineBanking\/WebUI.build\" target=\"build\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<nant buildfile=\"ExternalServices\/ExternalServices.build\" \/>\u00a0\u00a0\u00a0<\/target><\/project>\n\nThis script doesn\u2019t actually do anything \u2013 instead it just makes calls to each of the four solutions.\u00ad\u00ad\u00ad Each solution gets its own build file, which contains all the code required to compile and prepare its part of the application.\n\nNow let's take a look at a build script for one of these solutions. Each solution follows the same basic steps: prepare, compile, and stage. Here is a basic build script for ServiceLayer.build - the syntax for this is pretty straightforward:\n\n<project name=\"ServiceLayer\">\u00a0\u00a0\u00a0<property name=\"msbuildExe\" value=\"c:\\windows\\microsoft.net\\framework\\v4.0.30319\\msbuild.exe\" \/>\u00a0\u00a0\u00a0<target name=\"build\">\u00a0\u00a0\u00a0\u00a0\u00a0<call target=\"prepare\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<call target=\"compile\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<call target=\"stage\" \/>\u00a0\u00a0\u00a0<\/target>\u00a0\u00a0\u00a0<target name=\"prepare\">\u00a0\u00a0\u00a0\u00a0\u00a0<!-- Implementation omitted -->\u00a0\u00a0\u00a0<\/target>\u00a0\u00a0\u00a0<target name=\"compile\">\u00a0\u00a0\u00a0\u00a0\u00a0<exec program=\"\\${msbuildExe}\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<arg value=\"ServiceLayer.sln\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<arg value=\"\/p:Configuration=Release\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<arg value=\"\/t:rebuild\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<\/exec>\u00a0\u00a0\u00a0<\/target>\u00a0\u00a0\u00a0<target name=\"stage\">\u00a0\u00a0\u00a0\u00a0\u00a0<copy todir=\"..\/deploy\/BankWcf\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<include name=\"WcfServices\/**\/*.svc\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<include name=\"WcfServices\/**\/*.asax\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<include name=\"WcfServices\/**\/*.config\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<include name=\"WcfServices\/**\/*.dll\" \/>\u00a0\u00a0\u00a0\u00a0\u00a0<\/copy>\u00a0\u00a0\u00a0<\/target><\/project>\n\nThe preparation steps may involve building an AssemblyInfo file, or rebuilding proxies, or any number of other things. The compilation step in this case is simply calling MSBuild, the build engine Visual Studio uses to compile a solution. Finally, after everything builds successfully, the last step copies out the appropriate files into a staging area, to be picked up later.\n\nWe do the same thing for the other three solutions, modifying them appropriately based on the different project types.\n\nWriting your build scripts is just like writing any other kind of code \u2013 there are endless ways of accomplishing the same result. You can use the command-line compiler executables directly instead of MSBuild. You can build the projects individually rather than building the full solution. You can use MSDeploy to stage out or deploy your application instead of defining a filter and copying files. In the end, it's all about what you're comfortable with. As long as your scripts produce consistent output, there's no wrong way to write them.\n\n## Continuous Integration\n\nNow that we have build scripts, we need something that will call them. We could run our build scripts from the command line \u2013 but since we're trying to automate everything, we need a machine to run the scripts when appropriate. This is where continuous integration comes in.\n\nLet's use TeamCity, a product by JetBrains, for our CI. It has a very reasonable pricing model, and offers a fantastic user experience for setting up projects. After an easy installation on our new build server, we're ready to get started.\n\nIn TeamCity, your first step is setting up a project. The project consists of a name, along with a collection of build configurations. Let\u2019s create a project called \u201c3rd National Bank\u201d.\n\nWe\u2019re going to want to set up a build template, which will represent the settings used for the mainline as well as any branches we want to put under CI. We\u2019ll set up our version control settings, selecting Mercurial as our source code repo, the default branch, credentials, and a place to check the files out to. Next is a build step, selecting NAnt and our master build file. If we have a unit test project, we\u2019ll simply add another build step to run NUnit or MSTest or whatever we\u2019re using. Finally, we\u2019ll select a build trigger tied to our version control, which means the build will run every time someone pushes code to the main repository.\n\nThere are lots of other useful things you can do in TeamCity, like defining failure conditions based on build output, dependencies on other builds, and custom parameters, which you can explore as you need. But we\u2019ve got all we need for a basic build now.\n\nLet\u2019s create a new build configuration from this template, called \u201cMain Line\u201d. This will represent the top of the version control tree, where stable production-ready code lives. Since we also have feature branches out there, we can create as many more build configurations from the template as we need, one for each feature, and we should only need to make minor tweaks to the source control settings. We now have not only our mainline, but every open feature building automatically upon code checkin, all in just a couple minutes.\n\nWhen a feature is done and merged into the mainline for release, the build configuration for that branch can simply be deleted.\n\n## Deployment\n\nNow that our CI system has built our code, run our tests, and staged out a release, we can talk about deployment. Like anything else, there are many different strategies you can use to deploy your applications. Here are a few basic strategies you may want to use to deploy a web application in IIS:\n\n\u2022 Simply backup your existing applications and copy the new code on top. You never have to worry about touching your configuration.\n\u2022 Copy your code out to a brand new versioned directory on your web server. You can do this in advance. When you are ready, re-point IIS to the new directory. You can take an extra step of having a \"staging\" web application in IIS that you point to the new version, with which you can run some preliminary tests against prior to making the switch.\n\u2022 Don\u2019t hide the versions; include them in your URL: http:\/\/example.com\/v3.4\/ and http:\/\/example.com\/v3.5\/ - the root application of http:\/\/example.com\/ will send you to the newest application using a simple config setting or IIS setting. 3.4 will remain alive, and only customers going to the root application will see the new one. This gives you the opportunity to not interrupt existing sessions. After an hour or so, when all the sessions on 3.4 are gone, you can safely remove v3.4 from IIS.\n\nYour team can determine what's best for you - it depends on your organization's policy toward outage windows and uptime requirements, as well as your database design strategy. For our example, we'll assume we have a 1-hour weekly outage window, so we'll pick the simple file copy strategy which gives us time to back up and deploy the code and database and test it, prior to turning things back on.\n\nYour CI system has staged out your release, so it's simply a matter of getting these files out to your production servers and deploying your database changes. If you have file system access to your build and web servers from your desktop, copying the files can be as simple as executing a batch file that looks like:\n\nrem BACKUProbocopy \/E \\\\web1\\www\\BankWeb \\\\web1\\Backups\\BankWeb\\%DATETIME%robocopy \/E \\\\web1\\www\\BankRest \\\\web1\\Backups\\BankRest\\%DATETIME%robocopy \/E \\\\svc1\\www\\BankWcf \\\\svc1\\Backups\\BankWcf\\%DATETIME%rem DEPLOYrobocopy \/E \\\\build1\\BankWeb\\deploy \\\\web1\\www\\BankWebrobocopy \/E \\\\build1\\BankRest\\deploy \\\\web1\\www\\BankRestrobocopy \/E \\\\build1\\BankWcf\\deploy \\\\svc1\\www\\BankWcf\n...etc...\n\nIf you don't have full file system access, you'll need to find a more creative way to deploy your files. Of course, you can Remote Desktop into your server or an intermediary server to execute a batch file or manually copy files, but remember we're trying to automate this, so the fewer steps, the better. Ideally, you'll have a trusted system in the middle, which has the ability to deploy files to the web servers after it authenticates you. DubDubDeploy is one option that copies files from a trusted server over HTTP to allow you to deploy without access to the web server's file system.\n\nDeploying a database can be done many ways, again depending on your organization. In this example, we are using a database project, so it is as simple as executing a single command which takes the project, automatically compares to your production database, and executes the change script, along with any custom SQL you\u2019ve written that builds seed data. If you're comfortable letting the system do this on its own, it's just a matter of executing a command:\n\nmsbuild.exe \/target:Deploy \/p:TargetDatabase=3rdNational;TargetConnectionString=\u201dServer=db1;Integrated Security=True\u201d;Configuration=release BankDb.sqlproj\n\nOf course, you can execute this any number of ways - you can add it to your NAnt script as a target, or add it to TeamCity, or run it manually, or put it in a deployment batch file.\n\n## Conclusion\n\nWe've come a long way; when we started building, running tests, staging, backing up, and deploying our code was all done manually. Now there are scripts that compile our code, a system that continuously and consistently executes these scripts and runs our unit tests, and a simple repeatable deployment task.\n\nIf you don't have the time to set up everything at once, you don't have to. You can do this a little at a time, and still benefit from each step. For example, you can probably put together a basic build script for your entire set of applications in less than an hour. Maybe another hour to test and debug it to ensure it builds things the same way you're used to. Now, without even adding CI or other automation, you've made it easier to build and stage your app, so next time you deploy manually, you don't even have to open your IDE. Maybe you'll have time next week or next month to create a simple CI project, which you can improve the following month. Before you know it, you'll have a fully automated process.\n\nI've specifically used .NET, NAnt, and TeamCity for these examples, but the fundamental principles can be applied to anything. Whatever operating system, programming languages, server technologies, source control strategies, and team structure you have, automation is possible, affordable, and well worth the effort.\n\n## About the Author\n\nJoe Enos is a software engineer and entrepreneur, with 10 years\u2019 experience working in .NET software environments. His primary focus is automation and process improvement, both inside and outside the software world. He has spoken at software events across the United States about build automation for small software development teams, introducing the topics of build scripts and continuous integration.\n\nHis company\u2019s first software product, DubDubDeploy, was recently released - the first in a series of products to help improve software teams manage their build and deployment process. His team is currently working on fully automating the .NET build cycle.\n\nStyle\n\n## Hello stranger!\n\nYou need to Register an InfoQ account or or login to post comments. But there's so much more behind being registered.\n\nGet the most out of the InfoQ experience.\n\nAllowed html: a,b,br,blockquote,i,li,pre,u,ul,p\n\nby \u9ad8 \u7fcc\u7fd4,\n\n\u2022 ##### Suggest adding a hyperlink to \"The first part\"\n\nby \u9ad8 \u7fcc\u7fd4,\n\nYour message is awaiting moderation. Thank you for participating in the discussion.\n\nThe hyperlink of \"The first part\" is \"http:\/\/www.infoq.com\/articles\/Automated-Builds\".\n\nAllowed html: a,b,br,blockquote,i,li,pre,u,ul,p\n\nAllowed html: a,b,br,blockquote,i,li,pre,u,ul,p","date":"2021-11-29 06:45:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.27959275245666504, \"perplexity\": 2315.7389496187884}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964358688.35\/warc\/CC-MAIN-20211129044311-20211129074311-00438.warc.gz\"}"}
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Historical New Westminster was founded in 1858 and was the lower mainland's largest city until the early 20th century when Vancouver surpassed it. New Westminster was the capital of the Colony of British Columbia before the colonies merged and became part of Canada. Today, New Westminster remains an important member of the Greater Vancouver Regional District, with a population over 66,000. New Westminster's historical downtown area and Quayside Market are still a tourist attraction today, and many Hollywood movies use the area to double as New York, Chicago and London. The city boasts a beautiful view of the Fraser River, where tugboats work away and trains pass through constantly. With the SkyTrain line passing through downtown New Westminster, it is indeed a picturesque location, situated in the centre of Greater Vancouver, right between Surrey, Coquitlam, Burnaby and Richmond.
New Westminster dentists are family-focused, continually accepting new patients of all ages and dental needs. The 123Dentist Network includes 2 dental clinics in New Westminster. Our experienced, professional dental teams have staff on hand who speak languages other than English. So, if you've got a loved one who's reluctant to visit the dentist due to a language barrier, use the search filters below to find a local dentist who speaks their language. At the heart of the city is its commitment to community and family. Middle class families make up the majority of the population in New Westminster, and the city residentially dense core is more welcoming than ever.
New Westminster dentists are very supportive of local initiatives and charities like 'Purpose,' which provides dental supplies to families in need, and also by making an effort to serve a wide array of customers with different backgrounds. Columbia Square Dental has made it a habit of donating dental supplies to families in need during the winter holidays. Our dentists have fun celebrating community events and holidays, dressing up and decorating for Halloween, gathering donations for food drives at Christmas, and offering free oral cancer exams during Oral Cancer Awareness Month. That longstanding program is run by 123 Dentist and consistently raises awareness and funding for cancer research.
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"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/schulte-mecklenbeck.com\/post\/2017-11-13-professor-priming-or-not\/","text":"# Professor priming - or not\n\nThis was my first contribution to a\u00a0Registered Replication Report (RRR). Being one of 40 participating labs was an interesting exercise \u2013 it might seem straightforward to run the same study in different labs, but we learned that such small things as \u00fc, \u00e4 and \u00f6 can generate a huge amount of problems and work (read this if you are into these kind of things).\n\n <p>\nHere is one of the central results:\n<\/p>\n\n<p>\n<img class=\"aligncenter size-full wp-image-579\" src=\"\/uploads\/\/2017\/11\/Screen-Shot-2017-11-13-at-22.19.39.png\" alt=\"\" width=\"718\" height=\"609\" srcset=\"2017\/11\/Screen-Shot-2017-11-13-at-22.19.39.png 718w, 2017\/11\/Screen-Shot-2017-11-13-at-22.19.39-300x254.png 300w, 2017\/11\/Screen-Shot-2017-11-13-at-22.19.39-589x500.png 589w, 2017\/11\/Screen-Shot-2017-11-13-at-22.19.39-500x424.png 500w\" sizes=\"(max-width: 718px) 100vw, 718px\" \/>\n<\/p>\n\n<p>\nSo overall not a lot of action … our lab was actually the one with larges effect size (in the predicted direction).\n<\/p>\n\n<p>\nHere is the abstract of the whole paper and here the\u00a0<a href=\"https:\/\/www.psychologicalscience.org\/redesign\/wp-content\/uploads\/2017\/11\/Dijksterhuis_RRRcommentary_ACPT.pdf\">Commentary by\u00a0Ap Dijksterhuis<\/a>\u00a0naturally, he sees things a bit different: Dijksterhuis and van Knippenberg (1998) reported that participants primed with an intelligent category (\u201cprofessor\u201d) subsequently performed 13.1% better on a trivia test than participants primed with an unintelligent category (\u201csoccer hooligans\u201d). Two unpublished replications of this study by the original authors, designed to verify the appropriate testing procedures, observed a smaller difference between conditions (2-3%) as well as a gender difference: men showed the effect (9.3% and 7.6%) but women did not (0.3% and -0.3%). The procedure used in those replications served as the basis for this multi-lab Registered Replication Report (RRR). A total of 40 laboratories collected data for this project, with 23 laboratories meeting all inclusion criteria. Here we report the meta-analytic result of those 23 direct replications (total N = 4,493) of the updated version of the original study, examining the difference between priming with professor and hooligan on a 30-item general knowledge trivia task (a supplementary analysis reports results with all 40 labs, N = 6,454). We observed no overall difference in trivia performance between participants primed with professor and those primed with hooligan (0.14%) and no moderation by gender.\n<\/p>\n<\/div>\n\n##### Michael Schulte-Mecklenbeck\n###### Associate Professor\n\nI study human decision making, process tracing methods, food choice and traffic behavior.","date":"2021-01-20 09:17:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.25792136788368225, \"perplexity\": 7101.967433237792}, \"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-2021-04\/segments\/1610703519984.9\/warc\/CC-MAIN-20210120085204-20210120115204-00294.warc.gz\"}"}
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625 S. Fourth St., Louisville, KY
www.louisvillepalace.com/
World Dance,Modern
Bring It! Live: The Dance Battle Tour Join Coach D and the Dancing Dolls from the hit Lifetime show, "Bring It!", as they take the dance battle from your television screen to stages across the nation. This brand-new interactive competition combines jaw-dropping performances from the infamous Dancing Dolls and the high-pressure competitions seen on the television show--with a twist. You don't want to miss the dance event of the summer! Buckle yo' seatbelts--klik klak!
• Mon 7/22/19 at - 7:30 PM
WDW is a new boy band, with the emphasis on "boy." Their single, "Taking You," is making inroads with girls of a certain age.
John Cusack & Screening of "Say Anything"
Film,Talks/Lectures
The 1989 comedy/drama "Say Anything" brought popular culture the love story of Lloyd Dobler (Cusack) and Diane Court (Ione Skye). An eternal optimist seeks the heart of a brain trapped in the body of a game show hostess. One of the great modern movie romances of the late 20th century, "Say Anything" made a star of Cusack. He went on to success in "High Fidelity," "Grosse Point Blank," and "Being John Malkovich," among others. Tonight, join him for a screening of "Say Anything" followed by a conversation regarding his career and the making of the movie.
Jason Bonham's Led Zeppelin Evening
Jason Bonham's Led Zeppelin Evening is a spectacular rock concert with iconic Led Zeppelin songs, stunning atmospheric video and light effects, which highlight the unique history that Jason Bonham shares–in commemoration of his father John Bonham–with the legendary rock n' roll band. The band's powerful live performance of Led Zeppelin's classics takes concert-goers through a mesmerizing visual and aural journey as giant backdrops display iconic art, and Bonham's own historical video footage, photos and stories give the show an intimate feel.
• Sat 8/10/19 at - 8:00 PM
Rock,Pop
Matchbox Twenty frontman Rob Thomas made quite an impression with his guest vocal on Santana's 1999 Grammy winner, 'Supernatural.' Rob won three Grammys for Pop Collaboration with Vocals along with Santana, and Song of the Year with Itaal Shur - all for 'Smooth.' Thomas has since gone solo.
• Wed 9/4/19 at - 8:00 PM
TESLA is a multi-platinum-selling rock band from Northern California known for their melodic songs and down to earth appeal. Thanks to their die-hard, loyal fan base and their younger generation offspring, TESLA continues to tour to sold-out crowds around the world. TESLA's line-up consists of four of its original members: vocalist Jeff Keith, guitarist Frank Hannon, bassist Brian Wheat, and drummer Troy Luccketta and new guitarist Dave Rude who has brought a new energy and creativity to the band.
The popular comedian has been a pioneer in the comedy world's use of social media to develop a following. He has millions of Twitter followers. His performances online and on stage have often veered into the obnoxious; he had to apologize for a joke he made about the 2012 shootings in Aurora. Colo. Cook has released many comedy albums, appeared on both Comedy Central and HBO and guest starred on an episode "Louie." He also starred in a 2012 Los Angeles production of "The Producers," for which he received good reviews.
• Fri 9/20/19 at - 7:30 PM
Emmy-nominated actor Kevin James, who starred in the hit CBS comedy, "King of Queens," for nine seasons and has also appeared in such hit movies as "Paul Blart: Mall Cop," "Hitch," "Grown Ups" and more, first made his mark in the stand-up world and is now returning to his roots with this national stand-up comedy tour.
Electronic,Pop,Rap/Hip Hop
Lizzo (Melissa Jefferson) is a hip-hop artist based in Minneapolis.
• Wed 10/2/19 at - 8:00 PM
Rock,Blues,Country,R&B/Soul
Little Feat was formed by singer-songwriter, vocalist and guitarist Lowell George and keyboard player Bill Payne in 1969 in Los Angeles. The band gained popularity in the 1970s with an eclectic style, bringing together elements of blues, R&B, country and rock 'n' roll. Members have come and gone, but the music keeps to the same genre. Phish covered Little Feat's album 'Waiting for Columbus' at a Halloween concert in 2010, which resulted in a wider and younger audience of listeners for the '70s band.
• Fri 10/11/19 at - 8:00 PM
RuPaul's Drag Race: Werq The World Tour
Fashion Show/Bridal
Your favorite queens are here to "WERQ Your World" in the RuPauls Drag Race Official World Tour. Hosted by Michelle Visage with sickening performances by Alyssa Edwards, Detox, Kim Chi, Latrice Royale, Peppermint, Shangela, Valentina and Violet Chachki.
• Sun 10/13/19 at - 8:00 PM
Vince Gill
Perennially one of the most popular artists in country music, Vince Gill has sold more than 26 million albums, earned 19 Grammys and taken home 18 Country Music Association Awards. He is also known as one of country music's most generous humanitarians, having lent his talent to hundreds of charitable events.
• Thu 10/24/19 at - 7:30 PM
Roy Orbison & Buddy Holly Rock'n'roll Dream Tour
Pop,Rock,Multimedia
Accompanied by a live band and back-up singers, this cutting-edge, holographic performance with remastered audio will transport audiences back in time for an unforgettable evening of Roy & Buddys greatest hits onstage. Performing (virtually) together for the first time, this once in a lifetime show will feature chartbuster favorites including Roy Orbison's Oh, Pretty Woman, You Got It, Only the Lonely and Buddy Hollys Oh Boy!, Not Fade Away and That'll Be the Day, among many others.
• Wed 10/30/19 at - 8:00 PM
Nick Offerman is an American film and television actor, known for his role as Ron Swanson on 'Parks and Recreation.'
• Fri 11/8/19 at - 7:30 PM
Rock,Alternative,Country
Bridging the gaps between Americana, orchestral pop and alternative rock, Wilco has built a loyal following and earned a boatload of critical acclaim. The group has six albums, including "Yankee Hotel Foxtrot," "Summerteeth," "A Ghost Is Born," and "Sky Blue Sky."
Pop,Rock
Elvis Costello has followed his musical curiosity in a career spanning four decades. He's perhaps best known for his performances with The Attractions, The Imposters and for concert appearances with pianist Steve Nieve. His prolific output includes such hits as "Allison," "Radio Radio," "'Watching the Detectives," "'Everyday I Write the Book" and "Veronica." He evolved from being typecast in the late '70s as an angry young man at the forefront of whatever new wave was, to a musical renaissance man who has recorded albums with Burt Bacharach, Anne-Sophie Mutter, The Brodsky Quartet and The Roots. He's also been a presence on TV, having appeared several times on "Saturday Night LIve" (starting with his notorious 1977 performance), hosted the Channel 4 (UK)/CTV music series "Spectacle: Elvis Costello With ...," and had a stint as fill-in guest host on "The Late Show With David Letterman."
Wild Kratts Live 2.0: Activate Creature Power
For the Family,Children,Children's Theatre
Animated Kratt Brothers, Martin and Chris, "come to real life" in a classically Wild Kratts story! Experience live the "creature" fundamentals, and the excitement and inspiring quest of the Kratt Brothers that make their hit television series "Wild Kratts" so popular with kids and families.
Classical,Pop
Hailing from Utah, The Piano Guys became an Internet sensation by way of their series of strikingly original self-made music videos.
• Sat 11/23/19 at - 8:00 PM
We Will Rock You: National Tour
Resistance is growing. Underneath the gleaming cities, down in the lower depths live the Bohemians. Rebels who believe that there was once a Golden Age when the kids formed their own bands and wrote their own songs. They call that time, The Rhapsody. Open your eyes, look up to the skies and see. Legend persists that somewhere on iPlanet instruments still exist. Somewhere, the mighty axe of a great and hairy guitar god lies buried deep in rock. The Bohemians need a hero to find this axe and draw it from stone. Is the one who calls himself Galileo that man? He's just a poor boy. From a poor family. Since 2002 over 15 million theatergoers in 17 countries have been thrilled by this awe-inspiring production which is based on the songs of Queen with a book by Ben Elton (The Young Ones, Blackadder, Popcorn). Elton fashioned this hilarious futurist comedy around more than 24 of Queen's biggest hit songs.
• Tue 11/26/19 at - 8:00 PM
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 9,308
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Escapist is the eighth studio album by Australian singer-songwriter Stephen Cummings. The album was released in September 1996.
Cummings said "The story behind the recording of The Escapist is interesting, I can only liken it to having a 21st birthday party. You invite along all your friends and abruptly realise just how dissimilar your friends are. You start to despair and fret they won't come together and be friendly. In the end, common sense prevailed."
At the ARIA Music Awards of 1996 the album was nominated for ARIA Award for Best Adult Contemporary Album.
Reception
Mitchell Peters from Time Off, gave the album 3.5 out of 5 criticising the lead single but said "The rest of the album is beautifully played and drips with maturity, grace and a strange hypnotic style." Chris Johnston from Rolling Stone Australia gave the album 3.5 out of 5 saying "To follow up last year's acclaimed and adventurous Falling Swinger, the strictly-Melbourne Stephen Cummings releases Escapist, his eighth solo album. Escapist continues the experimentation of Falling Swinger by again enlisting errant Church main man Steve Kilbey as producer and collaborator." Antonino Tati from Beat magazine gave the album 9 out of 10 saying "Cummings' compositions... are awe-inspiring with their lyrical themes often found to be down in the dumps, but their musical themes as celestial as an endless bright blue sky." Barry Divola from Who Weekly gave the album a score of B− saying "Downbeat and simple, Escapist isn't always the quiet achiever it should be. Cummings has one of those conspiratorial voices that whispers in your ear, but some of the songs are too flat, and need more of a sting to grab the listener." Jeff Jenkins from TV Week gave the album 4 out of 5 saying "Stephen Cummings remains Australia's finest singer and songwriter on Escapist, his eighth solo album." adding "He's following his own eclectic path, and it's a journey that has jazz, country and dance influences." Scott Howlett from X Press Magazine said "Escapist... is a rich album of 'songs' – good songs – but none which immediately register as essential cuts. Sadly, for Cummings, it's another good album which unfortunately is not 'great' enough to impress as being 'the breakthrough' from the somewhat obscure plateau he's been standing upon since the demise of The Sports in 1981."
Track listing
Release history
References
1996 albums
Stephen Cummings albums
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 3,635
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Q: Liferay Display Page: can I have a private one AND a public one? For a given Web Content, I can choose a private Display Page. Or a public one.
Question: Is there any way to have a private Display Page AND a private Display Page?
*
*The private one would be displayed to users who belong to the site.
*The public one would be displayed to the others.
If I understand correctly the Display Page is referenced by its classUuid being stored as the layoutUuid field in the JournalArticle table.
A: No. There's exactly one display page per article.
However, you can build that one page with dynamic content, so that people see different content based on their identity (e.g. by using Asset Publisher or even a custom Web Content Display portlet, where you can configure multiple articles for the portlets "surrounding" the displayed article in question.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,272
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Brazil goalkeeper Alisson, center, fails to stop Belgium's first goal during the quarterfinal match at the 2018 football World Cup in the Kazan Arena, in Kazan, Russia, Friday, July 6, 2018. (AP Photo/Thanassis Stavrakis).
Belgium booked a World Cup semi-final against France as Brazil's hopes of winning a sixth title were ended by a 2-1 defeat in Kazan on Friday.
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{
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| 6,993
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ACCEPTED
#### According to
The Catalogue of Life, 3rd January 2011
#### Published in
Brittonia 49:3. 1997
#### Original name
Swartzia fistuloides Harms
### Remarks
null
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,281
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January 3, 2018–January 12, 2020
In this series, the Museum presented original research and insights on a new object each month, selected from the Museum's robust permanent collection.
Marcia Clark has been painting landscapes since the 1960s, when she discovered the paintings of Hudson River School founder Thomas Cole, whose work inspired her to look deeper. Clark credits the development of her unique, multi-dimensional style, on view in Butterville Road Intersection, to her daily experience in New Paltz, New York, where she lived from 1969 to 1976. She states that "living with this view was pivotal in turning my attention as a painter to the challenges of getting a panoramic sweep onto a two-dimensional format."
The December Object of the Month is displayed in memory of Robert "Bob" Floyd Judd (1956–2019), musicologist and good friend to the Museum. We are grateful that Bob's world travels, alongside Sarah Lawrence College President Cristle Collins Judd, allowed us to know him.
Women have played basketball since 1892, less than a year after the sport was invented. Senda Berenson Abbott (1868–1954), head of physical education at Smith College, introduced the sport there, and other women's colleges and YWCAs were quick to adopt it. Berenson believed that "Basketball is the game above all others that has proved of the greatest value to [women]. It develops physical and moral courage, self-reliance and self-control, and the ability to meet success and defeat with dignity." The full history of this rare textile, Pillow Top with Woman Basketball Player, remains elusive. Pillow tops were popular as premiums redeemable with numerous proofs of product purchase, often tobacco. Though we know S. M. Schwab Jr. & Co. printed this design, there is no indication of the artist or the company that placed the order. Most tobacco textiles are silk, not cotton, and feature montages of several images. However mysterious, the textile is of great historical interest for the technical quality of its printing, and especially for its subject matter.
Help us with our research: If you recognize the embroidered design on this woman's collar, please contact curatorial@hrm.org.
William Hahn (German, 1823–1887) studied painting in the renowned international art centers of Dresden and Düsseldorf in the 1850s. The subject matter of Woman in Kitchen (Kitchen Maid)—an older woman working beside a kitchen fire—relates to similar figural paintings by the artist dating from the 1860s, of ordinary people at work or play. Hahn revels in detailing his subjects' surroundings, from the broom and small pile of swept hay, to a lantern on the wall and a spinning wheel discernible in the next room. His dark palette and meticulous painting technique, as well as his domestic content, reflect German academic art traditions of the mid-nineteenth century.
In the early twentieth century, railroad posters like The Century in the Highlands of the Hudson tempted travelers with colorful and dynamic compositions of idyllic landscapes, featuring powerful locomotives. This poster, which offers a dramatic perspective of sun gleaming on the streamlined form of a steam engine, is an excellent example of the machine-age aesthetic of the Art Deco period. Leslie Ragan (1897–1972) was born in Iowa, studied at the Art Institute of Chicago, and went on to become one of the most celebrated commercial artists of the twentieth century, creating more than one hundred posters for the New York Central Railway alone. The 20th Century Limited was one of the most famous passenger trains in America at the time, offering luxury accommodations and express travel between New York and Chicago. The first Century train ran in 1902, taking passengers to their destination within twenty hours.
Between 1969 and 1975, Andy Warhol (American, 1928–1987) prefigured the smartphone by documenting his life with thousands of instant Polaroid photographs, including Untitled (Peter Beard with dog) in Red Book #124. He meticulously catalogued the images and stored them in individual red Holson Polaroid albums. Subjects ranged from friends and celebrities to his dogs and the landscape. Many of these photos served as source material for his silkscreen paintings. The Polaroid camera spoke to his fascination with the nature of modern consumerism and the photograph as a readymade, a machine-made object presented as a work of art.
Rudolf Eickemeyer, Sr. was a prominent figure in Yonkers commerce during the second half of the nineteenth century. As an engineer and factory owner, he obtained more than one hundred patents ranging from industrial sewing and harvesting machines to electromagnets and electric trolleys. Many of his inventions were hat-making machines, and the Museum owns several models he kept for his own use. This piece, Model: Combined Stretcher for Hat Making, performed two separate steps in forming a hat, which started as a damp cone of felt; it pulled out the brim and shaped the crown. His son, Rudolf Eickemeyer, Jr. was a well known photographer. He began his artistic career by working as a draftsman for his father and took his first photographs to document the firm's inventions and momentous occasions, such as the contract signing pictured here, Rudolph Eickemeyer, Sr. and his British Agent.
One of the premiere American artists of the twenty-first century, Mickalene Thomas explores issues of race, gender, and beauty through striking portraits of women in studio sets filled with color, pattern, and popular culture references. Thomas works in photography, mixed media painting, and digital prints, often combining these elements to lush effect, as seen in Clarivel with Black Blouse with White Ribbon. Her work reveals an admiration for the abstracted figural collages of African American artist Romare Bearden (American, 1911–1988).
On this fiftieth anniversary of the Stonewall Uprising in Greenwich village, the June Object of the Month is dedicated the brave individuals who stood up for their civil liberties—our civil liberties—and boldly ushered in the Gay Rights Movement. #HRMPride
Thomas J. Hill (American, b. England, 1857–1886)
Glenwood Station
Oil on cardboard
Gift of Mrs. John Stevenson Watt, 1929 (29.189)
Between 1850 and 1900, the railroad became a source of industrial and suburban commuter traffic along the Hudson River, and Yonkers transformed from a village into one of the largest cities in New York State. Thomas J. Hill's painting—among the smallest in the Museum's collection—depicts one of the only known views of the first Glenwood train station, which stood at the bottom of Point Street, two blocks south of Trevor Park. The stop was always called Glenwood, referring to the neighborhood not Glenwood Avenue, where the current station was rebuilt in the early-twentieth century.
The May Object of the Month is dedicated in fond memory of Docent Marilyn Maloff, who welcomed thousands of children with an open heart over the course of twenty-six years of service to the Museum.
Attributed to Helena de Kay Gilder
Cover Design for "The New Day: A Poem in Songs and Sonnet" by Richard Watson Gilder
New York: Scribner, Armstrong & Company, 1875
Collection of the Hudson River Museum (INV.10731)
Helena de Kay Gilder was a pioneering artist, book designer, and suffragette. She helped to organize the Art Students League and took lessons from Winslow Homer and John La Farge at the Tenth Street Studio Building in Manhattan, where she had a studio of her own. Homer painted a portrait of the artist in the early 1870s, illustrated in the label to the right. He also painted Moonlight, 1874, on view downstairs in The Color of the Moon: Lunar Paintings in American Art, the very same year Helena de Kay married writer poet, journalist, critic, and editor Richard Watson Gilder. The Gilders became a power couple in the Gilded Age art world of New York. In 1878, they helped to found the Society of American Artists, a progressive group of largely European-trained artists who found the strictures of the National Academy of Design stultifying. Their marriage was defined by artistic collaboration between partners.
The April Object of the Month is dedicated in fond memory of Docent Judith Auerbach, whose service and commitment over the course of thirty years made the Hudson River Museum a better place.
Mary Frey (American, b. 1948)
Girls Sunbathing
From the Domestic Rituals series, 1979–83
Gift of the artist, 1984 (84.22)
© Mary Frey
Fascinated with the role of documentary images in modern culture, Frey began the Domestic Rituals series after getting her MFA at the Yale University School of Art in 1979. As she described, "the pictures, which have a quasi-documentary look about them, resemble a kind of tableau-vivant." During her four-year project, she won the first of two photography fellowships from the National Endowment for the Arts. In 1984, she exhibited the series at the Hudson River Museum. In The New York Times, William Zimmer described how, through Frey's lens, "we are always brought back to earth by a specific, though universal, incident."
Derrick Adams (American, b.1970)
Orbiting Us #18
Mixed-media collage on paper
Museum Purchase, 2018 (2018.07)
Drawing from such divergent sources as Star Trek and Sun Ra's Afrofuturist science fiction film Space is the Place (1974), Adams deploys modern associations with space travel while also incorporating ancient motifs and imagery. Orbiting Us #18, with its silver mat frame, can be read as a spaceship window offering us a view of the galaxy. In front of the planet Jupiter, the crowned head of the Egyptian King Amen-em-hat III floats in space with a space suit and NASA technician facing it. The modern scene, captioned "Top Sergeant Julie Barrows prepares a pressure suit for presidential inspection," was culled from a 1960s article about the space race in Ebony Magazine. With this pairing, the artist collapses ancient mythical and modern American conceptions of space within the portal window of an imaginary vessel.
Harold Knickerbocker Faye (American, 1910–1980)
ca. 1935–38
Intaglio on Rives paper
Gift of Helen S. Faye, 1990 (90.10.8)
Moonlight strikes an industrial scene, where gravel or sand covers the ground. The artist, Harold Knickerbocker Faye, was born to a wealthy family—his father was vice president of Western Pacific Railway—but spent his short yet dynamic career depicting the other side of the tracks. While many artists of the period romanticized poverty, Faye turned a Realist eye towards New York City during the Depression, seeking formal beauty in otherwise bleak, depopulated scenes.
Dora Wheeler Keith (American, 1856–1940)
Publisher: Louis Prang & Co. (Boston, Massachusetts)
Christmas Card: Shout with Joy
Chromolithograph, silk fringe
Trained under the prominent artist and instructor William Merritt Chase, Dora Wheeler was a painter and tapestry designer for the decorative firm American Artists, founded in 1883 by her mother. In 1893, she executed a large series of murals for the ceiling of the Library of the Woman's Building for the World's Columbian Exposition in Chicago. Along with Mary Cassatt, who also painted murals for the Chicago fair, this commission was a pioneering achievement for a female artist in the nineteenth century, an era where social boundaries largely determined the type of artwork created by women.
Jacob Lawrence (American, 1917–2000)
Silkscreen; edition 51 of 60
One of the leading lights of the Harlem Renaissance, Lawrence was exceptional in his ability to forge a successful art career during the early years of the civil rights movement. His art concerned the genesis, exodus, and eventual apotheosis of the Black subject, and this late print bears traces of this long output. His best known work, The Migration Series, represented the movement of African Americans from the rural South to the industrial North.
Elihu Vedder (American, 1836–1923)
Stella Funestra (The Evil Star)
Pastel and charcoal on paper
Gift of the American Academy of Arts and Letters, 1955 (55.24c)
Elihu Vedder's Stella Funesta (The Evil Star) captures the somber ambivalence of the Gilded Age. By the last decade of the nineteenth century, America enjoyed unprecedented material comfort, rapidly growing cities, and an expanded border westward. This prosperous and modernizing culture was also haunted by the ghosts of the recent past. The rift left by the Civil War, the Plains Indian Wars, economic and social upheaval, and a crisis of faith in the age of Darwin contributed to a mood of nostalgic melancholy in this period, called The American Renaissance.
Jasper Francis Cropsey (American, 1823–1900)
Greenwood Lake, New Jersey
Anonymous Gift, 2017 (2017.07)
Jasper Francis Cropsey was a surprisingly versatile artist. One of the foremost painters in the Hudson River School of landscape painters, he was also an architect. Born in Rossville, Staten Island, Cropsey received his early artistic training as an architect's apprentice, where he learned oil and watercolor techniques for architectural drafting. In 1843, the young artist exhibited for the first time at the National Academy of Design with a painting titled Landscape Composition. This early training and recognition led to a long and varied career as a painter, peaking in the 1850s.
Robert Motherwell (American, 1915–1991)
Summertime in Italy (with Lines)
Gift of Arthur Zankel, 1991 (91.3.1)
Robert Motherwell's lithograph Summertime in Italy (with Lines), 1966, fuses exterior observation and interior reflection central to his artmaking. As a founding member of The New York School, or the Abstract Expressionists, Motherwell was interested in the subconscious associations with abstract shapes, but also in their formal qualities.
Kimbel and Cabus, New York
Secretary Cabinet
Ebonized hardwood, tiles, glass, brass hardware
Gift of Mrs. Joseph Lippman, 1973 (73.39)
The New York-based firm of Kimbel & Cabus led the way in Modern Gothic furniture design through the second half of the nineteenth century. Formed in 1862 by German-born cabinet maker Anthony Kimbel (ca. 1821–1895) and French-born cabinet maker Joseph Cabus (1824–1894), the firm represented the positive synergy between immigrant craftsmen, many of whom arrived around 1848, and the daring tastes of post-Civil War northeastern industrialists and financiers. The enthusiastic market rewarded their experimentation in tastefully eclectic objects like this cabinet.
Louise Nevelson (American, b. Ukraine, 1899–1988)
Sky Enclosure
Gift of John I. H. Baur, 1985 (85.16.1)
Louise Nevelson, born Leah Berliawsky in present-day Ukraine, moved to Maine with her family as a child. As a young adult, she moved to New York City in 1920 to pursue her childhood passion of studying art at the Art Students League. Nevelson lacked the money for fine art materials throughout the 1940s and 1950s, and used wood out of necessity. During a period of urban redevelopment in mid-twentieth century, her practice of reusing architectural salvage was fostered by the demolition of many row houses and tenement buildings in New York City. At first, Nevelson found wood scraps on the street. By the 1960s, she often purchased secondhand furniture, and by the 70s she had wooden objects made to her specifications. The evolving sources of Nevelson's materials illustrate her ascent as one of the leading sculptors in America.
Samuel Colman (American, 1832–1920)
Moonlight in Venice
Ink and wash on board
Gift of the Estate of H. Armour Smith, 1961 (61.13.55)
Born in Portland, Maine in 1832, Samuel Colman moved to New York at an early age, growing up in a literary and artistic environment fueled by his father's business as a book dealer. His uncle sold art supplies, and it was likely through his family that Colman met Asher B. Durand, under whom he studied painting. At age 22, Colman was elected as an associate member of the National Academy of Design and was firmly established as one of the foremost second-generation Hudson River School painters.
Robert Indiana (American, 1928–2018)
Demuth American Dream, No. 5
Gift of Mr. Andrew Lanyi, 1981 (81.11.6a-e)
Robert Indiana, who is best known for the LOVE insignia, has always had an interest in combining visual art and the written word. In 1980, the artist issued a print series based on his painting, Demuth American Dream, No. 5 (1963), an homage to Charles Demuth's painting, I Saw the Figure 5 in Gold (1928), as well as William Carlos Williams's earlier poem, The Great Figure. This tribute, repeating Demuth's visual language with slight variations, devotes each panel to a different word: EAT, HUG, DIE, and ERR.
Artist and Designer: Winslow Homer (American, 1836—1910)
Engraver: John Filmer (American, active 1863—1882)
The Fishing Party
Supplement to Appleton's Journal of Literature, Science, and Art
Gift of Dr. Howard Simon, 2002 (2002.11.03)
Considered one of the foremost painters of nineteenth-century America, Winslow Homer did not benefit from formal academic training early in his career. Instead, his professional experience as an artist was rooted in freelance illustration work for periodicals such as Harper's Weekly, Century Magazine, and Appleton's Journal of Literature, Science, and Art, from which this 1869 image comes. An avid angler, Homer made the depiction of fishing a lifelong artistic pursuit. From his earliest days as an illustrator in the popular press, to his watercolors of fisherwomen along the northern coast of England, to his late oil paintings of the sea, Homer kept his eye trained on fishing themes.
Joseph Cornell (American, 1903–1972)
Untitled (Hôtel de l'Etoil)
Mixed media collage construction
Gift of the C & B Foundation, 1975 (75.22.2)
Joseph Cornell was born in Nyack, NY, and lived most of his life with his family in a small, wood-framed house on Utopia Parkway in Queens. In the early 1950s, Cornell began a series of works that explored the associations of grand hotels in Europe, a subject that proved quite potent for Cornell, who never owned a passport. Cornell's practice of showcasing delicate vignettes in small, neat spaces that were suggestive, not explanatory, was influential to numerous future artists and filmmakers, from Robert Rauschenberg to Wes Anderson.
National Chicle Company
Sky Birds
Commercial color lithograph on paper
Collection of the Hudson River Museum
Between 1933 and 1934, the National Chicle Company produced the Sky Birds card collection. In a format previously reserved for baseball players, heroes of aviation took center stage, with their daring exploits listed on the back of each card. Distributed in one-cent packs (roughly 18 cents in today's currency), a total of 144 different cards were produced. The first two dozen cards featured World War I pilots, many of whom were members of the Lafayette Escadrille, a notoriously reckless volunteer brigade of American pilots in support of the French. The National Chicle Sky Birds set was a uniquely dramatic and entertaining way to illustrate the recent history of aviation.
Marcia Clark (American, born 1938). Butterville Road Intersection , 2011. Mixed media. Gift of the artist, 2018 (2018.03)
Unknown artist. Pillow Top with Woman Basketball Player, ca. 1900. Cotton, printed by S. M. Schwab Jr. & Co., New York, NY. Gift of Henry S. Hacker, 1995 (95.8.005).
William Hahn (German, 1823–1887). Woman in Kitchen (Kitchen Maid), 1863. Oil on canvas. Gift of Mr. Lombardi, By Exchange, 1963 (63.17).
Leslie Darrell Ragan (American, 1897–1972). The Century in the Highlands of the Hudson, ca. 1938. Chromolithographic poster. Gift of Henry S. Hacker, 1996 (96.14.9).
Andy Warhol (American, 1928–1987). Untitled (Peter Beard with dog) in Red Book #124, July 1972. Photo album filled with twenty Polaroid photographs. Gift of The Andy Warhol Foundation for the Visual Arts, Inc., 2013 (2013.04k-l). © 2019 The Andy Warhol Foundation for the Visual Arts, Inc. / Licensed by Artists Rights Society (ARS), New York. Courtesy of Peter Beard Studio and www.peterbeard.com
Left: Rudolf Eickemeyer, Sr. (American, b. Germany, 1831–1895). Model: Combined Stretcher for Hat Making, ca. 1869. Bronze, iron. Gift of Rudolf Eickemeyer, Jr., 1928 (28.213). Right: Rudolf Eickemeyer, Jr. (American, 1862–1932). Rudolph Eickemeyer, Sr. and his British Agent, 1890. Albumen photograph. Gift of Mrs. R. W. Eickemeyer, 1942 (42.91b).
Mickalene Thomas (American, b. 1971). Clarivel with Black Blouse with White Ribbon, 2016. Epson Inkjet print with HDR Ultrachrome inks. Published by the Museum of Contemporary African Diasporan Arts in consortium with the Benefit Print Project, Edition of 25. Museum Purchase, 2019 (2019.01). © 2019 Mickalene Thomas / Artists Rights Society (ARS), New York
Thomas J. Hill (American, b. England, 1857–1886). Glenwood Station, 1882. Oil on cardboard. Gift of Mrs. John Stevenson Watt, 1929 (29.189).
Attributed to Helena de Kay Gilder. Cover Design for "The New Day: A Poem in Songs and Sonnet" by Richard Watson Gilder. New York: Scribner, Armstrong & Company, 1875. Museum Collection (INV.10731).
Mary Frey. Girls Sunbathing, from the Domestic Rituals Series, 1979–83. Gelatin silver print. Gift of the Artist, 1984 (84.22).
Derrick Adams (American, b. 1970). Orbiting Us #18, 2018. Mixed-media collage on paper. Museum Purchase, 2018 (2018.07).
Harold Knickerbocker Faye (American, 1910–1980). Moonlight, ca. 1935–38. Intaglio on Rives paper. Gift of Helen S. Faye, 1990 (90.10.8).
Dora Wheeler Keith (American, 1856–1940). Christmas Card: Shout with Joy, 1882. Chromolithograph, silk fringe. Collection of the Hudson River Museum (INV.10397).
Jacob Lawrence. The Studio, 1996. Lithograph. Museum Purchase, 2018 (2018.05). © 2018 The Jacob and Gwendolyn Knight Lawrence Foundation, Seattle / Artists Rights Society (ARS), New York. Reproduction, including downloading of Jacob Lawrence works is prohibited by copyright laws and international conventions without the express written permission of Artists Rights Society (ARS), New York.
Elihu Vedder (American, 1836–1923). Stella Funesta (The Evil Star), ca. 1892. Pastel and charcoal on paper. Gift of the American Academy of Arts and Letters, 1955 (55.24c).
Jasper Francis Cropsey (American, 1823–1900). Greenwood Lake, New Jersey, 1897. Watercolor on paper. Anonymous Gift, 2017 (2017.07).
Goings on About Town: Above & Beyond The New Yorker (January 26, 2018)
|
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| null | null |
package examples.csci567.lecture7;
import android.app.AlarmManager;
import android.app.PendingIntent;
import android.content.BroadcastReceiver;
import android.content.Context;
import android.content.Intent;
import android.util.Log;
/**
* Created by bryandixon on 2/5/15.
*/
public class AlarmReceiver extends BroadcastReceiver {
private AlarmManager alarmMgr;
private PendingIntent alarmIntent;
private static final String TAG = "Lecture7-Alarm";
@Override
public void onReceive(Context context, Intent intent) {
Log.d(TAG, "ALARM");
}
public void setAlarm(Context context){
alarmMgr = (AlarmManager)context.getSystemService(Context.ALARM_SERVICE);
Intent intent = new Intent(context, AlarmReceiver.class);
alarmIntent = PendingIntent.getBroadcast(context, 0, intent, 0);
/*Calendar calendar = Calendar.getInstance();
calendar.setTimeInMillis(System.currentTimeMillis());
// Set the alarm's trigger time to 8:30 a.m.
calendar.set(Calendar.HOUR_OF_DAY, 8);
calendar.set(Calendar.MINUTE, 30);*/
/*
* If you don't have precise time requirements, use an inexact repeating alarm
* the minimize the drain on the device battery.
*
* The call below specifies the alarm type, the trigger time, the interval at
* which the alarm is fired, and the alarm's associated PendingIntent.
* It uses the alarm type RTC_WAKEUP ("Real Time Clock" wake up), which wakes up
* the device and triggers the alarm according to the time of the device's clock.
*
* Alternatively, you can use the alarm type ELAPSED_REALTIME_WAKEUP to trigger
* an alarm based on how much time has elapsed since the device was booted. This
* is the preferred choice if your alarm is based on elapsed time--for example, if
* you simply want your alarm to fire every 60 minutes. You only need to use
* RTC_WAKEUP if you want your alarm to fire at a particular date/time. Remember
* that clock-based time may not translate well to other locales, and that your
* app's behavior could be affected by the user changing the device's time setting.
*
* Here are some examples of ELAPSED_REALTIME_WAKEUP:
*
* // Wake up the device to fire a one-time alarm in one minute.
* alarmMgr.set(AlarmManager.ELAPSED_REALTIME_WAKEUP,
* SystemClock.elapsedRealtime() +
* 60*1000, alarmIntent);
*
* // Wake up the device to fire the alarm in 30 minutes, and every 30 minutes
* // after that.
* alarmMgr.setInexactRepeating(AlarmManager.ELAPSED_REALTIME_WAKEUP,
* AlarmManager.INTERVAL_HALF_HOUR,
* AlarmManager.INTERVAL_HALF_HOUR, alarmIntent);
*/
// Set the alarm to fire at approximately 8:30 a.m., according to the device's
// clock, and to repeat once a day.
//alarmMgr.setInexactRepeating(AlarmManager.RTC_WAKEUP,
// calendar.getTimeInMillis(), AlarmManager.INTERVAL_DAY, alarmIntent);
//Fire Repeating Alarm every one minute.
alarmMgr.setInexactRepeating(AlarmManager.ELAPSED_REALTIME_WAKEUP, AlarmManager.INTERVAL_FIFTEEN_MINUTES/15/4,
AlarmManager.INTERVAL_FIFTEEN_MINUTES/15/4, alarmIntent);
}
/**
* Cancels the alarm.
* @param context
*/
public void cancelAlarm(Context context) {
// If the alarm has been set, cancel it.
if (alarmMgr!= null) {
alarmMgr.cancel(alarmIntent);
}
}
}
|
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| 3,235
|
Home Our Firm Defence Services
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Taylor Vaughan are Regulated by Solicitors' Regulation Authority
Defence Services: Crown Court Representation
A person can find themselves before the Crown Court for many reasons as a result of:-
Electing Trial with a Judge and jury where their case falls within a group of crimes known as "either way offences".
Facing Trial for an indictable only offence where only the Crown Court can hear such Trials, e.g. Murder, Rape, Robbery etc.
Where Magistrates decline jurisdiction on a case they feel is too serious to be dealt with in their court.
Appealing a decision, the conviction or sentence of the lower courts.
Where having pleaded guilty in the Magistrates' Court, the matter has been committed for sentence where sentencing powers were insufficient.
Representation at the Crown Court is by Barrister or Solicitor Advocate. Helen Sechiari practiced at the Bar from Manchester House Chambers for many years before switching to become a solicitor and then with Roger Taylor setting up the firm of Taylor Vaughan. Helen continues her Crown Court advocacy as Solicitor Advocate preparing and running many of our Crown Court Trials from the start of the case to its conclusion.
With the higher court's sentencing powers, the dock at the Crown Court can be a cold and lonely place and it helps to have confidence with an experienced and capable legal team working for you. We don't keep clients in the dark and prefer straight talking to legal mumbo jumbo. We attempt to be direct whenever possible, keeping clients informed of exactly where they are and what they may face. We can't predict the future, beware of people who do, but experience does count for a great deal which clients prefer rather than facing the ordeal in uninformed ignorance.
Please feel free to contact us at anytime by email or using our 24 hr helpline: 0161 763 1066
© 2015 - 2020 Taylor Vaughan Solicitors | All Rights Reserved | Website Design By Irax Ltd
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,020
|
typedef void (*sa_sigaction_t)(int, siginfo_t *, void *);
namespace __sanitizer {
u32 GetUid() {
return getuid();
}
uptr GetThreadSelf() {
return (uptr)pthread_self();
}
void ReleaseMemoryPagesToOS(uptr beg, uptr end) {
uptr page_size = GetPageSizeCached();
uptr beg_aligned = RoundUpTo(beg, page_size);
uptr end_aligned = RoundDownTo(end, page_size);
if (beg_aligned < end_aligned)
// In the default Solaris compilation environment, madvise() is declared
// to take a caddr_t arg; casting it to void * results in an invalid
// conversion error, so use char * instead.
madvise((char *)beg_aligned, end_aligned - beg_aligned,
SANITIZER_MADVISE_DONTNEED);
}
void SetShadowRegionHugePageMode(uptr addr, uptr size) {
#ifdef MADV_NOHUGEPAGE // May not be defined on old systems.
if (common_flags()->no_huge_pages_for_shadow)
madvise((char *)addr, size, MADV_NOHUGEPAGE);
else
madvise((char *)addr, size, MADV_HUGEPAGE);
#endif // MADV_NOHUGEPAGE
}
bool DontDumpShadowMemory(uptr addr, uptr length) {
#if defined(MADV_DONTDUMP)
return madvise((char *)addr, length, MADV_DONTDUMP) == 0;
#elif defined(MADV_NOCORE)
return madvise((char *)addr, length, MADV_NOCORE) == 0;
#else
return true;
#endif // MADV_DONTDUMP
}
static rlim_t getlim(int res) {
rlimit rlim;
CHECK_EQ(0, getrlimit(res, &rlim));
return rlim.rlim_cur;
}
static void setlim(int res, rlim_t lim) {
struct rlimit rlim;
if (getrlimit(res, const_cast<struct rlimit *>(&rlim))) {
Report("ERROR: %s getrlimit() failed %d\n", SanitizerToolName, errno);
Die();
}
rlim.rlim_cur = lim;
if (setrlimit(res, const_cast<struct rlimit *>(&rlim))) {
Report("ERROR: %s setrlimit() failed %d\n", SanitizerToolName, errno);
Die();
}
}
void DisableCoreDumperIfNecessary() {
if (common_flags()->disable_coredump) {
setlim(RLIMIT_CORE, 0);
}
}
bool StackSizeIsUnlimited() {
rlim_t stack_size = getlim(RLIMIT_STACK);
return (stack_size == RLIM_INFINITY);
}
void SetStackSizeLimitInBytes(uptr limit) {
setlim(RLIMIT_STACK, (rlim_t)limit);
CHECK(!StackSizeIsUnlimited());
}
bool AddressSpaceIsUnlimited() {
rlim_t as_size = getlim(RLIMIT_AS);
return (as_size == RLIM_INFINITY);
}
void SetAddressSpaceUnlimited() {
setlim(RLIMIT_AS, RLIM_INFINITY);
CHECK(AddressSpaceIsUnlimited());
}
void SleepForSeconds(int seconds) {
sleep(seconds);
}
void SleepForMillis(int millis) {
usleep(millis * 1000);
}
void Abort() {
#if !SANITIZER_GO
// If we are handling SIGABRT, unhandle it first.
// TODO(vitalybuka): Check if handler belongs to sanitizer.
if (GetHandleSignalMode(SIGABRT) != kHandleSignalNo) {
struct sigaction sigact;
internal_memset(&sigact, 0, sizeof(sigact));
sigact.sa_sigaction = (sa_sigaction_t)SIG_DFL;
internal_sigaction(SIGABRT, &sigact, nullptr);
}
#endif
abort();
}
int Atexit(void (*function)(void)) {
#if !SANITIZER_GO
return atexit(function);
#else
return 0;
#endif
}
bool SupportsColoredOutput(fd_t fd) {
return isatty(fd) != 0;
}
#if !SANITIZER_GO
// TODO(glider): different tools may require different altstack size.
static const uptr kAltStackSize = SIGSTKSZ * 4; // SIGSTKSZ is not enough.
void SetAlternateSignalStack() {
stack_t altstack, oldstack;
CHECK_EQ(0, sigaltstack(nullptr, &oldstack));
// If the alternate stack is already in place, do nothing.
// Android always sets an alternate stack, but it's too small for us.
if (!SANITIZER_ANDROID && !(oldstack.ss_flags & SS_DISABLE)) return;
// TODO(glider): the mapped stack should have the MAP_STACK flag in the
// future. It is not required by man 2 sigaltstack now (they're using
// malloc()).
void* base = MmapOrDie(kAltStackSize, __func__);
altstack.ss_sp = (char*) base;
altstack.ss_flags = 0;
altstack.ss_size = kAltStackSize;
CHECK_EQ(0, sigaltstack(&altstack, nullptr));
}
void UnsetAlternateSignalStack() {
stack_t altstack, oldstack;
altstack.ss_sp = nullptr;
altstack.ss_flags = SS_DISABLE;
altstack.ss_size = kAltStackSize; // Some sane value required on Darwin.
CHECK_EQ(0, sigaltstack(&altstack, &oldstack));
UnmapOrDie(oldstack.ss_sp, oldstack.ss_size);
}
static void MaybeInstallSigaction(int signum,
SignalHandlerType handler) {
if (GetHandleSignalMode(signum) == kHandleSignalNo) return;
struct sigaction sigact;
internal_memset(&sigact, 0, sizeof(sigact));
sigact.sa_sigaction = (sa_sigaction_t)handler;
// Do not block the signal from being received in that signal's handler.
// Clients are responsible for handling this correctly.
sigact.sa_flags = SA_SIGINFO | SA_NODEFER;
if (common_flags()->use_sigaltstack) sigact.sa_flags |= SA_ONSTACK;
CHECK_EQ(0, internal_sigaction(signum, &sigact, nullptr));
VReport(1, "Installed the sigaction for signal %d\n", signum);
}
void InstallDeadlySignalHandlers(SignalHandlerType handler) {
// Set the alternate signal stack for the main thread.
// This will cause SetAlternateSignalStack to be called twice, but the stack
// will be actually set only once.
if (common_flags()->use_sigaltstack) SetAlternateSignalStack();
MaybeInstallSigaction(SIGSEGV, handler);
MaybeInstallSigaction(SIGBUS, handler);
MaybeInstallSigaction(SIGABRT, handler);
MaybeInstallSigaction(SIGFPE, handler);
MaybeInstallSigaction(SIGILL, handler);
MaybeInstallSigaction(SIGTRAP, handler);
}
bool SignalContext::IsStackOverflow() const {
// Access at a reasonable offset above SP, or slightly below it (to account
// for x86_64 or PowerPC redzone, ARM push of multiple registers, etc) is
// probably a stack overflow.
#ifdef __s390__
// On s390, the fault address in siginfo points to start of the page, not
// to the precise word that was accessed. Mask off the low bits of sp to
// take it into account.
bool IsStackAccess = addr >= (sp & ~0xFFF) && addr < sp + 0xFFFF;
#else
// Let's accept up to a page size away from top of stack. Things like stack
// probing can trigger accesses with such large offsets.
bool IsStackAccess = addr + GetPageSizeCached() > sp && addr < sp + 0xFFFF;
#endif
#if __powerpc__
// Large stack frames can be allocated with e.g.
// lis r0,-10000
// stdux r1,r1,r0 # store sp to [sp-10000] and update sp by -10000
// If the store faults then sp will not have been updated, so test above
// will not work, because the fault address will be more than just "slightly"
// below sp.
if (!IsStackAccess && IsAccessibleMemoryRange(pc, 4)) {
u32 inst = *(unsigned *)pc;
u32 ra = (inst >> 16) & 0x1F;
u32 opcd = inst >> 26;
u32 xo = (inst >> 1) & 0x3FF;
// Check for store-with-update to sp. The instructions we accept are:
// stbu rs,d(ra) stbux rs,ra,rb
// sthu rs,d(ra) sthux rs,ra,rb
// stwu rs,d(ra) stwux rs,ra,rb
// stdu rs,ds(ra) stdux rs,ra,rb
// where ra is r1 (the stack pointer).
if (ra == 1 &&
(opcd == 39 || opcd == 45 || opcd == 37 || opcd == 62 ||
(opcd == 31 && (xo == 247 || xo == 439 || xo == 183 || xo == 181))))
IsStackAccess = true;
}
#endif // __powerpc__
// We also check si_code to filter out SEGV caused by something else other
// then hitting the guard page or unmapped memory, like, for example,
// unaligned memory access.
auto si = static_cast<const siginfo_t *>(siginfo);
return IsStackAccess &&
(si->si_code == si_SEGV_MAPERR || si->si_code == si_SEGV_ACCERR);
}
#endif // SANITIZER_GO
bool IsAccessibleMemoryRange(uptr beg, uptr size) {
uptr page_size = GetPageSizeCached();
// Checking too large memory ranges is slow.
CHECK_LT(size, page_size * 10);
int sock_pair[2];
if (pipe(sock_pair))
return false;
uptr bytes_written =
internal_write(sock_pair[1], reinterpret_cast<void *>(beg), size);
int write_errno;
bool result;
if (internal_iserror(bytes_written, &write_errno)) {
CHECK_EQ(EFAULT, write_errno);
result = false;
} else {
result = (bytes_written == size);
}
internal_close(sock_pair[0]);
internal_close(sock_pair[1]);
return result;
}
void PlatformPrepareForSandboxing(__sanitizer_sandbox_arguments *args) {
// Some kinds of sandboxes may forbid filesystem access, so we won't be able
// to read the file mappings from /proc/self/maps. Luckily, neither the
// process will be able to load additional libraries, so it's fine to use the
// cached mappings.
MemoryMappingLayout::CacheMemoryMappings();
}
static bool MmapFixed(uptr fixed_addr, uptr size, int additional_flags,
const char *name) {
size = RoundUpTo(size, GetPageSizeCached());
fixed_addr = RoundDownTo(fixed_addr, GetPageSizeCached());
uptr p =
MmapNamed((void *)fixed_addr, size, PROT_READ | PROT_WRITE,
MAP_PRIVATE | MAP_FIXED | additional_flags | MAP_ANON, name);
int reserrno;
if (internal_iserror(p, &reserrno)) {
Report("ERROR: %s failed to "
"allocate 0x%zx (%zd) bytes at address %zx (errno: %d)\n",
SanitizerToolName, size, size, fixed_addr, reserrno);
return false;
}
IncreaseTotalMmap(size);
return true;
}
bool MmapFixedNoReserve(uptr fixed_addr, uptr size, const char *name) {
return MmapFixed(fixed_addr, size, MAP_NORESERVE, name);
}
bool MmapFixedSuperNoReserve(uptr fixed_addr, uptr size, const char *name) {
#if SANITIZER_FREEBSD
if (common_flags()->no_huge_pages_for_shadow)
return MmapFixedNoReserve(fixed_addr, size, name);
// MAP_NORESERVE is implicit with FreeBSD
return MmapFixed(fixed_addr, size, MAP_ALIGNED_SUPER, name);
#else
bool r = MmapFixedNoReserve(fixed_addr, size, name);
if (r)
SetShadowRegionHugePageMode(fixed_addr, size);
return r;
#endif
}
uptr ReservedAddressRange::Init(uptr size, const char *name, uptr fixed_addr) {
base_ = fixed_addr ? MmapFixedNoAccess(fixed_addr, size, name)
: MmapNoAccess(size);
size_ = size;
name_ = name;
(void)os_handle_; // unsupported
return reinterpret_cast<uptr>(base_);
}
// Uses fixed_addr for now.
// Will use offset instead once we've implemented this function for real.
uptr ReservedAddressRange::Map(uptr fixed_addr, uptr size, const char *name) {
return reinterpret_cast<uptr>(
MmapFixedOrDieOnFatalError(fixed_addr, size, name));
}
uptr ReservedAddressRange::MapOrDie(uptr fixed_addr, uptr size,
const char *name) {
return reinterpret_cast<uptr>(MmapFixedOrDie(fixed_addr, size, name));
}
void ReservedAddressRange::Unmap(uptr addr, uptr size) {
CHECK_LE(size, size_);
if (addr == reinterpret_cast<uptr>(base_))
// If we unmap the whole range, just null out the base.
base_ = (size == size_) ? nullptr : reinterpret_cast<void*>(addr + size);
else
CHECK_EQ(addr + size, reinterpret_cast<uptr>(base_) + size_);
size_ -= size;
UnmapOrDie(reinterpret_cast<void*>(addr), size);
}
void *MmapFixedNoAccess(uptr fixed_addr, uptr size, const char *name) {
return (void *)MmapNamed((void *)fixed_addr, size, PROT_NONE,
MAP_PRIVATE | MAP_FIXED | MAP_NORESERVE | MAP_ANON,
name);
}
void *MmapNoAccess(uptr size) {
unsigned flags = MAP_PRIVATE | MAP_ANON | MAP_NORESERVE;
return (void *)internal_mmap(nullptr, size, PROT_NONE, flags, -1, 0);
}
// This function is defined elsewhere if we intercepted pthread_attr_getstack.
extern "C" {
SANITIZER_WEAK_ATTRIBUTE int
real_pthread_attr_getstack(void *attr, void **addr, size_t *size);
} // extern "C"
int my_pthread_attr_getstack(void *attr, void **addr, uptr *size) {
#if !SANITIZER_GO && !SANITIZER_MAC
if (&real_pthread_attr_getstack)
return real_pthread_attr_getstack((pthread_attr_t *)attr, addr,
(size_t *)size);
#endif
return pthread_attr_getstack((pthread_attr_t *)attr, addr, (size_t *)size);
}
#if !SANITIZER_GO
void AdjustStackSize(void *attr_) {
pthread_attr_t *attr = (pthread_attr_t *)attr_;
uptr stackaddr = 0;
uptr stacksize = 0;
my_pthread_attr_getstack(attr, (void**)&stackaddr, &stacksize);
// GLibC will return (0 - stacksize) as the stack address in the case when
// stacksize is set, but stackaddr is not.
bool stack_set = (stackaddr != 0) && (stackaddr + stacksize != 0);
// We place a lot of tool data into TLS, account for that.
const uptr minstacksize = GetTlsSize() + 128*1024;
if (stacksize < minstacksize) {
if (!stack_set) {
if (stacksize != 0) {
VPrintf(1, "Sanitizer: increasing stacksize %zu->%zu\n", stacksize,
minstacksize);
pthread_attr_setstacksize(attr, minstacksize);
}
} else {
Printf("Sanitizer: pre-allocated stack size is insufficient: "
"%zu < %zu\n", stacksize, minstacksize);
Printf("Sanitizer: pthread_create is likely to fail.\n");
}
}
}
#endif // !SANITIZER_GO
pid_t StartSubprocess(const char *program, const char *const argv[],
fd_t stdin_fd, fd_t stdout_fd, fd_t stderr_fd) {
auto file_closer = at_scope_exit([&] {
if (stdin_fd != kInvalidFd) {
internal_close(stdin_fd);
}
if (stdout_fd != kInvalidFd) {
internal_close(stdout_fd);
}
if (stderr_fd != kInvalidFd) {
internal_close(stderr_fd);
}
});
int pid = internal_fork();
if (pid < 0) {
int rverrno;
if (internal_iserror(pid, &rverrno)) {
Report("WARNING: failed to fork (errno %d)\n", rverrno);
}
return pid;
}
if (pid == 0) {
// Child subprocess
if (stdin_fd != kInvalidFd) {
internal_close(STDIN_FILENO);
internal_dup2(stdin_fd, STDIN_FILENO);
internal_close(stdin_fd);
}
if (stdout_fd != kInvalidFd) {
internal_close(STDOUT_FILENO);
internal_dup2(stdout_fd, STDOUT_FILENO);
internal_close(stdout_fd);
}
if (stderr_fd != kInvalidFd) {
internal_close(STDERR_FILENO);
internal_dup2(stderr_fd, STDERR_FILENO);
internal_close(stderr_fd);
}
for (int fd = sysconf(_SC_OPEN_MAX); fd > 2; fd--) internal_close(fd);
execv(program, const_cast<char **>(&argv[0]));
internal__exit(1);
}
return pid;
}
bool IsProcessRunning(pid_t pid) {
int process_status;
uptr waitpid_status = internal_waitpid(pid, &process_status, WNOHANG);
int local_errno;
if (internal_iserror(waitpid_status, &local_errno)) {
VReport(1, "Waiting on the process failed (errno %d).\n", local_errno);
return false;
}
return waitpid_status == 0;
}
int WaitForProcess(pid_t pid) {
int process_status;
uptr waitpid_status = internal_waitpid(pid, &process_status, 0);
int local_errno;
if (internal_iserror(waitpid_status, &local_errno)) {
VReport(1, "Waiting on the process failed (errno %d).\n", local_errno);
return -1;
}
return process_status;
}
bool IsStateDetached(int state) {
return state == PTHREAD_CREATE_DETACHED;
}
} // namespace __sanitizer
#endif // SANITIZER_POSIX
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,357
|
What Does the Egyptian Revolution Mean for the United States Government?
By Hillary Mann Leverett (Posted Feb 12, 2011)
RevolutionsEgypt, Iraq, Israel, United StatesNews
The US has not supported democratization in Egypt, or really anywhere else in the Middle East, because US policymakers would not like the outcome of democratic processes. Policies made by governments that are freely elected by the people would not reflect, would not support, let alone enforce, the US polices that are unpopular, whether that's the siege of Gaza, whether that's the invasions and occupations of Iraq and Afghanistan, the people in Egypt are not going to continue to support, let alone enforce, these very, very unpopular policies. . . . I think we need to really look at what President Obama has done, what he is responsible for on his watch, and how he is going to possibly . . . respond to this in a constructive way. . . . If he continues to pursue US policy as he has, he is going to be at odds with the people on the ground not only in Egypt but, I think, throughout the Middle East. . . . I think Barack Obama, Vice President Biden, and Secretary Clinton were on the phone, as much as they possibly could, with Omar Suleiman, trying to orchestrate his takeover after they realized that Mubarak wasn't going to be able to carry on. They wanted to have Omar Suleiman, the CIA's point man in Egypt, the person responsible for the rendition program that brought Egyptians home to be tortured; they did everything they could to orchestrate his continued rule over Egypt. . . .
I think [the Egyptian revolution] is a good situation for the American people. It certainly puts the American administration, President Obama, in a very difficult dilemma. He has to now deal with the fact that he would be dealing with a government, however this turns out, that is going to have to reflect more the will of the people, and the will of the people is not to support the blockade of Gaza, is not to support the unending occupations of Iraq and Afghanistan, is not, potentially, to support the bombing of Iran.
Hillary Mann Leverett is CEO of Strategic Energy and Global Analysis (STRATEGA), a political risk consultancy. She is also Senior Lecturer and Senior Research Fellow at Yale University's Jackson Institute for Global Affairs. With Flynt Leverett, she keeps a blog The Race for Iran: <www.raceforiran.com>. The video above compiles Al Jazeera's interviews with Hillary Mann Leverett on 11 February 2011. The text above is an edited partial transcript of the interview. Cf. Hassan Nasrallah, "On the Egyptian Revolution and the American Strategy" (7 February 2011); and "[T]he White House and the State Department were already discussing setting aside new funds to bolster the rise of secular political parties. . . . [O]ther officials have acknowledged privately that if Egypt turns into a noisy democracy that includes the Muslim Brotherhood, there will undoubtedly be political debate in Egypt about whether the 1979 peace accord with Israel should remain in force" (David E. Sanger, "Obama Presses Egypt's Military on Democracy," New York Times, 11 February 2011).
On the Egyptian Revolution and the American Strategy
Integration instead of a Clash of Cultures: An Open Letter regarding the "LSE German Symposium 2011 — Integration Debate"
Also by Hillary Mann Leverett
Iranian "Plots" and American Hubris by Flynt Leverett, Hillary Mann Leverett October 14, 2011
The Key to Progress in Nuclear Diplomacy with Iran by Flynt Leverett, Hillary Mann Leverett August 20, 2011
The Race with Iran: Saudi Arabia's Sectarian Card by Flynt Leverett, Hillary Mann Leverett August 12, 2011
Listening to What Iranians Say about Their Nuclear Program Instead of Relying on "Intelligence" and Agenda-driven "Analysis" by Flynt Leverett, Hillary Mann Leverett August 08, 2011
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,512
|
Q: How to increase waiting time for HttpClient request in angular 5? I am using HttpClient to make a sever call and wanted to wait until it respond back. This particular service call takes more than 2 minutes.
Following code only waits for 2 minutes even though I have added timeout of 3 minutes.
this.http.get('serverUrl')
.timeout(180000)
.catch(() => {
return Observable.throw('Error');
});
I also tried implementing timeout with pipe operator. But this approach also timeout after 2 minutes of wait.
this.http.get('serverUrl')
.pipe(timeout(180000))
.catch(() => {
return Observable.throw('Error');
});
Using timeout, I was able to decrease the service call waiting time but could not increase the waiting time.
I followed some of the previous posts regarding the similar problem, but those solutions did not work.
Can't have a timeout of over 2 minutes with this.http.get?
Increase timeout in Angular 2+ and ASP.NET Core WebAPI application
The expectation is to increase the waiting time more than 2 minutes. I tested in Chrome and IE browser and found that service call using HttpClient is only waiting for 2 minutes for each service call.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,075
|
Q: Not returning from accept when trying to listen to bluetooth communication - Android I'm trying to establish a bluetooth communication between an android phone/tablet (4.0.3), and a bluetooth device, which is an earring reader (Destron Fearring DTR3E, in case you want to know, which I don't suppose you do).
I paired the phone with the reader (the reader has the pairing passcode on a tag) from the bluetooth settings, bluetooth is on of course, and now I'm trying to listen to reads from the device, by means of BluetoothServerSocket. The problem is that the accept call never returns, so obviously I am doing something wrong. The communication is done using RFCOMM.
Code:
private class AcceptThread extends Thread {
private final BluetoothServerSocket mmServerSocket;
public AcceptThread() {
// Use a temporary object that is later assigned to mmServerSocket,
// because mmServerSocket is final
BluetoothServerSocket tmp = null;
try {
// MY_UUID is the app's UUID string, also used by the client code
String uuid = "00001101-0000-1000-8000-00805F9B34FB";
tmp = bluetoothAdapter.listenUsingInsecureRfcommWithServiceRecord("pdfParserServer", UUID.fromString(uuid));
} catch (Exception e) {
e.printStackTrace();
}
mmServerSocket = tmp;
}
public void run() {
BluetoothSocket socket = null;
// Keep listening until exception occurs or a socket is returned
while (true) {
try {
socket = mmServerSocket.accept();
} catch (IOException e) {
break;
}
// If a connection was accepted
if (socket != null) {
// Do work to manage the connection (in a separate thread)
try {
mmServerSocket.close();
} catch (IOException e) {
e.printStackTrace();
}
break;
}
}
}
/** Will cancel the listening socket, and cause the thread to finish */
public void cancel() {
try {
mmServerSocket.close();
} catch (IOException e) { }
}
}
Is there something I am missing?
Thank you!
A: The only reason that could cause the code never to come back from accept is that, the device "Destron Fearring DTR3E" you are trying to connect to, has actually a bluetoothserver socket and not a bluetooth client, hence, the device might be waiting for you to actually connect to it, in stead of you creating a bluetoothserver socket and waiting for it to connect to your android device, you should read the specs on the device and make sure that actually is you the one that has to open a connection on "Destron Fearring DTR3E" socket...
Hope this helps...
Regards!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,515
|
Q: Make file that compiles gcc and puts the objects in a separate folder I am trying to create a make file that will work accordingly
OBJECT_DIRECTORY := out/obj
C_SOURCE_FILES = (Path and files fetched from different make files)
CFLAGS += -mcpu=$(CPU) -mthumb -mabi=aapcs -mfloat-abi=soft
CFLAGS += -Wall -Werror
CFLAGS += -D$(DEVICE) -std=gnu99
$(OBJECT_DIRECTORY)/%.o: %.c
$(CC) $(C_SOURCE_FILES) $(CFLAGS) -c -o $@ $<
I verified that the C_SOURCE_FILES variable actually contains the c source files. This since I am able to compile using this:
$(OBJECT_DIRECTORY)/%.o:
$(CC) $(C_SOURCE_FILES) $(CFLAGS) -c
Problem with that is that the object files are not placed in the folder where I need them for the linking.
By the way, I am executing the make file from the Eclipse C/C++ IDE
I would be extremely happy if someone could help me solve this problem.
A: Try this:
$(OBJECT_DIRECTORY)/%.o: %.c
$(CC) $(CFLAGS) -c -o $@ $<
You don't need the C_SOURCE_FILES variable here. This recipe will create out/obj/{file}.o for every file called {file}.c.
You don't show it, but the dependency list for the executable being created from the object files must explicitly call out each object file.
A: I solved the problem, I think... By creating a list of target files to be used in the compilation.
# Create the object file names needed for the target. Make needs these file names as target for the compilation
TEMP := $(notdir $(C_SOURCE_FILES:.c=.o))
C_OBJECT_FILES := $(addprefix $(OBJECT_DIRECTORY)/, $(TEMP:.c=.o) )
TEMP_1 := $(notdir $(ASSEMBLER_SOURCE_FILES:.s=.o))
ASSEMBLER_OBJECT_FILES := $(addprefix $(OBJECT_DIRECTORY)/,$(TEMP_1:.s=.o) )
# Tells make to compile all the files in the C_OBJECTS_VARIABLE
all: $(C_OBJECT_FILES)
# Do the compilation of the c files and place the object files in the out/obj folder
$(C_OBJECT_FILES): $(C_SOURCE_FILES)
$(CC) $(CFLAGS) -M $< -MF "$(@:.o=.d)" -MT $@
$(CC) $(CFLAGS) -c -o $@ $<
# Tells make to compile all the files in the ASSEMBLER_OBJECTS_VARIABLE
all: $(ASSEMBLER_OBJECT_FILES)
# Do the compilation of the assembler files and place the object files in the out/obj folder
$(ASSEMBLER_OBJECT_FILES): $(ASSEMBLER_SOURCE_FILES)
$(CC) $(ASMFLAGS) -c -o $@ $<
This works well and consistently. Issue I have now is that the rule for linking is not executed at all...
## Link C and assembler objects to an .out file
$(BINARY_DIRECTORY)/$(OUTPUT_FILENAME).out: $(C_OBJECT_FILES) $(ASSEMBLER_OBJECT_FILES) $(LIBRARIES)
$(CC) $(LDFLAGS) -o $(BINARY_DIRECTORY)/$(OUTPUT_FILENAME).out
I am wondering if it has anything to do with "all:"
A: Found the solution to that problem as well. I created a couple of new rules to make sure that the linking is getting executed. Now the make file works well.
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Édouard Adolphe Casimir Joseph Mortier (13. února 1768 v Le Cateau-Cambrésis – 28. července 1835 v Paříži) byl francouzský maršál a později získal titul vévoda z Trevisa.
Život
Mortier působil v období prvního císařství, restaurace a červencové monarchie. Byl synem bohatého obchodníka suknem, který i vstoupil do politiky a účastnil se za třetí stav v roce 1789 zasedání generálních stavů. Budoucí maršál studoval v Douai na Collège Irlandais a roku 1789 vstoupil do národní gardy. Roku 1791 pak do pluku dobrovolníků francouzské armády a byl zvolen kapitánem. Bojoval roku 1792 u Quévrainu, a Jemappes v následujícím roce u Neerwinden, Hondschoote, roku 1794 u Fleurus. Byl při obsazení Namuru (1793) a Maastrichtu (1794) pak v roce 1795 se vyznamenal pod maršálem Lefebvrem a generálem Kléberem u Altenkirchen a Friedbergu.
V roce 1799 povýšen na brigádního generála a bojoval u Liptringen a Stockachu a poté převelen do Švýcarska k Massenově armádě, kde po bitvě u Curychu byl povýšen na divizního generála (1799) a velel 4. pěší divizi. Roku 1800 pak velel jedné z pařížských posádek. Po vypuknutí válečného stavu s Velkou Británií roku 1803 obsadil Hannoversko a dalším rokem se stal se vrchním velitelem dělostřelectva. Téhož roku 1804 byl jmenován maršálem císařství. V roce 1805 během postupu podél Dunaje vedl 2. sbor (levobřežní) a zanedbal průzkum což se mu vymstilo u Dürnsteinu, kde s Gazanovou divizí byl nucen svést ústupový boj proti čtyřnásobné převaze Rusů (Kutuzov, Miloradovič, Dochturov).
V době bitvy u Slavkova velel silám, kryjícím Vídeň. Roku 1806 obsadil 1. listopadu Hesensko a poté hansovní města a v roce 1807 porazil švédský sbor u Anklamu 16. a 17. dubna. V bitvě u Friedlantu 14. června 1807 velel levému křídlu francouzské armády. V roce 1808 obdržel titul vévoda z Trevisa a byl odvelen do Španělska, kde se svým sborem bojoval u Somosiery a roku 1809 s Lannesem dobyl Zaragozu, zvítězil v bitvě u Ocasia (19. listopadu 1809) a podpořil Soulta u Bajadozu. Později zvítězil v bitvě u Gebory (19. února 1811) aby roku 1812 se účastnil ruského tažení, kde při ústupu svými útoky získal čas k dokončení Napoleonova manévru, umožňujícímu záchranu armády na Berezině. V následujícím roce pak velel mladé gardě a v jejím čele bojoval u Lützenu, Budyšína, Drážďan, Lipska a Hanau. V roce 1814 bránil s Marmontem Paříž. 8. dubna 1814 se podřídil Ludvíkovi XVIII. a byl jmenován pairem.
Při návratu Napoleona z Elby doprovodil Mortier krále do Gentu, byl jím zbaven přísahy a ještě v březnu 1814 se opět přidal k Napoleonovi. Z jeho rozkazu provedl inspekci pevnostních zařízení na severní a západní hranici. Při druhé restauraci byl jmenován členem soudního tribunálu, který měl soudit maršála Neye, což však odmítl. Byl za to zbaven titulu paira, ale od roku 1819 opět v komoře pairů zasedal. Po červencovém převratu byl v letech 1830-1831 vyslancem Francie v Sankt Petěrburgu. Roku 1833 byl králem, Ludvíkem Filipem jmenován velkokancléřem Čestné legie a 18. listopadu 1834 se stal ministrem války Francie a současně ministerským předsedou. Ve funkci však zůstal jen tři měsíce (20. února 1835 demise). Při přehlídce vojska 28. července 1835 byl členem doprovodu krále, na kterého spáchal Joseph Fieschi na Boulevardu du Temple atentát pekelným strojem. Výbuchem bylo zabito 12 lidí (další zemřeli následkem zranění). Král byl jen lehce raněn na čele, ale mezi mrtvými byl maršál Mortier. Je pohřben v dómu Invalidovny blízko Napoleonovy hrobky. V jeho rodném městě stojí Mortierův památník a v Lille je jeho socha. Maršál Mortier byl ze všech Napoleonových maršálů nejvyšší postavy – měřil 195 cm a byl v armádě znám pod přezdívkou Grand mortier – Velký moždíř.
Odkazy
Reference
Literatura
Externí odkazy
Francouzští maršálové
Narození v roce 1768
Úmrtí v roce 1835
Osobnosti napoleonských válek
Oběti atentátů
Premiéři Francie
Červencová monarchie
Muži
Pohřbení na Père Lachaise
Vévodové Prvního Francouzského císařství
Zavraždění politici
Narození 13. února
Úmrtí 28. července
Zavraždění vojáci
Jména vepsaná pod Vítězným obloukem v Paříži
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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The Commando è un film thriller d'azione del 2022 diretto da Asif Akbar e interpretato da Mickey Rourke e Michael Jai White.
È stato distribuito negli Stati Uniti il 7 gennaio 2022 da Saban Films. È stato acquistato per l'Italia dalla Variety Distribution, che lo ha reso disponibile on demand.
Trama
Una squadra SWAT della DEA, guidata dall'agente d'élite James Baker, assalta il laboratorio di droga di un cartello messicano. I cattivi vengono eliminati nello scontro a fuoco che ne consegue, ma Baker uccide inavvertitamente tre ostaggi. A causa di allucinazioni e incubi dovuti alla PTSD, causati dall'uccisione di innocenti, Baker viene mandato a casa per riprendersi.
In questo periodo, la sua famiglia fa una scoperta inaspettata nella loro casa: una scorta di denaro del valore di 3 milioni di dollari. Si dà il caso che Baker viva, con la moglie Lisa e le due figlie adolescenti, nella casa in cui il criminale di professione Johnny ha nascosto il suo bottino. La famiglia Baker si trova presto ad affrontare il pericolo e la minaccia di Johnny, appena scarcerato, che si riunisce rapidamente alla sua vecchia banda per riprendersi i 3 milioni di dollari rubati che aveva nascosto prima del suo arresto. Johnny è un vero e proprio duro: poco prima del suo rilascio, ha "trattato" con tre detenuti che hanno tentato di fargli la pelle proprio prima di essere liberati.
Alla fine, James e Lisa partono per un tranquillo weekend insieme, lasciando le due figlie a casa da sole. La più giovane organizza subito una festa in casa. È durante questa festa che gli scagnozzi di Johnny inscenano una violazione di domicilio.
Johnny e i suoi scagnozzi faranno di tutto per recuperare il denaro, compreso il rapimento delle figlie di Baker. La posta in gioco è alta in questa battaglia testa a testa: Baker non si ferma davanti a nulla per proteggere la sua famiglia dai criminali assetati di denaro. Ci sono molti scontri, ma nessuno tra Baker e Johnny, fino ai minuti finali.
Uno dei momenti salienti è quello in cui un liceale viene colpito da un proiettile mentre fa la pipì addosso al suo assassino.
Produzione
Nel settembre 2020 è stato annunciato il ruolo di Rourke nel film. Nell'ottobre 2020 White si è unito al cast del film.
Distribuzione
Il film è stato acquistato nel 2022 dalla Variety Distribution per l'Italia ed è distribuito doppiato in lingua italiana sulla piattaforma on demand.
Note
Collegamenti esterni
Film thriller d'azione
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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'Allow me to join Naxals': AP Dalit man writes to President of India after assault by cops
"Honorable President, please allow me to join the Naxal movement. Because when law and order has failed to do justice, I want to look for another way to protect my dignity." These triggering lines are from a letter written to Indian President Ramnath Kovind by a Dalit youngster from Andhra Pradesh, whose head was recently tonsured, allegedly at the Seetanagaram police station. The incident took place on July 20 when Indugumilli Prasad (23), a graduate, was summoned to the police station after an alleged tiff with men belonging to the Kapu community.
Prasad alleged that the Sub Inspector (SI) and other policemen acted at the behest of a local YSRCP MLA, as he was questioning illegal sand mining in the region, and humiliated him. Soon after the incident came to light, SI Sheikh Feroze Shah had been suspended and arrested before being sent to jail along with two constables, while a few other police personnel were facing "disciplinary action".
The victim alleges that except arresting the SI and two constables, there has been no action against the other accused. In the letter which he mailed to the Secretariat of President of India, he said " I am writing this as till now, no one else is arrested and the call details of the police officers are not made public, to hide the real criminals in the background."
He has also alleged that the East Godavari District Superintendent of Police and Collector too, had not taken the discrimination that was meted out to him seriously.
Speaking to TNM, he alleged that the SI was persuaded to humiliate him, by being offered a permanent posting in a police station of his choice, as he was under probation. According to him, all the accused in the case have ties to the ruling YSRCP in the state.
When asked about what triggered him to look towards Naxalism, he said "I was beaten up black and blue. I was pushed into a bathroom and my head was tonsured. Many officials, politicians assured me of justice but nothing happened. I feel if I join Naxalism, I will get justice. Neither did the government bother about me, nor did the police or politicians. Everyone has been unjust and I want justice. I feel that is the only path."
Prasad further claimed that a slanderous campaign was being carried out against him to tarnish his image in his area.
When TNM asked if he had written the letter to the President under the pressure or influence of any political party, Prasad denied this, saying, "There is nothing as such. All parties are coming to see me."
Responding to the row, Deputy Inspector General (DIG) KV Mohan Rao said that necessary steps had been taken for the protection of the complainant Prasad. He further added that a few people, with vested interests, were 'using' the complainant for political mileage and said that there would be action against such people. He claimed that there are a few politicians behind Prasad's statements.
"Those statements are in violation of public safety. Maoist organisations are banned in the state," he said, adding that such statements could encourage others to participate in illegal activities.
The senior police officer urged people to act within the purview of the Constitution and said that they should not misuse freedom of speech and equal rights granted by it.
Courtesy : TNM
A Bleak Week for Rights in the UK
Call For Release Of Chandrashekhar Azad 'Ravan' – India's Malcom X
The Legacy of Chief Justice Bobde- Evasion, Hypocrisy, and Duplicity
25th anniversary of the disappearance of Bhai Jaswant Singh Khalra Human Rights Activist #IfWeDoNo...
Advocacy, Announcements, Health Care, Human Rights, Justice, Kractivism, Law, Politics
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
The recent discovery of the anorganic spin--Peierls compounds
CuGeO$_3$ \cite{Hase93} and $\alpha'$--NaV$_2$O$_5$
\cite{Isobe96a,Weiden97a} has great\-ly stimulated interest in
low--dimensional magne\-tism. While many properties of these novel
materials can be described in terms of quasi {\em one}--dimensional
(1D) quantum spin--systems, clear evidence for a substantial degree of
{\em two}--dimensionality (2D) of their magnetism has been found -- most
noteworthy by inelastic neutron scattering (INS) which displays a
sizeable transverse dispersion of the magnetic excitations in
CuGeO$_3$ \cite{Nishi94,Regnault95a,Regnault96a}. In the present
study I will establish a simple framework to interpret the spin
dynamics of a frustrated and dimerized 2D quantum spin--model with a
particular focus on the low--temperature phase of CuGeO$_3$.
CuGeO$_3$ is an anorganic spin--Peierls system with a lattice
dimerization transition at a temperature $T_{SP}\simeq 14K$
\cite{Hase93,Nishi94,Regnault95a,Regnault96a,Martin96a,Pouget94}. Its
structure comprises of weakly coupled CuO$_2$ chains along the
c--axis, with copper in a spin--1/2 state
\cite{Voellenkle67,Hirota94a}. The nearest--neighbor ($n.n.$)
exchange--coupling between copper spins along the CuO$_2$ chains is
strongly reduced by almost orthogonal intermediate oxygen states
\cite{Braden96}. Therefore, next--nearest--neighbor ($n.n.n.$)
exchange in CuGeO$_3$ is relevant. Both, $n.n.$ and $n.n.n.$
exchange, are antiferromagnetic (AFM) \cite{Braden96,Geertsma96}
implying intra--chain frustration. In addition, $n.n.$, as well as
$n.n.n.$ {\em inter}--chain exchange is present which proceeds via the
O2 sites \cite{Braden96,Khomskii96a}. This exchange is believed to be
one order of magnitude less than the intra--chain coupling
\cite{Nishi94,Regnault95a,Regnault96a,Braden96,Khomskii96a} and
comparable to $T_{SP}$. Therefore the inter--chain coupling should be
relevant in the dimerized phase. This may be a key element in the INS
\cite{Nishi94,Regnault95a,Regnault96a,Martin96a} and magnetic Raman
scattering \cite{Kuroe94,Loosdrecht96,Lemmens96} data.
\begin{figure}[tb]
\vskip .2cm
\centerline{\psfig{file=bobofig1.eps,width=7cm}}
\vskip .1cm
\caption[l]{The $J$-$\lambda$-$\alpha$-$\mu$-$\beta$ model. Line
segments refer to exchange couplings for spins located at segment
vertices. Inter--chain couplings are shown for a single dimer site
only. ${\bf b}=b{\bf e}_b$ and ${\bf c}=c{\bf e}_c$ are the primitive
vectors.}
\label{bobofig1}
\end{figure}
A minimal model of CuGeO$_3$ which includes intra--, as well as
inter--chain interactions is the $J$-$\lambda$-$\alpha$-$\mu$-$\beta$
model \cite{Braden96,Uhrig97a} depicted in fig.~\ref{bobofig1}. The
various line segments label the coupling strengths
$J,J\lambda,J\lambda\alpha,J\mu$, and $J\mu\beta$ between the spins
located at the vertices in this figure. $J$ refers to the 'strongest'
or dimer--bond the left vertices of which form the dimer lattice ${\bf
l}\in {\cal D}$. Most important the dimerization in
fig.~\ref{bobofig1} is staggered along the b--axis. This is realized
both, in CuGeO$_3$ \cite{Hirota94a} as well as in
$\alpha'$--NaV$_2$O$_5$ \cite{Fujii96a}, and turns out to be relevant
for the magnon dispersion. In fig.~\ref{bobofig1} an additional,
so--called 'natural' labeling of the intra--chain parameters is
introduced, i.e $J_1,\tilde{\alpha}$, and $\delta$. This notation is
frequently used in the context of the 1D dimerized and frustrated
spin--chain limit. In CuGeO$_3$ $J_1$ is approximately $160K$
\cite{Castilla95,Riera95}. Consensus on the precise magnitude of the
intra--chain frustration--ratio $\tilde{\alpha}$ is still lacking.
Studies of the magnetic susceptibility, which has been compared only to
1D models, have resulted in $\tilde{\alpha}\approx 0.24$
\cite{Castilla95} as well as in $\tilde{\alpha}\approx 0.35$
\cite{Riera95}. This would place CuGeO$_3$ in the vicinity of the
critical value $\tilde{\alpha}_c \simeq 0.2411$ for the opening of a
spin gap solely due to frustration \cite{Okamoto92a}. $\delta$
resembles the lattice dimerization which is finite for $T<T_{SP}$
only. Values for the zero--temperature dimerization $\delta(T=0)$
ranging from $0.21$ to $0.012$ have been suggested
\cite{Braden96,Castilla95,Riera95,otherdeltas}. Knowledge on the
magnitude of $\mu$ and $\mu\beta$ is limited to $|\mu|,|\mu\beta|\ll 1$
\cite{Braden96,Khomskii96a}.
Magnetic excitations in CuGeO$_3$ are clearly distinct among the
uniform (U), i.e. $T>T_{SP}$, and the dimerized (D), i.e. $T<T_{SP}$,
phase. While the dynamic structure factor exhibits a gapless, c--axis
dispersive two--spinon continuum similar to that of the 1D Heisenberg
chain above $T_{SP}$ \cite{Arai96}, well defined magnon--like
excitations with sizeable c-- and b--axis dispersion have been
observed below $T_{SP}$ \cite{Nishi94,Regnault95a,Regnault96a,Ain97a}.
These magnons are gapful and are split off from a continuum which, at
zone--center, starts at roughly twice the magnon gap
\cite{Ain97a,Uhrig96a,Fledderjohann96,Uhrig97a}.
The aim of this work is to study the magnetic properties of the
$J$-$\lambda$-$\alpha$-$\mu$-$\beta$ model with a focus on the
D--phase of CuGeO$_3$. First I will describe a bond--spin
representation of the $J$-$\lambda$-$\alpha$-$\mu$-$\beta$ model
which, in turn, is treated by an appropriate linearization. Next,
results for the ground state energy, the spin gap, the magnon
dispersion, and the dynamic structure factor are contrasted against
other theoretical approaches and are compared with experimental
findings. Finally, details of an alternative mean--field approach
using the bond--spin representation are provided in appendix \ref{A}.
\section{Bond--Operator Theory}
In this section the properties of the
$J$-$\lambda$-$\alpha$-$\mu$-$\beta$ model are discussed by
representing the {\em site}--spin algebra in terms of {\em
bond}--spin operators \cite{Sachdev90a}. First, the essential features
of these operators are briefly restated. Consider any two spin--1/2
operators ${\bf S}_1$ and ${\bf S}_2$. The eigenstates of the related
total spin are a singlet $\left|s\right>$ and three triplets
$\left| t_\alpha^{\phantom{\dagger}} \right>$ with $\alpha= x,y,z$.
These can be created out of a vacuum $\left|0\right>$ by
applying the bosonic operators $s_{\phantom{x}}^\dagger$ and
$t_\alpha^\dagger$
\begin{equation}\label{1}
\begin{array}{lccr}
s_{\phantom{.}}^\dagger\left|0\right>
&\hat{=}&\left|s_{\phantom{.}}^{\phantom{\dagger}}\right>=&
(\left|\uparrow\downarrow\right>-\left|\downarrow\uparrow\right>)/
\sqrt{2}\\
t_x^\dagger \left|0\right>
&\hat{=}&\left|t_x^{\phantom{\dagger}}\right>=&
-(\left|\uparrow\uparrow\right>-\left|\downarrow\downarrow\right>)/
\sqrt{2}\\
t_y^\dagger \left|0\right>
&\hat{=}&\left|t_y^{\phantom{\dagger}}\right>=&
i(\left|\uparrow\uparrow\right>+\left|\downarrow\downarrow\right>)/
\sqrt{2}\\
t_z^\dagger \left|0\right>
&\hat{=}&\left|t_z^{\phantom{\dagger}}\right>=&
(\left|\uparrow\downarrow\right>+\left|\downarrow\uparrow\right>)/
\sqrt{2}
\end{array}
\;\;\;,\end{equation}
where $[s,s_{\phantom{\alpha}}^\dagger]=1$,
$[s_{\phantom{\alpha}}^{(\dagger)},t_\alpha^{(\dagger)}]=0$, and
$[t_\alpha^{\phantom{(\dagger)}},t_\beta^{\dagger}]=\delta_{\alpha
\beta}$.
The action of ${\bf S}_1$ and ${\bf S}_2$ on this space leads
to the representation
\begin{equation}\label{2}
S^\alpha_{\stackrel{{\scriptstyle 1}}{2}}\hat{=}
\frac{1}{2} ( \pm s_{\phantom{\alpha}}^\dagger t_\alpha^{\phantom{
\dagger}}
\pm t_\alpha^\dagger s
- i \varepsilon_{\alpha\beta\gamma} t_\beta^\dagger
t_\gamma^{\phantom{\dagger}} )
\;\;\;,\end{equation}
for the individual spin operators. Here
$\varepsilon_{\alpha\beta\gamma}$ is the Levi--Civita symbol and a
summation over repeated
indices is implied hereafter.
The upper(lower) subscript on the lhs. of (\ref{2}) refer to
upper(lower) sign on the rhs.. The bosonic Hilbert space
has to be restricted to the physical Hilbert space, i.e.
either one singlet or one triplet, by the constraint
\begin{equation}\label{3}
s_{\phantom{\alpha}}^{\dagger}s +
t_\alpha^{\dagger}t_\alpha^{\phantom{\dagger}} =1
\;\;\;.\end{equation}
Using (\ref{2}) and (\ref{3}) it is simple to check, that ${\bf S}_1$
and ${\bf S}_2$ satisfy a spin algebra indeed and moreover that
\begin{equation}\label{4}
S^\alpha_1 S^\alpha_2 = -\frac{3}{4} s_{\phantom{\alpha}}^{\dagger}s
+ \frac{1}{4} t_\alpha^{\dagger}t_\alpha^{\phantom{\dagger}}
\;\;\;.\end{equation}
In order to transform the $J$-$\lambda$-$\alpha$-$\mu$-$\beta$ model
into the boson representation a particular distribution '${\bf l}$' of
bonds, i.e. pairs of spins ${\bf S}_{{\bf l}\,1}$ and ${\bf S}_{{\bf
l}\,2}$, has to be selected. Here this selection will be based on the
limit of strong dimerization $(\lambda,\lambda\alpha)\rightarrow
(0,0)$, or equivalently $(\tilde{\alpha},\delta)\rightarrow (0,1)$,
and small inter--chain coupling $(\mu,\mu\beta)\rightarrow (0,0)$. In
this limit the ground state is a product of singlets on each
dimer--bond while the elementary excitations are composed of the
corresponding localized triplets. Therefore it is natural to place the
singlet and triplet bosons onto the dimer bonds, i.e. ${\bf S}_{{\bf
l}\,1}={\bf S}_{{\bf l}}$ and ${\bf S}_{{\bf l}\,2}={\bf S}_{{\bf
l+c}}$ with ${\bf l}\in{\cal D}$. The transformed Hamiltonian reads
\begin{eqnarray}\label{5}
H&=& H_0+H_1+H_2+H_3
\\
H_0&=& \sum_{{\bf l}\in{\cal D}} (
-\frac{3}{4} s_{{\bf l}}^{\dagger}
s_{{\bf l}}^{\phantom{\dagger}}
+\frac{1}{4} t_{{\bf l}\,\alpha}^{\dagger}
t_{{\bf l}\,\alpha}^{\phantom{\dagger}})
\nonumber \\
H_1&=& \sum_{{\bf l}\neq {\bf m}\in{\cal D}}
a({\bf l},{\bf m}) (
t_{{\bf l}\,\alpha}^{\dagger}
t_{{\bf m}\,\alpha}^{\phantom{\dagger}}
s_{{\bf m}}^{\dagger}
s_{{\bf l}}^{\phantom{\dagger}} +
t_{{\bf l}\,\alpha}^{\dagger}
t_{{\bf m}\,\alpha}^{\dagger}
s_{{\bf m}}^{\phantom{\dagger}}
s_{{\bf l}}^{\phantom{\dagger}} + h.c.)
\nonumber \\
H_2&=& \sum_{{\bf l}\neq {\bf m}\in{\cal D}}
b({\bf l},{\bf m}) (
i \varepsilon_{\alpha\beta\gamma}
t_{{\bf m}\,\alpha}^{\dagger}
t_{{\bf l}\,\beta}^{\dagger}
t_{{\bf l}\,\gamma}^{\phantom{\dagger}}
s_{{\bf m}}^{\phantom{\dagger}} + h.c.)
\nonumber \\
H_3&=& \sum_{{\bf l}\neq {\bf m}\in{\cal D}}
c({\bf l},{\bf m}) (
t_{{\bf l}\,\alpha}^{\dagger}
t_{{\bf m}\,\alpha}^{\dagger}
t_{{\bf m}\,\beta}^{\phantom{\dagger}}
t_{{\bf l}\,\beta}^{\phantom{\dagger}} -
t_{{\bf l}\,\alpha}^{\dagger}
t_{{\bf m}\,\beta}^{\dagger}
t_{{\bf m}\,\alpha}^{\phantom{\dagger}}
t_{{\bf l}\,\beta}^{\phantom{\dagger}})
\nonumber
\;\;\;,\end{eqnarray}
where each local Hilbert space is subject to the constraint (\ref{3})
and, if not explicitly stated otherwise, the unit of energy is $J$
hereafter. The inter--dimer matrix elements can be obtained from
fig.~\ref{1}
\begin{eqnarray}\label{6}
&&a({\bf l},{\bf m}) =-\frac{1}{4}[
t_1 \delta_{{\bf m}{\bf l}_{\scriptstyle 1}} +
t_2 (\delta_{{\bf m}{\bf l}_{\scriptstyle 2}}+\delta_{{\bf m}
{\bf l}_{\scriptstyle 3}})]
\\
&&b({\bf l},{\bf m}) =
\frac{1}{4}[\lambda
(\delta_{{\bf l}{\bf l}_{\scriptstyle 1}}-\delta_{{\bf m}{\bf l}_{
\scriptstyle 1}}) +
\mu (\delta_{{\bf m}{\bf l}_{\scriptstyle 3}}
-\delta_{{\bf m}{\bf l}_{\scriptstyle 2}}
+\delta_{{\bf l}{\bf l}_{\scriptstyle 2}}
-\delta_{{\bf l}{\bf l}_{\scriptstyle 3}})]
\nonumber \\
&&c({\bf l},{\bf m}) =
-\frac{1}{4}[
t_3 \delta_{{\bf m}{\bf l}_{\scriptstyle 1}} +
t_4 (\delta_{{\bf m}{\bf l}_{\scriptstyle 2}}+\delta_{{\bf m}
{\bf l}_{\scriptstyle 3}})]
\nonumber
\;\;\;,\end{eqnarray}
where ${\bf l}_{1,2,3}$ are defined in fig.~\ref{bobofig1} and $t_1=
\lambda(1-2\alpha)$, $t_2=\mu (1-2\beta)$, $t_3=\lambda (1+2\alpha)$, and
$t_4=\mu (1+2\beta)$. As anticipated, the inter--dimer matrix elements
$a({\bf l},{\bf m})$, $b({\bf l},{\bf m})$, and $c({\bf l}, {\bf m})$
vanish in the strong dimer limit leaving the Hamiltonian diagonal in
$s_{{\bf l}}$ and $t_{{\bf l}\,\alpha}$.
In order to treat the local constraint and the dimer interactions
approximations have to be made. To this end I will employ the
Holstein--Primakoff (HP) representation of the bond--operators which
has been detailed in \cite{Starykh96a,Chubokov89a,Chubukov91a}. In
this representation the constraint is treated by eliminating the
singlet operator via $s_{{\bf l}}^{\dagger}=s_{{\bf
l}}^{\phantom{\dagger}}=(1-t_{{\bf l}\,\alpha}^{ \dagger}t_{{\bf
l}\,\alpha}^{\phantom{\dagger}})^{-1/2}$. Moreover, after inserting
this into the Hamiltonian only terms up to second order in in the
triplet operators are retained. The latter procedure is analogous to
the linear spin--wave approximation in systems with broken
spin--rotational invariance. The linearized (LHP) Hamiltonian is
given by
\begin{eqnarray}\label{7}
H_{LHP}&&=-\frac{9}{4}D
\nonumber \\
&&+\frac{1}{2}\;\sum_{{\bf k}\in{\cal B}}
\Psi_{{\bf k}\,\alpha}^\dagger
\left[\begin{array}{cc}
1+\epsilon_{\bf k} & \epsilon_{\bf k} \\
\epsilon_{\bf k} & 1+\epsilon_{\bf k}
\end{array}\right]
\Psi_{{\bf k}\,\alpha}^{\phantom{\dagger}}
\\ \nonumber \\ \label{7a}
\epsilon_{\bf k}=&&-\frac{1}{2}[t_1 \cos(2k_c)
+2t_2\cos(k_b)\cos(k_c)]
\;\;\;,\end{eqnarray}
where $D$ is the number of dimers and ${\bf k}$ is a momentum
vector restricted to a Wigner--Seitz
cell ${\cal B}$ of the reciprocal lattice. For comparison with
experimental data ${\cal B}$ is oriented with respect to the
non--dimerized system, instead of the Brillouin zone of the dimer
lattice, i.e. ${\bf k}=(k_b,k_c)$ with $bk_b=0...2\pi$ and
$ck_c=0...\pi$ with $b,c$ set to unity hereafter. $\Psi_{{\bf
k}\,\alpha}^{(\dagger)}$ is a a spinor with $\Psi_{{\bf
k}\,\alpha}^\dagger=[t_{{\bf k}\,\alpha}^{\dagger} \; t_{-{\bf
k}\,\alpha}^{\phantom{\dagger}}]$ and $t_{{\bf l}\,\alpha
}^\dagger=1/\sqrt{D}\sum_{\bf k}e^{-i{\bf k}\cdot{\bf l}}t_{{\bf k}\,
\alpha}^\dagger$.
\begin{figure}[tb]
\vskip -3.3cm
\centerline{\hspace*{1cm}\psfig{file=bobofig2.eps,width=9.5cm}}
\vskip -3.3cm
\caption[l]{\begin{sloppypar}
Ground state energies in the dimerized--chain limit: LHP--theory
(solid) versus exact diagonalization \cite{Soos85a} (solid dots),
errors are less than marker size. Inset: small $\delta$
limit. $E^{\star}(\delta)/J_1=$ $-[(1+\delta) E_g((1-\delta)
/(1+\delta),0)-$ $E^{Bethe}_g]$.
\end{sloppypar}}
\label{bobofig2}
\end{figure}
Eqn. (\ref{7}) describes a threefold degenerate set of dispersive
triplets. The triplets are renormalized by ground--state
quantum--fluctuations. These are produced by the terms of type
$t_{{\bf k}\,\alpha}^\dagger t_{-{\bf k}\,\alpha}^\dagger$ and their
hermitian conjugate. The excitation spectrum $E_{\bf k}$ follows from
a Bogoliubov transformation
\begin{eqnarray}\label{8}
&&H_{LHP}=-\frac{9}{4}D+
\sum_{{\bf k}\in{\cal B},\alpha}
E_{\bf k}(a_{{\bf k}\,\alpha}^{\dagger}
a_{{\bf k}\,\alpha}^{\phantom{\dagger}}
+\frac{1}{2})
\\ \label{8a}
&&E_{\bf k}=\sqrt{1+2\epsilon_{\bf k}}
\;\;\;,\end{eqnarray}
where $a_{{\bf k}\,\alpha}^{(\dagger)}$ are the Bogoliubov
quasi--particles which are given by
\begin{eqnarray}\label{9}
&&\Psi_{{\bf k}\,\alpha}^{\phantom{\dagger}}=
\left[\begin{array}{cc}
g_{\bf k} & h_{\bf k} \\
h_{\bf k} & g_{\bf k}
\end{array}\right]
\Phi_{{\bf k}\,\alpha}^{\phantom{\dagger}}
\\ \nonumber \\
&&h^2_{\bf k}=
\frac{1}{2}\left(\frac{1+\epsilon_{\bf k}}{E_{\bf k}}-1\right)
\makebox[1cm][c]{;}
h_{\bf k}g_{\bf k}=-\frac{1}{2}\frac{\epsilon_{\bf k}}{E_{\bf k}}
\nonumber \\
&&g^2_{\bf k}=
\frac{1}{2}\left(\frac{1+\epsilon_{\bf k}}{E_{\bf k}}+1\right)
\nonumber
\;\;\;,\end{eqnarray}
where $\Phi_{{\bf k}\,\alpha}^{(\dagger)}$ is a a spinor with
$\Phi_{{\bf k}\,\alpha}^\dagger=[a_{{\bf k}\,\alpha}^{ \dagger} \;
a_{-{\bf k}\,\alpha}^{\phantom{\dagger}}]$.
To conclude this section, I note that instead of treating the
constraint by means of the HP representation one may also apply the so
called bond--operator mean--field theory (MFT) of
\cite{Sachdev90a}. The application of this method to the
$J$-$\lambda$-$\alpha$-$\mu$-$\beta$ model
is detailed in appendix \ref{A}. As will become evident in section
\ref{GapSection} this technique seems less well suited in the present
context.
\begin{figure}[tb]
\vskip -3.3cm
\centerline{\hspace*{.2cm}\psfig{file=bobofig3.eps,width=9.5cm}}
\vskip -3.3cm
\caption[l]{\begin{sloppypar}
Ground state energies in the frustrated--chain limit:
LHP--theory (solid) versus exact diagonalization \cite{Tonegawa87a}
(solid dots), and DMRG \cite{Chitra95a} (crosses), errors are less
than marker size. $E^{\star}(\alpha)/J_1=$ $-[E_g((1-2\alpha),0)-$
$E^{Bethe}_g]$.
\end{sloppypar}}
\label{bobofig3}
\end{figure}
\section{Results}
In the following sections the consequences of the LHP representation
of the 2D $J$-$\lambda$-$\alpha$-$\mu$-$\beta$ model will be
contrasted against other known results in the limiting case of the
1D $J_1$-$\tilde{\alpha}$-$\delta$ model as well as INS data observed
on CuGeO$_3$.
\subsection{Ground State Energy}
From (\ref{8}) it is obvious, that the ground state energy per {\em
lattice site}, $E_g$, is equal to $-3/8$ at the point of complete
dimerization. This is the proper energy gain for a bare singlet
formation. For arbitrary $t_1$ and $t_2$
\begin{eqnarray}\label{15}
\lefteqn{
E_g(t_1,t_2)=-\frac{9}{8}+\frac{3}{2\pi^2}\int_{0}^{\pi}dk_c\,
\{[1 - 2 t_2 \cos(k_c) }
\\
&& - t_1 \cos(2 k_c)]^{-1/2} \mbox{{\bf E}}\,(
\frac{-4 t_2 \cos(k_c)}{1 - 2 t_2 \cos(k_c) - t_1 \cos(2 k_c)}) \}
\nonumber
\;\;\;,\end{eqnarray}
where $\mbox{{\bf E}}$ is the complete elliptic integral of the second
kind. For vanishing inter--chain coupling this simplifies to
\begin{equation}\label{16}
E_g(t_1,0)= -9/8 + \frac{3}{2\pi} \sqrt{1 - t_1}\mbox{{\bf E}}(\frac{2
t_1}{t_1-1})
\;\;\;,\end{equation}
which can be compared to existing results in various regions of the
$(\tilde{\alpha},\delta)$--plane of the frustrated and dimerized
spin--1/2 chain. In particular at the isotropic Heisenberg point,
i.e. $(\tilde{\alpha},\delta)=(0,0)$, $E_g=3/(\sqrt{2}\pi)-9/8\approx
-0.4498$ which agrees reasonably well with the Bethe Ansatz result
$E^{Bethe}_g=1/4-\ln (2)\approx -0.4431$. In fig.~\ref{bobofig2} the
ground state energy along the $(\tilde{\alpha}=0,\delta)$--line,
i.e. for a dimerized chain, is contrasted against results from exact
diagonalization \cite{Soos85a}. Deviations are within the numerical
error of the diagonalization data for $\delta\gtrsim 0.05$. For
dimerizations below $0.05$ small differences are caused by an
unphysical extremum in the LHP energy which is visible in the
inset. While the agreement is encouraging along the
$(\tilde{\alpha}=0,\delta)$--line, only qualitative consistency is to
be expected along the $(\tilde{\alpha},\delta=0)$--line since this
region is more distant from the strong dimer limit
$(\tilde{\alpha},\delta)\sim (0,1)$. This is shown in
fig.~\ref{bobofig3} which compares $E_g(\tilde{\alpha})$ with exact
diagonalization \cite{Tonegawa87a} and density--matrix renormalization
group (DMRG) \cite{Chitra95a} data for the case of a frustrated
chain. This figure demonstrates that the LHP approach overestimates
the frustration induced loss of ground state energy at intermediate
$\tilde{\alpha}$.
\begin{figure}[tb]
\vskip -3.3cm
\centerline{\hspace*{1cm}\psfig{file=bobofig4.eps,width=9.5cm}}
\vskip -3.3cm
\caption[l]{\begin{sloppypar}
Spin gaps in the dimerized--chain limit: LHP--\-theory (solid)
versus 3rd--order perturbation theory \cite{Uhrig97a,Harris73a} ( dashed),
MFT (dotted), and exact diagonalization \cite{Soos85a} (solid dots),
errors are less than marker size.
\end{sloppypar}}
\label{bobofig4}
\end{figure}
\begin{figure}[tb]
\vskip -3.3cm
\centerline{\hspace*{1cm}\psfig{file=bobofig5.eps,width=9.5cm}}
\vskip -3.3cm
\caption[l]{\begin{sloppypar}
Spin gaps in the frustrated--chain limit: LHP--\-theory ( solid )
versus 3rd--order perturbation theory \cite{Uhrig97a} (dashed), MFT
(dotted), and two DMRG calculations \cite{Chitra95a} (crosses),
and \cite{White96a} (solid dots), errors are less than marker size.
\end{sloppypar}}
\label{bobofig5}
\end{figure}
\subsection{Spin Gap}\label{GapSection}
The LHP approximation does not break the spin--rotational invariance
and leaves the system in a quan\-tum--disordered ground state. This is
consistent with a spin gap $\Delta$ of the triplet dispersion which,
for positive $t_1$ and $t_2$, is given by $\Delta = \sqrt{1-t_1-2
t_2}$ and is situated at ${\bf k}=(0,0)$ and $(\pi,\pi)$. If $t_1$ and
$t_2$ are such that the triplet modes turn massless, i.e. $\Delta=0$,
a quantum phase--transition towards antiferromagnetism (AFM) will
occur at $T=0$. In terms of $\tilde{\alpha}$ and $\delta$, the AFM
instability--line is located at $\tilde{\alpha}+\delta=t_2(1+\delta)$.
Beyond this line the LHP approach breaks down.
Next the LHP spin gap at $t_2=0$ is compared to known results for the
frustrated and dimerized spin--1/2 chain. In fig.~\ref{bobofig4}
$\Delta(\delta)$ is contrasted against findings of exact
diagonalization \cite{Soos85a}, perturbation theory up to third order
in $\lambda$ \cite{Uhrig97a,Harris73a}, and a solution of the MFT equations
(\ref{12a}--\ref{13a}) which are discussed in appendix \ref{A}. This
figure displays reasonable agreement between the first three of these
approaches for all values of $\delta$. In addition it shows that the
MFT suffers from the inability to close the spin gap at the isotropic
Heisenberg point \cite{Zang95a,3rdNoZero}. This caveat renders the MFT
unsuitable for the case of a systems with nearly massless spin
excitations, e.g. CuGeO$_3$.
As noted in the previous section, analytic approaches based on the
strong dimer limit are less reliable if considered along the
$(\tilde{\alpha}, \delta=0)$--line. Nevertheless, a comparison of
$\Delta(\tilde{\alpha})$ as obtained from the LHP theory with various
other techniques is instructive and is shown in fig.~\ref{bobofig5}.
Qualitatively, all analytic methods depicted exhibit a tendency of the
spin gap to increase as $\tilde{\alpha}$ increases \cite{3rdHasMin}
however agreement with the DMRG data \cite{Chitra95a,White96a} is absent.
In particular, while 3rd--order perturbation theory and MFT show no
critical frustration--ratio $\tilde{\alpha}_c$ and have a finite gap for
all $0<\tilde{\alpha}<0.5$ the LHP representation leads to a critical
frustration $\tilde{\alpha}_c=0$. Quite remarkably there are also
substantial differences between the two DMRG results for
$\tilde{\alpha}\gtrsim 0.5$ \cite{Karen97a}.
\subsection{Triplet Dispersion}\label{DispSection}
In this section the relevance of the model with respect to the spin
excitations in CuGeO$_3$ will be assessed by comparison of $E_{\bf k}$
with INS data for the D-phase. In particular the role of
two--dimensionality will be considered. The essential effect of a
finite hopping amplitude $t_2$ is a {\em mixing} of the b-- and
c--axis dispersion. This mixing is due to the staggering of the
dimerization along the b--axis and leads to the
$t_2\cos(k_b)\cos(k_c)$--term in $\epsilon_{\bf k}$. Thus, $E_{\bf k}$
involves terms of different periodicity in $k_c$, i.e. $\cos(2k_c)$
and $\cos(k_c)$. Therefore, in contrast to the quasi 1D case, the
degeneracy of the triplets at the momenta $(0,0)$ and $(0,\pi)$ is
lifted. An identical reasoning based on the first--order contribution
of a 3rd--order perturbation theory has been given in \cite{Uhrig97a}.
Expanding $E_{\bf k}$ in terms of $t_1$ and $t_2$ I find
agreement up to first order with the dispersion given in
\cite{Uhrig97a}.
In fig.~\ref{bobofig6} INS data of the magnon dispersion in CuGeO$_3$
are shown for momenta along the edges of the reciprocal lattice cell
from ${\bf k}=(0,0)$ to $(0,\pi)$ as well as from $(0,0)$ to
$(\pi/2,0)$ \cite{Regnault96a}. Although data exactly at $(0,\pi)$ is
lacking, it seems very likely from this figure that the triplet
excitations in CuGeO$_3$ are {\em not} degenerate at $(0,0)$ and
$(0,\pi)$. In order to demonstrate that the
$J$-$\lambda$-$\alpha$-$\mu$-$\beta$ model can account for the
observed dispersion fig.~\ref{bobofig6} contains a comparison of
$E_{\bf k}$ with the INS data. Rather than performing this comparison
by a least--square fit, $E_{\bf k}$ has been identified with the
magnon energies only at ${\bf k}=(0,0)$, $(0,\pi/2)$, and
$(\pi/2,0)$. This fixes $J=11.5meV$, $t_1=0.859$, and $t_2=0.054$
unambiguously and limits the b-- to c--axis coupling--ratio to roughly
6\% which is consistent with
\cite{Nishi94,Regnault95a,Regnault96a,Khomskii96a}. While the
preceding leaves $\tilde{\alpha}$, $\delta$, $\beta$, and $\mu$
undetermined, $\tilde{\alpha}$ can be fixed using an additional input,
i.e. $\delta= 0.012$. This is within the range of values suggested in
the literature, i.e. $\delta=0.21 ... 0.012$
\cite{Braden96,Castilla95,Riera95,otherdeltas} and implies natural
parameters of $J_1=11.4meV=132K$ and $\tilde{\alpha}=0.059$. Allowing
for a larger dimerization leads to smaller $J_1$ and
$\tilde{\alpha}$. In particular $\tilde{\alpha}=0$ with $J_1=124K$ is
reached for $\delta=0.076$. Parameters with a slightly smaller(larger)
intra(inter)--chain exchange have been established in \cite{Uhrig97a},
i.e. $J_1=9.8meV=114K$, $\tilde{\alpha}=0$, and $t_2=0.12$ however
with $\delta=0.12$.
\begin{figure}[tb]
\vskip -3.2cm
\centerline{\psfig{file=bobofig6.eps,width=9.5cm}}
\vskip -2.4cm
\caption[l]{\begin{sloppypar}
INS data of c--axis (upper panel) and b--axis (lower panel) dispersion
of the triplet mode in the D--phase of CuGeO$_3$ at $T=1.8K$
\cite{Regnault96a} (solid diamonds) versus LHP--theory (solid) for
$J=11.5meV$, $t_1=0.859$ and $t_2=0.054$. ( $b=c=1$ ).
\end{sloppypar}}
\label{bobofig6}
\end{figure}
The agreement displayed in fig.~\ref{bobofig6} is satisfying and
demonstrates the main point of this section, i.e. that diagonal
triplet--hopping can account for the observed asymmetry of the INS
data in the D-phase of CuGeO$_3$. Moreover, the values of
$\tilde{\alpha}$ obtained suggests that the intra--chain frustration
in CuGeO$_3$ is significantly smaller than that derived from the purely
1D $J_1$-$\tilde{\alpha}$-$\delta$ model \cite{Castilla95,Riera95},
i.e. $0.24\lesssim\tilde{\alpha}\lesssim 0.36$. The determination of
model parameters however is in need of further studies to improve on
the quantitative reliability of the frustration--dependence of
presently available approaches for the 2D case.
This is in contrast to \cite{Uhrig97a} where
{\em quantitative} evidence for $\tilde{\alpha}\approx 0$, based on
3rd--order perturbation theory, has been suggested.
\begin{figure}[tb]
\vskip -2.9cm
\centerline{\psfig{file=bobofig7.eps,width=9.5cm}}
\vskip -3.3cm
\caption[l]{\begin{sloppypar}
Energy bounds, $\Omega_C({\bf q})$, for both triplet continua (dashed
dotted) versus triplet--mode energy, $E({\bf q})$, (solid) as well as
combined spectral weight of continua, $W_C({\bf q})$, (dotted) at
$T=0.05$ versus spectral weight of triplet mode, $W_T({\bf q})$,
(dashed) for momenta along $q_b=q_c$. ($b=c=1$, $J=1$ and $t_1,t_2$ as
in fig.~\ref{bobofig6}.)
\end{sloppypar}}
\label{bobofig7}
\end{figure}
\begin{figure}[tb]
\vskip -3.8cm
\centerline{\psfig{file=bobofig8.eps,width=9.5cm}}
\vskip -2.3cm
\caption[l]{\begin{sloppypar}
Triplet continua for various momenta along $q_b=q_c$ and for two
temperatures $T=0.05$ (a) and $T=0.15$ (b). Bottom--most curve
corresponds to ${\bf q}=0$ with $0.1$ incremental y--axis offset
for each consecutive momentum. Solid diamonds: bounds of triplet
continua. Solid square: triplet--mode energy. ($b=c=1$, $J=1$ and
$t_1,t_2$ as in fig.~\ref{bobofig6}.)
\end{sloppypar}}
\label{bobofig8}
\end{figure}
\subsection{Dynamic Structure Factor}\label{SqwSection}
Magnetic excitations are observed by measuring the dynamic structure
factor $S({\bf q},\omega)$ which is related by analytic continuation
and the fluctuation dissipation theorem $S({\bf q},\omega)=
\mbox{Im}[\chi({\bf q},\omega)]/(1-e^{-\omega/T})$ to the dynamic
spin susceptibility
\begin{equation}\label{17}
\chi_{\alpha\beta}({\bf q},\tau)= \langle T_\tau [S^\alpha_{\bf
q}(\tau) S^\beta_{\bf q}] \rangle
\;\;\;.\end{equation}
Here ${\bf q}$ is the momentum, $\tau$ the imaginary time, and
$T_\tau$ refers to time ordering. Since the physical spin is a
composite operator of the bond--bosons the information obtained from
the dynamic susceptibility is not restricted to the triplet dispersion
-- even at the LHP level. This will be clarified in the remainder of
this section.
Within the LHP representation the spin operator in momentum space is
\begin{eqnarray}\label{18}
S^\alpha_{\bf q}&=& \frac{1}{4}(1-e^{iq_c})
(t_{{\bf q}\,\alpha}^{\dagger}+
t_{{\bf -q}\,\alpha}^{\phantom{\dagger}})
\nonumber \\
&& -i\frac{1}{4\sqrt{D}}(1+e^{iq_c})
\sum_{\bf k\in {\cal B}}
\varepsilon_{\alpha\beta\gamma}
t_{{\bf k}+{\bf q}\,\beta}^{\dagger}
t_{{\bf k}\,\gamma}^{\phantom{\dagger}}
\;\;\;.\end{eqnarray}
Two qualitatively different excitations appear on the rhs.: a sharp
triplet mode due to the first term and a continuum of two--triplet
states due to the second. The momentum dependent form factors $(
1\mp\exp (iq_c)$ lead to a vanishing weight of the the triplet mode
(continuum) at $q_c=0$ ($q_c=\pi$). Since $H_{LHP}$ does not conserve the
bare triplet number, the continuum is divided in between two excitations
of different nature, i.e. virtual excitations at energies $E_{\bf k+q}+
E_{\bf k}$ and real excitations at energies $E_{\bf k+q}- E_{\bf
k}$. While the former are due to ground--state quantum--fluctuations
and occur at all temperatures, the latter result from excitations
across the spin gap and are present only at finite
temperatures. Evaluating the dynamic susceptibility by standard
methods I obtain
\begin{eqnarray}\label{19}
\lefteqn{
\chi_{\alpha\beta}({\bf q},\omega_n) = \delta_{\alpha\beta} \frac{1}{4}
\left\{(\cos(q_c) - 1)\frac{1}{(i\omega_n)^2-E^2_{\bf q}}
\right.}
\nonumber \\ \nonumber \\
&&+(\cos(q_c)+1) \frac{1}{2D} \sum_{{\bf k}\in {\cal B}} \left[
\frac{1+\epsilon_{{\bf k}+{\bf q}}+\epsilon_{\bf k} +
E_{{\bf k}+{\bf q}}E_{\bf k}}{E_{{\bf k}+{\bf q}}E_{\bf k}}\times \right.
\nonumber \\ &&
\phantom{(1+\cos(q_c))\frac{1}{2D}\sum_{{\bf k}\in {\cal B}}
\frac{1+\epsilon_{\bf k}}{E_{{\bf k}+{\bf q}}E_{\bf k}}}
\frac{n(E_{{\bf k}+{\bf q}})-n(E_{\bf k})}
{i\omega_n+E_{{\bf k}+{\bf q}}-E_{\bf k}}
\nonumber \\ \nonumber \\ &&
\phantom{(1}
+\frac{(1+\epsilon_{{\bf k}+{\bf q}}+\epsilon_{\bf k} -
E_{{\bf k}+{\bf q}}E_{\bf k})(E_{{\bf k}+{\bf q}}+E_{\bf k})}
{E_{{\bf k}+{\bf q}}E_{\bf k}}\times
\nonumber \\ &&
\phantom{(1\cos(q_c))\frac{1}{2D}\sum_{{\bf k}\in {\cal B}} [ }
\left.\left. \frac{n(E_{{\bf k}+{\bf q}})+n(E_{\bf k})+1}
{(i\omega_n)^2-(E_{{\bf k}+{\bf q}}+E_{\bf k})^2} \right] \right\}
\;\;\;,\end{eqnarray}
where $\omega_n=2n\pi T$ is a Bose Matsubara--frequency.
As anticipated the dynamical susceptibility (\ref{19}) exhibits finite
spectral intensity at $E_{\bf k}$, $E_{\bf k+q}-E_{\bf k}$, and
$E_{\bf k+q}+E_{\bf k}$. This is summarized in fig.~\ref{bobofig7} which
displays the two continua with respect to the triplet mode along a
particular momentum space direction. Parameters identical to those of
fig.~\ref{bobofig6} have been chosen. The triplet mode is situated in a
gap between the low--energy continuum due to thermal excitations and that
at high energies due to quantum fluctuations. At the zone center the
latter is gaped by $2 \Delta$. This is consistent with recent INS
data from the D--phase of CuGeO$_3$ \cite{Ain97a}. In addition to the
magnon these experiments indicate an 'unexpected' high--energy
continuum which is separated from the magnon by an additional gap.
The observed zone--center continuum--to--magnon gap--ratio is
approximately $2$. Deviations from the latter value of $2$
are possibly due to triplet--triplet interactions which are beyond the
LHP approach. Additionally, fig.~\ref{bobofig7} shows the weight of the
triplet mode, as well as that of the combined continua at $T=0.05J$.
At $T\ll J$ the continuum weight is almost completely due to quantum
fluctuations while, independent of temperature, the triplet weight is
given by
\begin{equation}\label{20}
W_T({\bf q})=\frac{\pi}{8} \frac{1-\cos(q_c)}{E_{\bf q}} \,
\stackrel{(\pi,\pi)}{=} \;\frac{\pi}{4\Delta}
\;\;\;,\end{equation}
The leftmost expression refers to the maximum of the triplet
weight. The corresponding momentum, i.e. $(\pi,\pi)$, indicates the
instability towards AMF as $\Delta\rightarrow 0$.
Figure~\ref{bobofig8} depicts the spectral intensity
$\mbox{Im}[\chi_{xx} ({\bf q},\omega_n\rightarrow
-i\omega+\nu)]$ of the continua as a function of frequency for two
temperatures and for various momenta along a direction in reciprocal
space identical to that of fig.~\ref{bobofig7}. The ${\bf k}$--sums
of (\ref{19}) have been performed numerically on a 600$\times$300
lattice. In order to obtain sufficient smoothing the frequency has
been shifted off the real axis by $\nu=0.05$. The parameters
correspond to those of fig.~\ref{bobofig6}. As a guide to the eye the
position of the triplet mode is labeled by solid squares while solid
diamonds in this figure label the exact locations of the spectral
bounds of the continua. Although smeared due to the imaginary
broadening, van--Hove singularities are clearly observable at the
spectral bounds as well as other characteristic energies within the
continua. Evidently the quantum fluctuations exhibit largest weight
at intermediate wave vectors while at higher temperatures additional
weight appears at smaller momentum due to thermal excitations.
Even though the intensity in fig.~\ref{bobofig8} decreases
both, as ${\bf q}$ approaches $(0,0)$ and $(\pi,\pi)$, only in the
latter case this is due to the form factor in (\ref{18}) while in the
former case this is a consequence of the conservation of the total spin.
\section{Conclusion}
In summary I have studied static and dynamic properties of a
frustrated and dimerized 2D quantum spin--model using the
bond--operator method. The ground state energy and the spin gap have
been gauged against known results from the 1D limiting cases of this
model. Effects of dimerization are found to be described almost
quantitatively while the influence of frustration is captured
qualitatively. The dynamic structure factor has been analyzed
and displays two characteristic features, i.e. a well defined magnon
excitation and a temperature dependent continuum. The magnon
dispersion shows a characteristic lifting of degeneracies, different
from purely one--dimensional models, and agrees very well with INS
data on CuGeO$_3$. The low--temperature continuum exhibits a
zone--center gap twice that of the magnon. This is also consistent
with INS experiments on CuGeO$_3$.
\acknowledgements
I am grateful to P. Fulde and the Max--Planck--Institut f\"ur Physik
komplexer Systeme for their kind hospitality. It is a pleasure to
thank G. Uhrig for helpful comments and for communicating his results
prior to publication. Stimulating discussions with B. B\"uchner and
E. M\"uller--Hartmann are acknowledged. This work has been supported
in part by the Deutsche Forschungsgemeinschaft through the SFB 341.
\begin{appendix}
\section{Bond--Operator Mean--Field Theory (MFT)}\label{A}
An alternative approach to the Hamiltonian (\ref{5}) arises by
introduction of a set of local Lagrange multipliers $\eta_{\bf l}$ to
enforce the constraint (\ref{3})
\begin{equation}\label{A1}
\tilde{H}=H - \sum_{{\bf l}\in{\cal D}}
\eta_{\bf l}(s_{{\bf l}}^{\dagger}
s_{{\bf l}}^{\phantom{\dagger}}
+t_{{\bf l}\,\alpha}^{\dagger}
t_{{\bf l}\,\alpha}^{\phantom{\dagger}}-1)
\;\;\;.\end{equation}
To treat this Hamiltonian one replaces the local constraint by a
global one, i.e. $\eta_{\bf l}=\eta$, and introduces a mean--field
(MF) decoupling of all quartic terms leading to an effective quadratic
Hamiltonian \cite{Sachdev90a}. This Hamiltonian has an overall {\em
negative} prefactor to the $s_{{\bf l}}^{\dagger} s_{{\bf l}}^{
\phantom{\dagger}}$--term which implies Bose condensation of the
singlets. Therefore $s_{{\bf l}}^{\dagger}=s_{{\bf
l}}^{\phantom{\dagger}}=\langle s_{{\bf l}}^{\phantom{
\dagger}}\rangle=s$ is assumed. Moreover it can be shown that
contributions from the triplic and quartic triplet--terms $H_2$ and
$H_3$ to the MFT can be neglected \cite{GopalanXXa,Brenig97b}. The MF
Hamiltonian reads
\begin{eqnarray}\label{10}
\lefteqn{
H_{MFT}=D(-\frac{3}{8}-\frac{3}{4}s^2-\eta s^2+\frac{5}{2}\eta)
}
\\
&&+\frac{1}{2}\;\sum_{{\bf k}\in{\cal B}}
\Psi_{{\bf k}\,\alpha}^\dagger
\left[\begin{array}{cc}
\frac{1}{4}-\eta+s^2\epsilon_{\bf k} & s^2\epsilon_{\bf k} \\
s^2\epsilon_{\bf k} & \frac{1}{4}-\eta+s^2\epsilon_{\bf k}
\end{array}\right]
\Psi_{{\bf k}\,\alpha}^{\phantom{\dagger}}
\nonumber
\;\;\;,\end{eqnarray}
with notations equivalent to (\ref{7}). Analogous to the latter equation
(\ref{10}) represents three dispersive triplet excitations, however,
with a modified dispersion relation. After a Bogoliubov transformation
one gets \cite{etag0}
\begin{eqnarray}\label{11}
H_{MFT}=&&D(-\frac{3}{8}-\frac{3}{4}s^2-\eta s^2+\frac{5}{2}\eta)
\nonumber \\
&&+\sum_{{\bf k}\in{\cal B},\alpha}
E^{MFT}_{\bf k}(b_{{\bf k}\,\alpha}^{\dagger}
b_{{\bf k}\,\alpha}^{\phantom{\dagger}}
+\frac{1}{2})
\\
\nonumber \\ \label{11a}
E^{MFT}_{\bf k}=&&(\frac{1}{4}-\eta)\sqrt{1+d \epsilon_{\bf k}}
\;\;\;,\end{eqnarray}
where $d=2s^2/(1/4-\eta)$. The $b_{{\bf k}\, \alpha}$ quasi--particles
follow from expressions identical to (\ref{9}) with $\epsilon_{\bf k}$,
$1+\epsilon_{\bf k}$, and $E_{\bf k}$ replaced by $s^2\epsilon_{\bf
k}$, $\frac{1}{4}-\eta+s^2\epsilon_{\bf k}$, and $E^{MFT}_{\bf k}$,
respectively. The Lagrange multiplier and the singlet amplitude have
to be determined by solving the saddle--point equations
$\langle\partial H_{MFT}/\partial\eta\rangle=0$ and $\langle\partial
H_{MFT}/\partial s\rangle=0$. This leads to
\begin{eqnarray}\label{12a}
0&=&s^2-\frac{2}{5}+\frac{3}{D}\sum_{{\bf k}\in{\cal B}}
\frac{1+d\epsilon_{\bf k}/2}{\sqrt{1+d \epsilon_{\bf k}}}
(\langle b_{{\bf k}\,x}^{\dagger}
b_{{\bf k}\,x}^{\phantom{\dagger}}\rangle +\frac{1}{2})
\\ \nonumber \\ \label{12b}
0&=&\frac{3}{4}+\eta-\frac{3}{D}\sum_{{\bf k}\in{\cal B}}
\frac{\epsilon_{\bf k}}{\sqrt{1+d \epsilon_{\bf k}}}
(\langle b_{{\bf k}\,x}^{\dagger}
b_{{\bf k}\,x}^{\phantom{\dagger}}\rangle +\frac{1}{2})
\;\;\;.\end{eqnarray}
These selfconsistency equations can be solved by combining them into a
single one for the variable $d$ only
\begin{eqnarray}\label{13a}
d&=&5-\frac{3}{D}\sum_{{\bf k}\in{\cal B}}
\frac{2}{\sqrt{1+d \epsilon_{\bf k}}} ( \langle b_{{\bf k}\,x}^{\dagger}
b_{{\bf k}\,x}^{\phantom{\dagger}}\rangle+\frac{1}{2} )
\\ \label{13b}
&\stackrel{(T=0)}{=}&
5-\frac{3}{D}\sum_{{\bf k}\in{\cal B}}
\frac{1}{\sqrt{1+d \epsilon_{\bf k}}}
\;\;\;,\end{eqnarray}
where $\eta$ follows by insertion of $d$ into (\ref{12b}). This
completes the description of the bond--operator MFT.
\end{appendix}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,453
|
Roman Razbeyko (born 9 March 1973) is a Russian track and field athlete. He competed in the men's decathlon at the 2000 Summer Olympics.
References
1973 births
Living people
Sportspeople from Rostov-on-Don
Russian decathletes
Olympic decathletes
Olympic athletes of Russia
Athletes (track and field) at the 2000 Summer Olympics
Russian Athletics Championships winners
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,053
|
Q: How do I add values of radio buttons to get a sum I have some radio inputs in a form which I have written code for when selected. However, when I try to do addition I get NaN.
Any ideas?
HTML
<label class="form-check-label ">
<input class="form-check-input row1" type="radio" name="inlineRadioOptions" id="r1" value="1">
</label>
</div>
<div class="form-check form-check-inline">
<label class="form-check-label ">
<input class="form-check-input row1" type="radio" name="inlineRadioOptions" id="r2" value="2">
</label>
</div>
Jquery
function sum(){
var one = $('input:checked.row1').val();
var two = $('input:checked.row2').val();
var three = $('input:checked.row3').val();
var four = $('input:checked.row4').val();
var five = $('input:checked.row5').val();
var six = $('input:checked.row6').val();
var seven = $('input:checked.row7').val();
var eight = $('input:checked.row8').val();
var nine = $('input:checked.row9').val();
}
A: Try something like the example below.
This will get the value of all checked inputs and calculate the total value.
I've used [class^=row] so it will take the value from any input that is checked and also have a class that starts with row
Note if you want to short down the code even more (compared to the snippet below) use :
function sum() {
var total = 0;
var all = $('input:checked[class^=row]').map(function() {
total += Number(this.value)
})
}
function sum() {
var all = $('input:checked[class^=row]').map(function() {
return this.value;
}).get().join(",");
var total = 0;
for (var i = 0; i < all.length; i++) {
total += all[i] << 0;
}
console.log(total)
}
$('input[class^=row]').change(function() {
sum();
})
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<input type="checkbox" class="row1" value="1" />
<input type="checkbox" class="row2" value="2" />
<input type="checkbox" class="row3" value="3" />
<input type="checkbox" class="row4" value="4" />
<input type="checkbox" class="row5" value="5" />
<input type="checkbox" class="row6" value="6" />
A: Loop over each input selected then parse those value to int.
function getsum() {
var sum= 0;
$('input:checked').each(function(){
sum += parseInt($(this).val());
});
return sum;
}
alert(getsum());
Fiddle
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,112
|
<meta http-equiv="refresh" content="0;url=https://homes.cs.washington.edu/~khzeng/">
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,003
|
INDONESIAN COMMUNITY COUNCIL OF NEW SOUTH WALES (ICC-NSW) Inc.
With the rapidly increasing numbers of the Indonesians living in New South Wales and Indonesian organizations either social, religious, ethnic, or certain interest, as a manifestation of their bond and empathy to Indonesia-the land of their origin and Australia-the land which welcomes and accommodates them, the need for an umbrella organization can no longer be ignored.
After a long-overdue process, starting with a meeting attended by the representatives of over 40 Indonesian community organizations, the declaration of the Indonesian Community Council (ICC-NSW) Incorporated was read in the presence of the diasporic Indonesian community on the 17th of August 2000 coinciding with the commemoration of the fifty-fifth anniversary of the Indonesian Independence Day. The ICC became an incorporated organization on the 4th of October 2000.
WHAT IS ICC-NSW Inc.
The Indonesian Community Council of New South Wales Incorporated is a nonprofit organization representing the diasporic Indonesians in New South Wales and other groups who are interested in Indonesian affairs.
The Association will consolidate to encourage all Indonesian organizations in New South Wales to participate as members and expect their commitment to support the Association.
The ICC will provide a common voice to link with the Government and community agencies at all levels and a forum for consultation and discussion on any issues of concern to the community. The Association honors the independence of each member organization in their internal affairs.
WHO ARE THE MEMBERS OF ICC-NSW Inc.
Members of ICC-NSW Inc. consists of organization in New South Wales, either based on socio-cultural, religious, economic, or special common interests, who are interested in Indonesian affairs with goodwill to foster a close relationship between Australia and Indonesia
A united diasporic Indonesians in New South Wales living in peace and harmony, creating a conducive situation for developing understanding between Australia and Indonesia with a mutual beneficial contribution.
Accommodating the voice of its members by consultation, coordination, consensus, and representation
Maximizing the significant cultural, social, political, and economic dividends arising from the diversity of Australia and Indonesia
Organization ICC - NSW
ICC New South Wales Inc
Address: PO Box 672– KINGSFORD – NSW 2032
Email: icc_nsw@hotmail.com
Phone: 0416476917, 0409840317, 0432432454
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,104
|
Nights of Fire: Kickstarter day at last!
It's taken a year and a half but today is the day – the designing is done, the testing is completed, the components are composed – and Nights of Fire is launching on Kickstarter!
Here are two videos to entice you, if you hadn't made up your mind already… and of course, if you had, these will make you feel good about doing it!
Paul Grogan explaining how to play the game. This should answer any questions you may have left about play mechanics, especially after reading the long interview I did with The Players Aid blog recently where I went into the sequence of play.
And here are Grant and Alexander from The Players Aid blog, giving you their impressions after a good play-through. I really appreciate the help and attention these guys have given me and my games over the past year and more. I wonder if we'll ever meet in person… but if we do, I owe them several beers!
And finally, here's what you've been waiting for: the link to Kickstarter!
Let's go, man, GO !!!
Edited to add: Sometime during the night we made the first goal! 0900 PST right now, 19 hours after the launch, and we are at over $29,000 US pledged, over the first goal of $25,000 and past the first stretch goal of $28,000, which sees two extra leader cards added. So the world will get this game, after all, and then some.
Twenty-seven days to go in the funding period… let's see how it goes.
With regards to questions regarding the combination of campaign mode and the minis in one product, this is related to economies of scale, and it is actually what makes that product possible.
If we were to separate the two, the mini pack would still need to sell at the current price point due to a high Minimum Order Quantity (MOQ), whilst the campaign mode, with the costs of new packaging design and material, additional warehousing and added costs in fulfilment and management of pledges, would need to sell for a separate 15-20$.
Because our game has no miniatures in the base game, and they are an optional add on, for us to be able to create them we need to commit to a very high comparative MOQ (when considering it is an optional add-on). This means that combining the two, and therefore increasing ever so slightly the attach rate, is what makes the product possible at all.
Thanks to this we feel we have achieved quite an aggressive pricing range for our entire Days of Ire line. The game is now cheaper than it was in the original Kickstarter and the Days and Nights pack is the best value Mini-Pack we have ever offered (in comparison, the Vengeance Saboteurs pack went for 45$ for fewer miniatures on KS).
This way, we are able to offer the Expansion for NoF AND the campaign mode, in a product that would essentially still cost the same without them. We feel this is much better way of doing this, as you, the backers are helping us reach that steep MOQ, while we offer you more content at no additional cost.
Furthermore, notice that our first few stretch goals (which we seem to be getting to soon!! :D) are specifically directed to adding value to the Days & Nights pack, with even more content at no extra cost as we reach more economical numbers.
Good for them for responding to customers, albeit a small number of vocal (real and potential) ones… though this does pose a small but real risk for them, because of the Minimum Order Quantity issue discussed in the publisher's quote above. Hopefully it will not come back to bite them in their fourth point of contact.
And the next day: we are at $37,000, and a new stretch goal has been revealed – if they make it to $45,000 you can read my designer's notes!
3 Responses to Nights of Fire: Kickstarter day at last!
Thanks for allowing us in to help you with promoting the game. We enjoyed our experience very much and always love the knowledge tucked away in the dark recesses of your mind Brian. Your games are really good and I love to not only play them but also to talk about them. Keep up the good work, eh!
I'm really happy that you enjoyed playing this game so much.
Kickstarter has been live for less than two hours and we are over 33% of the way to the first goal!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,614
|
Q: python google search API - not giving correct results ^
|
| this question DOES NOT HAVE an answer yet!
I am trying to do google search from python script.
xing = "site:xing.com inurl:profile intext:Systemadministrator AND UNIX AND Hamburg"
query = urllib.urlencode ( { 'q' : xing } )
response = urllib.urlopen ( 'http://ajax.googleapis.com/ajax/services/search/web?v=1.0&' + query ).read()
import json as m_json
json = m_json.loads ( response )
results = json [ 'responseData' ] [ 'results' ]
print len(results)
the problem is, it is giving me only 4 results. If i paste the search string in google, it is finding more than 100 results. what am i doing wrong here?
EDIT
i changed the code to this:
import urllib
import django.utils import simplejson
num_queries = 50*4
xing = "site:xing.com inurl:profile intext:Systemadministrator AND UNIX AND Hamburg"
query = urllib.urlencode({'q' : xing})
url = 'http://ajax.googleapis.com/ajax/services/search/web?v=1.0&%s' % query
for start in range(0, num_queries, 4):
request_url = '{0}&start={1}'.format(url, start)
search_results = urllib.urlopen(request_url)
json = simplejson.loads(search_results.read())
results = json['responseData']['results']
return HttpResponse(search_results)
i am receiving no data. search_results is empty. why is this?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,773
|
Bring your buyers to this spacious and nicely updated 4 bedroom 2 bath home, within blocks of downtown Phoenix. The home features a spacious open floorplan, new kitchen cabinets, new stainless steel appliances, quartz countertops, breakfast bar, new tile flooring in all living areas, new carpet in bedrooms, new energy efficient vinyl windows throughout, and new interior paint. The roof and HVAC systems were also recently replaced. Other features include dual closets in master bedroom, including a huge walk-in, and a large in-home laundry room with plenty of storage. Exterior features a large covered patio, fully fenced yard and 1 car garage with opener. All updates have been made, so your buyers can enjoy this home for years to come.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,957
|
{"url":"http:\/\/en.wikipedia.org\/wiki\/Compound_distribution","text":"# Compound probability distribution\n\n(Redirected from Compound distribution)\n\nIn probability and statistics, a compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with the parameters of that distribution being assumed to be themselves random variables. The compound distribution is the result of marginalizing over the intermediate random variables that represent the parameters of the initial distribution.\n\nAn important type of compound distribution occurs when the parameter being marginalized over represents the number of random variables in a summation of random variables.\n\n## Definition\n\nA compound probability distribution is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution $F$ with an unknown parameter \u03b8 or parameter vector \u03b8 that is distributed according to some other distribution G with hyperparameter \u03b1, and then determining the distribution that results from marginalizing over G (i.e. integrating the unknown parameter(s) out). The resulting distribution H is said to be the distribution that results from compounding F with G. Expressed mathematically for a scalar data point with scalar parameter and hyperparameter:\n\n$p_H(x|\\alpha) = {\\displaystyle \\int\\limits_\\theta p_F(x|\\theta)\\,p_G(\\theta|\\alpha) \\operatorname{d}\\!\\theta}$\n\nThe same formula applies if some or all of the variables are vectors. Here is the case for a vector data point with vector parameters and hyperparameters:\n\n$p_H(\\mathbf{x}|\\boldsymbol\\alpha) = {\\displaystyle \\int\\limits_\\boldsymbol\\theta p_F(\\mathbf{x}|\\boldsymbol\\theta)\\,p_G(\\boldsymbol\\theta|\\boldsymbol\\alpha) \\operatorname{d}\\!\\boldsymbol\\theta}$\n\nA compound distribution $H$ resembles in many ways the original distribution $F$ that generated it, but typically has greater variance, and often heavy tails as well. The support of $H$ is the same as the support of the $F$, and often the shape is broadly similar as well. The parameters of $H$ include the parameters of $G$ and any parameters of $F$ that are not marginalized out.\n\n## Examples\n\nCompounding a normal distribution with variance distributed according to an inverse gamma distribution (or equivalently, with precision distributed as a gamma distribution) yields a non-standardized Student's t-distribution. This distribution has the same symmetrical shape as a normal distribution with the same central point, but has greater variance and heavy tails (in fact, specifically fat tails).\n\nCompounding a binomial distribution with probability of success distributed according to a beta distribution yields a beta-binomial distribution. This distribution is discrete just as the binomial distribution is, with support over integers between 0 and n (the number of trials in the base binomial distribution). There are three parameters, a parameter $n$ (number of samples) from the binomial distribution and shape parameters $\\alpha$ and $\\beta$ from the beta distribution. The shape is the same as a binomial distribution when $\\alpha$ and $\\beta$ are high. (This makes sense because it indicates very high certainty that the prior probability is quite near a specific location. The limit, with all mass at a specific point, is the same as having no prior and just specifying the probability as a parameter, as in the plain, non-compounded binomial distribution.) When $\\alpha$ and $\\beta$ are quite low, however, the shape becomes closer and closer to the shape of the beta distribution.\n\nOther examples:\n\n## Application in Bayesian inference\n\nIn Bayesian inference, compound distributions arise when, in the notation above, F represents the distribution of future observations and G is the posterior distribution of the parameters of F, given the information in a set of observed data. This gives a posterior predictive distribution. Correspondingly, for the prior predictive distribution,F is the distribution of a new data point while G is the prior distribution of the parameters.\n\nAnother example is in collapsed Gibbs sampling,[citation needed] where \"collapsing\" a variable means marginalizing it out, and typically prior parameters are collapsed out.\n\n## In exponential families\n\nCompound distributions derived from exponential family distributions often have a closed form.[citation needed] See the article on the posterior predictive distribution for more information.\n\n## Random number of terms in a summation\n\nA related but slightly different concept of \"compound\" occurs when a random variable is constructed from a number of underling random variables, and where that number is itself a random variable. In one formulation of this, the compounding takes places over a distribution resulting from the convolution of N underlying distributions, in which N is itself treated as a random variable. The compound Poisson distribution results from considering a set of independent identically-distributed random variables distributed according to a given distribution and asking what the distribution of their sum is, if the number of variables is itself an unknown random variable $N$ distributed according to a Poisson distribution and independent of the variables being summed. In this case the random variable N is marginalized out much like \u03b8 above is marginalized out.\n\nMore general cases of this type have been considered. [3]\n\n## References\n\n1. ^ Teich, M. C.; Diament, P. (1989). \"Multiply stochastic representations for K distributions and their Poisson transforms\". Journal of the Optical Society of America A: Optics, Image Science and Vision 6 (1): 80\u201391. doi:10.1364\/JOSAA.6.000080.\n2. ^ Dubey, S. D. (1970). \"Compound gamma, beta and F distributions\". Metrika 16: 27\u201331. doi:10.1007\/BF02613934. edit\n3. ^ Grubbstr\u00f6m, Robert W.; Tang, Ou (2006). \"The moments and central moments of a compound distribution\". European Journal of Operational Research 170: 106\u2013119. doi:10.1016\/j.ejor.2004.06.012.","date":"2014-08-02 09:18:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 18, \"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\": 0.8665608167648315, \"perplexity\": 440.20065511372127}, \"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-2014-23\/segments\/1406510280868.21\/warc\/CC-MAIN-20140728011800-00031-ip-10-146-231-18.ec2.internal.warc.gz\"}"}
| null | null |
Great work in Butte and St. Labre last week, everyone!
12 pm Thursday -- Cross Country @ the Midland Roundtable Luncheon.
Just a way to show support for our sport in our community and find out about cross country throughout our city.
Athletes will all need yellow slips and will be excused to RIDE THE SCHOOL BUS to Riverfront at 1:30. Athletes are required to ride the bus to the meet. Athletes may choose to ride home with parents without an alternative travel form since this is an in-town meet.
Mtn. West team will not be excused from school and will practice together at normal time.
We have another busy XC week ahead of us with two meets. The JV run on Thursday at Riverfront Park and our Varsity will travel of Missoula on Friday for their Saturday meet. Thanks to all who helped bring last minute bus snacks on Saturday!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,523
|
# ALSO BY STEVE HAMILTON
NICK MASON
Exit Strategy
The Second Life of Nick Mason
ALEX MCKNIGHT
Let It Burn
Die a Stranger
Misery Bay
A Stolen Season
Ice Run
Blood is the Sky
North of Nowhere
The Hunting Wind
Winter of the Wolf Moon
A Cold Day in Paradise
STAND-ALONES
The Lock Artist
Night Work
G. P. PUTNAM'S SONS
_Publishers Since 1838_
An imprint of Penguin Random House LLC
375 Hudson Street
New York, New York 10014
Copyright © 2018 by Cold Day Productions, LLC
Penguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.
Library of Congress Cataloging-in-Publication Data
Names: Hamilton, Steve, author.
Title: Dead man running / Steve Hamilton.
Description: New York : G. P. Putnam's Sons, [2018] | Series: An Alex McKnight novel ; 11
Identifiers: LCCN 2018012928| ISBN 9780399574443 (hardcover) | ISBN 9780399574450 (ebook)
Subjects: LCSH: McKnight, Alex (Fictitious character)—Fiction. | Private investigators—Michigan—Upper Peninsula—Fiction. | Murder—Investigation—Fiction. | BISAC: FICTION / Suspense. | FICTION / Thrillers. | GSAFD: Mystery fiction.
Classification: LCC PS3558.A44363 D43 2018 | DDC 813/.54—dc23
LC record available at <https://lccn.loc.gov/2018012928>
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, businesses, companies, events, or locales is entirely coincidental.
Version_1
For Deborah Randall
# ACKNOWLEDGMENTS
AS ALWAYS, and more than ever, I am indebted to Shane Salerno, David Koll, and everyone at The Story Factory. Thanks, also, to everyone who's been with me since the beginning: Bill Keller and Frank Hayes, Maggie Griffin, Jan Long, Rob Brenner, and Nick Childs. And finally, my wife, Julia, who is everything to me; my son, Nicholas; and my daughter, Antonia. I will never be a good enough writer to express how blessed I am to have you as my family.
I found the following books incredibly helpful and highly recommend them:
_The Manhunter_ , by John Pascucci and Cameron Stauth
_The Anatomy of Motive_ , by John Douglas and Mark Olshaker
_Advanced Fugitive_ , by Kenn Abaygo
_The Devil's Dozen_ , by Katherine Ramsland, PhD
_Ted Bundy: Conversations with a Killer_ , by Stephen G. Michaud and Hugh Aynesworth
_Without Conscience_ , by Robert D. Hare, PhD
_The Sociopath Next Door_ , by Martha Stout, PhD
# CONTENTS
_Also by Steve Hamilton_
_Title Page_
_Copyright_
_Dedication_
_Acknowledgments_
_Epigraph_
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Eighteen
Chapter Nineteen
Chapter Twenty
Chapter Twenty-one
Chapter Twenty-two
Chapter Twenty-three
Chapter Twenty-four
Chapter Twenty-five
Chapter Twenty-six
Chapter Twenty-seven
Chapter Twenty-eight
Chapter Twenty-nine
Chapter Thirty
Chapter Thirty-one
Chapter Thirty-two
Chapter Thirty-three
Chapter Thirty-four
Epilogue
_About the Author_
_You don't understand me._
_You are not expected to._
_You are not capable of it._
_I am beyond your experience._
—RICHARD RAMIREZ, the Night Stalker
THE ACT WAS DONE in a moonless desert darkness, but seen in the light, half a world away.
It was just before 4 p.m. on a bright February day on the Mediterranean Sea when a man named Frank Thompson logged on to his laptop. He was one of more than three thousand passengers on the cruise ship, midway between Sardinia and Sicily. His home in Scottsdale, Arizona, was six thousand miles and eight time zones away. Now that the ship's Internet service had finally been restored, Frank just wanted to know two things: That the video cameras installed in his house were working. And that his house was safe and secure.
The man's wife was up on the deck. She didn't think he should be worrying about the house. She thought he should be relaxing and actually enjoying this cruise, after waiting so many years to do this. After spending twelve grand to make this trip happen.
_I'll start enjoying this,_ Frank said to himself, _when I can get a little peace of mind._
He checked the first video feed. It came from the X10 Internet camera mounted on the bookshelf next to the fireplace. It was positioned so that the lens would look through the legs of a wooden elephant, and it communicated wirelessly with the server in the study, which in turn fed the images to the Web. Available to see anytime, anywhere in the world. At least when the Internet was working.
The live image, as Frank hit the key to bring it up, showed the front door and half of the living room. Everything looked normal to him, and yet _not_ normal in a way he couldn't identify.
He kept looking at the image. The couch, the door, the little welcome mat to wipe the desert sand off your feet when you came in.
_What's wrong with this picture?_
Then it finally hit him. There was too much light.
It was early in the morning back home, which on most days would mean that sunlight should be coming through the big window in the kitchen. Just as it seemed to be doing here.
But when they had left the house, those curtains had been closed. Frank was sure of that.
He clicked the link to restart the video, which ran on a continuous eight-hour loop. The image jumped back eight hours to darkness, the only light a thin glow from the one lamp they had left on in the living room. He hit the fast-forward button and watched the image flicker, the minutes passing by in fast motion, with no movement.
Until there was.
It was just a flash. He backed the video up, then ran it at normal speed. The front door opened. How could it be unlocked? This video would never reveal that secret to him—he could only go back eight hours, and as of eight hours ago, the door was obviously unlocked and any goddamned person in the world could walk right into their house.
Like this stranger.
_Who was in their house._
Frank paused the video to get a better look. The man was tall and well built, a few years younger than Frank, with fair skin and long light brown hair that went down to his shoulders. He was dressed in black jeans and a black button-down shirt. Black shoes. Even a black baseball cap, his hair trailing down the back of his neck. Frank's mind caught on the hair first—he'd always hated long hair on men. Then on something else, a certain quality about the man himself, how he moved with complete composure. No rush. No nerves. Like he was actually comfortable being in another man's house after dark. Frank watched as the man crossed the room, moving from the front door toward the hallway.
Frank hit the pause button again, sat there going through a series of emotions. Shock, anger, surprise. And if he was being honest with himself, a slight tinge of excitement.
_This thing really works._
He switched to the kitchen feed, went back eight hours, ran it through at fast speed, watching for the same kind of flash. As he was doing this, his wife came back to their stateroom.
"Marion, look at this!" he said to her. "There's someone in our house!"
"What are you talking about?"
"Sit down." Frank went back to the living room feed, found the time stamp when the front door had been opened. Just after midnight, Arizona time. Eight in the morning on the ship's time.
Marion watched the video, looking confused and skeptical until she saw the man walking through her living room. Her eyes went wide.
"How did he get in? We have to call somebody!"
"Just a minute," he said. "Let's find out where he went. Let me see if he . . ."
Frank didn't finish the thought. He had switched to the bedroom feed now. The camera was mounted on top of the armoire in the master bedroom, partially covered by the arrangement of silk flowers in a basket. It looked down on the bed and the dresser with the jewelry box on top. All of Marion's diamonds were in that box. He hadn't let her bring any of them on this trip, a pronouncement he was already regretting. The apology was half formed on his lips when the screen went from black to something else.
A light had been turned on. In their bedroom.
The stranger stood in the doorway, looking at a woman who was already lying on the bed. Waiting for him.
"That's our bed," Marion said. _"That's our bed."_
The man stepped forward. He stood over the woman and looked down at her for a long time.
Frank was about to pause the video. No need to see what came next. But Marion stopped him.
"What's wrong with her?" she said. "Look at that woman, Frank . . ."
He looked closely. The woman was lying on her back, her hands folded together on her stomach. She wasn't moving.
Her skin . . .
White. Like wax. Her mouth was open. Her eyes . . .
Staring at nothing.
"Oh my God," Marion said. "That woman, she's . . ."
She didn't finish the thought.
She didn't have to.
Frank and Marion Thompson sat in their cabin and watched the stranger in their bedroom as he began to take off his clothes.
# CHAPTER ONE
THERE ARE SOME THINGS _a man should never have to see._
Roger Halliday had an older brother who'd done three tours in Vietnam. When he'd asked him what had happened over there, that's what his brother had told him. Those exact words, and nothing more.
Forty-plus years later, his brother long gone, Roger Halliday was an FBI agent taking his last lap before retirement. He'd started as a probie on an Evidence Response Team, put in some time with the Behavioral Analysis Unit before going mainline with the Criminal Investigative Division. So he'd seen his share of dead bodies, shot or stabbed or thrown off a thirty-two-story building onto the hood of a car.
But he'd never seen anything like this.
"How many times did you watch it?" his partner asked. Agent Juwan Cook, in the Bureau for eight years now. He was black, with a smooth head he shaved every morning and a thin mustache. Cook kept his eyes straight ahead as he drove.
"Three times," Halliday said. The first time, it had been just a matter of getting through it. When it was done, he had watched it again. Twice. Looking for details. Doing his job.
Cook shook his head and kept driving.
The house was on the north side of Scottsdale, up by the McDowell Sonoran Preserve. One of those places you drive by and imagine yourself living someday, if you work long enough and are smart with your money. Spanish architecture, the standard for any house in this neighborhood. Clay tiles on the roof, an in-ground pool in the back. Five thousand square feet, one story, easier to keep cool in the summer. A single strip of grass, just a token amount of green for the dog to walk on. Everything else typical Arizona—rock, gravel, and cactus.
Once they turned down the street, it wasn't hard to find the Thompsons' house. There were already half a dozen police vehicles lined up on the street, and there were so many men walking through the yard it was raising a dust cloud you could see from two miles away.
Halliday had gotten the call just after eleven thirty, the last link in a long chain that had started somewhere on the Mediterranean Sea. It was almost noon now, on a clear February day, seventy-six degrees on the car's digital thermometer, the kind of day that makes Arizona feel pretty much perfect. Halliday wished the call had come in sooner, that he had had the chance to get here first, to see the house, to see the victim. He wasn't looking forward to pulling rank when he finally got here—that was the kind of thing that made the local cops hate the "feebs," if they didn't already. But this was going to be his case. He could feel it.
After chasing him around California, Utah, Nevada, and now Arizona, this UNSUB, the unknown subject suspected in the murders of at least five other women, had finally left a victim in the wrong place.
_But why here?_
Halliday tried to see this house the way the UNSUB would have seen it. A quiet street, not many neighbors. They had driven by a guard shack on the main road, but the UNSUB would have known this was just for show. With all of the trails in the preserve running along the backs of these property lines . . . There were plenty of ways to get to the house. You do it at night, and nobody sees you.
But as easy as it would be to get in, he had to know this place would have something else: a level of internal security that would make it different from all of the other places he'd used before. Sure, maybe the owner of this house put too much trust in the guard shack and didn't have an alarm, but those Internet cameras he'd installed—the cameras that recorded everything that happened inside that house . . . The UNSUB should have suspected they were there. And even if he'd decided it was worth the risk, he should have found them and disabled them.
He'd never been this sloppy before.
_So maybe this is someone else,_ Halliday thought. But no. _No way._ He thought back to everything he'd seen on that video . . . _How many human beings on this planet are capable of doing something like that?_
It had to be the same man.
Cook pulled off the street and stopped behind a long line of squad cars. As Halliday led the way up the sidewalk to the front door, he was already getting the once-over from the uniformed locals. Everything about him, from the ground up, screamed _Fed_ : his shoes, his suit, the expression on his face. Nobody tried to stop him. He asked the uniform standing by the door for the name of the man in charge and was sent in to see Detective Millens from Scottsdale PD. He found the man standing alone in the kitchen, in plain clothes, a gold shield shining on his belt. He was young for a detective, around Cook's age, and he was busy talking into his cell phone. As soon as Millens saw Halliday, he ended the call.
Halliday got out his ID and showed it to him. Then he introduced Cook.
"Back of the house," the detective said. "Master bedroom."
As Halliday followed Millens, he looked down at the carpeting in the hallway and saw the shoeprints.
"How many of your men have been in here?"
"Two," the detective said, looking back at him. "Maybe three."
Halliday scanned the rest of the hallway with a frown as the detective led them into the bedroom. That was where he saw the body on the bed. It was covered by a white sheet.
Halliday took a moment to walk around the room. He looked out the window at the backyard—more rock, gravel, cactus—and at the road that looped around the lot and started climbing up toward Dixie Mountain. He stood motionless for a full minute, looking at every window in the neighboring houses, every vantage point from the road, every spot on the preserve where a man could use a set of high-powered binoculars to peer right into this room.
_He could be out there right now,_ Halliday thought, _watching us._
_But something tells me he's not there. Not now._
"Were these curtains open?"
"Yes," the detective said.
"Was the light on?"
"No, it wasn't."
Halliday left the window and went to the big jewelry box on the dresser, took out a pen, and used it to gently lift the lid. Nothing looked out of place. No conspicuous blank spots from which something might have been taken. So probably nothing stolen, even though, to his untrained eye, it looked like there were some real diamonds that would be worth putting in your pocket on the way out.
Halliday took a quick look in the bathroom, came back out and stood at the foot of the bed. The body lay in a straight line, perfectly centered on the bed, and even with the sheet over it he could tell that the arms had been folded neatly across the chest.
Five minutes in the house and he had already decided what needed to be done. Even if he knew it wouldn't be popular.
_"Everybody out!"_ he said, moving to the doorway so that anyone else in the house could hear him. _"Now_. _"_
"I understand if you need to take over the crime scene," Millens said, "but the ME's on his way right now and—"
"No," Halliday said. "We're not moving the body."
Millens took a moment to process what he was hearing. He looked back and forth between Halliday and Cook.
"You're not serious," he said.
"He'll come back again tonight," Halliday said. "She needs to be here. One more night."
He didn't feel like giving the detective a crash course on how a certain kind of "organized" serial killer works, the kind of killer who returns to the body several times after death for his further sexual gratification.
Like Gary Ridgway, the Green River Killer.
Like Ted Bundy.
"You're not leaving her here," Millens said. "You don't _have_ to. Think about it, Agent Halliday. If he comes back, you can still catch him."
"You don't understand," Halliday said. "He'll know if the body is gone. He won't come anywhere near this place."
"I'm not leaving her here," Millens said, nodding toward the body under the sheet. "It's that simple."
"Not your call to make," Cook said, standing next to his partner.
"Get all of your men out of here," Halliday said. "Make everything look _exactly_ like it did. I mean everything, right down to those footprints in the hallway. Which _never_ should have happened, Detective. Find a rake, a brush, whatever it takes. In thirty minutes, I want this place to look _untouched_."
Millens just stood there, looking back and forth between the agents.
"The sheet over the body," Halliday said, knowing this would be the hardest part. "I assume you did that."
"No way," Millens said. "No . . . fucking . . . way."
"I'll do it. After you've left."
"When they find out you didn't let us take her away—"
"We can't help her now," Halliday said. "What we _can_ do is stop this guy before he kills ten more."
Millens shook his head and left the room without saying another word. Halliday looked at his partner.
"Go," Halliday said.
"You really think he'd know if the body was gone?"
"You heard the man. The curtains were open."
"And the light was off," Cook said. "Even if he was on that road back there—"
Halliday put up his hand to stop him. He'd been taught long ago to trust his own instincts, the same instincts that went all the way back to the brief time in his career when he'd worked with the agents who had invented a new way to track serial killers. These agents studied them; they interviewed them; they took them through every case, going over every detail. They got inside their heads, even if that was the last place you'd ever want to be.
Thirty years later, Agent Halliday knew that the curtains had been left open for a reason. And that, lights on or off, the man they were looking for would never return to an empty house.
Cook knew his partner well enough not to press it any further. When he turned and left the room, Halliday was left alone with the body.
He had just talked to his daughter that morning. She was getting ready to come home from the hospital. If he had taken that retirement package, he'd be home right now with his wife, waiting to hold his first grandson.
But here he was. Two and a half hours after that phone call, standing in this bedroom.
He looked down at the still form on the bed. Then he lifted the sheet.
* * *
—
IT WAS ALMOST TWO A.M. They'd been waiting in the van for five hours. Halliday, Cook, Detective Millens, and another agent, named Pfeiffer, from the regional SWAT team. All four men were wearing tactical vests. Halliday's stomach burned from his third cup of coffee.
"The suspect was here around midnight last night," Millens said. "Don't these guys stick to a routine?"
_These guys._
"Yes," Halliday said. "Maybe all that commotion around the house today . . ."
Millens gave him a look, like _Don't even try to pin this on me or my police department_.
Halliday was already starting to regret his decision to let Millens sit in on the surveillance. The FBI had quickly installed its own cameras all over the house, and had patched those into its own secure video feed, all relayed to a series of monitors here inside the van, which was parked in the courtyard behind one of the neighbors' houses. They were smart enough not to park it on the street. Smart enough to make sure everything looked exactly the same. _Everything._
Millens was sitting next to Halliday, rocking back and forth with nervous energy. On the feed from the bedroom, the two men could see the body lying in the center of the bed, unnaturally still, the exposed skin glowing in the infrared light.
"I see something," Cook said. He was on the other side of the van, watching another monitor, with the feed from the external camera they'd mounted a mile away in the Sonoran Preserve parking lot. Halliday turned to look, but Cook's eyes were thirty years younger, so it took a few seconds for Halliday's to catch up.
Then he saw it.
The vague figure of a man, walking away from a vehicle, toward the camera they'd mounted on a light pole. As he passed under the light, his face was obscured by the baseball cap on his head. He was moving quickly. With purpose. Leaving the parking lot and entering the trail.
Heading toward the house.
There were two agents waiting in the closed-up rangers' building. They contacted Halliday now over the radio to let him know what he had just seen with his own eyes.
_He's coming your way_.
Another decision Halliday had made. _Let the man get to the house before we move. Let him come through the door. Don't try to pick him up in the parking lot._ It was a decision he now had twelve long minutes to wonder if he'd regret as he watched the next monitor with the video feed from the rooftop camera.
_The man should be on the street by now,_ he thought. _Even walking in the dark, it's less than a mile._
_He got spooked. Now he's running. We'll never see him again._
But then Cook pointed to the monitor. "There."
Another movement on the screen. The same figure, growing larger.
Halliday could feel the adrenaline pumping through his body. He could sense the same in the other three men in the van. The SWAT agent was on his feet, ready to open the back door.
Halliday keyed the mic on his radio.
"Nobody moves until he enters the house."
"Why are we waiting?" Millens said.
"He won't touch her again," Halliday said to him. "I promise. But I want him in that house."
They watched the figure as he came close to the front door, moving quickly but without rushing. Like a man with no doubts, no worries.
Then he stopped. He seemed to be looking right at the rooftop camera. Looking right at Halliday with the blurred face of a pale yellow ghost.
Halliday keyed the mic again.
"Everyone hold," he said.
Halliday watched the still figure, holding his breath.
_Open the door, God damn it._
_Open the fucking door._
The figure disappeared from the rooftop camera's view. Halliday quickly looked to the other monitor, saw him stepping into the living room and turning on the light.
_"Take him!"_
The back door of the van was thrown open. Halliday was last out of the vehicle, his partner already halfway to the house. He saw the other agents converging—from a darkened house on the other side of the street, from another house on the property behind. Three directions, twelve men in total, all armed, including eight more members of the regional SWAT team. Halliday tried to catch up to them, hoping the lead men would stop the UNSUB in time.
He wanted to keep his promise to Millens.
When Halliday made it to the front door, he had to take a moment, half doubled over, to catch his breath.
There was a man on the floor, his hands cuffed behind his back. Halliday bent down to see his face, pulling off the man's black baseball cap. Late forties maybe, long light brown hair, fair skin. He was wearing black jeans, black hiking shoes, and a black dress shirt that looked freshly ironed. Just in those two seconds, Halliday was already seeing a man who was smart, who was careful, who was so neat he made sure his shirt wasn't wrinkled when he broke into a house to visit a corpse.
The man looked back at him with something like a smile. No surprise. No anger. He was about to say something, but Halliday didn't want to hear it. Not yet. He left him there and went to the bedroom, thinking, _At least he didn't make it this far. He didn't even get a chance to turn on the light and see her again._
Halliday stood in the doorway. In the dim glow from the window, he saw the woman's body on the bed. He opened the closet door and found the white sheet he had taken away from her earlier that same day. With great care, he unfolded it and placed it over her body.
"Thank you," he said to her.
He stood over her for a while, listening to the men in the other room, knowing exactly what they were feeling—how you wait and wait for hours and then in the course of a few seconds you're in motion, everything a blur. Even after you take your man down, your heart is still beating fast. The adrenaline has nowhere else to go. These men would be up for the rest of the night, pacing back and forth in their bedrooms, or sitting with open bottles at their kitchen tables.
_We did it. We took down a serial killer._
For Halliday, the night would go differently. He'd go home for a couple of hours, just long enough to catch his breath, get up and make some coffee, put on some new clothes. Then come right back to the office. Years ago, he wouldn't have left the suspect, would have stayed up forty-eight hours in a row if he had to. But he knew he'd be back before anything important happened, and that his young partner would handle things in the meantime.
Besides, Halliday needed to be with his wife on a night like this, even if it was just slipping into the bed next to her for a few minutes. She'd wake up and ask him if everyone had made it home alive. After thirty years married to an agent, she knew that was the most important question to ask.
_Yes,_ he'd say.
_Did you catch him?_
_Yes._
_Good._ Just like that.
Then, while he stared at the ceiling, he would listen to her breathing as she went back to sleep.
* * *
—
BY THE TIME Halliday walked back into the FBI field office on Deer Valley Road, the UNSUB had a name: Martin T. Livermore. He was a robotics engineer—at least when he wasn't abducting women, killing them, and then violating their dead bodies for days at a time.
There were five open cases on Halliday's desk: two in California, one in Utah, one in Nevada, and one other case in Arizona, besides the one they'd just nailed him on. In all of the cases, it had been a woman between the ages of twenty-five and thirty-five. Three of those five women had been killed in their homes, two abducted and killed in nearby motels. In all of the cases, there were trace amounts of blood and semen found at the scene. And one other thing: rope fibers.
In all five cases, the women had been tied up. Here in the sixth case, where they finally had a body to examine, they could see the ligature marks crisscrossing her body, looped across her mouth like a gag, looped across her throat. The medical examiner's report on this woman would list asphyxiation as the official cause of death. There was evidence of sexual penetration, both before and after death.
She had been alive for six to eight hours after being captured.
Six to eight hours of torture before the ropes finally strangled her.
In each of the previous cases, the body had been moved to a second location, which had apparently been chosen with great care. Abandoned buildings, or houses that were unoccupied for weeks on end. In every case, this second location had been found eventually, by _someone_. If it was a house, the owners would finally come home and discover that someone had been sleeping in their bed—a perverse retelling of Goldilocks and the Three Bears, only this time with the evidence left behind by a killer. If it was an abandoned building, it would take longer, but eventually the location would be found. In one case, it had been a contractor visiting a long-vacant house, preparing an estimate for a renovation. He had found the victim's bloody clothing neatly folded next to the bathtub. In another case, a local police officer had gone into an old warehouse looking for a vagrant who'd broken in to steal copper wire. The officer had found a blanket neatly spread out on the second floor, near a window overlooking the street. Once again, there was bloody clothing nearby, neatly folded.
From each of these second locations, the FBI had gathered trace evidence and had matched that with the evidence from the kill site. A map was constructed for all five known victims, showing the progression from one to the next, moving from the first two victims in California to the single victim in Utah, then to Nevada, then Arizona. The second location would range from as little as twelve miles away from the kill site to over a hundred.
And then the bodies were never seen again.
When they reconstructed Livermore's movements through California, Utah, Nevada, and Arizona, they corresponded exactly with the dates and locations of the crime scenes. His DNA matched the samples taken from each location, in all five of the open cases.
Carolyn Kline, victim number six, had been killed in her own home on the north side of Phoenix four days before, had died on the floor of her living room before Livermore had brought her to the Thompsons' house in Scottsdale, the house he had returned to on the second night, at which point he was observed on the security video. If he hadn't been captured, he would have presumably followed the same protocol as the other five: after approximately one week of sexual contact with the dead body at the second location, the body would have been removed, taken to wherever he took them when he was done with them.
_We've never come close to catching him before._
That video recorder. He should have seen it.
To Halliday, it didn't make sense.
That's why he was so eager to talk to Livermore. So he could ask the man himself.
* * *
—
THE INTERVIEW TOOK PLACE on the top floor of the Maricopa County Fourth Avenue Jail. Livermore was being kept here while they figured out which federal detention center could best hold him. Fine with Halliday, because he knew this was the most secure jail in the state of Arizona. Maybe in the entire country. Halliday rode up in the elevator alone, getting his mind into the right place.
_Been a while since I sat across the table from someone like this,_ he said to himself. _I even let myself believe I'd be out of here before we caught another one._
That thought was followed by another, the same thought he'd been revisiting ever since he'd gotten the call about the body found in Scottsdale.
_There is no good reason to keep doing this to yourself. You've given this job enough._
The elevator door opened. Halliday walked down the hall. Two guards were waiting for him, and they nodded to him without saying a word. When they let him into the intermediate room, he had a moment to look through the small window in the door, set at eye level. He saw Livermore sitting at the interview table. His hair was pulled back and tied neatly behind his head. _He's just spent his first night in a jail cell,_ Halliday thought, _but he doesn't look rattled. Even from here, the man's eyes are clear and focused. Like he can't wait for this interview to begin._
Halliday thought back to the years he'd spent studying the work of Robert Ressler, the FBI agent who had essentially invented profiling. Then later John Douglas, who'd turned profiling into an art form. When Halliday had been invited to join the Behavioral Analysis Unit permanently, he'd told them he didn't want to move his family to Quantico. That had been the official excuse, anyway. The truth was he just didn't want to spend his career working with serial killers. He still kept in touch with BAU, and they had offered to send a man out here to talk to Livermore.
Maybe Halliday should have taken them up on the offer.
"This is what they do," his partner had said to him. "Why take this one yourself?"
Halliday hadn't given him an answer. Maybe he hadn't even known. Not really. Not until this moment, as he stood at the door and looked at Livermore through the little window.
_There's one reason I'm here,_ he said to himself.
_Carolyn Kline._
She was twenty-six years old, an optometrist's assistant, still taking classes at Arizona State University when she wasn't working. She was a daughter to two parents, a granddaughter to three surviving grandparents. A girlfriend to one boyfriend. But to Halliday she would always be the woman who helped him catch Livermore.
The second door was automatically unlocked with a loud buzzing sound. He took a breath and pushed it open. Livermore looked up at him with that same enigmatic smile he'd given him when he was first handcuffed at the crime scene. Now he was wearing the same orange jumpsuit every suspect was issued at intake, his hands not only cuffed but also attached to the chain that ran around his waist. His ankles were shackled with leg irons, with a ten-inch length of chain between them.
Halliday took a moment to gather his impressions of the man. To study him at close range, now that he had the opportunity. He saw a man who'd spent most of his life indoors. Fair skin. No sun damage. Not a surprise, given his job, but there was little else about the man to suggest an engineer. He didn't wear glasses. He wasn't soft around the edges. In fact, he looked like a man who took care of himself. A man most women would call handsome. Maybe even _striking_. Not musclebound, but lean and athletic. He could imagine this man doing an hour of light weightlifting, and then spending another hour on the treadmill. And then maybe one more hour looking at himself in the mirror.
Halliday had brought a folder into the room with him. He put that on the table and then sat down in the chair across from Livermore. There was a video camera mounted high in the corner of the room. The light was blinking. Every word, every movement, it would all be recorded.
"Congratulations," Livermore said. "I understand you have a new grandchild."
Halliday had just opened the folder, was about to take out six photographs. He stopped dead when he heard those words.
"How do you know that?" Halliday looked into Livermore's eyes, really _looked_ for the first time, and saw that he had green irises circled with bands so dark they were almost black. The eyes of an exotic animal.
Eyes you'd never forget.
Livermore shrugged at the question, gave Halliday that same half smile again.
_He overheard it in the hallway,_ Halliday said to himself. _One of those guards outside, talking about me before I got here._
_Ten seconds in this room and he's already trying to put me off balance._
_I will not let that happen._
He went back to the six photographs, taking them out of the folder one by one and putting them on the table.
Six photographs.
Six women.
"Every one of these women," Halliday said, "you selected for a reason. Every step, every movement . . . it was all thought out. Where you took them to kill them. Where you took them after that. You must have watched those houses, those buildings . . ."
Livermore shook his head, giving Halliday the little half smile.
"You were just as careful when choosing your locations," Halliday went on, "as you were when choosing your victims."
Halliday pushed one of the photographs forward. It was the last victim, Carolyn Kline. Livermore looked down at the woman's face for just a moment. Then his eyes returned to Halliday's. The half smile was still on his face.
"Until this one," Halliday said. "Why did you break the pattern?"
Livermore shook his head again, and this time he let out a snort of laughter.
"This funny to you?" Halliday said.
"In its own way, yes."
"Last time I checked, you're the one wearing the cuffs and leg irons, and heading to death row."
"Do you honestly think," Livermore said, each word chosen with deliberate care, "that I didn't know there were cameras in that house?"
It was the last thing Halliday expected to hear, but this time he kept his reaction hidden.
"They were as obvious as this one," Livermore said, tilting his head toward the video camera over his shoulder. "Or, for that matter, your interrogation techniques."
Halliday stayed silent, waiting for him to continue.
"The first camera was mounted on the bookshelf," Livermore said, "between the legs of the wooden elephant. The second was in the bedroom, on the armoire, in the silk flowers."
Halliday kept staring into the other man's strange green eyes. Kept waiting for more.
"I _wanted_ you to see me," Livermore said. "Did you enjoy the performance?"
Halliday didn't move.
"Tell me, Agent Halliday, is it strange for you now? Sitting here across from me? After such an . . . intimate experience as we shared? How many times did you watch it, anyway?"
_I will not react,_ Halliday thought. _I will not give him anything._
"I wonder if you'd even admit to yourself," Livermore said, "that in some primal part of your brain you may have enjoyed it."
_Nothing,_ Halliday said to himself. _I am made of stone._
"It's all right, Agent Halliday. It'll be our secret."
Halliday waited for another few heartbeats to pass. Then he reached down to the pile on the table and slowly pushed forward the other five photographs.
Livermore was still smiling as he studied the photographs, one after another. He had to bend his whole body sideways to touch one of them. Halliday was about to take it away, but then he saw that Livermore was simply straightening it, so that all of the photographs were in a neat, straight line.
_He has a meticulous, ordered mind,_ Halliday said to himself. _Compulsively neat. He_ _removes the bodies, takes them to a place where he can have his time with them. Someplace where he knows he won't be seen._
_Until this time._
"Where are the other bodies?" Halliday said.
"In a special place."
Halliday watched him for another moment, then took out the pad of paper from the folder and slid it across the table, with a felt-tip marker on top. Because you never give a killer a sharp object, not even a ballpoint pen. Not even when his wrists are cuffed to a chain around his waist.
"Write it down," Halliday said. "The exact location of this _special place_."
Livermore looked at the pad for a moment, as if actually considering it. Then he tilted his body sideways again, took the marker, and wrote down two words.
Halliday leaned over the table and read the two words:
_Alex McKnight._
"Is this supposed to mean something to me?"
"There is a small town in Michigan," Livermore said, "called Paradise. If you go there, you'll find this man living there. He's a retired police officer from Detroit. I would like you to bring him to me."
"Why do you want him here?"
"I understand Paradise is a very small town. He shouldn't be hard to find."
"If this is some kind of game, I'm not going to play it."
"I think you will," Livermore said.
"Why?"
"Because Alex McKnight is the only person I'll talk to."
Halliday sat there, watching the man. Waiting for more. When it didn't come, he stood up and started to gather the photographs and put them back in the folder.
Livermore wrote two more words on the pad.
Halliday stopped dead. He read the two words. Then he stared at the man across from him. That same cold half smile stayed on Livermore's face as he dropped the marker on the table and leaned back in his chair.
"You should sit back down," Livermore said. "I don't think we're done here."
# CHAPTER TWO
ALEX? IS THAT YOU?"
It was the last voice I ever thought I'd hear that night. Or any night, ever again. I was sitting at the Glasgow Inn, with a folder full of papers spread out on the bar top. A February night in the dead middle of a long winter. Jackie Connery, owner of the Glasgow Inn, was behind the bar, watching the Red Wings game on the television he'd hung from the ceiling. Vinnie LeBlanc, my closest neighbor, a member of the Bay Mills tribe, and a blackjack dealer at the casino, was sitting in one of the big overstuffed chairs by the fireplace. On most nights like this, I'd be sitting right across from him, warming my sore body after a long day of shoveling snow and chopping wood, nursing a Molson and listening to Jackie complain about driving all the way to Canada because I won't drink a fake American import.
But tonight I was sitting at the bar because I had all of these papers to look at and a stakeout date with my new partner in a couple of hours. My cell phone rang in my pocket and I took it out, expecting to hear Maureen's voice.
It was a woman, but not Maureen. It took me a moment to place the voice. That's how long it had been.
"Jeannie?"
"Yes, it's me."
I drew a blank on what to say next. I was counting up the years since I'd last talked to her, this woman I'd once been married to. I was just passing twenty years in my head when she broke the silence herself.
"I know it's been a long time," she said.
"Is everything okay?"
"Yes," she said. "It's . . ."
There was a long pause. I wasn't even sure if she was still on the line. The cell phone signal up here in the Upper Peninsula is always a crapshoot.
"It's good to hear your voice," she finally said. "I hope you've been well."
"Where are you living now?"
"I'm back in Grand Rapids. In my parents' house. How about you?"
"You remember those cabins my father built?"
"Up in Paradise?"
"That's where I am."
I had come up here a year after I had left the Detroit Police Department, just after Jeannie had left me. The most remote place I could find, maybe eight hundred full-time residents in this little town on the shores of Lake Superior, a good five-hour drive from Detroit. My stated reason was to sell off the cabins my father had built along an old logging road, just north of the one blinking light in town. The real reason was that I wanted to get away from everything else in the world, and I was wondering if Paradise would be the place to do it.
I'd been living here ever since.
"Listen," she said, "I know this is out of the blue, but there's a reason I'm calling you . . . You remember my grandmother?"
"I think so."
"She died, Alex. And she left some things . . . to both of us. You remember that place she had on the lake?"
As far as I could remember, I'd only been there once, right after we got married. I had a dim memory of a little house on an inland lake, maybe an hour north of Grand Rapids. The place had seemed like it was just about ready to fall down back then. Unless somebody had renovated it, I couldn't even imagine what it would look like now.
"Plus that old car she had, the big white Cadillac. Everything in the house. And some money. Not a whole lot . . . But it was everything she had."
"I don't understand. She left it for _us_?"
"Yes, to both of us. I don't know if she was just confused about the, um . . ."
She didn't have to say the word. The divorce. It was the only contact I ever had with her, after she had left. Those papers from her lawyer that had come in the mail. _Please sign and return._ Then it was done.
"Or I don't know," Jeannie said. "She was pretty sharp, right up to the end. Maybe she just had this naïve belief that it would be enough to get us back together."
"I'm sorry to hear about your grandmother," I said, trying hard to remember the old woman's face from the wedding. "But whatever you want to do with that house, and everything else . . . Just send me the papers and I'll sign whatever I need to. That stuff belongs to you."
"Gee, thanks," she said, and then she laughed. "That Cadillac is just what I need to drive to work."
Even after all of these years, it felt good to hear her laughing. When she stopped, there was a long silence.
"I have one more thing to tell you," she said. "It's something I should have said a long time ago . . ."
I waited.
"I never told you how sorry I was, Alex. All these years, I never got the chance to say it."
"It's all right, Jeannie. I know I didn't make it easy."
"That night you and Franklin both got shot," she said. "I remember being in the hospital, seeing you in that bed with all those tubes in your body . . . Not knowing if you'd ever open your eyes again. And then when I found out that Franklin was already dead . . . I just didn't know how to handle it, Alex. I walked away and I didn't even say good-bye. And I've never forgiven myself for that."
"I don't blame you for leaving," I said. "I let it go a long time ago. You should, too."
"Okay," she said. "I'll try."
There was another awkward silence, until she finally thanked me and said good-bye. I sat there on the barstool and looked at my phone, thinking about all the other things I should have asked her. This woman I once thought I'd spend the rest of my life with.
"Alex," Vinnie said from behind me, "what's going on?"
I turned and looked at him. Then at Jackie, who had stopped washing glasses and was watching me just as intently.
"That," I said, "was a phone call from another life."
I was still thinking about it as I left the Glasgow and got in my truck. It was twenty degrees, with two feet of snow on the ground and more on the way. I had just enough time to pick up Maureen and head over to Pickford, where we were hoping to find our man, a twice-convicted drug dealer who never showed up for his third court date in Iron Mountain. It was my latest case in my new job as a "fugitive recovery agent" for Superior Bail Bonds out of Marquette, a job that my former partner, Leon Prudell, once held—until he came home from one case with a knife wound. That was all his wife, Eleanor, needed to see. Now Leon was back working the copper kettles at the Soo Brewing Company full-time, and for some reason I had agreed to give the job a try in his place. It was a way to break up the long winter months, and I didn't have a wife waiting at home, worrying about me.
For liability reasons, Superior didn't let its agents carry firearms when recovering fugitives. As much as getting shot as a cop had put me off guns for the rest of my life, I still wasn't sure what to make of doing this job unarmed, especially when the fugitives weren't playing by the same rules. But tonight I was working with Maureen. She was a pro, and she really knew how to work the wives and girlfriends of the men we were tracking.
With a pair of handcuffs in my pocket, and no gun, I headed out into the cold Upper Peninsula night.
* * *
—
AGENT HALLIDAY watched his breath in the air, trying to remember if he'd ever felt air so cold it actually _hurt_ to breathe it. After seven hours on a plane, they were walking down the stairs, onto the tarmac. The pilot told them he'd de-ice the wings and keep everything ready to go.
They walked into the one-room Chippewa County International Airport, skipped the waiting room, and went right to the parking lot. The rental agent was waiting next to the black Jeep Cherokee, which was already running. They verified the directions to a town called Pickford, and then they were on the road.
It was fewer than twenty miles. Cook was driving, and Halliday could tell he'd never driven on ice before.
"You get us killed before we even get there," Halliday said, "that would not be a productive evening."
Cook shook his head and kept driving.
When they arrived in Pickford, they found two roads and a traffic light. A few buildings and wide-open fields with snow blowing across the road. Halliday had already contacted the owner of the Superior Bail Bonds company, Alex McKnight's current employer, and had made two things clear: he needed McKnight's location, and he needed the owner _not_ to warn McKnight they were coming.
"What else do we know about this guy?" Cook said.
"He was a cop in Detroit for eight years. Got shot on the job, came up here a year later. Rents out some cabins. He was a private investigator for a while . . ."
Cook looked over at him.
"Now he's a bounty hunter," Halliday said. "Excuse me, a fugitive recovery agent."
"Will he be armed?"
"According to his boss, no. But that's no guarantee."
Halliday checked his own weapon. He was one of the few agents still allowed to carry the old nine-millimeter Sig-Sauer P228, the same gun he swore he'd retire with.
"And as far as we know," Cook said, "McKnight has no connection to Livermore."
Halliday holstered his gun and looked out at the snow. " _That_ , partner, we're about to find out."
* * *
—
MAUREEN HAD ALREADY knocked on the door, told the woman who'd answered she was looking for our man, playing the "other girlfriend" angle, telling her she was about to throw all of his clothes out into the snow. She came back to the truck and told me this wouldn't take long.
Maureen was ten years younger than me. She had dirty-blond hair she kept tied behind her head, clear nail polish on rough hands, the hands of a woman who lived in the Upper Peninsula. She was a good partner, even if I didn't know much else about her yet.
"You're even less talkative than usual tonight," she said to me. We were waiting in the driveway next door to the girlfriend's house, the heater in my truck going full blast.
I was still thinking about the phone call, but I wasn't sure if I wanted to tell her about it. That's when a flash of headlights hit us both in the eyes. I put the truck in gear and lowered the plow, scraping against the asphalt as I drove forward. To whoever was coming down the street, I would be just another plowman, the UP's best winter disguise. The truck on the street was moving fast.
I felt the familiar rush of adrenaline, no different now than on my first day as a Detroit cop. Maureen had her cuffs in one hand, the other hand ready to open her door. If this went well, we'd take him right there in the girlfriend's driveway, before he even made it into the house. If it went badly, we could always hook his cuffs to the back of my truck bed, take him to the Chippewa County Jail, and see how frozen he was by the time we got there.
The man's truck skidded as it slowed in the street, then turned and finally came to a stop in the driveway next door. We both hit the ground at the same time, covered half the distance before the man even noticed us. It took another second for him to process what was about to happen to him.
That's when the second vehicle arrived and blew everything up.
It was a black Jeep Cherokee, coming too fast down the icy road and going right into a sidespin. Whoever was driving had no idea what he was doing. When the vehicle corrected itself, it came down the driveway we'd parked in, its headlights blinding us. The driver hit the brakes too late, sliding across the ice and slamming into the back bumper of my truck.
The vehicle backed up, spinning its tires and sliding sideways so that now all three of us were lit up in the headlights.
"We have a bond for your arrest!" Maureen said to our fugitive, who'd already stepped out of his truck and was now trying to climb back inside. He kicked out at her with a booted left foot, catching her in the stomach and sending her back into the snow. I grabbed the door and held it open with one hand, reached in for him with the other.
That was when another man caught me from behind. I threw an elbow, pure instinct at that point, figuring it must be one of the fugitive's buddies jumping me. I felt the impact all the way up my arm, then sensed the man stumbling away from me. As I turned to face him, he had already brought out a Glock from his belt. I didn't know the exact model, but the important thing was he had it pointed right at my chest. I heard the other truck's door closing, the engine starting.
"FBI," the man with the gun said. "Do not move."
"That's a fugitive!" I said. "Stop him!"
But it was already too late. The truck spun its way back out of the driveway, the tires whining against the ice as it slid back onto the road. I caught sight of him behind the wheel, for one perfect split second, as he gave me the middle finger and then flipped the truck into drive and tore away.
"What the hell's the matter with you?" I said. A black man with a smooth, shaved head, he still had the gun leveled at me, so I made a point of keeping both hands where he could see them. Maureen was up on one knee now, catching her breath and brushing herself off. Another man appeared behind her, white and much older than the one who'd grabbed me. I noticed for the first time that the two men were dressed in identical windbreakers.
"You just let a drug dealer get away," Maureen said, pushing away the older man's hand when he tried to help her up. "Who are you guys?"
"We're looking for Alex McKnight," the older man said.
I looked back and forth between them. They were both shivering in their windbreakers, comically underweight for a UP winter.
"What the hell is going on here?" I said. "Put that gun away."
The older agent came close to me. As he looked me in the eye, I knew exactly what he was doing, because I'd done it myself as a cop, more times than I could count. He wanted to see my immediate reaction to whatever he said next.
"We need to talk to you," he said. "About a man named Martin T. Livermore."
"Is that name supposed to mean something to me?"
"That's what we're here to find out."
He kept looking me in the eye until he nodded his head and put a hand on my back. "We'll talk about it on the plane," he said. "Let's go."
I pushed his arm away. "My partner and I are working here. You can talk to me tomorrow."
"I've got this," Maureen said. "Just go with them."
"I'm not going anywhere."
"This is not a request," the older agent said.
"Am I under arrest?"
"No," he said. "You are a person of interest."
_That's not something you want to hear an FBI agent tell you,_ I thought. _If I were sensible, now would be a great time to keep my mouth shut._
"I thought I was done with this a long time ago," I said, taking my usual pass on _sensible_. "Feebs messing up my whole life."
"Shut up and go with them," Maureen said. "Call me if you need me."
I took a long breath and watched it condense in the cold air.
"We've wasted enough time," the younger agent said. He had put his gun away, and by now there was a thin line of blood running from his nose. I didn't bother to apologize to him.
"Just go," Maureen said.
I shook my head and then finally got in the backseat of their vehicle. The younger agent got behind the wheel, spun the tires as he backed out of the driveway, and then took me away.
# CHAPTER THREE
I'D SEEN THE FACES of killers before. But never this face. Not until this moment, sitting on the plane.
"This is his mug shot," Agent Halliday said, putting the photograph on my tray table. "Taken less than twenty-four hours ago."
We had just lifted off in a small commuter jet, which had been waiting on the runway at the Chippewa County airport. There were no official FBI markings on the plane. There was probably an elaborate cover story behind that, unmarked planes the FBI used in special circumstances, but I didn't bother asking because I knew I wouldn't get an answer.
Halliday sat in the window seat. I was on the aisle. Across the aisle was Agent Cook. The airplane was still climbing in altitude, riding the rough air over the half-frozen lake. The photograph shifted back and forth on the tray table, until I finally put my hand on it to keep it in place.
I was still catching up to everything that had happened. Still trying to understand how this night had somehow led to me sitting here on this airplane with two FBI agents, and no idea where we were even going.
"Take your time," Halliday said. "It could be years since the last time you saw him."
I looked carefully at the man's face. He had close-set eyes, intense and focused, even here while holding a mug-shot board with his own name spelled out in white plastic letters. His head was slightly tilted, and there was a half smile on his face.
"No," I said. "I don't know this man."
"I said take your time."
I kept looking. He was about my age. Long hair combed back. Clean-shaven. There was an intensity in his eyes that maybe I wouldn't have noticed at first glance. On closer examination, I could see that he was a smart, focused man. The kind of man you'd expect to be successful at _something_.
"A man can remember ten thousand faces," Halliday said. "A woman, possibly more. If this man was significant in your life in any way . . ."
"This face means nothing to me."
"You were a cop for eight years," Halliday said. "Is it possible that—"
"Did he ever live in Detroit?"
"We can't put him in Detroit, no. He definitely has no arrest record there."
"Then I don't know what to tell you."
"The man who shot you . . ." Halliday said. "We don't see any connection with Livermore, but—"
"He was a crazy fuck who lived by himself," I said. "He wasn't connected to anybody."
Halliday left the mug shot on my tray table. He put another photograph on top of it. A young woman. "This was his first victim," he said. "Three years ago, in California."
I shook my head. It was another face that meant nothing to me.
"A second woman in California," he said, putting another photograph over the first.
I shook my head again. He showed me three more.
Utah.
Nevada.
Arizona.
All young women.
Women I'd never seen before.
He put down the sixth photograph. "Arizona again," he said. "Phoenix. Just a few days ago. Her name was Carolyn Kline."
"He abducted every one of these women," Cook said. "Tortured them. Then killed them. None of the bodies were ever found, until this one."
"He's a highly organized, highly sophisticated sociopath," Halliday said. "We don't know his exact methodology yet, but he's clearly very good at manipulating people. I sat in a room with him twelve hours ago."
I picked up all of the photographs, one by one, and looked at them again. Six faces of women who'd suffered and died, and then finally the one face who was responsible for all of it.
Martin T. Livermore.
A stranger.
"I'm going to make this question as simple as I can," I said. "What does any of this have to do with me?"
"He asked for you," Halliday said. "By name."
I let the words wash over me, trying to absorb their meaning but utterly failing.
"He says you're the only person he'll talk to."
"That makes no sense," I said. "Why would he want _me_?"
"That's a good question."
The two agents stayed silent for a while. I knew exactly what they were doing, because I had done it myself a thousand times. This was the exact right moment to wait. To let the silence build.
I picked up the mug shot, holding it closer.
I looked at the man's eyes one more time.
Nothing.
Absolutely nothing.
"I don't know what to tell you," I said.
"So then _what_ ," Cook said, "he just picked your name out of nowhere?"
"I have no idea."
He stared back at me, looking for the lie behind my eyes.
I wasn't lying.
Halliday put one more photograph on top of the pile. It was yet another attractive young woman, late twenties. This was more of a candid shot, taken at a party, the woman with a plastic cup in her hand, her other arm around another woman. A little silver ring in her eyebrow, and a big smile.
"Who is this?" I said.
"Her name is Stephanie Hyatt," Halliday said.
I shook my head. Another face I'd never seen before. "Is she the latest victim?"
Halliday watched me carefully as he composed his answer. I'd already figured out the dynamic between these two agents—Halliday was the one with the experience and the even temper. Cook had the raw energy, and he could obviously play the bad cop when it was needed. But now, as the overhead light painted every line in Halliday's face, he looked even older than I had first thought. Too old to be flying across the country and freezing his ass off in Michigan. Maybe too old to be doing this job at all.
"He claims to have met her two days ago," Halliday said. "I was not inclined to believe him, because it doesn't fit the pattern. He abducts one woman at a time, then he waits, sometimes for _months_ , until he moves on to his next victim."
"So why do you—"
"He had details," Halliday said. "Knew she was from Mesa. Knew she had a tattoo on her left shoulder blade. It all checked out. She was last seen two days ago."
"I don't mean to be insensitive," I said. "But you just told me most of those other bodies haven't been recovered, either. How is this one different?"
Neither man said anything. I looked back and forth between them, saw them exchanging something without words. Some horrible truth that they both shared. Another moment passed, and then it came to me.
The one good reason they'd fly two thousand miles across the country to find me.
"Are you telling me . . ." I said.
"Yes," Halliday said. "We have reason to believe she's still alive."
# CHAPTER FOUR
WHEN THE METAL DETECTOR went off, three different men drew their weapons on me. I froze and waited for them to pat me down. As if I'd actually be stupid enough to bring a gun into the jail. When they brought out the wand, it beeped when they waved it over my chest. That meant another pat-down, and eventually I ended up having to open my shirt and show them the scars.
"Twenty-two caliber slug," I said. "Next to my heart."
"You didn't set off the detector at the airport," Cook said from behind me.
"Airport detectors are set at a different level," I said. "This one's obviously cranked up to eleven, all right?"
We were in the Maricopa County Fourth Avenue Jail, where they'd driven me directly from the airport. It was early morning. The air was seventy degrees warmer than what I'd left behind in Michigan, even if I'd only felt it for the few yards between the airport and the car, and then the car and the jail.
I'd been surprised when we pulled up in front of the place, because you think of a jail as a place for drunk drivers to sleep it off, or for strung-out petty thieves to spend a week or two in holding cells, waiting for their trials because they can't scrape up the bail money. Not for one of the most notorious serial killers in recent history. But then two things hit me:
One, Martin T. Livermore—even though he was implicated in the murders of six women, and in the abduction of a seventh—was still officially an unconvicted suspect. That status would end about two seconds after his guilty verdicts were read in open court, but not before.
And two, this was Maricopa County, Arizona. So when it came to anything having to do with the apprehension and housing of criminals, right down to the county jail, these good folks did not fuck around.
As we got out of the car, I could see that the jail took up an entire city block, a hulking redbrick presence among the courthouses and other government buildings. There was a column of glass running four stories above the main entrance. The rest of the walls were broken only by the few thin slits that barely qualified as windows. It was as if the architect wanted to make sure anyone going into the place knew to leave all hope of ever seeing daylight again outside on the sidewalk.
"This is the most secure facility in the county system," Halliday had said as he'd closed the car door behind him. "The high-security floor is a vault."
We had come inside and met a half dozen of the detention officers who worked here. They all looked a little too amped up to me. But then here were two Feds, bringing in this civilian, this _stranger_ , right into the heart of their facility, to ride in their elevator all the way up to their most secure floor, to interview a serial killer. So I couldn't blame them for being a little on edge.
Of all the men waiting for us, the warden was the one in the suit and the bolo tie. "This is my jail," he said to all three of us, after he'd shaken every hand. "I make the rules. I'll be watching on the other side of the window at all times. Nobody will make any physical contact with the suspect. No objects will be passed to him. Do we all understand?"
He looked each of us in the eye, one by one, until he was satisfied. Then I was led through the metal detector first, and that was when everything went sideways. It looked like they were going to drag me all the way out here from Michigan just to let three Maricopa County detention officers gun me down in the lobby.
But when the sidearms were finally put away and I had buttoned up my shirt, the warden took us to the elevator, put his special key in the control panel, and took us up to the top floor. He led us down another long hall. Everything was clean and functional and harshly lit. Like any jail, anywhere in the country, and yet everything felt different here. Even in the air itself, this sense that something big was about to happen, something that everyone involved would remember for the rest of their lives.
_This is why you become a cop,_ I thought. _Or an agent. Or a guard or a warden, whatever you need to become to be a part of something like this. Whether you admit it to yourself or not._
When we got to the interview room, Agent Halliday told his partner to wait with the warden on the other side of the glass. Agent Cook didn't look happy about it, but then I hadn't seen him look happy about anything, from the moment he had run into the back of my truck in Michigan.
Agent Halliday and I were led into a small intermediate room, the door clanging shut behind us. We waited there for a long moment, until we heard the buzzing sound on the second door. Halliday pushed the door open, and we walked into the empty interview room.
There was a table in the center. Two chairs on the near side, one chair on the other. A single video camera was mounted high in one corner. The red light was blinking.
_Eight hours ago I was sitting in my truck,_ I said to myself, _freezing my ass off and watching for a petty drug dealer to show up at his girlfriend's house._
_Now I'm here._
Agent Halliday sat down. I took the chair next to him.
"Understand something," Halliday said to me. "I know you're an ex-cop. You think you know how to talk to a criminal . . ."
"I do."
"This man is something different."
"I'll be fine."
"I've interviewed psychopaths before," Halliday said, "but I'm still trying to understand Livermore. You're here because he asked for you. We want to know why. Let him go in that direction, but don't let him draw you in too deep."
"Just bring him in here," I said, feeling like I'd already waited too long for this. All of the confusion, all of the disorientation, it had been burned away on the long journey to reach this place. This moment. Now I just wanted to see this man face-to-face.
Another long minute passed. I heard the outer door opening, then the buzzing of the inner door. When it opened, I heard the rattling of chains. Halliday and I stood up at once, like some dignitary was entering the room. By the time I turned around, he was already through the doorway.
Martin T. Livermore.
He was wearing an orange jumpsuit. His hands were cuffed and attached to a chain around his waist, so his hands were effectively pinned to his sides. Legs shackled. He was my height, a little leaner. Long hair tied neatly behind his head. Smooth skin, even with a little stubble around his chin, everything about him perfectly composed, like he'd spent the last hour carefully grooming himself for an important appointment, instead of sitting by himself in an eight-by-ten jail cell, waiting to be interrogated.
I kept looking at him, waiting for a bell to go off in my mind. Some hint of recognition. But my mind remained silent. I _knew_ I had never seen him before, for the simple reason that having this man's intense green eyes on me was something I would never be able to forget.
I'd met plenty of killers before. I could remember running down at least two of them myself, literally chasing them and tackling them to the ground, putting on the handcuffs— _hooking_ them, as we used to say in Detroit—and then dragging them back to my car.
But this man . . . Thirty seconds in the room with him and I was already starting to wonder if maybe Halliday was right. Maybe this was another kind of man entirely, something I'd never seen before.
There were two guards with him, two men who obviously spent every free hour in the gym, the sleeves of their uniforms pulled tight over their biceps. They looked just as amped as everyone else in the building. Probably making double overtime and drinking in every second so they could tell the story at the bar.
"You can leave him here with us," Halliday said to the guards.
"That's not happening," one of them said. "We stay with him."
"There's no danger," Halliday said. "He's chained up. There are two other men outside watching us, including your warden."
"We stay with him," the guard said, sneaking a look at the one-way glass. He'd obviously been given an order, and he wasn't even going to _think_ about breaking it. "That's nonnegotiable."
Livermore had been watching the exchange with a slight smile on his face. He sat down on the other side of the table while the guards took their positions directly behind him. He didn't look comfortable—that would have been physically impossible—but he did his best to lean back in his chair as he kept looking at me. I'd sat across from five thousand men in interview rooms just like this one, and four thousand nine hundred ninety-nine of them had let their eyes wander to the ceiling, or had stared down at their own hands.
This was the first man who looked me square in the eye. Like he would somehow be the one leading this interview.
"Alex McKnight," Livermore said. "You'll excuse me for not shaking hands."
"Why am I here?" I said.
"Do you feel it? The gravity of this moment? The two of us finally sitting in the same room together?" He seemed to pick each word carefully, as if lifting it from a case lined in black felt.
I didn't answer him.
"Your whole life," he said, "has been leading to this moment."
His eyes remained locked on mine. The other men in the room didn't exist.
"What are you talking about?" I said.
"You and I are connected, Alex. Like two atomic particles hurtling in independent orbits. Until the inevitable day when they collide and everything changes."
"You're out of your mind."
"Everything I'm saying is quite lucid," Livermore said, and in that moment I saw a flash of something in his eyes. Something between impatience and outright anger. "Lucid and accurate. Our connection is real, Alex. It has shape, it has substance. A past and a future. You just haven't seen it yet."
"The only thing I see," I said, remembering what Halliday had told me on the plane, "is a lunatic who kills women and has sex with their dead bodies."
"I knew you'd say something like that. Eventually."
"You don't know a goddamned thing about me, Livermore."
I felt Halliday's hand on my back. I ignored it.
Livermore leaned forward and smiled at me.
"Four years in the minor leagues," he said. "Never made it to the show. Your strikeouts-to-walks ratio was never good enough. A classic case of overprotecting the plate, which I suppose says something about your personality. Your best batting average was .249."
That's when the room started to tilt. I could feel the other three men watching me. Agent Halliday, the two detention officers, measuring those words, watching for my reaction. A batting average is a hard number. It's right or it's wrong.
Livermore had it right.
"Eight years as a police officer in Detroit," he went on. "You and your partner answered a call one night. An emotionally disturbed individual, who ended up shooting both of you. Your partner died. You survived. They took out two slugs, left one in your chest."
He kept watching me, studying me like I was an open book to him, like he could see everything I'd ever done.
"You've been living with that night ever since," he said. "A cop's greatest failure, letting his partner get killed. Unfortunately for young Officer Franklin and his family, you couldn't protect your partner as well as you protected the plate."
I tightened my grip on the edge of the table. My partner's name on this man's lips . . . It was an obscenity.
"All right," Halliday said, "that's enough."
"You were married for nine years," Livermore said, ignoring him. "Your wife left you after you were shot. You ran away from what was left of your life, and you've been hiding in a little cabin on the edge of the world."
_I came two thousand miles,_ I thought, _to have my whole life laid out on this table._
_By this psychopath I've never met before._
"You hunt bail jumpers now. You pretend that this matters, even though you're no better than a janitor, collecting human refuse. Tracking down petty drug dealers who'll be replaced the next day. It's such a pale imitation of what you once were, Alex. Being a police officer may not be much, but at least you were _something_ then. You were a man who could actually change things in the world."
I didn't move. I kept staring into his eyes.
"Look at your hands, Alex. You chop wood, and you shovel snow. You have no other purpose now. And a man with no purpose is hardly a man at all."
He waited for me to look down at my hands. I didn't move my eyes.
"You should be thanking me," he said. "I am giving you a great gift today."
_He's trying to get under my skin,_ I thought. _I'm not going to let that happen._
"As much as you might deny it," he said, "there is some essential part of you that is _excited_ to be here with me."
"We brought him here like you asked," Halliday said. "Now tell us where Stephanie Hyatt is."
Livermore's eyes flashed again, like he was annoyed by the interruption to his little soliloquy about my worthless life. But then a half second later, his composure had returned.
"I was clear on this point," he said. "I will take Alex to her."
"You're not taking him anywhere," Halliday said. "You're dealing with me now."
"As of this moment, she has been tied up for seventy-two hours. I was careful not to constrict her circulation, but after so many hours in one position, it becomes unavoidable. The pain at this point must be unimaginable."
The words hung in the air. I could sense the tension building in Agent Halliday's body. Even the two guards were looking down on the back of Livermore's head like they wouldn't mind choking him right here in the interview room.
"She's out of water by now," Livermore went on. "I left her one container, with a straw. That means she's already gone through several stages of extreme thirst. Her muscles are cramping, which adds to the pain. Her tongue is swelling. If she tries to cry out for help, she'll barely make a sound."
He paused for a moment, gauging my reaction. I stared back at him without giving him anything at all.
"She'll soon have a severe headache," Livermore said, "as her brain literally begins to shrink inside her head. She'll have hallucinations. Then seizures. Her vital organs will shut down, one by one."
He regarded everyone in the room, even the guards behind him, with an imperious look on his face, like a military officer reviewing his troops and finding them lacking. Finally, he settled on Halliday.
"She is alone, Agent Halliday. Alone and scared. She knows she's been abandoned, and that she will soon die. And the saddest part of all . . ."
His eyes left Halliday and came back to me.
"She can't even cry about it," he said. "Because there is no water left in her body to make tears."
In that moment, I wanted nothing more than to go over the table. I would have done anything to him, would have water-boarded him, would have driven nails into his hands, whatever it took to make him tell us where this woman was.
"There is still hope for her," Livermore said to me. "These men all want to find her, but they have no idea how. You are the key."
"What do you think's going to happen here?" Halliday said. "We're going to open up the front door and let you out?"
"I understand you'll need to have a number of men involved," Livermore said. "As long as Alex is one of them."
"How many more men would you need?" I asked Halliday.
"Alex, we have to talk about this."
"How many?"
"My partner and I," he said. "The jail will want two men, three if you include a driver. The state police . . ."
"You already have a plan," I said. "In case you decided this was real."
He hesitated. "Yes."
But something still didn't feel right to me. I would have gotten my gold shield if I had stayed on the force, and if I had, my instincts would have guided me for the rest of my career. Serial killer, psychopath, it wouldn't have mattered. It all would have come down to one question.
_Is he telling the truth?_
I leaned forward across the table, so close it startled the guards behind him. He kept looking me in the eye, without moving a muscle.
"I don't think this woman is alive," I said.
He smiled. "Are you willing to take that chance?"
Halliday stood up and tried to grab me by the shoulder. I pushed him away. It was becoming a bad habit, physically resisting a federal agent, but at that point I didn't care anymore.
"Livermore," I said, "what the hell do I have to do with any of this?"
"I know I'm not answering your questions," Livermore said to me. "But that's not why I brought you here. You just have to play your part, Alex. You don't have to know how it ends."
"It ends with you strapped to a table," I said. "You know that, right?"
"I think we're done," he said with another smile. "For now."
He startled the guards again when he stood up. Each man grabbed one of his arms. A pained look crossed Livermore's face, like this was just one more annoyance. He looked nothing like a man who'd have to start accepting treatment like this for the rest of his life. As the guards pushed him toward the door, he stopped just long enough to meet my eyes one more time.
He nodded to me, gave me one last half smile. Then he walked out the door.
# CHAPTER FIVE
WE NEED TO BE ready for anything," Agent Halliday said. "I don't want any surprises."
"What the hell's he going to do?" Agent Cook said. "His arms and legs will be chained. With seven armed men watching him."
He had left me out of the equation, the eighth man along for the ride, but I wasn't going to say anything about it. One hour had passed since the jailhouse interview. One hour of me replaying everything Livermore had said to me, trying to make it all add up into some kind of sense. But it still wasn't coming together.
_I shouldn't be here,_ I said to myself. _I should be back home in Paradise, plowing snow._
_But if there really is a woman out there somewhere . . ._
We were all in the FBI sedan, with me in the backseat again. Cook was driving, Halliday was riding shotgun. We'd been up for almost thirty hours straight by now. I could see it weighing heavy on Halliday as he turned to look at me.
"Alex," he said, "I want you to be ready for an audible."
"What are you talking about?"
"You're the X factor here," he said. "These other men don't know you."
_You don't know me, either,_ I thought. But I let it go. If I were in their shoes, I'd be just as mystified by this stranger from Michigan, who didn't seem to know anything about Livermore, even though he could sit there and recite my batting average.
But still, what would the plan be? Cook was right—with Livermore's hands cuffed and chained to his belt . . . In their wildest imagination, how did they think one stranger could change that equation?
"Everything's covered," Cook said to his partner. "He'll be tied up like a Christmas turkey."
Halliday shook his head and looked out the passenger's-side window as the city of Phoenix passed by. I could feel the nervous energy coming off of him in waves, even if his partner didn't share it. I knew the feeling, despite every reason not to, because I had it myself. Something about this whole setup didn't feel right to me, and the feeling got stronger with each passing mile.
"Where are we going?" I finally said. "Unless that's classified information . . ."
"Little mining town called Bagdad," Halliday said. "Up in Yavapai County, about two and a half hours away. DPS is going to run the first vehicle, then Livermore will be in a prison van behind that, with two guards. We'll follow behind."
"Who's DPS?"
"Department of Public Safety. That's the state police here."
I nodded and sat back in my seat. We seemed to be alone on the road, but I knew the other vehicles were probably up ahead somewhere.
"There's only one road up here," Halliday said a few minutes later. "US 93. That's got everybody a little nervous."
"Livermore's a loner," Cook said. "Can you really see him with a crew of men up here waiting to ambush us?"
"No," Halliday said. "I can't see that."
It looked like he wanted to say more, but he kept it to himself.
"There they are," Cook said as the prison van came into view. Ahead of that, I saw the state police car with its lights flashing. We settled in at the tail end of the parade just as we were leaving Phoenix, heading through Glendale, Surprise, and Sun City West. Heading northwest, out into the great nothingness of central Arizona. The road itself was a straight line drawn on a flat plain, with railroad tracks to the right and telephone wires to the left. Beyond those, as far as I could see in any direction, it was nothing but brown earth covered by a thin layer of green and gray vegetation. Low mountains in the far distance and a bright blue sky above us. The view didn't change for an hour.
By the end of the second hour, the vegetation was growing thicker, the mountains coming closer with each passing mile. We were in a valley between the Poachie Range to the west and the Santa Marias to the east. There were great piles of red rocks on either side of the road. A voice broke through on the radio.
_"Lead to Halliday and Cook. Are we sure this road is secure?"_
Halliday looked over at his partner for a moment, then picked up the mic and keyed it.
"We had an advance team sweep through here a few minutes ago," he said, looking out the window. "Nothing up there but rocks."
But as he put the mic back, I could see him staring out the window, like we'd suddenly been transported to Afghanistan. Even Agent Cook, the man who'd been playing it cool all the way up here, hunched forward at the wheel to look out at the road.
_All because of one man riding in that van ahead of us,_ I thought, _sitting between two guards and tied up, as Agent Cook had said, like a Christmas turkey._
A few minutes later, just as we got to the turnoff for Bagdad, the same voice broke over the radio.
_"Stopping ahead. Pull in behind us."_
We made the turn and saw the other two vehicles pulled over to the side of the road. Cook stopped behind them and got out. The two state troopers from the lead car were standing behind the prison van, next to the van driver, and now Cook joined the conference and they stood around talking about something for a few minutes, before Cook finally gave me a wave to get out.
Halliday got out with me. It was coming up on noon now, the morning sun warming up the day, well into the eighties. I held up a hand against the glare. As I came closer to the party, I could feel the two troopers watching me carefully, measuring everything about me.
Halliday told me to wait a few yards away while he went to confer with the others. Every single man kept looking back at me until Halliday shook his head and came back to give me the news:
"They want you in the van."
"You're kidding me, right?"
"As of now, it's not our show anymore. The state guys, the jail guys, they all want you to be safe as we get close."
_"Safe,"_ I said, "as in locked up like a criminal."
_This is the "audible" he was talking about,_ I thought. But before he could say anything else, I left him there and approached the rest of the party.
"Go ahead," I said. "Put me in the van if that's what you need to do."
"Those are the orders," the van driver said.
When he opened the back door, I could see Livermore through the wire mesh. He was sitting on one of the side benches, with the same two guards who had stood behind him in the interview room early this morning. His own personal attachment.
"Got some company," the van driver said as he unlocked the inner door.
The two troopers stepped up to me then. First they asked for my cell phone, promising to give it back when this whole thing was over. I took it out and handed it to them. Then they asked me to put my hands on the side of the van before going inside.
"This gets better and better," I said, but it was clear we'd all just stand out there in the sun until I let them frisk me.
_It's not enough they drag me all the way out here and treat me like a goddamned suspect every step of the way . . ._
"For all we know," one of the officers said, "you could be working with this guy."
"Doing what?" I said. "What the hell could I do?"
The officer didn't say another word. He waited for me to assume the position, and then he gave me a thorough pat-down. He nodded to the van driver, and I was allowed the courtesy of climbing up into the van.
When I sat down across from Livermore, he looked at me and smiled. He was in the same jailhouse orange, with the same cuffs, chains, and shackles.
"This is an unexpected pleasure," he said.
"Just shut up," one of the guards said. They were both unarmed. Standard protocol. You didn't want any weapon inside the van that could be taken away and used against you.
The van started moving again. I could hear the voice of the van driver through the metal partition as he communicated with the rest of the convoy. Livermore kept watching me, that same smile on his face. I stared right back at him. I wasn't about to look away.
"We'll take a right turn soon," Livermore said without taking his eyes away from mine.
The van driver relayed this to the other vehicles. A few minutes later, we came to a brief stop, then we made a slow turn and started going downward. There were no windows to see out of, but I could tell from the rough ride that we were on a dirt road.
"Where the hell are we going?" I heard the van driver say into his radio.
Halliday's voice came back. _"Keep going. Nice and slow."_
Livermore kept watching me. "The Japanese have a saying," he said. " _Ame futte chi katamaru._ Literally, it means, 'After the rain, the earth hardens.'"
"I told you to shut up," the guard said.
"What it really means," Livermore said, "is that you and I are both going to find out just how _hardened_ you really are."
"We're in the goddamned desert," I said. "I don't think it's going to rain today."
"Alex . . ." he said, and then he paused before saying the five words that would become burned into my mind. "You think this ends _today_?"
The guard to Livermore's left leaned forward and looked at both of us. "Everybody," he said, _"shut the fuck up right now."_
Livermore stopped talking, but he kept smiling at me, and the vague feeling I'd had all morning started to take on its own color and shape. _This man is running the whole show,_ I said to myself. _Even though he's locked up tight, guarded by seven men . . ._
_Livermore is calling every shot._
The van came to a stop. A few seconds later, the back door opened. I was told to get out first. As I did, I saw that all three vehicles were parked where the road ended in a cul-de-sac, and a few yards beyond that there was an old mining shed that looked like it had been left for decades and forgotten about. More red rock was piled high all around us. The sun was almost directly overhead now. It beat down on us in that dead-end bowl, as the dust from the vehicles' braking hung in the air.
When the guards finally led out Livermore, he stood there blinking in the sunlight. The whole thing should have looked ridiculous to me—the two state guys with their tactical vests and their shotguns, the three guards from the jail and the two agents from the FBI. All to watch over this one man in orange with his hands cuffed to his belt.
But no matter how chained-up Livermore looked . . . no matter how many men were surrounding him . . . no matter how many guns . . .
"Where is the air support?" Livermore said as if reading my mind. "You only have eight men here. Surely you need a helicopter . . ."
"Where's the woman?" Halliday said. "In that shed?"
"Down this trail," Livermore said. "She's not far."
"This is not believable," Halliday said to him. "You drove her all the way down this dead-end road and then what, you made her walk down the trail with you?"
"She was tied up, Agent Halliday. I carried her."
"How far can you carry an adult woman, Livermore?"
Livermore looked at him with that little half smile. I'm sure Halliday hated it by now, maybe even as much as I did.
"I'm stronger than you think," he said.
Agent Cook came up close to him. "Just show us where you took her, you sick fuck."
"That's why we're here."
He led us toward the shed. As we got closer, a narrow trail appeared, leading through a break in the red rock.
I took Halliday aside. It was time to say something to the one man who I knew would listen to me.
"I've got a bad feeling about this," I said. "I think you do, too."
"What do you think's going to happen?"
"I don't know," I said, looking down the narrow path. "But you're doing everything according to his playbook. He got you to fly me all the way across the country. Now he's got all of us stumbling around out here in the middle of the desert . . ."
"So what do you want us to do? Just leave her out here?"
"There's nobody here to find," I said. "Think about it. What's he getting out of this? Did you take the death penalty off the table? Promise him he wouldn't go into general population?"
"I don't know what's driving him. Maybe he's just _that fucking crazy_ , Alex. But if there's _any_ chance she's out here . . ."
He looked over at Livermore. This one man in orange, his legs so hobbled he could barely walk.
"What if this was your daughter?" Halliday said. "Or your wife?"
I knew there was no answer that would satisfy him. That was the ultimate trump card. Then Halliday took out his semiautomatic, as if that alone would be enough to convince me I had nothing to worry about.
"Let's go find out," he said.
I shook my head and followed him. The path squeezed through the break in the rocks, then opened up just enough to give us all a little more room to breathe. But the ground still rose on either side of the trail, more naked rock with that same scrubby green and gray vegetation, the great saguaro cactuses all lined up on the very tops of the ridges, like they were spectators looking down at us.
Or waving their arms to warn us.
We walked in silence for several minutes, making our way over the rough ground. Besides the shackles, Livermore was wearing his standard jailhouse shoes with no laces and struggling even harder than the rest of us to find his footing on the trail.
The sun was beating down on us, and I was already getting thirsty. One of the state men had a backpack, and he stopped to pass out bottles of water. I drained half of mine at once.
"Are you going to let me die of thirst?" Livermore said.
It made me remember everything he had said about what the thirst was doing to that woman's body.
"You'll get water after she does," the officer said. Then he closed up his pack and we were on our way again. The sun got hotter, and my head started to hurt again. But we kept moving forward, as the trail started rising and winding its way between taller hills with more cactuses looking down on us.
"You did _not_ come this far," Halliday said. "Not in the dark."
"She's here," Livermore said, raising his head into the air as if he could smell her. "A hundred yards away."
Halliday hesitated. I could tell he was thinking hard about it. Finally, he nodded, and the line of men continued around a bend in the trail, until we found ourselves at the entrance to a small side canyon, the walls rising straight up on both sides, thirty feet high.
"She's in this canyon," Livermore said. "There's a small shelter at the end. Go ahead and look."
"You first," the van driver said, pushing Livermore ahead of him. "Move."
We followed them, each man looking up at the walls that seemed to close in on top of us. The two FBI agents were at the end of the line with me between them.
I didn't have a gun. That was the difference. I didn't have that simple, deadly arrogance I'd seen so many times before, the belief that a gun in your hand is enough to control everything around you.
The van driver took the radio from his belt and looked at it. When he turned up the volume, the white noise reverberated through the canyon. "What the fuck," the man said. "The radios are . . ."
He didn't finish the thought. Every man in the canyon stopped, and as Livermore turned, he caught my eye and smiled.
The next three seconds filled an eternity. With that one simple act, Livermore had shifted all of the attention to me. Every other man was watching me now, waiting for whatever came next.
"It's time, Alex!"
Livermore's words hung in the still air of the canyon as seven different brains raced to process their meaning. I reached my own conclusion just as quickly, but it was too late.
"Wait . . ." I said, already feeling two hands grabbing my shoulders from behind. I had no chance to react as everything tilted, and I felt myself being brought down to the ground. A man's knee grinding into my back, a classic cop takedown, driving the air from my lungs as everything else in the world came apart at once.
The first explosion shook the walls around us, a sound so loud it was a piercing physical impact against my ears. As I looked up I saw the men going down in front of me, saw blood sprayed against both sides of the canyon. If there were screams, I did not hear them. The second explosion was a wave of heat and concussion. The weight shifted on my back, moving backward, as I started to push myself up, just in time for one more blinding, noiseless flash and a sudden sting in my left knee and across my left biceps. The man who had been standing in front of me seemed to turn in the air in an oddly graceful pirouette, so that I could see how his upper body had been turned into a tangled mass of blood and fabric. He hung there for a moment, a marionette held up by invisible strings. Then he fell to the ground, his lifeless face inches from mine.
Everything was still. Another eternal three seconds until I saw a figure slowly raising himself to a standing position, twenty feet ahead of me.
Livermore.
He shuffled toward me, his ankles still shackled together, until he came to an officer who was still moving on the ground. I saw the shotgun barrel being raised and pointed at Livermore, then Livermore calmly taking the gun away. He had just enough play in the handcuffs attached to his waist to get the index finger of his right hand onto the trigger. He pointed the gun back down at the man, and then there was a flash from the barrel and a dull sound like the dial tone on an old phone as the man's head was blown apart against the ground.
Livermore came to the next man, looked down at him and, seemingly satisfied with the body's condition, kept moving until he came to the next man.
_You need to get up,_ I told myself. _You need to get out of here._
But I couldn't move yet. I stayed there, feeling everything spin around me, until another dial tone hit my ears a fraction of a second before I felt the spray of blood on my face.
_I'm next. He's going to do the same thing to me._
When I looked up, he was close to me. His eyes were locked on mine as he stopped three feet in front of me.
_This is it. His will be the last face I see._
I looked him in the eye and wondered if I'd even be able to hear the gun go off. There was another flash from the barrel, another dull sound that barely registered in my ears. I let out a breath, waiting for my body to react to the gunshot. For the pain to reach my brain.
It never came.
He stood there looking down at me. Then his lips moved like he was saying something to me, but there was no sound left in the world. Then with a sudden painful pop, my ears cleared, just enough to hear what he was saying.
"I told you," he said. "This does not end today. Not for us."
I clawed my way up to my hands and knees and reached for him, but he took one step backward and my head was spinning again. I tried to breathe. When I looked back up, he had already gone down the line of dead men, taking a weapon from each. A shotgun from the other trooper. A sidearm from the van driver. I felt around for something to hit him with. A rock, a stick, anything. It was the only thought I could form in my head.
_Stop him_. _Don't let him get away._
I reached behind me and felt something warm.
Blood. A body.
I turned and looked. It was Agent Cook. He was lying against the rock wall. His chest was soaked. I saw the blood coming in a steady stream from the hole in his neck.
When Livermore had raised that shotgun . . . When I was waiting to die . . .
He had shot Cook instead.
The agent's windbreaker was still tied around his waist. I took that off, wadded it up, and pressed it against his wound. But it was already too late. He kept looking at me, and then the light went out behind his eyes.
When I finally gave up, I turned around and saw Livermore at the far end of the canyon. He lingered there for a moment, a dark figure against the bright sunlight that gleamed from the other side.
He nodded to me, one more time.
Then he turned and disappeared.
# CHAPTER SIX
I WAS SURROUNDED by death. Surrounded by blood. The sun had moved to a new angle, making the red streaks on the rocks all around me glow in the light. I tried to stand up, felt everything spinning around me, and I went down hard on my knees. I stayed there for a long time, taking one breath at a time, waiting for everything to stop turning.
The explosions were still pounding in my ears as I stared down at the dirt on the canyon floor.
_Breathe in._
_Breathe out._
When the spinning walls finally slowed down, then settled back into place, I looked up at the far end of the canyon, the last spot Livermore had stood before turning to leave. I put one foot on the ground to push myself up, feeling a sudden pain rip through my left knee. Blood had soaked through my pants, and I saw a half dozen small round holes in the fabric. There were three more holes in my left shirtsleeve, more blood soaking through. Whatever had been in those explosives, I'd been grazed on the left side, and somehow nowhere else.
I tried to stand up again, starting with the right leg this time. When I was finally upright, I leaned against the canyon wall for a while, testing out my left leg, slowly putting weight on it until I was sure I could walk without going back down again.
I knew Cook was dead. I checked on the other men in front of me, one by one. Livermore had destroyed them, every single man, ripped apart their bodies with the shotgun.
_Right here,_ I thought as I came to a hole in the canyon wall. There was a metal pipe set into the rock, four inches in diameter. About five feet above the ground, the perfect height for a head-and-neck shot. The metal was blackened now and still smoking. If you set the right kind of charge in here, it would blow horizontally. And if you knew to hit the ground in time, and cover your ears . . .
_How much time did he spend setting all of this up? How many days did it take for him to put this all together, before he was captured?_
_The ultimate escape plan._
I picked up the radio the van driver had dropped. It still had power, but I heard nothing but static. I went back to the trooper who had taken my cell phone, pulled it out of his pocket and checked the signal. It was just as useless.
_He planned this so carefully,_ I thought, _not only does he tripwire enough explosives to kill seven men, he also triggers some sort of battery-operated radio jammer._
I stood against the wall for another minute, looking up and down the canyon, trying to see it from his point of view, trying to imagine how any man could see this place and come up with such a plan. And then pull it off so flawlessly.
_Until he got to you,_ I said to myself. _He had that shotgun pointed right at you. All he had to do was pull the trigger._
That was when it hit me.
_Wait a minute . . ._
I did a quick count of the dead bodies I had checked.
_Where's Halliday?_
I remembered him being at the back end of the line. I backtracked through the canyon to the point where it broke into a slight curve.
Halliday.
He was sitting with his back against the wall, looking at me. There was a great streak of blood across his face, more blood on his chest from injuries I could only guess at. As he raised one hand, I could see that he was missing the last two fingers.
I went down on my right knee beside him, tried the radio again. Tried my cell phone. Still nothing. The road was half an hour away.
"We have to get you out of here," I said.
He shook his head, but I wasn't going to leave him there. I went back and found another man's jacket, or what was left of it, came back and wrapped it around his hand, tying it as tight as I could. Then I went back and opened the state man's pack and looked inside. The explosives had ripped right through the bottles. I found one that had an inch of water left, came back and made Halliday drink it. I put one arm around him, holding myself steady with my other hand against the wall.
"Come on," I said. "You have to get up."
He cried out in pain as I half pulled, half pushed his body into a standing position. He slumped forward against me and took what sounded like a man's last breath. Then he seemed to gather some kind of strength from God-knows-where, and he looked in my eyes as he leaned his head back against the wall.
"Can you walk?"
He didn't respond. But it was time to try. I guided him away from the wall, watching him take one slow step, then another. It took us a full minute to get out of the canyon, back into the sunlight. I felt it burning on the back of my neck.
We took another step. Then another. The blood dripped from his ruined hand. I lifted up his shirt for one moment, saw a hundred small holes spread across his chest and abdomen. Like mine, each hole was bleeding. But he had a hell of a lot more of them. There was no way I could stop them all.
"We're going to make it," I told him. "The road's not far."
We both knew that was a lie. But we kept going.
"Just had a grandson," he said to me. His breathing was ragged and shallow.
"Don't talk. Tell me later."
"No." He leaned against me, so hard he almost pushed us both over. "You have to tell him . . . And my daughter . . . Tell them I'm sorry. Tell them I love them."
We made it another twenty yards. Then he slid to the ground. I tried to pull him up again. He shook his head.
"You tell them," he said. He coughed up blood, and it fell from his chin, onto his neck. "Promise me."
"I will," I said to him. "I promise."
I moved over so that my body was sheltering his face from the sun. I held his wounded hand tight, trying to stop the bleeding. But his shirt was soaked with more fresh blood now. Every movement he made, it made his heart pump more blood from his body. He was struggling to breathe.
I sat with him like that for another minute. Maybe two. He looked up at me one more time and gave me something almost like a smile.
Then he died.
I laid him back on the ground and closed his eyes. I stayed there with him for a few more minutes. Then I finally got to my feet.
I made it about five steps before I had to stop and throw up in the dirt. I emptied myself until there was nothing more than a dry convulsion coiling through my body. I stood up, wiped off my mouth, and kept walking.
My head was pounding, making me dizzy enough to wonder if I was even going in the right direction. It was all rocks and dirt and scrubby little cactuses and nothing else.
_I could get lost here. Make it out of that canyon alive, but then die trying to get back out to the road._
I found what I thought was the trail, but now my left knee was radiating with pain. I didn't want to lift up my pants or take off my shirt. I knew it would be just a minor version of what the explosion had done to Halliday's chest.
The sun had already started to grind me down again and turn my throat to dust. I wondered where Livermore was at that moment. Somewhere on the other side of the canyon. If he was smart enough to set up that ambush, then he was smart enough to leave himself a cache of water for his way out. Some new clothes so he could ditch the orange. Money. A cell phone with a separate battery to recharge it if it was dead. It would be so easy to hide just about anything in these rock piles. He might even have a vehicle hidden nearby, within walking distance. That was how empty it was out here.
The sun kept beating down on me with every step, making me more thirsty than I'd ever been in my life. My knee kept throbbing, and I started to get dizzy again. But I kept moving. I heard nothing but the silence of the desert and my own footsteps.
And the explosions. Still roaring in my ears.
The road finally came into view, waves of heat rising from the vehicles that were still parked there, exactly where we had left them. I went to the agents' car and tried the doors. They were locked. The van was locked. The state car was locked.
I went and found a rock the size of a softball, went back to the agents' car, and hit the driver's-side window, right in the center. It exploded into a thousand glass pebbles. I reached inside and hit the door lock, opened the door, and sat down behind the wheel. I grabbed the radio receiver and hit the transmit button.
"Code thirty, code thirty," I said, bringing back from my memory the one code you never wanted to hear on a Detroit police radio. "Seven officers down."
I let up the button and waited for an answer. The airwaves crackled.
"Please respond," I said, hitting the button one more time. "I repeat. Seven officers down."
# CHAPTER SEVEN
THE RECOIL FROM the shotgun was still tingling in his hands as Livermore walked from the canyon. He could still see McKnight's face, could still feel the moment, that shotgun pointed at his chest. The power he'd felt, knowing he could end his life with one slight movement of his finger.
But of course he hadn't. It wasn't time for that yet.
Not for McKnight.
He found his pack where he had hidden it behind a thick juniper bush. The fabric was the exact color of the brown-red rocks around it. You could stand five feet away and not see it. Livermore pushed his way through the bush and pulled it out, zipped it open and took out the hacksaw that was inside. Priority number one, cut his hands free. That took a minute. Priority number two, cut the shackles on his legs, increase his mobility. Another minute. Then he was on the move. He ducked into another side canyon and took off the orange jumpsuit. Priority number three. He put on the hiking clothes—the shorts, the shirt, the boots. He put on the baseball cap and stuffed the orange jumpsuit in a plastic bag, left that behind as he looped the pack around his back and kept walking. Five minutes out of the canyon and he already looked like just another hiker, out on the trail on a perfect February day in Arizona.
Priority number four, he took out the metal bottle of water from the pack and drained it. It was warm, but his body needed hydration. He'd get fresh water soon enough.
That left priority number five.
He walked up the hill through the barren red rocks, taking a quick look behind him now and then, just to make sure McKnight wasn't following him. He saw a thin haze of smoke still rising from the canyon, this place he had taken so much care to choose, a couple miles down the trail from the end of the abandoned road, next to a mine that hadn't been used in years. He had known the explosives would still be intact, even after several days. The weather was dry, and he'd packed them just right. He had known the angles would be correct, and that he would have enough time to get himself safely onto the ground. Above all, he had known that the wiring would perform exactly as designed, and that the battery-powered jammer would activate in time to transmit on a broad spectrum of radio frequencies, temporarily overwhelming the service on any two-way radios nearby. And of course on any cell phone, because a cell phone is nothing but another kind of radio.
He kept walking. When he got to the road, he took it northeast, walking a little over a mile and a half to the old junkyard just outside of Bagdad. It was a graveyard of old vehicles, all stored out here in the dry weather for parts. He picked his way through the ancient Fords and Chevys, and the occasional piece of copper-mining equipment, until he came to the SUV on the far edge of the yard. It was pulled in between two panel trucks, so you had to know exactly where to find it. And even if you did, you'd see that it was covered with decades' worth of dust and sand. You'd walk right by and probably never give this vehicle another thought.
But Livermore found the key in the tailpipe, opened up the back hatch, and pulled out the broom. He climbed onto the top and started brushing off the sand he had poured on the vehicle himself. Then he opened the driver's-side door. It didn't look like an old vehicle once you got inside, because of course it wasn't. The thing barely had thirty thousand miles on it. A Nissan Pathfinder, from Japan, of course—a land where the designers knew enough to let robotic arms, machines that Livermore himself had helped design, build their vehicles with a care and a precision that no human being could ever match. Bought with cash, never registered, the Pathfinder had a pair of stolen Arizona plates from a vehicle that was up on blocks behind a barn, the registration tab still current, but the plates unlikely to be noticed missing anytime soon.
Livermore popped the hood, reconnected the battery, and started it up. Then he cranked the wipers to push off the last of the sand on the windshield, put the vehicle in gear, and drove out of the yard.
When he hit the road, he went south for a few miles. At that point, he could have turned west and driven to Las Vegas. That would have been three and a half hours. An obvious choice. _Too_ obvious, because everybody runs to Vegas. The lights call to you from the desert. _Come and get lost here._ Anyone looking at a map would choose that as the first place to look for him—and there was really only one road to Vegas, US 93 all the way. They'd be all over it.
Flagstaff, on the other hand, was about the same distance to the east, with three different ways to get there. The superior choice for a superior mind.
So of course he wasn't going there, either.
When you have a mind that is beyond superior, you go in the last direction anyone would expect. That was why he was driving directly back toward Phoenix, but not before looping around to the east so he could come in from a new direction.
He settled in for the drive, already thinking about where he'd eat that night. Real food, after all of those hours spent in the Fourth Avenue Jail, his first and only experience having a metal tray slid through an opening in cell bars.
Livermore's pack was on the passenger's seat next to him. Inside was ten thousand dollars in cash, another change of clothes, his laptop, his cell phone with a charger, a set of ropes, and his Walther PPS semiautomatic with a box of nine-millimeter shells.
There were more supplies in the back of the vehicle, the things he'd packed several days ago. More ammunition for the pistol. More clothes. Two five-gallon cans of gasoline, sealed tight to prevent any fumes. A large plastic box filled with electronic components. Another plastic box filled with explosive material.
Because he needed to be ready for anything.
He turned on the AM radio, started scanning the dial for the news reports. An escaped fugitive is on the run, six foot two, with long, light brown hair. Armed and dangerous. A two-million-dollar reward for anyone who provides information leading to his capture.
_Free less than an hour,_ Livermore thought, _and I'm already worth two million dollars._
As he drove, he kept his mind busy by working through the odds. After another day, then another week of being free, of doing whatever he wanted to whomever he wanted to do it to, how high could that number go?
Finally, his mind settled on the last part of his plan. Almost an afterthought, the diversion he had set up to make sure the FBI and everyone else would stay busy looking in exactly the wrong direction.
They'd all be off chasing a ghost, leaving him alone with Alex McKnight.
# CHAPTER EIGHT
AS I WAITED, I kept replaying the scene in my mind. Watching those men die. Watching the blood fly against the canyon walls.
A squad car responded to my radio call, followed by more squad cars, ambulances, fire trucks, pretty much every official vehicle in central Arizona, until there wasn't room to maneuver on that road anymore. When the two FBI agents showed up, they took me from the ambulance, where someone had cut away my left pant leg and wrapped my knee, then cut away my left sleeve to do the same for my arm. The agents put me in the back of their sedan and drove me all the way back to Phoenix, that same two and a half hours down that same lonely road, never saying a word to me. Not that I felt like talking.
They took me to a hospital just off the expressway and checked me in to the emergency room. I sat there for another hour until a doctor and a nurse finally came in and looked me over. The nurse took my vitals while the doctor asked me about my ears and how much water I'd had in the past few hours. Then he unwrapped my knee and got to work on the shrapnel, taking out little round hunks of metal with a long set of tweezers and dropping them with a loud clang into a metal bowl.
He did my arm next. Three more clangs in the metal bowl. When both injuries were wrapped up with fresh bandages, they left me sitting there for another half hour, until I'd finally had enough of the place. I got up and limped out. I was expecting to see the agents in the waiting room, but they weren't there. All I saw was the usual assortment of the city's underclass, something I'd seen on a thousand different occasions back in Detroit. They came here because it was the one place that wouldn't turn them away.
When the agents finally caught up to me, I was standing in the parking lot, replaying the whole thing in my head and trying to find some way I could have made it come out different in the end. I noticed that my hands were shaking, and I almost wished I was a smoker then, so I could have a familiar ritual to calm myself down. The agents came out and found me, and one of them made the mistake of grabbing me by my wrapped-up arm, like I was some kind of fugitive trying to escape.
"I've had a bad enough day," I said to the man. "You grab me again, you'll have to shoot me."
He let go of me, but he didn't apologize. They put me in their vehicle and took me over to the FBI building on Deer Valley Road, parked me in an empty conference room, where I waited some more, until Agent Madison came in and sat across from me. He was another old-timer, same vintage as Halliday. Gray hair, dark suit, all business. He looked at me, took a long breath, then he went right to the question of the day.
_"Why are you alive?"_
I replayed those few seconds in my head one more time, right before the first explosion ripped through the canyon. How Livermore had stopped and turned, how he had called out to me . . . And then how someone had taken me to the ground.
Agent Cook. He'd been acting purely by instinct, neutralizing a perceived threat when Livermore had turned all of the attention to me.
"It was Agent Cook," I said. "He saved my life."
And then the next thought, which hit me even harder. _Livermore knew someone would take me down. All he had to do was wait for it, then go down himself._
_That was part of his plan, from the beginning. He wanted me to survive._
"Take me through it," Madison said. "Everything that happened."
I went over it as well as I could remember. Every detail, from the drive to the canyon to Livermore standing over me with the shotgun. Then leaving me there, with Halliday. Until Holliday couldn't go any farther.
"We're losing valuable time," Madison said. "He could be three hundred miles away right now, in any direction."
_Basic fugitive-hunting,_ I thought. _Time equals distance. Keep the time line as short as possible, or you'll never catch him._
"You have no idea where he is right now," Madison said.
"No."
"Or what he may be planning next."
"How the hell would I know that?" I said. "Why are you asking me this?"
He put both of his hands on the table. "You still can't tell us how you're connected to—"
"No," I said, doing my absolute goddamned best to stay calm. "If I had an answer, don't you think I'd give it to you?"
He sat there across the table and looked at me for a long time without saying anything.
"Agent Cook's got a couple of kids," he finally said. "Twelve and thirteen years old. He just recently transferred here. Roger, I knew a lot longer. We came up together. He and Louise . . ."
He stopped to clear his throat.
"Their daughter just had their first grandchild. I mean, literally, he was just born a few days ago."
"I know," I said. "He told me."
Madison took a moment to let that sink in. At some point, I'd want to deliver that message to his daughter and grandson, as I had promised. But today probably wasn't the day.
"You're staying in Phoenix for a while," Madison said. "We'll keep you at the same hotel, and I'll introduce you to the whole team, as soon as we have it together."
"How long will I be here?"
"I can't answer that right now. It's not totally up to me, anyway. There are other parties waiting to talk to you. Starting with the state police."
That wasn't a surprise. I knew I'd be answering the same questions, again and again.
"I know you were once a cop," he said, "so you know this is the worst day ever."
"Been there, yes."
My mind flashed to the funeral of my own partner, a million years ago. I only saw photographs in the newspaper, because at the time I was still in a hospital bed.
"Along with the grief, there's a lot of anger."
I nodded.
"DPS lost two men. They want somebody to be accountable for that. The jail lost three men. Same feeling. Everybody's looking at each other right now."
"Looking for somebody to blame. I get it. And I'm right in the middle of everything."
"Yes," he said. "You are."
"So tell me everything you know about this guy. Show me his file. If there _is_ a connection with me, that's the only way I'm going to see it."
"I told you, there are other people waiting to talk to you . . ."
"By the end of the day, he could be another three hundred miles away. Show me the file."
He thought about it for another few seconds. Then he got up and left the room. He came back carrying a box.
"Most recent first," he said, pulling out one file. "Not counting Stephanie Hyatt, who at this point is still missing."
I started paging through the reports for Carolyn Kline, the woman who had been taken to the house in Scottsdale. They were all from different local police departments, all written in that same bloodless cop language that never seemed to change, no matter who was writing it.
_Victim is a twenty-five-year-old female, five foot five, one hundred and thirty pounds._ _Last seen around 2300 hrs. on 11-07-2017 in vicinity of Skyline Tavern, 2200 N. Central Ave., Phoenix. Abducted at her home by persons unknown._
Everything the writer of the report could think of saying—but in the end just a collection of details, with no ultimate answers. Then the ME's report, which used a more obscure kind of detached language to describe this woman's last few hours on earth. The trauma, the violation, the rope ligatures.
Using every word except the one word that would have fit best.
_Torture._
"It says here there's a video," I said, moving on to the Scottsdale PD report. "I want to see it."
He hesitated. "It's not an easy thing to see."
"Show it to me."
He took out his laptop and opened it. A few seconds later, I was watching the silent video of Martin T. Livermore and a dead woman in a Scottsdale bedroom.
Approaching her body on the bed.
Taking off his clothes.
Reaching out with one hand to touch her skin.
Climbing onto the bed to lie next to her.
There was an unreality to everything I was seeing, and the fact that there was no audio made it even worse. The silence should have given the images a certain distance, but somehow it did exactly the opposite. I knew I'd be seeing this, running in a continuous loop in my mind, for the rest of my life.
When it was done, Madison closed the lid on his laptop. Before he could say another word, I got to my feet, feeling stiff and uneasy. My left knee was throbbing again, but I barely noticed it. I had to leave the building, had to breathe clean air.
As soon as I was outside, as soon as I had taken three deep breaths and made myself stand still, I looked across the road at the empty desert.
I remembered everything Livermore had said to me in that interview room. How he had taken my life apart, piece by piece. Showing me every failure, every reason to believe it accounted for nothing. I had resisted the thought at the time, but I knew when I finally closed my eyes that night, I'd lie awake and wonder if there was some small grain of truth in what he was telling me.
_There's one thing I can do,_ I said to him. _One act that will balance out everything else in my life._
_I'm going to find you_ , _wherever you are, no matter how far away, no matter how long it takes . . ._
_And I'm going to kill you._
# CHAPTER NINE
LIVERMORE STARED at the stranger's face in the mirror.
He'd already stopped at a drugstore in Anthem, a few miles north of Phoenix. One more advantage of coming back down this way, he could be off the road a lot sooner. Spend a few hours here, let them start fanning out. Take his time and then follow behind them.
After the drugstore, he had checked into a motel, paid his ninety bucks for the room, and got to work in the bathroom. He stood in front of the mirror and ran the electric razor over his head, watching the light brown hair collect in the sink. He made a point of putting it in a bag so he could get rid of it when he left. When he had all of his hair trimmed down close, he mixed up the dye and slathered it on his head. It didn't take much, because he didn't have much hair left. He waited the twenty minutes suggested on the package, then five minutes more. Then he rinsed it out and looked at himself. His whiskers were starting to come in, now that he had stopped shaving for a couple of days. He got out the eyebrow pencil and darkened in his eyebrows, then his mustache. He took the pair of low-strength reading glasses he'd bought and put them on.
He spent a long time looking at himself, studying himself from every angle. He tried to imagine what other people would think when they saw his face, what assumptions they would make and how Livermore could use those assumptions to his advantage.
He still looked good. The same smooth skin, the strong jawline. The piercing eyes. And, with or without the glasses, the intelligence still burning in his eyes. A light that he could never dim, even if he wanted to. But with his hair shorter and darker . . . Yes, he looked younger now. That could be useful. The older man with the long hair, the man whose description was already on the radio, the face that would soon be broadcast around the country . . .
That man was gone.
Now it was time to leave this place. Too risky to stay the night, not with a drugstore cashier who had seen the old version of Livermore buying his supplies just down the street, and a motel clerk who had taken his money and given him the key. As he repacked his bag, he noticed the antiseptic smell that every motel room in the country seemed to share. It took him back to another day, the last time he'd stayed in a motel, and everything that had happened afterward. Everything about Liana and how that had led to him standing here, six months later.
* * *
—
HE'D BEEN ON HIS WAY back to Phoenix, after taking Sandra to be with the others. He came back through Flagstaff, after so many hours driving, stopping at a restaurant for dinner, in no rush to go those last two hours down to his apartment. It was a Monday night, he remembered, two restaurants closed before he finally found the little steakhouse. He went in and ordered a drink, watched the woman bring it to him. Some men might not have liked the little bump in her nose or the way her eyebrows came a fraction of an inch too close together. But there was something about her that appealed to him.
_Not yet,_ he told himself. _It is too soon._
But he kept watching her and the surprising way she moved like a dancer around the tables, her hair pinned up, a white dress shirt and black skirt showing off her body. She smiled at him when she brought the drink over, and he smiled back. The way he'd taught himself. The science of robotics is, after all, a science of imitating human movement. Even, if you take that idea to the next step, human behavior. Which meant Martin T. Livermore was a trained expert in watching humans, observing what they did well and what they did not. Designing a machine to do things better.
_I can be a machine myself,_ he had realized. It was an insight that came to him after spending so much time in Japan, where _kaizen_ , the continuous pursuit of improvement, was such an important part of the culture. _I can design myself the same way I design a robotic arm to assemble an automobile, can fine-tune my own programming in the same way, to be even more effective today than I was yesterday._
He started going to bars. Not to try to pick up women. Not yet. He went there to observe, to listen, to study. To learn what worked and what didn't.
Then he started to develop his own style, using what he had learned, adding his own touches, turning up the shining light of his own raw intelligence just the right amount, confident but not arrogant. He was surprised at how well women responded to his hair, especially when he wore it loose.
"What do you really do?" he said to her. "No way this is your true calling."
Picking his words carefully, using the exact phrases that had worked in the past. _Your true calling_. Women loved that. They loved it even more when you listened to the answer like it was the most important information you'd hear all day.
"I'm a guide at the Grand Canyon on weekends," she said.
He leaned forward and watched her carefully, watching her eyes, watching her mouth.
"I love the Grand Canyon," he said. "That must be fascinating."
She smiled again. Responding to him, just as he knew she would.
"I get a little tired of saying the same thing over and over," she said. _"The canyon is two hundred and seventy-seven miles long. At its widest point, eighteen miles across. Most of it more than a mile deep. There are two billion years' worth of rock exposed . . . Unless you're one of those nutcases who thinks this whole thing was made in a flood six thousand years ago."_
He laughed at that and asked her how many tours she typically did in one weekend.
"Seven or eight," she said. "We fly down to the floor in a helicopter. Which is what I really want to do."
He opened his eyes wide in surprise. "Fly helicopters? _Really?_ "
"This job is just to pay for flight lessons."
"I've always wondered . . ." he said. "Do you have to learn to fly a plane first? Or can you go right to helicopters?"
She got called away to another table before she could answer. But he knew he had her hooked. He took his time over dinner and ended up sitting at the bar until closing. She came over and sat next to him when she was done with her shift, finally able to answer his question.
"Yes, you can go right to helicopters," she said as soon as she sat down. "But most people don't. I'd love to fly planes, too, so why not learn it all?"
He raised a toast to her plans, then answered some questions about what he did. Robotics engineering, and the time he'd spent in Japan. He knew that would work. A man who travels the world, who can make his way in a foreign country for years at a time . . .
"I worked for Seiko Epson," he said. "At the Nagano robotics center."
"Wow, you must be really smart."
"I helped design a new robotic arm. With a six-axis range of motion, instead of four. But you don't want to hear about that."
"I bet that was amazing," she said, moving a little closer. "But why Japan?"
"There were seven hundred thousand industrial robots in the world then. Five hundred thousand of them were in Japan."
He saw her eyes wander, just for a moment.
_Don't talk about numbers,_ he reminded himself. _Women do not care about numbers._
"But it was the culture that really turned me on," he said. "The Japanese people seem so . . . restrained. And yet underneath, there's this amazing sensuality . . ."
_Yes, she's reengaged._
"Why did you come back?" she said.
"I missed American steakhouses."
She laughed and moved even closer. He took out his wallet to pay the bill, the wallet with the American Express Platinum Card displayed prominently. Left a hundred-dollar bill with clearly no expectation of getting any change back. Another move that never failed.
He finally got her name. Liana. He told her it was the loveliest name he'd ever heard.
"I know you'll be closing soon," he said to her after another beat. "Where else can a man get a drink in this town?"
"There's a place down the street. They make the best martini in town . . ."
He looked at her and smiled. "I'll drive."
He followed her outside, opened the passenger's-side door to his Mercedes, made sure she was safely inside and buckled up before closing the door and going around to the driver's side.
"I don't usually get in cars with men I don't know," she said when he was behind the wheel.
"This time," he said, "I'm glad you did."
They lingered over the martinis and another hour of conversation, Livermore remembering to make it all about her without seeming obsessive about it, slipping in little references to himself when he felt the timing was right. A little bit more about his time in Japan, how tiresome it was to travel around the world so often, how hard it was to have a relationship with such a busy life, how he was finally ready to settle down and find something _real_.
Another thing women loved. _Something real._
They went back to her apartment, just outside of town.
"This place must have a great view in the morning," he said to her. Presumptuous, and yet he knew it was time to make that move. To push without really pushing. He could see from her shy smile that it was working.
They went inside and had another drink, listened to some music. Some "good old Motown," as she called it, and of course he was quick to say that was his favorite music ever—even though the truth was he found the subtle bass note hum inside his own head preferable to any music he'd ever heard. He waited for her to move closer, then finally kissed her, making her wait for it. Even when they went to bed, he made sure to do everything right. Focusing on her pleasure first, making her feel like she was the most beautiful, most desirable woman on the planet.
It was good. Livermore was almost surprised by it. That certain _something_ he had seen in her . . . How she had reminded him of the one woman who had set the standard so long ago. And how she was still living up to everything that _something_ had promised. Even when, toward the end, he had taken her wrists, had drawn them together over her head and held them down tight . . . How he felt her responding to this, how he could feel it rippling all the way through her body. This hunger inside her, something almost like desperation.
_Yes,_ he said to himself as he lay next to her. _This could finally be the one._
Then she lit up a cigarette.
In her own bedroom, with Livermore right there in the bed next to her, this woman _lit up a cigarette_.
"That's a nasty habit," he said to her, keeping his voice even.
"I know," she said, taking another long drag and looking at the cigarette. "I'm trying to quit."
He got out of the bed and put his pants on.
"Hey, come on, where are you going?" she said.
He kept getting dressed. His shirt, then his belt.
"To get a breath of fresh air," he said.
"Okay, look," she said, stubbing out the cigarette in a coffee mug next to the bed. "No more smoking. I didn't know it would bother you."
He stopped and looked down at her. At this woman who smoked in her own bed and put out her cigarettes in the same coffee mug she'd be drinking out of the next day. Somewhere in his mind a single green light flickered, then went off. After a moment of darkness, another light came on in its place.
This light was red.
Livermore smiled at her.
"Don't worry," he said. "I'll be right back."
She kept watching him as he left the room. When he was in the living room again, alone, he took a long look at the place. There were details that he hadn't noticed until that moment, indicators that marked this woman's life. From the dirty dishes in her sink to the pile of junk mail left on the table. A disorganized mind. Even for a woman.
He went to the bookshelf and turned his head sideways to read the titles. He stopped when he came to the erotic novels, or at least what certain people would consider to be erotic. Whitewashed and Americanized, huge bestsellers—but the kind of trash that would be laughed out of any Japanese bookstore.
_You should have seen these signs,_ he told himself, _before you let yourself get involved with her._ _She used the human part of you to make you believe she was different._
_She tricked you._
He left the apartment and went out to his Mercedes, came back to the door with his leather bag. Liana was standing in the hallway when he came back inside the apartment. She had put on a thin black robe, decorated with blood-red flowers.
She looked uncertain when she saw Livermore coming toward her. "What's in the bag?"
He didn't answer her. He put the bag on the floor and opened it. Then he reached inside and took out one of his ropes.
He had them in different lengths, from four meters to fifty, but each one was exactly six millimeters thick, and each one was a fine three-stranded jute that he had boiled himself, then dried for several days before finishing with a thin coat of mineral oil.
Her eyes went wide when she saw it.
"What are you going to do with that?"
She kept looking at the rope, mesmerized, as he slowly wound it into a tight coil.
He stayed silent as he finished making the coil. It was part of the process. The way you quiet your mind as you prepare for what comes next.
"I've heard about this," she said. "What's it called, _shibari_?"
_"Kinbaku,"_ he said. "Only Americans call it _shibari_."
"I don't know," she said, taking a small step backward. "I've never done this."
Her eyes stayed fixed on the rope as he came closer to her. He reached out and touched the coil to her face, gently brushing her skin.
Her face flushed. "Are you going to punish me?"
"Turn around," he said.
She hesitated for a moment, then obeyed. He gave her a slight push, and she went to the center of the room. Her face was still turned away from him.
"Take off your robe," he said.
"Yes." Her voice was low, and she was already breathing hard. She closed her eyes. Then she opened her robe and let it fall to the floor.
He felt himself stirring again, willed himself back into the quiet darkness inside his head, where only the single red light burned.
"Put your wrists together," he said. "Behind you."
When she did, he tied them, then he looped the rope around her body, twice beneath her breasts, twice above. He joined these loops to her wrists. Then he kept weaving the ropes around her, forming an intricate pattern.
The _kikkou_. The tortoise shell.
Around and around her body, reversing the ropes behind and in front, until she was wearing the rope like the tightest, most intimate clothing ever made.
"It feels nice," she whispered. "These ropes are soft."
He didn't answer her. There was no more need to talk now that she was tied. When she tried to speak again, he quickly looped the rope across her open mouth. He knew this would work better than any gag. She shook her head against it, her whole body suddenly going tense. Her eyes were wide open now, her mouth working against the gag. A muffled sound came out.
He looked in her eyes, still silent. His smile was gone.
She kept shaking her head, kept trying to speak. When she tried to step away from him, she nearly fell over, because the ropes had been tied all the way down her thighs, to her knees. Livermore caught her and held her upright. When her eyes locked on his again, that was the exact moment.
The moment she knew.
The moment she could see it in his face. See the man he really was.
See what was about to happen to her.
_"Only dirty whores smoke in bed,"_ he said, and then with one push he sent her to the floor.
She tried to scream. Another muffled sound that would go no farther than this room. She kept fighting against the ropes, writhing across the carpet. He stopped her and tied her ankles together. Then he looped the rope around her neck.
Once.
Twice.
He tied these ropes to her ankles, drawing her whole body back into an arch, as if he were stringing a bow. Her face was already turning a deeper shade of red. Tears streamed down her face.
Livermore stood up and watched her for a long time, watched her struggling against the basic horrible truth of her situation, as the slightest movement of her body drew the ropes even tighter. She had to actively work against it, to keep her back arched and her ankles raised as high as they could go. To relax would mean strangling herself.
He knelt down on the floor next to her, watching the color in her face change from red to purple. Her eyelids fluttered as her body began to go slack.
When he loosened the knot linking her ankles to her neck, the color in her face lightened and she opened her eyes again. He kept watching her, watching the life come back to her, then leaving again, then coming back.
Until it didn't.
That was when he got undressed.
* * *
—
WHEN HE WAS DONE with her, he sat with his back against the couch. He took the laptop from his bag and checked his email.
_She told me she lived here alone,_ he said to himself. But he knew that didn't make this a safe place. Not for long. A friend could come over, or a nosy neighbor, or even a boyfriend she had neglected to tell him about. He would have to find another place to move her to.
It was a familiar routine to him by now. A ritual.
And yet another part of the ritual, he finished with his email and went out onto the Internet. Looking up traces from another part of his life, from a memory that would come back to him whenever he spent a night like this one. It had been amazing to him how much social media had exploded in America while he'd been in Japan, how a person could splash their entire life all over the Internet for anyone in the world to see. But not everyone did that, as Livermore soon learned. He would never do it himself, so he understood. But this time, as he once again went through every website he could think of . . .
He found something.
One image, a single flash in time from how many years ago, and yet here it was recollected in the pixels that glowed on his laptop screen. He scanned one face, then another, then another. Until he settled on the face of a man. Composed, self-assured, athletic.
_This man._
He felt a surge of electricity flowing through his body, a wave of heat and energy that he could barely contain.
_What am I going to do about this man?_
He worked it over in his head for an hour, coming back to the same face, over and over again.
That was when the idea finally came to him. How he would test himself against this man. How he could prove himself.
He thought about it for a second hour, working out every detail in his mind. Until it was finally time to wrap up Liana in a sheet and, after checking to make sure nobody was in the parking lot, take her to his vehicle.
All of the preparations, the designs, the planning . . . It would all come together on this night, every star in the universe collecting around this one perfect idea, and all of it leading to this one man. This ex-cop living in a little town called Paradise.
Six months after the night the idea was born, everything was in motion now.
For Martin T. Livermore.
For Alex McKnight.
And it had only just begun.
# CHAPTER TEN
THIS IS ALEX MCKNIGHT," Agent Madison said to a roomful of men and women who all were staring a hole right through me. It was the next morning, after a sleepless night. "He is the lone survivor from yesterday's events."
_A hell of a way to put it,_ I thought. We were in the same conference room where Madison had originally questioned me, but now there were a dozen other agents all sitting around the table. Ten men and two women, all dressed in dark suits. There was a special phone for conference calls on the table. I had no idea how many more people were listening in on that.
"Mr. McKnight was a police officer in Detroit. He's been retired for a number of years."
"What is his connection to Livermore?" one of the agents in the back of the room asked. Because five minutes had passed without someone asking that question.
"At this point, we don't know."
I could sense everyone in the room exchanging looks with one another, then returning their focus to me.
"Mr. McKnight has been a person of interest in this case since Livermore first mentioned his name," Madison said. "He will continue to cooperate with us."
That didn't seem to satisfy anyone in the room. Not that I could blame them. Two agents they had worked with every day were dead. I was alive.
Madison moved on to credit cards, bank accounts, vehicle information—anything a person could use that would leave a trail. All of the markers that someone would leave if they were actively moving, because a person who's moving needs to sleep somewhere, needs to eat, needs to refuel his vehicle. Or whatever Livermore was now driving, anyway. His vehicle of record was the black Mercedes found in the parking lot at the Sonoran Preserve the night he was arrested.
Ten thousand dollars in cash had been taken out of his account a week before, but no further activity had been seen since then. This man had apparently designed this part of his plan just as well as the escape itself. He was staying completely off the grid.
_He's invisible,_ I thought. _But as every fugitive hunter knows, a man can't stay invisible forever._
Madison then led the team through everything else that had happened in the past eighteen hours, from the coordination with every local police department, as well as the state police in every neighboring state, to the interface with the media. He put up the FBI website on the screen. Somewhere out there was a man named Jason Derek Brown, who was wanted for first-degree murder, armed robbery, and unlawful flight to avoid prosecution. He'd been number one on the FBI's Ten Most Wanted list the week before, with a five-hundred-thousand-dollar reward for information leading to his arrest. Now Brown was number two behind Martin T. Livermore, a man whose capture was worth an even two million dollars.
Madison's words faded into the background as I stared at Livermore's face on the screen. It burned in my mind, even as the image was replaced by a map of the United States. Madison ran through Livermore's background, highlighting each location on the map. Born and raised in Columbus, Ohio. Attended Ohio State University, then Rensselaer Polytechnic Institute in Upstate New York. Worked for three different companies in the Northeast before he took a job with Seiko Epson in Long Beach, California. Transferred to the robotics center in Nagano, Japan. Lived there for nine years. Back in the States for three years.
Long enough to kill six women. Possibly seven.
Madison highlighted the two cities in California where his victims had last been seen. Then the cities in Utah and Nevada. Finally the two cities in Arizona: Flagstaff. Phoenix.
"What were his movements in Japan?" one of the agents said. A woman, sitting close to me. "If he never killed anyone until he came back . . ."
"We're in contact with the NPA offices in Nagano," Madison said. "So far his time there is one big blank. He went to work every day. Otherwise kept to himself."
"Does he have any family?"
"No brothers or sisters. Father died when Livermore was seventeen years old. Mother two years ago. She lived in Columbus, in the same house Martin grew up in, although we don't believe he's been back there for several years. The house has been empty since the mother died. We're processing it today, just in case."
"What about his current residence?"
"His address of record is a single-bedroom apartment on Buckeye Road," Madison said, "but we have reason to believe he used that only as a mail drop. The apartment is completely empty right now. There's not even a bed."
"Are you saying that—"
"I'm saying that nobody in that building has ever seen him before. So finding his real primary residence is one of our first priorities today."
He was about to continue when another agent entered the room and waved Madison over. He excused himself and went out to the hallway to talk to him. I could feel every eye in the room on me again as we all waited.
When Madison came back in, he asked me if I would excuse myself.
"Are you going to tell me why?" I said.
"We need to discuss some sensitive information. If you go with Agent Larkin, he'll take care of you until we're done."
_So much for being part of the team,_ I thought. I got up and left the room with the young agent right behind me.
"We'll go to my office," Larkin said. He was built solid, like an athlete, with a military buzz cut. "I have the case files if you want to look through them again."
I didn't answer him.
_End of the day,_ I told myself. _As long as I can get something useful out of these guys, I'll stick around. But then I'm on my own._
_Because I am going to find Livermore, one way or the other._
I saw the agent's full name on his office door. Matt Larkin. He didn't look like he'd seen his thirtieth birthday yet, and I knew he wasn't the man I needed to be pissed off at. I tried to cool myself down a notch as I sat in his guest chair.
_End of the day._
I spent the next hour going through the case files again, one by one, trying to put everything else out of my head. At one point, I took out the photographs of the six women Livermore had killed and put them in a line across the table. The same photos that Halliday and Cook had shown me on the flight out here.
I looked at the faces, carefully moving from one to the next. The hair color went from blond to brunette to black to redhead, back to brunette, back to blond. Two of the women might have been a little overweight. Two wore glasses.
"As far as we know," Larkin said, "these women have nothing in common. No connections from school or from work. His victim selection appears to be completely random."
I knew that was an important term in the Bureau. His _victim selection_. But there was something here that didn't look quite random to me. I _knew_ there was something here. I just couldn't see it yet.
"How long would it have taken you guys to catch him?" I said. "If he hadn't given himself away with that video?"
"I can't answer that."
"I'm sure there was a profile."
"You're right, but—"
"But profiles are bullshit."
He looked at me. "I wouldn't say that."
"What was the profile for the Beltway Sniper?" I said. "A white loner."
"At first . . ."
"Until you caught him. Turned out to be a black man riding around with a kid. How many roadblocks did they slip through?"
He didn't bother answering.
"Then the Unabomber," I said. "I read you had two profiles at one point. One guy says he's an academic, which turned out to be right, while the other guy says he's an airplane mechanic. How many years was that case open?"
Larkin shifted in his chair. "Seventeen."
"Biggest manhunt in history," I said. "Every agency in the country. How many agents did they have working that case, full-time? And then how did they finally catch him?"
"His brother."
"One person," I said. "One connection. That's what it takes."
"So for Livermore, that connection is apparently you."
_He's got me on that one,_ I said to myself. _If I only knew what that connection was._
I'd been keeping a lid on it all that morning, the raw anger that came up every time I thought about Livermore, every time I replayed any single moment from the past twenty-four hours.
_He's out there right now,_ I thought. _Somewhere._ _Wearing that same smile._
"So what are they talking about in there?" I said, nodding toward the conference room.
"Even if I knew . . ." Larkin said, leaving the thought hanging. "But when they're done with the meeting—"
"We'll be gone," I said. "Let's get out of here."
"And go where?"
"Take me to the most recent crime scenes," I said. "I want to see every place he's been."
"You know I can't do that."
"I'm pretty sure you can."
He looked at his doorway as if hoping for some help.
"Look," I said. "Either I'm under arrest or I'm free to leave at any time. I just read the addresses of those crime scenes. Now I'm going to go out and look at them."
When I stood up and walked out, the only question was whether I'd hear his footsteps behind me.
* * *
—
IT WAS SOMETHING I'd learned as a cop in Detroit. You can't capture a crime scene with words on paper, can't capture it with photographs. You need to go there, stand in the middle of it. _Feel_ the place, use all your senses at once. That's your only chance of finding something that someone else might have missed.
The first place Larkin took me was a little house over on the north side of town. This was the home of Carolyn Kline, the woman from the video. It was a small one-story house on a lot decorated with gravel and cactuses. I told Larkin to park on the street for a few minutes so I could watch the place. It was a busy street. In the middle of the day there was a steady stream of traffic.
"We believe he walked in through the front door," Larkin said. "There was no forced entry."
I nodded, trying to imagine it. Trying to put myself in his place, watching this house, waiting for the right moment. I wasn't trying to figure out _who_ , _where_ , or _when_ at that point. Those questions had already been answered before I got there. I was more interested in the _how_ , and even the _why_ if I could get to it.
The _why_ would put me closer to the mind of the man himself. Which would maybe help me understand why I was here in Phoenix, with bandages on my arm and knee, with the sounds of explosions still ringing in my ears.
"Let's go inside," I said.
We got out and went to the front door. There was no crime scene tape, but a sign posted on the locked door warned against entry by anyone but authorized law enforcement. Larkin had brought along a key. He opened up the door and let me inside. I stopped dead in the front room and looked at a section of the floor that had been marked with red tape.
"They found trace evidence right here," Larkin said. "Hair. No semen this time."
That didn't make sense to me. In all of the other cases, he'd had sexual contact with the body at the kill site, then again at the second location.
"What about the rope fibers?" I said.
"A few fibers, yes. But we found nothing in the bedroom."
I looked out the window at the busy street. Easy enough to close the curtains, but why not go to the quieter room?
_Because it wasn't about physical gratification at all. Not this time._
I bent down to look at a photograph of Carolyn Kline, making sure not to touch it. Even though the crime scene had already been processed, it just wouldn't have felt right to me.
I recognized the face from the file back at the FBI office, but here she was out in a boat, with sunglasses and a big smile. An older man was sitting next to her.
Her father.
A man who had to receive a visit from two police officers, just a few days ago. Because as a cop you always want your partner with you when you go deliver the news that will destroy a family.
I took a quick look through the rest of the house. There was nothing else to see, except a back door that led to a small backyard. There were neighbors on both sides, another in back, but if you left this place at night . . .
"He pulls up his car and takes her out this door," I said. "Then he takes her to the house in Scottsdale."
_Where he suddenly remembers why he kills these women in the first place . . ._
_It doesn't fit._
I didn't want to see the Thompson house yet. I'd seen the video and already knew what had happened there. Standing in the same room would make it even more real. Take me even farther into the man's mind. Farther than I was prepared to go.
"Take me to Stephanie Hyatt's place first," I said. As we drove through the midday Phoenix traffic, I tried to stop the endless loop that was running in my head, what Livermore had done in that bedroom, until we finally crossed over into the town of Mesa and came to a trailer park. A long line of metal boxes, baking in the sun.
"Taken just before Livermore was arrested," Larkin said, driving to the trailer on the farther end of the road. "The case is still being developed."
This trailer looked out of place to me. Once again, I tried to imagine Livermore sitting here on this road, watching the woman who lived here.
_Why the hell would he come here?_
We got out. Another front door with another sign. We went inside, to a small, cluttered trailer that was at least one hundred degrees. The smell was a mixture of cigarette smoke and cheap fast food, with a certain unmistakable top note, the sweet tang of marijuana.
"You found no trace evidence here," I said. "Or if you did, there was so little you might have missed it if you didn't know he was involved."
Larkin looked at me. "Yes."
I pictured Livermore standing in this room, in this exact spot, by the pile of dishes on the counter and the beer cans on the little dining room table. In that same first interview with Halliday, Livermore had told him that all of his first five victims were together somewhere. _In a special place._
"Wherever those other victims are," I said, "this one won't be with them. Stephanie Hyatt was the throwaway."
I felt bad standing in her trailer, calling her that. Like I was diminishing her value, even in death. But I knew it was the truth. I knew that's how Livermore would have seen her.
"He never abducted two women so close together before," I said.
"True."
"And he could only be with one at a time."
Larkin nodded.
"Carolyn Kline was just for the cameras," I said. "And this one . . ."
I looked around the place, trying to find another way to say it.
"This one was just the bait to set the trap. Nothing more."
"So where is she?" Larkin said. "Obviously not in that canyon."
I shook my head. _She's wherever you put things you don't need anymore,_ I thought as I looked at another photograph. This one mounted on the wall, on one of those cheap pieces of stained wood with the clear sheet of Plexiglas. Stephanie Hyatt with another young woman on either side of her. They all had the same hairstyle. All came from the same place. Sisters.
Another family that had to answer the door and see two cops standing on the other side, their hats in their hands.
At this point, I had more of the _how_ , but still wasn't close to the _why_. Still wasn't close to whatever it was that drove him.
"There's one more crime scene in town," Larkin said to me. "His fifth victim, Liana Massey, she lived up in Flagstaff, but after he killed her he brought her down here and kept her in a condo out in Fountain Hills."
"Liana Massey," I said, remembering the file. Remembering the face. "Six months ago."
A half hour later, we were on the eastern side of town, fighting our way through the seasonal traffic, with seemingly a million identical brown stucco buildings all around us, each one divided into eight or ten different condo units. It was February, so the part-time residents had left every cold-weather state in the country to come fight the same traffic and live in these cheap plaster boxes, where they could swear at the traffic instead of the snow.
"He kept her in their bed," Larkin said when we pulled up to the place. The unit in question was close to the end of the street, as secluded as you could get in a place like this, especially if you broke in during the summer.
"This place makes sense," I said as I got out of the car and looked up and down the street. I could picture Livermore parked here, watching the place, choosing the one perfect unit. The construction was so cheap, he could have gotten through any of the windows.
_You're pretty smart,_ I thought, _when you don't want to get caught._
"The wife didn't even realize someone had been there until she got in bed that first night. There were dried fluids from the victim's body. Some hair. Some skin that had sloughed off the body. They eventually found the victim's clothing under the bed."
We were about to head inside the building when Larkin was interrupted by his cell phone. He answered the call and listened for a moment, then he looked over at me.
"Mr. McKnight left the building," he said. "There was no way to stop him."
He closed his eyes as he listened to a few more words from whoever was on the other end. Probably Madison, and he probably wasn't happy.
"Yes, sir," Larkin said. "Right away."
"I take it we're going back to the office," I said.
He didn't answer me. He got back behind the wheel and waited for me to get in beside him. It was a long, silent ride back to Deer Valley Road. Agent Madison was standing outside the front door, blinking in the sun, as we pulled up.
"This is on me," I said to him as I got out. "Don't blame Agent Larkin."
"Mr. McKnight, you were not supposed to leave this facility."
"I'll ask you the same question I asked him," I said. "Am I under arrest or am I—"
"You're a person of interest, you know that."
"That means nothing to me. Other than getting thrown out of meetings as soon as they get interesting."
"I told you at the time, the information we needed to discuss was classified."
"You found the bodies," I said, watching him carefully. "Why wouldn't I be able to hear that?"
I didn't see anything change in his face. I thought about what else could have come to light today, what other piece of information was missing . . .
"You found his home," I said. "Where he really lives."
This time I could tell I was right.
"So what did you find?" I said. "Something connected to me?"
"No," he said. "It has nothing to do with you."
"Then why are you shutting me out? If you really want me to help you—"
"That was my call at the time," he said. "You can trust me or not. I don't care."
I shook my head and looked over at Larkin. He was doing a professional job of trying not to be noticed by his boss.
"That's the thing about FBI vehicles," I said, nodding to the dozen black sedans in the lot. "I can go rent a car, wait outside here and just follow the next one that leaves. Or hell, I bet if I just drive around town, I'll see them all parked somewhere."
"Don't even bother," he said, looking me in the eye. "I've made my decision. I'm taking you there right now."
# CHAPTER ELEVEN
LIVERMORE LINED UP the crosshairs on Alex McKnight's face.
"Stop moving," he whispered to him, from a hundred feet away. "Just stand still."
He waited for McKnight to obey him. A few seconds later, McKnight stood up straight and tall. Livermore had the shot he wanted.
He pushed the button, and the camera's shutter snapped open and closed. He pressed it a few more times, because even though he was sure the first shot was perfect, an engineer knows it's good to have redundancy.
He'd been following Alex all day, from the moment he had walked out of that hotel right down the street from the FBI offices, the place that should have had a sign reading FBI GUESTHOUSE. While he had watched Alex getting into the young agent's car, he'd sensed a police car pulling up next to him in the hotel parking lot. Phoenix PD, one uniform behind the wheel. The officer glanced over at Livermore for a moment. Looked right at him.
Livermore smiled. _Believe in the disguise,_ he told himself as the cop nodded to him and drove away. The short dark hair, the mustache that would grow in better with each passing day. The glasses. _Have absolute trust in yourself and reflect that trust to the rest of the world._
He waited down the street from the office, picked up Alex again when he went out with the young agent, making stops all over town. The house in Mesa, then the trailer park, now here in Fountain Hills. Livermore was parked down the street in his Japanese SUV, holding the new Japanese digital camera he had bought, a Nikon with a seventy-to-three-hundred-millimeter zoom lens.
He watched Alex get out of the car, take a few steps down the street, and look up at the condo's bedroom window.
_You can see why I chose this place . . ._
* * *
—
HE HAD BEEN two hours from Flagstaff, driving in the dead middle of the night, with Liana's body in the backseat. He'd kept to the speed limit. Used his turn signals. Everything straight and correct, still seeing that face from the laptop screen. Still working through the new plan in his head.
He'd brought her to this empty condo, already chosen. Close to the end of the street, with one window on the back that could not be seen from any of the neighboring units. A quiet street during the summer, but not so quiet that his vehicle would be suspicious. Far enough away from his own apartment on the west side of town, but not so far as to make it inconvenient.
He went through the window and unlocked the front door, then spent another hour back on the street, watching the place to make sure he hadn't tripped any silent alarms. When he was satisfied, he brought her out, still wrapped in the sheet and slung over his shoulder, took her inside through the front door, and then arranged her carefully on the queen-sized bed in the master suite. Where he would keep her for exactly eight days.
This was the ritual. Eight days, coming back every night to be with her. To watch how she changed. Her fingernails turning black. Her skin changing color.
From pink.
To purple.
To gray.
It still amazed him how quickly it happened, how he could almost imagine the color changing right before his eyes. Like in the bamboo forests he'd seen in Japan, how they had told him that if you stood still enough, you could actually watch the plant growing. It was the same thing. Only it was the timeless changes of death he was watching, not of life.
It aroused him.
It was a simple fact, something he had learned about himself from the very first time, had observed in himself with the same clinical precision he brought to everything else in his life. He was a man, after all. He was flesh and blood, born with a hard wiring every bit as well defined as the wiring in a robotic arm.
He'd been aroused when he'd taken that first woman's life from her, more aroused an hour later when her body was still warm. Even more aroused the next day when her body was cold. Part of it was the all-consuming sense of power he felt over her. Another part of it was the sense of uniqueness: _This is something that most men do not experience. _Cannot _experience. It is something left only to the rare few. To the superior beings who walk this earth once in a millennium._
And yet a final part of it was just a mystery to him.
_I don't know why I respond this way. I just do._
_Therefore it is right._
After eight days of being with her, of watching her, touching her, allowing her to give him pleasure even after her death, Livermore had taken Liana home. Not back to the empty apartment where he slept. He had taken her _home_ , where he could keep her with the others.
With Arlene, Theresa, Claire, and Sandra.
All five of them together, even now.
They had all taken advantage of that one human part of him. They had fooled him into believing, for one moment, that they could live up to the standard that another woman had set.
That they could somehow take her place.
And they had all paid for this mistake.
# CHAPTER TWELVE
TWENTY-FOUR HOURS after watching Livermore kill seven men, I was standing at his doorstep.
The apartment they had found was on the north side of town, in one of a dozen tall, gleaming buildings that rose thirty stories above the street. His was two blocks from the ramp to I-17. The perfect place for a man to stay anonymously comfortable, with a quick escape if he needed it.
Madison badged the security guard at the welcome desk, taking the passcard from him and leading Larkin and me to the elevator. He slid the card into the slot, and we rode up to the top floor. When the door opened, I saw a private vestibule, with access to only one apartment.
Martin T. Livermore lived in the goddamned penthouse suite.
The door was propped open, and I saw half a dozen agents already moving around inside. They were all wearing crime scene gloves and shoe covers. Madison grabbed a pair of each from the cardboard box outside the door and gave them to me, then to Larkin. The two men slipped theirs on quickly and waited for me to do the same. Then we stepped inside.
"We're just starting our work here," Madison said to me. "I'm sure I don't have to tell you not to disturb anything."
There were two agents setting up a floor grid, long white strings that would run parallel across the hardwood floor, with another set of strings running perpendicular. The result would be a perfect set of one-foot-by-one-foot squares covering the entire apartment. They would use this to catalog any trace materials found on the floor.
As Madison and Larkin went to talk to one of the other agents, I looked the rest of the place over. Everything was sleek and modern, with an open plan that provided a sight line all the way through the kitchen. There were granite countertops with brushed-steel side plates. White cabinets with European-style handles. The living room was bigger than my entire cabin back in Paradise, with black leather furniture on chrome frames. There were two huge Japanese prints on the walls, a crane standing in a pond, and a twisted tree in fog with a pagoda in the distance. No human figures anywhere. No television.
Another agent was busy setting up a series of numbered yellow evidence tags, small tents that would be placed next to anything of possible interest. He'd already put one next to a neat stack of mail on the dining room table, and another tag next to the newspaper, carefully folded next to the mail. I went to the big picture window and looked out over the city of Phoenix. The Superstition Mountains rose in the eastern distance.
As I went into the living room, I scanned the bookshelf. There were thirty or forty engineering reference books, a few books on Japan. Nothing else.
I stepped around another agent who was putting an evidence tag next to a slight indentation in the hardwood floor. Probably just an old dent in the wood, but I knew they'd run DNA tests on everything they could find, trying to determine if anyone else had been here. They'd also dust for prints, and they'd spray luminol under a black light to test for blood.
My gut told me they'd find nothing. As luxurious as this place was, there was something oddly sterile about it. Something almost impersonal.
_This is where he lives,_ I said to myself, _but he doesn't really_ live _here._
When I went down the hallway, I saw that the first door was closed. I could hear voices coming from behind it. I continued down to the bedroom, to the bathroom, back to the kitchen. Everything was just as immaculate. Just as sterile.
"McKnight," Agent Madison said. I turned to see him standing before the first door, which was now open. "In here."
I followed him into the room. It was Livermore's home office. Filing cabinets, a computer station, and a separate writing desk stacked with papers.
"Give us one moment," Madison said to Larkin and the other two agents in the room. His voice was all business.
The three men left the room. I stepped closer to the desk and looked at the papers. It took me a while to process what I was seeing. On top was a large map of San Francisco, then beneath that a set of blueprints with neatly drawn red arrows pointing to several different parts of the building.
"This is the classified information you were talking about at the meeting," I said. "I don't understand why I couldn't—"
"This is national security, McKnight."
I looked up at him. "Are you kidding me?"
As I carefully lifted the blueprints, I saw another map below it, this one of the larger Bay Area, and beneath that I saw the corner of a brochure, with a list of times and what looked to be names of events. _Welcome Cocktail Party. Opening Ceremony._
I put everything back in its original place. "You don't believe this bullshit, do you?"
"Give me a reason not to."
"What's he going to do?" I said, reading the text on the blueprints. "You think he's really going to blow up the Moscone Center?"
"This evidence points that way. We have to take this seriously."
"He's a killer," I said. "He's not a terrorist."
"You don't know _what_ he is, McKnight. You don't even know why he brought you here, remember?"
"Look at this apartment," I said. "Think about the man who lives here. Everything you know about him so far. Would he really leave all his master plans out here for you to see? Unless he _wanted_ you to?"
"So what are we supposed to do? Ignore this?"
"Yes, you're supposed to ignore it. You know why? Because you're too smart to fall for this. You're supposed to go back out there and find him."
"We'll find him. I promise you."
"Not in San Francisco you won't."
He took a beat, then nodded to the wall behind me. "Look at that time line."
I turned and looked at it. Several sheets of paper neatly stapled to the wall, with carefully drawn lines in a dozen different colors, each labeled with the name of a convention event. Another line read _Travel time, Moscone to San Mateo_. Another, _Enter and exit building_.
"The only thing missing is 'Dear FBI,'" I said.
"In five days, there'll be twenty thousand people there. How many—"
"But not Livermore."
" _Listen to me._ Twenty thousand people. God knows how many more in San Mateo. All in the same weekend. He can hit both venues within five hours."
"No," I said. "That's the diversion. You'll be in San Francisco while he's off killing ten more people somewhere else."
"If this is all fake, why did he keep this place a secret?"
"He didn't," I said. "Come on. He _knew_ you'd find this place."
"I don't know about that. He kept his computer pretty damned secure. We took a heavily encrypted hard drive out of his computer, and as soon as we crack it—"
"Don't you understand what's going on here? He'll tie up your computer people for a week. Along with every agent you've got. You're going to spend all this time, all this manpower . . . On what? On an _illusion_."
"Maybe," he said, looking me in the eye. "But I'm not going to have fifty thousand deaths on my hands. This is the same man who killed seven cops yesterday."
"I know," I said, "I was there, remember?"
"We have no idea what he's capable of, McKnight. We have an obligation to—"
"You have an obligation to catch him," I said. "Nothing else matters. How many more people does he have to kill before you realize that?"
"We're going to catch him," Madison said, after taking a beat. "But this isn't just a manhunt anymore. It's a potential _massive casualty disaster_."
I took a beat of my own, looking down at the pile of papers on the desk.
_You're not going to win this argument,_ I said to myself. _They're going down the wrong road, but you can't stop them. They're going to throw everything they've got at this, because after 9/11 you can't be the one person who didn't take a threat seriously._
_Which is exactly what Livermore is counting on._
"So just tell me one thing," I said, looking him in the eye again. "What does he get out of this?"
"Whatever a psychopath gets out of anything. Revenge. Fame."
"No," I said. "That's not the man I sat across the table from. He doesn't need any of that."
"Then what _does_ he need?"
I took a moment to think about it. "I don't know yet," I finally said, looking around the rest of the room. "But none of this is real. He's not here."
"So where is he?"
"Somewhere else," I said. "Somewhere he doesn't want us to see."
"Well, until you figure that out, we're going to follow the evidence we actually have. Agent Larkin will take you back to the office."
"No, thanks," I said.
"McKnight—"
"You can go chase your terrorist if you want," I said, taking off the gloves and handing them to him on my way out of the room. "But I'm going to go find Livermore."
A few minutes later, Larkin and I were alone in the car. His face was red, and he was gripping the steering wheel like he was about to tear it right off its column.
"If you were him," I said, "where would you go to be alone?"
Larkin didn't look over at me. He kept driving.
"You need to be able to work. Maybe make some noise. Someplace _clean_. That's important to him."
Still nothing.
"A storage unit," I said. "Think about it. Metal walls, concrete floor. Electrical outlets. Plenty of privacy. And best of all, you can pay for the place with cash."
"We're going back to the office," he said, keeping his eyes straight ahead.
"Take me somewhere I can rent a car," I said. "I'll do this on my own."
He shook his head.
"Or you can come with me. Actually help me find this guy."
"Mr. McKnight, I'm already—"
"Call me Alex. I'm not that old."
"Alex . . ."
"It's time to decide," I said.
He let out a breath and shook his head again.
"You either tell them I walked away," I said, "or you tell them you helped me find out where this guy really _lives_. It's up to you."
# CHAPTER THIRTEEN
IT WAS LIVERMORE'S personal conception of hell.
He was standing in line, in this forgotten little run-down gas station that smelled like the bottom of an old oil drum, after being on the road for seven hours straight. He was somewhere in the middle of New Mexico, trapped in this place, looking it up and down and seeing one horror after another, from the broken linoleum floor to the dust-covered vents on the stand-up cooler to the sagging yellow ceiling tiles. He had passed the bathroom door on the way in, locked up tight. You probably had to ask for the key, and when they handed it to you it would be chained to a urine-stained block of wood. Livermore would go outside and piss against the wall before he did that.
The man behind the register looked like he was eighty years old, an obsolete machine that should have been left out in a field years ago. Livermore had already waited for the man to verify that his hundred-dollar bill was the real thing, taking out his currency pen, drawing a line across the face, then holding it up to the light, the whole operation taking as much time as a U.S. Treasury agent breaking up an international counterfeiting ring. Now he was slowly counting out the change, one crumpled bill at a time, with his old, stained fingers. Livermore's mind drifted to different ways he could persuade the man to move a little faster, maybe starting with the box cutter hanging in its plastic container on the dusty wall behind him. Start with one ear, see if that motivated the man. Then move to the other. The man was probably already mostly deaf, anyway.
"The reward's up to two million dollars," another man said from behind him. "Wouldn't mind running into him, no matter how dangerous they say he is."
Livermore came out of his reverie and listened to what the man was saying.
"I'd knock him right out," the man said, speaking to whoever was standing in line behind him. "Throw him in the back of my truck like a goddamned buck. Go collect my reward."
Livermore turned around and looked at him, another old man, a foot shorter and starved down to skin, bones, ligaments, and an Adam's apple that bobbed up and down as he rambled on.
"Better believe I'd give him a taste of his own medicine. Get out my tire iron, you know what I'm saying? Give him a once-over. Make him suffer, like he done to those poor women."
As Livermore took his change and left, he overheard the same man prepaying on pump number seven.
A minute later, the man came back out to his truck and opened up the front door just long enough for Livermore to pop the gas cap. When he shut the door and turned around, he nearly jumped out of his skin. Livermore was standing right in front of him.
"Let me help you out there, sir," Livermore said, taking the nozzle from the pump.
The man raised a hand to stop him, but Livermore was already squeezing the handle to start the flow of gasoline.
"I overheard what you were saying in there," Livermore said.
The man just stood there, still confused.
"You really think you could do that?" Livermore said as the numbers on the ancient pump clicked by slowly. "You think you're capable?"
"Look here," the man said, "I don't need you to pump my gas . . ."
"This man you were talking about . . ." Livermore said. "This man who knows all about pain . . . Who spent his whole life studying it . . . Maybe even turning it into an art form. I'm not sure how impressed he'd be by your little 'once-over' with the tire iron."
The man was shifting his weight back and forth from one foot to the other, scanning the other vehicles at the pumps, as if hoping for a friendly face. Someone who might come to his rescue.
"You think it would be your cold metal against his flesh and bones," Livermore said. "That you'd be the one in control. But you're wrong. You'd have to meet this man where he lives. To beat him, _you'd have to become just like him_."
Another man walked by then. Livermore smiled to him and gave him a friendly nod.
"I see you work in a shop," Livermore said, reading the lettering on the side of the truck's door. "So I'm sure you know the value of picking the right tool for the job. Let's say you used your acetylene torch . . . Can you even imagine what that would be like? Watching parts of your own body being melted away?"
"Listen, buddy . . ." The man was licking his lips and wavering on his feet like he might get sick. His eyes kept darting to the numbers on the pump, as if willing the gas to flow faster into the tank.
"But be careful," Livermore said as the nozzle finally hit the automatic shutoff point. "By the time you're done, you'll be a different man. You might even find you have a real taste for it. Then maybe it'll be _you_ with the two-million-dollar reward on your head."
The man kept standing next to his truck as Livermore returned the nozzle to its place beside the pump. The entire exchange would have been a foolish move for most men. Pure insanity to draw attention to himself, to make this man remember him, here at this gas station in the middle of New Mexico. How easy it would be for him to remember Livermore, to recount all of the things that he'd said and to sit with the sketch artist and re-create the face that had become burned into his mind.
How easy, assuming that the man lived to see the end of this day.
"You have a good afternoon," Livermore said, taking another look at the name of the business on the door. "Mr. Henderson of Edgewood, New Mexico."
Livermore watched the man get behind the wheel of his truck and quickly slam the door shut. He gave a long look over his shoulder as he cranked the truck to life and put it in gear.
_If you're going in my direction,_ Livermore thought, _then maybe I'll make a stop in Edgewood so we can finish your education._
He got in his own vehicle and watched the truck go back to the expressway. It turned west. Livermore was going east. He put the man out of his mind and kept driving.
# CHAPTER FOURTEEN
AGENT LARKIN made his choice. He was coming with me.
We came up empty on the first two storage facilities. As soon as I saw the third facility, I knew we had a shot.
It was at the end of the street, set back a good hundred yards once you made your way through the gate and down to the main building. I could see the large doors on the units as soon as we got close, and beyond the back fence there was nothing but the desert and the Salt River in the distance.
We walked into the office and had to wait a few minutes, until the manager peeked around the little partition and looked surprised to see us standing there.
"Help you?"
Agent Larkin showed him his badge and told him he was looking for a unit that might have been rented by a man named Livermore. The manager shook his head at the name until Larkin pulled out the photograph and showed it to him. Then the color drained from his face.
"I take it he was here," Larkin said.
"Yes," the man said, "but he didn't call himself Livermore."
The manager went through a little box of index cards, until he finally found the one he was looking for.
"Gene Lamont," he said. "Yeah, that's it."
"Wait a minute," I said. "What was the name? How did he spell that?"
He showed me the card.
_Gene Lamont._
"That mean something to you?" Larkin said.
"He's a bench coach for the Tigers," I said. "Managed a couple teams, too."
Larkin just looked at me and waited for more.
"We were teammates in Toledo," I said. "Both catchers. My last year there, Gene got the September call-up, and I didn't."
"That's either a hell of a coincidence—"
"Or he was already messing with my head when he rented this place months ago."
"Any chance we could get into that unit?" Larkin asked the manager.
"If you want to," he said. "But it's empty now. I made him pack up and leave."
"Why is that?"
"Strict rule," the man said. "You can run power tools, anything you want, but no open flames."
"Open flames?" I said. "What the hell was he doing?"
"He had this whole grill thing set up, with a huge pot of water. First I thought he was cooking meth or something. I mean, what do _I_ know, right? Wouldn't be the first time. But turns out he was just boiling water."
"For what?"
The manager shrugged. "Hell if I know. He'd just moved everything in, all these crates and boxes, and then I notice he's got this big pot of water going. I told him he can't do that and he, um . . ."
The man stopped talking and looked away from us.
"What happened?" Larkin said.
"Nothing," the man said a little too loudly, like he was trying to convince himself. "He just said some things and he left."
"What did he say?"
"Something about rules being made for men with small minds, then something about the Air Force, how it was obviously the right place for me. Not sure how he even knew I was a veteran, let alone an airman."
Larkin shot me a quick look. This sounded like the Livermore I had met in that interview room, a man who somehow knew your whole life story.
"Then he said something else . . ."
We both waited for him to take a moment, clear his throat, and continue.
"He said he'd read about a man who got trapped in a storage unit for nine weeks straight. Guy was lucky, he said, because he had some food and water in there. But he was still almost dead when they found him."
The man paused for a moment, bringing it all back.
"He said, even though he was alive when they got him out of there, he never really came back from it. The heat, the darkness, the isolation . . . The man's mind just snapped. And of course, he said, this happened in New York, so just imagine if it was Arizona, how hot it would get. That man would have been cooked alive."
He paused again, swallowing hard.
"I told him to get the hell out of here," he said, "but he just stood there, right where you're standing now, and he told me that man was locked in accidentally. Because who would do something like that on purpose? And the way he kept looking at me as he drove off . . . I still think about it. Like I wouldn't be surprised at all if he came back someday."
"I'm going to give you this," Larkin said, taking a card out of his wallet and putting it down on the counter. "Just in case you do ever see him again. You call me right away."
_If you're alive to make the call,_ I thought. _This may be the last man on this earth you'd ever want to piss off._
"What was the exact date?" Larkin said. "The day this all happened?"
"Let's see," he said, looking at the file card again. "It had to be, what, October thirteenth? Fourteenth, maybe?"
"If he couldn't work here," Larkin said, looking back at me, "he must have gone somewhere else."
"Pretty sure most places have the same rules," the manager said. "But why are you looking for him, anyway? Should I be worried?"
Larkin thought about his answer. "If you watch the news on television," he finally said, "you may see his face. But he's probably long gone by now, so don't worry. You just keep that card by your phone and call me if you need to. There's a reward for information leading to his capture, too."
The manager still didn't look like a man who'd be sleeping well that night. But we thanked him and left. When we were back outside, I saw an old Lincoln Continental parked beside the building, the dark blue paint peeling in the harsh sunlight. I looked it over until I finally spotted the sticker in the corner of the windshield. Faded by the same sunlight, the faint outline of the Air Force symbol, the star with two wings.
_I never would have noticed this if I wasn't looking for it,_ I thought. _But Livermore does this automatically. He looks for information about everyone he meets, then uses it to his advantage. Even if his only goal is to unnerve you._
"I don't understand," Larkin said. "What does boiling water have to do with making explosives?"
"Whatever he was doing, he took it somewhere else. Middle of October. We could keep checking other storage units . . ."
"You heard the guy."
"Or we could look somewhere else," I said. "What would be his next choice?"
"Something private. Someplace he could do whatever he wanted."
"If you're him," I said, "where do you find that?"
A few minutes later, we were sitting in a coffee shop. Larkin had his laptop open and was using the Internet archives to go through the Craigslist ads for October. He wrote down the contact information for half a dozen people who had advertised space for rent.
"This one looks interesting," I said. "Five hundred square feet of workspace on a secluded road, metal building with good ventilation . . ."
"It's miles out of town. Almost in Maricopa."
"Call the number. See if the space got rented."
Larkin took out his cell phone and dialed the number. He asked whoever answered the phone if the space had been rented. He looked at me and nodded. He wasn't even halfway into his next question when the call ended.
"Doesn't want to talk about it," he said.
I nodded. "Sounds like somebody we need to go see."
# CHAPTER FIFTEEN
THE SUN WAS SETTING behind him when Livermore hit the Texas panhandle. He found an electronics store in Amarillo and went to the photography counter, gave the woman the flash card from his camera, asked for eight-by-ten prints in the best quality she had. While he waited, he went through the rest of the store until he found the GPS trackers.
He settled on a SilverCloud Sync GPS with a magnetic mount. Two inches square, it would fit in any vehicle's wheel well and operate for days at a time, at temperatures as low as twenty below zero.
When he took the tracker to the counter, the woman was looking at the prints she had made. The half dozen shots of Alex McKnight, walking out of the hotel lobby.
"Are these . . . surveillance pictures?"
She was smiling as she said it. It was obviously the most interesting thing that had happened to her all day.
Livermore put a smile on his own face to mirror hers.
"Yes," he said. "They are."
She was in her late twenties, maybe thirty years old. Her hair was dyed a shade of beet red that didn't look quite natural. A shade she probably should have left behind in her teenage years, Livermore thought. She wore a lot of makeup around her eyes, and her fingernails were painted the same shade of red as her hair.
_She thinks this makes her interesting,_ Livermore said to himself. _Painting her hair, painting her nails . . ._
"Why were you watching him?" she asked as she paged through them again, one after another. "Is he wanted for something?"
He took the photos from her gently, noting the fingerprints she was leaving on the glossy surface of the paper. He would have to clean them now.
"I can't talk about it here," he said.
"I get it. Top secret stuff." She pantomimed zipping her lips shut and throwing away the key.
"You could say that."
"So are you a cop? No, wait, let me guess."
She backed up a step, looking him up and down. Livermore waited patiently. He took a quick glance toward the front of the store, then the back.
The place was empty. They were alone.
"I don't think you're local," she said. "You sure don't _sound_ like you live around here."
_She has a good eye for detail,_ he thought. _And a good ear._
"So I'm thinking you're something federal. FBI, maybe, or a marshal."
_She'll remember everything about me. And she'll remember these photographs._
"I'm going to say a U.S. marshal. That's my guess."
He smiled again. "What's your name?"
"Irene."
He nodded. "So, Irene, let me guess what _your_ job is."
"No, see, this is just until my—"
"Until your band hits it big," Livermore said. "And you can stop driving around in your little Hyundai."
She looked at him with surprise. "How did you—"
He nodded to the set of keys resting on the stool behind her, next to her purse. There was a guitar pick with a hole drilled through it attached to the ring, and the Hyundai symbol was prominent on the biggest key.
"In the meantime," he said, "you sell electronics to people who don't understand how they work. And you develop photographs."
Her smiled faded slightly. "Yes . . ."
"And when you develop those photographs," he said, taking a half step closer to the counter, "sometimes you break up the monotony by looking at them, by trying to imagine the story behind them."
"I didn't mean to do anything like that. I was just—"
"You do this with all of the photographs you develop. To pass the time."
She looked down and cleared her throat. "No. Not really."
"Someday a man will walk in here with some pictures," he said. "Pictures you aren't supposed to see."
She looked toward the front of the store, like she was hoping someone else would come in. But there was nothing but the darkness outside.
"You won't try to imagine the story behind those photographs. You'll try to forget them. Wish you never saw them. But he'll know."
"Okay," she said, clearing her throat again. "Can I get you anything else?"
"You need to be careful with a man like that. The fact that he's bringing his photographs here . . . It means he has an appreciation for the finished product. He wants a good-quality print. But it also shows a lack of self-control. He's taking a risk by letting them pass through your hands. A risk he's prepared to deal with if he has to."
She looked back at him without saying a word. The entire store was silent.
"You don't want a man like that walking into this store," Livermore said. "Not if you're here all by yourself."
She looked down at the phone, then at the front of the store again. Then back to Livermore. "The manager is usually in the back."
"Usually. But not right now."
She didn't answer. She put her hands together to stop them from shaking.
Livermore checked his watch. Then he smiled at her one more time.
"You'll be closing the store soon," he said. "Looks like you'll be safe for tonight."
Like the man in the gas station back in New Mexico, she would remember him. She would be able to describe him exactly, re-create his face, transcribe virtually every word he had said to her. Once again, a foolish move for any other man. A tightrope act that Livermore would not trust to anyone else.
He put the GPS tracker on the counter, next to the envelope containing the prints, and paid for it all with cash. She rang up the transaction quickly, and her hands were still shaking as she put everything in a bag.
"Enjoy the rest of your evening," he said.
Then he went out to his car to wait for her.
# CHAPTER SIXTEEN
THIS LOOKS LIKE something out of a Mad Max movie," Larkin said as we got out of the car.
We had driven south, past the edge of the city. No more water, no more attempts to claim land from the desert. The Sierra Estrellas loomed to the west as we went twenty more miles, until we came to a huge sand-and-gravel pit. Then another mile to a turnoff, leading down a long dusty road that seemed to go nowhere. Until we finally came to a homestead and a small cluster of buildings, all glowing in the last light of the day.
We went to the front door and knocked. I looked around the place, at the old tractors and tools and a million other scraps of metal. An older man answered, a man wearing work pants and a filthy undershirt, with skin slightly less sun-worn than an Egyptian mummy. Larkin had his badge out as soon as the door opened.
"Who rented the space on your property?" I said.
"My daughter did the ad," he said. "A man answered it."
"That doesn't answer the question."
"He paid cash. He said he'd give me extra if I made sure nobody ever went inside."
"Name."
The old man had to think about it. I was expecting him to say _Gene Lamont_ again. But he didn't.
"Tim Hosley."
Another old teammate of mine, from that same team in Toledo. Another young player who got a September call-up.
_It's like he knew I'd be standing right here,_ I thought. _Right here in this spot. Asking this exact question._
"We're going to have to see inside that building," Larkin said.
"It's got a padlock on it. Don't have the key."
"That's what bolt cutters are for," I said. "I'm sure you have a pair."
The man didn't bother to lie. He closed the door for a moment, came back with a huge pair of bolt cutters, and we followed him down a long driveway that passed by other buildings, a few old cars, another tractor, another few million scraps of junk metal, until we got to the end.
_Gene Lamont,_ I thought. _Tim Hosley._
I'd been keeping myself focused on the job at hand, hunting down the leads, from one location to the next . . . But now that I was here, I could feel myself getting closer to him, to the place where the real Livermore lived. Those names from my past, they just fed the flames, made me want to find him even more.
It felt like he was watching us. At that very moment, as we walked down that last dusty road to the place that held his secrets.
It felt like he was laughing at us.
"Alex," Larkin said. "We should call the office first."
"We're here," I said. "Let's see what's inside."
The building was made of tin, every wall visibly warped from the heat. It was about five hundred square feet, as advertised, but with no windows. It looked a hundred years old, the kind of ramshackle shack that would have been claimed by rust a long time ago if it wasn't in the middle of a desert. There was a round circulating fan positioned on the corrugated roof, but it wasn't spinning. A horrible place to do anything in, I was sure, but I could imagine Livermore seeing this godforsaken shack on the end of a forgotten desert road and knowing he'd found the perfect home.
We watched the man line up the bolt cutters and squeeze the handles together. When the shank was cut clean through, he pulled off the lock and yanked open the door. I had a sudden thought, a moment too late, that the man who knew I'd probably be standing here, waiting for this door to open, was the same man who could set up a remote ambush in a canyon to kill seven men.
But when the door swung open, there was no explosion—just a wave of trapped heat coming out at us. I let out my breath as we both looked inside.
The first things we saw were the ropes. Dozens of them, hanging from the rafters in two parallel rows and creating a sort of hallway that led into the room.
"What the hell," the man said.
I walked between the rows, the ropes so close that they brushed against my shoulders as I passed through. Some of them were just a few feet long. Some of them were so long they had to be looped several times from the floor to the ceiling. In the dense heat, there was a strange, earthy smell to them.
The back of the building was obscured in the darkness. That was when the man turned the overhead light on.
"I don't know nothing about anything that went on in here," he said. "So whatever you find . . ."
"Don't touch anything else," Agent Larkin said to him. "Just step back."
The light was filtered through the ropes, casting a hundred thin shadows on either side of the room. It was really more like two separate rooms, divided by the hallway. I chose the right side first, pushing myself through the ropes, into what was clearly Livermore's laboratory. There was a metal worktable running along the wall. On top of that was a large tabletop grill, with a single iron pot that could have held twenty gallons of water. Beside that was a large bottle of mineral oil.
Agent Larkin pushed through the ropes to stand next to me. He'd already pulled out his cell phone and started dialing.
There were semi-clear boxes of wires stacked neatly on the workbench. Other boxes contained circuits, relays, connectors, other electrical parts too obscure for me to recognize.
"This is where he made his explosives," Larkin said, nodding toward the other side of the workbench, past the grill and the big iron pot. There was a set of glass beakers there, a rack of tubes, and a Bunsen burner. And several unmarked plastic containers of powders and liquids.
I moved to the metal utility shelves that sat next to the table. On one shelf there was a large collection of iron pipes, of all different sizes. On another shelf a half dozen pressure cookers.
"He was experimenting," Larkin said. "Pipe bombs, pressure cookers . . ."
I was already feeling numb, already overwhelmed by the scope of what this man had done here. All of the time and effort that had gone into these instruments of murder.
There was a box of fuses among the pipes. Slow burn, fast burn . . . Other fuses I couldn't identify, all homemade from matches or black gunpowder, even a few from Christmas tree lights.
_This is what he did every day. This was his hobby, finding new ways to kill people._
"Just like the Boston Marathon," Larkin said. "You still think he—"
"No," I said. "This doesn't make him a terrorist. All this stuff, it still feels . . . too personal."
But then I looked at everything in front of me again, all of these lethal tools spread out on these tables. And I started doubting every gut instinct I'd ever had about this man.
_Maybe Larkin is right,_ I thought. _Maybe we really don't know what this man is capable of._
I moved down to another shelf, where several clear jars sat in a neat line, all filled with a pale yellow jelly. Larkin bent down to look closely at the jars without touching them.
"This is . . . napalm."
I looked at him. "Are you kidding me?"
"He could have used this in the canyon instead, but he was probably worried about leaving it in the heat."
"So he blew us apart with shrapnel," I said, looking at another shelf filled with boxes of lead shot, everything from tiny birdshot to ball bearings a good inch in diameter.
Leaning against the shelves was a large industrial drill.
"That's a core drill," I said. "With a diamond-tipped bit, he could have cut right through the rock in the canyon. Set those pipes in, like little howitzers."
I kept moving, to the far corner of the room. There was a full-sized refrigerator standing there. It was humming, and there was a rusty metal latch bolted onto the door to keep it closed.
"That's for the ammonium nitrate," Larkin said. "You freeze it before you filter it. I can smell a little ammonia, can't you?"
I could only smell the ropes, but I took his word for it. I was looking at the latch, wondering why it was necessary . . .
"They track that stuff now," Larkin said. "Livermore went to a lot of trouble here, to stay off the grid. Or maybe he just liked making all of this stuff himself."
As I happened to look down at the floor in front of the refrigerator, I saw a constellation of red dots.
Drops of blood.
I picked up a neatly folded kitchen towel, used that to undo the latch, then grab the handle of the refrigerator and pull it open.
"Alex," Larkin said, "don't open that."
But it was too late. The door swung open, and the cold air hit us in the face.
Then we saw the body.
It was a woman, curled up on the floor of the refrigerator, like she had been trying to keep herself warm. She was naked except for a pair of blue panties. There was a butterfly tattoo on her left shoulder blade.
There were gouges all along the walls. From her fingernails.
"That's her," Larkin said. "Stephanie Hyatt."
He pushed the door closed and got back on his cell phone. I just stood there for a while, thinking about what could possibly come next.
_Six women wasn't enough,_ I thought. _You had to lock up number seven in a fucking refrigerator._
I wanted to pick up the drill and start swinging it, knock over every box, every jar, break everything I could find.
"Easy, Alex." Larkin stood next to me, waiting for his call to go through.
I walked away from him, through the rope hallway, pushing my way to the other side. Where there was the worktable and all of the supplies on the first side, here the back wall was dominated by three large hanging tapestries. All Japanese art, but not like the prints in Livermore's apartment. These depicted something else entirely: in one, a naked woman was blindfolded and tied to a chair, in another a man in an ornate robe dripped hot wax from a candle onto a woman's chest.
_No, not wax. It's burning right through her skin. It's hot metal._
On the second tapestry, a man was tied up and hanging upside down with what looked like a great iron wagon wheel around his neck. The third might have been the most disturbing of all, a man suspended over several bamboo stakes with the pointed tips piercing his body. As if to bring that image to life, there was a glass case beneath the image with a grow light shining down on a dozen short stalks of bamboo.
I went closer to the case and looked inside, then moved to the desk next to it, where parchment paper was kept rolled up in several bins. One paper was laid out flat, with a pen and ink nearby. There were Japanese symbols drawn on the paper, which of course meant nothing to me. But I would have bet anything the meaning of the symbols was as dark as the artwork hanging on the walls.
I felt like I had just walked from one half of Livermore's brain to the other, from the logical, analytical side to whatever the hell you'd even call the rest of it. The creative, the intuitive . . . I didn't know if you could even use words like those to describe the inside of this man's head.
Next to the desk was another glass case, this one containing an elaborate metal birdcage—except that the bottom of the cage was open and there were leather straps attached to it, as if you'd actually have a reason to strap a cage to someone's body. I didn't have to know anything else to know that this was something evil.
"Alex!"
Larkin's voice came to me from the other side of the ropes. I went to the last corner of the room, where a bookshelf stood. I looked at the spines. More Japanese symbols. There were larger books on the lower shelf. I still had the kitchen towel in my hand, and I used that to pull out a book and open it.
In the dim light I could make out the elaborate ink drawings, some with text, others spread across two facing pages. A man in an ornate robe tying a woman's wrists together. Then wrapping ropes around her body, above and below her breasts. I went through the pages quickly, each one more and more elaborate, a woman tied up and hanging in the air, then another with her legs doubled back into a painful hogtie. Yet another hanging from one leg, the other leg folded together and tied tightly, her arms trussed behind her back.
_The ropes,_ I thought. _The ligatures found on the victim in Scottsdale, the fibers at the other kill sites . . ._
_This is what he does to them._
_This is why he needs all of these ropes._
I put that book back and took out another. Another set of drawings to see, these even more sickening. This book was more general, with no artistic rope designs, just basic human suffering inflicted by every means I could ever imagine. Men and women spread out on racks, violated with instruments, pierced with needles and stakes and barbs.
"Alex, what are you doing?"
I took a breath and kept paging through the book. A man's body slowly crushed by an elephant's massive foot. A man with a cage tied to his stomach . . .
I looked back at the cage in the glass case.
_You tie this to your victim. Open the little door and put the rats inside._
It washed over me in a hot, sick wave.
_This is Livermore._
_This is what he dreams about._
I closed the book and put it back in its place. I was glad to be holding it with the kitchen towel. Never mind the crime scene, or the handling of evidence. I just didn't want any of this to touch my skin.
Before I stood up, I saw one more book on the bottom shelf. It was the only book without Japanese symbols. As I pulled it out, I saw that it was a scrapbook.
I took it over to the desk so I could see it in better light. As I put it down, I heard Agent Larkin behind me. He had pushed his way through the ropes to join me on this side. He stood in the same spot I had, looking up at the images on the wall.
"Holy fuck," he said.
Still holding the scrapbook with the towel, I opened it to the first page.
"I've got agents on the way," Larkin said. "We'll wait for them outside."
I saw a face.
_My face._
It was a photograph of me from my high school yearbook. I flipped the page, not even bothering to be careful now. One page after another, it was all here.
My whole life.
There were old photographs from the _Detroit News_ and the _Detroit Free Press_ , some of them reprints done on modern paper, some of them originals on yellowing newspaper stock. My entire career as a high school baseball player—high school tournaments, all-star teams, year-end awards. On the next page, my minor league career began. More pictures. Team statistics. Even box scores. This was where he got my .249 average at Toledo. This was where he got Gene Lamont's name. And Tim Hosley's.
I kept going, feeling another stream of ice water run down my back with every turn of the page. After baseball, my time in college. My official portrait from the Henry Ford College yearbook. No surprise, he'd actually gotten his hands on a copy of my college yearbook, too—just to cut out my picture and paste it into this book.
Another page and I was looking at the coverage of my engagement in the newspapers. Then my wedding. This was back in the day when the papers would actually send a photographer to take pictures at weddings, especially if the groom was a semi-famous local athlete. All of those clippings were here.
Then my career as a cop. My graduation picture from the police academy. And finally, what I knew was coming next . . .
The front pages of both local papers, showing my partner's picture next to the grainy image of me being carried on a stretcher into the ambulance.
TWO OFFICERS SHOT
One Dead, One in Critical Condition
I felt Larkin grabbing my arm, trying to pull me away from the scrapbook. But I didn't want to leave. I didn't want to go outside, into the light, into the fresh air. I didn't want to move until somebody explained to me why this was happening.
"Who are you, you crazy piece of shit?" I said. Because there was no explanation coming. No answers. Not yet.
_"Who are you?"_
# CHAPTER SEVENTEEN
CREEPIEST CUSTOMER EVER _._
Irene Murphy texted these three words to her best friend, Sarah. Still trying to laugh it off, but her hands were shaking as she keyed in each letter. She kept looking out at the darkness just beyond the windows, wondering if he'd come back, this man who had seemed kind of interesting and mysterious at first, until she had looked into his green eyes one time too many and had seen something else there. Something that scared her. Something that made her keep the phone in her hand while she counted down the last few minutes.
But he never came back.
She texted Sarah again, asking her about the next practice for their band. Still no response. She pictured her standing behind the bar, mixing a half dozen drinks at once, the phone buzzing away on the bar top, drowned out by the noise.
When the clock hit nine, she went to the front of the store and flipped the sign to CLOSED. Then she locked the door and spent the next half hour running the receipts. She activated the alarm, turned out the lights, then opened the door again, just long enough to step outside. She locked the door behind her and walked through the parking lot. There were a dozen cars in the lot, but that wasn't unusual. Not when your store was next to a restaurant that stayed open until midnight. Still, she made a point of looking through every windshield as she walked by. Her car was parked on the far side of the lot, beneath the last light.
She hurried to get in the car and lock her door. Letting out a breath, almost laughing to herself. It felt like when she was a kid, and she'd run up the stairs from the basement after she'd turned off the lights, trying to outrun the tentacles that were coming up after her.
She drove home to her apartment on the other side of Amarillo. Not much of a place, but she'd be able to move soon, maybe, if the band ever got some gigs. Or if she just found a better job, the kind where she didn't have to deal with creepy customers.
At one point, she saw a pair of headlights behind her. They were still there when she made one turn, then another. For a moment, she thought about driving right over to Sarah's bar. Running inside and telling her to call the police. Then she made another turn, and the car behind her kept going.
She let out another long breath.
When she got to her apartment building, she heard her phone beep. She picked it up, saw the text back from Sarah, finally.
_are you okay? come to the bar!_
She texted back, _Home now, call me when you get off_. Then she got out of her car and went inside. She went up the stairs to the third floor. Her apartment was the first one on the left. She took her key out and put it in the lock, just as she heard something behind her.
Before she could even turn, she felt a hand closing around her right wrist. The scream had barely formed in her throat when another hand came across her mouth. She tried to kick at the man's legs, but he was already pushing her inside her apartment. Her arm was twisted painfully behind her back, until she tried to scream again, biting at the hand on her mouth and tasting blood. As he cursed and let go of her, she spun and tried to rake his face with her keys. A self-defense lesson she had practiced in her head a thousand times, walking through that empty parking lot. But he caught her wrist, and his other hand shot forward, fast as a snake striking, to latch around her throat. He stood with his face close to hers.
_It's him. The man from the store._
He still had the same dead-calm expression with the strange half smile, even as he tightened his grip on her neck. She tried to suck in air, to fill her lungs with just enough to scream one more time, but he slammed her against the wall. She felt the impact on the back of her head and then everything went out of focus.
"Please," she said with her last breath. "Don't."
"You brought this on yourself."
The voice echoed in her head as she felt the grip tighten around her neck again. She clawed at the hand with her fingernails until she started to feel herself sliding down the wall. Lower and lower.
Until she had fallen all the way into nothing.
* * *
—
SHE OPENED HER EYES AGAIN. It took her a moment to realize where she was. Lying on her stomach, but somehow raised above the ground, looking down at the carpet. She tried to speak, but her mouth wouldn't open. As she moved her arms, she felt a sharp pain in her shoulders. Her wrists were bound together, behind her back. Her feet were bound, too. She couldn't lift her head more than a few inches. She turned as far as she could, to see what he had done to her.
_I'm on the coffee table,_ she said to herself. _He's tied me down._
_No, he's_ taped _me down._
The tape was looped across her mouth, then wrapped several times around her neck. She felt it against her back, against her legs and her ankles. From one end of her body to the other, the loops disappearing under the table. She even saw the empty roll, discarded a few feet away. As she strained to see the rest of her apartment, she didn't spot him anywhere. A full minute passed. She fought down the panic, made herself be still. Another minute.
_He's gone,_ she thought. _He left me here. I have to get free._
She worked against the tape, rocking back and forth and nearly turning the table over. She screamed inside her head, losing the battle against the panic, fighting and fighting until she had used up the last of her strength. She made herself be still again, made herself breathe.
Then the door opened, and the man came back inside. He was carrying a large shopping bag.
"Don't make this worse for yourself," he said, coming close and setting down the bag.
_You don't have to kill me,_ she thought, willing the words to life, trying to transmit them through the air to him. _You can do anything you want. Just don't kill me._
"You will not scream," he said as he bent down even closer and started to pull the tape from her mouth. "You will say exactly what I tell you to say. Do you understand?"
When her mouth was free, she gathered in her breath, but the scream died in her throat as he pulled her arms away from her back, igniting the dull ache in her shoulders into a white-hot blast of heat.
"That's approximately ten pounds of force," he said to her. "From this angle, it would take twenty to dislocate the shoulder. Do you want to know what thirty pounds feels like?"
She shook her head frantically. He repeated the words he wanted her to say, then he pointed the phone at her face and told her to start speaking. She strained to hear them, to remember them. Then she said the words, one by one, not even knowing what she was saying. None of it made sense. But she didn't want to feel that pain again.
When she was done, she closed her eyes and started to cry.
_This is really happening,_ she said to herself.
_I am going to die._
He said some words of his own into the phone, more words that made no sense to her.
_I am going to die._
He put the phone in his pocket. Then he bent down to the shopping bag and took out a small combination safe. He set the safe on the floor, directly in front of her. She could see that the door was slightly open, but she couldn't see what was inside.
_I am going to die._
She had already given up when he reached into the bag one last time.
_I am going to die._
_I am going to die._
# CHAPTER EIGHTEEN
I HAD BEEN A SPLIT SECOND too late to save my partner, back on that hot summer night in Detroit. Now I was God knows how many _days_ too late to save this woman in the refrigerator. I stood there trying to imagine what she must have gone through in her last hours as I watched them take her out in a body bag, along with everything else from that metal shed at the end of that desert road.
They took down over a hundred ropes. _Why had he boiled so many of them?_ Maybe it had become a ritual for him. What was more interesting to the techs was the chemistry set, the electronics, everything he had used to build his explosives.
They took the torture books, bringing each one of them out into the daylight. Some of them looked old. Rare, probably worth a lot of money to a collector, if there were such people in the world. Which I was sure there were.
They took down the wall hangings, brought out the glass cases with the bamboo and the rat cage. All of the parchment, every last piece of Livermore's life.
And then, in the midst of all this madness, for reasons I still didn't understand . . .
_My life._
They took the Alex McKnight scrapbook to the FBI offices to be held and processed as evidence. I spent an hour in the lab, watching them go through every page. When they got to the end and I had to see the clipping from the Detroit newspaper again—from the night I watched my partner die—that was when I'd had enough.
I left the place without bothering to tell anyone, went outside, and walked down the street. The sun was down, the air cooler. It was the first time I'd been truly alone since I'd arrived in Arizona, and somebody was probably already looking for me, but I didn't care anymore.
The FBI office was right next to the little regional airport on the north side of town, so I went into the rental car office and drove off the lot in the car I'd been threatening to rent all day. I made my way across town for a while, not even sure where I was going. Until I saw the gun shop.
It was lit up bright and cheery and it occupied half a city block, as only an Arizona gun shop could. There were full-color pictures of guns showcased in the windows as if they were fine pieces of jewelry, everything from handguns to assault rifles. I went inside and saw enough firepower to arm a small country, in glass cases that ran along three sides of the store, with more guns hanging on the walls.
I had carried a gun for eight years as a cop, only tried to fire it once—and that was the night I took it out a beat too slow. I had bought another gun when I got my PI license, but ended up throwing that into Lake Superior. I had sworn that day I'd never hold a gun again.
But now, as I picked up a Glock 22, the last weapon I had carried in Detroit, I sighted down the barrel and imagined Martin T. Livermore standing on the other end of it.
_I would kill you right now if I could. I would pull this trigger without a second thought._
I was about to fill out my ATF form 4473 and wait for the shop owner to run me through the NICS database, but when he asked me for my state of residence, I had to tell him Michigan.
"Gotta be an Arizona resident to purchase a handgun here," he said. I told him I was an ex-cop. He still couldn't sell me the gun, but he did direct me to a local gun owner a mile down the road who could help me out. A half hour later, I had a similar model Glock and a box of .40 caliber ammunition.
I bought some clothes and a bag to carry everything in, put it all in the trunk of the rental car, and drove back to the FBI office. Larkin seemed to be the only agent who'd even noticed that I had left. He didn't ask where I had gone.
He had a copy of the scrapbook pages to give me, and I spent the next couple of hours going through them. Every photograph, every article. If my own mother had lived past my eighth birthday, she couldn't have kept a more complete record of my life.
_But why?_
I was still no closer to the answer.
* * *
—
AFTER I DROVE BACK to my hotel room, I took apart the gun, cleaned and oiled it, put it back together again. Then I loaded it. An old ritual from another lifetime, but it made me feel better just having it in my hand.
His words came back to me, what he'd said to me in that van . . .
_Do you think this ends today?_
_No,_ I thought as I sat there in a hotel room with a freshly loaded Glock in my right hand. _It's not over._
I put the gun in the bedside table drawer, next to the Gideon Bible. Then I spread the scrapbook pages out on the bed and looked at them again. I might have slept for a while. If I did, it was a shallow sleep that didn't begin to ease my exhaustion. Then at some point in the middle of the night, I felt my phone vibrate. I turned on the light and looked at it. I had a new email message.
I shook myself fully awake and opened the email. It was from an address I didn't recognize—in fact, it looked like nothing more than a random string of letters and numbers. I opened the video that was attached to the email. It was hard to make out what I was seeing at first. Just a blurry, cheap video taken on somebody's cell phone, but then the image stabilized and I saw a woman's face. Young, maybe late twenties, with a nose ring, and even with the washed color on the video I could see that her hair had been dyed a bright shade of red. She was lying down on something . . .
A table. Something about her arms didn't look right. Like they were twisted behind her. I stayed right there on the edge of the bed as I watched her, my heart pounding.
"Say it," a voice said. A man's voice. A voice I knew.
Livermore.
"Say exactly what I told you to say."
The woman started crying. A shudder ran through her body. I could only imagine what was going through her mind. The fear, the pain, all of it at once.
"Alex," she said, choking back sobs. "Alex McKnight. You have to come save me. Please. It's all up to you. Please hurry. Please . . ."
She started crying again. Then her eyes went wide as duct tape was pulled tight across her mouth. The video stayed focused on the woman's face, but it was Livermore who spoke next. "Come alone," he said. "If I see anyone else . . . _anyone_ . . . she dies."
"Where are you?" I said to the video. _"Where the fuck are you?"_
"This is between you and me," he went on. "If you tell anyone else, I'll know. And she will suffer more than you can imagine."
I kept watching and waiting, holding the phone tight in my hand.
"Start driving to Albuquerque," he said. "Right now."
Then the video ended.
* * *
—
I WAS DRESSED within two minutes, in my rental car and on the road within three. It was just past four in the morning, the roads almost empty. I headed north, trying not to drive recklessly, trying to keep myself steady. It was a long way to Albuquerque.
_He sends everyone else to San Francisco,_ I thought. _Then he brings me east. I shouldn't be surprised._
_It really was a diversion all along._
As I drove up through northern Arizona, into the higher elevation, the air got colder and condensed against my windshield. I started to see traces of snow. By the time I hit Flagstaff, there were piles of it on either side of the road. I stopped and got some gas, breathing in the cold air. The sun was coming up, casting a thin light that didn't even begin to warm me.
_You have to stay alive,_ I said to the woman in the video. _Whoever you are, you have to stay alive._
I hadn't heard from Livermore again. I just had to keep going. It was the only thing I could even think of doing. For hours.
And hours.
I was in New Mexico, driving east, pushing the rental car hard, doing ninety in the left lane. As I got close to Albuquerque, my phone vibrated and I pulled over to see what had landed in my email.
It was another video. The same woman on the same table. She was looking up at me, her eyes glazed. Livermore's voice came from off camera again.
"Did I say Albuquerque? My mistake. I meant Amarillo."
The video ended.
I got back on the road, kept driving around the city, more hours until I was in the flat red dirt lands that told me I was getting close to Texas.
_It's been eight hours,_ I told myself. _Eight hours taped up like that. No human being could endure it._
But I kept going. Fighting through my own exhaustion, the road starting to go double in my eyes as I crossed the Texas border. It was just after noon now, the low February sun making my head hurt. I felt the phone vibrate in my pocket again, almost drove off the road trying to get to it. I pulled over and opened up the next email.
The woman's eyes were open, but she didn't move. I wasn't even sure if she was still breathing, but then she blinked.
"I hope you're not too late, Alex." Livermore's voice. I had a sudden vision of my hands around his throat, of me choking the life out of him so that nobody would ever have to hear that voice again.
"Here is the address," Livermore said. He said a number, a street, and an apartment number. I grabbed a pen from my bag and wrote it down on the back of my rental receipt. Then I plugged the address into my phone. I was still twenty miles away.
I left the expressway and made my way through town, second-guessing myself the whole way. Maybe I should have called somebody, even though I didn't have an address until just now. But the police could have been out searching for her. Or now that I _do_ have the address . . . I should call them _now_.
I kept up the argument with myself until I was close enough to the address to put it aside. _Time to think,_ I said to myself. _Time to be sharp._
_You're here._
I parked next to the building, took the gun out of the glove compartment, and went inside. It was three stories high, maybe seven or eight apartments per floor. There was no way to look through a window. I went up the stairs and found apartment 301, stood outside the door for a moment.
_Now what?_
I pictured him on the other side of the door, waiting for me. _If I open it, I'm walking right into a trap._
_No, it's your only chance to surprise him._
I tried to put myself in his place, imagine what I'd do . . .
I put my hand on the doorknob and turned it slowly. It was unlocked.
_No, you can't do it this way._
I took my hand away from the door, waited another few seconds. Listening. Then I knocked.
"Livermore!" I said. "I'm here!"
I moved back to the stairwell, put my back against the wall, and watched the door.
Nothing happened.
I moved back slowly, careful not to make any noise. This time I put my ear against the door. I heard a sound on the other side. It was faint. Muffled.
_She's still alive._
Her face flashed through my mind, the face of that woman, taped down to the table. _Open the door,_ I thought. _But then step back, wait for him to move, or shoot, or whatever the hell he's going to do._
I took a breath, lined myself up, and then kicked the door, just above the knob. It was cheap wood, both the frame and the door itself, so it flew open as I fell backward. I came back up with the gun drawn, just long enough to smell the gasoline. _No, not gasoline, something else._ There was a loud whoosh, and then a great wave of heat blew me back to the floor. From one second to the next, the doorway had turned into an inferno, the black smoke already billowing out into the hallway. I tried to wave it away, tried to take a step into the apartment, but I felt the flames on my shoes and on my pants and in another instant it felt like I was on fire myself. Then a strong pair of hands was grabbing me from behind and pulling me away from the door, just as I heard the scream coming from inside. It lasted for only a second, but it was a scream I'd hear for the rest of my life, as the hands pulled me back down the hallway with the fire trailing after me, as if chasing me.
I pushed the man away and tried to go back, but another blast of heat hit me square in the face, singeing my hair as the flames kept burning on the floor below me.
"Alex," a voice said. "You're on fire!"
I turned and saw Agent Larkin's face, didn't even have time to think about what he was doing here as he tackled me and put his jacket over my legs. The jacket caught on fire and he threw it away, then he grabbed me by the shoulders again and dragged me to the stairwell.
"It's too late!" he said, just as the sprinklers came on above us and the fire alarm started to ring. Other doors opened, up and down the hallway, as Larkin half dragged, half threw me down the stairs.
I knew he was right. I knew it was too late. And I knew something else, something that would stay with me forever. One terrible fact that I'd never be able to erase or explain away.
Whoever that woman was, Livermore had just made me burn her alive.
# CHAPTER NINETEEN
AS LIVERMORE watched Alex go into the building, he felt something that no drug could ever give him. Something like the way tying a perfect knot against white skin would feel, or drawing a perfect _kanji_ on white parchment. The perfect execution of a perfect idea. He could see the compound symbol for _kanpeki_ in his mind, meaning something so much more profound than any English word could capture. The symbol for _complete_ above the symbol for _jade_.
A perfect jewel.
That's what this moment was.
He pictured Alex going up the stairs. Going to the door marked 301. The hesitation he must have felt at that moment, wondering what to do next. Livermore wished he could see it. Alex weighing his options, not even knowing that no matter what he did, it would be wrong.
He had taken five gallon jars of the napalm he'd made in his laboratory. Styrofoam dissolved into gasoline, stirred with great patience until it turned into jelly. When lit, napalm is virtually inextinguishable. It is liquid fire, clinging to whatever it touches. Spread across the floor of that woman's apartment, with a simple electronic igniter wired to the door. A Christmas tree bulb with the tip removed, a small amount of the napalm put inside, to come in contact with the filament. Two nine-volt batteries. Such a simple device, not nearly as complicated as his explosives in the canyon. But safe, stable, foolproof. There was no danger of the napalm igniting until the exact moment when the door was pushed open.
Now as he waited for the second half of the show, Livermore got out of his vehicle and took a red metal canister with him to the side of the building. Straight gasoline this time. He found a good spot on the grass and carefully drew his design. The grass was already brown in the Texas winter, but he knew there'd be no misreading the char pattern. When he was done, he struck a match and watched the flames come to life.
He set down the gasoline container and went back to his vehicle. He didn't bother wiping down the fingerprints. He was past that now.
There was nothing to hide anymore. They would all know that Martin T. Livermore had been here.
A black sedan pulled up to the building, just as Livermore was about to turn the corner. He stayed on the other side of the building and watched the man get out and run inside. It was the young agent he had seen in Phoenix.
_I told you to come alone, Alex. I set up everything so carefully, created my little sleight-of-hand diversion just to make sure everyone else was looking in another direction . . ._
_Just to keep this between us, Alex. Like the great samurai Musashi's duel with Sasaki on the island of Funajima, how Musashi waited until everything had turned in his favor—the tides, the angle of the sun, even the rising impatience of Sasaki and how that drained his energy—until the moment came for Musashi to face him._
_That moment is coming, Alex._
_It is coming._
Livermore waited another beat, then he went around the building to McKnight's car and slipped the magnetic case containing the GPS tracker above the right rear wheel well. He didn't even have to stop, just a quick movement of the hand as he walked by, feeling the magnet come in contact with the metal, even as the first person came running out of the building, then the second, then a great tidal wave as every resident streamed out of every door. Nobody noticed the stranger walking back to his vehicle.
He watched the first fire truck arrive. The first police car. Then finally Alex McKnight himself, being pulled from the building by the young agent. Livermore took out his Nikon binoculars from the black case and focused in on Alex. The young agent was holding him by both arms, talking to him. As Alex broke away from him, the look of anguish on his face was another moment of _kanpeki_.
Livermore tilted the binoculars down to see the burnt material on Alex's pant legs, where the napalm had stuck to him. His shoes were still smoking.
The agent approached him again as Alex stood half bent over, like he was about to throw up. Larkin put a hand on his back, and Alex straightened and then walked away. That was how he ended up at the other side of the building.
One of the apartment windows blew out, and the smoke billowed out into the sky, as black as death. Firemen dragged one hose into the building while another hose was trained at the window from outside.
But Livermore saw none of it. He kept his eyes on Alex McKnight, even as he put his vehicle in gear and started to drive away. One last moment of _kanpeki_ as he watched Alex standing over the message he had left for him, _just for him_ , burned into the dead grass.
# CHAPTER TWENTY
WHATEVER MADNESS had driven Livermore to coat that apartment, _to coat the woman herself_ , with _napalm_ . . . it was the same madness that had driven him to create this line in the grass, ten feet long, with two shorter lines joining it at the point.
An arrow.
Drawn with great care, the lines smooth and even and straight. Lines drawn by an engineer.
_Lines drawn for me._
I could still feel the heat on my legs, where the napalm had stuck to my pants and to my shoes. I could still hear that woman's scream in my head as I played it back, over and over, trying to make it come out differently. If I had known somehow, if I had been more careful, or quicker, or slower, or God knows what. In the background, there were windows blowing out, firemen yelling at one another and running hoses inside and out of the building. But I didn't hear any of it.
I heard only that one last scream.
As I looked back down at the arrow, I followed its line across the street that ran behind the apartment building, directly to an office building. Even now the building was being cleared out by the FBI, working with Amarillo PD and the Texas DPS, and they were searching it from top to bottom.
My gut told me they would find nothing. This arrow was pointing at something else, beyond the building.
_But what?_
Maybe he was pointing it at himself somehow, wherever he would be next, and that was the only thought I could hold on to at that moment.
_Tell me exactly where to_ _go, you sick fuck. Bring me right to you. I_ _don't care if you kill me. As long as you go down with me._
"You all right?"
I turned to see Agent Larkin standing behind me.
"What are you doing here?" I said.
"You shouldn't have come here alone, Alex."
"What do you want?"
He looked down at the plastic evidence bag he was holding. "You need to see this."
He came closer and showed it to me. It was an eight-by-ten photograph.
Of me.
"There was a fireproof safe in the apartment," he said. "This was inside. Looks like it was taken yesterday. That's the roof of my car."
I looked at the tired face in the photograph. The condos behind me, just as I was about to get back into the car with Larkin. He was right, it was from yesterday, when we were out retracing Livermore's crime scenes.
It didn't surprise me that he would have followed us, that he would have taken this photograph. That he would bring it with him and leave it here next to the woman.
I was beyond surprise by now.
I turned it over and looked at the back. Livermore had drawn an elaborate Japanese symbol, like the symbols I had seen in that storage shed.
"What does this mean?" I said.
"We don't know yet. I'm taking it to the Amarillo FBI unit."
I nodded and handed it back to him. This young agent who must have stayed up all night, keeping watch at the hotel in Phoenix, then driving all day to follow me here. To a burning doorway. Where he had probably saved my life.
"I'm taking you, too."
* * *
—
I SPENT ANOTHER two hours sitting in another interview room. This one was a lot smaller, with no windows. I felt the walls closing in on me as yet another FBI special agent in another dark suit asked me the same questions and I gave him the same answers.
_I have no connection to this woman._
_I have no connection to Martin T. Livermore._
_I have no idea where he'll go next._
"They've got the state police on every road leading out of town," Larkin said to me when the interview was over. "They're knocking on every door in the neighborhood, looking for a witness. If we can get a description of his vehicle, at least . . ."
"Let me see a map," I said.
He showed me a map of northern Texas. There were six major roads leading out of Amarillo, the land crisscrossed with two dozen smaller roads, heading in all directions.
"You guys still think he's on his way to San Francisco?" I said.
"We can't rule it out, Alex. Maybe that's still the real plan and _this_ is the diversion."
I looked at him. "You don't believe that."
He didn't answer me. So I just got up and left the room. Larkin caught up to me in the parking lot. The sun was down, the air turning cold again.
"Where are you going?" he said.
"I don't know."
"I've been told to bring you back tomorrow. It's the only reason you're not in custody right now."
"You said you were going to help me."
"I'm trying to, Alex. But you're putting me in a tough spot."
"I need a drink."
We went into the first bar we found. We sat at the rail, and he watched me down one shot of Wild Turkey while he nursed a beer. I knew a second shot would lead to a third and a fourth. It was tempting, because I wanted to be numb. But I knew I needed to stay sharp, so I turned my glass over.
When Larkin's cell phone rang, he took it out of his pocket and looked at me while he talked to whoever was on the other end. Probably Agent Madison.
"I'm with him right now," he said. "Yes, we will. Tomorrow morning."
Someone turned on the jukebox, and he had to cover his other ear.
"It means _what_?"
He got up and went outside to finish the call. While he was gone, I sat there thinking about Livermore, wondering if he was still here in town. Maybe even taking more photographs.
_Come find me,_ I said to him. _You want me so bad, I'm right here._
Larkin came back inside and sat down next to me.
"The symbol on the back of the photograph," he said after a few long seconds of hesitation. "It's Japanese. It means _failure_."
I nodded, staring down at my overturned shot glass.
"He's just trying to fuck with your head," Larkin said. "You know that."
I picked up the glass, weighing it in my hand.
"Look," Larkin said, leaning in close to my right ear. "You were a cop before I was even born, so what the fuck am I even going to say to you? That you're not responsible for that woman's death? _You already know that, too._ "
I didn't try to answer.
"You come back with me, Alex, and we all work together. We catch him. End of story."
I turned and looked at him, and that was all it took for him to shut his mouth for a while. There were two seconds of silence, then the jukebox cranked out another bad country song, even louder than the first. Someone came up behind me and leaned over the bar to order a drink.
"I know that smell," he said with a heavy Texas accent, saturated in alcohol. "You're a nozzle jockey."
I ignored him.
"Y'all must'a just rolled," he said. "Smells like the Old Testament in here."
"If you don't mind, sir," Larkin said to him. "We're not firemen."
He looked back and forth between us, and then down at my burnt shoes and pant legs. He swayed back and forth for a moment, then caught his balance by grabbing me by the shoulder.
"Hank," the bartender said to him, with an accent just as heavy, if less soaked with cheap whiskey. "Leave 'em be."
"I'm just trying to understand," he said, his hand still on my shoulder. "One of 'em's half burned up, and they say they're not firemen. So what the hell were you guys doing?"
"None of your business," I said without looking at him. "And take your hand off my shoulder."
"Alex," Larkin said, "take it easy."
"It's okay." The man lifted his hand off my shoulder and then gave it a quick flick of his fingers like he was brushing away the dirt from home plate. "You boys ain't from around here, are ya . . ."
"Do you have a problem with that?"
"Hank," the bartender said again. "Go sit down, all right?"
"I got no problem," he said, putting a hand on the bar to steady himself now. "We're a friendly bunch around here. We like everybody. Even people who got no business being here."
I closed my eyes and waited for him to say something. Just one more sentence.
"I just wanna know why I wasn't invited to the barbecue."
I came off the barstool and grabbed him with both hands, driving him backward toward the jukebox, still playing that same bad song too loud, until I put his back against the glass and the whole thing rocked for a moment and then went silent.
I felt Larkin pulling me away from him, then he and the bartender both pushing me outside. The man trailed behind and stood in the doorway, an easy place to be brave with two other men between him and me.
"That's right," he said, "why don't you boys go back where you came from!"
I pushed the two men away from me and went to my car.
"Alex, wait."
He grabbed me from behind, just as I was opening my car door. I spun around to face him, and as I did I caught sight of the man still standing in the doorway, looking like he'd just won some sort of victory.
That's when it came to me.
_The arrow._
_I know what it means._
* * *
—
IT WAS FIVE IN THE MORNING when Larkin found me outside the motel, putting my bag in the backseat of my car.
"Going somewhere?"
"First rule of tracking a fugitive," I said. "They always go home."
"Livermore lives in Phoenix," Larkin said, his arms folded against his chest.
"Before that."
"California."
"No," I said. _"Home."_
"That arrow was pointing northeast . . ."
"Columbus, Ohio."
I closed the door and stood there, waiting for him to try to stop me.
"You can't go there," he said. "I told you, I have orders to—"
"I slipped out when you weren't looking. Nothing you could have done."
He looked at me. "Why would I let you do that?"
"Because you know I'm right. And because you're not an asshole."
"I'll call Columbus," he said. "We'll watch for him."
"I thought all your agents will be in San Francisco."
"I'll make this happen, Alex. Someone will be watching his house."
"Go ahead. I'll be there, too."
He stopped me before I could get into the car.
"Here's where I either put you in handcuffs," he said, "or risk my whole career and let you go after him."
"Once again," I told him, "it's time to make your choice."
* * *
—
A FEW MINUTES LATER, I hit the roadblock they'd set up on I-40. I waited in the line, wondering how busy it would get once the sun came up. Finally, I pulled up to the trooper standing there with a shotgun and waited for him to take a quick look inside my car. I was a male, around the right age, and traveling alone, so he checked my license. Then I was on my way again.
_Livermore can beat that,_ I said to myself. _I know it._
I passed through Springfield, Missouri, and then a long stretch through the middle of the state as the sun went down on another short February day. More hours of driving, after so many hours already spent chasing Livermore. There was nothing but the open road now, and exit signs leading to small towns I'd never heard of. I knew Livermore might be almost as exhausted as I was at that point, and I didn't think he'd try to make it all the way to Columbus. I tried to imagine him coming down this very same road, deciding where to stop for the night. I couldn't see him taking one of these exits yet. My gut told me he'd go all the way to St. Louis before stopping.
When I got to the city, I got off the expressway and drove down the dark empty streets, finally ending up over by Union Station, just on the edge of downtown. When I was finally out of the car, stretching my legs, the air was cold enough for me to see my own breath.
Larkin pulled in right behind me. I hadn't seen him in hours.
"What did your boss say?"
"He's not happy. I'm still on your trail. I won't go back until I find you."
"Tell him you lost me. You don't have to do this."
"I'm here," he said. "Let's get something to eat."
We checked in to our hotel, came back downstairs, and walked down Market Street. We didn't see anyone else, aside from a few homeless people stirring in a little park across the street. It was one of those cities that brought people downtown to work during the day, and then sent them home at night, leaving nothing but dark buildings and a single lamppost burning on every corner.
I kept looking around me, like I expected to see him.
We found the one sign of life down the street, a restaurant that probably relied on the hotel for all of its business. Larkin set up his laptop on a table and checked in with the office. I sat there and thought about how far I was from home. How I should have been sitting by the fire at the Glasgow Inn, having some of Jackie's beef stew.
"Amarillo developed some more information," he said. "Found out that Irene Murphy sent a text to a friend just before leaving the store last night. She was disturbed by an encounter she had with a customer."
"Livermore."
He nodded. "That picture he left in the safe, looks like he had it printed there. We can't see any other reason why he would have chosen her. Wrong place, wrong time."
There wasn't much else to say. I shook my head and looked up at the three big HD screens over the bar, each tuned to a different sport that nobody in the place seemed to be watching. Until the one screen in the middle broke into some kind of news report and suddenly Livermore's face was looking down at me.
The sound was off, but the caption read MANHUNT CONTINUES FOR SUSPECTED SERIAL KILLER.
Larkin and I both looked up at the mug shot, at the face of the man as he seemed to look back at us, with that half smile.
"He could walk in here right now," Larkin said. "Think anyone would recognize him?"
"That video he sent me," I said. "I didn't see his face, but I'm sure I'd know him in a second, no matter what he's done to himself."
Larkin nodded. There was nothing else to say. We finished eating, paid our tabs, and left.
"You got a family?" I said as we started walking back to the hotel. I kept looking around me, still expecting to see Livermore right behind us.
"Not yet. Just got engaged. You?"
I shook my head.
"Why the Bureau?" I asked him, not bothering with the follow-up: _Why would you want to be hated by so many other cops?_ I'd worked with plenty of agents back when I was a cop in Detroit, and I knew what my fellow officers thought of them. I remembered one informant who got killed because a feeb made him go back into a drug house to get a refund on some drugs that turned out to be fake. The feeb didn't want to go back and tell his boss the money was gone. It was something that had happened in another time, in another city. But I knew it was still true. You become a feeb, you set yourself apart.
"It was always hockey or the FBI," he said with a shrug. "Ever since I was a kid."
The former train station loomed high above us as we kept walking. It was something from another era, a full city block of ornate architecture, limestone with a red roof, and dominated by the clock tower at one corner. The streets seemed even emptier now. The homeless men in the park were either gone or wrapped up tight against the cold, huddled around small fires.
I said good night to him. When I was back in my room, I looked through the scrapbook pages again. I took the gun from my bag and set it on the bedside table. I imagined Livermore sleeping in another hotel room, maybe in the same city.
Or maybe he wasn't sleeping at all.
I opened up the window. The night air came into the room, mixed with the faint smell of smoke from the homeless men's fires still burning across the street.
I hadn't slept in so long. The smoke stayed with me as I finally drifted off, turning into a raging fire that burned inside my head as the woman trapped inside the flames kept calling my name.
# CHAPTER TWENTY-ONE
WHEN A MAN _tries to get into your head, he leaves the door to his own open._
The thought lingered in Livermore's mind as he sat in the line leading up to the roadblock, using the opportunity to watch the progress made by a little orange blip on his laptop.
_You think you know where I'm going,_ he thought. _You might even try to guess where I will spend the night._
_The best part of that is, the absolute genius of that idea . . . is that wherever you guess . . ._
_You will be right._
When he pulled up to the officer with the shotgun, Livermore couldn't help but flash back to the canyon, taking a similar weapon off that Arizona cop and using it to shoot that agent. The feel of it in his hands, the lethal power capable of not just killing but destroying. Of _deconstructing_. He'd only fired small arms before that, but how many hours had passed since the canyon and he could still feel the almost sensual tingling of that shotgun's recoil in his fingers. He put the thought out of his mind as he rolled down his window.
_I'm an airplane pilot,_ he told himself, _from Wayzata, Minnesota, just outside of Minneapolis. I fly 727s exclusively. This is my brother-in-law's car, which is why it has Arizona plates. He's letting me use it to drive up to the Grand Canyon. My wife is not with me because we're currently separated, but I have every hope that we'll be back together someday soon._
It was all necessary, every detail of his story. He needed to become that other person completely, even if it was just for one minute.
"License, please," the officer said as he gave the whole vehicle a quick scan.
"What's going on?" Livermore said.
"Looking for a fugitive." He was studying the license, looking back and forth between Livermore and the photo.
Livermore waited.
"Minnesota," the officer said. "Long way from home."
"Yes."
"You have a good trip back," he said, handing Livermore the license.
"Thank you. I hope you catch your man."
Livermore pulled away from the roadblock, speeding up smoothly as he hit the open road ahead of him. He tucked the license back in his shirt pocket.
He'd used Minnesota's non-enhanced license template, one of the last few that didn't comply with the new federal standards. Editing the image, adding the barcode and the encoded magnetic strip, printing it on Teslin synthetic paper, then laminating it. Twenty dollars and maybe two hours of his time. All well spent.
He settled in for the long drive across Oklahoma and Missouri. When he got close to St. Louis, he found a hardware store that was still open and bought his supplies. He checked his laptop again. The orange GPS dot was just ahead of him. He watched as it left the expressway and made a few turns downtown. Then it stopped.
Livermore started moving again. When he pulled up on Market, he saw two men walking down the dark street. From behind, it looked like Alex and the young agent.
_Let him come along for the ride,_ he thought. _It's all the same to me now._
He watched them go into a bar at the end of the block, then waited until they came back out, maybe forty-five minutes later. He watched them go back to the hotel, and then started counting down in his head as soon as they walked through the front door.
_Cross the lobby. Get in the elevator. Get out of the elevator and walk down the hall._
When the lights came on in two adjacent windows on the fifth floor, he waited another minute, until the curtains opened on the window to the left. He caught a quick glimpse of the young agent. That meant Alex was in the other room.
Fifth floor. Fourth room from the end.
He took his bag from the vehicle and locked it. He was about to cross the street, then stopped himself when he saw another car pull up to the front. A man and a woman got out, taking out two suitcases and putting them on the sidewalk. The man left the woman there, got back in the car, and drove away. Going off to park the car, but the woman was obviously uncomfortable standing out on the empty sidewalk in the darkness. Livermore stayed hidden behind his vehicle, watching her. He wondered why she didn't just go inside and wait there. Just because she didn't want to leave the two suitcases outside? Were the contents of those bags more valuable to her than her own life?
_I could pull up next to her right now._ _Pretend I'm another guest checking in, give her a smile, take a quick look up and down the street . . ._
He felt a warmth spreading through his body as he saw it in his head, every movement perfectly choreographed, as if it was written down on a script.
_Push her into the backseat, drive off before she can make a sound. The man would come around the corner a minute later, wondering where his wife had gone._
_But at least the suitcases would be safe._
He waited another minute, watching the woman shivering on the sidewalk, until the man came back from parking the car around the corner. The woman had something to say to the man, and they stood out there talking about it for a few seconds. The argument ended, and he hugged her. Then they went into the lobby.
Livermore crossed the street again, carrying his bag. There was one man behind the check-in desk. He was talking to the couple. Livermore put his head down and opened the door. He walked through the lobby without so much as a sideways glance. A man who belonged there. He headed right for the door marked STAIRS, opened it, stepped through, and went up the stairway.
When he got to the sixth floor, he counted down four rooms. He knocked softly on the door, then waited a few seconds, listening carefully. Then he took a quick look down the hallway in both directions, got down on one knee, and opened up his bag. Purchase number one at the hardware store was a long, thin strip of metal, eight feet long and bent into quarters. Livermore unfolded the strip and then refolded one end into a hook, about six inches in length. Then he placed the strip against the door, noting the exact distance between the floor and the door latch. He made another fold, then another to accommodate the thickness of the door. Then finally one more hook to use as a handle.
He worked the strip under the door, careful to preserve the folds, then turned it so that the hook inside the door would be extended upward. He worked the strip sideways, moving carefully until he felt it make contact with the door latch. He maintained the tension as he pulled down . . . a little more . . . a little more . . .
Until he heard the lock open.
He pushed the door and stepped inside.
The door wasn't stopped by the safety latch. If it had been, he would have had to refold the strip, running it higher up the door to push the latch open first. He turned on the light to make sure the room was unoccupied. The bed was made. There were no suitcases in the room.
Perfect.
Livermore put his bag down on the bed. Then he went back out into the hallway, careful to leave the door ajar, and walked to the stairwell door, then down the stairs to the ground floor. As he looked through the little window in the door he saw that the clerk had been replaced by a young woman. Maybe the clerk was helping the couple bring their suitcases up to their room. He didn't know. It didn't matter. He paused just long enough to put a smile on his face.
_I'm the airplane pilot from Minnesota,_ he told himself. _Only now I'm on a layover. I fly for Southwest Airlines. I stay at a lot of hotels._
He opened the door and went into the lobby. The woman behind the desk looked up at him and smiled right back.
"How are you tonight, sir?" She spoke with a slight Latin accent, and she looked a little heavier now that he was closer to her. But he liked the way her dark hair fell to her shoulders.
"Happy to be on the ground," he said, taking a quick look around the lobby. They were alone.
"We get a lot of pilots here," she said, nodding her head. "Must be doing something right."
He smiled again and waited a beat. She was very good at her job, Livermore could see. Very good at making someone feel like she would do anything to make sure you had a wonderful stay in her hotel.
"I hope everything's satisfactory with your room," she said. "Can I get you anything else?"
"Some extra towels would be nice. Room 604."
"Yes, of course," she said. "If you'll give me a moment, I'll bring those right up."
"No rush," he said with another smile. Then he went back to the stairwell.
"Sir," she said from behind him. "The elevator . . ."
He turned to look at her.
"I know it's slow," she said, shrugging her shoulders like _What are you gonna do?_ "But it's coming back down now."
Livermore looked at the lighted number above the elevator, as the _3_ turned into a _2_. The door would open and the other clerk would step out, the man who'd taken the suitcases up to the couple's room.
"I like the stairs," Livermore said. "Gets your blood flowing."
He smiled one more time and went through the door, just as he heard the elevator open behind him. He went back upstairs and into room 604. He had just enough time to get up onto the bed and tap the ceiling, to find where the joist ran above the plaster. He took out the cordless drill he had bought at the hardware store, and one of the two bits. After drilling the pilot hole into the ceiling, through the plaster and into the wood, he took the large metal hook and screwed that into the pilot hole. He brushed the sawdust from the bedcover. Then he took the twenty feet of grade-30 galvanized anchor chain he had bought and hung that from the hook. He pushed the bed away from the center of the room and tested the hook with his own weight.
It wouldn't be a work of art, but just like in Amarillo, he didn't want to waste his ropes here, didn't want to have to leave them behind when he left. And this chain would be much quicker, too.
He took out the last hardware store purchase from his bag. Another drill bit, this one bigger and almost comically long. It was used by electricians to drill holes all the way through thick walls, but he knew it would work here.
It would work exactly as he wanted.
He put it down on the bed, just as he heard a knock. He opened the door and saw the same woman from downstairs standing outside, holding a small pile of towels.
"Is this the right room, sir? I didn't see a record of—"
She stopped when she saw the chain hanging from the hook in the ceiling.
Another long moment passed as her eyes went from the ceiling to Livermore's face. She dropped the towels, caught in that one long moment, torn between screaming and trying to run away.
"Yes," Livermore said calmly as he grabbed her by the wrist and pulled her inside. "This is the right room."
# CHAPTER TWENTY-TWO
A BELL RANG IN THE DARKNESS.
A fire alarm. An air raid siren. A sound as loud and as jarring as that first explosion in the canyon.
I opened my eyes. It was dark.
It rang again.
This time, I pushed myself up from the bed. I didn't even know where I was for a moment, until I finally put it all back together.
_I'm in St. Louis. I'm in a hotel._
My curtains were closed, with only the thinnest beam of light coming into the room between them. I'd been so tired, after I don't even know how many nights of not sleeping. I had collapsed on this bed, and now . . .
The phone rang again. I needed to either smash it against the wall or answer it.
I answered it.
"How many more women are you going to let die, Alex?"
That voice.
"Where are you?" I said.
"Right above you."
"What are you talking about?"
The line went dead. I fumbled with the switch on the lamp, almost knocked the damned thing over, got it upright again, finally turned it on. I blinked in the sudden light, everything a blur until my eyes focused and I saw what was on the bed.
Red.
Bright red.
Against the white bedcover, a great scarlet stain, all around me. No, it was _on_ me, too. On my pants, on my shirt. That familiar coppery smell . . .
_Blood._
My first animal reaction was that the blood must be mine. _I'm shot. I'm cut. I'm bleeding._
Then the next thought: _No, it's not my blood._
Then the next, as I looked around the empty room, then finally up at the ceiling.
_A hole._
The rim of the hole . . . it was red. Another drop collected and fell onto my face.
_There was blood coming from this hole in the ceiling._
I grabbed the gun off the nightstand and threw open the door to my room, ran down the empty hallway to the stairwell. I went up one flight, paused for one half second as I checked the sixth-floor hallway. It was empty.
I ran to the room above mine, 604, and kicked open the door, my gun drawn. I could barely process what I saw next, the woman wrapped up and hanging by a chain from the ceiling. She slowly turned toward me, as if to greet me, and I saw the jagged line across her throat, the two streams of blood running down either side of her face, into her hair, and then one final stream dripping down onto the floor.
I went to her and was about to put two fingers to her throat to check her pulse. Pure muscle memory in the face of madness, until I saw her lifeless eyes and realized that most of the blood from her body was either on the floor or on my bed. Or on me. I wheeled around with the gun still extended, made sure there was nobody waiting for me in the blind spot behind the door or hiding in the bathroom.
The phone rang. I picked it up.
"I'm outside, Alex."
I left the room, went down the hallway to the stairwell, and pounded my way down all six floors to the ground level. When I opened the door, the lobby was empty. I had no idea where the clerk was, until I saw the lighted panel above the elevator. He was on his way upstairs. God help the man if he walked into room 604, but I couldn't stop him now.
I stepped outside, into the cold air. The old train station loomed behind me in the darkness. I didn't see anyone, in any direction.
I pictured my cell phone, sitting up on the desk in my room, plugged in and charging. But I wasn't about to go back for it. I wasn't about to do anything except keep moving forward, keep trying to find him. I went up to Market Street and looked west. It was just one lonely streetlamp after another, as far as I could see. When I looked east, toward the river, I saw more streetlamps, more darkness, more emptiness. But then something else, in the park across the street. A movement.
_It could be one of the homeless men,_ I thought. _Or it could be_ him _._
I ran across the street, into the park. It was two city blocks long, a great tree-lined rectangle, with a single pathway crossing through the middle. I could make out a number of fires spread out along the perimeter. Homeless men in small groups, huddled against the cold.
I stayed on the tree line, going from one tree to the next, looking into the interior of the park. I came upon three homeless men all wrapped up in blankets and gathered around a fire in a small metal bucket with holes poked into the sides. I scanned their faces.
"Did you see someone come this way?" I said.
"That's blood!" one of them said, and as I looked down at myself I realized my clothing was still soaked and the blood was splattered all over my arms and probably my face. I didn't have a coat on. I was stumbling around in the cold darkness, looking like I'd either been stabbed half to death or had done the stabbing myself.
The men all ran away from me, scattering in every direction. As I followed the progress of the man who'd pointed me out, I looked past him and saw, a block down, a silhouette standing under a streetlamp. Leaning against the pole, as if waiting for me. As soon as I started running, the figure vanished.
_It's suicide,_ said a small voice in my head as I got closer to the streetlamp. _You're running into the light, and you might as well paint a target on your chest._ But I was past reason now. Past all of the training I'd received as a cop, past any amount of common sense I'd ever had. It was only the madness now, everything I'd seen, the images coming back to me one after another as I ran. Agents blown apart in a canyon, the shotgun wound in Agent Cook's neck, the blood pumping from a hundred holes in Agent Halliday's chest, his face looking up at me as he asked me to send one last message to his daughter and grandson.
A woman taped up and left to die. Left to burn.
Another woman hung from the ceiling by a chain, bled like an animal on a killing floor.
I made it to the streetlamp, stopped in the light for a moment, and put one hand against the post to catch my breath. My wrapped-up left knee was throbbing.
_"Where are you?"_ I yelled into the night as soon as I had my wind back.
I saw him another block down Market Street, standing under another streetlamp, this one at the base of the steps leading up to a huge courthouse. Behind him, far down the street but lit up and positioned perfectly over his head like a frame, was the Gateway Arch.
I ran down the street toward the courthouse, but by the time I got there, he was gone again, and I thought I saw him standing at the very top of the steps, between two of the great columns. I took the steps two at a time, still holding the gun in my right hand, forgetting everything I'd ever learned about trigger discipline as I tripped and almost squeezed off a shot. I pushed myself back up and kept climbing, limping now, until I was at the very top. He wasn't there. I tried pulling on the great glass doors leading into the courthouse, but they were locked tight.
I turned around and looked everywhere, at each cone of light under each streetlamp. Then I heard the crack of a gunshot and one of the glass doors behind me shattered. I ducked down behind a concrete rampart.
"I'm right here, Alex!" A voice coming from the darkness, maybe a block away. "Are you that old? Are you that slow?"
I knew the gunshot would bring the police eventually. The only smart play was to stay behind the rampart. But I saw another movement across the street and I took off down the steps. When I hit the street I saw the long concrete wall on the other side, and beyond that a parking lot. The perfect place to wait for me, safe behind the wall, another easy shot as soon as I came to a stop.
_I should be dead already._ _He's had at least three shots at me, and the only time he fired he took out the glass behind me._
_He's toying with me._
But that just made me want to kill him even more.
When I got to the wall, I stood with the gun in both hands, peering down the line in one direction, then the other. The movement I'd seen was left to right. He was moving farther down Market Street, toward the arch.
I started running again, my left knee a riot of pain with every step, my lungs screaming. There was another streetlamp, another silhouette. This time I actually stopped and leveled my gun at him, half a block away. Trying to steady my hands, trying to aim . . . I knew it was an impossible shot.
_This is insanity,_ I told myself. _You can't shoot at a shadow from fifty yards away._
"Alex!"
A different voice this time. I turned and saw Agent Larkin racing to catch up to me.
_"He's there!"_ I yelled. _"That's him!"_
I pointed back toward the silhouette, just as he disappeared around another building. I kept running, one block, then another, stopping just long enough to look for him. There was another small park, just before the arch.
I saw a movement in a break in the trees, kept running until I reached it. Livermore was gone. Before I could move again, Larkin intercepted me and did everything but tackle me to the ground to stop me.
"Alex," he said, putting his face close to mine. _"What are you doing?"_
I tried to answer him. But I was gasping for air.
"The police are coming," he said. "Give me the gun."
"That woman . . ." I said, still fighting for air. Then there was another gunshot. I could practically feel the bullet passing over my head. As I pulled Agent Larkin down to the sidewalk, he drew his own weapon.
I pulled him away from the light, toward the line of trees. We stopped with our backs against two thick trunks, a few feet away from each other.
"Think about what you're doing," he said. "What he's _making_ you do."
_Breathe,_ I told myself. _Breathe and get ready._
"Do you hear me?"
In the distance, the faint sound of a police siren.
Then Livermore's voice again, from behind us. Not far.
_"Alex! Where are you?"_
I turned and ran toward the voice. I didn't care if he shot me anymore. He could put a bullet right through me. It wouldn't even be the first time in my life. _Shoot me as many times as you want, you evil piece of shit, and I'll keep coming._
I heard Agent Larkin calling my name behind me as I saw the figure moving. Then it stopped. There was one more line of trees behind him, one more square of open ground, then the last empty road and the arch glowing high in the sky. Beyond that nothing but the dark water of the river and the lights from another city on the far banks, looking so far away it might as well have been in outer space.
_I've got you. There's nowhere else you can go._
I didn't have any strength left in my legs, or any air in my lungs, but I kept running, getting closer and closer, waiting for the shot.
_Go ahead. Try to kill me. It's your last chance._
He was standing next to another campfire, the embers glowing at his feet. I didn't slow down. I didn't aim. I jumped over the fire and grabbed him by the shirt with my left hand, put the gun to his head with my right.
I looked the man in the eye. He stared back at me, with no understanding of what was happening to him. All he could see was a blood-soaked stranger leaping at him from the darkness. I could feel him shivering.
It wasn't Livermore.
I half doubled over, drawing the air into my lungs.
"Right here, Alex."
Livermore's voice. I turned and pointed the gun.
Right at Agent Larkin.
He was on the other side of the fire. Standing there, not moving, both of his hands empty. It didn't make any sense for a second, until I saw the man behind him. There was a homeless man's blanket wrapped around him. It fell to the ground.
I saw that face over the agent's right shoulder. Most of the hair was gone from his head. He had whiskers and a mustache. Everything was different, and yet he was the same man.
Livermore.
"Drop your gun, Alex."
I didn't. I tried to hold it steady. I had three inches of clearance. Thirty feet away.
"Do what he says," Larkin said. I could see his face clearly in the firelight. He looked scared and embarrassed at the same time.
The sirens got louder.
I dropped the gun to the ground.
"Let him go," I said. "He's got nothing to do with this."
"I let you live in that canyon," Livermore said. "I'm going to let you live again. Remember that, Alex. Remember that when we get to the end of this."
"The end of what? I don't even know what this is."
"You haven't figured it out yet? How we're connected?"
The sirens, louder and louder. The red and blue lights, flashing through the trees.
"It's over," I said to him.
"Not even close."
Then he shot Agent Larkin in the back.
As Larkin went down in slow motion, Livermore turned and ran. I had a gun at my feet, and a young man dying in front of me. I made my choice, the only choice there was. I went to him and tried to stop the bleeding, took off my shirt and held it against the entry wound in his back, turned him over and saw the exit wound.
Matt Larkin looked up at me. _It was always hockey_ _or the FBI._ That was what he had told me. A kid's dream that went one way instead of the other, leading up to this moment, bleeding on the ground as I yelled at him to hold on. For the second time in a week, I was about to watch a federal agent die in my arms.
"Fuck that," I said, picking him up off the ground. I staggered for a few steps, found my balance, and started moving toward the street.
"Alex . . ."
"I'm going to get you some help. Just stay with me."
When I finally got to the street, a police car came to a skidding stop right in front of me.
_"Ambulance!"_ I yelled as soon as the officer came out of his door. He had his gun drawn, but he holstered it when he saw me.
"Where's the shooter?"
"That way," I said, jerking my head toward the park. But Livermore had at least a full minute's head start by now. This was a man who had escaped from seven armed men while handcuffed. Then he'd gotten out of Amarillo, driving right through the roadblocks.
I knew he'd walk away from this, too.
"The ambulance will be here in two minutes," the officer said to me. Then he picked up his radio to communicate with the other cars.
"Ten thirty-two, Fourth Street and Market," he said. He repeated the information and then gave the ambulance a ten fifty-two for the same location. I kept cradling Larkin, looking into his eyes to make sure they were still open.
"You're going to make it," I said to him. "Just hold on."
When the ambulance finally arrived, they wheeled around a gurney with collapsible wheels and took him from me. He gave out a loud groan as they laid him down and wheeled him toward the back of the unit.
"One shot through the back," I told them. I didn't have to say anything about the exit wound, because the blood had already soaked through his shirt, to his coat.
"Alex . . ." he said again as they lifted him into the ambulance. I jumped in behind them, waiting for someone to stop me.
"Go with him," the cop said to me. "I'll catch up to you at the hospital."
I nodded, and they closed the doors. Then I watched the two medical techs working to stabilize Larkin, compressing the exit wound, checking his vitals, starting an IV.
"What's his name?" one of them asked me. He gave me a towel to wipe the blood from my face.
"Matt Larkin. He's an FBI agent."
They kept talking to him, told him he'd be at the hospital soon. His eyelids fluttered, and I thought we were going to lose him. But then he coughed and waved me over to him.
I came closer, struggling to keep my footing as the ambulance took a tight corner.
"You have to get him," he said to me.
"I will." It was the one thing he didn't have to tell me.
"Don't talk," the tech said to him. "Just relax. We're almost there."
"No," he said, looking me in the eye. "They'll try to stop you. You have to _go_."
I sat back down on the ledge, thinking it over. He was right. As soon as we got to the hospital, I'd be detained. At least overnight, possibly longer. Hell, I could picture Agent Madison on his way to St. Louis to personally escort me back to Phoenix. In handcuffs, if necessary.
Meanwhile, Livermore was still out there. Still moving.
"I'll be okay," Larkin said. "I promise."
I put my hand on his leg and gave it a squeeze. When the ambulance pulled up next to the hospital, they threw open the back doors and wheeled him inside. I saw two police cars, just a few yards away. I didn't see any officers yet. In about ten seconds, someone would find me.
I turned and moved away as fast as my wrecked body would let me, slipping into the night.
# CHAPTER TWENTY-THREE
LIVERMORE PULLED UP to the house, to the place where everything would end. He saw the man there, already waiting for him.
Livermore stayed in his vehicle for a few seconds, watching the man walk around the house. A cold wind picked up and the man turned away from it, hunched over and clutching at his coat. There were two inches of new snow on the ground.
When he got out of the vehicle, Livermore took a moment to reset himself. Then he approached the man and gave him a smile.
"You must be the inspector," he said.
The man was forty or fifty pounds overweight. He wore thick glasses, and his coat was tattered and stained with a decade's worth of coffee spills and sawdust.
"I am," the man said, taking off his glove for one moment to shake Livermore's hand. "Finally feels like February, eh?"
"I wouldn't know. I just came from Arizona."
"Lucky you," the inspector said. "You should have stayed there."
"I think I'd already overstayed my welcome."
The inspector smiled and nodded and looked up at the roof of the house. "I got started with the exterior. Doesn't look like the place gets used much these days."
"I only come back here from time to time," Livermore said. "Special occasions."
"Whose name is this place in, anyway?" the inspector said, looking at his sheet. "I was looking at the records and—"
"That wasn't necessary." This man wasn't here to ask questions like this. Somewhere in the town hall there was a piece of paper with his mother's maiden name on it, a name that had never been corrected by the barely competent small-town clerk who sent out the tax bills. Livermore paid the bills and never corrected the error, because he knew it gave this place an important layer of protection.
"I understand," the inspector said, "but if you're going to sell it . . ."
"I'm not selling. I just wanted to have a thorough inspection done. So please continue. You started with the exterior . . ."
The inspector nodded, back on track, back on familiar ground. "I've already covered some of the basics here. The grading, drainage away from the house, landscaping, walkway . . ."
"You have a list."
"Yes, of course." The inspector showed him the dozen pages on his clipboard. "No tree branches or crawling vines touching the house, no standing water, the downspouts direct the water in the right direction, the deck around back doesn't appear to have any termite damage . . ."
"So everything looks good."
"Just getting started," he said. "Paint looks decent. No cracks or peeling. But how old are those shingles?"
"At least thirty years."
"See, that might be an issue," the inspector said. "They look like they might need replacing soon."
"Sounds expensive."
The inspector shrugged, then flipped to the next page.
"Ridge lines straight and level . . . No bowing . . . Windows and doorframes square . . ."
He kept walking around the house, asking himself the questions and putting a tick on his paper, depending on whichever answer he seemed to settle on.
Livermore watched him carefully. He wanted to learn how this man walked around a house. How he flipped through his pages. Every nuance of every single movement.
"Drip caps on the windows," he said, turning and shaking his head. "That might be another issue. Some of these older homes . . ."
He kept moving down his list. Gutters, chimney, soffits, fascia. Livermore stopped him when he started talking about the building code for attic venting.
"You gotta have a one-to-one-hundred-fifty ratio," the inspector said, clearly one of those men who can talk for hours about the one thing in this world they know about. "What they call the _net free ventilation area_ to the total area vented. Unless you've got the right kind of vapor barrier, in which case you can go to a one-to-three-hundred."
_An arcane detail,_ Livermore thought. _Perfect._
Before they went inside, Livermore fired up the generator.
"You're not hooked up to the grid?" the inspector asked.
"Afraid not."
The inspector shook his head again, making another mark on his page.
When they were finally inside, Livermore watched the inspector start in the attic, checking for stains on the underside of the roof and the depth of the insulation. Then they moved through each of the rooms, starting upstairs. Ceilings, floors, walls, windows. Lights and electrical outlets. They came downstairs and spent several minutes in the kitchen.
"You need GFCI on any outlet within six feet of the sink," the inspector said, explaining that it stood for ground-fault circuit interrupter, and then taking out his tape measure to verify the outlet was five and a half feet away. He made another check mark on his sheet.
_GFCI,_ Livermore said to himself. _Another excellent detail._
The inspector checked all of the plumbing in the kitchen and the downstairs bathroom. He tested the smoke detectors, then he lit a match and tested the draw in the fireplace, and recommended a carbon monoxide detector even though the local ordinances didn't require one.
"You know your way around a house," Livermore said. "Where did you get your training?"
"It's a set of courses given by ASHI. The American Society of Home Inspectors."
"How many years have you been on the job?"
"Twenty, twenty-one . . ."
"Do you work for the county or the state?"
"I work for the county," he said. "They do have a statewide bureau. That's a different job."
Livermore nodded. Still absorbing every word. Every gesture.
"If I mess up bad enough, the state guy will be here to have my ass in a sling. Those guys are real pricks, too. Every single one of them."
_A state inspector,_ Livermore thought. _Bad news for everyone._
"The basement is last," the inspector said. "Gotta warn you, that's usually where we find the most problems."
"What kind of problems?"
"Moisture, usually. The water table's fairly high here. I already noticed your basement sits pretty low. You ever have any flooding?"
"Not that I can remember."
"We'll see," the inspector said. "If you've had water, I'll be able to tell."
Livermore stopped him. "Are you suggesting I'm lying to you now?"
"No, no," the man said, putting up one hand. "I'm just saying, this house has been here a long time. Longer than you and I, probably."
"I understand," Livermore said, giving him another smile to put him at ease. "And how about dead bodies? Do you have an entry on your list for that?"
"Depends on how many," the inspector said, giving him a wink.
"How about five?"
"Five dead bodies," the inspector said, pretending to write this down on his clipboard. "See, that's even worse than the moisture."
Livermore kept smiling as the inspector waited at the door to the basement. After a few long seconds, the man scratched the back of his head and cleared his throat.
"So can we take a look in the basement now?" he finally said.
"No," Livermore said as his smile disappeared. "I think this inspection has just ended."
# CHAPTER TWENTY-FOUR
I DROVE THROUGH the rest of the night, on what I hoped would be the final leg of this journey. To the place where this monster had come from.
Columbus, Ohio.
I didn't know what I'd find when I got there. I didn't know if he would be waiting for me again, or what he would have planned for me if he was. But the sun was up by the time I got there, the light filtered through the falling snow, after six hours of hard driving across Illinois and Indiana.
I had to try to remember the address of his childhood home from the files I had seen. The name of the street was in that file I'd seen in Agent Larkin's office. I had to bring it back, because I was on my own now, with nobody to help me.
I could see the name. A Canadian province.
Ontario.
That was it. Ontario Street.
I pulled over at the next gas station and filled up, went inside and bought a map of Columbus. When I saw the reaction from the man at the counter as he looked at me, I realized I was still covered in dried blood, both from the woman in the hotel and from Agent Larkin, aside from already looking like a man who hadn't had a real night's sleep in several days. I went into the bathroom to clean myself up as well as I could, stood there and looked at my own face in the mirror for a long moment. It was the face of a man who had already seen too much.
_Find your second wind,_ I told myself. _Or the third wind or the fourth wind or whatever the hell this is by now._
_Find something fast, because this isn't over yet._
I went back out and put on the winter coat I'd brought with me from Michigan. I sat in the car studying the map until I found Ontario Street. I took a breath, put the car in gear, and headed out onto the road.
When I got downtown, I hooked north, found Ontario Street just east of the expressway. It ran south to north across several blocks, with little one-story houses on small lots packed tight on either side, the street itself crumbling from the four seasons of hard weather. A real working-class neighborhood, in contrast to the penthouse suite in Phoenix.
I figured I'd eventually have to get out and start asking people where the old Livermore house was, until I kept going and finally saw one of the bigger houses in the neighborhood. Two stories high, standing above the little ranch homes on either side of it, with old wooden siding covered with peeling gray paint. A metal FOR SALE sign was stuck in the middle of the small front yard. The news about Martin T. Livermore had obviously already reached Columbus, and several people with rocks and spray cans remembered he had once lived here, because several of the windows were broken and the house had been tagged with graffiti. I could make out the word MURDERER in bright orange paint, just above the front door. It would have taken the whole day to figure out the rest of it, the messages scrawled on this house by the people who were drawn to it, people who felt the need to leave their mark on it, like some primitive totem that would keep the evil from spreading. I'd seen it before, on the houses owned by murderers in Detroit. First you tag the house, then you break in and vandalize it.
Then, at least in Detroit, you burn it down.
I parked my car on the street and got out. I saw an old man walking down the sidewalk with a little dog on a leash. He had his coat wrapped up tight around him as he walked into the cold wind.
"Excuse me," I said to him, nodding toward the house. "How long has this place been empty?"
"Mrs. Livermore's been gone, what, two years now," the old man said. "Place is such a wreck, they can't give it away. Then when the news broke about her son . . ."
"Did you know him?"
"Haven't seen him since he was a kid," the man said, shaking his head.
I thanked him and let him get back to his dog-walking. He went a few yards before he turned to me one more time. "I tell ya," he said, "I knew that Martin was a strange one. Even back then."
He didn't wait for anything else from me. He kept going down the street, leaving me to stand there alone. And to figure out what the hell to do next.
_You drew a goddamned arrow pointing to your hometown,_ I said to the wind. _So here I am._
I was about to go to the front door, but then I remembered what Agent Madison had said about the agents already processing this place, and what Agent Larkin had said about calling ahead to Columbus, in case he came back here. They had to be watching, so I stayed on the sidewalk, looking up at the house like just another curious bystander. Then I got back in my car and drove down the street. I clocked the sedan parked half a block down, the man behind the wheel looking at a newspaper, doing a professional job of not looking back at me.
I kept going, circling the block and coming to the house that was directly behind the Livermores'. I parked on that street, giving all of the cars a quick scan. I didn't see anybody else doing surveillance, so I went down the driveway and hopped over the back fence, onto the Livermores' property.
There was a detached garage, in even worse shape than the house. I looked through the window. There were no cars inside, just ratty old lawn furniture and tools and a lawn mower.
I crossed the yard to the back door, leaving my footprints in the newly fallen snow. There was a sign posted on the door, warning that any trespassing would be aggressively prosecuted. It was signed by the Columbus Division of Police and the FBI. When I peeked through the window in the door, I saw a kitchen with mid-century appliances and a tile floor with a road map of cracks running in every direction. I tried the doorknob. It didn't turn. I took a quick look around, knew I was hidden from the man parked out front, saw nothing else but empty backyards on either side of me. I hit the lowest glass panel with the heel of my hand, then reached inside, careful not to cut myself on the broken glass, and turned the knob from the inside.
Then I stepped into the childhood home of Martin T. Livermore.
There was a stale smell of mothballs and cigarette smoke, misery and insanity and God knows what else. I flipped on the light switch for a moment. A single bare bulb, hanging from an open fixture, cast a greenish light on the kitchen. Every cabinet was open and empty.
I turned off the light. Thirty seconds in this place, and I already wanted to be back outside, breathing the cold, fresh air. But I kept going.
I went into the living room, saw my own reflection in the mirror above the fireplace. There was a couch that should have been dragged out to the curb years ago, a coffee table with a half dozen scars along one edge, places where cigarettes had been left to burn. An old console television. The curtains were drawn shut, keeping everything in near darkness. I didn't want to turn on any more lights, not if they could be seen from the front of the house, but I did go close to the fireplace mantel and look at the framed photographs that were stacked on the floor, left here by whoever had been given the task of cleaning this place out. Probably throwing everything else away, but when you come to something like this, a half dozen photographs from someone else's life, it's the one thing you can't bring yourself to toss in the Dumpster.
The first was a black-and-white wedding picture. Mr. and Mrs. Livermore. A small, timid-looking woman in a white gown, a man twice her size in an Army dress uniform. One look at the unsmiling face of this man, a man now dead and buried, and I was already getting a small glimpse into the life that Martin T. Livermore had lived here in this house. An only child, with an Army father who probably domineered his wife and disciplined his son severely . . .
_No,_ I told myself. _Don't even start down that road. None of this turns a man into a serial killer. You can look for a reason for the rest of your life, and it will never be enough._
There was another photograph of the two parents with a young child in a baby carriage. The man still wasn't smiling. Then one more photograph of Martin as a three-year-old, wearing a little Army uniform, looking up at the camera and squinting in the sun.
And then nothing after that. No photographs from his teenage years. Or his adulthood. Maybe someone had taken those photographs. Or maybe Livermore's mother was trying to stop time, trying to remember her son only as a toddler, and nothing beyond that.
_She died before he was ever arrested,_ I thought. _But had she known that her son was a monster?_
_How could she_ not _know?_
I kept walking through the house. A bathroom with an old clawfoot tub, a master bedroom, another table scarred with cigarette burns. More closed curtains. The darkness and the silence, it was unnerving.
_Show me why you brought me here. There has to be something else . . ._
I went up the stairs, hearing them creak with each step. Whoever had cleaned out the first floor had come up these same stairs, and for whatever reason had given up on the job, leaving everything exactly as it had been on the day Livermore's mother had died. There was a sewing room with baby clothes still spread out on a table, then a baby's nursery that looked like something preserved in a museum. The room next to that was apparently Martin's. It was half the size of the nursery. Meaning what, I don't know. His mother had been either waiting for Martin's brother or sister, who never came, or hell . . . I didn't even want to think about it anymore. I kept the light off, but there was enough light to see the bed, neatly made up with a patchwork quilt. The desk and the dresser. The shelf hanging on the wall, with a collection of plastic models. Everything still the way it was. Cars, airplanes . . . Livermore's young mind already thinking like an engineer. And then the body of a man supported by a stand, his skin transparent so that all of his organs were visible. Next to that the body of a woman, with the same transparent skin.
A lot of kids had these, I knew, but here, in this room, these bodies with the clear plastic skin looked obscene.
I opened up the drawers to the dresser, found nothing but old clothes. In the desk drawer there was nothing but pens and paper clips and everything else you'd expect to find. In the closet, more clothes hanging, and leaning against one wall was a large telescope.
This room was almost as sterile and impersonal as his apartment in Phoenix. It wasn't until I'd gone out and found that metal shed he had rented . . . That was where I had seen the _real_ Livermore . . .
_So where's the real Livermore in this house?_
The FBI had already been here. I knew that. They'd gone through every room, seen everything that I'd seen. But they didn't expect to find anything.
I did.
I knew there was something else.
When I went back out into the hallway, I saw the attic access door above my head. A rope hung from one end, with a red wooden ball on the end. I grabbed the ball and pulled down, hearing the whole thing screech like a wounded animal as the stairs were extended down to the floor.
I took a moment to recover from the shattered silence, then I climbed up the stairs and into the attic, smelling the dust and the mildew. There was barely enough room for me to stand up straight, if I kept to the center. There were two small windows, one on either end. Both were covered with curtains. I went to one end, opened the curtains and looked outside at the backyard. I saw my own footprints in the snow.
As I turned, I saw boxes and a standing clothes hamper with an old Army uniform sealed in plastic. Then at the other end of the attic, another small window. As I moved closer, I could see the car still parked across the street, the same man behind the wheel. I stood up and walked back through the attic, slowing myself down, looking at everything carefully. Someone had built out this room for storage, the rough ceiling running from the roofline down each slope until it hit a short wall on either side of the room, maybe three feet high. It made sense to cut off the tight angle, but it also left a small space behind the wall. Perfect for a young Livermore to hide things.
I pushed my way through some boxes and saw the old wooden paneling that was tacked to the short wall. I tapped on it, and it felt solid. Moving down the course of the wall, I tried to find a spot that might look or sound different, until I came to an old wooden trunk that had been pushed back against the wall. A hundred years old, and it weighed a ton as I tried to move it. There were old caster wheels on the bottom, but they had seized with age, and they made a loud scraping sound on the wood floor as I pulled the trunk away from the wall. I stopped and pushed it back a few inches, got down on my knees and looked at the grooves the old wheels were making in the floor. I saw the old tracks, from years of use, running roughly along the same line. The lines I had just made looked fresh, the wood newly gouged. That told me one thing: if someone else had been up here recently, looking for something, they hadn't moved this trunk.
I pulled the trunk away from the wall again, far enough for me to kneel down behind it. As I felt along the paneling, nothing felt different, but it was that old, cheap paneling that came in sheets, with grooves running vertically to make it look like actual boards. A perfect way to cut a door into the sheet without making it obvious. I tapped along the wall until I came to the hollow spot, then felt along the grooves.
_Yes. A thin line cut here. There's something hidden behind this._
I didn't have much light to see what I was doing, and I wished I had a knife to pry open the piece of paneling. I worked along the edges with my fingers until I finally gave up and started tapping it hard on one side. It finally separated on the opposite edge. Then I pulled the little door open and looked into the darkness behind it. Another moment of apprehension as I imagined a young Livermore booby-trapping his hiding place, maybe setting some kind of spring-loaded spike that would impale the hand of anyone foolish enough to put his hand in there. _The hell with it,_ I thought, and reached inside and pulled out a shoebox.
I took it back to the little window to give myself enough light to see what was I was doing. Then I opened the box.
_This is it,_ I thought. _This is where he shows himself._
I pulled out the first photograph, an old-fashioned film snapshot with white borders, the kind you'd pick up at the drugstore a few days after you'd dropped off the film. I didn't even recognize the figure I saw in the first photograph. I moved to the second.
Everything stopped.
My heart. The spinning of the earth. Time itself.
_No,_ I said to myself. _No way._
I went to the third photograph.
Then the fourth.
I took the fifth photograph out of the box, held it up close so I could see it clearly.
That face.
"This can't be right," I said, breaking the silence.
After days of chasing Martin T. Livermore across the country, wondering how I could be connected to him . . .
I finally had my answer.
# CHAPTER TWENTY-FIVE
AS LIVERMORE watched the house, his mind went to that dark and quiet place again. The single red light inside his head flickered. Then it turned green.
He'd been parked on the street for an hour, watching the workmen replacing the roof. They had already taken off all the old shingles by the time he pulled up. A hard, dirty job, made all the harder by the cold weather. There was a Dumpster sitting next to the house, and it looked like they had spent the entire morning filling it. Now they were putting on the new shingles, starting from the crown of the roof and working their way down. They had a good three-man system going—one man bringing the shingles up the ladder, the second man holding them in place while the third man worked the nail gun. If they kept a good pace, even on a short February day like this one, they would have the new roof completely installed by nightfall.
It was Livermore's job to make sure that didn't happen.
He grabbed his bag and his clipboard and walked up the driveway. He was wearing a good sturdy winter coat, work boots, a new pair of reading glasses. When he got to the house he stood there looking up at the roof, until one of the men came down the ladder for more shingles.
"What can I do for you?" the man said.
"You can stop working. Immediately."
"And you are?"
"The state building inspector."
The worker scratched his head for a moment, then he called up to the other two men on the roof. A minute later, all three of them were standing on the ground. None of them looked happy.
"I'm sure we can work this all out," the man said. He gave the other two men a look, like _This is just what we need, huh?_ "Just tell me what you need so we can get back to work."
"I need to see that the vapor barrier has been properly installed," Livermore said. "As well as the flashing around the chimney. I need to see that you're using the correct grade of nails, and that your gun is set at the right pressure."
"Wait a minute . . ."
"And as long as you're making renovations to the roof, you need to bring everything else up to code, including the net free ventilation area."
"Just hold on," the man said, putting his hands up in surrender. "The net _what_?"
"What's going on?" a woman's voice said.
Livermore turned to look at the homeowner. She had grabbed a coat to wrap around herself, but was still shivering as she stood on the front porch.
_This is why I practiced,_ he told himself, _to be ready for anything._ He had total confidence that his heart rate was unchanged. That the command in his voice had not diminished by one degree.
"This is a state inspector," the head worker said to her. "We were expecting our guy to stop by, but—"
Livermore put up a hand to stop him and addressed the woman directly. "Are you currently living here, ma'am?"
"No," she said. "I came up for the day."
"These men need to stop working," Livermore said. "If I go over everything, they should be able to start again tomorrow morning."
"We need to get this roof done," the man said. "We can't just leave it."
"My call."
All three men threw their hands up in exasperation, and the homeowner turned away from all of them, muttering something under her breath.
"All right," the man said. "For God's sake . . . We're gonna clean some of this stuff up first. I mean, if that's all right with you."
"By all means," Livermore said.
All three men moved away and started picking up the last of the old shingles.
"Is this really necessary?" the homeowner said.
"I'll check out the rest of the house," Livermore said, "as long as they're still here."
She shrugged her shoulders and turned to go inside. Livermore followed her.
"Knock yourself out," she said, taking off her coat. She was wearing a thick sweater underneath, because the only heat in the house came from a kerosene lantern set up in the center of the kitchen.
She saw him looking at it. "I hope this is acceptable," she said, "unless you want me to freeze to death."
"It's fine. Is there anyone else staying here with you?"
"No, just me."
He nodded. "Maybe we can start upstairs . . ."
She shook her head again and led Livermore up to the second floor. He went through each bedroom carefully, making check marks on his clipboard. He took a look out the window and saw the men outside, putting away their tools and covering the new shingles that were still stacked on the ground.
Livermore worked his way back down to the ground floor, taking a few minutes with the fireplace, then finally coming back into the kitchen to inspect the outlets and the plumbing. Through the window, he could see the men still finishing up outside.
"GFCI outlets within six feet," he said, making another check mark.
"Is that a good thing or a bad thing?" she said.
"Is there a full basement?"
She nodded toward the door. Her arms were folded, and she let out a long, tired breath.
"Oil burner?"
"Yes," she said. "But no oil in the tank."
"Where's the emergency shutoff?"
She let out another breath and went down the stairs. Livermore followed her. When they were at the bottom of the stairs, Livermore could hear a cell phone ringing in the kitchen.
"It's right here," she said, ignoring the phone. She showed him the red switch mounted on the wall.
"This should be at the top of the stairs."
She turned away from him, shaking her head.
"The workmen can move that for you. When they come back tomorrow."
"Sure," she said, rolling her eyes. "Why not?"
They both went back upstairs. Livermore went to the living room and looked out the window. The men were finally in their truck and backing out of the driveway.
The cell phone rang again.
"Excuse me," she said as she left the room.
Livermore took the gun from his bag and, moving quietly, followed her into the kitchen.
# CHAPTER TWENTY-SIX
I HELD THE PHOTOGRAPH in my hand, trying to breathe. Trying to convince myself that it was real.
Everything that had happened to me that week, from the moment I was taken onto that plane and flown to Phoenix, the trip to the canyon where seven men died, then two more dead women as I chased this man across the country, another agent with a bullet blown right through him, his blood still on my clothes . . .
It all came down to this.
To this faded image hidden in a shoebox behind an attic wall in Columbus, Ohio.
A young woman standing at the shore of a lake.
Jeannie McDonald, who would someday become Jeannie McKnight.
My ex-wife.
She looked like she was maybe sixteen years old in this picture. A few years before I'd met her, but there was no mistaking that face.
The next photograph was another candid shot of Jeannie, from around the same time. She was looking up, like she was surprised by the shot.
Then another photograph of Jeannie sitting on the edge of the dock, like she didn't even know someone was looking at her. Then more like that one, distant shots of a young Jeannie, seemingly taken without her even knowing it.
I took out my cell phone and looked through the history. She had called me just before I'd started any of this. That night I was sitting in the Glasgow, listening to her voice from across the miles and the years. Something about the stuff her grandmother had left her . . .
A car. A house.
_A house._
I stopped for a second and grabbed that first photograph again. I had only been there once, not long after we were married. A little house on an inland lake, north of Grand Rapids. I couldn't even remember where it was exactly.
_But this is the house,_ I said to myself as I looked at the porch and the siding, and the yard that sloped down toward the lake. _This was taken at that same house when she was just a teenager._
I went back to my phone, kept looking until I found her number. I dialed and waited for it to ring. Once, twice, three times. Then it went to voicemail.
"Jeannie," I said. "It's Alex. Call me right away. As soon as you get this."
I hung up and hit the redial button.
It rang once, twice. This time, she answered.
"Jeannie," I said. "Are you okay?"
"Alex? Is that you? What's going on? You sound like—"
"Listen to me carefully," I said. "You may be in danger. There is a man named Martin T. Livermore. You knew him when you were younger . . ."
"Wait, what?"
" _Martin T. Livermore._ He may be coming after you."
"Slow down," she said. "That name's familiar . . ."
"At your grandmother's house. On the—"
"Yes! He was that boy on the other side of the lake. That one summer I spent here . . ."
"What do you mean, _here_? Where are you right now?"
"At the house on the lake. I'm having some work done so I can—"
"Jeannie," I said, my gut already tightening into a knot, "who else is there right now? Are you alone?"
"No," she said, "there's a whole crew of men here. They're working on the roof."
_A whole crew of men._ I let out my breath. She was safe, at least for the moment.
"I mean, they _were_ here," she said, "until the inspector came and stopped them. Hold on . . ."
I heard her footsteps over the line, the sound coming to me through the cell phone signal, over the span of three hundred miles. Then her voice came back.
"They just left," she said. "It's only the inspector now."
_One man,_ I thought. _She's alone with one man._
"I know it's been a long time," I said, "and you may not recognize him now . . ."
I flashed back to the park in St. Louis, seeing his face over Agent Larkin's shoulder, just before he shot him.
"He's about six foot two," I said. "His hair is short now. Dyed black. And he's growing in his mustache. When I saw him, he was wearing glasses . . ."
There was a long silence.
"Jeannie, are you there?"
"Alex," she said. "He's here."
# CHAPTER TWENTY-SEVEN
JEANNIE FELT the phone being taken from her hand. She was still processing the shock from that, how a stranger in her kitchen could do that . . . On top of the shock she was already feeling . . . Alex calling her . . . Everything he had said to her . . .
Until she finally looked at the man standing in the kitchen with her.
Really _looked_ at him.
She hadn't seen him in years. In decades. But he matched the description, all of the details Alex had given to her. More important, now that he had taken off his glasses, she could see his eyes, see across the years to that summer, so long ago . . .
It was him.
It was the boy from the other side of the lake.
"The last time I told you to come alone," she heard him say into the phone, "that agent was right behind you. If anything like that happens again, _anything_ , Alex . . ."
He paused to look at her.
"Then she will die in a way you can't even imagine."
He ended the call and put her phone in his pocket.
"What are you . . ." She didn't even know how to finish the question. She was still trying to put everything into place, to reconcile him being here, in the kitchen. Inspecting her house . . .
He smiled at her as she looked down at the gun in his hand. He held it pointed toward the floor, like it was just another _thing_ a man would carry around with him. A wrench, or a pencil.
"I'll put this away," he said as he lifted up his shirt and tucked it into his belt. She saw the hair on his tight stomach. The muscles and the flesh of this real man standing in front of her . . .
_It's really him._
That was when Jeannie came back to herself, as one second passed into the next, like a hypnotist snapping his fingers in front of her face, every emotion and impulse catching up to her at once.
She bolted from the room. The door was right in front of her, close enough for her to feel the metal brush against her fingertips, but then she felt one strong hand grabbing her by the back of the arm. She was spun around to face him again.
"Jeannie," he said. "We have a lot to talk about."
As she pulled her arm free and backed away from him, she saw something change in his expression, like a storm cloud suddenly forming in the sky. She kept backing away, thinking about her next move, trying to keep her panic under control, because she knew that panic would not help her.
He stepped toward her, matching every movement she made with his own. She was back in the kitchen now. There was another door behind her. If she went for it, he'd grab her again. She needed to distract him.
"I remember you," she said. "That summer . . ."
His face changed again, and now it was something that scared her on such a deep, primal level, she forgot all about panic and distracting him and everything else in her head except _getting out of there right now_ , as she made a break for the door and felt his hand on her arm again, and then her head pulled back as he grabbed on to her hair. She turned and swung her fists at him, kicking with both feet and screaming.
He threw her down and she slid across the kitchen floor. She saw the phone on the wall as she pulled herself up, grabbed the receiver, and tried to start dialing. But of course the rational part of her mind knew that the phone had been disconnected, and she only got to the nine anyway, one button before he was on top of her again. He grabbed the phone from her, threw it across the room, then took the base unit and ripped it from the wall. He left it swinging by a single wire as she ran back into the front room, got the door halfway open before he put one hand on it and slammed it shut.
She put a knee into his groin, heard him let out a yell of pain, and then she was running up the stairs. Nowhere to go from there, she knew that, but it was the only way to get away from him.
She went into her grandmother's bedroom, picked up the phone, and of course it was dead—why in God's name was she trying to use a phone when she _knew_ it was dead, because what she really needed was to open up a window and jump out, no matter how high it was, no matter how badly she would hurt herself.
_No,_ she told herself. _Get a hold of yourself and think._
She went into the bathroom and pulled open every drawer, looking for some kind of weapon. She pulled out a nail file, then a little pair of nail scissors. Not nearly enough, and then she remembered her grandfather's old hunting rifle, and that one day she'd watched him load it and how he'd told her that it would be hers someday, because he didn't have a grandson to give it to. But she hadn't seen that damned thing in years. Where would it be, if it was even in the house anymore?
She heard footsteps on the stairs, grabbed the scissors, and left the bathroom. She went out into the hallway, saw the top of his head as she went down to the other bedroom. When she pushed the door open, it caught against the boxes that were stacked inside. It was just a storage room now, used by her grandmother for a lifetime of clothes and furniture and whatever else. She locked the door behind her and pushed the boxes against it. Then she grabbed everything else she could—an old bowling ball in a bag, more clothes, a box of books, so heavy she could barely lift it—and she piled it all in front of the door.
She went to the window and opened it. The cold air rushed inside as she stuck her head out and looked down at the ground. There wasn't enough snow to break her fall. She would fracture both of her ankles, but at least she would be out of this house.
_And then he'd just walk right outside and grab you. You wouldn't be able to run._
"Help!" she yelled. "Help me!" Into the wind, again and again, until her voice started to break and her throat hurt. There was another house fifty yards away, through the woods, one more house another fifty yards beyond that one, but those houses and every house on the lake had been abandoned and locked up tight for the winter.
There was nobody to hear her. She was alone with this man, who even now was turning the doorknob. She looked for more heavy objects she could put on the pile, but it was useless. He would get this door open eventually.
_Another weapon,_ she thought as she rummaged through the rest of the boxes, finding a heavy wooden picture frame with an old photograph of her great-grandparents behind thick glass.
_I can use this._ _As soon as he puts his head through the door, I can hit him._
She put it near the door, even as she heard him throwing his weight against it from the other side.
_Something better,_ she thought, _there must be something . . ._
She kicked over another box, more old photographs, movies, letters, everything from her grandparents' life, all of it useless to Jeannie right now. More clothes. A table in the corner, with more boxes.
The sound of him kicking the door now, every thump resonating through the floor, right into her bones.
She threw open another box, then another, until she'd finally made her way to the closet. She wedged open the door and worked her way inside. A foolish move, she knew that, like a child running away from danger, thinking that she could hide from it, but she couldn't stop herself from huddling in the corner, in the darkness, the tears streaming down her face now, another scream building in her throat.
_No, I have to be quiet._
She folded herself into a ball, listening to the assault on the door. He was kicking and kicking and finally she heard the splinter of wood and then the sound of boxes falling away from the door.
"Jeannie," he said, his voice back to a dead calm now. "Where are you?"
Her eyes were adjusting to the dark. She pushed aside the old coats and dresses to see what else was in the closet. The old movie projector, a fold-up screen . . .
And a gun case.
_Here it is,_ she said to herself. _My grandfather's rifle._
She pulled the case over and opened it. She'd fired a pistol before, more than once. Alex had made her learn gun basics, back when they were living in Redford, just a few blocks from the Detroit city line, and there was always a gun in the house. Then a few years later she had even owned a little semiautomatic of her own before her friends talked her into getting rid of it. But she had never fired a rifle.
She smelled the gun oil as she took the rifle out of the case. It would have brought back a pleasant memory in any other circumstances, her grandfather loading it, how he'd promised her that it would be hers someday, but telling her that in the meantime she must never, _ever_ touch it. She tried to think back to that day, watching how he'd loaded it. Because right now that was the only thing she needed to remember.
"You have nothing to be afraid of," the voice said from miles away.
Then the sound of another box falling away from the door. He was in the room.
_It's a muzzleloader,_ she thought, pushing Livermore's voice out of her head, trying to focus on that day. Her grandfather's words coming back to her from across the decades. _You have to load this thing by hand, honeybunch. The way it's been done for hundreds of years._
She willed her heart to stop pounding, so she could breathe, so she could _think_ about what she was doing. There were a few steps to the process. Starting with the gunpowder. _Black powder,_ he'd called it. She found the bottle of black powder, and the metal tube she had seen him pour it into. She couldn't remember how much it took, but she could hear him warning her about putting in too much. _Just enough, not too much. Just a hundred grains, no more._ Whatever the hell that meant. She opened the bottle and poured some of the powder into the tube, then tipped that into the barrel.
"Come out," the voice said. "Right now."
There was a little cup that went in next, she thought. A _shell cup_ , he had called it. She rummaged through the other supplies in the case, found the plastic cup, about three inches long, just the right diameter to fit down the barrel. Had he put that in first? Before putting in the birdshot?
_Yes, he did. I can remember the sound of the shot going down the barrel. But first he used the ramrod thing to push the shell cup all the way down . . ._
She pulled the ramrod out of its holder next to the barrel, put in the shell cup, and then she tapped it down with the rod.
_He used another cap to measure the shot . . . Filled it up, poured that down the barrel . . ._
She opened the bottle of number six birdshot and poured it into the second shell. When it overflowed, it made a sound as it hit the floor, a hundred little pieces of metal drumming against the hardwood.
"What are you doing in there, Jeannie?"
She poured the shot down the barrel. Then she found the little plug and put that down the barrel.
_One more time with the ramrod. And then I'm ready._
She pushed the rod down the barrel, feeling the plug hit the bottom. Then she threw the ramrod aside.
"Jeannie . . ." She could hear the anger coming back into his voice.
_Wait._ _There was one more thing._
She tried to put herself back in that day, watching her grandfather. One more thing he'd done before the gun was ready to fire.
_What am I missing?_
She went through the rest of the supplies, found the little metal piece, tried to think back to where that went.
She heard the boxes being moved around on the other side of the closet door, made herself ignore everything else but that one day and that last step, her grandfather opening up the breech and putting in that last piece, something about a _metal jacket_ and how he put that little metal piece on the front end of the shell before closing it all up and telling her it was ready to fire.
She fumbled with the little metal jacket, dropped it on the floor and picked it up again. She slid the breech open and put it inside, closed it up, and thought about the safety and whether it was on or off or how to even work it, but it was too late now, anyway. The last box was moved, and she could hear his footsteps on the other side of the door, just inches away from her.
He wasn't talking anymore. He was standing there, waiting to open the door.
Waiting to take her.
She slowly pushed herself to her feet, the rifle across her chest, ready to swing it, ready to point it at his chest and fire.
_I hope to God I did everything right._ _Please, God, just let me get out of here._
She took a deep breath. She waited.
Nothing happened.
She could feel the panic building inside her again, filling up her stomach, her lungs, her throat . . .
_I can't just keep standing here. I have to do something._
One more breath, then she kicked the door open, heard it slam into him and drive him backward. In the next instant she was out and pointing the rifle at him.
"Get back," she said, a sudden resolve coming to her from somewhere inside her. "I'll kill you."
_Yes,_ she thought. _Even if this gun doesn't work . . . It doesn't need to . . . As long as he believes it might . . ._
"Jeannie, what are you doing?"
"I'll do it," she said, inching forward, the barrel still leveled right at his chest. "Get away from me."
"You do _not_ want to do this," he said, but he took another step backward.
"Sit down. Right now."
"No."
"Sit down or I'll kill you," she said. "I swear to God."
"You're not going to do that."
_"Sit down!"_
He held up his free hand to stop her, looked behind him to see where he would sit, then he started to bend down.
She saw him picking up the old electric coffeepot and throwing it at her, but it hit the barrel of the rifle a tenth of a second after the signal from her brain reached her trigger finger and the world erupted in a flash and a sound that obliterated everything else, the window shattering behind him as she tried to pull the trigger again but of course there was only one shot, only one chance, and she felt the rifle being pulled from her hands as she twisted away from him, tried to run, fell over a box, got up and took another step to the open door. He was right behind her. She grabbed the big wooden picture frame, turned and swung it at him, felt it connecting against his head, the glass cracking and the man going down, just long enough for her to get out into the hallway.
Down the stairs, hearing his footsteps behind her.
Throwing open the front door, letting out another scream that would be heard by nobody, nothing but the woods and empty houses all around her.
Except him, still behind her, getting closer.
She slipped in the snow as she ran to her car, touched the cold metal of the hood, turned the corner, and was about to grab for the door handle. But then the hands came around her again, catching her around the waist, pulling her backward.
She remembered the little pair of scissors she had taken from the bathroom, pulled them out of her pocket and slashed him across the face with them. He let out a yell, a sound like something from an animal, as he grabbed her hand and bent it, making the scissors fall into the snow. She tried to kick at his legs, slipped in the snow again, fell down hard and hit her face against the ground, tasted the blood in her mouth as she felt everything fading.
The last thing she remembered was being dragged along the snowy ground, back to the house.
# CHAPTER TWENTY-EIGHT
I WAS THREE HUNDRED MILES away when I heard Livermore's voice coming from Jeannie's house on the lake. Three hundred miles away when he hung up her phone.
Three hundred miles away from whatever he did next.
I went out the back door at a dead run, and as I did I heard the man coming up behind me. The man who'd been sitting in the car, watching the house, not that I even cared who he was or why he was suddenly right behind me.
"Stop!" he said. "FBI!"
I went over the back fence, waiting to feel the bullet ripping through my back. The shot never came, and I ran up the driveway, almost falling in the snow, catching myself as I got to my car. I started it, put it in gear, and took off, spraying snow behind me. The man appeared in my rearview mirror as I made the turn and gunned it back down Ontario Street, toward the expressway.
I heard a siren in the distance, then another coming from another direction. I caught the lights flashing just as I made the final turn, burying the accelerator as I merged onto I-270.
_You'll need a barricade to stop me._ _You'll have to blow out my tires and then shoot me when I come out of the car._
I saw another police car racing up behind me as I got off the expressway and hit US 23. The car blew past as I made the connection. I let out a breath and kept going, knowing that this was the fastest road to Grand Rapids, but knowing at the same time that this was a secondary highway, with slower traffic, and that more snow was starting to fall.
_God damn it,_ I said to myself. _Why did I rent this little shitbox car,_ _anyway?_ Even though I already knew the answer: because I never thought I'd drive it all the way across the country and then have to push through yet another three hundred miles to get to Jeannie.
I'd settle in at around eighty miles an hour, until I'd feel the tires starting to slip and I'd have to back off. Then after a few minutes I'd be back to full speed. I watched a hundred miles go on the odometer, racing through the empty fields of central Ohio. Then another hundred miles until I reached I-75 and took that through Toledo. The traffic got heavier as I hit the late-afternoon hours. People on their way home from work. But I picked my way through the cars, weaving from one lane into the other.
When I hit the Michigan state line, I knew I still had a long way to go. I had to stop myself from imagining what Livermore could be doing to Jeannie at that moment. Had to shut out every other thought from my mind but keeping the car on the road and getting to her as quickly as possible.
I came up behind two trucks driving side by side, leaned on my horn and flashed my lights until one truck finally pulled ahead of the other and I was clear. A few minutes later I cut between two cars with not enough room to spare, and I actually felt my driver's-side door brushing up against the front corner of his bumper. The driver swerved and fought to keep control, and for one second I thought he was going to go right off the road, but then he got all four wheels under him again and I left him behind.
I picked up my phone and hit the redial button, hoping by some crazy chance that she'd answer. Or even Livermore. But it rang through. I threw the phone on the passenger's seat and kept going.
The snow was falling harder. Nothing by Upper Peninsula standards, but enough to make everyone around me drive even slower. I was on I-96 now, heading northwest, passing through Lansing as the sun went down.
_This will be the last day,_ I promised myself. _Whatever happens, if you kill him or he kills you . . ._
Then I saw the flashing lights behind me.
_I'm not stopping._
It was a Michigan State Police car, one of the new blue Dodge Chargers. I would never be able to outrun it. The car stayed behind me for a half mile, finally pulling up next to me. I could see the red face of the trooper, and as he tried to wave me over I could see exactly how the rest of the scene would play out. I can usually talk my way out of just about anything—in the state of Michigan, at least. A Detroit cop who took three bullets on the job, I can drive ninety miles an hour anywhere in the UP, and even the troopers will let me get away with it. But I was a long way from the UP, and I was sure the FBI had put me out on the wire.
I knew that as soon as I pulled over, this trooper would come out of his car with his gun drawn. A felony stop, telling me to put my hands out the window. To open the door from the outside, stay facing away from him, move backward to the sound of his voice. Then get down on my knees with my hands interlaced on my head.
_You'll never talk your way out of this one,_ I thought. _And you can't outrun him._
But then, as I looked ahead, I saw the exit sign about a half mile down the road. Grand Rapids. The biggest city in western Michigan, which meant a lot of streets to get lost on, as soon as I was off the expressway.
I started to slow down, watching my rearview mirror as the trooper settled in behind me, his lights still flashing. There was maybe twenty feet between us.
_You need to find some ice,_ I told myself. _It's your only chance._
I kept my car rolling. The trooper stayed behind me, and through his windshield I could see his face turning an even deeper shade of red. I tested my brakes, hit a little patch of ice and slid, tested them again. There was enough snow on the ground that it was hard to see just how much ice might be hidden beneath it. I hit my brakes one more time and felt the car start to go sideways. I turned into the skid, pure instinct after God knows how many winters on Upper Peninsula roads, until I finally felt the tires hitting solid ground.
That was when I hit the gas, pulling away from the trooper just as he hit the patch of ice I had left behind me. I could see his tires spinning as he used all of his car's superior power at exactly the wrong time. He went completely sideways, and his front wheels were off the road. I kept pushing it as hard as I could, being careful not to go off the road myself. Fifty yards ahead of him now. Then a hundred.
I checked the rearview mirror and saw him backing out onto the highway and finally getting himself pointed in the right direction, but by then I had hit the exit ramp.
He had already closed half the distance when I hit the cross street and took the right, practically putting my car up on two wheels.
_I have to keep this separation,_ I thought. _Just enough to lose him for ten seconds._
I weaved my way through the traffic, watching him in my mirror, until I came to a curve in the road and he disappeared behind me. There was a gas station on the next corner, so I pulled in behind it, making sure I was out of the sight line. A few seconds later, I heard him blasting through the same intersection, heading north. I gave him a few more beats, summoning the patience from God-knows-where to make myself wait long enough. Then I went back out and headed west.
I kept my eye out for him, or anyone else in an official vehicle, as I made my way over to US 37. It was a smaller, secondary road that eventually went down to one lane in each direction. I was back to thinking about Jeannie, now that I had lost the trooper, and I drove with a new sense of purpose. Because I knew I was getting close.
I passed one car after another, cutting over into the other lane, driving toward the oncoming traffic, then cutting back. I had more close calls than I could count, until it was finally just a blur of speed and more honking horns. The plows hadn't hit this road yet, and one icy spot nearly put me in the ditch. As I straightened the car out, I realized I had an even bigger problem:
I couldn't remember where the house was.
It was a small lake, in the middle of absolutely nothing, like any of a thousand other lakes in this state. It was right after we were married, how many goddamned years ago, that one time we drove up to this place . . .
Up this road, to a town with a funny name. Then west. That was all I could remember. But I didn't have time to stop and think about it. I just had to trust that I'd know the place when I came to it.
I drove through Sparta, Kent City, Casnovia . . . Little towns with stoplights that I blew through, barely slowing down enough to make sure I didn't hit another car. Then Bailey, Ashland, Grant, Newaygo . . . It felt like I'd been driving forever.
_It can't be this far,_ I said to myself. _You missed the goddamned town._
But then I saw the sign for White Cloud, Michigan, and it all came back to me. Driving down this road as a much younger man, with my new wife.
I slid through the stoplight and took the hard left onto the narrow county road. Over one river, past Alley Lake . . .
Robinson Lake was next. Just another half mile.
Jeannie's lake.
As I drove down that last stretch of road, already seeing a single light coming from one of those houses on the edge of the lake, I could only wonder if I was too late.
# CHAPTER TWENTY-NINE
WHEN SHE OPENED her eyes again, Jeannie had no idea where she was.
She was staring at the ceiling. A ceiling she didn't recognize at first, until she tried to lift her head and felt everything spinning. It all started to come back to her, piece by piece. The lake house. The inspection.
Livermore.
She sat up on the couch, feeling the rough cloth against her arms. He had taken off her coat and her sweater. As she put her feet to the floor, she felt the cold wood. He had taken her shoes and socks, too. Her face was wet and numb from the snow, and she felt a raw scrape across her chin.
As she slowly got to her feet, holding the arm of the couch for balance, she felt the warmth coming from the fireplace. She looked over and saw the logs burning, then shuffled carefully over to stand in front of it. The heat radiated through her body, making her forget everything else.
Then she heard the noises from the kitchen. Chopping, water boiling on the stove.
_He's still here._ She looked down at the iron rack that held the fireplace tools. But the poker was gone.
_The door. I have to get out of here._
"You took a bad spill out there." The voice came from behind her, strangely calm.
She turned and saw him standing in the doorway. He was holding one of the kitchen towels to his face.
"You have to be careful on that ice," he said. "Come sit down. Dinner's almost ready."
She looked back at the front door, measuring the distance, estimating her chances.
"You don't want to go outside again," he said. "You'll freeze to death."
She hesitated for a moment, then she felt herself moving toward the kitchen, almost against her will. She stopped when she saw the table. It had been set with two plates. Water glasses, silverware. Everything in its perfect place. As if they were two normal people actually about to sit down to dinner.
"Why are you doing this?" she said, her lips trembling.
"Because this is a very special occasion."
"No," she said, shaking her head and looking around the kitchen. "Please . . ."
She had to fight down the urge to try to run again. She knew she wouldn't make it more than halfway across the room, even if she surprised him.
And he was right. Even if she got outside, she would freeze to death. There was nowhere to go. Just empty houses in either direction. Her car keys were in her coat, and that was gone.
"I'm making your favorite," he said. "It'll be ready in a moment."
_My favorite? How does he know that? How does he know anything about me?_
She looked at the butcher-block knife holder on the counter, just a few feet away from him. There were a half dozen knives in the block.
_If I can just get to them. That one long knife . . ._
"I sharpened your knives," he said, turning and watching her eyes. He held up the knife he'd been using to cut tomatoes. "Any chef will tell you, dull knives are more dangerous than sharp ones."
"Where are my clothes?" She heard her own voice breaking.
"They were wet," he said. "We don't want you to be . . . uncomfortable."
"I'm cold." Another shiver ran through her body.
"The food will warm you up."
He went back to his chopping. She stared at his back, wondering what to do next. She wanted to go back to the fire, but she didn't know what would happen if she tried to leave the kitchen.
He turned and looked at her, still holding the knife. "Sit down, Jeannie."
Jeannie swallowed hard and sat down. She massaged her legs, trying to rub some warmth into them.
A minute passed. The only sounds came from the stove or from the settling of the logs in the fireplace. Livermore drained the pasta in the sink, visibly wincing as the steam rose and gathered around his face. As he turned to her, she could see the jagged, red gash on his cheek.
_I slashed him with the scissors,_ she thought, _but he's not saying anything about it._
Somehow that was the most frightening thing of all.
He shook off whatever pain he was feeling, regained his composure, and brought over the pasta in the strainer. As he got close to her again, she could smell the odd, antiseptic odor that came from him, mixed with something else. Fire . . . smoke . . .
_Pure evil._ The words came into her mind, lit up in neon. She had to fight down the panic again.
"I want tonight to be perfect," he said as he put the rest of the pasta on his own plate. "You have no idea what I've gone through to make this night happen."
He went back to the stove and brought over the saucepan, ladled out some sauce on her pasta, then he did the same on his own. She watched him, strangely transfixed by his movements. Wondering again how any of this could be happening.
"I always hated this lake," he said as he sat down.
His cheek twitched as a thin line of blood dripped down onto his plate.
"Until that last summer," he said. "The summer we were together."
The words washed over her. She'd been sixteen years old back then, her parents sending her up here to spend a month with her grandparents. The last thing young Jeannie had wanted to do, spend four weeks in this stuffy little house that smelled like liniment and cigarette smoke, with nobody else around less than four times older than she was, without a television even.
And then on top of that, there was the strange boy across the lake.
Watching her.
Stalking her.
Taking pictures.
"You remember . . ." he said. The same boy, grown into a man, sitting across from her now. She would have never recognized him.
Until she saw those eyes. That same unblinking stare that had sent a cold chill through her body even then, as she sat on the edge of the dock, refusing to move. Refusing to give in to this stranger.
Until she'd look up and see him again, impossibly close to her, the camera around his neck. Wondering how he'd been able to sneak up on her, wondering how long he'd been standing there. That smile he'd have on his face when their eyes met. And how she'd finally break down and go inside, just to get away from him.
_I never said a word to you._
_Not once._
"Jeannie . . ." His voice went lower as he put his fork down. She closed her eyes and tried to stop shivering.
"Eat your dinner."
She kept her eyes closed.
_"I SAID EAT YOUR FUCKING DINNER!"_
He banged both fists on the table as he yelled, rattling the plates. The shock sent her back in her chair like a slap across her face. She fumbled for the fork, held it in her hand like she couldn't even remember how to make it work.
"Martin . . ." she said. The name sounded strange on her own lips. Almost obscene.
_"This is not how tonight is supposed to go!"_ he yelled, taking out the gun and slamming it on the table. _"You're ruining it!"_
"I'm sorry," she said, so softly she could barely hear the words herself.
"Listen," he said, fighting to control himself, measuring every word carefully. "I don't want you to be afraid of me. But you have to understand something, Jeannie. You have not made this easy for me. I think I've been more than patient."
She could see the veins standing out in his arm as he gripped his fork. She kept waiting for him to scream again. To come over the table at her. She could practically feel his hands around her throat.
"All this time, Jeannie . . . All these years. I kept thinking about you. Searching for you . . ."
Her whole body was going numb.
"And then I saw that picture from your wedding day," he said. "The _whole world_ saw that picture. Do you have any idea how that made me feel?"
She could feel herself slipping away now, into some deep recess in her own mind. His voice sounded like it was coming from someplace else. Another room in another house. Something about a picture. And a wedding day. The last blink of recognition before she slipped away even farther. That old photo her friend Lisa had put on that Facebook page she had set up for her. _I told you not to do that, Lisa. Who's dumb enough to put a divorced woman's wedding photo on Facebook?_
"You belonged to me, Jeannie. Not to that baseball player. Not to that _cop._ "
The voice driving her deeper into herself. The last remaining place where she could be safe.
"You were married to him for nine years. Over three thousand days of your life."
There was a movement, just a flickering shadow she could barely see. Then she felt the fork being taken from her hand.
"It was a mistake. But it's not too late, Jeannie. Even now, it's not too late for us."
Something touching her face now. Like a towline, bringing her back to the room. Bringing her back to her own self.
_No. I don't want to be here._
"I want to believe that," he said, his voice in her ear. "I _have_ to believe that."
She was back now. In this room, feeling his breath against her face, the cold tiles on her feet, the hard wooden chair against her back.
"You have no idea what will happen to you," he said, staring into her eyes, "if you can't make me believe."
She let out her breath as he took a step back. Then from one moment to the next, another kind of relief, as she let go of her bladder and the warmth spread out beneath her on the chair and then moved down her legs. She didn't care anymore. It felt strangely comforting.
"Now eat your dinner," he said. "Before it gets cold."
As the tears started coming down her cheeks, she found her voice again. "What are you going to do?"
"You'll see," he said as he returned the gun to his belt. "As soon as Alex gets here."
# CHAPTER THIRTY
I _'_ M HERE _,_ JEANNIE _._ PLEASE BE ALIVE _._
I had driven this cheap little car from Phoenix to Michigan, chasing a monster. Now I had finally arrived at the lake, passing one house, dark and abandoned on the edge of the frozen lake, then another, just as dark and abandoned. It was February, a season that had no purpose for these houses. But there was a light coming from the next house, streaming out onto the snow. I saw two vehicles parked in the driveway. A Nissan Pathfinder and a Subaru station wagon.
_This is the place,_ I said to myself, bringing back the memory from decades before. The house looked exactly the same, except that now the shingles had all been taken off and only half of them had been replaced.
I stopped on the street and watched the house for a few seconds. I didn't see any movement, didn't hear any sounds at all. There was no other plan in my head except walking up to the front door and knocking it down, trying to be ready for whatever happened next. But then I remembered what had happened in Amarillo, how I had opened one door and set off an apocalypse. For once in my life, it was time to think about what I was doing before I did it.
I got out and approached the house carefully, knowing that the snow would muffle my footsteps, but knowing just as well that whoever was inside would probably see me coming. I didn't go to the front door, but to the window looking out from the front room. When I put my face close to the glass, I could see furniture covered with white sheets. A box sealed up with strapping tape. Then as I moved over to get a better angle . . .
_Jeannie._
I could barely see down the hallway, but there she was, in the kitchen. She was sitting on a chair, against the wall, her head slumped forward. It hit me right in the stomach, how long it had been since I had last seen her. And how horrible it was to be seeing her under these circumstances.
I tried to see if her eyes were open. If she was still even alive. But I couldn't quite get the right angle to see her face.
_Move, Jeannie. Show me that you're alive._
I looked all around me. I hadn't seen Livermore inside the house, but I knew that meant nothing. He could have been outside at that moment, watching me from behind one of the trees, waiting for me to go in.
As I looked back in the window, I saw Jeannie stirring. Something like a shudder rolling through her body.
_What did he do to you?_
I had to fight the urge to break down the front door right then, finally convincing myself to move around to the back of the house. As I did, I took a quick look in each window. I didn't see Livermore.
When I got to the back door, leading into the kitchen, I could see only the side of her head. I tried the doorknob. It was unlocked.
I went inside. Jeannie turned and saw me. "Alex!"
I took one step toward her, then felt the whole world crashing down onto my head, driving me to the floor.
And then nothing.
* * *
—
WHEN I OPENED my eyes again, I saw a familiar face looking down at me.
"You made good time," Livermore said.
As I tried to get up, I felt the cold sting of metal against my wrists. I was lying on my back on the kitchen floor, my arms stretched out past my head. He had handcuffed me to the drainpipe under the sink. I was bleeding from a fresh cut in my forehead. The blood was running into my eyes, making it hard to see.
But there was Jeannie, still sitting on the kitchen chair. After all the years that had passed, to finally see her again, this close . . . She looked cold and she was somewhere beyond scared, in jeans and a tank top, with no coat, no shoes. There was a raw scrape on her chin. Livermore stood next to her, one hand on her shoulder.
"Let her go," I said to him. "This is between you and me."
"On the contrary, Alex. This has been between all three of us from the beginning."
"Jeannie," I said, "are you all right?"
"I'm sorry," she said, with a voice so weak I could barely hear her. "I'm so sorry."
I shook more blood from my eyes, rattling the cuffs against the drainpipe. As I gave it a yank, the cuffs bit deeper into my wrists.
"Look at this man," Livermore said to Jeannie. "You actually let him touch you. You shared a bed with him. Every night."
As she looked down at me, I could see her crying. I could only wonder how many tears she had already shed before I got here.
"You chose _this_ ," Livermore said, coming close enough to kick my left leg. "Over me."
_Do that again,_ I thought. _Come close enough for me to reach you . . ._
Jeannie didn't answer him. She kept looking at my face, while tears rolled down her own.
"I'll never understand it," Livermore said, stepping back to her and putting a hand on her shoulder again. "But you have one chance to make it right."
_What the hell is he talking about?_
She kept looking at me, her eyes glazed over. I didn't even know if she was hearing a word he said anymore.
_Wake up,_ I said to her in my mind. _Wake up and play along. It's your only chance to get out of this alive._
Livermore took Jeannie's chin and tilted her face away from mine, toward his. I rattled the cuffs again as I pulled at the drainpipe.
"Don't touch her," I said.
He looked down at me and laughed, then he turned back to Jeannie.
"Stand up," he said. "Stand up for the most important moment of your life."
She kept looking back down at me as she did, swaying and then catching her balance as she got to her feet.
I nodded to her, willing her to see the only way out. The only way to at least buy some more time.
_You have to go along with this, Jeannie. Whatever it is, just play along._
But then I saw Livermore take the gun from his belt.
I rattled the cuffs again, remembering how he'd stood behind Agent Larkin and shot him right in the back. Jeannie saw the gun but didn't even react. She just stared at it.
"Let me tell you what's going to happen," he said.
"Please don't kill him," she said.
"I could have killed him ten times already, if that's what I wanted."
She kept staring at the gun. Her chest rose and fell with each breath.
"No, I'm not going to shoot Alex," he said.
He handed her the gun.
_"You are."_
She took it from him, looking down at it like she had never seen one before. But I knew she could handle a pistol. I had taught her myself.
"Right now," he said. "This is where you prove yourself."
She looked at him, down at the gun again, then at me.
"Go ahead." He took a step backward.
She stared at the gun, finally wrapping her right hand around the handle, and then stabilizing her grip with her left. Just like I had taught her.
She wasn't looking at me anymore. She was concentrating on the gun, and in that moment I had a sudden doubt . . . Had she been traumatized enough that she'd actually kill me?
_I can't blame her if she does,_ I thought. _If that's the way this has to end . . ._
She looked at me, raised the barrel, put her finger on the trigger.
I waited.
She looked back at him for one moment, reset her grip on the gun.
She pulled back the hammer.
Raised the barrel.
Closed one eye.
Then she turned and pointed the gun at Livermore.
He made no move to stop her. He put his hands in the air.
She squeezed the trigger.
_Click._
She reracked and squeezed again. And again.
_Click._
Livermore stepped forward and took the gun from her. He didn't say a word, and the silence hung in the room as he slowly tucked the gun into his waistband. I held my breath, waiting for whatever would come next. Jeannie stood motionless, looking at nothing, until her eyes finally drifted to Livermore's face.
His back was turned to me, but Jeannie saw something new in his eyes that seemed to bring her back to life. She backed away from him, and as soon as he took one step toward her, I pulled at the cuffs, ignoring the damage to my wrists.
"Livermore!" I said. "Over here! I made her do that! Take it out on me!"
But he wasn't hearing me. In that moment, he was aware of nobody else but Jeannie, as she kept backing away until the wall stopped her.
He advanced until he was close enough to reach out his left arm, to close his hand around her neck. She looked down at me, the panic spreading across her face.
"Livermore!" I yelled. "You coward! I'm right here!"
He still couldn't hear me. He had her pinned against the wall, every ounce of force pressed into her neck. Then he raised his right hand and brought it across her face.
"Livermore!"
He did it again. Then again. She would have fallen to the floor if he hadn't kept holding her up. Then he looked down at the phone unit that was still hanging from the wall by one wire, bent over just enough to let Jeannie start sliding down the wall, then pushed her back upright as he stood with the phone in his right hand. Held it like he was going to hit her in the head with it.
_Think of something,_ I told myself. _Something to break the spell._
"I took her from you!" I yelled at him.
He froze.
"That's right, Livermore! I took Jeannie from you. How does that feel?"
He let out a breath, and then he looked down at me, as if suddenly remembering I was there.
"She used to talk about you all the time," I said. "The boy from the lake. For years, Livermore. I knew she was still thinking about you."
He let the phone drop. It crashed against the wall, dancing on the end of the wire. Then he let go of Jeannie's neck and she slid to the floor, her eyes half open, her left cheek glowing red from where he had slapped her.
_That's it. Stay focused on me._
"Every day I was married to her," I told him, "every night when I took her to bed . . . I laughed in your face."
_Come over here, you son of a bitch. Come closer._
He was standing over me now. Then he took one step sideways so he could aim a kick at my rib cage. I tensed up as I tried to absorb it, but it hit me like a knife right in the side, knocking the wind out of me. He kicked at me again and again. It was my turn to see the unhinged fury on his face. My turn to see the evil. But it was me now and not Jeannie, and that was all that mattered.
I tried to time his motion so that I could swing my legs around and trip him. It was Jeannie's last chance to get away, if I could get him on the ground, tie him up just long enough for her to make it out the front door. She was already on her hands and knees and crawling into the living room.
When I tried to catch him, he brought his foot down hard on my sore left knee, sending a jolt all the way up my body. Then he turned and went after her. I pictured her crawling to the front door, getting to her feet and running out into the darkness. But then I heard her scream as he captured her and dragged her back into the kitchen, pulling her by her hair.
"No!" I yelled, straining at the cuffs, waiting for him to start hitting her again. To end her life right in front of me.
"Let her go," I said. "Livermore, you piece of shit, _let her go_."
But he kept pulling Jeannie across the tile floor. He took one more look down at me. Then without another word spoken, he threw open the kitchen door and dragged her outside.
# CHAPTER THIRTY-ONE
I AM GOING TO DIE _._
She heard those five words in her head as she felt herself being pulled from the house, out into the cold air. The snow was a sudden shock against her bare feet. And yet it was drowned out by those five simple words, echoing over and over again.
_I am going to die._
"Please," she said, already shivering. "Martin . . ."
She went down on her hands and knees in the snow. He pulled her up and started to push her from behind, gathering her tank top in his fist and driving her forward. Through the trees, past the dark empty house next door, through more trees, past another house. All closed up for the winter. There was nobody to help her. Nobody but the man left behind in her kitchen, handcuffed to the drainpipe.
_I am going to die._
She tried to resist him, tried to find some kind of leverage to pull away, but she kept slipping in the snow. Her knees and elbows were bleeding. A cold wind came off the frozen lake and sliced through her bare skin.
_I am going to die._
She reached around and grabbed his wrist, tried to twist the thumb, a distant memory coming back from a self-defense class in college. His grip loosened just enough for her to break free, but then he pushed her hard from behind, and she went right out onto the ice and fell onto her back. Another sudden shock as the ice and the snow bit into her skin.
She rolled over and tried to push herself up. He stayed there on the edge, a shadow, not moving, as she kept slipping and falling back down, each time another cold shock, another scrape of her skin. In the end, she settled on her hands and knees, pulling herself into a ball, making herself as small as possible to protect herself against the wind.
_This is it,_ she said to herself. That same calm voice from a thousand miles away. _This is the end. I'm going to die right here on this lake, and they'll find my frozen body tomorrow. Or a month from now. Or in the spring . . ._
He came out onto the ice and grabbed her again, dragging her back to the shore. As she stood up straight she was close enough to see his eyes reflecting the dim ambient light. He hadn't said a word since she'd pointed the gun at him, and how much more terrifying was this silent disjointed face that looked back at her. There was something fundamentally different about him now, as if some basic human quality had been left behind in that house, some essential gear in his mind stripped and spinning free.
She tried to scream again, but he slapped one open hand against her cheek, making everything explode in a white flash of heat and pain. As he grabbed her arm, she went down to the ground, and he dragged her across the snow like a father pulling a child on a sled. The ice and the rough ground cut at her skin, until she finally managed to scramble to her feet. They continued around the lake this way, Livermore half pulling, half dragging, past more empty houses, past the part of the road that came near the lake, where Jeannie desperately hoped for one last chance, one pair of headlights coming from White Cloud. One vehicle she could wave to, could throw herself in front of.
But the road was just as dark and empty as the lake, and he kept pulling her toward the single light that loomed ahead of them. The Livermore house on the other side of the lake.
As they got closer, some primal part of her longed to be inside it, out of this cold air, sheltered from this wind. A dim light came through the back door and spilled out onto the snow. As he opened the door and pulled her inside, she blinked in the sudden glare and went down on the floor. She saw the skin on her arms, how red it was, and all the cuts and scrapes that were bleeding. She couldn't feel her feet anymore, and she was still shivering uncontrollably.
She saw drops of blood on the floor. Dried stains that had already been there for God knows how long, her own blood dripping from her face and arms to mix with it.
She didn't bother to wonder who else had bled in this room, or when. She was past caring. Past comprehending. When Livermore left the room, she looked up and made one last reach for the door. Her hands were just as numb as her feet, and she couldn't even work the knob to turn it.
"No," Livermore said as he came back into the room and threw a blanket at her. "You're not leaving."
He was talking to her again, but his voice sounded like the flat, emotionless drone of a machine. She grabbed the blanket and wrapped it around her as tight as she could, taking one breath at a time, staring at the floor, watching the snow melting and dripping from her hair, the drops of water mixing with the blood from her arms.
"There were others," he said, standing above her.
She didn't even try to move away from him.
"Women who had to pay the price."
She had nothing left. No strength, no fight.
_Please stop talking,_ she thought. _If you're going to kill me, kill me . . ._
"It's time for you to meet them."
She was still trying to comprehend what those words could even mean as he pulled her back to her feet and led her down the stairs.
# CHAPTER THIRTY-TWO
I KEPT CALLING Jeannie's name, long after Livermore had dragged her from the house. I wanted my voice to reach her, to let her know that I wasn't giving up hope. I couldn't let her believe that I wouldn't follow them, couldn't let her believe that she'd never see me, or anyone else, again.
_But you've got nothing,_ I said to myself. _No chance at all. Livermore will come back eventually and finish you off._
I rattled the handcuffs again, feeling the frustration washing over me, overwhelming me. My wrists were shredded, my arms sore from pulling against the pipe. My forehead was still bleeding, and there was no way to clear the blood that dripped into my eyes.
_I'm not going to die on this floor._
One more rattle, the cuffs biting into my skin. Then I made myself stop. I made myself think.
_I'm going to find a way out of this._
_I have to find a way._
I pulled myself deeper under the sink, looked at everything that was around me. But it was just cleaners and sponges and a bottle of bleach. I curled up into a ball so that I could bring one foot against the side of the top drawer. I pushed the drawer out with my foot, feeling it stick as it came to the end of the track. I kicked it a few times, and the whole thing came crashing onto the floor. It was a junk drawer, old batteries, flashlights, keys, hardware. But nothing with a blade. Nothing that would be of any use to me.
I worked on the next drawer, pushed it out and heard the silverware rattling onto the floor. I kicked the drawer clear and tried to see what kind of knives I could get to.
_There, a steak knife._
I worked it closer to me with my foot, but it got stuck against the lip of the under-sink cupboard I was lying in. I worked it back away from the lip until I could get one foot around each side of it. It took a few tries, but I was finally able to lift it and fling it at my own head. I was past caring if it hit me.
I pulled myself as close to the drainpipe as I could and twisted my arms around until I could get one hand on the knife. It wasn't much of a blade, but it was all I had. I twisted my arms back and tried to get an angle on the chain between the cuffs. Then I started sawing.
I had no leverage, could hardly put any force at all behind the blade. But I picked one spot and worked at it. After five minutes, stopping whenever I had to shake my head to clear the blood from my eyes, I had a small nick worn into the chain link. I took a breath and refocused. Then I kept sawing back and forth, willing the blade to cut into the metal. It wasn't working. I was just dulling the blade more and more with each stroke, and then eventually it slipped right out of my hand and fell behind the back of the bottom drawer. I let out a yell of frustration, brought my foot up again and slid that drawer open, finally kicking it hard enough to send it sliding across the kitchen floor. Even with the drawer out, there was no way I could get to that knife.
I pictured Jeannie being dragged from the house again, could only imagine how cold she must have been feeling at that moment, wherever she was. Or what Livermore was doing to her.
I could see only one choice left.
_You're going to have to find another knife._ _Only this time, you're going to have to cut your wrist._
_You lose your left hand. You find something to stop the bleeding. You go find Jeannie._
_It's the only way you're going to save her._
I cleared my eyes one more time and looked through the rest of the silverware on the kitchen floor. None of the knives looked like they could cut through anything. Not the metal chain on the cuffs. Not even my own flesh and bones—unless I was prepared to saw away at myself for the next three hours.
I yanked on the cuffs again, felt them cutting even deeper. Curling myself into a ball again, I brought my feet around and tried to set them against the back wall of the cupboard. There was a time when I would bend my legs like this a few hundred times a day, every time I put on the mask and crouched down behind the plate. Sometimes doubleheaders. Now I could barely get myself halfway into that same position. But I took a deep breath, and I forced my legs to bend . . . farther . . . farther . . . twisting my whole body around, the cuffs burning on my wrists . . .
Until _there_ , I had one foot on the wall. Now the other . . .
_You can do this,_ I told myself. _You have to do this. No matter what it takes._
I kept bending, pulling against the cuffs, willing my sore left knee to bend, until I finally had my other foot braced against the wall. I was completely twisted over like a pretzel now, so tight I couldn't have gotten myself free again even if I had wanted to. I tried to grab at the drainpipe, to take some of the pressure off my wrists. My hands kept slipping.
_More, you piece of shit. You need to move more._
I folded myself up the last inch, until I could get my fingers around the pipe, almost interlacing them on the other side. Just enough to keep the cuffs from cutting me any deeper.
Now I had to pull with my hands and push with my legs at the same time. I gritted my teeth and bore down on it. Straining against the pipe.
_You have to explode._ _You're a fucking rocket ship now. Blasting out of this place._
I kept straining and finally felt the drainpipe move a quarter of an inch. I got a new grip, tried again.
_One more explosion._
_Move, you son of a bitch. Move!_
Another breath. Another reset on the grip. I kept my eyes closed against the blood, pushed against the wall, pulled at the pipe. Pulled and pushed and pulled and I felt it give another quarter inch, and suddenly I was back in my own mind, a million years ago, seeing Jeannie walking across that campus for the first time, talking to her for the first time, kissing her for the first time, finally walking down the aisle to marry her, this woman who I'd lost and hadn't seen in how many goddamned years.
I screamed out with the pain, giving it everything I had left inside, going deep enough to find more. More than I'd ever had before.
Livermore's face. My hands around his throat.
_I'm going to kill you. I'm going to kill you. I'm going to kill you._
I felt the pipe give a little more and started to time my efforts into short bursts of strength, rocking the pipe back and forth like a car stuck in the snow, give and pull, give and pull, work it, take a breath, work it, take a breath.
It gave one more time, and I heard the sound of wood cracking around the sink and I thought I had it but then it was stuck solid again. I bent over one more time, laced my fingers together, and gave out one more scream as I used every ounce of my strength, concentrating on Jeannie's face.
_Come on, you piece of shit drainpipe!_
_Now!_
I pulled one more time, felt the whole thing coming down on me, too late to stop it, too late to protect myself. The edge of the sink caught me in the head and drove me backward. I slipped under it just enough to avoid most of the weight crushing my head into the floor, but then the bowl of the sink hit me in the jaw like the biggest sucker punch that was ever thrown by any heavyweight, and I had to stay down on the floor for God knows how long, waiting for everything to come back into focus. When it finally did, I realized that my hands were free. Free from the drainpipe, at least, if not from the cuffs.
I grabbed the side of what was left of the counter, the one half that had remained attached to the wall when the sink had fallen over. I felt for my cell phone. It was gone. Then I saw the collection of knives in the butcher-block holder, took out the heavy meat cleaver and tried to line it up with the chain between the cuffs. It was almost impossible to get any force on it, holding it with one hand and trying to swing it backward, to the space between my wrists, but I raised both hands and brought everything down on the counter at once, slamming the cleaver into the chain. Raised my hands and did it again.
The chain wouldn't break.
_I can't waste any more time._
I went through the other knives, pulling out the longest, sharpest knife in the block. Then I grabbed one of the flashlights that had been inside the top drawer I'd spilled on the floor, tested it, wiped my eyes with a kitchen towel, took one more breath. And went out the door.
The two cars were still parked in their spots in the driveway. I didn't know why Livermore hadn't taken one of them. Looking out at the dark road, I still didn't see any other way for him to get away from here, and as I shined the flashlight on the driveway itself, I saw no footprints in the snow.
I reversed direction, struggling to keep my balance on the slippery ground with my hands still cuffed together in front of me, and went into the backyard, finally picking up the footprints leading down to the lake.
_She was barefoot,_ I said to myself. _Wearing a tank top. She could die out here from the exposure alone._
When I went down closer to the lake, I saw the footprints leading along the shoreline. I followed them, keeping my flashlight trained on the ground, watching great drops of blood falling into the snow—from my face, or my wrists, or God even knows where else I was bleeding from. I flashed back to that woman in the hotel room above mine, the blood dripping through the ceiling and onto the white bedspread.
_I've seen enough blood for a lifetime. I don't want to see any more, unless it's Livermore's._
The footprints eventually led right to the lake, and as I shined the flashlight I saw the dark patches out on the ice showing from where the snow had been wiped away. My heart jumped into my throat as I imagined her going down through this ice, into the water below. She wouldn't last more than a minute. But as I scanned the ice I didn't see any holes, and I finally picked up the footprints as they continued along the shoreline.
_Where did he take her?_
I turned off the flashlight long enough to let my eyes adjust to the dark, and to see the dim light coming from a house on the other side. The only other house on this lake that showed any signs of life.
_There._
I hurried along the shoreline, seeing where the road came close and hoping that someone would come by so I could send for help. But there was nobody to help me. I would need to do this alone.
I continued to the house, saw the one light coming from the kitchen door. This was where he first saw her, I realized, a long time ago, when they were both teenagers. Not that it even mattered now. But it was an answer to my question.
I kept the flashlight off, moving in the dim light that reached across the yard. I went to the first window, looked inside and saw the empty kitchen lit by a single bulb in the ceiling, moved around to the next window, saw another empty room, this one dark. I went to the back porch, closed up for the winter. Around to the far side, another window looking in on a bedroom. Completing the circle, seeing nobody at all. There were no lights coming from the upstairs rooms, as far as I could see.
_Where are they?_
I got down on my hands and knees, looked through the narrow basement window that sat just a few inches off the ground. Through the cobwebs, I could make out one faint light that seemed to come from a little desk lamp. I wiped the blood from my eyes again, closed them for a long moment to give myself the best shot of night vision I could manage, and then looked through the window again.
I saw the top of Livermore's head. I didn't see Jeannie.
Standing back up, I weighed the long kitchen knife in my right hand. Still cuffed to my left, so I knew it would be hard to use the knife against him. But it was all I had.
Then I finally noticed the generator sitting next to the house. It was humming away, supplying this house with its only power. I looked it over carefully, and as I moved to the other side I saw a pair of boots first, then two legs, then the body of a man slumped against the far side. In the dim light, he seemed to be looking up at me, but as I touched his face I felt no warmth in his body. There was nothing left of his throat, just a dark frozen mass of blood and tissue that spread down onto his coat.
_I don't know how long you've been out here,_ I thought. _You're one more victim I was too late to help._
_I just pray to God I'm not too late for Jeannie._
# CHAPTER THIRTY-THREE
I DIDN'T WANT IT TO END THIS WAY."
Jeannie was huddled on the concrete floor of the basement, the blanket still wrapped tight around her. Livermore stood at the workbench, with a single light casting a pale glow that barely reached to the dark corners of the room. He had a box of shells on the bench, and as he talked to her he refilled the magazine of his semiautomatic.
"I gave you a chance," Livermore said. "You didn't take it. That was a big mistake."
Jeannie stared at the concrete floor. She had given up trying to play along.
She had given up on everything.
"This may surprise you," he said, "but I've made mistakes, too. I've let other women get close to me. Women I never should have trusted. If you think about it, it all goes back to _you_."
She didn't respond. The words were just a buzzing in her ears now.
"You were the prototype for me, Jeannie. You were the alpha. I should have known there could never be a beta."
The floor was cold against her skin. No matter how hard she clutched at the blanket, she could not stop shivering. Livermore slid the magazine back into his gun and slipped it into his belt. She heard him moving behind her, rummaging through the plastic storage boxes that were stacked against the wall. Then sliding one of the boxes across the floor.
"The first was Arlene," he said, looking back toward the other boxes against the wall. "Then Theresa."
The words started to break through. _What is he saying?_
"Then Claire, from Utah. Then Sandra, who I met in Las Vegas."
He took off the lid from the plastic box next to her. A box that anyone else in the world would use to keep Christmas decorations in.
"And this is Liana."
She still couldn't comprehend what she was hearing, because even after everything that had happened to her, there was only so much madness she could take in at once. But as he lifted the lid from the box and the _smell_ came washing out over her like the hot breath from an animal, breaking right through her terror and her shock, it all came together in that single moment and turned this _thing_ in front of her into a reality.
_This is real._
_It's a dead body._
She couldn't even scream at that point, not that anyone would have heard her, anyway. She slid away from the box, from the man who had killed this woman and had put her here. Had kept her in this box.
"She's the most recent," he said. As he tilted the box toward her, she saw an arm. Flesh still on the bone. A purplish liquid oozing from it. A black swarm, moving.
_Insects._
It was all pure animal reaction now, as she tried to scream, her voice so hoarse she could barely make a sound.
"This is the price they paid," he said. "Do you understand what I'm trying to tell you? I didn't want you to have to do the same."
Before she could stop herself, she looked down one more time and saw the whole woman's body, the liquefied organs at the bottom of the box, with clumps of hair and the flesh that still clung to the bones. What was left of the woman's face, her mouth wide open as if still screaming.
He stood up and pushed the box back toward the wall. Jeannie stopped trying to make any noise. Stopped trying to think. There was no strength left in her body. If he had pushed her over, she would have stayed there and never moved again.
"Stand up," he told her.
She stayed still.
"I said, stand up."
The words didn't register. She felt the cold concrete against her hands and knees, the blanket on her back. The rest of her mind was white noise.
"You're making me angry again, Jeannie."
More words that meant nothing to her. Until she felt the smooth fibers of a rope against her neck. She reached for it, pure instinct as it tightened against her windpipe. She clawed at the rope with her fingers, but it was pulled tighter and tighter until she finally felt herself being lifted from the floor. She struggled to her knees, then to her feet, feeling the rope go slack for just an instant. But before she could slip it from her throat, her right wrist was caught in another loop of rope. Then her left. Both hands were pulled away from her body, like the wings of a bird, or of an angel, and as she looked around her she saw both ropes leading to one of the exposed ceiling joists above her head, along with the third rope still wrapped tight around her neck. All three coming together in the hands of the man standing in front of her.
He moved behind her with the ropes, and she felt the tension increase on all three at once, drawing her up onto the balls of her feet, which she could still barely feel against the cold concrete floor.
He came around to face her again, the ropes all apparently tied to something behind her now. Keeping her suspended in this position she would not be able to hold on for long. The tears started to run down her cheeks again, but she didn't say a word.
He kept watching her, his face unreadable in the dim light coming from the desk lamp behind him. She felt herself weakening, felt herself leaning back against the ropes holding her upright, felt the center rope tightening against her throat with every slightest movement.
She wanted to say something now. One more utterance while she still could. Her last words on earth. But before she could make a sound, the lights went out, and they were both left in utter darkness.
# CHAPTER THIRTY-FOUR
I SLIPPED INTO the dark house, ready to die if I had to. There was a knife in my right hand, a flashlight in my left, but both hands were still cuffed together.
Livermore had a gun.
And he had Jeannie.
I paused in his kitchen for a moment, waiting to hear something. The house was quiet. I took a step forward and heard the floor squeak beneath my foot. Old wooden boards, no way to avoid it. He would hear me coming, no matter how carefully I moved.
_You need some kind of edge,_ I told myself. _Something to surprise him, to distract him, to get him away from Jeannie._
I took another step and felt my feet slipping from under me. When I caught myself against a table, I had to take a few seconds to stand still and let my head clear.
_You've lost too much blood. You don't have much time left before you pass out for good._
I covered most of the flashlight with my hand, turning it on just long enough to see the general outline of the room. I saw blood on the floor. My own, maybe Jeannie's. Maybe from the man outside. There was a door about twenty feet ahead of me. It had to lead down to the basement. When I opened it, it was too dark to see the stairs, but I could smell the basement's dampness. And something else . . .
It was a smell I knew, taking me right back to the first time I ever responded to a senior wellness check, in that old house in Detroit. My partner and I had found the woman on her bathroom floor, where she'd been lying for the past four days.
It was the smell of death.
I wanted to call out to Jeannie, to tell her that I was here, that I would make sure she was safe. But that would have been suicide for both of us. Instead, I crouched down at the top of the stairs and I listened. I waited. I gave my own gut instincts a chance to show me my next step.
There was nothing but silence. And darkness. And that sickening smell.
Then I heard a sharp intake of breath from somewhere below me. A muffled cry.
I took one step down the staircase, hearing the wood creaking, actually feeling the whole thing shifting under my foot.
_Fuck your instincts._ _Fuck your training, fuck everything you ever learned about how to approach a possibly armed suspect._
_Fuck Martin T. Livermore because I am not going to wait one more second._
I flipped the flashlight on for a fraction of second. Just long enough to see the stairs, to see where they started, to count how many steps I would need to take. I kept myself low and moved as fast as I could. Down the second step, to the third.
Then a flash of light and sound both exploded at once, freezing everything in that one brilliant instant, every last detail, my shoes on the stairs, Livermore with the gun raised and Jeannie standing off to the side, her eyes closed, her body stiffly upright, her arms reaching out into a letter _Y_.
Another two steps down the stairs, then another flash and another image burned into eternity. I was closer to him, but still too far away.
Two more steps and I felt the concrete beneath my feet. I turned on the flashlight to blind him, but the meager light was consumed by the third flash, and then I felt the sudden jolt in my right shoulder, my own body remembering that night so many years ago, another jolt just like this one, then the same burning sensation that came right after it as I watched my partner dying on the floor next to me. My whole right arm went numb in an instant, and the knife I was holding went sliding across the rough concrete.
A fourth flash lit up the room again, but by that time my momentum had already taken me into his chest and the gun went off right next to my ear. My shoulder was on fire but I had my hands on him, even though they were still cuffed together. After all this time, all those miles chasing him . . .
_I will not let go until you're dead._
He knocked the flashlight from my other hand, but at the same time I was able to grab at the gun, using both hands together, the cold metal twisting away from his fingers and clattering to the floor as we both fell hard against the workbench behind him, tools rattling and the breath coming out of him as I drove his back into the hard wooden edge.
The flashlight was on the floor somewhere, giving the room just enough light to make out the dark outline of his body. He tried to push me away, but I redoubled my grip on his shirt. Then he sucker-punched me right in the gut and folded me in half. I felt him slipping away from me, and then he stepped aside and tried to drive my head into the workbench. I ducked down just in time to avoid the blow, came up looking for him, but he was gone.
"Jeannie," I said. "Are you all right?"
I couldn't see her now. But I remembered the image that had burned into my mind. The unnatural way she had been standing, her arms spread out wide.
"Jeannie!"
She didn't answer me. Her body was nothing but a dark shadow against the wall.
_No, she's not dead. She can't be._
"JEANNIE!"
"It's too late," Livermore said, his voice strangely detached in the darkness. "For both of you."
My right arm was dead. I knew I was running out of time. As I took a step toward his voice, he picked up something from the top of a pile of plastic storage boxes that were stacked against the wall. I saw the glint of metal, but otherwise had no idea what he had just armed himself with. Something hard and heavy, something that would put me down for good if he caught me with it.
I took another step and he was gone again. He had moved deeper into the darkness of the room, choosing another corner to wait for me. I tried to quiet down my own breathing, my own mind, but as I took a step sideways he came out at me and swung the metal object right at my head. I ducked just in time, trying to move inside to tie him up, but he slipped away.
"Give up," he said. "Give up and die."
I shook the blood from my eyes, got myself low, ready to try again, ready to take my one last chance if it came.
"Come on," I said to him, wherever he was, trying to draw him out. "Don't be a fucking coward. Fight me like a man."
I heard a sound to my left, took a step, and ducked again as he swung at me.
_You have to time this just right. He swings, you drive before he can slip away. You bury your head right in his chest._
I waited and listened. I heard him breathing, heard him moving from one corner of the room to another, like an animal hunting its prey. I took another step as he somehow came up from behind me, and I felt the hard metal glancing off my temple. I had no chance to go after him. He was too fast, and everything was starting to fade.
_Come on, put me away. Step out and take a big swing at me._
"The minutes are working against you," he said. "You don't have many left."
I turned and moved toward the sound of his voice again, walking right into him and feeling another glancing blow against my forehead. I went down and rolled away from him. When I got back to my feet the room was spinning, and it took a long moment to determine up from down.
I took one more step backward, then another, until I sensed the stairwell right behind me, giving him his opportunity, hoping for one last chance to draw him out of the shadows.
_Come on, you've got me cornered here. Finish me off._
He stepped forward, a dark silhouette against the dim glow from the flashlight.
_That's right. Give me one more shot. Just one more._
He took a swing, and I felt one more blow across the top of my head. I went down to one knee again, and I knew he had me on the next swing. But then he paused. To line me up better, or to say one last thing, I didn't even know or care because it was an opening, and as I came up and put my good shoulder into his stomach, I drove him across the room until both of our bodies crashed into the stack of plastic boxes. There was a sound like many brittle sticks all breaking at once as I clasped both hands into a double fist and hit him in the face, putting everything I had left into it. He let out a cry of pain as I hit him again, then again, until he grabbed for my right shoulder and it lit me up like a fifty-thousand-volt shot from a Taser.
I felt his fingers grabbing at my face, until I tore them away with both hands and bent them back, trying to break as many as I could. He let out another cry of pain and swung with his other arm, so hard I heard it before I felt it, just under my left eye, making everything go white.
A moment later, I was on my back, looking up at the rough wooden ceiling and the cobwebs that shone in the dim light. Then I saw Livermore's face, looking down at me. He came closer, and I felt his weight on my chest.
I tried to reach for him, but he had my arms pinned down, making my right shoulder burn with a white-hot pain. He leaned more weight into it, and I felt myself losing consciousness. Then he eased back.
I spit blood in his face, tried to get free again until he put his weight back on my shoulder. As he bent down closer to my face, I could feel his breath on me, see his eyes and that little half smile that I hated so much.
He picked up the flashlight and shined it in my face. Then he raised the weapon above his head. I still couldn't see what he was holding.
He raised his hand higher. I could see it in his eyes. This was it.
"This is how it ends, Alex."
_After everything I've been through. Every mile I've chased him._
_This is how it ends._
But then his body stiffened. The expression on his face went from smug satisfaction to surprise, and a thin trickle of blood leaked from his mouth. Jeannie's face appeared over his shoulder, pale and streaked with tears.
As he turned to look at her, I saw the knife sticking out of his back.
"What did you do?" he said to her, and then everything froze. Jeannie stood there with her hands over her mouth. There was a rope around her neck.
_"You bitch!"_ he said as he reached out for her, the blood running down his back. _"You stupid whore!"_
He grabbed the rope and pulled her toward him. She screamed, and as he lifted his weight from my body I rose and brought my hands up together, looping them around his head and bringing the chain between the handcuffs against his throat. I pulled back and felt him falling against me, my head hitting the concrete again, but I kept the chain against his neck, pulling as hard as I could and working the knife deeper into his back.
A strangling sound came from his open mouth as I kept pulling the chain against his throat, feeling the handcuffs shredding through the last of the skin on my wrists, but it didn't matter anymore because everything I had was focused on those few links of chain that were stretched across his throat, as he flailed his arms back at me and caught me in the face with one fist after another, but I didn't let up.
I held on as he threw his body from one side of me to the other, one last desperate chance to break free, to breathe.
I held on as he tried to hit me in the face again. The punches getting weaker and weaker.
I held on for Jeannie, for the woman hanging in the hotel room, for the woman tied up and burned alive. For the woman in the refrigerator and the woman he violated in that bedroom while a video camera recorded every second. For all of the other women he'd killed. For Agent Halliday and the other men in the canyon. For Agent Larkin. I thought about each one of them as I held on tight, feeling the convulsions rippling through Livermore's body. Even as he stopped breathing, I held on.
_I'm not letting go, Livermore. Not until I can see you burning in hell._
I would have held on for another hour, just to make sure he was dead, that he was really gone, but my arms finally gave out, and my whole body went limp.
I couldn't move for a while, until I finally heard Jeannie crying softly. I looped my arms back from Livermore's head and pushed him off me. He rolled onto his back, his eyes still open. Staring back at me.
I crawled over to Jeannie, found her sitting on the blanket. I pulled the rope from her neck, saw more ropes around each wrist, took those off and let them drop to the floor. She was shivering so hard now. I tried to wrap myself around her, but I couldn't make the convulsions stop. I sat there with her, catching my breath, until the shaking in her body finally eased and I was able to stand up and pull her to her feet. I wrapped the blanket around her, and as I grabbed the flashlight from the floor, the beam settled on the contents of the box we had fallen into.
_Bones._
When I moved the beam, I saw another box that had been knocked over. Another skeleton. Then another and another until I finally came to the half-decomposed body in the fifth box. I took the light away from it and had to bend over and hold my knees.
"Get me out of here," Jeannie said, the words barely audible through her trembling lips. "Please, Alex . . ."
"I got you," I said, putting my left arm around her. "Let's go. Right now."
We went up the stairs, taking each one carefully. I could feel fresh blood coming from my right shoulder. More blood was running from my forehead into my eyes.
When we got upstairs, I did a quick scan of the kitchen with the flashlight, saw Livermore's coat draped over one of the chairs. I checked the pockets and found my phone.
Jeannie sat down on the floor. I didn't want to stay in this house anymore, but we didn't have a choice. I wasn't going to make her walk through the snow, and I didn't think I could carry her, not all the way back to her house.
I sat down next to her and dialed 911. When the operator answered, I gave her every piece of information I could. Robinson Lake, west of White Cloud. Then I told her to have the responders work their way around from Jeannie's house, the only house with lights on. I knew they would find us eventually.
When I put the phone down, Jeannie looked at me. Really looked at me for the first time since I'd gotten there, and as I looked back I finally got the chance to see that she was still the same person I had married all those years ago. Those same eyes, the same mouth. Everything I had fallen in love with, just a few years older.
"He shot you," she said, looking down at my shoulder. "You're bleeding bad."
"I'll be fine," I said. "We'll both be fine."
I turned off the flashlight and pulled her closer. We sat there together in the darkness, leaning against each other, waiting for the rest of the world to find us.
# EPILOGUE
ON OUR SECOND NIGHT in the hospital, Jeannie found me.
I was still recovering from the surgery, my right shoulder wrapped up and immobilized. They'd taken out the nine-millimeter slug from Livermore's Walther PPS. It was the same shoulder that had been hit on that summer night in Detroit. I also needed seventeen stitches to close up the cut in my forehead, another dozen or so in each wrist, and my left knee still hurt whenever I tried to put weight on it.
Jeannie had frostbite on her toes and fingers, severe scrapes and bruises on her elbows, knees, and chin. But that was the least of it. Most of the trauma she'd suffered had been inside her own head. The kind of wounds you can't sew up with stitches. The kind of wounds nobody will ever see.
And will take a lifetime to heal.
She stood in the doorway and watched me for a while. When I opened my eyes, she was there. I didn't know if her room was down the hall or on the opposite end of the hospital, but I was glad to see her. They'd even let her put on real clothes. I was still in my gown, still had an IV stuck in my arm.
"Every time I close my eyes," she said, "I see him."
I knew what she was talking about. I had seen him myself, more times than I could count. That face would wake me in the middle of the night, and I'd sit up trying to reach for him, feeling the pain ripping through my shoulder.
"Come here," I said.
She came to the bed, and I pulled her close. She collapsed against me, the dam breaking all at once. I held her while she cried, and we both ended up falling asleep together, right there in my hospital bed. When I woke up the next morning, there was a nurse in the room.
For the second time in my life, Jeannie was gone.
* * *
—
FIVE DAYS LATER, I was out of the hospital and sitting on an airplane. A commercial flight, back to Phoenix, in the daytime, surrounded by normal people escaping the cold weather. Agent Madison was waiting for me when I crossed over the security line.
We spent the rest of the afternoon at the FBI office on Deer Valley Road, sitting in the same conference room where he had first interviewed me.
"Agent Larkin is still recovering in St. Louis," he told me after I'd gone through everything, every detail I could remember. "He'll have to live with some of the internal damage for the rest of his life, but it could have been a lot worse."
I remembered that night, how he'd told me to walk away before the cops caught up to me. To go after Livermore on my own. I knew I'd have to find him someday, to thank him for that.
"There's someone else who would like to see you," Madison said.
I couldn't imagine who it would be, until he took me to another room and I saw a young couple with a baby fast asleep in a carrier. Madison introduced me to Agent Halliday's daughter, his son-in-law, and his new grandson.
I remembered the message he had given me. _Tell them I'm sorry. Tell them I love them._
I wasn't sure what else I could say to them. The baby woke up and started crying. I shook the son-in-law's hand and nodded to the woman as she took the baby out of the carrier and left the room with him. She didn't seem to want anything else from me, and I couldn't blame her for wanting to leave when I was done delivering her father's message.
Madison took me back to the airport, driving in silence most of the way.
"I'm on my way to Japan tomorrow," he finally said. "NPA came back to us with three missing-persons cases. The methodology is different, but that may have evolved over time. The time frame fits on all three cases. And close enough to Nagano. Maybe it was Livermore. Maybe not."
I knew what that meant. Three families in Japan who might never get answers.
But I couldn't do anything about that.
"You call us," he said after another few minutes of silence, "we take him alive."
"I call you, Jeannie's dead. You'll never convince me otherwise."
He didn't bother to argue. He dropped me off at the airport, told me he'd be in touch if he had any more questions.
"One more thing," he said. "There was a two-million-dollar reward on Livermore's head."
"That's for information leading to his arrest," I said. "Not for killing him."
"I believe it still qualifies. And you _are_ a fugitive recovery agent."
"Not anymore," I said. "If there's money coming, give it to the families of Livermore's victims."
I watched him think that one over for a few moments.
"I'm serious," I said to him. "Tell me you'll make that happen."
When he finally nodded, I got out of the car and closed the door behind me. As I walked into the airport, I heard him driving away.
* * *
—
JEANNIE WAS BACK HOME in Grand Rapids. She'd just started a new job as a paralegal in a law office. A long way from her major in art history, but what the hell, it probably paid a little better. I was back in Paradise, plowing out my road, getting the cabins ready for a new batch of snowmobilers. I went down to the Glasgow Inn every evening and sat by the fire. Jackie would have the Red Wings on the television over the bar as usual, the sound turned down low. Vinnie would come by after his shift at the casino and sit in the other chair by the fire.
My shoulder and my wrists were still taped up. My left knee still hurt whenever I tried to get up. They left me alone, let me drink my Canadians and stare into the flames.
My old partner Leon came by one night, just to catch up with me. That's when I told all three men what had happened. Sat them down by the fire and gave them the whole story, from the plane ride on a cold night, all the way to the desert, to the basement of a house on a tiny downstate lake they'd never heard of. I told the story once, leaving nothing out, so I'd never have to tell it again.
Two weeks after Jeannie and I said goodbye to each other, my phone rang late at night. It was Jeannie. She asked me how I was. I told her I was fine.
She didn't believe me, and we both knew it.
"I've been talking to somebody about what we went through," she said. "You should probably do the same."
"They told me that the _first_ time I got shot," I said. "Sent me to the department shrink."
"Alex . . ."
"I'm fine, Jeannie. But it's good to talk to you."
She didn't press me on it. I was glad I didn't have to have an argument with her, because I knew she was right. I had spent a whole year destroying my own life after getting shot and watching my partner die on the floor next to me. My own life and my marriage to this woman. She told me she was still seeing Livermore's face when she closed her eyes at night. I didn't even have to tell her I was having the same experience. Seeing the same face, and hearing his voice whenever the wind blew outside my lonely cabin in the woods.
I would still wake up in the middle of the night, reaching for him.
Trying to kill him, again and again.
The Livermore who lived inside my head would never die.
"Someone came into the office today," she said. "This woman, Alex, I could tell as soon as I saw her. She's living in fear. Believe me, I know what that feels like now."
"Could the lawyers help her?"
"No, they couldn't."
"What is she afraid of?"
"I don't know, Alex. She wouldn't tell me."
I could hear something in her voice. And after what she'd been through, I knew it would take a hell of a lot to rattle her.
"But you know it's bad," I said.
"Yes."
"And that's why you're calling me."
"You might be able to help her," she said. "Who else am I going to call? And besides . . ."
"Yeah?"
"We promised to stay in touch," she said. "You haven't called me yet. A man should keep his promises."
I was sitting in my little cabin in the woods, three hundred miles away from her. But it felt good to have this woman in my life again.
"So tell me everything else you know about her," I said. Because she was right, this woman I had once promised to watch over for the rest of my life.
A man should keep his promises.
# ABOUT THE AUTHOR
**Steve Hamilton** is the _New York Times_ \--bestselling author of fourteen novels, most recently _Exit Strategy_ and _The Second Life of Nick Mason_. His debut, _A Cold Day in Paradise_ , won both an Edgar Award and a Shamus Award for Best First Novel. His stand-alone novel _The Lock Artist_ was a _New York Times_ Notable Crime Book and won an Alex Award and the Edgar Award for Best Novel. Hamilton attended the University of Michigan, where he won the Hopwood Award for writing, and now lives in Cottekill, New York, with his wife and their two children.
# _What's next on
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## Contents
1. Cover
2. Also by Steve Hamilton
3. Title Page
4. Copyright
5. Dedication
6. Acknowledgments
7. Contents
8. Epigraph
9. Chapter One
10. Chapter Two
11. Chapter Three
12. Chapter Four
13. Chapter Five
14. Chapter Six
15. Chapter Seven
16. Chapter Eight
17. Chapter Nine
18. Chapter Ten
19. Chapter Eleven
20. Chapter Twelve
21. Chapter Thirteen
22. Chapter Fourteen
23. Chapter Fifteen
24. Chapter Sixteen
25. Chapter Seventeen
26. Chapter Eighteen
27. Chapter Nineteen
28. Chapter Twenty
29. Chapter Twenty-one
30. Chapter Twenty-two
31. Chapter Twenty-three
32. Chapter Twenty-four
33. Chapter Twenty-five
34. Chapter Twenty-six
35. Chapter Twenty-seven
36. Chapter Twenty-eight
37. Chapter Twenty-nine
38. Chapter Thirty
39. Chapter Thirty-one
40. Chapter Thirty-two
41. Chapter Thirty-three
42. Chapter Thirty-four
43. Epilogue
44. About the Author
1. Table of Contents
2. Cover
3. Cover
4. Title Page
5. Start
1. i
2. ii
3. iii
4. iv
5. v
6. vi
7. vii
8. viii
9. ix
10. x
11. xi
12. xii
13. xiii
14. xiv
15.
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"redpajama_set_name": "RedPajamaBook"
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es1602 — Organisation Release
New ESO Supernova Partner: Evans & Sutherland
Constellation partnership signed with global planetarium company
Evans & Sutherland (E&S) has signed a three-year partnership agreement with ESO as a Constellation Partner in the ESO Supernova Planetarium & Visitor Centre. Having developed advanced computer graphical technologies for over forty years, E&S was the world's first computer graphics company, and now specialises in a world-leading digital planetarium system, Digistar. The ESO Supernova Planetarium & Visitor Centre is a cooperation between the European Southern Observatory (ESO) and the Heidelberg Institute for Theoretical Studies (HITS). The building is a donation from the Klaus Tschira Stiftung (KTS), a German foundation, and ESO runs the facility.
Through their collaboration with ESO, E&S will be providing this cutting-edge planetarium technology for use in the forthcoming ESO Supernova, and will be the first to receive daily astronomy news and data directly from research carried out in observatories around the world through the Data2Dome project. Data2Dome, which ESO is spearheading with support from the International Planetarium Society's Data and Visualization task force, is being developed and refined in a close collaboration between the two organisations, in order to provide E&S planetariums worldwide with the most up-to-date astronomy information.
The ESO Supernova is due to open on the ESO Headquarters site just outside Garching, Munich in November 2017, and construction work is currently nearing completion. Collaboration between ESO and E&S is already ongoing, however, as developments advance in the Digistar Cloud service — Digistar's revolutionary built-in cloud sharing capability that will connect Digistar customers worldwide.
Companies, institutes and individual donors who wish to support the ESO Supernova and give back to the local community by supporting educational programmes aimed at encouraging young people towards STEM careers, may do so via several types of partnership. The partnerships bring with them corresponding levels of benefits, ranging from name and logo exposure in our centre, on print products, during planetarium shows and more to private use of parts of the building. In-kind contributions and individual donations are also possible. Both partners' and donors' contributions help us to ensure that the ESO Supernova can remain a free resource.
The ESO Supernova Planetarium & Visitor Centre is a cutting-edge astronomy centre for the public and an educational facility, located at the site of the ESO Headquarters in Garching bei München. The centre hosts the largest tilted planetarium in Germany, Austria and Switzerland and an interactive exhibition, sharing the fascinating world of astronomy and ESO to inspire coming generations to appreciate and understand the Universe around us. All content is provided in English and German and entrance is free, but requires prior booking. For more details visit: supernova.eso.org
ESO Supernova is proudly supported by: Evans & Sutherland and Energie-Wende-Garching.
The Klaus Tschira Stiftung (KTS) was created in 1995 by the physicist and SAP co-founder Klaus Tschira (1940-2015). It is one of Europe's largest privately funded non-profit foundations. The Foundation promotes the advancement of the natural sciences, mathematics, and computer science, and strives to raise appreciation for these fields. The Foundation's commitment begins in kindergarten and continue in schools, universities, and research facilities. The Foundation champions new methods of scientific knowledge transfer, and supports both development and intelligible presentation of research findings.
The Heidelberg Institute for Theoretical Studies (HITS gGmbH) was established in 2010 by the physicist and SAP co-founder Klaus Tschira (1940-2015) and the Klaus Tschira Foundation as a private, non-profit research institute. HITS conducts basic research in the natural sciences, mathematics, and computer science, with a focus on the processing, structuring, and analyzing large amounts of data. The research fields range from molecular biology to astrophysics. The shareholders of HITS are the HITS Stiftung, which is a subsidiary of the Klaus Tschira Foundation, Heidelberg University and the Karlsruhe Institute of Technology (KIT). HITS also cooperates with other universities and research institutes and with industrial partners. The base funding of HITS is provided by the HITS Stiftung with funds received from the Klaus Tschira Foundation. The primary external funding agencies are the Federal Ministry of Education and Research (BMBF), the German Research Foundation (DFG), and the European Union.
ESO is the foremost intergovernmental astronomy organisation in Europe and the world's most productive ground-based astronomical observatory by far. It is supported by 16 countries: Austria, Belgium, Brazil, the Czech Republic, Denmark, France, Finland, Germany, Italy, the Netherlands, Poland, Portugal, Spain, Sweden, Switzerland and the United Kingdom, along with the host state of Chile. ESO carries out an ambitious programme focused on the design, construction and operation of powerful ground-based observing facilities enabling astronomers to make important scientific discoveries. ESO also plays a leading role in promoting and organising cooperation in astronomical research. ESO operates three unique world-class observing sites in Chile: La Silla, Paranal and Chajnantor. At Paranal, ESO operates the Very Large Telescope, the world's most advanced visible-light astronomical observatory and two survey telescopes. VISTA works in the infrared and is the world's largest survey telescope and the VLT Survey Telescope is the largest telescope designed to exclusively survey the skies in visible light. ESO is a major partner in ALMA, the largest astronomical project in existence. And on Cerro Armazones, close to Paranal, ESO is building the 39-metre European Extremely Large Telescope, the E-ELT, which will become "the world's biggest eye on the sky".
ESO Supernova fact sheet (PDF)
Evans & Sutherland website
Evans & Sutherland Data2Dome press release
Cell: +49 170 867 5293
ESO ePOD Community Coordinator & Communication Strategy Officer
Release No.: es1602
PR Image es1602a
Evans & Sutherland logo
Press Releases on eso.org
Press Releases on iau.org
Press Releases on spacetelescope.org
|
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Spaelotis caliginea är en fjärilsart som beskrevs av Butler 1878. Spaelotis caliginea ingår i släktet Spaelotis och familjen nattflyn. Inga underarter finns listade i Catalogue of Life.
Källor
Nattflyn
caliginea
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{"url":"https:\/\/ham.stackexchange.com\/questions\/17821\/why-do-psk-modes-have-bandwidth","text":"# Why do PSK modes have \u201cbandwidth\u201d?\n\nMy understanding of a naive PSK scheme is that you have some frequency(s), f(n), at baseband, and you modulate information by shifting the phase (what \"point in time\") the signal is at. Isn't the phase shift instantaneous? In this case, isn't the frequency constant? Why do PSK modes look vaguely like MFSK in a waterfall?\n\nBinary PSK with instantaneous phase shifts would be equivalent to multiplying a sine wave (the carrier) with a square wave with values at 1 or -1.\n\nWhen two signals are multiplied, this forms a frequency mixer. A mixer with inputs at frequencies $$f_1$$ and $$f_2$$ creates outputs at $$f_1 + f_2$$ and $$|f_1 - f_2|$$.\n\nA sine wave is just one frequency, let's call that $$f_c$$ for the carrier frequency. And the square wave will be at the symbol rate, which for PSK31 is 31.25 symbols per second. A square wave is a series of odd harmonics. More specifically, a square wave at frequency $$f$$ is equivalent to the infinite sum:\n\n$$\\sin(2\\pi f) + {1\\over 3} \\sin(3\\pi f) +{1 \\over 5} \\sin(5\\pi f) + \\dots$$\n\nThis means a square wave at 31.25 Hz has frequency components at:\n\n\u2022 31.25 Hz\n\u2022 93.75 Hz (31.25 * 3)\n\u2022 156.25 Hz (31.25 * 5)\n\u2022 187.5 Hz (31.25 * 7)\n\u2022 ...\n\nSo say you're transmitting PSK at 14.075 MHz at a symbol rate of 31.25 per second. This means you'll be emitting power on frequencies:\n\n\u2022 $$14.075\\:\\mathrm{MHz} \\pm 31.25\\:\\mathrm{Hz}$$\n\u2022 $$14.075\\:\\mathrm{MHz} \\pm 93.75\\:\\mathrm{Hz}$$\n\u2022 $$14.075\\:\\mathrm{MHz} \\pm 156.25\\:\\mathrm{Hz}$$\n\u2022 $$14.075\\:\\mathrm{MHz} \\pm 187.5\\:\\mathrm{Hz}$$\n\u2022 $$\\dots$$\n\nAs you can see, the bandwidth extends out to infinity. The power diminishes as you get away from the carrier frequency, but not very rapidly, and it never reaches zero. If you're transmitting with 1 kW then you'll be spewing significant harmonics over the entire band, and even outside it.\n\nConsequently, except for very low power, cheap radios you might find in part 15 devices, the phase shifts are not instantaneous but gradual. For example PSK31 uses a cosine envelope, meaning in the case of alternating between phases it multiplies the carrier not by a square wave, but rather by a cosine. Since a cosine consists of just one frequency component, this generates not an infinite series of frequency components in the output of the mixer, but just two: the carrier frequency, plus and minus 31.25 Hz.\n\nThings get a little worse when the phase isn't strictly alternating between states, because the first derivative of phase is discontinuous. This does generate an infinite series of harmonics (I have a graph in another answer) but one which decreases much more rapidly than the square wave case before. It should be noted the technical design of PSK31 isn't especially good, and professionally design PSK implementations often use a root-raised cosine pulse shaping filter which is better in this regard.\n\nIn general, the only thing which occupies just one frequency is a sinusoid with no start and no end which isn't modulated at all. Changing the amplitude or phase in any way will cause the signal to occupy more bandwidth. It's rather easy to demonstrate why this must be true intuitively: were it possible to transmit information with just one frequency, signals could be crammed infinitely close together, so an infinite number of users could be crammed in a finite amount of bandwidth. There'd be no need to license or sell spectrum because there'd always be room to add more users. Also, we could fit infinite information bandwidth in any slice of spectrum, so we wouldn't need more bandwidth signals for higher data rates.\n\nThe more gradually amplitude or phase change, the less bandwidth will be occupied. Ideally the derivative of amplitude and phase are continuous functions, as well as the second, third, and so on derivatives. A gaussian function higher-order derivatives are all continuous, which is why you see gaussian functions come up in modulations like GMSK.\n\nIsn't the phase shift instantaneous?\n\nIdeally yes, in practice it is not.\n\nIn this case, isn't the frequency constant?\n\nNo, phase and frequency are related. A shift in phase is equivalent to a shift in frequency. People found that looking for a phase shift instead of a frequency shift can take less RF bandwidth for the same data throughput.\n\nWhy do PSK modes look vaguely like MFSK in a waterfall?\n\nBecause, in a manner of speaking, PSK is a lot like MFSK. Nyquist-Shannon says that data transmission takes bandwidth. There is a minimum bandwidth required for any data to move at a given rate. The more noise on the data path the greater the bandwidth is needed to overcome this. Because noise is, roughly speaking, correlated to the bandwidth of the channel it helps to minimize the bandwidth to minimize noise.\n\nWith a noiseless wire that has no resistance or capacitance the bandwidth needed for infinite data throughput is zero. Since we don't live in an ideal world data takes bandwidth.\n\nI don't know if I'm helping here since my vocabulary may have some nuanced differences to yours. Looking up the theory on Shannon and Nyquist bandwidth will help. As will the relationships between phase, frequency, and amplitude.\n\n\u2022 I would say an instantaneous phase shift is very far from ideal. Probably also illegal, as it would generate spurious emissions on many out-of-band harmonics. \u2013\u00a0Phil Frost - W8II Jan 6 at 18:24\n\u2022 \" I would say an instantaneous phase shift is very far from ideal. Probably also illegal, as it would generate spurious emissions on many out-of-band harmonics. \" Ideally anything outside of the intended channel would be filtered, or ideally nobody would care about out of band harmonics. \u2013\u00a0MacGuffin Jan 6 at 19:00\n\u2022 If you filter the out of band harmonics, you aren't changing phase instantaneously anymore. And if you are imagining a world where no one cares about out of band harmonics so there's no need to filter, that isn't an idealized world, just a simplified one. \u2013\u00a0Phil Frost - W8II Jan 6 at 19:06\n\u2022 \"the bandwidth needed for infinite data throughput is zero\" The Shannon-Hartley theorem is $C=B \\log_2(1+{S\/N})$. Even if $S\/N$ approaches infinity, the capacity is zero if the bandwidth is zero. Transmission of information with zero bandwidth is impossible, even in the absence of noise. \u2013\u00a0Phil Frost - W8II Jan 6 at 20:02\n\u2022 @Expectator probably the relevant thing is not the Nyquist-Shannon sampling theorem but rather the Shannon-Hartley theorem. A PSK signal is an analog thing in the electromagnetic field -- sampling there is not directly relevant to your question. \u2013\u00a0Phil Frost - W8II Jan 6 at 20:36\n\nFor a mathematical sinusoid, instantaneous frequency is the first derivative of the phase of that sinusoid with respect to time. So if the phase isn't changing at a constant rate with respect to time, the first derivative will change, and thus so will the instantaneous frequency.\n\nAlso, in the real world, there can be no instantaneous discontinuous phase changes, as all capacitors (including all the parasitic ones and in the wires) require finite time to charge up or down in order to change signal levels. Any band-limiting filters reduce the rate of change even further.","date":"2021-05-10 05:32:07","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\": 12, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7348237633705139, \"perplexity\": 857.0069810221977}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989030.87\/warc\/CC-MAIN-20210510033850-20210510063850-00112.warc.gz\"}"}
| null | null |
Netflix's Stuck Apart Review: Existential Crisis
Entrenched in a midlife crisis, Aziz seeks solace from his mundane job, lonesome friends and rowdy family while pretending to have his act together in Stuck Apart.
Netflix's Tony Parker: The Final Shot Review: A Success Story
Tony Parker: The Final Shot is a documentary about the famous French NBA player's success that steers clear of any depth.
EntertainmentMoviesReviews
Zee5's Ka Pae Ranasingam Review: A Relevant Movie, Albeit Too Long
Nilanjana Chatterjee - January 16, 2021 0
Archi Sengupta - January 16, 2021 0
Archi Sengupta
Ka Pae Ranasingam is a Tamil-language drama film directed by P. Virumaandi in his directorial debut and stars Vijay Sethupathi and Aishwarya Rajesh. The film released on a pay-per-view platform after it was scrapped for a theatrical release due to the COVID-19 pandemic.
3-hours of politics
Ka Pae Ranasingam follows the injustices and the corruption in the bureaucracy. After her husband Ranasingam's death, Ariyanachi must fight with everything she has to get his body back from Dubai.
The movie tries to point out everything that happens to the common people in India. Right from water shortage to the delay in getting a dead body back, Ka Pae Ranasingam focuses on various social injustices that the Indian common people have to face everyday.
Taking place in a village, Ranasingam is the villagers' favourite. He can convince people very well, and his urge to fight for what is right makes Ariyanachi fall for him. The film starts off strong and sweet, focusing on both Ranasingam as a warrior for the people, and as a sweet and lovable romantic. His character is strong and resilient, and someone to truly look up to.
However, the hero of this movie is Ariyanachi. It totally focuses on her fight for justice and to get her husband's body back. She has to run from pillar to post and has to fight the system to achieve it. For a film that is essentially women-centric, Ka Pae Ranasingam doesn't offer much space for Ariyanachi. Sure, you see her in almost all the scenes, but you still feel that Ranasingam is the real hero here, although he dies in the first 10 minutes or so. His character looms in the background of the movie, however, and you never really dive too deep into Ariyanachi.
Additionally, let me just mention here that Ka Pae Ranasingam is 3 hours long. Yes, 3 hours. The movie is valid and it is engaging most of the time, but 3 hours is a huge amount of time. The movie focuses on a lot of problems in India which are valid and commendable, but you'll feel the weight of the film an hour and a half in. The plight of the people and of Ariyanachi will break your heart, and will make you angry too, but there's still some amount of immediacy missing in the narrative. The government's inaction is infuriating, but it gets sluggish after a while.
Ka Pae Ranasingam follows a non-linear narrative. The film jumps from present to past and then back and gives us a look at what was and its contrast now. Vijay Sethupathi as Ranasingam is commendable in his performance. Whenever he is on-screen, you feel his presence and he commands the entire movie with his strong aura. Aishwarya Rajesh is good too and does well with her character. She's seen the most in the movie and gives us a relatable and grounded Ariyanachi who is looking to get his husband back. You can see her power and helplessness all at the same time.
Summing up: Ka Pae Ranasingam
Ka Pae Ranasingam is a political drama that has its heart in the right place. It tries to showcase the helplessness of the commoners when it comes to dealing with matters related to the government. It feels sadder due to the parallels it draws between situations past and present. However, the 3-hour-long film somewhere starts to feel tedious, even while displaying absolutely heart-breaking incidents.
There's a scene in the movie where it showcases how actor Sridevi's death made a huge impact on the people and how her family received her body within hours. However, Ariyanachi, helpless for 10 months, is still trying to find answers. It's scenes like these where the movie shines and shows us how humanity is probably lost.
Ka Pae Ranasingam is streaming on Zee5. You can rent it for Rs 199.
Like the Ka Pae Ranasingam review? Read our other reviews here.
Ka Pae Ranasingam talks about everything wrong with India's bureaucracy and tackles some very heartbreaking topics. It's engaging for the most part, but the runtime becomes tedious after a while.
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Horror Movies + Cats > People
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Over The Moon is Netflix's first in-house animated feature film that too in the grand Disney style! Stream it Or Skip It? Read Here!
Netflix's The Cabin with Bert Kreischer Review: Lackadaisical but Therapeutic
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Beyonce Speaks About Racism and Sexism in the Music Industry
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Netflix's All Together Now Review: Sweet But Unrealistic!
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'All Together Now' is a new dramedy that is trying too hard to make you emotional with its extra comfortable and sweet plot. Do not expect a lot from this film.
Netflix's The Trial Of Chicago 7 Review: Unserved Justice and it's Shocking Relevance
Sanskriti Srivastava - October 17, 2020 0
The Trial Of Chicago 7 is a movie based on protests in Chicago at the 1968 Democratic National Convention that turned into riots. It's a re-enactment of what the protestors went through in the absurd trial that followed.
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| 4,407
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{"url":"https:\/\/de.zxc.wiki\/wiki\/Lagerhaltung","text":"# Warehousing\n\n The items warehousing and storage overlap thematically. Help me to better differentiate or merge the articles (\u2192\u00a0 instructions ) . To do this, take part in the relevant redundancy discussion . Please remove this module only after the redundancy has been completely processed and do not forget to include the relevant entry on the redundancy discussion page{{ Done | 1 = ~~~~}}to mark. SwissChocolateSC ( discussion ) 05:38, Nov. 1, 2019 (CET)\nAutomatic small parts warehouse\nHigh-bay warehouse\nExternal warehouse with clay pipes from Heinrich Taxis GmbH + Co. KG in the 1920s.\n\nIn production and logistics, storage is the storage of material as a sub-task of materials management .\n\nStorage means the deliberate interruption of the operational flow of materials , i. That is, there are deliberately formed stocks . Warehousing requires a warehouse , i. H. a room , a building ( logistics property ) or an area in which goods can be stored and is a central theme of inventory management .\n\nWarehousing is called storage as an economic branch ; it falls under the systematic NACE Section I, Transport and Storage (Rev 2, 2010).\n\n## Subject of storage\n\nStorage objects can be\n\nPersonnel is not stored, but at most kept internally or externally. Therefore, the characteristics, strategies and optimization methods of storage are not applicable to personnel.\n\n## Functions and tasks of warehousing\n\nBasically, warehouse logistics distinguishes between the following warehouse functions or tasks:\n\n\u2022 Backup function\n\u2022 Bridging function\n\u2022 Finishing function\n\u2022 Transformation function\n\u2022 Speculative function\n\n### Backup function\n\nWith the security function , the warehouse serves to ensure production and delivery and is comparable to private hoarding .\n\nThe backup function comes into play when insufficient information is available in the company about future quantity requirements, delivery times and requirements. This can be the case in particular with products that are subject to seasonal fluctuations - and thus delivery bottlenecks - but which actually have to be available at all times.\n\nIn order to guarantee this constant availability, sufficient buffer quantities are defined by iron stocks , minimum quantities and reporting stocks , which are calculated from the delivery time of a possible order quantity at the procurement sources. As a result, the required quantities are always kept available - in at least sufficient quantity and quality . (see section on stocks )\n\nWith the provision or assortment function , storage contributes to a range in the assortment . In this respect, the supply function supplements the balancing function, as it bridges the parts of the range where there is a discrepancy between the quantities of procurement and sales .\n\n### Bridging function\n\nIf the procurement quantity is greater than the production quantity , the material that has not yet been used for production is stored by the compensation function . A warehouse is used as an intermediate storage facility in order to keep the material flow stable in the volume flows. The bridging function also serves to prepare finished goods for delivery before they are delivered. A storage in the sense of the bridging function can also take place dynamically if, for. B. Goods on continuous conveyors (treadmills, taxiways, etc.) are parked or moved there in waiting loops.\n\n### Finishing function\n\nThe refining function is also called the warehouse's production function, which only enables subsequent processing. The refinement function describes storage that leads to the desired change in the product and is therefore part of the production process. This is particularly true of\n\n### Transformation function\n\nIn particular in the warehouses of trading companies , the warehouse performs transformation tasks; for example, the goods are transferred to salable packaging and labeled. A supplementary sorting task can also be defined here, in which non-salable goods are sorted out and disposed of.\n\n### Speculative function\n\nReasons for the speculative function of storage can be foreseeable extreme price fluctuations on the procurement market or particularly low cost prices . In addition, you can speculate with stock items by ordering large quantities and receiving discounts . However, if the price of the goods falls, this can also have a negative effect. B. in hardware .\n\nThe warehouse fulfills a size-degression function because it enables orders for several items and can thus reduce the order costs per unit.\n\nThe environmental protection function includes\n\nIn particular, it is also stored for training purposes; camps are set up that have no function other than training; the pieces stored in it are then neither used for production nor for trade.\n\n## Storage types\n\nThe camp can be classified according to various criteria; The following aspects are among the most important distinguishing features (with overlaps):\n\nThe aim of planning a warehouse must be that it can fulfill the required warehouse function; in particular that in the sense of the security function z. B. Production sites can be continuously supplied with the required materials. The warehouse planning refers to the planning of the warehouse organization, the warehouse and transport technology, the storage units to be stored and the warehouse layout. Systematic warehouse planning should, among other things, reduce storage costs. In addition, warehouse planning can help increase the level of mechanization and automation.\n\nWhen choosing warehouse locations, it must be decided whether the warehouse will be managed centrally or decentrally. When deciding on the degree of centralization, the spatial aspect is often decisive:\n\nCentral warehousing means the spatial combination of all warehousing functions and all stored goods under uniform management. The advantages that result from the central storage are a simplification of the acceptance of goods, care, maintenance, inventory determination and inspection. Other points are the low capital commitment of current assets , lower inventories and lower space costs .\n\nWith decentralized warehousing , the input materials are stored at the location of the user in the form of intermediate storage (buffer storage). The main advantages of this storage method are the higher flexibility, the more precise disposition of the individual materials in the production areas and the shorter transport routes .\n\n## Storage systems\n\nWith fixed storage location allocation , fixed storage locations are provided for each article, which are reserved for these articles only (\"same to same\"). The advantage lies in the simple determination of the storage space. Due to the fluctuating stock levels for each item over time, part of the permanently allocated storage space is not used, which leads to poor utilization of the storage capacity.\n\nWith the dynamic storage location allocation (open warehouse system) , the articles are stored in a free storage location. Storage takes place at will or according to specified parameters. One advantage of this method is that if faults occur in one storage aisle, for example, the same material can be removed from another storage aisle. The greatest advantage, however, is a very high utilization of the storage capacity. However, in order to guarantee later access to the stock items, the storage locations must be precisely documented. This can be done with a storage compartment card , which then stores the data in the EDP system, or the storage location is specified by a warehouse management system. In the chaotic storage of hazardous materials (. Eg certain adhesives, chemicals) are restrictions and Storage of incompatible materials to be considered so dangerous reactions are avoided in case of incidents.\n\n## Stock\n\nAn inventory as such is the quantity of a good in the warehouse at a certain point in time. The following special stocks are important:\n\n\u2022 Minimum stock\n\u2022 Reorder point (order point)\n\u2022 Maximum or maximum stock\n\nThe minimum stock (in practice also safety stock or outdated iron stock \/ reserve ) is the stock level that must not be fallen below in order to be able to maintain the readiness for delivery even in emergencies. The minimum stock varies according to the material and \/ or supplier. It generally covers the risk of the supplier failing to meet deadlines or quality. When the reorder point is reached through withdrawals from the inventory, a message to the purchasing department to replenish the inventory - by placing an order - is triggered during the automatic disposition . The reorder point thus determines the due time of the requirement. See also: sediment analysis , ordering policy\n\n${\\ displaystyle {\\ text {Reorder level}} = ({\\ text {Daily requirement}} \\ cdot {\\ text {Lead time}}) + {\\ text {Minimum stock}}}$\n\nThe maximum or maximum stock is the maximum stock that may be available in the warehouse in order to prevent high costs , high capital commitment and too high a storage risk .\n\n${\\ displaystyle {\\ text {maximum stock}} = {\\ text {minimum stock}} + {\\ text {optimal order quantity}}}$\n\noptimal inventory = the optimal inventory enables a smooth operation and causes low storage costs. The optimal stock level must be coordinated with the optimal order quantity.\n\nAs part of the cumulative quantity is the minimum stock and maximum stock of a bearing dynamic sizes of storage using lead time are recalculated. Because if the material requirement in production decreases (increases), then the demand in the warehouse also decreases (increases) and can even be \"zero\". Therefore makes the cumulative quantity principle a fixed reorder no sense. However, in order to discover possible errors or to avoid possible bottlenecks, an upper and lower reporting limit can be defined; As soon as this is reached, an alert is issued so that it can be checked whether the under-stock \/ over-limit is correct or whether there is an error (e.g. in the data acquisition or the requirement calculation).\n\n## Storage procedure\n\nWith the FIFO principle ( First In - First Out ), the goods stored first are also retrieved first.\n\nWith the Lifo principle ( Last In - First Out ), the last stored supplies are removed first. As a rule, this is undesirable, but sometimes inevitably the consequence of the bearing design.\n\nIn the food sector, for medicines or sterile goods, storage is also carried out according to the best-before date ( First Expired - First Out , FEFO), as this can also differ from the storage sequence.\n\nOther withdrawal methods are the Hifo principle ( Highest In - First Out ) or Lofo principle ( Lowest In - First Out ), these are used less often.\n\n## Warehouse codes\n\n### Stock intensity\n\n${\\ displaystyle {\\ text {Cost of goods}} = {\\ text {Sales of goods (in pieces)}} \\ cdot {\\ text {Reference price}}}$\n\nThe stock intensity measures the ratio of stocks to sales or to business assets .\n\n### Average inventory\n\nThe average inventory indicates how high the inventory is on average over the course of a fiscal year. It can be calculated as a quantity or a value.\n\n${\\ displaystyle \\ varnothing {\\ text {Stock}} = {\\ frac {{\\ text {Stock at the beginning of the year}} + 12 \\, {\\ text {Stock at the end of the month}}} {13}}}$\n\nIf only a company's published balance sheets are available, the following, less precise formula is often used. The formula only considers the stocks available on the balance sheet date and is therefore very inaccurate. It is mainly used by external analysts in their balance sheet analysis .\n\n${\\ displaystyle \\ varnothing {\\ text {Inventory}} = {\\ frac {{\\ text {Starting inventory}} + {\\ text {Ending inventory}}} {2}}}$\n\n### Inventory turnover\n\nThe inventory turnover rate indicates the ratio of consumption \/ time unit and the average inventory, and therefore shows how often a warehouse has been completely filled and emptied within a certain time unit. The key figure can be determined in terms of quantity or value. Low values \u200b\u200bmean that the material remains in the warehouse for a long time and is an indication of high safety stocks. These have a negative effect on the capital commitment .\n\n${\\ displaystyle {\\ text {Inventory turnover rate}} = {\\ frac {\\ text {Out of stock}} {\\ varnothing {\\ text {Inventory}}}}}$\n\nor\n\n${\\ displaystyle {\\ text {Inventory turnover}} = {\\ frac {\\ text {Cost of goods}} {\\ varnothing {\\ text {Inventory at cost prices}}}}}$\n\n### Average storage time\n\nThe average storage duration provides information about the situation in the warehouse and the development of the capital commitment in the warehouse. It shows how long the stocks - and thus of course the capital required for them - are tied up in the warehouse on average. The shorter the storage period of a product \/ component, the better, since storage causes running costs, requires space and this can make the products more expensive.\n\n${\\ displaystyle \\ varnothing {\\ text {Storage period}} = {\\ frac {360 \\, {\\ text {days}} \\ cdot \\ varnothing {\\ text {Inventory}}} {\\ text {Consumption (per year)}} }}$\n\nor\n\n${\\ displaystyle \\ varnothing {\\ text {Storage period}} = {\\ frac {360 \\, {\\ text {days}}} {\\ text {Inventory turnover rate}}}}$\n\n### Inventory rate\n\nThe storage interest rate indicates what percentage of interest the capital tied up in the average inventory costs during the average storage period.\n\n${\\ displaystyle {\\ text {Storage interest rate}} = {\\ frac {{\\ text {Interest rate (pa)}} \\ cdot \\ varnothing {\\ text {Storage duration (in days)}}} {360 \\, {\\ text {days }}}}}$\nBeispiel:\n\n(10\u00a0% \u00b7 200 Tage) \/ 360 Tage = 5,55\u00a0% f\u00fcr 200 Tage\n\n\nThe storage interest or the storage interest indicate how much interest the entrepreneur misses during the storage period. The capital is tied up in the warehouse and therefore cannot be invested with interest. The storage interest rate is used to calculate the storage interest.\n\n${\\ displaystyle {\\ text {Storage interest}} = {\\ frac {\\ varnothing {\\ text {Inventory}} \\ cdot {\\ text {Inventory interest}}} {100}}}$\nBeispiel:\n\nLagerzinsen = Wert des durchschnittlichen Lagerbestandes \u00b7 Lagerzinssatz\n\n5000\u00a0\u20ac \u00b7 5,55\u00a0% = 277,50\u00a0\u20ac f\u00fcr 200 Tage\n\n\n### Stock range \/ readiness for delivery\n\nThe days' supply indicates how long the average inventory will last with an average consumption per period. The days' supply can also be calculated for a specific key date (e.g. start of a quarter).\n\n${\\ displaystyle {\\ text {Storage range}} = {\\ frac {{\\ text {Existing or}} \\ varnothing {\\ text {Inventory}}} {\\ varnothing {\\ text {Consumption per period}}}}}$\nBeispiel:\n\n20 St\u00fcck \/ 0,117 St\u00fcck pro Tag = 171 Tage\n\nDurchschnittlicher Verbrauch pro Tag = (Wareneinsatz \/ 360 Tage)\n42 St\u00fcck \/ 360 Tage ~ 0,117\n\n\n### Stock quota\n\nThe stocking quota indicates the ratio of the number of stocked to the total number of procured material items.\n\nStock quota = number of stored articles \/ total number of articles procured\n\n### Storage utilization rate\n\nThe degree of storage utilization shows the ratio of space used to available space. The indicator reveals both bottlenecks (overcrowding) and insufficient utilization (overcapacity).\n\nBeispiel:\n\nFl\u00e4chennutzungsgrad = (genutzte Lagerfl\u00e4che \/ verf\u00fcgbare Lagerfl\u00e4che)\n\nRaumnutzungsgrad = (genutzter Lagerraum \/ verf\u00fcgbarer Lagerraum)\n\n\nThe service level of the bearing can in the average time period to be the material (external service level), or measured by the average time between the instruction to paging and the location output (internal service level) between the demand request and the provision.\n\n### Capital commitment\n\nThe capital commitment is a key figure for the non-liquid assets in a company. For example: stocks, machines and systems.\n\n${\\ displaystyle \\ varnothing {\\ text {tied up capital}} = \\ varnothing {\\ text {stock level}} \\ cdot {\\ frac {\\ text {procurement costs}} {\\ text {quantity}}}}$\n\n### Key figures of the means of transport use\n\n${\\ displaystyle {\\ text {Degree of deployment}} = {\\ frac {\\ text {Time of deployment}} {\\ text {Working time}}}}$\n${\\ displaystyle {\\ text {Degree of failure}} = {\\ frac {\\ text {Downtime}} {\\ text {Working time}}}}$\n\nKey figures are to be created for the company in coordination with the management and management. Insignificant key figures from a company perspective should be omitted. It is important to derive or specify budget figures from the key figures so that deviation analyzes can be produced and corrections can be initiated.\n\nEssentially, according to Hartmann, Hoppe and Schwalbach, the following extended storage parameters can be added to the storage key figures mentioned above. These warehouse parameters are used to describe your own warehouse and are less suitable for comparison with other companies or warehouses.\n\n\u2022 Value in euros of stagnant stocks with a turnover of less than two per year\n\u2022 Consignment stocks at suppliers in euros and as a percentage of the total stock\n\u2022 External inventories of the suppliers in-house in euros\n\u2022 No longer usable inventories in euros and percent\n\u2022 Age structure in percent of the supplies per time class\n\u2022 Share of new materials and discontinued materials\n\u2022 Sediment parameter : the part of the stock that has not been moved over a certain period of time.\n\u2022 Incoming value parameters : Valuated stock = delivered quantity \u00d7 valuation price.\n\u2022 Parameter security cushion: formed from the key figures range of the average access and range of the average stock upon access .\n\u2022 Safety stock parameter : Comparison of the value of the mean stock at access to (quotient of mean stock at access and safety stock, which should be approximately 1).\n\u2022 Parameter cost size: range of mean access and value of mean access compared.\n\u2022 Analysis according to the volume of the parts: division of the materials into large-volume, medium-volume and small-volume parts.\n\n## Approaches to warehouse, inventory and supply analysis\n\n### Conflicting goals in warehousing\n\nBased on Hartmann, corporate development and management must face a changed market landscape. These include ever shorter delivery times, higher demands on readiness to deliver, greater flexibility on the part of suppliers, more product diversity, shorter product life cycles and fundamentally changed requirement profiles for suppliers due to globalization and offshoring. The way to do everything by setting up stocks inevitably leads to high capital commitment and decreasing liquidity. Further disadvantages of excessively high inventories result from the risk of changes in products, the spoilage of the products (e.g. food) and the costs of storing and managing the inventories.\n\nThe areas of tension arising from the departmental interests can be compared below:\n\n1. Procurement objective:\nSecurity of supply for large order lots, safety stocks\nRequirement: Reduction in stocks of\nsmall order lots, short procurement times\n2. Production\ngoal : uniform utilization of capacities, large production batches , buffer storage\nRequirement: Stock reduction,\ncapacity reserves , batch size 1, needs-based production, production and supply according to the production plan\n3. Sales targets: Fulfillment of customer requirements,\nextensive range, high level of readiness for delivery\nRequirement: Reduction in stocks of\nsmall ranges, fewer product variants\n\n### Determination of the necessary inventory\n\n#### The industry comparison\n\nIn order to determine and interpret where the company will be positioned with its stock, it is useful to know the industry comparison. The table from Hartmann presents a study from 1995 in which inventories are shown in relation to sales, broken down according to industries. The following percentages of inventories in relation to sales were given for the industries mentioned:\n\nBranch Stock share\nmechanical engineering 24.70%\nElectrical engineering 19.70%\nTotal manufacturing industry 19.30%\nTextile industry 18.90%\nMetal products 18.10%\nChemical industry 12.20%\nconstruction industry 10.40%\nVehicle technology 8.60%\n\nMore recent data was presented in the Harting publication in 2005. He states that the average inventory value is 14 percent of the turnover of German companies.\n\n#### Lot size half\n\nA calculation of the \u201cideal stock level\u201d is i. d. Usually carried out according to the bottom-up approach. This is based on the question of how far the inventory can be reduced without jeopardizing the readiness for delivery. The theoretically optimal stock is determined from the safety stock plus half the optimal lot size. The bottom-up approach shows general inventory strategies.\n\n${\\ displaystyle {\\ text {Target stock}} = \\ left ({\\ frac {\\ text {Lot size}} {2}} + {\\ text {Safety stock}} \\ right) \\ cdot {\\ frac {\\ text {Price} } {\\ text {piece}}}}$\n\n#### Low value principle\n\nThe ideal stock level can also be calculated using the top-down approach.\n\n${\\ displaystyle {\\ text {target stock}} = ({\\ text {lowest stock value within a period)}} \\ cdot {\\ frac {\\ text {price}} {\\ text {pieces}}}}$\n\nTo determine this, the IT system takes the past inventory values \u200b\u200bfrom a time period and selects the lowest value for each article. Then multiply the lowest value from the time period by the price \/ piece. You get the lowest stock value within the selected time period per article. Then the individual results for all articles are added together to produce the ideal stock based on the low value principle.\n\n### The warehouse, inventory and supply analysis\n\nThe relevant specialist literature names many theses on the formation of stocks. The following list is based on research by Hartmann, Hoppe and Schwalbach.\n\n\u2022 Missing thematization of the inventory by the management and subordinate entrepreneurial inventory goals\n\u2022 Transition from in-house production to external procurement (offshoring)\n\u2022 Insufficient qualification of the dispatchers\n\u2022 Incorrect lot sizes\n\u2022 Incorrect safety stocks and added safety thinking in areas\n\u2022 Lack of coordination between supply chains and internal and inter-company business processes\n\u2022 Missing integration of the data technology\n\u2022 Inadequate delivery capacity of the suppliers\n\u2022 Failure to work on the inventory due to lack of knowledge of the inventory figures\n\u2022 Insufficient quality of the disposition procedures\n\u2022 Great depth and breadth of the range\n\u2022 Decoupling of planning and control\n\u2022 Fuzzy sales planning, ignorance of sales changes and poor forecast quality\n\u2022 Organizational deficiencies\n\nThe following three-way relationship is established as a field of tension.\n\n+ Sales data that are unreliable in terms of quality or time\n+ Suppliers who are unreliable in terms of quality or time\n= Fuzzy decision-making basis for the dispatcher\n\nIn other words, poor input data leads to poor decisions by decision makers.\n\n## Approaches to destocking, inventory reduction and inventory reduction\n\nExisting problems cannot be solved in isolation, but rather through the interaction of all those involved in the process . Furthermore, it can be stated that many who are responsible for the inventory in the company are not responsible for the consequences of their actions in relation to stocks or are not aware of it. The responsibility and co-responsibility for the inventory is often not attributable to a single person. An assignment of joint responsibility for the supply and the determining triggering elements is missing. On the other hand, the stocks will hide suboptimal states because weak points in the processes are concealed or not revealed. There is no need or challenge to constantly work on the best process design.\n\nIt is therefore necessary to address the stocks in order to initiate the willingness to actively participate in the causes of stocks. The order should be given by the company management, which must also support team building. The influence of a lack of supply chain management (SCM) and the resulting effects of excessive inventories on company results became clear to companies in the 1970s. Due to the high level of interest rates prevailing at the time, the companies were blocked by the debt service, based on the debt financing of the inventories. During this time, companies reacted with the first studies of the effects of the supply chain (= supplier chain) and came to the conclusion that an optimal inventory policy depends on the design of internal and inter-company business processes.\n\nThe following approaches to lowering have been shortened and essentially taken from Schwalbach's remarks.\n\n1. The product management, the construction department\nEvery new product development is connected with the chance to influence the inventory in the long term. In the process steps of product and presentation development, as long as the final product approval has not yet taken place, the standard or the use of existing materials can be influenced.\nArticle creation according to standards.\nIntroduction of mature articles.\nReduction of the number of articles\n2. Sales Sales\nplanning is planning that is fraught with uncertainties. Often the plan specifications and planning results do not match.\nRolling planning\nfacts from market observation and demand\nplanning teams promoting\nsales\n3. Disposition & work\npreparation Fields for parameter maintenance are available to the dispatchers and work planners in the available ERP system.\nSelection of the disposition\nprocedure Qualification of the employees\nImproved disposition tools\nChange of the disposition\nlot sizes Security parameters and warehouse service level\nAs an interface and negotiating partner, purchasing has a large number of very effective instruments at its disposal. Purchasing represents the improvement in the performance of suppliers.\nPartnership cooperation with suppliers\nSupply chain management \/ supply chain\nSupplier selection, development and maintenance\nSupplier contracts\n5. The organization\nA key point in reducing the inventory is the organizational design.\nClear allocation of competencies and responsibilities.\nEstablishment of the working group, inventory\nreduction, target agreements\n\n## attachment\n\n### Individual evidence\n\n1. a b Baumann, Baumgart, K\u00e4hler, Lewerenz, Schliebner: Logistic processes - jobs in warehouse logistics ; ISBN 978-3-441-00360-1\n2. ^ Kummer, Gr\u00fcn, Jammernegg: Fundamentals of procurement, production and logistics , Pearson, 2006, p. 216f.\n3. Christof Schulte: Logistics - Ways to optimize the supply chain , Vahlen, 5th edition, 2009, p. 228 f.\n4. Holistic warehouse planning: What do you have to consider? Retrieved April 16, 2018 .\n5. a b Horst Hartmann (1999): Inventory management and controlling , Dt. Betriebswirte Verlag GmbH; Gernsbach.\n6. ^ A b Marc Hoppe (2005): Inventory optimization with SAP Galileo Press GmbH, Bonn.\n7. a b c d Lutz Schwalbach (2006): Inventory and stock reduction. Determination of potential, structured analyzes and functional solution images , BoD Verlag, Nordersted, ISBN 978-3-8334-6715-8 .\n8. ^ Hartmann, Horst: Inventory management and controlling , Gernsbach: Dt. Betriebswirte Verlag GmbH 1999, p. 30.\n9. Horst Hartmann (1999): Inventory management and controlling , Dt. Betriebswirte Verlag, Gernsbach, p. 17.\n10. Detlef Harting (2005): Inventory and inventory management slim and modern , BA procurement current, issue 6, Konradin Verlag Robert Kohlhammer GmbH, Leinfelden, p. 26.\n11. See R\u00fcggeberg, Christian: Supply Chain Management as a Challenge for the Future , Wiesbaden: Deutscher Universit\u00e4ts-Verlag \/ GWV Fachverlage GmbH, 2003, p. 4 and p. 17.\n\n### literature\n\n\u2022 J\u00f6rg Becker, Axel Winklmann (2008): Retail Controlling . 2nd edition, Springer, Berlin, ISBN 3-540-29611-5\n\u2022 Horst Tempelmeier (2006): Inventory Management in Supply Chains. 2nd Edition. Norderstedt, Books on Demand, ISBN 3-8334-5032-0 .\n\u2022 Lutz Schwalbach (2006): Inventory and stock reduction. Determination of potential, structured analyzes and functional solution images. BoD Verlag: Norderstedt, ISBN 978-3-8334-6715-8\n\u2022 Marc Hoppe: Inventory optimization with SAP. Bonn: Galileo Press GmbH 2005, ISBN 978-3-89842-611-4\n\u2022 Gerhard Oeldorf (2004): Materials Management . Ludwigshafen, Kiel, ISBN 3-470-54141-8\n\u2022 Wolfram Fischer (2004): Material flow and logistics. Springer, Berlin, ISBN 3-540-40187-3\n\u2022 Wolfgang Vry (2004): Procurement and Warehousing. Ludwigshafen, Kiehl, ISBN 3-470-63127-1\n\u2022 Christian R\u00fcggeberg (2003): Supply Chain Management as a Challenge for the Future , Wiesbaden: Deutscher Universit\u00e4ts-Verlag \/ GWV Fachverlage GmbH, p. 4 and p. 17\n\u2022 Rainer Weber (2003): Contemporary materials management with storage. expert-Verlag, Renningen, ISBN 3-8169-2269-4\n\u2022 Horst Hartmann (1999): inventory management and controlling , Gernsbach: Dt. Betriebswirte Verlag GmbH, ISBN 978-3-88640-083-6\n\u2022 WEKA Media GmbH (since 1999): warehouse planning, organization and optimization. ISBN 3-8111-6822-3 (loose-leaf collection with CD-ROM)\n\u2022 Hans Arnolds et al. \/ Heege, Franz \/ Tussing, Werner: Materials Management and Purchasing , 10th Edition, Wiesbaden: Gabler Verlag 1998, ISBN 978-3-409-35160-7","date":"2022-01-26 04:25:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 17, \"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\": 0.394400954246521, \"perplexity\": 3825.727632415031}, \"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-2022-05\/segments\/1642320304915.53\/warc\/CC-MAIN-20220126041016-20220126071016-00564.warc.gz\"}"}
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